Perry's Chemical Engineers' Handbook

PERRY’S CHEMICAL ENGINEERS’ HANDBOOK SEVENTH EDITION McGraw-Hill New York San Francisco Washington, D.C. Auckland Bogotá Caracas Lisbon London Madrid Mexico City Milan Montreal New Delhi San Juan Singapore Sydney Tokyo Toronto

Prepared by a staff of specialists under the editorial direction of Late Editor Robert H. Perry

Editor Don W. Green Deane E. Ackers Professor of Chemical and Petroleum Engineering, University of Kansas Associate Editor James O. Maloney Professor Emeritus of Chemical Engineering, University of Kansas

Library of Congress Cataloging-in-Publication Data Perry’s chemical engineers’ handbook. — 7th ed. / prepared by a staff of specialists under the editorial direction of late editor Robert H. Perry : editor, Don W. Green : associate editor, James O’Hara Maloney. p. cm. Includes index. ISBN 0-07-049841-5 (alk. paper) 1. Chemical engineering—Handbooks, manuals, etc. I. Perry, Robert H., date. II. Green, Don W. III. Maloney, James O. TP151.P45 1997 660—dc21 96-51648 CIP

Copyright © 1997, 1984, 1973, 1963, 1950, 1941, 1934 by The McGrawHill Companies, Inc. Copyright renewed 1962, 1969 by Robert H. Perry. All rights reserved. Printed in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher. 1 2 3 4 5 6 7 8 9 0

DOW/DOW 9 0 2 1 0 9 8 7

ISBN 0-07-049841-5

INTERNATIONAL EDITION Copyright © 1997. Exclusive rights by The McGraw-Hill Companies, Inc., for manufacture and export. This book cannot be re-exported from the country to which it is consigned by McGraw-Hill. The International Edition is not available in North America. When ordering this title, use ISBN 0-07-115448-5.

The sponsoring editors for this book were Zoe Foundotos and Robert Esposito, the editing supervisor was Marc Campbell, and the production supervisor was Pamela A. Pelton. It was set in Caledonia by North Market Street Graphics. Printed and bound by R. R. Donnelley & Sons Company. This book was printed on acid-free paper. Information contained in this work has been obtained by The McGrawHill Companies, Inc. (“McGraw-Hill”) from sources believed to be reliable. However, neither McGraw-Hill nor its authors guarantee the accuracy or completeness of any information published herein, and neither McGraw-Hill nor its authors shall be responsible for any errors, omissions, or damages arising out of use of this information. This work is published with the understanding that McGraw-Hill and its authors are supplying information but are not attempting to render engineering or other professional services. If such services are required, the assistance of an appropriate professional should be sought.

Michael M. Ahbott, Ph.D., Howard P. Isermann Department of Chemical Engineering, Rensselaer Polytechnic Institute; Member, American Institute of Chemical Engineers (Section 4, Thermodynamics) Terry Allen, Ph.D., Senior Research Associate (retired), Du~ont Central Research and Development \Section 20, Size Reduction and Size Enlargement) John D. Dacha, Ph.D., Consulting Scientist, Chevron Products Company; Member, ASTM (American Society for TestiI1cgand Materials), Committee D02 on Petroleufll Products and Lubri"cants; American Chemical Society; International Association for Stability and Handling of Liquid Fuels, Steering Committee (Section 27, Energy Resources, Conversion, and Utilization) Glenn w. Baldwin, M.S., P.E., Staff Engineer, Union Carbide Corporatio~; Membe!, American Institute of Chemical Engineers (Section 12, Psychrometry, Evaporative Cooling, and Solids Drying) Scott D. Barnicki,

Ph.D., Senior Resear~h Chemical Engineer, Eastman Chemical Company

(Section 13, Distillation) Kenneth J. Bell, Ph.D., P.E., Regents Professor Emeritus, School of Chemical Engineering, Oklahoma State University; Member, American Institute of Chemical Engineers (Section 11, Heat- Transfer Equipment) Richard C. Bennett, B.S., Ch.E., Registered Professional Engineer, Illinois; Member, American Institute of Chemical Engineers (AIChE); President of Crystallization Technology, Inc.; FoJmer Presideptof Swenson Process Equipme~t, Inc. (Section 13, Liquid-Solid Operations and Equipment) Charles E. Benson, M.Eng., M.E., Director, Combustion Technology, Arthur D. Little, lIic.; Member, American Society of Mechanical Engineers; Comb~tion Institute (Section 27, Energy Resources, Conversion, and Utilization) Patrick M. Bernhagen, P.E., B.S.M.E., Senior Mechanical Engineer, Foster wheeler USA Corporation, American Society of Mechanical EngineeFs ( Section 11, Heat -Transfer Equipment) c

ix

x

LIST OF CONTRIBUTORS

Heinz P. Bloch, P.E., B.S.M.E., M.S.M.E., Consulting Engineer, Process Machinery Consulting; American Society of Mechanical Engineers, Vibration Institute; Registered Professional Engineer (New Jersey, Texas) (Section 29, Process Machinery Drives) Frank T. Bodurtha, Sc.D., E.I. DuPont de Nemours and Co., Inc. (retired), Wilmington, Delaware (retired); Consultant, Frank T. Bodurtha, Inc. (Section 26, Process Safety) Meherwan P. Boyce, P.E., Ph.D., President, Boyce Engineering International; ASME Fellow; Registered Professional Engineer (Texas, Oklahoma) (Section 10, Transport and Storage of Fluids; Section 29, Process Mochinery Drives), Laurence G. Britton, Process Safety)

Ph.D.,

Research Scientist, Union Carbide Corporation

(Section 26,

Evan Buck, M.S.Ch.E., Manager, Thermophysical Property Skill Center, Central Technology, Union Carbide Corporation (Section 2, Physical and Chemical Data) Henry R. Bungay, P.E., Ph.D., Professor of Chemical and Environmental Engineering, Rensselaer Polytechnic Institute; Member, American Institute of Chemical Engineers, American Chemical Society, American Society for Microbiology, American Society for Engineering Education, Society for General Microbiology (Section 24, Biochemical Engineering) Anthony I. Buonicore, M.Ch.E., P.E., Diplomate AAEE, CEO, Environmental Data Resources, Inc.; Member, American Institute of Chemical Engineers, Air and Waste Management Association (Section 25, Waste Management) Michael M. Calistrat, B.S.M.E., M.S.M.E., Owner, Michael Calistrat and Associates; Member, American Society of Mechanical Engineers (Section 29, Process Machinery Drives) , Giorgio Carta, Ph.D., Professor, Department of Chemical Engineering, U niversity of Virginia; Member, American Institute of Chemical Engineers, American Chemical Society, International Adsorption Society (Section 16, Adsorption and Ion Exchange) Vincent Conrad, Ph.D., Group Leader, Technical Services Development Laboratory,CONSOL, Inc.; Member, Spectroscopy Society of Pittsburgh, Society for Analytical Chemistry of Pittsburgh, Society for Applied Spectroscopy (Section 27, Energy Resources, Conversion, and Utilization) Harrison Cooper, ph.D., Harrison R. Cooper Systems, Inc., Salt take City, Utah (Section 19, Solid-Solid Operations and Equipment) B. B. Crocker, S.M., P.E., Consulting Chemical Engineer; Fellow, American Institute of chemical Engineers; Member, Air Pollution Control Association (Section 14, Gas Absorption and GasLiquid System Design) Daniel A. Crowl, Ph.D., Professor of Chemical Engineering, Chemical Engineering Department, Michigan Technological University; Member, American Institute of Chemical Engineers, American Chemical Society (Section 26, Process Safety) Roger W. Cusack, Vice President, Glitsch Process Systems, Inc.; Member, American Insi:itute of Chemical Engineers (Section 15, Liquid-Liquid Extraction Operations and Equipment) Donald A. Dahlstrom, Ph.D., Research Professor, Chemical and Fuels Engineering Department and Metallurgical Engineering Department, University of Utah; Member, National Academy of Engineering, American Institute of Chemical Engineers (AIChE), American Chemical Society (ACS), Society of Mining, Metallurgic Exploration (SME) of the American Institute of Mining, Metallurgical and Petrol~um Engine()r~ (AIME), AmericaI) Society of Engineering Education (Section 18, Liquid-Solid °?erations and Equipment) Thomas E. Daubert, Ph.D., Professor, Department of Chemical Engineering, The Pennsylvania State University (Section 2, Physic~ and Chemical Data) R. H. Daugherty, Ph.D., Consulting Engineer, Research Center~ Reliance Electric Company; Member, Institute of Electrical and Electronics Engineers (Section 29, Process Machinery Drives)

LIST OF CONTRIBUTORS

xi

James F. DaVis, Ph.D., Professor of Chemical Engineering, Ohio State University (Section 3, Mathematics) James B. Dunson, B.S., Principal Consultant, E. I. duPont de Nemours & Co.; Member Americanlnstitute of Chemical Engineers; Registered Professional Engineer \Delaware) (Section 17, Gas-Solid Operation and Equipment) Thomas F. Edgar, Ph.p., Professor of Chemical Engineering, y niversity o£Texas, Austin, Texas (Section8, Process Control) Robert C. Emmet, Jr., B,.5., Ch.E., Senior Process Consultant, EIMCO Process Equipment Co.; Member, American Institute of Chemical Engineers (AIChE), American Institute of Mining, Metallurgical and Petroleum Engineers (AIME), Society of Mining, Metallurgical and Exploration Engineers (SME) (Section 18, Liquid-Solid Operations and Equipment) Stanley M. Englund, M.S., Ch.E., Fellow, American Institute of Chemical Engineers; Process Consultant, The Dow Chemical Company (retired) (Section 26, Process Safety) BryanJ. Ennis, Ph.D., President, E&G Associates, and Adjunct Professor of Chemical Engineering; Vanderbilt University; Member and Chair of Powder Technology Programming Group of the Particle Technology Forum, American Institute of Chemical Engineers (Section 20, Size Reduction and Size Enlargement) William Eyka~p, Ph.D., Adjunc~ Professor of Chemical Engineering, Tufts University; Formerly President, Koch Membrane Systems; Member, American Institute of Chemical Engineers, American Chemical Society, American Association for the Advancement of Science, North American Membrane Society, European Society of Membrane Science and Technology (Section 22, Altemative Separation Processes) James R. Fair, Ph.D., P.E., Professor of Chemical Engineering, University of Texas; National Academy of Engineering; Fello'Y, American Institute of Chemical Engineers; Member, American Chemical Society, American S,ociety for Engineering Education, N atio~al Society of Professional Engineers

(Section

14, Gas Ab~orption

and

., Gas-Liquid

System

Design)

, Bruce A. Finlayson, Ph.D., Rehnberg Professor and Chair, Department of Chemical Engineering, University ofWashington; Member, National Academy of Engineering (Section 3, Mathematics) Thomas M. FI~n, Ph.D" P.E., Cryogenic Engineer, President CRYOCO, Louisville, CoJorado; Member, American Institute of Chemical Engineers (Section 11, Heat-Trans£er Equipment) Anthony G.,Fonseca, Ph,P., Director, Coal Utiltzation, CONSOL, Inc,; Member, American Chemical Society, Society for Mining, Metallurgy, and Extraction ( Section 27, Energy Resources, Conversion, and Utilization) D. G. Friend,

National Institutes of Standards and Technology, Boulder, Colorado (Section 2,

Physical and Chemical Data) George W. Gassman, B.S.M.E., Senior Research Specialist, Final Control Systems, Fisher Controls International, Inc., Marshalltown, Iowa (Section 8, Process Control) Fred K. Geitner,

P.Eng., B.S.M.E.,

M.S.M.E.,

Consulting Engineer; Registered Professional

Engineer (Ontario, Canada) (Section 29, Process Machinery Drives) Victor M. Goldschmidt,

Ph.D., P.E., Professor of Mechanical Engineering, Purdue University,

West Lafayette, Indiana (Section 11, Heat-Trallsfer Equipment) Stanley Grossel, President, Process Safety & Design, Inc.; Fellow, American Institute of chemical Engineers; Member, American Chemical Society; Member, The Combustion Institute; Member,Explosion Protection Systems Committee of NFPA (Section 26, Process Safety) Peter Harriott, Ph.D., Professor, School of Chemical Engineering, Cornell University; Member, American Institute of Chemical Engineering, American Chemical Society (ACS) (Section 18; Liquid-Solid Operations and Equipment)

xii

LIST OF CONTRIBUTORS

T. Alan Hatton, Ph.D., Ralph Landau Professor and Director of the DavidH. Koch School of Chemical Engineering Practice, Massachusetts Institute of Technology; Founding Fellow, American Institute of Medical and Biological Engineering; Member. American Institute of Chemical Engineers, American Chemical Society, International Association of Colloid and Interface scientists, American Association for the Advancement of Science, N eutt:on Scattering Society of America (Section 22, Alternative Separation Processes) Joseph D. Henry, Jr., Ph.D., P.E., Senior Fellow, Department of Engineering and Public Policy, Carnegie Mellon University; Member, American InStitute of Chemical Engineers, American Society for Engineering Education (Section 22, Alternative Separation Processes) W. G. High, C.Eng., B.Sc., F.I.Mech.E., tion 26, Process Safety)

Burgoyne Consultants Ltd., W. Yorks, England (Sec-

Richard Hogg, Ph.D., Professor, Department of Mineral Engineering, The Pennsylvania State University, University Park, PA (Section 19, Solid-Solid Operations and Equipment) F. A. Holland, D.Sc., Ph.D., Consultant in Heat Energy Recycling; Research Professor, University of Salford, England; Fellow; Institution of Chemical Engineers, London (Section 9, Process Economics) Hoyt C. Hottel,

S.M., Professor Emeritus of Chemical Engineering, Massachusetts Institute of

Technology; Member, National Academy of Sciences, American Academy of Arts and Sciences, American Institute of Chemical Engineers, American Chemical Society, Combustion Institute (Section 5, Heat and Mass Transfer) Colin S.Howat, Ph.D., P.E., John E. & Winfred E. Sharp Professor, Department of Chemical and Petroleum Engineering, University of Kansas; Member, American Institute of Chemical Engineers; Member, American Society of Engineering Education (Section 30, Analysis ofPlant Performance) Predrag S. Hrnjak, Ph.D., V.Res., Assistant Professor, University of Illinois at Urbana champaign and Principal Investigator-U. of I. Air Conditioning and Refrigeration Center, Assistant Professor, University of Belgrade; Member, International Institute of Refrigeration, American Society of Heating, Refrigeration and Air Conditioning (Section 11, Heat-Transfer Equipment) Arthur

E. Humphrey,

Ph.D., Retired, Professor of Chemical Engineering, Pennsylvania State

University; Member, U.S. National Academy of Engineering, American Institute of Chemical Engineers, American Chemical Society, American Society for Microbiology (Section 24, Biochemical Engineering) Eric Jenett, M.S.Ch.E.,

Manager, Process Engineering, Brown & Root, Inc.; Associate Member,

AIChE, Project Management Institute; Registered Professional Engineer (Texas) (Section 29, Process Machinery Drives) John S. Jeris, Sc.D., P.E., Professor of Environmental Engineering, Manhattan College; Environmental Consultant; Member, American Water Works Association, Water Environment Federation Section Director (Section 25, Waste Management) T. L. P. Jespen, M.S., MiD. Proc;, Metallurgical Engineer, Basic, Inc;; Gabbs, Nevada (Section 19, Solid-Solid Operations and Equipment) Keith P. Johnston, Ph.D., P.E., Professor of Chemical Engineering, University of Texas (Austin); Member, American Institute of Chemical Engineers, American Chemical Society, U niversity of Texas Separations Research Program (Section 22, Alternative Separation Processes) Trevor A. Kletz, D.Sc., Senior Visiting Research Fellow, Department of Chemical Engineering, Loughborough University, U.K.; Fellow, American Institute of Chemical Engineers, Royal Academy of Engineers (U .K.), Institution of Chemical Enginee!s (U .K.), and Royal Society of chemistry (U.K.) (Section 26, Process Safety) Edgar B. Klunder, Ph.D., Project Manager, Energy Technol
U,~.

LIST OF CONTRIBUTORS

xiii

Kent S. Knaebel, Ph.D., President, Adsorption Research, Inc.; Member, American Il;lstitute of Chemical Engineers, American Chemical Society, International Adsorption Society. Professional Engineer (Ohio) (Section 5, Heat and Mass Transfer) Frank Knoll, M.S., Min. Proc., President, Carpco, Inc., Jacksonville, Florida (Section 19, SolidSolid Operations and Equipment) James G.Knudsen, Ph.D.,Professor Emeritus of Chemical Engineering, Oregon State university; Member, American Institute of Chemical Engineers, American Chemical Society; Registered Professional Engineer (Oregon) (Section 5, Heat and Mass Transfer) Michael Krumpelt, Ph.D., Manager, Fuel Cell TechnolQgy, Argonne National Laboratory; Member, American Institute of Chemical Engineers, American Chemical Society, Electrochemical Society (Section 27, Energy Resources, Conversion, and Utilization) Irwin J. Kugelman, Sc.D., Professor of Civil Engineering, Lehigh University; Member, American Society of Civil Engineering, Water Environmental Federation (Section 25, Waste Management) rim Laros, M.S. Mineral Processing, Senior Process Consultant, EIMCO Process Equipment Co,; Member, Society of Mining, Metallurgy and Exploration (SME of AIME) (Section 18, Liquid-Solid Operations and Equipment) Richard M. Lemert, Ph.D., P.E., Assistant Professor of Chemical Engineering, University of Toledo; Member, American Institute of Chemical Engineers, American Chemical Society, Society qf Mining Engineers, American Society for Engineering Education (Section 22, Alternative Separation Processes) Robert Lemlich, Ph.D., P.E., Professor of Chemical Engineering Emeritus, University of Cincinnati; Fellow, American Institute of Chemical Engineers; Member, American Chemical Society, American Society for Engineering Education, American Chemical Society (Sectiqn 22, Alternative Separation Processes) Wallace Leung, Sc.D., Director, Process Technology, Bird Machine Company; Member, American Filtration and Separation Society (Director) (Section 18, Liquid-Solid Operations and Equipment) M. Douglas LeVan, Ph.D., Professor, Department of Chemical Engineering, University of Virginia; Member, American Institute of Chemical Engineers, American Chemical Society, Intemational Adsorption Society (Section 16, Adsorption and Ion Exchange) Peter E. Liley, Ph.D., D.I.C., School of Mechanical Engineering, Purdue University (Section 2, Physical and Chemical Data) James D. Litster, Ph.D., Associate Professor, Department of Chemical Engineering, University of Queensland; Member, Institute of Chemical Engiiieers-Australia (Section 20, Size Reduction and Size Enlargement) Peter J. Lofuis, D. Phil., Arthur D. Little, Inc.; Member, American Society of Mechanical Engineers (Section 27, Energy Resources, Conversion, and Utilization) Hsue-peng Loh, Ph.D., P.E., Federal Energy Technology Center (Morgantown), U .S. Department of Energy; Member, American Institute of Chemical Engineers, American Society of Information Sciences (Section 27, Energy Resources, Conversion, and Utilization) Douglas E. Lowenhaupt, M.S., Group Leader, Coke Laboratory, CONSOL, Inc.; Member, American Society for Testing and Materials, Iron and Steel Making Society, International Committee for Coal Petrology (Section 27, Energy Resources, Conversion, and Utilization) James 0. Maloney, Ph.D., P.E., Emeritus Professor of Chemical Engineering, University of Kansas; Fellow, American Institute of Chemical Engineering; Fellow, American Association for the Advancement of Science; Member, American Chemical Society, American Society for Engineering Education (Section 1, Conversion Factors and Mathematical Symbols)

Thomas J. McAvoy, Ph.D., Professor of Chemical Engineering, University of Maryland, College Park, Maryland (Section 8, Process Control) Chad McCleary, EIMCO Process Equipment Company, Process Consultant (Section 18, Liquid-Solid Operations and Equipment) Thomas F. McGowan, P.E., Senior Consultant, RMT/Four Nines; Member, American Institute of Chemical Engineers, American Society of Mechanical Engineers, Air and Waste Management Association (Section 25, Waste Management) Howard G. Mcllvried, III, Ph.D., Senior Engineer, Burns and Roe Services Corporation, Federal Energy Technology Center (Pittsburgh), Member, American Chemical Society, American Institute of Chemical Engineers (Section 27, Energy Resources, Conversion, and Utilization) John D. McKenna, Ph.D., President and Chairman, ETS International, Inc., Member, American Institute of Chemical Engineers, Air and Waste Management Association (Section 25, Waste Management) Shelhy A. Miller, Ph.D., P.E., Resident Retired Senior Engineer; Argonne National Laboratory; American Association for the Advancement of Science (Fellow), American Chemical Society, American Institute of Chemical Engineers (Fellow), American Institutes of Chemists (Fellow), Filtration Society, New York Academy of Sciences, Society of Chemical Industry (Section 18; Liqmd.; Solid Operations and Equipment; Section 27, Energy Resources, Conversion, and Utilization) Booker Morey, Ph.D., Senior Consultant, SRI International; Member; Society of Mining, Metallurgy and Exploration (SME of AIME), The Filtration Society, Air and Waste Management Association; Registered Professional Engineer (California and Massachusetts) (Section 18, LiquidSolid Operations and Equipment) Charles G. Moyers, Ph.D., P.E., Principal Engineer, Union Carbide Corporation; Fellow, American Institute of Chemical Engineers (Section 12, Psychrometry, Evaporative Cooling, and Solids Drying; Section 22, Alternative Separation Processes) John Newman, Ph.D., Professor of Chemical Engineering, University of California, Berkeley; Principle Investigator; Inorganic Materials Research Division, Lawrence Berkeley Laboratory (Section 22, Alternative Separation Processes) James Y. Oldshue, Ph.D., President, Oldshue Technologies International, Inc.; Member, National Academy of Engineering; Adjunct Professor of Chemical Engineering atBeijing Institute of Chemical Technology, Beijing, China; Member; American Chemical Society (ACE), American Institute of Chemical Engineering (AIChE), Traveler Century Club, Executive Committee on the Transfer of Appropriate Technology for the World Federation of Engineering Organizations (Section 18, Liquid-Solid Operations and Equipment) Robert W. Ormsby, M.S., Ch.E. P.E., Manager of Safety, Chemical Group, Air Products and Chemicals, Inc.; Air Products Corp.; Fellow, American Institute of Chemical Engineers (Section 26, Process Safety) John E. Owens, B.E.E., Electrostatic Consultant, Condux, Inc.; Member, Institute of Electrical and Electronics Engineers, Electrostatics Society of America (Section 26, Process Safety) Bhupendra Parekb, Ph.D., Associate Director, Center for Applied Energy Research, University of Kentucky, Lexington, Kentucky (Section 19, Solid-Solid Operations and Equipment) Mel Pen, Ph.D., Senior Consultant, E. I. duPont de Nemours & Co.; Fellow, American Institute of Chemical Engineers; Registered Professional Engineer (Delaware) (Section 17, Gas-Solid Operations and Equipment) W. R. Penney, Ph.D., P.E., Professor of Chemical Engineering, University of Arkansas; Member, American Institute of Chemical Engineers (Section 14, Gas Absorption and Gas-Liquid System Design)

LIST OF CONTRIBUTORS

xv

WalterF. Podolski, Ph.D., Chemical Engineer, Electrochemical Technology Program, Argonne National Laboratory; Member, American Institute of Chemical Engineers (Section 27, Energy Resources, Conversion, and Utilization) Herbert A. Pohl, Ph.D. (deceased), Professor of Physics, Oklahoma State University (Section 22, Alternative Separation Processes) Kent Pollock, Ph.D., Member of Technical Staff, Group 91, Space Surveillance Techniques, MIT Lincoln Laboratory (Section 22, Alternative Separation Processes) George Priday,B.S., Ch.E., EIMCO Process Equipment Company; Member, American Institute of Chemical Engineering {AIChE), Instrument Society of America (ISA) (Section 18, LiquidSolid Operations and Equipment) Michael E. Prudich, Ph.D" Professor and Chair of Chemical Engineering, Ohio U~iversity; Member, American Institute of Chemical Engineers, American Chemical Society, Society of Mining Engineers, American Society for Engineering Education (Section 22, Alternative Skparation Processes) Raj K. Rajamani, Ph.D., Professor, Department of Metallurgy and Metallurgical Engineering, University of Utah, Salt Lake City, Utah (Section 19, Solid-Solid Operations and Equipment) Lawrence K. Rath, B.S., P.E., Federal Energy Technology Center (Morgantown), U .S. Department of Energy; Member, American Institute of Chemical Engineers (Section 27, Energy Resources, Conversion, and Utilization) Grantges J. Raymus, M.E., M.S., President, Raymus Associates, Incorporated, Packaging consultants; Adjunct Professor and Program Coordinator, Center for Packaging Science and Engineering, College of Engineering, Rutgers, The State University of New Jersey; formerly Manager of Packaging Engineering, Union Carbide Corporation; Registered Professional Engineer, california; Member, Institute of Packaging Professionals, ASME (Section 21, Handling of Bulk Solids and Packagi?g of Solids and Liquids) Lanny A. Robbins, Ph.D., Research Fellow, Dow Chemical Company; Member, American Institute of Chemical Engineers (Section 15, Liquid-Liquid Extraction Operations and Equipment) Joseph J. Santoleri, P.E.,Senior Consultant, RMT/Four Nines; Member, American Institute of Chemical Engineers, American Society of Mechanical Engineers, Air and Waste Management Association (Section 25, Waste Management) Adel F. Sarofim, Sc.D., Lammot DuPont Professor of Chemical Engineering and Assistant Director, Fuels Research Laboratory, Massachusetts Institute of Technology; Member, American Institute of Chemical Engineers, American Chemical Society, Combustion Institute (Section 5, Heat and Mass Transfer) Kalanadh v. S. Sastry, Ph.D., Professor, Department of Materials Science and Mineral Engineering, University of California, Berkeley, CA; Member, American Institute of Chemical Engineers, Society for Mining, Metallurgy and Exploration (Section 19, Solid-Solid Operations and Equipment) Paul J. Schafbuch, Ph.D" Senior Research Specialist, ~inal Control Systems, Fisher Controls International, Inc., Marshalltown, Iowa (Section 8, Process Control) Carl A. Schiappa, B.S., Ch.E., Process Engineering Associate, Michigan Division Engineering, The Dow Chemical Company; Member, AIChE and CCPS (Section 26, Process Safety) David K. Schmalzer, Ph.D., P.E., Fossil Energy Program Manager, Argonne National Laboratory; Member, American Chemical Society, Americanlnstitute of Chemical Engineers (Section 27, Energy Resources, Conversion, and Utilization) J. D. Seader, Ph.D., Professor of Chemical Engineering, University of Utah, Salt Lake City, Utah; Fellow, American Institute of Chemical Engineers; Member, American Chemical Society; Member. American SocietY for Enl!ineerin!! Education (Section 13. Distillation)

xvi

LIST OF CONTRIBUTORS

Dale E. Seborg, Ph.D., Professor of Chemical Engineering, University of California, Santa Barbara, California (Section 8, Process Control) Richard L. Shilling, Fintube Company-a

P.E., B.S.M., B.E.M.E., Manager of Engineering Development, Brown Koch Engineering Company; Member, American Society of Mechanical

Engineers (Section 11, Heat-Transfer Equipment) F. Greg Shinskey, B.S.Ch.E.,

Consultant (retired from Foxboro Co.), North Sandwich, New

Hampshire (Section 8, Process Control) Oliver w. Siebert, P.E., B.S.M.E., Washington University, Graduate Metallurgical Engineering, Sever Institute of Technology; Professor, Department of Chemical Engineering, Washington University, St. Louis, Missouri; President, Siebert Materials Engineering, Inc., St, Louis, Missouri; Senior Engineering Fellow (retired), Monsanto Co.; Mechanical Designer, Sverdrup Corp.; Metallurgist, Carondelet Foundry; United Nations Consultant to the People's Republic of China; Fellow, American Institute of Chemical Engineers; Life Fellow, American Society of Mechanical Engineers; Past Elected Director and Fellow, N ational Association of Corrosion Engineers, Int'I; American Society for Metals, Int'I; American Welding Society; Pi Tau Sigma, Sigma Xi, and Tau Beta Pi (Section 28, Materials of Construction) Jeffrey J. Siirola, Ph.D., Research Fellow, Eastman Chemical Company; Member, National Academy of Engineering; Fellow, American Institute ofChemical Engineers, American Chemical Society, American Association for Artificial Intelligence, American Society for Engineering Education (Section 13, Distillation) Charles E. Silverblatt, M.S., Ch.E., Peregrine International Associates, Inc.; Consultant to WesTech Engineering, Inc., American Institute of Mining, Metallurgical and Petroleum Engines (AIME) (Section 18, Liquid-Solid Operations and Equipment) Richard

Siwek, M.S., Explosion Protection Manager, Corporate Unit Safety and Environment,

Ciba-Geigy Ltd., Basel, Switzerland (Section 26, Process Safety) J. Stephen

Slottee, M.S., Ch.E., Manager, Technology and Development, EIMCO

Equipment

Co.; Member, American Institute of Chemical Engineers (AIChE)

Process

(Section 18,

Liquid-Solid Operations and Equipment) Cecil L. Smith,

Ph.D.,

Principal, Cecil L. Smith Inc., Baton Rouge, Louisiana (Section 8,

Process Control) Julian C. Smith, B. Chem., Ch.E., Professor Emeritus Chemical Engineering, Cornell University; Member, American Chemical Society (ACS), American Institute of Chemical Engineers (AIChE) (Section 18, Liquid-Solid Operations and Equipment) Richard H. Snow, Ph.D., Engineering Advisor, lIT Research Institute; Member, American Chemical Society, Sigma Xi; Fellow, American Institute of Chemical Engineers (Section 20, Size Reduction and Size Enlargement) Thomas Sorenson,

M.B.A.,

MiD. Eng., President, Galigher Ash (Canada) Ltd. (Section 19,

Solid-Solid Operations and Equipment) Rameshwar D. Srivastava, Ph.D., Fuels Group Manager, Burns and Roe Services Corporation, Federal Energy Technology Center (Pittsburgh) (Section 27, Energy Resources, Conversion, and Utilization ) F. C. Standiford,

M.S., P.E., Member, American Institute of Chemical Engineers, American

Chemical Society (Section 11, Heat-Transfer Equipment) D. E. SteiDmeyer, M.A., M.S., P.E., Distinguished Fellow, Monsanto Company; Fellow; American Institute of Chemical Engineers; Member, American Chemical Society (Section 14, Gas Absorption and Gas-Liquid System Design) Gary J. Stiegel, M.S., P.E., Program Coordinator, Federal Energy Technology Center (Pittsburgh), U .S. Department of Energy (Section 27, Energy Resources, Conversion, and Utilization)

LIST OF CONTRIBUTORS

xvii

John G. Stoecker II, B.S.M.E., University of Missouri School of Mines and Metallurgy; principal Consultant, Stoecker & Associates, St. Louis, Missouri; Principal Materials Engineering Specialist (retired), Monsanto Co.; High-Temperature Design/Application Engineer, Abex Corporation; Member, NACE International, ASM International (Section 27, Energy Resources, Conversion, and Utilization) Judson S. Swearingen,

Ph.D., Retired President, Rotoflow Corporation (Section 29, Process

Machinery Drives) Louis Theodore, Sc.D., Professor of Chemical Engineering, Manhattan College; Member, Air and Waste Management Association (Section 25, Waste Management) Michael P. Thien, Sc.D., Senior Research Fellow, Merck & Co., Inc.; Member, American Institute of Chemical Engineers, American Chemical Society, International Society for Pharmaceutical Engineers (Section 22, Alternative Separation Processes) George H. Thomson, and Chemical Data)

AIChE Design Institute for Physical Property Data (Section 2, Physical

James N. Tilton, Ph.D., P.E., Senior Consultant, Process Engineering, E. I. duPont de Nemours & Co.; Member, American Institute of Chemical Engineers; Registered Professional Engineer (Delaware) (Section 6, Fluid and Particle Dynamics) Klaus D. Timmerhaus, Ph.D., P.E., Professor and President's Teaching Scholar, University of Colorado, Boulder, Colorado; Fellow, American Institute of Chemical Engineers, American Society for Engineering Education, American Association for the Advancement of Science; Member, American Astronautical Society, N ational Academy of Engineering, Austrian Academy of Science, International Institute of Refrigeration, American Society of Heating, Refrigerating and Air Conditioning Engineers, American Society of Environmental Engineers, Engineering Society for Advancing Mobility on Land, Sea, Air, and Space, Sigma Xi, The Research Society (Section 11, Heat-Transfer Equipment) David B. Todd, Ph.D., President, Todd Engineering; Member, American Association for the Advancement of Science (AAAS), American Chemical Society (ACS), American Institute of Chemical Engineering (AIChE), American Oil Chemists Society (AOCS), Society of Plastics Engineers (SPE), and Society of the Plastics Industry (SPI); Registered Professional Engineer, Michigan (Section 18, Liquid-Solid Operations and Equipment) George T. Tsao, Ph.D., Director, Laboratory for Renewable Resource Engineering, Purdue University; Member, American Institute of Chemical Engineers, American Chemical Society, American Society for Microbiology (Section 24, Biochemical Engineering) Hendrick c. Van Ness, D.Eng., Howard P. Isermann Department of Chemical Engineering, Rensselaer Polytechnic Institute; Fellow, American Institute of Chemical Engineers; Member, American Chemical Society (Section 4, Thermodynamics) Stanley M. Walas, Ph.D., Professor Emeritus, Department of Chemical and Petroleum Engineering, University of Kansas; Fellow, American Institute of Chemical Engineers (Section 7, Reaction Kinetics; Section 23, Chemical Reactors) Phillip C. Wankat, Ph.D., Professor of Chemical Engineering, Purdue University; Member, American Institute of Chemical Engineers, American Chemical Society, International Adsorption Society (Section 5, Heat and Mass Transfer) Ionel Wechsler, M.S., MiD. and Met., Vice President, Sala Magnetics, Inc., Cambridge, Massachusetts (Section 19, Solid-Solid Operations and Equipment) Arthur W. Westerberg, Ph.D., Swearingen University Professor of Chemical Engineering, Carnegie Mellon University; Member, National Academy of Engineering (Section 3, Mathematics) John M. Wheeldon, Ph.D., Electric Power Research Institute (Section 27, Energy Resources, Conversion. and Utilization)

xviii

LIST OF CONTRIBUTORS

Robert E. White, Ph.D., Principal Engineer; Chemistry and Chemical Engineering Division, Southwest Research Institute (Section 26, Process Safety) J. K. Wilkinson, M.Sc., Consultant Chemical Engineer; Fellow, Institution of Chemical Engineers, London (Section 9, Process Economics) David Winegarder, Ph.D., Engineering Associate, Michigan Division Engineering, The Dow Chemical Company; Member AIChE and CCPS (Section 26, Process Safety) John L. Woodward,

Ph.D., Principal, DNV Technica, Inc. (Section 26, Process Safety)

Yoshiyuki Yamashita, Ph.D., Associate Professor of Chemical Engineering, Tohoku University, Sendai, Japan (Section 3, Mathematics) Cannen M. Yon, M.S., DevelopmentAssociate, UOP, Des Plaines, Illinois; Member, American Institute of Chemical Engineers (Section 16, Adsorption and Ion Exchange)

Preface to the Seventh Edition

Perry’s has been an important source for chemical engineering information since 1934. The significant contributions of the editors who have guided preparation of the previous editions is acknowledged. These include John H. Perry (first to third editions), Robert H. Perry (fourth to sixth editions), Cecil H. Chilton (fourth and fifth editions), and Sidney D. Kirkpatrick (fourth edition). Ray Genereaux (DuPont) contributed to each of the first six editions, and Shelby Miller (Argonne National Lab) worked on the second through the seventh. The current editors directed both the sixth and seventh editions. Advances in the technology of chemical engineering have continued as we have moved toward the twenty-first century, and this edition will carry us into that century. The Handbook has been reorganized. The first group of sections focuses on chemical and physical property data and the fundamentals of chemical engineering. The second and largest group of sections deals with processes, generally divided as heat transfer operations, distillation, kinetics, liquid-liquid, liquid-solid, and so on. The last group treats auxiliary information such as materials of construction, process machinery drives, waste management, and process safety. All sections have been revised and updated, and several sections are entirely new or have been extensively revised. Examples of these sections are mathematics, mass transfer, reaction kinetics, process control, transport and storage of fluids, alternative separation processes, heat-transfer equipment, chemical reactions, liquid-solid operations and equipment, process safety, and analysis of plant performance. Significant new information has also been included in the physical and chemical data sections. Several section editors and contributors worked on this seventh edition, and these persons and their affiliations are listed as a part of the front material. Approximately one-half of the section editors are fellows of the AIChE. In addition, the following chemical engineering students at the University of Kansas assisted in the preparation of the index: Jason Canter, Pau Ying Chong, Mei Ling Chuah, Li Phoon Hor, Siew Pouy Ng, Francis J. Orzulak, Scott C. Renze, Page B. Surbaugh, and Stephen F. Weller. Shari L. Gladman and Sarah Smith provided extensive secretarial assistance. Much of Bob Perry’s work carries over into this edition and his influence is both recognized and remembered. DON W. GREEN JAMES O. MALONEY University of Kansas April, 1997 xix

For the detailed contents of any section, consult the title page of that section. See also the alphabetical index in the back of the handbook.

Section

Conversion Factors and Mathematical Symbols

Jam£s 0. Maloney

1

Physical and Chemical Data Peter E. Liley, George H. Thomson, D. G. Friend, Thomas E. Daubert, Evan Buck

2

Mathematics Bruce A. Finlayson, Jam£s F Davis, Arthur W Westerberg, Yoshiyuki Yamashita

3

Thermodynamics

4

Hendrick C. Van Ness, Michael M. Abbott

Heat and Mass Transfer Jam£s G. Knudsen, Hoyt C. Hottel, Adel F Sarofim, Phillip c. Wankat, Kent S. Knaebel

5

Fluid and Particle Dynamics

6

Reaction Kinetics

Jam£s N. Tilton

Stanley M. Walas

7

Process Control Thomas F Edgar; Cecil L. Smith, F Greg Shinskey, George W Gassman, Paul J. Schajbuch, Thomas J. McAvoy, Dale E. Seborg

8

Process Economics

9

F A. Holland, J. K Wilkinson

Transport and Storage of Fluids

Meherwan 1'. Royce

10

Heat- Transfer Equipment Richard L. Shilling, Kenneth J. Bell, Patrick M. Bemhagen, Thomas M. Flynn, Victor M. Goldschmidt, Predrag S. Hrnjak, F C. Standiford, Klaus D. Timmerhaus

11

Psychrometry, Evaporative Cooling, and Solids Drying Glenn W Baldwin

12

Distillation

Charles G. Moyers,

J. D. Seader;Jeffrey J. Siirola, Scott D. Bamicki

13

vii

viii

CONTENTS

Gas Absorption and Gas-Liquid System Design W R. Penny, B. B. Crocker

james R. Fair; D. E. Steinmeyer;

Liquid-Liquid Extraction Operations and Equipment Roger W Cusack Adsorption and Ion Exchange

14 Lanny A. Robbins,

M. Douglas LeVan, Giorgio Carta, Carmen M. Yon

Gas-Solid Operations and Equipment

Mel Pell, james B. Dunson

15 16 17

Liquid-Solid Operations and Equipment Donald A. Dahlstrom, Richard C. Bennett, Robert G, Emmet, Peter Harriott, Tim Laros, Wallace Leung, Shelby A. Miller; Brooker Morey, james Y. Oldshue, George Priday, Charles E. Silverblatt, I. Stephen Slottee, Iulian c. Smith

18

Solid-Solid Operations and Equipment Kalanadh V S. Sastry, Harrison Cooper; Richard Hogg, T L. p Iespen, Frank Knoll, Bhupendra Parekh, Raj K Rajamani, Thomas Sorenson, Ionel Wechsler; Chad McCleary, David B. Todd

19

Size Reduction and Size Enlargement Bryan I. Ennis, james D. Litster

Richard L. Snow, Terry Allen,

Handling of Bulk Solids and Packaging of Solids and Liquids

Grantges I. Raymus

Alternative Separation Processes joseph D. Henry, Ir:, Michael E. Prudich, William Eykamp, T Alan Hatton, Keith p johnston, Richard M. Lemert, Robert Lemlich, Charles G. Moyers, john Newman, Herbert A. Pohl, Kent Pollock, Michael p Thien Chemical Reactors

Stanley M. Walas

Biochemical Engineering

Henry R. Bungay, Arthur E. Humphrey, George T Tsao

20 21

22 23 24

Waste Management Louis Theodore, Anthony I. Buonicore, John D. McKenna, Irwin I. Kugelman, john s. Ieris, joseph I. Santoleri, Thomas R McGowan

25

Process Safety Stanley M. Englund, Frank T Bodurtha, Laurence G. Britton, Daniel A. Crowl, Stanley Grossel, W G. High, Trevor A. Kletz, Robert W Ormsby, john E. Owens, Carl A. Schiappa, Richard Siwek, Robert E. White, David Winegardner; john L. Woodward

26

Energy Resources, Conversion, and Utilization Walter R Podolski, Shelby A. Miller; David K Schnialzer; Anth9ny G. Fonseca, Vincent Conrad, Douglas E. Lowenhaupt, john Bacha, Lawrence K Rath, Hsue-peng Loh, EdgarB. Klunder; Howard G. McIlvried, III, Gary I. Stiegel, Rameshwar D. Srivastava, Peter I. Loftus, Charles E. Benson, john M. Wheeldon, Michael Krumpelt

27

Materials of Construction

28

Oliver W Siebert, john G. Stoecker

Process Machinery Drives Heinz p Bloch, R. H. Daugherty, Fred K Geitner; Meherwan p Boyce, judson S. Swearingen, Eric jennet, Michael M. Calistrat Analysis of Plant Performance Index follows Section 30.

Colin S. Howat

29 30

Section 1

Conversion Factors and Mathematical Symbols*

James O. Maloney, Ph.D., P.E., Emeritus Professor of Chemical Engineering, University of Kansas; Fellow, American Institute of Chemical Engineering; Fellow, American Association for the Advancement of Science; Member, American Chemical Society, American Society for Engineering Education

Fig. 1-1 Table 1-1 Table 1-2a Table 1-2b Table 1-3 Table 1-4 Table 1-5 Table 1-6 Table 1-7 Table 1-8 Table 1-9

CONVERSION FACTORS Graphic Relationships of SI Units with Names . . . . . . . . . SI Base and Supplementary Quantities and Units. . . . . . . Derived Units of SI that Have Special Names. . . . . . . . . . Additional Common Derived Units of SI . . . . . . . . . . . . . SI Prefixes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conversion Factors: U.S. Customary and Commonly Used Units to SI Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . Metric Conversion Factors as Exact Numerical Multiples of SI Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alphabetical Listing of Common Conversions . . . . . . . . . Common Units and Conversion Factors . . . . . . . . . . . . . . Kinematic-Viscosity Conversion Formulas . . . . . . . . . . . . Values of the Gas-Law Constant. . . . . . . . . . . . . . . . . . . . .

1-2 1-3 1-3 1-3 1-3 1-4 1-13 1-15 1-18 1-18 1-18

Table 1-10 United States Customary System of Weights and Measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 1-11 Temperature Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . Table 1-12 Specific Gravity, Degrees Baumé, Degrees API, Degrees Twaddell, Pounds per Gallon, Pounds per Cubic Foot . . . Table 1-13 Wire and Sheet-Metal Gauges . . . . . . . . . . . . . . . . . . . . . . Table 1-14 Fundamental Physical Constants . . . . . . . . . . . . . . . . . . . .

1-19 1-19 1-20 1-21 1-22

CONVERSION OF VALUES FROM U.S. CUSTOMARY UNITS TO SI UNITS Table 1-15 Table 1-16

MATHEMATICAL SYMBOLS Mathematical Signs, Symbols, and Abbreviations . . . . . . . Greek Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1-24 1-24

* Much of the material was taken from Sec. 1. of the fifth edition. The contribution of Cecil H. Chilton in developing that material is acknowledged. 1-1

FIG. 1-1

1-2

Graphic relationships of SI units with names (U.S. National Bureau of Standards, LC 1078, December 1976.)

TABLE 1-1

SI Base and Supplementary Quantities and Units

Quantity or “dimension” Base quantity or “dimension” length mass time electric current thermodynamic temperature amount of substance luminous intensity Supplementary quantity or “dimension” plane angle solid angle

SI unit

SI unit symbol (“abbreviation”); Use roman (upright) type

meter kilogram second ampere kelvin mole* candela

m kg s A K mol cd

radian steradian

rad sr

* When the mole is used, the elementary entities must be specified; they may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles.

TABLE 1-2a Derived Units of SI that Have Special Names Quantity

Unit

Symbol

Formula

frequency (of a periodic phenomenon) force pressure, stress energy, work, quantity of heat power, radiant flux quantity of electricity, electric charge electric potential, potential difference, electromotive force capacitance electric resistance conductance magnetic flux magnetic-flux density inductance luminous flux illuminance activity (of radionuclides) absorbed dose

hertz newton pascal joule watt coulomb volt

Hz N Pa J W C V

l/s (kg⋅m)/s2 N/m2 N⋅m J/s A⋅s W/A

farad ohm siemens weber tesla henry lumen lux becquerel gray

F Ω S Wb T H lm lx Bq Gy

C/V V/A A/V V⋅s Wb/m2 Wb/A cd⋅sr lm/m2 l/s J/kg

TABLE 1-2b Additional Common Derived Units of SI Quantity

Unit

acceleration angular acceleration angular velocity area concentration (of amount of substance) current density density, mass electric-charge density electric-field strength electric-flux density energy density entropy heat capacity heat-flux density, irradiance luminance magnetic-field strength molar energy molar entropy molar-heat capacity moment of force permeability permittivity radiance

meter per second squared radian per second squared radian per second square meter mole per cubic meter

m/s2 rad/s2 rad/s m2 mol/m3

ampere per square meter kilogram per cubic meter coulomb per cubic meter volt per meter coulomb per square meter joule per cubic meter joule per kelvin joule per kelvin watt per square meter

A/m2 kg/m3 C/m3 V/m C/m2 J/m3 J/K J/K W/m2

candela per square meter ampere per meter joule per mole joule per mole-kelvin joule per mole-kelvin newton-meter henry per meter farad per meter watt per square-metersteradian watt per steradian joule per kilogram-kelvin joule per kilogram joule per kilogram-kelvin cubic meter per kilogram newton per meter watt per meter-kelvin meter per second pascal-second square meter per second cubic meter 1 per meter

cd/m2 A/m J/mol J/(mol⋅K) J/(mol⋅K) N⋅m H/m F/m W/(m2⋅sr)

radiant intensity specific-heat capacity specific energy specific entropy specific volume surface tension thermal conductivity velocity viscosity, dynamic viscosity, kinematic volume wave number

TABLE 1-3

Symbol

W/sr J/(kg⋅K) J/kg J/(kg⋅K) m3/kg N/m W/(m⋅K) m/s Pa⋅s m2/s m3 1/m

SI Prefixes

Multiplication factor

Prefix

Symbol

000 = 1018 000 = 1015 000 = 1012 000 = 109 000 = 106 000 = 103 100 = 102 10 = 101 0.1 = 10−1 0.01 = 10−2 0.001 = 10−3 0.000 001 = 10−6 0.000 000 001 = 10−9 0.000 000 000 001 = 10−12 0.000 000 000 000 001 = 10−15 0.000 000 000 000 000 001 = 10−18

exa peta tera giga mega kilo hecto* deka* deci* centi milli micro nano pico femto atto

E P T G M k h da d c m µ n p f a

1 000 000 000 1 000 000 1 000 1

000 000 000 000 1

000 000 000 000 000 1

*Generally to be avoided.

1-3

TABLE 1-4

Conversion Factors: U.S. Customary and Commonly Used Units to SI Units Customary or commonly used unit

Quantity

SI unit

Alternate SI unit

Conversion factor; multiply customary unit by factor to obtain SI unit

Space,† time Length

naut mi mi chain link fathom yd ft

1.852* E + 00 1.609 344* E + 00 2.011 68* E + 01 2.011 68* E − 01 1.828 8* E + 00 9.144* E − 01 3.048* E − 01 3.048* E + 01 2.54* E + 01 2.54 E + 00 2.54* E + 01

in in mil

km km m m m m m cm mm cm µm

Length/length

ft/mi

m/km

1.893 939 E − 01

Length/volume

ft/U.S. gal ft/ft3 ft/bbl

m/m3 m/m3 m/m3

8.051 964 E + 01 1.076 391 E + 01 1.917 134 E + 00

Area

mi2 section acre ha yd2 ft2 in2

km2 ha ha m2 m2 m2 mm2 cm2

2.589 988 E + 00 2.589 988 E + 02 4.046 856 E − 01 1.000 000* E + 04 8.361 274 E − 01 9.290 304* E − 02 6.451 6* E + 02 6.451 6* E + 00

Area/volume

ft2/in3 ft2/ft3

m2/cm3 m2/m3

5.699 291 E − 03 3.280 840 E + 00

Volume

cubem acre⋅ft

km3 m3 ha⋅m m3 m3 m3 dm3 m3 dm3 m3 dm3 dm3 dm3 dm3 cm3 cm3 cm3

4.168 182 1.233 482 1.233 482 7.645 549 1.589 873 2.831 685 2.831 685 4.546 092 4.546 092 3.785 412 3.785 412 1.136 523 9.463 529 4.731 765 2.841 307 2.957 353 1.638 706

yd3 bbl (42 U.S. gal) ft3 U.K. gal U.S. gal U.K. qt U.S. qt U.S. pt U.K. fl oz U.S. fl oz in3

L L L L L L

E + 00 E + 03 E − 01 E − 01 E − 01 E − 02 E + 01 E − 03 E + 00 E − 03 E + 00 E + 00 E − 01 E − 01 E + 01 E + 01 E + 01

Volume/length (linear displacement)

bbl/in bbl/ft ft3/ft U.S. gal/ft

m3m m3/m m3/m m3/m L/m

6.259 342 E + 00 5.216 119 E − 01 9.290 304* E − 02 1.241 933 E − 02 1.241 933 E + 01

Plane angle

rad deg (°) min (′) sec (″)

rad rad rad rad

1 1.745 329 E − 02 2.908 882 E − 04 4.848 137 E − 06

Solid angle

sr

sr

1

Time

year week h

a d s min s h ns

1 7.0* 3.6* 6.0* 6.0* 1.666 667 1

E + 00 E + 03 E + 01 E + 01 E − 02

1.016 047 9.071 847 5.080 234 4.535 924 4.535 924 3.110 348 2.834 952 6.479 891

E + 00 E − 01 E + 01 E + 01 E − 01 E + 01 E + 01 E + 01

min mµs Mass, amount of substance Mass

1-4

U.K. ton U.S. ton U.K. cwt U.S. cwt lbm oz (troy) oz (av) gr

Mg Mg kg kg kg g g mg

t t

TABLE 1-4

Conversion Factors: U.S. Customary and Commonly Used Units to SI Units (Continued ) Customary or commonly used unit

Quantity Amount of substance

lbm⋅mol std m3(0°C, 1 atm) std ft3 (60°F, 1 atm)

SI unit

Alternate SI unit

Conversion factor; multiply customary unit by factor to obtain SI unit 4.535 924 E − 01 4.461 58 E − 02 1.195 30 E − 03

kmol kmol kmol

Enthalpy, calorific value, heat, entropy, heat capacity Calorific value, enthalpy (mass basis)

Btu/lbm cal/g cal/lbm

MJ/kg kJ/kg kWh/kg kJ/kg J/kg

Caloric value, enthalpy (mole basis)

kcal/(g⋅mol) Btu/(lb⋅mol)

kJ/kmol kJ/kmol

Calorific value (volume basis—solids and liquids)

Btu/U.S. gal

MJ/m3 kJ/m3 kWh/m3 MJ/m3 kJ/m3 kWh/m3 MJ/m3 kJ/m3 kWh/m3 MJ/m3 kJ/m3

kJ/dm3

Btu/U.K. gal Btu/ft3

cal/mL (ft⋅lbf)/U.S. gal

J/g J/g

2.326 000 2.326 000 6.461 112 4.184* 9.224 141

E − 03 E + 00 E − 04 E + 00 E + 00

4.184* E + 03 2.326 000 E + 00

kJ/dm3 kJ/dm3

2.787 163 2.787 163 7.742 119 2.320 800 2.320 800 6.446 667 3.725 895 3.725 895 1.034 971 4.184* 3.581 692

E − 01 E + 02 E − 02 E − 01 E + 02 E − 02 E − 02 E + 01 E − 02 E + 00 E − 01

Calorific value (volume basis—gases)

cal/mL kcal/m3 Btu/ft3

kJ/m3 kJ/m3 kJ/m3 kWh/m3

J/dm3 J/dm3 J/dm3

4.184* 4.184* 3.725 895 1.034 971

E + 03 E + 00 E + 01 E − 02

Specific entropy

Btu/(lbm⋅°R) cal/(g⋅K) kcal/(kg⋅°C)

kJ/(kg⋅K) kJ/(kg⋅K) kJ/(kg⋅K)

J/(g⋅K) J/(g⋅K) J/(g⋅K)

4.186 8* 4.184* 4.184*

E + 00 E + 00 E + 00

Specific-heat capacity (mass basis)

kWh/(kg⋅°C) Btu/(lbm⋅°F) kcal/(kg⋅°C)

kJ/(kg⋅K) kJ/(kg⋅K) kJ/(kg⋅K)

J/(g⋅K) J/(g⋅K) J/(g⋅K)

3.6* 4.186 8* 4.184*

E + 03 E + 00 E + 00

Specific-heat capacity (mole basis)

Btu/(lb⋅mol⋅°F) cal/(g⋅mol⋅°C)

kJ/(kmol⋅K) kJ/(kmol⋅K)

4.186 8* 4.184*

E + 00 E + 00

Temperature (absolute)

°R K

K K

5/9 1

Temperature (traditional)

°F

°C

5/9(°F − 32)

Temperature (difference)

°F

K, °C

5/9

Pressure

atm (760 mmHg at 0°C or 14,696 psi)

µmHg (0°C) µ bar mmHg = torr (0°C) cmH2O (4°C) lbf/ft2 (psf) mHg (0°C) bar dyn/cm2

MPa kPa bar MPa kPa MPa kPa bar kPa kPa kPa kPa kPa Pa Pa Pa

1.013 250* E − 01 1.013 250* E + 02 1.013 250* E + 00 1.0* E − 01 1.0* E + 02 6.894 757 E − 03 6.894 757 E + 00 6.894 757 E − 02 3.376 85 E + 00 2.488 4 E − 01 1.333 224 E − 01 9.806 38 E − 02 4.788 026 E − 02 1.333 224 E − 01 1.0* E + 05 1.0* E − 01

Vacuum, draft

inHg (60°F) inH2O (39.2°F) inH2O (60°F) mmHg (0°C) = torr cmH2O (4°C)

kPa kPa kPa kPa kPa

3.376 85 2.490 82 2.488 4 1.333 224 9.806 38

E + 00 E − 01 E − 01 E − 01 E − 02

Liquid head

ft in

m mm cm

3.048* 2.54* 2.54*

E − 01 E + 01 E + 00

Pressure drop/length

psi/ft

kPa/m

2.262 059 E + 01

Temperature, pressure, vacuum

bar mmHg (0°C) = torr

1-5

TABLE 1-4

Conversion Factors: U.S. Customary and Commonly Used Units to SI Units (Continued ) Customary or commonly used unit

Quantity

SI unit

Alternate SI unit

Conversion factor; multiply customary unit by factor to obtain SI unit

Density, specific volume, concentration, dosage kg/m3 g/m3 kg/m3 g/cm3 kg/m3 kg/m3 g/cm3 kg/m3 kg/m3

1.601 846 1.601 846 1.198 264 1.198 264 9.977 633 1.601 846 1.601 846 1.0* 1.601 846

E + 01 E + 04 E + 02 E − 01 E + 01 E + 01 E − 02 E + 03 E + 01

ft /lbm U.K. gal/lbm U.S. gal/lbm

m3/kg m3/g dm3/kg dm3/kg dm3/kg

6.242 796 6.242 796 6.242 796 1.002 242 8.345 404

E − 02 E − 05 E + 01 E + 01 E + 00

Specific volume (mole basis)

L/(g⋅mol) ft3/(lb⋅mol)

m3/kmol m3/kmol

1 6.242 796 E − 02

Specific volume

bbl/U.S. ton bbl/U.K. ton

m3/t m3/t

1.752 535 E − 01 1.564 763 E − 01

Yield

bbl/U.S. ton bbl/U.K. ton U.S. gal/U.S. ton U.S. gal/U.K. ton

dm3/t dm3/t dm3/t dm3/t

Concentration (mass/mass)

wt % wt ppm

kg/kg g/kg mg/kg

lbm/bbl g/U.S. gal g/U.K. gal lbm/1000 U.S. gal lbm/1000 U.K. gal gr/U.S. gal gr/ft3 lbm/1000 bbl mg/U.S. gal gr/100 ft3

kg/m3 kg/m3 kg/m3 g/m3 g/m3 g/m3 mg/m3 g/m3 g/m3 mg/m3

ft3/ft3 bbl/(acre⋅ft) vol% U.K. gal/ft3 U.S. gal/ft3 mL/U.S. gal mL/U.K. gal vol ppm U.K. gal/1000 bbl U.S. gal/1000 bbl U.K. pt/1000 bbl

m3/m3 m3/m3 m3/m3 dm3/m3 dm3/m3 dm3/m3 dm3/m3 cm3/m3 dm3/m3 cm3/m3 cm3/m3 cm3/m3

Concentration (mole/volume)

(lb⋅mol)/U.S. gal (lb⋅mol)/U.K. gal (lb⋅mol)/ft3 std ft3 (60°F, 1 atm)/bbl

kmol/m3 kmol/m3 kmol/m3 kmol/m3

Concentration (volume/mole)

U.S. gal/1000 std ft3 (60°F/60°F) bbl/million std ft3 (60°F/60°F)

dm3/kmol dm3/kmol

Throughput (mass basis)

U.K. ton/year U.S. ton/year U.K. ton/day

Density

lbm/ft3 lbm/U.S. gal lbm/U.K. gal lbm/ft3 g/cm3 lbm/ft3

Specific volume

ft3/lbm 3

Concentration (mass/volume)

Concentration (volume/volume)

cm3/g cm3/g

L/t L/t L/t L/t

g/dm3 g/L mg/dm3 mg/dm3 mg/dm3 mg/dm3 mg/dm3

L/m3 L/m3 L/m3 L/m3 L/m3

L/kmol L/kmol

1.752 535 1.564 763 4.172 702 3.725 627

E + 02 E + 02 E + 00 E + 00

1.0* 1.0* 1

E − 02 E + 01

2.853 010 2.641 720 2.199 692 1.198 264 9.977 633 1.711 806 2.288 351 2.853 010 2.641 720 2.288 351

E + 00 E − 01 E − 01 E + 02 E + 01 E + 01 E + 03 E + 00 E − 01 E + 01

1 1.288 931 1.0* 1.605 437 1.336 806 2.641 720 2.199 692 1 1.0* 2.859 403 2.380 952 3.574 253

E − 04 E − 02 E + 02 E + 02 E − 01 E − 01 E − 03 E + 01 E + 01 E + 00

1.198 264 9.977 644 1.601 846 7.518 21

E + 02 E + 01 E + 01 E − 03

3.166 91 1.330 10

E + 00 E − 01

1.016 047 9.071 847 1.016 047 4.233 529 9.071 847 3.779 936 1.016 047 9.071 847 4.535 924

E + 00 E − 01 E + 00 E − 02 E − 01 E − 02 E + 00 E − 01 E − 01

Facility throughput, capacity

U.S. ton/day U.K. ton/h U.S. ton/h lbm/h

1-6

t/a t/a t/d t/h t/d t/h t/h t/h kg/h

TABLE 1-4

Conversion Factors: U.S. Customary and Commonly Used Units to SI Units (Continued ) Customary or commonly used unit

Quantity Throughput (volume basis)

bbl/day 3

ft /day bbl/h ft3/h U.K. gal/h U.S. gal/h U.K. gal/min U.S. gal/min

SI unit

Alternate SI unit

Conversion factor; multiply customary unit by factor to obtain SI unit E + 01 E − 01 E − 03 E − 01 E − 02 E − 03 E − 03 E − 03 E − 03 E − 01 E − 02 E − 01 E − 02

t/a m3/d m3/h m3/h m3/h m3/h L/s m3/h L/s m3/h L/s m3/h L/s

5.803 036 1.589 873 1.179 869 1.589 873 2.831 685 4.546 092 1.262 803 3.785 412 1.051 503 2.727 655 7.576 819 2.271 247 6.309 020

kmol/h kmol/s

4.535 924 E − 01 1.259 979 E − 04

Throughput (mole basis)

(lbm⋅mol)/h

Flow rate (mass basis)

U.K. ton/min U.S. ton/min U.K. ton/h U.S. ton/h U.K. ton/day U.S. ton/day million lbm/year U.K. ton/year U.S. ton/year lbm/s lbm/min lbm/h

kg/s kg/s kg/s kg/s kg/s kg/s kg/s kg/s kg/s kg/s kg/s kg/s

1.693 412 1.511 974 2.822 353 2.519 958 1.175 980 1.049 982 5.249 912 3.221 864 2.876 664 4.535 924 7.559 873 1.259 979

E + 01 E + 01 E − 01 E − 01 E − 02 E − 02 E + 00 E − 05 E − 05 E − 01 E − 03 E − 04

Flow rate (volume basis)

bbl/day

U.K. gal/h U.S. gal/h U.K. gal/min U.S. gal/min ft3/min ft3/s

m3/d L/s m3/d L/s m3/s L/s m3/s L/s dm3/s dm3/s dm3/s dm3/s dm3/s dm3/s

1.589 873 1.840 131 2.831 685 3.277 413 4.416 314 4.416 314 7.865 791 7.865 791 1.262 803 1.051 503 7.576 820 6.309 020 4.719 474 2.831 685

E − 01 E − 03 E − 02 E − 04 E − 05 E − 02 E − 06 E − 03 E − 03 E − 03 E − 02 E − 02 E − 01 E + 01

Flow rate (mole basis)

(lb⋅mol)/s (lb⋅mol)/h million scf/D

kmol/s kmol/s kmol/s

4.535 924 E − 01 1.259 979 E − 04 1.383 45 E − 02

Flow rate/length (mass basis)

lbm/(s⋅ft) lbm/(h⋅ft)

kg/(s⋅m) kg/(s⋅m)

1.488 164 E + 00 4.133 789 E − 04

Flow rate/length (volume basis)

U.K. gal/(min⋅ft) U.S. gal/(min⋅ft) U.K. gal/(h⋅in) U.S. gal/(h⋅in) U.K. gal/(h⋅ft) U.S. gal/(h⋅ft)

m2/s m2/s m2/s m2/s m2/s m2/s

Flow rate/area (mass basis)

lbm/(s⋅ft2) lbm/(h⋅ft2)

kg/(s⋅m2) kg/(s⋅m2)

Flow rate/area (volume basis)

ft3/(s⋅ft2) ft3/(min⋅ft2) U.K. gal/(h⋅in2) U.S. gal/(h⋅in2) U.K. gal/(min⋅ft2) U.S. gal/(min⋅ft2) U.K. gal/(h⋅ft2) U.S. gal/(h⋅ft2)

m/s m/s m/s m/s m/s m/s m/s m/s

Flow rate

3

ft /day bbl/h ft3/h

L/s L/s L/s L/s L/s L/s

m3/(s⋅m) m3/(s⋅m) m3/(s⋅m) m3/(s⋅m) m3/(s⋅m) m3/(s⋅m)

2.485 833 2.069 888 4.971 667 4.139 776 4.143 055 3.449 814

E − 04 E − 04 E − 05 E − 05 E − 06 E − 06

4.882 428 E + 00 1.356 230 E − 03 m3/(s⋅m2) m3/(s⋅m2) m3/(s⋅m2) m3/(s⋅m2) m3/(s⋅m2) m3/(s⋅m2) m3/(s⋅m2) m3/(s⋅m2)

3.048* 5.08* 1.957 349 1.629 833 8.155 621 6.790 972 1.359 270 1.131 829

E − 01 E − 03 E − 03 E − 03 E − 04 E − 04 E − 05 E − 05

1-7

TABLE 1-4

Conversion Factors: U.S. Customary and Commonly Used Units to SI Units (Continued ) Customary or commonly used unit

Quantity

SI unit

Alternate SI unit

Conversion factor; multiply customary unit by factor to obtain SI unit

Energy, work, power Energy, work

therm

E + 02 E + 05 E + 01 E + 01 E + 00 E + 03 E − 01 E + 00 E + 03 E − 01 E + 00 E + 03 E + 00 E − 04 E + 00 E − 04 E + 00 E − 03 E − 03 E − 03 E − 03 E − 05 E − 07

kcal cal ft⋅lbf lbf⋅ft J (lbf⋅ft2)/s2 erg

MJ kJ kWh MJ MJ kJ kWh MJ kJ kWh MJ kJ kJ kWh kJ kWh kJ kJ kJ kJ kJ kJ J

1.055 056 1.055 056 2.930 711 1.431 744 2.684 520 2.684 520 7.456 999 2.647 780 2.647 780 7.354 999 3.6* 3.6* 1.899 101 5.275 280 1.055 056 2.930 711 4.184* 4.184* 1.355 818 1.355 818 1.0* 4.214 011 1.0*

Impact energy

kgf⋅m lbf⋅ft

J J

9.806 650* E + 00 1.355 818 E + 00

Surface energy

erg/cm2

mJ/m2

1.0*

Specific-impact energy

(kgf⋅m)/cm2 (lbf⋅ft)/in2

J/cm2 J/cm2

9.806 650* E − 02 2.101 522 E − 03

Power

million Btu/h ton of refrigeration Btu/s kW hydraulic horsepower—hhp hp (electric) hp [(550 ft⋅lbf)/s] ch or CV Btu/min (ft⋅lbf)/s kcal/h Btu/h (ft⋅lbf)/min

MW kW kW kW kW kW kW kW kW kW W W W

2.930 711 3.516 853 1.055 056 1 7.460 43 7.46* 7.456 999 7.354 999 1.758 427 1.355 818 1.162 222 2.930 711 2.259 697

Power/area

Btu/(s⋅ft2) cal/(h⋅cm2) Btu/(h⋅ft2)

kW/m2 kW/m2 kW/m2

1.135 653 E + 01 1.162 222 E − 02 3.154 591 E − 03

Heat-release rate, mixing power

hp/ft3 cal/(h⋅cm3) Btu/(s⋅ft3) Btu/(h⋅ft3)

kW/m3 kW/m3 kW/m3 kW/m3

2.633 414 1.162 222 3.725 895 1.034 971

Cooling duty (machinery)

Btu/(bhp⋅h)

W/kW

Specific fuel consumption (mass basis)

lbm/(hp⋅h)

mg/J kg/kWh

kg/MJ

1.689 659 E − 01 6.082 774 E − 01

Specific fuel consumption (volume basis)

m3/kWh U.S. gal/(hp⋅h) U.K. pt/(hp⋅h)

dm3/MJ dm3/MJ dm3/MJ

mm3/J mm3/J mm3/J

2.777 778 E + 02 1.410 089 E + 00 2.116 806 E − 01

Fuel consumption

U.K. gal/mi U.S. gal/mi mi/U.S. gal mi/U.K. gal

dm3/100 km dm3/100 km km/dm3 km/dm3

L/100 km L/100 km km/L km/L

2.824 807 2.352 146 4.251 437 3.540 064

U.S. tonf⋅mi hp⋅h ch⋅h or CV⋅h kWh Chu Btu

1-8

E + 00

E − 01 E + 00 E + 00 E − 01 E − 01 E − 01 E − 01 E − 02 E − 03 E + 00 E − 01 E − 02

E + 01 E + 00 E + 01 E − 02

3.930 148 E − 01

E + 02 E + 02 E − 01 E − 01

TABLE 1-4

Conversion Factors: U.S. Customary and Commonly Used Units to SI Units (Continued ) Customary or commonly used unit

Quantity

SI unit

Alternate SI unit

Conversion factor; multiply customary unit by factor to obtain SI unit

in/s in/min

km/h km/h m/s cm/s m/s mm/s mm/s m/d mm/s mm/s

1.852* E + 00 1.609 344* E + 00 3.048* E − 01 3.048* E + 01 5.08* E − 03 8.466 667 E − 02 3.527 778 E − 03 3.048* E − 01 2.54* E + 01 4.233 333 E − 01

Corrosion rate

in/year (ipy) mil/year

mm/a mm/a

2.54* 2.54*

Rotational frequency

r/min

r/s rad/s

1.666 667 E − 02 1.047 198 E − 01

Acceleration (linear)

ft/s2

m/s2 cm/s2

3.048* 3.048*

Acceleration (rotational)

rpm/s

rad/s2

1.047 198 E − 01

Momentum

(lbm⋅ft)/s

(kg⋅m)/s

1.382 550 E − 01

Force

U.K. tonf U.S. tonf kgf (kp) lbf dyn

kN kN N N mN

9.964 016 E + 00 8.896 443 E + 00 9.806 650* E + 00 4.448 222 E + 00 1.0 E − 02

Bending moment, torque

U.S. tonf⋅ft kgf⋅m lbf⋅ft lbf⋅in

kN⋅m N⋅m N⋅m N⋅m

2.711 636 E + 00 9.806 650* E + 00 1.355 818 E + 00 1.129 848 E − 01

Bending moment/length

(lbf⋅ft)/in (lbf⋅in)/in

(N⋅m)/m (N⋅m)/m

5.337 866 E + 01 4.448 222 E + 00

Moment of inertia

lbm⋅ft2

Velocity (linear), speed

knot mi/h ft/s ft/min ft/h ft/day

MPa MPa MPa MPa kPa Pa

E − 01 E + 01

4.214 011 E − 02

kg⋅m2 2

E + 01 E − 02

N/mm2 N/mm2 N/mm2 N/mm2

1.378 951 E + 01 9.806 650* E + 00 9.576 052 E − 02 6.894 757 E − 03 4.788 026 E − 02 1.0* E − 01

Stress

U.S. tonf/in kgf/mm2 U.S. tonf/ft2 lbf/in2 (psi) lbf/ft2 (psf) dyn/cm2

Mass/length

lbm/ft

kg/m

1.488 164 E + 00

Mass/area structural loading, bearing capacity (mass basis)

U.S. ton/ft2 lbm/ft2

Mg/m2 kg/m2

9.764 855 E + 00 4.882 428 E + 00

Diffusivity

ft2/s m2/s ft2/h

m2/s mm2/s m2/s

9.290 304* E − 02 1.0* E + 06 2.580 64* E − 05

Thermal resistance

(°C⋅m2⋅h)/kcal (°F⋅ft2⋅h)/Btu

(K⋅m2)/kW (K⋅m2)/kW

8.604 208 E + 02 1.761 102 E + 02

Heat flux

Btu/(h⋅ft2)

kW/m2

3.154 591 E − 03

Thermal conductivity

(cal⋅cm)/(s⋅cm ⋅°C) (Btu⋅ft)/(h⋅ft2⋅°F)

W/(m⋅K) W/(m⋅K) (kJ⋅m)/(h⋅m2⋅K) W/(m⋅K) W/(m⋅K) W/(m⋅K)

4.184* 1.730 735 6.230 646 1.162 222 1.442 279 1.162 222

E + 02 E + 00 E + 00 E + 00 E − 01 E − 01

kW/(m2⋅K) kW/(m2⋅K) kW/(m2⋅K) kW/(m2⋅K) kJ/(h⋅m2⋅K) kW/(m2⋅K) kW/(m2⋅K)

4.184* 2.044 175 1.162 222 5.678 263 2.044 175 5.678 263 1.162 222

E + 01 E + 01 E − 02 E − 03 E + 01 E − 03 E − 03

Miscellaneous transport properties

2

(kcal⋅m)/(h⋅m2⋅°C) (Btu⋅in)/(h⋅ft2⋅°F) (cal⋅cm)/(h⋅cm2⋅°C) Heat-transfer coefficient

cal/(s⋅cm2⋅°C) Btu/(s⋅ft2⋅°F) cal/(h⋅cm2⋅°C) Btu/(h⋅ft2⋅°F) Btu/(h⋅ft2⋅°R) kcal/(h⋅m2⋅°C)

1-9

TABLE 1-4

Conversion Factors: U.S. Customary and Commonly Used Units to SI Units (Continued ) Customary or commonly used unit

Quantity

SI unit

Alternate SI unit

kW/(m3⋅K) kW/(m3⋅K)

Conversion factor; multiply customary unit by factor to obtain SI unit

Volumetric heat-transfer coefficient

Btu/(s⋅ft3⋅°F) Btu/(h⋅ft3⋅°F)

6.706 611 E + 01 1.862 947 E − 02

Surface tension

dyn/cm

mN/m

Viscosity (dynamic)

(lbf⋅s)/in2 (lbf⋅s)/ft2 (kgf⋅s)/m2 lbm/(ft⋅s) (dyn⋅s)/cm2 cP lbm/(ft⋅h)

Pa⋅s Pa⋅s Pa⋅s Pa⋅s Pa⋅s Pa⋅s Pa⋅s

Viscosity (kinematic)

ft2/s in2/s m2/h ft2/h cSt

m2/s mm2/s mm2/s m2/s mm2/s

9.290 304* E − 02 6.451 6* E + 02 2.777 778 E + 02 2.580 64* E − 05 1

Permeability

darcy millidarcy

µm2 µm2

9.869 233 E − 01 9.869 233 E − 04

Thermal flux

Btu/(h⋅ft2) Btu/(s⋅ft2) cal/(s⋅cm2)

W/m2 W/m2 W/m2

3.152 1.135 4.184

E + 00 E + 04 E + 04

Mass-transfer coefficient

(lb⋅mol)/[h⋅ft2(lb⋅mol/ft3)] (g⋅mol)/[s⋅m2(g⋅mol/L)]

m/s m/s

8.467 1.0

E − 05 E + 01

Admittance

S

S

1

Capacitance

µF

µF

1

Charge density

C/mm3

C/mm3

1

Conductance

S⍀

S S

1 1

1 (N⋅s)/m2 (N⋅s)/m2 (N⋅s)/m2 (N⋅s)/m2 (N⋅s)/m2 (N⋅s)/m2 (N⋅s)/m2

6.894 757 E + 03 4.788 026 E + 01 9.806 650* E + 00 1.488 164 E + 00 1.0* E − 01 1.0* E − 03 4.133 789 E − 04

Electricity, magnetism

(mho)

Conductivity

S/m ⍀ /m ⍀ m /m

S/m S/m mS/m

1 1 1

Current density

A/mm2

A/mm2

1

2

Displacement

C/cm

C/cm2

1

Electric charge

C

C

1

Electric current

A

A

1

Electric-dipole moment

C⋅m

C⋅m

1

Electric-field strength

V/m

V/m

1

Electric flux

C

C

1

Electric polarization

C/cm2

C/cm2

1

Electric potential

V mV

V mV

1 1

Electromagnetic moment

A⋅m2

A⋅m2

1

Electromotive force

V

V

1

Flux of displacement

C

C

1

Frequency

cycles/s

Hz

1

Impedance

Ω

Ω

1

Linear-current density

A/mm

A/mm

1

Magnetic-dipole moment

Wb⋅m

Wb⋅m

1

Magnetic-field strength

A/mm Oe gamma

A/mm A/m A/m

1 7.957 747 E + 01 7.957 747 E − 04

Magnetic flux

mWb

mWb

1

1-10

TABLE 1-4

Conversion Factors: U.S. Customary and Commonly Used Units to SI Units (Continued ) Customary or commonly used unit

Quantity Magnetic-flux density

mT G gamma

SI unit mT T nT

Alternate SI unit

Conversion factor; multiply customary unit by factor to obtain SI unit 1 1.0* 1

Magnetic induction

mT

mT

1

Magnetic moment

A⋅m2

A⋅m2

1

Magnetic polarization

mT

mT

1

Magnetic potential difference

A

A

1

Magnetic-vector potential

Wb/mm

Wb/mm

1

Magnetization

A/mm

A/mm

1

Modulus of admittance

S

S

1

Modulus of impedance

Ω

Ω

1

Mutual inductance

H

H

1

Permeability

µH/m

µH/m

1

E − 04

Permeance

H

H

1

Permittivity

µF/m

µF/m

1

Potential difference

V

V

1

Quantity of electricity

C

C

1

Reactance

Ω

Ω

1

Reluctance

H−1

H−1

1

Resistance

Ω

Ω

1

Resistivity

Ω⋅cm Ω⋅m

Ω⋅cm Ω⋅m

1 1

Self-inductance

mH

mH

1

Surface density of change

mC/m2

mC/m2

1

Susceptance

S

S

1

Volume density of charge

C/mm3

C/mm3

1

Absorbed dose

rad

Gy

1.0*

Acoustical energy

J

J

1

Acoustical intensity

W/cm2

W/m2

1.0*

Acoustical power

W

W

1

Sound pressure

N/m2

N/m2

1.0*

Illuminance

fc

lx

1.076 391 E + 01

Illumination

fc

lx

1.076 391 E + 01

Irradiance

W/m2

W/m2

1

Light exposure

fc⋅s

lx⋅s

1.076 391 E + 01

Luminance

cd/m2

cd/m2

1

Acoustics, light, radiation

Luminous efficacy

lm/W

lm/W

1

Luminous exitance

lm/m2

lm/m2

1

Luminous flux

lm

lm

1

Luminous intensity

cd

cd

1

Radiance

W/m2⋅sr

W/m2⋅sr

1

Radiant energy

J

J

1

Radiant flux

W

W

1

Radiant intensity

W/sr

W/sr

1

Radiant power

W

W

1

E − 02 E + 04

1-11

TABLE 1-4

Conversion Factors: U.S. Customary and Commonly Used Units to SI Units (Concluded ) Customary or commonly used unit

Quantity

SI unit

Alternate SI unit

Conversion factor; multiply customary unit by factor to obtain SI unit

Wavelength

Å

nm

1.0*

E − 01

Capture unit

10−3 cm−1

m−1

E + 01

m−1

m−1

1.0* 1 1

Ci

Bq

3.7*

E + 10

Radioactivity

10−3 cm−1

* An asterisk indicates that the conversion factor is exact. † Conversion factors for length, area, and volume are based on the international foot. The international foot is longer by 2 parts in 1 million than the U.S. Survey foot (land-measurement use). NOTE: The following unit symbols are used in the table: Unit symbol

Name

Unit symbol

Name

A a Bq C cd Ci d °C ° dyn F fc G g gr Gy H h ha Hz J K L, ᐉ, l

ampere annum (year) becquerel coulomb candela curie day degree Celsius degree dyne farad footcandle gauss gram grain gray henry hour hectare hertz joule kelvin liter

lm lx m min ′ N naut mi Oe Ω Pa rad r S s ″ sr St T t V W Wb

lumen lux meter minute minute newton U.S. nautical mile oersted ohm pascal radian revolution siemens second second steradian stokes tesla tonne volt watt weber

NOTE:

1-12

Copyright SPE-AIME, The SI Metric System of Units and SPE’s Tentative Metric Standard, Society of Petroleum Engineers, Dallas, 1977.

TABLE 1-5 Metric Conversion Factors as Exact Numerical Multiples of SI Units The first two digits of each numerical entry represent a power of 10. For example, the entry “−02 2.54” expresses the fact that 1 in = 2.54 × 10−2 m. To convert from abampere abcoulomb abfarad abhenry abmho abohm abvolt acre ampere (international of 1948) angstrom are astronomical unit atmosphere bar barn barrel (petroleum 42 gal) barye British thermal unit (ISO/ TC 12) British thermal unit (International Steam Table) British thermal unit (mean) British thermal unit (thermochemical) British thermal unit (39°F) British thermal unit (60°F) bushel (U.S.) cable caliber calorie (International Steam Table) calorie (mean) calorie (thermochemical) calorie (15°C) calorie (20°C) calorie (kilogram, International Steam Table) calorie (kilogram, mean) calorie (kilogram, thermochemical) carat (metric) Celsius (temperature) centimeter of mercury (0°C) centimeter of water (4°C) chain (engineer’s) chain (surveyor’s or Gunter’s) circular mil cord coulomb (international of 1948) cubit cup curie day (mean solar) day (sidereal) degree (angle) denier (international) dram (avoirdupois) dram (troy or apothecary) dram (U.S. fluid) dyne electron volt erg Fahrenheit (temperature)

Multiply by

To convert from

To

Multiply by

ampere coulomb farad henry mho ohm volt meter2 ampere

To

+01 1.00 +01 1.00 +09 1.00 −09 1.00 +09 1.00 −09 1.00 −08 1.00 +03 4.046 856 −01 9.998 35

meter meter2 meter newton/meter2 newton/meter2 meter2 meter3 newton/meter2 joule

−10 1.00 +02 1.00 +11 1.495 978 +05 1.013 25 +05 1.00 −28 1.00 −01 1.589 873 −01 1.00 +03 1.055 06

joule

+03 1.055 04

joule joule

+03 1.055 87 +03 1.054 350

joule joule meter3 meter meter joule

+03 1.059 67 +03 1.054 68 −02 3.523 907 +02 2.194 56 −04 2.54 +00 4.1868

joule joule joule joule joule

+00 4.190 02 +00 4.184 +00 4.185 80 +00 4.181 90 +03 4.186 8

joule joule

+03 4.190 02 +03 4.184

kilogram kelvin newton/meter2 newton/meter2 meter meter

−04 2.00 tK = tc + 273.15 +03 1.333 22 +01 9.806 38 +01 3.048 +01 2.011 68

meter3 meter meter newton/meter2 lumen/meter2 candela/meter2 meter meter/second2 meter3 meter3 meter3 tesla tesla ampere turn meter3 meter3 degree (angular) radian kilogram kilogram meter meter2 henry meter3 watt watt watt watt watt watt second (mean solar) second (mean solar) kilogram kilogram meter newton/meter2 newton/meter2 newton/meter2 newton/meter2 joule 1/meter joule

−05 2.957 352 −01 3.048 −01 3.048 006 +03 2.988 98 +01 1.076 391 +00 3.426 259 +02 2.011 68 −02 1.00 −03 4.546 087 −03 4.404 883 −03 3.785 411 −09 1.00 −04 1.00 −01 7.957 747 −04 1.420 652 −04 1.182 941 −01 9.00 −02 1.570 796 −05 6.479 891 −03 1.00 −01 1.016 +04 1.00 +00 1.000 495 −01 2.384 809 +02 7.456 998 +03 9.809 50 +02 7.46 +02 7.354 99 +02 7.457 +02 7.460 43 +03 3.60 +03 3.590 170 +01 5.080 234 +01 4.535 923 −02 2.54 +03 3.386 389 +03 3.376 85 +02 2.490 82 +02 2.4884 +00 1.000 165 +02 1.00 +03 4.186 74

meter2 meter3 coulomb

−10 5.067 074 +00 3.624 556 −01 9.998 35

meter meter3 disintegration/second second (mean solar) second (mean solar) radian kilogram/meter kilogram kilogram meter3 newton joule joule kelvin

fluid ounce (U.S.) foot foot (U.S. survey) foot of water (39.2°F) footcandle footlambert furlong gal (galileo) gallon (U.K. liquid) gallon (U.S. dry) gallon (U.S. liquid) gamma gauss gilbert gill (U.K.) gill (U.S.) grad grad grain gram hand hectare henry (international of 1948) hogshead (U.S.) horsepower (550 ft lbf/s) horsepower (boiler) horsepower (electric) horsepower (metric) horsepower (U.K.) horsepower (water) hour (mean solar) hour (sidereal) hundredweight (long) hundredweight (short) inch inch of mercury (32°F) inch of mercury (60°F) inch of water (39.2°F) inch of water (60°F) joule (international of 1948) kayser kilocalorie (International Steam Table) kilocalorie (mean) kilocalorie (thermochemical) kilogram mass kilogram-force (kgf) kilopond-force kip knot (international) lambert lambert langley lbf (pound-force, avoirdupois) lbm (pound-mass, avoirdupois) league (British nautical) league (international nautical) league (statute) light-year link (engineer’s) link (surveyor’s or Gunter’s) liter lux maxwell meter micrometer mil mile (U.S. statute) mile (U.K. nautical) mile (international nautical) mile (U.S. nautical) millibar millimeter of mercury (0°C)

joule joule kilogram newton newton newton meter/second candela/meter2 candela/meter2 joule/meter2 newton

+03 4.190 02 +03 4.184 +00 1.00 +00 9.806 65 +00 9.806 65 +03 4.448 221 −01 5.144 444 +04 1/π +03 3.183 098 +04 4.184 +00 4.448 221

kilogram

−01 4.535 923

meter meter

+03 5.559 552 +03 5.556

meter meter meter meter meter3 lumen/meter2 weber wavelengths Kr 86 meter meter meter meter meter meter newton/meter2 newton/meter2

+03 4.828 032 +15 9.460 55 −01 3.048 −01 2.011 68 −03 1.00 +00 1.00 −08 1.00 +06 1.650 763 −06 1.00 −05 2.54 +03 1.609 344 +03 1.853 184 +03 1.852 +03 1.852 +02 1.00 +02 1.333 224

Fahrenheit (temperature)

Celsius

farad (international of 1948) faraday (based on carbon 12) faraday (chemical) faraday (physical) fathom fermi (femtometer)

farad coulomb

−01 4.572 −04 2.365 882 +10 3.70 +04 8.64 +04 8.616 409 −02 1.745 329 −07 1.111 111 −03 1.771 845 −03 3.887 934 −06 3.696 691 −05 1.00 −19 1.602 10 −07 1.00 tK = (5/9)(tF + 459.67) tc = (5/9)(tF − 32) −01 9.995 05 +04 9.648 70

coulomb coulomb meter meter

+04 9.649 57 +04 9.652 19 +00 1.828 8 −15 1.00

1-13

TABLE 1-5 Metric Conversion Factors as Exact Numerical Multiples of SI Units (Concluded ) The first two digits of each numerical entry represent a power of 10. For example, the entry “−02 2.54” expresses the fact that 1 in = 2.54 × 10−2 To convert from minute (angle) minute (mean solar) minute (sidereal) month (mean calendar) nautical mile (international) nautical mile (U.S.) nautical mile (U.K.) oersted ohm (international of 1948) ounce-force (avoirdupois) ounce-mass (avoirdupois) ounce-mass (troy or apothecary) ounce (U.S. fluid) pace parsec pascal peck (U.S.) pennyweight perch phot pica (printer’s) pint (U.S. dry) pint (U.S. liquid) point (printer’s) poise pole pound-force (lbf avoirdupois) pound-mass (lbm avoirdupois) pound-mass (troy or apothecary) poundal quart (U.S. dry) quart (U.S. liquid) rad (radiation dose absorbed) Rankine (temperature) rayleigh (rate of photon emission) rhe rod roentgen rutherford second (angle)

1-14

To

Multiply by

radian second (mean solar) second (mean solar) second (mean solar) meter meter meter ampere/meter ohm newton kilogram kilogram meter3 meter meter newton/meter2 meter3 kilogram meter lumen/meter2 meter meter3 meter3 meter (newton-second)/meter2 meter newton

−04 2.908 882 +01 6.00 +01 5.983 617 +06 2.628 +03 1.852 +03 1.852 +03 1.853 184 +01 7.957 747 +00 1.000 495 −01 2.780 138 −02 2.834 952 −02 3.110 347 −05 2.957 352 −01 7.62 +16 3.083 74 +00 1.00 −03 8.809 767 −03 1.555 173 +00 5.0292 +04 1.00 −03 4.217 517 −04 5.506 104 −04 4.731 764 −04 3.514 598 −01 1.00 +00 5.0292 +00 4.448 221

kilogram

−01 4.535 923

kilogram

−01 3.732 417

newton meter3 meter3 joule/kilogram

−01 1.382 549 −03 1.101 220 −04 9.463 529 −02 1.00

kelvin 1/second-meter2

tK = (5/9)tR +10 1.00

meter2/(newtonsecond) meter coulomb/kilogram disintegration/second radian

+01 1.00 +00 5.0292 −04 2.579 76 +06 1.00 −06 4.848 136

To

Multiply by

second (ephemeris) second (mean solar)

To convert from

second second (ephemeris)

second (sidereal) section scruple (apothecary) shake skein slug span statampere statcoulomb statfarad stathenry statmho statohm statute mile (U.S.) statvolt stere stilb stoke tablespoon teaspoon ton (assay) ton (long) ton (metric) ton (nuclear equivalent of TNT) ton (register) ton (short, 2000 lb) tonne torr (0°C) township unit pole volt (international of 1948) watt (international of 1948) yard year (calendar) year (sidereal) year (tropical) year 1900, tropical, Jan., day 0, hour 12 year 1900, tropical, Jan., day 0, hour 12

second (mean solar) meter2 kilogram second meter kilogram meter ampere coulomb farad henry mho ohm meter volt meter3 candela/meter2 meter2/second meter3 meter3 kilogram kilogram kilogram joule meter3 kilogram kilogram newton/meter2 meter2 weber volt watt meter second (mean solar) second (mean solar) second (mean solar) second (ephemeris)

+00 1.000 000 Consult American Ephemeris and Nautical Almanac −01 9.972 695 +06 2.589 988 −03 1.295 978 −08 1.00 +02 1.097 28 +01 1.459 390 −01 2.286 −10 3.335 640 −10 3.335 640 −12 1.112 650 +11 8.987 554 −12 1.112 650 +11 8.987 554 +03 1.609 344 +02 2.997 925 +00 1.00 +04 1.00 −04 1.00 −05 1.478 676 −06 4.928 921 −02 2.916 666 +03 1.016 046 +03 1.00 +09 4.20 +00 2.831 684 +02 9.071 847 +03 1.00 +02 1.333 22 +07 9.323 957 −07 1.256 637 +00 1.000 330 +00 1.000 165 −01 9.144 +07 3.1536 +07 3.155 815 +07 3.155 692 +07 3.155 692

second

+07 3.155 692

TABLE 1-6

Alphabetical Listing of Common Conversions

To convert from Acres Acres Acres Acre-feet Ampere-hours (absolute) Angstrom units Angstrom units Angstrom units Atmospheres Atmospheres Atmospheres Atmospheres Atmospheres Atmospheres Atmospheres Atmospheres Bags (cement) Barrels (cement) Barrels (oil) Barrels (oil) Barrels (U.S. liquid) Barrels (U.S. liquid) Barrels per day Bars Bars Bars Board feet Boiler horsepower Boiler horsepower B.t.u. B.t.u. B.t.u. B.t.u. B.t.u. B.t.u. B.t.u. B.t.u. B.t.u. B.t.u. B.t.u. per cubic foot B.t.u. per hour B.t.u. per minute B.t.u. per pound B.t.u. per pound per degree Fahrenheit B.t.u. per pound per degree Fahrenheit B.t.u. per second B.t.u. per square foot per hour B.t.u. per square foot per minute B.t.u. per square foot per second for a temperature gradient of 1°F. per inch

To

Multiply by

To convert from

To

Multiply by

Square feet Square meters Square miles Cubic meters Coulombs (absolute) Inches Meters Microns Millimeters of mercury at 32°F Dynes per square centimeter Newtons per square meter Feet of water at 39.1°F Grams per square centimeter Inches of mercury at 32°F Pounds per square foot Pounds per square inch Pounds (cement) Pounds (cement) Cubic meters Gallons Cubic meters Gallons Gallons per minute Atmospheres Newtons per square meter Pounds per square inch Cubic feet B.t.u. per hour Kilowatts Calories (gram) Centigrade heat units (c.h.u. or p.c.u.) Foot-pounds Horsepower-hours Joules Liter-atmospheres Pounds carbon to CO2 Pounds water evaporated from and at 212°F Cubic foot-atmospheres Kilowatt-hours Joules per cubic meter Watts Horsepower Joules per kilogram Calories per gram per degree centigrade Joules per kilogram per degree Kelvin Watts Joules per square meter per second Kilowatts per square foot Calories, gram (15°C.), per square centimeter per second for a temperature gradient of 1°C. per centimeter

43,560 4074 0.001563 1233 3600 3.937 × 10−9 1 × 10−10 1 × 10−4 760 1.0133 × 106 101,325 33.90 1033.3 29.921 2116.3 14.696 94 376 0.15899 42 0.11924 31.5 0.02917 0.9869 1 × 105 14.504 1 ⁄12 33,480 9.803 252 0.55556 777.9 3.929 × 10−4 1055.1 10.41 6.88 × 10−5

B.t.u. (60°F.) per degree Fahrenheit Bushels (U.S. dry) Bushels (U.S. dry) Calories, gram Calories, gram Calories, gram Calories, gram Calories, gram Calories, gram, per gram per degree C. Calories, kilogram Calories, kilogram per second Candle power (spherical) Carats (metric) Centigrade heat units Centimeters Centimeters Centimeters Centimeters Centimeters Centimeters of mercury at 0°C. Centimeters of mercury at 0°C. Centimeters of mercury at 0°C Centimeters of mercury at 0°C. Centimeters of mercury at 0°C. Centimeters per second Centimeters of water at 4°C. Centistokes Circular mils Circular mils Circular mils Cords Cubic centimeters Cubic centimeters Cubic centimeters Cubic centimeters Cubic feet Cubic feet Cubic feet Cubic feet Cubic feet Cubic feet Cubic foot-atmospheres Cubic foot-atmospheres Cubic feet of water (60°F.) Cubic feet per minute Cubic feet per minute Cubic feet per second Cubic feet per second Cubic inches Cubic yards Curies Curies Degrees Drams (apothecaries’ or troy)

Calories per degree centigrade Cubic feet Cubic meters B.t.u. Foot-pounds Joules Liter-atmospheres Horsepower-hours Joules per kilogram per degree Kelvin Kilowatt-hours Kilowatts Lumens Grams B.t.u. Angstrom units Feet Inches Meters Microns Atmospheres Feet of water at 39.1°F. Newtons per square meter Pounds per square foot Pounds per square inch Feet per minute Newtons per square meter Square meters per second Square centimeters Square inches Square mils Cubic feet Cubic feet Gallons Ounces (U.S. fluid) Quarts (U.S. fluid) Bushels (U.S.) Cubic centimeters Cubic meters Cubic yards Gallons Liters Foot-pounds Liter-atmospheres Pounds Cubic centimeters per second Gallons per second Gallons per minute Million gallons per day Cubic meters Cubic meters Disintegrations per minute Coulombs per minute Radians Grams

453.6 1.2444 0.03524 3.968 × 10−3 3.087 4.1868 4.130 × 10−2 1.5591 × 10−6 4186.8 0.0011626 4.185 12.556 0.2 1.8 1 × 108 0.03281 0.3937 0.01 10,000 0.013158 0.4460 1333.2 27.845 0.19337 1.9685 98.064 1 × 10−6 5.067 × 10−6 7.854 × 10−7 0.7854 128 3.532 × 10−5 2.6417 × 10−4 0.03381 0.0010567 0.8036 28,317 0.028317 0.03704 7.481 28.316 2116.3 28.316 62.37 472.0 0.1247 448.8 0.64632 1.6387 × 10−5 0.76456 2.2 × 1012 1.1 × 1012 0.017453 3.888

0.001036 0.3676 2.930 × 10−4 37,260 0.29307 0.02357 2326 1 4186.8 1054.4 3.1546 0.1758 1.2405

1-15

TABLE 1-6

Alphabetical Listing of Common Conversions (Concluded )

To convert from Drams (avoirdupois) Dynes Ergs Faradays Fathoms Feet Feet per minute Feet per minute Feet per (second)2 Feet of water at 39.2°F. Foot-poundals Foot-poundals Foot-poundals Foot-pounds Foot-pounds Foot-pounds Foot-pounds Foot-pounds Foot-pounds Foot-pounds force Foot-pounds per second Foot-pounds per second Furlongs Gallons (U.S. liquid) Gallons Gallons Gallons Gallons Gallons Gallons per minute Gallons per minute Grains Grains Grains per cubic foot Grains per gallon Grams Grams Grams Grams Grams Grams Grams per cubic centimeter

1-16

To Grams Newtons Joules Coulombs (abs.) Feet Meters Centimeters per second Miles per hour Meters per (second)2 Newtons per square meter B.t.u. Joules Liter-atmospheres B.t.u. Calories, gram Foot-poundals Horsepower-hours Kilowatt-hours Liter-atmospheres Joules Horsepower Kilowatts Miles Barrels (U.S. liquid) Cubic meters Cubic feet Gallons (Imperial) Liters Ounces (U.S. fluid) Cubic feet per hour Cubic feet per second Grams Pounds Grams per cubic meter Parts per million Drams (avoirdupois) Drams (troy) Grains Kilograms Pounds (avoirdupois) Pounds (troy) Pounds per cubic foot

Multiply by 1.7719 1 × 10−5 1 × 10−7 96,500 6 0.3048 0.5080 0.011364 0.3048 2989 3.995 × 10−5 0.04214 4.159 × 10−4 0.0012856 0.3239 32.174 5.051 × 10−7 3.766 × 10−7 0.013381 1.3558 0.0018182 0.0013558 0.125 0.03175 0.003785 0.13368 0.8327 3.785 128 8.021 0.002228 0.06480 1⁄ 7000 2.2884 17.118 0.5644 0.2572 15.432 0.001 0.0022046 0.002679 62.43

To convert from Horsepower (British) Horsepower (metric) Horsepower (metric) Hours (mean solar) Inches Inches of mercury at 60°F Inches of water at 60°F Joules (absolute) Joules (absolute) Joules (absolute) Joules (absolute) Joules (absolute) Joules (absolute) Kilocalories Kilograms Kilograms force Kilograms per square centimeter Kilometers Kilowatt-hours Kilowatt-hours Kilowatts Knots (international) Knots (nautical miles per hour) Lamberts Liter-atmospheres Liter-atmospheres Liters Liters Liters Lumens Micromicrons Microns Microns Miles (nautical) Miles (nautical) Miles Miles Miles per hour Miles per hour Milliliters Millimeters

To Pounds water evaporated per hour at 212°F Foot-pounds per second Kilogram-meters per second Seconds Meters Newtons per square meter Newtons per square meter B.t.u. (mean) Calories, gram (mean) Cubic foot-atmospheres Foot-pounds Kilowatt-hours Liter-atmospheres Joules Pounds (avoirdupois) Newtons Pounds per square inch Miles B.t.u. Foot-pounds Horsepower Meters per second Miles per hour Candles per square inch Cubic foot-atmospheres Foot-pounds Cubic feet Cubic meters Gallons Watts Microns Angstrom units Meters Feet Miles (U.S. statute) Feet Meters Feet per second Meters per second Cubic centimeters Meters

Multiply by 2.64 542.47 75.0 3600 0.0254 3376.9 248.84 9.480 × 10−4 0.2389 0.3485 0.7376 2.7778 × 10−7 0.009869 4186.8 2.2046 9.807 14.223 0.6214 3414 2.6552 × 106 1.3410 0.5144 1.1516 2.054 0.03532 74.74 0.03532 0.001 0.26418 0.001496 1 × 10−6 1 × 104 1 × 10−6 6080 1.1516 5280 1609.3 1.4667 0.4470 1 0.001

TABLE 1-6

Alphabetical Listing of Common Conversions (Concluded )

To convert from Grams per cubic centimeter Grams per liter Grams per liter Grams per square centimeter Grams per square centimeter Hectares Hectares Horsepower (British) Horsepower (British) Horsepower (British) Horsepower (British) Horsepower (British) Horsepower (British) Horsepower (British) Pounds (avoirdupois) Pounds (avoirdupois) Pounds (avoirdupois) Pounds per cubic foot Pounds per cubic foot Pounds per square foot Pounds per square foot Pounds per square inch Pounds per square inch Pounds per square inch Pounds force Pounds force per square foot Pounds water evaporated from and at 212°F. Pound-centigrade units (p.c.u.) Quarts (U.S. liquid) Radians Revolutions per minute Seconds (angle) Slugs Slugs Slugs Square centimeters

To Pounds per gallon Grains per gallon Pounds per cubic foot Pounds per square foot Pounds per square inch Acres Square meters B.t.u. per minute B.t.u. per hour Foot-pounds per minute Foot-pounds per second Watts Horsepower (metric) Pounds carbon to CO2 per hour Grains Kilograms Pounds (troy) Grams per cubic centimeter Kilograms per cubic meter Atmospheres Kilograms per square meter Atmospheres Kilograms per square centimeter Newtons per square meter Newtons Newtons per square meter Horsepower-hours B.t.u. Cubic meters Degrees Radians per second Radians Gee pounds Kilograms Pounds Square feet

Multiply by 8.345 58.42 0.0624 2.0482 0.014223 2.471 10,000 42.42 2545 33,000 550 745.7 1.0139 0.175 7000 0.45359 1.2153 0.016018 16.018 4.725 × 10−4 4.882 0.06805 0.07031 6894.8 4.4482 47.88 0.379 1.8 9.464 × 10−4 57.30 0.10472 4.848 × 10−6 1 14.594 32.17 0.0010764

To convert from

To

Millimeters of mercury at 0°C. Millimicrons Mils Mils Minims (U.S.) Minutes (angle) Minutes (mean solar) Newtons Ounces (avoirdupois) Ounces (avoirdupois) Ounces (U.S. fluid) Ounces (troy) Pints (U.S. liquid) Poundals

Newtons per square meter Microns Inches Meters Cubic centimeters Radians Seconds Kilograms Kilograms Ounces (troy) Cubic meters Ounces (apothecaries’) Cubic meters Newtons

Square feet Square feet per hour Square inches Square inches Square yards Stokes Tons (long) Tons (long) Tons (metric) Tons (metric) Tons (metric) Tons (short) Tons (short) Tons (refrigeration) Tons (British shipping) Tons (U.S. shipping) Torr (mm. mercury, 0°C.) Watts Watts Watts Watt-hours Yards

Square meters Square meters per second Square centimeters Square meters Square meters Square meters per second Kilograms Pounds Kilograms Pounds Tons (short) Kilograms Pounds B.t.u. per hour Cubic feet Cubic feet Newtons per square meter B.t.u. per hour Joules per second Kilogram-meters per second Joules Meters

Multiply by 133.32 0.001 0.001 2.54 × 10−5 0.06161 2.909 × 10−4 60 0.10197 0.02835 0.9115 2.957 × 10−5 1.000 4.732 × 10−4 0.13826 0.0929 2.581 × 10−5 6.452 6.452 × 10−4 0.8361 1 × 10−4 1016 2240 1000 2204.6 1.1023 907.18 2000 12,000 42.00 40.00 133.32 3.413 1 0.10197 3600 0.9144

1-17

TABLE 1-7 Mass (M)

Length (L)

Common Units and Conversion Factors* 1 pound mass = 453.5924 grams = 0.45359 kilograms = 7000 grains 1 slug = 32.174 pounds mass 1 ton (short) = 2000 pounds mass 1 ton (long) = 2240 pounds mass 1 ton (metric) = 1000 kilograms = 2204.62 pounds mass 1 pound mole = 453.59 gram moles = 30.480 centimeters = 0.3048 meters 1 inch = 2.54 centimeters = 0.0254 meters 1 mile (U.S.) = 1.60935 kilometers 1 yard = 0.9144 meters 1 foot

Area (L2)

Volume (L3)

1 square foot = 929.0304 square centimeters = 0.09290304 square meters 1 square inch = 6.4516 square centimeters 1 square yard = 0.836127 square meters 1 cubic foot

1 gallon Time (θ) 1 hour

= 28,316.85 cubic centimeters = 0.02831685 cubic meters = 28.31685 liters = 7.481 gallons (U.S.) = 3.7853 liters = 231 cubic inches = 60 minutes = 3600 seconds

Temperature (T) 1 centigrade or Celsius degree = 1.8 Fahrenheit degree Temperature, Kelvin = T°C + 273.15 Temperature, Rankine = T°F + 459.7 Temperature, Fahrenheit = 9/5 T°C + 32 Temperature, centigrade or Celsius = 5/9 (T°F − 32) Temperature, Rankine = 1.8 T K Force (F) 1 pound force = 444,822.2 dynes = 4.448222 Newtons = 32.174 poundals Pressure (F/L2) Normal atmospheric pressure

1 atm = 760 millimeters of mercury at 0°C (density 13.5951 g/cm3) = 29.921 inches of mercury at 32°F = 14.696 pounds force/square inch = 33.899 feet of water at 39.1°F = 1.01325 × 106 dynes/square centimeter = 1.01325 × 105 Newtons/square meter Density (M/L3) 1 pound mass/cubic foot = 0.01601846 grams/cubic centimeter = 16.01846 kilogram/cubic meter Energy (H or FL) 1 British thermal unit = 251.98 calories = 1054.4 joules = 777.97 foot-pounds force = 10.409 liter-atmospheres = 0.2930 watt-hour 2 Diffusivity (L /θ) 1 square foot/hour = 0.258 cm2/s = 2.58 × 10−5 m2/s Viscosity (M/Lθ) 1 pound mass/foot hour = 0.00413 g/cm s 0.000413 kg/m s 1 centipoise = 0.01 poise = 0.01 g/cm s = 0.001 kg/m s = 0.000672 lbm/ft s = 0.0000209 lbfs/ft2 Thermal conductivity [H/θL2(T/L)] 2 1 Btu/hr ft (°F/ft) = 0.00413 cal/s cm2 (°C/cm) = 1.728 J/s m2 (°C/m) Heat transfer coefficient 1 Btu/hr ft2 °F = 5.678 J/s m2 °C Heat capacity (H/MT) 1 Btu/lbm °F = 1 cal/g °C = 4184 J/kg °C Gas constant 1.987 Btu/lbm mole °R = 1.987 cal/mol K = 82.057 atm cm3/mol K = 0.7302 atm ft3/lb mole °F = 10.73 (lbf /in.2) (ft3)/lb mole °R = 1545 (lbf /ft2) (ft3)/lb mole °R = 8.314 (N/m2) (m3)/mol K Gravitational acceleration g = 9.8066 m/s2 = 32.174 ft/s2

NOTE: U.S. customary units; or British units, on left and SI units on right. *Adapted from Faust et al., Principles of Unit Operations, John Wiley and Sons, 1980.

TABLE 1-8

Kinematic-Viscosity Conversion Formulas

Viscosity scale Saybolt Universal Saybolt Furol Redwood No. 1 Redwood Admiralty Engler

Range of t, sec 32 < t < 100 t > 100 25 < t < 40 t > 40 34 < t < 100 t > 100

Kinematic viscosity, stokes 0.00226t − 1.95/t 0.00220t − 1.35/t 0.0224t − 1.84/t 0.0216t − 0.60/t 0.00260t − 1.79/t 0.00247t − 0.50/t 0.027t − 20/t 0.00147t − 3.74/t

TABLE 1-9 Temp. scale

Values of the Gas-Law Constant Press. units

Vol. units

Kelvin atm. atm. mm. Hg bar kg/cm2 atm mm Hg

cm3 liters liters liters liters ft3 ft3

atm in Hg mm Hg lb/in2abs lb/ft2abs

ft3 ft3 ft3 ft3 ft3

Rankine

1-18

Wt. units

Energy units

R

g-moles g-moles g-moles g-moles g-moles g-moles g-moles g-moles lb-moles lb-moles lb-moles lb-moles lb-moles lb-moles lb-moles lb-moles lb-moles lb-moles lb-moles

calories joules (abs) joules (int) atm cm3 atm liters mm Hg-liters bar-liters kg/(cm2)(liters) atm-ft3 mm Hg-ft3 chu or pcu Btu hp-hr kw-hr atm-ft3 in Hg-ft3 mm Hg-ft3 (lb)(ft3)/in2 ft-lb

1.9872 8.3144 8.3130 82.057 0.08205 62.361 0.08314 0.08478 1.314 998.9 1.9872 1.9872 0.0007805 0.0005819 0.7302 21.85 555.0 10.73 1545.0

TABLE 1-10 United States Customary System of Weights and Measures Linear Measure 12 inches (in) or (″) = 1 foot (ft) or (′) 3 feet = 1 yard (yd) 16.5 feet = 1 rod (rd) 5.5 yards

冧 冧

5280 feet = 1 mile (mi) 320 rods 1 mil = 0.001 inch Nautical:

TABLE 1-11

Temperature Conversion Formulas

°F = (°C × 5/9) + 32 °C = (°F − 32) × 5/9 °R = °F + 459.69 °K = °C + 273.15 °K = °R × 5/9 Temperature difference, ⌬T °F = °C × 9/5 NOTE: An extensive table of temperature conversions may be found in the sixth edition of the Handbook (Table 1-12).

6080.2 feet = 1 nautical mile 6 feet = 1 fathom 120 fathoms = 1 cable length 1 knot = 1 nautical mile per hour 60 nautical miles = 1° of latitude

Square Measure 144 sq. inches (sq. in) or (in2) or (ⵧ″) = 1 sq. foot (ft2) or (ⵧ′) 9 sq. feet (ft2) (ⵧ′) = 1 sq. yard (yd2) 30.25 sq. yards = 1 sq. rod, pole, or perch 10 sq. chains 160 sq. rods = = 1 acre 43,560 sq. ft 640 acres = 1 sq. mile = 1 section 1 circular inch (area of circle of 1 inch diameter) = 0.7854 sq. inch 1 sq. inch = 1.2732 circular inch 1 circular mil = area of circle of 0.001 inch diameter 1,000,000 circular mils = 1 circular inch

冦

冧

Circular Measure 60 seconds (″) (sec) = 1 minute (min) or (′) 60 minutes (′) = 1 degree (°) 90 degrees (°) = 1 quadrant 360 degrees (°) = 1 circumference = 1 radian (rad.) 57.29578 degrees = 57° 17′ 44.81″

冦

Volume Measure Solid:

1728 cubic in (cu. in) (in3) = 1 cubic foot (cu. ft)(ft3) 27 cu. ft = 1 cubic yard (cu. yd) Dry Measure: 2 pints = 1 quart 8 quarts = 1 peck 4 pecks = 1 bushel 1 United States Winchester bushel = 2150.42 cubic inches Liquid: 4 gills = 1 pint (pt) 2 pints = 1 quart (qt) 4 quarts = 1 gallon (gal) 7.4805 gallons = 1 cubic foot Apothecaries’ Liquid: 60 minims (min. or ) = 1 fluid dram or drachm 8 drams ( ) = 1 fluid ounce 16 ounces (oz. ) = 1 pint Avoirdupois Weight 16 drams = 437.5 grains = 1 ounce (oz) 16 ounces = 7000 grains = 1 pound (lb) 100 pounds = 1 hundredweight (cwt) 2000 pounds = 1 short ton: 2240 pounds = 1 long ton Troy Weight 24 grains = 1 pennyweight (dwt) 20 pennyweights = 1 ounce (oz) 12 ounces = 1 pound (lb) Apothecaries’ Weight 20 grains (gr) = 1 scruple ( ) 3 scruples = 1 dram ( ) 8 drams = 1 ounce ( ) 12 ounces = 1 pound (lb)

1-19

TABLE 1-12

Specific Gravity, Degrees Baumé, Degrees API, Degrees Twaddell, Pounds per Gallon, Pounds per Cubic Foot* 145 140 sp gr 60°/60°F − 1 141.5 °Bé = 145 − ᎏ (heavier than H2O); °Bé = ᎏ − 130 (lighter than H2O); °Tw = ᎏᎏ °API = ᎏ − 131.5 sp gr sp gr 0.005 sp gr Lb per gal at 60°F wt in air

Lb per ft3 at 60°F wt in air

Sp gr 60°/ 60°

°Bé

104.33 102.38 100.47 98.58 96.73

4.9929 5.0346 5.0763 5.1180 5.1597

37.350 37.662 37.973 38.285 38.597

0.700 .705 .710 .715 .720

70.00 68.58 67.18 65.80 64.44

94.00 92.22 90.47 88.75 87.05

94.90 93.10 91.33 89.59 87.88

5.2014 5.2431 5.2848 5.3265 5.3682

39.910 39.222 39.534 39.845 40.157

.725 .730 .735 .740 .745

.650 .655 .660 .665 .670

85.38 83.74 82.12 80.53 78.96

86.19 84.53 82.89 81.28 79.69

5.4098 5.4515 5.4932 5.5349 5.5766

40.468 40.780 41.092 41.404 41.716

.675 .680 .685 .690 .695

77.41 75.88 74.38 72.90 71.44

78.13 76.59 75.07 73.57 72.10

5.6183 5.6600 5.7017 5.7434 5.7851

42.028 42.340 42.652 42.963 43.275

Lb per ft3 at 60°F wt in air

Sp gr 60°/ 60°

°Bé

Sp gr 60°/ 60°

°Bé

°API

0.600 .605 .610 .615 .620

103.33 101.40 99.51 97.64 95.81

.625 .630 .635 .640 .645

°API

Lb per gal at 60°F wt in air

Lb per ft3 at 60°F wt in air

Sp gr 60°/ 60°

°Bé

°API

70.64 69.21 67.80 66.40 65.03

5.8268 5.8685 5.9101 5.9518 5.9935

43.587 43.899 44.211 44.523 44.834

0.800 .805 .810 .815 .820

45.00 43.91 42.84 41.78 40.73

45.38 44.28 43.19 42.12 41.06

63.10 61.78 60.48 59.19 57.92

63.67 62.34 61.02 59.72 58.43

6.0352 6.0769 6.1186 6.1603 6.2020

45.146 45.458 45.770 46.082 46.394

.825 .830 .835 .840 .845

39.70 38.67 37.66 36.67 35.68

.750 .755 .760 .765 .770

56.67 55.43 54.21 53.01 51.82

57.17 55.92 54.68 53.47 52.27

6.2437 6.2854 6.3271 6.3688 6.4104

46.706 47.018 47.330 47.642 47.953

.850 .855 .860 .865 .870

.775 .780 .785 .790 .795

50.65 49.49 48.34 47.22 46.10

51.08 49.91 48.75 47.61 46.49

6.4521 6.4938 6.5355 6.5772 6.6189

47.265 48.577 48.889 49.201 49.513

.875 .880 .885 .890 .895

Lb per ft3 at 60°F wt in air

Sp gr 60°/ 60°

Lb per gal at 60°F wt in air

Lb per ft3 at 60°F wt in air

Sp gr 60°/ 60°

°Bé

°API

Lb per gal at 60°F wt in air

6.6606 6.7023 6.7440 6.7857 6.8274

49.825 50.137 50.448 50.760 51.072

0.900 .905 .910 .915 .920

25.56 24.70 23.85 23.01 22.17

25.72 24.85 23.99 23.14 22.30

7.4944 7.5361 7.5777 7.6194 7.6612

56.062 56.374 56.685 56.997 57.310

40.02 38.98 37.96 36.95 35.96

6.8691 6.9108 6.9525 6.9941 7.0358

51.384 51.696 52.008 52.320 52.632

.925 .930 .935 .940 .945

21.35 20.54 19.73 18.94 18.15

21.47 20.65 19.84 19.03 18.24

7.7029 7.7446 7.7863 7.8280 7.8697

57.622 57.934 58.246 58.557 58.869

34.71 33.74 32.79 31.85 30.92

34.97 34.00 33.03 32.08 31.14

7.0775 7.1192 7.1609 7.2026 7.2443

52.943 53.255 53.567 53.879 54.191

.950 .955 .960 .965 .970

17.37 16.60 15.83 15.08 14.33

17.45 16.67 15.90 15.13 14.38

7.9114 7.9531 7.9947 8.0364 8.0780

59.181 59.493 59.805 60.117 60.428

30.00 29.09 28.19 27.30 26.42

30.21 29.30 28.39 27.49 26.60

7.2860 7.3277 7.3694 7.4111 7.4528

54.503 54.815 55.127 55.438 55.750

.975 .980 .985 .990 .995 1.000

13.59 12.86 12.13 11.41 10.70 10.00

13.63 12.89 12.15 11.43 10.71 10.00

8.1197 8.1615 8.2032 8.2449 8.2866 8.3283

60.740 61.052 61.364 61.676 61.988 62.300 Lb per ft3 at 60°F. wt. in air

°Tw

Lb per gal at 60°F wt in air

°Tw

Lb per gal at 60°F wt in air

°Bé

°Tw

Lb per gal at 60°F wt in air

°Bé

°Tw

Lb per gal at 60°F wt in air

1.005 1.010 1.015 1.020 1.025

0.72 1.44 2.14 2.84 3.54

1 2 3 4 5

8.3700 8.4117 8.4534 8.4950 8.5367

62.612 62.924 63.236 63.547 63.859

1.255 1.260 1.265 1.270 1.275

29.46 29.92 30.38 30.83 31.27

51 52 53 54 55

10.4546 10.4963 10.5380 10.5797 10.6214

78.206 78.518 78.830 79.141 79.453

1.505 1.510 1.515 1.520 1.525

48.65 48.97 49.29 49.61 49.92

101 102 103 104 105

12.5392 12.5809 12.6226 12.6643 12.7060

93.800 94.112 94.424 94.735 95.047

1.755 1.760 1.765 1.770 1.775

62.38 62.61 62.85 63.08 63.31

151 152 153 154 155

14.6238 14.6655 14.7072 14.7489 14.7906

109.394 109.705 110.017 110.329 110.641

1.030 1.035 1.040 1.045 1.050

4.22 4.90 5.58 6.24 6.91

6 7 8 9 10

8.5784 8.6201 8.6618 8.7035 8.7452

64.171 64.483 64.795 65.107 65.419

1.280 1.285 1.290 1.295 1.300

31.72 32.16 32.60 33.03 33.46

56 57 58 59 60

10.6630 10.7047 10.7464 10.7881 10.8298

79.765 80.077 80.389 80.701 81.013

1.530 1.535 1.540 1.545 1.550

50.23 50.54 50.84 51.15 51.45

106 107 108 109 110

12.7477 12.7894 12.8310 12.8727 12.9144

95.359 95.671 95.983 96.295 96.606

1.780 1.785 1.790 1.795 1.800

63.54 63.77 63.99 64.22 64.44

156 157 158 159 160

14.8323 14.8740 14.9157 14.9574 14.9990

110.953 111.265 111.577 111.889 112.200

1.055 1.060 1.065 1.070 1.075

7.56 8.21 8.85 9.49 10.12

11 12 13 14 15

8.7869 8.8286 8.8703 8.9120 8.9537

65.731 66.042 66.354 66.666 66.978

1.305 1.310 1.315 1.320 1.325

33.89 34.31 34.73 35.15 35.57

61 62 63 64 65

10.8715 10.9132 10.9549 10.9966 11.0383

81.325 81.636 81.948 82.260 82.572

1.555 1.560 1.565 1.570 1.575

51.75 52.05 52.35 52.64 52.94

111 112 113 114 115

12.9561 12.9978 13.0395 13.0812 13.1229

96.918 97.230 97.542 97.854 98.166

1.805 1.810 1.815 1.820 1.825

64.67 64.89 65.11 65.33 65.55

161 162 163 164 165

15.0407 15.0824 15.1241 15.1658 15.2075

112.512 112.824 113.136 113.448 113.760

1.080 1.085 1.090 1.095 1.100

10.74 11.36 11.97 12.58 13.18

16 17 18 19 20

8.9954 9.0371 9.0787 9.1204 9.1621

67.290 67.602 67.914 68.226 68.537

1.330 1.335 1.340 1.345 1.350

35.98 36.39 36.79 37.19 37.59

66 67 68 69 70

11.0800 11.1217 11.1634 11.2051 11.2467

82.884 83.196 83.508 83.820 84.131

1.580 1.585 1.590 1.595 1.600

53.23 53.52 53.81 54.09 54.38

116 117 118 119 120

13.1646 13.2063 13.2480 13.2897 13.3313

98.478 98.790 99.102 99.414 99.725

1.830 1.835 1.840 1.845 1.850

65.77 65.98 66.20 66.41 66.62

166 167 168 169 170

15.2492 15.2909 15.3326 15.3743 15.4160

114.072 114.384 114.696 115.007 115.318

1.105 1.110 1.115 1.120 1.125

13.78 14.37 14.96 15.54 16.11

21 22 23 24 25

9.2038 9.2455 9.2872 9.3289 9.3706

68.849 69.161 69.473 69.785 70.097

1.355 1.360 1.365 1.370 1.375

37.99 38.38 38.77 39.16 39.55

71 72 73 74 75

11.2884 11.3301 11.3718 11.4135 11.4552

84.443 84.755 85.067 85.379 85.691

1.605 1.610 1.615 1.620 1.625

54.66 54.94 55.22 55.49 55.77

121 122 123 124 125

13.3730 13.4147 13.4564 13.4981 13.5398

100.037 100.349 100.661 100.973 101.285

1.855 1.860 1.865 1.870 1.875

66.83 67.04 67.25 67.46 67.67

171 172 173 174 175

15.4577 15.4993 15.5410 15.5827 15.6244

115.630 115.943 116.255 116.567 116.879

1.130 1.135 1.140 1.145 1.150

16.68 17.25 17.81 18.36 18.91

26 27 28 29 30

9.4123 9.4540 9.4957 9.5374 9.5790

70.409 70.721 71.032 71.344 71.656

1.380 1.385 1.390 1.395 1.400

39.93 40.31 40.68 41.06 41.43

76 77 78 79 80

11.4969 11.5386 11.5803 11.6220 11.6637

86.003 86.315 86.626 86.938 87.250

1.630 1.635 1.640 1.645 1.650

56.04 56.32 56.59 56.85 57.12

126 127 128 129 130

13.5815 13.6232 13.6649 13.7066 13.7483

101.597 101.909 102.220 102.532 102.844

1.880 1.885 1.890 1.895 1.900

67.87 68.08 68.28 68.48 68.68

176 177 178 179 180

15.6661 15.7078 15.7495 15.7912 15.8329

117.191 117.503 117.814 118.126 118.438

1.155 1.160 1.165 1.170 1.175

19.46 20.00 20.54 21.07 21.60

31 32 33 34 35

9.6207 9.6624 9.7041 9.7458 9.7875

71.968 72.280 72.592 72.904 73.216

1.405 1.410 1.415 1.420 1.425

41.80 42.16 42.53 42.89 43.25

81 82 83 84 85

11.7054 11.7471 11.7888 11.8304 11.8721

87.562 87.874 88.186 88.498 88.810

1.655 1.660 1.665 1.670 1.675

57.39 57.65 57.91 58.17 58.43

131 132 133 134 135

13.7900 13.8317 13.8734 13.9150 13.9567

103.156 103.468 103.780 104.092 104.404

1.905 1.910 1.915 1.920 1.925

68.88 69.08 69.28 69.48 69.68

181 182 183 184 185

15.8746 15.9163 15.9580 15.9996 16.0413

118.740 119.062 119.374 119.686 119.998

1.180 1.185 1.190 1.195 1.200

22.12 22.64 23.15 23.66 24.17

36 37 38 39 40

9.8292 9.8709 9.9126 9.9543 9.9960

73.528 73.840 74.151 74.463 74.775

1.430 1.435 1.440 1.445 1.450

43.60 43.95 44.31 44.65 45.00

86 87 88 89 90

11.9138 11.9555 11.9972 12.0389 12.0806

89.121 89.433 89.745 90.057 90.369

1.680 1.685 1.690 1.695 1.700

58.69 58.95 59.20 59.45 59.71

136 137 138 139 140

13.9984 14.0401 14.0818 14.1235 14.1652

104.715 105.027 105.339 105.651 105.963

1.930 1.935 1.940 1.945 1.950

69.87 70.06 70.26 70.45 70.64

186 187 188 189 190

16.0830 16.1247 16.1664 16.2081 16.2498

120.309 120.621 120.933 121.245 121.557

1.205 1.210 1.215 1.220 1.225

24.67 25.17 25.66 26.15 26.63

41 42 43 44 45

10.0377 10.0793 10.1210 10.1627 10.2044

75.087 75.399 75.711 76.022 76.334

1.455 1.460 1.465 1.470 1.475

45.34 45.68 46.02 46.36 46.69

91 92 93 94 95

12.1223 12.1640 12.2057 12.2473 12.2890

90.681 90.993 91.305 91.616 91.928

1.705 1.710 1.715 1.720 1.725

59.96 60.20 60.45 60.70 60.94

141 142 143 144 145

14.2069 14.2486 14.2903 14.3320 14.3737

106.275 106.587 106.899 107.210 107.522

1.955 1.960 1.965 1.970 1.975

70.83 71.02 71.21 71.40 71.58

191 192 193 194 195

16.2915 16.3332 16.3749 16.4166 16.4583

121.869 122.181 122.493 122.804 123.116

1.230 1.235 1.240 1.245 1.250

27.11 27.59 28.06 28.53 29.00

46 47 48 49 50

10.2461 10.2878 10.3295 10.3712 10.4129

76.646 76.958 77.270 77.582 77.894

1.480 1.485 1.490 1.495 1.500

47.03 47.36 47.68 48.01 48.33

96 97 98 99 100

12.3307 12.3724 12.4141 12.4558 12.4975

92.240 92.552 92.864 93.176 93.488

1.730 1.735 1.740 1.745 1.750

61.18 61.34 61.67 61.91 62.14

146 147 148 149 150

14.4153 14.4570 14.4987 14.5404 14.5821

107.834 108.146 108.458 108.770 109.082

1.980 1.985 1.990 1.995 2.000

71.77 71.95 72.14 72.32 72.50

196 197 198 199 200

16.5000 16.5417 16.5833 16.6250 16.6667

123.428 123.740 124.052 124.364 124.676

Sp gr 60°/ 60°

°Bé

Lb per ft3 at 60°F wt in air

Sp gr 60°/ 60°

Lb per ft3 at 60°F. wt. in air

*Prepared by Lewis V. Judson, Ph.D., Chief of Length Section of National Bureau of Standards with the advice and assistance of E. L. Peffer, B.S., A.M., late Chief of Capacity and Density Section, National Bureau of Standards.

1-20

TABLE 1-13

Wire and Sheet-Metal Gauges* Values in approximate decimals of an inch As a number of gauges are in use for various shapes and metals, it is advisable to state the thickness in thousandths when specifying gauge number.

Gauge number

American (AWG) or Brown & Sharpe (B & S) (for nonferrous wire and sheet)†

U.S. Steel Wire (Stl WG) or Washburn & Moen or Roebling or Am. Steel & Wire Co. [A. (steel) WG] (for steel wire)

Birmingham (BWG) (for steel wire) or Stubs Iron Wire (for iron or brass wire)‡

U.S. Standard (for sheet and plate metal, wrought iron)

0000000 000000 00000 0000 000 00 0

— — — 0.460 .410 .365 .325

0.4900 .4615 .4305 .3938 .3625 .3310 .3065

— — — 0.454 .425 .380 .340

1 2 3 4 5

.289 .258 .229 .204 .182

.2830 .2625 .2437 .2253 .2070

6 7 8 9 10

.162 .144 .128 .114 .102

11 12 13 14 15

U.S. Steel Wire (Stl WG) or Washburn & Moen or Roebling or Am. Steel & Wire Co. [A. (steel) WG] (for steel wire)

Birmingham (BWG) (for steel wire) or Stubs Iron Wire (for iron or brass wire)‡

U.S. Standard (for sheet and plate metal, wrought iron)

Standard Birmingham (BG) (for sheet and hoop metal)

Imperial Standard and Wire Gauge (SWG) (British legal standard)

Gauge number

Gauge number

American (AWG) or Brown & Sharpe (B & S) (for nonferrous wire and sheet)†

Standard Birmingham (BG) (for sheet and hoop metal)

Imperial Standard and Wire Gauge (SWG) (British legal standard)

Gauge number

— — — — — — —

0.6666 .6250 .5883 .5416 .5000 .4452 .3964

0.500 .464 .432 .400 .372 .348 .324

0000000 000000 00000 0000 000 00 0

26 27 28 29 30

0.0159 .0142 .0126 .0113 .0100

0.0181 .0173 .0162 .0150 .0140

0.018 .016 .014 .013 .012

0.0188 .0172 .0156 .0141 .0125

0.0196 .0175 .0156 .0139 .0123

0.018 .0164 .0148 .0136 .0124

26 27 28 29 30

.300 .284 .259 .238 .220

— — 0.239 .224 .209

.3532 .3147 .2804 .2500 .2225

.300 .276 .252 .232 .212

1 2 3 4 5

31 32 33 34 35

.0089 .0080 .0071 .0063 .0056

.0132 .0128 .0118 .0104 .0095

.010 .009 .008 .007 .005

.0109 .0102 .0094 .0086 .0078

.0110 .0098 .0087 .0077 .0069

.0116 .0108 .0100 .0092 .0084

31 32 33 34 35

.1920 .1770 .1620 .1483 .1350

.203 .180 .165 .148 .134

.194 .179 .164 .150 .135

.1981 .1764 .1570 .1398 .1250

.192 .176 .160 .144 .128

6 7 8 9 10

36 37 38 39 40

.0050 .0045 .0040 .0035 .0031

.0090 .0085 .0080 .0075 .0070

.004 — — — —

.0070 .0066 .0062 — —

.0061 .0054 .0048 .0043 .0039

.0076 .0068 .0060 .0052 .0048

36 37 38 39 40

.091 .081 .072 .064 .057

.1205 .1055 .0915 .0800 .0720

.120 .109 .095 .083 .072

.120 .105 .090 .075 .067

.1113 .0991 .0882 .0785 .0699

.116 .104 .092 .080 .072

11 12 13 14 15

41 42 43 44 45

— — — — —

.0066 .0062 .0060 .0058 .0055

— — — — —

— — — — —

.0034 .0031 .0027 .0024 .0022

.0044 .0040 .0036 .0032 .0028

41 42 43 44 45

16 17 18 19 20

.051 .045 .040 .036 .032

.0625 .0540 .0475 .0410 .0348

.065 .058 .049 .042 .035

.060 .054 .0478 .0418 .0359

.0625 .0556 .0495 .0440 .0392

.064 .056 .048 .040 .036

16 17 18 19 20

46 47 48 49 50

— — — — —

.0052 .0050 .0048 .0046 .0044

— — — — —

— — — — —

.0019 .0017 .0015 .0014 .0012

.0024 .0020 .0016 .0012 .0010

46 47 48 49 50

21 22 23 24 25

.0285 .0253 .0226 .0201 .0179

.0317 .0286 .0258 .0230 .0204

.032 .028 .025 .022 .020

.0329 .0299 .0269 .0239 .0209

.0349 .0313 .0278 .0248 .0220

.032 .028 .024 .022 .020

21 22 23 24 25

Metric wire gauge is 10 times the diameter in millimeters. *Courtesy of Dr. Lewis V. Judson with I. H. Fullmer, National Bureau of Standards. † Sometimes used for iron wire. ‡ Sometimes used for copper plate and for steel plate 12 gauge and heavier and for steel tubes.

1-21

TABLE 1-14

Fundamental Physical Constants

1 sec = 1.00273791 sidereal seconds g0 = 9.80665 m/sec2 1 liter = 0.001 cu. m 1 atm = 101,325 newtons/sq m 1 mm Hg (pressure) = (1⁄ 760) atm = 133.3224 newtons/sq m 1 int ohm = 1.000495 ⫾ 0.000015 abs ohm 1 int amp = 0.999835 ⫾ 0.000025 abs amp 1 int coul = 0.999835 ⫾ 0.000025 abs coul 1 int volt = 1.000330 ⫾ 0.000029 abs volt 1 int watt = 1.000165 ⫾ 0.000052 abs watt 1 int joule = 1.000165 ⫾ 0.000052 abs joule T0°C = 273.150 ⫾ 0.010°K (PV)0°CP=0 = (RT)0°C = 2271.16 ⫾ 0.04 abs joule/mole = 22,414.6 ⫾ 0.4 cu. cm atm/mole = 22.4146 ⫾ 0.0004 liter atm/mole R = 8.31439 ⫾ 0.00034 abs joule/deg mole = 1.98719 ⫾ 0.00013 cal/deg mole = 82.0567 ⫾ 0.0034 cu. cm atm/deg mole = 0.0820567 ⫾ 0.0000034 liter atm/deg mole ln 10 = 2.302585 R ln 10 = 19.14460 ⫾ 0.00078 abs joule/deg mole = 4.57567 ⫾ 0.00030 cal/deg mole N = (6.02283 ⫾ 0.0022) × 1023/mole h = (6.6242 ⫾ 0.0044) × 10−34 joule sec c = (2.99776 ⫾ 0.00008) × 108 m/sec (h2/8π2k) = (4.0258 ⫾ 0.0037) × 10−39 g sq cm deg (h/8π2c) = (2.7986 ⫾ 0.0018) × 10−39 g cm Z = Nhc = 11.9600 ⫾ 0.0036 abs joule cm/mole = 2.85851 ⫾ 0.0009 cal cm/mole (Z/R) = (hc/k) = c2 = 1.43847 ⫾ 0.00045 cm deg Ᏺ = 96,501.2 ⫾ 10.0 int coul/g-equiv or int joule/int volt g-equiv = 96,485.3 ⫾ 10.0 abs coul/g-equiv or abs joule/abs volt g-equiv = 23,068.1 ⫾ 2.4 cal/int volt g-equiv = 23,060.5 ⫾ 2.4 cal/abs volt g-equiv e = (1.60199 ⫾ 0.00060) × 10−19 abs coul = (1.60199 ⫾ 0.00060) × 10−20 abs emu = (4.80239 ⫾ 0.00180) × 10−10 abs esu 1 int electron-volt/molecule = 96,501.2 ⫾ 10 int joule/mole = 23,068.1 ⫾ 2.4 cal/mole 1 abs electron-volt/molecule = 96,485.3 ⫾ 10. abs joule/mole = 23,060.5 ⫾ 2.4 cal/mole 1 int electron-volt = (1.60252 ⫾ 0.00060) × 10−12 erg 1 abs electron-volt = (1.60199 ⫾ 0.00060) × 10−12 erg hc = (1.23916 ⫾ 0.00032) × 10−4 int electron-volt cm = (1.23957 ⫾ 0.00032) × 10−4 abs electron-volt cm k = (8.61442 ⫾ 0.00100) × 10−5 int electron-volt/deg = (8.61727 ⫾ 0.00100) × 10−5 abs electron-volt/deg = (R/N) = (1.38048 ⫾ 0.00050) × 10−23 joule/deg 1 IT cal = (1⁄ 860) = 0.00116279 int watt-hr = 4.18605 int joule = 4.18674 abs joule = 1.000654 cal 1 cal = 4.1840 abs joule = 4.1833 int joule = 41.2929 ⫾ 0.0020 cu. cm atm = 0.0412929 ⫾ 0.0000020 liter atm 1 IT cal/g = 1.8 Btu/lb 1 Btu = 251.996 IT cal = 0.293018 int watt-hr = 1054.866 int joule = 1055.040 abs joule = 252.161 cal 1 horsepower = 550 ft-lb (wt)/sec = 745.578 int watt = 745.70 abs watt 1 in = (1/0.3937) = 2.54 cm 1 ft = 0.304800610 m 1 lb = 453.5924277 g 1 gal = 231 cu. in = 0.133680555 cu. ft = 3.785412 × 10−3 cu. m = 3.785412 liter

1-22

sec = mean solar second Definition: g0 = standard gravity Definition: atm = standard atmosphere mm Hg (pressure) = standard millimeter mercury int = international; abs = absolute amp = ampere coul = coulomb

Absolute temperature of the ice point, 0°C PV product for ideal gas at 0°C R = gas constant per mole

ln = natural logarithm (base e) N = Avogadro number h = Planck constant c = velocity of light Constant in rotational partition function of gases Constant relating wave number and moment of inertia Z = constant relating wave number and energy per mole c2 = second radiation constant Ᏺ = Faraday constant

e = electronic charge

Constant relating wave number and energy per molecule k = Boltzmann constant Definition of IT cal: IT = International steam tables cal = thermochemical calorie Definition: cal = thermochemical calorie

Definition of Btu: Btu = IT British Thermal Unit

cal = thermochemical calorie Definition of horsepower (mechanical): lb (wt) = weight of 1 lb at standard gravity Definition of in: in = U.S. inch ft = U.S. foot (1 ft = 12 in) Definition; lb = avoirdupois pound Definition; gal = U.S. gallon

CONVERSION OF VALUES FROM U.S. CUSTOMARY UNITS TO SI UNITS American engineers are probably more familiar with the magnitude of physical entities in U.S. customary units than in SI units. Consequently, errors made in the conversion from one set of units to the other may go undetected. The following six examples will show how to convert the elements in six dimensionless groups. Proper conversions will result in the same numerical value for the dimensionless number. The dimensionless numbers used as examples are the Reynolds, Prandtl, Nusselt, Grashof, Schmidt, and Archimedes numbers. Table 1-7 provides a number of useful conversion factors. To make a conversion of an element in U.S. customary units to SI units, one multiplies the value of the U.S. customary unit, found on the left side in the table, by the equivalent value on the right side. For example, to convert 10 British thermal units to joules, one multiplies 10 by 1054.4 to obtain 10544 joules. In each example, the initial values of the factors are expressed in U.S. customary units, and the dimensionless value is calculated. Then the factors are converted to SI units, and the dimensionless value is recalculated. The two dimensionless values will be approximately the same. (Small variations occur due to the number of significant figures carried in the solution.)

Example 1. Calculation of a Reynolds Number DVρ NRe = ᎏ µ U.S. customary units D = 3 in. = 3⁄12 ft V = 6 ft/s ρ = 0.08 lbm/ft3 µ = 0.015 cp = (0.015)(0.000672) lbm/ft⋅s (3/12)(6)(0.08) NRe = ᎏᎏ = 11,904 (0.015)(0.000672) SI units D = (3)(0.0254) m V = (6)(0.3048) m/s ρ = (0.08)(16.018) kg/m3 µ = (0.015)(0.001) kg/m⋅s

(Difference due to rounding)

Example 4. Calculation of a Grashof Number NGr = L3ρ2gβ(∆T)/µ2 U.S. Customary units L = 3 ft ρ = 0.0725 lbm/ft3 g = 32.174 ft/s2 β = 0.00168/°R ∆T = 99 °R µ = 0.019 centipoise = 0.019 × 0.000672 lbm/ft⋅s = 1.277 × 10−5 lbm/ft⋅s (33) (0.0725)2(32.174) (0.00168) (99) = 4.66 × 109 NGr = ᎏᎏᎏᎏ (1.277 × 10−5)2 SI units L = (3)(0.3048) = 0.9144 m ρ = (0.0725)(16.018) = 1.1613 kg/m3 g = 9.807 m/s2 β = (0.00168)/(1.8) = 0.000933/°K ∆T = (99)(1.8) = 178.2 °K µ = (0.019)(0.001) = 1.9 × 10−5 kg/m⋅s (0.9144)3(1.1613)2(9.807)(0.000933)(178.2) = 4.66 × 109 NGr = ᎏᎏᎏᎏᎏ (1.9 × 10−5)2

Example 5. Calculation of a Schmidt Number µ NSc = ᎏ ρD

(3 × 0.0254) (6 × 0.3048) (0.08 × 16.018) NRe = ᎏᎏᎏᎏᎏ = 11,904 (0.015) (0.001)

Example 2. Calculation of a Prandtl Number Cp µ NPr = ᎏ k U.S. customary units γp = 0.47 Btu/lbm °F µ = 15 centipoise = (15) (0.000672) (3600) lbm/ft⋅hr k = 0.065 Btu/hr⋅ft2 (°F/ft) (0.47) (15 × 0.000672 × 3600) NPr = ᎏᎏᎏᎏ = 262.4 0.065 SI units γ = (0.47)(4184) J/kg °C µ = (15)(0.001) kg/m⋅s k = (0.065)(1.728) J/s⋅m2 (°C/m) (0.47) (4184) (15) (0.001) NPr = ᎏᎏᎏ = 262.6 (0.065) (1.728) (Difference due to rounding)

Example 3. Calculation of a Nusselt Number hD NNu = ᎏ k U.S. customary units h = 200 Btu/hr⋅ft2⋅°F D = 1.5 in. = 1.5/12 ft k = 0.07 Btu/hr⋅ft2 (°F/ft) (200)(1.5/12) NNu = ᎏᎏ = 357.1 0.07 SI units h = (200)(5.678) J/(s⋅m2⋅°C) D = (1.5)(0.0254) m k = (0.07)(1.728) J/s⋅m2 (°C/m)

(200) (5.678) (1.5) (0.0254) NNu = ᎏᎏᎏ = 357.7 (0.07) (1.728)

U.S. customary units µ = 0.02 centipoise = (0.02)(2.42) lbm/ft⋅hr ρ = 0.08 lbm/ft3 D = 1.0 ft2/hr (diffusivity) (0.02) (2.42) NSc = ᎏᎏ = 0.605 (0.08)(1.0) SI units µ = (0.02)(0.001) kg/m⋅s ρ = (0.08)(16.02) kg/m2 D = (1.0)(2.58 × 10−5) m2/s (0.02) (0.001) = 0.605 NSc = ᎏᎏᎏᎏ (0.08)(16.02)(1.0) (2.58 × 10−5)

Example 6. Calculation of an Archimedes Number d 3ρf(ρp − ρf)g NAr = ᎏᎏ µ2 U.S. customary units d = 2 mm = 2/[(1000)(0.3048)] = 0.00656 ft ρf = 0.0175 lbm/ft3 ρp = 168.5 lbm/ft3 g = 32.174 ft/s2 µ = 0.04 centipoise = 0.04 × 0.000672 = 2.688−5 lbm/ft⋅s (0.00656)3 (0.0175) (168.5 − 0.017) (32.174) = 37,064 NAr = ᎏᎏᎏᎏᎏ (2.688 × 10−5)2 SI units d = 2/1000 m ρp = 168.5 × 16.02 = 2699.37 kg/m3 ρf = 0.0175 × 16.02 = 0.2804 g/m3 g = 9.807 m/s2 µ = 0.04 × 0.001 = 4 × 10−5 kg/m⋅s (2/1000)3 (0.2804) (2699.37 − 0.28) (9.807) = 37,118 NAr = ᎏᎏᎏᎏᎏ (4 × 10−5)2 (Difference due to rounding)

1-23

MATHEMATICAL SYMBOLS TABLE 1-15

Mathematical Signs, Symbols, and Abbreviations

⫾ (⫿) : ⬋ < ⬏ > ⬐ ⬵ ∼ ⬑ ≠ ⬟ ∝ ∞ ∴ 兹苶苶 3 苶苶 兹 n 兹苶苶 ⬔ ⊥ 储 |x| log or log10 loge or ln e a° a′ a a″ a sin cos tan ctn or cot sec csc vers covers exsec sin−1 sinh cosh tanh sinh−1 f(x) or φ(x) ∆x 冱 dx dy/dx or y′ d2y/dx2 or y″ dny/dxn ∂y/∂x ∂ny/∂xn ∂ny ᎏ ∂x∂y

冮

冕

plus or minus (minus or plus) divided by, ratio sign proportional sign less than not less than greater than not greater than approximately equals, congruent similar to equivalent to not equal to approaches, is approximately equal to varies as infinity therefore square root cube root nth root angle perpendicular to parallel to numerical value of x common logarithm or Briggsian logarithm natural logarithm or hyperbolic logarithm or Naperian logarithm base (2.178) of natural system of logarithms an angle a degrees prime, an angle a minutes double prime, an angle a seconds, a second sine cosine tangent cotangent secant cosecant versed sine coversed sine exsecant anti sine or angle whose sine is hyperbolic sine hyperbolic cosine hyperbolic tangent anti hyperbolic sine or angle whose hyperbolic sine is function of x increment of x summation of differential of x derivative of y with respect to x second derivative of y with respect to x nth derivative of y with respect to x partial derivative of y with respect to x nth partial derivative of y with respect to x nth partial derivative with respect to x and y integral of

b

integral between the limits a and b

a

y˙ y¨ ∆ or ∇2

δ

冖

1-24

first derivative of y with respect to time second derivative of y with respect to time the “Laplacian” ∂2 ∂2 ∂2 ᎏ2 + ᎏ2 + ᎏ2 ∂x ∂y ∂z sign of a variation sign for integration around a closed path

冢

冣

TABLE 1-16 Alpha Beta Gamma Delta Epsilon Zeta Eta Theta Iota Kappa Lambda Mu

Greek Alphabet

= Α, α = A, a = Β, β = B, b = Γ, γ = G, g = ∆, δ = D, d = Ε, ε = E, e = Ζ, ζ = Z, z = Η, η = E, e = Θ, θ = Th, th = Ι, ␫ = I, i = Κ, κ = K, k = Λ, λ = L, l = Μ, µ = M, m

Nu Xi Omicron Pi Rho Sigma Tau Upsilon Phi Chi Psi Omega

= Ν, ν = N, n = Ξ, ξ = X, x = Ο, ο = O, o = Π, π = P, p = Ρ, ρ = R, r = Σ, σ = S, s = Τ, τ = T, t = ⌼, υ = U, u = Φ, φ = Ph, ph = Χ, χ = Ch, ch = Ψ, ψ = Ps, ps = Ω, ω = O, o

Section 2

Physical and Chemical Data*

Peter E. Liley, Ph.D., D.I.C., School of Mechanical Engineering, Purdue University. (physical and chemical data) George H. Thomson, AIChE Design Institute for Physical Property Data. (Tables 2-6, 2-30, 2-164, 2-193, 2-196, 2-198, 2-221) D.G. Friend, National Institutes of Standards and Technology, Boulder, CO. (Tables 2-333, 2-334, Figs. 2-25, 2-26) Thomas E. Daubert, Ph.D., Professor, Department of Chemical Engineering, The Pennsylvania State University. (Prediction and Correlation of Physical Properties) Evan Buck, M.S.Ch.E., Manager, Thermophysical Property Skill Center, Central Technology, Union Carbide Corporation. (Prediction and Correlation of Physical Properties)

GENERAL REFERENCES PHYSICAL PROPERTIES OF PURE SUBSTANCES Tables 2-1 2-2

Physical Properties of the Elements and Inorganic Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physical Properties of Organic Compounds. . . . . . . . . . . . . .

VAPOR PRESSURES OF PURE SUBSTANCES Units Conversions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-3 Vapor Pressure of Water Ice from −15 to 0°C . . . . . . . . . . . . 2-4 Vapor Pressure of Liquid Water from −16 to 0°C . . . . . . . . . 2-5 Vapor Pressure of Liquid Water from 0 to 100°C . . . . . . . . . 2-6 Vapor Pressure of Inorganic and Organic Liquids. . . . . . . . . 2-6a Alphabetical Index to Substances in Tables 2-6, 2-30, 2-164, 2-193, 2-196, 2-198, and 2-221 . . . . . . . . . . . . . . . . . 2-7 Vapor Pressures of Inorganic Compounds, up to 1 atm . . . . 2-8 Vapor Pressures of Organic Compounds, up to 1 atm. . . . . . VAPOR PRESSURES OF SOLUTIONS Units Conversions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables and Figures 2-9 Partial Pressures of Water over Aqueous Solutions of HCl . . 2-10 Partial Pressures of HCl over Aqueous Solutions of HCl . . .

2-11 2-7 2-28

2-12 2-13

2-48 2-48 2-48 2-48 2-49 2-50 2-55 2-57 2-61

2-76 2-76 2-76

2-14 2-15 2-16 2-17 2-18 2-19 2-20 2-21 2-22 2-23

Vapor Pressures of H3PO4 Aqueous: Partial Pressure of H2O Vapor (Fig. 2-1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vapor Pressures of H3PO4 Aqueous: Weight of H2O in Saturated Air (Fig. 2-2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Pressures of H2O and SO2 over Aqueous Solutions of Sulfur Dioxide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Water Partial Pressure, bar, over Aqueous Sulfuric Acid Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sulfur Trioxide Partial Pressure, bar, over Aqueous Sulfuric Acid Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sulfuric Acid Partial Pressure, bar, over Aqueous Sulfuric Acid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total Pressure, bar, of Aqueous Sulfuric Acid Solutions . . . . Partial Pressures of HNO3 and H2O over Aqueous Solutions of HNO3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Pressures of H2O and HBr over Aqueous Solutions of HBr at 20 to 55°C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Pressures of HI over Aqueous Solutions of HI at 25°C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vapor Pressures of the System: Water-Sulfuric Acid-Nitric Acid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total Vapor Pressures of Aqueous Solutions of CH3COOH . . Partial Pressures of H2O over Aqueous Solutions of HN3 . . Vapor Pressure of Aqueous Diethylene Glycol Solutions (Fig. 2-3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mole Percentages of H2O over Aqueous Solutions of NH3 . . Partial Pressures of NH3 over Aqueous Solutions of NH3 . . .

2-77 2-77 2-77 2-78 2-80 2-82 2-83 2-84 2-85 2-85 2-85 2-85 2-85 2-85 2-86 2-87

* The contributions of J.K. Fink, Argonne National Laboratory; U. Grigull, Tech. Universität, Munich, Germany; and H. Sato, Keio University, Japan, are acknowledged. 2-1

2-2 2-24 2-25 2-26 2-27

PHYSICAL AND CHEMICAL DATA Total Vapor Pressures of Aqueous Solutions of NH3 . . . . . . . Partial Pressures of H2O over Aqueous Solutions of Sodium Carbonate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Pressures of H2O and CH3OH over Aqueous Solutions of Methyl Alcohol . . . . . . . . . . . . . . . . . . . . . . . . . Partial Pressures of H2O over Aqueous Solutions of Sodium Hydroxide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

WATER-VAPOR CONTENT OF GASES Chart for Gases at High Pressures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Water Content of Air (Fig. 2-4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DENSITIES OF PURE SUBSTANCES Units Conversions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-28 Density (kg/m3) of Water from 0 to 100°C. . . . . . . . . . . . . . . 2-29 Density (kg/m3) of Mercury from 0 to 350°C. . . . . . . . . . . . . 2-30 Densities of Inorganic and Organic Liquids . . . . . . . . . . . . . DENSITIES OF AQUEOUS INORGANIC SOLUTIONS Units and Units Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-31 Aluminum Sulfate [A12(SO4)3] . . . . . . . . . . . . . . . . . . . . . . . . . 2-32 Ammonia (NH3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-33 Ammonium Acetate (CH3COONH4) . . . . . . . . . . . . . . . . . . . . 2-34 Ammonium Bichromate [(NH4)2Cr2O7]. . . . . . . . . . . . . . . . . . 2-35 Ammonium Chloride (NH4Cl) . . . . . . . . . . . . . . . . . . . . . . . . . 2-36 Ammonium Chromate [(NH4)2CrO4]. . . . . . . . . . . . . . . . . . . . 2-37 Ammonium Nitrate (NH4NO3). . . . . . . . . . . . . . . . . . . . . . . . . 2-38 Ammonium Sulfate [(NH4)2SO4] . . . . . . . . . . . . . . . . . . . . . . . 2-39 Arsenic Acid (H3A3O4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-40 Barium Chloride (BaCl2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-41 Cadmium Nitrate [Cd(NO3)2]. . . . . . . . . . . . . . . . . . . . . . . . . . 2-42 Calcium Chloride (CaCl2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-43 Calcium Hydroxide [Ca(OH)2] . . . . . . . . . . . . . . . . . . . . . . . . . 2-44 Calcium Hypochlorite (CaOCl2). . . . . . . . . . . . . . . . . . . . . . . . 2-45 Calcium Nitrate [Ca(NO3)2] . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-46 Chromic Acid (CrO3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-47 Chromium Chloride (CrCl3) . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-48 Copper Nitrate [Cu(NO3)2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-49 Copper Sulfate (CuSO4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-50 Cuprous Chloride (Cu2Cl2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-51 Ferric Chloride (FeCl3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-52 Ferric Sulfate [Fe2(SO4)3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-53 Ferric Nitrate [Fe(NO3)3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-54 Ferrous Sulfate (FeSO4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-55 Hydrogen Bromide (HBr). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-56 Hydrogen Cyanide (HCN) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-57 Hydrogen Chloride (HCl). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-58 Hydrogen Fluoride (HF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-59 Hydrogen Peroxide (H2O2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-60 Hydrofluosilic Acid (H2SiF6). . . . . . . . . . . . . . . . . . . . . . . . . . . 2-61 Magnesium Chloride (MgCl2) . . . . . . . . . . . . . . . . . . . . . . . . . 2-62 Magnesium Sulfate (MgSO4) . . . . . . . . . . . . . . . . . . . . . . . . . . 2-63 Nickel Chloride (NiCl2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-64 Nickel Nitrate [Ni(NO3)2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-65 Nickel Sulfate (NiSO4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-66 Nitric Acid (HNO3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-67 Perchloric Acid (HClO4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-68 Phosphoric Acid (H3PO4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-69 Potassium Bicarbonate (KHCO3) . . . . . . . . . . . . . . . . . . . . . . . 2-70 Potassium Bromide (KBr). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-71 Potassium Carbonate (K2CO3) . . . . . . . . . . . . . . . . . . . . . . . . . 2-72 Potassium Chromate (K2CrO4) . . . . . . . . . . . . . . . . . . . . . . . . . 2-73 Potassium Chlorate (KClO3) . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-74 Potassium Chloride (KCl) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-75 Potassium Chrome Alum [K2Cr2(SO4)4]. . . . . . . . . . . . . . . . . . 2-76 Potassium Hydroxide (KOH) . . . . . . . . . . . . . . . . . . . . . . . . . . 2-77 Potassium Nitrate (KNO3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-78 Potassium Dichromate (K2Cr2O7). . . . . . . . . . . . . . . . . . . . . . . 2-79 Potassium Sulfate (K2SO4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-80 Potassium Sulfite (K2SO3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-81 Sodium Acetate (NaC2H3O2) . . . . . . . . . . . . . . . . . . . . . . . . . . 2-82 Sodium Arsenate (Na3AsO4) . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-83 Sodium Bichromate (Na2Cr2O7) . . . . . . . . . . . . . . . . . . . . . . . . 2-84 Sodium Bromide (NaBr). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-85 Sodium Formate (HCOONa) . . . . . . . . . . . . . . . . . . . . . . . . . . 2-86 Sodium Carbonate (Na2CO3) . . . . . . . . . . . . . . . . . . . . . . . . . . 2-87 Sodium Chlorate (NaClO3) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2-88 2-89 2-89 2-89

2-90 2-90

2-91 2-91 2-93 2-94

2-99 2-99 2-99 2-99 2-99 2-99 2-99 2-99 2-99 2-99 2-99 2-100 2-100 2-100 2-100 2-100 2-100 2-100 2-100 2-100 2-100 2-100 2-100 2-101 2-101 2-101 2-101 2-101 2-101 2-101 2-101 2-101 2-101 2-101 2-101 2-101 2-101 2-102 2-103 2-103 2-103 2-103 2-104 2-104 2-104 2-104 2-104 2-104 2-104 2-104 2-104 2-104 2-104 2-104 2-104 2-104 2-104 2-105 2-105

2-88 2-89 2-90 2-91 2-92 2-93 2-94 2-95 2-96 2-97 2-98 2-99 2-100 2-101 2-102 2-103 2-104 2-105

Sodium Chloride (NaCl) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sodium Chromate (Na2CrO4) . . . . . . . . . . . . . . . . . . . . . . . . . Sodium Hydroxide (NaOH) . . . . . . . . . . . . . . . . . . . . . . . . . . Sodium Nitrate (NaNO3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sodium Nitrite (NaNO2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sodium Silicates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sodium Sulfate (Na2SO4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sodium Sulfide (Na2S) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sodium Sulfite (Na2SO3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sodium Thiosulfate (Na2S2O3) . . . . . . . . . . . . . . . . . . . . . . . . Sodium Thiosulfate Pentahydrate (Na2S2O3⋅5H2O). . . . . . . . Stannic Chloride (SnCl4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stannous Chloride (SnCl2) . . . . . . . . . . . . . . . . . . . . . . . . . . . Sulfuric Acid (H2SO4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zinc Bromide (ZnBr2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zinc Chloride (ZnCl2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zinc Nitrate [Zn(NO3)2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zinc Sulfate (ZnSO4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

DENSITIES OF AQUEOUS ORGANIC SOLUTIONS Units and Units Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-106 Formic Acid (HCOOH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-107 Acetic Acid (CH3COOH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-108 Oxalic Acid (H2C2O4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-109 Methyl Alcohol (CH3OH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-110 Ethyl Alcohol (C2H5OH). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-111 Densities of Mixtures of C2H5OH and H2O at 20°C . . . . . . . 2-112 Specific Gravity (60°/60°F [(15.56°/15.56°C)]) of Mixtures by Volume of C2H5OH and H2O . . . . . . . . . . . . . . 2-113 n-Propyl Alcohol (C3H7OH) . . . . . . . . . . . . . . . . . . . . . . . . . . 2-114 Isopropyl Alcohol (C3H7OH) . . . . . . . . . . . . . . . . . . . . . . . . . 2-115 Glycerol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-116 Hydrazine (N2H4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-117 Densities of Aqueous Solutions of Miscellaneous Organic Compounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DENSITIES OF MISCELLANEOUS MATERIALS Tables 2-118 Approximate Specific Gravities and Densities of Miscellaneous Solids and Liquids. . . . . . . . . . . . . . . . . . . . . 2-119 Density (kg/m3) of Selected Elements as a Function of Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SOLUBILITIES Units Conversions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-120 Solubilities of Inorganic Compounds in Water at Various Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-121 Acetylene (C2H2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-122 Air. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-123 Ammonia (NH3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-124 Ammonia (NH3)—Low Pressures . . . . . . . . . . . . . . . . . . . . . 2-125 Carbon Dioxide (CO2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-126 Carbon Monoxide (CO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-127 Carbonyl Sulfide (COS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-128 Chlorine (Cl2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-129 Chlorine Dioxide (ClO2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-130 Ethane (C2H6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-131 Ethylene (C2H4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-132 Helium (He) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-133 Hydrogen (H2)—Temperature . . . . . . . . . . . . . . . . . . . . . . . . 2-134 Hydrogen (H2)—Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-135 Hydrogen Chloride (HCl). . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-136 Hydrogen Sulfide (H2S) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-137 Methane (CH4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-138 Nitrogen (N2)—Temperature . . . . . . . . . . . . . . . . . . . . . . . . . 2-139 Nitrogen (N2)—Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-140 Oxygen (O2)—Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 2-141 Oxygen (O2)—Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-142 Ozone (O3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-143 Propylene (C3H6). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-144 Sulfur Dioxide (SO2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . THERMAL EXPANSION Units Conversions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Expansion of Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-145 Linear Expansion of the Solid Elements . . . . . . . . . . . . . . . .

2-105 2-105 2-105 2-105 2-105 2-105 2-106 2-106 2-106 2-106 2-106 2-106 2-106 2-107 2-109 2-109 2-109 2-109

2-109 2-109 2-110 2-111 2-111 2-112 2-113 2-114 2-115 2-115 2-116 2-116 2-117

2-119 2-120 2-120 2-121 2-125 2-125 2-125 2-125 2-125 2-125 2-125 2-126 2-126 2-126 2-126 2-126 2-126 2-127 2-127 2-127 2-127 2-127 2-127 2-127 2-127 2-128 2-128 2-128

2-128 2-128 2-128 2-129

PHYSICAL AND CHEMICAL DATA 2-146 2-147 2-148

Linear Expansion of Miscellaneous Substances. . . . . . . . . . . Cubical Expansion of Liquids . . . . . . . . . . . . . . . . . . . . . . . . . Cubical Expansion of Solids . . . . . . . . . . . . . . . . . . . . . . . . . .

JOULE-THOMSON EFFECT Units Conversions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-149 Additional References Available for the Joule-Thomson Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-150 Approximate Inversion-Curve Locus in Reduced Coordinates (Tr = T/Tc; Pr = P/Pc) . . . . . . . . . . . . . . . . . . . . . 2-151 Joule-Thomson Data for Air . . . . . . . . . . . . . . . . . . . . . . . . . . 2-152 Approximate Inversion-Curve Locus for Air . . . . . . . . . . . . . 2-153 Joule-Thomson Data for Argon . . . . . . . . . . . . . . . . . . . . . . . 2-154 Approximate Inversion-Curve Locus for Argon. . . . . . . . . . . 2-155 Joule-Thomson Data for Carbon Dioxide . . . . . . . . . . . . . . . 2-156 Approximate Inversion-Curve Locus for Carbon Dioxide . . 2-157 Approximate Inversion-Curve Locus for Deuterium . . . . . . 2-158 Approximate Inversion-Curve Locus for Ethane. . . . . . . . . . 2-159 Joule-Thomson Data for Helium . . . . . . . . . . . . . . . . . . . . . . 2-160 Approximate Inversion-Curve Locus for Normal Hydrogen. . 2-161 Approximate Inversion-Curve Locus for Methane . . . . . . . . 2-162 Joule-Thomson Data for Nitrogen . . . . . . . . . . . . . . . . . . . . . 2-163 Approximate Inversion-Curve Locus for Propane. . . . . . . . . CRITICAL CONSTANTS Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 2-164 Critical Constants and Acentric Factors of Inorganic and Organic Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . COMPRESSIBILITIES Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Units Conversions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-165 Compressibility Factors for Air. . . . . . . . . . . . . . . . . . . . . . . . 2-166 Compressibility Factors for Argon . . . . . . . . . . . . . . . . . . . . . 2-167 Compressibility Factors for Carbon Dioxide . . . . . . . . . . . . . 2-168 Compressibility Factors for Carbon Monoxide . . . . . . . . . . . 2-169 Compressibility Factors for Ethanol. . . . . . . . . . . . . . . . . . . . 2-170 Compressibility Factors for Ethylene. . . . . . . . . . . . . . . . . . . 2-171 Compressibility Factors for Normal Hydrogen . . . . . . . . . . . 2-172 Compressibility Factors for KLEA 60 . . . . . . . . . . . . . . . . . . 2-173 Compressibility Factors for KLEA 61 . . . . . . . . . . . . . . . . . . 2-174 Compressibility Factors for KLEA 66 . . . . . . . . . . . . . . . . . . 2-175 Compressibility Factors for Krypton . . . . . . . . . . . . . . . . . . . 2-176 Compressibility Factors for Methane (R50) . . . . . . . . . . . . . 2-177 Compressibility Factors for Methanol . . . . . . . . . . . . . . . . . . 2-178 Compressibility Factors for Neon. . . . . . . . . . . . . . . . . . . . . . 2-179 Compressibility Factors for Nitrogen. . . . . . . . . . . . . . . . . . . 2-180 Compressibility Factors for Oxygen . . . . . . . . . . . . . . . . . . . . 2-181 Compressibility Factors for Refrigerant 32 . . . . . . . . . . . . . . 2-182 Compressibility Factors for Refrigerant 123 . . . . . . . . . . . . . 2-183 Compressibility Factors for Refrigerant 124 . . . . . . . . . . . . . 2-184 Compressibility Factors for Refrigerant 134a . . . . . . . . . . . . 2-185 Compressibility Factors for Water Substance (fps units) . . . 2-186 Compressibility Factors of Water Substance (SI units) . . . . . 2-187 Compressibility Factors for Xenon . . . . . . . . . . . . . . . . . . . . . 2-188 Compressibilities of Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . 2-189 Compressibilities of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . LATENT HEATS Units Conversions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-190 Heats of Fusion and Vaporization of the Elements and Inorganic Compounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-191 Heats of Fusion of Miscellaneous Materials . . . . . . . . . . . . . 2-192 Heats of Fusion of Organic Compounds . . . . . . . . . . . . . . . . 2-193 Heats of Vaporization of Inorganic and Organic Compounds SPECIFIC HEATS OF PURE COMPOUNDS Units Conversions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-194 Heat Capacities of the Elements and Inorganic Compounds 2-195 Specific Heat [kJ/(kg·K)] of Selected Elements. . . . . . . . . . . 2-196 Heat Capacities of Inorganic and Organic Liquids . . . . . . . . 2-197 Specific Heats of Organic Solids. . . . . . . . . . . . . . . . . . . . . . . 2-198 Heat Capacities of Inorganic and Organic Compounds in the Ideal Gas State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2-130 2-131 2-131 2-132 2-132 2-133 2-133 2-133 2-133 2-134 2-134 2-134 2-134 2-134 2-135 2-135 2-135 2-135 2-135 2-136 2-136 2-140 2-140 2-140 2-140 2-141 2-141 2-141 2-142 2-142 2-142 2-143 2-143 2-143 2-144 2-144 2-144 2-145 2-145 2-145 2-146 2-146 2-146 2-147 2-148 2-149 2-149 2-150 2-150 2-151 2-153 2-154 2-156 2-161 2-161 2-161 2-169 2-170 2-175 2-178

2-199 Cp /Cv: Ratios of Specific Heats of Gases at 1-atm Pressure . . 2-200 Specific Heat Ratio, Cp /Cv, for Air . . . . . . . . . . . . . . . . . . . . .

2-3 2-183 2-183

SPECIFIC HEATS OF AQUEOUS SOLUTIONS Units Conversions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-201 Acetic Acid (at 38°C). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-202 Ammonia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-203 Aniline (at 20°C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-204 Copper Sulfate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-205 Ethyl Alcohol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-206 Glycerol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-207 Hydrochloric Acid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-208 Methyl Alcohol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-209 Nitric Acid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-210 Phosphoric Acid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-211 Potassium Chloride . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-212 Potassium Hydroxide (at 19°C) . . . . . . . . . . . . . . . . . . . . . . . 2-213 Normal Propyl Alcohol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-214 Sodium Carbonate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-215 Sodium Chloride . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-216 Sodium Hydroxide (at 20°C) . . . . . . . . . . . . . . . . . . . . . . . . . 2-217 Sulfuric Acid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-218 Zinc Sulfate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2-184 2-184 2-184 2-184 2-184 2-184 2-184 2-184 2-184 2-184 2-185 2-185 2-185 2-185 2-185 2-185 2-185 2-185

SPECIFIC HEATS OF MISCELLANEOUS MATERIALS Tables 2-219 Specific Heats of Miscellaneous Liquids and Solids . . . . . . . 2-219a Oils (Animal, Vegetable, Mineral Oils). . . . . . . . . . . . . . . . . .

2-186 2-186

HEATS AND FREE ENERGIES OF FORMATION Units Conversions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 2-220 Heats and Free Energies of Formation of Inorganic Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HEATS OF COMBUSTION Table 2-221 Enthalpies and Gibbs Energies of Formation, Entropies, and Net Enthalpies of Combustion of Inorganic and Organic Compounds at 298.15 K . . . . . . . . . . . . . . . . . . . . . 2-222 Ideal Gas Sensible Enthalpies, hT − h298 (kJ/kgmol), of Combustion Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-223 Ideal Gas Entropies, s°, kJ/kgmol⋅K, of Combustion Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2-184 2-184

2-186 2-187

2-195 2-199 2-200

HEATS OF SOLUTION Tables 2-224 Heats of Solution of Inorganic Compounds in Water . . . . . . 2-201 2-225 Heats of Solution of Organic Compounds in Water (at Infinite Dilution and Approximately Room Temperature). . . . . . . . 2-204 THERMODYNAMIC PROPERTIES Explanation of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Units Conversions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-226 Thermophysical Properties of Saturated Acetone . . . . . . . . . 2-227 Saturated Acetylene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-228 Saturated Air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-229 Thermophysical Properties of Compressed Air . . . . . . . . . . . 2-230 Enthalpy and Psi Functions for Ideal-Gas Air . . . . . . . . . . . Temperature-Entropy Diagram for Air (Fig. 2-5) . . . . . . . . . 2-231 Air. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-232 Saturated Ammonia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy–Log-Pressure Diagram for Ammonia (Fig. 2-6) . . Enthalpy-Concentration Diagram for Aqueous Ammonia (Fig. 2-7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-233 Saturated Argon (R740) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-234 Thermodynamic Properties of Compressed Argon . . . . . . . . 2-235 Liquid-Vapor Equilibrium Data for the Argon-NitrogenOxygen System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-236 Thermodynamic Properties of the International Standard Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-237 Saturated Benzene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-238 Saturated Bromine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-239 Saturated Normal Butane (R600). . . . . . . . . . . . . . . . . . . . . . 2-240 Superheated Normal Butane . . . . . . . . . . . . . . . . . . . . . . . . .

2-205 2-205 2-205 2-206 2-206 2-207 2-208 2-212 2-213 2-214 2-214 2-215 2-216 2-217 2-217 2-218 2-221 2-221 2-222 2-222 2-223

2-4 2-241 2-242 2-243 2-244 2-245 2-246 2-247 2-248 2-249 2-250 2-251 2-252 2-253 2-254 2-255 2-256 2-257 2-258 2-259 2-260 2-261 2-262 2-263 2-264 2-265 2-266 2-267 2-268 2-269 2-270 2-271 2-272 2-273 2-274 2-275 2-276 2-277 2-278 2-279 2-280 2-281 2-282 2-283 2-284 2-285 2-286 2-287 2-288 2-289 2-290 2-291 2-292 2-293 2-294

2-295 2-296 2-297 2-298 2-299 2-300 2-301 2-302

PHYSICAL AND CHEMICAL DATA Saturated Carbon Dioxide . . . . . . . . . . . . . . . . . . . . . . . . . . . Superheated Carbon Dioxide . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Carbon Monoxide. . . . . . . . . . . . . . . . . . . . . . . . . . Temperature-Entropy Diagram for Carbon Monoxide (Fig. 2-8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermophysical Properties of Saturated Carbon Tetrachloride . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Carbon Tetrafluoride (R14) . . . . . . . . . . . . . . . . . . Saturated Cesium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermophysical Properties of Saturated Chlorine . . . . . . . . Enthalpy–Log-Pressure Diagram for Chlorine (Fig. 2-9) . . . Saturated Chloroform (R20) . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Decane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Normal Deuterium. . . . . . . . . . . . . . . . . . . . . . . . . Saturated Deuterium Oxide . . . . . . . . . . . . . . . . . . . . . . . . . . Deuterium Oxide Gas at 1-kg/cm3 Pressure . . . . . . . . . . . . . Saturated Diphenyl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Ethane (R170) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Superheated Ethane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Ethanol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy-Concentration Diagram for Aqueous Ethyl Alcohol (Fig. 2-10). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Ethylene (Ethene—R1150) . . . . . . . . . . . . . . . . . . Compressed Ethylene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Fluorine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluorine Gas at Atmospheric Pressure. . . . . . . . . . . . . . . . . . Flutec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Halon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Helium3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Helium4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Superheated Helium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Helium4 Gas at Atmospheric Pressure . . . . . . . . . . . . . . . . . . Saturated n-Heptane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hexane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Hydrazine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated n-Hydrogen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compressed n-Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated para-Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Hydrogen Peroxide. . . . . . . . . . . . . . . . . . . . . . . . . Hydrogen Sulfide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy-Concentration Diagram for Aqueous Hydrogen Chloride at 1 atm (Fig. 2-11) . . . . . . . . . . . . . . . . . . . . . . . . Saturated Isobutane (R600a) . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Krypton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compressed Krypton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Lithium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lithium Bromide—Water Solutions. . . . . . . . . . . . . . . . . . . . Saturated Mercury. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Methane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy–Log-Pressure Diagram for Mercury (Fig. 2-12) . . Superheated Methane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermophysical Properties of Saturated Methanol . . . . . . . . Thermodynamic Properties of Compressed Methanol . . . . . Saturated Methyl Chloride . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Neon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compressed Neon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Nitrogen (R728) . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature-Entropy Diagram for Nitrogen (Fig. 2-13) . . . Thermophysical Properties of Nitrogen (R728) at Atmospheric Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Nitrogen Tetroxide . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Nitrous Oxide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mollier Diagram for Nitrous Oxide (Fig. 2-14) . . . . . . . . . . . Nonane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Octane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Oxygen (R732) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature-Entropy Chart for Oxygen (Fig. 2-15) . . . . . . . Enthalpy-Concentration Diagram for Oxygen-Nitrogen Mixture at 1 atm (Fig. 2-16) . . . . . . . . . . . . . . . . . . . . . . . . . Pentane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Potassium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mollier Diagram for Potassium (Fig. 2-17) . . . . . . . . . . . . . . Saturated Propane (R290). . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Propylene (Propene, R1270) . . . . . . . . . . . . . . . . . Compressed Propylene (Propene, R1270). . . . . . . . . . . . . . . Saturated Refrigerant 11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy–Log-Pressure Diagram for Refrigerant 11 (Fig. 2-18) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant 12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant 13. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy–Log-Pressure Diagram for Refrigerant 12 (Fig. 2-19) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2-224 2-225 2-225

2-303 2-304 2-305

2-226 2-227 2-227 2-228 2-229 2-230 2-231 2-231 2-232 2-232 2-232 2-233 2-233 2-234 2-235 2-235 2-236 2-237 2-239 2-239 2-239 2-239 2-239 2-240 2-241 2-241 2-242 2-242 2-243 2-243 2-244 2-246 2-246 2-246 2-247 2-248 2-248 2-249 2-249 2-249 2-250 2-251 2-252 2-253 2-254 2-255 2-256 2-256 2-257 2-257 2-258 2-259 2-259 2-259 2-260 2-261 2-261 2-262 2-263 2-264 2-264 2-264 2-265 2-266 2-267 2-268 2-269 2-269 2-270 2-270

2-306 2-307 2-308 2-309 2-310 2-311 2-312 2-313 2-314 2-315 2-316 2-317 2-318 2-319 2-320 2-321 2-322 2-323 2-324 2-325 2-326 2-327 2-328 2-329 2-330 2-331 2-332 2-333 2-334 2-335 2-336 2-337 2-338 2-339 2-340 2-341 2-342 2-343 2-344 2-345

2-346 2-347

2-348 2-349 2-350 2-351 2-352 2-353 2-354 2-355 2-356

2-271

Saturated Refrigerant 13B1 . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant 21. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant 22. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy–Log-Pressure Diagram for Refrigerant 22 (Fig. 2-20) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermophysical Properties of Compressed R22 . . . . . . . . . . Saturated Refrigerant 23. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermophysical Properties of Saturated Difluoromethane (R32) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specific Heat at Constant Pressure, Thermal Conductivity, Viscosity, and Prandtl of R32 Gas . . . . . . . . . . . . . . . . . . . . . Saturated SUVA MP 39. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SUVA MP 39 at Atmospheric Pressure . . . . . . . . . . . . . . . . . Enthalpy–Log-Pressure Diagram for Refrigerant 32 (Fig. 2-21) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermodynamic Properties of Saturated KLEA 60 . . . . . . . Thermodynamic Properties of Saturated KLEA 61 . . . . . . . Enthalpy–Log-Pressure Diagram for KLEA 60 (Fig. 2-22). . Enthalpy–Log-Pressure Diagram for KLEA 61 (Fig. 2-23). . Saturated SUVA HP 62 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SUVA HP 62 at Atmospheric Pressure. . . . . . . . . . . . . . . . . . Thermodynamic Properties of Saturated KLEA 66 . . . . . . . Enthalpy–Log-Pressure Diagram for KLEA 66 (Fig. 2-24) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated SUVA MP 66. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SUVA MP 66 at Atmospheric Pressure . . . . . . . . . . . . . . . . . Saturated SUVA HP 80 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SUVA HP 80 at Atmospheric Pressure. . . . . . . . . . . . . . . . . . Saturated SUVA HP 81 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SUVA HP 81 at Atmospheric Pressure. . . . . . . . . . . . . . . . . . Saturated Refrigerant 113. . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant 114. . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant 115. . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermodynamic Properties of Refrigerant 123 . . . . . . . . . . . Saturated Refrigerant 124. . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy–Log-Pressure Diagram for Refrigerant 123 (Fig. 2-25) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermophysical Properties of Saturated Refrigerant 125 . . . Thermophysical Properties of Refrigerant 134a . . . . . . . . . . Enthalpy–Log-Pressure Diagram for Refrigerant 125 (Fig. 2-26) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermophysical Properties of Compressed Gaseous Refrigerant 134a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Refrigerant 141b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy–Log-Pressure Diagram for Refrigerant 134a (Fig. 2-27) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Refrigerant 142b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant R143a . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant R152a . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant 216. . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant 245. . . . . . . . . . . . . . . . . . . . . . . . . . . . Refrigerant C 318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant 500. . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant 502. . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant 503. . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant 504. . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermodynamic Properties of Refrigerant 507 . . . . . . . . . . . Saturated Rubidium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermophysical Properties of Saturated Seawater . . . . . . . . Saturated Sodium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mollier Diagram for Sodium (Fig. 2-28) . . . . . . . . . . . . . . . . Enthalpy-Concentration Diagram for Aqueous Sodium Hydroxide at 1 atm (Fig. 2-29) . . . . . . . . . . . . . . . . . . . . . . . Saturated Sulfur Dioxide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermodynamic Properties of Saturated Sulfur Hexafluoride (SF6). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy-Concentration Diagram for Aqueous Sulfuric Acid at 1 atm (Fig. 2-30) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy–Log-Pressure Diagram for Sulfur Hexafluoride (Fig. 2-31) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated SUVA AC 9000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Toluene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Solid/Vapor Water. . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Water Substance–Temperature (fps units) . . . . . . Saturated Water Substance–Temperature (SI units) . . . . . . . Saturated Liquid Water–Miscellaneous Properties . . . . . . . . Thermodynamic Properties of Compressed Steam . . . . . . . . Density, Specific Heats at Constant Pressure and at Constant Volume and Velocity of Sound for Compressed Water, 1–1000 bar, 0–150°C . . . . . . . . . . . . . . . . . . . . . . . . . Specific Heat and Other Thermophysical Properties of Water Substance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2-272 2-272 2-273 2-273 2-274 2-274 2-275 2-276 2-276 2-276 2-277 2-278 2-278 2-279 2-280 2-281 2-281 2-281 2-282 2-283 2-283 2-283 2-284 2-284 2-284 2-285 2-286 2-286 2-287 2-287 2-288 2-289 2-289 2-290 2-291 2-292 2-293 2-294 2-294 2-295 2-295 2-296 2-296 2-297 2-297 2-298 2-298 2-298 2-299 2-299 2-300 2-301 2-302 2-302 2-303 2-303 2-303 2-304 2-304 2-304 2-305 2-306 2-308 2-309 2-311 2-313

PHYSICAL AND CHEMICAL DATA 2-357 Thermodynamic Properties of Water Substance along the Melting Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-358 Saturated Xenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-359 Compressed Xenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-360 Surface Tension (N/m) of Saturated Liquid Refrigerants . . . 2-361 Velocity of Sound (m/s) in Gaseous Refrigerants at Atmospheric Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-362 Velocity of Sound (m/s) in Saturated Liquid Refrigerants. . . TRANSPORT PROPERTIES Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Units Conversions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-363 Transport Properties of Selected Gases at Atmospheric Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-364 Viscosities of Gases: Coordinates for Use with Fig. 2-32. . . . Nomograph for Determining (a) Absolute Viscosity of a Gas as a Function of Temperature Near Ambient Pressure and (b) Relative Viscosity of a Gas Compared with Air (Fig. 2-32) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-365 Viscosities of Liquids: Coordinates for Use with Fig. 2-33 . . Nomograph for Viscosities of Liquids at 1 atm (Fig. 2-33) . . 2-366 Viscosity of Sucrose Solutions . . . . . . . . . . . . . . . . . . . . . . . . . Nomograph for Thermal Conductivity of Organic Liquids (Fig. 2-34) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-367 Thermal Conductivity Nomograph Coordinates . . . . . . . . . . 2-368 Prandtl Number of Air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-369 Prandtl Number of Liquid Refrigerants. . . . . . . . . . . . . . . . . 2-370 Thermophysical Properties of Miscellaneous Saturated Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-371 Diffusivities of Pairs of Gases and Vapors (1 atm) . . . . . . . . . 2-372 Diffusivities in Liquids (25°C) . . . . . . . . . . . . . . . . . . . . . . . . 2-373 Thermal Conductivities of Some Building and Insulating Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-374 Thermal-Conductivity-Temperature Table for Metals . . . . . 2-375 Thermal Conductivity of Chromium Alloys . . . . . . . . . . . . . . 2-376 Thermal Conductivity of Some Alloys at High Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-377 Thermal Conductivities of Some Materials for Refrigeration and Building Insulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-378 Thermal Conductivities of Insulating Materials at High Temperatures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-379 Thermal Conductivities of Insulating Materials at Moderate Temperatures (Nusselt). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-380 Thermal Conductivities of Insulating Materials at Low Temperatures (Gröber) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-381 Thermal Diffusivity (m2/s) of Selected Elements . . . . . . . . . 2-382 Thermophysical Properties of Selected Nonmetallic Solid Substances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pure Component Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Critical Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Critical Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 1 Estimate the Critical Temperature and Critical Pressure of 2-Butanol Using the Ambrose Method . . . . . . . . . . . . . . . . . . . . . Example 2 Estimate the Critical Temperature and Critical Pressure of 2-Butanol, Which Has an Experimental Normal Boiling Point of 372.7 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Critical Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 3 Estimate the Critical Volume of 2-Butanol . . . . . . . . . . . Critical Compressibility Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Normal Freezing Temperature (Melting Point) . . . . . . . . . . . . . . . . . Normal Boiling Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 4 Estimate the Normal Boiling Point of 2-Butanol. . . . . . . Acentric Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miscellaneous Characterizing Constants . . . . . . . . . . . . . . . . . . . . . . . Vapor Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Correlation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Prediction Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 5 Estimate the Vapor Pressure of 1-Butene at 98°C. . . . . . Example 6 Estimate the Vapor Pressure of Tetralin at 150°C . . . . . . Example 7 Estimate the Vapor Pressure of Thiophene at 500 K. . . . Example 8 Estimate the Vapor Pressure of Acetaldehyde at 0°C . . . Ideal Gas Thermal Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2-315 2-315 2-316 2-317 2-317 2-318

2-318 2-318 2-318 2-319 2-320

2-321 2-322 2-323 2-324 2-324 2-324 2-325 2-325 2-326 2-328 2-330 2-333 2-334 2-335 2-335 2-335 2-335 2-336 2-336 2-336 2-337

2-337 2-337 2-338 2-339 2-340 2-340 2-340 2-342 2-342 2-344 2-344 2-344 2-344 2-344 2-345 2-345 2-345 2-345 2-345 2-346 2-346 2-346 2-347 2-347 2-347

Heat Capacity, C po . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 9 Using Eq. 2-48 to Estimate the Ideal Gas Heat Capacity of Acetone (C3H6O) at 600 K . . . . . . . . . . . . . . . . . . . . . . . Enthalpy of Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 10
H°f 298 of 2-Butanol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gibbs Energy of Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 11
G°f 298 of Phenol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Entropy of Formation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy of Vaporization and Fusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy of Vaporization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 12 Estimate
Hv of Propionaldehyde at 350 K . . . . . . . . . . Example 13 Estimate
Hv of Ethyl Acetate . . . . . . . . . . . . . . . . . . . . Example 14 Estimate
Hv of Ethyl Acetate at 450 K. . . . . . . . . . . . . Enthalpy of Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solid and Liquid Heat Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solid Heat Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 15 Estimate Solid Heat Capacity of Dinenzothiophene. . . Liquid Heat Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 16 Estimate Liquid Heat Capacity of 2-Butanol. . . . . . . . . Example 17 Estimate Liquid Heat Capacity of 1,4 Pentadiene . . . . Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Correlation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vapor Density Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 18 Estimate the Molar Volume of Isobutane at 155°C and 1.0 MPa Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 19 Estimate the Molar Volume of Isobutane at 155°C and 8.6 MPa Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid Density Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 20 Estimate the Density of Saturated Liquid Propane at 0°C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 21 Estimate the Liquid Density of n-Nonane at 104.5°C and 6.893 MPa Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solid Density Prediction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Viscosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Correlation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vapor Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 22 Estimate the Vapor Viscosity of Propane at 101.3 kPa and 80°C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 23 Estimate the Vapor Viscosity of a Mixture of Propane and Methane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 24 Estimate the Vapor Viscosity of Isopropyl Alcohol at 251°C and Atmospheric Pressure . . . . . . . . . . . . . . . . . . . . . . . . . Example 25 Estimate the Vapor Viscosity of Carbon Dioxide at 350 K and a Total Pressure of 20 MPa. . . . . . . . . . . . . . . . . . . . . . . . Liquid Viscosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 26 Estimate the Liquid Viscosity of cis-1,4dimethylcyclohexane at 0°C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vapor and Liquid Thermal Conductivity. . . . . . . . . . . . . . . . . . . . . . . . . Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 27 Estimate Thermal Conductivity for n-Hexane. . . . . . . . Example 28 Estimate Thermal Conductivity of Carbon Dioxide at 370 K and Low Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 29 Estimate the Thermal Conductivity of Carbon Dioxide at 370 K and 10 MPa Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 30 Estimate Thermal Conductivity of a Mixture. . . . . . . . . Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 31 Estimate Thermal Conductivity of n-Octane at 373.15 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 32 Estimate Thermal Conductivity of n-Octane at 473.15 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 33 Estimate Thermal Conductivity of n-Butanol . . . . . . . . Example 34 Estimate the Thermal Conductivity of n-Propionaldehyde . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 35 Estimate Thermal Conductivity of n-Butanol . . . . . . . . Example 36 Estimate Thermal Conductivity of a Mixture. . . . . . . . . Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gas Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 37 Estimate the Diffusivity of Benzene Vapor Diffusing into Air at 30°C and 96.5 kPa Total Pressure . . . . . . . . . . . . . . . . . . Example 38 Estimate the Diffusivity of Hydrogen (1) in Nitrogen (2) at 60°C and 17.23 MPa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 39 Estimate the Infinite Dilution Diffusivity of Propane (1) in Chlorobenzene (2) at 25°C. . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Surface Tension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 40 Estimate Surface Tension for Mercaptan. . . . . . . . . . . . Example 41 Estimate Surface Tension for Isobutytic Acid . . . . . . . . Example 42 Estimate Surface Tension of a Mixture . . . . . . . . . . . . . Example 43 Estimate Surface Tension of a Water-Methanol Mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flammability Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2-5 2-347 2-348 2-348 2-348 2-348 2-348 2-348 2-349 2-349 2-350 2-350 2-350 2-350 2-351 2-351 2-351 2-351 2-351 2-354 2-355 2-355 2-355 2-355 2-357 2-358 2-359 2-360 2-361 2-361 2-362 2-362 2-363 2-363 2-363 2-364 2-365 2-366 2-367 2-367 2-367 2-367 2-368 2-368 2-368 2-368 2-369 2-369 2-369 2-369 2-370 2-370 2-370 2-370 2-371 2-371 2-371 2-372 2-372 2-372 2-372 2-373 2-374

GENERAL REFERENCES Considerations of reader interest, space availability, the system or systems of units employed, copyright considerations, etc., have all influenced the revision of material in previous editions for the present edition. Reference is made at numerous places to various specialized works and also, when appropriate, to more general works. A listing of general works may be useful to readers in need of further information. ASHRAE Handbook—Fundamentals, IP and SI editions, ASHRAE, Atlanta, various dates; Beaton, C.F. and G.F. Hewitt, Physical Property Data for the Design Engineer, Hemisphere, New York, 1989 (394 pp.); Benedek, P. and F. Olti, Computer-Aided Chemical Thermodynamics of Gases and Liquids, Wiley, New York, 1985 (731 pp.); Daubert, T.E. and R.P. Danner, Physical and Thermodynamic Properties of Pure Chemicals, 4 vols., Hemisphere, New York, 1989 (2030 pp.); suppl. 1, 1991 (456 pp.); suppl. 2, 1992 (736 pp.); Gmehling, J., Azeotropic Data, 2 vols., VCH Weinheim, Germany, 1994 (1900 pp.); Kaye, S.M., Encyclopedia of Explosives and Related Items, U.S. Army R&D command, Dover, NJ, 1980; King, M.B., Phase Equilibrium in Mixtures, Pergamon, Oxford, 1969; Lyman, W.J., W.F. Reehl et al., Handbook of Chemical Property Estimation Methods, McGraw-Hill, N.Y., 1982 (929 pp.); Ohse, R.W., Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Sci. Pubs., Oxford, England, 1985 (1020 pp.); Reid, R.C., J.M. Prausnitz et al., The Properties of Gases and Liquids, McGraw-Hill, New York, 1987 (742 pp.); Sterbacek, Z., B. Biskup et al., Calculation of Properties Using Corresponding States Methods, Elsevier, Amsterdam, 1979. Compilations of critical data include Ambrose, D., “Vapor-Liquid Critical Properties,” N.P.L. Teddington, Middx. rept. Chem 107, 1980 (62 pp.); Brule, M.R., L.L Lee et. al., Chem. Eng., 86, 25 (Nov. 19, 1979) 155–164; Kudchaker,

2-6

A.P., G.H. Alani et al., Chem. Revs., 68 (1968) 659–735; Matthews, J.F., Chem. Revs., 72 (1972) 71–100; Reid, R.C., J.M. Prausnitz et al., The Properties of Gases and Liquids, 4th ed., McGraw-Hill, New York, 1987 (741 pp.); Ohse, R.W. and H. von Tippelskirch, High Temp.—High Press., 9 (1977) 367–385; Young, D.A., “Phase Diagrams of the Elements,” UCRL Rept. 51902, 1975 (64 pp.); republished in expanded form by the University of California Press, 1991. Rothman, D. et al., Max Planck Inst. f. Stromungsforschung, Ber 6, 1978 (77 pp.). PUBLICATIONS ON THERMOCHEMISTRY Pedley, J.B., Thermochemical Data and Structures of Organic Compounds, 1, Thermodyn. Res. Ctr., Texas A&M Univ., 1994 (976 pp., 3000 cpds.); Frenkel, M., G.J. Kabo et al., Thermodynamics of Organic Compounds in the Gas State, 2 vols., Thermodyn. Res. Ctr., Texas A&M Univ., 1994 (1825 pp., 2000 cpds.); Barin, I., Thermochemical Data of Pure Substances, 2 vols., 2d ed., VCH Weinheim, Germany 1993 (1834 pp., 2400 substances); and Gurvich, L.V., I.V. Veyts et al., Thermodynamic Properties of Individual Substances, 3 vols., 4th ed., Hemisphere, New York, 1989, 1990, and 1993 (2520 pp.). See also Lide, D.R. and G.W.A. Milne, Handbook of Data on Organic Compounds, 7 vols., 3d ed., Chemical Rubber, Miami, 1993 (7000 pp.); Daubert, T.E., R.P. Danner et al., Physical and Thermodynamic Properties of Pure Chemicals: Data Compilation, extant 1995, Taylor & Francis, Bristol, PA, 1995; Database 11, N.I.S.T. Gaithersburg, MD. U.S. Bureau of Mines publications include Bulletins 584, 1960 (232 pp.); 592, 1961 (149 pp.); 595, 1961 (68 pp.); 654, 1970 (26 pp.) Chase, M.W., C.A. Davies et al., JANAF Thermochemical Tables, 3d ed., J. Phys. Chem. Ref. Data 14 suppl 1., 1986 (1896 pp.).

PHYSICAL PROPERTIES OF PURE SUBSTANCES TABLE 2-1

Physical Properties of the Elements and Inorganic Compounds* Abbreviations Used in the Table

a., acid A., specific gravity with reference to air = 1 abs., absolute ac., acetic acid act., acetone al., 95 percent ethyl alcohol alk, alkali (i.e., aq. NaOH or KOH) am., amyl (C5H11) amor., amorphous anh., anhydrous aq., aqueous or water aq. reg., aqua regia

atm., atmosphere or 760 mm. of mercury pressure bk., black brn., brown bz., benzene c., cold cb., cubic cc, cubic centimeter chl., chloroform col., colorless or white conc., concentrated cr., crystals or crystalline d., decomposes D., specific gravity with reference to hydrogen = 1

d. 50, decomposes at 50°C; 50 d., melts at 50°C with decomposition delq., deliquescent dil., dilute dk., dark eff., effloresces or efflorescent et., ethyl ether expl., explodes gel., gelatinous gly., glycerol (glycerin) gn., green h., hot hex., hexagonal

hyg., hygroscopic i., insoluble ign., ignites lq., liquid lt., light m. al., methyl alcohol mn., monoclinic nd., needles NH3, liquid ammonia NH4OH, ammonium hydroxide solution oct., octahedral or., orange pd., powder

Formula weights are based upon the International Atomic Weights of 1941 and are computed to the nearest hundredth. Refractive index, where given for a uniaxial crystal, is for the ordinary (ω) ray; where given for a biaxial crystal, the index given is for the median (β) value. Unless otherwise specified, the index is given for the sodium D-line (λ = 589.3 mµ). Specific gravity values are given at room temperatures (15° to 20°C) unless otherwise indicated by the small figures which follow the value: thus, “5.6 18° 4 ” indicates a specific gravity of 5.6 for the substance at 18°C referred to water at 4°C. In this table the values for the specific gravity of gases are given with reference to air (A) = 1, or hydrogen (D) = 1. Melting point is recorded in a certain case as “82 d.” and in some other case as “d. 82,” the distinction being made in this manner to indicate that the former is a melting point with decomposition at 82°C, while in the latter decomposition only occurs at 82°C. Where a value such as “−2H2O, 82” is given it indicates loss of 2 moles of water per formula weight of the compound at a temperature of 82°C. Boiling point is given at atmospheric pressure (760 mm. of mercury) unless otherwise indicated; thus, “8215 mm.” indicates the boiling point is 82°C when the pressure is 15 mm.

Name Aluminum acetate, normal acetate, basic bromide bromide carbide chloride chloride fluoride (fluellite) fluoride hydroxide nitrate nitride oxide oxide (corundum) phosphate

Formula

Formula weight

Color, crystalline form and refractive index

Al Al(C2H3O2)3 Al(OH)(C2H3O2)2 AlBr3 AlBr3·6H2O Al4C3 AlCl3

26.97 204.10 162.07 266.72 374.82 143.91 133.34

silv., cb. wh. pd. wh., amor. trig. col., delq. cr. yel., hex., 2.70 wh., delq., hex.

AlCl3·6H2O AlF3·H2O Al2F6·7H2O Al(OH)3 Al(NO3)3·9H2O Al2N2 Al2O3 Al2O3 AlPO4

241.44 101.99 294.05 77.99 375.14 81.96 101.94 101.94 121.95

col., delq., trig., 1.560 col., rhb., 1.490 wh., cr. pd. wh., mn. rhb., delq. yel., hex. col., hex., 1.67–8 wh., trig., 1.768 col., hex.

pl., plates pr., prisms or prismatic pyr., pyridine rhb., rhombic (orthorhombic) s., soluble satd., saturated sl., slightly soln., solution subl., sublimes sulf., sulfides tart. a., tartaric acid tet., tetragonal tr., transition tri., triclinic

trig., trigonal v., very vac., in vacuo vl., violet volt., volatile or volatilizes wh., white yel., yellow ∞, soluble in all proportions <, less than >, greater than 42, about or near 42 −3H2O, 100, loses 3 moles of water per formula weight at 100°C

Solubility is given in parts by weight (of the formula shown at the extreme left) per 100 parts by weight of the solvent; the small superscript indicates the temperature. In the case of gases the solubility is often expressed in some manner as “510° cc” which indicates that at 10°C, 5 cc. of the gas are soluble in 100 g. of the solvent. The symbols of the common mineral acids: H2SO4, HNO3, HCl, etc., represent dilute aqueous solutions of these acids. See also special tables on Solubility. REFERENCES: The information given in this table has been collected mainly from the following sources: Mellor, A Comprehensive Treatise on Inorganic and Theoretical Chemistry, Longmans, New York, 1922. Abegg, Handbuch der anorganischen Chemie, S. Hirzel, Leipzig, 1905. Gmelin-Kraut, Handbuch der anorganischen Chemie, 7th ed., Carl Winter, Heidelberg; 8th ed., Verlag Chemie, Berlin, 1924. Friend, Textbook of Inorganic Chemistry, Griffin, London, 1914. Winchell, Microscopic Character of Artificial Inorganic Solid Substances or Artificial Minerals, Wiley, New York, 1931. International Critical Tables, McGraw-Hill, New York, 1926. Tables annuelles internationales de constants et donnes numeriques, McGraw-Hill, New York. Annual Tables of Physical Constants and Numerical Data, National Research Council, Princeton, N.J., 1943. Comey and Hahn, A Dictionary of Chemical Solubilities, Macmillan, New York, 1921. Seidell, Solubilities of Inorganic and Metal Organic Compounds, Van Nostrand, New York, 1940.

Specific gravity 2.7020° 3.01 25° 4 2.95 2.44 25° 4 2.17 2.42 3.05 3.99 4.00 2.59

25° 4

Melting point, °C 660 d. 200 d. 97.5 d. 100 d. >2200 1945.2atm. d. −4H2O, 120 −2H2O, 300 73 21504atm. 1999 to 2032 1999 to 2032

Boiling point, °C 2056 268 752mm

182.7 ; subl. 178 −6H2O, 250 d. 134 d. >1400 2210

Solubility in 100 parts Cold water i. s. i. s. s. d. to CH4 69.8715° 400 sl. s. i. 0.00010418° v. s. d. slowly i. i. i.

Hot water i. d.

Other reagents s. HCl, H2SO4, alk.

s. d.

s.a.; i. NH4 salts s.al., act., CS2 s. al., CS2 s. a.; i. act. s. et., chl., CCl4; i. bz.

v. s.

50 al.; s. et.

s.

sl. s. i. v. s. d. i. i. i.

s. a., alk.; i. a. s. al., CS2 s. alk. d. v. sl. s. a., alk. v. sl. s. a., alk. s. a., alk.; i. ac.

*By N. A. Lange, Ph.D., Handbook Publishers, Inc., Sandusky, Ohio. Abridged from table of Physical Constants of Inorganic Compounds in Lange, “Handbook of Chemistry.” 2-7

TABLE 2-1

Physical Properties of the Elements and Inorganic Compounds (Continued )

Name Aluminum (Cont.) potassium silicate (muscovite) potassium silicate (orthoclase) Aluminum potassium tartrate sodium fluoride (cryolite) sodium silicate sulfate Alum, ammonium (tschermigite) ammonium chrome ammonium iron potassium (kalinite) potassium chrome sodium Ammonia† Ammonium acetate auricyanide bicarbonate bromide carbonate carbonate, carbamate

Formula 3Al2O3·K2O·6SiO2· 2H2O Al2O3·K2O·6SiO2 AlK(C4H4O6)2 AlF3·3NaF Al2O3·Na2O·6SiO2 Al2(SO4)3 Al2(SO4)3·(NH4)2SO4· 24H2O Cr2(SO4)3·(NH4)2SO4· 24H2O Fe2(SO4)3·(NH4)2SO4· 24H2O Al2(SO4)3·K2SO4· 24H2O Cr2(SO4)3·K2SO4· 24H2O Al2(SO4)3·Na2SO4· 24H2O NH3

Formula weight

Color, crystalline form and refractive index

Boiling point, °C

Solubility in 100 parts Cold water

mn., 1.590

2.9

d.

i.

556.49 362.21 209.96 524.29 342.12 906.64

col., mn., 1.524 col. wh., mn., 1.3389 col., tri., 1.529 wh. cr. col., oct., 1.4594

2.56

1450 (1150)

2.90 2.61 2.71 1.64 20° 4

1000 1100 d. 770 93.5

i. s. sl. s. i. 31.30° 3.90°

956.72

gn. or vl., oct., 1.4842

1.72

964.40

vl., oct., 1.485

1.71

40

col., mn., 1.4564

1.76 26° 4

92

998.84

red or gn., cb., 1.4814

1.83

89

916.56

col., oct., 1.4388

17.03

col. gas, 1.325 (lq.)

77.08 337.33 79.06 97.96 114.11 157.11

wh., hyg. cr. pl. mn. or rhb., 1.5358 col., cb., 1.7108 col. pl. wh. cr.

272.22

wh.

53.50 444.05 373.14 367.52 152.09 44.06 252.10 392.21 37.04 57.05 63.06

hydrosulfide hydroxide molybdate molybdate, heptanitrate (α), stable −16° to 32° nitrate (β), stable 32° to 84°

NH4HS NH4OH (NH4)2MoO4 (NH4)6Mo7O24·4H2O‡ NH4NO3 NH4NO3

51.11 35.05 196.03 1235.95 80.05 80.05

col., rhb. in soln. only mn. col., mn. col., tet., 1.611 col., rhb. or mn.

nitrite osmochloride oxalate oxalate, acid perchlorate persulfate phosphate, monobasic

NH4NO2 (NH4)2OsCl6 (NH4)2C2O4·H2O NH4HC2O4·H2O NH4ClO4 (NH4)2S2O8 NH4H2PO4

64.05 439.02 142.12 125.08 117.50 228.20 115.04

wh. nd. cb. col., rhb. col., trimetric col., rhb., 1.4833 wh., mn., 1.5016 col., tet., 1.5246

wh., cb., 1.639, 1.6426 yel., cb. tet. pink., cb. yel., mn. col., cb. or., mn. mn. wh., hex. wh., rhb., 1.390 col., mn., delq.

−20H2O, 120; −24H2O, 200

100 d.

948.76

chloride (salammoniac) chloroplatinate chloroplatinite chlorostannate chromate cyanide dichromate ferrocyanide fluoride fluoride, acid formate

2-8

Melting point, °C

796.40

NH4C2H3O2 NH4CN·Au(CN)3·H2O NH4HCO3 NH4Br (NH4)2CO3·H2O NH4HCO3· NH2CO2NH4‡ (NH4)2CO3· 2NH4HCO3·H2O NH4Cl (NH4)2PtCl6 (NH4)2PtCl4 (NH4)2SnCl6 (NH4)2CrO4 NH4CN (NH4)2Cr2O7 (NH4)4Fe(CN)6·6H2O NH4F NH4F·HF HCO2NH4

carbonate, sesqui-

Specific gravity

20° 4

1.675

−79°

0.817 0.5971 (A) 1.073 1.573 2.327 15° 4

1.53 3.065

2.4 1.91712° 0.79100° (A) 2.15

i. 89100° ∞ 100°

25°

124 −18H2O, 64.5

i. al.

50

106.4

0°

i. al. 45°

121.7

−77.7

−33.4

89.9

114 d. 200 d. 35–60 subl. 542 d. 58 subl.

d.

1484° s. 11.90° 6810° 10015° 2515°

v. s. 2730° 145.6100°

2015°

5049°

d. 350 d. d.

subl. 520

d. 180 36 d. 185 d.

12° 2.21 12 1.266

114–116

2.27

d.

1.66 25° 4 1.725 25° 4

169.6

1.69 2.93 20° 4 1.501 1.556 1.95 1.98 1.803 19° 4

expl.

d.

d. 180; subl. in vac. subl. 120

d. 210 d. 210

0°

0°

29.4 0.715° s. 33.315° 40.530° s. 47.230° s. v. s. v. s. 1020°

i. al.

7.4

96°

14.820° al.; s. et. s. al.; sl. s. act. i. al. i. al. s. al., et., act. i. al., CS2, NH3

6765°

77.3100° 1.25100° v. s.

s. NH3; sl. s. al., m. al. 0.005 al.

d. v. s. v. s. d.

sl. s. act., NH3; i. al. s. al. s. al.; i. act. i. al. s. al.; i. NH3

53180°

s. al.

d.

i. al., NH3 i. al.

v. s. s. d. 4425° 118.30° 365.835°

241.830° 58080°

s.

d.

3.820° al., 17.120° m. al.; v. s. NH3 s. al.

11.850°

sl. s. al.; i. NH3

46.9100° d. 173.2100°

220° al.; s. act.; i. et.

0°

d. d. d. 120

i. al.

∞93°

20

61

i. HCl d. a.

s. al.

25°

5.70°

Other reagents

s.

21.2

d. 17°

Hot water

2.5 s. 10.90° 58.20° 22.70°

s. al.

i. ac.

phosphate, dibasic phosphate, metaAmmonium phosphomolybdate silicofluoride sulfamate sulfate (mascagnite) sulfate, acid sulfide sulfide, pentasulfite sulfite, acid tartrate thiocyanate vanadate, metaAntimony chloride, tri- (butter of antimony)* oxide, tri- (valentinite) oxide, tri- (senarmontite) sulfide, tri- (stibnite)

(NH4)2HPO4 (NH4)4P4O12 (NH4)3PO4·12MoO3· 3H2O (?) (NH4)2SiF6 NH4·SO3NH2 (NH4)2SO4 NH4HSO4 (NH4)2S (NH4)2S5 (NH4)2SO3·H2O NH4HSO3 (NH4)2C4H4O6 NH4CNS NH4VO3 Sb

132.07 388.08 1930.55 178.14 114.12 132.14 115.11 68.14 196.38 134.16 99.11 184.15 76.12 116.99 121.76

col., mn., 1.53 col., mn. yel.

13115° s. 0.0315°

1.619 2.21 d.

cb., 1.3696 col. pl. col., rhb., 1.5230 col., rhb., 1.480 yel.-wh. or.-red pr. col., mn. rhb. col., mn. col., mn., 1.685 col. cr. tin wh., trig.

2.01

i.

17.5°

s. alk.; i. al., HNO3

1380

18.5 1340° 70.60° 100 v. s. s. 10012° s. 450° 1200° 0.4418° i.

87 17020° 3.0570° i.

73.4

220.2

601.60°

∞72°

1570

v. sl. s.

1.769 20° 4 1.78

132 235 d. 146.9 d.

1.41 2.03 12° 4 1.60 1.305 2.326 6.68425°

d. d. d. 149.6 d. 630.5

SbCl3

228.13

col., rhb., delq.

3.14 20° 4

Sb2O3 Sb2O3 Sb2S3

291.52 291.52 339.70

rhb., 2.35 cb., 2.087 bk., rhb., 4.046

5.67 5.2 4.64

656 652 550

subl. d. 160

i. act.

490

d. 170

0.00017

55.5 35750° 103.3100°

s. al.; i. act. i. al., act., CS2 v. sl. s. al.; i. act. 12025° NH3 i. al., act.

60°

sl. s. 18°

d.

sl. s. al. s. al., act., NH3, SO2 i. al., NH4Cl s. aq. reg., h. conc. H2SO4 s. al., HCl, HBr, H2C4H4O6 s. HCl, KOH, H2C4H4O6 s. HCl; alk., NH4HS, K2S; i. ac. s. HCl, alk., NH4HS

sulfide, pentatelluride, triAntimonyl potassium tartrate (tartar emetic) sulfate, normal sulfate, basic Argon

Sb2S5 Sb2Te3

403.82 626.35

golden gray

4.1200°

−2S, 135 629

i.

i.

(SbO)KC4H4O6·a H2O (SbO)2SO4 (SbO)2SO4·Sb2(OH)4 A

333.94 371.58 683.13 39.94

wh., rhb. wh. pd. wh. pd. col. gas

2.60 4.89

−a H2O, 100 −189.2

−185.7

35.7100° d. d. 2.2350° cc

s. gly.; i. al.

Arsenic (crystalline) (α) Arsenic (black) (β)

As4 As4

299.64 299.64

met., hex. bk., amor.

1.65−288°; 1.402−185.7°; 1.38 (A) 5.72714° 4.720°

5.268.7° d. i. 5.60° cc

81436atm.

subl. 615

i. i.

i. i.

s. HNO3 s. HNO3, aq. reg., aq. Cl2, h. alk.

Arsenic (yellow)(γ) acid, orthoacid, metaacid, pyropentoxide sulfide, di- (realgar)

As4 H3AsO4·a H2O HAsO3 H4As2O7 As2O5 As2S2

299.64 150.94 123.92 265.85 229.82 213.94

yel., cb. col., hyg. wh., hyg. col. wh., amor. red, mn., 2.68

−H2O, 160

50 H3AsO4 H3AsO4 76.7100° d.

s. alk.

d. 565

16.7 d. to form d. to form 59.50° i.

sulfide, pentaArsenious chloride (butter of arsenic) hydride (arsine) oxide (arsenolite) oxide (claudetite) oxide

As2S5

310.12

yel.

d. 500

0.0001360°

i.

s. HNO3, alk.

AsCl3 AsH3 As2O3 As2O3 As2O3

181.28 77.93 197.82 197.82 197.82

oily lq. col. gas col., cb., fibrous, 1.755 col., mn., 1.92 amor. or vitreous

130 −55; d. 230

d. 20 cc sl. s. sl. s. 1.210°

d. sl. s. sl. s. sl. s. 2.9340°

Auric chloride

AuCl3·2H2O

339.60

or. cr.

v. s.

v. s.

cyanide Aurous chloride cyanide Cf. also under Gold Barium acetate acetate bromide

Au(CN)3·6H2O AuCl AuCN

383.35 232.66 223.22

yel. cr. yel. cr.

7.4

v. s. d. i.

v. s. d. i.

s. HCl, HBr, PCl3 sl. s. alk. i. al., et. i. al., et. s. HCl, alk., Na2CO3; i. al., et. s. HCl, al., et.; sl. s. NH3 s. al. s. HCl, HBr; d. al. s. KCN; i. al., et.

Ba Ba(C2H3O2)2 Ba(C2H3O2)2·H2O BaBr2

137.36 255.45 273.46 297.19

silv. met. col. wh., tri. pr., 1.517 col.

3.5 2.468 2.19 4.781 24° 4

d. 58.80° 7530°(anh.) 980°

d. 75.0100° 7940°(anh.) 149100°

2.020° 2.0–2.5 4.086 (α)3.50619°; (β)3.25419° lq. 2.163 2.695 (A) 3.865 25° 4 3.85 3.738

d. 358 35.5 d. d. 206 (α)tr. 267; (β)307 −18 −113.5 subl. subl. 315 d. d. 50 AuCl3, 170 d.

d. 290

850

1140

−H2O, 41 847

d.

5.1515° gly. 2425° cc al.

s. alk., al. s. K2S, NaHCO3

s. a.; d. al. i. al. v. s. m. al.; v. sl. s. act.

*Usually the solution. †See special tables. ‡Usual commercial form. 2-9

TABLE 2-1

Physical Properties of the Elements and Inorganic Compounds (Continued )

Name Barium (Cont.) bromide carbonate (witherite) carbonate (α) carbonate (β) Barium chlorate chlorate chloride chloride chloride hydroxide hydroxide nitrate (nitrobarite) oxalate oxide peroxide peroxide phosphate, monobasic phosphate, dibasic phosphate, tribasic phosphate, pyrosilicofluoride sulfate (barite, barytes) sulfide, monosulfide, trisulfide, tetraBeryllium (glucinum) Bismuth carbonate, subchloride, dichloride, trinitrate nitrate, suboxide, trioxide, trioxide, trioxychloride

Formula

Formula weight

Color, crystalline form and refractive index

Specific gravity

Melting point, °C −2H2O, 100 tr. 811 to α tr. 982 to β 174090atm 414 d. 120 tr. 925 962 −2H2O, 100

BaBr2·2H2O BaCO3 BaCO3 BaCO3 Ba(ClO3)2 Ba(ClO3)2·H2O* BaCl2 BaCl2 BaCl2·2H2O† Ba(OH)2 Ba(OH)2·8H2O Ba(NO3)2 BaC2O4 BaO

333.22 197.37 197.37 197.37 304.27 322.29 208.27 208.27 244.31 171.38 315.50 261.38 225.38 153.36

col., mn., 1.7266 wh., rhb., 1.676 wh., hex. wh. col. col., mn., 1.577 col., mn., 1.7361 col., cb. col., mn., 1.646 col., mn. col., mn., 1.5017 col., cb., 1.572 wh. cr. col., cb., 1.98

BaO2* BaO2·8H2O BaH4(PO4)2 BaHPO4 Ba3(PO4)2 Ba2P2O7 BaSiF6 BaSO4

169.36 313.49 331.35 233.35 602.04 448.68 279.42 233.42

gray or wh. pd. pearly sc. tri. wh., rhb. nd., 1.635 wh., cb. wh., rhb. pr. col., rhb., 1.636

BaS BaS3 BaS4·2H2O Be(Gl) Bi

169.42 233.54 301.63 9.02 209.00

Bi2O3·CO2·H2O BiCl2(?) BiCl3* Bi(NO3)3·5H2O BiONO3·H2O Bi2O3 Bi2O3 Bi2O3 BiOCl

528.03 279.91 315.37 485.10 305.02 466.00 466.00 466.00 260.46

col., cb., 2.155 yel.-gn. red, rhb. gray, met., hex. silv. wh. or reddish, hex. wh. pd. bk. nd. wh. cr. col., tri. hex. pl. yel., rhb. yel., tet. yel., cb. wh., amor.

6.86 4.86 4.75 2.82 4.92815° 8.9 8.55 8.20 7.7215°

d. 163 230 d. 30 d. 260 820 860 tr. 704 185 d.

3.69 4.29

3.179 3.856 24° 4 3.097 24° 4 4.495 16° 2.188 3.24428° 2.658 5.72 4.958 4°

2.9 4.16515° 4.116° 3.920° 4.27915° 4.49915°

Boiling point, °C d. d. 1450

1560 1560

2.988 1.816 9.8020°

Hot water v. s. 0.0065100°

s. al. s. a.; i. al.

0.002218° 20.350° s. 310°

0.0065100° 84.880° s. 59100°

s. a.; i. al.

76.8100° 101.480°

sl. s. HCl, HNO3; i. al.

77.9 592

−8H2O, 550 d.

1923

2000

d. d. d.

tr. to mn. 1149

v. sl. s. 0.168 d. 0.015 i. 0.01 0.02617° 0.0001150°

2767 1450

d. s. 4115° i. i.

d. s. v. s. sl. s. d. i.

i. d. d. d. i. i. i. i. sl. s.

i.

i. i. i. i. sl. s.

2.660°

40.2100°

−O, 800 −8H2O, 100

1580 d. d. 400 d. 200 1284 271

d. 300 447 −5H2O, 80 1900

Other reagents

v. s. 0.002218°

39.30° 1.670° 5.615° 5.00° 0.00168° 1.50°

4.2515° 20°

Solubility in 100 parts Cold water

34.2100° 0.002424° 90.880°

0.09100° 0.00028530°

sl. s. al., act. sl. s. HCl, HNO3; i. al.

v. sl. s. al.; i. et. sl. s. a.; i. al. s. a., NH4Cl; i. al. s. HCl, HNO3, abs. al.; i. NH3, act. s. dil. a.; i. act. s. dil. a.; i. al., et., act. s. a. s. a., NH4 salts s. a. s. a., NH4 salts sl. s. HCl, NH4Cl; i. al. s. conc. H2SO4; 0.006, 3% HCl d. HCl; i. al. i. al., CS2 s. dil. a., alk. s. aq. reg., conc. H2SO4, HNO3 s. a. s. al. 4219° act.; s. a.; i. al. s. a. s. a. s. a. s. a. s. a.; i. act., NH3, H2C4H4O6 22.220° gly., 0.2425° et.; s. al. s. HNO3; i. al.

Boric acid

H3BO3

61.84

wh., tri.

1.43515°

Boron

B

10.82

2300

2550

i.

i.

B4C B2O3 B2O3·3H2O HBrO3 Br2

55.29 69.64 123.69 128.92 159.83

gray or bk., amor. or mn. bk. cr. col. glass, 1.459 tri., 1.456 col.; in soln. only rhb., or red lq.

2.32

carbide oxide oxide (sassolite) Bromic acid Bromine

2.54 1.85 1.49

2450 577 d. d. 100 −7.2

>3500 >1500

i. 1.10° sl. s. v. s. 4.220°

i. 15.7100° s. d. 3.1330°

i. a. s. a., al., gly.

hydrate Cadmium acetate acetate carbonate

Br2·10H2O Cd Cd(C2H3O2)2 Cd(C2H3O2)2·2H2O* CdCO3

339.99 112.41 230.50 266.53 172.42

red, oct. silv. met., hex. col. col., mn. wh., trig.

8.6520° 2.341 2.01 4.2584°

i.

wh., cb.

4.047 25° 4

s. a., NH4NO3 s. m. al. s. al. s. a., KCN, NH4 salts; i. NH3 1.5215° al.; i. et., act.

chloride

CdCl2

183.32

3.11920°; 5.87 (A)

d. 6.8 320.9 256 −H2O, 130 d. <500 568

58.78 767 d.

960

s. i. v. s. v. s. i. 0°

90

i. 147100°

s. al., et., alk., CS2

chloride cyanide hydroxide nitrate nitrate oxide oxide oxide, subCadmium sulfate sulfate sulfate sulfate sulfate sulfide (greenockite) Calcium acetate aluminate aluminum silicate (anorthite) arsenate bromide carbonate (aragonite) carbonate (calcite) chloride (hydrophilite) chloride chloride citrate cyanamide ferrocyanide fluoride (fluorite) formate hydride hydroxide hypochlorite hypophosphate lactate magnesium carbonate (dolomite) magnesium silicate (diopside) nitrate (nitrocalcite) nitrate nitride nitrite oxalate oxalate oxide

CdCl2.2a H2O Cd(CN)2 Cd(OH)2 Cd(NO3)2 Cd(NO3)2.4H2O* CdO CdO Cd2O CdSO4 CdSO4.H2O 3CdSO4.8H2O* CdSO4.H2O CdSO4.7H2O CdS Ca Ca(C2H2O2)2.H2O Ca(AlO2)2 CaO.Al2O3.2SiO2 Ca3(AsO4)2 CaBr2 CaCO3 CaCO3 CaCl2* CaCl2.H2O CaCl2.6H2O Ca3(C6H5O7)2.4H2O CaCN2 Ca2Fe(CN)6.12H2O CaF2 Ca(HCO2)2 CaH2 Ca(OH)2 Ca(ClO)2.4H2O Ca2P2O6.2H2O Ca(C3H5O3)2.5H2O

228.36 164.45 146.43 236.43 308.49 128.41 128.41 240.82 208.47 226.49 769.54 280.53 334.58 144.47 40.08 176.18 158.02 278.14 398.06 199.91 100.09 100.09 110.99 129.01 219.09 570.50 80.11 508.31 78.08 130.12 42.10 74.10 215.06 274.15 308.30

col., mn., 1.6513

3.327

wh., trig. col. col. nd. brn., cb. brn., amor, 2.49 gn., amor. rhb. mn. col., mn., 1.565 col. mn. yel.-or., hex., 2.506 silv. met., cb. wh. nd. col., rhb. or mn. tri., 1.5832 wh. pd. delq. nd. col., rhb., 1.6809 col., hex., 1.550 wh., delq., cb, 1.52 col., delq. col., trig., 1.417 col. nd. col., rhombohedral yel., tri., 1.5818 wh., cb., 1.4339 col., rhb. wh. cr. or pd. col., hex., 1.574 wh., feathery cr. granular col., eff.

4.79 15° 4

CaO.MgO.2CO2 CaO.MgO.2SiO2 Ca(NO3)2 Ca(NO3)2.4H2O* Ca3N2 Ca(NO2)2.H2O CaC2O4 CaC2O4.H2O CaO

184.42 216.52 164.10 236.16 148.26 150.11 128.10 146.12 56.08

trig., 1.68174 wh., mn. col., cb. col., mn., 1.498 brn. cr. delq., hex. col., cb. col. col., cb., 1.837

peroxide phosphate, monobasic phosphate, dibasic phosphate, tribasic phosphate, metaphosphate, pyrophosphate, pyro- (brushite) phosphide silicate (α) (pseudowollastonite)

CaO2.8H2O CaH4(PO4)2.H2O CaHPO4.2H2O Ca3(PO4)2 Ca(PO3)2 Ca2P2O7 Ca2P2O7.5H2O Ca3P2 CaSiO3

216.21 252.09 172.10 310.20 198.04 254.12 344.20 182.20 116.14

silicate (β) (wollastonite) sulfate (anhydrite)

CaSiO3 CaSO4

116.14 136.14

pearly, tet. wh., tri. wh., mn. pl. wh., amor. wh., tet., 1.588 col., biaxial, 1.60 wh., mn. red cr. col., pseudo hex., 1.6150 or mn.(?) col., mn., 1.610 col., rhb., 1.576, or mn, 1.50

2.455 17° 4 8.15 6.95 8.192 18° 4 4.691 24° 4 3.78620° 3.09 3.05 2.48 20° 4 4.58 20° 1.55

tr. 34 d. >200 d. 300 350 59.4

132

d. 900–1000 d. 1000 tr. 108 tr. 41.5

3.6720° 2.765

tr. 4 1750100atm 810 d. 1600 1551

3.353 25° 4 2.93 25° 2.711 4 2.152 15° 4

760 d. 825 1339103atm. 772

1810

1.6817°

29.92 −2H2O, 130

−6H2O, 200 −4H2O, 185

1.7 3.18020° 2.015 1.7 2.2

2.872 3.3 2.36 1.82 2.6317° 2.2334° 2.24° 2.2 3.32 2.220 16° 4 2.306 16° 4 3.14 2.82 3.09 2.25 2.5115° 2.905 2.915 2.96

subl. in N2, 980 1200  30

>1600

1330 d. d. 675 −H2O, 580 d. −2H2O, 200 −3H2O, 100

−8H2O, 100 −H2O, 100 d. 1670 975 1230

2850 expl. 275 d. 200

>1600 1540 tr. 1190 to α 1450(mn.)

180100°

76.50° s. 114.20° s. 350−5° 0.000001 d. 520° d.

60.8100° s. 127.660° s.

0.01325° 1250° 0.001220°† 0.001425° 59.50° s. v. s. 0.08518° s. d. s. 0.001618° 16.10° d. 0.1850° delq.; d. i. 10.5

32659.5° i. i.

Colloidal d. 45.580° i. 312105° 0.002100° 0.002100° 347260° s. v. s. 0.09626° d. 15090° 0.001726° 18.4100 0.077100° d. ∞

2.0515° m. al. s. a.; NH4OH, KCN s. a., NH4 salts; i. alk. v. s. a. s. al., NH3; i. HNO3 s. a., NH4 salts; i. alk. s. a., NH4 salts; i. alk. d. a., alk. i.act., NH3 i. al. i. al. i. al. s. a.; v. s. NH4OH s, a.; sl. s. al. sl. s. al. s. HCl s. dil. a. s. al., act.; sl. s. NH3 s. a., NH4Cl s. a., NH4Cl s. al. s. al. s. al. 0.006518° al. i. al. sl. s. a. i. al., et. d. a.; i. bz. s. NH4Cl d. a. s. HCl, H4P2O6 ∞h. al.; i. et.

18°

d. 730–760 1391 561 42.7 900 d. −H2O, 200 2570

16820° 0.024718° 0.0002625° 109.70° 2150° i. i.

tr. 1193 to rhb.

0.032 i. 1020° 2660° d. 770° 0.0006713° i. Forms Ca(OH)2 sl. s. 0.0224.5° 0.0025 i. i. sl. s. d. 0.009517° 0.29820°

i. 376151° v. s. d. 41790° 0.001495° i. d. d. 0.075100° d. i.

0.1619100°

1415° al.; s. amyl al., NH3 s. dil. a.; i. abs. al. s. 90% al. s. a.; i. ac. s. a.; i. ac s. a.; i. al. s. a. d.; i. al., et. s. a.; i. al., ac. i. a. s. a. s. a.; i. NH4Cl s. dil. a.; i. al., et. s. HCl s. a., Na2S2O3, NH4 salts

*Usual commercial form. †The solubility of CaCO3 in H2O is greatly increased by increasing the amount of CO2 in the H2O. 2-11

Physical Properties of the Elements and Inorganic Compounds (Continued )

TABLE 2-1

Name Calcium (Cont.) sulfate (gypsum)

Formula

Formula weight

Color, crystalline form and refractive index

CaSO4·2H2O

172.17

col., mn., 1.5226

Ca(SH)2·6H2O CaS CaSO3·2H2O CaC4H4O6·4H2O Ca(CNS)2·3H2O CaS2O3·6H2O CaWO4

214.31 72.14 156.17 260.22 210.28 260.30 288.00

col. pr. col., cb. wh., cr., 1.595 col., rhb. wh., delq. cr. col., tri., 1.56 wh., tet., 1.9200

C C C CO2

12.01 12.01 12.01 44.01

bk., amor. col., cb., 2.4195 bk., hex. col. gas

disulfide

CS2

76.13

col. lq.

monoxide

CO

28.01

col., poisonous, odorless gas

oxychloride (phosgene) oxysulfide

COCl2 COS

98.92 60.07

poisonous gas gas

sulfhydrate sulfide (oldhamite) sulfite tartrate thiocyanate thiosulfate tungstate (scheelite) Carbon, cf. table of organic compounds Carbon, amorphous Carbon, diamond Carbon, graphite dioxide

suboxide thionyl chloride Ceric hydroxide hydroxynitrate oxide sulfate Cerium

C3O2 CSCl2 2CeO2·3H2O Ce(OH)(NO3)3·3H2O CeO2 Ce(SO4)2·4H2O Ce

68.03 114.98 398.31 397.21 172.13 404.31 140.13

Cerous sulfate sulfate Cesium Chloric acid Chlorine

Ce2(SO4)3 Ce2(SO4)3·8H2O Ce HClO3·7H2O Cl2

568.44 712.57 132.91 210.58 70.91

hydrate Chloroplatinic acid Chlorostannic acid Chlorosulfonic acid Chromic acetate chloride chloride fluoride hydroxide

Cl2·8H2O H2PtCl6·6H2O H2SnCl6·6H2O HO·SO2·Cl Cr2(C2H3O2)6·2H2O CrCl3 CrCl3·6H2O* CrF3 Cr(OH)3

215.04 518.08 441.55 116.52 494.32 158.38 266.48 109.01 103.03

Cr(OH)3·2H2O Cr(NO3)3·9H2O* Cr(NO3)3·7a H2O Cr2O3 Cr2(SO4)3 Cr2(SO4)3·5H2O Cr2(SO4)3·15H2O Cr2(SO4)3·18H2O

139.07 400.18 373.15 152.02 392.20 482.28 662.44 716.49

hydroxide nitrate nitrate oxide sulfate sulfate sulfate sulfate

gas yel.-red lq. yel., gelatinous red, mn. wh. or pa. yel., cb. yel., rhb. steel gray, cb. or hex. wh., mn. or rhb. tri. silv. met., hex. lq. rhb., or gn.-yel. gas rhb. red-brn., delq. delq. col. lq. gn. pink, trig. vl. or gn., hex. pl. gn., rhb. gn. or blue, gelatinous gn. purple pr. purple, mn. dark gn., hex. rose pd. gn. vl. vl., cb., 1.564

Specific gravity 2.32 2.815°

Melting point, °C

Boiling point, °C

−1a H2O, 128

−2H2O, 163

d. 15 −2H2O, 100 d.

d. 650

Solubility in 100 parts Cold water

Hot water

0.2230°

0.25750°

v. s. d. 0.004318° 0.0370° s. 71.29° 0.2

v. s. d. 0.002790° 0.2285° v. s. d.

Other reagents s. a., gly., Na2S2O3, NH4 salts s. al. s. a. s. H2SO3 sl. s. al. v. s. al. i. al. s. NH4Cl; i. a.

1.87316° 6.06

d.

1.8–2.1 3.5120° 2.2620° lq. 1.101−87°; 1.53 (A); solid 1.56−79° 22° lq. 1.261 20 ; 2.63 (A) lq. 0.814−195° 4 ; 0.968 (A) 1.392 19° 4 lq. 1.24−87°; 2.10 (A) lq. 1.1140° 1.50915°

>3500 >3500 >3500 −56.65.2atm.

4200 4200 4200 subl. −78.5

i. i. i. 179.70° cc

i. i. i. 90.120° cc

i. a., alk. i. a., alk. i. a., alk. s. a., alk.

−108.6

46.3

0.20°

0.01450°

s. al.; et.

−207

−192

0.00440°; 3.50° cc

0.001850° 2.3220° cc

s. al., Cu2Cl2

−104 −138.2

8.2756mm −50.2760mm

v. s. sl. d. 1330° cc

d. 40.330° cc

s. ac., CCl4, bs.; d.a. v. s. alk., al.

−107

7761mm 73.5

d.

s. et. s. a.; sl. s. alk. carb.; i. alk

7.3 3.91 6.920° cb.; 6.7 hex. 3.91 2.88617° 1.9020° 1.28214.2° lq. 1.56−33.6°; 2.490° (A) 1.23 2.431 1.97128° 1.78725° 2.75715° 1.835 25° 4 3.8

5.21 3.012 1.86717° 1.722°

1950 645

1400

d. i. s. d. i.

i.

0°

−8H2O, 630 28.5 <−20 −101.6 d. 9.6 60 19.2 −80

670 d. 40 −34.6

151.5765mm 1200–1500 d.

subl. 83 >1000

d.

−2H2O, 100 36.5 100 1900

d. 100 d.

100

−10H2O, 100 −12H2O, 100

18.98 250° d. v. s. 1.460°; 31010° cc s. v. s. s. d. s. i.§ v. s. d. i. i. i. s. s. i. i.† s. s. 12020°

Slowly oxidized 0.4100° 7.640°

s. H2SO4, HCl s. dil. H2SO4 s. dil. a.; i. al.

s. a., al., NH3 0.57 ; 17730° cc 30°

v. s.

sl. s.

i. s. s. i. d. 67° d.

s. alk. s. alk. s. al., et. d. al.; i. CS2 4.7615° m. al. i. a., act., CS2 s. al.; i. et. sl. s. a.; i. al., NH3 s. a., alk.; sl. s. NH3 s. a., alk. s. a., alk., al., act. sl. s. a. i. a. s. al., H2SO4 sl. s. al. s. al.

sulfide Chromium

Cr2S3 Cr

200.02 52.01

brn.-bk. pd. gray, met., cb.

3.7719° 7.1

−S, 1350 1615

trioxide (chromic acid) Chromous chloride hydroxide oxide sulfate sulfide (daubrelite) Chromyl chloride Cobalt carbonyl sulfide, diCobaltic chloride chloride, dichro chloride, luteo chloride, praseo Cobaltic chloride, purpureo chloride, roseo hydroxide oxide sulfate sulfide Cobalto-cobaltic oxide Cobaltous acetate chloride chloride nitrate

CrO3 CrCl2 Cr(OH)2 CrO CrSO4·7H2O CrS CrO2Cl2 Co Co(CO)4 CoS2 CoCl3 Co(NH3)3Cl3·H2O Co(NH3)6Cl3 Co(NH3)4Cl3·H2O Co(NH3)5Cl3 Co(NH3)5Cl3·H2O Co(OH)3 Co2O3 Co2(SO4)3 Co2S3 Co3O4 Co(C2H3O2)2·4H2O CoCl2 CoCl2·6H2O* Co(NO3)2·6H2O

100.01 122.92 86.03 68.01 274.18 84.07 154.92 58.94 170.98 123.06 165.31 234.42 267.50 251.46 250.47 268.49 109.96 165.88 406.06 214.06 240.82 249.09 129.85 237.95 291.05

red, rhb. wh., delq. yel.-brn. bk. pd. blue bk. pd. dark red lq. silv. met., cb. or. cr. bk., cb. red cr.

2.70 2.75

197 d.

or., mn. gn., rhb. rhb. brick red bk. bk. blue cr. bk. cr. bk., cb. red-vl., mn., 1.542 blue cr. red, mn. red, mn., 1.4

1.7016 1.847 25° 1.819 25

CoO CoSO4 CoSO4·H2O

74.94 155.00 173.02

brn., cb. red pd. red pd., mn.(?), 1.639 red, mn., 1.483 brn. nd. yel.-red met., cb.

oxide sulfate sulfate sulfate (biebeorite) sulfide (syeporite) Copper Cupric acetate acetate aceto-arsenite (Paris green)

d. 3.97 1.92 8.920° 1.7318° 4.269 2.94

1550 −96.5 1480 51 subl.

5.18

d. 100 −1a H2O, 100 d. 900

4.8 6.07 1.705318.7° 3.356 25° 1.924 25 25° 1.883 25

−4H2O, 140 subl. 86 <100

5.68 3.71025° 3.13

d. 1800 d. 880 d.

25° 1.948 25 5.4518° 20° 8.92 1.930 20° 4 1.882

96.8 >1100 1083

−7H2O, 420

115

240 d.

1.98

d. i.

164.90° v. s. d. i. 12.350 i. d. i. i. i. s. s. 4.260° v. s. 0.2320° 16.120° i. i. d. i. i. s. 457° 116.50° 84.030°(anh.)

206.7100° v. s.

i. 25.60° s.

i.

i. d.

s. h. HNO3 s. HCl, dil. H2SO4; i. HNO3 s. H2SO4, al., et. sl. s. al.; i. et. s. conc. a. i. dil. HNO3 sl. s. al. v. s. a. s. et. s. a. s. al., et., CS2 s. HNO3, aq. reg.

s. 12.7446.5° 1.03146.5° 24.8716° i. i. i. s. 10596° 17780° 334.990° (anh.) i. 83100° s.

s. a.; al. i. al., NH4OH s. a.; i. al. i. al. sl. s. HCl s. a.; i. al. s. a. s. H2SO4 d. a. s. H2SO4; i. HCl, HNO3 s. a., al. 31 al.; 8.6 act. v. s. et., act. 10012.5° al.; s. act.; sl. s. NH3 s. a., NH4OH; i. al. 1.0418° m. al.; i. NH8

s.

20

7 al.; s. et.; gly. s. a., NH4OH

d. 110

33.80°

99.380°

s. a.

1.81 3.88

d. 150 d. 220

18.0521.5° i.

d. d.

i. 70.70°

d. 107.9100°

i. al. s. NH4OH, h. aq. NaHCO3 s. KCN; 0.03 aq. CO 5315° al.; 6815° m. al.

110.40° i. i. sl. s. i. i. 12.5 i. 16.7 38140° 243.70°

192.4100°

ammonium sulfate carbonate, basic (azurite)

CuSO4·4NH3·H2O 2CuCO3·Cu(OH)2

245.77 344.75

blue, tet., 1.670, 1.744 blue, rhb. blue, mn., 1.758

carbonate, basic (malachite) chloride (eriochalcite)

CuCO3·Cu(OH)2 CuCl2

221.17 134.48

dark gn., mn., 1.875 brn.-yel. pd.

3.9 3.054

d. 498

chloride chromate, basic cyanide dichromate ferricyanide ferrocyanide formate hydroxide lactate nitrate nitrate

CuCl2·2H2O CuCrO4·2CuO·2H2O Cu(CN)2 CuCr2O7·2H2O Cu3[Fe(CN)6]2 Cu2Fe(CN)6·7H2O Cu(HCO2)2 Cu(OH)2 Cu(C3H5O3)2·2H2O Cu(NO3)2·3H2O* Cu(NO3)2·6H2O

170.52 374.75 115.61 315.62 614.63 465.21 153.61 97.59 277.74 241.63 295.68

gn., rhb., 1.684 yel.-brn. yel.-gn. bk., tri. yel.-gn. red-brn. blue, mn. blue, gelatinous dark blue, mn. blue, delq. blue, rhb.

2.3922.4°

−2H2O, 110 −2H2O, 260 d. −2H2O, 100

277.51

1049 −6H2O, 110 d.

i. i.

3380° 0.0003818° i. s. 7.2 i.

ammonium chloride

dark gn., mn. gn.

117.6 2900 d. 52

20°

CoSO4·7H2O* CoS Cu Cu(C2H3O2)2 Cu(C2H3O2)2·H2O (CuOAs2O3)3· Cu(C2H3O2)2* CuCl2·2NH4Cl·2H2O

281.11 91.00 63.57 181.66 199.67 1013.83

2200

2.28618° 1.831 3.368

−H2O

2.0473.9° 2.074

114.5 −3H2O, 26.4

2300

Forms Cu2Cl2 993 d.

−HNO3, 170

i.

d. i. d. d. 45100° 66680° ∞

2.58° al. s. a., aq. reg. s. HNO3, h. H2SO4

s. al.; et., NH4Cl s. HNO3, NH4OH s. KCN, C5H5N s. a.; NH4OH s. NH4OH; i. HCl s. NH4OH; i. a., NH8 0.25 al. s. a., NH4OH, KCN, al. sl. s. al. 10012.5° al. s. al.

*Usual commercial form. †Also a soluble modification.

2-13

TABLE 2-1

Physical Properties of the Elements and Inorganic Compounds (Continued )

Name Cupric acetate (Cont.) oxide (paramelaconite) oxide (tenorite) oxychloride phosphide sulfate (hydrocyanite) sulfate (blue vitriol or chalcanthite) sulfide (covellite)

Formula

Specific gravity

249.71 95.63

tartate Cuprous ammonium iodide carbonate chloride (nantokite) cyanide

CuC4H4O6·3H2O CuI·NH4I·H2O Cu2CO3 Cu2Cl2 Cu2(CN)2

265.69 353.47 187.15 198.05 179.16

ferricyanide ferrocyanide fluoride hydroxide oxide (cuprite) Cuprous phosphide sulfide (chalcocite) sulfide Cyanogen

Cu3Fe(CN)6 Cu4Fe(CN)6 Cu2F2 CuOH Cu2O Cu6P2 Cu2S Cu2S C2N2

402.67 466.24 165.14 80.58 143.14 443.38 159.20 159.20 52.02

brn.-red brn.-red red cr. yel. red, cb., 2.705 gray-bk. bk., rhb. bk., cb. poisonous gas

Fe(OH)(C2H3O2)2

190.95

brn., amor.

FeCl3 FeCl3·6H2O* Fe4[Fe(CN)6]3

162.22 270.32 859.27

bk.-brn., hex. delq. red-yel., delq. dark blue

2.80411°

282 37 d.

Fe(OH)3 Fe(C3H5O3)3 Fe(NO3)3·6H2O Fe2O3

106.87 323.06 349.97 159.70

red-brn. brn., amor., delq. rhb., delq. red or bk., trig., 3.042 rhb., 1.814 yel., trig. yel., delq. gn. bk., cb., 2.42

3.4 to 3.9

−1a H2O, 500

sulfate sulfate (coquimbite) Ferroso-ferric chloride ferricyanide (Prussian green) oxide (magnetite; magnetic iron oxide) oxide, hydrated Ferrous ammonium sulfate

Fe2(SO4)3 Fe2(SO4)3·9H2O FeCl2·2FeCl3·18H2O Fe′′′ 4 Fe′′ 3 [Fe(CN)6]6 Fe3O4

399.88 562.02 775.49 1662.70 231.55 303.61 392.15

chloride (lawrencite)

Fe3O4·4H2O FeSO4·(NH4)2SO4· 6H2O FeCl2

chloroplatinate ferricyanide (Turnbull’s blue) ferrocyanide formate hydroxide nitrate oxide

FePtCl6·6H2O Fe3[Fe(CN)6]2 Fe2Fe(CN)6 Fe(HCO2)2·2H2O Fe(OH)2 Fe(NO3)2·6H2O FeO

571.92 591.47 323.66 181.92 89.87 287.96 71.85

2-14

126.76

bk. blue-gn., mn., 1.4915 gn.-yel., hex., 1.567 yel., hex. dark blue blue-wh., amor.

6.40 6.45

Melting point, °C

CuSO4·5H2O* CuS

hydroxide lactate nitrate oxide (hematite)

79.57 79.57 365.69 252.67 159.63

Color, crystalline form and refractive index bk., cb. bk., tri., 2.63 blue-gn. bk. gn.-wh., rhb., 1.733 blue, tri., 1.5368 blue, hex. or mn., 1.45 1 gn. pd. rhb. pl. yel. wh., cb., 1.973 wh., mn.

Cyanogen compounds, cf. table of organic compounds Ferric acetate, basic ammonium sulfate, cf. Alum chloride (molysite) chloride ferrocyanide (Prussian blue)

CuO CuO CuCl2·2CuO·4H2O Cu3P2 CuSO4

Formula weight

6.35 3.60615° 2.286 15.6° 4 4.6

d. 1026 d. 1026 −3H2O, 140 d. d. >600 −4H2O, 110 tr. 103

4.4 3.53 2.9

d. 422 474.5

3.4 6.0 6.4 to 6.8 5.6 5.80 lq. 0.866−17.2°; 1.806 (A)

Solubility in 100 parts Cold water i. i. i. i. 14.30°

Hot water i. i. 75.4100°

Other reagents s. a.; KCN, NH4Cl s. a., KCN, NH4Cl s. a. s. HNO3; i. HCl i. al.

Forms CuO, 650 −5H2O, 250 d. 220

24.3 0.00003318°

205100°

1.18° al. s. HNO3, KCN

0.1485°

1366 d.

0.0215° d. i. 1.5225° i.

s. a., KOH s. NH4I s. a., NH4OH s. HCl, NH4OH, al. s. KCN, HCl, NH4OH; sl. s. NH3 s. NH4OH; i. HCl s. NH4OH; i. NH4Cl s. HF, HCl, HNO3; i. al. s. a., NH4OH s. HCl, NH4Cl, NH4OH s. HNO3; i. HCl s. HNO3, NH4OH; i. act. s. HNO3, NH4OH; i. act. 230020° cc al.; 50018° cc et.

d.

908 −a H2O, 360 1235

−O, 1800

1100 1130 −34.4

−20.5

subl. 1100

0°

i. i. i. i. i. i. 0.000518° 0.000518° 45020° cc

i. i.

i. i.

i.

s. a.; al. v. s. al.; et. +HCl s. al., act., gly. s. HCl, conc. H2SO4; i. al., et. s. a.; i. al., et. i. et. s. al., act. s. HCl

74.40° 2460° i.

535.8100° ∞ d.

i. v. s. 1500° i.

i. v. s. ∞ d. d. s.

5.2

d. 50 d. 180 1538 d.

sl. s. 440 s. i. i.

i.

s. d. h. HCl i. al.

1.864

d. d.

i. 180°

i. 10075°

s. a. i. al.

64.410°

105.7100°

100 al.; s. act.; i. et.

v. s. i. i. sl. s. 0.00067 2000° i.

v. s.

20°

1.684 5.12

35 1560 d.

3.09718° 2.1

d. 480

2.7 2.714 d. 3.4 5.7

315 280

d.

delq.

d. lt. gn. cr. bk.

Boiling point, °C

60.5 1420

i. H2SO4, NH3 s. abs. al.

i. dil. a., al.

30025° i.

s. a., NH4Cl s. a.; i. alk.

phosphate (vivianite)

Fe3(PO4)2·8H2O

501.64

silicate sulfate (siderotilate) sulfate (copperas) sulfide cf. also under iron Fluoboric acid Fluorine

FeSiO3 FeSO4·5H2O FeSO4·7H2O* FeS

131.91 241.99 278.02 87.91

HBF4 F2

87.83 38.00

Fluosilicic acid Gadolinium Gallium bromide Glucinum cf. Beryllium Gold Gold, colloidal Gold salts cf. under Auric and Aurous Hafnium Helium Hydrazine formate hydrate hydrochloride hydrochloride, dinitrate nitrate, disulfate sulfate Hydrazoic acid (azoimide) Hydriodic acid Hydriodic acid Hydriodic acid Hydriodic acid Hydriodic acid Hydrobromic acid Hydrobromic acid

H2SiF6 Gd GaBr3

144.08 156.9 309.47

Au Au

197.20 197.20

yel. met., cb. blue to vl.

19.3

1063

2600

Hf He N2H4 N2H4·2HCO2H N2H4·H2O N2H4·HCl N2H4·2HCl N2H4·HNO3 N2H4·2HNO3 N2H4·a H2SO4 N2H4·H2SO4 HN3 HI HI·H2O HI·2H2O HI·3H2O HI·4H2O HBr HBr·H2O

178.6 4.00 32.05 124.10 50.06 68.51 104.98 95.06 158.08 81.09 130.12 43.03 127.93 145.94 163.96 181.98 199.99 80.92 98.94

hex. col. gas col. lq. cb. col. yel. lq. wh., cb. cr. nd. delq. pl. rhb. col. lq. col. gas col. lq. col. lq. col. lq. col. lq. col. gas; 1.325 (lq.) col. lq.

12.1 0.1368 (A) 1.011 15° 4

>1700 <−272.2 1.4 128 −40

>3200(?) −268.9 113.5

Hydrobromic acid Hydrobromic acid Hydrochloric acid Hydrochloric acid Hydrochloric acid Hydrochloric acid Hydrocyanic acid (prussic acid)

HBr (47.8% in H2O) HBr·2H2O HCl† HCl (45.2% in H2O) HCl·2H2O HCl·3H2O HCN

80.92 116.96 36.47 36.47 72.50 90.51 27.03

Hydrofluoric acid Hydrofluoric acid Hydrogen

HF HF (35.35% in H2O) H2

col. lq. wh. cr. col. gas; 1.256 (lq.) col. lq. col. lq. col. lq. poisonous gas or col. lq., 1.254 gas or col. lq. col. lq. col. gas or cb.

peroxide selenide sulfide Hydroxylamine hydrochloride nitrate sulfate

H2O2‡ H2Se H2S NH2OH NH2OH·HCl NH2OH·HNO3 NH2OH·a H2SO4

20.01 20.01 2.016 34.02 81.22 34.08 33.03 69.50 96.05 82.07

blue, mn., 1.592, 1.603 mn. gn., tri., 1.536 blue-gn., mn. bk., hex. col. lq. gn.-yel. gas

2.58 3.5 2.2 1.89914.8° 4.84 lq. 1.51−187°; 1.3115° (A)

1550

col. lq., 1.333 col. gas col. gas rhb., delq. col., mn. col. cr. col., mn.

1.0321° 1.42

1.378 4.40° (A) 1.715°

2.710° (A) 1.78 1.486 2.11−15° 1.2680° (A) 1.48 1.46 −18.3° 4 18°

0.697

0.98813.6° 1.15 lq. 0.0709−252.7° 0.06948 (A) 1.438 20° 4 2.12−42° 1.1895 (A) 1.3518° 1.6717°

i.

s. a.; i. ac.

64 1193

−5H2O, 300 −7H2O, 300 d.

s. 32.80° 0.00061618°

s. 14950°

i. al. i. al. s. a.; i. NH3

−223

130 d. −187

∞ d.

∞

s. al.

s.

s.

s.

s.

i. s.

i.

s. aq. reg., KCN; i. a. s. aq. reg., KCN; i. a.

0.970° cc ∞ s. ∞ v. s. s. 174.910° v. s. v. s. 3.05522° ∞ 4250010° cc ∞ ∞ ∞ ∞ 2210°

1.0850° cc ∞

Absorbed by Pt s. al.

∞ v. s. v. s. v. s.

∞ al.; i. et. sl. s. al. s. al.

delq. cr. 20°

i.

198 70.7 104 85 254 −80 −50.8 −43 −48 −36.5 −86

−11 −111 −15.35 0 −24.4 −14 −83 −35 −259.1 −0.89 −64 −82.9 34 151 48 170 d.

118.5739.5mm subl. 140 d. 37 −35.5 127774mm

−67

d. d. 26

∞ s. 82.30° ∞ ∞ ∞ ∞

19.4 120 −252.7

∞ 0° to 19.4° v. s. 2.10° cc

126 −85

760mm

151.4 −42 −59.6 56.522mm d. d. <100

∞ 3774° cc 4370° cc s. 83.317° v. s. 32.90°

27.6560° ∞ v. s.

130100°

s. 56.160°

i. al. v. sl. s. abs. al. ∞ al. s. al. ∞ al. s. al. s. al. s. al. s. al. Stable at −15.5° and 1 atm., and at −11.3° and 2.5 atm. s. al. s. al., et. s. al. s. al. s. al. ∞ al., et.

v. s. 0.8580° cc

sl. s. Fe, Pd, Pt

27022.5° cc 18640° cc d. v. s. d. 68.590°

s. a., et.; i. petr. et s. CS2, COCl2 9.5415° cc al.; s. CS2 s. a., al. s. al.; i. et. v. s. abs. al. v. sl. s. al.; i. et., abs. al.

*Usual commercial form. †Usual commercial form about 31 percent. ‡Usual commercial forms 3 or 30 percent. 2-15

TABLE 2-1

Physical Properties of the Elements and Inorganic Compounds (Continued )

Name

Formula

Hypobromous acid Illinium Indium Iodic acid

HBrO Il In HIO8

Iodine oxide, pentaIodoplatinic acid Iridium Iron, cast† pure steel white pig wrought carbide (cementite) carbonyl nitride

I2 I2O5 H2PtI6·9H2O Ir Fe Fe Fe Fe Fe Fe2C Fe(CO)5 Fe2N

silicide sulfide, di- (marcasite) sulfide, di- (pyrite) sulfide (pyrrhotite) Cf. also under ferric and ferrous Krypton Lanthanum Lead

FeSi FeS2 FeS2 Fe7S8

acetate acetate (sugar of lead) acetate acetate, basic acetate, basic acetate, basic arsenate, monobasic arsenate, dibasic (schultenite) arsenate, metaarsenate, pyroLead azide bromide

Formula weight 96.92 146(?) 114.76 175.93

Color, crystalline form and refractive index

Specific gravity

Melting point, °C

yel.

Boiling point, °C

Solubility in 100 parts Cold water

Hot water

4050mm

s.

d.

soft, tet. met. col., rhb.

7.320° 4.6290°

155 110 d.

1450

i. 2860°

i. 576101°

blue-bk., rhb. wh., trimetric brn., delq. mn. wh. met., cb. gray silv. met., cb. silv. gray gray gray pseudo hex. pa. yel. lq. gray

4.9320° 4.799 25° 4

113.5 d. 300

184.35

0.0956660°

22.420° 7.03 7.8620° 7.6 to 7.8 7.6 to 7.8 7.86 7.4 1.45721° 6.35

2350 1275 1535 1375 1075 1505 1837 −21 d. >560

>4800

0.01620° 187.412° s. d. i. i. i. i. i. i. i. i. d.

83.91 119.97 119.97 647.43

yel.-gray, oct. yel., rhb. yel., cb. hex.

6.1 20° 4 4.87 5.0 4.6 20° 4

d. d.

i. 0.00049 0.0005 i.

i.

tr. 450 1171 d. >700

Kr La Pb

83.70 138.92 207.21

col. gas lead gray silv. met., cb.

2.818 (A) 6.1520° 20° 11.337 20

−169 826 327.5

−151.8 1800 1620

11.050° cc d. i.

3.5760° cc

Pb(C2H3O2)2 Pb(C2H3O2)2·3H2O† Pb(C2H3O2)2IOH2O Pb2(C2H3O2)3OH Pb(C2H3O2)2· Pb(OH)2·H2O Pb(C2H3O2)2· 2Pb(OH)2 PbH4(AsO4)2 PbHAsO4 Pb(AsO3)2 Pb2As2O7 PbN6 PbBr2

325.30 379.35 505.46 608.56 584.54

wh. cr. wh., mn. wh., rhb. wh. wh. nd.

3.251 20° 4 2.55 1.689

280 −3H2O, 75 22

807.75

wh. nd.

489.06 347.13 453.03 676.24 291.26 367.05

tri., 1.82 wh., mn., 1.9097 hex. rhb., 2.03 col. nd. col., rhb.

4.46 5.94 6.4215° 15° 6.85 15 6.66

802 expl. 350 373

253.84 333.84 1120.91 193.10 55.85 55.85 55.85 55.85 55.85 179.56 195.90 125.71

15°

d. 140 d. >200

3000

102.5760mm

−H2O, 280

918

i. i. i. i. i. i. i.

i.

19.70° 45.6415° s. v. s. v. s.

22150° 200100° s.

5.55

18.2

d. i. d. i. i. 0.45540°

d. 0.05100° 4.75100°

sl. s.

PbCO3

267.22

wh., rhb., 2.0763

6.6

d. 315

0.0001120°

d.

2PbCO3·Pb(OH)2† PbCl2 PbCrO4 PbCrO4·PbO Pb(HCO2)2 3PbO·H2O Pb(NO3)2

775.67 278.12 323.22 546.43 297.25 687.65 331.23

6.14 5.80 6.12

d. 400 501 844

4.56 7.592 4.53

d. 190 −H2O, 130 d. 470

i. 0.6730° 0.00000720° i. 1.616° 0.014 38.80°

i. 3.34100° i. i. 18100° d.

oxide, suboxide, mono- (litharge)

Pb2O PbO

430.42 223.21

wh., hex. wh., rhb., 2.2172 yel., mn., 2.42 or.-yel. nd. wh., rhb. cb. col., cb. or mn., 1.7815 bk., amor. yel., tet.

8.34 9.53

d. red heat 888

i. 0.006818°

i.

oxide, mono (massicotite)

PbO

223.21

yel., rhb., 2.61

8.0

2-16

s. a. v. s. 87% al.; i. abs. al. et., chl. s. al., KI, et. i. abs. al., et., chl. sl. s. aq. reg., aq. Cl2 s. a.; i. alk. s. a.; i. alk. s. a.; i. alk. s. a.; i. alk. s. a.; i. alk. s. a. s. al., H2SO4, alk. s. HCl, H2SO4 i. aq. reg. i. dil. a. i. dil. a.

sl. s. al., bz. s. a. s. HNO3; i. c. HCl, H2SO4 s. gly.; v. sl. s. al. s. gly.; sl. s. al. sl. s. al. s. al.

carbonate (cerussite) carbonate, basic (hydrocerussite; white lead) chloride (cotunnite) chromate (crocoite) chromate, basic formate hydroxide nitrate

954760mm d.

Other reagents

138.8100°

s. al. s. HNO3 s. HNO3, NaOH s. HNO3 s. HCl, HNO3; i. sc. v. s. ac.; i. NH4OH s. a., KBr.; sl. s. NH3; i. al. s. a., alk.; i. NH3, al. s. ac.; sl. s. aq. CO2 sl. s. dil. HCl, NH3, i. al. s. a., alk.; i. NH3, ac. s. a., alk. i. al. s. a., alk. 8.822° al. s. a., alk. s. alk., PbAc, NH4Cl, CaCl2

oxide, mono-

PbO

223.21

amor.

9.2 to 9.5

oxide, red (minium) oxide, sesquioxide, di- (plattnerite) silicate sulfate (anglesite)

Pb3O4 Pb2O3 PbO2 PbSiO3 PbSO4

685.63 462.42 239.21 283.27 303.27

9.1

Pb(HSO4)2·H2O PbSO4·PbO PbS Pb(CNS)2 Li LiC7H5O2 LiBr

419.36 526.48 239.27 323.37 6.94 128.05 86.86

LiBr·2H2O Li2CO3 LiCl

122.89 73.89 42.40

citrate fluoride formate hydride hydroxide hydroxide nitrate nitrate oxide phosphate, monobasic phosphate, tribasic phosphate, tribasic salicylate sulfate sulfate sulfate, acid Lutecium Magnesium acetate acetate aluminate (spinel)

Li3C6H5O7·4H2O LiF LiHCO2·H2O LiH LiOH LiOH·H2O LiNO3 LiNO3·3H2O Li2O LiH2PO4 Li3PO4 Li3PO4·12H2O LiC7H5O3 Li2SO4 Li2SO4·H2O† LiHSO4 Lu Mg Mg(C2H3O2)2 Mg(C2H3O2)2·4H2O† MgO·Al2O3

281.98 25.94 69.97 7.95 23.95 41.96 68.95 123.00 29.88 103.94 115.80 331.99 144.05 109.94 127.96 104.01 174.99 24.32 142.41 214.47 142.26

red, amor. red-yel., amor. brn., tet., 2.229 col., mn., 1.961 wh., mn. or rhb., 1.8823 cr. col., mn. lead gray, cb., 3.912 col., mn. silv. met. cb. wh. leaflets wh., delq., cb., 1.784 wh. pr. col., mn., 1.567 wh., delq., cb., 1.662 wh. cr. wh., cb., 1.3915 col., rhb. wh., cb. wh. cr. col., mn. col., trig., 1.735 col. col., 1.644 col. wh., rhb. wh., trig. col. col., mn., 1.465 col., mn., 1.477 pr.

ammonium chloride ammonium phosphate (struvite) ammonium sulfate (boussingaultite) benzoate carbonate (magnesite) carbonate (nesquehonite) carbonate, basic (hydromagnesite) Magnesium chloride (chloromagnesite) chloride (bischofite) hydroxide (brucite) nitride oxide (magnesia; periclase) perchlorate

MgCl2·NH4Cl·6H2O MgNH4PO4·6H2O

256.83 245.44

sulfate, acid sulfate, basic (lanarkite) sulfide (galena) thiocyanate Lithium benzoate bromide bromide carbonate chloride

*See also a table of alloys. †Usual commercial form.

MgSO4·(NH4)2SO4· 6H2O Mg(C7H5O2)2·3H2O MgCO3 MgCO3·3H2O 3MgCO3·Mg(OH)2·3H2O MgCl2 MgCl2·6H2O† Mg(OH)2 Mg3N2 MgO Mg(ClO4)2†

i.

i.

d. 500 d. 360 d. 290 766

i. i. i. i.

i. i. i.

1336  5

3.464 25° 4

547

1265

0.00280° 0.000118° 0.004418° 0.0000918° 0.0520° d. 3325° 1430° (2H2O)

0.005640°

6.92 7.5 3.82 0.5320°

1170 d. 977 1120 d. 190 186

2.110° 2.068 25° 4

44 618 614

d. 1360

24620° 1.540° 670°

9.375 6.49 6.2

2.29521.5° 1.46 0.820 2.54 1.83 2.38 2.013 25° 4

d. 870 −H2O, 94 680 445 261 29.88

2.22 2.06 2.12313°

>100 837 100 d. 860 −H2O, 130 170.5

silv. met., hex. wh. wh., mn. pr., 1.491 col. cb., 1.718–23

1.7420° 1.42 1.454 3.6

651 323 80 2135

wh., rhb., delq. col., rhb., 1.496

1.456 1.715

−4H2O, 195 d. 100

2.461 2.53717.5° 1.645

1670 925 d. subl. <1000

1110

61.215° 0.2718° 49.20° d. 12.70° 22.310° 53.40° v. s. forms LiOH

s. alk., PbAc, NH4Cl, CaCl2 s. ac., h. HCl s. a., alk. s. ac., h. alk.; i. al. s. a.

i. s. d. 40100° 266100° (1H2O) 0.72100° 127.5100° 66.7100° 0.13535° 346.6104° 17.5100° 26.880° 19470° ∞

0.03418° v. sl. s. 12826° 35.340° 43.60° d.

v. sl. s. v. sl. s.

i. v. s. v. s. i.

sl. s. d. v. s. v. s.

16.7 0.02310°

s. 0.019580°

29.9100° 35100°

s. conc. a., NH4 salts; i. al. sl. s. H2SO4 sl. s. H2SO4 s. a.; i. alk. s. KCNS, HNO3 s. a., NH3 7.725°, 1078° al. s. al., act. s. al. s. dil. a.; i. al., act., NH3 2.4815° al.; s. et. sl. s. al., et. s. HF; i. act. sl. s. al., et. i. et. sl. s. al. sl. s. al. s. al., NH3

s. a., NH4Cl; i. act. v. s. al. i. act., 80% al. i. 80% al. s. a., NH4 salts 5.2515° m. al. v. s. al. v. sl. s. dil. HCl; i. dil. HNO3 s. a.; i. al.

360.62

col., mn.

1.72

>120

16.86

320.59 83.43 138.38 365.37

wh. pd. wh., trig. 1.700 col., rhb., 1.501 wh., rhb., 1.530

3.037 1.852 2.16

−3H2O, 110 d. 350 −H2O, 100 d.

4.525° (anh.) 0.0106 0.151819° 0.04

d. 0.011

s. act. s. a., aq. CO2; i. act., NH3 s. a., aq. CO2 s. a., NH4 salts; i. al.

95.23

col., hex., 1.675

2.32525°

712

52.80°

73100°

50 al.

203.33 58.34 100.98 40.32 223.23

wh., delq., mn., 1.507 wh., trig., 1.5617 gn.-yel., amor. col., cb., 1.7364 wh., delq.

1.56 2.4 3.65 2.6025°

118 d. d. d. 2800 d.

1412 d. 3600

0°

0°

281 0.000918° i. 0.00062 99.625°

100°

130 s.

100°

918 d. v. s.

50 al. s. NH4 salts, dil. a. s. a.; i. al. s. a., NH4 salts; i. al. 2425 al., 51.825° m. al.; 0.29 et.

Physical Properties of the Elements and Inorganic Compounds (Continued )

TABLE 2-1

Name

Formula

Formula weight

Color, crystalline form and refractive index

Magnesium chloride (Cont.) peroxide phosphate, pyrophosphate, pyropotassium chloride (carnallite) potassium sulfate (picromerite) silicofluoride sodium chloride sulfate sulfate (epsom salt; epsomite) Manganese acetate acetate carbonate (rhodocrosite)

MgO2 Mg2P2O7 Mg2P2O7·3H2O MgCl2·KCl·6H2O MgSO4·K2SO4·6H2O MgSiF6·6H2O MgCl2·NaCl·H2O MgSO4 MgSO4·7H2O* Mn Mn(C2H3O2)2 Mn(C2H3O2)2·4H2O* MnCO3

56.32 222.60 276.65 277.88 402.73 274.48 171.70 120.38 246.49 54.93 173.02 245.08 114.94

wh. pd. col., mn., 1.604 wh., amor. delq., rhb., 1.475 mn., 1.4629 col., trig., 1.3439 col. col. col., rhb., 1.4554 gray-pink met.

chloride (scacchite) chloride

MnCl2 MnCl2·4H2O*

125.84 197.91

chloride, perhydroxide (ous) (pyrochroite) hydroxide (ic) (manganite) nitrate oxide (ous) (manganosite) oxide (ic) oxide, di- (pyrolusite; polianite) sulfate (ous) sulfate (ous) (szmikite)

MnCl4 Mn(OH)2 Mn2O3·H2O Mn(NO3)2·6H2O MnO Mn2O3 MnO2*

196.76 88.95 175.88 287.04 70.93 157.86 86.93

rose, delq., cb. rose red, delq., mn. 1.575 gn. wh., trig. brn., rhb., 2.24 rose red, mn. gray-gn., cb., 2.16 brn.-bk., cb. bk., rhb.

MnSO4 MnSO4·H2O

150.99 169.01

red-wh. pa. pink, mn., 1.595

sulfate (ous)

MnSO4·2H2O

187.02

2.52615° 15°

pa. pink, mn. rose, trig., 1.817

Specific gravity

Melting point, °C

2.59822° 2.56 1.60 19.4° 4 2.15 1.788 17.5° 4

expl. 275 1383 −3H2O, 100 265 d. 72 d.

2.66 1.68 7.220° 1.74 20° 4 1.589 3.125

1185 70 d. 1260

2.977 25° 4 2.01

650 58.0

3.25818° 3.258 1.8221° 5.18 4.81 5.026

d. d. 25.8 1650 −0, 1080 −0, >230

3.235 2.87

sulfate (ous)

MnSO4·7H2O

277.10

pink, mn. or rhb.

2.092

sulfate (ic)

Mn2(SO4)3

398.04

gn., delq. cr.

3.24

Ma Hg(C2H3O2)2 HgBr2 HgCO3·2HgO HgCl2 Hg(CNO)2 Hg(OH)2 HgO HgCl2·3HgO HgSiF6·HgO·3H2O HgSO4 HgSO4·2HgO HgC2H3O2 HgBr Hg2CO3

98–99.5 318.70 360.44 693.84 271.52 284.65 234.63 216.61 921.35 613.33 296.67 729.89 259.65 280.53 461.23

11.5 3.270 6.053

2300 (?) d. 237

5.44 4.42

277 expl. −H2O, 175 d. 100 d. 260

MnSO4·3H2O

205.04

sulfate (ous)

MnSO4·4H2O*

223.05

sulfate (ous)

MnSO4·5H2O

241.07

sulfate (ous)

MnSO4·6H2O

259.09

Masurium Mercuric acetate bromide carbonate, basic chloride (corrosive sublimate) fulminate hydroxide oxide (montroydite) oxychloride (kleinite) silicofluoride, basic sulfate sulfate, basic (turpeth) Mercurous acetate bromide carbonate

2-18

2.356 pink, rhb. or mn., 1.518 pink, tri., 1.508

wh. pl. wh., rhb. brn.-red wh., rhb., 1.859 cb. yel. or red, rhb., 2.5 yel., hex. yel. nd. wh., rhb. yel., tet. wh. sc. wh., tet. yel. pd.

2.107 15°

2.103

11.14 7.93 6.47 6.44 7.307

d. d. subl. 345 d. 130

Solubility in 100 parts Cold water

Hot water

Other reagents

i. i. i. 64.519° d. 19.260° 64.817.5° s. 26.90° 72.40° d. s. s. 0.006525°

i. i. sl. s. d. 81.775° s. s. 68.3100° 17840°

63.40° 1518°

123.8100° ∞

s. al., m. al. s. aq. CO2, dil. a.; l. NH3, al. s. al.; i. et., NH3 s. al.; i. et.

129.5

s. 0.00220° i. 4260° i. i. i.

s. i. i. ∞ i. i. i.

s. al., et. s. a., NH4 salts; i. alk. s. h. H2SO4 v. s. al. s. a., NH4Cl s. a.; i. act. s. HCl; i. HNO3, act.

d. 850

530° 98.4748°

7350° 79.77100°

s. al.; i. et.

85.2735°

106.855°

1900

d.

700 Stable 57 to 117 Stable 40 to 57 Stable 30 to 40 Stable 18 to 30 Stable 8 to 18 Stable −5 to +8 Stable −10 to −5; 19 d. d. 160

sulfate (ous)

Boiling point, °C

1190 −H2O, 106; −4H2O, 200

5°

−4H2O, 450

74.22

99.3157°

13616°

16950°

5°

−7H2O, 280

322 304

s. 64.550°

200

2040°

2479°

176

d. HF s. al. s. al. s. dil. a.

i. al.

35°

142

0°

s. a. s. a.; i. alk. s. a.; i. al. d. al.

25114°

v. s.

d.

s. HCl, dil. H2SO4; l. conc. H2SO4, HNO3

2510° 0.520° i. 3.60° sl. s. i. 0.005225° i. d. d. 0.005 0.7513° 7 × 10−9 i.

100100° 25100°

s. al. sl. d. 25.20° al.; v. sl. s. et. s. aq. CO2, NH4Cl 3325° 99% al.; 33 et. s. NH4OH, al. s. a. s. a.; i. al. s. HCl s. a. s. a.; i. al., act., NH8 s. a.; i. al. s. H2SO4, HNO3; i. al. s. a.; i. al., act. s. NH4Cl

61.3100° i. 0.041100° d. 0.167100° d. i. d.

HgCl

236.07

wh., tet., 1.9733

7.150

302

383.7

0.00140°

0.000743°

iodide nitrate Mercurous oxide

HgI HgNO3·H2O Hg2O

327.53 280.63 417.22

yel., tet. wh. mn. bk.

7.70 4.7853.9° 9.8

290 d. 70 d. 100

subl. 140; 310d. expl.

2 × 10−8 v. s. i.

v. sl. s. d. 0.0007

sulfate Mercury† Molybdenum

Hg2SO4 Hg Mo

497.28 200.61 95.95

wh., mn. silv. lq. or hex.(?) gray, cb.

7.56 13.54620° 10.2

d. −38.87 2620  10

0.05516.5° i. i.

0.092100° i. i.

MoCl2

166.85

yel., amor.

3.714 25° 4

d.

i.

i.

3.578 25° 4

d.

chloride (calomel)

chloride, dichloride, tri-

MoCl3

202.32

dark red pd.

chloride, tetra-

MoCl4

237.78

brn., delq.

chloride, pentaoxide, tri- (molybdite) sulfide, di- (molybdenite) sulfide, trisulfide, tetraMolybdic acid Molybdic acid Neodymium Neon Neptunium Nickel

MoCl5 MoO3 MoS2 MoS3 MoS4 H2MoO4 H2MoO4·H2O Nd Ne 239

273.24 143.95 160.07 192.13 224.19 161.97 179.98 144.27 20.18

Np Ni

239 58.69 176.78 291.20 394.99 422.62 218.52 272.57 320.71 841.51 118.70 587.58

carbonyl chloride chloride

Ni(C2H3O2)2 NiCl2·NH4Cl·6H2O NiSO4·(NH4)2SO4· 6H2O Ni(BrO3)2·6H2O NiBr2 NiBr2·3H2O NiBr2·6NH3 NiPtBr6·6H2O NiCO3 2NiCO3·3Ni(OH)2· 4H2O Ni(CO)4 NiCl2 NiCl2·6H2O*

chloride, ammonia cyanide dimethylglyoxime

NiCl2·6NH3 Ni(CN)2·4H2O NiC8H14O4N4

231.80 182.79 288.91

formate hydroxide (ic) hydroxide (ous) nitrate nitrate, ammonia oxide, mono- (bunsenite) potassium cyanide sulfate

Ni(HCO2)2·2H2O Ni(OH)3 Ni(OH)2·d H2O Ni(NO3)2·6H2O Ni(NO3)2·4NH3·2H2O NiO Ni(CN)2·2KCN·H2O NiSO4

184.76 109.71 97.21 290.80 286.87 74.69 258.97 154.75

acetate ammonium chloride ammonium sulfate bromate bromide bromide bromide, ammonia bromoplatinate carbonate carbonate, basic

170.73 129.60 237.70

bk. cr. col., rhb. bk., hex., 4.7 red-brn. brn. pd. yel-wh., hex. yel., mn. yellowish col. gas

2.928 25° 4 19.5°

4.50 4.80114°

3.12415° 6.920° lq. 1.204−245.9° 0.674 (A)

s. lq. O2, al., act., bz.

i.

s. dil. HNO3; sl. s. H2SO4, HCl; i. NH3 i. al.

s.

d.

194

268

s.

795 1185 d. d. d. 115 −H2O, 70 840 −248.67

subl.

−2H2O, 200 −245.9

d. 18°

0.107 i. sl. s. i. v. sl. s. 0.13318° d. 2.60° cc 238

Produced by Neutron bombardment of U 1452 2900 i.

gn. pr. gn., delq., mn. blue-gn., mn., 1.5007 gn., cb. yel., delq. gn., delq. vl. pd. trig. lt. gn., rhb. lt. gn.

1.798 1.645 1.923

d.

2.575 4.64 28° 4

16.6 15025° 2.53.5°

v. s. 39.288°

v. sl. s. (NH4)2SO4

d. d. −3H2O, 200

28 112.80° 1999° v. s.

156100° 316100° d.

s. NH4OH s. al., et., NH4OH s. al., et., NH4OH i. c. NH4OH

d. d.

0.009325° i.

i. d.

s. a. s. a., NH4 salts

0.0189.8° 53.80° 180

i. 87.6100° v. s.

s. aq. reg., HNO3, al., et. s. NH4OH, al.; i. NH3 v. s. al.

s. i. i.

d. i. i.

s. NH4OH; i. al. s. KCN; i. dil. KCl s. abs. al., a.; i. ac., NH4OH

i. v. sl. s. ∞56.7°

s. a., NH4OH, NH4Cl s. a., NH4OH; i. alk. s. NH4OH; i. abs. al. i. al. s. a., NH4OH d. a. i. al., et., act.

1.837 3.715

gn. pl. scarlet red cr.

1.145° cc

d.

d.

8.9020

1.3117° 3.544

2.10679° i. s. i. sl. s. 2.1370°

i.

volt.

silv. met., cb.

lq. yel., delq. gn., delq., mn., 1.57

356.9 3700

s. aq. reg., Hg(NO3)2; sl. s. HNO3, HCl; i. al., etc. s. KI; i. al. s. HNO3; i. al., et. s. h. ac.; i. alk., dil. HCl, NH3 s. H2SO4, HNO3 s. HNO3; i. HCl s. h. conc. H2SO4; i. HCl, HF, NH3, dil. H2SO4, Hg s. HCl, H2SO4, NH4OH, al., et. s. HNO3, H2SO4; v. sl. s. al., et. s. HNO3, H2SO4; sl. s. al., et. s. HNO3, H2SO4; i. abs. al., et. s. a., NH4OH s. H2SO4, aq. reg. s. alk. sulfides s. alk. sulfides; i. NH3 s. NH4OH, H2SO4; i. NH s. a., NH4OH, NH4, salts

−25 subl.

43751mm 973

−4H2O, 200 subl. 250

d.

gn. cr. bk. lt. gn. gn., mn.

2.154 4.36 2.05

d. d. d. 56.7

gn.-bk., cb., 2.37 red yel., mn. yel., cb.

7.45 1.87511° 3.68

Forms Ni2O3 at 400 −H2O, 100 −SO3, 840

136.7

s. i. v. sl. s. 243.00° v. s. i. s. 27.20°

i. 76.7100°

*Usual commercial form. †See also Tables 2-28 and 2-280. 2-19

Physical Properties of the Elements and Inorganic Compounds (Continued )

TABLE 2-1

Name Nickel (Cont.) sulfate

Formula NiSO4·6H2O*

sulfate (morenosite) Nitric acid Nitric acid Nitric acid Nitro acid sulfite Nitrogen

NiSO4·7H2O HNO3 HNO3·H2O HNO3·3H2O NO2HSO3 N2

Nitrogen oxide, mono- (ous)

N2O

oxide, di- (ic)

NO or (NO)2

oxide, tri-

N2O3

oxide, tetra- (per- or di-)

NO2 or (NO2)2

Formula weight

Color, crystalline form and refractive index

262.85

gn. mn. or blue, tet., 1.5109 gn., rhb., 1.4893 col. lq. col. lq. col. lq. col., rhb. col. gas or cb. cr.

280.86 63.02 81.03 117.06 127.08 28.02 44.02

col. gas

30.01 (60.02) 76.02

col. gas red-brn. gas or blue lq. or solid yel. lq., col. solid, red-brn. gas wh., rhb.

oxide, penta-

N2O5

46.01 (92.02) 108.02

oxybromide oxychloride

NOBr NOCl

109.92 65.47

brn. lq. red-yel. lq. or gas

Nitroxyl chloride Osmium chloride, dichloride, trichloride, tetraOxygen

NO2Cl Os OsCl2 OsCl3 OsCl4 O2

81.47 190.2 261.11 296.57 332.03 32.00

yel.-brn. gas blue, hex. gn., delq. brn., cb. red-yel. nd. col. gas or hex. solid

Ozone

O3

Palladium bromide (ous) chloride chloride cyanide hydride Palladous dichlorodiammine Perchloric acid Perchloric acid Perchloric acid Periodic acid Periodic acid Permanganic acid Permolybdic acid Persulfuric acid Phosphamic acid Phosphatomolybdic acid Phosphine Phosphonium chloride 2-20

48.00

col. gas

Pd

106.70

silv. met., cb.

PdBr2 PdCl2 PdCl2·2H2O Pd(CN)2

266.53 177.61 213.65 158.74

brn. brn., cb. brn. pr. yel.

Pd2H Pd(NH3)2Cl2 HClO4 HClO4·H2O HClO4·2H2O* 73.6% anh. HIO4 HIO4·2H2O HMnO4 HMoO4·2H2O H2S2O8 PONH2·(OH)2 H7P(Mo2O7)6·28H2O PH3

214.41 211.68 100.46 118.48 136.50

met. red or yel., tet. unstable, col. lq fairly stable nd. stable lq., col.

PH4Cl

191.93 227.96 119.94 196.99 194.14 97.02 2365.88 34.00 70.47

wh. cr. delq., mn. exists only in solution wh. cr. hyg. cr. cb. yel. cb. col. gas wh., cb.

Specific gravity 2.07 1.948 1.502

Melting point, °C

1.448

18°

Solubility in 100 parts Cold water

tr. 53.3

−6H2O, 280

13150°

98–100 −42 −38 −18.5 73 d. −209.86

−6H2O, 103 86

0°

Hot water 280100° 30°

117.8 ∞ ∞ ∞

−195.8

63.5 ∞ ∞ 263−20° d. 2.350° cc

−90.7

130.520° cc

−151

7.340° cc

60.8224° cc 0.0100°

3.5

s.

−9.3

21.3

d.

1.026−252.5° 0.808−195.8° 12.50° (D) lq. 1.226−89° −102.3 1.530 (A) lq. 1.269−150.2° −161 1.0367 (A) 1.4472° −102 20°

Boiling point, °C

1.5520° cc

Other reagents v. s. NH4OH, al. s. al. expl. with al. d. al. d. al. s. H2SO4 sl. s. al. s. H2SO4, al. 26.6 cc al.; 3.5 cc H2SO4; s. aq. FeSO4 s. a., et. s. HNO3, H2SO4, chl., CS2

1.63

30

47

s.

>1.0 1.417−12° 2.31 (A) lq. 1.3214° 22.4820°

−55.5 −64.5

−2 −5.5

d. d.

<−30 2700

5 >5300

1.14−188° 1.426−252.5° 1.1053 (A) 1.71−183° 3.03−80° 1.658 (A) 12.020° 111550°

−218.4

−183

d i. s. d. sl. s. s. d. 4.890° cc

−251

−112

0.4940° cc

060° cc

s. oil turp., oil cinn.

1555

2200

i.

i.

i. s. s. i.

i. s. s. i.

s. aq. reg., h. H2SO4; i. NH3 s. HBr s. HCl, act., al. s. HCl, act., al. s. HCN, KCN, NH4OH; i. dil. a.

d. 560–600

500 d. d. 11.06 2.5 1.768 22° 4 1.88 1.71 25° 4

Forms HNO3 s. fuming H2SO4 i.

2.630° cc 1.7100° cc

sl. s. aq. reg., HNO3; i. NH3 s. NaCl, al., et. s. a., alk., al.; sl. s. et. s. HCl, al. sl. s. al., s. fused Ag

d.

−90°

lq. 0.746 1.146 (A)

−112 50 −17.8

1618mm d. 200

d. 138 d. 110

subl. 110

<60 d. 78 −132.5 46atm

28

s. s. s. v. s.

−25H2O, 140 −85

s. v. s. v. s. v. s. v. s. v. s. s. 2617° cc

subl.

d.

s. a., NH4OH s. al. v. s. d. v. s. d. v. s. d. 100°

i.

sl. s. al., et. d. al. i. al. s. HNO8 s. Cu2Cl2, al., et.

Phosphoric acid, hypoPhosphoric acid, meta-

H4P2O6 HPO3

161.99 79.99

cr. vitreous, delq.

2.2–2.5

55 subl.

d. 70

s. s.

Phosphoric acid, orthoPhosphoric acid, pyro-

H3PO4† H4P2O7

98.00 177.99

col., rhb. wh. nd.

1.834

42.35 61

−a H2O, 213

2340 80028°

Phosphorous acid, hypoPhosphorous acid, orthoPhosphorous acid, pyroPhosphorus, black Phosphorus, red Phosphorus, yellow

H3PO2 H3PO3 H4P2O6 P4 P4 P4

66.00 82.00 145.99 123.92 123.92 123.92

syrupy col. nd. rhombohedral red, cb. yel., hex., 2.1168

1.49318.8° 1.65121.2°

26.5 74 38

d. d. 200 d. 130 ign. in air, 400 ign. in air, 725 280

∞ 307.30° d. i. i. 0.0003

PCl3 PCl5

137.35 208.27

col., fuming lq. delq., tet.

−111.8 148 under pressure subl. 250 2

75.95760mm subl. 160

d. d.

21.4520° lq. 191755°

1755

4300

chloride, trichloride, pentaoxide, pentaoxychloride Phosphotungstic acid Platinum

P2O5 POCl2 P2O5·2WO3·42H2O Pt

141.96 153.35 3681.67 195.23

wh., delq., amor. col., fuming lq. yel.-gn. cr. silv. met., cb.

18.2°

2.69 2.2020° 1.8220°; lq. 1.74544.5° 1.574 20.8° 4 solid 1.6; 3.60295° (A) 2.387 1.675

59043atm 44.1; ign. 34

26°

Forms H3PO4 d. s. i.

107.2760mm

25°

45062° Forms H3PO4 v. s. Forms H3PO4 ∞ 73040°

i. CS2 s. alk.; i. CS2, NH3, et. 0.4 al.; 100010° CS2; 1.50°, 1081° bs.; s. NH3 s. et., chl., CS2 s. CS2, C6H5COCl

v. s.

s. H2SO4; i. NH3, act. d. al. s. al., et. s. aq. reg., fused alk.

i.

PtCl4

337.06

brn.

d. 370

140

v. s.

chloride (ous)

PtCl2

266.14

brn.

5.8711°

d. 581

i.

i.

PtCl4·8H2O Pt(CN)2 Pu Pu K

481.19 247.27 238 239 39.10

red, mn. yel.-brn.

2.43

−4H2O, 100

v. s. i.

v. s. i.

d.

KC2H3O2 KH(C2H3O2)2 K2(AlO2)2·3H2O KNH2 KH2AsO4 KAu(CN)4·1·5H2O KAu(CN)2 KHCO3 KHSO4 KBrO3 KBr K2CO3 K2CO3·2H2O 2K2CO3·3H3O KClO3 KCl K2PtCl6 K2CrO4 KCNO KCN K2Cr2O7 K3Fe(CN)6 K4Fe(CN)6·3H2O KHCO2 KH KHS KOH KIO3 KI

98.14 158.19 250.18 55.12 180.02 367.39 288.33 100.11 136.16 167.01 119.01 138.20 174.23 330.45 122.56 74.56 486.16 194.20 81.11 65.11 294.21 329.25 422.39 84.11 40.10 72.16 56.10 214.02 166.02

wh. pd. delq. nd. or pl. cr. yel.-grn. col., tet., 1.5674 pl. rhb. mn., 1.482 rhb., or mn., 1.480 trig. col., cb., 1.5594 wh., delq. pd., 1.531 rhb. mn. col., mn., 1.5167 col., cb., 1.4904 yel., cb., 1.825 yel., rhb., 1.7261 wh., tet. wh., cb., delq., 1.410 red, tri. red, mn. pr., 1.5689 yel., mn., 1.5772 col., rhb. cb., 1.453 wh., delq., rhb. wh., delq., rhb. col., mn. wh., cb., 1.6670

Forms KOH 39690°

acetate acetate, acid aluminate amide arsenate (monobasic) auricyanide aurocyanide bicarbonate bisulfate bromate bromide carbonate carbonate carbonate chlorate chloride (sylvite) chloroplatinate chromate (tarapacaite) cyanate cyanide dichromate ferricyanide ferrocyanide formate hydride hydrosulfide hydroxide iodate iodide

silv. met., cb.

238

Produced by deuteron bombardment on U Produced by neutron bombardment on U238 0.8620° 62.3 760 lq. 0.8342° 1.8 292 148 d. 200 2.867 2.17 2.35 3.2717.5° 2.7525° 2.29 2.043 2.13 2.32 1.988 3.499 2.73218° 2.048 1.5216° 2.69 1.84 1.85317° 1.91 0.80 2.0 2.044 3.89 3.13

338 288 d. 200 d. 100–200 210 370 d. 730 891 368 790 d. 250 975 634.5 398 d. −3HO2, 70 167.5 d. 455 380 560 723

subl. 400

d. 1380 d. d. 400 1500

d. d. 1320 1330

2170° d. s. d. 18.876° s. 14.3 22.40° 36.30° 3.110° 53.50° 105.50° 1830° 129.40° 3.30° 27.60° 0.740° 58.00° s. s. 4.90° 334.4° 27.812.2° 33118° d. s. 970° 4.730° 127.50°

s. al. v. s. al., et.

i. i. sl. s.

chloride (ic)

chloride (ic) cyanide (ous) Plutonium Plutonium Potassium

i. lq. CO2

d. v. s. v. s. 200100° 6060° 121.6100° 49.75100° 104100° 156100° 331100° 268100° 57100° 56.7100° 5.2100° 75.6100° d. 122.2108.8° 80100° 77.5100° 90.696.8° 65790° s. d. 178100° 32.2100° 208100°

s. al., act.; sl. s. NH2; i. et. s. HCl, NH4OH; sl. s. NH3; i. al., et. s. al., et. i. alk. s. a., al., Hg 33 al.; i. et. s. ac. s. alk.; i al. d. al.; 3.625° NH3 i. al. s. al. sl. s. al.; i. et. i. satd. K2CO3, al. d. al. sl. s. al.; i. act. sl. s. al., et. i. al. 0.83 al.; s. alk. s. al., alk. i. al., et. i. al. v. sl. s. al. s. gly.; 0.919.5° al.; 1.3 h. al. i. al. s. act.; sl. s. al.; i. NH3 s. act.; i. NH3, al., et. sl. s. al.; i. et. i. et., bz., CS2 s. al. v. s. al., et.; i. NH3 s. KI; i. al., NH3 420° al.; s. NH3; sl. s. et.

*One commercial form 70 to 72 per cent. †Common commercial form 85 per cent H3PO4 in aqueous solution. 2-21

TABLE 2-1

Physical Properties of the Elements and Inorganic Compounds (Continued )

Formula

Formula weight

Color, crystalline form and refractive index

Potassium (Cont.) iodide, triiodoplatinate manganate metabisulfite nitrate (saltpeter) nitrite oxalate oxalate, acid oxalate, acid oxide perchlorate permanganate persulfate phosphate, monobasic

KI3 K2PtI6 K2MnO4 K2S2O5 KNO3 KNO2 K2C2O4·H2O KHC2O4* KHC2O4·a H2O K2O KClO4 KMnO4 K2S2O3 KH2PO4

419.86 1034.94 197.12 222.31 101.10 85.10 184.23 128.12 137.13 94.19 138.55 158.03 270.31 136.09

phosphate, dibasic phosphate, tribasic phosphate, metaphosphate, metaphosphate, pyrophthalate, acid platinocyanide silicate silicate, tetrasulfate (arcanite) Potassium sulfate, pyrosulfide, monosulfite sulfite, acid tartrate tartrate, acid thiocyanate

K2HPO4 K3PO4 KPO3 K4P4O12·2H2O K4P2O7·3H2O KHC8H4O4 K2Pt(CN)4·3H2O K2SiO3 K2Si4O9·H2O K2SO4 K2S2O7 K2S·5H2O K2SO3·2H2O KHSO3 K2C4H4O6·a H2O KHC4H4O6* KCNS

174.18 212.27 118.08 508.34 384.39 204.22 431.54 154.25 352.45 174.25 254.31 200.33 194.28 120.16 235.27 188.18 97.17

thiosulfate thiosulfate Praseodymium Radium bromide Radon (Niton)

K2S2O3 3K2S2O3·H2O Pr Ra RaBr2 Rn

190.31 588.95 140.92 226.05 385.88 222.0

dark blue, delq., mn. cb. gn., rhb. mn., pl. col., rhb., 1.5038 pr. wh., mn. mn., 1.545 trimetric wh., cb. col., rhb., 1.4737 purple, rhb. wh., tri., 1.4669 col., delq., tet., 1.5095 wh., delq. wh., rhb. wh. pd. amor. delq. wh. cr. yel., rhb., 1.62 hyg. 1.521 rhb., 1.530 col., rhb., 1.4947 col. rhb., delq. wh., rhb. wh., mn. col., mn., 1.526 col., rhb. col., delq., mn., 1.660 col., cb. delq., mn. yel. wh., met. wh., mn. gas

Rhenium

Re

186.31

hex.

Rhodium chloride chloride Rubidium

Rh RhCl3 RhCl3·4H2O Rb

102.91 209.28 281.35 85.48

gray-wh., cb. red dark red silv. wh.

Ruthenium Ruthenium Samarium Scandium Selenic acid Selenic acid Selenium Selenium

Ru Ru Sm (also Sa) Sc H2SeO4 H2SeO4·H2O Se8 Se8

101.70 101.70 150.43 45.10 144.98 162.99 631.68 631.68

bk., porous gray, hex.

Name

2-22

hex. pr. nd. red pd., amor., 2.92 gray, trig., 3.00; red, hex.

Specific gravity 3.498 5.18 2.1110.6° 1.915 2.13 2.0 2.32 20° 4 2.524 11° 4 2.703 2.338 2.56417° 2.25814.5° 2.26414.5° 2.33 1.63 2.4516° 2.417 2.662 2.277

1.98 1.956 1.886 2.23 6.520° 5? 5.79 lq. 5.5; 111 (D)

Melting point, °C 45 d. 190 d. 150 tr. 129; 333 297 d. d. d.

Boiling point, °C d. 225

d. 400 d. 350

d. 400 d. <240 d. <100 256 d. 1340 tr. 450; 798 −2H2O, 100 −2H2O, 180 d. 976 d. 400 tr. 588 300 60 d. d. 190

1320 d. −3H2O, 300

−3H2O, 150 d.

172.3 d. 400 −H2O, 180 940 960 728 −71

d. 500

Solubility in 100 parts Cold water v. s. s. d. 250° 13.30° 2810° 28.70° 14.350° 2.20° Forms KOH 0.750° 2.830° 1.770° 14.80°

Hot water

s. KI, al. 12094° 246100° 413100° 83.2100° 48.1100° 51.5100° v. s. 21.8100° 32.3575° 1040° 83.590°

3325° 193.125° s. s. s. 10.225° sl. s. s. s. 7.350° s. s. 100 45.515° 12.517.5° 0.370° 1770°

v. s. v. s. s. 83 v. s. 36 v. s. s. s. 24.1100° d.

96.10°

311.290°

>100 91.575° 278100° 6.1100° 21720°

d. 1140 subl. 900 −62

1955 d. 450

>2500 subl. 800

lq. 1.47588.5; 1.5320° 8.6 12.220° 7.7 2.5? 2.950 15° 4 2.627 15° 4 4.2625° 4.80; 4.50

38.5 >1950 2450 >1300 1200 58 26 50 220

s. KOH sl. s. al.; i. et. 0.130° al.; i. et. v. s. NH3; sl. s. al.

s. al., et. 0.10520° m. al.; i. et. s. H2SO4; d. al. i. al. i. al. sl. s. al. i. al. s. a. i. al. s. al., et. i. al. i. al. i. al., act., CS2 s. al., gly.; i. et. sl. s. al.; i. NH3 i. abs. al. sl. s. al. s. a., alk.; i. al., ac. 20.822° act.; s. al. i. al.

d. d. +H2 7020° 510° cc

s. 8.560° cc

3440 12.5

Other reagents

d. a. s. al. i. HF, HCl; s. H2SO4; HNO3 sl. s. aq. reg., a. v. sl. s. alk.; i. aq. reg., a. s. HCl, al.; i. et. s. a., al.

i. i.

700

i. i. v. s. d.

>2700

i. i.

i. i.

sl. s. aq. reg., a.

2400 260 205 688 688

130030° v. s. i. i.

∞60°

s. H2SO4; d. al.; i. NH3

i. i.

s. CS2, H2SO4, CH2I2 s. CS2, H2SO4

Selenium Selenous acid Silicic acid, metaSilicic acid, orthoSilicon, crystalline

Se8 H2SeO3 H2SiO3 H4SiO4 Si

631.68 128.98 78.08 96.09 28.06

4.825° 3.004 15° 4 2.1–2.3 1.57617° 2.420°

cr. brn., amor. blue-bk., trig., 2.654 lf. or lq. col., fuming lq., 1.412 gas col. gas iridescent, amor. col., cb. or tet., 1.487

217 d.

688

2600

i. 900° i. sl. s. i.

i. 40090° i. sl. s. i.

1420

2.0–2.5

2600

i.

i.

2 3.17 1.580° 1.50

>2700 −1 −70

2600 subl. 2200 144760mm 57.6

i. i. d. d.

i. i.

3.57 (A) lq. 0.68−185° 2.2 2.32

−95.7 −185 1600–1750 1710

−651810mm −112760mm subl. 1750 2230

v. s. d. i. i. i.

Silicon, graphitic

Si

Silicon, amorphous carbide chloride, trichloride, tetra-

Si SiC Si2Cl6 SiCl4

28.06 40.07 268.86 169.89

SiF4 SiH4 SiO2·xH2O SiO2

104.06 32.09

SiO2 SiO2 SiO2 Ag AgBr

60.06 60.06 60.06 107.88 187.80

hex., 1.5442 trig., rhb., 1.469 silv. met., cb. pa. yel., cb., 2.252

2.20 2.65020° 2.26 10.520° 6.473 25° 4

tr. <1425 tr. 1670 960.5 434

2230 2230 2230 1950 d. 700

i. i. i. i. 0.0000220°

i. 0.00037100°

carbonate chloride (cerargyrite)

Ag2CO3 AgCl

275.77 143.34

yel. pd. wh., cb., 2.071

6.077 5.56

218 d. 455

1550

0.00320° 0.00008910°

0.05100° 0.00217100°

cyanide nitrate (lunar caustic) Sodium

AgCN AgNO3 Na

133.90 169.89 22.997

wh., 1.685 col., rhb., 1.744 silv. met, cb.

3.95 4.352 19° 4 0.9720°

−(CN)2, 320 212 97.5

444 d. 880

acetate acetate aluminate amide Sodium ammonium phosphate antimonate, metaarsenate arsenate, acid (monobasic) arsenate, acid (dibasic) arsenate, acid (dibasic) arsenite, acid benzoate bicarbonate bifluoride bisulfate bisulfite borate, tetraborate, tetra

NaC2H3O2 NaC2H3O2·3H2O NaAlO2 NaNH2 NaNH4HPO4·4H2O 2NaSbO3·7H2O Na3AsO4·12H2O NaH2AsO4·H2O Na2HAsO4·7H2O* Na2HAsO4·12H2O Na2HAs03 NaC7H5O2 NaHCO3 NaHF2 NaHSO4 NaHSO3 Na2B4O7 Na2B4O7·5H2O

82.04 136.09 81.97 39.02 209.09 511.63 424.09 181.94 312.02 402.10 169.91 144.11 84.01 62.00 120.06 104.06 201.27 291.35

wh., mn., 1.464 wh., mn. amor. olive gn. col., mn. cb. hex., 1.4589 rhb., 1.5535 col., mn., 1.4658 mn., 1.4496 col. col. cr. wh., mn., 1.500 col. cr. col., tri. col., mn., 1.526

1.528 1.45

324 58 1650 210 79 d.

col., rhb., 1.461

2.742 1.48 2.367 1.815

Na2B4O7·10H2O*

381.43

wh., mn., 1.4694

1.73

fluoride hydride (silane) oxide, di- (opal) oxide, di- (cristobalite) oxide, di- (lechatelierite) oxide, di- (quartz) oxide, di- (tridymite) Silver bromide (bromyrite)

borate, tetra- (borax)

28.06

steel gray hex. amor., 1.41 amor. gray, cb., 3.736

60.06

1.574

−3H2O, 120 400

1.759 2.535 1.871 1.72 1.87

86.3 d. 100 125 28

2.20

−CO2, 270 d. >315 d. 741

d., −H2O

75

−10H2O, 200

17.5°

bromate bromide bromide

NaBrO3 NaBr NaBr·2H2O

150.91 102.91 138.95

col., cb. col., cb., 1.6412 col., mn.

3.339 3.20517.5° 2.176

381 755 50.7

carbonate (soda ash) carbonate

Na2CO3 Na2CO3·H2O

106.00 124.02

2.533 1.55

851 −H2O, 100

carbonate carbonate (sal soda)

Na2CO3·7H2O Na2CO3·10H2O

232.12 286.16

wh. pd., 1.535 wh., rhb., 1.506– 1.509 rhb. or trig. wh., mn., 1.425

1.51 1.46

d. 35.1

−7H2O, 100 −12H2O, 100

0.00002220° 1220° d., forms NaOH 46.520° v. s. s. d. 16.7 0.03112.8° 26.717° s. 6115° 5.590.1° v. s. 62.525° 6.90° 3.720° 500° sl. s. 1.30° 2262° (anh.)

i. i. i. i.

952100° 170100° v. s. v. s. 100

s. HF; i. alk. s. HF; i. alk. s. HF; i. alk. s. HNO3, h. H2SO4; i. alk. 0.5118° NH4OH; s. KCN, Na2S2O3 s. NH4OH, Na2S2O3; i. al. s. NH4OH, KCN; sl. s. HCl s. NH4OH, KCN, HNO3 s. gly.; v. sl. s. al. i. bz.; d. al. 2.118° al. 7.825° abs. al. i. al. d. al. i. al. sl. s. al., NH4 salts; i. ac. 1.67 al., 5015° gly. sl. s. al. sl. s. al. 2.325°, 8.378° al. i. al.

1390

27.5 9020° 79.50° (anh.)

d.

7.10° s. s. 21.50°

s. 23830°

0°

s. HNO3, al., et. i. al., et.; d. KOH s. HF, h. alk., fused CaCl2 s. HF; i. alk.

v. s. 140.730° 76.9100° 16.460° s. 100100° s. 8.7940° 52.3100° (anh.) 20.380° (anh.) 90.9100° 121100° 118.380° (anh.) 48.5104° s.

1.30.5 (anh).

i. CS2; s. H2SO4 v. s. al.; i. NH3 s. alk.; i. NH4Cl s. alk.; i. NH4Cl s. HNO3 + HF, Ag; sl. s. Pb, Zn; i. HF s. HNO3 + HF, fused alk.; i. HF. s. HF, KOH s. fused alk.; i. a. d. alk. d. conc. H2SO4, al.

d. al.; i. NH3 i. al., act. i. al. s. gly.; i. abs. al. i. al. sl. s. al. sl. s. al. i. al., et. s. gly.; i. al., et. i. al.

*Usual commercial form. 2-23

Physical Properties of the Elements and Inorganic Compounds (Continued )

TABLE 2-1

Name Sodium ammonium phosphate (Cont.) carbonate, sesqui- (trona) chlorate

Formula

Na3H(CO3)2·2H2O NaClO3

Formula weight

Specific gravity

NaCl Na2CrO4 Na2CrO4·10H2O 2Na3C6H5O7·11H2O NaCN Na2Cr2O7·2H2O

58.45 162.00 342.16 714.36 49.02 298.05

ferricyanide ferrocyanide

Na3Fe(CN)6·H2O Na4Fe(CN)6·10H2O

298.97 484.11

red, delq. yel., mn.

1.458

fluoride (villiaumite) formate hydride

NaF NaHCO2 NaH

tet., 1.3258 wh., mn. silv. nd., 1.470

2.79 1.919 0.92

992 253 d. 800

hydrosulfide hydrosulfide hydrosulfite hydroxide hydroxide hypochlorite iodide iodide lactate nitrate (soda niter) nitrite

NaSH·2H2O NaSH·3H2O Na2S2O4·2H2O NaOH NaOH·3a H2O NaOCl NaI* NaI·2H2O NaC3H5O3 NaNO3 NaNO2

2.130

d. 22 d. 318.4 15.5 d. 651

1300

2.257 2.1680°

d. 308 271

d. 380 d. 320

oxide

Na2O

2.27

subl.

2-24

NaBO3·H2O NaClO4 NaClO4·H2O Na2O2* Na2O2·8H2O NaH2PO4·H2O* NaH2PO4·2H2O Na2HPO4·7H2O Na2HPO4·12H2O Na3PO4 Na3PO4·12H2O* Na4P4O12 Na4P2O7* Na4P2O7·10H2O Na2H2P2O7 Na2H2P2O7·6H2O NaKC4H4O6·4H2O Na2SiO3 Na2SiO3·9H2O Na4SiO4 Na2SiF6 Na2SnO3·3H2O Na2SO4 Na2SO4

92.10 110.11 210.15 40.00 103.06 74.45 149.92 185.95 112.07 85.01 69.01 61.99 99.83 122.45 140.47 77.99 222.12 138.01 156.03 268.09 358.17 163.97 380.16 407.91 265.95 446.11 221.97 330.07 282.23 122.05 284.20 184.05 188.05 266.74 142.05 142.05

col., delq., nd. rhb. col. cr. wh., delq. mn. pa. yel., in soln. only col., cb., 1.7745 col., mn. col., amor. col., trig., 1.5874 pa. yel., rhb. wh., delq. wh. pd. rhb., 1.4617 hex. yel.-wh. pd. wh., hex. col., rhb., 1.4852 col., rhb., 1.4629 col., mn., 1.4424 col., mn., 1.4361 wh. wh., trig., 1.4458 col. wh. mn., 1.4525 col., mn., 1.510 col., mn., 1.4645 rhb., 1.493 col., rhb., 1.520 rhb. col., hex., 1.530 wh., hex., 1.312 hex. tablets col., rhb., 1.477 col., mn.

2.163 2.723 1.483 1.857 23.5° 4 18°

2.52

3.6670° 2.448

2.02 2.805 2.040 1.91 1.679 1.52 2.53717.5° 1.62 2.476 2.45 1.82 1.862 1.848 1.790

2.679 2.698

d. 248

Boiling point, °C

chloride chromate chromate citrate cyanide dichromate

42.00 68.01 24.005

2.112 2.49015°

Melting point, °C

wh., mn., 1.5073 wh., cb., or trig., 1.5151 col., cb., 1.5443 yel., rhb. yel., delq., mn. wh., rhb. wh., cb., 1.452 red, mn., 1.6994

perborate perchlorate perchlorate peroxide peroxide phosphate, monobasic phosphate, monobasic phosphate, dibasic phosphate, dibasic phosphate, tribasic phosphate, tribasic phosphate, metaphosphate, pyrophosphate, pyrophosphate (pyrodisodium) phosphate (pyrodisodium) potassium tartrate silicate, metaSodium silicate, metasilicate, orthosilicofluoride stannate sulfate (thenardite) sulfate

226.05 106.45

Color, crystalline form and refractive index

800.4 392 19.9 −11H2O, 150 563.7 −2H2O, 84.6; 356 (anh.)

d. 40 482 d. d. 130 d. d. 30 −H2O, 100 60 d. 34.6 1340 73.4 616 d. 988 d. d. 220 70 to 80 1088 47 1018 d. d. 140 tr. 100 to mn. tr. 500 to hex.

d. 1413 d. 1496 d. 400

Solubility in 100 parts Cold water

130° 790°

42100° 230100° 0°

1390

d. 200 −12H2O, 180 −11H2O, 100

−4H2O, 215 −6H2O, 100

100°

35.7 320° v. s. 9125° 4810° 2380°

39.8 126100° ∞ 250100° 8235° 50880°

18.90° 17.920° (anh.)

67100° 6398.5° (anh.) 5100° 160100°

40° 440° d. d.

Hot water

s. s. 2220° 420° s. 260° 158.70° v. s. v. s. 730° 72.10°

s. s. d. 347100° v. s. 15856° 302100° v. s. v. s. 180100° 163.2100°

Forms NaOH sl. s. 1700° 20915° s. d. s. d. 710° 91.10° 18540° 4.30° 4.50° 28.315° s. 2.260° 5.40° 4.50° 6.90° 260° s. v. s. s. 0.440° 500° 50° 48.840°

d. 320100° 28450° d. d. 39083° 30840° 2000100° 76.730° 77100° ∞ s. 4596° 93100° 2140° 3640° 6626° s. d. v. s. s. 2.45100° 6750° 42100° 42.5100°

Other reagents

s. al. sl. s. al.; i. conc. HCl sl. s. al. i. al. s. NH3; sl. s. al. i. al. i. al. v. sl. s. al. sl. s. al.; i. et. i. bz., CS2, CCl4, NH3; s. molten metal s. al.; d. a. s. al.; d. a. i. al. v. s. al., et., gly.; i. act. v. s. al., act. v. s. NH3 s. al.; i. et. s. NH3; sl. s. gly., al. 0.320° et.; 0.3 abs. al.; 4.420° m. al.; v. s. NH3 d. al. s. gly., alk. s. al.; 51 m. al.; 52 act.; i. et. s. al. s. dil. a. i. al. i. al. i. CS2 s. a., alk. d. a. i. al., NH3 sl. s. al. i. Na or K salts, al. 2918°, a N NaOH i. al. i. al., act. i. al. d. HI; s. H2SO4

sulfate sulfate sulfate (Glauber’s salt) sulfide, monosulfide, tetrasulfide, pentasulfite sulfite tartrate thiocyanate thiosulfate thiosulfate (hypo) tungstate tungstate tungstate, parauranate vanadate vanadate, pyroStannic chloride

Na2SO4 Na2SO4.7H2O Na2SO4.10H2O Na2S Na2S4 Na2S5 Na2SO3 Na2SO3.7H2O Na2C4H4O6.2H2O NaCNS Na2S2O3 Na2S2O3.5H2O* Na2WO4 Na2WO4.2H2O* Na6W7O24.16H2O Na2UO4 Na3VO4.16H2O Na4V2O7 SnCl4

142.05 268.17 322.21 78.05 174.23 206.29 126.05 252.17 230.10 81.08 158.11 248.19 293.91 329.95 2097.68 348.06 472.20 305.89 260.53

col., hex. tet. col., mn., 1.396 pink or wh., amor. yel., cb. yel. hex. pr., 1.565 mn. rhb. delq., rhb., 1.625 mn. mn. pr., 1.5079 wh., rhb. wh., rhb. wh., tri. yel. col. nd. hex. col., fuming lq.

884 1.464 1.856 2.633 15° 4 1.561 1.818

2.226

866 (anh.) 654 −30.2

7.0

1127

SnO2

150.70

wh., tet., 1.9968

346.85

col., delq., hex.

SnBr2 SnCl2 SnCl2.2H2O* SnSO4 Sr

278.53 189.61 225.65 214.76 87.63

yel., rhb. wh., rhb. wh., tri. wh. cr. silv. met.

2.71

acetate carbonate (strontianite) chloride chloride hydroxide hydroxide

Sr(C2H3O2)2 SrCO3 SrCl2 SrCl2.6H2O* Sr(OH)2 Sr(OH)2.8H2O*

205.72 147.64 158.54 266.64 121.65 265.77

wh. cr. wh., rhb., 1.664 wh., cb., 1.6499 wh., rhb., 1.5364 wh., delq. col., tet., 1.499

2.099 3.70 3.052 1.93317° 3.625 1.90

nitrate nitrate oxide (strontia)

Sr(NO3)2* Sr(NO3)2.4H2O SrO

211.65 283.71 103.63

col., cb., 1.5878 wh., mn. col., cb., 1.870

2.986 2.2 4.7

SrO2 SrO2.8H2O SrSO4 Sr(HSO4)2 NH2SO3H S S8 S8 S2Br2 S2Cl2 SCl2 SCl4 SO2

119.63 263.76 183.69 281.77 97.09 32.06 256.48 256.48 223.95 135.03 102.97 173.89 64.06

wh. pd. wh. cr. col., rhb., 1.6237 col., granular wh., rhb. pa. yel. pd., 2.0–2.9 pa. yel., mn. pa. yel., rhb. red, fuming lq. red-yel. lq. dark red fuming lq. yel.-brn. lq. col. gas

oxide, tri-(β) Sulfuric acid Sulfuric acid

SO3 (SO3)2 H2SO4* H2SO4.H2O

80.06 160.12 98.08 116.09

col. pr. col., silky, nd. col., viscous lq. pr. or lq.

d.

d. 48.0 692 −2H2O, 100 −16H2O, 300

Sn(SO4)2.2H2O

oxide, tri-(α)

275 251.8 d. −7H2O, 150 287

sulfate

peroxide peroxide sulfate (celestite) sulfate, acid Sulfamic acid Sulfur, amorphous Sulfur, monoclinic Sulfur, rhombic Sulfur bromide, monochloride, monochloride, dichloride, tetraoxide, di-

−10H2O, 100

1.667 1.685 4.179 3.245 3.98714°

oxide (cassiterite)

Stannous bromide chloride chloride (tin salt) sulfate Strontium

32.4

5.1217° 15.5°

2.6

3.96 2.03 12° 4 2.046 1.96 2.07 2.635 1.687 1.621 15° 15 lq., 1.4340°; 2.264 (A) lq., 1.923; 2.75 (A) 1.9720° 1.834 18° 4 1.842 15° 4

215.5 246.8 37.7 −SO2, 360 800 149760atm 873 −4H2O, 61 375 −7H2O in dry air 570

114.1

620 623 d. 1150 d. −CO2, 1350 −6H2O, 100

19.420° 44.90° 3615° 15.410° s. s. 13.90° 34.72° 296° 11010° 500° 74.70° 57.580° 880° 8 i. v. s. s. s. i.

i. d.

s. 83.90° 118.70° 1919° d.

d. 269.815° ∞ 18100° Forms Sr(OH)2 36.497° 0.065100° 100.8100° 19840° 21.83100° 47.7100°

36.90° 0.001118° 43.50° 1040° 0.410° 0.900°

444.6 444.6 444.6 540.18mm 138 59 d. > −20 −10.0

16.83

44.6

d.

50 10.49 8.62

d. 340 290

Forms H2SO4 ∞ ∞

d. −8H2O, 100 1580 d. d. 205 d. 120 119.0 112.8 −46 −80 −78 −30 −75.5

d.

d.

v. s.

400° 62.20° Forms Sr(OH)2 0.00820° 0.01820° 0.01130° d. 200° i. i. i. d. d. d. d. 22.80°

2430

45.360° 202.626° 41234° 57.390° s. s. 28.384° 67.818° 6643° 225100° 23180° 301.860° 97100° 123.5100° d. i. d.

i. al. sl. s. al.; i. et. s. al. s. al. i. al., NH i. al. i. al. v. s. al. s. NH3; v. sl. s. al. sl. s. NH3; i. a., al. s. alk. carb., dil. a. i. al. i. al. s. abs. al., act., NH3; s. ∞ CS2 s. conc. H2SO4; i. alk.; NH4OH, NH3 s. dil. H2SO4, HCl; d. abs. al. s. C6H5N s. alk., abs. al., et. s. tart. a., alk., al. s. H2SO4 s. al., a. 0.2615° m. al. s. a., NH4 salts, aq. CO2 v. sl. s. act., abs. al.; i. NH3 s. NH4Cl s. NH4Cl; i. act.

10089° 12420°

s. NH3; 0.012 abs. al. i. HNO3 sl. s. al.; i. et.

d. d. 0.011432°

s. al., NH4Cl; i. act. s. al.; i. NH4OH sl. s. a.; i. dil. H2SO4, al. 1470° H2SO4 sl. s. al., act.; i. et. sl. s. CS2 s. CS2, al. 240°, 18155° CS2

4070° i. i. i.

s. CS2, et., bz. d. al. 4.550°

s. H2SO4; al., ac. s. H2SO4

∞ ∞

s. H2SO4 d. al. d. al.

*Usual commercial form. 2-25

Physical Properties of the Elements and Inorganic Compounds (Concluded )

TABLE 2-1

Name

Formula

Formula weight

Color, crystalline form and refractive index

Specific gravity

Melting point, °C

Boiling point, °C

Solubility in 100 parts Cold water

Hot water

Sulfuric acid Sulfuric acid, pyroSulfuric oxychloride Sulfurous oxybromide oxychloride Tantalum

H2SO4·2H2O H2S2O7 SO2Cl2 SOBr2 SOCl2 Ta

134.11 178.14 134.97 207.89 118.97 180.88

col. lq. cr. col. lq. or.-yel. lq. col. lq. bk.-gray, cb.

1.650 40° 1.920° 1.667 20° 4 2.6818° 1.638 16.6

−38.9 35 −54.1 −50 −104.5 2850

167 d. 69.1760mm 6840mm 78.8 >4100

∞ d. d. d. d. i.

∞

Tellurium

Te

127.61

met., hex.

(α) 6.24; (β) 6.00

452

1390

i.

i.

Terbium Thallium acetate chloride, monochloride, sesquichloride, trichloride, trisulfate (ic) sulfate (ous) sulfate, acid Thio, cf. sulfo or sulfur Thorium

Tb Tl TlC2H3O2 TlCl Tl2Cl3 TlCl3 TlCl3·4H2O Tl2(SO4)3·7H2O Tl2SO4 TlHSO4

159.20 204.39 263.43 239.85 515.15 310.76 382.83 823.07 504.84 301.46

blue-wh., tet. silky nd. wh., cb. yel., hex. hex. pl. nd. lf. col., rhb., 1.8671 trimorphous

11.85 3.68 7.00 5.9

303.5 110 430 400–500 25 37 −6H2O, 200 632 115 d.

1650 806 d. d. −4H2O, 100 d. d.

i. v. s. 0.210° 0.2615° v. s. 86.217° d. 2.700°

Th

232.12

cb.

11.2

1845

>3000

i.

i.

oxide, di- (thorianite) sulfate sulfate Thulium Tin

ThO2 Th(SO4)2 Th(SO4)2·9H2O Tm Sn

264.12 424.24 586.38 169.40 118.70

wh., cb.

9.69 4.22517° 2.77

>2800

4400

silv. met., tet.

7.31

231.85

2260

i. 0.740° sl. s. i. i.

5.2250° sl. s. i. i.

Tin

Sn

118.70

gray, cb.

5.750

Stable −163 to +18

2260

i.

i.

Tin salts, cf. stannic and stannous Titanic acid

H2TiO3

97.92

wh. pd.

i.

i.

i. d.

d.

s. s. i.

s. d. i.

i.

i.

Titanium chloride, di-

mn. pr.

Ti TiCl2

47.90 118.81

dark gray, cb. bk., delq.

chloride, trichloride, tetraoxide, di- (anatase)

TiCl3 TiCl4* TiO2

154.27 189.73 79.90

oxide, di- (brookite)

TiO2

vl., delq. col. lq. brn. or bk., tet., 2.534–2.564 brn. or bk., rhb., 2.586 col. if pure, tet., 2.615 gray-bk., cb.

oxide, di- (rutile) Tungsten carbide carbide

TiO2 W

79.90 79.90 183.92

6.77

17.5°

4.50

lq., 1.726 3.84

136.4

s. HCl, H2SO; sl. s. HNO3; i. HF, alk. s. h. H2SO4; i. alk.

s. HCl, H2SO4, dil. HNO3 h. aq KOH s. a., h. alk. solns. s. alk.; v. sl. s. dil. a.; i. al. s. a. i. CS2, et., chl. s. dil. HCl sl. s. alk.

1640 d.

<3000

i.

i.

s. H2SO4, alk.

3370

5900

i.

i.

6000 6000

i. i.

i. i.

i. i.

i. sl. s.

s. h. conc. KOH; sl. s. NH3, HNO3, aq. reg. s. F2; i. a. s. h. HNO3; sl. s. HCl, H2SO4 s. alk.; i. a. s. HF, alk., NH3

i.

i.

s. a., alk. carb.; i. alk.

i. d. i.

i.

s. a.; i. alk. d. a. s. HNO3, conc. H2SO4

18°

15.7 16.0618°

2777 2877

oxide, triTungstic acid (tungstite)

WO3 H2WO4

231.92 249.94

yel., rhb. yel., rhb. 2.24

7.16 5.5

Uranic acid

H2UO4

304.09

yel. pd.

5.92615°

wh. cr. cr. bk., rhb.

18.485 13° 4

>2130 −a H2O, 100; 1473 −H2O, 250 to 300 1133 2400 2176

2-26

s. al., et. s. al., et. s. dil. H2SO4

19.3

gray pd., cb. iron gray

238.07 512.14 270.07

d. d. 18.45100°

s. HNO3, H2SO4; i. NH3 v. s. al. sl. s. HCl; i. al., NH4OH

4.26

195.93 379.85

U U2C3 UO2

>3000

4.17

WC W2C

Uranium carbide oxide, di- (uraninite)

i. 1.8100° 1.9100°

v. sl. s. dil. H2SO4

−9H2O, 400

1800 Unstable in air d. 440 −30

i.

Other reagents d. al. d. al. s. ac.; d. al. s. bz., CS2, CCl4; d. act. s. bz., chl. s. fused alk., HF; i. HCl, HNO3, H2SO4 s. H2SO4, HNO3, KCN, KOH, aq. reg.; i. CS2

11.28 10.9

3500

i.

oxide (pitchblende) sulfate (ous) Uranyl acetate carbonate (rutherfordine) nitrate sulfate Vanadic acid, metaVanadic acid, pyroVanadium chloride, dichloride, trichloride, tetraoxide, dioxide, trioxide, tetraoxide, pentaoxychloride, monoVanadyl chloride chloride, dichloride, triWater†

U3O8 U(SO4)2·4H2O UO2(C2H3O2)2·2H2O UO2CO3 UO2(NO3)2·6H2O UO2SO4·3H2O HVO3 H4V2O7 V VCl2 VCl3 VCl4 V2O2 V2O3 V2O4 V2O5 VOCl (VO)2Cl VOCl2 VOCl3 H2O

842.21 502.25 424.19 330.08 502.18 420.18 99.96 217.93 50.95 121.86 157.23 192.78 133.90 149.90 165.90 181.90 102.41 169.36 137.86 173.32 18.016

Water, heavy Xenon

D2O Xe

20.029 131.30

olive gn. gn., rhb. yel., rhb. tet. yel., rhb., 1.4967 yel. cr. yel. scales pa. yel., amor. lt. gray, cb. gn., hex., delq. pink, tabular, delq. red lq. lt. gray cr. bk. cr. blue cr. red-yel., rhb. brn. pd. yel. cr. gn., delq. yel. lq. col. lq., 1.3330020°; hex. solid, 1.309 col. lq., 1.3284420° col. gas

Ytterbium Yttrium Zinc acetate acetate bromide carbonate

Yb Y Zn Zn(C2H3O2)2 Zn(C2H3O2)2·2H2O* ZnBr2 ZnCO3

173.04 88.92 65.38 183.47 219.50 225.21 125.39

dark gray, hex. silv. met., hex. mn. wh., mn., 1.494 rhb. wh., trig., 1.818

ZnCl2

136.29

wh., delq., 1.687, uniaxial col., rhb. col., rhb. cb.

chloride cyanide hydroxide iodide

Zn(CN)2 Zn(OH)2 ZnI2

117.42 99.40 319.22

nitrate oxide (zincite) oxide peroxide phosphide silicate

Zn(NO3)2·6H2O ZnO ZnO ZnO2 Zn3P2 ZnSiO3

297.49 81.38 81.38 97.38 258.10 141.44

sulfate (zincosite)

ZnSO4

sulfate sulfate sulfate (goslarite) sulfide (α) (wurzite) sulfide (β) (sphalerite)

ZnSO4·H2O ZnSO4·6H2O ZnSO4·7H2O* ZnS ZnS

161.44 179.46 269.54 287.55 97.44 97.44

sulfide (blende) sulfite Zirconium oxide, di- (baddeleyite) oxide, di- (free from Hf)

ZnS ZnSO3·2a H2O Zr ZrO2 ZrO2

97.44 190.48 91.22 123.22 123.22

col., tet. wh., hex., 2.004 wh., amor. yel. steel gray, cb. hex. or rhb.; glass, 1.650 wh., rhb., 1.669 col. mn. rhb., 1.4801 wh., hex., 2.356 wh., cb.; glass (?) 2.18–2.25 wh., granular mn. cb., pd. ign. easily yel. or brn., mn., 2.19 wh., mn.

7.31 15°

2.89 5.6 2.807 3.2816.5° 5.96 3.2318° 3.0018° 1.81630° 3.64 4.87 18° 4 4.399 3.357 18° 4 2.824 3.64 2.8813° 1.829 1.004° (lq.); 0.9150° (ice) 1.10720° lq., 3.06−109.1 2.7−140° 4.53 (A)

d. −4H2O, 300 −2H2O, 110 60.2 d. 100

118

1710

3000

d. −109 ign. 1970 1967 800

148.5755mm

d. 1750

d. in air

i. 2311° 9.217°

i. 963° d.

s. HNO3, H2SO4 s. dil. a. s. al., act.

170.30° 18.913.2° i. i. i. s. s. s. d. i. sl. s. i. 0.820° i. i. d. s. d.

∞60° 23025°

v. s. ac., al., et.; i. dil., alk. 4 al.; s. a. s. a., alk.; i. NH3 s. a., alk., NH4OH s. HNO3, H2SO4; i. aq., alk. s. al., et. s. abs. al., et. s. abs. al., et., chl., ac. s. a. s. HNO3, HF, alk. s. a., alk. s. a., alk.; i. abs. al. v. s. HNO3 s. HNO3 s. abs. al., dil. HNO3 s. al., et., ∞Br2 ∞ al.; sl. s. et.

i. d. d. i. s. i.

<−15 0

127.19 100

3.82 −140

101.42 −109.1

∞ 24.20° cc

∞ 7.350° cc

∞ al.; sl. s. et.

5.51 7.140 1.840 1.735 4.2194° 4.42

1490 419.4 242 237 394 −CO2, 300

2500 907 subl. in vac. −2H2O, 100 650

sl. d. i. 3025° 4025° 3900° 0.00115°

d. i. 44.6100° 66.6100° 670100°

2.91 25° 4

283

732

43225°

615100°

v. s. dil. a., h. KOH s. a., ac., alk. 2.825°, 16679° al. v. s. al. v. s. NH4OH, al., et. s. a., alk., NH4 salts; i. act., NH3 10012.5° al.; v. s. et.; i. NH3 s. KCN, NH3, alk.; i. al. s. a., alk., NH4OH s. a., al., NH3, aq. (NH4)2CO3 v. s. al. s. a., alk., NH4Cl; i. NH3

3.053 4.666 14.2° 4

d. 80 d. 125 446

2.065 14° 4 5.606 5.47 1.571 4.55 13° 4 3.52

36.4 >1800 >1800 expl. 212 >420 1437

3.74 15° 4 3.28 15° 4 2.072 15° 4 1.96616.5° 4.087 4.102 25° 4

d. 740 d. 238 −5H2O, 70 tr. 39 1850150atm tr. 1020

4.04 6.4 5.49 5.73

−2a H2O, 100 1700 2700

18°

624 −6H2O, 105

1100

−7H2O, 280 subl. 1185

d. 200 >2900 4300

0.0005 0.0005218° 4300°

sl. s.

324.5 0.0004218° 0.0004218° 0.0022 i. i.

∞36.4°

420° s. s. 115.20° 0.0006918° i.

61100° 89.5100° s. 653.6100° i. i.

sl. s. al.; s. gly.

i. 0.16 i. i. i.

i. d. i. i. i.

v. s. a.; i. ac. s. H2SO3, NH4OH; i. al. s. HF, aq. reg.; sl. s. a. s. H2SO4, HF s. H2SO4, HF

510100°

i. NH4OH; d. a. s. dil. a.

sl. s. al.; i. act.; NH3 sl. s. al.; i. act.; NH3 v. s. a.; i. ac. s. a.

*Usual commercial form. †Cf. special tables on water and steam, Tables 2-3, 2-4, 2-5, 2-185, 2-186 and 2-351 through 2-357. NOTE: °F = 9⁄ 5 °C + 32.

2-27

TABLE 2-2

Physical Properties of Organic Compounds* Abbreviations Used in the Table

(A), density referred to air al., ethyl alcohol amor., amorphous aq., aqua, water brn., brown bz., benzene c., cubic cc., cubic centimeter chl., chloroform col., colorless

cr., crystalline d., decomposes d-, dextrorotatory dl-, dextro-laevorotatory et., ethyl ether expl., explodes gn., green h., hot hex., hexagonal

i-, iso-, containing the group (CH3)2CHi., insoluble ign., ignites l-, laevorotatory lf., leaflets lq., liquid m-, meta mn., monoclinic n-, normal

nd., needles o-, ortho or., orange p-, para pd., powder pet., petroleum ether pl., plates pr., prisms rhb., rhombic s., soluble

This table of the physical properties includes the organic compounds of most general interest. For the properties of other organic compounds, reference must be made to larger tables in Lange’s Handbook of Chemistry (Handbook Publishers), Handbook of Chemistry and Physics (Chemical Rubber Publishing Co.), Van Nostrand’s Chemical Annual, International Critical Tables (McGraw-Hill), and similar works. The molecular weights are based on the 1941 atomic weight values. The densities are given for the temperature indicated and are usually referred to water at 4°C, e.g., 1.02895/4 a density of 1.028 at 95°C Name Abietic acid Acenaphthene Acetal Acet-aldehyde -aldehyde, par-aldehyde ammonia -amide -anilide -phenetidide (o-) (m-) -toluidide (o-) (p-) Acetic acid anhydride nitrile Acetone Acetonyl urea Acetophenone benzoyl hydride Acetyl-chloride -phenylenediamine (-p) Acetylene dichloride (cis) (trans) Aconitic acid Acridine Acrolein ethylene aldehyde Acrylic acid nitrile Adipic acid amide nitrile Adrenaline (1-) (3,4,1) Alanine (α) (dl-) Aldol acetaldol Alizarin Allyl alcohol bromide chloride thiocyanate (i) thiourea Aluminum ethoxide Amino-anthraquinone (α) (β) -azobenzene -benzoic acid (m-) (p-)

2-28

Synonym sylvic acid, abietinic acid naphthylene ethylene acetaldehyde diethylacetal ethanal paraldehyde ethanamide antifebrin o-ethoxyacetanilide acetyl-m-phenetidine N-tolylacetamide N-tolylacetamide ethanoic acid, vinegar acid acetyl oxide, acetic oxide methyl cyanide propanone, dimethyl ketone dimethyl hydantoin methyl-phenyl ketone ethanoyl chloride amino-acetanilide (p) ethyne; ethine 1,2-dichloroethene dioform equisetic acid; citridic acid acrylic aldehyde, propenal propenoic acid vinyl cyanide hexandioc acid, adipinic acid tetramethylene 1-suprarenine 2-hydroxybutyraldehyde Anthraquinoic acid propen-1-ol-3,propenyl alcohol 3-bromo-propene-1 3-chloro-propene-1 mustard oil thiosinamide

aminodracylic acid

s-, sec-, secondary silv., silvery sl., slightly subl., sublimes sym., symmetrical t-, tertiary tet., tetragonal tri., triclinic uns., unsymmetrical v., very

v. s., very soluble v. sl. s., very slightly soluble wh., white yel., yellow (+), right rotation >, greater than <, less than ∞, infinitely

referred to water at 4°C, the 4 being omitted when it is not clear whether the reference is to water at 4°C or at the temperature indicated by the upper figure. The melting and boiling points given have been selected from available data as probably the most accurate. The solubility is given in grams of the substance in 100 g. of the solvent. In the case of gases, the solubility is often expressed in some manner as “510 cc.” which indicates that, at 10°C, 5 cc. of the gas are soluble in 100 g. of the solvent.

Formula

Formula weight

Form and color

C20H30O2 C10H6(CH2)2 CH3CH(OC2H5)2 CH3CHO (C2H4O)3 CH3CHOHNH2 CH3CONH2 C6H5NHCOCH3 CH3CONHC6H4OC2H5 CH3CONHC6H4OC2H5 CH3C6H4NHCOCH3 CH3C6H4NHCOCH3 CH3CO2H (CH3CO)2O CH3CN CH3COCH3 (CH3)2 CH3COC6H5 CH3COCl C2H3ONHC6H4NH2 HC⯗CH CHCl:CHCl CHCl:CHCl C3H3(CO2H)3 C6H4 < (CH)(N) > C6H4 CH2:CHCHO CH2:CHCO2H CH2:CHCN (CH2CH2CO2H)2 (CH2CH2CONH2)2 (CH2CH2CN)2 C6H3(OH)2(CHOHCH2NHCH3) CH3CH(NH2)CO2H CH3CH(OH)CH2CO2H C6H4(CO)2C6H2(OH)2 CH2:CHCH2OH CH2:CHCH2Br CH2:CHCH2Cl CH2:CHCH2NCS CH2:CHCH2NHCSNH2 Al(OCH2CH3)3 C6H4(CO)2C6H3NH2 C6H4(CO)2C6H3NH2 C6H5N:NC6H4NH2 H2NC6H4CO2H H2NC6H4CO2H

302.44 154.20 118.17 44.05 132.16 61.08 59.07 135.16 179.21 179.21 149.19 149.19 60.05 102.09 41.05 58.08 128.13 120.14 78.50 150.18 26.04 96.95 96.95 174.11 179.21 56.06 72.06 53.06 146.14 144.17 108.14 183.20 89.09 88.10 240.20 58.08 120.99 76.53 99.15 116.18 164.15 223.22 223.22 197.23 137.13 137.13

lf. rhb./al. lq. col. lq. col. cr. col. cr. col. cr. rhb./al. lf./al. lf./al. rhb. rhb. or mn. col. lq. col. lq. col. lq. col. lq. tri./al. lf. col. lq. nd./aq. col. gas col. lq. col. lq. cr./aq. rhb./aq. al. col. lq. col. lq. col. lq. mn. pr. cr. pd. col. oil col. pd. nd./aq. col. lq. red rhb. col. lq. lq. col. lq. col. oil col. pr. pd. red nd. red nd. yel. mn. nd./aq. mn. pr.

Specific gravity 1.06995/95 0.82122/4 0.78318/4 0.99420/4 1.159 1.214 1.16815 1.21215 1.04920/4 1.08220/4 0.78320/4 0.79220/4 1.03315/15 1.10520/4 (A) 0.906 1.29115/4 1.26515/4 0.84120/4 1.06216/4 0.81120 1.36025/4 19/19

0.951

1.10320/4 0.85420/4 1.39820/4 0.93820/4 1.01320/4 1.21920/20 1.14220/0

1.5114°

Melting point, °C 182 95 −123.5 10.5–12 97 81(69.4) 113–4 79 96–7 110 153 16.7 −73 −41 −94.6 175 20.5 −112.0 162 −81.5891 −80.5 −50 192 d. 110–1 −87.7 12–13 −82 151–3 226–7 1 d. 207–11 295 d. 289–90 −129 −119.4 −136.4 −80 77–8 150–60 256 302 126–7 173–4 187–8

Boiling point, °C 278–9 102.2 20.2 124.4752 100–10 d. 222 305 >250 296 306–7 118.1 139.6 81.6–2.0 56.5 subl. 202.3749 51–2 −84760 60.3 48.4 346 52.5 141–2 78–9 26510 295 subl. >200 8320 430 96.6 70–1753 44.6 152 200–510 subl. subl. 225120

Solubility in 100 parts Water

Alcohol

Ether

i. i. 625 ∞ 1213 v. s. s. 0.56 i. sl. s. 0.8619 0.0922 ∞ 12 c. ∞ ∞ s. i. d. s. h. 100 cc.18 0.3520 0.6320 3315 sl. s. h. 40 ∞ s. 1.415 0.412 v. sl. s. 0.0320 2217 ∞ 0.03100 ∞ i. <0.1 0.2 30 d. i. i. sl. s. h. v. sl. s. 0.313

v. s. s. h. ∞ ∞ ∞ v. s. s. 2120 s. s. s. 1025 ∞ ∞ ∞ ∞ s. s. d. v. s. 600 cc.18 ∞ ∞ sl. s. s. s. ∞

v. s. s. chl. ∞ ∞ ∞ sl. s. v. sl. s. 7 25

v. s.

0.615

s. v. sl. s. v. sl. s. ∞ v. s. ∞ ∞ ∞ ∞ s. i. s. s. s. h. 210 1110

v. sl. s. i. i. s. v. s. ∞ ∞ ∞ ∞ v. sl. s. v. sl. s. s. i. s. 1.86 8.26

s. s. ∞ ∞ ∞ ∞ s. s. ∞ v. s. ∞ ∞ v. sl. s. s. s.

Amino-diphenylamine (p-) -G-acid (2-)(6-,8-), Na2 salt -mono-potassium salt -sodium salt -J-acid (2-)(5-,7-) -mono-potassium salt -naphthol sulfonic (1-,2-,4-)(α-) (1-,8-,4-) -phenol (o-) (m-) (p-) -toluene sulfonic acid (1-,2-,3-) (1-,4-,2-) (1-,4-,3-) (1-,2-,5-) Amyl acetate (n-) (i-) (s-) (s-) (t-) alcohol (n-) fusel oil, (s-,n-) methyl-propyl carbinol, (prim.-,i-) isobutyl carbinol, (s-,i-) (t-) (d-) -amine (n-) (s-,n-) (i-) (t-)

aniline (i-) benzoate (i-) bromide (n-) (i-) (t-) n-butyrate (n-) (i-) (t-) i-butyrate (i-) chloride (n-) (s-) (s-) (i-) (s-,i-) (t-) i-cyanide (i-) formate (n-) (i-) iodide (n-) (i-) (s-,n-) (t-) mercaptan (n-) (n-) (i-) phenol (t-)(p-) propionate (n-) (i-) (act.) salicylate (n-) Amyl i-valerate (i) (t-)

2-aminophenol 3-aminophenol p-hydroxyaniline

common amyl acetate α-Me-Bu-acetate di Et-carbinol acetate pentanol-1 pentanol-2 2-methyl-butanol-4 2-methyl-butanol-3 2-methyl-butanol-2 active amyl alcohol

1-NH2-2-Me-butane 3-amino pentane 3-NH2-2-Me-butane 1-bromopentane 4-Br-2-Me-butane 2-Br-2-Me-butane

1-chloropentane 2-chloropentane 3-chloropentane 4-Cl-2-Me-butane 3-Cl-2-Me-butane 2-Cl-2-Me-butane 1-Cl-2-Me-butane iso-caproic iso-nitrile 1-iodopentane 4-I-2-Me-butane 2-iodopentane 2-I-2-Me-butane pentanthiol-1 pentanthiol-3 2-Me-butanthiol-4 pentaphen

H2NC6H4NHC6H5 C10H5(NH2)(SO3Na)2 C10H5(NH2)S2O6HK C10H5(NH2)S2O6HNa C10H5(NH2)(SO3H)2 C10H5(NH2)S2O6HK C10H5OHNH2SO3Ha H2O NH2(OH)C10H5SO3H H2NC6H4OH H2NC6H4OH H2NC6H4OH C6H3(CH3)(NH2)SO3H C6H3(CH3)(NH2)SO3HH2O C6H3(CH3)(NH2)SO3Ha H2O C6H3(CH3)(NH2)SO3HH2O CH3CO2CH2(CH2)3CH3 CH3CO2CH2CH2CH(CH3)2 CH3CO2CH2CH(CH3)C2H5 CH3CO2CH(CH3)CH2C2H5 CH3CO2CH(C2H5)2 CH3CO2C(CH3)2C2H5 CH3(CH2)3CH2OH C2H5CH2CH(OH)CH3 (CH3)2CHCH2CH2OH (C2H5)2CHOH (CH3)2CHCH(OH)CH3 (CH3)2C(OH)C2H5 (CH3)3CCH2OH C2H5CH(CH3)CH2OH CH3(CH2)4NH2 (C3H7)(CH3)CHNH2 (CH3)2CH(CH2)2NH2 (C2H5)(CH3)2CNH2 C2H5CH(CH3)CH2NH2 (C2H5)2CHNH2 (CH3)2CHCH(CH3)NH2 C6H5NHC5H11 C6H5CO2C5H11 CH3(CH2)3CH2Br (CH3)2CH(CH2)2Br (CH3)2C(Br)C2H5 C2H5CH2CO2(CH2)4CH3 C2H5CH2CO2C5H11 C3H7CO2C(CH3)2C2H5 (CH3)2CHCO2C5H11 CH3(CH2)3CH2Cl C2H5CH2CHClCH3 (C2H5)2CHCl (CH3)2CH(CH2)2Cl (CH3)CHCHClCH3 (CH3)2CClC2H5 (CH3)(C2H5)CHCH2Cl (CH3)2CH(CH2)2NC HCO2CH2(CH2)3CH3 HCO2CH2CH2CH(CH3)2 CH3(CH2)3CH2I (CH3)2CHCH2CH2I C2H5CH2CHICH3 (CH3)2CIC2H5 C2H5CH(CH3)CH2I CH3(CH2)3CH2SH (C2H5)2CHSH (CH3)2CH(CH2)2SH C5H11C6H4OH C2H5CO2(CH3)4CH3 C2H5CO2(CH2)2CH(CH3)2 C2H5CO2C5H11 HOC6H4CO2C5H11 C4H9CO2C5H11 C4H9CO2C5H11

184.23 347.28 341.39 325.29 303.30 341.39 248.25 239.24 109.12 109.12 109.12 187.21 205.23 196.22 205.23 130.18 130.18 130.18 130.18 130.18 130.18 88.15 88.15 88.15 88.15 88.15 88.15 88.15 88.15 87.16 87.16 87.16 87.16 87.16 87.16 87.16 163.25 192.25 151.05 151.05 151.05 158.23 158.23 158.23 158.23 106.60 106.60 106.60 106.60 106.60 106.60 106.60 97.16 116.16 116.16 198.06 198.06 198.06 198.06 198.06 104.21 104.21 104.21 164.24 144.21 144.21 144.21 208.25 172.26 172.26

nd./aq. al.

67

354

col. nd. pr. lf. nd. mn. nd. tri./aq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. cr. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. lq. col. lq. col. lq. col. lq. lq. col. lq. col. lq. col. lq. lq. col. lq. lq. col. lq. col. lq. lq. lq. lq. lq. lq. lq. lq. lq. lq. lq. lq. lq. col. lq. lq. cr. lq. col. lq. col. lq. lq. col. lq. col. lq.

173 122–3 184–6 d.

subl. subl.

d. 0.87920/20 0.87615/4 0.88013 0.9220 0.87120/4 0.87419 0.817 20/20 0.81020/20 0.81315/4 0.815 25/4 0.81919 0.809 20/4 0.81620/4 0.76619 0.749 20/4 0.75118/4 0.73125/4 0.75518 0.749 20/4 0.75718 0.92815/4 0.99214/14 1.21820/4 1.22017/15 1.21619/0 0.87115/4 0.86619/15 0.86515/0 0.8760/4 0.878 20/4 0.870 20/4 0.895 21 0.89320/4 0.8830 0.87120/4 0.88117.5 0.9020 0.882 20/4 1.51020/4 1.515 18/4 1.507 17/4 1.47119/15 1.524 20/4 0.857 20

−H2O, 120 −70.8

−78.5 −117.2 −11.9 52–3 −55 −105

−95 −73.2

−99

−72.9 −73.5 −93.5 −86

0.83520/4 0.87615/4 0.870 20/4 0.866 20/4 1.06515 0.85820/15 0.86114/0

93 −73.1

148.4737 142757 141–2 133.5 133 124.5 749 137.9 119.5 132.0 115.6 113–4 102 113–4 128 103–4 91–2 95 77–8 95–6 90–1 83–4 254.5 261746 129.7 120745 108765 186.4 178.6 164 168.8 108.4 96.7 97.3 99.7 758 91753 85.7 98–9 137–9 132 123.5 157.0 147 765 144–5 127 765 148 126767 105 120 265–7 168.7 160.2 5816 265 194 173–4

sl. s. v. sl. s. 12.820 2.718 10.0 20 3.418 v. s. v. sl. s. 1.7 0 2.6 0 1.10 0.97 11 0.5 20 0.47 311 v. sl. s. 0.315 v. sl. s. sl. s. sl. s. v. sl. s. 2.7 22 420 214 5.5 30 2.830 sl. s. sl. s. 3.630 s. ∞ ∞ ∞ ∞ ∞ ∞ i. i. i. 0.0216 i. 0.05 50 i. sl. s. i. i. i. i. i. i. i. i. i. v. sl. s. 0.322 i. i. i. i. i. i. i. i. sl. s. i. 0.125 v. sl. s. i. v. sl. s. sl. s.

s.

s.

4.30 s. 40

v. s. sl. s. i. bz.

i. i. ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ s. ∞ ∞ s. ∞ ∞ ∞ ∞ ∞ ∞ ∞ s. s. s. ∞ ∞ ∞ s. s. s. ∞ s. s. s. s. s. ∞ ∞ s. ∞ ∞ ∞ ∞ ∞ ∞ ∞ s. ∞ ∞ ∞ ∞ ∞ s.

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ s. ∞ ∞ s. ∞ ∞ ∞ ∞ ∞ ∞ ∞ s. s. ∞ ∞ ∞ s. s. s. ∞ ∞ s. s. s. s. ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ s. ∞ ∞ ∞ ∞ ∞ s.

*By N. A. Lange, Ph.D., Handbook Publishers, Inc., Sandusky, Ohio. Abridged from table of Physical Constants of Organic Compounds in Lange’s “Handbook of Chemistry.” 2-29

TABLE 2-2

Physical Properties of Organic Compounds (Continued )

Name Amylene (n-)(α-) (i-) (α-) (-n)(β-) (i-)(β-) Anethole (p-) Anhydroformald-aniline Aniline hydrochloride nitrate sulfate Anisal-acetone (p-) Anisic acid (p-) aldehyde (p-) Anisidine (o-) (m-) (p-) Anisole Anthracene Anthramine (α) (β) Anthranil Anthranilic acid (o-) Anthrapurpurin (1-,2-,7-) Anthraquinone disulfonate Na2 (1-,5-) (1-,8-) (2-,6-) (2-,7-) sulfonate Na (1-) (2-) Anthrarufin (1-,5-) Antipyrene Apiole Arabinose (α)(d- or l-) (dl-) Arachidic acid Arsanilic acid (p-) Asparagine (l-) Aspirin (o-) Atropic acid Auramine Aurine, coralline (4-,4′-) Azo-anisole (2-,2′-) benzene Azoxybenzene Barbituric acid Benzal acetone Benzaldehyde Benzamide Benzanilide Benzene sulfinic acid sulfonic acid sulfonic amide sulfonic chloride Benzidine (4-,4′-) disulfonic acid (2-,2′-) (3-,3′-) Benzil Benzoic acid anhydride nitrile

2-30

Synonym pentene-1 2-methyl-butene-3 2-methyl-butene-1 pentene-2 2-methyl-butene-2 p-propenyl anisole methylene aniline amino benzene, phenyl amine, cyanol aniline salt, aniline chloride MeO-benzalacetone 2-amino-anisole MeO-aniline(m) 4-amino anisole methyl phenyl ether paranaphthalene, anthracin green oil α-amino-anthracene β-amino-anthracene

diphenyleneketone, dihydrodiketoanthracene ρ-anthraquinone disulfonate x-anthraquinone disulfonate

1-ph-2,3-diMepyrazolone-5 1-allyl-2, 5-diMeO-3,4 methylenedioxybenzene eicosanoic acid

α-phenyl acrylic acid 4,4′-dimethylaminobenzophenomide diMeO-azobenzene diphenyldiimide malonyl urea Me-cinnamyl ketone artificial almond oil benzol, phenyl hydride, cyclohexatriene benzene sulfonamide benzene sulfonyl chloride

dibenzoyl phenyl cyanide

Formula weight

Form and color

Specific gravity

C2H5CH2CH:CH2 (CH3)2CHCH:CH2 (C2H5)(CH3)C:CH2 C2H5CH:CHCH3 (CH3)2C:CHCH3 CH3CH:CHC6H4OCH3 (CH2NC6H5)3 C6H5NH2

70.13 70.13 70.13 70.13 70.13 148.20 315.40 93.12

lq. col. lq. col. lq. col. lq. col. lq. lf./al. pr./al. col. oil

0.644 20 0.63215 0.667 0/0 0.650 20/4 0.66319/4 0.99120/20

C6H5NH2HCl C6H5NH2HNO3 (C6H5NH2)2H2SO4 CH3OC6H4CH:CHCOCH3 CH3OC6H4CO2H CH3OC6H4CHO CH3OC6H4NH2 CH3OC6H4NH2 CH3OC6H4NH2 CH3OC6H5 C6H4:(CH)2:C6H4

129.59 156.14 284.32 176.22 152.14 136.14 123.15 123.15 123.15 108.13 178.22

cr. rhb. lf./al. lf./et. mn./aq. col. oil col. lq. oil pl./aq. col. lq. col. mn.

1.222 4 1.356 4 1.377 4

C6H4:(CH)2:C6H3NH2 C6H4:(CH)2:C6H3NH2 C6H4:(NH)CO H2NC6H4CO2H C14H5O2(OH)3 C6H4:(CO)2:C6H4

193.24 193.24 119.12 137.13 256.20 208.20

yel./al. yel./al. col. oil col. rhb. or. nd./al. yel. rhb.

C14H6O2(SO3Na)25H2O C14H6O2(SO3Na)24H2O C14H6O2(SO3Na)27H2O C14H6O2(SO3Na)24H2O C14H7O2SO3Na C14H7O2SO3Na C14H6O3(OH)2 C11H12ON2 C12H14O2

502.38 484.37 538.41 484.37 310.25 310.25 240.20 188.22 222.23

yel. lf. yel. pr. col. cr. cr. yel. lf. silv. lf. yel. lf. mn./aq. col. nd.

1.088113/4 1.0220/4

CH2OH(CHOH)3CHO CH2OH(CHOH)3CHO CH3(CH2)18CO2H H2NC6H4.AsO3H2 HO2CC2H3(NH2)CONH2 CH3CO2C6H4OH C6H5C(:CH2)CO2H [(CH3)2NC6H4]2C:NH

150.13 150.13 312.52 217.04 132.12 180.15 148.15 267.36

rhb. pr.

1.585 20/4

col. lf. nd./aq. rhb. nd./aq. nd./aq. col./al.

(HOC6H4)2C:C6H4:O (CH3OC6H4N:)2 C6H5N:NC6H5 (C6H5)2N2O CO:(NHCO)2:CH22H2O C6H5CH:CHCOCH3 C6H5CHO C6H5CONH2 C6H5CONHC6H5 C6H6

290.30 242.27 182.22 198.22 164.12 146.18 106.12 121.13 197.23 78.11

red or. pr. or. mn. yel. rhb. col./aq. pl. col. lq. col. pr. lf./al. col. lq.

C6H5SO2H C6H5SO3H C6H5SO2NH2 C6H5SO2Cl NH2C6H4C6H4NH2 (C6H3(NH2)SO3H)23H2O (C6H3(NH2)SO3H)2 C6H5COCOC6H5 C6H5CO2H (C6H5CO)2O C6H5CN

142.17 158.17 157.18 176.62 184.23 398.40 344.35 210.22 122.12 226.22 103.12

pr./aq. col. nd. mn./aq. cr. cr./aq. pr./aq.

1.38415/15

83–4 65–6 156 14.5 128–9 d. >175

pr. mn. pr. rhb./et. col. lq.

1.2315 1.26615/4 1.19915/4 1.00125/6

95 121.7 42 −12.9

Formula

1.02220/4

1.385 4 1.123 20/4 1.09815/15 1.096 20/4 1.089 55/55 0.990 22/4 1.25 27/4

1.18715/4 1.438 20/4

1.543

15/4

1.20320/4 1.248 20/20 1.035 20/20 1.046 20/4 1.341 1.314 0.879 20/4

Melting point, °C −135 −139 −124 22.5 143 −6.2

Boiling point, °C

Solubility in 100 parts Water

Alcohol

Ether

30–1 20.5 771 31–2758 36.4 37–8 235.3 185 184.4

i. i. i. v. sl. s. i. v. sl. s. i. 3.618

∞ ∞ ∞ ∞ s. s. sl. s. ∞

∞ ∞ ∞ ∞ ∞ ∞ s. ∞

245

s. s. sl. s. v. s. v. s. ∞ ∞ s. s. s. 1.520

i. sl. s. i. v. s. v. s. ∞ ∞ s. s. s.

198 d. 190 d. 73–4 184.2 2.5 5.2 <−12 57.2 −37.3 217–8

275–80 247–8 225 251 243 154–5 340–2

1815 s. 514 i. 0.0319 v. sl. s. v. sl. s. v. sl. s. s. h. i. i.

130 238 <−18 144–5 369 286

subl. d. >215 subl. 462 379–81

i. i. sl. s. h. 0.35 14 sl. s. h. i.

s. sl. s. s. 1110 v. s. h. 0.0518

sl. s. s. 167 sl. s. v. sl. s.

i.

i.

subl. 319174 294

v. s. sl. s. 3.920 30.5 20 0.5320 0.8425 i. 100 25 i

v. sl. s. i. i. sl. s. 100 s.

i. i. i. s. sl. s. s.

460 16.910 i. v. s. h. 3.128 137 0.1 c. i.

0.59°

i.

s. h. v. s. h. i. c. s. s. 720

v. s. i.

i. i. i. i. s. h. i. 0.3 1.35 25 i. 0.07 22

s. s. 4.220 11.415 sl. s. s. ∞ 1725 430 s.

s. s. ∞ sl. s. sl. s. ∞

v. s. h. v. s. 0.4316 i. 1 h. 0.09 25 v. sl. s. i. 0.217 i. 1100

v. s. v. s. v. s. v. s. 1 h. i.

v. s. i. v. s. s. 2 i.

v. s. 4615 s. ∞

v. s. 6615 s. ∞

280 113(109) 30 159.5 164.5 77 232 227–35 135–6 106–7 136 310 d. 153 68 36 d. 245 41–2 −26 130 163 5.5

328 d. 235 267 d.

297 d. 260–2 179 290 117–910 80.1 d. > 100 d. 251.5 400740 348 d. 249.2 360 190.7

520 s. 2.320 s. s.

Benzoin (dl-) Benzophenone Benzotrichloride Benzoyl-benzoic acid (o-) -chloride -peroxide Benzyl acetate alcohol amine aniline benzoate butyrate chloride ether formate propionate Berberonic acid (2-,4-,5-) Biuret Borneol (dl-) (d- or l-) (iso-) Bornyl acetate (d-) Bromo-aniline (p-) -benzene -camphor (3-)(d-) -diphenyl (p-) -naphthalene (α-) (β-) -phenol (o-) (m-) (p-) -styrene (ω)(1) (2) -toluene (o-) (m-) (p-) Bromoform Butadiene (1-,2-) (1-,3-) Butadienyl acetylene Butane (i-) Butyl acetate (n-) (s-) (i-) (tert-) alcohol (n-) (s-) (i-) (tert-) amine (n-) (s-) (i-) (t-) p-aminophenol (N)(n) (N)(i-) aniline (n-) (i-) arsonic acid (n-) benzoate (n-) (i-) bromide (n-) (s-) (i-) (t-) butyrate (n-)(n-) (n-)(i-) (i-)(i-) caproate carbamate (i-) cellosolve (n-)

diphenyl ketone phenyl chloroform

phenyl carbinol ω-amino toluene phenyl-benzylamine ω-chlorotoluene dibenzyl ether

allophanamide

phenyl bromide α-bromocamphor α-naphthyl bromide β-naphthyl bromide

o-tolyl bromide tribromo-methane methyl-allene erythrene diethyl trimethyl-methane

butanol-1 butanol-2 2-methyl-propanol-1 2-methyl-propanol-2

1-bromo-butane 2-bromo-butane 1-Br-2-Me-propane 2-Br-2-Me-propane

2-BuO-ethanol-1

C6H5COCHOHC6H5 C6H5COC6H5 C6H5CCl3 C6H5COC6H4CO2HH2O C6H5COCl (C6H5CO)2O2 CH3CO2CH2C6H5 C6H5CH2OH C6H5CH2NH2 C6H5CH2NHC6H5 C6H5CO2CH2C6H5 C2H5CH2CO2CH2C6H5 C6H5CH2Cl (C6H5CH2)2O HCO2CH2C6H5 C2H5CO2CH2C6H5 C5H2N(CO2H)32H2O NH(CONH2)2 C10H17OH C10H17OH C10H17OH CH3CO2C10H17 BrC6H4NH2 C6H5Br BrC10H15O BrC6H4C6H5 C10H7Br C10H7Br BrC6H4OH BrC6H4OH BrC6H4OH C6H5CH:CHBr C6H5CH:CHBr CH3C6H4Br CH3C6H4Br CH3C6H4Br CHBr3 CH3CH:C:CH2 CH2:CHCH:CH2 CH2:(CH)2:CHC⯗CH CH3CH2CH2CH3 (CH3)2CHCH3 CH3CO2(CH2)2C2H5 CH3CO2CH(CH3)C2H5 CH3CO2CH2CH(CH3)2 CH3CO2C(CH3)3 C2H5CH2CH2OH C2H5CH(OH)CH3 (CH3)2CHCH2OH (CH3)3COH C2H5CH2CH2NH2 C2H5CH(NH2)CH3 (CH3)2CHCH2NH2 (CH3)3CNH2 C4H9NHC6H4OH C4H9NHC6H4OH C4H9NHC6H5 C4H9NHC6H5 C4H9AsO(OH)2 C6H5CO2C4H9 C6H5CO2C4H9 C2H5CH2CH2Br C2H5CH(Br)CH3 (CH3)2CHCH2Br (CH3)3CBr C2H5CH2CO2CH2CH2C2H5 C2H5CH2CO2CH2CH(CH3)2 (CH3)2CHCO2CH2CH(CH3)2 CH3(CH2)4CO2C4H9 NH2CO2CH2CH(CH3)2 C4H9OCH2CH2OH

212.24 182.21 195.48 244.24 140.57 242.22 150.17 108.13 107.15 183.24 212.24 178.22 126.58 198.25 136.14 164.20 247.16 103.08 154.24 154.24 154.24 196.28 172.03 157.02 231.14 233.11 207.07 207.07 173.02 173.02 173.02 183.05 183.05 171.04 171.04 171.04 252.77 54.09 54.09 78.11 58.12 58.12 116.16 116.16 116.16 116.16 74.12 74.12 74.12 74.12 73.14 73.14 73.14 73.14 165.23 165.23 149.23 149.23 182.04 178.22 178.22 137.03 137.03 137.03 137.03 144.21 144.21 144.21 172.26 117.15 118.17

mn. col. rhb. col. lq. tri./aq. col. lq. rhb./et. col. lq. col. lq. lq. mn. pr. nd. col. lq. col. lq. lq. col. lq. lq. tri. nd./al. col. cr. col. cr. col. cr. rhb./pet. rhb. col. lq. cr. cr./al. col. oil lf./al. col. lq. cr. tet. cr. lq. lq. col. lq. col. lq. cr./al. col. lq. lq. col. gas col. lq. col. gas col. gas col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. lq. col. lq. col. lq. col. lq. col. lq. lq. oil col. lf. col. oil col. oil lq. lq. lq. lq. col. lq. col. lq. col. lq. col. lq. col. lf. col. lq.

1.08354 1.38014 1.21220/4 1.05717 1.04320/4 0.98220/4 1.065 25/25 1.1220/4 1.01616/18 1.100 20/20 1.03616 1.08123 1.03616/17 20/4

1.011 1.01120/4 0.99115 1.820 1.495 20/4 1.449 20/4 1.48220/4 1.605 0 1.55380 1.588 80 1.42220/4 1.427 20/4 1.42220/4 1.410 20/4 1.390 20/4 2.890 20/4 0.62120/4 0.773 20/4 0.600 0.600 0.882 20 0.865 25/4 0.87120/4 0.866 20/4 0.810 20/4 0.808 20/4 0.80517.5 0.779 26 0.739 25/4 0.724 20/4 0.73220/20 0.69818/4

133–7 48.5 −4.75 93(128) −0.5 108 d. −51.5 −15.3 37–8 21 238–40 −39 3.6 243 192–3 d. 210.5 208–9 212 29 63–4 −30.6 77–8 90–1 5–6 59 5.6 32–3 63.5 7 −7.5 −28 −39.8 28.5 8–9 −108.9 −135 −145 −76.3 −98.9 −79.9 −114.7 −108 25.5 −50 −104 −85 −67.5 71 79

0.940 20/4 1.005 25/25 0.997 25/25 1.277 20/4 1.25125/4 1.258 25/4 1.21120/4 0.87220/20 0.86318/4 0.875 0/4 0.8820/0 0.95676/4 0.90320/4

158–9 −22 −112.4 −112 −118.5 −16.2 −80.7 65

344768 305.4 220.7 197.2 expl. 213.5 204.7 184.5 306750 323–4 i. 179.4 295–8 202–3747 220–2 subl. 212–3 226–7 156.2 274 310 281.1 281–2 194–5 236–7 238 221 10826 181.8 183.7 184–5 150.5 18–9 −4.41 83–6 −0.6 −10 125 740 112744 118 95–6760 117 99.5 107–8 82.9 77.8 66772 68–9 45.2 235720 231–2 249–50 241.5 101.6 91.3 91.5 73.3 165.7736 156.9 148–9 204.3 206–7 171.2

v. sl. s. i. i. sl. s. d. i. i. 417 ∞ i. i. v. s. i. i. i. i. v. sl. s. 1.30 v. sl. s. v. sl. s. i. i. i. c. i. i. i. i. i. s. 1.415 i. i. i. i. i. 0.1 c. i. i. i. i. i. 0.7 i. 0.625 i. 915 12.520 1015 ∞ ∞ ∞ ∞ i. i. i. 0.0115 s. i. i. 0.0616 i. 0.0618 i. i. i. i. i. i. ∞

s. h. 6.515 s.

sl. s. 1513 s.

d. h. s. h. ∞ ∞ ∞

∞ s. ∞ ∞ ∞ s. ∞

∞ v. s. ∞ s. h. s.

∞ s. ∞

sl. s. h. s.

i.

v. s.

v. s.

s. v. s. s. 2026 s. s. 620 s. s. v. s. ∞ ∞ s. s. s. ∞ ∞ ∞

s. v. s. ∞ v. s. 34 25 ∞ v. s. ∞ s. v. s. ∞ ∞ ∞25 s. ∞25 ∞ ∞ ∞

s. s. ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

s. s. ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

v. s. v. s. s. s. ∞ ∞

v. s. v. s. i. s. ∞ ∞

∞ ∞ ∞ ∞ ∞

∞ ∞ ∞ ∞ ∞

s. ∞

s. ∞

2-31

TABLE 2-2

Physical Properties of Organic Compounds (Continued )

Name chloride (n-) (s-) (i-) (t-) dimethylbenzene (t-)(1-,3-,5-) formate (n-) (s-) (i-) furoate (n-) iodide (n-) (s-) (i-) (t-) lactate (n-) mercaptan (n-) (i-) (t-) methacrylate (n-) (i-) phenol (p-)(t-) propionate (n-) (s-) (i-) stearate (n-) (i-) iso-thiocyanate (n-) (i-) (s-)(d-) (t-) valerate (n-)(n-) (i-)(n-) (i-)(s-) (i-)(i-) Butylene (α-) (β-) Butyraldehyde (n-) (i-) Butyric acid (n-) (i-) amide (n-) (i-) anhydride (n-) (i-) anilide (n-) Caffeic acid (3-,4-) Caffeine Camphene (dl-) (d- or l-) Camphor (d-) Camphoric acid (d-) Cantharidine Capric acid Caproic acid (n-) (i-) Caprylic acid (n-) Carbazole Carbitol Carbon disulfide monoxide suboxide tetrabromide tetrachloride tetrafluoride Carbonyl sulfide Carminic acid Carvacrol (1-,2-,4-)

2-32

Synonym 1-chloro-butane 2-chloro-butane 1-Cl2-2-Me-propane 2-Cl2-2-Me-propane

1-iodo-butane 2-iodo-butane 1-iodo-2-Me-propane 2-iodo-2-Me-propane butanthiol-1 2-Me-propanthiol-1

butyl mustard oil iso-Bu mustard oil

butene-1 butene-2 2-Me-propanol butanoic acid 2-Me-propanoic acid n-butyramide iso-butyramide n-butyranilide

decanoic acid hexanoic acid 2-Me-pentanoic-5 acid octanoic acid diphenylenelimine, dibenzopyrrole diethylene glycol mono-Et ether

tetrabromomethane tetrachloromethane tetrafluoromethane

Formula C2H5CH2CH2Cl C2H5CHClCH3 (CH3)2CHCH2Cl (CH3)3CCl (CH3)3CC6H3:(CH3)2 HCO2CH2CH2C2H5 HCO2CH(CH3)C2H5 HCO2CH2CH(CH3)2 OC4H3CO2C4H9 C2H5CH2CH2I C2H5CHICH3 (CH3)2CHCH2I (CH3)3CI CH3CH(OH)CO2C4H3 C2H5CH2CH2SH (CH3)2CHCH2SH (CH3)3CSH CH2:C(CH3)CO2C4H9 CH2:C(CH3)CO2C4H9 (CH3)3CC6H4OH C2H5CO2C4H9 C2H5CO2C4H9 C2H5CO2C4H9 CH3(CH2)16CO2C4H9 CH3(CH2)16CO2C4H9 C2H5CH2CH2N:CS (CH3)2CHCH2N:CS C4H9N:CS (CH3)3CN:CS CH3(CH2)3CO2(CH2)3CH3 (CH3)2CHCH2CO2(CH2)3CH3 (CH3)2CHCH2CO2C4H9 C4H9CO2C4H9 C2H5CH:CH2 CH3CH:CHCH3 CH3CH2CH2CHO (CH3)2CHCHO C2H5CH2CO2H (CH3)2CHCO2H C2H5CH2CONH2 (CH3)2CHCONH2 (C2H5CH2CO)2O [(CH3)2CHCO]2O C3H7CONHC6H5 (HO)2C6H3C2H2CO2H C8H10O2N4H2O C10H16 C10H16 C10H16O C8H14(CO2H)2 C10H12O4 CH3(CH2)8CO2H CH3(CH2)4CO2H (CH3)2CH(CH2)2CO2H CH3(CH2)6CO2H (C6H4)2NH C2H5O(CH2)2O(CH2)2OH CS2 CO OC:C:CO CBr4 CCl4 CF4 COS C22H20O13 CH3C6H3(OH)CH(CH3)2

Formula weight

Form and color

Specific gravity

Melting point, °C

Boiling point, °C

92.57 92.57 92.57 92.57 162.26 102.13 102.13 102.13 168.19 184.03 184.03 184.03 184.03 146.18 90.18 90.18 90.18 142.19 142.19 150.21 130.18 130.18 130.18 340.57 340.57 115.19 115.19 115.19 115.19 158.23 158.23 158.23 158.23 56.10 56.10 72.10 72.10 88.10 88.10 87.12 87.12 158.19 158.19 163.21 180.15 212.21 136.23 136.23 152.23 200.23 196.20 172.26 116.16 116.16 144.21 167.20 134.17 76.13 28.01 68.03 331.67 153.84 88.01 60.07 492.40 150.21

col. lq. col. lq. col. lq. col. lq. col. lq. lq. lq. lq. col. lq. lq. lq. lq. lq. col. lq. col. lq. lq. lq. lq. lq. nd./aq. col. lq. col. lq. col. lq. col. lq. wax lq. lq. lq. lq. lq. lq. col. lq. col. lq. col. gas col. gas col. lq. col. lq. col. lq. col. lq. rhb. mn. pl. col. lq. col. lq. mn. pr. yel./aq. nd./al. cr. cr. trig. mn. cr. col. nd. oily lq. col. oil col. lf. lf. col. lq. col. lq. col. gas gas col. mn. col. lq. gas col. gas red pd. col. lq.

0.887 20 0.87120/4 0.88415 0.84715

−123.1 −131 −131.2 −26.5

77.9763 67.8767 68.9 51–2 200–2147 106.9 97 98.2 118–2025 129.9 118–9 120 99 75–66 97–8 88 65–7 155 155 236–8 146 132.5 136.8 220–525

0.9110 0.88220/4 0.885 20/4 1.056 20/4 1.617 20/4 1.595 20 1.606 20/4 1.370 19/15 0.968 0.837 25/4 0.836 20/4 0.889 15.6 0.889 15.6 0.908 112/4 0.88315 0.866 20/4 0.888 0/4 0.855 25/25 0.95611 0.96414/4 0.943 20/4 0.91910 0.87015/4 0.862 25/4 0.848 20/4 0.8740/4 0.69 0.817 20/4 0.79420/4 0.96420/4 0.949 20/4 1.032 1.013 0.968 20/20 0.950 25/4 1.134 1.2319 0.82278 0.845 50/4 0.999 9/9 1.186 0.889 87 0.922 20/4 0.925 20/4 0.910 20/4 0.990 20/20 1.263 20/4 0.81−195/4 1.1140 3.42 1.595 20/4 1.24−87 0.977 20/4

−95.3 −103.5 −104 −90.7 −34 −116 <−79

99 −89.55 −71.4 27.5 25

10.5 −93

−130 −127 −99 −65.9 −4.7 −47 115–6 129–30 −75 −53.5 92 195–213 237 50 42.7 178–9 187 212 31.5 −1.5 −35 16 244.8 −108.6 −207 −107 90.1(48) −22.6 −138.2 d. 136 0.5

165724 162 159–63 140770 186 168.8 163–4752 168.7 −5758 3746 75.7 64757 163.5757 154.5 216 216–20 199.5 181.5734 18915 d. subl. 160 159.6 209.1759 268–70 202761 207.7 237.5 354.8 201.9 46.3 −192 7761 189.5 76.8 −128 −50.2760 238

Solubility in 100 parts Water 0.0718 i. i. i. i. v. sl. s. sl. s. 1.122 i. i. i. i. i. sl. s. sl. s. v. sl. s. i. i. sl. s. i. i. i. 0.3 25 i. i. i. i. i. v. sl. s. i. i. i. i. 4 1120 ∞ 20 20 16.315 v. s. d. d. i. s. h. 2 i. i. 0.1 0.612 0.003 0.003 1.120 v. sl. s. 0.0715 i. ∞ 0.20 3.50 cc. d. 0.0230 0.0820 sl. s. 8014 cc. s. v. sl. s.

Alcohol

Ether

∞ ∞ ∞ ∞

∞ ∞ ∞ ∞

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ v. s. s.

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ v. s. s.

s. ∞ ∞ ∞ s.

s. ∞ ∞ ∞ s.

s. s. s. s. ∞ ∞ ∞ ∞ v. s.

s. s. s. s. ∞ ∞ ∞ ∞ v. s.

∞ ∞ ∞ ∞ s. s. d d s. s. 2 s. s. 12012 s.

∞ ∞ ∞ ∞ sl. s. sl. s. ∞ ∞ s. sl. s. 0.3 s. s. v. s.

s. s. s. s. 0.9214 v. s. ∞ s.

s. s. s. s. sl. s. s. ∞

s. ∞

s. s. ∞

s. s. ∞

s. v. sl. s. ∞

Carvacrylamine (2-,1-,4-) Carvone (d-) Cellosolve acetate Cellulose Cetyl acetate alcohol Chloral hydrate Chloranil Chloretone Chloro-acetanilide (p-) -acetic acid -acetone -acetophenone (ω-) -acetyl chloride -aniline (o-) (m-) (p-) -anthraquinone (1-) (2-) -benzaldehyde (o-) (m-) (p-) -benzene -benzoic acid (o-) (m-) (p-) -buta-1,3-diene (2-) (1-) -buta-1,2-diene (4-) -dimethylhydantoin -dinitrobenzene (α)(1-,2-)(4-) (α)(1-,3-)(4-) -diphenyl (o-) (m-) (p-) -hydroquinone -naphthalene (α-) (β-) -nitrobenzene (o-) (m-) (p-) -nitrotoluene (2-,4-) (2-,6-) -phenol (o-) (m-) (p-) -propionic acid (α)(dl-) -toluene (o-) (m-) (p-) Chloroform Chlorophyll (α-) Chloropicrin Cholesterol Chrysene Chrysoidine (2-,4-) Chrysophanic acid Cinchomeronic acid (3-,4-) Cineole, eucalyptole Cinnamic acid (cis-) (trans-) aldehyde Cinnamyl alcohol cinnamate Citraconic acid (cis-) Citral (α) Citric acid Citronellal (d-) Citronellol (d-) Coniine (d-)(2-)

H2NC6H3(CH3)C3H7 C10H14O C2H5O(CH2)2OH CH3CO2CH2CH2OC2H5 (C6H10O5)x CH3CO2(CH2)15CH3 CH3(CH2)14CH2OH CCl3CHO CCl3CH(OH)2 OC:(CClCCl)2:CO Cl3CC(OH)(CH3)2 CH3CO2NHC6H4Cl ClCH2CO2H CH3COCH2Cl C6H5COCH2Cl ClCH2COCl ClC6H4NH2 ClC6H4NH2 ClC6H4NH2 C6H4(CO)2C6H3Cl C6H4(CO)2C6H3Cl ClC6H4CHO ClC6H4CHO ClC6H4CHO C6H5Cl ClC6H4CO2H ClC6H4CO2H ClC6H4CO2H CH2:CClCH:CH2 CH2:CHCH:CHCl CH2:C:CHCH2Cl  C(CH3)2N(Cl)CON(Cl)CO ClC6H3(NO2)2 ClC6H3(NO2)2 C6H5C6H4Cl C6H5C6H4Cl C6H5C6H4Cl ClC6H3(OH)2 C10H7Cl C10H7Cl ClC6H4NO2 ClC6H4NO2 ClC6H4NO2 CH3C6H3(NO2)(Cl) CH3C6H3(NO2)(Cl) ClC6H4OH ClC6H4OH ClC6H4OH CH3CHClCO2H CH3C6H4Cl CH3C6H4Cl CH3C6H4Cl CHCl3 C55H72O5N4Mg Cl3CNO2 C27H45OHH2O C18H12 C6H5N:NC6H3(NH2)2 C14H5(OH)2(CH3)O2 C5H3N(CO2H)2 C10H18O C6H5CH:CHCO2H C6H5CH:CHCO2H C6H5CH:CHCHO C6H5CH:CHCH2OH C8H7CO2C9H9 CH3C(CO2H):CHCO2H C9H15CHO C3H4(OH)(CO2H)3 C9H17CHO C10H20O C3H7C5H10N

149.23 150.21 90.12 132.16 162.14 284.47 242.43 147.40 165.42 245.89 177.47 169.61 94.50 92.53 154.59 112.95 127.57 127.57 127.57 242.65 242.65 140.57 140.57 140.57 112.56 156.57 156.57 156.57 88.54 88.54 88.54 197.03 202.56 202.56 188.65 188.65 188.65 144.56 162.61 162.61 157.56 157.56 157.56 171.56 171.56 128.56 128.56 128.56 108.53 126.58 126.58 126.58 119.39 893.48 164.39 404.65 228.28 212.25 254.23 167.12 154.24 148.15 148.15 132.15 134.17 264.31 130.10 152.23 192.12 154.24 156.26 127.22

oil col. lq. col. lq. col. lq. amor. nd. lf. col. lq. mn. pr. yel./bz. col. cr. rhb. col. cr. col. lq. rhb. col. lq. lq. lq. rhb. yel. nd. nd./al. nd. pr. pr. col. lq. mn./aq. pr. tri. col. lq. col. lq. col. lq. cr./et. rhb./et. cr. cr. lf. mn. col. lq. lf./al. mn. nd. yel./al. mn. pr. cr. cr. col. lq. nd. nd. col. lq. col. lq. col. lq. col. lq. col. lq. lq. rhb./al. col. rhb. yel. cr. yel./al. cr./HCl col. oil mn. pr. mn. pr. lq. nd. nd. or pr. nd. col. oil cr. col. oil col. oil col. lq.

0.99420 0.96120/4 0.93120/4 0.975 20/4 1.3–1.4 0.858 20 0.818 50/4 1.505 25/4 1.619 50/4 1.385 22 1.58 20/20 1.16216 1.32415 1.498 20/20 1.21320/4 1.216 20/4 1.42719 1.298 1.25015 1.196 61 1.107 20/4 1.544 25/4 1.496 25/4 1.54124 0.958 20/20 0.965 20/20 0.99120/20 1.5 20/20 1.697 22

1.194 20/4 1.26616 1.305 80/4 1.34350/4 1.298 91 1.256 80 1.24118/15 1.268 25 1.306 20/4 1.306 9 1.082 20/4 1.07220/4 1.070 20/4 1.489 20 1.65123/4 1.067

0.927 20 1.2844 1.245 1.110 20/20 1.040 35/35 1.08516.5 1.617 0.89017/4 1.54220/4 0.85517.5 0.84820/4 0.84717

−16 −70 22–3 49–50 −57 51.7 290 97 175–6 61.2 −44.5 58–9 0 −10.4 70–1 162 208–9 11 17–8 47.8 −45.2 141–2 158 242–3

130 39(36) 53(43) 34 89 77.5 106 −20 56–7 32.5 44.4(24) 83–4 38.2 37.5 7(0) 32–3 41–3 <−20 −34 −47.8 7.5 −63.5 d. −64 149–51 253–4 117.5 195 258–9 d. 1.5 68 133 −7.5 33 44 92–3 153 −2

241 230766 135.1 156.3 i. 20015 189.515 97.6768 d. 98 subl. 167 189.5 121 245–7 105 210.5 230767 230–1 subl. 208748 213–4 213748 132.1 subl. 59.4 69 88 315 d. 315 d. 267–8 284–5 282 263 sl. d. 259.3 264751 245.5753 235.6 242761 240718 238 175–6 214 217 186 159.5 161.6 162.2 61.2 i. 112.3766 subl. 448 subl. subl. d. 176–7 12519 300 252 sl. d. 257.5 229 d. 204–8 224–5 166–7

v. sl. s. i. ∞ 22 i. i. i. v. s. 47417 i. 0.8 c. sl. s. v. s. ∞ 0.11 d. i. i. s. h. i. i. v. sl. s. v. sl. s. s. h. 0.049 20 0.20825 0.04125 0.00825 v. sl. s. v. sl. s. d. 0.2125 i. i. i. i. i. v. s. i. i. i. i. i. i. i. 2.8520 2.6020 2.7120 ∞ i. i. i. 0.8220 s. 0.1718 0.2620 i. sl. s. h. i. c. v. sl. s. 1.915

s. ∞ ∞ ∞ i. v. sl. s. c. s. ∞ v. s. i. c. 111 s. s. ∞ v. s. d.

0.0418 v. sl. s. sl. s. i. 36025 i. 207.725 v. sl. s. v. sl. s. 1.1

s. sl. s. h.

s. ∞ ∞ ∞ s. ∞ s. i. c. s. v. s. s. ∞ v. s. s. s. s.

v. s. v. s. v. s. ∞ s. s. s. ∞ ∞

v. s. v. s. v. s. ∞ s. s. s. ∞ ∞

v. s. h. s. h.

v. s. s.

v. s. s. v. s. s. h. v. s. h. v. s. h.

v. s. ∞ v. s. s. v. s. v. s.

s. s. v. s. ∞ s. s. s. ∞

s. s. v. s. ∞ ∞ ∞ ∞ ∞

s. 1.117 0.116 s. s. h. sl. s. ∞

s. 18 v. sl. s. s. sl. s. i. ∞

2420 s. v. s. 4 c. s. ∞ 7615 ∞ ∞ v. s.

v. s. ∞ v. s. 33 s. ∞ 215 ∞ ∞ v. s.

2-33

TABLE 2-2

Physical Properties of Organic Compounds (Continued ) Name

Coumaric acid (o-) (p-) Coumarin Coumarone Creatine Creatinine Creosol (3-,1-,4-) Cresidine (1-,2-,4-) Cresol (o-) (m-) (p-) Cresyl benzoate (o-) (m-) (p-) Crotonic acid (α-) acid (β-)(cis-) aldehyde (α) Cumene Cumic acid (p-) Cumidine (p-) Cyanamide Cyanic acid Cyanoacetic acid Cyanogen bromide chloride Cyanuric acid Cyclo-butane -heptane -hexane -hexanol -hexanone -hexene -hexyl acetate amine bromide chloride -pentadiene (1-,3-) -pentane -pentanone -propane Cymene (o-) (m-) (p-) Cystine (l-) Dambose Decahydronaphthalene (cis-) (trans-) Decane (n-) Decyl alcohol Dextrin Diacetone alcohol Diamino-benzophenone (4-,4′-) -diphenylamine (4-,4′-) -diphenylmethane (4-,4′-) -diphenylurea (4-,4′-) Diamyl-amine (i-) ether (n-) (i-) Diamyl ketone (i-) phthalate (n-) (i-) tartrate (i-) Dianisidine (o-)(4-,3-)2 Diazo-aminobenzene -aminotoluene (2-,2′-) -methane

2-34

Formula HOC6H4CH:CHCO2H HOC6H4CH:CHCO2H C9H6O2 C8H6O C4H9N3O2H2O C4H7N3O CH3OC6H3(CH3)OH CH3(NH2)C6H3OCH3 CH3C6H4OH CH3C6H4OH CH3C6H4OH C6H5CO2C6H4CH3 C6H5CO2C6H4CH3 C6H5CO2C6H4CH3 CH3CH:CHCO2H CH3CH:CHCO2H CH3CH:CHCHO C6H5CH(CH3)2 (CH3)2CHC6H4CO2H (CH3)2CHC6H4NH2 H2NCN HOCN or HNCO CH2(CN)CO2H (CN)2 BrCN ClCN C3H3O3N32H2O CH2 < (CH2)2 > CH2 CH2 < (CH2CH2CH2)2 > CH2 < (CH2CH2)2 > CH2 CH2 < (CH2CH2)2 > CHOH CH2 < (CH2CH2)2 > CO (CH2CH2CH:)2 CH3CO2C6H11 CH2 < (CH2CH2)2 > CHNH2 CH2 < (CH2CH2)2 > CHBr CH2 < (CH2CH2)2 > CHCl CH2 < (CH:CH)2 > CH2 < (CH2CH2)2 > < (CH2CH2)2 > CO < CH2CH2CH2 > CH3C6H3CH(CH3)2 CH3C6H4CH(CH3)2 CH3C6H4CH(CH3)2 [SCH2CH(NH2)CO2H]2 C6H6(OH)6 C10H18 C10H18 CH3(CH2)8CH3 CH3(CH2)8CH2OH (C6H10O5)x (CH3)2C(OH)CH2COCH3 H2NC6H4COC6H4NH2 H2NC6H4NHC6H4NH2 H2NC6H4CH2C6H4NH2 (H2NC6H4NH)2CO [(CH3)2CHCH2CH2]2NH (C2H5CH2CH2CH2)2O [(CH3)2CH(CH2)2]2O [(CH3)2CHCH2CH2]2CO C6H4(CO2C5H11)2 C6H4(CO2C5H11)2 (HOCHCO2C5H11)2 [NH2(OCH3)C6H3]2 C6H5N:NNHC6H5 C7H7N:NNHC7H7 CH2:N2

Formula weight

Form and color

164.15 164.15 146.14 118.13 149.15 113.12 138.16 137.18 108.13 108.13 108.13 212.24 212.24 212.24 86.09 86.09 70.09 120.19 164.20 135.20 42.04 43.03 85.06 52.04 105.93 61.48 165.11 56.10 98.18 84.16 100.16 98.14 82.14 142.19 99.17 163.06 118.61 66.10 70.13 84.11 42.08 134.21 134.21 134.21 240.29 180.16 138.24 138.24 142.28 158.28 162.14 116.16 212.24 199.25 198.26 242.28 157.29 158.28 158.28 170.29 306.39 306.39 290.35 244.28 197.23 225.28 42.04

nd./aq. cr./aq. rhb./et. oil mn./aq. mn. pr. nd./pet. cr. lq. pr. lq. cr. cr. col. mn. nd. col. lq. col. lq. tri. lq. col. nd. gas col. lq. col. gas nd. gas mn./aq. col. gas oil col. lq. col. nd. col. oil lq. oil col. lq. col. lq. col. lq. col. lq. col. oil col. oil col. gas col. lq. col. lq. col. lq. pl. mn./aq. lq. lq. col. lq. col. oil amor. lq. yel. nd. lf./aq. nd./aq. cr. col. lq. col. lq. col. lq. yel. oil col. lq. col. lq. lq. col. lf. yel. lf. or. cr. gas

Specific gravity

0.93520/4 1.07815/15 1.09220/20 1.04820/4 1.03420/4 1.03520/4

0.96479.7 1.03115/4 0.85320/20 0.86220/4 1.1624 0.953 1.07348/4 1.1400 0.86617 2.01520/4 1.2220 1.7680/4 0.7030/4 0.81020/4 0.77920/4 0.96220/4 0.94719/4 0.81020/4 0.9850/4 0.86520/0 1.32420/20 0.97718/4 0.80519/4 0.74520/4 0.94820 0.720−79 0.87520/4 0.86220 0.85720/4 1.752 0.89518/4 0.87220/4 0.7302 0.83020/4 1.038 0.93125

0.76721/4 0.77420/4 0.77720/4 0.82125/4

Melting point, °C

Boiling point, °C

207–8 206–7 d. 70 <−18 295 260 d. 5.5 93–4 30.8 10.9 35–6

subl.

55 71.5 72 15.5 −69 −96.9 116–7 <−20 44–5 −80 65–6 −34.4 52 −6.5 >360 −50 −12 6.5 23.9 −45 −103.7

−43.9 −85 −93.3 −58.2 −126.6 <−25 −73.5 d. 258–61 253 −51 −32 −29.7 7 −47 237–9 158 93–4 subl. 310 −44 −69 14.6

1.03 1.06315/4 131.5 96–8 51 −145

290–1 173–4 221–2765 235 190.8 202.8 202 308 314 316 189 170–1 d. 102.2 152.5 subl. 225761 14019 −640 1080.2 −21 61.3750 12.5–13 d. 11–12726 118–20 80–1 160–1 155–6 83.3 174750 134 165714 142 41–2 49–50 129–30 −34749 177 175–6 176–7 31915 193.3 185.3 174.0 232.9 167.9 d. 249–5315 188–90 190 173.4 228 204–611 22540 19516 expl. −23

Solubility in 100 parts Water sl. s. c. s. h. 0.3 c. i. 1.418 8.716 v. sl. s. v. sl. s. 2.5 0.5 1.8 i. i. i. 8.315 ∞25 18 i. 0.0225 i. v. s. sl. s. s. 45020 cc. s. 250020 cc. 0.2717 i. i. i. 3.620 s. v. sl. s. i. i. i. i. i. i. v. sl. s. i. i. i. i. 0.0119 212 i. i. i. i. s. ∞ sl. s. h. sl. s. sl. s. c. v. sl. s. sl. s. i. i. i. i. i. i. i. 0.05 d.

Alcohol

Ether

s. v. s. h. v. s.

v. sl. s. v. s. s. s. i.

0.0117 116 ∞ s. ∞30 ∞ ∞36

∞ s. ∞30 ∞ ∞36

s. ∞ ∞ s.

∞ ∞ s.

v. s. s. 230020 cc. s. v. s. 0.122 v. s. ∞ s. s. v. s. ∞ s. s. ∞ s. s. s. s. i. i. s. s. ∞ s. i. ∞ s. s. s.

v. s. s. s. 50020 cc. s. 500020 cc.

∞ s. s. v. s. ∞ s. ∞ ∞ s. s. s. s. i. s. s. ∞ i. ∞ s. s. s.

s. ∞ ∞ s.

∞ ∞ ∞ s.

s.

s.

s. s. h.

s. v. s. s.

Dibenzothiazyl-disulfide (2-,2′-) Dibensoyl methane Dibensyl-amine -aniline ketone phthalate (o-) succinate Dibromo-benzene (o-) (m-) (p-) -diphenyl (4-,4′-) Dibutyl-adipate (n-) (i-) -amine (n-) (i-) -p-aminophenol (s-) -aniline (n-) carbonate (n-) (i-) (s-) ether (n-) (i-) (s-) ketone (n-) (i-) malate (l-)(n-) oxalate (n-) phthalate (n-) tartrate (d-)(n-) (d-)(i-) Dichloro-acetic acid -acetone (αα-) -aniline (2-,5-) -anthraquinone (1-,3-) (1-,4-) (1-,5-) (1-,6-) (1-,8-) (2-,3-) (2-,6-) (2-,7-) -benzene (o-) (m-) (p-) -butane (n-)(1-,4-) -diphenyl (4-,4′-) -ethane (1-,2-) -naphthalene (β-)(1-,4-) (γ-)(1-,5-) -nitrobenzene (2-,5-) -pentane (1-,5-) -phenol (2-,4-) Dichloramine T (p-) Dicyandiamide Diethanolamine Diethyl adipate -amine -aminophenol (m-) -aniline sulfonic acid (m-) carbonate diethyl malonate Diethyl dimethyl malonate glutarate ketone malonate -malonic acid -naphthylamine (α-) (β-) oxalate phthalate (o-) sulfate sulfide

(C6H4NSC)2S2 (C6H5CO)2CH2 (C6H5CH2)2NH C6H5N(CH2C6H5)2 (C6H5CH2)2CO C6H4(CO2CH2C6H5)2 (CH2CO2CH2C6H5)2 C6H4Br2 C6H4Br2 C6H4Br2 BrC6H4C6H4Br (CH2CH2CO2C4H9)2 (CH2CH2CO2C4H9)2 (C2H5CH2CH2)2NH [(CH3)2CHCH2]2NH (C4H9)2NC6H4OH C6H5N(C4H9)2 CO(OC4H9)2 CO(OC4H9)2 CO(OC4H9)2 (C2H5CH2CH2)2O [(CH3)2CHCH2]2O [C2H5(CH3)CH]2O (C2H5CH2CH2)2CO [(CH3)2CHCH2]2CO C2H4O(CO2C4H9)2 (CO2C4H9)2 C6H4(CO2C4H9)2 (CHOHCO2C4H9)2 (CHOHCO2C4H9)2 Cl2CHCO2H Cl2CHCOCH3 Cl2C6H3NH2 C6H4:(CO)2:C6H2Cl2 C6H4:(CO)2:C6H2Cl2 C6H3Cl:(CO)2:C6H3Cl C6H3Cl:(CO)2:C6H3Cl C6H3Cl:(CO)2:C6H3Cl C6H4:(CO)2:C6H2Cl2 C6H3Cl:(CO)2:C6H3Cl C6H3Cl:(CO)2:C6H3Cl C6H4Cl2 C6H4Cl2 C6H4Cl2 ClCH2(CH2)2CH2Cl ClC6H4C6H4Cl ClCH2CH2Cl C10H6Cl2 C10H6Cl2 Cl2C6H3NO2 ClCH2(CH2)3CH2Cl Cl2C6H3OH CH3C6H4SO2NCl2 H2NC(:NH)NHCN HN(CH2CH2OH)2 (CH2CH2CO2C2H5)2 (C2H5)2NH (C2H5)2NC6H4OH (C2H5)2NC6H5 (C2H5)2NC6H4SO3H OC(OC2H5)2 (C2H5)2C(CO2C2H5)2 (CH3)2C(CO2C2H5)2 CH2(CH2CO2C2H5)2 (C2H5)2CO CH2(CO2C2H5)2 (C2H5)2C(CO2H)2 C10H7N(C2H5)2 C10H7N(C2H5)2 (CO2C2H5)2 C6H4(CO2C2H5)2 O2S(OC2H5)2 (C2H5)2S

232.46 224.25 197.27 273.36 210.26 346.36 298.32 235.92 235.92 235.92 312.02 258.35 258.35 129.24 129.24 221.33 205.33 174.23 174.23 174.23 130.22 130.22 130.22 142.23 142.23 246.30 202.24 278.34 262.30 262.30 128.95 126.98 162.02 277.10 277.10 277.10 277.10 277.10 277.10 277.10 277.10 147.01 147.01 147.01 127.02 223.10 98.97 197.06 197.06 192.01 141.04 163.01 240.11 84.08 105.14 202.24 73.14 165.23 149.23 229.29 118.13 216.27 188.22 188.22 86.13 160.17 160.17 199.28 199.28 146.14 222.23 154.18 90.18

cr. rhb./al. col. oil pr./al. cr. pr./al. lf./al. col. lq. col. lq. pl./al. mn. pr. col. lq. col. lq. col. lq. col. lq. lq. lq. col. lq. col. lq. col. lq. lq. lq. lq. lq. oil lq. col. lq. col. lq. pr. cr. lq. lq. nd. yel. nd. yel. nd. yel. nd. yel. nd. yel. nd. yel. nd. yel. nd. yel. nd. col. lq. col. lq. col. mn. lq. pr. col. lq. nd./al. lf./al. tri./al. col. lq. nd. cr. mn. pl. pr. col. lq. col. lq. rhb. oil cr. col. lq. col. lq. col. lq. syrup col. lq. col. lq. pr./aq. col. oil col. oil col. lq. col. lq. col. lq. col. lq.

1.50 1.028 25/25

20/4

1.956 1.952 20/4 2.26118 1.897 0.965 20/4 0.950 25 0.768 20/20 0.74125/4

180 78 −26 70–1 34–5 42–3 45–6 1.8 −6.9 87–8 164–5 −38 −20 −70

0.924 20/4 0.91915 0.769 20/20 0.76215 0.756 21 0.82718/4 0.805 21/4 1.038 20/4 0.986 20/4 1.045 21 1.09815 1.03175/4 1.560 25/25 1.23415

1.305 20/4 1.288 20/4 1.458 21 1.442 0/4 1.256 20/20 1.300 76/4 1.669 22 1.094 25/4 1.383 60/25 1.4014 1.09720/4 1.00920/4 0.712 15/15 0.934 20/4 20/4

0.975 0.985 20/4 0.994 25/25 1.025 21 0.816 19/4 1.055 20/4 1.005 1.026 1.079 20/4 1.121 25/25 1.172 25/4 0.837 20/4

−98 −5.9 −29.6 22–2.5 73–4 9.7(−4) 50 208–9 187.5 251 203–4 202–3 268–70 282 210–11 −17.6 −24.8 53 −38.7 148 −35.3 67–8 107 54.6 45 83 207–8 28 −21 −38.9 78 −34.4 270 d. −43 −24 −42 −49.8 125 −40.6 −25 −99.5

d. 219–2118 268–71250 >300 330.6 27412 23814 221–2 219755 218.6758 355–60 18314 278–80 159761 139–40 17010 262.8 207740 190 178–80 142.4 122.5 121 187.7 168.1 170–118 245.5 340 200–318 323–5 194.4 120 251

179 172766 174764 161–3 315–9 83.7 286–7 740 subl. 266 180–1 209–10 d. 270 748 239–41761 55.5759 276–80 216 759

126 230 196.7 237 101.7 198.9 d. 170–80 285–90 318 186 298–9 210 92–3 754

i. i. i. i. i. v. sl. s. i. i. i. i. i. i. i. ∞ v. sl. s. i. i. i. i.

4.420 s. v. s. h.

s. s. s.

s. s. s. s. 1.6 v. sl. s. h. ∞

s. s. s. s. 7125 ∞

∞ s.

∞ s.

∞ s.

∞

<0.05 i. i. i. <0.06 v. sl. s. i. 0.04 25 i. v. sl. s. ∞ v. sl. s. v. sl. s. i. i. i. i. i. i. i. i. i. i. i.

∞ ∞ ∞ s. ∞

∞ ∞ ∞ v. s. ∞

s. ∞

s. ∞

∞ s. s. i. v. sl. s. sl. s.

∞ s. s.

∞ s. v. s.

∞ s. v. s.

i. 0.90 i. i. i. i. 0.45 20 sl. s. 2.318 ∞ 0.4380 v. s. s. 1.412 s. i. i. i. 0.88 20 4.7 20 2.08 20 65 16 i. i. v. sl. s. i. i. 0.3120

v. sl. s. ∞ v. sl. s. s. v. s. h. s. v. s.

425 ∞

1.318 ∞ s. ∞

0.0118 v. sl. s. s. ∞

s.

s.

∞ ∞ ∞ v. s. ∞ ∞ v. s. ∞ ∞ ∞ ∞ s. ∞

∞ ∞ ∞ s. ∞ ∞ v. s. ∞ ∞ ∞ ∞ ∞ ∞

v. sl. s.

sl. s. sl. s.

s. s. v. s.

2-35

TABLE 2-2

Physical Properties of Organic Compounds (Continued ) Name

tartrate (d-) -toluidine (o-) (m-) (p-) Diethyleneglycol dinitrate Difluorodichloromethane Diglycerol Dihydroxy-dinaphthyl (α-) (-2,-2′,-1,-1′) -diphenyl (4-,4′-) -ethyl formal (β-) -naphthalene (1-,5-) (1-,8-) Dimethoxy-benzene (p-) -diphenylamine (4-,4′-) -ethyl adipate Dimethyl adipate -amine -aminoasobenzene (p-) -aminoethanol -aminophenol (m-) -aniline sulfonic acid (m-) (p-) carbonate ether -formamide fumarate glutarate glyoxime -naphthalene (1-,4-) (2-,3-) -naphthylamine (α-) (β-) oxalate phthalate (o-) sulfate sulfide tartrate (d-) -vinyl-ethenyl carbinol Dinaphthyl (αα-) -methane (αα′-) (β,β′-) Dinitro-anisole (1-)(2-,4-) -benzene (o-) (m-) (p-) sulfonic acid (2-,4-)(1-) -benzoic acid (2-,4-) (3-,5-) -benzophenone (4-,4′-) -diphenyl (4-,4′-) (2-,4′-) -naphthalene (1-,5-) (1-,8-) Dinitro-phenol (2-,3-) (2-,4-) (2-,6-) -salicylic acid (3-,5-) -stilbene (4-,4′-) -toluene (2-,4-) (3-,4-) (3-,5-) Dioxane Dipentene

2-36

Formula (CHOHCO2C2H5)2 CH3C6H4N(C2H5)2 CH3C6H4N(C2H5)2 CH3C6H4N(C2H5)2 O(CH2CH2ONO2)2 F2CCl2 [(HO)2C3H5]2O (HOC10H6)2 (HOC10H6)2 (HOC6H4)2 CH2(OCH2CH2OH)2 C10H6(OH)2 C10H6(OH)2 (CH3O)2C6H4 HN(C6H4OCH3)2 (CH2)4(CO2C2H4OCH3)2 [(CH2)2CO2CH3]2 (CH3)2NH C6H5N:NC6H4N(CH3)2 (CH3)2NCH2CH2OH (CH3)2NC6H4OH (CH3)2NC6H5 (CH3)2NC6H4SO3H (CH3)2NC6H4SO3HH2O OC(OCH3)2 CH3OCH3 HCON(CH3)2 (:CHCO2CH3)2 (CH2)3(CO2CH3)2 (CH3C:NOH)2 C10H6(CH3)2 C10H6(CH3)2 C10H7N(CH3)2 C10H7N(CH3)2 (CO2CH3)2 C6H4(CO2CH3)2 (CH3O)2SO2 (CH3)2S (CHOHCO2CH3)2 (CH3)2COHC⯗CCH:CH2 C10H7C10H7 (C10H7)2CH2 (C10H7)2CH2 CH3OC6H3(NO2)2 C6H4(NO2)2 C6H4(NO2)2 C6H4(NO2)2 (NO2)2C6H3SO3H3H2O (NO2)2C6H3CO2H (NO2)2C6H3CO2H (NO2C6H4)2CO (NO2C6H4)2 (NO2C6H4)2 C10H6(NO2)2 C10H6(NO2)2 (NO2)2C6H3OH (NO2)C6H3OH (NO2)C6H3OH (NO2)2C6H2(OH)CO2HH2O (NO2C6H4CH:)2 (NO2)2C6H3CH3 (NO2)2C6H3CH3 (NO2)2C6H3CH3 O < (CH2CH2)2 > O C13H10

Formula weight

Form and color

Specific gravity

Melting point, °C

Boiling point, °C

206.19 163.25 163.25 163.25 196.12 120.92 166.17 286.31 286.31 186.20 136.15 160.16 160.16 138.16 229.26 262.30 174.19 45.08 225.28 89.14 137.18 121.18 201.24 219.25 90.08 46.07 73.09 144.12 160.17 116.12 156.22 156.22 171.23 171.23 118.09 194.18 126.13 62.13 178.14 110.15 254.31 268.34 268.34 198.13 168.11 168.11 168.11 302.22 212.12 212.12 272.21 244.20 244.20 218.16 218.16 184.11 184.11 184.11 246.13 270.24 182.13 182.13 182.13 88.10 136.23

lq. lq. lq. lq. lq. gas lq. pl./al. nd./al. rhb./al. lq. pr./aq. nd. lf. cr. lq. col. lq. col. lq. yel./al. col. lq. nd. yel. lq. cr. pr. col. lq. gas lq. col. tri. lq. col. cr. lq. lf./al. col. oil col. cr. col. mn. col. lq. col. oil oil cr. lq. lf./al. pr./al. nd./al. col. mn. col. mn. col. rhb. col. mn. pr. cr./aq. mn. pr. col. nd. nd./al. mn. nd. rhb. yel. mn. yel. rhb. yel. rhb. pl./aq. yel. lf. nd. nd. mn. pr. col. lq. col. lq.

1.204 20/4

17

0.924 15.5 1.377 25/4 1.486 −30

−11.3 −155

280 208–9755 231–2 228–9

1.25 1.154 25 1.053

55/55

1.075 15.6 1.063 20/4 0.680 0/4 0.887 20/4 0.956 20/4 1.070 20/4 0.94525 1.08915.6 1.016 20/4 1.042 20 1.03970/70 1.14854 1.18925/25 1.3520/4 0.84621/4 1.32820/4 0.887 20/4

1.341 20 1.5918 1.575 20/4 1.625 18

1.445 1.474 1.681 20 1.683 24

1.321 71 1.259 111 1.277 111 1.033 20/4 0.865 18

300 218 270–2 −5.3 258–60 140 56 103 10–1 −96 116–7 85 2.5 d. 266 257 0.5 −138.5 −58.3 102 −37 240–6 <−18 104 46 54 −26.8 −83.2 61.5 160 109 92 94–5 117–8 89.8 173–4 106–8 179–80 204–5 189 233 93.5 216 170–2 144–5 114–5 63–4 173 d. 210–6 70 60–1 92–3 9.5–10.5

−29.2 220–3010 subl. subl. 264 d. 212.6 145–502 11518 7.4 d. 135756 265–8 193 89–90 −23.7 152.8 192 13050 264–6 265767 274.5711 304–5 163.3 280734 188.3 37.3 280 150 240–412 >360 319774 300–2 299777 subl.

subl. d. subl.

300 subl. 101.1 178

Solubility in 100 parts Water sl. s. i. i. i. i. 5.7 cc.26 s. h. i. i. sl. s. ∞ sl. s. sl. s. h. v. sl. s. i. 5 i. v. s. i. ∞ sl. s. h. i. s. s. h. i. 3700 cc.18 ∞ i. 20

0.06 i. i. i. i. 6 0.43 v. sl. s. i. s. 620 i. i. i. sl. s. h. 0.01 c. 0.399 0.18100 s. 1.8525 s h. i. i. i. i. i. sl. s. 0.5 c. s. h. s. c. i. 0.0322 i. sl. s. ∞ i.

Alcohol

Ether

∞ s. s.

∞ s. s.

s.

s. i. v. s. v. s. v. s.

s. s. v. s. s. v. s.

v. s. v. s. v. s.

s. s.

s. s.

s. s.

s. s.

v. sl. s. ∞ s.

v. sl. s. ∞ s.

sl. s.

sl. s.

v. s.

v. s.

sl. s. s. s. s.

s. s. s.

∞ s. 20015

∞ s.

s. h. 0.8 c. s. 1.520 1.921 320 0.1821 s. s. v. s.

s. v. s.

v. sl. s. sl. s.

1.520 v. s. h. 0.2 c. v. s. h. 420 s. h. v. s. v. sl. s. 1.215

v. s. v. s. h. s. v. s. v. sl. s. 916

s. h. s.

s. s.

Diphenyl -amine carbonate -chloroarsine -ethane ether guanidine -methane phenylenediamine (p-) succinate sulfide sulfone urea (uns.) Diphenylene oxide Dipropyl adipate (n-) -amine (n-) (i-) aniline (n-) carbonate (n-) ether (n-) (i-) ketone (n-) (i-) oxalate (n-) (i-) Disalicylal ethylenediamine Ditolyl guanidine (o-) Divinyl acetylene Docosane (n-) Dodecane (n-) Dulcitol Durene (1-,2-,4-,5-) Elaidic acid Eosine Ephedrine (l-) Epichlorhydrin (α-) Epidichlorohydrin (α-) Erythritol (dl-) tetranitrate Ethane Ethanol-amine formamide Ether Ethyl abietate acetate acetoacetate alcohol -amine hydrochloride aniline sulfonic acid (m-) anisate (p-) anthranilate (o-) benzene benzoate -benzyl-aniline bromide butyrate (n-) (i-) caprate (n-) Ethyl caproate (n-) caprylate (n-) chloride chloroacetate chlorocarbonate cinnamate (trans-) cyanoacetate formate furoate (α) heptoate hypochlorite iodide lactate

C6H5C6H5 C6H5NHC6H5 O(COC6H5)2 (C6H5)2AsCl (C6H5CH2)2 C6H5OC6H5 (C6H5NH)2C:NH (C6H5)2CH2 (C6H5NH)2C6H4 (CH2CO2C6H5)2 (C6H5)2S (C6H5)2SO2 (C6H5)2NCONH2 < (C6H4)2O (CH2CH2CO2C3H7)2 (C2H5CH2)2NH [(CH3)2CH]2NH C6H5N(C3H7)2 O(COCH2C2H5)2 (C2H5CH2)2O [(CH3)2CH]2O (C2H5CH2)2CO [(CH3)2CH]2CO (CO2CH2C2H5)2 [CO2CH(CH3)2]2 [HOC6H4CH:NCH2]2 (C7H7NH)2C:NH (H2C:CHC⯗)2 CH3(CH2)20CH3 CH3(CH2)10CH3 CH2OH(CHOH)4CH2OH (CH3)4C6H2 C8H17CH:CH(CH2)7CO2H C20H8O5Br4 C6H5CHOHCH(CH3)NHCH3 C2H3OCH2Cl CH2:CClCH2Cl CH2OH(CHOH)2CH2OH C4H6(ONO2)4 CH3CH3 HOCH2CH2NH2 HCONHCH2CH2OH (CH3CH2)2O C19H29CO2C2H5 CH3CO2C2H5 CH3COCH2CO2C2H5 CH3CH2OH C2H5NH2 C2H5NH2HCl C6H5NHC2H5 C2H5NHC6H4SO3H CH3OC6H4CO2C2H5 NH2C6H4CO2C2H5 C6H5C2H5 C6H5CO2C2H5 C6H5N(C2H5)CH2C6H5 C2H5Br C2H5CH2CO2C2H5 (CH3)2CHCO2C2H5 CH3(CH2)8CO2C2H5 CH3(CH2)4CO2C2H5 CH3(CH2)6CO2C2H5 CH3CH2Cl ClCH2CO2C2H5 ClCO2CH2CH3 C6H5CH:CHCO2H CH2(CN)CO2C2H5 HCO2CH2CH3 OC4H3CO2C2H5 CH3(CH2)5CO2C2H5 ClOCH2CH3 CH3CH2I CH3CH(OH)CO2C2H5

154.20 169.22 214.21 264.57 182.25 170.20 211.26 168.23 260.32 270.27 186.26 218.26 212.24 168.18 230.30 101.19 101.19 177.28 146.18 102.17 102.17 114.18 114.18 174.19 174.19 268.30 239.31 78.11 310.59 170.33 182.17 134.21 282.45 647.93 165.23 92.53 110.98 122.12 302.12 30.07 61.08 89.09 74.12 330.49 88.10 130.14 46.07 45.08 81.55 121.18 201.24 180.20 165.19 106.16 150.17 211.29 108.98 116.16 116.16 200.31 144.21 172.26 64.52 122.55 108.53 176.21 113.11 74.08 140.13 158.23 80.52 155.98 118.13

col. mn. col. mn. nd./al. rhb. col. pr. col. rhb. mn./al. col. pr. cr. lf./al. col. lq. nd./aq. rhb. lf./al. col. lq. col. lq. col. lq. yel. oil col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. cr. cr. lq. cr. lq. mn. mn. lf./al. col. cr. cr./et. lq. col. lq. tet. pr. lf./al. col. gas col. oil lq. col. lq. lq. col. lq. col. lq. col. lq. col. lq. mn. lq. nd./aq. lq. cr. col. lq. col. lq. yel. oil col. lq. col. lq. col. lq. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. lf. col. lq. yel. lq. col. lq. oil

0.992 73/4 1.160 20/20 1.272 14 1.583 40 0.978 50/50 1.073 20 1.001 26/4 1.119 15/15 1.248 25/4 1.276 0.979 20/4 0.739 20/4 0.722 22 0.910 20 0.968 22 0.744 21/0 0.725 21/0 0.822 20/4 0.806 20/4 1.038 0/0 1.34 1.1020/4 0.776 20/4 0.778 44/4 0.751 20/4 1.466 15 0.838 81/4 0.851 79/4 1.183 25/25 1.204 25 1.451 20/4 −88

0.546 1.022 20 1.169 25 0.708 25/4 1.020 20/20 0.901 20/4 1.025 20/4 0.789 20/4 0.689 15/15 1.216 0.963 20/4 1.103 25/25 1.117 20/4 0.867 20/4 1.052 15/15 1.034 18.5 1.431 20/4 0.879 20/4 0.871 20/4 0.859 28 0.873 20/20 0.878 17 0.917 6/6 1.159 20/4 1.138 20/4 1.049 20/4 1.062 20/4 0.923 20/4 1.117 21/4 0.872 20/20 1.013 −6/4 1.933 20/4 1.030 25/4

69–70 52.9 80 43–4 52–3 27 147–8 26–7 152 122–3 <−40 128–9 189 86–7 −20.3 −39.6 −61 −122 −60 −32.6 −51.7

254.9 302 302–6 d. 327 284 259 d. > 170 265 330 296–7 379 287–8 143–510 110–1 83.5743 245.4 168.2 91 69 144.2 123.7 213.5 190

125–6 178–9 44.5 −9.6 189 79–80 51–2 40 −25.6 126 61 −172 10.5 <−40 −116.3 −82.4 −45 −112 −80.6 108–9 −63.5 d. 294 7–8 13 −94.4 −34.6 −117.8 −93.3 −88.2 −20 −67.5 −45 −139 −26 −80.6 12 −22.5 −79 34 −66.1 expl. −105

85 224.515 214.5 290–53 193–5 288100 255 117756 94 329–31 expl. −88.6 171757 d. 34.6 2004 77.1 180755 78.4 16.6 204 269–70 266–8 136.2 211–2 28510 38.4 120–1 110–1 244.6758 165–6736 207–8753 13 144 94–5 271 208753 54760 195766 187–8 36752 72.4 155

i. 0.0325 i. 0.2 d. i. v. sl. s. v. sl. s. i. i. i. i. sl. s. h. v. sl. s. i. i. s. s. i. v. sl. s. sl. s. 0.2 0.43 v. sl. s. d. h. 0.0328 v. sl. s. i. i. i. 3.215 i. i. i. 5 <5 i. 60 i. c. 4.7 cc.20 ∞ ∞ 7.520 i. 8.515 1317 ∞ ∞ 24017 i. 2.1515 i. v. sl. s. 0.0115 0.0820 i. 1.060 0.6825 sl. s. 0.00220 i. i. 0.450 i. d. i. 225 1118 i. 0.02920 0.420 ∞

1020 5619.5 v. s. 20 s. s. 920 v. s.

6.620 s. s. s. v. s. ∞ sl. s. v. s. s. ∞

s. h. s. h. s. s. h. s. ∞ s. s.

s. v. s. s. ∞

∞ ∞ ∞ ∞

∞ ∞ ∞ ∞

s. h.

s.

4 h. v. s. v. sl. s. s. v. s. s. 500 ∞ ∞ sl. s. c. s. 150 cc. ∞

v. s. v. s. i. s. v. s.

s.

s. ∞ ∞ i. s. 1

∞ ∞ ∞ ∞ v. s. ∞ s. s. ∞ ∞ 18 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

∞ ∞ ∞ ∞ i. ∞ s. s. ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

2-37

TABLE 2-2

Physical Properties of Organic Compounds (Continued ) Name

Formula

laurate mercaptan methacrylate naphthylamine (α-) naphthyl ether (α-) nitrate nitrite oleate palmitate pelargonate propionate salicylate (o-) stearate toluate (o-) (m-) toluene sulfonate (p-) toluidine (o-) (p-) urea valerate (n-) (i-) Ethylal Ethylene bromide bromohydrin chlorobromide chlorohydrin diamine oxide Ethylidene diacetate Eugenol (1-,4-,3-) (i-)(1-,3-,4-) Fenchyl alcohol (dl-) (d-)(α-) (i-)(l-) Ferric dimethyl-dithiocarbamate Fluorene Fluorescein Fluoro-dichloromethane -trichloromethane Formaldehyde (m-) (p-) Formamide Formanilide Formic acid Fructose Fuchsin Fulminic acid Fumaric acid (trans-) Furfural Furfuran Furfuryl acetate alcohol butyrate propionate Furoic acid G-acid, K salt (2-)(6-,8-) Na salt (2-)(6-,8-) Galactose (d-)(α-) Gallic acid (3-,4-,5-) Gamma acid (2-,8-,6-) Geraniol Glucose (d-)(α-) (d-)(β-) Glucuronic acid Glutam(in)ic acid (dl-)

CH3(CH2)10CO2C2H5 CH3CH2SH CH2:C(CH3)CO2C2H5 C10H7NHC2H5 C10H7OC2H5 C2H5ONO2 C2H5ONO C17H33CO2C2H5 CH3(CH2)14CO2C2H5 CH3(CH2)7CO2C2H5 CH3CH2CO2C2H5 HOC6H4CO2C2H5 CH3(CH2)16CO2C2H5 CH3C6H4CO2C2H5 CH3C6H4CO2C2H5 CH3C6H4SO3C2H5 CH3C6H4NHC2H5 CH3C6H4NHC2H5 C2H5NHCONH2 CH3(CH2)3CO2C2H5 (CH3)2CH(CH2)2CO2C2H5 CH2(OC2H5)2 H2C:CH2 BrCH2CH2Br BrCH2CH2OH ClCH2CH2Br ClCH2CH2OH H2NCH2CH2NH2 < (CH2)2 > O CH3CH(O2CCH3)2 C3H5C6H3(OH)OCH3 C3H5C6H3(OCH3)OH C10H17OH C10H17OH C10H17OH Fe[SSCN(CH3)2]3 (C6H4)2 > CH2 C20H12O5 FCHCl2 Cl3CF HCHO (CH2O)3 (CH2O)xxH2O HCONH2 HCONHC6H5 HCO2H CH2OH(CHOH)3COCH2OH C20H19N3HCl C:NOH HO2CCH:CHCO2H C4H3OCHO C4H4O CH3CO2CH2C4H3O C4H3OCH2OH C3H7CO2CH2C4H3O C2H5CO2CH2C4H3O C4H3OCO2H HOC10H5(SO3K)2 HOC10H5(SO3Na)2 C5H11O5CHO (HO)3C6H2CO2HH2O C10H5(NH2)(OH)SO3H C9H15CH2OH C5H11O5CHO C6H12O6H2O CHO(CHOH)4CO2H [CHNH2(CH2)2](CO2H)2

2-38

Formula weight

Form and color

Specific gravity

Melting point, °C

Boiling point, °C

228.36 62.13 114.14 171.23 172.22 91.07 75.07 310.50 284.47 186.29 102.13 166.17 312.52 164.20 164.20 200.25 135.20 135.20 88.11 130.18 130.18 104.15 28.05 187.88 124.98 143.43 80.52 60.10 44.05 146.14 164.20 164.20 154.24 154.24 154.24 416.47 166.21 332.30 102.93 137.38 30.03 90.08 (30.03) 45.04 121.13 46.03 180.16 337.84 43.03 116.07 96.08 68.07 140.13 98.10 168.19 154.16 112.08 380.46 348.26 180.16 188.13 239.24 154.24 180.16 198.17 194.14 147.13

oil lq. col. lq. oil cr. col. lq. lq. oil col. nd. col. lq. col. lq. col. lq. col. cr. lq. lq. pr./al. lq. lq. nd. col. lq. col. lq. lq. col. gas col. lq. col. lq. lq. col. lq. col. lq. lq. col. lq. oil oil col. cr. col. pr. col. cr. cr. cr./al. yel. red gas col. lq. gas wh. amor. lq. mn. col. lq. nd./aq. red

0.868 13/4 0.839 20/4 0.913 15.6 1.060 20/4 1.061 20/20 1.100 25/4 0.900 15.5 0.867 25 0.858 25/4 0.866 17.5 0.891 20/4 1.136 15/4 0.848 36.3 1.032 25/25 1.030 20/20 1.166 48/4 0.948 25/4 0.942 25/4 1.213 18 0.877 20 0.867 20/4 0.824 25/4 0.57−102/4 2.180 20/4 1.772 20/4 1.689 19 1.213 20/4 0.900 20/20 0.887 7/4 1.061 12 1.070 15/15 1.091 15/15 0.935 40 0.964 20/4 0.961

−10.7 −121

269 36–7 118 303723 276.4 87–8 17 216–815 19110 227–8757 99.1 233–4 20110 227 231750 221.3 215–6 217

col. pr. lq. col. lq. col. oil oil col. lq. col. lq. mn. pr. cr. cr. pr. mn./aq. cr. col. lq. rhb. cr. cr. cr./aq.

1.635 20/4 1.159 20/4 0.937 20/4 1.118 20/4 1.129 25/4 1.053 20/4 1.109 20/4

1.203 0/4 1.4260 1.494 17.2 0.815 −20 1.1765 1.139 20/4 1.147 15/15 1.220 20/4 1.669 17.5 1.22

5.5 −102 <−15 24–5 −44.5 −72.6 1.3 33.4(31) <−10 33–4 <−15 92 −91.2 −99.3 −66.5 −169 10 −16.6 −69 8.5 −111.3 18.85 10.3 −10 35 45–7 61–2 d. 100–30 115–6 d. > 290 −127 −92 64 150–60 2 47 8.6 95–105 d. >200 286–7 −38.7

133–4

1.694 4/4 0.883 15 1.544 25 1.562 18/4 1.460

145.5 135 89 −103.9 131.5 150.3 106.7 128.8 117.2 13.5747 168740 253.5 267.5 201 201–2 201–2 ign. >150 293–5 14.5 24.9 −21 114.5759 subl. 193 216120 100.8

290 161.7760 31–2756 175–7 169.5752 212–3 195–6 230–2

165.5 d. 220 <−15 146 150 154 199 d.

230 d.

Solubility in 100 parts Water

Alcohol

Ether

i. 1.5 i. i. i. 1.355 v. sl. s. i. i. i. 2.420 i. i. i. i. i. i. i. v. s. 0.2425 0.1720 918 26 cc.0 0.4380 sl. s. 0.6980 ∞ ∞ ∞ sl. s. v. sl. s. v. sl. s. sl. s. sl. s. i. v. sl. s. i. v. sl. s. h. i. i. v. s. 2125 20–3018 ∞ sl. s. ∞ v. s. 0.3

s. s. s. s. s. ∞ ∞ ∞ s. ∞ ∞ ∞ s. ∞ ∞ s.

∞ s. s. s. s. ∞ ∞ ∞ s. ∞ ∞ ∞ s. ∞ ∞ s.

80 ∞ ∞ ∞ 360 cc. ∞ s.

i. ∞ ∞ ∞ s. ∞

∞ ∞ ∞ ∞ ∞ ∞

∞ 0.3 v. s.

s.

s.

s. h. s. h. s. ∞ v. s. s. i. ∞ v. s. ∞ 818 s.

s.

0.717 9.113 i. i. ∞ v. sl. s. v. sl. s. 3.615 825 3420 10.30 113

5.830 ∞ s. s. s. s. s. s.

0.725 ∞ s. s. s. ∞ ∞ s.

0.640 2815

2.515

i. 8217.5 15415 v. s. 1.5 20

∞ sl. s.

∞ i.

v. sl. s.

v. sl. s.

∞ ∞

s. ∞ v. s. s. i. v. sl. s. s. i.

Glutaric acid Glycerol acetate (mono-) (di-) nitrate (mono-) (α-) (β-) dinitrate (1-,3-) Glyceryl triacetate tribenzoate tributyrate tricaprate tricaproate tricaprylate trilaurate trimyristate trinitrate trinitrite trioleate tripalmitate tristearate Glycide Glycine, Glycocoll Glycol diacetate dibenzoate dibutyrate dicaprylate diformate dilaurate dinitrate dinitrite dipalmitate dipropionate ether formal formate (mono-) Glycolic acid Guaiacol (o-) Guanidine H-acid, Na salt (1-,8-,3-,6-) Heptacosane (n-) Heptane (n-) (i-)

Heptoic acid aldehyde Heptyl acetate (n-) alcohol (n-) mercaptan Hexachloro-benzene -ethane Hexacosane (n-) Hexadecane (n-) Hexaethylbenzene Hexamethylbenzene Hexamethylene-diamine -diisocyanate -glycol tetramine Hexane (n-) (i-) (neo-)

CH2(CH2CO2H)2 CH2OHCHOHCH2OH C5H10O4 (CH3CO2)2C3H5OH CH2OHCHOHCH2NO3 CH2OHCHNO3CH2OH CHOH(CH2ONO2)2 (CH3CO2)3C3H5 (C6H5CO2)3C3H5 (C2H5CH2CO2)3C3H5 [CH3(CH2)8CO2]3C3H5 [CH3(CH2)4CO2]3C3H5 [CH3(CH2)6CO2]3C3H5 [CH3(CH2)10CO2]3C3H5 [CH3(CH2)12CO2]3C3H5 CH2NO3CHNO3CH2NO3 CH2NO2CHNO2CH2NO2 (C17H33CO2)3C3H5 [CH3(CH2)14CO2]3C3H5 [CH3(CH2)16CO2]3C3H5 C2H3OCH2OH NH2CH2CO2H CH2OHCH2OH (CH3CO2CH2)2 (C6H5CO2CH2)2 (C3H7CO2CH2)2 (C7H15CO2CH2)2 (HCO2CH2)2 (C11H23CO2CH2)2 (O2NOCH2)2 (ONOCH2)2 (C15H31CO2CH2)2 (C2H5CO2CH2)2 (HOCH2CH2)2O < OCH2CH2OCH2 > HCO2CH2CH2OH HOCH2CO2H CH3OC6H4OH NH:C(NH2)2 C10H8O7NS2Na1a H2O CH3(CH2)25CH3 CH3(CH2)5CH3 (CH3)2CH(CH2)3CH3 C3H7CH(CH3)C2H5 (CH3)3CCH2C2H5 [(CH3)2CH]2CH2 (CH3)2C(C2H5)2 (C2H5)3CH (CH3)3CCH(CH3)2 CH3(CH2)5CO2H CH3(CH2)5CHO CH3CO2CH2(CH2)5CH3 CH3(CH2)5CH2OH [(CH3)2CH]2CHOH (C2H5CH2CH2)2CHOH CH3CH(SH)C5H11 C6Cl6 CCl3CCl3 CH3(CH2)24CH3 CH3(CH2)14CH3 C6(C2H5)6 C6(CH3)6 NH2(CH2)6NH2 OCN(CH2)6NCO HO(CH2)6OH (CH2)6N4 CH3(CH2)4CH3 (CH3)2CH(CH2)2CH3 (CH3)3CC2H5 (CH3)2CHCH(CH3)2 (C2H5)2CHCH3

132.11 92.09 134.13 176.17 137.09 137.09 182.09 218.20 404.40 302.36 554.83 386.51 470.67 638.98 723.14 227.09 179.09 885.40 807.29 891.45 74.08 75.07 62.07 146.14 270.27 202.24 314.45 118.09 426.66 152.07 120.07 538.87 174.19 106.12 74.08 90.08 76.05 124.13 59.07 368.31 380.72 100.20 100.20 100.20 100.20 100.20 100.20 100.20 100.20 130.18 114.18 158.24 116.20 116.20 116.20 132.26 284.80 236.76 366.69 226.43 246.42 162.26 116.20 168.19 118.17 140.19 86.17 86.17 86.17 86.17 86.17

col. cr. col. lq. col. oil col. lq. col. pr. lf. oil col. lq. nd. col. lq. col. cr. col. lq. col. lq. col. nd. lf. yel. oil yel. lq. col. oil col. nd. col. pr. col. lq. mn. col. lq. col. lq. rhb./et. col. lq. lq. lq. amor. yel. lq. lq. nd. lq. lq. lq. lq. nd./aq. pr. col. cr. cr. col. cr. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. lq. lq. mn. rhb. cr. lf. pr./al. pl./al. lf. lq. nd./aq. col. rhb. col. lq. lq. lq. lq. lq.

1.429 15 1.260 50/4 1.20 20/4 1.178 15/15 1.4015 1.4015 1.4715 1.16117/4 1.228 12 1.032 20/4 0.921 40/4 0.987 20/4 0.954 20/4 0.894 60/4 0.885 60/6 1.601 15 1.291 10/16 0.915 15 0.866 80/4 0.862 80/4 1.114 16/16 1.161 1.113 19/4 1.109 14/4 1.024 0

97.5 17.9 40 58–9 54 <−30 −78 75–6 <−75 31(25) −25 8.3(−21) 45–6 56.5 13.3(2) −4 65.1 70.8(55)

20020 290 158165 175–640 155–60 155–60 146–815 258–9 d. 305–9

16015 150 sl. d. 24018 310–200.1 166 sl. d.

232–6 d. −15.6 −31 73–4

197.4 190.5 >360 240

22 1.482 21/2 1.216 0 1.045 25 1.118 20/20 1.060 20/4 1.199 15/4 1.140 15/15 0.780 60/4 0.684 20/4 0.679 20/4 0.687 20/4 0.674 20/4 0.675 20/4 0.693 20/4 0.698 20/4 0.690 20/4 0.918 20 0.850 20/ᐉ 0.874 16/16 0.824 20/4 0.829 20/4 0.820 20/4 0.835 20 2.044 24 2.091 20/4 0.779 57/4 0.774 20/4 0.831 130/4 1.0428 20/4

0.659 0.654 20/4 0.649 20/20 0.662 20/4 0.664 20/4

52–4 −20 <−15 71–2 −10.5 79(63) 28.3 50 59.5 −90.6 −118.2 −119.4 −125 −119.4 −135.0 −118.7 −25 −10 −42 34.6 −37 228–31 186–7 56.6 18.5 130 166 42 42 subl. −94 −153.7 −98.2 −129.8 −118

174 18820 expl. 114 96–8 2600.1 211–2 244.8 75–6 180 d. 205 27015 98.4760 90.0 91.8 79.1 80.8 86.0 93.5 80.8 221–2 155 191.5 759 175 756 140 156 174–5 765 309742 186777 26215 287.5 298.3 265 204–5 143–420 250 69 60.2 49.7 58.0760 63.2

63.920 ∞ v. s. s. 7015 7.1715 i. i. i. i. i. i. i. 0.1820 d. i. i. i. ∞ 23 c. ∞ 14.322 i. i. i. v. sl. s. i. 0.9225 i. i. sl. s. ∞ ∞ ∞ ∞ 1.715 v. s. 0.1720 i. 0.00515 i. i. i. i. i. i. i. 0.2515 0.0220 i. 0.18 25 v. sl. s. i. i. i. 0.00522 i. i. i. i. v. s. d. s. 8112 0.01415 i. i. i. i.

v. s. ∞ v. s. s. v. s. v. s. v. s. ∞ s. h. s. s. h. s. s. sl. s. c. s. 5020 d. sl. s. 0.00421 s. h. ∞ 0.1 c. ∞ ∞

v. s. i. sl. s. sl. s. v. sl. s. sl. s. v. s. ∞ s. s. v. s. s. s. v. s.

v. s.

∞ s. v. s. v. s. s. h. ∞ i. 1.0 ∞ s. v. s.

v. s. s. s. d. s. ∞ ∞

v. s. ∞ s. s. ∞ i.

9025 v. s. s.

v. s. v. s.

sl. s. s. s. s. s. s. s. s. s. ∞ s. ∞ ∞ s.

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ s. ∞ s. ∞ ∞ s.

v. sl. s. h. v. s. v. sl. s. ∞ 0.7525 0.20 s. d. s. 3 5033

s. h. v. s. ∞ 825 v. s. sl. s. h. v. sl. s. ∞ s. s. s. s.

TABLE 2-2

Physical Properties of Organic Compounds (Continued ) Name

Hexyl acetate (n-) alcohol (n-) formate (n-) resorcinol (2-,4-) Hippuric acid Histidine (l-) Homophthalic acid (o-) Hydracrylic acid Hydro-cyanic acid -quinone (p-) Hydroxy-benzaldehyde (p-) -benzanilide (o-) -quinoline (2-)(α-) (8-)(o-) Indigo White Indole Indoxyl Iodo-benzene -phenol (p-) Iodoform Ionone (α-) (β-) Irone (β-) Isatin Isoprene Ketene Koch acid (1-)(3-,6-,8-) Lactic acid (dl-) anhydride Lactide (dl-) Lactose Lauric acid Laurone Lauryl alcohol Lead tetraethyl tetramethyl Lecithin (protagon) Lepidine (py-4) Leucine (l-) Levulinic acid Limonene (d- or l-) Linalool (d- or l-) Linalyl acetate Linoleic acid Maleic acid anhydride Malic acid (dl-) (d- or l-) Malonic acid Maltose Mandelic acid (dl-) Mannitol (d-) Mannose (d-) Margaric acid Mellitic acid Menthol (l-)(α-) Mercapto-benzothiazole (2-) -thiazoline (2-) Mercuric cyanide fulminate Mesityl oxide Mesitylene (1-,3-,5-) Metanilic acid (m-) Methane

2-40

Formula CH3CO2(CH2)5CH3 CH3(CH2)4CH2OH (CH3)2CHC(CH3)2OH (CH3)2COHCH2C2H5 HCO2CH2(CH2)4CH3 CH3(CH2)5C6H3(OH)2 C6H5CONHCH2CO2H C6H9O2N3 HO2CC6H4CH2CO2H HOCH2CH2CO2H HCN C6H4(OH)2 HOC6H4CHO HOC6H4CONHC6H5 C9H6NOH C9H6NOH [C6H4(CO)(NH)C:]2 C16H12O2N2 C8H7N C8H6NOH C6H5I IC6H4OH HCI3 C10H16:CHCOCH3 C10H16:CHCOCH3 C14H22O C6H4 < (CO)(N) > COH CH2:CHC(CH3):CH2 H2C:CO C10H4(NH2)S3O9HNa2 CH3CH(OH)CO2H C6H10O5 C6H8C4 C12H22O11H2O CH3(CH2)10CO2H [CH3(CH2)10]2CO CH3(CH2)10CH2OH Pb(CH2CH3)4 Pb(CH3)4 C42H48O9PN C9H6NCH3 (CH3)2CHCH2CH(NH2)CO2H CH3CO(CH2)2CO2H C10H16 C10H17OH CH3CO2C10H17 C17H31CO2H HO2CCH:CHCO2H < (CHCO)2 > O HO2CCH2CH(OH)CO2H HO2CCH2CH(OH)CO2H H2C(CO2H)2 C12H22O11H2O C6H5CH(OH)CO2H CH2OH(CHOH)4CH2OH CH2OH(CHOH)4CHO CH3(CH2)15CO2H C6(CO2H)6 C10H19OH < C6H4N:C(SH)S > < CH2N:C(SH)SCH2 > Hg(CN)2 Hg(ONC)2a H2O (CH3)2C:CHCOCH3 C6H3(CH3)3 H2NC6H4SO3H CH4

Formula weight

Form and color

Specific gravity

144.21 102.17 102.17 102.17 130.18 194.26 179.17 155.16 180.15 90.08 27.03 110.11 122.12 213.23 145.15 145.15 262.26 264.27 117.14 133.14 204.02 220.02 393.78 192.29 192.29 206.32 147.13 68.11 42.04 427.34 90.08 162.14 144.12 360.31 200.31 338.60 186.33 323.45 267.35 778.08 143.18 131.17 116.11 136.23 154.24 196.28 280.44 116.07 98.06 134.09 134.09 104.06 360.31 152.14 182.17 180.16 270.44 342.17 156.26 167.24 119.20 252.65 293.65 98.14 120.19 173.18 16.04

col. lq. col. lq. lq. lq. lq. col. nd. rhb. lf./aq. cr./aq. syrup lq. cr. nd./aq. pr./al. pr./al. pr. cr. gray lf./aq. yel. pr. col. lq. nd./aq. yel. hex. col. oil col. oil col. oil yel. red col. lq. col. gas cr. hyg. yel. oil tri./al. col. rhb. col. nd. pl. lf. col. lq. col. lq. wax lq. cr. lf. lq. col. oil col. lq. yel. oil mn. cr. col. cr. col. cr. col. tri. col. nd. rhb./aq. col. rhb. rhb. col. pl. nd./al. col. cr. nd. cr. cr. cr./aq. lq. col. lq. col. nd. gas

0.890 0/0 0.820 20/20 0.821 20/0 0.809 20/4 0.898 0 1.371 20/4

0.697 18 1.332 15 1.129 130

1.35

1.824 25/4 1.857 112 4.008 17 0.930 20 0.944 20 0.939 20

Melting point, °C −51.6 −14 −107 68–70 187–8 d. 287 175–80 −12 170.3 116–7 135 199–200 75–6 390–2 52 85 −28.5 93–4 119

0.681 20/4

200–1 −120 −151

1.249 15/4

16.8

10/4

0.862 1.525 20 0.869 50/4 0.809 69/4 0.831 24/4 1.659 18/4 1.995 20/4 1.086 20 1.29318 1.140 20/20 0.842 20/4 0.868 20 0.895 20 0.903 18/4 1.609 1.5 1.601 20/4 1.595 20/4 1.631 15 1.540 17 1.300 20/4 1.489 20/4 1.539 20/4 0.853 60 15/15

0.890 1.4220/4 1.50 4.003 22 4.4 0.858 20/4 0.865 20/4

0.415 −164

124.5 202 48(44) 69–70 24 −136 −27.5 150–200 d. 9–10 295 33.5 −96.9 −9.5 130.5 57–60 128–9 99–100 130–5 d. d. 118.1 166 132 60–1 286–8 42–3 179 106 d. 320 expl. −59 −45(−52) d. −182.6

Boiling point, °C 169.2 157.2 120–1 123762 153.6 1797 d. d. 25–6 285730 subl. d. subl. 266.6752 subl. 253–4 110 188.6 d. subl. 136.117 14018 14416 subl. 34 −56 12214 d. 250 255757 d. 225100 255–9 152291 110760 261–3 subl. 245–6 177 198–200 220762 d. 229–3016 135 d. 202 150 d. 140 d. d. 290–53 227100 d. 212 d.

130750 164.8 −161.4

Solubility in 100 parts Water i. 0.620 v. sl. s. v. sl. s. 0.05 0.420 s. s. h. ∞ 615 1.3831 v. sl. s. h. s. h. v. sl. s. c. i. i. s. h. s. 0.03420 sl. s. 0.0125 sl. s. sl. s. v. sl. s. s. h. i. d. 7.220 ∞ v. sl. s. v. sl. s. 1710 i. i. i. i. i. i. sl. s. 2.218 v. s. i. v. sl. s. v. sl. s. i. 7925 16.380 14426 v. s. 13816 10825 1620 1314 24817 i. v. s. 0.04 c. i. 1.660 12.515 0.0712 320 i. 215 0.420 cc.

Alcohol

Ether

v. s. ∞ ∞ ∞ ∞ v. s. s. h. v. sl. s. v. s.

v. s. ∞ ∞ ∞ ∞ s. 0.2518 i. sl. s.

∞ v. s.

∞ v. s.

s. v. s. s. i. s. s. h. s. s. v. s. 1.517 ∞ ∞ v. s. v. s. h. ∞ d.

s. v. s. sl. s. i. s. s. s. ∞ v. s. 13.625 ∞ ∞ v. s. sl. s. ∞ s.

∞ s. v. sl. s. c. i. s. i. c. s. sl. s. ∞ s. h. ∞

∞ s.

s. ∞ ∞ s. h. ∞

v. s. ∞ s. ∞ ∞ 7030

v. s. ∞ ∞ ∞ ∞ 825

v. s. v. s. 4225 v. sl. s. c. s. 0.0114 v. sl. s. 3228 v. s. v. s. s.

v. s. 8.415 815 i. s. i. i. v. s.

s. ∞ s. v. sl. s. 4720 cc.

i. s.

v. s. sl. s.

∞ ∞ v. sl. s. 10410 cc.

Methoxy-methoxyethanol Methyl acetate acrylic acid (α-) alcohol -amine -amine hydrochloride aniline anthracene (α-) (β-) anthranilate (o-) anthraquinone (2-) benzoate benzylaniline bromide butyrate (n-) (i-) caprate caproate (n-) caprylate cellosolve chloride chloroacetate chloroformate cinnamate cyclohexane ethyl carbonate ethyl ketone ethyl oxalate formate furoate glucamine glycolate heptoate hypochlorite iodide lactate laurate mercaptan methacrylate myristate naphthalene (α-) (β-) nitrate nitrite nonyl ketone (n-) oleate orange palmitate phosphine propionate propyl ketone (n-) salicylate (o-) stearate toluate (o-) (m-) (p-) Methyl toluidine (o-) (m-) (p-) valerate (n-) (i-) vinyl ketone Methylal Methylene-bis-(phenyl-4-isocyanate) bromide chloride dianiline iodide Michler’s hydrol (p-,p′-) ketone Morphine Mucic acid

CH3(OCH2)2CH2OH CH3CO2CH3 CH2:C(CH3)CO2H CH3OH CH3NH2 CH3NH2HCl C6H5NHCH3 C6H4:(CH)2:C6H3CH3 C6H4:(CH)2:C6H3CH3 NH2C6H4CO2CH3 C6H4:(CO)2:C6H3CH3 C6H5CO2CH3 C6H5N(CH3)CH2C6H5 CH3Br CH3(CH2)2CO2CH3 (CH3)2CHCO2CH3 CH3(CH2)8CO2CH3 CH3(CH2)4CO2CH3 CH3(CH2)6CO2CH3 CH3OCH2CH2OH CH3Cl ClCH2CO2CH3 ClCO2CH3 C6H5CH:CHCO2CH3 CH2 < (CH2CH2)2 > CHCH3 CH3OCOOC2H5 CH3.COC2H5 CH3OCOCO2C2H5 HCO2CH3 C4H3OCO2CH3 CH2OH(CHOH)4CH2NHCH3 HOCH2CO2CH3 CH3(CH2)5CO2CH3 ClOCH3 CH3I CH3CH(OH)CO2CH3 CH3(CH2)10CO2CH3 CH3SH CH2:C(CH3)CO2CH3 CH3(CH2)12CO2CH3 C10H7CH3 C10H7CH3 CH3ONO2 CH3ONO CH3(CH2)8COCH3 C17H33CO2CH3 (CH3)2NC6H4N2C6H4SO3Na CH3(CH2)14CO2CH3 CH3PH2 CH3CH2CO2CH3 CH3COCH2CH2CH3 HOC6H4CO2CH3 CH3(CH2)16CO2CH3 CH3C6H4CO2CH3 CH3C6H4CO2CH3 CH3C6H4CO2CH3 CH3C6H4NHCH3 CH3C6H4NHCH3 CH3C6H4NHCH3 CH3(CH2)3CO2CH3 (CH3)2CHCH2CO2CH3 CH3COCH:CH2 HCH(OCH3)2 (OCNC6H4)2CH2 CH2Br2 CH2Cl2 (C6H5NH)2CH2 CH2I2 [(CH3)2NC6H4]2CHOH [(CH3)2NC6H4]2CO C17H19O3NH2O (CHOHCHOHCO2H)2

106.12 74.08 86.09 32.04 31.06 67.52 107.15 192.25 192.25 151.16 222.23 136.14 197.27 94.95 102.13 102.13 186.29 130.18 158.23 76.09 50.49 108.53 94.50 162.18 98.18 104.10 72.10 132.11 60.05 126.11 195.21 90.08 144.21 66.49 141.95 104.10 214.34 48.10 100.11 242.39 142.19 142.19 77.04 61.04 170.29 296.48 327.33 270.44 48.03 88.10 86.13 152.14 298.49 150.17 150.17 150.17 121.18 121.18 121.18 116.16 116.16 70.09 76.09 250.25 173.86 84.94 198.26 267.87 270.36 268.35 303.35 210.14

lq. col. lq. pr. col. lq. col. gas pl./al. lq. lf./al. col. lf. col. lq. col. nd. col. lq. lq. gas col. lq. col. lq. lq. col. lq. col. lq. col. lq. gas col. lq. col. lq. cr. col. lq. lq. col. lq. lq. lq. col. lq. lq. lq. gas col. lq. lq. lq. gas lq. cr./al. oil mn. lq. gas col. oil oil red pd. col. cr. gas col. lq. col. lq. col. lq. col. cr. col. lq. col. lq. cr. lq. lq. lq. lq. col. lq. lq. col. lq. lq. col. lq. col. lq. cr. col. lq. gn. lf./al. pr./al. pd.

1.038 25 0.924 20/4 1.015 20/4 0.792 20/4 0.699 −11 1.23 0.989 20/4 1.047 99.4 1.181 0/4 1.168 19/4 1.087 25/25 1.732 0/0 0.898 20/4 0.891 20/4 0.904 0/0 0.887 18 0.965 20/4 0.952 0 1.236 20/4 1.236 15 1.042 36/0 0.769 20/4 1.002 27 0.805 20/4 1.156 0/0 0.974 20/4 1.179 21/4

<−70 −98.7 15–16 −97–8 −92.5 226–8 −57 86 207 24 176–7 −12.5 9.2 −93 <−95 −84.7 −18 −40 −97.7 −32.7 33.4 −126.3 −14.5 −85.9 −99.8

18

1.168 0.881 15/4 20/4

2.279 1.090 19

0.896 0 0.950 15.6 1.025 14/4 0.994 40/4 1.203 25 0.991 15 0.828 20/20 0.879 18

−64.4 5 −121 −48 18–9 −19 35–6 expl. 13.5 30–1

20/4

0.915 0.812 15/15 1.182 25/25 1.073 15 1.066 15 0.973 15 0.935 55/4 0.895 15/4 0.881 20/4 0.836 20/4 0.866 15/4 1.222 30 2.495 20/4 1.336 20/4 3.325 20/4 1.317

−87.5 −77.8 −8.3 38–9 <−50 33–4

−91 −104.8 −52.8 −96.7 65 5.7 96–7 174 254 d. 206–14

167.5 57.1 161–3 64.7 −6.7 758 23015 195.5 135.5 15 subl. 198–9 305–6 4.5 758 102.3 92.6 223–4 149.5 192–4 124–5 −24 130740 71–2 263 101 109.2 79.6 173.7 32 181.3 151.2 172–3 12726 42.4 144.8 14818 5.8752 100.3 295715 244.6 241–2 65 −12 228 190–110 19615 −14759 79.7 102 222.2 21515 213 215 217 206–7 206–7 211761 127.3 116.7764 81 42–3 210–213 98.5756 40–1 208–9 d. 180 d. >360 d.

∞ 3322 s. h. ∞ v. s. v. s. 0.0125 i. i. sl. s. i. 0.0230 i. v. sl. s. 1.7 v. sl. s. i. i. i. ∞ 28016 cc. v. sl. s. d. i. i. i. 3510 i. 3020 i.

∞ ∞ ∞ v. s. 23 h. s.

∞ ∞ ∞

v. sl. s. s. s. ∞ s. s. ∞ ∞ ∞ ∞ ∞ ∞ v. s. ∞ ∞ v. s. s. ∞ ∞ v. s. ∞ ∞

v. sl. s. s. s. ∞ s. s. ∞ ∞ ∞ ∞ ∞ ∞ v. s. ∞ ∞ v. s. s. ∞ ∞ v. s.

∞ s.

∞ s.

v. s.

v. s.

v. s. v. s. s. s. s. ∞

v. s. v. s. s. s. s. ∞

s. sl. s. ∞ ∞ ∞ s. ∞

∞ ∞ ∞ s. ∞

v. s. ∞ ∞ ∞ ∞ ∞

v. s. ∞ ∞ ∞ ∞ ∞

∞ d. ∞ ∞ s. ∞ s. h. sl. s. sl. s. i.

∞

i. ∞

∞

i. 1.815 ∞ i. s. i. i. i. i. sl. s. i. i. 0.2 c. i. i. 0.520 v. sl. s. 0.0730 i. i. i. i. i. i. i. v. sl. s. v. sl. s. >85 33 d. 1.170 220 i. 1.420 i. i. 0.0220 0.3314

s.

∞ ∞ s. ∞ s. v. sl. s. s. i.

TABLE 2-2

Physical Properties of Organic Compounds (Continued ) Name

Mustard gas Myricyl alcohol Myristic acid Myristyl alcohol Naphthalene disulfonic acid (1-,5-) (1-,6-) sulfonic acid (α-) (β-) Naphthasultam (1-,8-) disulfonate Na (1-,8-) (2-,4-) Naphthoic acid (α-) (β-) Naphthol (α-) (β-) sulfonic acid (α-)(1-,2-) (β-)(2-,6-) Naphthyl acetate (α-) (β-) amine (α-) (β-) amine hydrochloride (α-) (β-) amine sulfonic acid (1-,4-) (1-,5-) (1-,7-) (1-,8-) (2-,5-) (2-,6-) (2-,7-) isocyanate (α-) Nicotine Nicotinic acid (3-) (i-)(4-) Nitro-acetanilide (p-) -acetophenone (m-) -aminoanisole (4-,1-,2-) (5-,1-,2-) (3-,1-,4-) -aminophenol (4-,2-,1-) -aniline (o-) (m-) (p-) -anisole (o-) (p-) -anthraquinone (α-) -anthraquinone sulfonic acid (1-,5-) -benzal chloride (m-) -benzaldehyde (m-) Nitro-benzene -benzidine (2-) -benzoic acid (o-) (m-) (p-) -benzyl alcohol (m-) -benzyl bromide (p-) -chlorotoluene (1-,2-,6-) -cresol (1-,3-,4-) -cymene (1-,2-,4-) -dimethylaniline (o-) (m-) (p-) -diphenyl (o-) (p-) -diphenylamine (o-) -guanidine

2-42

Formula (ClCH2CH2)2S C31H63OH(?) CH3(CH2)12CO2H CH3(CH2)12CH2OH C10H8 C10H6(SO3H)2 C10H6(SO3H)2 C10H7SO3H2H2O C10H7SO2HH2O C10H7O2NS C10H5O8NS3Na22H2O C10H4O8NS3Na38a H2O C10H7CO2H C10H7CO2H C10H7OH C10H7OH HOC10H6SO3H HOC10H6SO3H CH3CO2C10H7 CH3CO2C10H7 C10H7NH2 C10H7NH2 C10H7NH2HCl C10H7NH2HCl NH2C10H6SO3H NH2C10H6SO3HH2O NH2C10H6SO3HH2O NH2C10H6SO3HH2O NH2C10H6SO3H NH2C10H6.SO3HH2O NH2C10H6SO3HH2O C10H7N:CO C10H14N2 C5H4NCO2H C5H4NCO2H CH3CONHC6H4NO2 CH3COC6H4NO2 NO2C6H3(OCH3)NH2 NO2C6H3(OCH3)NH2 NO2C6H3(OCH3)NH2 NO2C6H3(NH2)OH NO2C6H4NH2 NO2C6H4NH2 NO2C6H4NH2 CH3OC6H4NO2 CH3OC6H4NO2 C6H4:(CO)2:C6H3NO2 NO2C14H6O2SO3H NO2C6H4CHCl2 NO2C6H4CHO C6H5NO2 NH2C6H4C6H3(NH2)NO2 NO2C6H4CO2H NO2C6H4CO2H NO2C6H4CO2H NO2C6H4CH2OH NO2C6H4CH2Br CH3C6H3(NO2)Cl CH3C6H3(NO2)OH CH3C6H3(NO2)CH(CH3)2 NO2C6N4NHCH3 NO2C6H4NHCH3 NO2C6H4NHCH3 C6H5C6H4NO2 C6H5C6H4NO2 C6H5NHC6H4NO2 H2NC(NH)NHNO2

Formula weight

Form and color

Specific gravity

Melting point, °C

Boiling point, °C

159.08 452.82 228.36 214.38 128.16 288.28 288.28 244.26 226.24 205.22 445.35 584.45 172.17 172.17 144.16 144.16 224.22 224.22 186.20 186.20 143.18 143.18 179.65 179.65 223.24 241.26 241.26 241.26 223.24 241.26 241.26 169.17 162.23 123.11 123.11 180.16 165.14 168.15 168.15 168.15 154.12 138.12 138.12 138.12 153.13 153.13 253.20 333.26 206.03 151.12 123.11 229.23 167.12 167.12 167.12 153.13 216.04 171.58 153.13 179.21 166.18 166.18 166.18 199.20 199.20 214.22 104.07

oil cr. col. lf. cr. pl./al. lf. cr. cr. cr. nd. cr. lf. nd. mn. mn. mn. pl./aq. lf. nd./al. nd./al. rhb. lf./aq. nd. lf. nd. cr. cr. cr. cr. cr. cr. col. lq. oil nd./al. nd./aq. rhb. nd. red nd. yel. nd. red or. pr. yel. rhb. yel. rhb. yel. mn. col. cr. pr./al. nd. yel. cr. mn. nd./aq. yel. lq. red nd. tri./aq. mn. yel. mn. cr. nd./al. cr. yel. oil yel. oil red mn. yel. nd. rhb. nd./al. or. cr. nd./aq.

1.275 20/4 0.777 95 0.853 70/4 0.824 38/4 1.145 20/4

13–4 88 57–8 38 80.2 d. d. 125 90 125 177–8

217

1.077 100/4 1.224 4 1.217 4

1.123 25/25 1.061 98/4

160–1 184 96 122–3 >250 125 46–9 69–70 50 111–2

250.5100 16715 217.9

300 >300 278–80 285–6

300.8 306.1 subl.

d.

1.18 1.009 20/4

1.207 156 1.211 156 1.442 15 1.43 1.437 14 1.254 20/4 1.233 20

1.205

18/4

1.575 4/4 1.494 4/4 1.550 22/4

89/4

1.240 1.067 20/4 1.179 20/4 1.313 17 1.44

<−80 235.2 317 215–6 80–1 118 139–40 123 142–3 71.5 114 146–7 9.4 54 230 65 58 5.7 143 147.5 140–1 240–2 27 99–100 37.5 32 60–1 163–4 37 113–4 75–6 246–7

269–70 246730 subl. d. 202

284.1 306.4 331.7 272–3 274 2707 16423 210.9

subl. 175–803 238 12522 15215 151–380 280–5 320 340

Solubility in 100 parts Water 0.0725 i. i. <0.02 0.00325 10220 16420 v. s. 7730 s. h. v. s. v. s. v. sl. s. h. 0.00725 sl. s. h. 0.07425 v. s. h. v. s. sl. s. h. i. 0.17 c. v. s. h. 3.820 v. s. 0.2100 sl. s. 0.4625 0.42100 0.08 0.38100 0.28100 d. s. s. h. s. h. s. h. i. i.

Alcohol

Ether

s. v. sl. s. v. s. sl. s. 9.520 s. s. v. s.

s. v. s. v. s. s. v. s. i. i. sl. s.

sl. s.

s.

sl. s. s. h. s. v. s. v. s. v. s. s. s. v. s. s. s. v. s. i.

s. s. v. s. v. s. i. s. s. v. s. s. s. i.

s. ∞ s. h. sl. s. h. s. s. s.

s. ∞ v. sl. s. v. sl. s. s.

sl. s. sl. s. c. s. h. 0.1120 0.0819 0.1730 0.0630 i. s. i. 1.95112 0.1920 sl. s. h. 0.6520 0.24165 0.0215

s. v. s. v. s. 7.120 5.820 ∞ v. s. sl. s. i. v. s. h. v. s. h. v. s.

s. v. s. v. s. 7.920 6.120 ∞ v. s. v. sl. s. i. v. s. v. s. ∞

2811 3112 0.910

2211 2510 2.218

i. i. v. sl. s. i. v. sl. s. i. i. i. i.

219

v. s.

v. s.

v. s.

v. s. s. s. h. s. sl. s. c.

v. s. s.

sl. s.

v. sl. s.

9100

v. s. v. s.

-naphthalene (α-) (β-) -phenol (o-) (m-) (p-) -phenol sulfonic acid (1-,4-,2-) (1-,2-,4-) -phthalic acid (3-) (4-) -toluene (o-) (m-) (p-) -toluene sulfonic acid (1-,4-,2-) -toluidine (4-,1-,2-) (3-,1-,4-) Nitron Nitroso-dimethylaniline (p-) -naphthol (β-)(1-) Nonadecane (n-) Nonane (n-) Octadecane (n-) Octane (n-) (iso-) Octyl acetate (n-) (sec-) alcohol (n-) (sec-) Octylene (n-) Oleic acid Orcinol (1-,3-,5-) Oxalic acid Palmitic acid Pelargonic acid Penta-chloroethane -decane (n-) -erythritol Pentandiol Pentane (n-) (i-) (neo-) Phenacetin Phenanthrene Phenetidine (o-) (p-) Phenetole Phenol -phthalein -sulfonic acid (o-) Phenyl acetaldehyde acetic acid -acetylene aniline (o-) (p-) Phenyl-ethyl alcohol -glycine -hydrazine -hydrazine sulfonic acid (p-) isocyanate -methylpyrazolone (3-)(N-) -mustard oil naphthalene (α-) (β-) naphthylamine (α-) (β-) phenol (o-) (p-) propyl alcohol (γ-) quinoline (2-)(α-) (8-)(0-) salicylate, salol stearate urethane

C10H7NO2 C10H7NO2 NO2C6H4OH NO2C6H4OH NO2C6H4OH HOC6H3(NO2)SO3H3H2O HOC6H3(NO2)SO3H3H2O NO2C6H3(CO2H)2 NO2C6H3(CO2H)2 CH3C6H4NO2 CH3C6H4NO2 CH3C6H4NO2 CH3C6H3(NO2)SO3H2H2O NO2C6H3(CH3)NH2 NO2C6H3(CH3)NH2 C20H16N4 ONC6H4N(CH3)2 ONC10H6OH CH3(CH2)17CH3 CH3(CH2)7CH3 CH3(CH2)16CH3 CH3(CH2)6CH3 (CH3)3CCH2CH(CH3)2 CH3CO2CH2(CH2)6CH3 CH3CO2CH(CH3)C6H13 CH3(CH2)6CH2OH CH3(CH2)5CH(OH)CH3 CH3(CH2)5CH:CH2 C8H17CH:CH(CH2)7CO2H (HO)2C6H3CH3 HO2CCO2H2H2O CH3(CH2)14CO2H CH3(CH2)7CO2H CHCl2CCl3 CH3(CH2)13CH3 C(CH2OH)4 HOCH2(CH2)3CH2OH CH3(CH2)3CH3 (CH3)2CHCH2CH3 (CH3)2C(CH3)2 C2H5OC6H4NHCOCH3 < (C6H4CH)2 > C2H5OC6H4NH2 C2H5OC6H4NH2 C2H5OC6H5 C6H5OH C20H14O4 HOC6H4SO3He H2O C6H5CH2CHO C6H5CH2CO2H C6H5C:CH C6H5C6H4NH2 C6H5C6H4NH2 C6H5CH2CH2OH C6H5NHCH2CO2H C6H5NHNH2 H2NNHC6H4SO3H C6H5N:CO C4H5ON2C6H5 C6H5N:CS C10H7C6H5 C10H7C6H5 C10H7NHC6H5 C10H7NHC6H5 C6H5C6H4OH C6H5C6H4OH C6H5(CH2)3OH C6H5C9H6N C6H5C9H6N HOC6H4CO2C6H5 CH3(CH2)16CO2C6H5 C6H5NHCO2C2H5

173.16 173.16 139.11 139.11 139.11 273.22 273.22 211.13 211.13 137.13 137.13 137.13 253.23 152.15 152.15 312.36 150.18 173.16 268.51 128.25 254.48 114.22 114.22 172.26 172.26 130.22 130.22 112.21 282.45 124.13 126.07 256.42 158.23 202.31 212.41 136.15 104.15 72.15 72.15 72.15 179.21 178.22 137.18 137.18 122.16 94.11 318.31 187.68 120.14 136.14 102.13 169.22 169.22 122.16 151.16 108.14 188.20 119.12 174.20 135.18 204.26 204.26 219.27 219.27 170.20 170.20 136.19 205.25 205.25 214.21 360.56 165.19

yel./al. col./al. yel. mn. col. mn. yel. pr. nd. nd./aq. yel./aq. yel. cr. yel. lq. lq. rhb. pl./aq. yel. mn. red mn. yel. lf. gn. tri. brn. pr. cr. col. lq. cr. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. lq. col. nd. pr./bz. col. mn. col. pl. col. oil col. lq. col. lq. cr. lq. col. lq. col. lq. col. lq. col. mn. pl./al. oil lq. col. lq. col. nd. col. rhb. cr. lq. lf. col. lq. cr. lf. col. oil cr. yel. oil cr./al. lq. pr./aq. col. lq. waxy lf./al. pr./al. rhb. nd. nd. oil nd. lq. rhb./al. cr. pl./al.

1.223 62 45

1.295 1.485 20 1.479 20

1.163 20/4 1.160 18/4 1.139 55/55 1.365 15 1.312 17

0.777 32/4 0.718 20/4 0.775 28/4 0.703 20/4 0.692 20/4 0.885 0/4 0.863 14/4 0.827 20/4 0.822 20/4 0.721 18/4 0.854 78/4 1.290 4 1.653 19/4 0.849 70/4 0.906 20/4 1.671 25/4 0.770 20/4 0.994 20/4 0.630 18/4 0.621 19 0.613 20/4 1.179 25 1.061 15 0.967 20/4 1.071 25/4 1.299 25/4 1.025 20 1.081 80/4 0.930 20/4 1.023 18/4 1.097 23/4 1.096 20/4 1.138 15/15 1.17 1.18 1.008 20/4 1.250 20/4 1.106

30/4

59–60 79 44–5 96–7 113–4 d. 110 51.5 222 164–5 −4.1 15–16 51.9 130 105–7 116–7 189–90 d. 86–7 109.5 32 −53.7 28 −56.5 −107.4 −38.5 −16 −38.6 14 107–8 101.5 63–4 12.5 −22 10 262 −129.7 −160.0 −20 134–5 99–100 <−21 3–4 −30.2 42–3 261–2 50 d. 76–7 −43 45–6 50–2 127 19.6 286 128 −21 45 102.5 62 107–8 56–7 164–5 <−18 86 42–3 52 52–3

304 16515 214.5 19470 subl.

222.3 230–1 237.7

330 150.5759 317 125.7 99.3760 210 195 194–5 179–80 126 285–6100 287–90 subl. 271.5100 253–4 162 270.5 27630 239.4 36.3 27.95 9.5 d. 340 228–9 254–5 172 181.4 193–4 265.5 142–3 299760 302 219–21750 243.5 166769 19117 219–20 336–7 345–6 335258 399.5 275 305–8 235–7 363 283187 172–312 26715 237–8

i. i. 1.08100 1.3520 1.625 v. s. v. s. 2.0525 v. s. 0.0780 0.0580 0.0480 47.728 v. sl. s. sl. s. h. i. i. 0.120 i. i. i. 0.00216 i. i. i. 0.05425 0.09625 i. i. v. s. s. i. v. sl. s. 0.0520 i. 5.615 ∞ 0.03616 i. i. 0.720 i. i. i. i. 8.215 0.220 v. s. v. sl. s. 1.6620 i. v. sl. s. s. h. 1.620 s. sl. s. h. 0.612 d. 120 i. i. i. 0.0860 0.460 i. i. sl. s. sl. s. sl. s. 0.01525 i. i. c.

s. v. s. v. s. v. s. v. s. v. s. v. s. v. s. h. v. s. ∞ ∞ 8.615 v. s. s. s. s. h. s. 2.418 sl. s. sl. s. sl. s. sl. s. sl. s. s. s. ∞ ∞ ∞ ∞ v. s. s. 920 s. ∞ v. s. v. sl. s. ∞ ∞ s. 40 h. 10 h. s. s. ∞ ∞ 1025 v. s. ∞ v. s. ∞ s. s. s. s. ∞ sl. s. d. v. s. h. s. v. s. sl. s. s. v. s. h. s. s. ∞ s. h. s. v. s. s.

s. v. s. v. s. s. v. s. sl. s. sl. s. s. ∞ ∞ 80.815 v. s. s. v. sl. s. s. s. s. s. s. s. s. s. ∞ ∞ ∞ ∞ v. s. 1.3 s. s. ∞ v. s. i. ∞ ∞ s. 1.625 v. s. s. s. ∞ ∞ 5.9 c. ∞ v. s. ∞ s. s. ∞ sl. s. ∞ v. s. v. sl. s. s. v. s. sl. s. s. v. s. h. s. s. ∞ s. s. s. s.

2-43

TABLE 2-2

Physical Properties of Organic Compounds (Continued ) Name

Phenylene-diamine (o-) (m-) (p-) Phloroglucinol (1-,3-,5-) Phorone Phosgene Phthalic acid (o-) (m-)(iso-) anhydride (o-) nitrile (o-) Phthalide Phthalimide (o-) Picoline (α-) (β-) (γ-) Picramic acid (1-,2-,4-,6-) Picric acid (2-,4-,6-) Picryl chloride (2-,4-,6-) Pinacol Pinacoline Pinene (α-)(dl-) hydrochloride Pinol (dl-) Piperidine carboxylic acid (α-)(dl-) Piperidinium pentamethylene dithiocarbamate Propane Propionic acid aldehyde anhydride Propyl acetate (n-) (i-) alcohol (n-) (i-) amine (n-) (i-) aniline (n-) benzoate (n-) (i-) bromide (n-) (i-) n-butyrate (n-) i-butyrate (n-) n-butyrate (i-) i-butyrate (i-) chloride (n-) (i-) Propyl formate (n-) (i-) furoate (n-) lactate (n-) (i-) mercaptan (n-) (i-) propionate (n-) (i-) thiocyanate (i-) n-valerate (n-) i-valerate (n-) i-valerate (i-) Propylene bromide chlorohydrin chloride glycol oxide Protocatechuic acid (3-,4-)

2-44

Formula C6H4(NH2)2 C6H4(NH2)2 C6H4(NH2)2 C6H3(OH)32H2O [(CH3)2C:CH]2CO OCCl2 C6H4(CO2H)2 C6H4(CO2H)2 C6H4 < (CO)2 > O C6H4(CN)2 C6H4(CH2)(CO) > O C6H4 < (CO)2 > NH C5H4NCH3 C5H4NCH3 C5H4NCH3 HOC6H2(NH2)(NO2)2 HOC6H2(NO2)3 ClC6H2(NO2)3 [(CH3)2COH]2 CH3COC(CH3)3 C10H16 C10H17Cl C10H16O CH2 < (CH2CH2)2 > NH HO2CCH < (CH2CH2)2 > NH (CH2)5CS2HHN(CH2)5 CH3CH2CH3 CH3CH2CO2H CH3CH2CHO (CH3CH2CO)2O CH2CO2CH2CH2CH3 CH3CO2CH(CH3)2 CH3CH2CH2OH (CH3)2CHOH CH3CH2CH2NH2 (CH3)2CHNH2 C6H5NHCH2CH2CH3 C6H5CO2CH2CH2CH3 C6H5CO2CH(CH3)2 CH3CH2CH2Br (CH3)2CHBr C2H5CH2CO2CH2C2H5 (CH3)2CHCO2CH2C2H5 C2H5CH2CO2CH(CH3)2 (CH3)2CHCO2CH(CH3)2 CH3CH2CH2Cl (CH3)2CHCl HCO2CH2CH2CH3 HCO2CH(CH3)2 C4H3OCO2C3H7 CH3CH(OH)CO2CH2C2H5 CH3CH(OH)CO2CH(CH3)2 CH3CH2CH2SH (CH3)2CHSH C2H5CO2CH2C2H5 C2H5CO2CH(CH3)2 (CH3)2CHCNS CH3(CH2)3CO2CH2C2H5 (CH3)2CHCH2CO2C3H7 (CH3)2CHCH2CO2C3H7 CH3CH:CH2 CH3CHBrCH2Br CH3CHClCH2OH CH3CHClCH2Cl CH3CH(OH)CH2OH CH3(CHCH2)O (HO)2C6H3CO2HH2O

Formula weight

Form and color

108.14 108.14 108.14 162.14 138.20 98.92 166.13 166.13 148.11 128.13 134.13 147.13 93.12 93.12 93.12 199.12 229.11 247.56 118.17 100.16 136.23 172.69 152.23 85.15 129.16 232.41 44.09 74.08 58.08 130.14 102.13 102.13 60.09 60.09 59.11 59.11 135.20 164.20 164.20 123.00 123.00 130.18 130.18 130.18 130.18 78.54 78.54 88.10 88.10 154.16 132.16 132.16 76.15 76.15 116.16 116.16 101.16 144.21 144.21 144.21 42.08 201.91 94.54 112.99 6.09 58.08 172.13

lf./aq. rhb. mn. rhb. yel. pr. gas mn./aq. nd./aq. rhb. cr. nd./aq. cr./et. col. lq. col. lq. lq. red nd. yel. rhb. yel. mn. col. nd. col. lq. col. lq. lf. lq. lq. cr. cr. gas col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. lq. lq. col. lq. col. lq. lq. lq. col. lq. col. lq. gas col. lq. col. lq. col. lq. col. oil col. lq. nd./aq.

Specific gravity 1.139 15/15 0.885 20/4 1.392 19/4 1.593 20/4 1.527

4

1.164 99/4 0.950 15/4 0.961 15/4 0.957 15/4 1.763 20/4 1.797 20 0.967 15 0.800 16 0.878 20/4 0.953 20/20 0.860 20/4 1.13 0.585 −45/4 0.992 20/4 0.807 20/4 1.012 20/4 0.886 20/4 0.874 20/20 0.804 20/4 0.78920/4 0.718 20/20 0.694 15/4 0.949 18 1.021 25/25 1.010 25/25 1.353 20/4 1.310 20/4 0.879 15 0.884 0/4 0.865 18 0.869 0/4 0.890 20/4 0.859 20 0.901 20/4 0.873 20/4 1.075 26/4 0.836 25/4 0.809 25/4 0.883 20/4 0.893 0 0.963 20 0.874 15 0.863 20/4 0.854 17 0.609 −47/4 1.933 20/4 1.103 20 1.159 20/20 1.040 19.4 0.831 20/20 1.542 4/4

Melting point, °C

Boiling point, °C

103–4 62.8 140 117 28 −104 208 330 130.8 141 73(65) 238 −70

256–8 284–7 267 subl. 197.2743 8.2756 d. subl. 284.5

169 121.8 83 43(38) −52.5 −55 131–2 −9 264 175 −187.1 −22 −81 −45 −92.5 −73.4 −127 −85.8 −83 −101 −51.6 −109.9 −89 −95.2

−122.8 −117 −92.9

−112 −130.7 −76 −70.7 −185 −55.5 <−70 199 d.

290 subl. 128.8 143.5 143.1 expl. d. 171–2789 106.2 154–6 207–8 183–4 106 −42.2 141.1 49.5740 168.8780 101.6 88.4 97.8 82.5 49–50761 33–4 222 231 218.5 70.8 60 142.7 134–5 128 120.8 46.4 36.5 81.3 68-71751 211 122-3150 167.5 67-8 58-60 122-3 109-11750 152-3754 67.5 155.9 142756 −48749 141.6 133-4 96.8 188-9 35

Solubility in 100 parts Water 73381 35.125 669107 1.1325 0.150 v. sl. s. 0.7025 0.2100 v. sl. s. sl. s. c. v. sl. s. 0.0425 v. s. ∞ ∞ 0.1422 1.2320 0.01815 sl. s. c. 2.515 v. sl. s. i.

Alcohol

Ether

v. s. v. s. s. v. s. s.

v. s. s. s. v. s. s.

1218 s. s.

0.6815

s. 5 ∞ ∞ ∞ s. 620 4.817 v. s. s. s. 33 s. ∞

sl. s. s. h. ∞ ∞ ∞ sl. s. 113 717 v. s. s. ∞ s. s.

∞ s. 628 6.518 cc. ∞ 2020 d. 1.616 320 ∞ ∞ ∞ ∞ i. i. i. 0.2520 0.3220 0.1717 v. sl. s. v. sl. s. v. sl. s. 0.2720 0.3120 12.222 2.122 v. sl. s. s. s. v. sl. s. v. sl. s. 0.5625 0.625 i. i. i.

s. ∞ ∞ d. ∞ ∞ ∞ ∞ ∞ ∞ v. s. s. s. ∞ ∞ ∞

v. s. ∞ ∞

∞ ∞ ∞ ∞ s. s. s. s. ∞ ∞ ∞ ∞ ∞ ∞

∞ ∞ ∞ ∞ ∞ s. s. s. ∞ ∞ ∞ ∞ ∞ ∞

44.6 cc. 0.2520 s. 0.2720 ∞ 3320 1.8214

1200 cc. s. s. v. s. ∞ ∞ v. s.

v. s. s. v. s. 8 ∞ s.

∞ ∞ ∞ ∞ ∞ ∞ v. s. s. s. ∞ ∞ ∞

Pulegol (iso-)(d-) Pulegone Pyrazole Pyrazoline Pyrazolone Pyrene Pyridazine Pyridine Pyrocatechol (o-) Pyrogallol (1-,2-,3-) Pyrone Pyrrole Pyrrolidine Pyrroline Pyruvic acid Quercitrin Quinaldine (py-2) Quinoline (iso-) -diol (1-,3-) Quinone (p-) R-acid Ca salt (2-)(3-,6-) K salt Na salt Raffinose Resorcinol (m-) Retene Rhamnose (β-) Ricinoleic acid Rosaniline Rosolic acid Saccharin Safrole (1-,3-,4-) (iso-)(1-,3-,4-) Salicylic acid (o-) aldehyde (o-) Saligenin Schaeffer’s salt, Ca K Na Semicarbazide hydrochloride Skatole (3-) Sodium methylate Sorbitol Sorbose (d- or l-) Starch Stearic acid amide Styrene Suberic acid Succinic acid Sucrose Sulfanilic acid (p-) Sylvestrene (d-) Tartaric acid (meso-) (racemic) (d- or l-) Tartronic acid Terephthalic acid (p-) Terpin hydrate (cis-) Terpineol (α-)(d- or l-) (dl-) Terpinyl acetate (α-)(dl-) Tetrabromo-ethane (sym) (uns) Tetrachloro-ethane (sym) (uns) -ethylene Tetracosane (n-) Tetradecane (n-) Tetraethyl-thiuram disulfide

C10H17OH C10H16O NHN:CHCH:CH NHN:CHCH2CH2 NHCOCH2CH:N C16H10 N2 < (CHCH)2 > CH < (CHCH)2 > N C6H4(OH)2 C6H3(OH)3 CO < (CHCH)2 > O < (CH:CH)2 > NH < (CH2CH2)2 > NH < (CHCH2)2 > NH CH3COCO2H C21H20O112H2O CH3C9H6N C9H7N C9H7N C6H4CH:C(OH)N:C(OH) CO < (CHCH)2 > CO HOC10H5(SO3)2Ca HOC10H5(SO3K)2 HOC10H5(SO3Na)2 C18H32O165H2O C6H4(OH)2 C18H18 CH3(CHOH)4CHOH2O C17H32(OH)CO2H C20H21ON3 C20H16O3 C6H4(CO)(SO2) > NH CH2:CHCH2C6H3:O2CH2 CH3CH:CHC6H3:O2CH2 HOC6H4CO2H HOC6H4CHO HOC6H4CH2OH (HOC10H6SO3)2Ca5H2O HOC10H6SO3K HOC10H6SO3Na NH2CONHNH2 NH2CONHNH3Cl CH3C8H6N CH3ONa [CH2OH(CHOH)2]2 C6H12O6 (C6H10O5)x CH3(CH2)16CO2H CH3(CH2)16CONH2 C6H5CH:CH2 HO2C(CH2)6CO2H HO2C(CH2)2CO2H C12H22O11 H2NC6H4SO3H C10H16 (CHOHCO2H)2 (CHOHCO2H)2H2O (CHOHCO2H)2 CH(OH)(CO2H)2a H2O C6H4(CO2H)2 C10H20O2H2O C10H18O C10H18O CH3CO2C10H17 Br2CHCHBr2 Br3CCH2Br Cl2CHCHCl2 Cl3CCH2Cl Cl2C:CCl2 CH3(CH2)22CH3 CH3(CH2)12CH3 [(C2H5)2NCS]2S2

154.24 152.23 68.08 70.09 84.08 202.24 80.09 79.10 110.11 126.11 96.08 67.09 71.12 69.10 88.06 484.40 143.18 129.15 129.15 161.15 108.09 342.35 380.46 348.26 594.52 110.11 234.32 182.17 298.45 319.39 304.33 183.18 162.18 162.18 138.12 122.12 124.13 576.59 262.31 246.21 75.07 111.54 131.17 54.03 182.17 180.16 162.14 284.47 283.48 104.14 174.19 118.09 342.30 173.18 136.23 150.09 168.10 150.09 129.07 166.13 190.28 154.24 154.24 196.28 345.70 345.70 167.86 167.86 165.85 338.64 198.38 296.52

col. lq. col. lq. nd./et. lq. nd. yel. pr. lq. col. lq. nd./aq. nd. cr. lq. lq. lq. col. lq. yel. nd. lq. lq. pl. cr. yel. mn. cr. cr. cr. cr./aq. col. rhb. lf./al. col. mn. lq. col. nd. red lf. mn. col. mn. col. lq. mn. col. oil rhb./aq. cr. cr. cr. pr./al. pr. lf. pd. cr. rhb. amor. mn. col. cr. col. lq. nd./aq. col. mn. col. mn. col. cr. lq. cr. tri. mn. pr./aq. cr. rhb. col. cr. col. cr. lq. col. lq. col. lq. col. lq. lq. col. lq. cr. col. lq. cr.

0.911 20/4 0.932 20/20 70 1.277 0/4 1.107 20/4 0.982 20/4 1.344 4 1.453 4 1.190 40.3 0.948 20/4 0.852 22.5 0.910 20/4 1.267 20/4 1.059 20/4 1.095 20 1.099 21/4 1.318 20/4

1.465 0 1.272 15 1.1316 1.47120/4 0.954 16

1.100 20/4 1.122 20/4 1.443 20/4 1.153 25/4 1.161 25

1.654 15 1.5021 0.847 69.3 0.903 20/4 1.266 25/4 1.572 25/4 1.588 15 0.863 20/4 1.737 1.697 20/4 1.760 20/4 1.510 0.935 15 0.935 20/20 0.966 20/4 2.964 20/4 2.875 20/4 1.600 20/4 1.588 20/4 1.624 15/4 0.779 51/4 0.765 20/4 1.17

165 149-50 −8 −42 104-5 133-4 32.5

13.6 182-5 −1 −15 24.6 237 115.7

119 110.7 98-9 126 4-5 186 d. 308-10 d. 225-8 11.2 6-7 159 -7 86-7

96 173 d. 95 d. 300 110-2 165 d. 70-1 108-9 −31 140-4 189-90 170-86 d. d. > 280

86-9 10 224754 186-8 144 subl. d. >360 208 115-6 240-5 309 215-7 131 87-8 90-1 165 244-5750 237.1747 240.5763 subl.

d. 130 276.5 390-4 226-810 subl. 233-4 252-3 21120 196.5 subl.

755

265-6

291110 25112 145-6 279100 235 d. 176-7

159-60 205-6 168-70 d. 155-8 subl. 117 38-40 35 < −50 −1.0 0 −36 −19 51.1 5.5 70

d. subl. d. 219-21 218-9752 220 d. 15154 10413 146.3 129-30 120.8 324 252.5

v. sl. s. i. s. ∞ s. i. ∞ ∞ 45.120 40 13 v. sl. s. i. ∞ v. s. ∞ 0.04 20 v. sl. s. 6 sl. s. v. sl. s. sl. s. h. 30.6 25 29.5 25 25.2 25 14.3 20 14712 i. 60.8 21 i. v. sl. s. 0.1225 0.4 25 i. i. 0.223 1.7 86 6.615 4.7620 3.4625 6.2925 v. s. v. s. 0.05 c. d. v. s. 5517 i. 0.0325 i. v. sl. s. 0.1416 6.820 1790 0.810 12015 20.620 13920 v. s. 0.001 c. 0.415 i. i. i. i. 20

0.29 i. 0.0220 i. i.

∞ s. ∞ v. s. 3 h. s. ∞ v. s. s. s. s. ∞ ∞ ∞ s. ∞ s.

0.120 v. s. 69 h.

∞ s. sl. s. v. sl. s. v. s. s. s. v. s. s. v. s. s. ∞ ∞ ∞ sl. s. s. ∞ s. s.

∞ sl. s. v. s. h. 3.1 c. s. ∞ 4915 ∞ v. s.

v. s. v. s. h. i. ∞ i. sl. s. 1.05 c. ∞ ∞ 5115 ∞ v. s.

v. s. sl. s. s.

i. i. s.

v. s. h. sl. s. i. 220 s. h. ∞ s. 9.915 0.9 v. sl. s.

i. 6g s. h: ∞ 0.815 1.215 i. v. sl. s.

20 2515 v. s. sl. s. h. 1015 v. s. v. s. 20 ∞ s. ∞ ∞ ∞ v. s.

0.09 0.415 i. i. 115 v. s. v. s. ∞ ∞ ∞ ∞ s. v. s.

2-45

TABLE 2-2

Physical Properties of Organic Compounds (Concluded ) Name

Tetrafluoro-ethylene Tetrahydro-furan -furfuryl alcohol -pyran Tetralin Tetramethyl-thiuram disulfide Tetryl (2-,4-,6-) Theobromine Thio-acetic acid -aniline (4-, 4′-) -carbanilide -naphthol (β-) -phenol -salicylic acid (o-) -urea Thiophene Thymol (5-,2-,1-) Tolidine (0-)(3-,3′-,4-,4′-) Toluene sulfonic acid (o-) (p-) sulfonic amide (p-) sulfonic chloride (p-) Toluic acid (o-) (m-) (p-) Toluidine (o-) (m-) (p-) hydrochloride (o-) sulfonic acid (1-,2-,3-) Toluylenediamine (1-,2-,4-) Tolylene diisocyanate (1-,2-,4-) Trehalose Triamylamine (n-) (i-) Tributyl-amine (n-) phosphite Trichloro-acetic acid -benzene (s-)(1-,3-,5-) -ethane (1-,1-,1-) -ethylene -phenol Tricosane (n-) Tricresyl phosphate (o-) Tridecane (n-) Triethanol amine Triethyl-amine -benzene (1-,3-,5-) (1-,2-,4-) borate citrate Triethylene glycol Trifluoro-chloromethane chloroethylene -trichloroethane Trimethoxybutane (1-,3-,3-) Trimethylamine Trimethylene bromide chloride glycol Trinitro-benzene (1-,3-,5-) -benzoic acid (2-,4-,6-) -tert-butylxylene -naphthalene (α-)(1-,3-,5-) (β-)(1-,3-,8-) (γ-)(1-,4-,5-)

2-46

Formula F2C:CF2 CH2(CH2)2CH2O C4H7OCH2OH CH2(CH2)3CH2O C6H4CH2(CH2)2CH2 [(CH3)2NCS]2S2 (NO2)3C6H2N(CH3)NO2 C7H8O2N4 CH3COSH (NH2C6H4)2S (C6H5NH)2CS C10H7SH C6H5SH HSC6H4CO2H NH2CSNH2 < (CH:CH)2 > S (CH3)(C3H7)C6H3OH [CH3(NH2)C6H3]2 C6H5CH3 CH3C6H4SO3H2H2O CH3C6H4SO3HH2O CH3C6H4SO2NH2 CH3C6H4SO2Cl CH3C6H4CO2H CH3C6H4CO2H CH3C6H4CO2H CH3C6H4NH2 CH3C6H4NH2 CH3C6H4NH2 CH3C6H4NH3Cl CH3(NH2)C6H3SO3H CH3C6H3(NH2)2 CH3C6H3(NCO)2 C12H22O112H2O [CH3(CH2)3CH2]3N [(CH3)2CH(CH2)2]3N [CH3(CH2)2CH2]3N [CH3(CH2)3O]3P Cl3CCO2H C6H3Cl3 Cl3CCH3 Cl2C:CHCl Cl3C6H2OH CH3(CH2)21CH3 OP(OC6H4CH3)3 CH3(CH2)11CH3 (HOCH2CH2)3N (CH3CH2)3N (C2H5)3C6H3 (C2H5)3C6H3 B(OCH2CH3)3 HOC3H4(CO2C2H5)3 (CH2OCH2CH2OH)2 CF3Cl F2C:CFCl Cl2CFCClF2 CH2(OCH3)CH2C(OCH3)2CH3 (CH3)3N BrCH2CH2CH2Br ClCH2CH2CH2Cl HOCH2CH2CH2OH C6H3(NO2)3 (NO2)3C6H2CO2H (NO2)3C6(CH3)2C4H9 C10H5(NO2)3 C10H5(NO2)3 C10H5(NO2)3

Formula weight

Form and color

Specific gravity

Melting point, °C

Boiling point, °C

100.02 72.10 102.13 86.13 132.20 240.41 287.15 180.17 76.11 216.29 228.30 160.22 110.17 154.18 76.12 84.13 150.21 212.28 92.13 208.23 190.21 171.21 190.64 136.14 136.14 136.14 107.15 107.15 107.15 143.62 187.21 122.17 174.15 378.33 227.42 227.42 185.34 250.32 163.40 181.46 133.42 131.40 197.46 324.61 368.36 184.35 149.19 101.19 162.26 162.26 146.00 276.28 150.17 104.47 116.48 187.39 148.20 59.11 201.91 112.99 76.09 213.11 257.12 297.26 263.16 263.16 263.16

gas col. lq. col. lq. lq. col. lq. cr. yel. mn. rhb. yel. lq. nd./aq. rhb./al. cr./al. col. lq. yel. nd. rhb./al. col. lq. cr. lf. col. lq. cr. mn. mn. tri. cr./aq. pr./aq. cr./aq. col. lq. col. lq. cr. mn. pr. cr. rhb. lq. rhb./al. lq. col. lq. col. lq. lq. cr. nd. lq. col. lq. nd. lf. lq. col. lq. col. lq. col. oil lq. lq. lq. oil col. lq. gas gas lq. lq. gas lq. lq. oil col. rhb. rhb./aq. nd./al. rhb. cr./al. yel. cr.

1.58−78 0.88821/4 1.05020/4 0.88120/4 0.97318/4 1.29 1.5719

−142.5 −65

−76.3 65-6 177-8743 88 206764

1.07410 1.324 1.07423/4 1.40520/4 1.07015/4 0.97225/25 20/4

0.866

1.062115/4 1.054112/4 0.99920/4 0.98920/4 1.04620/4

1.2328

−31 155-6 130.5 330 < −17 108 154 81 164 180-2 −30 51.5 128-9 −95 d. 104-5 137 69 104-5 110-1 179-80 −16.3 −31.5 44-5 218-20 99

expl. 93 d. 286-8 168-9 subl. d. 84 232752 110.8 128.80 146-70 134.510 259751 263 274-5 199.7 203.3 200.3 242 283-5 134.520

97 0.78620/4 0.77820/20 0.92520/4 1.61746/15 1.32526/4 1.46620/20 1.49075/4 0.77948/4 0.75720/4 1.12620/20 0.72920/20 0.86120/4 0.88217/4 0.86420/20 1.13720/4 1.12520/20 1.726−130 1.57620/4 0.932 0.662 −5 1.987 15/4 1.201 15 1.060 20/4 1.688 20/4

58 63.5 −73 68-9 47.7 −6.2 20-1 −114.8

−5 −182 −157.5 −35 −124 −34.4 121 210-20 d. 110 122-3 218-9 148-9

240-5 235 216.5761 122-312 195.5754 208.5764 74.1 87.2 246 23415 234 277-9150 89.4 215 217-8755 120 294 290 −80 −27.9 47.6 63-525 3.5 167.5 123-5 214 d.

Solubility in 100 parts Water 0.0130 s. ∞ s. i. i. i. 0.0615 s. sl. s. h. i. v. sl. s. v. sl. s. sl. s. h. 9.213 i. 0.0919 v. sl. s. 0.0516 v. s. v. s. 0.29 i. 2.17100 1.6100 1.3100 1.525 sl. s. 0.7421 s. 0.9711 s. h. d. s. h. i. i. i. i. 12025 i. i. 0.125 0.0925 i. i. i. ∞ ∞ > 190 i. i. d. i. ∞ d. i. d. 4119 0.1730 0.2725 ∞ 0.0315 2.0524 i. i. 0.02100 i.

Alcohol

Ether

s. ∞

s. ∞

s.

s.

s. h. 0.06 c. ∞ s. v. s. v. s. v. s. s. s. s. v. s. s. s. s. s. 7.45 s. v. s. v. s. v. s. ∞ ∞ v. s. sl. s.

s. 0.03 h. ∞ s. v. s. v. s. ∞ sl. s. v. s. s. ∞

s. v. s. v. s. ∞ ∞ v. s.

s. d. sl. s. h.

i.

s.

s.

∞

s. sl. s. ∞ ∞ v. s.

∞ ∞ v. s.

v. s. ∞ ∞ s. s.

v. s. sl. s. ∞ s. s.

∞ ∞

∞ v. sl. s.

∞

∞

s. s. s. ∞ 1.918

s. s. s. 1.518

sl. s. s. 0.0523 0.1119

0.1315 0.419

s.

s.

-phenol (2-,3-,6-) -toluene (β-)(2-,3-,4-) (γ-)(2-,4-,5-) (α-)(2-,4-,6-) Trional Triphenyl-arsine carbinol guanidine (α-) methane methyl phosphate Tripropylamine (n-) Undecane (n-) Urea nitrate Uric acid Valeric acid (n-) (i-) aldehyde (n-) (i-) amide (n-) (i-) Vanillic acid (3-,4-,1-) alcohol (3-,4-,1-) hyl-thiuram disulfide Vanillin (3-,4-,1-) Veratrole (o-) Vinyl acetate (poly-) acetic acid acetylene alcohol (poly-) chloride propionate Xylene (o-) (m-) (p-) sulfonic acid (1-,4-,2-) Xylidine (1:2)(3-) (1:2)(4-) (1:3)(2-) (1:3)(4-) (1:3)(5-) (1:4)(2-) Xylose (l-)(+) Xylylene dichloride (p-) Zinc diethyl dimethyl dimethyl-dithiocarbamate NOTE:

(NO2)3C6H2OH CH3C6H2(NO2)3 CH3C6H2(NO2)3 CH3C6H3(NO2)3 C2H5(CH3)C(SO2C2H5)3 (C6H5)3As (C6H5)3COH C6H5N:C(NHC6H5)2 (C6H5)3CH (C6H5)3C . . . OP(OC6H5)3 (CH3CH2CH2)3N CH3(CH2)3CH3 H2NCONH2 CO(NH2)2HNO3 C5H4O3N4 C2H5CH2CH2CO2H (CH3)2CHCH2CO2H C2H5CH2CH2CHO (CH3)2CHCH2CHO C2H5CH2CH2CONH2 (CH3)2CHCH2CONH2 CH3O(OH)C6H3CO2H CH3O(OH)C6H3CH2OH [(C2H5)2NCS]2S2 CH3O(OH)C6H3CHO C6H4(OCH3)2 CH3CO2CH:CH2 (CH3CO2CH:CH2)x CH2:CHCH2CO2H CH2:CHC:CH CH2:CHOH (CH2:CHOH)x CH2:CHCl C2H5CO2CH:CH2 C6H4(CH3)2 C6H4(CH3)2 C6H4(CH3)2 (CH3)2C6H3SO3H2H2O (CH3)2C6H3NH2 (CH3)2C6H3NH2 (CH3)2C6H3NH2 (CH3)2C6H3NH2 (CH3)2C6H3NH2 (CH3)2C6H3NH2 CH2OH(CHOH)3CHO C6H4(CH2Cl)2 Zn(CH2CH3)2 Zn(CH3)2 Zn[S2CN(CH3)2]2

229.11 227.13 227.13 227.13 242.34 306.21 260.32 287.35 244.32 243.31 326.28 143.27 156.30 60.06 123.07 168.11 102.13 102.13 86.13 86.13 101.15 101.15 168.14 154.16 296.52 152.14 138.16 86.09 (86.09) 86.09 52.07 44.06 (44.06) 62.50 100.11 106.16 106.16 106.16 222.25 121.18 121.18 121.18 121.18 121.18 121.18 150.13 175.06 123.50 95.45 305.79

nd. cr. yel. pl. cr./al. pl./al. pl. cr. rhb./al. cr. col. cr. pr./al. col. lq. col. lq. col. pr. col. mn. cr. col. lq. col. lq. lq. col. lq. mn. pl. mn. nd./aq. mn./aq. cr. mn. cr. col. lq. col. lq. gas gas lq. col. lq. col. lq. col. lq. col. lf. lq. pr. lq. lq. oil oil nd. mn. col. lq. col. lq.

1.620 20/4 1.620 20/4 1.654 1.199 85/4 1.306 1.188 20/4 1.13 1.014 99/4 1.206 58/4 0.757 20/4 0.741 20/4 1.335 20/4 1.893 20 0.939 20/4 0.931 20/20 0.819 11 0.803 17 1.023 0.965 20/4 1.17 1.056 1.091 15/15 0.932 20/4 1.1920 1.013 15/15 0.705 1.5

117-8 112 104 80.8 76 59-60 162.5 144-5 93.4 145-7 49-50 −93.5 −25.6 132.7 152 d. d. −34.5 −37.6 −92 −51 106 135-7 207 115 70 81-2 22.5 < −60 100-25 −39

1.320 0.908 25/25

d. >200 −160

0.881 20/4 0.867 17/4 0.861 20/4

−25 −47.4 13.2 86 < −15 49-50

0.991 15 1.076 17.5 0.980 15 0.978 20/4 0.972 20/4 0.979 21/4 1.535 0 1.417 0 1.182 18 1.386 11 2.0040/4

15.5 153-4 100.5 −28 −40 248-50

expl. expl. expl. d. >360 >360 d. 359754 d. 24511 156.5 194.5 d. 187 176 103.4 92.5 232 subl. d. 285 207.1 72-3 163 5.5 −12 93-5 144 139.3 138.5 1490.1 223 224-6 216-7 213-4 221-2 215789 240-5 d. 118 46

s. h. i. i. 0.0120 0.315 i. i. i. i. i. i. v. sl. s. i. 10017 v. s. h. 0.06 h. 3.316 4.220 v. sl. s. sl. s. v. s. s. 0.1214 v. s. h. i. 114 v. sl. s. 220 i. s. 0.670.6 s. sl. s. v. sl. s. i. i. i. s. v. sl. s. v. sl. s. v. sl. s. v. sl. s. v. sl. s. v. sl. s. 11720 i. d. d. i.

v. s. sl. s. c. s. h. 1.522 50 s. v. s. 40 v. s. h. sl. s. h. 15525 ∞ ∞ 2020 s. i. ∞ ∞ s. s. v. s. s. v. s. v. s.

v. s. s. v. s. 533 6.615 v. s. v. s.

v. s. s. ∞

v. s. s. ∞

∞

∞

v. s. v. s. ∞ ∞ sl. s. i. ∞ ∞ s. s. v. s. s. v. s. v. s.

s.

v. s.

s. s. s.

∞ ∞ v. s.

s.

s.

v. sl. s. s. d. d.

i. v. sl. s.

°F = 9⁄5 °C + 32.

2-47

2-48

PHYSICAL AND CHEMICAL DATA

VAPOR PRESSURES OF PURE SUBSTANCES UNITS CONVERSIONS

ADDITIONAL REFERENCES

For this subsection, the following units conversions are applicable:

Additional compilations of vapor-pressure data include Boublik, Fried, and Hala, The Vapor Pressures of Pure Substances, Elsevier, Amsterdam, 1984. See also Hirata, Ohe, and Nagahama, Computer Aided Data Book of Vapor-Liquid Equilibria, Kodansha/Elsevier, Tokyo, 1975; Weishaupt, Landolt-Börnstein New Series Group IV, vol. 3; Thermodynamic Equilibria of Boiling Mixtures, Springer-Verlag, Berlin, 1975; Wichterle, Linek, and Hala, Vapor-Liquid Equilibrium Data Bibliography, Elsevier, Amsterdam, 1973; suppl. 1, 1976; suppl. 2, 1982.

°F = 9⁄5 °C + 32. To convert millimeters of mercury to pounds-force per square inch, multiply by 0.01934.

TABLE 2-3

Vapor Pressure of Water Ice from -15 to 0°C* mmHg

TABLE 2-4

Vapor Pressure of Liquid Water from -16 to 0°C* mmHg

t, °C

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

t, °C

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

−14 −13 −12 −11 −10

1.361 1.490 1.632 1.785 1.950

1.348 1.477 1.617 1.769 1.934

1.336 1.464 1.602 1.753 1.916

1.324 1.450 1.588 1.737 1.899

1.312 1.437 1.574 1.722 1.883

1.300 1.424 1.559 1.707 1.866

1.288 1.411 1.546 1.691 1.849

1.276 1.399 1.532 1.676 1.833

1.264 1.386 1.518 1.661 1.817

1.253 1.373 1.504 1.646 1.800

−15 −14 −13 −12 −11

1.436 1.560 1.691 1.834 1.987

1.425 1.547 1.678 1.819 1.971

1.414 1.534 1.665 1.804 1.955

1.402 1.522 1.651 1.790 1.939

1.390 1.511 1.637 1.776 1.924

1.379 1.497 1.624 1.761 1.909

1.368 1.485 1.611 1.748 1.893

1.356 1.472 1.599 1.734 1.878

1.345 1.460 1.585 1.720 1.863

1.334 1.449 1.572 1.705 1.848

−9 −8 −7 −6 −5

2.131 2.326 2.537 2.765 3.013

2.112 2.306 2.515 2.742 2.987

2.093 2.285 2.493 2.718 2.962

2.075 2.266 2.472 2.695 2.937

2.057 2.246 2.450 2.672 2.912

2.039 2.226 2.429 2.649 2.887

2.021 2.207 2.408 2.626 2.862

2.003 2.187 2.387 2.603 2.838

1.985 2.168 2.367 2.581 2.813

1.968 2.149 2.346 2.559 2.790

−10 −9 −8 −7 −6

2.149 2.326 2.514 2.715 2.931

2.134 2.307 2.495 2.695 2.909

2.116 2.289 2.475 2.674 2.887

2.099 2.271 2.456 2.654 2.866

2.084 2.254 2.437 2.633 2.843

2.067 2.236 2.418 2.613 2.822

2.050 2.219 2.399 2.593 2.800

2.034 2.201 2.380 2.572 2.778

2.018 2.184 2.362 2.553 2.757

2.001 2.167 2.343 2.533 2.736

−4 −3 −2 −1 −0

3.280 3.568 3.880 4.217 4.579

3.252 3.539 3.848 4.182 4.542

3.225 3.509 3.816 4.147 4.504

3.198 3.480 3.785 4.113 4.467

3.171 3.451 3.753 4.079 4.431

3.144 3.422 3.722 4.045 4.395

3.117 3.393 3.691 4.012 4.359

3.091 3.364 3.660 3.979 4.323

3.065 3.336 3.630 3.946 4.287

3.039 3.308 3.599 3.913 4.252

−5 −4 −3 −2 −1

3.163 3.410 3.673 3.956 4.258

3.139 3.384 3.647 3.927 4.227

3.115 3.359 3.620 3.898 4.196

3.092 3.334 3.593 3.871 4.165

3.069 3.309 3.567 3.841 4.135

3.046 3.284 3.540 3.813 4.105

3.022 3.259 3.514 3.785 4.075

3.000 3.235 3.487 3.757 4.045

2.976 3.211 3.461 3.730 4.016

2.955 3.187 3.436 3.702 3.986

*For data at 0(0.2)−30(2)−98°C see p. 2324, Handbook of Chemistry and Physics, 40th ed., Chemical Rubber Publishing Co.

−0

4.579 4.546 4.513 4.480 4.448 4.416 4.385 4.353 4.320 4.289

*Computed from the above table with the aid of the thermodynamic equation pw −1.1489t log10  =  − 1.330 × 10−5t2 + 9.084 × 10−8t3 273.1 + t pi

TABLE 2-5

Vapor Pressure of Liquid Water from 0 to 100°C* mmHg

t, °C

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1 2 3 4

4.579 4.926 5.294 5.685 6.101

4.613 4.962 5.332 5.725 6.144

4.647 4.998 5.370 5.766 6.187

4.681 5.034 5.408 5.807 6.230

4.715 5.070 5.447 5.848 6.274

4.750 5.107 5.486 5.889 6.318

4.785 5.144 5.525 5.931 6.363

4.820 5.181 5.565 5.973 6.408

4.855 5.219 5.605 6.015 6.453

4.890 5.256 5.645 6.058 6.498

5 6 7 8 9

6.543 7.013 7.513 8.045 8.609

6.589 7.062 7.565 8.100 8.668

6.635 7.111 7.617 8.155 8.727

6.681 7.160 7.669 8.211 8.786

6.728 7.209 7.722 8.267 8.845

6.775 7.259 7.775 8.323 8.905

6.822 7.309 7.828 8.380 8.965

6.869 7.360 7.882 8.437 9.025

6.917 7.411 7.936 8.494 9.086

6.965 7.462 7.990 8.551 9.147

10 11 12 13 14

9.209 9.844 10.518 11.231 11.987

9.271 9.910 10.588 11.305 12.065

9.333 9.976 10.658 11.379 12.144

9.395 10.042 10.728 11.453 12.223

9.458 10.109 10.799 11.528 12.302

9.521 10.176 10.870 11.604 12.382

9.585 10.244 10.941 11.680 12.462

9.649 10.312 11.013 11.756 12.543

9.714 10.380 11.085 11.833 12.624

9.779 10.449 11.158 11.910 12.706

15 16 17 18 19

12.788 13.634 14.530 15.477 16.477

12.870 13.721 14.622 15.575 16.581

12.953 13.809 14.715 15.673 16.685

13.037 13.898 14.809 15.772 16.789

13.121 13.987 14.903 15.871 16.894

13.205 14.076 14.997 15.971 16.999

13.290 14.166 15.092 16.071 17.105

13.375 14.256 15.188 16.171 17.212

13.461 14.347 15.284 16.272 17.319

13.547 14.438 15.380 16.374 17.427

20 21 22 23 24

17.535 18.650 19.827 21.068 22.377

17.644 18.765 19.948 21.196 22.512

17.753 18.880 20.070 21.324 22.648

17.863 18.996 20.193 21.453 22.785

17.974 19.113 20.316 21.583 22.922

18.085 19.231 20.440 21.714 23.060

18.197 19.349 20.565 21.845 23.198

18.309 19.468 20.690 21.977 23.337

18.422 19.587 20.815 22.110 23.476

18.536 19.707 20.941 22.243 23.616

25 26 27 28 29

23.756 25.209 26.739 28.349 30.043

23.897 25.359 26.897 28.514 30.217

24.039 25.509 27.055 28.680 30.392

24.182 25.660 27.214 28.847 30.568

24.326 25.812 27.374 29.015 30.745

24.471 25.964 27.535 29.184 30.923

24.617 26.117 27.696 29.354 31.102

24.764 26.271 27.858 29.525 31.281

24.912 26.426 28.021 29.697 31.461

25.060 26.582 28.185 29.870 31.642

30 31 32 33 34

31.824 33.695 35.663 37.729 39.898

32.007 33.888 35.865 37.942 40.121

32.191 34.082 36.068 38.155 40.344

32.376 34.276 36.272 38.369 40.569

32.561 34.471 36.477 33.584 40.796

32.747 34.667 36.683 38.801 41.023

32.934 34.864 36.891 39.018 41.251

33.122 35.062 37.099 39.237 41.480

33.312 35.261 37.308 39.457 41.710

33.503 35.462 37.518 39.677 41.942

35 36 37 38 39

42.175 44.563 47.067 49.692 52.442

42.409 44.808 47.324 49.961 52.725

42.644 45.054 47.582 50.231 53.009

42.880 45.301 47.841 50.502 53.294

43.117 45.549 48.102 50.774 53.580

43.355 45.799 48.364 51.048 53.867

43.595 46.050 48.627 51.323 54.156

43.836 46.302 48.891 51.600 54.446

44.078 46.556 49.157 51.879 54.737

44.320 46.811 49.424 52.160 55.030

40 41 42 43 44

55.324 58.34 61.50 64.80 68.26

55.61 58.65 61.82 65.14 68.61

55.91 58.96 62.14 65.48 68.97

56.21 59.27 62.47 65.82 69.33

56.51 59.58 62.80 66.16 69.69

56.81 59.90 63.13 66.51 70.05

57.11 60.22 63.46 66.86 70.41

57.41 60.54 63.79 67.21 70.77

57.72 60.86 64.12 67.56 71.14

58.03 61.18 64.46 67.91 71.51

45 46 47 48 49

71.88 75.65 79.60 83.71 88.02

72.25 76.04 80.00 84.13 88.46

72.62 76.43 80.41 84.56 88.90

72.99 76.82 80.82 84.99 89.34

73.36 77.21 81.23 85.42 89.79

73.74 77.60 81.64 85.85 90.24

74.12 78.00 82.05 86.28 90.69

74.50 78.40 82.46 86.71 91.14

74.88 78.80 82.87 87.14 91.59

75.26 79.20 83.29 87.58 92.05

t, °C

0

1

2

3

4

5

6

7

8

9

50 60 70 80

92.51 149.38 233.7 355.1

97.20 156.43 243.9 369.7

102.09 163.77 254.6 384.9

107.20 171.38 265.7 400.6

112.51 179.31 277.2 416.8

118.04 187.54 289.1 433.6

123.80 196.09 301.4 450.9

129.82 204.96 314.1 468.7

136.08 214.17 327.3 487.1

142.60 223.73 341.0 506.1

90 91 92 93 94

525.76 546.05 566.99 588.60 610.90

527.76 548.11 569.12 590.80 613.17

529.77 550.18 571.26 593.00 615.44

531.78 552.26 573.40 595.21 617.72

533.80 554.35 575.55 597.43 620.01

535.82 556.44 577.71 599.66 622.31

537.86 558.53 579.87 601.89 624.61

539.90 560.64 582.04 604.13 626.92

541.95 562.75 584.22 606.38 629.24

544.00 564.87 586.41 608.64 631.57

95 96 97 98 99 100 101

633.90 657.62 682.07 707.27 733.24 760.00 787.57

636.24 660.03 684.55 709.83 735.88 762.72 790.37

638.59 662.45 687.04 712.40 738.53 765.45 793.18

640.94 664.88 689.54 714.98 741.18 768.19 796.00

643.30 667.31 692.05 717.56 743.85 770.93 796.82

645.67 669.75 694.57 720.15 746.52 773.68 801.66

648.05 672.20 697.10 722.75 749.20 776.44 804.50

650.43 674.66 699.63 725.36 751.89 779.22 807.35

652.82 677.12 702.17 727.98 754.58 782.00 810.21

655.22 679.69 704.71 730.61 757.29 784.78 813.06

*From the Physikalisch-technische Reichsanstalt, Holborn, Scheel, and Henning, Wärmetabellen, Friedrich Vieweg & Sohn, Brunswick, 1909. By permission. For data at 50(0.2)101.8°C, see Handbook of Chemistry and Physics, 40th ed., p. 2326, Chemical Rubber Publishing Co. For a tabulation of temperature for pressures 700(1)779 mm Hg, see Atack, Handbook of Chemical Data, p. 117, Reinhold, New York, 1957. For a tabulation of pressure for 105(5)200(10)370°C, see Atack, p. 134, and for 100(1)374°C, see Handbook of Chemistry and Physics, 40th ed., pp. 2328–2330, Chemical Rubber Publishing Co.

2-49

TABLE 2-6 Cmpd. no.

Vapor Pressure of Inorganic and Organic Liquids CAS no.

C1

C2

C3

C4

C5

Tmin, K

Ps at Tmin

Tmax, K

Ps at Tmax

1 2 3 4 5 6 7 8 9 10

Methane Ethane Propane n-Butane n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane

CH4 C2H6 C3H8 C4H10 C5H12 C6H14 C7H16 C8H18 C9H20 C10H22

74828 74840 74986 106978 109660 110543 142825 111659 111842 124185

39.205 51.857 59.078 66.343 78.741 104.65 87.829 96.084 109.35 112.73

−1324.4 −2598.7 −3492.6 −4363.2 −5420.3 −6995.5 −6996.4 −7900.2 −9030.4 −9749.6

−3.4366 −5.1283 −6.0669 −7.046 −8.8253 −12.702 −9.8802 −11.003 −12.882 −13.245

3.1019E−05 1.4913E−05 1.0919E−05 9.4509E−06 9.6171E−06 1.2381E−05 7.2099E−06 7.1802E−06 7.8544E−06 7.1266E−06

2 2 2 2 2 2 2 2 2 2

90.69 90.35 85.47 134.86 143.42 177.83 182.57 216.38 219.66 243.51

1.1687E+04 1.1273E+00 1.6788E−04 6.7441E−01 6.8642E−02 9.0169E−01 1.8269E−01 2.1083E+00 4.3058E−01 1.3930E+00

190.56 305.32 369.83 425.12 469.7 507.6 540.2 568.7 594.6 617.7

4.5897E+06 4.8522E+06 4.2135E+06 3.7699E+06 3.3642E+06 3.0449E+06 2.7192E+06 2.4673E+06 2.3054E+06 2.0908E+06

11 12 13 14 15 16 17 18 19 20

n-Undecane n-Dodecane n-Tridecane n-Tetradecane n-Pentadecane n-Hexadecane n-Heptadecane n-Octadecane n-Nonadecane n-Eicosane

C11H24 C12H26 C13H28 C14H30 C15H32 C16H34 C17H36 C18H38 C19H40 C20H42

1120214 112403 629505 629594 629629 544763 629787 593453 629925 112958

131 137.47 137.45 140.47 135.57 156.06 156.95 157.68 182.54 203.66

−15.855 −16.698 −16.543 −16.859 −16.022 −18.941 −18.966 −18.954 −22.498 −25.525

8.1871E−06 8.0906E−06 7.1275E−06 6.5877E−06 5.6136E−06 6.8172E−06 6.4559E−06 5.9272E−06 7.4008E−06 8.8382E−06

2 2 2 2 2 2 2 2 2 2

247.57 263.57 267.76 279.01 283.07 291.31 295.13 301.31 305.04 309.58

4.0836E−01 6.1534E−01 2.5096E−01 2.5268E−01 1.2884E−01 9.2265E−02 4.6534E−02 3.3909E−02 1.5909E−02 9.2574E−03

639 658 675 693 708 723 736 747 758 768

1.9493E+06 1.8223E+06 1.6786E+06 1.5693E+06 1.4743E+06 1.4106E+06 1.3438E+06 1.2555E+06 1.2078E+06 1.1746E+06

21 22 23 24 25 26 27

2-Methylpropane 2-Methylbutane 2,3-Dimethylbutane 2-Methylpentane 2,3-Dimethylpentane 2,3,3-Trimethylpentane 2,2,4-Trimethylpentane

C4H10 C5H12 C6H14 C6H14 C7H16 C8H18 C8H18

75285 78784 79298 107835 565593 560214 540841

100.18 72.35 77.235 77.36 78.282 83.105 87.868

−4841.9 −5010.9 −5695.9 −5791.7 −6347 −6903.7 −6831.7

−13.541 −7.883 −8.5109 −8.4912 −8.502 −9.1858 −9.9783

2.0063E−02 8.9795E−06 8.0163E−06 7.7939E−06 6.4169E−06 6.4703E−06 7.7729E−06

1 2 2 2 2 2 2

113.54 113.25 145.19 119.55 160 172.22 165.78

1.4051E−02 1.1569E−04 1.5081E−02 9.2204E−06 1.2631E−02 1.6820E−02 1.6187E−02

408.14 460.43 499.98 497.5 537.35 573.5 543.96

3.6199E+06 3.3709E+06 3.1255E+06 3.0192E+06 2.8823E+06 2.8116E+06 2.5630E+06

28 29 30 31 32 33 34 35

Ethylene Propylene 1-Butene cis-2-Butene trans-2-Butene 1-Pentene 1-Hexene 1-Heptene

C2H4 C3H6 C4H8 C4H8 C4H8 C5H10 C6H12 C7H14

74851 115071 106989 590181 624646 109671 592416 592767

74.242 57.263 68.49 102.62 70.589 120.15 85.3 92.68

−2707.2 −3382.4 −4350.2 −5260.3 −4530.4 −6192.4 −6171.7 −7055.2

−9.8462 −5.7707 −7.4124 −13.764 −7.7229 −16.597 −9.702 −10.679

2.2457E−02 1.0431E−05 1.0503E−05 1.9183E−02 1.0928E−05 2.1922E−02 8.9604E−06 8.4459E−06

1 2 2 1 2 1 2 2

104 87.89 87.8 134.26 167.62 107.93 133.39 154.27

1.2361E+02 9.3867E−04 7.1809E−07 2.4051E−01 7.4729E+01 3.5210E−06 2.5272E−04 1.2810E−03

282.34 365.57 419.95 435.58 428.63 464.78 504.03 537.29

5.0296E+06 4.6346E+06 4.0391E+06 4.2388E+06 4.0811E+06 3.5557E+06 3.1397E+06 2.8225E+06

36 37 38 39 40 41 42 43 44

1-Octene 1-Nonene 1-Decene 2-Methylpropene 2-Methyl-1-butene 2-Methyl-2-butene 1,2-Butadiene 1,3-Butadiene 2-Methyl-1,3-butadiene

C8H16 C9H18 C10H20 C4H8 C5H10 C5H10 C4H6 C4H6 C5H8

111660 124118 872059 115117 563462 513359 590192 106990 78795

97.57 144.45 78.808 102.5 97.33 82.605 39.714 73.522 79.656

−7836 −9676.2 −8367.9 −5021.8 −5631.8 −5606.6 −3769.9 −4564.3 −5239.6

−11.272 −19.446 −7.9553 −13.88 −12.589 −9.4236 −2.6407 −8.1958 −9.4314

7.7267E−06 1.8031E−02 8.7442E−18 2.0296E−02 1.5395E−02 1.0512E−05 6.9379E−18 1.1580E−05 9.5850E−03

2 1 6 1 1 2 6 2 1

171.45 191.78 206.89 132.81 135.58 139.39 136.95 164.25 127.27

2.7570E−03 8.5514E−03 1.7308E−02 6.2213E−01 1.9687E−02 1.9447E−02 4.4720E−01 6.9110E+01 2.4768E−03

566.65 593.25 616.4 417.9 465 471 452 425.17 484

2.5735E+06 2.3308E+06 2.2092E+06 3.9760E+06 3.4544E+06 3.3769E+06 4.3613E+06 4.3041E+06 3.8509E+06

45 46 47 48 49 50

Acetylene Methylacetylene Dimethylacetylene 3-Methyl-1-butyne 1-Pentyne 2-Pentyne

C2H2 C3H4 C4H6 C5H8 C5H8 C5H8

74862 74997 503173 598232 627190 627214

172.06 119.42 66.592 69.459 82.805 137.29

−5318.5 −5364.5 −4999.8 −5250 −5683.8 −7447.1

−27.223 −16.81 −6.8387 −7.1125 −9.4301 −19.01

5.4619E−02 2.5523E−02 6.6793E−06 7.9289E−17 1.0767E−05 2.1415E−02

1 1 2 6 2 1

192.4 170.45 240.91 183.45 167.45 163.83

1.2603E+05 3.7264E+02 6.1212E+03 4.3551E+01 2.3990E+00 2.0462E−01

308.32 402.39 473.2 463.2 481.2 519

6.1467E+06 5.6206E+06 4.8699E+06 4.1986E+06 4.1701E+06 4.0198E+06

51 52 53 54 55 56

1-Hexyne 2-Hexyne 3−Hexyne 1-Heptyne 1-Octyne Vinylacetylene1

C6H10 C6H10 C6H10 C7H12 C8H14 C4H4

693027 764352 928494 628717 629050 689974

133.2 123.71 47.091 66.447 82.353 55.682

−7492.9 −7639 −5104 −6395.6 −7240.6 −4439.3

−18.405 −16.451 −3.6371 −6.3848 −9.1843 −5.0136

2.2062Ε−02 1.6495Ε−02 5.1621Ε−04 1.1250E−17 5.8038E−03 1.9650E−17

1 1 1 6 1 6

141.25 183.65 170.05 192.22 193.55 173.15

3.9157Ε−04 5.4026Ε−01 2.1950Ε−01 6.7026E−01 1.0092E−01 6.6899E+01

516.2 549 544 559 585 454

3.6352Ε+06 3.5301Ε+06 3.5397Ε+06 3.1343E+06 2.8202E+06 4.8874E+06

2-50

Name

Formula

−11143 −11976 −12549 −13231 −13478 −15015 −15557 −16093 −17897 −19441

57 58 59 60 61 62 63 64 65 66

Cyclopentane Methylcyclopentane Ethylcyclopentane Cyclohexane Methylcyclohexane 1,1-Dimethylcyclohexane Ethylcyclohexane Cyclopentene 1-Methylcyclopentene Cyclohexene

C5H10 C6H12 C7H14 C6H12 C7H14 C8H16 C8H16 C5H8 C6H10 C6H10

287923 96377 1640897 110827 108872 590669 1678917 142290 693890 110838

51.434 79.673 88.622 116.51 92.611 81.184 80.208 49.88 52.732 88.184

−4770.6 −6086.6 −7011 −7103.3 −7077.8 −6927 −7203.2 −4649.7 −5286.9 −6624.9

−4.3515 −8.7933 −10.038 −15.49 −10.684 −8.8498 −8.6023 −4.1191 −4.4509 −10.059

1.9605E−17 7.4046E−06 7.4481E−06 1.6959E−02 8.1239E−06 5.4580E−06 4.5901E−06 1.9564E−17 1.0883E−17 8.2566E−06

6 2 2 1 2 2 2 6 6 2

179.28 130.73 134.71 279.69 146.58 239.66 161.84 138.13 146.62 169.67

9.4420E+00 6.7059E−05 3.7061E−06 5.3802E+03 1.5256E−04 6.0584E+01 3.5747E−04 1.6884E−02 3.9787E−03 1.0377E−01

511.76 532.79 569.52 553.58 572.19 591.15 609.15 507 542 560.4

4.5028E+06 3.7808E+06 3.3970E+06 4.0958E+06 3.4828E+06 2.9387E+06 3.0411E+06 4.8062E+06 4.1303E+06 4.3922E+06

67 68 69 70 71 72 73

Benzene Toluene o-Xylene m-Xylene p-Xylene Ethylbenzene Propylbenzene

C6H6 C7H8 C8H10 C8H10 C8H10 C8H10 C9H12

71432 108883 95476 108383 106423 100414 103651

83.918 80.877 90.356 84.782 85.475 88.09 136.83

−6517.7 −6902.4 −7948.7 −7598.3 −7595.8 −7688.3 −9544.8

−9.3453 −8.7761 −10.081 −9.2612 −9.378 −9.7708 −18.190

7.1182E−06 5.8034E−06 5.9756E−06 5.5445E−06 5.6875E−06 5.8844E−06 1.6590E−02

2 2 2 2 2 2 1

278.68 178.18 247.98 225.3 286.41 178.15 324.18

4.7620E+03 4.2348E−02 2.1968E+01 3.2099E+00 5.8144E+02 4.0140E−03 2.0014E+03

562.16 591.8 630.33 617.05 616.23 617.2 638.32

4.8819E+06 4.1012E+06 3.7424E+06 3.5286E+06 3.4984E+06 3.5968E+06 3.2001E+06

74 75 76 77 78 79 80 81

1,2,4-Trimethylbenzene Isopropylbenzene 1,3,5-Trimethylbenzene p-Isopropyltoluene Naphthalene Biphenyl Styrene m-Terphenyl

C9H12 C9H12 C9H12 C10H14 C10H8 C12H10 C8H8 C18H14

95636 98828 108678 99876 91203 92524 100425 92068

60.658 143.62 48.603 107.71 62.447 76.811 105.93 88.044

−7260.4 −9687.7 −6545.2 −9402.7 −8109 −9878.5 −8685.9 −13367

−5.3772 −19.305 −3.6412 −12.545 −5.5571 −7.4384 −12.42 −8.6482

4.5816E−18 1.7703E−02 1.9307E−18 6.6661E−06 2.0800E−18 2.0436E−18 7.5583E−06 8.7874E−19

6 1 6 2 6 6 2 6

229.33 177.14 228.42 205.25 353.43 342.2 242.54 360

7.9735E−01 3.8034E−04 1.1889E+00 9.9261E−03 9.9229E+02 9.3752E+01 1.0613E+01 1.0112E+00

649.13 631.1 637.36 653.15 748.35 789.26 636 924.85

3.2533E+06 3.1837E+06 3.1119E+06 2.7957E+06 3.9941E+06 3.8615E+06 3.8234E+06 3.5297E+06

82 83 84 85 86 87 88

Methanol Ethanol 1-Propanol 1-Butanol 2-Butanol 2-Propanol 2-Methyl-2-propanol

CH4O C2H6O C3H8O C4H10O C4H10O C3H8O C4H10O

67561 64175 71238 71363 78922 67630 75650

81.768 74.475 88.134 93.173 152.54 76.964 172.31

−6876 −7164.3 −8498.6 −9185.9 −11111 −7623.8 −11590

−8.7078 −7.327 −9.0766 −9.7464 −19.025 −7.4924 −22.118

7.1926E−06 3.1340E−06 8.3303E−18 4.7796E−18 1.0426E−05 5.9436E−18 1.3709E−05

2 2 6 6 2 6 2

175.47 159.05 146.95 184.51 158.45 185.28 298.97

1.1147E−01 4.8459E−04 3.0828E−07 5.7220E−04 1.1323E−06 3.6606E−02 5.9356E+03

512.64 513.92 536.78 563.05 536.05 508.3 506.21

8.1402E+06 6.1171E+06 5.1214E+06 4.3392E+06 4.2014E+06 4.7908E+06 3.9910E+06

89 90 91 92 93 94 95 96

1-Pentanol 2-Methyl-1-butanol 3-Methyl-1-butanol 1-Hexanol 1-Heptanol Cyclohexanol Ethylene glycol 1,2-Propylene glycol

C5H12O C5H12O C5H12O C6H14O C7H16O C6H12O C2H6O2 C3H8O2

71410 137326 123513 111273 111706 108930 107211 57556

168.96 410.44 107.02 117.31 160.08 135.01 79.276 212.8

−12659 −20262 −10237 −11239 −14095 −12238 −10105 −15420

−21.366 −62.366 −11.695 −13.149 −19.211 −15.702 −7.521 −28.109

1.1591E−05 6.3353E−02 6.8003E−18 9.3676E−18 1.7043E−17 1.0349E−17 7.3408E−19 2.1564E−05

2 1 6 6 6 6 6 2

195.56 203 155.95 228.55 239.15 296.6 260.15 213.15

3.1816E−04 3.7992E−04 2.1036E−08 3.7401E−02 1.6990E−02 7.9382E+01 2.4834E−01 9.2894E−05

586.15 565 577.2 611.35 631.9 650 719.7 626

3.8657E+06 3.8749E+06 3.9013E+06 3.4557E+06 3.1810E+06 4.2456E+06 7.7100E+06 6.0413E+06

97 98 99 100

Phenol o-Cresol m-Cresol p-Cresol

C6H6O C7H8O C7H8O C7H8O

108952 95487 108394 106445

95.444 210.88 95.403 118.53

−10113 −13928 −10581 −11957

−10.09 −29.483 −10.004 −13.293

6.7603E−18 2.5182E−02 4.3032E−18 8.6988E−18

6 1 6 6

314.06 304.19 285.39 307.93

1.8798E+02 6.5326E+01 5.8624E+00 3.4466E+01

694.25 697.55 705.85 704.65

6.0585E+06 5.0583E+06 4.5221E+06 5.1507E+06

101 102 103 104 105 106 107 108 109 110 111 112

Dimethyl ether Methyl ethyl ether Methyl n-propyl ether Methyl isopropyl ether Methyl-n-butyl ether Methyl isobutyl ether Methyl tert-butyl ether Diethyl ether Ethyl propyl ether Ethyl isopropyl ether Methyl phenyl ether Diphenyl ether

C2H6O C3H8O C4H10O C4H10O C5H12O C5H12O C5H12O C4H10O C5H12O C5H12O C7H8O C12H10O

115106 540670 557175 598538 628284 625445 1634044 60297 628320 625547 100663 101848

44.704 205.79 50.83 55.096 102.04 58.165 55.875 136.9 143.11 57.723 128.06 59.969

−3.4444 −28.739 −4.1773 −4.8689 −12.278 −5.2568 −4.9604 −19.254 −18.751 −5.2136 −16.693 −5.1538

5.4574E−17 3.5317E−05 9.4076E−18 2.9518E−17 1.2131E−05 2.0194E−17 1.9123E−17 2.4508E−02 2.0620E−05 2.2998E−17 1.4919E−02 1.9983E−18

6 2 6 6 2 6 6 1 2 6 1 6

131.65 160 133.97 127.93 157.48 150 164.55 156.85 145.65 140 235.65 300.03

3.0496E+00 5.3423E−01 4.8875E−03 2.4971E−03 1.9430E−02 1.9801E−02 5.3566E−01 3.9545E−01 7.3931E−04 4.3092E−03 2.4466E+00 7.0874E+00

400.1 437.8 476.3 464.5 510 497 497.1 466.7 500.23 489 645.6 766.8

5.2735E+06 4.4658E+06 3.7721E+06 3.8892E+06 3.3089E+06 3.4130E+06 3.4106E+06 3.6412E+06 3.3729E+06 3.4145E+06 4.2731E+06 3.0971E+06

−3525.6 −9834.5 −4781.7 −4793.2 −6954.9 −5362.1 −5131.6 −6954.3 −8353.7 −5236.9 −9307.7 −8585.5

2-51

TABLE 2-6 Cmpd. no.

Vapor Pressure of Inorganic and Organic Liquids (Continued ) CAS no.

C1

C2

C3

C4

C5

Tmin, K

Ps at Tmin

Tmax, K

Ps at Tmax

113 114 115 116 117 118 119 120 121 122

Formaldehyde Acetaldehyde 1-Propanal 1-Butanal 1-Pentanal 1-Hexanal 1-Heptanal 1-Octanal 1-Nonanal 1-Decanal

CH2O C2H4O C3H6O C4H8O C5H10O C6H12O C7H14O C8H16O C9H18O C10H20O

50000 75070 123386 123728 110623 66251 111717 124130 124196 112312

101.51 193.69 80.581 99.33 149.58 81.507 107.17 250.25 337.71 201.64

−4917.2 −8036.7 −5896.1 −7083.6 −8890 −7776.8 −9070.3 −16162 −18506 −15133

−13.765 −29.502 −8.9301 −11.733 −20.697 −8.4516 −12.503 −33.927 −50.224 −26.264

2.2031E−02 4.3678E−02 8.2236E−06 1.0027E−05 2.2101E−02 1.5143E−17 7.4446E−06 2.2349E−05 4.7345E−02 1.4625E−05

1 1 2 2 1 6 2 2 1 2

181.15 150.15 170 176.75 182 217.15 229.8 246 255.15 267.15

8.8700E+02 3.2320E−01 1.3133E+00 3.1699E−01 5.2282E−02 1.2473E+00 1.1177E+00 4.1640E−01 3.4172E−01 4.8648E−01

408 466 504.4 537.2 566.1 591 617 638.1 658 674.2

6.5935E+06 5.5652E+06 4.9189E+06 4.3232E+06 3.9685E+06 3.4607E+06 3.1829E+06 2.9704E+06 2.7430E+06 2.5989E+06

123 124 125 126 127 128 129 130 131 132 133 134

Acetone Methyl ethyl ketone 2-Pentanone Methyl isopropyl ketone 2-Hexanone Methyl isobutyl ketone 3-Methyl-2-pentanone 3-Pentanone Ethyl isopropyl ketone Diisopropyl ketone Cyclohexanone Methyl phenyl ketone

C3H6O C4H8O C5H10O C5H10O C6H12O C6H12O C6H12O C5H10O C6H12O C7H14O C6H10O C8H8O

67641 78933 107879 563804 591786 108101 565617 96220 565695 565800 108941 98862

69.006 72.698 84.635 308.74 65.841 153.23 64.641 44.286 206.77 96.919 95.118 62.688

−5599.6 −6143.6 −7078.4 −13693 −7042 −10055 −6457.4 −5415.1 −12537 −8014.2 −8300.4 −8088.8

−7.0985 −7.5779 −9.3 −47.557 −6.1376 −19.848 −6.218 −3.0913 −27.894 −11.093 −10.796 −5.5434

6.2237E−06 5.6476E−06 6.2702E−06 5.7002E−02 7.2196E−18 1.6426E−05 3.4543E−06 1.8580E−18 2.2462E−05 7.3452E−06 6.5037E−06 2.0774E−18

2 2 2 1 6 2 2 6 2 2 2 6

178.45 186.48 196.29 181.15 217.35 189.15 167.15 234.18 200 204.81 242 292.81

2.7851E+00 1.3904E+00 7.5235E−01 2.2648E−02 1.5111E+00 3.3536E−02 3.2662E−03 7.3422E+01 6.0339E−02 3.9036E−01 6.9667E+00 3.5899E+01

508.2 535.5 561.08 553 587.05 571.4 573 560.95 567 576 653 709.5

4.7091E+06 4.1201E+06 3.7062E+06 3.8413E+06 3.3120E+06 3.2659E+06 3.3213E+06 3.6993E+06 3.3424E+06 3.0606E+06 4.0126E+06 3.8451E+06

135 136 137 138 139 140 141

Formic acid Acetic acid Propionic acid n-Butyric acid Isobutyric acid Benzoic acid Acetic anhydride

CH2O2 C2H4O2 C3H6O2 C4H8O2 C4H8O2 C7H6O2 C4H6O3

64186 64197 79094 107926 79312 65850 108247

50.323 53.27 54.552 93.815 110.38 88.513 100.95

−5378.2 −6304.5 −7149.4 −9942.2 −10540 −11829 −8873.2

−4.203 −4.2985 −4.2769 −9.8019 −12.262 −8.6826 −11.451

3.4697E−06 8.8865E−18 1.1843E−18 9.3124E−18 1.4310E−17 2.3248E−19 6.1316E−06

2 6 6 6 6 6 2

281.45 289.81 252.45 267.95 227.15 395.45 200.15

2.4024E+03 1.2769E+03 1.3142E+01 6.7754E+00 7.8244E−02 7.9550E+02 2.1999E−02

588 591.95 600.81 615.7 605 751 606

5.8074E+06 5.7390E+06 4.6080E+06 4.0705E+06 3.6834E+06 4.4691E+06 3.9702E+06

142 143 144 145 146 147 148

Methyl formate Methyl acetate Methyl propionate Methyl n-butyrate Ethyl formate Ethyl acetate Ethyl propionate

C2H4O2 C3H6O2 C4H8O2 C5H10O2 C3H6O2 C4H8O2 C5H10O2

107313 79209 554121 623427 109944 141786 105373

77.184 61.267 70.717 71.87 73.833 66.824 105.64

−5606.1 −5618.6 −6439.7 −6885.7 −5817 −6227.6 −8007

−8.392 −5.6473 −6.9845 −7.0944 −7.809 −6.41 −12.477

7.8468E−06 2.1080E−17 2.0129E−17 1.4903E−17 6.3200E−06 1.7914E−17 9.0000E−06

2 6 6 6 2 6 2

174.15 175.15 185.65 187.35 193.55 189.6 199.25

6.8808E+00 1.0170E+00 6.3409E−01 1.3435E−01 1.8119E+01 1.4318E+00 7.7988E−01

487.2 506.55 530.6 554.5 508.4 523.3 546

5.9829E+06 4.6948E+06 4.0278E+06 3.4797E+06 4.7080E+06 3.8502E+06 3.3365E+06

149 150 151 152 153 154 155

Ethyl n-butyrate n-Propyl formate n-Propyl acetate n-Butyl acetate Methyl benzoate Ethyl benzoate Vinyl acetate

C6H12O2 C4H8O2 C5H10O2 C6H12O2 C8H8O2 C9H10O2 C4H6O2

105544 110747 109604 123864 93583 93890 108054

57.661 104.08 115.16 71.34 82.976 53.024 57.406

−6346.5 −7535.9 −8433.9 −7285.8 −9226.1 −7676.8 −5702.8

−5.032 −12.348 −13.934 −6.9459 −8.4427 −4.1593 −5.0307

8.2534E−18 9.6020E−06 1.0346E−05 9.9895E−18 5.9115E−18 1.6850E−18 1.1042E−17

6 2 2 6 6 6 6

175.15 180.25 178.15 199.65 260.75 238.45 180.35

1.0390E−02 2.1101E−01 1.7113E−02 1.4347E−01 1.8653E+00 1.4385E−01 7.0586E−01

571 538 549.73 579.15 693 698 519.13

2.9352E+06 4.0310E+06 3.3657E+06 3.1097E+06 3.5896E+06 3.2190E+06 3.9298E+06

156 157 158 159 160 161

Methylamine Dimethylamine Trimethylamine Ethylamine Diethylamine Triethylamine

CH5N C2H7N C3H9N C2H7N C4H11N C6H15N

74895 124403 75503 75047 109897 121448

75.206 71.738 134.68 81.56 49.314 56.55

−5082.8 −5302 −6055.8 −5596.9 −4949 −5681.9

−8.0919 −7.3324 −19.415 −9.0779 −3.9256 −4.9815

8.1130E−06 6.4200E−17 2.8619E−02 8.7920E−06 9.1978E−18 1.2363E−17

2 6 1 2 6 6

179.69 180.96 156.08 192.15 223.35 158.45

1.7671E+02 7.5575E+01 9.9206E+00 1.5183E+02 3.7411E+02 1.0646E−02

430.05 437.2 433.25 456.15 496.6 535.15

7.4139E+06 5.2583E+06 4.1020E+06 5.5937E+06 3.6744E+06 3.0373E+06

162 163 164 165 166

n-Propylamine di-n-Propylamine Isopropylamine Diisopropylamine Aniline

C3H9N C6H15N C3H9N C6H15N C6H7N

107108 142847 75310 108189 62533

58.398 54 136.66 462.84 66.287

−5312.7 −6018.5 −7201.5 −18227 −8207.1

−5.2876 −4.4981 −18.934 −73.734 −6.0132

1.9913E−06 9.9684E−18 2.2255E−02 9.2794E−02 2.8414E−18

2 6 1 1 6

188.36 210.15 177.95 176.85 267.13

1.3004E+01 3.6942E+00 7.7251E+00 4.4724E−03 7.1322E+00

496.95 550 471.85 523.1 699

4.7381E+06 3.1113E+06 4.5404E+06 3.1987E+06 5.3514E+06

2-52

Name

Formula

167 168

N-Methylaniline N,N-Dimethylaniline

C7H9N C8H11N

100618 121697

70.843 51.352

−8517.5 −7160

−6.7007 −4.0127

5.6411E−18 8.1481E−07

6 2

216.15 275.6

1.0207E−02 1.7940E+01

701.55 687.15

5.1935E+06 3.6262E+06

169 170 171 172

Ethylene oxide Furan Thiophene Pyridine

C2H4O C4H4O C4H4S C5H5N

75218 110009 110021 110861

91.944 74.738 89.171 82.154

−5293.4 −5417 −6860.3 −7211.3

−11.682 −8.0636 −10.104 −8.8646

1.4902E−02 7.4700E−06 7.4769E−06 5.2528E−06

1 2 2 2

160.65 187.55 234.94 231.51

7.7879E+00 5.0026E+01 1.8538E+02 2.0535E+01

469.15 490.15 579.35 619.95

7.2553E+06 5.5497E+06 5.7145E+06 5.6356E+06

173 174 175 176 177 178 179 180

Formamide N,N-Dimethylformamide Acetamide N-Methylacetamide Acetonitrile Propionitrile n-Butyronitrile Benzonitrile

CH3NO C3H7NO C2H5NO C3H7NO C2H3N C3H5N C4H7N C7H5N

75127 68122 60355 79163 75058 107120 109740 100470

100.3 82.762 125.81 79.128 58.302 82.699 66.32 55.463

−10763 −7955.5 −12376 −9523.9 −5385.6 −6703.5 −6714.9 −7430.8

−10.946 −8.8038 −14.589 −7.7355 −5.4954 −9.1506 −6.3087 −4.548

3.8503E−06 4.2431E−06 5.0824E−06 3.1616E−18 5.3634E−06 7.5424E−06 1.3516E−17 1.7501E−18

2 2 2 6 2 2 6 6

275.6 212.72 353.33 301.15 229.32 180.26 161.25 260.4

1.0350E+00 1.9532E−01 3.3637E+02 2.8618E+01 1.8694E+02 1.6936E−01 6.1777E−04 5.1063E+00

771 649.6 761 718 545.5 564.4 582.25 699.35

7.7514E+06 4.3653E+06 6.5688E+06 4.9973E+06 4.8517E+06 4.1906E+06 3.7870E+06 4.2075E+06

181 182 183 184 185 186 187 188 189

Methyl mercaptan Ethyl mercaptan n-Propyl mercaptan n-Butyl mercaptan Isobutyl mercaptan sec-Butyl mercaptan Dimethyl sulfide Methyl ethyl sulfide Diethyl sulfide

CH4S C2H6S C3H8S C4H10S C4H10S C4H10S C2H6S C3H8S C4H10S

74931 75081 107039 109795 513440 513531 75183 624895 352932

54.15 65.551 62.165 65.382 61.736 60.649 83.485 79.07 60.867

−4337.7 −5027.4 −5624 −6262.4 −5909.2 −5785.9 −5711.7 −6114.1 −5969.6

−4.8127 −6.6853 −5.8595 −6.2585 −5.7554 −5.6113 −9.4999 −8.631 −5.5979

4.5000E−17 6.3208E−06 2.0597E−17 1.4943E−17 1.5119E−17 1.5877E−17 9.8449E−06 6.5333E−06 1.4530E−17

6 2 6 6 6 6 2 2 6

150.18 125.26 159.95 157.46 128.31 133.02 174.88 167.23 169.2

3.1479E+00 1.1384E−03 6.5102E−02 2.3532E−03 4.7502E−06 3.3990E−05 7.9009E+00 2.2456E−01 4.3401E−02

469.95 499.15 536.6 570.1 559 554 503.04 533 557.15

7.2309E+06 5.4918E+06 4.6272E+06 3.9730E+06 4.0603E+06 4.0598E+06 5.5324E+06 4.2610E+06 3.9629E+06

190 191 192 193 194 195 196 197

Fluoromethane Chloromethane Trichloromethane Tetrachloromethane Bromomethane Fluoroethane Chloroethane Bromoethane

CH3F CH3Cl CHCl3 CCl4 CH3Br C2H5F C2H5Cl C2H5Br

593533 74873 67663 56235 74839 353366 75003 74964

59.123 64.697 146.43 78.441 72.586 56.639 70.159 62.217

−3043.7 −4048.1 −7792.3 −6128.1 −4698.6 −3576.5 −4786.7 −5113.3

−6.1845 −6.8066 −20.614 −8.5766 −7.9966 −5.5801 −7.5387 −5.9761

1.6637E−05 1.0371E−05 2.4578E−02 6.8465E−06 1.1553E−05 9.8969E−06 9.3370E−06 4.7174E−17

2 2 1 2 2 2 2 6

131.35 175.43 207.15 250.33 179.47 129.95 134.8 154.55

4.3287E+02 8.7091E+02 5.2512E+01 1.1225E+03 1.9544E+02 8.3714E+00 1.1658E−01 3.7155E−01

317.42 416.25 536.4 556.35 467 375.31 460.35 503.8

5.8754E+06 6.6905E+06 5.5543E+06 4.5436E+06 7.9972E+06 5.0060E+06 5.4578E+06 6.2903E+06

198 199 200 201 202 203 204 205

1-Chloropropane 2-Chloropropane 1,1-Dichloropropane 1,2-Dichloropropane Vinyl chloride Fluorobenzene Chlorobenzene Bromobenzene

C3H7Cl C3H7Cl C3H6Cl2 C3H6Cl2 C2H3Cl C6H5F C6H5Cl C6H5Br

540545 75296 78999 78875 75014 462066 108907 108861

79.24 46.854 83.495 65.955 91.432 51.915 54.144 63.749

−5718.8 −4445.5 −6661.4 −6015.6 −5141.7 −5439 −6244.4 −7130.2

−8.789 −3.6533 −9.2386 −6.5509 −10.981 −4.2896 −4.5343 −5.879

8.4486E−06 1.3260E−17 6.7652E−06 4.3172E−06 1.4318E−05 8.7527E−18 4.7030E−18 5.2136E−18

2 6 2 2 2 6 6 6

150.35 155.97 200 172.71 119.36 230.94 227.95 242.43

6.9630E−02 9.0844E−01 4.5248E+00 8.2532E−02 1.9178E−02 1.5142E+02 8.4456E+00 7.8364E+00

503.15 489 560 572 432 560.09 632.35 670.15

4.5812E+06 4.5097E+06 4.2394E+06 4.2319E+06 5.7495E+06 4.5437E+06 4.5293E+06 4.5196E+06

206 207 208 209 210 211 212 213 214 215 216 217 218

Air3 Hydrogen Helium-44 Neon Argon Fluorine Chlorine Bromine Oxygen Nitrogen Ammonia Hydrazine Nitrous oxide

H2 He Ne Ar F2 Cl2 Br2 O2 N2 NH3 N2H4 N2O

132259100 1333740 7440597 7440019 7440371 7782414 7782505 7726956 7782447 7727379 7664417 302012 10024972

21.662 12.69 11.533 29.755 42.127 42.393 71.334 108.26 51.245 58.282 90.483 76.858 96.512

−692.39 −94.896 −8.99 −271.06 −1093.1 −1103.3 −3855 −6592 −1200.2 −1084.1 −4669.7 −7245.2 −4045

−0.39208 1.1125 0.6724 −2.6081 −4.1425 −4.1203 −8.5171 −14.16 −6.4361 −8.3144 −11.607 −8.22 −12.277

4.7574E−03 3.2915E−04 2.7430E−01 5.2700E−04 5.7254E−05 5.7815E−05 1.2378E−02 1.6043E−02 2.8405E−02 4.4127E−02 1.7194E−02 6.1557E−03 2.8860E−05

1 2 1 2 2 2 1 1 1 1 1 1 2

59.15 13.95 1.76 24.56 83.78 53.48 172.12 265.85 54.36 63.15 195.41 274.69 182.3

5.6421E+03 7.2116E+03 1.4625E+03 4.3800E+04 6.8721E+04 2.5272E+02 1.3660E+03 5.8534E+03 1.4754E+02 1.2508E+04 6.1111E+03 4.0847E+02 8.6908E+04

132.45 33.19 5.2 44.4 150.86 144.12 417.15 584.15 154.58 126.2 405.65 653.15 309.57

3.7934E+06 1.3154E+06 2.2845E+05 2.6652E+06 4.8963E+06 5.1674E+06 7.7930E+06 1.0276E+07 5.0206E+06 3.3906E+06 1.1301E+07 1.4731E+07 7.2782E+06

2-53

TABLE 2-6 Cmpd. no. 219 220 221 222 223 224 225 226 227 228 229 230 231

Vapor Pressure of Inorganic and Organic Liquids (Concluded ) Name Nitric oxide Cyanogen Carbon monoxide Carbon dioxide Carbon disulfide Hydrogen fluoride Hydrogen chloride Hydrogen bromide2 Hydrogen cyanide Hydrogen sulfide Sulfur dioxide Sulfur trioxide Water

Formula NO C2N2 CO CO2 CS2 HF HCl HBr HCN H2S SO2 SO3 H2O

CAS no.

C1

C2

C3

C4

C5

Tmin, K

Ps at Tmin

Tmax, K

Ps at Tmax

10102439 460195 630080 124389 75150 7664393 7647010 10035106 74908 7783064 7446095 7446119 7732185

72.974 88.589 45.698 140.54 67.114 59.544 104.27 29.315 36.75 85.584 47.365 180.99 73.649

−2650 −5059.9 −1076.6 −4735 −4820.4 −4143.8 −3731.2 −2424.5 −3927.1 −3839.9 −4084.5 −12060 −7258.2

−8.261 −10.483 −4.8814 −21.268 −7.5303 −6.1764 −15.047 −1.1354 −2.1245 −11.199 −3.6469 −22.839 −7.3037

9.7000E−15 1.5403E−05 7.5673E−05 4.0909E−02 9.1695E−03 1.4161E−05 3.1340E−02 2.3806E−18 3.8948E−17 1.8848E−02 1.7990E−17 7.2350E−17 4.1653E−06

6 2 2 1 1 2 1 6 6 1 6 6 2

109.5 245.25 68.15 216.58 161.11 189.79 158.97 185.15 259.83 187.68 197.67 289.95 273.16

2.1956E+04 7.3385E+04 1.5430E+04 5.1867E+05 1.4944E+00 3.3683E+02 1.3522E+04 2.9501E+04 1.8687E+04 2.2873E+04 1.6743E+03 2.0934E+04 6.1056E+02

180.15 400.15 132.92 304.21 552 461.15 324.65 363.15 456.65 373.53 430.75 490.85 647.13

6.5156E+06 5.9438E+06 3.4940E+06 7.3896E+06 8.0408E+06 6.4872E+06 8.3564E+06 8.4627E+06 5.3527E+06 8.9988E+06 7.8596E+06 8.1919E+06 2.1940E+07

All substances are listed in alphabetical order in Table 2-6a. Compiled from Daubert, T. E., R. P. Danner, H. M. Sibul, and C. C. Stebbins, DIPPR Data Compilation of Pure Compound Properties, Project 801 Sponsor Release, July, 1993, Design Institute for Physical Property Data, AIChE, New York, NY; and from Thermodynamics Research Center, “Selected Values of Properties of Hydrocarbons and Related Compounds,” Thermodynamics Research Center Hydrocarbon Project, Texas A&M University, College Station, Texas (extant 1994). Temperatures are in K; vapor pressures are in Pa. Pa × 9.869233E−06 = atm; Pa × 1.450377E−04 = psia; vapor pressure = exp [C1 + (C2/T) + C3 × ln (T) + C4 × T C5 ]. 1 Decomposes violently on heating. Forms explosive peroxides with air or oxygen. Polymerizes under pressure and heat. 2 Coefficients are hypothetical above the decomposition temperature. 3 At the bubble point. 4 Exhibits superfluid properties below 2.2 K.

2-54

VAPOR PRESSURES OF PURE SUBSTANCES

2-55

TABLE 2-6a Alphabetical Index to Substances in Tables 2-6, 2-30, 2-164, 2-193, 2-196, 2-198, and 2-221 Name Acetaldehyde Acetamide Acetic acid Acetic anhydride Acetone Acetonitrile Acetophenone Acetylene Air Ammonia Aniline Anisole Argon Benzene Benzoic acid Benzonitrile Biphenyl Bromine Bromobenzene Bromoethane Bromomethane 1,2-Butadiene 1,3-Butadiene n-Butane 1-Butanol 2-Butanol 1-Butene cis-2-Butene trans-2-Butene n-Butyl acetate n-Butyl mercaptan sec-Butyl mercaptan Butyraldehyde n-Butyric acid n-Butyronitrile Carbon dioxide Carbon disulfide Carbon monoxide Carbon tetrachloride Chlorine Chlorobenzene Chloroethane Chloroform Chloromethane 1-Chloropropane 2-Chloropropane o-Cresol m-Cresol p-Cresol Cumene Cyanogen Cyclohexane Cyclohexanol Cyclohexanone Cyclohexene Cyclopentane Cyclopentene p-Cymene 1-Decanal n-Decane 1-Decene 1,1-Dichloropropane 1,2-Dichloropropane Diethylamine Diethyl ether Diethyl sulfide Diisopropylamine Diisopropyl ketone Dimethylacetylene Dimethylamine N,N-Dimethylaniline 2,3-Dimethylbutane 1,1-Dimethylcyclohexane Dimethyl ether

Synonym Ethanal Ethanoic acid 2-Propanone Methyl cyanide Methyl phenyl ketone

Methyl phenyl ether

Phenyl cyanide 1,1′-Biphenyl Ethyl bromide Methyl bromide

sec-Butyl alcohol Z-2-Butene E-2-Butene 1-Butanethiol 2-Butanethiol Butanal Propyl cyanide

Tetrachloromethane Ethyl chloride Trichloromethane Methyl chloride Propyl chloride Isopropyl chloride 2-Methylphenol 3-Methylphenol 4-Methylphenol Isopropylbenzene Cyclohexyl alcohol Cyclohexyl ketone

p-Isopropyltoluene

Ethyl ether Ethyl sulfide 2,4-Dimethyl-3-pentanone 2-Butyne N,N-Dimethylbenzamine Diisopropyl Methyl ether

Cmpd. no.

Formula

114 175 136 141 123 177 134 45 206 216 166 111 210

C2H4O C2H5NO C2H4O2 C4H6O3 C3H6O C2H3N C8H8O C2H2

67 140 180 79 213 205 197 194 42 43 4 85 86 30 31 32 152 184 186 116 138 179

C6H6 C7H6O2 C7H5N C12H10 Br2 C6H5Br C2H5Br CH3Br C4H6 C4H6 C4H10 C4H10O C4H10O C4H8 C4H8 C4H8 C6H12O2 C4H10S C4H10S C4H8O C4H8O2 C4H7N

222 223 221 193 212 204 196 192 191 198 199 98 99 100 75 220 60 94 133 66 57 64 77

CO2 CS2 CO CCl4 Cl2 C6H5Cl C2H5Cl CHCl3 CH3Cl C3H7Cl C3H7Cl C7H8O C7H8O C7H8O C9H12 C2N2 C6H12 C6H12O C6H10O C6H10 C5H10 C5H8 C10H14

122 10 38 200 201 160 108 189 165 132 47 157 168 23 62 101

C10H20O C10H22 C10H20 C3H6Cl2 C3H6Cl2 C4H11N C4H10O C4H10S C6H15N C7H14O C4H6 C2H7N C8H11N C6H14 C8H16 C2H6O

NH3 C6H7N C7H8O Ar

Name

Synonym

1-Hexene 1-Hexyne 2-Hexyne 3-Hexyne Hydrazine Hydrogen Hydrogen bromide Hydrogen chloride Hydrogen cyanide Hydrogen fluoride Hydrogen sulfide

Cmpd. no.

Formula

34 51 52 53 217 207 226 225 227 224 228

C6H12 C6H10 C6H10 C6H10 N2H4 H2 HBr HCl HCN HF H2S

Isobutyl mercaptan Isobutyric acid Isooctane Isoprene Isopropylamine

2-Methyl-1-propanethiol 2-Methylpropanoic acid 2,2,4-Trimethylpentane 2-Methyl-1,3-butadiene

185 139 27 44 164

C4H10S C4H8O2 C8H18 C5H8 C3H9N

Mesitylene Methane Methanol N-Methylacetamide Methyl acetate Methylacetylene Methylamine N-Methylaniline Methyl benzoate 2-Methylbutane 2-Methyl-1-butanol 3-Methyl-1-butanol 2-Methyl-1-butene 2-Methyl-2-butene Methyl butyl ether 3-Methyl-1-butyne Methyl butyrate Methylcyclohexane Methylcyclopentane 1-Methylcyclopentene Methyl ethyl ether Methyl ethyl ketone Methyl ethyl sulfide Methyl formate Methyl isobutyl ether Methyl isobutyl ketone Methyl isopropyl ether Methyl isopropyl ketone Methyl mercaptan 2-Methylpentane 3-Methyl-2-pentanone 2-Methylpropane 2-Methyl-2-propanol 2-Methylpropene Methyl propionate Methyl propyl ether Methyl tert-butyl ether

1,3,5-Trimethylbenzene

76 1 82 176 143 46 156 167 153 22 90 91 40 41 105 48 145 61 58 65 102 124 188 142 106 128 104 126 181 24 129 21 88 39 144 103 107

C9H12 CH4 CH4O C3H7NO C3H6O2 C3H4 CH5N C7H9N C8H8O2 C5H12 C5H12O C5H12O C5H10 C5H10 C5H12O C5H8 C5H10O2 C7H14 C6H12 C6H10 C3H8O C4H8O C3H8S C2H4O2 C5H12O C6H12O C4H10O C5H10O CH4S C6H14 C6H12O C4H10 C4H10O C4H8 C4H8O2 C4H10O C5H12O

78 209 219 215 218 19 121 9 37

C10H8 Ne NO N2 N2O C19H40 C9H18O C9H20 C9H18

18 120 8 36 55 214

C18H38 C8H16O C8H18 C8H16 C8H14 O2

15 117 5 89

C15H32 C5H10O C5H12 C5H12O

Naphthalene Neon Nitric oxide Nitrogen Nitrous oxide n-Nonadecane 1-Nonanal n-Nonane 1-Nonene n-Octadecane 1-Octanal n-Octane 1-Octene 1-Octyne Oxygen n-Pentadecane 1-Pentanal n-Pentane 1-Pentanol

Methyl alcohol

Isopentane Isoamyl alcohol Amylene

2-Butanone 2-Thiabutane

Methanethiol Isohexane Methyl sec-butyl ketone Isobutane tert-Butyl alcohol Isobutene

n-Nonaldehyde

n-Octaldehyde

Valeraldehyde n-Amyl alcohol

2-56

PHYSICAL AND CHEMICAL DATA

TABLE 2-6a Alphabetical Index to Substances in Tables 2-6, 2-30, 2-164, 2-193, 2-196, 2-198, and 2-221 (Concluded ) Name N,N-Dimethylformamide 2,3-Dimethylpentane Dimethyl sulfide Diphenyl ether n-Dodecane n-Eicosane Ethane Ethanol Ethyl acetate Ethylamine Ethylbenzene Ethyl benzoate Ethyl butyrate Ethylcyclohexane Ethylcyclopentane Ethylene Ethylene glycol Ethylene oxide Ethyl formate Ethyl isopropyl ether Ethyl isopropyl ketone Ethyl mercaptan Ethyl propionate Ethyl propyl ether Fluorine Fluorobenzene Fluoroethane Fluoromethane Formaldehyde Formamide Formic acid Furan Helium-4 n-Heptadecane 1-Heptanal n-Heptane 1-Heptanol 1-Heptene 1-Heptyne n-Hexadecane 1-Hexanal n-Hexane 1-Hexanol 2-Hexanone

Synonym

Methyl sulfide

Ethyl alcohol Phenylethane

1,2-Ethanediol 1,2-Epoxyethane

Ethanethiol

Ethyl fluoride Methyl fluoride Methanal Methanoic acid

n-Heptaldehyde n-Heptyl alcohol

Caproaldehyde n-Hexyl alcohol Methyl n-butyl ketone

Cmpd. no.

Formula

174 25 187 112 12

C3H7NO C7H16 C2H6S C12H10O C12H26

20 2 83 147 159 72 154 149 63 59 28 95 169 146 110 131 182 148 109

C20H42 C2H6 C2H6O C4H8O2 C2H7N C8H10 C9H10O2 C6H12O2 C8H16 C7H14 C2H4 C2H6O2 C2H4O C3H6O2 C5H12O C6H12O C2H6S C5H10O2 C5H12O

211 203 195 190 113 173 135 170

F2 C6H5F C2H5F CH3F CH2O CH3NO CH2O2 C4H4O

208 17 119 7 93 35 54 16 118 6 92 127

He C17H36 C7H14O C7H16 C7H16O C7H14 C7H12 C16H34 C6H12O C6H14 C6H14O C6H12O

Name

Synonym

Cmpd. no.

Formula

Methyl n-propyl ketone Diethyl ketone

125 130 33 49 50 97 115 3 84 87 137 178 151 162 163 73 29 96 150 183 172

C5H10O C5H10O C5H10 C5H8 C5H8 C6H6O C3H6O C3H8 C3H8O C3H8O C3H6O2 C3H5N C5H10O2 C3H9N C6H15N C9H12 C3H6 C3H8O2 C4H8O2 C3H8S C5H5N

Styrene Sulfur dioxide Sulfur trioxide

80 229 230

C8H8 SO2 SO3

m-Terphenyl n-Tetradecane Thiophene Toluene n-Tridecane Triethylamine Trimethylamine 1,2,4-Trimethylbenzene 2,3,3-Trimethylpentane

81 14 171 68 13 161 158 74 26

C18H14 C14H30 C4H4S C7H8 C13H28 C6H15N C3H9N C9H12 C8H18

11

C11H24

Vinyl acetate Vinylacetylene Vinyl chloride

155 56 202

C4H6O2 C4H4 C2H3Cl

Water

231

H2O

69 70 71

C8H10 C8H10 C8H10

2-Pentanone 3-Pentanone 1-Pentene 1-Pentyne 2-Pentyne Phenol 1-Propanal n-Propane 1-Propanol 2-Propanol n-Propionic acid n-Propionitrile n-Propyl acetate n-Propylamine di-n-Propylamine n-Propylbenzene Propylene 1,2-Propylene glycol n-Propyl formate n-Propyl mercaptan Pyridine

Propionaldehyde n-Propyl alcohol Isopropyl alcohol Ethyl cyanide

1,2-Propanediol Propanethiol

n-Undecane

o-Xylene m-Xylene p-Xylene

1,2-Dimethylbenzene 1,3-Dimethylbenzene 1,4-Dimethylbenzene

VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-7

2-57

Vapor Pressures of Inorganic Compounds, up to 1 atm* Compound

Pressure, mm Hg 1

Name Aluminum borohydride bromide chloride fluoride iodide oxide Ammonia heavy Ammonium bromide carbamate chloride cyanide hydrogen sulfide iodide Antimony tribromide trichloride pentachloride triiodide trioxide Argon Arsenic Arsenic tribromide trichloride trifluoride pentafluoride trioxide Arsine Barium Beryllium borohydride bromide chloride iodide Bismuth tribromide trichloride Diborane hydrobromide Borine carbonyl triamine Boron hydrides dihydrodecaborane dihydrodiborane dihydropentaborane tetrahydropentaborane tetrahydrotetraborane Boron tribromide trichloride trifluoride Bromine pentafluoride Cadmium chloride fluoride iodide oxide Calcium Carbon (graphite) dioxide disulfide monoxide oxyselenide oxysulfide selenosulfide subsulfide tetrabromide tetrachloride tetrafluoride Cesium bromide chloride fluoride iodide

5

10

20

Al Al(BH4)3 AlBr3 Al2Cl6 AlF3 AlI3 Al2O3 NH3 ND3 NH4Br N2H6CO2 NH4Cl NH4CN NH4HS NH4I Sb SbBr3 SbCl3 SbCl5 SbI3 Sb4O6 A As AsBr3 AsCl3 AsF3 AsF5 As2O3 AsH3 Ba Be(BH4)2 BeBr2 BeCl2 BeI2 Bi BiBr3 BiCl3 B2H5Br BH3CO B3N3H6 B10H14 B2H6 B5H9 B5H11 B4H10 BBr3 BCl3 BF3 Br2 BrF5 Cd CdCl2 CdF2 CdI2 CdO Ca C CO2 CS2 CO COSe COS CSeS C3S2 CBr4 CCl4 CF4 Cs CsBr CsCl CsF CsI

40

60

100

200

400

760

Melting point, °C

1749 −3.9 176.1 152.0 1422 294.5 2665 −68.4 −67.4 320.0 26.7 271.5 −0.5 0.0 331.8 1223 203.5 143.3 114.1 303.5 957 −200.5 518 145.2 70.9 13.2 −84.3 332.5 −98.0 1301 58.6 405 411 411 1271 360 343 −29.0 −95.3 +4.0

1844 +11.2 199.8 161.8 1457 322.0 2766 −57.0 −57.0 345.3 37.2 293.2 +9.6 +10.5 355.8 1288 225.7 165.9

1947 28.1 227.0 171.6 1496 354.0 2874 −45.4 −45.4 370.8 48.0 316.5 20.5 21.8 381.0 1364 250.2 192.2

2056 45.9 256.3 180.2 1537 385.5 2977 −33.6 −33.4 396.0 58.3 337.8 31.7 33.3 404.9 1440 275.0 219.0

660 −64. 97. 192.4 1040

333.8 1085 −195.6 548 167.7 89.2 26.7 −75.5 370.0 −87.2 1403 69.0 427 435 435 1319 392 372 −15.4 −85.5 18.5

368.5 1242 −190.6 579 193.6 109.7 41.4 −64.0 412.2 −75.2 1518 79.7 451 461 461 1370 425 405 0.0 −74.8 34.3

401.0 1425 −185.6 610 220.0 130.4 56.3 −52.8 457.2 −62.1 1638 90.0 474 487 487 1420 461 441 +16.3 −64.0 50.6

142.3 −120.9 +9.6 20.1 −28.1 33.5 −32.4 −123.0 +9.3 −4.5 611 797 1486 640 1341 1207 4373 −100.2 −5.1 −205.7 −61.7 −85.9 28.3 109.9 119.7 23.0 −150.7 509 1072 1069 1025 1055

163.8 −111.2 24.6 34.8 −14.0 50.3 −18.9 −115.9 24.3 +9.9 658 847 1561 688 1409 1288 4516 −93.0 +10.4 −201.3 −49.8 −75.0 45.7 130.8 139.7 38.3 −143.6 561 1140 1139 1092 1124

−99.6 40.8 51.2 +0.8 70.0 −3.6 −108.3 41.0 25.7 711 908 1651 742 1484 1388 4660 −85.7 28.0 −196.3 −35.6 −62.7 65.2

−86.5 58.1 67.0 16.1 91.7 +12.7 −100.7 58.2 40.4 765 967 1751 796 1559 1487 4827 −78.2 46.5 −191.3 −21.9 −49.9 85.6

163.5 57.8 −135.5 624 1221 1217 1170 1200

189.5 76.7 −127.7 690 1300 1300 1251 1280

Temperature, °C

Formula 1284 81.3 100.0 1238 178.0 2148 −109.1

1421 −52.2 103.8 116.4 1298 207.7 2306 −97.5

1487 −42.9 118.0 123.8 1324 225.8 2385 −91.9

1555 −32.5 134.0 131.8 1350 244.2 2465 −85.8

1635 −20.9 150.6 139.9 1378 265.0 2549 −79.2

198.3 −26.1 160.4 −50.6 −51.1 210.9 886 93.9 49.2 22.7 163.6 574 −218.2 372 41.8 −11.4

234.5 −10.4 193.8 −35.7 −36.0 247.0 984 126.0 71.4 48.6 203.8 626 −213.9 416 70.6 +11.7

252.0 −2.9 209.8 −28.6 −28.7 263.5 1033 142.7 85.2 61.8 223.5 666 −210.9 437 85.2 +23.5

270.6 +5.3 226.1 −20.9 −20.8 282.8 1084 158.3 100.6 75.8 244.8 729 −207.9 459 101.3 36.0

−117.9 212.5 −142.6

−108.0 242.6 −130.8 984 19.8 325 328 322 1099 261 242 −75.3 −127.3 −45.0

−103.1 259.7 −124.7 1049 28.1 342 346 341 1136 282 264 −66.3 −121.1 −35.3

−98.0 279.2 −117.7 1120 36.8 361 365 361 1177 305 287 −56.4 −114.1 −25.0

290.0 14.0 245.0 −12.6 −12.3 302.8 1141 177.4 117.8 91.0 267.8 812 −204.9 483 118.7 50.0 −2.5 −92.4 299.2 −110.2 1195 46.2 379 384 382 1217 327 311 −45.4 −106.6 −13.2

1684 −13.4 161.7 145.4 1398 277.8 2599 −74.3 −74.0 303.8 19.6 256.2 −7.4 −7.0 316.0 1176 188.1 128.3 101.0 282.5 873 −202.9 498 130.0 58.7 +4.2 −88.5 310.3 −104.8 1240 51.7 390 395 394 1240 340 324 −38.2 −101.9 −5.8

3586 −134.3 −73.8 −222.0 −117.1 −132.4 −47.3 14.0

80.8 −149.5 −40.4 −29.9 −73.1 −20.4 −75.2 −145.4 −32.8 −51.0 455 618 1231 481 1100 926 3828 −124.4 −54.3 −217.2 −102.3 −119.8 −26.5 41.2

90.2 −144.3 −30.7 −19.9 −64.3 −10.1 −66.9 −141.3 −25.0 −41.9 484 656 1286 512 1149 983 3946 −119.5 −44.7 −215.0 −95.0 −113.3 −16.0 54.9

100.0 −138.5 −20.0 −9.2 −54.8 +1.5 −57.9 −136.4 −16.8 −32.0 516 695 1344 546 1200 1046 4069 −114.4 −34.3 −212.8 −86.3 −106.0 −4.4 69.3

−50.0 −184.6 279 748 744 712 738

−30.0 −174.1 341 838 837 798 828

−19.6 −169.3 375 887 884 844 873

−8.2 −164.3 409 938 934 893 923

117.4 −131.6 −8.0 +2.7 −44.3 14.0 −47.8 −131.0 −8.0 −21.0 553 736 1400 584 1257 1111 4196 −108.6 −22.5 −210.0 −76.4 −98.3 +8.6 85.6 96.3 +4.3 −158.8 449 993 989 947 976

127.8 −127.2 −0.4 10.2 −37.4 22.1 −41.2 −127.6 −0.6 −14.0 578 762 1436 608 1295 1152 4273 −104.8 −15.3 −208.1 −70.2 −93.0 17.0 96.0 106.3 12.3 −155.4 474 1026 1023 980 1009

+1.0 289 291 283 1021 −93.3 −139.2 −63.0 60.0 −159.7 −50.2 −90.9 −41.4 −91.5 −154.6 −48.7 −69.3 394 1112 416 1000

*Compiled from the extended tables published by D. R. Stull in Ind. Eng. Chem., 39, 517 (1947).

2050 −77.7 −74.0 520 36 630.5 96.6 73.4 2.8 167 656 −189.2 814 −18 −5.9 −79.8 312.8 −116.3 850 123 490 405 488 271 218 230 −104.2 −137.0 −58.2 99.6 −169 −47.0 −119.9 −45 −107 −126.8 −7.3 −61.4 320.9 568 520 385 851 −57.5 −110.8 −205.0 −138.8 −75.2 +0.4 90.1 −22.6 −183.7 28.5 636 646 683 621

2-58

PHYSICAL AND CHEMICAL DATA

TABLE 2-7

Vapor Pressures of Inorganic Compounds, up to 1 atm (Continued ) Compound

Name Chlorine fluoride trifluoride monoxide dioxide heptoxide Chlorosulfonic acid Chromium carbonyl oxychloride Cobalt chloride nitrosyl tricarbonyl Columbium fluoride Copper Cuprous bromide chloride iodide Cyanogen bromide chloride fluoride Deuterium cyanide Fluorine oxide Germanium bromide chloride hydride Trichlorogermane Tetramethylgermane Digermane Trigermane Gold Helium para-Hydrogen Hydrogen bromide chloride cyanide fluoride iodide oxide (water) sulfide disulfide selenide telluride Iodine heptafluoride Iron pentacarbonyl Ferric chloride Ferrous chloride Krypton Lead bromide chloride fluoride iodide oxide sulfide Lithium bromide chloride fluoride iodide Magnesium chloride Manganese chloride Mercury Mercuric bromide chloride iodide Molybdenum hexafluoride oxide

Pressure, mm Hg 1

5

10

20

−118.0 −98.5

−106.7 −143.4 −80.4 −81.6

−45.3 32.0 1616 36.0 −18.4

−23.8 53.5 1768 58.0 +3.2

−101.6 −139.0 −71.8 −73.1 −59.0 −13.2 64.0 1845 68.3 13.8

−93.3 −134.3 −62.3 −64.3 −51.2 −2.1 75.3 1928 79.5 25.7

86.3 1879 718 702 656 −76.8 −10.0 −53.8 −118.5 −46.7 −214.1 −182.3 56.8 −15.0 −145.3 −13.0 −45.2 −60.1 −0.9 2154 −271.3 −261.3 −121.8 −135.6 −47.7 −65.8 −102.3 11.2 −116.3 −15.2 −97.9 −75.4 73.2 −63.0 2039 +4.6 235.5 700 −187.2 1162 610 648 904 571 1085 975 881 888 932 1211 841 743 930 1505 778 184.0 179.8 180.2 204.5 3535 −40.8 814

−1.3 103.0 1970 777 766 716 −70.1 −1.0 −46.1 −112.8 −38.8 −211.0 −177.8 71.8 −4.1 −139.2 −3.0 −35.0 −49.9 +11.8 2256 −271.1 −260.4 −115.4 −130.0 −39.7 −56.0 −94.5 22.1 −109.7 −5.1 −91.8 −67.8 84.7 −54.5 2128 16.7 246.0 737 −182.9 1234 646 684 950 605 1134 1005 940 939 987 1270 883 789 988 1583 825 204.6 194.3 195.8 220.0 3690 −32.0 851

60

100

200

400

760

Melting point, °C

−71.7 −120.8 −34.7 −39.4 −29.4 29.1 105.3 2139 108.0 58.0 843 29.0 148.5 2207 951 960 907 −51.8 22.6 −24.9 −97.0 −17.5 −202.7 −165.8 113.2 27.5 −120.3 26.5 −6.3 −20.3 47.9 2521 −270.3 −257.9 −97.7 −114.0 −17.8 −28.2 −72.1 51.6 −91.6 22.0 −74.2 −45.7 116.5 −31.9 2360 50.3 272.5 842 −171.8 1421 745 784 1080 701 1265 1108 1097 1076 1129 1425 993 909 1142 1792 960 261.7 237.8 237.0 261.8 4109 −8.0 955

−60.2 −114.4 −20.7 −26.5 −17.8 44.6 120.0 2243 121.8 75.2 904 44.4 172.2 2325 1052 1077 1018 −42.6 33.8 −14.1 −89.2 −5.4 −198.3 −159.0 135.4 44.4 −111.2 41.6 +8.8 −4.7 67.0 2657 −269.8 −256.3 −88.1 −105.2 −5.3 −13.2 −60.3 66.5 −82.3 35.3 −65.2 −32.4 137.3 −20.7 2475 68.0 285.0 897 −165.9 1519 796 833 1144 750 1330 1160 1178 1147 1203 1503 1049 967 1223 1900 1028 290.7 262.7 256.5 291.0 4322 +4.1 1014

−47.3 −107.0 −4.9 −12.5 −4.0 62.2 136.1 2361 137.2 95.2 974 62.0 198.0 2465 1189 1249 1158 −33.0 46.0 −2.3 −80.5 +10.0 −193.2 −151.9 161.6 63.8 −100.2 58.3 26.0 +13.3 88.6 2807 −269.3 −254.5 −78.0 −95.3 +10.2 +2.5 −48.3 83.0 −71.8 49.6 −53.6 −17.2 159.8 −8.3 2605 86.1 298.0 961 −159.0 1630 856 893 1219 807 1402 1221 1273 1226 1290 1591 1110 1034 1316 2029 1108 323.0 290.0 275.5 324.2 4553 17.2 1082

−33.8 −100.5 +11.5 +2.2 +11.1 78.8 151.0 2482 151.0 117.1 1050 80.0 225.0 2595 1355 1490 1336 −21.0 61.5 +13.1 −72.6 26.2 −187.9 −144.6 189.0 84.0 −88.9 75.0 44.0 31.5 110.8 2966 −268.6 −252.5 −66.5 −84.8 25.9 19.7 −35.1 100.0 −60.4 64.0 −41.1 −2.0 183.0 +4.0 2735 105.0 319.0 1026 −152.0 1744 914 954 1293 872 1472 1281 1372 1310 1382 1681 1171 1107 1418 2151 1190 357.0 319.0 304.0 354.0 4804 36.0 1151

−100.7 −145 −83 −116 −59 −91 −80 1615

Temperature, °C

Formula Cl2 ClF ClF3 Cl2O ClO2 Cl2O7 HSO3Cl Cr Cr(CO)6 CrO2Cl2 CoCl2 Co(CO)3NO CbF5 Cu Cu2Br2 Cu2Cl2 Cu2I2 C2N2 CNBr CNCl CNF DCN F2 F2O GeBr4 GeCl4 GeH4 GeHCl3 Ge(CH3)4 Ge2H6 Ge3H8 Au He H2 HBr HCl HCN H2F2 HI H2O H2S HSSH H2Se H2Te I2 IF7 Fe Fe(CO)5 Fe2Cl6 FeCl2 Kr Pb PbBr2 PbCl2 PbF2 PbI2 PbO PbS Li LiBr LiCl LiF LiI Mg MgCl2 Mn MnCl2 Hg HgBr2 HgCl2 HgI2 Mo MoF6 MoO3

40

1628 572 546 −95.8 −35.7 −76.7 −134.4 −68.9 −223.0 −196.1 −45.0 −163.0 −41.3 −73.2 −88.7 −36.9 1869 −271.7 −263.3 −138.8 −150.8 −71.0 −123.3 −17.3 −134.3 −43.2 −115.3 −96.4 38.7 −87.0 1787 194.0 −199.3 973 513 547 479 943 852 723 748 783 1047 723 621 778 1292 126.2 136.5 136.2 157.5 3102 −65.5 734

1795 666 645 610 −83.2 −18.3 −61.4 −123.8 −54.0 −216.9 −186.6 43.3 −24.9 −151.0 −22.3 −54.6 −69.8 −12.8 2059 −271.5 −261.9 −127.4 −140.7 −55.3 −74.7 −109.6 +1.2 −122.4 −24.4 −103.4 −82.4 62.2 −70.7 1957 −6.5 221.8 −191.3 1099 578 615 861 540 1039 928 838 840 880 1156 802 702 877 1434 736 164.8 165.3 166.0 189.2 3393 −49.0 785

−84.5 −128.8 −51.3 −54.3 −42.8 +10.3 87.6 2013 91.2 38.5 770 +11.0 121.5 2067 844 838 786 −62.7 +8.6 −37.5 −106.4 −30.1 −207.7 −173.0 88.1 +8.0 −131.6 +8.8 −23.4 −38.2 26.3 2363 −270.7 −259.6 −108.3 −123.8 −30.9 −45.0 −85.6 34.0 −102.3 +6.0 −84.7 −59.1 97.5 −45.3 2224 30.3 256.8 779 −178.4 1309 686 725 1003 644 1189 1048 1003 994 1045 1333 927 838 1050 1666 879 228.8 211.5 212.5 238.2 3859 −22.1 892

−79.0 −125.3 −44.1 −48.0 −37.2 +18.2 95.2 2067 98.3 46.7 801 18.5 133.2 2127 887 886 836 −57.9 14.7 −32.1 −102.3 −24.7 −205.6 −170.0 98.8 16.2 −126.7 16.2 −16.2 −30.7 35.5 2431 −270.6 −258.9 −103.8 −119.6 −25.1 −37.9 −79.8 41.5 −97.9 12.8 −80.2 −53.7 105.4 −39.4 2283 39.1 263.7 805 −175.7 1358 711 750 1036 668 1222 1074 1042 1028 1081 1372 955 868 1088 1720 913 242.0 221.0 222.2 249.0 3964 −16.2 917

735 −11 75.5 1083 504 422 605 −34.4 58 −6.5 −12 −223 −223.9 26.1 −49.5 −165 −71.1 −88 −109 −105.6 1063 −259.1 −87.0 −114.3 −13.2 −83.7 −50.9 0.0 −85.5 −89.7 −64 −49.0 112.9 5.5 1535 −21 304 −156.7 327.5 373 501 855 402 890 1114 186 547 614 870 446 651 712 1260 650 −38.9 237 277 259 2622 17 795

VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-7

2-59

Vapor Pressures of Inorganic Compounds, up to 1 atm (Continued ) Compound

Name Neon Nickel carbonyl chloride Nitrogen Nitric oxide Nitrogen dioxide Nitrogen pentoxide Nitrous oxide Nitrosyl chloride fluoride Osmium tetroxide (yellow) (white) Oxygen Ozone Phosgene Phosphorus (yellow) (violet) tribromide trichloride pentachloride Phosphine Phosphonium bromide chloride iodide Phosphorus trioxide pentoxide oxychloride thiobromide thiochloride Platinum Potassium bromide chloride fluoride hydroxide iodide Radon Rhenium heptoxide Rubidium bromide chloride fluoride iodide Selenium dioxide hexafluoride oxychloride tetrachloride Silicon dioxide tetrachloride tetrafluoride Trichlorofluorosilane Iodosilane Diiodosilane Disiloxan Trisilane Trisilazane Tetrasilane Octachlorotrisilane Hexachlorodisiloxane Hexachlorodisilane Tribromosilane Trichlorosilane Trifluorosilane Dibromosilane Difluorosilane Monobromosilane Monochlorosilane Monofluorosilane Tribromofluorosilane Dichlorodifluorosilane Trifluorobromosilane

Pressure, mm Hg 1

5

10

20

40

−257.3 1810

−255.5 1979

−254.6 2057

−253.7 2143

671 −226.1 −184.5 −55.6 −36.8 −143.4

731 −221.3 −180.6 −42.7 −23.0 −133.4

759 −219.1 −178.2 −36.7 −16.7 −128.7

789 −216.8 −175.3 −30.4 −10.0 −124.0

−132.0 3.2 −5.6 −219.1 −180.4 −92.9 76.6 237 7.8 −51.6 55.5

−120.3 22.0 +15.6 −213.4 −168.6 −77.0 111.2 271 34.4 −31.5 74.0

−114.3 31.3 26.0 −210.6 −163.2 −69.3 128.0 287 47.8 −21.3 83.2

−107.8 41.0 37.4 −207.5 −157.2 −60.3 146.2 306 62.4 −10.2 92.5

−43.7 −91.0 −25.2 384

−28.5 −79.6 −9.0 39.7 424

50.0 −18.3 2730 341 795 821 885 719 745 −144.2 212.5 297 781 792 921 748 356 157.0 −118.6 34.8 74.0 1724

72.4 +4.6 3007 408 892 919 988 814 840 −132.4 237.5 358 876 887 982 839 413 187.7 −105.2 59.8 96.3 1835

−63.4 −144.0 −92.6

−44.1 −134.8 −76.4 −53.0 3.8 −95.8 −49.7 −49.9 −6.2 74.7 17.8 27.4 −8.0 −62.6 −142.7 −40.0 −136.0 −85.7 −104.3 −145.5 −25.4 −110.5

−21.2 −74.0 −1.1 53.0 442 2.0 83.6 16.1 3146 443 940 968 1039 863 887 −126.3 248.0 389 923 937 1016 884 442 202.5 −98.9 71.9 107.4 1888 1732 −34.4 −130.4 −68.3 −47.7 18.0 −88.2 −40.0 −40.4 +4.3 89.3 29.4 38.8 +3.4 −53.4 −138.2 −29.4 −130.4 −77.3 −97.7 −141.2 −15.1 −102.9

−13.3 −68.0 +7.3 67.8 462 13.6 95.5 29.0 3302 483 994 1020 1096 918 938 −119.2 261.0 422 975 990 1052 935 473 217.5 −92.3 84.2 118.1 1942 1798 −24.0 −125.9 −59.0 −33.4 34.1 −79.8 −29.0 −30.0 15.8 104.2 41.5 51.5 16.0 −43.8 −132.9 −18.0 −124.3 −68.3 −90.1 −136.3 −3.7 −94.5

−252.6 2234 −23.0 821 −214.0 −171.7 −23.9 −2.9 −118.3 −60.2 −100.3 51.7 50.5 −204.1 −150.7 −50.3 166.7 323 79.0 +2.3 102.5 −129.4 −5.0 −61.5 16.1 84.0 481 27.3 108.0 42.7 3469 524 1050 1078 1156 976 995 −111.3 272.0 459 1031 1047 1096 991 506 234.1 −84.7 98.0 130.1 2000 1867 −12.1 −120.8 −48.8 −21.8 52.6 −70.4 −16.9 −18.5 28.4 121.5 55.2 65.3 30.0 −32.9 −127.3 −5.2 −117.6 −57.8 −81.8 −130.8 +9.2 −85.0

100

200

400

760

Melting point, °C

−251.0 2364 −6.0 866 −209.7 −166.0 −14.7 7.4 −110.3 −46.3 −88.8 71.5 71.5 −198.8 −141.0 −35.6 197.3 349 103.6 21.0 117.0 −118.8 7.4 −52.0 29.3 108.3 510 47.4 126.3 63.8 3714 586 1137 1164 1245 1064 1080 −99.0 289.0 514 1114 1133 1168 1072 554 258.0 −73.9 118.0 147.5 2083 1969 +5.4 −113.3 −33.2 −4.4 79.4 −55.9 +1.6 −1.1 47.4 146.0 75.4 85.4 51.6 −16.4 −118.7 14.1 −107.3 −42.3 −68.5 −122.4 28.6 −70.3

−249.7 2473 +8.8 904 −205.6 −162.3 −5.0 15.6 −103.6 −34.0 −79.2 89.5 89.5 −194.0 −132.6 −22.3 222.7 370 125.2 37.6 131.3 −109.4 17.6 −44.0 39.9 129.0 532 65.0 141.8 82.0 3923 643 1212 1239 1323 1142 1152 −87.7 307.0 563 1186 1207 1239 1141 594 277.0 −64.8 134.6 161.0 2151 2053 21.0 −170.2 −19.3 +10.7 101.8 −43.5 17.8 +14.0 63.6 166.2 92.5 102.2 70.2 −1.8 −111.3 31.6 −98.3 −28.6 −57.0 −115.2 45.7 −58.0 −69.8

−248.1 2603 25.8 945 −200.9 −156.8 +8.0 24.4 −96.2 −20.3 −68.2 109.3 109.3 −188.8 −122.5 −7.6 251.0 391 149.7 56.9 147.2 −98.3 28.0 −35.4 51.6 150.3 556 84.3 157.8 102.3 4169 708 1297 1322 1411 1233 1238 −75.0 336.0 620 1267 1294 1322 1223 637 297.7 −55.2 151.7 176.4 2220 2141 38.4 −100.7 −4.0 27.9 125.5 −29.3 35.5 31.0 81.7 189.5 113.6 120.6 90.2 +14.5 −102.8 50.7 −87.6 −13.3 −44.5 −106.8 64.6 −45.0 −55.9

−246.0 2732 42.5 987 −195.8 −151.7 21.0 32.4 −85.5 −6.4 −56.0 130.0 130.0 −183.1 −111.1 +8.3 280.0 417 175.3 74.2 162.0 −87.5 38.3 −27.0 62.3 173.1 591 105.1 175.0 124.0 4407 774 1383 1407 1502 1327 1324 −61.8 362.4 679 1352 1381 1408 1304 680 317.0 −45.8 168.0 191.5 2287 2227 56.8 −94.8 +12.2 45.4 149.5 −15.4 53.1 48.7 100.0 211.4 135.6 139.0 111.8 31.8 −95.0 70.5 −77.8 +2.4 −30.4 −98.0 83.8 −31.8 −41.7

−248.7 1452 −25 1001 −210.0 −161 −9.3 30 −90.9 −64.5 −134 56 42 −218.7 −251 −104 44.1 590 −40 −111.8

Temperature, °C

Formula Ne Ni Ni(CO)4 NiCl2 N2 NO NO2 N2O5 N 2O NOCl NOF OsO4 OsO4 O2 O3 COCl2 P P PBr3 PCl3 PCl5 PH3 PH4Br PH4Cl PH4I P4O6 P4O10 POCl3 PSBr3 PSCl3 Pt K KBr KCl KF KOH KI Rn Re2O7 Rb RbBr RbCl RbF RbI Se SeO2 SeF6 SeOCl2 SeCl4 Si SiO2 SiCl4 SiF4 SiFCl3 SiH3I SiH2I2 (SiH3)2O Si3H8 (SiH3)3N Si4H10 Si3Cl3 (SiCl3)2O Si2Cl6 SiHBr3 SiHCl3 SiHF3 SiH2Br2 SiH2F2 SiH3Br SiH3Cl SiH3F SiFBr3 SiF2Cl2 SiF3Br

60

−112.5 −68.9 −68.7 −27.7 46.3 −5.0 +4.0 −30.5 −80.7 −152.0 −60.9 −146.7 −117.8 −153.0 −46.1 −124.7

−251.9 2289 −15.9 840 −212.3 −168.9 −19.9 +1.8 −114.9 −54.2 −95.7 59.4 59.4 −201.9 −146.7 −44.0 179.8 334 89.8 10.2 108.3 −125.0 +0.3 −57.3 21.9 94.2 493 35.8 116.0 51.8 3574 550 1087 1115 1193 1013 1030 −106.2 280.0 482 1066 1084 1123 1026 527 244.6 −80.0 106.5 137.8 2036 1911 −4.8 −117.5 −42.2 −14.3 64.0 −64.2 −9.0 −11.0 36.6 132.0 63.8 73.9 39.2 −25.8 −123.7 +3.2 −113.3 −51.1 −76.0 −127.2 17.4 −78.6

−132.5 −28.5 22.5 569 2 38 −36.2 1755 62.3 730 790 880 380 723 −71 296 38.5 682 715 760 642 217 340 −34.7 8.5 1420 1710 −68.8 −90 −120.8 −57.0 −1.0 −144.2 −117.2 −105.7 −93.6 −33.2 −1.2 −73.5 −126.6 −131.4 −70.2 −93.9 −82.5 −139.7 −70.5

2-60

PHYSICAL AND CHEMICAL DATA

TABLE 2-7

Vapor Pressures of Inorganic Compounds, up to 1 atm (Concluded ) Compound

Pressure, mm Hg

Name

Formula

Trifluorochlorosilane Hexafluorodisilane Dichlorofluorobromosilane Dibromochlorofluorosilane Silane Disilane Silver chloride iodide Sodium bromide chloride cyanide fluoride hydroxide iodide Strontium Strontium oxide Sulfur monochloride hexafluoride Sulfuryl chloride Sulfur dioxide trioxide (α) trioxide (β) trioxide (γ) Tellurium chloride fluoride Thallium Thallous bromide chloride iodide Thionyl bromide Thionyl chloride Tin Stannic bromide Stannous chloride Stannic chloride iodide hydride Tin tetramethyl trimethyl-ethyl trimethyl-propyl Titanium chloride Tungsten Tungsten hexafluoride Uranium hexafluoride Vanadyl trichloride Xenon Zinc chloride fluoride diethyl Zirconium bromide chloride iodide

SiF3Cl Si2F6 SiFCl2Br SiFClBr2 SiH4 Si2H6 Ag AgCl AgI Na NaBr NaCl NaCN NaF NaOH NaI Sr SrO S S2Cl2 SF6 SO2Cl2 SO2 SO3 SO3 SO3 Te TeCl4 TeF6 Tl TlBr TlCl TlI SOBr2 SOCl2 Sn SnBr4 SnCl2 SnCl4 SnI4 SnH4 Sn(CH3)4 Sn(CH3)3·C2H5 Sn(CH3)3·C3H7 TiCl4 W WF6 UF6 VOCl3 Xe Zn ZnCl2 ZnF2 Zn(C2H5)2 ZrBr4 ZrCl4 ZrI4

1

5

10

20

40

−144.0 −81.0 −86.5 −65.2 −179.3 −114.8 1357 912 820 439 806 865 817 1077 739 767

−133.0 −68.8 −68.4 −45.5 −168.6 −99.3 1500 1019 927 511 903 967 928 1186 843 857 847 2198 223.0 +15.7 −120.6 −35.1 −83.0 −23.7 −19.2 −2.0 605

−127.0 −63.1 −59.0 −35.6 −163.0 −91.4 1575 1074 983 549 952 1017 983 1240 897 903 898 2262 243.8 27.5 −114.7 −24.8 −76.8 −16.5 −12.3 +4.3 650 233 −92.4 983 522 517 531 31.0 −21.9 1703 72.7 391 +10.0 175.8 −118.5 −20.6 +3.8 21.8 21.3 4507 −49.2 −13.8 12.2 −152.8 593 508 1086 +11.7 250 230 311

−120.5 −57.0 −48.8 −24.5 −156.9 −82.7 1658 1134 1045 589 1005 1072 1046 1300 953 952 953 2333 264.7 40.0 −108.4 −13.4 −69.7 −9.1 −4.9 11.1 697 253 −86.0 1040 559 550 567 44.1 −10.5 1777 88.1 420 22.0 196.2 −111.2 −9.3 16.1 34.0 34.2 4690 −41.5 −5.2 26.6 −147.1 632 536 1129 24.2 266 243 329

−112.8 −50.6 −37.0 −12.0 −150.3 −72.8 1743 1200 1111 633 1063 1131 1115 1363 1017 1005 1018 2410 288.3 54.1 −101.5 −1.0 −60.5 −1.0 +3.2 17.9 753 273 −78.4 1103 598 589 607 58.8 +2.2 1855 105.5 450 35.2 218.8 −102.3 +3.5 30.0 48.5 48.4 4886 −33.0 +4.4 40.0 −141.2 673 566 1175 38.0 281 259 344

100

200

400

760

Melting point, °C

−108.2 −46.7 −29.0 −4.7 −146.3 −66.4 1795 1242 1152 662 1099 1169 1156 1403 1057 1039 1057

−101.7 −41.7 −19.5 +6.3 −140.5 −57.5 1865 1297 1210 701 1148 1220 1214 1455 1111 1083 1111

−91.7 −34.2 −3.2 23.0 −131.6 −44.6 1971 1379 1297 758 1220 1296 1302 1531 1192 1150 1192

−81.0 −26.4 +15.4 43.0 −122.0 −29.0 2090 1467 1400 823 1304 1379 1401 1617 1286 1225 1285

−70.0 −18.9 35.4 59.5 −111.5 −14.3 2212 1564 1506 892 1392 1465 1497 1704 1378 1304 1384

305.5 63.2 −96.8 +7.2 −54.6 +4.0 8.0 21.4 789 287 −73.8 1143 621 612 631 68.3 10.4 1903 116.2 467 43.5 234.2 −96.6 11.7 38.4 57.5 58.0 5007 −27.5 10.4 49.8 −137.7 700 584 1207 47.2 289 268 355

327.2 75.3 −90.9 17.8 −46.9 10.5 14.3 28.0 838 304 −67.9 1196 653 645 663 80.6 21.4 1968 131.0 493 54.7 254.2 −89.2 22.8 50.0 69.8 71.0 5168 −20.3 18.2 62.5 −132.8 736 610 1254 59.1 301 279 369

359.7 93.5 −82.3 33.7 −35.4 20.5 23.7 35.8 910 330 −57.3 1274 703 694 712 99.0 37.9 2063 152.8 533 72.0 283.5 −78.0 39.8 67.3 88.0 90.5 5403 −10.0 30.0 82.0 −125.4 788 648 1329 77.0 318 295 389

399.6 115.4 −72.6 51.3 −23.0 32.6 32.6 44.0 997 360 −48.2 1364 759 748 763 119.2 56.5 2169 177.7 577 92.1 315.5 −65.2 58.5 87.6 109.6 112.7 5666 +1.2 42.7 103.5 −117.1 844 689 1417 97.3 337 312 409

444.6 138.0 −63.5 69.2 −10.0 44.8 44.8 51.6 1087 392 −38.6 1457 819 807 823 139.5 75.4 2270 204.7 623 113.0 348.0 −52.3 78.0 108.8 131.7 136.0 5927 17.3 55.7 127.2 −108.0 907 732 1497 118.0 357 331 431

−142 −18.6 −112.3 −99.3 −185 −132.6 960.5 455 552 97.5 755 800 564 992 318 651 800 2430 112.8 −80 −50.2 −54.1 −73.2 16.8 32.3 62.1 452 224 −37.8 3035 460 430 440 −52.2 −104.5 231.9 31.0 246.8 −30.2 144.5 −149.9

60

Temperature, °C

2068 183.8 −7.4 −132.7 −95.5 −39.0 −34.0 −15.3 520 −111.3 825 440 −6.7 −52.9 1492 316 −22.7 −140.0 −51.3 −30.0 −12.0 −13.9 3990 −71.4 −38.8 −23.2 −168.5 487 428 970 −22.4 207 190 264

−98.8 931 490 487 502 +18.4 −32.4 1634 58.3 366 −1.0 156.0 −125.8 −31.0 −7.6 +10.7 +9.4 4337 −56.5 −22.0 +0.2 −158.2 558 481 1055 0.0 237 217 297

−30 3370 −0.5 69.2 −111.6 419.4 365 872 −28 450 437 499

VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-8

2-61

Vapor Pressures of Organic Compounds, up to 1 atm* Pressure, mm Hg Compound

1

Name

Formula

Acenaphthalene Acetal Acetaldehyde Acetamide Acetanilide Acetic acid anhydride Acetone Acetonitrile Acetophenone Acetyl chloride Acetylene Acridine Acrolein (2-propenal) Acrylic acid Adipic acid Allene (propadiene) Allyl alcohol (propen-1-ol-3) chloride (3-chloropropene) isopropyl ether isothiocyanate n-propyl ether 4-Allylveratrole iso-Amyl acetate n-Amyl alcohol iso-Amyl alcohol sec-Amyl alcohol (2-pentanol) tert-Amyl alcohol sec-Amylbenzene iso-Amyl benzoate bromide (1-bromo-3-methylbutane) n-butyrate formate iodide (1-iodo-3-methylbutane) isobutyrate Amyl isopropionate iso-Amyl isovalerate n-Amyl levulinate iso-Amyl levulinate nitrate 4-tert-Amylphenol Anethole Angelonitrile Aniline 2-Anilinoethanol Anisaldehyde o-Anisidine (2-methoxyaniline) Anthracene Anthraquinone Azelaic acid Azelaldehyde Azobenzene Benzal chloride (α,α-Dichlorotoluene) Benzaldehyde Benzanthrone Benzene Benzenesulfonylchloride Benzil Benzoic acid anhydride Benzoin Benzonitrile Benzophenone Benzotrichloride (α,α,α-Trichlorotoluene) Benzotrifluoride (α,α,α-Trifluorotoluene) Benzoyl bromide chloride nitrile Benzyl acetate alcohol

C12H10 C6H14O2 C2H4O C2H5NO C8H9NO C2H4O2 C4H6O3 C3H6O C2H3N C8H8O C2H3OCl C2H2 C13H9N C3H4O C3H4O2 C6H10O4 C3H4 C3H6O C3H5Cl C6H12O C4H5NS C6H12O C11H14O2 C7H14O2 C5H12O C5H12O C5H12O C5H12O C11H16 C12H16O2 C5H11Br C9H18O2 C6H12O2 C5H11I C9H18O2 C8H16O2 C10H20O2 C10H18O3 C10H18O3 C5H11NO3 C11H16O C10H12O C5H7N C6H7N C8H11NO C8H8O2 C7H9NO C14H10 C14H8O2 C9H16O4 C9H18O C12H10N2 C7H6Cl2 C7H6O C17H10O C6H6 C6H5ClO2S C14H10O2 C7H6O2 C14H10O3 C14H12O2 C7H5N C13H10O C7H5Cl3 C7H5F3 C7H5BrO C7H5ClO C8H5NO C9H10O2 C7H8O

5

10

20

114.8 −2.3 −65.1 92.0 146.6 +6.3 24.8 −40.5 −26.6 64.0 −35.0 −133.0 165.8 −46.0 27.3 191.0 −108.0 +0.2 −52.0 −23.1 +25.3 −18.2 113.9 +23.7 34.7 30.9 22.1 +7.2 55.8 104.5 +2.1 47.1 +5.4 +21.9 40.1 33.7 54.4 110.0 104.0 28.8 109.8 91.6 +15.0 57.9 134.3 102.6 88.0 173.5 219.4 210.4 58.4 135.7 64.0 50.1 274.5 −19.6 96.5 165.2 119.5 180.0 170.2 55.3 141.7 73.7 −10.3 75.4 59.1 71.7 73.4 80.8

131.2 +8.0 −56.8 105.0 162.0 17.5 36.0 −31.1 −16.3 78.0 −27.6 −128.2 184.0 −36.7 39.0 205.5 −101.0 10.5 −42.9 −12.9 38.3 −7.9 127.0 35.2 44.9 40.8 32.2 17.2 69.2 121.6 13.6 59.9 17.1 34.1 52.8 46.3 68.6 124.0 118.8 40.3 125.5 106.0 28.0 69.4 149.6 117.8 101.7 187.2 234.2 225.5 71.6 151.5 78.7 62.0 297.2 −11.5 112.0 183.0 132.1 198.0 188.1 69.2 157.6 87.6 −0.4 89.8 73.0 85.5 87.6 92.6

148.7 19.6 −47.8 120.0 180.0 29.9 48.3 −20.8 −5.0 92.4 −19.6 −122.8 203.5 −26.3 52.0 222.0 −93.4 21.7 −32.8 −1.8 52.1 +3.7 142.8 47.8 55.8 51.7 42.6 27.9 83.8 139.7 26.1 74.0 30.0 47.6 66.6 60.0 83.8 139.7 134.4 53.5 142.3 121.8 41.0 82.0 165.7 133.5 116.1 201.9 248.3 242.4 85.0 168.3 94.3 75.0 322.5 −2.6 129.0 202.8 146.7 218.0 207.0 83.4 175.8 102.7 12.2 105.4 87.6 100.2 102.3 105.8

40

60

100

200

400

760

197.5 50.1 −22.6 158.0 227.2 63.0 82.2 +7.7 27.0 133.6 +3.2 −107.9 256.0 +2.5 86.1 265.0 −72.5 50.0 −4.5 29.0 89.5 35.8 183.7 83.2 85.8 80.7 70.7 55.3 124.1 186.8 60.4 113.1 65.4 84.4 104.4 97.6 125.1 180.5 177.0 88.6 189.0 164.2 77.5 119.9 209.5 176.7 155.2 250.0 285.0 286.5 123.0 216.0 138.3 112.5 390.0 26.1 174.5 255.8 186.2 270.4 258.0 123.5 224.4 144.3 45.3 147.7 128.0 141.0 144.0 141.7

222.1 66.3 −10.0 178.3 250.5 80.0 100.0 22.7 43.7 154.2 16.1 −100.3 284.0 17.5 103.3 287.8 −61.3 64.5 10.4 44.3 108.0 52.6 204.0 101.3 102.0 95.8 85.7 69.7 145.2 210.2 78.7 133.2 83.2 103.8 124.2 117.3 146.1 203.1 198.1 106.7 213.0 186.1 96.3 140.1 230.6 199.0 175.3 279.0 314.6 309.6 142.1 240.0 160.7 131.7 426.5 42.2 198.0 283.5 205.8 299.1 284.4 144.1 249.8 165.6 62.5 169.2 149.5 161.3 165.5 160.0

250.0 84.0 +4.9 200.0 277.0 99.0 119.8 39.5 62.5 178.0 32.0 −92.0 314.3 34.5 122.0 312.5 −48.5 80.2 27.5 61.7 129.8 71.4 226.2 121.5 119.8 113.7 102.3 85.7 168.0 235.8 99.4 155.3 102.7 125.8 146.0 138.4 169.5 227.4 222.7 126.5 239.5 210.5 117.7 161.9 254.5 223.0 197.3 310.2 346.2 332.8 163.4 266.1 187.0 154.1

277.5 102.2 20.2 222.0 303.8 118.1 139.6 56.5 81.8 202.4 50.8 −84.0 346.0 52.5 141.0 337.5 −35.0 96.6 44.6 79.5 150.7 90.5 248.0 142.0 137.8 130.6 119.7 101.7 193.0 262.0 120.4 178.6 123.3 148.2 168.8 160.2 194.0 253.2 247.9 147.5 266.0 235.3 140.0 184.4 279.6 248.0 218.5 342.0 379.9 356.5 185.0 293.0 214.0 179.0

60.6 224.0 314.3 227.0 328.8 313.5 166.7 276.8 189.2 82.0 193.7 172.8 185.0 189.0 183.0

80.1 251.5 347.0 249.2 360.0 343.0 190.6 305.4 213.5 102.2 218.5 197.2 208.0 213.5 204.7

Temperature, °C −23.0 −81.5 65.0 114.0 −17.2 1.7 −59.4 −47.0 37.1 −50.0 −142.9 129.4 −64.5 +3.5 159.5 −120.6 −20.0 −70.0 −43.7 −2.0 −39.0 85.0 0.0 +13.6 +10.0 +1.5 −12.9 29.0 72.0 −20.4 21.2 −17.5 −2.5 14.8 +8.5 27.0 81.3 75.6 +5.2 62.6 −8.0 34.8 104.0 73.2 61.0 145.0 190.0 178.3 33.3 103.5 35.4 26.2 225.0 −36.7 65.9 128.4 96.0 143.8 135.6 28.2 108.2 45.8 −32.0 47.0 32.1 44.5 45.0 58.0

168.2 31.9 −37.8 135.8 199.6 43.0 62.1 −9.4 +7.7 109.4 −10.4 −116.7 224.2 −15.0 66.2 240.5 −85.2 33.4 −21.2 +10.9 67.4 16.4 158.3 62.1 68.0 63.4 54.1 38.8 100.0 158.3 39.8 90.0 44.0 62.3 81.8 75.5 100.6 155.8 151.7 67.6 160.3 139.3 55.8 96.7 183.7 150.5 132.0 217.5 264.3 260.0 100.2 187.9 112.1 90.1 350.0 +7.6 147.7 224.5 162.6 239.8 227.6 99.6 195.7 119.8 25.7 122.6 103.8 116.6 119.6 119.8

181.2 39.8 −31.4 145.8 211.8 51.7 70.8 −2.0 15.9 119.8 −4.5 −112.8 238.7 −7.5 75.0 251.0 −78.8 40.3 −14.1 18.7 76.2 25.0 169.6 71.0 75.5 71.0 61.5 46.0 110.4 171.4 48.7 99.8 53.3 71.9 91.7 85.2 110.3 165.2 162.6 76.3 172.6 149.8 65.2 106.0 194.0 161.7 142.1 231.8 273.3 271.8 110.0 199.8 123.4 99.6 368.8 15.4 158.2 238.2 172.8 252.7 241.7 109.8 208.2 130.0 34.0 133.4 114.7 127.0 129.8 129.3

Melting point, °C 95 −123.5 81 113.5 16.7 −73 −94.6 −41 20.5 −112.0 −81.5 110.5 −87.7 14 152 −136 −129 −136.4 −80

−117.2 −11.9

93 22.5 −6.2 2.5 5.2 217.5 286 106.5 68 −16.1 −26 174 +5.5 14.5 95 121.7 42 132 −12.9 48.5 −21.2 −29.3 0 −0.5 33.5 −51.5 −15.3

*Compiled from the extended tables published by D. R. Stull in Ind. Eng. Chem., 39, 517 (1947). For information on fuels see Hibbard, N.A.C.A. Research Mem. E56I21, 1956. For methane see Johnson (ed.), WADD-TR-60-56, 1960.

2-62

PHYSICAL AND CHEMICAL DATA

TABLE 2-8

Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name

Benzylamine Benzyl bromide (α-bromotoluene) chloride (α-chlorotoluene) cinnamate Benzyldichlorosilane Benzyl ethyl ether phenyl ether isothiocyanate Biphenyl 1-Biphenyloxy-2,3-epoxypropane d-Bornyl acetate Bornyl n-butyrate formate isobutyrate propionate Brassidic acid Bromoacetic acid 4-Bromoanisole Bromobenzene 4-Bromobiphenyl 1-Bromo-2-butanol 1-Bromo-2-butanone cis-1-Bromo-1-butene trans-1-Bromo-1-butene 2-Bromo-1-butene cis-2-Bromo-2-butene trans-2-Bromo-2-butene 1,4-Bromochlorobenzene 1-Bromo-1-chloroethane 1-Bromo-2-chloroethane 2-Bromo-4,6-dichlorophenol 1-Bromo-4-ethyl benzene (2-Bromoethyl)-benzene 2-Bromoethyl 2-chloroethyl ether (2-Bromoethyl)-cyclohexane 1-Bromoethylene Bromoform (tribromomethane) 1-Bromonaphthalene 2-Bromo-4-phenylphenol 3-Bromopyridine 2-Bromotoluene 3-Bromotoluene 4-Bromotoluene 3-Bromo-2,4,6-trichlorophenol 2-Bromo-1,4-xylene 1,2-Butadiene (methyl allene) 1,3-Butadiene n-Butane iso-Butane (2-methylpropane) 1,3-Butanediol 1,2,3-Butanetriol 1-Butene cis-2-Butene trans-2-Butene 3-Butenenitrile iso-Butyl acetate n-Butyl acrylate alcohol iso-Butyl alcohol sec-Butyl alcohol tert-Butyl alcohol iso-Butyl amine n-Butylbenzene iso-Butylbenzene sec-Butylbenzene tert-Butylbenzene iso-Butyl benzoate n-Butyl bromide (1-bromobutane) iso-Butyl n-butyrate carbamate Butyl carbitol (diethylene glycol butyl ether) n-Butyl chloride (1-chlorobutane) iso-Butyl chloride

1

5

10

20

29.0 32.2 22.0 173.8 45.3 26.0 95.4 79.5 70.6 135.3 46.9 74.0 47.0 70.0 64.6 209.6 54.7 48.8 +2.9 98.0 23.7 +6.2 −44.0 −38.4 −47.3 −39.0 −45.0 32.0 −36.0 −28.8 84.0 30.4 48.0 36.5 38.7 −95.4

70.0

54.8 59.6 47.8 206.3 70.2 52.0 127.7 107.8 101.8 169.9 75.7 103.4 74.8 99.8 93.7 241.7 81.6 77.8 27.8 133.7 45.4 30.0 −23.2 −17.0 −27.0 −17.9 −24.1 59.5 −18.0 −7.0 115.6 42.5 76.2 63.2 66.6 −77.8 22.0 117.5 135.4 42.0 49.7 50.8 47.5 146.2 65.0 −72.7 −87.6 −85.7 −94.1 67.5 132.0 −89.4 −81.1 −84.0 +2.9 +1.4 +23.5 +20.0 +11.6 +7.2 −3.0 −31.0 48.8 40.5 44.2 39.0 93.6 −11.2 30.0 83.7 95.7

67.7 73.4 60.8 221.5 83.2 65.0 144.0 121.8 117.0 187.2 90.2 118.0 89.3 114.0 108.0 256.0 94.1 91.9 40.0 150.6 55.8 41.8 −12.8 −6.4 −16.8 −7.2 −13.8 72.7 −9.4 +4.1 130.8 74.0 90.5 76.3 80.5 −68.8 34.0 133.6 152.3 55.2 62.3 64.0 61.1 163.2 78.8 −64.2 −79.7 −77.8 −86.4 85.3 146.0 −81.6 −73.4 −76.3 14.1 12.8 35.5 30.2 21.7 16.9 +5.5 −21.0 62.0 53.7 57.0 51.7 108.6 −0.3 42.2 96.4 107.8

81.8 88.3 75.0 239.3 96.7 79.6 160.7 137.0 134.2 205.8 106.0 133.8 104.0 130.0 123.7 272.9 108.2 107.8 53.8 169.8 67.2 54.2 −1.4 +5.4 −5.3 +4.6 −2.4 87.8 0.0 16.0 147.7 90.2 105.8 90.8 95.8 −58.8 48.0 150.2 171.8 69.1 76.0 78.1 75.2 181.8 94.0 −54.9 −71.0 −68.9 −77.9 100.0 161.0 −73.0 −64.6 −67.5 26.6 25.5 48.6 41.5 32.4 27.3 14.3 −10.3 76.3 67.8 70.6 65.6 124.2 +11.6 56.1 110.1 120.5

97.3 104.8 90.7 255.8 111.8 95.4 180.1 153.0 152.5 226.3 123.7 150.7 121.2 147.2 140.4 290.0 124.0 125.0 68.6 190.8 79.5 68.2 +11.5 18.4 +7.2 17.7 +10.5 103.8 +10.4 29.7 165.8 108.5 123.2 106.6 113.0 −48.1 63.6 170.2 193.8 84.1 91.0 93.9 91.8 200.5 110.6 −44.3 −61.3 −59.1 −68.4 117.4 178.0 −63.4 −54.7 −57.6 40.0 39.2 63.4 53.4 44.1 38.1 24.5 +1.3 92.4 83.3 86.2 80.8 141.8 24.8 71.7 125.3 135.5

C4H9Cl C4H9Cl

−49.0 −53.8

−28.9 −34.3

−18.6 −24.5

−7.4 −13.8

+5.0 −1.9

60

100

200

400

760

107.3 115.6 100.5 267.0 121.3 105.5 192.6 163.8 165.2 239.7 135.7 161.8 131.7 157.6 151.2 301.5 133.8 136.0 78.1 204.5 87.0 77.3 19.8 27.2 15.4 26.2 18.7 114.8 17.0 38.0 177.6 121.0 133.8 116.4 123.7 −41.2 73.4 183.5 207.0 94.1 100.0 104.1 102.3 213.0 121.6 −37.5 −55.1 −52.8 −62.4 127.5 188.0 −57.2 −48.4 −51.3 48.8 48.0 72.6 60.3 51.7 45.2 31.0 8.8 102.6 93.3 96.0 90.6 152.0 33.4 81.3 134.6 146.0

120.0 129.8 114.2 281.5 133.5 118.9 209.2 177.7 180.7 255.0 149.8 176.4 145.8 172.2 165.7 316.2 146.3 150.1 90.8 221.8 97.6 89.2 30.8 38.1 26.3 37.5 29.9 128.0 28.0 49.5 193.2 135.5 148.2 129.8 138.0 −31.9 85.9 198.8 224.5 107.8 112.0 117.8 116.4 229.3 135.7 −28.3 −46.8 −44.2 −54.1 141.2 202.5 −48.9 −39.8 −42.7 60.2 59.7 85.1 70.1 61.5 54.1 39.8 18.8 116.2 107.0 109.5 103.8 166.4 44.7 94.0 147.2 159.8

140.0 150.8 134.0 303.8 152.0 139.6 233.2 198.0 204.2 280.4 172.0 198.0 166.4 194.2 187.5 336.8 165.8 172.7 110.1 248.2 112.1 107.0 47.8 55.7 42.8 54.5 46.5 149.5 44.7 66.8 216.5 156.5 169.8 150.0 160.0 −17.2 106.1 224.2 251.0 127.7 133.6 138.0 137.4 253.0 156.4 −14.2 −33.9 −31.2 −41.5 161.0 222.0 −36.2 −26.8 −29.7 78.0 77.6 104.0 84.3 75.9 67.9 52.7 32.0 136.9 127.2 128.8 123.7 188.2 62.0 113.9 165.7 181.2

161.3 175.2 155.8 326.7 173.0 161.5 259.8 220.4 229.4 309.8 197.5 222.2 190.2 218.2 211.2 359.6 186.7 197.5 132.3 277.7 128.3 126.3 66.8 75.0 61.9 74.0 66.0 172.6 63.4 86.0 242.0 182.0 194.0 172.3 186.2 −1.1 127.9 252.0 280.2 150.0 157.3 160.0 160.2 278.0 181.0 +1.8 −19.3 −16.3 −27.1 183.8 243.5 −21.7 −12.0 −14.8 98.0 97.5 125.2 100.8 91.4 83.9 68.0 50.7 159.2 149.6 150.3 145.8 212.8 81.7 135.7 186.0 205.0

184.5 198.5 179.4 350.0 194.3 185.0 287.0 243.0 254.9 340.0 223.0 247.0 214.0 243.0 235.0 382.5 208.0 223.0 156.2 310.0 145.0 147.0 86.2 94.7 81.0 93.9 85.5 196.9 82.7 106.7 268.0 206.0 219.0 195.8 213.0 +15.8 150.5 281.1 311.0 173.4 181.8 183.7 184.5 305.8 206.7 18.5 −4.5 −0.5 −11.7 206.5 264.0 −6.3 +3.7 +0.9 119.0 118.0 147.4 117.5 108.0 99.5 82.9 68.6 183.1 172.8 173.5 168.5 237.0 101.6 156.9 206.5 231.2

13.0 +5.9

24.0 16.0

40.0 32.0

58.8 50.0

77.8 68.9

Temperature, °C

Formula C7H9N C7H7Br C7H7Cl C16H14O2 C7H8Cl2Si C9H12O C13H12O C8H7NS C12H10 C15H14O2 C12H20O2 C14H24O2 C11H18O2 C14H24O2 C13H22O2 C22H42O2 C2H3BrO2 C7H7BrO C6H5Br C12H9Br C4H9BrO C4H7BrO C4H7Br C4H7Br C4H7Br C4H7Br C4H7Br C6H4BrCl C2H4BrCl C2H4BrCl C6H3BrCl2O C8H9Br C8H9Br C4H8BrClO C8H15Br C2H3Br CHBr3 C10H7Br C12H9BrO C5H4BrN C7H7Br C7H7Br C7H7Br C6H2BrCl3O C8H9Br C4H6 C4H6 C4H10 C4H10 C4H10O2 C4H10O3 C4H8 C4H8 C4H8 C4H5N C6H12O2 C7H12O2 C4H10O C4H10O C4H10O C4H10O C4H11N C10H14 C10H14 C10H14 C10H14 C11H14O2 C4H9Br C8H16O2 C5H11NO2 C8H18O3

40

84.2 100.0 16.8 24.4 14.8 10.3 112.4 37.5 −89.0 −102.8 −101.5 −109.2 22.2 102.0 −104.8 −96.4 −99.4 −19.6 −21.2 −0.5 −1.2 −9.0 −12.2 −20.4 −50.0 22.7 14.1 18.6 13.0 64.0 −33.0 +4.6

Melting point, °C −4 −39 39

69.5 29

61.5 49.5 12.5 −30.7 90.5

−100.3 −133.4 −111.2 −114.6 16.6 −16.6 68 −45.0

−138 8.5 5.5 95 −28 39.8 28.5 +9.5 −108.9 −135 −145 77 −130 −138.9 −105.4 −98.9 −64.6 −79.9 −108 −114.7 25.3 −85.0 −88.0 −51.5 −75.5 −58 −112.4 65 −123.1 −131.2

VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-8

2-63

Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name

Formula

sec-Butyl chloride (2-Chlorobutane) tert-Butyl chloride sec-Butyl chloroacetate 2-tert-Butyl-4-cresol 4-tert-Butyl-2-cresol iso-Butyl dichloroacetate 2,3-Butylene glycol (2,3-butanediol) 2-Butyl-2-ethylbutane-1,3-diol 2-tert-Butyl-4-ethylphenol n-Butyl formate iso-Butyl formate sec-Butyl formate sec-Butyl glycolate iso-Butyl iodide (1-iodo-2-methylpropane) isobutyrate isovalerate levulinate naphthylketone (1-isovaleronaphthone) 2-sec-Butylphenol 2-tert-Butylphenol 4-iso-Butylphenol 4-sec-Butylphenol 4-tert-Butylphenol 2-(4-tert-Butylphenoxy)ethyl acetate 4-tert-Butylphenyl dichlorophosphate

C4H9Cl C4H9Cl C6H11ClO2 C11H16O C11H16O C6H10Cl2O2 C4H10O2 C10H22O2 C12H15O C5H10O2 C5H10O2 C5H10O2 C6H12O3 C4H9I C8H16O2 C9H18O2 C9H16O3 C15H16O C10H14O C10H14O C10H14O C10H14O C10H14O C14H20O3 C10H13Cl2 O2P C11H14O C7H14O2 C12H18O C12H18O C12H18O C12H18O C4H8O2 C4H8O2 C4H7N C11H14O C10H16 C10H16O2 C10H16O C10H19N C10H20O C10H20O2 C6H12O2 C6H12O2 C6H10O2 C6H11N C8H18O C8H16O C8H16O2 C8H15N C12H9N CO2 CS2 CO COSe COS CBr4 CCl4 CF4 C10H14O C10H14O C10H12O2 C2HCl3O C2H3Cl3O2

tert-Butyl phenyl ketone (pivalophenone) iso-Butyl propionate 4-tert-Butyl-2,5-xylenol 4-tert-Butyl-2,6-xylenol 6-tert-Butyl-2,4-xylenol 6-tert-Butyl-3,4-xylenol Butyric acid iso-Butyric acid Butyronitrile iso-Valerophenone Camphene Campholenic acid d-Camphor Camphylamine Capraldehyde Capric acid n-Caproic acid iso-Caproic acid iso-Caprolactone Capronitrile Capryl alcohol (2-octanol) Caprylaldehyde Caprylic acid (octanoic acid) Caprylonitrile Carbazole Carbon dioxide disulfide monoxide oxyselenide (carbonyl selenide) oxysulfide (carbonyl sulfide) tetrabromide tetrachloride tetrafluoride Carvacrol Carvone Chavibetol Chloral (trichloroacetaldehyde) hydrate (trichloroacetaldehyde hydrate) Chloranil Chloroacetic acid anhydride 2-Chloroaniline 3-Chloroaniline 4-Chloroaniline Chlorobenzene 2-Chlorobenzotrichloride (2-α,α,α-tetrachlorotoluene)

C6Cl4O2 C2H3ClO2 C4H4Cl2O3 C6H6ClN C6H6ClN C6H6ClN C6H5Cl C7H4Cl4

1

5

10

20

40

−60.2

−39.8

−29.2

−17.7

17.0 70.0 74.3 28.6 44.0 94.1 76.3 −26.4 −32.7 −34.4 28.3 −17.0 +4.1 16.0 65.0 136.0 57.4 56.6 72.1 71.4 70.0 118.0 96.0

41.8 98.0 103.7 54.3 68.4 122.6 106.2 −4.7 −11.4 −13.3 53.6 +5.8 28.0 41.2 92.1 167.9 86.0 84.2 100.9 100.5 99.2 150.0 129.6

54.6 112.0 118.0 67.5 80.3 136.8 121.0 +6.1 −0.8 −3.1 66.0 17.0 39.9 53.8 105.9 184.0 100.8 98.1 115.5 114.8 114.0 165.8 146.0

68.2 127.2 134.0 81.4 93.4 151.2 137.0 18.0 +11.0 +8.4 79.8 29.8 52.4 67.7 120.2 201.6 116.1 113.0 130.3 130.3 129.5 183.3 164.0

−5.0 −19.0 83.6 143.9 150.8 96.7 107.8 167.8 154.0 31.6 24.1 21.3 94.2 42.8 67.2 82.7 136.2 219.7 133.4 129.2 147.2 147.8 146.0 201.5 184.3

57.8 −2.3 88.2 74.0 70.3 83.9 25.5 14.7 −20.0 58.3

85.7 +20.9 119.8 103.9 100.2 113.6 49.8 39.3 +2.1 87.0

97.6 41.5 45.3 51.9 125.0 71.4 66.2 38.3 9.2 32.8 73.4 92.3 43.0

125.7 68.6 74.0 78.8 142.0 89.5 83.0 66.4 34.6 57.6 92.0 114.1 67.6

99.0 32.3 135.0 119.0 115.0 127.0 61.5 51.2 13.4 101.4 47.2 139.8 82.3 83.7 92.0 152.2 99.5 94.0 80.3 47.5 70.0 101.2 124.0 80.4

114.3 44.8 151.0 135.0 131.0 143.0 74.0 64.0 25.7 116.8 60.4 153.9 97.5 97.6 106.3 165.0 111.8 107.0 95.7 61.7 83.3 110.2 136.4 94.6

130.4 58.5 169.8 152.2 148.5 159.7 88.0 77.8 38.4 133.8 75.7 170.0 114.0 112.5 122.2 179.9 125.0 120.4 112.3 76.9 98.0 120.0 150.6 110.6

60

100

200

400

760

+3.4 −11.4 93.0 153.7 161.7 106.6 116.3 178.0 165.4 39.8 32.4 29.6 104.0 51.8 75.9 92.4 147.0 231.5 143.9 140.0 157.0 157.9 156.0 212.8 197.2

14.2 −1.0 105.5 167.0 176.2 119.8 127.8 191.9 179.0 51.0 43.4 40.2 116.4 63.5 88.0 105.2 160.2 246.7 157.3 153.5 171.2 172.4 170.2 228.0 214.3

31.5 +14.6 124.1 187.8 197.8 139.2 145.6 212.0 200.3 67.9 60.0 56.8 135.5 81.0 106.3 124.8 181.8 269.7 179.7 173.8 192.1 194.3 191.5 250.3 240.0

50.0 32.6 146.0 210.0 221.8 160.0 164.0 233.5 223.8 86.2 79.0 75.2 155.6 100.3 126.3 146.4 205.5 294.0 203.8 196.3 214.7 217.6 214.0 277.6 268.2

68.0 51.0 167.8 232.6 247.0 183.0 182.0 255.0 247.8 106.0 98.2 93.6 177.5 120.4 147.5 168.7 229.9 320.0 228.0 219.5 237.0 242.1 238.0 304.4 299.0

140.8 67.6 180.3 163.6 158.2 170.0 96.5 86.3 47.3 144.6 85.0 180.0 124.0 122.0 132.0 189.8 133.3 129.6 123.2 86.8 107.4 126.0 160.0 121.2 248.2 −104.8 −15.3 −208.1 −70.2 −93.0 106.3 12.3 −155.4 155.3 143.8 170.7 29.1 46.2

154.0 79.5 195.0 176.0 172.0 184.0 108.0 98.0 59.0 158.0 97.9 193.7 138.0 134.6 145.3 200.0 144.0 141.4 137.2 99.8 119.8 133.9 172.2 134.8 265.0 −100.2 −5.1 −205.7 −61.7 −85.9 119.7 23.0 −150.7 169.7 157.3 185.5 40.2 55.0

Temperature, °C

175.0 197.7 220.0 97.0 116.4 136.8 217.5 241.3 265.3 196.0 217.8 239.8 192.3 214.2 236.5 204.5 226.7 249.5 125.5 144.5 163.5 115.8 134.5 154.5 76.7 96.8 117.5 180.1 204.2 228.0 117.5 138.7 160.5 212.7 234.0 256.0 157.9 182.0 209.2 153.0 173.8 195.0 164.8 186.3 208.5 217.1 240.3 268.4 160.8 181.0 202.0 158.3 181.0 207.7 157.8 182.1 207.0 119.7 141.0 163.7 138.0 157.5 178.5 145.4 156.5 168.5 190.3 213.9 237.5 155.2 179.5 204.5 292.5 323.0 354.8 −93.0 −85.7 −78.2 +10.4 28.0 46.5 −201.3 −196.3 −191.3 −49.8 −35.6 −21.9 −75.0 −62.7 −49.9 139.7 163.5 189.5 38.3 57.8 76.7 −143.6 −135.5 −127.7 191.2 213.8 237.0 179.6 203.5 227.5 206.8 229.8 254.0 57.8 77.5 97.7 68.0 82.1 96.2

Melting point, °C −131.3 −26.5

22.5

−95.3 −90.7 −80.7

99

−71

−74 −47 50 178.5 31.5 −1.5 −35 −38.6 16 244.8 −57.5 −110.8 −205.0

−134.3 −73.8 −222.0 −117.1 −132.4

−124.4 −54.3 −217.2 −102.3 −119.8

−119.5 −44.7 −215.0 −95.0 −113.3

−114.4 −34.3 −212.8 −86.3 −106.0

−50.0 −184.6 70.0 57.4 83.6 −37.8 −9.8

−30.0 −174.1 98.4 86.1 113.3 −16.0 +10.0

−19.6 −169.3 113.2 100.4 127.0 −5.0 19.5

−8.2 −164.3 127.9 116.1 143.2 +7.2 29.2

−108.6 −22.5 −210.0 −76.4 −98.3 96.3 +4.3 −158.8 145.2 133.0 159.8 20.2 39.7

70.7 43.0 67.2 46.3 63.5 59.3 −13.0

89.3 68.3 94.1 72.3 89.8 87.9 +10.6

97.8 81.0 108.0 84.8 102.0 102.1 22.2

106.4 94.2 122.4 99.2 116.7 117.8 35.3

116.1 109.2 138.2 115.6 133.6 135.0 49.7

122.0 118.3 148.0 125.7 144.1 145.8 58.3

129.5 130.7 159.8 139.5 158.0 159.9 70.7

140.3 149.0 177.8 160.0 179.5 182.3 89.4

151.3 169.0 197.0 183.7 203.5 206.6 110.0

162.6 189.5 217.0 208.8 228.5 230.5 132.2

290 61.2 46 0 −10.4 70.5 −45.2

69.0

101.8

117.9

135.8

155.0

167.8

185.0

208.0

233.0

262.1

28.7

−138.8 90.1 −22.6 −183.7 +0.5 −57 51.7

2-64

PHYSICAL AND CHEMICAL DATA

TABLE 2-8

Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name

2-Chlorobenzotrifluoride (2-chloro-α,α,α-trifluorotoluene) 2-Chlorobiphenyl 4-Chlorobiphenyl α-Chlorocrotonic acid Chlorodifluoromethane Chlorodimethylphenylsilane 1-Chloro-2-ethoxybenzene 2-(2-Chloroethoxy) ethanol bis-2-Chloroethyl acetacetal 1-Chloro-2-ethylbenzene 1-Chloro-3-ethylbenzene 1-Chloro-4-ethylbenzene 2-Chloroethyl chloroacetate 2-Chloroethyl 2-chloroisopropyl ether 2-Chloroethyl 2-chloropropyl ether 2-Chloroethyl α-methylbenzyl ether Chloroform (trichloromethane) 1-Chloronaphthalene 4-Chlorophenethyl alcohol 2-Chlorophenol 3-Chlorophenol 4-Chlorophenol 2-Chloro-3-phenylphenol 2-Chloro-6-phenylphenol Chloropicrin (trichloronitromethane) 1-Chloropropene 2-Chloropyridine 3-Chlorostyrene 4-Chlorostyrene 1-Chlorotetradecane 2-Chlorotoluene 3-Chlorotoluene 4-Chlorotoluene Chlorotriethylsilane 1-Chloro-1,2,2-trifluoroethylene Chlorotrifluoromethane Chlorotrimethylsilane trans-Cinnamic acid Cinnamyl alcohol Cinnamylaldehyde Citraconic anhydride cis-α-Citral d-Citronellal Citronellic acid Citronellol Citronellyl acetate Coumarin o-Cresol (2-cresol; 2-methylphenol) m-Cresol (3-cresol; 3-methylphenol) p-Cresol (4-cresol; 4-methylphenol) cis-Crotonic acid trans-Crotonic acid cis-Crotononitrile trans-Crotononitrile Cumene 4-Cumidene Cuminal Cuminyl alcohol 2-Cyano-2-n-butyl acetate Cyanogen bromide chloride iodide Cyclobutane Cyclobutene Cyclohexane Cyclohexaneethanol Cyclohexanol Cyclohexanone 2-Cyclohexyl-4,6-dinitrophenol Cyclopentane Cyclopropane Cymene

1

5

10

20

60

100

200

400

760

88.3 197.0 212.5 155.9 −76.4 124.7 141.8 139.5 150.7 110.0 113.6 116.0 140.0 115.8 125.6 164.8 10.4 180.4 188.1 106.0 143.0 150.0 237.0 237.1 53.8 −15.1 104.6 121.2 122.0 215.5 94.7 96.3 96.6 82.3 −66.7 −111.7 +6.0 232.4 177.8 177.7 145.4 160.0 140.1 195.4 159.8 161.0 216.5 127.4 138.0 140.0 116.3 128.0 50.1 62.8 88.1 158.0 160.0 176.2 133.8 −51.8 22.6 −24.9 97.6 −32.8 −41.2 25.5 142.7 103.7 90.4 229.0 −1.3 −70.0 110.8

108.3 219.6 237.8 173.8 −65.8 145.5 162.0 157.2 169.8 130.2 133.8 137.0 159.8 135.7 146.3 186.3 25.9 204.2 210.0 126.4 164.8 172.0 261.3 261.6 71.8 +1.3 125.0 142.2 143.5 240.3 115.0 116.6 117.1 101.6 −55.0 −102.5 21.9 253.3 199.8 199.3 165.8 181.8 160.0 214.5 179.8 178.8 240.0 146.7 157.3 157.3 133.9 146.0 68.0 81.1 107.3 180.0 182.8 197.9 152.2 −42.6 33.8 −14.1 111.5 −18.9 −27.8 42.0 161.7 121.7 110.3 248.7 +13.8 −59.1 131.4

130.0 243.8 264.5 193.2 −53.6 168.6 185.5 176.5 190.5 152.2 156.7 159.8 182.2 156.5 169.8 210.8 42.7 230.8 234.5 149.8 188.7 196.0 289.4 289.5 91.8 18.0 147.7 165.7 166.0 267.5 137.1 139.7 139.8 123.6 −41.7 −92.7 39.4 276.7 224.6 222.4 189.8 205.0 183.8 236.6 201.0 197.8 264.7 168.4 179.0 179.4 152.2 165.5 88.0 101.5 129.2 203.2 206.7 221.7 173.4 −33.0 46.0 −2.3 126.1 −3.4 −12.2 60.8 183.5 141.4 132.5 269.8 31.0 −46.9 153.5

152.2 267.5 292.9 212.0 −40.8 193.5 208.0 196.0 212.6 177.6 181.1 184.3 205.0 180.0 194.1 235.0 61.3 259.3 259.3 174.5 214.0 220.0 317.5 317.0 111.9 37.0 170.2 190.0 191.0 296.0 159.3 162.3 162.3 146.3 −27.9 −81.2 57.9 300.0 250.0 246.0 213.5 228.0 206.5 257.0 221.5 217.0 291.0 190.8 202.8 201.8 171.9 185.0 108.0 122.8 152.4 227.0 232.0 246.6 195.2 −21.0 61.5 +13.1 141.1 +12.9 +2.4 80.7 205.4 161.0 155.6 291.5 49.3 −33.5 177.2

Temperature, °C

Formula C7H4ClF3 C12H9Cl C12H9Cl C4H5ClO2 CHClF2 C8H11ClSi C8H9ClO C4H9ClO2 C6H12Cl2O2 C8H9Cl C8H9Cl C8H9Cl C4H6Cl2O2 C5H10Cl2O C5H10Cl2O C10H13ClO CHCl3 C10H7Cl C8H9ClO C6H5ClO C6H5ClO C6H5ClO C12H9ClO C12H9ClO CCl3NO2 C3H5Cl C5H4ClN C8H7Cl C8H7Cl C14H29Cl C7H7Cl C7H7Cl C7H7Cl C6H15ClSi C2ClF3 CClF3 C3H9ClSi C9H8O2 C9H10O C9H8O C5H4O3 C10H16O C10H18O C10H18O2 C10H20O C12H22O2 C9H6O2 C7H8O C7H8O C7H8O C4H6O2 C4H6O2 C4H5N C4H5N C9H12 C9H13N C10H12O C10H14O C7H11NO2 C2N2 CBrN CClN CIN C4H8 C4H6 C6H12 C8H16O C6H12O C6H10O C12H14N2O5 C5H10 C3H6 C10H14

40

0.0 89.3 96.4 70.0 −122.8 29.8 45.8 53.0 56.2 17.2 18.6 19.2 46.0 24.7 29.8 62.3 −58.0 80.6 84.0 12.1 44.2 49.8 118.0 119.8 −25.5 −81.3 13.3 25.3 28.0 98.5 +5.4 +4.8 +5.5 −4.9 −116.0 −149.5 −62.8 127.5 72.6 76.1 47.1 61.7 44.0 99.5 66.4 74.7 106.0 38.2 52.0 53.0 33.5

24.7 109.8 129.8 95.6 −110.2 56.7 72.8 78.3 83.7 43.0 45.2 46.4 72.1 50.1 56.5 91.4 −39.1 104.8 114.3 38.2 72.0 78.2 152.2 153.7 −3.3 −63.4 38.8 51.3 54.5 131.8 30.6 30.3 31.0 +19.8 −102.5 −139.2 −43.6 157.8 102.5 105.8 74.8 90.0 71.4 127.3 93.6 100.2 137.8 64.0 76.0 76.5 57.4

−29.0 −19.5 +2.9 60.0 58.0 74.2 42.0 −95.8 −35.7 −76.7 25.2 −92.0 −99.1 −45.3 50.4 21.0 +1.4 132.8 −68.0 −116.8 17.3

−7.1 +3.5 26.8 88.2 87.3 103.7 68.7 −83.2 −18.3 −61.4 47.2 −76.0 −83.4 −25.4 77.2 44.0 26.4 161.8 −49.6 −104.2 43.9

37.1 134.7 146.0 108.0 −103.7 70.0 86.5 90.7 97.6 56.1 58.1 60.0 86.0 63.0 70.0 106.0 −29.7 118.6 129.0 51.2 86.1 92.2 169.7 170.7 +7.8 −54.1 51.7 65.2 67.5 148.2 43.2 43.2 43.8 32.0 −95.9 −134.1 −34.0 173.0 117.8 120.0 88.9 103.9 84.8 141.4 107.0 113.0 153.4 76.7 87.8 88.6 69.0 80.0 +4.0 15.0 38.3 102.2 102.0 118.0 82.0 −76.8 −10.0 −53.8 57.7 −67.9 −75.4 −15.9 90.0 56.0 38.7 175.9 −40.4 −97.5 57.0

50.6 151.2 164.0 121.2 −96.5 84.7 101.5 104.1 112.2 70.3 73.0 75.5 100.0 77.2 84.8 121.8 −19.0 134.4 145.0 65.9 101.7 108.1 186.7 189.8 20.0 −44.0 65.8 80.0 82.0 166.2 56.9 57.4 57.8 45.5 −88.2 −128.5 −23.2 189.5 133.7 135.7 103.8 119.4 99.8 155.6 121.5 126.0 170.0 90.5 101.4 102.3 82.0 93.0 16.4 27.8 51.5 117.8 117.9 133.8 96.2 −70.1 −1.0 −46.1 68.6 −58.7 −66.6 −5.0 104.0 68.8 52.5 191.2 −30.1 −90.3 71.1

65.9 169.9 183.8 135.6 −88.6 101.2 117.8 118.4 127.8 86.2 89.2 91.8 116.0 92.4 101.5 139.6 −7.1 153.2 162.0 82.0 118.0 125.0 207.4 208.2 33.8 −32.7 81.7 96.5 98.0 187.0 72.0 73.0 73.5 60.2 −79.7 −121.9 −11.4 207.1 151.0 152.2 120.3 135.9 116.1 171.9 137.2 140.5 189.0 105.8 116.0 117.7 96.0 107.8 30.0 41.8 66.1 134.2 135.2 150.3 111.8 −62.7 +8.6 −37.5 80.3 −48.4 −56.4 +6.7 119.8 83.0 67.8 206.7 −18.6 −82.3 87.0

75.4 182.1 196.0 144.4 −83.4 111.5 127.8 127.5 138.0 96.4 99.6 102.0 126.2 102.2 111.8 150.0 +0.5 165.6 173.5 92.0 129.4 136.1 219.6 220.0 42.3 −25.1 91.6 107.2 108.5 199.8 81.8 83.2 83.3 69.5 −74.1 −117.3 −4.0 217.8 162.0 163.7 131.3 146.3 126.2 182.1 147.2 149.7 200.5 115.5 125.8 127.0 104.5 116.7 38.5 50.9 75.4 145.0 146.0 161.7 121.5 −57.9 14.7 −32.1 88.0 −41.8 −50.0 14.7 129.8 91.8 77.5 216.0 −11.3 −77.0 97.2

Melting point, °C −6.0 34 75.5 −160

−80.2 −53.3 −62.6

−63.5 −20 7 32.5 42 +6 −64 −99.0 −15.0 +0.9 +7.3 −157.5 133 33 −7.5

70 30.8 10.9 35.5 15.5 72 −96.0

−34.4 58 −6.5 −50 +6.6 23.9 −45.0 −93.7 −126.6 −68.2

VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-8

2-65

Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name

cis-Decalin trans-Decalin Decane Decan-2-one 1-Decene Decyl alcohol Decyltrimethylsilane Dehydroacetic acid Desoxybenzoin Diacetamide Diacetylene (1,3-butadiyne) Diallyldichlorosilane Diallyl sulfide Diisoamyl ether oxalate sulfide Dibenzylamine Dibenzyl ketone (1,3-diphenyl2-propanone) 1,4-Dibromobenzene 1,2-Dibromobutane dl-2,3-Dibromobutane meso-2,3-Dibromobutane 1,2-Dibromodecane Di(2-bromoethyl) ether α,β-Dibromomaleic anhydride 1,2-Dibromo-2-methylpropane 1,3-Dibromo-2-methylpropane 1,2-Dibromopentane 1,2-Dibromopropane 1,3-Dibromopropane 2,3-Dibromopropene 2,3-Dibromo-1-propanol Diisobutylamine 2,6-Ditert-butyl-4-cresol 4,6-Ditert-butyl-2-cresol 4,6-Ditert-butyl-3-cresol 2,6-Ditert-butyl-4-ethylphenol 4,6-Ditert-butyl-3-ethylphenol Diisobutyl oxalate 2,4-Ditert-butylphenol Dibutyl phthalate sulfide Diisobutyl d-tartrate Dicarvacryl-mono-(6-chloro-2-xenyl) phosphate Dicarvacryl-2-tolyl phosphate Dichloroacetic acid 1,2-Dichlorobenzene 1,3-Dichlorobenzene 1,4-Dichlorobenzene 1,2-Dichlorobutane 2,3-Dichlorobutane 1,2-Dichloro-1,2-difluoroethylene Dichlorodifluoromethane Dichlorodiphenyl silane Dichlorodiisopropyl ether Di(2-chloroethoxy) methane Dichloroethoxymethylsilane 1,2-Dichloro-3-ethylbenzene 1,2-Dichloro-4-ethylbenzene 1,4-Dichloro-2-ethylbenzene cis-1,2-Dichloroethylene trans-1,2-Dichloro ethylene Di(2-chloroethyl) ether Dichlorofluoromethane 1,5-Dichlorohexamethyltrisiloxane Dichloromethylphenylsilane 1,1-Dichloro-2-methylpropane 1,2-Dichloro-2-methylpropane 1,3-Dichloro-2-methylpropane 2,4-Dichlorophenol 2,6-Dichlorophenol

1

5

10

20

22.5 −0.8 16.5 44.2 14.7 69.5 67.4 91.7 123.3 70.0 −82.5 +9.5 −9.5 18.6 85.4 43.0 118.3 125.5

50.1 +30.6 42.3 71.9 40.3 97.3 96.4 122.0 156.2 95.0 −68.0 34.8 +14.4 44.3 116.0 73.0 149.8 159.8

64.2 47.2 55.7 85.8 53.7 111.3 111.0 137.3 173.5 108.0 −61.2 47.4 26.6 57.0 131.4 87.6 165.6 177.6

79.8 65.3 69.8 100.7 67.8 125.8 126.5 153.0 192.0 122.6 −53.8 61.3 39.7 70.7 147.7 102.7 182.2 195.7

97.2 85.7 85.5 117.1 83.3 142.1 144.0 171.0 212.0 138.2 −45.9 76.4 54.2 86.3 165.7 120.0 200.2 216.6

C6H4Br2 61.0 C4H8Br2 7.5 C4H8Br2 +5.0 C4H8Br2 +1.5 C10H20Br2 95.7 C4H8Br2O 47.7 C4H2Br2O3 50.0 C4H8Br2 −28.8 C4H8Br2 14.0 C5H10Br2 19.8 C3H6Br2 −7.0 C3H6Br2 +9.7 C3H4Br2 −6.0 C3H6Br2O 57.0 C8H19N −5.1 C15H24O 85.8 C15H24O 86.2 C15H24O 103.7 C16H26O 89.1 C16H26O 111.5 C10H18O4 63.2 C14H22O 84.5 C16H22O4 148.2 C8H18S +21.7 C12H22O6 117.8 C32H34ClO4P 204.2

79.3 33.2 30.0 26.6 123.6 75.3 78.0 −3.0 40.0 45.4 +17.3 35.4 +17.9 84.5 +18.4 116.2 117.3 135.2 121.4 142.6 91.2 115.4 182.1 51.8 151.8 234.5

87.7 46.1 41.6 39.3 137.3 88.5 92.0 +10.5 53.0 58.0 29.4 48.0 30.0 98.2 30.6 131.0 132.4 150.0 137.0 157.4 105.3 130.0 198.2 66.4 169.0 249.3

103.6 60.0 56.4 53.2 151.0 103.6 106.7 25.7 67.5 72.0 42.3 62.1 43.2 113.5 43.7 147.0 149.0 167.0 154.0 174.0 120.3 146.0 216.2 80.5 188.0 264.5

C27H33O4P 180.2 C2H2Cl2O2 44.0 C6H4Cl2 20.0 C6H4Cl2 12.1 C6H4Cl2 C4H8Cl2 −23.6 C4H8Cl2 −25.2 C2Cl2F2 −82.0 CCl2F2 −118.5 C12H10Cl2Si 109.6 C6H12Cl2O 29.6 C5H10Cl2O2 53.0 C8H8Cl2OSi −33.8 C8H8Cl2 46.0 C8H8Cl2 47.0 C8H8Cl2 38.5 C2H2Cl2 −58.4 C2H2Cl2 −65.4 C4H8Cl2O 23.5 CHCl2F −91.3 C6H18Cl2 26.0 O2Si3 C7H8Cl2Si 35.7 C4H8Cl2 −31.0 C4H8Cl2 −25.8 C4H8Cl2 −3.0 C6H4Cl2O 53.0 C6H4Cl2O 59.5

209.3 69.8 46.0 39.0 −0.3 −3.0 −65.6 −104.6 142.4 55.2 80.4 −12.1 75.0 77.2 68.0 −39.2 −47.2 49.3 −75.5 52.0

221.8 82.6 59.1 52.0 54.8 +11.5 +8.5 −57.3 −97.8 158.0 68.2 94.0 −1.3 90.0 92.3 83.2 −29.9 −38.0 62.0 −67.5 65.1

63.5 −8.4 −4.2 +20.6 80.0 87.6

77.4 +2.6 +6.7 32.0 92.8 101.0

60

Melting point, °C

100

200

400

760

108.0 98.4 95.5 127.8 93.5 152.0 154.3 181.5 224.5 148.0 −41.0 86.3 63.7 96.0 177.0 130.6 212.2 229.4

123.2 114.6 108.6 142.0 106.5 165.8 169.5 197.5 241.3 160.6 −34.0 99.7 75.8 109.6 192.2 145.3 227.3 246.6

145.4 136.2 128.4 163.2 126.7 186.2 191.0 219.5 265.2 180.8 −20.9 119.4 94.8 129.0 215.0 166.4 249.8 272.3

169.9 160.1 150.6 186.7 149.2 208.8 215.5 244.5 293.0 202.0 −6.1 142.0 116.1 150.3 240.0 191.0 274.3 301.7

194.6 186.7 174.1 211.0 172.0 231.0 240.0 269.0 321.0 223.0 +9.7 165.3 138.6 173.4 265.0 216.0 300.0 330.5

−43.3 −30.7 −29.7 +3.5

120.8 76.0 72.0 68.0 167.4 119.8 123.5 42.3 83.5 87.4 57.2 77.8 57.8 129.8 57.8 164.1 167.4 185.3 172.1 192.3 137.5 164.3 235.8 96.0 208.5 280.5

131.6 86.0 82.0 78.0 177.5 130.0 133.8 53.7 93.7 97.4 66.4 87.8 67.0 140.0 67.0 175.2 179.0 196.1 183.9 204.4 147.8 175.8 247.8 105.8 221.6 290.7

146.5 99.8 95.3 91.7 190.2 144.0 147.7 68.8 107.4 110.1 78.7 101.3 79.5 153.0 79.2 190.0 194.0 211.0 198.0 218.0 161.8 190.0 263.7 118.6 239.5 304.9

168.5 120.2 115.7 111.8 209.6 165.0 168.0 92.1 117.8 130.2 97.8 121.7 98.0 173.8 97.6 212.8 217.5 233.0 220.0 241.7 183.5 212.5 287.0 138.0 264.7 323.8

192.5 143.5 138.0 134.2 229.8 188.0 192.0 119.8 150.6 151.8 118.5 144.1 119.5 196.0 118.0 237.6 243.4 257.1 244.0 264.6 205.8 237.0 313.5 159.0 294.0 342.0

218.6 166.3 160.5 157.3 250.4 212.5 215.0 149.0 174.6 175.0 141.6 167.5 141.2 219.0 139.5 262.5 269.3 282.0 268.6 290.0 229.5 260.8 340.0 182.0 324.0 361.0

87.5 −64.5

237.0 96.3 73.4 66.2 69.2 24.5 21.2 −48.3 −90.1 176.0 82.2 109.5 +11.3 105.9 109.6 99.8 −19.4 −28.0 76.0 −58.6 79.0

251.5 111.8 89.4 82.0 84.8 37.7 35.0 −38.2 −81.6 195.5 97.3 125.5 24.4 123.8 127.5 118.0 −7.9 −17.0 91.5 −48.8 94.8

260.3 121.5 99.5 92.2 95.2 47.8 43.9 −31.8 −76.1 207.5 106.9 135.8 32.6 135.0 139.0 129.0 −0.5 −10.0 101.5 −42.6 105.0

272.5 134.0 112.9 105.0 108.4 60.2 56.0 −23.0 −68.6 223.8 119.7 149.6 44.1 149.8 153.3 144.0 +9.5 −0.2 114.5 −33.9 118.2

290.0 152.3 133.4 125.9 128.3 79.7 74.0 −10.0 −57.0 248.0 139.0 170.0 61.0 172.0 176.0 166.2 24.6 +14.3 134.0 −20.9 138.3

309.8 173.7 155.8 149.0 150.2 100.8 94.2 +5.0 −43.9 275.5 159.8 192.0 80.3 197.0 201.7 191.5 41.0 30.8 155.4 −6.2 160.2

330.0 194.4 179.0 173.0 173.9 123.5 116.0 20.9 −29.8 304.0 182.7 215.0 100.6 222.1 226.6 216.3 59.0 47.8 178.5 +8.9 184.0

92.4 14.6 18.7 44.8 107.7 115.5

109.5 28.2 32.0 58.6 123.4 131.6

120.0 37.0 40.2 67.5 133.5 141.8

134.2 48.2 51.7 78.8 146.0 154.6

155.5 65.8 68.9 96.1 165.2 175.5

180.2 85.4 87.8 115.4 187.5 197.7

205.5 106.0 108.0 135.0 210.0 220.0

Temperature, °C

Formula C10H18 C10H18 C10H22 C10H20O C10H20 C10H22O C13H30Si C8H8O4 C14H12O C4H7NO2 C4H2 C6H10Cl2Si C6H10S C10H22O C12H22O4 C10H22S C14H15N C15H14O

40

+7 60 78.5 −34.9 −83

−26 34.5

−34.5

−70.3 −55.5 −34.4 −70

−79.7 73.5

9.7 −17.6 −24.2 53.0 −80.4 −112

−40.8 −76.4 −61.2 −80.5 −50.0 −135 −53.0

45.0

2-66

PHYSICAL AND CHEMICAL DATA

TABLE 2-8

Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name

α,α-Dichlorophenylacetonitrile Dichlorophenylarsine 1,2-Dichloropropane 2,3-Dichlorostyrene 2,4-Dichlorostyrene 2,5-Dichlorostyrene 2,6-Dichlorostyrene 3,4-Dichlorostyrene 3,5-Dichlorostyrene 1,2-Dichlorotetraethylbenzene 1,4-Dichlorotetraethylbenzene 1,2-Dichloro-1,1,2,2-tetrafluoroethane Dichloro-4-tolylsilane 3,4-Dichloro-α,α,α-trifluorotoluene Dicyclopentadiene Diethoxydimethylsilane Diethoxydiphenylsilane Diethyl adipate Diethylamine N-Diethylaniline Diethyl arsanilate 1,2-Diethylbenzene 1,3-Diethylbenzene 1,4-Diethylbenzene Diethyl carbonate cis-Diethyl citraconate Diethyl dioxosuccinate Diethylene glycol Diethyleneglycol-bis-chloroacetate Diethylene glycol dimethyl ether Di(2-methoxyethyl) ether glycol ethyl ether Diethyl ether ethylmalonate fumarate glutarate Diethylhexadecylamine Diethyl itaconate ketone (3-pentanone) malate maleate malonate mesaconate oxalate phthalate sebacate 2,5-Diethylstyrene Diethyl succinate isosuccinate sulfate sulfide sulfite d-Diethyl tartrate dl-Diethyl tartrate 3,5-Diethyltoluene Diethylzinc 1-Dihydrocarvone Dihydrocitronellol 1,4-Dihydroxyanthraquinone Dimethylacetylene (2-butyne) Dimethylamine N,N-Dimethylaniline Dimethyl arsanilate Di(α-methylbenzyl) ether 2,2-Dimethylbutane 2,3-Dimethylbutane Dimethyl citraconate 1,1-Dimethylcyclohexane cis-1,2-Dimethylcyclohexane trans-1,2-Dimethylcyclohexane trans-1,3-Dimethylcyclohexane cis-1,3-Dimethylcyclohexane cis-1,4-Dimethylcyclohexane trans-1,4-Dimethylcyclohexane

1

5

10

20

56.0 61.8 −38.5 61.0 53.5 55.5 47.8 57.2 53.5 105.6 91.7 −95.4 46.2 11.0 −19.1 111.5 74.0

84.0 100.0 −17.0 90.1 82.2 83.9 75.7 86.0 82.2 138.7 126.1 −80.0 71.7 38.3 34.1 +2.4 142.8 106.6

49.7

78.0

98.1 116.0 −6.1 104.6 97.4 98.2 90.0 100.4 97.4 155.0 143.8 −72.3 84.2 52.2 47.6 13.3 157.6 123.0 −33.0 91.9

113.8 133.1 +6.0 120.5 111.8 114.0 105.5 116.2 111.8 172.5 162.0 −63.5 97.8 67.3 62.0 25.3 174.3 138.3 −22.6 107.2

130.0 151.0 19.4 137.8 129.2 131.0 122.4 133.7 129.2 192.2 183.2 −53.7 113.2 84.0 77.9 38.0 193.2 154.6 −11.3 123.6

38.0 22.3 20.7 20.7 −10.1 59.8 70.0 91.8 148.3

62.6 48.7 46.8 47.1 +12.3 88.3 98.0 120.0 180.0

74.8 62.0 59.9 60.3 23.8 103.0 112.0 133.8 195.8

88.0 76.4 74.5 74.7 36.0 118.2 126.8 148.0 212.0

C6H14O3 C6H14O3 C4H10O C9H16O4 C8H12O4 C9H16O4 C20H43N C9H14O4 C5H10O C8H14O5 C8H12O4 C7H12O4 C9H14O4 C6H10O4 C12H14O4 C14H26O4 C12H16 C8H14O4 C8H14O4 C4H10O4S C4H10S C4H10O3S C8H14O6 C8H14O6 C11H16 C4H10Zn C10H16O C10H22O C14H8O4 C4H6 C2H7N C8H11N C8H12AsNO3 C16H18O C6H14 C6H14 C7H10O4 C8H16 C8H16 C8H16 C8H16 C8H16 C8H16 C8H16

13.0 45.3 −74.3 50.8 53.2 65.6 139.8 51.3 −12.7 80.7 57.3 40.0 62.8 47.4 108.8 125.3 49.7 54.6 39.8 47.0 −39.6 10.0 102.0 100.0 34.0 −22.4 46.6 68.0 196.7 −73.0 −87.7 29.5 15.0 96.7 −69.3 −63.6 50.8 −24.4 −15.9 −21.1 −19.4 −22.7 −20.0 −24.3

37.6 72.0 −56.9 77.8 81.2 94.7 175.8 80.2 +7.5 110.4 85.6 67.5 91.0 71.8 140.7 156.2 78.4 83.0 66.7 74.0 −18.6 34.2 133.0 131.7 61.5 0.0 75.5 91.7 239.8 −57.9 −72.2 56.3 39.6 128.3 −50.7 −44.5 78.2 −1.4 +7.3 +1.7 +3.4 0.0 +3.2 −1.7

50.0 85.8 −48.1 91.6 95.3 109.7 194.0 95.2 17.2 125.3 100.0 81.3 105.3 83.8 156.0 172.1 92.6 96.6 80.0 87.7 −8.0 46.4 148.0 147.2 75.3 +11.7 90.0 103.0 259.8 −50.5 −64.6 70.0 51.8 144.0 −41.5 −34.9 91.8 +10.3 18.4 13.0 14.9 +11.2 14.5 +10.1

63.0 100.3 −38.5 106.0 110.2 125.4 213.5 111.0 27.9 141.2 115.3 95.9 120.3 96.8 173.6 189.8 108.5 111.7 94.7 102.1 +3.5 59.7 164.2 163.8 90.2 24.2 106.0 115.0 282.0 −42.5 −56.0 84.8 65.0 160.3 −31.1 −24.1 106.5 23.0 31.1 25.6 27.4 23.6 27.1 22.6

60

100

200

400

760

141.0 163.2 28.0 149.0 140.0 142.0 133.3 144.6 140.0 204.8 195.8 −47.5 122.6 95.0 88.0 46.3 205.0 165.8 −4.0 133.8

154.5 178.9 39.4 163.5 153.8 155.8 147.6 158.2 153.8 220.7 212.0 −39.1 135.5 109.2 101.7 57.6 220.0 179.0 +6.0 147.3

176.2 202.8 57.0 185.7 176.0 178.0 169.0 181.5 176.0 245.6 238.5 −26.3 153.5 129.0 121.8 74.2 243.8 198.2 21.0 168.2

199.5 228.8 76.0 210.0 200.0 202.5 193.5 205.7 200.0 272.8 265.8 −12.0 175.2 150.5 144.2 93.2 259.7 219.1 38.0 192.4

223.5 256.5 96.8 235.0 225.0 227.0 217.0 230.0 225.0 302.0 296.5 +3.5 196.3 172.8 166.6 113.5 296.0 240.0 55.5 215.5

102.6 92.5 90.4 91.1 49.5 135.7 143.8 164.3 229.0

111.8 102.6 100.7 101.3 57.9 146.2 153.7 174.0 239.5

123.8 116.2 114.4 115.3 69.7 160.0 167.7 187.5 252.0

141.9 136.7 134.8 136.1 86.5 182.3 188.0 207.0 271.5

161.0 159.0 156.9 159.0 105.8 206.5 210.8 226.5 291.8

181.0 183.5 181.1 183.8 125.8 230.3 233.5 244.8 313.0

77.5 116.7 27.7 122.4 126.7 142.8 235.0 128.2 39.4 157.8 131.8 113.3 137.3 110.6 192.1 207.5 125.8 127.8 111.0 118.0 16.1 74.2 182.3 181.7 107.0 38.0 123.7 127.6 307.4 −33.9 −46.7 101.6 79.7 179.6 −19.5 −12.4 122.6 37.3 45.3 39.7 41.4 37.5 41.1 36.5

86.8 126.8 −21.8 132.4 137.7 153.2 248.5 139.9 46.7 169.0 142.4 123.0 147.9 119.7 204.1 218.4 136.8 138.2 121.4 128.6 24.2 83.8 194.0 193.2 117.7 47.2 134.7 136.7 323.3 −27.8 −40.7 111.9 88.6 191.5 −12.1 −4.9 132.7 45.7 54.4 48.7 50.4 46.4 50.1 45.4

99.5 140.3 −11.5 146.0 151.1 167.8 265.5 154.3 56.2 183.9 156.0 136.2 161.6 130.8 219.5 234.4 151.0 151.1 134.8 142.5 35.0 96.3 208.5 208.0 131.7 59.1 149.7 145.9 344.5 −18.8 −32.6 125.8 101.0 206.8 −2.0 +5.4 145.8 57.9 66.8 61.0 62.5 58.5 62.3 57.6

118.0 159.0 +2.2 166.0 172.2 189.5 292.8 177.5 70.6 205.3 177.8 155.5 183.2 147.9 243.0 255.8 173.2 171.7 155.1 162.5 51.3 115.8 230.4 230.0 152.4 77.0 171.8 160.2 377.8 −5.0 −20.4 146.5 119.8 229.7 +13.4 21.1 165.8 76.2 85.6 79.6 81.0 76.9 80.8 76.0

138.5 180.3 17.9 188.7 195.8 212.8 324.6 203.1 86.3 229.5 201.7 176.8 205.8 166.2 267.5 280.3 198.0 193.8 177.7 185.5 69.7 137.0 254.8 254.3 176.5 97.3 197.0 176.8 413.0 +10.6 −7.1 169.2 140.3 254.8 31.0 39.0 188.0 97.2 107.0 100.9 102.1 97.8 101.9 97.0

159.8 201.9 34.6 211.5 218.5 237.0 355.0 227.9 102.7 253.4 225.0 198.9 229.0 185.7 294.0 305.5 223.0 216.5 201.3 209.5 88.0 159.0 280.0 280.0 200.7 118.0 223.0 193.5 450.0 27.2 +7.4 193.1 160.5 281.0 49.7 58.0 210.5 119.5 129.7 123.4 124.4 120.1 124.3 119.3

Temperature, °C

Formula C8H5Cl2N C6H5AsCl2 C3H6Cl2 C8H6Cl2 C8H6Cl2 C8H6Cl2 C8H6Cl2 C8H6Cl2 C8H6Cl2 C14H20Cl2 C14H20Cl2 C2Cl2F4 C7H8Cl2Si C7H3Cl2F3 C10H8 C6H16O2Si C16H20O2Si C10H18O4 C4H11N C10H15N C10H16As NO3 C10H14 C10H14 C10H14 C5H10O3 C9H14O4 C8H10O6 C4H10O3 C8H12Cl2O5

40

Melting point, °C

−94 −12.1 32.9 −21 −38.9 −34.4 −31.4 −83.9 −43.2 −43

−116.3 +0.6

−42 −49.8 −40.6 1.3 −20.8 −25.0 −99.5 17 −28 194 −32.5 −96 +2.5 −99.8 −128.2 −34 −50.0 −88.0 −92.0 −76.2 −87.4 −36.9

VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-8

2-67

Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name

Dimethyl ether 2,2-Dimethylhexane 2,3-Dimethylhexane 2,4-Dimethylhexane 2,5-Dimethylhexane 3,3-Dimethylhexane 3,4-Dimethylhexane Dimethyl itaconate 1-Dimethyl malate Dimethyl maleate malonate trans-Dimethyl mesaconate 2,7-Dimethyloctane Dimethyl oxalate 2,2-Dimethylpentane 2,3-Dimethylpentane 2,4-Dimethylpentane 3,3-Dimethylpentane 2,3-Dimethylphenol (2,3-xylenol) 2,4-Dimethylphenol (2,4-xylenol) 2,5-Dimethylphenol (2,5-xylenol) 3,4-Dimethylphenol (3,4-xylenol) 3,5-Dimethylphenol (3,5-xylenol) Dimethylphenylsilane Dimethyl phthalate 3,5-Dimethyl-1,2-pyrone 4,6-Dimethylresorcinol Dimethyl sebacate 2,4-Dimethylstyrene 2,5-Dimethylstyrene α,α-Dimethylsuccinic anhydride Dimethyl sulfide d-Dimethyl tartrate dl-Dimethyl tartrate N,N-Dimethyl-2-toluidine N,N-Dimethyl-4-toluidine Di(nitrosomethyl) amine Diosphenol 1,4-Dioxane Dipentene Diphenylamine Diphenyl carbinol (benzhydrol) chlorophosphate disulfide 1,2-Diphenylethane (dibenzyl) Diphenyl ether 1,1-Diphenylethylene trans-Diphenylethylene 1,1-Diphenylhydrazine Diphenylmethane Diphenyl sulfide Diphenyl-2-tolyl thiophosphate 1,2-Dipropoxyethane 1,2-Diisopropylbenzene 1,3-Diisopropylbenzene Dipropylene glycol Dipropyleneglycol monobutyl ether isopropyl ether Di-n-propyl ether Diisopropyl ether Di-n-propyl ketone (4-heptanone) Di-n-propyl oxalate Diisopropyl oxalate Di-n-propyl succinate Di-n-propyl d-tartrate Diisopropyl d-tartrate Divinyl acetylene (1,5-hexadiene-3-yne) 1,3-Divinylbenzene Docosane n-Dodecane 1-Dodecene n-Dodecyl alcohol Dodecylamine Dodecyltrimethylsilane Elaidic acid

1

5

10

20

−115.7 −29.7 −23.0 −26.9 −26.7 −25.8 −22.1 69.3 75.4 45.7 35.0 46.8 +6.3 20.0 −49.0 −42.0 −48.0 −45.9 56.0 51.8 51.8 66.2 62.0 +5.3 100.3 78.6 49.0 104.0 34.2 29.0 61.4 −75.6 102.1 100.4 28.8 50.1 +3.2 66.7 −35.8 14.0 108.3 110.0 121.5 131.6 86.8 66.1 87.4 113.2 126.0 76.0 96.1 159.7 −38.8 40.0 34.7 73.8 64.7 46.0 −43.3 −57.0 23.0 53.4 43.2 77.5 115.6 103.7 −45.1 32.7 157.8 47.8 47.2 91.0 82.8 91.2 171.3

−101.1 −7.9 −1.1 −5.3 −5.5 −4.4 +0.2 94.0 104.0 73.0 59.8 74.0 30.5 44.0 −28.7 −20.8 −27.4 −25.0 83.8 78.0 78.0 93.8 89.2 30.3 131.8 107.6 76.8 139.8 61.9 55.9 88.1 −58.0 133.2 131.8 54.1 74.3 27.8 95.4 −12.8 40.4 141.7 145.0 160.5 164.0 119.8 97.8 119.6 145.8 159.3 107.4 129.0 179.8 −10.3 67.8 62.3 102.1 92.0 72.8 −22.3 −37.4 44.4 80.2 69.0 107.6 147.7 133.7 −24.4 60.0 195.4 75.8 74.0 120.2 111.8 122.1 206.7

−93.3 +3.1 +9.9 +5.2 +5.3 +6.1 11.3 106.6 118.3 86.4 72.0 87.8 42.3 56.0 −18.7 −10.3 −17.1 −14.4 97.6 91.3 91.3 107.7 102.4 42.6 147.6 122.0 90.7 156.2 75.8 69.0 102.0 −49.2 148.2 147.5 66.2 86.7 40.0 109.0 −1.2 53.8 157.0 162.0 182.0 180.0 136.0 114.0 135.0 161.0 176.1 122.8 145.0 201.6 +5.0 81.8 76.0 116.2 106.0 86.2 −11.8 −27.4 55.0 93.9 81.9 122.2 163.5 148.2 −14.0 73.8 213.0 90.0 87.8 134.7 127.8 137.7 223.5

−85.2 15.0 22.1 17.2 17.2 18.2 23.5 119.7 133.8 101.3 85.0 102.1 55.8 69.4 −7.5 +1.1 −5.9 −2.9 112.0 105.0 105.0 122.0 117.0 56.2 164.0 136.4 105.8 175.8 90.8 84.0 116.3 −39.4 164.3 164.0 80.2 100.0 53.7 124.0 +12.0 68.2 175.2 180.9 203.8 197.0 153.7 130.8 151.8 179.8 194.0 139.8 162.0 215.5 22.3 96.8 91.2 131.3 120.4 100.8 0.0 −16.7 66.2 108.6 95.6 138.0 180.4 164.0 −2.8 88.7 233.5 104.6 102.4 150.0 141.6 153.8 242.3

60

100

200

400

760

−62.7 48.2 56.0 50.6 50.5 52.5 57.7 153.7 175.1 140.4 121.9 141.5 93.9 104.8 23.9 33.3 25.4 29.3 152.2 143.0 143.0 161.0 156.0 94.2 210.0 177.5 147.3 222.6 132.3 124.7 155.3 −12.0 208.8 209.5 118.1 140.3 90.3 165.6 45.1 108.3 222.8 227.5 265.0 241.3 202.8 178.8 198.6 227.4 242.5 186.3 211.8 252.5 74.2 138.7 132.3 169.9 159.8 140.3 33.0 13.7 96.0 148.1 132.6 180.3 227.0 207.3 29.5 130.0 286.0 146.2 142.3 192.0 182.1 199.5 288.0

−50.9 65.7 73.8 68.1 68.0 70.0 75.6 171.0 196.3 160.0 140.0 161.0 114.0 123.3 40.3 50.1 41.8 46.2 173.0 161.5 161.5 181.5 176.2 114.2 232.7 198.0 167.8 245.0 153.2 145.6 175.8 +2.6 230.5 232.3 138.3 161.6 110.0 186.2 62.3 128.2 247.5 250.0 299.5 262.6 227.8 203.3 222.8 251.7 267.2 210.7 236.8 270.3 103.8 159.8 153.7 189.9 180.0 160.0 50.3 30.0 111.2 168.0 151.2 202.5 250.1 228.2 46.0 151.4 314.2 167.2 162.2 213.0 203.0 222.0 312.4

−37.8 85.6 94.1 88.2 87.9 90.4 96.0 189.8 219.5 182.2 159.8 183.5 136.0 143.3 59.2 69.4 60.6 65.5 196.0 184.2 184.2 203.6 197.8 136.4 257.8 221.0 192.0 269.6 177.5 168.7 197.5 18.7 255.0 257.4 161.5 185.4 131.3 209.5 81.8 150.5 274.1 275.6 337.2 285.8 255.0 230.7 249.8 278.3 294.0 237.5 263.9 290.0 140.0 184.3 177.6 210.5 203.8 183.1 69.5 48.2 127.3 190.3 171.8 226.5 275.6 251.8 64.4 175.2 343.5 191.0 185.5 235.7 225.0 248.0 337.0

−23.7 106.8 115.6 109.4 109.1 112.0 117.7 208.0 242.6 205.0 180.7 206.0 159.7 163.3 79.2 89.8 80.5 86.1 218.0 211.5 211.5 225.2 219.5 159.3 283.7 245.0 215.0 293.5 202.0 193.0 219.5 36.0 280.0 282.0 184.8 209.5 153.0 232.0 101.1 174.6 302.0 301.0 378.0 310.0 284.0 258.5 277.0 306.5 322.2 264.5 292.5 310.0 180.0 209.0 202.0 231.8 227.0 205.6 89.5 67.5 143.7 213.5 193.5 250.8 303.0 275.0 84.0 199.5 376.0 216.2 208.0 259.0 248.0 273.0 362.0

Temperature, °C

Formula C2H6O C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C7H10O4 C6H10O5 C6H8O4 C5H8O4 C7H10O4 C10H22 C4H6O4 C7H16 C7H16 C7H16 C7H16 C8H10O C8H10O C8H10O C8H10O C8H10O C8H12Si C10H10O4 C7H8O2 C8H10O2 C12H22O4 C10H12 C10H12 C6H8O3 C2H6S C6H10O6 C6H10O6 C9H13N C9H13N C2H5N3O2 C10H16O2 C4H8O2 C10H16 C12H11N C13H12O C12H10ClPO3 C12H10S2 C14H14 C12H10O C14H12 C14H12 C12H12N2 C13H12 C12H10S C18H17O3PS C8H18O2 C12H18 C12H18 C6H14O3 C10H22O3 C9H20O3 C6H14O C6H14O C7H14O C8H14O4 C8H14O4 C10H18O4 C10H18O6 C10H18O6 C6H6 C10H10 C22H46 C12H26 C12H24 C12H26O C12H27N C15H34Si C18H34O2

40 −76.2 28.2 35.6 30.5 30.4 31.7 37.1 133.7 150.1 117.2 100.0 118.0 71.2 83.6 +5.0 13.9 +6.5 +9.9 129.2 121.5 121.5 138.0 133.3 71.4 182.8 152.7 122.5 196.0 107.7 100.2 132.3 −28.4 182.4 182.4 95.0 116.3 68.2 141.2 25.2 84.3 194.3 200.0 227.9 214.8 173.7 150.0 170.8 199.0 213.5 157.8 182.8 230.6 42.3 114.0 107.9 147.4 136.3 117.0 +13.2 −4.5 78.1 124.6 110.5 154.8 199.7 181.8 +10.0 105.5 254.5 121.7 118.6 167.2 157.4 172.1 260.8

−70.4 36.7 44.2 39.0 38.9 40.4 45.8 142.6 160.4 127.1 109.7 127.8 80.8 92.8 13.0 22.1 14.5 18.1 139.5 131.0 131.0 148.0 143.5 81.3 194.0 163.8 133.2 208.0 118.0 110.7 142.4 −21.4 193.8 193.8 105.2 126.4 77.7 151.3 33.8 94.6 206.9 212.0 244.2 226.2 186.0 162.0 183.4 211.5 225.9 170.2 194.8 240.4 55.8 124.3 118.2 156.5 146.3 126.8 21.6 +3.4 85.8 134.8 120.0 166.0 211.7 192.6 18.1 116.0 268.3 132.1 128.5 177.8 168.0 184.2 273.0

Melting point, °C −138.5

−90.7 38 −62 −52.8 −123.7 −135 −119.5 −135.0 75 25.5 74.5 62.5 68 51.5 38

−83.2 61.5 89 −61

10 52.9 68.5 61 51.5 27 124 44 26.5

−105

−122 −60 −32.6

−66.9 44.5 −9.6 −31.5 24 51.5

2-68

PHYSICAL AND CHEMICAL DATA

TABLE 2-8

Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound

1

5

10

20

−16.5 −69.0 206.7 52.6 −159.5 36.3 −50.9 167.0 −43.4 28.5 −92.5 −29.5 47.0 −29.0 −31.3 −82.3 52.0 38.5 29.7 33.7 33.5 −9.8 44.0 107.6 −74.3 10.6 −18.4 −24.3 118.2 11.0 107.8 133.2 −89.8 +1.0 −5.1 +6.6 87.6 28.3 31.5 67.8 −14.5 −32.2 9.6 76.0 98.3 −168.3 112.0 −4.0 −11.0 −27.0 −44.5 53.0 −33.5

+5.6 −50.0 239.7 80.0 −148.5 63.1 −31.0 198.2 −23.5 54.0 −76.7 −8.7 70.7 −6.4 −12.0 −66.4 80.0 66.4 55.9 60.3 60.2 +13.9 72.0 136.4 −56.4 35.8 +4.0 −2.4 149.8 35.8 65.8 131.8 168.2 −73.9 25.4 +18.0 30.2 108.5 55.5 58.4 93.5 +9.2 −10.8 34.0 106.3 130.2 −158.3 142.4 +19.0 +10.5 +4.7 −24.0 79.7 −10.2

16.6 −40.3 254.5 93.7 −142.9 76.2 −20.7 213.5 −13.5 67.3 −68.7 +2.0 82.0 +5.0 −2.3 −58.3 93.8 80.6 69.0 73.9 73.9 25.9 86.0 150.3 −47.5 48.0 15.3 +8.4 165.0 48.0 77.8 143.7 186.0 −65.8 37.5 29.9 41.9 134.0 68.8 72.0 106.0 20.6 −0.1 46.3 121.7 146.0 −153.2 158.0 30.3 21.5 18.6 −13.6 92.1 +1.6

29.0 −29.5 270.6 108.4 −136.7 91.0 −9.8 230.0 −3.0 81.1 −59.9 13.0 94.4 17.7 +8.0 −48.6 109.0 96.0 83.1 88.5 88.5 38.6 101.4 166.8 −37.8 61.8 27.8 20.6 181.8 61.7 91.0 155.5 205.5 −56.8 50.4 42.0 54.3 150.3 83.6 86.7 119.8 33.4 +11.7 59.5 137.7 162.8 −147.6 173.5 42.5 33.0 32.7 −2.4 105.8 14.7

42.0 −17.3 289.1 124.6 −129.8 107.2 +3.7 247.0 +9.1 96.2 −50.0 26.0 108.1 31.8 19.0 −39.8 125.7 113.2 98.8 104.8 104.7 52.8 118.2 181.8 −26.7 77.0 41.5 33.8 199.8 76.3 105.6 168.8 226.5 −47.0 65.2 56.0 68.2 169.2 99.9 103.3 133.8 47.6 25.0 74.0 154.4 182.0 −141.3 191.0 56.0 45.8 48.0 +10.0 120.0 29.7

50.6 −9.7 300.2 135.2 −125.4 127.5 11.5 258.3 16.6 106.0 −43.4 33.5 116.7 40.6 26.0 −33.4 136.0 123.6 109.0 115.5 115.4 61.8 129.0 191.9 −19.5 86.7 50.1 42.3 211.5 85.8 114.8 177.3 239.8 −40.6 74.0 65.2 77.3 181.2 110.2 113.8 142.1 56.7 33.4 83.6 166.0 193.7 −137.3 201.8 64.1 53.8 57.9 18.1 129.5 39.0

62.0 +1.2 314.4 148.5 −119.3 131.4 22.1 273.5 27.0 118.5 −34.9 44.5 127.5 53.0 34.9 −25.1 149.8 137.3 122.3 129.2 128.4 74.1 143.2 205.0 −10.0 99.8 62.0 53.5 226.6 98.4 126.2 187.9 256.8 −32.0 86.0 76.6 89.3 196.0 124.3 127.2 152.8 69.0 45.0 96.1 180.3 209.8 −131.8 215.0 75.0 62.5 70.4 29.4 141.8 51.8

C4H10O2

−48.0

−26.2

−15.3

−3.0

+10.7

19.7

31.8

50.0

70.8

93.0

C3H8O2

−13.5

+10.2

22.0

34.3

47.8

56.4

68.0

85.3

104.3

124.4

−89.7 40.5 −117.0 −60.5 37.6 14.3 −20.0 50.0 −60.7 −112.5 −54.4 27.8 47.3 −76.7 26.5 −91.0

−73.8 67.3 −103.8 −42.2 63.8 38.8 +2.1 77.7 −41.9 −98.4 −34.3 57.3 74.0 −59.1 51.0 −75.6

−65.7 80.2 −97.7 −33.0 77.1 50.5 12.8 91.8 −32.3 −91.7 −24.3 72.1 87.3 −50.2 63.2 −67.8

−56.6 94.6 −90.0 −22.7 91.5 63.9 25.0 106.3 −21.9 −84.1 −13.1 88.0 101.8 −40.7 76.1 −59.1

−46.9 110.3 −81.8 −11.5 107.5 78.1 38.5 123.7 −10.2 −75.8 −0.9 106.0 117.7 −29.8 91.0 −49.4

−40.7 120.6 −76.4 −4.3 117.5 87.6 47.1 134.0 −2.9 −70.4 +7.2 117.8 127.6 −22.4 100.0 −43.3

−32.1 133.8 −69.3 −5.4 130.4 99.8 58.9 147.9 +7.2 −63.2 18.0 131.8 141.3 −13.0 112.0 −34.8

−19.5 153.2 −58.0 20.0 150.1 117.8 76.7 168.2 22.4 −52.0 34.1 149.8 160.2 +1.5 130.0 −22.0

−4.9 175.6 −45.5 37.1 172.5 138.0 97.0 192.2 39.8 −39.5 52.3 167.3 183.0 17.7 149.8 −7.8

+10.7 198.0 −32.0 54.3 195.0 158.2 118.5 216.0 57.4 −26.5 72.4 184.0 206.2 35.0 170.0 +7.5

Name

Formula

Epichlorohydrin 1,2-Epoxy-2-methylpropane Erucic acid Estragole (p-methoxy allyl benzene) Ethane Ethoxydimethylphenylsilane Ethoxytrimethylsilane Ethoxytriphenylsilane Ethyl acetate acetoacetate Ethylacetylene (1-butyne) Ethyl acrylate α-Ethylacrylic acid α-Ethylacrylonitrile Ethyl alcohol (ethanol) Ethylamine 4-Ethylaniline N-Ethylaniline 2-Ethylanisole 3-Ethylanisole 4-Ethylanisole Ethylbenzene Ethyl benzoate benzoylacetate bromide α-bromoisobutyrate n-butyrate isobutyrate Ethylcamphoronic anhydride Ethyl isocaproate carbamate carbanilate Ethylcetylamine Ethyl chloride chloroacetate chloroglyoxylate α-chloropropionate trans-cinnamate 3-Ethylcumene 4-Ethylcumene Ethyl cyanoacetate Ethylcyclohexane Ethylcyclopentane Ethyl dichloroacetate N,N-diethyloxamate N-Ethyldiphenylamine Ethylene Ethylene-bis-(chloroacetate) Ethylene chlorohydrin (2-chloroethanol) diamine (1,2-ethanediamine) dibromide (1,2-dibromethane) dichloride (1,2-dichloroethane) glycol (1,2-ethanediol) glycol diethyl ether (1,2-diethoxyethane) glycol dimethyl ether (1,2-dimethoxyethane) glycol monomethyl ether (2-methoxyethanol) oxide Ethyl α-ethylacetoacetate fluoride formate 2-furoate glycolate 3-Ethylhexane 2-Ethylhexyl acrylate Ethylidene chloride (1,1-dichloroethane) fluoride (1,1-difluoroethane) Ethyl iodide Ethyl l-leucinate Ethyl levulinate Ethyl mercaptan (ethanethiol) Ethyl methylcarbamate Ethyl methyl ether

C3H5ClO C4H8O C22H42O2 C10H12O C2H6 C10H16OSi C5H14OSi C20H20OSi C4H8O2 C6H10O3 C4H6 C5H8O2 C5H8O2 C5H7N C2H6O C2H7N C8H11N C8H11N C9H12O C9H12O C9H12O C8H10 C9H10O2 C11H12O3 C2H5Br C6H11BrO2 C6H12O2 C6H12O2 C11H16O5 C8H16O2 C3H7NO2 C9H11NO2 C18H39N C2H5Cl C4H7ClO2 C4H5ClO3 C5H9ClO2 C11H12O2 C11H16 C11H16 C5H7NO2 C8H16 C7H14 C4H6Cl2O2 C8H15NO3 C14H15N C2H4 C6H8Cl2O4 C2H5ClO C2H8N2 C2H4Br2 C2H4Cl2 C2H6O2 C6H14O2

C2H4O C8H14O3 C2H5F C3H6O2 C7H8O3 C4H8O3 C8H18 C11H20O2 C2H4Cl2 C2H4F2 C2H5I C8H17NO2 C7H12O3 C2H6S C4H9NO2 C3H8O

40

60

100

200

400

760

Temperature, °C 79.3 98.0 117.9 17.5 36.0 55.5 336.5 358.8 381.5 168.7 192.0 215.0 −110.2 −99.7 −88.6 151.5 175.0 199.5 38.1 56.3 75.7 295.0 319.5 344.0 42.0 59.3 77.1 138.0 158.2 180.8 −21.6 −6.9 +8.7 61.5 80.0 99.5 144.0 160.7 179.2 71.6 92.2 114.0 48.4 63.5 78.4 −12.3 +2.0 16.6 170.6 194.2 217.4 156.9 180.8 204.0 142.1 164.2 187.1 149.7 172.8 196.5 149.2 172.3 196.5 92.7 113.8 136.2 164.8 188.4 213.4 223.8 244.7 265.0 +4.5 21.0 38.4 119.7 141.2 163.6 79.8 100.0 121.0 71.0 90.0 110.0 248.5 272.8 298.0 117.8 139.2 160.4 144.2 164.0 184.0 203.8 220.0 237.0 283.3 313.0 342.0 −18.6 −3.9 +12.3 103.8 123.8 144.2 94.5 114.7 135.0 107.2 126.2 146.5 219.3 245.0 271.0 145.4 168.2 193.0 148.3 171.8 195.8 169.8 187.8 206.0 87.8 109.1 131.8 62.4 82.3 103.4 115.2 135.9 156.5 202.8 226.5 252.0 233.0 258.8 286.0 −123.4 −113.9 −103.7 237.3 259.5 283.5 91.8 110.0 128.8 81.0 99.0 117.2 89.8 110.1 131.5 45.7 64.0 82.4 158.5 178.5 197.3 71.8 94.1 119.5

Melting point, °C −25.6 33.5 −183.2

−82.4 −45 −130 −71.2 −112 −80.6 −4 −63.5

−94.9 −34.6 −117.8 −93.3 −88.2 49 52.5 −139 −26 12

−111.3 −138.6

−169 −69 8.5 10 −35.3 −15.6

−111.3 −79 34

−96.7 −117 −105 −121

VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-8

2-69

Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name

1-Ethylnaphthalene Ethyl α-naphthyl ketone (1-propionaphthone) Ethyl 3-nitrobenzoate 3-Ethylpentane 4-Ethylphenetole 2-Ethylphenol 3-Ethylphenol 4-Ethylphenol Ethyl phenyl ether (phenetole) Ethyl propionate Ethyl propyl ether Ethyl salicylate 3-Ethylstyrene 4-Ethylstyrene Ethylisothiocyanate 2-Ethyltoluene 3-Ethyltoluene 4-Ethyltoluene Ethyl trichloroacetate Ethyltrimethylsilane Ethyltrimethyltin Ethyl isovalerate 2-Ethyl-1,4-xylene 4-Ethyl-1,3-xylene 5-Ethyl-1,3-xylene Eugenol iso-Eugenol Eugenyl acetate Fencholic acid d-Fenchone dl-Fenchyl alcohol Fluorene Fluorobenzene 2-Fluorotoluene 3-Fluorotoluene 4-Fluorotoluene Formaldehyde Formamide Formic acid trans-Fumaryl chloride Furfural (2-furaldehyde) Furfuryl alcohol Geraniol Geranyl acetate Geranyl n-butyrate Geranyl isobutyrate Geranyl formate Glutaric acid Glutaric anhydride Glutaronitrile Glutaryl chloride Glycerol Glycerol dichlorohydrin (1,3-dichloro-2-propanol) Glycol diacetate Glycolide (1,4-dioxane-2,6-dione) Guaicol (2-methoxyphenol) Heneicosane Heptacosane Heptadecane Heptaldehyde (enanthaldehyde) n-Heptane Heptanoic acid (enanthic acid) 1-Heptanol Heptanoyl chloride (enanthyl chloride) 2-Heptene Heptylbenzene Heptyl cyanide (enanthonitrile) Hexachlorobenzene Hexachloroethane Hexacosane Hexadecane 1-Hexadecene n-Hexadecyl alcohol (cetyl alcohol)

1

5

10

20

70.0

101.4

116.8

133.8

152.0

C13H12O C9H9NO4 C7H16 C10H14O C8H10O C8H10O C8H10O C8H10O C5H10O2 C5H12O C9H10O3 C10H12 C10H12 C3H5NS C9H12 C9H12 C9H12 C4H5Cl3O2 C5H14Si C5H14Sn C7H14O2 C10H14 C10H14 C10H14 C10H12O2 C10H12O2 C12H14O3 C10H16O2 C10H16O C10H18O C13H10 C6H5F C7H7F C7H7F C7H7F CH2O CH3NO CH2O2 C4H2Cl2O2 C5H4O2 C5H6O2 C10H18O C12H20O2 C14H24O2 C14H24O2 C11H18O2 C5H8O4 C5H6O3 C5H6N2 C5H6Cl2O2 C3H8O3 C3H6Cl2O

124.0 108.1 −37.8 48.5 46.2 60.0 59.3 18.1 −28.0 −64.3 61.2 28.3 26.0 13.2 9.4 7.2 7.6 20.7 −60.6 −30.0 −6.1 25.7 26.3 22.1 78.4 86.3 101.6 101.7 28.0 45.8 −43.4 −24.2 −22.4 −21.8

155.5 140.2 −17.0 75.7 73.4 86.8 86.5 43.7 −7.2 −45.0 90.0 55.0 52.7 +10.6 34.8 32.3 32.7 45.5 −41.4 −7.6 +17.0 52.0 53.0 48.8 108.1 117.0 132.3 128.7 54.7 70.3 129.3 −22.8 −2.2 −0.3 +0.3

70.5 −20.0 +15.0 18.5 31.8 69.2 73.5 96.8 90.9 61.8 155.5 100.8 91.3 56.1 125.5 28.0

96.3 −5.0 38.5 42.6 56.0 96.8 102.7 125.2 119.6 90.3 183.8 133.3 123.7 84.0 153.8 52.2

171.0 155.0 −6.8 89.5 87.0 100.2 100.2 56.4 +3.4 −35.0 104.2 68.3 66.3 22.8 47.6 44.7 44.9 57.7 −31.8 +3.8 28.7 65.6 66.4 62.1 123.0 132.4 148.0 142.3 68.3 82.1 146.0 −12.4 +8.9 +11.0 11.8 −88.0 109.5 +2.1 51.8 54.8 68.0 110.0 117.9 139.0 133.0 104.3 196.0 149.5 140.0 97.8 167.2 64.7

188.1 173.6 +4.7 103.8 101.5 114.5 115.0 70.3 14.3 −24.0 119.3 82.8 80.8 36.1 61.2 58.2 58.5 70.6 −21.0 16.1 41.3 79.8 80.6 76.5 138.7 149.0 164.2 155.8 83.0 95.6 164.2 −1.2 21.4 23.4 24.0 −79.6 122.5 10.3 65.0 67.8 81.0 125.6 133.0 153.8 147.9 119.8 210.5 166.0 156.5 112.3 182.2 78.0

206.9 192.6 17.5 119.8 117.9 130.0 131.3 86.6 27.2 −12.0 136.7 99.2 97.3 50.8 76.4 73.3 73.6 85.5 −9.0 30.0 55.2 96.0 97.2 92.6 155.8 167.0 183.0 171.8 99.5 110.8 185.2 +11.5 34.7 37.0 37.8 −70.6 137.5 24.0 79.5 82.1 95.7 141.8 150.0 170.1 164.0 136.2 226.3 185.5 176.4 128.3 198.0 93.0

C6H10O4 C4H4O4 C7H8O2 C21H44 C27H56 C17H36 C7H14O C7H16 C7H14O2 C7H16O C7H13ClO C7H14 C13H20 C7H13N C6Cl6 C2Cl6 C26H54 C16H34 C16H32 C16H34O

38.3

64.1 103.0 79.1 188.0 248.6 145.2 32.7 −12.7 101.3 64.3 54.6 −14.1 94.6 47.8 149.3 49.8 240.0 135.2 131.7 158.3

77.1 116.6 92.0 205.4 266.8 160.0 43.0 −2.1 113.2 74.7 64.6 −3.5 110.0 61.6 166.4 73.5 257.4 149.8 146.2 177.8

90.8 132.0 106.0 223.2 284.6 177.7 54.0 +9.5 125.6 85.8 75.0 +8.3 126.0 76.3 185.7 87.6 275.8 164.7 162.0 197.8

106.1 148.6 121.6 243.4 305.7 195.8 66.3 22.3 139.5 99.8 86.4 21.5 144.0 92.6 206.0 102.3 295.2 181.3 178.8 219.8

100

200

400

760

Melting point, °C

164.1

180.0

204.6

230.8

258.1

−27

218.2 205.0 25.7 129.8 127.9 139.8 141.7 95.4 35.1 −4.0 147.6 109.6 107.6 59.8 86.0 82.9 83.2 94.4 −1.2 38.4 64.0 106.2 107.4 103.0 167.3 178.2 194.0 181.5 109.8 120.2 197.8 19.6 43.7 45.8 46.5 −65.0 147.0 32.4 89.0 91.5 104.0 151.5 160.3 180.2 174.0 147.2 235.5 196.2 189.5 139.1 208.0 102.0

233.5 220.3 36.9 143.5 141.8 152.0 154.2 108.4 45.2 +6.8 161.5 123.2 121.5 71.9 99.0 95.9 96.3 107.4 +9.2 50.0 75.9 120.0 121.2 116.5 182.2 194.0 209.7 194.0 123.6 132.3 214.7 30.4 55.3 57.5 58.1 −57.3 157.5 43.8 101.0 103.4 115.9 165.3 175.2 193.8 187.7 160.7 247.0 212.5 205.5 151.8 220.1 114.8

255.5 244.6 53.8 163.2 161.6 171.8 175.0 127.9 61.7 23.3 183.7 144.0 142.0 90.0 119.0 115.5 116.1 125.8 25.0 67.3 93.8 140.2 141.8 137.4 204.7 217.2 232.5 215.0 144.0 150.0 240.3 47.2 73.0 75.4 76.0 −46.0 175.5 61.4 120.0 121.8 133.1 185.6 196.3 214.0 207.6 182.6 265.0 236.5 230.0 172.4 240.0 133.3

280.2 270.6 73.0 185.7 184.5 193.3 197.4 149.8 79.8 41.6 207.0 167.2 165.0 110.1 141.4 137.8 136.4 146.0 42.8 87.6 114.0 163.1 164.4 159.6 228.3 242.3 257.4 237.8 166.8 173.2 268.6 65.7 92.8 95.4 96.1 −33.0 193.5 80.3 140.0 141.8 151.8 207.8 219.8 235.0 228.5 205.8 283.5 261.0 257.3 195.3 263.0 153.5

306.0 298.0 93.5 208.0 207.5 214.0 219.0 172.0 99.1 61.7 231.5 191.5 189.0 131.0 165.1 161.3 162.0 167.0 62.0 108.8 134.3 186.9 188.4 183.7 253.5 267.5 282.0 264.1 191.0 201.0 295.0 84.7 114.0 116.0 117.0 −19.5 210.5 100.6 160.0 161.8 170.0 230.0 243.3 257.4 251.0 230.0 303.0 287.0 286.2 217.0 290.0 174.3

115.8 158.2 131.0 255.3 318.3 207.3 74.0 30.6 148.5 108.0 93.5 30.0 154.8 103.0 219.0 112.0 307.8 193.2 190.8 234.3

128.0 173.2 144.0 272.0 333.5 223.0 84.0 41.8 160.0 119.5 102.7 41.3 170.2 116.8 235.5 124.2 323.2 208.5 205.3 251.7

147.8 194.0 162.7 296.5 359.4 247.8 102.0 58.7 179.5 136.6 116.3 58.6 193.3 137.7 258.5 143.1 348.4 231.7 226.8 280.2

168.3 217.0 184.1 323.8 385.0 274.5 125.5 78.0 199.6 155.6 130.7 78.1 217.8 160.0 283.5 163.8 374.6 258.3 250.0 312.7

190.5 240.0 205.0 350.5 410.6 303.0 155.0 98.4 221.5 175.8 145.0 98.5 244.0 184.6 309.4 185.6 399.8 287.5 274.0 344.0

60

Temperature, °C

Formula C12H12

40

52.4 152.6 211.7 115.0 12.0 −34.0 78.0 42.4 34.2 −35.8 64.0 21.0 114.4 32.7 204.0 105.3 101.6 122.7

47 −118.6 −45 −4 46.5 −30.2 −72.6 1.3 −5.9 −95.5

−99.3

−10 295 19 5 35 113 −42.1 −80 −110.8 −92 8.2

97.5

17.9 −31 97 28.3 40.4 59.5 22.5 −42 −90.6 −10 34.6

230 186.6 56.6 18.5 4 49.3

2-70

PHYSICAL AND CHEMICAL DATA

TABLE 2-8

Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name

Formula

n-Hexadecylamine (cetylamine) Hexaethylbenzene n-Hexane 1-Hexanol 2-Hexanol 3-Hexanol 1-Hexene n-Hexyl levulinate n-Hexyl phenyl ketone (enanthophenone) Hydrocinnamic acid Hydrogen cyanide (hydrocyanic acid) Hydroquinone 4-Hydroxybenzaldehyde α-Hydroxyisobutyric acid α-Hydroxybutyronitrile 4-Hydroxy-3-methyl-2-butanone 4-Hydroxy-4-methyl-2-pentanone 3-Hydroxypropionitrile Indene Iodobenzene Iodononane 2-Iodotoluene α-Ionone Isoprene Lauraldehyde Lauric acid Levulinaldehyde Levulinic acid d-Limonene Linalyl acetate Maleic anhydride Menthane 1-Menthol Menthyl acetate benzoate formate Mesityl oxide Methacrylic acid Methacrylonitrile Methane Methanethiol Methoxyacetic acid N-Methylacetanilide Methyl acetate acetylene (propyne) acrylate alcohol (methanol) Methylamine N-Methylaniline Methyl anthranilate benzoate 2-Methylbenzothiazole α-Methylbenzyl alcohol Methyl bromide 2-Methyl-1-butene 2-Methyl-2-butene Methyl isobutyl carbinol (2-methyl4-pentanol) n-butyl ketone (2-hexanone) isobutyl ketone (4-methyl-2-pentanone) n-butyrate isobutyrate caprate caproate caprylate chloride chloroacetate cinnamate α-Methylcinnamic acid Methylcyclohexane Methylcyclopentane Methylcyclopropane Methyl n-decyl ketone (n-dodecan-2-one) dichloroacetate N-Methyldiphenylamine

C16H35N C18H30 C6H14 C6H14O C6H14O C6H14O C6H12 C11H20O3 C13H18O C9H10O2 CHN C6H6O2 C7H6O2 C4H8O3 C5H9NO C5H10O2 C6H12O2 C3H5NO C9H8 C6H5I C9H19I C7H7I C13H20O C5H8 C12H24O C12H24O2 C5H8O2 C5H8O3 C10H16 C12H20O2 C4H2O3 C10H20 C10H20O C12H22O2 C17H24O2 C11H20O2 C6H10O C4H6O2 C4H5N CH4 CH4S C3H6O3 C9H11NO C3H6O2 C3H4 C4H6O2 CH4O CH5N C7H9N C8H9NO2 C8H8O2 C8H7NS C8H10O CH3Br C5H10 C5H10 C6H14O C6H12O C6H12O C5H10O2 C5H10O2 C11H22O2 C7H14O2 C9H18O2 CH3Cl C3H5ClO2 C10H10O2 C10H10O2 C7H14 C6H12 C4H8 C12H24O C3H4Cl2O2 C13H13N

1

5

10

20

40

60

100

123.6

157.8 134.3 −34.5 47.2 34.8 25.7 −38.0 120.0 130.3 133.5 −55.3 153.3 153.2 98.5 65.8 69.3 46.7 87.8 44.3 50.6 96.2 65.9 108.8 −62.3 108.4 150.6 54.9 128.1 40.4 82.5 63.4 35.7 83.2 85.8 154.2 75.8 +14.1 48.5 −23.3 −199.0 −75.3 79.3 103.8 −38.6 −97.5 −23.6 −25.3 −81.3 62.8 109.0 64.4 97.5 75.2 −80.6 −72.8 −57.0

176.0 150.3 −25.0 58.2 45.0 36.7 −28.1 134.7 145.5 148.7 −47.7 163.5 169.7 110.5 77.8 81.0 58.8 102.0 58.5 64.0 109.0 79.8 123.0 −53.3 123.7 166.0 68.0 141.8 53.8 96.0 78.7 48.3 96.0 100.0 170.0 90.0 26.0 60.0 −12.5 −195.5 −67.5 92.0 118.6 −29.3 −90.5 −13.5 −16.2 −73.8 76.2 124.2 77.3 111.2 88.0 −72.8 −64.3 −47.9

195.7 168.0 −14.1 70.3 55.9 49.0 −17.2 150.2 161.0 165.0 −39.7 174.6 186.8 123.8 90.7 94.0 72.0 117.9 73.9 78.3 123.0 95.6 139.0 −43.5 140.2 183.6 82.7 154.1 68.2 111.4 95.0 62.7 110.3 115.4 186.3 105.8 37.9 72.7 −0.6 −191.8 −58.8 106.5 135.1 −19.1 −82.9 −2.7 −6.0 −65.9 90.5 141.5 91.8 125.5 102.1 −64.0 −54.8 −37.9

215.7 187.7 −2.3 83.7 67.9 62.2 −5.0 167.8 178.9 183.3 −30.9 192.0 206.0 138.0 104.8 108.2 86.7 134.1 90.7 94.4 138.1 112.4 155.6 −32.6 157.8 201.4 98.3 169.5 84.3 127.7 111.8 78.3 126.1 132.1 204.3 123.0 51.7 86.4 +12.8 −187.7 −49.2 122.0 152.2 −7.9 −74.3 +9.2 +5.0 −56.9 106.0 159.7 107.8 141.2 117.8 −54.2 −44.1 −26.7

228.8 199.7 +5.4 92.0 76.0 70.7 +2.8 179.0 189.8 194.0 −25.1 203.0 217.5 146.4 113.9 117.4 96.0 144.7 100.8 105.0 147.7 123.8 166.3 −25.4 168.7 212.7 108.4 178.0 94.6 138.1 122.0 88.6 136.1 143.2 215.8 133.8 60.4 95.3 21.5 −185.1 −43.1 131.8 164.2 −0.5 −68.8 17.3 12.1 −51.3 115.8 172.0 117.4 150.4 127.4 −48.0 −37.3 −19.4

245.8 216.0 15.8 102.8 87.3 81.8 13.0 193.6 204.2 209.0 −17.8 216.5 233.5 157.7 125.0 129.0 108.2 157.7 114.7 118.3 159.8 138.1 181.2 −16.0 184.5 227.5 121.8 190.2 108.3 151.8 135.8 102.1 149.4 156.7 230.4 148.0 72.1 106.6 32.8 −181.4 −34.8 144.5 179.8 +9.4 −61.3 28.0 21.2 −43.7 129.8 187.8 130.8 163.9 140.3 −39.4 −28.0 −9.9

+22.1 28.8 +19.7 −5.5 −13.0 93.5 30.0 61.7 −99.5 19.0 108.1 155.0 −14.0 −33.8 −80.6 106.0 26.7 134.0

33.3 38.8 30.0 +5.0 −2.9 108.0 42.0 74.9 −92.4 30.0 123.0 169.8 −3.2 −23.7 −72.8 120.4 38.1 149.7

45.4 50.0 40.8 16.7 +8.4 123.0 55.4 89.0 −84.8 41.5 140.0 185.2 +8.7 −12.8 −64.0 136.0 50.7 165.8

58.2 62.0 52.8 29.6 21.0 139.0 70.0 105.3 −76.0 54.5 157.9 201.8 22.0 −0.6 −54.2 152.4 64.7 184.0

67.0 69.8 60.4 37.4 28.9 148.6 79.7 115.3 −70.4 63.0 170.0 212.0 30.5 +7.2 −48.0 163.8 73.6 195.4

78.0 79.8 70.4 48.0 39.6 161.5 91.4 128.0 −63.0 73.5 185.8 224.8 42.1 17.9 −39.3 177.5 85.4 210.1

200

400

760

Temperature, °C

−53.9 24.4 14.6 +2.5 −57.5 90.0 100.0 102.2 −71.0 132.4 121.2 73.5 41.0 44.6 22.0 58.7 16.4 24.1 70.0 37.2 79.5 −79.8 77.7 121.0 28.1 102.0 14.0 55.4 44.0 +9.7 56.0 57.4 123.2 47.3 −8.7 25.5 −44.5 −205.9 −90.7 52.5 −57.2 −111.0 −43.7 −44.0 −95.8 36.0 77.6 39.0 70.0 49.0 −96.3 −89.1 −75.4 −0.3 +7.7 −1.4 −26.8 −34.1 63.7 +5.0 34.2 −2.9 77.4 125.7 −35.9 −53.7 −96.0 77.1 3.2 103.5

272.2 300.4 330.0 241.7 268.5 298.3 31.6 49.6 68.7 119.6 138.0 157.0 103.7 121.8 139.9 98.3 117.0 135.5 29.0 46.8 66.0 215.7 241.0 266.8 225.0 248.3 271.3 230.8 255.0 279.8 −5.3 +10.2 25.9 238.0 262.5 286.2 256.8 282.6 310.0 175.2 193.8 212.0 142.0 159.8 178.8 146.5 165.5 185.0 126.8 147.5 167.9 178.0 200.0 221.0 135.6 157.8 181.6 139.8 163.9 188.6 179.0 199.3 219.5 160.0 185.7 211.0 202.5 225.2 250.0 −1.2 +15.4 32.6 207.8 231.8 257.0 249.8 273.8 299.2 142.0 164.0 187.0 208.3 227.4 245.8 128.5 151.4 175.0 173.3 196.2 220.0 155.9 179.5 202.0 122.7 146.0 169.5 168.3 190.2 212.0 178.8 202.8 227.0 253.2 277.1 301.0 169.8 194.2 219.0 90.0 109.8 130.0 123.9 142.5 161.0 50.0 70.3 90.3 −175.5 −168.8 −161.5 −22.1 −7.9 +6.8 163.5 184.2 204.0 202.3 227.4 253.0 24.0 40.0 57.8 −49.8 −37.2 −23.3 43.9 61.8 80.2 34.8 49.9 64.7 −32.4 −19.7 −6.3 149.3 172.0 195.5 212.4 238.5 266.5 151.4 174.7 199.5 183.2 204.5 225.5 159.0 180.7 204.0 −26.5 −11.9 +3.6 −13.8 +2.5 20.2 +4.9 21.6 38.5 94.9 94.3 85.6 64.3 55.7 181.6 109.8 148.1 −51.2 90.5 209.6 245.0 59.6 34.0 −26.0 199.0 103.2 232.8

113.5 111.0 102.0 83.1 73.6 202.9 129.8 170.0 −38.0 109.5 235.0 266.8 79.6 52.3 −11.3 222.5 122.6 257.0

131.7 127.5 119.0 102.3 92.6 224.0 150 193.0 −24.0 130.3 263.0 288.0 100.9 71.8 +4.5 246.5 143.0 282.0

Melting point, °C 130 −95.3 −51.6 −98.5 48.5 −13.2 170.3 115.5 79 −47 −2 −28.5

−146.7 44.5 48 33.5 −96.9 58 42.5 54.5 −59 15 −182.5 −121 102 −98.7 −102.7 −97.8 −93.5 −57 24 −12.5 15.4 −93 −135 −133 −56.9 −84.7 −84.7 −18 −40 −97.7 −31.9 33.4 −126.4 −142.4

−7.6

VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-8

2-71

Vapor Pressures of Organic Compounds, up to 1 atm* (Continued ) Pressure, mm Hg Compound Name

Formula

Methyl n-dodecyl ketone (2-tetradecanone) Methylene bromide (dibromomethane) chloride (dichloromethane) Methyl ethyl ketone (2-butanone) 2-Methyl-3-ethylpentane 3-Methyl-3-ethylpentane Methyl fluoride formate α-Methylglutaric anhydride Methyl glycolate 2-Methylheptadecane 2-Methylheptane 3-Methylheptane 4-Methylheptane 2-Methyl-2-heptene 6-Methyl-3-hepten-2-ol 6-Methyl-5-hepten-2-ol 2-Methylhexane 3-Methylhexane Methyl iodide laurate levulinate methacrylate myristate α-naphthyl ketone (1-acetonaphthone) β-naphthyl ketone (2-acetonaphthone) n-nonyl ketone (undecan-2-one) palmitate n-pentadecyl ketone (2-heptdecanone) 2-Methylpentane 3-Methylpentane 2-Methyl-1-pentanol 2-Methyl-2-pentanol Methyl n-pentyl ketone (2-heptanone) phenyl ether (anisole) 2-Methylpropene Methyl propionate 4-Methylpropiophenone 2-Methylpropionyl bromide Methyl propyl ether n-propyl ketone (2-pentanone) isopropyl ketone (3-Methyl-2-butanone) 2-Methylquinoline Methyl salicylate α-Methyl styrene 4-Methyl styrene Methyl n-tetradecyl ketone (2-hexadecanone) thiocyanate isothiocyanate undecyl ketone (2-tridecanone) isovalerate Monovinylacetylene (butenyne) Myrcene Myristaldehyde Myristic acid (tetradecanoic acid) Naphthalene 1-Naphthoic acid 2-Naphthoic acid 1-Naphthol 2-Naphthol 1-Naphthylamine 2-Naphthylamine Nicotine 2-Nitroaniline 3-Nitroaniline 4-Nitroaniline 2-Nitrobenzaldehyde 3-Nitrobenzaldehyde Nitrobenzene Nitroethane Nitroglycerin Nitromethane 2-Nitrophenol 2-Nitrophenyl acetate

C14H28O CH2Br2 CH2Cl2 C4H8O C8H18 C8H18 CH3F C2H4O2 C6H8O3 C3H6O3 C18H38 C8H18 C8H18 C8H18 C8H16 C8H16O C8H16O C7H16 C7H16 CH3I C13H26O2 C6H10O3 C5H8O2 C15H30O2 C12H10O C12H10O C11H22O C17H34O2 C17H34O C6H14 C6H14 C6H14O C6H14O C7H14O C7H8O C4H8 C4H8O2 C10H12O C4H7BrO C4H10O C5H10O C5H10O C10H9N C8H8O3 C9H10 C9H10 C16H32O C2H3NS C2H3NS C13H26O C6H12O2 C4H4 C10H16 C14H28O C14H28O2 C10H8 C11H8O2 C11H8O2 C10H8O C10H8O C10H9N C10H9N C10H14N2 C6H6N2O2 C6H6N2O2 C6H6N2O2 C7H5NO3 C7H5NO3 C6H5NO2 C2H5NO2 C3H5N3O9 CH3NO2 C6H5NO3 C8H7NO4

1

5

10

20

40

99.3 −35.1 −70.0 −48.3 −24.0 −23.9 −147.3 −74.2 93.8 +9.6 119.8 −21.0 −19.8 −20.4 −16.1 41.6 41.9 −40.4 −39.0

130.0 −13.2 −52.1 −28.0 −1.8 −1.4 −137.0 −57.0 125.4 33.7 152.0 +1.3 +2.6 +1.5 +6.7 65.0 66.0 −19.5 −18.1 −55.0 117.9 66.4 −10.0 145.7 146.3 152.3 95.5 166.8 161.6 −41.7 −39.8 38.0 +16.8 43.6 30.0 −96.5 −21.5 89.3 38.4 −54.3 +8.0 −1.0 104.0 81.6 34.0 42.0

145.5 −2.4 −43.3 −17.7 +9.5 +9.9 −131.6 −48.6 141.8 45.3 168.7 12.3 13.3 12.4 17.8 76.7 77.8 −9.1 −7.8 −45.8 133.2 79.7 +1.0 160.8 161.5 168.5 108.9 184.3 178.0 −32.1 −30.1 49.6 27.6 55.5 42.2 −81.9 −11.8 103.8 50.6 −45.4 17.9 +8.3 119.0 95.3 47.1 55.1

161.3 +9.7 −33.4 −6.5 21.7 22.3 −125.9 −39.2 157.7 58.1 186.0 24.4 25.4 24.5 30.4 89.3 90.4 +2.3 +3.6 −35.6 149.0 93.7 11.0 177.8 178.4 185.7 123.1 202.0 196.4 −21.4 −19.4 61.6 38.8 67.7 55.8 −73.4 −1.0 120.2 64.1 −35.4 28.5 18.3 134.0 110.0 61.8 69.2

179.8 23.3 −22.3 +6.0 35.2 36.2 −119.1 −28.7 177.5 72.3 204.8 37.9 38.9 38.0 44.0 102.7 104.0 14.9 16.4 −24.2 166.0 109.5 25.5 195.8 196.8 203.8 139.0

151.5 +9.8 −8.3 117.0 +2.9 −77.7 40.0 132.0 174.1 74.2 184.0 189.7 125.5 128.6 137.7 141.6 91.8 135.7 151.5 177.6 117.7 127.4 71.6 +1.5 167 −7.9 76.8 128.0

167.3 21.6 +5.4 131.8 14.0 −70.0 53.2 148.3 190.8 85.8 196.8 202.8 142.0 145.5 153.8 157.6 107.2 150.4 167.8 194.4 133.4 142.8 84.9 12.5 188 +2.8 90.4 142.0

184.6 34.5 20.4 147.8 26.4 −61.3 67.0 166.2 207.6 101.7 211.2 216.9 158.0 161.8 171.6 175.8 123.7 167.7 185.5 213.2 150.0 159.0 99.3 24.8 210 14.1 105.8 155.8

60

100

200

400

760

191.4 31.6 −15.7 14.0 43.9 45.0 −115.0 −21.9 189.9 81.8 216.3 46.6 47.6 46.6 52.8 111.5 112.8 23.0 24.5 −16.9 176.8 119.3 34.5 207.5 208.6 214.7 148.6

206.0 42.3 −6.3 25.0 55.7 57.1 −109.0 −12.9 205.0 93.7 231.5 58.3 59.4 58.3 64.6 122.6 123.8 34.1 35.6 −7.0 190.8 133.0 47.0 222.6 223.8 229.8 161.0

228.2 58.5 +8.0 41.6 73.6 75.3 −99.9 +0.8 229.1 111.8 254.5 76.0 77.1 76.1 82.3 139.5 140.0 50.8 52.4 +8.0

253.3 79.0 24.1 60.0 94.0 96.2 −89.5 16.0 255.5 131.7 279.8 96.2 97.4 96.3 102.2 156.6 156.6 69.8 71.6 25.3

278.0 98.6 40.7 79.6 115.6 118.3 −78.2 32.0 282.5 151.5 306.5 117.6 118.9 117.7 122.5 175.5 174.3 90.0 91.9 42.4

153.4 63.0 245.3 246.7 251.6 181.2

175.8 82.0 269.8 270.5 275.8 202.3

197.7 101.0 295.8 295.5 301.0 224.0

214.3 −9.7 −7.3 74.7 51.3 81.2 70.7 −63.8 +11.0 138.0 79.4 −24.3 39.8 29.6 150.8 126.2 77.8 85.0

226.7 −1.9 +0.1 83.4 58.8 89.8 80.1 −57.7 18.7 149.3 88.8 −17.4 47.3 36.2 161.7 136.7 88.3 95.0

242.0 +8.1 10.5 94.2 69.2 100.0 93.0 −49.3 29.0 164.2 101.6 −8.1 56.8 45.5 176.2 150.0 102.2 108.6

265.8 24.1 26.5 111.3 85.0 116.1 112.3 −36.7 44.2 187.4 120.5 +6.0 71.0 59.0 197.8 172.6 121.8 128.7

291.7 41.6 44.2 129.8 102.6 133.2 133.8 −22.2 61.8 212.7 141.7 22.5 86.8 73.8 211.7 197.5 143.0 151.2

319.5 60.3 63.3 147.9 121.2 150.2 155.5 −6.9 79.8 238.5 163.0 39.1 103.3 88.9 246.5 223.2 165.4 175.0

203.7 49.0 38.2 165.7 39.8 −51.7 82.6 186.0 223.5 119.3 225.0 231.5 177.8 181.7 191.5 195.7 142.1 186.0 204.2 234.2 168.8 177.7 115.4 38.0 235 27.5 122.1 172.8

215.0 58.1 47.5 176.6 48.2 −45.3 92.6 198.3 237.2 130.2 234.5 241.3 190.0 193.7 203.8 208.1 154.7 197.8 216.5 245.9 180.7 189.5 125.8 46.5 251 35.5 132.6 181.7

230.5 70.4 59.3 191.5 59.8 −37.1 106.0 214.5 250.5 145.5 245.8 252.7 206.0 209.8 220.0 224.3 169.5 213.0 232.1 261.8 196.2 204.3 139.9 57.8

254.4 89.8 77.5 214.0 77.3 −24.1 126.0 240.4 272.3 167.7 263.5 270.3 229.6 234.0 244.9 249.7 193.8 236.3 255.3 284.5 220.0 227.4 161.2 74.8

279.8 110.8 97.8 238.3 96.7 −10.1 148.3 267.9 294.6 193.2 281.4 289.5 255.8 260.6 272.2 277.4 219.8 260.0 280.2 310.2 246.8 252.1 185.8 94.0

307.0 132.9 119.0 262.5 116.7 +5.3 171.5 297.8 318.0 217.9 300.0 308.5 282.5 288.0 300.8 306.1 247.3 284.5 305.7 336.0 273.5 278.3 210.6 114.0

46.6 146.4 194.1

63.5 167.6 213.0

82.0 191.0 233.5

101.2 214.5 253.0

Temperature, °C

87.8 39.8 −30.5 115.0 115.6 120.2 68.2 134.3 129.6 −60.9 −59.0 15.4 −4.5 19.3 +5.4 −105.1 −42.0 59.6 13.5 −72.2 −12.0 −19.9 75.3 54.0 7.4 16.0 109.8 −14.0 −34.7 86.8 −19.2 −93.2 14.5 99.0 142.0 52.6 156.0 160.8 94.0 104.3 108.0 61.8 104.0 119.3 142.4 85.8 96.2 44.4 −21.0 127 −29.0 49.3 100.0

Melting point, °C −52.8 −96.7 −85.9 −114.5 −90 −99.8

−109.5 −120.8 −121.1

−118.2 −64.4 5 18.5 55.5 15 30 −154 −118 −103 −37.3 −140.3 −87.5

−77.8 −92 −1 −8.3 −23.2

−51 35.5 28.5

23.5 57.5 80.2 160.5 184 96 122.5 50 111.5 71.5 114 146.5 40.9 58 +5.7 −90 11 −29 45

2-72

PHYSICAL AND CHEMICAL DATA

TABLE 2-8

Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name

1-Nitropropane 2-Nitropropane 2-Nitrotoluene 3-Nitrotoluene 4-Nitrotoluene 4-Nitro-1,3-xylene (4-nitro-m-xylene) Nonacosane Nonadecane n-Nonane 1-Nonanol 2-Nonanone Octacosane Octadecane n-Octane n-Octanol (1-octanol) 2-Octanone n-Octyl acrylate iodide (1-Iodooctane) Oleic acid Palmitaldehyde Palmitic acid Palmitonitrile Pelargonic acid Pentachlorobenzene Pentachloroethane Pentachloroethylbenzene Pentachlorophenol Pentacosane Pentadecane 1,3-Pentadiene 1,4-Pentadiene Pentaethylbenzene Pentaethylchlorobenzene n-Pentane iso-Pentane (2-methylbutane) neo-Pentane (2,2-dimethylpropane) 2,3,4-Pentanetriol 1-Pentene α-Phellandrene Phenanthrene Phenethyl alcohol (phenyl cellosolve) 2-Phenetidine Phenol 2-Phenoxyethanol 2-Phenoxyethyl acetate Phenyl acetate Phenylacetic acid Phenylacetonitrile Phenylacetyl chloride Phenyl benzoate 4-Phenyl-3-buten-2-one Phenyl isocyanate isocyanide Phenylcyclohexane Phenyl dichlorophosphate m-Phenylene diamine (1,3-phenylenediamine) Phenylglyoxal Phenylhydrazine N-Phenyliminodiethanol 1-Phenyl-1,3-pentanedione 2-Phenylphenol 4-Phenylphenol 3-Phenyl-1-propanol Phenyl isothiocyanate Phorone iso-Phorone Phosgene (carbonyl chloride) Phthalic anhydride Phthalide Phthaloyl chloride 2-Picoline Pimelic acid α-Pinene β-Pinene

1

5

10

20

40

C3H7NO2 −9.6 C3H7NO2 −18.8 C7H7NO2 50.0 C7H7NO2 50.2 C7H7NO2 53.7 C8H9NO2 65.6 C29H60 234.2 C19H40 133.2 C9H20 +1.4 C9H20O 59.5 C9H18O 32.1 C28H58 226.5 C18H38 119.6 C8H18 −14.0 C8H18O 54.0 C8H16O 23.6 C11H20O2 58.5 C8H17I 45.8 C18H34O2 176.5 C16H32O 121.6 C16H32O2 153.6 C16H31N 134.3 C9H18O2 108.2 C6HCl5 98.6 C2HCl5 +1.0 C8H5Cl5 96.2 C6HCl5O C25H52 194.2 C15H32 91.6 C5H8 −71.8 C5H8 −83.5 C16H26 86.0 C16H25Cl 90.0 C5H12 −76.6 C5H12 −82.9 C5H12 −102.0 C5H12O3 155.0 C5H10 −80.4 C10H16 20.0 C14H10 118.2 C8H10O2 58.2 C8H11NO 67.0 C6H6O 40.1 C8H10O2 78.0 C10H12O3 82.6 C8H8O2 38.2 C8H8O2 97.0 C8H7N 60.0 C8H7ClO 48.0 C13H10O2 106.8 C10H10O 81.7 C7H5NO 10.6 C7H5N 12.0 C12H16 67.5 C6H5Cl2O2P 66.7

+13.5 +4.1 79.1 81.0 85.0 95.0 269.8 166.3 25.8 86.1 59.0 260.3 152.1 +8.3 76.5 48.4 87.7 74.8 208.5 154.6 188.1 168.3 126.0 129.7 27.2 130.0

25.3 15.8 93.8 96.0 100.5 109.8 286.4 183.5 38.0 99.7 72.3 277.4 169.6 19.2 88.3 60.9 102.0 90.0 223.0 171.8 205.8 185.8 137.4 144.3 39.8 148.0

230.0 121.0 −53.8 −66.2 120.0 123.8 −62.5 −65.8 −85.4 189.3 −63.3 45.7 154.3 85.9 94.7 62.5 106.6 113.5 64.8 127.0 89.0 75.3 141.5 112.2 36.0 37.0 96.5 95.9

248.2 135.4 −45.0 −57.1 135.8 140.7 −50.1 −57.0 −76.7 204.5 −54.5 58.0 173.0 100.0 108.6 73.8 121.2 128.0 78.0 141.3 103.5 89.0 157.8 127.4 48.5 49.7 111.3 110.0

37.9 28.2 109.6 112.8 117.7 125.8 303.6 200.8 51.2 113.8 87.2 295.4 187.5 31.5 101.0 74.3 117.8 105.9 240.0 190.0 223.8 204.2 149.8 160.0 53.9 166.0 192.2 266.1 150.2 −34.8 −47.7 152.4 158.1 −40.2 −47.3 −67.2 220.5 −46.0 72.1 193.7 114.8 123.7 86.0 136.0 144.5 92.3 156.0 119.4 103.6 177.0 143.8 62.5 63.4 126.4 125.9

51.8 41.8 126.3 130.7 136.0 143.3 323.2 220.0 66.0 129.0 103.4 314.2 207.4 45.1 115.2 89.8 135.6 123.8 257.2 210.0 244.4 223.8 163.7 178.5 69.9 186.2 211.2 285.6 167.7 −23.4 −37.0 171.9 178.2 −29.2 −36.5 −56.1 239.6 −34.1 87.8 215.8 130.5 139.9 100.1 152.2 162.3 108.1 173.6 136.3 119.8 197.6 161.3 77.7 78.3 144.0 143.4

C6H8N2 C8H6O2 C6H8N2 C10H15NO2 C11H12O2 C12H10O C12H10O C9H12O C7H5NS C9H14O C9H14O CCl2O C8H4O3 C8H6O2 C8H4Cl2O2 C6H7N C7H12O4 C10H16 C10H16

71.8 145.0 98.0 100.0

131.2 75.0 101.6 179.2 128.5 131.6

74.7 47.2 42.0 38.0 −92.9 96.5 95.5 86.3 −11.1 163.4 −1.0 +4.2

102.4 75.6 68.3 66.7 −77.0 121.3 127.7 118.3 +12.6 196.2 +24.6 30.0

147.0 87.8 115.8 195.8 144.0 146.2 176.2 116.0 89.8 81.5 81.2 −69.3 134.0 144.0 134.2 24.4 212.0 37.3 42.3

163.8 100.7 131.5 213.4 159.9 163.3 193.8 131.2 115.5 95.6 96.8 −60.3 151.7 161.3 151.0 37.4 229.3 51.4 58.1

182.5 115.5 148.2 233.0 178.0 180.3 213.0 147.4 122.5 111.3 114.5 −50.3 172.0 181.0 170.0 51.2 247.0 66.8 71.5

60

200

400

760

60.5 50.3 137.6 142.5 147.9 153.8 334.8 232.8 75.5 139.0 113.8 326.8 219.7 53.8 123.8 99.0 145.6 135.4 269.8 222.6 256.0 236.6 172.3 190.1 80.0 199.0 223.4 298.4 178.4 −16.5 −30.0 184.2 191.0 −22.2 −29.6 −49.0 249.8 −27.1 97.6 229.9 141.2 149.8 108.4 163.2 174.0 118.1 184.5 147.7 129.8 210.8 172.6 87.7 88.0 154.2 153.6

72.3 62.0 151.5 156.9 163.0 168.5 350.0 248.0 88.1 151.3 127.4 341.8 236.0 65.7 135.2 111.7 159.1 150.0 286.0 239.5 271.5 251.5 184.4 205.5 93.5 216.0 239.6 314.0 194.0 −6.7 −20.6 200.0 208.0 −12.6 −20.2 −39.1 263.5 −17.7 110.6 249.0 154.0 163.5 121.4 176.5 189.2 131.6 198.2 161.8 143.5 227.8 187.8 100.6 101.0 169.3 168.0

90.2 80.0 173.7 180.3 186.7 191.7 373.2 271.8 107.5 170.5 148.2 364.8 260.6 83.6 152.0 130.4 180.2 173.3 309.8 264.1 298.7 277.1 203.1 227.0 114.0 241.8 261.8 339.0 216.1 +8.0 −6.7 224.1 230.3 +1.9 −5.9 −23.7 284.5 −3.4 130.6 277.1 175.0 184.0 139.0 197.6 211.3 151.2 219.5 184.2 163.8 254.0 211.0 120.8 120.8 191.3 189.8

110.6 99.8 197.7 206.8 212.5 217.5 397.2 299.8 128.2 192.1 171.2 388.9 288.0 104.0 173.8 151.0 204.0 199.3 334.7 292.3 326.0 304.5 227.5 251.6 137.2 269.3 285.0 365.4 242.8 24.7 +8.3 250.2 257.2 18.5 +10.5 −7.1 307.0 +12.8 152.0 308.0 197.5 207.0 160.0 221.0 235.0 173.5 243.0 208.5 186.0 283.5 235.4 142.7 142.3 214.6 213.0

131.6 120.3 222.3 231.9 238.3 244.0 421.8 330.0 150.8 213.5 195.0 412.5 317.0 125.6 195.2 172.9 227.0 225.5 360.0 321.0 353.8 332.0 253.5 276.0 160.5 299.0 309.3 390.3 270.5 42.1 26.1 277.0 285.0 36.1 27.8 +9.5 327.2 30.1 175.0 340.2 219.5 228.0 181.9 245.3 259.7 195.9 265.5 233.5 210.0 314.0 261.0 165.6 165.0 240.0 239.5

−108 −93 −4.1 15.5 51.9 +2 63.8 32 −53.7 −5 −19 61.6 28 −56.8 −15.4 −16

194.0 124.2 158.7 245.3 189.8 192.2 225.3 156.8 133.3 121.4 125.6 −44.0 185.3 193.5 182.2 59.9 258.2 76.8 81.2

209.9 136.2 173.5 260.6 204.5 205.9 240.9 170.3 147.7 134.0 140.6 −35.6 202.3 210.0 197.8 71.4 272.0 90.1 94.0

233.0 153.8 195.4 284.5 226.7 227.9 263.2 191.2 169.6 153.5 163.3 −22.3 228.0 234.5 222.0 89.0 294.5 110.2 114.1

259.0 173.5 218.2 311.3 251.2 251.8 285.5 212.8 194.0 175.3 188.7 −7.6 256.8 261.8 248.3 108.4 318.5 132.3 136.1

285.5 193.5 243.5 337.8 276.5 275.0 308.0 235.0 218.5 197.2 215.2 +8.3 284.5 290.0 275.8 128.8 342.1 155.0 158.3

62.8 73 19.5

Temperature, °C

Formula

99.8

Melting point, °C

100

−45.9 14 34 64.0 31 12.5 85.5 −22 188.5 53.3 10

−129.7 −159.7 −16.6

99.5 40.6 11.6 −6.7 76.5 −23.8 70.5 41.5 +7.5

56.5 164.5 −21.0 28 −104 130.8 73 88.5 −70 103 −55

VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-8

2-73

Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name

Piperidine Piperonal Propane Propenylbenzene Propionamide Propionic acid anhydride Propionitrile Propiophenone n-Propyl acetate iso-Propyl acetate n-Propyl alcohol (1-propanol) iso-Propyl alcohol (2-propanol) n-Propylamine Propylbenzene Propyl benzoate n-Propyl bromide (1-bromopropane) iso-Propyl bromide (2-bromopropane) n-Propyl n-butyrate isobutyrate iso-Propyl isobutyrate Propyl carbamate n-Propyl chloride (1-chloropropane) iso-Propyl chloride (2-chloropropane) iso-Propyl chloroacetate Propyl chloroglyoxylate Propylene Propylene glycol (1,2-Propanediol) Propylene oxide n-Propyl formate iso-Propyl formate 4,4′-iso-Propylidenebisphenol n-Propyl iodide (1-iodopropane) iso-Propyl iodide (2-iodopropane) n-Propyl levulinate iso-Propyl levulinate Propyl mercaptan (1-propanethiol) 2-iso-Propylnaphthalene iso-Propyl β-naphthyl ketone (2-isobutyronaphthone) 2-iso-Propylphenol 3-iso-Propylphenol 4-iso-Propylphenol Propyl propionate 4-iso-Propylstyrene Propyl isovalerate Pulegone Pyridine Pyrocatechol Pyrocaltechol diacetate (1,2-phenylene diacetate) Pyrogallol Pyrotartaric anhydride Pyruvic acid Quinoline iso-Quinoline Resorcinol Safrole Salicylaldehyde Salicylic acid Sebacic acid Selenophene Skatole Stearaldehyde Stearic acid Stearyl alcohol (1-octadecanol) Styrene Styrene dibromide [(1,2-dibromoethyl) benzene] Suberic acid Succinic anhydride Succinimide Succinyl chloride α-Terpineol Terpenoline

1

5

10

20

87.0 −128.9 17.5 65.0 4.6 20.6 −35.0 50.0 −26.7 −38.3 −15.0 −26.1 −64.4 6.3 54.6 −53.0 −61.8 −1.6 −6.2 −16.3 52.4 −68.3 −78.8 +3.8 9.7 −131.9 45.5 −75.0 −43.0 −52.0 193.0 −36.0 −43.3 59.7 48.0 −56.0 76.0

−7.0 117.4 −115.4 43.8 91.0 28.0 45.3 −13.6 77.9 −5.4 −17.4 +5.0 −7.0 −46.3 31.3 83.8 −33.4 −42.5 +22.1 +16.8 +5.8 77.6 −50.0 −61.1 28.1 32.3 −120.7 70.8 −57.8 −22.7 −32.7 224.2 −13.5 −22.1 86.3 74.5 −36.3 107.9

+3.9 132.0 −108.5 57.0 105.0 39.7 57.7 −3.0 92.2 +5.0 −7.2 14.7 +2.4 −37.2 43.4 98.0 −23.3 −32.8 34.0 28.3 17.0 90.0 −41.0 −52.0 40.2 43.5 −112.1 83.2 −49.0 −12.6 −22.7 240.8 −2.4 −11.7 99.9 88.0 −26.3 123.4

15.8 148.0 −100.9 71.5 119.0 52.0 70.4 +8.8 107.6 16.0 +4.2 25.3 12.7 −27.1 56.8 114.3 −12.4 −22.0 47.0 40.6 29.0 103.2 −31.0 −42.0 53.9 55.6 −104.7 96.4 −39.3 −1.7 −12.1 255.5 +10.0 0.0 114.0 102.4 −15.4 140.3

29.2 165.7 −92.4 87.7 134.8 65.8 85.6 22.0 124.3 28.8 17.0 36.4 23.8 −16.0 71.6 131.8 −0.3 −10.1 61.5 54.3 42.4 117.7 −19.5 −31.0 68.7 68.8 −96.5 111.2 −28.4 +10.8 −0.2 273.0 23.6 +13.2 130.1 118.1 −3.2 159.0

60

100

200

400

760

37.7 177.0 −87.0 97.8 144.3 74.1 94.5 30.1 135.0 37.0 25.1 43.5 30.5 −9.0 81.1 143.3 +7.5 −2.5 70.3 63.0 51.4 126.5 −12.1 −23.5 78.0 77.2 −91.3 119.9 −21.3 18.8 +7.5 282.9 32.1 21.6 140.6 127.8 +4.6 171.4

49.0 191.7 −79.6 111.7 156.0 85.8 107.2 41.4 149.3 47.8 35.7 52.8 39.5 +0.5 94.0 157.4 18.0 +8.0 82.6 73.9 62.3 138.3 −2.5 −13.7 90.3 88.0 −84.1 132.0 −12.0 29.5 17.8 297.0 43.8 32.8 154.0 141.8 15.3 187.6

66.2 214.3 −68.4 132.0 174.2 102.5 127.8 58.2 170.2 64.0 51.7 66.8 53.0 15.0 113.5 180.1 34.0 23.8 101.0 91.8 80.2 155.8 +12.2 +1.3 108.8 104.7 −73.3 149.7 +2.1 45.3 33.6 317.5 61.8 50.0 175.6 161.6 31.5 211.8

85.7 238.5 −55.6 154.7 194.0 122.0 146.0 77.7 194.2 82.0 69.8 82.0 67.8 31.5 135.7 205.2 52.0 41.5 121.7 112.0 100.0 175.8 29.4 18.1 128.0 123.0 −60.9 168.1 17.8 62.6 50.5 339.0 81.8 69.5 198.0 185.2 49.2 238.5

106.0 263.0 −42.1 179.0 213.0 141.1 167.0 97.1 218.0 101.8 89.0 97.8 82.5 48.5 159.2 231.0 71.0 60.0 142.7 133.9 120.5 195.0 46.4 36.5 148.6 150.0 −47.7 188.2 34.5 81.3 68.3 360.5 102.5 89.5 221.2 208.2 67.4 266.0

Temperature, °C

Formula C5H11N C8H6O3 C3H8 C9H10 C3H7NO C3H6O2 C6H10O3 C3H5N C9H10O C5H10O2 C5H10O2 C3H8O C3H8O C3H9N C9H12 C10H12O2 C3H7Br C3H7Br C7H14O2 C7H14O2 C7H14O2 C4H9NO2 C3H7Cl C3H7Cl C5H9ClO2 C5H7ClO3 C3H6 C3H8O2 C3H6O C4H8O2 C4H8O2 C15H16O2 C3H7I C3H7I C8H14O3 C8H14O3 C3H8S C13H14

40

C14H14O C9H12O C9H12O C9H12O C6H12O2 C11H14 C8H16O2 C10H16O C5H5N C6H6O2

133.2 56.6 62.0 67.0 −14.2 34.7 +8.0 58.3 −18.9

165.4 83.8 90.3 94.7 +8.0 62.3 32.8 82.5 +2.5 104.0

181.0 97.0 104.1 108.0 19.4 76.0 45.1 94.0 13.2 118.3

197.7 111.7 119.8 123.4 31.6 91.2 58.0 106.8 24.8 134.0

215.6 127.5 136.2 139.8 45.0 108.0 72.8 121.7 38.0 150.6

227.0 137.7 146.6 149.7 53.8 118.4 82.3 130.2 46.8 161.7

242.3 150.3 160.2 163.3 65.2 132.8 95.0 143.1 57.8 176.0

264.0 170.1 182.0 184.0 82.7 153.9 113.9 162.5 75.0 197.7

288.2 192.6 205.0 206.1 102.0 178.0 135.0 189.8 95.6 221.5

313.0 214.5 228.0 228.2 122.4 202.5 155.9 221.0 115.4 245.5

C10H10O4 C6H6O3 C5H6O3 C3H4O3 C9H7N C9H7N C6H6O2 C10H10O2 C7H6O2 C7H6O3 C10H18O4 C4H4Se C9H9N C18H36O C18H36O2 C18H36O C8H8

98.0 69.7 21.4 59.7 63.5 108.4 63.8 33.0 113.7 183.0 −39.0 95.0 140.0 173.7 150.3 −7.0

129.8 151.7 99.7 45.8 89.6 92.7 138.0 93.0 60.1 136.0 215.7 −16.0 124.2 174.6 209.0 185.6 +18.0

145.7 167.7 114.2 57.9 103.8 107.8 152.1 107.6 73.8 146.2 232.0 −4.0 139.6 192.1 225.0 202.0 30.8

161.8 185.3 130.0 70.8 119.8 123.7 168.0 123.0 88.7 156.8 250.0 +9.1 154.3 210.6 243.4 220.0 44.6

179.8 204.2 147.8 85.3 136.7 141.6 185.3 140.1 105.2 172.2 268.2 24.1 171.9 230.8 263.3 240.4 59.8

191.6 216.3 158.6 94.1 148.1 152.0 195.8 150.3 115.7 182.0 279.8 33.8 183.6 244.2 275.5 252.7 69.5

206.5 232.0 173.8 106.5 163.2 167.6 209.8 165.1 129.4 193.4 294.5 47.0 197.4 260.0 291.0 269.4 82.0

228.7 255.3 196.1 124.7 186.2 190.0 230.8 186.2 150.0 210.0 313.2 66.7 218.8 285.0 316.5 293.5 101.3

253.3 281.5 221.0 144.7 212.3 214.5 253.4 210.0 173.7 230.5 332.8 89.8 242.5 313.8 343.0 320.3 122.5

278.0 309.0 247.4 165.0 237.7 240.5 276.5 233.0 196.5 256.0 352.3 114.3 266.2 342.5 370.0 349.5 145.2

C8H8Br2 C8H14O4 C4H4O3 C4H5NO2 C4H4Cl2O2 C10H18O C10H16

86.0 172.8 92.0 115.0 39.0 52.8 32.3

115.6 205.5 115.0 143.2 65.0 80.4 58.0

129.8 219.5 128.2 157.0 78.0 94.3 70.6

145.2 238.2 145.3 174.0 91.8 109.8 84.8

161.8 254.6 163.0 192.0 107.5 126.0 100.0

172.2 265.4 174.0 203.0 117.2 136.3 109.8

186.3 279.8 189.0 217.4 130.0 150.1 122.7

207.8 300.5 212.0 240.0 149.3 171.2 142.0

230.0 322.8 237.0 263.5 170.0 194.3 163.5

254.0 345.5 261.0 287.5 192.5 217.5 185.0

Melting point, °C −9 37 −187.1 −30.1 79 −22 −45 −91.9 21 −92.5 −127 −85.8 −83 −99.5 −51.6 −109.9 −89.0 −95.2

−122.8 −117 −185 −112.1 −92.9 −98.8 −90 −112

15.5 26 61 −76

−42 105 133 13.6 −15 24.6 110.7 11.2 −7 159 134.5 95 63.5 69.3 58.5 −30.6 142 119.6 125.5 17 35

2-74

PHYSICAL AND CHEMICAL DATA

TABLE 2-8

Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name

Formula

1,1,1,2-Tetrabromoethane 1,1,2,2-Tetrabromoethane Tetraisobutylene Tetracosane 1,2,3,4-Tetrachlorobenzene 1,2,3,5-Tetrachlorobenzene 1,2,4,5-Tetrachlorobenzene 1,1,2,2-Tetrachloro-1,2-difluoroethane 1,1,1,2-Tetrachloroethane 1,1,2,2-Tetrachloroethane 1,2,3,5-Tetrachloro-4-ethylbenzene Tetrachloroethylene 2,3,4,6-Tetrachlorophenol 3,4,5,6-Tetrachloro-1,2-xylene Tetradecane Tetradecylamine Tetradecyltrimethylsilane Tetraethoxysilane 1,2,3,4-Tetraethylbenzene Tetraethylene glycol Tetraethylene glycol chlorohydrin Tetraethyllead Tetraethylsilane Tetralin 1,2,3,4-Tetramethylbenzene 1,2,3,5-Tetramethylbenzene 1,2,4,5-Tetramethylbenzene 2,2,3,3-Tetramethylbutane Tetramethylene dibromide (1,4-dibromobutane) Tetramethyllead Tetramethyltin Tetrapropylene glycol monoisopropyl ether Thioacetic acid (mercaptoacetic acid) Thiodiglycol (2,2′-thiodiethanol) Thiophene Thiophenol (benzenethiol) α-Thujone Thymol Tiglaldehyde Tiglic acid Tiglonitrile Toluene Toluene-2,4-diamine 2-Toluic nitrile (2-tolunitrile) 4-Toluic nitrile (4-tolunitrile) 2-Toluidine 3-Toluidine 4-Toluidine 2-Tolyl isocyanide 4-Tolylhydrazine Tribromoacetaldehyde 1,1,2-Tribromobutane 1,2,2-Tribromobutane 2,2,3-Tribromobutane 1,1,2-Tribromoethane 1,2,3-Tribromopropane Triisobutylamine Triisobutylene 2,4,6-Tritertbutylphenol Trichloroacetic acid Trichloroacetic anhydride Trichloroacetyl bromide 2,4,6-Trichloroaniline 1,2,3-Trichlorobenzene 1,2,4-Trichlorobenzene 1,3,5-Trichlorobenzene 1,2,3-Trichlorobutane 1,1,1-Trichloroethane 1,1,2-Trichloroethane Trichloroethylene Trichlorofluoromethane 2,4,5-Trichlorophenol 2,4,6-Trichlorophenol

C2H2Br4 C2H2Br4 C16H32 C24H50 C6H2Cl4 C6H2Cl4 C6H2Cl4 C2Cl4F2 C2H2Cl4 C2H2Cl4 C8H6Cl4 C2Cl4 C6H2Cl4O C8H6Cl4 C14H30 C14H31N C17H38Si C8H20O4Si C14H22 C8H18O5 C8H17ClO4 C8H20Pb C8H20Si C10H12 C10H14 C10H14 C10H14 C8H18 C4H8Br2 C4H12Pb C4H12Sn C15H32O5 C2H4O2S C4H10O2S C4H4S C6H6S C10H16O C10H14O C5H8O C5H8O2 C5H7N C7H8 C7H10N2 C8H7N C8H7N C7H9N C7H9N C7H9N C8H7N C7H10N2 C2HBr3O C4H7Br3 C4H7Br3 C4H7Br3 C2H3Br3 C3H5Br3 C12H27N C12H24 C18H30O C2HCl3O2 C4Cl6O3 C2BrCl3O C6H4Cl3N C6H3Cl3 C6H3Cl3 C6H3Cl3 C4H7Cl3 C2H3Cl3 C2H3Cl3 C2HCl3 CCl3F C6H3Cl3O C6H3Cl3O

1

5

10

20

58.0 65.0 63.8 183.8 68.5 58.2

83.3 95.5 93.7 219.6 99.6 89.0

95.7 110.0 108.5 237.6 114.7 104.1

108.5 126.0 124.5 255.3 131.2 121.6

−37.5 −16.3 −3.8 77.0 −20.6 100.0 94.4 76.4 102.6 120.0 16.0 65.7 153.9 110.1 38.4 −1.0 38.0 42.6 40.6 45.0 −17.4

−16.0 +7.4 +20.7 110.0 +2.4 130.3 125.0 106.0 135.8 150.7 40.3 96.2 183.7 141.8 63.6 +23.9 65.3 68.7 65.8 65.0 +3.2

−5.0 19.3 33.0 126.0 13.8 145.3 140.3 120.7 152.0 166.2 52.6 111.6 197.1 156.1 74.8 36.3 79.0 81.8 77.8 74.6 13.5

32.0 −29.0 −51.3 116.6 60.0 42.0 −40.7 18.6 38.3 64.3 −25.0 52.0 −25.5 −26.7 106.5 36.7 42.5 44.0 41.0 42.0 25.2 82.4 18.5 45.0 41.0 38.2 32.6 47.5 32.3 18.0 95.2 51.0 56.2 −7.4 134.0 40.0 38.4

58.8 −6.8 −31.0 147.8 87.7 96.0 −20.8 43.7 65.7 92.8 −1.6 77.8 −2.4 −4.4 137.2 64.0 71.3 69.3 68.0 68.2 51.0 110.0 45.0 73.5 69.0 66.0 58.0 75.8 57.4 44.0 126.1 76.0 85.3 +16.7 157.8 70.0 67.3 63.8 27.2 −32.0 −2.0 −22.8 −67.6 102.1 105.9

72.4 +4.4 −20.6 163.0 101.5 128.0 −10.9 56.0 79.3 107.4 +10.0 90.2 +9.2 +6.4 151.7 77.9 85.8 81.4 82.0 81.8 64.0 123.8 58.0 87.8 83.2 79.8 70.6 90.0 69.8 56.5 142.0 88.2 99.6 29.3 170.0 85.6 81.7 78.0 40.0 −21.9 +8.3 −12.4 −59.0 117.3 120.2

40

60

100

200

400

760

Temperature, °C

+0.5 −52.0 −24.0 −43.8 −84.3 72.0 76.5

Melting point, °C

+6.7 32.1 46.2 143.7 26.3 161.0 156.0 135.6 170.0 183.5 65.8 127.7 212.3 172.6 88.0 50.0 93.8 95.8 91.0 88.0 24.6

123.2 144.0 142.2 276.3 149.2 140.0 146.0 19.8 46.7 60.8 162.1 40.1 179.1 174.2 152.7 189.0 201.5 81.1 145.8 228.0 190.0 102.4 65.3 110.4 111.5 105.8 104.2 36.8

132.0 155.1 152.6 288.4 160.0 152.0 157.7 28.1 56.0 70.0 175.0 49.2 190.0 185.8 164.0 200.2 213.3 90.7 156.7 237.8 200.5 111.7 74.8 121.3 121.8 115.4 114.8 44.5

144.0 170.0 167.5 305.2 175.7 168.0 173.5 38.6 68.0 83.2 191.6 61.3 205.2 200.5 178.5 215.7 227.8 103.6 172.4 250.0 214.7 123.8 88.0 135.3 135.7 128.3 128.1 54.8

161.5 192.5 190.0 330.5 198.0 193.7 196.0 55.0 87.2 102.2 215.3 79.8 227.2 223.0 201.8 239.8 250.0 123.5 196.0 268.4 236.5 142.0 108.0 157.2 155.7 149.9 149.5 70.2

181.0 217.5 214.6 358.0 225.5 220.0 220.5 73.1 108.2 124.0 243.0 100.0 250.4 248.3 226.8 264.6 275.0 146.2 221.4 288.0 258.2 161.8 130.2 181.8 180.0 173.7 172.1 87.4

200.0 243.5 240.0 386.4 254.0 246.0 245.0 92.0 130.5 145.9 270.0 120.8 275.0 273.5 252.5 291.2 300.0 168.5 248.0 307.8 281.5 183.0 153.0 207.2 204.4 197.9 195.9 106.3

87.6 16.6 −9.3 179.8 115.8 165.0 0.0 69.7 93.7 122.6 23.2 103.8 22.1 18.4 167.9 93.0 101.7 95.1 96.7 95.8 78.2 138.6 72.1 103.2 98.6 94.6 84.2 105.8 83.0 70.0 158.0 101.8 114.3 42.1 182.6 101.8 97.2 93.7 55.0 −10.8 21.6 −1.0 −49.7 134.0 135.8

104.0 30.3 +3.5 197.7 131.8 210.0 +12.5 84.2 110.0 139.8 37.0 119.0 36.7 31.8 185.7 110.0 109.5 110.0 113.5 111.5 94.0 154.1 87.8 120.2 116.0 111.8 100.0 122.8 97.8 86.7 177.4 116.3 131.2 57.2 195.8 119.8 114.8 110.8 71.5 +1.6 35.2 +11.9 −39.0 151.5 152.2

115.1 39.2 11.7 209.0 142.0 240.5 20.1 93.9 120.2 149.8 45.8 127.8 46.0 40.3 196.2 120.8 130.0 119.8 123.8 121.5 104.0 165.0 97.5 131.6 127.0 122.2 110.0 134.0 107.3 96.7 188.0 125.9 141.8 66.7 204.5 131.5 125.7 121.8 82.0 9.5 44.0 20.0 −32.3 162.5 163.5

128.7 50.8 22.8 223.3 154.0 285 30.5 106.6 134.0 164.1 57.7 140.5 58.2 51.9 211.5 135.0 145.2 133.0 136.7 133.7 117.7 178.0 110.2 146.0 141.8 136.3 123.5 148.0 119.7 110.0 203.0 137.8 155.2 79.5 214.6 146.0 140.0 136.0 96.2 20.0 55.7 31.4 −23.0 178.0 177.8

149.8 68.8 39.8 245.0

173.8 89.0 58.5 268.3

197.5 110.0 78.0 292.7

−20 −27.5

46.5 125.8 154.2 185.5 75.4 158.0 77.8 69.5 232.8 156.0 167.3 153.0 157.6 154.0 137.8 198.0 130.0 167.8 163.5 157.8 143.5 170.0 138.0 130.2 226.2 155.4 176.2 98.4 229.8 168.2 162.0 157.7 118.0 36.2 73.3 48.0 −9.1 201.5 199.0

64.7 146.7 177.8 209.2 95.5 179.2 99.7 89.5 256.0 180.0 193.0 176.2 180.6 176.9 159.9 219.5 151.6 192.0 188.0 182.2 165.4 195.0 157.8 153.0 250.6 175.2 199.8 120.2 246.4 193.5 187.7 183.0 143.0 54.6 93.0 67.0 +6.8 226.5 222.5

84.4 168.0 201.0 231.8 116.4 198.5 122.0 110.6 280.0 205.2 217.6 199.7 203.3 200.4 183.5 242.0 174.0 216.2 213.8 206.5 188.4 220.0 179.0 179.0 276.3 195.6 223.0 143.0 262.0 218.5 213.0 208.4 169.0 74.1 113.9 86.7 23.7 251.8 246.0

−38.3

51.1 46.5 54.5 139 26.5 −68.7 −36 −19.0 69.5 5.5

11.6 −136 −31.0 −6.2 −24.0 79.5 −102.2

−16.5

51.5 64.5 −95.0 99 −13 29.5 −16.3 −31.5 44.5 65.5

−26 16.5 −22 57 78 52.5 17 63.5 −30.6 −36.7 −73 62 68.5

VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-8

2-75

Vapor Pressures of Organic Compounds, up to 1 atm (Concluded ) Pressure, mm Hg Compound Name

Tri-2-chlorophenylthiophosphate 1,1,1-Trichloropropane 1,2,3-Trichloropropane 1,1,2-Trichloro-1,2,2-trifluoroethane Tricosane Tridecane Tridecanoic acid Triethoxymethylsilane Triethoxyphenylsilane 1,2,4-Triethylbenzene 1,3,4-Triethylbenzene Triethylborine Triethyl camphoronate citrate Triethyleneglycol Triethylheptylsilane Triethyloctylsilane Triethyl orthoformate phosphate Triethylthallium Trifluorophenylsilane Trimethallyl phosphate 2,3,5-Trimethylacetophenone Trimethylamine 2,4,5-Trimethylaniline 1,2,3-Trimethylbenzene 1,2,4-Trimethylbenzene 1,3,5-Trimethylbenzene 2,2,3-Trimethylbutane Trimethyl citrate Trimethyleneglycol (1,3-propanediol) 1,2,4-Trimethyl-5-ethylbenzene 1,3,5-Trimethyl-2-ethylbenzene 2,2,3-Trimethylpentane 2,2,4-Trimethylpentane 2,3,3-Trimethylpentane 2,3,4-Trimethylpentane 2,2,4-Trimethyl-3-pentanone Trimethyl phosphate 2,4,5-Trimethylstyrene 2,4,6-Trimethylstyrene Trimethylsuccinic anhydride Triphenylmethane Triphenylphosphate Tripropyleneglycol Tripropyleneglycol monobutyl ether Tripropyleneglycol monoisopropyl ether Tritolyl phosphate Undecane Undecanoic acid 10-Undecenoic acid Undecan-2-ol n-Valeric acid iso-Valeric acid γ-Valerolactone Valeronitrile Vanillin Vinyl acetate 2-Vinylanisole 3-Vinylanisole 4-Vinylanisole Vinyl chloride (1-chloroethylene) cyanide (acrylonitrile) fluoride (1-fluoroethylene) Vinylidene chloride (1,1-dichloroethene) 4-Vinylphenetole 2-Xenyl dichlorophosphate 2,4-Xyaldehyde 2-Xylene (2-xylene) 3-Xylene (3-xylene) 4-Xylene (4-xylene) 2,4-Xylidine 2,6-Xylidine

1

5

10

20

217.2

231.2

246.7

261.7

−7.0 33.7 −49.4 206.3 98.3 166.3 +22.8 98.8 74.2 76.0

+4.2 46.0 −40.3 223.0 104.0 181.0 34.6 112.6 88.5 90.2 −148.0 166.0 144.0 158.1 114.6 120.6 40.5 82.1 51.7 +0.8 149.8 122.3 −73.8 109.0 55.9 50.7 47.4 −18.8 160.4 100.6 84.6 80.5 +3.9 −4.3 +6.9 +7.1 46.4 67.8 91.6 79.7 97.4 197.0 249.8 140.5 147.0 127.3 198.0 73.9 149.0 156.3 112.8 79.8 71.3 79.8 30.0 154.0 −18.0 81.0 83.0 85.7 −83.7 −20.3 −132.2 −51.2 105.6 187.0 99.0 32.1 28.3 27.3 93.0 87.0

16.2 59.3 −30.0 242.0 120.2 195.8 47.2 127.2 104.0 105.8 −140.6 183.6 171.1 174.0 130.3 137.7 53.4 97.8 67.7 12.3 169.8 137.5 −65.0 123.7 69.9 64.5 61.0 −7.5 177.2 115.5 99.7 96.0 16.0 +7.5 19.2 19.3 57.6 83.0 107.1 94.8 113.8 206.8 269.7 155.8 161.8 143.7 213.2 85.6 166.0 172.0 127.5 93.1 84.0 95.2 43.3 170.5 −7.0 94.7 97.2 100.0 −75.7 −9.0 −125.4 −41.7 120.3 205.0 114.0 45.1 41.1 40.1 107.6 102.7

29.9 74.0 −18.5 261.3 137.7 212.4 61.7 143.5 121.7 122.6 −131.4 201.8 190.4 191.3 148.0 155.7 67.5 115.7 85.4 25.4 192.0 154.2 −55.2 139.8 85.4 79.8 76.1 +5.2 194.2 131.0 106.0 113.2 29.5 20.7 33.0 32.9 69.8 100.0 124.2 111.8 131.0 215.5 290.3 173.7 179.8 161.4 229.7 104.4 185.6 188.7 143.7 107.8 98.0 101.9 57.8 188.7 +5.3 110.0 112.5 116.0 −66.8 +3.8 −118.0 −31.1 136.3 223.8 129.7 59.5 55.3 54.4 123.8 120.2

60

100

200

400

760

271.5

283.8

302.8

322.0

341.3

38.3 83.6 −11.2 273.8 148.2 222.0 70.4 153.2 132.2 133.4 −125.2 213.5 202.5 201.5 158.2 168.0 76.0 126.3 95.7 33.2 207.0 165.7 −48.8 149.5 95.3 89.5 85.8 13.3 205.5 141.1 126.3 123.8 38.1 29.1 41.8 41.6 77.3 110.0 135.5 122.3 142.2 221.2 305.2 184.6 190.2 173.2 239.8 115.2 197.2 199.5 153.7 116.6 107.3 122.4 66.9 199.8 13.0 119.8 122.3 126.1 −61.1 11.8 −113.0 −24.0 146.4 236.0 139.8 68.8 64.4 63.5 133.7 131.5

50.0 96.1 −1.7 289.8 162.5 236.0 82.7 167.5 146.8 147.7 −116.0 228.6 217.8 214.6 174.0 184.3 88.0 141.6 112.1 44.2 225.7 179.7 −40.3 162.0 108.8 102.8 98.9 24.4 219.6 153.4 140.3 137.9 49.9 40.7 53.8 53.4 87.6 124.0 149.8 136.8 156.5 228.4 322.5 199.0 204.4 187.8 252.2 128.1 212.5 213.5 167.2 128.3 118.9 136.5 78.6 214.5 23.3 132.3 135.3 139.7 −53.2 22.8 −106.2 −15.0 159.8 251.5 152.2 81.3 76.8 75.9 146.8 146.0

67.7 115.6 +13.5 313.5 185.0 255.2 101.0 188.0 168.3 168.3 −101.0 250.8 242.2 235.2 196.0 208.0 106.0 163.7 136.0 60.1 255.0 201.3 −27.0 182.3 129.0 122.7 118.6 41.2 241.3 172.8 160.3 158.4 67.8 58.1 72.0 71.3 102.2 145.0 171.8 157.8 179.8 239.7 349.8 220.2 224.4 209.7 271.8 149.3 237.8 232.8 187.7 146.0 136.2 157.7 97.7 237.3 38.4 151.0 154.0 159.0 −41.3 38.7 −95.4 −1.0 180.0 275.3 172.3 100.2 95.5 94.6 166.4 168.0

87.5 137.0 30.2 339.8 209.4 276.5 121.8 210.5 193.7 193.2 −81.0 276.0 267.5 256.6 221.0 235.0 125.7 187.0 163.5 78.7 288.5 224.3 −12.5 203.7 152.0 145.4 141.0 60.4 264.2 193.8 184.5 183.5 88.2 78.0 92.7 91.8 118.4 167.8 196.1 182.3 205.5 249.8 379.2 244.3 247.0 232.8 292.7 171.9 262.8 254.0 209.8 165.0 155.2 182.3 118.7 260.0 55.5 172.1 175.8 182.0 −28.0 58.3 −84.0 +14.8 202.8 301.5 194.1 121.7 116.7 115.9 188.3 193.7

108.2 158.0 47.6 366.5 234.0 299.0 143.5 233.5 218.0 217.5 −56.2 301.0 294.0 278.3 247.0 262.0 146.0 211.0 192.1 98.3 324.0 247.5 +2.9 234.5 176.1 169.2 164.7 80.9 287.0 214.2 208.1 208.0 109.8 99.2 114.8 113.5 135.0 192.7 221.2 207.0 231.0 259.2 413.5 267.2 269.5 256.6 313.0 195.8 290.0 275.0 232.0 184.4 175.1 207.5 140.8 285.0 72.5 194.0 197.5 204.5 −13.8 78.5 −72.2 31.7 225.0 328.5 215.5 144.4 139.1 138.3 211.5 217.9

Temperature, °C

Formula C18H12Cl3O3 188.2 PS C3H5Cl3 −28.8 C3H5Cl3 +9.0 C2Cl3F3 −68.0 C23H48 170.0 C13H28 59.4 C13H26O2 137.8 C7H18O3Si −1.5 C12H20O3Si 71.0 C12H18 46.0 C12H18 47.9 C6H15B C15H26O6 C12H20O7 107.0 C6H14O4 114.0 C13H30Si 70.0 C14H32Si 73.7 C7H16O3 +5.5 C6H15O4P 39.6 C6H15Tl +9.3 C6H5F3Si −31.0 C12H21PO4 93.7 C11H14O 79.0 C3H9N −97.1 C9H13N 68.4 C9H12 16.8 C9H12 13.6 C9H12 9.6 C7H16 C9H14O7 106.2 C3H8O2 59.4 C11H16 43.7 C11H16 38.8 C8H18 −29.0 C8H18 −36.5 C8H18 −25.8 C8H18 −26.3 C8H16O 14.7 C3H9O4P 26.0 C11H14 48.1 C11H14 37.5 C7H10O3 53.5 C19H16 169.7 C18H15O4P 193.5 C9H20O4 96.0 C13H28O4 101.5 C12H26O4 82.4 C21H21O4P 154.6 C11H24 32.7 C11H22O2 101.4 C11H20O2 114.0 C11H24O 71.1 C5H10O2 42.2 C5H10O2 34.5 C5H8O2 37.5 C5H9N −6.0 C8H8O3 107.0 C4H6O2 −48.0 C9H10O 41.9 C9H10O 43.4 C9H10O 45.2 C2H3Cl −105.6 C3H3N −51.0 C2H3F −149.3 C2H2Cl2 −77.2 C10H12O 64.0 C12H9Cl2PO 138.2 C9H10O 59.0 C8H10 −3.8 C8H10 −6.9 C8H10 −8.1 C8H11N 52.6 C8H11N 44.0

40

150.2 138.7 144.0 99.8 104.8 29.2 67.8 37.6 −9.7 131.0 108.0 −81.7 95.9 42.9 38.3 34.7 146.2 87.2 71.2 67.0 −7.1 −15.0 −3.9 −4.1 36.0 53.7 77.0 65.7 82.6 188.4 230.4 125.7 131.6 112.4 184.2 59.7 133.1 142.8 99.0 67.7 59.6 65.8 +18.1 138.4 −28.0 68.0 69.9 72.0 −90.8 −30.7 −138.0 −60.0 91.7 171.1 85.9 +20.2 +16.8 +15.5 79.8 72.6

Melting point, °C

−77.7 −14.7 −35 47.7 −6.2 41

135

−63.0

−117.1 67 −25.5 −44.1 −44.8 −25.0 78.5

−112.3 −107.3 −101.5 −109.2

93.4 49.4

−25.6 29.5 24.5 −34.5 −37.6 81.5

−153.7 −82 −160.5 −122.5 75 −25.2 −47.9 +13.3

2-76

PHYSICAL AND CHEMICAL DATA

VAPOR PRESSURES OF SOLUTIONS UNITS CONVERSIONS

To convert cubic feet to cubic meters, multiply by 0.02832. To convert bars to pounds-force per square inch, multiply by 14.504. To convert bars to kilopascals, multiply by 1 × 102.

For this subsection, the following units conversions are applicable: °F = 9⁄5°C + 32. To convert millimeters of mercury to pounds-force per square inch, multiply by 0.01934.

TABLE 2-9

Partial Pressures of Water over Aqueous Solutions of HCl* log10 pmm = A − B/T, (T in K), which, however, agrees only approximately with the table. The table is more nearly correct. Partial pressure of H2O, mmHg, °C

% HCl

A

B

0°

5°

10°

15°

20°

25°

30°

35°

40°

45°

50°

60°

6 10 14 18 20

8.99156 8.99864 8.97075 8.98014 8.97877

2282 2295 2300 2323 2334

4.18 3.84 3.39 2.87 2.62

6.04 5.52 4.91 4.21 3.83

8.45 7.70 6.95 5.92 5.40

11.7 10.7 9.65 8.26 7.50

15.9 14.6 13.1 11.3 10.3

21.8 20.0 18.0 15.4 14.1

29.1 26.8 24.1 20.6 19.0

39.4 35.5 31.9 27.5 25.1

50.6 47.0 42.1 36.4 33.3

66.2 61.5 55.3 47.9 43.6

86.0 80.0 72.0 62.5 57.0

139 130 116 102 93.5

22 24 26 28 30

9.02708 8.96022 9.01511 8.97611 9.00117

2363 2356 2390 2395 2422

2.33 2.05 1.76 1.50 1.26

3.40 3.04 2.60 2.24 1.90

4.82 4.31 3.71 3.21 2.73

6.75 6.03 5.21 4.54 3.88

9.30 8.30 7.21 6.32 5.41

12.6 11.4 9.95 8.75 7.52

17.1 15.4 13.5 11.8 10.2

22.8 20.4 18.0 15.8 13.7

30.2 27.1 24.0 21.1 18.4

39.8 35.7 31.7 27.9 24.3

52.0 46.7 41.5 36.5 32.0

32 34 36 38 40 42

9.03317 9.07143 9.11815 9.20783 9.33923 9.44953

2453 2487 2526 2579 2647 2709

1.04 0.85 0.68 0.53 0.41 0.31

1.57 1.29 1.03 0.81 0.63 0.48

2.27 1.87 1.50 1.20 0.94 0.72

3.25 2.70 2.19 1.75 1.37 1.06

4.55 3.81 3.10 2.51 2.00 1.56

6.37 5.35 4.41 3.60 2.88 2.30

11.7 9.95 8.33 6.92 5.68 4.60

15.7 13.5 11.4 9.52 7.85 6.45

21.0 18.1 15.4 13.0 10.7 8.90

27.7 24.0 20.4 17.4 14.5 12.1

8.70 7.32 6.08 5.03 4.09 3.28

70°

80°

90°

100° 110°

220 204 185 162 150

333 310 273 248 230

492 463 425 374 345

715 677 625 550 510

960 892 783 729

85.6 77.0 69.0 60.7 53.5

138 124 112 99.0 87.5

211 194 173 154 136

317 290 261 234 207

467 426 387 349 310

670 611 555 499 444

46.5 40.5 34.8 29.6 25.0 21.2

76.5 66.5 57.0 49.1 42.1 35.8

120 104 90.0 77.5 67.3 57.2

184 161 140 120 105 89.2

275 243 212 182 158 135

396 355 311 266 230 195

*Accuracy, ca. 2 percent for solutions of 15 to 30 percent HCl between 0 and 100°; for solutions of > 30 percent HCl the accuracy is ca. 5 percent at the lower temperatures and ca. 15 percent at the higher temperatures. Below 15 percent HCl, the accuracy is ca. 5 percent at the lower temperatures and higher strengths to ca. 15 to 20 percent at the lower strengths and perhaps 15 to 20 percent at the higher temperatures and lower strengths.

TABLE 2-10

Partial Pressures of HCl over Aqueous Solutions of HCl*

log10 pmm = A − B/T, (T in K), which, however, agrees only approximately with the table. The table is more nearly correct. mmHg, °C

% HCl

A

2 4 6 8 10

11.8037 11.6400 11.2144 11.0406 10.9311

4736 0.0000117 0.000023 0.000044 0.000084 0.000151 0.000275 0.00047 0.00083 0.00140 0.00380 0.0100 0.0245 0.058 0.132 0.280 4471 0.000018 0.000036 .000069 .000131 .00024 .00044 .00077 .00134 .0023 .00385 .0064 .0165 .0405 .095 .21 .46 .93 4202 .000066 .000125 .000234 .000425 .00076 .00131 .00225 .0038 .0062 .0102 .0163 .040 .094 .206 .44 .92 1.78 4042 .000118 .000323 .000583 .00104 .00178 .0031 .00515 .0085 .0136 .022 .0344 .081 .183 .39 .82 1.64 3.10 3908 .00042 .00075 .00134 .00232 .00395 .0067 .0111 .0178 .0282 .045 .069 .157 .35 .73 1.48 2.9 5.4

12 14 16 18 20

10.7900 10.6954 10.6261 10.4957 10.3833

3765 3636 3516 3376 3245

22 24 26 28 30

10.3172 10.2185 10.1303 10.0115 9.8763

3125 .0734 2995 .175 2870 .41 2732 1.0 2593 2.4

.119 .277 .64 1.52 3.57

.187 .43 .98 2.27 5.23

.294 .66 1.47 3.36 7.60

.45 1.00 2.17 4.90 10.6

.68 1.49 3.20 7.05 15.1

1.02 2.17 4.56 9.90 21.0

32 34 36 38 40

9.7523 9.6061 9.5262 9.4670 9.2156

2457 2316 2229 2094 1939

5.7 13.1 29.0 63.0 130

8.3 18.8 41.0 87.0 176

11.8 26.4 56.4 117 233

16.8 36.8 78 158 307

23.5 50.5 105.5 210 399

32.5 68.5 142 277 515

44.5 92 188 360 627

42 44 46

8.9925 1800 253 8.8621 1681 510 940

332 655

430 840

560

709

900

B

0°

.00099 .0024 .0056 .0135 .0316

5°

.00175 .00415 .0095 .0225 .052

10°

.00305 .0071 .016 .037 .084

15°

.0052 .0118 .0265 .060 .132

20°

.0088 .0196 .0428 .095 .205

25°

.0145 .0316 .0685 .148 .32

30°

.0234 .050 .106 .228 .48

35°

.037 .078 .163 .345 .72

40°

45°

50°

60°

70°

80°

90°

100°

110°

.058 .121 .247 .515 1.06

.091 .185 .375 .77 1.55

.136 .275 .55 1.11 2.21

.305 .60 1.17 2.3 4.4

.66 1.25 2.40 4.55 8.5

1.34 2.50 4.66 8.6 15.6

2.65 4.8 8.8 15.7 28.1

5.1 9.0 16.1 28 49

9.3 16.0 28 48 83

1.50 3.14 6.50 13.8 28.6

2.18 4.5 9.2 19.1 39.4

3.14 6.4 12.7 26.4 53

4.42 8.9 17.5 35.7 71

8.6 16.9 32.5 64 124

16.3 31.0 58.5 112 208

29.3 54.5 100 188 340

52 94 169 309 542

90 157 276 493 845

146 253 436 760

60.0 122 246 465 830

81 161 322 598

107 211 416 758

141 273 535 955

238 450 860

390 720

623

970

*Accuracy, ca. 2 percent for solutions of 15 to 30 percent HCl between 0 and 100°; for solutions of > 30 percent HCl the accuracy is ca. 5 percent at the lower temperatures and ca. 15 percent at the higher temperatures. Below 15 percent HCl, the accuracy is ca. 5 percent at the lower temperatures and higher strengths to ca. 15 to 20 percent at the lower strengths and perhaps 15 to 20 percent at the higher temperatures and lower strengths.

VAPOR PRESSURES OF SOLUTIONS

FIG. 2-1 Vapor pressures of H3PO4 aqueous: partial pressure of H2O vapor. (Courtesy of Victor Chemical Works, Stauffer Chemical Company; measurements by W. H. Woodstock.)

TABLE 2-11 g SO2 / 100 g H2O

Temperature, ∞C 10

20

0.01 0.05 0.10 0.15 0.20

0.02 0.38 1.15 2.10 3.17

0.04 0.66 1.91 3.44 5.13

0.07 1.07 3.03 5.37 7.93

0.25 0.30 0.40 0.50 1.00

4.34 5.57 8.17 10.9 25.8

6.93 8.84 12.8 17.0 39.5

8.00 10.00 15.00 20.00

Vapor pressures of H3PO4 aqueous: weight of H2O in saturated air. (Courtesy of Victor Chemical Works, Stauffer Chemical Company; measurements by W. H. Woodstock.)

FIG. 2-2

Partial Pressures of H2O and SO2 over Aqueous Solutions of Sulfur Dioxide* Partial pressures of H2O and SO2, mmHg, °C 0

2.00 3.00 4.00 5.00 6.00

2-77

10.6 13.5 19.4 25.6 58.4

30

40

50

60

90

120

0.12 1.68 4.62 8.07 11.8

0.19 2.53 6.80 11.7 17.0

0.29 3.69 9.71 16.5 23.8

0.43 5.24 13.5 22.7 32.6

1.21 12.9 31.7 52.2 73.7

2.82 27.0 63.9 104 145

15.7 19.8 28.3 37.1 83.7

58.6 93.2 129 165 202

88.5 139 192 245 299

129 202 277 353 430

183 285 389 496 602

275 351 542 735

407 517 796

585 741

818

22.5 28.2 40.1 52.3 117

31.4 39.2 55.3 72.0 159

42.8 53.3 74.7 96.8 212

95.8 118 164 211 454

253 393 535 679 824

342 530 720

453 700

955

*Extracted with permission from J. Chem Eng. Data 8, 1963: 333–336. Copyright 1963 American Chemical Society.

186 229 316 404 856

2-78

PHYSICAL AND CHEMICAL DATA

TABLE 2-12 °C 0 10 20 30 40 50 60 70 80 90

Water Partial Pressure, bar, over Aqueous Sulfuric Acid Solutions* Weight percent, H2SO4

10.0 .582E−02 .117E−01 .223E−01 .404E−01 .703E−01 .117 .189 .296 .449 .664

20.0 .534E−02 .107E−01 .205E−01 .373E−01 .649E−01 .109 .175 .275 .417 .617

30.0 .448E−02 .909E−02 .174E−01 .319E−01 .558E−01 .939E−01 .152 .239 .365 .542

40.0 .326E−02 .670E−02 .130E−01 .241E−01 .427E−01 .725E−01 .119 .188 .290 .434

50.0

60.0

.193E−02 .405E−02 .802E−02 .151E−01 .272E−01 .470E−01 .782E−01 .126 .196 .298

.836E−03 .180E−02 .367E−02 .710E−02 .131E−01 .232E−01 .395E−01 .651E−01 .104 .161

75.0

80.0

85.0

.207E−03 .467E−03 .995E−03 .201E−02 .387E−02 .715E−02 .127E−01 .217E−01 .360E−01 .578E−01

70.0

.747E−04 .175E−03 .388E−03 .811E−03 .162E−02 .309E−02 .565E−02 .997E−02 .170E−01 .281E−01

.197E−04 .490E−04 .115E−04 .253E−03 .531E−03 .106E−02 .204E−02 .376E−02 .668E−02 .115E−01

.343E−05 .952E−05 .245E−04 .589E−04 .133E−03 .286E−03 .584E−03 .114E−02 .213E−02 .383E−02

.905E−01 .138 .206 .301 .481 .605 .837 1.138 1.525 2.017

.452E−01 .708E−01 .108 .162 .236 .339 .478 .662 .902 1.212

.192E−01 .312E−01 .493E−01 .760E−01 .115 .170 .246 .350 .489 .673

.666E−02 .112E−01 .183E−01 .291E−01 .451E−01 .682E−01 .101 .147 .208 .291

100 110 120 130 140 150 160 170 180 190

.957 1.349 1.863 2.524 3.361 4.404 5.685 7.236 9.093 11.289

.891 1.258 1.740 2.361 3.149 4.132 5.342 6.810 8.571 10.658

.786 1.113 1.544 2.101 2.810 3.697 4.793 6.127 7.731 9.640

.634 .904 1.264 1.732 2.333 3.090 4.031 5.185 6.584 8.259

.441 .638 .903 1.253 1.708 2.289 3.021 3.930 5.045 6.397

.244 .360 .519 .734 1.020 1.392 1.870 2.475 3.233 4.169

200 210 220 230 240 250 260 270 280 290

13.861 16.841 20.264 24.160 28.561 33.494 38.984 45.055 51.726 59.015

13.107 15.951 19.225 22.960 27.188 31.939 37.240 43.116 49.590 56.681

11.887 14.505 17.529 20.992 24.927 29.364 34.334 39.865 45.984 52.715

10.245 12.576 15.287 18.414 21.992 26.056 30.642 35.784 41.514 47.865

8.020 9.948 12.217 14.864 17.929 21.452 25.472 30.030 35.168 40.926

5.312 6.696 8.354 10.322 12.641 15.351 18.496 22.121 26.274 31.003

2.632 3.395 4.331 5.466 6.831 8.458 10.382 12.640 15.269 18.311

1.606 2.101 2.714 3.467 4.381 5.480 6.788 8.333 10.142 12.242

.913 1.220 1.609 2.096 2.699 3.435 4.326 5.395 6.663 8.155

.401 .542 .724 .952 1.237 1.587 2.012 2.525 3.136 3.857

300 66.934 310 75.495 320 84.705 330 94.567 340 105.083 350 116.251

64.407 72.781 81.816 91.518 101.894 112.946

60.081 68.100 76.792 86.172 96.252 107.043

54.868 62.553 70.947 80.077 89.969 100.646

47.346 54.470 62.337 70.988 80.463 90.802

36.360 42.395 49.164 56.721 65.123 74.426

21.808 25.804 30.343 35.473 41.240 47.692

14.665 17.438 20.591 24.153 28.154 32.622

9.897 11.912 14.227 16.867 19.855 23.217

4.701 5.680 6.806 8.093 9.551 11.193

*Vermeulen, Dong, Robinson, Nguyen, and Gmitro, AIChE meeting, Anaheim, Calif., 1982; and private communication from Prof. Theodore Vermeulen, Chemical Engineering Dept., University of California, Berkeley.

VAPOR PRESSURES OF SOLUTIONS TABLE 2-12

2-79

Water Partial Pressure, bar, over Aqueous Sulfuric Acid Solutions (Concluded ) Weight percent, H2SO4

°C

90.0

92.0

94.0

96.0

97.0

98.0

98.5

99.0

99.5

100.0

0 10 20 30 40 50 60 70 80 90

.518E−06 .159E−05 .448E−05 .117E−04 .285E−04 .652E−04 .141E−03 .290E−03 .569E−03 .107E−02

.242E−06 .762E−06 .220E−05 .587E−05 .146E−04 .341E−04 .754E−04 .158E−03 .316E−03 .606E−03

.107E−06 .344E−06 .101E−05 .275E−05 .696E−05 .166E−04 .372E−04 .795E−04 .162E−03 .315E−03

.401E−07 .130E−06 .390E−06 .108E−05 .278E−05 .672E−05 .154E−04 .334E−04 .691E−04 .137E−03

.218E−07 .713E−07 .215E−06 .598E−06 .155E−05 .379E−05 .875E−05 .192E−04 .400E−04 .801E−04

.980E−08 .323E−07 .978E−07 .275E−06 .720E−06 .177E−05 .413E−05 .912E−05 .192E−04 .388E−04

.569E−08 .188E−07 .572E−07 .161E−06 .424E−06 .105E−05 .245E−05 .544E−05 .115E−04 .234E−04

.268E−08 .888E−08 .271E−07 .766E−07 .202E−06 .503E−06 .118E−05 .263E−05 .559E−05 .114E−04

.775E−09 .258E−08 .789E−08 .224E−07 .595E−07 .149E−06 .350E−06 .784E−06 .168E−05 .343E−05

.196E−09 .655E−09 .201E−08 .575E−08 .153E−07 .384E−07 .910E−07 .205E−06 .439E−06 .903E−06

100 110 120 130 140 150 160 170 180 190

.194E−02 .338E−02 .571E−02 .938E−02 .150E−01 .233E−01 .354E−01 .526E−01 .766E−01 .110

.112E−02 .198E−02 .341E−02 .569E−02 .923E−02 .146E−01 .225E−01 .340E−01 .502E−01 .729E−01

.590E−03 .107E−02 .186E−02 .315E−02 .519E−02 .832E−02 .130E−01 .199E−01 .298E−01 .438E−01

.261E−03 .479E−03 .851E−03 .146E−02 .245E−02 .399E−02 .633E−02 .983E−02 .149E−01 .222E−01

.154E−03 .285E−03 .511E−03 .886E−03 .149E−02 .245E−02 .393E−02 .614E−02 .941E−02 .141E−01

.752E−04 .141E−03 .254E−03 .445E−03 .757E−03 .125E−02 .202E−02 .319E−02 .492E−02 .744E−02

.455E−04 .855E−04 .155E−03 .278E−03 .467E−03 .776E−03 .126E−02 .199E−02 .309E−02 .469E−02

.223E−04 .420E−04 .766E−04 .135E−03 .232E−03 .387E−03 .629E−03 .999E−03 .155E−02 .236E−02

.674E−05 .128E−04 .233E−04 .414E−04 .711E−04 .119E−03 .194E−03 .309E−03 .482E−03 .735E−03

.178E−05 .339E−05 .623E−05 .111E−04 .191E−04 .321E−04 .526E−04 .840E−04 .131E−03 .201E−03

.631E−01 .894E−01 .125 .171 .232 .310 .409 .534 .689 .880

.325E−01 .467E−01 .660E−01 .918E−01 .126 .170 .227 .300 .391 .505

.208E−01 .300E−01 .427E−01 .598E−01 .825E−01 .112 .151 .200 .263 .341

.110E−01 .161E−01 .230E−01 .325E−01 .451E−01 .618E−01 .835E−01 .111 .147 .192

.698E−02 .102E−01 .147E−01 .208E−01 .290E−01 .398E−01 .540E−01 .723E−01 .957E−01 .125

.352E−02 .516E−02 .743E−02 .105E−01 .147E−01 .202E−01 .274E−01 .366E−01 .485E−01 .634E−01

.110E−02 .161E−02 .232E−02 .329E−02 .460E−02 .633E−02 .858E−02 .115E−01 .152E−01 .199E−01

.300E−03 .442E−03 .638E−03 .906E−03 .127E−02 .174E−02 .237E−02 .317E−02 .420E−02 .548E−02

.248 .316 .400 .502 .624 .770

.162 .208 .264 .331 .413 .511

.820E−01 .105 .133 .167 .208 .256

.257E−01 .328E−01 .415E−01 .520E−01 .646E−01 .795E−01

.708E−02 .905E−02 .114E−01 .143E−01 .178E−01 .218E−01

200 210 220 230 240 250 260 270 280 290

.154 .213 .290 .389 .514 .673 .870 1.112 1.407 1.763

.104 .146 .201 .273 .366 .485 .635 .822 1.052 1.335

300 310 320 330 340 350

2.190 2.696 3.292 3.990 4.801 5.738

1.676 2.088 2.578 3.159 3.843 4.641

1.112 1.394 1.732 2.133 2.608 3.164

.646 .817 1.025 1.274 1.571 1.922

.437 .556 .701 .875 1.083 1.331

2-80

PHYSICAL AND CHEMICAL DATA

TABLE 2-13

Sulfur Trioxide Partial Pressure, bar, over Aqueous Sulfuric Acid Solutions* Weight percent, H2SO4

°C

10.0

20.0

30.0

40.0

50.0

60.0

70.0

75.0

80.0

85.0

0 10 20 30 40 50 60 70 80 90

.644E−29 .149E−27 .278E−26 .426E−25 .549E−24 .602E−23 .573E−22 .477E−21 .352E−20 .233E−19

.103E−27 .223E−26 .394E−25 .577E−24 .714E−23 .757E−22 .699E−21 .567E−20 .410E−19 .266E−18

.205E−26 .395E−25 .626E−24 .832E−23 .941E−22 .921E−21 .789E−20 .599E−19 .408E−18 .250E−17

.688E−25 .113E−23 .156E−22 .181E−21 .181E−20 .158E−19 .122E−18 .843E−18 .524E−17 .296E−16

.368E−23 .522E−22 .621E−21 .630E−20 .555E−19 .429E−18 .294E−17 .181E−16 .101E−15 .516E−15

.341E−21 .415E−20 .426E−19 .376E−18 .288E−17 .195E−16 .118E−15 .643E−15 .319E−14 .145E−13

.784E−19 .796E−18 .685E−17 .509E−16 .331E−15 .191E−14 .985E−14 .461E−13 .197E−12 .775E−12

.174E−17 .158E−16 .121E−15 .808E−15 .473E−14 .246E−13 .116E−12 .492E−12 .192E−11 .693E−11

.531E−16 .417E−15 .280E−14 .164E−13 .851E−13 .395E−12 .165E−11 .634E−11 .223E−10 .731E−10

.229E−14 .141E−13 .767E−13 .371E−12 .162E−11 .643E−11 .234E−10 .791E−10 .249E−09 .734E−09

100 110 120 130 140 150 160 170 180 190

.139E−18 .756E−18 .377E−17 .174E−16 .743E−16 .297E−15 .111E−14 .393E−14 .131E−13 .415E−13

.157E−17 .844E−17 .418E−16 .191E−15 .815E−15 .325E−14 .122E−13 .430E−13 .144E−12 .458E−12

.140E−16 .719E−16 .340E−15 .150E−14 .615E−14 .237E−13 .862E−13 .296E−12 .967E−12 .301E−11

.153E−15 .730E−15 .323E−14 .133E−13 .517E−13 .188E−12 .649E−12 .212E−11 .622E−11 .197E−10

.242E−14 .105E−13 .424E−13 .160E−12 .569E−12 .191E−11 .608E−11 .184E−10 .532E−10 .147E−09

.606E−13 .236E−12 .858E−12 .293E−11 .943E−11 .287E−10 .833E−10 .231E−09 .610E−09 .155E−08

.283E−11 .961E−11 .307E−10 .922E−10 .262E−09 .710E−09 .183E−08 .453E−08 .107E−07 .246E−07

.232E−10 .729E−10 .215E−09 .601E−09 .159E−08 .403E−08 .974E−08 .226E−07 .505E−07 .109E−06

.223E−09 .641E−09 .174E−08 .446E−08 .109E−07 .256E−07 .575E−07 .125E−06 .260E−06 .527E−06

.204E−08 .538E−08 .135E−07 .324E−07 .745E−07 .165E−06 .351E−06 .725E−06 .145E−05 .282E−05

200 210 220 230 240 250 260 270 280 290

.125E−12 .362E−12 .100E−11 .265E−11 .678E−11 .167E−10 .399E−10 .920E−10 .206E−09 .449E−09

.139E−11 .404E−11 .112E−10 .301E−10 .777E−10 .193E−09 .466E−09 .109E−08 .247E−08 .545E−08

.893E−11 .254E−10 .695E−10 .183E−09 .465E−09 .114E−08 .272E−08 .628E−08 .141E−07 .308E−07

.561E−10 .154E−09 .405E−09 .103E−08 .253E−08 .602E−08 .139E−07 .312E−07 .683E−07 .145E−06

.391E−09 .100E−08 .246E−08 .587E−08 .135E−07 .303E−07 .660E−07 .140E−06 .288E−06 .580E−06

.379E−08 .894E−08 .204E−07 .450E−07 .965E−07 .201E−06 .408E−06 .807E−06 .156E−05 .295E−05

.542E−07 .116E−06 .240E−06 .482E−06 .944E−06 .180E−05 .336E−05 .612E−05 .109E−04 .191E−04

.228E−06 .462E−06 .911E−06 .175E−05 .328E−05 .600E−05 .108E−04 .189E−04 .326E−04 .553E−04

.103E−05 .198E−05 .368E−05 .668E−05 .119E−04 .206E−04 .352E−04 .590E−04 .973E−04 .158E−03

.534E−05 .986E−05 .178E−04 .314E−04 .543E−04 .923E−04 .154E−03 .253E−03 .408E−03 .649E−03

300 310 320 330 340 350

.953E−09 .197E−08 .397E−08 .782E−08 .151E−07 .285E−07

.117E−07 .245E−07 .502E−07 .100E−06 .196E−06 .376E−06

.657E−07 .136E−06 .277E−06 .551E−06 .107E−05 .204E−05

.302E−06 .614E−06 .122E−05 .237E−05 .452E−05 .846E−05

.114E−05 .220E−05 .414E−05 .766E−05 .139E−04 .246E−04

.546E−05 .990E−05 .176E−04 .308E−04 .529E−04 .893E−04

.329E−04 .556E−04 .923E−04 .151E−03 .243E−03 .387E−03

.921E−04 .151E−03 .245E−03 .391E−03 .617E−03 .963E−03

.253E−03 .398E−03 .621E−03 .956E−03 .145E−02 .219E−02

.102E−02 .158E−02 .242E−02 .367E−02 .550E−02 .815E−02

*Vermeulen, Dong, Robinson, Nguyen, and Gmitro, AIChE meeting, Anaheim, Calif., 1982; and private communication from Prof. Theodore Vermeulen, Chemical Engineering Dept., University of California, Berkeley.

VAPOR PRESSURES OF SOLUTIONS TABLE 2-13

2-81

Sulfur Trioxide Partial Pressure, bar, over Aqueous Sulfuric Acid Solutions (Concluded ) Weight percent, H2SO4

°C

90.0

92.0

94.0

96.0

97.0

98.0

98.5

99.0

99.5

0 10 20 30 40 50 60 70 80 90

.671E−13 .345E−12 .159E−11 .664E−11 .254E−10 .897E−10 .294E−09 .904E−09 .261E−08 .712E−08

.216E−12 .107E−11 .475E−11 .192E−10 .709E−10 .242E−09 .771E−09 .230E−08 .643E−08 .171E−07

.677E−12 .326E−11 .141E−10 .557E−10 .201E−09 .669E−09 .207E−08 .602E−08 .165E−07 .426E−07

.240E−11 .114E−10 .482E−10 .186E−09 .655E−09 .214E−08 .647E−08 .184E−07 .492E−07 .124E−06

.500E−11 .234E−10 .986E−10 .376E−09 .131E−08 .424E−08 .127E−07 .357E−07 .946E−07 .237E−06

.124E−10 .578E−10 .241E−09 .911E−09 .315E−08 .101E−07 .299E−07 .833E−07 .218E−06 .541E−06

.224E−10 .104E−09 .433E−09 .163E−08 .562E−08 .179E−07 .528E−07 .146E−06 .381E−06 .940E−06

.502E−10 .232E−09 .961E−09 .360E−08 .123E−07 .391E−07 .115E−06 .316E−06 .820E−06 .201E−05

.182E−09 .839E−09 .346E−08 .129E−07 .440E−07 .139E−06 .405E−06 .111E−05 .286E−05 .698E−05

.755E−09 .347E−08 .142E−07 .528E−07 .179E−06 .560E−06 .163E−05 .444E−05 .114E−04 .276E−04

100 110 120 130 140 150 160 170 180 190

.184E−07 .456E−07 .108E−06 .244E−06 .533E−06 .112E−05 .229E−05 .453E−05 .870E−05 .163E−04

.430E−07 .103E−06 .238E−06 .526E−06 .112E−05 .230E−05 .459E−05 .886E−05 .166E−04 .304E−04

.105E−06 .247E−06 .555E−06 .120E−05 .250E−05 .504E−05 .983E−05 .186E−04 .343E−04 .615E−04

.300E−06 .689E−06 .152E−05 .321E−05 .656E−05 .129E−04 .247E−04 .459E−04 .829E−04 .146E−03

.565E−06 .128E−05 .280E−05 .586E−05 .118E−04 .231E−04 .438E−04 .806E−04 .144E−03 .252E−03

.127E−05 .287E−05 .619E−05 .128E−04 .257E−04 .497E−04 .932E−04 .170E−03 .301E−03 .520E−03

.220E−05 .494E−05 .106E−04 .219E−04 .435E−04 .837E−04 .156E−03 .283E−03 .499E−03 .859E−03

.470E−05 .105E−04 .224E−04 .459E−04 .910E−04 .174E−03 .324E−03 .586E−03 .103E−02 .177E−02

.162E−04 .359E−04 .764E−04 .156E−03 .308E−03 .588E−03 .109E−02 .196E−02 .343E−02 .587E−02

.638E−04 .141E−03 .298E−03 .606E−03 .119E−02 .226E−02 .416E−02 .746E−02 .130E−01 .222E−01

200 210 220 230 240 250 260 270 280 290

.297E−04 .528E−04 .919E−04 .157E−03 .261E−03 .428E−03 .690E−03 .109E−02 .170E−02 .261E−02

.543E−04 .946E−04 .161E−03 .269E−03 .441E−03 .708E−03 .112E−02 .174E−02 .266E−02 .401E−02

.108E−03 .185E−03 .309E−03 .508E−03 .819E−03 .130E−02 .202E−02 .309E−02 .466E−02 .694E−02

.251E−03 .422E−03 .694E−03 .112E−02 .178E−02 .276E−02 .423E−02 .638E−02 .948E−02 .139E−01

.429E−03 .714E−03 .117E−02 .187E−02 .293E−02 .453E−02 .688E−02 .103E−01 .152E−01 .221E−01

.878E−03 .145E−02 .235E−02 .373E−02 .582E−02 .891E−02 .134E−01 .200E−01 .293E−01 .423E−01

.144E−02 .237E−02 .383E−02 .605E−02 .939E−02 .143E−01 .215E−01 .319E−01 .465E−01 .670E−01

.296E−02 .486E−02 .781E−02 .123E−01 .191E−01 .291E−01 .437E−01 .646E−01 .943E−01 .136

.981E−02 .161E−01 .258E−01 .405E−01 .627E−01 .955E−01 .143 .212 .309 .444

.370E−01 .603E−01 .965E−01 .152 .234 .356 .532 .786 1.144 1.646

300 310 320 330 340 350

.395E−02 .589E−02 .868E−02 .126E−01 .181E−01 .258E−01

.595E−02 .873E−02 .126E−01 .181E−01 .255E−01 .357E−01

.102E−01 .148E−01 .211E−01 .299E−01 .418E−01 .578E−01

.201E−01 .287E−01 .405E−01 .565E−01 .780E−01 .107

.318E−01 .451E−01 .632E−01 .877E−01 .120 .164

.604E−01 .852E−01 .119 .164 .224 .303

.953E−01 .134 .186 .256 .348 .470

.193 .272 .378 .520 .708 .956

.632 .889 1.236 1.703 2.323 3.142

100.0

2.339 3.289 4.575 6.303 8.603 11.640

TABLE 2-14

Sulfuric Acid Partial Pressure, bar, over Aqueous Sulfuric Acid* Weight Percent, H2SO4

°C

10.0

20.0

30.0

40.0

50.0

60.0

70.0

75.0

80.0

85.0

0 10 20 30 40 50 60 70 80 90

.576E−21 .634E−20 .588E−19 .468E−18 .324E−17 .197E−16 .107E−15 .526E−15 .235E−14 .960E−14

.843E−20 .874E−19 .769E−18 .584E−17 .389E−16 .229E−15 .121E−14 .581E−14 .254E−13 .102E−12

.141E−18 .131E−17 .104E−16 .721E−16 .441E−15 .241E−14 .119E−13 .535E−13 .221E−12 .844E−12

.344E−17 .276E−16 .193E−15 .119E−14 .649E−14 .320E−13 .144E−12 .592E−12 .225E−11 .798E−11

.109E−15 .769E−15 .474E−14 .259E−13 .127E−12 .562E−12 .228E−11 .851E−11 .295E−10 .956E−10

.438E−14 .273E−13 .149E−12 .725E−12 .317E−11 .126E−10 .462E−10 .156E−09 .492E−09 .145E−08

.249E−12 .135E−11 .649E−11 .278E−10 .108E−09 .380E−09 .124E−08 .373E−08 .105E−07 .279E−07

.200E−11 .101E−10 .447E−10 .178E−09 .643E−09 .212E−08 .646E−08 .183E−07 .485E−07 .121E−06

.161E−10 .743E−10 .305E−09 .113E−08 .379E−08 .117E−07 .334E−07 .888E−07 .222E−06 .522E−06

.121E−09 .490E−09 .179E−08 .594E−08 .181E−07 .513E−07 .135E−06 .336E−06 .786E−06 .175E−05

100 110 120 130 140 150 160 170 180 190

.353E−13 .127E−12 .418E−12 .129E−11 .375E−11 .103E−10 .272E−10 .682E−10 .164E−09 .378E−09

.381E−12 .132E−11 .432E−11 .132E−10 .385E−10 .106E−09 .279E−09 .702E−09 .170E−08 .394E−08

.300E−11 .997E−11 .312E−10 .924E−10 .259E−09 .694E−09 .178E−08 .436E−08 .103E−07 .234E−07

.264E−10 .824E−10 .243E−09 .678E−09 .181E−08 .460E−08 .112E−07 .264E−07 .599E−07 .131E−06

.291E−09 .835E−09 .227E−08 .589E−08 .146E−07 .346E−07 .789E−07 .174E−06 .369E−06 .760E−06

.402E−08 .106E−07 .264E−07 .631E−07 .144E−06 .316E−06 .670E−06 .137E−05 .271E−05 .521E−05

.698E−07 .166E−06 .375E−06 .814E−06 .169E−05 .340E−05 .659E−05 .124E−04 .225E−04 .400E−04

.287E−06 .644E−06 .138E−05 .285E−05 .565E−05 .108E−04 .200E−04 .359E−04 .627E−04 .107E−03

.117E−05 .249E−05 .508E−05 .995E−05 .188E−04 .343E−04 .608E−04 .104E−03 .175E−03 .286E−03

.371E−05 .752E−05 .147E−04 .277E−04 .503E−04 .889E−04 .152E−03 .255E−03 .416E−03 .663E−03

200 210 220 230 240 250 260 270 280 290

.842E−09 .181E−08 .376E−08 .758E−08 .148E−07 .283E−07 .526E−07 .954E−07 .169E−06 .294E−06

.883E−08 .191E−07 .401E−07 .817E−07 .162E−06 .312E−06 .588E−06 .108E−05 .194E−05 .342E−05

.514E−07 .109E−06 .226E−06 .455E−06 .889E−06 .170E−05 .316E−05 .577E−05 .103E−04 .180E−04

.278E−06 .573E−06 .115E−05 .224E−05 .427E−05 .793E−05 .144E−04 .257E−04 .450E−04 .771E−04

.152E−05 .295E−05 .559E−05 .103E−04 .186E−04 .329E−04 .569E−04 .965E−04 .161E−03 .263E−03

.975E−05 .178E−04 .316E−04 .549E−04 .935E−04 .156E−03 .255E−03 .411E−03 .650E−03 .101E−02

.691E−04 .117E−03 .193E−03 .311E−03 .494E−03 .770E−03 .118E−02 .178E−02 .265E−02 .389E−02

.177E−03 .288E−03 .459E−03 .717E−03 .110E−02 .166E−02 .247E−02 .362E−02 .524E−02 .750E−02

.457E−03 .715E−03 .110E−02 .166E−02 .245E−02 .358E−02 .516E−02 .733E−02 .103E−01 .143E−01

.104E−02 .159E−02 .239E−02 .354E−02 .515E−02 .740E−02 .105E−01 .147E−01 .203E−01 .278E−01

300 310 320 330 340 350

.500E−06 .834E−06 .137E−05 .220E−05 .349E−05 .544E−05

.591E−05 .100E−04 .167E−04 .273E−04 .440E−04 .698E−04

.309E−04 .522E−04 .865E−04 .141E−03 .227E−03 .360E−03

.130E−03 .215E−03 .352E−03 .565E−03 .895E−03 .140E−02

.424E−03 .672E−03 .105E−02 .162E−02 .246E−02 .369E−02

.156E−02 .236E−02 .352E−02 .519E−02 .757E−02 .109E−01

.563E−02 .805E−02 .114E−01 .159E−01 .221E−01 .303E−01

.106E−01 .148E−01 .205E−01 .281E−01 .382E−01 .516E−01

.196E−01 .266E−01 .359E−01 .480E−01 .636E−01 .836E−01

.376E−01 .504E−01 .670E−01 .883E−01 .116 .150

°C

90.0

92.0

94.0

96.0

97.0

98.0

98.5

99.0

99.5

100.0

0 10 20 30 40 50 60 70 80 90

.534E−09 .200E−08 .677E−08 .211E−07 .607E−07 .163E−06 .411E−06 .976E−06 .220E−05 .473E−05

.803E−09 .296E−08 .993E−08 .306E−07 .870E−07 .231E−06 .575E−06 .135E−05 .302E−05 .642E−05

.112E−08 .409E−08 .136E−07 .415E−07 .117E−06 .309E−06 .765E−06 .179E−05 .396E−05 .835E−05

.148E−08 .540E−08 .179E−07 .543E−07 .153E−06 .400E−06 .985E−06 .229E−05 .504E−05 .106E−04

.167E−08 .609E−08 .201E−07 .611E−07 .171E−06 .449E−06 .110E−05 .256E−05 .562E−05 .118E−04

.187E−08 .679E−08 .224E−07 .680E−07 .191E−06 .498E−06 .122E−05 .283E−05 .622E−05 .130E−04

.196E−08 .714E−08 .236E−07 .714E−07 .200E−06 .523E−06 .128E−05 .297E−05 .652E−05 .136E−04

.206E−08 .750E−08 .247E−07 .749E−07 .210E−06 .548E−06 .134E−05 .310E−05 .681E−05 .143E−04

.217E−08 .788E−08 .260E−07 .786E−07 .220E−06 .574E−06 .140E−05 .325E−05 .712E−05 .149E−04

.228E−08 .827E−08 .273E−07 .824E−07 .230E−06 .600E−06 .147E−05 .339E−05 .743E−05 .155E−04

100 110 120 130 140 150 160 170 180 190

.973E−05 .192E−04 .366E−04 .672E−04 .120E−03 .207E−03 .348E−03 .572E−03 .917E−03 .144E−02

.131E−04 .256E−04 .482E−04 .879E−04 .155E−03 .266E−03 .444E−03 .723E−03 .115E−02 .179E−02

.169E−04 .328E−04 .614E−04 .111E−03 .195E−03 .332E−03 .550E−03 .889E−03 .140E−02 .217E−02

.213E−04 .412E−04 .767E−04 .138E−03 .241E−03 .408E−03 .673E−03 .108E−02 .170E−02 .262E−02

.237E−04 .457E−04 .849E−04 .153E−03 .266E−03 .449E−03 .740E−03 .119E−02 .186E−02 .286E−02

.261E−04 .503E−04 .935E−04 .168E−03 .292E−03 .493E−03 .810E−03 .130E−02 .204E−02 .312E−02

.274E−04 .527E−04 .977E−04 .175E−03 .304E−03 .514E−03 .844E−03 .135E−02 .212E−02 .325E−02

.285E−04 .549E−04 .102E−03 .182E−03 .316E−03 .534E−03 .876E−03 .140E−02 .220E−02 .336E−02

.298E−04 .572E−04 .106E−03 .190E−03 .329E−03 .554E−03 .909E−03 .145E−02 .227E−02 .348E−02

.310E−04 .595E−04 .110E−03 .197E−03 .341E−03 .574E−03 .941E−03 .150E−02 .235E−02 .359E−02

200 210 220 230 240 250 260 270 280 290

.221E−02 .333E−02 .494E−02 .719E−02 .103E−01 .146E−01 .203E−01 .279E−01 .380E−01 .510E−01

.273E−02 .408E−02 .601E−02 .869E−02 .124E−01 .174E−01 .240E−01 .329E−01 .444E−01 .592E−01

.329E−02 .490E−02 .715E−02 .103E−01 .146E−01 .203E−01 .279E−01 .380E−01 .510E−01 .676E−01

.395E−02 .585E−02 .850E−02 .122E−01 .171E−01 .238E−01 .326E−01 .441E−01 .589E−01 .778E−01

.431E−02 .637E−02 .924E−02 .132E−01 .186E−01 .257E−01 .352E−01 .475E−01 .633E−01 .835E−01

.470E−02 .693E−02 .100E−01 .143E−01 .201E−01 .278E−01 .380E−01 .513E−01 .683E−01 .900E−01

.488E−02 .720E−02 .104E−01 .149E−01 .209E−01 .289E−01 .394E−01 .531E−01 .706E−01 .930E−01

.505E−02 .744E−02 .108E−01 .153E−01 .215E−01 .297E−01 .405E−01 .545E−01 .725E−01 .954E−01

.522E−02 .768E−02 .111E−01 .158E−01 .221E−01 .305E−01 .416E−01 .560E−01 .744E−01 .978E−01

.538E−02 .791E−02 .114E−01 .162E−01 .227E−01 .314E−01 .427E−01 .574E−01 .762E−01 .100

300 310 320 330 340 350

.678E−01 .892E−01 .116 .150 .192 .243

.782E−01 .102 .132 .170 .216 .272

.888E−01 .115 .149 .190 .240 .301

.102 .132 .169 .214 .270 .337

.109 .141 .180 .228 .287 .358

.117 .151 .193 .245 .307 .383

.121 .156 .199 .252 .317 .394

.124 .160 .204 .258 .328 .402

.127 .164 .209 .263 .330 .410

.130 .167 .213 .269 .386 .417

Weight percent, H2SO4

*Vermeulen, Dong, Robinson, Nguyen, and Gmitro, AIChE meeting, Anaheim, CA, 1982; and private communication from Prof. Theodore Vermeulen, Chemical Engineering Dept., University of California, Berkeley. 2-82

TABLE 2-15

Total Pressure, bar, of Aqueous Sulfuric Acid Solutions* Weight percent, H2SO4

°C 0 10 20 30 40 50 60 70 80 90

10.0 .582E−02 .117E−01 .223E−01 .404E−01 .703E−01 .117 .189 .296 .449 .664

20.0 .534E−02 .107E−01 .205E−01 .373E−01 .649E−01 .109 .175 .275 .417 .617

30.0 .448E−02 .909E−02 .174E−01 .319E−01 .558E−01 .939E−01 .152 .239 .365 .542

40.0 .326E−02 .670E−02 .130E−01 .241E−01 .427E−01 .725E−01 .119 .188 .290 .434

50.0 .193E−02 .405E−02 .802E−02 .151E−01 .272E−01 .470E−01 .782E−01 .126 .196 .298

60.0 .836E−03 .180E−02 .367E−02 .710E−02 .131E−01 .232E−01 .395E−01 .651E−01 .104 .161

70.0

75.0

80.0

85.0

.207E−03 .467E−03 .995E−03 .201E−02 .387E−02 .715E−02 .127E−01 .217E−01 .360E−01 .578E−01

.747E−04 .175E−03 .388E−03 .811E−03 .162E−02 .309E−02 .565E−02 .997E−01 .170E−01 .281E−01

.197E−04 .490E−04 .115E−03 .253E−03 .531E−03 .106E−02 .204E−02 .376E−02 .668E−02 .115E−01

.343E−05 .952E−05 .245E−04 .589E−04 .134E−03 .286E−03 .584E−03 .114E−02 .213E−02 .383E−02

.905E−01 .138 .206 .301 .431 .605 .837 1.138 1.525 2.017

.452E−01 .708E−01 .108 .162 .236 .339 .478 .662 .902 1.212

.192E−01 .312E−01 .493E−01 .760E−01 .115 .170 .246 .350 .489 .673

.666E−02 .112E−01 .183E−01 .291E−01 .451E−01 .683E−01 .101 .147 .209 .292

100 110 120 130 140 150 160 170 180 190

.957 1.349 1.863 2.524 3.361 4.404 5.685 7.236 9.093 11.289

.891 1.258 1.740 2.361 3.149 4.132 5.342 6.810 8.571 10.658

.786 1.113 1.544 2.101 2.810 3.697 4.793 6.127 7.731 9.640

.634 .904 1.264 1.732 2.333 3.090 4.031 5.185 6.584 8.259

.441 .638 .903 1.253 1.708 2.289 3.021 3.930 5.045 6.397

.244 .360 .519 .734 1.020 1.392 1.870 2.475 3.233 4.169

200 210 220 230 240 250 260 270 280 290

13.861 16.841 20.264 24.160 28.561 33.494 38.984 45.055 51.726 59.015

13.107 15.951 19.225 22.960 27.188 31.939 37.240 43.116 49.590 56.681

11.887 14.505 17.529 20.992 24.927 29.364 34.334 39.865 45.984 52.715

10.245 12.576 15.287 18.414 21.992 26.056 30.642 35.784 41.514 47.866

8.020 9.948 12.217 14.864 17.929 21.452 25.472 30.030 35.168 40.926

5.312 6.696 8.354 10.322 12.641 15.351 18.496 22.122 26.275 31.004

2.633 3.396 4.331 5.466 6.832 8.459 10.384 12.642 15.272 18.315

1.606 2.101 2.715 3.468 4.382 5.481 6.791 8.337 10.147 12.250

.913 1.221 1.610 2.098 2.701 3.439 4.332 5.402 6.673 8.170

.402 .544 .726 .956 1.242 1.594 2.023 2.540 3.157 3.886

300 310 320 330 340 350

66.934 75.495 84.705 94.567 105.083 116.251

64.407 72.781 81.816 91.518 101.894 112.947

60.081 68.101 76.792 86.172 96.252 107.043

54.869 62.553 70.947 80.078 89.970 100.647

47.347 54.470 62.338 70.990 80.466 90.806

36.361 42.398 49.168 56.727 65.130 74.437

21.814 25.812 30.355 35.489 41.262 47.723

14.675 17.453 20.611 24.182 28.193 32.674

9.916 11.939 14.264 16.916 19.920 23.303

4.740 5.732 6.876 8.185 9.672 11.351

Weight percent, H2SO4 °C

90.0

92.0

94.0

96.0

97.0

98.0

98.5

99.0

99.5

100.0

0 10 20 30 40 50 60 70 80 90

.518E−06 .159E−05 .449E−05 .117E−04 .385E−04 .653E−04 .141E−03 .291E−03 .571E−03 .107E−02

.243E−06 .765E−06 .221E−05 .590E−05 .147E−04 .344E−04 .759E−04 .159E−03 .319E−03 .612E−03

.109E−06 .348E−06 .102E−05 .279E−05 .708E−05 .169E−04 .380E−04 .813E−04 .166E−03 .324E−03

.416E−07 .136E−06 .407E−06 .113E−05 .293E−05 .712E−05 .164E−04 .357E−04 .742E−04 .148E−03

.235E−07 .774E−07 .235E−06 .659E−06 .173E−05 .425E−05 .987E−05 .218E−04 .458E−04 .921E−04

.117E−07 .391E−07 .121E−06 .344E−06 .914E−06 .228E−05 .538E−05 .120E−04 .257E−04 .524E−04

.768E−08 .261E−07 .812E−07 .234E−06 .630E−06 .159E−05 .379E−05 .856E−05 .184E−04 .390E−04

.479E−08 .166E−07 .528E−07 .155E−06 .425E−06 .109E−05 .264E−05 .605E−05 .132E−04 .277E−04

.313E−08 .113E−07 .373E−07 .114E−06 .323E−06 .861E−06 .216E−05 .514E−05 .117E−04 .253E−04

.323E−08 .124E−07 .435E−07 .141E−06 .425E−06 .120E−05 .319E−05 .804E−05 .193E−04 .441E−04

100 110 120 130 140 150 160 170 180 190

.195E−02 .340E−02 .575E−02 .944E−02 .151E−01 .235E−01 .357E−01 .532E−01 .775E−01 .111

.113E−02 .201E−02 .346E−02 .578E−02 .939E−02 .149E−01 .230E−01 .347E−01 .514E−01 .747E−01

.607E−03 .110E−02 .192E−02 .327E−02 .539E−02 .866E−02 .136E−01 .208E−01 .312E−01 .460E−01

.283E−03 .521E−03 .929E−03 .161E−02 .270E−02 .441E−02 .703E−02 .110E−01 .167E−01 .250E−01

.178E−03 .332E−03 .598E−03 .104E−02 .177E−02 .293E−02 .471E−02 .741E−02 .114E−01 .172E−01

.103E−03 .194E−03 .354E−03 .626E−03 .107E−02 .180E−02 .293E−02 .466E−02 .726E−02 .111E−01

.751E−04 .143E−03 .263E−03 .470E−03 .815E−03 .137E−02 .226E−02 .363E−02 .571E−02 .880E−02

.555E−04 .107E−03 .201E−03 .363E−03 .639E−03 .109E−02 .183E−02 .299E−02 .478E−02 .749E−02

.527E−04 .106E−03 .206E−03 .387E−03 .708E−03 .126E−02 .219E−02 .372E−02 .619E−02 .101E−01

.966E−04 .204E−03 .414E−03 .314E−03 .155E−02 .287E−02 .516E−02 .905E−02 .155E−01 .260E−01

.665E−01 .944E−01 .132 .182 .247 .331 .439 .575 .744 .954

.367E−01 .530E−01 .752E−01 .105 .145 .197 .264 .351 .460 .597

.255E−01 .371E−01 .531E−01 .749E−01 .104 .143 .193 .258 .341 .446

.166E−01 .245E−01 .354E−01 .505E−01 .710E−01 .985E−01 .135 .153 .245 .324

.133E−01 .198E−01 .289E−01 .417E−01 .592E−01 .830E−01 .115 .157 .213 .285

.115E−01 .175E−01 .260E−01 .382E−01 .553E−01 .790E−01 .112 .156 .215 .295

.161E−01 .253E−01 .392E−01 .596E−01 .895E−01 .132 .193 .279 .398 .562

.427E−01 .687E−01 .109 .169 .258 .389 .577 .846 1.225 1.751

200 210 220 230 240 250 260 270 280 290

.156 .216 .295 .396 .525 .688 .881 1.141 1.447 1.817

.107 .150 .207 .282 .379 .503 .660 .856 1.099 1.398

300 310 320 330 340 350

2.261 2.791 3.417 4.153 5.011 6.006

1.761 2.199 2.723 3.347 4.084 4.949

1.211 1.524 1.901 2.353 2.889 3.523

.767 .977 1.234 1.545 1.919 2.366

.578 .742 .944 1.191 1.491 1.852

.425 .553 .713 .911 1.156 1.456

.379 .498 .649 .840 1.078 1.374

.399 .536 .714 .944 1.239 1.614

.785 1.085 1.486 2.018 2.718 3.631

2.476 3.465 4.800 6.586 8.957 12.079

*Vermeulen, Dong, Robinson, Nguyen, and Gmitro, AIChE meeting, Anaheim, Calif., 1982; and private communication from Prof. Theodore Vermeulen, Chemical Engineering Dept., University of California, Berkeley. 2-83

2-84

PHYSICAL AND CHEMICAL DATA

TABLE 2-16

Partial Pressures of HNO3 and H2O over Aqueous Solutions of HNO3 mmHg Percentages are weight % HNO3 in solution. 20%

°C

25%

HNO3

H2O

HNO3

30% H2O

HNO3

35% H2O

0 5 10 15 20

4.1 5.7 8.0 10.9 15.2

3.8 5.4 7.6 10.3 14.2

3.6 5.0 7.1 9.7 13.2

25 30 35 40 45

20.6 27.6 36.5 47.5 62

19.2 25.7 33.8 44 57.5

17.8 23.8 31.1 41 53

0.09

0.11 .17

40%

HNO3

H2O

45%

HNO3

H2O

3.3 4.6 6.5 8.9 12.0 16.2 21.7 28.3 37.7 48

0.09 .13 .20 .28

0.12 .17 .25 .36 .52

50%

HNO3

H2O

HNO3

3.0 4.2 5.8 8.0 10.8

H2O

0.10 .15

2.6 3.6 5.0 6.9 9.4

0.12 .18 .27

2.1 3.0 4.2 5.8 7.9

14.6 19.5 25.5 33.5 43

.23 .33 .48 .68 .96

12.7 16.9 22.3 29.3 38.0

.39 .56 .80 1.13 1.57

10.7 14.4 19.0 25.0 32.5

49.5 62.5 80 100 126

2.18 2.95 4.05 5.46 7.25

42.5 54 70 88 110

50 55 60 65 70

0.09 .13 .19 .27

80 100 128 162 200

.13 .18 .28 .40 .54

75 94 121 151 187

.25 .35 .51 .71 1.00

69 87 113 140 174

.42 .59 .85 1.18 1.63

63 79 102 127 159

.75 1.04 1.48 2.05 2.80

56 71 90 114 143

1.35 1.83 2.54 3.47 4.65

75 80 85 90 95

.38 .53 .74 1.01 1.37

250 307 378 458 555

.77 1.05 1.44 1.95 2.62

234 287 352 426 517

1.38 1.87 2.53 3.38 4.53

217 267 325 393 478

2.26 3.07 4.15 5.50 7.32

198 243 297 359 436

3.80 5.10 6.83 9.0 11.7

178 218 268 325 394

6.20 8.15 10.7 13.7 17.8

158 195 240 292 355

9.6 12.5 16.3 20.9 26.8

138 170 211 258 315

100 105 110 115 120

1.87 2.50

675 800

3.50 4.65

628 745

6.05 7.90

580 690

530 631 755

15.5 20.0 25.7 32.5

480 573 688 810

23.0 29.2 37.0 46

430 520 625 740

34.2 43.0 54.5 67 84

383 463 560 665 785

°C

HNO3

H2O

HNO3

H2O

HNO3

H2O

HNO3

H2O

0 5 10 15 20

0.14 .21 .31 .45

1.8 2.5 3.5 4.9 6.7

0.19 .28 .41 .59 .84

1.5 2.1 3.0 4.1 5.6

0.41 .60 .86 1.21 1.68

1.3 1.8 2.6 3.5 4.9

0.79 1.12 1.58 2.18 3.00

1.1 1.6 2.2 3.0 4.1

25 30 35 40 45

.66 .93 1.30 1.82 2.50

9.1 12.2 16.1 21.3 28.0

1.21 1.66 2.28 3.10 4.20

7.7 10.3 13.6 18.1 23.7

2.32 3.17 4.26 5.70 7.55

6.6 8.8 11.6 15.5 20.0

4.10 5.50 7.30 9.65 12.6

5.5 7.4 9.8 12.8 16.7

50 55 60 65 70

3.41 4.54 6.15 8.18 10.7

36.3 46 60 76 95

5.68 7.45 9.9 13.0 16.8

31 39 51 64 81

10.0 12.8 16.8 21.7 27.5

26.0 33.0 43.0 54.5 68

16.5 21.0 27.1 34.5 43.3

21.8 27.3 35.3 44.5 56

75 80 85 90 95

13.9 18.0 23.0 29.4 37.3

120 148 182 223 272

21.8 27.5 34.8 43.7 55.0

102 126 156 192 233

35.0 43.5 54.5 67.5 83.5

100 105 110 115 120 125

47 58.5 73 90 110

331 400 485 575 685

69.5 84.5 103 126 156 187

285 345 417 495 590 700

55%

60%

9.7 12.7 16.5

65%

103 124 152 181 218 260

70%

80% HNO3

90%

100%

H2O

HNO3

H2O

HNO3

2 3 4 6 8

1.2 1.7 2.4

5.5 8 11 15 20

10.5 14 18.5 24.5 32

3.2 4 5.5 7 9.5

27 36 47 62 80

1 1.3 1.8 2.4 3

57 77 102 133 170

11 15 22 30 42

41 52 67 85 106

12 15 20 25 31

103 127 157 192 232

4 5 6.5 8 10

215 262 320 385 460 540 625 720 820

86 106 131 160 195

54.5 67.5 83 103 125

70 86 107 130 158

130 158 192 230 278

38 48 60 73 89

282 338 405 480 570

13 16 20 24 29

238 288 345 410 490 580

152 183 221 262 312 372

192 231 278 330 393 469

330 392 465 545 640

108 129 155 185 219

675 790

35 42

VAPOR PRESSURES OF SOLUTIONS TABLE 2-17 Partial Pressures of H2O and HBr over Aqueous Solutions of HBr at 20 to 55°C mmHg 20°C

% HBr

HBr

32 34 36 38 40 42 44 46 48 50 52 54 56 58 60

25°C H2O

0.09 .23 .71 2.2 6.8 21

6.2 4.5 3.3 2.4 1.7 1.3

HBr

50°C H2O

0.0016 .0022 .0033 .0061 .011 .023 .048 .10 .13 .37 1.1 3.2 9.3 27

HBr

8.2 6.1 4.5 3.3 2.4 1.9

1.3 3.2 7.2 17 40 91

TABLE 2-20 Total Vapor Pressures of Aqueous Solutions of CH3COOH Percentages of weight % acetic acid in the solution mmHg

55°C

H2O

HBr

30.2 24.3 19.3 16.0 13.3 10.4

H2O

2.0 4.6 10.2 23.0 51 115 260

2-85

°C

25%

50%

75%

20 25 30 35 40

16.3 22.1 29.6 39.4 51.7

15.7 21.4 28.8 38.3 50.2

15.3 20.8 27.8 36.6 48.1

45 50 55 60 65

67.0 87.2 110 141 178

65.0 85.0 107 138 172

62.0 80.1 102 130 162

70 75 80 85 90

223 277 342 419 510

216 269 331 407 497

203 251 310 376 458

95 100

618 743

602 725

550 666

38 31 25 21 18 14 11.4

TABLE 2-18 Partial Pressures of HI over Aqueous Solutions of HI at 25°C mmHg %HI pHI

4 0.00064

46 0.0010

48 0.0022

50 0.0050

52 0.013

54 0.035

56 0.10

TABLE 2-19 Vapor Pressures of the System: Water-Sulfuric Acid-Nitric Acid For these data reference must be made to the graphs of International Critical Tables, vol. 3, pp. 306–308.

TABLE 2-21

Vapor pressure of aqueous diethylene glycol solutions. (Courtesy of Carbide and Carbon Chemicals Corp.)

FIG. 2-3

Partial Pressures of H2O over Aqueous Solutions of NH3* Pressures are in pounds per square inch absolute Molal concentration of ammonia in the solutions in percentages (Weight concentration of ammonia in the solution in percentages)

0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

t, °F

(0)

32 40 50 60 70

0.09 0.084 .12 .115 .18 .17 .26 .24 .36 .34

0.079 .108 .16 .23 .32

0.074 .101 .15 .21 .30

0.070 .095 .14 .20 .28

0.065 .089 .13 .19 .26

0.060 .083 .12 .17 .25

0.056 .076 .11 .16 .23

0.051 .070 .10 .15 .21

0.047 .064 .094 .13 .19

0.042 .058 .085 .12 .17

0.038 .052 .076 .11 .15

0.034 .046 .068 .097 .14

0.030 .040 .059 .085 .12

0.025 .035 .051 .073 .10

0.021 .029 .042 .061 .086

0.017 .023 .034 .049 .069

0.013 .015 .025 .037 .052

0.008 .012 .017 .024 .034

0.004 .006 .008 .012 .017

80 90 100 110 120

.51 .48 .70 .66 .95 .90 1.27 1.20 1.69 1.60

.45 .63 .85 1.14 1.51

.42 .58 .79 1.07 1.42

.40 .55 .74 1.00 1.33

.37 .51 .69 .93 1.24

.34 .47 .64 .86 1.15

.32 .44 .59 .80 1.06

.29 .40 .55 .73 .97

.27 .37 .50 .67 .89

.24 .33 .45 .60 .80

.22 .30 .41 .54 .72

.19 .26 .36 .48 .64

.17 .23 .31 .42 .56

.14 .20 .27 .36 .48

.12 .16 .22 .30 .40

.096 .13 .18 .24 .32

.072 .10 .13 .18 .24

.048 .066 .090 .120 .160

.024 .033 .045 .061 .081

130 140 150 160 170

2.22 2.89 3.72 4.74 5.99

2.10 2.73 3.51 4.48 5.66

1.98 2.57 3.31 4.22 5.34

1.86 2.42 3.11 3.97 5.02

1.74 2.26 2.91 3.71 4.70

1.62 2.11 2.72 3.46 4.38

1.51 1.96 2.52 3.22 4.07

1.39 1.81 2.33 2.97 3.75

1.28 1.66 2.14 2.73 3.45

1.17 1.52 1.95 2.49 3.15

1.05 1.37 1.76 2.25 2.84

.95 1.23 1.59 2.02 2.56

.84 1.10 1.41 1.80 2.28

.74 .96 1.24 1.58 1.99

.63 .82 1.06 1.35 1.71

.53 .69 .88 1.12 1.42

.42 .55 .71 .90 1.13

.32 .41 .53 .67 1.85

.210 .270 .350 .450 .570

.100 .140 .180 .220 .300

180 7.51 7.10 190 9.34 8.83 200 11.53 10.90 210 14.12 13.35 220 17.19 16.25

6.69 8.32 10.27 12.58 15.32

6.30 7.82 9.65 11.82 14.39

5.89 7.32 9.04 11.07 13.48

5.49 6.83 8.43 10.32 12.57

5.10 6.34 7.83 9.59 11.67

4.71 5.86 7.23 8.86 10.78

4.33 5.38 6.64 8.13 9.90

3.94 4.91 6.06 7.42 9.03

3.57 4.44 5.48 6.71 8.17

3.21 3.99 4.93 6.04 7.31

2.85 3.55 4.38 5.34

2.50 3.10 3.81

2.14 2.65

1.77

1.42

1.06

230 20.78 19.64 240 24.97 23.60 250 29.83 28.20

18.51 22.25 26.58

17.40 20.91 25.00

16.29 19.58 23.39

15.19 18.26 21.82

14.11 16.95 20.25

13.03 15.66 18.71

11.97 14.38 17.18

10.91 13.12 15.67

9.87 11.86

(4.74) (9.50) (14.29) (19.10) (23.94) (28.81) (33.71) (38.64) (43.59) (48.57) (53.58) (58.62) (63.69) (68.79) (73.91) (79.07) (84.26) (89.47) (94.72)

*Wilson, Univ. Ill., Eng. Expt. Sta. Bull. 146.

TABLE 2-22

Mole Percentages of H2O over Aqueous Solutions of NH3* Molal concentration of ammonia in the solutions in percentages (Weight concentration of ammonia in the solutions in percentages)

t, °F

0 (0)

5 (4.74)

10 (9.50)

15 (14.29)

20 (19.10)

25 (23.94)

30 (28.81)

35 (33.71)

40 (38.64)

45 (43.59)

50 (48.57)

55 (53.58)

60 (58.62)

65 (63.69)

70 (68.79)

75 (73.91)

80 (79.07)

85 (84.26)

90 (89.47)

95 (94.72)

32 40 50 60 70

100 100 100 100 100

24.3 25.3 26.6 27.9 29.1

13.2 14.1 15.2 16.2 17.4

7.63 8.15 9.09 9.50 10.30

4.43 4.73 5.24 5.69 6.14

2.50 2.74 3.03 3.42 3.65

1.43 1.59 1.78 1.97 2.27

0.856 .943 1.060 1.210 1.390

0.514 .581 .652 .777 .873

0.335 .372 .434 .481 .569

0.216 .248 .290 .331 .383

0.151 .172 .202 .238 .266

0.109 .124 .148 .172 .205

0.0816 .0914 .1095 .1290 .1510

0.0585 .0706 .0838 .0986 .112

0.0457 .0533 .0630 .0754 .0882

0.0345 .0395 .0477 .0566 .0656

0.0249 .0243 .0332 .0406 .0474

0.0146 .0185 .0215 .0251 .0296

0.00689 .00879 .00959 .01125 .0135

80 90 100 110 120

100 100 100 100 100

31.6 32.7 34.4 35.9 37.5

18.5 20.0 21.0 22.2 23.4

11.20 12.00 12.90 13.80 14.70

6.89 7.40 7.92 8.59 9.22

4.08 4.47 4.85 5.29 5.75

2.45 2.73 3.00 3.30 3.63

1.550 1.730 1.890 2.110 2.320

.978 1.100 1.250 1.370 1.520

.659 .742 .834 .932 1.044

.444 .505 .574 .644 .714

.323 .366 .420 .466 .529

.230 .267 .307 .347 .395

.1750 .2020 .2290 .2640 .3020

.130 .157 .179 .208 .233

.103 .115 .135 .157 .180

.0772 .0884 .104 .118 .135

.0528 .0647 .0714 .0846 .0970

.0351 .0408 .0473 .0540 .0619

.0167 .0194 .0226 .0262 .0300

130 140 150 160 170

100 100 100 100 100

39.0 40.7 42.3 44.1 45.6

24.5 25.8 27.1 28.3 29.6

15.60 16.50 17.50 18.40 19.40

9.85 10.50 11.20 11.90 12.70

6.18 6.69 7.19 7.69 8.22

3.95 4.28 4.63 5.01 5.38

2.550 2.790 3.080 3.300 3.580

1.690 1.860 2.040 2.230 2.430

1.160 1.286 1.410 1.550 1.700

.811 .906 1.004 1.110 1.220

.596 .663 .741 .818 .904

.444 .501 .558 .617 .689

.3430 .3840 .4320 .4800 .5300

.263 .297 .334 .372 .414

.205 .232 .257 .287 .320

.154 .175 .197 .218 .242

.1117 .124 .140 .154 .174

.0703 .0786 .0892 .1005 .112

.0339 .0385 .0439 .0499 .0567

180 190 200 210 220

100 100 100 100 100

47.3 48.7 50.4 52.1 53.7

30.9 32.2 33.4 34.7 36.1

20.40 21.40 22.30 23.40 24.40

13.40 14.10 14.90 15.70 16.40

8.76 9.31 9.88 10.45 11.05

5.78 6.18 6.59 7.03 7.48

3.870 4.160 4.470 4.780 5.100

2.640 2.860 3.080 3.310 3.560

1.850 2.020 2.190 2.360 2.540

1.340 1.460 1.580 1.720 1.860

.994 1.087 1.187 1.272 1.390

.756 .830 .907 .983

.5860 .6420 .7010

.456 .501

.352

.268

.192

230 240 250

100 100 100

55.2 56.8 58.4

37.3 38.6 39.8

25.40 26.50 27.50

17.30 18.00 18.80

11.63 12.24 12.88

7.91 8.36 8.82

5.440 5.780 6.120

3.810 4.060 4.340

2.730 2.920 3.120

2.000 2.150

*Wilson, Univ. Ill., Eng. Expt. Sta. Bull. 146.

2-86

100 (100.00) 0.00

VAPOR PRESSURES OF SOLUTIONS TABLE 2-23

2-87

Partial Pressures of NH3 over Aqueous Solutions of NH3* Pressures are in pounds per square inch absolute Molal concentration of ammonia in the solutions in percentages (Weight concentration of ammonia in the solutions in percentages)

t, °F

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 (4.74) (9.50) (14.29) (19.10) (23.94) (28.81) (33.71) (38.64) (43.59) (48.57) (53.58) (58.62) (63.69) (68.79) (73.91) (79.07) (84.26) (89.47) (94.72)

32 40 50 60 70

0.26 .33 .47 .62 .83

0.52 .66 .89 1.19 1.52

0.90 1.14 1.50 2.00 2.60

1.51 1.92 2.53 3.21 4.28

2.67 3.16 4.16 5.36 6.87

4.27 5.13 6.63 8.48 10.76

6.54 7.98 10.24 13.06 16.33

8.93 11.98 15.24 19.15 23.84

14.13 17.14 21.56 26.92 33.20

19.36 23.33 29.17 36.14 44.25

25.12 30.15 37.46 46.12 56.29

31.13 37.15 45.86 56.22 68.32

36.74 43.69 53.79 65.81 79.42

42.69 49.56 60.82 73.99 89.26

45.92 54.40 66.63 80.90 97.42

49.26 58.31 71.26 86.44 104.01

52.13 61.62 75.22 91.04 109.55

54.89 64.77 79.05 95.67 114.83

58.01 68.31 83.40 100.65 120.61

80 90 100 110 120

1.04 1.36 1.72 2.14 2.67

1.98 2.52 3.20 4.00 4.95

3.34 4.25 5.34 6.65 8.21

5.45 6.88 8.60 10.64 13.09

8.69 10.89 13.53 16.65 20.30

13.52 16.76 20.68 25.21 30.54

20.29 25.04 30.57 37.01 44.56

29.40 35.94 43.57 52.43 62.62

40.69 49.45 59.49 71.20 84.44

53.84 64.99 77.85 92.59 109.40

67.97 81.61 97.27 115.16 135.48

82.36 98.35 116.81 137.62 161.44

95.52 113.79 134.70 158.42 185.14

107.06 127.22 150.23 176.18 205.81

116.42 138.18 162.94 190.85 222.28

124.20 147.02 173.22 203.02 236.05

130.57 154.46 181.97 212.71 247.14

136.35 161.74 190.13 222.22 258.24

143.70 169.73 199.17 232.79 270.02

130 140 150 160 170

3.28 3.97 4.78 5.68 6.75

6.09 7.41 8.92 10.70 12.67

10.05 12.21 14.70 17.57 20.85

15.93 19.23 23.09 27.45 32.41

24.58 29.43 35.09 41.56 48.89

36.74 43.77 51.91 61.03 71.48

53.16 62.97 74.28 86.91 101.09

74.27 87.53 102.51 119.37 138.30

99.69 116.72 136.15 157.71 181.95

128.45 149.93 173.64 200.45 230.36

158.45 184.17 212.91 244.98 280.54

188.16 218.18 251.24 288.38 329.42

215.14 248.70 286.00 327.82 373.61

238.70 275.33 316.24 361.75 411.59

257.87 297.12 340.82 389.08 442.28

272.88 314.45 360.39 411.30 466.67

286.08 328.99 376.57 429.73 487.85

298.46 342.93 392.45 447.35 507.63

311.80 358.46 409.62 466.38 528.50

180 7.90 190 9.23 200 10.70 210 12.26 220 14.02

14.96 17.55 20.45 23.68 27.15

24.56 28.78 33.49 38.76 44.61

38.13 44.49 51.58 59.65 68.43

57.19 66.49 76.90 88.48 101.24

83.07 96.22 110.85 126.83 144.74

116.97 134.89 154.58 176.24 200.46

159.37 182.72 208.56 236.97 268.30

208.66 238.39 270.94 307.08 346.07

263.43 299.86 340.02 383.99 431.43

319.89 363.11 410.17 462.36 518.19

374.25 424.15 478.62 537.56

424.10 479.40 539.79

466.26 526.15

500.63

528.08

551.24

230 15.95 240 17.92 250 20.12

31.09 35.40 40.09

51.06 58.00 65.74

78.14 89.02 100.69

115.45 130.94 147.66

164.17 185.79 209.37

226.67 255.26 286.89

302.53 339.72 380.42

389.29 435.78 486.73

483.53 540.44

*Wilson, Univ. Ill., Eng. Expt. Sta. Bull. 146.

TABLE 2-24

Total Vapor Pressures of Aqueous Solutions of NH3* Pressures are in pounds per square inch absolute Molal concentration of ammonia in the solutions in percentages (Weight concentration of ammonia in the solutions in percentages)

t, °F

0 (0)

5 (4.74)

10 (9.50)

15 (14.29)

20 (19.10)

25 (23.94)

30 (28.81)

35 (33.71)

40 (38.64)

45 (43.59)

50 (48.57)

55 (53.58)

60 (58.62)

65 (63.69)

70 (68.79)

75 (73.91)

80 (79.07)

85 (84.26)

90 (89.47)

95 (94.72)

100 (100.00)

32 40 50 60 70

0.09 .12 .18 .26 .36

0.34 .45 .64 .86 1.17

0.60 .77 1.05 1.42 1.84

0.97 1.24 1.65 2.21 2.90

1.58 2.01 2.67 3.51 4.56

2.60 3.25 4.29 5.55 7.13

4.20 5.21 6.75 8.65 11.01

6.54 8.06 10.35 13.22 16.56

9.93 12.05 15.34 19.30 24.05

14.18 17.20 21.65 27.05 33.39

19.40 23.39 29.26 36.26 44.42

25.16 30.20 37.54 46.23 56.44

31.16 37.20 45.93 56.32 68.46

36.77 43.73 53.85 65.90 79.54

42.72 49.60 60.87 74.06 89.36

45.94 54.43 66.67 80.96 97.51

49.28 58.33 71.29 86.49 104.08

52.14 61.64 75.25 91.08 109.60

54.90 64.78 79.07 95.69 114.86

58.01 68.32 83.41 100.66 120.63

62.29 73.32 89.19 107.6 128.8

80 90 100 110 120

.51 .70 .95 1.27 1.69

1.52 2.02 2.62 3.34 4.27

2.43 3.15 4.05 5.14 6.46

3.76 4.83 6.13 7.72 9.63

5.85 7.43 9.34 11.64 14.42

9.06 11.40 14.22 17.58 21.54

13.86 17.23 21.32 26.07 31.69

20.61 25.48 31.16 37.81 45.62

29.69 36.34 44.12 53.16 63.59

40.96 49.82 59.99 71.87 85.33

54.08 65.32 78.30 93.19 110.2

68.19 81.91 97.68 115.7 136.2

82.55 98.61 117.17 138.10 162.08

95.69 114.02 135.01 158.84 185.70

107.20 127.42 150.50 176.54 206.29

116.54 138.34 163.16 191.15 222.68

124.30 147.15 173.40 203.26 236.37

130.64 154.56 182.10 212.89 247.38

136.40 161.81 190.22 222.34 258.40

143.72 169.76 199.22 232.85 270.1

153.0 180.6 211.9 247.0 286.4

130 140 150 160 170

2.22 2.89 3.72 4.74 5.99

5.38 6.70 8.29 10.16 12.41

8.07 9.98 12.23 14.92 18.01

11.91 14.63 17.81 21.54 25.87

17.67 21.49 26.00 31.16 37.11

26.20 31.54 37.81 45.02 53.27

38.25 45.73 54.43 64.25 75.55

54.55 64.78 76.61 89.88 104.84

75.55 89.19 104.65 122.10 141.75

100.86 118.24 138.1 160.2 185.1

129.5 151.3 175.4 202.7 233.2

159.0 185.4 214.5 247.0 283.1

189.00 219.28 252.65 290.18 331.7

215.88 249.66 287.24 329.4 375.6

239.33 276.15 317.3 363.1 413.3

258.40 297.81 341.7 390.2 443.7

273.3 315.0 361.1 412.2 467.8

286.4 329.4 377.1 430.4 488.7

298.67 343.2 392.8 447.8 508.2

311.9 358.6 409.8 466.6 528.8

330.3 379.1 432.2 492.8 558.4

180 190 200 210 220

7.51 9.34 11.53 14.12 17.19

15.00 18.06 21.60 25.61 30.27

21.65 25.87 30.72 36.26 42.47

30.86 36.60 43.14 50.58 59.00

44.02 51.81 60.62 70.72 81.91

62.68 73.32 85.33 98.80 113.81

88.17 102.56 118.68 136.42 156.41

121.68 140.75 161.81 185.10 211.24

163.7 188.1 215.2 245.1 278.2

212.6 243.3 277.0 314.5 355.1

267.0 304.3 345.5 390.7 439.6

323.1 367.1 415.1 468.4 525.5

377.1 427.7 483.0 542.9

426.6 482.5 543.6

468.4 528.8

502.4

529.5

552.3

230 240 250

20.78 24.97 29.83

35.59 41.52 48.32

49.60 57.65 66.67

68.46 78.91 90.74

94.43 108.60 124.08

130.64 149.20 169.48

178.28 202.74 229.62

239.70 270.92 305.60

314.5 354.1 397.6

400.2 448.9 502.4

493.4 552.3

*Wilson, Univ. Ill., Eng. Expt. Sta. Bull. 146.

2-88

VAPOR PRESSURES OF SOLUTIONS TABLE 2-25 Partial Pressures of H2O over Aqueous Solutions of Sodium Carbonate mmHg %Na2CO3 t, °C

0

5

10

15

0 10 20 30 40 50 60 70 80 90 100

4.5 9.2 17.5 31.8 55.3 92.5 149.5 239.8 355.5 526.0 760.0

4.5 9.0 17.2 31.2 54.2 90.7 146.5 235 348 516 746

8.8 16.8 30.4 53.0 88.7 143.5 230.5 342 506 731

16.3 29.6 57.6 86.5 139.9 225 334 494 715

20

28.8 50.2 84.1 136.1 219 325 482 697

25

27.8 48.4 81.2 131.6 211.5 315 467 676

30

26.4 46.1 77.5 125.7 202.5 301 447 648

2-89

TABLE 2-26 Partial Pressures of H2O and CH3OH over Aqueous Solutions of Methyl Alcohol* 39.9°C

Mole fraction CH3OH

PH2O, mmHg

PCH3OH , mmHg

0 14.99 17.85 21.07 27.31 31.06 40.1 47.0 55.8 68.9 86.0 100.0

54.7 39.2 38.5 37.2 35.8 34.9 32.8 31.5 27.3 20.7 10.1 0

0 66.1 75.5 85.2 100.6 108.8 127.7 141.6 158.4 186.6 225.2 260.7

59.4°C

Mole fraction CH3OH

PH2O, mmHg

PCH3OH , mmHg

0 22.17 27.40 33.24 39.80 47.08 55.5 69.2 78.5 85.9 100.0

145.4 106.9 102.2 96.6 91.7 84.8 76.9 57.8 43.8 30.1 0

0 210.1 240.2 272.1 301.9 335.6 373.7 439.4 486.6 526.9 609.3

*International Critical Tables, vol. 3, McGraw-Hill, p. 290.

TABLE 2-27 Conc. g NaOH/ 100 g H2O 0 5 10 20 30 40 50 60 70 80 90 100 120 140 160 180 200 250 300 350 400 500 700 1000 2000 4000 8000

Partial Pressures of H2O over Aqueous Solutions of Sodium Hydroxide mmHg Temperature, °C 0

20

40

60

80

100

120

160

200

250

300

350

4.6 4.4 4.2 3.6 2.9 2.2

17.5 16.9 16.0 13.9 11.3 8.7 6.3 4.4 3.0 2.0 1.3 0.9

55.3 53.2 50.6 44.2 36.6 28.7 20.7 15.5 10.9 7.6 5.2 3.6 1.7

149.5 143.5 137.0 120.5 101.0 81.0 62.5 47.0 34.5 24.5 17.5 12.5 6.3 3.0 1.5

355.5 341.5 325.5 288.5 246.0 202.0 160.5 124.0 94.0 70.5 53.0 38.5 20.5 11.0 6.0 3.5 2.0 0.5 0.1

760.0 730.0 697.0 621.0 537.0 450.0 368.0 294.0 231.0 179.0 138.0 105.0 61.0 35.5 20.5 12.0 7.0 2.0 0.5

1,489 1,430 1,365 1,225 1,070 920 770 635 515 415 330 262 164 102 63 40 25 8 2.7 0.9

4,633 4,450 4,260 3,860 3,460 3,090 2,690 2,340 2,030 1,740 1,490 1,300 915 765 470 340 245 110 50 23 11

11,647 11,200 10,750 9,800 8,950 8,150 7,400 6,750 6,100 5,500 5,000 4,500 3,650 2,980 2,430 1,980 1,620 985 610 380 240 100

29,771 28,600 27,500 25,300 23,300 21,500 19,900 18,400 17,100 15,800 14,700 13,650 11,800 10,300 8,960 7,830 6,870 5,000 3,690 2,750 2,080 1,210 440

64,200 61,800 59,300 54,700 50,800 47,200 44,100 41,200 38,700 36,300 34,200 32,200 28,800 25,900 23,300 21,200 19,200 15,400 12,500 10,300 8,600 6,100 3,300 1,470 150

123,600 118,900 114,100 105,400 98,000 91,600 85,800 80,700 76,000 71,900 68,100 64,600 58,600 53,400 49,000 45,100 41,800 35,000 29,800 25,700 22,400 17,500 11,500 6,800 1,760 120 7

2-90

PHYSICAL AND CHEMICAL DATA

WATER-VAPOR CONTENT OF GASES CHART FOR GASES AT HIGH PRESSURES The accompanying figure is useful in determining the water-vapor content of air at high pressure in contact with liquid water.

Water content of air, °C = (°F − 32) × 5⁄ 9. (Landsbaum, Dadds, and Stutzman. Reprinted from vol. 47, January 1955 issue of Ind. Eng. Chem. [p. 192]. Copyright 1955 by the American Chemical Society and reproduced by permission of the copyright owner.)

FIG. 2-4

DENSITIES OF PURE SUBSTANCES

2-91

DENSITIES OF PURE SUBSTANCES UNITS CONVERSIONS For this subsection, the following units conversions are applicable:

To convert kilograms per cubic meter to pounds per cubic foot, multiply by 0.06243.

°F = 9⁄ 5 °C + 32.

TABLE 2-28

Density (kg/m3) of Water from 0 to 100°C* ρ, kg/m3

t, °C

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1 2 3 4

999.839 999.898 999.940 999.964 999.972

999.846 999.903 999.943 999.966 999.972

999.852 999.908 999.946 999.967 999.972

999.859 999.913 999.949 999.968 999.971

999.865 999.917 999.952 999.969 999.971

999.871 999.921 999.954 999.970 999.970

999.877 999.925 999.956 999.971 999.969

999.882 999.929 999.959 999.971 999.968

999.888 999.933 999.961 999.972 999.967

999.893 999.936 999.962 999.972 999.965

5 6 7 8 9

999.964 999.940 999.901 999.848 999.781

999.962 999.937 999.897 999.842 999.773

999.960 999.934 999.892 999.836 999.765

999.958 999.930 999.887 999.829 999.758

999.956 999.926 999.882 999.823 999.750

999.954 999.923 999.877 999.816 999.742

999.951 999.919 999.871 999.809 999.734

999.949 999.915 999.866 999.802 999.725

999.946 999.910 999.860 999.795 999.717

999.943 999.906 999.854 999.788 999.708

10 11 12 13 14

999.699 999.605 999.497 999.377 999.244

999.691 999.595 999.486 999.364 999.230

999.682 999.584 999.474 999.351 999.216

999.672 999.574 999.462 999.338 999.202

999.663 999.563 999.451 999.325 999.188

999.654 999.553 999.439 999.312 999.173

999.644 999.542 999.426 999.299 999.159

999.635 999.531 999.414 999.285 999.144

999.625 999.520 999.402 999.272 999.129

999.615 999.509 999.389 999.258 999.114

15 16 17 18 19

999.099 998.943 998.775 998.595 998.405

999.084 998.926 998.757 998.577 998.385

999.069 998.910 998.740 998.558 998.366

999.054 999.894 998.722 998.539 998.346

999.038 998.877 998.704 998.520 998.326

999.022 998.860 998.686 998.502 998.306

999.007 998.843 998.668 998.482 998.286

998.991 998.826 998.650 998.463 998.265

998.975 998.809 998.632 998.444 998.245

998.958 998.792 998.614 998.425 998.224

20 21 22 23 24

998.204 997.992 997.770 997.538 997.296

998.183 997.971 997.747 997.515 997.272

998.162 997.949 997.725 997.491 997.247

998.141 997.927 997.702 997.467 997.222

998.120 997.905 997.679 997.443 997.197

998.099 997.883 997.656 997.419 997.172

998.078 997.860 997.632 997.394 997.146

998.057 997.838 997.609 997.370 997.121

998.035 997.816 997.585 997.345 997.096

998.014 997.793 997.562 997.321 997.070

25 26 27 28 29

997.045 996.783 996.513 996.233 995.945

997.019 996.757 996.485 996.205 995.915

996.993 996.730 996.458 996.176 995.886

996.967 996.703 996.430 996.148 995.856

996.941 996.676 996.402 996.119 995.827

996.915 996.649 996.374 996.090 995.797

996.889 996.622 996.346 996.061 995.767

996.863 996.595 996.318 996.032 995.737

996.836 996.568 996.290 996.003 995.707

996.810 996.540 996.262 995.974 995.677

30 31 32 33 34

995.647 995.341 995.026 994.703 994.371

995.617 995.310 994.997 994.670 994.338

995.586 995.278 994.962 994.637 994.304

995.556 995.247 994.930 994.604 994.270

995.526 995.216 994.898 994.571 994.236

995.495 995.184 994.865 994.538 994.202

995.464 995.153 994.833 994.505 994.168

995.433 995.121 994.801 994.472 994.134

995.403 995.090 994.768 994.438 994.100

995.372 995.058 994.735 994.405 994.066

35 36 37 38 39

994.032 993.684 993.328 992.965 992.594

993.997 993.648 993.292 992.928 992.557

993.963 993.613 993.256 992.891 992.519

993.928 993.578 993.220 992.855 992.481

993.893 993.543 993.184 992.818 992.444

993.859 993.507 993.148 992.780 992.406

993.824 993.471 993.111 992.743 992.368

993.789 993.436 993.075 992.706 992.330

993.754 993.400 993.038 992.669 992.292

993.719 993.364 993.002 992.631 992.254

40 41 42 43 44

992.215 991.830 991.436 991.036 990.628

992.177 991.791 991.396 990.995 990.587

992.139 991.751 991.357 990.955 990.546

992.100 991.712 991.317 990.914 990.504

992.062 992.673 991.277 990.873 990.463

992.023 991.634 991.237 990.833 990.421

991.985 991.594 991.197 990.792 990.380

991.946 991.555 991.157 990.751 990.338

991.907 991.515 991.116 990.710 990.297

992.868 991.476 991.076 990.669 990.255

45 46 47 48 49

990.213 989.792 989.363 988.928 988.485

990.171 989.749 989.320 988.884 988.441

990.129 989.706 989.276 988.840 988.396

990.087 989.664 989.233 988.796 988.352

990.045 989.621 989.190 988.752 988.307

990.003 989.578 989.146 988.707 988.262

989.961 989.535 989.103 988.663 988.217

989.919 989.492 989.059 988.619 988.172

989.876 989.449 989.015 988.574 988.127

989.834 989.406 988.971 988.530 988.082

*From “Water: Density at Atmospheric Pressure and Temperatures from 0 to 100°C,” Tables of Standard Handbook Data, Standartov, Moscow, 1978. To conserve space, only a few tables of density values are given. The reader is reminded that density values may be found as the reciprocal of the specific volume values tabulated in the “Thermodynamic Properties: Tables” subsection.

2-92

PHYSICAL AND CHEMICAL DATA

TABLE 2-28

Density (kg/m3) of Water from 0 to 100°C (Concluded ) ρ, kg/m3

t, °C

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

50 51 52 53 54

988.037 987.581 987.120 986.652 986.177

987.992 987.536 987.073 986.604 986.129

987.946 987.490 987.027 986.557 986.081

987.901 987.444 986.980 986.510 986.033

987.844 987.398 986.933 986.463 985.985

987.810 987.351 986.886 986.415 985.937

987.764 987.305 986.840 986.368 985.889

987.719 987.259 986.793 986.320 985.841

987.673 987.213 986.746 986.272 985.793

987.627 987.166 986.699 986.225 985.745

55 56 57 58 59

985.696 985.219 984.716 984.217 983.712

985.648 985.160 984.666 984.167 983.661

985.599 985.111 984.617 984.116 983.610

985.551 985.062 984.567 984.066 983.559

985.502 985.013 984.517 984.016 983.508

985.454 984.963 984.467 983.965 983.457

985.405 984.914 984.417 983.914 983.406

985.356 984.865 984.367 983.864 983.354

985.307 984.815 984.317 983.813 983.303

985.258 984.766 984.267 983.762 983.252

60 61 62 63 64

983.200 982.683 982.160 981.631 981.097

983.149 982.631 982.108 981.578 981.043

983.097 982.579 982.055 981.525 980.989

983.046 982.527 982.002 981.472 980.935

982.994 982.475 981.949 981.418 980.881

982.943 982.422 981.897 981.365 980.827

982.891 982.370 981.844 981.311 980.773

982.839 982.318 981.791 981.258 980.719

982.787 982.265 981.738 981.204 980.665

982.735 982.213 981.685 981.151 980.611

65 66 67 68 69

980.557 980.011 979.459 978.902 978.339

980.502 979.956 979.403 978.846 978.283

980.443 979.901 979.348 978.790 978.226

980.393 979.846 979.293 978.734 978.170

980.339 979.791 979.237 978.678 978.113

980.284 979.736 979.181 978.621 978.056

980.230 979.680 979.126 978.565 977.999

980.175 979.625 979.070 978.509 977.942

980.120 979.570 979.014 978.452 977.885

980.065 979.515 978.958 978.396 977.828

70 71 72 73 74

977.771 977.198 976.619 976.035 975.445

977.714 977.140 976.561 975.976 975.386

977.657 977.082 976.503 975.917 975.327

977.600 977.025 976.444 975.858 975.267

977.543 976.967 976.386 975.800 975.208

977.485 976.909 976.327 975.741 975.148

977.428 976.851 976.269 975.682 975.089

977.370 976.793 976.211 975.623 975.029

977.313 976.735 976.152 975.564 974.970

977.255 976.677 976.093 975.504 974.910

75 76 77 78 79

974.850 974.250 973.645 973.025 972.419

974.791 974.190 973.584 972.974 972.358

974.731 974.130 973.524 972.912 972.296

974.671 974.069 973.463 972.851 972.234

974.611 974.009 973.402 972.789 972.172

974.551 973.948 973.341 972.728 972.110

974.491 973.888 973.280 972.666 972.048

974.431 973.827 973.218 972.605 971.986

974.371 973.767 973.157 972.543 971.923

974.311 973.706 973.096 972.481 971.861

80 81 82 83 84

971.799 971.173 970.543 969.907 969.267

971.737 971.110 970.479 969.843 969.202

971.674 971.048 970.416 969.772 969.138

971.612 970.985 970.353 969.715 969.073

971.549 970.922 970.289 969.652 969.009

971.487 970.859 970.226 969.587 968.944

971.424 970.796 970.162 969.523 968.880

971.361 970.732 970.098 969.459 968.815

971.299 970.669 970.035 969.395 968.751

971.236 970.606 969.971 969.331 968.686

85 86 87 88 89

968.621 967.971 967.316 966.656 965.991

968.556 967.906 967.250 966.589 965.924

968.491 967.840 967.184 966.523 965.857

968.427 967.775 967.118 966.457 965.790

968.362 967.709 967.052 966.390 965.723

968.297 967.641 966.986 966.324 965.656

969.232 967.578 966.920 966.257 965.589

968.166 967.513 966.854 966.191 965.522

968.101 967.447 966.788 966.124 965.455

968.036 967.381 966.722 966.057 965.388

90 91 92 93 94

965.321 964.647 963.967 963.284 962.595

965.254 954.579 963.899 963.215 962.526

965.187 964.511 963.831 963.146 962.457

965.119 964.443 963.763 963.077 962.387

965.052 964.376 963.694 963.009 962.318

964.984 964.308 963.626 962.940 962.249

964.917 964.240 963.558 962.871 962.180

964.849 964.172 963.489 962.802 962.110

964.782 964.104 963.421 962.733 962.041

964.714 964.036 963.352 962.664 961.971

95 96 97 98 99

961.902 961.204 960.501 959.794 959.082

961.832 961.134 960.431 959.723 959.010

961.762 961.064 960.360 959.652 958.939

961.693 960.993 960.289 959.581 958.867

961.693 960.923 960.219 959.510 958.796

961.553 960.853 960.148 959.438 958.724

961.483 960.783 960.077 959.367 958.653

961.414 960.712 960.006 959.296 958.581

961.344 960.642 959.936 959.225 958.509

961.274 960.572 959.865 959.153 958.431

100

958.365

DENSITIES OF PURE SUBSTANCES TABLE 2-29

2-93

Density (kg/m3) of Mercury from 0 to 350°C* Density, kg /m3

t, °C

2

3

4

5

6

7

8

9

0 10 20 30 40

13595.08 13570.44 13545.87 13521.36 13496.92

0

13592.61 13567.98 13543.41 13518.91 13494.48

1

13590.14 13565.52 13540.96 13516.47 13492.04

13587.68 13563.06 13538.51 13514.02 13489.60

13585.21 13560.60 13536.06 13511.58 13487.16

13582.75 13558.14 13533.61 13509.13 13484.72

13580.29 13555.69 13531.16 13506.69 13482.29

13577.82 13553.23 13528.71 13504.25 13479.85

13575.36 13550.78 13526.26 13501.80 13477.41

13572.90 13548.32 13523.81 13499.36 13474.98

50 60 70 80 90

13472.54 13448.22 13423.96 13399.75 13375.59

13470.11 13445.80 13421.54 13397.34 13373.18

13467.67 13443.37 13419.12 13394.92 13370.77

13465.24 13440.94 13416.69 13392.50 13368.36

13462.81 13438.51 13414.27 13390.08 13365.94

13460.38 13436.09 13411.85 13387.67 13363.53

13457.94 13433.66 13409.43 13385.25 13361.12

13455.51 13431.23 13407.01 13382.84 13358.71

13453.08 13428.81 13404.59 13380.42 13356.30

13450.65 13426.39 13402.17 13378.01 13353.89

100 110 120 130 140

13351.5 13327.4 13303.4 13279.4 13255.4

13349.1 13325.0 13301.0 13277.0 13253.0

13346.7 13322.6 13298.6 13274.6 13250.6

13344.3 13320.2 13296.2 13272.2 13248.2

13341.9 13317.8 13293.8 13269.8 13245.8

13339.4 13315.4 13291.4 13267.4 13243.4

13337.0 13313.0 13288.9 13265.0 13241.0

13334.6 13310.6 13286.6 13262.6 13238.7

13332.2 13308.2 13284.2 13260.2 13236.3

13329.8 13305.8 13281.8 13257.8 13233.9

150 160 170 180 190

13231.5 13207.6 13183.7 13159.8 13136.0

13229.1 13205.2 13181.3 13157.4 13133.6

13226.7 13202.8 13178.9 13155.0 13131.2

13224.3 13200.4 13176.5 13152.6 13128.3

13221.9 13198.0 13174.1 13150.3 13126.4

13219.5 13195.6 13171.7 13147.9 13124.0

13217.1 13193.2 13169.4 13145.5 13121.7

13214.7 13190.8 13167.0 13143.1 13119.3

13212.4 13188.5 13164.6 13140.7 13116.9

13210.0 13186.1 13162.2 13138.3 13114.5

200 210 220 230 240

13112.1 13088.3 13064.5 13040.6 13016.8

13109.7 13085.9 13062.1 13038.3 13014.5

13107.4 13083.5 13059.7 13035.9 13012.1

13105.0 13081.1 13057.3 13033.5 13009.7

13102.6 13078.8 13054.9 13031.1 13007.3

13100.2 13076.4 13052.6 13028.7 13004.9

13097.8 13074.0 13050.2 13026.4 13002.5

13095.4 13071.6 13047.8 13024.0 13000.2

13093.1 13069.2 13045.4 13021.6 12997.8

13090.7 13066.8 13043.0 13019.2 12995.4

250 260 270 280 290

12993.0 12969.2 12945.4 12921.5 12897.7

12990.6 12966.8 12943.0 12919.1 12895.3

12988.3 12964.4 12940.6 12916.7 12892.9

12985.9 12962.0 12938.2 12914.4 12890.5

12983.5 12959.7 12935.8 12912.0 12888.1

12981.1 12957.3 12933.4 12909.6 12885.7

12978.7 12954.9 12931.1 12907.2 12883.3

12976.3 12952.5 12928.7 12904.8 12880.9

12974.0 12950.1 12926.3 12902.4 12878.5

12971.6 12947.7 12923.9 12900.0 12876.2

300 310 320 330 340

12873.8 12849.9 12825.9 12801.9 12777.8

12871.4 12847.5 12823.5 12799.5 12775.4

12869.0 12845.1 12821.1 12797.1 12773.0

12866.6 12842.7 12818.7 12794.7 12770.6

12864.2 12840.3 12816.3 12792.3 12768.2

12861.8 12837.9 12813.9 12789.9 12765.8

12859.4 12835.5 12811.5 12787.5 12763.4

12857.0 12833.1 12809.1 12785.1 12761.0

12854.6 12830.7 12806.7 12782.7 12758.6

12852.2 12828.3 12804.3 12780.2 12756.1

350

12753.7

*From “Mercury—Density and Thermal Expansion at Atmospheric Pressure and Temperatures from 0 to 350°C,” Tables of Standard Handbook Data, Standartov, Moscow, 1978. The density values obtainable from those cited for the specific volume of the saturated liquid in the “Thermodynamic Properties” subsection show minor differences. No attempt was made to adjust either set.

TABLE 2-30 Cmpd. no.

Densities of Inorganic and Organic Liquids Formula

CAS no.

Mol. wt.

C1

C2

C3

C4

Tmin, K

Density at Tmin

Tmax, K

Density at Tmax

1 2 3 4 5 6 7 8 9 10

Methane Ethane Propane n-Butane n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane

CH4 C2H6 C3H8 C4H10 C5H12 C6H14 C7H16 C8H18 C9H20 C10H22

74828 74840 74986 106978 109660 110543 142825 111659 111842 124185

16.043 30.070 44.097 58.123 72.150 86.177 100.204 114.231 128.258 142.285

2.9214 1.9122 1.3757 1.0677 0.84947 0.70824 0.61259 0.53731 0.48387 0.42831

0.28976 0.27937 0.27453 0.27188 0.26726 0.26411 0.26211 0.26115 0.26147 0.25745

190.56 305.32 369.83 425.12 469.7 507.6 540.2 568.7 594.6 617.7

0.28881 0.29187 0.29359 0.28688 0.27789 0.27537 0.28141 0.28034 0.28281 0.28912

90.69 90.35 85.47 134.86 143.42 177.83 182.57 216.38 219.66 243.51

28.18 21.64 16.583 12.62 10.474 8.747 7.6998 6.6558 6.007 5.3811

190.56 305.32 369.83 425.12 469.7 507.6 540.2 568.7 594.6 617.7

10.082 6.845 5.011 3.927 3.178 2.682 2.337 2.058 1.851 1.664

11 12 13 14 15 16 17 18 19 20

n-Undecane n-Dodecane n-Tridecane n-Tetradecane n-Pentadecane n-Hexadecane n-Heptadecane n-Octadecane n-Nonadecane n-Eicosane

C11H24 C12H26 C13H28 C14H30 C15H32 C16H34 C17H36 C18H38 C19H40 C20H42

1120214 112403 629505 629594 629629 544763 629787 593453 629925 112958

156.312 170.338 184.365 198.392 212.419 226.446 240.473 254.500 268.527 282.553

0.39 0.35541 0.3216 0.30545 0.28445 0.26807 0.2545 0.23864 0.22451 0.21624

0.25678 0.25511 0.2504 0.2535 0.25269 0.25287 0.254 0.25272 0.25133 0.25287

639 658 675 693 708 723 736 747 758 768

0.2913 0.29368 0.3071 0.30538 0.30786 0.31143 0.31072 0.31104 0.3133 0.31613

247.57 263.57 267.76 279.01 283.07 291.31 295.13 301.31 305.04 309.58

4.9362 4.5132 4.2035 3.8924 3.6471 3.4187 3.2241 3.0466 2.8933 2.7496

21 22 23 24 25 26 27

2-Methylpropane 2-Methylbutane 2,3-Dimethylbutane 2-Methylpentane 2,3-Dimethylpentane 2,3,3-Trimethylpentane 2,2,4-Trimethylpentane

C4H10 C5H12 C6H14 C6H14 C7H16 C8H18 C8H18

75285 78784 79298 107835 565593 560214 540841

58.123 72.150 86.177 86.177 100.204 114.231 114.231

1.0463 0.9079 0.76929 0.73335 0.7229 0.6028 0.5886

0.27294 0.2761 0.27524 0.2687 0.28614 0.27446 0.27373

408.14 460.43 499.98 497.5 537.35 573.5 543.96

0.27301 0.28673 0.27691 0.28361 0.2713 0.2741 0.2846

113.54 113.25 145.19 119.55 160.00 172.22 165.78

28 29 30 31 32 33 34 35

Ethylene Propylene 1-Butene cis-2-Butene trans-2-Butene 1-Pentene 1-Hexene 1-Heptene

C2H4 C3H6 C4H8 C4H8 C4H8 C5H10 C6H12 C7H14

74851 115071 106989 590181 624646 109671 592416 592767

28.054 42.081 56.108 56.108 56.108 70.134 84.161 98.188

2.0961 1.4094 1.0972 1.1609 1.1426 0.9038 0.7389 0.63734

0.27657 0.26465 0.2649 0.27104 0.27095 0.26648 0.26147 0.26319

282.34 365.57 419.95 435.58 428.63 464.78 504.03 537.29

0.29147 0.295 0.29043 0.2816 0.2854 0.2905 0.2902 0.27375

36 37 38 39 40 41 42 43 44

1-Octene 1-Nonene 1-Decene 2-Methylpropene 2-Methyl-1-butene 2-Methyl-2-butene 1,2-Butadiene 1,3-Butadiene 2-Methyl-1,3-butadiene1

C8H16 C9H18 C10H20 C4H8 C5H10 C5H10 C4H6 C4H6 C5H8

111660 124118 872059 115117 563462 513359 590192 106990 78795

112.215 126.242 140.269 56.108 70.134 70.134 54.092 54.092 68.119

0.5871 0.4945 0.44244 1.1454 0.91619 0.93322 1.187 1.2384 0.95673

0.27005 0.26108 0.25838 0.2725 0.26752 0.27251 0.26114 0.2725 0.26488

566.65 593.25 616.4 417.9 465 471 452 425.17 484

45 46 47 48 49 50

Acetylene Methylacetylene Dimethylacetylene 3-Methyl-1-butyne 1-Pentyne 2-Pentyne

C2H2 C3H4 C4H6 C5H8 C5H8 C5H8

74862 74997 503173 598232 627190 627214

26.038 40.065 54.092 68.119 68.119 68.119

2.4091 1.6086 1.1717 0.94575 0.8491 0.92099

0.27223 0.26448 0.25895 0.26008 0.2352 0.25419

51 52 53

1-Hexyne 2-Hexyne 3-Hexyne1

C6H10 C6H10 C6H10

693027 764352 928494

82.145 82.145 82.145

0.84427 0.76277 0.78045

0.27185 0.25248 0.26065

2-94

Name

639 658 675 693 708 723 736 747 758 768

1.519 1.393 1.284 1.205 1.126 1.060 1.002 0.944 0.893 0.855

12.575 10.776 9.0343 9.2041 7.8746 7.0934 6.9163

408.14 460.43 499.98 497.5 537.35 573.5 543.96

3.833 3.288 2.795 2.729 2.526 2.196 2.150

104.00 87.89 87.80 134.26 167.62 107.93 133.39 154.27

23.326 18.143 14.326 13.895 13.1 11.543 9.6388 8.1759

282.34 365.57 419.95 435.58 428.63 464.78 504.03 537.29

7.579 5.326 4.142 4.283 4.217 3.392 2.826 2.422

0.27187 0.27319 0.28411 0.28186 0.28164 0.26031 0.3065 0.28813 0.28571

171.45 191.78 206.89 132.81 135.58 139.39 136.95 164.25 127.27

7.1247 6.333 5.7131 13.506 11.332 11.218 15.123 14.061 12.205

566.65 593.25 616.4 417.9 465 471 452 425.17 484

2.174 1.894 1.712 4.203 3.425 3.425 4.546 4.545 3.612

308.32 402.39 473.2 463.2 481.2 519

0.28477 0.279 0.27289 0.30807 0.353 0.31077

192.40 170.45 240.91 183.45 167.45 163.83

23.692 19.027 13.767 11.519 12.532 12.24

308.32 402.39 473.2 463.2 481.2 519

8.850 6.082 4.525 3.636 3.610 3.623

516.2 549 544

0.2771 0.31611 0.28571

141.25 183.65 170.05

10.23 10.133 10.021

516.2 549 544

3.106 3.021 2.994

54 55 56

1-Heptyne 1-Octyne Vinylacetylene2

C7H12 C8H14 C4H4

628717 629050 689974

96.172 110.199 52.076

0.67366 0.59229 1.2703

0.26003 0.26118 0.26041

559 585 454

0.29804 0.29357 0.297

192.22 193.55 173.15

8.4987 7.478 15.664

559 585 454

2.591 2.268 4.878

57 58 59 60 61 62

C5H10 C6H12 C7H14 C6H12 C7H14 C8H16

287923 96377 1640897 110827 108872 590669

70.134 84.161 98.188 84.161 98.188 112.215

1.124 0.84798 0.7193 0.8908 0.735 0.55873

0.28859 0.27042 0.26936 0.27396 0.27041 0.25143

511.76 532.79 569.52 553.58 572.19 591.15

0.2506 0.28276 0.2777 0.2851 0.2927 0.27758

179.28 130.73 134.71 279.69 146.58 239.66

11.883 10.492 9.018 9.3797 9.018 7.3417

511.76 532.79 569.52 553.58 572.19 591.15

3.895 3.136 2.670 3.252 2.718 2.222

63 64 65 66

Cyclopentane Methylcyclopentane Ethylcyclopentane Cyclohexane Methylcyclohexane 1,1-Dimethylcyclohexane Ethylcyclohexane Cyclopentene 1-Methylcyclopentene Cyclohexene

C8H16 C5H8 C6H10 C6H10

1678917 142290 693890 110838

112.215 68.119 82.145 82.145

0.61587 1.1035 0.88824 0.92997

0.26477 0.27035 0.26914 0.27056

609.15 507 542 560.4

0.28054 0.28699 0.27874 0.28943

161.84 138.13 146.62 169.67

7.8679 13.47 10.98 11.16

609.15 507 542 560.4

2.326 4.082 3.300 3.437

67 68 69 70 71 72 73

Benzene Toluene o-Xylene m-Xylene p-Xylene Ethylbenzene Propylbenzene

C6H6 C7H8 C8H10 C8H10 C8H10 C8H10 C9H12

71432 108883 95476 108383 106423 100414 103651

78.114 92.141 106.167 106.167 106.167 106.167 120.194

1.0162 0.8488 0.69883 0.69555 0.6816 0.6952 0.57695

0.2655 0.26655 0.26113 0.26204 0.25963 0.26037 0.25395

562.16 591.8 630.33 617.05 616.23 617.2 638.32

0.28212 0.2878 0.27429 0.27602 0.2768 0.2844 0.283

278.68 178.18 247.98 225.30 286.41 178.15 183.15

11.421 10.495 8.6285 8.6505 8.1616 9.0568 7.8942

562.16 591.8 630.33 617.05 616.23 617.2 638.32

3.828 3.184 2.676 2.654 2.625 2.670 2.272

74 75 76 77 78 79 80 81

1,2,4-Trimethylbenzene Isopropylbenzene 1,3,5-Trimethylbenzene p-Isopropyltoluene Naphthalene6 Biphenyl Styrene m-Terphenyl

C9H12 C9H12 C9H12 C10H14 C10H8 C12H10 C8H8 C18H14

95636 98828 108678 99876 91203 92524 100425 92068

120.194 120.194 120.194 134.221 128.174 154.211 104.152 230.309

0.60394 0.604 0.59879 0.51036 0.61674 0.5039 0.7397 0.30826

0.25955 0.25912 0.25916 0.25383 0.25473 0.25273 0.2603 0.23669

649.13 631.1 637.36 653.15 748.35 789.26 636 924.85

0.27716 0.2914 0.27968 0.28816 0.27355 0.281 0.3009 0.29678

229.33 177.14 228.42 205.25 333.15 342.20 242.54 360.00

7.6895 7.9496 7.6154 6.8779 7.7543 6.4395 9.1088 4.5223

649.13 631.1 637.36 653.15 748.35 789.26 636 924.85

2.327 2.331 2.311 2.011 2.421 1.994 2.842 1.302

82 83 84 85 86 87 88

Methanol Ethanol 1-Propanol 1-Butanol 2-Butanol 2-Propanol 2-Methyl-2-propanol

CH4O C2H6O C3H8O C4H10O C4H10O C3H8O C4H10O

67561 64175 71238 71363 78922 67630 75650

32.042 46.069 60.096 74.123 74.123 60.096 74.123

2.288 1.648 1.235 0.965 0.966 1.24 0.9212

0.2685 0.27627 0.27136 0.2666 0.26064 0.27342 0.2544

512.64 513.92 536.78 563.05 536.05 508.3 506.21

0.2453 0.2331 0.24 0.24419 0.2746 0.2353 0.276

175.47 159.05 146.95 184.51 158.45 185.28 298.97

27.912 19.413 15.231 12.016 12.57 14.547 10.555

512.64 513.92 536.78 563.05 536.05 508.3 506.21

8.521 5.965 4.551 3.620 3.706 4.535 3.621

89 90 91 92 93 94 95 96

1-Pentanol 2-Methyl-1-butanol 3-Methyl-1-butanol 1-Hexanol 1-Heptanol Cyclohexanol Ethylene glycol 1,2-Propylene glycol

C5H12O C5H12O C5H12O C6H14O C7H16O C6H12O C2H6O2 C3H8O2

71410 137326 123513 111273 111706 108930 107211 57556

88.150 88.150 88.150 102.177 116.203 100.161 62.068 76.095

0.8164 0.82046 0.837 0.70617 0.60481 0.8243 1.3151 1.0923

0.2673 0.26829 0.27375 0.26901 0.2632 0.26546 0.25125 0.26106

586.15 565 577.2 611.35 631.9 650 719.7 626

0.2506 0.2322 0.22951 0.2479 0.273 0.2848 0.2187 0.20459

195.56 203.00 155.95 228.55 239.15 296.60 260.15 213.15

10.057 10.017 10.204 8.4506 7.421 9.4693 18.31 14.363

586.15 565 577.2 611.35 631.9 650 719.7 626

3.054 3.058 3.058 2.625 2.298 3.105 5.234 4.184

97 98 99 100

Phenol o-Cresol m-Cresol p-Cresol

C6H6O C7H8O C7H8O C7H8O

108952 95487 108394 106445

94.113 108.140 108.140 108.140

1.3798 1.0861 0.9061 1.1503

0.31598 0.30624 0.28268 0.31861

694.25 697.55 705.85 704.65

0.32768 0.30587 0.2707 0.30104

314.06 304.19 285.39 307.93

11.244 9.5751 9.6115 9.4494

694.25 697.55 705.85 704.65

4.367 3.547 3.205 3.610

101 102 103 104 105 106 107 108

Dimethyl ether Methyl ethyl ether Methyl-n-propyl ether Methyl isopropyl ether Methyl-n-butyl ether Methyl isobutyl ether1 Methyl tert-butyl ether Diethyl ether

C2H6O C3H8O C4H10O C4H10O C5H12O C5H12O C5H12O C4H10O

115106 540670 557175 598538 628284 625445 1634044 60297

46.069 60.096 74.123 74.123 88.150 88.150 88.150 74.123

1.5693 1.2635 1.0124 1.0318 0.8281 0.8252 0.82157 0.9554

0.2679 0.27878 0.27942 0.28478 0.27245 0.27282 0.27032 0.26847

400.1 437.8 476.3 464.5 510 497 497.1 466.7

0.2882 0.2744 0.2555 0.2444 0.2827 0.2857 0.2829 0.2814

131.65 160.00 133.97 127.93 157.48 150.00 164.55 156.85

18.95 13.995 11.696 11.568 9.8068 9.7673 9.7682 11.487

400.1 437.8 476.3 464.5 510 497 497.1 466.7

5.858 4.532 3.623 3.623 3.040 3.025 3.039 3.559

2-95

TABLE 2-30 Cmpd. no.

Densities of Inorganic and Organic Liquids (Continued ) CAS no.

Mol. wt.

C1

C2

C3

C4

Tmin, K

Density at Tmin

Tmax, K

Density at Tmax

109 110 111 112

Ethyl propyl ether Ethyl isopropyl ether Methyl phenyl ether Diphenyl ether

C5H12O C5H12O C7H8O C12H10O

628320 625547 100663 101848

88.150 88.150 108.140 170.211

0.7908 0.82049 0.77488 0.52133

0.266 0.26994 0.26114 0.26218

500.23 489 645.6 766.8

0.292 0.30381 0.28234 0.31033

145.65 140.00 235.65 300.03

9.8474 9.9117 9.6675 6.2648

500.23 489 645.6 766.8

2.973 3.040 2.967 1.988

113 114 115 116 117 118 119 120 121 122

Formaldehyde3 Acetaldehyde 1-Propanal 1-Butanal 1-Pentanal 1-Hexanal 1-Heptanal 1-Octanal 1-Nonanal 1-Decanal

CH2O C2H4O C3H6O C4H8O C5H10O C6H12O C7H14O C8H16O C9H18O C10H20O

50000 75070 123386 123728 110623 66251 111717 124130 124196 112312

30.026 44.053 58.080 72.107 86.134 100.161 114.188 128.214 142.241 156.268

1.9415 1.6994 1.296 1.0361 0.83871 0.71899 0.62649 0.56833 0.49587 0.46802

0.22309 0.26167 0.26439 0.26731 0.26252 0.26531 0.26376 0.26939 0.26135 0.27146

408 466 504.4 537.2 566.1 591 617 638.1 658 674.2

0.28571 0.2913 0.29471 0.28397 0.29444 0.27628 0.29221 0.26975 0.30736 0.26869

181.15 150.15 170.00 176.75 182.00 217.15 229.80 246.00 255.15 267.15

30.945 21.499 15.929 12.589 10.534 8.7243 7.6002 6.6637 6.0165 5.3834

408 466 504.4 537.2 566.1 591 617 638.1 658 674.2

8.703 6.494 4.902 3.876 3.195 2.710 2.375 2.110 1.897 1.724

123 124 125 126

C3H6O C4H8O C5H10O C5H10O

67641 78933 107879 563804

58.080 72.107 86.134 86.134

1.2332 0.93767 0.90411 0.8374

0.25886 0.25035 0.27207 0.26204

508.2 535.5 561.08 553

0.2913 0.29964 0.30669 0.2857

178.45 186.48 196.29 181.15

15.683 12.663 10.398 10.565

508.2 535.5 561.08 553

4.764 3.745 3.323 3.196

127 128 129 130 131 132 133 134

Acetone Methyl ethyl ketone 2-Pentanone Methyl isopropyl ketone1 2-Hexanone Methyl isobutyl ketone 3-Methyl-2-pentanone1 3-Pentanone Ethyl isopropyl ketone Diisopropyl ketone Cyclohexanone Methyl phenyl ketone

C6H12O C6H12O C6H12O C5H10O C6H12O C7H14O C6H10O C8H8O

591786 108101 565617 96220 565695 565800 108941 98862

100.161 100.161 100.161 86.134 100.161 114.188 98.145 120.151

0.70659 0.71791 0.6969 0.71811 0.66469 0.56213 0.8663 0.64417

0.26073 0.26491 0.2587 0.24129 0.24527 0.23385 0.26941 0.24863

587.05 571.4 573 560.95 567 576 653 709.5

0.2963 0.28544 0.2857 0.27996 0.34305 0.2618 0.2977 0.28661

217.35 189.15 167.15 234.18 200.00 204.81 242.00 292.81

8.7505 8.8579 9.1722 10.102 9.0933 8.7779 10.081 8.5581

587.05 571.4 573 560.95 567 576 653 709.5

2.710 2.710 2.694 2.976 2.710 2.404 3.216 2.591

135 136 137 138 139 140 141

Formic acid Acetic acid Propionic acid n-Butyric acid Isobutyric acid Benzoic acid1 Acetic anhydride

CH2O2 C2H4O2 C3H6O2 C4H8O2 C4H8O2 C7H6O2 C4H6O3

64186 64197 79094 107926 79312 65850 108247

46.026 60.053 74.079 88.106 88.106 122.123 102.090

1.938 1.4486 1.1041 0.89213 0.88575 0.71587 0.86852

0.24225 0.25892 0.25659 0.25938 0.25736 0.24812 0.25187

588 591.95 600.81 615.7 605 751 606

0.24435 0.2529 0.26874 0.24909 0.26265 0.2857 0.31172

281.45 289.81 252.45 267.95 227.15 395.45 200.15

26.806 17.492 13.933 11.087 11.42 8.8935 11.643

588 591.95 600.81 615.7 605 751 606

8.000 5.595 4.303 3.440 3.442 2.885 3.448

142 143 144 145 146 147 148

Methyl formate Methyl acetate Methyl propionate Methyl n-butyrate Ethyl formate Ethyl acetate Ethyl propionate

C2H4O2 C3H6O2 C4H8O2 C5H10O2 C3H6O2 C4H8O2 C5H10O2

107313 79209 554121 623427 109944 141786 105373

60.053 74.079 88.106 102.133 74.079 88.106 102.133

1.525 1.13 0.9147 0.76983 1.1343 0.8996 0.7405

0.2634 0.2593 0.2594 0.26173 0.26168 0.25856 0.25563

487.2 506.55 530.6 554.5 508.4 523.3 546

0.2806 0.2764 0.2774 0.26879 0.2791 0.278 0.2795

174.15 175.15 185.65 187.35 193.55 189.60 199.25

18.811 14.475 11.678 9.7638 14.006 11.478 9.6317

487.2 506.55 530.6 554.5 508.4 523.3 546

5.790 4.358 3.526 2.941 4.335 3.479 2.897

149 150 151 152 153 154 155

Ethyl n-butyrate n-Propyl formate n-Propyl acetate n-Butyl acetate Methyl benzoate Ethyl benzoate Vinyl acetate

C6H12O2 C4H8O2 C5H10O2 C6H12O2 C8H8O2 C9H10O2 C4H6O2

105544 110747 109604 123864 93583 93890 108054

116.160 88.106 102.133 116.160 136.150 150.177 86.090

0.63566 0.915 0.73041 0.669 0.53944 0.4883 0.9591

0.25613 0.26134 0.25456 0.26028 0.23519 0.23878 0.2593

571 538 549.73 579.15 693 698 519.13

0.27829 0.28 0.27666 0.309 0.2676 0.28487 0.27448

175.15 180.25 178.15 199.65 260.75 238.45 180.35

8.4912 11.59 9.7941 8.3747 8.2133 7.2924 12.287

571 538 549.73 579.15 693 698 519.13

2.482 3.501 2.869 2.570 2.294 2.045 3.699

156 157 158 159 160

Methylamine Dimethylamine Trimethylamine Ethylamine Diethylamine

CH5N C2H7N C3H9N C2H7N C4H11N

74895 124403 75503 75047 109897

31.057 45.084 59.111 45.084 73.138

1.39 1.5436 1.0116 1.1477 0.85379

0.21405 0.27784 0.25683 0.23182 0.25675

430.05 437.2 433.25 456.15 496.6

0.2275 0.2572 0.2696 0.26053 0.27027

179.69 180.96 156.08 192.15 223.35

25.378 16.964 13.144 17.588 10.575

430.05 437.2 433.25 456.15 496.6

6.494 5.556 3.939 4.951 3.325

2-96

Name

Formula

161 162 163 164 165 166 167 168

Triethylamine n-Propylamine di-n-Propylamine Isopropylamine Diisopropylamine Aniline N-Methylaniline N,N-Dimethylaniline

C6H15N C3H9N C6H15N C3H9N C6H15N C6H7N C7H9N C8H11N

121448 107108 142847 75310 108189 62533 100618 121697

101.192 59.111 101.192 59.111 101.192 93.128 107.155 121.182

0.7035 0.9195 0.659 1.2801 0.6181 1.0405 0.6527 0.4923

0.27386 0.23878 0.26428 0.2828 0.25786 0.2807 0.24324 0.22868

535.15 496.95 550 471.85 523.1 699 701.55 687.15

0.2872 0.2461 0.2766 0.2972 0.271 0.29236 0.25374 0.2335

158.45 188.36 210.15 177.95 176.85 267.13 216.15 275.60

8.2843 13.764 7.9929 13.561 8.0541 11.176 9.7244 7.9705

535.15 496.95 550 471.85 523.1 699 701.55 687.15

2.569 3.851 2.494 4.527 2.397 3.707 2.683 2.153

169 170 171 172

Ethylene oxide Furan Thiophene Pyridine

C2H4O C4H4O C4H4S C5H5N

75218 110009 110021 110861

44.053 68.075 84.142 79.101

1.836 1.1339 1.2875 0.9815

0.26024 0.24741 0.28195 0.24957

469.15 490.15 579.35 619.95

0.2696 0.2612 0.3077 0.29295

160.65 187.55 234.94 231.51

23.477 15.702 13.431 13.193

469.15 490.15 579.35 619.95

7.055 4.583 4.566 3.933

173 174

CH3NO C3H7NO

75127 68122

45.041 73.095

1.2486 0.89615

0.20352 0.23478

771 649.6

0.25178 0.28091

275.60 212.72

25.488 13.954

771 649.6

6.135 3.817

175 176 177 178 179 180

Formamide5 N,N-Dimethylformamide Acetamide N-Methylacetamide Acetonitrile Propionitrile n-Butyronitrile Benzonitrile

C2H5NO C3H7NO C2H3N C3H5N C4H7N C7H5N

60355 79163 75058 107120 109740 100470

59.068 73.095 41.053 55.079 69.106 103.123

1.016 0.88268 1.3064 1.0224 0.87533 0.73136

0.21845 0.23568 0.22597 0.23452 0.24331 0.24793

761 718 545.5 564.4 582.25 699.35

0.26116 0.27379 0.28678 0.2804 0.28586 0.2841

353.33 301.15 229.32 180.26 161.25 260.40

16.936 13.012 20.628 16.027 13.047 10.009

761 718 545.5 564.4 582.25 699.35

4.651 3.745 5.781 4.360 3.598 2.950

181 182 183 184 185 186 187 188 189

Methyl mercaptan Ethyl mercaptan n-Propyl mercaptan n-Butyl mercaptan Isobutyl mercaptan sec-Butyl mercaptan Dimethyl sulfide Methyl ethyl sulfide Diethyl sulfide

CH4S C2H6S C3H8S C4H10S C4H10S C4H10S C2H6S C3H8S C4H10S

74931 75081 107039 109795 513440 513531 75183 624895 352932

48.109 62.136 76.163 90.189 90.189 90.189 62.136 76.163 90.189

1.9323 1.3047 1.0714 0.89458 0.88801 0.89137 1.4029 1.067 0.82413

0.28018 0.2694 0.27214 0.27463 0.27262 0.27365 0.27991 0.27101 0.26333

469.95 499.15 536.6 570.1 559 554 503.04 533 557.15

0.28523 0.27866 0.29481 0.28512 0.29522 0.2953 0.2741 0.29363 0.27445

150.18 125.26 159.95 157.46 128.31 133.02 174.88 167.23 169.20

21.564 16.242 12.716 10.585 10.851 10.761 15.556 12.672 10.476

469.95 499.15 536.6 570.1 559 554 503.04 533 557.15

6.897 4.843 3.937 3.257 3.257 3.257 5.012 3.937 3.130

190 191 192 193 194 195 196 197

Fluoromethane Chloromethane Trichloromethane Tetrachloromethane Bromomethane Fluoroethane Chloroethane Bromoethane

CH3F CH3Cl CHCl3 CCl4 CH3Br C2H5F C2H5Cl C2H5Br

593533 74873 67663 56235 74839 353366 75003 74964

34.033 50.488 119.377 153.822 94.939 48.060 64.514 108.966

2.1854 1.817 1.0841 0.99835 1.6762 1.6525 2.176 1.1908

0.24725 0.25877 0.2581 0.274 0.26141 0.27099 0.3377 0.25595

317.42 416.25 536.4 556.35 467 375.31 460.35 503.8

0.27558 0.2833 0.2741 0.287 0.28402 0.2442 0.3361 0.29152

131.35 175.43 209.63 250.33 179.47 129.95 134.80 154.55

29.526 22.347 13.702 10.843 20.64 19.785 16.934 15.833

317.42 416.25 536.4 556.35 467 375.31 460.35 503.8

8.839 7.022 4.200 3.644 6.412 6.098 6.444 4.653

198 199 200 201 202 203 204 205

1-Chloropropane 2-Chloropropane 1,1-Dichloropropane1 1,2-Dichloropropane Vinyl chloride Fluorobenzene Chlorobenzene Bromobenzene

C3H7Cl C3H7Cl C3H6Cl2 C3H6Cl2 C2H3Cl C6H5F C6H5Cl C6H5Br

540545 75296 78999 78875 75014 462066 108907 108861

78.541 78.541 112.986 112.986 62.499 96.104 112.558 157.010

1.087 1.1202 0.91064 0.89833 1.5115 1.0146 0.8711 0.8226

0.26832 0.27669 0.26561 0.26142 0.2707 0.27277 0.26805 0.26632

503.15 489 560 572 432 560.09 632.35 670.15

0.28055 0.27646 0.28571 0.2868 0.2716 0.28291 0.2799 0.2821

150.35 155.97 200.00 172.71 119.36 230.94 227.95 242.43

13.328 12.855 11.03 11.526 18.481 11.374 10.385 9.9087

503.15 489 560 572 432 560.09 632.35 670.15

4.051 4.049 3.429 3.436 5.584 3.720 3.250 3.089

206 207 208 209 210 211 212 213 214 215 216 217

Air Hydrogen Helium-44 Neon Argon Fluorine Chlorine Bromine Oxygen Nitrogen Ammonia Hydrazine

H2 He Ne Ar F2 Cl2 Br2 O2 N2 NH3 N2H4

132259100 1333740 7440597 7440019 7440371 7782414 7782505 7726956 7782447 7727379 7664417 302012

28.951 2.016 4.003 20.180 39.948 37.997 70.905 159.808 31.999 28.014 17.031 32.045

2.8963 5.414 7.2475 7.3718 3.8469 4.2895 2.23 2.1872 3.9143 3.2091 3.5383 1.0516

0.26733 0.34893 0.41865 0.3067 0.2881 0.28587 0.27645 0.29527 0.28772 0.2861 0.25443 0.16613

132.45 33.19 5.2 44.4 150.86 144.12 417.15 584.15 154.58 126.2 405.65 653.15

0.27341 0.2706 0.24096 0.2786 0.29783 0.28776 0.2926 0.3295 0.2924 0.2966 0.2888 0.1898

59.15 13.95 2.20 24.56 83.78 53.48 172.12 265.85 54.35 63.15 195.41 274.69

33.279 38.487 37.115 61.796 35.491 44.888 24.242 20.109 40.77 31.063 43.141 31.934

132.45 33.19 5.2 44.4 150.86 144.12 417.15 584.15 154.58 126.2 405.65 653.15

10.834 15.516 17.312 24.036 13.353 15.005 8.067 7.408 13.605 11.217 13.907 6.330 2-97

TABLE 2-30 Cmpd. no. 218 219 220 221 222 223 224 225 226 227 228 229 230 231

Densities of Inorganic and Organic Liquids (Concluded ) Name

Nitrous oxide Nitric oxide Cyanogen Carbon monoxide Carbon dioxide Carbon disulfide Hydrogen fluoride Hydrogen chloride Hydrogen bromide1 Hydrogen cyanide Hydrogen sulfide Sulfur dioxide Sulfur trioxide Water7

Formula N2O NO C2N2 CO CO2 CS2 HF HCl HBr HCN H2S SO2 SO3 H2O

CAS no.

Mol. wt.

C1

C2

C3

C4

Tmin, K

10024972 10102439 460195 630080 124389 75150 7664393 7647010 10035106 74908 7783064 7446095 7446119 7732185

44.013 30.006 52.036 28.010 44.010 76.143 20.006 36.461 80.912 27.026 34.082 64.065 80.064 18.015

2.781 5.246 1.0761 2.897 2.768 1.7968 2.5635 3.342 2.832 1.3413 2.7672 2.106 1.4969 5.459

0.27244 0.3044 0.20984 0.27532 0.26212 0.28749 0.1766 0.2729 0.2832 0.18589 0.27369 0.25842 0.19013 0.30542

309.57 180.15 400.15 132.92 304.21 552 461.15 324.65 363.15 456.65 373.53 430.75 490.85 647.13

0.2882 0.242 0.20635 0.2813 0.2908 0.3226 0.3733 0.3217 0.28571 0.28206 0.29015 0.2895 0.4359 0.081

182.30 109.50 245.25 68.15 216.58 161.11 189.79 158.97 185.15 259.83 187.68 197.67 289.95 273.16

Density at Tmin 27.928 44.487 18.513 30.18 26.828 19.064 60.203 34.854 27.985 27.202 29.13 25.298 24.241 55.583

Tmax, K

Density at Tmax

309.57 180.15 400.15 132.92 304.21 552 461.15 324.65 363.15 456.65 373.53 430.75 490.85 333.15

10.208 17.234 5.128 10.522 10.560 6.250 14.516 12.246 10.000 7.216 10.111 8.150 7.873 54.703

All substances are listed in alphabetical order in Table 2-6a. Compiled from Daubert, T. E., R. P. Danner, H. M. Sibul, and C. C. Stebbins, DIPPR Data Compilation of Pure Compound Properties, Project 801 Sponsor Release, July, 1993, Design Institute for Physical Property Data, AIChE, New York, NY; and from Thermodynamics Research Center, “Selected Values of Properties of Hydrocarbons and Related Compounds,” Thermodynamics Research Center Hydrocarbon Project, Texas A&M University, College Station, Texas (extant 1994). Temperatures are in kelvins. Liquid densities are in kmol/m3. Density formulas: kmol/m3 × (mol. wt./1E+03) = g/cm3; kmol/m3 × (mol. wt./1.601846E+01) = lb/ft3. C The liquid density equation is C1/C2[1 + (1 − T/C3) 4] unless otherwise noted. 1 The modified Rackett equation, density = (Pc /RTc)/ZRA1 + [1 − (T/Tc)]2/7, was used. See Spencer, C. F., and R. P. Danner, “Improved Equation for Prediction of Saturated Liquid Density,” J. Chem. Eng. Data 17, 236 (1972). 2 Decomposes violently on heating. Forms explosive peroxides with air or oxygen. Polymerizes under pressure and heat. 3 For the hypothetical pure liquid. 4 Exhibits superfluid properties below 2.2 K. 5 Coefficients are hypothetical above the decomposition temperature. 6 Lower limit is for the undercooled liquid. 7 For the temperature range 333.15 to 403.15 K, use the coefficients: C1 = 4.9669E+00, C2 = 2.7788E−01, C3 = 6.4713E+02, C4 = 1.8740E−01. For the temperature range 403.15 to 647.13 K, use C1 = 4.3910E+00, C2 = 2.4870E−01, C3 = 6.4713E+02, C4 = 2.5340E−01.

2-98

DENSITIES OF AQUEOUS INORGANIC SOLUTIONS

2-99

DENSITIES OF AQUEOUS INORGANIC SOLUTIONS UNITS AND UNITS CONVERSIONS Densities are given in grams per cubic centimeter. To convert to pounds per cubic foot, multiply by 62.43. °F = 9⁄ 5 °C + 32. ADDITIONAL REFERENCES For more detailed data on densities see International Critical Tables: tabular index, vol. 3, p. 1; abrasives, vol. 2, p. 87; air, moist, vol. 1, p. 71; building stones, vol. 2, p. 52; clays, vol. 2, p. 56; coals, vol. 2, p. 135; compounds, vol. 1. pp. 106, 176, 313, 341; elements, vol. 1, pp. 102, 340; fibers, vol. 2, p. 237; gases and vapors, vol. 3, pp. 3, 345; glass, vol. 2, p. 93; liquids and vitreous solids, vol. 3, p. 22; vol. 1, pp. 102, 340; vol. 2, pp. 456, 463; vol. 3, pp. 20, 35; liquid coolants and saturated TABLE 2-31

Aluminum Sulfate [Al2(SO4)3] d 15 4

%

d 15 4

1 2 4 8 12

1.0093 1.0195 1.0404 1.0837 1.1293

16 20 24 26

1.1770 1.2272 1.2803 1.3079

0°C

5°C

10°C

20°C

25°C

d 15 4

%

0.9943 0.9954 0.9959 0.9958 0.9955 0.9939 0.993 32 0.889 .9906 .9915 .9919 .9917 .9913 .9895 .988 36 .877 .9834 .9840 .9842 .9837 .9832 .9811 .980 40 .865 0.970 .9701 .9701 .9695 .9686 .9677 .9651 .964 45 .849 .958 .9576 .9571 .9561 .9548 .9534 .9501 .948 50 .832 .947 .9461 .9450 .9435 .9420 .9402 .9362 .934 60 .796 .9353 .9335 .9316 .9296 .9275 .9229 70 .755 .9249 .9226 .9202 .9179 .9155 .9101 80 .711 .9150 .9122 .9094 .9067 .9040 .8980 90 .665 .9101 .9070 .9040 .9012 .8983 .8920 100 .618

TABLE 2-33 Ammonium Acetate* (CH3COONH4)

TABLE 2-34 Ammonium Bichromate [(NH4)2Cr2O7]

%

d425

%

d412

1 2 4 8 12 16 20 24 28 30 35 40 45

0.9992 1.0013 1.0055 1.0136 1.0216 1.0294 1.0368 1.0439 1.0507 1.0540 1.0618 1.0691 1.0760

1 2 4 8 12 16 20

1.0051 1.0108 1.0223 1.0463 1.0715 1.0981 1.1263

*For data at 16°C for 3(1)52 percent see Atack Handbook of Chemical Data, p. 33, Reinhold, New York, 1957. TABLE 2-35

%

°C

d 4t

3.80 10.52 19.75 28.04

20 13 13.7 19.6

1.0219 1.0627 1.1189 1.1707

Ammonia (NH3)

% −15°C −10°C −5°C 1 2 4 8 12 16 20 24 28 30

TABLE 2-36 Ammonium Chromate [(NH4)2CrO4]

%

TABLE 2-32

vapors are available from WADC-TR-59-598, 1959; plastics are collected in the Handbook of Chemistry and Physics, Chemical Rubber Publishing Co.: solid helium, neon, argon, fluorine, and methane data are given by Johnson (ed.), WADD-TR-60-56, 1960; temperatures of maximum solubility, vol. 3, p. 107; metals, vol. 2, p. 463; oils, fats, and waxes, vol. 2, p. 201; orthobaric, vol. 3, pp. 202, 228, 237, 244; petroleums, vol. 2, pp. 137, 144; plastics, vol. 2, p. 296; porcelains, vol. 2, pp. 68, 75; refrigerating brines, vol. 2, p. 327; rubber, vol. 2, pp. 255, 259; soaps, vol. 5, p. 447; metallic solid solutions, vol. 2, p. 358; solids, vol. 3, pp. 43, 45; vol. 2, p. 456; vol. 3, p. 21; solutions and mixtures, vol. 3, pp. 17, 51, 95, 104, 107, 111, 125, 130; woods, vol. 2, p. 1. Also see the Handbook of Chemistry and Physics, Chemical Rubber Publishing Co., 40th ed., etc.

TABLE 2-37

Ammonium Nitrate (NH4NO3)

%

0°C

10°C

25°C

40°C

60°C

80°C

1.0 2.0 4.0 8.0 12.0 16.0 20.0 24.0 28.0 30.0 40.0 50.0

1.0043 1.0088 1.0178 1.0358 1.0539 1.0721 1.0905 1.1090 1.1277 1.1371 1.1862 1.2380

1.0039 1.0082 1.0168 1.0340 1.0515 1.0691 1.0870 1.1051 1.1234 1.1327 1.1810 1.2320

1.0011 1.0051 1.0132 1.0297 1.0464 1.0633 1.0806 1.0982 1.1161 1.1252 1.1727 1.2229

0.9961 1.0000 1.0079 1.0238 1.0400 1.0565 1.0734 1.0907 1.1082 1.1171 1.1640 1.2136

0.9870 .9908 .9985 1.0142 1.0301 1.0462 1.0627 1.0796 1.0968 1.1055 1.1515 1.2006

0.9755 .9793 .9869 1.0024 1.0181 1.0342 1.0506 1.0673 1.0844 1.0931 1.1385 1.1868

TABLE 2-38

Ammonium Sulfate [(NH4)2SO4]

%

0°C

20°C

40°C

80°C

100°C

1 2 4 8 12 16 20 24 28 35 40 50

1.0061 1.0124 1.0248 1.0495 1.0740 1.0980 1.1215 1.1448 1.1677 1.2072 1.2350 1.2899

1.0041 1.0101 1.0220 1.0456 1.0691 1.0924 1.1154 1.1383 1.1609 1.2800 1.2277 1.2825

0.9980 1.0039 1.0155 1.0387 1.0619 1.0849 1.1077 1.1304 1.1529 1.1919 1.2196 1.2745

0.9777 .9836 .9953 1.0187 1.0421 1.0653 1.0883 1.1111 1.1338 1.1731 1.2011 1.2568

0.9644 .9705 .9826 1.0066 1.0303 1.0539 1.0772 1.1003 1.1232 1.1629 1.1910 1.2466

Ammonium Chloride (NH4Cl)

%

0°C

10°C

20°C

30°C

50°C

80°C

100°C

1 2 4 8 12 16 20 24

1.0033 1.0067 1.0135 1.0266 1.0391 1.0510 1.0625 1.0736

1.0029 1.0062 1.0126 1.0251 1.0370 1.0485 1.0596 1.0705

1.0013 1.0045 1.0107 1.0227 1.0344 1.0457 1.0567 1.0674

0.9987 1.0018 1.0077 1.0195 1.0310 1.0422 1.0532 1.0641

0.9910 .9940 .9999 1.0116 1.0231 1.0343 1.0454 1.0564

0.9749 .9780 .9842 .9963 1.0081 1.0198 1.0312 1.0426

0.9617 .9651 .9718 .9849 .9975 1.0096 1.0213 1.0327

TABLE 2-39

Arsenic Acid (H3A3O4)

%

d 15 4

%

d 15 4

1 2 6 10 16

1.0057 1.0124 1.0398 1.0681 1.1128

20 30 40 50 60 70

1.1447 1.2331 1.3370 1.4602 1.6070 1.7811

2-100

PHYSICAL AND CHEMICAL DATA

TABLE 2-40

Barium Chloride (BaCl2)

TABLE 2-46

Chromic Acid (CrO3)

%

0°C

20°C

40°C

60°C

80°C

100°C

%

d 15 4

%

d 15 4

2 4 8 12 16 20 24 26

1.0181 1.0368 1.0760 1.1178 1.1627 1.2105

1.0159 1.0341 1.0721 1.1128 1.1564 1.2031 1.2531 1.2793

1.0096 1.0275 1.0648 1.1047 1.1478 1.1938 1.2430 1.2688

1.0004 1.0181 1.0551 1.0948 1.1373 1.1828 1.2316 1.2571

0.9890 1.0066 1.0434 1.0827 1.1249 1.1702 1.2186 1.2440

0.9755 .9931 1.0299 1.0692 1.1113 1.1563 1.2045 1.2298

1 2 6 10 16

1.006 1.014 1.045 1.076 1.127

20 26 30 40 50 60

1.163 1.220 1.260 1.371 1.505 1.663

TABLE 2-41

TABLE 2-47

d 18 4

%

d 18 4

2 4 8 12 16

1.0154 1.0326 1.0683 1.1061 1.1468

20 25 30 40 50

1.1904 1.2488 1.3124 1.4590 1.6356

TABLE 2-42 % −5°C 2 4 8 12 16 20 25 30 35 40

Cadmium Nitrate [Cd(NO3)2]

%

1.0708 1.1083 1.1471 1.1874

Chromium Chloride (CrCl3) d 18 4

%

Violet

Green

Equilibrium mixture of violet and green

1 2 4 8 12 14

1.0076 1.0166 1.0349 1.0724 1.1114 1.1316

1.0071 1.0157 1.0332 1.0691 1.1065

1.0075 1.0165 1.0347 1.0722 1.1111

Calcium Chloride (CaCl2)

0°C

20°C

30°C

40°C

60°C

1.0171 1.0346 1.0703 1.1072 1.1454 1.1853 1.2376 1.2922

1.0148 1.0316 1.0659 1.1015 1.1386 1.1775 1.2284 1.2816 1.3373 1.3957

1.0120 1.0286 1.0626 1.0978 1.1345 1.1730 1.2236 1.2764 1.3316 1.3895

1.0084 1.0249 1.0586 1.0937 1.1301 1.1684 1.2186 1.2709 1.3255 1.3826

0.9994 1.0158 1.0492 1.0840 1.1202 1.1581 1.2079 1.2597 1.3137 1.3700

80°C 100°C 120°C* 140°C 0.9881 1.0046 1.0382 1.0730 1.1092 1.1471 1.1965 1.2478 1.3013 1.3571

0.9748 .9915 1.0257 1.0610 1.0973 1.1352 1.1846 1.2359 1.2893 1.3450

0.9596 .9765 1.0111 1.0466 1.0835 1.1219

0.9428 .9601 .9954 1.0317 1.0691 1.1080

TABLE 2-48

Copper Nitrate [Cu(NO3)2]

%

d 20 4

%

d 20 4

1 2 4 8

1.007 1.015 1.032 1.069

12 16 20 25

1.107 1.147 1.189 1.248

*Corrected to atmospheric pressure.

TABLE 2-43 Calcium Hydroxide [Ca(OH)2]

TABLE 2-44 Calcium Hypochlorite* (CaOCl2)

TABLE 2-49 (CuSO4)

Copper Sulfate

TABLE 2-50 Cuprous Chloride (Cu2Cl2)

%

d 15 4

d 425

% total salt

d 15 4

%

0°C

20°C

40°C

%

0°C

20°C

40°C

0.05 .10 .15

0.99979 1.00044 1.00110

0.99773 .99838 .99904

2 4 6 8 10 12

1.0169 1.0345 1.0520 1.0697 1.0876 1.1060

1 4 8 12 16 18

1.0104 1.0429 1.0887 1.1379

1.0086 1.0401 1.084 1.1308 1.180 1.206

1.0024 1.0332 1.0764 1.1222

1 4 8 12 16 20

1.0095 1.0387 1.0788 1.1208 1.1653 1.2121

1.0072 1.036 1.0754 1.1165 1.1595 1.2052

1.002 1.0305 1.0682 1.107 1.151 1.1953

*CaOCl2 = 89.15% CaCl2 = 7.31% Ca(ClO3)2 = 0.26% Ca(OH)2 = 2.92%.

TABLE 2-45

Calcium Nitrate [Ca(NO3)2]

TABLE 2-51

Ferric Chloride (FeCl3)

%

6°C

18°C

25°C

30°C

%

0°C

10°C

20°C

30°C

2* 4 8 12 16 20 25 30 35 40 45 68*

1.0157 1.0316 1.0641 1.0979 1.1330 1.1694 1.2168

1.0137 1.0291 1.0608 1.0937 1.1279 1.1636 1.2106 1.260 1.311 1.365 1.422 1.747

1.0120 1.0272 1.0585 1.0911 1.1250 1.1602 1.2065

1.0105 1.0256 1.0565 1.0887 1.1224 1.1575 1.2032

1.741

1.736

1 2 4 8 12 16 20 25 30 35 40 45 50

1.0086 1.0174 1.0347 1.0703 1.1088 1.1475 1.1870 1.2400 1.2970 1.3605 1.4280

1.0084 1.0168 1.0341 1.0692 1.1071 1.1449 1.1847 1.2380 1.2950 1.3580 1.4235 1.4920 1.5610

1.0068 1.0152 1.0324 1.0669 1.1040 1.1418 1.1820 1.2340 1.2910 1.3530 1.4175 1.4850 1.5510

1.0040 1.0122 1.0292 1.0636 1.1006 1.1386 1.1786 1.2290 1.2850 1.3475 1.4115

*Supercooled tetrahydrate (m.p. 41.4°C).

DENSITIES OF AQUEOUS INORGANIC SOLUTIONS TABLE 2-52 [Fe2(SO4)3]

Ferric Sulfate

TABLE 2-53 [Fe(NO3)3]

17.5

%

d4

1 2 4 8 12 16 20 30 40 50 60

1.0072 1.0157 1.0327 1.0670 1.1028 1.1409 1.1811 1.3073 1.4487 1.6127 1.7983

TABLE 2-54 Ferrous Sulfate (FeSO4) % 0.2 0.4 0.8 1.0 4.0 8.0 12.0 16.0 20.0

Ferric Nitrate

d4

d4

%

d4

%

d4

1 2 4 8 12 16 20 25

1.0065 1.0144 1.0304 1.0636 1.0989 1.1359 1.1748 1.2281

5 10 20 30 40 50 60 70 80 90 95 100

1.020 1.040 1.080 1.119 1.159 1.198 1.235 1.258 1.259 1.178 1.089 1.0005

1.017 1.035 1.070 1.101 1.130 1.155

1 2 4 6 8 10 12 14 16 18 20 22 24

1.0022 1.0058 1.0131 1.0204 1.0277 1.0351 1.0425 1.0499 1.0574 1.0649 1.0725 1.0802 1.0880

26 28 30 35 40 45 50 55 60 70 80 90 100

1.0959 1.1040 1.1122 1.1327 1.1536 1.1749 1.1966 1.2188 1.2416 1.2897 1.3406 1.3931 1.4465

4

10

25

TABLE 2-60

20°C

%

d4

d4

d4

%

1.0090 1.0380 1.0790 1.1235 1.1690 1.2150

1.00068 1.00275 1.00645 1.0085 1.0375 1.0785 1.1220 1.1675 1.2135

1.0002 1.0022 1.0062 1.0082

1.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0 30.0 40.0 50.0 60.0 65.0

1.0073 1.0146 1.0295 1.0448 1.0604 1.0764 1.0928 1.1097 1.1272 1.1453 1.1640 1.1832 1.2030 1.2235 1.2446 1.2663 1.3877 1.5305 1.6950 1.7854

1.0068 1.0139 1.0285 1.0435 1.0589 1.0747 1.0910 1.1078 1.1251 1.1430 1.1615 1.1806 1.2003 1.2206 1.2415 1.2630 1.3838 1.5257 1.6892 1.7792

1.0041 1.0111 1.0255 1.0402 1.0552 1.0707 1.0867 1.1032 1.1202 1.1377 1.1557 1.1743 1.1935 1.2134 1.2340 1.2552 1.3736 1.5127 1.6731 1.7613

1 2 4 8 12

15

1 2 4 8 12 16 82 90 100

0.998 .996 .993 .984 .971 .956 .752 .724 .691

TABLE 2-57

18

%

18

18°C

d4

18

d4

15°C

%

0

%

TABLE 2-55 Hydrogen Bromide (HBr)

TABLE 2-56 Hydrogen Cyanide (HCN)

TABLE 2-59 Hydrogen Peroxide (H2O2)

TABLE 2-58 Hydrogen Fluoride (HF) 20

2-101

Hydrofluosilic Acid (H2SiF6) 17.5

d4

1.0080 1.0161 1.0324 1.0661 1.1011

TABLE 2-61

Hydrogen Chloride (HCl)

−5°C

0°C

10°C

20°C

40°C

60°C

80°C

100°C

1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

1.0048 1.0104 1.0213 1.0321 1.0428 1.0536 1.0645 1.0754 1.0864 1.0975 1.1087 1.1200 1.1314 1.1426 1.1537 1.1648

1.0052 1.0106 1.0213 1.0319 1.0423 1.0528 1.0634 1.0741 1.0849 1.0958 1.1067 1.1177 1.1287 1.1396 1.1505 1.1613

1.0048 1.0100 1.0202 1.0303 1.0403 1.0504 1.0607 1.0711 1.0815 1.0920 1.1025 1.1131 1.1238 1.1344 1.1449 1.1553

1.0032 1.0082 1.0181 1.0279 1.0376 1.0474 1.0574 1.0675 1.0776 1.0878 1.0980 1.1083 1.1187 1.1290 1.1392 1.1493 1.1593 1.1691 1.1789 1.1885 1.1980

0.9970 1.0019 1.0116 1.0211 1.0305 1.0400 1.0497 1.0594 1.0692 1.0790 1.0888 1.0986 1.1085 1.1183 1.1280 1.1376

0.9881 .9930 1.0026 1.0121 1.0215 1.0310 1.0406 1.0502 1.0598 1.0694 1.0790 1.0886 1.0982 1.1076 1.1169 1.1260

0.9768 0.9819 0.9919 1.0016 1.0111 1.0206 1.0302 1.0398 1.0494 1.0590 1.0685 1.0780 1.0874 1.0967 1.1058 1.1149

0.9636 .9688 .9791 .9892 .9992 1.0090 1.0188 1.0286 1.0383 1.0479 1.0574 1.0668 1.0761 1.0853 1.0942 1.1030

d4

16 20 25 30 34

1.1373 1.1748 1.2235 1.2742 1.3162

Magnesium Chloride (MgCl2)

%

0°C

20°C

40°C

60°C

80°C

100°C

2 4 8 12 16 20 25 30

1.0168 1.0338 1.0683 1.1035 1.1395 1.1764 1.2246 1.2754

1.0146 1.0311 1.0646 1.0989 1.1342 1.1706 1.2184 1.2688

1.0084 1.0248 1.0580 1.0921 1.1272 1.1635 1.2111 1.2614

0.9995 1.0159 1.0493 1.0836 1.1188 1.1552 1.2031 1.2535

0.9883 1.0050 1.0388 1.0735 1.1092 1.1460 1.1942 1.2451

0.9753 .9923 1.0269 1.0622 1.0984 1.1359 1.1847 1.2360

TABLE 2-62

%

17.5

%

Magnesium Sulfate (MgSO4)

%

0°C

20°C

30°C

40°C

50°C

60°C

80°C

2 4 8 12 16 20 26

1.0210 1.0423 1.0858 1.1309 1.1777 1.2264 1.3032

1.0186 1.0392 1.0816 1.1256 1.1717 1.2198 1.2961

1.0158 1.0362 1.0782 1.1220 1.1679 1.2159 1.2922

1.0123 1.0326 1.0743 1.1179 1.1637 1.2117 1.2879

1.0081 1.0283 1.0700 1.1135 1.1592 1.2072 1.2836

1.0032 1.0234 1.0650 1.1083

0.9916 1.0118 1.0534 1.0968

TABLE 2-63 Nickel Chloride (NiCl2) 18

TABLE 2-64 Nickel Nitrate [Ni(NO3)2] 20

TABLE 2-65 Nickel Sulfate (NiSO4 ) 18

%

d4

%

d4

%

d4

1 2 4 8 12 16 20 30

1.0082 1.0179 1.0375 1.0785 1.1217 1.1674 1.2163 1.353

1 2 4 8 12 16 20 30 35

1.0065 1.0150 1.0325 1.0688 1.1070 1.1480 1.191 1.311 1.377

1 2 4 8 12 16 18

1.0091 1.0198 1.0415 1.0852 1.1325 1.1825 1.2090

2-102

PHYSICAL AND CHEMICAL DATA

TABLE 2-66 %

Nitric Acid (HNO3)

0°C

5°C

10°C

15°C

20°C

25°C

30°C

40°C

50°C

60°C

80°C

100°C

1 2 3 4

1.0058 1.0117 1.0176 1.0236

1.00572 1.01149 1.01730 1.02315

1.00534 1.01099 1.01668 1.02240

1.00464 1.01018 1.01576 1.02137

1.00364 1.00909 1.01457 1.02008

1.00241 1.00778 1.01318 1.01861

1.0009 1.0061 1.0114 1.0168

0.9973 1.0025 1.0077 1.0129

0.9931 .9982 1.0033 1.0084

0.9882 .9932 .9982 1.0033

0.9767 .9816 .9865 .9915

0.9632 .9681 .9730 .9779

5 6 7 8 9

1.0296 1.0357 1.0418 1.0480 1.0543

1.02904 1.03497 1.0410 1.0471 1.0532

1.02816 1.03397 1.0399 1.0458 1.0518

1.02702 1.03272 1.0385 1.0443 1.0502

1.02563 1.03122 1.0369 1.0427 1.0485

1.02408 1.02958 1.0352 1.0409 1.0466

1.0222 1.0277 1.0333 1.0389 1.0446

1.0182 1.0235 1.0289 1.0344 1.0399

1.0136 1.0188 1.0241 1.0295 1.0349

1.0084 1.0136 1.0188 1.0241 1.0294

.9965 1.0015 1.0066 1.0117 1.0169

.9829 .9879 .9929 .9980 1.0032

10 11 12 13 14

1.0606 1.0669 1.0733 1.0797 1.0862

1.0594 1.0656 1.0718 1.0781 1.0845

1.0578 1.0639 1.0700 1.0762 1.0824

1.0561 1.0621 1.0681 1.0742 1.0803

1.0543 1.0602 1.0661 1.0721 1.0781

1.0523 1.0581 1.0640 1.0699 1.0758

1.0503 1.0560 1.0618 1.0676 1.0735

1.0455 1.0511 1.0567 1.0624 1.0681

1.0403 1.0458 1.0513 1.0568 1.0624

1.0347 1.0401 1.0455 1.0509 1.0564

1.0221 1.0273 1.0326 1.0379 1.0432

1.0083 1.0134 1.0186 1.0238 1.0289

15 16 17 18 19

1.0927 1.0992 1.1057 1.1123 1.1189

1.0909 1.0973 1.1038 1.1103 1.1168

1.0887 1.0950 1.1014 1.1078 1.1142

1.0865 1.0927 1.0989 1.1052 1.1115

1.0842 1.0903 1.0964 1.1026 1.1088

1.0818 1.0879 1.0940 1.1001 1.1062

1.0794 1.0854 1.0914 1.0974 1.1034

1.0739 1.0797 1.0855 1.0913 1.0972

1.0680 1.0737 1.0794 1.0851 1.0908

1.0619 1.0675 1.0731 1.0787 1.0843

1.0485 1.0538 1.0592 1.0646 1.0700

1.0341 1.0393 1.0444 1.0496 1.0547

20 21 22 23 24

1.1255 1.1322 1.1389 1.1457 1.1525

1.1234 1.1300 1.1366 1.1433 1.1501

1.1206 1.1271 1.1336 1.1402 1.1469

1.1178 1.1242 1.1306 1.1371 1.1437

1.1150 1.1213 1.1276 1.1340 1.1404

1.1123 1.1185 1.1247 1.1310 1.1374

1.1094 1.1155 1.1217 1.1280 1.1343

1.1031 1.1090 1.1150 1.1210 1.1271

1.0966 1.1024 1.1083 1.1142 1.1201

1.0899 1.0956 1.1013 1.1070 1.1127

1.0754 1.0808 1.0862 1.0917 1.0972

1.0598 1.0650 1.0701 1.0753 1.0805

25 26 27 28 29

1.1594 1.1663 1.1733 1.1803 1.1874

1.1569 1.1638 1.1707 1.1777 1.1847

1.1536 1.1603 1.1670 1.1738 1.1807

1.1503 1.1569 1.1635 1.1702 1.1770

1.1469 1.1534 1.1600 1.1666 1.1733

1.1438 1.1502 1.1566 1.1631 1.1697

1.1406 1.1469 1.1533 1.1597 1.1662

1.1332 1.1394 1.1456 1.1519 1.1582

1.1260 1.1320 1.1381 1.1442 1.1503

1.1185 1.1244 1.1303 1.1362 1.1422

1.1027 1.1083 1.1139 1.1195 1.1251

1.0857 1.0910 1.0963 1.1016 1.1069

30 31 32 33 34

1.1945 1.2016 1.2088 1.2160 1.2233

1.1917 1.1988 1.2059 1.2131 1.2203

1.1876 1.1945 1.2014 1.2084 1.2155

1.1838 1.1906 1.1974 1.2043 1.2113

1.1800 1.1867 1.1934 1.2002 1.2071

1.1763 1.1829 1.1896 1.1963 1.2030

1.1727 1.1792 1.1857 1.1922 1.1988

1.1645 1.1708 1.1772 1.1836 1.1901

1.1564 1.1625 1.1687 1.1749 1.1812

1.1482 1.1542 1.1602 1.1662 1.1723

1.1307 1.1363 1.1419 1.1476 1.1533

1.1122 1.1175 1.1228 1.1281 1.1335

35 36 37 38 39

1.2306 1.2375 1.2444 1.2513 1.2581

1.2275 1.2344 1.2412 1.2479 1.2546

1.2227 1.2294 1.2361 1.2428 1.2494

1.2183 1.2249 1.2315 1.2381 1.2446

1.2140 1.2205 1.2270 1.2335 1.2399

1.2098 1.2163 1.2227 1.2291 1.2354

1.2055 1.2119 1.2182 1.2245 1.2308

1.1966 1.2028 1.2089 1.2150 1.2210

1.1876 1.1936 1.1995 1.2054 1.2112

1.1784 1.1842 1.1899 1.1956 1.2013

1.1591 1.1645 1.1699 1.1752 1.1805

1.1390 1.1440 1.1490 1.1540 1.1589

40 41 42 43 44

1.2649 1.2717 1.2786 1.2854 1.2922

1.2613 1.2680 1.2747 1.2814 1.2880

1.2560 1.2626 1.2692 1.2758 1.2824

1.2511 1.2576 1.2641 1.2706 1.2771

1.2463 1.2527 1.2591 1.2655 1.2719

1.2417 1.2480 1.2543 1.2606 1.2669

1.2370 1.2432 1.2494 1.2556 1.2618

1.2270 1.2330 1.2390 1.2450 1.2510

1.2170 1.2229 1.2287 1.2345 1.2403

1.2069 1.2126 1.2182 1.2238 1.2294

1.1858 1.1911 1.1963 1.2015 1.2067

1.1638 1.1687 1.1735 1.1783 1.1831

45 46 47 48 49

1.2990 1.3058 1.3126 1.3194 1.3263

1.2947 1.3014 1.3080 1.3147 1.3214

1.2890 1.2955 1.3021 1.3087 1.3153

1.2836 1.2901 1.2966 1.3031 1.3096

1.2783 1.2847 1.2911 1.2975 1.3040

1.2732 1.2795 1.2858 1.2921 1.2984

1.2680 1.2742 1.2804 1.2867 1.2929

1.2570 1.2630 1.2690 1.2750 1.2811

1.2461 1.2519 1.2577 1.2635 1.2693

1.2350 1.2406 1.2462 1.2518 1.2575

1.2119 1.2171 1.2223 1.2275 1.2328

1.1879 1.1927 1.1976 1.2024 1.2073

50 51 52 53 54

1.3327 1.3391 1.3454 1.3517 1.3579

1.3277 1.3339 1.3401 1.3462 1.3523

1.3215 1.3277 1.3338 1.3399 1.3459

1.3157 1.3218 1.3278 1.3338 1.3397

1.3100 1.3160 1.3219 1.3278 1.3336

1.3043 1.3102 1.3160 1.3218 1.3275

1.2987 1.3045 1.3102 1.3159 1.3215

1.2867 1.2923 1.2978 1.3033 1.3087

1.2748 1.2802 1.2856 1.2909 1.2961

1.2628 1.2680 1.2731 1.2782 1.2833

1.2377 1.2425 1.2473 1.2521 1.2568

1.2118 1.2163 1.2208 1.2252 1.2296

55 56 57 58 59

1.3640 1.3700 1.3759 1.3818 1.3875

1.3583 1.3642 1.3700 1.3757 1.3813

1.3518 1.3576 1.3634 1.3691 1.3747

1.3455 1.3512 1.3569 1.3625 1.3680

1.3393 1.3449 1.3505 1.3560 1.3614

1.3331 1.3386 1.3441 1.3495 1.3548

1.3270 1.3324 1.3377 1.3430 1.3482

1.3141 1.3194 1.3246 1.3298 1.3348

1.3013 1.3064 1.3114 1.3164 1.3213

1.2883 1.2932 1.2981 1.3029 1.3077

1.2615 1.2661 1.2706 1.2751 1.2795

1.2339 1.2382 1.2424 1.2466 1.2507

60 61 62 63 64

1.3931 1.3986 1.4039 1.4091

1.3868 1.3922 1.3975 1.4027 1.4078

1.3801 1.3855 1.3907 1.3958 1.4007

1.3734 1.3787 1.3838 1.3888 1.3936

1.3667 1.3719 1.3769 1.3818 1.3866

1.3600 1.3651 1.3700 1.3748 1.3795

1.3533 1.3583 1.3632 1.3679 1.3725

1.3398 1.3447 1.3494 1.3540

1.3261 1.3308 1.3354 1.3398

1.3124 1.3169 1.3213 1.3255

1.2839 1.2881 1.2922 1.2962

1.2547 1.2587 1.2625 1.2661

DENSITIES OF AQUEOUS INORGANIC SOLUTIONS Nitric Acid (HNO3) (Concluded )

TABLE 2-66 %

0°C

5°C

10°C

15°C

20°C

25°C

30°C

65 66 67 68 69

1.4128 1.4177 1.4224 1.4271 1.4317

1.4055 1.4103 1.4150 1.4196 1.4241

1.3984 1.4031 1.4077 1.4122 1.4166

1.3913 1.3959 1.4004 1.4048 1.4091

1.3841 1.3887 1.3932 1.3976 1.4019

1.3770 1.3814 1.3857 1.3900 1.3942

70 71 72 73 74

1.4362 1.4406 1.4449 1.4491 1.4532

1.4285 1.4328 1.4371 1.4413 1.4454

1.4210 1.4252 1.4294 1.4335 1.4376

1.4134 1.4176 1.4218 1.4258 1.4298

1.4061 1.4102 1.4142 1.4182 1.4221

1.3983 1.4023 1.4063 1.4103 1.4142

75 76 77 78 79

1.4573 1.4613 1.4652 1.4690 1.4727

1.4494 1.4533 1.4572 1.4610 1.4647

1.4415 1.4454 1.4492 1.4529 1.4565

1.4337 1.4375 1.4413 1.4450 1.4486

1.4259 1.4296 1.4333 1.4369 1.4404

1.4180 1.4217 1.4253 1.4288 1.4323

80 81 82 83 84

1.4764 1.4800 1.4835 1.4869 1.4903

1.4683 1.4718 1.4753 1.4787 1.4820

1.4601 1.4636 1.4670 1.4704 1.4737

1.4521 1.4555 1.4589 1.4622 1.4655

1.4439 1.4473 1.4507 1.4540 1.4572

1.4357 1.4391 1.4424 1.4456 1.4487

85 86 87 88 89

1.4936 1.4968 1.4999 1.5029 1.5058

1.4852 1.4883 1.4913 1.4942 1.4970

1.4769 1.4799 1.4829 1.4858 1.4885

1.4686 1.4716 1.4745 1.4773 1.4800

1.4603 1.4633 1.4662 1.4690 1.4716

1.4518 1.4548 1.4577 1.4605 1.4631

90 91 92 93 94

1.5085 1.5111 1.5136 1.5156 1.5177

1.4997 1.5023 1.5048 1.5068 1.5088

1.4911 1.4936 1.4960 1.4979 1.4999

1.4826 1.4850 1.4873 1.4892 1.4912

1.4741 1.4766 1.4789 1.4807 1.4826

1.4656 1.4681 1.4704 1.4722 1.4741

95 96 97 98 99 100

1.5198 1.5220 1.5244 1.5278 1.5327 1.5402

1.5109 1.5130 1.5152 1.5187 1.5235 1.5310

1.5019 1.5040 1.5062 1.5096 1.5144 1.5217

1.4932 1.4952 1.4974 1.5008 1.5056 1.5129

1.4846 1.4867 1.4889 1.4922 1.4969 1.5040

1.4761 1.4781 1.4802 1.4835 1.4881 1.4952

TABLE 2-67 15

%

d4

1 2 4 6 8 10 12 14 16 18 20 22 24 26

1.0050 1.0109 1.0228 1.0348 1.0471 1.0597 1.0726 1.0589 1.0995 1.1135 1.1279 1.1428 1.1581 1.1738

2-103

Perchloric Acid (HClO4) 20

d4

TABLE 2-69

40°C

50°C

d4

%

d4

d4

d4

°C

1%

2%

4%

1.0020 1.0070 1.0169 1.0270 1.0372 1.0475

0.9933 0.9986 0.9906 1.0205 1.0320 1.0440 1.0560 1.0680 1.0810 1.0940 1.1070 1.1205 1.1345 1.1490

28 30 32 34 36 38 40 45 50 55 60 65 70

1.1900 1.2067 1.2239 1.2418 1.2603 1.2794 1.2991 1.3521 1.4103 1.4733 1.5389 1.6059 1.6736

1.1851 1.2013 1.2183 1.2359 1.2542 1.2732 1.2927 1.3450 1.4018 1.4636 1.5298 1.5986 1.6680

1.1645 1.1800 1.1960 1.2130 1.2310 1.2490 1.2680 1.3180 1.3730 1.4320 1.4950 1.5620 1.6290

0 10 15 20 30 40 50 60 80 100

1.0066 1.0064 1.0058 1.0049 1.0024 0.9990 .9949 .9901 .9786 .9653

1.0134 1.0132 1.0125 1.0117 1.0092 1.0058 1.0017 0.9969 .9855 .9722

1.0270 1.0268 1.0260 1.0252 1.0228 1.0195 1.0154 1.0106 0.9993 .9860

1.1697

50

15

20

50

80°C

100°C

6%

8%

10%

1.0396

1.0534

1.0674

Potassium Bicarbonate (KHCO3)

d4

25

60°C

TABLE 2-70 Potassium Bromide (KBr) TABLE 2-68

Phosphoric Acid (H3PO4)

°C

2%

6%

14%

0 10 20 30 40

1.0113 1.0109 1.0092 1.0065 1.0029

1.0339 1.0330 1.0309 1.0279 1.0241

1.0811 1.0792 1.0764 1.0728 1.0685

20%

26%

35%

50%

75%

1.1192 1.1167 1.1567 1.221 1.341 1.1134 1.1529 1.216 1.335 1.579 1.1094 1.1484 1.211 1.329 1.572 1.1048

100%

1.870 1.862

%

d 20 4

1 2 6 12 20 30 40

1.0054 1.0127 1.0426 1.0903 1.1601 1.2593 1.3746

2-104

PHYSICAL AND CHEMICAL DATA

TABLE 2-71

Potassium Carbonate (K2CO3)

TABLE 2-77

Potassium Nitrate (KNO3)

%

0°C

10°C

20°C

40°C

60°C

80°C

100°C

%

0°C

10°C

20°C

40°C

60°C

80°C

100°C

1 2 4 8 12 16 20 24 28 30 35 40 45 50

1.0094 1.0189 1.0381 1.0768 1.1160 1.1562 1.1977 1.2405 1.2846 1.3071 1.3646 1.4244 1.4867 1.5517

1.0089 1.0182 1.0369 1.0746 1.1131 1.1530 1.1941 1.2366 1.2804 1.3028 1.3600 1.4195 1.4815 1.5462

1.0072 1.0163 1.0345 1.0715 1.1096 1.1490 1.1898 1.2320 1.2756 1.2979 1.3548 1.4141 1.4759 1.5404

1.0010 1.0098 1.0276 1.0640 1.1013 1.1399 1.1801 1.2219 1.2652 1.2873 1.3440 1.4029 1.4644 1.5285

0.9919 1.0005 1.0180 1.0538 1.0906 1.1290 1.1690 1.2106 1.2538 1.2759 1.3324 1.3913 1.4528 1.5169

0.9803 .9889 1.0063 1.0418 1.0786 1.1170 1.1570 1.1986 1.2418 1.2640 1.3206 1.3795 1.4408 1.5048

0.9670 .9756 .9951 1.0291 1.0663 1.1049 1.1451 1.1869 1.2301 1.2522 1.3089 1.3678 1.4290 1.4928

1 2 4 8 12 16 20 24

1.00654 1.01326 1.02677 1.05419 1.08221

1.00615 1.01262 1.02566 1.05226 1.07963

1.00447 1.01075 1.02344 1.04940 1.07620 1.10392 1.13261 1.16233

0.99825 1.00430 1.01652 1.04152 1.06740 1.09432 1.12240 1.15175

0.9890 .9949 1.0068 1.0313 1.0567 1.0831 1.1106 1.1391

0.9776 .9834 .9951 1.0192 1.0442 1.0703 1.0974 1.1256

0.9641 .9699 .9816 1.0056 1.0304 1.0562 1.0831 1.1110

TABLE 2-72 Potassium Chromate (K2Cr O4)

TABLE 2-73 (KClO3)

Potassium Chlorate

%

d4

d4

°C

1%

2%

3%

4%

1 2 4 8 12 16 20 24 28 30

1.0073 1.0155 1.0321 1.0659 1.1009

1.0066 1.0147 1.0311 1.0647 1.0999 1.1366 1.1748 1.2147 1.2566 1.2784

0 10 20 30 40 60 80 100

1.0061 1.0059 1.0045 1.0020 0.9986 .9895 .9781 .9646

1.0124 1.0122 1.0109 1.0085 1.0051 0.9959 .9845 .9709

1.0189 1.0187 1.0174 1.0151 1.0116 1.0024 0.9910 .9774

1.0256 1.0254 1.0241 1.0218 1.0183 1.0091 0.9977 .9840

15

TABLE 2-74

18

Potassium Chloride (KCl)

%

0°C

20°C

25°C

40°C

60°C

80°C

100°C

1.0 2.0 4.0 8.0 12.0 16.0 20.0 24.0 28.0

1.00661 1.01335 1.02690 1.05431 1.08222 1.11068 1.13973

1.00462 1.01103 1.02391 1.05003 1.07679 1.10434 1.13280 1.16226

1.00342 1.00977 1.02255 1.04847 1.07506 1.10245 1.13072 1.15995

0.99847 1.00471 1.01727 1.04278 1.06897 1.09600 1.12399 1.15299 1.18304

0.9894 .9956 1.0080 1.0333 1.0592 1.0861 1.1138 1.1425 1.1723

0.9780 .9842 .9966 1.0219 1.0478 1.0746 1.1024 1.1311 1.1609

0.9646 .9708 .9634 1.0888 1.0350 1.0619 1.0897 1.1185 1.1483

%

110°C

120°C

130°C

140°C

3.79 7.45 13.62

0.9733 .9978 1.0388

0.9663 .9899 1.0313

0.9583 .9827 1.0238

0.9502 .9745 1.0159

TABLE 2-75 Potassium Chrome Alum [K2Cr2(SO4)4]

TABLE 2-78 Potassium Dichromate (K2Cr2O7) 20

d 15 4

%

d 15 4

1 2 6 10 14 20 30 40 50

1.007 1.016 1.052 1.089 1.129 1.193 1.315 1.456 1.615

1.0 2.0 4.0 6.0 8.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 51.7

1.0083 1.0175 1.0359 1.0544 1.0730 1.0918 1.1396 1.1884 1.2387 1.2905 1.3440 1.3991 1.4558 1.5143 1.5355 (sat’d. soln.)

20

%

d4

%

d4

1 2 4 6 8 10

1.0052 1.0122 1.0264 1.0408 1.0554 1.0703

1 2 4 6 8 10

1.0063 1.0145 1.0310 1.0477 1.0646 1.0817

TABLE 2-80 Potassium Sulfite (K2SO3) 15

TABLE 2-81 Sodium Acetate (NaC2H3O2) 20

%

d4

%

d4

1 2 4 8 12 16 20 24 26

1.0073 1.0155 1.0322 1.0667 1.1026 1.1402 1.1793 1.2197 1.2404

1 2 4 8 12 18 20 26 28

1.0033 1.0084 1.0186 1.0392 1.0598 1.0807 1.1021 1.1351 1.1462

TABLE 2-82 Sodium Arsenate (Na3AsO4) 17

TABLE 2-83 Sodium Bichromate (Na2Cr2O7) 15

%

d4

%

d4

1 2 4 8 10 12

1.0097 1.0207 1.0431 1.0892 1.1130 1.1373

1 2 4 8 12 16 20 24 28 30 35 40 45 50

1.006 1.013 1.027 1.056 1.084 1.112 1.140 1.166 1.193 1.207 1.244 1.279 1.312 1.342

TABLE 2-76 Potassium Hydroxide (KOH)

%

TABLE 2-79 Potassium Sulfate (K2SO4)

TABLE 2-84 Sodium Bromide (NaBr) 17

TABLE 2-85 Sodium Formate (HCOONa) 25

%

d4

%

d4

1 2 4 8 10 12 20 30 40

1.0060 1.0139 1.0298 1.0631 1.0803 1.0981 1.1745 1.2841 1.4138

1 2 4 8 12 16 20 24 28 30 35 40

1.003 1.009 1.022 1.048 1.074 1.100 1.127 1.155 1.184 1.199 1.236 1.274

DENSITIES OF AQUEOUS INORGANIC SOLUTIONS TABLE 2-86

Sodium Carbonate (Na2CO3)

TABLE 2-90

2-105

Sodium Hydroxide (NaOH)

%

0°C

10°C

20°C

30°C

40°C

60°C

80°C

100°C

%

0°C

15°C

20°C

40°C

60°C

80°C

100°C

1 2 4 8 12 14 16 18 20 24 28 30

1.0109 1.0219 1.0439 1.0878 1.1319 1.1543

1.0103 1.0210 1.0423 1.0850 1.1284 1.1506

1.0086 1.0190 1.0398 1.0816 1.1244 1.1463

1.0058 1.0159 1.0363 1.0775 1.1200 1.1417 1.1636 1.1859 1.2086 1.2552 1.3031 1.3274

1.0022 1.0122 1.0323 1.0732 1.1150 1.1365

0.9929 1.0027 1.0223 1.0625 1.1039 1.1251

0.9814 .9910 1.0105 1.0503 1.0914 1.1125

0.9683 .9782 .9980 1.0380 1.0787 1.0996

1 2 4 8 12 16 20 24 28 32 36 40 44 48 50

1.0124 1.0244 1.0482 1.0943 1.1399 1.1849 1.2296 1.2741 1.3182 1.3614 1.4030 1.4435 1.4825 1.5210 1.5400

1.01065 1.02198 1.04441 1.08887 1.13327 1.17761 1.22183 1.26582 1.3094 1.3520 1.3933 1.4334 1.4720 1.5102 1.5290

1.0095 1.0207 1.0428 1.0869 1.1309 1.1751 1.2191 1.2629 1.3064 1.3490 1.3900 1.4300 1.4685 1.5065 1.5253

1.0033 1.0139 1.0352 1.0780 1.1210 1.1645 1.2079 1.2512 1.2942 1.3362 1.3768 1.4164 1.4545 1.4922 1.5109

0.9941 1.0045 1.0254 1.0676 1.1101 1.1531 1.1960 1.2388 1.2814 1.3232 1.3634 1.4027 1.4405 1.4781 1.4967

0.9824 .9929 1.0139 1.0560 1.0983 1.1408 1.1833 1.2259 1.2682 1.3097 1.3498 1.3889 1.4266 1.4641 1.4827

0.9693 .9797 1.0009 1.0432 1.0855 1.1277 1.1700 1.2124 1.2546 1.2960 1.3360 1.3750 1.4127 1.4503 1.4690

TABLE 2-87

Sodium Chlorate (NaClO3) 18

18

%

d4

%

d4

1 2 4 6 8 10 12 14 16

1.0053 1.0121 1.0258 1.0397 1.0538 1.0681 1.0827 1.0977 1.1131

18 20 22 24 26 28 30 32 34

1.1288 1.1449 1.1614 1.1782 1.1953 1.2128 1.2307 1.2491 1.2680

TABLE 2-88

TABLE 2-91

Sodium Chloride (NaCl)

%

0°C

10°C

25°C

40°C

60°C

80°C

100°C

1 2 4 8 12 16 20 24 26

1.00747 1.01509 1.03038 1.06121 1.09244 1.12419 1.15663 1.18999 1.20709

1.00707 1.01442 1.02920 1.05907 1.08946 1.12056 1.15254 1.18557 1.20254

1.00409 1.01112 1.02530 1.05412 1.08365 1.11401 1.14533 1.17776 1.19443

0.99908 1.00593 1.01977 1.04798 1.07699 1.10688 1.13774 1.16971 1.18614

0.9900 .9967 1.0103 1.0381 1.0667 1.0962 1.1268 1.1584 1.1747

0.9785 .9852 .9988 1.0264 1.0549 1.0842 1.1146 1.1463 1.1626

0.9651 .9719 .9855 1.0134 1.0420 1.0713 1.1017 1.1331 1.1492

Sodium Nitrate (NaNO3)

%

0°C

20°C

40°C

60°C

80°C

100°C

1 2 4 8 12 16 20 24 28 30 35 40 45

1.0071 1.0144 1.0290 1.0587 1.0891 1.1203 1.1526 1.1860 1.2204 1.2380 1.2834 1.3316

1.0049 1.0117 1.0254 1.0532 1.0819 1.1118 1.1429 1.1752 1.2085 1.2256 1.2701 1.3175 1.3683

0.9986 1.0050 1.0180 1.0447 1.0724 1.1013 1.1314 1.1629 1.1955 1.2122 1.2560 1.3027 1.3528

0.9894 .9956 1.0082 1.0340 1.0609 1.0892 1.1187 1.1496 1.1816 1.1980 1.2413 1.2875 1.3371

0.9779 .9840 .9964 1.0218 1.0481 1.0757 1.1048 1.1351 1.1667 1.1830 1.2258 1.2715 1.3206

0.9644 .9704 .9826 1.0078 1.0340 1.0614 1.0901 1.1200 1.1513 1.1674 1.2100 1.2555 1.3044

TABLE 2-89 Sodium Chromate (Na2CrO4)

TABLE 2-93

TABLE 2-92 Sodium Nitrite (NaNO2)

18

15

%

d4

%

d4

1 2 4 8 12 16 20 24 26

1.0074 1.0164 1.0344 1.0718 1.1110 1.1518 1.1942 1.2383 1.2611

1 2 4 8 12 16 20

1.0058 1.0125 1.0260 1.0535 1.0816 1.1103 1.1394

Sodium Silicates Concentration, % 1

2

4

8

10

14

Na2O/3.9SiO2 Na2O/3.36SiO2 Na2O/2.40SiO2 Na2O/2.44SiO2 Na2O/2.06SiO2 Na2O/1.69SiO2

20

24

30

36

40

45

50

1.445 1.450

1.520

1.594

20

Formula

d4 1.006 1.006 1.007

1.014 1.014 1.016

1.030 1.030 1.034

1.063 1.065 1.071

1.080 1.083 1.090

1.116 1.120 1.130

1.172 1.179

1.211 1.222

1.275 1.290

1.365

1.007 1.007

1.016 1.017

1.035 1.036

1.073 1.077

1.093 1.098

1.134 1.141

1.200 1.210

1.247 1.259

1.309 1.321 1.337

1.387 1.397 1.424

2-106

PHYSICAL AND CHEMICAL DATA TABLE 2-94 %

0°C

1 2 4 8 12 16 20 24

1.0094 1.0189 1.0381 1.0773 1.1174 1.1585 1.2008 1.2443

TABLE 2-95 Sodium Sulfide (Na2S)

Sodium Sulfate (Na2SO4) 20°C

30°C

40°C

60°C

80°C

100°C

1.0073 1.0164 1.0348 1.0724 1.1109 1.1586 1.1915 1.2336

1.0046 1.0135 1.0315 1.0682 1.1062 1.1456 1.1865 1.2292

1.0010 1.0098 1.0276 1.0639 1.1015 1.1406 1.1813 1.2237

0.9919 1.0007 1.0184 1.0544 1.0915 1.1299 1.1696

0.9805 .9892 1.0068 1.0426 1.0795 1.1176 1.1569

0.9671 .9758 .9934 1.0292 1.0661 1.1042

TABLE 2-96 Sodium Sulfite (Na2SO3)

TABLE 2-97 Sodium Thiosulfate (Na2S2O3)

%

d 18 4

%

d 19 4

%

d 20 4

1 2 4 8 12 16 18

1.0098 1.0211 1.0440 1.0907 1.1388 1.1885 1.2140

1 2 4 8 12 16 18

1.0078 1.0172 1.0363 1.0751 1.1146 1.1549 1.1755

1 2 4 8 12 16 20 24 28 30 35 40

1.0065 1.0148 1.0315 1.0654 1.1003 1.1365 1.1740 1.2128 1.2532 1.2739 1.3273 1.3827

TABLE 2-98 Sodium Thiosulfate Pentahydrate (Na2S2O3⋅5H2O) %

d 19 4

1 2 4 8 12 16 20 24 28 30 40 50

1.0052 1.0105 1.0211 1.0423 1.0639 1.0863 1.1087 1.1322 1.1558 1.1676 1.2297 1.2954

TABLE 2-99 Stannic Chloride (SnCl4) %

d 15 4

1 2 4 8 12 16 20 24 28 30 35 40 45 50 55 60 65 70

1.007 1.015 1.031 1.064 1.099 1.135 1.173 1.212 1.255 1.278 1.337 1.403 1.475 1.555 1.644 1.742 1.851 1.971

TABLE 2-100 Stannous Chloride (SnCl2) %

d 15 4

1 2 4 8 12 16 20 24 28 30 35 40 45 50 55 60 65

1.0068 1.0146 1.0306 1.0638 1.0986 1.1353 1.1743 1.2159 1.2603 1.2837 1.3461 1.4145 1.4897 1.5729 1.6656 1.7695 1.8865

DENSITIES OF AQUEOUS INORGANIC SOLUTIONS TABLE 2-101 %

2-107

Sulfuric Acid (H2SO4)

0°C

10°C

15°C

20°C

25°C

30°C

40°C

50°C

60°C

80°C

100°C

1 2 3 4

1.0074 1.0147 1.0219 1.0291

1.0068 1.0138 1.0206 1.0275

1.0060 1.0129 1.0197 1.0264

1.0051 1.0118 1.0184 1.0250

1.0038 1.0104 1.0169 1.0234

1.0022 1.0087 1.0152 1.0216

0.9986 1.0050 1.0113 1.0176

0.9944 1.0006 1.0067 1.0129

0.9895 .9956 1.0017 1.0078

0.9779 .9839 .9900 .9961

0.9645 .9705 .9766 .9827

5 6 7 8 9

1.0364 1.0437 1.0511 1.0585 1.0660

1.0344 1.0414 1.0485 1.0556 1.0628

1.0332 1.0400 1.0469 1.0539 1.0610

1.0317 1.0385 1.0453 1.0522 1.0591

1.0300 1.0367 1.0434 1.0502 1.0571

1.0281 1.0347 1.0414 1.0481 1.0549

1.0240 1.0305 1.0371 1.0437 1.0503

1.0192 1.0256 1.0321 1.0386 1.0451

1.0140 1.0203 1.0266 1.0330 1.0395

1.0022 1.0084 1.0146 1.0209 1.0273

.9888 .9950 1.0013 1.0076 1.0140

10 11 12 13 14

1.0735 1.0810 1.0886 1.0962 1.1039

1.0700 1.0773 1.0846 1.0920 1.0994

1.0681 1.0753 1.0825 1.0898 1.0971

1.0661 1.0731 1.0802 1.0874 1.0947

1.0640 1.0710 1.0780 1.0851 1.0922

1.0617 1.0686 1.0756 1.0826 1.0897

1.0570 1.0637 1.0705 1.0774 1.0844

1.0517 1.0584 1.0651 1.0719 1.0788

1.0460 1.0526 1.0593 1.0661 1.0729

1.0338 1.0403 1.0469 1.0536 1.0603

1.0204 1.0269 1.0335 1.0402 1.0469

15 16 17 18 19

1.1116 1.1194 1.1272 1.1351 1.1430

1.1069 1.1145 1.1221 1.1298 1.1375

1.1045 1.1120 1.1195 1.1271 1.1347

1.1020 1.1094 1.1168 1.1243 1.1318

1.0994 1.1067 1.1141 1.1215 1.1290

1.0968 1.1040 1.1113 1.1187 1.1261

1.0914 1.0985 1.1057 1.1129 1.1202

1.0857 1.0927 1.0998 1.1070 1.1142

1.0798 1.0868 1.0938 1.1009 1.1081

1.0671 1.0740 1.0809 1.0879 1.0950

1.0537 1.0605 1.0674 1.0744 1.0814

20 21 22 23 24

1.1510 1.1590 1.1670 1.1751 1.1832

1.1453 1.1531 1.1609 1.1688 1.1768

1.1424 1.1501 1.1579 1.1657 1.1736

1.1394 1.1471 1.1548 1.1626 1.1704

1.1365 1.1441 1.1517 1.1594 1.1672

1.1335 1.1410 1.1486 1.1563 1.1640

1.1275 1.1349 1.1424 1.1500 1.1576

1.1215 1.1288 1.1362 1.1437 1.1512

1.1153 1.1226 1.1299 1.1373 1.1448

1.1021 1.1093 1.1166 1.1239 1.1313

1.0885 1.0957 1.1029 1.1102 1.1176

25 26 27 28 29

1.1914 1.1996 1.2078 1.2160 1.2243

1.1848 1.1929 1.2010 1.2091 1.2173

1.1816 1.1896 1.1976 1.2057 1.2138

1.1783 1.1862 1.1942 1.2023 1.2104

1.1750 1.1829 1.1909 1.1989 1.2069

1.1718 1.1796 1.1875 1.1955 1.2035

1.1653 1.1730 1.1808 1.1887 1.1966

1.1588 1.1665 1.1742 1.1820 1.1898

1.1523 1.1599 1.1676 1.1753 1.1831

1.1388 1.1463 1.1539 1.1616 1.1693

1.1250 1.1325 1.1400 1.1476 1.1553

30 31 32 33 34

1.2326 1.2409 1.2493 1.2577 1.2661

1.2255 1.2338 1.2421 1.2504 1.2588

1.2220 1.2302 1.2385 1.2468 1.2552

1.2185 1.2267 1.2349 1.2432 1.2515

1.2150 1.2232 1.2314 1.2396 1.2479

1.2115 1.2196 1.2278 1.2360 1.2443

1.2046 1.2126 1.2207 1.2289 1.2371

1.1977 1.2057 1.2137 1.2218 1.2300

1.1909 1.1988 1.2068 1.2148 1.2229

1.1771 1.1849 1.1928 1.2008 1.2088

1.1630 1.1708 1.1787 1.1866 1.1946

35 36 37 38 39

1.2746 1.2831 1.2917 1.3004 1.3091

1.2672 1.2757 1.2843 1.2929 1.3016

1.2636 1.2720 1.2805 1.2891 1.2978

1.2599 1.2684 1.2769 1.2855 1.2941

1.2563 1.2647 1.2732 1.2818 1.2904

1.2526 1.2610 1.2695 1.2780 1.2866

1.2454 1.2538 1.2622 1.2707 1.2793

1.2383 1.2466 1.2550 1.2635 1.2720

1.2311 1.2394 1.2477 1.2561 1.2646

1.2169 1.2251 1.2334 1.2418 1.2503

1.2027 1.2109 1.2192 1.2276 1.2361

40 41 42 43 44

1.3179 1.3268 1.3357 1.3447 1.3538

1.3103 1.3191 1.3280 1.3370 1.3461

1.3065 1.3153 1.3242 1.3332 1.3423

1.3028 1.3116 1.3205 1.3294 1.3384

1.2991 1.3079 1.3167 1.3256 1.3346

1.2953 1.3041 1.3129 1.3218 1.3308

1.2880 1.2967 1.3055 1.3144 1.3234

1.2806 1.2893 1.2981 1.3070 1.3160

1.2732 1.2819 1.2907 1.2996 1.3086

1.2589 1.2675 1.2762 1.2850 1.2939

1.2446 1.2532 1.2619 1.2707 1.2796

45 46 47 48 49

1.3630 1.3724 1.3819 1.3915 1.4012

1.3553 1.3646 1.3740 1.3835 1.3931

1.3515 1.3608 1.3702 1.3797 1.3893

1.3476 1.3569 1.3663 1.3758 1.3854

1.3437 1.3530 1.3624 1.3719 1.3814

1.3399 1.3492 1.3586 1.3680 1.3775

1.3325 1.3417 1.3510 1.3604 1.3699

1.3251 1.3343 1.3435 1.3528 1.3623

1.3177 1.3269 1.3362 1.3455 1.3549

1.3029 1.3120 1.3212 1.3305 1.3399

1.2886 1.2976 1.3067 1.3159 1.3253

50 51 52 53 54

1.4110 1.4209 1.4310 1.4412 1.4515

1.4029 1.4128 1.4228 1.4329 1.4431

1.3990 1.4088 1.4188 1.4289 1.4391

1.3951 1.4049 1.4148 1.4248 1.4350

1.3911 1.4009 1.4109 1.4209 1.4310

1.3872 1.3970 1.4069 1.4169 1.4270

1.3795 1.3893 1.3991 1.4091 1.4191

1.3719 1.3816 1.3914 1.4013 1.4113

1.3644 1.3740 1.3837 1.3936 1.4036

1.3494 1.3590 1.3687 1.3785 1.3884

1.3348 1.3444 1.3540 1.3637 1.3735

55 56 57 58 59

1.4619 1.4724 1.4830 1.4937 1.5045

1.4535 1.4640 1.4746 1.4852 1.4959

1.4494 1.4598 1.4703 1.4809 1.4916

1.4453 1.4557 1.4662 1.4768 1.4875

1.4412 1.4516 1.4621 1.4726 1.4832

1.4372 1.4475 1.4580 1.4685 1.4791

1.4293 1.4396 1.4500 1.4604 1.4709

1.4214 1.4317 1.4420 1.4524 1.4629

1.4137 1.4239 1.4342 1.4446 1.4551

1.3984 1.4085 1.4187 1.4290 1.4393

1.3834 1.3934 1.4035 1.4137 1.4240

60 61 62 63 64

1.5154 1.5264 1.5375 1.5487 1.5600

1.5067 1.5177 1.5287 1.5398 1.5510

1.5024 1.5133 1.5243 1.5354 1.5465

1.4983 1.5091 1.5200 1.5310 1.5421

1.4940 1.5048 1.5157 1.5267 1.5378

1.4898 1.5006 1.5115 1.5225 1.5335

1.4816 1.4923 1.5031 1.5140 1.5250

1.4735 1.4842 1.4950 1.5058 1.5167

1.4656 1.4762 1.4869 1.4977 1.5086

1.4497 1.4602 1.4708 1.4815 1.4923

1.4344 1.4449 1.4554 1.4660 1.4766

2-108

PHYSICAL AND CHEMICAL DATA

TABLE 2-101

Sulfuric Acid (H2SO4) (Concluded )

%

0°C

10°C

15°C

20°C

25°C

30°C

40°C

50°C

60°C

80°C

100°C

65 66 67 68 69

1.5714 1.5828 1.5943 1.6059 1.6176

1.5623 1.5736 1.5850 1.5965 1.6081

1.5578 1.5691 1.5805 1.5920 1.6035

1.5533 1.5646 1.5760 1.5874 1.5989

1.5490 1.5602 1.5715 1.5829 1.5944

1.5446 1.5558 1.5671 1.5785 1.5899

1.5361 1.5472 1.5584 1.5697 1.5811

1.5277 1.5388 1.5499 1.5611 1.5724

1.5195 1.5305 1.5416 1.5528 1.5640

1.5031 1.5140 1.5249 1.5359 1.5470

1.4873 1.4981 1.5089 1.5198 1.5307

70 71 72 73 74

1.6293 1.6411 1.6529 1.6648 1.6768

1.6198 1.6315 1.6433 1.6551 1.6670

1.6151 1.6268 1.6385 1.6503 1.6622

1.6105 1.6221 1.6338 1.6456 1.6574

1.6059 1.6175 1.6292 1.6409 1.6526

1.6014 1.6130 1.6246 1.6363 1.6480

1.5925 1.6040 1.6155 1.6271 1.6387

1.5838 1.5952 1.6067 1.6182 1.6297

1.5753 1.5867 1.5981 1.6095 1.6209

1.5582 1.5694 1.5806 1.5919 1.6031

1.5417 1.5527 1.5637 1.5747 1.5857

75 76 77 78 79

1.6888 1.7008 1.7128 1.7247 1.7365

1.6789 1.6908 1.7026 1.7144 1.7261

1.6740 1.6858 1.6976 1.7093 1.7209

1.6692 1.6810 1.6927 1.7043 1.7158

1.6644 1.6761 1.6878 1.6994 1.7108

1.6597 1.6713 1.6829 1.6944 1.7058

1.6503 1.6619 1.6734 1.6847 1.6959

1.6412 1.6526 1.6640 1.6751 1.6862

1.6322 1.6435 1.6547 1.6657 1.6766

1.6142 1.6252 1.6361 1.6469 1.6575

1.5966 1.6074 1.6181 1.6286 1.6390

80 81 82 83 84

1.7482 1.7597 1.7709 1.7815 1.7916

1.7376 1.7489 1.7599 1.7704 1.7804

1.7323 1.7435 1.7544 1.7649 1.7748

1.7272 1.7383 1.7491 1.7594 1.7693

1.7221 1.7331 1.7437 1.7540 1.7639

1.7170 1.7279 1.7385 1.7487 1.7585

1.7069 1.7177 1.7281 1.7382 1.7479

1.6971 1.7077 1.7180 1.7279 1.7375

1.6873 1.6978 1.7080 1.7179 1.7274

1.6680 1.6782 1.6882 1.6979 1.7072

1.6493 1.6594 1.6692 1.6787 1.6878

85 86 87 88 89

1.8009 1.8095 1.8173 1.8243 1.8306

1.7897 1.7983 1.8061 1.8132 1.8195

1.7841 1.7927 1.8006 1.8077 1.8141

1.7786 1.7872 1.7951 1.8022 1.8087

1.7732 1.7818 1.7897 1.7968 1.8033

1.7678 1.7763 1.7842 1.7914 1.7979

1.7571 1.7657 1.7736 1.7809 1.7874

1.7466 1.7552 1.7632 1.7705 1.7770

1.7364 1.7449 1.7529 1.7602 1.7669

1.7161 1.7245 1.7324 1.7397 1.7464

1.6966 1.7050 1.7129 1.7202 1.7269

90 91 92 93 94

1.8361 1.8410 1.8453 1.8490 1.8520

1.8252 1.8302 1.8346 1.8384 1.8415

1.8198 1.8248 1.8293 1.8331 1.8363

1.8144 1.8195 1.8240 1.8279 1.8312

1.8091 1.8142 1.8188 1.8227 1.8260

1.8038 1.8090 1.8136 1.8176 1.8210

1.7933 1.7986 1.8033 1.8074 1.8109

1.7829 1.7883 1.7932 1.7974 1.8011

1.7729 1.7783 1.7832 1.7876 1.7914

1.7525 1.7581 1.7633 1.7681

1.7331 1.7388 1.7439 1.7485

95 96 97 98 99 100

1.8544 1.8560 1.8569 1.8567 1.8551 1.8517

1.8439 1.8457 1.8466 1.8463 1.8445 1.8409

1.8388 1.8406 1.8414 1.8411 1.8393 1.8357

1.8337 1.8355 1.8364 1.8361 1.8342 1.8305

1.8286 1.8305 1.8314 1.8310 1.8292 1.8255

1.8236 1.8255 1.8264 1.8261 1.8242 1.8205

1.8137 1.8157 1.8166 1.8163 1.8145 1.8107

1.8040 1.8060 1.8071 1.8068 1.8050 1.8013

1.7944 1.7965 1.7977 1.7976 1.7958 1.7922

%

d 5.96 4

%

d 13.00 4

d 18.00 4

0.005 .01 .02 .03 .04

1.000 0140 1.000 0576 1.000 1434 1.000 2276 1.000 3104

0.05 .1 .2 .3 .4

0.999 810 1.000 185 1.000 912 1.001 623 1.002 326

0.999 028 .999 400 1.000 119 1.000 820 1.001 512

.05 .06 .07 .08 .09

1.000 3920 1.000 4726 1.000 5523 1.000 6313 1.000 7098

.5 .6 .8 1.0 1.2

1.003 023 1.003 716 1.005 090 1.006 452 1.007 807

1.002 197 1.002 877 1.004 227 1.005 570 1.006 909

.10 .15 .20 .25 .30

1.000 7880 1.001 1732 1.001 5514 1.001 9254 1.002 2961

1.4 1.6 1.8 2.0 2.2

1.009 159 1.010 510 1.011 860 1.013 209 1.014 557

1.008 247 1.009 583 1.010 918 1.012 252 1.013 586

.35 .40 .45 .50

1.002 6639 1.003 0292 1.003 3923 1.003 7534

2.4

1.015 904

1.014 919

DENSITIES OF AQUEOUS ORGANIC SOLUTIONS TABLE 2-102

Zinc Bromide (ZnBr2)

%

0°C

20°C

40°C

60°C

80°C

100°C

2 4 8 12 16

1.0188 1.0381 1.0777 1.1186 1.1609

1.0167 1.0354 1.0738 1.1135 1.1544

1.0102 1.0285 1.0660 1.1046 1.1445

1.0008 1.0187 1.0554 1.0932 1.1320

0.9890 1.0065 1.0422 1.0789 1.1169

0.9751 0.9921 1.0270 1.0629 1.1000

20 30 40 50 60 65

1.2043 1.3288 1.477 1.661 1.891 2.026

1.1965 1.3170 1.462 1.643 1.869 2.002

1.1855 1.3030 1.445 1.623 1.845 1.976

1.1720 1.2868 1.427 1.602 1.822 1.951

1.1560 1.2688 1.406 1.579 1.797 1.924

1.1382 1.2489 1.385 1.555 1.771 1.898

TABLE 2-103

TABLE 2-104 [Zn(NO3)2]

Zinc Nitrate

TABLE 2-105 (ZnSO4)

2-109

Zinc Sulfate

%

18°C

%

18°C

%

20°C

2 4 6 8 10 12 14 16

1.0154 1.0322 1.0496 1.0675 1.0859 1.1048 1.1244 1.1445

18 20 25 30 35 40 45 50

1.1652 1.1865 1.2427 1.3029 1.3678 1.4378 1.5134 1.5944

2 4 6 8 10 12 14 16

1.019 1.0403 1.0620 1.0842 1.1071 1.1308 1.1553 1.1806

Zinc Chloride (ZnCl2)

%

0°C

20°C

40°C

60°C

80°C

100°C

2 4 8 12 16

1.0192 1.0384 1.0769 1.1159 1.1558

1.0167 1.0350 1.0715 1.1085 1.1468

1.0099 1.0274 1.0624 1.0980 1.1350

1.0003 1.0172 1.0508 1.0853 1.1212

0.9882 1.0044 1.0369 1.0704 1.1055

0.9739 .9894 1.0211 1.0541 1.0888

20 30 40 50 60 70

1.1970 1.3062 1.4329 1.5860

1.1866 1.2928 1.4173 1.5681 1.749 1.962

1.1736 1.2778 1.4003 1.5495

1.1590 1.2614 1.3824 1.5300

1.1428 1.2438 1.3637 1.5097

1.1255 1.2252 1.3441 1.4892

DENSITIES OF AQUEOUS ORGANIC SOLUTIONS* From International Critical Tables, vol. 3, pp. 115–129. All compositions are in weight percent in vacuo. All density values are d4t = g/mL in vacuo.

UNITS AND UNITS CONVERSIONS Unless otherwise noted, densities are given in grams per cubic centimeter. To convert to pounds per cubic foot, multiply by 62.43. °F = 9⁄ 5 °C + 32 TABLE 2-106

*For gasoline and aircraft fuels see Hibbard, NACA Res. Mem. E56I21 (declassified 1958).

Formic Acid (HCOOH)

%

0°C

15°C

20°C

30°C

%

0°C

15°C

20°C

30°C

%

0°C

15°C

20°C

30°C

%

0°C

15°C

20°C

30°C

0 1 2 3 4

0.9999 1.0028 1.0059 1.0090 1.0120

0.9991 1.0019 1.0045 1.0072 1.0100

0.9982 1.0019 1.0044 1.0070 1.0093

0.9957 0.9980 1.0004 1.0028 1.0053

25 26 27 28 29

1.0706 1.0733 1.0760 1.0787 1.0813

1.0627 1.0652 1.0678 1.0702 1.0726

1.0609 1.0633 1.0656 1.0681 1.0705

1.0540 1.0564 1.0587 1.0609 1.0632

50 51 52 53 54

1.1349 1.1374 1.1399 1.1424 1.1448

1.1225 1.1248 1.1271 1.1294 1.1318

1.1207 1.1223 1.1244 1.1269 1.1295

1.1098 1.1120 1.1142 1.1164 1.1186

75 76 77 78 79

1.1953 1.1976 1.1999 1.2021 1.2043

1.1794 1.1816 1.1837 1.1859 1.1881

1.1769 1.1785 1.1801 1.1818 1.1837

1.1636 1.1656 1.1676 1.1697 1.1717

5 6 7 8 9

1.0150 1.0179 1.0207 1.0237 1.0266

1.0124 1.0151 1.0177 1.0204 1.0230

1.0115 1.0141 1.0170 1.0196 1.0221

1.0075 1.0101 1.0125 1.0149 1.0173

30 31 32 33 34

1.0839 1.0866 1.0891 1.0916 1.0941

1.0750 1.0774 1.0798 1.0821 1.0844

1.0729 1.0753 1.0777 1.0800 1.0823

1.0654 1.0676 1.0699 1.0721 1.0743

55 56 57 58 59

1.1472 1.1497 1.1523 1.1548 1.1573

1.1341 1.1365 1.1388 1.1411 1.1434

1.1320 1.1342 1.1361 1.1381 1.1401

1.1208 1.1230 1.1253 1.1274 1.1295

80 81 82 83 84

1.2065 1.2088 1.2110 1.2132 1.2154

1.1902 1.1924 1.1944 1.1965 1.1985

1.1806 1.1876 1.1896 1.1914 1.1929

1.1737 1.1758 1.1778 1.1798 1.1817

10 11 12 13 14

1.0295 1.0324 1.0351 1.0379 1.0407

1.0256 1.0281 1.0306 1.0330 1.0355

1.0246 1.0271 1.0296 1.0321 1.0345

1.0197 1.0221 1.0244 1.0267 1.0290

35 36 37 38 39

1.0966 1.0993 1.1018 1.1043 1.1069

1.0867 1.0892 1.0916 1.0940 1.0964

1.0847 1.0871 1.0895 1.0919 1.0940

1.0766 1.0788 1.0810 1.0832 1.0854

60 61 62 63 64

1.1597 1.1621 1.1645 1.1669 1.1694

1.1458 1.1481 1.1504 1.1526 1.1549

1.1424 1.1448 1.1473 1.1493 1.1517

1.1317 1.1338 1.1360 1.1382 1.1403

85 86 87 88 89

1.2176 1.2196 1.2217 1.2237 1.2258

1.2005 1.2025 1.2045 1.2064 1.2084

1.1953 1.1976 1.1994 1.2012 1.2028

1.1837 1.1856 1.1875 1.1893 1.1910

15 16 17 18 19

1.0435 1.0463 1.0491 1.0518 1.0545

1.0380 1.0405 1.0430 1.0455 1.0480

1.0370 1.0393 1.0417 1.0441 1.0464

1.0313 1.0336 1.0358 1.0381 1.0404

40 41 42 43 44

1.1095 1.1122 1.1148 1.1174 1.1199

1.0988 1.1012 1.1036 1.1060 1.1084

1.0963 1.0990 1.1015 1.1038 1.1062

1.0876 1.0898 1.0920 1.0943 1.0965

65 66 67 68 69

1.1718 1.1742 1.1766 1.1790 1.1813

1.1572 1.1595 1.1618 1.1640 1.1663

1.1543 1.1565 1.1584 1.1604 1.1628

1.1425 1.1446 1.1467 1.1489 1.1510

90 91 92 93 94

1.2278 1.2297 1.2316 1.2335 1.2354

1.2102 1.2121 1.2139 1.2157 1.2174

1.2044 1.2059 1.2078 1.2099 1.2117

1.1927 1.1945 1.1961 1.1978 1.1994

20 21 22 23 24

1.0571 1.0598 1.0625 1.0652 1.0679

1.0505 1.0532 1.0556 1.0580 1.0604

1.0488 1.0512 1.0537 1.0561 1.0585

1.0427 1.0451 1.0473 1.0496 1.0518

45 46 47 48 49

1.1224 1.1249 1.1274 1.1299 1.1324

1.1109 1.1133 1.1156 1.1179 1.1202

1.1085 1.1108 1.1130 1.1157 1.1185

1.0987 1.1009 1.1031 1.1053 1.1076

70 71 72 73 74

1.1835 1.1858 1.1882 1.1906 1.1929

1.1685 1.1707 1.1729 1.1751 1.1773

1.1655 1.1677 1.1702 1.1728 1.1752

1.1531 1.1552 1.1573 1.1595 1.1615

95 96 97 98 99

1.2372 1.2390 1.2408 1.2425 1.2441

1.2191 1.2208 1.2224 1.2240 1.2257

1.2140 1.2158 1.2170 1.2183 1.2202

1.2008 1.2022 1.2036 1.2048 1.2061

100

1.2456

1.2273

1.2212

1.2073

2-110

PHYSICAL AND CHEMICAL DATA

TABLE 2-107 %

Acetic Acid (CH3COOH)

0°C

10°C

15°C

20°C

25°C

30°C

40°C

%

0°C

10°C

15°C

20°C

25°C

30°C

40°C

0 1 2 3 4

0.9999 1.0016 1.0033 1.0051 1.0070

0.9997 1.0013 1.0029 1.0044 1.0060

0.9991 1.0006 1.0021 1.0036 1.0051

0.9982 .9996 1.0012 1.0025 1.0040

0.9971 .9987 1.0000 1.0013 1.0027

0.9957 .9971 .9984 .9997 1.0011

0.9922 .9934 .9946 .9958 .9970

50 51 52 53 54

1.0729 1.0738 1.0748 1.0757 1.0765

1.0654 1.0663 1.0671 1.0679 1.0687

1.0613 1.0622 1.0629 1.0637 1.0644

1.0575 1.0582 1.0590 1.0597 1.0604

1.0534 1.0542 1.0549 1.0555 1.0562

1.0492 1.0499 1.0506 1.0512 1.0518

1.0408 1.0414 1.0421 1.0427 1.0432

5 6 7 8 9

1.0088 1.0106 1.0124 1.0142 1.0159

1.0076 1.0092 1.0108 1.0124 1.0140

1.0066 1.0081 1.0096 1.0111 1.0126

1.0055 1.0069 1.0083 1.0097 1.0111

1.0041 1.0055 1.0068 1.0081 1.0094

1.0024 1.0037 1.0050 1.0063 1.0076

.9982 .9994 1.0006 1.0018 1.0030

55 56 57 58 59

1.0774 1.0782 1.0790 1.0798 1.0805

1.0694 1.0701 1.0708 1.0715 1.0722

1.0651 1.0658 1.0665 1.0672 1.0678

1.0611 1.0618 1.0624 1.0631 1.0637

1.0568 1.0574 1.0580 1.0586 1.0592

1.0525 1.0531 1.0536 1.0542 1.0547

1.0438 1.0443 1.0448 1.0453 1.0458

10 11 12 13 14

1.0177 1.0194 1.0211 1.0228 1.0245

1.0156 1.0171 1.0187 1.0202 1.0217

1.0141 1.0155 1.0170 1.0184 1.0199

1.0125 1.0139 1.0154 1.0168 1.0182

1.0107 1.0120 1.0133 1.0146 1.0159

1.0089 1.0102 1.0115 1.0127 1.0139

1.0042 1.0054 1.0065 1.0077 1.0088

60 61 62 63 64

1.0813 1.0820 1.0826 1.0833 1.0838

1.0728 1.0734 1.0740 1.0746 1.0752

1.0684 1.0690 1.0696 1.0701 1.0706

1.0642 1.0648 1.0653 1.0658 1.0662

1.0597 1.0602 1.0607 1.0612 1.0616

1.0552 1.0557 1.0562 1.0566 1.0571

1.0462 1.0466 1.0470 1.0473 1.0477

15 16 17 18 19

1.0262 1.0278 1.0295 1.0311 1.0327

1.0232 1.0247 1.0262 1.0276 1.0291

1.0213 1.0227 1.0241 1.0255 1.0269

1.0195 1.0209 1.0223 1.0236 1.0250

1.0172 1.0185 1.0198 1.0210 1.0223

1.0151 1.0163 1.0175 1.0187 1.0198

1.0099 1.0110 1.0121 1.0132 1.0142

65 66 67 68 69

1.0844 1.0850 1.0856 1.0860 1.0865

1.0757 1.0762 1.0767 1.0771 1.0775

1.0711 1.0716 1.0720 1.0725 1.0729

1.0666 1.0671 1.0675 1.0678 1.0682

1.0621 1.0624 1.0628 1.0631 1.0634

1.0575 1.0578 1.0582 1.0585 1.0588

1.0480 1.0483 1.0486 1.0489 1.0491

20 21 22 23 24

1.0343 1.0358 1.0374 1.0389 1.0404

1.0305 1.0319 1.0333 1.0347 1.0361

1.0283 1.0297 1.0310 1.0323 1.0336

1.0263 1.0276 1.0288 1.0301 1.0313

1.0235 1.0248 1.0260 1.0272 1.0283

1.0210 1.0222 1.0233 1.0244 1.0256

1.0153 1.0164 1.0174 1.0185 1.0195

70 71 72 73 74

1.0869 1.0874 1.0877 1.0881 1.0884

1.0779 1.0783 1.0786 1.0789 1.0792

1.0732 1.0736 1.0738 1.0741 1.0743

1.0685 1.0687 1.0690 1.0693 1.0694

1.0637 1.0640 1.0642 1.0644 1.0645

1.0590 1.0592 1.0594 1.0595 1.0596

1.0493 1.0495 1.0496 1.0497 1.0498

25 26 27 28 29

1.0419 1.0434 1.0449 1.0463 1.0477

1.0375 1.0388 1.0401 1.0414 1.0427

1.0349 1.0362 1.0374 1.0386 1.0399

1.0326 1.0338 1.0349 1.0361 1.0372

1.0295 1.0307 1.0318 1.0329 1.0340

1.0267 1.0278 1.0289 1.0299 1.0310

1.0205 1.0215 1.0225 1.0234 1.0244

75 76 77 78 79

1.0887 1.0889 1.0891 1.0893 1.0894

1.0794 1.0796 1.0797 1.0798 1.0798

1.0745 1.0746 1.0747 1.0747 1.0747

1.0696 1.0698 1.0699 1.0700 1.0700

1.0647 1.0648 1.0648 1.0648 1.0648

1.0597 1.0598 1.0598 1.0598 1.0597

1.0499 1.0499 1.0499 1.0498 1.0497

30 31 32 33 34

1.0491 1.0505 1.0519 1.0532 1.0545

1.0440 1.0453 1.0465 1.0477 1.0489

1.0411 1.0423 1.0435 1.0446 1.0458

1.0384 1.0395 1.0406 1.0417 1.0428

1.0350 1.0361 1.0372 1.0382 1.0392

1.0320 1.0330 1.0341 1.0351 1.0361

1.0253 1.0262 1.0272 1.0281 1.0289

80 81 82 83 84

1.0895 1.0895 1.0895 1.0895 1.0893

1.0798 1.0797 1.0796 1.0795 1.0793

1.0747 1.0745 1.0743 1.0741 1.0738

1.0700 1.0699 1.0698 1.0696 1.0693

1.0647 1.0646 1.0644 1.0642 1.0638

1.0596 1.0594 1.0592 1.0589 1.0585

1.0495 1.0493 1.0490 1.0487 1.0483

35 36 37 38 39

1.0558 1.0571 1.0584 1.0596 1.0608

1.0501 1.0513 1.0524 1.0535 1.0546

1.0469 1.0480 1.0491 1.0501 1.0512

1.0438 1.0449 1.0459 1.0469 1.0479

1.0402 1.0412 1.0422 1.0432 1.0441

1.0371 1.0380 1.0390 1.0399 1.0408

1.0298 1.0306 1.0314 1.0322 1.0330

85 86 87 88 89

1.0891 1.0887 1.0883 1.0877 1.0872

1.0790 1.0787 1.0783 1.0778 1.0773

1.0735 1.0731 1.0726 1.0721 1.0715

1.0689 1.0685 1.0680 1.0675 1.0668

1.0635 1.0630 1.0626 1.0620 1.0613

1.0582 1.0576 1.0571 1.0564 1.0557

1.0479 1.0473 1.0467 1.0460 1.0453

40 41 42 43 44

1.0621 1.0633 1.0644 1.0656 1.0667

1.0557 1.0568 1.0578 1.0588 1.0598

1.0522 1.0532 1.0542 1.0551 1.0561

1.0488 1.0498 1.0507 1.0516 1.0525

1.0450 1.0460 1.0469 1.0477 1.0486

1.0416 1.0425 1.0433 1.0441 1.0449

1.0338 1.0346 1.0353 1.0361 1.0368

90 91 92 93 94

1.0865 1.0857 1.0848 1.0838 1.0826

1.0766 1.0758 1.0749 1.0739 1.0727

1.0708 1.0700 1.0690 1.0680 1.0667

1.0661 1.0652 1.0643 1.0632 1.0619

1.0605 1.0597 1.0587 1.0577 1.0564

1.0549 1.0541 1.0530 1.0518 1.0506

1.0445 1.0436 1.0426 1.0414 1.0401

45 46 47 48 49

1.0679 1.0689 1.0699 1.0709 1.0720

1.0608 1.0618 1.0627 1.0636 1.0645

1.0570 1.0579 1.0588 1.0597 1.0605

1.0534 1.0542 1.0551 1.0559 1.0567

1.0495 1.0503 1.0511 1.0518 1.0526

1.0456 1.0464 1.0471 1.0479 1.0486

1.0375 1.0382 1.0389 1.0395 1.0402

95 96 97 98 99

1.0813 1.0798 1.0780 1.0759 1.0730

1.0714

1.0652 1.0632 1.0611 1.0590 1.0567

1.0605 1.0588 1.0570 1.0549 1.0524

1.0551 1.0535 1.0516 1.0495 1.0468

1.0491 1.0473 1.0454 1.0431 1.0407

1.0386 1.0368 1.0348 1.0325 1.0299

100

1.0697

1.0545

1.0498

1.0440

1.0380

1.0271

DENSITIES OF AQUEOUS ORGANIC SOLUTIONS TABLE 2-108

TABLE 2-109

2-111

Oxalic Acid (H2C2O4)

%

d 17.5 4

%

d 17.5 4

1 2 4

1.0035 1.0070 1.0140

8 10 12

1.0280 1.0350 1.0420

Methyl Alcohol (CH3OH)*

%

0°C

10°C

20°C

15°C

%

0°C

10°C

20°C

15°C

%

0°C

10°C

20°C

15°C

0 1 2 3 4

0.9999 .9981 .9963 .9946 .9930

0.9997 .9980 .9962 .9945 .9929

15.56°C 0.9990 .9973 .9955 .9938 .9921

0.9982 .9965 .9948 .9931 .9914

0.99913 .99727 .99543 .99370 .99198

35 36 37 38 39

0.9534 .9520 .9505 .9490 .9475

0.9484 .9469 .9453 .9437 .9420

15.56°C 0.9456 .9440 .9422 .9405 .9387

0.9433 .9416 .9398 .9381 .9363

0.94570 .94404 .94237 .94067 .93894

70 71 72 73 74

0.8869 .8847 .8824 .8801 .8778

0.8794 .8770 .8747 .8724 .8699

15.56°C 0.8748 .8726 .8702 .8678 .8653

0.8715 .8690 .8665 .8641 .8616

0.87507 .87271 .87033 .86792 .86546

5 6 7 8 9

.9914 .9899 .9884 .9870 .9856

.9912 .9896 .9881 .9865 .9849

.9904 .9889 .9872 .9857 .9841

.9896 .9880 .9863 .9847 .9831

.99029 .98864 .98701 .98547 .98394

40 41 42 43 44

.9459 .9443 .9427 .9411 .9395

.9403 .9387 .9370 .9352 .9334

.9369 .9351 .9333 .9315 .9297

.9345 .9327 .9309 .9290 .9272

.93720 .93543 .93365 .93185 .93001

75 76 77 78 79

.8754 .8729 .8705 .8680 .8657

.8676 .8651 .8626 .8602 .8577

.8629 .8604 .8579 .8554 .8529

.8592 .8567 .8542 .8518 .8494

.86300 .86051 .85801 .85551 .85300

10 11 12 13 14

.9842 .9829 .9816 .9804 .9792

.9834 .9820 .9805 .9791 .9778

.9826 .9811 .9796 .9781 .9766

.9815 .9799 .9784 .9768 .9754

.98241 .98093 .97945 .97802 .97660

45 46 47 48 49

.9377 .9360 .9342 .9324 .9306

.9316 .9298 .9279 .9260 .9240

.9279 .9261 .9242 .9223 .9204

.9252 .9234 .9214 .9196 .9176

.92815 .92627 .92436 .92242 .92048

80 81 82 83 84

.8634 .8610 .8585 .8560 .8535

.8551 .8527 .8501 .8475 .8449

.8503 .8478 .8452 .8426 .8400

.8469 .8446 .8420 .8394 .8366

.85048 .84794 .84536 .84274 .84009

15 16 17 18 19

.9780 .9769 .9758 .9747 .9736

.9764 .9751 .9739 .9726 .9713

.9752 .9738 .9723 .9709 .9695

.9740 .9725 .9710 .9696 .9681

.97518 .97377 .97237 .97096 .96955

50 51 52 53 54

.9287 .9269 .9250 .9230 .9211

.9221 .9202 .9182 .9162 .9142

.9185 .9166 .9146 .9126 .9106

.9156 .9135 .9114 .9094 .9073

.91852 .91653 .91451 .91248 .91044

85 86 87 88 89

.8510 .8483 .8456 .8428 .8400

.8422 .8394 .8367 .8340 .8314

.8374 .8347 .8320 .8294 .8267

.8340 .8314 .8286 .8258 .8230

.83742 .83475 .83207 .82937 .82667

20 21 22 23 24

.9725 .9714 .9702 .9690 .9678

.9700 .9687 .9673 .9660 .9646

.9680 .9666 .9652 .9638 .9624

.9666 .9651 .9636 .9622 .9607

.96814 .96673 .96533 .96392 .96251

55 56 57 58 59

.9191 .9172 .9151 .9131 .9111

.9122 .9101 .9080 .9060 .9039

.9086 .9065 .9045 .9024 .9002

.9052 .9032 .9010 .8988 .8968

.90839 .90631 .90421 .90210 .89996

90 91 92 93 94

.8374 .8347 .8320 .8293 .8266

.8287 .8261 .8234 .8208 .8180

.8239 .8212 .8185 .8157 .8129

.8202 .8174 .8146 .8118 .8090

.82396 .82124 .81849 .81568 .81285

25 26 27 28 29

.9666 .9654 .9642 .9629 .9616

.9632 .9618 .9604 .9590 .9575

.9609 .9595 .9580 .9565 .9550

.9592 .9576 .9562 .9546 .9531

.96108 .95963 .95817 .95668 .95518

60 61 62 63 64

.9090 .9068 .9046 .9024 .9002

.9018 .8998 .8977 .8955 .8933

.8980 .8958 .8936 .8913 .8890

.8946 .8924 .8902 .8879 .8856

.89781 .89563 .89341 .89117 .88890

95 96 97 98 99

.8240 .8212 .8186 .8158 .8130

.8152 .8124 .8096 .8068 .8040

.8101 .8073 .8045 .8016 .7987

.8062 .8034 .8005 .7976 .7948

.80999 .80713 .80428 .80143 .79859

30 31 32 33 34

.9604 .9590 .9576 .9563 .9549

.9560 .9546 .9531 .9516 .9500

.9535 .9521 .9505 .9489 .9473

.9515 .9499 .9483 .9466 .9450

.95366 .95213 .95056 .94896 .94734

65 66 67 68 69

.8980 .8958 .8935 .8913 .8891

.8911 .8888 .8865 .8842 .8818

.8867 .8844 .8820 .8797 .8771

.8834 .8811 .8787 .8763 .8738

.88662 .88433 .88203 .87971 .87739

100

.8102

.8009

.7959

.7917

.79577

*It should be noted that the values for 100 percent do not agree with some data available elsewhere, e.g., American Institute of Physics Handbook, McGraw-Hill, New York, 1957. Also, see Atack, Handbook of Chemical Data, Reinhold, New York, 1957.

2-112

PHYSICAL AND CHEMICAL DATA

TABLE 2-110

Ethyl Alcohol (C2H5OH)*

%

10°C

15°C

20°C

25°C

30°C

35°C

40°C

%

10°C

15°C

20°C

25°C

30°C

35°C

40°C

0 1 2 3 4

0.99973 785 602 426 258

0.99913 725 542 365 195

0.99823 636 453 275 103

0.99708 520 336 157 .98984

0.99568 379 194 014 .98839

0.99406 217 031 .98849 672

0.99225 034 .98846 663 485

50 51 52 53 54

0.92126 .91943 723 502 279

0.91776 555 333 110 .90885

0.91384 160 .90936 711 485

0.90985 760 534 307 079

0.90580 353 125 .89896 667

0.90168 .89940 710 479 248

0.89750 519 288 056 .88823

5 6 7 8 9

098 .98946 801 660 524

032 .98877 729 584 442

.98938 780 627 478 331

817 656 500 346 193

670 507 347 189 031

501 335 172 009 .97846

311 142 .97975 808 641

55 56 57 58 59

055 .90831 607 381 154

659 433 207 .89980 752

258 031 .89803 574 344

.89850 621 392 162 .88931

437 206 .88975 744 512

016 .88784 552 319 085

589 356 122 .87888 653

10 11 12 13 14

393 267 145 026 .97911

304 171 041 .97914 790

187 047 .97910 775 643

043 .97897 753 611 472

.97875 723 573 424 278

685 527 371 216 063

475 312 150 .96989 829

60 61 62 63 64

.89927 698 468 237 006

523 293 062 .88830 597

113 .88882 650 417 183

699 446 233 .87998 763

278 044 .87809 574 337

.87851 615 379 142 .86905

417 180 .86943 705 466

15 16 17 18 19

800 692 583 473 363

669 552 433 313 191

514 387 259 129 .96997

334 199 062 .96923 782

133 .96990 844 697 547

.96911 760 607 452 294

670 512 352 189 023

65 66 67 68 69

.88774 541 308 074 .87839

364 130 .87895 660 424

.87948 713 477 241 004

527 291 054 .86817 579

100 .86863 625 387 148

667 429 190 .85950 710

227 .85987 747 407 266

20 21 22 23 24

252 139 024 .96907 787

068 .96944 818 689 558

864 729 592 453 312

639 495 348 199 048

395 242 087 .95929 769

134 .95973 809 643 476

.95856 687 516 343 168

70 71 72 73 74

602 365 127 .86888 648

187 .86949 710 470 229

.86766 527 287 047 .85806

340 100 .85859 618 376

.85908 667 426 184 .84941

470 228 .84986 743 500

025 .84783 540 297 053

25 26 27 28 29

665 539 406 268 125

424 287 144 .95996 844

168 020 .95867 710 548

.95895 738 576 410 241

607 442 272 098 .94922

306 133 .94955 774 590

.94991 810 625 438 248

75 76 77 78 79

408 168 .85927 685 442

.85988 747 505 262 018

564 322 079 .84835 590

134 .84891 647 403 158

698 455 211 .83966 720

257 013 .83768 523 277

.83809 564 319 074 .82827

30 31 32 33 34

.95977 823 665 502 334

686 524 357 186 011

382 212 038 .94860 679

067 .94890 709 525 337

741 557 370 180 .93986

403 214 021 .93825 626

055 .93860 662 461 257

80 81 82 83 84

197 .84950 702 453 203

.84772 525 277 028 .83777

344 096 .83848 599 348

.83911 664 415 164 .82913

473 224 .82974 724 473

029 .82780 530 279 027

578 329 079 .81828 576

35 36 37 38 39

162 .94986 805 620 431

.94832 650 464 273 079

494 306 114 .93919 720

146 .93952 756 556 353

790 591 390 186 .92979

425 221 016 .92808 597

051 .92843 634 422 208

85 86 87 88 89

.83951 697 441 181 .82919

525 271 014 .82754 492

095 .82840 583 323 062

660 405 148 .81888 626

220 .81965 708 448 186

.81774 519 262 003 .80742

322 067 .80811 552 291

40 41 42 43 44

238 042 .93842 639 433

.93882 682 478 271 062

518 314 107 .92897 685

148 .92940 729 516 301

770 558 344 128 .91910

385 170 .91952 733 513

.91992 774 554 332 108

90 91 92 93 94

654 386 114 .81839 561

227 .81959 688 413 134

.81797 529 257 .80983 705

362 094 .80823 549 272

.80922 655 384 111 .79835

478 211 .79941 669 393

028 .79761 491 220 .78947

45 46 47 48 49

226 017 .92806 593 379

.92852 640 426 211 .91995

472 257 041 .91823 604

085 .91868 649 429 208

692 472 250 028 .90805

291 069 .90845 621 396

.90884 660 434 207 .89979

95 96 97 98 99

278 .80991 698 399 094

.80852 566 274 .79975 670

424 138 .79846 547 243

.79991 706 415 117 .78814

555 271 .78981 684 382

114 .78831 542 247 .77946

670 388 100 .77806 507

100

.79784

360

.78934

506

075

641

203

*For data from −78° to 78°C, see p. 2-142, Table 2N-5, American Institute of Physics Handbook, McGraw-Hill, New York, 1957.

DENSITIES OF AQUEOUS ORGANIC SOLUTIONS TABLE 2-111 % alcohol by weight

2-113

Densities of Mixtures of C2H5OH and H2O at 20°C g/mL Tenths of %

0

1

2

3

4

5

6

7

0 1 2 3 4

0.99823 636 453 275 103

804 618 435 257 087

785 599 417 240 070

766 581 399 222 053

748 562 381 205 037

729 544 363 188 020

710 525 345 171 003

692 507 327 154 *987

5 6 7 8 9

.98938 780 627 478 331

922 765 612 463 316

906 749 597 449 301

890 734 582 434 287

874 718 567 419 273

859 703 553 404 258

843 688 538 389 244

827 673 523 374 229

10 11 12 13 14

187 047 .97910 775 643

172 033 896 761 630

158 019 883 748 617

144 006 869 735 604

130 *992 855 722 591

117 *978 842 709 578

103 *964 828 696 565

089 *951 815 683 552

15 16 17 18 19

514 387 259 129 .96997

501 374 246 116 984

488 361 233 103 971

475 349 220 089 957

462 336 207 076 944

450 323 194 063 931

438 310 181 050 917

425 297 168 037 904

412 284 155 024 891

20 21 22 23 24

864 729 592 453 312

850 716 578 439 297

837 702 564 425 283

823 688 551 411 269

810 675 537 396 254

796 661 523 382 240

783 647 509 368 225

769 634 495 354 211

756 620 481 340 196

25 26 27 28 29

168 020 .95867 710 548

153 005 851 694 532

139 *990 836 678 516

124 *975 820 662 499

109 *959 805 646 483

094 *944 789 630 466

080 *929 773 613 450

30 31 32 33 34

382 212 038 .94860 679

365 195 020 842 660

349 178 003 824 642

332 161 *985 806 624

315 143 *967 788 605

298 126 *950 770 587

35 36 37 38 39

494 306 114 .93919 720

475 287 095 899 700

456 268 075 879 680

438 249 056 859 660

419 230 036 840 640

40 41 42 43 44

518 314 107 .92897 685

498 294 086 876 664

478 273 065 855 642

458 253 044 834 621

45 46 47 48 49

472 257 041 .91823 604

450 236 019 801 582

429 214 *997 780 560

408 193 *976 758 538

Tenths of % 0

1

2

3

4

5

6

7

50 51 52 53 54

0.91384 160 .90936 711 485

361 138 914 689 463

339 116 891 666 440

317 093 869 644 417

295 071 846 621 395

272 049 824 598 372

250 026 801 576 349

228 206 183 004 *981 *959 779 756 734 553 531 508 327 304 281

796 642 493 345 201

55 56 57 58 59

258 031 .89803 574 344

236 008 780 551 321

213 *985 757 528 298

190 *962 734 505 275

167 *939 711 482 252

145 *917 688 459 229

122 *894 665 436 206

099 076 054 *871 *848 *825 643 620 597 413 390 367 183 160 137

075 061 *937 *923 801 788 670 657 539 526

60 61 62 63 64

113 .88882 650 417 183

090 859 626 393 160

067 836 603 370 136

044 812 580 347 113

021 789 557 323 089

*998 766 533 300 066

*975 743 510 277 042

*951 *928 *905 720 696 673 487 463 440 253 230 206 019 *995 *972

400 272 142 010 877

65 66 67 68 69

.87948 713 477 241 004

925 689 454 218 *981

901 666 430 194 *957

878 642 406 170 *933

854 619 383 147 *909

831 595 359 123 *885

807 572 336 099 *862

784 760 737 548 524 501 312 288 265 075 052 028 *838 *814 *790

742 606 467 326 182

70 71 72 73 74

.86766 527 287 047 .85806

742 503 263 022 781

718 479 239 *998 757

694 455 215 *974 733

671 431 191 *950 709

647 407 167 *926 685

623 383 143 *902 661

599 575 551 339 335 311 119 095 071 *878 *854 *830 636 612 588

065 *914 757 597 433

050 035 *898 *883 742 726 581 565 416 400

75 76 77 78 79

564 322 079 .84835 590

540 297 055 811 566

515 273 031 787 541

491 467 443 419 394 370 346 249 225 200 176 152 128 103 006 *982 *958 *933 *909 *884 *860 762 738 713 689 664 640 615 517 492 467 443 418 393 369

281 108 *932 752 568

264 091 *914 734 550

247 230 074 056 *896 *878 715 697 531 512

80 81 82 83 84

344 096 .83848 599 348

319 072 823 574 323

294 047 798 549 297

270 022 773 523 272

245 *997 748 498 247

220 *972 723 473 222

196 *947 698 448 196

171 146 121 *923 *898 *873 674 649 624 423 398 373 171 146 120

400 211 017 820 620

382 192 *997 800 599

363 172 *978 780 579

344 325 153 134 *958 *939 760 740 559 539

85 86 87 88 89

095 .82840 583 323 062

070 815 557 297 035

044 789 531 271 009

019 763 505 245 *983

*994 738 479 219 *956

*968 712 453 193 *930

*943 686 427 167 *903

*917 *892 *866 660 635 609 401 375 349 140 114 088 *877 *850 *824

437 232 023 812 600

417 212 002 791 579

396 191 *981 770 557

376 170 *960 749 536

356 335 149 129 *939 *918 728 707 515 493

90 91 92 93 94

.81797 529 257 .80983 705

770 502 230 955 677

744 475 203 928 649

717 448 175 900 621

690 421 148 872 593

664 394 120 844 565

637 366 093 817 537

386 171 *954 736 516

365 150 *932 714 494

343 128 *910 692 472

322 106 *889 670 450

300 279 085 063 *867 *845 648 626 428 406

95 96 97 98 99

424 138 .79846 547 243

395 109 816 517 213

367 080 787 487 182

338 051 757 456 151

310 281 253 224 195 166 022 *993 *963 *934 *905 *875 727 698 668 638 608 578 426 396 365 335 305 274 120 089 059 028 *997 *966

100

.78934

*Indicates change in the first two decimal places.

8

% alcohol by weight

9

673 655 489 471 310 292 137 120 *971 *954 811 658 508 360 215

610 339 066 789 509

8

583 312 038 761 480

9

556 285 010 733 452

2-114

PHYSICAL AND CHEMICAL DATA

TABLE 2-112 % alcohol by volume at 60°F 0 1 2 3 4

Specific Gravity (60°/60°F [(15.56°/15.56°C)]) of Mixtures by Volume of C2H5OH and H2O

Tenths of % 0

1

1.00000 *985 0.99850 835 703 688 559 545 419 405

2

3

4

5

6

7

8

9

*970 820 674 531 391

*955 806 659 516 378

*940 791 645 502 364

*925 776 630 488 350

*910 761 616 474 336

*895 747 602 460 323

*880 732 587 446 309

865 717 573 432 296

176 163 047 035 *923 *911 803 791 684 672

% alcohol by volume at 60°F

Tenths of % 0

1

2

3

4

5

6

50 51 52 53 54

0.93426 230 031 .92830 626

407 210 011 810 605

387 190 *991 789 585

368 171 *971 769 564

348 151 *951 749 544

328 131 *931 728 523

309 111 *911 708 502

55 56 57 58 59

419 210 .91999 784 565

398 189 978 762 543

377 168 956 741 521

357 147 935 719 499

336 126 914 697 477

315 105 892 675 455

294 084 871 653 433

7

8

9

289 270 250 091 071 051 *890 *870 *850 688 667 647 482 461 440

5 6 7 8 9

282 150 022 .98899 779

269 137 009 887 767

255 124 *997 875 755

242 111 *984 863 743

228 098 *972 851 731

215 085 *960 838 720

202 073 *947 826 708

189 060 *935 814 696

10 11 12 13 14

661 544 430 319 210

649 532 419 308 200

637 521 408 297 190

625 509 396 286 179

614 498 385 275 168

602 487 374 264 157

590 475 363 254 147

579 464 352 243 136

567 452 341 232 125

556 441 330 221 115

60 61 62 63 64

344 120 .90893 664 434

322 097 870 641 411

299 075 847 618 388

277 052 825 595 365

255 030 802 572 341

232 007 779 549 318

210 *984 756 526 295

15 16 17 18 19

104 .97998 895 794 694

093 988 885 784 684

083 977 875 774 674

072 967 864 764 664

062 956 854 754 654

051 946 844 744 645

040 936 834 734 635

030 925 824 724 625

019 915 814 714 615

009 905 804 704 605

65 66 67 68 69

202 .89967 729 489 245

179 943 705 465 220

155 920 681 441 196

132 896 657 416 171

108 872 633 392 147

085 848 609 368 122

061 825 585 343 098

038 801 561 319 073

014 *991 777 753 537 513 295 270 048 024

20 21 22 23 24

596 496 395 293 189

586 486 385 283 179

576 476 375 272 168

566 466 365 262 158

556 456 354 252 147

546 446 344 241 137

536 436 334 231 126

526 425 324 221 116

516 415 313 210 105

506 405 303 200 095

70 71 72 73 74

.88999 751 499 244 .87987

974 725 474 218 961

950 700 448 193 935

925 675 423 167 910

900 650 397 141 884

875 625 372 116 858

850 600 346 090 832

825 574 321 064 806

801 549 296 039 780

25 26 27 28 29

084 .96978 870 760 648

073 967 859 749 637

063 957 848 738 625

052 946 837 727 614

042 935 826 715 603

031 924 815 704 591

020 914 804 693 580

010 903 793 682 568

*999 *988 892 881 782 771 671 659 557 546

75 76 77 78 79

728 465 199 .86929 656

702 439 172 902 629

676 412 145 875 601

650 386 118 847 574

623 359 092 820 546

597 332 065 793 518

571 306 038 766 491

545 518 492 279 252 226 011 *984 *957 738 711 684 463 435 408

30 31 32 33 34

534 418 296 170 041

522 406 284 157 028

511 394 271 144 015

499 382 259 132 002

488 370 246 119 *988

476 358 234 106 *975

464 346 221 093 *962

453 334 209 080 *948

441 429 321 309 196 183 067 054 *935 *921

80 81 82 83 84

380 100 .85817 531 240

352 072 789 502 211

324 044 760 473 181

296 015 732 444 152

269 *987 703 415 122

241 *959 674 386 093

213 *931 646 357 063

185 157 129 *902 *874 *846 617 588 560 328 299 270 033 004 *974

35 36 37 38 39

.95908 770 628 482 332

894 756 614 467 317

881 742 599 452 302

867 728 585 437 286

854 714 570 423 271

840 700 556 408 256

826 685 541 393 240

812 671 526 378 225

784 643 497 347 194

85 86 87 88 89

.84944 642 336 025 .83707

914 612 305 *994 675

884 581 274 *962 643

854 551 243 *930 610

824 520 212 *899 578

794 490 181 *867 545

764 459 150 *835 513

734 703 673 428 398 367 119 088 056 *803 *771 *739 480 447 415

40 41 42 43 44

178 020 .94858 693 524

162 004 842 676 507

147 *988 825 660 490

131 *972 809 643 473

115 *956 792 626 455

100 *940 776 609 438

084 *923 759 592 421

068 *907 743 575 403

052 036 *891 *875 726 710 558 541 386 369

90 91 92 93 94

382 049 .82705 351 .81984

349 015 670 315 947

315 *981 635 279 909

282 *947 600 243 871

249 *913 565 206 834

216 *879 529 170 796

183 *845 494 133 757

150 116 083 *810 *776 *741 458 423 387 096 059 022 719 681 642

45 46 47 48 49

351 174 .93993 808 619

334 156 975 789 600

316 138 956 771 581

298 120 938 752 562

281 102 920 733 543

263 084 901 714 523

245 066 883 695 504

228 048 864 676 485

95 96 97 98 99

603 206 .80792 356 .79889

564 165 750 311 841

525 125 707 265 792

486 084 664 219 743

446 042 620 173 693

407 001 577 127 643

367 *960 533 080 593

327 287 247 *918 *876 *834 489 445 401 033 *985 *937 543 492 441

100

389

*Indicates change in first two decimal places.

798 657 512 362 209

210 030 845 657 465

192 011 827 638 446

273 062 849 631 410

252 041 827 610 388

231 020 806 588 366

188 165 143 *962 *939 *916 733 710 687 503 480 457 272 249 225

776 524 270 013 754

DENSITIES OF AQUEOUS ORGANIC SOLUTIONS

2-115

TABLE 2-113 n-Propyl Alcohol (C3H7OH) %

0°C

15°C

30°C

%

0°C

15°C

30°C

%

0°C

15°C

30°C

%

0°C

15°C

30°C

%

0°C

15°C

30°C

0 1 2 3 4

0.9999 .9982 .9967 .9952 .9939

0.9991 .9974 .9960 .9944 .9929

0.9957 .9940 .9924 .9908 .9893

20 21 22 23 24

0.9789 .9776 .9763 .9748 .9733

0.9723 .9705 .9688 .9670 .9651

0.9643 .9622 .9602 .9583 .9563

40 41 42 43 44

0.9430 .9411 .9391 .9371 .9352

0.9331 .9310 .9290 .9269 .9248

0.9226 .9205 .9184 .9164 .9143

60 61 62 63 64

0.9033 .9013 .8994 .8974 .8954

0.8922 .8902 .8882 .8861 .8841

0.8807 .8786 .8766 .8745 .8724

80 81 82 83 84

0.8634 .8614 .8594 .8574 .8554

0.8516 .8496 .8475 .8454 .8434

0.8394 .8373 .8352 .8332 .8311

5 6 7 8 9

.9926 .9914 .9904 .9894 .9883

.9915 .9902 .9890 .9877 .9864

.9877 .9862 .9848 .9834 .9819

25 26 27 28 29

.9717 .9700 .9682 .9664 .9646

.9633 .9614 .9594 .9576 .9556

.9543 .9522 .9501 .9481 .9460

45 46 47 48 49

.9332 .9311 .9291 .9272 .9252

.9228 .9207 .9186 .9165 .9145

.9122 .9100 .9079 .9057 .9036

65 66 67 68 69

.8934 .8913 .8894 .8874 .8854

.8820 .8800 .8779 .8759 .8739

.8703 .8682 .8662 .8641 .8620

85 86 87 88 89

.8534 .8513 .8492 .8471 .8450

.8413 .8393 .8372 .8351 .8330

.8290 .8269 .8248 .8227 .8206

10 11 12 13 14

.9874 .9865 .9857 .9849 .9841

.9852 .9840 .9828 .9817 .9806

.9804 .9790 .9775 .9760 .9746

30 31 32 33 34

.9627 .9608 .9589 .9570 .9550

.9535 .9516 .9495 .9474 .9454

.9439 .9418 .9396 .9375 .9354

50 51 52 53 54

.9232 .9213 .9192 .9173 .9153

.9124 .9104 .9084 .9064 .9044

.9015 .8994 .8973 .8952 .8931

70 71 72 73 74

.8835 .8815 .8795 .8776 .8756

.8719 .8700 .8680 .8659 .8639

.8600 .8580 .8559 .8539 .8518

90 91 92 93 94

.8429 .8408 .8387 .8364 .8342

.8308 .8287 .8266 .8244 .8221

.8185 .8164 .8142 .8120 .8098

15 16 17 18 19

.9833 .9825 .9817 .9808 .9800

.9793 .9780 .9768 .9752 .9739

.9730 .9714 .9698 .9680 .9661

35 36 37 38 39

.9530 .9511 .9491 .9471 .9450

.9434 .9413 .9392 .9372 .9351

.9333 .9312 .9289 .9269 .9247

55 56 57 58 59

.9132 .9112 .9093 .9073 .9053

.9023 .9003 .8983 .8963 .8942

.8911 .8890 .8869 .8849 .8828

75 76 77 78 79

.8736 .8716 .8695 .8675 .8655

.8618 .8598 .8577 .8556 .8536

.8497 .8477 .8456 .8435 .8414

95 96 97 98 99

.8320 .8296 .8272 .8248 .8222

.8199 .8176 .8153 .8128 .8104

.8077 .8054 .8031 .8008 .7984

100

.8194

.8077

.7958

TABLE 2-114

Isopropyl Alcohol (C3H7OH)

%

0°C

15°C*

15°C*

20°C

30°C

%

0°C

0 1 2 3 4

0.9999 .9980 .9962 .9946 .9930

0.9991 .9973 .9956 .9938 .9922

0.99913 .9972 .9954 .9936 .9920

0.9982 .9962 .9944 .9926 .9909

0.9957 .9939 .9921 .9904 .9887

35 36 37 38 39

0.9557 .9536 .9514 .9493 .9472

5 6 7 8 9

.9916 .9902 .9890 .9878 .9866

.9906 .9892 .9878 .9864 .9851

.9904 .9890 .9875 .9862 .9849

.9893 .9877 .9862 .9847 .9833

.9871 .9855 .9839 .9824 .9809

40 41 42 43 44

.9450 .9428 .9406 .9384 .9361

10 11 12 13 14

.9856 .9846 .9838 .9829 .9821

.9838 .9826 .9813 .9802 .9790

.98362 .9824 .9812 .9800 .9788

.9820 .9808 .9797 .9876 .9776

.9794 .9778 .9764 .9750 .9735

45 46 47 48 49

15 16 17 18 19

.9814 .9806 .9799 .9792 .9784

.9779 .9768 .9756 .9745 .9730

.9777 .9765 .9753 .9741 .9728

.9765 .9754 .9743 .9731 .9717

.9720 .9705 .9690 .9675 .9658

20 21 22 23 24

.9777 .9768 .9759 .9749 .9739

.9719 .9704 .9690 .9675 .9660

.97158 .9703 .9689 .9674 .9659

.9703 .9688 .9669 .9651 .9634

25 26 27 28 29

.9727 .9714 .9699 .9684 .9669

.9643 .9626 .9608 .9590 .9570

.9642 .9624 .9605 .9586 .9568

30 31 32 33 34

.9652 .9634 .9615 .9596 .9577

.9551

.95493 .9530 .9510 .9489 .9468

15°C*

15°C*

20°C

30°C

%

0°C

15°C*

15°C*

20°C

30°C

0.9446 .9424 .9401 .9379 .9356

0.9419 .9399 .9377 .9355 .9333

0.9338 .9315 .9292 .9269 .9246

70 71 72 73 74

0.8761 .8738 .8714 .8691 .8668

0.8639 .8615 .8592 .8568 .8545

0.86346 .8611 .8588 .8564 .8541

0.8584 .8560 .8537 .8513 .8489

0.8511 .8487 .8464 .8440 .8416

.93333 .9311 .9288 .9266 .9243

.9310 .9287 .9264 .9239 .9215

.9224 .9201 .9177 .9154 .9130

75 76 77 78 79

.8644 .8621 .8598 .8575 .8551

.8521 .8497 .8474 .8450 .8426

.8517 .8493 .8470 .8446 .8422

.8464 .8439 .8415 .8391 .8366

.8392 .8368 .8344 .8321 .8297

.9338 .9315 .9292 .9270 .9247

.9220 .9197 .9174 .9150 .9127

.9191 .9165 .9141 .9117 .9093

.9106 .9082 .9059 .9036 .9013

80 81 82 83 84

.8528 .8503 .8479 .8456 .8432

.8403 .8379 .8355 .8331 .8307

.83979 .8374 .8350 .8326 .8302

.8342 .8317 .8292 .8268 .8243

.8273 .8248 .8224 .8200 .8175

50 51 52 53 54

.9224 .9201 .9178 .9155 .9132

.91043 .9081 .9058 .9035 .9011

.9069 .9044 .9020 .8996 .8971

.8990 .8966 .8943 .8919 .8895

85 86 87 88 89

.8408 .8384 .8360 .8336 .8311

.8282 .8259 .8234 .8209 .8184

.8278 .8254 .8229 .8205 .8180

.8219 .8194 .8169 .8145 .8120

.8151 .8127 .8201 .8078 .8053

.9642 .9624 .9606 .9587 .9569

55 56 57 58 59

.9109 .9086 .9063 .9040 .9017

.8988 .8964 .8940 .8917 .8893

.8946 .8921 .8896 .8874 .8850

.8871 .8847 .8823 .8800 .8777

90 91 92 93 94

.8287 .8262 .8237 .8212 .8186

.8161 .8136 .8110 .8085 .8060

.81553 .8130 .8104 .8079 .8052

.8096 .8072 .8047 .8023 .7998

.8029 .8004 .7979 .7954 .7929

.9615 .9597 .9577 .9558 .9540

.9549 .9529 .9509 .9488 .9467

60 61 62 63 64

.8994 .8970 .8947 .8924 .8901

0.8829 .8805 .8781

.88690 .8845 .8821 .8798 .8775

.8825 .8800 .8776 .8751 .8727

.8752 .8728 .8704 .8680 .8656

95 96 97 98 99

.8160 .8133 .8106 .8078 .8048

.8034 .8008 .7981 .7954 .7926

.8026 .7999 .7972 .7945 .7918

.7973 .7949 .7925 .7901 .7877

.7904 .7878 .7852 .7826 .7799

.9520 .9500 .9481 .9460 .9440

.9446 .9426 .9405 .9383 .9361

65 66 67 68 69

.8878 .8854 .8831 .8807 .8784

.8757 .8733 .8710 .8686 .8662

.8752 .8728 .8705 .8682 .8658

.8702 .8679 .8656 .8632 .8609

.8631 .8607 .8583 .8559 .8535

100

.8016

.7896

.78913

.7854

.7770

*Two different observers; see International Critical Tables, vol. 3, p. 120.

2-116

PHYSICAL AND CHEMICAL DATA

TABLE 2-115

Glycerol* Density

Density

Glycerol, %

15°C

15.5°C

20°C

25°C

30°C

Glycerol, %

15°C

15.5°C

20°C

25°C

30°C

Glycerol, % 15°C

100 99 98 97 96

1.26415 1.26160 1.25900 1.25645 1.25385

1.26381 1.26125 1.25865 1.25610 1.25350

1.26108 1.25850 1.25590 1.25335 1.25080

1.15802 1.25545 1.25290 1.25030 1.24770

1.25495 1.25235 1.24975 1.24710 1.24450

65 64 63 62 61

1.17030 1.16755 1.16480 1.16200 1.15925

1.17000 1.16725 1.16445 1.16170 1.15895

1.16750 1.16475 1.16205 1.15930 1.15655

1.16475 1.16200 1.15925 1.15655 1.15380

1.16195 1.15925 1.15650 1.15375 1.15100

30 29 28 27 26

95 94 93 92 91

1.25130 1.24865 1.24600 1.24340 1.24075

1.25095 1.24830 1.24565 1.24305 1.24040

1.24825 1.24560 1.24300 1.24035 1.23770

1.24515 1.24250 1.23985 1.23725 1.23460

1.24190 1.23930 1.23670 1.23410 1.23150

60 59 58 57 56

1.15650 1.15370 1.15095 1.14815 1.14535

1.15615 1.15340 1.15065 1.14785 1.14510

1.15380 1.15105 1.14830 1.14555 1.14280

1.15105 1.14835 1.14560 1.14285 1.14015

1.14830 1.14555 1.14285 1.14010 1.13740

90 89 88 87 86

1.23810 1.23545 1.23280 1.23015 1.22750

1.23775 1.23510 1.23245 1.22980 1.22710

1.23510 1.23245 1.22975 1.22710 1.22445

1.23200 1.22935 1.22665 1.22400 1.22135

1.22890 1.22625 1.22360 1.22095 1.21830

55 54 53 52 51

1.14260 1.13980 1.13705 1.13425 1.13150

1.14230 1.13955 1.13680 1.13400 1.13125

1.14005 1.13730 1.13455 1.13180 1.12905

1.13740 1.13465 1.13195 1.12920 1.12650

85 84 83 82 81

1.22485 1.22220 1.21955 1.21690 1.21425

1.22445 1.22180 1.21915 1.21650 1.21385

1.22180 1.21915 1.21650 1.21380 1.21115

1.21870 1.21605 1.21340 1.21075 1.20810

1.21565 1.21300 1.21035 1.20770 1.20505

50 49 48 47 46

1.12870 1.12600 1.12325 1.12055 1.11780

1.12845 1.12575 1.12305 1.12030 1.11760

1.12630 1.12360 1.12090 1.11820 1.11550

80 79 78 77 76

1.21160 1.20885 1.20610 1.20335 1.20060

1.21120 1.20845 1.20570 1.20300 1.20025

1.20850 1.20575 1.20305 1.20030 1.19760

1.20545 1.20275 1.20005 1.19735 1.19465

1.20240 1.19970 1.19705 1.19435 1.19170

45 44 43 42 41

1.11510 1.11235 1.10960 1.10690 1.10415

1.11490 1.11215 1.10945 1.10670 1.10400

75 74 73 72 71

1.19785 1.19510 1.19235 1.18965 1.18690

1.19750 1.19480 1.19205 1.18930 1.18655

1.19485 1.19215 1.18940 1.18670 1.18395

1.19195 1.18925 1.18650 1.18380 1.18110

1.18900 1.18635 1.18365 1.18100 1.17830

40 39 38 37 36

1.10145 1.09875 1.09605 1.09340 1.09070

70 69 68 67 66

1.18415 1.18135 1.17860 1.17585 1.17305

1.18385 1.18105 1.17830 1.17555 1.17275

1.18125 1.17850 1.17575 1.17300 1.17025

1.17840 1.17565 1.17295 1.17020 1.16745

1.17565 1.17290 1.17020 1.16745 1.16470

35 34 33 32 31

1.08800 1.08530 1.08265 1.07995 1.07725

20°C

25°C

30°C

1.07455 1.07195 1.06935 1.06670 1.06410

1.07435 1.07175 1.06915 1.06655 1.06390

1.07270 1.07010 1.06755 1.06495 1.06240

1.07070 1.06815 1.06560 1.06305 1.06055

1.06855 1.06605 1.06355 1.06105 1.05855

25 24 23 22 21

1.06150 1.05885 1.05625 1.05365 1.05100

1.06130 1.05870 1.05610 1.05350 1.05090

1.05980 1.05720 1.05465 1.05205 1.04950

1.05800 1.05545 1.05290 1.05035 1.04780

1.05605 1.05350 1.05100 1.04850 1.04600

1.13470 1.13195 1.12925 1.12650 1.12380

20 19 18 17 16

1.04840 1.04590 1.04335 1.04085 1.03835

1.04825 1.04575 1.04325 1.04075 1.03825

1.04690 1.04440 1.04195 1.03945 1.03695

1.04525 1.04280 1.04035 1.03790 1.03545

1.04350 1.04105 1.03860 1.03615 1.03370

1.12375 1.12110 1.11840 1.11575 1.11310

1.12110 1.11845 1.11580 1.11320 1.11055

15 14 13 12 11

1.03580 1.03330 1.03080 1.02830 1.02575

1.03570 1.03320 1.03070 1.02820 1.02565

1.03450 1.03200 1.02955 1.02705 1.02455

1.03300 1.03055 1.02805 1.02560 1.02315

1.03130 1.02885 1.02640 1.02395 1.02150

1.11280 1.11010 1.10740 1.10470 1.10200

1.11040 1.10775 1.10510 1.10240 1.09975

1.10795 1.10530 1.10265 1.10005 1.09740

10 9 8 7 6

1.02325 1.02085 1.01840 1.01600 1.01360

1.02315 1.02075 1.01835 1.01590 1.01350

1.02210 1.01970 1.01730 1.01495 1.01255

1.02070 1.01835 1.01600 1.01360 1.01125

1.01905 1.01670 1.01440 1.01205 1.00970

1.10130 1.09860 1.09590 1.09320 1.09050

1.09930 1.09665 1.09400 1.09135 1.08865

1.09710 1.09445 1.09180 1.08915 1.08655

1.09475 1.09215 1.08955 1.08690 1.08430

5 4 3 2 1

1.01120 1.00875 1.00635 1.00395 1.00155

1.01110 1.00870 1.00630 1.00385 1.00145

1.01015 1.00780 1.00540 1.00300 1.00060

1.00890 1.00655 1.00415 1.00180 0.99945

1.00735 1.00505 1.00270 1.00035 0.99800

1.08780 1.08515 1.08245 1.07975 1.07705

1.08600 1.08335 1.08070 1.07800 1.07535

1.08390 1.08125 1.07860 1.07600 1.07335

1.08165 1.07905 1.07645 1.07380 1.07120

0

0.99913 0.99905 0.99823 0.99708 0.99568

*Bosart and Snoddy, Ind. Eng. Chem., 20, (1928): 1378.

TABLE 2-116

Density 15.5°C

Hydrazine (N2H4)

%

d 15 4

%

d 15 4

1 2 4 8 12 16 20 24 28

1.0002 1.0013 1.0034 1.0077 1.0121 1.0164 1.0207 1.0248 1.0286

30 40 50 60 70 80 90 100

1.0305 1.038 1.044 1.047 1.046 1.040 1.030 1.011

DENSITIES OF AQUEOUS ORGANIC SOLUTIONS TABLE 2-117

2-117

Densities of Aqueous Solutions of Miscellaneous Organic Compounds*

d (resp., dw, ds) = density of the solution (resp., water; resp., the pure liquid solute) in g/mL; ps (resp., pw) = wt % of solute (resp., water) in the solution; range = range of applicability of the equation. Section A Name

Formula

t, °C

Acetaldehyde Acetamide

C2H4O C2H5NO

Acetone

C3H6O

Acetonitrile Allyl alcohol Benzenepentacarboxylic acid Butyl alcohol (n-)

C2H3N C3H6O C11H6O10 C4H10O

Butyric acid (n-)

C4H8O2

Chloral hydrate

C2H3Cl3O2

Chloroacetic acid

C2H3ClO2

Citric acid (hydrate)

C6H3O7 + H2O

Dichloroacetic acid

C2H2Cl2O2

Diethylamine hydrochloride Ethylamine hydrochloride

C4H12ClN C2H8ClN

Ethylene glycol

C2H6O2

Ethyl ether tartrate Formaldehyde Formamide

C4H10O C8H14O6 CH2O CH3NO

Furfural Isoamyl alcohol

C5H4O2 C5H12O

Isobutyl alcohol

C4H10O




Isobutyric acid

C4H8O2




Isovaleric acid Lactic acid Maleic acid

C5H10O2 C3H6O C4H4O4

Malic acid

C4H6O5

Malonic acid Methyl acetate

C3H4O4 C3H6O2

glucoside (α-)

C7H14O6

Nicotine Nitrophenol (p-)

C10H14N2 C6H5NO3

Oxalic acid

C2H2O4

Phenol

C6H6O

Phenylglycolic acid Picoline (α-) (β-)

C8H8O3 C6H7N C6H7N

Propionic acid

C3H6O2

Pyridine Resorcinol Succinic acid

C5H5N C6H6O2 C4H6O4

Tartaric acid (d, l, or dl)

C4H6O6

*From “International Critical Tables,” vol. 3, pp. 111–114.





























18 15 0 4 15 20 25 15 0 25 20 18 25 0 15 30 20 25 18 20 25 21 21 0 15 20 25 15 15 25 20 25 20 15 20 15 18 25 25 25 25 20 25 20 20 0 30 20 15 0 15 17.5 20 25 15 80 25 25 25 18 25 25 18 25 15 17.5 20 30 40 50 60

d = d w + Aps + Bp s2 + Cp3s Range, ps

A

B

0– 30 0– 6 0–100 0–100 0–100 0–100 0–100 0– 16 0– 89 0– 0.6 0– 7.9 0– 10 0– 62 0– 70 0– 78 0– 90 0– 32 0– 86 0– 50 0– 30 0– 97 0– 36 0– 65 0–100 0– 6 0– 5 0– 4.5 0– 95 0– 40 22– 96 0– 8 0– 8 0– 2.5 0– 8 0– 8 0– 9 0– 9 0– 12 0– 5 0– 9 0– 40 0– 40 0– 40 0– 40 0– 20 26– 51 26– 51 0– 60 0– 1.5 0– 4 0– 4 0– 9 0– 4 0– 4 0– 5 0– 65 0– 11 0– 70 0– 60 0– 10 0– 40 0– 60 0– 52 0– 5.5 0– 15 0– 50 0– 50 0– 50 0– 50 0– 50 0– 50

+0.03255 +0.03639 −0.03856 −0.027648 −0.021009 −0.021233 −0.021171 −0.021175 −0.033729 +0.025615 −0.021651 +0.03414 +0.035135 +0.024489 +0.024455 +0.024401 +0.023648 +0.023602 +0.023824 +0.024427 +0.024427 +0.0334 +0.021193 +0.021483 +0.02133 −0.02221 −0.02221 +0.022367 +0.022518 +0.021217 +0.021827 +0.021664 +0.02155 −0.02146 −0.02169 +0.0352 +0.0345 +0.0337 +0.03253 +0.02231 +0.0234 +0.023933 +0.023736 +0.02389 +0.0340 +0.023336 +0.023151 +0.03642 +0.023216 +0.025898 +0.02494 +0.02494 +0.025264 +0.025108 +0.02111 +0.03462 +0.02207 −0.04386 −0.04683 +0.0395 +0.039245 +0.03229 +0.02201 +0.02304 +0.024482 +0.024455 +0.024432 +0.024335 +0.024265 +0.024205 +0.024155

−0.0516 +0.04171 −0.05449 −0.041193 −0.059682 −0.053529 −0.05904 −0.042024 −0.041232 −0.02117 +0.04285 +0.04131 −0.04166 +0.042802 +0.042198 +0.041887 +0.05302 +0.05552 +0.041141 +0.05537 +0.05537 +0.0676 −0.05307 +0.052992 −0.05108 +0.0448 +0.0435 +0.05358 −0.05658 +0.053199 +0.05366 +0.0421 +0.043 +0.056 +0.0438

−0.04282 +0.05186 +0.0575 +0.05957 +0.04175 +0.041066 −0.0574 +0.05996 +0.05975 +0.05454 −0.0455 −0.033185 −0.058 −0.058 −0.031996 −0.031607 −0.04283 −0.0686 +0.0423 −0.051405 −0.0513 −0.04172 −0.0599 −0.05204 +0.05519 +0.04185 +0.04185 +0.041837 +0.04185 +0.04185 +0.04185 +0.04185

C

−0.07588 +0.08272 −0.08624 −0.075327 −0.0856 +0.072984

+0.0611 −0.071291 +0.074366 +0.076549 +0.0722 +0.0717 +0.077534 +0.077534 −0.0747 −0.075248

−0.076005 +0.06542 −0.072529

+0.01544 +0.08978 −0.07687 +0.0441 +0.04254 +0.04208

−0.074167 +0.07361 −0.0828 −0.0819

2-118

PHYSICAL AND CHEMICAL DATA

TABLE 2-117

Densities of Aqueous Solutions of Miscellaneous Organic Compounds (Concluded ) d = d w + Aps + Bp s2 + Cp3s (Cont.)

Section A Name

t, °C

Formula

Tetraethyl ammonium chloride Thiourea

C8H20ClN CH4N2S

Trichloroacetic acid

C2HCl3O2

Triethylamine hydrochloride

C6H16ClN

Trimethyl carbinol

C4H10O

Urea

CH4N2O

Urethane Valeric acid (n-)

C3H7NO2 C5H10O2

Name







Formula C4H10O C4H8O2 C4H10O

Isobutyl alcohol

C4H10O

Isobutyric acid Nicotine Picoline (α-) (β-) Pyridine Trimethyl carbinol

C4H8O2 C10H14N2 C6H7N C6H7N C5H5N C4H10O




Name

Chloral hydrate

C2H3Cl3O2

Ethyl tartrate

C7H14O6

Furfural

C5H4O2

Pyridine

C5H5N

B

C

+0.056 +0.05374 +0.04153 +0.041387 +0.056119 +0.05558 −0.041908 −0.04176 −0.044802 +0.051552 +0.053712 −0.041817 −0.05245 −0.0427

+0.07122

t, °C

Range, pw

A

B

20 25 25 0 15 26 20 25 25 25 20

0–20 0–38 0– 1.1 0–14 0–16 0–80 0–40 0–30 0–40 0–40 0–20

+0.022103 +0.021854 +0.0234 +0.022437 +0.02224 +0.021808 +0.02199 +0.022715 +0.021925 +0.021157 +0.022287

−0.04113 −0.042314 +0.0336 −0.04285 −0.04129 −0.042358 −0.04331 −0.04393 −0.04352 −0.05536 +0.05275




+0.061038 −0.0869 +0.07957 +0.07887 +0.051216 +0.072573 −0.072285 +0.051379 −0.073437

d = ds + Apw + Bp 2w + Cp 3w

ds

C

+0.061253 +0.07315 +0.0625 −0.062

dt = do + At + Bt2

ps

do

Range, °C

A

B

76.60 80.95 2.00 10.00 5.00 10.00 25.00 4.62 5.69 6.56 9.34 21.20 29.50 40.40

0.9122 0.8614 1.0094 1.0476 1.0150 1.0270 1.0665 1.0125 1.0140 1.0155 1.0055 1.0115 1.0145 1.0182

0–45 0–43 7–80 7–80 15–80 15–80 15–80 22–74 22–74 22–74 11–73 14–73 12–72 9–74

−0.038 −0.037292 −0.042597 −0.047955 −0.032103 −0.032116 −0.03401 −0.03232 −0.03221 −0.03211 −0.03171 −0.03378 −0.03463 −0.03605

−0.0527 −0.0675 −0.054313 −0.054253 −0.052544 −0.062929 −0.0523 −0.05254 −0.05268 −0.05290 −0.053615 −0.05248 −0.05235 −0.05167

Formula C3H6O C4H10O

A +0.031884 +0.022995 +0.02499 +0.025053 +0.025051 +0.046 −0.02117 −0.021286 +0.023213 +0.022718 +0.022702 +0.022728 +0.021278 +0.0334

0.8097 0.9534 0.7077 0.8170 0.8055 0.9425 1.0093 0.9404 0.9515 0.9776 0.7856 Section C

Allyl alcohol Butyl alcohol (n-)

0– 63 0– 7 0– 61 10– 30 0– 94 0– 54 0–100 0–100 0– 12 0– 51 0– 35 0– 10 0– 56 0– 3




Section B Butyl alcohol (n-) Butyric acid (n-) Ethyl ether

Range, ps

21 15 12.5 20 25 21 20 25 14.8 18 20 25 20 25








DENSITIES OF MISCELLANEOUS MATERIALS

2-119

DENSITIES OF MISCELLANEOUS MATERIALS TABLE 2-118

Approximate Specific Gravities and Densities of Miscellaneous Solids and Liquids*

Water at 4°C and normal atmospheric pressure taken as unity. For more detailed data on any material, see the section dealing with the properties of that material.

Substance

Sp. gr.

Aver. weight lb/ft 3

Sp. gr.

Aver. weight lb/ft 3

Metals, Alloys, Ores Aluminum, cast-hammered bronze Brass, cast-rolled Bronze, 7.9 to 14% Sn phosphor

2.55–2.80 7.7 8.4–8.7 7.4–8.9 8.88

165 481 534 509 554

Timber, Air-dry Apple Ash, black white Birch, sweet, yellow Cedar, white, red

0.66–0.74 0.55 0.64–0.71 0.71–0.72 0.35

44 34 42 44 22

Copper, cast-rolled ore, pyrites German silver Gold, cast-hammered coin (U.S.)

8.8–8.95 4.1–4.3 8.58 19.25–19.35 17.18–17.2

556 262 536 1205 1073

Cherry, wild red Chestnut Cypress Elm, white Fir, Douglas

0.43 0.48 0.45–0.48 0.56 0.48–0.55

27 30 29 35 32

Iridium Iron, gray cast cast, pig wrought spiegeleisen

21.78–22.42 7.03–7.13 7.2 7.6–7.9 7.5

1383 442 450 485 468

balsam Hemlock Hickory Locust Mahogany

0.40 0.45–0.50 0.74–0.80 0.67–0.77 0.56–0.85

25 29 48 45 44

ferro-silicon ore, hematite ore, limonite ore, magnetite slag

6.7–7.3 5.2 3.6–4.0 4.9–5.2 2.5–3.0

437 325 237 315 172

Maple, sugar white Oak, chestnut live red, black

0.68 0.53 0.74 0.87 0.64–0.71

43 33 46 54 42

Lead ore, galena Manganese ore, pyrolusite Mercury

11.34 7.3–7.6 7.42 3.7–4.6 13.6

710 465 475 259 849

white Pine, Norway Oregon red Southern white

0.77 0.55 0.51 0.48 0.61–0.67 0.43

48 34 32 30 38–42 27

8.97 8.9 21.5 10.4–10.6 7.83 7.80 7.70–7.73 7.2–7.5 6.4–7.0 19.22

555 537 1330 656 489 487 481 459 418 1200

Poplar Redwood, California Spruce, white, red Teak, African Indian Walnut, black Willow

0.43 0.42 0.45 0.99 0.66–0.88 0.59 0.42–0.50

27 26 28 62 48 37 28

6.9–7.2 3.9–4.2

440 253

Various Solids Cereals, oats, bulk barley, bulk corn, rye, bulk wheat, bulk Cork

49 50 75 94 112

26 39 45 48 15

0.789 0.796 1.20 1.50 1.80

0.51 0.62 0.73 0.77 0.22–0.26

Various Liquids Alcohol, ethyl (100%) methyl (100%) Acid, muriatic, 40% nitric, 91% sulfuric, 87% Chloroform Ether Lye, soda, 66% Oils, vegetable mineral, lubricants

1.500 0.736 1.70 0.91–0.94 0.88–0.94

95 46 106 58 57

Cotton, flax, hemp Fats Flour, loose pressed Glass, common

1.47–1.50 0.90–0.97 0.40–0.50 0.70–0.80 2.40–2.80

93 58 28 47 162

2.45–2.72 2.90–3.00 3.2–4.7 0.32 0.86–1.02

161 184 247 20 59

54 62.428 59.830 56 8

plate or crown crystal dint Hay and straw, bales Leather

0.861–0.867 1.0 0.9584 0.88–0.92 0.125 1.02–1.03

64

Paper Potatoes, piled Rubber, caoutchouc goods Salt, granulated, piled

0.70–1.15 0.67 0.92–0.96 1.0–2.0 0.77

58 44 59 94 48

Ashlar Masonry Bluestone Granite, syenite, gneiss Limestone Marble Sandstone

2.3–2.6 2.4–2.7 2.1–2.8 2.4–2.8 2.0–2.6

153 159 153 162 143

Saltpeter Starch Sulfur Wool

1.07 1.53 1.93–2.07 1.32

67 96 125 82

Rubble Masonry Bluestone Granite, syenite, gneiss Limestone Marble Sandstone

2.2–2.5 2.3–2.6 2.0–2.7 2.3–2.7 1.9–2.5

147 153 147 156 137

Monel metal, rolled Nickel Platinum, cast-hammered Silver, cast-hammered Steel, cold-drawn machine tool Tin, cast-hammered cassiterite Tungsten Zinc, cast-rolled blende

Substance

Turpentine Water, 4°C max. density 100°C ice snow, fresh fallen sea water

*From Marks, Mechanical Engineers’ Handbook, McGraw-Hill.

Sp. gr.

Aver. weight lb/ft 3

Dry Rubble Masonry Granite, syenite, gneiss Limestone, marble Sandstone, bluestone

1.9–2.3 1.9–2.1 1.8–1.9

130 125 110

Brick Masonry Hard brick Medium brick Soft brick Sand-lime brick

1.8–2.3 1.6–2.0 1.4–1.9 1.4–2.2

128 112 103 112

Concrete Masonry Cement, stone, sand slag, etc. cinder, etc.

2.2–2.4 1.9–2.3 1.5–1.7

144 130 100

0.64–0.72 1.5 0.85–1.00 1.4–1.9 2.08–2.25

40–45 94 53–64 103 94–135

Portland cement Slags, bank slag bank screenings machine slag slag sand

3.1–3.2 1.1–1.2 1.5–1.9 1.5 0.8–0.9

196 67–72 98–117 96 49–55

Earth, etc., Excavated Clay, dry damp plastic and gravel, dry Earth, dry, loose dry, packed moist, loose moist, packed mud, flowing mud, packed Riprap, limestone

1.0 1.76 1.6 1.2 1.5 1.3 1.6 1.7 1.8 1.3–1.4

63 110 100 76 95 78 96 108 115 80–85

1.4 1.7 1.4–1.7 1.6–1.9 1.89–2.16

90 105 90–105 100–120 126

1.28 1.44 0.96 1.00 1.12 1.00

80 90 60 65 70 65

2.1–2.8 4.50 2.7–3.2 2.55 2.5–2.6

153 281 184 159 159

Borax Chalk Clay, marl Dolomite Feldspar, orthoclase

1.7–1.8 1.8–2.8 1.8–2.6 2.9 2.5–2.7

109 143 137 181 162

Gneiss Granite Greenstone, trap Gypsum, alabaster Hornblende Limestone Marble Magnesite Phosphate rock, apatite Porphyry

2.7–2.9 2.6–2.7 2.8–3.2 2.3–2.8 3.0 2.1–2.86 2.6–2.86 3.0 3.2 2.6–2.9

175 165 187 159 187 155 170 187 200 172

Substance

Various Building Materials Ashes, cinders Cement, Portland, loose Lime, gypsum, loose Mortar, lime, set Portland cement

Riprap, sandstone Riprap, shale Sand, gravel, dry, loose gravel, dry, packed gravel, wet Excavations in Water Clay River mud Sand or gravel and clay Soil Stone riprap Asbestos Barytes Basait Bauxite Bluestone

Minerals

2-120

PHYSICAL AND CHEMICAL DATA

TABLE 2-118

Approximate Specific Gravities and Densities of Miscellaneous Solids and Liquids (Concluded )

Water at 4°C and normal atmospheric pressure taken as unity. For more detailed data on any material, see the section dealing with the properties of that material.

Substance

Sp. gr.

Minerals (Cont.) Pumice, natural Quartz, flint Sandstone Serpentine Shale, slate Soapstone, talc Syenite Stone, Quarried, Piled Basalt, granite, gneiss Greenstone, hornblende Limestone, marble, quartz Sandstone Shale NOTE:

Aver. weight lb/ft3

0.37–0.90 2.5–2.8 2.0–2.6 2.7–2.8 2.6–2.9

40 165 143 171 172

2.6–2.8 2.6–2.7

169 165

1.5 1.7 1.5 1.3 1.5

96 107 95 82 92

Substance

Aver. weight lb/ft 3

Sp. gr.

Bituminous Substances Asphaltum Coal, anthracite bituminous lignite peat, turf, dry

1.1–1.5 1.4–1.8 1.2–1.5 1.1–1.4 0.65–0.85

81 97 84 78 47

charcoal, pine charcoal, oak coke Graphite Paraffin

0.28–0.44 0.47–0.57 1.0–1.4 1.64–2.7 0.87–0.91

23 33 75 135 56

Substance

Sp. gr.

Aver. weight lb/ft3

Bituminous Substances (Cont.) Petroleum refined (kerosene) benzine gasoline Pitch Tar, bituminous

0.87 0.78–0.82 0.73–0.75 0.70–0.75 1.07–1.15 1.20

54 50 46 45 69 75

Coal and Coke, Piled Coal, anthracite bituminous, lignite peat, turf charcoal coke

0.75–0.93 0.64–0.87 0.32–0.42 0.16–0.23 0.37–0.51

47–58 40–54 20–26 10–14 23–32

To convert pounds per cubic foot to kilograms per cubic meter, multiply by 16.02. °F = 9⁄ 5 °C + 32.

TABLE 2-119

Density (kg/m3) of Selected Elements as a Function of Temperature Element symbol

Temperature, K*

Al

Be†

Cr

Cu

Au

Ir

Fe

Pb

Mo

Ni

Pt

Ag

Zn†

50 100 150 200 250

2736 2732 2726 2719 2710

3650 3640 3630 3620 3610

7160 7155 7150 7145 7140

9019 9009 8992 8973 8951

19,490 19,460 19,420 19,380 19,340

22,600 22,580 22,560 22,540 22,520

7910 7900 7890 7880 7870

11,570 11,520 11,470 11,430 11,380

10,260 10,260 10,250 10,250 10,250

8960 8950 8940 8930 8910

21,570 21,550 21,530 21,500 21,470

10,620 10,600 10,575 10,550 10,520

7280 7260 7230 7200 7170

300 400 500 600 800

2701 2681 2661 2639 2591

3600 3580 3555 3530

7135 7120 7110 7080 7040

8930 8885 8837 8787 8686

19,300 19,210 19,130 19,040 18,860

22,500 22,450 22,410 22,360 22,250

7860 7830 7800 7760 7690

11,330 11,230 11,130 11,010 10,430

10,240 10,220 10,210 10,190 10,160

8900 8860 8820 8780 8690

21,450 21,380 21,330 21,270 21,140

10,490 10,430 10,360 10,300 10,160

7135 7070 7000 6935 6430

1000 1200 1400 1600 1800

2365 2305 2255

7000 6945 6890 6760 6700

8568 8458 7920 7750 7600

18,660 18,440 17,230 16,950

22,140 22,030 21,920 21,790 21,660

7650 7620 7520 7420 7320

10,190 9,940

10,120 10,080 10,040 10,000 9,950

8610 8510 8410 8320 7690

21,010 20,870 20,720 20,570 20,400

10,010 9,850 9,170 8,980

6260

21,510

7030

9,900

7450

20,220

2000

7460

NOTE:

Above the horizontal line the condensed phase is solid; below the line, it is liquid. *°R = 9⁄ 5 K. †Polycrystalline form tabulated. Similar tables for an additional 45 elements appear in the Handbook of Heat Transfer, 2d ed., McGraw-Hill, New York, 1984.

SOLUBILITIES UNITS CONVERSIONS For this subsection, the following units conversions are applicable: °F = 9⁄5 °C + 32. To convert cubic centimeters to cubic feet, multiply by 3.532 × 10−5.

To convert millimeters of mercury to pounds-force per square inch, multiply by 0.01934. To convert grams per liter to pounds per cubic foot, multiply by 6.243 × 10−2.

TABLE 2-120

Solubilities of Inorganic Compounds in Water at Various Temperatures*

This table shows the amount of anhydrous substance that is soluble in 100 g of water at the temperature in degrees Celsius as indicated; when the name is followed by †, the value is expressed in grams of substance in 100 cm3 of saturated solution. Solid phase gives the hydrated form in equilibrium with the saturated solution. Solid phase

Substance

Formula

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Aluminum chloride sulfate Ammonium aluminum sulfate bicarbonate bromide chloride chloroplatinate chromate chromium sulfate dichromate dihydrogen phosphite hydrogen phosphate iodide magnesium phosphate manganese phosphate nitrate oxalate perchlorate† persulfate sulfate thiocyanate vanadate (meta) Antimonious fluoride sulfide Arsenic oxide Arsenious sulfide

AlCl3 Al2(SO4)3 (NH4)2Al2(SO4)4 NH4HCO3 NH4Br NH4Cl (NH4)2PtCl6 (NH4)2CrO4 (NH4)2Cr2(SO4)4 (NH4)2Cr2O7 NH4H2PO3 (NH4)2HPO4 NH4I NH4MgPO4 NH4MnPO4 NH4NO3 (NH4)2C2O4 NH4ClO4† (NH4)2S2O8 (NH4)2SO4 NH4CNS NH4VO3 SbF3 Sb2S3 As2O5 As2S3

27 28 29

Barium acetate acetate carbonate

Ba(C2H3O2)2 Ba(C2H3O2)2 BaCO3

3H2O 1H2O

30 31 32 33 34 35 36 37 38

chlorate chloride chromate hydroxide iodide iodide nitrate nitrite oxalate

Ba(ClO3)2 BaCl2 BaCrO4 Ba(OH)2 BaI2 BaI2 Ba(NO3)2 Ba(NO2)2 BaC2O4

1H2O 2H2O

3H2O

39 40 41 42 43 44 45 46 47 48 49 50 51

perchlorate sulfate Beryllium sulfate sulfate sulfate Boric acid Boron oxide Bromine Cadmium chloride chloride chloride cyanide hydroxide

Ba(ClO4)2 BaSO4 BeSO4 BeSO4 BeSO4 H3BO3 B2O3 Br2 CdCl2 CdCl2 CdCl2 Cd(CN)2 Cd(OH)2

52 53 54

sulfate Calcium acetate acetate

CdSO4 Ca(C2H3O2)2 Ca(C2H3O2)2

6H2O 18H2O 24H2O

0°C 31.2 2.1 11.9 60.6 29.4

10°C 33.5 4.99 15.8 68 33.3 0.7

10.7825°

24II2O 171 6H2O 7H2O 1H2O

154.2 0.023 118.3 2.2 11.56 58.2 70.6 119.8

163.2

3.1 73.0 144

384.7

8H2O 6H2O 2H2O

20°C 69.8615° 36.4 7.74 21 75.5 37.2

59.5 5.17×10−5 at 18° 59

62.1 63 0.00168°

20.34 31.6 0.0002 1.67 170.2 5.0

1H2O

26.95 33.3 0.00028 2.48 185.7 7.0 0.00168°

205.8 1.15 × 10−4

2.0 × 10−4

19014.5° 13115 172.3 0.052 0 192 4.4 20.85 75.4 170 0.48 444.7 0.00017518° 65.8

4H2O 2aH2O 1H2O

2H2O 1H2O

3.57 1.5 3.4 125.1

50°C

60°C

70°C

80°C

90°C

100°C

46.1 14.88

52.2 20.10

59.2 26.70

66.1

73.0

80.8

89.0 109.796°

91.1 45.8

99.2 50.4

107.8 55.2

116.8 60.2

126 65.6

135.6 71.3

145.6 77.3 1.25

190.5 0.036 0 297.0 8.0 30.58

199.6 0.030

208.9 0.040 0 421.0

218.7 0.016 0.005 499.0

228.8 0.019 0.007 580.0

40.4 47.17 26031° 181.4 241.8 5.9 78.0 207.7 0.84 563.6

81.0

69.5

71.2

1.32

344.0 10.3

39.05

740.0

48.19

88.0 1.78

250.3 871.0 57.01

95.3

103.3

75.1

76.7

3.05 73.0

0.002218° 33.80 35.7 0.00037 3.89 203.1 9.2 67.5 0.002218° 289.1 2.4 × 10−4

5.04 2.2 3.20

75 0.0024 at 24.2° 41.70 38.2 0.00046 5.59 219.6 11.6

79

77

74

74

49.61 40.7

43.6

66.81 46.4

49.4

8.22

13.12

20.94

231.9 14.2

17.1

358.7

426.3

75 84.84 52.4

104.9 58.8

101.4

247.3 20.3

261.0 27.0 205.8

271.7 34.2 300

0.0024 at 24.2° 2.85 × 10−4 52 43.78 6.60

495.2

562.3

60.67 46.74 8.72 4.0

62 11.54

14.81 6.2

16.73

84.76 23.75 9.5

83 98 30.38

100 110 40.25 15.7

3.13 132.1

135.1

76.48 37.4

40.4 10.94 27 83.2 41.4

40°C

71

6H2O 4H2O 2H2O 2.66 1.1 4.22 97.59 90.01

30°C

76.00 36.0

134.5 1.715° 76.60 34.7

135.3

136.5

140.4

147.0

2.6 × 10−4 at 25° 33.8

78.54 33.2

83.68 32.7

63.13

60.77

31.1

29.7

33.5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54

*By N. A. Lange. Abridged from “Table of Solubilities of Inorganic Compounds in Water at Various Temperatures” in Lange, Handbook of Chemistry, 10th ed., McGraw-Hill, New York, 1961. For tables of the solubility of gases in water at various temperatures, Atack (Handbook of Chemical Data, Reinhold, New York, 1957) gives values at closer temperature intervals, usually 1 or 5°C, than are tabulated here. For materials marked by ‡, additional data are given in tables subsequent to this one. For the solubility of various hydrocarbons in water at high pressures see J. Chem. Eng. Data, 4, 212 (1959).

2-121

TABLE 2-120

Solubilities of Inorganic Compounds in Water at Various Temperatures (Continued )

Substance

Formula

1 2 3 4 5 6 7 8 9 10 11

Calcium bicarbonate chloride chloride fluoride hydroxide nitrate nitrate nitrate nitrite nitrite oxalate

Ca(HCO3)2 CaCl2 CaCl2 CaF2 Ca(OH)2 Ca(NO3)2 Ca(NO3)2 Ca(NO3)2 Ca(NO2)2 Ca(NO2)2 CaC2O4

12 13 14 15 16 17 18 19 20 21 22 23 24

sulfate Carbon dioxide, 760 mm ‡ monoxide, 760 mm ‡ Cesium chloride nitrate sulfate Chlorine, 760 mm ‡ Chromic anhydride Cuprio chloride nitrate nitrate sulfate sulfide

CaSO4 CO2 CO CsCl CsNO3 Cs2SO4 Cl2 CrO3 CuCl2 Cu(NO3)2 Cu(NO3)2 CuSO4 CuS

25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62

Cuprous chloride Ferric chloride Ferrous chloride chloride nitrate sulfate sulfate Hydrobromic acid, 760 mm Hydrochloric acid, 760 mm Iodine Lead acetate bromide carbonate chloride chromate fluoride nitrate sulfate Magnesium bromide chloride hydroxide nitrate sulfate sulfate sulfate Manganous sulfate sulfate sulfate sulfate Mercurous chloride Molybdic oxide Nickel chloride nitrate nitrate sulfate sulfate Nitric oxide, 760 mm Nitrous oxide

CuCl FeCl3 FeCl2 FeCl2 Fe(NO3)2 FeSO4 FeSO4 HBr HCl I2 Pb(C2H3O2)2 PbBr2 PbCO3 PbCl2 PbCrO4 PbF2 Pb(NO3)2 PbSO4 MgBr2 MgCl2 Mg(OH)2 Mg(NO3)2 MgSO4 MgSO4 MgSO4 MnSO4 MnSO4 MnSO4 MnSO4 HgCl MoO3 NiCl2 Ni(NO3)2 Ni(NO3)2 NiSO4 NiSO4 NO N2O

2-122

Solid phase 6H2O 2H2O 4H2O 3H2O

0°C 16.15 59.5 0.185 102.0

10°C 65.0 0.176 115.3

20°C 16.60 74.5 0.001618° 0.165 129.3

30°C

40°C

50°C

17.05

60°C

70°C

17.50

80°C

90°C

17.95

18.40

102 0.001726° 0.153 152.6

136.8 0.141 195.9 237.5

0.128

0.116

141.7 0.106

147.0 0.094

152.7 0.085

62.07

2H2O

0.1759 0.3346 0.0044 161.4 9.33 167.1 1.46 164.9 70.7 81.8

2H2O 6H2O 3H2O 5H2O

6.7 × 10−4 at 13° 0.1928 0.2318 0.0035 174.7 14.9 173.1 0.980 73.76 95.28 17.4

74.4

81.9 64.5

6.8 × 10−4 at 25° 0.1688 0.0028 186.5 23.0 178.7 0.716 77.0 125.1 20.7 3.3 × 10−5 at 18° 1.5225° 91.8

9.5 × 10−4 at 50° 0.2090 0.1257 0.0024 197.3 33.9 184.1 0.562

132.6

80.34

14 × 10−4 at 95° 0.2097 0.0973 0.0021 208.0 47.2 189.9 0.451 174.0 83.8

25

159.8 28.5

33.3

77.3

315.1 82.5

73.0

0.0761 0.0018 218.5 64.4 194.9 0.386 182.1 87.44

0.2047 0.0576 0.0015 229.7 83.8 199.9 0.324

151.9

363.6 244.8

0.1966 0.0013 239.5 107.0 205.0 0.274

0.0010 250.0 134.0 210.3 0.219

91.2

99.2

178.8 40

207.8 55

88.7

525.8 100

0.0006 260.1 163.0 214.9 0.125 217.5

71.02 15.65

20.51

83.8 26.5

535.7

210.3

0.029 0.4554 0.6728

6H2O 6H2O 6H2O 7H2O 6H2O 1H2O 7H2O 5H2O 4H2O 1H2O

38.8 0.0028 91.0 52.8

0.060 48.3 0.0035 94.5 53.5

40.2

48.6

0.85 0.00011 0.99 7 × 10−6 0.064 56.5 0.0041 96.5 54.5 0.000918°

40.8 53.23

30.9 42.2 60.01 59.5

53.9 79.58

59.5

27.22

32

1.53

1.94

2.36

3.34

4.75

1.20

1.45

1.70

1.98

2.62

3.34

0.068 66 0.0049 99.2

40.8 45.3

62.9 64.5

67.76 66.44 0.264 68.9

75 0.0056 101.6 57.5

130 56.1

85

95

104.1

107.5 61.0

0.00757 0.1705

115

68.8 0.0007 0.476 73.3 122.2

38.8

50.4

53.5

59.5

64.2 62.9

69.0

74.0 68.3

72.6 58.17

55.0

52.0

48.0

42.5

34.0

0.687 78.3

1.206 82.2

2.055 85.2

113.7 66.0

84.74 45.6

120.2 73.0 137.0

163.1 0.00984

37.3

63.3 0.056

35.5 44.5

0.0002 0.138 64.2 96.31

43.6

67.3 0.04 55.0425° 1.15

171.5 59.6 0.078

66.55

0.00014 2H2O 6H2O 6H2O 3H2O 7H2O 6H2O

32.9

198

3H2O

105.8

165.6 50.9

221.2 82.3

0.1619 0 0 270.5 197.0 220.3 0 206.8 107.9 75.4

105.3 6H2O 7H2O 1H2O

0.077

76.68

14.3

4H2O

159

281.5 358.7

4H2O 2H2O

100°C

2.106 87.6

169.1

235.1

42.46 0.00618 0.1211

0.00517

0.00440

50.15 0.00376

54.80 0.00324

59.44 0.00267

63.17 0.00199

0.00114

76.7 0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

Potassium acetate acetate alum bicarbonate bisulfate bitartrate carbonate chlorate chloride chromate dichromate ferricyanide hydroxide hydroxide nitrate nitrite perchlorate permanganate persulfate† sulfate thiocyanate Silver cyanide nitrate sulfate Sodium acetate acetate bicarbonate carbonate carbonate chlorate chloride chromate chromate chromate dichromate dichromate dihydrogen phosphate dihydrogen phosphate dihydrogen phosphate hydrogen arsenate hydrogen phosphate hydrogen phosphate hydrogen phosphate hydrogen phosphate hydroxide hydroxide hydroxide hydroxide nitrate nitrite oxalate phosphate, tripyrophosphate sulfate sulfate sulfate sulfide sulfide sulfide sulfite sulfite tetraborate tetraborate vanadate (meta)

KC2H3O2 KC2H3O2 K2SO4⋅Al2(SO4)3 KHCO3 KHSO4 KHC4H4O6 K2CO3 KClO3 KCl K2CrO4 K2Cr2O7 K3Fe(CN)6 KOH KOH KNO3 KNO2 KClO4 KMnO4 K2S2O8† K2SO4 KCNS AgCN AgNO3 Ag2SO4 NaC2H3O2 NaC2H3O2 NaHCO3 Na2CO3 Na2CO3 NaClO8 NaCl Na2CrO4 Na2CrO4 Na2CrO4 Na2Cr2O7 Na2Cr2O7 NaH2PO4 NaH2PO4 NaH2PO4 Na2HAsO4 Na2HPO4 Na2HPO4 Na2HPO4 Na2HPO4 NaOH NaOH NaOH NaOH NaNO3 NaNO2 Na2C2O4 Na3PO4 Na4P2O7 Na2SO4 Na2SO4 Na2SO4 Na2S Na2S Na2S Na2SO3 Na2SO3 Na2B4O7 Na2B4O7 NaVO8

1aH2O aH2O 24H2O

2H2O

2H2O 1H2O

†

3H2O 10H2O 1H2O 10H2O 4H2O

216.7 3.0 22.4 36.3 0.32 105.5 3.3 27.6 58.2 5 31 97

233.9 4.0 27.7 0.40 108 5 31.0 60.0 7 36 103

255.6 5.9 33.2 51.4 0.53 110.5 7.4 34.0 61.7 12 43 112

13.3 278.8 0.75 2.83 1.62 7.35 177.0

20.9

122 0.573 36.3 119 6.9 7

170 0.695 40.8 121 8.15 12.5

31.6 298.4 1.80 6.4 4.49 11.11 217.5 2.2 × 10−5 222 0.796 46.5 123.5 9.6 21.5

79 35.7 31.70

89 35.8 50.17

101 36.0 88.7

1.05 4.4 2.60 9.22

283.8 8.39 39.1 0.90 113.7 10.5 37.0 63.4 20 50 126 45.8 2.6 9.0 7.19 12.97 300 0.888 54.5 126 11.1 38.8 50.5 113 36.3 88.7

2H2O

163.0

2H2O 1H2O

57.9

323.3 11.70 45.4 67.3 1.32 116.9 14 40.0 65.2 26 60 63.9 334.9 4.4 12.56 9.89 14.76

337.3 17.00

350 24.75 60.0

1.83 121.2 19.3 42.6 66.8 34

2.46 126.8 24.5 45.5 68.6 43 66

140 85.5

110.0

364.8 40.0

133.1 48.3 70.4 52

138

380.1 71.0 4.6 139.8 38.5 51.1 72.1 61

169

396.3 109.0

147.5 54.0 73.9 70

202

6.5 16.89

9 22.2

11.8

14.8

18

16.50

18.17

19.75

21.4

22.8

376 0.979 65.5 129.5 12.7

455 1.08 83 134 14.45

525 1.15 139 139.5 16.4

1.22

669 1.30

48.5 126 36.6

140 37.0

46.4 155 37.3

104

114.6

95.96

177.8

244.8

146

153

172 37.8

45.8 189 38.4

123.0 316.7

124.8 376.2

1.36 161

39.0

121.6 6.95 155.7 57 56.7 75.6 80 82.6104 178 246 412.8 21.8 24.1 952 1.41 170 45.5 230 39.8 125.9 426.3

12H2O 12H2O 7H2O 2H2O

69.9

85.2

106.5

138.2

15.5 3.6

26.5 7.7

37 20.8

47

158.6 7.3 1.67

179.3 65

190.3

207.3 85

225.3

82.9

88.1

92.4

102.9

246.6

51.8 80.2

102.2 4H2O 3aH2O 1H2O

42 51.5 109

119

129

145

174

104 98.4

114 104.1

124

313

12H2O 10H2O 10H2O 7H2O 9H2O 5aH2O 6H2O 7H2O

73 72.1

80 78.0

1.5 3.16 5.0 19.5

13.9

96 91.6

4.1 3.95 9.0 30

88 84.5 3.7 11 6.23 19.4 44

15.42

18.8

22.5

20

26.9

36

20 9.95 40.8

31 13.50

43 17.45

55 21.83

81 30.04

347 180 163.2 6.33 108 40.26

48.8 28.5

46.7

45.3

43.7

42.5

39.82 36.4

42.69 39.1

28.2 10.5

28.8 20.3

28 10H2O 5H2O 2H2O

1.3

1.6

2.7 15.325°

3.9

148 132.6

45.73 43.31

68.4

59.23 57.28

28.3 24.4

30.2

51.40 49.14

31.5

41

52.5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

2-123

TABLE 2-120

Solubilities of Inorganic Compounds in Water at Various Temperatures (Concluded )

Substance 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Sodium vanadate (meta) Stannous chloride sulfate Strontium acetate acetate chloride chloride nitrate nitrate nitrate sulfate Sulfur dioxide, 760 mm † Thallium sulfate Thorium sulfate sulfate sulfate sulfate Zinc chlorate chlorate nitrate nitrate sulfate sulfate sulfate

2-124

Formula NaVO3 SnCl2 SnSO4 Sr(C2H3O2)2 Sr(C2H3O2)2 SrCl2 SrCl2 Sr(NO3)2 Sr(NO3)2 Sr(NO3)2 SrSO4 SO2 Tl2SO4 Th(SO4)2 Th(SO4)2 Th(SO4)2 Th(SO4)2 ZnClO3 ZnClO3 Zn(NO3)2 Zn(NO3)2 ZnSO4 ZnSO4 ZnSO4

Solid phase

0°C

10°C

9H2O 8H2O 6H2O 4H2O 6H2O 4H2O 6H2O 3H2O 7H2O 6H2O 1H2O

36.9 43.5

43.61 42.95 47.7

41.6 52.9

52.7 40.1

64.0 70.5

0.0113 22.83 2.70 0.74 1.0 1.50

0.0114 11.29 4.87 1.38 1.62 1.90

145.0

30°C

21.1025° 269.815° 19

83.9 4H2O aH2O 6H2O 2H2O 1H2O 4H2O

20°C

16.21 3.70 0.98 1.25

40°C

50°C

26.23

60°C

70°C

80°C

32.97

36.9

38.875°

36.24

36.10

85.9

90.5

90°C

100°C

18 39.5 58.7

88.6 0.0114 7.81 6.16 1.995

65.3

37.35 72.4

81.8

83.8

97.2

90.1

2.998

5.41

4.5 9.21 5.22

4.04

2.54

2.45

36.4 130.4

100.8 139

93.8

96

98

100

10.92

12.74

14.61

16.53

18.45

6.64 1.63

1.09

86.6

83.7

80.8

152.5 200.3 118.3

94.78

209.2

223.2

273.1

206.9 41.9

47

54.4 70.1

76.8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

SOLUBILITIES The H in solubility tables (2-121 to 2-144) is the proportionality constant for the expression of Henry’s law, p = Hx, where x = mole fraction of the solute in the liquid phase; p = partial pressure of the solute in the gas phase, expressed in atmospheres; and H = a proportionality constant expressed in units of atmospheres of solute pressure in the gas phase per unit concentration of the solute in the liquid phase. (The unit of concentration of the solute in the liquid phase is moles solute per mole solution.)

TABLE 2-124

Ammonia (NH3)—Low Pressures

Weight NH3 per 100 weights H2O

0.105

0.244

0.32

0.38

0.576

0.751

1.02

Partial pressure NH3, mm. Hg, at 25°C

0.791

1.83

2.41

2.89

4.41

5.80

7.96

Weight NH3 per 100 weights H2O

1.31

1.53

1.71

1.98

2.11

2.58

2.75

10.31

11.91

13.46

15.75

16.94

20.86

22.38

Partial pressure NH3, mm. Hg, at 25°C TABLE 2-121

Acetylene (C2H2)

t, °C

0

5

10

15

20

25

30

10−3 × H*

0.72

0.84

0.96

1.08

1.21

1.33

1.46

2-125

“Landolt-Börnstein Physikalische-chemische Tabellen,” Eg. I, p. 303, 1927. Phase-equilibrium data for the binary system NH3-H2O are given by Clifford and Hunter, J. Phys. Chem., 37, 101 (1933).

International Critical Tables, vol. 3, p. 260, McGraw-Hill, 1928. *H. See footnote for Table 2-122. TABLE 2-125

TABLE 2-122

Carbon Dioxide (CO2)

Weight of CO2 per 100 weights of H2O* Total pressure, atm 12°C 18°C 25°C 31.04°C 35°C 40°C 50°C 75°C 100°C

Air

t, °C

0

5

10

15

20

25

30

35

10−4 × H*

4.32

4.88

5.49

6.07

6.64

7.20

7.71

8.23

t, °C

40

45

50

60

70

80

90

100

10−4 × H*

8.70

9.11

9.46

10.1

10.5

10.7

10.8

10.7

International Critical Tables, vol. 3, p. 257. *H is calculated from the absorption coefficients of O2 and N2, taking into consideration the correction for constant argon content.

25 50 75 100 150 200 300 400 500 700

7.03 7.18 7.27 7.59

3.86 6.33 6.69 6.72 7.07

7.86 8.12

7.35 7.77

5.38 6.17 6.28

7.54

2.80 4.77 5.80 5.97 6.25 6.48

2.56 4.39 5.51 5.76 6.03 6.29

2.30 4.02 5.10 5.50 5.81 6.28

7.27 7.65

7.06 7.51

6.89 7.26

1.92 3.41 4.45 5.07 5.47 5.76 6.20 6.58

1.35 2.49 3.37 4.07 4.86 5.27 5.83 6.30

1.06 2.01 2.82 3.49 4.49 5.08 5.84 6.40

7.58

7.43

7.61

*In the original, concentration is expressed in cubic centimeters of CO2 (reduced to 0°C and 1 atm) dissolved in 1 g of water. TABLE 2-123 Weight NH3 per 100 weights H2O 100 90 80 70 60 50 40 30 25 20 15 10 7.5 5 4 3 2.5 2 1.6 1.2 1.0 0.5

Ammonia (NH3) Partial pressure of NH3, mm. Hg 0°C

947 785 636 500 380 275 190 119 89.5 64 42.7 25.1 17.7 11.2

10°C

20°C

987 1450 780 1170 600 945 439 686 301 470 190 298 144 227 103.5 166 70.1 114 41.8 69.6 29.9 50.0 19.1 31.7 16.1 24.9 11.3 18.2 15.0 12.0

*Extrapolated values.

25°C 30°C

23.5 19.4 15.3 12.0 9.1 7.4 3.4

719 454 352 260 179 110 79.7 51.0 40.1 29.6 24.4 19.3 15.3 11.5

40°C

3300 2760 2130 1520 1065 692 534 395 273 167 120 76.5 60.8 45 (37.6)* (30.0) (24.1) (18.3) (15.4)

50°C 60°C

825 596 405 247 179 115 91.1 67.1 (55.7) (44.5) (35.5) (26.7) (22.2)

834 583 361 261 165 129.2 94.3 77.0 61.0 48.7 36.3 30.2

TABLE 2-126

Carbon Monoxide (CO) 10−4 × H

Partial pressure of CO, mm Hg

17.7°C

19.0°C

900 2000 3000 4000 5000 6000 7000 8000

4.77 4.77 4.77 4.78 4.80 4.82 4.86 4.88

4.88 4.91 4.93 4.95 4.97 4.98 5.02 5.08

International Critical Tables, vol. 3, p. 260.

TABLE 2-127

Carbonyl Sulfide (COS)

t °C

0

5

10

15

20

25

30

10−3 × H

0.92

1.17

1.48

1.82

2.19

2.59

3.04

International Critical Tables, vol. 3, p. 261.

2-126

PHYSICAL AND CHEMICAL DATA

TABLE 2-128 Partial pressure of Cl2, mm Hg

Chlorine (Cl2)

TABLE 2-129

Solubility, g of Cl2 per liter 10°C

20°C

30°C

40°C

50°C

5 10 30 50 100

0.488 .679 1.221 1.717 2.79

0.451 .603 1.024 1.354 2.08

0.438 .575 .937 1.210 1.773

0.424 .553 .873 1.106 1.573

0.412 .532 .821 1.025 1.424

0.398 .512 .781 .962 1.313

150 200 250 300 350

3.81 4.78 5.71

2.73 3.35 3.95 4.54 5.13

2.27 2.74 3.19 3.63 4.06

1.966 2.34 2.69 3.03 3.35

1.754 2.05 2.34 2.61 2.86

1.599 1.856 2.09 2.31 2.53

400 450 500 550 600

5.71 6.26 6.85 7.39 7.97

4.48 4.88 5.29 5.71 6.12

3.69 3.98 4.30 4.60 4.91

3.11 3.36 3.61 3.84 4.08

2.74 2.94 3.14 3.33 3.52

650 700 750 800 900

8.52 9.09 9.65 10.21

6.52 6.90 7.29 7.69 8.46

5.21 5.50 5.80 6.08 6.68

4.32 4.54 4.77 4.99 5.44

3.71 3.89 4.07 4.27 4.62

9.27 10.84 13.23 17.07 21.0

7.27 8.42 10.14 13.02 15.84

5.89 6.81 8.05 10.22 12.32

4.97 5.67 6.70 8.38 10.03

18.73 21.7 24.7 27.7 30.8

14.47 16.62 18.84 20.7 23.3

11.70 13.38 15.04 16.75 18.46

Cl2.8H2O2 separates

3000 3500 4000 4500 5000 Partial pressure of Cl2, mm Hg

70°C

80°C

90°C

1 3 5 7 10 11 12 13 14 15 16

100°C

110°C

5 10 30 50 100

0.383 .492 .743 .912 1.228

0.369 .470 .704 .863 1.149

0.351 .447 .671 .815 1.085

0.339 .431 .642 .781 1.034

0.326 .415 .627 .747 .987

0.316 .402 .598 .722 .950

150 200 250 300 350

1.482 1.706 1.914 2.10 2.28

1.382 1.580 1.764 1.932 2.10

1.294 1.479 1.642 1.793 1.940

1.227 1.396 1.553 1.700 1.831

1.174 1.333 1.480 1.610 1.736

1.137 1.276 1.413 1.542 1.661

400 450 500 550 600

2.47 2.64 2.80 2.97 3.13

2.25 2.41 2.55 2.69 2.83

2.08 2.22 2.35 2.47 2.59

1.965 2.09 2.21 2.32 2.43

1.854 1.972 2.08 2.19 2.29

1.773 1.880 1.986 2.09 2.19

650 700 750 800 900

3.29 3.44 3.59 3.75 4.04

2.97 3.10 3.23 3.37 3.63

2.72 2.84 2.96 3.08 3.30

2.55 2.66 2.76 2.87 3.08

2.41 2.50 2.60 2.69 2.89

2.28 2.37 2.47 2.56 2.74

1000 1200 1500 2000 2500

4.36 4.92 5.76 7.14 8.48

3.88 4.37 5.09 6.26 7.40

3.53 3.95 4.58 5.63 6.61

3.28 3.67 4.23 5.17 6.05

3.07 3.43 3.95 4.78 5.59

2.91 3.25 3.74 4.49 5.25

3000 3500 4000 4500 5000

9.83 11.22 12.54 13.88 15.26

8.52 9.65 10.76 11.91 13.01

7.54 8.53 9.52 10.46 11.42

6.92 7.79 8.65 9.49 10.35

6.38 7.16 7.94 8.72 9.48

5.97 6.72 7.42 8.13 8.84

0°C

5°C

10°C

15°C

20°C

30°C

40°C

2.00 6.00 10.0 14.0 20.0

1.50 4.7 7.8 10.9 15.5 17.0 18.6 20.3

1.25 3.85 6.30 8.95 12.8 14.0 15.3 16.6 18.0 19.2 20.3

1.00 3.20 5.25 7.35 10.5 11.7 12.8 13.8 14.9 16.0 17.0

0.90 2.70 4.30 6.15 8.80 9.70 10.55 11.5 12.3 13.2 14.2

0.60 1.95 3.20 4.40 6.30 7.00 7.50 8.20 8.80 9.50 10.1

0.46 1.30 2.25 3.20 4.50 5.00 5.45 5.85 6.35 6.80 7.20

Ishi, Chem. Eng. (Japan), 22, 153 (1958).

TABLE 2-130

Ethane (C2H6)

t, °C

0

5

10

15

20

25

30

35

10−4 × H

1.26

1.55

1.89

2.26

2.63

3.02

3.42

3.83

t, °C

40

45

50

60

70

80

90

100

10−4 × H

4.23

4.63

5.00

5.65

6.23

6.61

6.87

6.92

International Critical Tables, vol. 3, p. 261.

TABLE 2-131

Solubility, g of Cl2 per liter 60°C

Weight of ClO2, grams per liter of solution

Vol % of ClO2 in gas phase

0°C

1000 1200 1500 2000 2500

Chlorine Dioxide (ClO2)

Ethylene (C2H4)

t, °C

0

5

10

15

20

25

30

10−3 × H

5.52

6.53

7.68

8.95

10.2

11.4

12.7

International Critical Tables, vol. 3, p. 260.

TABLE 2-132

Helium (He)

t, °C

0

10

20

30

40

50

10−4 × H

12.9

12.6

12.5

12.4

12.1

11.5

See also Pray, Schweickert, and Minnich, Ind. Eng. Chem., 44, 1146 (1952).

TABLE 2-133

Hydrogen (H2)—Temperature

t, °C

0

5

10

15

20

25

30

35

10−4 × H

5.79

6.08

6.36

6.61

6.83

7.07

7.29

7.42

t, °C

40

45

50

60

70

80

90

100

10−4 × H

7.51

7.60

7.65

7.65

7.61

7.55

7.51

7.45

“International Critical Tables,” vol. 3, p. 256. See also Pray, Schweickert, and Minnich, Ind. Eng. Chem., 44, 1146 (1952).

SOLUBILITIES TABLE 2-134

Hydrogen (H2)—Pressure

TABLE 2-137

10−4 × H

Partial pressure H2, mm Hg

19.5°C

900 1100 2000 3000 4000 5000 6000 7000 8200 8250

23°C

7.42 7.75 7.76 7.77 7.81 7.89 8.00 8.16 8.41

7.42 7.43 7.47 7.56 7.70 7.87

2-127

Methane (CH4)

t, °C

0

5

10

15

20

25

30

35

10−4 × H

2.24

2.59

2.97

3.37

3.76

4.13

4.49

4.86

t, °C

40

45

50

60

70

80

90

100

10−4 × H

5.20

5.51

5.77

6.26

6.66

6.82

6.92

7.01

International Critical Tables, vol. 3, p. 260.

TABLE 2-138

8.17

International Critical Tables, vol. 3, p. 256.

Nitrogen (N2)—Temperature*

t, °C

0

5

10

15

20

25

30

35

10−4 × H

5.29

5.97

6.68

7.38

8.04

8.65

9.24

9.85

t, °C

40

45

50

60

70

80

90

100

10−4 × H

10.4

10.9

11.3

12.0

12.5

12.6

12.6

12.6

“International Critical Tables,” vol. 3, p. 256. See also Pray, Schweickert, and Minnich, Ind. Eng. Chem., 44, 1146 (1952). *Atmospheric nitrogen = 98.815 vol. % N2 + 1.185 vol. % A. TABLE 2-135

Hydrogen Chloride (HCl)

Weights of HCl per 100 weights of H2O

TABLE 2-139

Partial pressure of HCl, mm Hg 0°C

78.6 66.7 56.3 47.0 38.9 31.6 25.0 19.05 13.64 8.70 4.17 2.04

10°C

510 130 29.0 5.7 1.0 0.175 .0316 .0056 .00099 .000118 .000018

Weights of HCl per 100 weights of H2O

20°C

840 233 56.4 11.8 2.27 0.43 .084 .016 .00305 .000583 .000069 .0000117

30°C

399 105.5 23.5 4.90 1.00 0.205 .0428 .0088 .00178 .00024 .000044

Nitrogen (N2)—Pressure

627 188 44.5 9.90 2.17 0.48 .106 .0234 .00515 .00077 .000151

900 2000 3000 4000 5000 6000 7000 8100 8200

80°C

535 141 35.7 8.9 2.21 0.55 .136 .0344 .0064 .00140

110°C

623 188 54.5 15.6 4.66 1.34 0.39 .095 .0245

760 253 83 28 9.3 3.10 0.93 .280

Oxygen (O2)—Temperature

t, °C

0

5

10

15

20

25

30

35

10−4 × H

2.55

2.91

3.27

3.64

4.01

4.38

4.75

5.07

t, °C

40

45

50

60

70

80

90

100

10−4 × H

5.35

5.63

5.88

6.29

6.63

6.87

6.99

7.01

International Critical Tables, vol. 3, p. 257. Pray, Schweickert, and Minnich [Ind. Eng. Chem., 44, 1146 (1952)] give H = 4.46 × 10−4 at 25°C and other values up to 343°C.

Hydrogen Sulfide (H2S)

t, °C

0

5

10

15

20

25

30

35

10−2 × H

2.68

3.15

3.67

4.23

4.83

5.45

6.09

6.76

t, °C

40

45

50

60

70

80

90

100

7.45

9.08 9.15 9.25 9.38 9.49 9.62 9.75 9.91

TABLE 2-141

Oxygen (O2)—Pressure

Partial pressure of O2, mm Hg

10 × H

24.9°C

8.24 8.32 8.41 8.49 8.59 8.74 8.86 9.04

See also Goodman and Krase [Ind. Eng. Chem., 23, 401 (1931)] for values up to 169°C and 300 atm.

TABLE 2-140

Enthalpy and phase-equilibrium data for the binary system HCl-H2O are given by Van Nuys, Trans. Am. Inst. Chem. Engrs., 39, 663 (1943).

−2

19.4°C

Partial pressure of HCl, mm Hg 50°C

78.6 66.7 56.3 47.0 38.9 31.6 25.0 19.05 13.64 8.70 4.17 2.04

TABLE 2-136

10−4 × H

Partial pressure of N2, mm Hg

8.14

8.84

10.3

International Critical Tables, vol. 3, p. 259.

11.9

13.5

14.4

14.8

800 900 2000 3000 4000 5000 6000 7000 8150 8200

10−4 × H 23°C

25.9°C 4.79

4.58 4.59 4.60 4.68 4.73 4.80 4.88 4.98

4.80 4.83 4.88 4.92 4.98 5.05 5.16

International Critical Tables, vol. 3, p. 257. See also Trans. Am. Soc. Mech. Engrs., 76, 69 (1954) for solubility of O2 for 100°F < T < 650°F, 300 < P < 2000 lb/in2.

2-128

PHYSICAL AND CHEMICAL DATA

TABLE 2-142 t, °C

0

10−3 × H 1.94

Ozone (O3)

TABLE 2-143

5

10

15

20

25

30

35

2.18

2.48

2.88

3.76

4.57

5.98

8.18

40

50

12.0 27.4

International Critical Tables, vol. 3, p. 257.

TABLE 2-144

Propylene (C3H6)

t, °C

2

6

10

14

18

10−3 × H

3.04

3.84

4.46

5.06

5.69

International Critical Tables, vol. 3, p. 260.

Partial Vapor Pressure of Sulfur Dioxide over Water, mm Hg Temperature, °C

g SO2 / 100 g H2O

0

10

0.01 0.05 0.10 0.15 0.20

0.02 0.38 1.15 2.10 3.17

0.04 0.66 1.91 3.44 5.13

0.25 0.30 0.40 0.50 1.00

4.34 5.57 8.17 10.9 25.8

6.93 8.84 12.8 17.0 39.5

2.00 3.00 4.00 5.00 6.00 8.00 10.00 15.00 20.00

20 0.07 1.07 3.03 5.37 7.93 10.6 13.5 19.4 25.6 58.4

30

40

50

60

90

120

0.12 1.68 4.62 8.07 11.8

0.19 2.53 6.80 11.7 17.0

0.29 3.69 9.71 16.5 23.8

0.43 5.24 13.5 22.7 32.6

1.21 12.9 31.7 52.2 73.7

2.82 27.0 63.9 104 145

15.7 19.8 28.3 37.1 83.7

58.6 93.2 129 165 202

88.5 139 192 245 299

129 202 277 353 430

183 285 389 496 602

275 351 542 735

407 517 796

585 741

818

22.5 28.2 40.1 52.3 117

31.4 39.2 55.3 72.0 159

42.8 53.3 74.7 96.8 212

95.8 118 164 211 454

253 393 535 679 824

342 530 720

453 700

955

186 229 316 404 856

Condensed from Rabe, A. E. and Harris, J. F., J. Chem. Eng. Data, 8 (3), 333–336, 1963. Copyright © American Chemical Society and reproduced by permission of the copyright owner.

THERMAL EXPANSION UNITS CONVERSIONS For this subsection, the following units conversion is applicable:

2, p. 93; metals, vol. 2, p. 459; petroleums, vol. 2, p. 145; porcelains, vol. 2, pp. 70, 78; refractory materials, vol. 2, p. 83; solid insulators, vol. 2, p. 310.

°F = 9⁄ 5 °C + 32. THERMAL EXPANSION OF GASES ADDITIONAL REFERENCES The tables given under this subject are reprinted by permission from the Smithsonian Tables. For more detailed data on thermal expansion, see International Critical Tables: tabular index, vol. 3, p. 1; abrasives, vol. 2, p. 87; alloys, vol. 2, p. 463; building stones, vol. 2, p. 54; carbons, vol. 2, p. 303; elements, vol. 1, p. 102; enamels, vol. 2, p. 115; glass, vol.

No tables of the coefficients of thermal expansion of gases are given in this edition. The coefficient at constant pressure, 1/υ(∂υ/∂T)p, for an ideal gas is merely the reciprocal of the absolute temperature. For a real gas or liquid, both it and the coefficient at constant volume, 1/p (∂p/∂T)v, should be calculated either from the equation of state or from tabulated PVT data.

THERMAL EXPANSION TABLE 2-145

2-129

Linear Expansion of the Solid Elements*

C is the true expansion coefficient at the given temperature; M is the mean coefficient between given temperatures; where one temperature is given, the true coefficient at that temperature is indicated; α and β are coefficients in formula lt = l0(1 + αt + βt2); l0 is length at 0°C (unless otherwise indicated, when, if x is the reference temperature, lt = lx[1 + α(t − tx) + β(t − tx)2]; lt is length at t°C). Element

Temp. °C

C × 104

Aluminum Aluminum Antimony Arsenic Bismuth Cadmium Cadmium Carbon, diamond graphite Chromium Cobalt Copper Copper Gold Gold Indium Iodine Iridium Iridium Iron, soft cast wrought steel Lead (99.9)

20 300 20 20 20 0 0 50 50

0.224 0.284 0.136 0.05 0.014 0.54 0.20⊥ 0.012 0.06

20 20 200 20

0.123 0.162 0.170 0.140

40

0.417

20

0.065

40 20 20 20

0.1210 0.118 0.119 0.114

Magnesium

100 280 20

0.291 0.343 0.254

Manganese

20

0.233

Molybdenum†

20

0.053

Nickel

20

0.126

Osmium Palladium

40 20

0.066 0.1173

Platinum

20 20

0.0887 0.0893

40 40 0 40 20 20

0.0850 0.0963 0.439 0.0763 0.1846 0.195

Potassium Rhodium Ruthenium Selenium Silicon Silver Sodium Steel, 36.4Ni Tantalum†

20

Tellurium Thallium Tin

20 40 20 20 27 20‡ 20‡ 20

Tungsten† Zinc

0.065 0.016 0.302 0.214 0.305 0.0444 0.643 0.125⊥ 0.358

Temp. range, °C

M × 104

100 500 20

0.235 0.311 0.080⊥

20 −180, −140 −180, −140

0.103⊥ 0.59 0.117⊥

20, 100

0.068

100 300 17, 100 −191, 17

0.166 0.175 0.143 0.132

−190,

0.837

17

Temp. range, °C

α × 104

β × 106

0,

500

0.22

0.009

20, 20,

100 100

0.526 0.214⊥

20, 6, 0,

500 121 625

0.086 0.121 0.161

0.0064 0.0040

0,

520

0.142

0.0022

0.0636 0.0679

0.0032 0.0011

0, 80 1070, 1720 0, 100

20, 100 20, 200 −100, 20, 0, −190, 0, 25, 25, 0,

0, 6,

+ 20 100 100 0 100 100 500 100

0.11

0.291 0.300 0.240 0.260 0.228 0.159 0.052 0.049 0.055 0.130

50 21

0.83 0.0876

0, 100 −3, +18 0, 100

0.660 0.0249 0.197

−190, 20, 20, −78, 0,

−17 260 340 0 100 20

0.622 0.031 0.055 0.059 0.0655 0.272⊥

20 100 −100 100 100

0.154⊥ 0.045 0.656 0.639 0.141⊥

0, 0, 0, 100,

750 750 750 240

0.1158 0.1170 0.1118 0.269

0.0053 0.0053 0.0053 0.011

+ 20,

500

0.2480

0.0096

20, 300 −142, 19 19, +305

0.216 0.0515 0.0501

0.0121 0.0057 0.0014

−190, + 20 + 20, +300 500, 1000

0.1308 0.1236 0.1346

0.0166 0.0066 0.0033

−190, 0, −190, 0, 0,

+100 1000 −100 + 80 1000

0.1152 0.1167 0.0875 0.0890 0.0887

0.00517 0.0022 0.00314 0.00121 0.00132

−75, −112

0.0746

−75, −67 0, 875 20, 500 0, 50 260, 500 340, 500 20, 400

0.0182 0.1827 0.1939 0.72 0.144 0.136 0.0646

8, 0, −140, +20, +20,

0.00479 0.00295

0.0009

95

0.2033

0.0263

−105, +502 + 0, 400

0.0428 0.354

0.00058 0.010

*Smithsonian Tables. For more complete tabulations see Table 142, Smithsonian Physical Tables, 9th ed., 1954; Handbook of Chemistry and Physics, 40th ed., pp. 2239–2245. Chemical Rubber Publishing Co.; Goldsmith, and Waterman, WADC-TR-58-476, 1959; Johnson (ed.), WADD-TR-60-56, 1960, etc. †Molybdenum, 300° to 2500°C; lt = l300[1 + 5.00 × 10−6(t − 300) + 10.5 × 10−10(t − 300)2] Tantalum, 300° to 2800°C; lt = l300[1 + 6.60 × 10−6(t − 300) + 5.2 × 10−10(t − 300)2] Tungsten, 300° to 2700°C; lt = l300[1 + 4.44 × 10−6(t − 300) + 4.5 × 10−10(t − 300)2] Beryllium, 20° to 100°C; 12.3 × 10−6 per °C. Columbium, 0° to 100°C; 7.2 × 10−6 per °C. Tantalum, 20° to 100°C; 6.6 × 10−6 per °C. ‡Two errors in the data of zinc have been corrected. These values were taken from Grüneisen and Goens, Z. Physik., 29, 141 (1924).

2-130

PHYSICAL AND CHEMICAL DATA

TABLE 2-146

Linear Expansion of Miscellaneous Substances*

The coefficient of cubical expansion may be taken as three times the linear coefficient. In the following table, t is the temperature or range of temperature, and C, the coefficient of expansion. Substance

t°C

Amber

0−30 0−09 20−60

Bakelite, bleached Brass: Cast 0−100 Wire 0−100 Wire 0−100 71.5 Cu + 27.7 Zn + 0.3 Sn + 0.5 Pb 40 71 Cu + 29 Zn 0−100 Bronze: 3 Cu + 1 Sn 16.6−100 3 Cu + 1 Sn 16.6−350 3 Cu + 1 Sn 16.6−957 86.3 Cu + 9.7 Sn + 4 Zn 40 97.6 Cu + hard 0−80 2.2 Sn + soft 0−80 0.2 P Caoutchouc Caoutchouc 16.7−25.3 Celluloid 20−70 Constantan 4−29 Duralumin, 94Al 20−100 20−300 Ebonite 25.3−35.4 Fluorspar, CaF2 0−100 German silver 0−100 Gold-platinum, 2 Au + 1 Pt 0−100 Gold-copper, 2 Au + 1 Cu 0−100 Glass: Tube 0−100 Tube 0−100 Plate 0−100 Crown (mean) 0−100 Crown (mean) 50−60 Flint 50−60 Jena ther- 16III 0−100 mometer normal

{

}

C × 104 0.50 0.61 0.22 0.1875 0.1930 0.1783 to 0.193 0.1859 0.1906 0.1844 0.2116 0.1737 0.1782 0.1713 0.1708 0.657 to 0.686 0.770 1.00 0.1523 0.23 0.25 0.842 0.1950 0.1836 0.1523 0.1552 0.0833 0.0828 0.0891 0.0897 0.0954 0.0788 0.081

Substance

t°C

C × 104

Jena thermometer 59III Jena thermometer 59III Gutta percha Ice Iceland spar: Parallel to axis Perpendicular to axis Lead tin (solder) 2 Pb + 1 Sn Limestone Magnalium Manganin Marble Monel metal

0−100 −191 to +16 20 −20 to −1

0.058 0.424 1.983 0.51

0−80 0−80

0.2631 0.0544

0−100 25−100 12−39 15−100 25−100 25−600 0−16 16−38 38−49

0.2508 0.09 0.238 0.181 0.117 0.14 0.16 1.0662 1.3030 4.7707

40

0.0884

0−100 20−790 1000−1400

0.1523 0.0413 0.0553

0−80 −190 to + 16 0−80 −190 to + 16 16 to 500 16 to 1000 40 0 −160 0−100 25−100 25−600

0.0797 0.0521 0.1337 −0.0026 0.0057 0.0058 0.4040 0.691 0.300 0.1933 0.037 0.136

Paraffin Paraffin Paraffin Platinum-iridium, 10 Pt + 1 Ir Platinum-silver, 1 Pt + 2 Ag Porcelain Porcelain Bayeux Quartz: Parallel to axis Parallel to axis Perpend. to axis Quartz glass Quartz glass Quartz glass Rock salt Rubber, hard Rubber, hard Speculum metal Steel, 0.14 C, 34.5 Ni

Substance

t°C

Topas: Parallel to lesser horizontal axis 0−100 Parallel to greater horizontal axis 0−100 Parallel to vertical axis 0−100 Tourmaline: Parallel to longitudinal axis 0−100 Parallel to horizontal axis 0−100 Type metal 16.6−254 Vulcanite 0−18 Wedgwood ware 0−100 Wood: Parallel to fiber: Ash 0−100 Beech 2.34 Chestnut 2.34 Elm 2.34 Mahogany 2.34 Maple 2.34 Oak 2.34 Pine 2.34 Walnut 2.34 Across the fiber: Beech 2.34 Chestnut 2.34 Elm 2.34 Mahogany 2.34 Maple 2.34 Oak 2.34 Pine 2.34 Walnut 2.34 Wax white 10−26 Wax white 26−31 Wax white 31−43 Wax white 43−57

C × 104

0.0832 0.0836 0.0472 0.0937 0.0773 0.1952 0.6360 0.0890 0.0951 0.0257 0.0649 0.0565 0.0361 0.0638 0.0492 0.0541 0.0658 0.614 0.325 0.443 0.404 0.484 0.544 0.341 0.484 2.300 3.120 4.860 15.227

*Smithsonian Tables. For a more complete tabulation see Tables 143, 144. Smithsonian Physical Tables. 9th ed., 1954, also reprinted in American Institute of Physics Handbook, McGraw-Hill, New York, 1957; Handbook of Chemistry and Physics, 40th ed., pp. 2239–2245, Chemical Rubber Publishing Co. For data on many solids prior to 1926, see Gruneisen, Handbuch der Physik, vol. 10, pp. 1–52, 1926, translation available as N.A.S.A. RE 2-18-59W, 1959. For eight plastic solids below 300 K, see Scott, Cryogenic Engineering, p. 331, Van Nostrand, Princeton, NJ, 1959. For 11 other materials to 300 K, see Scott, loc. cit., p. 333. For quartz and silica, see Cook, Brit. J. Appl. Phys., 7, 285 (1956).

THERMAL EXPANSION TABLE 2-147

Cubical Expansion of Liquids*

TABLE 2-148

If V0 is the volume at 0°, then at t° the expansion formula is Vt = V0(1 + αt + βt2 + γ t3). The table gives values of α, β, and γ, and of C, the true coefficient of cubical expansion at 20° for some liquids and solutions. The temperature range of the observation is ∆t. Values for the coefficient of cubical expansion of liquids can be derived from the tables of specific volumes of the saturated liquid given as a function of temperature later in this section. Liquid

Range

α × 103

β × 106

Acetic acid Acetone Alcohol: Amyl Ethyl, 30% by volume Ethyl, 50% by volume Ethyl, 99.3% by volume Ethyl, 500 atm. pressure Ethyl, 3000 atm. pressure Methyl Benzene Bromine Calcium chloride: 5.8% solution 40.9% solution Carbon disulfide 500 atm. pressure 3000 atm. pressure Carbon tetrachloride Chloroform Ether Glycerin Hydrochloric acid, 33.2% solution Mercury Olive oil Pentane Potassium chloride, 24.3% solution Phenol Petroleum, 0.8467 density Sodium chloride, 20.6% solution Sodium sulfate, 24% solution Sulfuric acid: 10.9% solution 100.0% Turpentine Water

16−107 0−54

1.0630 1.3240

0.12636 3.8090

− 15−80 18−39 0−39

0.9001 0.2928 0.7450

0.6573 10.790 1.85

27−46

1.012

2.20

0−40

0.866

0−40 0−61 11−81 0−59

0.524 1.1342 1.17626 1.06218

18−25 17−24 −34−60 0−50 0−50 0−76 0−63 −15−38 0−33 0−100

γ × 108

C × 108 at 20°

1.0876 1.071 − 0.87983 1.487 1.18458 0.902 −11.87 0.730 1.12

1.3635 1.27776 1.87714

0.8741 1.199 0.80648 1.237 −0.30854 1.132

0.07878 0.42383 1.13980 0.940 0.581 1.18384 1.10715 1.51324 0.4853

4.2742 0.8571 1.37065

0.250 0.458 1.91225 1.218

0.89881 4.66473 2.35918 0.4895

1.35135 1.236 − 1.74328 1.273 4.00512 1.656 0.505

0−33

0.4460 0.18182 0.6821 1.4646

0.215 0.0078 1.1405 3.09319

− 0.539 1.6084

16−25 36−157

0.2695 0.8340

2.080 0.10732

0.4446

24−120

0.8994

1.396

0.955

0−29

0.3640

1.237

0.414

11−40

0.3599

1.258

0.410

0−30 0.2835 2.580 0−30 0.5758 −0.432 − 9−106 0.9003 1.9595 0−33 −0.06427 8.5053

0.455 0.18186 0.721 1.608 0.353 1.090

0.387 0.558 − 0.44998 0.973 − 6.7900 0.207

*Smithsonian Tables, Table 269. For a detailed discussion of mercury data, see Cook, Brit. J. Appl. Phys., 7, 285 (1956). For data on nitrogen and argon, see Johnson (ed.), WADD-TR-60-56, 1960. Bromoform1 7.7 − 50°C. Vt = 0.34204[1 + 0.00090411(t − 7.7) + 0.0000006766(t − 7.7)2] 0.34204 in the specific volume of bromoform at 7.7°C. Glycerin2 −62 to 0°C. Vt = V0(1 + 4.83 × 10−4t − 0.49 × 10−6t2) 0 − 80°C. Vt = V0(1 + 4.83 × 10−4t + 0.49 × 10−6t2) 3 Mercury 0 − 300°C. Vt − V0[1 + 10−8(18153.8t + 0.7548t2 + 0.001533t2 + 0.00000536t4)] 1 Sherman and Sherman, J. Am. Chem. Soc., 50, 1119 (1928). (An obvious error in their equation has been corrected.) 2 Samsoen, Ann. phys., (10) 9, 91 (1928). 3 Harlow, Phil. Mag., (7) 7, 674 (1929).

2-131

Cubical Expansion of Solids*

If v2 and v1 are the volumes at t2 and t1, respectively, then v2 = v1(1 + C∆t), C being the coefficient of cubical expansion and ∆t the temperature interval. Where only a single temperature is stated, C represents the true coefficient of cubical expansion at that temperature. Substance

t or ∆t

C × 104

Antimony Beryl Bismuth Copper† Diamond Emerald Galena Glass, common tube hard Jena, borosilicate 59 III pure silica Gold Ice Iron Lead† Paraffin Platinum Porcelain, Berlin chloride nitrate sulfate Quartz Rock salt Rubber Silver Sodium Stearic acid Sulfur, native Tin Zinc†

0−100 0−100 0−100 0−100 40 40 0−100 0−100 0−100 20−100 0−80 0−100 −20 to −1 0−100 0−100 20 0−100 20 0−100 0−100 20 0−100 50−60 20 0−100 20 33.8−45.4 13.2−50.3 0−100 0−100

0.3167 0.0105 0.3948 0.4998 0.0354 0.0168 0.558 0.276 0.214 0.156 0.0129 0.4411 1.1250 0.3550 0.8399 5.88 0.265 0.0814 1.094 1.967 1.0754 0.3840 1.2120 4.87 0.5831 2.13 8.1 2.23 0.6889 0.8928

*Smithsonian Tables, Table 268. †See additional data below. Aluminum1 100 − 530°C. V = V0(1 + 2.16 × 10−5t + 0.95 × 10−8t2) 1 Cadmium 130 − 270°C. V = V0(1 + 8.04 × 10−5t + 5.9 × 10−8t2) 1 Copper 110 − 300°C. V = V0(1 + 1.62 × 10−5t + 0.20 × 10−8t2) Colophony2 0 − 34°C. V = V0(1 + 2.21 × 10−4t + 0.31 × 10−6t2) 34 − 150°C. V = V34[1 + 7.40 × 10−4(t − 34) + 5.91 × 10−6(t − 34)2] 1 Lead 100 − 280°C. V = V0(1 + 1.60 × 10−5t + 3.2 × 10−8t2) 2 Shellac 0 − 46°C. V = V0(1 + 2.73 × 10−4t + 0.39 × 10−6t2) 46 − 100°C. V = V46[1 + 13.10 × 10−4(t − 46) + 0.62 × 10−6(t − 46)2] Silica (vitreous)3 0 − 300°C. Vt = V0[1 + 10−8(93.6t + 0.7776t2 − 0.003315t2 + 0.000005244t4) Sugar (cane, amorphous)2 0 − 67°C. Vt = V0(1 + 2.34 × 10−4t + 0.14 × 10−6t2) 67 − 160°C. Vt = V67[1 + 5.02 × 10−4(t − 67) + 0.43 × 10−6(t − 67)2] Zinc1 120 − 360°C. Vt = V0(1 + 8.50 × 10−5t + 3.9 × 10−8t2) 1 2 3

Uffelmann, Phil. Mag., (7) 10, 633 (1930). Samsoen, Ann. phys., (10) 9, 83 (1928). Harlow, Phil. Mag., (7) 7, 674 (1929).

2-132

PHYSICAL AND CHEMICAL DATA

JOULE-THOMSON EFFECT UNITS CONVERSIONS For this subsection, the following units conversions are applicable: To convert the Joule-Thomson coefficient, µ, in degrees Celsius per atmosphere to degrees Fahrenheit per atmosphere, multiply by 1.8.

TABLE 2-149

°F = 9⁄ 5 °C + 32; °R = 9⁄ 5 K To convert bars to pounds-force per square inch, multiply by 14.504; to convert bars to kilopascals, multiply by 1 × 102.

Additional References Available for the Joule-Thomson Coefficient Temp. range, °C

Pressure range, atm Gas Air Ammonia Argon Benzene Butane Carbon dioxide Carbon monoxide Deuterium Dowtherm A Ethane Ethylene Helium Hydrogen

>200

<0

0–10

10–50

50–200

12, 15, 16 19, 35 28 39 31 26 7, 8, 28 37 17

12, 15, 19 35

15, 19, 35

19, 35

39 31 26 7, 8, 37

39 31

39

7, 8, 37

7, 8, 37

1,* 22, 24 25

17 1,* 22, 24, 25

46 45 1, 38 24, 30

17 22, 24, 25 1* 46 45 1, 38 22, 24, 25 30 6

6 33 13, 40

Methane Mixtures Natural gas Nitrogen

13, 28, 40

13, 40

Nitrous oxide Pentane Propane Steam

26, 34, 44 41 28, 29, 42

34 43 29, 42, 47

38 24, 30

34 42, 47

1, 38 22, 24, 25 30 33 13

33 13, 40

0–300 12, 15, 16 19, 35 28 39 31 26 7, 8, 9, 10 37 17 46 45 9, 10 38 24 6 9, 11 33 9, 10, 13 28, 40 9, 10 26, 34, 44 43 28, 29, 42 45

>300

Unclassified 3, 4, 18 2, 3

31

46 48

13

19

29, 42, 47

29, 47

*See also 14 (generalized chart); 18 (review, to 1919); 20–22; 23 (review, to 1948); 27 (review, to 1905); 32, 36, 41, 50. REFERENCES: 1. Baehr. Z. Elektrochem., 60, 515 (1956). 2. Beattie, J. Math. Phys., 9, 11 (1930). 3. Beattie, Phys. Rev., 35, 643 (1930). 4. Bradley and Hale, Phys. Rev., 29, 258 (1909). 5. Brown and Dean, Bur. Stand. J. Res., 60, 161 (1958). 6. Budenholzer, Sage, et al., Ind. Eng. Chem., 29, 658 (1937). 7. Burnett, Phys. Rev., 22, 590 (1923). 8. Burnett, Univ. Wisconsin Bull. 9(6), 1926. 9. Charnley, Ph.D. thesis. University of Manchester, 1952. 10. Charnley, Isles, et al., Proc. R. Soc. (London), A217, 133 (1953). 11. Charnley, Rowlinson, et al., Proc. R. Soc. (London), A230, 354 (1955). 12. Dalton, Commun. Phys. Lab. Univ. Leiden, no. 109c, 1909. 13. Deming and Deming, Phys. Rev., 48, 448 (1935). 14. Edmister, Pet. Refiner, 28, 128 (1949). 15. Eucken, Clusius, et al., Z. Tech. Phys., 13, 267 (1932). 16. Eumorfopoulos and Rai, Phil. Mag., 7, 961 (1926). 17. Huang, Lin, et al., Z. Phys., 100, 594 (1936). 18. Hoxton, Phys. Rev., 13, 438 (1919). 19. Ishkin and Kaganev, J. Tech. Phys. U.S.S.R., 26, 2323 (1956). 20. Isles, Ph.D. thesis, Leeds University. 21. Jenkin and Pye, Phil. Trans. R. Soc. (London), A213, 67 (1914); A215, 353 (1915). 22. Johnston, J. Am. Chem. Soc., 68, 2362 (1946). 23. Johnston, Trans. Am. Soc. Mech. Eng., 70, 651 (1948). 24. Johnston, Bezman, et al., J. Am. Chem. Soc., 68, 2367 (1946). 25. Johnston, Swanson, et al., J. Am. Chem. Soc., 68, 2373 (1946). 26. Kennedy, Sage, et al., Ind. Eng. Chem., 28, 718 (1936). 27. Kester, Phys. Rev., 21, 260 (1905). 28. Keyes and Collins, Proc. Nat. Acad. Sci., 18, 328 (1932). 29. Kleinschmidt, Mech. Eng., 45, 165 (1923); 48, 155 (1926). 30. Koeppe, Kältetechnik, 8, 275 (1956). 31. Lindsay and Brown, Ind. Eng. Chem., 27, 817 (1935). 32. Noell, dissertation, Munich, 1914, Forschungsdienst, 184, p. 1, 1916. 33. Palienko, Tr. Inst. Ispol’ z. Gaza, Akad. Nauk Ukr. SSR, no. 4, p. 87, 1956. 34. Pattee and Brown, Ind. Eng. Chem., 26, 511, (1934). 35. Roebuck, Proc. Am. Acad. Arts Sci., 60, 537 (1925); 64, 287 (1930). 36. Roebuck, see 49 below, 37. Roebuck and Murrell, Phys. Rev., 55, 240 (1939). 38. Roebuck and Osterberg, Phys. Rev., 37, 110 (1931); 43, 60 (1933). 39. Roebuck and Osterberg, Phys. Rev., 46, 785 (1934). 40. Roebuck and Osterberg, Phys. Rev., 48, 450 (1935). 41. Roebuck, Murrell, et al., J. Am. Chem. Soc., 64, 400 (1942). 42. Sage, unpublished data, California Institute of Technology, 1959. 43. Sage and Lacy, Ind. Eng. Chem., 27, 1484 (1934). 44. Sage, Kennedy, et al., Ind. Eng. Chem., 28, 601 (1936). 45. Sage, Webster, et al., Ind. Eng. Chem., 29, 658 (1937). 46. Ullock, Gaffert, et al., Trans. Am. Inst. Chem. Eng., 32, 73 (1936). 47. Yang, Ind. Eng. Chem., 45, 786 (1953). 48. Zelmanov, J. Phys. U.S.S.R., 3, 43 (1940). 49. Roebuck, recalculated data. 50. Michels et al., van der Waals laboratory publications. Gunn, Cheuh, and Prausnitz, Cryogenics, 6, 324 (1966), review equations relating the inversion temperatures and pressures. The ability of various equations of state to relate these was also discussed by Miller, Ind. Eng. Chem. Fundam., 9, 585 (1970); and Juris and Wenzel, Am. Inst. Chem. Eng. J., 18, 684 (1972). Perhaps the most detailed review is that of Hendricks, Peller, and Baron. NASA Tech. Note D 6807, 1972.

JOULE-THOMSON EFFECT

2-133

TABLE 2-150 Approximate Inversion-Curve Locus in Reduced Coordinates (Tr = T/Tc; Pr = P/Pc)* Pr

0

0.5

1

1.5

2

2.5

3

4

TrL TrU

0.782 4.984

0.800 4.916

0.818 4.847

0.838 4.777

0.859 4.706

0.880 4.633

0.903 4.550

0.953 4.401

Pr

5

6

7

8

9

10

11

11.79

TrL TrU

1.01 4.23

1.08 4.06

1.16 3.88

1.25 3.68

1.35 3.45

1.50 3.18

1.73 2.86

2.24 2.24

*Calculated from the best three-constant equation recommended by Miller, Ind. Eng. Chem. Fundam., 9, 585 (1970). TrL refers to the lower curve, and TrU, to the upper curve.

TABLE 2-151

Joule-Thomson Data for Air* t, °C

P, atm 1 20 60 100 140 180 200

−150

−100

−75

−50

−25

0

25

50

75

100

150

200

250

0.0450 .0185 − .0070 − .0255 − .0330

0.5895 .5700 .4820 .2775 .1360 .0655 .0440

0.4795 .4555 .3835 .2880 .1855 .1136 .0855

0.3910 .3690 .3195 .2505 .1825 .1270 .1065

0.3225 .3010 .2610 .2130 .1650 .1240 .1090

0.2745 .2580 .2200 .1820 .1450 .1100 .0950

0.2320 .2173 .1852 .1550 .1249 .0959

0.1956 .1830 .1571 .1310 .1070 .0829

0.1614 .1508 .1293 .1087 .0889 .0707

0.1355 .1258 .1062 .0884 .0726 .0580

0.0961 .0883 .0732 .0600 .0482 .0376

0.0645 .0580 .0453 .0343 .0250 .0174

0.0409 .0356 .0254 .0165 .0092 .0027

*Free of water and CO2. Extracted from Table 261, Smithsonian Physical Tables, 9th rev. ed., Washington, DC, 1954. These data are corrected from earlier publications. µ in °C/atm.

TABLE 2-152 P, bar TL, K TU, K

Approximate Inversion-Curve Locus for Air

0 (112)* 653

25

50

75

100

125

150

175

200

225

114 641

117 629

120 617

124 606

128 594

132 582

137 568

143 555

149 541

P, bar

250

275

300

325

350

375

400

425

432

TL, K TU, K

156 526

164 509

173 491

184 470

197 445

212 417

230 386

265 345

300 300

*Hypothetical low-pressure limit.

TABLE 2-153

Joule-Thomson Data for Argon* Pressure, atm

t, °C

1

−150 −125 −100 −75 −50

20

60

100

140

180

200

1.812 1.112 0.8605 .7100 .5960

1.102 0.8485 .6895 .5720

−0.0025 .1250 .6900 .5910 .4963

−0.0277 .0415 .2820 .4225 .3970

−0.0403 .0090 .1137 .2480 .2840

−0.0595 −.0100 .0560 .1537 .2037

−0.0640 − .0165 .0395 .1215 .1860

−25 0 25 50 75

.5045 .4307 .3720 .3220 .2695

.4805 .4080 .3490 .3015 .2557

.4210 .3600 .3077 .2650 .2285

.3460 .3010 .2628 .2297 .1993

.2763 .2505 .2213 .1947 .1710

.2140 .2050 .1890 .1700 .1505

.1950 .1883 .1745 .1580 .1415

100 125 150 200 250

.2413 .2105 .1845 .1377 .0980

.2277 .1980 .1720 .1280 .0910

.1975 .1707 .1485 .1102 .0785

.1715 .1480 .1285 .0950 .0665

.1490 .1300 .1123 .0823 .0555

.1320 .1153 .0998 .0715 .0485

.1255 .1100 .0945 .0675 .0468

300

.0643

.0607

.0530

.0445

.0370

.0370

.0276

*Extracted from Table 263, Smithsonian Physical Tables, 9th rev. ed., Washington, DC, 1954. These data are corrected from an earlier publication. µ in °C/atm.

2-134

PHYSICAL AND CHEMICAL DATA TABLE 2-154

Approximate Inversion-Curve Locus for Argon

P, bar

0

25

50

75

100

125

150

175

200

225

TL, K TU, K

94 765

97 755

101 744

105 736

109 726

113 716

118 705

123 694

128 683

134 671

P, bar

250

275

300

325

350

375

400

425

450

475

TL, K TU, K

141 657

148 643

158 627

170 610

183 591

201 569

222 544

248 515

288 478

375 375

TABLE 2-155

Joule-Thomson Data for Carbon Dioxide* Pressure, atm

t, °C

1

20

60

73

100

140

180

200

−75 −50 0 50 100

2.4130 1.2900 0.8950 .6490

−0.0200 − .0140 1.4020 .8950 .6375

−0.0200 − .0150 .0370 .8800 .6080

−0.0232 − .0165 .0310 .8225 .5920

−0.0228 − .0160 .0215 .5570 .5405

−0.0240 − .0183 .0115 .1720 .4320

−0.0250 − .0228 .0085 .1025 .3000

−0.0290 − .0248 .0045 .0930 .2555

125 150 200 250 300

.5600 .4890 .3770 .3075 .2650

.5450 .4695 .3575 .2885 .2425

.5160 .4430 .3400 .2625 .2080

.5068 .4380 .3325 .2565 .2002

.4750 .4155 .3150 .2420 .1872

.4130 .3760 .2890 .2235 .1700

.3230 .3102 .2600 .2045 .1540

.2915 .2910 .2455 .1975 .1505

*Extracted from Table 266, Smithsonian Physical Tables, 9th rev. ed., Washington, DC, 1954. These data are corrected from an earlier publication. µ in °C/atm.

TABLE 2-156

Approximate Inversion-Curve Locus for Carbon Dioxide*

P, bar

50

100

150

200

250

300

350

400

450

TL, K TU, K

243 1290

251 1261

258 1233

266 1205

272 1175

283 1146

293 1111

302 1076

312 1045

P, bar

500

550

600

650

700

750

800

850

884

TL, K TU, K

325 1015

338 983

351 950

365 914

383 878

403 840

441 796

496 739

608 608

*Interpolated from Vukalovich and Altunin’s interpolation of data of Price, Ind. Eng. Chem., 47, 1691 (1955). TL = lower inversion temperature, and TU = upper inversion temperature.

TABLE 2-157 P, bar TL, K TU, K

Approximate Inversion-Curve Locus for Deuterium

0 (31)* 216

25

50

75

100

125

150

175

194

34 202

38 189

43 178

49 168

56 157

65 146

77 131

108 108

*Hypothetical low-pressure limit.

TABLE 2-158 P, bar

0

TL, K

Approximate Inversion-Curve Locus for Ethane 25

50

75

100

125

150

175

200

225

249

255

262

269

275

282

290

297

306

P, bar

250

275

300

325

350

375

400

425

450

475

TL, K

315

325

335

345

357

370

383

398

415

432

P, bar

500

525

550

575

600

TL, K

453

477

505

545

626

JOULE-THOMSON EFFECT TABLE 2-159

2-135

Joule-Thomson Data for Helium*

T, K

160

180

200

220

240

260

280

300

µ

−0.0574

−0.0587

−0.0594

−0.0601

−0.0608

−0.0614

−0.0619

−0.0625

T, K

320

340

360

380

400

420

440

460

µ

−0.0629

−0.0634

−0.0637

−0.0640

−0.0643

−0.0645

−0.0645

−0.0643

T, K

480

500

520

540

560

580

600

µ

−0.0640

−0.0636

−0.0630

−0.0622

−0.0611

−0.0587

−0.0540

*Interpolated and converted from data in Table 262, Smithsonian Physical Tables, 9th rev. ed., Washington, DC, 1954. These data are corrected from those in an earlier publication. µ is in °C/atm. Below about 200 atm, little change in the coefficient with pressure occurs.

TABLE 2-160

Approximate Inversion-Curve Locus for Normal Hydrogen

P, bar

0

25

50

75

100

125

150

164

TL, K TU, K

(28)* 202

32 193

38 183

44 171

52 157

61 141

73 119

92 92

*Hypothetical low-pressure limit.

TABLE 2-161 P, bar

Approximate Inversion-Curve Locus for Methane

25

TL, K

50

75

100

125

150

175

200

225

250

275

300

161

166

172

176

182

189

195

202

209

217

225

P, bar

325

350

375

400

425

450

475

500

525

534

TL, K TU, K

234

243

254

265

277

292

309 505

331 474

365 437

400 400

TABLE 2-162

Joule-Thomson Data for Nitrogen* Pressure, atm

t, °C

1

20

33.5

60

100

140

180

200

−150 −125 −100 −75 −50

1.2659 0.8557 .6490 .5033 .3968

1.1246 0.7948 .5958 .4671 .3734

0.1704 .7025 .5494 .4318 .3467

0.0601 .4940 .4506 .3712 .3059

0.0202 .1314 .2754 .2682 .2332

−0.0056 .0498 .1373 .1735 .1676

−0.0211 .0167 .0765 .1026 .1120

−0.0284 .0032 .0587 .0800 .0906

−25 0 25 50 75

.3224 .2656 .2217 .1855 .1555

.3013 .2494 .2060 .1709 .1421

.2854 .2377 .1961 .1621 .1336

.2528 .2088 .1729 .1449 .1191

.2001 .1679 .1400 .1164 .0941

.1506 .1316 .1105 .0915 .0740

.1101 .1015 .0874 .0732 .0583

.0932 .0891 .0779 .0666 .0543

100 125 150 200 250

.1292 .1070 .0868 .0558 .0331

.1173 .0973 .0776 .0472 .0256

.1100 .0904 .0734 .0430 .0230

.0975 .0786 .0628 .0372 .0160

.0768 .0621 .0482 .0262 .0071

.0582 .0459 .0348 .0168 .0009

.0462 .0347 .0248 .0094 −.0037

.0419 .0326 .0228 .0070 −.0058

300

.0140

.0096

.0050

−.0013

−.0075

−.0129

−.0160

−.0171

*Extracted from Table 264, Smithsonian Physical Tables, 9th rev. ed., Washington, DC, 1954. These data are corrected from an earlier publication. µ in °C/atm.

TABLE 2-163

Approximate Inversion-Curve Locus for Propane

P, bar

0

25

50

75

100

125

150

175

200

225

250

275

TL, K

(296)*

303

311

318

327

336

345

355

365

374

389

403

P, bar

300

325

350

375

400

425

450

475

500

525

541

TL, K

418

435

452

473

495

521

551

586

628

686

780

*Hypothetical low-pressure limit.

2-136

PHYSICAL AND CHEMICAL DATA

CRITICAL CONSTANTS ADDITIONAL REFERENCES Other data and estimation techniques for the elements are contained in Gates and Thodos, Am. Inst. Chem. Eng. J., 6 (1960):50–54; and Ohse and von Tippelskirch, High Temperatures—High Pressures, 9 TABLE 2-164 Cmpd. no.

(1977):367–385. For inorganic substances see Mathews, Chem. Rev., 72 (1972):71–100; for organics see Kudchaker, Alani, and Zwolinski, Chem. Rev., 68 (1968):659–735; and for fluorocarbons see Advances in Fluorine Chemistry, App. B, Butterworth. Washington, 1963, pp. 173–175.

Critical Constants and Acentric Factors of Inorganic and Organic Compounds Name

Formula

CAS no.

Mol. wt.

Tc, K

Pc × 1E-06 Pa

Vc, m3/Kmol

Zc

Acentric factor

1 2 3 4 5 6 7 8 9 10

Methane Ethane Propane n-Butane n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane

CH4 C2H6 C3H8 C4H10 C5H12 C6H14 C7H16 C8H18 C9H20 C10H22

74828 74840 74986 106978 109660 110543 142825 111659 111842 124185

16.043 30.070 44.097 58.123 72.150 86.177 100.204 114.231 128.258 142.285

190.564 305.32 369.83 425.12 469.7 507.6 540.2 568.7 594.6 617.7

4.59 4.85 4.21 3.77 3.36 3.04 2.72 2.47 2.31 2.09

0.099 0.146 0.200 0.255 0.315 0.373 0.428 0.486 0.540 0.601

0.286 0.279 0.273 0.272 0.271 0.269 0.259 0.254 0.252 0.245

0.011 0.098 0.149 0.197 0.251 0.304 0.346 0.396 0.446 0.488

11 12 13 14 15 16 17 18 19 20

n-Undecane n-Dodecane n-Tridecane n-Tetradecane n-Pentadecane n-Hexadecane n-Heptadecane n-Octadecane n-Nonadecane n-Eicosane

C11H24 C12H26 C13H28 C14H30 C15H32 C16H34 C17H36 C18H38 C19H40 C20H42

1120214 112403 629505 629594 629629 544763 629787 593453 629925 112958

156.312 170.338 184.365 198.392 212.419 226.446 240.473 254.500 268.527 282.553

639 658 675 693 708 723 736 747 758 768

1.95 1.82 1.68 1.57 1.47 1.41 1.34 1.26 1.21 1.17

0.658 0.718 0.779 0.830 0.888 0.943 0.998 1.059 1.119 1.169

0.242 0.239 0.233 0.226 0.222 0.221 0.219 0.214 0.215 0.215

0.530 0.577 0.617 0.643 0.685 0.721 0.771 0.806 0.851 0.912

21 22 23 24 25 26 27

2-Methylpropane 2-Methylbutane 2,3-Dimethylbutane 2-Methylpentane 2,3-Dimethylpentane 2,3,3-Trimethylpentane 2,2,4-Trimethylpentane

C4H10 C5H12 C6H14 C6H14 C7H16 C8H18 C8H18

75285 78784 79298 107835 565593 560214 540841

58.123 72.150 86.177 86.177 100.204 114.231 114.231

408.14 460.43 499.98 497.5 537.35 573.5 543.96

3.62 3.37 3.13 3.02 2.88 2.81 2.56

0.261 0.304 0.358 0.366 0.396 0.455 0.465

0.278 0.268 0.269 0.267 0.255 0.268 0.264

0.177 0.226 0.246 0.279 0.292 0.289 0.301

28 29 30 31 32 33 34 35

Ethylene Propylene 1-Butene cis-2-Butene trans-2-Butene 1-Pentene 1-Hexene 1-Heptene

C2H4 C3H6 C4H8 C4H8 C4H8 C5H10 C6H12 C7H14

74851 115071 106989 590181 624646 109671 592416 592767

28.054 42.081 56.108 56.108 56.108 70.134 84.161 98.188

282.34 365.57 419.95 435.58 428.63 464.78 504.03 537.29

5.03 4.63 4.04 4.24 4.08 3.56 3.14 2.82

0.132 0.188 0.241 0.233 0.237 0.295 0.354 0.413

0.283 0.286 0.279 0.273 0.272 0.271 0.265 0.261

0.086 0.137 0.190 0.204 0.216 0.236 0.280 0.330

36 37 38 39 40 41 42 43 44

1-Octene 1-Nonene 1-Decene 2-Methylpropene 2-Methyl-1-butene 2-Methyl-2-butene 1,2-Butadiene 1,3-Butadiene 2-Methyl-1,3-butadiene

C8H16 C9H18 C10H20 C4H8 C5H10 C5H10 C4H6 C4H6 C5H8

111660 124118 872059 115117 563462 513359 590192 106990 78795

112.215 126.242 140.269 56.108 70.134 70.134 54.092 54.092 68.119

566.65 593.25 616.4 417.9 465 471 452 425.17 484

2.57 2.33 2.21 3.98 3.45 3.38 4.36 4.30 3.85

0.460 0.528 0.584 0.238 0.292 0.292 0.220 0.220 0.277

0.251 0.249 0.252 0.272 0.261 0.252 0.255 0.268 0.265

0.377 0.417 0.478 0.192 0.237 0.272 0.166 0.192 0.158

45 46 47 48 49 50

Acetylene Methylacetylene Dimethylacetylene 3-Methyl-1-butyne 1-Pentyne 2-Pentyne

C2H2 C3H4 C4H6 C5H8 C5H8 C5H8

74862 74997 503173 598232 627190 627214

26.038 40.065 54.092 68.119 68.119 68.119

308.32 402.39 473.2 463.2 481.2 519

6.15 5.62 4.87 4.20 4.17 4.02

0.113 0.164 0.221 0.275 0.277 0.276

0.271 0.276 0.274 0.300 0.289 0.257

0.188 0.216 0.239 0.308 0.290 0.174

51 52 53 54 55 56

1-Hexyne 2-Hexyne 3-Hexyne 1-Heptyne 1-Octyne Vinylacetylene

C6H10 C6H10 C6H10 C7H12 C8H14 C4H4

693027 764352 928494 628717 629050 689974

82.145 82.145 82.145 96.172 110.199 52.076

516.2 549 544 559 585 454

3.64 3.53 3.54 3.13 2.82 4.89

0.322 0.331 0.334 0.386 0.441 0.205

0.273 0.256 0.261 0.260 0.256 0.265

0.335 0.221 0.219 0.272 0.323 0.109

CRITICAL CONSTANTS TABLE 2-164

2-137

Critical Constants and Acentric Factors of Inorganic and Organic Compounds (Continued ) Mol. wt.

Tc, K

Pc × 1E-06 Pa

287923 96377 1640897 110827 108872 590669 1678917 142290 693890 110838

70.134 84.161 98.188 84.161 98.188 112.215 112.215 68.119 82.145 82.145

511.76 532.79 569.52 553.58 572.19 591.15 609.15 507 542 560.4

4.50 3.78 3.40 4.10 3.48 2.94 3.04 4.81 4.13 4.39

C6H6 C7H8 C8H10 C8H10 C8H10 C8H10 C9H12

71432 108883 95476 108383 106423 100414 103651

78.114 92.141 106.167 106.167 106.167 106.167 120.194

562.16 591.8 630.33 617.05 616.23 617.2 638.32

1,2,4-Trimethylbenzene Isopropylbenzene 1,3,5-Trimethylbenzene p-Isopropyltoluene Naphthalene Biphenyl Styrene m-Terphenyl

C9H12 C9H12 C9H12 C10H14 C10H8 C12H10 C8H8 C18H14

95636 98828 108678 99876 91203 92524 100425 92068

120.194 120.194 120.194 134.221 128.174 154.211 104.152 230.309

82 83 84 85 86 87 88

Methanol Ethanol 1-Propanol 1-Butanol 2-Butanol 2-Propanol 2-Methyl-2-propanol

CH4O C2H6O C3H8O C4H10O C4H10O C3H8O C4H10O

67561 64175 71238 71363 78922 67630 75650

89 90 91 92 93 94 95 96

1-Pentanol 2-Methyl-1-butanol 3-Methyl-1-butanol 1-Hexanol 1-Heptanol Cyclohexanol Ethylene glycol 1,2-Propylene glycol

C5H12O C5H12O C5H12O C6H14O C7H16O C6H12O C2H6O2 C3H8O2

97 98 99 100

Phenol o-Cresol m-Cresol p-Cresol

C6H6O C7H8O C7H8O C7H8O

101 102 103 104 105 106 107 108 109 110 111 112

Dimethyl ether Methyl ethyl ether Methyl n-propyl ether Methyl isopropyl ether Methyl n-butyl ether Methyl isobutyl ether Methyl tert-butyl ether Diethyl ether Ethyl propyl ether Ethyl isopropyl ether Methyl phenyl ether Diphenyl ether

113 114 115 116 117 118 119 120 121 122

Formaldehyde Acetaldehyde 1-Propanal 1-Butanal 1-Pentanal 1-Hexanal 1-Heptanal 1-Octanal 1-Nonanal 1-Decanal

Cmpd. no.

Name

Zc

Acentric factor

57 58 59 60 61 62 63 64 65 66

Cyclopentane Methylcyclopentane Ethylcyclopentane Cyclohexane Methylcyclohexane 1,1-Dimethylcyclohexane Ethylcyclohexane Cyclopentene 1-Methylcyclopentene Cyclohexene

C5H10 C6H12 C7H14 C6H12 C7H14 C8H16 C8H16 C5H8 C6H10 C6H10

0.257 0.319 0.374 0.308 0.368 0.450 0.430 0.245 0.303 0.291

0.272 0.272 0.269 0.274 0.269 0.269 0.258 0.279 0.278 0.274

0.196 0.230 0.271 0.212 0.236 0.233 0.246 0.196 0.232 0.216

67 68 69 70 71 72 73

Benzene Toluene o-Xylene m-Xylene p-Xylene Ethylbenzene Propylbenzene

4.88 4.10 3.74 3.53 3.50 3.60 3.20

0.261 0.314 0.374 0.377 0.381 0.375 0.440

0.273 0.262 0.267 0.259 0.260 0.263 0.265

0.209 0.262 0.311 0.325 0.320 0.301 0.344

74 75 76 77 78 79 80 81

649.13 631.1 637.36 653.15 748.35 789.26 636 924.85

3.25 3.18 3.11 2.80 3.99 3.86 3.82 3.53

0.430 0.429 0.433 0.497 0.413 0.502 0.352 0.768

0.259 0.260 0.254 0.256 0.265 0.295 0.254 0.352

0.380 0.322 0.397 0.366 0.296 0.367 0.295 0.561

32.042 46.069 60.096 74.123 74.123 60.096 74.123

512.64 513.92 536.78 563.05 536.05 508.3 506.21

8.14 6.12 5.12 4.34 4.20 4.79 3.99

0.117 0.168 0.220 0.276 0.270 0.221 0.276

0.224 0.240 0.252 0.256 0.254 0.250 0.262

0.566 0.643 0.617 0.585 0.574 0.670 0.613

71410 137326 123513 111273 111706 108930 107211 57556

88.150 88.150 88.150 102.177 116.203 100.161 62.068 76.095

586.15 565 577.2 611.35 631.9 650 719.7 626

3.87 3.87 3.90 3.46 3.18 4.25 7.71 6.04

0.327 0.327 0.327 0.381 0.435 0.322 0.191 0.239

0.260 0.270 0.266 0.259 0.263 0.253 0.246 0.277

0.592 0.678 0.586 0.572 0.592 0.371 0.487 1.102

108952 95487 108394 106445

94.113 108.140 108.140 108.140

694.25 697.55 705.85 704.65

6.06 5.06 4.52 5.15

0.229 0.282 0.312 0.277

0.240 0.246 0.240 0.244

0.438 0.438 0.444 0.507

C2H6O C3H8O C4H10O C4H10O C5H12O C5H12O C5H12O C4H10O C5H12O C5H12O C7H8O C12H10O

115106 540670 557175 598538 628284 625445 1634044 60297 628320 625547 100663 101848

46.069 60.096 74.123 74.123 88.150 88.150 88.150 74.123 88.150 88.150 108.140 170.211

400.1 437.8 476.3 464.5 510 497 497.1 466.7 500.23 489 645.6 766.8

5.27 4.47 3.77 3.89 3.31 3.41 3.41 3.64 3.37 3.41 4.27 3.10

0.171 0.221 0.276 0.276 0.329 0.331 0.329 0.281 0.336 0.329 0.337 0.503

0.271 0.271 0.263 0.278 0.257 0.273 0.272 0.264 0.273 0.276 0.268 0.244

0.192 0.229 0.264 0.280 0.335 0.310 0.264 0.281 0.347 0.306 0.353 0.441

CH2O C2H4O C3H6O C4H8O C5H10O C6H12O C7H14O C8H16O C9H18O C10H20O

50000 75070 123386 123728 110623 66251 111717 124130 124196 112312

30.026 44.053 58.080 72.107 86.134 100.161 114.188 128.214 142.241 156.268

408 466 504.4 537.2 566.1 591 617 638.1 658 674.2

6.59 5.57 4.92 4.32 3.97 3.46 3.18 2.97 2.74 2.60

0.115 0.154 0.204 0.258 0.313 0.369 0.421 0.474 0.527 0.580

0.223 0.221 0.239 0.250 0.264 0.260 0.261 0.265 0.264 0.269

0.282 0.292 0.256 0.278 0.347 0.387 0.427 0.474 0.514 0.582

Formula

CAS no.

Vc, m3/Kmol

2-138

PHYSICAL AND CHEMICAL DATA

TABLE 2-164

Critical Constants and Acentric Factors of Inorganic and Organic Compounds (Continued ) Mol. wt.

Tc, K

Pc × 1E-06 Pa

67641 78933 107879 563804 591786 108101 565617 96220 565695 565800 108941 98862

58.080 72.107 86.134 86.134 100.161 100.161 100.161 86.134 100.161 114.188 98.145 120.151

508.2 535.5 561.08 553 587.05 571.4 573 560.95 567 576 653 709.5

4.71 4.12 3.71 3.84 3.31 3.27 3.32 3.70 3.34 3.06 4.01 3.85

CH2O2 C2H4O2 C3H6O2 C4H8O2 C4H8O2 C7H6O2 C4H6O3

64186 64197 79094 107926 79312 65850 108247

46.026 60.053 74.079 88.106 88.106 122.123 102.090

588 591.95 600.81 615.7 605 751 606

Methyl formate Methyl acetate Methyl propionate Methyl n-butyrate Ethyl formate Ethyl acetate Ethyl propionate

C2H4O2 C3H6O2 C4H8O2 C5H10O2 C3H6O2 C4H8O2 C5H10O2

107313 79209 554121 623427 109944 141786 105373

60.053 74.079 88.106 102.133 74.079 88.106 102.133

149 150 151 152 153 154 155

Ethyl n-butyrate n-Propyl formate n-Propyl acetate n-Butyl acetate Methyl benzoate Ethyl benzoate Vinyl acetate

C6H12O2 C4H8O2 C5H10O2 C6H12O2 C8H8O2 C9H10O2 C4H6O2

105544 110747 109604 123864 93583 93890 108054

156 157 158 159 160 161

Methylamine Dimethylamine Trimethylamine Ethylamine Diethylamine Triethylamine

CH5N C2H7N C3H9N C2H7N C4H11N C6H15N

162 163 164 165 166 167 168

n-Propylamine di-n-Propylamine Isopropylamine Diisopropylamine Aniline N-Methylaniline N,N-Dimethylaniline

169 170 171 172

Cmpd. no.

Name

Formula

Zc

Acentric factor

123 124 125 126 127 128 129 130 131 132 133 134

Acetone Methyl ethyl ketone 2-Pentanone Methyl isopropyl ketone 2-Hexanone Methyl isobutyl ketone 3-Methyl-2-pentanone 3-Pentanone Ethyl isopropyl ketone Diisopropyl ketone Cyclohexanone Methyl phenyl ketone

C3H6O C4H8O C5H10O C5H10O C6H12O C6H12O C6H12O C5H10O C6H12O C7H14O C6H10O C8H8O

0.210 0.267 0.301 0.313 0.369 0.369 0.371 0.336 0.369 0.416 0.311 0.386

0.234 0.247 0.239 0.261 0.250 0.254 0.259 0.267 0.262 0.266 0.230 0.252

0.307 0.320 0.345 0.349 0.395 0.389 0.386 0.340 0.394 0.411 0.308 0.365

135 136 137 138 139 140 141

Formic acid Acetic acid Propionic acid n-Butyric acid Isobutyric acid Benzoic acid Acetic anhydride

5.81 5.74 4.61 4.07 3.68 4.47 3.97

0.125 0.179 0.232 0.291 0.291 0.347 0.290

0.148 0.208 0.214 0.231 0.213 0.248 0.229

0.317 0.463 0.574 0.682 0.612 0.603 0.450

142 143 144 145 146 147 148

487.2 506.55 530.6 554.5 508.4 523.3 546

5.98 4.69 4.03 3.48 4.71 3.85 3.34

0.173 0.229 0.284 0.340 0.231 0.287 0.345

0.255 0.256 0.259 0.257 0.257 0.254 0.254

0.254 0.326 0.349 0.378 0.282 0.363 0.391

116.160 88.106 102.133 116.160 136.150 150.177 86.090

571 538 549.73 579.15 693 698 519.13

2.94 4.03 3.37 3.11 3.59 3.22 3.93

0.403 0.286 0.349 0.389 0.436 0.489 0.270

0.249 0.257 0.257 0.251 0.272 0.271 0.246

0.399 0.310 0.390 0.410 0.421 0.477 0.348

74895 124403 75503 75047 109897 121448

31.057 45.084 59.111 45.084 73.138 101.192

430.05 437.2 433.25 456.15 496.6 535.15

7.41 5.26 4.10 5.59 3.67 3.04

0.154 0.180 0.254 0.202 0.301 0.389

0.319 0.260 0.289 0.298 0.268 0.266

0.279 0.293 0.210 0.283 0.300 0.316

C3H9N C6H15N C3H9N C6H15N C6H7N C7H9N C8H11N

107108 142847 75310 108189 62533 100618 121697

59.111 101.192 59.111 101.192 93.128 107.155 121.182

496.95 550 471.85 523.1 699 701.55 687.15

4.74 3.11 4.54 3.20 5.35 5.19 3.63

0.260 0.401 0.221 0.417 0.270 0.373 0.465

0.298 0.273 0.256 0.307 0.248 0.332 0.295

0.280 0.446 0.276 0.388 0.381 0.480 0.403

Ethylene oxide Furan Thiophene Pyridine

C2H4O C4H4O C4H4S C5H5N

75218 110009 110021 110861

44.053 68.075 84.142 79.101

469.15 490.15 579.35 619.95

7.26 5.55 5.71 5.64

0.142 0.218 0.219 0.254

0.264 0.297 0.260 0.278

0.201 0.205 0.195 0.239

173 174 175 176 177 178 179 180

Formamide N,N-Dimethylformamide Acetamide N-Methylacetamide Acetonitrile Propionitrile n-Butyronitrile Benzonitrile

CH3NO C3H7NO C2H5NO C3H7NO C2H3N C3H5N C4H7N C7H5N

75127 68122 60355 79163 75058 107120 109740 100470

45.041 73.095 59.068 73.095 41.053 55.079 69.106 103.123

771 649.6 761 718 545.5 564.4 582.25 699.35

7.75 4.37 6.57 5.00 4.85 4.19 3.79 4.21

0.163 0.262 0.215 0.267 0.173 0.229 0.278 0.339

0.197 0.212 0.223 0.224 0.185 0.205 0.217 0.245

0.410 0.312 0.419 0.437 0.340 0.325 0.371 0.352

181 182 183 184 185 186 187 188 189

Methyl mercaptan Ethyl mercaptan n-Propyl mercaptan n-Butyl mercaptan Isobutyl mercaptan sec-Butyl mercaptan Dimethyl sulfide Methyl ethyl sulfide Diethyl sulfide

CH4S C2H6S C3H8S C4H10S C4H10S C4H10S C2H6S C3H8S C4H10S

74931 75081 107039 109795 513440 513531 75183 624895 352932

48.109 62.136 76.163 90.189 90.189 90.189 62.136 76.163 90.189

469.95 499.15 536.6 570.1 559 554 503.04 533 557.15

7.23 5.49 4.63 3.97 4.06 4.06 5.53 4.26 3.96

0.145 0.206 0.254 0.307 0.307 0.307 0.200 0.254 0.320

0.268 0.273 0.263 0.257 0.268 0.271 0.264 0.244 0.273

0.158 0.188 0.232 0.272 0.253 0.251 0.194 0.209 0.294

CAS no.

Vc, m3/Kmol

CRITICAL CONSTANTS TABLE 2-164 Cmpd. no.

2-139

Critical Constants and Acentric Factors of Inorganic and Organic Compounds (Concluded ) Name

Formula

CAS no.

Mol. wt.

Tc, K

Pc × 1E-06 Pa

Vc, m3/Kmol

Zc

Acentric factor

190 191 192 193 194 195 196 197

Fluoromethane Chloromethane Trichloromethane Tetrachloromethane Bromomethane Fluoroethane Chloroethane Bromoethane

CH3F CH3Cl CHCl3 CCl4 CH3Br C2H5F C2H5Cl C2H5Br

593533 74873 67663 56235 74839 353366 75003 74964

34.033 50.488 119.377 153.822 94.939 48.060 64.514 108.966

317.42 416.25 536.4 556.35 467 375.31 460.35 503.8

5.88 6.69 5.55 4.54 8.00 5.01 5.46 6.29

0.113 0.142 0.238 0.274 0.156 0.164 0.155 0.215

0.252 0.275 0.296 0.270 0.321 0.263 0.221 0.323

0.198 0.154 0.228 0.191 0.192 0.218 0.206 0.259

198 199 200 201 202 203 204 205

1-Chloropropane 2-Chloropropane 1,1-Dichloropropane 1,2-Dichloropropane Vinyl chloride Fluorobenzene Chlorobenzene Bromobenzene

C3H7Cl C3H7Cl C3H6Cl2 C3H6Cl2 C2H3Cl C6H5F C6H5Cl C6H5Br

540545 75296 78999 78875 75014 462066 108907 108861

78.541 78.541 112.986 112.986 62.499 96.104 112.558 157.010

503.15 489 560 572 432 560.09 632.35 670.15

4.58 4.51 4.24 4.23 5.75 4.54 4.53 4.52

0.247 0.247 0.292 0.291 0.179 0.269 0.308 0.324

0.270 0.274 0.266 0.259 0.287 0.262 0.265 0.263

0.228 0.196 0.253 0.256 0.106 0.247 0.251 0.251

206 207 208 209 210 211 212 213 214 215 216 217 218

Air Hydrogen Helium-4 Neon Argon Fluorine Chlorine Bromine Oxygen Nitrogen Ammonia Hydrazine Nitrous oxide

H2 He Ne Ar F2 Cl2 Br2 O2 N2 NH3 N2H4 N2O

132259100 1333740 7440597 7440019 7440371 7782414 7782505 7726956 7782447 7727379 7664417 302012 10024972

28.951 2.016 4.003 20.180 39.948 37.997 70.905 159.808 31.999 28.014 17.031 32.045 44.013

132.45 33.19 5.2 44.4 150.86 144.12 417.15 584.15 154.58 126.2 405.65 653.15 309.57

3.79 1.32 0.23 2.67 4.90 5.17 7.79 10.28 5.02 3.39 11.30 14.73 7.28

0.092 0.064 0.058 0.042 0.075 0.067 0.124 0.135 0.074 0.089 0.072 0.158 0.098

0.318 0.307 0.305 0.300 0.292 0.287 0.279 0.286 0.287 0.288 0.241 0.429 0.277

0.000 −0.215 −0.388 −0.038 0.000 0.053 0.073 0.128 0.020 0.037 0.253 0.315 0.143

219 220 221 222 223 224 225 226 227 228 229 230 231

Nitric oxide Cyanogen Carbon monoxide Carbon dioxide Carbon disulfide Hydrogen fluoride Hydrogen chloride Hydrogen bromide Hydrogen cyanide Hydrogen sulfide Sulfur dioxide Sulfur trioxide Water

NO C2N2 CO CO2 CS2 HF HCl HBr HCN H2S SO2 SO3 H2O

10102439 460195 630080 124389 75150 7664393 7647010 10035106 74908 7783064 7446095 7446119 7732185

30.006 52.036 28.010 44.010 76.143 20.006 36.461 80.912 27.026 34.082 64.065 80.064 18.015

180.15 400.15 132.92 304.21 552 461.15 324.65 363.15 456.65 373.53 430.75 490.85 647.13

6.52 5.94 3.49 7.39 8.04 6.49 8.36 8.46 5.35 9.00 7.86 8.19 21.94

0.058 0.195 0.095 0.095 0.160 0.069 0.082 0.100 0.139 0.099 0.123 0.127 0.056

0.252 0.348 0.300 0.277 0.280 0.117 0.253 0.280 0.195 0.287 0.269 0.255 0.228

0.585 0.276 0.048 0.224 0.118 0.383 0.134 0.069 0.407 0.096 0.244 0.423 0.343

All substances are listed in alphabetical order in Table 2-6a. Compiled from Daubert, T. E., R. P. Danner, H. M. Sibul, and C. C. Stebbins, DIPPR Data Compilation of Pure Compound Properties, Project 801 Sponsor Release, July, 1993, Design Institute for Physical Property Data, AIChE, New York, NY; and from Ambrose, D. “Vapour-Liquid Critical Properties”, Report Chem 107, National Physical Laboratory, Teddington, UK, October, 1979. In order to ensure thermodynamic consistency, in almost all cases these properties are calculated from Tc and the vapor pressure and liquid density correlation coefficients listed in those tables. This means that there will be slight differences between the values listed here and those in the DIPPR tables. Most of the differences are less than 1%, and almost all the rest are less than the estimated accuracy of the quantity in question. The atomic weights used, taken from J. Phys. Chem. Ref. Data 22(6), 1993, are C = 12.011, H = 1.00794, O = 15.9994, N = 14.00674, S = 32.066, F = 18.9984, Cl = 35.4527, Br = 79.904, and I = 126.90447. The value of the gas constant, R, used here is 8314.51 J/(kmol·K), as given by E. R. Cohen and B. N. Taylor in J. Phys. Chem. Ref. Data 17, 1988. K − 273.15 = °C; 1.8 × K − 459.67 = °F; Pa × 9.869233E-06 = atm; Pa × 1.450377E-04 = psia j; m3/kmol × (1E + 03/mol. wt.) = cm3/g; m3/kmol × (1.601846E + 01/mol wt) = ft3/lb.

2-140

PHYSICAL AND CHEMICAL DATA

COMPRESSIBILITIES reminded that compressibilities can be calculated from the pressure—volume (or density)—temperature tables of the subsection “Thermodynamic Properties.”

INTRODUCTION The increasing ranges of pressure and temperature of interest to technology for an ever-increasing number of substances would necessitate additional tables in this subsection as well as in the subsection “Thermodynamic Properties.” Space restrictions preclude this. Hence, in the present revision, an attempt was made to update the fluidcompressibility tables for selected fluids and to omit tables for other fluids. The reader is thus referred to the fourth edition for tables on miscellaneous gases at 0°C, acetylene, ammonia, ethane, ethylene, hydrogen-nitrogen mixtures, and methyl chloride. The reader is also

TABLE 2-165

UNITS CONVERSIONS For this subsection, the following units conversions are applicable: °R = 9⁄5 K. To convert bars to pounds-force per cubic inch, multiply by 14.504. To convert bars to kilopascals, multiply by 1 × 102.

Compressibility Factors for Air* Pressure, bar

Temp., K

1

5

10

20

40

60

80

100

150

200

250

300

400

500

75 80 90 100 120

0.0052 0.9764 0.9797 0.9880

0.0260 0.0250 0.0236 0.8872 0.9373

0.0519 0.0499 0.0471 0.0453 0.8660

0.1036 0.0995 0.0940 0.0900 0.6730

0.2063 0.1981 0.1866 0.1782 0.1778

0.3082 0.2958 0.2781 0.2635 0.2557

0.4094 0.3927 0.3686 0.3498 0.3371

0.5099 0.4887 0.4581 0.4337 0.4132

0.7581 0.7258 0.6779 0.6386 0.5964

1.0025 0.9588 0.8929 0.8377 0.7720

1.1931 1.1098 1.0395 0.9530

1.4139 1.3110 1.2227 1.1076

1.7161 1.5937 1.5091

2.1105 1.9536 1.7366

140 160 180 200 250

0.9927 0.9951 0.9967 0.9978 0.9992

0.9614 0.9748 0.9832 0.9886 0.9957

0.9205 0.9489 0.9660 0.9767 0.9911

0.8297 0.8954 0.9314 0.9539 0.9822

0.5856 0.7803 0.8625 0.9100 0.9671

0.3313 0.6603 0.7977 0.8701 0.9549

0.3737 0.5696 0.7432 0.8374 0.9463

0.4340 0.5489 0.7084 0.8142 0.9411

0.5909 0.6340 0.7180 0.8061 0.9450

0.7699 0.7564 0.7986 0.8549 0.9713

0.9114 0.8840 0.9000 0.9311 1.0152

1.0393 1.0105 1.0068 1.0185 1.0702

1.3202 1.2585 1.2232 1.2054 1.1990

1.5903 1.4970 1.4361 1.3944 1.3392

300 350 400 450 500

0.9999 1.0000 1.0002 1.0003 1.0003

0.9987 1.0002 1.0012 1.0016 1.0020

0.9974 1.0004 1.0025 1.0034 1.0034

0.9950 1.0014 1.0046 1.0063 1.0074

0.9917 1.0038 1.0100 1.0133 1.0151

0.9901 1.0075 1.0159 1.0210 1.0234

0.9903 1.0121 1.0229 1.0287 1.0323

0.9930 1.0183 1.0312 1.0374 1.0410

1.0074 1.0377 1.0533 1.0614 1.0650

1.0326 1.0635 1.0795 1.0913 1.0913

1.0669 1.0947 1.1087 1.1183 1.1183

1.1089 1.1303 1.1411 1.1463 1.1463

1.2073 1.2116 1.2117 1.2090 1.2051

1.3163 1.3015 1.2890 1.2778 1.2667

600 800 1000

1.0004 1.0004 1.0004

1.0022 1.0020 1.0018

1.0039 1.0038 1.0037

1.0081 1.0077 1.0068

1.0164 1.0157 1.0142

1.0253 1.0240 1.0215

1.0340 1.0321 1.0290

1.0434 1.0408 1.0365

1.0678 1.0621 1.0556

1.0920 1.0844 1.0744

1.1172 1.1061 1.0948

1.1427 1.1283 1.1131

1.1947 1.1720 1.1515

1.2475 1.2150 1.1889

*Calculated from values of pressure, volume (or density), and temperature in Vasserman, Kazavchinskii, and Rabinovich, Thermophysical Properties of Air and Air Components, Moscow, Nauka, 1966, and NBS-NSF Trans. TT 70-50095, 1971; and Vasserman and Rabinovich, Thermophysical Properties of Liquid Air and Its Components, Moscow, 1968, and NBS-NSF Trans. 69-55092, 1970.

TABLE 2-166

Compressibility Factors for Argon*

Temp., K

1

5

10

20

40

60

80

100

200

300

400

500

100 150 200 250 300

0.9773 0.9932 0.9972 0.9988 0.9995

0.0183 0.9647 0.9857 0.9935 0.9969

0.0366 0.9273 0.9713 0.9869 0.9941

0.0729 0.8447 0.9419 0.9741 0.9884

0.1449 0.6101 0.8810 0.9494 0.9777

0.2162 0.2249 0.8208 0.9263 0.9686

0.2867 0.2781 0.7624 0.9056 0.9611

0.3567 0.3324 0.7121 0.8877 0.9552

0.6975 0.5934 0.6870 0.8590 0.9533

1.0267 0.8387 0.8360 0.9207 0.9950

1.3470 1.0732 1.0051 1.0262 1.0673

1.6932 1.2995 1.1982 1.1479 1.1786

400 500 600 800 1000

1.0001 1.0002 1.0003 1.0003 1.0002

0.9997 1.0007 1.0012 1.0012 1.0013

0.9998 1.0012 1.0025 1.0023 1.0022

0.9999 1.0034 1.0046 1.0050 1.0050

1.0004 1.0071 1.0094 1.0102 1.0096

1.0018 1.0113 1.0143 1.0151 1.0142

1.0031 1.0154 1.0198 1.0205 1.0193

1.0056 1.0205 1.0250 1.0258 1.0239

1.0280 1.0501 1.0553 1.0532 1.0484

1.0656 1.0874 1.0904 1.0830 1.0736

1.1157 1.1301 1.1291 1.1147 1.0999

1.1976 1.1997 1.1933 1.1707 1.1497

Pressure, bar

*Calculated from PVT values tabulated in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standard Press, Moscow, 1976. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.).

COMPRESSIBILITIES TABLE 2-167

2-141

Compressibility Factors for Carbon Dioxide* Pressure, bar

Temp., °C

1

5

10

20

40

60

80

100

200

300

400

500

0 50 100 150 200

0.9933 0.9964 0.9977 0.9985 0.9991

0.9658 0.9805 0.9883 0.9927 0.9953

0.9294 0.9607 0.9764 0.9853 0.9908

0.8496 0.9195 0.9524 0.9705 0.9818

0.8300 0.9034 0.9416 0.9640

0.7264 0.8533 0.9131 0.9473

0.5981 0.8022 0.8854 0.9313

0.4239 0.7514 0.8590 0.9170

0.5891 0.7651 0.8649

0.6420 0.7623 0.8619

0.8235 0.8995

0.9098 0.9621

250 300 350 400 450

0.9994 0.9996 0.9998 0.9999 1.0000

0.9971 0.9982 0.9991 0.9997 1.0000

0.9943 0.9967 0.9983 0.9994 1.0003

0.9886 0.9936 0.9964 0.9989 1.0005

0.9783 0.9875 0.9938 0.9982 1.0013

0.9684 0.9822 0.9914 0.9979 1.0023

0.9593 0.9773 0.9896 0.9979 1.0038

0.9511 0.9733 0.9882 0.9984 1.0056

0.9253 0.9640 0.9895 1.0073 1.0070

0.9294 0.9746 1.0053 1.0266 1.0412

0.9508 1.0030 1.0340 1.0559 1.0709

1.0096 1.0464 1.0734 1.0928 1.1067

500 600 700 800 900

1.0000 1.0000 1.0003 1.0002 1.0002

1.0004 1.0007 1.0010 1.0009 1.0009

1.0008 1.0013 1.0017 1.0019 1.0020

1.0015 1.0030 1.0036 1.0040 1.0041

1.0035 1.0062 1.0073 1.0082 1.0083

1.0056 1.0093 1.0161 1.0122 1.0128

1.0079 1.0129 1.0155 1.0168 1.0171

1.0107 1.0168 1.0198 1.0212 1.0221

1.0282 1.0386 1.0436 1.0458 1.0463

1.0522 1.0648 1.0707 1.0731 1.0726

1.0820 1.0948 1.1000 1.1016 1.1012

1.1165 1.1277 1.1318 1.1324 1.1303

1000

1.0002

1.0009

1.0021

1.0042

1.0084

1.0128

1.0172

1.0218

1.0460

1.0725

1.0725

1.1274

*Calculated from density-pressure-temperature data in Vukalovitch and Altunin, Thermophysical Properties of Carbon Dioxide, Atomizdat, Moscow, 1965, and Collet’s, London, 1968, translation.

TABLE 2-168

Compressibility Factors for Carbon Monoxide*

Temp., K

1

4

7

10

40

70

100

200 250 300 350 400

0.9973 0.9989 0.9997 1.0000 1.0002

0.9893 0.9957 0.9987 1.0002 1.0010

0.9813 0.9926 0.9977 1.0003 1.0017

0.9734 0.9896 0.9968 1.0005 1.0025

0.9632 0.9907 1.0042 1.0042

0.9896 1.0112 1.0112

0.9935 1.0216 1.0216

450 500 600 700 800

1.0003 1.0004 1.0005 1.0005 1.0004

1.0014 1.0016 1.0018 1.0018 1.0017

1.0025 1.0029 1.0032 1.0032 1.0030

1.0035 1.0041 1.0045 1.0045 1.0044

1.0152 1.0172 1.0186 1.0183 1.0175

1.0285 1.0314 1.0332 1.0325 1.0309

1.0433 1.0469 1.0485 1.0470 1.0445

900 1000 1500 2000 2500

1.0004 1.0004 1.0003 1.0002 1.0002

1.0017 1.0016 1.0012 1.0009 1.0007

1.0029 1.0027 1.0021 1.0016 1.0013

1.0041 1.0039 1.0029 1.0022 1.0018

1.0166 1.0156 1.0115 1.0088 1.0071

1.0291 1.0273 1.0200 1.0155 1.0124

1.0418 1.0391 1.0286 1.0221 1.0178

3000

1.0002

1.0006

1.0010

1.0015

1.0059

1.0104

1.0148

Pressure, atm

*From Hilsenrath et al., N.B.S. Circ. 564, 1955. Some of the above values have been rounded to four decimal places. Values at 10-K increments below 1000 K and at 50 K increments for higher temperatures appear in the original, also for pressures below atmospheric.

TABLE 2-169

Compressibility Factors for Ethanol

Temp., K

0.1

0.5

1.013

10

20

50

100

250

500

300 350 400 450 500

0.0022

0.0023

0.0024

0.999 1.000 1.000

0.993 0.997 0.997

0.986 0.991 0.994

0.0229 0.0215 0.0204 0.908 0.941

0.0458 0.0411 0.0408 0.874

0.114 0.107 0.101 0.101 0.122

0.228 0.208 0.201 0.198 0.214

0.565 0.509 0.490 0.472 0.473

1.11 1.03 0.95 0.898 0.868

600 700 800 900 1000

1.000 1.000 1.000 1.000 1.000

0.998 0.999 1.000 1.000 1.000

0.997 0.999 0.999 1.000 1.000

0.972 0.985 0.992 0.996 0.998

0.943 0.971 0.984 0.992 0.997

0.948 0.973 0.988 0.993

0.672 0.902 0.953 0.981 0.990

0.470 0.760 0.890 0.962 1.002

0.868 0.921 0.988 1.04 1.08

Pressure, bar

Rounded and interpolated from Thermodynamics Research Center tables, Texas A&M University.

2-142

PHYSICAL AND CHEMICAL DATA TABLE 2-170

Compressibility Factors for Ethylene Temperature, K

Pressure, bar

110

150

200

250

300

350

400

450

500

1 5 10 15 20

0.0047 0.0237 0.0472 0.0710 0.0946

0.0038 0.0189 0.0378 0.0566 0.0754

0.9808 0.0162 0.0323 0.0484 0.0644

0.9902 0.9495 0.8946 0.8320 0.7578

0.9944 0.9717 0.9425 0.9121 0.8804

0.9966 0.9828 0.9659 0.9479 0.9299

0.9979 0.9894 0.9785 0.9679 0.9574

0.9986 0.9935 0.9867 0.9749 0.9734

0.9991 0.9959 0.9919 0.9876 0.9833

30 40 60 80 100

0.1418 0.1889 0.2831 0.3767 0.4702

0.1129 0.1504 0.2251 0.2994 0.3734

0.0963 0.1280 0.1910 0.2533 0.3150

0.0950 0.1251 0.1838 0.2410 0.2968

0.8122 0.7342 0.5235 0.3302 0.3480

0.8936 0.8560 0.7791 0.7023 0.6359

0.9357 0.9144 0.8730 0.9056 0.9220

0.9603 0.9477 0.9231 0.9009 0.8825

0.9754 0.9677 0.9541 0.9428 0.9321

150 200 250 300 400

0.7030 0.9337 1.1636 1.3917 1.8441

0.5567 0.7382 0.9179 1.0960 1.4475

0.4671 0.6161 0.7630 0.9075 1.1910

0.4324 0.5630 0.6904 0.8148 1.0565

0.4528 0.5641 0.6740 0.7816 0.9909

0.5842 0.6347 0.7110 0.7969 0.9726

0.7483 0.7499 0.7895 0.8479 0.9849

0.8523 0.8494 0.8710 0.9095 1.0142

0.9167 0.9184 0.9343 0.9631 1.0450

500

solid

1.7934

1.4679

1.2908

1.1932

1.1468

1.1304

1.1341

1.1436

Calculated from Jacobsen, R.T., M. Jahangiri, et al., Ethylene, Blackwell Sci. Publs., Oxford, 1988 (299 pp.).

TABLE 2-171

Compressibility Factors for Normal Hydrogen* Pressure, bar

Temp., K

1

10

20

40

60

80

100

200

400

600

800

1000

20 40 60 80 100

0.0169 0.9848 0.9955 0.9986 0.9998

0.1680 0.8340 0.9562 0.9776 0.9979

0.3302 0.6311 0.9169 0.9763 0.9976

0.6430 0.5240 0.8608 0.9655 1.0022

0.9434 0.6627 0.8498 0.9676 1.0133

1.2346 0.8118 0.8832 0.9842 1.0280

1.5166 0.9590 0.9432 1.0138 1.0528

2.844 1.650 1.347 1.257 1.225

2.878 2.158 1.834 1.659

3.993 2.902 2.389 2.095

5.034 3.598 2.907 2.512

6.019 4.263 3.404 2.902

200 300 400 500 600

1.0007 1.0005 1.0004 1.0004 1.0003

1.0066 1.0059 1.0048 1.0040 1.0034

1.0134 1.0117 1.0096 1.0080 1.0068

1.0275 1.0236 1.0192 1.0160 1.0136

1.0422 1.0357 1.0289 1.0240 1.0204

1.0575 1.0479 1.0386 1.0320 1.0272

1.0734 1.0603 1.0484 1.0400 1.0340

1.163 1.124 1.098 1.080 1.068

1.355 1.253 1.196 1.159 1.133

1.555 1.383 1.293 1.236 1.197

1.753 1.510 1.388 1.311 1.259

1.936 1.636 1.481 1.385 1.320

800 1000 2000

1.0002 1.0002 1.0009

1.0026 1.0021 1.0013

1.0052 1.0042 1.0023

1.0104 1.0084 1.0044

1.0156 1.0126 1.0065

1.0208 1.0168 1.0086

1.0259 1.0209 1.0107

1.051 1.041 1.021

1.100 1.080 1.040

1.147 1.117 1.057

1.193 1.153 1.073

1.237 1.187 1.088

*Calculated from PVT tables of McCarty, Hord, and Roder, NBS Monogr. 168, 1981.

TABLE 2-172

Compressibility Factors for KLEA 60

Temp., K

1

Pressure, bar 5

10

15

20

25

30

250 260 270 280 290

0.9687 0.9780 0.9803 0.9824 0.9848

0.9099 0.9199

300 310 320 330 340

0.9867 0.9872 0.9884 0.9894 0.9905

0.9284 0.9359 0.9425 0.9484 0.9537

0.8459 0.8637 0.8790 0.8908 0.9026

0.7800 0.8066 0.8299 0.8488

0.7577 0.7888

0.6700 0.7184

0.6305

350

0.9920

0.9582

0.9139

0.8663

0.8145

0.7570

0.6908

Zsat Tsat

0.9712 234.0

0.9022 273.1

0.8361 295.0

0.7777 309.5

0.7224 320.7

0.6677 329.8

0.6118 337.6

Zsat

Psat

0.9494 0.9315 0.9098 0.8839 0.8538

2.08 3.11 4.49 6.30 8.62

0.8175 0.7756 0.7261 0.6666

11.55 15.19 19.66 25.10

Converted and interpolated from “Thermodynamic Properties of KLEA 60,” British units, © ICI Chemicals and Polymers, 1993 (20 pp.). Reproduced by permission. KLEA 60 is R32/125/134a (20/40/40 wt %).

COMPRESSIBILITIES TABLE 2-173

Compressibility Factors for KLEA 61

Temp., K

1

2-143

Pressure, bar 5

10

15

20

25

30

250 260 270 280 290

0.9746 0.9773 0.9798 0.9787 0.9838

0.9067 0.9185

300 310 320 330 340

0.9854 0.9868 0.9881 0.9892 0.9903

0.9270 0.9348 0.9416 0.9481 0.9529

0.8431 0.8615 0.8772 0.8909 0.9027

0.7755 0.8042 0.8280 0.8484

0.7148 0.7518 0.7934

0.6659 0.7174

0.6312

350

0.9917

0.9577

0.9131

0.8653

0.8134

0.7565

0.6916

Zsat Tsat

0.9686 230.0

0.8944 269.0

0.8237 290.9

0.7602 305.5

0.7003 316.7

0.6399 325.9

0.5780 333.7

Zsat

Psat

0.9381 0.9172 0.8920 0.8622 0.8272

2.46 3.63 5.18 7.19 9.75

0.7868 0.7377 0.6801 0.6087

12.21 16.88 21.68 27.50

Converted and interpolated from “Thermodynamic Properties of KLEA 61,” British units, © ICI Chemicals and Polymers, 1993 (23 pp.). Reproduced by permission. KLEA 61 is R32/125/134a (10/70/20 wt %).

TABLE 2-174

Compressibility Factors for KLEA 66

Temp., K

1

Pressure, bar 5

10

15

20

25

30

250 260 270 280 290

0.974 0.9772 0.9796 0.9838 0.9858

0.9089 0.9209

300 310 320 330 340

0.9872 0.9883 0.9896 0.9907 0.9917

0.9287 0.9359 0.9431 0.9490 0.9540

0.8461 0.8663 0.8786 0.8910 0.9035

0.8056 0.8292 0.8492

0.7551 0.7878

0.7147

350

0.9926

0.9588

0.9137

0.8659

0.8127

0.7542

0.6843

Zsat Tsat

0.9719 236.1

0.9044 275.5

0.8397 297.6

0.7827 312.1

0.7289 323.5

0.6759 332.6

0.6220 340.4

Zsat

Psat

0.9541 0.9374 0.9172 0.8931 0.8645

1.89 2.84 4.12 5.81 7.98

0.8328 0.7920 0.7462 0.6918 0.6255

10.73 14.15 18.37 23.50 29.73

Converted and interpolated from “Thermodynamic properties of KLEA 66,” British units, © ICI Chemicals and Polymers, 1993 (20 pp.). Reproduced by permission. KLEA 66 is R32/125/134a (23/25/52 wt %).

TABLE 2-175

Compressibility Factors for Krypton*

Temp., K

1

5

10

20

40

60

80

100

200

300

400

500

150 200 250 300 350

0.9837 0.9933 0.9966 0.9982 0.9989

0.9155 0.9648 0.9841 0.9899 0.9949

0.0310 0.9278 0.9635 0.9800 0.9897

0.0618 0.8459 0.9265 0.9595 0.9793

0.1227 0.6039 0.8468 0.9197 0.9522

0.1829 0.1870 0.7605 0.8807 0.9415

0.2423 0.2393 0.6680 0.8437 0.9250

0.3012 0.2903 0.5810 0.8097 0.9110

0.5875 0.5313 0.5785 0.7337 0.8774

0.8636 0.7568 0.7461 0.7954 0.8992

1.1315 0.9730 0.9197 0.9302 0.9799

1.3932 1.1820 1.0891 1.0627 1.0664

400 450 500 600 800

0.9993 0.9998 0.9998 1.0000 1.0002

0.9967 0.9985 0.9992 1.0003 1.0010

0.9933 0.9969 0.9984 1.0005 1.0020

0.9867 0.9939 0.9970 1.0012 1.0041

0.9746 0.9886 0.9942 1.0025 1.0079

0.9635 0.9838 0.9921 1.0043 1.0122

0.9539 0.9800 0.9910 1.0064 1.0170

0.9459 0.9774 0.9906 1.0091 1.0214

0.9323 0.9663 1.0019 1.0301 1.0475

0.9570 1.0011 1.0311 1.0618 1.0779

1.0150 1.0543 1.0732 1.1000 1.1112

1.0910 1.1142 1.1258 1.1431 1.1147

1000

1.0002

1.0013

1.0023

1.0045

1.0091

1.0135

1.0184

1.0230

1.0486

1.0767

1.1063

1.1369

Pressure, bar

*Calculated from PVT values tabulated in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standards Press, Moscow, 1976. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.).

2-144

PHYSICAL AND CHEMICAL DATA

TABLE 2-176

Compressibility Factors for Methane (R50)*

Temp., K

1

5

10

20

40

60

80

100

200

300

400

500

150 200 250 300 350

0.9854 0.9936 0.9965 0.9983 0.9991

0.9225 0.9676 0.9838 0.9915 0.9954

0.8275 0.9339 0.9680 0.9830 0.9911

0.0714 0.8599 0.9352 0.9667 0.9825

0.1411 0.6784 0.8682 0.9343 0.9662

0.2093 0.3559 0.8020 0.9047 0.9520

0.2763 0.3172 0.7386 0.8783 0.9401

0.3423 0.3618 0.6854 0.8556 0.9306

0.6599 0.6141 0.6899 0.8280 0.9227

0.9623 0.8568 0.8554 0.9154 0.9800

1.2537 1.0887 1.0359 1.0432 1.0723

1.5363 1.3122 1.2155 1.1829 1.1804

400 450 500 600 800

0.9995 0.9997 0.9999 1.0000 1.0003

0.9977 0.9989 0.9997 1.0009 1.0017

0.9953 0.9979 0.9995 1.0020 1.0034

0.9912 0.9963 0.9995 1.0039 1.0068

0.9835 0.9935 0.9996 1.0081 1.0130

0.9772 0.9917 1.0005 1.0125 1.0197

0.9726 0.9911 1.0022 1.0171 1.0263

0.9696 0.9916 1.0048 1.0217 1.0330

0.9779 1.0098 1.0285 1.0540 1.0678

1.0245 1.0528 1.0699 1.0969 1.1068

1.0986 1.1152 1.1248 1.1470 1.1496

1.1859 1.1899 1.1899 1.2019 1.1951

1000

1.0004

1.0014

1.0035

1.0071

1.0141

1.0207

1.0274

1.0342

1.0678

1.1033

1.1400

1.1790

Pressure, bar

*Calculated from PVT values tabulated in Goodwin, NBS Tech. Note 653, 1974, for temperatures up to 500 K, and from PVT values tabulated in Zhuravlev. Thermophysical Properties of Gaseous and Liquid Methane, Standartov, Moscow, 1969, and NBS-NSF transl. TT 70-50097, 1970.

TABLE 2-177

Compressibility Factors for Methanol Pressure, bar

Temp., K

0.1

0.5

1.0133

10

20

50

100

150

200

250

300

400

500

200 250 300 350 400

0.0002 0.0002 0.9792 0.9844 0.9872

0.0011 0.0009 0.0008 0.9713 0.9795

0.0022 0.0019 0.0017 0.9551 0.9722

0.0219 0.0185 0.0164 0.0150 0.0142

0.0438 0.0370 0.0327 0.0298 0.0283

0.1091 0.0923 0.0813 0.0742 0.0702

0.2174 0.1837 0.1617 0.1473 0.1386

0.3250 0.2743 0.2413 0.2193 0.2056

0.4319 0.3643 0.3201 0.2904 0.2714

0.5381 0.4535 0.3981 0.3606 0.3362

0.6437 0.5422 0.4755 0.4301 0.4000

0.8531 0.7176 0.6284 0.5671 0.5253

1.6030 0.8909 0.7791 0.7016 0.6478

450 500 600 700 800

0.9890 0.9903 0.9922 0.9934 0.9964

0.9835 0.9859 0.9889 0.9907 0.9920

0.9792 0.9828 0.9867 0.9889 0.9904

0.9145 0.9525 0.9756 0.9816 0.9838

0.7989 0.9081 0.9643 0.9778 0.9818

0.0701 0.6799 0.9042 0.9541 0.9711

0.1366 0.1505 0.7629 0.8932 0.9411

0.2007 0.2110 0.6275 0.8392 0.9156

0.2629 0.2699 0.5255 0.8027 0.9025

0.3238 0.3271 0.4921 0.7797 0.8994

0.3834 0.3829 0.5010 0.7675 0.9026

0.4997 0.4912 0.5606 0.7713 0.9205

0.6128 0.5959 0.6358 0.7993 0.9485

Goodwin, R.D., J. Phys. Chem. Ref. Data, 16 (4), 799, 1987.

TABLE 2-178

Compressibility Factors for Neon*

Temp., K

1

5

10

20

40

60

80

100

200

300

400

500

50 100 150 200 250

0.9913 0.9993 1.0002 1.0003 1.0001

0.9472 0.9970 1.0017 1.0023 1.0022

0.9083 0.9949 1.0036 1.0049 1.0045

0.8013 0.9913 1.0078 1.0100 1.0097

0.3810 0.9854 1.0162 1.0204 1.0198

0.4398 0.9245 1.0262 1.0318 1.0295

0.4984 0.9864 1.0375 1.0427 1.0403

0.5850 0.9930 1.0497 1.0551 1.0502

0.9864 1.0796 1.1236 1.1191 1.1057

1.3659 1.2197 1.2131 1.1909 1.1633

1.7289 1.3796 1.3113 1.2655 1.2223

2.0794 1.5473 1.4150 1.3422 1.2822

300 400 500 600 800

1.0000 1.0000 1.0000 1.0000 1.0000

1.0020 1.0017 1.0014 1.0012 1.0009

1.0041 1.0036 1.0029 1.0024 1.0018

1.0091 1.0074 1.0058 1.0049 1.0043

1.0181 1.0151 1.0124 1.0107 1.0081

1.0277 1.0216 1.0188 1.0160 1.0123

1.0369 1.0301 1.0252 1.0214 1.0163

1.0469 1.0376 1.0316 1.0267 1.0206

1.0961 1.0771 1.0641 1.0542 1.0413

1.1476 1.1172 1.0963 1.0814 1.0622

1.1997 1.1575 1.1291 1.1091 1.0829

1.2520 1.1981 1.1621 1.1369 1.1039

1000

1.0000

1.0007

1.0014

1.0034

1.0068

1.0098

1.0132

1.0165

1.0330

1.0500

1.0670

1.0836

Pressure, bar

*Calculated from PVT values tabulated in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standards Press, Moscow, 1976. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.).

COMPRESSIBILITIES

2-145

TABLE 2-179

Compressibility Factors for Nitrogen*

Temp., K

1

5

10

20

40

60

80

100

200

300

400

500

70 80 90 100 120

0.0057 0.9593 0.9722 0.9798 0.9883

0.0287 0.0264 0.0251 0.8910 0.9397

0.0573 0.0528 0.0500 0.0487 0.8732

0.1143 0.1053 0.0996 0.0966 0.7059

0.2277 0.2093 0.1975 0.1905 0.1975

0.3400 0.3122 0.2938 0.2823 0.2822

0.4516 0.4140 0.3888 0.3720 0.3641

0.5623 0.5148 0.4826 0.4605 0.4438

1.1044 1.0061 0.9362 0.8840 0.8188

1.6308 1.4797 1.3700 1.2852 1.1684

Solid 1.9396 1.7890 1.6707 1.5015

Solid 2.3879 2.1962 2.0441 1.8223

140 160 180 200 250

0.9927 0.9952 0.9967 0.9978 0.9992

0.9635 0.9766 0.9846 0.9897 0.9960

0.9253 0.9529 0.9690 0.9791 0.9924

0.8433 0.9042 0.9381 0.9592 0.9857

0.6376 0.8031 0.8782 0.9212 0.9741

0.4251 0.7017 0.8125 0.8882 0.9655

0.4278 0.6304 0.7784 0.8621 0.9604

0.4799 0.6134 0.7530 0.8455 0.9589

0.7942 0.8107 0.8550 0.9067 1.0048

1.0996 1.0708 1.0669 1.0760 1.1143

1.3920 1.3275 1.2893 1.2683 1.2501

1.6726 1.5762 1.5105 1.4631 1.3962

300 350 400 450 500

0.9998 1.0001 1.0002 1.0003 1.0004

0.9990 1.0007 1.0011 1.0018 1.0020

0.9983 1.0011 1.0024 1.0033 1.0040

0.9971 1.0029 1.0057 1.0073 1.0081

0.9964 1.0069 1.0125 1.0153 1.0167

0.9973 1.0125 1.0199 1.0238 1.0257

1.0000 1.0189 1.0283 1.0332 1.0350

1.0052 1.0271 1.0377 1.0430 1.0451

1.0559 1.0810 1.0926 1.0973 1.0984

1.1422 1.1560 1.1609 1.1606 1.1575

1.2480 1.2445 1.2382 1.2303 1.2213

1.3629 1.3405 1.3216 1.3043 1.2881

600 800 1000

1.0004 1.0004 1.0003

1.0021 1.0017 1.0015

1.0040 1.0036 1.0034

1.0084 1.0074 1.0067

1.0173 1.0157 1.0136

1.0263 1.0237 1.0205

1.0355 1.0320 1.0275

1.0450 1.0402 1.0347

1.0951 1.0832 1.0714

1.1540 1.1264 1.1078

1.2028 1.1701 1.1449

1.2657 1.2140 1.1814

Pressure, bar

*Computed from pressure-volume-temperature tables in the Vasserman monographs referenced under Table 2-165. TABLE 2-180

Compressibility Factors for Oxygen* Pressure, bar

Temp., K

1

5

10

20

40

60

80

100

200

300

400

500

75 80 90 100 120

0.0043 0.0041 0.0038 0.9757 0.9855

0.0213 0.0203 0.0188 0.0177 0.9246

0.0425 0.0406 0.0376 0.0354 0.8367

0.0849 0.0811 0.0750 0.0705 0.0660

0.1693 0.1616 0.1494 0.1404 0.1302

0.2533 0.2418 0.2233 0.2096 0.1935

0.3368 0.3214 0.2966 0.2783 0.2558

0.4200 0.4007 0.3696 0.3464 0.3173

0.8301 0.7912 0.7281 0.6798 0.6148

1.2322 1.1738 1.0780 1.0040 0.8999

1.6278 1.5495 1.4211 1.3206 1.1762

2.0175 1.9196 1.7580 1.6309 1.4456

140 160 180 200 250

0.9911 0.9939 0.9960 0.9970 0.9987

0.9535 0.9697 0.9793 0.9853 0.9938

0.9034 0.9379 0.9579 0.9705 0.9870

0.7852 0.8689 0.9134 0.9399 0.9736

0.1334 0.6991 0.8167 0.8768 0.9477

0.1940 0.3725 0.7696 0.8140 0.9237

0.2527 0.2969 0.5954 0.7534 0.9030

0.3099 0.3378 0.5106 0.6997 0.8858

0.5815 0.5766 0.6043 0.6720 0.8563

0.8374 0.8058 0.8025 0.8204 0.9172

1.0832 1.0249 0.9990 0.9907 1.0222

1.3214 1.2364 1.1888 1.1623 1.1431

300 350 400 450 500

0.9994 0.9998 1.0000 1.0002 1.0002

0.9968 0.9990 1.0000 1.0007 1.0011

0.9941 0.9979 1.0000 1.0015 1.0022

0.9884 0.9961 1.0000 1.0024 1.0038

0.9771 0.9919 1.0003 1.0048 1.0075

0.9676 0.9890 1.0011 1.0074 1.0115

0.9597 0.9870 1.0022 1.0106 1.0161

0.9542 0.9870 1.0045 1.0152 1.0207

0.9560 1.0049 1.0305 1.0445 1.0523

0.9972 1.0451 1.0718 1.0859 1.0927

1.0689 1.1023 1.1227 1.1334 1.1380

1.1572 1.1722 1.1816 1.1859 1.1866

600 800 1000

1.0003 1.0003 1.0003

1.0014 1.0014 1.0013

1.0024 1.0026 1.0026

1.0052 1.0055 1.0053

1.0102 1.0109 1.0101

1.0153 1.0164 1.0149

1.0207 1.0219 1.0198

1.0266 1.0271 1.0253

1.0582 1.0565 1.0507

1.0961 1.0888 1.0783

1.1374 1.1231 1.1072

1.1803 1.1582 1.1369

*Calculated from pressure-volume-temperature tables in the Vasserman monographs listed under Table 2-165. TABLE 2-181

Compressibility Factors for Refrigerant 32* Pressure, bar

Temp., K

1

5

10

15

20

25

30

40

50

230 240 250 260 270

0.9656 0.9711 0.9755 0.9791 0.9819

0.8865 0.9036

280 290 300 310 320

0.9844 0.9864 0.9880 0.9894 0.9904

0.9180 0.9285 0.9376 0.9453 0.9518

0.8210 0.8476 0.8686 0.8358 0.8998

0.7899 0.8197 0.8436

0.7439 0.7812

0.7089

330 340 350

0.9914 0.9923 0.9932

0.9573 0.9619 0.9655

0.9118 0.9203 0.9296

0.8628 0.8790 0.8932

0.8102 0.8338 0.8534

0.7518 0.7846 0.8115

0.6851 0.7316 0.7671

0.6021 0.6675

0.5312

0.9595 221.2

0.8843 258.8

0.8202 279.8

0.7670 293.8

0.7191 304.6

0.6722 313.5

0.6303 321.1

0.5427 333.9

0.4467 344.3

Zsat Tsat

Zsat

Psat

0.9453 0.9278 0.9062 0.8811 0.8522

1.54 2.40 3.60 5.22 7.34

0.8194 0.7822 0.7401 0.6922 0.6370

10.07 13.51 17.76 22.95 29.21

0.5719 0.4905 0.3702

36.72 45.66 56.35

*Converted and interpolated from British units shown in Thermodynamic properties of KLEA 32, ICI Chemicals and Polymers, 1993. Reproduced by permission.

2-146

PHYSICAL AND CHEMICAL DATA

TABLE 2-182

Compressibility Factors for Refrigerant 123

Temp., °C

1

40 50 60 70 80

0.9639 0.9682 0.9717 0.9745 0.9766

0.9248 0.9327 0.9401

100 120 140 160 180

0.9804 0.9839 0.9861 0.9886 0.9908

0.9501 0.9591 0.9650 0.9714 0.9762

0.9197 0.9146 0.9282 0.9406 0.9518

0.8355 0.8667 0.8915 0.9077 0.9254

0.8140 0.8503 0.8747 0.8970

0.8023 0.8398 0.8709

0.7479 0.8026 0.8402

0.6916 0.7600 0.8072

0.7134 0.7712

0.6553 0.7346

200 225 250

0.9924 0.9938 0.9954

0.9806 0.9846 0.9885

0.9602 0.9692 0.9

0.9388 0.9526 0.9651

0.9174 0.9378 0.9528

0.8931 0.9170 0.9382

0.8688 0.8972 0.9229

0.8422 0.8800 0.9101

0.8163 0.8566 0.8930

Zsat T

0.9575 27.5

0.9210 55.4

0.9730 80.8

0.8292 97.9

0.7947 111.1

0.7654 122.0

0.7229 131.4

0.7110 139.7

Pressure, bar 2.5

5

7.5

10

12.5

15

17.5

20

22.5

25

Zsat

Psat

0.9427 0.9294 0.9134 0.8950 0.8727

1.54 2.13 2.96 3.78 4.90

0.6841

0.8262 0.7640 0.6890 0.5820 0.3926

7.87 12.01 17.59 24.92 34.54

0.7882 0.8401 0.8771

0.7539 0.8157 0.8581

— — —

— — —

0.6564 147.2

0.6206 154.0

0.5821 160.2

— —

— —

20

22.5

25

Dashes indicate inaccessible states; blanks indicate no available data. TABLE 2-183

Compressibility Factors for Refrigerant 124

Temp., °C

1

Pressure, bar 2.5

−20 −10 0 10 20

0.9573 0.9641 0.9693

30 40 50 60 80

0.9736 0.9675 0.9798 0.9820 0.9854

0.9313 0.9396 0.9473 0.9534 0.9633

100 120 140 160 180

0.9880 0.9899 0.9917

0.9700 0.9749 0.9794 0.9825

5

7.5

0.9468 −12.4

0.9071 11.9

12.5

15

17.5

0.8728 0.8889 0.9017 0.9226

0.8229 0.8462 0.8820

0.8366

0.9370 0.9478 0.9575 0.9645 0.9690

0.9040 0.9206 0.9357 0.9464 0.9536

0.8710 0.8935 0.9138 0.9247 0.9382

0.8314 0.8634 0.8884 0.9061 0.9213

0.7918 0.8329 0.8641 0.8868 0.9056

0.7463 0.8022 0.8391 0.8644 0.8857

0.6950 0.7682 0.8105 0.8489 0.8634

0.6380 0.7285 0.7896 0.8285 0.8574

0.6878 0.7647 0.7868 0.8379

0.9601

0.9471 0.9589 0.9650

0.9338 0.9488 0.9573

0.9211 0.9391 0.9443

0.9042 0.9223 0.9412

0.8951 0.9160 0.9333

0.8783 0.9040 0.9252

0.8647 0.8947 0.9174

0.8185 48.7

0.7830 60.2

0.7488 69.6

0.7157 77.8

0.6825 85.0

0.6484 91.4

0.6279 96.3

0.5788 102.6

200 225 250 Zsat T

10

0.8605 34.0

0.7251

Zsat

Psat

0.9562 0.9431 0.9284 0.9243 0.8920

0.72 1.10 1.63 2.34 3.27

0.8828 0.8427 0.8151 0.7803 0.7024

4.45 5.93 7.75 9.96 15.74

0.5955 0.3912

23.75 34.70

Dashes indicate inaccessible states; blanks indicate no available data. TABLE 2-184

Compressibility Factors for Refrigerant 134a Pressure, bar

Temp., °C

1

−10 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 satn. sat. T

0.9622 0.9710 0.9752 0.9778 0.9817 0.9839 0.9857 0.9872 0.9886 0.9897 0.9908 0.9916 0.9920 0.9924 0.9927 0.9929 0.9931 0.9567 −26.37

5

0.8819 0.8973 0.9098 0.9206 0.9296 0.9376 0.9442 0.9495 0.9543 0.9592 0.9638 0.9673 0.9691 0.9727 0.8741 15.74

10

0.8005 0.8280 0.8449 0.8678 0.8828 0.8954 0.9062 0.9151 0.9235 0.9308 0.9370 0.9428 0.7989 39.39

15

0.7361 0.7917 0.8137 0.8390 0.8555 0.8630 0.8802 0.8949 0.9040 0.8877 0.7017 55.23

Dashes indicate inaccessible states; blanks indicate no available data.

20

0.6853 0.7327 0.7682 0.7965 0.8144 0.8386 0.8553 0.8694 0.8817 0.6704 67.49

25

0.6290 0.6860 0.7335 0.7630 0.7915 0.8165 0.8350 0.8495 0.6094 77.57

30

0.5832 0.6557 0.7046 0.7418 0.7716 0.7964 0.8173 0.5415 86.20

40

0.5732 0.6249 0.6771 0.7169 0.7489 0.4442 100.35

50

Zsat

Psat

0.4530 0.4885 0.5645 0.6303 0.6783 — —

0.9316 0.9119 0.8888 0.8621 0.8314 0.7963 0.7560 0.7098 0.6562 0.5911 0.5054 0.3462 — — — — — — —

2.005 2.926 4.144 5.716 7.701 10.17 13.18 16.82 21.17 26.38 32.45 39.72 — — — — — — —

TABLE 2-185 Compressibility Factors for Water Substance (fps units)* Pressure, lb/in2 abs.

Temp., °F 400

600

800

1000

1200

1400

1600

1800

2000

2200

2400

2600

2800

3000

3200

3400

3600

3800

4000

10 15 20 40 60

0.9965 0.9943 0.9930 0.9861 0.9788

0.9989 0.9972 0.9970 0.9940 0.9910

0.9992 0.9986 0.9981 0.9967 0.9951

0.9995 0.9993 0.9991 0.9981 0.9973

0.9999 0.9997 0.9995 0.9990 0.9984

0.9999 0.9998 0.9996 0.9994 0.9991

0.9999 0.9999 0.9998 0.9996 0.9994

1.0000 0.9999 0.9999 0.9998 0.9997

1.0000 1.0000 1.0000 0.9999 0.9999

1.0000 1.0000 1.0000 0.9999 0.9999

1.0001 1.0001 1.0001 1.0001 1.0001

1.0006 1.0004 1.0003 1.0003 1.0003

1.0012 1.0012 1.0011 1.0010 1.0009

1.0024 1.0022 1.0020 1.0018 1.0018

1.0053 1.0042 1.0036 1.0028 1.0024

1.0084 1.0072 1.0065 1.0054 1.0048

1.0145 1.0124 1.0112 1.0090 1.0080

1.0211 1.0188 1.0173 1.0139 1.0120

1.0332 1.0295 1.0269 1.0214 1.0186

80 100 150 200 400

0.9714 0.9469 0.9435 0.9216

0.9878 0.9848 0.9770 0.9690 0.9356

0.9935 0.9919 0.9879 0.9839 0.9675

0.9963 0.9954 0.9931 0.9908 0.9817

0.9979 0.9974 0.9960 0.9947 0.9893

0.9987 0.9985 9.9976 0.9968 0.9935

0.9992 0.9990 0.9985 0.9980 0.9960

0.9996 0.9995 0.9993 0.9991 0.9982

0.9998 0.9998 0.9997 0.9996 0.9992

0.9999 0.9999 0.9998 0.9998 0.9998

1.0001 1.0001 1.0001 1.0001 1.0002

1.0003 1.0004 1.0004 1.0005 1.0007

1.0008 1.0007 1.0006 1.0007 1.0011

1.0016 1.0015 1.0014 1.0015 1.0017

1.0023 1.0022 1.0021 1.0021 1.0023

1.0044 1.0042 1.0039 1.0037 1.0033

1.0073 1.0067 1.0059 1.0055 1.0049

1.0108 1.0099 1.0087 1.0080 1.0070

1.0170 1.0157 1.0137 1.0126 1.0105

0.8989 0.8586 0.8138 0.6702

0.9509 0.9336 0.9162 0.8695 0.8188

0.9725 0.9633 0.9540 0.9305 0.9067

0.9839 0.9790 0.9733 0.9600 0.9468

0.9904 0.9872 0.9841 0.9764 0.9687

0.9942 0.9925 0.9905 0.9859 0.9813

0.9973 0.9964 0.9955 0.9932 0.9900

0.9988 0.9985 0.9981 0.9971 0.9958

0.9997 0.9996 0.9994 0.9992 0.9990

1.0002 1.0003 1.0004 1.0007 1.0010

1.0008 1.0010 1.0012 1.0017 1.0023

1.0014 1.0016 1.0019 1.0026 1.0034

1.0019 1.0022 1.0025 1.0033 1.0042

1.0026 1.0029 1.0032 1.0040 1.0049

1.0034 1.0036 1.0039 1.0048 1.0058

1.0048 1.0049 1.0052 1.0059 1.0068

1.0066 1.0065 1.0066 1.0072 1.0082

1.0097 1.0094 1.0092 1.0096 1.0104

0.5608

0.8060 0.7042 0.6185 0.5699

0.8942 0.8442 0.8003 0.7657

0.9392 0.9121 0.8883 0.8693

0.9647 0.9497 0.9371 0.9274

0.9836 0.9771 0.9714 0.9668

0.9930 0.9907 0.9895 0.9890

0.9989 0.9991 1.0004 1.0025

1.0024 1.0048 1.0075 1.0105

1.0050 1.0081 1.0118 1.0158

1.0069 1.0110 1.0152 1.0196

1.0082 1.0128 1.0172 1.0220

1.0093 1.0139 1.0188 1.0240

1.0106 1.0152 1.0204 1.0258

1.0118 1.0165 1.0216 1.0271

1.0132 1.0179 1.0229 1.0284

1.0149 1.0195 1.0242 1.0298

600 800 1,000 1,500 2,000 4,000 6,000 8,000 10,000

*Calculated by P. E. Liley from various steam tables for the lower temperatures and from Paper B-11 by P. H. Kesselman and Yu. I. Blank, 7th. Int. Conf. Properties of Steam, Tokyo, 1968, for the higher temperatures.

2-147

TABLE 2-186 Temperature, K

Compressibility Factors of Water Substance (SI units)* Pressure, bar 1

5

10

15

20

25

30

40

50

60

80

100

150

200

250

300

400

500

600

800

400 450 500 550 600 650

0.990 0.993 0.996 0.997 0.998 0.999

0.003 0.003 0.980 0.985 0.990 0.992

0.006 0.006 0.958 0.969 0.979 0.984

0.009 0.009 0.930 0.956 0.970 0.977

0.012 0.012 0.901 0.939 0.961 0.968

0.014 0.014 0.878 0.922 0.948 0.959

0.017 0.016 0.016 0.904 0.935 0.958

0.023 0.022 0.021 0.865 0.910 0.937

0.029 0.027 0.026 0.822 0.885 0.919

0.035 0.033 0.031 0.773 0.858 0.902

0.046 0.043 0.042 0.042 0.798 0.864

0.058 0.054 0.052 0.052 0.726 0.824

0.086 0.080 0.077 0.077 0.082 0.702

0.114 0.107 0.102 0.102 0.107 0.514

0.143 0.134 0.127 0.126 0.131 0.177

0.171 0.159 0.152 0.150 0.155 0.183

0.227 0.206 0.201 0.181 0.201 0.221

0.282 0.255 0.249 0.198 0.246 0.260

0.336 0.304 0.297 0.289 0.290 0.303

0.445 0.402 0.390 0.378 0.375 0.383

1000 0.552 0.498 0.482 0.464 0.457 0.460

700 750 800 850 900 950

1.000 1.000 1.000 1.000 1.000 1.000

0.994 0.996 0.997 0.997 0.998 0.998

0.988 0.991 0.993 0.995 0.997 0.997

0.984 0.988 0.991 0.992 0.993 0.994

0.976 0.981 0.985 0.989 0.992 0.994

0.967 0.975 0.982 0.984 0.989 0.993

0.966 0.971 0.976 0.981 0.986 0.991

0.952 0.961 0.970 0.977 0.982 0.985

0.941 0.955 0.966 0.973 0.979 0.983

0.929 0.945 0.957 0.967 0.974 0.980

0.900 0.927 0.945 0.957 0.965 0.973

0.876 0.907 0.929 0.946 0.958 0.967

0.800 0.856 0.892 0.917 0.936 0.950

0.716 0.801 0.853 0.889 0.915 0.933

0.618 0.743 0.813 0.860 0.893 0.916

0.503 0.682 0.773 0.831 0.872 0.901

0.326 0.557 0.693 0.775 0.830 0.867

0.316 0.465 0.620 0.715 0.792 0.839

0.340 0.435 0.568 0.679 0.760 0.816

0.406 0.456 0.538 0.631 0.714 0.780

0.476 0.509 0.561 0.629 0.700 0.761

1000 1200 1400 1600 1800 2000

1.000 1.000 1.000 1.000 1.001 1.003

0.999 1.000 1.000 1.000 1.001 1.002

0.998 0.999 1.000 1.000 1.001 1.002

0.995 0.998 1.000 1.000 1.000 1.002

0.995 0.998 1.000 1.000 1.000 1.002

0.994 0.997 1.000 1.000 1.000 1.002

0.993 0.997 1.000 1.000 1.000 1.002

0.990 0.995 0.999 1.000 1.000 1.002

0.987 0.994 0.998 1.000 1.000 1.002

0.985 0.994 0.998 1.000 1.001 1.003

0.978 0.992 0.998 1.000 1.002 1.003

0.973 0.990 0.997 1.000 1.003 1.004

0.960 0.986 0.996 1.001 1.003 1.004

0.948 0.982 0.995 1.002 1.004 1.006

0.935 0.975 0.995 1.002 1.005 1.008

0.923 0.968 0.994 1.004 1.008 1.011

0.900 0.961 0.993 1.006 1.011 1.014

0.878 0.957 0.992 1.009 1.014 1.018

0.859 0.949 0.994 1.012 1.017 1.021

0.831 0.942 0.996 1.015 1.021 1.032

0.816 0.937 0.998 1.020 1.031 1.043

*Calculated by P. E. Liley from various steam tables for the lower temperatures and from Pap. B-11 by P. H. Kesselman and Yu. I. Blank, 7th Internal Conference on the Properties of Steam, Tokyo, 1968, for the higher temperatures.

2-148

COMPRESSIBILITIES TABLE 2-187 Temperature, K

2-149

Compressibility Factors for Xenon* Pressure, bar 1

5

10

20

40

60

80

100

200

300

400

500

200 250 300 350 400

0.9831 0.9911 0.9949 0.9967 0.9977

0.9088 0.9545 0.9736 0.9834 0.9892

0.0293 0.9052 0.9465 0.9669 0.9183

0.0584 0.7887 0.8885 0.9322 0.9562

0.1162 0.1114 0.7517 0.8473 0.9128

0.1733 0.1642 0.5492 0.7840 0.8696

0.2300 0.2158 0.2794 0.7039 0.8278

0.2861 0.2663 0.3016 0.6249 0.7888

0.5601 0.5074 0.5021 0.5645 0.6916

0.8253 0.7355 0.6997 0.7124 0.7642

1.0833 0.9546 0.8886 0.8706 0.8850

1.3356 1.1670 1.0707 1.0269 1.0148

450 500 600 800 1000

0.9989 0.9982 0.9996 1.0000 1.0000

0.9928 0.9951 0.9979 0.9998 1.0004

0.9856 0.9902 0.9957 1.0002 1.0015

0.9714 0.9810 0.9917 1.0004 1.0031

0.9429 0.9623 0.9841 1.0012 1.0144

0.9163 0.9452 0.9772 1.0020 1.0101

0.8911 0.9293 0.9715 1.0034 1.0133

0.8679 0.9156 0.9667 1.0054 1.0172

0.7335 0.8774 0.9596 1.0213 1.0394

0.8331 0.8953 0.9791 1.0476 1.0669

0.9187 0.9572 1.0211 1.0818 1.0979

1.0224 1.0412 1.0799 1.1222 1.1331

*Calculated from PVT values tabulated in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standards Press, Moscow, 1976. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.).

TABLE 2-188

Compressibilities of Liquids*

At the constant temperature T, the compressibility β = (1/V 0)(dV/dP). In general as P increases, β decreases rapidly at first and then slowly; the change of β with T is large at low pressures but very small at pressures above 1000 to 2000 megabars. 1 megabar = 0.987 atm. = 106 dynes/cm2 based upon the older usage, 1 bar = 1 dyne/cm2. The use of the bar as a pressure unit is not encouraged.

Substance

Temp., °C

Pressure, megabars

Acetone Acetone Acetone Acetone Amyl alcohol alcohol, iso. alcohol, iso. alcohol, n alcohol, n alcohol, n alcohol, n Benzene Benzene Benzene Bromine Bromine Butyl alcohol, iso alcohol, iso alcohol, iso alcohol, iso alcohol, iso alcohol, iso Carbon bisulfide bisulfide bisulfide bisulfide tetrachloride tetrachloride Chloroform Chloroform Dichloroethylsulfide Dichloroethylsulfide Ethyl acetate acetate

14 20 20 40 14 20 20 20 20 20 40 17 20 20 20 20 18 20 20 20 20 20 16 20 20 20 20 20 20 20 32 32 13 20

23 500 1,000 12,000 23 200 400 500 1,000 12,000 12,000 5 200 400 200 400 8 200 400 500 1,000 12,000 21 500 1,000 12,000 200 400 200 400 1,000 2,000 23 200

Compressibility per megabar β × 106

Substance

111 61 52 9 88 84 70 61 46 8 8 89 77 67 56 51 97 81 64 56 46 8 86 57 48 6 86 73 83 70 34 24 103 90

Ethyl acetate alcohol alcohol alcohol alcohol bromide bromide bromide bromide bromide chloride chloride chloride chloride ether ether ether ether iodide iodide iodide iodide iodide Gallium Glycerol Hexane Hexane Kerosene Kerosene Kerosene Mercury Mercury Mercury Mercury

Temp., °C

Pressure, megabars

Compressibility per megabar β × 106

Substance

20 14 20 20 20 20 20 20 20 20 15 20 20 20 25 20 20 20 20 20 20 20 20 30 15 20 20 20 20 20 20 22 22 22

400 23 500 1,000 12,000 200 400 500 1,000 12,000 23 500 1,000 12,000 23 500 1,000 12,000 200 400 500 1,000 12,000 300 5 200 400 500 1,000 12,000 300 500 1,000 12,000

75 100 63 54 8 100 82 70 54 8 151 102 66 8 188 84 61 10 81 69 64 50 8 3.97 22 117 91 55 45 8 3.95 3.97 3.91 2.37

Methyl alcohol alcohol alcohol alcohol alcohol alcohol Nitric acid Oils: Almond Castor Linseed Olive Rapeseed Phosphorus trichloride trichloride trichloride trichloride Propyl alcohol (n) alcohol (n) alcohol (n?) alcohol (n?) alcohol (n?) Toluene Toluene Turpentine Water Water Water Water Water Water Water Xylene, meta meta

Temp., °C

Pressure, megabars

Compressibility per megabar β × 106

15 20 20 20 20 20 0

23 200 400 500 1,000 12,000 17

103 95 80 65 54 8 32

15 15 15 15 20 10 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 40 40 40 20 20

5 5 5 5

53 46 51 55 59 71 63 47 8 77 67 65 47 7 74 64 74 49 43 41 39 38 33 9 69 60

250 500 1,000 12,000 200 400 500 1,000 12,000 200 400 13 200 400 500 500 1,000 12,000 200 400

* Smithsonian Tables, Table 106. Scott (Cryogenic Engineering, Van Nostrand, Princeton, NJ, 1959) gives data for liquid nitrogen (p. 283), oxygen (p. 276), and hydrogen (p. 303). For a convenient index to the high-pressure work of Bridgman, see American Institute of Physics Handbook, p. 2-163, McGraw-Hill, New York, 1957.

2-150

PHYSICAL AND CHEMICAL DATA TABLE 2-189

Compressibilities of Solids

Many data on the compressibility of solids obtained prior to 1926 are contained in Gruneisen, Handbuch der Physik, vol. 10, Springer, Berlin, 1926, pp. 1–52; also available as translation, NASA RE 2-18-59W, 1959. See also Tables 271, 273, 276, 278, and other material in Smithsonian Physical Tables, 9th ed., 1954. For a review of high-pressure work to 1946, see Bridgman, Rev. Mod. Phys., 18, 1 (1946).

LATENT HEATS UNITS CONVERSIONS For this subsection, the following units conversions are applicable: °F = 9⁄ 5 °C + 32. To convert calories per gram-mole to British thermal units per

pound-mole, multiply by 1.799; to convert calories per gram to British thermal units per pound, multiply by 1.799. To convert millimeters of mercury to pounds-force per square inch, multiply by 1.934 × 10−2.

LATENT HEATS TABLE 2-190

2-151

Heats of Fusion and Vaporization of the Elements and Inorganic Compounds*

Unless stated otherwise, the values have been taken from the compilations by K. K. Kelley on “Heats of Fusion of Inorganic Compounds,” U.S. Bur. Mines Bull. 393 (1936), and “The Free Energies of Vaporization and Vapor Pressures of Inorganic Substances,” U.S. Bur. Mines Bull. 383 (1935).

Substance Aluminum Al Al2Br6 Al2Cl6 AlF3·3NaF Al2I6 Al2O3 Antimony Sb SbBr3 SbCl3 SbCl5 Sb4O6 Sb4S6 Argon A Arsenic As AsBr3 AsCl3 AsF5 As4O6 Barium Ba BaBr2 BaCl2 BaF2 Ba(NO3)2 Ba3(PO4)2 BaSO4 Beryllium Be Bismuth Bi BiBr3 BiCl3 Bi2O3 Bi2S5 Boron BBr3 BCl3 BF3 B2H6 B3H10 B5H9 B5H11 B10H14 B2H5Br B3N3H6 Bromine Br2 BrF5 Cadmium Cd CdBr2 CdCl2 CdF2 CdI2 CdO CdSO4 Calcium Ca CaBr2 CaCO3 CaCl2 CaF2 Ca(NO3)2 CaO CaO·Al2O3·2SiO2 CaO·MgO·2SiO2 CaO·SiO2 CaSO4 Carbon C (graphite) CBr4 CCl4 CF4 CH4 C2N2 CNBr CNCl

mp, °C

Heat of fusion,a,b cal/mole

660.0 97.5 192.5 1000 191.0 2045

2,550 5,420 16,960 16,380 7,960 (26,000)

2057 256.4 180.2c

61,020 10,920 26,750c

385.5 3000

15,360

630.5 97 73.4 4 655 546

4,770 3,510 3,030 2,400 (27,000) 11,200

1440

46,670

219 172d 1425

10,360 11,570 17,820

−185.8

1,590

610c

31,000c

122 −52.8 457.2

7,570 4,980 14,300

−189.3 814 31 −16 −80.7 313

290 (6,620) 2,810 2,420 2,800 8,000

704 847 960 1287 595 1730 1350

(1,400)e 6,000 5,370 3,000 (5,980) 18,600 9,700

1280

2,500e

271.3

2,505

224 817 747

2,600 6,800 8,900

−128 −165.5 −119.8 −46.9

480

99.7 −104 −58

7,800

−7.2 −61.3

2,580 1,355

320.9 568 568 1110 387

1,460 (5,000) 5,300 (5,400) 3,660

1000

4,790

851 730 1282 782 1392 561 2707 1550 1392 1512 1297

2,230 4,180 (12,700) 6,100 4,100 5,120 (12,240) 29,400 (18,200) 13,400 6,700

bp at 1 atm, °C

1638

35,670

1420 461 441

18,020 17,350

91.3 12.5 −100.9 −92.4 16 58 67 f 16 50.4

7,300 5,680 4,620 3,685 6,470 7,700 8,500 11,600 6,230 7,670

58.0 40.4

7,420 7,470

765

23,870

967

29,860

796 1559c

25,400 53,820c

1487

36,580

3600 90 −24.0

11,000e 1,050 644

−182.5 −27.8 52 −5

224 1,938u

77 −127.9 −161.4 −21.1

2,240

13

*See also subsection “Thermodynamic Properties.”

Heat of vaporization,a,b cal/mole

7,280 3,110 2,040 5,576u 11,010c 6,300

Substance Carbon (Cont.) CNF CNI CO CO2 COS COCl2 CS2 Cerium Ce Cesium Cs CsBr CsCl CsF CsI CsNO3 Chlorine Cl2 ClF ClF3 Cl2O ClO2 Cl2O7 Chromium Cr CrO2Cl2 Cobalt Co CoCl2 Copper Cu Cu2Br2 Cu2Cl2 CuI Cu2(CN)2 Cu2O CuO Cu2S Fluorine F2 F2O Gallium Ga Germanium Ge GeH4 Ge2H6 Ge3H8 GeHCl3 GeBr4 GeCl4 Ge(CH3)4 Gold Au Helium He Hydrogen H2 HBr HCl HCN HF (HF)6 HI H2O H22O (= D2O) H2O2 HNO3 H3PO2 H3PO3 H3PO4 H4P2O6 H2S H2S2 H2SO4 H2Se H2SeO4 H2Te Indium In

mp, °C

Heat of fusion,a,b cal/mole

−205.0 −57.5 −138.8

200 1,900 1,129 k

−112.0

1,049 l

775

2,120

28.4

500

642 715

3,600 (2,450)

407

3,250

−101.0

1,531m

1550

3,930

1490 727

3,660 7,390

1083.0

3,110

430

4,890

473 1230 1447 1127

(5,400) (13,400) 2,820 5,500

−223 29.8

bp at 1 atm, °C −72.8 141 −191.5 −78.4c −50.2 8.0

5,780c 13,980c 1,444 6,030 c, r 4,423 k 5,990

690 1300 1300 1251 1280

16,320 35,990 35,690 34,330 35,930

−34.1 −101 11.3 2.0 10.9 79

5,890 6,280 7,100 8,480

2475 117

8,250

1050

27,170

2595 1355 1490 1336

72,810 16,310 11,920 15,940

−188.2 −144.8 1,336

959 −165 −109 −105.6 −71 26.1 −49.5 −88

(8,300)

1063.0

3,030

−271.4 −259.2 −86.9 −114.2 −13.2 −83.0

28 575 476 2,009i 1,094

−50.8 0.0 3.8 −2 −47 17.4 74 42.4 55 −85.5 −87.6 10.5

686 1,436 1,501s 2,520c 600 2,310 3,070 2,520 8,300 568t 1,805 2,360

58 −48.9

3,450 1,670

156.4

781

Heat of vaporization,a,b cal/mole

4,878 m

1,640 2,650

2071 −89.1 31.4 110.6 75g 189 84 44 2966

3,580 5,900 7,550 8,000 8,560 7,030 6,460 81,800

−268.4

22

−252.7 −66.7 −85.0 25.7 33.3 51.2

216 4,210 3,860 6,027i 7,460 5,020

100.0 101.4 158

9,729 h,q 9,945 r,q 10,270

−60.3

4,463 t

−41.3

4,880

−2.2

5,650

2-152

PHYSICAL AND CHEMICAL DATA

TABLE 2-190

Substance Iodine I2 ICl(α) ICl(β) IF7 Iron Fe FeCl2 Fe2Cl6 Fe(CO)5 FeO FeS Krypton Kr Lead Pb PbBr2 PbCl2 PbF2 PbI2 PbMoO4 PbO PbS PbSO4 PbWO4 Lithium Li LiBO2 LiBr LiCl LiF LiI LiOH Li2MoO4 LiNO3 Li2SiO3 Li4SiO4 Li2SO4 Li2WO4 Magnesium Mg MgBr2 MgCl2 MgF2 MgO Mg3(PO4)2 MgSiO3 MgSO4 MgZn2 Manganese Mn MnCl2 MnSiO3 MnTiO3 Mercury Hg HgBr2 HgCl2 HgI2 HgSO4 Molybdenum Mo MoF6 MoO3 Neon Ne Nickel Ni NiCl2 Ni(CO)4 Ni2S Ni3S2 Nitrogen N2 NF3 NH3 NH4CNS NH4NO3 N2O NO N2O4 N2O5 NOCl Osmium OsF8 OsO4 (yellow) OsO4 (white) Oxygen O2 O3

Heats of Fusion and Vaporization of the Elements and Inorganic Compounds (Continued ) mp, °C 113.0 17.2 13.9

Heat of fusion,a,b cal/mole

bp at 1 atm, °C

Heat of vaporization,a,b cal/mole

3,650 2,660 2,270

183 4c

7,460c

1530 677 304 −21 1380 1195

3,560 7,800 20,590 3,250 (7,700) 5,000

2735 1026 319 105

84,600 30,210 12,040 9,000

−157

360e

152.9

10,390

2,310e

327.4 488 498 824 412 1065 890 1114 1087 1123

1,224 4,290 5,650 1,860 5,970 (25,800) 2,820 4,150 9,600 (15,200)

1744 914 954 1293 872

42,060 27,700 29,600 38,300 24,850

1472 1281

51,310 (50,000)

179 845 552 614 847 440 462 705

1,100 (5,570) 2,900 3,200 (2,360) (1,420) 2,480 4,200

1372

32,250

1310 1382 1681 1171

35,420 35,960 50,970 40,770

1177 1249 857 742

7,210 7,430 3,040 (6,700)

650 711 712 1221 2642 1184 1524 1127 589

2,160 8,300 8,100 5,900 18,500 (11,300) 14,700 3,500 (8,270)

1220 650 1274 1404

3,450 7,340 (8,200) (7,960)

2152 1190

557 3,960 4,150 4,500 (1,440)

361 319 304 354

13,980 14,080 14,080 14,260

(6,660) 2,500 (2,500)

(4800) 36 1151

(128,000) 6,000

−38.9 241 277 250 850 2622 17 745 −248.5

1107 1418

32,520 32,690

55,150 29,630

77

−246.0

440e

1455

4,200

2730 987c 42.5

87,300 48,360c 7,000

645 790

(2,980) 5,800 −195.8 −129.0 −33.4

1,336 3,000 5,581n

−88.5 −151.7 30 32.4 −6.4

3,950 3,307 7,040 13,800c 6,140

−210.0 −77.7 146 169.6 −90.8 −163.6 −13

56 42 −218.9

172 1,352n (4,700) 1,460 1,563 550 5,540

4,060 2,340 106

47.4 130

6,840 9,450

−183.0 −111

1,629 2,880

Substance Palladium Pd Phosphorus P4 (yellow) P4 (violet) P4 (black) PCl3 PH3 P4O6 P4O10(α) P4O10(β) POCl3 P2S3 Platinum Pt Potassium K KBO2 KBr KCl KCN KCNS K2CO3 K2CrO4 K2Cr2O7 KF KI K2MoO4 KNO3 KOH KPO3 K3PO4 K4P2O7 K2SO4 K2TiO3 K2WO4 Praseodymium Pr Radon Rn Rhenium Re Re2O7 Re2O8 Rubidium Rb RbBr RbCl RbF RbI RbNO3 Selenium Se2 Se6 SeF6 SeO2 SeOCl2 Silicon Si SiCl4 Si2Cl6 Si3Cl8 (SiCl3)2O SiF4 Si2F6 SiF3Cl SiF2Cl2 SiH4 Si2H6 Si3H8 Si4H10 SiH3Br SiH2Br2 SiHCl3 (SiH3)3N (SiH3)2O SiO2 (quartz) SiO2 (cristobalite) Silver Ag AgBr AgCl AgCN AgI AgNO3 Ag2S Ag2SO4 Sodium Na NaBO2

mp, °C

Heat of fusion,a,b cal/mole

1554

4,120

bp at 1 atm, °C

44.2

615

−133.8 23.8 569

270o 3,360 17,080

1.1

3,110

1773.5

4,700

(4400)

(107,000)

63.5 947 742 770 623 179 897 984 398 857 682 922 338 360 817 1340 1092 1074 810 927

574 (5,700) 5,000 6,410 (3,500) 2,250 7,800 6,920 8,770 6,500 4,100 (4,000) 2,840 (2,000) 2,110 8,900 14,000 8,100 (10,600) (4,400)

776

18,920

1383 1407

37,060 38,840

1324

34,690

1327

30,850

932

2,700

−71 (3000) 296 147

15,340 3,800

39.1 677 717 833 638 305

525 3,700 4,400 4,130 2,990 1,340

217

1,220

10

1,010

1427 −67.6 −1

9,470 1,845

−33

280 417c 453c 74.2 −87.7 174 591 358c 105.1 508

Heat of vaporization,a,b cal/mole

12,520 25,600c 33,100 7,280 3,489 o 10,380 20,670 8,380

−61.8

4,010

362.4

18,060

679 1352 1381 1408 1304 753 736 −45.8c 317c 168 2290 56.8 139 211.4 135.6 −94.8c −18.9c −70.1 −31.5 −111.6 −14.3 53.1 100 2.4 70.5 31.8 48.7 −15.4 2230

18,110 37,120 36,920 39,510 35,960 25,490 20,600 6,350c 20,900

6,860 12,340 8,820 6,130c 10,400c 4,460 5,080 2,960 5,110 6,780 8,890 5,650 6,840 6,360 6,850 5,350

−18.5 −138 −144 −185 −132.5 −117 −93.5 −93.8 −70.0 −126.5 −105.6 −144 1470 1700

3,900

960.5 430 455 350 557 209 842 657

2,700 2,180 3,155 2,750 2,250 2,755 3,360 (4,300)

2212

60,720

1564

42,520

1506

34,450

97.7 966

630 8,660

914

23,120

3,400 2,100

LATENT HEATS TABLE 2-190

2-153

Heats of Fusion and Vaporization of the Elements and Inorganic Compounds (Concluded )

Substance Sodium (Cont.) NaBr NaCl NaClO3 NaCN NaCNS Na2CO3 NaF NaI Na2MoO4 NaNO3 NaOH aNa2O·aAl2O3·3SiO2 NaPO3 Na4P2O7 Na2S Na2SiO3 Na2Si2O5 Na2SO4 Na2WO4 Strontium Sr SrBr2 SrCl2 SrF2 Sr3(PO4)2 Sulfur S (rhombic) S (monoclinic) S2Cl2 SF6 SO2 SO3(α) SO3(β) SO3(γ) SOBr2 SOCl2 SO2Cl2 Tellurium Te TeCl4 TeF6

mp, °C

Heat of fusion,a,b cal/mole

bp at 1 atm, °C

Heat of vaporization,a,b cal/mole

747 800 255 562 323 854 992 662 687 310 322 1107 988 970 920 1087 884 884 702

6,140 7,220 5,290 (4,400) 4,450 7,000 7,000 5,240 3,600 3,760 2,000 13,150 (5,000) (13,700) (1,200) 10,300 8,460 5,830 5,800

1392 1465

37,950 40,810

1500

37,280

757 643 872 1400 1770

2,190 4,780 4,100 4,260 18,500

1384

112.8 119.2 −75.5 17 32.4 62.2

453

1,769p 2,060 2,890 6,310

3,230

1704

53,260

1378

33,610

444.6

2,200

138 −63.5c −5.0 44.8

8,720 5,600c 5,960p 10,190

139.5 75.4 69.2

9,920 7,600 7,760

1090 392 −38.6c

16,830 6,700c

Substance Thallium Tl TlBr TlCl Tl2CO3 TlI TlNO3 Tl2S Tl2SO4 Tin Sn4 SnBr2 SnBr4 SnCl2 SnCl4 Sn(CH3)4 SnH4 SnI4 Titanium TiBr4 TiCl4 TiO2 Tungsten W WF6 Uranium UF6 Xenon Xe Zinc Zn ZnCl2 Zn(C2H5)2 ZnO ZnS Zirconium ZrBr4 ZrCl4 ZrI4 ZrO2

a

k

b

l

c

m

Values in parentheses are uncertain. For the freezing point or the normal boiling point unless otherwise stated. Sublimation. d Decomposes at about 75°C; value obtained by extrapolation. e Bichowsky and Rossini, “Thermochemistry of the Chemical Substances,” Reinhold, New York (1936). f Decomposes before the normal boiling point is reached. g Decomposes at about 40°C; value obtained by extrapolation. h See also pp. 2-304 through 2-307 on steam table. i Giauque and Ruehrwein, J. Am. Chem. Soc., 61 (1939): 2626. j Giauque and Egan, J. Chem. Phys., 5 (1937): 45.

TABLE 2-191

mp, °C

Alloys 30.5 Pb + 69.5 Sn 36.9 Pb + 63.1 Sn 63.7 Pb + 36.3 Sn 77.8 Pb + 22.2 Sn 1 Pb + 9 Sn 24 Pb + 27.3 Sn + 48.7 Bi 25.8 Pb + 14.7 Sn + 52.4 Bi + 7 Cd Silicates Anorthite (CaAl2Si2O8) Orthoclase (KAlSi2O8) Microcline (KAlSi3O8) Wollastonite (CaSiO8) Malacolite (Ca8MgSi4O12) Diopside (CaMgSi2O4) Olivine (Mg2SiO4) Fayalite (Fe2SiO4) Spermaceti Wax (bees’)

bp at 1 atm, °C

Heat of vaporization,a,b cal/mole

302.5 460 427 273 440 207 449 632

1,030 5,990 4,260 4,400 3,125 2,290 3,000 5,500

1457 819 807

38,810 23,800 24,420

823

25,030

231.8 232 30 247 −33.2

1,720 (1,700) 3,000 3,050 2,190

2270

68,000

−149.8 143.5

(4,300)

38.2 −23 1825

(2,060) 2,240 (11,400)

136

3390 −0.4

(8,400) 1,800

(5900) 17.3

(176,000) 6,350

55.1c

9,990c

−111.5 419.5 283 1975 1645

2715

740 1,595 (5,500)

623 113 78.3 −52.3

20,740 8,330 7,320 4,420

8,350

−108.0

3,110

907 732 118

27,430 28,710 8,960

357c 311c 431c

25,800c 25,290c 29,030c

4,470 (9,000)

20,800

Kemp and Giauque, J. Am. Chem. Soc., 59 (1937): 79. Brown and Manov, J. Am. Chem. Soc., 59 (1937): 500. Giauque and Powell, J. Am. Chem. Soc. 61 (1939): 1970. n Overstreet and Giauque, J. Am. Chem. Soc 59 (1937): 254. o Stephenson and Giauque, J. Chem. Phys., 5 (1937): 149. p Giauque and Stephenson, J. Am. Chem. Soc., 60 (1938): 1389. q Osborne, Stimson, and Ginnings, Bur. Standards J. Research, 23, 197 (1939): 261. r Miles and Menzies, J. Am. Chem. Soc., 58 (1936): 1067. s Long and Kemp, J. Am. Chem. Soc., 58 (1936): 1829. t Giauque and Blue, J. Am. Chem. Soc., 58 (1936): 831. u Ruehrwein and Giauque, J. Am. Chem. Soc., 61 (1939): 2940.

Heats of Fusion of Miscellaneous Materials

Material

Heat of fusion,a,b cal/mole

mp, °C

Heat of fusion, cal/g

183 179 177.5 176.5 236 98.8 75.5

17 15.5 11.6 9.54 28 6.85 8.4

43.9 61.8

100 100 83 100 94 100 130 85 37.0 42.3

2-154

PHYSICAL AND CHEMICAL DATA

TABLE 2-192

Heats of Fusion of Organic Compounds

The values for the hydrocarbons are from the tables of the American Petroleum Institute Research Project 44 at the National Bureau of Standards, with some from Parks and Huffman, Ind. Eng. Chem., 23, 1138 (1931). The values for the nonhydrocarbon compounds were recalculated from data in International Critical Tables, vol. 5. Hydrocarbon compounds Paraffins Methane Ethane Propane n-Butane 2-Methylpropane n-Pentane 2-Methylbutane 2,2-Dimethylpropane n-Hexane 2-Methylpentane 2,2-Dimethylbutane 2,3-Dimethylbutane n-Heptane 2-Methylhexane 3-Ethylpentane 2,2-Dimethylpentane 2,4-Dimethylpentane 3,3-Dimethylpentane 2,2,3-Trimethylbutane n-Octane 2-Methylheptane 3-Methylpentane 4-Methylheptane 2,2-Dimethylhexane 2,5-Dimethylhexane 3,3-Dimethylhexane 2-Methyl-3-ethylpentane 3-Methyl-3-ethylpentane 2,2,3-Trimethylpentane 2,2,4-Trimethylpentane 2,3,3-Trimethylpentane 2,3,4-Trimethylpentane 2,2,3,3-Tetramethylbutane n-Nonane n-Decane n-Undecane n-Dodecane Eicosane Pentacosane Tritriacontane Aromatics Benzene Methylbenzene (Toluene) Ethylbenzene o-Xylene m-Xylene p-Xylene n-Propylbenzene Isopropylbenzene 1-Methyl-2-ethylbenzene

Formula

mp, °C

Heat of fusion, cal/g

CH4 C2H6 C3H8 C4H10 C4H10 C5H12 C5H12 C5H12 C6H14 C6H14 C6H14 C6H14 C7H16 C7H16 C7H16 C7H16 C7H16 C7H16 C7H16 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C9H20 C10H22 C11H24 C12H26 C20H42 C25H52 C33H68

−182.48 −183.23 −187.65 −138.33 −159.60 −129.723 −159.890 −16.6 −95.320 −153.680 −99.73 −128.41 −90.595 −118.270 −118.593 −123.790 −119.230 −134.46 −24.96 −56.798 −109.04 −120.50 −120.955 −121.18 −91.200 −126.10 −114.960 −90.870 −112.27 −107.365 −100.70 −109.210 +100.69 −53.9 −30.0 −25.9 −9.6 +36.4 +53.3 +71.1

14.03 22.712 19.100 19.167 18.668 27.874 17.076 10.786 36.138 17.407 1.607 2.251 33.513 21.158 22.555 13.982 15.968 16.856 5.250 43.169 21.458 23.795 22.692 24.226 26.903 14.9 23.690 22.657 18.061 19.278 3.204 19.392 14.900 41.2 48.3 34.1 51.3 52.0 53.6 54.0

C6H6 C7H8 C8H10 C8H10 C8H10 C8H10 C9H12 C9H12 C9H12

+5.533 −94.991 −94.950 −25.187 −47.872 +13.263 −99.500 −96.028 −80.833

30.100 17.171 20.629 30.614 26.045 38.526 16.97 19.22 21.13 Heat of fusion, cal/g

Formula

mp, °C

Acetic acid Acetone Acrylic acid Allo-cinnamic acid Aminobenzoic acid (o-) (m-) (p-) Amyl alcohol Anethole Aniline Anthraquinone Apiol Azobenzene Azoxybenzene

C2H4O2 C3H6O C3H4O2 C9H8O2 C7H7NO2 C7H7NO2 C7H7NO2 C5H12O C10H12O C6H5NH2 C14H8O2 C12H14O4 C12H10N2 C12H10N2O

16.7 −95.5 12.3 68 145 179.5 188.5 −78.9 22.5 −6.3 284.8 29.5 67.1 36

46.68 23.42 37.03 27.35 35.48 38.03 36.46 26.65 25.80 27.09 37.48 25.80 28.91 21.62

Benzil Benzoic acid Benzophenone Benzylaniline Bromocamphor Bromochlorbenzene (o-) (m-) (p-) Bromoiodobenzene (o-) (m-) (p-) Bromol hydrate Bromophenol (p-) Bromotoluene (p-)

C14H10O2 C7H8O2 C13H10O C13H13N C10H15BrO C6H4BrCl C6H4BrCl C6H4BrCl C6H4BrI C6H4BrI C6H4BrI C2H3Br3O2 C6H5BrO C7H7Br

95.2 122.45 47.85 32.37 78 −12.6 −21.2 64.6 21 9.3 90.1 46 63.5 28

22.15 33.90 23.53 21.86 41.57 15.41 15.29 23.41 12.18 10.27 16.60 16.90 20.50 20.86

Nonhydrocarbon compounds

Hydrocarbon compounds Aromatics—(Cont.) 1-Methyl-3-ethylbenzene 1-Methyl-4-ethylbenzene 1,2,3-Trimethylbenzene 1,2,4-Trimethylbenzene 1,3,5-Trimethylbenzene Naphthalene Camphene Durene Isodurene Prehnitene p-Cymene n-Butyl benzene tert-Butyl benzene β-Methyl naphthalene Diphenyl Hexamethyl benzene Diphenyl methane Anthracene Phenanthrene Tolane Stilbene Dibenzil Triphenyl methane Alkyl cyclohexanes Cyclohexane Methylcyclohexane Alkyl cyclopentanes Cyclopentane Methylcyclopentane Ethylcyclopentane 1,1-Dimethylcyclopentane cis-1,2-Dimethylcyclopentane trans-1,2-Dimethylcyclopentane trans-1,3-Dimethylcyclopentane Monoolefins Ethene (Ethylene) Propene (Propylene) 1-Butene cis-2-Butene trans-2-Butene 2-Methylpropene (isobutene) 1-Pentene cis-2-pentene trans-2-pentene 2-Methyl-1-butene 3-Methyl-1-butene 2-Methyl-2-butene Acetylenes Acetylene 2-Butyne (dimethylacetylene) Nonhydrocarbon compounds

mp, °C

Heat of fusion, cal/g

C9H12 C9H12 C9H12 C9H12 C9H12 C10H8 C10H12 C10H14 C10H14 C10H14 C10H14 C10H14 C10H14 C11H10 C12H10 C12H18 C13H12 C14H10 C14H10 C14H10 C14H12 C14H14 C19H16

−95.55 −62.350 −25.375 −43.80 −44.720 +80.0 +51 +79.3 −24.0 −7.7 −68.9 −88.5 −58.1 +34.1 +68.6 +165.5 +25.2 +216.5 +96.3 +60 +124 +51.4 +92.1

15.14 25.29 16.64 24.54 18.97 36.0 57 37.4 23.0 20.0 17.1 19.5 14.9 20.1 28.8 30.4 26.4 38.7 25.0 28.7 40.0 30.7 21.1

C6H12 C7H14

+6.67 −126.58

7.569 16.429

C5H10 C6H12 C7H14 C7H14 C7H14 C7H14 C7H14

−93.80 −142.445 −138.435 −69.73 −53.85 −117.57 −133.680

2.068 19.68 11.10 3.36 3.87 15.68 17.93

C2H4 C3H6 C4H8 C4H8 C4H8 C4H8 C5H10 C5H10 C5H10 C5H10 C5H10 C5H10

−169.15 −185.25 −185.35 −138.91 −105.55 −140.35 −165.27 −151.363 −140.235 −137.560 −168.500 −133.780

28.547 17.054 16.393 31.135 41.564 25.265 16.82 24.239 26.536 26.879 18.009 25.738

C2H2 C4H6

−81.5 −132.23

23.04 40.808

Formula

Formula

mp, °C

Heat of fusion, cal/g

Butyl alcohol (n-) (t-) Butyric acid (n-)

C4H10O C4H10O C4H8O2

−89.2 25.4 −5.7

29.93 21.88 30.04

Capric acid (n-) Caprylic acid (n-) Carbazole Carbon tetrachloride Carvoxime (d-) (l-) (dl-) Cetyl alcohol Chloracetic acid (α-) (β-) Chloral alcoholate hydrate Chloroaniline (p-) Chlorobenzoic acid (o-) (m-) (p-) Chloronitrobenzene (m-) (p-) Cinnamic acid anhydride Cresol (p-) Crotonic acid (α-) (cis-) Cyanamide Cyclohexanol

C10H20O2 C8H16O2 C12H9N CCl4 C10H15NO C10H15NO C10H15NO C16H34O C2H3ClO2 C2H3ClO2 C4H7Cl3O2 C2H3Cl3O2 C6H6ClN C7H5ClO2 C7H5ClO2 C7H5ClO2 C6H4ClNO2 C6H4ClNO2 C9H8O2 C18H14O3 C7H8O C4H6O2 C4H6O2 CH2N2 C6H12O

31.99 16.3 243 −22.8 71.5 71 91 49.27 61.2 56 9 47.4 71 140.2 154.25 239.7 44.4 83.5 133 48 34.6 72 71.2 44 25.46

38.87 35.40 42.05 41.57 23.29 23.41 24.61 33.80 31.06 35.12 24.03 33.18 37.15 39.30 36.41 49.21 29.38 31.51 36.50 28.14 26.28 25.32 34.90 49.81 4.19

LATENT HEATS TABLE 2-192

2-155

Heats of Fusion of Organic Compounds (Concluded ) Heat of fusion, cal/g

Formula

mp, °C

Dibromobenzene (o-) (m-) (p-) Dibromophenol (2, 4-) Dichloroacetic acid Dichlorobenzene (o-) (m-) ( p-) Dihydroxybenzene (o-) (m-) (p-) Di-iodobenzene (o-) (m-) (p-) Dimethyl tartrate (dl-) (d-) pyrone Dinitrobenzene (o-) (m-) ( p-) Dinitrotoluene (2, 4-) Dioxane Diphenyl amine

C6H4Br2 C6H4Br2 C6H4Br2 C6H4Br2O C2H2Cl2O2 C6H4Cl2 C6H4Cl2 C6H4Cl2 C6H6O2 C6H6O2 C6H6O2 C6H4I2 C6H4I2 C6H4I2 C6H10O6 C6H10O6 C7H8O2 C6H4N2O4 C6H4N2O4 C6H4N2O4 C7H6N2O4 C4H8O2 C12H11N

1.8 −6.9 86 12 −4(?) −16.7 −24.8 53.13 104.3 109.65 172.3 23.4 34.2 129 87 49 132 116.93 89.7 173.5 70.14 11.0 52.98

12.78 13.38 20.55 13.97 14.21 21.02 20.55 29.67 49.40 46.20 58.77 10.15 11.54 16.20 35.12 21.50 56.14 32.25 24.70 39.99 26.40 34.85 25.23

Elaidic acid Ethyl acetate alcohol Ethylene dibromide Ethyl ether

C18H34O2 C4H8O2 C2H6O C2H4Br2 C4H10O

44.4 83.8 −114.4 10.012 −116.3

52.08 28.43 25.76 13.52 23.54

Formic acid

CH2O2

Glutaric acid Glycerol Glycol, ethylene

Nonhydrocarbon compounds

8.40

58.89

C6H8O4 C3H8O3 C2H6O2

97.5 18.07 −11.5

37.39 47.49 43.26

Hydrazo benzene Hydrocinnamic acid Hydroxyacetanilide

C12H12N2 C9H10O2 C8H9NO2

134 48 91.3

22.89 28.14 33.59

Iodotoluene (p-) Isopropyl alcohol ether

C7H7I C3H8O C6H14O

34 −88.5 −86.8

18.75 21.08 25.79

Lauric acid (n-) Levulinic acid

C12H24O2 C5H8O3

43.22 33

43.72 18.97

Menthol (l-) (α) Methyl alcohol Myristic acid Methyl cinnamate fumarate oxalate phenylpropiolate succinate

C10H20O CH4O C14H28O2 C10H10O2 C6H8O4 C4H6O4 C10H8O2 C6H10O4

43.5 −97.8 53.86 36 102 54.35 18 19.5

18.63 23.7 47.49 26.53 57.93 42.64 22.86 35.72

Formula

mp, °C

Heat of fusion, cal/g

Naphthol (α-) (β-) Naphthylamine (α-) Nitroaniline (o-) (m-) (p-) Nitrobenzene Nitrobenzoic acid (o-) (m-) (p-) Nitronaphthalene Nitrophenol (o-)

C10H8O C10H8O C10H9N C6H6N2O2 C6H6N2O2 C6H6N2O2 C6H5NO2 C7H5NO4 C7H5NO4 C7H5NO4 C10H7NO2 C6H5NO3

95.0 120.6 50 71.2 114.0 147.3 5.85 145.8 141.1 239.2 56.7 45.13

38.94 31.30 22.34 27.88 40.97 36.46 22.52 40.06 27.59 52.80 25.44 26.76

Palmitic acid Paraldehyde Pelargic acid (n-) (β-) Pelargonic acid (n-) (α-) Phenol Phenylacetic acid Phenylhydrazine Propyl ether (n)

C16H32O2 C6H12O3 C9H18O2 C9H18O2 C6H6O C8H8O2 C6H8N2 C6H14O

61.82 10.5 12.35 40.92 76.7 19.6 −126.1

39.18 25.02 39.04 30.63 29.03 25.44 36.31 20.66

Quinone

C6H4O2

115.7

40.85

Stearic acid Succinic anhydride Succinonitrile

C18H30O2 C4H4O3 C4H4N2

68.82 119 54.5

47.54 48.74 11.71

Tetrachloroxylene (o-) (p-) Thiophene Thiosinamine Thymol Toluic acid (o-) (m-) (p-) Toluidine (p-) Tribromophenol (2, 4, 6-) Trichloroacetic acid Trinitroglycerol Trinitrotoluene (2, 4, 6-) Tristearin

C8H6Cl4 C8H6Cl4 C4H4S C4H8N2S C10H14O C8H8O2 C8H8O2 C8H8O2 C7H9N C6H3Br3O C2HCl3O2 C3H5N3O9 C7H5N3O6 C57H110O6

86 95 −39.4 77 51.5 103.7 108.75 179.6 43.3 93 57.5 12.3 80.83 70.8, 54.5

21.02 22.10 14.11 33.45 27.47 35.40 27.59 39.90 39.90 13.38 8.60 23.02 22.34 45.63

Undecylic acid (α-) (n-) (β-) (n-) Urethane

C11H22O2 C11H22O2 C3H7NO2

28.25 48.7

32.20 42.91 40.85

Veratrol

C8H10O2

22.5

27.45

Xylene dibromide (o-) (m-) dichloride (o-) (m-) (p-)

C8H8Br2 C8H8Br2 C8H8Cl2 C8H8Cl2 C8H8Cl2

Nonhydrocarbon compounds

95 77 55 34 100

24.25 21.45 29.03 26.64 32.73

TABLE 2-193 Cmpd. no.

Heats of Vaporization of Inorganic and Organic Compounds Name

Formula

CAS no.

Mol wt

C1 × 1E−07

C2

C3

C4

Tmin, K

∆Hv at Tmin × 1E−07

Tmax, K

∆Hv at Tmax

1 2 3 4 5 6 7 8 9 10

Methane Ethane Propane n-Butane n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane

CH4 C2H6 C3H8 C4H10 C5H12 C6H14 C7H16 C8H18 C9H20 C10H22

74828 74840 74986 106978 109660 110543 142825 111659 111842 124185

16.043 30.070 44.097 58.123 72.150 86.177 100.204 114.231 128.258 142.285

1.0194 2.1091 2.9209 3.6238 3.9109 4.4544 5.0014 5.5180 6.0370 6.6126

0.26087 0.60646 0.78237 0.8337 0.38681 0.39002 0.38795 0.38467 0.38522 0.39797

−0.14694 −0.55492 −0.77319 −0.82274 0 0 0 0 0 0

0.22154 0.32799 0.39246 0.39613 0 0 0 0 0 0

90.69 90.35 85.47 134.86 143.42 177.83 182.57 216.38 219.66 243.51

0.8724 1.7879 2.4787 2.8684 3.3968 3.7647 4.2619 4.5898 5.0545 5.4168

190.56 305.32 369.83 425.12 469.7 507.6 540.2 568.7 594.6 617.7

0 0 0 0 0 0 0 0 0 0

11 12 13 14 15 16 17 18 19 20

n-Undecane n-Dodecane n-Tridecane n-Tetradecane n-Pentadecane n-Hexadecane n-Heptadecane n-Octadecane n-Nonadecane n-Eicosane

C11H24 C12H26 C13H28 C14H30 C15H32 C16H34 C17H36 C18H38 C19H40 C20H42

1120214 112403 629505 629594 629629 544763 629787 593453 629925 112958

156.312 170.338 184.365 198.392 212.419 226.446 240.473 254.500 268.527 282.553

7.2284 7.7337 8.4339 9.0539 9.6741 10.1560 10.4730 10.9690 11.6740 12.8600

0.40607 0.40681 0.4257 0.44467 0.45399 0.45726 0.4374 0.44327 0.45865 0.50351

0 0 0 0 0 0 0 0 0 0.32986

0 0 0 0 0 0 0 0 0 −0.42184

247.57 263.57 267.76 279.01 283.07 291.31 295.13 301.31 305.04 309.58

5.9240 6.2802 6.8015 7.2002 7.6728 8.0225 8.3699 8.7246 9.2185 9.5933

639 658 675 693 708 723 736 747 758 768

0 0 0 0 0 0 0 0 0 0

21 22 23 24 25 26 27

2-Methylpropane 2-Methylbutane 2,3-Dimethylbutane 2-Methylpentane 2,3-Dimethylpentane 2,3,3-Trimethylpentane 2,2,4-Trimethylpentane

C4H10 C5H12 C6H14 C6H14 C7H16 C8H18 C8H18

75285 78784 79298 107835 565593 560214 540841

58.123 72.150 86.177 86.177 100.204 114.231 114.231

3.1667 3.7700 4.1404 4.2780 4.6536 4.9910 4.7721

0.3855 0.3952 0.38124 0.384 0.37579 0.383 0.37992

0 0 0 0 0 0 0

0 0 0 0 0 0 0

113.54 113.25 145.19 119.55 160 172.22 165.78

2.7927 3.3720 3.6328 3.8495 4.0747 4.3530 4.1565

408.14 460.43 499.98 497.5 537.35 573.5 543.96

0 0 0 0 0 0 0

28 29 30 31 32 33 34 35

Ethylene Propylene 1-Butene cis-2-Butene trans-2-Butene 1-Pentene 1-Hexene 1-Heptene

C2H4 C3H6 C4H8 C4H8 C4H8 C5H10 C6H12 C7H14

74851 115071 106989 590181 624646 109671 592416 592767

28.054 42.081 56.108 56.108 56.108 70.134 84.161 98.188

2.8694 3.2300 3.4190 3.3320 3.7740 4.3236 4.8120

0.3746 0.8375 0.3747 0.3754 0.3736 0.37647 0.3788 0.3685

0 −0.9216 0 0 0 0 0 0

0 0.5012 0 0 0 0 0 0

104 87.89 87.8 134.26 167.62 107.93 133.39 154.27

1.6025 2.4031 2.9582 2.9773 2.7684 3.4166 3.8483 4.2478

282.34 365.57 419.95 435.58 428.63 464.78 504.03 537.29

0 0 0 0 0 0 0 0

36 37 38 39 40 41 42 43 44

1-Octene 1-Nonene 1-Decene 2-Methylpropene 2-Methyl-1-butene 2-Methyl-2-butene 1,2-Butadiene 1,3-Butadiene 2-Methyl-1,3-butadiene

C8H16 C9H18 C10H20 C4H8 C5H10 C5H10 C4H6 C4H6 C5H8

111660 124118 872059 115117 563462 513359 590192 106990 78795

112.215 126.242 140.269 56.108 70.134 70.134 54.092 54.092 68.119

5.3980 5.9940 6.4898 3.2720 3.9091 3.9121 3.5220 3.2580 3.9310

0.3835 0.3953 0.39187 0.383 0.39866 0.3634 0.395 0.373 0.425

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

171.45 191.78 206.89 132.81 135.58 139.39 136.95 164.25 127.27

4.7013 5.1366 5.5289 2.8262 3.4072 3.4437 3.0540 2.7155 3.4529

566.65 593.25 616.4 417.9 465 471 452 425.17 484

0 0 0 0 0 0 0 0 0

45 46 47 48 49 50

Acetylene Methylacetylene Dimethylacetylene 3-Methyl-1-butyne 1-Pentyne 2-Pentyne

C2H2 C3H4 C4H6 C5H8 C5H8 C5H8

74862 74997 503173 598232 627190 627214

26.038 40.065 54.092 68.119 68.119 68.119

2.3795 3.2775 3.8560 3.7920 3.9540 4.4158

0.375 0.3997 0.3737 0.3565 0.3512 0.44347

0 0 0 0 0 0

0 0 0 0 0 0

192.4 170.45 240.91 183.45 167.45 163.83

1.6488 2.6297 2.9557 3.1681 3.4025 3.7321

308.32 402.39 473.2 463.2 481.2 519

0 0 0 0 0 0

51 52

1-Hexyne 2-Hexyne

C6H10 C6H10

693027 764352

82.145 82.145

4.5740 4.9110

0.3698 0.4392

0 0

0 0

141.25 183.65

4.0640 4.1067

516.2 549

0 0

2-156

53 54 55 56

3-Hexyne 1-Heptyne 1-Octyne Vinylacetylene1

C6H10 C7H12 C8H14 C4H4

928494 628717 629050 689974

82.145 96.172 110.199 52.076

4.8080 5.0514 5.6306 3.6490

0.436 0.41163 0.4148 0.4

0 0 0 0.043

0 0 0 0

170.05 192.22 193.55 173.15

4.0831 4.2470 4.7663 2.9876

544 559 585 454

0 0 0 0

57 58 59 60 61 62 63 64 65 66

Cyclopentane Methylcyclopentane Ethylcyclopentane Cyclohexane Methylcyclohexane 1,1-Dimethylcyclohexane Ethylcyclohexane Cyclopentene 1-Methylcyclopentene Cyclohexene

C5H10 C6H12 C7H14 C6H12 C7H14 C8H16 C8H16 C5H8 C6H10 C6H10

287923 96377 1640897 110827 108872 590669 1678917 142290 693890 110838

70.134 84.161 98.188 84.161 98.188 112.215 112.215 68.119 82.145 82.145

3.8900 4.3600 4.8288 4.4940 4.7534 5.0402 5.3832 3.8107 4.3541 4.4405

0.361 0.38531 0.37809 0.3974 0.39461 0.4036 0.41763 0.3543 0.36805 0.37479

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

179.28 130.73 134.71 279.69 146.58 239.66 161.84 138.13 146.62 169.67

3.3292 3.9118 4.3604 3.3977 4.2295 4.0862 4.7318 3.4046 3.8769 3.8791

511.76 532.79 569.52 553.58 572.19 591.15 609.15 507 542 560.4

0 0 0 0 0 0 0 0 0 0

67 68 69 70 71 72 73

Benzene Toluene o-Xylene m-Xylene p-Xylene Ethylbenzene Propylbenzene

C6H6 C7H8 C8H10 C8H10 C8H10 C8H10 C9H12

71432 108883 95476 108383 106423 100414 103651

78.114 92.141 106.167 106.167 106.167 106.167 120.194

4.7500 5.0144 5.5330 5.4600 5.3740 5.4640 5.7663

0.45238 0.3859 0.377 0.3726 0.3656 0.392 0.3956

0.0534 0 0 0 0 0 −8.9129E−03

−0.1181 0 0 0 0 0 0

278.68 178.18 247.98 225.3 286.41 178.15 215.03

3.4909 4.3670 4.5826 4.6097 4.2761 4.7811 5.0574

562.16 591.8 630.33 617.05 616.23 617.2 574.54

0 0 0 0 0 0 2.4695E+07

74 75 76 77 78 79 80 81

1,2,4-Trimethylbenzene Isopropylbenzene 1,3,5-Trimethylbenzene p-Isopropyltoluene Naphthalene Biphenyl Styrene m-Terphenyl

C9H12 C9H12 C9H12 C10H14 C10H8 C12H10 C8H8 C18H14

95636 98828 108678 99876 91203 92524 100425 92068

120.194 120.194 120.194 134.221 128.174 154.211 104.152 230.309

5.9126 5.7950 6.0380 6.3314 7.0510 7.5736 5.7260 10.1230

0.35632 0.3956 0.37999 0.40289 0.4612 0.3975 0.4055 0.3767

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

229.33 177.14 228.42 205.25 353.43 342.2 242.54 360

5.0621 5.0869 5.1010 5.4387 5.2508 6.0420 4.7128 8.4070

649.13 631.1 637.36 653.15 748.35 789.26 636 924.85

0 0 0 0 0 0 0 0

82 83 84 85 86 87 88

Methanol Ethanol 1-Propanol 1-Butanol 2-Butanol 2-Propanol 2-Methyl-2-propanol

CH4O C2H6O C3H8O C4H10O C4H10O C3H8O C4H10O

67561 64175 71238 71363 78922 67630 75650

32.042 46.069 60.096 74.123 74.123 60.096 74.123

5.2390 5.6900 6.3300 6.7390 7.2560 6.3080 7.7320

0.3682 0.3359 0.3575 0.173 0.4774 0.3921 0.5645

0 0 0 0.2915 0 0 0

0 0 0 0 0 0 0

175.47 159.05 146.95 184.51 158.45 185.28 298.97

4.4900 5.0245 5.6460 6.0575 6.1383 5.2807 4.6703

512.64 513.92 536.78 563.05 536.05 508.3 506.21

0 0 0 0 0 0 0

89 90 91 92 93 94 95 96

1-Pentanol 2-Methyl-1-butanol 3-Methyl-1-butanol 1-Hexanol 1-Heptanol Cyclohexanol Ethylene glycol 1,2-Propylene glycol

C5H12O C5H12O C5H12O C6H14O C7H16O C6H12O C2H6O2 C3H8O2

71410 137326 123513 111273 111706 108930 107211 57556

88.150 88.150 88.150 102.177 116.203 100.161 62.068 76.095

8.3100 7.7839 8.0815 8.5980 9.6900 9.2440 8.2900 8.0700

0.511 0.45313 0.50185 0.513 0.572 0.64825 0.4266 0.295

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

195.56 203 155.95 228.55 239.15 296.6 260.15 213.15

6.7533 6.3619 6.8999 6.7623 7.3822 6.2273 6.8461 7.1374

586.15 565 577.2 611.35 631.9 650 719.7 626

0 0 0 0 0 0 0 0

97 98 99 100

Phenol o-Cresol m-Cresol p-Cresol

C6H6O C7H8O C7H8O C7H8O

108952 95487 108394 106445

94.113 108.140 108.140 108.140

7.3060 7.1979 8.0082 8.4942

0.4246 0.40317 0.45314 0.50234

0 0 0 0

0 0 0 0

314.06 304.19 285.39 307.93

5.6577 5.7135 6.3326 6.3649

694.25 697.55 705.85 704.65

0 0 0 0

101 102 103 104 105 106 107

Dimethyl ether Methyl ethyl ether Methyl n-propyl ether Methyl isopropyl ether Methyl-n-butyl ether Methyl isobutyl ether Methyl tert-butyl ether

C2H6O C3H8O C4H10O C4H10O C5H12O C5H12O C5H12O

115106 540670 557175 598538 628284 625445 1634044

46.069 60.096 74.123 74.123 88.150 88.150 88.150

2.9940 3.5300 3.9795 3.9305 4.5328 4.2678 4.2024

0.3505 0.376 0.3729 0.3711 0.3824 0.37995 0.37826

0 0 0 0 0 0 0

0 0 0 0 0 0 0

131.65 160 133.97 127.93 157.48 150 164.55

2.6032 2.9751 3.5184 3.4876 3.9358 3.7232 3.6096

400.1 437.8 476.3 464.5 510 497 497.1

0 0 0 0 0 0 0 2-157

TABLE 2-193 Cmpd. no.

Heats of Vaporization of Inorganic and Organic Compounds (Continued ) Name

Formula

CAS no.

Mol wt

C1 × 1E−07

C2

C3

C4

Tmin, K

∆Hv at Tmin × 1E−07

Tmax, K

∆Hv at Tmax

108 109 110 111 112

Diethyl ether Ethyl propyl ether Ethyl isopropyl ether Methyl phenyl ether Diphenyl ether

C4H10O C5H12O C5H12O C7H8O C12H10O

60297 628320 625547 100663 101848

74.123 88.150 88.150 108.140 170.211

4.0600 5.4380 4.2580 5.8662 6.8243

0.3868 0.60624 0.37221 0.37127 0.30877

0 0 0 0 0

0 0 0 0 0

156.85 145.65 140 235.65 300.03

3.4651 4.4140 3.7556 4.9560 5.8546

466.7 500.23 489 645.6 766.8

0 0 0 0 0

113 114 115 116 117 118 119 120 121 122

Formaldehyde Acetaldehyde 1-Propanal 1-Butanal 1-Pentanal 1-Hexanal 1-Heptanal 1-Octanal 1-Nonanal 1-Decanal

CH2O C2H4O C3H6O C4H8O C5H10O C6H12O C7H14O C8H16O C9H18O C10H20O

50000 75070 123386 123728 110623 66251 111717 124130 124196 112312

30.026 44.053 58.080 72.107 86.134 100.161 114.188 128.214 142.241 156.268

3.0760 4.6070 4.1492 4.6403 5.1478 5.6661 6.1299 6.8347 7.3363 7.9073

0.2954 0.62 0.36751 0.3849 0.37541 0.38533 0.37999 0.41039 0.41735 0.4129

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

181.15 150.15 170 176.75 182 217.15 229.8 246 255.15 267.15

2.5863 3.6199 3.5675 3.9797 4.4502 4.7495 5.1353 5.5966 5.9779 6.4201

408 466 504.4 537.2 566.1 591 617 638.1 658 674.2

0 0 0 0 0 0 0 0 0 0

123 124 125 126 127 128 129 130 131 132 133 134

Acetone Methyl ethyl ketone 2-Pentanone Methyl isopropyl ketone 2-Hexanone Methyl isobutyl ketone 3-Methyl-2-pentanone 3-Pentanone Ethyl isopropyl ketone Diisopropyl ketone Cyclohexanone Methyl phenyl ketone

C3H6O C4H8O C5H10O C5H10O C6H12O C6H12O C6H12O C5H10O C6H12O C7H14O C6H10O C8H8O

67641 78933 107879 563804 591786 108101 565617 96220 565695 565800 108941 98862

58.080 72.107 86.134 86.134 100.161 100.161 100.161 86.134 100.161 114.188 98.145 120.151

4.2150 4.6220 5.1740 5.1400 5.6770 5.4000 5.1130 5.2359 5.3880 5.5980 5.5500 6.6104

0.3397 0.355 0.39422 0.3858 0.3817 0.383 0.3395 0.40465 0.40616 0.3774 0.3538 0.37425

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

178.45 186.48 196.29 250 217.35 189.15 167.15 234.18 200 204.81 242 292.81

3.6390 3.9704 4.3663 4.0753 4.7584 4.6294 4.5480 4.2075 4.5154 4.7426 4.7114 5.4166

508.2 535.5 561.08 553 587.05 571.4 573 560.95 567 576 653 709.5

0 0 0 0 0 0 0 0 0 0 0 0

135 136 137 138 139 140 141

Formic acid Acetic acid Propionic acid n-Butyric acid Isobutyric acid Benzoic acid2 Acetic anhydride

CH2O2 C2H4O2 C3H6O2 C4H8O2 C4H8O2 C7H6O2 C4H6O3

64186 64197 79094 107926 79312 65850 108247

46.026 60.053 74.079 88.106 88.106 122.123 102.090

2.3700 2.0265 2.7290 7.4996 4.4967 10.1900 6.3520

1.999 0.11911 0.06954 2.333 1.1615 0.478 0.3986

−5.1503 −1.3487 −1.0423 −3.8644 −2.4573 0 0

3.331 1.4227 1.1152 2.016 1.5823 0 0

281.45 289.81 252.45 267.95 227.15 395.45 200.15

1.9532 2.3185 2.9964 4.1566 3.6179 7.1277 5.4139

588 591.95 600.81 615.7 605 751 606

0 0 0 0 0 0 0

142 143 144 145 146 147 148

Methyl formate Methyl acetate Methyl propionate Methyl n-butyrate Ethyl formate Ethyl acetate Ethyl propionate

C2H4O2 C3H6O2 C4H8O2 C5H10O2 C3H6O2 C4H8O2 C5H10O2

107313 79209 554121 623427 109944 141786 105373

60.053 74.079 88.106 102.133 74.079 88.106 102.133

4.1030 4.4920 5.0080 5.3781 4.5909 4.9330 5.3325

0.3825 0.3685 0.3959 0.39523 0.4123 0.3847 0.401

0 0 0 0 0 0 0

0 0 0 0 0 0 0

174.15 175.15 185.65 187.35 193.55 189.6 199.25

3.4644 3.8418 4.2231 4.5694 3.7679 4.1490 4.4449

487.2 506.55 530.6 554.5 508.4 523.3 546

0 0 0 0 0 0 0

149 150 151 152 153 154 155

Ethyl n-butyrate n-Propyl formate n-Propyl acetate n-Butyl acetate Methyl benzoate Ethyl benzoate Vinyl acetate

C6H12O2 C4H8O2 C5H10O2 C6H12O2 C8H8O2 C9H10O2 C4H6O2

105544 110747 109604 123864 93583 93890 108054

116.160 88.106 102.133 116.160 136.150 150.177 86.090

5.6419 4.9687 5.4327 5.7800 6.9650 6.3400 4.7700

0.37985 0.4025 0.407 0.3935 0.4061 0.2911 0.3765

0 0 0 0 0 0 0

0 0 0 0 0 0 0

175.15 180.25 178.15 199.65 260.75 238.45 180.35

4.9090 4.2162 4.6322 4.8943 5.7500 5.6137 4.0619

571 538 549.73 579.15 693 698 519.13

0 0 0 0 0 0 0

156 157 158

Methylamine Dimethylamine Trimethylamine

CH5N C2H7N C3H9N

74895 124403 75503

31.057 45.084 59.111

3.8580 4.0900 3.3050

0.404 0.42005 0.354

0 0 0

0 0 0

179.69 180.96 156.08

3.1006 3.2678 2.8216

430.05 437.2 433.25

0 0 0

2-158

159 160 161

Ethylamine Diethylamine Triethylamine

C2H7N C4H11N C6H15N

75047 109897 121448

45.084 73.138 101.192

4.2750 4.6133 4.6640

0.5857 0.42628 0.3663

−0.332 0 0

0.169 0 0

192.15 223.35 158.45

3.2955 3.5761 4.1011

456.15 496.6 535.15

0 0 0

162 163 164 165 166 167 168

n-Propylamine di-n-Propylamine Isopropylamine Diisopropylamine Aniline N-Methylaniline N,N-Dimethylaniline

C3H9N C6H15N C3H9N C6H15N C6H7N C7H9N C8H11N

107108 142847 75310 108189 62533 100618 121697

59.111 101.192 59.111 101.192 93.128 107.155 121.182

4.4488 5.4280 4.4041 5.0070 7.1950 6.3860 6.7900

0.39494 0.3665 0.43325 0.4362 0.458 0.3104 0.4053

0 0 0 0 0 0 0

0 0 0 0 0 0 0

188.36 210.15 177.95 176.85 267.13 216.15 275.6

3.6857 4.5500 3.5874 4.1823 5.7710 5.6961 5.5162

496.95 550 471.85 523.1 699 701.55 687.15

0 0 0 0 0 0 0

169 170 171 172

Ethylene oxide Furan Thiophene Pyridine

C2H4O C4H4O C4H4S C5H5N

75218 110009 110021 110861

44.053 68.075 84.142 79.101

3.6652 4.0050 4.5793 5.1740

0.37878 0.3995 0.38557 0.38865

0 0 0 0

0 0 0 0

160.65 196.29 234.94 231.51

3.1271 3.2647 3.7472 4.3144

469.15 490.15 579.35 619.95

0 0 0 0

173 174 175 176 177 178 179 180

Formamide3 N,N-Dimethylformamide Acetamide N-Methylacetamide Acetonitrile Propionitrile n-Butyronitrile Benzonitrile

CH3NO C3H7NO C2H5NO C3H7NO C2H3N C3H5N C4H7N C7H5N

75127 68122 60355 79163 75058 107120 109740 100470

45.041 73.095 59.068 73.095 41.053 55.079 69.106 103.123

7.3580 5.9217 8.1070 7.3402 4.3511 4.9348 5.2200 6.2615

0.3564 0.37996 0.42 0.38974 0.34765 0.41873 0.165 0.35427

0 0 0 0 0 0 0.6692 0

0 0 0 0 0 0 −0.539 0

275.7 212.72 353.15 301.15 229.32 180.26 161.25 260.4

6.2844 5.0931 6.2386 5.9384 3.5996 4.2005 4.7223 5.3091

771 649.6 761 718 545.5 564.4 582.25 699.35

0 0 0 0 0 0 0 0

181 182 183 184 185 186 187 188 189

Methyl mercaptan Ethyl mercaptan n-Propyl mercaptan n-Butyl mercaptan Isobutyl mercaptan sec-Butyl mercaptan Dimethyl sulfide Methyl ethyl sulfide Diethyl sulfide

CH4S C2H6S C3H8S C4H10S C4H10S C4H10S C2H6S C3H8S C4H10S

74931 75081 107039 109795 513440 513531 75183 624895 352932

48.109 62.136 76.163 90.189 90.189 90.189 62.136 76.163 90.189

3.4448 3.8440 4.4782 4.9702 4.7420 4.6432 3.8690 4.4740 4.7182

0.37427 0.37534 0.41073 0.41199 0.40535 0.399 0.3694 0.4097 0.3643

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

150.18 125.26 159.95 157.46 128.31 133.02 174.88 167.23 169.2

2.9825 3.4489 3.8723 4.3505 4.2664 4.1614 3.3042 3.8344 4.1353

469.95 499.15 536.6 570.1 559 554 503.04 533 557.15

0 0 0 0 0 0 0 0 0

190 191 192 193 194 195 196 197

Fluoromethane Chloromethane Trichloromethane Tetrachloromethane Bromomethane Fluoroethane Chloroethane Bromoethane

CH3F CH3Cl CHCl3 CCl4 CH3Br C2H5F C2H5Cl C2H5Br

593533 74873 67663 56235 74839 353366 75003 74964

34.033 50.488 119.377 153.822 94.939 48.060 64.514 108.966

2.4708 2.9745 4.1860 4.3252 3.1690 2.7617 3.5240 3.9004

0.37014 0.353 0.3584 0.37688 0.3015 0.32162 0.3652 0.38012

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

131.35 175.43 209.63 250.33 179.47 129.95 134.8 154.55

2.0276 2.4520 3.5047 3.4528 2.7379 2.4089 3.1052 3.3933

317.42 416.25 536.4 556.35 467 375.31 460.35 503.8

0 0 0 0 0 0 0 0

198 199 200 201 202 203 204 205

1-Chloropropane 2-Chloropropane 1,1-Dichloropropane 1,2-Dichloropropane Vinyl chloride Fluorobenzene Chlorobenzene Bromobenzene

C3H7Cl C3H7Cl C3H6Cl2 C3H6Cl2 C2H3Cl C6H5F C6H5Cl C6H5Br

540545 75296 78999 78875 75014 462066 108907 108861

78.541 78.541 112.986 112.986 62.499 96.104 112.558 157.010

3.9890 3.8871 4.7740 4.6750 3.4125 4.5820 5.1480 5.5520

0.37956 0.38043 0.39204 0.36529 0.4513 0.3717 0.36614 0.37694

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

150.35 155.97 200 172.71 119.36 230.94 227.95 242.43

3.4862 3.3586 4.0147 4.0997 2.9491 3.7605 4.3707 4.6875

503.15 489 560 572 432 560.09 632.35 670.15

0 0 0 0 0 0 0 0

206 207 208 209 210 211 212 213

Air Hydrogen Helium-4 Neon Argon Fluorine Chlorine Bromine

H2 He Ne Ar F2 Cl2 Br2

132259100 1333740 7440597 7440019 7440371 7782414 7782505 7726956

28.951 2.016 4.003 20.180 39.948 37.997 70.905 159.808

0.8474 0.1013 0.0125 0.2389 0.8731 0.8876 3.0680 4.0000

0.3822 0.698 1.3038 0.3494 0.3526 0.34072 0.8458 0.351

0 −1.817 −2.6954 0 0 0 −0.9001 0

0 1.447 1.7098 0 0 0 0.453 0

59.15 13.95 2.2 24.56 83.78 53.48 172.12 265.85

0.6759 0.0913 0.0097 0.1803 0.6561 0.7578 2.2878 3.2323

132.45 33.19 5.2 44.4 150.86 144.12 417.15 584.15

0 0 0 0 0 0 0 0 2-159

TABLE 2-193 Cmpd. no.

Heats of Vaporization of Inorganic and Organic Compounds (Concluded ) Name

Formula

CAS no.

Mol wt

C1 × 1E−07

C2

C3

C4

Tmin, K

∆Hv at Tmin × 1E−07

Tmax, K

∆Hv at Tmax

214 215 216 217 218

Oxygen Nitrogen Ammonia Hydrazine Nitrous oxide

O2 N2 NH3 N2H4 N2O

7782447 7727379 7664417 302012 10024972

31.999 28.014 17.031 32.045 44.013

0.9008 0.7491 3.1523 5.9794 2.3215

0.4542 0.40406 0.3914 0.9424 0.384

−0.4096 −0.317 −0.2289 −1.398 0

0.3183 0.27343 0.2309 0.8862 0

54.36 63.15 195.41 274.69 182.3

0.7742 0.6024 2.5298 4.5238 1.6502

154.58 126.2 405.65 653.15 309.57

0 0 0 0 0

219 220 221 222 223 224 225 226 227 228 229 230 231

Nitric oxide Cyanogen Carbon monoxide Carbon dioxide Carbon disulfide Hydrogen fluoride Hydrogen chloride Hydrogen bromide Hydrogen cyanide2 Hydrogen sulfide Sulfur dioxide Sulfur trioxide Water

NO C2N2 CO CO2 CS2 HF HCl HBr HCN H2S SO2 SO3 H2O

10102439 460195 630080 124389 75150 7664393 7647010 10035106 74908 7783064 7446095 7446119 7732185

30.006 52.036 28.010 44.010 76.143 20.006 36.461 80.912 27.026 34.082 64.065 80.064 18.015

2.1310 3.3840 0.8585 2.1730 3.4960 13.4510 2.2093 2.4850 3.3490 2.5676 3.6760 7.3370 5.2053

0.4056 0.3707 0.4921 0.382 0.2986 13.36 0.3466 0.39 0.2053 0.37358 0.4 0.5647 0.3199

0 0 −0.326 −0.4339 0 −23.383 0 0 0 0 0 0 −0.212

0 0 0.2231 0.42213 0 10.785 0 0 0 0 0 0 0.25795

109.5 245.25 68.13 216.58 161.11 277.56 158.97 185.15 259.83 187.68 197.67 289.95 273.16

1.4578 2.3803 0.6517 1.5202 3.1537 0.7104 1.7498 1.8817 2.8176 1.9782 2.8753 4.4303 4.4733

180.15 400.15 132.5 304.21 552 461.15 324.65 363.15 456.65 373.53 430.75 490.85 647.13

0 0 915280 0 0 0 0 0 0 0 0 0 0

All substances are listed in alphabetical order in Table 2-6a. Compiled from Daubert, T. E., R. P. Danner, H. M. Sibul, and C. C. Stebbins, DIPPR Data Compilation of Pure Compound Properties, Project 801 Sponsor Release, July, 1993, Design Institute for Physical Property Data, AIChE, New York, NY; and from Thermodynamics Research Center, “Selected Values of Properties of Hydrocarbons and Related Compounds,” Thermodynamics Research Center Hydrocarbon Project, Texas A&M University, College Station, Texas (extant 1994). Temperatures are expressed in kelvins; heats of vaporization, in J/kmol. J/ kmol × 2.390E−04 = cal/gmol; J/ kmol × 4.302106E−04 = Btu/lbmol. The heat of vaporization equation used is ∆Hv = C1 × (1 − Tr)C2 + C3 × Tr + C4 × Tr × Tr. Tr is the reduced temperature, T/Tc. 1 Coefficients are hypothetical; compound decomposes violently on heating. 2 For the monomer. 3 Equation coefficients are hypothetical above the decomposition temperature. 2-160

SPECIFIC HEATS OF PURE COMPOUNDS

2-161

SPECIFIC HEATS OF PURE COMPOUNDS To convert kilojoules per kilogram-kelvin to British thermal units per pound-degree Rankine, multiply by 0.2388.

UNITS CONVERSIONS For this subsection, the following units conversions are applicable: °F = 9⁄ 5 °C + 32 °F = 1.8 K

ADDITIONAL REFERENCES

To convert calories per gram-kelvin to British thermal units per pound-degree Rankine, multiply by 1.0; to convert calories per grammole-kelvin to British thermal units per pound-mole-degree Rankine, multiply by 1.0. TABLE 2-194

Heat Capacities of the Elements and Inorganic Compounds* Heat capacity at constant pressure (T = K; 0°C = 273.1 K), cal/deg mol

Range of temperature, K

Uncertainty, %

c l c l c l c c c c l c l c c, sillimanite c, disthene c, andalusite c, mullite c c c

4.80 + 0.00322T 7.00 18.74 + 0.01866T 29.5 13.25 + 0.02800T 31.2 76 19.3 50.5 38.63 + 0.04760T − 449200/T 2 142 16.88 + 0.02266T 28.8 22.08 + 0.008971T − 522500/T 2 40.79 + 0.004763T − 992800/T 2 41.81 + 0.005283T − 1211000/T 2 43.96 + 0.001923T − 1086000/T 2 59.65 + 0.0670T 113.2 + 0.0652T 63.5 235

273–931 931–1273 273–370 370–407 273–465 465–504 288–327 288–326 288–326 273–1273 1273–1373 273–464 464–480 273–1973 273–1573 273–1673 273–1573 273–576 273–575 273–373 288–325

1 5 3 5 3 3 ? ? ? 2 ? 3 5 3 3 2 3 5 3 ? ?

c l c c c c c

5.51 + 0.00178T 7.15 17.2 + 0.0293T 10.3 + 0.0511T 19.1 + 0.0171T 22.6 + 0.0162T 24.2 + 0.0132T

273–903 903–1273 273–370 273–346 273–929 273–1198 273–821

2 5 ? ? ? ? ?

Substance Aluminum1 Al AlBr3 AlCl3 AlCl3·6H2O AlF3 AlF3·3aH2O AlF3·3NaF AlI3 Al2O3 Al2O3·SiO2 3Al2O3·2SiO2 4Al2O3·3SiO2 Al2(SO4)3 Al2(SO4)3·18H2O Antimony Sb SbBr3 SbCl3 Sb2O3 Sb2O4 Sb2S3 Argon2 A Arsenic As AsCl3 As2O3 As2S3 Barium BaCl2 BaCl2·H2O BaCl2·2H2O Ba(ClO3)2·H2O BaCO3 BaMoO4 Ba(NO3)2 BaSO4 Beryllium3,4 Be BeO BeO·Al2O3 BeSO4

Additional data are contained in the subsection “Thermodynamic Properties.” Data on water are also contained in that subsection. Additional tables for water are found in Eng. Sci. Data Item 68008, 251 Regent Street, London, England, which contains about 5000 values from 1 to 1000 bar, 0 to 1500°C.

State†

g

4.97

c l c c

5.17 + 0.00234T 31.9 8.37 + 0.0486T 25.8

273–1168 286–371 273–548 293–373

All

5 ? ? ?

0

c c c c c, α c, β c c c

17.0 + 0.00334T 28.2 37.3 51 17.26 + 0.0131T 30.0 34 39.8 21.35 + 0.0141T

273–1198 273–307 273–307 289–320 273–1083 1083–1255 273–297 285–371 273–1323

? ? ? ? 5 15 ? ? 5

c c c c

4.698 + 0.001555T − 121000/T 2 8.69 + 0.00365T − 313000/T 2 25.4 20.8

273–1173 273–1175 273–373 273–373

1 5 ? ?

*From Kelley, U.S. Bur. Mines Bull. 371, 1934. For a revision see Kelley, U.S. Bur. Mines Bull. 477, 1948. Data for many elements and compounds are given by Johnson (ed.), WADD-TR-60-56, 1960, for cryogenic temperatures. Tabulated data for gases can be obtained from many of the references cited in the “Thermodynamic Properties” subsection and other tables in this section. Thinh, Duran, et al., Hydrocarbon Process., 50, 98 (January 1971), review previous equation fits and give newer fits for 408 hydrocarbons and related compounds. Later publications include Duran, Thinh, et al., Hydrocarbon Process., 55, 153 (August 1976); Thompson, J. Chem. Eng. Data, 22(4), 431 (1977); and Passut and Danner, Ind. Eng. Chem. Process Des. Dev., 11, 543 (1972); 13, 193 (1974). † The symbols in this column have the following meaning; c, crystal; l, liquid; g, gas; gls, glass.

TABLE 2-194

Heat Capacities of the Elements and Inorganic Compounds (Continued )

Substance Bismuth4 Bi Bi2O3 Bi2S3 Boron B B2O3 BN Bromine Br2 Cadmium Cd CdO CdS CdSO4·8/3H2O Calcium Ca CaCl2 CaCO3 CaF2 CaMg(CO3)2 CaMoO4 CaO Ca(OH)2 CaO·Al2O3·2SiO2 CaO·MgO·2SiO2 CaO·SiO2 CaP2O6 CaSO4 CaSO4·2H2O CaWO4 Carbon5 C CH4 CO6 CO2 CS2 Cerium Ce CeO2 Ce2(MoO4)3 Ce2(SO4)3 Ce2(SO4)3·5H2O Cesium Cs CsBr CsCl CsF CsI Chlorine Cl2 Chromium4 Cr CrCl3 Cr2O3 CrSb CrSb2 Cr2(SO4)3 Cobalt4 Co CoAs2·CoS2 CoSb Co2Sn CoS CoSO4·7H2O 2-162

State†

Heat capacity at constant pressure (T = K; 0°C = 273.1 K), cal/deg mol

Range of temperature, K

Uncertainty, %

c l c c

5.38 + 0.00260T 7.60 23.27 + 0.01105T 30.4

273–544 544–1273 273–777 284–372

3 3 2 ?

c gls gls c

1.54 + 0.00440T 5.14 + 0.0320T 30.4 1.61 + 0.00400T

273–1174 273–513 513–623 273–1173

5 3 3 5

g

9.00

300–2000

5

c l c c c

5.46 + 0.002466T 7.13 9.65 + 0.00208T 12.9 + 0.00090T 51.3

273–594 594–973 273–2086 273–1273 293

1 5 ? ? ?

c c c c c c c c c c, anorthite gls c, diopside gls c, wollastonite c, pseudowollastonite gls c c c c

5.31 + 0.00333T 6.29 + 0.00140T 16.9 + 0.00386T 19.68 + 0.01189T − 307600/T 2 14.7 + 0.00380T 40.1 33 10.00 + 0.00484T − 108000/T 2 21.4 63.13 + 0.01500T − 1537000/T 2 67.41 + 0.01048T − 1874000/T 2 54.46 + 0.005746T − 1500000/T 2 51.68 + 0.009724T − 1308000/T 2 27.95 + 0.002056T − 745600/T 2 25.48 + 0.004132T − 488100/T 2 23.16 + 0.009672T − 487100/T 2 39.5 18.52 + 0.02197T − 156800/T 2 46.8 27.9

273–673 673–873 273–1055 273–1033 273–1651 299–372 273–297 273–1173 276–373 273–1673 273–973 273–1573 273–973 273–1573 273–1673 273–973 287–371 273–1373 282–373 292–322

2 2 ? 3 ? ? ? 2 ? 1 1 1 1 1 1 1 ? 5 ? ?

c, graphite c, diamond g g g l

2.673 + 0.002617T − 116900/T 2 2.162 + 0.003059T − 130300/T 2 5.34 + 0.0115T 6.60 + 0.00120T 10.34 + 0.00274T − 195500/T 2 18.4

273–1373 273–1313 273–1200 273–2500 273–1200 293

2 3 2 1a 1a ?

c c c c c

5.88 + 0.00123T 15.1 96 66.4 131.6

273–908 273–373 273–297 273–373 273–319

? ? ? ? ?

c l g c c c c

1.96 + 0.0182T 8.00 4.97 12.6 + 0.00259T 11.7 + 0.00309T 11.3 + 0.00285T 11.6 + 0.00268T

273–301 302 All 273–909 273–752 273–957 273–894

3 3 0 ? ? ? ?

g

8.28 + 0.00056T

273–2000

1a

c l c c c c c

4.84 + 0.00295T 9.70 23 26.0 + 0.00400T 12.3 + 0.00120T 19.2 + 0.00184T 67.4

273–1823 1823–1923 286–319 273–2263 273–1383 273–949 273–373

5 10 ? ? ? ? ?

c l c c c c c

5.12 + 0.00333T 8.40 32.9 11.7 + 0.00156T 15.83 + 0.00950T 10.6 + 0.00251T 96

273–1763 1763–1873 283–373 273–1464 273–903 273–1373 286–303

5 5 ? ? 2 ? ?

TABLE 2-194

Heat Capacities of the Elements and Inorganic Compounds (Continued )

Substance Copper7 Cu CuAl CuAl2 Cu3Al CuI CuI2 CuO CuO·SiO2·H2O CuS Cu2S CuS·FeS Cu2Sb Cu2Sb Cu2Se Cu3Si CuSO4 CuSO4·H2O CuSO4·3H2O CuSO4·5H2O Fluorine8 F2 Gallium Ga2O3 Ga2(SO4)3 Germanium4 Ge Gold Au AuSb2 Helium9 He Hydrogen10 H H2 HBr HCl HI H2O H2S H2S2O7 Indium In Iodine I2 Iridium Ir Iron4 Fe

FeAs2 Fe3C FeCO3 FeO Fe2O3 Fe3O4 Fe2O3·3H2O FeS FeS2 FeSi Fe2SiO4 FeSO4 Fe2(SO4)3 FeSO4·4H2O FeSO4·7H2O Krypton Kr

State†

Heat capacity at constant pressure (T = K; 0°C = 273.1 K), cal/deg mol

Range of temperature, K

Uncertainty, %

273–1357 1357–1573 273–733 273–773 273–775 273–675 274–328 273–810 293–323 273–1273 273–376 376–1173 292–321 273–573 273–693 273–383 383–488 273–1135 282 282 282 282

1 3 2 2 2 ? ? 2 ? ? 3 2 ? 2 2 5 5 ? ? ? ? ?

c l c c c c c c c c c, α c, β c c c c, α c, β c c c c c

5.44 + 0.001462T 7.50 9.88 + 0.00500T 16.78 + 0.00366T 19.61 + 0.01054T 12.1 + 0.00286T 20.1 10.87 + 0.003576T − 150600/T 2 29 10.6 + 0.00264T 9.38 + 0.0312T 20.9 24 13.73 + 0.01350T 21.79 + 0.00900T 20.85 20.35 20.3 + 0.00587T 24.1 31.3 49.0 67.2

g

6.50 + 0.00100T

300–3000

5

c c

18.2 + 0.0252T 62.4

273–923 273–373

? ?

273–1336 1336–1573 273–628 628–713

2 5 1 ?

c c l c, α c, βγ

5.61 + 0.00144T 7.00 17.12 + 0.00465T 11.47 + 0.01756T

g

4.97

g g g g g l g g c l

4.97 6.62 + 0.00081T 6.80 + 0.00084T 6.70 + 0.00084T 6.93 + 0.00083T See Tables 2-355 through 2-357 8.22 + 0.00015T + 0.00000134T 2 7.20 + 0.00360T 27 58

All 273–2500 273–2000 273–2000 273–2000

0 2 2 1a 2

300–2500 300–600 281 308

? 8 ? ?

g

9.00

300–2000

5

c

5.50 + 0.00148T

273–1873

1

c, α c, β c, γ c, δ l c c c c c c c c, α c, β c c c c c c c

4.13 + 0.00638T 6.12 + 0.00336T 8.40 10.0 8.15 17.8 25.17 + 0.00223T 22.7 12.62 + 0.001492T − 76200/T 2 24.72 + 0.01604T − 423400/T 2 41.17 + 0.01882T − 979500/T 2 47.8 2.03 + 0.0390T 12.05 + 0.00273T 10.7 + 0.01336T 10.54 + 0.00458T 33.57 + 0.01907T − 879700/T 2 22 66.2 63.6 96

273–1041 1041–1179 1179–1674 1674–1803 1803–1873 283–373 273–1173 293–368 273–1173 273–1097 273–1065 286–373 273–411 411–1468 273–773 273–903 273–1161 293–373 273–373 282 291–319

3 3 5 5 5 ? 10 ? 2 2 2 ? 5 3 ? 2 2 ? ? ? ?

g

4.97

All

0

c

All

0 2-163

2-164

PHYSICAL AND CHEMICAL DATA TABLE 2-194

Heat Capacities of the Elements and Inorganic Compounds (Continued )

Substance Lanthanum La La2O3 La2(MoO4)3 La2(SO4)3 La2(SO4)3·9H2O Lead4 Pb Pb3(AsO4)2 PbB2O4 PbB4O7 PbBr2 PbCl2 2PbCl2·NH4Cl PbCO3 PbCrO4 PbF2 PbI2 PbMoO4 Pb(NO3)2 PbO PbO2 Pb2P2O7 PbS PbSO4 PbS2O3 PbWO4 Lithium Li LiBr LiBr·H2O LiCl LiCl·H2O LiF LiI LiI·H2O LiI·2H2O LiI·3H2O LiNO3 Magnesium4 Mg MgAg Mg4Al3 MgAu Mg2Au Mg3Au MgCl2 MgCl2·6H2O MgCO3 MgCu2 Mg2Cu MgNi2 MgO MgO·Al2O3 MgO·SiO2 6MgO·MgCl2·8B2O3 Mg(OH)2 Mg3Sb2 Mg2Si MgSO4 MgSO4·H2O MgSO4·6H2O MgSO4·7H2O

State†

Heat capacity at constant pressure (T = K; 0°C = 273.1 K), cal/deg mol

Range of temperature, K

Uncertainty, %

c c c c c

5.91 + 0.00100T 22.6 + 0.00544T 86 66.9 152

273–1009 273–2273 273–307 273–373 273–319

? ? ? ? ?

c l c c c c l c l c c c c c l c c c c c c c c c

5.77 + 0.00202T 6.8 65.5 26.5 41.4 18.13 + 0.00310T 27.4 15.88 + 0.00835T 27.2 53.1 21.1 29.1 16.5 + 0.00412T 18.66 + 0.00293T 32.3 30.4 36.4 10.33 + 0.00318T 12.7 + 0.00780T 48.3 10.63 + 0.00401T 26.4 29 35

273–600 600–1273 286–370 288–371 289–371 273–761 761–860 273–771 771–851 293 286–320 292–323 273–1091 273–648 648–776 292–322 286–320 273–544 273–? 284–371 273–873 293–372 293–373 273–297

2 5 ? ? ? 2 10 2 10 ? ? ? ? 2 20 ? ? 2 ? ? 3 ? ? ?

c g c c c c c c c c c c l

0.68 + 0.0180T 4.97 11.5 + 0.00302T 22.6 11.0 + 0.00339T 23.6 8.20 + 0.00520T 12.5 + 0.00208T 23.6 32.9 43.2 9.17 + 0.0360T 26.8

273–459 All 273–825 278–318 273–887 279–360 273–1117 273–723 277–359 277–345 277–347 273–523 523–575

10 0 ? ? ? ? ? ? ? ? ? 5 5

c l c c c c c c c c c c c c c c, amphibole c, pyroxene gls c, α c, β c c c c c c c

6.20 + 0.00133T − 67800/T 2 7.4 10.58 + 0.00412T 34.4 + 0.0198T 11.3 + 0.00189T 16.2 + 0.00451T 21.2 + 0.00614T 17.3 + 0.00377T 77.1 16.9 14.96 + 0.00776T 15.5 + 0.00652T 15.87 + 0.00692T 10.86 + 0.001197T − 208700/T 2 28 25.60 + 0.004380T − 674200/T 2 23.35 + 0.008062T − 558800/T 2 23.30 + 0.007734T − 542000/T 2 58.7 + 0.408T 107.2 + 0.2876T 18.2 28.2 + 0.00560T 15.4 + 0.00415T 26.7 33 80 89

273–923 923–1048 273–905 273–736 273–1433 273–1073 273–1103 273–991 292–342 290 273–903 273–843 273–903 273–2073 288–319 273–1373 273–773 273–973 273–538 538–623 292–323 273–1234 273–1343 296–372 282 282 291–319

1 10 2 ? ? ? ? ? ? ? 3 ? 2 2 ? 1 1 1 5 5 ? ? ? ? ? ? ?

SPECIFIC HEATS OF PURE COMPOUNDS TABLE 2-194

Heat Capacities of the Elements and Inorganic Compounds (Continued ) Heat capacity at constant pressure (T = K; 0°C = 273.1 K), cal/deg mol

Range of temperature, K

Uncertainty, %

c, α c, β c, γ l c c c c c c c c c c

3.76 + 0.00747T 5.06 + 0.00395T 4.80 + 0.00422T 11.0 16.2 + 0.00520T 7.79 + 0.0421T + 0.0000090T 2 7.43 + 0.01038T − 0.00000362T 2 10.33 + 0.0530T − 0.0000257T 2 19.25 + 0.0538T − 0.0000209T 2 1.92 + 0.0471T − 0.0000297T 2 31 10.21 + 0.00656T − 0.00000242T 2 27.5 78

273–1108 1108–1317 1317–1493 1493–1673 273–923 273–773 273–1923 273–1173 273–1773 273–773 291–322 273–1883 293–373 290–319

5 5 5 10 ? ? ? ? ? ? ? ? ? ?

l g g c c c c c, α c, β c c c

6.61 4.97 9.00 11.05 + 0.00370T 15.3 + 0.0103T 25 11.4 + 0.00461T 17.4 + 0.004001T 20.2 11.5 10.9 + 0.00365T 31.0

273–630 All 300–2000 273–798 273–553 285–319 273–563 273–403 403–523 278–371 273–853 273–307

1 0 5 ? ? ? ? 3 3 ? ? ?

c c c

5.69 + 0.00188T − 50300/T 2 15.1 + 0.0121T 19.7 + 0.00315T

273–1773 273–1068 273–729

5 ? ?

g

4.97

c, α c, β l c c c c c c c c

4.26 + 0.00640T 6.99 + 0.000905T 8.55 11.3 + 0.00215T 9.25 + 0.00640T 15.8 + 0.00329T 10.0 + 0.00312T 20.78 + 0.0102T 33.4 82 11.00 + 0.00433T

g g c c, α c, β c c c g

Substance Manganese Mn

MnCl2 MnCO3 MnO Mn2O3 Mn3O4 MnO2 Mn2O3·H2O MnS MnSO4 MnSO4·5H2O Mercury11 Hg Hg2 HgCl HgCl2 Hg(CN)2 HgI HgI2 HgO HgS Hg2SO4 Molybdenum Mo MoO3 MoS2 Neon12 Ne Nickel4 Ni NiO NiS Ni2Si NiSi Ni3Sn NiSO4 NiSO4·6H2O NiTe Nitrogen13 N2 NH3 NH4Br NH4Cl NH4I NH4NO3 (NH4)2SO4 NO Osmium Os Oxygen14 O2 Palladium Pd Phosphorus P PCl3 P4O10 Platinum4 Pt Potassium K

State†

All

0

273–626 626–1725 1725–1903 273–1273 273–597 273–1582 273–1273 273–904 293–373 291–325 273–700

2 5 10 ? 3 ? ? 2 ? ? 2

6.50 + 0.00100T 6.70 + 0.00630T 22.8 9.80 + 0.0368T 5.0 + 0.0340T 17.8 31.8 51.6 8.05 + 0.000233T − 156300/T 2

300–3000 300–800 274–328 273–457 457–523 273–328 273–293 275–328 300–5000

3 1a ? 5 5 ? ? ? 2

c

5.686 + 0.000875T

273–1877

1

g

8.27 + 0.000258T − 187700/T

300–5000

1

2

c

5.41 + 0.00184T

273–1822

2

c, yellow c, red l l c g

5.50 0.21 + 0.0180T 6.6 28.7 15.72 + 0.1092T 73.6

273–317 273–472 317–373 284–371 273–631 631–1371

5 10 10 ? 2 3

c

5.92 + 0.00116T

273–1873

1

c l

5.24 + 0.00555T 7.7

273–336 336–373

5 5

2-165

2-166

PHYSICAL AND CHEMICAL DATA TABLE 2-194

Heat Capacities of the Elements and Inorganic Compounds (Continued )

Substance Potassium—(Cont.) K K2 KAsO3 KBO2 K2B4O7 KBr KCl KClO3 KClO4 2KCl·CuCl2.2H2O 2KCl·PtCl4 2KCl·SnCl4 2KCl·ZnCl2 2KCN·Zn(CN)2 K2CO3 K2CrO4 K2Cr2O7 KF K4Fe(CN)6 K4Fe(CN)6·3H2O KH2AsO4 KH2PO4 KHSO4 KMnO4 KNO3 K2O·Al2O3·3SiO2

K4P2O7 K2SO4 K2S2O3 K2SO4·Al2(SO4)3·24H2O K2SO4·Cr2(SO4)3·24H2O K2SO4·MgSO4·6H2O K2SO4·NiSO4·6H2O K2SO4·ZnSO4·6H2O Prometheum Pr Radon Rn Rhenium Re Rhodium Rh Rubidium Rb RbBr RbCl Rb2CO3 RbF RbI Scandium Sc2O3 Sc2(SO4)3 Selenium Se Silicon Si SiC SiCl4 SiO2

Silver4 Ag

State† g g c c c c c c c c c c c c c c c l c c c c c c c c c l c, orthoclase gls, orthoclase c, microcline gls, microcline c c c c c c c c

Heat capacity at constant pressure (T = K; 0°C = 273.1 K), cal/deg mol 4.97 9.00 25.3 12.6 + 0.0126T 51.3 11.49 + 0.00360T 10.93 + 0.00376T 25.7 26.3 63 55 54.5 43.4 57.4 29.9 35.9 42.80 + 0.0410T 96.9 10.8 + 0.00284T 80.1 114.5 32 28.3 30 28 6.42 + 0.0530T 28.8 29.5 69.26 + 0.00821T − 2331000/T 2 69.81 + 0.01053 − 2403000/T 2 65.65 + 0.01102T − 1748000/T 2 64.83 + 0.01438T − 1641000/T 2 63.1 33.1 37 352 324 106 107 120

Range of temperature, K

Uncertainty, %

All 300–2000 290–372 273–1220 290–372 273–543 273–1043 289–371 287–318 292–323 286–319 292–323 279–319 277–319 296–372 289–371 273–671 671–757 273–1129 273–319 273–310 289–319 290–320 292–324 287–318 273–401 401–611 611–683 273–1373 273–1373 273–1373 273–1373 290–371 287–371 293–373 292–322 292–324 292–323 289–319 293–317

0 5 ? ? ? 2 2 ? ? ? ? ? ? ? ? ? 5 5 ? ? ? ? ? ? ? 10 5 10 1a 1a 1a 1a ? ? ? ? ? ? ? ?

c g

4.97

c

6.30 + 0.00053T

273–2273

All

0 ?

c

5.40 + 0.00219T

273–1877

2

c l c c c c c

3.27 + 0.0131T 7.85 11.6 + 0.00255T 11.5 + 0.00249T 28.4 11.3 + 0.00256T 11.6 + 0.00263T

273–312 312–373 273–954 273–987 291–320 273–1048 273–913

2 5 ? ? ? ? ?

c c

21.1 62.0

273–373 273–373

? ?

c l

4.53 + 0.00550T 8.35

273–490 490–570

2 3

c c l c, quartz, α c, quartz, β c, cristobalite, α c, cristobalite, β gls

5.74 + 0.000617T − 101000/T 2 8.89 + 0.00291T − 284000/T 2 32.4 10.87 + 0.008712T − 241200/T 2 10.95 + 0.00550T 3.65 + 0.0240T 17.09 + 0.000454T − 897200/T 2 12.80 + 0.00447T − 302000/T 2

273–1174 273–1629 293–373 273–848 848–1873 273–523 523–1973 273–1973

2 2 ? 1 3a 2a 2 3a

c l

5.60 + 0.00150T 8.2

273–1234 1234–1573

1 3

SPECIFIC HEATS OF PURE COMPOUNDS TABLE 2-194

Heat Capacities of the Elements and Inorganic Compounds (Continued )

Substance Silver—(Cont.) Ag3Al Ag2Al AgAl12 AgBr AgCl AgCNO AgI AgNO3 Ag3PO4 Ag2S Ag3Sb Ag2Se Sodium15 Na NaBO2 Na2B4O7 Na2B4O7·10H2O NaBr NaCl NaClO3 NaCNO Na2CO3 NaF Na2HPO4·7H2O Na2HPO4·12H2O NaI NaNO3 Na2O·Al2O3·3SiO2 NaPO3 Na4P2O7 Na2SO4 Na2S2O3 Na2S2O3·5H2O Sodium-potassium alloys15 Strontium SrBr2 SrBr2·H2O SrBr2·6H2O SrCl2 SrCl2·H2O SrCl2·2H2O SrCO3 SrI2 SrI2·H2O SrI2·2H2O SrI2·6H2O SrMoO4 Sr(NO3)2 SrSO4 Sulfur16 S S2 S2Cl2 SO2 Tantalum Ta Tellurium Te Thallium Tl

State†

Heat capacity at constant pressure (T = K; 0°C = 273.1 K), cal/deg mol

Range of temperature, K

Uncertainty, %

273–902 273–903 273–768 273–703 703–836 273–728 728–806 273–353 273–423 273–433 433–482 482–541 293–325 273–448 448–597 273–694 273–406 406–460

2 2 5 6 5 2 5 ? 6 2 5 5 ? 5 5 5 5 5

273–371 371–451 All 273–1239 289–371 292–323 273–543 273–1074 1073–1205 273–528 528–572 273–353 288–371 273–1261 275–307 275–307 273–936 273–583 583–703 273–1373 273–1173 290–319 290–371 289–371 273–307 273–307

1a 2 0 ? ? ? 2 2 3 3 5 ? ? ? ? ? ? 5 10 1 1 ? ? ? ? ?

c c c c l c l c c, α c, α c, β l c c, α c, β c c, α c, β

22.56 + 0.00570T 16.85 + 0.00450T 58.62 + 0.0575T 8.58 + 0.0141T 14.9 9.60 + 0.00929T 14.05 18.7 8.58 + 0.0141T 18.83 + 0.0160T 25.7 30.2 37.5 18.8 21.8 19.53 + 0.0160T 20.2 20.4

c l g c c c c c l c l c c c c c c c l c, albite gls c c c c c l

5.01 + 0.00536T 7.50 4.97 10.4 + 0.0199T 47.9 147 11.74 + 0.00233T 10.79 + 0.00420T 15.9 9.48 + 0.0468T 31.8 13.1 28.9 10.4 + 0.00289T 86.6 133.4 12.5 + 0.00162T 4.56 + 0.0580T 37.2 63.78 + 0.01171T − 1678000/T 2 61.25 + 0.01768T − 1545000/T 2 22.1 60.7 32.8 34.9 86.2

c c c c c c c c c c c c c c

18.1 + 0.00311T 28.9 82.1 18.2 + 0.00244T 28.7 38.3 21.8 18.6 + 0.00304T 28.5 39.1 84.9 37 38.3 26.2

273–923 277–370 276–327 273–1143 276–365 277–366 281–371 273–783 276–363 275–336 275–333 273–297 290–320 293–369

? ? ? ? ? ? ? ? ? ? ? ? ? ?

c, rhombic c, monoclinic g l g

3.63 + 0.00640T 4.38 + 0.00440T 8.58 + 0.00030T 27.5 7.70 + 0.00530T − 0.00000083T 2

273–368 368–392 300–2500 273–332 300–2500

3 3 5 ? 2a

c

5.91 + 0.00099T

273–1173

2

c

5.19 + 0.00250T

273–600

3

c, α c, β

5.32 + 0.00385T 8.12

273–500 500–576

1 1

2-167

2-168

PHYSICAL AND CHEMICAL DATA TABLE 2-194

Heat Capacities of the Elements and Inorganic Compounds (Concluded )

Substance Thallium—(Cont.) Tl TlBr TlCl Thorium Th ThO2 Th(SO4)2 Tin4 Sn SnAu SnCl2 SnCl4 SnO SnO2 SnPt SnS SnS2 Titanium Ti TiCl4 TiO2 Tungsten W WO3 Uranium U U3O8 Vanadium V Xenon Xe Zinc4 Zn ZnCl2 ZnO ZnS ZnSb ZnSO4 ZnSO4·H2O ZnSO4·6H2O ZnSO4·7H2O Zirconium ZrO2 ZrO2·SiO2 1

State†

Heat capacity at constant pressure (T = K; 0°C = 273.1 K), cal/deg mol

Range of temperature, K

Uncertainty, %

l c l c l

7.12 12.53 + 0.00100T 16.0 12.56 + 0.00088T 14.2

576–773 273–733 733–800 273–700 700–803

3 10 10 5 10

c c c

6.40 14.6 + 0.00507T 41.2

273–373 273–1273 273–373

? ? ?

c l c c l c c c c c

5.05 + 0.00480T 6.6 11.79 + 0.00233T 16.2 + 0.00926T 38.4 9.40 + 0.00362T 13.94 + 0.00565T − 252000/T 2 11.49 + 0.00190T 12.1 + 0.00165T 20.5 + 0.00400T

273–504 504–1273 273–581 273–520 286–371 273–1273 273–1373 273–1318 273–1153 273–873

2 10 1 ? ? ? ? 1 ? ?

c l c

8.91 + 0.00114T − 433000/T 2 35.7 11.81 + 0.00754T − 41900/T 2

273–713 285–372 273–713

3 ? 3

c c

5.65 + 0.00866 16.0 + 0.00774T

273–2073 273–1550

1 ?

c c

6.64 59.8

273–372 276–314

? ?

c

5.57 + 0.00097T

273–1993

?

g

4.97

c l c c c c c c c c

5.25 + 0.00270T 7.59 + 0.00055T 15.9 + 0.00800T 11.40 + 0.00145T − 182400/T 2 12.81 + 0.00095T − 194600/T 2 11.5 + 0.00313T 28 34.7 80.8 100.2

273–692 692–1122 273–638 273–1573 273–1173 273–810 293–373 282 282 273–307

1 3 ? 1 5 ? ? ? ? ?

c c

11.62 + 0.01046T − 177700/T 2 26.7

273–1673 297–372

5 ?

All

0

See also Table 2-195. Data to 298 K are also given by Scott, Cryogenic Engineering, Van Nostrand, Princeton, N.J., 1959. For liquid and gas data, see Johnson (ed.), WADD-TR-60-56, 1960. 3 Stalder, NACA Tech. Note 4141, 1957 (Fig. 5), gives data from 400 to 2600°R. 4 See also Table 2-195. 5 For data from 400 to 5500°R see Stalder, NACA Tech. Note 4141, 1975 (Fig. 4). 6 For solid, liquid, and gas data, see Johnson (ed.), WADD-TR-60-56, 1960. 7 For data from 400 to 2350°R see Stalder, NACA Tech. Note 4141, 1957. 8 For solid, liquid, and gas data, see Johnson (ed.), WADD-TR-60-56, 1960. 9 For liquid and gas data, see Johnson (ed.), WADD-TR-60-56, 1960. 10 For solid, liquid, and gas data, see Johnson (ed.), WADD-TR-60-56, 1960. 11 See also Table 2-195; Douglas, Ball, et al., Bur. Stand. J. Res., 46 (1951): 334; Busey and Giaque, J. Am. Chem. Soc., 75 (1953): 806; Sheldon, ASME Pap. 49-A-30, 1949. 12 For solid, liquid, and gas data, see Johnson (ed.), WADD-TR-56-60, 1960. 13 For solid, liquid, and gas data, see Johnson (ed.), WADD-TR-56-60, 1960. 14 For solid, liquid, and gas data, see Johnson (ed.), WADD-TR-56-60, 1960. Ozone: For liquid see Brabets and Waterman, J. Chem. Phys., 28 (1958): 1212. 15 For data on liquid Na-K alloys to 1500°F and for liquid Na to 1460°F, see Lubarsky and Kaufman, NACA Rep. 1270, 1956. 16 See also Evans and Wagman, Bur. Stand. J. Res. 49 (1952): 141; Gratch, OTS PB 124957, 1950; Guthrie, Scott, et al., J. Am. Chem. Soc., 76 (1954): 1488. 2

SPECIFIC HEATS OF PURE COMPOUNDS TABLE 2-195

2-169

Specific Heat [kJ/(kg·K)] of Selected Elements Temperature, K

Symbol

4

6

8

10

20

40

60

80

100

200

250

300

400

600

800

Al Be Bi Cr Co

0.00026 0.00008 0.00054 0.00016 0.00036

0.00050

0.00088

0.214

0.357

0.00541 0.00050 0.00085

0.0089 0.0014 0.0340 0.0021 0.0048

0.0775

0.00220 0.00029 0.00059

0.00140 0.00028 0.01040 0.00081 0.00121

0.0729 0.0107 0.0404

0.092 0.059 0.110

0.102 0.127 0.184

0.481 0.195 0.109 0.190 0.234

0.797 1.109 0.120 0.382 0.376

0.859 1.537 0.121 0.424 0.406

0.902 1.840 0.122 0.450 0.426

0.949 2.191 0.123 0.501 0.451

1.042 2.605 0.142 0.565 0.509

1.134 2.823 0.136 0.611 0.543

Cu Ge Au Ir Fe

0.00011

0.00024

0.00018

0.00047

0.00048 0.00037 0.00126

0.137 0.108 0.084

0.203 0.153 0.100

0.00061

0.00090

0.0076 0.0129 0.0163 0.0021 0.0039

0.059 0.0619 0.0569

0.00038

0.00086 0.00081 0.00255 0.00032 0.00127

0.0276

0.086

0.154

0.254 0.192 0.109 0.090 0.216

0.357 0.286 0.124 0.122 0.384

0.377 0.305 0.127 0.128 0.422

0.386 0.323 0.129 0.131 0.450

0.396 0.343 0.131 0.133 0.491

0.431 0.364 0.136 0.140 0.555

0.448 0.377 0.141 0.146 0.692

Pb Mg Hg Mo Ni

0.00075 0.00034 0.00417 0.00011 0.00054

0.00242 0.00080 0.01420 0.00019 0.00086

0.00747 0.00155 0.01820 0.00032 0.00121

0.01350 0.00172 0.02250 0.00050 0.00178

0.0531 0.0148 0.0515 0.0029 0.0058

0.0944 0.138 0.0895 0.0236 0.0380

0.108 0.336 0.107 0.061 0.103

0.114 0.513 0.116 0.105 0.173

0.118 0.648 0.121 0.140 0.232

0.125 0.929 0.136 0.223 0.383

0.127 0.985 0.141 0.241 0.416

1.129 1.005 0.139 0.248 0.444

0.132 1.082 0.136 0.261 0.490

0.142 1.177 0.135 0.280 0.590

1.263 0.104 0.292 0.530

Pt Ag Sn Zn

0.00019 0.00016 0.00024 0.00011

0.00028 0.00035 0.00127 0.00029

0.00067 0.00093 0.00423 0.00096

0.00112 0.00186 0.00776 0.00250

0.0077 0.0159 0.0400 0.0269

0.0382 0.0778 0.108 0.123

0.069 0.133 0.149 0.205

0.088 0.166 0.173 0.258

0.101 0.187 0.189 0.295

0.127 0.225 0.214 0.366

0.132 0.232 0.220 0.380

0.134 0.236 0.222 0.389

0.136 0.240 0.245 0.404

0.140 0.251 0.257 0.435

0.146 0.264 0.257 0.479

TABLE 2-196 Cmpd. no.

Heat Capacities of Inorganic and Organic Liquids Name

Formula

CAS no.

Mol wt

C1

C2

C3

C4

C5

Tmin, K

Cp at Tmin × 1E−05

Tmax, K

Cp at Tmax × 1E−05

1 2 3 4 5 6 7 8 9 10

Methane (eqn. 2) Ethane (eqn. 2) Propane (eqn. 2) n-Butane (eqn. 2) n-Pentane n-Hexane n-Heptane (eqn. 2) n-Octane n-Nonane n-Decane

CH4 C2H6 C3H8 C4H10 C5H12 C6H14 C7H16 C8H18 C9H20 C10H22

74828 74840 74986 106978 109660 110543 142825 111659 111842 124185

16.043 30.070 44.097 58.123 72.150 86.177 100.204 114.231 128.258 142.285

6.5708E+01 4.4009E+01 6.2983E+01 6.4730E+01 1.5908E+05 1.7212E+05 6.1260E+01 2.2483E+05 3.8308E+05 2.7862E+05

3.8883E+04 8.9718E+04 1.1363E+05 1.6184E+05 −2.7050E+02 −1.8378E+02 3.1441E+05 −1.8663E+02 −1.1398E+03 −1.9791E+02

−2.5795E+02 9.1877E+02 6.3321E+02 9.8341E+02 9.9537E−01 8.8734E−01 1.8246E+03 9.5891E−01 2.7101E+00 1.0737E+00

6.1407E+02 −1.8860E+03 −8.7346E+02 −1.4315E+03 0 0 −2.5479E+03 0 0 0

0 0 0 0 0 0 0 0 0 0

90.69 92 85.47 134.86 143.42 177.83 182.57 216.38 219.66 243.51

0.5361 0.6855 0.8488 1.1380 1.4076 1.6750 1.9989 2.2934 2.6348 2.9409

190 290 360 420 390 460 520 460 325 460

11 12 13 14 15 16 17 18 19 20

n-Undecane n-Dodecane n-Tridecane n-Tetradecane n-Pentadecane n-Hexadecane n-Heptadecane n-Octadecane n-Nonadecane n-Eicosane

C11H24 C12H26 C13H28 C14H30 C15H32 C16H34 C17H36 C18H38 C19H40 C20H42

1120214 112403 629505 629594 629629 544763 629787 593453 629925 112958

156.312 170.338 184.365 198.392 212.419 226.446 240.473 254.500 268.527 282.553

2.9398E+05 5.0821E+05 3.5018E+05 3.5314E+05 3.4691E+05 3.7035E+05 3.7697E+05 3.9943E+05 3.4257E+05 3.5272E+05

−1.1498E+02 −1.3687E+03 −1.0470E+02 2.9130E+01 2.1954E+02 2.3147E+02 3.4782E+02 3.7464E+02 7.6208E+02 8.0732E+02

9.6936E−01 3.1015E+00 1.0022E+00 8.6116E−01 6.5632E−01 6.8632E−01 5.7895E−01 5.8156E−01 2.0481E−01 2.1220E−01

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

247.57 263.57 267.76 279.01 283.07 291.31 295.13 301.31 305.04 309.58

3.2493 3.6292 3.9400 4.2831 4.6165 4.9602 5.3005 5.6511 5.9409 6.2299

433.42 330 508.62 526.73 543.84 560.01 575.3 589.86 603.05 616.93

4.2624 3.9429 5.5619 6.0741 6.6042 7.1521 7.6869 8.2276 8.7663 9.3154

21 22 23 24 25 26 27

2-Methylpropane 2-Methylbutane 2,3-Dimethylbutane 2-Methylpentane 2,3-Dimethylpentane1 2,3,3-Trimethylpentane 2,2,4-Trimethylpentane

C4H10 C5H12 C6H14 C6H14 C7H16 C8H18 C8H18

75285 78784 79298 107835 565593 560214 540841

58.123 72.150 86.177 86.177 100.204 114.231 114.231

1.7237E+05 1.0830E+05 1.2945E+05 1.4222E+05 1.4642E+05 3.8862E+05 9.5275E+04

−1.7839E+03 1.4600E+02 1.8500E+01 −4.7830E+01 5.9200E+01 −1.4395E+03 6.9670E+02

1.4759E+01 −2.9200E−01 6.0800E−01 7.3900E−01 6.0400E−01 3.2187E+00 −1.3765E+00

−4.7909E−02 1.5100E−03 0 0 0 0 2.1734E−03

5.8050E−05 0 0 0 0 0 0

113.54 113.25 145.19 119.55 90 280 165.78

0.9961 1.2328 1.4495 1.4706 1.5664 2.3791 1.8285

380 310 331.13 333.41 380 320 520

2.0725 1.7048 2.0224 2.0842 2.5613 2.5757 3.9095

28 29 30 31 32 33 34 35

Ethylene Propylene 1-Butene cis-2-Butene trans-2-Butene 1-Pentene 1-Hexene 1-Heptene

C2H4 C3H6 C4H8 C4H8 C4H8 C5H10 C6H12 C7H14

74851 115071 106989 590181 624646 109671 592416 592767

28.054 42.081 56.108 56.108 56.108 70.134 84.161 98.188

2.4739E+05 1.1720E+05 1.3589E+05 1.2668E+05 1.1276E+05 1.5467E+05 1.9263E+05 1.8997E+05

−4.4280E+03 −3.8632E+02 −4.7739E+02 −6.5470E+01 −1.0470E+02 −4.2600E+02 −5.7116E+02 −1.5670E+02

4.0936E+01 1.2348E+00 2.1835E+00 −6.4000E−01 5.2140E−01 1.9640E+00 2.4004E+00 3.4300E−01

−1.6970E−01 0 −2.2230E−03 2.9120E−03 0 −1.8038E−03 −1.9758E−03 1.5222E−03

2.6816E−04 0 0 0 0 0 0 0

103.97 87.89 87.8 134.26 167.62 107.93 133.39 154.27

0.7013 0.9279 1.0930 1.1340 1.0986 1.2930 1.5446 1.7955

252.7 298.15 300 350 274.03 310 336.63 330

0.9758 1.1178 1.2917 1.5022 1.2322 1.5761 1.9700 2.3032

36 37 38 39 40 41 42 43 44

1-Octene 1-Nonene 1-Decene 2-Methylpropene 2-Methyl-1-butene2 2-Methyl-2-butene2 1,2-Butadiene 1,3-Butadiene 2-Methyl-1,3-butadiene

C8H16 C9H18 C10H20 C4H8 C5H10 C5H10 C4H6 C4H6 C5H8

111660 124118 872059 115117 563462 513359 590192 106990 78795

112.215 126.242 140.269 56.108 70.134 70.134 54.092 54.092 68.119

3.7930E+05 2.5875E+05 3.1950E+05 8.7680E+04 1.4951E+05 1.5160E+05 1.3515E+05 1.2886E+05 1.4148E+05

−2.1175E+03 −3.5450E+02 −5.7621E+02 2.1710E+02 −2.4763E+02 −2.6672E+02 −3.1114E+02 −3.2310E+02 −2.8870E+02

8.2362E+00 1.3126E+00 1.7087E+00 −9.1530E−01 9.1849E−01 9.0847E−01 9.7007E−01 1.0150E+00 1.0910E+00

−9.0093E−03 0 0 2.2660E−03 0 0 −1.5230E−04 3.2000E−05 0

0 0 0 0 0 0 0 0 0

171.45 191.78 206.89 132.81 135.58 139.39 136.95 165 130.32

2.1295 2.3904 2.7343 1.0568 1.3282 1.3207 1.1034 1.0333 1.2239

315 420.02 443.75 343.15 304.31 311.71 290 350 307.2

2.4793 3.4142 4.0027 1.4596 1.5921 1.5673 1.2279 1.4148 1.5575

45 46 47 48 49 50

Acetylene Methylacetylene Dimethylacetylene 3-Methyl-1-butyne 1-Pentyne 2-Pentyne

C2H2 C3H4 C4H6 C5H8 C5H8 C5H8

74862 74997 503173 598232 627190 627214

26.038 40.065 54.092 68.119 68.119 68.119

2.0011E+05 7.9791E+04 8.8153E+04 1.0520E+05 8.6200E+04 6.8671E+04

−1.1988E+03 8.9490E+01 1.2416E+02 1.9110E+02 2.5660E+02 2.4666E+02

3.0027E+00 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

192.4 200 240.91 200 200 200

0.8061 0.9769 1.1806 1.4342 1.3752 1.1800

250 249.94 300.13 299.49 313.33 329.27

0.8808 1.0216 1.2542 1.6243 1.6660 1.4989

2-170

14.9780 1.2444 2.6079 5.0822 2.0498 2.7534 4.0657 3.4189 2.9890 4.1478

51 52 53 54 55 56

1-Hexyne 2-Hexyne 3-Hexyne 1-Heptyne 1-Octyne Vinylacetylene3

C6H10 C6H10 C6H10 C7H12 C8H14 C4H4

693027 764352 928494 628717 629050 689974

82.145 82.145 82.145 96.172 110.199 52.076

9.3000E+04 9.4860E+04 8.2795E+04 8.5122E+04 9.1748E+04 6.8720E+04

3.2600E+02 2.5415E+02 2.8340E+02 4.0247E+02 4.7140E+02 1.3500E+02

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

200 300 300 192.22 193.55 200

1.5820 1.7110 1.6781 1.6248 1.8299 0.9572

344.48 357.67 354.35 372.93 399.35 278.25

2.0530 1.8576 1.8322 2.3522 2.8000 1.0628

57 58 59 60 61 62 63 64 65 66

Cyclopentane Methylcyclopentane Ethylcyclopentane Cyclohexane Methylcyclohexane 1,1-Dimethylcyclohexane Ethylcyclohexane Cyclopentene 1-Methylcyclopentene Cyclohexene

C5H10 C6H12 C7H14 C6H12 C7H14 C8H16 C8H16 C5H8 C6H10 C6H10

287923 96377 1640897 110827 108872 590669 1678917 142290 693890 110838

70.134 84.161 98.188 84.161 98.188 112.215 112.215 68.119 82.145 82.145

1.2253E+05 1.5592E+05 1.7852E+05 −2.2060E+05 1.3134E+05 1.3450E+05 1.3236E+05 1.2538E+05 5.3271E+04 1.0585E+05

−4.0380E+02 −4.9000E+02 −5.1835E+02 3.1183E+03 −6.3100E+01 8.7650E+00 7.2740E+01 −3.4970E+02 3.2792E+02 −6.0000E+01

1.7344E+00 2.1383E+00 2.3255E+00 −9.4216E+00 8.1250E−01 8.1151E−01 6.4738E−01 1.1430E+00 0 6.8000E−01

−1.0975E−03 −1.5585E−03 −1.6818E−03 1.0687E−02 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

179.28 130.73 134.71 279.69 146.58 239.66 161.84 138.13 200 169.67

0.9956 1.2492 1.4678 1.4836 1.3955 1.8321 1.6109 0.9888 1.1885 1.1525

322.4 366.48 301.82 400 320 392.7 404.95 317.38 348.64 356.12

1.3584 1.8682 1.8767 2.0323 1.9435 2.6309 2.6798 1.2953 1.6760 1.7072

67 68 69 70 71 72 73

Benzene Toluene o-Xylene m-Xylene p-Xylene Ethylbenzene Propylbenzene

C6H6 C7H8 C8H10 C8H10 C8H10 C8H10 C9H12

71432 108883 95476 108383 106423 100414 103651

78.114 92.141 106.167 106.167 106.167 106.167 120.194

1.2944E+05 1.4014E+05 3.6500E+04 1.7555E+05 −3.5500E+04 1.3316E+05 2.3477E+05

−1.6950E+02 −1.5230E+02 1.0175E+03 −2.9950E+02 1.2872E+03 4.4507E+01 −8.0022E+02

6.4781E−01 6.9500E−01 −2.6300E+00 1.0880E+00 −2.5990E+00 3.9645E−01 3.4037

0 0 3.0200E−03 0 2.4260E−03 0 −3.1739E−03

0 0 0 0 0 0 0

278.68 178.18 248 225.3 286.41 178.15 173.59

1.3251 1.3507 1.7315 1.6330 1.7697 1.5367 1.8182

353.24 500 415 360 600 409.35 370

1.5040 2.3774 2.2166 2.0873 3.2520 2.1781 2.4389

74 75 76 77 78 79 80 81

1,2,4-Trimethylbenzene Isopropylbenzene 1,3,5-Trimethylbenzene p-Isopropyltoluene Naphthalene Biphenyl Styrene m-Terphenyl

C9H12 C9H12 C9H12 C10H14 C10H8 C12H10 C8H8 C18H14

95636 98828 108678 99876 91203 92524 100425 92068

120.194 120.194 120.194 134.221 128.174 154.211 104.152 230.309

1.7880E+05 1.8290E+05 1.4805E+05 1.4560E+05 2.9800E+04 1.2177E+05 1.1334E+05 1.9567E+05

−1.2847E+02 −1.7400E+02 1.9700E+01 2.4870E+02 5.2750E+02 4.2930E+02 2.9020E+02 5.9407E+02

8.3741E−01 9.1200E−01 6.2260E−01 1.8700E−01 0 0 −6.0510E−01 0

0 0 0 0 0 0 1.3567E−03 0

0 0 0 0 0 0 0 0

229.33 177.14 228.42 205.25 353.43 342.2 242.54 360

1.9338 1.8069 1.8503 2.0452 2.1623 2.6868 1.6749 4.0954

350 500 350 450.28 491.14 533.37 418.31 650

2.3642 3.2390 2.3121 2.9550 2.8888 3.5075 2.2816 5.8182

82 83 84 85 86 87 88

Methanol Ethanol 1-Propanol 1-Butanol 2-Butanol 2-Propanol 2-Methyl-2-propanol

CH4O C2H6O C3H8O C4H10O C4H10O C3H8O C4H10O

67561 64175 71238 71363 78922 67630 75650

32.042 46.069 60.096 74.123 74.123 60.096 74.123

1.0580E+05 1.0264E+05 1.5876E+05 1.9120E+05 2.0670E+05 7.2355E+05 −9.2546E+05

−3.6223E+02 −1.3963E+02 −6,3500E+02 −7.3040E+02 −1.0204E+03 −8.0950E+03 7.8949E+03

9.3790E−01 −3.0341E−02 1.9690E+00 2.2998E+00 3.2900E+00 3.6662E+01 −1.7661E+01

0 2.0386E−03 0 0 0 −6.6395E−02 1.3617E−02

0 0 0 0 0 4.4064E−05 0

175.47 159.05 146.95 184.51 158.45 185.28 298.96

0.7112 0.8787 1.0797 1.3473 1.2762 1.1189 2.2016

400 390 400 390.81 372.7 480 460

1.1097 1.6450 2.1980 2.5701 2.8340 2.8122 2.9455

89 90 91 92 93 94 95 96

1-Pentanol 2-Methyl-1-butanol 3-Methyl-1-butanol 1-Hexanol 1-Heptanol Cyclohexanol Ethylene glycol 1,2-Propylene glycol

C5H12O C5H12O C5H12O C6H14O C7H16O C6H12O C2H6O2 C3H8O2

71410 137326 123513 111273 111706 108930 107211 57556

88.150 88.150 88.150 102.177 116.203 100.161 62.068 76.095

2.0120E+05 8.2937E+04 −5.3777E+04 4.8466E+05 4.3790E+05 −4.0000E+04 3.5540E+04 5.8080E+04

−6.5130E+02 4.5998E+02 8.8342E+02 −2.7613E+03 −2.0947E+03 8.5300E+02 4.3678E+02 4.4520E+02

2.2750E+00 0.0000E+00 0 6.5555E+00 5.2090E+00 0 −1.8486E−01 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

200.14 250 295.52 228.55 239.15 296.6 260.15 213.15

1.6198 1.9793 2.0729 1.9599 2.3487 2.1300 1.3666 1.5297

389.15 401.85 350 320 370 434 493.15 460.75

2.9227 2.6778 2.5542 2.7233 3.7597 3.3020 2.0598 2.6321

Phenol o-Cresol m-Cresol p-Cresol

C6H6O C7H8O C7H8O C7H8O

108952 95487 108394 106445

94.113 108.140 108.140 108.140

1.0172E+05 −1.8515E+05 −2.4670E+05 2.5998E+05

3.1761E+02 3.1480E+03 3.2568E+03 −1.1123E+03

0 −8.0367E+00 −7.4202E+00 4.9427E+00

0 7.2540E−03 6.0467E−03 −5.4367E−03

0 0 0 0

314.06 304.2 285.39 307.93

2.0147 2.3297 2.1895 2.2740

425 400 400 400

2.3670 2.5243 2.5578 2.5794

97 98 99 100

2-171

TABLE 2-196

Heat Capacities of Inorganic and Organic Liquids (Continued )

Cmpd. no.

Name

Formula

CAS no.

Mol wt

C1

C2

C3

101 102 103 104 105 106 107 108 109 110 111 112

Dimethyl ether Methyl ethyl ether Methyl-n-propyl ether Methyl isopropyl ether Methyl-n-butyl ether Methyl isobutyl ether Methyl tert-butyl ether Diethyl ether Ethyl propyl ether Ethyl isopropyl ether Methyl phenyl ether Diphenyl ether

C2H6O C3H8O C4H10O C4H10O C5H12O C5H12O C5H12O C4H10O C5H12O C5H12O C7H8O C12H10O

115106 540670 557175 598538 628284 625445 1634044 60297 628320 625547 100663 101848

46.069 60.096 74.123 74.123 88.150 88.150 88.150 74.123 88.150 88.150 108.140 170.211

1.1010E+05 1.2977E+05 1.4411E+05 1.4344E+05 1.7785E+05 5.1380E+04 1.4012E+05 4.4400E+04 1.0368E+05 1.0625E+05 1.5094E+05 1.3416E+05

−1.5747E+02 −3.3196E+02 −1.0209E+02 −1.5407E+02 −1.7157E+02 4.5040E+02 −9.0000E+00 1.3010E+03 7.2630E+02 2.9215E+02 9.3455E+01 4.4767E+02

5.1853E−01 1.3869E+00 5.8113E−01 7.2550E−01 7.4379E−01 0 5.6300E−01 −5.5000E+00 −2.6047E+00 0 2.3602E−01 0

113 114 115 116 117 118 119 120 121 122

Formaldehyde4 Acetaldehyde 1-Propanal 1-Butanal 1-Pentanal 1-Hexanal 1-Heptanal 1-Octanal 1-Nonanal 1-Decanal

CH2O C2H4O C3H6O C4H8O C5H10O C6H12O C7H14O C8H16O C9H18O C10H20O

50000 75070 123386 123728 110623 66251 111717 124130 124196 112312

30.026 44.053 58.080 72.107 86.134 100.161 114.188 128.214 142.241 156.268

6.1900E+04 1.1510E+05 9.9306E+04 6.5682E+04 1.1205E+05 1.1770E+05 2.2236E+05 1.3065E+05 1.3682E+05 1.5046E+05

2.8300E+01 −4.3300E+02 1.1573E+02 1.3291E+03 2.5778E+02 3.2952E+02 −1.0517E+02 4.6361E+02 5.3129E+02 5.8663E+02

123 124 125 126 127 128 129 130 131 132 133 134

Acetone Methyl ethyl ketone 2-Pentanone Methyl isopropyl ketone 2-Hexanone Methyl isobutyl ketone 3-Methyl-2-pentanone 3-Pentanone Ethyl isopropyl ketone Diisopropyl ketone Cyclohexanone Methyl phenyl ketone

C3H6O C4H8O C5H10O C5H10O C6H12O C6H12O C6H12O C5H10O C6H12O C7H14O C6H10O C8H8O

67641 78933 107879 563804 591786 108101 565617 96220 565695 565800 108941 98862

58.080 72.107 86.134 86.134 100.161 100.161 100.161 86.134 100.161 114.188 98.145 120.151

1.3560E+05 1.3230E+05 1.9459E+05 1.8361E+05 2.7249E+05 1.2492E+05 9.9815E+04 1.9302E+05 8.3630E+04 1.7927E+05 1.0980E+05 7.2692E+04

135 136 137 138 139 140 141

Formic acid Acetic acid Propionic acid n-Butyric acid Isobutyric acid Benzoic acid Acetic anhydride

CH2O2 C2H4O2 C3H6O2 C4H8O2 C4H8O2 C7H6O2 C4H6O3

64186 64197 79094 107926 79312 65850 108247

46.026 60.053 74.079 88.106 88.106 122.123 102.090

142 143 144 145 146 147 148

Methyl formate Methyl acetate Methyl propionate Methyl-n-butyrate Ethyl formate Ethyl acetate Ethyl propionate

C2H4O2 C3H6O2 C4H8O2 C5H10O2 C3H6O2 C4H8O2 C5H10O2

107313 79209 554121 623427 109944 141786 105373

60.053 74.079 88.106 102.133 74.079 88.106 102.133

2-172

C5

Tmin, K

Cp at Tmin × 1E−05

Tmax, K

Cp at Tmax × 1E−05

0 0 0 0 0 0 0 8.7630E−03 4.0957E−03 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

131.65 218.9 133.97 127.93 157.48 300 164.55 156.92 145.65 298.15 298.15 300.03

0.9836 1.2356 1.4086 1.3560 1.6928 1.8650 1.5388 1.4698 1.6686 1.9335 1.9978 2.6847

250 328.35 312.2 310 343.35 370 328.35 460 320 326.15 484.2 570

1.0314 1.7030 1.6888 1.6540 2.0663 2.1803 1.9786 3.3202 2.0358 2.0153 2.5153 3.8933

0 1.4250E+00 0 −7.1579E+00 0 0 6.5074E−01 0 0 0

0 0 0 1.2755E−02 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

204 150.15 200 176.75 200 217.15 229.8 246 255.15 267.15

0.6767 0.8221 1.2245 1.4741 1.6361 1.8926 2.3256 2.4470 2.7238 3.0718

234 294 328.75 300 376.15 401.45 381.25 447.15 468.15 488.15

0.6852 1.1097 1.3735 1.6459 2.0901 2.4999 2.7685 3.3795 3.8554 4.3682

−1.7700E+02 2.0087E+02 −2.6386E+02 −2.6885E+02 −7.9070E+02 3.0410E+02 3.4672E+02 −1.7643E+02 3.9900E+02 2.8370E+01 2.6150E+02 3.3783E+02

2.8370E−01 −9.5970E−01 7.6808E−01 8.6080E−01 2.5834E+00 0 0 5.6690E−01 0 5.3750E−01 0 3.5572E−01

6.8900E−04 1.9533E−03 0 0 −2.0040E−03 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

178.45 186.48 196.29 181.15 220.87 298.15 298.15 234.18 298.15 204.81 290 298.2

1.1696 1.4905 1.7239 1.6316 2.0228 2.1559 2.0319 1.8279 2.0259 2.0763 1.8563 2.0506

329.44 373.15 375.46 367.55 382.62 390 390.55 375.14 425 410 486.5 532.12

1.3271 1.7511 2.0380 2.0108 2.3590 2.4352 2.3523 2.0661 2.5320 2.8126 2.3702 3.5318

7.8060E+04 1.3964E+05 2.1366E+05 2.3770E+05 1.2754E+05 −5.4800E+03 3.6600E+04

7.1540E+01 −3.2080E+02 −7.0270E+02 −7.4640E+02 −6.5350E+01 6.4712E+02 5.1100E+02

0 8.9850E−01 1.6605E+00 1.8290E+00 8.2867E−01 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

281.45 289.81 252.45 267.95 270 395.45 250

0.9820 1.2213 1.4209 1.6902 1.7031 2.5042 1.6435

380 391.05 414.32 436.42 427.65 450 350

1.0525 1.5159 2.0756 2.6031 2.5114 2.8572 2.1545

1.3020E+05 6.1260E+04 7.1140E+04 1.0293E+05 8.0000E+04 2.2623E+05 7.6330E+04

−3.9600E+02 2.7090E+02 3.3550E+02 1.2910E+02 2.2360E+02 −6.2480E+02 4.0010E+02

1.2100E+00 0 0 6.2516E−01 0 1.4720E+00 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

174.15 253.4 300 277.25 254.2 189.6 298.15

1.2991 1.7179 1.8678 1.3684 1.6068 1.9562

304.9 373.4 390 415.87 374.2 350.21 410

1.2195 1.6241 2.0198 2.6474 1.6367 1.8796 2.4037

C4

149 150 151 152 153 154 155

Ethyl-n-butyrate n-Propyl formate n-Propyl acetate n-Butyl acetate Methyl benzoate Ethyl benzoate Vinyl acetate

C6H12O2 C4H8O2 C5H10O2 C6H12O2 C8H8O2 C9H10O2 C4H6O2

105544 110747 109604 123864 93583 93890 108054

116.160 88.106 102.133 116.160 136.150 150.177 86.090

8.2434E+04 7.5700E+04 8.3400E+04 1.1730E+05 1.1950E+05 1.2450E+05 1.3630E+05

4.2245E+02 3.2610E+02 3.8410E+02 3.5220E+02 2.9400E+02 3.7060E+02 −1.0617E+02

2.0992E−01 0 0 0 0 0 7.5175E−01

156 157 158 159 160 161

Methylamine Dimethylamine Trimethylamine Ethylamine Diethylamine Triethylamine

CH5N C2H7N C3H9N C2H7N C4H11N C6H15N

74895 124403 75503 75047 109897 121448

31.057 45.084 59.111 45.084 73.138 101.192

9.2520E+04 −2.1487E+05 1.3605E+05 1.2170E+05 1.0133E+05 1.1148E+05

3.7450E+01 3.7872E+03 −2.8800E+02 3.8993E+01 2.4318E+02 3.6813E+02

0 −1.3781E+01 9.9130E−01 0 0 0

162 163 164 165 166 167 168

n-Propylamine di-n-Propylamine Isopropylamine Diisopropylamine Aniline N-Methylaniline N,N-Dimethylaniline

C3H9N C6H15N C3H9N C6H15N C6H7N C7H9N C8H11N

107108 142847 75310 108189 62533 100618 121697

59.111 101.192 59.111 101.192 93.128 107.155 121.182

1.3953E+05 4.9120E+04 −3.2469E+04 9.8434E+04 1.4150E+05 1.2850E+05 4.1860E+04

7.8000E+01 5.6224E+02 1.9771E+03 4.2904E+02 1.7120E+02 1.0020E+02 5.2750E+02

169 170 171 172

Ethylene oxide Furan Thiophene Pyridine

C2H4O C4H4O C4H4S C5H5N

75218 110009 110021 110861

44.053 68.075 84.142 79.101

1.4471E+05 1.1437E+05 8.1350E+04 1.0785E+05

173 174 175 176 177 178 179 180

Formamide5 N,N-Dimethylformamide Acetamide N-Methylacetamide Acetonitrile Propionitrile n-Butyronitrile Benzonitrile

CH3NO C3H7NO C2H5NO C3H7NO C2H3N C3H5N C4H7N C7H5N

75127 68122 60355 79163 75058 107120 109740 100470

45.041 73.095 59.068 73.095 41.053 55.079 69.106 103.123

181 182 183 184 185 186 187 188 189

Methyl mercaptan Ethyl mercaptan n-Propyl mercaptan n-Butyl mercaptan Isobutyl mercaptan sec-Butyl mercaptan2 Dimethyl sulfide Methyl ethyl sulfide Diethyl sulfide

CH4S C2H6S C3H8S C4H10S C4H10S C4H10S C2H6S C3H8S C4H10S

74931 75081 107039 109795 513440 513531 75183 624895 352932

190 191 192 193 194 195 196 197

Fluoromethane2 Chloromethane Trichloromethane Tetrachloromethane Bromomethane Fluoroethane Chloroethane Bromoethane

CH3F CH3Cl CHCl3 CCl4 CH3Br C2H5F C2H5Cl C2H5Br

593533 74873 67663 56235 74839 353366 75003 74964

0 0 0 0 0 0 0

0 0 0 0 0 0 0

285.5 298.15 274.7 289.58 260.75 238.45 259.56

2.2015 1.7293 1.8891 2.1929 1.9616 2.1287 1.5939

428.25 398.15 404.7 429.58 472.65 486.55 389.35

3.0185 2.0554 2.3885 2.6860 2.5846 3.0482 2.0892

0 1.6924E−02 0 0 0 0

0 0 0 0 0 0

179.69 180.96 156.08 192.15 223.35 200

0.9925 1.1947 1.1525 1.2919 1.5564 1.8511

266.82 298.15 276.02 289.73 328.6 361.92

1.0251 1.3779 1.3208 1.3300 1.8124 2.4471

0 0 −7.0145E+00 0 0 3.7400E−01 0

0 0 8.6913E−03 0 0 0 0

0 0 0 0 0 0 0

188.36 277.9 177.95 275 267.13 216.15 343.58

1.5422 2.0537 1.4621 2.1642 1.8723 1.6763 2.2310

340 407.9 320 357.05 457.15 469.02 513.58

1.6605 2.7846 1.6671 2.5162 2.1976 2.5777 3.1277

−7.5887E+02 −2.1569E+02 1.2980E+02 −3.4787E+01

2.8261E+00 7.2691E−01 −3.9000E−03 3.9565E−01

−3.0640E−03 0 0 0

0 0 0 0

160.65 187.55 234.94 231.51

0.8303 0.9949 1.1163 1.2100

283.85 304.5 357.31 388.41

0.8693 1.1609 1.2723 1.5403

6.3400E+04 1.4790E+05 1.0230E+05 6.2600E+04 9.7582E+04 1.1819E+05 1.0400E+05 7.6900E+04

1.5060E+02 −1.0600E+02 1.2870E+02 2.4340E+02 −1.2220E+02 −1.2098E+02 1.7400E+02 3.1420E+02

0 3.8400E−01 0 0 3.4085E−01 4.2075E−01 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

292 273.82 354.15 359 229.32 180.26 161.25 260.4

1.0738 1.4767 1.4788 1.4998 0.8748 1.1005 1.3206 1.5872

493 466.44 571 538.5 354.75 370.5 390.75 464.15

1.3765 1.8200 1.7579 1.9367 0.9713 1.3112 1.7199 2.2274

48.109 62.136 76.163 90.189 90.189 90.189 62.136 76.163 90.189

1.1530E+05 1.3467E+05 1.6733E+05 2.3219E+05 1.7336E+05 1.9789E+05 1.4695E+05 1.6124E+05 2.3852E+05

−2.6323E+02 −2.3439E+02 −3.1910E+02 −8.0435E+02 −2.1732E+02 −4.9154E+02 −3.8006E+02 −2.8861E+02 −1.0384E+03

6.0412E−01 5.9656E−01 8.1270E−01 2.7063E+00 7.0933E−01 1.7219E+00 1.2035E+00 7.8179E−01 4.0587E+00

0 0 0 −2.3017E−03 0 −1.2499E−03 −8.4787E−04 0 −4.4691E−03

0 0 0 0 0 0 0 0 0

150.18 125.26 159.95 157.46 128.31 133.02 174.88 167.23 181.95

0.8939 1.1467 1.3708 1.6365 1.5715 1.6003 1.1276 1.3484 1.5703

298.15 315.25 340.87 390 361.64 370 310.48 339.8 322.08

0.9052 1.2007 1.5299 1.9359 1.8754 1.8844 1.1959 1.5344 1.7579

34.033 50.488 119.377 153.822 94.939 48.060 64.514 108.966

7.4746E+04 9.6910E+04 1.2485E+05 −7.5270E+05 1.2973E+05 8.3303E+04 1.2790E+05 9.4364E+04

−1.3232E+02 −2.0790E+02 −1.6634E+02 8.9661E+03 −5.9654E+02 6.5454E+01 −3.4515E+02 −1.0912E+02

5.3772E−01 3.7456E−01 4.3209E−01 −3.0394E+01 2.1600E+00 0 9.1500E−01 4.4032E−01

0 4.8800E−04 0 3.4455E−02 −2.4234E−03 0 0 0

0 0 0 0 0 0 0 0

140 175.43 233.15 250.33 184.45 200 134.8 160

0.6676 0.7460 1.0956 1.2763 0.7798 0.9639 0.9800 0.8818

220 373.15 366.48 388.71 276.71 281.48 340 320

0.7166 0.9684 1.2192 1.6374 0.7870 1.0173 1.1632 1.0453

2-173

TABLE 2-196 Cmpd. no.

Heat Capacities of Inorganic and Organic Liquids (Concluded ) Name

Formula

CAS no.

Mol wt

C1

C2

C3

540545 75296 78999 78875 75014 462066 108907 108861

78.541 78.541 112.986 112.986 62.499 96.104 112.558 157.010

9.6344E+04 6.9362E+04 7.0010E+04 1.1094E+05 −1.0320E+04 −9.9120E+05 −1.3075E+06 1.2160E+05

1.1752E+02 2.1501E+02 2.6660E+02 8.3496E+00 3.2280E+02 1.1734E+04 1.5338E+04 −9.4500E+00

0 0 0 4.7218E−01 0 −4.0669E+01 −5.3974E+01 3.5800E−01

C5

Tmin, K

Cp at Tmin × 1E−05

Tmax, K

Cp at Tmax × 1E−05

0 0 0 0 0 4.7333E−02 6.3483E−02 0

0 0 0 0 0 0 0 0

230 200 280 286 200 239.99 227.95 293.15

1.2337 1.1236 1.4466 1.5195 0.5424 1.3675 1.3617 1.4960

319.67 308.85 420 429 400 319.99 360 495.08

1.3391 1.3577 1.8198 2.0142 1.1880 1.5018 1.8101 2.0467

C4

198 199 200 201 202 203 204 205

1-Chloropropane 2-Chloropropane 1,1-Dichloropropane 1,2-Dichloropropane Vinyl chloride Fluorobenzene Chlorobenzene Bromobenzene

C3H7Cl C3H7Cl C3H6Cl2 C3H6Cl2 C2H3Cl C6H5F C6H5Cl C6H5Br

206 207 208 209 210 211 212 213 214 215 216 217 218

Air Hydrogen (eqn. 2) Helium-46 Neon Argon Fluorine Chlorine Bromine Oxygen Nitrogen Ammonia (eqn. 2) Hydrazine Nitrous oxide

H2 He Ne Ar F2 Cl2 Br2 O2 N2 NH3 N2H4 N 2O

132259100 1333740 7440597 7440019 7440371 7782414 7782505 7726956 7782447 7727379 7664417 302012 10024972

28.951 2.016 4.003 20.180 39.948 37.997 70.905 159.808 31.999 28.014 17.031 32.045 44.013

−2.1446E+05 6.6653E+01 3.8722E+05 1.0341E+06 1.3439E+05 −9.4585E+04 6.3936E+04 3.7570E+04 1.7543E+05 2.8197E+05 6.1289E+01 7.9815E+04 6.7556E+04

9.1851E+03 6.7659E+03 −4.6557E+05 −1.3877E+05 −1.9894E+03 7.5299E+03 4.6350E+01 3.2850E+02 −6.1523E+03 −1.2281E+04 8.0925E+04 5.0929E+01 5.4373E+01

−1.0612E+02 −1.2363E+02 2.1180E+05 7.1540E+03 1.1043E+01 −1.3960E+02 −1.6230E−01 −6.7000E−01 1.1392E+02 2.4800E+02 7.9940E+02 4.3379E−02 0

4.1616E−01 4.7827E+02 −4.2494E+04 −1.6255E+02 0 1.1301E+00 0 0 −9.2382E−01 −2.2182E+00 −2.6510E+03 0 0

0 0 3.2129E+03 1.3841E+00 0 −3.3241E−03 0 0 2.7963E−03 7.4902E−03 0 0 0

75 13.95 2.2 24.56 83.78 58 172.12 265.9 54.36 63.15 203.15 274.69 182.3

0.5307 0.1262 0.1087 0.3666 0.4523 0.5541 0.6711 0.7755 0.5365 0.5593 0.7575 0.9708 0.7747

115 32 4.6 40 135 98 239.12 305.37 142 112 401.15 653.15 200

0.7132 1.3122 0.2965 0.6980 0.6708 0.5966 0.6574 0.7541 0.9066 0.7960 4.1847 1.3158 0.7843

219 220 221 222 223 224 225 226 227 228 229 230 231

Nitric oxide Cyanogen Carbon monoxide (eqn. 2) Carbon dioxide Carbon disulfide Hydrogen fluoride Hydrogen chloride Hydrogen bromide Hydrogen cyanide Hydrogen sulfide (eqn. 2) Sulfur dioxide Sulfur trioxide Water

NO C2N2 CO CO2 CS2 HF HCl HBr HCN H 2S SO2 SO3 H 2O

10102439 460195 630080 124389 75150 7664393 7647010 10035106 74908 7783064 7446095 7446119 7732185

30.006 52.036 28.010 44.010 76.143 20.006 36.461 80.912 27.026 34.082 64.065 80.064 18.015

−2.9796E+06 3.1322E+06 6.5429E+01 −8.3043E+06 8.5600E+04 6.2520E+04 4.7300E+04 5.7720E+04 9.5398E+04 6.4666E+01 8.5743E+04 2.5809E+05 2.7637E+05

7.6602E+04 −2.4320E+04 2.8723E+04 1.0437E+05 −1.2200E+02 −2.2302E+02 9.0000E+01 9.9000E+00 −1.9752E+02 4.9354E+04 5.7443E+00 0.0000E+00 −2.0901E+03

−6.5259E+02 4.8844E+01 −8.4739E+02 −4.3333E+02 5.6050E−01 6.2970E−01 0 0 3.8830E−01 2.2493E+01 0 0 8.1250E+00

1.8879E+00 0 1.9596E+03 6.0052E−01 −1.4520E−03 0 0 0 0 −1.6230E+03 0 0 −1.4116E−02

0 0 0 0 2.0080E−06 0 0 0 0 0 0 0 9.3701E−06

109.5 245.25 68.15 220 161.11 189.79 165 185.15 259.83 187.68 197.67 303.15 273.16

0.6229 1.0557 0.5912 0.7827 0.7577 0.4288 0.6215 0.5955 0.7029 0.6733 0.8688 2.5809 0.7615

150 300 132 290 552 292.67 185 206.45 298.85 370 350 303.15 533.15

1.9909 2.3216 6.4799 1.6603 1.3125 0.5119 0.6395 0.5976 0.7105 4.9183 0.8775 2.5809 0.8939

All substances are listed in alphabetical order in Table 2-6a. Compiled from Daubert, T. E., R. P. Danner, H. M. Sibul, and C. C. Stebbins, DIPPR Data Compilation of Pure Compound Properties, Project 801 Sponsor Release, July, 1993, Design Institute for Physical Property Data, AIChE, New York, NY; and from Thermodynamics Research Center, “Selected Values of Properties of Hydrocarbons and Related Compounds,” Thermodynamics Research Center Hydrocarbon Project, Texas A&M University, College Station, Texas (extant 1994). Temperatures are expressed in kelvins; liquid heat capacities are in J/kmol-K. J/(kmol·K) × 2.390E−04 = cal/(gmol·°C); J/(kmol·K) × 2.390059E−04 = Btu/(lbmol·°F). Equation 1, heat capacity = C1 + C2 × T + C3 × T 2 + C4 × T 3 + C5 × T 4, should be used except as otherwise specified. Equation 2 is heat capacity = C12/t + C2 − (2 × C1 × C3)t − (C1 × C4)t2 − (C32/3)t3 − (C3 × C4/2)t4 − (C42/5)t5. t = (1 − Tr) and Tr is the reduced temperature, T/Tc. 1 Coefficients are for the monomer and are hypothetical above 473 K. 2 For the saturated heat capacity. 3 Coefficients are hypothetical; compound decomposes violently on heating. 4 Coefficients are hypothetical and are based on predicted data. 5 Coefficients are hypothetical. 6 Exhibits superfluid properties below 2.2 K. 2-174

SPECIFIC HEATS OF PURE COMPOUNDS TABLE 2-197

Specific Heats of Organic Solids Recalculated from International Critical Tables, vol. 5, pp. 101–105

Compound

Formula

Acetic acid Acetone Aminobenzoic acid (o-) (m-) (p-) Aniline Anthracene

C2H4O2 C3H6O C7H7NO2 C7H7NO2 C7H7NO2 C6H7N C14H10

Anthraquinone Apiol Azobenzene

C14H8O2 C12H14O4 C12H10N2

Benzene

C6H6

Benzoic acid Benzophenone

C7H6O2 C13H10O

Betol

C17H12O3

Bromoiodobenzene (o-) (m-) (p-) Bromonaphthalene (β-) Bromophenol

C6H4BrI C6H4BrI C6H4BrI C10H7Br C6H5BrO

Camphene Capric acid Caprylic acid Carbon tetrachloride

C10H16 C10H20O2 C8H16O2 CCl4

Cerotic acid Chloral alcoholate hydrate Chloroacetic acid Chlorobenzoic acid (o-) (m-) (p-) Chlorobromobenzene (o-) (m-) (p-) Crotonic acid Cyamelide Cyanamide Cyanuric acid

C27H54O2 C4H7Cl3O2 C2H3Cl3O2 C2H3ClO2 C7H5ClO2 C7H5ClO2 C7H5ClO2 C6H4BrCl C6H4BrCl C6H4BrCl C4H6O2 C3H3N3O3 CH2N2 C3H3N3O3

Dextrin Dextrose

(C6H10O5)x C6H12O6

Dibenzyl Dibromobenzene (o-) (m-) (p-) Dichloroacetic acid Dichlorobenzene (o-) (m-) (p-) Dicyandiamide

C14H14 C6H4Br2 C6H4Br2 C6H4Br2 C2H2Cl2O2 C6H4Cl2 C6H4Cl2 C6H4Cl2 C2H4N4

Temperature, °C −200 to +25 −210 to −80 85 to mp 120 to mp 128 to mp

sp ht, cal/g °C

50 100 150 0 to 270 10 28

0.330 + 0.00080t 0.540 + 0.0156t 0.254 + 0.00136t 0.253 + 0.00122t 0.287 + 0.00088t 0.741 0.308 0.350 0.382 0.258 + 0.00069t 0.299 0.330

−250 −225 −200 −150 −100 −50 0 20 to mp −150 −100 −50 0 +20 −150 −100 0 +50 −50 to 0 −75 to −15 −40 to 50 41 32

0.0399 0.0908 0.124 0.170 0.227 0.299 0.375 0.287 + 0.00050t 0.115 0.172 0.220 0.275 0.303 0.129 0.167 0.248 0.308 0.143 + 0.00025t 0.143 0.116 + 0.00032t 0.260 0.263

35 8 −2 −240 −200 −160 −120 −80 −40 15 78 32 60 80 to mp 94 to mp 180 to mp −34 −52 −40 38 to 70 40 20 40

0.380 0.695 0.628 0.013 0.081 0.131 0.162 0.182 0.201 0.387 0.509 0.213 0.363 0.228 + 0.00084t 0.232 + 0.00073t 0.242 + 0.00055t 0.192 0.150 0.150 0.520 + 0.00020t 0.263 0.547 0.318

0 to 90 −250 −200 −100 0 20 28 −36 −25 −50 to +50

0.291 + 0.00096t 0.016 0.077 0.160 0.277 0.300 0.363 0.248 0.134 0.139 + 0.00038t 0.406 0.185 0.186 0.219 + 0.0021t 0.456

−48.5 −52 −50 to +53 0 to 204

2-175

2-176

PHYSICAL AND CHEMICAL DATA TABLE 2-197

Specific Heats of Organic Solids (Continued ) Recalculated from International Critical Tables, vol. 5, pp. 101–105

Compound

Formula

Dihydroxybenzene (o-) (m-) (p-)

C6H6O2 C6H6O2 C6H6O2

Di-iodobenzene (o-) (m-) (p-) Dimethyl oxalate Dimethylpyrene Dinitrobenzene (o-) (m-) (p-) Diphenyl Diphenylamine Dulcitol

C6H4I2 C6H4I2 C6H4I2 C4H6O4 C7H8O2 C6H4N2O4 C6H4N2O4 C6H4N2O4 C12H10 C12H11N C6H14O6

Erythritol Ethyl alcohol

C4H10O4 C2H6O (crystalline)

(vitreous)

Temperature, °C

sp ht, cal/g °C

−163 to mp −160 to mp −250 −240 −220 −200 −150 to mp −50 to +15 −52 to −42 −50 to +80 10 to 50 50 −160 to mp −160 to mp 119 to mp 40 26 20

0.278 + 0.00098t 0.269 + 0.00118t 0.025 0.038 0.061 0.081 0.268 + 0.00093t 0.109 + 0.00026t 0.100 + 0.00026t 0.101 + 0.00026t 0.212 + 0.0044t 0.368 0.252 + 0.00083t 0.248 + 0.00077t 0.259 + 0.00057t 0.385 0.337 0.282

60 −190 −180 −160 −140 −130 −190 −180 −175 −170 −190 to −40

0.351 0.232 0.248 0.282 0.318 0.376 0.260 0.296 0.380 0.399 0.366 + 0.00110t

Ethylene glycol

C2H6O2

Formic acid

CH2O2

−22 0

0.387 0.430

Glutaric acid Glycerol

C5H8O4 C3H8O3

20 −265 −260 −250 −220 −200 −100 0

0.299 0.009 0.022 0.047 0.085 0.115 0.217 0.330

Hexachloroethane Hexadecane Hydroxyacetanilide

C2Cl6 C16H34 C8H9NO2

Iodobenzene Isopropyl alcohol

C6H5I C3H8O

Lactose Lauric acid Levoglucosane Levulose

C12H22O11 C12H22O11·H2O C12H24O2 C6H10O5 C6H12O6

Malonic acid Maltose Mannitol Melamine Myristic acid Naphthalene Naphthol (α-) (β-) Naphthylamine (α-) Nitroaniline (o-) (m-) (p-) Nitrobenzoic acid (o-) (m-) (p-) Nitronaphthalene

25 41 to mp

0.174 0.495 0.249 + 0.00154t

40 −200 to −160

0.191 0.051 + 0.00165t

20 20 −30 to +40 40 20

0.287 0.299 0.430 + 0.000027t 0.607 0.275

C3H4O4 C12H22O11 C6H14O6 C3H6N6 C14H28O2

20 20 0 to 100 40 0 to 35

0.275 0.320 0.313 + 0.00025t 0.351 0.381 + 0.00545t

C10H8 C10H8O C10H8O C10H9N C6H6N2O2 C6H6N2O2 C6H6N2O2 C7H5NO4 C7H5NO4 C7H5NO4 C10H7NO2

−130 to mp 50 to mp 61 to mp 0 to 50 −160 to mp −160 to mp −160 to mp −163 to mp 66 to mp −160 to mp 0 to 55

0.281 + 0.00111t 0.240 + 0.00147t 0.252 + 0.00128t 0.270 + 0.0031t 0.269 + 0.000920t 0.275 + 0.000946t 0.276 + 0.001000t 0.256 + 0.00085t 0.258 + 0.00091t 0.247 + 0.00077t 0.236 + 0.00215t

SPECIFIC HEATS OF PURE COMPOUNDS TABLE 2-197

Specific Heats of Organic Solids (Concluded ) Recalculated from International Critical Tables, vol. 5, pp. 101–105

Compound

Formula

Temperature, °C

sp ht, cal/g °C

Oxalic acid

C2H2O4 C2H2O4.2H2O

−200 to +50 −200 −100 0 +50 100

0.259 + 0.00076t 0.117 0.239 0.338 0.385 0.416

Palmitic acid

C16H32O2

Phenol Phthalic acid Picric acid

C6H6O C8H6O4 C6H3N3O7

Propionic acid Propyl alcohol (n-)

C3H6O2 C3H8O

Pyrotartaric acid

C6H8O4

−180 −140 −100 −50 0 +20 14 to 26 20 −100 0 +50 100 120 −33 −200 −175 −150 −130 20

0.167 0.208 0.251 0.306 0.382 0.430 0.561 0.232 0.165 0.240 0.263 0.297 0.332 0.726 0.170 0.363 0.471 0.497 0.301

Quinhydrone

C12H10O4

Quinone

C6H4O2

−250 −225 −200 −100 0 −250 −225 −200 −150 to mp

0.017 0.061 0.098 0.191 0.256 0.031 0.082 0.113 0.282 + 0.00083t

Salol Stearic acid Succinic acid Sucrose Sugar (cane)

C13H10O3 C18H36O2 C4H6O4 C12H22O11 C12H22O11

32 15 0 to 160 20 22 to 51

0.289 0.399 0.248 + 0.00153t 0.299 0.301

Tartaric acid Tartaric acid

C4H6O6 C4H6O6·H2O

Tetrachloroethylene Tetryl

C2Cl4 C7H5N5O8

1 Tetryl + 1 picric acid 1 Tetryl + 2 TNT

C13H8N8O15 C21H15N11O20

Thymol Toluic acid (o-) (m-) (p-) Toluidine (p-)

C10H14O C8H8O2 C8H8O2 C8H8O2 C7H9N

Trichloroacetic acid Trimethyl carbinol Trinitrotoluene

C2HCl3O2 C4H10O C7H5N3O6

36 −150 −100 −50 0 +50 −40 to 0 −100 −50 0 +100 −100 to +100 −100 0 +50 0 to 49 54 to mp 54 to mp 130 to mp 0 20 40 solid −4 −100 −50 0 +100 −185 to +23 20 to 50 0 to 91

0.287 0.112 0.170 0.231 0.308 0.366 0.198 + 0.00018t 0.182 0.199 0.212 0.236 0.253 + 0.00072t 0.172 0.280 0.325 0.315 + 0.0031t 0.277 + 0.00120t 0.239 + 0.00195t 0.271 + 0.00106t 0.337 0.387 0.440 0.459 0.559 0.170 0.253 0.311 0.385 0.241 0.423 0.189 + 0.0027t

Trinitroxylene

C8H7N3O6

Triphenylmethane

C19H16

Urea

CH4N2O

20

0.320

2-177

TABLE 2-198 Cmpd. no.

Heat Capacities of Inorganic and Organic Compounds in the Ideal Gas State Name

Formula

CAS no.

Mol wt.

C1 × 1E−05

C2 × 1E−05

C3 × 1E−03

C4 × 1E−05

C5

Tmin, K

Cp at Tmin × 1E−05

Tmax, K

Cp at Tmax × 1E−05

1 2 3 4 5 6 7 8 9 10

Methane Ethane Propane n-Butane n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane

CH4 C2H6 C3H8 C4H10 C5H12 C6H14 C7H16 C8H18 C9H20 C10H22

74828 74840 74986 106978 109660 110543 142825 111659 111842 124185

16.043 30.070 44.097 58.123 72.150 86.177 100.204 114.231 128.258 142.285

0.3330 0.4033 0.5192 0.7134 0.8805 1.0440 1.2015 1.3554 1.5175 1.6720

0.7993 1.3422 1.9245 2.4300 3.0110 3.5230 4.0010 4.4310 4.9150 5.3530

2.0869 1.6555 1.6265 1.6300 1.6502 1.6946 1.6766 1.6356 1.6448 1.6141

0.4160 0.7322 1.1680 1.5033 1.8920 2.3690 2.7400 3.0540 3.4700 3.7820

991.96 752.87 723.6 730.42 747.6 761.6 756.4 746.4 749.6 742

50 200 200 200 200 200 200 200 200 200

0.3330 0.4256 0.5632 0.7673 0.9404 1.1117 1.2828 1.4529 1.6257 1.7967

1500 1500 1500 1500 1500 1500 1500 1500 1500 1500

0.8890 1.4562 2.0556 2.6602 3.2927 3.8620 4.4283 4.9764 5.5407 6.0932

11 12 13 14 15 16 17 18 19 20

n-Undecane n-Dodecane n-Tridecane n-Tetradecane n-Pentadecane n-Hexadecane n-Heptadecane n-Octadecane n-Nonadecane n-Eicosane

C11H24 C12H26 C13H28 C14H30 C15H32 C16H34 C17H36 C18H38 C19H40 C20H42

1120214 112403 629505 629594 629629 544763 629787 593453 629925 112958

156.312 170.338 184.365 198.392 212.419 226.446 240.473 254.500 268.527 282.553

1.9529 2.1295 2.1496 2.3082 2.4679 2.6283 2.7878 2.9502 3.1062 3.2481

6.0998 6.6330 7.3045 7.8678 8.4212 8.9733 9.5247 10.0340 10.5750 11.0900

1.7087 1.7155 1.6695 1.6823 1.6865 1.6912 1.6935 0.7711 0.7679 1.6360

4.1302 4.5161 4.9998 5.4486 5.8537 6.2640 6.6651 −4.3012 −4.5661 7.4500

775.4 777.5 741.02 743.1 743.6 744.41 744.57 916.73 −912.03 −726.27

200 200 200 200 200 200 200 200 200 200

2.0594 2.2442 2.3156 2.4864 2.6586 2.8312 3.0034 3.1800 3.3533 3.5235

1500 1500 1500 1500 1500 1500 1500 1500 1500 1500

6.8342 7.4325 8.0251 8.6225 9.2209 9.8182 10.4160 11.0160 11.6130 12.2110

21 22 23 24 25 26 27

2-Methylpropane 2-Methylbutane 2,3-Dimethylbutane 2-Methylpentane 2,3-Dimethylpentane 2,3,3-Trimethylpentane 2,2,4-Trimethylpentane

C4H10 C5H12 C6H14 C6H14 C7H16 C8H18 C8H18

75285 78784 79298 107835 565593 560214 540841

58.123 72.150 86.177 86.177 100.204 114.231 114.231

0.6549 0.7460 0.7772 0.9030 0.8544 0.9820 1.1390

2.4776 3.2650 4.0320 3.8010 4.5772 5.4020 5.2860

1.5870 1.5450 1.5440 1.6020 1.5181 1.5310 1.5940

1.5750 1.9230 2.5080 2.4530 2.9740 3.4930 3.3510

−706.99 666.7 −649.95 −691.6 641.01 639.9 677.94

200 200 200 200 200 200 200

0.7218 0.8546 0.9363 1.0192 1.0550 1.2194 1.3139

1500 1500 1500 1500 1500 1500 1500

2.6656 3.3792 4.0353 3.9617 4.5983 5.3754 5.3769

28 29 30 31 32 33 34 35

Ethylene Propylene 1-Butene cis-2-Butene trans-2-Butene 1-Pentene 1-Hexene 1-Heptene

C2H4 C3H6 C4H8 C4H8 C4H8 C5H10 C6H12 C7H14

74851 115071 106989 590181 624646 109671 592416 592767

28.054 42.081 56.108 56.108 56.108 70.134 84.161 98.188

0.3338 0.4339 0.5998 0.5765 0.6592 0.7595 0.9180 1.0775

0.9479 1.5200 2.0846 2.1150 2.0700 2.5525 3.0220 3.4900

1.5960 1.4250 1.5884 1.6299 1.6733 1.5820 1.5742 1.5705

0.5510 0.7860 1.2940 1.2872 1.2510 1.6660 2.0320 2.4030

740.8 623.9 707.3 739.1 742.2 713 715 717.4

60 130 200 200 200 200 200 200

0.3338 0.4388 0.6547 0.6199 0.7004 0.8273 0.9995 1.1723

1500 1500 1500 1500 1500 1500 1500 1500

1.0987 1.6836 2.2853 2.2715 2.2904 2.8467 3.4088 3.9706

36 37 38 39 40 41 42 43 44

1-Octene 1-Nonene 1-Decene 2-Methylpropene 2-Methyl-1-butene 2-Methyl-2-butene 1,2-Butadiene 1,3-Butadiene 2-Methyl-1,3-butadiene

C8H16 C9H18 C10H20 C4H8 C5H10 C5H10 C4H6 C4H6 C5H8

111660 124118 872059 115117 563462 513359 590192 106990 78795

112.215 126.242 140.269 56.108 70.134 70.134 54.092 54.092 68.119

1.2355 1.3950 1.7573 0.6125 0.8703 0.8192 0.5750 0.5095 0.6527

3.9570 4.4255 5.1710 2.0660 2.5556 2.6038 1.6476 1.7050 2.2993

1.5640 1.5624 1.7664 1.5450 1.7757 1.7593 1.5270 1.5324 1.4943

2.7669 3.1370 3.6210 1.2057 1.7636 1.7195 0.9900 1.3370 1.5164

718.17 719.6 803.02 676 807.82 800.93 677.3 685.6 −647.15

200 200 200 200 200 200 200 200 200

1.3440 1.5168 1.8333 0.6763 0.9060 0.8559 0.6269 0.5756 0.7508

1500 1500 1500 1500 1500 1500 1500 1500 1500

4.5322 5.0938 5.8682 2.2814 2.8923 2.8709 1.9202 1.9555 2.5571

45 46 47 48 49 50

Acetylene Methylacetylene Dimethylacetylene 3-Methyl-1-butyne 1-Pentyne 2-Pentyne

C2H2 C3H4 C4H6 C5H8 C5H8 C5H8

74862 74997 503173 598232 627190 627214

26.038 40.065 54.092 68.119 68.119 68.119

0.3199 0.4478 0.6534 0.8274 0.7530 0.7074

0.5424 1.0917 1.6179 2.1377 2.0905 2.2229

1.5940 1.5508 1.7837 1.7550 1.5307 1.5570

0.4325 0.6750 1.0242 1.5149 1.3780 1.3125

607.1 658.2 821.4 782 672.8 690.78

200 200 200 200 200 200

0.3566 0.4882 0.6721 0.8646 0.8276 0.7700

1500 1500 1500 1500 1500 1500

0.7575 1.3293 1.9148 2.5255 2.4754 2.5052

51 52 53 54

1-Hexyne 2-Hexyne 3-Hexyne 1-Heptyne

C6H10 C6H10 C6H10 C7H12

693027 764352 928494 628717

82.145 82.145 82.145 96.172

0.9129 1.0360 0.9376 1.0712

2.5577 3.0090 3.0150 3.0258

1.5290 2.1160 1.9057 1.5273

1.7370 2.1060 1.9860 2.0975

683 902.4 817 689.62

200 300 300 200

1.0004 1.2215 1.1909 1.1721

1500 1500 1500 1500

3.0371 3.1894 3.1889 3.5985

2-178

55 56

1-Octyne Vinylacetylene

C8H14 C4H4

629050 689974

110.199 52.076

1.2307 0.5598

3.4942 1.2141

1.5280 1.6102

2.4617 0.8908

694.81 −710.4

200 200

1.3448 0.5967

1500 1500

4.1604 1.5590

57 58 59 60 61 62 63 64 65 66

Cyclopentane Methylcyclopentane Ethylcyclopentane Cyclohexane Methylcyclohexane 1,1-Dimethylcyclohexane Ethylcyclohexane Cyclopentene 1-Methylcyclopentene Cyclohexene

C5H10 C6H12 C7H14 C6H12 C7H14 C8H16 C8H16 C5H8 C6H10 C6H10

287923 96377 1640897 110827 108872 590669 1678917 142290 693890 110838

70.134 84.161 98.188 84.161 98.188 112.215 112.215 68.119 82.145 82.145

0.4160 0.6646 0.8205 0.4320 0.9227 1.0776 1.1059 0.4807 0.6941 0.5817

3.0140 3.5070 4.0342 3.7350 4.1150 4.6718 4.6306 2.5159 3.0209 3.1717

1.4617 1.5892 1.5670 1.1920 1.6504 1.6540 1.6628 1.5803 1.6903 1.5435

1.8095 2.3526 2.6697 1.6350 2.9006 3.3397 3.2990 1.7454 2.1209 2.1273

−668.8 727.13 715.52 −530.1 779.48 792.5 781.1 718.37 781.56 701.62

100 200 200 100 200 200 200 150 200 150

0.4165 0.7510 0.9272 0.4366 0.9953 1.1535 1.1875 0.4918 0.7464 0.5978

1500 1500 1500 1500 1500 1500 1500 1500 1500 1500

2.9298 3.5495 4.1472 3.6516 4.3180 4.9543 4.9184 2.5619 3.1496 3.2132

67 68 69 70 71 72 73

Benzene Toluene o-Xylene m-Xylene p-Xylene Ethylbenzene Propylbenzene (eqn. 3)

C6H6 C7H8 C8H10 C8H10 C8H10 C8H10 C9H12

71432 108883 95476 108383 106423 100414 103651

78.114 92.141 106.167 106.167 106.167 106.167 120.194

0.4442 0.5814 0.8521 0.7568 0.7512 0.7844 −21.4827

2.3205 2.8630 3.2954 3.3924 3.3970 3.3990 3.8070

1.4946 1.4406 1.4944 1.4960 1.4928 1.5590 54701

1.7213 1.8980 2.1150 2.2470 2.2470 2.4260 −0.001713

−678.15 −650.43 -675.8 −675.9 −675.1 −702 0

200 200 200 200 200 200 200

0.5340 0.7016 0.9643 0.8759 0.8710 0.8912 1.0802

1500 1500 1500 1500 1500 1500 1500

2.4169 3.0029 3.5965 3.5920 3.5923 3.6147 4.1537

74 75 76 77 78 79 80 81

1,2,4-Trimethylbenzene Isopropylbenzene 1,3,5-Trimethylbenzene p-Isopropyltoluene Naphthalene Biphenyl Styrene m-Terphenyl

C9H12 C9H12 C9H12 C10H14 C10H8 C12H10 C8H8 C18H14

95636 98828 108678 99876 91203 92524 100425 92068

120.194 120.194 120.194 134.221 128.174 154.211 104.152 230.309

1.0106 1.0810 0.9154 1.3186 0.6805 0.9060 0.8930 1.6397

3.8314 3.7932 3.9270 4.3036 3.5494 4.2634 2.1503 6.0125

1.5010 1.7505 1.4980 1.7734 1.4262 1.4553 0.7720 1.6902

2.3950 3.0027 2.5090 3.2570 2.5984 3.1550 0.9990 5.1314

678.3 794.8 676.9 811.9 650.1 661.2 2442 757.5

200 200 200 200 200 200 100 298.15

1.1354 1.1480 1.0474 1.3825 0.8454 1.0913 0.8931 2.4618

1500 1500 1500 1500 1500 1500 1500 1500

4.1854 4.1808 4.1807 4.7952 3.7359 4.5581 3.2416 6.6678

82 83 84 85 86 87 88

Methanol Ethanol 1-Propanol 1-Butanol 2-Butanol 2-Propanol 2-Methyl-2-propanol

CH4O C2H6O C3H8O C4H10O C4H10O C3H8O C4H10O

67561 64175 71238 71363 78922 67630 75650

32.042 46.069 60.096 74.123 74.123 60.096 74.123

0.3925 0.4920 0.6190 0.7454 0.8202 0.5723 0.7704

0.8790 1.4577 2.0213 2.5907 2.5220 1.9100 2.5390

1.9165 1.6628 1.6293 1.6073 1.6010 1.4210 1.5502

0.5365 0.9390 1.2956 1.7320 1.5864 1.2155 1.6690

896.7 744.7 727.4 712.4 −704.15 626 −679.3

200 200 200 200 200 150 200

0.3980 0.5224 0.6665 0.8162 0.8890 0.5924 0.8567

1500 1500 1500 1500 1500 1500 1500

1.0533 1.6576 2.2458 2.8509 2.8513 2.1792 2.8508

89 90 91 92 93 94 95 96

1-Pentanol 2-Methyl-1-butanol 3-Methyl-1-butanol 1-Hexanol 1-Heptanol Cyclohexanol Ethylene glycol 1,2-Propylene glycol

C5H12O C5H12O C5H12O C6H14O C7H16O C6H12O C2H6O2 C3H8O2

71410 137326 123513 111273 111706 108930 107211 57556

88.150 88.150 88.150 102.177 116.203 100.161 62.068 76.095

0.9060 1.0890 1.1060 1.0625 1.2215 0.9043 0.8200 2.0114

3.0620 2.1850 2.2100 3.5210 3.9910 2.5771 1.2780 0.8082

1.6054 0.8530 0.8760 1.5835 1.5800 0.7882 1.6980 1.8656

2.1150 1.4000 1.2200 2.4620 2.8350 1.3068 0.9290 −2.4404

−717.97 2906 2940 715.75 717.7 1952.2 −754 279.98

200 298.15 298.15 200 200 200 200 298.15

0.9890 1.3247 1.3213 1.1607 1.3330 0.9648 0.8481 1.0218

1500 1500.1 1200.15 1500 1500 1500 1500 1000.15

3.4133 3.4718 3.1770 3.9726 4.5346 3.8251 1.8521 2.1175

97 98 99 100

Phenol o-Cresol m-Cresol p-Cresol

C6H6O C7H8O C7H8O C7H8O

108952 95487 108394 106445

94.113 108.140 108.140 108.140

0.4340 0.7988 0.7515 0.7384

2.4450 2.8530 2.0900 2.9080

1.1520 1.4765 0.6666 1.4559

1.5120 2.0420 1.2120 2.0910

−507 −664.7 2214 −650.42

100 200 200 200

0.4401 0.9158 0.8701 0.8707

1500 1500 1500 1500

2.6045 3.2163 3.2075 3.2102

101 102 103 104 105 106

Dimethyl ether Methyl ethyl ether Methyl-n-propyl ether Methyl isopropyl ether Methyl-n-butyl ether Methyl isobutyl ether

C2H6O C3H8O C4H10O C4H10O C5H12O C5H12O

115106 540670 557175 598538 628284 625445

46.069 60.096 74.123 74.123 88.150 88.150

0.5148 0.6868 0.9215 0.8923 0.8205 0.7284

1.4420 1.9959 2.3943 2.4765 3.0869 3.1713

1.6034 1.5534 1.6936 1.6960 1.3864 1.3520

0.7747 1.1168 1.4896 1.5598 1.7886 1.8948

725.4 692.04 797.79 791.4 613.87 585.14

200 200 298 200 300 300

0.5436 0.7396 1.1251 0.9280 1.3300 1.3200

1500 1500 1200 1500 1200 1200

1.6581 2.2931 2.6391 2.8696 3.1994 3.1987 2-179

TABLE 2-198 Cmpd. no.

Heat Capacities of Inorganic and Organic Compounds in the Ideal Gas State (Continued ) Mol wt.

C1 × 1E−05

C2 × 1E−05

C3 × 1E−03

C4 × 1E−05

C5

1634044 60297 628320 625547 100663 101848

88.150 74.123 88.150 88.150 108.140 170.211

0.9933 0.8621 1.1320 1.0953 0.7637 1.0985

3.0667 2.5510 2.9400 3.0032 2.9377 4.3412

1.7426 1.5413 1.8270 1.7988 1.6051 1.6222

2.0764 1.4370 2.0550 2.1311 2.1700 3.6455

795.59 −688.9 −852 817.35 751.2 743.62

CH2O C2H4O C3H6O C4H8O C5H10O C6H12O C7H14O C8H16O C9H18O C10H20O

50000 75070 123386 123728 110623 66251 111717 124130 124196 112312

30.026 44.053 58.080 72.107 86.134 100.161 114.188 128.214 142.241 156.268

0.3327 0.4451 0.7174 0.8966 1.0743 1.2320 1.4040 1.6088 1.7347 1.9641

0.4954 1.0687 1.9140 2.3731 2.8363 2.2146 2.5907 4.2180 4.5115 5.1412

1.8666 1.6141 2.0144 1.9754 1.9549 0.8400 0.8315 1.9126 1.7120 1.8989

0.2808 0.6135 1.1708 1.5866 2.0146 1.2190 1.3120 3.2780 3.3256 4.1278

Acetone Methyl ethyl ketone 2-Pentanone Methyl isopropyl ketone 2-Hexanone Methyl isobutyl ketone 3-Methyl-2-pentanone 3-Pentanone Ethyl isopropyl ketone Diisopropyl ketone Cyclohexanone Methyl phenyl ketone

C3H6O C4H8O C5H10O C5H10O C6H12O C6H12O C6H12O C5H10O C6H12O C7H14O C6H10O C8H8O

67641 78933 107879 563804 591786 108101 565617 96220 565695 565800 108941 98862

58.080 72.107 86.134 86.134 100.161 100.161 100.161 86.134 100.161 114.188 98.145 120.151

0.5704 0.7840 0.9005 1.5914 1.0940 1.2270 1.0028 0.9690 1.2400 1.0869 0.5776 0.8540

1.6320 2.1032 2.7085 1.7640 1.8070 2.1950 3.3169 2.4907 3.2000 4.0540 3.3535 2.3340

1.6070 1.5488 1.6592 1.2076 0.6890 0.8420 1.6900 1.4177 1.9670 1.7802 1.2202 0.8310

135 136 137 138 139 140 141

Formic acid1 Acetic acid2 Propionic acid2 n-Butyric acid2 Isobutyric acid2 Benzoic acid Acetic anhydride

CH2O2 C2H4O2 C3H6O2 C4H8O2 C4H8O2 C7H6O2 C4H6O3

64186 64197 79094 107926 79312 65850 108247

46.026 60.053 74.079 88.106 88.106 122.123 102.090

0.3381 0.4020 0.6959 1.4880 0.7469 0.7759 0.7130

0.7593 1.3675 1.7778 1.3522 2.4356 2.6455 2.2220

142 143 144 145 146 147 148

Methyl formate Methyl acetate Methyl propionate Methyl n-butyrate Ethyl formate Ethyl acetate Ethyl propionate

C2H4O2 C3H6O2 C4H8O2 C5H10O2 C3H6O2 C4H8O2 C5H10O2

107313 79209 554121 623427 109944 141786 105373

60.053 74.079 88.106 102.133 74.079 88.106 102.133

0.5060 0.5550 0.7765 0.8940 0.5370 0.9981 0.9370

149 150 151 152 153 154 155

Ethyl n-butyrate n-Propyl formate n-Propyl acetate n-Butyl acetate Methyl benzoate Ethyl benzoate Vinyl acetate

C6H12O2 C4H8O2 C5H10O2 C6H12O2 C8H8O2 C9H10O2 C4H6O2

105544 110747 109604 123864 93583 93890 108054

116.160 88.106 102.133 116.160 136.150 150.177 86.090

156 157 158 159

Methylamine Dimethylamine Trimethylamine Ethylamine

CH5N C2H7N C3H9N C2H7N

74895 124403 75503 75047

31.057 45.084 59.111 45.084

Name

Formula

107 108 109 110 111 112

Methyl tert-butyl ether Diethyl ether Ethyl propyl ether Ethyl isopropyl ether Methyl phenyl ether Diphenyl ether

C5H12O C4H10O C5H12O C5H12O C7H8O C12H10O

113 114 115 116 117 118 119 120 121 122

Formaldehyde Acetaldehyde 1-Propanal 1-Butanal 1-Pentanal 1-Hexanal 1-Heptanal 1-Octanal 1-Nonanal 1-Decanal

123 124 125 126 127 128 129 130 131 132 133 134

2-180

CAS no.

Cp at Tmin × 1E−05

Tmax, K

Cp at Tmax × 1E−05

200 200 298.15 298.15 300 300

1.0394 0.9316 1.3538 1.3620 1.1302 1.7298

1500 1500 1500 1200 1200 1200

3.4321 2.9244 3.4535 3.2289 3.0226 4.5143

934.9 737.8 930.6 904.13 890.44 2205 2201 869 810.96 862.51

50 200 200 200 200 200 200 200 200 200

0.3327 0.4660 0.7266 0.9119 1.0960 1.2672 1.4479 1.6504 1.8005 2.0192

1500 1500 1500 1500 1500 1500 1500 1500 1500 1500

0.7113 1.2994 2.1149 2.6775 3.2404 3.7314 4.2863 4.9286 5.4439 6.0539

0.9680 1.1855 1.8012 −407.4000 1.4740 1.1910 2.3000 1.3010 2.3460 2.9786 1.5700 0.7730

731.5 693 743.96 10.503 1772 2460 770.7 646.7 896 791.6 586.92 2227

200 200 200 300 200 298.15 300 200 298.15 300 200 298.15

0.6049 0.8397 0.9591 1.1291 1.1815 1.4755 1.3604 1.0536 1.4479 1.5102 0.7321 1.1313

1500 1500 1500 1500 1200 1500.15 1200 1500 1200 1500 1500 1500

1.8820 2.4816 3.0797 2.9991 3.3207 3.6532 3.4275 3.0358 3.4234 4.3093 3.4870 3.2797

1.1925 1.2620 1.7098 1.1460 1.7150 1.7925 1.6203

0.3180 0.7003 1.2654 −678.0000 1.8484 2.2382 1.6760

550 569.7 −763.78 6.98 757.75 835.9 746.5

50 50 298.15 298.15 298.15 200 200

0.3381 0.4020 0.8938 1.1533 1.0427 0.8126 0.7665

1500 1500 1500 1200.1 1200 1500 1500

0.9933 1.5756 2.1248 2.4716 2.5383 2.9712 2.5675

1.2190 1.7820 2.4420 2.9100 1.8860 2.0931 2.8290

1.6370 1.2600 1.7140 1.5700 1.2070 2.0226 1.6480

0.8940 0.8530 1.8180 2.0730 0.8640 1.8030 2.1550

743 562 716 678.3 496 928.05 724.7

250 298 300 298 100 200 300

0.5888 0.8489 1.1242 1.3461 0.5412 1.0126 1.3377

1500 1500 1200 1200 1500 1500 1200

1.5109 2.0754 2.5276 3.0766 2.1485 2.6594 3.0569

1.1150 0.8710 1.7994 1.1684 0.9396 1.0944 0.5360

3.3910 2.4470 1.7530 3.7690 2.5590 4.1794 2.1190

1.6705 1.9254 1.1960 1.9560 0.8250 0.8838 1.1980

2.5180 1.8880 −4.1200 2.8180 1.3600 −1.6090 1.1470

298 298.15 298.15 300 300 300 100

1.5583 1.1022 1.3594 1.5358 1.2586 1.4598 0.5404

1200 1500 1500 1200 1200 1500 1500

3.6213 2.7484 3.2024 3.6724 3.3569 4.2540 2.3750

0.4100 0.5565 0.7107 0.5940

1.0578 1.6384 1.5051 1.6180

1.7080 1.7341 0.7966 1.8120

0.6836 1.0899 0.8454 1.0780

150 200 200 200

0.4136 0.5812 0.7439 0.6139

1500 1500 1500 1500

1.2388 1.8585 2.4322 1.8528

733.6 −821.3 108.2 811.2 3000 −1183.1 510 735 793.04 2187.6 820

Tmin, K

160 161

Diethylamine Triethylamine

C4H11N C6H15N

109897 121448

73.138 101.192

0.9102 1.2766

2.6740 2.5559

1.7190 0.8094

1.7926 1.4829

794.94 2231.7

200 200

0.9502 1.3278

1500 1500

3.0519 4.2046

162 163 164 165 166 167 168

n-Propylamine di-n-Propylamine Isopropylamine Diisopropylamine Aniline N-Methylaniline N,N-Dimethylaniline

C3H9N C6H15N C3H9N C6H15N C6H7N C7H9N C8H11N

107108 142847 75310 108189 62533 100618 121697

59.111 101.192 59.111 101.192 93.128 107.155 121.182

0.7608 1.2114 0.6855 1.1384 0.6533 0.7796 0.8742

2.1049 2.6127 2.1876 2.5747 2.5192 3.0280 2.7204

1.7256 0.7896 1.5831 0.7384 1.4608 1.5203 0.7242

1.3936 1.6903 1.3855 1.6200 1.8870 2.3280 1.1300

789.03 2394.4 691.76 2143 −653.1 699.8 1949

200 300 200 300 200 300 300

1500 1500 1500 1500 1500 1500 1500

2.4353 4.2484 2.4540 4.1941 2.8047 3.3641 3.8844

169 170 171 172

Ethylene oxide Furan Thiophene Pyridine

C2H4O C4H4O C4H4S C5H5N

75218 110009 110021 110861

44.053 68.075 84.142 79.101

0.3346 0.3727 0.4040 0.4413

1.2116 1.6606 1.6270 2.0830

1.6084 1.5112 1.4564 1.4783

0.8241 1.3145 1.3212 1.5330

737.3 686 649 676.8

50 200 200 200

0.7933 1.5900 0.7510 1.5995 0.7705 1.2602 1.3903 0.0000 0.3346 0.4376 0.4884 0.5220

1500 1500 1500 1500

1.3297 1.7940 1.8097 2.2194

173 174 175 176 177 178 179 180

Formamide N,N-Dimethylformamide Acetamide N-Methylacetamide Acetonitrile Propionitrile n-Butyronitrile Benzonitrile

CH3NO C3H7NO C2H5NO C3H7NO C2H3N C3H5N C4H7N C7H5N

75127 68122 60355 79163 75058 107120 109740 100470

45.041 73.095 59.068 73.095 41.053 55.079 69.106 103.123

0.3822 0.7220 0.3420 0.6116 0.4191 0.5357 0.6906 0.7186

0.9300 1.7830 1.2940 2.0290 0.8876 1.4617 1.9996 2.2700

1.8450 1.5320 1.0750 1.7683 1.5818 1.5530 1.5494 1.4669

0.6900 1.3100 0.6400 1.3302 0.5032 0.9120 1.3146 1.6930

850 762 502 835.5 699.8 678.2 675 −680.77

150 200 100 300 100 200 200 200

0.3833 0.7594 0.3448 0.7698 0.4192 0.5832 0.7607 0.8053

1500 1500 1500 1500 1500 1500 1500 1500

1.1203 2.2596 1.4997 2.2209 1.1285 1.7235 2.3273 2.6706

181 182 183 184 185 186 187 188 189

Methyl mercaptan Ethyl mercaptan n-Propyl mercaptan n-Butyl mercaptan Isobutyl mercaptan sec-Butyl mercaptan Dimethyl sulfide Methyl ethyl sulfide Diethyl sulfide

CH4S C2H6S C3H8S C4H10S C4H10S C4H10S C2H6S C3H8S C4H10S

74931 75081 107039 109795 513440 513531 75183 624895 352932

48.109 62.136 76.163 90.189 90.189 90.189 62.136 76.163 90.189

0.4146 0.5576 0.7474 0.9248 0.9142 0.9237 0.6037 0.7508 0.9429

0.8307 1.3617 1.9523 2.7795 2.4513 2.5166 1.3747 1.9577 2.6863

1.5890 1.5221 1.6310 1.6837 1.6265 1.6109 1.6410 1.6424 1.7624

0.4612 0.8073 1.2112 1.5974 1.6157 1.5641 0.7988 1.1949 1.6752

716.7 687.5 750.92 758.68 745.8 739.2 −743.5 749.19 −798.3

200 200 200 200 200 200 200 273.16 200

0.4329 0.5970 0.7848 0.9714 0.9660 0.9763 0.6298 0.9004 0.9794

1500 1500 1500 1500 1500 1500 1500 1500 1500

1.0781 1.6729 2.3216 3.1008 2.9095 2.9615 1.6949 2.3178 3.0338

190 191 192 193 194 195 196 197

Fluoromethane Chloromethane Trichloromethane Tetrachloromethane Bromomethane Fluoroethane Chloroethane Bromoethane

CH3F CH3Cl CHCl3 CCl4 CH3Br C2H5F C2H5Cl C2H5Br

593533 74873 67663 56235 74839 353366 75003 74964

34.033 50.488 119.377 153.822 94.939 48.060 64.514 108.966

0.3329 0.3409 0.3942 0.3758 0.3377 0.4437 0.4568 0.4719

0.7399 0.7246 0.6573 0.7054 0.7150 1.3119 1.2967 1.2787

1.8639 1.7230 0.9280 0.5121 1.5780 1.6422 1.5992 1.5957

0.4608 0.4480 0.4930 0.4850 0.4175 0.8544 0.8590 0.8517

891.16 780.5 399.6 236.1 691.4 738.77 708.8 703.87

50 150 100 100 100 200 100 200

0.3329 0.3424 0.4048 0.4730 0.3378 0.4726 0.4569 0.5089

1500 1500 1500 1500 1500 1500 1500 1500

0.9024 0.9097 1.0063 1.0662 0.9107 1.5008 1.5112 1.5121

198 199 200 201 202 203 204 205

1-Chloropropane 2-Chloropropane 1,1-Dichloropropane 1,2-Dichloropropane Vinyl chloride Fluorobenzene Chlorobenzene Bromobenzene

C3H7Cl C3H7Cl C3H6Cl2 C3H6Cl2 C2H3Cl C6H5F C6H5Cl C6H5Br

540545 75296 78999 78875 75014 462066 108907 108861

78.541 78.541 112.986 112.986 62.499 96.104 112.558 157.010

0.6210 0.6181 0.7145 0.7866 0.4236 0.6265 0.8011 0.7210

1.8430 1.8023 1.7344 1.7429 0.8735 2.1646 2.3100 2.0640

1.6290 1.5438 1.5240 1.7157 1.6492 1.5640 2.1570 1.6504

1.2337 1.1893 1.2230 1.2627 0.6556 1.7278 2.0460 1.6870

724 685.93 674.2 765.1 739.07 −724.29 −897.6 765.3

200 200 150 200 200 200 200 200

0.6674 0.6768 0.7268 0.8217 0.4457 0.6914 0.8219 0.7679

1500 1500 1500 1500 1500 1500 1500 1500

2.1126 2.1023 2.1609 2.1894 1.1423 2.4736 2.5327 2.4628

206 207 208 209 210 211

Air Hydrogen3 Helium-4 (eqn 2) Neon Argon Fluorine

H2 He Ne Ar F2

132259100 1333740 7440597 7440019 7440371 7782414

28.951 2.016 4.003 20.180 39.948 37.997

0.2896 0.2762 0.2079 0.2079 0.2079 0.2912

0.0939 0.0956 0 0 0 0.1013

3.0120 2.4660 0 0 0 1.4530

0.0758 0.0376 0 0 0 0.0941

1484 567.6 0 0 0 662.91

50 250 100 100 100 50

0.2896 0.2843 0.2079 0.2079 0.2079 0.2912

1500 1500 1500 1500 1500 1500

0.3496 0.3225 0.2079 0.2079 0.2079 0.3812 2-181

TABLE 2-198 Cmpd. no.

Heat Capacities of Inorganic and Organic Compounds in the Ideal Gas State (Concluded ) Name

Formula

CAS no.

Mol wt.

C1 × 1E−05

C2 × 1E−05

C3 × 1E−03

C4 × 1E−05

C5

Tmin, K

Cp at Tmin × 1E−05

Tmax, K

Cp at Tmax × 1E−05

212 213 214 215 216 217 218

Chlorine Bromine Oxygen Nitrogen Ammonia Hydrazine Nitrous oxide

Cl2 Br2 O2 N2 NH3 N2H4 N2O

7782505 7726956 7782447 7727379 7664417 302012 10024972

70.905 159.808 31.999 28.014 17.031 32.045 44.013

0.2914 0.3011 0.2910 0.2911 0.3343 0.3871 0.2934

0.0918 0.0801 0.1004 0.0861 0.4898 0.8576 0.3236

0.9490 0.7514 2.5265 1.7016 2.0360 1.7228 1.1238

0.1003 0.1078 0.0936 0.0010 0.2256 0.5664 0.2177

425 314.6 1153.8 909.79 882 733.53 479.4

50 100 50 50 100 200 100

0.2914 0.3090 0.2910 0.2911 0.3343 0.4070 0.2948

1500 1500 1500 1500 1500 1500 1500

0.3793 0.3794 0.3653 0.3484 0.6647 1.0571 0.5828

219 220 221 222 223 224 225 226 227 228 229 230 231

Nitric oxide (eqn 2) Cyanogen Carbon monoxide Carbon dioxide Carbon disulfide Hydrogen fluoride Hydrogen chloride Hydrogen bromide Hydrogen cyanide Hydrogen sulfide Sulfur dioxide Sulfur trioxide Water

NO C2N2 CO CO2 CS2 HF HCl HBr HCN H2S SO2 SO3 H2O

10102439 460195 630080 124389 75150 7664393 7647010 10035106 74908 7783064 7446095 7446119 7732185

30.006 52.036 28.010 44.010 76.143 20.006 36.461 80.912 27.026 34.082 64.065 80.064 18.015

0.3498 0.3545 0.2911 0.2937 0.3010 0.2913 0.2916 0.2912 0.3013 0.3329 0.3338 0.3341 0.3336

−3.5320E−04 0.5015 0.0877 0.3454 0.3338 0.0933 0.0905 0.0953 0.3171 0.2609 0.2586 0.4968 0.2679

7.7290E−05 1.0570 3.0851 1.4280 0.8960 2.9050 2.0938 2.1420 1.6102 0.9134 0.9328 0.8732 2.6105

−5.7357E−10 0.4520 0.0846 0.2640 0.2893 0.0020 −0.0011 0.0157 0.2179 −0.1798 0.1088 0.2856 0.0890

1.4526E−08 −396 1538.2 588 374.7 1326 120 1400 626 949.4 423.7 393.74 1169

100 100 60 50 100 50 50 50 100 100 100 100 100

0.3217 0.3648 0.2911 0.2937 0.3100 0.2913 0.2914 0.2912 0.3014 0.3329 0.3354 0.3408 0.3336

1500 1500 1500 5000 1500 1500 1500 1500 1500 1500 1500 1500 2273.15

0.3586 0.8100 0.3521 0.6335 0.6148 0.3224 0.3406 0.3479 0.5522 0.5143 0.5695 0.7967 0.5276

All substances are listed in alphabetical order in Table 2-6a. Compiled from Daubert, T. E., R. P. Danner, H. M. Sibul, and C. C. Stebbins, DIPPR Data Compilation of Pure Compound Properties, Project 801 Sponsor Release, July, 1993, Design Institute for Physical Property Data, AIChE, New York, NY; and from Thermodynamics Research Center, “Selected Values of Properties of Hydrocarbons and Related Compounds,” Thermodynamics Research Center Hydrocarbon Project, Texas A&M University, College Station, Texas (extant 1994). Temperatures are expressed in kelvins; heat capacities, in J/kmol-K. J/(kmol·K) × 2.390E−04 = cal/(gmol·°C); J/(kmol·K) × 2.390059E−04 = Btu/(lbmol·°F). cosh    unless otherwise specified.     + C4 T  T

C3 C3 Use heat capacity = C1 + C2  sinh  T T

2

C5

C5

2

Equation 2 is heat capacity = C1 + C2 × T + C3 × T 2 + C4 × T 3 + C5 × T 4. Equation 3 is heat capacity = C1 + C2 × ln T + C3/T + C4 × T. 1 For the monomer. Monomer and dimer are in equilibrium below 600 K. 2 For the monomer. 3 For equilibrium mixture of ortho and para hydrogen. 2-182

SPECIFIC HEATS OF PURE COMPOUNDS

2-183

TABLE 2-199 Cp /Cv : Ratios of Specific Heats of Gases at 1-atm Pressure*

Compound

Formula

Acetaldehyde Acetic acid Acetylene

C2H4O C2H4O2 C2H2

Air

Ammonia Argon

NH3 A

Temperature, °C

Ratio of specific heats, (γ) = Cp /Cv

30 136 15 −71 925 17 −78 −118 15 15 −180 0–100

1.14 1.15 1.26 1.31 1.36 1.403 1.408 1.415 1.310 1.668 1.76 (?) 1.67

Benzene Bromine

C6H6 Br2

90 20–350

1.10 1.32

Carbon dioxide

CO2

disulfide monoxide

CS2 CO

1.304 1.37 1.21 1.404 1.41 1.355 1.15 1.256 1.08

Chlorine Chloroform Cyanogen Cyclohexane

Cl2 CHCl3 (CN)2 C6H12

15 −75 100 15 −180 15 100 15 80

Dichlorodifluormethane

CCl2F2

25

1.139

Ethane

C2H6

Ethyl alcohol ether

C2H6O C4H10O

Ethylene

C2H4

100 15 −82 90 35 80 100 15 −91

1.19 1.22 1.28 1.13 1.08 1.086 1.18 1.255 1.35

Helium Hexane (n-) Hydrogen

He C6H14 H2

bromide chloride

HBr HCl

cyanide

HCN

−180 80 15 −76 −181 20 15 100 65 140 210

1.660 1.08 1.410 1.453 1.597 1.42 1.41 1.40 1.31 1.28 1.24

Compound

Formula

Hydrogen (Cont.) iodide sulfide

HI H2S

Iodine Isobutane

I2 C4H10

Krypton

Kr

Mercury Methane

Hg CH4

Methyl acetate alcohol ether Methylal

C3H6O2 CH4O C2H6O C3H8O2

Neon Nitric oxide

Ne NO

Nitrogen

N2

Nitrous oxide

N2O

Oxygen

O2

Pentane (n-) Phosphorus Potassium

C5H12 P K

Sodium Sulfur dioxide

Na SO2

Xenon

Xe

Temperature, °C

Ratio of specific heats, (γ) = Cp /Cv

20–100 15 −45 −57

1.40 1.32 1.30 1.29

185 15

1.30 1.11

19

1.68

360 600 300 15 −80 −115 15 77 6–30 13 40

1.67 1.113 1.16 1.31 1.34 1.41 1.14 1.203 1.11 1.06 1.09

19 15 −45 −80 15 −181 100 15 −30 −70

1.64 1.400 1.39 1.38 1.404 1.47 1.28 1.303 1.31 1.34

15 −76 −181

1.401 1.415 1.45

86 300 850

1.086 1.17 1.77

750–920 15

1.68 1.29

19

1.66

*From International Critical Tables, vol. 5, pp. 80–82.

TABLE 2-200

Specific Heat Ratio, Cp /Cv , for Air Pressure, bar

Temperature, K

1

10

20

40

60

80

100

150

200

250

300

400

500

600

800

1000

150 200 250 300 350

1.410 1.406 1.403 1.402 1.399

1.510 1.452 1.429 1.418 1.411

1.668 1.505 1.457 1.436 1.422

2.333 1.630 1.517 1.470 1.446

4.120 1.781 1.577 1.505 1.467

3.973 1.943 1.640 1.537 1.488

3.202 2.093 1.699 1.570 1.509

2.507 2.274 1.816 1.640 1.553

2.243 2.236 1.877 1.687 1.589

2.091 2.140 1.896 1.716 1.612

1.988 2.050 1.885 1.730 1.627

1.851 1.920 1.836 1.727 1.640

1.768 1.832 1.782 1.707 1.638

1.712 1.771 1.743 1.683 1.629

1.654 1.682 1.681 1.645 1.605

1.639 1.619 1.636 1.619 1.585

400 450 500 600 800

1.395 1.392 1.387 1.377 1.353

1.404 1.397 1.391 1.378 1.355

1.412 1.404 1.395 1.382 1.357

1.429 1.416 1.406 1.386 1.359

1.444 1.428 1.414 1.392 1.361

1.460 1.438 1.421 1.398 1.365

1.472 1.449 1.430 1.403 1.366

1.505 1.471 1.448 1.413 1.372

1.529 1.490 1.463 1.423 1.375

1.548 1.505 1.474 1.432 1.381

1.563 1.518 1.484 1.439 1.384

1.579 1.533 1.499 1.448 1.392

1.584 1.541 1.507 1.457 1.397

1.580 1.542 1.510 1.461 1.401

1.567 1.537 1.510 1.465 1.406

1.555 1.528 1.504 1.466 1.409

1000

1.336

1.337

1.338

1.339

1.342

1.343

1.343

1.345

1.348

1.350

1.354

1.358

1.361

1.365

1.368

1.372

Calculated from Cp, Cv values of Sychev, V. V., A. A. Vasserman, et al., “Thermodynamic Properties of Air,” Standartov, Moscow, 1978 and Hemisphere, New York, 1988 (276 pp.).

2-184

PHYSICAL AND CHEMICAL DATA

SPECIFIC HEATS OF AQUEOUS SOLUTIONS UNITS CONVERSIONS

ADDITIONAL REFERENCES

For this subsection, the following units conversions are applicable: °F = 9⁄5 °C + 32. To convert calories per gram-degree Celsius to British thermal units per pound-degree Fahrenheit, multiply by 1.0.

For additional data, see International Critical Tables, vol. 5, pp. 115–116, 122–125.

TABLE 2-201

TABLE 2-208

Acetic Acid (at 38°C)

Mole % acetic acid Cal/g °C

TABLE 2-202

0 1.0

6.98 0.911

30.9 0.73

54.5 0.631

100 0.535

Ammonia Specific heat, cal/g °C

Mole % NH3

2.4°C

20.6°C

41°C

61°C

0 10.5 20.9 31.2 41.4

1.01 0.98 .96 .956 .985

1.0 0.995 .99 1.0

0.995 1.06 1.03

1.0 1.02

TABLE 2-203

TABLE 2-204

100 0.497

5°C

20°C

40°C

5.88 12.3 27.3 45.8 69.6 100

1.02 0.975 .877 .776 .681 .576

1.0 0.982 .917 .811 .708 .60

0.995 .98 .92 .83 .726 .617

TABLE 2-209

95 0.52

90.5 0.53

82.3 0.56

75.2 0.581

Copper Sulfate

Composition CuSO4 + 50H2O CuSO4 + 200H2O CuSO4 + 400H2O TABLE 2-205

Specific heat, cal/g °C Mole % CH3OH

Aniline (at 20°C)

Mol % aniline Cal/g °C

Methyl Alcohol

Temperature

Specific heat, cal/g °C

12° to 15°C 12° to 14°C 13° to 17°C

0.848 .951 .975

Nitric Acid

% HNO3 by Weight

Specific Heat at 20°C, cal/g °C

0 10 20 30 40 50 60 70 80 90

1.000 0.900 .810 .730 .675 .650 .640 .615 .575 .515

Ethyl Alcohol Specific heat, cal/g °C

Mole % C2H5OH 4.16 11.5 37.0 61.0 100.0 TABLE 2-206

3°C

23°C

41°C

1.05 1.02 0.805 .67 .54

1.02 1.03 0.86 .727 .577

1.02 1.03 0.875 .748 .621

Glycerol Specific heat, cal/g °C

Mole % C3H5(OH)3

15°C

32°C

2.12 4.66 11.5 22.7 43.9 100.0

0.961 .929 .851 .765 .67 .555

0.960 .924 .841 .758 .672 .576

TABLE 2-207

Hydrochloric Acid Specific heat, cal/g °C

Mole % HCl 0.0 9.09 16.7 20.0 25.9

0°C

10°C

20°C

40°C

60°C

1.00 0.72 .61 .58 .55

0.72 .605 .575

0.74 .631 .591

0.75 .645 .615

0.78 .67 .638 .61

TABLE 2-210

Phosphoric Acid*

%H2PO4

Cp at 21.3°C cal/g °C

%H3PO4

Cp at 21.3°C cal/g °C

2.50 3.80 5.33 8.81 10.27 14.39 16.23 19.99 22.10 24.56 25.98 28.15 29.96 32.09 33.95 36.26 38.10 40.10 42.08 44.11 46.22 48.16 49.79

0.9903 .9970 .9669 .9389 .9293 .8958 .8796 .8489 .8300 .8125 .8004 .7856 .7735 .7590 .7432 .7270 .7160 .7024 .6877 .6748 .6607 .6475 .6370

50.00 52.19 53.72 56.04 58.06 60.23 62.10 64.14 66.13 68.14 69.97 69.50 71.88 73.71 75.79 77.69 79.54 80.00 82.00 84.00 85.98 88.01 89.72

0.6350 .6220 .6113 .5972 .5831 .5704 .5603 .5460 .5349 .5242 .5157 .5160 .5046 .4940 .4847 .4786 .4680 .4686 .4593 .4500 .4419 .4359 .4206

*Z. Physik. Chem., A167, 42 (1933).

SPECIFIC HEATS OF AQUEOUS SOLUTIONS TABLE 2-211

Potassium Chloride

TABLE 2-215

Sodium Chloride Specific heat, cal/g °C

Specific heat, cal/g °C Mole % KCl

6°C

20°C

33°C

40°C

Mole % NaCl

0.99 3.85 5.66 7.41

0.945 .828 .77

0.947 .831 .775 .727

0.947 .835 .778

0.947 .837 .775

0.249 .99 2.44 9.09

TABLE 2-212

Potassium Hydroxide (at 19°C)

Mole % KOH Cal/g °C

TABLE 2-213

0 1.0

0.497 0.975

1.64 0.93

TABLE 2-216 4.76 0.814

9.09 0.75

Normal Propyl Alcohol

Mole % NaOH Cal/g °C

TABLE 2-217

Mole % C3H7OH

5°C

20°C

40°C

1.55 5.03 11.4 23.1 41.2 73.0 100.0

1.03 1.07 1.035 0.877 .75 .612 .534

1.02 1.06 1.032 0.90 .78 .645 .57

1.01 1.03 0.99 .91 .815 .708 .621

% Na2CO3 by weight 0.000 1.498 2.000 2.901 4.000 5.000 6.000 8.000 10.000 13.790 13.840 20.000 25.000

Sodium Carbonate* Temperature, °C 17.6

30.0

76.6

98.0

0.9992 .9807

0.9986

1.0098

1.0084

.9786 .9597

20°C

33°C

57°C

0.96 .91 .805

0.99 .97 .915 .81

0.97 .915 .81

0.923 .82

0 1.0

0.5 0.985

1.0 0.97

9.09 0.835

16.7 0.80

28.6 0.784

37.5 0.782

Sulfuric Acid*

%H2SO4

Cp at 20°C, cal/g °C

%H2SO4

Cp at 20°C, cal/g °C

0.34 0.68 1.34 2.65 3.50 5.16 9.82 15.36 21.40 22.27 23.22 24.25 25.39 26.63 28.00 29.52 30.34 31.20 33.11

0.9968 .9937 .9877 .9762 .9688 .9549 .9177 .8767 .8339 .8275 .8205 .8127 .8041 .7945 .7837 .7717 .7647 .7579 .7422

35.25 37.69 40.49 43.75 47.57 52.13 57.65 64.47 73.13 77.91 81.33 82.49 84.48 85.48 89.36 91.81 94.82 97.44 100.00

0.7238 .7023 .6770 .6476 .6153 .5801 .5420 .5012 .4628 .4518 .4481 .4467 .4408 .4346 .4016 .3787 .3554 .3404 .3352

*Vinal and Craig, Bur. Standards J. Research, 24, 475 (1940).

.9594 .9428

6°C

Sodium Hydroxide (at 20°C)

Specific heat, cal/g °C

TABLE 2-214

2-185

0.9761 .9392

.9183 .9086 .8924

*J. Chem. Soc. 3062–3079 (1931).

.9452 .8881 .8631

TABLE 2-218 .8936 .8615

0.8911

Zinc Sulfate

Composition

Temperature

Specific heat, cal/g °C

ZnSO4 + 50H2O ZnSO4 + 200H2O

20° to 52°C 20° to 52°C

0.842 .952

2-186

PHYSICAL AND CHEMICAL DATA

SPECIFIC HEATS OF MISCELLANEOUS MATERIALS TABLE 2-219 and Solids

Specific Heats of Miscellaneous Liquids

Material Alumina Alundum Asbestos Asphalt Bakelite Brickwork Carbon (gas retort) (see under Graphite) Cellulose Cement, Portland Clinker Charcoal (wood) Chrome brick Clay Coal tar oils Coal tars Coke Concrete Cryolite Diamond Fireclay brick Fluorspar Gasoline Glass (crown) (flint) (pyrex) (silicate) wool Granite Graphite Gypsum Kerosene Limestone Litharge Magnesia Magnesite brick Marble Porcelain, fired Berlin Porcelain, green Berlin Porcelain, fired earthenware Porcelain, green earthenware

Specific heat, cal/g °C 0.2 (100°C); 0.274 (1500°C) 0.186 (100°C) 0.25 0.22 0.3 to 0.4 About 0.2 0.168 (26° to 76°C) 0.314 (40° to 892°C) 0.387 (56° to 1450°C) 0.204 0.32 0.186 0.242 0.17 0.224 0.26 to 0.37 0.34 (15° to 90°C) 0.35 (40°C); 0.45 (200°C) 0.265 (21° to 400°C) 0.359 (21° to 800°C) 0.403 (21° to 1300°C) 0.156 (70° to 312°F); 0.219 (72° to 1472°F) 0.253 (16° to 55°C) 0.147 0.198 (100°C); 0.298 (1500°C) 0.21 (30°C) 0.53 0.16 to 0.20 0.117 0.20 0.188 to 0.204 (0 to 100°C) 0.24 to 0.26 (0 to 700°C) 0.157 0.20 (20° to 100°C) 0.165 (26° to 76°C); 0.390 (56° to 1450°C) 0.259 (16° to 46°C) 0.47 0.217 0.055 0.234 (100°C); 0.188 (1500°C) 0.222 (100°C); 0.195 (1500°C) 0.21 (18°C) 0.189 (60°C) 0.185 (60°C) 0.186 (60°C) 0.181 (60°C)

TABLE 2-219 Specific Heats of Miscellaneous Liquids and Solids (Concluded ) Specific heat, cal/g °C

Material Pyrex glass Pyrites (copper) Pyrites (iron) Pyroxylin plastics Quartz Rubber (vulcanized) Sand Silica Silica brick Silicon carbide brick Silk Steel Stone Stoneware (common) Turpentine Wood (Oak) Woods, miscellaneous Wool Zirconium oxide

0.20 0.131 (30°C) 0.136 (30°C) 0.34 to 0.38 0.17 (0°C); 0.28 (350°C) 0.415 0.191 0.316 0.202 (100°C); 0.195 (1500°C) 0.202 (100°C) 0.33 0.12 about 0.2 0.188 (60°C) 0.42 (18°C) 0.570 0.45 to 0.65 0.325 0.11 (100°C); 0.179 (1500°C)

TABLE 2-219a Oils (Animal, Vegetable, Mineral Oils) 15 Cp[cal/(g ⋅ °C) = A/  d 4 + B(t − 15)

where d = density, g/cm3. °F = 9⁄5 °C + 32; to convert calories per gram-degree Celsius to British thermal units per pound-degree Fahrenheit, multiply by 1.0; to convert grams per cubic centimeter to pounds per cubic foot, multiply by 62.43. Oils

A

Castor Citron Fatty drying non-drying semidrying oils (except castor) Naphthene base Olive Paraffin base Petroleum oils

0.500

HEATS AND FREE ENERGIES OF FORMATION UNITS CONVERSIONS °F = 9⁄5 °C + 32; to convert kilocalories per gram-mole to British thermal units per pound-mole, multiply by 1.799 × 10−3.

B

0.0007 (0.438 at 54°C) 0.440 0.0007 0.450 0.0007 0.445 0.0007 0.450 0.0007 0.405 0.0009 (0.47 at 7°C) 0.425 0.0009 0.415 0.0009

HEATS AND FREE ENERGIES OF FORMATION TABLE 2-220

2-187

Heats and Free Energies of Formation of Inorganic Compounds

The values given in the following table for the heats and free energies of formation of inorganic compounds are derived from (a) Bichowsky and Rossini, “Thermochemistry of the Chemical Substances,” Reinhold, New York, 1936; (b) Latimer, “Oxidation States of the Elements and Their Potentials in Aqueous Solution,” PrenticeHall, New York, 1938; (c) the tables of the American Petroleum Institute Research Project 44 at the National Bureau of Standards; and (d) the tables of Selected Values of Chemical Thermodynamic Properties of the National Bureau of Standards. The reader is referred to the preceding books and tables for additional details as to methods of calculation, standard states, and so on.

Compound Aluminum Al AlBr3 Al4C3 AlCl3 AlF3 AlI3 AlN Al(NH4)(SO4)2 Al(NH4)(SO4)2·12H2O Al(NO3)3·6H2O Al(NO3)3·9H2O Al2O3 Al(OH)3 Al2O3·SiO2 Al2O3·SiO2 Al2O3·SiO2 3Al2O3·2SiO2 Al2S3 Al2(SO4)3 Al2(SO4)3·6H2O Al2(SO4)3·18H2O Antimony Sb SbBr3 SbCl3 SbCl5 SbF3 SbI3 Sb2O3 Sb2O4 Sb2O5 Sb2S3 Arsenic As AsBr3 AsCl3 AsF3 AsH3 AsI3 As2O3 As2O5 As2S3 Barium Ba BaBr2 BaCl2 Ba(ClO3)2 Ba(ClO4)2 Ba(CN)2 Ba(CNO)2 BaCN2 BaCO3 BaCrO4

State† c c aq c c aq, 600 c aq c aq c c c c c c, corundum c c, sillimanite c, disthene c, andalusite c, mullite c c aq c c c c c l c c c, I, orthorhombic c, II, octahedral c c c, black

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole 0.00 −123.4 −209.5 −30.8 −163.8 −243.9 −329 −360.8 −72.8 −163.4 −57.7 −561.19 −1419.36 −680.89 −897.59 −399.09 −304.8 −648.7 −642.4 −642.0 −1874 −121.6 −820.99 −893.9 −1268.15 −2120 0.00 −59.9 −91.3 −104.8 −216.6 −22.8 −165.4 −166.6 −213.0 −230.0 −38.2

c c l l g c c c c amorphous

0.00 −45.9 −80.2 −223.76 43.6 −13.6 −154.1 −217.9 −20 −34.76

c c aq, 400 c aq, 300 c aq, 1600 c aq, 800 c c aq c c, witherite c

0.00 −180.38 −185.67 −205.25 −207.92 −176.6 −170.0 −210.2

*For footnotes see end of table.

−48 −212.1 −63.6 −284.2 −342.2

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole 0.00 −189.2 −29.0 −209.5 −312.6 −152.5 −50.4 −486.17 −1179.26 −526.32 −376.87 −272.9

−739.53 −759.3 −1103.39

Compound Barium (Cont.) BaF2 BaH2 Ba(HCO3)2 BaI2 Ba(IO3)2 BaMoO4 Ba3N2 Ba(NO2)2 Ba(NO3)2 BaO Ba(OH)2 BaO·SiO2 Ba3(PO4)2 BaPtCl6 BaS BaSO3 BaSO4 BaWO4 Beryllium Be BeBr2

0.00 −77.8

BeCl2 BeI2

−146.0 −186.6 −196.1 −36.9 0.00 −70.5 −212.27 37.7 −134.8 −183.9 −20 0.00 −183.0 −196.5 −134.4 −155.3 −180.7 −271.4

Be3N2 BeO Be(OH)2 BeS BeSO4 Bismuth Bi BiCl3 BiI3 BiO Bi2O3 Bi(OH)3 Bi2S3 Bi2(SO4)3 Boron B BBr3 BCl3 BF3 B2H6 BN B2O3 B(OH)3 B2S3 Bromine Br2 BrCl

State†

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole

c aq, 1600 c aq c aq, 400 c aq c c c aq c aq, 600 c c aq, 400 c c c c c c c

−287.9 −284.6 −40.8 −459 −144.6 −155.17 −264.5 −237.50 −370 −90.7 −184.5 −179.05 −236.99 −227.74 −133.0 −225.9 −237.76 −363 −992 −284.9 −111.2 −282.5 −340.2 −402

c c aq c aq c aq c c c c c aq

0.00 −79.4 −142 −112.6 −163.9 −39.4 −112 −134.5 −145.3 −215.6 −56.1 −281

c c aq c aq c c c c c

0.00 −90.5 −101.6 −24 −27 −49.5 −137.1 −171.1 −43.9 −607.1

c l g g g g c c gls c c

0.00 −52.7 −44.6 −94.5 −265.2 7.5 −32.1 −302.0 −297.6 −260.0 −56.6

l g g

0.00 7.47 3.06

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole −265.3 −31.5 −414.4 −158.52 −198.35

−150.75 −189.94

−209.02

−313.4 0.00 −127.9 −141.4 −103.4 −122.4 −138.3

−254.8 0.00 −76.4

−43.2 −117.9 −39.1 0.00 −50.9 −90.8 −261.0 19.9 −27.2 −282.9 −280.3 −229.4 0.00 0.931 −0.63

2-188

PHYSICAL AND CHEMICAL DATA

TABLE 2-220

Heats and Free Energies of Formation of Inorganic Compounds (Continued )

Compound Cadmium Cd CdBr2 CdCl2 Cd(CN)2 CdCO3 CdI2 Cd3N2 Cd(NO3)2 CdO Cd(OH)2 CdS CdSO4 Calcium Ca CaBr2 CaC2 CaCl2 CaCN2 Ca(CN)2 CaCO3 CaCO3·MgCO3 CaC2O4 Ca(C2H3O2)2 CaF2 CaH2 CaI2 Ca3N2 Ca(NO3)2 Ca(NO3)2·2H2O Ca(NO3)2·3H2O Ca(NO3)2·4H2O CaO Ca(OH)2 CaO·SiO2 CaS CaSO4 CaSO4·aH2O CaSO4·2H2O CaWO4 Carbon C CO CO2 Cerium Ce CeN Cesium Cs CsBr CsCl

State†

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole

c c aq, 400 c aq, 400 c c c aq, 400 c aq, 400 c c c c aq, 400

0.00 −75.8 −76.6 −92.149 −96.44 36.2 −178.2 −48.40 −47.46 39.8 −115.67 −62.35 −135.0 −34.5 −222.23 −232.635

c c aq, 400 c c aq c c aq c, calcite c, aragonite c c c aq c aq c c aq, 400 c c aq, 400 c c c c c aq, 800 c, II, wollastonite c, I, pseudowollastonite c c, insoluble form c, soluble form α c, soluble form β c c c

0.00 −162.20 −187.19 −14.8 −190.6 −209.15 −85 −43.3 −289.5 −289.54 −558.8 −332.2 −356.3 −364.1 −290.2 −286.5 −46 −128.49 −156.63 −103.2 −224.05 −228.29 −367.95 −439.05 −509.43 −151.7 −235.58 −239.2 −377.9 −376.6 −114.3 −338.73 −336.58 −335.52 −376.13 −479.33 −387

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole 0.00 −70.7 −67.6 −81.889 −81.2 −163.2 −43.22 −71.05 −55.28 −113.7 −33.6 −194.65 0.00 −181.86 −16.0 −179.8 −195.36 −54.0 −270.8 −270.57

−311.3 −264.1 −35.7 −157.37 −88.2 −177.38 −293.57 −351.58 −409.32 −144.3 −213.9 −207.9 −357.5 −356.6 −113.1 −311.9 −309.8 −308.8 −425.47

c, graphite c, diamond g g

0.00 0.453 −26.416 −94.052

0.00 0.685 −32.808 −94.260

c c

0.00 −78.2

0.00 −70.8

0.00 −97.64 −91.39 −106.31 −102.01

0.00

c c aq, 500 c aq, 400

−94.86 −101.61

Compound Cesium (Cont.) Cs2CO3 CsF CsH CsHCO3 CsI CsNH2 CsNO3 Cs2O CsOH Cs2S Cs2SO4 Chlorine Cl2 ClF ClO ClO2 ClO3 Cl2O Cl2O7 Chromium Cr CrBr3 Cr3C2 Cr4C CrCl2 CrF2 CrF3 CrI2 CrO3 Cr2O3 Cr2(SO4)3 Cobalt Co CoBr2 Co3C CoCl2 CoCO3 CoF2 CoI2 Co(NO3)2 CoO Co3O4 Co(OH)2 Co(OH)3 CoS Co2S3 CoSO4 Columbium Cb Cb2O5 Copper Cu CuBr CuBr2 CuCl CuCl2

State† c c aq, 400 c c aq, 2000 c aq, 400 c c aq, 400 c c aq, 200 c c aq g g g g g g g c aq c c c aq c c c aq c c aq

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole −271.88 −131.67 −140.48 −12 −230.6 −226.6 −83.91 −75.74 −28.2 −121.14 −111.54 −82.1 −100.2 −117.0 −87 −344.86 −340.12 0.00 −25.7 33 24.7 37 18.20 63 0.00 −21.008 −16.378 −103.1 −152 −231 −63.7 −139.3 −268.8

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole

−135.98 −7.30 −210.56 −82.61 −96.53 −107.87 −316.66 0.00 29.5 22.40 0.00 −122.7 −21.20 −16.74 −93.8 −102.1

−64.1 −249.3 −626.3

c c aq c c aq, 400 c aq c aq c aq c c c c c c c aq, 400

0.00 −55.0 −73.61 9.49 −76.9 −95.58 −172.39 −172.98 −24.2 −43.15 −102.8 −114.9 −57.5 −196.5 −131.5 −177.0 −22.3 −40.0 −216.6

c c

0.00 −462.96

0.00

0.00 −26.7 −34.0 −42.4 −31.4 −48.83 −64.7

0.00 −23.8

c c c aq c c aq, 400

0.00 −61.96 7.08 −66.6 −75.46 −155.36 −144.2 −37.4 −65.3 −108.9 −142.0 −19.8 −188.9

−33.25 −24.13

HEATS AND FREE ENERGIES OF FORMATION TABLE 2-220

Heats and Free Energies of Formation of Inorganic Compounds (Continued )

Compound Copper (Cont.) CuClO4 Cu(ClO3)2 Cu(ClO4)2 CuI CuI2 Cu3N Cu(NO3)2 CuO Cu2O Cu(OH)2 CuS Cu2S CuSO4 Cu2SO4 Erbium Er Er(OH)3 Fluorine F2 F2O Gallium Ga GaBr3 GaCl3 GaN Ga2O Ga2O3 Germanium Ge Ge3N4 GeO2 Gold Au AuBr AuBr3 AuCl AuCl3 AuI Au2O3 Au(OH)3 Hafnium Hf HfO2 Hydrogen H3AsO3 H3AsO4 HBr HBrO HBrO3 HCl HCN HClO HClO3 HClO4 HC2H3O2 H2C2O4 HCOOH

2-189

State† aq aq, 400 aq c c aq c c aq, 200 c c c c c c aq, 800 c aq

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole −28.3 −17.8 −4.8 −11.9 17.78 −73.1 −83.6 −38.5 −43.00 −108.9 −11.6 −18.97 −184.7 −200.78 −179.6

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole 1.34 15.4 −5.5 −16.66 −8.76 −36.6 −31.9 −38.13 −85.5 −11.69 −20.56 −158.3 −160.19 −152.0

Compound Hydrogen (Cont.) H2CO3 HF HI HIO HIO3 HN3 HNO3 HNO3·H2O HNO3·3H2O H2O H2O2 H3PO2

c c

0.00 −326.8

0.00

g g

0.00 5.5

0.00 9.7

H3PO3 H3PO4 H2S

c c c c c c

0.00 −92.4 −125.4 −26.2 −84.3 −259.9

0.00

c c c

0.00 −15.7 −128.6

0.00

c c c aq c c aq c c c

0.00 −3.4 −14.5 −11.0 −8.3 −28.3 −32.96 0.2 11.0 −100.6

0.00

c c

0.00 −271.1

0.00 −258.2

aq c aq g aq, 400 aq aq g aq, 400 g aq, 100 aq, 400 aq aq, 660 l aq, 400 c aq, 300 l aq, 200

−175.6 −214.9 −214.8 −8.66 −28.80 −25.4 −11.51 −22.063 −39.85 31.1 24.2 −28.18 −23.4 −31.4 −116.2 −116.74 −196.7 −194.6 −97.8 −98.0

−153.04

H2S2 H2SO3 H2SO4 H2Se H2SeO3 H2SeO4

24.47

H2SiO3 H4SiO4 H2Te H2TeO3

4.21 −0.76 18.71

H2TeO4 Indium In InBr3 InCl3

−183.93 −12.72 −24.58 −19.90 5.00 −22.778 −31.330 27.94 26.55 −19.11 −0.25 −10.70 −93.56 −96.8 −165.64 −82.7 −85.1

InI3 InN In2O3 Iodine I2 IBr ICl ICl3 I2O5 Iridium Ir IrCl IrCl2 IrCl3 IrF6 IrO2 Iron Fe FeBr2

State†

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole

aq g aq, 200 g aq, 400 aq c aq g g l aq, 400 l l g l l aq, 200 c aq c aq c aq, 400 g aq, 2000 l aq, 200 l aq, 400 g aq c aq c aq, 400 c c g c aq aq

−167.19 −64.2 −75.75 6.27 −13.47 −38 −56.77 −54.8 70.3 −31.99 −41.35 −49.210 −112.91 −252.15 −57.7979 −68.3174 −45.16 −45.80 −145.5 −145.6 −232.2 −232.2 −306.2 −309.32 −4.77 −9.38 −3.6 −146.88 −193.69 −212.03 20.5 18.1 −126.5 −122.4 −130.23 −143.4 −267.8 −340.6 36.9 −145.0 −145.0 −165.6

c c aq c aq c aq c c

0.00 −97.2 −112.9 −128.5 −145.6 −56.5 −67.2 −4.8 −222.47

c g g g c c

0.00 14.88 10.05 4.20 −21.8 −42.5

0.00 4.63 1.24 −1.32 −6.05

c c c c l c

0.00 −20.5 −40.6 −60.5 −130 −40.14

0.00 −16.9 −32.0 −46.5

c, α c aq, 540

0.00 −57.15 −78.7

−149.0 −64.7 0.365 −12.35 −23.33 −32.25 78.50 −17.57 −19.05 −78.36 −193.70 −54.6351 −56.6899 −28.23 −31.47 −120.0 −204.0 −270.0 −7.85 −128.54 17.0 18.4 −101.36 −247.9 33.1 −115.7

0.00 −97.2 −117.5 −60.5

0.00 −69.47

2-190

PHYSICAL AND CHEMICAL DATA

TABLE 2-220

Heats and Free Energies of Formation of Inorganic Compounds (Continued )

Compound Iron (Cont.) FeBr3 Fe3C Fe(CO)5 FeCO3 FeCl2 FeCl3 FeF2 FeI2 FeI3 Fe4N Fe(NO3)2 Fe(NO3)3 FeO Fe2O3 Fe3O4 Fe(OH)2 Fe(OH)3 FeO·SiO2 Fe2P FeSi FeS FeS2 FeSO4 Fe2(SO4)3 FeTiO3 Lanthanum La LaCl3 La3H8 LaN La2O3 LaS2 La2S3 La2(SO4)3 Lead Pb PbBr2 PbCO3 Pb(C2H3O2)2 PbC2O4 PbCl2 PbF2 PbI2 Pb(NO3)2 PbO PbO2 Pb3O4 Pb(OH)2 PbS PbSO4 Lithium Li LiBr LiBrO3 Li2C2 LiCN LiCNO

State† aq c l c, siderite c aq c aq, 2000 aq, 1200 c aq aq c aq aq, 800 c c c c c c c c c c, pyrites c, marcasite c aq, 400 aq, 400 c, ilmenite

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole −95.5 5.69 −187.6 −172.4 −81.9 −100.0 −96.4 −128.5 −177.2 −24.2 −47.7 −49.7 −2.55 −118.9 −156.5 −64.62 −198.5 −266.9 −135.9 −197.3 −273.5 −13 −19.0 −22.64 −38.62 −33.0 −221.3 −236.2 −653.3 −295.51

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole −76.26 4.24 −154.8 −72.6 −83.0 −96.5 −151.7 −45 −39.5 0.862 −72.8 −81.3 −59.38 −179.1 −242.3 −115.7 −166.3

Compound Lithium (Cont.) LiC2H3O2 Li2CO3 LiCl LiClO3 LiClO4 LiF LiH LiHCO3 LiI LiIO3 Li3N LiNO3 Li2O Li2O2 LiOH

−23.23 −35.93 −195.5 −196.4 −533.4 −277.06

c c aq c c c c c aq

0.00 −253.1 −284.7 −160 −72.0 −539 −148.3 −351.4 −972

0.00

c c aq c, cerussite c aq, 400 c c aq c c c aq, 400 c, red c, yellow c c c c c

0.00 −66.24 −56.4 −167.6 −232.6 −234.2 −205.3 −85.68 −82.5 −159.5 −41.77 −106.88 −99.46 −51.72 −50.86 −65.0 −172.4 −123.0 −22.38 −218.5

0.00 −62.06 −54.97 −150.0

c c aq, 400 aq c aq aq

0.00 −83.75 −95.40 −77.9 −13.0 −31.4 −101.2

0.00

LiOH·H2O Li2O·SiO2 Li2Se Li2SO4 Li2SO4·H2O Magnesium Mg Mg(AsO4)2 MgBr2

−64.6

−184.40 −75.04 −68.47 −148.1 −41.47 −58.3 −45.53 −43.88 −52.0 −142.2 −102.2 −21.98 −192.9

−95.28 −65.70 −31.35 −94.12

Mg(CN)2 MgCN2 Mg(C2H3O2)2 MgCO3 MgCl2 MgCl2·H2O MgCl2·2H2O MgCl2·4H2O MgCl2·6H2O MgF2 MgI2 MgMoO4 Mg3N2 Mg(NO3)2 Mg(NO3)2·2H2O Mg(NO3)2·6H2O MgO MgO·SiO2 Mg(OH)2 MgS MgSO4 MgTe MgWO4 Manganese Mn MnBr2 Mn3C

State†

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole

aq c aq, 1900 c aq, 278 aq aq c aq, 400 c aq, 2000 c aq, 400 aq c c aq, 400 c c aq c aq, 400 c gls c aq c aq, 400 c

−183.9 −289.7 −293.1 −97.63 −106.45 −87.5 −106.3 −145.57 −144.85 −22.9 −231.1 −65.07 −80.09 −121.3 −47.45 −115.350 −115.88 −142.3 −151.9 −159 −116.58 −121.47 −188.92 −374 −84.9 −95.5 −340.23 −347.02 −411.57

c c aq c aq, 400 aq c aq c c aq, 400 c c c c c c aq, 400 c c c aq, 400 c c c c c, ppt. c, brucite c aq c aq, 400 c c

0.00 −731.3 −749 −123.9 −167.33 −39.7 −61 −344.6 −261.7 −153.220 −189.76 −230.970 −305.810 −453.820 −597.240 −263.8 −86.8 −136.79 −329.9 −115.2 −188.770 −209.927 −336.625 −624.48 −143.84 −347.5 −221.90 −223.9 −84.2 −108 −304.94 −325.4 −25 −345.2

c, α c aq c

0.00 −91 −106 1.1

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole −160.00 −269.8 −267.58 −102.03 −70.95 −81.4 −136.40 −210.98 −83.03 −102.95 −37.33 −96.95 −138.0 −106.44 −108.29

−105.64 −314.66 −375.07 0.00 −630.14 −156.94 −29.08 −286.38 −241.7 −143.77 −205.93 −267.20 −387.98 −505.45 −132.45 −100.8 −140.66 −160.28 −496.03 −136.17 −326.7 −200.17 −193.3 −277.7 −283.88

0.00 −97.8 1.26

HEATS AND FREE ENERGIES OF FORMATION TABLE 2-220

Heats and Free Energies of Formation of Inorganic Compounds (Continued )

Compound Manganese (Cont.) Mn(C2H3O2)2 MnCO3 MnC2O4 MnCl2 MnF2 MnI2 Mn5N2 Mn(NO3)2 Mn(NO3)2.6H2O MnO MnO2 Mn2O3 Mn3O4 MnO.SiO2 Mn(OH)2 Mn(OH)3 Mn3(PO4)2 MnSe MnS MnSO4 Mn2(SO4)3 Mercury Hg HgBr HgBr2 Hg(C2H3O2)2 HgCl2 HgCl Hg2Cl2 Hg(CN)2 HgC2O4 HgH HgI2 HgI Hg2I2 Hg(NO3)2 Hg2(NO3)2 HgO Hg2O HgS HgSO4 Hg2SO4 Molybdenum Mo Mo2C Mo2N MoO2 MoO3 MoS2 MoS3 Nickel Ni NiBr2 Ni3C Ni(C2H3O2)2 Ni(CN)2 NiCl2

2-191

State†

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole

c aq c c c aq, 400 aq, 1200 c aq c c aq, 400 c c c c c c c c c c c, green c aq, 400 c aq

−270.3 −282.7 −211 −240.9 −112.0 −128.9 −206.1 −49.8 −76.2 −57.77 −134.9 −148.0 −557.07 −92.04 −124.58 −229.5 −331.65 −301.3 −163.4 −221 −736 −26.3 −47.0 −254.18 −265.2 −635 −657

l g c aq c aq c aq g c c aq, 1110 c g c, red g c aq aq c, red c, yellow ppt. c c, black c c

0.00 23 −40.68 −38.4 −196.3 −192.5 −53.4 −50.3 19 −63.13 62.8 66.25 −159.3 57.1 −25.3 33 −28.88 −56.8 −58.5 −21.6 −20.8 −21.6 −10.7 −166.6 −177.34

c c c c c c c

0.00 4.36 −8.3 −130 −180.39 −56.27 −61.48

c c aq c aq aq c

0.00 −53.4 −72.6 9.2 −249.6 230.9 −75.0

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole

aq, 400 c aq c aq c aq, 200 c c c c c aq, 200

−94.34 −157.5 −171.6 −22.4 −42.0 −101.5 −113.5 −58.4 −129.8 −163.2 −20.4 −216 −231.3

−74.19

g g g aq, 200 c aq c aq, 400 c aq c aq aq c aq c aq, 400 c aq c aq c aq c aq c aq, 500 aq aq, 400 c aq, 400 l l c g g g g c l g

0.00 −27 −10.96 −19.27 −64.57 −60.27 −148.1 −148.58 −0.7 3.6 −17.8 −12.3 −223.4 −266.3 −260.6 −75.23 −71.20 −69.4 −63.2 −276.9 −271.3 −111.6 −110.2 −48.43 −44.97 −87.40 −80.89 −87.59 −55.21 −281.74 −279.33 12.06 −57.96 −232.2 19.55 21.600 7.96 2.23 −10.0 11.6 12.8

Compound

State†

Nickel (Cont.) −227.2 −192.5 −102.2 −180.0 −73.3 −46.49 −101.1 −441.2 −86.77 −111.49 −209.9 −306.22 −282.1 −143.1 −190 −27.5 −48.0 −228.41

NiF2 NiI2 Ni(NO3)2 NiO Ni(OH)2 Ni(OH)3 NiS NiSO4 Nitrogen N2 NF3 NH3 NH4Br NH4C2H3O2 NH4CN NH4CNS

0.00 18 −38.8 −9.74 −139.2 −42.2 −23.25 14

(NH4)2CO3 (NH4)2C2O4 NH4Cl NH4ClO4 (NH4)2CrO4 NH4F NH4I NH4NO3

52.25 −24.0 23 −26.53 −13.09 −15.65 −13.94 −12.80 −8.80 −149.12 0.00 2.91 −118.0 −162.01 −54.19 −57.38 0.00 −60.7 8.88 −190.1 66.3

NH4OH (NH4)2S (NH4)2SO4 N2H4 N2H4·H2O N2H4·H2SO4 N2O NO NO2 N2O4 N2O5 NOBr NOCl Osmium Os OsO4 Oxygen O2 O3 Palladium Pd PdO Phosphorus P P

−142.9 −36.2 −64.0 −51.7 −105.6

−187.6 0.00 −3.903 −43.54 −108.26 20.4 4.4 −164.1 −196.2 −48.59 −21.1 −209.3 −84.7 −31.3

−14.50 −215.06 −214.02

24.82 20.719 12.26 23.41 19.26 16.1

c c g

0.00 −93.6 −80.1

0.00 −70.9 −68.1

g g

0.00 33.88

0.00 38.86

c c

0.00 −20.40

0.00

c, white (“yellow”) c, red (“violet”) g

0.00 −4.22 150.35

0.00 −1.80 141.88

2-192

PHYSICAL AND CHEMICAL DATA

TABLE 2-220

Heats and Free Energies of Formation of Inorganic Compounds (Continued )

Compound Phosphorus (Cont.) P2 P4 PBr3 PBr5 PCl3 PCl5 PH3 PI3 P2O5 POCl3 Platinum Pt PtBr4 PtCl2 PtCl4 PtI4 Pt(OH)2 PtS PtS2 Potassium K K3AsO3 K3AsO4 KH2AsO4 KBr KBrO3 KC2H3O2 KCl KClO3 KClO4 KCN KCNO KCNS K2CO3 K2C2O4 K2CrO4 K2Cr2O7 KF K3Fe(CN)6 K4Fe(CN)6 KH KHCO3 KI KIO3 KIO4 KMnO4 K2MoO4

State† g g l c g l g g c c g c c aq c c aq c c c c c aq aq c c aq, 400 c aq, 1667 c aq, 400 c aq, 400 c aq, 400 c aq, 400 c aq, 400 c aq c aq, 400 c aq, 400 c aq, 400 c aq, 400 c aq, 400 c aq, 180 c aq c aq c c aq, 2000 c aq, 500 c aq, 400 aq c aq, 400 aq, 880

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole 33.82 13.2 −45 −60.6 −70.0 −76.8 −91.0 2.21 −10.9 −360.0 −138.4 0.00 −40.6 −50.7 −34 −62.6 −82.3 −18 −87.5 −20.18 −26.64 0.00 −323.0 −390.3 −271.2 −94.06 −89.19 −81.58 −71.68 −173.80 −177.38 −104.348 −100.164 −93.5 −81.34 −103.8 −101.14 −28.1 −25.3 −99.6 −94.5 −47.0 −41.07 −274.01 −280.90 −319.9 −315.5 −333.4 −328.2 −488.5 −472.1 −134.50 −138.36 −48.4 −34.5 −131.8 −119.9 −10 −229.8 −224.85 −78.88 −73.95 −121.69 −115.18 −98.1 −192.9 −182.5 −364.2

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole 24.60 5.89 −65.2 −63.3 −73.2 −1.45 −127.2 0.00

Compound Potassium (Cont.) KNH2 KNO2 KNO3 K2O K2O·Al2O3·SiO2 K2O·Al2O3·SiO2 KOH K3PO3 K3PO4 KH2PO4 K2PtCl4 K2PtCl6

−67.9 −18.55 −24.28 0.00 −355.7 −236.7 −90.8 −92.0 −60.30 −156.73 −97.76 −98.76 −69.30 −72.86 −28.08 −90.85 −44.08

K2Se K2SeO4 K2S K2SO3 K2SO4 K2SO4·Al2(SO4)3 K2SO4·Al2(SO4)3· 24H2O K2S2O6 Rhenium Re ReF6 Rhodium Rh RhO Rh2O Rh2O3 Rubidium Rb RbBr

−264.04

RbCN Rb2CO3

−293.1

RbCl

−306.3 −440.9 −133.13

−5.3 −207.71 −77.37 −79.76 −101.87 −99.68 −169.1 −168.0 −342.9

RbF RbHCO3 RbI RbNH2 RbNO3 Rb2O Rb2O2 RbOH Ruthenium Ru RuS2 Selenium Se

State†

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole

c aq c aq, 400 c c, leucite gls c, adularia c, microcline gls c aq, 400 aq aq c c aq c aq, 9400 c aq aq c aq, 400 c aq c aq, 400 c

−28.25 −86.0 −118.08 −109.79 −86.2 −1379.6 −1368.2 −1784.5 −1784.5 −1747 −102.02 −114.96 −397.5 −478.7 −362.7 −254.7 −242.6 −299.5 −286.1 −74.4 −83.4 −267.1 −121.5 −110.75 −267.7 −269.7 −342.65 −336.48 −1178.38

c c

−2895.44 −418.62

−2455.68

c g

0.00 −274

0.00

c c c c

0.00 −21.7 −22.7 −68.3

0.00

0.00 −95.82 −45.0 −90.54 −25.9 −273.22 −282.61 −105.06 −53.6 −101.06 −133.23 −139.31 −230.01 −225.59 −81.04 −31.2 −74.57 −27.74 −119.22 −110.52 −82.9 −107 −101.3 −115.8

0.00

c c g aq, 500 aq c aq, 220 c g aq, ∞ c aq, 400 c aq, 2000 c g aq, 400 c c aq, 400 c c c aq, 200 c c c, I, hexagonal

−75.9 −94.29 −93.68

−105.0 −443.3 −326.1 −226.5 −263.6 −99.10 −240.0 −111.44 −251.3 −314.62 −310.96 −1068.48

−52.50 −93.38 −263.78 −98.48 −57.9 −100.13 −134.5 −209.07 −40.5 −81.13 −95.05

−106.39

0.00 −46.99

0.00 −44.11

0.00

0.00

HEATS AND FREE ENERGIES OF FORMATION TABLE 2-220

Compound

Heats and Free Energies of Formation of Inorganic Compounds (Continued )

State†

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole

SiF4 SiH4 SiI4 Si3N4 SiO2

Silver Ag AgBr Ag2C2 AgC2H3O2 AgCN Ag2CO3 Ag2C2O4 AgCl AgF AgI AgIO3 AgNO2 AgNO3 Ag2O Ag2S Ag2SO4 Sodium Na Na3AsO3 Na3AsO4 NaBr NaBrO NaBrO3 NaC2H3O2 NaCN NaCNO NaCNS Na2CO3 NaCO2NH2 Na2C2O4 NaCl NaClO3 NaClO4

Compound

State†

Sodium (Cont.)

Selenium (Cont.) Se2Cl2 SeF6 SeO2 Silicon Si SiBr4 SiC SiCl4

2-193

c, II, red, monoclinic l g c

0.2 −22.06 −246 −56.33

−13.73 −222

c l c l g g g c c c, cristobalite, 1600° form c, cristobalite, 1100° form c, quartz c, tridymite

0.00 −93.0 −28 −150.0 −142.5 −370 −14.8 −29.8 −179.25 −202.62

0.00

c c c c aq c c c c c aq, 400 c c c aq c aq, 6500 c c c aq

0.00 −23.90 84.5 −95.9 −91.7 33.8 −119.5 −158.7 −30.11 −48.7 −53.1 −15.14 −42.02 −11.6 −2.9 −29.4 −24.02 −6.95 −5.5 −170.1 −165.8

c aq, 500 c aq, 500 c aq, 400 aq aq, 400 c aq, 400 c aq, 200 c aq c aq, 400 c aq, 1000 c c aq, 600 c aq, 400 c aq, 400 c

0.00 −314.61 −366 −381.97 −86.72 −86.33 −78.9 −68.89 −170.45 −175.450 −22.47 −22.29 −96.3 −91.7 −39.94 −38.23 −269.46 −275.13 −142.17 −313.8 −309.92 −98.321 −97.324 −83.59 −78.42 −101.12

−27.4 −133.9 −133.0 −360 −9.4 −154.74

Na2Cr2O7 NaF NaH NaHCO3 NaI NaIO3 Na2MoO4 NaNO2 NaNO3

−202.46 −203.35 −203.23

Na2CrO4

−190.4 0.00 −23.02 −70.86 38.70 −103.0 −25.98 −47.26 −16.17 −24.08 3.76 9.99 −7.66 −7.81 −2.23 −7.6 −146.8 −139.22 0.00 −341.17 −87.17 −57.59 −152.31 −23.24 −86.00 −39.24 −249.55 −251.36 −283.42 −91.894 −93.92 −62.84

Na2O Na2O2 Na2O·SiO2 Na2O·Al2O3·3SiO2 Na2O·Al2O3·4SiO2 NaOH Na3PO3 Na3PO4 Na2PtCl4 Na2PtCl6 Na2Se Na2SeO4 Na2S Na2SO3 Na2SO4 Na2SO4·10H2O Na2WO4 Strontium Sr SrBr2 Sr(C2H3O2)2 Sr(CN)2 SrCO3 SrCl2 SrF2 Sr(HCO3)2 SrI2 Sr3N2 Sr(NO3)2 SrO SrO·SiO2 SrO2 Sr2O Sr(OH)2 Sr3(PO4)2 SrS

aq, 476 c aq, 800 aq, 1200 c aq, 400 c c aq c aq, ∞ aq, 400 c aq c aq c aq, 400 c c c c, natrolite c c aq, 400 aq, 1000 c aq, 400 aq c aq c aq, 440 c aq, 800 c aq, 400 c aq, 800 c aq, 1100 c c aq c c aq, 400 c aq aq c c aq, 400 c aq c aq, 400 c c aq, 400 c gls c c c aq, 800 c aq c

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole

−97.66 −319.8 −323.0 −465.9 −135.94 −135.711 −14 −226.0 −222.1 −69.28 −71.10 −112.300 −364 −358.7 −86.6 −83.1 −111.71 −106.880 −99.45 −119.2 −383.91 −1180 −1366 −101.96 −112.193 −389.1 −457 −471.9 −237.2 −272.1 −280.9 −59.1 −78.1 −254 −261.5 −89.8 −105.17 −261.2 −264.1 −330.50 −330.82 −1033.85 −391 −381.5

−73.29

0.00 −171.0 −187.24 −358.0 −364.4 −59.5 −290.9 −197.84 −209.20 −289.0 −459.1 −136.1 −156.70 −91.4 −233.2 −228.73 −140.8 −364 −153.3 −153.6 −228.7 −239.4 −980 −985 −113.1

−296.58 −431.18 −129.0 −128.29 −9.30 −202.66 −202.87 −74.92 −94.84 −333.18 −71.04 −87.62 −88.84 −90.06 −105.0 −361.49 −90.60 −100.18 −428.74 −216.78

−89.42 −230.30 −101.76 −240.14 −241.58 −302.38 −301.28 −870.52 −345.18 0.00 −182.36 −311.80 −54.50 −271.9 −195.86 −413.76 −157.87 −76.5 −185.70 −133.7 −139.0 −208.27 −881.54

2-194

PHYSICAL AND CHEMICAL DATA

TABLE 2-220

Heats and Free Energies of Formation of Inorganic Compounds (Continued )

Compound

State†

Strontium (Cont.) SrSO4 SrWO4 Sulfur S

S2 S6 S8 S2Br2 SCl4 S2Cl2 S2Cl4 SF6 SO SO2 SO3

SO2Cl2 Tantalum Ta TaN Ta2O5 Tellurium Te TeBr4 TeCl4 TeF6 TeO2 Thallium Tl TlBr TlCl TlCl3 TlF TlI TlNO3 Tl2O Tl2O3 TlOH Tl2S Tl2SO4 Thorium Th ThBr4 ThC2 ThCl4 ThI4 Th3N4 ThO2 Th(OH)4 Th(SO4)2

aq c aq, 400 c

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole −120.4 −345.3 −345.0 −393

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole −109.78 −309.30

0.00 −0.071 0.257

0.00 0.023 0.072 0.071 43.57 19.36 13.97 12.770

53.25 31.02 27.78 27.090 −4 −13.7 −14.2 −24.1 −262 19.02 −70.94 −94.39 −103.03 −105.09 −105.92 −109.34 −82.04 −89.80

−237 12.75 −71.68 −88.59 −88.28 −88.22 −88.34 −88.98 −74.06 −75.06

c c c

0.00 −51.2 −486.0

0.00 −45.11 −453.7

c c c g c

0.00 −49.3 −77.4 −315 −77.56

0.00

c c aq c aq c aq aq c aq c aq c c c aq c c aq, 800

0.00 −41.5 −28.0 −49.37 −38.4 −82.4 −91.0 −77.6 −31.1 −12.7 −58.2 −48.4 −43.18 −120 −57.44 −53.9 −22 −222.8 −214.1

c c aq c c aq aq c c c, “soluble” c aq

0.00 −281.5 −352.0 −45.1 −335 −392 −292.0 −309.0 −291.6 −336.1 −632 −668.1

−5.90

−57.4 −292 −64.66 0.00 −39.43 −32.34 −44.46 −39.09 −44.25 −73.46 −31.3 −20.09 −36.32 −34.01 −45.54 −45.35 −197.79 −191.62 0.00

SnBr2

SnCl2 SnCl4 SnI2 SnO SnO2 Sn(OH)2 Sn(OH)4 SnS Titanium Ti TiC TiCl4 TiN TiO2 Tungsten W WO2 WO3 WS2 Uranium U UC2 UCl3 UCl4 U3N4 UO2 UO2(NO3)2·6H2O UO3 U3O8 Vanadium V VCl2 VCl3 VCl4 VN V2O2 V2O3 V2O4 V2O5 Zinc Zn ZnSb ZnBr2 Zn(C2H3O2)2 Zn(CN)2 ZnCO3 ZnCl2

−295.31

ZnF2 ZnI2

−322.32 −246.33 −282.3 −280.1

Zn(NO3)2 ZnO ZnO·SiO2 Zn(OH)2 ZnS ZnSO4

−549.2

c, II, tetragonal c, III, “gray,” cubic c aq c aq c aq l aq c aq c c c c c

0.00 0.6 −61.4 −60.0 −94.8 −110.6 −83.6 −81.7 −127.3 −157.6 −38.9 −33.3 −67.7 −138.1 −136.2 −268.9 −18.61

−68.94 −110.4 −124.67

c c l c c, III, rutil amorphous

0.00 −110 −181.4 −80.0 −225.0 −214.1

0.00 −109.2 −165.5 −73.17 −211.9 −201.4

c c c c

0.00 −130.5 −195.7 −84

0.00 −118.3 −177.3

c c c c c c c c c

0.00 −29 −213 −251 −274 −256.6 −756.8 −291.6 −845.1

0.00

c c l l c c c c c

0.00 −147 −187 −165 −41.43 −195 −296 −342 −373

c c c aq, 400 c aq, 400 c c c aq, 400 aq c aq aq, 400 c, hexagonal c c, rhombic c, wurtzite c aq, 400

0.00 −3.6 −77.0 −93.6 −259.4 −269.4 17.06 −192.9 −99.9 −115.44 −192.9 −50.50 −61.6 −134.9 −83.36 −282.6 −153.66 −45.3 −233.4 −252.12

Compound Tin Sn

SnBr4 c, rhombic c, monoclinic l, λ l, λµ equilibrium g g g g l l l l g g g g l c, α c, β c, γ g l

State†

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole 0.00 1.1 −55.43 −97.66

−30.95 −60.75 −123.6 −115.95 −226.00

−249.6 −242.2 −617.8

0.00

−35.08 −277 −316 −342 0.00 −3.88 −72.9 −214.4 −173.5 −88.8 −166.6 −49.93 −87.7 −76.19 −44.2 −211.28

HEATS OF COMBUSTION TABLE 2-220

2-195

Heats and Free Energies of Formation of Inorganic Compounds (Concluded )

Compound

State†

Zirconium Zr ZrC ZrCl4 ZrN

c c c c

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole 0.00 −29.8 −268.9 −82.5

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole 0.00 −34.6 −75.9

Compound Zirconium (Cont.) ZrO2 Zr(OH)4 ZrO(OH)2

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole

State† c, monoclinic c c

−258.5 −411.0 −337

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole −244.6 −307.6

† The physical state is indicated as follows: c, crystal (solid); l, liquid; g, gas; gls, glass or solid supercooled liquid; aq, in aqueous solution. A number following the symbol aq applies only to the values of the heats of formation (not to those of free energies of formation); and indicates the number of moles of water per mole of solute; when no number is given, the solution is understood to be dilute. For the free energy of formation of a substance in aqueous solution, the concentration is always that of the hypothetical solution of unit molality. ‡ The increment in heat content, ∆H, in the reaction of forming the given substance from its elements in their standard states. When ∆H is negative, heat is evolved in the process, and, when positive, heat is absorbed. § The heat of solution in water of a given solid, liquid, or gaseous compound is given by the difference in the value for the heat of formation of the given compound in the solid, liquid, or gaseous state and its heat of formation in aqueous solution. The following two examples serve as an illustration of the procedure: (1) For NaCl(c) and NaCl(aq, 400H2O), the values of ∆H(formation) are, respectively, −98.321 and −97.324 kg-cal per mole. Subtraction of the first value from the second gives ∆H = 0.998 kg-cal per mole for the reaction of dissolving crystalline sodium chloride in 400 moles of water. When this process occurs at a constant pressure of 1 atm, 0.998 kg-cal of energy are absorbed. (2) For HCl(g) and HCl(aq, 400H2O), the values for ∆H(formation) are, respectively, −22.06 and −39.85 kg-cal per mole. Subtraction of the first from the second gives ∆H = −17.79 kg-cal per mole for the reaction of dissolving gaseous hydrogen chloride in 400 moles of water. At a constant pressure of 1 atm, 17.79 kg-cal of energy are evolved in this process.  The increment in the free energy, ∆F, in the reaction of forming the given substance in its standard state from its elements in their standard states. The standard states are: for a gas, fugacity (approximately equal to the pressure) of 1 atm; for a pure liquid or solid, the substance at a pressure of 1 atm; for a substance in aqueous solution, the hypothetical solution of unit molality, which has all the properties of the infinitely dilute solution except the property of concentration. ¶ The free energy of solution of a given substance from its normal standard state as a solid, liquid, or gas to the hypothetical one molal state in aqueous solution may be calculated in a manner similar to that described in footnote § for calculating the heat of solution.

HEATS OF COMBUSTION TABLE 2-221 Enthalpies and Gibbs Energies of Formation, Entropies, and Net Enthalpies of Combustion of Inorganic and Organic Compounds at 298.15 K

Mol wt

Ideal gas enthalpy of formation, J/kmol × 1E-07

Methane Ethane Propane n-Butane n-Pentane n-Hexane n-Heptane n-Octane n-Nonane

CH4 C2H6 C3H8 C4H10 C5H12 C6H14 C7H16 C8H18 C9H20

74828 74840 74986 106978 109660 110543 142825 111659 111842

16.043 30.070 44.097 58.123 72.150 86.177 100.204 114.231 128.258

−7.4520 −8.3820 −10.4680 −12.5790 −14.6760 −16.6940 −18.7650 −20.8750 −22.8740

−5.0490 −3.1920 −2.4390 −1.6700 −0.8813 −0.0066 0.8165 1.6000 2.4980

1.8627 2.2912 2.7020 3.0991 3.4945 3.8874 4.2798 4.6723 5.0640

−0.8026 −1.4286 −2.0431 −2.6573 −3.2449 −3.8551 −4.4647 −5.0742 −5.6846

10 11 12 13 14 15 16 17 18 19

n-Decane n-Undecane n-Dodecane n-Tridecane n-Tetradecane n-Pentadecane n-Hexadecane n-Heptadecane n-Octadecane n-Nonadecane

C10H22 C11H24 C12H26 C13H28 C14H30 C15H32 C16H34 C17H36 C18H38 C19H40

124185 1120214 112403 629505 629594 629629 544763 629787 593453 629925

142.285 156.312 170.338 184.365 198.392 212.419 226.446 240.473 254.500 268.527

−24.9460 −27.0430 −29.0720 −31.1770 −33.2440 −35.3110 −37.4170 −39.4450 −41.5120 −43.5790

3.3180 4.1160 4.9810 5.7710 6.5990 7.4260 8.2160 9.0830 9.9100 10.7400

5.4570 5.8493 6.2415 6.6337 7.0259 7.4181 7.8102 8.2023 8.5945 8.9866

−6.2942 −6.9036 −7.5137 −8.1229 −8.7328 −9.3424 −9.9515 −10.5618 −11.1715 −11.7812

20 21 22 23 24 25 26 27 28 29

n-Eicosane 2-Methylpropane 2-Methylbutane 2,3-Dimethylbutane 2-Methylpentane 2,3-Dimethylpentane 2,3,3-Trimethylpentane 2,2,4-Trimethylpentane Ethylene Propylene

C20H42 C4H10 C5H12 C6H14 C6H14 C7H16 C8H18 C8H18 C2H4 C3H6

112958 75285 78784 79298 107835 565593 560214 540841 74851 115071

282.553 58.123 72.150 86.177 86.177 100.204 114.231 114.231 28.054 42.081

−45.6460 −13.4180 −15.3700 −17.6800 −17.4550 −19.4100 −21.8450 −22.4010 5.2510 1.9710

11.5700 −2.0760 −1.4050 −0.3125 −0.5338 0.5717 1.8280 1.3940 6.8440 6.2150

9.3787 2.9539 3.4374 3.6592 3.8089 4.1455 4.2702 4.2296 2.1920 2.6660

−12.3908 −2.6490 −3.2395 −3.8476 −3.8492 −4.4608 −5.0688 −5.0653 −1.3230 −1.9257

Cmpd. no. 1 2 3 4 5 6 7 8 9

Name

Formula

CAS no.

Ideal gas Gibbs energy Ideal gas Standard net enthalpy of formation, entropy, of combustion, J/kmol × 1E-07 J/(kmol·K) × 1E-05 J/kmol × 1E-09

2-196

PHYSICAL AND CHEMICAL DATA

TABLE 2-221 Enthalpies and Gibbs Energies of Formation, Entropies, and Net Enthalpies of Combustion of Inorganic and Organic Compounds (Continued ) Cmpd. no.

Name

Formula

CAS no.

Mol wt

Ideal gas enthalpy of formation, J/kmol × 1E-07

Ideal gas Gibbs energy Ideal gas Standard net enthalpy of formation, entropy, of combustion, J/kmol × 1E-07 J/(kmol·K) × 1E-05 J/kmol × 1E-09

30 31 32 33 34 35 36 37 38 39

1-Butene cis-2-Butene trans-2-Butene 1-Pentene 1-Hexene 1-Heptene 1-Octene 1-Nonene 1-Decene 2-Methylpropene

C4H8 C4H8 C4H8 C5H10 C6H12 C7H14 C8H16 C9H18 C10H20 C4H8

106989 590181 624646 109671 592416 592767 111660 124118 872059 115117

56.108 56.108 56.108 70.134 84.161 98.188 112.215 126.242 140.269 56.108

−0.0540 −0.7400 −1.1000 −2.1300 −4.2000 −6.2800 −8.3600 −10.4000 −12.4700 −1.7100

7.0270 6.5360 6.3160 7.8450 8.7390 9.4830 10.3000 11.1500 11.9800 5.8080

3.0775 3.0120 2.9650 3.4699 3.8389 4.2549 4.6469 5.0399 5.4319 2.9309

−2.5408 −2.5339 −2.5303 −3.1296 −3.7394 −4.3489 −4.9606 −5.5684 −6.1781 −2.5242

40 41 42 43 44 45 46 47 48 49

2-Methyl-1-butene 2-Methyl-2-butene 1,2-Butadiene 1,3-Butadiene 2-Methyl-1,3-butadiene Acetylene Methylacetylene Dimethylacetylene 3-Methyl-1-butyne 1-Pentyne

C5H10 C5H10 C4H6 C4H6 C5H8 C2H2 C3H4 C4H6 C5H8 C5H8

563462 513359 590192 106990 78795 74862 74997 503173 598232 627190

70.134 70.134 54.092 54.092 68.119 26.038 40.065 54.092 68.119 68.119

−3.5300 −4.1800 16.2300 10.9240 7.5730 22.8200 18.4900 14.5700 13.8000 14.4400

6.6680 6.0450 19.8600 14.9720 14.5896 21.0680 19.3840 18.4900 20.7200 21.0300

3.3950 3.3860 2.9300 2.7889 3.1564 2.0081 2.4836 2.8330 3.1890 3.2980

−3.1159 −3.1088 −2.4617 −2.4090 −2.9842 −1.2570 −1.8487 −2.4189 −3.0460 −3.0510

50 51 52 53 54 55 56 57 58 59

2-Pentyne 1-Hexyne 2-Hexyne 3-Hexyne 1-Heptyne 1-Octyne Vinylacetylene Cyclopentane Methylcyclopentane Ethylcyclopentane

C5H8 C6H10 C6H10 C6H10 C7H12 C8H14 C4H4 C5H10 C6H12 C7H14

627214 693027 764352 928494 628717 629050 689974 287923 96377 1640897

68.119 82.145 82.145 82.145 96.172 110.199 52.076 70.134 84.161 98.188

12.5100 12.3700 10.5000 10.6000 10.3000 8.2300 30.4600 −7.7030 −10.6200 −12.6900

19.0700 21.8500 19.9000 19.9000 22.7000 23.5000 30.6000 3.8850 3.6300 4.4800

3.3084 3.6940 3.7200 3.7600 4.0850 4.4780 2.7940 2.9290 3.3990 3.7830

−3.0291 −3.6610 −3.6400 −3.6400 −4.2717 −4.8815 −2.3620 −3.0709 −3.6741 −4.2839

60 61 62 63 64 65 66 67 68 69

Cyclohexane Methylcyclohexane 1,1-Dimethylcyclohexane Ethylcyclohexane Cyclopentene 1-Methylcyclopentene Cyclohexene Benzene Toluene o-Xylene

C6H12 C7H14 C8H16 C8H16 C5H8 C6H10 C6H10 C6H6 C7H8 C8H10

110827 108872 590669 1678917 142290 693890 110838 71432 108883 95476

84.161 98.188 112.215 112.215 68.119 82.145 82.145 78.114 92.141 106.167

−12.3300 −15.4800 −18.1000 −17.1500 3.3100 −0.3800 −0.4600 8.2880 5.0170 1.9080

3.1910 2.7330 3.5229 3.9550 11.0500 10.3800 10.7700 12.9600 12.2200 12.2000

2.9728 3.4330 3.6501 3.8260 2.9127 3.2640 3.1052 2.6930 3.2099 3.5383

−3.6560 −4.2571 −4.8639 −4.8705 −2.9393 −3.5340 −3.5320 −3.1360 −3.7340 −4.3330

70 71 72 73 74 75 76 77 78 79

m-Xylene p-Xylene Ethylbenzene Propylbenzene 1,2,4-Trimethylbenzene Isopropylbenzene 1,3,5-Trimethylbenzene p-Isopropyltoluene Naphthalene Biphenyl

C8H10 C8H10 C8H10 C9H12 C9H12 C9H12 C9H12 C10H14 C10H8 C12H10

108383 106423 100414 103651 95636 98828 108678 99876 91203 92524

106.167 106.167 106.167 120.194 120.194 120.194 120.194 134.221 128.174 154.211

1.7320 1.8030 2.9920 0.7910 −1.3800 0.4000 −1.5900 −2.9000 15.0580 18.2420

11.8760 12.1400 13.0730 13.8090 11.7100 13.7900 11.8100 13.3520 22.4080 28.0230

3.5854 3.5223 3.6063 3.9843 3.9610 3.8600 3.8560 4.2630 3.3315 3.9367

−4.3318 −4.3330 −4.3450 −4.9542 −4.9307 −4.9510 −4.9291 −5.5498 −4.9809 −6.0317

80 81 82 83 84 85 86 87 88 89

Styrene m-Terphenyl Methanol Ethanol 1-Propanol 1-Butanol 2-Butanol 2-Propanol 2-Methyl-2-propanol 1-Pentanol

C8H8 C18H14 CH4O C2H6O C3H8O C4H10O C4H10O C3H8O C4H10O C5H12O

100425 92068 67561 64175 71238 71363 78922 67630 75650 71410

104.152 230.309 32.042 46.069 60.096 74.123 74.123 60.096 74.123 88.150

14.7400 27.6600 −20.0940 −23.4950 −25.5200 −27.4600 −29.2900 −27.2700 −31.2400 −29.8737

21.3900 42.3000 −16.2320 −16.7850 −15.9900 −15.0300 −16.9600 −17.3470 −17.7600 −14.6022

3.4510 5.2630 2.3988 2.8064 3.2247 3.6148 3.6469 3.0920 3.2630 4.0250

−4.2190 −9.0530 −0.6382 −1.2350 −1.8438 −2.4560 −2.4408 −1.8300 −2.4239 −3.0605

90 91 92 93 94 95 96

2-Methyl-1-butanol 3-Methyl-1-butanol 1-Hexanol 1-Heptanol Cyclohexanol Ethylene glycol 1,2-Propylene glycol

C5H12O C5H12O C6H14O C7H16O C6H12O C2H6O2 C3H8O2

137326 123513 111273 111706 108930 107211 57556

88.150 88.150 102.177 116.203 100.161 62.068 76.095

−30.2085 −30.2100 −31.6500 −33.6400 −28.6200 −38.7500 −42.1500

−14.6709 −14.5000 −13.4400 −12.5300 −10.9500 −30.2600 −30.4000

3.9351 3.8770 4.4010 4.7919 3.2770 3.2350 3.5200

−3.0620 −3.0623 −3.6766 −4.2860 −3.4639 −1.0590 −1.6476

HEATS OF COMBUSTION

2-197

TABLE 2-221 Enthalpies and Gibbs Energies of Formation, Entropies, and Net Enthalpies of Combustion of Inorganic and Organic Compounds (Continued ) Cmpd. no.

Name

Formula

CAS no.

Mol wt

Ideal gas enthalpy of formation, J/kmol × 1E-07

Ideal gas Gibbs energy Ideal gas Standard net enthalpy of formation, entropy, of combustion, J/kmol × 1E-07 J/(kmol·K) × 1E-05 J/kmol × 1E-09

97 98 99

Phenol o-Cresol m-Cresol

C6H6O C7H8O C7H8O

108952 95487 108394

94.113 108.140 108.140

−9.6399 −12.8570 −13.2300

−3.2637 −3.5430 −4.0190

3.1481 3.5259 3.5604

−2.9210 −3.5280 −3.5278

100 101 102 103 104 105 106 107 108 109

p-Cresol Dimethyl ether Methyl ethyl ether Methyl n-propyl ether Methyl isopropyl ether Methyl n-butyl ether Methyl isobutyl ether Methyl tert-butyl ether Diethyl ether Ethyl propyl ether

C7H8O C2H6O C3H8O C4H10O C4H10O C5H12O C5H12O C5H12O C4H10O C5H12O

106445 115106 540670 557175 598538 628284 625445 1634044 60297 628320

108.140 46.069 60.096 74.123 74.123 88.150 88.150 88.150 74.123 88.150

−12.5350 −18.4100 −21.6400 −23.8200 −25.2000 −25.8100 −26.6000 −28.3500 −25.2100 −27.2200

−3.1660 −11.2800 −11.7100 −11.1000 −12.1800 −10.1700 −10.7000 −11.7500 −12.2100 −11.5200

3.5075 2.6670 3.0881 3.5200 3.4160 3.9010 3.8100 3.5780 3.4230 3.8810

−3.5226 −1.3284 −1.9314 −2.5174 −2.5311 −3.1282 −3.1220 −3.1049 −2.5035 −3.1200

110 111 112 113 114 115 116 117 118 119

Ethyl isopropyl ether Methyl phenyl ether Diphenyl ether Formaldehyde Acetaldehyde 1-Propanal 1-Butanal 1-Pentanal 1-Hexanal 1-Heptanal

C5H12O C7H8O C12H10O CH2O C2H4O C3H6O C4H8O C5H10O C6H12O C7H14O

625547 100663 101848 50000 75070 123386 123728 110623 66251 111717

88.150 108.140 170.211 30.026 44.053 58.080 72.107 86.134 100.161 114.188

−28.5800 −6.7900 5.2000 −10.8600 −16.6200 −18.6300 −20.7000 −22.7800 −24.8600 −26.9400

−12.6400 2.2700 17.5000 −10.2600 −13.3100 −12.4600 −11.6300 −10.7100 −10.0050 −9.1910

3.8000 3.6100 4.1300 2.1866 2.6420 3.0440 3.4365 3.8289 4.2214 4.6138

−3.1030 −3.6072 −5.8939 −0.5268 −1.1045 −1.6857 −2.3035 −2.9100 −3.5200 −4.1360

120 121 122 123 124 125 126 127 128 129

1-Octanal 1-Nonanal 1-Decanal Acetone Methyl ethyl ketone 2-Pentanone Methyl isopropyl ketone 2-Hexanone Methyl isobutyl ketone 3-Methyl-2-pentanone

C8H16O C9H18O C10H20O C3H6O C4H8O C5H10O C5H10O C6H12O C6H12O C6H12O

124130 124196 112312 67641 78933 107879 563804 591786 108101 565617

128.214 142.241 156.268 58.080 72.107 86.134 86.134 100.161 100.161 100.161

−29.0200 −31.0900 −33.1700 −21.5700 −23.9000 −25.9200 −26.2400 −27.9826 −28.8000 −28.1000

−8.3770 −7.5530 −6.7390 −15.1300 −14.7000 −13.8300 −13.9000 −13.0081 −13.5000 −12.9000

5.0063 5.3988 5.7912 2.9540 3.3940 3.7860 3.6990 4.1786 4.0700 4.1200

−4.7400 −5.3500 −5.9590 −1.6590 −2.2680 −2.8796 −2.8770 −3.4900 −3.4900 −3.4900

130 131 132 133 134 135 136 137 138 139

3-Pentanone Ethyl isopropyl ketone Diisopropyl ketone Cyclohexanone Methyl phenyl ketone Formic acid Acetic acid Propionic acid n-Butyric acid Isobutyric acid

C5H10O C6H12O C7H14O C6H10O C8H8O CH2O2 C2H4O2 C3H6O2 C4H8O2 C4H8O2

96220 565695 565800 108941 98862 64186 64197 79094 107926 79312

86.134 100.161 114.188 98.145 120.151 46.026 60.053 74.079 88.106 88.106

−25.7900 −28.6100 −31.1400 −22.6100 −8.6700 −37.8600 −43.2800 −45.3500 −47.5800 −48.4100

−13.4400 −13.3000 −13.2000 −8.6620 −0.1364 −35.1000 −37.4600 −36.6700 −36.0000 −36.2100

3.7000 4.0690 4.5700 3.2200 3.8450 2.4870 2.8250 3.2300 3.6200 3.4120

−2.8804 −3.4860 −4.0950 −3.2990 −3.9730 −0.2115 −0.8146 −1.3950 −2.0077 −2.0004

140 141 142 143 144 145 146 147 148 149

Benzoic acid Acetic anhydride Methyl formate Methyl acetate Methyl propionate Methyl n-butyrate Ethyl formate Ethyl acetate Ethyl propionate Ethyl n-butyrate

C7H6O2 C4H6O3 C2H4O2 C3H6O2 C4H8O2 C5H10O2 C3H6O2 C4H8O2 C5H10O2 C6H12O2

65850 108247 107313 79209 554121 623427 109944 141786 105373 105544

122.123 102.090 60.053 74.079 88.106 102.133 74.079 88.106 102.133 116.160

−29.4100 −57.2500 −35.2400 −41.1900 −42.7500 −45.0700 −38.8300 −44.4500 −46.3600 −48.5500

−21.4200 −47.3400 −29.5000 −32.4200 −31.1000 −30.5300 −30.3100 −32.8000 −31.9300 −31.2200

3.6900 3.8990 2.8520 3.1980 3.5960 3.9880 3.2820 3.5970 4.0250 4.4170

−3.0951 −1.6750 −0.8924 −1.4610 −2.0780 −2.6860 −1.5070 −2.0610 −2.6740 −3.2840

150 151 152 153 154 155 156 157 158 159

n-Propyl formate n-Propyl acetate n-Butyl acetate Methyl benzoate Ethyl benzoate Vinyl acetate Methylamine Dimethylamine Trimethylamine Ethylamine

C4H8O2 C5H10O2 C6H12O2 C8H8O2 C9H10O2 C4H6O2 CH5N C2H7N C3H9N C2H7N

110747 109604 123864 93583 93890 108054 74895 124403 75503 75047

88.106 102.133 116.160 136.150 150.177 86.090 31.057 45.084 59.111 45.084

−40.7600 −46.4800 −48.5600 −28.7900 −32.6000 −31.4900 −2.2970 −1.8450 −2.4310 −4.7150

−29.3600 −32.0400 −31.2600 −18.1000 −19.0500 −22.7900 3.2070 6.8390 9.8990 3.6160

3.6780 4.0230 4.4250 4.1400 4.5500 3.2800 2.4330 2.7296 2.8700 2.8480

−2.0410 −2.6720 −3.2800 −3.7720 −4.4100 −1.9500 −0.9751 −1.6146 −2.2449 −1.5874

160 161 162

Diethylamine Triethylamine n-Propylamine

C4H11N C6H15N C3H9N

109897 121448 107108

73.138 101.192 59.111

−7.1420 −9.5800 −7.0500

7.3080 11.4100 4.1700

3.5220 4.0540 3.2420

−2.8003 −4.0405 −2.1650

2-198

PHYSICAL AND CHEMICAL DATA

TABLE 2-221 Enthalpies and Gibbs Energies of Formation, Entropies, and Net Enthalpies of Combustion of Inorganic and Organic Compounds (Continued ) Cmpd. no.

Name

Formula

CAS no.

Mol wt

Ideal gas enthalpy of formation, J/kmol × 1E-07

Ideal gas Gibbs energy Ideal gas Standard net enthalpy of formation, entropy, of combustion, J/kmol × 1E-07 J/(kmol·K) × 1E-05 J/kmol × 1E-09

163 164 165 166 167 168 169

di-n-Propylamine Isopropylamine Diisopropylamine Aniline N-Methylaniline N,N-Dimethylaniline Ethylene oxide

C6H15N C3H9N C6H15N C6H7N C7H9N C8H11N C2H4O

142847 75310 108189 62533 100618 121697 75218

101.192 59.111 101.192 93.128 107.155 121.182 44.053

−11.6000 −8.3800 −15.0000 8.7100 8.8000 10.0500 −5.2630

8.6800 3.1920 5.7900 16.6800 20.2000 24.7728 −1.3230

4.2900 3.1240 4.1200 3.1980 3.4100 3.6600 2.4299

−4.0189 −2.1566 −3.9900 −3.2390 −3.9000 −4.5250 −1.2180

170 171 172 173 174 175 176 177 178 179

Furan Thiophene Pyridine Formamide N,N-Dimethylformamide Acetamide N-Methylacetamide Acetonitrile Propionitrile n-Butyronitrile

C4H4O C4H4S C5H5N CH3NO C3H7NO C2H5NO C3H7NO C2H3N C3H5N C4H7N

110009 110021 110861 75127 68122 60355 79163 75058 107120 109740

68.075 84.142 79.101 45.041 73.095 59.068 73.095 41.053 55.079 69.106

−3.4800 11.5440 14.0370 −19.2200 −19.1700 −23.8300 −24.0000 7.4040 5.1800 3.4058

0.0823 12.6620 19.0490 −14.7100 −8.8400 −15.9600 −13.5000 9.1868 9.7495 10.8658

2.6714 2.7865 2.8278 2.4857 3.2600 2.7220 3.2000 2.4329 2.8614 3.2543

−1.9959 −2.4352 −2.6721 −0.5021 −1.7887 −1.0741 −1.7100 −1.1904 −1.8007 −2.4148

180 181 182 183 184 185 186 187 188 189

Benzonitrile Methyl mercaptan Ethyl mercaptan n-Propyl mercaptan n-Butyl mercaptan Isobutyl mercaptan sec-Butyl mercaptan Dimethyl sulfide Methyl ethyl sulfide Diethyl sulfide

C7H5N CH4S C2H6S C3H8S C4H10S C4H10S C4H10S C2H6S C3H8S C4H10S

100470 74931 75081 107039 109795 513440 513531 75183 624895 352932

103.123 48.109 62.136 76.163 90.189 90.189 90.189 62.136 76.163 90.189

21.8823 −2.2900 −4.6300 −6.7500 −8.7800 −9.6900 −9.6600 −3.7240 −5.9600 −8.3470

26.0872 −0.9800 −0.4814 0.2583 1.1390 0.5982 0.5120 0.7302 1.1470 1.7780

3.2104 2.5500 2.9610 3.3650 3.7520 3.6280 3.6670 2.8585 3.3320 3.6800

−3.5224 −1.1517 −1.7366 −2.3458 −2.9554 −2.9490 −2.9490 −1.7449 −2.3531 −2.9607

190 191 192 193 194 195 196 197 198 199

Fluoromethane Chloromethane Trichloromethane Tetrachloromethane Bromomethane Fluoroethane Chloroethane Bromoethane 1-Chloropropane 2-Chloropropane

CH3F CH3Cl CHCl3 CCl4 CH3Br C2H5F C2H5Cl C2H5Br C3H7Cl C3H7Cl

593533 74873 67663 56235 74839 353366 75003 74964 540545 75296

34.033 50.488 119.377 153.822 94.939 48.060 64.514 108.966 78.541 78.541

−23.4300 −8.1960 −10.2900 −9.5810 −3.7700 −26.4400 −11.2260 −6.3600 −13.3180 −14.4770

−21.0400 −5.8440 −7.0100 −5.3540 −2.8190 −21.2300 −6.0499 −2.5820 −5.2610 −6.1360

2.2273 2.3418 2.9560 3.0991 2.4580 2.6440 2.7578 2.8730 3.1547 3.0594

−0.5219 −0.6754 −0.3800 −0.2653 −0.7054 −1.1270 −1.2849 −1.2850 −1.8670 −1.8630

200 201 202 203 204 205 206 207 208 209

1,1-Dichloropropane 1,2-Dichloropropane Vinyl chloride Fluorobenzene Chlorobenzene Bromobenzene Air Hydrogen Helium-4 Neon

C3H6Cl2 C3H6Cl2 C2H3Cl C6H5F C6H5Cl C6H5Br H2 He Ne

78999 78875 75014 462066 108907 108861 132259100 1333740 7440597 7440019

112.986 112.986 62.499 96.104 112.558 157.010 28.951 2.016 4.003 20.180

−15.0800 −16.2800 2.8450 −11.6566 5.1090 10.5018 0 0 0 0

−6.5200 −8.0180 4.1950 −6.9036 9.8290 13.8532 0 0 0 0

3.4480 3.5480 2.7354 3.0263 3.1403 3.2439 1.9900 1.3057 1.2604 1.4622

−1.7200 −1.7070 −1.1780 −2.8145 −2.9760 −3.0192 0 −0.2418 0 0

210 211 212 213 214 215 216 217 218 219

Argon Fluorine Chlorine Bromine Oxygen Nitrogen Ammonia Hydrazine Nitrous oxide Nitric oxide

Ar F2 Cl2 Br2 O2 N2 NH3 N2H4 N2O NO

7440371 7782414 7782505 7726956 7782447 7727379 7664417 302012 10024972 10102439

39.948 37.997 70.905 159.808 31.999 28.014 17.031 32.045 44.013 30.006

0 0 0 3.0910 0 0 −4.5898 9.5353 8.2050 9.0250

0 0 0 0.3140 0 0 −1.6400 15.9170 10.4160 8.6570

1.5474 2.0268 2.2297 2.4535 2.0504 1.9150 1.9266 2.3861 2.1985 2.1060

0 0 0 0 0 0 −0.3168 −5.3420 −0.0820 −0.0902

220 221 222 223 224 225 226 227

Cyanogen Carbon monoxide Carbon dioxide Carbon disulfide Hydrogen fluoride Hydrogen chloride Hydrogen bromide Hydrogen cyanide

C2N2 CO CO2 CS2 HF HCl HBr HCN

460195 630080 124389 75150 7664393 7647010 10035106 74908

52.036 28.010 44.010 76.143 20.006 36.461 80.912 27.026

30.9072 −11.0530 −39.3510 11.6900 −27.3300 −9.2310 −3.6290 13.5143

29.7553 −13.7150 −39.4370 6.6800 −27.5400 −9.5300 −5.3340 12.4725

2.4146 1.9756 2.1368 2.3790 1.7367 1.8679 1.9859 2.0172

−1.0961 −0.2830 0 −1.0769 0.1524 −0.0286 −0.0690 −0.6233

HEATS OF COMBUSTION

2-199

TABLE 2-221 Enthalpies and Gibbs Energies of Formation, Entropies, and Net Enthalpies of Combustion of Inorganic and Organic Compounds (Concluded ) Cmpd. no. 228 229 230 231

Name

Formula

Hydrogen sulfide Sulfur dioxide Sulfur trioxide Water

H2S SO2 SO3 H2O

CAS no. 7783064 7446095 7446119 7732185

Mol wt

Ideal gas enthalpy of formation, J/kmol × 1E-07

34.082 64.065 80.064 18.015

−2.0630 −29.6840 −39.5720 −24.1814

Ideal gas Gibbs energy Ideal gas Standard net enthalpy of formation, entropy, of combustion, J/kmol × 1E-07 J/(kmol·K) × 1E-05 J/kmol × 1E-09 −3.3440 −30.0120 −37.0950 −22.8590

−0.5180 0 0.0989 0

2.0560 2.4810 2.5651 1.8872

All substances are listed in alphabetical order in Table 2-6a. Compiled from Daubert, T. E., R. P. Danner, H. M. Sibul, and C. C. Stebbins, DIPPR Data Compilation of Pure Compound Properties, Project 801 Sponsor Release, July, 1993, Design Institute for Physical Property Data, AIChE, New York, NY; and from Thermodynamics Research Center, “Selected Values of Properties of Hydrocarbons and Related Compounds,” Thermodynamics Research Center Hydrocarbon Project, Texas A&M University, College Station, Texas (extant 1994). The compounds are considered to be formed from the elements in their standard states at 298.15 K and 101,325 Pa. These include C (graphite) and S (rhombic). Enthalpy of combustion is the net value for the compound in its standard state at 298.15K and 101,325 Pa. Products of combustion are taken to be CO2 (gas), H2O (gas), F2 (gas), Cl2 (gas), Br2 (gas), I2 (gas), SO2 (gas), N2 (gas), H3PO4 (solid), and SiO2 (crystobalite). J/kmol × 2.390E-04 = cal/gmol; J/kmol × 4.302106E-04 = Btu/lbmol. J/(kmol·K) × 2.390E-04 = cal/(gmol·°C); J/(kmol·K) × 2.390059E-04 = Btu/(lbmol·°F). TABLE 2-222

Ideal Gas Sensible Enthalpies, hT - h298 (kJ/kgmol), of Combustion Products

Temperature, K

CO

CO2

H

OH

H2

N

NO

NO2

N2

N2O

O

O2

SO2

H2O

200 240 260 280 298.15

−2858 −1692 −1110 −529 0

−3414 −2079 −1383 −665 0

−2040 −1209 −793 −377 0

−2976 −1756 −1150 −546 0

−2774 −1656 −1091 −522 0

−2040 −1209 −793 −378 0

−2951 −1743 −1142 −543 0

−3495 −2104 −1392 −672 0

−2857 −1692 −1110 −528 0

−3553 −2164 −1438 −692 0

−2186 −1285 −840 −398 0

−2868 −1703 −1118 −533 0

−3736 −2258 −1496 −718 0

−3282 −1948 −1279 −609 0

300 320 340 360 380

54 638 1221 1805 2389

69 823 1594 2382 3184

38 454 870 1285 1701

55 654 1251 1847 2442

53 630 1209 1791 2373

38 454 870 1286 1701

55 652 1248 1845 2442

68 816 1571 2347 3130

54 636 1219 1802 2386

72 854 1654 2470 3302

41 478 913 1346 1777

54 643 1234 1828 2425

74 881 1702 2538 3387

62 735 1410 2088 2769

400 420 440 460 480

2975 3563 4153 4643 5335

4003 4835 5683 6544 7416

2117 2532 2948 3364 3779

3035 3627 4219 4810 5401

2959 3544 4131 4715 5298

2117 2533 2949 3364 3780

3040 3638 4240 4844 5450

3927 4735 5557 6392 7239

2971 3557 4143 4731 5320

4149 5010 5884 6771 7670

2207 2635 3063 3490 3918

3025 3629 4236 4847 5463

4250 5126 6015 6917 7831

3452 4139 4829 5523 6222

500 550 600 650 700

5931 7428 8942 10477 12023

8305 10572 12907 15303 17754

4196 5235 6274 7314 8353

5992 7385 8943 10423 11902

5882 6760 8811 10278 11749

4196 5235 6274 7314 8353

6059 7592 9144 10716 12307

8099 10340 12555 14882 17250

5911 7395 8894 10407 11937

8580 10897 13295 15744 18243

4343 5402 6462 7515 8570

6084 7653 9244 10859 12499

8758 11123 13544 16022 18548

6925 8699 10501 12321 14192

750 800 850 900 950

13592 15177 16781 18401 20031

20260 22806 25398 28030 30689

9392 10431 11471 12510 13550

13391 14880 16384 17888 19412

13223 14702 16186 17676 19175

9329 10431 11471 12510 13550

13919 15548 17195 18858 20537

19671 22136 24641 27179 29749

13481 15046 16624 18223 19834

20791 23383 26014 28681 31381

9620 10671 11718 12767 13812

14158 15835 17531 19241 20965

21117 23721 26369 29023 31714

16082 18002 19954 21938 23954

1000 1100 1200 1300 1400

21690 25035 28430 31868 35343

33397 38884 44473 50148 55896

14589 16667 18746 20824 22903

20935 24024 27160 30342 33569

20680 23719 26797 29918 33082

14589 16667 18746 20824 22903

22229 25653 29120 32626 36164

32344 37605 42946 48351 53808

21463 24760 28109 31503 34936

34110 39647 45274 50976 56740

14860 16950 19039 21126 23212

22703 26212 29761 33344 36957

34428 39914 45464 51069 56718

26000 30191 34506 38942 43493

1500 1600 1700 1800 1900

38850 42385 45945 49526 53126

61705 67569 73480 79431 85419

24982 27060 29139 31217 33296

36839 40151 43502 46889 50310

36290 39541 42835 46169 49541

24982 27060 29139 31218 33296

39729 43319 46929 50557 54201

59309 64846 70414 76007 81624

38405 41904 45429 48978 52548

62557 68420 74320 80254 86216

25296 27381 29464 31547 33630

40599 44266 47958 51673 55413

62404 68123 73870 79642 85436

48151 52908 57758 62693 67706

2000 2100 2200 2300 2400

56744 60376 64021 67683 71324

91439 97488 103562 109660 115779

35375 37453 39532 41610 43689

53762 57243 60752 64285 67841

52951 56397 59876 63387 66928

35375 37454 39534 41614 43695

57859 61530 65212 68904 72606

87259 92911 98577 104257 109947

56137 59742 63361 66995 70640

92203 98212 104240 110284 116344

35713 37796 39878 41962 44045

59175 62961 66769 70600 74453

91250 97081 102929 108792 114669

72790 77941 83153 88421 93741

2500 2600 2700 2800 2900

74985 78673 82369 86074 89786

121917 128073 134246 140433 146636

45768 47846 49925 52004 54082

71419 75017 78633 82267 85918

70498 74096 77720 81369 85043

45777 47860 49945 52033 54124

76316 80034 83759 87491 91229

115648 121357 127075 132799 138530

74296 77963 81639 85323 89015

122417 128501 134596 140701 146814

46130 48216 50303 52391 54481

78328 82224 86141 90079 94036

120559 126462 132376 138302 144238

99108 104520 109973 115464 120990

3000 3500 4000 4500 5000

93504 112185 130989 149895 168890

152852 184109 215622 247354 279283

56161 66554 75947 87340 97733

89584 108119 126939 145991 165246

88740 107555 126874 146660 166876

56218 66769 77532 88614 100111

94973 113768 132671 151662 170730

144267 173020 201859 230756 259692

92715 111306 130027 148850 167763

152935 183636 214453 245348 276299

56574 67079 77675 88386 99222

98013 118165 188705 159572 180749

150184 180057 210145 240427 270893

126549 154768 183552 212764 242313

Converted and usually rounded off from JANAF Thermochemical Tables, NSRDS-NBS-37, 1971 (1141 pp.)

2-200

PHYSICAL AND CHEMICAL DATA

TABLE 2-223 Temperature, K

Ideal Gas Entropies, s°, kJ/kgmol·K, of Combustion Products CO

CO2

H

OH

H2

N

NO

NO2

N2

N2O

O

O2

SO2

H2O

200 240 260 280 298.15

186.0 191.3 193.7 195.3 197.7

200.0 206.0 208.8 211.5 213.8

106.4 110.1 111.8 113.3 114.7

171.6 177.1 179.5 181.8 183.7

119.4 124.5 126.8 129.2 130.7

145.0 148.7 150.4 151.9 153.3

198.7 204.1 206.6 208.8 210.8

225.9 232.2 235.0 237.7 240.0

180.0 185.2 187.6 189.8 191.6

205.6 211.9 214.8 217.5 220.0

152.2 156.2 158.0 159.7 161.1

193.5 198.7 201.1 203.3 205.1

233.0 239.9 242.8 245.8 248.2

175.5 181.4 184.1 186.6 188.8

300 320 340 360 380

197.8 199.7 201.5 203.2 204.7

214.0 216.5 218.8 221.0 223.2

114.8 116.2 117.4 118.6 119.7

183.9 185.9 187.7 189.4 191.0

130.9 132.8 134.5 136.2 137.7

153.4 154.8 156.0 157.2 158.3

210.9 212.9 214.7 216.4 218.0

240.3 242.7 245.0 247.2 249.3

191.8 193.7 195.5 197.2 198.7

220.2 222.7 225.2 227.5 229.7

161.2 162.6 163.9 165.2 166.3

205.3 207.2 209.0 210.7 212.5

248.5 251.1 253.6 256.0 258.2

189.0 191.2 193.3 195.2 197.1

400 420 440 460 480

206.2 207.7 209.0 210.4 211.6

225.3 227.3 229.3 231.2 233.1

120.8 121.8 122.8 123.7 124.6

192.5 194.0 195.3 196.6 197.9

139.2 140.6 141.9 143.2 144.5

159.4 160.4 161.4 162.3 163.1

219.5 221.0 222.3 223.7 225.0

251.3 253.2 255.1 257.0 258.8

200.2 201.5 202.9 204.2 205.5

231.9 234.0 236.0 238.0 239.9

167.4 168.4 169.4 170.4 171.3

213.8 215.3 216.7 218.0 219.4

260.4 262.5 264.6 266.6 268.5

198.8 200.5 202.0 203.6 205.1

500 550 600 650 700

212.8 215.7 218.3 220.8 223.1

234.9 239.2 243.3 247.1 250.8

125.5 127.5 129.3 131.0 132.5

199.1 201.8 204.4 206.8 209.0

145.7 148.6 151.1 153.4 155.6

164.0 166.0 167.8 169.4 171.0

226.3 229.1 231.9 234.4 236.8

260.6 264.7 268.8 272.6 276.0

206.7 209.4 212.2 214.6 216.9

241.8 246.2 250.4 254.3 258.0

172.2 174.2 176.1 177.7 179.3

220.7 223.7 226.5 229.1 231.5

270.5 274.9 279.2 283.1 286.9

206.5 210.5 213.1 215.9 218.7

750 800 850 900 950

225.2 227.3 229.2 231.1 232.8

255.4 257.5 260.6 263.6 266.5

133.9 135.2 136.4 137.7 138.8

211.1 213.0 214.8 216.5 218.1

157.6 159.5 161.4 163.1 164.7

172.5 173.8 175.1 176.3 177.4

239.0 241.1 243.0 245.0 246.8

279.3 282.5 285.5 288.4 291.3

219.0 221.0 223.0 224.8 226.5

261.5 264.8 268.0 271.1 274.0

180.7 182.1 183.4 184.6 185.7

233.7 235.9 237.9 239.9 241.8

290.4 293.8 297.0 300.1 303.0

221.3 223.8 226.2 228.5 230.6

1000 1100 1200 1300 1400

234.5 237.7 240.7 243.4 246.0

269.3 274.5 279.4 283.9 288.2

139.9 141.9 143.7 145.3 146.9

219.7 222.7 225.4 228.0 230.3

166.2 169.1 171.8 174.3 176.6

178.5 180.4 182.2 183.9 185.4

248.4 251.8 254.8 257.6 260.2

293.9 298.9 303.6 307.9 311.9

228.2 231.3 234.2 236.9 239.5

276.8 282.1 287.0 291.5 295.8

186.8 188.8 190.6 192.3 193.8

243.6 246.9 250.0 252.9 255.6

305.8 311.0 315.8 320.3 324.5

232.7 236.7 240.5 244.0 247.4

1500 1600 1700 1800 1900

248.4 250.7 252.9 254.9 256.8

292.2 296.0 299.6 303.0 306.2

148.3 149.6 150.9 152.1 153.2

232.6 234.7 236.8 238.7 240.6

178.8 180.9 182.9 184.8 186.7

186.9 188.2 189.5 190.7 191.8

262.7 265.0 267.2 269.3 271.3

315.7 319.3 322.7 325.9 328.9

241.9 244.1 246.3 248.3 250.2

299.8 303.6 307.2 310.6 313.8

195.3 196.6 197.9 199.1 200.2

258.1 260.4 262.7 264.8 266.8

328.4 332.1 335.6 338.9 342.0

250.6 253.7 256.6 259.5 262.2

2000 2100 2200 2300 2400

258.7 260.5 262.2 263.8 265.4

309.3 312.2 315.1 317.8 320.4

154.3 155.3 156.3 157.2 158.1

242.3 244.0 245.7 247.2 248.7

188.4 190.1 191.7 193.3 194.8

192.9 193.9 194.8 195.8 196.7

273.1 274.9 276.6 278.3 279.8

331.8 334.5 337.2 339.7 342.1

252.1 253.8 255.5 257.1 258.7

316.9 319.8 322.6 325.3 327.9

201.3 202.3 203.2 204.2 205.0

268.7 270.6 272.4 274.1 275.7

345.0 347.9 350.6 353.2 355.7

264.8 267.3 269.7 272.0 274.3

2500 2600 2700 2800 2900

266.9 268.3 269.7 271.0 272.3

322.9 325.3 327.6 329.9 332.1

158.9 159.7 160.5 161.3 162.0

250.2 251.6 253.0 254.3 255.6

196.2 197.7 199.0 200.3 201.6

197.5 198.3 199.1 199.9 200.6

281.4 282.8 284.2 285.6 286.9

344.5 346.7 348.9 350.9 352.9

260.2 261.6 263.0 264.3 265.6

330.4 332.7 335.0 337.3 339.4

205.9 206.7 207.5 208.3 209.0

277.3 278.8 280.3 281.7 283.1

358.1 360.4 362.6 364.8 366.9

276.5 278.6 380.7 282.7 284.6

3000 3500 4000 4500 5000

273.6 279.4 284.4 288.8 292.8

334.2 343.8 352.2 359.7 366.4

162.7 165.9 168.7 171.1 173.3

256.8 262.5 267.6 272.1 276.1

202.9 208.7 213.8 218.5 222.8

201.3 204.6 207.4 210.1 212.5

288.2 294.0 299.0 303.5 307.5

354.9 363.8 371.5 378.3 384.4

266.9 272.6 277.6 282.1 286.0

341.5 350.9 359.2 366.5 373.0

209.7 212.9 215.8 218.3 220.6

284.4 290.7 296.2 301.1 305.5

368.9 378.1 386.1 393.3 399.7

286.5 295.2 302.9 309.8 316.0

Usually rounded off from JANAF Thermochemical Tables, NSRDS-NBS-37, 1971 (1141 pp.). Equilibrium constants can be calculated by combining ∆h°f values from Table 2-221, hT − h298 from Table 2-222, and s° values from the above, using the formula ln kp = −∆G/(RT), where ∆G = ∆h°f + (hT − h298) − T °. s

HEATS OF SOLUTION

2-201

HEATS OF SOLUTION TABLE 2-224

Heats of Solution of Inorganic Compounds in Water

Heat evolved, in kilogram-calories per gram formula weight, on solution in water at 18°C. Computed from data in Bichowsky and Rossini, Thermochemistry of Chemical Substances, Reinhold, New York, 1936.

Substance

Dilution*

Formula

Heat, kg-cal/ g-mole

Aluminum bromide chloride

aq 600 600 aq aq aq aq aq aq aq aq ∞ aq 600 aq ∞ aq ∞ 800 aq aq aq aq aq

AlBr3 AlCl3 AlCl3·6H2O AlF3 AlF3·aH2O AlF3·3aH2O AlI3 Al2(SO4)3 Al2(SO4)3·6H2O Al2(SO4)3·18H2O NH4Br NH4Cl (NH4)2CrO4 (NH4)2Cr2O7 NH4I NH4NO3 NH4BO3·H2O (NH4)2SO4 NH4HSO4 (NH4)2SO3 (NH4)2SO3·H2O SbF3 SbI3 H3AsO4

+85.3 +77.9 +13.2 +31 +19.0 −1.7 +89.0 +126 +56.2 +6.7 −4.45 −3.82 −5.82 −12.9 −3.56 −6.47 −9.0 −2.75 +0.56 −1.2 −4.13 −1.7 −0.8 −0.4

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ aq aq aq ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ aq aq aq aq aq aq aq aq aq

Ba(BrO3)2·H2O BaBr2 BaBr2·H2O BaBr2·2H2O Ba(ClO3)2 Ba(ClO3)2·H2O BaCl2 BaCl2·H2O BaCl2.2H2O Ba(CN)2 Ba(CN)2·H2O Ba(CN)2·2H2O Ba(IO3)2 Ba(IO3)2·H2O BaI2 BaI2·H2O BaI2·2H2O BaI2·2aH2O BaI2·7H2O Ba(NO3)2 Ba(ClO4)2 Ba(ClO4)2·3H2O BaS BeBr2 BeCl2 BeI2 BeSO4 BeSO4·H2O BeSO4·2H2O BeSO4·4H2O BiI3 H3BO3

−15.9 +5.3 −0.8 −3.87 −6.7 −10.6 +2.4 −2.17 −4.5 +1.5 −2.4 −4.9 −9.1 −11.3 +10.5 +2.7 +0.14 −0.58 −6.61 −10.2 −2.8 −10.5 +7.2 +62.6 +51.1 +72.6 +18.1 +13.5 +7.9 +1.1 +3 −5.4

400 400 400 400 400 400 400 400 400 400 ∞ ∞

CdBr2 CdBr2·4H2O CdCl2 CdCl2·H2O CdCl2·2aH2O Cd(NO3)2·H2O Cd(NO3)2·4H2O CdSO4 CdSO4·H2O CdSO4·2wH2O Ca(C2H3O2)2 Ca(C2H3O2)2·H2O

+0.4 −7.3 +3.1 +0.6 −3.00 +4.17 −5.08 +10.69 +6.05 +2.51 +7.6 +6.5

fluoride iodide sulfate Ammonium bromide chloride chromate dichromate iodide nitrate perborate sulfate sulfate, acid sulfite Antimony fluoride iodide Arsenic acid Barium bromate bromide chlorate chloride cyanide iodate iodide

nitrate perchlorate sulfide Beryllium bromide chloride iodide sulfate

Bismuth iodide Boric acid Cadmium bromide chloride nitrate sulfate Calcium acetate

Substance Calcium—(Cont.) bromide

Dilution*

Formula

Heat, kg-cal/ g-mole

∞ ∞ ∞ ∞ ∞ ∞ ∞ 400 ∞ ∞ ∞ ∞ ∞ ∞ ∞ aq aq ∞ ∞ ∞ aq

+24.86 −0.9 +4.9 +12.3 +12.5 +2.4 −4.11 +0.7 +28.0 +1.8 +4.1 +0.7 −3.2 −4.2 −7.99 −0.6 −1 +5.1 +3.6 −0.18 +18.6 +5.3 +2.0 +5.7 +18.4 −1.25 +18.5 +9.8 −2.9 +18.8 +15.0 −1.4 −3.6 +2.4 +0.5 +10.3 −2.6 −10.7 +15.9 +9.3 +3.65 −2.85 +11.6

Cuprous sulfate

aq

CaBr2 CaBr2·6H2O CaCl2 CaCl2·H2O CaCl2·2H2O CaCl2·4H2O CaCl2·6H2O Ca(CHO2)2 CaI2 CaI2·8H2O Ca(NO3)2 Ca(NO3)2·H2O Ca(NO3)2·2H2O Ca(NO3)2·3H2O Ca(NO3)2·4H2O Ca(H2PO4)2·H2O CaHPO4·2H2O CaSO4 CaSO4·aH2O CaSO4·2H2O CrCl2 CrCl2·3H2O CrCl2·4H2O CrI2 CoBr2 CoBr2·6H2O CoCl2 CoCl2·2H2O CoCl2·6H2O CoI2 CoSO4 CoSO4·6H2O CoSO4·7H2O Cu(C2H3O2)2 Cu(CHO2)2 Cu(NO3)2 Cu(NO3)2·3H2O Cu(NO3)2·6H2O CuSO4 CuSO4·H2O CuSO4·3H2O CuSO4·5H2O Cu2SO4

Ferric chloride

1000 1000 1000 800 aq 400 400 400 aq 400 400 400 400

FeCl3 FeCl3·2aH2O FeCl3·6H2O Fe(NO3)3·9H2O FeBr2 FeCl2 FeCl2·2H2O FeCl2·4H2O FeI2 FeSO4 FeSO4·H2O FeSO4·4H2O FeSO4·7H2O

+31.7 +21.0 +5.6 −9.1 +18.0 +17.9 +8.7 +2.7 +23.3 +14.7 +7.35 +1.4 −4.4

400 400 aq aq aq 400 ∞ ∞ ∞ ∞ ∞

Pb(C2H3O2)2 Pb(C2H3O2)2·3H2O PbBr2 PbCl2 Pb(CHO2)2 Pb(NO3)2 LiBr LiBr·H2O LiBr·2H2O LiBr·3H2O LiCl

+1.4 −5.9 −10.1 −3.4 −6.9 −7.61 +11.54 +5.30 +2.05 −1.59 +8.66

chloride

formate iodide nitrate

phosphate, monodibasic sulfate Chromous chloride iodide Cobaltous bromide chloride iodide sulfate Cupric acetate formate nitrate sulfate

nitrate Ferrous bromide chloride iodide sulfate

Lead acetate bromide chloride formate nitrate Lithium bromide

chloride

aq aq aq 400 400 400 aq 400 400 400 aq aq 200 200 200 800

*The numbers represent moles of water used to dissolve 1 g formula weight of substance; ∞ means “infinite dilution”; and aq means “aqueous solution of unspecified dilution.”

2-202

PHYSICAL AND CHEMICAL DATA

TABLE 2-224

Heats of Solution of Inorganic Compounds in Water (Continued )

Substance Lithium—(Cont.)

fluoride hydroxide iodide

nitrate sulfate Magnesium bromide chloride

iodide nitrate phosphate sulfate

sulfide Manganic nitrate sulfate Manganous acetate bromide chloride formate iodide

sulfate Mercuric acetate bromide chloride nitrate Mercurous nitrate Nickel bromide Nickel chloride

iodide nitrate sulfate

Dilution*

Formula

Heat, kg-cal/ g-mole

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

LiCl·H2O LiCl·2H2O LiCl·3H2O LiF LiOH LiOH·fH2O LiOH·H2O LiI LiI·aH2O LiI·H2O LiI·2H2O LiI·3H2O LiNO3 LiNO3·3H2O Li2SO4 Li2SO4·H2O

+4.45 +1.07 −1.98 −0.74 +4.74 +4.39 +9.6 +14.92 +10.08 +6.93 +3.43 −0.17 +0.466 −7.87 +6.71 +3.77

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ aq ∞ ∞ ∞ ∞ ∞ ∞ aq 400 400 400 aq aq aq aq aq aq 400 400 400 aq aq aq aq aq aq aq 400 400 400 aq aq aq aq aq

MgBr2 MgBr2·H2O MgBr2·6H2O MgCl2 MgCl2·2H2O MgCl2·4H2O MgCl2·6H2O MgI2 Mg(NO3)2·6H2O Mg3(PO4)2 MgSO4 MgSO4·H2O MgSO4·2H2O MgSO4·4H2O MgSO4·6H2O MgSO4·7H2O MgS Mn(NO3)2 Mn(NO3)2·3H2O Mn(NO3)2·6H2O Mn2(SO4)3 Mn(C2H3O2)2 Mn(C2H3O2)2·4H2O MnBr2 MnBr2·H2O MnBr2·4H2O MnCl2 MnCl2·2H2O MnCl2·4H2O Mn(CHO2)2 Mn(CHO2)2·2H2O MnI2 MnI2·H2O MnI2·2H2O MnI2·4H2O MnI2·6H2O MnSO4 MnSO4·H2O MnSO4·7H2O Hg(C2H3O2)2 HgBr2 HgCl2 Hg(NO3)2·aH2O Hg2(NO3)2·2H2O

+43.7 +35.9 +19.8 +36.3 +20.8 +10.5 +3.4 +50.2 −3.7 +10.2 +21.1 +14.0 +11.7 +4.9 +0.55 −3.18 +25.8 +12.9 −3.9 −6.2 +22 +12.2 +1.6 +15 +14.4 +16.1 +16.0 +8.2 +1.5 +4.3 −2.9 +26.2 +24.1 +22.7 +19.9 +21.2 +13.8 +11.9 −1.7 −4.0 −2.4 −3.3 −0.7 −11.5

aq aq 800 800 800 800 aq 200 200 200 200

NiBr2 NiBr2·3H2O NiCl2 NiCl2·2H2O NiCl2·4H2O NiCl2·6H2O NiI2 Ni(NO3)2 Ni(NO3)2·6H2O NiSO4 NiSO4·7H2O

+19.0 +0.2 +19.23 +10.4 +4.2 −1.15 +19.4 +11.8 −7.5 +15.1 −4.2

Substance

Dilution*

Phosphoric acid, orthopyroPotassium acetate aluminum sulfate

∞ 400 aq aq aq ∞ 800 ∞ aq aq ∞ aq ∞

+2.79 −0.1 +25.9 +4.65 +3.55 +48.5 +26.6 −10.1 −5.1 −10.13 −5.13 +6.58 +4.25 −0.43 −10.31 −4.404 −4.9 +55 +42 +33 +7 −9.5 −3.0 −17.8 +3.96 −1.85 −6.05 +0.86 +1.21 +12.91 +4.27 +3.48 +0.86 −6.93 −5.23 −8.633 −4.6 −7.5 −12.94 −10.4 +4.7 −11.0 −10.22 −6.32 −3.10 −11.0 +1.8 +1.37 −6.08 −13.0 −4.5

aq 200 ∞ ∞ 500 500 1800 900 900 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 800 800 800 200 200

AgC2H3O2 AgNO3 NaC2H3O2 NaC2H3O2·3H2O Na3AsO4 Na3AsO4·12H2O NaHCO3 Na2B4O7 Na2B4O7·10H2O NaBr NaBr·2H2O Na2CO3 Na2CO3·H2O Na2CO3·7H2O Na2CO3·10H2O NaClO3 NaCl Na2CrO4 Na2CrO4·4H2O Na2CrO4·10H2O NaCN NaCN·aH2O

−5.4 −4.4 +4.085 −4.665 +15.6 −12.61 −4.1 +10.0 −16.8 −0.58 −4.57 +5.57 +2.19 −10.81 −16.22 −5.37 −1.164 +2.50 −7.52 −16.0 −0.37 −0.92

2000 ∞ ∞ ∞

chlorate chloride chromate chrome sulfate

∞ ∞ 2185 600

cyanide dichromate fluoride

200 1600 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 400

hydroxide

iodate iodide nitrate oxalate perchlorate permanganate phosphate, dihydrogen pyrosulfite sulfate sulfate, acid sulfide sulfite thiocyanate thionate, dithiosulfate Silver acetate nitrate Sodium acetate arsenate bicarbonate borate, tetrabromide carbonate

chlorate chloride chromate cyanide

Heat, kg-cal/ g-mole

H3PO4 H3PO4·aH2O H4P2O7 H4P2O7·1aH2O KC2H3O2 KAl(SO4)2 KAl(SO4)2·3H2O KAl(SO4)2·12H2O KHCO3 KBrO3 KBr K2CO3 K2CO3·aH2O K2CO3·1aH2O KClO3 KCl K2CrO4 KCr(SO4)2 KCr(SO4)2·H2O KCr(SO4)2·2H2O KCr(SO4)2·6H2O KCr(SO4)2·12H2O KCN K2Cr2O7 KF KF·2H2O KF·4H2O KHS KHS·dH2O KOH KOH·eH2O KOH·H2O KOH·7H2O KIO3 KI KNO3 K2C2O4 K2C2O4·H2O KClO4 KMnO4 KH2PO4 K2S2O5 K2S2O5·aH2O K2SO4 KHSO4 K2S K2SO3 K2SO3·H2O KCNS K2S2O6 K2S2O3

400 400 aq aq ∞ 600 600

bicarbonate bromate bromide carbonate

hydrosulfide

Formula

HEATS OF SOLUTION TABLE 2-224

Heats of Solution of Inorganic Compounds in Water (Concluded )

Substance

Dilution*

Formula

200 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 600 ∞ aq ∞ 1600 1600 1600 1600 1600 1600 600 600 800 800 1600 1600 1200 1200 ∞ ∞ 800 800 ∞ ∞ ∞ ∞ ∞ ∞ ∞

NaCN·2H2O NaF NaHS NaHS·2H2O NaOH NaOH·aH2O NaOH·wH2O NaOH·eH2O NaOH·H2O NaI NaI·2H2O NaPO3 NaNO3 NaNO2 NaClO4 Na2HPO4 Na3PO4 Na3PO4·12H2O Na2HPO4·2H2O Na2HPO4·7H2O Na2HPO4·12H2O NaH2PO3 NaH2PO3·2aH2O Na2HPO3 Na2HPO3·5H2O Na4P2O7 Na4P2O7·10H2O Na2H2P2O7 Na2H2P2O7·6H2O Na2SO4 Na2SO4·10H2O NaHSO4 NaHSO4·H2O Na2S Na2S·4aH2O Na2S·5H2O Na2S·9H2O Na2SO3 Na2SO3·7H2O NaCNS

Sodium—(Cont.) fluoride hydrosulfide Sodium hydroxide

iodide metaphosphate nitrate nitrite perchlorate phosphate di triphosphate di diphosphite, monodipyrophosphate disulfate sulfate, acid sulfide

sulfite thiocyanate NOTE:

2-203

Heat, kg-cal/ g-mole −4.41 −0.27 +4.62 −1.49 +10.18 +8.17 +7.08 +6.48 +5.17 +1.57 −3.89 +3.97 −5.05 −3.6 −4.15 +5.21 +13 −15.3 −0.82 −12.04 −23.18 +0.90 −5.29 +9.30 −4.54 +11.9 −11.7 −2.2 −14.0 +0.28 −18.74 +1.74 +0.15 +15.2 +0.09 −6.54 −16.65 +2.8 −11.1 −1.83

Substance Sodium—(Cont.) thionate, diSodium thiosulfate Stannic bromide Stannous bromide iodide Strontium acetate bromide

chloride

iodide

nitrate sulfate Sulfuric acid, pyroZinc acetate bromide chloride iodide nitrate sulfate

To convert kilocalories per gram-mole to British thermal units per pound-mole, multiply by 1.799 × 10−3.

Dilution*

Formula

Heat, kg-cal/ g-mole

aq aq aq aq aq aq aq ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

Na2S2O6 Na2S2O6·2H2O Na2S2O3 Na2S2O3·5H2O SnBr4 SnBr2 SnI2 Sr(C2H3O2)2 Sr(C2H3O2)2·aH2O SrBr2 SrBr2·H2O SrBr2·2H2O SrBr2·4H2O SrBr2·6H2O SrCl2 SrCl2·H2O SrCl2·2H2O SrCl2·6H2O SrI2 SrI2·H2O SrI2·2H2O SrI2·6H2O Sr(NO3)2 Sr(NO3)2·4H2O SrSO4 H2S2O7

−5.80 −11.86 +2.0 −11.30 +15.5 −1.6 −5.8 +6.2 +5.9 +16.4 +9.25 +6.5 +0.4 −6.1 +11.54 +6.4 +2.95 −7.1 +20.7 +12.65 +10.4 −4.5 −4.8 −12.4 +0.5 −18.08

400 400 400 400 400 aq 400 400 400 400 400 400

Zn(C2H3O2)2 Zn(C2H3O2)2·H2O Zn(C2H3O2)2·2H2O ZnBr2 ZnCl2 ZnI2 Zn(NO3)2·3H2O Zn(NO3)2·6H2O ZnSO4 ZnSO4·H2O ZnSO4·6H2O ZnSO4·7H2O

+9.8 +7.0 +3.9 +15.0 +15.72 +11.6 −5 −6.0 +18.5 +10.0 −0.8 −4.3

2-204

PHYSICAL AND CHEMICAL DATA TABLE 2-225 Heats of Solution of Organic Compounds in Water (at Infinite Dilution and Approximately Room Temperature) Recalculated and rearranged from International Critical Tables, vol. 5, pp. 148–150. (g⋅cal)/(g⋅mol) = Btu/(lb⋅mol) × 1.799.

Solute Acetic acid (solid), C2H4O2 Acetylacetone, C5H8O2 Acetylurea, C3H6N2O2 Aconitic acid, C6H6O6 Ammonium benzoate, C7H9NO2 picrate succinate (n-) Aniline, hydrochloride, C6H8ClN Barium picrate Benzoic acid, C7H6O2 Camphoric acid, C10H16O4 Citric acid, C6H8O7 Dextrin, C12H20O10 Fumaric acid, C4H4O4 Hexamethylenetetramine, C6H12N4 Hydroxybenzamide (m-), C7H7NO2 (m-), (HCl) (o-), C7H7NO2 (p-) Hydroxybenzoic acid (o-), C7H6O3 (p-), C7H6O3 Hydroxybenzyl alcohol (o-), C7H8O2 Inulin, C36H62O31 Isosuccinic acid, C4H6O4 Itaconic acid, C5H6O4 Lactose, C12H22O11·H2O Lead picrate (2H2O) Magnesium picrate (8H2O) Maleic acid, C4H4O4 Malic acid, C4H6O5 Malonic acid, C3H4O4 Mandelic acid, C8H2O3 Mannitol, C6H14O6 Menthol, C10H20O Nicotine dihydrochloride, C10H16Cl2N2 Nitrobenzoic acid (m-), C7H5NO4 (o-), C7H5NO4 (p-), C7H5NO4 Nitrophenol (m-), C6H5NO3 (o-), C6H5NO3 (p-), C6H5NO3

Heat of Solution, G-cal/g-mole Solute* −2,251 −641 −6,812 −4,206 −2,700 −8,700 −3,489 −2,732 −4,708 −6,501 −502 −5,401 268 −5,903 4,780 −4,161 −7,003 −4,340 −5,392 −6,350 −5,781 −3,203 −96 −3,420 −5,922 −3,705 −7,098 −13,193 14,699 −15,894 −4,441 −3,150 −4,493 −3,090 −5,260 0 6,561 −5,593 −5,306 −8,891 −5,210 −6,310 −4,493

Solute Oxalic acid, C2H2O4 (2H2O) Phenol (solid), C6H6O Phthalic acid, C8H6O4 Picric acid, C6H3N3O7 Piperic acid, C12H10O4 Piperonylic acid, C8H6O4 Potassium benzoate citrate tartrate (n-) (0.5 H2O) Pyrogallol, C6H6O3 Pyrotartaric acid Quinone Raffinose, C18H32O16 (5H2O) Resorcinol, C6H6O2 Silver malonate (n-) Sodium citrate (tri-) picrate potassium tartrate (4H2O) succinate (n-) (6H2O) tartrate (n-) (2H2O) Strontium picrate (6H2O) Succinic acid, C4H6O4 Succinimide, C4H5NO2 Sucrose, C12H22O11 Tartaric acid (d-) Thiourea, CH4N2S Urea, CH4N2O acetate formate nitrate oxalate Vanillic acid Vanillin Zinc picrate (8H2O)

Heat of Solution, G-cal/g-mole Solute* −2,290 −8,485 −2,605 −4,871 −7,098 −10,492 −9,106 −1,506 2,820 −5,562 −3,705 −5,019 −3,991 −9,703 −3,960 −9,799 5,270 −6,441 −1,817 −12,342 2,390 −10,994 −1,121 −5,882 7,887 −14,412 −6,405 −4,302 −1,319 −3,451 −5,330 −3,609 −8,795 −7,194 −10,803 −17,806 −5,160 −5,210 −11,496 −15,894

* + denotes heat evolved, and − denotes heat absorbed. All values are positive unless otherwise noted. The data in the International Critical Tables were calculated by E. Anderson.

THERMODYNAMIC PROPERTIES

2-205

THERMODYNAMIC PROPERTIES EXPLANATION OF TABLES The following subsection presents information on the thermodynamic properties of a number of fluids. In some cases transport properties are also included. Notation cp = specific heat e = specific internal energy h = enthalpy k = thermal conductivity p = pressure s = specific entropy t = temperature T = absolute temperature u = specific internal energy µ = viscosity v = specific volume f = subscript denoting saturated liquid g = subscript denoting saturated vapor UNITS CONVERSIONS For this subsection, the following units conversions are applicable: cp, specific heat: To convert kilojoules per kilogram-kelvin to British thermal units per pound–degree Fahrenheit, multiply by 0.23885. e, internal energy: To convert kilojoules per kilogram to British thermal units per pound, multiply by 0.42992. g, gravity acceleration: To convert meters per second squared to feet per second squared, multiply by 3.2808. h, enthalpy: To convert kilojoules per kilogram to British thermal units per pound, multiply by 0.42992. k, thermal conductivity: To convert watts per meter-kelvin to British thermal unit–feet per hour–square foot–degree Fahrenheit, multiply by 0.57779. p, pressure: To convert bars to kilopascals, multiply by 1 × 102; to convert bars to pounds-force per square inch, multiply by 14.504; and to convert millimeters of mercury to pounds-force per square inch, multiply by 0.01934. s, entropy: to convert kilojoules per kilogram-kelvin to British thermal units per pound–degree Rankine, multiply by 0.23885.

t, temperature: °F = 9⁄5 °C + 32. T, absolute temperature: °R = 9⁄ 5 K. u, internal energy: to convert kilojoules per kilogram to British thermal units per pound, multiply by 0.42992. µ, viscosity: to convert pascal-seconds to pound-force–seconds per square foot, multiply by 0.020885; to convert pascal-seconds to cp, multiply by 1000. v, specific volume: to convert cubic meters per kilogram to cubic feet per pound, multiply by 16.018. ρ, density: to convert kilograms per cubic meter to pounds per cubic foot, multiply by 0.062428. ADDITIONAL REFERENCES Bretsznajder, Prediction of Transport and Other Physical Properties of Fluids, Pergamon, New York, 1971. D’Ans and Lax, Handbook for Chemists and Physicists (in German), 3 vols., Springer-Verlag, Berlin. Engineering Data Book, Natural Gas Processors Suppliers Association, Tulsa, Okla. Ganic, Hartnett, and Rohsenow, Handbook of Heat Transfer, 2d ed., McGraw-Hill, New York, 1984. Gray, American Institute of Physics Handbook, 3d ed., McGraw-Hill, New York, 1972. Kay and Laby, Tables of Physical and Chemical Constants, Longman, London, various editions and dates. Landolt-Börnstein Tables, many volumes and dates, Springer-Verlag, Berlin. Lange, Handbook of Chemistry, McGraw-Hill, New York, various editions and dates. Partington, Advanced Treatise on Physical Chemistry, 5 vols., Longman, London, 1950. Raznjevic, Handbook of Thermodynamic Tables and Charts, McGraw-Hill, New York, 1976 and other editions. Reynolds, Thermodynamic Properties in SI, Department of Mechanical Engineering, Stanford University, 1979. Stephan and Lucas, Viscosity of Dense Fluids, Plenum, New York and London, 1979. Selected Values of Properties of Chemical Compounds and Selected Values of the Properties of Hydrocarbons and Related Compounds, Thermodynamics Research Center, Texas A&M University, College Station, looseleaf, intermittent publication. Vargaftik, Tables of the Thermophysical Properties of Gases and Liquids, Wiley, New York, 1975. Vargaftik, Filippov, Tarzimanov, and Totskiy, Thermal Conductivity of Liquids and Gases (in Russian), Standartov, Moscow, 1978. Weast, Handbook of Chemistry and Physics, Chemical Rubber Co., Boca Raton, FL, annually.

2-206

PHYSICAL AND CHEMICAL DATA

TABLE 2-226

Thermophysical Properties of Saturated Acetone

Temperature, K

Pressure, bar

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/kg⋅K

sg, kJ/kg⋅K

c pf, kJ/kg⋅K

µ f, 10−6 Pa⋅s

k f, W/m⋅K

Pr

300 310 320 329.3 b 330

0.318 0.482 0.710 1.013 1.040

0.001 0.001 0.001 0.001 0.001

261 285 309 333 335

1.415 0.942 0.645 0.456 0.448

−67 −46 −22 0 2

466 476 490 506 506

−0.213 −0.144 −0.068 0 0.003

1.561 1.540 1.531 1.537 1.521

2.29 2.29

232 231

0.141 0.141

3.77 3.75

340 350 360 370 380

1.52 2.04 2.74 3.60 4.52

0.001 0.001 0.001 0.001 0.001

359 383 408 435 464

0.311 0.237 0.179 0.138 0.110

25 51 78 103 127

509 529 543 554 566

0.075 0.150

1.514 1.516

2.33 2.38 2.43 2.48 2.53

212 200 187 176 165

0.137 0.132 0.128 0.124 0.119

3.61 3.61 3.55 3.52 3.51

390 400 410 420 430

5.87 7.31 8.94 10.82 13.64

0.001 0.001 0.001 0.001 0.001

495 528 564 604 647

0.0854 0.0684 0.0556 0.0454 0.0356

151 184 207 231 256

577 588 598 608 618

2.59 2.65 2.73 2.82 2.92

153 141 130 119 109

0.115 0.111 0.107 0.103 0.099

3.45 3.37 3.32 3.26 3.21

440 450 460 470 480

16.37 19.42 22.79 27.52 32.52

0.001 0.001 0.001 0.001 0.001

695 748 81 88 98

0.0292 0.0240 0.0199 0.0159 0.0130

281 308 337 365 396

625 632 637 641 638

3.03 3.15 3.29 3.45 3.76

99 90 80 71 64

0.095 0.092 0.088 0.083 0.077

3.16 3.08 2.99 2.95 3.13

490 500 508.2c

37.73 43.08 47.61

0.002 15 0.002 46 0.003 67

vf, m3/kg

0.0091 0.0063 0.0037

b = normal boiling point; c = critical point P, v, h, and s interpolated and converted from Heat Exchanger Design Handbook, vol. 5, Hemisphere, Washington, DC, 1983 and reproduced in Beaton, C. F. and G. F. Hewitt, Physical Property Data for the Design Engineer, Hemisphere, New York, 1989 (394 pp.). Other values compiled by P. E. Liley An enthalpy-pressure diagram to 1000 psia, 250–500 °F appears in J. Chem. Eng. Data 7, 1 (1962): 75–78.

TABLE 2-227

Saturated Acetylene*

Temperature, K

Pressure, bar

162.0 169.3 173.9 180.0 184.3

vcond, m3/kg

vg, m3/kg

hcond, kJ/kg

hg, kJ/kg

scond, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

0.101 0.203 0.304 0.507 0.709

5.081 2.644 1.805 1.116 0.810

158 173 182 194 203

983 994 999 1007 1011

2.967 3.039 3.095 3.161 3.216

8.062 7.889 7.797 7.672 7.596

189.1 192.4t

1.013 1.283

0.5780 0.4617

214 221

1015 1018

3.272 3.312

7.511 7.455

192.4t 200.9 209.4

1.283 2.027 3.040

0.00164 0.00165 0.00169

0.4617 0.3011 0.2074

378 411 445

1018 1027 1035

4.127 4.296 4.461

7.455 7.362 7.280

221.5 230.4 240.7 253.2 263.0

5.066 7.093 10.13 15.20 20.27

0.00174 0.00179 0.00186 0.00195 0.00204

0.1264 0.0907 0.0635 0.0420 0.0309

493 528 565 602 628

1046 1052 1058 1061 1061

4.684 4.837 4.990 5.133 5.231

7.180 7.111 7.037 6.947 6.878

271.6 278.9 284.9 290.4 300.0

25.33 30.40 35.46 40.53 50.66

0.00213 0.00223 0.00232 0.00242 0.00270

0.0240 0.0193 0.0159 0.0133 0.0093

654 680 704 727 778

1060 1057 1051 1041 1017

5.326 5.414 5.494 5.576 5.737

6.822 6.767 6.716 6.658 6.534

307.8 308.7c

60.80 62.47

0.00335 0.00434

0.0061 0.0043

850 908

968 908

5.965 6.158

6.351 6.158

*Values recalculated into SI units from those of Din. Thermodynamic Functions of Gases, vol. 2, Butterworth, London, 1956. Above the solid line the condensed phase is solid; below the line it is liquid. t = triple point; c = critical point.

THERMODYNAMIC PROPERTIES TABLE 2-228 T, K

2-207

Saturated Air*

Pf, bar

Pg, bar

vf, m3/kg

vg, m3/kg

h f, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

5.55 3.73 2.57 1.82 1.313

−159.2 −155.2 −151.4 −147.8 −144.2

59.7 61.7 63.6 65.5 67.4

2.528 2.585 2.641 2.696 2.747

6.255 6.164 6.080 6.002 5.929

cpf, kJ/(kg⋅K)

µ f, 10−4 Pa⋅s

k f, W/(m⋅K)

3.25 2.98 2.75 2.54 2.36

0.180 0.176 0.173 0.169 0.166

60 62 64 66 68

0.123 0.174 0.239

0.071 0.104 0.147

1.040.–3 1.050.–3 1.060.–3 1.070.–3 1.080.–3

70 72 74 76 78

0.323 0.429 0.560 0.721 0.915

0.205 0.280 0.376 0.495 0.644

1.089.–3 1.101.–3 1.113.–3 1.125.–3 1.136.–3

0.968 0.728 0.556 0.431 0.339

−140.6 −137.1 −133.5 −129.9 −126.3

69.2 71.0 72.8 74.5 76.2

2.797 2.847 2.895 2.941 2.988

5.862 5.799 5.740 5.685 5.634

1.817 1.827 1.838 1.849 1.861

2.21 2.07 1.95 1.84 1.74

0.163 0.160 0.156 0.152 0.148

80 82 84 86 88

1.146 1.420 1.741 2.114 2.544

0.825 1.043 1.305 1.614 1.976

1.146.–3 1.160.–3 1.173.–3 1.187.–3 1.201.–3

0.270 0.217 0.177 0.145 0.120

−122.6 −118.8 −115.0 −111.2 −107.4

77.8 79.4 80.9 82.3 83.6

3.034 3.079 3.123 3.167 3.209

5.585 5.540 5.496 5.454 5.414

1.873 1.885 1.898 1.912 1.927

1.65 1.58 1.51 1.44 1.38

0.145 0.142 0.139 0.135 0.132

90 92 94 96 98

3.036 3.596 4.229 4.940 5.736

2.397 2.884 3.441 4.075 4.792

1.216.–3 1.231.–3 1.247.–3 1.265.–3 1.283.–3

0.1002 0.0843 0.0713 0.0607 0.0520

−103.5 −99.5 −95.5 −91.5 −87.5

84.8 85.9 87.0 87.9 88.7

3.251 3.293 3.335 3.376 3.416

5.376 5.340 5.304 5.270 5.236

1.944 1.962 1.982 2.003 2.027

1.32 1.27 1.23 1.18 1.14

0.128 0.125 0.121 0.117 0.114

100 105 110 115 120

6.621 9.265 12.59 16.68 21.61

5.599 8.056 11.22 15.21 20.14

1.302.–3 1.355.–3 1.418.–3 1.495.–3 1.596.–3

0.0447 0.0312 0.0222 0.0159 0.0115

−83.3 −72.8 −61.9 −50.3 −37.5

89.3 90.2 90.1 88.4 84.8

3.456 3.553 3.649 3.747 3.850

5.204 5.124 5.045 4.964 4.877

2.053 2.137 2.264 2.477 2.916

1.10 1.02 0.95 0.87 0.75

0.110 0.102 0.093 0.084 0.076

125 130 132.55c

27.43 34.16

26.14 33.32 37.69

1.757.–3 2.075.–3 3.196.–3

0.0081 0.0054 0.0032

−22.0 0.4 37.4

78.2 66.1 37.4

3.969 4.136 4.410

4.776 4.644 4.410

4.585

0.42

∞

0.067 ∞

*Liquid properties extracted or converted from Vasserman and Rabinovich, Thermophysical Properties of Liquid Air and Its Components, Moscow, 1968, and NBSNSF transl. TT 69-55092, 1970. Copyrighted material. Reproduced by permission. Vapor properties extracted or converted from Vasserman, Kazavchinskii, and Rabinovich, Thermophysical Properties of Air and Its Components, Nauka, Moscow, 1966, and NBS-NSF transl. TT 70-50095, 1971. Copyrighted material. Reproduced by permission. Note that on pages 150–151 of the TT 69-55092 publication certain values of TT 70-50095 were adjusted. As a complete retabulation was not given, the tables here are based upon the two separate publications, as indicated. See also Table 2-235 for the argon-oxygen-nitrogen equilibrium data. c = critical point. The notation 1.040.–3 signifies 1.040 × 10−3.

2-208

PHYSICAL AND CHEMICAL DATA

TABLE 2-229 Pressure, bar 1v h s Cp µ k

Thermophysical Properties of Compressed Air* Temperature, K 80

Mix

90

100

120

140

160

180

200

220

240

260

280

300

0.251 87.9 5.650 1.044 0.064 0.0084

0.281 98.3 5.759 1.032 0.071 0.0093

0.340 118.8 5.946 1.020 0.085 0.0112

0.399 139.1 6.103 1.014 0.097 0.0129

0.457 159.3 6.238 1.010 0.109 0.0147

0.515 179.5 6.357 1.008 0.121 0.0164

0.537 199.7 6.463 1.007 0.133 0.0181

0.631 219.8 6.559 1.006 0.144 0.0198

0.688 239.9 6.647 1.006 0.154 0.0214

0.746 260.0 6.727 1.006 0.165 0.0231

0.803 280.2 6.802 1.006 0.175 0.0247

0.861 300.3 6.871 1.007 0.185 0.0263

5v h s Cp µ k

0.00115 −122.3 3.031 1.868 1.794 0.146

0.00122 −103.3 3.250 1.941 1.163 0.128

0.0509 90.6 5.246 1.212 0.077 0.0103

0.0646 113.6 5.455 1.107 0.087 0.0119

0.0773 135.3 5.623 1.065 0.098 0.0135

0.0895 156.4 5.763 1.045 0.110 0.0151

0.102 177.1 5.885 1.033 0.122 0.0168

0.114 197.7 5.994 1.025 0.134 0.0185

0.125 218.1 6.092 1.020 0.145 0.0201

0.137 238.5 6.180 1.017 0.155 0.0217

0.149 258.8 6.262 1.015 0.165 0.0234

0.160 279.1 6.337 1.013 0.175 0.0250

0.172 299.4 6.406 1.013 0.185 0.0265

10 v h s Cp µ k

0.00115 −122.0 3.028 1.863 1.816 0.146

0.00121 −103.1 3.246 1.932 1.177 0.128

0.00130 −83.2 3.452 2.041 0.838 0.111

0.0298 106.2 5.214 1.270 0.089 0.0126

0.0370 130.2 5.398 1.146 0.101 0.0141

0.0436 152.5 5.548 1.093 0.112 0.0157

0.0499 174.1 5.675 1.065 0.124 0.0173

0.0561 195.2 5.786 1.049 0.135 0.0189

0.0621 216.1 5.885 1.038 0.146 0.0205

0.0681 236.7 5.975 1.031 0.156 0.0221

0.0741 257.3 6.058 1.026 0.166 0.0237

0.0800 277.8 6.134 1.023 0.176 0.0253

0.0859 298.3 6.204 1.201 0.186 0.0268

20 v h s Cp µ k

0.00114 −121.3 3.022 1.853 1.859 0.147

0.00121 −102.5 3.239 1.916 1.205 0.130

0.00129 −82.9 3.442 2.010 0.857 0.112

0.0116 85.2 4.882 2.237 0.098 0.0152

0.0167 118.5 5.140 1.390 0.106 0.0157

0.0206 144.3 5.312 1.215 0.116 0.0169

0.0241 167.7 5.450 1.141 0.127 0.0182

0.0274 190.1 5.568 1.101 0.137 0.0197

0.0306 211.9 5.672 1.076 0.148 0.0212

0.0337 233.2 5.765 1.061 0.158 0.0228

0.0368 254.3 5.849 1.050 0.168 0.0243

0.0398 275.2 5.927 1.042 0.178 0.0258

0.0428 296.0 5.998 1.037 0.187 0.0273

40 v h s Cp µ k

0.00114 −120.0 3.011 1.834 1.943 0.149

0.00120 −101.4 3.225 1.886 1.261 0.132

0.00128 −82.2 3.424 1.958 0.896 0.115

0.00153 −39.8 3.807 2.432 0.516 0.0814

0.0058 83.6 4.745 3.193 0.132 0.0460

0.0090 125.3 5.025 1.610 0.129 0.0201

0.0114 154.3 5.196 1.335 0.135 0.0206

0.0131 179.7 5.330 1.221 0.144 0.0217

0.0148 203.5 5.444 1.159 0.154 0.0229

0.0165 226.3 5.543 1.122 0.163 0.0242

0.0182 248.5 5.632 1.097 0.172 0.0256

0.0198 270.2 5.712 1.081 0.182 0.0270

0.0214 291.7 5.786 1.068 0.191 0.0284

60 v h s Cp µ k

0.00113 −118.6 3.000 1.818 2.028 0.150

0.00119 −100.3 3.211 1.860 1.318 0.134

0.00126 −81.4 3.407 1.915 0.936 0.117

0.00147 −40.8 3.773 2.205 0.559 0.0861

0.00222 22.8 4.260 4.808 0.277 0.0480

0.00505 90.0 4.798 2.338 0.153 0.0360

0.00687 132.6 5.020 1.594 0.149 0.0238

0.00833 163.9 5.174 1.361 0.154 0.0240

0.00963 191.1 5.298 1.249 0.161 0.0248

0.0108 216.1 5.404 1.186 0.169 0.0258

0.0120 240.0 5.497 1.146 0.178 0.0270

0.0131 263.1 5.581 1.119 0.186 0.0283

0.0142 285.6 5.657 1.100 0.195 0.0296

80 v h s Cp µ k

0.00113 −117.2 2.989 1.802 2.12 0.152

0.00119 −99.1 3.198 1.838 1.38 0.134

0.00126 −80.4 3.391 1.881 0.977 0.120

0.00145 −41.3 3.745 2.078 0.597 0.0901

0.00188 9.0 4.138 2.992 0.356 0.0599

0.00327 78.4 4.597 3.029 0.194 0.0420

0.00480 125.3 4.875 1.887 0.167 0.0278

0.00601 158.7 5.051 1.510 0.166 0.0268

0.00706 187.1 5.186 1.342 0.170 0.0269

0.00803 212.9 5.299 1.250 0.177 0.0276

0.00894 237.3 5.396 1.194 0.184 0.0286

0.00981 260.8 5.484 1.156 0.191 0.0296

0.0107 283.7 5.562 1.130 0.200 0.0308

100 v h s Cp µ k

0.00112 −115.8 2.978 1.789 2.21 0.154

0.00118 −97.8 3.186 1.818 1.44 0.137

0.00125 −79.4 3.376 1.852 1.02 0.122

0.00142 −41.3 3.721 1.992 0.631 0.0936

0.00174 3.9 4.076 2.506 0.405 0.0669

0.00252 61.7 4.457 2.874 0.249 0.0500

0.00366 111.8 4.753 2.114 0.193 0.0327

0.00467 148.8 4.949 1.650 0.181 0.0299

0.00556 179.4 5.095 1.431 0.181 0.0293

0.00637 206.7 5.214 1.311 0.185 0.0295

0.00713 232.2 5.315 1.239 0.191 0.0302

0.00785 256.4 5.406 1.191 0.198 0.0311

0.00855 279.9 5.486 1.158 0.205 0.0320

150 v h s Cp µ k

0.00111 −112.2 2.954 1.789 2.44 0.157

0.00116 −94.5 3.157 1.818 1.60 1.142

0.00122 −76.6 3.342 1.852 1.13 0.127

0.00137 −40.1 3.673 1.992 0.709 0.101

0.00158 0.5 3.988 2.506 0.490 0.0785

0.00194 45.2 4.287 2.874 0.349 0.0588

0.00247 89.5 4.548 2.114 0.266 0.0455

0.00309 129.2 4.757 1.650 0.229 0.0389

0.00369 163.2 4.919 1.431 0.215 0.0360

0.00425 193.4 5.051 1.311 0.211 0.0348

0.00478 221.0 5.161 1.239 0.212 0.0346

0.00529 247.0 5.257 1.267 0.215 0.0349

0.00578 271.8 5.343 1.220 0.220 0.0354

200 v h s Cp µ k

0.00110 −108.5 2.930 1.733 2.70 0.161

0.00115 −91.2 3.130 1.747 1.78 0.146

0.00120 −73.6 3.312 1.761 1.25 0.132

0.00133 −38.0 3.634 1.809 0.782 0.107

0.00150 0.2 3.931 1.905 0.561 0.0868

0.00174 40.2 4.198 1.988 0.420 0.0691

0.00206 79.8 4.432 1.953 0.331 0.0559

0.00245 117.6 4.631 1.814 0.279 0.0476

0.00287 152.2 4.796 1.643 0.253 0.0429

0.00328 183.6 4.932 1.501 0.241 0.0405

0.00368 212.5 5.048 1.396 0.236 0.0393

0.00407 239.6 5.149 1.321 0.235 0.0389

0.00446 265.5 5.238 1.266 0.237 0.0389

250 v h s Cp µ k

0.00109 −104.8 2.909 1.712 2.96 0.165

0.00114 −87.6 3.106 1.722 1.97 0.150

0.00119 −70.3 3.285 1.733 1.39 0.137

0.00130 −35.4 3.601 1.767 0.855 0.113

0.00144 1.3 3.886 1.824 0.625 0.0935

0.00162 38.9 4.138 1.854 0.476 0.0769

0.00186 75.8 4.355 1.831 0.385 0.0641

0.00214 111.7 4.544 1.748 0.327 0.0552

0.00244 145.6 4.706 1.635 0.292 0.0495

0.00276 177.1 4.843 1.522 0.272 0.0460

0.00307 206.6 4.961 1.427 0.262 0.0441

0.00338 234.3 5.064 1.353 0.257 0.0430

0.00368 260.8 5.155 1.297 0.256 0.0426

*For sources, units, and remarks, see Table 2-228. v = specific volume, m3/kg; h = specific enthalpy, kJ/kg; s = specific entropy, kJ/(kg⋅K); cp = specific heat at constant pressure, kJ/(kg⋅K); µ = viscosity, 10−4 Pa⋅s; and k = thermal conductivity, W/(m⋅K). For specific heat ratio, see Table 2-200; for Prandtl number, see Table 2-369.

THERMODYNAMIC PROPERTIES

2-209

Temperature, K 350

400

450

500

600

800

1000

1200

1400

1600

1800

2000

2500

1.005 350.7 7.026 1.009 0.208 0.0301

1.148 401.2 7.161 1.014 0.230 0.0336

1.292 452.1 7.282 1.021 0.251 0.0371

1.436 503.4 7.389 1.030 0.270 0.0404

1.723 607.5 7.579 1.051 0.306 0.0466

2.297 822.5 7.888 1.099 0.370 0.0577

2.872 1046.8 8.138 1.141 0.424 0.0681

3.446 1278 8.349 1.175 0.473 0.0783

4.020 1515 8.531 1.207 0.527 0.0927

4.594 1764 8.695 1.248 0.584 0.106

5.168 2017 8.844 1.286 0.637 0.120

5.743 2279 8.983 1.337 0.689 0.137

7.200 3011 9.308 1.665 0.818 0.222

0.201 350.0 6.563 1.014 0.208 0.0303

0.230 400.8 6.698 1.017 0.230 0.0338

0.259 451.8 6.818 1.024 0.251 0.0372

0.288 503.2 6.927 1.032 0.270 0.0405

0.345 607.4 7.116 1.053 0.306 0.0467

0.460 822.6 7.426 1.100 0.370 0.0578

0.575 1046.9 7.676 1.142 0.425 0.0681

0.690 1279 7.887 1.175 0.473 0.0783

0.805 1516 8.069 1.208 0.527 0.0927

0.920 1764 8.233 1.248 0.584 0.106

1.034 2017 8.382 1.285 0.637 0.120

1.149 2278 8.520 1.326 0.689 0.136

1.438 2981 8.832 1.516 0.818 0.195

0.101 349.2 6.361 1.019 0.209 0.0305

0.115 400.2 6.497 1.021 0.231 0.0340

0.130 451.4 6.618 1.027 0.252 0.0374

0.144 502.9 6.727 1.034 0.271 0.0407

0.173 607.3 6.917 1.055 0.306 0.0469

0.231 822.7 7.226 1.100 0.370 0.0579

0.288 1047.2 7.477 1.142 0.425 0.0682

0.345 1279 7.688 1.175 0.473 0.0784

0.403 1516 7.870 1.208 0.527 0.0927

0.460 1765 8.034 1.248 0.584 0.106

0.518 2018 8.183 1.284 0.637 0.120

0.575 2279 8.321 1.324 0.689 0.135

0.720 2974 8.630 1.481 0.817 0.187

0.0503 347.7 6.158 1.030 0.210 0.0309

0.0577 399.1 6.295 1.029 0.232 0.0344

0.0650 450.7 6.417 1.033 0.253 0.0377

0.0723 502.4 6.526 1.039 0.272 0.0410

0.0868 607.2 6.716 1.057 0.307 0.0471

0.116 823.0 7.027 1.102 0.371 0.0581

0.145 1047.7 0.277 1.143 0.425 0.0685

0.173 1280 7.489 1.176 0.474 0.0787

0.202 1517 7.671 1.209 0.527 0.0928

0.231 1766 7.835 1.249 0.584 0.106

0.260 2019 7.984 1.284 0.637 0.120

0.288 2279 8.121 1.322 0.689 0.135

0.360 2970 8.428 1.456 0.817 0.181

0.0252 344.6 5.950 1.051 0.213 0.0318

0.0290 397.0 6.090 1.044 0.235 0.0351

0.0327 449.2 6.212 1.044 0.255 0.0384

0.0364 501.5 6.323 1.049 0.274 0.0416

0.0438 606.9 6.515 1.063 0.309 0.0476

0.0583 823.7 6.826 0.105 0.372 0.0584

0.0728 1048.8 7.077 1.145 0.426 0.0687

0.0872 1281 7.289 1.177 0.474 0.0789

0.102 1519 7.473 1.210 0.527 0.0928

0.116 1768 7.636 1.249 0.584 0.106

0.130 2021 7.785 1.284 0.637 0.120

0.145 2281 7.922 1.322 0.689 0.135

0.181 2969 8.229 1.438 0.817 0.177

0.0169 340.4 5.824 1.072 0.217 0.0328

0.0194 394.0 5.967 1.059 0.237 0.0359

0.0220 447.1 6.091 1.055 0.257 0.0391

0.0245 500.6 6.202 1.057 0.275 0.0422

0.0294 606.8 6.396 1.069 0.310 0.0481

0.0392 824.3 6.708 1.108 0.373 0.0588

0.0489 1050.0 6.960 1.147 0.427 0.0690

0.0585 1283 7.172 1.178 0.475 0.0790

0.0681 1521 7.355 1.210 0.527 0.0929

0.0776 1770 7.520 1.249 0.584 0.106

0.0872 2023 7.669 1.286 0.637 0.120

0.0968 2284 7.806 1.322 0.689 0.134

0.1207 2969 8.112 1.430 0.817 0.176

0.0127 339.0 5.733 1.091 0.220 0.0337

0.0147 393.1 5.878 1.073 0.240 0.0368

0.0166 446.5 6.004 1.066 0.259 0.0398

0.0185 499.8 6.116 1.065 0.278 0.0428

0.0223 606.7 6.311 1.075 0.312 0.0486

0.0296 825.1 6.624 1.111 0.374 0.0592

0.0369 1051.1 6.877 1.149 0.428 0.0693

0.0442 1284 7.089 1.180 0.475 0.0793

0.0513 1522 7.273 1.210 0.527 0.0929

0.0585 1772 7.437 1.249 0.584 0.106

0.0657 2025 7.586 1.286 0.637 0.120

0.0729 2285 7.723 1.322 0.689 0.134

0.0908 2971 8.029 1.426 0.817 0.175

0.0102 336.5 5.661 1.110 0.224 0.0347

0.0118 391.3 5.807 1.087 0.243 0.0376

0.0134 445.3 5.935 1.076 0.262 0.0405

0.0149 499.0 6.048 1.073 0.280 0.0434

0.0180 606.6 6.244 1.080 0.314 0.0491

0.0239 825.8 6.559 1.114 0.375 0.0595

0.0298 1052.4 6.812 1.151 0.429 0.0696

0.0356 1286 7.024 1.181 0.477 0.0795

0.0413 1524 7.208 1.211 0.527 0.0930

0.0470 1774 7.373 1.250 0.584 0.106

0.0528 2027 7.522 1.288 0.637 0.120

0.0584 2288 7.659 1.323 0.689 0.134

0.0729 2972 7.964 1.423 0.817 0.175

0.00695 330.9 5.525 1.151 0.235 0.0374

0.00806 387.5 5.677 1.117 0.252 0.0398

0.00914 442.9 5.807 1.099 0.270 0.0424

0.0102 497.5 5.922 1.092 0.286 0.0451

0.0123 606.6 6.121 1.093 0.318 0.0504

0.0163 827.8 6.439 1.121 0.379 0.0605

0.0202 1055.5 6.693 1.155 0.431 0.0703

0.0241 1290 6.906 1.184 0.478 0.0801

0.0279 1529 7.092 1.213 0.527 0.0932

0.0317 1779 7.256 1.252 0.584 0.106

0.0356 2033 7.405 1.290 0.637 0.120

0.0394 2294 7.543 1.325 0.689 0.133

0.0490 2977 7.848 1.418

0.00534 326.5 5.426 1.184 0.248 0.0400

0.00620 384.5 5.581 1.141 0.262 0.0420

0.00702 440.9 5.715 1.119 0.278 0.0423

0.00783 496.6 5.831 1.108 0.293 0.0467

0.00940 607.0 6.033 1.104 0.324 0.0517

0.0125 829.9 6.353 1.128 0.382 0.0614

0.0154 1058.7 6.608 1.160 0.434 0.0711

0.0184 1294 6.822 1.187 0.481 0.0808

0.0212 1533 7.009 1.214 0.528 0.0934

0.0241 1783 7.173 1.254 0.585 0.106

0.0269 2038 7.323 1.292 0.638 0.120

0.0298 2299 7.460 1.326

0.0370 2982 7.765 1.415

0.00440 323.2 5.348 1.208 0.262 0.0429

0.00509 382.3 5.506 1.161 0.273 0.0443

0.00576 439.6 5.641 1.135 0.286 0.0462

0.00642 496.0 5.760 1.121 0.301 0.0484

0.00770 607.6 5.963 1.115 0.329 0.0531

0.0102 832.2 6.286 1.135 0.386 0.0624

0.0126 1062.0 6.542 1.164 0.437 0.0718

0.0149 1298 6.757 1.190 0.483 0.0814

0.0172 1538 6.944 1.216 0.528 0.0937

0.0195 1789 7.108 1.256 0.585 0.106

0.0218 2043 7.258 1.294

0.0241 2304 7.396 1.328

0.0298 2988 7.701 1.414

2-210

PHYSICAL AND CHEMICAL DATA

TABLE 2-229 Pressure, bar

Thermophysical Properties of Compressed Air (Concluded ) Temperature, K 90

100

120

140

160

180

200

220

240

260

280

300

0.00112 −84.0 3.083 1.703 2.18 0.154

0.00117 −67.0 3.260 1.713 1.53 0.141

0.00127 −32.4 3.572 1.740 0.932 0.118

0.00139 3.1 3.849 1.769 0.687 0.0996

0.00155 39.2 4.090 1.777 0.529 0.0836

0.00173 74.5 4.298 1.751 0.433 0.0710

0.00195 109.0 4.480 1.689 0.370 0.0619

0.00219 142.0 4.637 1.607 0.329 0.0555

0.00243 173.2 4.773 1.518 0.303 0.0514

0.00269 202.7 4.891 1.438 0.288 0.0487

0.00294 230.8 4.995 1.370 0.280 0.0471

0.00318 257.7 5.088 1.316 0.276 0.0462

400 v h s Cp µ k

0.00110 −76.6 3.042 1.674 2.63 0.161

0.00114 −59.8 3.216 1.686 1.86 0.149

0.00123 −25.9 3.523 1.704 1.10 0.127

0.00133 8.3 3.788 1.702 0.802 0.110

0.00145 42.4 4.016 1.685 0.631 0.0946

0.00158 75.8 4.214 1.654 0.500 0.0823

0.00173 108.5 4.386 1.607 0.446 0.0729

0.00189 140.1 4.537 1.550 0.397 0.0660

0.00206 170.5 4.669 1.490 0.364 0.0610

0.00224 199.7 4.786 1.431 0.341 0.0574

0.00242 227.8 4.890 1.378 0.325 0.0550

0.00260 254.8 4.983 1.331 0.316 0.0533

500 v h s Cp µ k

0.00109 −69.0 3.005 1.655 3.13 0.167

0.00112 −52.3 3.177 1.670 2.24 0.156

0.00120 −18.7 3.482 1.686 1.31 0.135

0.00128 14.4 3.743 1.667 0.924 0.119

0.00138 47.4 3.966 1.644 0.710 0.104

0.00148 79.8 4.151 1.598 0.0560 0.0916

0.00160 111.4 4.317 1.557 0.512 0.0822

0.00173 142.0 4.463 1.509 0.459 0.0749

0.00186 171.7 4.593 1.461 0.420 0.0694

0.00199 200.5 4.708 1.415 0.391 0.0653

0.00213 228.4 4.811 1.371 0.370 0.0622

0.00227 255.4 4.905 1.331 0.356 0.0599

0.00151 116.0 2.263 1.525

0.00161 146.1 4.406 1.480 0.516 0.0828

0.00172 175.3 4.533 1.438 0.472 0.0769

0.00183 203.6 4.646 1.398 0.439 0.0724

0.00194 231.2 4.749 1.361 0.414 0.0689

0.00205 258.1 4.842 1.327 0.396 0.0662

0.00147 157.4 4.318 1.445

0.00155 185.9 4.442 1.406

0.0964

0.0901

0.00163 213.7 4.553 1.372 0.529 0.0850

0.00171 240.3 4.653 1.342 0.497 0.0809

0.00179 267.3 4.745 1.314 0.473 0.0776

0.00151 226.4 4.482 1.355

0.00157 253.2 4.582 1.327

0.0961

0.0916

0.00163 279.5 4.672 1.303 0.546 0.0878

300 v h s Cp µ k

600 v h s Cp µ k 800 v h s Cp µ k 1000 v h s Cp µ k

80 0.00108 −101.0 2.888 1.694 3.24 0.168

0.0903

THERMODYNAMIC PROPERTIES

2-211

Temperature, K 350

400

450

500

600

800

1000

1200

1400

1600

1800

2000

2500

0.00379 320.9 5.283 1.226 0.276 0.0457

0.00437 380.9 5.443 1.176 0.284 0.0466

0.00493 438.9 5.580 1.148 0.296 0.0481

0.00548 495.9 5.700 1.133 0.308 0.0501

0.00656 608.5 5.906 1.124 0.335 0.0544

0.00864 834.5 6.230 1.140 0.390 0.0634

0.0107 1065.3 6.488 1.168 0.440 0.0726

0.0126 1302 6.703 1.193 0.485 0.0820

0.0145 1542 6.891 1.217 0.529 0.0940

0.0164 1794 7.056 1.257

0.0183 2049 7.206 1.298

0.0202 2310 7.344 1.330

0.0250 2993 7.648 1.413

0.00304 319.1 5.181 1.246 0.307 0.0513

0.00348 380.0 5.344 1.195 0.308 0.0512

0.00390 439.0 5.483 1.166 0.315 0.0521

0.00432 496.8 5.605 1.149 0.325 0.0535

0.00514 611.0 5.813 1.138 0.348 0.0571

0.00673 839.4 6.142 1.151 0.398 0.0653

0.00826 1072.0 6.401 1.176 0.446 0.0740

0.00977 1310 6.618 1.199 0.490 0.0832

0.0111 1552 6.808 1.222

0.0126 1804 6.972 1.258

0.0140 2059 7.123 1.301

0.0155 2321 7.261 1.333

0.0190 3004 7.566 1.412

0.00262 319.9 5.103 1.255 0.338 0.0568

0.00296 381.3 5.267 1.206 0.333 0.0557

0.00330 440.8 5.408 1.176 0.336 0.0560

0.00364 499.1 5.531 1.159 0.343 0.0569

0.00430 614.3 5.741 1.148 0.361 0.0598

0.00558 844.6 6.072 1.159 0.407 0.0672

0.00683 1078.8 6.333 1.183 0.452 0.0755

0.00804 1318 6.550 1.205 0.495 0.0844

0.00911 1561 6.743 1.226

0.0103 1814 6.907 1.265

0.0114 2070 7.058 1.306

0.0126 2332 7.196 1.337

0.0154 3015 7.501 1.412

0.00234 322.6 5.041 1.258 0.370 0.0620

0.00262 384.2 5.205 1.211 0.359 0.0602

0.00290 444.0 5.346 1.182 0.358 0.0598

0.00318 502.6 5.470 1.166 0.361 0.0603

0.00374 618.5 5.681 1.154 0.375 0.0625

0.00481 850.1 6.014 1.166 0.416 0.0691

0.00586 1085.5 6.277 1.189 0.459 0.0770

0.00689 1326 6.495 1.210 0.501 0.0857

0.00776 1570 6.690 1.231

0.00873 1824 6.854 1.267

0.00970 2080 7.005 1.310

0.0107 2343 7.144 1.341

0.0130 3026 7.449 1.412

0.00200 331.6 4.943 1.257 0.432 0.0718

0.00221 393.8 5.108 1.216 0.411 0.0688

0.00242 453.4 5.250 1.188 0.402 0.0673

0.00263 512.3 5.374 1.172 0.399 0.0669

0.00304 625.8 5.586 1.161 0.405 0.0679

0.00385 862.0 5.922 1.175 0.436 0.0730

0.00465 1099.3 6.136 1.198 0.474 0.0800

0.00544 1341 6.407 1.219 0.512 0.0881

0.00608 1588 6.605 1.240

0.00681 1844 6.769 1.275

0.00754 2101 6.921 1.318

0.00826 2365 7.060 1.347

0.0101 3049 7.366 1.412

0.00180 343.4 4.869 1.254 0.494 0.0810

0.00196 405.1 5.034 1.217 0.463 0.0768

0.00213 465.3 5.176 1.192 0.446 0.0744

0.00230 524.4 5.300 1.175 0.438 0.0733

0.00262 641.2 5.513 1.164 0.435 0.0732

0.00328 875.1 5.850 1.179 0.456 0.0768

0.00392 1113.3 6.115 1.204 0.489 0.0830

0.00455 1356 6.337 1.225 0.524 0.0906

0.00507 1606 6.539 1.248

0.00565 1863 6.703 1.283

0.00624 2121 6.856 1.325

0.00681 2386 6.995 1.354

0.00825 3071 7.302 1.413

2-212

PHYSICAL AND CHEMICAL DATA TABLE 2-230

Enthalpy and Psi Functions for Ideal-Gas Air*

T, K

h, kJ/kg

Ψ

T, K

h, kJ/kg

Ψ

T, K

h, kJ/kg

Ψ

200 210 220 230 240

200.0 210.0 220.0 230.1 240.1

−0.473 −0.400 −0.329 −0.262 −0.197

650 660 670 680 690

659.8 670.5 681.1 691.8 702.5

1.339 1.364 1.388 1.412 1.436

1200 1220 1240 1260 1280

1278 1301 1325 1349 1372

2.376 2.406 2.435 2.463 2.491

250 260 270 280 290

250.1 260.1 270.1 280.1 290.2

−0.135 −0.076 −0.018 0.037 0.090

700 710 720 730 740

713.3 724.0 734.8 745.6 756.4

1.459 1.482 1.505 1.528 1.550

1300 1320 1340 1360 1380

1396 1420 1444 1467 1491

2.519 2.547 2.574 2.601 2.627

300 310 320 330 340

300.2 310.3 320.3 330.4 340.4

0.142 0.191 0.240 0.286 0.332

750 760 770 780 790

767.3 778.2 789.1 800.0 811.0

1.572 1.594 1.615 1.637 1.658

1400 1420 1440 1460 1480

1515 1539 1563 1587 1612

2.653 2.679 2.705 2.730 2.755

350 360 370 380 390

350.5 360.6 370.7 380.8 390.9

0.376 0.419 0.461 0.502 0.541

800 810 820 830 840

821.9 832.9 844.0 855.0 866.1

1.679 1.699 1.720 1.740 1.760

1500 1520 1540 1560 1580

1636 1660 1684 1709 1738

2.779 2.803 2.827 2.851 2.875

400 410 420 430 440

401.0 411.2 421.3 431.5 441.7

0.580 0.618 0.655 0.691 0.727

850 860 870 880 890

877.2 888.3 899.4 910.6 921.8

1.780 1.800 1.819 1.838 1.857

1600 1620 1640 1660 1680

1758 1782 1806 1831 1855

2.898 2.921 2.944 2.966 2.988

450 460 470 480 490

451.8 462.1 472.3 482.5 492.8

0.761 0.795 0.829 0.861 0.893

900 910 920 930 940

933.0 944.2 955.4 966.7 978.0

1.876 1.895 1.914 1.932 1.950

1700 1720 1740 1760 1780

1880 1905 1929 1954 1979

3.010 3.032 3.054 3.075 3.096

500 510 520 530 540

503.1 513.4 523.7 534.0 544.4

0.925 0.956 0.986 1.016 1.045

950 960 970 980 990

989.3 1000.6 1011.9 1023.3 1034.7

1.969 1.987 2.004 2.022 2.039

1800 1820 1840 1860 1880

2003 2028 2053 2078 2102

3.117 3.138 3.158 3.178 3.198

550 560 570 580 590

554.8 565.2 575.6 586.1 596.5

1.074 1.102 1.130 1.158 1.185

1000 1020 1040 1060 1080

1046.1 1068.9 1091.9 1114.9 1138.0

2.057 2.091 2.125 2.158 2.190

1900 1920 1940 1960 1980

2127 2152 2177 2202 2227

3.218 3.238 3.258 3.277 3.296

600 610 620 630 640

607.0 617.5 628.1 638.6 649.2

1.211 1.238 1.264 1.289 1.314

1100 1120 1140 1160 1180

1161.1 1184.3 1207.6 1230.9 1254.3

2.223 2.254 2.285 2.316 2.346

2000 2050 2100 2150 2200

2252 2315 2377 2440 2504

3.215 3.362 3.408 3.453 3.496

*Values rounded off from Chappell and Cockshutt, Nat. Res. Counc. Can. Rep. NRC LR 759 (NRC No. 14300), 1974. This source tabulates values of seven thermodynamic functions at 1-K increments from 200 to 2200 K in SI units and at other increments for two other unit systems. An earlier report (NRC LR 381, 1963) gives a more detailed description of an earlier fitting from 200 to 1400 K. In the above table h = specific enthalpy, kJ/kg, and Ψ2 − Ψ1 = log10 (P2 /P1)s for an isentrope. In terms of the Keenan and Kaye function φ, Ψ = (log10 e/R) ⋅ φ.

THERMODYNAMIC PROPERTIES

FIG. 2-5 Temperature-entropy diagram for air. [Landsbaum, Dadds, Stevens, et al., Am. Inst. Chem. Eng. J., 1(3), 303 (1955). Reproduced by permission of the authors and of the editor, American Institute of Chemical Engineers.]

2-213

2-214

PHYSICAL AND CHEMICAL DATA

TABLE 2-231

Air

Other tables include Stewart, R. B., S. G. Penoncello, et al., University of Idaho CATS report, 85-5, 1985 (0.1–700 bar, 85–750 K), and a revision is in process of publication. Tables including reactions with hydrocarbons include Gordon, S., NASA Techn. Paper 1907, 4 vols., 1982. See also Gupta, R. N., K-P. Lee, et al., NASA RP 1232, 1990 (89 pp.) and RP 1260, 1991 (75 pp.). Analytic expressions for high temperatures were given by Matsuzaki, R., Jap. J. Appl. Phys., 21, 7 (1982): 1009–1013 and Japanese National Aerospace Laboratory report NAL TR 671, 1981 (45 pp.). Functions from 1500 to 15000 K were tabulated by Hilsenrath, J. and M. Klein, AEDC-TR-65-58 = AD 612 301, 1965 (333 pp.). Tables from 10000 to 10,000,000 K were authored by Gilmore, F. R., Lockheed rept. 3-27-67-1, vol 1., 1967 (340 pp.), also published as Radiative Properties of Air, IFI/Plenum, New York, 1969 (648 pp.). Saturation and superheat tables and a chart to 7000 psia, 660°R appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity see Thermophysical Properties of Refrigerants, ASHRAE, 1993. AIR, MOIST An ASHRAE publication, Thermodynamic Properties of Dry Air and Water and S. I. Psychrometric Charts, 1983 (360 pp.), extensively reviews moist air properties. Gandiduson, P., Chem. Eng., Oct. 29, 1984 gives on page 118 a nomograph from 50 to 120°F, while equations in SI units were given by Nelson, B., Chem. Eng. Progr. 76, 5 (May 1980): 83–85. Liley, P. E., 2000 Solved Problems in M.E. Thermodynamics, McGraw-Hill, New York, 1989, gives four simple equations with which most calculations can be made. Devres, Y.O., Appl. Energy 48 (1994): 1–18 gives equations with which three known properties can be used to determine four others. Klappert, M. T. and G. F. Schilling, Rand RM-4244-PR = AD 604 856, 1984 (40 pp.) gives tables from 100 to 270 K, while programs from −60 to 2°F are given by Sando, F. A., ASHRAE Trans., 96, 2 (1990): 299–308. Viscosity references include Kestin, J. and J. H. Whitelaw, Int. J. Ht. Mass Transf. 7, 11 (1964): 1245–1255; Studnokov, E. L., Inz.-Fiz. Zhur. 19, 2 (1970): 338–340; Hochramer, D. and F. Munczak, Setzb. Ost. Acad. Wiss II 175, 10 (1966): 540–550. For thermal conductivity see, for instance, Mason, E. A. and L. Monchick, Humidity and Moisture Control in Science and Industry, Reinhold, New York, 1965 (257–272).

TABLE 2-232

Saturated Ammonia*

P, bar

vf, m3/kg

vg, m3/kg

h f, kJ/kg

h g, kJ/kg

s f, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

c pf, kJ/(kg⋅K)

µ f, 10−4 Pa⋅s

k f, W/(m⋅K)

195.5t 200 210 220 230

0.0608 0.0865 0.1775 0.3381 0.6044

1.327.–3 1.372.–3 1.394.–3 1.417.–3 1.442.–3

15.648 11.237 5.729 3.135 1.822

−1110.1 −1088.8 −1044.1 −1000.6 −957.0

380.1 388.5 406.7 424.1 440.7

4.203 4.311 4.529 4.731 4.925

11.827 11.698 11.438 11.207 11.002

4.73 4.61 4.38 4.35 4.38

4.25 4.07 3.69 3.34 3.02

0.715 0.709 0.685 0.661 0.638

240 250 260 270 280

1.0226 1.6496 2.5529 3.8100 5.5077

1.468.–3 1.495.–3 1.524.–3 1.551.–3 1.589.–3

1.115 0.712 0.472 0.324 0.228

−912.9 −868.2 −823.1 −777.3 −730.9

456.2 470.6 483.8 495.6 506.0

5.113 5.294 5.471 5.643 5.811

10.817 10.650 10.498 10.358 10.228

4.43 4.48 4.54 4.60 4.66

2.73 2.45 2.20 1.97 1.76

0.615 0.592 0.569 0.546 0.523

T, K

290 300 310 320 330

7.741 10.61 14.24 18.72 24.20

1.626.–3 1.666.–3 1.710.–3 1.760.–3 1.815.–3

0.165 0.121 0.091 0.069 0.053

−683.8 −636.0 −587.2 −537.5 −486.7

514.7 521.5 526.1 528.2 527.5

5.975 6.135 6.293 6.448 6.602

10.108 9.994 9.885 9.779 9.675

4.73 4.82 4.91 5.02 5.17

1.58 1.41 1.26 1.13 1.02

0.500 0.477 0.454 0.431 0.408

340 350 360 370 380

30.79 38.64 47.90 58.74 71.35

1.878.–3 1.952.–3 2.039.–3 2.148.–3 2.291.–3

0.0410 0.0319 0.0249 0.0194 0.0149

−434.3 −380.0 −323.2 −262.6 −196.5

523.3 515.1 501.8 481.9 452.7

6.755 6.908 7.063 7.222 7.391

9.571 9.465 9.354 9.235 9.100

5.37 5.64 6.04 6.68 7.80

0.92 0.83 0.75 0.69 0.61

0.385 0.361 0.337 0.313 0.286

85.98 103.0 113.0

2.499.–3 2.882.–3 4.255.–3

0.0113 0.0077 0.0043

−120.9 −23.5 142.7

408.1 329.0 142.7

7.578 7.813 8.216

8.935 8.694 8.216

0.50 0.39 0.25

0.254 0.21 ∞

390 400 405.4c

10.3 21. ∞

*P, v, h, and s values condensed from ASHRAE Handbook, 1981: Fundamentals. Copyright 1981 by the American Society of Heating, Refrigerating and Air Conditioning Engineers, Inc., and reproduced by permission of the copyright owner. cp, µ, and k values are interpolated and converted from Thermophysical Properties of Refrigerants, ASHRAE, New York, 1976. t = triple point; c = critical point. The notation 1.327.–3 signifies 1.327 × 10−3. At 195.5 K, the viscosity of the saturated liquid is 4.25 × 10−4 Pa⋅s. Most recent tabulations of ammonia properties are based upon the extensive tabulation to 5000 bar, 750 K of Haar, L. and J. S. Gallagher, J. Phys. Chem. Ref. Data, 7, 3 (1978): 635–792, which does, however, neglect dissociation. For tables to 70,000 psia, 920°F, see Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). A chart in fps units corresponding with these tables appears on page 17.34 of the ASHRAE 1989 Fundamentals Handbook. Simmons, A. L., C. E. Miller III, et al., Tables and Charts of Equilibrium Thermodynamic Properties of Ammonia for Temperatures from 500 to 50000 K, NASA SP 3099, 1976 (255 pp.), tabulates ρ, h, s, cp, cv, Z, and so on, from 0.01 to 400 bar and also 18 species of decomposition products. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives material for integral degrees Celsius with temperatures on the ITS 90 scale for saturation temperatures from −77.66 to 132.22 °C. The same diagram reproduced here appears in that source.

THERMODYNAMIC PROPERTIES

FIG. 2-6 Enthalpy–log-pressure diagram for ammonia. 1 MPa = 10 bar. (Copyright 1981 by the American Society of Heating, Refrigerating and Air-Conditioning Engineers and reproduced by permission of the copyright owner.)

2-215

2-216

FIG. 2-7 Enthalpy-concentration diagram for aqueous ammonia. From Thermodynamic and Physical Properties NH3 –H2O, Int. Inst. Refrigeration, Paris, France, 1994 (88 pp.). Reproduced by permission. In order to determine equilibrium compositions, draw a vertical from any liquid composition on any boiling line (the lowest plots) to intersect the appropriate auxiliarly curve (the intermediate curves). A horizontal then drawn from this point to the appropriate dew line (the upper curves) will establish the vapor composition. The Int. Inst. Refrigeration publication also gives extensive P-v-x tables from −50 to 316°C. Other sources include Park, Y. M. and Sonntag, R. E., ASHRAE Trans., 96, 1 (1990): 150–159 (x, h, s, tables, 360 to 640 K); Ibrahim, O. M. and S. A. Klein, ASHRAE Trans., 99, 1 (1993): 1495–1502 (Eqs., 0.2 to 110 bar, 293 to 413 K); Smolen, T. M., D. B. Manley, et al., J. Chem. Eng. Data, 36 (1991): 202–208 (p-x correlation, 0.9 to 450 psia, 293–413 K); Ruiter, J. P., Int. J. Refrig., 13 (1990): 223–236 gives ten subroutines for computer calculations.

TABLE 2-233 T, K

Saturated Argon (R740)* vf, m3/kg

P, bar

10 20 30 40 50

vg, m3/kg

5.646.–4 5.666.–4 5.707.–4 5.763.–4 5.831.–4

hf, kJ/kg

hg, kJ/kg

0.20 2.20 6.12 11.30 17.26

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

0.0266 0.1559 0.3129 0.4610 0.5937

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

0.083 0.306 0.466 0.560 0.627

60 70 80 83.8t 83.8t

0.082 0.406 0.687 0.687

5.912.–4 6.008.–4 6.125.–4 6.178.–4 7.068.–4

2.1800 0.3918 0.2434 0.2434

23.85 31.08 39.07 42.34 71.88

229.08 232.88 235.06 235.06

0.7138 0.8250 0.9316 0.9720 1.333

3.415 3.364 3.280 3.280

0.687 0.752 0.836 0.877 1.050

2.93

0.134

85 87.3 90 95 100

0.790 1.013 1.338 2.137 3.247

7.107.–4 7.174.–4 7.269.–4 7.440.–4 7.628.–4

0.2145 0.1710 0.1327 0.0864 0.0588

73.16 75.61 78.55 84.15 89.85

235.55 236.39 237.37 238.91 240.20

1.348 1.375 1.403 1.462 1.520

3.258 3.216 3.168 3.091 3.023

1.058 1.073 1.091 1.124 1.158

2.81 2.60 2.40 2.08 1.82

0.132 0.128 0.124 0.116 0.109

110 115 120 125 130

6.665 9.107 12.13 15.81 20.23

8.064.–4 8.322.–4 8.618.–4 8.965.–4 9.620.–4

0.0299 0.0221 0.0166 0.0126 0.0096

101.83 108.11 114.62 121.50 128.79

241.66 241.78 241.33 240.30 238.41

1.632 1.685 1.738 1.792 1.846

2.903 2.848 2.794 2.743 2.690

1.229 1.274 1.336 1.427 1.550

1.46 1.32 1.21 1.12 1.01

0.096 0.090 0.084 0.078 0.072

135 140 145 150 150.9

25.49 31.68 38.93 47.39 48.98

9.906.–4 1.061.–3 1.172.–3 1.468.–3 1.867.–3

0.0074 0.0056 0.0041 0.0026 0.0019

136.76 145.58 155.73 174.64 189.94

234.60 230.74 223.09 204.35 189.94

1.902 1.961 2.026 2.133 2.201

2.633 2.570 2.490 2.331 2.201

1.752

0.89 0.75 0.60 0.45 0.28

0.066 0.060 0.054 ∞

*Values extracted and in some cases rounded off from those cited in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standards Press, Moscow, 1976. This source contains values for the compressed state for pressures up to 1000 bar, etc. t = triple point. Above the solid line the condensed phase is solid; below it, it is liquid. The notation 5.646.–4 signifies 5.646 × 10−4. At 83.8 K, the viscosity of the saturated liquid is 2.93 × 10−4 Pa⋅s = 0.000293 Ns/m2. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.). TABLE 2-234

Thermodynamic Properties of Compressed Argon* Pressure, bar

T, K

1

100

200

300

400

500

600

700

800

900

v 100 h s

0.2035 243.4 3.299

7.420.–4 93.6 1.494

7.255.–4 97.9 1.464

7.120.–4 102.5 1.438

7.006.–4 107.2 1.414

6.907.–4 112.0 1.393

6.819.–4 116.8 1.372

6.050.–4 91.1 1.037

6.009.–4 96.1 1.026

5.976.–4 101.0 1.016

5.935.–4 106.0 1.007

1000

v 200 h s

0.4151 296.4 3.667

2.96.–3 250.2 2.538

1.430.–5 217.1 2.276

1.159.–3 209.1 2.173

1.045.–3 207.9 2.112

9.778.–4 209.2 2.068

9.312.–4 211.9 2.033

8.962.–4 215.1 2.004

8.683.–4 218.9 1.979

8.454.–4 223.0 1.957

8.260.–4 227.4 1.936

v 300 h s

0.6241 348.6 3.879

5.96.–3 330.9 2.872

2.976.–3 316.3 2.686

2.071.–3 306.6 2.572

1.666.–3 301.4 2.493

1.443.–3 299.3 2.435

1.304.–3 299.2 2.389

1.207.–3 300.5 2.352

1.136.–3 302.7 2.320

1.081.–3 305.6 2.293

1.037.–3 310.0 2.269

v 400 h s

0.8326 400.7 4.028

8.37.–3 391.3 3.048

4.279.–3 383.6 2.881

2.957.–3 378.4 2.780

2.322.–3 375.2 2.707

1.955.–3 373.8 2.651

1.719.–3 373.8 2.603

1.557.–3 374.8 2.565

1.435.–3 376.6 2.533

1.344.–3 379.2 2.505

1.271.–3 382.0 3.480

v 500 h s

1.0409 452.8 4.145

1.062.–2 447.7 3.174

5.464.–3 444.3 3.018

3.772.–3 442.0 2.924

2.940.–3 440.9 2.854

2.448.–3 440.6 2.801

2.124.–3 441.4 2.755

1.899.–3 422.9 2.718

1.730.–3 444.7 2.685

1.607.–3 447.1 2.658

1.506.–3 449.9 2.633

v 600 h s

1.2489 504.9 4.240

1.280.–2 502.4 3.274

6.589.–3 501.6 3.122

4.539.–3 501.4 3.031

3.525.–3 501.8 2.966

2.922.–3 503.0 2.914

2.522.–3 504.6 2.870

2.238.–3 506.6 2.834

2.023.–3 508.7 2.801

1.866.–3 511.2 2.774

1.736.–3 513.9 2.750

v 700 h s

1.4569 556.9 4.320

1.495.–2 556.5 3.356

7.686.–3 556.9 3.207

5.281.–3 558.0 3.118

4.088.–3 559.8 3.054

3.377.–3 561.8 3.005

2.906.–3 564.2 2.963

2.570.–3 566.9 2.928

2.317.–3 569.6 2.897

2.123.–3 527.5 2.870

1.966.–3 575.3 2.845

v 800 h s

1.6659 609.9 4.389

1.708.–2 609.8 3.427

8.768.–3 611.0 3.279

6.011.–3 612.9 3.191

4.640.–3 615.2 3.129

3.822.–3 618.1 3.081

3.280.–3 621.2 3.039

2.893.–3 624.5 3.005

2.603.–3 627.8 2.975

2.376.–3 631.3 2.948

2.196.–3 634.8 2.924

v 900 h s

1.8739 661.0 4.451

1.920.–2 662.7 3.490

9.841.–3 664.6 3.342

6.732.–3 667.2 3.255

5.183.–3 670.1 3.193

4.259.–3 673.3 3.145

3.646.–3 676.8 3.105

3.209.–3 680.7 3.071

2.881.–3 684.4 3.042

2.626.–3 688.3 3.016

2.423.–3 692.3 2.992

v 1000 h s

2.0819 713.1 4.506

2.131.–2 715.4 3.545

1.091.–2 717.9 3.398

7.448.–3 720.9 3.312

5.723.–3 724.3 3.250

4.692.–3 727.8 3.203

4.008.–3 731.5 3.163

3.520.–3 735.6 3.129

3.156.–3 739.8 3.100

2.872.–3 744.1 3.074

2.645.–3 748.5 3.051

*Values extracted and in some cases rounded off from those cited in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standards Press, Moscow, 1976. v = specific volume, m3/kg; h = specific enthalpy, kJ/kg; s = specific entropy, kJ/(kg⋅K). This source contains an exhaustive tabulation of values. The notation 7.420.–4 signifies 7.420 × 10−4. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.). The 1993 ASHRAE Handbook—Fundamentals (SI ed.) has a thermodynamic chart for pressures from 1 to 2000 bar, temperatures from 90 to 700 K. Saturation and superheat tables and a chart to 50,000 psia, 1220 °R appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity see Thermophysical Properties of Refrigerants, ASHRAE, 1993. Extensive tables for 10 properties from 0.9–100 bar, 86–400 K are given by Jacques, A., Fermi Accelerator Lab., Batavia, IL, rept TM 1517, 1988 (201 pp.). In Hilsenrath, J., C. G. Messina, et al., AEDC-TR-66-248 = AD 644 081, 1966 (121 pp.), thermodynamic properties and chemical composition from 2400 to 35,000 K are tabulated. See also Drellishak, K. S. et al., AEDC-TDR-63-146, 1963; AEDC-TDR-64-12 = AD 427839, 1964. 2-217

2-218

PHYSICAL AND CHEMICAL DATA

TABLE 2-235

Liquid-Vapor Equilibrium Data for the Argon-Nitrogen-Oxygen System*

Liquid mole fraction N2 /N2 + O2

Vapor mole fraction Ar

N2

Ar

O2

Temperature,°R

Relative volatility N2 /Ar

N2 /O2

Ar/O2

Pressure activity coefficient

Enthalpy, Btu/ (lb·mol)

Heat capacity, Btu/(lb·mol·°R)

N2

Ar

O2

Liquid

Vapor

Liquid

Vapor

Pressure, 1 atm 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

0. 0.0154 0.0306 0.0456 0.0603 0.0748 0.1031 0.1439 0.2687 0.4796 0.6605 0.8293 0.9136

1.0000 0.9845 0.9694 0.9544 0.9397 0.9253 0.8970 0.8561 0.7313 0.5204 0.3395 0.1707 0.0865

162.4 162.3 162.2 162.1 162.0 161.9 161.7 161.5 160.7 159.4 158.5 157.7 157.5

2.575 2.581 2.586 2.592 2.597 2.602 2.613 2.629 2.682 2.786 2.888 2.991 3.042

4.010 4.007 4.004 4.001 3.998 3.995 3.989 3.979 3.941 3.852 3.746 3.632 3.572

1.557 1.553 1.548 1.544 1.540 1.535 1.526 1.513 1.469 1.382 1.297 1.214 1.174

1.118 1.117 1.115 1.113 1.112 1.110 1.107 1.103 1.091 1.076 1.075 1.087 1.099

1.165 1.161 1.158 1.155 1.151 1.148 1.142 1.132 1.104 1.058 1.026 1.008 1.003

0.999 1.000 1.000 1.000 1.001 1.001 1.002 1.003 1.010 1.034 1.072 1.127 1.162

−1841. −1844. −1847. −1850. −1852. −1855. −1860. −1868. −1893. −1938. −1978. −2015. −2032.

1093. 1087. 1082. 1076. 1071. 1066. 1056. 1041. 997. 924. 862. 807. 779.

13.2 13.1 13.1 13.1 13.0 13.0 12.9 12.9 12.6 11.9 11.3 10.7 10.4

7.406 7.374 7.342 7.311 7.281 7.251 7.192 7.107 6.847 6.406 6.026 5.669 5.491

0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.3135 0.3095 0.3056 0.3017 0.2978 0.2939 0.2863 0.2752 0.2399 0.1759 0.1169 0.0595 0.0303

0. 0.0119 0.0237 0.0354 0.0470 0.0585 0.0812 0.1145 0.2207 0.4170 0.6036 0.7933 0.8937

0.6865 0.6786 0.6707 0.6630 0.6553 0.6476 0.6325 0.6103 0.5394 0.4072 0.2795 0.1471 0.0762

157.7 157.6 157.6 157.6 157.6 157.5 157.5 157.4 157.2 157.0 156.9 156.9 157.1

2.621 2.626 2.631 2.636 2.641 2.645 2.655 2.669 2.717 2.812 2.906 3.001 3.048

4.111 4.106 4.100 4.095 4.090 4.085 4.074 4.058 4.003 3.887 3.766 3.640 3.576

1.568 1.563 1.558 1.554 1.549 1.544 1.534 1.520 1.473 1.382 1.296 1.213 1.173

1.103 1.102 1.100 1.099 1.098 1.096 1.094 1.090 1.080 1.070 1.072 1.086 1.099

1.168 1.164 1.161 1.157 1.154 1.151 1.144 1.135 1.106 1.061 1.029 1.009 1.004

1.012 1.012 1.012 1.013 1.013 1.013 1.014 1.015 1.022 1.045 1.082 1.134 1.166

−1834. −1837. −1839. −1842. −1844. −1846. −1851. −1858. −1882. −1926. −1969. −2009. −2029.

1060. 1057. 1053. 1049. 1045. 1042. 1034. 1024. 990. 928. 871. 813. 783.

13.2 13.1 13.1 13.1 13.1 13.0 13.0 12.9 12.5 11.9 11.3 10.7 10.4

7.410 7.386 7.361 7.337 7.313 7.289 7.242 7.173 6.951 6.540 6.147 5.746 5.534

0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.5095 0.5042 0.4990 0.4938 0.4886 0.4834 0.4732 0.4580 0.4083 0.3123 0.2162 0.1144 0.0593

0. 0.0096 0.0192 0.0288 0.0383 0.0477 0.0666 0.0946 0.1866 0.3680 0.5550 0.7602 0.8744

0.4905 0.4861 0.4818 0.4775 0.4731 0.4688 0.4602 0.4474 0.4051 0.3197 0.2288 0.1254 0.0663

154.0 154.0 154.0 154.1 154.1 154.1 154.1 154.2 154.3 154.8 155.4 156.2 156.7

2.641 2.646 2.651 2.655 2.660 2.665 2.674 2.688 2.735 2.829 2.921 3.009 3.052

4.155 4.149 4.143 4.137 4.131 4.125 4.112 4.094 4.032 3.907 3.779 3.647 3.580

1.573 1.568 1.563 1.558 1.553 1.548 1.538 1.523 1.474 1.381 1.294 1.212 1.173

1.085 1.084 1.083 1.082 1.081 1.080 1.078 1.075 1.068 1.062 1.068 1.086 1.099

1.171 1.168 1.164 1.161 1.158 1.154 1.148 1.139 1.110 1.064 1.032 1.011 1.006

1.026 1.026 1.026 1.027 1.027 1.027 1.028 1.030 1.036 1.058 1.093 1.140 1.169

−1814. −1816. −1819. −1821. −1824. −1826. −1831. −1839. −1863. −1911. −1958. −2004. −2026.

1035. 1032. 1029. 1027. 1024. 1021. 1016. 1008. 981. 930. 877. 819. 787.

13.2 13.2 13.1 13.1 13.1 13.0 13.0 12.9 12.6 11.9 11.3 10.7 10.4

7.422 7.402 7.382 7.362 7.342 7.322 7.283 7.224 7.031 6.648 6.252 5.817 5.574

0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.7333 0.7279 0.7226 0.7172 0.7118 0.7064 0.6956 0.6794 0.6244 0.5075 0.3743 0.2121 0.1141

0. 0.0070 0.0140 0.0210 0.0280 0.0350 0.0491 0.0703 0.1426 0.2977 0.4776 0.7010 0.8382

0.2667 0.2651 0.2635 0.2619 0.2602 0.2586 0.2553 0.2503 0.2331 0.1948 0.1482 0.0870 0.0477

148.7 148.8 148.8 148.9 148.9 149.0 149.1 149.3 149.9 151.2 152.8 154.8 155.9

2.629 2.634 2.640 2.645 2.650 2.656 2.667 2.683 2.737 2.841 2.939 3.025 3.063

4.124 4.119 4.114 4.108 4.103 4.098 4.087 4.072 4.018 3.907 3.788 3.657 3.586

1.569 1.564 1.558 1.553 1.548 1.543 1.533 1.517 1.468 1.375 1.289 1.209 1.171

1.050 1.049 1.049 1.048 1.048 1.047 1.047 1.046 1.044 1.048 1.062 1.084 1.099

1.187 1.183 1.179 1.176 1.172 1.169 1.162 1.152 1.122 1.074 1.039 1.016 1.008

1.065 1.065 1.065 1.066 1.066 1.066 1.066 1.067 1.071 1.088 1.116 1.154 1.177

−1748. −1751. −1754. −1757. −1760. −1763. −1770. −1779. −1810. −1871. −1932. −1991. −2020.

997. 996. 994. 992. 991. 989. 986. 981. 964. 928. 885. 829. 794.

13.3 13.3 13.2 13.2 13.2 13.1 13.1 13.0 12.6 11.9 11.3 10.7 10.4

7.452 7.437 7.422 7.407 7.392 7.377 7.347 7.301 7.145 6.811 6.423 5.944 5.651

0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.8569 0.8521 0.8472 0.8424 0.8375 0.8326 0.8227 0.8076 0.7557 0.6391 0.4938 0.2961 0.1647

0. 0.0056 0.0111 0.0167 0.0224 0.0280 0.0394 0.0566 0.1163 0.2507 0.4191 0.6500 0.8047

0.1431 0.1424 0.1416 0.1409 0.1402 0.1395 0.1380 0.1357 0.1280 0.1102 0.0871 0.0539 0.0306

144.9 145.0 145.1 145.1 145.2 145.3 145.5 145.7 146.5 148.4 150.6 153.5 155.2

2.575 2.582 2.589 2.595 2.602 2.609 2.622 2.642 2.707 2.833 2.945 3.037 3.071

3.993 3.991 3.988 3.985 3.983 3.980 3.975 3.966 3.937 3.867 3.777 3.662 3.592

1.551 1.546 1.541 1.536 1.531 1.526 1.516 1.501 1.454 1.365 1.283 1.206 1.170

1.024 1.024 1.024 1.024 1.024 1.023 1.023 1.023 1.025 1.036 1.056 1.083 1.098

1.218 1.214 1.210 1.206 1.202 1.198 1.190 1.179 1.145 1.090 1.049 1.021 1.010

1.126 1.125 1.125 1.124 1.124 1.123 1.122 1.121 1.120 1.126 1.143 1.169 1.185

−1663. −1667. −1672. −1676. −1680. −1684. −1692. −1704. −1744. −1824. −1903. −1978. −2014.

970. 969. 968. 967. 966. 965. 963. 960. 948. 923. 888. 837. 801.

13.6 13.5 13.5 13.4 13.4 13.4 13.3 13.2 12.8 12.0 11.3 10.7 10.4

7.483 7.471 7.459 7.446 7.434 7.421 7.396 7.357 7.224 6.926 6.556 6.055 5.723

0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.9384 0.9340 0.9296 0.9252 0.9207 0.9162 0.9071 0.8934 0.8452 0.7339 0.5869 0.3690 0.2117

0. 0.0047 0.0094 0.0142 0.0189 0.0237 0.0334 0.0481 0.0994 0.2177 0.3740 0.6059 0.7736

0.0616 0.0613 0.0610 0.0607 0.0604 0.0601 0.0595 0.0586 0.0554 0.0483 0.0391 0.0252 0.0147

142.0 142.0 142.1 142.2 142.3 142.4 142.6 142.8 143.8 146.0 148.7 152.2 154.5

2.501 2.509 2.517 2.525 2.533 2.540 2.556 2.580 2.658 2.809 2.943 3.045 3.078

3.811 3.811 3.812 3.812 3.813 3.813 3.814 3.814 3.814 3.797 3.751 3.662 3.595

1.524 1.519 1.515 1.510 1.506 1.501 1.492 1.478 1.435 1.352 1.275 1.202 1.168

1.013 1.013 1.013 1.013 1.013 1.013 1.012 1.013 1.015 1.028 1.051 1.082 1.098

1.273 1.268 1.263 1.257 1.258 1.247 1.238 1.224 1.181 1.112 1.062 1.026 1.013

1.214 1.212 1.210 1.209 1.207 1.205 1.202 1.197 1.185 1.173 1.173 1.184 1.193

−1570. −1575. −1580. −1585. −1590. −1595. −1606. −1621. −1671. −1772. −1870. −1964. −2007.

949. 948. 947. 947. 946. 945. 944. 942. 934. 916. 889. 843. 806.

14.0 13.9 13.9 13.8 13.8 13.7 13.6 13.5 13.0 12.2 11.4 10.7 10.3

7.514 7.503 7.492 7.481 7.470 7.459 7.437 7.403 7.284 7.013 6.662 6.153 5.790

*Calculated values, from Wilson, Silverberg, and Zellner, USAF Aero Propulsion Laboratory. Rep. APL TDR 64-64 (AD 603 151), 1964. Relative volatility = α i − j = (yi /xi)(x j /yj), where x = liquid composition, y = vapor composition. Pressure activity coefficient = yi P/x i p0, where p0 = vapor pressure. These data were confirmed by the analyses of Bender, Cryogenics, 13, 11 (1973); and by Elshayal and Lu, J. Chem. Eng. Data, 16, 31 (1971). See also Armstrong, G. T., J. M. Goldstein, et al., J. Res. N.B.S., 55, 5 (1955): 265–277; Bender, E., Cryogenics, 13, 1 (1973): 11–18; Elshayal, I. M. and B. C-y Lu, J. Chem. Eng. Data, 16, 1 (1971): 31–37; Funada, I., S. Yoshimura, et al., Advan. Cryog. Engng., 7 (1982): 893–901; and Hwang, S-C., Fluid Phase Equila., 37 (1987): 153–167.

THERMODYNAMIC PROPERTIES TABLE 2-235

Liquid-Vapor Equilibrium Data for the Argon-Nitrogen-Oxygen System (Continued )

Liquid mole fraction N2 /N2 + O2

2-219

Vapor mole fraction Ar

N2

Ar

O2

Temperature,°R

Relative volatility N2 /Ar

N2 /O2

Pressure activity coefficient

Ar/O2

Enthalpy, Btu/ (lb·mol)

Heat capacity, Btu/(lb·mol·°R)

N2

Ar

O2

Liquid

Vapor

Liquid

Vapor

Pressure, 1 atm (Cont.) 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.9709 0.9667 0.9624 0.9581 0.9538 0.9494 0.9407 0.9274 0.8808 0.7723 0.6262 0.4018 0.2339

0. 0.0044 0.0088 0.0133 0.0177 0.0222 0.0312 0.0450 0.0931 0.2048 0.3552 0.5860 0.7589

0.0291 0.0289 0.0288 0.0286 0.0285 0.0284 0.0281 0.0276 0.0261 0.0229 0.0186 0.0122 0.0072

140.6 140.7 140.8 140.9 141.0 141.1 141.3 141.6 142.6 144.9 147.8 151.7 154.2

2.459 2.468 2.476 2.485 2.493 2.502 2.519 2.545 2.629 2.793 2.938 3.048 3.082

3.710 3.712 3.714 3.716 3.718 3.720 3.724 3.729 3.743 3.755 3.733 3.660 3.597

1.509 1.504 1.500 1.496 1.491 1.487 1.478 1.465 1.424 1.344 1.270 1.201 1.167

1.015 1.014 1.014 1.013 1.013 1.013 1.013 1.012 1.014 1.026 1.050 1.081 1.098

1.311 1.305 1.299 1.293 1.287 1.281 1.270 1.254 1.204 1.126 1.069 1.029 1.014

1.271 1.268 1.265 1.263 1.260 1.257 1.252 1.245 1.226 1.200 1.190 1.192 1.197

−1522. −1527. −1533. −1538. −1544. −1550. −1561. −1578. −1633. −1745. −1853. −1956. −2004.

939. 938. 938. 937. 937. 936. 935. 934. 928. 912. 889. 846. 809.

14.2 14.2 14.1 14.1 14.0 14.0 13.8 13.7 13.2 12.3 11.4 10.7 10.3

7.530 7.519 7.509 7.498 7.488 7.477 7.456 7.423 7.310 7.050 6.708 6.197 5.822

0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.9916 0.9874 0.9832 0.9790 0.9748 0.9705 0.9619 0.9488 0.9032 0.7965 0.6513 0.4236 0.2489

0. 0.0042 0.0085 0.0127 0.0170 0.0213 0.0300 0.0432 0.0893 0.1969 0.3433 0.5728 0.7489

0.0084 0.0084 0.0083 0.0083 0.0083 0.0082 0.0081 0.0080 0.0076 0.0066 0.0054 0.0036 0.0021

139.8 139.9 140.0 140.1 140.2 140.3 140.5 140.8 141.8 144.2 147.2 151.3 153.9

2.429 2.438 2.447 2.456 2.465 2.474 2.492 2.519 2.608 2.780 2.934 3.050 3.084

3.638 3.641 3.644 3.647 3.650 3.653 3.658 3.667 3.691 3.722 3.718 3.658 3.597

1.498 1.494 1.489 1.485 1.481 1.477 1.468 1.456 1.415 1.339 1.267 1.199 1.167

1.018 1.018 1.017 1.017 1.016 1.016 1.015 1.014 1.014 1.025 1.049 1.081 1.098

1.342 1.335 1.329 1.322 1.315 1.309 1.296 1.278 1.224 1.137 1.075 1.031 1.015

1.316 1.313 1.309 1.306 1.302 1.299 1.293 1.284 1.257 1.220 1.202 1.198 1.200

−1488. −1494. −1500. −1505. −1511. −1517. −1529. −1547. −1606. −1725. −1841. −1951. −2002.

933. 932. 932. 931. 931. 930. 929. 928. 923. 910. 889. 847. 810.

14.4 14.4 14.3 14.2 14.2 14.1 14.0 13.9 13.3 12.3 11.4 10.7 10.3

7.541 7.531 7.520 7.510 7.500 7.489 7.469 7.437 7.327 7.073 6.737 6.226 5.844

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

1.0000 0.9959 0.9917 0.9875 0.9833 0.9791 0.9705 0.9576 0.9122 0.8063 0.6616 0.4327 0.2553

0. 0.0041 0.0083 0.0125 0.0167 0.0209 0.0295 0.0424 0.0878 0.1937 0.3384 0.5674 0.7447

0. 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0. 0. 0.

139.4 139.5 139.6 139.7 139.8 139.9 140.1 140.4 141.5 143.9 147.0 151.1 153.8

2.416 2.425 2.434 2.443 2.452 2.462 2.480 2.507 2.598 2.774 2.932 3.050 3.085

3.607 3.611 3.614 3.617 3.621 3.624 3.630 3.640 3.668 3.708 3.712 3.657 3.598

1.493 1.489 1.485 1.480 1.476 1.472 1.464 1.452 1.412 1.337 1.266 1.199 1.166

1.021 1.020 1.019 1.019 1.018 1.018 1.017 1.016 1.015 1.025 1.048 1.080 1.098

1.357 1.350 1.343 1.336 1.329 1.322 1.309 1.290 1.232 1.142 1.077 1.032 1.016

1.338 1.334 1.330 1.326 1.322 1.318 1.311 1.301 1.271 1.230 1.208 1.201 1.201

−1473. −1479. −1485. −1491. −1497. −1503. −1516. −1534. −1595. −1716. −1836. −1949. −2001.

930. 929. 929. 929. 928. 928. 927. 926. 921. 909. 888. 848. 811.

14.5 14.4 14.4 14.3 14.3 14.2 14.1 13.9 13.4 12.4 11.4 10.7 10.3

7.546 7.535 7.525 7.515 7.505 7.495 7.474 7.443 7.333 7.082 6.749 6.838 5.853

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

0. 0.0137 0.0272 0.0405 0.0537 0.0668 0.0924 0.1299 0.2471 0.4548 0.6410 0.8190 0.9084

1.0000 0.9863 0.9728 0.9595 0.9463 0.9332 0.9076 0.8701 0.7529 0.5452 0.3590 0.1811 0.0916

190.6 190.5 190.4 190.3 190.2 190.1 189.9 189.6 188.8 187.5 186.5 185.8 185.5

2.020 2.023 2.026 2.030 2.033 2.037 2.043 2.053 2.086 2.148 2.207 2.264 2.292

2.776 2.775 2.773 2.772 2.770 2.768 2.765 2.759 2.738 2.688 2.627 2.560 2.524

1.375 1.372 1.369 1.366 1.362 1.359 1.353 1.344 1.313 1.251 1.190 1.131 1.101

0.987 0.986 0.985 0.985 0.984 0.983 0.982 0.980 0.974 0.967 0.967 0.976 0.983

1.111 1.109 1.106 1.104 1.102 1.100 1.096 1.089 1.070 1.038 1.016 1.003 1.000

1.000 1.001 1.001 1.002 1.002 1.003 1.004 1.005 1.013 1.034 1.066 1.110 1.137

−1466. −1470. −1473. −1477. −1480. −1484. −1490. −1500. −1533. −1593. −1647. −1698. −1723.

1224. 1218. 1212. 1207. 1201. 1196. 1185. 1169. 1121. 1037. 964. 896. 862.

13.4 13.4 13.4 13.4 13.4 13.3 13.3 13.3 13.1 12.7 12.3 11.9 11.8

8.122 8.094 8.065 8.037 8.010 7.982 7.929 7.850 7.605 7.166 6.771 6.389 6.196

0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.2393 0.2363 0.2334 0.2305 0.2277 0.2248 0.2192 0.2109 0.1843 0.1353 0.0896 0.0452 0.0229

0. 0.0116 0.0230 0.0344 0.0457 0.0569 0.0792 0.1120 0.2174 0.4150 0.6045 0.7961 0.8957

0.7607 0.7521 0.7436 0.7351 0.7267 0.7183 0.7017 0.6772 0.5983 0.4497 0.3058 0.1588 0.0814

186.2 186.1 186.1 186.0 186.0 186.0 185.9 185.8 185.5 185.2 185.0 185.1 185.2

2.063 2.066 2.069 2.072 2.075 2.078 2.083 2.092 2.119 2.173 2.224 2.273 2.296

2.831 2.828 2.825 2.822 2.820 2.817 2.811 2.802 2.772 2.707 2.638 2.564 2.526

1.372 1.369 1.365 1.362 1.359 1.356 1.349 1.340 1.308 1.246 1.186 1.128 1.100

0.994 0.994 0.993 0.992 0.991 0.990 0.988 0.986 0.979 0.971 0.971 0.978 0.985

1.120 1.117 1.115 1.113 1.110 1.108 1.104 1.097 1.077 1.044 1.020 1.006 1.002

1.020 1.021 1.021 1.021 1.022 1.022 1.023 1.024 1.030 1.049 1.078 1.117 1.141

−1452. −1455. −1459. −1462. −1465. −1469. −1475. −1485. −1517. −1578. −1637. −1693. −1720.

1193. 1188. 1184. 1180. 1175. 1171. 1163. 1150. 1110. 1037. 968. 900. 865.

13.7 13.6 13.6 13.6 13.6 13.5 13.5 13.4 13.2 12.8 12.4 12.0 11.8

8.150 8.126 8.102 8.078 8.054 8.031 7.984 7.915 7.691 7.269 6.860 6.444 6.226

0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.4170 0.4126 0.4082 0.4038 0.3994 0.3951 0.3864 0.3735 0.3316 0.2506 0.1706 0.0883 0.0452

0. 0.0099 0.0198 0.0297 0.0395 0.0493 0.0688 0.0979 0.1932 0.3808 0.5714 0.7742 0.8834

0.5831 0.5776 0.5721 0.5666 0.5611 0.5557 0.5448 0.5286 0.4752 0.3686 0.2580 0.1376 0.0715

182.4 182.4 182.4 182.4 182.4 182.4 182.4 182.5 182.6 183.0 183.6 184.3 184.8

2.094 2.097 2.100 2.102 2.105 2.107 2.112 2.120 2.145 2.194 2.239 2.281 2.301

2.861 2.857 2.854 2.851 2.847 2.844 2.837 2.827 2.792 2.720 2.645 2.568 2.528

1.366 1.363 1.359 1.356 1.353 1.349 1.343 1.333 1.301 1.240 1.181 1.126 1.099

0.997 0.996 0.995 0.994 0.993 0.992 0.990 0.988 0.982 0.974 0.974 0.980 0.986

1.128 1.125 1.123 1.120 1.118 1.116 1.111 1.104 1.083 1.049 1.024 1.008 1.003

1.041 1.041 1.042 1.042 1.042 1.042 1.043 1.044 1.049 1.065 1.090 1.124 1.145

−1431. −1434. −1438. −1441. −1445. −1448. −1455. −1465. −1498. −1562. −1625. −1687. −1717.

1166. 1162. 1159. 1155. 1152. 1149. 1142. 1132. 1099. 1035. 971. 904. 867.

13.8 13.8 13.8 13.8 13.7 13.7 13.7 13.6 13.3 12.9 12.4 12.0 11.8

8.189 8.167 8.146 8.125 8.104 8.083 8.041 7.978 7.771 7.363 6.944 6.497 6.255

Pressure, 4 atm

2-220

PHYSICAL AND CHEMICAL DATA

TABLE 2-235

Liquid-Vapor Equilibrium Data for the Argon-Nitrogen-Oxygen System (Concluded )

Liquid mole fraction N2 /N2 + O2

Vapor mole fraction Ar

N2

Ar

O2

Temperature,°R

Relative volatility N2 /Ar

N2 /O2

Pressure activity coefficient

Ar/O2

Enthalpy, Btu/ (lb·mol)

Heat capacity, Btu/(lb·mol·°R)

N2

Ar

O2

Liquid

Vapor

Liquid

Vapor

Pressure, 4 atm (Cont.) 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.6560 0.6506 0.6453 0.6400 0.6347 0.6293 0.6186 0.6025 0.5482 0.4348 0.3104 0.1684 0.0882

0. 0.0077 0.0155 0.0232 0.0310 0.0387 0.0543 0.0778 0.1575 0.3259 0.5141 0.7334 0.8595

0.3441 0.3417 0.3393 0.3369 0.3345 0.3320 0.3271 0.3197 0.2943 0.2393 0.1756 0.0982 0.0523

176.3 176.4 176.4 176.5 176.6 176.6 176.8 177.0 177.7 179.2 180.9 182.9 184.1

2.124 2.127 2.129 2.132 2.135 2.137 2.143 2.151 2.176 2.223 2.264 2.297 2.309

2.859 2.856 2.853 2.850 2.846 2.843 2.837 2.827 2.794 2.725 2.652 2.573 2.531

1.346 1.343 1.340 1.337 1.333 1.330 1.324 1.315 1.284 1.226 1.171 1.120 1.096

0.992 0.991 0.991 0.990 0.990 0.989 0.988 0.986 0.982 0.978 0.979 0.985 0.989

1.146 1.143 1.140 1.138 1.135 1.133 1.128 1.120 1.098 1.061 1.033 1.013 1.006

1.090 1.090 1.090 1.090 1.090 1.090 1.090 1.090 1.091 1.100 1.116 1.139 1.153

−1372. −1376. −1380. −1384. −1387. −1391. −1399. −1410. −1449. −1525. −1600. −1674. −1710.

1121. 1119. 1117. 1115. 1112. 1110. 1106. 1099. 1077. 1029. 975. 910. 872.

14.1 14.0 14.0 14.0 13.9 13.9 13.9 13.8 13.5 13.0 12.5 12.0 11.8

8.277 8.260 8.242 8.224 8.206 8.188 8.153 8.098 7.915 7.527 7.095 6.597 6.312

0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.8076 0.8023 0.7971 0.7918 0.7865 0.7812 0.7704 0.7542 0.6980 0.5739 0.4261 0.2413 0.1293

0. 0.0064 0.0127 0.0192 0.0256 0.0321 0.0451 0.0649 0.1332 0.2848 0.4667 0.6963 0.8367

0.1924 0.1913 0.1902 0.1891 0.1879 0.1868 0.1845 0.1810 0.1688 0.1413 0.1073 0.0625 0.0340

171.7 171.8 171.9 172.0 172.1 172.2 172.4 172.6 173.7 175.9 178.5 181.6 183.4

2.120 2.123 2.127 2.130 2.133 2.137 2.143 2.153 2.184 2.239 2.283 2.311 2.317

2.798 2.796 2.794 2.792 2.790 2.788 2.784 2.778 2.757 2.708 2.648 2.576 2.533

1.320 1.317 1.314 1.311 1.308 1.305 1.299 1.290 1.262 1.209 1.160 1.115 1.093

0.986 0.986 0.985 0.985 0.985 0.984 0.984 0.983 0.981 0.980 0.984 0.989 0.992

1.173 1.170 1.167 1.164 1.161 1.159 1.153 1.144 1.119 1.076 1.043 1.019 1.009

1.154 1.153 1.152 1.152 1.151 1.150 1.149 1.148 1.143 1.141 1.145 1.155 1.161

−1296. −1301. −1305. −1310. −1315. −1319. −1329. −1343. −1389. −1482. −1573. −1661. −1704.

1086. 1084. 1083. 1082. 1080. 1079. 1076. 1072. 1056. 1021. 976. 916. 876.

14.2 14.2 14.1 14.1 14.1 14.0 14.0 13.9 13.6 13.0 12.5 12.0 11.8

8.369 8.354 8.338 8.322 8.306 8.289 8.257 8.208 8.038 7.666 7.228 6.690 6.367

0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.9152 0.9102 0.9052 0.9001 0.8950 0.8899 0.8795 0.8638 0.8087 0.6826 0.5231 0.3077 0.1684

0. 0.0055 0.0110 0.0165 0.0221 0.0277 0.0390 0.0562 0.1162 0.2535 0.4274 0.6624 0.8150

0.0848 0.0843 0.0839 0.0834 0.0829 0.0825 0.0815 0.0801 0.0751 0.0638 0.0496 0.0299 0.0166

168.0 168.1 168.2 168.3 168.4 168.5 168.7 169.1 170.3 173.1 176.4 180.3 182.7

2.095 2.099 2.103 2.107 2.112 2.116 2.124 2.136 2.175 2.244 2.295 2.323 2.325

2.699 2.699 2.699 2.698 2.698 2.698 2.698 2.697 2.693 2.674 2.636 2.576 2.535

1.288 1.286 1.283 1.280 1.278 1.275 1.270 1.263 1.238 1.192 1.149 1.109 1.090

0.986 0.986 0.986 0.985 0.985 0.984 0.984 0.983 0.981 0.983 0.988 0.993 0.994

1.216 1.212 1.209 1.205 1.201 1.198 1.190 1.180 1.148 1.095 1.055 1.024 1.012

1.239 1.237 1.235 1.234 1.232 1.230 1.227 1.222 1.208 1.188 1.176 1.171 1.169

−1209. −1215. −1220. −1226. −1232. −1237. −1249. −1266. −1322. −1434. −1543. −1647. −1698.

1057. 1056. 1055. 1054. 1053. 1052. 1050. 1047. 1037. 1012. 976. 920. 880.

14.3 14.2 14.2 14.2 14.1 14.1 14.0 14.0 13.7 13.1 12.5 12.0 11.8

8.462 8.447 8.432 8.417 8.402 8.387 8.356 8.309 8.148 7.786 7.347 6.777 6.420

0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.9596 0.9547 0.9497 0.9448 0.9397 0.9347 0.9245 0.9090 0.8546 0.7287 0.5659 0.3387 0.1873

0. 0.0051 0.0103 0.0155 0.0208 0.0260 0.0367 0.0529 0.1096 0.2407 0.4102 0.6467 0.8045

0.0404 0.0402 0.0399 0.0397 0.0395 0.0393 0.0388 0.0381 0.0358 0.0305 0.0239 0.0146 0.0082

166.3 166.4 166.5 166.7 166.8 166.9 167.2 167.5 168.8 171.8 175.3 179.7 182.4

2.077 2.081 2.086 2.091 2.095 2.100 2.109 2.122 2.166 2.242 2.299 2.328 2.329

2.641 2.641 2.642 2.643 2.644 2.645 2.646 2.649 2.653 2.651 2.627 2.575 2.536

1.271 1.269 1.267 1.264 1.262 1.260 1.255 1.248 1.225 1.182 1.143 1.106 1.089

0.990 0.990 0.989 0.989 0.988 0.988 0.987 0.986 0.984 0.985 0.990 0.995 0.995

1.245 1.241 1.236 1.232 1.228 1.224 1.215 1.203 1.167 1.107 1.062 1.027 1.013

1.292 1.289 1.287 1.284 1.281 1.279 1.274 1.267 1.246 1.215 1.193 1.179 1.173

−1163. −1169. −1175. −1181. −1188. −1194. −1206. −1225. −1287. −1409. −1527. −1640. −1694.

1044. 1043. 1042. 1042. 1041. 1040. 1039. 1036. 1028. 1007. 975. 922. 882.

14.3 14.3 14.2 14.2 14.2 14.1 14.1 14.0 13.7 13.1 12.5 12.0 11.8

8.509 8.494 8.479 8.464 8.449 8.434 8.404 8.358 8.198 7.840 7.400 6.818 6.446

0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.10 0.20 0.40 0.60 0.80 0.90

0.9882 0.9834 0.9784 0.9735 0.9685 0.9635 0.9534 0.9380 0.9380 0.8840 0.7584 0.5940 0.3597 0.2003

0. 0.0050 0.0099 0.0149 0.0200 0.0250 0.0353 0.0509 0.0509 0.1056 0.2327 0.3990 0.6361 0.7972

0.0118 0.0117 0.0116 0.0116 0.0115 0.0114 0.0113 0.0111 0.0111 0.0104 0.0089 0.0070 0.0043 0.0024

165.2 165.3 165.5 165.6 165.7 165.8 166.1 166.5 166.5 167.9 170.9 174.6 179.3 182.1

2.062 2.067 2.072 2.077 2.082 2.087 2.097 2.111 2.111 2.158 2.240 2.302 2.332 2.331

2.598 2.599 2.601 2.602 2.604 2.605 2.608 2.612 2.612 2.624 2.633 2.620 2.574 2.537

1.260 1.257 1.255 1.253 1.251 1.248 1.244 1.237 1.237 1.216 1.176 1.138 1.104 1.088

0.995 0.994 0.994 0.993 0.992 0.992 0.991 0.989 0.989 0.987 0.986 0.991 0.996 0.996

1.269 1.264 1.259 1.254 1.250 1.245 1.236 1.222 1.222 1.182 1.116 1.067 1.030 1.014

1.334 1.330 1.327 1.324 1.321 1.317 1.311 1.302 1.302 1.276 1.235 1.206 1.185 1.176

−1130. −1136. −1143. −1149. −1156. −1163. −1176. −1195. −1195. −1261. −1390. −1516. −1635. −1692.

1035. 1035. 1034. 1033. 1033. 1032. 1031. 1029. 1029. 1022. 1003. 974. 923. 883.

14.3 14.3 14.2 14.2 14.2 14.2 14.1 14.0 14.0 13.7 13.1 12.5 12.0 11.8

8.541 8.527 8.512 8.497 8.482 8.467 8.437 8.391 8.391 8.233 7.877 7.436 6.846 6.464

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

1.0000 0.9951 0.9902 0.9853 0.9803 0.9753 0.9653 0.9499 0.8960 0.7706 0.6055 0.3684 0.2058

0. 0.0049 0.0098 0.0147 0.0197 0.0247 0.0347 0.0501 0.1040 0.2294 0.3945 0.6316 0.7942

0. 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0. 0. 0.

164.7 164.9 165.0 165.1 165.3 165.4 165.7 166.1 167.4 170.6 174.4 179.1 182.0

2.057 2.061 2.066 2.071 2.076 2.081 2.091 2.106 2.154 2.239 2.303 2.333 2.332

2.580 2.581 2.583 2.584 2.586 2.588 2.591 2.596 2.610 2.626 2.617 2.574 2.537

1.254 1.252 1.250 1.248 1.246 1.243 1.239 1.233 1.212 1.173 1.136 1.103 1.088

0.997 0.997 0.996 0.995 0.995 0.994 0.993 0.991 0.988 0.987 0.992 0.996 0.996

1.280 1.275 1.270 1.265 1.260 1.255 1.245 1.231 1.189 1.120 1.069 1.030 1.015

1.352 1.349 1.346 1.342 1.338 1.335 1.328 1.319 1.289 1.244 1.211 1.188 1.177

−1115. −1122. −1129. −1136. −1142. −1149. −1163. −1183. −1250. −1382. −1511. −1633. −1691.

1032. 1031. 1031. 1030. 1029. 1029. 1028. 1026. 1019. 1002. 973. 923. 884.

14.3 14.3 14.3 14.2 14.2 14.2 14.1 14.0 13.7 13.1 12.5 12.0 11.8

8.555 8.541 8.526 8.511 8.496 8.481 8.451 8.405 8.247 7.892 7.451 6.858 6.471

THERMODYNAMIC PROPERTIES TABLE 2-236

2-221

Thermodynamic Properties of the International Standard Atmosphere* T, K

P, bar

ρ, kg/m3

g, m/s2

M

a, m/s

µ, Pa⋅s

k, W/(m⋅K)

λ, m

0 1,000 2,000 3,000 4,000

288.15 281.65 275.15 268.66 262.17

1.01325 0.89876 0.79501 0.70121 0.61660

1.2250 1.1117 1.0066 0.90925 0.81935

9.80665 9.8036 9.8005 9.7974 9.7943

28.964 28.964 28.964 28.964 28.964

340.29 336.43 332.53 328.58 324.59

1.79.−5 1.76.−5 1.73.−5 1.69.−5 1.66.−5

2.54.−5 2.49.−5 2.43.−5 2.38.−5 2.33.−5

6.63.−8 7.31.−8 8.07.−8 8.94.−8 9.92.−8

0 1,000 2,999 2,999 3,997

5,000 6,000 7,000 8,000 9,000

255.68 249.19 242.70 236.22 229.73

0.54048 0.47217 0.41105 0.35651 0.30800

0.73643 0.66011 0.59002 0.52579 0.46706

9.7912 9.7882 9.7851 9.7820 9.7789

28.964 28.964 28.964 28.964 28.964

320.55 316.45 312.31 308.11 303.85

1.63.−5 1.59.−5 1.56.−5 1.53.−5 1.49.−5

2.28.−5 2.22.−5 2.17.−5 2.12.−5 2.06.−5

1.10.−7 1.23.−7 1.38.−7 1.55.−7 1.74.−7

4,996 5,994 6,992 7,990 8,987

10,000 15,000 20,000 25,000 30,000

223.25 216.65 216.65 221.55 226.51

0.26499 0.12111 0.05529 0.02549 0.01197

0.41351 0.19476 0.08891 0.04008 0.01841

9.7759 9.7605 9.7452 9.7300 9.7147

28.964 28.964 28.964 28.964 28.964

299.53 295.07 295.07 298.39 301.71

1.46.−5 1.42.−5 1.42.−5 1.45.−5 1.48.−5

2.01.−5 1.95.−5 1.95.−5 1.99.−5 2.04.−5

1.97.−7 4.17.−7 9.14.−7 2.03.−6 4.42.−6

9,984 14,965 19,937 24,902 29,859

40,000 50,000 60,000 70,000 80,000

250.35 270.65 247.02 219.59 198.64

2.87.−3 8.00.−4 2.20.−4 5.22.−5 1.05.−5

4.00.−3 1.03.−3 3.10.−4 8.28.−5 1.85.−5

9.6844 9.6542 9.6241 9.5942 9.5644

28.964 28.964 28.964 28.964 28.964

317.19 329.80 315.07 297.06 282.54

1.60.−5 1.70.−5 1.58.−5 1.44.−5 1.32.−5

2.23.−5 2.40.−5 2.21.−5 1.98.−5 1.80.−5

2.03.−5 7.91.−5 2.62.−4 9.81.−4 4.40.−3

39,750 49,610 59,439 69,238 79,006

90,000 100,000 150,000 200,000 250,000

186.87 195.08 634.39 854.56 941.33

1.84.−6 3.20.−7 4.54.−9 8.47.−10 2.48.−10

3.43.−6 5.60.−7 2.08.−9 2.54.−10 6.07.−11

9.5348 9.5052 9.3597 9.2175 9.0785

28.95 28.40 24.10 21.30 19.19

2.37.−2 0.142 33 240 890

88,744 98,451 146,542 193,899 240,540

300,000 400,000 500,000 600,000 800,000

976.01 995.83 999.24 999.85 999.99

8.77.−11 1.45.−11 3.02.−12 8.21.−13 1.70.−13

1.92.−11 2.80.−12 5.22.−13 1.14.−13 1.14.−14

8.9427 8.6799 8.4286 8.1880 7.7368

17.73 15.98 14.33 11.51 5.54

2600 1.6.+4 7.7.+4 2.8.+5 1.4.+6

286,480 376,320 463,540 548,252 710,574

1,000,000

1000.00

7.51.−14

3.56.−15

7.3218

3.94

3.1.+6

864,071

Z, m

H, m

*Extracted from U.S. Standard Atmosphere, 1976, National Oceanic and Atmospheric Administration, National Aeronautics and Space Administration and the U.S. Air Force, Washington, 1976. Z = geometric altitude, T = temperature, P = pressure, g = acceleration of gravity, M = molecular weight, a = velocity of sound, µ = viscosity, k = thermal conductivity, λ = mean free path, ρ = density, and H = geopotential altitude. The notation 1.79.−5 signifies 1.79 × 10−5. TABLE 2-237 T, K

Saturated Benzene*

P, bar

vf, m3/kg –3

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

290 300 310 320 330

0.0860 0.1382 0.2139 0.3206 0.4665

1.133.−10 1.147.−10–3 1.162.−10–3 1.176.−10–3 1.192.−10–3

3.569.−10 2.292.−10 1.525.−10 1.046.−10 7.379.−10–1

371.1 388.3 405.9 423.8 442.1

810.3 820.4 830.8 841.5 852.4

2.172 2.229 2.286 2.344 2.400

3.686 3.670 3.657 3.650 3.643

1.719 1.746 1.774 1.804 1.836

6.75 5.80 5.14 4.52 3.95

0.147 0.144 0.141 0.138 0.135

340 350 360 370 380

0.6615 0.9162 1.2419 1.6517 2.1588

1.207.−10–3 1.224.−10–3 1.241.−10–3 1.259.−10–3 1.277.−10–3

5.332.−10–1 3.938.−10–1 2.965.−10–1 2.233.−10–1 1.767.−10–1

460.8 479.6 498.7 518.1 537.7

863.6 875.0 886.7 898.6 910.6

2.455 2.510 2.564 2.617 2.669

3.641 3.641 3.642 3.646 3.651

1.868 1.890 1.920 1.950 1.989

3.55 3.23 2.99 2.72 2.46

0.132 0.129 0.126 0.123 0.120

390 400 410 420 430

2.7774 3.5228 4.4091 5.4540 6.6739

1.297.−10–3 1.318.−10–3 1.340.−10–3 1.363.−10–3 1.388.−10–3

1.393.−10–1 1.112.−10–1 8.972.−10–2 7.309.−10–2 6.003.−10–2

557.6 577.9 598.6 619.7 641.3

922.9 935.2 947.8 960.4 973.0

2.592 2.644 2.823 2.873 2.924

3.657 3.665 3.674 3.684 3.695

2.030 2.070 2.110 2.160 2.210

2.24 2.05 1.88 1.73 1.60

0.117 0.114 0.111 0.107 0.104

440 450 460 470 480

8.0861 9.7088 11.451 13.660 16.028

1.415.−10–3 1.444.−10–3 1.475.−10–3 1.510.−10–3 1.548.−10–3

4.965.−10–2 4.131.−10–2 3.455.−10–2 2.901.−10–2 2.441.−10–2

663.5 686.3 709.7 733.8 758.6

985.6 998.2 1010.7 1022.9 1034.9

2.974 3.025 3.075 3.126 3.179

3.706 3.718 3.730 3.742 3.753

2.260 2.320 2.380 2.450 2.519

1.48 1.37 1.28 1.10 1.12

0.101 0.098 0.095 0.092 0.089

490 500 510 520 530

18.685 21.651 24.952 28.613 32.669

1.591.−10–3 1.640.−10–3 1.697.−10–3 1.765.−10–3 1.849.−10–3

2.059.−10–2 1.736.−10–2 1.462.−10–2 1.226.−10–2 1.020.−10–2

784.3 810.9 838.5 867.2 897.2

1046.4 1057.3 1067.5 1076.6 1084.3

3.230 3.284 3.336 3.391 3.446

3.765 3.777 3.785 3.794 3.800

2.590 2.670 2.750 2.839 2.941

1.05 0.98 0.91 0.84 0.77

0.086 0.083

540 550 560 562.2

37.161 42.144 47.696 48.979

2.126.−10–3 2.258.−10–3 2.512.−10–3 3.290.−10–3

8.349.−10–3 6.616.−10–3 4.696.−10–3 3.290.−10–3

928.8 963.2 1007.3 1043.0

1089.5 1090.4 1077.6 1043.0

3.504 3.565 3.642 3.706

3.802 3.797 3.769 3.706

0.70 0.65 0.60

*Converted from a tabulation by Counsell, Lawrenson, and Lees, Nat. Phys. Lab. Teddington (U.K.) Rep. Chem. 52, 1976. Another tabulation by Kesselman et al., in Vargaftik (ed.), Tables on the Thermophysical Properties of Liquids and Gases, Hemisphere, Washington and London, 1975, shows some differences. The notation 1.133.−6 signifies 1.133 × 10−6. Other tables are given by Goodwin, R. D., J. Phys. Chem. Ref. Data, 17, 4 (1988): 1541–1636.

2-222

PHYSICAL AND CHEMICAL DATA

TABLE 2-238

Saturated Bromine* vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

260 280 300 320 340

0.042 0.124 0.310 0.680 1.330

3.106.−4 3.168.−4 3.232.−4 3.311.−4 3.385.−4

3.195 1.169 0.5002 0.2425 0.1309

−147.2 −138.9 −131.6 −124.2 −112.3

51.8 56.2 60.6 64.8 71.1

0.903 0.933 0.956 0.978 1.004

1.669 1.629 1.597 1.570 1.539

0.486 0.479 0.475 0.473 0.471

13.4 11.5 9.3 7.8 6.7

0.131 0.127 0.122 0.118 0.114

360 380 400 420 440

2.384 4.010 6.390 9.730 14.25

3.464.−4 3.550.−4 3.647.−4 3.752.−4 3.885.−4

0.0767 0.0477 0.0311 0.0211 0.0148

−108.6 −100.6 −93.4 −85.8 −77.7

73.1 76.9 80.6 84.0 87.1

1.026 1.048 1.063 1.084 1.103

1.531 1.515 1.501 1.488 1.477

0.470 0.471 0.475 0.480 0.489

5.7 5.0 4.5 4.0 3.7

0.109 0.104 0.099 0.094 0.089

460 480 500 520 540

20.17 27.75 37.21 48.81 62.80

4.023.−4 4.179.−4 4.378.−4 4.623.−4 4.938.−4

0.0107 0.00786 0.00589 0.00445 0.00337

−69.0 −59.7 −49.3 −37.7 −24.0

89.9 92.2 94.0 95.0 94.8

1.122 1.142 1.161 1.183 1.207

1.467 1.457 1.448 1.438 1.428

0.503 0.527 0.595 0.710 0.860

3.3 3.1 2.8 2.6 2.5

0.084 0.079 0.073 0.066 0.059

79.41 98.90 103.4

5.368.−4 6.250.−4 8.475.−4

0.00251 0.00167 0.00085

−7.1 18.8 64.8

92.5 82.5 64.8

1.237 1.280 1.356

1.414 1.390 1.356

1.063 2.31 ∞

2.3 2.2 2.1

0.050 0.035 ∞

T, K

P, bar

560 580 584.2c

*Reproduced or converted from a tabulation by Seshadri, Viswanath, and Kuloor, Ind. J. Technol., 6 (1970): 191–198. c = critical point.

TABLE 2-239

Saturated Normal Butane (R600)* hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

0.00 9.95 29.44 49.10 68.94

494.21 499.96 511.39 523.13 535.16

2.3056 2.3778 2.5121 2.6389 2.7592

5.9702 5.8779 5.7251 5.6016 5.5017

1.946 1.953 1.970 1.985 2.001

15.8 14.4 12.0 9.94 8.26

0.181 0.179 0.175 0.171 0.167

88.97 109.22 129.71 150.45 171.49

547.48 560.07 572.93 586.06 599.42

2.8738 2.9835 3.0887 3.1900 3.2879

5.4211 5.3564 5.3048 5.2643 5.2331

2.018 2.035 2.055 2.077 2.101

6.87 5.71 4.83 4.15 3.61

0.163 0.160 0.156 0.152 0.148

2.31 1.40 0.893 0.592 0.406

192.83 214.50 236.52 258.92 281.72

613.02 626.83 640.82 654.97 669.24

3.3828 3.4749 3.5647 3.6523 3.7380

5.2097 5.1929 5.1818 5.1755 5.1732

2.128 2.158 2.192 2.231 2.274

3.18 2.83 2.55 2.31 2.10

0.144 0.140 0.136 0.132 0.128

1.686.−3 1.718.−3 1.752.−3 1.790.−3 1.830.−3

0.286 0.207 0.1533 0.1156 0.0885

309.94 328.62 352.77 377.46 402.71

683.60 697.99 712.36 726.67 740.84

3.8220 3.9046 3.9860 4.0663 4.1458

5.1744 5.1783 5.1846 5.1928 5.2025

2.323 2.377 2.437 2.503 2.577

1.93 1.77 1.62 1.47 1.34

0.124 0.120 0.116 0.113 0.109

5.9179 7.5354 9.4573 11.72 14.35

1.874.−3 1.923.−3 1.978.−3 2.041.−3 2.114.−3

0.0687 0.0539 0.0427 0.0340 0.0272

428.61 455.25 482.74 511.22 540.88

754.80 768.49 781.79 794.60 806.72

4.2248 4.3035 4.3822 4.4613 4.5412

5.2132 5.2248 5.2367 5.2485 5.2597

2.657 2.746 2.842 2.947 3.062

1.21 1.08 0.97 0.87 0.78

0.105 0.101 0.097 0.093 0.089

380 390 400 410 420

17.40 20.90 24.92 29.54 34.86

2.200.−3 2.307.−3 2.447.−3 2.652.−3 3.048.−3

0.0218 0.0174 0.0138 0.0106 0.0075

571.94 604.76 639.85 678.30 723.89

817.86 827.56 834.95 838.10 830.34

4.6225 4.7058 4.7922 4.8842 4.9903

5.2696 5.2771 5.2800 5.2740 5.2437

3.20 3.34 3.50 3.69 3.84

0.69 0.62 0.55 0.49 0.44

0.085 0.081 0.077 0.074 0.072

425.2c

37.96

4.405.−3

0.0044

783.50

783.50

5.1290

5.1290

∞

P, bar

vf, m3/kg

134.9 140 150 160 170

6.7.−6 1.7.−5 8.7.−5 3.5.−4 1.17.−3

1.360.−3 1.369.−3 1.387.−3 1.405.−3 1.424.−3

180 190 200 210 220

3.37.−3 8.53.−3 1.94.−2 4.05.−2 7.81.−2

1.443.−3 1.463.−3 1.484.−3 1.505.−3 1.528.−3

230 240 250 260 270

0.1411 0.2408 0.3915 0.6100 0.9155

1.551.−3 1.575.−3 1.601.−3 1.628.−3 1.656.−3

280 290 300 310 320

1.3297 1.8765 2.5811 3.4706 4.5731

330 340 350 360 370

T, K t

vg, m3/kg 28630 11635 2470 654 207 76.4 31.8 14.7 7.39 4.00

∞

*Values rounded and reproduced or converted from Goodwin, NBSIR 79-1621, 1979. t = triple point; c = critical point. The notation 6.7.−6 signifies 6.7 × 10−6.

THERMODYNAMIC PROPERTIES TABLE 2-240

2-223

Superheated Normal Butane* Temperature, K

P, bar

150

200

250

300

350

400

450

500

600

700

v 1.013 h s

0.00139 29.6 2.512

0.00148 129.8 3.088

0.00160 236.6 3.564

0.4106 718.9 5.334

0.4847 810.7 5.616

0.5575 913.1 5.889

0.6297 1026.0 6.155

0.7013 1149.0 6.414

0.8440 1423 6.913

0.9861 1730 7.386

v 5h s

0.00139 30.0 2.511

0.00148 130.2 3.088

0.00160 237.0 3.563

0.00175 352.9 3.985

0.0909 798.5 5.363

0.1078 904.3 5.645

0.1238 1019.3 5.916

0.1393 1143.7 6.178

0.1693 1420 6.680

0.1988 1728 7.155

v 10 h s

0.00139 30.6 2.510

0.00148 130.8 3.087

0.00160 237.4 3.562

0.00175 353.3 3.983

0.00198 482.7 4.382

0.0502 891.9 5.524

0.0593 1010.3 5.803

0.0677 1136.8 6.069

0.0835 1415 6.575

0.0987 1725 7.052

v 20 h s

0.00138 31.7 2.509

0.00148 131.8 3.085

0.00160 238.4 3.560

0.00174 354.0 3.980

0.00196 482.6 4.376

0.0205 860.0 5.364

0.0268 990.1 5.670

0.0318 1122.0 5.948

0.0406 1406 6.464

0.0487 1718 6.945

v 30 h s

0.00138 32.8 2.507

0.00148 132.9 3.082

0.00159 239.3 3.557

0.00174 354.7 3.976

0.00195 482.6 4.370

0.00240 637.3 4.783

0.0156 965.5 5.570

0.0198 1105.9 5.866

0.0263 1396 6.394

0.0320 1711 6.880

v 40 h s

0.00138 33.9 2.505

0.00148 134.0 3.080

0.00159 240.3 3.555

0.00173 355.4 3.973

0.00194 482.7 4.365

0.00234 633.6 4.768

0.0097 932.2 5.468

0.0137 1088.1 5.797

0.0192 1387 6.341

0.0237 1705 6.832

v 50 h s

0.00138 35.0 2.503

0.00148 135.0 3.078

0.00159 241.3 3.552

0.00173 356.2 3.970

0.00193 428.8 4.360

0.00229 631.0 4.755

0.00549 877.0 5.329

0.0101 1068.2 5.734

0.0149 1377 6.297

0.0188 1699 6.792

v 60 h s

0.00138 36.2 2.501

0.00148 136.1 3.076

0.00159 242.3 3.550

0.00172 356.9 3.967

0.00192 483.1 4.355

0.00255 629.1 4.745

0.00352 825.1 5.204

0.00764 1046.4 5.673

0.0121 1367 6.258

0.0155 1692 6.759

v 80 h s

0.00138 38.4 2.498

0.00147 138.3 3.072

0.00158 244.2 3.545

0.00172 358.5 3.960

0.00190 483.7 4.346

0.00219 626.5 4.727

0.00286 798.1 5.130

0.00482 1001.5 5.559

0.00868 1347 6.191

0.0114 1680 6.704

v 100 h s

0.00138 40.6 2.495

0.00147 140.4 3.069

0.00158 246.2 3.540

0.00171 360.1 3.954

0.00188 484.5 4.337

0.00214 624.9 4.712

0.00264 787.9 5.095

0.00368 971.3 5.310

0.00669 1329 6.134

0.00901 1668 6.658

v 200 h s

0.00137 51.9 2.478

0.00146 151.3 3.049

0.00156 257.9 3.518

0.00167 368.8 3.927

0.00178 490.3 4.301

0.00200 624.4 4.660

0.00225 773.3 5.010

0.00258 933.7 5.348

0.00349 1270 5.960

0.00460 1623 6.849

v 300 h s

0.00136 63.2 2.462

0.00145 162.2 3.032

0.00154 266.7 3.498

0.00164 378.3 3.903

0.00176 498.0 4.273

0.00191 629.2 4.623

0.00209 773.4 4.962

0.00231 928.4 5.288

0.00284 1255 5.884

0.00345 1603 6.419

v 400 h s

0.00136 74.5 2.447

0.00144 173.3 3.015

0.00152 277.4 3.479

0.00162 388.2 3.882

0.00173 506.8 4.248

0.00185 636.2 4.593

0.00200 778.0 4.927

0.00217 930.2 5.247

0.00255 1253 5.836

0.00298 1600 6.366

v 500 h s

0.00136 85.8 2.432

0.00143 184.4 2.999

0.00151 288.1 3.461

0.00160 398.4 3.863

0.00170 516.3 4.226

0.00181 644.5 4.569

0.00193 784.8 4.898

0.00207 935.3 5.215

0.00240 1256 5.799

0.00272 1599 6.328

*Converted and rounded from tables of Goodwin, NBSIR 79-1621, 1979. Saturation and superheat tables and a diagram to 100 bar, 580 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For material to 10,000 psia, 640°F, see Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.

2-224

PHYSICAL AND CHEMICAL DATA

TABLE 2-241

Saturated Carbon Dioxide*

T, K

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

216.6 220 225 230 235

5.180 5.996 7.357 8.935 10.75

8.484.−4 8.574.−4 8.710.−4 8.856.−4 9.011.−4

0.0712 0.0624 0.0515 0.0428 0.0357

386.3 392.6 401.8 411.1 420.5

731.5 733.1 735.1 736.7 737.9

2.656 2.684 2.723 2.763 2.802

4.250 4.232 4.204 4.178 4.152

1.707 1.761

240 245 250 255 260

12.83 15.19 17.86 20.85 24.19

9.178.−4 9.358.−4 9.554.−4 9.768.−4 1.000.−3

0.0300 0.0253 0.0214 0.0182 0.0155

430.2 440.1 450.3 460.8 471.6

738.9 739.4 739.6 739.4 738.7

2.842 2.882 2.923 2.964 3.005

4.128 4.103 4.079 4.056 4.032

270 275 280 290 300

32.03 36.59 41.60 53.15 67.10

1.056.−3 1.091.−3 1.130.−3 1.241.−3 1.470.−3

0.0113 0.0097 0.0082 0.0058 0.0037

494.4 506.5 519.2 547.6 585.4

735.6 732.8 729.1 716.9 690.2

3.089 3.132 3.176 3.271 3.393

3.981 3.954 3.925 3.854 3.742

304.2c

73.83

2.145.−3

0.0021

636.6

636.6

3.558

3.558

1.879

µf, 10−4 Pa⋅s

1.64

kf, W/(m⋅K) 0.182 0.178 0.171 0.164 0.160

1.933

1.45

1.992

1.28

0.156 0.148 0.140 0.134 0.128

2.125

1.14

2.410

1.02

2.887 3.724

0.91 0.79 0.60

0.116 0.109 0.102 0.088 0.074

∞

0.31

∞

*c = critical point. The notation 8.484.−4 signifies 8.484 × 10−4. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives material for integral degrees Celsius with temperatures on the LPTS 68 scale for saturation temperatures from −56.57 to 30.98 degrees Celsius. The thermodynamic diagram from 4 to 1000 bar extends to 420°C. Saturation and superheat tables and a chart to 15,000 psia, 840°F appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see ASHRAE Thermophysical Properties of Refrigerants, 1993. Saturation and superheat tables and a diagram to 200 bar, 1000 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). Holste, J. C., D. M. Bailey, et al., Energy Progr., 6, 2 (1986): 125–130, give properties mainly in the range 0–100 bar, 200–450 K for the superheated vapor. Compare these with Angus, S., B. Armstrong, et al., International Tables of the Fluid State—Carbon Dioxide, Pergamon, Oxford, 1976 (377 pp.). In Miller, C. E. III and S. E. Wilder, NASA SP 3097, 1976 (489 pp.), many properties and decomposition products are tabulated for pressures from 10−7 to 104 atm., 100–25,000 K. For the range to 50 kb, 400–2100 K, see Bottinga, Y. and P. Richet, Amer. J. Sci., 281 (1981): 615–660.

THERMODYNAMIC PROPERTIES TABLE 2-242

2-225

Superheated Carbon Dioxide* Temperature, K

P, bar

300

350

400

450

500

600

700

800

900

1000

v 1h s

0.5639 809.3 4.860

0.6595 853.1 4.996

0.7543 899.1 5.118

0.8494 947.1 5.231

0.9439 997.0 5.337

1.1333 1102 5.527

1.3324 1212 5.697

1.5115 1327 5.850

1.7005 1445 5.990

1.8894 1567 6.120

v 5h s

0.1106 805.5 4.548

0.1304 850.3 4.686

0.1498 897.0 4.810

0.1691 945.5 4.925

0.1882 995.8 50.31

0.2264 1101 5.222

0.2645 1211 5.392

0.3024 1326 5.546

0.3403 1445 5.685

0.3782 1567 5.814

v 10 h s

0.0539 800.7 4.405

0.0642 846.9 4.548

0.0742 894.4 4.674

0.0841 943.5 4.790

0.0938 994.1 4.897

0.1131 1100 5.089

0.1322 1211 5.260

0.1513 1326 5.414

0.1703 1445 5.555

0.1893 1567 5.683

v 20 h s

0.0255 790.2 4.249

0.0311 839.8 4.402

0.0364 889.3 4.534

0.0416 939.4 4.653

0.0466 990.8 4.762

0.0564 1098 4.955

0.0661 1209 5.127

0.0757 1325 5.282

0.0853 1444 5.423

0.0948 1567 5.551

v 30 h s

0.0159 778.5 4.144

0.0201 832.4 4.341

0.0238 883.8 4.447

0.0274 935.2 4.569

0.0309 987.3 4.679

0.0375 1096 4.876

0.0441 1208 5.049

0.0505 1324 5.204

0.0570 1444 5.346

0.0633 1566 5.474

v 40 h s

0.0110 764.9 4.055

0.0146 824.6 4.239

0.0175 878.3 4.380

0.0203 931.1 4.507

0.0230 984.3 4.619

0.0281 1094 4.818

0.0331 1205 4.993

0.0379 1323 5.148

0.0428 1443 5.291

0.0476 1566 5.419

v 50 h s

0.0080 748.2 3.968

0.0112 816.3 4.179

0.0138 872.6 4.330

0.0161 926.9 4.457

0.0183 981.1 4.572

0.0224 1091 4.773

0.0265 1205 4.948

0.0304 1322 5.104

0.0343 1443 5.247

0.0382 1566 5.377

v 60 h s

0.0058 726.9 3.878

0.0090 807.7 4.126

0.0113 866.9 4.314

0.0133 922.7 4.416

0.0151 977.8 4.532

0.0187 1089 4.736

0.0221 1204 4.912

0.0254 1321 5.069

0.0286 1442 5.212

0.0318 1565 5.341

v 80 h s

0.0062 788.4 4.029

0.0081 855.1 4.208

0.0097 914.2 4.347

0.0112 971.3 4.468

0.0140 1085 4.675

0.0166 1201 4.854

0.0191 1320 5.011

0.0216 1441 5.155

0.0240 1565 5.286

v 100 h s

0.0045 766.2 3.936

0.0062 843.0 4.144

0.0076 905.7 4.290

0.0089 964.9 4.417

0.0111 1081 4.627

0.0133 1198 4.808

0.0153 1318 4.967

0.0173 1440 5.111

0.0193 1564 5.241

v 150 h s

0.0023 704.5 3.716

0.0038 811.9 4.005

0.0049 884.8 4.177

0.0058 949.4 4.313

0.0074 1072 4.536

0.0089 1192 4.722

0.0103 1314 4.884

0.0117 1437 5.030

0.0130 1562 5.162

v 200 h s

0.0017 670.0 3.591

0.0027 783.2 3.894

0.0035 865.2 4.088

0.0043 934.9 4.234

0.0056 1063 4.468

0.0067 1186 4.668

0.0078 1310 4.824

0.0088 1435 4.970

0.0099 1561 5.104

v 300 h s

0.0017 745.3 3.747

0.0023 834.0 3.956

0.0029 910.6 4.118

0.0038 1047 4.367

0.0046 1176 4.573

0.0053 1303 4.743

0.0060 1431 4.886

0.0067 1559 5.021

v 400 h s

0.0015 728.1 3.663

0.0018 814.6 3.867

0.0022 893.3 4.033

0.0029 1035 4.292

0.0035 1168 4.497

0.0041 1298 4.671

0.0047 1428 4.824

0.0052 1558 4.960

0.0016 803.5 3.805

0.0018 881.9 3.970

0.0024 1027 4.234

0.0029 1162 4.443

0.0034 1294 4.620

0.0038 1426 4.774

0.0043 1557 4.913

v 500 h s

*Interpolated and rounded from Vukalovich and Altunin, Thermophysical Properties of Carbon Dioxide, Atomizdat, Moscow, 1965; and Collett, England, 1968. TABLE 2-243

Saturated Carbon Monoxide*

T, K

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

81.62 83.36 88.25 96.16 101.51

1.01 1.52 2.03 4.05 6.08

1.268.−3 1.295.−3 1.317.−3 1.385.−3 1.440.−3

0.0666 0.0631 0.0606 0.0547 0.0513

150.25 158.56 165.00 182.76 195.0

365.30 368.07 370.00 374.21 375.98

3.005 3.104 3.178 3.368 3.489

5.640 5.559 5.501 5.359 5.271

105.69 109.17 116.08 121.48 125.97

8.12 10.13 15.20 20.27 25.33

1.489.−3 1.535.−3 1.651.−3 1.778.−3 1.936.−3

0.0318 0.0253 0.0163 0.0116 0.0085

204.8 213.2 231.0 246.3 261.2

376.6 376.6 374.5 370.2 363.6

3.580 3.656 3.807 3.918 4.041

5.206 5.152 5.043 4.948 4.854

129.84 132.91c

30.40 34.96

2.168.−3 3.337.−3

0.0063 0.0033

277.6

313.15

4.161

4.747

*Pressure and volume values converted, and enthalpy and entropy values reproduced, from Hust and Stewart, NBS Tech. Note 202, 1963. This source gives values at and above 72.373 K at closer pressure intervals. c = critical point. The notation 1.268.−3 signifies 1.268 × 10−3. Goodwin, R. D., J. Phys. Chem. Ref. Data, 14, 4 (1985): 849–932, gives properties to 1000 bar, 68–1000 K.

2-226

PHYSICAL AND CHEMICAL DATA

FIG. 2-8 Temperature-entropy diagram for carbon monoxide. Pressure P, in atmospheres; density ρ, in grams per cubic centimeter; enthalpy H, in joules per gram. (From Hust and Stewart, NBS Tech. Note 202, 1963.)

THERMODYNAMIC PROPERTIES TABLE 2-244

2-227

Thermophysical Properties of Saturated Carbon Tetrachloride hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf , 10−6 Pa·s

kf , W/(m⋅K)

Pr

280 290 300 310 320

0.064 0.105 0.165 0.251 0.370

0.000 0.000 0.000 0.000 0.000

619 625 633 641 649

2.414 1.495 0.971 0.669 0.463

205.5 212.9 220.9 228.8 236.9

420.7 425.7 430.9 436.1 441.3

1.018 1.042 1.068 1.095 1.121

1.787 1.775 1.768 1.764 1.760

0.835 0.844 0.853 0.863 0.874

1042 892 774 679 603

0.1043 0.1020 0.0998 0.0975 0.0952

8.34 7.38 6.62 6.01 5.54

330 340 350 360 370

0.531 0.743 1.017 1.361 1.795

0.000 0.000 0.000 0.000 0.000

657 666 674 684 694

0.3306 0.2407 0.1802 0.1370 0.1053

246.0 254.5 263.1 271.8 280.8

446.4 451.5 456.6 461.7 466.6

1.149 1.174 1.199 1.224 1.248

1.756 1.754 1.752 1.751 1.751

0.885 0.897 0.910 0.924 0.939

539 486 441 402 368

0.0930 0.0907 0.0884 0.0861 0.0839

5.13 4.81 4.54 4.31 4.12

380 390 400 410 420

2.327 2.970 3.735 4.642 5.700

0.000 0.000 0.000 0.000 0.000

704 715 727 739 753

0.0820 0.0651 0.0525 0.0426 0.0350

289.7 298.1 307.9 317.1 326.0

471.5 475.8 481.2 485.8 490.4

1.272 1.295 1.319 1.341 1.363

1.750 1.751 1.752 1.753 1.754

0.954 0.970 0.987 1.010 1.034

338 311 287 265 246

0.0816 0.0794 0.0771 0.0749 0.0726

3.95 3.80 3.67 3.57 3.50

430 440 450 460 470

6.927 8.342 9.958 11.792 13.869

0.000 0.000 0.000 0.000 0.000

766 780 796 801 834

0.02899 0.02413 0.02020 0.01692 0.01425

335.2 344.3 353.6 363.1 372.8

494.9 499.2 503.4 507.3 511.1

1.384 1.405 1.426 1.446 1.467

1.756 1.757 1.759 1.760 1.761

1.060 1.094 1.141 1.207 1.240

227 211 195 180 167

0.0704 0.0682 0.0660 0.0638 0.0666

3.42 3.38 3.37 3.36 3.36

480 490 500 510 520

16.21 18.83 21.77 25.02 28.68

0.000 0.000 0.000 0.000 0.000

856 880 858 945 987

0.01205 0.01011 0.00858 0.00722 0.00607

382.6 392.0 402.5 412.9 424.3

514.6 517.5 520.2 522.6 524.2

1.487 1.507 1.526 1.546 1.568

1.762 1.763 1.762 1.761 1.760

1.278 1.320 1.375 1.44 1.52

156 145 133

0.0594 0.0511 0.0549

3.36 3.35 3.35

530 540 550 556.4c

32.71 37.18 44.12 45.60

0.001 0.001 0.001 0.001

041 121 248 792

0.00500 0.00400 0.00309 0.00179

436.4 448.3 463.4 494.4

524.5 522.7 518.2 494.4

1.590 1.614 1.638 1.692

1.756 1.749 1.738 1.692

T, K

vf, m3/kg

P, bar

vg, m3/kg

c = critical point. Base points: hf = 200 at 273.15 K = 0°C = hA − 300 kJ/kg; sf = 1.000 at 273.15 K = 0°C = sA − 4.000 kJ/(kg⋅K). Values mostly rounded and converted from Altunin, V. V., V. Z. Geller, et al., Thermophysical Properties of Freons, vol. 9, Hemisphere, Washington, DC, 1987 (243 pp.). Some irregularities exist in these data.

TABLE 2-245

Saturated Carbon Tetrafluoride (R14)*

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

100 110 120 130 140

0.0089 0.0286 0.0924 0.2986 0.6901

5.370.−4 5.515.−4 5.668.−4 5.834.−4 6.018.−4

10.77 3.648 1.228 0.4051 0.1855

495.8 502.7 510.4 518.8 527.7

648.4 652.9 657.1 661.1 664.8

5.487 5.556 5.624 5.691 5.757

7.003 6.919 6.847 6.786 6.736

0.887 0.887 0.890 0.896 0.904

3.56

0.136 0.128 0.119 0.111 0.104

150 160 170 180 190

1.4074 2.598 4.426 7.067 10.702

6.225.−4 6.460.−4 6.733.−4 7.055.−4 7.449.−4

0.0951 0.0532 0.0318 0.0200 0.0131

537.2 549.4 557.6 568.2 579.3

668.3 671.4 674.0 676.1 677.4

5.822 5.885 5.947 6.007 6.066

6.696 6.662 6.629 6.607 6.583

0.922 0.975 1.031 1.104 1.203

3.28 3.03 2.80 2.59 2.39

0.097 0.089 0.081 0.072 0.064

200 210 220 227.5c

15.531 21.794 29.269 37.45

7.957.−4 8.674.−4 9.931.−4 1.598.−3

0.0087 0.0058 0.0036 0.0016

591.0 603.5 618.5 646.9

677.8 676.4 671.4 646.9

6.124 6.182 6.233 6.371

6.558 6.536 6.490 6.371

1.334 1.506 1.73 ∞

2.19 2.01 1.85

0.057 0.049 0.042 ∞

T, K

*P, v, h, and s values interpolated, extrapolated, and converted from Oguchi, Reito, 52 (1977): 869–889. c = critical point. The notation 5.370.−4 signifies 5.370 × 10−4. Equations and constants approximated to ASHRAE tables are given by Mecaryk, K. and M. Masaryk, Heat Recovery Systems and CHP, 11, 2–3 (1991). The 1993 ASHRAE Handbook—Fundamentals (S.I. ed.) contains a saturation table from −140 to −45.65°C and an enthalpy–log-pressure diagram from 0.1 to 300 bar, −140 to 300°C. For properties to 1000 bar from 90 to 420 K, see Rublo, R. G., J. A. Zollweg, et al., J. Chem. Eng. Data, 36 (1991): 171–184. Saturation and superheat tables and a diagram to 80 bar, 600 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). Chari, Ph.D. thesis, University of Michigan, 1960, presents saturation-temperature tables in fps units for 1°F increments from −270 to −51°F. Thermodynamic and transport properties, equations, and computer code and tables at constant entropy from 89 to 845 K are given by Hunt, J. L. and Boney, L. R., NASA TN D-7181, 1973 (105 pp.), largely based upon the Chari data.

2-228

PHYSICAL AND CHEMICAL DATA

TABLE 2-246 T, K 301.6m 400 500 600 700 800 900 1000 1200 1500

Saturated Cesium* P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

2.66.−9 3.83.−6 3.11.−4 5.65.−3 0.0440

5.444.−4 5.615.−4 5.800.−4 5.999.−4 6.215.−4

7.01.+7 6.54.+4 1001 65.63 9.671

74.6 98.5 122.0 144.9 167.0

637.6 651.9 666.1 678.4 688.9

0.696 0.765 0.817 0.859 0.893

2.563 2.148 1.905 1.748 1.638

0.245 0.240 0.232 0.224 0.219

0.2029 0.6620 1.693 6.790 27.6

6.443.−4 6.689.−4 6.954.−4 7.628.−4 8.84.−4

188.7 210.6 233.2 281.1 358.8

698.3 707.3 716.4 736.1 772.2

0.922 0.975 0.972 1.015 1.072

1.559 1.500 1.455 1.394 1.345

0.217 0.222 0.231 0.248 0.275

2.353 0.796 0.335 0.097 0.029

*Converted from tables in Vargaftik, Tables of the Thermophysical Properties of Liquids and Gases, Nauka, Moscow, 1972, and Hemisphere, Washington, 1975. m = melting point. The notation 2.66.−9 signifies 2.66 × 109. Many of the Vargaftik values also appear in Ohse, R. W., Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Sci. Pubs., Oxford, 1985 (1020 pp.). This source contains superheat data. Saturation and superheat tables and a diagram to 30 bar, 1550 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For a Mollier diagram from 0.1 to 327 psia, 1300−2700°R, see Weatherford, W. D., J. C. Tyler, et al., WADD-TR-61-96, 1961. An extensive review of properties of the solid and the saturated liquid was given by Alcock, C. B., M. W. Chase, et al., J. Phys. Chem. Ref. Data, 23, 3 (1994): 385–497.

TABLE 2-247

Thermophysical Properties of Saturated Chlorine

T, °C

P, bar

vf , m3/kg

vg , m3/kg

hf , kJ/ kg

hg , kJ/kg

sf , kJ/(kg·K)

sg , kJ/(kg·K)

cpf , kJ/(kg·K)

cpg , kJ/(kg·K)

µf , 10−6 Pa· s

µg , 10−6 Pa·s

kf ,W/(m ·K)

kg,W/(m· K)

Prf

Prg

−50 −40 −30 −20 −10

0.475 0.773 1.203 1.802 2.608

0.000 0.000 0.000 0.000 0.000

623 634 645 656 668

0.5448 0.3481 0.2314 0.1593 0.1134

221.5 231.0 240.6 250.3 260.0

518.2 522.2 526.1 529.9 533.9

1.7650 1.8074 1.8480 1.8869 1.9243

3.0946 3.0562 3.0223 2.9921 2.9649

0.9454 0.9474 0.9496 0.9520 0.9547

0.476 0.484 0.497 0.513 0.532

565 520 483 452 422

10.3 10.8 11.4 11.9 12.4

0.1684 0.1650 0.1613 0.1573 0.1527

0.0061 0.0065 0.0069 0.0074 0.0078

3.17 2.99 2.85 2.74 2.64

0.809 0.815 0.820 0.826 0.841

0 10 20 30 40

3.664 5.014 6.702 8.774 11.27

0.000 0.000 0.000 0.000 0.000

681 695 710 726 744

0.0829 0.0619 0.0471 0.0364 0.0286

269.7 279.4 289.2 299.0 308.8

537.4 540.5 543.3 545.7 548.0

1.9604 1.9953 2.0291 2.0622 2.0946

2.9402 2.9177 2.8924 2.8777 2.8593

0.9579 0.9618 0.9667 0.9728 0.9816

0.554 0.579 0.607 0.638 0.674

393 368 348 333 318

13.0 13.5 14.1 14.7 15.2

0.1478 0.1427 0.1378 0.1327 0.1282

0.0083 0.0088 0.0093 0.0099 0.0104

2.55 2.48 2.45 2.44 2.43

0.864 0.888 0.918 0.950 0.985

50 60 70 80 90

14.25 17.76 21.85 26.65 32.17

0.000 0.000 0.000 0.000 0.000

763 784 808 834 865

0.02276 0.01827 0.01481 0.01202 0.00972

318.6 329.1 340.0 351.4 364.1

549.8 551.2 552.1 552.5 552.4

2.1264 2.1578 2.1892 2.2207 2.2528

2.8417 2.8245 2.8074 2.7900 2.7714

0.9968 1.022 1.054 1.124 1.253

0.720 0.786 0.885 1.017 1.205

304 290 278 267 256

15.8 16.4 17.1 17.9 18.7

0.1230 0.1171 0.1122 0.1050 0.0986

0.0110 0.0117 0.0126 0.0137 0.0149

2.46 2.53 2.61 2.85 3.26

1.034 1.107 1.201 1.331 1.510

100 110 120 130 140

38.44 45.54 53.57 62.68 72.84

0.000 0.000 0.001 0.001 0.001

901 956 016 121 335

0.00789 0.00639 0.00508 0.00392 0.00282

377.8 391.3 407.1 426.1 451.1

551.0 548.8 543.7 535.0 517.3

2.2860 2.3207 2.3590 2.4032 2.4595

2.7502 2.7317 2.7064 2.6733 2.6198

1.418 1.632 1.891

1.434 1.696 1.960

247 238 230

19.5 20.6 22.2

0.0916 0.0850 0.0775

0.0163 0.0178 0.0195

3.82 4.57 5.61

1.700 1.96 2.23

144c

77.10

0.001 77

0.00177

483.1

483.1

2.5365

2.5365

c = critical point. Values interpolated and converted from Martin, J. J., 1977 (private communication), and from Heat Exchanger Design Handbook, vol. 5, Hemisphere, Washington, DC, 1983. Values of Ziegler, Chem.-Ing.Tech., 22 (1950): 229, apparently were also used in Landolt-Bornstein, IVa, (1967): 238–239, and in Ullmans Enzyklopädie der technische Chemie, 9, Verlag Chemie, Weinheim, 1975 (317–372). 2-229

PHYSICAL AND CHEMICAL DATA 10 Chlorine

8 6

200°C 180

4

2.

6

160

140 2. 7

2 120

100°C 2. 8

10 8 80 6

9

2.

0 3.

4

Satu rate d va por

Pressure, bar

2-230

2

kJ

/kg

•

K

60

0

3. t En

p ro

y

40

20

1

3.

1 2

3.

0.8 0°C 0.6 –20

3.3

0.4

–40 3.4

0.2 –60

0.1

FIG. 2-9

500 Enthalpy–log-pressure diagram for chlorine.

3.5

550 Enthalpy, kJ/kg

600

THERMODYNAMIC PROPERTIES TABLE 2-248 T, K

2-231

Saturated Chloroform (R20) vf, m3/kg

P, bar

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg·K)

sg, kJ/(kg·K)

280 300 320 340 360

0.115 0.293 0.620 1.224 2.255

0.000 0.000 0.000 0.000 0.000

660 678 695 715 739

1.689 0.714 0.358 0.190 0.107

−46.0 −32.6 −13.4 5.2 23.3

219.5 230.6 241.1 252.1 263.0

−0.165 −0.105 −0.041 0.015 0.065

0.798 0.773 0.754 0.741 0.731

380 400 420 440 460

3.830 6.039 9.058 13.39 18.80

0.000 0.000 0.000 0.000 0.000

765 795 822 871 921

0.0653 0.0425 0.0288 0.0195 0.0137

41.7 61.4 82.8 106.1 131.6

273.7 284.2 294.2 303.6 311.2

0.114 0.165 0.217 0.270 0.325

480 500 520 530 536.6c

26.00 34.66 44.68 50.44 54.72

0.000 0.001 0.001 0.001 0.002

980 059 193 328 00

0.00962 0.00673 0.00467 0.00359 0.00200

157.4 186.2 219.6 242.7 284.1

316.5 320.8 321.3 315.7 284.1

0.380 0.436 0.499 0.540 0.602

cpf, kJ/(kg·K)

µf, 10−6 Pa·s

kf, W/(m·K)

Prf

1.03

748 587 468 381 319

0.120 0.114 0.109 0.103 0.095

3.35

0.725 0.722 0.721 0.719 0.716

1.07 1.11 1.15 1.21 1.32

273 237 206 177 155

0.0921 0.0863 0.0808 0.0750 0.0694

3.17 3.04 2.93 2.86 2.95

0.711 0.706 0.694 0.678 0.602

1.43 1.59

129.6 105.5 81.2 67.7

0.0641 0.0584 0.0518 0.0461

2.89 2.87

c = critical point. hf = sf = 0 at n.b.p., 334.5 K. P, v, h, and s interpolated from Altunin, V. V., V. Z. Geller, et al., Thermophysical Properties of Freons, U.S.S.R. N.S.R.D.S. series, vol. 9., Hemisphere.

TABLE 2-249

Saturated Decane*

P, bar

vf, m3/kg

243.5 260 280 300 320

0.00001 0.00006 0.00042 0.00197 0.00720

1.319.−3 1.334.−3 1.356.−3 1.381.−3 1.410.−3

340 360 380 400 420

0.02155 0.05522 0.1248 0.2549 0.4789

1.442.−3 1.478.−3 1.515.−3 1.552.−3 1.591.−3

440 447.3 460 480 500

0.8387 1.0133 1.3852 2.1745 3.2690

T, K m

vg, m3/kg

µf, 10−4 Pa·s

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg·K)

sg, kJ/(kg·K)

cpf, kJ/(kg·K)

418.1 452.7 495.3 539.0 584.0

812.5 836.3 866.9 899.2 933.2

2.561 2.699 2.856 3.007 3.153

4.092 4.120 4.158 4.200 4.246

2.119 2.109 2.155 2.217 2.286

8.883 3.763 1.750 0.892 0.490

631.1 680.1 730.7 782.0 835.6

968.9 1006.2 1045.0 1085.0 1126.2

3.303 3.443 3.581 3.712 3.842

4.296 4.350 4.408 4.469 4.534

5.2 4.16 3.52 2.98 2.54

1.632.−3 1.650.−3 1.682.−3 1.735.−3 1.797.−3

0.290 0.243 0.178 0.115 0.0759

889.6 909.4 944.5 1002.6 1062.7

1168.4 1184.0 1211.4 1255.2 1299.4

3.968 4.014 4.089 4.213 4.335

4.602 4.627 4.670 4.739 4.808

2.23 2.09

20750 3300. 443. 88.74 22.73

520 540 560 580 600

4.733 6.633 9.062 12.16 16.12

1.868.−3 1.952.−3 2.067.−3 2.255.−3 2.588.−3

0.0525 0.0369 0.0248 0.0154 0.0093

1124.5 1190.1 1256.1 1318.5 1384.5

1344.4 1389.5 1432.2 1468.1 1495.6

4.456 4.573 4.698 4.802 4.913

4.879 4.949 5.011 5.060 5.098

617.5c

20.97

4.238.−3

0.0042

1483.2

1483.2

5.073

5.073

kf, W/(m · K)

25.0 16.6 11.3 8.2 6.5

*Values converted from Das and Kuloor, Ind. J. Technol., 5 (1967): 75. m = melting point; c = critical point. The notation 1.319.−3 signifies 1.319 × 10−3.

0.149 0.144 0.139 0.134 0.129 0.124 0.119 0.116 0.110

2-232

PHYSICAL AND CHEMICAL DATA TABLE 2-250

Saturated Normal Deuterium*

T, K

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

18.71 19 20 21 22

0.1709 0.1944 0.2944 0.4297 0.6072

0.005752 0.005771 0.005840 0.005914 0.005993

2.232 1.988 1.365 0.968 0.705

−161.1 −160.0 −152.8 −145.9 −138.7

158.6 159.1 163.9 167.6 170.6

4.54 4.68 4.97 5.30 5.63

21.62 21.48 20.81 20.23 19.69

23 24 25 26 27

0.8344 1.1192 1.4694 1.8932 2.3989

0.00608 0.00617 0.00627 0.00638 0.00650

0.5256 0.3995 0.3088 0.2421 0.1921

−131.4 −123.8 −116.1 −108.2 −100.2

173.0 174.6 175.5 175.7 175.1

5.95 6.26 6.57 6.87 7.16

19.18 18.70 18.23 17.79 17.36

28 29 30 31 32

2.995 3.690 4.493 5.412 6.457

0.00663 0.00678 0.00694 0.00713 0.00735

0.1540 0.1246 0.1015 0.0831 0.0683

−92.0 −83.6 −74.9 −65.9 −56.5

173.8 171.7 168.7 165.0 160.3

7.44 7.72 8.00 8.27 8.54

16.94 16.52 16.12 15.72 15.32

33 34 35 36 37

7.455 8.962 10.44 12.09 13.91

0.00761 0.00793 0.00834 0.00890 0.00976

0.0563 0.0465 0.0382 0.0311 0.0249

−46.4 −35.5 −23.2 −8.6 10.0

154.7 148.0 140.0 130.1 117.1

8.83 9.12 9.45 9.82 10.28

14.92 14.52 14.11 13.67 13.17

38 38.34c

15.92 16.65

0.01158 0.01433

0.0185 0.0143

39.7 69.2

95.0 69.2

11.01 11.76

12.47 11.76

*Condensed and converted from tables of Prydz, NBS Rep. 9276, 1967. c = critical point. For equations and T-s and Z charts from 0.1 to 100 atm, 20–300 K, see also Prydz, R. and K. D. Timmerhaus, Advan. Cryog. Eng., 13 (1968): 384–396. TABLE 2-251

Saturated Deuterium Oxide*

P, bar

vf, m3/kg

vg, m3/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

277.0 278.2 283.2 288.2 293.2

0.00668 0.00720 0.01030 0.01449 0.02011

9.047.−4 9.045.−4 9.042.−4 9.043.−4 9.047.−4

172.2 160.4 114.1 82.48 60.45

0.0 5.0 25.9 46.9 67.8

2320.9 2322.5 2330.9 2339.3 2347.6

0.000 0.0188 0.0920 0.166 0.239

8.380 8.351 8.233 8.122 8.016

298.2 303.2 308.2 313.2 318.2

0.02758 0.03730 0.04990 0.06598 0.08638

9.054.−4 9.063.−4 9.075.−4 9.091.−4 9.108.−4

44.88 33.71 25.59 19.66 15.24

88.7 109.6 130.5 151.5 172.4

2356.0 2364.0 2372.3 2380.7 2388.6

0.311 0.382 0.450 0.518 0.585

7.915 7.818 7.725 7.637 7.550

323.2 333.2 353.2 373.2 398.2

0.1120 0.1831 0.4439 0.9646 2.2427

9.127.−4 9.170.−4 9.274.−4 9.403.−4 9.599.−4

11.93 7.52 3.27 1.58 0.72

193.3 234.7 318.4 402.0 507.5

2396.6 2413.3 2445.1 2474.8 2509.6

0.650 0.776 1.020 1.253 1.527

7.468 7.315 7.042 6.807 6.555

T, K t

423.2 448.2 473.2 498.2 523.2 548.2 573.2 598.2 623.2 644.7c

4.653 8.806 15.46 25.52 39.99 60.04 86.97 122.4 168.3 218.4

hf, kJ/kg

9.835.−4 1.012.−3 1.044.−3 1.082.−3 1.133.−3

0.362 0.198 0.115 0.0704 0.0447

612.5 718.8 826.8 938.5 1055.2

2541.8 2569.4 2585.7 2597.0 2598.7

1.781 2.020 2.256 2.483 2.707

6.341 6.149 5.973 5.812 5.658

1.200.−3 1.276.−3 1.392.−3 1.596.−3 2.950.−3

0.0290 0.0191 0.0124 0.0075 0.0030

1177.4 1306.7 1445.6 1607.1

2587.0 2555.6 2492.4 2366.5

2.930 3.153 3.356 3.631

5.501 5.332 5.132 4.850

*Extracted or converted from values in Kazavchinskii, Kesselman, et al., Thermophysical Properties of Heavy Water, Moscow and Leningrad, 1963; NBS-NSF transl. 70-50094, 1971. t = triple point; c = critical point. The notation 9.047.−4 signifies 9.047 × 10−4. Hill, P. G., MacMillan, R. D. and others give extensive tables for 0–1000 bar, 4–800°C in Atomic Energy of Canada, Chalk River rept. AECL-7531, 1981 (196 pp.). See also J. Phys. Chem. Ref. Data, 11, 1 (1982): 1–14; 19, 5 (1990): 1233–1274. TABLE 2-252

Deuterium Oxide Gas at 1-kg/cm3 Pressure

T, K

400

450

500

550

600

650

700

750

v, m3/ kg h, kJ/ kg s, kJ/(kg⋅K)

1.676 2525 6.931

1.895 2619 7.151

2.112 2712 7.349

2.322 2807 7.529

2.535 2904 7.697

2.747 3002 7.855

2.960 3102 8.003

3.172 3205 8.153

THERMODYNAMIC PROPERTIES TABLE 2-253

2-233

Saturated Diphenyl* hg, kJ/ kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

0.0 13.0 30.0 47.2 65.0

444.2 444.2 446.7 449.7 454.5

0.000 0.036 0.084 0.130 0.178

1.298 1.266 1.236 1.213 1.200

1.760 1.782 1.813 1.844 1.875

15.0 13.5 11.7 10.3 9.1

0.139 0.138 0.136 0.135 0.133

82.7 99.3 139.9 180.3 222.7

462.7 461.2 499.0 532.4 569.7

0.224 0.273 0.358 0.451 0.545

1.194 1.202 1.228 1.267 1.378

1.906 1.936 1.998 2.060 2.122

8.1 7.3 6.0 5.0 4.3

0.132 0.130 0.127 0.125 0.122

0.9594 0.4452 0.3652 0.2261 0.1447

267.6 314.9 361.5 404.5 457.2

611.6 651.8 687.8 723.8 762.7

0.652 0.746 0.824 0.915 1.032

1.367 1.424 1.477 1.529 1.582

2.184 2.246 2.308 2.370 2.432

3.7 3.3 2.7 2.4 2.2

0.119 0.116 0.113 0.110 0.107

1.258.−3 1.291.−3 1.326.−3 1.366.−3 1.412.−3

0.0977 0.0685 0.0504 0.0381 0.0301

522.3 563.7 630.4 689.1 745.9

801.7 842.4 886.4 930.9 977.1

1.125 1.223 1.316 1.375 1.457

1.635 1.688 1.740 1.748 1.791

2.494 2.556 2.618 2.680 2.741

1.90 1.71 1.54 1.39 1.24

0.105 0.102 0.099 0.096 0.093

2.803 2.865 2.93 3.00

1.10 0.97

0.090 0.087

T, K

P, bar

vf, m3/kg

vg, m3/kg

343 350 360 370 380

0.0010 0.0016 0.0029 0.0049 0.0064

1.010.−3 1.014.−3 1.021.−3 1.030.−3 1.037.−3

252.5 156.1 85.0 49.9 29.9

390 400 420 440 460

0.0129 0.0200 0.0432 0.0879 0.1694

1.046.−3 1.054.−3 1.072.−3 1.092.−3 1.112.−3

480 500 520 540 560

0.3112 0.5218 0.8375 1.290 1.941

1.132.−3 1.154.−3 1.177.−3 1.204.−3 1.230.−3

580 600 620 640 660

2.818 3.926 5.408 7.328 9.572

18.3 11.7 5.84 3.021 1.652

hf, kJ/kg

680 700 720 740 760

12.05 15.21 19.14 23.93 28.71

1.465.−3 1.529.−3 1.56.−3 1.70.−3 1.95.−3

0.0236 0.0186 0.0147 0.0113 0.0085

802.8 860.1 917.5 975.2 1033.1

1024.9 1073.1 1116.7 1152.8 1182.5

1.585 1.663 1.746 1.822 1.901

1.856 1.951 2.003 2.058 2.099

780 800

34.83 42.46

2.16.−3 3.18.−3

0.0058 0.0032

1091.2 1148.4

1163.0 1148.4

1.977 2.047

2.107 2.047

*Interpolated by P. E. Liley from the Landolt-Börnstein band IVa, p. 557, 1967 tables based on Technical Data on Fuel, British National Committee, World Energy Conference, London.

TABLE 2-254

Saturated Ethane (R170)* hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

176.8 198.7 221.5 244.4 267.4

769.4 782.4 795.0 807.2 819.3

2.560 2.790 3.008 3.207 3.391

9.113 8.627 8.222 7.897 7.637

2.260 2.274 2.284 2.292 2.302

14.19 9.37 6.57 4.89 3.81

0.215 0.208 0.201 0.194 0.187

10.08 4.263 2.039 1.075 0.6139

290.5 313.7 337.2 360.9 384.9

831.4 843.5 855.6 867.6 879.4

3.562 3.722 3.873 4.017 4.154

7.426 7.254 7.113 6.998 6.901

2.316 2.333 2.355 2.383 2.417

3.07 2.55 2.17 1.88 1.65

0.180 0.174 0.167 0.160 0.153

1.862 1.908 1.958 2.014 2.076

0.3738 0.2395 0.1602 0.1109 0.0789

409.3 434.2 459.7 485.9 512.8

890.8 901.7 911.9 921.4 929.6

4.285 4.412 4.535 4.655 4.773

6.819 6.750 6.689 6.635 6.585

2.458 2.508 2.568 2.640 2.730

1.47 1.33 1.21 1.11 1.03

0.147 0.140 0.133 0.126 0.119

9.670 13.01 17.12 22.10 28.06

2.148 2.231 2.330 2.452 2.613

0.0573 0.0423 0.0316 0.0237 0.0177

540.8 569.9 600.7 633.6 669.3

936.6 941.9 945.4 946.4 943.6

4.890 5.006 5.123 5.233 5.370

6.539 6.493 6.449 6.392 6.350

2.843 2.991 3.214 3.511 4.011

0.96 0.82 0.73 0.64 0.55

0.112 0.106 0.099 0.092 0.085

35.14 43.54 48.71

2.847 3.295 4.891

0.0129 0.0087 0.0048

709.8 761.6 841.2

934.7 910.8 841.2

5.502 5.669 5.919

6.278 6.166 5.919

5.089 9.919 ∞

0.44 0.31

0.078 0.067

P, bar

vf, m3/kg

90.4 100 110 120 130

1.131.−5 1.110.−4 7.467.−3 3.545.−3 1.291.−2

1.534.−3 1.546 1.573 1.615 1.644

140 150 160 170 180

3.831.−2 9.672.−2 0.2146 0.4290 0.7874

1.675 1.708 1.743 1.780 1.819

190 200 210 220 230

1.347 2.174 3.340 4.922 7.004

240 250 260 270 280 290 300 305.3c

T, K t

vg, m3/kg 21945 2484.5 407.0 93.61 27.83

*Values reproduced or converted from Goodwin, Roder, and Straty, NBS Tech. Note 684, 1976. t = triple point; c = critical point. The notation 1.131.−5 signifies 1.131 × 10−5.

2-234

PHYSICAL AND CHEMICAL DATA

TABLE 2-255

Superheated Ethane* Temperature, K

P, bar

100

150

200

250

300

350

400

450

500

600

700

v 1.013 h s

0.00156 198.9 2.790

0.00171 313.8 3.722

0.5310 909.3 6.993

0.6725 984.7 7.330

0.8118 1068.3 7.634

0.9500 1161.5 7.921

1.0877 1265.3 8.198

1.2250 1379.8 8.467

1.3622 1504.6 8.730

1.6360 1783 9.237

1.9096 2097 9.720

v 5h s

0.00156 199.4 2.789

0.00171 314.3 3.720

0.00191 434.5 4.411

0.1288 973.3 6.858

0.1595 1060.3 7.175

0.1890 1155.6 7.468

0.2178 1260.7 7.748

0.2464 1376.1 8.020

0.2747 1501.5 8.284

0.3308 1781 8.793

0.3867 2096 9.227

v 10 h s

0.00156 200.0 2.788

0.00171 314.9 3.719

0.00190 435.0 4.408

0.0590 956.5 6.618

0.0765 1050.0 6.959

0.0923 1148.2 7.262

0.1073 1255.0 7.547

0.1220 1371.5 7.821

0.1365 1497.9 8.087

0.1650 1777 8.598

0.1933 2094 9.083

v 20 h s

0.00156 201.3 2.785

0.00170 316.1 3.715

0.00190 435.9 4.404

0.00222 569.8 4.999

0.0346 1026.1 6.710

0.0438 1132.3 7.038

0.0521 1243.3 7.334

0.0599 1362.4 7.614

0.0674 1490.5 7.884

0.0822 1774 8.399

0.0966 2090 8.886

v 40 h s

0.00155 203.9 2.780

0.00170 318.5 3.709

0.00189 437.9 4.394

0.00219 569.9 4.982

0.0118 947.9 6.309

0.0193 1096.2 6.770

0.0244 1218.6 7.097

0.0288 1343.8 7.391

0.0329 1475.9 7.670

0.0407 1764 8.194

0.0482 2083 8.686

v 60 h s

0.00155 206.5 2.775

0.00170 321.0 3.702

0.00188 439.8 4.385

0.00217 570.3 4.966

0.00290 738.1 5.574

0.0109 1050.9 6.557

0.0132 1192.0 6.934

0.0185 1324.8 7.247

0.0215 1461.2 7.535

0.0270 1754 8.068

0.0321 2077 8.564

v 80 h s

0.00155 209.1 2.769

0.00169 323.4 3.696

0.00188 441.9 4.377

0.00215 570.9 4.951

0.00273 728.1 5.522

0.00667 993.8 6.345

0.0106 1163.6 6.800

0.0134 1305.5 7.135

0.0158 1446.7 7.432

0.0201 1745 7.975

0.0459 2070 8.476

v 100 h s

0.00155 211.7 2.764

0.00169 325.8 3.690

0.00187 443.9 4.368

0.00213 571.8 4.938

0.00263 722.7 5.486

0.00465 924.4 6.166

0.00791 1134.7 6.682

0.0104 1286.3 7.040

0.0124 1432.4 7.348

0.0160 1736 7.900

0.0193 2064 8.406

v 150 h s

0.00155 218.1 2.752

0.00168 332.0 3.674

0.00185 449.2 4.348

0.00209 574.6 4.907

0.00247 716.4 5.423

0.00328 887.4 5.955

0.00488 1075.2 6.457

0.00655 1242.3 6.851

0.00805 1399.3 7.182

0.0107 1715 7.758

0.0130 2050 8.274

v 200 h s

0.00154 224.6 2.738

0.00167 338.2 3.660

0.00184 454.7 4.329

0.00205 578.2 4.880

0.00237 714.8 5.377

0.00291 870.5 5.863

0.00383 1041.7 6.320

0.00495 1210.2 6.717

0.00605 1327.3 7.059

0.00806 1697 7.651

0.00986 2038 8.176

v 300 h s

0.00153 237.6 2.715

0.00166 350.6 3.632

0.00181 465.9 4.294

0.00200 586.8 4.833

0.00225 715.9 5.309

0.00259 860.9 5.757

0.00307 1014.9 6.168

0.00367 1175.5 6.547

0.00433 1338.7 6.891

0.00563 1671 7.496

0.00686 2019 8.032

v 400 h s

0.00153 250.6 2.692

0.00165 363.2 3.605

0.00179 477.6 4.262

0.00195 596.6 4.793

0.00216 723.7 5.257

0.00244 861.6 5.688

0.00276 1008.3 6.080

0.00316 62.5 6.443

0.00361 1322.7 6.780

0.00454 1657 7.388

0.00545 2008 7.930

v 500 h s

0.00152 263.5 2.670

0.00163 375.8 3.580

0.00176 489.3 4.234

0.00192 607.1 4.758

0.00210 732.0 5.213

0.00232 866.5 5.634

0.00258 1009.3 6.015

0.00288 1159.3 6.369

0.00322 1316.9 6.00

0.00392 1650 7.306

0.00465 2003 7.851

*Converted and rounded off from the tables of Goodwin, Roder, and Straty, NBS Tech. Note 684, 1976. v = specific volume, m3/kg; h = specific enthalpy, kJ/ kg; s = specific entropy, kJ/(kg⋅K). Saturation and superheat tables and a diagram to 300 bar, 580 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). Saturation and superheat tables and a chart to 10,000 psia, 640°F appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) contains a thermodynamic diagram from 0.1 to 700 bar for temperatures to 600 K.

THERMODYNAMIC PROPERTIES TABLE 2-256 T, K

P, bar

2-235

Saturated Ethanol vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−6 Pa·s

kf, W/(m·K)

Prf

250 260 270 280 290

0.0027 0.0059 0.0128 0.025 0.048

0.001 0.001 0.001 0.001 0.001

184 196 208 220 233

2.113 2.167 2.227 2.294 2.369

295 229 193 156 127

0.177 0.175 0.173 0.171 0.170

35.2 28.4 24.8 20.9 17.7

300 310 320 330 340

0.088 0.151 0.253 0.406 0.632

0.001 0.001 0.001 0.001 0.001

246 260 274 288 304

2.45 2.54 2.64 2.75 2.86

104 86 72 61 52

0.168 0.165 0.162 0.159 0.157

15.2 13.2 11.7 10.6 9.5

350 360 370 380 390

0.956 1.409 2.023 2.837 3.897

0.001 0.001 0.001 0.001 0.001

318 337 357 379 403

0.7656 0.5052 0.3555 0.2556 0.1873

199.9 230.1 262.2 295.1 329.1

1161.9 1178.4 1193.9 1208.4 1221.5

2.99 3.12 3.27 3.42 3.58

45.0 39.0 34.2 30.0 26.1

0.155 0.153 0.151 0.149 0.147

8.7 8.0 7.4 6.9 6.3

400 410 420 430 440

5.251 6.954 9.063 11.64 14.72

0.001 0.001 0.001 0.001 0.001

430 461 495 532 574

0.1398 0.1058 0.0812 0.0631 0.0493

364.2 400.8 435.7 472.2 512.7

1233.6 1244.2 1254.2 1262.3 1269.2

3.74 3.99 4.26 4.55 4.88

22.7 20.0 17.6 15.3 13.9

0.145 0.144 0.142 0.140 0.139

5.9 5.5 5.3 5.0 4.9

450 460 470 480 490

18.33 22.61 27.66 33.55 40.39

0.001 0.001 0.001 0.001 0.001

623 682 752 832 950

0.0389 0.0308 0.0243 0.0193 0.0148

557.2 605.0 653.7 704.5 757.7

1274.2 1275.5 1271.1 1262.3 1250.2

5.23

12.5

0.137

4.8

500 510 516.3c

48.28 57.32 63.90

0.002 091

0.0110

818.9

1232.7

c = critical point. Values interpolated and converted from Heat Exchanger Design Handbook, vol. 5, Hemisphere, Washington, DC, 1983, and from various literature sources.

FIG. 2-10 Enthalpy-concentration diagram for aqueous ethyl alcohol. Reference states: Enthalpies of liquid water and ethyl alcohol at 0°C are zero. NOTE: In order to interpolate equilibrium compositions, a vertical may be erected from any liquid composition on the boiling line and its intersection with the auxiliary line determined. A horizontal from this intersection will establish the equilibrium vapor composition on the dew line. (Bosnjakovic, Technische Thermodynamik, T. Steinkopff, Leipzig, 1935.)

2-236

PHYSICAL AND CHEMICAL DATA

TABLE 2-257

Saturated Ethylene (Ethene—R1150)

Temperature, K

Pressure, bar

104.0t 110 120 130 140

0.00123 0.00334 0.01380 0.04456 0.1191

0.001 0.001 0.001 0.001 0.001

527 545 576 609 644

150 160 170 180 190

0.2747 0.5636 1.0526 1.8207 2.9574

0.001 0.001 0.001 0.001 0.001

681 721 763 810 861

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

251.36 97.57 25.75 8.62 3.46

−323.81 −309.54 −284.17 −259.13 −234.80

244.36 251.47 263.23 274.87 286.28

−1.9901 −1.8571 −1.6362 −1.4358 −1.2554

3.4730 3.2431 2.9255 2.6717 2.4663

2.497 2.500 2.539 2.465 2.405

1.5977 0.8232 0.4625 0.2784 0.1770

−210.90 −187.12 −163.23 −139.05 −114.46

297.37 308.00 318.04 327.35 335.79

−1.0908 −0.9378 −0.7935 −0.6559 −0.5244

2.2977 2.1566 2.0375 1.9352 1.7812

2.377 2.377 2.395 2.427 2.472

200 210 220 230 240

4.560 6.730 9.575 13.206 17.742

0.001 0.001 0.002 0.002 0.002

918 981 054 139 241

0.1177 0.0810 0.0573 0.0413 0.0302

−89.33 −63.52 −36.84 −9.04 20.23

343.21 349.41 354.18 357.17 357.90

−0.3967 −0.2730 −0.1515 −0.0314 0.0088

1.7659 1.6932 1.6258 1.5609 1.4957

2.531 2.608 2.711 2.852 3.055

250 260 270 280 282.3c

23.307 30.046 38.132 47.834 50.403

0.002 0.002 0.002 0.003 0.004

369 541 804 442 669

0.02222 0.01624 0.01152 0.00720 0.00467

51.55 85.91 125.79 183.40 234.55

355.37 348.68 333.71 292.83 234.55

0.2114 0.3397 0.4819 0.6803 0.8585

1.4276 1.3503 1.3054 1.0711 0.8585

3.372 3.945 5.40 20.0

t = triple point; c = critical point. hf = sf = 0 at 233.15 K = −40°C. Converted from Jacobsen, R. T., M. Jahangiri, et al., Ethylene—Intl. Thermodyn. Tables of the Fluid State—10, Blackwell Sci. Publ., Oxford, U.K., 1988 (299 pp.). Saturation and superheat tables and a diagram to 100 bar, 460 K are given by Reynolds, W. C., Thermodynamic properties in S.I., Stanford Univ. publ., 1979 (173 pp.). Saturation and superheat tables and a chart to 6000 psia, 360°F appear in Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) has a thermodynamic chart for pressures from 0.1 to 400 bar and temperatures up to 460 K.

THERMODYNAMIC PROPERTIES TABLE 2-258

Compressed Ethylene Temperature, K

Pressure, bar

110

125

150

175

200

225

250

275

v (m3/kg) 1 h (kJ/kg) s (kJ/kg⋅K)

0.001 545 −309.4 −1.858

0.001 592 −271.4 −1.534

0.001 681 −210.8 −1.091

0.5036 324.8 2.091

0.5814 357.0 2.264

0.6580 389.9 2.419

0.7337 424.0 2.562

0.8091 459.7 2.698

v (m3/kg) 5 h (kJ/kg) s (kJ/kg⋅K)

0.001 544 −308.9 −1.859

0.001 591 −271.0 −1.535

0.001 680 −210.4 −1.093

0.001 785 −150.8 −0.726

0.001 917 −89.3 −0.397

0.1240 378.4 1.907

0.1407 415.0 2.061

0.1569 452.3 2.203

v (m3/kg) 10 h (kJ/kg) s (kJ/kg⋅K)

0.001 543 −308.3 −1.860

0.001 591 −270.4 −1.537

0.001 679 −209.8 −1.095

0.001 783 −150.3 −0.728

0.001 914 −89.0 −0.400

0.05643 361.2 1.646

0.06672 402.4 1.820

0.07525 442.3 1.973

v (m3/kg) 20 h (kJ/kg) s (kJ/kg⋅K)

0.001 542 −307.1 −1.863

0.001 589 −269.2 −1.540

0.001 676 −208.7 −1.098

0.001 780 −149.4 −0.733

0.001 908 −88.2 −0.406

0.002 084 −23.0 −0.099

0.02810 370.3 1.520

0.03405 419.7 1.708

v (m3/kg) 30 h (kJ/kg) s (kJ/kg⋅K)

0.001 541 −305.9 −1.866

0.001 588 −268.0 −1.543

0.001 674 −207.6 −1.102

0.001 776 −148.4 −0.737

0.001 903 −87.5 −0.412

0.002 072 −22.8 −0.107

0.002 347 50.5 0.201

0.01978 390.7 1.508

v (m3/kg) 40 h (kJ/kg) s (kJ/kg⋅K)

0.001 540 −304.7 −1.869

0.001 587 −266.8 −1.546

0.001 672 −206.5 −1.106

0.001 773 −147.4 −0.741

0.001 897 −86.7 −0.418

0.002 062 −22.5 −0.115

0.002 318 49.1 0.186

0.01163 344.7 1.284

v (m3/kg) 50 h (kJ/kg) s (kJ/kg⋅K)

0.001 539 −303.5 −1.872

0.001 585 −265.7 −1.550

0.001 670 −205.4 −1.110

0.001 770 −146.4 −0.746

0.001 892 −85.9 −0.423

0.002 052 −22.2 −0.123

0.002 293 48.1 0.173

0.002 846 139.8 0.521

v (m3/kg) 60 h (kJ/kg) s (kJ/kg⋅K)

0.001 538 −302.3 −1.875

0.001 584 −264.5 −1.553

0.001 668 −204.2 −1.113

0.001 767 −145.4 −0.750

0.001 887 −85.1 −0.428

0.002 043 −21.8 −0.130

0.002 270 47.4 0.161

0.002 723 132.3 0.484

v (m3/kg) 80 h (kJ/kg) s (kJ/kg⋅K)

0.001 535 −299.8 −1.881

0.001 581 −262.1 −1.559

0.001 664 −202.0 −1.120

0.001 761 −143.4 −0.759

0.001 877 −83.5 −0.439

0.002 025 −20.9 −0.145

0.002 232 46.5 0.139

0.002 585 124.1 0.434

v (m3/kg) 100 h (kJ/kg) s (kJ/kg⋅K)

0.001 533 −297.4 −1.887

0.001 579 −259.7 −1.565

0.001 660 −199.7 −1.127

0.001 754 −141.2 −0.767

0.001 867 −81.8 −0.449

0.002 009 −19.9 −0.158

0.002 199 46.1 0.120

0.002 495 119.6 0.400

v (m3/kg) 150 h (kJ/kg) s (kJ/kg⋅K)

0.001 528 −291.3 −1.901

0.001 571 −253.7 −1.580

0.001 650 −194.0 −1.145

0.001 740 −136.0 −0.787

0.001 846 −77.3 −0.473

0.001 973 −16.7 −0.188

0.002 136 46.6 0.079

0.002 356 114.4 0.337

v (m3/kg) 200 h (kJ/kg) s (kJ/kg⋅K)

0.001 522 −285.3 −1.914

0.001 565 −247.7 −1.595

0.001 641 −188.3 −1.161

0.001 727 −130.7 −0.806

0.001 826 −72.5 −0.495

0.001 943 −13.0 −0.215

0.002 086 48.6 0.045

0.002 268 113.2 0.291

v (m3/kg) 250 h (kJ/kg) s (kJ/kg⋅K)

0.001 517 −279.2 −1.928

0.001 559 −241.7 −1.610

0.001 633 −182.5 −1.177

0.001 715 −125.2 −0.824

0.001 809 −67.6 −0.516

0.001 918 −8.9 −0.240

0.002 046 51.4 0.015

0.002 203 113.9 0.253

v (m3/kg) 300 h (kJ/kg) s (kJ/kg⋅K)

0.001 512 −273.0 −1.942

0.001 552 −235.7 −1.623

0.001 625 −174.5 −1.192

0.001 704 −119.6 −0.841

0.001 793 −62.5 −0.536

0.001 895 −4.4 −0.262

0.002 012 54.9 −0.012

0.002 151 115.9 0.220

v (m3/kg) 400 h (kJ/kg) s (kJ/kg⋅K)

0.001 503 −260.8 −1.968

0.001 542 −223.6 −1.650

0.001 609 −164.8 −1.221

0.001 683 −108.3 −0.873

0.001 765 −51.9 −0.572

0.001 855 5.1 −0.303

0.001 957 63.0 −0.059

0.002 072 122.1 0.166

v (m3/kg) 500 h (kJ/kg) s (kJ/kg⋅K)

0.001 499 −246.9 −1.978

0.001 531 −211.4 −1.676

0.001 596 −152.9 −1.249

0.001 665 −96.8 −0.906

0.001 740 −40.9 −0.605

0.001 823 15.3 −0.339

0.001 913 72.3 −0.099

0.002 01 130.1 0.121

Converted from Jacobsen, R. T., M. Jahangiri, et al., Ethylene—Intl. Thermodyn. Tables of the Fluid State—10, Blackwell Sci. Publ., Oxford, 1988 (299 pp.). sf = hf = 0 at 233.15 K = −40°C.

2-237

2-238

PHYSICAL AND CHEMICAL DATA TABLE 2-258

Compressed Ethylene (Concluded ) Temperature, K

Pressure, bar

300

325

350

375

400

425

450

v (m3/kg) 1 h (kJ/kg) s (kJ/kg⋅K)

0.8842 497.3 2.829

0.9591 536.9 2.956

1.0339 578.8 3.079

1.1084 622.8 3.201

1.1830 668.9 3.320

1.2575 717.2 3.437

1.3319 767.9 3.553

v (m3/kg) 5 h (kJ/kg) s (kJ/kg⋅K)

0.1728 491.0 2.338

0.1884 531.5 2.467

0.2039 574.1 2.593

0.2193 618.6 2.716

0.2346 665.2 2.836

0.2499 713.9 2.954

0.2650 764.9 3.071

v (m3/kg) 10 h (kJ/kg) s (kJ/kg⋅K)

0.08380 482.8 2.113

0.09207 542.5 2.247

0.1002 568.0 2.375

0.1081 613.3 2.500

0.1160 660.5 2.622

0.1238 709.7 2.742

0.1316 761.1 2.859

v (m3/kg) 20 h (kJ/kg) s (kJ/kg⋅K)

0.03914 465.0 1.866

0.04379 509.8 2.009

0.04823 555.5 2.144

0.05257 602.4 2.274

0.05675 650.9 2.399

0.06088 701.2 2.521

0.06491 753.5 2.640

v (m3/kg) 30 h (kJ/kg) s (kJ/kg⋅K)

0.02404 444.7 1.696

0.02763 493.8 1.853

0.03090 542.3 1.996

0.03400 591.2 2.131

0.03700 641.2 2.261

0.03990 692.6 2.387

0.04270 745.9 2.508

v (m3/kg) 40 h (kJ/kg) s (kJ/kg⋅K)

0.01630 420.6 1.550

0.01947 476.3 1.728

0.02220 528.3 1.882

0.02473 579.5 2.023

0.02710 631.2 2.157

0.02938 688.9 2.286

0.03160 738.2 2.409

v (m3/kg) 50 h (kJ/kg) s (kJ/kg⋅K)

0.01140 390.4 1.404

0.01451 456.9 1.617

0.01697 513.4 1.784

0.01916 567.5 1.933

0.02119 621.1 2.072

0.02311 675.1 2.207

0.02495 730.5 2.330

v (m3/kg) 60 h (kJ/kg) s (kJ/kg⋅K)

0.007 757 347.8 1.230

0.01116 435.1 1.510

0.01347 497.7 1.696

0.01546 555.2 1.854

0.01725 610.9 1.999

0.01892 666.3 2.135

0.02052 722.9 2.263

v (m3/kg) 80 h (kJ/kg) s (kJ/kg⋅K)

0.003 672 238.7 0.832

0.006 864 382.8 1.295

0.009 136 463.5 1.534

0.01085 529.4 1.717

0.01237 590.1 1.874

0.01374 648.5 2.016

0.01502 707.6 2.151

v (m3/kg) 100 h (kJ/kg) s (kJ/kg⋅K)

0.003 094 210.2 0.715

0.004 698 330.2 1.098

0.006 596 427.5 1.387

0.008 163 503.1 1.596

0.009 492 569.3 1.768

0.01068 630.9 1.918

0.01177 692.7 2.059

v (m3/kg) 150 h (kJ/kg) s (kJ/kg⋅K)

0.002 684 188.8 0.596

0.003 223 272.7 0.864

0.004 040 361.4 1.126

0.004 983 445.5 1.359

0.005 914 521.6 1.556

0.006 765 592.1 1.722

0.007 578 658.1 1.878

v (m3/kg) 200 h (kJ/kg) s (kJ/kg⋅K)

0.002 508 181.7 0.529

0.002 840 255.0 0.763

0.003 292 332.4 0.992

0.003 838 410.6 1.208

0.004 445 487.0 1.406

0.005 058 560.1 1.580

0.005 664 629.5 1.742

v (m3/kg) 250 h (kJ/kg) s (kJ/kg⋅K)

0.002 397 179.2 0.480

0.002 644 247.7 0.698

0.003 024 319.1 0.910

0.003 327 392.1 1.111

0.003 743 465.5 1.301

0.004 190 538.2 1.476

0.004 648 608.5 1.639

v (m3/kg) 300 h (kJ/kg) s (kJ/kg⋅K)

0.002 317 179.1 0.440

0.002 517 244.6 0.670

0.002 578 312.6 0.850

0.003 037 382.2 1.043

0.003 351 452.9 1.226

0.003 690 524.1 1.398

0.004 042 593.9 1.558

v (m3/kg) 400 h (kJ/kg) s (kJ/kg⋅K)

0.002 203 182.7 0.377

0.002 352 244.9 0.576

0.002 522 308.8 0.764

0.002 711 374.3 0.946

0.002 919 441.3 1.119

0.003 413 509.8 1.285

0.003 382 578.3 1.442

v (m3/kg) 500 h (kJ/kg) s (kJ/kg⋅K)

0.002 122 189.1 0.326

0.002 245 250.7 0.523

0.002 379 312.8 0.707

0.002 524 376.1 0.882

0.002 678 440.4 1.048

0.002 847 505.8 1.206

0.003 022 572.0 1.358

THERMODYNAMIC PROPERTIES TABLE 2-259

2-239

Saturated Fluorine* hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

−158.6 −153.5 −149.1 −141.8 −134.4

40.9 42.0 45.8 49.6 53.2

1.602 1.642 1.768 1.885 1.995

5.314 5.235 5.004 4.816 4.666

1.446 1.442 1.437 1.442 1.450

8.8 8.0 6.0 4.7 3.8

0.186 0.184 0.177 0.170 0.162

0.583 0.309 0.176 0.108 0.069

−127.0 −119.5 −111.9 −104.3 −96.5

56.8 60.1 63.3 66.1 68.6

2.097 2.194 2.285 2.372 2.455

4.540 4.433 4.342 4.262 4.191

1.460 1.474 1.498 1.535 1.555

3.19 2.71 2.33 2.00 1.76

0.154 0.146 0.137 0.129 0.120

7.193.−4 7.412.−4 7.659.−4 7.948.−4 8.283.−4

0.0466 0.0323 0.0231 0.0168 0.0125

−88.6 −80.5 −72.2 −63.6 −54.5

70.7 72.4 73.6 74.1 73.9

2.535 2.612 2.688 2.763 2.837

4.127 4.068 4.012 3.959 3.906

1.585 1.630 1.692 1.782 1.888

1.53 1.36 1.21 1.08 0.96

0.112 0.103 0.095 0.087 0.080

8.696.−4 9.223.−4 9.963.−4 1.119.−3 1.743.−3

0.0093 0.0069 0.0051 0.0036 0.0017

−44.9 −34.5 −22.7 −8.4 23.9

72.7 70.2 65.6 56.9 23.9

2.912 2.989 3.073 3.170 3.388

3.864 3.795 3.727 3.636 3.388

2.05 2.33 2.90 3.64 ∞

0.86 0.74 0.63 0.49

0.073 0.066 0.070 0.105 ∞

T, K

P, bar

vf, m3/kg

vg, m3/kg

53.5t 55 60 65 70

0.0025 0.0041 0.0155 0.0477 0.1230

5.866.−4 5.898.−4 6.005.−4 6.119.−4 6.240.−4

46.2 17.1 8.46 2.93 1.24

75 80 85 90 95

0.276 0.555 1.019 1.740 2.802

6.369.−4 6.508.−4 6.657.−4 6.819.−4 6.997.−4

100 105 110 115 120

4.280 6.280 8.885 12.20 16.33

125 130 135 140 144.3c

21.37 27.48 34.72 43.47 52.15

*Values reproduced or converted from Prydz and Straty, NBS Tech. Note 392, rev., September 1973. t = triple point; c = critical point. The notation 5.866.−4 signifies 5.866 × 10−4. TABLE 2-260

Fluorine Gas at Atmospheric Pressure*

T, K

84.95

90

100

120

140

160

180

200

220

240

260

280

300

v, m3/kg h, kJ/kg s, kJ/(kg⋅K)

0.1776 63.22 4.342

0.1892 67.30 4.390

0.2118 75.27 4.474

0.2562 90.96 4.616

0.3002 106.53 4.737

0.3439 122.06 4.840

0.3874 137.62 4.932

0.4309 153.2 5.014

0.4744 169.0 5.090

0.5176 184.9 5.158

0.5610 201.0 5.221

0.6043 217.2 5.282

0.6476 233.7 5.340

*Extracted from Prydz and Straty, NBS Tech. Note 392, 1970. This source is recommended for other pressures and temperatures. Other information is contained in J. Chem. Phys., 53 (1970): 2359; and J. Res. NBS, 74A (1970): 499, 661, 747. TABLE 2-261

TABLE 2-262

Flutec

Proprietary name for a series of fluorocarbons produced by the Imperial Smelting Corp., Avonmouth, Bristol, UK. Bulletins of thermodynamic properties include PP1 (C6F14), PP2 (C7F14), PP3 (C8F16), PP5 (C10F18), PP9 (C11F20), and PP50, usually for 0.1–100 kg/m2, 0–500°C. See also Green, S. W., Chem. & Ind. (1969): 63–67.

TABLE 2-263

Halon

A series of fire-extinguishing fluids. Halon 1211 is produced by ICI, and Halon 1301, by duPont, the latter issuing a bulletin with thermodynamic properties and a diagram for the range 0.6–600 psia, −160–460°F.

Saturated Helium3*

T, K

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

1.0 1.1 1.2 1.3 1.4

0.0122 0.0182 0.0274 0.0370 0.0517

0.01222 0.01224 0.01227 0.01231 0.01236

1.72 1.33 1.02 0.805 0.649

−5.69 −5.49 −5.26 −5.01 −4.75

6.75 7.40 8.03 8.65 9.27

2.28 2.65 2.95 3.20 3.40

14.72 14.34 14.02 13.70 13.41

1.5 1.6 1.7 1.8 1.9

0.0659 0.0871 0.107 0.137 0.163

0.01241 0.01247 0.01254 0.01262 0.01271

0.526 0.437 0.363 0.308 0.260

−4.47 −4.17 −3.84 −3.47 −3.07

9.88 10.46 11.04 11.60 12.15

3.60 3.80 3.91 4.01 4.13

13.13 12.88 12.53 12.38 12.14

2.0 2.1 2.2 2.3 2.4

0.202 0.237 0.284 0.326 0.385

0.01282 0.01294 0.01308 0.01324 0.01343

0.222 0.189 0.164 0.142 0.124

−2.64 −2.17 −1.55 −0.99 −0.34

12.68 13.19 13.67 14.13 14.57

4.26 4.40 4.55 4.71 4.87

11.91 11.69 11.47 11.25 11.04

2.5 2.6 2.7 2.8 2.9

0.438 0.508 0.576 0.653 0.732

0.01365 0.01390 0.01419 0.01456 0.01497

0.109 0.096 0.085 0.074 0.064

0.36 1.16 2.01 2.96 4.01

14.98 15.37 15.89 16.40 16.37

5.03 5.20 5.38 5.57 5.77

10.84 10.64 10.41 10.17 9.92

3.0 3.1 3.2 3.3 3.32c

0.803 0.907 1.023 1.128 1.165

0.01549 0.01614 0.01720 0.01902 0.02394

0.055 0.047 0.039 0.028 0.024

5.28 6.70 8.44 10.66 13.25

16.32 16.20 15.98 14.50 13.25

6.00 6.24 6.54 6.96 7.50

9.66 9.34 8.90 8.35 7.50

*Converted and smoothed from a tabulation of Gibbons and Nathan, USAF Rep. AFML-TR-67-175, 1967. c = critical point. Kelly, D. P. and W. K. Haubach, in AEC R&D rept. MLM 1161, 1963 (56 pp.), give a comprehensive graphical comparison of the properties of He3 and He4.

2-240

PHYSICAL AND CHEMICAL DATA

TABLE 2-264

Saturated Helium4

Temperature, K

Pressure, bar

vf, m3/kg

0.8 0.9 1.0 1.1 1.2

1.475.−5 5.379.−5 1.557.−4 3.800.−4 8.148.−4

0.00689 0.00689 0.00689 0.00689 0.00689

1.3 1.4 1.5 1.6 1.7

0.00158 0.00282 0.00472 0.00746 0.01128

0.00689 0.00689 0.00689 0.00688 0.00688

1.8 1.9 2.0 2.1 2.2

0.01638 0.02299 0.03129 0.04141 0.05335

0.00688 0.00687 0.00686 0.00685 0.00684

2.3 2.4 2.5 2.6 2.7

0.06730 0.08354 0.01023 0.1237 0.1481

2.8 2.9 3.0 3.1 3.2

vg, m3/kg

hf, kJ/kg

h g, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf , kJ/(kg⋅K)

cpg, kJ/(kg⋅K)

0.0019 0.0054 0.0127 0.0268 0.0518

19.42 19.94 20.44 20.95 21.44

0.0047 0.0087 0.0163 0.0296 0.0510

23.94 21.86 20.44 18.82 17.67

0.022 0.050 0.100 0.185 0.318

5.210 5.230 5.262 5.305 5.360

0.0932 0.1579 0.2543 0.3923 0.5836

21.92 22.40 22.87 23.32 23.77

0.0836 0.1308 0.1962 0.2839 0.3981

16.70 15.86 15.13 14.49 13.93

0.511 0.780 1.138 1.602 2.193

5.424 5.496 5.574 5.654 5.736

2.1993 1.6420 1.2601 0.9921 0.7994

0.8422 1.186 1.642 2.261 3.090

24.20 24.63 25.04 25.45 25.85

0.5437 0.7270 0.9578 1.256 1.638

13.43 12.98 12.58 12.23 11.92

2.938 3.893 5.187 7.244 4.222

5.818 5.898 5.975 6.046 6.111

0.00685 0.00687 0.00690 0.00693 0.00695

0.6566 0.5470 0.4608 0.3923 0.3367

3.418 3.678 3.922 4.161 4.408

26.24 26.63 27.00 27.37 27.72

1.780 1.886 1.980 2.068 2.155

11.65 11.40 11.17 10.96 10.76

2.685 2.375 2.284 2.320 2.351

6.170 6.228 6.285 6.344 6.406

0.1755 0.2063 0.2405 0.2784 0.3201

0.00699 0.00703 0.00707 0.00713 0.00717

0.2913 0.2537 0.2223 0.1958 0.1728

4.662 4.923 5.195 5.483 5.787

28.06 28.38 28.69 28.98 29.26

2.240 2.324 2.408 2.494 2.581

10.57 10.39 10.22 10.05 9.90

2.403 2.486 2.597 2.740 2.896

6.470 6.540 6.616 6.700 6.792

3.3 3.4 3.5 3.6 3.7

0.3659 0.4159 0.4704 0.5296 0.5935

0.00723 0.00728 0.00735 0.00742 0.00749

0.1542 0.1376 0.1232 0.1107 0.0997

6.108 6.448 6.806 7.184 7.581

29.52 29.76 29.97 30.17 30.34

2.670 2.780 2.852 2.946 3.042

9.747 9.600 9.458 9.318 9.181

3.061 3.273 3.413 3.601 3.801

6.897 7.015 7.150 7.305 7.484

3.8 3.9 4.0 4.1 4.2

0.6625 0.7366 0.8162 0.9014 0.9923

0.00758 0.00766 0.00776 0.00786 0.00797

0.0900 0.0814 0.0738 0.0669 0.0606

7.998 8.437 8.899 9.387 9.901

30.48 30.60 30.68 30.73 30.74

3.140 3.239 3.341 3.444 3.551

9.046 8.911 8.776 8.641 8.504

4.017 4.254 4.519 4.820 5.170

7.694 7.942 8.238 8.641 9.033

4.3 4.4 4.5 4.6 4.7

1.089 1.193 1.303 1.419 1.543

0.00810 0.00824 0.00841 0.00860 0.00881

0.0550 0.0499 0.0452 0.0408 0.0367

10.45 11.02 11.64 12.31 13.04

30.71 30.62 30.47 30.24 29.91

3.661 3.775 3.893 4.018 4.151

8.363 8.218 8.067 7.906 7.732

5.587 6.097 6.742 7.590 8.763

4.8 4.9 5.0 5.1 5.195c

1.674 1.813 1.960 2.116 2.275

0.00907 0.00941 0.00986 0.01056 0.01436

0.0329 0.0291 0.0252 0.0207 0.0145

13.85 14.76 15.85 17.26

29.45 28.80 27.83 26.08

4.296 4.458 4.649 4.898

7.539 7.327 7.041 6.624

1125.9 347.1 133.0 59.8 30.4 16.93 10.17 6.49 4.35 3.04

c = critical pt. From Arp, V. D. and R. D. McCarty, N.I.S.T. TN 1334, 1989 (142 pp.).

10.51 13.38 19.02 34.60

9.58 10.29 11.22 12.50 14.37 17.32 22.64 34.93 95.84

THERMODYNAMIC PROPERTIES TABLE 2-265

Superheated Helium*

P, bars

0

2-241

Temp., °C 1 v h s

5.677 0.327 0.0116

100

200

300

400

500

600

800

1000

7.754 519.6 1.620

9.831 1039 2.853

11.908 1558 3.849

13.985 2078 4.684

16.063 2597 5.403

18.140 3116 6.035

22.294 4155 7.106

26.448 5193 7.993

5 v h s

1.138 1.636 −3.343

1.553 520.9 −1.723

1.968 1040 −0.490

2.384 1560 0.506

2.799 2079 1.341

3.215 2598 2.060

3.630 3117 2.692

4.461 4156 3.763

5.291 5194 4.650

10 v h s

0.5704 3.272 −4.782

0.780 522.5 −3.162

0.986 1042 −1.929

1.193 1561 −0.934

1.401 2080 −0.098

1.609 2600 0.621

1.816 3119 1.252

2.232 4157 2.323

2.647 5196 3.211

20 v h s

0.2867 6.544 −6.221

0.3904 525.8 −4.601

0.4942 1045 −3.368

0.5979 1564 −2.373

0.7017 2083 −1.537

0.9093 2603 −0.818

1.1169 3122 −0.187

1.3245 4160 0.884

1.8435 5199 1.771

50 v h s

0.1164 16.360 −8.121

0.1579 535.5 −6.501

0.1993 1055 −5.268

0.2408 1574 −4.273

0.2822 2093 −3.438

0.3257 2612 −2.719

0.3652 3131 −2.088

0.4481 4169 −1.017

0.5311 5207 −0.130

100 v h s

0.0597 37.720 −9.555

0.0803 551.7 −7.936

0.1010 1071 −6.703

0.1217 1590 −5.709

0.1424 2108 −4.874

0.1631 2627 −4.155

0.1838 3146 −3.524

0.2252 4184 −2.454

0.2666 5222 −1.567

150 v h s

0.0407 49.080 −10.391

0.0545 567.9 −8.773

0.0682 1087 −7.541

0.0820 1605 −6.546

0.0958 2124 −5.712

0.1095 2643 −4.994

0.1233 3161 −4.363

0.1509 4199 −3.293

0.1785 5236 −2.407

200 v h s

0.0312 65.440 −10.983

0.0416 584.1 −9.635

0.0518 1103 −8.134

0.0622 1621 −7.139

0.0725 2140 −6.306

0.0828 2658 −5.588

0.0931 3176 −4.957

0.1137 4213 −3.888

0.1344 5250 −3.002

*Extracted from Tsederberg, Popov, et al., Thermodynamic and Thermophysical Properties of Helium, Atomizdat, Moscow, 1969, and NBS-NSF TT 50096, 1971. Copyright material. Reproduced by permission. This source contains entries for many more temperatures and pressures than can be reproduced here. v = volume, m3/kg; h = enthalpy, kJ/ kg; s = entropy, kJ/(kg·K). The 1993 ASHRAE Handbook—Fundamentals (SI ed.) has a thermodynamic chart for pressures from 0.1 to 50 bar and temperatures from 2.5 to 15 K. Saturation and superheat tables to 9000 psia, 800°R; and a chart to 700 psia, 40°R appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993. A useful compilation of properties is given by Betts, D. S., Cryogenics, 16, 1 (1976): 3–16. A 32-term equation of state for the range up to 20,000 bar, 2–1500 K is given by McCarty, R. D. and V. D. Arp, Advan. Cryog. Engng., 35 (1990): 1465–1475.

TABLE 2-266

Helium4 Gas at Atmospheric Pressure*

T, K

4.224

5

10

20

30

40

50

75

100

200

300

400

500

600

800

1000

v, m3/kg h, kJ/kg s, kJ/(kg⋅K)

0.0591 30.30 8.327

0.0834 36.18 9.614

0.1612 64.91 13.369

0.4094 117.95 17.321

0.6161 170.24 19.442

0.8218 222.4 20.94

1.0273 274.4 22.10

1.5403 404.4 24.21

2.053 534.2 25.71

4.102 1054 29.30

6.154 1573 31.41

8.191 2092 32.90

10.24 2612 34.06

12.31 3131 35.01

16.40 4170 36.50

20.50 5208 37.66

*From McCarty, NBS Rep. 9762, 1970. Reproduced by permission. The source contains values for further temperatures and for other functions, usually to additional significant figures.

2-242

PHYSICAL AND CHEMICAL DATA

TABLE 2-267

Saturated n-Heptane*

P, bar

vf, m3/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µ f, 10−4 Pa⋅s

k f, W/(m⋅K)

182.6t 200 220 240 250

0.00002 0.00019 0.00133 0.00303

1.292.−3 1.316.−3 1.344.−3 1.374.−3 1.389.−3

284.1 319.4 359.7 400.5 421.3

722.6 757.1 791.4 808.3

2.260 2.441 2.636 2.814 2.899

4.457 4.442 4.443 4.447

2.025 2.011 2.026 2.063 2.088

39.4 21.0 12.6 8.52 7.23

0.150 0.148 0.145 0.142 0.140

260 270 280 290 300

0.00635 0.01316 0.02347 0.03997 0.06674

1.405.−3 1.422.−3 1.440.−3 1.457.−3 1.475.−3

3.744

442.3 463.6 485.2 507.2 529.6

824.9 841.2 857.8 874.8 891.9

2.981 3.061 3.140 3.217 3.293

4.453 4.460 4.471 4.485 4.501

2.117 2.147 2.180 2.216 2.252

6.52 5.46 4.83 4.29 3.85

0.137 0.135 0.132 0.129 0.126

310 320 330 340 350

0.1070 0.1656 0.2461 0.3614 0.5130

1.494.−3 1.514.−3 1.534.−3 1.555.−3 1.578.−3

2.412 1.596 1.101 0.7650 0.5510

552.3 575.4 598.8 622.8 647.0

908.9 926.0 943.3 961.2 979.1

3.367 3.441 3.513 3.584 3.655

4.517 4.537 4.557 4.579 4.604

2.291 2.329 2.370 2.412 2.454

3.48 3.17 2.89 2.66 2.45

0.123 0.121 0.119 0.116 0.114

360 370 371.6 380 390

0.712 0.967 1.013 1.289 1.689

1.601.−3 1.625.−3 1.629.−3 1.651.−3 1.678.−3

0.4058 0.3036 0.2904 0.2308 0.1781

671.9 697.1 701.9 723.9 750.4

997.5 1016.1 1019.8 1035.4 1054.2

3.725 3.794 3.805 3.864 3.932

4.629 4.656 4.660 4.684 4.711

2.500 2.548 2.556 2.60 2.65

2.24 2.04 2.01 1.86 1.71

0.111 0.109 0.108 0.107 0.105

400 420 440 460 480

2.180 3.471 5.268 7.691 10.92

1.708.−3 1.775.−3 1.853.−3 1.954.−3 2.065.−3

0.1388 0.0734 0.0576 0.0389 0.0265

777.2

1073.2

4.000

4.740

2.70 2.81 2.93 3.05 3.19

1.58 1.35 1.15 0.97 0.82

0.103 0.099 0.095 0.091 0.087

500 520 540.1c

15.10 20.43 27.35

2.235.−3 2.52.−3 4.3.−3

3.38 3.7

0.67

0.080

0.0043

T, K

vg, m3/kg

h f, kJ/kg

0.0178

*Values of P and v interpolated and converted from tables in Vargaftik, Handbook of Thermophysical Properties of Gases and Liquids, Hemisphere, Washington, and McGraw-Hill, New York, 1975. Values of h and s calculated from API tables published by the Thermodynamics Research Center, Texas A&M University, College Station. t = triple point; c = critical point. Saturation and superheat tables and a diagram to 200 bar, 680 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.).

TABLE 2-268

Hexane

Saturation and superheat tables and a diagram to 100 bar, 680 K, are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.).

THERMODYNAMIC PROPERTIES TABLE 2-269

2-243

Saturated Hydrazine

Temperature, K

vf, m3/kg

vg, m3/kg

h f, kJ/kg

hg, kJ/kg

s f, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

1.013 1.135 1.560 2.102 2.786

0.001 053 0.001 060 0.001 081 0.001 104 0.001 127

0.9833 0.8850 0.6579 0.4994 0.3850

−105.9 −104.8 −101.4 −97.6 −93.9

65.4 66.0 68.2 70.6 73.0

0.5994 0.6029 0.6120 0.6211 0.6300

1.0426 1.0409 1.0360 1.0314 1.0275

4.732 7.610 11.76 17.42 29.59

0.001 178 0.001 235 0.001 299 0.001 374 0.001 460

0.2355 0.1500 0.1005 0.0690 0.0407

−86.1 −76.9 −67.1 −57.3 −47.8

77.6 82.1 86.6 90.8 94.6

0.6492 0.6707 0.6916 0.7124 0.7320

1.0212 1.0163 1.0118 1.0086 1.0058

Pressure, bar

386.6 390 400 410 420 440 460 480 500 520 540 560 580 600 620

34.75 47.09 62.44 81.17 102.7

0.001 563 0.001 681 0.001 835 0.002 045 0.002 320

0.0353 0.0263 0.0196 0.0142 0.0106

−36.0 −25.2 −12.4 5.2 23.2

97.7 101.2 103.6 104.2 103.6

0.7566 0.7762 0.8002 0.8335 0.8671

1.0042 1.0020 1.0002 0.9988 0.9967

640 653 c

128.1 146.9

0.002 86 0.004 33

0.0074 0.0043

45.9 83.7

98.1 83.7

0.9035 0.9715

0.9906 0.9715

Converted from E. F. Fricke, Republic Aviation Co. rept. F-5028-101. c = critical point.

TABLE 2-270

Saturated n-Hydrogen*

T, K

P, bar

vf, m3/kg

v g, m3/kg

hf, kJ/kg

h g, kJ/kg

s f, kJ/(kg⋅K)

s g, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µ f, 10−4 Pa⋅s

k f, W/(m⋅K)

13.95t 14 15 16 17

0.072 0.074 0.127 0.204 0.314

0.01298 0.01301 0.01316 0.01332 0.01348

7.974 7.205 4.488 2.954 2.032

218.3 219.6 226.4 233.8 241.6

667.4 669.3 678.2 686.7 694.7

14.079 14.173 14.640 15.104 15.568

46.635 46.301 44.763 43.418 42.227

6.36 6.47 6.91 7.36 7.88

0.255 0.248 0.218 0.194 0.175

0.073 0.075 0.083 0.089 0.093

18 19 20 21 22

0.461 0.654 0.901 1.208 1.585

0.01366 0.01387 0.01407 0.01430 0.01455

1.449 1.064 0.8017 0.6177 0.4828

249.9 258.8 268.3 278.4 289.2

702.1 708.8 714.8 720.2 724.4

16.032 16.498 16.966 17.440 17.919

41.158 40.188 39.299 38.485 37.710

8.42 8.93 9.45 10.13 10.82

0.159 0.146 0.135 0.125 0.116

0.095 0.097 0.098 0.100 0.101

23 24 25 26 27

2.039 2.579 3.213 3.950 4.800

0.01483 0.01515 0.01551 0.01592 0.01639

0.3829 0.3072 0.2489 0.2032 0.1667

300.8 313.3 326.7 341.2 357.0

727.6 729.8 730.7 730.2 728.0

18.405 18.901 19.408 19.929 20.473

36.973 36.266 35.579 34.900 34.221

11.69 12.52 13.44 14.80 16.17

0.108 0.101 0.094 0.088 0.082

0.101 0.101 0.100 0.098 0.096

28 29 30 31 32

5.770 6.872 8.116 9.510 11.07

0.01696 0.01765 0.01854 0.01977 0.02174

0.1370 0.1125 0.0919 0.0738 0.0571

374.3 393.6 415.4 441.3 474.7

723.7 716.6 705.9 689.7 663.2

21.041 21.650 22.315 23.075 24.032

33.524 32.795 32.002 31.091 29.926

18.48 22.05 26.59 36.55 65.37

0.076 0.070 0.065 0.058 0.051

0.094 0.091 0.087 0.086 0.092

33.18c

13.13

0.03182

0.0318

565.4

565.4

26.680

26.680

∞

*Values extracted and occasionally rounded off from McCarty, Hord, and Roder, NBS Monogr. 168, 1981. t = triple point; c = critical point.

∞

2-244

PHYSICAL AND CHEMICAL DATA

TABLE 2-271

Compressed n-Hydrogen* Temperature, K

Pressure, bar

15

20

30

40

50

60

80

100

150

200

6.076 679.2 46.02

8.176 731.6 49.04

12.333 835.5 53.25

16.473 938.9 56.23

20.606 1042.3 58.53

24.736 1146 60.43

32.991 1356 63.45

41.244 1575 65.89

61.870 2172 70.68

82.495 2826 74.46

1 v h s

0.0131 227.3 14.62

0.0141 268.3 16.96

1.196 826.0 43.56

1.625 932.7 46.63

2.046 1037.9 48.98

2.463 1143 50.89

3.295 1354 53.93

4.123 1574 56.38

6.190 2172 61.17

8.254 2826 64.96

5 v h s

0.0131 231.7 14.57

0.0140 272.1 16.88

0.2006 775.0 35.80

0.3039 903.4 39.52

0.3958 1017.6 42.07

0.4839 1128 44.07

0.6553 1345 47.20

0.8238 1568 49.68

1.241 2170 54.66

1.655 2826 58.31

10 v h s

0.0130 237.2 14.50

0.0138 277.0 16.77

0.0181 412.1 22.09

0.1376 861.8 35.95

0.1895 991.1 38.85

0.2366 1109 40.99

0.3255 1334 44.23

0.4116 1560 46.75

0.6221 2167 51.63

0.8303 2826 55.44

20 v h s

0.0129 248.2 14.37

0.0136 286.9 16.58

0.0167 406.5 21.33

0.0521 752.0 31.07

0.0866 934.7 35.19

0.1135 1070 37.67

0.1611 1312 41.15

0.2057 1546 43.76

0.3129 2163 48.71

0.4179 2826 52.55

40 v h s

0.0133 307.3 16.26

0.0155 413.5 20.50

0.0216 589.3 25.49

0.0376 823.5 30.73

0.0533 997 33.91

0.0796 1271 37.87

0.1033 1521 40.65

0.1586 2155 45.75

0.2119 2826 49.64

60 v h s

0.0130 328.0 15.98

0.0147 427.2 19.95

0.0182 570.1 24.03

0.0254 757.0 28.19

0.0351 940 31.54

0.0532 1237 35.82

0.0697 1499 38.76

0.1073 2149 43.99

0.1433 2828 47.92

80 v h s

0.0127 348.9 15.74

0.0142 443.5 19.53

0.0167 572.3 23.21

0.0211 732.8 26.78

0.0273 905 29.93

0.0406 1210 34.34

0.0531 1482 37.37

0.0818 2146 42.72

0.1090 2831 46.69

100 v h s

0.0125 369.8 15.53

0.0138 461.1 19.19

0.0158 581.5 22.63

0.0190 727.4 25.88

0.0233 888 28.80

0.0335 1192 33.19

0.0434 1469 36.28

0.0666 2144 41.73

0.0885 2835 45.73

200 v h s

0.0117 474.4 14.71

0.0125 556.1 17.99

0.0136 658.7 20.93

0.0150 776.9 23.56

0.0167 908 25.94

0.0207 1182 29.88

0.0253 1458 32.97

0.0368 2156 38.59

0.0480 2869 42.72

400 v h s

0.0113 751.0 16.59

0.0119 841.9 19.20

0.0126 945.4 21.50

0.0134 1059 23.58

0.0151 1303 27.07

0.0171 1560 29.94

0.0225 2249 35.48

0.0279 2973 39.67

600 v h s

0.0106 941.5 15.68

0.0110 1027 18.14

0.0115 1124 20.29

0.0120 1231 22.24

0.0131 1463 25.57

0.0144 1709 28.31

0.0178 2385 33.74

0.0214 3107 37.92

800 v h s

0.0104 1209 17.35

0.0107 1302 19.43

0.0111 1405 21.30

0.0120 1628 24.50

0.0130 1870 27.20

0.0155 2535 32.54

0.0181 3255 36.70

1000 v h s

0.0099 1387 16.72

0.0102 1478 18.75

0.0106 1578 20.58

0.0112 1796 23.70

0.0120 2032 26.33

0.0140 2692 31.63

0.0160 3403 35.75

0.1 v h s

*Values extracted and sometimes rounded off from the tables of McCarty, Hord, and Roder, NBS Monogr. 168, 1981. This source contains an exhaustive tabulation of property values for both the normal and the para forms of hydrogen. v = specific volume, m3/kg; h = specific enthalpy, kJ/kg; s = specific entropy, kJ/(kg⋅K). The 1993 ASHRAE Handbook—Fundamentals (SI ed.) has a thermodynamic diagram for 0.1 to 500 bar for temperatures up to 100 K. Tables and a Mollier chart from 10−4 to 1000 atm, 300–20,000 K are given by Kubin, R. F. and L. L. Presley, NASA SP 3002, 1964. Liebenberg, D. H., R. L. Mills, and others, in LA-6645-MS, 1977 (26 pp.), give properties from 75 to 307 K for pressures from 2 to 20 kbar. See also Baker, J. R. and H. F. Swift, J. Appl. Phys., 43, 3 (1972): 950–953. An extensive collection of data for H2, D2, T2, and so on below 30 K is given by Souers, P. C., UCRL 52628, 1979 (91 pp.); and for temperatures below 40 K by Roder, H. M., G. E. Childs, et al., NBS TN 641, 1973 (114 pp.). Saturation and superheat tables to 10,000 psia, 900°R and a chart to 180°R appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For viscosity, thermal conductivity, and specific heat, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.

THERMODYNAMIC PROPERTIES

2-245

Temperature, K 250

300

350

400

450

500

600

700

800

900

1000

103.12 3517 77.53

123.23 4227 80.13

144.35 4945 82.34

164.97 5668 84.27

185.60 6393 85.98

206.22 7118 87.51

247.46 8571 90.15

288.70 10028 92.40

329.94 11493 94.36

371.18 12969 96.10

412.43 14458 97.66

10.32 3517 68.03

12.38 4227 70.63

14.44 4946 72.85

16.50 5669 74.78

18.57 6393 76.48

20.63 7118 78.01

24.75 8571 80.66

28.88 10029 82.91

33.00 11494 84.86

37.13 12969 86.60

41.25 14459 88.17

2.069 3518 61.39

2.482 4229 63.99

2.895 4948 66.21

3.307 5671 68.14

3.720 6396 69.84

4.132 7121 71.37

4.957 8574 74.02

5.782 10032 76.27

6.607 11497 78.23

7.432 12973 79.96

8.257 14462 81.53

1.038 3519 58.52

1.245 4231 61.12

1.451 4951 63.34

1.658 5674 65.28

1.864 6399 66.98

2.070 7125 68.51

2.483 8578 71.16

2.896 10036 73.41

3.308 11501 75.37

3.720 12977 77.10

4.133 14467 78.67

0.522 3522 55.65

0.6259 4235 58.26

0.7294 4956 60.48

0.8328 5680 62.41

0.9361 6406 64.12

1.040 7132 65.65

1.246 8586 68.30

1.452 10044 70.55

1.658 11509 72.51

1.865 12985 74.24

2.071 14475 75.81

0.2644 3527 52.76

0.3166 4244 55.38

0.3685 4967 57.61

0.4204 5692 59.55

0.4721 6419 61.26

0.5238 7146 62.79

0.6271 8601 65.44

0.7303 10059 67.69

0.8335 11525 69.65

0.9366 13002 71.39

1.040 14492 72.95

0.1786 3533 51.05

0.2136 4253 53.69

0.2483 4978 55.92

0.2829 5705 57.86

0.3174 6432 59.58

0.3519 7160 61.11

0.4209 8616 63.76

0.4897 10075 66.02

0.5585 11542 67.97

0.6273 13018 70.51

0.6961 14508 71.28

0.1357 3540 49.84

0.1621 4263 52.49

0.1882 4989 54.73

0.2142 5718 56.67

0.2401 6446 58.39

0.2660 7174 59.92

0.3177 8631 62.57

0.3694 10091 64.83

0.4120 11558 66.79

0.4726 13035 68.52

0.5242 14525 70.09

0.1099 3547 48.89

0.1312 4273 51.55

0.1521 5001 53.79

0.1730 5731 55.74

0.1937 6460 57.46

0.2145 7189 59.00

0.2559 8647 61.65

0.2972 10107 63.90

0.3385 11574 65.87

0.3798 13051 67.60

0.4211 14542 69.17

0.0588 3594 45.94

0.0695 4329 48.62

0.0801 5064 50.89

0.0905 5798 52.85

0.1001 6531 54.58

0.1114 7263 56.12

0.1321 8724 58.78

0.1528 10187 61.04

0.1734 11656 63.00

0.1941 13134 64.74

0.2147 14625 66.31

0.0334 3716 42.98

0.0388 4458 45.68

0.0441 5202 47.97

0.0493 5943 49.95

0.0545 6681 51.69

0.0597 7416 53.24

0.0701 8883 55.91

0.0804 10349 58.17

0.0908 11820 60.14

0.1011 13300 61.88

0.1114 14792 63.45

0.0249 3854 41.24

0.0285 4600 43.95

0.0321 5349 46.26

0.0355 6095 48.26

0.0390 6836 50.00

0.0425 7574 51.56

0.0494 9045 54.24

0.0562 10513 56.50

0.0631 11985 58.47

0.0700 13466 60.21

0.0768 14958 61.78

0.0207 4003 40.03

0.0234 4748 42.73

0.0260 5501 45.05

0.0286 6249 47.05

0.0312 6993 48.81

0.0338 7734 50.37

0.0390 9207 53.05

0.0441 10677 55.32

0.0492 12150 57.29

0.0543 13631 59.03

0.0594 15124 60.60

0.0181 4156 39.10

0.0202 4898 41.79

0.0223 5654 44.12

0.0244 6405 46.12

0.0265 7151 47.88

0.0286 7893 49.45

0.0327 9370 52.14

0.0367 10842 54.41

0.0408 12316 56.38

0.0449 13797 58.12

0.0490 15289 59.69

2-246

PHYSICAL AND CHEMICAL DATA

TABLE 2-272

Saturated para-Hydrogen*

T, K

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf , kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

13.8t 14 15 16 17

0.070 0.079 0.134 0.216 0.329

0.0130 0.0130 0.0132 0.0133 0.0135

7.97 7.20 4.49 2.96 2.03

−308.9 −307.6 −300.9 −293.4 −285.6

140.3 142.1 151.1 159.6 167.6

4.97 5.06 5.53 5.99 6.45

37.52 37.19 36.65 34.31 33.11

6.37 6.47 6.91 7.36 7.88

0.255 0.248 0.218 0.194 0.175

0.073 0.075 0.082 0.089 0.092

18 19 20 21 22

0.482 0.682 0.935 1.250 1.634

0.0137 0.0139 0.0141 0.0143 0.0146

1.45 1.07 0.802 0.618 0.483

−277.3 −268.4 −258.9 −248.8 −237.9

175.0 181.7 187.7 193.0 197.3

6.92 7.38 7.85 8.32 8.80

32.05 31.08 30.19 29.37 28.60

8.42 8.93 9.45 10.13 10.82

0.159 0.146 0.135 0.125 0.116

0.095 0.097 0.098 0.100 0.101

23 24 25 26 27

2.096 2.645 3.288 4.035 4.892

0.0148 0.0152 0.0155 0.0159 0.0164

0.383 0.307 0.249 0.203 0.167

−226.3 −213.9 −200.4 −185.9 −170.2

200.5 202.7 203.6 203.1 200.9

9.29 9.78 10.29 10.81 11.36

27.86 27.15 26.46 25.79 25.11

11.69 12.52 13.44 14.81 16.18

0.108 0.101 0.094 0.088 0.082

0.101 0.100 0.099 0.098 0.096

0.076 0.070 0.065 0.058 0.051

0.094 0.091 0.087 0.088 0.092

28 29 30 31 32

5.88 6.98 8.23 9.63 11.20

0.0170 0.0177 0.0185 0.0198 0.0217

0.137 0.113 0.092 0.074 0.057

−152.9 −133.6 −111.7 −85.8 −52.4

196.5 189.5 178.8 162.6 136.1

11.93 12.54 13.20 13.96 14.92

24.41 23.68 22.89 21.98 20.81

18.5 22.1 26.6 36.6 65.4

33c

12.93

0.0318

0.032

38.3

38.3

17.56

17.56

∞

∞

*Values extracted and occasionally rounded off from McCarty, Hord, and Roder, NBS Monogr. 168, 1981. t = triple point; c = critical point. Saturation and superheat tables to 12,000 psia, 900°R and a chart to 180°R appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) has a thermodynamic chart for pressures from 0.1 to 1000 bar for temperatures up to 100 K.

TABLE 2-273 T, K 273 300 350 400 450

Saturated Hydrogen Peroxide*

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

0.0004 0.0031 0.0564 0.4521 2.143

0.00068 0.00069 0.00072 0.00076 0.00081

1672 235 15.1 2.12 0.487

−5577 −5510 −5376 −5238 −5091

−4027 −3995 −3933 −3878 −3820

2.990 3.224 3.631 4.032 4.346

8.662 8.269 7.758 7.440 7.172

1.45 1.48 1.54 1.61 1.68

18.0 11.3 4.3 2.2 1.3

0.483 0.481 0.474 0.464 0.453

1.75 1.82 1.90

500 550 600 650 700

7.126 18.56 40.75 79.27 141.7

0.00088 0.00095 0.00107 0.00125 0.00171

0.155 0.0605 0.0268 0.0125 0.0048

−4945 −4794 −4635 −4463 −4195

−3777 −3745 −3731 −3746 −3860

4.656 4.941 5.209 5.485 5.682

6.992 6.846 6.720 6.582 6.339

708.5c

155.3

0.00284

0.0028

−4012

−4012

5.732

5.732

0.89 0.65 0.50

0.443 0.431 0.416

*Values reproduced or converted from a tabulation by Tsykalo and Tabachnikov in V. A. Rabinovich (ed.), Thermophysical Properties of Gases and Liquids, Standartov, Moscow, 1968; NBS-NSF transl. TT 69-55091, 1970. The reader may be reminded that very pure hydrogen peroxide is very difficult to obtain owing to its decomposition or instability. c = critical point. The FMC Corp., Philadelphia, PA tech. bull. 67, 1969 (100 pp.) contains an enthalpy-pressure diagram to 3000 psia, 1100 K.

TABLE 2-274

Hydrogen Sulfide

West, J. R., Chem. Eng. Progr., 44, 4 (1948): 207–292 gives tables and a chart for the range 1–90 atm., −76 to 1300°F while properties from 10 to 330 bar, 300 to 500 K were tabulated by Lui, C-H., D. M. Bailey, et al., Hydroc. Proc., 65, 7 ( July 1986): 41–43.

THERMODYNAMIC PROPERTIES

Enthalpy-concentration diagram for aqueous hydrogen chloride at 1 atm. Reference states: enthalpy of liquid water at 0°C is zero; enthalpy of pure saturated HCl vapor at 1 atm (−85.03°C) is 8000 kcal/mol. NOTE: It should be observed that the weight basis includes the vapor, which is particularly important in the two-phase region. Saturation values may be read at the ends of the tie lines. [Van Nuys, Trans. Am. Inst. Chem. Eng., 39, 663 (1943).]

FIG. 2-11

2-247

2-248

PHYSICAL AND CHEMICAL DATA

TABLE 2-275

Saturated Isobutane (R600a)* µf , 10−4 Pa⋅s

kf, W/(m⋅K)

1.78 1.87 1.93 1.99

9.46

0.163 0.158 0.149

4.984 4.916 4.878 4.863 4.861

2.05 2.12 2.19 2.28 2.33

6.06 4.21 3.11 2.40 2.14

0.142 0.134 0.127 0.120 0.117

3.617 3.700 3.783 3.865 3.946

4.863 4.867 4.874 4.882 4.891

2.39 2.46 2.53 2.61 2.70

1.93 1.75 1.59 1.46 1.35

0.113 0.110 0.106 0.102 0.099

T, K

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

113.6t 120 140 160 180

1.9.−7 9.3.−7 4.8.−5 8.2.−4 0.0070

1.349.−3 1.360.−3 1.396.−3 1.435.−3 1.476.−3

8.60.+6 1.84.+6 4210 278.2 36.66

0.0 11.0 46.0 82.1 119.5

485.3 491.1 510.1 530.8 533.0

1.863 1.957 2.226 2.467 2.688

6.136 5.957 5.541 5.272 5.097

200 220 240 260 270

0.0369 0.1374 0.3989 0.9600 1.4081

1.520.−3 1.568.−3 1.621.−3 1.680.−3 1.712.−3

7.723 2.265 0.8432 0.3738 0.2617

158.5 199.0 241.4 285.8 308.8

576.7 601.5 627.4 654.2 667.7

2.893 3.086 3.270 3.446 3.532

280 290 300 310 320

2.0020 2.7686 3.7365 4.934 6.392

1.746.−3 1.784.−3 1.824.−3 1.868.−3 1.916.−3

0.1882 0.1385 0.1040 0.0794 0.0614

332.3 356.4 381.1 406.4 432.4

681.3 694.9 708.4 721.7 734.8

330 340 350 360 370

8.140 10.21 12.64 15.46 18.72

1.971.−3 2.032.−3 2.103.−3 2.187.−3 2.289.−3

0.0481 0.0380 0.0301 0.0240 0.0190

459.2 486.9 515.7 545.6 577.1

747.7 760.0 771.8 782.7 792.3

4.028 4.109 4.191 4.273 4.357

4.902 4.912 4.923 4.932 4.939

2.81 2.92 3.04 3.17 3.31

1.25 1.15 1.05 0.95 0.85

0.095 0.092 0.088 0.083 0.080

380 390 400 408.0c

22.48 26.82 31.86 36.55

2.420.−3 2.604.−3 2.920.−3 4.464.−3

0.0150 0.0115 0.0083 0.0045

610.6 647.1 689.6 752.5

799.8 803.7 799.6 752.5

4.444 4.536 4.639 4.791

4.942 4.937 4.915 4.791

3.45 3.62 3.85 ∞

0.75 0.63 0.51

0.076 0.071 0.065 ∞

*Values reproduced or converted from Goodwin, NBSIR 79-1612, 1979. t = triple point; c = critical point. The notation 1.9.−7 signifies 1.9 × 10−7. Slightly different values for the range 0.5 to 34.5 bar, 250–404 K appear in Waxman, M. and J. S. Gallagher, J. Chem. Eng. Data, 28, (1983): 224–241. This source also contains superheat tables for 1–400 bar, 250–600 K. Saturation and superheat tables and a diagram to 200 bar, 600 K are given by Reynolds, W. C., Thermodynamic properties in S.I., Stanford Univ. publ., 1979 (173 pp.). Saturation and superheat tables and a chart to 10,000 psia, 640°F appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993. Equations and data for thermal conductivity and viscosity are given by Nieuwoldt, J. C., B. LeNeindre, et al., J. Chem. Eng. Data, 32, (1987): 1–8.

TABLE 2-276 T, K

Saturated Krypton*

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

10 20 30 40 50

3.235.−4 3.246.−4 3.265.−4 3.288.−4 3.313.−4

0.22 1.59 3.84 6.49 9.37

0.0256 0.1141 0.2034 0.2791 0.3431

0.070 0.188 0.247 0.276 0.295

60 70 80 90 100

3.341.−4 3.372.−4 3.407.−4 3.446.−4 3.492.−4

12.40 15.57 18.97 22.58 26.42

0.3982 0.4471 0.4925 0.5353 0.5765

0.311 0.327 0.345 0.366 0.389

110 115.76 115.76 119.76 120

0.732 1.013 1.032

3.544.−4 3.579.−4 4.090.−4 4.143.−4 4.146.−4

0.1529 0.1136 0.1116

30.52 33.18 52.78 54.99 55.09

161.8 162.6 162.6

0.6165 0.6390 0.8095 0.8279 0.8291

1.751 1.726 1.724

0.414 0.427 0.547 0.545 0.544

3.72

0.0900

130 140 150 160 170

2.112 3.878 6.552 10.37 15.57

4.284.−4 4.440.−4 4.619.−4 4.831.−4 5.091.−4

0.0578 0.0330 0.0201 0.0130 0.0086

60.55 66.02 71.58 77.34 83.48

164.1 165.3 166.1 166.4 166.0

0.8724 0.9124 0.9499 0.9859 1.022

1.669 1.622 1.580 1.543 1.507

0.542 0.546 0.559 0.587 0.641

3.16 2.64 2.20 1.87 1.54

0.0828 0.0756 0.0688 0.0625 0.0558

180 190 200 209.39

22.41 31.20 42.23 54.96

5.423.−4 5.882.−4 6.641.−4 1.098.−3

0.0059 0.0040 0.0026 0.0011

90.26 98.19 108.40 133.90

164.6 161.8 156.0 133.9

1.058 1.098 1.147 1.262

1.472 1.433 1.386 1.262

0.734 0.905 1.515 ∞

1.28 1.05 0.80

0.0494 0.0433 0.0348 ∞

*Values extracted and in some cases rounded off from those cited in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standards Press, Moscow, 1976. This source contains values for the compressed state for pressures up to 1000 bar, etc. The notation 3.235.−4 signifies 3.235 × 10−4. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.).

THERMODYNAMIC PROPERTIES TABLE 2-277 Temperature, K

2-249

Compressed Krypton* Pressure, bar 1

10

20

40

60

80

100

200

400

600

800

1000

100

v h s

3.49.−4 26.42 0.5765

3.49.−4 26.69 0.5760

3.49.−4 26.99 0.5755

3.48.−4 27.59 0.5745

3.48.−4 28.18 0.5735

3.47.−4 28.78 0.5724

3.47.−4 29.38 0.5714

3.45.−4 32.38 0.5667

3.42.−4 38.42 0.5580

3.39.−4 44.47 0.5503

3.36.−4 50.52 0.5432

3.33.−4 56.57 0.5366

200

v h s

0.1971 183.1 1.859

0.0184 179.3 1.618

8.39.−3 174.5 1.533

3.00.−3 159.4 1.405

6.19.−4 105.6 1.129

5.94.−4 104.4 1.116

5.76.−4 103.6 1.106

5.27.−4 102.7 1.073

4.83.−4 105.3 1.037

4.58.−4 109.7 1.013

4.41.−4 114.9 0.993

4.28.−4 120.3 0.977

v 300 h s

0.2971 208.1 1.961

0.0292 206.3 1.728

0.0143 204.2 1.654

6.84.−3 200.0 1.575

4.37.−3 195.7 1.525

3.14.−3 191.2 1.485

2.41.−3 186.6 1.451

1.09.−3 166.8 1.333

6.92.−4 155.1 1.239

5.94.−4 155.2 1.196

5.44.−4 158.2 1.169

5.13.−4 162.3 1.149

400

v h s

0.3966 233.0 2.032

0.0394 231.9 1.802

0.0196 230.7 1.730

9.67.−3 228.3 1.657

6.37.−3 225.9 1.612

4.73.−3 223.6 1.579

3.75.−3 221.3 1.552

1.85.−3 211.4 1.463

1.01.−3 199.7 1.368

7.79.−4 196.8 1.317

6.76.−4 197.9 1.284

6.14.−4 200.8 1.259

500

v h s

0.4960 257.8 2.088

0.0495 257.1 1.858

0.0247 256.3 1.788

0.0123 254.9 1.716

8.20.−3 253.3 1.673

6.15.−3 251.9 1.642

4.91.−3 250.5 1.617

2.49.−3 244.5 1.537

1.33.−3 236.9 1.451

9.81.−4 234.2 1.400

8.22.−4 234.7 1.365

7.29.−4 237.4 1.340

600

v h s

0.5953 282.7 2.133

0.0596 282.2 1.904

0.0298 281.7 1.834

0.0149 280.7 1.763

9.96.−3 279.7 1.721

7.49.−3 278.8 1.691

6.01.−3 277.9 1.667

3.07.−3 274.2 1.591

1.64.−3 269.6 1.511

1.18.−3 268.1 1.462

9.67.−4 269.1 1.428

8.44.−4 271.7 1.403

700

v h s

0.6946 307.5 2.171

0.0696 307.2 1.942

0.0348 306.9 1.873

0.0175 306.2 1.803

0.0117 305.6 1.761

8.80.−3 305.1 1.732

7.07.−3 304.5 1.708

3.62.−3 302.2 1.634

1.93.−3 299.8 1.557

1.38.−3 299.6 1.511

1.11.−3 301.1 1.478

9.56.−4 304.0 1.453

800

v h s

0.7939 332.3 2.204

0.0795 332.2 1.975

0.0399 331.9 1.906

0.0200 331.6 1.837

0.0134 331.2 1.795

0.0101 330.9 1.766

8.11.−3 330.5 1.743

4.16.−3 329.3 1.671

2.21.−3 328.6 1.596

1.57.−3 329.4 1.551

1.25.−3 331.5 1.518

1.07.−3 334.6 1.494

900

v h s

0.8931 357.1 2.233

0.0895 357.0 2.005

0.0448 356.9 1.936

0.0225 356.8 1.866

0.0151 356.6 1.825

0.0114 356.4 1.796

9.13.−3 356.3 1.773

4.68.−3 355.8 1.702

2.48.−3 356.3 1.628

1.75.−3 358.0 1.584

1.39.−3 360.7 1.553

1.18.−3 364.2 1.528

1000

v h s

0.9924 381.9 2.260

0.0994 381.9 2.031

0.0498 381.9 1.962

0.0250 381.8 1.893

0.0168 381.8 1.852

0.0126 381.8 1.823

0.0102 381.8 1.800

5.20.−3 381.9 1.729

2.74.−3 383.4 1.657

1.93.−3 385.9 1.614

1.53.−3 389.0 1.583

1.29.−3 392.8 1.559

*Values extracted and in some cases rounded off from those cited in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standards Press, Moscow, 1976. This source contains an exhaustive tabulation of values. v = specific volume, m3/kg; h = specific enthalpy, kJ/kg; s = specific entropy, kJ/(kg⋅K). The notation 3.49.−4 signifies 3.49 × 10−4. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.).

TABLE 2-278 T, K 453.7m 500 600 700 800 900 1000 1200 1400 1500

Saturated Lithium*

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

1.78.−13 8.21.−12 4.18.−9 3.51.−7 9.57.−6

1.912.−3 1.946.−3 1.988.−3 2.028.−3 2.070.−3

2.40.+7 9.94.+5

1703 1905 2334 2697 3174

24259 24390 24674 24869 25162

6.776 7.199 7.983 8.633 9.192

56.492 52.169 45.216 40.307 36.678

4.30 4.34 4.23 4.19 4.17

1.24.−4 9.60.−4 0.0204 0.1794 0.4269

2.114.−3 2.160.−3 2.262.−3 2.370.−3 2.433.−3

8.55.+4 1.22.+4 669.3 86.06 38.17

3590 4006 4835 5668 6088

25341 25477 25654 25778 25845

9.682 10.120 10.876 11.518 11.808

33.850 31.591 28.225 25.882 24.979

4.16 4.16 4.14 4.19 4.20

*Converted from tables in Vargaftik, Tables of the Thermophysical Properties of Liquids and Gases, Nauka, Moscow, 1972, and Hemisphere, Washington, 1975. m = melting point. The notation 1.78.−13 signifies 1.78 × 10−13. Many of the Vargaftik values also appear in Ohse, R. W., Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Sci. Pubs., Oxford, 1985 (1020 pp.). This source contains superheat data. Saturation and superheat tables and a diagram to 14 bar, 2200 K are given by Reynolds, W. C., Thermodynamic properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For a Mollier diagram from 0.1 to 140 psia, 2100–3600°R, see Weatherford, P. M., J. C. Tyler, et al., WADD-TR-61-96, 1961. An extensive review of properties of the solid and the saturated liquid was given by Alcock, C. B., M. W. Chase, et al., J. Phys. Chem. Ref. Data, 23, 3 (1994): 385–497.

TABLE 2-279

Lithium Bromide—Water Solutions

Ruiter, J. P., Rev. Int. Froid = Int. J. Refrig., 13 (1990): 223–236 gives subroutines for computer calculations. See also ASHRAE Handbook—Fundamentals.

2-250

PHYSICAL AND CHEMICAL DATA

TABLE 2-280

Saturated Mercury*

T, K

P, bar

vf × 105, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

hfg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

203.15 213.15 223.15 233.15 243.15

2.298 ⋅ 10−11 1.288 ⋅ 10−10 6.169 ⋅ 10−10 2.580 ⋅ 10−9 9.573 ⋅ 10−9

7.26239 7.27570 7.28900 7.30231 7.31563

3.665 ⋅ 109 6.862 ⋅ 108 1.499 ⋅ 108 3.746 ⋅ 107 1.053 ⋅ 107

33.131 34.567 35.997 37.422 38.842

342.637 343.674 344.710 345.746 346.782

309.506 309.107 308.713 308.324 307.940

0.32434 0.33124 0.33780 0.34404 0.35001

1.84787 1.78142 1.72123 1.66647 1.61647

253.15 263.15 273.15 283.15 293.15

3.198 ⋅ 10−8 9.736 ⋅ 10−8 2.728 ⋅ 10−7 7.101 ⋅ 10−7 1.729 ⋅ 10−6

7.32896 7.34229 7.35563 7.36898 7.38234

3.281 ⋅ 106 1.120 ⋅ 106 4.150 ⋅ 105 1.653 ⋅ 105 7.026 ⋅ 104

40.258 41.668 43.074 44.476 45.874

347.819 348.855 349.891 350.927 351.964

307.561 307.187 306.817 306.451 306.090

0.35571 0.36118 0.36642 0.37146 0.37631

1.57065 1.52852 1.48967 1.45375 1.42045

303.15 313.15 323.15 333.15 343.15

3.968 ⋅ 10−6 8.626 ⋅ 10−6 1.786 ⋅ 10−5 3.356 ⋅ 10−5 6.724 ⋅ 10−5

7.39572 7.40911 7.42252 7.43594 7.44938

3.167 ⋅ 104 1.505 ⋅ 104 7.501 ⋅ 103 3.905 ⋅ 103 2.115 ⋅ 103

47.268 48.659 50.046 51.430 52.810

353.000 354.036 355.072 356.108 357.145

305.732 305.377 305.026 304.678 304.335

0.38099 0.38550 0.38986 0.39408 0.39816

1.38951 1.36068 1.33378 1.30862 1.28505

353.15 363.15 373.15 383.15

1.232 ⋅ 10−4 2.182 ⋅ 10−4 3.745 ⋅ 10−4 6.247 ⋅ 10−4

7.46285 7.47633 7.48984 7.50337

1.188 ⋅ 103 6.899 ⋅ 102 413.0 254.2

54.188 55.563 56.936 58.306

358.181 359.217 360.253 361.289

303.993 303.654 303.317 302.983

0.40212 0.40596 0.40969 0.41331

1.26292 1.24213 1.22255 1.20408

393.15 403.15 413.15 423.15 433.15

1.015 ⋅ 10−3 1.608 ⋅ 10−3 2.491 ⋅ 10−3 3.778 ⋅ 10−3 5.618 ⋅ 10−3

7.51693 7.53052 7.55415 7.55780 7.57148

153.6 103.9 68.75 46.43 31.96

59.674 61.039 62.403 63.765 65.125

362.326 363.362 364.397 365.433 366.469

302.652 302.323 301.994 301.668 301.344

0.41684 0.42027 0.42361 0.42687 0.43004

1.18665 1.17017 1.15456 1.13978 1.12575

443.15 453.15 463.15 473.15 483.15

8.204 ⋅ 10−3 1.178 ⋅ 10−2 1.664 ⋅ 10−2 2.315 ⋅ 10−2 3.177 ⋅ 10−2

7.58520 7.59897 7.61277 7.62662 7.64051

66.484 67.842 69.198 70.553 71.908

367.504 368.539 369.574 370.609 371.642

301.020 300.697 300.376 300.056 299.734

0.43314 0.43617 0.43913 0.44203 0.44486

1.11242 1.09975 1.08768 1.07619 1.06524

493.15 503.15 513.15 523.15 533.15

4.304 ⋅ 10−2 5.758 ⋅ 10−2 7.614 ⋅ 10−2 9.959 ⋅ 10−2 0.12892

7.65444 7.66843 7.68247 7.69656 7.71071

4.748 3.621 2.793 2.176 1.7132

73.261 74.614 75.967 77.319 78.671

372.676 373.708 374.740 375.771 376.800

299.415 299.094 298.773 298.452 298.129

0.44763 0.45035 0.45301 0.45562 0.45818

1.05478 1.04479 1.03525 1.02611 1.01737

543.15 553.15 563.15 573.15 583.15

0.16527 0.20993 0.26435 0.33015 0.40910

7.72491 7.73918 7.75351 7.7679 7.7823

1.3613 1.0912 0.88213 0.71874 0.59002

80.023 81.375 82.728 84.080 85.434

377.829 378.855 379.880 380.904 381.925

297.806 297.480 297.152 296.824 296.491

0.46069 0.46316 0.46558 0.46796 0.47030

1.00899 1.00095 0.99324 0.98584 0.97893

593.15 603.15 613.15 623.15 633.15

0.50320 0.61460 0.74567 0.89896 1.0772

7.7969 7.8115 7.8262 7.8409 7.8558

0.48779 0.40600 0.34008 0.28660 0.24291

86.788 88.143 89.499 90.856 92.215

382.944 383.960 384.973 385.984 386.991

296.156 295.817 295.474 295.128 294.776

0.47260 0.47487 0.47709 0.47929 0.48145

0.97190 0.96532 0.95899 0.95289 0.94702

643.15 653.15 663.15 673.15 683.15

1.2834 1.5207 1.9725 2.1024 2.454

7.8707 7.8858 7.9008 7.9160 7.9313

0.20702 0.17735 0.15269 0.13207 0.11476

93.575 94.937 96.300 97.666 99.033

387.994 388.994 389.989 390.980 391.966

294.419 294.057 293.689 293.314 292.933

0.48358 0.48568 0.48774 0.48978 0.49180

0.94135 0.93589 0.93061 0.92552 0.92059

693.15 703.15 713.15 723.15 733.15

2.852 3.299 3.801 4.362 4.986

7.9467 7.9622 7.9778 7.9935 8.0094

0.10014 0.08775 0.07719 0.06815 0.06039

100.403 101.775 103.150 104.528 105.908

392.947 393.923 394.893 395.858 396.816

292.544 292.148 291.743 291.330 290.908

0.49378 0.49574 0.49768 0.49959 0.50148

0.91583 0.91123 0.90677 0.90245 0.89827

743.15 753.15 763.15 773.15 783.15

5.679 6.446 7.292 8.222 9.242

8.0252 8.0413 8.0574 8.074 8.090

0.05369 0.04789 0.04285 0.03846 0.03462

107.292 108.679 110.069 111.463 112.861

397.767 398.711 399.649 400.579 401.501

290.475 290.032 289.580 289.116 288.640

0.50335 0.50519 0.50702 0.50882 0.51061

0.89422 0.89029 0.88647 0.88277 0.87917

793.15 803.15 813.15 823.15 833.15

10.358 11.576 12.901 14.340 15.899

8.106 8.123 8.140 8.157 8.174

0.03124 0.02827 0.02565 0.02333 0.02126

114.262 115.668 117.078 118.492 119.911

402.415 403.321 404.218 405.106 405.985

288.153 287.653 287.140 286.614 286.074

0.51238 0.51412 0.51586 0.51757 0.51927

0.87568 0.87228 0.86898 0.86576 0.86263

22.39 15.95 11.54 8.469 6.301

*From Vukalovich, Ivanov, Fokin, and Yakovlev, Thermophysical Properties of Mercury, Standartov, Moscow, 1971. For the saturated liquid the specific volume at 203.15 K is 7.26239 × 10−5 m3/kg, etc. All the tabular values for 203.15 K, 213.15 K, 223.15 K, and 233.15 K represent a metastable equilibrium between the subcooled liquid and the saturated vapor. Saturation and superheat tables and a diagram to 100 bar, 1600 K are given by Reynolds, W. C., Thermodynamic properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For a Mollier diagram from 1 to 8200 psia and 2700°R, see Weatherford, W. D., J. C. Tyler, et al., WADD-TR-61-96, 1961.

THERMODYNAMIC PROPERTIES

2-251

Saturated Mercury* (Concluded )

TABLE 2-280 T, K

P, bar

vf × 105, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

hfg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

843.15 853.15 863.15 873.15 883.15

17.584 19.403 21.36 23.46 25.72

8.191 8.209 8.226 8.244 8.262

0.019426 0.017785 0.016317 0.015000 0.013815

121.335 122.763 124.197 125.636 127.080

406.855 407.715 408.565 409.405 410.235

285.520 284.952 284.368 283.769 283.155

0.52095 0.52262 0.52427 0.52591 0.52753

0.85959 0.85662 0.85372 0.85090 0.84815

893.15 903.15 913.15 923.15 933.15

28.14 30.72 33.47 36.41 39.53

8.280 8.298 8.316 8.335 8.353

0.012748 0.011784 0.010911 0.010120 0.009401

128.530 129.986 131.448 132.915 134.389

411.054 411.861 412.658 413.444 414.218

282.524 281.875 281.210 280.529 279.829

0.52914 0.53074 0.53232 0.53389 0.53545

0.84546 0.84284 0.84028 0.83777 0.83533

943.15 953.15 963.15 973.15 983.15

42.85 46.36 50.09 54.03 58.20

8.372 8.391 8.410 8.430 8.450

0.008746 0.008150 0.007604 0.007105 0.006648

135.869 137.356 138.850 140.350 141.858

414.980 415.731 416.469 417.195 417.909

279.111 278.375 277.619 276.845 276.051

0.53700 0.53854 0.54006 0.54158 0.54308

0.83294 0.83060 0.82831 0.82606 0.82387

993.15 1003.15 1013.15 1023.15 1033.15

62.59 67.22 72.10 77.22 82.60

8.468 8.488 8.508 8.529 8.550

0.006228 0.005842 0.005487 0.005159 0.004856

143.372 144.894 146.424 147.961 149.506

418.610 419.298 419.974 420.636 421.286

275.238 274.404 273.550 272.675 271.780

0.54458 0.54607 0.54754 0.54901 0.55047

0.82172 0.81961 0.81754 0.81552 0.81353

1043.15 1053.15 1063.15 1073.15

88.25 94.17 100.37 106.85

8.570 8.590 8.612 8.632

0.004576 0.004317 0.004077 0.003854

151.059 152.619 154.188 155.766

421.923 422.546 423.156 423.752

270.864 269.927 268.968 267.986

0.55192 0.55336 0.55479 0.55621

0.81158 0.80966 0.80778 0.80593

TABLE 2-281

Saturated Methane*

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

90.7 95 100 105 110

0.117 0.198 0.345 0.565 0.884

2.215.−3 2.244.−3 2.278.−3 2.316.−3 2.353.−3

3.976 2.463 1.479 0.940 0.625

216.4 232.5 246.3 263.2 280.1

759.9 769.0 776.9 785.7 794.5

4.231 4.406 4.556 4.719 4.882

10.225 10.034 9.862 9.710 9.558

3.288 3.318 3.369 3.425 3.478

2.02 1.71 1.56 1.33 1.22

0.225 0.215 0.206 0.197 0.189

115 120 125 130 135

1.325 1.919 2.693 3.681 4.912

2.396.−3 2.438.−3 2.487.−3 2.536.−3 2.594.−3

0.430 0.306 0.223 0.167 0.127

297.7 315.3 333.5 351.7 370.6

802.5 810.4 817.3 824.1 829.5

5.035 5.188 5.332 5.476 5.614

9.436 9.314 9.062 8.810 8.871

3.525 3.570 3.620 3.679 3.755

1.09 0.98 0.89 0.81 0.73

0.181 0.173 0.165 0.158 0.150

140 145 150 155 160

6.422 8.246 10.41 12.97 15.94

2.652.−3 2.722.−3 2.792.−3 2.882.−3 2.971.−3

0.098 0.077 0.061 0.049 0.039

389.5 409.5 429.4 450.8 472.1

834.8 844.4 853.9 848.5 843.0

5.751 5.885 6.019 6.151 6.283

8.932 8.891 8.849 8.725 8.601

3.849 3.965 4.101 4.27 4.47

0.66 0.61 0.56 0.51 0.46

0.143 0.136 0.129 0.122 0.115

165 170 175 180 185

19.39 23.81 27.81 32.86 38.59

3.095.−3 3.218.−3 3.419.−3 3.619.−3 3.979.−3

0.032 0.026 0.020 0.016 0.012

495.4 518.6 545.8 572.9 605.4

840.0 837.0 827.6 818.1 797.7

6.417 6.551 6.697 6.843 7.017

8.513 8.424 8.315 8.205 8.049

4.75 5.16 5.89 7.27 11.1

0.42 0.38 0.34 0.30 0.25

0.108 0.101 0.094 0.088 0.085

190 190.6c

45.20 45.99

4.900.−3 6.233.−3

0.008 0.006

661.6 704.4

750.7 704.4

7.293 7.516

7.762 7.516

70. ∞

0.19 0.17

0.090 ∞

T, K t

*Values reproduced or converted from Goodwin, NBS Tech. Note 653, 1974. t = triple point; c = critical point. The notation 2.215.−3 signifies 2.215 × 10−3.

2-252

PHYSICAL AND CHEMICAL DATA

FIG. 2-12

2-280.)

Enthalpy-log-pressure diagram for mercury. (Drawn from tabular data in footnote reference to Table

THERMODYNAMIC PROPERTIES TABLE 2-282

2-253

Superheated Methane* Temperature, K

P, bar

100

150

200

250

300

350

400

450

500

1

v h s

0.00228 246.4 4.555

0.7661 879.0 10.152

1.0299 984.3 10.757

1.2915 1090.4 11.230

1.5521 1199.8 11.629

1.8122 1314.8 11.983

2.0719 1437.4 12.310

2.3669 1568.8 12.618

2.5911 1708.9 12.914

5

v h s

0.00228 247.0 4.553

0.1434 865.0 9.256

0.2006 976.1 9.896

0.2549 1084.7 10.381

0.3083 1195.5 10.785

0.3611 1311.5 11.142

0.4136 1434.7 11.471

0.4657 1566.6 11.781

0.5181 1706.9 12.066

10

v h s

0.00227 247.8 4.549

0.0643 843.6 8.797

0.0968 965.5 9.501

0.1254 1077.9 10.002

0.1528 1190.6 10.414

0.1798 1307.9 10.775

0.2063 1432.0 11.106

0.2327 1564.1 11.417

0.2590 1705.3 11.715

20

v h s

0.00227 249.4 4.542

0.00277 429.8 6.003

0.0446 941.9 9.059

0.0606 1063.6 9.603

0.0751 1180.7 10.030

0.0891 1300.6 10.400

0.1027 1426.5 10.736

0.1162 1560.3 11.050

0.1295 1702.1 11.349

40

v h s

0.00226 252.5 4.528

0.00274 430.8 5.973

0.0176 879.3 8.465

0.0281 1032.9 9.155

0.0363 1160.5 9.621

0.0438 1286.0 10.008

0.0510 1415.7 10.354

0.0579 1552.1 10.674

0.0648 1696.0 10.978

60

v h s

0.00226 255.7 4.515

0.00271 432.2 5.946

0.00615 734.0 7.623

0.0173 999.8 8.847

0.0234 1140.0 9.359

0.0287 1271.7 9.765

0.0338 1405.1 10.121

0.0386 1544.2 10.440

0.0432 1690.0 10.756

80

v h s

0.00225 258.9 4.502

0.00268 433.8 5.920

0.00411 660.5 7.209

0.0119 964.4 8.590

0.0171 1119.7 9.158

0.0213 1257.7 9.584

0.0252 1394.9 9.951

0.0289 1536.6 10.283

0.0324 1684.4 10.595

100

v h s

0.00224 262.1 4.489

0.00266 435.5 5.897

0.00375 644.5 7.090

0.00888 928.5 8.364

0.0133 1099.6 8.991

0.0169 1244.2 9.437

0.0201 1385.2 9.814

0.0231 1529.4 10.153

0.0260 1679.0 10.469

150

v h s

0.00223 270.2 4.458

0.00261 440.7 5.843

0.00337 630.2 6.930

0.00555 860.0 7.953

0.00852 1054.1 8.664

0.0111 1213.1 9.155

0.0134 1362.8 9.555

0.0155 1513.0 9.907

0.0175 1667.0 10.233

200

v h s

0.00221 278.3 4.429

0.00256 446.5 5.796

0.00318 626.5 6.829

0.00447 825.0 7.719

0.00644 1019.8 8.426

0.00837 1187.2 8.944

0.0101 1343.8 9.362

0.0118 1498.9 9.727

0.0133 1656.9 10.060

300

v h s

0.00218 294.7 4.373

0.00249 459.6 5.714

0.00296 629.2 6.690

0.00369 804.4 7.471

0.00474 982.9 8.122

0.00593 1153.6 8.649

0.00708 1316.8 9.085

0.00818 1478.5 9.465

0.00924 1642.2 9.811

400

v h s

0.00244 473.8 5.645

0.00282 637.7 6.588

0.00336 802.4 7.323

0.00406 970.1 7.935

0.00486 1137.8 8.451

0.00569 1303.0 8.893

0.00560 1467.7 9.280

0.00729 1634.7 9.633

500

v h s

0.00239 488.8 5.584

0.00272 648.9 6.507

0.00315 807.7 7.215

0.00368 969.0 7.802

0.00428 1132.8 8.307

0.00492 1297.8 8.748

0.00555 1464.2 9.139

0.00616 1633.2 9.496

*Converted and rounded off from the tables of Goodwin, NBS Tech. Note 654, 1974. v = specific volume, m3/kg; h = specific enthalpy, kJ/kg; s = specific entropy, kJ/(kg⋅K). For a thermodynamic diagram from 0.1 to 400 bar and 620°C, see the 1993 ASHRAE Handbook—Fundamentals (SI ed.). Saturation and superheat tables and a chart to 6000 psia, 680°F appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993. See also Friend, D. G., J. F. Ely, et al., J. Phys. Chem. Ref. Data. 18, 2 (1989): 583–638.

2-254

PHYSICAL AND CHEMICAL DATA

TABLE 2-283 Pressure, bar

Thermophysical Properties of Saturated Methanol kf, W/(m⋅K)

Prf

2.531 2.554 2.669 2.777

625 525 401 329

0.204 0.196 0.193 0.189

7.75 6.84 5.55 4.83

7.5379 7.4836 7.4398 7.4051 7.3474

2.845 2.894 2.946 2.984 3.050

288 268 242 227 204

0.186 0.184 0.182 0.181 0.179

4.41 4.22 3.92 3.74 3.48

4.7307 4.7836 4.8678 4.9366 5.0708

7.2992 7.2624 7.1988 7.1471 7.0461

3.117 3.176 3.265 3.349 3.540

187 174 156 141 117

0.178 0.177 0.175 0.173 0.171

3.27 3.12 2.91 2.73 2.42

1553.8 1486.4 1474.7 1450.1 1423.2

5.1744 5.2605 5.3355 5.4650 5.5793

6.9677 6.9017 6.8435 6.7409 6.6475

3.72 3.91 4.12 4.67 5.55

102 92 84 72 63

0.169 0.167 0.165 0.160 0.154

2.25 2.15 2.10 2.10 2.27

1391.8 1318.7 1186.8

5.6889 5.8803 6.0979

6.5543 6.3791 6.0979

hf, kJ/kg

hg, kJ/kg

0.001 0.001 0.001 0.001 0.001

057 257 276 307 336

1700000 7.309 3.801 1.599 0.819

0.0 261.0 293.9 345.0 391.7

1303.1 1440.3 1455.4 1476.2 1492.1

2.8114 3.9383 4.0493 4.2117 4.3516

10.2328 8.0281 7.9032 7.7386 7.6104

1.5 2.0 2.5 3.0 4.0

348.0 356.0 362.5 368.0 377.1

0.001 0.001 0.001 0.001 0.001

356 371 385 396 417

0.5632 0.4276 0.3443 0.2893 0.2188

421.0 444.2 463.6 479.8 507.8

1500.3 1505.8 1509.8 1512.4 1515.9

4.4361 4.5014 4.5536 4.5992 4.6728

5 6 8 10 15

384.5 390.8 401.3 409.8 426.3

0.001 0.001 0.001 0.001 0.001

434 450 479 504 560

0.17569 0.14683 0.11015 0.08783 0.05761

529.7 549.6 582.7 610.3 665.8

1517.4 1518.4 1518.0 1516.1 1507.9

20 25 30 40 50

438.9 449.3 458.2 472.9 484.9

0.001 0.001 0.001 0.001 0.001

611 666 710 814 934

0.04224 0.03290 0.02661 0.01863 0.01373

710.5 749.0 783.8 846.7 905.2

60 80 80.95c

495.1 508.1 512.6

0.002 086 0.002 507 0.003 715

0.01032 0.00642 0.00372

963.3 1065.3 1186.8

t

Temp., K

vf, m3/kg

µf, 10−6 Pa·s

vg, m3/kg

175.6 288.4 301.7 320.7 337.7

4 × 10−6 0.1 0.2 0.5 1.013

sf, kJ/(kg⋅K) sg, kJ/(kg⋅K) cpf, kJ/(kg⋅K)

t = triple point; c = critical point. v, h, s, and cp interpolated and converted from Goodwin, R. D., J. Phys. Chem. Ref. Data, 16, 4 (1987): 799–891.

THERMODYNAMIC PROPERTIES TABLE 2-284

2-255

Thermodynamic Properties of Compressed Methanol Temperature, K

Pressure, bar

200

250

300

350

v (m3/ kg) 0.1 h (kJ/ kg) s (kJ/ kg·K)

0.001137 57.5 3.096

400

450

500

550

600

0.001203 169.8 3.597

7.630 1456.7 8.081

8.942 1529.5 8.305

10.23 1607.5 8.514

11.56 1691.5 8.711

12.84 1781.7 8.901

1878.1 9.085

15.45 1980.2 9.263

v (m3/ kg) 0.5 h (kJ/ kg) s (kJ/ kg·K)

0.001137 57.5 3.096

0.001202 169.9 3.597

0.001274 290.5 4.038

1.764 1522.7 7.877

2.033 1603.0 8.091

2.296 1687.9 8.291

2.558 1778.9 8.482

2.818 1875.3 8.666

3.078 1977.4 8.844

v (m3/ kg) 1.013 h (kJ/ kg) s (kJ/ kg·K)

0.001137 57.6 3.096

0.001202 169.9 3.597

0.001274 290.5 4.038

0.8560 1514.0 7.675

0.9958 1598.7 7.902

1.1283 1685.1 8.105

1.2843 1795.4 8.117

1.3870 1873.5 8.482

1.5157 1975.8 8.660

v (m3/ kg) 10 h (kJ/ kg) s (kJ/ kg·K)

0.001136 58.4 3.095

0.001201 170.7 3.596

0.001272 291.2 4.036

0.001357 427.4 4.451

0.001474 578.8 4.857

0.1068 1638.1 7.427

0.1236 1751.5 7.667

0.1381 1858.0 7.870

0.1519 1965.2 8.056

v (m3/ kg) 15 h (kJ/ kg) s (kJ/ kg·K)

0.001136 58.8 3.094

0.001201 171.1 3.595

0.001272 291.6 4.035

0.001356 427.7 4.450

0.001472 578.9 4.856

0.0673 1601.9 7.253

0.0806 1735.6 7.536

0.0911 1849.4 7.752

0.1007 1960.2 7.946

v (m3/ kg) 20 h (kJ/ kg) s (kJ/ kg·K)

0.001135 59.2 3.094

0.001200 171.6 3.595

0.001271 292.0 4.035

0.001355 428.1 4.449

0.001469 579.0 4.854

0.0466 1565.3 7.087

0.0589 1717.7 7.431

0.0675 1840.0 7.664

0.0751 1954.8 7.864

v (m3/ kg) 30 h (kJ/ kg) s (kJ/ kg·K)

0.001134 60.1 3.092

0.001199 172.4 3.593

0.001269 292.9 4.036

0.001355 428.8 4.447

0.001465 579.4 4.851

0.001659 751.3 5.264

0.0367 1675.4 7.253

0.0436 1818.7 7.526

0.0492 1942.7 7.743

v (m3/ kg) 40 h (kJ/ kg) s (kJ/ kg·K)

0.001133 61.0 3.091

0.001198 173.3 3.592

0.001268 293.7 4.032

0.001350 429.5 4.445

0.001461 579.8 4.849

0.001649 750.4 5.258

0.0251 1623.0 7.088

0.0314 1794.2 7.414

0.0361 1928.8 7.650

v (m3/ kg) 50 h (kJ/ kg) s (kJ/ kg·K)

0.001133 61.9 3.090

0.001197 174.2 3.591

0.001266 294.5 4.030

0.001348 430.2 4.443

0.001457 580.2 4.846

0.001637 749.7 5.579

0.0176 1556.4 6.912

0.0239 1766.7 7.314

0.0282 1913.4 7.570

v (m3/ kg) 60 h (kJ/ kg) s (kJ/ kg·K)

0.001131 62.8 3.089

0.001196 175.0 3.589

0.001265 295.3 4.029

0.001346 430.9 4.442

0.001453 580.6 4.843

0.001628 749.1 5.248

0.0120 1461.8 6.692

0.0188 1736.1 7.220

0.0228 1896.6 7.500

v (m3/ kg) 75 h (kJ/ kg) s (kJ/ kg·K)

0.001130 64.1 3.087

0.001194 176.3 3.587

0.001263 296.6 4.027

0.001343 431.9 4.439

0.001448 581.2 4.839

0.001614 748.3 5.241

0.002084 982.1 5.718

0.01359 1683.9 7.081

0.0174 1869.1 7.405

v (m3/ kg) 100 h (kJ/ kg) s (kJ/ kg·K)

0.001128 66.3 3.084

0.001191 178.5 3.584

0.001259 298.6 4.023

0.001337 433.8 4.435

0.001439 582.4 4.833

0.001595 747.5 5.230

0.001952 964.8 5.673

0. 1572.9 6.829

0.01188 1818.8 7.261

v (m3/ kg) 150 h (kJ/ kg) s (kJ/ kg·K)

0.001125 70.7 3.078

0.001186 182.8 3.578

0.001252 302.8 4.016

0.001328 437.4 4.426

0.001423 584.9 4.822

0.001562 746.8 5.211

0.001825 948.4 5.622

1248.8 6.302

0.006513 1704.3 6.997

v (m3/ kg) 200 h (kJ/ kg) s (kJ/ kg·K)

0.001121 75.1 3.071

0.001182 187.2 3.571

0.001246 307.0 4.009

0.001317 441.2 4.418

0.001408 587.8 4.811

0.001535 747.0 5.194

0.001751 939.9 5.587

0.002314 1223.5 6.125

0.004091 1583.5 6.752

v (m3/ kg) 300 h (kJ/ kg) s (kJ/ kg·K)

0.001113 83.9 3.060

0.001172 195.9 3.559

0.001234 315.4 3.996

0.001302 448.9 4.403

0.001384 593.8 4.791

0.001492 749.4 5.166

0.001656 932.0 5.537

0.001957 1173.4 5.996

0.002600 1443.5 6.466

v (m3/ kg) 400 h (kJ/ kg) s (kJ/ kg·K)

0.001107 92.7 3.048

0.001164 204.7 3.548

0.001223 324.0 3.983

0.001288 456.9 4.388

0.001363 600.5 4.774

0.001459 753.4 5.142

0.001593 929.6 5.500

0.001808 1154.4 5.926

0.002182 1388.1 6.335

v (m3/ kg) 500 h (kJ/ kg) s (kJ/ kg·K)

0.001101 101.5 3.037

0.001156 213.4 3.536

0.001213 332.6 3.971

0.001274 465.1 4.375

0.001345 607.7 4.757

0.001431 758.4 5.121

0.001546 930.4 5.470

0.001716 1145.6 5.880

0.001980 1360.6 6.254

Converted and interpolated from Goodwin, R. D., J. Phys. Chem. Ref. Data, 16, 4 (1987): 799–891. These extensive tables extend to 700 bar for temperatures from 175.6 to 800 K. Another extensive compilation is by deReuck, K. M. and R. J. B. Craven, Methanol, C.R.C. Press, 1993 (320 pp.). Equations and diagrams to 30 bar, 200°C are given by Eicholz, H. D., S. Schulz, et al., Kalte u Klim., no. 9, (1981) 322–331. For pressures to 1040 bar, 298–489 K, see Machado, J. R. S. and W. B. Street, J.Chem.Eng.Data, 28 (1983): 218–223; to 2800 bar from 273 to 333 K, see Sun, T., S. N. Biswas, et al., J. Chem. Eng. Data, 33 (1988): 395–398. Dissociation was considered by Yerlett, T. K. and C. J. Wormald, J. Chem. Thermo., 18 (1986): 719–726, and by Kazarnovskii, Ya. S. and E. V. Pavlova, Russ. J. Phys. Chem., 56, 6 (1982): 847–851.

TABLE 2-285

Saturated Methyl Chloride* µf, 10−4 Pa⋅s

kf, W/(m⋅K)

1.483 1.486 1.489 1.492 1.496

4.44 4.27 4.11 3.96 3.82

0.241 0.236 0.232 0.228 0.224

5.896 5.866 5.845 5.822 5.786

1.500 1.504 1.508 1.513 1.518

3.69 3.57 3.46 3.35 3.25

0.219 0.215 0.211 0.207 0.202

4.050 4.080 4.110 4.139 4.168

5.762 5.740 5.720 5.699 5.680

1.523 1.528 1.533 1.539 1.546

3.16 3.08 3.00 2.92 2.85

0.198 0.194 0.190 0.186 0.182

824.4 826.8 829.0 831.2 833.2

4.197 4.225 4.253 4.280 4.308

5.662 5.644 5.628 5.612 5.597

1.554 1.565 1.574 1.583 1.594

2.78 2.72 2.66 2.61 2.56

0.177 0.173 0.169 0.165 0.160

461.2 469.3 477.4 485.6 493.8

835.2 837.0 838.8 840.5 841.9

4.334 4.361 4.388 4.414 4.440

5.581 5.567 5.553 5.540 5.527

1.605 1.617 1.631 1.644 1.658

2.51 2.46 2.42 2.37 2.33

0.156 0.152 0.148 0.143 0.139

0.0386 0.0343 0.0282 0.0228 0.0186

502.1 510.4 518.8 538.3 562.9

843.3 844.5 846.4 847.5 847.6

4.465 4.491 4.542 4.592 4.643

5.516 5.504 5.481 5.457 5.434

2.30 2.27 2.12 1.99 1.87

0.135 0.131 0.124 0.117 0.110

13.47.−4 14.11.−4 14.67.−4 15.66.−4 16.48.−4

0.0151 0.0117 0.0096 0.0075 0.0063

581.6 602.8 622.9 643.6 663.2

845.9 842.6 837.4 826.4 819.1

4.694 4.747 4.805 4.870 4.904

5.398 5.382 5.358 5.323 5.289

1.77 1.67 1.59 1.51

0.103 0.095 0.086 0.075

17.97.−4 21.10.−4 27.40.−4

0.0052 0.0038 0.0027

677.3 714.1 749.3

807.1 778.6 749.3

4.954 5.025 5.116

5.256 5.200 5.116

T, K

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

175 180 185 190 195

0.0117 0.0165 0.0233 0.0327 0.0462

8.84.−4 8.91.−4 8.97.−4 9.04.−4 9.10.−4

27.90 19.85 14.12 10.12 7.208

274.5 280.9 287.5 294.5 301.7

764.3 767.7 771.0 774.3 777.5

3.529 3.570 3.603 3.647 3.684

6.328 6.274 6.222 6.172 6.124

1.469 1.472 1.475 1.477 1.480

200 205 210 215 220

0.0653 0.0919 0.1315 0.181 0.243

9.17.−4 9.25.−4 9.33.−4 9.40.−4 9.48.−4

5.137 3.835 2.656 1.975 1.505

309.0 316.3 323.7 331.0 338.4

780.7 783.9 787.0 790.1 793.2

3.722 3.756 3.791 3.825 3.859

6.080 6.038 5.998 5.961 5.928

225 230 235 240 245

0.319 0.417 0.539 0.688 0.866

9.56.−4 9.65.−4 9.73.−4 9.81.−4 9.89.−4

1.168 0.911 0.718 0.572 0.462

345.7 353.1 360.5 368.0 375.6

796.3 799.3 802.3 805.3 808.2

3.892 3.925 3.957 3.988 4.019

250 255 260 265 270

1.076 1.328 1.627 1.970 2.364

9.98.−4 10.08.−4 10.18.−4 10.27.−4 10.36.−4

0.377 0.311 0.257 0.215 0.1807

383.2 390.7 398.3 406.0 413.7

811.1 814.0 816.8 819.4 822.0

275 280 285 290 295

2.830 3.347 3.936 4.612 5.361

10.46.−4 10.57.−4 10.68.−4 10.79.−4 10.91.−4

0.1524 0.1301 0.1115 0.0960 0.0830

421.5 429.4 437.3 445.2 453.2

300 305 310 315 320

6.189 7.110 8.111 9.243 10.47

11.03.−4 11.15.−4 11.27.−4 11.40.−4 11.55.−4

0.0723 0.0632 0.0556 0.0489 0.0433

325 330 340 350 360

11.78 13.27 16.52 20.53 25.29

11.70.−4 11.86.−4 12.17.−4 12.54.−4 12.97.−4

370 380 390 400 405

30.74 36.99 44.05 52.29 56.6

410 415 416c

61.5 67.4 69.0

*Interpolated by P. E. Liley from the Landolt-Börnstein band IVa, p. 677, 1967 tables by Steinle/Dienemann. c = critical point. The notation 8.84.−4 signifies 8.84 × 10−4. TABLE 2-286

Saturated Neon*

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

10 20 24.6m 24.6m 26

0.434 0.718

6.654.−4 6.823.−4 6.696.−4 8.012.−4 8.172.−4

0.2266 0.1429

0.75 6.78 11.96 28.22 30.90

117.0 118.1

0.0992 0.4906 0.7257 1.388 1.494

5.006 4.846

0.278 0.945 1.345 1.802 1.868

1.57 1.37

0.146 0.132

28 30 32 34 36

1.321 2.238 3.552 5.352 7.728

8.413.−4 8.687.−4 9.001.−4 9.370.−4 9.820.−4

0.0817 0.0501 0.0323 0.0217 0.0149

34.75 38.80 43.06 47.57 52.34

119.3 120.1 120.6 120.6 119.9

1.634 1.771 1.905 2.036 2.166

4.653 4.483 4.329 4.184 4.043

1.955 2.052 2.163 2.302 2.506

1.16 1.00 0.84 0.71 0.59

0.124 0.115 0.106 0.097 0.088

1.039.−3 1.116.−3 1.232.−3 1.538.−3 2.070.−3

0.0104 0.0073 0.0050 0.0031 0.0021

57.52 63.33 69.82 80.83 92.50

118.4 115.8 111.8 103.0 92.5

2.297 2.435 2.582 2.812 3.062

3.900 3.749 3.582 3.316 3.062

2.825 3.436 5.26 25.0 ∞

0.48 0.38 0.31 0.25

0.078 0.069 0.059

T, K

38 40 42 44 44.4c

10.78 14.62 19.39 25.22 26.53

∞

*Values extracted and in some cases rounded off from those cited in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standards Press, Moscow, 1976. m = melting point; c = critical point. The notation 6.654.−4 signifies 6.654 × 10−4. This source contains values for the compressed state up to 1000 bar, etc. This book was published in English translation by Hemisphere, New York 1988 (604 pp.). Saturation and superheat tables and a diagram to 200 bar, 320 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). Saturation and superheat tables to 60,000 psia, 900°R and a chart to 4000 psia, 560°R appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993. 2-256

THERMODYNAMIC PROPERTIES TABLE 2-287

2-257

Compressed Neon* Pressure, bar

Temperature, K

1

10

20

40

60

80

100

200

400

600

800

1000

v 100 h s

0.4117 195.4 6.129

0.0410 194.0 5.168

0.0204 192.4 4.869

0.0102 189.4 4.556

6.76.−3 186.6 4.363

5.08.−3 184.0 4.221

4.09.−3 181.6 4.106

2.22.−3 174.1 3.739

1.42.−3 173.3 3.386

1.18.−3 180.0 3.197

1.06.−3 189.2 3.066

9.74.−4 199.2 2.964

v 200 h s

0.8243 298.5 6.844

0.0828 298.4 5.893

0.0416 298.4 5.605

0.0210 298.2 5.315

0.0142 298.2 5.143

0.0107 298.2 5.020

8.69.−3 298.3 4.924

4.61.−3 299.4 4.620

2.61.−3 304.6 4.308

1.95.−3 312.4 4.124

1.63.−3 321.8 3.994

1.43.−3 332.1 3.893

v 300 h s

1.236 401.6 7.262

0.1241 401.8 6.312

0.0624 402.2 6.026

0.0315 402.8 5.739

0.0212 403.5 5.570

0.0160 404.1 5.450

0.0129 404.9 5.357

6.77.−3 408.8 5.065

3.71.−3 417.8 4.769

2.69.−3 427.8 4.593

2.18.−3 438.5 4.469

1.87.−3 449.7 4.372

v 400 h s

1.648 504.6 7.558

0.1654 505.0 6.609

0.0830 505.4 6.323

0.0418 506.4 6.037

0.0281 507.4 5.896

0.0212 508.3 5.750

0.0171 509.3 5.657

8.88.−3 514.4 5.369

4.77.−3 525.3 5.078

3.40.−3 536.7 4.907

2.72.−3 548.4 4.785

2.30.−3 560.2 4.690

v 500 h s

2.060 607.6 7.788

0.2066 608.1 6.839

0.1036 608.6 6.553

0.0521 609.7 6.267

0.0350 610.8 6.100

0.0264 611.9 5.981

0.0213 613.0 5.889

0.0110 618.8 5.601

5.82.−3 630.7 5.313

4.10.−3 642.9 5.144

3.24.−3 655.2 5.023

2.73.−3 667.5 4.929

v 600 h s

2.472 710.6 7.975

0.2478 711.1 7.027

0.1242 711.7 6.741

0.0625 712.9 6.455

0.0419 714.1 6.288

0.0316 715.3 6.169

0.0254 716.5 6.077

0.0130 722.5 5.791

6.85.−3 735.0 5.504

4.80.−3 747.8 5.335

3.77.−3 760.5 5.215

3.15.−3 773.2 5.122

v 700 h s

2.884 813.5 8.134

0.2890 814.1 7.186

0.1449 814.7 6.900

0.0728 816.0 6.614

0.0487 817.2 6.447

0.0367 818.5 6.328

0.0295 819.7 6.236

0.0151 826.0 5.950

7.89.−3 838.9 5.664

5.49.−3 851.9 5.496

4.29.−3 865.0 5.376

3.57.−3 878.1 5.284

v 800 h s

3.296 916.5 8.272

0.3302 917.1 7.323

0.1655 917.7 7.038

0.0831 919.0 6.752

0.0556 920.3 6.585

0.0419 921.6 6.466

0.0336 922.9 6.374

0.0172 929.3 6.088

8.92.−3 942.4 5.802

6.18.−3 955.7 5.634

4.81.−3 969.0 5.515

3.98.−3 982.2 5.423

v 900 h s

3.708 1020 8.393

0.3714 1020 7.444

0.1861 1021 7.159

0.0934 1022 6.873

0.0625 1023 6.706

0.0470 1025 6.588

0.0378 1026 6.496

0.0192 1033 6.210

9.96.−3 1046 5.924

6.87.−3 1059 5.756

5.32.−3 1073 5.637

4.40.−3 1086 5.545

v 1000 h s

4.120 1123 8.502

0.4126 1123 7.553

0.2067 1124 7.267

0.1037 1125 6.982

0.0693 1126 6.815

0.0522 1128 6.696

0.0419 1129 6.604

0.0213 1136 6.318

0.0110 1149 6.032

7.56.−3 1163 5.856

5.84.−3 1176 5.746

4.81.−3 1190 5.654

*Values extracted and in some cases rounded off from those cited in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standards Press, Moscow, 1976. This source contains an exhaustive tabulation of values. v = specific volume, m3/kg; h = specific enthalpy, kJ/kg; s = specific entropy, kJ/(kg⋅K). The notation 6.76.−3 signifies 6.76 × 10−3. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.). TABLE 2-288

Saturated Nitrogen (R728)*

T, K

P, bar

vf, 10–3 m3/kg

63.15t 65 70 75 77.35

0.1253 0.1743 0.3859 0.7609 1.0133

1.155 1.165 1.193 1.224 1.239

80 85 90 95 100

1.369 2.287 3.600 5.398 7.775

1.258 1.297 1.340 1.390 1.447

vg, m3/kg 1477 1091 525.6 281.8 216.9 164.0 101.7 66.28 44.87 31.26

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

−148.5 −144.9 −135.2 −125.4 −120.8

64.1 65.8 70.5 74.9 76.8

2.459 2.516 2.657 2.789 2.849

5.826 5.757 5.595 5.460 5.404

1.928 1.930 1.937 1.948 1.955

2.74 2.17 1.77 1.60

0.170 0.160 0.151 0.141 0.136

−115.6 −105.7 −95.6 −85.2 −74.5

78.9 82.3 85.0 86.8 87.7

2.913 3.032 3.147 3.256 3.363

5.345 5.244 5.152 5.067 4.985

1.964 1.989 2.028 2.086 2.176

1.48 1.27 1.10 0.97 0.87

0.132 0.123 0.114 0.105 0.097

0.79 0.71 0.60 0.48 0.32

0.088 0.080 0.071 0.063 0.052

105 110 115 120 125

10.83 14.67 19.40 25.15 32.05

1.514 1.597 1.714 1.892 2.324

22.23 15.98 11.47 8.031 5.016

−63.8 −51.4 −38.1 −21.4 5.1

87.4 85.6 81.8 74.3 57.2

3.469 3.575 3.687 3.821 4.024

4.904 4.820 4.729 4.619 4.444

2.319 2.566 3.063

126.25c

33.96

3.289

3.289

34.8

34.8

4.252

4.252

∞

∞

*Reproduced and converted from Vasserman and Rabinovich, Thermophysical Properties of Liquid Air and Its Components, Standartov, Moscow, 1968; and Israel Program for Scientific Translations, TT 69-55092, 1970. t = triple point; c = critical point. Other extensive tables are given by Angus, S., International Thermodynamic Tables of the Fluid State—6. Nitrogen, Pergamon, 1977 (244 pp.); Hanley, H. J. M., R. D. McCarty, et al., J. Phys. Chem. Ref. Data, 3 (1974): 979–1019. Saturation and superheat tables to 30,000 psia and a chart to 10,000 psia, all to 860°R, appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) has a thermodynamic chart for pressures from 0.1 to 800 bar and temperatures from 80 to 500 K.

2-258

FIG. 2-13

4729.)

PHYSICAL AND CHEMICAL DATA

Temperature-entropy diagram for nitrogen. Section of T-S diagram for nitrogen by E. S. Burnett, 1950. (Reprinted from U.S. Bur. Mines Rep. Invest.

TABLE 2-289

Thermophysical Properties of Nitrogen (R728) at Atmospheric Pressure

T (K) v (m3/ kg) h (kJ/ kg) s (kJ/ kg·K) cp (kJ/ kg·K) Z vs (m/s) η (10−6 Pa·s) k (W/m·K) NPr

77.4b 0.2164 76.7 5.403 1.341 0.9545 172 5.0 0.0074 0.913

80 0.2252 80.0 5.446 1.196 0.9610 177 5.2 0.0077 0.811

100 0.2871 101.9 5.690 1.067 0.9801 202 6.7 0.0098 0.728

120 0.3474 123.1 5.884 1.056 0.9883 222 8.0 0.0117 0.727

140 0.4071 144.2 6.046 1.050 0.9927 240 9.3 0.0136 0.723

160 0.4664 165.2 6.186 1.047 0.9952 257 10.6 0.0154 0.721

180 0.5255 186.1 6.309 1.045 0.9967 273 11.8 0.0171 0.720

200 0.5845 207.0 6.419 1.043 0.9977 288 12.9 0.0187 0.719

220 0.6434 227.8 6.519 1.043 0.9984 302 14.0 0.0203 0.718

240 0.7023 248.7 6.609 1.042 0.9990 316 15.0 0.0218 0.717

T (K) v (m3/ kg) h (kJ/ kg) s (kJ/ kg·K) cp (kJ/ kg·K) Z vs (m/s) η (10−6 Pa·s) k (W/m·K) NPr

260 0.7611 269.5 6.693 1.042 0.9994 329 16.0 0.0232 0.717

280 0.8199 290.3 6.770 1.041 0.9997 341 17.0 0.0247 0.716

300 0.8786 311.2 6.842 1.041 0.9998 359 17.9 0.0260 0.716

320 0.9371 332.0 6.909 1.042 0.9999 365 18.8 0.0273 0.717

340 0.9960 352.8 6.972 1.042 1.0000 376 19.7 0.0286 0.717

360 1.0546 373.7 7.032 1.043 1.0001 387 20.5 0.0299 0.717

380 1.1134 394.5 7.088 1.044 1.0002 397 21.4 0.0311 0.717

400 1.1719 411.5 7.142 1.045 1.0002 408 22.2 0.0324 0.717

420 1.2305 436.3 7.193 1.047 1.0002 417 23.0 0.0336 0.717

440 1.2892 457.3 7.242 1.048 1.0003 427 23.8 0.0347 0.717

T (K) v (m3/ kg) h (kJ/ kg) s (kJ/ kg·K) cp (kJ/ kg·K) Z vs (m/s) η (10−6 Pa·s) k (W/m·K) NPr

460 1.3481 478.3 7.288 1.051 1.0003 437 24.5 0.0359 0.718

480 1.4065 499.3 7.333 1.053 1.0004 446 25.3 0.0371 0.718

500 1.4654 520.4 7.376 1.056 1.0004 455 26.0 0.0383 0.718

600 1.758 626.9 7.570 1.075 1.000 496 29.5 0.0440 0.722

700 2.052 735.6 7.738 1.098 1.000 534 32.8 0.0496 0.726

800 2.344 846.6 7.886 1.122 1.000 568 35.9 0.0551 0.730

900 2.636 960.0 8.019 1.146 1.000 601 38.8 0.0606 0.734

1000 2.931 1075.7 8.141 1.167 1.001 631 41.6 0.658 0.737

1500 4.396 1680.5 8.630 1.244 1.001 765

2000 5.862 2313.5 8.995 1.284 1.001 879

b = normal boiling point. TABLE 2-290

Saturated Nitrogen Tetroxide vg, m3/kg

Mf

Mg

299.32 309.57 326.66 337.43 345.45

0.000 0.000 0.000 0.000 0.000

694 711 733 749 762

0.2996 0.1630 0.0876 0.0608 0.0469

91.857 91.886 91.766 91.625 91.488

79.157 76.503 73.538 71.748 70.480

10 15 20 30 40

351.88 364.09 373.17 386.57 396.52

0.000 0.000 0.000 0.000 0.000

774 800 822 863 903

0.0382 0.0262 0.0199 0.0133 0.0098

91.346 90.979 90.601 89.823 89.018

69.483 67.742 66.547 64.997 64.099

50 60 80 100

404.50 411.20 422.07 430.76

0.000 0.000 0.001 0.001

945 993 129 577

0.00761 0.00607 0.00394 0.00209

88.191 87.344 85.602 83.817

63.532 63.181 62.959 63.366

Pressure, bar

Temperature, K

1.0133 2 4 6 8

vf, m3/kg

Condensed from McCarty, R. D., H-U. Steurer, et al., NBS IR 86 - 3054, 1986 (106 pp.). M = mol wt for the reaction N2O4 A 2NO2 A 2NO + O2. No derived thermodynamic functions were tabulated due to unduly large differences in literature values, but 92 references are given. TABLE 2-291

Saturated Nitrous Oxide

Temp., °F

Pressure, psia

vf, ft3/lbm

vg, ft3/lbm

h f, Btu/lbm

h g, Btu/lbm

s f, Btu/lbm°R

sg, Btu/lbm°R

−127.2 −100 −80 −60 −40

14.70 33.68 56.79 90.29 136.68

0.01310 0.01358 0.01398 0.01444 0.01495

5.069 2.374 1.463 0.939 0.648

0.0 11.7 20.8 30.2 40.3

161.7 165.9 168.8 171.5 173.7

0.0000 0.0304 0.0534 0.0782 0.1044

0.4864 0.4591 0.4433 0.4315 0.4222

−20 0 20 40 60

198.62 278.97 380.88 507.51 662.69

0.01555 0.01625 0.01711 0.01819 0.01968

0.450 0.316 0.227 0.164 0.117

50.6 60.5 70.2 80.3 91.9

175.3 176.2 176.2 175.0 172.1

0.1296 0.1518 0.1718 0.1920 0.2145

0.4133 0.4036 0.3928 0.3815 0.3687

0.0222 0.0247 0.0354

0.0792 0.0611 0.0354

105.7 114.7 136.4

165.0 157.5 136.4

0.2382 0.2523 0.2890

0.3480 0.3302 0.2890

80 90 97.6c

851.5 961.0 1052.2

Rounded and condensed from Couch, E. J. and K. A. Kobe, Univ. Texas Rep., Cont. DAI-23-072-ORD-685, June 1, 1956. c = critical point. 2-259

2-260

PHYSICAL AND CHEMICAL DATA

Mollier diagram for nitrous oxide. (Fig. 9, Univ. Texas Rep., Cont. DAI-23-072-ORD685, June 1, 1956, by Couch and Kobe. Reproduced by permission.) Some irregularity in the compressibility factors from 80 to 160 atm, 50 to 100°C exists (Couch, private communication, 1967). See Couch et al., J. Chem. Eng. Data, 6, (1961) for PVT data.

FIG. 2-14

THERMODYNAMIC PROPERTIES TABLE 2-292

2-261

Nonane* h g, kJ/kg

s f, kJ/(kg⋅K)

s g, kJ/(kg⋅K)

c pf, kJ/(kg⋅K)

µ f, 10−4 Pa⋅s

k f, W/(m⋅K)

358.4 359.2 400.6 442.2 484.8

828.7 859.4

2.424 2.427 2.607 2.774 2.932

4.210 4.243

2.07 2.07 2.08 2.10 2.16

33.5 33.0 17.9 12.1 8.7

0.150 0.150 0.145 0.140 0.134

30.35 10.19 4.00 1.80 0.894

528.6 573.8 622.0 671.3 722.5

891.7 925.6 961.1 998.2 1036.5

3.083 3.229 3.370 3.511 3.650

4.282 4.324 4.368 4.419 4.476

1.591.−3 1.637.−3 1.690.−3 1.748.−3 1.815.−3

0.485 0.286 0.161 0.104 0.069

776.7 833.3 890.2 950.3 1012.1

1076.0 1116.6 1157.1 1199.2 1241.3

3.788 3.927 4.053 4.186 4.316

4.536 4.601 4.660 4.727 4.794

5.309 7.437 10.20 13.76 18.02

1.895.−3 2.00.−3 2.13.−3 2.35.−3 2.78.−3

0.045 0.030 0.021 0.013 0.008

1076.2 1141.3 1207.7 1275.4 1342.9

1282.9 1324.5 1363.8 1338.7 1318.1

4.444 4.569 4.691 4.811 4.927

4.857 4.921 3.980 5.029 5.056

22.90

4.23.−3

0.004

1305.2

1305.2

5.032

5.032

vf, m3/kg

T, K

P, bar

219.7t 220 240 260 280

2.6.−6 2.7.−6 3.74.−5 2.97.−4 1.61.−3

300 320 340 360 380

6.40.−3 0.0203 0.0547 0.1279 0.2678

1.404.−3 1.436.−3 1.471.−3 1.508.−3 1.548.−3

400 420 440 460 480

0.513 0.911 1.521 2.401 3.639

500 520 540 560 580 594.6c

vg, m3/kg

h f, kJ/kg

2.22 2.30

6.53 5.13 4.16 3.44 2.91

0.129 0.123 0.118 0.112 0.107

2.50 2.18

0.101 0.096 0.092 0.089 0.085 0.082

*Values of p and v interpolated and converted from tables in Vargaftik, Handbook of Thermophysical Properties of Gases and Liquids, Hemisphere, Washington, and McGraw-Hill, New York, 1975. Values of h and s calculated from API tables published by Texas A&M University, College Station. t = triple point; c = critical point.

TABLE 2-293

Octane* h g, kJ/kg

s f, kJ/(kg⋅K)

s g, kJ/(kg⋅K)

c pf, kJ/(kg⋅K)

µ f, 10−3 Pa⋅s

k f, W/(m⋅K)

365.9 373.2 414.1 455.8 498.4

811.4 842.1 873.5

2.487 2.520 2.698 2.865 3.023

4.207 4.259 4.312

2.033 2.035 2.059 2.105 2.165

2.25 2.01 1.24 0.87 0.65

0.149 0.148 0.143 0.138 0.133

10.7 4.01 1.752 0.844 0.448

542.4 589.8 637.9 687.1 737.7

906.2 939.8 974.6 1010.4 1047.3

3.175 3.325 3.471 3.611 3.747

4.366 4.419 4.461 4.509 4.562

2.231

0.504 0.405 0.334 0.282 0.244

0.128 0.123 0.118 0.112 0.107

1.632.−3 1.685.−3 1.747.−3 1.818.−3 1.904.−3

0.252 0.155 0.100 0.066 0.045

790.1 843.1 897.5 954.8 1013.5

1084.8 1123.6 1162.5 1202.0 1241.8

3.881 4.010 4.137 4.264 4.388

4.617 4.677 4.740 4.802 4.864

0.200 0.167 0.143 0.121 0.103

0.102 0.099 0.095 0.091 0.087

2.013.−3 2.16.−3 2.37.−3 2.81.−3 4.26.−3

0.031 0.021 0.014 0.008 0.004

1072.8 1136.0 1201.5 1276.7 1331.7

1281.2 1318.6 1352.4 1370.4 1331.7

4.508 4.629 4.749 4.880 4.977

4.924 4.980 5.028 5.048 4.977

0.086 0.072 0.058 0.044

0.083

vf, m3/kg

T, K

P, bar

216.4t 220 240 260 280

1.49.−5 2.41.−5 2.18.−4 0.0014 0.0061

1.353.−3 1.368.−3 1.384.−3

300 320 340 360 380

0.0207 0.0575 0.1384 0.3000 0.5856

1.420.−3 1.457.−3 1.495.−3 1.536.−3 1.582.−3

400 420 440 460 480

1.0507 1.758 2.797 4.246 6.201

500 520 540 560 568.8c

8.785 12.15 16.46 21.98 24.97

vg, m3/kg

700 125 31.9

h f, kJ/kg

*Values of p and v interpolated and converted from tables in Vargaftik, Handbook of Thermophysical Properties of Gases and Liquids, Hemisphere, Washington, and McGraw-Hill, New York, 1975. Values of h and s calculated from API tables published by Texas A&M University, College Station. t = triple point; c = critical point. Saturation and superheat tables and a diagram to 100 bar, 680 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.).

2-262

PHYSICAL AND CHEMICAL DATA

TABLE 2-294

Saturated Oxygen (R732)*

T, K

P, bar

vf, 10–3 m3/kg

54.35 t 55 60 65 70

0.0015 0.0018 0.0073 0.0233 0.0624

0.776 0.778 0.790 0.802 0.816

75 80 85 90 90.18

0.1448 0.3003 0.5677 0.9943 1.0133

0.827 0.845 0.862 0.880 0.881

1.634 2.547 3.794 5.443 7.559

0.899 0.920 0.944 0.970 0.998

95 100 105 110 115

vg, 10–3 m3/kg 93980 77920 21240 7200 2894 1330 680.7 379.7 227.1 223.2 143.9 95.46 65.81 46.81 34.15

cpf, kJ/(kg⋅K)

µ f, 10−4 Pa⋅s

kf , W/(m⋅K)

5.642 5.510 5.397 5.301 5.297

1.570 1.589 1.607 1.625 1.626

3.04 2.54 2.16 1.88 1.87

0.170 0.164 0.157 0.151 0.151

3.045 3.113 3.196 3.276 3.354

5.216 5.141 5.073 5.009 4.950

1.645 1.672 1.706 1.752 1.814

1.66 1.51 1.34 1.20 1.07

0.144 0.138 0.131 0.125 0.118

h f, kJ/kg

hg, kJ/kg

s f, kJ/(kg⋅K)

s g, kJ/(kg⋅K)

−189.8 −188.9 −181.1 −173.3 −165.5

48.9 49.5 53.8 58.1 62.4

2.156 2.172 2.308 2.432 2.545

6.548 6.507 6.223 5.992 5.801

−159.2 −149.7 −141.7 −133.7 −133.4

66.6 70.8 74.9 78.8 78.9

2.631 2.754 2.849 2.940 2.943

−125.4 −117.1 −108.6 −99.9 −90.0

82.4 85.7 88.5 90.8 92.6

120 125 130 135 140

10.21 13.48 17.44 22.19 27.82

1.031 1.070 1.116 1.170 1.237

25.42 19.21 14.67 11.25 8.612

−81.6 −71.8 −61.5 −50.6 −38.9

93.6 93.9 93.3 91.6 88.4

3.432 3.510 3.588 3.667 3.748

4.892 4.836 4.779 4.720 4.657

1.896 2.004 2.148 2.341 2.629

0.97 0.86 0.78 0.70 0.60

0.111 0.103 0.096 0.088 0.080

145 150 154.77c

34.45 42.23 50.87

1.332 1.487 2.464

6.499 4.705 2.464

−25.9 −10.8 35.2

82.9 73.1 35.2

3.833 3.928 4.219

4.583 4.487 4.219

3.141 3.935 ∞

0.52

0.072 ∞

*Reproduced and converted from Vasserman and Rabinovich, Thermophysical Properties of Liquid Air and Its Components, Standartov, Moscow, 1968; and Israel Program for Scientific Translations, TT 69-55092, 1970. t = triple point; c = critical point. Other tables are given by Sytchev, V. V., A. A. Vasserman, et al., Thermodynamic Properties of Oxygen, Hemisphere, New York, 1987 (307 pp.); Stewart, R. B., R. T. Jacobsen, et al., J. Phys. Chem. Ref. Data, 20, 5 (1991): 917–1021; For fps units, see Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993. See also Roder, H. M., Transport Properties of Oxygen, NASA Ref. Publ. 1102, 1983 (87 pp.); Laesecke, A., K. Krauss, et al., J. Phys. Chem. Ref. Data, 19, 5 (1990): 1089–1122.

THERMODYNAMIC PROPERTIES

FIG. 2-15 Temperature-entropy chart for oxygen. Pressure P, atm; density ρ, (g⋅mol)/L; temperature, K; enthalpy H, J/(g⋅mol); entropy, J/(g⋅mol⋅K). (NBS Chart D-56. Reproduced by permission.)

2-263

2-264

PHYSICAL AND CHEMICAL DATA

Enthalpy-concentration diagram for oxygen-nitrogen mixture at 1 atm. Reference states: Enthalpies of liquid oxygen and liquid nitrogen at the normal boiling point of nitrogen are zero. (Dodge, Chemical Engineering Thermodynamics, McGraw-Hill, New York, 1944.) Wilson, Silverberg, and Zellner, AFAPL TDR 64-64 (AD 603151), 1964, p. 314, present extensive vapor-liquid equilibrium data for the three-component system argon-nitrogen-oxygen as well as for binary systems including oxygen-nitrogen.

FIG. 2-16

TABLE 2-295

Pentane

Canjar and Manning (Thermodynamic Properties and Reduced Correlations for Gases, Gulf, Houston, 1967) give extensive tables and an enthalpy–log-pressure diagram, based upon Brydon, Walen, and Canjar [Chem. Eng. Prog. Symp. Ser., 49, 7, (1951): 151–157]. For isopentane, Arnold, Liou, and Eldridge [ J. Chem. Eng. Data, 10, 88 (1965)] used the Benedict-WebbRubin equation to generate information to 600°F and 60 atm. Das and Kuloor used the same equation in Ind. J. Technol., 5, 46 (1967) to calculate information up to 1500 K and 1000 atm. Saturation and superheat tables and a diagram to 200 bar, 600 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For equations, see Grigoryev, B. A., Yu. L. Rastorguyev, et al., Int. J. Thermophys., 11, 3 (1990): 487–502.

TABLE 2-296 T, K m

336.4 400 500 600 700 800 1000 1200 1400 1500

Saturated Potassium*

P, bar

vf, m3/kg

1.37.−9 1.84.−7 3.13.−5 9.26.−4 0.01022

0.001208 0.001229 0.001266 0.001304 0.001346

0.06116 0.7322 3.913 12.44 20.0

0.001389 0.001488 0.001605 0.001742 0.001816

vg, m3/kg

h f, kJ/kg

h g, kJ/kg

sf, kJ/(kg⋅K)

s g, kJ/(kg⋅K)

c pf, kJ/(kg⋅K)

4.64.+6 3.39.+4 3164 142.3

93.8 145.5 225.1 302.7 379.4

2327 2342 2390 2433 2468

1.928 2.068 2.246 2.388 2.506

8.567 7.559 6.576 5.937 5.490

0.822 0.805 0.785 0.771 0.762

455.5 609.7 773.5 948.0 1040.0

2498 2552 2610 2679 2718

2.608 2.780 2.929 3.063 3.123

5.161 4.722 4.459 4.299 4.209

0.761 0.792 0.846 0.899 0.924

26.75 2.691 0.584 0.207 0.132

*Converted from tables in Vargaftik, Tables of the Thermophysical Properties of Liquids and Gases, Nauka, Moscow, 1972; and Hemisphere, Washington, 1975. m = melting point. The notation 1.37.−9 signifies 1.37 × 10−9. Many of the Vargaftik values also appear in Ohse, R. W., Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Sci. Pubs., Oxford, 1985 (1020 pp.). This source contains superheat data. Saturation and superheat tables and a diagram to 30 bar, 1650 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For a Mollier diagram from 0.1 to 250 psia, 1300 to 2700°R, see Weatherford, W. D., J. C. Tyler, et al., WADD-TR61-96, 1961. An extensive review of properties of the solid and the saturated liquid is given by Alcock, C. B., M. W. Chase, et al., J. Phys. Chem. Ref. Data, 23, 3 (1994): 385–497.

THERMODYNAMIC PROPERTIES

2-265

FIG. 2-17 Mollier diagram for potassium. Basis: enthalpy = 0.0 cal/g atom at 298 K; entropy = 15.8 cal/(g atom·K) at 298 K. (Aerojet-General Rep. AGN8194, vol. 2, 1967. Reproduced by permission.)

2-266

PHYSICAL AND CHEMICAL DATA

TABLE 2-297

Saturated Propane (R290)* vg, m3/kg

µ f, 10−4 Pa⋅s

k f, W/(m⋅K)

1.96 1.98 2.00 2.02 2.04

6.61 5.54 4.67

0.191 0.183 0.175

5.9862 5.9114 5.8502 5.8005 5.7603

2.07 2.10 2.13 2.16 2.20

3.97 3.27 2.98 2.65 2.36

0.166 0.158 0.150 0.143 0.136

3.8631 3.9605 4.0563 4.1505 4.2433

5.7280 5.7022 5.6817 5.6656 5.6528

2.25 2.29 2.34 2.41 2.48

2.07 1.86 1.69 1.53 1.40

0.129 0.123 0.117 0.111 0.106

906.03 916.54 926.41 935.45 943.38

4.3349 4.4257 4.5160 4.6062 4.6971

5.6426 5.6343 5.6270 5.6200 5.6124

2.56 2.65 2.76 2.89 3.06

1.29 1.19 1.10 0.93 0.82

0.100 0.096 0.091 0.086 0.082

949.79 953.92 954.23 946.56 879.20

4.7896 4.8850 4.9861 5.0997 5.3300

5.6030 5.5896 5.5681 5.5277 5.3300

3.28 3.62 4.23 5.98 ∞

0.72 0.62 0.52 0.40 0.29

0.078 0.073 0.069 0.066 ∞

T, K

P, bar

v f, m3/kg

h f, kJ/kg

h g, kJ/kg

s f, kJ/(kg⋅K)

s g, kJ/(kg⋅K)

c pf, kJ/(kg⋅K)

85.5t 90 100 110 120

3.0.−9 1.5.−8 3.2.−7 3.9.−6 3.1.−5

1.364.−3 1.373.−3 1.392.−3 1.412.−3 1.432.−3

5.37.+7 1.12.+7 5.85.+5 53275 7350

124.92 133.56 152.74 172.03 191.46

690.02 693.58 702.23 711.71 721.78

1.8738 1.9723 2.1743 2.3581 2.5271

8.3548 8.0953 7.6163 7.2377 6.9343

1.92 1.92 1.93 1.94 1.95

130 140 150 160 170

1.8.−4 7.7.−4 2.74.−3 8.22.−3 0.0214

1.453.−3 1.475.−3 1.497.−3 1.521.−3 1.545.−3

1400 344 103 36.8 15.0

211.03 230.77 250.67 270.78 291.10

732.27 743.07 754.12 765.37 776.80

2.6838 2.8300 2.9674 3.0971 3.2202

6.6885 6.4881 6.3237 6.1886 6.0775

180 190 200 210 220

0.0495 0.1035 0.1993 0.3574 0.6031

1.570.−3 1.597.−3 1.625.−3 1.654.−3 1.686.−3

6.84 3.43 1.868 1.087 0.669

311.66 332.48 353.61 375.07 396.90

788.40 800.15 812.03 824.01 836.04

3.3377 3.4503 3.5586 3.6631 3.7645

230 240 250 260 270

0.9661 1.4800 2.1819 3.1118 4.3120

1.719.−3 1.754.−3 1.792.−3 1.833.−3 1.878.−3

0.432 0.290 0.2020 0.1445 0.1059

419.16 442.07 465.58 489.70 514.45

848.08 860.07 871.94 883.62 895.02

280 290 300 310 320

5.8278 7.7063 9.9973 12.75 16.03

1.927.−3 1.982.−3 2.044.−3 2.115.−3 2.200.−3

0.0791 0.0600 0.0461 0.0357 0.0279

539.88 566.06 593.11 621.18 650.49

330 340 350 360 369.8c

19.88 24.36 29.56 35.55 42.42

2.301.−3 2.430.−3 2.607.−3 2.896.−3 4.566.−3

0.0218 0.0170 0.0130 0.0095 0.0046

681.37 714.38 750.52 792.50 879.20

*Values converted and mostly rounded off from those of Goodwin, NBSIR 77-860, 1977. t = triple point; c = critical point. The notation 3.0.−9 signifies 3.0 × 10−9. Later tables for the same temperature range for saturation and for the superheat state from 0.1 to 1000 bar, 85.5 to 600 K, were published by Younglove, B. A. and J. F. Ely, J. Phys. Chem. Ref. Data, 16, 4 (1987): 685–721, but the lower temperature saturation tables contain some errors. Saturation and superheat tables and a chart to 10,000 psia, 800°F appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For thermodynamic properties for 0.1 to 1000 bar, 100 to 700 K, see Sychev, V. V., A. A. Vasserman, et al., Thermodynamic Properties of Propane, Hemisphere, New York, NY, 1991 (275 pp.). Saturation and superheat tables and a diagram to 200 bar, 600 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.

THERMODYNAMIC PROPERTIES TABLE 2-298

2-267

Saturated Propylene (Propene, R1270)

T, K

P, bar

v f, m3/kg

vg, m3/kg

h f, kJ/kg

hg, kJ/ kg

s f, kJ/(kg⋅K)

s g, kJ/(kg⋅K)

c pf, kJ/(kg⋅K)

87.9t 90 100 110 120

9.54.−9 2.05.−8 4.81.−7 6.08.−6 4.88.−5

0.001 301 0.001 305 0.001 325 0.001 346 0.001 367

1.82.+7 8.66.+6 411 165 35 753 4 856

−290.1 −285.1 −265.4 −247.7 −229.8

279.2 281.1 290.2 299.6 309.3

−1.923 −1.867 −1.659 −1.490 −1.335

4.554 4.424 3.897 3.488 3.158

1.695 1.760 1.820

130 140 150 160 170

2.77.−4 1.20.−3 4.17.−3 0.0122 0.0309

0.001 389 0.001 411 0.001 434 0.001 458 0.001 483

927.0 230.91 71.043 25.903 10.842

−211.4 −192.4 −172.9 −153.1 −133.1

319.3 329.4 339.8 350.4 361.2

−1.187 −1.046 −0.912 −0.784 −0.663

2.895 2.681 2.506 2.363 2.245

180 190 200 210 220

0.0697 0.1425 0.2686 0.4727 0.7849

0.001 508 0.001 535 0.001 563 0.001 593 0.001 624

5.080 2.613 1.452 0.860 0.538

−112.7 −92.2 −71.4 −50.3 −28.8

372.1 383.1 394.2 405.3 416.3

−0.547 −0.436 −0.329 −0.226 −0.127

225.5 b 230 240 250 260

1.0133 1.2401 1.8775 2.7401 3.8737

0.001 642 0.001 657 0.001 693 0.001 732 0.001 774

0.4241 0.3515 0.2388 0.1674 0.1206

−16.9 −7.0 15.3 38.0 61.3

422.2 427.1 437.8 448.2 458.2

270 280 290 300 310

5.3269 7.1499 9.3954 12.12 15.38

0.001 820 0.001 872 0.001 929 0.001 995 0.002 071

0.0888 0.0666 0.0507 0.0390 0.0303

85.2 109.9 135.3 161.6 189.0

320 330 340 350 360

19.23 23.75 29.01 35.12 42.20

0.002 162 0.002 273 0.002 418 0.002 628 0.003 038

0.0236 0.0184 0.0142 0.0107 0.0075

365.6 c

46.65

0.004 476

0.0045

µ f, 10−6 Pa·s

k f, W/(m⋅K)

Pr

2017 1526 1185

0.214 0.209 0.204

15.98 12.85 10.57

1.875 1.923 1.964 1.996 2.020

941 735 587 478 397

0.198 0.193 0.188 0.183 0.178

8.91 7.32 6.13 5.21 4.50

2.147 2.066 1.999 1.943 1.896

2.044 2.067 2.094 2.128 2.162

334.5 286.1 244.9 212.7 187.0

0.173 0.168 0.162 0.157 0.152

3.95 3.52 3.17 2.88 2.66

−0.073 −0.030 0.064 0.157 0.247

1.874 1.857 1.825 1.797 1.774

2.182 2.199 2.243 2.298 2.369

175.0 166.2 149.2 135.0 123.0

0.149 0.147 0.142 0.137 0.131

2.56 2.49 2.36 2.26 2.22

467.8 476.9 485.3 492.8 499.3

0.336 0.425 0.512 0.600 0.688

1.753 1.735 1.719 1.704 1.688

2.418 2.494 2.584 2.693 2.842

112.9 106.6 100.0 93.0 85.6

0.126 0.121 0.116 0.112 0.109

2.17 2.20 2.23 2.24 2.23

217.7 248.2 280.9 317.6 364.1

504.3 507.4 507.6 502.8 486.0

0.776 0.867 0.961 1.062 1.188

1.672 1.652 1.627 1.592 1.527

3.007 3.335 3.723 4.669

77.8 69.6 61.0

0.104 0.097 0.090 0.082

2.25 2.39 2.52

433.3

433.3

1.374

1.374

t = triple point; b = normal boiling point; c = critical point. The notation 9.54.−9 signifies 9.54 × 10−9. hf = sf = 0 at 233.15 K = −40°C. Converted from Angus, S., B. Armstrong, et al., Intnl. Thermodynamic Properties of the Fluid State—7. Propylene (Propene) R1270, Pergamon Press, Oxford, 1980 (401 pp.).

2-268

PHYSICAL AND CHEMICAL DATA

TABLE 2-299

Compressed Propylene (Propene, R1270) Temperature, K

Pressure, bar

225

250

275

300

325

350

375

400

425

450

v (m3/ kg) 1 h (kJ/ kg) s (kJ/ kg⋅K)

0.00164 −17.9 −0.0779

0.4817 455.5 2.0169

0.5334 491.2 2.1530

0.5846 529.1 2.2847

0.6354 569.1 2.4128

0.6858 611.4 2.5380

0.7361 656.0 2.6610

0.7861 702.8 2.7821

0.8361 751.9 2.9010

0.8859 803.2 3.0183

v (m3/ kg) 10 h (kJ/ kg) s (kJ/ kg⋅K)

0.00164 −17.2 −0.0810

0.00173 38.5 0.1535

0.00184 97.6 0.3788

0.04986 501.2 1.7631

0.05670 547.0 1.9653

0.06295 593.2 2.0466

0.06889 640.8 2.1774

0.07460 689.7 2.3042

0.08014 740.6 2.4273

0.08557 793.3 2.5480

v (m3/ kg) 20 h (kJ/ kg) s (kJ/ kg⋅K)

0.00163 −16.3 −0.0841

0.00172 39.3 0.1497

0.00184 98.1 0.3736

0.00198 161.5 0.5941

0.02324 512.7 1.6920

0.02773 568.2 1.8567

0.03149 621.5 2.0024

0.03489 673.5 2.1381

0.03803 726.5 2.2674

0.04104 781.5 2.3923

v (m3/ kg) 40 h (kJ/ kg) s (kJ/ kg⋅K)

0.00163 −14.4 −0.0908

0.00172 40.8 0.1419

0.00182 99.0 0.3638

0.00196 161.3 0.5806

0.00216 230.0 0.8001

0.00256 313.3 1.0466

0.01465 633.6 1.9256

0.01682 695.4 2.0758

0.01872 755.6 2.2131

v (m3/ kg) 60 h (kJ/ kg) s (kJ/ kg⋅K)

0.00162 −12.6 −0.0970

0.00171 42.3 0.1345

0.00181 100.1 0.3546

0.00194 161.5 0.5684

0.00211 228.2 0.7816

0.00240 303.8 1.0055

0.00743 575.4 1.7272

0.00944 656.7 1.9250

0.01126 726.1 2.0832

v (m3/ kg) 80 h (kJ/ kg) s (kJ/ kg⋅K)

0.00162 −10.7 −0.1031

0.00170 44.0 0.1274

0.00180 101.3 0.3458

0.00192 162.0 0.5570

0.00208 227.2 0.7657

0.00231 299.0 0.9781

0.00402 499.7 1.5107

0.00605 607.7 1.7795

0.00757 693.3 1.9693

v (m3/ kg) 100 h (kJ/ kg) s (kJ/ kg⋅K)

0.00161 −8.8 −0.1091

0.00169 45.6 0.1202

0.00179 102.6 0.3374

0.00190 162.7 0.5466

0.00204 226.7 0.7514

0.00224 296.1 0.9570

0.00256 373.5 1.1704

0.00316 464.9 1.4061

0.00426 567.8 1.6456

0.00551 660.1 1.8669

v (m3/ kg) 150 h (kJ/ kg) s (kJ/ kg⋅K)

0.00160 −4.1 −0.1236

0.00167 49.9 0.1038

0.00176 106.1 0.3180

0.00186 165.0 0.5228

0.00198 227.0 0.7214

0.00214 292.8 0.9163

0.00234 362.9 1.1100

0.00262 438.5 1.3049

0.00300 519.5 1.5021

0.00354 604.6 1.6958

v (m3/ kg) 200 h (kJ/ kg) s (kJ/ kg⋅K)

0.00159 0.8 −0.1371

0.00166 54.4 0.0884

0.00174 110.0 0.3004

0.00183 167.9 0.5021

0.00194 228.7 0.6963

0.00207 292.3 0.8852

0.00222 359.2 1.0701

0.00242 429.9 1.2521

0.00266 504.4 1.4319

0.00296 581.4 1.6086

v (m3/ kg) 300 h (kJ/ kg) s (kJ/ kg⋅K)

0.00157 10.9 −0.1625

0.00163 63.8 0.0601

0.00170 118.5 0.2688

0.00178 175.2 0.4660

0.00187 234.2 0.6549

0.00197 295.5 0.8367

0.00208 359.3 1.0127

0.00221 425.7 1.1839

0.00236 494.4 1.3507

0.00253 565.6 1.5133

v (m3/ kg) 400 h (kJ/ kg) s (kJ/ kg⋅K)

0.00155 21.2 −0.1863

0.00161 73.6 0.0347

0.00167 127.7 0.2407

0.00174 183.6 0.4351

0.00182 241.5 0.6207

0.00190 301.5 0.7985

0.00199 363.7 0.9705

0.00209 428.0 1.1362

0.00220 494.4 1.2972

0.00232 562.9 1.4536

v (m3/ kg) 500 h (kJ/ kg) s (kJ/ kg⋅K)

0.00153 31.6 −0.2082

0.00159 83.7 0.0112

0.00165 137.3 0.2155

0.00171 192.6 0.4080

0.00178 249.8 0.5910

0.00185 309.0 0.7664

0.00193 370.1 0.9351

0.00201 433.3 1.0981

0.00210 498.7 1.2559

0.00220 565.4 1.4092

Converted and interpolated from Angus, S., B. Armstrong, et al., International Thermodynamic Tables of the Fluid State—7. Propylene, Pergamon, Oxford, 1980 (401 pp.). The 1993 ASHRAE Handbook—Fundamentals (SI ed.) has a thermodynamic chart for pressures from 0.1 to 1000 bar for temperatures up to 580 K. Saturation and superheat tables and a diagram to 30,000 psia, 580°F appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.

THERMODYNAMIC PROPERTIES TABLE 2-300

2-269

Saturated Refrigerant 11*

P, bar

vf, m3/kg

vg, m3/kg

h f, kJ/kg

h g, kJ/kg

s f, kJ/(kg⋅K)

s g, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µ f, 10−4 Pa⋅s

k f, W/(m⋅K)

200 220 240 260 270

0.0043 0.0417 0.0768 0.2215 0.3514

5.901.−4 6.061.−4 6.225.−4 6.398.−4 6.491.−4

28.06 6.272 1.882 0.703 0.458

−14.37 −8.20 4.97 21.01 29.53

186.30 195.89 205.85 216.06 221.23

−0.0651 −0.0361 0.0210 0.0851 0.1172

0.9431 0.8925 0.8581 0.8353 0.8272

0.815 0.828 0.842 0.856 0.863

1.674 1.142 0.831 0.635 0.563

0.115 0.110 0.104 0.098 0.095

280 290 300 310 320

0.5364 0.7917 1.1341 1.5821 2.1556

6.587.−4 6.688.−4 6.794.−4 6.908.−4 7.027.−4

0.309 0.216 0.154 0.113 0.0847

38.25 47.10 56.06 65.10 74.22

226.40 231.58 236.73 241.83 246.88

0.1489 0.1799 0.2102 0.2397 0.2686

0.8209 0.8160 0.8124 0.8099 0.8081

0.870 0.878 0.887 0.897 0.907

0.504 0.454 0.413 0.377 0.346

0.093 0.090 0.087 0.084 0.081

330 340 350 360 380

2.876 3.764 4.845 6.142 9.487

7.156.−4 7.293.−4 7.442.−4 7.603.−4 7.974.−4

0.0645 0.0500 0.0392 0.0311 0.0201

83.42 92.72 102.12 111.64 131.12

251.84 256.69 261.40 265.95 274.40

0.2967 0.3243 0.3513 0.3778 0.4298

0.8071 0.8065 0.8064 0.8065 0.8069

0.917 0.928 0.939 0.950 0.975

0.320 0.297 0.276 0.259 0.229

0.079 0.076 0.073 0.070 0.065

8.435.−4 9.042.−4 9.930.−4 1.167.−3 1.799.−3

0.0134 0.0090 0.0059 0.0036 0.0018

151.38 172.76 196.01 223.85 258.70

281.69 287.20 289.72 285.36 258.70

0.4808 0.5317 0.5840 0.6435 0.7162

0.8066 0.8041 0.7970 0.7773 0.7162

1.004 1.04 1.09 1.19 ∞

0.203 0.169 0.131 0.084 0.033

0.059 0.053 0.048 0.037 ∞

T, K

400 420 440 460 471.2c

14.02 19.98 27.65 37.36 44.09

*Values reproduced or converted from Table 1, p. 17.75, ASHRAE Handbook, 1981: Fundamentals, American Society of Heating, Refrigerating and AirConditioning Engineers, Atlanta, 1981. Copyright material. Reproduced by permission of the copyright owner. c = critical point. The notation 5.901.−4 signifies 5.901 × 10−4. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives material for integral degrees Celsius with temperatures on the ITS 90 scale. For experimental isochores for the compressed liquid from 12 to 301 bar, 254 to 453 K, see Blanke, W. and R. Weiss, PTB Bericht W 30, Braunschweig, Germany, 1992 (54 pp.). Equations and constants approximated to 1985 ASHRAE tables are given by Mecarik, K. and M. Masaryk, Heat Recovery Systems and CHP, 11, 2/3 (1991): 193–197. For tables and a chart to 3000 psia, 460°F, see Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For similar material to 80 bar, 650 K, see Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For specific heat at constant pressure, thermal conductivity, and viscosity in both SI and fps units, see Liley, P. E., Thermophysical Properties of Refrigerants, ASHRAE, Atlanta, GA, 1993.

FIG. 2-18 Enthalpy–log-pressure diagram for Refrigerant 11. 1 MPa = 10 bar. (Copyright 1981 by the American Society of Heating, Refrigerating and Air-Conditioning Engineers and reproduced by permission of the copyright owner.) This chart, redrawn with a different zero point and temperatures in Celsius, appears in ASHRAE Handbook—Fundamentals (SI ed.), Atlanta, GA, 1993.

2-270

PHYSICAL AND CHEMICAL DATA

TABLE 2-301

Saturated Refrigerant 12* h f, kJ/kg

h g, kJ/kg

s f, kJ/(kg⋅K)

s g, kJ/(kg⋅K)

c pf, kJ/(kg⋅K)

µ f, 10−4 Pa⋅s

k f, W/(m⋅K)

294.6 302.3 310.3 318.3 326.5

496.0 500.2 504.5 508.9 513.5

3.492 3.543 3.591 3.637 3.681

4.835 4.780 4.734 4.696 4.665

0.808 0.817 0.827 0.836 0.845

18.9 15.1 12.1 9.69 7.94

0.123 0.119 0.116 0.113 0.109

1.370 0.7589 0.4476 0.2784 0.1811

334.8 343.2 351.8 360.6 369.5

518.1 522.7 527.4 531.1 536.8

3.724 3.765 3.805 3.844 3.881

4.640 4.620 4.603 4.590 4.579

0.855 0.864 0.873 0.882 0.891

6.64 5.65 4.88 4.26 3.77

0.105 0.102 0.098 0.094 0.090

6.810.−4 6.970.−4 7.112.−4 7.282.−4 7.470.−4

0.1225 0.08559 0.06147 0.04543 0.03888

378.0 387.7 397.0 406.5 416.1

541.5 546.1 550.7 555.1 559.4

3.918 3.954 3.989 4.023 4.057

4.570 4.563 4.558 4.554 4.551

0.902 0.913 0.926 0.942 0.959

3.37 3.03 2.75 2.52 2.31

0.087 0.083 0.080 0.076 0.072

T, K

P, bar

vf, m3/kg

v g, m3/kg

150 160 170 180 190

0.00091 0.00305 0.00871 0.02178 0.04877

5.767.−4 5.849.−4 5.926.−4 6.024.−4 6.118.−4

179.12 36.05 13.40 5.666 2.665

200 210 220 230 240

0.0996 0.1879 0.3317 0.5531 0.8781

6.217.−4 6.139.−4 6.431.−4 6.549.−4 6.675.−4

250 260 270 280 290

1.3359 1.959 2.784 3.825 5.184

300 310 320 330 340

6.840 8.860 11.29 14.17 17.58

7.678.−4 7.912.−4 8.173.−4 8.478.−4 8.840.−4

0.02582 0.01992 0.01553 0.01218 0.00957

426.0 436.0 446.2 456.8 467.8

563.5 567.3 570.9 574.0 576.5

4.090 4.122 4.154 4.186 4.218

4.548 4.546 4.543 4.541 4.538

0.979 1.005 1.041 1.093 1.166

2.14 2.00 1.86 1.74 1.60

0.069 0.065 0.061 0.058 0.054

350 360 370 380 385c

21.57 26.19 31.56 37.76 41.31

9.286.−4 9.868.−4 1.072.−3 1.237.−3 1.876.−3

0.00750 0.00582 0.00439 0.00305 0.00188

479.4 492.1 506.4 524.7 551.1

578.2 578.7 577.2 571.2 551.1

4.250 4.285 4.322 4.369 4.437

4.533 4.525 4.514 4.900 4.437

1.264 1.39 1.55

1.45 1.28 1.06 0.75 0.31

0.050 0.046 0.041

∞

∞

*P, v, h, and s data interpolated from Perelshteyn (ed.), Tables and Diagrams of the Thermodynamic Properties of Refrigerants 12, 13, and 22, Moscow, 1971. cp, µ, and k data interpolated and converted from Thermophysical Properties of Refrigerants, American Society of Heating, Refrigerating and Air-Conditioning Engineers, New York, 1976. c = critical point. The notation 5.767.−4 signifies 5.767 × 10−4.

TABLE 2-302

Saturated Refrigerant 13*

P, bar

vf, m3/kg

91 100 110 120 130

3.817.−6 3.418.−5 2.563.−4 0.00137 0.00571

5.367.−4 5.448.−4 5.538.−4 5.635.−4 5.739.−4

140 150 160 170 180

0.01895 0.05250 0.1258 0.2680 0.5186

5.850.−4 5.969.−4 6.095.−4 6.231.−4 6.380.−4

190 200 210 220 230

0.9269 1.5507 2.456 3.712 5.396

T, K

vg, m3/kg

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

4.610 4.558 4.516 4.482 4.454

0.826 0.845 0.865 0.884

6.83 5.60 4.59 3.83

0.114 0.109 0.104 0.099

3.642 3.688 3.732 3.777 3.820

4.431 4.413 4.397 4.385 4.374

0.898 0.910 0.924 0.943 0.972

3.26 2.82 2.48 2.20 1.97

0.093 0.088 0.083 0.078 0.072

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

238.1 243.8 251.1 258.2 265.8

424.9 429.0 433.1 437.2 441.3

3.080 3.140 3.205 3.267 3.327

5.133 4.990 4.860 4.759 4.677

5.865 2.2617 1.0019 0.4962 0.2689

273.7 281.7 290.0 298.4 307.1

455.3 449.3 453.5 457.6 461.8

3.385 3.441 3.494 3.545 3.594

6.536.−4 6.709.−4 6.899.−4 7.110.−4 7.346.−4

0.1567 9.69.−2 6.28.−2 4.24.−2 2.95.−2

315.9 325.0 334.3 343.8 353.6

465.9 469.9 473.8 477.5 481.0

19557 2392 347.0 70.25 18.15

240 250 260 270 280

7.589 10.37 13.85 18.13 23.32

7.615.−4 7.928.−4 8.302.−4 8.769.−4 9.320.−4

2.11.−2 1.53.−2 1.13.−2 8.28.−3 6.10.−3

363.5 373.9 384.7 396.2 408.8

484.1 486.1 489.1 490.5 490.6

3.862 3.903 3.944 3.986 4.029

4.364 4.355 4.346 4.336 4.323

1.014 1.072 1.151 1.255 1.386

1.79 1.63 1.50 1.34 1.14

0.067 0.062 0.057 0.051 0.045

290 300 302.0c

29.57 37.05 38.70

1.035.−3 1.284.−3 1.808.−3

4.34.−3 2.60.−3 1.81.−3

423.6 445.3 463.1

488.3 477.5 463.1

4.080 4.151 4.209

4.303 4.257 4.209

1.549 1.75 ∞

0.87 0.52 0.29

0.038 ∞

*P, v, h, and s data interpolated from Perelshteyn (ed.), Tables and Diagrams of the Thermodynamic Properties of Refrigerants 12, 13 and 22, Moscow, 1971. cp, µ, and k data interpolated and converted from Thermophysical Properties of Refrigerants, American Society of Heating, Refrigerating and Air-Conditioning Engineers, New York, 1976. c = critical point. The notation 3.817.−6 signifies 3.817 × 10−6. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) contains a table at closer temperature increments and also an enthalpy–log-pressure diagram from 0.1 to 70 bar, −100 to 240°C. Equations and constants approximated to 1985 ASHRAE tables are given by Mecarik, K. and M. Masaryk, Heat Recovery Systems and CHP, 11, 2/3 (1991): 193–197. Saturation and superheat tables and a diagram to 60 bar, 600 K are given by Reynolds, W. C., Thermodynamic Properties in SI, Stanford Univ. publ., 1979 (173 pp.). For tables and a chart to 1000 psia, 520°F, see Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.

THERMODYNAMIC PROPERTIES

2-271

Enthalpy–log-pressure diagram for Refrigerant 12. 1 MPa = 10 bar. (Copyright 1981 by the American Society of Heating, Refrigerating and AirConditioning Engineers and reproduced by permission of the copyright owner.) This chart, redrawn for integral Celsius temperatures with a different zero point, appears on p. 17.4 of the 1993 ASHRAE Handbook—Fundamentals (SI ed.). This handbook gives material for integral degrees Celsius with temperatures on the ITS 90 scale. For experimental isochores for the compressed liquid from 10 to 302 bar, 122 to 462 K, see Blanke, W. and R. Weiss, PTB Bericht W30, Braunschweig, Germany, 1992 (54 pp.). Equations and constants approximated to 1985 ASHRAE tables are given by Mecarik, K. and M. Masaryk, Heat Recovery Systems and CHP, 11, 2/3 (1991): 193–197. Tables at 2°C increments to 240°C, 50 bar are given by Watson, J. T. R., Thermophysical Properties of Refrigerant 12, H.M.S.O., Edinburgh, Scotland, 1975 (183 pp.). Saturation and superheat tables and a diagram to 40 bar, 620 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ., 1979 (173 pp.). Tables and a chart to 1100 psia, 480°F are given by Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.

FIG. 2-19

2-272

PHYSICAL AND CHEMICAL DATA

TABLE 2-303

Saturated Refrigerant 13B1*

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

170 180 190 200 210

0.059 0.127 0.250 0.455 0.777

4.594.−4 4.677.−4 4.765.−4 4.860.−4 4.961.−4

1.6015 0.7840 0.4190 0.2407 0.1467

−40.90 −34.75 −28.51 −22.17 −15.68

90.95 94.37 97.83 101.32 104.82

−0.2033 −0.1682 −0.1345 −0.1020 −0.0704

0.5723 0.5491 0.5305 0.5154 0.5033

0.597 0.618 0.634 0.648 0.663

9.54 7.60 6.20 5.13 4.33

0.101 0.096 0.091 0.086 0.082

215.4 220 230 240 250

1.013 1.254 1.933 2.863 4.096

5.020.−4 5.071.−4 5.190.−4 5.321.−4 5.466.−4

0.1147 0.0940 0.0628 0.0433 0.0308

−12.09 −9.02 −2.19 4.83 12.03

106.70 108.28 111.68 114.99 118.16

−0.0536 −0.0396 −0.0094 0.0202 0.0494

0.4978 0.4936 0.4857 0.4793 0.4739

0.670 0.676 0.690 0.703 0.721

3.97 3.71 3.22 2.83 2.51

0.079 0.077 0.073 0.068 0.063

260 270 280 290 300

5.690 7.703 10.20 13.25 16.91

5.627.−4 5.809.−4 6.018.−4 6.264.−4 6.562.−4

0.0224 0.0166 0.0124 0.0094 0.0072

19.44 27.06 34.94 43.11 51.68

121.16 123.93 126.41 128.51 130.09

0.0781 0.1064 0.1345 0.1625 0.1908

0.4693 0.4652 0.4612 0.4570 0.4522

0.742 0.767 0.800 0.842 0.891

2.25 2.04 1.84 1.69 1.57

0.059 0.054 0.049 0.045 0.040

310 320 330 340.2c

21.28 26.44 32.48 39.64

6.940.−4 7.458.−4 8.295.−4 1.344.−3

0.0055 0.0041 0.0030 0.0013

60.81 70.80 82.42 108.70

130.97 130.76 128.59 108.70

0.2197 0.2503 0.2845 0.3605

0.4460 0.4376 0.4245 0.3605

0.951 1.09 1.29 ∞

1.45 1.26 0.99 0.35

0.035 0.030 0.026 ∞

T, K

*Values reproduced or converted from Table 4, p. 17.83, ASHRAE Handbook, 1981: Fundamentals, American Society of Heating, Refrigerating and Air-Conditioning Engineers, Atlanta, 1981. Copyright material. Reproduced by permission of the copyright owner. c = critical point. The notation 4.594.−4 signifies 4.594 × 10−4. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) contains a table at closer temperature increments and also an enthalpy–log-pressure diagram from 0.1 to 35 bar, −80 to 220°C. For tables and a chart to 500 psia, 480°F, see Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.

Refrigerant 14 (tetrafluoromethane) See Carbon Tetrafluoride (Table 2-245).

TABLE 2-304

Refrigerant 20 See Chloroform (Table 2-248).

Saturated Refrigerant 21

Temperature, K

Pressure, bar

250 260 270 280 290

0.2415 0.3953 0.6200 0.9364 1.3682

0.000 0.000 0.000 0.000 0.000

300 310 320 330 340

1.9417 2.6849 3.6279 4.8022 6.2409

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

677 687 698 709 722

0.8292 0.5247 0.3455 0.2355 0.1654

16.6 26.5 36.6 46.7 57.1

274.8 279.9 284.9 290.0 295.0

0.0687 0.1076 0.1454 0.1824 0.2186

1.1015 1.0820 1.0653 1.0511 1.0389

0.000 0.000 0.000 0.000 0.000

735 748 763 778 794

0.1192 0.0879 0.0661 0.0505 0.0391

67.7 78.4 89.5 100.7 112.3

300.0 304.8 309.5 314.1 318.4

0.2543 0.2894 0.3242 0.3586 0.3927

1.0286 1.0196 1.0119 1.0051 0.9989

sg, kJ⋅(kg⋅K)

350 360 370 380 390

7.978 10.049 12.489 15.337 18.630

0.000 0.000 0.000 0.000 0.000

812 830 850 870 893

0.0307 0.0243 0.0194 0.0155 0.0125

124.1 136.2 148.6 161.2 173.9

322.4 326.1 329.3 331.9 333.8

0.4266 0.4602 0.4935 0.5264 0.5587

0.9932 0.9877 0.9820 0.9758 0.9688

400 410 420 430 440

22.41 26.72 31.60 37.10 43.26

0.000 0.000 0.000 0.001 0.001

918 944 972 002 034

0.01011 0.00820 0.00672 0.00564 0.00491

186.4 198.3 208.7 216.4 221.1

334.8 334.7 333.7 332.4 332.3

0.5896 0.6180 0.6418 0.6587 0.6682

0.9605 0.9506 0.9394 0.9286 0.9208

Reproduced and rounded from unpublished Center for Applied Thermodynamic Studies, Moscow ID report, 1981. For a thermodynamic diagram to 350 bar, 370°C, see Rombusch, U. K., Allgem. Warme., 11, 3 (1962).

TABLE 2-305

Saturated Refrigerant 22*

P, bar

vf, m3/kg

vg, m3/kg

150 160 170 180 190

0.0017 0.0054 0.0150 0.0369 0.0821

6.209.−4 6.293.−4 6.381.−4 6.474.−4 6.573.−4

83.40 28.20 10.85 4.673 2.225

200 210 220 230 240

0.1662 0.3116 0.5470 0.9076 1.4346

6.680.−4 6.794.−4 6.917.−4 7.050.−4 7.195.−4

250 260 270 280 290

2.174 3.177 4.497 6.192 8.324

300 310 320 330 340 350 360 369.3c

T, K

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

1.059 1.058 1.057 1.058 1.060

0.770 0.647 0.554

0.161 0.156 0.151 0.146 0.141

4.921 4.885 4.854 4.828 4.805

1.065 1.071 1.080 1.091 1.105

0.481 0.424 0.378 0.340 0.309

0.136 0.131 0.126 0.121 0.117

3.898 3.942 3.986 4.029 4.071

4.785 4.768 4.752 4.738 4.725

1.122 1.143 1.169 1.193 1.220

0.282 0.260 0.241 0.225 0.211

0.112 0.107 0.102 0.097 0.092

612.8 615.1 616.7 617.3 616.5

5.113 4.153 4.194 4.235 4.278

4.713 4.701 4.688 4.674 4.658

1.257 1.305 1.372 1.460 1.573

0.198 0.186 0.176 0.167 0.151

0.087 0.082 0.077 0.072 0.067

613.3 605.5 570.0

4.324 4.378 4.501

4.637 4.605 4.501

1.718 1.897 ∞

0.130 0.106

0.062

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

268.2 278.2 288.3 298.7 308.6

547.3 552.1 557.0 561.9 566.8

3.355 3.430 3.494 3.551 3.605

5.215 5.141 5.075 5.013 4.963

1.145 0.6370 0.3772 0.2352 0.1532

318.8 329.1 339.7 350.6 361.7

571.6 576.5 581.2 585.9 590.5

3.657 3.707 3.756 3.804 3.852

7.351.−4 7.523.−4 7.733.−4 7.923.−4 8.158.−4

0.1037 0.07237 0.05187 0.03803 0.02838

373.0 384.5 396.3 408.2 420.4

594.9 599.0 603.0 606.6 610.0

10.956 14.17 18.02 22.61 28.03

8.426.−4 8.734.−4 9.096.−4 9.535.−4 1.010.−3

0.02148 0.01643 0.01265 9.753.−3 7.479.−3

432.7 445.5 458.6 472.4 487.2

34.41 41.86 49.89

1.086.−3 1.212.−3 2.015.−3

5.613.−3 4.036.−3 2.015.−3

503.7 523.7 570.0

*P, v, h, and s data interpolated from Perelshteyn (ed.), Tables and Diagrams of the Thermodynamic Properties of Refrigerants 12, 13 and 22, Moscow, 1971. cp, µ, and k data interpolated and converted from Thermophysical Properties of Refrigerants, American Society of Heating, Refrigerating and Air-Conditioning Engineers, New York, 1976. c = critical point. The notation 6.209.−4 signifies 6.209 × 10−4. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives a saturation table from −150 to 96.14°C and an enthalpy–log-pressure diagram from 0.1 to 150 bar, −60 to 200°C. For experimental isochores for the compressed liquid from 12 to 297 bar, 120 to 378 K, see Blanke, W. and R. Weiss, PTB Bericht W 30, Braunschweig, Germany, 1992 (54 pp.). Equations and constants approximated to 1985 ASHRAE tables are given by Mecarik, K. and M. Masaryk, Heat Recovery Systems and CHP, 11 2/3 (1991): 193–197. Saturation and superheat tables and a diagram to 100 bar, 620 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (179 pp.). For tables and a chart to 2000 psia, 480°F, see Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.

FIG. 2-20 Enthalpy–log-pressure diagram for Refrigerant 22. 1 MPa = 10 bar. (Copyright 1981 by the American Society of Heating, Refrigerating and Air-Conditioning Engineers and reproduced by permission of the copyright owner.)

2-273

2-274

PHYSICAL AND CHEMICAL DATA

TABLE 2-306

Thermophysical Properties of Compressed R22 Temperature, K

Pressure, bar

Property

275

300

325

350

375

400

425

450

1

cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m·K) Pr

0.639 11.8 0.0091 0.829

0.653 12.8 0.0106 0.793

0.689 13.9 0.0121 0.787

0.714 14.9 0.0136 0.782

0.739 15.8 0.0151 0.773

0.758 16.7 0.0166 0.762

0.781 17.7 0.0181 0.765

0.806 18.7 0.0196 0.769

5

cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m·K) Pr

0.725 11.8 0.0096 0.887

0.728 12.8 0.0107 0.871

0.744 13.8 0.0123 0.852

0.759 15.0 0.0138 0.839

0.766 16.2 0.0153 0.803

0.775 17.0 0.0170 0.775

0.791 18.0 0.0184 0.773

0.816 18.8 0.0199 0.771

10

cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m·K) Pr

1.166 211 0.0954 2.58

0.847 13.7 0.0121 0.959

0.810 14.4 0.0128 0.901

0.799 15.1 0.0144 0.838

0.797 16.1 0.0160 0.802

0.803 17.1 0.0175 0.785

0.814 18.1 0.0190 0.775

0.828 19.0 0.0205 0.767

20

cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m·K) Pr

0.164 211 0.0963 2.55

1.237 159 0.0849 2.32

0.949 16.5 0.0157 0.997

0.889 17.3 0.0172 0.894

0.865 18.0 0.0184 0.846

0.858 18.8 0.0199 0.811

0.859 19.6 0.0214 0.787

40

cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m·K) Pr

1.152 218 0.0980 2.56

1.217 164 0.0872 2.29

1.359 123 0.0767 2.18

1.373 20.7 0.0219 1.30

1.089 20.5 0.0210 1.063

0.996 20.7 0.0217 0.950

0.956 21.2 0.0233 0.870

60

cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m·K) Pr

1.142 221 0.0993 2.54

1.191 170 0.0889 2.28

1.311 128 0.0786 2.14

1.460 94.6

1.767 24.7 0.0305 1.431

1.221 24.2 0.0287 1.030

1.089 23.9 0.0268 0.971

80

cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m·K) Pr

1.132 226 0.1003 2.55

1.177 175 0.0904 2.28

1.277 133 0.0803 2.12

1.444 101 0.0690 2.12

1.861 73.6 0.0523 2.62

1.396 29.9 0.0374 1.12

1.262 27.7 0.0337 1.04

100

cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m·K) Pr

1.122 230 0.1013 2.55

1.154 179 0.0916 2.26

1.247 138 0.0817 2.11

1.361 108 0.0716 2.04

1.564 83.2 0.0607 2.14

1.923 37.6 0.0421 1.72

1.471 32.4 0.0378 1.26

2.073 55.5 0.0504 2.28

Some values are approximate as significant differences exist in the literature.

TABLE 2-307

Saturated Refrigerant 23

Temp., K

Pressure, bar

180 190 191.1b 200 210

0.510 0.950 1.013 1.652 2.709

0.000 0.000 0.000 0.000 0.000

220 230 240 250 260

4.298 6.312 9.091 12.69 17.25

270 280 290 299.1c

22.94 29.98 38.68 48.36

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

678 693 695 710 729

0.4088 0.2279 0.2139 0.1353 0.0845

−66.0 −54.4 −53.1 −42.6 −30.3

181.1 185.3 185.7 189.1 192.4

−0.3179 −0.2554 −0.2485 −0.1948 −0.1353

1.0549 1.0062 1.0011 0.9635 0.9254

0.000 0.000 0.000 0.000 0.000

751 777 807 844 889

0.0551 0.0372 0.0259 0.0183 0.0132

−17.5 −4.3 9.4 23.6 38.1

195.4 197.8 199.6 200.7 200.9

−0.0764 −0.0182 0.0392 0.0957 0.1512

0.8913 0.8602 0.8314 0.8042 0.7773

0.000 0.001 0.001 0.001

948 031 169 905

0.0095 0.0068 0.0046 0.0019

53.5 70.5 92.0 143.0

199.8 196.4 188.1 143.0

0.2071 0.2665 0.3387 0.5062

0.7493 0.7162 0.6698 0.5062

cpf, kJ/(kg⋅K)

µf, 10−6 Pa·s

kf, W/(m⋅K)

Pr

0.710 1.043 1.289 1.497

170.1 150.4 131.2 113.0

0.105 0.098 0.091 0.084

1.15 1.56 1.85 2.00

b = normal boiling point; c = critical point. hf = sf = 0 at 233.15 K = −40°C. Interpolated and converted from ASHRAE Handbook—Fundamentals, 1993. Experimental P−ρ−T data from 95 to 413 K reported in J. Phys. Chem., 89 (1985): 4637–4646 were used by Rubio, R. G., J. A. Zollweg, et al., J. Chem. Eng. Data, 36, (1991): 171–184, to calculate properties up to 1000 bar from 126 to 332° K. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives a saturation table from −100 to 25.92°C and an enthalpy–log-pressure diagram from 0.1 to 80 bar, −100 to 280°C. Equations and constants approximated to the 1985 ASHRAE tables are given by Mecaryk, K. and M. Masaryk, Heat Recovery Systems and CHP, 11, 2/3 (1991): 193–197. For an enthalpy–log-pressure diagram from 0.005 to 200 bar, −140 to 180°C, see Morsy, T. E., Kaltetechnik—Klimat., 18, 9 (1966): 347–349. Saturation and superheat tables and a diagram to 100 bar, 600 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For tables and a chart to 1000 psia, 560°F, see Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (21 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.

TABLE 2-308 Temp., K

Thermophysical Properties of Saturated Difluoromethane (R32)

Pressure, bar

vf , m3/kg

vg , m3/kg

hf , kJ/kg

hg , kJ/kg

sf , kJ/(kg⋅K)

sg , kJ/(kg⋅K)

cpf , kJ/(kg⋅K)

cpg , kJ/(kg⋅K)

µf , 10−6 Pa·s

µg , 10−6 Pa·s

200 210 220 230 240

0.2960 0.5440 0.9384 1.5345 2.3963

7.845.−4 8.025.−4 8.208.−4 8.402.−4 8.611.−4

1.0580 0.5990 0.3593 0.2261 0.1483

−52.340 −37.750 −20.690 −4.984 10.947

351.160 356.880 361.970 366.773 371.129

−0.2418 −0.1652 −0.0909 −0.0215 0.0459

1.7757 1.7098 1.6485 1.5948 1.5468

1.557 1.580 1.613

0.799 0.839 0.894

283.8 247.8 220.4

10.30 10.37 10.46

250 260 270 280 290

3.5966 5.2160 7.3423 10.070 13.502

8.842.−4 9.096.−4 9.376.−4 9.696.−4 1.006.−3

0.1005 0.07020 0.05009 0.03643 0.02687

374.971 378.224 380.786 382.525 383.262

0.1117 0.1763 0.2397 0.3029 0.3654

1.5030 1.4624 1.4247 1.3886 1.3534

1.642 1.682 1.730 1.786 1.863

0.963 1.043 1.138 1.244 1.375

198.1 177.9 159.1 141.8 126.1

300 310 320 330 340

17.749 22.931 29.186 36.675 45.603

1.049.−3 1.100.−3 1.166.−3 1.243.−3 1.394.−3

0.02001 0.01497 0.01117 0.00822 0.00581

115.754 135.801 157.212 180.724 208.262

382.737 380.576 376.163 368.357 354.460

0.4283 0.4919 0.5574 0.6264 0.7047

1.3182 1.2815 1.2415 1.1950 1.1347

1.955 2.084 2.282 2.620 3.560

1.560 1.810 2.16 2.62 4.21

112.0 98.8 86.1 75.1 65.4

350 351.4c

56.336 57.927

0.00237

0.00317 0.00237

274.640 286.675

337.933 286.675

0.8927 0.9269

1.0735 0.9269

27.1778 43.786 60.849 78.456 96.713

kf , W/(m⋅K)

kg , W/(m⋅K)

Prf

Prg

10.66 10.95 11.31 11.70 12.21

0.1646 0.1562 0.1487 0.1403 0.1308

0.0097 0.0106 0.0115 0.0125 0.0136

1.98 1.92 1.85 1.81 1.80

1.06 1.08 1.12 1.16 1.23

12.82 13.71 14.4 15.3 17.3

0.1228 0.1155 0.1073 0.0990

0.0149 0.0165 0.0184 0.0205 0.0236

1.78 1.78 1.83 1.99

1.34 1.50 1.69 1.96 3.10

c = critical point. The notation 7.845.−4 signifies 7.845 × 10−4. P, v, T, h, s, and cp converted and extrapolated from Defibaugh, D. R., G. Morrison, et al., J. Chem. Eng. Data, 39 (1994): 333–340. Saturated liquid and vapor viscosities from smooth curve fits of Oliveira, C. M. B. P. and W. A. Wakeham, Int. J. Thermophys., 14, 6 (1993): 1131–1143. Thermal conductivity values based upon papers by Geller, V. Z. and M. E. Perlaitis, and by Gross, Proc. 10th Symp. Thermophys. Props., Boulder, CO, 1994. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives a saturation table to 78.41°C and a diagram to 200 bar, 200°C.

2-275

2-276

PHYSICAL AND CHEMICAL DATA

TABLE 2-309

Specific Heat at Constant Pressure, Thermal Conductivity, Viscosity, and Prandtl Number of R32 Gas P, bar

Temp., K 250

260

270

280

290

300

5

P, bar

Property

1

10

cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr

0.805 10.55 0.0094 0.908

cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr

0.810 11.00 0.0100 0.890

1.025 10.96 0.0104 1.080

cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr

0.818 11.42 0.0107 0.873

0.991 11.37 0.0111 1.015

cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr

0.825 11.82 0.0116 0.860

0.969 11.77 0.0118 0.967

1.238 11.68 0.0125 1.157

cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr

0.837 12.28 0.0121 0.849

0.959 12.22 0.0125 0.938

1.161 12.17 0.0131 1.079

cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr

0.849 12.70 0.0128 0.842

0.951 12.69 0.0132 0.914

1.118 12.66 0.0138 1.026

15

Temp., K 310

320

330

340

350

1

5

10

15

20

0.861 13.12 0.0135 0.878

0.945 13.10 0.0139 0.891

1.084 13.08 0.0144 0.985

1.279 13.07 0.0150 1.114

1.560 13.09 0.0159 1.284

25

30

40

0.873 13.54 0.0142 0.836

0.944 13.54 0.0146 0.875

1.059 13.55 0.0150 0.957

1.207 13.56 0.0156 1.049

1.400 13.60 0.0164 1.161

1.704 13.75 0.0173 1.354

0.885 13.96 0.0148 0.834

0.942 13.96 0.0152 0.865

1.038 13.98 0.0156 0.930

1.158 14.01 0.0162 1.001

1.301 14.15 0.0169 1.089

1.508 14.28 0.0177 1.217

1.837 14.52 0.0187 1.426

0.897 14.38 0.0155 0.832

0.937 14.40 0.0159 0.849

1.020 14.43 0.0163 0.903

1.135 14.47 0.0168 0.978

1.242 14.53 0.0175 1.031

1.388 14.65 0.0182 1.117

1.612 14.85 0.0190 1.260

2.488 16.00 0.0217 1.834

0.910 14.80 0.0162 0.831

0.934 14.82 0.0165 0.839

1.004 14.84 0.0169 0.882

1.118 14.87 0.0174 0.955

1.200 14.92 0.0180 0.995

1.308 15.06 0.0186 1.060

1.440 15.21 0.0194 1.130

1.914 16.16 0.0216 1.432

1.370 12.62 0.0144 1.201

Some values read from charts may be approximate. cp values interpolated and converted from Thermodynamic Properties of KLEA 32, I.C.I., 1993 (47 pp.). Viscosity interpolated from Takahashi, M., C. Yokoyama, et al., Proc. 14th Symp. Thermophys. Props., Japan, 1993 (pp. 427–430). Thermal conductivities are taken from Geller, V. Z. and M. E. Perlaitis, and from Gross, Proc. 10th Symp. Thermophys. Props., Boulder, CO, 1994.

TABLE 2-310 Temp., °C

Saturated SUVA MP 39

Pf, bar

Pg, bar

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−6 Pa·s

kf , W/(m·K)

Prf

−40 −30 −20 −10 0

0.733 1.155 1.748 2.553 3.615

0.533 0.871 1.361 2.043 2.965

0.000 0.000 0.000 0.000 0.000

712 728 744 762 781

0.3778 0.2391 0.1576 0.1075 0.0755

154.0 164.9 176.2 188.6 200.0

385.0 390.6 396.3 401.8 407.3

0.8188 0.8647 0.9099 0.9577 1.0000

1.8244 1.8059 1.7907 1.7781 1.7675

1.078 1.109 1.137 1.165 1.197

351 323 291 266 241

0.1209 0.1154 0.1107 0.1057 0.1012

3.13 3.06 2.99 2.93 2.85

10 20 30 40 50

4.984 6.712 8.857 11.475 14.628

4.177 5.733 7.697 10.133 13.112

0.000 0.000 0.000 0.000 0.000

803 826 851 878 909

0.0544 0.0399 0.0298 0.0225 0.0172

212.7 225.3 238.3 252.0 266.4

412.6 417.6 422.2 426.5 430.1

1.0454 1.0884 1.1316 1.1752 1.2194

1.7587 1.7510 1.7439 1.7372 1.7304

1.233 1.277 1.329 1.392 1.468

221 202 186 170 157

0.0967 0.0922 0.0877 0.0830 0.0781

2.82 2.80 2.83 2.85 2.95

60 70 80 90 100

18.378 22.79 27.92 33.83 40.53

16.711 21.01 26.12 32.13 39.22

0.000 0.000 0.001 0.001 0.001

944 988 028 084 140

0.01313 0.01005 0.00764 0.00570 0.00403

281.6 297.9 315.9 336.2 361.4

433.0 434.9 435.4 433.5 426.9

1.2647 1.3118 1.3616 1.4163 1.4820

1.7228 1.7138 1.7022 1.6858 1.6584

1.564 1.652 1.802 1.958 2.16

143 131 122 115 110

0.0737 0.0684 0.0631 0.0577 0.0533

3.04 3.16 3.48 3.90 4.46

108.0c

46.04

46.04

0.001 96

0.00196

397

397

vf, m3/kg

c = critical point. SUVA MP 39 = R401A = CHClF2 (R22) 53% wt + CH3CHF2 (R 152a) 13% wt + CHClFCF3 (R124) 34% wt, near-azeotropic blend. Some values read from charts are approximate. Material used by permission of DuPont Fluoroproducts.

TABLE 2-311

SUVA MP 39 at Atmospheric Pressure

Temp., °C

−27.01

−20

0

20

40

60

80

100

120

140

v (m3/ kg) h (kJ/ kg) s (kJ/ kg⋅K) cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr Z

0.2102 351.7 1.8009 0.648 10.17 0.00878 0.750 0.9829

0.2167 396.9 1.8193 0.669 10.43 0.00921 0.758 0.9852

0.2351 410.4 1.8706 0.698 11.18 0.01041 0.750 0.9906

0.2534 424.5 1.9204 0.727 11.93 0.01161 0.749 0.9949

0.2715 439.2 1.9689 0.757 12.68 0.01282 0.749 0.9979

0.2896 454.4 2.0161 0.787 13.42 0.01404 0.748 1.0005

0.3076 470.3 2.0623 0.811 14.17 0.01536 0.748 1.0025

0.3256 486.6 2.1073 0.836 14.89 0.01668 0.748 1.0043

0.3435 503.5 2.1513 0.859 15.61 0.01796 0.747 1.0056

0.3613 521.2 2.1943 0.883 16.32 0.01929 0.747 1.0060

For composition see footnote to Table 2-310. Some values read from charts are approximate. Material used by permission of DuPont Fluoroproducts.

50 R 32

20

1.

40

30

0

1.3

350 20

g

.40

•

340

K

y1

p tro

0.95

0.90

k kJ/

330 320

En

310 300 K

0

1.5

Satu rated vapor

Pressure, bar

10 8

6

290

1.6

0

280 270

4

0

1.7

260

2

0 1.8 250 240 230

1

FIG. 2-21

360

380

400 Enthalpy, kJ/kg

420

440

Enthalpy–log-pressure diagram for Refrigerant 32. 2-277

2-278

PHYSICAL AND CHEMICAL DATA TABLE 2-312 Pressure, bar

Thermodynamic Properties of Saturated KLEA 60 Tf , K

Tg , K

vg , m3/kg

hf , kJ/kg

hg , kJ/kg

227.3 236.1 242.8 248.3 253.0

234.0 242.5 249.1 254.5 259.1

0.000 0.000 0.000 0.000 0.000

7118 7263 7381 7483 7573

0.2097 0.1433 0.1093 0.0885 0.0744

−7.80 3.92 12.89 20.27 26.57

229.64 235.02 239.07 242.35 245.08

0.9965 0.9833 0.9744 0.9679 0.9629

4 5 6 8 10

260.7 267.3 272.9 282.1 289.8

266.8 273.1 278.5 287.5 295.0

0.000 0.000 0.000 0.000 0.000

7735 7880 8012 8254 8480

0.0564 0.0442 0.0384 0.0286 0.0228

37.23 46.12 53.84 67.02 78.23

249.54 253.07 255.24 260.70 263.86

0.9552 0.9496 0.9450 0.9378 0.9318

12.5 15 17.5 20 22.5

297.9 304.8 311.0 316.5 321.4

302.8 309.5 315.4 320.7 325.5

0.000 0.000 0.000 0.000 0.000

8750 9017 9290 9613 9884

0.01802 0.01481 0.01247 0.01069 0.00928

90.50 101.51 111.64 121.18 130.31

266.95 269.12 270.58 271.46 271.79

0.9257 0.9190 0.9128 0.9065 0.8999

25 27.5 30

326.1 330.4 334.5

329.8 333.9 337.6

0.001 023 0.001 063 0.001 115

0.00828 0.00717 0.00635

139.17 147.89 156.58

271.63 270.97 269.81

0.8927 0.8850 0.8765

1 1.5 2 2.5 3

vf , m3/kg

sf , kJ/(kg⋅K)

sg , kJ/(kg⋅K)

hf = sf = 0 at 233.15 K = −40°C. Converted and interpolated from Thermodynamic Properties of Klea 60 (British units, 20 pp.), copyright ICI Chemicals and Polymers Limited, 1993. Reproduced by permission. Tf = bubble point temperature; Tg = dew point temperature.

TABLE 2-313 Pressure, bar

Thermodynamic Properties of Saturated KLEA 61 vf , m3/kg

vg , m3/kg

hf , kJ/kg

hg , kJ/kg

6852 6994 7110 7211 7301

0.1800 0.1230 0.0937 0.0758 0.0637

−9.45 2.52 9.72 16.59 22.47

191.64 196.90 200.88 204.10 206.80

0.8433 0.8341 0.8282 0.8245 0.8215

0.000 0.000 0.000 0.000 0.000

7463 7607 7740 7985 8214

0.04831 0.03888 0.03249 0.02435 0.01936

32.43 40.76 48.00 59.82 70.98

211.22 214.74 217.65 222.21 225.63

0.8172 0.8141 0.8123 0.8080 0.8048

298.7 305.5 311.4 316.7 321.5

0.000 0.000 0.000 0.000 0.000

8491 8768 9053 9353 9680

0.01528 0.01251 0.01049 0.00896 0.00774

82.59 93.02 102.67 111.79 120.55

228.80 231.08 232.64 233.60 233.99

0.8010 0.7971 0.7929 0.7882 0.7829

325.9 330.0 333.7

0.001 005 0.001 048 0.001 102

0.00674 0.00590 0.00518

129.11 137.62 146.21

233.85 233.16 231.84

0.7769 0.7700 0.7619

Tf , K

Tg , K

225.6 234.3 241.8 246.4 251.1

230.0 238.5 245.0 250.4 254.9

0.000 0.000 0.000 0.000 0.000

4 5 6 8 10

258.9 265.4 270.9 280.2 287.8

262.6 269.0 274.4 283.4 290.9

12.5 15 17.5 20 22.5

295.8 302.8 308.8 314.3 319.3

25 27.5 30

323.9 328.1 332.1

1 1.5 2 2.5 3

sf , kJ/(kg⋅K)

sg , kJ/(kg⋅K)

Converted and interpolated from Thermodynamic Properties of Klea 61 (British units, 20 pp.), copyright ICI Chemicals and Polymers Limited, 1993. Reproduced by permission. Tf = bubble-point temperature; Tg = dew-point temperature. hf = sf = 0 at 233.15 K = −40 °C.

THERMODYNAMIC PROPERTIES

2-279

30 0.9 0

5

0.70 0.75

0 .9

20

R 60

0.80

0

425

1.0

15

0.90

400

1.1 0

10

1.0

5

kJ/ kg

•

K

0.85

8

5

375

350

1.2

6

0

1.1

Pressure, bar

0.95

1.3

0

325

1.2

5

4

1.

35

300 K

40

2

5

1.

275

1

FIG. 2-22

1.4

1.5

250

Enthalpy–log-pressure diagram for KLEA 60.

300 Enthalpy (h), kJ/kg

350

400

2-280

PHYSICAL AND CHEMICAL DATA

30 0.80

0.70 0.75

0.8 5

0.80

20

450

0.9 0

0.85 0.90 15

0.9

5

0.95

425

1.0 0k J/k g

•

K

400

10

5 1.0

rated

0

375

350

1.1

6

Satu

Pressure, bar

vapo

r

8

1.1

5

4

1.2

0

325

1.2

5

300

30

2

1.3

5

1.

275 1.5 250 K

1 200 FIG. 2-23

250 Enthalpy–log-pressure diagram for KLEA 61.

300 Enthalpy (h), kJ/kg

350

THERMODYNAMIC PROPERTIES TABLE 2-314 Temp., °C

2-281

Saturated SUVA HP 62

Pf, bar

Pg, bar

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−6 Pa·s

kf, W/(m⋅K)

−50 −40 −30 −20 −10

0.852 1.367 2.095 3.087 4.404

0.821 1.325 2.041 3.018 4.321

0.000 0.000 0.000 0.000 0.000

761 779 799 820 843

0.2244 0.1434 0.0953 0.0656 0.0463

133.1 145.6 159.9 172.8 186.1

337.3 343.8 350.3 356.5 362.6

0.7318 0.7862 0.8460 0.8975 0.9487

1.6487 1.6380 1.6301 1.6245 1.6202

0.0970

1.220 1.260 1.302

370 318 276 238 207

0.0868 0.0834 0.0801

3.88 3.60 3.37

0 10 20 30 40

6.111 8.278 10.977 14.287 18.292

6.013 8.165 10.851 14.150 18.148

0.000 0.000 0.000 0.000 0.001

868 898 933 977 037

0.03338 0.02444 0.01809 0.01348 0.01003

200.0 214.5 229.9 246.2 263.8

368.3 373.6 378.3 382.2 385.0

1.0000 1.0515 1.1038 1.1574 1.2130

1.6188 1.6138 1.6106 1.6065 1.6005

1.351 1.412 1.489 1.592 1.753

181 158 138 122 106

0.0767 0.0733 0.0698 0.0663 0.0624

3.19 3.04 2.94 2.93 2.98

50 60 70 72.1c

23.08 28.75 35.58 37.32

22.94 28.63

0.001 122 0.001 261

386.1 384.2 375.9 361

1.5910 1.5742

2.09

91 76 61

0.0583 0.0535

3.26

0.002 06

283.2 305.8 339.8 361

1.2723 1.3389

37.32

0.00739 0.00527 0.00285 0.00206

vf, m3/kg

Prf

c = critical point. SUVA HP 62 = CHF2CF3 (R125) 44% wt + CH3CF3 (R143a) 52% wt + CH2FCF3 (R134a) 4% wt, near-azeotropic blend. Material used by permission of DuPont Fluoroproducts. Some values read from charts may be approximate.

TABLE 2-315

SUVA HP 62 at Atmospheric Pressure

Temp., °C

−45.63

−40

−20

0

20

40

60

80

100

120

v (m3/ kg) h (kJ/ kg) s (kJ/ kg⋅K) cp (kJ/ kg⋅K) µ (10−6 Pa⋅s) k (W/m⋅K) Pr Z

0.1866 336.0 1.6599 0.732 9.47 0.00860 0.806 0.9755

0.1921 344.4 1.6636 0.738 9.68 0.00932 0.767 0.9800

0.2100 359.9 1.7274 0.781 10.45 0.01059 0.771 0.9867

0.2278 376.2 1.7891 0.821 11.22 0.01186 0.777 0.9919

0.2455 393.1 1.8491 0.860 11.99 0.01313 0.785 0.9961

0.2630 410.9 1.9076 0.897 12.76 0.01440 0.795 0.9989

0.2805 429.3 1.9646 0.933 13.53 0.01568 0.805 1.0014

0.2980 448.4 2.0203 0.967 14.30 0.01695 0.816 1.0037

0.3153 468.2 2.0747 1.000 15.07 0.01827 0.827 1.0050

0.3325 488.7 2.1278 1.032 15.84 0.01949 0.839 1.0060

v,h, and s from DuPont bull. T—HP62—SI, June 1993 (17 pp.). cp and k from DuPont bull. ART 18, June 1993 (37 pp.). Some values read from charts may be approximate. Material used by permission of DuPont Fluoroproducts.

TABLE 2-316

Thermodynamic Properties of Saturated KLEA 66

Pressure, bar

Tf, K

Tg, K

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

0.69 1 1.5 2 2.5

221.46 228.89 237.69 244.45 249.99

228.77 236.05 244.69 251.33 256.76

0.000 0.000 0.000 0.000 0.000

7122 7237 7382 7501 7600

0.31325 0.22131 0.15140 0.11537 0.09104

−16.16 −5.89 6.22 15.51 23.12

241.25 245.91 251.38 255.52 258.95

1.0729 1.0580 1.0430 1.0330 1.0258

3 4 5 6 8

254.36 262.60 269.12 274.70 284.03

261.39 269.14 275.51 280.98 290.08

0.000 0.000 0.000 0.000 0.000

7695 7857 8001 8133 8375

0.07855 0.05964 0.04806 0.04021 0.03022

29.62 40.60 49.74 57.69 71.22

261.63 266.16 269.74 272.68 277.25

1.0201 1.0114 1.0055 0.9993 0.9913

10 12.5 15 17.5 20

291.74 299.87 306.87 313.05 318.60

297.56 305.44 312.18 318.10 323.40

0.000 0.000 0.000 0.000 0.000

8599 8867 9131 9400 9680

0.02410 0.01910 0.01571 0.01324 0.01137

82.73 95.32 106.59 116.97 126.73

280.64 283.74 285.93 287.28 288.26

0.9834 0.9770 0.9701 0.9633 0.9564

22.5 25 27.5 30

323.7 328.3 332.7 336.7

328.2 332.5 336.6 340.4

0.000 0.001 0.001 0.001

9981 032 072 125

0.00988 0.00883 0.00766 0.00703

136.0 145.1 153.9 162.7

288.6 288.4 287.8 286.6

0.9493 0.9418 0.9338 0.9251

vf, m3/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

Converted and interpolated from Thermodynamic Properties of Klea 66 (British units, 22 pp.), copyright ICI Chemicals and Polymers Limited, 1993. Reproduced by permission. Tf = bubble-point temperature; Tg = dew-point temperature. hf = sf = 0 at 233.15 K = −40°C.

2-282

PHYSICAL AND CHEMICAL DATA

0 1 .2

5 1 .1

0

450 1 .1

1.0 5

0.9

5

0.75

1.0 0

30

K

0.80 20

1.2 5

15

kJ

425

0.90

/kg

•

0.85

Satu ra

8

1 .3

375

1. 35

Pressure, bar

400 K ted v apor

10

0

Qua

lity

0.95

6

350

py En

tro

325

1.

40

4

1.4

5

300

2 275

1.5

0

1.5 250 K 1 250 FIG. 2-24

300 Enthalpy–log-pressure diagram for KLEA 66.

350 Enthalpy (h), kJ/kg

400

THERMODYNAMIC PROPERTIES TABLE 2-317 Temp., °C

2-283

Saturated SUVA MP 66

Pf, bar

Pg, bar

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−6 Pa·s

kf, W/(m⋅K)

Prf

−40 −30 −20 −10 0

0.788 1.239 1.872 2.726 3.850

0.585 0.952 1.479 2.212 3.198

0.000 0.000 0.000 0.000 0.000

710 725 740 758 778

0.3498 0.2224 0.1471 0.1008 0.0710

153.8 164.8 176.0 188.6 200.0

386.0 391.6 397.1 402.6 407.8

0.8184 0.8643 0.9095 0.9577 1.0000

1.8291 1.8100 1.7940 1.7807 1.7694

1.078 1.109 1.137 1.165 1.197

349 313 282 257 236

0.1209 0.1154 0.1106 0.1057 0.1012

3.11 3.01 2.90 2.83 2.79

10 20 30 40 50

5.297 7.120 9.379 12.133 15.444

4.491 6.146 8.229 10.808 13.955

0.000 0.000 0.000 0.000 0.000

801 827 858 895 939

0.05124 0.03771 0.02818 0.02131 0.01625

212.6 225.1 238.2 251.9 266.3

412.9 417.7 422.1 426.1 429.4

1.0450 1.0879 1.1311 1.1747 1.2190

1.7598 1.7512 1.7433 1.7357 1.7278

1.233 1.277 1.329 1.392 1.468

217 198 181 168 151

0.0967 0.0922 0.0877 0.0830 0.0781

2.77 2.74 2.74 2.82 2.84

60 70 80 90 100

19.378 24.00 29.37 35.55 42.30

17.750 22.28 27.64 33.96

0.000 0.001 0.001 0.001

994 066 164 313

0.01244 0.00951 0.00721 0.00534

281.6 298.1 316.3 337.2

431.9 433.4 433.2 430.4

1.2645 1.3120 1.3625 1.4187

1.7191 1.7088 1.6956 1.6768

1.564 1.652 1.802

139 127 116

0.0737 0.0684 0.0631 0.0577 0.0533

2.95 3.07 3.31

106.1c

46.82

46.82

0.001 95

0.00195

389

389

vf, m3/kg

c = critical point. SUVA MP 66 = R401 = CHClF2 (R22) 61% wt + CH3CHF2 (R152a) 11% wt + CHClFCF3 (R124) 28% wt, near-azeotropic blend. Material used by permission of DuPont Fluoroproducts. Some values read from charts are approximate.

TABLE 2-318

SUVA MP 66 at Atmospheric Pressure

Temp., °C

−28.63b

−20

0

20

40

60

80

100

120

140

v (m3/ kg) h (kJ/ kg) s (kJ/ kg⋅K) cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr Z

0.2086 392.2 1.8081 0.641 9.78 0.00817 0.767 0.9652

0.2177 397.9 1.8299 0.652 10.43 0.00921 0.738 0.9730

0.2362 411.2 1.8804 0.688 11.18 0.01041 0.737 0.9783

0.2545 425.1 1.9295 0.716 11.93 0.01161 0.736 0.9822

0.2727 439.6 1.9772 0.744 12.68 0.01282 0.735 0.9852

0.2908 454.6 2.0237 0.771 13.42 0.01404 0.735 0.9876

0.3089 470.1 2.0690 0.796 14.17 0.01536 0.734 0.9896

0.3269 486.2 2.1132 0.822 14.89 0.01668 0.734 0.9912

0.3449 502.7 2.1564 0.844 15.61 0.01796 0.733 0.9925

0.3629 519.4 2.1986 0.866 16.32 0.01929 0.733 0.9937

v, h, and s from DuPont bull. T—MP 66—SI, Jan. 1993 (17 pp.). cp, µ, and k from DuPont bull. ART 10, Jan. 1993 (27 pp.). Some values read from charts may be approximate. Material used by permission of DuPont Fluoroproducts. b = normal boiling point.

TABLE 2-319 Temp., °C

Saturated SUVA HP 80 vf, m3/kg

cpf, kJ/(kg⋅K)

µf, 10−6 Pa·s

kf, W/(m⋅K)

1.6327 1.6206 1.6110 1.6034 1.5972

0.0970

1.193 1.217 1.236

377 317 283 247 215

0.0880 0.0849 0.0813

3.84 3.54 3.27

1.0000 1.0461 1.0927 1.1403 1.1897

1.5919 1.5870 1.5820 1.5762 1.5690

1.253 1.286 1.340 1.412 1.512

188 165 146 128 113

0.0778 0.0743 0.0708 0.0672 0.0634

3.03 2.86 2.76 2.69 2.70

1.2420 1.2998

1.5589 1.5433

1.64 1.81

98 83 68

0.0593 0.0551

2.71 2.79

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

679 695 713 733 757

0.2033 0.1303 0.0869 0.0598 0.0423

139.6 150.8 163.1 174.9 187.6

334.1 339.9 345.6 351.1 356.4

0.7578 0.8070 0.8584 0.9053 0.9541

785 819 860 911 977

0.03060 0.02248 0.01671 0.01250 0.00936

200.0 213.0 226.7 241.2 256.8

361.3 365.9 369.8 373.1 375.4

24.04 29.97

0.001 070 0.001 212

0.00696 0.00505

273.9 293.6

376.2 374.6

41.35

0.001 850

0.00185

340

340

Pf, bar

Pg, bar

−50 −40 −30 −20 −10

0.962 1.520 2.305 3.370 4.776

0.872 1.403 2.156 3.188 4.560

0.000 0.000 0.000 0.000 0.000

0 10 20 30 40

6.588 8.877 11.720 15.195 19.388

6.336 8.592 11.404 14.855 19.034

0.000 0.000 0.000 0.000 0.000

50 60 70 75.5c

24.39 30.30 41.35

Prf

c = critical point. SUVA HP 80 = R402 = CHF2CF3 (R125) 60% wt + CH3CH2CH3 (R290) 2% wt + CHClF2 (R22) 38% wt, near-azeotropic blend. Material used by permission of DuPont Fluoroproducts. Some values, read from charts, may be approximate.

2-284

PHYSICAL AND CHEMICAL DATA

TABLE 2-320

SUVA HP 80 at Atmospheric Pressure

Temp., °C

−46.95b

−40

−20

0

20

40

60

80

100

120

v (m3/ kg) h (kJ/ kg) s (kJ/ kg⋅K) cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr Z

0.1768 335.9 1.6286 0.648 9.42 0.00888 0.687 0.9673

0.1827 340.5 1.6490 0.654 9.69 0.00932 0.680 0.9697

0.1996 354.3 1.7055 0.687 10.45 0.01059 0.678 0.9758

0.2164 368.6 1.7599 0.721 11.22 0.01186 0.681 0.9804

0.2331 383.5 1.8124 0.749 11.99 0.01313 0.685 0.9840

0.2497 398.7 1.8633 0.779 12.75 0.01440 0.690 0.9868

0.2663 414.9 1.9128 0.807 13.52 0.01568 0.696 0.9892

0.2828 431.4 1.9610 0.836 14.29 0.01695 0.703 0.9910

0.2992 448.5 2.0081 0.863 15.06 0.01822 0.713 0.9923

0.3155 466.1 2.0541 0.890 15.82 0.01949 0.722 0.9932

b = normal boiling pt. v, h, and s from DuPont bull. T—HP 80—SI, Jan. 1993 (17 pp.). cp, µ, and k from DuPont bull. ART 18, June 1993 (37 pp.). Some values read from charts may be approximate. Material used by permission of DuPont Fluoroproducts.

TABLE 2-321 Temp., °C

Saturated SUVA HP 81 vf, m3/kg

cpf, kJ/(kg⋅K)

µf, 10−6 Pa·s

kf, W/(m⋅K)

Prf

1.7122 1.6957 1.6820 1.6706 1.6611

1.178 1.191 1.204

383 333 290 253 223

0.1031 0.0983 0.0941 0.0900 0.0863

3.63 3.35 3.11

1.0000 1.0450 1.0905 1.1367 1.1842 1.2339

1.6528 1.6451 1.6376 1.6299 1.6211 1.6104

1.221 1.288 1.313 1.37 1.75 2.07

195 173 151 137 122 106

0.0818 0.0790 0.0753 0.0715 0.0676 0.0633

2.91 2.82 2.63 2.49 3.16 3.47

1.2873 1.3164

1.5961 1.5866

91 75

0.0586 0.0544

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

687 702 719 739 761

0.2425 0.1548 0.1028 0.0707 0.0499

140.3 151.4 163.3 174.9 187.8

351.7 357.2 362.7 368.0 373.0

0.7606 0.8092 0.8589 0.9054 0.9550

787 817 854 899 955 030

0.03610 0.02656 0.01980 0.01490 0.01125 0.00848

200.0 212.7 226.0 240.1 255.1 271.4

377.8 382.2 386.0 389.3 391.5 392.8

28.03 34.60

0.001 136 0.001 307

0.00632 0.00456

289.5 299.6

392.2 390.9

44.45

0.001 88

0.00188

351

351

Pf, bar

Pg, bar

−50 −40 −30 −20 −10

0.883 1.403 2.135 3.132 4.451

0.787 1.273 1.967 2.923 4.198

0.000 0.000 0.000 0.000 0.000

0 10 20 30 40 50

6.153 8.307 10.984 14.261 18.216 22.93

5.852 7.959 10.591 13.827 17.750 22.45

0.000 0.000 0.000 0.000 0.000 0.001

60 70 80 82.6c

28.50 35.01 44.45

c = critical point. SUVA HP 81 = R402 (38/2/60) = CHF2CF3 (R125) 38% wt + CH3CH2CH3 (R290) 2% wt + CHClF2 (R22) 60% wt, near-azeotropic blend. Material used by permission of DuPont Fluoroproducts. Some values read from charts may be approximate.

TABLE 2-322

SUVA HP 81 at Atmospheric Pressure

Temp., °C

−44.87b

−40

−20

0

20

40

60

80

100

120

v (m3/ kg) h (kJ/ kg) s (kJ/ kg⋅K) cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr Z

0.1903 354.7 1.7032 1.187 10.16 0.00739 1.632 0.9622

0.1960 357.7 1.7169 1.177 10.33 0.00768 1.583 0.9703

0.2142 370.8 1.7711 1.169 11.10 0.00902 1.439 0.9766

0.2322 384.6 1.8232 1.159 11.86 0.01036 1.327 0.9811

0.2500 398.8 1.8735 1.149 12.62 0.01170 1.239 0.9843

0.2678 413.6 1.9222 1.143 13.39 0.01304 1.174 0.9870

0.2856 428.9 1.9696 1.134 14.15 0.01438 1.124 0.9894

0.3032 444.7 2.0158 1.128 14.78 0.01572 1.061 0.9909

0.3209 461.0 2.0607 1.124 15.54 0.01706 1.024 0.9926

0.3386 477.7 2.1047 1.120 16.30 0.01840 0.992 0.9940

b = normal boiling point. v, h, and s from DuPont bull. T—HP 81—SI, Jan. 1993 (17 pp.). cp, µ, and k from DuPont bull. ART 18, June 1993 (37 pp.). Some values, read from charts, may be approximate. Material used by permission of DuPont Fluoroproducts.

THERMODYNAMIC PROPERTIES TABLE 2-323

2-285

Saturated Refrigerant 113*

P, bar

v f, m3/kg

vg, m3/kg

h f, kJ/kg

hg, kJ/kg

s f, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

c pf, kJ/(kg⋅K)

µ f, 10−4 Pa⋅s

k f, W/(m⋅K)

240 250 260 270 280

0.0233 0.0435 0.0767 0.1290 0.2076

5.908.−4 5.986.−4 6.066.−4 6.150.−4 6.237.−4

4.548 2.537 1.492 0.9189 0.5893

5.70 14.19 22.83 31.65 40.63

171.97 178.06 184.22 190.46 196.75

0.0241 0.0587 0.0926 0.1259 0.1585

0.7169 0.7142 0.7134 0.7141 0.7161

0.845 0.877 0.895 0.916 0.933

17.9 14.8 12.3 10.4 8.9

0.087 0.084 0.083 0.081 0.079

290 300 310 320 330

0.3217 0.4817 0.6999 0.9897 1.3657

6.328.−4 6.422.−4 6.522.−4 6.626.−4 6.737.−4

0.3917 0.2687 0.1895 0.1370 0.1012

49.77 59.07 68.51 78.09 87.80

203.08 209.44 215.80 222.17 228.53

0.1906 0.2221 0.2530 0.2833 0.3131

0.7192 0.7233 0.7281 0.7336 0.7396

0.946 0.958 0.971 0.983 0.992

7.6 6.6 5.9 5.2 4.7

0.077 0.075 0.073 0.071 0.069

340 350 360 370 380

1.8347 2.4406 3.174 4.062 5.123

6.854.−4 6.979.−4 7.112.−4 7.255.−4 7.411.−4

0.0762 0.0584 0.0454 0.0357 0.0284

97.64 107.58 117.65 127.82 138.11

234.86 241.16 247.41 253.59 259.70

0.3424 0.3711 0.3993 0.4270 0.4542

0.7460 0.7528 0.7598 0.7669 0.7742

1.000 1.013 1.029 1.042 1.059

4.2 3.8 3.4 3.2 2.9

0.066 0.065 0.062 0.060 0.058

T, K

390 400 410 420 430

6.379 7.849 9.556 11.52 13.78

7.580.−4 7.767.−4 7.975.−4 8.211.−4 8.483.−4

0.0229 0.0185 0.0151 0.0124 0.0102

148.52 159.07 169.78 180.69 191.85

265.71 271.59 277.31 282.83 288.09

0.4810 0.5075 0.5336 0.5595 0.5853

0.7815 0.7888 0.7958 0.8027 0.8091

1.084 1.109 1.14 1.18 1.22

2.7 2.46 2.28 2.10 1.93

0.056 0.054 0.052 0.050 0.047

440 450 460 470 480

16.35 19.26 22.56 26.29 30.52

8.806.−4 9.201.−4 9.713.−4 1.044.−3 1.174.−3

0.0083 0.0068 0.0055 0.0044 0.0032

203.35 215.31 227.97 241.79 258.16

292.98 297.38 301.03 303.41 303.00

0.6112 0.6375 0.6645 0.6933 0.7264

0.8149 0.8198 0.8234 0.8244 0.8198

1.27 1.32 1.38 1.45 1.54

1.75 1.58 1.33 1.07 0.77

0.045 0.042 0.039 0.035 0.031

487.5c

34.11

1.754.−3

0.0018

288.10

288.10

0.7828

0.7828

∞

0.30

∞

*Values reproduced or converted from Table 8, p. 17.91, ASHRAE Handbook, 1981: Fundamentals, American Society of Heating, Refrigerating and AirConditioning Engineers, Atlanta, 1981. Copyright material. Reproduced by permission of the copyright owner. c = critical point. The notation 5.908.−4 signifies 5.908 × 10−4. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives a saturation table from −30 to 214.4°C and an enthalpy–log-pressure diagram from 0.1 to 60 bar, 0 to 260°C. Equations and constants approximated to the 1985 ASHRAE tables were given by Mecaryk, K. and M. Masaryk, Heat Recovery Systems and CHP, 11, 2/3 (1991): 193–197. For experimental isochores for the compressed liquid from 21 to 304 bar, 266 to 453 K, see Blanke, W. and R. Weiss, PTB Bericht W 30, Braunschweig, Germany, 1992 (54 pp.). For tables to 300 bar, 460 K, see Geller, V. Z. and V. A. Rabinovich (ed.), Thermophysical Properties of Substances and Materials, Standartov, Moscow, 7 (1973): 135–154. Mastroianni, M. J., R. F. Stahl, et al., J. Chem. Eng. Data, 23, 2 (1978): 113–118 give a diagram to 1000 psia, 600°F. Tables and a diagram to 800 psia, 520°F are given by Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.

2-286

PHYSICAL AND CHEMICAL DATA

TABLE 2-324

Saturated Refrigerant 114*

P, bar

vf, m3/kg

vg, m3/kg

h f, kJ/kg

h g, kJ/kg

s f, kJ/(kg⋅K)

s g, kJ/(kg⋅K)

c pf, kJ/(kg⋅K)

µ f , 10−4 Pa⋅s

k f, W/(m⋅K)

190 200 210 220 230

0.0058 0.0137 0.029 0.059 0.109

6.326.−4 6.344.−4 6.366.−4 6.391.−4 6.421.−4

15.823 7.094 3.465 1.822 1.021

−42.58 −31.87 −21.48 −11.37 −1.50

125.78 131.01 136.41 141.95 147.61

−0.2091 −0.1542 −0.1035 −0.0565 −0.0126

0.6794 0.6648 0.6541 0.6466 0.6419

0.765 0.787 0.810 0.831 0.854

23.9 18.2 14.3 11.5 9.4

0.093 0.090 0.088 0.085 0.082

240 250 260 270 280

0.190 0.317 0.505 0.773 1.143

6.457.−4 6.500.−4 6.554.−4 6.619.−4 6.700.−4

0.604 0.375 0.2431 0.1633 0.1132

8.18 17.74 27.22 36.71 46.27

153.36 159.18 165.05 170.95 176.85

0.0286 0.0676 0.1047 0.1405 0.1751

0.6393 0.6387 0.6396 0.6418 0.6452

0.877 0.900 0.923 0.946 0.967

7.9 6.61 5.66 4.96 4.30

0.080 0.077 0.075 0.072 0.069

290 300 310 320 330

1.636 2.279 3.096 4.116 5.366

6.799.−4 6.918.−4 7.060.−4 7.224.−4 7.412.−4

0.0807 0.0590 0.0440 0.0334 0.0257

55.95 65.79 75.79 85.92 96.16

182.75 188.61 194.44 200.19 205.84

0.2090 0.2422 0.2748 0.3067 0.3379

0.6494 0.6543 0.6598 0.6657 0.6719

0.991 1.015 1.038 1.062 1.087

3.80 3.35 3.02 2.69 2.48

0.067 0.064 0.061 0.059 0.056

340 350 360 370 380

6.877 8.683 10.82 13.32 16.24

7.624.−4 7.863.−4 8.135.−4 8.453.−4 8.836.−4

0.0201 0.0158 0.0125 0.0099 0.0079

106.49 116.96 127.63 138.60 149.99

211.37 216.71 221.82 226.57 230.84

0.3685 0.3984 0.4280 0.4575 0.4872

0.6781 0.6843 0.6903 0.6957 0.7002

1.111 1.136 1.160 1.185 1.210

2.27 2.07 1.91 1.76 1.59

0.054 0.051 0.048 0.045 0.042

390 400 410 419.0c

19.62 23.52 28.00 32.61

9.324.−4 1.001.−3 1.118.−3 1.795.−3

0.0062 0.0048 0.0035 0.0018

162.01 175.03 190.13 219.90

234.36 236.61 236.20 219.90

0.5176 0.5496 0.5857 0.6559

0.7032 0.7036 0.6980 0.6559

1.236 1.261 1.5 ∞

1.39 1.17 0.87 0.34

0.038 0.034 0.030 ∞

T, K

*Values reproduced or converted from Table 9, p. 17.93, ASHRAE Handbook, 1981: Fundamentals, American Society of Heating, Refrigerating and AirConditioning Engineers, Atlanta, 1981. Copyright material. Reproduced by permission of the copyright owner. c = critical point. The notation 6.326.−4 signifies 6.326 × 10−4. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives a saturation table from −80 to 145.88°C and an enthalpy–log-pressure diagram from 0.1 to 100 bar, −20 to 220°C. Equations and constants approximated to the 1985 ASHRAE tables were given by Mecaryk, K. and M. Masaryk, Heat Recovery Systems and CHP., 11, 2/3 (1991): 193–197. Saturation and superheat tables and a diagram to 60 bar, 540 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (179 pp.). Tables and a chart to 1500 psia, 480°F are given by Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.

TABLE 2-325 Temp., °F −100 −80 −60 −40 −20

Saturated Refrigerant 115* Volume, ft3/lb

Enthalpy, Btu/lb

Entropy, Btu/(lb)(°F)

Pressure, lb/in2 abs.

Liquid

Vapor

Liquid

Vapor

Liquid

Vapor

2.327 4.573 8.306 14.13 22.74

0.00966 0.00986 0.01009 0.01033 0.01060

10.57 5.624 3.218 1.953 1.245

−13.07 −8.78 −4.43 0.00 4.50

45.83 48.39 50.96 53.53 56.07

−0.0335 −0.0219 −0.0108 0.0000 0.0104

0.1302 0.1286 0.1278 0.1275 0.1277

0 20 40 60 80

34.94 51.59 73.65 102.1 138.1

0.01090 0.01123 0.01161 0.01204 0.01255

0.8257 0.5657 0.3979 0.2857 0.2081

9.09 13.76 18.54 23.45 28.54

58.56 61.00 63.35 65.60 67.71

0.0206 0.0305 0.0401 0.0496 0.0591

0.1282 0.1290 0.1298 0.1308 0.1317

100 120 140 160 170

182.7 237.3 303.2 382.0 427.0

0.01316 0.01393 0.01496 0.01664 0.01838

0.1530 0.1125 0.0817 0.0567 0.0444

33.85 39.50 45.67 52.76 56.56

69.63 71.24 72.36 72.42 71.33

0.0686 0.0782 0.0884 0.0996 0.1055

0.1325 0.1330 0.1329 0.1314 0.1290

175.89 c

457.6

0.0261

0.0261

64.30

64.30

0.1175

0.1175

*Unpublished data of General Chemicals Division, Allied Chemical Company. Used by permission. c = critical temperature. No material in SI units appears in the 1993 ASHRAE Handbook—Fundamentals (SI ed.). Tables and a chart to 50 ata, 200°C are given by Mathias, H. and H. J. Loffler, Techn. Univ. Berlin rept., 1966 (42 pp.). A chart to 1500 psia, 500°F was given by Mears, W. H., E. Rosenthal, et al., J. Chem. Eng. Data, 11, 3 (1966): 338–343.

THERMODYNAMIC PROPERTIES TABLE 2-326

2-287

Thermodynamic Properties of Refrigerant 123

Pressure, bar

Temp., K

vg, m3/kg

h f, kJ/kg

h g, kJ/kg

s f, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

c pf, kJ/(kg⋅K)

k f, W/(m⋅K)

µ f, 10−6 Pa·s

Prf

0.1 0.5 1.0 1.013 1.5

249.49 282.87 300.62 300.99 312.25

0.000 0.000 0.000 0.000 0.000

6315 6664 6862 6868 7008

1.3430 0.2993 0.1567 0.1546 0.1070

13.25 41.72 58.62 58.99 70.51

198.51 218.53 229.20 229.43 236.52

0.0548 0.1610 0.2195 0.2208 0.2582

0.7977 0.7863 0.7869 0.7870 0.7892

0.849 0.923 1.000 1.001 1.038

0.0908 0.0811 0.0759 0.0758 0.0726

798.7 503.7 409.8 408.1 361.5

7.46 5.73 5.40 5.39 5.17

2.0 2.5 3.0 4.0 5.0

321.18 328.50 334.79 345.29 353.95

0.000 0.000 0.000 0.000 0.000

7126 7230 7323 7490 7640

0.08139 0.06546 0.05525 0.03836 0.03358

79.90 87.76 94.59 106.16 115.83

241.76 246.20 249.96 256.17 261.17

0.2877 0.3118 0.3323 0.3661 0.3935

0.7917 0.7942 0.7965 0.8006 0.8042

1.063 1.079 1.091 1.108 1.120

0.0696 0.0678 0.0660 0.0630 0.0605

329.6 306.1 287.4 259.7 239.1

5.03 4.87 4.75 4.57 4.43

6 8 10 15 20

361.41 373.92 384.19 404.54 420.30

0.000 0.000 0.000 0.000 0.000

7779 8038 8280 8874 9512

0.02799 0.02090 0.01675 0.01062 0.00751

124.23 138.48 150.35 174.49 194.19

265.34 272.04 277.18 286.01 291.01

0.4168 0.4551 0.4860 0.5462 0.5928

0.8073 0.8124 0.8162 0.8218 0.8232

1.130 1.148 1.168 1.234 1.345

25 30 36.68*

433.33 444.10 456.83

0.001 030 0.001 136 0.001 818

0.00549 0.00408 0.00182

212.00 228.26 264.54

293.05 291.27 264.54

0.6334 0.6692 0.7393

0.8203 0.8112 0.7393

1.559 2.005

vf, m3/kg

h f = s f = 0 at −40°C = 233.15 K. s f, s g, cp,f units: kJ/kg⋅K. Interpolated and converted from 1993 ASHRAE Handbook—Fundamentals (SI ed.) saturation table from −40 to 183.68°C. This source also contains an enthalpy–log-pressure diagram from 0.1 to 200 bar, −40 to 320°C.

TABLE 2-327 Temp., °C

Saturated Refrigerant 124 Pressure, bar

vf, m3/kg

vg, m3/kg

h f, kJ/kg

h g, kJ/kg

s f, kJ/(kg⋅K)

s g, kJ/(kg⋅K)

−40 −30 −20 −10 0

0.2680 0.4499 0.7197 1.1044 1.6348

0.000 644 0.000 655 0.000 668 0.000 681 0.000 696

0.5173 0.3185 0.2049 0.1369 0.0945

159.1 169.3 179.5 189.7 200.0

334.9 340.6 346.2 351.8 357.4

0.8384 0.8813 0.9222 0.9616 1.0000

1.5927 1.5856 1.5808 1.5777 1.5762

10 20 30 40 50

2.3447 3.2710 4.4529 5.9320 7.7521

0.000 711 0.000 728 0.000 747 0.000 768 0.000 791

0.06703 0.04867 0.03604 0.02713 0.02069

210.5 221.3 282.3 243.7 255.4

363.0 368.5 373.9 379.2 384.4

1.0376 1.0747 1.1115 1.1480 1.1843

1.5760 1.5768 1.5785 1.5808 1.5836

60 70 80 90 100

9.9599 12.605 15.742 19.432 23.749

0.000 818 0.000 849 0.000 887 0.000 935 0.000 999

0.01594 0.01236 0.00961 0.00744 0.00569

267.5 280.1 293.2 307.0 321.9

389.3 393.9 398.0 401.3 403.4

1.2207 1.2572 1.2942 1.3318 1.3710

1.5864 1.5890 1.5909 1.5915 1.5894

110 120 122.5c

28.787 34.702 36.340

0.001 098 0.001 338 0.001 810

0.00420 0.00269 0.00181

338.4 360.6 378.5

403.0 394.9 378.5

1.4133 1.4685

1.5820 1.5558

c = critical point. Bull. T—124—SI, Jan. 1993 (28 pp.). Used by permission of DuPont Fluoroproducts. The 1993 ASHRAE Handbook— Fundamentals (SI ed.) gives a saturation table to 122.47°C and a diagram to 200 bar, 320°C.

FIG. 2-25

2-288

Enthalpy–log-pressure diagram for Refrigerant 123.

THERMODYNAMIC PROPERTIES TABLE 2-328

2-289

Thermophysical Properties of Saturated Refrigerant 125 µf , 10−6 Pa·s

Temp., K

Pressure, bar

vf, m3/kg

vg , m3/kg

h f, kJ/kg

h g , kJ/kg

s f, kJ/(kg⋅K)

s g, kJ/(kg·K)

172.5* 180 190 200 210

0.035 0.064 0.133 0.257 0.465

0.000591 0.000599 0.000611 0.000624 0.000638

3.48 1.958 0.986 0.5312 0.3057

−30.5 −23.3

140.8 146.7

−0.1386 −0.1024

0.7183 0.7067

220 224.9† 230 240 250

0.794 1.013 1.290 2.005 3.000

0.000653 0.000660 0.000669 0.000686 0.000705

0.1854 0.1475 0.1175 0.0775 0.0527

−13.9 −8.8 −3.4 7.7 19.3

152.6 155.5 158.4 164.2 170.0

−0.0604 −0.0386 −0.0147 0.0324 0.0800

0.6981 0.6948 0.6919 0.6875 0.6847

1.077 1.139 1.184

445.8 411.2 379.6 326.6 282.8

260 270 280 290 300

4.336 6.078 8.298 11.068 14.476

0.000725 0.000749 0.000776 0.000809 0.000848

0.0369 0.0264 0.0193 0.0143 0.0106

31.3 43.7 56.5 69.7 83.5

175.3 180.5 185.4 189.9 194.2

0.1274 0.1743 0.2206 0.2666 0.3126

0.6831 0.6822 0.6819 0.6815 0.6805

1.221 1.257 1.299 1.356 1.437

245.8 213.6 185.3 159.7 136.3

310 320 330 339.4‡

18.62 23.63 29.65 35.95

0.000898 0.000969 0.001088 0.00175

0.0079 0.0059 0.0041 0.0018

98.1 113.9 132.2 169.0

196.9 198.5 197.4 169.0

0.3597 0.4079 0.4621 0.5699

0.6774 0.6726 0.6639 0.5699

1.57 1.82

115.0 95.4

cpf, kJ/(kg⋅K)

644 531

* = triple point; † = normal boiling point; ‡ = critical point. Converted, extrapolated and interpolated from 1993 ASHRAE Handbook—Fundamentals (SI ed.) hf = sf = 0 at 233.15 K = −40°C. This source also contains an enthalpy–log-pressure diagram from 0.3 to 100 bar, −65 to 175°C. An apparently identical diagram but a different saturation table is contained in Duarte-Garza, H.A., Hwang, C.A. et al., ASHRAE Trans., 99, 2 (1993): 649–664. R124: The 1993 ASHRAE Handbook—Fundamentals (SI ed.) contains a saturation table from −60 to 122.47°C.

TABLE 2-329

Thermophysical Properties of Refrigerant 134a Temp., K

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

s f, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf , kJ/(kg·K)

0.0039t 0.5 0.6 0.8 1.0

169.85 232.69 236.22 242.04 246.80

0.0006285 0.0007062 0.0007113 0.0007199 0.0007272

35.263 0.3692 0.3015 0.2375 0.1924

−76.68 −0.57 3.85 11.15 17.14

186.50 225.27 227.52 231.19 234.15

−0.3830 −0.0025 0.0161 0.0467 0.0713

1.1665 0.9669 0.9636 0.9560 0.9507

1.147 1.242 1.248 1.258 1.267

1.013 1.5 2.0 2.5 3.0

247.03 256.03 263.09 268.88 273.82

0.0007276 0.0007421 0.0007543 0.0007648 0.0007743

0.1902 0.1312 0.0999 0.0806 0.0677

17.50 28.96 38.13 45.75 52.33

234.33 239.86 244.14 247.60 250.50

0.0728 0.1181 0.1533 0.1819 0.2059

0.9503 0.9419 0.9364 0.9326 0.9297

1.268 1.288 1.306 1.322 1.337

4.0 5 6 8 10

282.08 288.89 294.72 304.47 312.53

0.0007912 0.0008063 0.0008203 0.0008460 0.0008703

0.0512 0.04116 0.03434 0.02565 0.02035

63.50 72.87 81.04 95.00 106.86

255.22 258.99 262.09 267.01 270.74

0.2458 0.2784 0.3062 0.3522 0.3901

0.9256 0.9232 0.9208 0.9171 0.9144

12 14 16 18 20

319.47 325.57 330.11 336.04 340.63

0.0008938 0.0009170 0.0009362 0.0009555 0.0009894

0.01675 0.01414 0.01247 0.01059 0.00931

117.34 126.80 134.00 143.68 151.39

273.65 275.92 277.40 279.01 279.95

0.4227 0.4515 0.4729 0.5013 0.5236

25 30 35 40 40.56c

350.73 359.37 366.89 373.50 374.18

0.0010585 0.001144 0.001270 0.001606 0.001948

0.00695 0.00528 0.00399 0.00255 0.00195

169.30 185.05 203.19 229.24 241.22

280.64 278.32 273.52 257.12 241.22

0.5738 0.6212 0.6657 0.7292 0.7620

Pressure, bar

µ f, 10−6 Pa·s

k f, W/(m⋅K)

Prf

0.1121 0.1105 0.1078 0.1056

5.61 5.42 5.12 4.90

406 358.7 326.6 303.2 285.1

0.1054 0.1013 0.0980 0.0954 0.0931

4.89 4.56 4.35 4.20 4.09

1.363 1.387 1.410 1.454 1.497

257.7 237.5 221.6 197.6 179.5

0.0893 0.0861 0.0835 0.0790 0.0753

3.93 3.83 3.74 3.64 3.57

0.9120 0.9095 0.9073 0.9041 0.9010

1.541 1.589 1.631 1.698 1.764

165.1 153.0 144.3 133.2 124.8

0.0721 0.0693 0.0672 0.0645 0.0623

3.53 3.51 3.50 3.51 3.53

0.8913 0.8807 0.8574 0.8038 0.7620

1.987 2.418

106.6 90.4

0.0577 0.0538

3.67 4.06

2187 506 480 438 408

t = triple point, c = critical point. hf = sf = 0 at −40°C = 233.15 K. T, v, h, and s interpolated and converted from Refrigerant 134a—Thermodynamic and Physical Properties, Int. Inst. Refrig., Paris, France, 1992 (28 pp.). Other properties from this source and from Oliveira, C. M. B. P. and W. A. Wakeham, Int. J. Thermophys., 14, 1 (1993): 33–44; Krauss, R., J. Luettmer-Strathmann, et al., Int. J. Thermophys., 14, 4 (1993): 951–988; ASHRAE Handbook—Fundamentals, Atlanta, GA, 1993; ICI KLEA 134a bulletin, 1993 (43 pp.); and R134a— Thermodynamic and Physical Properties, Int. Inst. Refrig., Paris, France, 1992 (28 pp.). Papers giving polynomial curve fits and similar simple equations include Cleland, A. C., Rev. Int. Froid = Int. J. Refrig., 17, 4 (1994): 245–249; Dobrokhotov, A., A. Grebenkov, et al., Proc. 14th Japan Symp. Thermophys. Props., (1993): 271–274; Huber, M. L. and J. F. Ely, Rev. Int. Froid = Int. J. Refrig., 17, 1 (1994): 18–31 (includes extensive list of vapor pressure and liquid density sources for many refrigerants); Modic, J., Proc. 11th Int. Symp. Htg, Refrig and Air-Condg., Zagreb, (1991): 174–185; and Kabelac, S., Int. J. Refrig., 14, (1991): 217–222. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives saturation data for integral degrees Celsius with temperatures on the ITS 90 scale from −103.03°C to 101.03°C. The thermodynamic diagram from 0.1 to 200 bar extends to 320°C.

FIG. 2-26

2-290

Enthalpy–log-pressure diagram for Refrigerant 125.

THERMODYNAMIC PROPERTIES TABLE 2-330

2-291

Thermophysical Properties of Compressed Gaseous Refrigerant 134a Pressure, bar

Temp., K

0

1

2

3

4

5

6

cp (kJ/ kg⋅K) µ (10−6 Pa·s) 230 k (W/m⋅K) Pr

— — — —

— — — —

— — — —

— — — —

— — — —

— — — —

cp (kJ/ kg⋅K) −6 240 µ (10 Pa·s) k (W/m⋅K) Pr

— — — —

— — — —

— — — —

— — — —

— — — —

— — — —

cp (kJ/ kg⋅K) −6 250 µ (10 Pa·s) k (W/m⋅K) Pr

0.7437 10.11 0.0096 0.783

0.7953 10.15 0.0097 0.797

— — — —

— — — —

— — — —

— — — —

— — — —

cp (kJ/ kg⋅K) −6 260 µ (10 Pa·s) k (W/m⋅K) Pr

0.7627 10.47 0.0105 0.761

0.8048 10.51 0.0107 0.790

— — — —

— — — —

— — — —

— — — —

— — — —

cp (kJ/ kg) −6 270 µ (10 Pa·s) k (W/m⋅K) Pr

0.7813 10.84 0.0117 0.724

0.8158 10.88 0.0118 0.761

0.8557 10.94 0.0118 0.793

— — — —

— — — —

— — — —

— — — —

cp (kJ/ kg⋅K) −6 280 µ (10 Pa·s) k (W/m⋅K) Pr

0.7996 11.22 0.0122 0.735

0.8283 11.26 0.0123 0.757

0.8604 11.29 0.0123 0.790

— — — —

— — — —

— — — —

cp (kJ/ kg⋅K) −6 290 µ (10 Pa·s) k (W/m⋅K) Pr

0.8176 11.62 0.0130 0.731

0.8412 11.65 0.0131 0.748

0.8673 11.68 0.0131 0.773

0.8938 11.71 0.0133 0.787

0.9335 11.74

cp (kJ/ kg⋅K) −6 300 µ (10 Pa·s) k (W/m⋅K) Pr

0.8354 12.05 0.0139 0.730

0.8556 12.06 0.0139 0.742

0.8771 12.08 0.0140 0.757

0.8972 12.10 0.0141 0.770

0.9277 12.15 0.0142 0.792

0.0143 0.816

cp (kJ/ kg⋅K) −6 310 µ (10 Pa·s) k (W/m⋅K) Pr

0.8530 12.44 0.0145 0.730

0.8703 12.45 0.0146 0.742

0.8875 12.47 0.0147 0.753

0.9046 12.49 0.0148 0.763

0.9292 12.52 0.0149 0.781

0.9546 12.54 0.0150 0.798

cp (kJ/ kg⋅K) −6 320 µ (10 Pa·s) k (W/m⋅K) Pr

0.8703 12.83 0.0153 0.730

0.8843 12.84 0.0153 0.740

0.8993 12.86 0.0154 0.751

0.9163 12.88 0.0155 0.761

0.9356 12.90 0.0156 0.774

0.9548 12.93 0.0157 0.786

0.9750 12.97 0.0158 0.800

cp (kJ/ kg⋅K) −6 330 µ (10 Pa·s) k (W/m⋅K) Pr

0.8874 13.22 0.0160 0.729

0.8996 13.23 0.0161 0.739

0.9114 13.25 0.0161 0.750

0.9268 13.27 0.0162 0.759

0.9398 13.29 0.0163 0.766

0.9569 13.32 0.0164 0.777

0.9750 13.35 0.0165 0.789

cp (kJ/ kg⋅K) −6 340 µ (10 Pa·s) k (W/m⋅K) Pr

0.9042 13.61 0.0169 0.728

0.9152 13.62 0.0169 0.738

0.9262 13.64 0.0169 0.748

0.9372 13.66 0.0170 0.755

0.9502 13.68 0.0170 0.765

0.9632 13.70 0.0171 0.772

0.9770 13.73 0.0171 0.780

cp (kJ/ kg⋅K) −6 350 µ (10 Pa·s) k (W/m⋅K) Pr

0.9208 13.98 0.0175 0.730

0.9307 13.99 0.0176 0.740

0.9406 14.01 0.0176 0.749

0.9505 14.03 0.0177 0.754

0.9607 14.05 0.0177 0.763

0.9695 14.07 0.0178 0.767

0.9830 14.10 0.0179 0.774

— — — — 0.9606

0.9976

0.9827 0.0152

Dashes indicate unavailable states; blanks indicate no data. Note that “profound differences” presently exist in the transport properties of R134a according to Chemistry International, 16(6), 233, Nov. 1994, and 18(2) 44–47, 1996.

2-292

PHYSICAL AND CHEMICAL DATA

TABLE 2-330

Thermophysical Properties of Compressed Gaseous R134a (Concluded ) Pressure, bar

Temp., K

8

10

12.5

15

17.5

20

22.5

cp (kJ/ kg⋅K) 300 µ (10−6 Pa·s) k (W/m⋅K) Pr

— — — —

— — — —

— — — —

— — — —

— — — —

— — — —

— — — —

cp (kJ/ kg⋅K) 310 µ (10−6 Pa·s) k (W/m⋅K) Pr

1.053

— — — —

— — — —

— — — —

— — — —

— — — —

— — — —

cp (kJ/ kg⋅K) 320 µ (10−6 Pa·s) k (W/m⋅K) Pr

1.028 13.05 0.0161 0.833

1.097 13.13

— — — —

— — — —

— — — —

— — — —

— — — —

cp (kJ/ kg⋅K) 330 µ (10−6 Pa·s) k (W/m⋅K) Pr

1.015 13.41 0.0168 0.810

1.065 13.49 0.0171 0.840

1.151 13.64 0.0177 0.887

1.276 13.86 0.0184 0.961

— — — —

— — — —

— — — —

cp (kJ/ kg⋅K) 340 µ (10−6 Pa·s) k (W/m⋅K) Pr

1.008 13.79 0.0174 0.799

1.049 13.86 0.0177 0.821

1.107 13.98 0.181 0.855

1.187 14.17 0.0187 0.899

1.319

— — — —

— — — —

cp (kJ/ kg⋅K) 350 µ (10−6 Pa·s) k (W/m⋅K) Pr

1.008 14.15 0.0181 0.788

1.040 14.22 0.0183 0.828

1.086 14.34 0.0186 0.837

1.148 14.49 0.0192 0.866

1.225

1.340 14.97 0.0205

0.0155

TABLE 2-331

0.0198

Refrigerant 141b

The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives saturation data to 150°C and a diagram to 20 bar, 150°C. For equation of state including decomposition, see Weber, L. A., paper 69, Proc. 18th Int. Congr. Refrig., Montreal, 1991.

1.525 0.0215

THERMODYNAMIC PROPERTIES

2-293

40 410

5

5

0.

6

400 K

1 .7

0.

0.

8

Q ua

lit

y

1 .8

=

20

0

0.

7

25

1 .7

30

0

0.4

1. 25

390

90

0

10 9 8 7

1.

1.

6 95

py

5

tro

1.

En

Pressure (P ), bar

0. 9

1.

kJ /k g

•

85

K

15

4

00

2.

380

3

370 360

2.5 340 330

2 310 1.5

290 260

1

270

320

05

2.

300 K

280 10

2.

250 400

450 Enthalpy (h), kJ/kg

FIG. 2-27

350 K

Enthalpy–log-pressure diagram for Refrigerant 134a.

500

2-294

PHYSICAL AND CHEMICAL DATA

TABLE 2-332

Refrigerant 142b* µ f, 10−4 Pa⋅s

k f, W/(m⋅K)

1.15 1.17 1.18 1.19

0.517

0.123 0.118 0.114 0.111 0.109

0.9341 0.9264 0.9204 0.9155 0.9116

1.21 1.22 1.24 1.26 1.28

0.466 0.422 0.385 0.355 0.329

0.103 0.099 0.095 0.091 0.088

1.30 1.32 1.34

0.305 0.285 0.267 0.241 0.216

0.084 0.080 0.075 0.072 0.068

0.192

0.064 0.060 0.056 0.052 0.048

T, K

P, bar

vf, m3/kg

vg, m3/kg

h f, kJ/kg

h g, kJ/kg

s f, kJ/(kg⋅K)

s g, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

200 210 220 230 240

0.0380 0.0728 0.1314 0.2252 0.3691

7.505.−4 7.626.−4 7.751.−4 7.883.−4 8.019.−4

4.337 2.374 1.373 0.833 0.527

−24.36 −17.48 −10.21 −2.52 9.92

200.49 206.82 213.28 219.82 229.74

−0.1123 −0.0788 −0.0450 −0.0109 0.0414

1.0119 0.9893 0.9708 0.9558 0.9387

250 260 270 280 290

0.5815 0.8846 1.3046 1.8714 2.6184

8.164.−4 8.317.−4 8.480.−4 8.653.−4 8.843.−4

0.346 0.234 0.162 0.115 0.0838

14.32 23.54 33.32 43.68 54.60

233.05 239.66 246.18 252.57 258.77

0.0592 0.0952 0.1320 0.1695 0.2076

300 310 320 330 340

3.583 4.803 6.324 8.187 10.44

9.047.−4 9.273.−4 9.525.−4 9.810.−4 1.014.−3

0.0619 0.0464 0.0353 0.0271 0.0210

66.07 78.07 90.55 103.45 116.71

264.69 270.26 275.40 280.01 283.99

0.2462 0.2851 0.3243 0.3634 0.4024

0.9082 0.9051 0.9020 0.8985 0.8943

350 360 370 380 390

13.13 16.30 20.01 24.29 29.20

1.052.−3 1.099.−3 1.157.−3 1.235.−3 1.348.−3

0.0164 0.0129 0.0102 0.0080 0.0062

130.30 144.18 158.45 173.45 190.16

287.23 289.61 291.01 291.22 289.77

0.4409 0.4791 0.5170 0.5557 0.5974

0.8893 0.8831 0.8753 0.8656 0.8528

400 410c

34.78 41.5

1.541.−3 2.300.−3

0.0046 0.0023

212.57 255.00

284.04 255.00

0.6521

0.8307

0.044 ∞

*Values reproduced and converted from Table 10, p. 17.95, ASHRAE Handbook, 1981: Fundamentals, American Society of Heating, Refrigerating and AirConditioning Engineers, Atlanta, 1981. Copyright material. Reproduced by permission of the copyright owner. c = critical point. The notation 7.505.−4 signifies 7.505 × 10−4. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives material for integral degrees Celsius with temperatures on the IPTS 68 scale from −50 to 125°C. The thermodynamic diagram from 0.1 to 35 bar extends to 180°C. For experimental isochores for the compressed liquid from 6 to 298 bar, 147 to 432 K, see Blanke, W. and R. Weiss, PTB Bericht W 30, Braunschweig, Germany, 1992 (54 pp.). Tables and a diagram to 500 psia, 400°F are given in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.

TABLE 2-333 Temp., K t

161.82 170 180 190 200 210 220 225.92 230 240 250 260 270 280 290 300 310 320 330 340 346.75c

Saturated Refrigerant R143a*

Pressure, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

h g, kJ/kg

s f , kJ/(kg⋅K)

s g, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µ f , 10−4 Pa⋅s

k f, W/(m⋅K)

0.01124 0.02497 0.05914 0.126 0.2458 0.4455 0.7586 1.01325 1.225 1.89 2.806 4.027 5.613 7.629 10.14 13.23 16.98 21.48 26.85 33.25 38.32

0.000752 0.000764 0.000778 0.000793 0.000809 0.000826 0.000845 0.000856 0.000865 0.000886 0.000910 0.000936 0.000966 0.000999 0.001038 0.001084 0.001140 0.001214 0.001321 0.001514 0.002311

14.22 6.709 2.991 1.474 0.7898 0.4532 0.2754 0.2098 0.1755 0.1164 0.07975 0.05617 0.04045 0.02964 0.02200 0.01646 0.01234 0.009182 0.006678 0.004520 0.002311

53.2 63.0 75.3 87.7 100.3 113.1 126.1 133.9 139.3 152.8 166.6 180.8 195.3 210.3 225.9 242.1 259.2 277.4 297.5 321.8 360.6

320.5 325.5 331.8 338.1 344.5 350.8 357.1 360.8 363.3 369.4 375.3 380.9 386.2 391.1 395.4 399.0 401.6 402.7 401.0 393.4 360.6

0.3181 0.3774 0.4474 0.5147 0.5792 0.6415 0.7018 0.7367 0.7604 0.8176 0.8736 0.9287 0.983 1.037 1.091 1.144 1.199 1.255 1.315 1.385 1.471

1.970 1.922 1.872 1.832 1.800 1.774 1.752 1.741 1.734 1.720 1.708 1.698 1.690 1.682 1.675 1.668 1.659 1.647 1.629 1.595 1.471

1.188 1.215 1.235 1.252 1.268 1.287 1.308 1.323 1.333 1.362 1.394 1.431 1.475 1.527 1.593 1.679 1.804 2.006 2.421 4.021 —

6.011 5.366 4.692 4.121 3.636 3.221 2.864 2.676 2.556 2.288 2.055 1.852 1.673 1.496 1.334 1.192 1.067 0.9569 0.8321 0.6560 —

0.1416 0.1403 0.1372 0.1331 0.1281 0.1226 0.1168 0.1133 0.1108 0.1047 0.09862 0.09272 0.08682 0.08098 0.07521 0.06951 0.06381 0.05803 0.05202 0.04480 —

*Values calculated from NIST Thermodynamic Properties of Refrigerants and Refrigerant Mixtures Database (REFPROP, Version 5). Thermodynamic properties are from 32-term MBWR equation of state; transport properties are from extended corresponding states model. t = triple point; c = critical point.

THERMODYNAMIC PROPERTIES TABLE 2-334

2-295

Saturated Refrigerant R152a*

Temp., K

Pressure, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

h g, kJ/kg

s f , kJ/(kg⋅K)

s g, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µ f , 10−4 Pa⋅s

k f, W/(m⋅K)

154.56t 160 170 180 190 200 210 220 230 240 249.12 250 260 270 280 290 300 310 320 330 340 350 360 370 380 386.41c

0.000641 0.001297 0.004145 0.01141 0.02775 0.06088 0.1224 0.2284 0.4004 0.6647 1.01325 1.053 1.603 2.354 3.354 4.650 6.297 8.351 10.87 13.92 17.57 21.90 27.00 32.97 39.97 45.17

0.000839 0.000846 0.000859 0.000873 0.000887 0.000902 0.000918 0.000935 0.000952 0.000971 0.000989 0.000991 0.001012 0.001035 0.001060 0.001087 0.001118 0.001152 0.001190 0.001235 0.001289 0.001355 0.001440 0.001563 0.001785 0.002717

303.6 155.2 51.59 19.82 8.588 4.110 2.138 1.193 0.7064 0.4397 0.2961 0.2855 0.1922 0.1334 0.09500 0.06916 0.05126 0.03857 0.02935 0.02251 0.01735 0.01335 0.01020 0.007603 0.005274 0.002717

14.0 22.2 37.2 52.4 67.6 82.9 98.3 113.8 129.5 145.5 160.2 161.6 178.0 194.7 211.7 229.1 246.9 265.2 284.1 303.7 324.2 345.7 368.8 394.3 425.4 477.3

419.8 423.5 430.6 437.9 445.3 452.8 460.4 468.0 475.6 483.1 489.8 490.5 497.7 504.7 511.4 517.8 523.8 529.3 534.1 538.2 541.3 542.9 542.5 538.5 526.2 477.3

0.1130 0.1647 0.2560 0.3425 0.4247 0.5032 0.5784 0.6507 0.7205 0.7881 0.8481 0.8538 0.9178 0.9805 1.042 1.103 1.162 1.222 1.281 1.340 1.400 1.460 1.523 1.591 1.671 1.778

2.738 2.673 2.570 2.484 2.413 2.353 2.303 2.261 2.225 2.195 2.171 2.169 2.147 2.129 2.112 2.098 2.085 2.073 2.062 2.051 2.038 2.024 2.006 1.980 1.936 1.778

1.492 1.500 1.510 1.517 1.525 1.535 1.547 1.562 1.580 1.600 1.622 1.624 1.651 1.681 1.716 1.756 1.803 1.859 1.928 2.015 2.131 2.299 2.573 3.143 5.407 —

10.85 9.614 8.058 6.940 6.012 5.236 4.582 4.028 3.556 3.153 2.834 2.805 2.500 2.231 1.995 1.789 1.607 1.447 1.305 1.180 1.069 1.005 0.9225 0.8191 0.6638 —

0.1932 0.1894 0.1822 0.1753 0.1685 0.1618 0.1552 0.1487 0.1423 0.1361 0.1304 0.1299 0.1239 0.1179 0.1121 0.1064 0.1008 0.09526 0.08986 0.08457 0.07939 0.07336 0.06666 0.06032 0.05176 —

*Values calculated from NIST Thermodynamic Properties of Refrigerants and Refrigerant Mixtures Database (REFPROP, Version 5). Thermodynamic properties are from 32-term MBWR equation of state; transport properties are from extended corresponding states model. t = triple point; c = critical point.

TABLE 2-335 Temp., °F

Saturated Refrigerant 216* Pressure, lb/in2 abs.

Volume, ft3/lb

Enthalpy, Btu/lb

Entropy, Btu/(lb)(°F)

Liquid

Vapor

Liquid

Vapor

Liquid

Vapor

−40 −20 0 20 40

0.339 0.713 1.382 2.497 4.247

0.00927 0.00942 0.00958 0.00974 0.00992

59.957 29.749 15.986 9.184 5.582

0.000 4.778 9.541 14.298 19.056

62.415 65.276 68.208 71.199 74.239

0.0000 0.0111 0.0217 0.0318 0.0415

0.1487 0.1487 0.1493 0.1504 0.1520

60 80 100 120 140

6.862 10.612 15.797 22.753 31.845

0.01010 0.01030 0.01050 0.01073 0.01097

3.558 2.361 1.6215 1.1462 0.8304

23.821 28.598 33.391 38.205 43.049

77.319 80.429 83.559 86.701 89.845

0.0509 0.0599 0.0686 0.0770 0.0852

0.1538 0.1559 0.1582 0.1607 0.1632

160 180 200 220 240

43.468 58.046 76.033 97.913 124.21

0.01124 0.01153 0.01186 0.01223 0.01266

0.6142 0.4623 0.3529 0.2725 0.2121

47.930 52.861 57.857 62.939 68.132

92.981 96.099 99.186 102.225 105.196

0.0931 0.1009 0.1085 0.1161 0.1235

0.1658 0.1685 0.1712 0.1739 0.1765

260 280 300 320 340

155.50 192.40 235.63 286.03 344.81

0.01317 0.01378 0.01458 0.01570 0.01764

0.1660 0.1300 0.1013 0.0776 0.0565

73.474 79.015 84.835 91.089 98.234

108.066 110.789 113.282 115.373 116.538

0.1309 0.1384 0.1460 0.1539 0.1628

0.1790 0.1813 0.1834 0.1851 0.1856

355.98c

399.45

0.02771

0.0277

110.248

110.248

0.1773

0.1773

*From published data, Chemicals Division, Union Carbide Corporation. Used by permission. The paper describing these data is by Shank, ASHRAE J., 7 (1965): 94–101. c = critical temperature. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives material for integral degrees Celsius with temperatures on the ITS 90 scale from −118.59 to 113.26°C. The thermodynamic diagram from 0.1 to 30 bar extends to 180°C. For tables and a diagram to 400 psia, 360°F, see Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993. Thermal conductivity data as a function of pressure and temperature are reported by Krauss, R. and K. Stephan, Proc. 12th Symp. Thermophys. Props., Boulder, CO, 1994.

2-296

PHYSICAL AND CHEMICAL DATA TABLE 2-336

Saturated Refrigerant 245*

P, bar

v f, m3/kg

vg, m3/kg

172 180 190 200 210

0.0034 0.0076 0.0190 0.0425 0.0870

6.46.−4 6.57.−4 6.70.−4 6.83.−4 6.97.−4

31.49 14.63 6.20 2.91 1.48

220 230 240 250 260

0.1654 0.2946 0.4958 0.7946 1.2204

7.11.−4 7.25.−4 7.40.−4 7.55.−4 7.72.−4

270 280 290 300 310

1.806 2.584 3.600 4.888 6.491

320 330 340 350 360 370 375 380.1c

T, K

h f, kJ/kg

h g, kJ/kg

s f, kJ/(kg⋅K)

s g, kJ/(kg⋅K)

−63.4 −55.9 −46.2 −36.0 −25.7

133.8 138.7 145.1 151.7 158.5

−0.3131 −0.2707 −0.2182 −0.1666 −0.1157

0.8327 0.8099 0.7885 0.7725 0.7612

0.822 0.475 0.292 0.192 0.125

−14.8 −3.6 8.0 19.9 32.3

165.4 172.5 179.6 186.8 194.0

−0.0654 −0.0156 0.0337 0.0824 0.1305

0.7539 0.7500 0.7487 0.7497 0.7525

7.89.−4 8.08.−4 8.30.−4 8.53.−4 8.80.−4

0.0862 0.0611 0.0443 0.0327 0.0246

44.9 57.9 71.1 84.6 98.4

201.1 208.3 215.3 222.2 228.9

0.1781 0.2249 0.2711 0.3161 0.3614

0.7567 0.7621 0.7683 0.7751 0.7822

8.456 10.83 13.67 17.04 21.02

9.11.−4 9.48.−4 9.93.−4 0.00105 0.00113

0.0186 0.0143 0.0111 0.0084 0.0063

112.6 127.1 142.1 157.2 174.7

235.3 241.4 246.9 251.5 254.8

0.4057 0.4497 0.4937 0.5382 0.5844

0.7893 0.7960 0.8018 0.8060 0.8071

25.71 28.46 31.37

0.00125 0.00137 0.00204

0.0045 0.0036 0.0020

193.6 205.2 231.8

255.2 252.5 231.8

0.6349 0.6649 0.7341

0.8013 0.7953 0.7341

*Values converted from tables of Shank, Thermodynamic Properties of UCON 245 Refrigerant, Union Carbide Corporation, New York, 1966. See also Shank, J. Chem. Eng. Data, 12, 474–480 (1967). c = critical point. The notation 6.46.−4 signifies 6.46 × 10−4.

TABLE 2-337

Refrigerant C 318*

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

200 210 220 230 240

0.0216 0.0449 0.0875 0.1608 0.2810

5.507.−4 5.593.−4 5.683.−4 5.778.−4 5.879.−4

3.810 1.931 1.038 0.588 0.349

353.5 361.0 369.2 377.6 386.4

498.0 500.1 502.2 504.4 510.9

3.909 3.947 3.984 4.022 4.060

4.560 4.564 4.569 4.574 4.578

0.98 1.00

11.7 9.55

0.088 0.085

250 260 270 280 290

0.466 0.741 1.133 1.672 2.392

5.988.−4 6.106.−4 6.234.−4 6.375.−4 6.529.−4

0.2166 0.1401 0.0938 0.0647 0.0458

395.6 405.2 415.1 425.8 436.2

517.4 524.0 530.7 537.3 543.9

4.097 4.133 4.172 4.210 4.247

4.584 4.592 4.599 4.609 4.618

1.02 1.03 1.05 1.07 1.09

7.90 6.63 5.64 4.85 4.22

0.082 0.078 0.075 0.071 0.068

300 310 320 330 340

3.325 4.522 6.007 7.826 10.018

6.694.−4 6.893.−4 7.115.−4 7.365.−4 7.666.−4

0.0332 0.0245 0.0184 0.0139 0.0106

447.3 458.7 470.5 482.7 495.2

550.4 556.9 563.3 569.4 575.4

4.284 4.322 4.359 4.396 4.433

4.626 4.638 4.648 4.659 4.669

1.12 1.15 1.18 1.23 1.27

3.70 3.20 2.94 2.66 2.33

0.065 0.061 0.058 0.054 0.051

350 360 370 380 388.5c

12.632 15.71 19.33 23.59 27.83

8.034.−4 8.508.−4 9.172.−4 1.031.−3 1.613.−3

0.0082 0.0062 0.0047 0.0033 0.0016

508.1 521.5 535.6 551.4 577.2

581.0 585.8 589.9 591.5 577.2

4.469 4.507 4.544 4.585 4.651

4.678 4.685 4.691 4.691 4.651

1.32 1.39

2.00

0.048

T, K

*Values of P, v, h, and s interpolated, extrapolated, and converted from tables of Oguchi, Reito, 52 (1977): 869–889. Values of cp, µ, and k interpolated and converted from tables in Thermophysical Properties of Refrigerants, American Society of Heating, Refrigerating and Air-Conditioning Engineers, New York, 1976. c = critical point. Saturation and superheat tables and a diagram to 80 bar, 580 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For equations, see Cipollone, R., ASHRAE Trans., 97, 2 (1991): 262–267.

THERMODYNAMIC PROPERTIES TABLE 2-338

2-297

Saturated Refrigerant 500*

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

200 210 220 230 240

0.1219 0.2258 0.3936 0.6511 1.0291

6.966.−4 7.090.−4 7.222.−4 7.361.−4 7.509.−4

1.360 0.766 0.457 0.286 0.187

−29.56 −21.03 −12.17 −2.97 6.58

185.87 191.25 196.63 201.96 207.23

−0.1363 −0.0948 −0.0536 −0.0130 0.0277

0.9408 0.9161 0.8955 0.8782 0.8638

1.044 1.018 0.997 0.987 0.987

6.11 5.15 4.42 3.85 3.42

0.113 0.109 0.106 0.102 0.098

250 260 270 280 290

1.5632 2.2932 3.2624 4.5172 6.1064

7.668.−4 7.839.−4 8.024.−4 8.226.−4 8.450.−4

0.1261 0.0879 0.0628 0.0459 0.0342

16.50 26.78 37.44 48.48 59.91

212.40 217.45 222.35 227.06 231.56

0.0680 0.1082 0.1481 0.1878 0.2275

0.8517 0.8415 0.8329 0.8257 0.8194

0.997 1.017 1.048 1.089 1.140

3.04 2.74 2.48 2.26 2.08

0.094 0.090 0.086 0.082 0.078

300 310 320 330 340

8.0809 10.49 13.40 16.86 20.93

8.699.−4 8.981.−4 9.306.−4 9.690.−4 1.016.−3

0.0259 0.0198 0.0154 0.0119 0.0093

71.76 84.05 96.83 110.17 124.20

235.79 239.69 243.19 246.14 248.36

0.2671 0.3067 0.3464 0.3864 0.4271

0.8139 0.8088 0.8038 0.7985 0.7922

1.201 1.273 1.355 1.447 1.550

1.92 1.77 1.63 1.48 1.34

0.074 0.070 0.066 0.062 0.058

350 360 370 378.6c

25.70 31.25 37.72 44.26

1.077.−3 1.162.−3 1.307.−3 2.012.−3

0.0072 0.0055 0.0040 0.0020

139.18 155.66 175.59 219.50

249.47 248.71 244.26 219.50

0.4689 0.5135 0.5650 0.6729

0.7841 0.7721 0.7509 0.6729

1.663 1.919 2.07 ∞

T, K

*Values reproduced and converted from Table 12, p. 17.99, ASHRAE Handbook, 1981: Fundamentals, American Society of Heating, Refrigerating and Air-Conditioning Engineers, Atlanta, 1981. Copyright material. Reproduced by permission of the copyright owner. c = critical point. The notation 6.966.−4 signifies 6.966 × 10−4. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives material for integral degrees Celsius with temperatures on the IPTS 68 scale from −70 to 105.60°C. The thermodynamic diagram from 0.1 to 70 bar extends to 240°C. Equations and constants approximated to the 1985 ASHRAE tables were given by Mecaryk, K. and M. Masaryk, Heat Recovery Systems and CHP, 11, 2/3 (1991): 193–197. Saturation and superheat tables and a diagram to 80 bar, 560 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). Tables and a chart to 1000 psia, 480°F are given by Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). Specific heat and viscosity appear in Thermophysical Properties of Refrigerants, ASHRAE, 1993.

TABLE 2-339

Saturated Refrigerant 502*

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

200 210 220 230 240

0.2274 0.4098 0.6965 1.1251 1.7392

6.381.−4 6.507.−4 6.640.−4 6.783.−4 6.938.−4

0.646 0.374 0.228 0.146 0.0969

−29.04 −20.83 −12.15 −2.99 6.66

153.34 158.42 163.49 168.50 173.42

−0.1337 −0.0937 −0.0534 −0.0128 0.0280

0.7782 0.7599 0.7449 0.7328 0.7228

1.018 1.036 1.055 1.075 1.097

5.72 4.88 4.23 3.71 3.28

0.103 0.099 0.095 0.091 0.087

250 260 270 280 290

2.5867 3.7188 5.1893 7.0530 9.3660

7.105.−4 7.289.−4 7.492.−4 7.720.−4 7.979.−4

0.0665 0.0470 0.0340 0.0251 0.0188

16.78 27.36 38.36 49.77 61.55

178.20 182.81 187.21 191.35 195.16

0.0691 0.1102 0.1514 0.1923 0.2330

0.7148 0.7082 0.7027 0.6980 0.6937

1.120 1.144 1.170 1.197 1.225

2.94 2.65 2.41 2.18 1.99

0.083 0.079 0.075 0.072 0.068

T, K

300 310 320 330 340

12.19 15.57 19.60 24.35 29.95

8.280.−4 8.637.−4 9.081.−4 9.666.−4 1.053.−3

0.0143 0.0109 0.0084 0.0064 0.0048

73.68 86.17 99.06 112.53 127.13

198.56 201.43 203.57 204.62 203.71

0.2734 0.3134 0.3532 0.3933 0.4351

0.6896 0.6852 0.6798 0.6723 0.6604

1.254 1.285 1.317 1.351 1.386

1.79 1.59 1.40 1.23 1.07

0.064 0.060 0.056 0.052 0.048

350 355.3c

36.62 40.75

1.220.−3 1.786.−3

0.0033 0.0018

145.44 174.00

197.82 174.00

0.4859 0.5634

0.6355 0.5634

1.422

0.93

0.044

*Values reproduced and converted from Table 13, p. 17.101, ASHRAE Handbook, 1981: Fundamentals, American Society of Heating, Refrigerating and AirConditioning Engineers, Atlanta, 1981. Copyright material. Reproduced by permission of the copyright owner. c = critical point. The notation 6.381.−4 signifies 6.381 × 10−4. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives material for integral degrees Celsius with temperatures on the IPTS 68 scale from −70 to 82.2°C. The thermodynamic diagram from 0.1 to 80 bar extends to 180°C. Equations and constants approximated to 1985 ASHRAE tables are given by Mecaryk, K., and M. Masaryk, Heat Recovery Systems and CHP, 11, 2/3 (1991): 193–197. Saturation and superheat tables and a diagram to 20 bar, 515 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). Tables and a chart to 1000 psia, 400°F appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.

2-298

PHYSICAL AND CHEMICAL DATA

TABLE 2-340

Saturated Refrigerant 503*

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

150 160 170 180 190

0.0750 0.1798 0.3828 0.7395 1.3187

6.384.−4 6.478.−4 6.585.−4 6.700.−4 6.850.−4

1.894 0.837 0.414 0.224 0.130

−89.60 −79.73 −69.55 −59.08 −48.36

111.02 115.40 119.70 123.84 127.77

−0.4694 −0.4057 −0.3441 −0.2844 −0.2267

0.8681 0.8139 0.7691 0.7318 0.7003

0.482 0.554 0.620 0.682 0.747

6.12 5.05 4.16 3.43 2.94

0.128 0.123 0.116 0.111 0.105

200 210 220 230 240

2.1999 3.4713 5.2281 7.5713 10.61

7.014.−4 7.204.−4 7.426.−4 7.687.−4 8.001.−4

0.0803 0.0520 0.0350 0.0242 0.0172

−37.45 −26.36 −15.10 − 3.65 8.07

131.45 134.84 137.87 140.49 142.58

−0.1710 −0.1173 −0.0656 −0.0155 0.0334

0.6735 0.6503 0.6298 0.6112 0.5939

0.817 0.896 0.988 1.017 1.227

2.56 2.25 1.98 1.73 1.52

0.099 0.094 0.088 0.082 0.076

250 260 270 280 290

14.46 19.25 25.13 32.27 40.87

8.386.−4 8.874.−4 9.526.−4 1.050.−3 1.264.−3

0.0124 0.0090 0.0064 0.0045 0.0028

20.22 33.10 47.22 63.64 86.41

143.98 144.38 143.23 139.25 127.51

0.0817 0.1305 0.1816 0.2384 0.3131

0.5767 0.5585 0.5373 0.5085 0.4548

1.382 1.57 1.79 2.03 2.35

1.33 1.17 1.03 0.91

0.070 0.065 0.059 0.054

292.6c

43.57

1.773.−3

0.0018

110.20

110.20

0.3864

0.3864

∞

T, K

∞

*P, v, h, and s values reproduced and converted from Table 14, p. 17.103, ASHRAE Handbook, 1981: Fundamentals, American Society of Heating, Refrigerating and Air-Conditioning Engineers, Atlanta, 1981. Copyright material. Reproduced by permission of the copyright owner. cp, µ, and k values interpolated and converted from Thermophysical Properties of Refrigerants, American Society of Heating, Refrigerating and Air-Conditioning Engineers, New York, 1976. c = critical point. The notation 6.384.−4 signifies 6.384 × 10−4. Saturation and superheat tables and a diagram to 80 bar, 600 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). Tables and a chart to 1000 psia, 460°F are given by Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat and viscosity see Thermophysical Properties of Refrigerants, ASHRAE, 1993. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives material for integral degrees Celsius with temperatures on the IPTS 68 scale for saturation conditions from −125 to 19.50°C. The thermodynamic diagram from 0.1 to 80 bar extends to 220°C. TABLE 2-341

Saturated Refrigerant 504* Volume, ft3/lb

Enthalpy, Btu/lb

Entropy, Btu/(lb)(°F)

Temp., °F

Pressure, lb/in2 abs.

Liquid

Vapor

Liquid

Vapor

Liquid

Vapor

−120 −100 −80 −60 −40

2.964 6.042 11.34 19.85 32.76

0.01095 0.01119 0.01146 0.01175 0.01206

15.31 7.874 4.372 2.585 1.609

−21.48 −16.39 −11.12 −5.65 0.00

86.69 89.31 91.84 94.25 96.50

−0.0565 −0.0420 −0.0277 −0.0137 0.0000

0.2609 0.2519 0.2435 0.2362 0.2299

−20 0 20 40 60

51.44 77.41 112.3 158.0 216.2

0.01242 0.01282 0.01328 0.01379 0.01443

1.045 0.7029 0.4859 0.3431 0.2458

5.85 11.91 18.22 24.81 31.78

98.58 100.45 102.09 103.44 104.41

0.0135 0.0269 0.0401 0.0533 0.0667

0.2244 0.2195 0.2150 0.2107 0.2065

80 100 120 140 150

289.2 379.1 488.3 618.1 692.2

0.01522 0.01629 0.01783 0.02083 0.02597

0.1773 0.1274 0.0893 0.0578 0.0394

39.25 47.43 56.78 69.97 76.96

104.85 104.49 102.72 97.70 89.76

0.0804 0.0948 0.1107 0.1322 0.1432

0.2020 0.1968 0.1899 0.1784 0.1642

*Unpublished data of Allied Chemical Company, 1970. Used by permission. TABLE 2-342

Thermodynamic Properties of Refrigerant 507*

Temp., K

Pressure, bar

230.5 240 250 260 270 280

1.013 1.59 2.42 3.54 4.95 6.70

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

0.000 0.000 0.000 0.000 0.000 0.000

574 602 627 658 695 738

0.1280 0.0826 0.0546 0.0377 0.0270 0.0198

−3.1 10.3 22.6 37.6 51.6 64.7

143.3 150.2 154.5 159.0 163.8 169.0

−0.015 0.042 0.092 0.149 0.202 0.250

0.620 0.623 0.619 0.617 0.618 0.620

290 300 310 320 330

8.85 11.52 14.74 18.76 23.65

0.000 0.000 0.000 0.001 0.001

787 839 903 006 221

0.0148 0.0112 0.0084 0.0062 0.0042

77.2 89.4 101.6 115.7 135.5

174.6 180.3 185.4 188.6 189.3

0.295 0.336 0.378 0.422 0.481

0.634 0.640 0.648 0.649 0.641

340 341.5c

29.57 32.67

0.001 618 0.001 97

0.0025 0.0020

161.7 172.7

179.9 172.7

0.557 0.590

0.611 0.590

*Azeotropic mixture of R152a and R218. hf = sf = 0 at 233.15 K = −40°C. Interpolated, extrapolated and converted from Lavrenchenko, G. K., M. G. Khmelnuk, et al., Int. J. Refrig., 17, 7 (1994): 461. Some values are tentative. This source also gives a ln P–h diagram from 0.6 to 30 bar, −50 to 70°C. Differences exist between the published diagram and tables. c = critical point.

THERMODYNAMIC PROPERTIES TABLE 2-343

Saturated Rubidium*

T, K

P, bar

vf, m3/kg

312.7m 400 500 600 700

2.46.−9 1.69.−6 1.73.−4 0.0037 0.0317

6.75.−4 6.98.−4 7.22.−4 7.46.−4 7.73.−4

0.1584 1.467 6.466 18.6 28.5

8.10.−4 8.65.−4 9.40.−4 1.03.−3 1.08.−3

800 1000 1200 1400 1500

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

2.3.+5 2790 156.6 20.75

118.7 151.6 188.8 225.4 261.3

1036 1057 1078 1096 1111

0.998 1.091 1.174 1.241 1.296

3.932 3.355 2.953 2.692 2.511

0.379 0.375 0.369 0.362 0.357

4.662 0.605 0.159

296.8 367.6 440.1

1124 1150 1179

1.343 1.422 1.490

2.378 2.205 2.104

0.353 0.360 0.385

*Converted from tables in Vargaftik, Tables of the Thermophysical Properties of Liquids and Gases, Nauka, Moscow, 1972, and Hemisphere, Washington, 1975. m = melting point. The notation 2.46.−9 signifies 2.46 × 10−9. Many of the Vargaftik values also appear in Ohse, R. W., Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Sci. Pubs., Oxford, 1985 (1020 pp.). This source contains superheat data. Saturation and superheat tables and a diagram to 40 bar, 1600 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For a Mollier diagram from 0.1 to 320 psia, 1200 to 2700°R, see Weatherford, W. D., J. C. Tyler, et al., WADD-TR-61-96, 1961. An extensive review of properties of the solid and the saturated liquid was given by Alcock, C. B., M. W. Chase, et al., J. Phys. Chem. Ref. Data, 23, 3 (1994): 385–497.

TABLE 2-344 Temp., °C

Thermophysical Properties of Saturated Seawater

Pressure, bar

v, (m3/kg)103

cp, kJ/(kg⋅K)

µ, Ns/m2

k, W/(m⋅K)

NPr

105κ, 1/bar

0 1 2 3 4

0.005993 0.006438 0.006916 0.007427 0.007970

1.000158 1.000099 1.000057 1.000033 1.000025

4.000 4.000 4.000 4.000 4.001

0.001884 0.001827 0.001772 0.001720 0.001669

0.560 0.563 0.565 0.567 0.569

13.46 12.98 12.55 12.13 11.74

5.06 5.02 4.98 4.95 4.92

5 6 7 8 9

0.008548 0.009163 0.009816 0.010511 0.011248

1.000033 1.000057 1.000096 1.000149 1.000261

4.001 4.001 4.002 4.002 4.002

0.001620 0.001574 0.001529 0.001486 0.001445

0.571 0.574 0.576 0.578 0.580

11.35 10.97 10.62 10.29 9.97

4.89 4.86 4.83 4.80 4.78

10 11 12 13 14

0.01203 0.01286 0.01374 0.01467 0.01566

1.000298 1.000392 1.000500 1.000620 1.000727

4.003 4.003 4.003 4.004 4.004

0.001405 0.001367 0.001330 0.001294 0.001259

0.582 0.584 0.586 0.588 0.590

9.70 9.37 9.09 8.81 8.54

4.76 4.74 4.72 4.70 4.68

15 16 17 18 19

0.01671 0.01781 0.01898 0.02022 0.02153

1.000899 1.001055 1.001224 1.001404 1.001595

4.005 4.005 4.006 4.006 4.007

0.001226 0.001195 0.001165 0.001136 0.001107

0.592 0.594 0.595 0.597 0.599

8.29 8.06 7.82 7.62 7.41

4.66 4.65 4.63 4.62 4.60

20 21 22 23 24

0.02291 0.02437 0.02591 0.02753 0.02924

1.001796 1.002009 1.002232 1.002465 1.002708

4.007 4.007 4.008 4.008 4.009

0.001080 0.001054 0.001029 0.001005 0.000981

0.600 0.602 0.604 0.605 0.607

7.21 7.02 6.82 6.66 6.48

4.59 4.57 4.56 4.55 4.54

25 26 27 28 29

0.03104 0.03294 0.03494 0.03705 0.03926

1.002961 1.003224 1.003496 1.003778 1.004069

4.009 4.009 4.010 4.010 4.011

0.000958 0.000936 0.000915 0.000895 0.000875

0.608 0.609 0.611 0.612 0.614

6.31 6.16 6.01 5.86 5.72

4.53 4.52 4.51 4.50 4.49

30

0.04159

1.004369

4.011

0.000855

0.615

5.58

4.48

κ = (−1/V)(∂v/∂p)T ⋅ 105. Thus, at 0°C, the compressibility is 5.06 × 10−5/bar. For further information see, for instance, Bromley, LeR. A., J. Chem. Eng. Data, 12, 2 (1967): 202–206; 13, 1 (1968): 60–62 and 13, 3: 399–402; 15, 2 (1970): 246–253; and A.I.Ch.E.J., 20, 2 (1974): 326–335. Thermal conductivity data sources include Castelli, V. J., E. M. Stanley, et al., Deep Sea Res., 211 (1974): 311–318; Levy, F. L., Int. J. Refrig., 5, 3 (1982): 155–159. For velocity of sound, see, for instance, U.S. Naval Oceanographic Office SP 58, 1962 (50 pp.). More recent information is contained in UNESCO technical papers. See Marine Science No. 38, 1981 (6 pp.) and No. 44, 1983 (53 pp.). For sea ice properties, see Fukusako, S., Int. J. Thermophys., 11, 2 (1990): 353–372.

2-299

TABLE 2-345 Saturated Sodium Temp., K

Pressure, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

cpg, kJ/(kg⋅K)

µf, 10−6 Pa·s

µg, 10−6 Pa·s

kf, W/(m⋅K)

kg, W/(m⋅K)

Prg

1.59.−10 1.80.−9 8.99.−7 5.57.−5 0.00105

0.001 0.001 0.001 0.001 0.001

078 088 115 144 174

8.54.+9 8.08.+8 1.99.+6 38022 2320

207 247 382 514 642

4739 4757 4817 4872 4921

2.259 2.920 3.222 3.462 3.661

14.475 14.195 12.092 10.745 10.631

1.383 1.372 1.334 1.301 1.277

0.86 1.25 1.80 2.28

688 599 415 321 264

800 900 1000 1100 1154.7

0.00941 0.05147 0.1995 0.6016 1.013

0.001 0.001 0.001 0.001 0.001

208 242 280 323 347

291.5 58.8 16.6 5.95 3.89

769 895 1020 1146 1215

4966 5007 5044 5079 5097

3.830 3.978 4.110 4.230 4.290

9.076 8.547 8.134 7.805 7.652

1.260 1.252 1.252 1.261 1.271

2.59 2.72 2.70 2.62 2.56

227 201 181 166 159

19.6 20.6 23.0 25.3 26.5

62.9 58.3 54.2 50.5 48.7

0.0343 0.0406 0.0455 0.0492 0.0522

0.0045 0.0043 0.0042 0.0042 0.0041

1.48 1.38 1.36 1.35 1.30

27.5 29.9 32.2 34.6 37.1

47.2 44.0 41.1 38.2 35.4

0.0547 0.0570 0.0592

0.0041 0.0042 0.0044 0.0046 0.0050

1.26 1.27 1.30

1200 1300 1400 1500 1600

1.50 3.26 6.30 11.13 18.28

0.001 0.001 0.001 0.001 0.001

366 416 471 531 597

2.54 1.24 0.676 0.400 0.253

1273 1402 1534 1671 1812

5111 5140 5168 5193 5217

4.340 4.444 4.542 4.636 4.727

7.538 7.319 7.138 6.984 6.855

1.279 1.305 1.340 1.384 1.437

2.51 2.43 2.39 2.36 2.34

153 143 135 128 122

1700 1800 1900 2000 2100

28.28 41.61 58.70 79.91 105.5

0.001 0.001 0.001 0.001 0.002

675 761 862 984 174

0.168 0.117 0.084 0.063 0.0472

1959 2113 2274 2444 2625

5238 5256 5268 5273 5265

4.816 4.904 4.992 5.079

6.745 6.650 6.568 6.494

1.500 1.574 1.661 1.764 1.926

2.41 2.46 2.53 2.66 2.91

117 112 108 104

2200 2300 2400 2500 2503.7c

135.7 170.6 210.3 254.7 256.4

0.002 0.002 0.002 0.004 0.004

320 584 985 19 57

0.0361 0.0275 0.0203 0.0098 0.0046

2822 3047 3331 3965 4294

5241 5188 5078 4617 4294

2.190 2.690 4.012 39.3

3.40 4.47 8.03 417.

89.4 87.2 80.1 73.7 68.0

Prf

371 400 500 600 700

32.6 29.7 26.6 23.2

0.0106 0.0094 0.0069 0.0057 0.0050

0.0054 0.0059 0.0067 0.0079

c = critical point. sf values converted from Cordfunke, E. H. P. and R. J. M. Konings, Thermochemical Data for Reactor Materials and Fission Products, North Holland Elsevier, NY, 1990. sg determined as sf + (hg − hf)/T. µg and kg values estimated by P. E. Liley. All other values are from Fink, J. K. and L. Leibowitz, Argonne Nat. Lab Rept. ANL/RE-95-2, 1995. The Fink and Leibowitz work also appeared in High Temp. Materials Sci., 35, 65–103, 1996. Saturation and superheat tables and a diagram to 14 bar, 1700 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For a Mollier diagram for 0.1–150 psia, 1500–2700°R, see Weatherford, P. M., J. C. Tyler, et al., WADD-TR-61-96, 1961.

2-300

THERMODYNAMIC PROPERTIES

2-301

2500 K 2400 6500

2300 2200 2100 2000 1900 1800

ar

1700

5

10 b

1600

1 ba

r

2

h, kJ/kg

6000

0.5

1500

0.2

1400

0.1

1300

5500

0.0

5

1200

0.0

2

1100

0.0 1

1000

5000 7.0

800

r

8.0

9.0 Entropy (s), kJ/kg • K

bar

vapo

01

te d

0.0

tura

0.0 02

Sa

0.0 05

900

700

10.0

FIG. 2-28 Mollier Diagram for Sodium. Drawn from the Vargaftik et al. values in Ohse, R. W., Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Sci. Pubs., Oxford, UK, 1985. These values are identical with those of Vargaftik, N. B., Handbook of Thermophysical Properties of Gases and Liquids, Moscow, 1972, and the Hemisphere translation, pp. 19. An apparent discontinuity exists between the superheat values and the saturation values, not reproduced here. For a Mollier diagram in f.p.s. units from 0.1 to 150 psia, 1500 to 2700°R, see Fig. 3-36, p. 3-232 of the 6th edition of this handbook. An extensive review of properties of the solid and the saturated liquid was given by Alcock, C. B., Chase, M. W. et al., J. Phys. Chem. Ref. Data, 23(3), 385–497, 1994.

2-302

PHYSICAL AND CHEMICAL DATA

FIG. 2-29 Enthalpy-concentration diagram for aqueous sodium hydroxide at 1 atm. Reference states: enthalpy of liquid water at 32°F and vapor pressure is zero; partial molal enthalpy of infinitely dilute NaOH solution at 64°F and 1 atm is zero. [McCabe, Trans. Am. Inst. Chem. Eng., 31, 129 (1935).]

TABLE 2-346

Saturated Sulfur Dioxide*

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

200 210 220 230 240

0.02056 0.04569 0.09997 0.1844 0.3202

6.189.−4 6.284.−4 6.384.−4 6.488.−4 6.596.−4

12.602 5.946 2.876 1.605 0.9602

7.4 9.1 28.6 43.5 56.5

433.3 446.1 453.8 459.5 464.5

0.033 0.041 0.123 0.198 0.254

2.212 2.159 2.075 2.001 1.952

1.280 1.284 1.288 1.293 1.299

12.3 10.6 8.37 7.03 5.97

250 260 270 280 290

0.5430 0.8778 1.3634 2.0402 2.9574

6.707.−4 6.819.−4 6.938.−4 7.057.−4 7.184.−4

0.5864 0.3745 0.2479 0.1699 0.1197

70.0 85.1 99.8 114.8 129.2

469.7 474.5 479.3 484.3 488.5

0.308 0.363 0.425 0.473 0.523

1.906 1.865 1.827 1.793 1.763

1.308 1.317 1.328 1.343 1.363

5.11 4.39 3.78 3.30 2.87

0.262 0.243 0.224 0.206 0.190

300 310 320 330 340

4.1675 5.7372 7.8226 10.301 13.229

7.312.−4 7.447.−4 7.590.−4 7.847.−4 8.066.−4

0.08647 0.06366 0.04707 0.03572 0.02792

143.1 157.1 170.1 183.0 196.0

492.5 496.3 498.9 501.2 502.5

0.568 0.612 0.649 0.690 0.731

1.732 1.706 1.678 1.654 1.633

1.389 1.422 1.459 1.499 1.546

2.51 2.19 1.91 1.67 1.46

0.174 0.162 0.151 0.139 0.128

350 360 370 380 390

16.759 21.01 26.01 31.92 38.76

8.303.−4 8.571.−4 8.877.−4 9.236.−4 9.671.−4

0.02209 0.01755 0.01399 0.01110 0.00877

211.2 223.7 239.9 257.9 277.7

502.9 503.1 502.9 502.7 500.7

0.781 0.817 0.862 0.910 0.962

1.614 1.593 1.573 1.555 1.534

1.603 1.68 1.75 1.84 1.97

1.27 1.11 0.96 0.84 0.73

0.117 0.108 0.098 0.089 0.081

400 410 420 425.1c

46.67 55.80 66.19 78.81

1.023.−3 1.098.−3 1.235.−3 1.906.−3

0.00685 0.00559 0.00387 0.00191

300.2 326.2 355.6 423.6

496.7 489.5 474.1 423.6

1.020 1.083 1.155 1.304

1.511 1.481 1.436 1.304

2.12

0.63 0.53 0.44

0.072 0.064 0.055

T, K

kf, W/(m⋅K)

*Values interpolated and converted from tables of Kang, McKetta, et al., Bur. Eng. Res. Repr. 59, University of Texas, Austin, 1961. See also J. Chem. Eng. Data, 6 (1961): 220–227; and Am. Inst. Chem. Eng. J., 7 (1961): 418. c = critical point. The notation 6.189.−4 signifies 6.189 × 10−4. The AIChE publication contains a Mollier diagram to 4500 psia, 480°F, while the reprint contains saturation and superheat tables.

THERMODYNAMIC PROPERTIES TABLE 2-347

2-303

Thermodynamic Properties of Saturated Sulfur Hexafluoride (SF6)*

Temp., K

Pressure, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

222.4 225 230 235 240 245 250 255 260 265 270 275 280 285 290 295 300 305 310 315 318.7

2.200 2.470 3.045 3.710 4.475 5.346 6.332 7.442 8.684 10.07 11.60 13.30 15.18 17.25 19.52 22.01 24.75 27.76 31.05 34.67 37.79

0.0005389 0.0005429 0.0005507 0.0005588 0.0005675 0.0005768 0.0005866 0.0005971 0.0006085 0.0006207 0.0006341 0.0006488 0.0006652 0.0006836 0.0007047 0.0007295 0.0007594 0.000798 0.000851 0.000949 0.001372

0.05428 0.04861 0.03978 0.03286 0.02737 0.02296 0.01939 0.01647 0.01406 0.01205 0.01035 0.00892 0.00769 0.00663 0.00571 0.00490 0.00418 0.00352 0.00291 0.00228 0.00137

−57.55 −54.41 −48.51 −42.67 −36.87 −31.13 −25.47 −19.87 −14.33 −8.85 −3.41 2.00 7.42 12.88 18.45 24.20 30.22 36.75 44.05 53.98 71.74

59.08 60.49 63.12 65.68 68.18 70.61 72.96 75.22 77.39 79.44 81.38 83.19 84.84 86.30 87.52 88.45 89.00 88.97 88.06 85.22 71.74

−0.2310 −0.2171 −0.1913 −0.1663 −0.1421 −0.1186 −0.0960 −0.0741 −0.0528 −0.0323 −0.0123 0.0071 0.0262 0.0451 0.0639 0.0829 0.1025 0.1233 0.1462 0.1769 0.2317

0.2935 0.2936 0.2940 0.2947 0.2956 0.2966 0.2977 0.2988 0.2999 0.3009 0.3017 0.3024 0.3027 0.3027 0.3021 0.3008 0.2984 0.2945 0.2881 0.2761 0.2317

cpf, kJ/(kg⋅K)

cpg, kJ/(kg⋅K)

0.409 0.631 0.870 1.17 1.63 2.48 ∞

0.579 0.583 0.592 0.602 0.613 0.626 0.640 0.656 0.674 0.695 0.720 0.748 0.783 0.827 0.882 0.941 1.070 1.26 1.63 2.40 ∞

*See also Oda, A., M. Uematsu, et al., Bull. JSME, 26, 219 (1983): 1590–1596. Ulybin, S.A., Thermodynamic Properties of Sulfur Hexafluoride, Moscow, 1977 (53 pp.). For thermal conductivity to 500 bar, see Rastorguev, Yu L., B. A. Grigorev, et al., Teploenergetika 24, 6 (1977): 78–81 and Bakulin, S. S. and S. A. Ulybin, Teplofiz. Vysok. Temp., 16, 1 (1978): 59–66. For viscosity to 400 bar, see Grigorev, B. A., A. S. Keramidi, et al., Teploenergetika, 24, 9 (1977): 85–87; and Ulybin, S. A. and V. I. Makanushkin, Teplofiz. Vysok. Temp., 15, 6 (1977): 1195–1201.

FIG. 2-30 Enthalpy-concentration diagram for aqueous sulfuric acid at 1 atm. Reference states: enthalpies of pure-liquid components at 32°F and vapor pressures are zero. NOTE: It should be observed that the weight basis includes the vapor, which is particularly important in the two-phase region. The upper ends of the tie lines in this region are assumed to be pure water. (Hougen and Watson, Chemical Process Principles, part I, Wiley, New York, 1943.)

FIG. 2-31

Enthalpy–log-pressure diagram for sulfur hexafluoride.

TABLE 2-348

Saturated SUVA AC 9000

DuPont bulletin T–AC–9000–SI, 1994 (16 pp.) gives tables and a chart to 100 bar, 235°C. With a stated composition of 23% wt CH2F2 (R23), 25% wt CHF2CF3 (R125), and 52% wt CH2FCH3 (R134a) this is apparently identical to KLEA 66, to which the reader is referred. TABLE 2-349

Saturated Toluene* hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

316.7 333.0 349.6 366.5 383.7

745.7 756.1 766.8 777.8 789.2

2.236 2.295 2.353 2.410 2.467

3.825 3.806 3.792 3.782 3.776

1.64 1.66 1.68 1.71 1.74

8.02 6.96 6.10 5.41 4.83

0.141 0.138 0.136 0.133 0.131

2.67 1.80 1.25 0.891 0.698

401.3 419.6 437.4 456.0 475.1

800.9 812.9 825.2 837.8 850.7

2.522 2.577 2.632 2.686 2.739

3.771 3.771 3.772 3.777 3.783

1.78 1.81 1.84 1.88 1.92

4.34 3.93 3.58 3.28 3.01

0.128 0.126 0.124 0.121 0.119

1.261.−3 1.277.−3 1.294.−3 1.312.−3 1.350.−3

0.481 0.364 0.279 0.218 0.137

494.6 514.4 534.7 555.4 598.1

863.8 877.2 890.9 904.8 933.1

2.792 2.846 2.898 2.950 3.054

3.791 3.801 3.811 3.824 3.852

1.96 2.01 2.05 2.09 2.17

2.78 2.56 2.37 2.19 1.89

0.117 0.114 0.112 0.110 0.105

1.64

0.101 0.096 0.091 0.086 0.082

P, bar

vf, m3/kg

vg, m3/kg

270 280 290 300 310

0.0076 0.0139 0.0246 0.0418 0.0682

1.127.−3 1.138.−3 1.150.−3 1.162.−3 1.175.−3

34.9 19.1 10.6 6.46 4.08

320 330 340 350 360

0.1072 0.1633 0.2416 0.3480 0.4894

1.188.−3 1.201.−3 1.215.−3 1.230.−3 1.245.−3

370 380 390 400 420

0.6736 0.9090 1.2049 1.5713 2.5589

T, K

440 460 480 500 520

3.965 5.892 8.451 11.76 15.96

1.393.−3 1.443.−3 1.499.−3 1.567.−3 1.651.−3

9.00.−2 6.11.−2 4.26.−2 3.03.−2 2.19.−2

642.3 688.1 735.5 784.4 834.9

962.0 991.3 1021.1 1051.3 1081.4

3.156 3.258 3.358 3.457 3.554

3.883 3.917 3.953 3.989 4.027

2.24 2.31 2.38 2.45 2.53

540 560 580 590 591.8c

21.99 27.65 35.56 40.16 41.04

1.761.−3 1.919.−3 2.213.−3 2.650.−3 3.432.−3

1.58.−2 1.13.−2 7.59.−3 5.28.−3 3.43.−3

887.3 942.8 1005.6 1050.2 1084.9

1109.6 1132.1 1142.3 1128.1 1084.9

3.651 3.750 3.857 3.932 3.989

4.062 4.088 4.093 4.063 3.989

2.65 2.82

0.078 0.074

*Values converted and mostly rounded off from the tables of Counsell, Lawrenson, and Lees, Nat. Phys. Lab., Teddington (U.K.) Rep. Chem. 52, 1976. c = critical point. The notation 1.127.−6 signifies 1.127 × 10−6. For other tables, see Goodwin, R. D., J. Phys. Chem. Ref. Data, 18, 4 (1989): 1565–1636. TABLE 2-350

Saturated Solid/Vapor Water* Volume, ft3/lb

Enthalpy, Btu/lb

Entropy, Btu/(lb)(°F)

Temp., °F

Pressure, lb/in2 abs.

Solid

Vapor

Solid

Vapor

Solid

Vapor

−160 −150 −140 −130 −120

4.949.−8 1.620.−7 4.928.−7 1.403.−6 3.757.−6

0.01722 0.01723 0.01724 0.01725 0.01726

3.607.+9 1.139.+9 3.864.+8 1.400.+8 5.386.+7

−222.05 −218.82 −215.49 −212.08 −208.58

990.38 994.80 999.21 1003.63 1008.05

−0.4907 −0.4801 −0.4695 −0.4590 −0.4485

3.5549 3.4387 3.3301 3.2284 3.1330

−110 −100 −90 −80 −70

9.517.−6 2.291.−5 5.260.−5 1.157.−4 2.443.−4

0.01728 0.01729 0.01730 0.01731 0.01732

2.189.+7 9.352.+6 4.186.+6 1.955.+6 9.501.+5

−204.98 −201.28 −197.49 −193.60 −189.61

1012.47 1016.89 1021.31 1025.73 1030.15

−0.4381 −0.4277 −0.4173 −0.4069 −0.3965

3.0434 2.9591 2.8796 2.8045 2.7336

−60 −50 −45 −40 −35

4.972.−4 9.776.−4 1.354.−3 1.861.−3 2.540.−3

0.01734 0.01735 0.01736 0.01737 0.01737

4.788.+5 2.496.+5 1.824.+5 1.343.+5 9.961.+4

−185.52 −181.34 −179.21 −177.06 −174.88

1034.58 1039.00 1041.21 1043.42 1045.63

−0.3862 −0.3758 −0.3707 −0.3655 −0.3604

2.6664 2.6028 2.5723 2.5425 2.5135

−30 −25 −20 −15 −10

3.440.−3 4.627.−3 6.181.−3 8.204.−3 1.082.−2

0.01738 0.01739 0.01739 0.01740 0.01741

7.441.+4 5.596.+4 4.237.+4 3.228.+4 2.475.+4

−172.68 −170.46 −168.21 −165.94 −163.65

1047.84 1050.05 1052.26 1054.47 1056.67

−0.3552 −0.3501 −0.3449 −0.3398 −0.3347

2.4853 2.4577 2.4308 2.4046 2.3791

−5 0 5 10 15

1.419.−2 1.849.−2 2.396.−2 3.087.−2 3.957.−2

0.01741 0.01742 0.01743 0.01744 0.01744

1.909.+4 1.481.+4 1.155.+4 9.060.+3 7.144.+3

−161.33 −158.98 −156.61 −154.22 −151.80

1058.88 1061.09 1063.29 1065.50 1067.70

−0.3295 −0.3244 −0.3193 −0.3142 −0.3090

2.3541 2.3297 2.3039 2.2827 2.2600

16 18 20 22 24

4.156.−2 4.581.−2 5.045.−2 5.552.−2 6.105.−2

0.01745 0.01745 0.01745 0.01746 0.01746

6.817.+3 6.210.+3 5.662.+3 5.166.+3 4.717.+3

−151.32 −150.34 −149.36 −148.38 −147.39

1068.14 1069.02 1069.90 1070.38 1071.66

−0.3080 −0.3060 −0.3039 −0.3019 −0.2998

2.2555 2.2466 2.2378 2.2291 2.2205

26 28 30 31 32

6.708.−2 7.365.−2 8.080.−2 8.461.−2 8.858.−2

0.01746 0.01746 0.01747 0.01747 0.01747

4.311.+3 3.943.+3 3.608.+3 3.453.+3 3.305.+3

−146.40 −145.40 −144.40 −143.90 −143.40

1072.53 1073.41 1074.29 1074.73 1075.16

−0.2978 −0.2957 −0.2937 −0.2927 −0.2916

2.2119 2.2034 2.1950 2.1908 2.1867

*Condensed from Fundamentals, American Society of Heating, Refrigerating and Air-Conditioning Engineers, 1967 and 1972. Reproduced by permission. The validity of many standard reference tables has been critically reviewed by Jancso, Pupezin, and van Hook, J. Phys. Chem., 74 (1970): 2984. This source is recommended for further study. The notation 4.949.−8, 3.607.+9, etc., means 4.949 × 10−8, 3.607 × 109, etc. 2-304

THERMODYNAMIC PROPERTIES TABLE 2-351 Temp., °F

Saturated Water Substance—Temperature (fps units)* Pressure, lb/in2 abs.

Volume, ft3/lb Liquid

Enthalpy, Btu/lb Vapor

Entropy, Btu/(lb)(°F)

Liquid

Vapor

Liquid

Vapor

32.018 35 40 45 50

0.08865 0.09991 0.12163 0.14744 0.17796

0.016022 0.016020 0.016019 0.016020 0.016023

3302.4 2948.1 2445.8 2037.8 1704.8

0.000 3.002 8.027 13.044 18.054

1075.5 1076.8 1079.0 1081.2 1083.4

0.0000 0.0061 0.0162 0.0262 0.0361

2.1872 2.1767 2.1594 2.1426 2.1262

55 60 65 70 75

0.21392 0.25611 0.30545 0.36292 0.42964

0.016027 0.016033 0.016041 0.016050 0.016060

1432.0 1207.6 1022.1 868.4 740.3

23.059 28.060 33.057 38.052 43.045

1085.6 1087.7 1089.9 1092.1 1094.3

0.0458 0.0555 0.0651 0.0745 0.0839

2.1102 2.0946 2.0794 2.0645 2.0500

80 85 90 95 100

0.50683 0.59583 0.69813 0.81534 0.94294

0.016072 0.016085 0.016099 0.016114 0.016130

633.3 543.6 468.1 404.4 350.4

48.037 53.027 58.018 63.008 67.999

1096.4 1098.6 1100.8 1102.9 1105.1

0.0932 0.1024 0.1115 0.1206 0.1295

2.0359 2.0221 2.0086 1.9954 1.9825

110 120 130 140 150

1.2750 1.6927 2.2230 2.8892 3.7184

0.016165 0.016204 0.016247 0.016293 0.016343

265.39 203.26 157.33 122.98 97.07

77.98 87.97 97.96 107.89 117.95

1109.3 1113.6 1117.8 1122.0 1126.1

0.1472 0.1646 0.1817 0.1985 0.2150

1.9577 1.9339 1.9112 1.8895 1.8686

160 170 180 190 200

4.7414 5.9926 7.5110 9.340 11.526

0.016395 0.016451 0.016510 0.016572 0.016637

77.27 62.06 50.225 40.957 33.639

127.96 137.97 148.00 158.04 168.09

1130.2 1134.2 1138.2 1142.1 1146.0

0.2313 0.2473 0.2631 0.2787 0.2940

1.8487 1.8295 1.8111 1.7934 1.7764

210 212 220 230 240

14.123 14.696 17.186 20.779 24.968

0.016705 0.016719 0.016775 0.016849 0.016926

27.816 26.799 23.148 19.381 16.321

178.15 180.17 188.23 198.33 208.45

1149.7 1150.5 1153.4 1157.1 1160.6

0.3091 0.3121 0.3241 0.3388 0.3533

1.7600 1.7568 1.7442 1.7290 1.7142

250 260 270 280 290

29.825 35.427 41.856 49.200 57.550

0.017066 0.017089 0.017175 0.017264 0.01736

13.819 11.762 10.060 8.644 7.4603

218.59 228.76 238.95 249.17 259.4

1164.0 1167.4 1170.6 1173.8 1167.8

0.3677 0.3819 0.3960 0.4098 0.4236

1.7000 1.6862 1.6729 1.6599 1.6473

300 320 340 360 380

67.005 89.643 117.992 153.01 195.73

0.01745 0.01766 0.01787 0.01811 0.01836

6.4658 4.9138 3.7878 2.9573 2.3353

269.7 290.4 311.3 332.3 353.6

1179.7 1185.2 1190.1 1194.4 1198.0

0.4372 0.4640 0.4902 0.5161 0.5416

1.6351 1.6116 1.5892 1.5678 1.5473

400 420 440 460 480

247.26 308.78 381.54 466.87 566.15

0.01864 0.01894 0.01926 0.01961 0.02000

1.8630 1.4997 1.2169 0.99424 0.81717

375.1 396.9 419.0 441.5 464.5

1201.0 1203.1 1204.4 1204.8 1204.1

0.5667 0.5915 0.6161 0.6405 0.6648

1.5274 1.5080 1.4890 1.4704 1.4518

500 520 540 560 580

680.86 812.53 962.79 1133.38 1326.17

0.02043 0.02091 0.02146 0.02207 0.02279

0.67492 0.55956 0.46513 0.38714 0.32216

487.9 512.0 536.8 562.4 589.1

1202.2 1199.0 1194.3 1187.7 1179.0

0.6890 0.7133 0.7378 0.7625 0.7876

1.4333 1.4146 1.3954 1.3757 1.3550

600 620 640 660 680

1543.2 1786.9 2059.9 2365.7 2708.6

0.02364 0.02466 0.02595 0.02768 0.03037

0.26747 0.22081 0.18021 0.14431 0.11117

617.1 646.9 679.1 714.9 758.5

1167.7 1153.2 1133.7 1107.0 1068.5

0.8134 0.8403 0.8686 0.8995 0.9365

1.3330 1.3092 1.2821 1.2498 1.2086

700 702 704 705.47

3094.3 3135.5 3177.2 3208.2

0.03662 0.03824 0.04108 0.05078

0.07519 0.06997 0.06300 0.05078

825.2 835.0 854.2 906.0

991.7 979.7 956.2 906.0

0.9924 1.0006 1.0169 1.0612

1.1359 1.1210 1.1046 1.0612

*Extracted and condensed from 1967 ASME Steam Tables. Copyright reserved. Reproduced by permission.

2-305

TABLE 2-352

Temp., K

Saturated Water Substance—Temperature (SI units)

Pressure, bar*

Volume, m3/ kg Vapor

Entropy, kJ/(kg⋅K)

Condensed† Vapor Condensed†

Specific heat, Cp, kJ/(kg⋅K) Vapor

Viscosity, Ns/m2 Condensed†

Vapor

Thermal conductivity, W/(m⋅K) Condensed†

Vapor

Prandtl no. Condensed†

Vapor

Surface tension, N/m Temp., Condensed† K

Vapor

Condensed†

150

6.30.−11

1.073.−3

9.55.+9

−539.6

2273

−2.187

16.54

1.155

3.73

150

160 170 180 190 200

7.72.−10 7.29.−9 5.38.−8 3.23.−7 1.62.−6

1.074.−3 1.076.−3 1.077.−3 1.078.−3 1.079.−3

9.62.+8 1.08.+8 1.55.+7 2.72.+6 5.69.+5

−525.7 −511.7 −497.8 −483.8 −467.5

2291 2310 2328 2347 2366

−2.106 −2.026 −1.947 −1.868 −1.789

15.49 14.57 13.76 13.03 12.38

1.233 1.311 1.389 1.467 1.545

3.52 3.34 3.18 3.04 2.91

160 170 180 190 200

210 220 230 240 250

7.01.−6 2.65.−5 8.91.−5 3.72.−4 7.59.−4

1.081.−3 1.082.−3 1.084.−3 1.085.−3 1.087.−3

1.39.+5 3.83.+4 1.18.+4 4.07.+3 1.52.+3

−451.2 −435.0 −416.3 −400.1 −381.5

2384 2403 2421 2440 2459

−1.711 −1.633 −1.555 −1.478 −1.400

11.79 11.20 10.79 10.35 9.954

1.623 1.701 1.779 1.857 1.935

2.79 2.69 2.59 2.50 2.42

210 220 230 240 250

255 260 265 270 273.15

1.23.−3 1.96.−3 3.06.−3 4.69.−3 6.11.−3

1.087.−3 1.088.−3 1.089.−3 1.090.−3 1.091.−3

956.4 612.2 400.4 265.4 206.3

−369.8 −360.5 −351.2 −339.6 −333.5

2468 2477 2486 2496 2502

−1.361 −1.323 −1.281 −1.296 −1.221

9.768 9.590 9.461 9.255 9.158

1.974 2.013 2.052 2.091 2.116

2.38 2.35 2.31 2.27 2.26

255 260 265 270 273.15

273.15 275 280 285 290

0.00611 0.00697 0.00990 0.01387 0.01917

1.000.−3 1.000.−3 1.000.−3 1.000.−3 1.001.−3

206.3 181.7 130.4 99.4 69.7

0.0 7.8 28.8 49.8 70.7

2502 2505 2514 2523 2532

0.000 0.028 0.104 0.178 0.251

9.158 9.109 8.980 8.857 8.740

4.217 4.211 4.198 4.189 4.184

1.854 1.855 1.858 1.861 1.864

1750.−6 1652.−6 1422.−6 1225.−6 1080.−6

8.02.−6 8.09.−6 8.29.−6 8.49.−6 8.69.−6

0.569 0.574 0.582 0.590 0.598

0.0182 0.0183 0.0186 0.0189 0.0193

12.99 12.22 10.26 8.81 7.56

0.815 0.817 0.825 0.833 0.841

0.0755 0.0753 0.0748 0.0743 0.0737

273.15 275 280 285 290

295 300 305 310 315

0.02617 0.03531 0.04712 0.06221 0.08132

1.002.−3 1.003.−3 1.005.−3 1.007.−3 1.009.−3

51.94 39.13 27.90 22.93 17.82

91.6 112.5 133.4 154.3 175.2

2541 2550 2559 2568 2577

0.323 0.393 0.462 0.530 0.597

8.627 8.520 8.417 8.318 8.224

4.181 4.179 4.178 4.178 4.179

1.868 1.872 1.877 1.882 1.888

959.−6 855.−6 769.−6 695.−6 631.−6

8.89.−6 9.09.−6 9.29.−6 9.49.−6 9.69.−6

0.606 0.613 0.620 0.628 0.634

0.0195 0.0196 0.0201 0.0204 0.0207

6.62 5.83 5.20 4.62 4.16

0.849 0.857 0.865 0.873 0.883

0.0727 0.0717 0.0709 0.0700 0.0692

295 300 305 310 315

320 325 330 335 340

0.1053 0.1351 0.1719 0.2167 0.2713

1.011.−3 1.013.−3 1.016.−3 1.018.−3 1.021.−3

13.98 11.06 8.82 7.09 5.74

196.1 217.0 237.9 258.8 279.8

2586 2595 2604 2613 2622

0.649 0.727 0.791 0.854 0.916

8.151 8.046 7.962 7.881 7.804

4.180 4.182 4.184 4.186 4.188

1.895 1.903 1.911 1.920 1.930

577.−6 528.−6 489.−6 453.−6 420.−6

9.89.−6 10.09.−6 10.29.−6 10.49.−6 10.69.−6

0.640 0.645 0.650 0.655 0.660

0.0210 0.0213 0.0217 0.0220 0.0223

3.77 3.42 3.15 2.88 2.66

0.894 0.901 0.908 0.916 0.925

0.0683 0.0675 0.0666 0.0658 0.0649

320 325 330 335 340

345 350 355 360 365

0.3372 0.4163 0.5100 0.6209 0.7514

1.024.−3 1.027.−3 1.030.−3 1.034.−3 1.038.−3

4.683 3.846 3.180 2.645 2.212

300.7 321.7 342.7 363.7 384.7

2630 2639 2647 2655 2663

0.977 1.038 1.097 1.156 1.214

7.729 7.657 7.588 7.521 7.456

4.191 4.195 4.199 4.203 4.209

1.941 1.954 1.968 1.983 1.999

389.−6 365.−6 343.−6 324.−6 306.−6

10.89.−6 11.09.−6 11.29.−6 11.49.−6 11.69.−6

0.665 0.668 0.671 0.674 0.677

0.0226 0.0230 0.0233 0.0237 0.0241

2.45 2.29 2.14 2.02 1.91

0.933 0.942 0.951 0.960 0.969

0.0641 0.0632 0.0623 0.0614 0.0605

345 350 355 360 365

370 373.15 375 380 385

0.9040 1.0133 1.0815 1.2869 1.5233

1.041.−3 1.044.−3 1.045.−3 1.049.−3 1.053.−3

1.861 1.679 1.574 1.337 1.142

405.8 419.1 426.8 448.0 469.2

2671 2676 2679 2687 2694

1.271 1.307 1.328 1.384 1.439

7.394 7.356 7.333 7.275 7.218

4.214 4.217 4.220 4.226 4.232

2.017 2.029 2.036 2.057 2.080

289.−6 279.−6 274.−6 260.−6 248.−6

11.89.−6 12.02.−6 12.09.−6 12.29.−6 12.49.−6

0.679 0.680 0.681 0.683 0.685

0.0245 0.0248 0.0249 0.0254 0.0258

1.80 1.76 1.70 1.61 1.53

0.978 0.984 0.987 0.995 1.004

0.0595 0.0589 0.0586 0.0576 0.0566

370 373.15 375 380 385

390 400 410 420 430

1.794 2.455 3.302 4.370 5.699

1.058.−3 1.067.−3 1.077.−3 1.088.−3 1.099.−3

0.980 0.731 0.553 0.425 0.331

490.4 532.9 575.6 618.6 661.8

2702 2716 2729 2742 2753

1.494 1.605 1.708 1.810 1.911

7.163 7.058 6.959 6.865 6.775

4.239 4.256 4.278 4.302 4.331

2.104 2.158 2.221 2.291 2.369

237.−6 217.−6 200.−6 185.−6 173.−6

12.69.−6 13.05.−6 13.42.−6 13.79.−6 14.14.−6

0.686 0.688 0.688 0.688 0.685

0.0263 0.0272 0.0282 0.0293 0.0304

1.47 1.34 1.24 1.16 1.09

1.013 1.033 1.054 1.075 1.10

0.0556 0.0536 0.0515 0.0494 0.0472

390 400 410 420 430

2-306

Condensed†

Enthalpy, kJ/kg

440 450 460 470 480

7.333 9.319 11.71 14.55 17.90

1.110.−3 1.123.−3 1.137.−3 1.152.−3 1.167.−3

0.261 0.208 0.167 0.136 0.111

705.3 749.2 793.5 838.2 883.4

2764 2773 2782 2789 2795

2.011 2.109 2.205 2.301 2.395

6.689 6.607 6.528 6.451 6.377

4.36 4.40 4.44 4.48 4.53

2.46 2.56 2.68 2.79 2.94

162.−6 152.−6 143.−6 136.−6 129.−6

14.50.−6 14.85.−6 15.19.−6 15.54.−6 15.88.−6

0.682 0.678 0.673 0.667 0.660

0.0317 0.0331 0.0346 0.0363 0.0381

1.04 0.99 0.95 0.92 0.89

1.12 1.14 1.17 1.20 1.23

0.0451 0.0429 0.0407 0.0385 0.0362

440 450 460 470 480

490 500 510 520 530

21.83 26.40 31.66 37.70 44.58

1.184.−3 1.203.−3 1.222.−3 1.244.−3 1.268.−3

0.0922 0.0766 0.0631 0.0525 0.0445

929.1 975.6 1023 1071 1119

2799 2801 2802 2801 2798

2.479 2.581 2.673 2.765 2.856

6.312 6.233 6.163 6.093 6.023

4.59 4.66 4.74 4.84 4.95

3.10 3.27 3.47 3.70 3.96

124.−6 118.−6 113.−6 108.−6 104.−6

16.23.−6 16.59.−6 16.95.−6 17.33.−6 17.72.−6

0.651 0.642 0.631 0.621 0.608

0.0401 0.0423 0.0447 0.0475 0.0506

0.87 0.86 0.85 0.84 0.85

1.25 1.28 1.31 1.35 1.39

0.0339 0.0316 0.0293 0.0269 0.0245

490 500 510 520 530

540 550 560 570 580

52.38 61.19 71.08 82.16 94.51

1.294.−3 1.323.−3 1.355.−3 1.392.−3 1.433.−3

0.0375 0.0317 0.0269 0.0228 0.0193

1170 1220 1273 1328 1384

2792 2784 2772 2757 2737

2.948 3.039 3.132 3.225 3.321

5.953 5.882 5.808 5.733 5.654

5.08 5.24 5.43 5.68 6.00

4.27 4.64 5.09 5.67 6.40

101.−6 97.−6 94.−6 91.−6 88.−6

18.1.−6 18.6.−6 19.1.−6 19.7.−6 20.4.−6

0.594 0.580 0.563 0.548 0.528

0.0540 0.0583 0.0637 0.0698 0.0767

0.86 0.87 0.90 0.94 0.99

1.43 1.47 1.52 1.59 1.68

0.0221 0.0197 0.0173 0.0150 0.0128

540 550 560 570 580

1.05 1.14 1.30 1.52 1.65

1.84 2.15 2.60 3.46 4.20

0.0105 0.0084 0.0063 0.0045 0.0035

590 600 610 620 625

0.0026 0.0015 0.0008 0.0001 0.0000

630 635 640 645 647.3‡

590 600 610 620 625

108.3 123.5 137.3 159.1 169.1

1.482.−3 1.541.−3 1.612.−3 1.705.−3 1.778.−3

0.0163 0.0137 0.0115 0.0094 0.0085

1443 1506 1573 1647 1697

2717 2682 2641 2588 2555

3.419 3.520 3.627 3.741 3.805

5.569 5.480 5.318 5.259 5.191

6.41 7.00 7.85 9.35 10.6

7.35 8.75 11.1 15.4 18.3

84.−6 81.−6 77.−6 72.−6 70.−6

21.5.−6 22.7.−6 24.1.−6 25.9.−6 27.0.−6

0.513 0.497 0.467 0.444 0.430

0.0841 0.0929 0.103 0.114 0.121

630 635 640 645 647.3‡

179.7 190.9 202.7 215.2 221.2

1.856.−3 1.935.−3 2.075.−3 2.351.−3 3.170.−3

0.0075 0.0066 0.0057 0.0045 0.0032

1734 1783 1841 1931 2107

2515 2466 2401 2292 2107

3.875 3.950 4.037 4.223 4.443

5.115 5.025 4.912 4.732 4.443

12.6 16.4 26 90 ∞

22.1 27.6 42

67.−6 64.−6 59.−6 54.−6 45.−6

28.0.−6 30.0.−6 32.0.−6 37.0.−6 45.0.−6

0.412 0.392 0.367 0.331 0.238

0.130 0.141 0.155 0.178 0.238

∞

2.0 2.7 4.2 12 ∞

4.8 6.0 9.6 26 ∞

*1 bar = 105 N/m2. †Above the solid line, the condensed phase is solid; below it, liquid. ‡Critical temperature. −11 −3 9 NOTE: The notations 6.30.−11, 1.073.−3, 9.55.+9, etc. signify 6.30 × 10 , 1.073 × 10 , 955 × 10 , etc. Tables 2-351 and 2-352 are provided for general use. Tables to higher precision are available over certain ranges and for various properties. The most current internationally accepted tables are found in Haar, L., J. S. Gallagher, and G. S. Kell, NBS/NRC Steam Tables, Hemisphere, Washington, DC, 1984 (320 pp.). These do not tabulate certain properties at saturation states. A revised release on the IAPWS Skeleton Tables 1985 for the thermodynamic properties of ordinary water substance, Sept. 1993 (15 pp), is apparently the latest international publication. In J. Phys. Chem. Ref. Data 17, 4 (1988): 1439–1540, H. Sato, M. Uematsu, and others review existing steam tables and present the 1985 formulation of skeleton tables. Property codes and programs include Cheng, S. C. and C. Nguyen, Modeling and Simulation on Microcomputers 1989 (R. W. Allen, ed.), S.C.S. Intl., San Diego, 1989 (pp. 138–141); Garland, W. J. and B. J. Hand, Nucl. Engng. & Des., 113, (1989): 21–34; Dickey, D. S., Chem. Eng. 98, 9 (1991): 207–8 and 98, 11: 235–6; Muneer, T. and S. M. Scott, Proc. Inst. Mech. Eng., 205, (1991): 25–29; and Energy Convsn. Mgmt., 31, 4 (1991): 315–325. Useful pictorial representations of 20 properties as a function of both temperature (to 800°C) and pressure (to 1000 bar) are given by Grigull, U., J. Bach, et al., Warme- u. Stoff., 1 (1968): 202–213. Property equations for the saturated liquid for the range 0–300°C are given by Charters, W. W. S. and H. A. Sadafi, Rev. Int. Froid, 10, (Mar. 1987): 105–6. Gordon, S., NASA Tech. Paper 1906, 1982 gives detailed tables for ice from 0 K. Ice and snow properties are reviewed by Fukusako, S., Int. J. Thermophys., 11, 2 (1990): 353–372. See also Wagner, W., A. Saul, et al., J. Phys. Chem. Ref. Data, 23, 3 (1994): 515–525, and Table 2-358.

2-307

2-308

PHYSICAL AND CHEMICAL DATA

TABLE 2-353

Saturated Liquid Water—Miscellaneous Properties

Temperature, °C

104 β

104 kT / bar

104 ks / bar

vs, m/s

µf , 10−6 Pa⋅s

cp, kJ/kg⋅K

k, W/m⋅K

Pr, bar

0 1 2 3 4

−0.681 −0.501 −0.327 −0.160 0.003

0.50885 0.50509 0.50151 0.49808 0.49481

0.50855 0.50493 0.50143 0.49806 0.49481

1402.4 1407.4 1412.2 1417.0 1421.6

1.793 1.732 1.675 1.621 1.569

4.2176 4.2140 4.2107 4.2077 4.2048

0.567 0.569 0.570 0.572 0.573

13.32 12.83 12.37 11.93 11.51

σ, N/m 0.07565 0.07551 0.07537 0.07522 0.07508

5 6 7 8 9

0.160 0.312 0.460 0.604 0.744

0.49169 0.48871 0.48587 0.48315 0.48056

0.49167 0.48865 0.48573 0.48291 0.48019

1426.2 1430.6 1434.9 1439.1 1443.3

1.520 1.474 1.429 1.387 1.346

4.2022 4.1999 4.1977 4.1956 4.1938

0.575 0.577 0.578 0.580 0.581

11.11 10.73 10.38 10.04 9.72

0.07494 0.07480 0.07465 0.07451 0.07436

10 11 12 13 14

0.880 1.012 1.141 1.267 1.389

0.47809 0.47573 0.47347 0.47133 0.46928

0.47757 0.47504 0.47260 0.47024 0.46797

1447.3 1451.2 1455.0 1458.7 1462.4

1.308 1.271 1.236 1.202 1.170

4.1921 4.1906 4.1892 4.1879 4.1867

0.5828 0.5844 0.5859 0.5875 0.5891

9.41 9.11 8.84 8.57 8.32

0.07422 0.07407 0.07393 0.07378 0.07364

15 16 17 18 19

1.509 1.626 1.740 1.852 1.961

0.46733 0.46548 0.46371 0.46203 0.46043

0.46578 0.46366 0.46162 0.45966 0.45776

1465.9 1469.4 1472.7 1476.0 1479.2

1.139 1.110 1.081 1.054 1.028

4.1856 4.1847 4.1838 4.1830 4.1823

0.5906 0.5922 0.5937 0.5953 0.5968

8.07 7.84 7.62 7.41 7.20

0.07349 0.07334 0.07319 0.07304 0.07289

20 21 22 23 24

2.068 2.173 2.275 2.376 2.475

0.45892 0.45748 0.45612 0.45484 0.45362

0.45593 0.45417 0.45248 0.45084 0.44927

1482.3 1485.3 1488.3 1491.2 1493.9

1.003 0.979 0.955 0.933 0.911

4.1817 4.1812 4.1807 4.1802 4.1798

0.5983 0.5999 0.6014 0.6029 0.6044

7.01 6.82 6.64 6.47 6.30

0.07274 0.07259 0.07244 0.07228 0.07213

25 26 27 28 30

2.572 2.667 2.761 2.852 3.032

0.45247 0.45139 0.45038 0.44943 0.44771

0.44776 0.44630 0.44490 0.44355 0.44102

1496.7 1499.3 1501.9 1504.3 1509.1

0.891 0.871 0.852 0.833 0.798

4.1795 4.1792 4.1790 4.1788 4.1785

0.6059 0.6074 0.6089 0.6104 0.6133

6.15 5.99 5.85 5.70 5.44

0.07198 0.07182 0.07167 0.07151 0.07120

32 34 36 38 40

3.206 3.375 3.539 3.698 3.853

0.44622 0.44496 0.44390 0.44305 0.44239

0.43869 0.43655 0.43459 0.43280 0.43118

1513.6 1517.8 1521.7 1525.4 1528.9

0.765 0.734 0.705 0.679 0.653

4.1783 4.1782 4.1783 4.1784 4.1786

0.6162 0.6190 0.6218 0.6246 0.6273

5.19 4.95 4.74 4.54 4.35

0.07089 0.07058 0.07025 0.06992 0.06960

42 44 46 48 50

4.004 4.152 4.296 4.438 4.576

0.44192 0.44162 0.44149 0.44153 0.44173

0.42972 0.42842 0.42726 0.42624 0.42535

1532.1 1535.0 1537.7 1540.3 1542.6

0.629 0.607 0.586 0.566 0.547

4.1789 4.1792 4.1797 4.1801 4.1807

0.6299 0.6315 0.6351 0.6375 0.6400

4.17 4.02 3.86 3.71 3.57

0.06927 0.06894 0.06861 0.06828 0.06795

55 60 65 70 75

4.910 5.231 5.539 5.837 6.128

0.44290 0.44496 0.44788 0.45162 0.45614

0.42370 0.42281 0.42262 0.42309 0.42418

1547.4 1551.0 1553.4 1554.8 1555.1

0.5043 0.4668 0.4338 0.4045 0.3784

4.1824 4.1844 4.1869 4.1897 4.1929

0.6457 0.6511 0.6561 0.6607 0.6649

3.267 3.000 2.768 2.565 2.386

0.06710 0.06624 0.06537 0.06449 0.06359

80 85 90 95 100

6.411 6.689 6.962 7.233 7.501

0.46143 0.46748 0.47429 0.48185 0.49019

0.42587 0.42812 0.43093 0.43429 0.43819

1554.4 1552.9 1550.5 1547.2 1543.1

0.3550 0.3340 0.3150 0.2979 0.2823

4.1965 4.2005 4.2050 4.2102 4.2164

0.6686 0.6721 0.6753 0.6779 0.6800

2.228 2.088 1.962 1.850 1.756

0.06268 0.06176 0.06083 0.05988 0.05892

Values mostly from Aleksandrov, A. A. and M. S. Trakhtenhertz, Thermophysical Properties of Water at Atmospheric Pressure, Standartov, Moscow, 1977 (99 pp.).

THERMODYNAMIC PROPERTIES TABLE 2-354 Temperature, K

2-309

Thermodynamic Properties of Compressed Steam* Pressure, bar 0.1

0.5

1

5

10

20

40

60

80

350

v h s

16.12 2644 8.327

1.027.−3 321.7 1.037

1.027.−3 231.8 1.037

1.027.−3 322.1 1.037

1.027.−3 322.5 1.037

1.026.−3 323.3 1.036

1.025.−3 324.9 1.035

1.024.−3 326.4 1.034

1.023.−3 328.1 1.032

1.023.−3 329.7 1.031

400

v h s

18.44 2739 8.581

3.67 2735 7.831

1.827 2730 7.502

1.067.−3 533.1 1.601

1.067.−3 533.4 1.600

1.066.−3 534.1 1.599

1.065.−3 535.4 1.597

1.064.−3 536.8 1.595

1.063.−3 538.2 1.593

1.061.−3 539.6 1.592

450

v h s

20.75 2835 8.811

4.14 2833 8.061

2.063 2830 7.736

0.410 2804 6.949

1.124.−3 749.0 2.110

1.123.−3 749.8 2.107

1.121.−3 750.8 2.105

1.119.−3 751.9 2.102

1.118.−3 753.0 2.099

1.116.−3 754.1 2.097

500

v h s

23.07 2932 9.012

4.61 2931 8.261

2.298 2929 7.944

0.452 2912.4 7.177

0.221 2891.2 6.823

0.104 2839.4 6.422

1.201.−3 975.9 2.578

1.198.−3 976.3 2.575

1.196.−3 976.8 2.571

1.193.−3 977.3 2.567

600

v h s

27.7 3131 9.374

5.53 3130 8.630

2.76 3129 8.309

0.548 3120 7.560

0.271 3109 7.223

0.133 3087 6.875

0.0630 3036 6.590

0.0396 2976 6.224

0.0276 2906 5.997

0.0201 2820 5.775

700

v h s

32.3 3335 9.692

6.46 3335 8.946

3.23 3334 8.625

0.643 3328 7.877

0.319 3322 7.550

0.158 3307 7.215

0.0769 3278 6.864

0.0500 3247 6.644

0.0346 3214 6.431

0.0283 3179 6.334

800

v h s

36.9 3547 9.971

7.38 3546 9.228

3.69 3546 8.908

0.736 3542 8.161

0.367 3537 7.837

0.182 3526 7.507

0.0889 3506 7.151

0.0589 3485 6.965

0.0436 3464 6.809

0.0343 3442 6.685

900

v h s

41.5 3765 10.228

8.31 3765 9.485

4.15 3764 9.165

0.829 3761 8.420

0.414 3757 8.097

0.206 3750 7.770

0.102 3737 7.462

0.0674 3719 7.237

0.0501 3704 7.092

0.0398 3688 6.975

1000

v h s

46.2 3990 10.466

9.23 3990 9.723

4.615 3990 9.402

0.921 3987 8.659

0.460 3984 8.336

0.229 3978 8.011

0.114 3967 7.682

0.0758 3955 7.486

0.0564 3944 7.345

0.0449 3935 7.233

1500

v h s

69.2 5231 11.47

13.9 5228 10.77

6.92 5227 10.40

1.385 5225 9.66

0.692 5224 9.34

0.341 5221 9.015

0.1730 5217 8.693

0.1153 5212 8.503

0.0865 5207 8.368

0.0692 5203 8.262

2000

v h s

93.0 6832 12.38

18.6 6734 11.58

9.26 6706 11.25

1.850 6662 10.48

0.925 6649 10.15

0.462 6639 9.828

0.231 6629 9.503

0.1543 6623 9.313

0.1157 6619 9.178

0.0926 6616 9.073

2500

v h s

123.7 10417 13.95

24.0 9330 12.73

11.90 9046 12.28

2.35 8621 11.35

1.171 8504 10.80

0.583 8413 10.62

0.291 8342 10.26

0.1942 8307 10.06

0.1457 8285 9.920

0.1166 8269 9.810

*v = specific volume, m3/kg; h = specific enthalpy, kJ/kg; s = specific entropy, kJ/(kg⋅K). The notation 1.027.−3 signifies 1.027 × 10−3.

100

2-310

PHYSICAL AND CHEMICAL DATA

TABLE 2-354

Thermodynamic Properties of Compressed Steam (Concluded ) Pressure, bar

Temperature, K

150

200

250

300

400

500

600

700

800

900

1000

v 350 h s

1.020.−3 333.7 1.028

1.018.−3 337.7 1.025

1.016.−3 341.7 1.022

1.014.−3 344.7 1.019

1.009.−3 353.8 1.013

1.005.−3 361.8 1.007

1.002.−3 369.7 1.001

9.977.−4 377.7 0.996

9.937.−4 385.7 0.991

9.900.−4 393.7 0.985

9.865.−4 401.7 0.979

400

v h s

1.059.−3 543.1 1.587

1.056.−3 546.5 1.583

1.053.−3 550.1 1.578

1.050.−3 553.5 1.574

1.045.−3 560.6 1.565

1.041.−3 567.8 1.557

1.035.−3 574.9 1.549

1.031.−3 582.1 1.541

1.027.−3 589.3 1.533

1.022.−3 596.5 1.526

1.018.−3 603.8 1.518

450

v h s

1.112.−3 756.8 2.088

1.108.−3 759.5 2.082

1.105.−3 762.3 2.076

1.101.−3 765.2 2.070

1.094.−3 771.0 2.060

1.088.−3 776.9 2.049

1.082.−3 783.0 2.039

1.076.−3 789.6 2.029

1.070.−3 795.3 2.019

1.065.−3 801.6 2.010

1.059.−3 807.9 2.002

500

v h s

1.187.−3 978.8 2.558

1.181.−3 980.3 2.549

1.175.−3 981.9 2.541

1.170.−3 983.7 2.533

1.160.−3 987.4 2.517

1.151.−3 991.5 2.502

1.142.−3 995.9 2.488

1.134.−3 1000.5 2.474

1.126.−3 1005.3 2.461

1.119.−3 1010.3 2.449

1.112.−3 1015.4 2.437

600

v h s

1.519.−3 1499 3.501

1.483.−3 1489 3.469

1.454.−3 1479 3.443

1.428.−3 1472 3.419

1.392.−3 1462 3.379

1.362.−3 1456 3.346

1.337.−3 1452 3.316

1.315.−3 1449 3.290

1.296.−3 1447 3.266

1.280.−3 1447 3.244

1.265.−3 1447 3.223

700

v h s

1.724.−2 3082 6.037

1.157.−2 2965 5.770

7.986.−3 2821 5.494

5.416.−3 2635 5.179

2.630.−3 2233 4.554

2.038.−3 2084 4.308

1.831.−3 2021 4.192

1.716.−3 1986 4.116

1.639.−3 1962 4.058

1.589.−3 1946 4.012

1.536.−3 1931 3.972

800

v h s

2.195.−2 3386 6.444

1.575.−2 3325 6.252

1.201.−2 3261 6.086

9.512.−3 3193 5.934

6.391.−3 3047 5.654

4.576.−3 2895 5.397

3.496.−3 2734 5.175

2.866.−3 2648 4.998

2.484.−3 2567 4.864

2.239.−3 2508 4.761

2.072.−3 2465 4.701

900

v h s

2.590.−2 3649 6.755

1.899.−2 3609 6.587

1.483.−2 3568 6.449

1.207.−2 3526 6.327

8.619.−3 3440 6.119

6.581.−3 3354 5.940

5.257.−3 3269 5.780

4.348.−3 3188 5.637

3.704.−3 3113 5.510

3.454.−3 3049 5.399

2.907.−3 2995 5.305

1000

v h s

2.954.−2 3904 7.023

2.186.−2 3874 6.867

1.726.−2 3845 6.741

1.420.−2 3816 6.633

1.038.−2 3756 6.453

8.102.−3 3697 6.302

6.605.−3 3640 6.172

5.557.−3 3584 6.055

4.792.−3 3532 5.951

4.212.−3 3482 5.856

3.763.−3 3435 5.727

1500

v h s

0.0461 5202 8.074

0.0346 5198 7.936

0.0277 5186 7.827

0.0231 5180 7.738

0.0173 5171 7.597

0.0139 5157 7.484

0.0116 5144 7.391

0.00993 5133 7.310

0.00871 5120 7.239

0.00776 5108 7.176

0.00700 5095 7.118

2000

v h s

0.0619 6613 8.883

0.0465 6610 8.748

0.0372 6608 8.642

0.0311 6605 8.555

0.0234 6599 8.418

0.0188 6595 8.310

0.0157 6590 8.222

0.0135 6585 8.147

0.0119 6581 8.082

0.0106 6577 8.024

0.0096 6574 7.971

2500

v h s

0.0778 8269 9.610

0.0584 8269 9.468

0.0468 8269 9.358

0.0391 8268 9.270

0.0294 8267 9.129

0.0236 8265 9.020

0.0197 8261 8.930

0.0170 2856 8.854

0.0149 8250 8.788

0.0133 8244 8.730

0.0120 8240 8.677

TABLE 2-355 Density, Specific Heats at Constant Pressure and at Constant Volume and Velocity of Sound for Compressed Water, 1–1000 bar, 0–150°C 0°C (ITS-90)

10°C (ITS-90)

20°C (ITS-90)

30°C (ITS-90)

P, bar

density, kg/m3

Cp, kJ/ (kg⋅K)

Cv, kJ/ (kg⋅K)

w, m/s

density, kg/m3

Cp, kJ/(kg⋅K)

Cv, kJ/(kg⋅K)

w, m/s

density, kg/m3

Cp, kJ/(kg⋅K)

Cv, kJ/(kg⋅K)

w, m/s

density, kg/m3

Cp, kJ/(kg⋅K)

Cv, kJ/(kg⋅K)

w, m/s

1 50 100 150 200 250 300 400 500 600 800 1000

999.702 1002.03 1004.38 1006.71 1009.01 1011.28 1013.53 1017.97 1022.31 1026.57 1034.85 1042.83

4.1923 4.174 4.156 4.139 4.123 4.108 4.093 4.065 4.040 4.018 3.979 3.948

4.1877 4.168 4.149 4.130 4.112 4.095 4.078 4.046 4.016 3.988 3.937 3.892

1447.3 1455 1464 1472 1480 1489 1497 1513 1529 1545 1577 1609

998.207 1000.44 1002.69 1004.93 1007.13 1009.32 1011.48 1015.74 1019.92 1024.02 1031.99 1039.68

4.1812 4.166 4.151 4.137 4.124 4.110 4.098 4.074 4.052 4.032 3.996 3.967

4.1538 4.137 4.119 4.103 4.087 4.071 4.056 4.027 3.999 3.974 3.926 3.884

1482.3 1491 1499 1507 1516 1524 1532 1548 1565 1581 1613 1644

995.650 997.82 1000.02 1002.19 1004.34 1006.47 1008.57 1012.72 1016.79 1020.79 1028.56 1036.06

4.1774 4.164 4.151 4.139 4.127 4.115 4.104 4.083 4.063 4.044 4.011 3.982

4.1148 4.099 4.084 4.069 4.055 4.041 4.027 4.001 3.976 3.952 3.908 3.869

1509.1 1517 1526 1534 1543 1551 1559 1576 1592 1608 1640 1671

992.217 994.36 996.52 998.66 1000.77 1002.87 1004.94 1009.03 1013.03 1016.97 1024.62 1032.00

4.1775 4.166 4.154 4.142 4.131 4.121 4.110 4.091 4.072 4.055 4.023 3.995

4.0715 4.058 4.044 4.031 4.018 4.005 3.993 3.969 3.946 3.924 3.884 3.847

1528.9 1537 1546 1554 1563 1571 1579 1596 1612 1628 1660 1692

P, bar

density, kg/m3

Cp, kJ/ (kg⋅K)

Cv, kJ/ (kg⋅K)

w, m/s

density, kg/m3

Cp, kJ/(kg⋅K)

Cv, kJ/(kg⋅K)

w, m/s

density, kg/m3

Cp, kJ/(kg⋅K)

Cv, kJ/(kg⋅K)

w, m/s

density, kg/m3

Cp, kJ/ (kg⋅K)

Cv, kJ/(kg⋅K)

w, m/s

1 50 100 150 200 250 300 400 500 600 800 1000

988.036 990.16 992.31 994.44 996.54 998.62 1000.68 1004.74 1008.72 1012.62 1020.21 1027.53

4.1799 4.169 4.158 4.147 4.137 4.126 4.117 4.098 4.080 4.063 4.033 4.005

4.0248 4.012 4.000 3.988 3.976 3.965 3.954 3.932 3.911 3.892 3.854 3.820

1542.6 1551 1560 1568 1577 1585 1594 1610 1627 1643 1676 1707

983.197 985.33 987.48 989.61 991.71 993.80 995.86 999.92 1003.90 1007.80 1015.38 1022.69

4.1840 4.173 4.163 4.152 4.142 4.132 4.123 4.105 4.087 4.071 4.041 4.013

3.9755 3.964 3.953 3.943 3.932 3.922 3.911 3.892 3.873 3.855 3.821 3.789

1551.0 1560 1568 1577 1586 1594 1603 1620 1637 1653 1686 1718

977.766 979.92 982.09 984.23 986.36 988.46 990.53 994.62 998.62 1002.54 1010.15 1017.48

4.1896 4.179 4.169 4.158 4.148 4.139 4.129 4.111 4.094 4.078 4.048 4.020

3.9246 3.915 3.905 3.895 3.885 3.876 3.867 3.849 3.832 3.815 3.784 3.754

1554.8 1564 1573 1582 1590 1599 1608 1625 1642 1659 1693 1726

971.791 973.98 976.18 978.35 980.51 982.63 984.74 988.87 992.92 996.88 1004.56 1011.94

4.1967 4.186 4.176 4.165 4.155 4.146 4.136 4.118 4.101 4.085 4.054 4.027

3.8727 3.864 3.855 3.846 3.838 3.829 3.821 3.805 3.789 3.774 3.745 3.717

1554.5 1564 1573 1582 1591 1600 1609 1627 1644 1662 1696 1730

50°C (ITS-90)

40°C (ITS-90)

70°C (ITS-90)

60°C (ITS-90)

Prepared by H. Sato, Keio University, Japan, Oct. 1994. Based upon “An equation of state for the thermodynamic properties of water in the liquid phase including the metastable state,” from Properties of Water and Steam,” Proc. 11th Int. Conf. Props. Steam (M. Pichal and O. Sifner, eds.), Hemisphere, New York, 1990 (551 pp.).

2-311

TABLE 2-355 Density, Specific Heats at Constant Pressure and at Constant Volume and Velocity of Sound for Compressed Water, 1–1000 bar, 0–150°C (Concluded ) 80°C (ITS-90)

90°C (ITS-90)

100°C (ITS-90)

110°C (ITS-90)

P, bar

density, kg/m3

Cp, kJ/ (kg⋅K)

Cv, kJ/ (kg⋅K)

w, m/s

density, kg/m3

Cp, kJ/(kg⋅K)

Cv, kJ/(kg⋅K)

w, m/s

density, kg/m3

Cp, kJ/(kg⋅K)

Cv, kJ/(kg⋅K)

w, m/s

density, kg/m3

Cp, kJ/(kg⋅K)

Cv, kJ/(kg⋅K)

w, m/s

1 50 100 150 200 250 300 400 500 600 800 1000

965.309 967.54 969.79 972.00 974.20 976.36 978.50 982.71 986.82 990.83 998.62 1006.08

4.2056 4.195 4.184 4.174 4.164 4.154 4.144 4.126 4.108 4.092 4.061 4.033

3.8206 3.813 3.805 3.797 3.789 3.782 3.774 3.759 3.745 3.731 3.704 3.678

1550.5 1560 1569 1579 1588 1597 1607 1625 1643 1661 1696 1731

958.348 960.64 962.94 965.21 967.45 969.67 971.85 976.15 980.34 984.43 992.34 999.92

4.2164 4.205 4.194 4.184 4.173 4.163 4.153 4.135 4.117 4.100 4.068 4.039

3.7689 3.762 3.755 3.748 3.741 3.734 3.727 3.714 3.701 3.688 3.663 3.639

1543.1 1553 1563 1572 1582 1592 1601 1620 1639 1658 1694 1730

950.927 953.28 955.65 957.99 960.30 962.57 964.82 969.21 973.50 977.69 985.76 993.47

4.2296 4.218 4.206 4.195 4.184 4.174 4.164 4.144 4.126 4.108 4.076 4.046

3.7181 3.712 3.706 3.699 3.693 3.687 3.681 3.669 3.657 3.645 3.621 3.598

1532.5 1543 1553 1563 1573 1583 1593 1613 1632 1652 1690 1727

943.059 945.50 947.95 950.36 952.74 955.09 957.40 961.92 966.32 970.61 978.87 986.73

4.2453 4.233 4.221 4.209 4.198 4.187 4.176 4.155 4.136 4.118 4.084 4.053

3.6684 3.663 3.657 3.652 3.646 3.641 3.635 3.624 3.613 3.602 3.579 3.556

1519.0 1530 1540 1551 1561 1572 1582 1603 1623 1644 1683 1723

P, bar

density, kg/m3

Cp, kJ/ (kg⋅K)

Cv, kJ/ (kg⋅K)

w, m/s

density, kg/m3

Cp, kJ/(kg⋅K)

Cv, kJ/(kg⋅K)

w, m/s

density, kg/m3

Cp, kJ/(kg⋅K)

Cv, kJ/(kg⋅K)

w, m/s

density, kg/m3

Cp, kJ/(kg⋅K)

Cv, kJ/(kg⋅K)

w, m/s

1 50 100 150 200 250 300 400 500 600 800 1000

934.749 937.28 939.83 942.33 944.79 947.22 949.61 954.27 958.81 963.22 971.68 979.72

4.2639 4.251 4.238 4.225 4.213 4.201 4.190 4.168 4.148 4.128 4.092 4.059

3.6201 3.615 3.610 3.605 3.600 3.595 3.590 3.580 3.569 3.558 3.536 3.512

1502.8 1514 1525 1536 1547 1558 1569 1591 1612 1634 1676 1717

925.997 928.64 931.29 933.90 936.46 938.97 941.45 946.28 950.96 955.51 964.20 972.44

4.2859 4.271 4.257 4.244 4.231 4.218 4.206 4.183 4.161 4.140 4.101 4.066

3.5733 3.569 3.564 3.560 3.555 3.550 3.545 3.535 3.525 3.514 3.491 3.466

1484.1 1496 1508 1519 1531 1543 1554 1577 1600 1623 1667 1710

916.797 919.57 922.34 925.06 927.73 930.35 932.92 937.94 942.79 947.48 956.43 964.88

4.3114 4.296 4.280 4.266 4.251 4.238 4.224 4.199 4.176 4.153 4.112 4.073

3.5279 3.524 3.520 3.516 3.511 3.506 3.501 3.491 3.480 3.469 3.444 3.417

1463.0 1475 1488 1501 1513 1525 1538 1562 1586 1610 1658 1704

907.143 910.06 912.97 915.82 918.61 921.34 924.03 929.24 934.27 939.13 948.36 957.04

4.3408 4.324 4.307 4.291 4.276 4.261 4.246 4.219 4.194 4.169 4.124 4.081

3.4848 3.481 3.477 3.473 3.469 3.464 3.459 3.448 3.436 3.423 3.395 3.364

1439.8 1453 1467 1480 1493 1507 1520 1546 1572 1598 1648 1698

120°C (ITS-90)

2-312

140°C (ITS-90)

130°C (ITS-90)

150°C (ITS-90)

TABLE 2-356

Specific Heat and Other Thermophysical Properties of Water Substance* Temperature, K

Pressure, bar

300

350

400

450

500

600

700

800

900

1000

1200

1400

1600

1800

µ 1 cp k Pr

8.57.−4 4.18 0.614 5.81

3.70.−4 4.19 0.668 2.32

1.32.−5 1.99 0.0268 0.980

1.52.−5 1.97 0.0311 0.967

1.73.−5 1.98 0.0358 0.955

2.15.−5 2.02 0.0464 0.936

2.57.−5 2.09 0.0581 0.920

2.98.−5 2.15 0.0710 0.906

3.39.−5 2.22 0.0843 0.891

3.78.−5 2.29 0.0981 0.881

4.48.−5 2.43 0.13 0.83

5.06.−5 2.58 0.16 0.80

5.65.−5 2.73 0.21 0.75

6.19.−5 3.02 0.33 0.57

6.70.−5 3.79 0.57 0.45

5

µ cp k Pr

8.57.−4 4.18 0.614 5.82

3.70.−4 4.19 0.668 2.32

2.17.−4 4.26 0.689 1.34

1.49.−5 2.21 0.0335 0.983

1.72.−5 2.10 0.0369 0.973

2.15.−5 2.07 0.0469 0.947

2.57.−5 2.11 0.0585 0.925

2.98.−5 2.16 0.0713 0.907

3.39.−5 2.23 0.0846 0.892

3.78.−5 2.29 0.0984 0.881

4.45.−5 2.43 0.13 0.83

5.06.−5 2.58 0.16 0.81

5.65.−5 2.73 0.20 0.77

6.19.−5 2.98 0.28 0.65

6.70.−5 3.40 0.43 0.53

10

µ cp k Pr

8.57.−4 4.18 0.615 5.82

3.70.−4 4.19 0.668 2.32

2.17.−4 4.25 0.689 1.34

1.51.−4 4.39 0.677 0.981

1.71.−5 2.29 0.0380 1.028

2.15.−5 2.13 0.0474 0.963

2.58.−5 2.13 0.0590 0.931

2.99.−5 2.18 0.0717 0.908

3.39.−5 2.24 0.0851 0.892

3.78.−5 2.30 0.0988 0.881

4.45.−5 2.44 0.13 0.84

5.06.−5 2.58 0.16 0.82

5.65.−5 2.73 0.20 0.78

6.19.−5 2.95 0.26 0.70

6.70.−5 3.29 0.39 0.57

20

µ cp k Pr

8.56.−4 4.17 0.616 5.80

3.71.−4 4.19 0.669 2.32

2.18.−4 4.25 0.689 1.34

1.51.−4 4.39 0.679 0.979

1.68.−5 2.84 0.0402 1.19

2.15.−5 2.26 0.0485 0.999

2.59.−5 2.19 0.0599 0.946

3.00.−5 2.21 0.0726 0.912

3.40.−5 2.26 0.0859 0.893

3.79.−5 2.32 0.0996 0.881

4.46.−5 2.45 0.13 0.84

5.06.−5 2.59 0.16 0.82

5.65.−5 2.73 0.20 0.79

6.19.−5 2.92 0.25 0.72

6.70.−5 3.21 0.36 0.60

40

µ cp k Pr

8.55.−4 4.17 0.617 5.78

3.71.−4 4.19 0.671 2.31

2.18.−4 4.25 0.690 1.34

1.52.−4 4.38 0.680 0.977

1.19.−4 4.65 0.644 0.862

2.15.−5 2.60 0.516 1.08

2.61.−5 2.32 0.0620 0.975

3.02.−5 2.28 0.0744 0.924

3.42.−5 2.30 0.0877 0.895

3.80.−5 2.34 0.101 0.881

4.47.−5 2.46 0.13 0.84

5.07.−5 2.59 0.16 0.82

5.65.−5 2.73 0.19 0.80

6.19.−5 2.90 0.24 0.73

6.70.−5 3.14 0.33 0.63

60

µ cp k Pr

8.54.−4 4.16 0.619 5.74

3.72.−4 4.18 0.672 2.31

2.19.−4 4.24 0.692 1.34

1.53.−4 4.37 0.682 0.976

1.20.−4 4.63 0.646 0.859

2.14.−5 3.11 0.0561 1.19

2.63.−5 2.47 0.0645 1.008

3.04.−5 2.35 0.0764 0.934

3.43.−5 2.34 0.0895 0.899

3.82.−5 2.37 0.103 0.879

4.48.−5 2.48 0.13 0.84

5.07.−5 2.60 0.16 0.82

5.66.−5 2.73 0.19 0.81

6.19.−5 2.89 0.24 0.74

6.70.−5 3.11 0.32 0.65

80

µ cp k Pr

8.53.−4 4.16 0.620 5.72

3.72.−4 4.18 0.674 2.31

2.19.−4 4.24 0.693 1.34

1.53.−4 4.36 0.684 0.975

1.20.−4 4.62 0.648 0.856

2.14.−5 3.88 0.0628 1.33

2.66.−5 2.65 0.0672 1.046

3.06.−5 2.43 0.0785 0.946

3.45.−5 2.39 0.0914 0.902

3.83.−5 2.40 0.105 0.877

4.48.−5 2.49 0.13 0.84

5.08.−5 2.61 0.16 0.83

5.66.−5 2.73 0.10 0.81

6.19.−5 2.88 0.24 0.74

6.70.−5 3.09 0.31 0.66

100

µ cp k Pr

8.52.−4 4.15 0.622 5.69

3.73.−4 4.17 0.675 2.31

2.20.−4 4.23 0.694 1.34

1.53.−4 4.35 0.685 0.975

1.21.−4 4.60 0.651 0.853

2.14.−5 5.22 0.0730 1.74

2.69.−5 2.85 0.0704 1.088

3.08.−5 2.52 0.0807 0.960

3.47.−5 2.44 0.0934 0.905

3.85.−5 2.44 0.107 0.876

4.49.−5 2.50 0.13 0.84

5.08.−5 2.62 0.16 0.83

5.66.−5 2.73 0.19 0.81

6.19.−5 2.88 0.24 0.74

6.70.−5 3.08 0.31 0.67

µ cp k Pr

8.51.−4 4.14 0.624 5.64

3.74.−4 4.16 0.678 2.30

2.22.−4 4.22 0.699 1.34

1.56.−4 4.34 0.693 0.974

1.22.−4 4.54 0.657 0.842

8.22.−5

150

2.72.−5 3.55 0.079 1.22

3.12.−5 2.74 0.086 0.994

3.51.−5 2.57 0.098 0.916

3.89.−5 2.53 0.110 0.891

4.52.−5 2.54 0.14 0.84

5.09.−5 2.65 0.16 0.83

5.67.−5 2.75 0.19 0.82

6.19.−5 2.88 0.23 0.76

6.70.−5 3.06

µ cp k Pr

8.50.−4 4.12 0.626 5.59

3.75.−4 4.15 0.681 2.29

2.24.−4 4.21 0.702 1.34

1.57.−4 4.32 0.697 0.974

1.23.−4 4.51 0.661 0.833

2.80.−5 4.67 0.095 1.38

3.17.−5 3.04 0.095 1.014

3.54.−5 2.71 0.104 0.925

3.93.−5 2.62 0.113 0.903

4.54.−5 2.57 0.14 0.84

5.11.−5 2.67 0.16 0.83

5.67.−5 2.76 0.19 0.82

2.88

3.05

250

µ cp k Pr

8.49.−4 4.12 0.627 5.57

3.76.−4 4.14 0.683 2.28

2.26.−4 4.20 0.705 1.34

1.59.−4 4.30 0.701 0.974

1.23.−4 4.49 0.672 0.826

8.41.−5 5.90 0.537 0.924

2.89.−5 6.16 0.112 1.590

3.24.−5 3.40 0.103 1.070

3.59.−5 2.86 0.110 0.940

3.98.−5 2.71 0.119 0.910

4.56.−5 2.61 0.136 0.85

5.12.−5 2.69 0.16 0.84

5.68.−5 2.77

2.89

3.04

300

µ cp k Pr

8.49.−4 4.10 0.629 5.53

3.77.−4 4.13 0.685 2.27

2.28.−4 4.19 0.708 1.34

1.60.−4 4.29 0.704 0.973

1.24.−4 4.44 0.675 0.820

8.50.−5 5.60 0.548 0.859

3.7.−5 10.20 0.173 2.18

3.4.−5 3.82 0.113 1.149

3.64.−5 3.03 0.113 0.976

4.02.−5 2.81 0.123 0.917

4.59.−5 2.65 0.14 0.87

5.14.−5 2.72

5.68.−5 2.78

2.90

3.04

400

µ cp k Pr

8.49.−4 4.08 0.631 5.49

3.80.−4 4.12 0.689 2.26

2.30.−4 4.16 0.714 1.34

1.62.−4 4.26 0.710 0.971

1.26.−4 4.42 0.676 0.817

8.64.−5 5.31 0.567 0.799

5.3.−5 13.20 0.327 2.14

3.6.−5 4.86 0.145 1.207

3.8.−5 3.39 0.129 0.999

4.1.−5 3.01 0.134 0.926

4.6.−5 2.70 0.15

5.17.−5 2.77

2.81

2.91

3.04

500

µ cp k Pr

8.50.−4 4.06 0.634 5.44

3.82.−4 4.10 0.695 2.25

2.31.−4 4.15 0.719 1.33

1.64.−4 4.23 0.717 0.971

1.28.−4 4.38 0.693 0.814

8.83.−5 5.08 0.583 0.773

5.8.−5 8.44 0.378 1.30

4.0.−5 5.70 0.186 1.225

4.0.−5 3.90 0.147 1.061

4.2.−5 3.21 0.145 0.932

4.7.−5 2.77

2.81

2.84

2.92

3.04

200

0.520 8.32.−5 0.525

2000

2-313

TABLE 2-356

Specific Heat and Other Thermophysical Properties of Water Substance (Concluded ) Temperature, K

Pressure, bar

300

350

400

450

500

600

700

800

900

1000

µ 600 cp k Pr

8.51.−4 4.04 0.639 5.38

3.85.−4 4.08 0.699 2.24

2.32.−4 4.13 0.725 1.32

1.66.−4 4.20 0.725 0.970

1.30.−4 4.33 0.700 0.812

9.17.−5 4.92 0.597 0.755

6.5.−5 6.93 0.420 1.073

4.4.−5 6.83 0.239 1.175

4.2.−5 4.19 0.170 1.035

4.4.−5 3.38 0.159 0.935

2.87

2.86

2.86

2.92

3.04

700

µ cp k Pr

8.52.−4 4.01 0.644 5.33

3.87.−4 4.07 0.706 2.23

2.33.−4 4.12 0.730 1.32

1.69.−4 4.17 0.732 0.970

1.33.−4 4.29 0.707 0.810

9.50.−5 4.78 0.614 0.739

6.9.−5 6.12 0.442 1.047

4.9.−5 6.26 0.279 1.098

4.5.−5 4.62 0.198 1.010

4.6.−5 3.59 0.177 0.935

2.94

2.91

2.88

2.93

3.05

800

µ cp k Pr

8.53.−4 3.99 0.648 5.28

3.90.−4 4.05 0.709 2.23

2.34.−4 4.10 0.735 1.31

1.72.−4 4.15 0.736 0.970

1.36.−4 4.26 0.714 0.808

9.82.−5 4.67 0.625 0.725

7.3.−5 5.60 0.478 0.855

5.4.−5 6.09 0.320 1.028

4.8.−5 4.77 0.228 1.003

4.8.−5 3.75 0.193 0.933

3.01

2.96

2.91

2.95

3.05

900

µ cp k Pr

8.54.−4 3.98 0.651 5.23

3.93.−4 4.03 0.713 2.22

2.35.−4 4.08 0.738 1.30

1.74.−4 4.13 0.742 0.969

1.38.−4 4.23 0.724 0.806

1.00.−4 4.57 0.636 0.712

7.6.−5 5.29 0.496 0.810

5.8.−5 5.86 0.351 0.968

5.1.−5 4.85 0.260 0.950

5.0.−5 3.86 0.210 0.919

3.08

3.00

2.94

2.97

3.06

1000

µ cp k Pr

8.56.−4 3.97 0.653 5.19

3.96.−4 4.02 0.717 2.22

2.36.−4 4.06 0.743 1.30

1.76.−4 4.11 0.747 0.968

1.40.−4 4.20 0.731 0.804

1.02.−4 4.47 0.650 0.701

7.9.−5 5.08 0.516 0.778

6.2.−5 5.51 0.372 0.918

5.4.−5 4.88 0.288 0.900

5.1.−5 3.96 0.228 0.886

3.16

3.05

2.97

2.98

3.07

*µ = viscosity, Ns/m2; cp = specific heat at constant pressure, kJ/(kg⋅K); k = thermal conductivity, W/(m⋅K); Pr = Prandtl number.

2-314

1200

1400

1600

1800

2000

THERMODYNAMIC PROPERTIES TABLE 2-357

Thermodynamic Properties of Water Substance along the Melting Line T, °C

103 v f , m3/kg

h f , kJ/kg

s f, kJ/kg⋅K

cpf, kJ/kg⋅K

cmelt, kJ/kg⋅K

106α f, K−1

106K f,T bar−1

6.117.−5t 1.01325 50 100 150

0.0100 0.0026 −0.3618 −0.7410 −1.1249

1.00021 1.00016 0.99770 0.99523 0.99278

0 0.0719 3.5140 6.9794 10.3964

0 −0.0001 −0.0054 −0.0110 −0.0167

4.219 4.218 4.196 4.174 4.152

3.969 3.970 3.997 4.023 4.047

−67.42 −67.17 −54.92 −42.52 −30.24

50.90 50.88 50.30 49.73 49.17

200 250 300 400 500

−1.5166 −1.9151 −2.3206 −3.1532 −4.0156

0.99037 0.98798 0.98562 0.98098 0.97643

13.7648 17.0843 20.3547 26.7472 32.9403

−0.0225 −0.0285 −0.0347 −0.0474 −0.0607

4.132 4.112 4.092 4.056 4.022

4.070 4.092 4.113 4.150 4.184

−18.05 −5.93 6.12 30.09 53.97

48.63 48.11 47.59 46.61 45.68

600 800 1000

−4.909 −6.790 −8.803

0.97196 0.96326 0.95493

38.932 50.300 60.836

−0.0747 −0.1046 −0.1371

3.992 3.937 3.893

4.215 4.270 4.320

77.87 126.18 175.98

44.80 43.19 41.74

P, bar

2-315

Condensed from U. Grigull, Private communication, January 18, 1995. Materials prepared at Technical University München, Germany by U. Grigull and S. Marek. For a table as a function of temperature, see Grigull, U. and S. Marek, Warme u. Stoff., 30 (1994): 1–8. t = the triple point (at 6.117 × 10−5 bar, 0.01°C); vf = 0.0010021 m3/kg: α f = −67.42 × 10−6/K. Other equations for properties are given by Jones, F. E. and G. L. Harris, J. Res. N.I.S.T., 97, 3 (1992): 335–340, and by Wagner, W. and A. Pruss, J. Phys. Chem. Ref. Data, 22, 3 (1993): 783–787. Steam tables include Walker, W. A., U.S. Naval Ordn. Lab. rept. NOLTR NOLTR-66-217 = AD 651105 (0–1000 bar, 0–150°C), 1967 (72 pp.); Grigull, U., J. Straub, et al., Steam Tables in S.I. Units (0.01–1000 bar, 0–1000°C), Springer-Verlag, Berlin, 1990 (133 pp.); Tseng, C. M., T. A. Hamp, et al., Atomic Energy of Canada rept. (30 props, sat liq & vap., 1–220 bar), AECL-5910 1977 (90 pp.). For dissociation, see e.g., Knonicek, V., Rozpr. Cesko Acad Ved., Rada techn ved (0.01–100 bar, 1000–5000 K). 77, 1 (1967). The proceedings of the 10th international conference on the properties of steam were edited by Sytchev, V. V. and A. A. Aleksandrov, Plenum, NY, 1984; and for the 11th conference by Pichal, M. and O. Sifner, Hemisphere, 1989 (550 pp.). For electrical conductivity, see e.g., Marshall, W. L., J. Chem. Eng. Data, 32 (1987): 221–226.

TABLE 2-358 T, K

Saturated Xenon*

P, bar

v f, m3/kg

vg, m3/kg

h f, kJ/kg

hg, kJ/kg

s f, kJ/(kg⋅K)

s g, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µ f, 10−4 Pa⋅s

k f, W/(m⋅K)

10 20 30 40 50

2.642.−4 2.650.−4 2.661.−4 2.675.−4 2.689.−4

0.19 1.21 2.74 4.47 6.31

0.0236 0.0901 0.1510 0.2003 0.2410

0.058 0.133 0.164 0.178 0.186

60 80 100 120 140

2.704.−4 2.737.−4 2.776.−4 2.820.−4 2.874.−4

8.21 12.14 16.30 20.81 25.67

0.2755 0.3319 0.3783 0.4197 0.4581

0.191 0.202 0.214 0.231 0.251

160 161.4m 161.4m 170 180

0.816 1.336 2.218

2.941.−4 2.946.−4 3.372.−4 3.439.−4 3.523.−4

0.1219 0.0776 0.0487

30.94 31.30 48.98 52.01 55.52

145.5 146.5 147.5

0.4946 0.4969 0.6072 0.6253 0.6452

1.206 1.181 1.156

0.270 0.271 0.350 0.349 0.349

4.50 3.99

0.0707 0.0663

190 200 210 220 230

3.480 5.212 7.504 10.45 14.16

3.615.−4 3.715.−4 3.828.−4 3.955.−4 4.100.−4

0.0321 0.0220 0.0156 0.0113 0.0084

59.04 62.61 66.25 70.00 73.91

148.3 148.9 149.2 149.4 149.2

0.6641 0.6820 0.6994 0.7163 0.7330

1.134 1.113 1.095 1.077 1.060

0.352 0.357 0.365 0.379 0.400

3.51 3.09 2.71 2.39 2.09

0.0622 0.0582 0.0542 0.0506 0.0468

240 250 260 270 280

18.72 24.25 30.87 38.69 47.86

4.271.−4 4.476.−4 4.730.−4 5.079.−4 5.689.−4

0.0063 0.0047 0.0036 0.0027 0.0019

78.05 82.54 87.52 93.30 100.6

148.5 147.5 145.7 142.8 138.0

0.7498 0.7671 0.7855 0.8058 0.8308

1.044 1.027 1.009 0.989 0.964

0.432 0.482 0.560 0.685 0.995

1.83 1.60 1.38 1.18 0.95

0.0429 0.0393 0.0355 0.0313 0.0275

289.7 c

58.21

9.091.−4

0.0009

120.0

120.0

0.8962

0.896

∞

∞

*Values extracted and in some cases rounded off from those cited in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standards Press, Moscow, 1976. This source contains values for the compressed state for pressures up to 1000 bar, etc. m = melting point; c = critical point. The notation 2.642.−4 signifies 2.642 × 10−4. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.).

2-316

PHYSICAL AND CHEMICAL DATA

TABLE 2-359

Compressed Xenon* Pressure, bar

T, K 100 v h s

1 2.776.−4 16.30 0.3783

100

200

300

400

500

600

700

2.764.−4 18.84 0.3762

2.752.−4 21.40 0.3742

2.742.−4 23.95 0.3723

2.731.−4 26.50 0.3704

2.721.−4 29.05 0.3686

2.711.−4 31.59 0.3669

2.702.−4 34.13 0.3652

2.693.−4 36.67 0.3636

800

2.684.−4 39.21 0.3621

900

2.675.−4 41.74 0.3802

1000

200 v h s

0.1245 151.8 1.228

3.623.−4 64.22 0.6727

3.547.−4 66.14 0.6643

3.484.−4 68.19 0.6570

3.430.−4 70.34 0.6505

3.383.−4 72.56 0.6446

3.342.−4 74.83 0.6391

3.304.−4 77.13 0.6340

3.270.−4 79.46 0.6292

3.240.−4 81.81 0.6247

3.211.−4 84.18 0.6204

300 v h s

0.1890 168.0 1.294

5.729.−4 106.4 0.8401

4.769.−4 101.6 0.8073

4.431.−4 101.3 0.7908

4.220.−4 102.0 0.7789

4.068.−4 103.3 0.7691

3.955.−4 104.9 0.7608

3.862.−4 106.7 0.7540

3.783.−4 108.5 0.7477

3.716.−4 110.6 0.7424

3.657.−4 112.8 0.7370

400 v h s

0.2527 183.9 1.340

1.998.−3 164.2 1.012

8.759.−4 145.4 0.9330

6.452.−4 137.4 0.8945

5.604.−4 134.7 0.8730

5.141.−4 134.1 0.8581

4.839.−4 134.5 0.8467

4.622.−4 135.5 0.8373

4.457.−4 136.8 0.8292

4.325.−4 138.3 0.8220

4.217.−4 140.0 0.8162

500 v h s

0.3163 199.8 1.375

2.899.−3 187.8 1.065

1.389.−3 177.1 1.004

9.449.−4 169.4 0.9664

7.577.−4 165.1 0.9409

6.593.−4 163.0 0.9228

5.986.−4 162.3 0.9088

5.570.−4 162.4 0.8975

5.268.−4 163.1 0.8881

5.038.−4 164.3 0.8801

4.859.−4 165.7 0.8731

600 v h s

0.3798 215.7 1.404

3.673.−3 207.4 1.101

1.823.−3 200.3 1.047

1.240.−3 194.8 1.013

9.699.−4 191.1 0.9885

8.206.−4 188.9 0.9700

7.273.−4 187.9 0.9555

6.636.−4 187.6 0.9435

6.172.−4 188.0 0.9334

5.820.−4 188.8 0.9247

5.545.−4 189.9 0.9172

700 v h s

0.4432 231.5 1.428

4.397.−3 225.6 1.129

2.217.−3 220.6 1.078

1.513.−3 216.7 1.047

1.175.−3 213.8 1.023

9.815.−4 212.2 1.006

8.583.−4 211.3 0.9916

7.734.−4 211.1 0.9797

7.115.−4 211.3 0.9695

6.642.−4 212.0 0.9606

6.268.−4 213.1 0.9528

800 v h s

0.5066 247.4 1.450

5.093.−3 243.0 1.152

2.587.−3 239.5 1.103

1.769.−3 236.7 1.073

1.370.−3 234.8 1.052

1.137.−3 233.6 1.035

9.870.−4 233.0 1.021

8.824.−4 232.9 1.009

8.057.−4 233.3 0.9988

7.469.−4 234.0 0.9901

7.005.−4 235.0 0.9823

900 v h s

0.5700 263.2 1.468

5.773.−3 260.1 1.172

2.944.−3 257.5 1.125

2.014.−3 255.7 1.096

1.557.−3 254.4 1.075

1.288.−3 253.6 1.058

1.112.−3 253.4 1.045

9.893.−4 253.7 1.033

8.989.−4 254.2 1.023

8.289.−4 254.9 1.015

7.737.−4 256.1 1.007

1000 v h s

0.6333 279.1 1.485

6.441.−3 276.8 1.190

3.291.−3 275.1 1.143

2.252.−3 273.9 1.115

1.738.−3 273.2 1.095

1.435.−3 272.9 1.079

1.235.−3 273.0 1.065

1.094.−3 273.4 1.054

9.899.−4 274.1 1.044

9.097.−4 275.1 1.036

8.461.−4 276.2 1.028

*Values extracted and in some cases rounded off from those cited in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon. Standards Press, Moscow, 1976. This source contains an exhaustive tabulation of values. v = specific volume, m3/kg; h = specific enthalpy, kJ/kg; s = specific entropy, kJ/(kg⋅K). The notation 2.776.−4 signifies 2.776 × 10−4. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.).

THERMODYNAMIC PROPERTIES

2-317

TABLE 2-360

Surface Tension (N/m) of Saturated Liquid Refrigerants*

R no.

−50

−25

0

25

50

75

100

125

150

0.0279 0.0188 0.0092 0.0197 0.0115

0.0244 0.0152 0.0056 0.0156 0.0065

0.0210 0.0118 0.0025 0.0117 0.0025

0.0178 0.0085 0.0002 0.0081

0.0146 0.0055 — 0.0047 —

0.0116 0.0029 — 0.0018 —

0.0087 0.0007 — — —

0.0060 — — — —

0.0036 — — — —

—

0.0231 0.0154

0.0069 0.0172 0.0109 0.0047 0.0082

0.0032 0.0144 0.0082 0.0022 0.0050

0.0002 0.0118 0.0056

0.0192

0.0201 0.0138 0.0075 0.0117

0.0021

— 0.0092 0.0033 — 0.0000

— 0.0067 0.0012 — —

— 0.0045 — — —

142b 152a 170 290 C318

0.0213 0.0201 0.0100

0.0178 0.0166 0.0051

—

0.0143

0.0145 0.0132 0.0032 0.0101 0.0113

0.0113 0.0100 0.0005 0.0082 0.0085

0.0083 0.0068 — 0.0041 0.0048

0.0055 0.0038 — 0.0016 0.0033

0.0029 0.0011 — — 0.0011

— — — —

— — — — —

502 503 600 600a 718

0.0159 0.0094

0.0121 0.0053 0.0180

0.0054 — 0.0122 0.0101 0.0720

0.0026 — 0.0094 0.0073 0.0680

— 0.0068 0.0047 0.0636

— — 0.0043 0.0024 0.0590

— — 0.0020 0.0005 0.0540

— — 0.0001 — 0.0488

0.0005

—

—

—

—

—

0.0070

0.0041

0.0014

Temperature, °C 11 12 13 22 23 32 113 114 115 134a

744 1150 1270

—

—

0.0086 0.0018 0.0150 0.0132 0.0755

0.0100 0.0171

0.0096 0.0055 0.0136

0.0044 0.0013 0.0102

*Dashes indicate inaccessible states; blanks indicate no available data. Values and equations were given by Srinivasan, K., Can. J. Chem. Eng. (27 liquids), 68 (1990): 493; Lielmezs, J. and T. A. Herrick, Chem. Eng. J. (34 liquids), 32 (1986): 165–169. Somayajulu, G. R., Int. J. Thermophys. (64 liquids); 9, 4 (1988): 559–566; Ibrahim, N. and S. Murad, Chem. Eng. Commun. (29 polar liquids), 79 (1979): 165–174; Yaws, C. L.; Morachevsky, A. G. and I. B. Sladkov, Physico-Chemical Properties of Molecular Inorganic Compounds (200 compounds), Khimiya, Leningrad, 1987, Jasper, J., J. Phys. Chem. Ref. Data (2200 compounds), 1, (1972): 841–1009; and Vargaftik, N. B., B. N. Volkov, et al., J. Phys. Chem. Ref. Data (water), 12, 3 (1983): 817–820. See also Escobedo, J. and Mansoori, G. R., AIChE J., 42(5), May 1996: 1425–1433.

TABLE 2-361

Velocity of Sound (m/s) in Gaseous Refrigerants at Atmospheric Pressure* Temperature, °C

−50

−25

0

25

50

75

100

125

150

11 12 13 14 22

— — 142 158 —

— 136 150 166 166

— 143 157 173 174

141 150 164 180 182

147 156 170 187 189

153 162 176 194 196

158 168 182 200 202

163 173 188 206 208

168 179 193 212 215

23 32 113 114 134a

179

188

197

205

212

220

227

234

240

— —

— — 146

— — 154

— 120 162

121 126 169

126 131 175

131 136 180

135 141 186

140 146 192

170 290 600 600a 718

272 — — — —

286 227 — — —

299 238 200 201 —

311 249 210 211 —

323 258 220 221 —

334 268 228 229 —

344 277 237 237 473

355 286 245 246 490

364 294 252 253 505

290 —

248 305 235

258 318 246

269 330 257

279 341 267

288 352 277

297 363 286

307 373 295

316 384 303

R. no.

744 1150 1270

*Dashes indicate inaccessible states; blanks indicate no available data.

2-318

PHYSICAL AND CHEMICAL DATA TABLE 3-362

Velocity of Sound (m/s) in Saturated Liquid Refrigerants* Temperature, °C

R. no.

−50

−25

0

25

50

75

100

125

150

11 12 13 14 22

933 829 602 182 899

843 695 444 — 790

772 564 302 — 682

705 434

639

569

493

— 571

— — 446

— — 319

— — —

408 — — — —

323 — — — —

538

348

191

—

—

— —

— —

— —

— 853

871 726

786 623 454

700 540 346

633 453 255

371

284 —

183 —

— —

858 1210 1290 1205 —

743 982 1163 1078 —

626 884 1031 947 1402

517 719 896 812 1495

387 551 759 661 1542

262 367 609 528 1554

105 — 477 378 1543

— — 325 208 1514

— — 142 — 1468

751 644 1022

525 372 859

272

—

—

—

—

—

874 1184

694

524

335

23 32 113 114 115 134a 290 600 600a 718 744 1150 1270

*Dashes indicate inaccessible states; blanks indicate no available data.

TRANSPORT PROPERTIES INTRODUCTION Extensive tables of the viscosity and thermal conductivity of air and of water or steam for various pressures and temperatures are given with the thermodynamic-property tables. The thermal conductivity and the viscosity for the saturated-liquid state are also tabulated for many fluids along with the thermodynamic-property tables earlier in this section. UNITS CONVERSIONS For this subsection the following units conversions are applicable: Diffusivity: to convert square centimeters per second to square feet per hour, multiply by 3.8750; to convert square meters per second to square feet per hour, multiply by 38,750. Pressure: to convert bars to pounds-force per square inch, multiply by 14.504. Temperature: °F = 9⁄5 °C + 32; °R = 9⁄5 K. Thermal conductivity: to convert watts per meter-kelvin to British thermal unit–feet per hour–square foot–degree Fahrenheit, multiply by 0.57779; and to convert British thermal unit–feet per hour–square foot–degree Fahrenheit to watts per meter-kelvin, multiply by 1.7307. Viscosity: to convert pascal-seconds to centipoises, multiply by 1000. ADDITIONAL REFERENCES An extensive coverage of the general pressure and temperature variation of thermal conductivity is given in the monograph by Vargaftik,

Filippov, Tarzimanov, and Totskiy, Thermal Conductivity of Liquids and Gases (in Russian), Standartov, Moscow, 1978, now published in English translation by CRC Press, Miami, FL. For a similar work on viscosity, see Stephan and Lucas, Viscosity of Dense Fluids, Plenum, New York and London, 1979. Tables and polynomial fits for refrigerants in both the gaseous and the liquid state are contained in ASHRAE Thermophysical Properties of Refrigerants, American Society of Heating, Refrigerating and Ventilating Engineers, Atlanta, GA, 1993. Other sources for viscosity include Fischer & Porter Co. catalog 10-A-94, “Fluid densities and viscosities,” 1953 (200 industrial fluids in 48 pp.) and van Velzen, D., R. L. Cardozo et al., EURATOM Ispra, Italy rept. 4735 e, 1972 (160 pp.). Liquid viscosity, 314 cpds, is summarized in I&EC Fundtls., 11 (1972): 20–26. Five hundred forty-nine binary and ternary systems are discussed in Skubla, P., Coll. Czech. Chem. Commun., 46 (1981): 303–339. See also Duhne, C. R., Chem. Eng. (NY), 86, 15 (July 16, 1979): 83–91 (equations and 326 liquids); and Rao, K. V. K., Chem. Eng. (NY), 90, 11 (May 30, 1983): 90–91 (nomograph, 87 liquids). For rheology, non-Newtonian behavior, and the like, see, for instance, Barnes, H., The Chem. Engr. (UK), (June 24, 1993): 17–23; Hyman, W. A., I&EC Fundtls., 16 (1976): 215–218; and Ferguson, J. and Z. Kemblowski, Applied Fluid Rheology, Elsevier, 1991 (325 pp.). Other sources for thermal conductivity include Ho, C. Y., R. W. Powell et al., J. Phys. Chem. Ref. Data, 1 (1972) and 3, suppl. 1 (1974); Childs, Ericks et al., N.B.S. Monogr. 131, 1973; Jamieson, D. T., J. B. Irving et al., Liquid Thermal Conductivity, H.M.S.O., Edinburgh, Scotland, 1975 (220 pp.).

TRANSPORT PROPERTIES TABLE 2-363

2-319

Transport Properties of Selected Gases at Atmospheric Pressure* Viscosity, 10−4 Pa⋅s Temperature, K

Thermal conductivity, W/(m⋅K) Temperature, K Substance

250

300

400

500

Acetone Acetylene Ammonia Argon Benzene

0.0080 0.0162 0.0197 0.0152 0.0077

0.0115 0.0213 0.0246 0.0177 0.0104

0.0201 0.0332 0.0364 0.0223 0.0195

0.0310 0.0452 0.0506 0.0264 0.0335

600

Bromine Butane CO2 CCl4 Chlorine

0.0038 0.0117 0.0129 0.0053 0.0071

0.0048 0.0160 0.0166 0.0067 0.0089

0.0067 0.0264 0.0244 0.0099 0.0124

0.0377 0.0323 0.0126 0.0156

Deuterium Ethane Ethylene Helium Heptane

0.122 0.0156 0.0152 0.134 0.0082

0.141 0.0218 0.0214 0.150 0.0120

0.176 0.0360 0.0342 0.180 0.0214

0.0516 0.0491 0.211 0.0325

0.0685 0.0653 0.247 0.0447

Hydrogen Methane Nitrogen Oxygen Pentane

0.156 0.0277 0.0222 0.0225 0.0107

0.182 0.0343 0.0260 0.0267 0.0152

0.221 0.0484 0.0325 0.0343 0.0250

0.256 0.0671 0.0386 0.0412 0.0362

0.291 0.0948 0.0441 0.0480

Propane Propylene R 11 R 12 R 13

0.0129 0.0114

0.0295 0.0226 0.0119 0.0151 0.0185

0.0417 0.0430

0.0072 0.0091

0.0183 0.0168 0.0078 0.0097 0.0121

0.0208 0.0248

R 21 R 22 SO2

0.0080 0.0078

0.0088 0.0109 0.0096

0.0135 0.0170 0.0143

0.0181 0.0230 0.0200

250

0.0561 0.0656 0.0301 0.0524

0.085 0.195

0.0403

0.126

0.0190

0.0580

0.0290 0.0256

Prandtl number, dimensionless Temperature, K

300

400

500

600

0.077 0.104 0.102 0.229 0.076

0.101 0.135 0.139 0.289 0.101

0.128 0.164 0.175 0.343 0.127

0.156

250

300

400

500

0.211 0.390 0.154

0.669

0.91 0.668

0.87 0.666

0.86 0.663

0.076 0.150 0.101 0.136

0.203 0.101 0.196 0.131 0.178

0.260 0.125 0.239 0.162 0.218

0.291 0.151 0.278 0.191 0.259

0.793

0.805 0.778

0.820 0.752

0.734

0.817 0.812 0.671

0.773 0.796 0.668

0.746 0.769 0.663

0.746 0.750 0.661

0.71 0.742 0.721

0.71 0.739 0.714

0.71 0.737 0.708

0.71 0.736 0.707

0.810 0.860

0.788 0.762 0.761 0.745 0.759

0.826

0.827 0.796

0.774 0.797 0.814 0.781 0.766

0.820

0.779 0.771

0.773 0.760

0.111 0.079 0.087 0.176

0.126 0.094 0.103 0.199

0.153 0.123 0.135 0.243 0.080

0.178 0.148 0.162 0.284 0.099

0.201 0.171 0.187 0.322 1.116

0.080 0.095 0.156 0.179

0.090 0.112 0.180 0.207

0.109 0.142 0.223 0.258

0.126 0.170 0.261 0.306

0.143 0.195 0.295 0.348

0.069 0.073 0.094 0.108 0.123

0.082 0.087 0.110 0.126 0.145

0.108 0.115 0.144 0.162 0.190

0.131 0.141

0.100 0.109

0.115 0.129 0.129

0.154 0.168 0.175

0.217

0.708 0.757

0.256

*An approximate interpolation scheme is to plot the logarithm of the viscosity or the thermal conductivity versus the logarithm of the absolute temperature. At 250 K the viscosity of gaseous argon is to be read as 1.95 × 10−5 Pa⋅s = 0.0000195 Ns/m2.

2-320

PHYSICAL AND CHEMICAL DATA

TABLE 2-364

Viscosities of Gases: Coordinates for Use with Fig. 2-32* µ × 107 p

Gas

X

Y

Acetic acid Acetone Acetylene Air Ammonia Amylene (β) Argon Arsine Benzene Bromine Butane (n) Butane (iso) Butyl acetate (iso) Butylene (α) Butylene (β) Butylene (iso) Butyl formate (iso) Cadmium Carbon dioxide Carbon disulfide Carbon monoxide Carbon oxysulfide Carbon tetrachloride Chlorine Chloroform Cyanogen Cyclohexane Cyclopropane Deuterium Diethyl ether Dimethyl ether Diphenyl ether Diphenyl methane Ethane Ethanol Ethyl acetate Ethyl chloride Ethylene Ethyl propionate Fluorine Freon-11 Freon-12 Freon-14 Freon-21 Freon-22 Freon-113 Freon-114 Helium Heptane (n) Hexane (n) Hydrogen Hydrogen-helium 10% H2, 90% He 25% H2, 75% He 40% H2, 60% He 60% H2, 40% He 81% H2, 19% He

7.0 8.4 9.3 10.4 8.4 8.6 9.7 8.6 8.7 8.8 8.6 8.6 5.7 8.4 8.7 8.3 6.6 7.8 8.9 8.5 10.5 8.2 8.0 8.8 8.8 8.2 9.0 8.3 11.0 8.8 9.0 8.6 8.0 9.0 8.2 8.4 8.5 9.5 12.0 7.3 8.6 9.0 9.5 9.0 9.0 11.0 9.4 11.3 8.6 8.4 11.3

14.6 13.2 15.5 20.4 16.0 12.2 22.6 20.0 13.2 19.4 13.2 13.2 16.3 13.5 13.1 13.9 16.0 22.5 19.1 15.8 20.0 17.9 15.3 18.3 15.7 16.2 12.2 14.7 16.2 12.7 15.0 10.4 10.3 14.5 14.5 13.4 15.6 15.2 12.4 23.8 16.2 17.4 20.4 16.7 17.7 14.0 16.4 20.8 10.6 12.0 12.4

825 (50°C) 735 1017 1812 1000 676 2215 1576 746 1495 735 744 778 761 746 786 840 5690 (500) 1463 990 1749 1220 966 1335 1000 1002 701 870 1240 730 925 610 (50) 605 (50) 915 835 743 978 1010 890 2250 1298 (93) 1496 (93) 1716 1389 (93) 1554 (93) 1166 (93) 1364 (93) 1946 618 (50) 644 880

11.0 11.0 10.7 10.8 10.5

20.5 19.4 18.4 16.7 15.0

1780 (0) 1603 (0) 1431 (0) 1227 (0) 1016 (0)

Ref.

Gas

X

Y

µ × 107 p

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 3 5 3 3 3 3 1 1 1 1 1

Hydrogen–sulfur dioxide 10% H2, 90% SO2 20% H2, 80% SO2 50% H2, 50% SO2 80% H2, 20% SO2 Hydrogen bromide Hydrogen chloride Hydrogen cyanide Hydrogen iodide Hydrogen sulfide Iodine Krypton Mercury Mercuric bromide Mercuric chloride Mercuric iodide Mesitylene Methane Methane (deuterated) Methanol Methyl acetate Methyl acetylene 3-Methyl-1-butene Methyl butyrate (iso) Methyl bromide Methyl chloride 3-Methylene-1-butene Methylene chloride Methyl formate Neon Nitric oxide Nitrogen Nitrous oxide Nonane (n) Octane (n) Oxygen Pentane (n) Pentane (iso) Phosphene Propane Propanol (n) Propanol (iso) Propyl acetate Propylene Pyridine Silane Stannic chloride Stannic bromide Sulfur dioxide Thiazole Thiophene Toluene 2,2,3-Trimethylbutane Trimethylethane Water Xenon Zinc

8.7 8.6 8.9 9.7 8.4 8.5 7.1 8.5 8.4 8.7 9.4 7.4 8.5 7.7 8.4 9.5 9.5 9.5 8.3 8.4 8.9 8.0 6.6 8.1 8.5 8.0 8.5 5.1 11.1 10.4 10.6 9.0 9.2 8.8 10.2 8.5 8.9 8.8 8.9 8.4 8.4 8.0 8.5 8.6 9.0 9.1 9.0 8.4 10.0 8.3 8.6 10.0 8.0 8.0 9.3 8.0

18.1 18.2 18.3 17.7 21.6 19.2 14.5 21.5 18.0 18.7 24.0 24.9 19.0 18.7 18.0 10.2 15.8 17.6 15.6 14.0 14.3 13.3 15.8 18.7 16.5 13.3 15.8 18.0 25.8 20.8 20.0 19.0 8.9 9.8 21.6 12.3 12.1 17.0 13.5 13.5 13.6 14.3 14.4 13.3 16.8 16.0 16.7 18.2 14.4 14.2 12.5 10.4 13.0 16.0 23.0 22.0

1259 (17) 1277 (17) 1332 (17) 1306 (17) 1843 1425 737 1830 1265 1730 (100) 2480 4500 (200) 2253 2200 (200) 2045 (200) 660 (50) 1092 1290 935 870 (50) 867 716 824 1327 1062 716 989 923 3113 1899 1766 1460 554 (50) 586 (50) 2026 668 685 1150 800 770 774 797 840 830 (50) 1148 1330 (100) 142 (100) 1250 958 901 (50) 686 691 (50) 686 1250 (100) 2255 5250 (500)

Ref. 4

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

*Viscosity at 20°C unless otherwise indicated. From Beerman, Meas. Control (June 1982): 154–157. References: 1. I. F. Golubev, Viscosity of Gases and Gas Mixtures, Moscow 1959; transl. U.S. Department of Commerce, Clearinghouse for Federal Scientific and Technical Information, Springfield, Va., TT 70-50022, ISPT Cat. No. 5680, Table 4, Jerusalem 1970. 2. R. H. Perry and C. H. Chilton, Chemical Engineers’ Handbook, 5th ed., McGraw-Hill, New York, 1973, pp. 3-210, 3-211. 3. Ibid., Table 3-282, p. 3-210. 4. By interpolation of data in Ref. 1. 5. Thermophysical Properties of Refrigerants, American Society of Heating, Refrigerating and Air-Conditioning Engineers, New York. 6. N. A. Lange, Handbook of Chemistry, 4th ed., Handbook Publishers, Sandusky, Ohio, 1941. For another alignment chart for 165 hydrocarbons from −100 to 500°C, see Sastry, R. C. and A. Satyanarayan, Chem. Industry Devs. (July 1978): 11–14.

TRANSPORT PROPERTIES

Nomograph for determining (a) absolute viscosity of a gas as a function of temperature near ambient pressure and (b) relative viscosity of a gas compared with air. For coordinates see Table 2-364. To convert poises to pascal-seconds, multiply by 0.1. [From Beerman, Meas. Control, 154–157 (June 1982).]

FIG. 2-32

2-321

2-322

PHYSICAL AND CHEMICAL DATA

TABLE 2-365

Viscosities of Liquids: Coordinates for Use with Fig. 2-33

Liquid

X

Y

Liquid

X

Y

Acetaldehyde Acetic acid, 100% Acetic acid, 70% Acetic anhydride Acetone, 100% Acetone, 35% Acetonitrile Acrylic acid Allyl alcohol Allyl bromide Allyl iodide Ammonia, 100% Ammonia, 26% Amyl acetate Amyl alcohol Aniline Anisole Arsenic trichloride Benzene Brine, CaCl2, 25% Brine, NaCl, 25% Bromine Bromotoluene Butyl acetate Butyl acrylate Butyl alcohol Butyric acid Carbon dioxide Carbon disulfide Carbon tetrachloride Chlorobenzene Chloroform Chlorosulfonic acid Chlorotoluene, ortho Chlorotoluene, meta Chlorotoluene, para Cresol, meta Cyclohexanol Cyclohexane Dibromomethane Dichloroethane Dichloromethane Diethyl ketone Diethyl oxalate Diethylene glycol Diphenyl Dipropyl ether Dipropyl oxalate Ethyl acetate Ethyl acrylate Ethyl alcohol, 100% Ethyl alcohol, 95% Ethyl alcohol, 40% Ethyl benzene Ethyl bromide 2-Ethyl butyl acrylate Ethyl chloride Ethyl ether Ethyl formate 2-Ethyl hexyl acrylate Ethyl iodide Ethyl propionate Ethyl propyl ether Ethyl sulfide Ethylene bromide Ethylene chloride Ethylene glycol Ethylidene chloride Fluorobenzene Formic acid Freon-11 Freon-12 Freon-21 Freon-22

15.2 12.1 9.5 12.7 14.5 7.9 14.4 12.3 10.2 14.4 14.0 12.6 10.1 11.8 7.5 8.1 12.3 13.9 12.5 6.6 10.2 14.2 20.0 12.3 11.5 8.6 12.1 11.6 16.1 12.7 12.3 14.4 11.2 13.0 13.3 13.3 2.5 2.9 9.8 12.7 13.2 14.6 13.5 11.0 5.0 12.0 13.2 10.3 13.7 12.7 10.5 9.8 6.5 13.2 14.5 11.2 14.8 14.5 14.2 9.0 14.7 13.2 14.0 13.8 11.9 12.7 6.0 14.1 13.7 10.7 14.4 16.8 15.7 17.2

4.8 14.2 17.0 12.8 7.2 15.0 7.4 13.9 14.3 9.6 11.7 2.0 13.9 12.5 18.4 18.7 13.5 14.5 10.9 15.9 16.6 13.2 15.9 11.0 12.6 17.2 15.3 0.3 7.5 13.1 12.4 10.2 18.1 13.3 12.5 12.5 20.8 24.3 12.9 15.8 12.2 8.9 9.2 16.4 24.7 18.3 8.6 17.7 9.1 10.4 13.8 14.3 16.6 11.5 8.1 14.0 6.0 5.3 8.4 15.0 10.3 9.9 7.0 8.9 15.7 12.2 23.6 8.7 10.4 15.8 9.0 5.6 7.5 4.7

Freon-113 Glycerol, 100% Glycerol, 50% Heptane Hexane Hydrochloric acid, 31.5% Iodobenzene Isobutyl alcohol Isobutyric acid Isopropyl alcohol Isopropyl bromide Isopropyl chloride Isopropyl iodide Kerosene Linseed oil, raw Mercury Methanol, 100% Methanol, 90% Methanol, 40% Methyl acetate Methyl acrylate Methyl i-butyrate Methyl n-butyrate Methyl chloride Methyl ethyl ketone Methyl formate Methyl iodide Methyl propionate Methyl propyl ketone Methyl sulfide Napthalene Nitric acid, 95% Nitric acid, 60% Nitrobenzene Nitrogen dioxide Nitrotoluene Octane Octyl alcohol Pentachloroethane Pentane Phenol Phosphorus tribromide Phosphorus trichloride Propionic acid Propyl acetate Propyl alcohol Propyl bromide Propyl chloride Propyl formate Propyl iodide Sodium Sodium hydroxide, 50% Stannic chloride Succinonitrile Sulfur dioxide Sulfuric acid, 110% Sulfuric acid, 100% Sulfuric acid, 98% Sulfuric acid, 60% Sulfuryl chloride Tetrachloroethane Thiophene Titanium tetrachloride Toluene Trichloroethylene Triethylene glycol Turpentine Vinyl acetate Vinyl toluene Water Xylene, ortho Xylene, meta Xylene, para

12.5 2.0 6.9 14.1 14.7 13.0 12.8 7.1 12.2 8.2 14.1 13.9 13.7 10.2 7.5 18.4 12.4 12.3 7.8 14.2 13.0 12.3 13.2 15.0 13.9 14.2 14.3 13.5 14.3 15.3 7.9 12.8 10.8 10.6 12.9 11.0 13.7 6.6 10.9 14.9 6.9 13.8 16.2 12.8 13.1 9.1 14.5 14.4 13.1 14.1 16.4 3.2 13.5 10.1 15.2 7.2 8.0 7.0 10.2 15.2 11.9 13.2 14.4 13.7 14.8 4.7 11.5 14.0 13.4 10.2 13.5 13.9 13.9

11.4 30.0 19.6 8.4 7.0 16.6 15.9 18.0 14.4 16.0 9.2 7.1 11.2 16.9 27.2 16.4 10.5 11.8 15.5 8.2 9.5 9.7 10.3 3.8 8.6 7.5 9.3 9.0 9.5 6.4 18.1 13.8 17.0 16.2 8.6 17.0 10.0 21.1 17.3 5.2 20.8 16.7 10.9 13.8 10.3 16.5 9.6 7.5 9.7 11.6 13.9 25.8 12.8 20.8 7.1 27.4 25.1 24.8 21.3 12.4 15.7 11.0 12.3 10.4 10.5 24.8 14.9 8.8 12.0 13.0 12.1 10.6 10.9

TRANSPORT PROPERTIES

Nomograph for viscosities of liquids at 1 atm. For coordinates see Table 2-365. To convert centipoises to pascalseconds, multiply by 0.001.

FIG. 2-33

2-323

2-324

PHYSICAL AND CHEMICAL DATA

TABLE 2-367

TABLE 2-366

Viscosity of Sucrose Solutions* Viscosity in centipoises

Percentage sucrose by weight

Temp., °C

20

40

0 5 10 15 20 25 30 35 40 45

3.818 3.166 2.662 2.275 1.967 1.710 1.510 1.336 1.197 1.074

14.82 11.60 9.830 7.496 6.223 5.206 4.398 3.776 3.261 2.858

60

Temp., °C

113.9 74.9 56.7 44.02 34.01 26.62 21.30 17.24

50 55 60 65 70 75 80 85 90 95

Percentage sucrose by weight 20

40

60

0.974 0.887 0.811 0.745 0.688 0.637 0.592 0.552

2.506 2.227 1.989 1.785 1.614 1.467 1.339 1.226 1.127 1.041

14.06 11.71 9.87 8.37 7.18 6.22 5.42 4.75 4.17 3.73

*International Critical Tables, vol. 5, p. 23. Bingham and Jackson, Bur. Standards Bull. 14 (1919): 59.

FIG. 2-34 and TABLE 2-367

Nomograph (right) for thermal conductivity of organic liquids.

TRANSPORT PROPERTIES TABLE 2-368

2-325

Prandtl Number of Air* Pressure, bar

Temperature, K

1

5

10

20

30

40

50

60

70

80

90

100

80 90 100 120 140

mix 0.796 0.786 0.773 0.763

2.31 1.76 0.872 0.813 0.782

2.32 1.77 1.54 0.89 0.82

2.35 1.78 1.53 1.44 0.94

2.37 1.79 1.53 1.65 1.20

2.40 1.81 1.53 1.54 1.59

2.42 1.82 1.53 1.48 2.14

2.45 1.83 1.53 1.43 2.43

2.48 1.85 1.53 1.40 2.07

2.51 1.87 1.54 1.38 1.78

2.54 1.89 1.54 1.36 1.62

2.57 1.91 1.55 1.34 1.52

160 180 200 240 280

0.754 0.745 0.738 0.724 0.710

0.765 0.754 0.743 0.727 0.711

0.78 0.763 0.749 0.729 0.713

0.84 0.792 0.766 0.737 0.717

0.92 0.830 0.788 0.746 0.721

1.03 0.876 0.812 0.756 0.726

1.13 0.932 0.841 0.767 0.731

1.25 1.00 0.87 0.78 0.737

1.37 1.07 0.90 0.80 0.742

1.65 1.14 0.95 0.81 0.75

1.83 1.20 0.97 0.81 0.75

1.72 1.25 1.00 0.82 0.76

300 350 400 450 500

0.705 0.699 0.694 0.691 0.689

0.707 0.699 0.694 0.691 0.689

0.708 0.699 0.694 0.691 0.689

0.712 0.701 0.695 0.691 0.689

0.715 0.703 0.696 0.692 0.689

0.717 0.705 0.697 0.692 0.690

0.721 0.707 0.698 0.693 0.690

0.725 0.709 0.699 0.693 0.690

0.728 0.711 0.700 0.694 0.690

0.732 0.712 0.701 0.695 0.691

0.737 0.714 0.703 0.695 0.691

0.742 0.716 0.704 0.696 0.691

600 700 800 900 1000

0.690 0.696 0.705 0.709 0.711

0.690 0.696 0.704 0.709 0.711

0.690 0.695 0.704 0.708 0.711

0.689 0.695 0.704 0.708 0.711

0.689 0.695 0.704 0.708 0.711

0.689 0.695 0.703 0.708 0.710

0.689 0.695 0.703 0.708 0.710

0.689 0.695 0.703 0.708 0.710

0.689 0.695 0.703 0.708 0.710

0.690 0.695 0.702 0.708 0.709

0.690 0.695 0.702 0.708 0.709

0.690 0.695 0.702 0.708 0.709

*Compiled by P. E. Liley from tables of specific heat at constant pressure, thermal conductivity, and viscosity given in SI units for integral kelvin temperatures and pressures in bars by Vasserman. Thermophysical Properties of Air and Its Components and Thermophysical Properties of Liquid Air and Its Components. Nauka, Moscow, and in translated form by the National Bureau of Standards, Washington. The number of significant figures given above reflects the similar numbers appearing for the constituent properties in the source references. While reasonable agreement occurs for atmospheric pressure with some other works, the fragmentary data available for the saturated, etc., states show large deviations.

TABLE 2-369

Prandtl Number of Liquid Refrigerants* Temperature, K Refrigerant

Trichlorofluoromethane Dichlorodifluoromethane Chlorotrifluoromethane Bromotrifluoromethane Dichlorofluoromethane Chlorodifluoromethane Methyl chloride Trichlorotrifluoroethane Dichlorotetrafluoroethane Chloropentafluoroethane Ethane Propane Octafluorocyclobutane Dichlorodifluoromethane/difluoroethane Chlorodifluoromethane/chloropentafluoroethane Trifluoromethane/chlorotrifluoromethane Methylene fluoride/chloropentafluoroethane Butane Isobutane (2-methyl propane) Ammonia Water Ethylene Propylene

No. 11 12 13 13B1 21 22 40 113 114 115 170 290 C318 500 502 503 504 600 600a 717 718 1150 1270

180

200

220

240

260

280

300

11.9 5.25 2.96 3.75

8.64 4.27 2.67 3.27 5.72

6.73 3.65 2.69 2.94 4.50

5.33 3.27 3.05 2.83 3.87

4.74 3.08 3.57 3.03 3.48

4.18 3.04 3.61 3.25

2.93 2.42

2.79 2.40

2.77 2.45

2.87 2.60

— 15.13 7.85

3.23 2.53 — 11.18 6.16

8.59 5.21

6.94 4.67

5.77 4.40

5.06 4.46

2.55 5.28 —

2.29 4.46 — 5.78 5.73

2.22 3.88 — 4.23 4.71

2.40 3.44 11.2 3.40 4.13

2.70 3.16 8.74 3.13 3.81

—

—

3.02 7.35 3.01

3.16 6.37 3.13

5.87 3.35

5.96 3.72

2.10

2.09 4.90 6.19 8.26

2.24 3.60 5.20 6.36

2.43 3.04 4.44 5.18 1.97

2.89 2.79 3.83 4.49 1.76

2.69 3.44 3.93 1.54

— 2.85 3.22 3.66 1.40

— 3.30 3.07 3.53 1.29

— — 3.02 3.53 1.24

— 1.74 2.24

— 1.78 1.88

— 2.07 1.71

— 2.70 1.71

10.3 4.4 1.88

5.69 — 2.24

3.65 — 3.91

2.60 — 4.73

7.00 4.80 4.68 — 25.7 —

8.35

3.76

— — 1.85 3.80

320

340

360

380

3.19 — 4.52 3.16

3.44 —

4.00 — —

— —

3.18 2.85 7.04 4.78 4.90

3.54

3.17

6.23 4.82

— 5.61

5.18

—

—

—

—

—

— — —

— —

— —

3.77 1.25

4.68 1.34

1.99 —

1.59 — —

*Dashes indicate inaccessible states. Average uncertainty is about 20 percent. Values derived from formulations for thermal conductivity, specific heat at constant pressure, and viscosity contained in Thermophysical Properties of Refrigerants. American Society of Heating, Refrigerating and Air-Conditioning Engineers, New York, 1976. For further details see M. W. Johnson, M.S.M.E. thesis, Purdue University, West Lafayette, Ind., 1976.

2-326

PHYSICAL AND CHEMICAL DATA

TABLE 2-370

Thermophysical Properties of Miscellaneous Saturated Liquids Temperature, °C

Substance

Property

Acetaldehyde ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr Acetic acid

−50

−40

−30

−20

−10

0

863 2.05 460 0.211 4.47

852 2.08 404 0.206 4.08

840 2.11 358 0.200 3.78

828 2.14 321 0.195 3.52

816 2.17 290 0.189 3.33

804 2.20 263 0.184 3.14

10

20

794 2.24 241 0.182 2.97

783 2.28 222 0.180 2.81

ρ (kg/m3) cp (kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

1049 2.031 1210 0.173 14.2

30

40

50

60

1039

1028

1018

1006

1102 0.170

1010 0.168

795 0.167

600 0.165

70

80

90

100

995

984

972

960

0.163

0.161

Aniline

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

— — — — —

— — — — —

— — — — —

— — — — —

— — — — —

1039 2.024 10200 0.186 111

1030 2.047 6500 0.184 72

1022 2.071 4400 0.182 50

1013 2.093 3160 0.180 36.7

1005 2.113 2370 0.177 28.3

996 2.132 1850 0.174 22.7

987 2.17 1510 0.171 19.2

978 2.20 1270 0.169 16.5

969 2.23 1090 0.168 14.5

960 2.27 935 0.167 12.7

951 2.32 825 0.167 11.5

Butanol

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

845 1.947 34700 0.175 3860

841 1.996 22400 0.174 2570

837 2.046 14700 0.173 1740

833 2.100 10300 0.172 1260

829 2.153 7400 0.171 930

825 2.202 5190 0.170 670

817 2.262 3870 0.168 120

810 2.345 2950 0.167 41

803 2.437 2300 0.166 33.8

797 2.524 1780 0.165 27.2

791 2.621 1410 0.164 22.5

784

776

768

760

753

1140 0.163

930 0.162

760 630 535 0.161 0.160 0.159

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

1362 0.988 630 0.194 3.21

1348 0.989 580 0.190 3.02

1334 0.990 535 0.186 2.85

1320 0.991 496 0.182 2.70

1306 0.993 463 0.178 2.58

1292 0.996 435 0.174 2.49

1278 1.004 405 0.170 2.39

1263 1.017 375 0.166 2.30

350 0.161

330 0.158

0.156

0.154

0.152

0.150

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

— — — — —

— — — — —

— — — — —

— — — — —

— — — — —

— — — — —

789 2.068 1175 0.122 19.9

779 2.081 980 0.120 17.0

769 2.094 820 0.119 14.4

759 2.106 710 0.118 12.7

750 2.119 605 0.117 11.0

740

731

721

540 0.116

0.114

0.112

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

2.01 6400 0.188 68.4

2.04 4790 0.186 52.5

2.08 3650 0.184 41.3

2.13 2825 0.181 33.2

2.19 2220 0.179 27.2

806 2.27 1770 0.177 22.7

798 2.35 1470 0.175 19.7

789 2.43 1200 0.173 16.9

781 2.52 1000 0.171 14.7

776 2.62 835 0.168 13.0

763 2.73 700 0.165 11.6

754 2.83 590 0.162 10.3

745 2.93 500 0.159 9.2

735 3.03 435 0.156 8.4

725 3.19 370 0.153 7.7

716 3.30 314 0.151 6.9

947

935

924

912

888

876

863

851

838

825

811

797

580

510

901 2.01 455 0.145 6.3

400 0.142

370 0.139

345 0.136

310 0.133

280 0.130

250 230 220 0.127 0.123 0.119

Carbon disulfide

Cyclohexane

Ethanol

Ethyl acetate

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) 1090 k (W/m⋅K) Pr

Ethylamine

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

761 2.95 580 0.204 8.39

750 2.97 500 0.201 7.39

739 2.98 435 0.199 6.51

729 3.00 390 0.196 5.97

718 3.01 350 0.194 5.43

707 3.03 320 0.191 5.08

695

683

671

658

646

633

620

607

Ethyl ether

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

790 2.135 550 0.159 7.39

780 2.156 470 0.155 6.54

769 2.179 410 0.151 5.92

758 2.205 365 0.147 5.48

747 2.233 330 0.144 5.12

736 2.265 290 0.140 4.69

725 2.299 265 0.139 4.38

714 2.332 233 0.134 4.05

702 2.36 214 0.129 3.92

689 2.39 197 0.125 3.77

676 2.43 181 0.120 3.67

666 2.47 166 0.116 3.54

653 2.51 153 0.112 3.43

640

625

611

140

129

118

Ethyl iodide

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

0.656 0.663 0.670 0.677 730 0.092 5.37

0.684 655 0.090 4.98

0.691 590 0.088 4.63

0.698 539 0.086 4.30

0.705 495 0.085 4.11

0.712 455 0.083 3.90

0.718 420 0.081 3.72

0.724 390 0.080 3.53

Ethylene glycol

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

1127 2.272 57000 0.254 510

1120 2.327 33300 0.255 305

1113 2.381 20200 0.256 190

1106 2.431 13400 0.258 126

1099 2.484 9100 0.259 87.3

1092 2.536 7070 0.260 69.0

1085 2.586 4000

1077 2.636 3450

1070 1063 1056 2.685 2.734 2.779 3000 2440 2000

Formic acid

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

1241

1231

1220

1209

1196

1184

1170

1156

1140

0.265

2260 0.261

1800 0.257

1470 0.257

1220 0.253

1030 0.250

890 0.246

780 0.243

680 615 550 0.240 0.236 0.232

1124

1108

TRANSPORT PROPERTIES TABLE 2-370

2-327

Thermophysical Properties of Miscellaneous Saturated Liquids (Concluded ) Temperature, °C

Substance Gasoline

Glycerol

Kerosine

Property

−50

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) 1710 k (W/m⋅K) 0.131 Pr ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

—

−40

1400 0.128 —

−30

−20

784 1.88 1170 990 0.125 0.123 15.1 —

—

−10

0

775 1.92 850 0.121 13.5 —

10

20

30

40

50

60

70

80

90

100

767 1.97 735 0.120 12.1

759 2.02 645 0.118 11.0

751 2.06 530 0.116 9.41

743 2.11 464 0.114 8.59

735 2.15 410 0.112 7.87

721 2.20 367 0.110 7.34

717 2.25 330 0.108 6.88

708 2.30 298 0.106 6.47

699 2.35 270 0.104 6.10

690 2.41 246 0.102 5.81

681 2.46 225 0.100 5.54

1276

1270

1260 2.393 1.5.+6 0.284 12650

1254 2.406

1248 2.457

1242 2.504

2.548

2.588

2.625 2.657 2.686

0.285

0.287

0.288

0.289

0.291

0.293 0.294 0.295

2.28 73

2.32 66

2.35 60

2.38 55

1.2.+7 4.0.+6

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) 1150 k (W/m⋅K) Pr

725

500

360

275

781 1.91 215 0.140 2.93

774 1.96 173 0.139 2.44

767 2.02 149 0.139 2.17

760 2.07 126 0.138 1.89

754 2.13 108 0.138 1.67

748 2.18 95 0.137 1.51

742 2.23 83 0.137 1.35

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k(W/m⋅K) Pr

2.30 2305 0.225 23.6

2.32 1800 0.222 18.8

2.35 1410 0.219 15.1

2.37 1170 0.216 12.9

2.40 975 0.212 11.0

2.42 820 0.209 9.53

2.45 692 0.206 8.23

2.47 590 0.203 7.18

783 2.49 510 0.199 6.38

774 2.52 455 0.195 5.88

766 2.55 400 0.192 5.31

756 2.65 355 0.189 4.98

746 2.78 315 0.187 4.68

736 2.94 271 0.184 4.34

725 3.13 240 0.182 4.13

711 3.30 218 0.180 3.99

Methyl formate

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

1069 1.84 830 0.217 7.04

1056 1.86 711 0.213 6.21

1043 1.88 618 0.209 5.56

1030 1.90 544 0.205 5.04

1017 1.92 481 0.200 4.62

1003 1.95 430 0.195 4.30

989 1.99 380 0.191 3.96

975 2.03 345 0.186 3.77

960 2.08 315 0.180 3.64

944

929

913

897

880

863

845

Oil, castor

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

Methanol

Oil, olive

Pentane

Propanol

Sulfuric acid

2,420,000 986,000 451,000 231,000 125,000 74,000 43,000 0.182 0.181 0.180 0.179 0.178 0.177 0.176 0.175 0.174 0.17

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

138,000 0.170

914 1.633 84,000 0.169 810

52,000 0.168

36,300 0.167

24,500 0.166

17,000 12,400 0.166 0.165 0.165 0.164 0.164

616

606

596

585

574

562

209 0.115

190 0.112

175 0.108

161 0.105

148 0.101

137 124 113 0.098 0.095 0.091

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

693 2.060 489 0.142 7.14

665 2.137 339 0.132 5.49

656 2.167 307 0.128 5.20

646 2.206 279 0.125 4.92

636 2.239 254 0.122 4.66

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

849 1.955 20,200 13,500 9500 6900 0.167 0.166 0.165 236

819 2.219 3900

811

814

796

788

779

770

761

752

5110

2900

2245

1720 0.171

1400 0.169

1130 0.168

921 0.167

760 0.165

630 508 447 0.164 0.163 0.162

1834 1.382 25,400

15,700

11,500

8820

7220

6090

5190

829 1.80 380 0.124 5.5

820 1.83 355 0.122 5.3

810 1.87 325 0.119 5.1

820

730

675

684 2.084 428 0.139 6.42

674 2.110 379 0.136 5.88

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

Toluene

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

Turpentine

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

48,400 35,200 0.314 932 1.514 2120 0.152 21.1

923 1.535 1670 0.149 17.8

913 1.556 1345 0.147 14.2

904 1.579 1100 0.144 12.1

895 1.602 915 0.142 10.3

626 2.273 234 0.119 4.47

886 1.633 770 0.139 9.0

876 1.652 670 0.137 8.1

867 1.675 590 0.134 7.4

858 1.701 520 0.132 6.7

848 1.73 470 0.129 6.3

839 1.76 420 0.126 5.9

1.72 2250 0.130 29.8

1.76 1780 0.129 24.3

1.80 1490 0.128 20.9

1270 0.127 18.4

1070 0.126 16.1

1.93 925 0.125 14.3

550

747

800 1.92 295 0.117 4.8

538

743

790 1.97 270 0.114 4.7

2-328

PHYSICAL AND CHEMICAL DATA

TABLE 2-371

Diffusivities of Pairs of Gases and Vapors (1 atm) Dv in cm2/s

Substance Acetic acid Acetone n-Amyl alcohol sec-Amyl alcohol Amyl butyrate Amyl formate i-Amyl formate Amyl isobutyrate Amyl propionate Aniline Anthracene Argon Benzene Benzidine Benzyl chloride n-Butyl acetate i-Butyl acetate n-Butyl alcohol i-Butyl alcohol Butyl amine i-Butyl amine i-Butyl butyrate i-Butyl formate i-Butyl isobutyrate i-Butyl proprionate i-Butyl valerate Butyric acid i-Butyric acid Cadmium Caproic acid i-Caproic acid Carbon dioxide

Carbon disulfide Carbon monoxide Carbon tetrachloride Chlorobenzene Chloroform Chloropicrin m-Chlorotoluene o-Chlorotoluene p-Chlorotoluene Cyanogen chloride Cyclohexane n-Decane Diethylamine 2,3-Dimethyl butane Diphenyl n-Dodecane Ethane Ethanol Ether (diethyl) Ethyl acetate Ethyl alcohol Ethyl benzene Ethyl n-butyrate Ethyl i-butyrate Ethylene Ethyl formate Ethyl propionate Ethyl valerate Eugenol Formic acid Helium n-Heptane n-Hexane Hexyl alcohol Hydrogen

Temp., °C 0 0 0 30 0 0 0 0 0 0 30 0 20 0 0 0 0 0 0 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 20 25 500‡ 0 0 450‡ 0 30 0 25 0 0 0 0 15 45 90 0 15 0 126 0 0 0 0 30 0 0 0 0 0 0 0 0 0 0 0 20 38 15 0 0 25 500

Air

A

H2

0.1064 .109 .0589 .072 .040 .0543 .058 .0419 .046 .0610 .075 .0421

0.416 .361 .235

.077 .0298 .066 .058 .0612 .0703 .088 .0727 .0821 .0853 .0468 .0705 .0457 .0529 .0424 .067 .0679

.306

O2

N2

CO2

N2O

CH4

C2H6

C2H4

n-C4H10

i-C4H10

0.0716 .0422

.171 .1914

.0347

0.194 0.0797

.0528

.2364 .2716

.0425 .0476

.2771

.0483

.185

.0327

.191 .203 .173 .264 .271

.0364 .0366 .0308 .0476 .0471 .17

.050 .0513 .138

.550

.139

0.096

0.153

.163 .0996*

.00215†

.9 .0892

.369 .651 .293

.185 1.0 0.0636

.319

.0744

.063 .137

0.116

.075 .091 .088 .054 .059 .051 .111 0.0719

.0760

.086 .306

.0841

.0884 .0657

.301

.0753

.0751

.0610 .308 .459 .377 .298 .273

.0778 .0715 .089 .102 .0658 .0579 .0591 .0840 .068 .0512 .0377 .1308

.0813 .0686 .0546 .0487

.375

.0685

.224 .229 .486 .337 .236 .205

.0407 .0413 .0573 .0450 .0367

.510

Ref. 8 6, 16 8 5 8 8 8 8 8 8 5 8 18 8, 15 8 8 8 8 8 5 8 8 8 8 8 8 8 8 8 8 13 8 8 8 19 1, 9 18 8 8 18 16, 17 5 6 10 8 8 8 10 3 6 3 8 3 8 3 8 20 7, 8 8 5 8 8 8 8 8 8 4, 8 8 8 8 8 19

.0874

.641 .705 .066§ .0663 .0499 .611

.290 .200

.0753 .697 4.2

.0757 .674

.0351 .550 .646

.535

.625

0.459 .537

0.486 .726

0.272

0.277

3 8 8 2 18

TRANSPORT PROPERTIES TABLE 2-371

2-329

Diffusivities of Pairs of Gases and Vapors (1 atm) (Concluded ) Dv in cm2/s

Substance

Temp., °C

Air

Hydrogen cyanide Hydrogen peroxide Iodine Mercury Mesitylene Methane Methyl acetate Methyl alcohol Methyl butyrate Methyl i-butyrate Methyl cyclopentane Methyl formate Methyl propionate Methyl valerate Naphthalene Nitrogen

0 60 0 0 0 500 0 0 0 0 15 0 0 0 0 0 25 0 0 30 0 0 0 0 0 0 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 25 0 30 0

.173 .188 .07 .112 .056

Nitrous oxide n-Octane Oxygen Phosgene Propionic acid Propyl acetate n-Propyl alcohol i-Propyl alcohol n-Propyl benzene i-Propyl benzene n-Propyl bromide i-Propyl bromide Propyl butyrate Propyl formate n-Propyl iodide i-Propyl iodide n-Propyl isobutyrate i-Propyl isobutyrate Propyl propionate Propyl valerate Safrol i-Safrol Sulfur hexafluoride Toluene Trimethyl carbinol 2,2,4-Trimethyl pentane 2,2,3-Trimethyl heptane n-Valeric acid i-Valeric acid Water

H2

O2

N2

CO2

N2O

CH4

C2H6

C2H4

n-C4H10

i-C4H10

.070 .13

.53 1.1

.084 .132 .0633 .0639 .0731 .0872 .0735 0.0569 .0513

.333 .506 .242 .257 .318

.0567 .0879 .0446 .0451 0.0742

.0758

.295

.0528 0.181 0.165 .096

0.535 .0505 0.0642 .178 .095 .0829 .067 .085 .0818 .101 .0481 .0489 .085 .0902 .0530 .0712 .079 .0802 .0549 .059 .057 .0466 .0434 .0455

.271 .697

.0705

0.0710 .181

.139

.330

.0588

.315

.0577

.206 .281

.0364 .0490

.212

.0388

.212 .189

.0395 .0341

.418 .076 .088 .087

30 90 0 0 0 450

A

.071

.0618

.288

.0688

.270 .050 .0544 .220

.212 .75

0.148

0.163

0.0960

0.0908

Ref. 10 11 8, 12, 14 8, 12, 13 8 18 8 8 8 8 3 8 8 8 8 8 2 8 8 3 8 10 8 8 8 8 5 8 8 8 8 8 8 8 8 8 8 8 8 8 8 2 4, 8 5 8

.0705

3

.0684

3 8 8 8, 20 18

.0376 .138 1.3

* 320 mm Hg. † 40 atm. ‡ Also at other temperatures. § Strong function of concentration. References 1 Amdur, Irvine, Mason, and Ross, J. Chem. Phys., 20, 436 (1952). 2 Boyd, Stein, Steingrimsson, and Rumpel, J. Chem. Phys., 19, 548 (1951). 3 Cummings and Ubbelohde, J. Chem. Soc. (London), 1953, p. 3751. 4 Fairbanks and Wilke, Ind. Eng. Chem., 42, 471 (1950). 5 Gilliland, Ind. Eng. Chem., 26, 681 (1934). 6 Gorynnova and Kuvskinskii, Zhur. Tekh. Fiz., 18, 1421 (1948). 7 Hansen, Dissertation, Jena, 1907. 8 “International Critical Tables,” vol. 5, p. 62. 9 Jeffries and Drickamer, J. Chem. Phys., 22, 436 (1954). 10 Klotz and Miller, J. Am. Chem. Soc., 69, 2557 (1947). 11 McMurtrie and Keyes, J. Am. Chem. Soc., 70, 3755 (1948). 12 Mullaly and Jacques, Phil. Mag., 48, 6, 1105 (1924). 13 Spier, Physica, 6 (1939): 453; 7, 381 (1940). 14 Topley and Whytlaw-Gray, Phil. Mag., 4, 873 (1927). 15 Trautz and Ludwig, Ann. Physik, 5, 5, 887 (1930). 16 Trautz and Muller, Ann. Physik, 22, 353 (1935). 17 Trautz and Ries, Ann. Physik, 8, 163 (1931). 18 Walker and Westenberg, J. Chem. Phys., 32, 136 (1960). 19 Westenberg and Walker, J. Chem. Phys., 26, 1753 (1957). 20 Winkelmann, Wied. Ann., 22, 152 (1884); 23, 203 (1884); 26, 105 (1885); 33, 445 (1888); 36, 92 (1889).

2-330

PHYSICAL AND CHEMICAL DATA

In this table are a representative selection of diffusion coefficients. The subsection “Prediction and Correlation of Physical Properties” should be consulted for estimation techniques. As general references, the works by Hirschfelder, Curtiss, and Bird, Molecular Theory of Gases and Liquids, Wiley, New York, 1964; Chapman and Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge, New York, 1970; Reid and Sherwood, The Properties of Gases and Liquids, TABLE 2-372

McGraw-Hill, New York, 1964; and Bretsznajder, Prediction of Transport and Other Physical Properties of Fluids, Pergamon, New York, 1971, may be found useful. The most exhaustive recent compilation for gases is by Mason and Marrero, J. Phys. Chem. Ref. Data, 1 (1972). Unfortunately, the Mason and Marrero work cites only equations and equation constants and not direct tabulations. For these, the LandoltBörnstein series is suggested.

Diffusivities in Liquids (25°C)

Dilute solutions and 1 atm unless otherwise noted; use DLµ/T = constant to estimate effect of temperature; * indicates that reference gives effect of concentration.

Solute

Solvent

DL × 105, sq cm/sec

Acetal* Acetamide* Acetamide* Acetic acid Acetic acid Acetic acid Acetic acid Acetic acid Acetic acid* Acetonitrile Acetylene Allyl alcohol* Allyl alcohol Ammonia* i-Amyl alcohol* i-Amyl alcohol Benzene Benzene (50 mole %) Benzene (50 mole %) Benzene (50 mole %) Benzene (50 mole %) Benzene (50 mole %) Benzene (50 mole %) Benzoic acid Benzoic acid Benzoic acid Benzoic acid Benzoic acid Bromine Bromine Bromine Bromobenzene Bromoform* Bromoform Bromoform Bromoform* Bromoform Bromoform n-Butanol Caffeine Carbon dioxide Carbon dioxide Carbon disulfide (50 mole %, 200 atm.) Carbon disulfide (50 mole %, 200 atm.) Carbon disulfide (50 mole %, 218 atm.) Carbon disulfide (50 mole %, 200 atm.) Carbon disulfide (50 mole %, 100 atm.) Carbon disulfide (50 mole %, 50 atm.) Carbon disulfide (50 mole %, 200 atm.) Carbon disulfide (50 mole %) Carbon tetrachloride Carbon tetrachloride* Carbon tetrachloride Carbon tetrachloride Carbon tetrachloride* Carbon tetrachloride Carbon tetrachloride Carbon tetrachloride Carbon tetrachloride Carbon tetrachloride Chloral* Chloral hydrate

Ethanol Ethanol Water Acetone Benzene Carbon tetrachloride Ethylene glycol Toluene Water Water Water Ethanol Water Water Ethanol Water Carbon tetrachloride n-Decane 2,4-Dimethyl pentane n-Dodecane n-Heptane n-Hexadecane n-Octadecane Acetone Benzene Carbon tetrachloride Ethylene glycol Toluene Benzene Carbon disulfide Water Benzene Acetone i-Amyl alcohol Ethanol Ethyl ether Methanol n-Propanol Water Water Ethanol Water n-Butanol i-Butanol Chlorobenzene 2,4-Dimethyl pentane n-Heptane Methyl cyclohexane n-Octane Toluene Benzene Cyclohexane Decalin Dioxane Ethanol n-Heptane Kerosene Methanol i-Octane Tetralin Ethanol Water

1.25 0.68 1.19 3.31 2.11 1.49 0.13 2.26 1.24 1.66 1.78, 2.11 1.06 1.19 1.7, 2.0, 2.3 0.87 1.0 1.53 1.72 2.49 1.40 2.47 0.96 0.86 2.62 1.38 0.91 0.043 1.49 2.7 4.1 1.3 2.30 2.90 0.53 1.08 3.62 2.20 0.94 0.96 0.63 4.0 1.96 3.57 2.42 3.00 3.63 3.0 3.5 3.10 2.06 2.04 1.49 0.776 1.02 1.50 3.17 0.961 2.30 2.57 0.735 0.68 0.77

Estimated possible, error,  %1 5 5 3

3 5 5 6 5 8

5

5 6 6 1

3 2 2 2 2 2 2 2 2 2 5 7

Ref. 11 11 11 4 1, 4 4 4 4 11 11 1, 24 11 11 1, 11 11 11, 25 7 26 26 26 26 26 26 4 4 4 4 4 11 11 11 25 11 11 11 11 23 11 1, 11, 18, 25 11 11 1, 3, 5, 20, 24, 28 14 14 14 14 14 14 14 14 7, 9 9, 10* 9 9 9, 10* 9 9 9 9 9 11 11

TRANSPORT PROPERTIES TABLE 2-372

Diffusivities in Liquids (25°C) (Continued )

Dilute solutions and 1 atm unless otherwise noted; use DL µ /T = constant to estimate effect of temperature; * indicates that reference gives effect of concentration.

Solute Chlorine Chlorobenzene Chloroform Chloroform Cinnamic acid Cinnamic acid Cinnamic acid Cinnamic acid 1,1′-Dichloropropanol Dicyanodiamide* Diethyl ether Diethyl ether 2,4-Dimethyl pentane (50 mole %) 2,4-Dimethyl pentane (50 mole %) Ethanol* Ethyl acetate Ethylene dichloride Formic acid Formic acid Formic acid Formic acid Formic acid Formic acid Glucose Glycerol Glycerol Glycerol* n-Heptane (50 mole %) n-Heptane (50 mole %) n-Heptane (50 mole %) n-Heptane (50 mole %) Hexamethylene tetramine Hydrogen chloride* Hydrogen Hydrogen sulfide Hydroquinone* Hydroquinone* Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine* Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodobenzene Lactose* Maltose* Mannitol* Methanol Nicotine* Nitric acid* Nitrobenzene Nitrogen Nitrous oxide Oxalic acid*

Solvent

DL × 105, sq cm/sec

Water Benzene Benzene Ethanol Acetone Benzene Carbon tetrachloride Toluene Water Water Benzene Water n-Dodecane n-Hexadecane Water Ethyl benzoate Benzene Acetone Benzene Carbon tetrachloride Ethylene glycol Toluene Water Water i-Amyl alcohol Ethanol Water n-Dodecane n-Hexadecane n-Octadecane n-Tetradecane Water Water Water Water Ethanol Water Acetic acid Anisole Benzene Bromobenzene Carbon disulfide Carbon tetrachloride Chloroform Cyclohexane Dioxane Ethanol Ethyl acetate Ethyl ether Ethylene bromide n-Heptane n-Hexane Mesitylene Methanol Methyl cyclohexane n-Octane Tetrabromoethane n-Tetradecane Toluene m-Xylene Ethanol Water Water Water Water Water Water Carbon tetrachloride Water Water Water

1.44 2.66 2.50 1.38 2.41 1.12 0.76 2.41 1.0 1.18 2.73 0.85 1.44 0.88 1.28 0.94 2.8 3.77 2.28 1.89 0.094 2.65 1.37 0.69 0.12 0.56 0.94 1.58 1.00 0.92 1.29 0.67 3.10 5.85 (4.4) 1.61 0.53 0.88, 1.12 1.13 1.25 1.98 1.25 3.2 1.45 2.30 1.80 1.07 1.30 2.2 3.61 0.93 3.4, 2.5 4.15 1.49 1.74 2.1 2.76 2.0 0.96 2.1 1.82 1.09 0.49 0.48 0.65 1.6 0.60 2.98 1.00 1.9 1.8 1.61

Estimated possible, error,  %1 4 6 3

6 4

4

10 6 6

3 5

10 8 3

3 5 5 5 8 2

2

Ref. 1, 28 25 1, 25 11 4 4 4 4 11 11 25 2 26 26 1, 7, 9,* 11,* 22 6 1, 25 4 4 4 4 11 11 11 11 1, 11* 26 26 26 26 11 4, 11,* 12* 1, 11, 24(?) 1 11 2, 11* 11 11 9, 19, 23 4, 11, 19 11, 19, 23 9, 11, 19 11, 23 4 9 4, 11* 11, 19 11 11 9, 11, 19 4, 9 9 19 4 4 11 4 11 9, 11 11 11 11 11 1, 7, 11 11 11 7 1, 24 1, 11 11

2-331

2-332

PHYSICAL AND CHEMICAL DATA TABLE 2-372

Diffusivities in Liquids (25°C) (Concluded )

Dilute solutions and 1 atm unless otherwise noted; use DL µ /T = constant to estimate effect of temperature; * indicates that reference gives effect of concentration.

Solute Oxygen Oxygen Oxygen Pentaerythritol* Phenol Phenol Phenol Phenol Phenol Phenol n-Propanol Pyridine* Pyridine Pyrogallol Raffinose* Resorcinol* Resorcinol* Saccharose* Stearic acid* Succinic acid* Sucrose Sulfur dioxide Sulfuric acid* Tartaric acid* 1,1,2,2-Tetrabromoethane Toluene Toluene Toluene Toluene Toluene Urea Urea Urethane Water

Solvent Glycerol*-water (106 poise) Sucrose*-water (125 poise) Water Water i-Amyl alcohol Benzene Carbon disulfide Chloroform Ethanol Ethyl ether Water Ethanol Water Water Water Ethanol Water Water Ethanol Water Water Water Water Water 1,1,2,2-Tetrachloroethane n-Decane n-Dodecane n-Heptane n-Hexane n-Tetradecane Ethanol Water Water Glycerol

DL × 105, sq cm/sec

Estimated possible, error,  %1

Ref.

0.24

13

0.25

13

2.5 0.77 0.2 1.68 3.7 2.0 0.89 3.9 1.1 1.24 0.76 0.74 0.41 0.46 0.87 0.49 0.65 0.94 0.56 1.7 1.97 0.80 0.61 2.09 1.38 3.72 4.21 1.02 0.73 1.37 1.06 0.021

References 1 Arnold, J. Am. Chem. Soc., 52, 3937 (1930). 2 Calvet, J. Chim. Phys., 44, 47 (1947). 3 Carlson, J. Am. Chem. Soc., 33, 1027 (1911). 4 Chang and Wilke, J. Phys. Chem., 59, 592 (1955). 5 Davidson and Cullen, Trans. Inst. Chem. Eng., 35, 51 (1957). 6 Dummer, Z. Anorg. Chem., 109, 31 (1949). 7 Gerlach, Ann. Phys. (Leipzig), 10, 437 (1931). 8 Gosting and Akeley, J. Am. Chem. Soc., 74, 2058 (1952). 9 Hammond and Stokes, Trans. Faraday Soc., 49, 890 (1953); 49, 886 (1953). 10 Hammond and Stokes, Trans. Faraday Soc., 52, 781 (1956). 11 International Critical Tables, vol. 5, p. 63. 12 James, Hollingshead, and Gordon, J. Chem. Phys., 7, 89 (1939); 7, 836 (1939). 13 Jordon, Ackermann, and Berger, J. Am. Chem. Soc., 78, 2979 (1956). 14 Koeller and Drickamer, J. Chem. Phys., 21, 575 (1953). 15 Kolthoff and Miller, J. Am. Chem. Soc., 63, 1013 (1941).

20 4

3 7 7 4 5 4 4 5 6 3 10 4

2

1, 3, 15, 21, 24 11 11 1 11 11 11 11 1, 7, 11 11 11 11 11 11 11 11 11 11 2, 27 15, 17 11 11 11 4 4 4 4 4 11 8, 11 11, 25 16

TRANSPORT PROPERTIES TABLE 2-373

2-333

Thermal Conductivities of Some Building and Insulating Materials* k = Btu/(h⋅ft2)(°F/ft)

Material

Apparent density ρ, lb/ft3 at room temperature

Aerogel, silica, opacified

8.5

Asbestos-cement boards Asbestos sheets Asbestos slate

120 55.5 112 112 29.3 29.3 36 36 36 36 43.5 43.5 0.2

Asbestos

Aluminum foil (7 air spaces per 2.5 in.) Ashes, wood Asphalt Boiler scale (Note 1) Bricks: Alumina (92–99% Al2O3 by wt.) fused Alumina (64–65% Al2O3 by wt.) (See also Bricks, fire clay) Building brick work Carbon Chrome brick (32% Cr2O3 by wt.) Diatomaceous earth, natural, across strata (Note 2) Diatomaceous, natural, parallel to strata (Note 2) Diatomaceous earth, molded and fired (Note 2) Diatomaceous earth and clay, molded and fired (Note 2) Diatomaceous earth, high burn, large pores (Note 3)

132

Kaolin insulating firebrick (Note 4) Magnesite (86.8% MgO, 6.3% Fe2O3, 3% CaO, 2.6% SiO2 by wt.)

Silicon carbide brick, recrystallized (Note 3)

Calcium carbonate, natural White marble Chalk Calcium sulfate (4H2O), artificial plaster (artificial) (building) Cambric (varnished) Carbon, gas Carbon stock Cardboard, corrugated

k

120 290 20 51 0 60 −200 0 0 100 200 400 −200 0 38 177 0–100 20

0.013 .026 .43 .096 .087 .114 .043 .090 .087 .111 .120 .129 .090 .135 .025 .038 .041 .43

427 1315 800 1100 20

96.7 200 200 200

200 650 1315

1.8 2.7 0.62 .63 .4 3.0 .67 .85 1.0

27.7 27.7

204 871

0.051 .077

27.7 27.7 38 38

204 871 204 871

.081 .106 .14 .18

42.3 42.3

204 871

.14 .19

27 27 19 19

200 1000 200 600 1000 1400 500 1150 200 760

.13 .34 .58 .85 .95 1.02 0.15 .26 .050 .113

158 158 158

204 650 1200

2.2 1.6 1.1

129 129 129 129 129 162

600 800 1000 1200 1400 30

10.7 9.2 8.0 7.0 6.3 1.3 1.7 0.4 .22 .43 .25 .091 2.0 0.55 3.6 0.037

115 115

37 37

Fire clay (Missouri)

Kaolin insulating brick (Note 3)

t, °C

96 84.6 132 77.9 94

40 75 25 38 0–100 −184 0

Material Cotton wool Cork board Cork (regranulated) (ground) Diatomaceous earth powder, coarse (Note 2) fine (Note 2) molded pipe covering (Note 2) 4 vol. calcined earth and 1 vol. cement, poured and fired (Note 2) Dolomite Ebonite Enamel, silicate Felt, wool Fiber insulating board Fiber, red (with binder, baked) Gas carbon Glass Borosilicate type Window glass Soda glass Granite Graphite, longitudinal powdered, through 100 mesh Gypsum (molded and dry) Hair felt (perpendicular to fibers) Ice Infusorial earth, see diatomaceous earth Kapok Lampblack Lava Leather, sole Limestone (15.3 vol. % H2O) Linen Magnesia (powdered) Magnesia (light carbonate) Magnesium oxide (compressed) Marble Mica (perpendicular to planes) Mill shavings Mineral wool

Apparent density ρ, lb/ft3 at room temperature

t, °C

k

5 10 8.1 9.4 20.0 20.0 17.2 17.2 26.0 26.0

30 30 30 30 38 871 204 871 204 871

0.024 .025 .026 .025 .036 .082 .040 .074 .051 .088

61.8 61.8 167

204 871 50

38 20.6 14.8 80.5

139

30 78 17 57.5 0.88 10 62.4 103

20 40

0 21 21 20 40 21

0.020 .038 .49 .092 .54 .05 .35 0.034 .32 1.2–1.7 0.25 0.033–0.05 0.0225 .024 .075 .14 3.4 2.9 0.88 .17 .14 .075 .087 .109 0.075–0.092 0.19 1.06 0.03

38 24–127 30 94 0 100 21 21 30 30

.026 .096 .064 .022 .86 .27 0.09–0.097 0.16 .028 .04 .034

49.7 13 49.9

24 30 47 21 20

9.4 19.7

30 30

50

Paper Paraffin wax Petroleum coke Porcelain Portland cement, see concrete Pumice stone Pyroxylin plastics Rubber (hard) (para) (soft) Sand (dry) Sandstone Sawdust Scale (Note 1) Silk varnished Slag, blast furnace Slag wool Slate Snow Sulfur (monoclinic) (rhombic) Wall board, insulating type Wall board, stiff paste board Wood shavings

.16 .23 1.0 0.10 0.5–0.75 30 0.03 21 .028 20 .27 20–97 .097 0–100 2.0 0.2–0.73 30–75 0.63 0.3–0.61 0.3–0.44 1.0–2.3 20 95 40 0.104 20 .25 30 .021 0 1.3

0 100 500 200 90 21–66 74.8 94.6 140 12 6.3 12 34.7 14.8 43 8.8

2-334

PHYSICAL AND CHEMICAL DATA

TABLE 2-373

Thermal Conductivities of Some Building and Insulating Materials* (Concluded ) k = Btu/(h⋅ft2)(°F/ft) Apparent density ρ, lb/ft3 at room temperature

Material Celluloid Charcoal flakes

87.3 11.9 15

Clinker (granular) Coke, petroleum Coke, petroleum (20–100 mesh) Coke (powdered) Concrete (cinder) (stone) (1:4 dry)

62

t, °C

k

30 80 80 0–700 100 500 400 0–100

.12 .043 .051 .27 3.4 2.9 0.55 .11 .20 .54 .44

Material Wood (across grain): Balsa Oak Maple Pine, white Teak White fir Wood (parallel to grain): Pine Wool, animal

Apparent density ρ, lb/ft3 at room temperature

t, °C

k

7–8 51.5 44.7 34.0 40.0 28.1

30 15 50 15 15 60

0.025–0.03 0.12 .11 .087 .10 .062

34.4 6.9

21 30

.20 .021

*Marks, Mechanical Engineers’ Handbook, 4th ed., McGraw-Hill, New York, 1941. International Critical Tables, McGraw-Hill, 1929, and other sources. Note 1: B. Kamp [Z. tech. Physik, 12, 30 (1931)] shows the effect of increased porosity in decreasing thermal conductivity of boiler scale. Partridge [University of Michigan, Eng. Research Bull., 15, 1930] has published a 170-page treatise on Formation and Properties of Boiler Scale. Note 2: Townshend and Williams, Chem. & Met., 39, 219 (1932). Note 3: Norton, Refractories, 2d ed., McGraw-Hill, New York, 1942. Note 4: Norton, private communication. TABLE 2-374

Thermal-Conductivity-Temperature Table for Metals* Thermal conductivities tabulated in watts per meter-kelvin Temperature, K

Substance

10

20

40

60

80

100

200

300

400

500

7 38,000 470 47 240

32 13,500 230 196 100

121 2,300 110 810 45

174 850 80 1,400 31

160 380 60 1,650 24

125 300 48 1,490 22

55 237 32 480 18

36 273 26 272 16

26 240 22 196 14

20 237 20 146 12

165 900 400 250 4

305 250 570 450 9

400 150 450 380 16

327 120 250 250 18

230 110 180 190 19

170 110 158 160 20

45 105 111 120 23

25 104 90 100 25

15 101 87 85 27

Copper Gallium Gold Graphite† Graphite‡

19,000 2,200 2,800 27 81

10,700 640 1,500 108 420

2,100 250 520 135 1,630

850 200 380 81 2,980

570 170 350 54 4,290

483 140 345 39 4,980

413 100 327 15 3,250

398 85 315 10 2,000

Hastelloy Inconel Iridium Iron Lead

1 2 1,300 710 175

3 4 1,900 1,000 57

4 8 750 560 43

5 10 360 270 42

6 11 230 170 41

7 11 172 132 40

9 14 147 94 37

10 15 145 80 35

11

13

143 69 34

140 61 33

Magnesium Magnesium oxide Manganese Manganin Mercury

1,200 1,100 2 2 54

1,300 3,100 2 4 40

620 2,200 4 9 35

290 950 5 11 33

190 460 5 13 33

169 260 6 13 32

159 75 7 17 32

156 48 8 22 8

153 36 9 28 10

Molybdenum Nickel Nylon Palladium Platinum

150 2,600 0.04 1,200 1,200

280 1,700 0.10 610 490

350 570 0.17 160 130

250 290 0.20 100 92

210 200 0.23 88 82

179 158 0.25 80 79

143 106 0.28 78 75

138 91 0.30 78 73

PTFE§ Pyrex Quartz Rhodium Rubber

0.94 0.12 1,200 2,900

1.43 0.20 480 3,900

1.94 0.33 82 1,000 0.13

2.1 0.42 40 370 0.15

2.15 0.51 30 250 0.16

2.16 0.57

2.20 0.88

2.25 1.1

190 0.17

160 0.20

Alumina Aluminum Antimony Beryllium oxide Bismuth Boron Cadmium Chromium Cobalt Constantan

Selenium (axis) Silica Silver Tantalum Tellurium Tin Titanium Tungsten Uranium Zinc Zirconium

1200

1400

16 232

10 220

8 93

7 99

6 105

111

70

47

33

25

12 99 85 70 30

81

71

65

62

61

392

388

383

371

357

342

312 7 1,460

309 5 1,140

304 4 930

292 3 680

278 3 530

262 2 440

2 370

55 31

43 19

33 22

28 24

31 26

151 27 9 34 11

149 21

146 13

84 10

98 8

112 7

40 12

13

14

134 80

130 72

126 66

118 67

112 72

105 76

100 80

78 72

80 72

72

73

78

78

81

358 61

62

2.3 1.6

2.5 2.1

150 0.22

145 0.24

140 0.25

140

57

25

15

10

8

6

16,500 108 300

5,200 146 93

1,100 88 29

630 68 17

500 62 13

430 59 11

425 58 6

4 1.34 424 57 4

3 1.52 420 58 3

2 1.70 413 58 3

14

320 28

130 39 880

100

110

59

101 37 330 20 150 42

90 33 310 22 135 38

84 31 280 23 130 34

72 26 190 26 123 25

67 21 180 28 120 23

62 20 170 30 116 22

60 20 150 32 110 21

600

1.87 405 59

800

2.22 389 59

1000

2.60 374 60

19 140 110 21

* Especially at low temperatures, the thermal conductivity can often be markedly reduced by even small traces of impurities. This table, for the highest-purity specimens available, should thus be used with caution in applications with commercial materials. From Perry, Engineering Manual, 3d ed., McGraw-Hill, New York, 1976. A more detailed table appears as Section 5.5.6 in the Heat Exchanger Design Handbook, Hemisphere Pub. Corp., Washington, DC, 1983. † Parallel to basal plane. ‡ Perpendicular to basal plane. § Also known as Teflon, etc.

TRANSPORT PROPERTIES TABLE 2-375

Thermal Conductivity of Chromium Alloys* k = Btu/(h⋅ft2)(°F/ft)

American Iron and Steel Institute Type No.

k at 212°F

k at 932°F

9.4 8.8 8.0 9.3 14.4 15.1 12.5 21.2

12.4 12.5 10.8 12.8 16.6 15.2 14.2 19.5

301, 302, 302B, 303, 304, 316† 308 309, 310 321, 347 403, 406, 410, 414, 416† 430, 430F† 442 501, 502†

* Table 3-322 is based on information from manufacturers. † Shelton and Swanger (National Bureau of Standards), Trans. Am. Soc. Steel Treat., 21, 1061–1078 (1933).

TABLE 2-376 Thermal Conductivity of Some Alloys at High Temperature*

TABLE 2-377 Thermal Conductivities of Some Materials for Refrigeration and Building Insulation* k = Btu/(h⋅ft2)(°F/ft) at approximately room temperature

Material Soft flexible materials in sheet form: Chemically treated wood fiber Eel grass between paper Felted cattle hair Flax fibers between paper Hair and asbestos fibers, felted Insulating hair, and jute Jute and asbestos fibers, felted Loose materials: Cork, regranulated, fine particles Charcoal, 6 mesh Diatomaceous earth, powdered Glass wool, curled Gypsum in powdered form Mineral wool, fibrous

Thermal conductivity, Btu/(ft)(hr)(°R) °R

Kovar

500 600 700 800 900

2-335

Advance

Monel

Hastelloy A

Inconel

Nichrome V

7.8 8.3 8.6 8.7 8.7

11.4 12.6 13.9 15.1

9.0 10.2 11.2 12.3 13.4

5.6 6.2 6.8 7.3 7.8

6.0 6.5 7.0 7.6 8.1

5.5 6.1 6.7 7.3 7.8

Sawdust Wood shavings, from planer Semiflexible materials in sheet form: Flax fiber Semirigid materials in board form: Corkboard

1000 1100 1200 1300 1400

8.9 9.2 9.5 9.8 10.2

16.4 17.6 18.8 20.0 21.2

14.4 15.4 16.5 17.6 18.7

8.4 9.0 9.5 10.1 10.7

8.6 9.1 9.7 10.2 10.8

8.4 9.0 9.5 10.1 10.7

Mineral wool, block, with binder Stiff fibrous materials in sheet form: Wood pulp Sugar-cane fiber Cellular gypsum

1500 1600 1700 1800 1900

10.5 10.8 11.1 11.3 11.5

22.5 23.8 25.0 26.2 27.4

19.8 20.8 21.9 23.0 24.0

11.3 11.8 12.3 12.9 13.4

11.3 11.8 12.4 13.0 13.6

11.3 11.9 12.4 13.0 13.5

2000 2100 2200

11.8 12.1 12.3

28.7 30.0

25.1 26.1 27.2

14.0 14.6 15.1

14.0 14.5 15.0

14.1 14.7 15.3

Apparent density, lb/cu ft room temp.

k

2.2 3.4–4.6 11–13 4.9 7.8 6.1–6.3 10.0

0.023 0.021–0.022 0.022 .023 .023 0.022–0.023 0.031

8–9 15.2 10.6 4–10 26–34 6 10 14 18 12 8.8

.025 .031 .026 .024 0.043–0.05 0.0217 .0225 .0233 .0242 .034 .034

13.0

.026

7.0 10.6 16.7

.0225 .025 .031

16.2–16.9 13.2–14.8 8 12 18 24 30

.028 .028 .029 .037 .049 .064 .083

*Abstracted from U.S. Bur. Standards Letter Circ. 227, Apr. 19, 1927.

*Silverman, J. Metals, 5, 631 (1953). Copyright American Institute of Mining, Metallurgical and Petroleum Engineers, Inc.

TABLE 2-378

Thermal Conductivities of Insulating Materials at High Temperatures* k = Btu/(h⋅ft2)(°F/ft)

Material Laminated asbestos felt (approx. 40 laminations per in) Laminated asbestos felt (approx. 20 laminations per in) Corrugated asbestos (4 plies per in) 85% magnesia (density, 13 lb/ft3) Diatomaceous earth, asbestos and bonding material Diatomaceous earth brick Diatomaceous earth brick Diatomaceous earth brick Diatomaceous earth powder (density, 18 lb/ft3) Rock wool

For temperatures, °F up to 700 500 300 600 1600 1600 2000 2500

Mean temperatures, °F 100

200

300

400

500

0.033 .045 .050 .034 .045 .054 .127 .128 .039 .030

0.037 .050 .058 .036 .047 .056 .130 .131 .042 .034

0.040 .055 .069 .038 .049 .058 .133 .135 .044 .039

0.044 .060

0.048 .065

.040 .050 .060 .137 .139 .048 .044

.053 .063 .140 .143 .051 .050

Asbestos cement, 1.2; 85% magnesia cement, 0.05; asbestos and rock wool cement, 0.075 approx. *Marks, “Mechanical Engineers’ Handbook,” 4th ed., McGraw-Hill, New York, 1941.

600

800

1000

0.055 .065 .143 .148 .054 .057

0.060 .069 .150 .155 .061

0.065 .073 .158 .163 .068

1500

2000

0.176 .183

0.203

2-336

PHYSICAL AND CHEMICAL DATA

TABLE 2-379 Thermal Conductivities of Insulating Materials at Moderate Temperatures (Nusselt)* k = Btu/(h⋅ft2)(°F/ft) Weight, lb/cu ft

Material Asbestos Burned infusorial earth for pipe coverings Insulating composition (loose) Cotton Silk hair Silk Wool Pulverized cork Infusorial earth (loose)

TABLE 2-380 Thermal Conductivities of Insulating Materials at Low Temperatures (Gröber)* k = Btu/(h⋅ft2)(°F/ft)

Temperatures, °F 32

100

200

300

400

600

800

36.0

0.087 0.097 0.110 0.117 0.121 0.125 0.130

12.5

.043 .046 .052 .057 .062 .073 .085

25.0 5.0 9.1 6.3 8.5 10.0 22.0

.040 .032 .026 .025 .022 .021 .035

.046 .035 .030 .028 .027 .026 .039

.050 .053 .055 .039 .034 .034 .033 .032 .045 .047 .050 .053

Temperatures, °F

Material

Weight, lb/cu ft

32

−50

−100

−200

−300

Asbestos Asbestos Cotton Silk

44.0 29.0 5.0 6.3

0.135 .0894 .0325 .0290

0.132 .0860 .0302 .0256

0.130 .0820 .0276 .0235

0.125 .0720 .0235 .0196

0.100 .0545 .0198 .0155

*Marks, Mechanical Engineers’ Handbook, 4th ed., McGraw-Hill, New York, 1941.

*Marks, Mechanical Engineers’ Handbook, 4th ed., McGraw-Hill, New York, 1941.

TABLE 2-381

Thermal Diffusivity (m2/s) of Selected Elements* Temperature, K

Element

20

40

60

80

100

Aluminum Beryllium Chromium Copper Gold

0.50

200

400

600

800

1000

0.012

0.0014

4.4. − 4

0.038 0.16 0.005

0.0037 0.0040 4.5. − 4

5.9. − 4 6.9. − 4 2.3. − 4

2.0. − 4 3.1. − 4 1.8. − 4

2.3. − 4 0.0036 1.2. − 4 2.2. − 4 1.5. − 4

1.1. − 4 1.5. − 4 4.1. − 5 1.3. − 4 1.3. − 4

9.4. − 5 4.0. − 5 2.6. − 5 1.1. − 4 1.2. − 4

8.4. − 5 2.6. − 5 2.0. − 5 1.0. − 4 1.2. − 4

7.4. − 5 2.1. − 5 1.7. − 5 9.0. − 5 1.1. − 4

6.6. − 5 1.7. − 5 1.4. − 5 9.0. − 5 9.8. − 5

Iridium Iron Lead Molybdenum Nickel

0.046 0.043 9.3. − 5 0.0095 0.033

3.2. − 3 3.9. − 5 0.0014 0.0017

4.9. − 4 3.3. − 5 4.0. − 4 3.1. − 4

1.6. − 4 3.1. − 5 2.0. − 4 1.3. − 4

8.4. − 5 8.2. − 5 2.9. − 5 1.3. − 4 8.0. − 5

5.6. − 5 3.1. − 5 2.6. − 5 6.3. − 5 3.1. − 5

4.8. − 5 1.8. − 5 2.3. − 5 5.1. − 5 1.9. − 5

4.4. − 5 1.3. − 5 2.0. − 5 4.5. − 5 1.3. − 5

4.1. − 5 1.1. − 5 1.3. − 5 4.2. − 5 1.4. − 5

3.5. − 5 1.0. − 5 1.5. − 5 3.8. − 5 1.5. − 5

Platinum Silver Zinc

0.0029 0.031 0.0046

1.6. − 4 0.0013 3.1. − 4

6.3. − 5 4.5. − 4 1.0. − 4

4.3. − 5 2.8. − 4 7.0. − 5

3.6. − 5 2.3. − 4 5.5. − 5

2.7. − 5 1.8. − 4 4.7. − 5

2.5. − 5 1.7. − 4 3.9. − 5

2.5. − 5 1.6. − 4 3.4. − 5

2.5. − 5 1.5. − 4 1.8. − 5

2.5. − 5 1.4. − 4 2.2. − 5

*Tables for up to 24 temperatures for 47 elements appear in the Handbook of Heat Transfer, 2d ed., McGraw-Hill, New York, 1984. The notation 3.2. − 4 signifies 2.3 × 10−4.

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-382

2-337

Thermophysical Properties of Selected Nonmetallic Solid Substances Density, kg/m3

Material Alumina Asphalt Bakelite Beryllia Brick

3975 2110 1300 3000 1925

Brick, fireclay Carbon, amorphous Clay Coal Cotton

2640 1950 1460 1350 80

Diamond Granite Hardboard Magnesite Magnesia

3500 2630 1000 3025 3635

Emissivity

Specific heat, kJ/(kg⋅K)

Thermal conductivity, W/(m⋅K)

0.82 0.93

0.765 0.920 1.465 1.030 0.835

36 0.06 1.4 270 0.72

11.9 0.03 0.74 88 0.45

0.960 0.724 0.880 1.26 1.30

1.0 1.6 1.3 0.26 0.06

0.39 1.13 1.01 0.15 0.58

0.509 0.775 1.38 1.13 0.943

2300 2.79 0.15 4.0 48

1290 1.37 0.11 1.2 14

0.93 0.86 0.91 0.80

0.38 0.72

Oak Paper Pine Plaster board Plywood

770 930 525 800 540

0.90 0.83 0.84 0.91

2.38 1.34 2.75

Pyrex Rubber Rubber, foam Salt Sandstone

2250 1150 70

0.835 2.00

2150

0.92 0.92 0.90 0.34 0.59

Silica Sapphire Silicon carbide Soil

3975 3160 2050

Teflon Thoria Urethane foam Vermiculite

2200 4160 70 120

Thermal diffusivity, m2/s × 106

0.18 0.011 0.12 0.17 0.12

0.10 0.01 0.54

0.74 0.09

0.854 0.745

1.4 0.2 0.03 7.1 2.9

0.79 0.48 0.86 0.38

0.743 0.765 0.675 1.84

1.3 46 490 0.52

15 230 0.14

0.92 0.28

0.35 0.71 1.05 0.84

0.26 14 0.03 0.06

0.34 4.7 0.36 0.60

1.22

0.18

1.8

NOTE: Difficulties of accurately characterizing many of the specimens mean that many of the values presented here must be regarded as being of order of magnitude only. For some materials, actual measurement may be the only way to obtain data of the required accuracy. To convert kilograms per cubic meter to pounds per cubic foot, multiply by 0.062428; to convert kilojoules per kilogram-kelvin to British thermal units per pound-degree Fahrenheit, multiply by 0.23885.

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES INTRODUCTION In the absence of reliable experimental data, the methods presented here provide physical property estimates that are sufficiently accurate for many engineering applications. These techniques have been selected on the basis of accuracy, generality, and, in most cases, simplicity; they are divided into 11 categories: (1) pure component constants: critical properties, normal freezing and boiling temperatures, acentric factor, radius of gyration, dipole moment, and van der Waals area and volume; (2) vapor pressure; (3) ideal gas thermal properties: heat capacity and enthalpy, Gibbs energy, and entropy of formation; (4) enthalpy of vaporization and fusion; (5) solid and liquid heat capacity; (6) vapor, liquid, and solid density; (7) vapor and liquid viscosity; (8) vapor and liquid thermal conductivity; (9) vapor and liquid diffusivity; (10) surface tension; and (11) flammability properties: flash point, flammability limits, and autoignition temperature. The definition of the property and limitations and accuracy of each method of correla-

tion or prediction are given for each property. Numerical examples are included for many of the methods. Equation symbols are listed under “Nomenclature,” and literature citations, indicated by superscript numbers, follow the nomenclature under “References.” Essentially all of the methods are derived from work on the American Petroleum Institute Technical Data Book 23 (hydrocarbon compounds and their mixtures), the AIChE Design Institute for Physical Property Data (DIPPR) Data Prediction Manual 22 (nonhydrocarbon compounds and their mixtures), and the DIPPR Data Compilation Project 24. UNITS Applicable dimensional units are shown individually with each equation. The International Metric System (SI) is used when feasible; otherwise commonly used U.S. engineering units are employed. The reader is referred to Sec. 1 for unit conversion factors.

2-338

PHYSICAL AND CHEMICAL DATA Nomenclature Symbol

Definition

SI units

U.S. customary units

B Cp Cv d D F k K ln M MeABP N NA P P sat [P] rel den R R  S T V x y Z

Second virial coefficient Heat capacity at constant pressure Heat capacity at constant volume Mass density Diffusivity Conductivity factor in Eq. (2-136) Thermal conductivity Watson/UOP characterization factor = (1.8Tb)1/3/rel den Denotes natural logarithm Molecular weight Mean average boiling point Carbon number Avogadro’s number Pressure Vapor pressure Parachor Relative density at 15°C and 0.1 MPa Universal gas constant; e.g., 8314 Pa·m3/kmol K Radius of gyration Absolute entropy Absolute temperature Molar volume Mole fraction of component in liquid phase Mole fraction of component in vapor phase Compressibility factor = PV/RT

m3/kmol J/(kmol·K) kg/m3 m2/s

ft3/lbmol Btu/(lbm °F) lbm/ft3 ft2/s

W/(m·K)

Btu/(h·ft °F)

Pa Pa

lbf/in2 lbf/in2

J/kg K m3/kmol

Btu/lbm R ft3/lbmole

∆Gf ∆Hf ∆Hfus ∆HV ∆Sf ∆Sfus ∆ZV

J/kg J/kg J/kg J/kg J/(kg·K) J/(kg·K)

Btu/lbm Btu/lbm Btu/lbm Btu/lbm Btu/lbm Btu/lbm

Θ λ µ ν ρ σ φ ω

Gibbs energy of formation Enthalpy (heat) of formation Enthalpy (heat) of fusion Enthalpy (heat) of vaporization Entropy of formation Entropy of fusion Difference of vapor and liquid compressibility factors defined in Eq. (2-55) Tb /Tc Dipole moment Absolute viscosity Kinematic viscosity Molar density Surface tension Volume fraction Acentric factor

Pa·s m2/s kmol/m3 N/m

lbm/(ft·s) ft2/s lb·mole/ft3 dyne/cm

r (0) (1) ° ′

At reference condition Simple spherical molecule, corresponding states Correction factor, corresponding states Of the ideal gas At atmospheric pressure

b bp c G HI i j L m mc mlt o pc r RA sat S V w

At normal boiling temperature At bubble point At critical point Of the gas/vapor Upper limit Component index, of the ith component Component index, of the jth component Of the liquid Of the mixture Mixture correspondence At melting temperature Of organic component Pseudocritical quantity Reduced quantity Rackett parameter Saturated Of the solid Of the vapor Of water

Greek symbols

Superscripts

Subscripts

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2-339

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Basic pure component constants required to characterize components or mixtures for calculation of other properties include the melting point, normal boiling point, critical temperature, critical pressure, critical volume, critical compressibility factor, acentric factor, and several other characterization properties. This section details for each property the method of calculation for an accurate technique of prediction for each category of compound, and it references other accurate techniques for which space is not available for inclusion. Critical Temperature The critical temperature of a compound is the temperature above which a liquid phase cannot be formed, no matter what the pressure on the system. The critical temperature is important in determining the phase boundaries of any compound and is a required input parameter for most phase equilibrium thermal property or volumetric property calculations using analytic equations of state or the theorem of corresponding states. Critical temperatures are predicted by various empirical methods according to the type of compound or mixture being considered. For pure hydrocarbons, the method of Ambrose2 is the most accurate and will also be useful for predicting critical pressure and volume. Equation (2-1) requires only the normal boiling point, Tb, and the molecular structure of the compound.



log Tc = A + B log10 (rel den) + C log Tb



(2-1)

Tc and Tb are the critical and normal boiling temperatures, respectively, expressed in kelvins. Values of ∆T from Table 2-383 are summed for each part of the molecule to yield ∆T (e.g., for isobutane, 3 × CH3 + 1 × CH). ∆(Platt no.) is equal to the Platt number of any alkyl chains in the molecule minus the Platt number of the n-alkane with the same number of carbon atoms. The Platt number is defined as the number of pairs of carbon atoms that are separated by three carbon-carbon bonds and is an indicator of branching (e.g., for 2,2,3trimethylpentane, the Platt number is 5). The Platt number of an n-alkane is the number of carbon atoms minus three (e.g., for n-octane, the Platt number is five). Errors in Tc average about 4 K for paraffins to C20 and other hydrocarbons to C14.

(2-2)

Tc and Tb are the critical and normal boiling temperatures, respectively, expressed in kelvins. The relative density (rel den) of the liquid at 15°C is 0.1 MPa. The regression constants A, B, and C are tabulated by family in Table 2-384. Errors average about 3 K. For pure nonhydrocarbon organics, the most accurate method for prediction of critical temperature for all compound groups is also the Ambrose2 method. Equation (2-1) applies to all nonhydrocarbon compounds except perfluorocarbons, where the constant 1.242 is replaced by 1.570. For compounds containing any of C, H, O, N, S, or halogens up to C13 and ranging in critical temperature from 228– 790 K, the average error is about 6 K. Alternate methods for nonhydrocarbon organics are the first order method of Lydersen63 with an average error of 9 K although the method of Ambrose is considerably better for alcohols and ketones. Equation (2-3) is the Lydersen equation for critical temperature and requires only the normal boiling point and the molecular structure for solution. Tb Tc =  [0.567 + ∆T − ( ∆T)2]

(2-3)

Contributions to ∆T are given in Table 2-385. For pure inorganic compounds, the method of Gambill31 was modified to yield Eq. (2-4) and only requires the normal boiling point as input. Tc = 1.64Tb

PURE COMPONENT CONSTANTS

1 Tc = Tb 1 +  1.242 + ∆T − 0.023∆(Platt no.)

Equation (2-2), another somewhat simpler method for estimating the critical temperature of pure hydrocarbons only, is the method of Nokay76 and requires the normal boiling point, the relative density, and the compound family.

(2-4)

Although this equation was tested with available experimental data (38 compounds), it can only be considered a rough approximation. Inorganic-organic and inorganic-halide compounds are predicted better by replacing the constant 1.64 in Eq. (2-4) by 1.55. For both hydrocarbons and nonhydrocarbon organic defined mixtures, the method of Li60 is used with a relatively simple volumetric average mixing rule as shown in Eq. (2-5) to calculate the true critical temperature. yjVc j Tcm =  Tc j (2-5) j

yiVc i



i



Tcm is the mixture critical temperature in K. Vc is the critical volume of a component, m3/kmole. The mole fraction of a component is y. The mixture contains i components. For hydrocarbon systems, the average error is about 3 K, while for systems containing nonhydrocarbons, the average error is 15 K with the highest errors occurring where simple gases are present. The method of Chueh and Prausnitz18 yields average errors slightly lower than the method of Li, but it is computationally more complex. Critical Pressure The critical pressure of a compound is the vapor pressure of the compound at the critical temperature. Below the critical temperature, any compound above its vapor pressure will be a liquid. The critical pressure is required for calculations discussed in the part of the section on critical temperature. For pure hydrocarbons, the method of Ambrose2 is the most accurate. Equation (2-6) requires only the molecular weight (M) and the molecular structure of the compound. 0.101325M Pc =  [0.339 + ∆P − 0.026∆(Platt no.)]2

(2-6)

Pc is the critical pressure, MPa. Values of ∆P from Table 2-383 are summed for each part of the molecule to yield ∆P. Calculation of the Platt number is discussed under “Critical Temperature.” Errors in Pc average 0.07 MPa and are less reliable for compounds with 12 or more carbon atoms.

TABLE 2-383

Group Increments for the Ambrose Method ∆T

∆P

∆V

Group description

CH3

0.138

0.2260

55.1

Aromatic Compounds (Cont.)

CH2

0.138

0.2260

55.1

CH

0.095

0.2200

47.1

C

0.018

0.1960

38.1

CH2

0.113

0.1935

45.1

CH

0.113

0.1935

45.1

C

0.070

0.1875

37.1

C

0.088

0.1610

35.1

CH

0.038

0.1410

35.1

C

0.038

0.1410

35.1

Group description

Ring Increments CH2

0.090

0.1820

44.5

CH

0.090

0.1820

44.5

CH

in fused ring *

0.030

0.1820

44.5

C

0.090

0.1820

44.5

CH

0.075

0.1495

37.0

C

0.075

0.1495

37.0

C

0.060

0.1170

29.5

Aromatic Compounds 0.458

0.448

0.488

0.9240

0.8940

0.9440

222

222

222

0.488

0.9440

222

0.438

0.8640

222

0.478

0.9140

0.8340

∆P

∆V

0.468

0.8840

222

0.468

0.8840

222

0.418

0.8040

222

0.368

0.7240

222

0.220

0.5150

148

in fused ring* Nonring Increments OH

Use Eq. (a), below Use Eq. (b), below

O

0.138

0.160

CO

0.220

0.282

0.220

0.220

0.578

0.450

1.156

0.900

CHO COOH CO

O

CO

O

OC

0.330

0.470

NO2

0.370

0.420

NH2

0.208

0.095

NH

0.208

0.135

N

0.088

0.170

CN

0.423

0.360

S

0.105

0.270

SH

0.090

0.270

Si

0.138

0.461

SiH

0.371

0.507

SiH3

0.195

—

0.159

0.725

0.131

0.663

222 Si

0.428

∆T

222

Si

O

O

cyclic

2-341

2-342

PHYSICAL AND CHEMICAL DATA

TABLE 2-383

Group Increments for the Ambrose Method (Concluded ) ∆T

∆P

−0.050 −0.200 0.180 (1st) 0.055 0.110 (1st) 0.055 0.110 (1st) 0.055

−0.065 −0.170 0.223 0.318 0.500

0.448 0.448 0.198 0.220 0.080

0.924 0.850 −0.025 0.515 0.183

Group description Double Bond Triple Bond F Cl Br Aromatic Compounds Corrections benzene pyridine OH C4H4 (fused ring) F *Group contributions for

CH and

∆V

∆T

∆P

0.080 0.080

0.318 0.600 0.850

0.010

0

0.030

0.020

−0.040 −0.080

−0.050 0

Group description Cl Br I The single or first substituent on an aromatic ring The second or subsequent ring substituents Each pair of ring substituents in ortho positions with respect to each other If one of the ortho pair is OH

∆V

in fused rings have been calculated from minimal data and may be less reliable than the other values.

Substituent increments do not apply to halogens. Eq. (a): ∆T (OH) = 0.87 − 0.11n + 0.003n2 Eq. (b): ∆p (OH) = 0.100 − 0.013n n = carbon number of compound. For branched alcohols, an effective carbon number can be determined by interpolation between the normal alcohol of the same carbon number and the immediately lower normal alcohol using normal boiling points. (See Example 1.)

TABLE 2-384

∆T = 2(.138) + .138 + .095 + .558 = 1.067 1 Tc = 372.7 1 +  1.242 + 1.067 − 0

Nokay Equation

Type compound

A

B

C

Paraffin Naphthene Olefin Acetylene Diolefin Aromatic

1.35940 0.65812 1.09534 0.74673 0.1384 1.0615

0.43684 −0.07165 0.27749 0.30381 −0.39618 0.22732

0.56224 0.81196 0.65563 0.79987 0.99481 0.66929





Tc = 534.1 K An accurate experimental value is 536.05 K.

∆P = 2(.226) + .226 + .220 + .0597 = 0.9577 .101325(74.12) Pc = 2 (.339 + .9577 − 0)

Example 1 Estimate the critical temperature and critical pressure of 2-butanol using the Ambrose method, Eqs. (2-1) and (2-6). The experimental normal boiling point is 372.7 K. 1 Tc = Tb 1 +  1.242 + ∆ T − 0.023∆ (Platt no.)





0.101325M Pc =  [0.339 + ∆P − 0.026∆ (Platt no.)]2 Determine group contributions from Table 2-383 for a structure CH3CHOHCH2CH3: Number of groups

∆T

∆P

CH3

2

0.138

0.2260

CH2

1

0.138

0.2260

CH–

1

0.095

0.2200

OH

1

(a)

(b)

Group

Pc = 4.467 MPa An accurate experimental value is 4.179 MPa. For pure nonhydrocarbon organics, the simplest accurate method for prediction of critical pressure is the method of Lydersen.63 Equation (2-7) requires the molecular weight (M) and the molecular structure of the compound. 0.101325M Pc =  (2-7) (0.34 + ∆P)2 Pc is the critical pressure, MPa. Values of ∆P from Table 2-385 are summed to yield ∆P. The average error in Pc is about 0.2 MPa when tested on compounds ranging in carbon number from C1 to C10.

Example 2 Estimate the critical temperature and critical pressure of 2-butanol, which has an experimental normal boiling point of 372.7 K. Use the Lydersen method, Eqs. (2-3) and (2-7). The structure of 2-butanol is CH3CHOHCH2CH3. Determine group contributions from Table 2-385. Group

Number of groups

∆T

∆P

CH3

2

0.020

0.227

CH2

1

0.020

0.227

To determine n, the normal boiling points of 1-propanol (370.3 K) and 1-butanol (390.9 K) are required:

CH

1

0.012

0.210

372.7 − 370.3 n = 3 +  (4 − 3) = 3.1 390.9 − 370.3

OH

1

0.082

0.06

∆T°(OH) = 0.87 − 0.11(n) + 0.003 n = 0.558 ∆p°(OH) = 0.100 − 0.013(n) = 0.0597 2





The Platt number of the compound is determined by substituting a CH3 group for the OH. CH3CH(CH3) − CH2CH3, Platt no. = 2 The Platt number of the n-alkane of the same carbon number is 5 − 3 = 2. Thus, ∆(Platt no.) = 2 − 2 = 0. Therefore:

0.154 Tb 372.7 Tc = 2 = 2 0.567 + ∆T − ( ∆T) .567 + .154 − (.154) Tc = 534.5 K

0.951

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-385

Group Increments for the Lydersen Method Incremental contributions

Group description Nonring Elements CH3, CH2 CH

∆T

∆P

∆V

Incremental contributions ∆P

∆V

0.085

(0.4)

0.080

0.047

0.47

0.080

(0.02)

(0.12)

(0.011)

0.018 0.017 0.010 0.012

0.224 0.320 (0.50) (0.83)

0.018 0.049 (0.070) (0.095)

0.031 0.031

0.095 0.135

0.028 (0.037)

(0.024)

(0.09)

(0.027)

0.014

0.17

(0.042)

(0.007)

(0.13)

(0.032)

CN

(0.060)

(0.36)

(0.080)

NO2

(0.055)

(0.42)

(0.078)

0.015 0.015 (0.008) (0.003)

0.27 0.27 (0.24) (0.24)

0.055 0.055 (0.045) (0.047)

Si

0.026

0.468

—

SiH

0.040

0.513

—

SiH3

0.027

—

—

0.025

0.730

—

0.027

0.668

—

Group description

C

0.020

0.227

0.055

0.012

0.210

0.051

0.00

C CH,

0.210

0.041

CH

0.018

0.198

0.045

C

0.00

0.198

0.036

0.005

0.153

(0.036)

0.013

0.184

0.0445

, C

Ring Increments CH2

∆T

Oxygen Increments (Cont.) COOH (acid) COO

CH2,

2-343

(ester)

O (except for combinations above) Halogen Increments F Cl Br I Nitrogen Increments NH2 NH (nonring) NH (ring)

0.012

CH

0.192

0.046 N

(−0.007)

C

0.154

(0.031) N

C

,

C

CH

0.011

0.154

0.036

0.011

0.154

0.037

0.066

0.924

Organometallic Increments

OH (alcohols)

0.082

0.06

(0.018)

OH (phenols)

0.031

(−0.02)

(0.003)

O

(nonring)

0.021

0.16

0.020

O

(ring)

(0.014)

(0.12)

(0.008)

O (nonring)

C

O (ring)

HC

O (aldehyde)

(ring)

Sulfur Increments SH S (nonring) S (ring) 苷S

Oxygen Increments

C

(nonring)

0.040

0.29

0.060

(0.033)

(0.2)

(0.050)

0.048

0.33

0.073

Values in parentheses are based on too few experimental points to be reliable.

Si

O

Si

O

cyclic

2-344

PHYSICAL AND CHEMICAL DATA TABLE 2-386 Atomic and Structural Contributions for the Fedors Method

The accurate experimental critical temperature is 536.05 K. 0.101325M (0.101325)(74.12) =  Pc =  (0.34 + ∆P)2 (0.34 + 0.951)2

Atomic increments

Pc = 4.506 MPa The accurate experimental critical pressure is 4.179 MPa. No known method is available to predict the critical pressure of inorganic compounds. For both hydrocarbon and nonhydrocarbon organic defined mixtures, the method of Kreglewski and Kay52 is recommended. The critical temperature, critical pressure, and acentric factor of each compound and the critical temperature of the mixture must be known or predicted from the methods of this section.



Tcm − Tpc

 x ω   T

Pcm = Ppc + Ppc 5.808 + 4.93

n

i

i

i=1

∆V

Feature

∆V

C H O O (alcohols) N N (amines) F

0.034426 0.009172 0.020291 0.018000 0.048855 0.047422 0.022242

Si Sisiloxane Sicyclic siloxane

0.086174 0.126483 0.126483

Cl Br I S 3-membered ring 4-membered ring 5-membered ring 6-membered ring double bond triple bond ring attached directly to another ring

0.052801 0.071774 0.096402 0.050866 −0.105824 −0.017247 −0.039126 −0.039508 +0.005028 +0.000797 +0.035524

(2-8)

pc

Use of Eq. (2-8) requires the pseudocritical properties defined by Eqs. (2-9) and (2-10)

Structural increments

Atom

n

Tpc = xiTc i

(2-9)

i=1 n

Ppc = xiPc i

(2-10)

i=1

Each component i of the mixture must have available its Tc, Pc, and ω. Tcm can be predicted from Eq. (2-5). For hydrocarbon systems, average errors in predicted critical pressures are about 0.2 MPa, except when organic gases are present and errors are unacceptably large. Errors for nonhydrocarbon organics not including inorganic gases average 0.5 MPa.

Critical Volume The critical volume of a compound is the volume occupied by a set mass of a compound at its critical temperature and pressure. While useful in itself, the critical volume is extensively used in equations for estimating volumetric fractions. For pure hydrocarbons, two methods are quite accurate. The Ambrose2 method used for Tc and Pc is also used for critical volume. Eq. (2-11) only requires the molecular structure of the compound. Vc = 10−3(40 + ∆V)

(2-11)

Vc is the critical volume, m3/kmole. Values of ∆V are given in Table 2-383. The average error for hydrocarbons of twelve or less carbon atoms is about 0.01 m3/kmole. The Riedel method90 requires the critical temperature (Tc), critical pressure (Pc), and acentric factor (ω) of the compound as given by Eqs. (2-12) and (2-13). If the gas constant is in Pa·m3/kmole·K, the critical volume will be in m3/kmole. RTc Vc =  (2-12) Pc[3.72 + 0.26(α − 7.00)] α = 5.811 + 4.919ω

(2-13)

The average error for paraffins up to C18 and other hydrocarbon families up to C11 is about 0.015 m3/kmole. If estimated values of Tc and Pc are used, errors may be higher. For pure nonhydrocarbon organics, the method of Fedors,29 which requires only molecular structure, is the most accurate. Equation (2-14) shows the method to depend only on the molecular structure. The resulting Vc will be in m3/kmole. Vc = 0.0266 + ∆V

(2-14)

Values for ∆V are given in Table 2-386. The average error for compounds up to C7 is about 0.007 m3/kmole, although a maximum error of 0.03 m3/kmole has been noted. No experimental data above C7 are available for comparison. Example 3 Estimate the critical volume of 2-butanol. Use the method of Fedors, Eq. (2-14). Vc = 0.0266 + ∆V The molecular formula is C4H10O. Using the atomic contribution values from Table 2-386.

∆V = 4(.034426) + 10(.009172) + 1(.018000)

∆V = 0.2474 Vc = 0.0266 + 0.2474 = 0.2740 m3/kmole The accurate experimental critical volume is 0.2690 m3/kmole.

The method of Lydersen63 may also be used for prediction of critical volume, but it is not so accurate as the method of Fedors. Equation (2-15) depends only on molecular structure and gives a critical volume in m3/kmole. Vc = 0.040 + ∆V

(2-15)

Group contributions for ∆V are given in Table 2-385. Errors average about 0.01 m3/kmole. There is no known method for predicting the critical volume of inorganic compounds. For both hydrocarbon and nonhydrocarbon organic defined mixtures, the method of Chueh and Prausnitz19 is useful. For hydrocarbon systems, the mixing rule is shown by Eq. (2-16) for binaries and by Eq. (2-17) for multicomponents. Equations (2-18) through (2-20) give the input parameters. The mixture critical volume Vcm is a function of the pure component critical volumes. The constant C is zero for hydrocarbon systems and 0.1559 for systems containing a nonhydrocarbon gas. Vcm = φ1Vc1 + φ2Vc2 + 2φ1φ2ν12 n

n

i

j

Vcm = φiφ jνi j

(i ≠ j)

2/3 cj

(2-16) (2-17)

xjV φj =  n

xiV 2/3ci

(2-18)

Vij(Vci + Vcj) νij =  2.0

(2-19)

i=1

Vci − Vcj (2-20) Vij = −1.4684  + C Vci + Vcj Errors average about 10 percent for systems containing hydrocarbons. Systems containing only organics or organics and gases may give very high errors. A specialized modification of the method (Chueh and Prausnitz18) is available for binary mixtures containing organics but may give errors over 20 percent.





Critical Compressibility Factor The critical compressibility factor of a compound is calculated from the experimental or predicted values of the critical properties by the definition, Eq. (2-21). PcVc Zc =  (2-21) RTc Critical compressibility factors are used as characterization parameters in corresponding states methods (especially those of Lydersen) to predict volumetric and thermal properties. The factor varies from about 0.23 for water to 0.26–0.28 for most hydrocarbons to slightly above 0.30 for light gases. Normal Freezing Temperature (Melting Point) The melting point is the temperature at which melting occurs at atmospheric pressure. In most cases, measurements are made in air, making values slightly lower than if the measurements were made in vacuum. Impurities can cause a substantial decrease in the measured melting point. The melting point is very slightly higher than the triple point temperature—the temperature at which equilibrium exists between solid, liquid, and vapor—for a pure compound. For practical purposes, the two temperatures are equal. Reliable methods for predicting melting points have not until recently been advanced. Constantinou and

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES Gani20 derived a group contribution method that shows promise. Initial evaluations show average errors of 5 to 10 percent (10–30 K) on a wide variety of compounds, but larger errors can occur. It is recommended that several compounds of known melting point in the same or a similar family be predicted in order to estimate the probable error. Normal Boiling Temperature The normal boiling temperature (point) is the temperature at which the vapor pressure equals exactly 101,325 Pa (1 atmosphere). Caution should be taken in using values from older references, where the temperature may be reported for the prevailing pressure (0.95–0.97 atm) rather than at 1 atmosphere. If at least two values of vapor pressure very close to 1 atmosphere are available, the normal boiling point can be interpolated or extrapolated on a plot of log P sat vs. 1/T. The section on vapor pressure discusses this in more detail. Various methods are available for estimation of the normal boiling point of organic compounds. Lyman et al.64 review and give calculational procedures for the methods of Meissner, Miller, and Lydersen/ Forman-Thodos. A more recent method that has been determined to be more accurate is the method of Pailhes,80 which requires one experimental vapor pressure point and Lydersen group contributions for critical temperature and critical pressure (Table 2-385). log Pc + (1 − Θ) log (1/p) Tb = T  (2-22) log Pc where T and p = the one low pressure vapor pressure point, in K and atm, respectively Θ = Tc /Tb calculated from Eq. (2-3) using Lydersen contributions Pc = critical pressure calculated from Eq. (2-7) using Lydersen contributions A recent study of the method on a wide variety of complex organics shows an overall average error of less than 2 percent (~10 K). If no vapor pressure point is available, the new group contribution method of Constantinou and Gani20 discussed under the section on melting point gives an overall average error of about 4 percent (~20 K) and may be useful. The method of Miller (Lyman et al.64), which requires only the molecular structure, has also been found to be relatively accurate for organics. Example 4 Estimate the normal boiling point of 2-butanol. One vapor pressure point of 0.802 psia at 100°F is available. Use the Pailhes method, Eq. (2-22). log Pc + (1 − θ) log (1/p) TB = T  log Pc From Example 2, Pc = 4.506 MPa; θ = Tc /Tb = 1.434; T = 100°F = 310.9 K; and p = 0.802 psia = 5528 Pa. log (4.506 × 106) + (1 − 1.434) log (1/5528) Tb = (310.9)  log (4.506 × 106) Tb = (310.9)(1.244) = 386.8 K An accepted experimental normal boiling point is 372.7 K. Note the error here is 3.8 percent, a value above the average. If the vapor pressure point available would have been closer to one atmosphere, the error would have been much lower.

Acentric Factor The acentric factor of a compound (ω) is primarily a measure of the shape of a molecule, though it also measures a molecule’s polarity. It is calculated from the reduced vapor pressure (Prsat ) at a reduced temperature of 0.7 by the definition, Eq. (2-23). ω = −log (Prsat)Tr = 0.7 − 1.000

(2-23)

Critical temperature and pressure are required and can be estimated from the methods of this section. Vapor pressure is predicted by the methods of the next section. Experimental values should be used if available. The acentric factor is used as a third parameter with Tc and Pc in Pitzer-type corresponding states methods to predict volumetric properties and in cubic equations of state such as the Redlich-KwongSoave and Peng-Robinson equations. For simple spherical molecules, the acentric factor is essentially zero, rising as branching and molecu-

2-345

lar weight increases. For compounds of similar size and shape the acentric factor increases slightly with increasing polarity. For mixtures, the acentric factor is usually taken as a simple molar average value of the n components of the mixture. n

ω = xiω i

(2-24)

i=1

Miscellaneous Characterizing Constants The radius of gyration (R ) is a simultaneous size-shape factor varying with the manner in which mass is distributed about the center of gravity of the molecule. For planar molecules, the radius of gyration is R= 

(AB) N 

 M 1/2

A

(2-25)

For three-dimensional molecules, it is R= 

(ABC) 2πN 

 M 1/3

A

(2-26)

AB and ABC are the products of the principal moments of inertia. Moments of inertia are calculated from bond angles and bond lengths. Many values are given by Landolt-Bornstein.53 NA is Avogadro’s number, and M is the molecular weight of the molecule. Stuper et al.105 give a computerized method for prediction of the radius of gyration. The dipole moment (λ) of a molecule is the first moment of the electric charge density of a molecule. Paraffins have dipole moments of zero, while dipole moments of almost all hydrocarbons are small. McClellan68 lists many dipole moments. The computer method of Dixon and Jurs27 is the most useful method for predicting dipole moments. Lyman et al.64 give other methods of calculation. The van der Waals volume and area are characterizing parameters relating molecular configurations. Bondi8 describes group contribution methods for their calculation. VAPOR PRESSURE Vapor pressure is the most important of the basic thermodynamic properties affecting liquids and vapors. The vapor pressure is the pressure exerted by a pure component at equilibrium at any temperature when both liquid and vapor phases exist and thus extends from a minimum at the triple point temperature to a maximum at the critical temperature, the critical pressure. This section briefly reviews methods for both correlating vapor pressure data and for predicting vapor pressure of pure compounds. Except at very high total pressures (above about 10 MPa), there is no effect of total pressure on vapor pressure. If such an effect is present, a correction, the Poynting correction, can be applied. The pressure exerted above a solid-vapor mixture may also be called vapor pressure but is normally only available as experimental data for common compounds that sublime. Correlation Methods Vapor pressure is correlated as a function of temperature by numerous methods mainly derived from the Clapeyron equation discussed in the section on enthalpy of vaporization. The classic simple equation used for correlation of low to moderate vapor pressures is the Antoine4 equation (2-27). B (2-27) ln Psat = A +  T+C A, B, and C are regression constants for the specific compound. The Antoine equation does not fit data accurately much above the normal boiling point. Thus, as regression by computer is now standard, more accurate expressions applicable to the critical point have become usable. The entire DIPPR Compilation24 is regressed with the modified Riedel89 equation (2-28) with constants available for over 1500 compounds. B ln P sat = A +  + C ln T + DT E (2-28) T A, B, C, and D are regression constants and E is an exponent equal to 1, 2, or 6 depending on which regression gives the most accurate fit of the data.

2-346

PHYSICAL AND CHEMICAL DATA

For purposes of the API Technical Data Book (Daubert and Danner23), another modified Riedel equation (2-29) was chosen and found to fit hydrocarbon data well over the entire pressure range. Coefficients are given for several hundred hydrocarbons. B E ln P sat = A +  + C ln T + DT 2 + 2 (2-29) T T Both equations (2-28) and (2-29) are also extrapolatable above the critical temperature where necessary for thermodynamic calculations. The other modern equation used for correlation is the modified and linearized Wagner124 equation (2-30), which has the advantage that it will match critical data exactly, although it cannot be extrapolated above the critical point. The equation is also included with coefficients to several hundred compounds in the Technical Data Book—Petroleum Refining. ln Pr = aX1 + bX2 + cX3 + dX4 where

(2-30)

1−T (1 − Tr)1.5 (1 − Tr)2.6 (1 − Tr)5 X1 = r , X2 =  , X3 =  , X4 =  Tr Tr Tr Tr Pr = P /Pc sat

Tr = T / Tc sat

(2-31)

(ln P ) = 5.92714 − 6.09648/Tr − 1.28862 ln Tr + 0.169347T 6r (2-32) sat (0) r

(1) (ln P sat = 15.2518 − 15.6875/Tr − 13.4721 ln Tr + 0.43577T 6r r ) (2-33)

The method is applicable at reduced temperatures above 0.30 or the freezing point, whichever is higher, and below the critical point. The method is most reliable when 0.5 < Tr < 0.95, where errors in prediction average 3.5 percent when experimental critical properties are known. Errors are higher for predicted criticals. The method is useful when solved iteratively with Eq. (2-23) to predict the acentric factor. Example 5 Estimate the vapor pressure of 1-butene at 98°C. Use Eq. (2-31): (1) sat (0) ln P sat + ω(ln P sat r = (ln P r ) r )

Pure component properties of 1-butene are Tc = 146.4°C, Pc = 4.02 MPa, and ω = 0.1867. 371.1 Tr =  = 0.885 419.5 From Eq. (2-32):

(ln Prsat)(0) = −0.7227

From Eq. (2-33):

(ln Prsat)(1) = −0.6190

3000.538X − 6.761560 log Psat =  43X − 0.987672

ln Prsat = −.7227 + (.1867)(−0.6190) ln Prsat = −0.8383 Prsat = 0.4325 P sat = PrsatPc = (.4325)(4.02) = 1.74 MPa An experimental value is 1.72 MPa. When criticals cannot be estimated with reasonable accuracy, the method of Maxwell and Bonnell67 is recommended. The normal boiling point and the specific gravity at 60°F (15.5°C) are required inputs. According to what vapor pressure range is expected, the vapor pressure is calculated from Eqs. (2-34), (2-35), or (2-36). If the wrong range is selected, the procedure will need to be repeated.

(2-34)

For 0.0013 ≤ X ≤ 0.0022 (2 mm Hg ≤ P ≤ 760 mm Hg): sat

2663.129X − 5.994296 log Psat =  95.76X − 0.972546

(2-35)

For X < 0.0013 (P sat > 760 mm Hg): 2770.085X − 6.412631 log Psat =  36X − 0.989679

(2-36)

X is calculated from Eq. (2-37) and T′b is calculated from Eq. (2-38). Iterative calculation may be required. T ′b  − 0.0002867(T′b) T  X = 748.1 − 0.2145(T′b) (2-37) Psat Tb − T′b = 2.5f(K − 12) log  760 where

Both Riedel and Wagner regressions usually fit data within a few tenths of a percent over the entire range between the triple point and the critical point. Prediction Methods Two methods have gained almost universal acceptance for prediction of the vapor pressure of pure hydrocarbons. The method of Lee and Kesler55 is the preferred method if the critical temperature and the critical pressure of the hydrocarbon is known or can be reasonably predicted by the methods of the first section. The corresponding states method is shown in equation (2-31) with the simple fluid and correction terms to be calculated from equations (2-32) and (2-33), respectively, for any Tr. (1) sat (0) ln P sat + ω(ln P sat r = (ln P r ) r )

For X > 0.0022 (Psat < 2 mm Hg):

(2-38)

Tb = normal boiling point, °R T ′b = normal boiling point corrected to K = 12, °R T = absolute temperature, °R f = correction factor. For all subatmospheric vapor pressures and for all substances having normal boiling points greater than 400°F, f = 1. For substances having normal boiling points less than 200°F, f = 0. For superatmospheric vapor pressures of substances having normal boiling points between 200°F and 400°F, f is given by (Tb − 659.7)/200 K = Watson characterization factor, T b1/3/sp gr

Evaluation of the method for pure hydrocarbons shows errors averaging 8 percent for vapor pressures above 1 mm Hg and 30 percent below 1 mm Hg. The method is also usable for narrow boiling (range up to 50°F) undefined hydrocarbon mixtures with the only change being that the mean average boiling point replaces the normal boiling point in all calculations.

Example 6 Estimate the vapor pressure of tetralin at 150°C (302°F). Its normal boiling point is 207.6°C (405.7°F) and its Watson characterization factor is 9.78. Use the Maxwell-Bonnell method, Eq. (2-35). 2663.129X − 5.994296 log Psat =  95.76X − 0.972546 At 150°C with a normal boiling point of 207.6°C, a vapor pressure between 2 and 760 mm Hg would be expected. Assume T b′ = Tb = 405.7°F as a first trial. From Eq. (2-37): (865.7/762) − 0.0002867(865.7) X =  748.1 − 0.2145(865.7) X = 0.001579 2663.129(0.001579) − 5.994296 log Psat =  = 2.179 95.76(0.001579) − 0.972546 Psat = 151.0 mm Hg Use Eq. (2-38) to calculate the correction. 151 Psat Tb − T′b = 2.5f(K − 12) log  = 2.5(1)(9.48 − 12) log  760 760 Tb − T′b = 4.4° Thus, for the second trial, T′b = 405.7 − 4.4 = 401.3°F. Using Eq. (2-37), recalculate X. (861.3/762) − 0.0002867(861.3) X =  = 0.001568 748.1 − 0.2145(861.3) From Eq. (2-35), log Psat = 2.214; Psat = 163.7 mm Hg. Use Eq. (2-38) to recalculate the correction. 163.7 Tb − T′b = 2.5(1)(9.48 − 12) log  = 4.2° 760 Thus, T′b = 401.5°F. If greater accuracy is desired, carry out a third trial. (An experimental vapor pressure is 161.8 mm Hg.) For nonhydrocarbon organics, vapor pressures above 15 kPa for com-

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES pounds of known or estimable normal boiling point are predicted using the method of Riedel89 given by Eq. (2-39). log P°r = φ(Tr) − (α − 7)ψ(Tr)

Use Eq. (2-44), determining parameters from Eqs. (2-45), (2-46), and (2-47). log Psat = m log Pwsat + C

(2-39)

at T1 = 256.55 K, P1sat = 2.00 × 104 Pa, log P1sat = 4.3010

Correlation functions φ(Tr), ψ(Tr), and ζ(Tr) are given by Eqs. (2-40), (2-41), and (2-42), respectively.

at T2 = 294.95 K, P2sat = 1.067 × 105 Pa, log P2sat = 5.0282 Use Eq. (2-45) to calculate the vapor pressure of water at T1 and T2.

φ(Tr) = 0.118ζ(Tr) − 7 log10 Tr

(2-40)

ψ(Tr) = 0.0364ζ(Tr) − log10 Tr

(2-41)

log Pωsat1 = 2.2282,

Pωsat1 = 169.12 Pa

ζ(Tr) = 36/Tr + 96.7 log10 Tr − 35 − Tr6

(2-42)

log Pωsat2 = 3.4213,

Pωsat2 = 2637.9 Pa

From Eq. (2-46): log P1sat − log P sat 4.3010 − 5.0282 2 m =  =  = 0.6093 log P wsat1 − log P wsat2 2.2282 − 3.4213 From Eq. (2-47):

The Riedel α is calculated from Eq. (2-43). 0.136ζ(Tr ) + log10 Pc − 5.01 α =  0.0364ζ(Tr ) − log10 Tr b

b

(2-43)

b

Critical properties, if not available, can be estimated from the methods of the previous section. Tr is the reduced temperature at the temperature of interest, while Tr is the reduced temperature at the normal boiling point. The method is accurate within 2 to 3 percent above 15 kPa, while errors increase to 10–30 percent at lower pressures. Care should be taken not to use the method below the freezing point temperature.

2-347

C = log P1sat − m log P wsat1 = 4.3010 − 0.6093(2.2282) C = 2.9434

b

T = 273.15 K (0°C) 3129.8 log P = 31.51 −  − 7.1385 log 273.15 + 1.757 × 10−6(273.15)2 273.15 log P sat w = 2.7907

at

sat w

Example 7 Estimate the vapor pressure of thiophene at 500 K. Pure component properties are Tc = 579.4 K, Pc = 5.694 MPa, and Tb = 357.5 K. Use the Riedel method, Eq. (2-39).

log Psat = (0.6093)(2.7907) + 2.9434 log Psat = 4.6438

log P sat r = −φ(Tr) − (α − 7) ψ (Tr) 500 357.5 Tr =  = 0.8629 Tr =  = 0.6170 579.4 579.4 From Eq. (2-42): (36) ζ(Tr ) =  + 96.7 log (0.6170) − 35 − (0.6170)6 = 3.01 (0.6170) From Eq. (2-43): 0.136(3.01) + log (5.694 × 106) − 5.01 α =  (0.0364)(3.01) − log (0.6170)

Psat = 44,030 Pa = 44.0 kPa

b

IDEAL GAS THERMAL PROPERTIES

b

α = 6.749 From Eq. (2-42), calculate ζ(Tr) and then calculate φ(Tr) and ψ(Tr) from Eqs. (2-40) and (2-41). log Prsat = −0.461 − (6.749 − 7)(0.068) log Prsat = −0.444 Prsat = 0.3598 Psat = PrsatPc = (.3598)(5.694) = 2.049 MPa An experimental value is 2.037 MPa. For nonhydrocarbon organics for which normal boiling points are unknown or expected vapor pressures are below 15 kPa, the reference substance method of Othmer and Yu78 as given by Eq. (2-44) is recommended. log Psat = m log P wsat + C

(2-44)

The vapor pressure of water P wsat may be calculated by Eq. (2-45). 3.1298 × 103 log Pwsat = 31.51 −  − 7.1385 log T + 1.757 × 10−6T 2 (2-45) T with temperatures in K and vapor pressures in Pa. Values of the compound specific constants m and c were originally derived by Othmer et al. and greatly expanded to over 600 common organics by Danner and Daubert.22 If constants are not available but any two vapor pressure data points are available, the constants m and C can be calculated using Eqs. (2-46) and (2-47). sat log P sat 1 − log P 2 m =  (2-46) log P wsat1 − log P wsat2 sat C = log P sat 1 − m log P w1

(2-47)

where the subscripts 1 and 2 refer to the two reference temperatures T1 and T2. Average errors at low pressures for compounds with tabulated m and C are within a few percent. When values of m and C are calculated from only two vapor pressure points, the method should be used only for interpolation and limited extrapolation. The method is usable from about 220 K (so long as it is above the freezing point of the compound) to the critical point of water (about 647 K).

Example 8 Estimate the vapor pressure of acetaldehyde at 0°C. Two vapor pressure points are 20.0 kPa at 256.55 K and 107.6 kPa at 294.85 K.

A substance is in the ideal gas state when the volume of its molecules is a zero fraction of the total volume taken up by the substance and when the individual molecules are far enough apart from each other so that there is no interaction between them. Although this only occurs at infinite volume and zero pressure, in practice, ideal gas properties can be used for gases up to a pressure of two atmospheres with little loss of accuracy. Thermal properties of ideal gas mixtures may be obtained by mole-fraction averaging the pure component values. Heat Capacity, Cpo Heat capacity is defined as the amount of energy required to change the temperature of a unit mass or mole one degree; typical units are J/kg·K or J/kmol·K. There are many sources of ideal gas heat capacities in the literature; e.g., Daubert et al.,24 Daubert and Danner,23 JANAF thermochemical tables,15 TRC thermodynamic tables,115,116 and Stull et al.104 If Cpo values are not in the preceding sources, there are several estimation techniques that require only the molecular structure. The methods of Thinh et al.113 and Benson et al.6,7 are the most accurate but are also somewhat complicated to use. The equation of Harrison and Seaton36 for Cpo between 300 and 1500 K is almost as accurate and easy to use: Cpo = a1 + a2C + a3H + a4O + a5N + a6S + a7F + a8Cl + a9I + a10Br + a11Si + a12Al + a13B + a14P + a15E where

(2-48)

C = ideal gas heat capacity, J/mol K a1–a15 = constant parameters obtained from Table 2-387 as a function of temperature C = number of carbon atoms in the molecule H = number of hydrogen atoms in the molecule O = number of oxygen atoms in the molecule N = number of nitrogen atoms in the molecule S = number of sulfur atoms in the molecule F = number of fluorine atoms in the molecule Cl = number of chlorine atoms in the molecule I = number of iodine atoms in the molecule Br = number of bromine atoms in the molecule Si = number of silicon atoms in the molecule Al = number of aluminum atoms in the molecule B = number of boron atoms in the molecule P = number of phosphorus atoms in the molecule E = number of atoms in the molecule excluding the 13 atom-types listed above o p

2-348

PHYSICAL AND CHEMICAL DATA

TABLE 2-387

Values of the Constant Parameters a1–a15 in Eq. (2-48) at Different Temperatures

Temp., K

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

300 400 500 600 800 1000 1500

4.86 0.864 −1.85 −4.61 −7.49 −8.53 −7.37

9.04 12.6 15.5 17.5 20.1 21.6 23.9

5.69 7.37 8.89 10.5 13.1 15.2 17.9

11.4 13.9 15.7 17.5 19.4 20.4 20.6

11.9 14.0 16.0 17.3 19.4 20.4 21.1

15.3 17.0 19.4 20.3 22.3 22.9 22.5

12.7 16.2 17.9 20.1 21.5 22.4 22.1

16.8 18.9 20.2 21.4 22.4 22.8 22.6

18.7 20.5 22.1 23.3 25.0 25.4 24.6

17.8 19.9 21.2 22.4 23.4 23.8 23.0

14.6 17.5 19.6 20.9 23.2 23.9 24.1

15.8 18.3 20.0 21.1 22.3 22.8 23.2

11.5 14.7 17.0 18.3 20.8 22.2 24.2

18.0 20.9 21.6 22.8 23.0 23.4 24.2

19.5 20.8 21.7 22.1 23.0 23.3 23.3

Parameter a

n = number of different atomic groups contained in the molecule Ni = number of atomic groups i contained in the molecule ∆Gi = numeric value of atomic group i obtained from Table 2-388

Results and parameters may be interpolated between temperatures. Average errors are between 2 and 6 percent, with the higher errors at the lower temperatures. Example 9 Using Eq. 2-48 to estimate the ideal gas heat capacity of acetone (C3H6O) at 600 K: Cp° = −4.61 + (17.5)(3) + (10.5)(6) + (17.5)(1) = 128.39 J/mol K

Average errors of 8 to 9 kJ/mol may be expected.

Daubert et al.24 report a value of 121.8 J/mol K.

Enthalpy of Formation The ideal gas standard enthalpy (heat) of formation (∆H°f 298) of a chemical compound is the increment of enthalpy associated with the reaction of forming that compound in the ideal gas state from the constituent elements in their standard states, defined as the existing phase at a temperature of 298.15 K and one atmosphere (101.3 kPa). Sources for data are Refs. 15, 23, 24, 104, 115, and 116. The most accurate, but again complicated, estimation method is that of Benson et al.6,7 A compromise between complexity and accuracy is based on the additive atomic group-contribution scheme of Joback44; his original units of kcal/mol have been converted to kJ/mol by the conversion 1 kcal/mol = 4.1868 kJ/mol:

Example 11 The ∆G°f 298 of phenol is estimated using Table 2-388. The molecular groups are

(2-49)

i=1

where ∆H°f 298 = enthalpy of formation at 298.15 K, kJ/mol n = number of different atomic groups contained in the molecule Ni = number of atomic groups i contained in the molecule ∆Hi = numeric value of atomic group i obtained from Table 2-388. Average expected errors are about 9 kJ/mol. For other temperatures: ∆H°f T = ∆H°f 298 +



T

C°p dT

(2-50)

298

C

1

OH (phenol)

(both ring)

∆G°f 298 = 53.88 + 5(11.30) + (54.05) + (−197.37) = −32.94 kJ/mol

∆G°f T = ∆H°f T − T ∆S°f T

where

(2-52)

∆G°f T = Gibbs energy of formation at T, kJ/mol ∆H°f T = enthalpy of formation at T, kJ/mol (see above) ∆S°f T = entropy of formation at T, kJ/mol K (see below)

Entropy of Formation The ideal gas standard entropy of formation (∆S°f 298) of a chemical compound is the increment of entropy associated with the reaction of forming that compound in the ideal gas state from the constituent elements in their standard state defined as the existing phase at a temperature of 298.15 K and one atmosphere (101.325 kPa). Thus: n

r ∆S°f 298 = S°compound − Ni Selement i

Example 10 The ∆H°f 298 of 2-butanol is estimated using Table 2-388. The molecular groups are 2CH3, 1CH2, 1CH (all nonring), and 1OH (alcohol). Therefore: ∆H°f 298 = 68.29 + 2(−76.45) + (−20.64) + (29.89) + (−208.04) = −283.40 kJ/mol The value from Daubert et al.24 is −292.9 kJ/mol.

Gibbs Energy of Formation The ideal gas standard Gibbs energy of formation (∆G°f 298) of a chemical compound is the increment of Gibbs energy associated with the reaction of forming that compound in the ideal gas state from the constituent elements in their standard state defined as the existing phase at a temperature of 298.15 K and one atmosphere (101.325 kPa). Refs. 15, 23, 24, 104, 115, and 116 are good sources of data. The additive atomic groupcontribution scheme of Joback44 may be used to estimate ∆G°f 298; his original units of kcal/mol have been converted to kJ/mol by the conversion 1 kcal/mol = 4.1868 kJ/mol: n

(2-51)

i=1

where

1

The value from Daubert et al.24 is −32.64 kJ/mol. For other temperatures, the exact Eq. (2-52) may be used at temperature T (K):

See above for discussion of the ideal gas heat capacity (C°p ).

∆G°f 298 = 53.88 + Ni ∆Gi

CH

Therefore,

n

∆H°f 298 = 68.29 + Ni ∆Hi

5

∆G°f 298 = Gibbs energy of formation at 298.15 K, kJ/mol

(2-53)

i=1

where

∆S°f 298 = entropy of formation at 298.15 K and 1 atm, J/mol K S°compound = ideal gas absolute entropy of the compound at 298.15 K and 1 atm, J/mol K n = number of different elements contained in the compound Ni = moles of element i contained in one mole of compound r Selement i = absolute entropy of element i in its standard state at 298.15 K and 1 atm, J/mol K.

Ideal gas absolute entropies of many compounds may be found in Daubert et al.,24 Daubert and Danner,23 JANAF Thermochemical Tables,15 TRC Thermodynamic Tables,115,116 and Stull et al.104 Otherwise, the estimation method of Benson et al.6,7 is reasonably accurate, with average errors of 1–2 J/mol K. Elemental standard-state absolute entropies may be found in Cox et al.21 Values from this source for some common elements are listed in Table 2-389. ∆S°f 298 may also be calculated from Eq. (2-52) if values for ∆H°f 298 and ∆G°f 298 are known.

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-388

Atomic Group Contributions to Estimate ⌬H f°298 and ⌬G f°298 ∆H

Nonring Increments CH3

∆G

−76.45

−43.96

CH2

−20.64

8.42

CH

29.89

58.36

∆H

∆G

Oxygen Increments (Cont.) CHO (aldehyde)

−162.03

−143.48

COOH (acid)

−426.72

−387.87

COO

−337.92

−301.95

−247.61

−250.83

−22.02

14.07

NH (nonring)

53.47

89.39

NH (ring)

31.65

75.61

(ester)

O (except for above) C

82.23

116.02

CH2

−9.63

3.77

CH

37.97

48.53

C

83.99

92.36

C CH C Ring Increments CH2 CH

C

123.34

163.16

(nonring)

23.61

—

79.30

77.71

N

(ring)

55.52

79.93

115.51

109.82

NH

93.70

119.66

−26.80

−3.68

CN

88.43

89.22

NO2

−66.57

−16.83

8.67

40.99

Sulfur Increments SH S (nonring) S (ring)

−17.33 41.87 39.10

−22.99 33.12 27.76

−251.92 −71.55 −29.48 21.06

−247.19 −64.31 −38.06 5.74

87.88

46.43

54.05

−208.04

−189.20

OH (phenol)

−221.65

−197.37

O

(nonring)

−132.22

−105.00

O

(ring)

−138.16

−98.22

C

O (nonring)

−133.22

−120.50

C

O (ring)

−164.50

−126.27

Halogen Increments F Cl Br I

where

ENTHALPY OF VAPORIZATION AND FUSION Enthalpy of Vaporization The enthalpy (heat) of vaporization ∆HV is defined as the difference of the enthalpies of a unit mole or mass of a saturated vapor and saturated liquid of a pure component; i.e., at a temperature (below the critical temperature) and corresponding vapor pressure. ∆HV is related to vapor pressure by the thermodynamically exact Clausius-Clapeyron equation: d ln Psat ∆Hv = −R ∆ZV  d (1/T)

(nonring)

N

11.30

(aromatic or cyclic olefin)

N 136.70

2.09

Oxygen Increments OH (alcohol)

Nitrogen Increments NH2

142.14

79.72

CH (aromatic or cyclic olefin) C

2-349

(2-54)

R = gas constant in energy units ∆ZV = ZG − ZL ZG = compressibility factor of the saturated vapor ZL = compressibility factor of the saturated liquid Psat = vapor pressure T = absolute temperature

(2-55)

If accurate ZG and ZL data are available, excellent ∆HV values can be obtained by differentiating a vapor pressure correlation and using Eq. (2-54). If not, ∆ZV may be esimated by Haggenmacher’s equation34:

2-350

PHYSICAL AND CHEMICAL DATA where Tr 1,2 = reduced temperature = T1 /Tc or T2 /Tc T1,2 = temperature, K Tc = critical temperature, K

TABLE 2-389 Standard-State Entropy of Elements at 298.15 K and 1 Atmosphere Element

State

Absolute entropy J/mol K

C H2 O2 N2 S F2 Cl2 Br2 I2

crystal (graphite) gas gas gas crystal (rhombic) gas gas liquid crystal

5.74 130.571 205.043 191.500 32.054 202.682 222.972 152.21 116.14





P 1/2 ∆Zv = 1 − r Tr where Pr = reduced pressure = P/Pc Tr = reduced temperature = T/Tc 3

Equation (2-59) works best between the normal boiling and critical temperatures, producing values of engineering accuracy.

Example 14 Estimate ∆Hv of ethyl acetate at 450 K, using the normal boiling point values as a basis (see Example 13). ∆Hv1 = 32.23 kJ/mol, Tr1 = 0.6692, and Tr2 = 450.0/523.3 = 0.8599. Substituting in Eq. (2-59): 1 − 0.8599 0.38 ∆Hv(450) = 32.23  = 23.25 kJ/mol 1 − 0.6692 A value of 23.16 kJ/mol is obtained from Daubert et al.24



(2-56)

However, Eq. (2-56) should be used only near or below the normal boiling point; even then, the accuracy of the resulting ∆HV is significantly reduced. The corresponding states approach suggested by Pitzer et al.82 requires only the critical temperature and acentric factor of the compound. For a close approximation, an analytical representation of this method proposed by Reid et al.86 for 0.6 < Tr < 1.0 is: ∆HV /RTc = 7.08(1 − Tr)0.354 + 10.95ω(1 − Tr)0.456

(2-57)

where ∆HV = enthalpy of vaporization, kJ/mol R = gas constant = 0.008314 kJ/mol K Tc = critical temperature, K Tr = reduced temperature, T/Tc T = temperature, K ω = acentric factor

Enthalpy of Fusion The enthalpy (heat) of fusion ∆Hfus is defined as the difference of the enthalpies of a unit mole or mass of a solid and liquid at its melting temperature and one atmosphere pressure of a pure component. There are no generally applicable estimation techniques that are very accurate. However, if the melting temperature is known, the atomic group contribution method of Chickos et al.16 yields approximate results: ∆Hfus = Tmlt∆Sfus where

a = 35.19 NR + 4.289 (NCH2 − 3 NR) where

Example 12 Estimate ∆HV of Propionaldehyde at 350 K. The required properties from Daubert et al.24 are Tc = 504.4 K and ω = 0.2559. Tr = 350.0/504.4 = 0.6939. Substituting in Eq. (2-57): ∆HV /RTc = (7.08)(1 − 0.6939)0.354 + (10.95)(0.2559)(1 − 0.6939)0.456 = 6.289 ∆HV = (6.289)(0.008314)(504.4) = 26.37 kJ/mol The reported value is 26.85 kJ/mol.24 The enthalpy of vaporization at the normal boiling temperature ∆Hvb (kJ/mol) can be estimated by an equation suggested by Riedel90: (ln [Pc /101.325] − 1) (2-58) ∆Hvb = 1.093 RTc Tbr  0.930 − Tbr where R = gas constant = 0.008314 kJ/mol K. Tc = critical temperature, K Tbr = reduced normal boiling temperature = Tb /Tr Tb = normal boiling temperature, K Pc = critical pressure, kPa



Average errors are about 2 percent.

Example 13 Estimate ∆HVb of Ethyl Acetate. The required properties for ethyl acetate are from Daubert et al.24: Tc = 523.3 K, Tb = 350.2 K, and Pc = 3880.0 kPa. Tbr = 350.2/523.3 = 0.6692. Substituting in Eq. (2-58): (ln [3880.0/101.325] − 1) ∆HVb = (1.093)(0.008314)(523.3) (0.6692)  0.930 − 0.6692







= 32.28 kJ/mol The value from Daubert et al.24 is 32.23 kJ/mol. The enthalpy of vaporization decreases with temperature and is zero at the critical point. If the value of an enthalpy of vaporization ∆Hv1 is known at temperature T1, this temperature dependency can be represented by the Watson relation127 to calculate another enthalpy of vaporization ∆Hv2 at any other temperature T2: 1 − Tr 0.38 (2-59) ∆Hv2 = ∆Hv1 2 1 − Tr1





(2-60)

∆Hfus = enthalpy of fusion at the melting temperature, J/mol Tmlt = melting temperature, K ∆Sfus = a + b = entropy of fusion at the melting temperature, J/mol K.

It should be noted that the methodology for a and b results in a ∆Sfus associated with the phase change from a solid at 0 K to the liquid at Tmlt. No entropy changes resulting from solid transitions are taken into account, and ∆Sfus for a substance that undergoes such a transition will be overestimated by this technique.

Maximum errors are in the order of 8 percent.





(2-61)

NR = number of nonaromatic rings NCH2 = number of CH2 atomic groups in nonaromatic ring(s) required to form a cyclic paraffin of the same ring size(s) as contained in the molecule of interest.

Example: For

O

, NCH 2 = 5; a = 0 if there are no nonaromatic rings

in the molecule of interest. If a nonaromatic ring in fact contains a CH2 atomic group, then no consideration of that group in the b term in Eq. (2-62) is required. ng

ns

nf

i=1

j=1

k=1

b = (Ng)i(∆s)i + (Ns)j(Cs)j(∆s)j + (Nf )k(Ct)k(∆s)k where

(2-62)

ng = number of different nonring or aromatic C-H atomic groups bonded to other carbon atoms in the molecule of interest (Ng)i = number of C-H atomic groups i bonded to other carbon atoms in the molecule of interest ns = number of different nonring or aromatic C-H atomic groups bonded to at least one functional group or atom in the molecule of interest (Ns)j = number of C-H atomic groups j bonded to at least one functional group or atom in the molecule of interest nf = number of different functional groups or atoms in the molecule of interest (Nf)k = number of functional groups or atoms k in the molecule of interest (Cs)j = coefficient for C-H atomic group j bonded to at least one functional group or atom in the molecule of interest; numeric values for C-H atomic groups are found in Table 2-390 Ct = coefficient for the functional group or atom k in the molecule of interest, where t = the total number of functional groups or atoms in the molecule of interest. Exception: Molecules containing any number of fluorine atoms are treated as having only one functional fluorine atom. Numeric values of C1–C4 are in Table 2-391 for functional groups or atoms (∆s)i, j, k = contribution of the atomic group or atom i, j, or k to the entropy of fusion, J/mol K. Numeric values for C-H atomic groups are in Table 2-390; values for functional groups or atoms are in Table 2-391

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-390 Cs and ⌬s Values for C-H Atomic Groups to Estimate ⌬Hfus Cs

∆s

CH3

1.0

18.33

CH2

1.0

9.41

CH

0.69

−16.19

C

0.67

−38.70

CH2

1.0

14.56

CH

3.23

4.85

C

1.0

−11.38

CH

1.0

10.88

C

1.0

2.18

1.0

6.44

Nonring

2-351

There are no reliable prediction methods for solid heat capacity as a function of temperature. However, the atomic element contribution method of Hurst and Harrison,41 which is a modification of Kopp’s Rule,50 provides estimations at 298.15 K and is easy to use: n

Aromatic CH C

(bonded to paraffinic C)

1.0

−10.33

C

(bonded to olefinic C or non-C)

1.0

−4.27

CpS = Ni ∆Ei

(2-63)

i=1

where CpS = solid heat capacity at 298.15 K, J/mol K n = number of different atomic elements in the compound Ni = number of atomic elements i in the compound ∆Ei = numeric value of the contribution of atomic element i found in Table 2-393 Average errors are in the 9–10 percent range. Example 15 Estimate solid heat capacity of dibenzothiophene, C12H8S. The required atomic element contributions from Table 2-393 are: C = 10.89, H = 7.56, and S = 12.36. Substituting in Eq. (2-63): CpS = (12)(10.89) + (8)(7.56) + (1)(12.36) = 203.52 J/mol K Daubert et al.24 report a value of 198.5 J/mol K.

Liquid Heat Capacity The two commonly used liquid heat capacities are either at constant pressure or at saturated conditions. There is negligible difference between them for most compounds up to a reduced temperature (temperature/critical temperature) of 0.7. Liquid heat capacity increases with increasing temperature, although a minimum occurs near the triple point for many compounds. There are a number of reliable estimating techniques for obtaining pure-component liquid heat capacity as a function of temperature, including Ruzicka and Dolmalski,93,94 Tarakad and Danner,112 and Lee and Kesler.55 These methods are somewhat complicated. The relatively simple atomic group contribution approach of Chueh and Swanson17 for liquid heat capacity at 293.15 K is presented here: n

CpL = Ni ∆cpi + 18.83m

(2-64)

i=1

1.0

−2.51

CH

0.76

−15.98

C

1.0

−32.97

CH

0.62

−4.35

C

0.86

−11.72

1.0

−5.36

C

(bonded to acetylenic C)

Ring

C or

C

where CpL = liquid heat capacity at 293.15 K, J/mol K. n = number of different atomic groups in the compound Ni = number of atomic groups i in the compound ∆cpi = numeric value of the contribution of atomic element i found in Table 2-394. The original units of cal/mol K have been converted to J/mol K by the conversion 1 cal/mol K = 4.184 J/mol K m = number of carbon groups requiring an additional contribution, which are those that are joined by a single bond to a carbon group, which in turn is connected to a third carbon group by a double or triple bond. If a carbon group meets this criterion in more than one way, m should be increased by one for each of the ways. Exceptions: CH3 groups or carbon groups in a ring never require an additional contribution; and the first additional contribution for a CH2 group is 10.46 J/mol K rather than 18.83 J/mol K. However, if the CH2 group meets the criterion in a second way, the second additional contribution reverts to the 18.83 J/mol K value (see Example 17, below).

Chickos et al.16 report an average error of 2050 J/mol for monofunctional molecules and 3180 J/mol for multifunctional molecules when using their method to estimate ∆Hfus. Four example estimations are shown in Table 2-392.

Errors should be less than 6 percent for all compounds except for acids, amines, and halides.

SOLID AND LIQUID HEAT CAPACITY

Example 16 Estimate the liquid heat capacity at 293.15 K of 2-butanol. The atomic groups are:

The heat capacity is defined as the amount of energy required to change the temperature of a unit mass or mole one degree; typical units are J/kg·K or J/kmol·K. Solid Heat Capacity Solid heat capacity increases with increasing temperature, with steep rises near the triple point for many compounds. When experimental data are available, a simple polynomial equation in temperature is often used to correlate the data. It should be noted that step changes in heat capacity occur if the compound undergoes crystalline state changes at different temperatures.

2

CH3

1

CH2

1 CHOH Substituting in Eq. (2-64) the atomic group contributions from Table 2-394 with m = 0: CpL = (2)(36.82) + (1)(30.38) + (1)(76.15) = 180.17 J/mol K

The value from Daubert et al.24 is 190.3 J/mol K.

TABLE 2-391 Ct and Ds Values for Functional Groups and Atoms to Estimate DHfus ∆s

C1

C2

C3

C4

OH (alcohol)

1.0

12.6

18.9

26.4

1.13

OH (phenol)

1.0

1.0

1.0

1.0

16.57

O

(ether, nonring)

1.0

1.0

1.0

1.0

1.09

O

(ether, ring)

1.0

1.0

1.0

1.0

1.34

C

O (ketone, nonring)

1.0

1.0

3.14

C

O (ketone, ring)

1.0

1.0

−1.88

CHO (aldehyde)

1.0

COOH (acid)

1.0

1.83

1.88

1.72

14.90

COO

(ester)

1.0

1.0

1.0

1.0

3.68

NH2 (aliphatic)

1.0

1.82

16.23

NH2 (aromatic)

1.0

1.0

15.48

NH (nonring)

1.0

1.0

−2.18

NH (ring)

1.0

1.84

N

(nonring)

1.0

−15.90

N

(ring)

1.0

1.0

−17.07

N

(ring)

1.0

1.0

1.67

N

(aromatic)

1.0

1.0

CN (nitrile)

1.0

1.4

NO2

1.0

1.0

1.0

1.0

26.19

1.0

1.0

−0.42

1.0

1.0

17.99

19.66

1.0

7.32 9.62

1.0

17.36

O C NH2 O C NH SH S S

(nonring)

1.0

(ring)

1.0

–

2-352

0.36 1.0

7.20 2.18

SO2 (nonring)

1.0

F (on

C)

1.0

1.0

1.0

1.0

14.73

F (on

C)

1.0

1.0

1.0

1.0

13.01

F (on ring C)

1.0

1.0

1.0

1.0

15.90

Cl

1.0

2.0

2.0

1.93

8.37

Br

1.0

1.0

1.0

0.82

17.95

I

1.0

1.0

3.26

16.95

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-392

2-353

Examples of Estimations of DHfus, J/mol Melting temp., K

Molecule

Atomic group cyclopentane

(35.19)(1) + (4.289) [5 − (3)(1)]

=

43.77

(1)(0.76)(−15.98)

=

−12.14

(1)(1.0)(17.99) ∆Sfus

= =

17.99 49.62

∆Hfus = (49.62)(155.4) ∆Hfus = (experimental)16

= =

(35.19) (1) + (4.289)[5 − (3)(1)]

=

43.77

CH (aromatic)

(4)(6.44)

=

25.76

C

(ring)

(1)(−11.72)

=

−11.72

C

(ring)

(1)(0.86)(−11.72)

=

−10.08

CH (ring)

(1)(−4.35)

=

−4.35

CH (ring)

(1)(0.62)(−4.35)

=

−2.69

(1)(1.0)(2.18) ∆Sfus

= =

2.18 42.87

∆Hfus = (42.87)(304.5) ∆Hfus = (experimental)16

= =

CH (aromatic)

(4)(6.44)

=

25.76

C

(aromatic)

(1)(1.0)(−4.27)

=

−4.27

C)

(1)(1.0)(13.01)

=

13.01

(1)(−10.33)

=

−10.33

CH

(1)(0.69)(−16.19)

=

−11.17

COOH

(1)(1.88)(14.90)

=

28.01

OH (alcohol)

(1)(18.9)(1.13) ∆Sfus

= =

21.36 62.37

∆Hfus = (62.37)(363.0) ∆Hfus = (experimental)16

= =

CH2

(2)(1.0) 9.41

=

18.82

CH

(2)(0.69)(−16.19)

=

−22.34

OH (alcohol)

(2)(26.4)(1.13)

=

59.66

Br

(2)(0.82)(17.95) ∆Sfus

= =

29.44 85.58

∆Hfus = (85.58)(338.2) ∆Hfus = (experimental)16

= =

CH SH

155.4

Contribution

(ring)

SH

(t = 1) cyclopentane

7710.9 7802

S 304.5 (t = 1)

S

(ring)

13054 11823

F CH

363.0

COOH

F (on

OH C

(aromatic)

(t = 3)

HO

CH2

H

Br

C

C

Br H (t = 4)

338.2 CH2

OH

22640 20959

28943 29291

2-354

PHYSICAL AND CHEMICAL DATA

TABLE 2-393 Atomic Element Contributions to Estimate Solid Heat Capacity at 298.15 K Atomic element

∆E

Atomic element

∆E

Atomic element

∆E

C H O N S F Cl Br I Al B

10.89 7.56 13.42 18.74 12.36 26.16 24.69 25.36 25.29 18.07 10.10

Ba Be Ca Co Cu Fe Hg K Li Mg Mn

32.37 12.47 28.25 25.71 26.92 29.08 27.87 28.78 23.25 22.69 28.06

Mo Na Ni Pb Si Sr Ti V W Zr All other

29.44 26.19 25.46 31.60 17.00 28.41 27.24 29.36 30.87 26.82 26.63

TABLE 2-394

Example 17 Estimate liquid heat capacity at 293.15 K of 1,4-pentadiene, CH2苷CHCH2CH苷CH2. The atomic groups are: 2

CH2

2

CH

1

CH2

The CH2 group is twice joined by a single bond to a carbon group, which in turn is connected to a third carbon group by a double bond, and m = 2. However, by the second exception, the first additional contribution is 10.46 J/mol K rather than 18.83 J/mol K. Substituting in Eq. (2-64) the atomic group contributions from Table 2-394: CpL = (2)(21.76) + (2)(21.34) + (1)(30.38) + 10.46 + 18.83 = 145.87 J/mol K

Atomic Group Contributions to Estimate Liquid Heat Capacity at 293.15 K ∆cp

Nonring Increments CH3

36.82

CH2

30.38

CH

20.92

∆cp Oxygen Increments (Cont.) CH2OH CHOH

7.36

CH2

21.76

CH

21.34

C

15.90

CH C Ring Increments CH2 CH

OH (except for above) ONO2 Nitrogen Increments NH2

C

CH

43.93

24.69

N

31.38

24.69

N

25.94

CN

(ring)

Sulfur Increments SH

12.13

Halogen Increments F

18.83 58.16

44.77 33.47

16.74

Cl (first or second on a carbon)

35.98

Cl (third or fourth on a carbon)

25.10

Br

37.66

I

35.98

22.18

Oxygen Increments O C

58.58

NH

18.41

or

44.77 119.24

S C

76.15

111.29

COH C

73.22

O

35.15 52.97

CHO (aldehyde)

52.97

COOH (acid)

79.91

COO

60.67

(ester)

Hydrogen Increment H (for formic acid, formates, hydrogen cyanide, etc.)

14.64

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES Daubert et al.24 report a value of 145.6 J/mol K. For liquid mixtures, the values of the pure components can be molefraction-averaged. This procedure neglects any heat of mixing effects.

DENSITY Density is defined as the mass of a substance contained in a unit volume. In the SI system of units, the ratio of the density of a substance to the density of water at 15°C is known as its relative density, while the older term specific gravity is the ratio relative to water at 60°F. Various units of density, such as kg/m3, lb-mass/ft3, and g/cm3, are commonly used. In addition, molar densities, or the density divided by the molecular weight, is often specified. This section briefly discusses methods of correlation of density as a function of temperature and presents the most common accurate methods for prediction of vapor, liquid, and solid density. Correlation Methods Vapor densities are not correlated as functions of temperature alone, as pressure and temperature are both important. At high temperatures and very low pressures, the ideal gas law can be applied; while at moderate temperature and low pressure, vapor density is usually correlated by the virial equation. Both methods will be discussed later. Molar liquid density (ρ) is best correlated by an equation adopted from the Rackett predictor. The equation has the form of Eq. (2-65): A ρ = D (2-65) T B 1 −  C The regression constants A, B, and D are determined from the nonlinear regression of available data, while C is usually taken as the critical temperature. The liquid density decreases approximately linearly from the triple point to the normal boiling point and then nonlinearly to the critical density (the reciprocal of the critical volume). A few compounds such as water cannot be fit with this equation over the entire range of temperature. Liquid density data to be regressed should be at atmospheric pressure up to the normal boiling point, above which saturated liquid data should be used. Constants for 1500 compounds are given in the DIPPR compilation.24 Solid density data are sparse and usually only available over a narrow temperature range, for which the general decrease in density with temperature is approximately linear. Vapor Density Prediction A myriad of methods exist for prediction of vapor density as a function of temperature and pressure. This section will only present the most accurate and generally used methods. For simple molecules at temperatures above the critical and at pressures no more than a few atmospheres, the ideal gas law, Eq. (2-66), may be used to estimate vapor density.





1 P ρ== V RT

(2-66)

At slightly higher pressures up to a reduced pressure of about 0.4, the truncated virial equation, Eq. (2-67), is commonly used for all types of organic fluids. PV B Z==1+ (2-67a) RT V P Z =  = 1 + Bρ ρRT

(2-67b)

Second virial coefficients, B, are a function of temperature and are available for about 1500 compounds in the DIPPR compilation.24 The second virial coefficient can be regressed from experimental PVT data or can be reasonably and accurately predicted. Tsonopoulos117 proposed a prediction method for nonpolar compounds that requires the critical temperature, critical pressure, and acentric factor. Equations (2-68) through (2-70) describe the method. BPc (2-68)  = B0 + ωB1 RTc



 

 



 

 

 

2-355

0.330 0.1385 0.0121 0.000607 B0 = 0.1445 −  −  −  −  Tr T r2 T 3r T r5

 (2-69)



0.0331 0.423 0.008 −  −  (2-70) B1 = 0.0637 +  T r2 T 3r T 8r For non-hydrogen-bonding polar compounds such as carbonyls and ethers, Tsonopoulos117 recommends that Eq. (2-68) be expanded to a third term that is a function of the reduced dipole moment (µr) as described by Eqs. (2-71) through (2-73): BPc (2-71)  = B0 + ωB1 + B2 RTc B2 = −0.0002410λr − 4.308 × 10−21λ8r (2-72) 105λ2p Pc λr =  (2-73) T c2 The dipole moment λp in Eq. (2-73) is in debyes, while Pc is in atm and Tc is in K. Units must be watched carefully. For hydrogen-bonding molecules, Eq. (2-71) can be used with a value of B2 calculated by Eq. (2-74). a b B2 = 6 − 8 (2-74) Tr Tr Variables a and b are specific constants reported by Tsonopoulos117 for some alcohols and water (e.g., methanol: a = 0.0878, b = 0.0560; and water: a = 0.0279, b = 0.0229). Tsonopoulos also gives specific prediction methods for haloalkanes118 and water pollutants.119 Example 18 Estimate the molar volume of isobutane at 155°C and 1.0 MPa pressure. Properties of isobutane are Tc = 135.0°C, Pc = 3.647 MPa, and ω = 0.1170. 1.0 155 + 273.1 Tr =  = 1.05 Pr =  = 0.274 135 + 273.1 3.647 Since reduced pressure is below 0.4, use virial equation (2-67a). Calculate B by the Tsonopoulos method, Eq. (2-68). BPc  = B0 + ωB1 RTc Using Eq. (2-69):



 

 

 

0.330 0.1385 0.0121 0.000607 −  −  B0 = 0.1445 −  −  1.05 1.052 1.053 1.055



B0 = 0.1445 − 0.3143 − 0.1256 − 0.0105 − 0.0005 = −0.3064 Using Eq. (2-70):



 

 

0.0331 0.423 0.008 −  −  B1 = 0.0637 +  1.052 1.053 1.058



B1 = 0.0637 + 0.0300 − 0.3654 − 0.0054 = −0.2771 [−0.3064 + (0.1770)(−0.2771)](8314)(408.1) B =  = −0.3307 (3.647 × 106) B PV =1+ RT V 106 V 0.3307  = 1 −  (8314)(428.1) V Trial and error or the quadratic formula can be used for the solution. If you opt for trial and error, start with the ideal gas value (B = 0) where V = 3.559 m3/ kmole [Eq. (2-66)]. Solving, V = 3.190 m3/kmole. For prediction of vapor density of pure hydrocarbon and nonpolar gases, the corresponding states method of Pitzer et al.82 is the most accurate method, with errors of less than 1 percent except in the critical region where errors of up to 30 percent can occur. The method correlates the compressibility factor by Eq. (2-75), after which the density can be calculated by Eq. (2-75): Z = Z(0) + ωZ(1) (2-75) P 1 ρ== (2-76) ZRT V Z(0) is the compressibility factor for the simple fluid, while Z(1) is the correction term for molecular acentricity, both of which are functions of Tr and Pr. Both

2-356

PHYSICAL AND CHEMICAL DATA

plots and detailed tabulations of the functions are available in the Technical Data Book.23 Critical temperature and pressure and the acentric factor from tabulations or as predicted are required. For hydrogen, Tc and Pc should be taken as 41.7 K and 2100 kPa, respectively. For approximate calculations, Figs. 2-35 and 2-36 should be used for calculating Z(0) and Z(1), respectively, for superheated vapors with 0.2 ≤ Pr ≤ 10. Z(0) will approach 1, and Z(1) will approach 0 for Pr < 0.2. More accurately, Eq. (2-77) can be used for Pr between 0 and 0.2.

FIG. 2-35

P Z = 1 + r [(0.1445 + 0.073ω) − (0.330 − 0.46ω)T r−1 Tr − (0.1385 + 0.50ω)T r−2 − (0.0121 + 0.097ω)T r−3 − 0.0073ωT r−8]

(2-77)

Extension of the pressure range to Pr = 14 is available in the Technical Data Book. For saturated vapor densities, the values of Z(0) and Z(1) are tabulated as a

Generalized compressibility factors—Pitzer Method, simple fluid term.

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES function of reduced pressure in Table 2-395. If the saturation temperature rather than the saturation pressure is known, the vapor pressure of the compound can be determined either from data or the vapor pressure prediction methods discussed earlier.

As high pressure, use Eq. (2-75) to calculate Z and then Eq. (2-76) to estimate the molar volume. The properties of isobutane necessary are Tc = 135.0°C, Pc = 3.647 MPa, and ω = 0.1770.

Example 19 Estimate the molar volume of isobutane at 155°C and 8.6 MPa pressure.

FIG. 2-36

2-357

Generalized compressibility factors—Pitzer Method, correction term.

155 + 273.1 Tr =  = 1.05 135 + 273.1

8.6 Pr =  = 2.36 3.647

2-358

PHYSICAL AND CHEMICAL DATA

TABLE 2-395

Saturated Vapor Density Parameters

Pr

Z (0)

Z (1)

Pr

Z (0)

Z (1)

1.00 0.99 0.98 0.97 0.96 0.95 0.94 0.92 0.90 0.85 0.80 0.75 0.70

0.291 0.35 0.38 0.40 0.41 0.42 0.43 0.45 0.47 0.50 0.53 0.56 0.59

−0.080 −0.083 −0.085 −0.087 −0.088 −0.089 −0.089 −0.090 −0.091 −0.090 −0.087 −0.081 −0.075

0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05

0.615 0.64 0.665 0.688 0.711 0.734 0.758 0.783 0.809 0.835 0.864 0.896 0.935

−0.069 −0.063 −0.056 −0.049 −0.041 −0.033 −0.025 −0.018 −0.012 −0.008 −0.005 −0.002 0.000

Using Fig. 2-35, Z(0) = 0.385. Using Fig. 2-36, Z(1) = −0.063. Z = Z(0) + ω Z(1) Z = 0.385 + 0.1770(−0.063) = 0.374 ZRT (0.374)(8314)(428.1) V =  =  P 8.6 × 106 V = 0.155 m3/kmol An experimental value for Z is 0.377. Note that use of the Lee-Kesler fit [Eq. (2-78)] would give a slightly more accurate answer than the graphical method, and this fit is used for any computer applications. Lee and Kesler55 fit the entire Pitzer method to equations, rewriting the basic Eq. (2-75) with respect to a heavy reference fluid (n-octane) as shown by Eq. (2-78). ω (h) Z = Z(0) +  (Z − Z(0)) ω(h)

(2-78)

where h specifies the heavy reference fluid with an acentric factor of 0.3978. The parameters in the equation are calculated for the simple fluid and the heavy reference fluid with an acentric factor of 0.3978. The parameters in the equation are calculated for the simple fluid and the heavy reference fluid from Eq. (2-79) PrVr B C D c4 γ −γ Z(i) =  = 1 +  + 2 + 5 +  β + 2 exp 2 Tr Vr V r Vr T r3 V 2r Vr Vr



  

(2-79)

     

b2 b3 b4 B = b1 −  − 2 − 3 Tr Tr Tr

   

c2 c3 C = c1 −  − 3 Tr Tr

 

d2 D = d1 +  Tr where

Z(i) = Z(0) when the constants in the equation correspond to the simple fluid and Z(h) when the constants in the equation correspond to the heavy reference fluid P = pressure, kPa Pc = critical pressure of the compound whose density is sought, kPa V = molar volume of the simple fluid or the heavy reference fluid, as the case may be, in m3/kmole. R = gas constant = 8.3140 m3 kPa/k mole K Tc = critical temperature of the compound whose density is sought, K T = temperature, K

Constant

Simple fluid

Heavy reference fluid

b1 b2 b3 b4 c1 c2 c3 c4 d1 × 104 d2 × 104 β γ

0.1181193 0.265728 0.154790 0.030323 0.0236744 0.0186984 0.0 0.042724 0.155488 0.623689 0.65392 0.060167

0.2026579 0.331511 0.027655 0.203488 0.0313385 0.0503618 0.016901 0.041577 0.48736 0.0740336 1.226 0.03754

For hydrocarbon and nonpolar gas mixtures, the Pitzer pure component method can be used to predict vapor density by replacing the true critical properties with pseudocritical properties defined in Eqs. (2-80) and (2-81) by Kay.47 n

Tpc = xiTci

(2-80)

i=1 n

Ppc = xiPci

(2-81)

i=1

The mixture acentric factor, Eq. (2-82), can also be used. n

ω = xiωi

(2-82)

i=1

Errors in compressibility factors tabulated for over 6500 data points rarely exceed 2 percent except in the critical region, where 15 percent errors may be expected and 50 percent errors can occur. For mixtures near the critical point, special techniques are available as discussed in the sixth chapter of the Technical Data Book. For pure organic vapors, the Lydersen et al.63 corresponding states method is the most accurate technique for predicting compressibility factors and, hence, vapor densities. Critical temperature, critical pressure, and critical compressibility factor defined by Eq. (2-21) are used as input parameters. Figure 2-37 is used to predict the compressibility factor at Zc = 0.27, and the result is corrected to the Zc of the desired fluid using Eq. (2-83). Z = Z@ Zc = 0.27 + Di(Zc − 0.27)

(2-83)

Di is equal to Da read from Fig. 2-38 if Zc > 0.27; and Di is equal to Db read from Fig. 2-39 if Zc < 0.27. At reduced temperatures less than 0.9, Di can be taken as 0. The density is then calculated from Eq. (2-76). All families of organic compounds except mercaptans and carboxylic acids are predicted within an average deviation of 5 percent. No specific mixing rules have been tested for predicting compressibility factors for defined organic mixtures. However, the Lydersen method using pseudocritical properties as defined in Eqs. (2-80), (2-81), and (2-82) in place of true critical properties will give a reasonable estimate of the compressibility factor and hence the vapor density. Vapor densities for pure compounds can also be predicted by cubic equations of state. For hydrocarbons, relatively accurate RedlichKwong-type equations such as the Soave98 and Peng-Robinson81 equations are often used. Both require only Tc, Pc, and ω as inputs. For organic compounds, the Lee-Erbar-Edmister54 equation (which requires the same input parameters) has been used with errors essentially equivalent to those determined for the Lydersen method. While analytical equations of state are not often used when only densities are required, values from equations of state are used as inputs to equation of state formulations for thermal and equilibrium properties. Liquid Density Prediction Methods for the prediction of pure saturated hydrocarbons and nonhydrocarbon organics, compressed hydrocarbon liquids, and defined and undefined hydrocarbon mixtures were evaluated. Only the most accurate and convenient methods are included here. The most convenient method for predicting the saturated liquid density of both pure hydrocarbons and pure organic liquids is the method of Rackett84 as modified by Spencer and Danner.99 Equation (2-84) is used to calculate the saturated liquid molar density at any temperature using input parameters of Tc, Pc, and ZRA. ZRA is a parameter regressed from experimental data. Values for some common substances are given in Table 2-396. Extensive tabulations are given in the Technical Data Book23 for hydrocarbons and nonhydrocarbons as well as organic and inorganic gases. Additional values are given in the Data Prediction Manual22 for nonhydrocarbons. RTc 1 (2-84)  = Vsat =  Z nRA ρsat Pc

 

n = 1.0 + (1.0 − Tr)2/7 Errors for hydrocarbons between the triple and critical points average about 0.7 percent, with organics averaging about 1.2 percent. The cor-

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES

FIG. 2-37

2-359

Generalized compressibility factor—Lydersen Method, Zc = 0.27.

relation is especially sensitive to the value of ZRA near the critical point. If no value of ZRA is available or derivable, the critical compressibility factor can be used in Eq. (2-84) as originally proposed by Rackett. Use of Zc increases the average error to about 3.0 percent.

Example 20 Estimate the density of saturated liquid propane at 0°C. Use Eq. (2-84).

1   Z 

Pc ρsat =  RTc

n RA

n = 1.0 + (1.0 − Tr)2/7

2-360

PHYSICAL AND CHEMICAL DATA Pure component properties of propane are M = 44.1, Tc = 96.7°C, and Pc = 4.246 MPa. From Table 2-388, ZRA = 0.2763. 273.1 Tr =  = 0.739, n = 1.0 + (1.0 − 0.739)2/7 = 1.6813 369.8 n = 1.6813 (4.246 × 106) 1 ρsat =   (8314)(369.8) 0.27631.6813





ρsat = 1.201 kmole/m = 529.6 kg/m3 3

An accepted experimental value is 531 kg/m3. An alternate method with approximately the same accuracy as the Rackett method is the COSTALD method of Hankinson and Thomson.35 The critical temperature, a characteristic volume near the critical volume, and an acentric factor optimized for vapor pressure prediction by the Soave98 equation of state are required input parameters. The method is detailed in the Technical Data Book.23 Prediction of the density of compressed pure liquid hydrocarbons and their defined mixtures are readily and accurately predicted by the method of Lu.62 One value of low pressure liquid density and the critical temperature and pressure are required to predict the density at higher pressures from Eq. (2-85).

 

C ρ2 = 2 ρ1 C1

(2-85)

The constants C1 and C2 are both obtained from Fig. 2-40: C1, usually from the saturated liquid line; and C2, at the higher pressure. Errors should be less than 1 percent for pure hydrocarbons except at reduced temperatures above 0.95 where errors of up to 10 percent may occur. The method can be used for defined mixtures substituting pseudocritical properties for critical properties. For mixtures, the Technical Data Book—Petroleum Refining gives a more complex and accurate mixing rule than merely using the pseudocritical properties. The saturated low pressure value should be obtained from experiment or from prediction procedures discussed in this section for both pure and mixed liquids. FIG. 2-38

Compressibility factor correction for Lydersen Method, Zc > 0.27.

Example 21 Estimate the liquid density of n-nonane at 104.5°C and 6.893 MPa pressure. A tabulated value of liquid density at 60°F (15.5°C) and 1 atm is 719.8 kg/m3. Pure component properties are Tc = 321.5°C and Pc = 2.288 MPa. Use Eq. (2-85) to correct a low pressure density. C ρ2 = 2 ρ1 C1 where ρ1 is a reference density. C2 and C1 are functions of Tr and Pr. 288.6 377.6 Tr2 =  = 0.635 Tr1 =  = 0.485 594.6 594.6

 

0.1013 6.893 Pr1 =  = 0.0443 Pr2 =  = 3.013 2.288 2.288 From Fig. 2-40, C1 = 1.08 and C2 = 0.998.





0.998 ρ2 =  (719.8) = 665.1 kg/m3 1.08 An accepted experimental value at this temperature is 658.5 kg/m3, a 1 percent difference. An analytical method for the prediction of compressed liquid densities was proposed by Thomson et al.114 The method requires the saturated liquid density at the temperature of interest, the critical temperature, the critical pressure, an acentric factor (preferably the one optimized for vapor pressure data), and the vapor pressure at the temperature of interest. All properties not known experimentally may be estimated. Errors range from about 1 percent for hydrocarbons to 2 percent for nonhydrocarbons. For prediction of the densities of a defined liquid mixture at its bubble point (ρbp), the method of Spencer and Danner100 is the simplest. The density is calculated from Eq. (2-86) using inputs from Eqs. (2-87) and (2-88). For hydrocarbons, Tmc is calculated by Eqs. (2-89) through (2-92) if high accuracy is desired or by Eq. (2-93) for a less accurate answer. 1 =R ρbp

 x P Z n

Tc

i

i

i=1

ci

[1 + (1 − Tr)2/7] RAm

(2-86)

n

ZRAm = xi ZRAi

(2-87)

i=1

T Tr =  Tmc n

FIG. 2-39

Compressibility factor correction for Lydersen Method, Zc < 0.27.

(2-88) n

Tmc = φiφjTcij i=1 j=i

(2-89)

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-396

2-361

The Modified Rackett Equation Input Parameters for Calculating Pure Saturated Liquid Densities

Liquid

ZRA

Hydrocarbons Methane Ethane Propane n-Butane 2-Methylpropane (isobutane) n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane n-Dodecane n-Tetradecane n-Hexadecane n-Octadecane n-Eicosane Cyclopentane Methylcyclopentane Cyclohexane Methylcyclohexane Ethene (ethylene) Propene (propylene) 1-Butene cis-2-Butene trans-2-Butene 2-Methylpropene (isobutylene) 1-Hexene 1,3-Butadiene 2-Methyl-1,3-butadiene Ethyne (acetylene) Propyne (methylacetylene) Benzene Methylbenzene (toluene) Ethylbenzene 1,2-Dimethylbenzene (o-xylene) 1,3-Dimethylbenzene (m-xylene) 1,4-Dimethylbenzene (p-xylene) Isopropylbenzene (cumene) Biphenyl Naphthalene

Liquid

0.2880 0.2819 0.2763 0.2730 0.2760 0.2685 0.2637 0.2610 0.2569 0.2555 0.2527 0.2471 0.2270 0.2386 0.2292 0.2281 0.2709 0.2712 0.2729 0.2702 0.2813 0.2783 0.2735 0.2705 0.2722 0.2727 0.2654 0.2713 0.2680 0.2707 0.2703 0.2696 0.2645 0.2619 0.2626 0.2594 0.2590 0.2616 0.2746 0.2611

xiVci φi =  n

xjVc j

(2-90)

j=1

 Tcij = T c Tc (1 − kij)

(2-91)

j

i



1/3 1/3  V V ci  cj 

kij = 1.0 −  (Vc1/3 + Vc1/3 )/2 i j



3

(2-92)

n

Tmc = xi Tci

(2-93)

i=1

Errors for binary hydrocarbon systems average about 2.5 percent except near the critical, where errors can approach 20 percent. If inorganic gases are included, errors average 4 percent except at high concentrations of carbon dioxide or hydrogen, where higher errors would be expected. If the simplified method for predicting Tmc is used, average errors of 5 to 7 percent should be expected for both binary hydrocarbon and nonhydrocarbon systems. No data are available to test the method for systems with more than two components. A similarly accurate but slightly more complex method for prediction of densities of defined liquid hydrocarbon mixtures at their bubble points was published by Hankinson and Thomson35 and was previously cited for prediction of pure liquid hydrocarbons. For undefined hydrocarbon mixtures, the liquid density may be predicted at any temperature (T) from the mean average boiling point (MeABP) and the specific gravity (sp gr) by Eq. (2-94), adopted from Ritter et al.92



P = 62.3636 (sp gr)2 (1.2655)(sp gr) − 0.5098 + 8.011 × 10−5MeABP(T − 519.67) 1/2 −  (2-94) MeABP 3 The density is calculated in lbm /ft if the temperatures are both in °R. Errors



ZRA

Organics Acetic acid Methanol Ethanol 2-Propanol Acetaldehyde Acetone Methyl ethyl ketone Methyl isobutyl ketone Ethylamine Aniline Methyl formate Methyl acetate Ethyl acetate Ethyl acrylate Methyl-n-butyl ether Diethyl ether Diisopropyl ether

0.2242 0.2340 0.2523 0.2508 0.2387 0.2448 0.2524 0.2589 0.2640 0.2607 0.2581 0.2553 0.2538 0.2583 0.2655 0.2643 0.2699

Halogen Compounds Methyl chloride Dichloromethane Chloroform Tetrachloromethane Chlorobenzene Propionitrile

0.2679 0.2619 0.2751 0.2721 0.2650 0.2156

Inorganics Ammonia Argon Carbon dioxide Carbon disulfide Carbon monoxide Chlorine Hydrogen Hydrogen chloride Hydrogen sulfide Nitrogen Oxygen Sulfur dioxide Sulfur trioxide

0.2466 0.2933 0.2729 0.2850 0.2898 0.2781 0.3218 0.2673 0.2818 0.2893 0.2890 0.2667 0.2513

average about 0.3 percent at atmospheric pressure. At high pressures, the liquid density of undefined hydrocarbon mixtures can be predicted from the low pressure value by the method of Wright131 fully outlined in the Technical Data Book.23

Solid Density Prediction The prediction of solid density is an inexact science and sometimes is taken as the liquid density at the triple point, although the solid density normally is higher than this value with a discontinuity at the triple point. Based on solid density data reviewed for the DIPPR compilation,24 the solid density at the triple point can be estimated for organic compounds as 1.17 times the liquid density at the triple point. As liquid density at low temperatures varies little with temperature, the density of the liquid at the lowest estimable point above the triple point can be used with little degradation of the result. As solid density only decreases very slightly with increasing temperature and very little data on solid density as a function of temperature exist, no methods have been developed for predicting the solid density vs. temperature. VISCOSITY Viscosity is defined as the shear stress per unit area at any point in a confined fluid divided by the velocity gradient in the direction perpendicular to the direction of flow. If this ratio is constant with time at a given temperature and pressure for any species, the fluid is called a Newtonian fluid. This section is limited to Newtonian fluids, which include all gases and most nonpolymeric liquids and their mixtures. Most polymers, pastes, slurries, waxy oils, and some silicate esters are examples of non-Newtonian fluids. The absolute viscosity (µ) is defined as the sheer stress at a point

2-362

PHYSICAL AND CHEMICAL DATA

FIG. 2-40

Densities of compressed pure liquid hydrocarbons and their defined mixtures.

divided by the velocity gradient at that point. The most common unit is the poise (1 g/cm sec). The SI unit is the Pa·sec (1 kg/m sec). As many common fluids have viscosities in the hundredths of a poise the centipoise (cp) is often used. One centipoise is then equal to one mPa sec. The kinematic viscosity (ν) is defined as the ratio of the absolute viscosity to density at the same temperature and pressure. The most common unit corresponding to the poise is the stoke (1 cm2/sec). The SI unit would be m2/sec. Correlation Methods This section briefly discusses methods for correlating viscosities as a function of temperature and presents the most common accurate methods for prediction of vapor and liquid viscosity. Vapor viscosity is accurately correlated as a function of temperature by Eq. (2-95). AT B µv =  (2-95) C D 1 +  + 2 T T

If data are available over a wide range, all four regression constants (A, B, C, and D) are usually used. Over narrow temperature ranges, only constants A and B are necessary. Liquid viscosity is accurately correlated as a function of temperature by the modified Riedel equation previously discussed for correlation of vapor pressure and shown by Eq. (2-96). B µᐉ = exp A +  + C ln (T) + DT E (2-96) T For most systems, only the first three terms are used. Only the first two terms are used for narrow ranges. If data are available in a wide range extending far above the normal boiling point, all four terms are used, with values of E varying in integers from −10 to 10 (excluding 0 and −1). Constants for about 1500 compounds for both viscosities are available in the DIPPR compilation.24 Vapor Viscosity Methods for prediction of vapor viscosity abound such that only the most accurate and generally used methods are included.





PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES For prediction of the vapor viscosity of pure hydrocarbons at low pressure (below Tr of 0.6), the method of Stiel and Thodos101 is the most accurate. Only the molecular weight, the critical temperature, and the critical pressure are required. Equation (2-97) with values of N from Eqs. (2-98) and (2-99) is used. NM1/2P c2/3 µv = 4.60 × 10−4  T c1/6 0.94 N = 0.0003400Tr for Tr ≤ 1.5 N = 0.0001778(4.58Tr − 1.67)0.625

(2-97) (2-98) for Tr > 1.5

(2-99)

The resultant viscosity is in centipoise (mPa·sec) if Tc and Pc are given in K and Pa, respectively. This method can also be used for light nonhydrocarbon gases except for hydrogen where, special N’s are required. For hydrocarbons below ten carbon atoms, average errors of about 3 percent can be expected, with errors increasing to 5–10 percent for heavier hydrocarbons. Example 22 Estimate the vapor viscosity of propane at 101.3 kPa and 80°C. Use Eq. (2-97).

Tc = 96.7°C,

Z = Z(0) + ωZ(1) 398.1 Tr =  = 1.62 245.5

Pc = 4.246 MPa,

and M = 44.1.

To determine whether Eq. (2-98) or (2-99) should be used to calculate N, calculate Tr. 80 + 273.1 Tr =  = 0.955 96.7 + 273.1 Use Eq. (2-98):

Example 23 Estimate the vapor viscosity of a mixture of propane and methane. Assume 60 mole percent methane and 40 mole percent propane at 125°C and 10.34 MPa total pressure. The low pressure viscosity is 0.0123 cp. Use Eq. (2-102): M1/2P 2/3 c µ − µo = 5.0 × 10−8  [exp (1.439ρr) − exp (−1.11ρ1.858 )] r T1/6 c Properties of the pure components are: Methane. M = 16.04, Tc = −110.4°C, Pc = 4.593 MPa Propane. M = 44.10, Tc = 96.7°C, Pc = 4.246 MPa For the mixture: Tpc = (0.60)(−110.4 + 273.1) + (0.40)(96.7 + 273.1) = 245.5 K Ppc = (0.60)(4.593) + (0.40)(4.246) = 4.454 MPa Mm = (0.60)(16.04) + (0.40)(44.10) = 27.26 To calculate ρr, use Eq. (2-76) to calculate ρ using Pitzer corresponding states first to calculate Z. Then calculate ρr = ρ/ρc = ρVpc Additional properties required are: Methane ω = 0.0115, Vc = 0.0986 m3/kmole Propane ω = 0.1523, Vc = 0.2002 m3/kmole For the mixture: ωm = (0.60)(0.0115) + (0.40)(0.1523) = 0.0678 Vpc = (0.60)(0.0986) + (0.40)(0.2002) = 0.1392 m3/kmole Using Eq. (2-58):

M1/2Pc2/3 µv = 4.60 × 10−4N  T c1/6

 

x 1 + Qij j xi j=1 j≠1

      

µi 1/2 Mj 1/4 2   Mi µj Qij =  Mi 1/2 8 1 +  Mj 1+

(2-100)

P (10.34) ρ =  =  ZRT (0.88)(8.314 × 10−3)(398.1)

µ − µo = 0.0037 µ = 0.0037 + 0.0123 = 0.0160 cp An experimental value of 0.0167 cp compares favorably. For pure nonhydrocarbon polar gases at low pressures, the viscosity can be estimated by the method of Reichenberg85 given by Eq. (2-103). ATr µ =  (2-103) [1 + 0.36Tr(Tr − 1)]1/6 For organic compounds: M1/2Tc A=

niCi

(2-101)

Errors, when tested against binary and multicomponent mixtures of both hydrocarbons and nonhydrocarbon gas mixtures, average about 3 percent. For prediction of the vapor viscosity of gaseous hydrocarbons and mixtures of hydrocarbons at high pressures (not applicable to nonhydrocarbon gases) above a Tr of 0.6, low pressure values are calculated from Eq. (2-97) and/or (2-100) and then corrected for pressure by the method of Dean and Stiel25 given by Eq. (2-102). M1/2P c2/3 [exp (1.439ρr) − exp (−1.11ρ1.858 )] µ − µo = 5.0 × 10−8  r T c1/6

Z = 0.87 + (0.0678)(.20) = 0.88

(5.08 × 10−8)(5.22)(27,085) µ − µo =  [2.0358 − 0.7413] (2.503)

An experimental value of 0.0095 cp compares favorably. For prediction of the vapor viscosity of gaseous mixtures of hydrocarbons and nonhydrocarbon gases at low pressures below a Tr of 0.6, the method of Bromley and Wilke13 is recommended. The mixing rule is given by Eq. (2-100) with the interaction parameter Q for each pair of components defined by Eq. (2-101). µi

Z(0)  0.20

(5.0 × 10−8)(27.26)1/2(4.454 × 106)2/3 1.439(.494) −1.11(.494)1.858 µ − µo =  [e −e ] (245.5)1/6

µv = 0.0097 cp

n

From Fig. 2-36:

ρ = 3.55 kmole/m3

(4.60 × 10−4)(3.255 × 10−4)(44.1)1/2(4.246 × 106)2/3 µv =  (369.8)1/6

n

Z(0) = 0.87

10.34 Pr =  = 2.32 4.454

ρr = ρVpc = (3.55)(.1392) = 0.494

Thus, N = 3.255 × 10−4.

i=1

From Fig. 2-35:

Hence:

N = 0.0003400Tr0.94

µm =

2-363

(2-102)

If critical pressure and critical temperature are given in Pa and K, respectively, viscosities in centipoise result. The variable µo is either the low pressure pure component or mixture viscosity according to whether a pure component or mixture is being considered. For mixtures, simple molar average pseudocritical temperature (Kay’s rule), pressure, and density, and molar average molecular weight are used. The vapor density can be predicted by the methods previously discussed. Errors of above 5 percent are common for hydrocarbons and their mixtures. Experimental densities will reduce the errors slightly.

For inorganic gases:

(2-104)





M1/2P c2/3 (2-105) A = 1.6104 × 10−10  T c1/6 Viscosities are calculated in Pa sec (103 cp = 1 Pa sec) with Tc in K and Pc in Pa. Group contributions based on atomic structure for organic compounds necessary for Eq. (2-104) are tabulated in Table 2-397. Errors average about 5 percent for most organics, with slightly higher errors for inorganic gases. For pure nonhydrocarbon nonpolar gases, an alternate method is the method of Yoon and Thodos,132 which requires the same input parameters as the Reichenberg method. The method, when evaluated for the Technical Data Manual, showed errors of about 3 percent for compounds with low dipole moments and requires special correlations for hydrogen and helium.

Example 24 Estimate the vapor viscosity of isopropyl alcohol at 251°C and atmospheric pressure. Use Eq. (2-103) with A determined from Eq. (2-104). ATr µ =  [1 + 0.36Tr(Tr − 1)]1/6 M1/2Tc A=

niCi

2-364

PHYSICAL AND CHEMICAL DATA

TABLE 2-397

Group Contribution Values for Eq. (2-104) Contribution Ci × 10−8 m⋅s⋅K/kg

Group CH3

0.904

CH2 (nonring)

0.647

CH

0.267

Pure component properties are Tc = 508.3 K and M = 60.1. Isopropyl alcohol group contributions for niCi are calculated using Table 2-397.

2 1 1

C

(nonring)

CH2

0.768

CH

0.553

(nonring)

C

(nonring)

0.178

CH

0.741 (nonring)

C

0.524

CH2 (ring)

0.691

group = (1)(.267 × 108) +

OH group = (1)(.796 × 108) = 2.871 × 108

60.11/2(508.3) A =  × 108 = 1.373 × 10−5 Pa sec 2.871 (1.373 × 10−5)(1.031) µ =  = 1.413 × 10−5 Pa sec [1 + 0.36(1.031)(.031)]1/6 (An experimental value of 1.380 × 10−5 Pa sec compares favorably.) For pure nonhydrocarbon polar and nonpolar gases at high pressure, a method of prediction attributed to Stiel and Thodos102 depending on the reduced density as a corrector to the low pressure gas viscosity (µo) takes various forms for polar gases according to the reduced density (ρr) as shown in Eqs. (2-106) through (2-109). ρr ≤ 0.1

(µ − µo)B = 1.656 × 10−7ρ1.111 r

(2-106)

0.1 ≤ ρr ≤ 0.9

(µ − µo)B = 6.07 × 10−9(9.045ρr + 0.63)1.739

(2-107)

0.9 ≤ ρr < 2.2

log {4 − log10 [(µ − µo)107B]} = 0.6439 − 0.1005ρr

(2-108)

2.2 ≤ ρr < 2.6

log {4 − log10 [(µ − µo)107B]} = 0.6439 − 0.1005ρr B = 2173.424 T

C

(ring)

CH C

0.023

(ring)

0.590

(ring)

0.359

F

0.446

Cl

1.006

Br

1.283

OH (alcohols)

0.796

O (nonring)

0.359

C

O (nonring)

1.202

CHO (aldehydes)

1.402

COOH (acids)

1.865

COO

(esters) or HCOO

(formates)

1.341 0.971

NH2 NH (nonring)

0.368

N

0.497

(ring)

− 4.75 × 10−4(ρ3r − 10.65)2

(2-109)

M−1/2 P c−2/3

(2-110)

0.116

(ring)

CH

CH

524.1 Tr =  = 1.031 508.3

−0.153

(nonring)

CH3 groups = (2)(.904 × 108) +

CN

1.813

S (ring)

0.886

1/6 c

For nonpolar gases, Jossi et al.45 extended the method as shown in Eq. (2-111) for 0.1 < ρr < 3.0. [(µ − µo)B107 + 1]1/4 = 1.0230 + 0.23364ρr + 0.58533ρr2 − 0.40758ρr3 + 0.093324ρr4 (2-111) In all cases viscosities are in Pa sec with Tc in K and Pc in Pa. For nonpolars, errors are very small; while for polars, average errors reach 11 percent.

Example 25 Estimate the vapor viscosity of carbon dioxide at 350 K and a total pressure of 20 MPa. An experimental low pressure viscosity at 350 K is 1.7386 × 10−5 Pa sec. Use Eq. (2-111) with B calculated by Eq. (2-110). The pure component properties necessary are Tc = 304.19 K, Pc = 7.3815 MPa, Vc = 0.094 m3/kmole, and M = 44.01. Using the Lee-Kesler form of the Pitzer method [Eq. (2-78)], Z = 0.4983. (0.094)(2 × 107) V Vc P ρr = c =  =  = 1.2965 V ZRT (0.4983)(8314)(350) (2173.424)(304.19)1/6 2173.424T1/6 c B =  =  M1/2P 2/3 (44.01)1/2(7.3815 × 106)2/3 c B = 2.2411 × 10−2 [(µ − µo)B107 + 1]1/4 = 1.0230 + 0.23364(1.2965) + 0.58533(1.2965)2 − 0.40758(1.2965)3 + 0.093324(1.2965)4 = 1.6852 (µ − 1.7386 × 10−5)(2.2411 × 10−2)(107) + 1 = 8.0659 µ − 1.7386 × 10−5 = 3.1529 × 10−5 µ = 4.89 × 10−5 Pa sec An experimental value of 4.73 × 10−5 Pa sec compares favorably. For both polar and nonpolar nonhydrocarbon gaseous mixtures at low pressures, the most accurate viscosity prediction method is the method of Brokaw.10,11 The method is quite accurate but requires the dipole moment and the Stockmayer energy parameter (ε /k) for polar components as well as pure component viscosities, molecular weights, the normal boiling point, and the liquid molar volume at the normal boiling point. The Technical Data Manual should be consulted for the full method. For nonpolar, nonhydrocarbon vapor mixtures at high pressures, the method of Dean and Stiel25 [Eq. (2-102)] discussed earlier can be used. The accuracy of the method is excellent and dependent on the pure component viscosity values used as input parameters.

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES Liquid Viscosity The viscosity of both pure hydrocarbon and pure nonhydrocarbon liquids are most accurately predicted by the method of van Velzen et al.122 The basic equation (2-112) depends on group contributions which are dependent on structure for the calculation of compound-specific constants B and To. 1 1 log µ = B  −  − 3.0 (2-112) T To Resultant viscosities are in Pa sec. If the −3.0 on the right is deleted,



TABLE 2-398



2-365

answers are in cp. To is calculated by Eq. (2-114) or (2-115) according to the value of an adjusted carbon number N* calculated by Eq. (2-113) using the actual carbon number N and group contributions from Table 2-398. N* = N + ∆Ni (2-113) i

N* ≤ 20

To = 28.86 + 37.439N* − 1.3547N*2 + 0.02076N*3

(2-114)

N* > 20

To = 8.164N* + 238.59

(2-115)

Group Contribution Values for Liquid Viscosity Prediction

Structures or functional group n-Alkanes Isoalkanes Saturated hydrocarbons with two methyl groups in isoposition n-Alkanes n-Alkadienes Isoalkanes Isoalkadienes Hydrocarbon with one double bond and two methyl groups in isoposition Hydrocarbon with two double bonds and two methyl groups in isoposition Cyclopentanes Cyclohexanes Alkyl benzenes Polyphenols Alcohols Primary Secondary Tertiary Diols (correction) Phenols (correction) OH on side chain to aromatic ring (correction) Acids Isoacids Acids with aromatic nucleus in structure (correction) Esters Esters with aromatic nucleus in structure (correction) Ketones Ketones with aromatic nucleus in structure (correction) Ethers Aromatic ethers Amines Primary Primary amine in side chain of aromatic compound (correction) Secondary Tertiary Primary amines with NH2 group on aromatic nucleus Secondary or tertiary amine with at least one aromatic group attached to amino nitrogen Nitro compounds 1-Nitro 2-Nitro 3-Nitro 4-Nitro; 5-Nitro Aromatic nitrocompounds Nitrile

∆ Ni

∆Bi

0 1.389 − 0.238N 2.319 − 0.238N

0 15.51 15.51

−0.152 − 0.042N −0.304 − 0.084N 1.237 − 0.280N 1.085 − 0.322N 2.626 − 0.518N

−44.94 + 5.410N* −44.94 + 5.410N* −36.01 + 5.410N* −36.01 + 5.410N* −36.01 + 5.410N*

2.474 − 0.560N

−36.01 + 5.410N*

0.205 + 0.069N 3.971 − 0.172N 1.48 6.517 − 0.311N 0.60 3.055 − 0.161N −5.340 + 0.815N

−45.96 + 2.224N* −339.67 + 23.135N* −272.85 + 25.041N* −272.85 + 25.041N* −140.04 + 13.869N* −140.04 + 13.869N* −188.40 + 9.558N*

10.606 − 0.276N 11.200 − 0.605N 11.200 − 0.605N See remarks 16.17 − N −0.16

−589.44 + 70.519N* 497.58 928.83 557.77 213.68 213.68

b

6.795 + 0.365N 10.71 See remarks

−249.12 + 22.449N* −249.12 + 22.449N* −249.12 + 22.449N*

4.81

−188.40 + 9.558N*

N < 11, not recommended for N = 1,2 N ≥ 11 Calculate ∆B as for straight-chain acid; calculate ∆N for straight-chain acid but reduce ∆N by 0.24 for each methyl group in isoposition

Remarks

For any additional CH3 groups in isoposition, increase ∆N by 1.389 − 0.238N For any additional CH3 groups in isoposition, increase ∆N by 1.389 − 0.238N N ≤ 16; not recommended for N = 5,6 N ≥ 16 N < 17; not recommended for N = 6,7 N ≥ 17 N < 16; not recommended for N = 6,7a,e,f N ≥ 16a,e, f

a

b b

For ∆N, use alcohol contributions and add N − 2.50

a,c,d

4.337 − 0.230N −1.174 + 0.376N

−149.13 + 18.695N* −140.04 + 13.869N*

If hydrocarbon groups have isoconfiguration, see e Add to values of ∆N, ∆B calculated for ester

3.265 − 0.122N 2.70

−117.21 + 15.781N* −760.65 + 50.478N*

If hydrocarbon groups have isoconfiguration, see e Add to values of ∆N, ∆B calculated for ketone

0.298 + 0.209N 11.5 − N

−9.39 + 2.848N* −140.04 + 13.869N*

If hydrocarbon groups have isoconfiguration, see e The ∆N value is not a correction to regular ether value, but the ∆B value is a correction to regular etherc

3.581 + 0.325N −0.16

25.39 + 8.744N* 0

If hydrocarbon groups have isoconfiguration, see e Corrections to be added to amine calculatione

1.390 − 0.461N 3.27 15.04 − N

25.39 + 8.744N* 25.39 + 8.744N* 0

If hydrocarbon groups have isoconfiguration, see e If hydrocarbon groups have isoconfiguration, see e The ∆N value is not a correction to regular amine value; to find ∆B, use primary amine valuea,c

f

7.812 − 0.236N 5.84 5.56 5.36 7.182 − 0.236N 4.039 − 0.0103N

f

−213.14 + 18.330N* −213.4 + 18.330N* −338.01 + 25.086N* −338.01 + 25.086N* −213.14 + 18.330N* −241.66 + 27.937N*

Note alkene contribution is necessary For aromatic correction, see f

2-366

PHYSICAL AND CHEMICAL DATA

TABLE 2-398

Group Contribution Values for Liquid Viscosity Prediction (Concluded ) ∆ Ni

∆Bi

−0.7228 + 0.1755N 2.321 − 0.2357N 10.452 − 1.1276N

286.26 − 31.009N* −26.063 − 11.516N* 3599.9 − 199.96N*

Structures or functional group Amines (Cont.) Isomethyl on nitrile Aromatic nitrile Dinitrile Halogenated compounds Fluoride Chloride Bromide Iodide Special configurations (corrections) C(Cl)x CClCCl C(Br)x CBrCBr CF3 In alcohols In other compounds Aldehydes Aldehydes with an aromatic nucleus in structure (correction) Anhydrides Anhydrides with an aromatic nucleus in structure (correction) Amides Amides with an aromatic nucleus in structure (correction) Sulfide Isomethyl on sulfide a

1.43 3.21 4.39 5.76

Remarks

5.75 −17.03 −101.97 + 5.954N* −85.32

1.91 − 1.459x 0.96 0.50 1.60

e, f e, f e, f

−26.38 0 81.34 − 86.850x −57.73

−3.93 −3.93 3.38 2.70

341.68 25.55 146.45 − 25.11N* −760.65 + 50.478N*

7.97 − 0.50N 2.70

−33.50 −760.65 + 50.478N*

13.12 + 1.49N 2.70

524.63 − 20.72N* −760.65 + 50.478N*

3.9965 − 0.1861N 0.1601

−76.676 + 8.1403N* −25.026

For substitutions on an aromatic nucleus in more than one position, additional corrections are required: Ortho

∆N = 0.51

Meta Para

∆N = 0.11 ∆N = −0.04




∆B = −571.94 (with OH) 54.84 (without OH) ∆B = 27.25 ∆B = 17.57

For alcohols, if there is a methyl group in the isoposition, increase ∆ N by 0.24 and ∆ B by 94.23. If the compound has an aromatic OH or NH2, or if there is an aromatic ether, use ∆N contribution in table but neglect other substituents on the ring such as halogen, CH3, NO2, and the like. For the calculation of ∆ B, however, such substituents must be taken into account. d For aromatic alcohols and compounds with an OH on a side chain, the alcohol contribution (primary, etc.) must be included. For example, o-chlorophenol: ∆ B = ∆B (primary alcohol) + ∆ B (chlorine) + ∆ B (phenol) + ∆ B (ortho correction—see footnote a) With N* = 16.17 (see footnote c): ∆B = (−589.44 + 70.519 × 16.17) + (−17.03) + (213.68) + (−571.94) = 175.56 Ba = 745.94 B = Ba + ∆B = 921.50 2-Phenylethanol: N = 8; ∆N = ∆N (primary alcohol) + ∆N (correction) = [10.606 − (0.276)(8)] + (−0.16) = 8.24 N* = N + ∆N = 8 + 8.24 = 16.24 ∆ B = ∆B (primary alcohol) + ∆ B (correction) = [−589.44 + (70.519)(16.24)] + 213.68 = 769.47 Ba = 747.43 B = Ba + ∆B = 1516.9 e For esters, alkylbenzenes, halogenated hydrocarbons, and ketones: If the hydrocarbon chain has a methyl group in an isoposition, decrease ∆N by 0.24 and increase ∆ B by 8.93 for each such grouping. For ethers and amines, decrease ∆ N by 0.50 and increase ∆ B by 8.93 for each isogroup. f For alkylbenzenes, nitrobenzenes, halogenated benzenes and for secondary or tertiary amines where at least one aromatic group is connected to an amino nitrogen, add the following corrections for each aromatic nucleus. If N < 16, increase ∆N by 0.60; if N ≥ 16, increase ∆ N by 3.055 − 0.161N for each aromatic group. For any N, increase ∆B by (−140.04 + 13.869N*). From van Velzen et al. (122). b c

B is calculated by Eq. (2-116) using values of Ba calculated from Eq. (2-117) or (2-118) according to the value of N* and group contributions from Table 2-398. B = Ba + ∆Bi

(2-116)

N* ≤ 20

Ba = 24.79 + 66.885N* − 1.3173N*2 − 0.00377N*3

(2-117)

N* > 20

Ba = 530.59 + 13.740N*

(2-118)

i

The method should not be used for the first member of a homologous series or for temperatures much above the normal boiling point (Tr ≈ 0.75). Errors for both hydrocarbons and nonhydrocarbons average 15 percent for a wide variety of compounds. Higher errors are noted for amines, diols, ethers, and fluorides. Table 2-398 gives ∆N and ∆B contributions for most common groups. Space prohibits examples for

each type of compound or inclusion of specialized cases that are more fully discussed in the Technical Data Manual. Example 26 Estimate the liquid viscosity of cis-1,4-dimethylcyclohexane at 0°C. Use Eq. (2-112) with N* calculated from Eq. (2-113) and B calculated from Eq. (2-116). Determine group contributions from Table 2-398.

Cyclohexanes n-Alkanes

(1) (2)

∆Ni

∆Bi

1.48 0

−272.85 + 25.041 N* 0

N* = N + ni∆Ni i

N* = 8 + (1)(1.48) + (2)(0) = 9.48

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES Use Eq. (2-114) to calculate To: To = 28.86 + 37.439(9.48) − 1.3547(9.48)2 + (.02076)(9.48)3 To = 279.72 B = Ba + ni∆Bi i

Use Eq. (2-117) to calculate Ba: Ba = 24.79 + 66.885(9.48) − 1.3173(9.48)2 − 0.00377(9.48)3 Ba = 537.26





1 1 log µ = B  −  − 3.0 T To





1 1 log µ = 501.80  −  − 3.0 273.15 279.72 log µ = −2.9569 µ = 0.001104 Pa sec = 1.104 cp An experimental value at 0°C is 1.224 cp. A mixing rule developed by Kendall and Monroe48 is useful for determining the liquid viscosity of defined hydrocarbon mixtures. Equation (2-119) depends only on the pure component viscosities at the given temperature and pressure and the mixture composition. µm =

 xµ  1/3 i

i

(2-119)

For mixtures of the same chemical family, errors average less than 3 percent, while errors overall average 5–6 percent, with errors of mixed families averaging from 10–15 percent. For estimating the liquid viscosity of defined nonhydrocarbon mixtures, a mixing rule shown by Eq. (2-120) was recommended by the Technical Data Manual. ln µm = xi ln µi

(2-120)

i

Errors average near 15 percent. For interpolating viscosities of hydrocarbon mixtures within a limited range knowing viscosities at two temperatures, ASTM Procedure D341-89, including both charts and equations, is recommended. Several recommended methods for predicting the viscosity of undefined hydrocarbon mixtures, such as petroleum fractions and coal liquids, are presented and evaluated in the Technical Data Book—Petroleum Refining. In addition, several methods for determining the liquid viscosity of blends of hydrocarbon mixtures and a method for liquid viscosity of pure hydrocarbons blended with undefined mixtures are given. The most accurate method for predicting the liquid viscosity of hydrocarbons containing fewer than 20 carbon atoms at high pressure is a corresponding states method developed by Graboski and included in the Technical Data Book—Petroleum Refining. Critical temperature, critical pressure, the acentric factor, and knowing or being able to calculate at least one viscosity at a reference temperature and pressure are required. Errors average about 5 percent. For high molecular weight hydrocarbons at high pressure, low pressure viscosities can be converted instead of using the method of Kouzel,51 which requires a low pressure liquid viscosity as input. Compounds of more than 20 carbon atoms and their mixtures are treated in this way. For predicting the liquid viscosity of pure hydrocarbon mixtures at high temperatures, the method of Letsou and Stiel58 is available. Error analyses with only a small amount of data shows errors averaging 34 percent in the reduced temperature range of 0.76 to 0.98. Equation (2-121) defines the method with inputs of Eqs. (2-122) and (2-123). T1/6 c µ 2173.424  = µ(0) + ωµ(1) (2-121) M1/2P 2/3 c





Cp may be assumed to be the ideal gas heat capacity, Cpo. Average errors can be expected to be less than 5 percent.

3

n

i=1



µ = 1.5174 × 10 − 2.135 × 10−5Tr + 7.5 × 10−6T r2

(2-122)

µ(1) = 4.2552 × 10−5 − 7.674 × 10−5Tr + 3.4 × 10−5T r2

(2-123)

(0)

Gases For pure component, low pressure (<350 kPa) hydrocarbon gases, Misic and Thodos70,71 recommend the following equations. For methane and cyclic compounds below reduced temperatures of 1.0: C kG = 4.45 × 10−7 Tr p (2-124) λ For these hydrocarbons above reduced temperatures of 1.0 and for other hydrocarbons at any temperature: C kG = 10−7 (14.52 Tr − 5.14)2/3 p (2-125) λ In these equations, 2/3 1/2 101.325 λ = T 1/6 (2-126)  c M Pc where kG = vapor thermal conductivity, W/m K Tr = reduced temperature, T/Tc T = temperature, K Tc = critical temperature, K Cp = heat capacity at constant pressure, J/kmol K M = molecular weight Pc = critical pressure, kPa

 

B = 537.26 + (1)[−272.85 + 25.041 (9.48)] B = 501.80



2-367

−5

Results are in Pa sec with inputs of Tc in K and Pc in Pa.

VAPOR AND LIQUID THERMAL CONDUCTIVITY Thermal conductivity describes the ease with which conductive heat can flow through a vapor, liquid, or solid layer of a substance. It is defined as the proportionality constant in Fourier’s law of heat conduction in units of energy ⋅ length/time ⋅ area ⋅ temperature; e.g., W/m K.

Example 27 Estimate thermal conductivity for n-hexane. For n-hexane at 373.15 K and low pressure, the required properties from Daubert et al.24 are: Tc = 507.6 K, M = 86.18, Pc = 3025.0 kPa, and Cpo = 1.721 × 105. Tr = 373.15/ 507.6 = 0.7351. Using Eq. (2-126): 101.325 2/3 λ = (507.6)1/6 (86.18)1/2  3025.0





λ = 2.724 Substituting into Eq. (2-125): 1.721 × 105 kG = 10−7 [(14.52)(0.7351) − 5.14]2/3  2.724





kG = 0.01977 W/m K The reported value is 0.02025 W/m K.24 For pure nonhydrocarbon gases at low pressures (up through 1 atm), the following equations may be used at temperature T (K): ηGCv Monatomic gases12: kG = 2.5  (2-127) M Linear molecules12:

ηG 2928.8 kG =  1.30 Cv + 14644.0 −  M Tr





ηG kG =  (1.15 Cv + 16903.36) M kG = vapor thermal conductivity, W/m K ηG = vapor viscosity, Pa·s Cv = heat capacity at constant volume, J/kmol K M = molecular weight Tr = reduced temperature, T/Tc Tc = critical temperature, K

Nonlinear molecules102: where

(2-128) (2-129)

Cv may be calculated as Cpo − R, where Cpo is the ideal gas heat capacity in J/kmol K and R is the gas constant, 8314 J/kmol K. Average errors are in the 8–10 percent range but may be higher for polar compounds. This method should not be used for molecules that associate; e.g., organic acids.

Example 28 Estimate thermal conductivity of carbon dioxide at 370 K and low pressure. For carbon dioxide (a linear molecule) at 370.0 K and low pressure, the required properties from Daubert et al.24 are: Tc = 304.2 K, ηG = 1.828 × 10−5 Pa·s, M = 44.01, and Cpo = 40520 J/kmol K. Tr = 370.0/304.2 = 1.2163, and Cv = 40520 − 8314 = 32206 J/kmol K. Substituting in Eq. (2-128): 1.828 × 10−5 2928.8 kG =  (1.30) (32206) + 14644.0 −  44.01 1.2163



kG = 0.02247 W/m K A value of 0.0220 W/m K is obtained from Vargaftik et al.124



2-368

PHYSICAL AND CHEMICAL DATA

For pure gases above atmospheric pressure, the method of Stiel and Thodos102 may be used: A × 10−4(eBρr + C) kG = k′G +  (2-130) Tc1/ 6 M1/2 5 Zc  2/3 Pc



ρr < 0.5 0.5 < ρr > 2.0 2.0 < ρr > 2.8

A = 2.702 A = 2.528 A = 0.574



B = 0.535 B = 0.670 B = 1.155

373.15 K and low pressure. The required pure component properties from Daubert et al.24 are: k1 = 0.02504 W/m K, k2 = 0.01587 W/m K, µ1 = 1.161 × 10−5 Pa·s, µ2 = 1.361 × 10−5 Pa·s, M1 = 46.07, M2 = 50.49, Tb1 = 248.3 K, and Tb2 = 248.9 K. By Eq. (2-134), S1 = (1.5)(248.3) = 372.45, and S2 = (1.5)(248.9) = 373.35. By Eq. (2-133), S12 = S21 = (1.0)[(372.45)(373.35)]1/2 = 372.90. Substituting in Eq. (2-132): A11 = A22 = 1.0

C = −1.000 C = −1.069 C = 2.016

where kG = vapor thermal conductivity at the temperature T (K) and pressure P of interest, W/m K k′G = vapor thermal conductivity at T and atmospheric pressure, W/m K ρr = reduced density = Vc /V Vc = critical molar volume, m3/kmol V = molar volume at T and P, m3/kmol Tc = critical temperature, K M = molecular weight Pc = critical pressure, MPa Zc = critical compressibility factor = PcVc /RTc R = gas constant = 0.008314 MPa m3/kmol K Errors in the range of 5–6 percent are typical with this method but may be higher for branched compounds.

Example 29 Estimate the thermal conductivity of carbon dioxide at

1.161 × 10−5

373.15 + 372.45  
 1.361 × 10  46.07   373.15 + 373.35  

1 A12 =  1 + 4

50.49

3/4

1/2 2

−5

373.15 + 372.90   = 0.956 373.15 + 372.45





373.15 + 373.35 1.361 × 10  
 1.161 × 10  50.49   373.15 + 372.45  

1 A21 =  1 + 4

−5

46.07

3/4

1/2 2

−5

373.15 + 372.90   = 1.047 373.15 + 373.35





Using Eq. (2-131): (0.23)(0.02504) (0.77)(0.01587) km =  +  = 0.01805 W/m K (0.23)(1.0) + (0.77)(0.956) (0.23)(1.047) + (0.77)(1.0) The experimental value is 0.01778 W/m K.66

24

370 K and 10 MPa pressure. The required properties from Daubert et al. are: Tc = 304.2 K, M = 44.01, Pc = 7.383 MPa, and Vc = 0.0940 m3/kmol; Zc = (7.383) × (0.0940)/(0.008314)(304.2) = 0.274. k′G = 0.0220 W/m K123, and V = 0.22809 m3/ kmol3; ρr = 0.0940/0.22809 = 0.4121. Substituting in Eq. (2-130) using the equation constants A, B, and C for ρr < 0.5: (2.702 × 10−4)(e(0.535)(0.4121) − 1.000) kG = 0.0220 +  = 0.0315 W/m K (304.2)1/6(44.01)1/2 (0.274)5  (7.383)2/3



124

Vargaftik et al.



report a value of 0.0308 W/m K.

The thermal conductivity of low pressure (1 atm or less) gas mixtures can be determined from the relation of Wassiljewa126: n yi ki km =  (2-131) n i=1

yj Aij j−1

where km = mixture thermal conductivity, W/m K n = number of components yi, j = mole fraction of component i or j in the vapor mixture ki = thermal conductivity of pure component i at the temperature of interest The binary interaction parameter Aij is obtained by the method of Lindsay and Bromley61: 1 µi Mj 3/4 T + Si 1/2 2 T + Sij Aij =  1 +   (2-132)   4 µj Mi T + Sj T + Si where µi, j = vapor viscosity of pure component i or j at the temperature T of interest and low pressure, Pa·s Mi, j = molecular weight of pure component i or j T = temperature, K Sij = Sji; see Eq. (2-133) C = 1.0 except when either or both components i and j are very polar; then C = 0.73 Si, j = 79 K for helium, hydrogen, and neon; for all others, see Eq. (2-134) Tbi, j = normal boiling temperature of pure component i or j, K


  

Sji = C(SiSj)1/2 Si,j = 1.5Tbi, j

  



(2-133) (2-134)

Expected errors for this method are 4–5 percent. At higher pressures, a pressure correction using Eq. (2-130) may be used. The mixture is treated as a hypothetical pure component with mixture critical properties obtained via Eqs. (2-5), (2-8), and (2-17) and with the molecular weight being mole-averaged. Example 30 Estimate thermal conductivity of a mixture of 0.23 mole fraction dimethylether (1) and 0.77 mole fraction methyl chloride (2) at

Liquids For pure component hydrocarbon liquids at reduced temperatures between 0.25 and 0.8 and at pressures below 3.4 MPa, an equation based on the methods of Pachaiyappan et al.79 and Riedel88 may be used: 3 + 20(1 − Tr)2/3 kL = CρM n  (2-135) 293.15 2/3 3 + 20 1 −  Tc









where kL = liquid thermal conductivity, W/m K M = molecular weight ρ = molar density at 293.15 K, kmol/m3 Tr = reduced temperature, T/Tc T = temperature, K Tc = critical temperature, K For unbranched, straight chain hydrocarbons, n = 1.001 and C = 1.811 × 10−4. For branched and cyclic hydrocarbons, n = 0.7717 and C = 4.407 × 10−4. Average errors are 5 percent when this equation is used. For pressures greater than 3.4 MPa, the thermal conductivity from Eq. (2-135) may be corrected by the technique suggested by Lenoir.57 The correction factor is the ratio of conductivity factors F/F′, where F is at the desired temperature and higher pressure, and F′ is at the same temperature and lower pressure (usually atmospheric). The conductivity factors are calculated from: 2.054T 2r F = 17.77 + 0.065Pr − 7.764Tr −  (2-136) e0.2 Pr where Tr = reduced temperature as in Eq. (2-135) Pr = reduced pressure, P/Pc P = pressure, MPa Pc = critical pressure, MPa The average error in the pressure correction alone is typically 3 percent. Example 31 Estimate thermal conductivity of n-octane at 373.15 K and pressures of 0.1 MPa and 20.0 MPa. The required properties from Daubert et 24 al. are: ρ at 293.15 K = 6.155 kmol/m3, M = 114.2, Tc = 568.7 K, and Pc = 2.490 MPa. Tr = 373.15/568.7 = 0.6561, n = 1.001, and C = 1.811 × 10−4. Substituting in Eq. (2-135) for the thermal conductivity at 0.1 MPa:



3 + (20)(1 − 0.6561)2/3 kL = (1.811 × 10−4)(6.155)(114.2)1.001  293.15 2/3 3 + (20) 1 −  568.7 = 0.107 W/m K







PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES The reported value is 0.111 W/m K.123 To correct to 20 MPa, Pr = 20.0/2.490 = 8.032, and P′r = 0.1/2.490 = 0.04016. Substituting in Eq. (2-136) to calculate F and F′: (2.054)(0.6561)2 F = 17.77 + (0.065)(8.032) − (7.764)(0.6561) −  e(0.2)(8.032) F = 13.02 (2.054)(0.6561)2 F′ = 17.77 + (0.065)(0.04016) − (7.764)(0.6561) −  e(0.2)(0.04016) F′ = 11.80 Thus, the thermal conductivity of n-octane at 373.15 K and 20.0 MPa is estimated to be (0.111)(13.02/11.80) = 0.122 W/m K as compared with an experimental value of 0.121 W/m K.123 For pure component hydrocarbon liquids above the normal boiling point and all pressures, the method of Kanitkar and Thodos46 is recommended: α eβρr P < 10,000 kPa (2-137) kL =  λ βρ 2.596 × 10−4 P 1.6 r + α e r kL =  P > 10,000 kPa (2-138) λ where kL = liquid thermal conductivity at the temperature T (K) and pressure P (kPa) of interest, W/m K α = 0.0112 β−3.322 (2-139) β = 0.40 + 0.986e−0.64λ (2-140)





101.325 2/3 1/2  λ = T 1/6 c M Pc ρr = reduced density = Vc /V Vc = critical molar volume, m3/kmol V = molar volume at T and P, m3/kmol Tc = critical temperature, K M = molecular weight Pc = critical pressure, kPa Pr = reduced pressure, P/Pc

(2-141)



β = 0.40 + 0.986e−(0.64)(3.639)

= 0.496

α = (0.0112)(0.496)

= 0.115

−3.322

Using Eq. (2-138): (2.596 × 10−4)(6.024)1.6 + 0.115e(0.496)(2.478) kL =  = 0.109 W/m K 3.639 The reported value is 0.106 W/m K.123 The thermal conductivity of pure component nonhydrocarbon liquids may be estimated by the method of Baroncini et al.,5 with a modification by Myers75 for silicon compounds, at reduced temperatures between 0.3 and 0.8 and at pressures below 3.5 MPa: abc (1 − Tr)0.38 kL =   (2-142) T 1/6 m r where kL = liquid thermal conductivity, W/m K Tr = reduced temperature, T/Tc T = temperature, K Tc = critical temperature, K a = constant parameter b = constant parameter = function of normal boiling temperature Tb, K c = constant paramerer = function of critical temperature Tc, K m = constant parameter = function of molecular weight M

 

Compound class

a

b

c

m

Acids* Alcohols, phenols Esters† Ethers Halides‡ Refrigerants R20–R23 Ketones Alkoxysilanes Alkyl-(aryl)-chlorosilanes

0.00319 0.00339 0.0415 0.0385 0.494 0.562 0.00383 0.00482 0.6510

T b6/5 T b6/5 T b6/5 T b6/5 1.0 1.0 T b6/5 T b6/5 1.0

T c−1/6 T c−1/6 T c−1/6 T c−1/6 T 1/6 c T 1/6 c T c−1/6 T c−1/6 T 1/6 c

M 1/2 M 1/2 M M M 1/2 M 1/2 M 1/2 M 1/2 M 1/2

*Do not use for formic, myristic, or oleic acids. †Do not use for butyl stearate. ‡Do not use for refrigerants R20–R23 (CHCl3, CHFCl2, CHClF2, or CHF3). 390.8 K. Tr = 360.0/563.0 = 0.6394. From Table 2-399, a = 0.00339, b = (Tb)6/5 = (390.8)6/5 = 1289.3, c = (Tc)−1/6 = (563.0)−1/6 = 0.3480, and m = M1/2 = (74.12)1/2 = 8.609. Substituting in Eq. (2-142): (0.00339)(1289.3)(0.3480) (1 − 0.6394)0.38 kL =   = 0.1292 W/m K 8.609 (0.6394)1/6 123 The reported value is 0.1429 W/m K. For pure component nonhydrocarbon liquids for which Eq. (2-142) is not applicable, the method of Missenard72,73 may be used at temperature T (K) and below pressures of 3.5 MPa: 3 + 20 (1 − Tr)2/3 kL = kLr  (2-143) 273.15 2/3 3 + 20 1 −  Tc













(Tbρr)1/2 Cpr kLr = 2.656 × 10−7  M1/2 N1/4

Example 32 Estimate thermal conductivity of n-octane at 473.15 K and 15,000 kPa. The required properties from Daubert et al.24 are: Tc = 568.7 K, M = 114.2, Pc = 2490.0 kPa, and Vc = 0.4860 m3/kmol. The specific volume at 473.15 K and 15,000 kPa is 0.001717 m3/kg123; V = (0.001717)(114.2) = 0.1961 m3/kmol. Thus, Pr = 15000/2490 = 6.024, and ρr = 0.4860/0.1961 = 2.478. Substituting in Eqs. (2-141), (2-140), and (2-139): 101.325 2/3 λ = (568.7)1/6 (114.2)1/2  = 3.639 2490



TABLE 2-399 Values of Constant Parameters in Eq. (2-142) for Various Compound Classes



Average errors can be expected to be in the order of 10 percent.



Values of a, b, c, and m for various compound classes are found in Table 2-399. Average errors are about 8 percent.

Example 33 Estimate thermal conductivity of n-butanol. The properties required to estimate the liquid thermal conductivity of n-butanol at 360.0 K and 0.1 MPa from Daubert et al.24 are: Tc = 563.0 K, M = 74.12, and Tb =

2-369

where

(2-144)

kL = liquid thermal conductivity, W/m K kLr = liquid thermal conductivity at 273.15 K, W/m K Tr = reduced temperature, T/Tc Tc = critical temperature, K Tb = normal boiling temperature, K ρr = molar density at 273.15 K, kmol/m3 Cpr = molar heat capacity at 273.15 K, J/kmol K M = molecular weight N = number of atoms in the molecule

Errors in the order of 8 percent can be expected.

Example 34 Estimate the thermal conductivity of n-propionaldehyde (CH3CH2CHO) at 318.15 K and low pressure (0.1 MPa); the necessary properties from Daubert et al.24 are: Tc = 504.4 K, Tb = 321.1 K, ρr = 14.11 kmol/m3, C pr = 1.309 × 105 J/kmol K, and M = 58.08. N = 10, and Tr = 318.15/504.4 = 0.6307. Substituting in Eq. (2-144): [(321.1) (14.11)]1/2 (1.309 × 105) kLr = (2.656 × 10−7)  = 0.1727 W/m K (58.08)1/2 (10)1/4 Using Eq. (2-143): 3 + 20 (1 − 0.6307)2/3 = 0.1542 W/m K kL = 0.1727  3 + 20 (1 − 273.15/504.4)2/3





A value from the literature is 0.1541 W/m K.24 For pressures greater than 3.5 MPa, the correction factor suggested by Missenard74 may be used to obtain the thermal conductivity of pure component nonhydrocarbon liquids. Thus: Pr kL = k′L 0.98 + 0.0079 PrT r1.4 + 0.63 T 1.2 (2-145)  r 30 + Pr where kL = liquid thermal conductivity at the desired temperature T (K) and pressure P (MPa), W/m K k′L = liquid thermal conductivity at T and pressure of 0.1 MPa, W/m K Pr = reduced pressure, P/Pc Pc = critical pressure, MPa Tr = reduced temperature, T/Tc Tc = critical temperature, K







Average errors are in the range of 5–20 percent.

Example 35 Estimate thermal conductivity of n-butanol. The required properties at 360 K and 15 MPa from Daubert et al.24 are: Tc = 563.0 K and

2-370

PHYSICAL AND CHEMICAL DATA

Pc = 4.423 MPa. Tr = 360.0/563.0 = 0.6394 and Pr = 15.0/4.423 = 3.391. k′L = 0.1429 W/m K.123 Substituting in Eq. (2-145): 3.391 kL = 0.1429 0.98 + (0.0079) (3.391) (0.6394)1.4 + (0.63) (0.6394)1.2  30 + 3.391







= 0.1474 W/m K Vargaftik123 reports a value of 0.1494 W/m K. For both aqueous and nonaqueous liquid mixtures, the method of Li59 is suggested for pressures below 3.5 MPa: n

km =

n

φ φ k i

j

ij

(2-146)

i=1 j=1

xi Vi φi =  n

xj Vj

(2-147)

Gas Diffusivity For prediction of the gas diffusivity of binary hydrocarbon-hydrocarbon gas systems at low pressures (below about 500 psia [3.5 MPa]) the method of Gilliland32 given by Eq. (2-150) is recommended. 1 1 0.5 0.1014T 1.5  +  M1 M2 (2-150) D12 =  1/3 2 P(V 1/3 1 + V2 )



Component 1 is the solute, while component 2 is the solvent. Units of T, P, and V are °R, psia, and cm3/gmole, respectively. Diffusivity is then in ft2/hr. The molar volumes V1 and V2 at the normal boiling point are estimated by Tyn and Calus,120 Eq. (2-151). Vi = 0.285Vc1.048 i

j=1

2 kij =  (2-148) (1/ki) + (1/kj) where km = mixture liquid thermal conductivity at temperature T (K), W/m K n = number of components xi, j = mole fraction of component i or j in the liquid mixture Vi, j = liquid molar volume of pure component i or j at temperature T, m3/kmol ki, j = liquid thermal conductivity of pure component i or j at temperature T, W/m K Expected errors are in the 4–6 percent range. At pressures greater than 3.5 MPa, a pressure correction using Eq. (2-145) may be used. The mixture is treated as a hypothetical pure component with mixture critical properties obtained via Eqs. (2-5) and (2-8).

Example 36 Estimate thermal conductivity of a mixture of 0.302 mole fraction diethyl ether (1) and 0.698 mole fraction methanol (2) at 273.15 K and 0.1 MPa. The required properties from Daubert et al.24 are: V1 = 0.1007 m3/kmol, V2 = 0.03942 m3/kmol, k1 = 0.1383 W/m K, and k2 = 0.2069 W/m K. Substituting in Eq. (2-148): 2 k12 =  = 0.1658 1/0.1383 + 1/0.2069 Then, using Eq. (2-147) to obtain φ1 and φ2: (0.302)(0.1007) 0.03041 φ1 =  =  = 0.525 (0.302)(0.1007) + (0.698)(0.03942) 0.05793

(2-151)

The method gives average errors of less than 4 percent. For prediction of the gas diffusivity of binary air-hydrocarbon or nonhydrocarbon gas mixtures at low pressures, the method of Fuller et al.30 given by Eq. (2-152) is recommended. 1 1 0.5 0.1013T 1.75  +  M1 M2 D12 =  (2-152) P[( v1)1/3 + ( v2)1/3]2





Units of T and P are K and Pa, respectively, with the resulting diffusivity in m2/sec. All vi are group contribution values for the subscript component summed over atoms, groups, and structural features given in Table 2-400. For air-hydrocarbon systems, average deviations do not exceed 9 percent. For general nonhydrocarbon gas systems, the average deviation is about 6 percent. Example 37 Estimate the diffusivity of benzene vapor diffusing into air at 30°C and 96.5 kPa total pressure. Use Eq. (2-152). 1 1 0.5 0.1013T1.75  +  M M 1 2 D12 =  P[( v1)1/3 + ( v2)1/3]2





T = 303.1 K, M1 = 78.1, M2 = 28.86, and P = 96,500 Pa.

(0.698)(0.03942) φ2 =  = 0.475 0.05793 Substituting in Eq. (2-146):

Now consult Table 2-400. For benzene (C6H6):

km = (0.525)2(0.1383) + 2(0.525)(0.475)(0.1658) + (0.475)2(0.2069) = 0.167 W/m K

For air, v2 = 20.1:

v1 = 6(16.5) + 6(1.98) − 20.2

v1 = 90.68





1 1 0.5 (.01013)303.11.75  +  78.1 28.86 D12 =  (96,500)(90.681/3 + 20.11/3)2

Jamieson and Hastings42 report a value of 0.173 W/m K for this mixture.

DIFFUSIVITY Diffusion is the molecular transport of mass without flow. The diffusivity (D) or diffusion coefficient is the proportionality constant between the diffusion and the concentration gradient causing diffusion. It is usually defined by Fick’s first law for one-dimensional, binary component diffusion for molecular transport without turbulence shown by Eq. (2-149) dC1 N1 (2-149)  = −D12  A dL The molar flow of species 1(N1) per unit area (A) is directly proportional to the change in concentration of species 1 (C1) per distance diffused (L). The usual units of diffusivity are m2/sec. In chemical engineering, the primary application of the diffusivity is to calculate the Schmidt number (µ/ρD) used to correlate mass transfer properties. This number is also used in reaction rate calculations involving transport to and away from catalyst surfaces. Experimental diffusion coefficients are scarce and not highly accurate, especially in the liquid phase, leading to prediction methods with marginal accuracy. However, use of the values predicted are generally suitable for engineering calculations. At concentrations above about 10 mole percent, predicted values should be used with caution. Diffusivities in liquids are 104–105 times lower than those in gases.



D12 = 9.68 × 10−6 m2/sec This value is within 1 percent of an available experimental value. TABLE 2-400

Atomic Diffusion Volumes for Use in Eq. (2-153)

Atomic and structural diffusion-volume increments ν C H O N

16.5 1.98 5.481 (5.69)

Cl S Aromatic ring Heterocyclic ring

(19.5) (17.0) −20.2 −20.2

Diffusion volumes for simple molecules H2 D2 He N2 O2 Air Ar Kr Xe

7.07 6.70 2.88 17.9 16.6 20.1 16.1 22.8 (37.9)

CO CO2 N2O NH3 H2O CCl2F2 SF6 CI2 Br2 SO2

18.9 26.9 35.9 14.9 12.7 (114.8) (69.7) (37.7) (67.2) (41.1)

Parentheses indicate that the value listed is based only on a few data points.

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-401

Parameters for Eq. (2-153)

Pr

(D 12 P)R

A

B

C

0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 4.0 5.0

1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.02 1.02 1.02 1.02 1.03 1.03 1.04 1.05 1.06 1.07

0.38042 0.067433 0.098371 0.137610 0.175081 0.216376 0.314051 0.385736 0.514553 0.599184 0.557725 0.593007 0.696001 0.790770 0.502100 0.837452 0.890390

1.52267 2.16794 2.42910 2.77605 2.98256 3.11384 0.50264 3.07773 3.54744 3.61216 3.41882 3.18415 3.3760 3.27984 2.39031 3.23513 3.13001

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.141211 0.278407 0.372683 0.504894 0.678469 0.0665702 0.0 0.602907 0.0 0.0

E

13.45454 14.00000 10.00900 8.57519 10.37483 11.21674

D′12 D12 = 1.013 × 105  (D12 P)R(1 − AT r−B)(1 − CT −E r ) P

6.19043

(2-153)

Example 38 Estimate the diffusivity of hydrogen (1) in nitrogen (2) at 60°C and 17.23 MPa. A value of the low pressure diffusivity obtained using Eq. (2-152) is D′12 = 9.2 × 10−5 m2/sec. Use Eq. (2-153): D′12 D12 = 1.013 × 105  (D12P)R(1 − AT r−B)(1 − CT −E r ) P For nitrogen, Tc = 126.2 K and Pc = 3.394 MPa. 17.23 333.2 Pr =  = 5.077 Tr =  = 2.640 126.2 3.394 From Table 2-401: (D12P)R = 1.07, A = 0.89039, B = 3.1300, C = 0 9.2 × 10−5 D12 = 1.013 × 105 7 (1.07)[1 − .89039(2.64)−3.13] 1.723 × 10



D12 = 5.54 × 10 m /sec

y D j

(1.1728 × 10−16)(298.15)(1.0 × 112.56)1/2 D12 =  (7.548 × 10−4)(0.0745)0.6 D12 = 2.33 × 10−9 m2/sec

Liquid Diffusivity Liquid diffusivities are in general not as accurately predicted as vapor diffusivities, and specialized methods have been developed. References to each method determined to be accurate are given, but only the most common methods will be presented. For predicting liquid diffusivities of binary nonpolar liquid systems at high solute dilution, Umesi120 developed a method that only depends on the viscosity of the solvent (2) and the radius of gyration of the solvent (2) and the solute (1). The Technical Data Book— Petroleum Refining gives the method and values of the radii of gyration for common hydrocarbons. Errors average 16 percent but reach 30 percent at times. For predicting diffusivities in binary polar or associating liquid systems at high solute dilution, the method of Wilke and Chang129 defined in Eq. (2-156) can be utilized. The Tyn and Calus equation (2-152) can be used to determine the molar volume of the solute at the normal boiling point. Errors average 20 percent, with occasional errors of 35 percent. The method is not considered to be accurate above a solute concentration of 5 mole percent. For concentrated binary nonpolar liquid systems (more than 5 mole percent solute), the diffusivity can be estimated by a molar average mixing rule developed by Caldwell and Babb,14 Eq. (2-156). D1m = x1D21 + (1 − x1)D12

(2-156)

D21 and D12 are dilute solution binary diffusivities. Errors depend on the procedure used to determine the dilute solution diffusivities. For multicomponent nonpolar liquid systems, Leffler and Cullinan56 developed a mixing rule, Eq. (2-157). 2

An available experimental value is 4.89 × 10−7 m2/sec. For prediction of gas phase diffusion coefficients in multicomponent hydrocarbon/nonhydrocarbon gas systems, the method of Wilke128 shown in Eq. (2-154) is used. 1 − yi

For chlorobenzene, M2 = 112.56, χ2 = 1.0, µ2 @ 25°C = 7.548 × 10−4 Pa sec, and V1 = 0.0745 m3/kmole.

D1mµm = (D12µ2)x (D13µ3)x . . .

2

Dim =

T(χ2M2)1/2 D12 = 1.1728 × 10−16  µ2V 10.6

An experimental value of 2.77 × 10−9 m2/sec is available.

D′12 is the low pressure diffusivity at the temperature of interest. (D12P)R is a reduced diffusivity pressure product at infinite reduced temperature; and A, B, C, and E are constants. All are a function of Pr tabulated in Table 2-401. Component 1 is the diffusing species, while component 2 is the concentrated species. Critical properties are for the solvent. The pressure is given in Pa. The diffusivity is in m2/sec. Errors from evaluation average near 15 percent.

−7

molecular weight are required input parameters. For the common solvents, χ decreases to 1 as polarity decreases, with values of 2.6 for water, 1.9 for methanol, 1.5 for ethanol, and 1.0 for less polar solvents. When tested with both hydrocarbon and nonhydrocarbon systems, average errors are about 25 percent—not excessive, considering the magnitude of the diffusivity.

Example 39 Estimate the infinite dilution diffusivity of propane (1) in chlorobenzene (2) at 25°C. Use Eq. (2-113):

An alternate method for gas diffusivity of binary gas mixtures at low pressures is the method of Hirschfelder et al.40 The method requires several molecular parameters and, when evaluated, gives an average absolute error of about 10 percent. The method is discussed in detail in the Data Prediction Manual. For predicting the diffusivity of binary gas mixtures at high pressures, the method of Takahashi,109 Eq. (2-153), applies.



2-371

(2-154)

ij

j j≠1

This mixing rule is used to determine the diffusivity of any component in a j + 1 component mixture and requires binary diffusivities of component i with all other components. It has been estimated that errors are about 5 percent greater than the greatest error in the binary diffusivities. Fairbanks and Wilke,28 using the same Eq. (2-154), made the same recommendation with essentially the same errors. For prediction of the diffusivity of a dilute dissolved gas (hydrocarbon or nonhydrocarbon) in a liquid, the standard method is that of Wilke and Chang129 shown by Eq. (2-155). T(χ2M2)1/2 (2-155) D12 = 1.1728 × 10−16  µ2V 0.6 1 Component 1 is the diffusing gas, while component 2 is the solvent. The solvent viscosity µ2 in Pa sec, the solute molar volume at the normal boiling point V1 in m3/kmole, and the solvent association parameter χ2 multiplied by the solvent

(2-157)

3

The diffusivity of solute 1 in the mixture is related to the binary infinite dilution diffusivities for each of the other components calculated from Eq. (2-155) or the Umesi method. The viscosities are calculated by the methods in the previous section. Errors are not quantifiable, as little experimental data exist, although these errors would be related to those assumed for the binary pairs. For concentrated binary liquid nonhydrocarbon systems, the method of Caldwell and Babb,14 Eq. (2-156) has been modified by introduction of a thermodynamic correction term as shown in Eq. (2-158). d ln γ1 D1mµm = [xiD21 + (1 − x1)D12] 1 +  d ln x1





(2-158)

The activity coefficient (γ) based corrector is calculated using any applicable activity correlating equation such as the van Laar (slightly polar) or Wilson (more polar) equations. The average absolute error is 20 percent. An alternate method for binary concentrated liquid systems where activity coefficients are not available or estimable is the method of Leffler and Cullinan56 previously given in Eq. (2-156). Absolute errors average 25 percent. For estimating the diffusivity of the dilute solute (10 mole percent) in water, the method of Hayduk and Laudie,37 Eq. (2-159), applies.

2-372

PHYSICAL AND CHEMICAL DATA 8.621 × 10−14 D12 =  µ1.14 V 0.589 2 1

Component 1 is the solute, while component 2 is water. The molar volume of the solute in m3/kmole is at the solute normal boiling point, while the viscosity of water in Pa sec is at the temperature of the system resulting in a diffusivity in m2/sec. The average error is about 9 percent when tested on 36 experimental systems. For estimating the diffusivity of a dilute solute (<10 mole percent) in any solvent except water, the method of King et al.,49 Eq. (2-160), applies. D12 = 4.4 × 10

−15

λ2  λ1

  

T V2   µ2 V1

1/6

1/2 1/2 ln (D1m µ1/2 m ) = x2 ln (D* 12 µ 2 ) + x3 ln (D* 13 µ 3 )



σ = a − bT

(2-162)

where a and b are listed for most of the substances. To obtain values at a higher temperature than the upper temperature limit indicated by Jasper, the following expression may be used: σ = d(1 − Tr)e

(2-163)

where d and e are determined such that σ and dσ/dT at the upper temperature limit THI have the same values when calculated from both Eqs. (2-162) and (2-163): b(Tc − THI) e =  (2-164) a − b THI d = (a − b THI)(1 − THI)−e



σ = 4.601 × 10−4(5.49 × 106)2/3(499.15)1/3(0.6170)(1 − 0.6073)11/9 σ = 22.36 mN/m The reported experimental value is 22.68 mN/m.43 For hydrocarbon and polar chemicals, the approach originally suggested by Macleod65 as further developed by Sugden106,107 can be used:




(2-165)

where Tc = critical temperature, °C Tr = reduced temperature = (T + 273.15)/(Tc + 273.15) Eq. (2-163) correctly predicts that the surface tension becomes zero at the critical point.87 For nonpolar, nonhydrocarbon chemicals not found in Jasper,43 use can be made of the corresponding states approach of Brock and Bird9 as modified by Miller69 at temperature T (K): σ = 4.601 × 10−4 P c2/3 T c1/3 Q (1 − Tr)11/9

(2-166)

Tbr(ln Pc − 11.5261) Q = 0.1207 1 +  − 0.281 1 − Tbr

(2-167)



[P] σ =  (ρL − ρG) 1000

(2-161)

The molecules in a gas-liquid interface are in tension and tend to contract to a minimum surface area. This tension may be quantified by the surface tension, which is defined as the force in the plane of the surface per unit length. Jasper43 has made a critical evaluation of experimental surface tension data for approximately 2200 pure chemicals. He correlates surface tension σ (mN/m = dyn/cm) with temperature T (°C) over a specified temperature range as

σ = surface tension, mN/m Pc = critical pressure, Pa

0.6173 (ln 5.49 × 106 − 11.5261) Q = 0.1207 1 +  − 0.281 1 − 0.6173 Q = 0.6170

SURFACE TENSION

where

Example 40 Estimate surface tension for ethyl mercaptan. The required properties from Daubert et al.24 are Pc = 5.49 × 106 Pa, Tc = 499.15 K, and Tb = 308.15 K. For T = 303.15 K, Tr = 303.15/499.15 = 0.6073, and Tbr = 308.15/ 499.15 = 0.6173. Substituting into Eqs. (2-167) and (2-166):

(2-160)

The solute 1 is dissolved in a solvent pair of 2 and 3. D* are infinite dilution binary diffusivities estimated by the proper method discussed previously. The mixture viscosity can be predicted by methods of the previous section. The average absolute error when tested on 40 systems is 25 percent. The method gives higher errors if the solute is gaseous.



Errors are usually less than 5 percent.

1/2

Component 1 is the solute, while component 2 is the solvent. The latent heats, λ, are at the normal boiling point, as are the molar volumes. Using T in K and µ in Pa sec yields a diffusivity in m2/sec. The average error is 21 percent when tested on 237 experimental systems. For prediction of the liquid diffusivity of a solute in a pair of mixed solvents, the method of Tang and Himmelblau,111 Eq. (2-161), is recommended.



Tc = critical temperature, K Tr = reduced temperature, T/Tc Tb = normal boiling temperature, K Tbr = reduced normal boiling temperature, Tb /Tc

(2-159)

where

4

(2-168)

σ = surface tension, mN/m ρL = liquid density, kmol/m3 ρG = vapor density, kmol/m3

The temperature-independent parachor [P] may be calculated by the additive scheme proposed by Quale.83 The atomic group contributions for this method, with contributions for silicon, boron, and aluminum from Myers,75 are shown in Table 2-402. At low pressures, where ρL >> ρG, the vapor density term may be neglected. Errors using Eq. (2-168) are normally less than 5 to 10 percent.

Example 41 Estimate surface tension for isobutyric acid. For isobutyric acid, the liquid density from Daubert et al.24 is 10.77 kmol/m3 at 293.15 K. [P] is determined from Table 2-402: O

CH3 CH

C

CH3

OH

is made up of the two groups CH3CH(CH3) and COOH. Therefore, [P] = 133.3 + 73.8 = 207.1. With Eq. (2-168), neglecting the vapor density,



 = 24.75 mN/m

207.1 σ =  (10.77) 1000

4

Jasper43 quotes a value of 25.04 mN/m at 293.15 K. In general, the surface tension of a liquid mixture is not a simple function of the pure component surface tensions because the composition of the mixture surface is not the same as the bulk. For nonaqueous solutions of n components, the method of Winterfeld, Scriven, and Davis130 is applicable: n n xi xi (2-169) σm = ρ2   (σiσj)1/2 ρLi ρLi i=1 j=1

  

n

1 xi (2-170) =  ρ i = 1 ρLi where σm = mixture surface tension, mN/m xi, j = mole fraction of component i or j in the liquid mixture ρLi, j = pure component liquid density of component i or j, kmol/m3 σi, j = pure component surface tension of component i or j, mN/m Accuracies of 3–4 percent average deviation are typical when using this method.

Example 42 Estimate surface tension of a mixture. At 298.15 K, Daubert et al.24 report the liquid density of n-pentane to be 8.617 kmol/m3 and its surface tension to be 15.47 mN/m. From the same source, the corresponding values for dichloromethane are 15.52 kmol/m3 and 27.22 mN/m. Using Eqs. (2-170) and (2-169) for a mixture of 0.1606 mole fraction n-pentane and 0.8394 mole fraction dichloromethane: 1 0.1606 0.8394 =+ ρ 8.617 15.52 ρ = 13.75 kmol/m3

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-402

2-373

Atomic Group Contributions for Calculation of the Parachor [P ]

Atomic group

[P]

Carbon-hydrogen C H CH3 (CH2)n n = 1 − 12 n > 12 CH3CH(CH3) CH3CH2  CH(CH3) CH3CH2CH2CH(CH3) CH3CH(CH3)CH2 CH3CH2CH(C2H5) CH3C(CH3)2 CH3CH2C(CH3)2 CH3CH(CH3)CH(CH3) CH3CH(CH3)C(CH3)2 C6H5

Atomic group

9.0 15.5 55.5 40.0 40.3 133.3 171.9 211.7 173.3 209.5 170.4 207.5 207.9 243.5 189.6

Special Groups H in OH H in HN O OH O2 in acids, esters COO COOH N NH2 S P Si Si (silanes)

10.0 12.5 19.8 29.8 54.8 63.8 73.8 17.5 42.5 49.1 40.5 30.3 43.3

[P]

Special Groups (Cont.) B Al F Cl Br I

13.2 34.9 26.1 55.2 68.0 90.3

Ethylenic Bond Terminal* 2,3-position 3,4-position†

19.1 17.7 16.3

Triple Bond

40.6

Ring Closure 3-membered 4-membered 5-membered 6-membered 7-membered

12.5 6.0 3.0 0.8 4.0

ⴝ0 (ketone) 3 carbon atoms 4 carbon atoms 5 carbon atoms 6 carbon atoms 7 carbon atoms 8 carbon atoms 9 carbon atoms 10 carbon atoms 11 carbon atoms

22.3 20.0 18.5 17.3 17.3 15.1 14.1 13.0 12.6

* Use the value for double bonds in cyclic compound. Assume 3 double bonds for the aromatic ring. † Use 16.3 for double bonds in the 3, 4 or higher positions.



 (15.47) 0.1606 0.8394 + 2(13.75)     [(15.47)(27.22)] 8.617 15.52 0.8394 + (13.75)    (27.22) 15.52

0.1606 σm = (13.75)2  8.617

2

2

1/2

σm = 23.89 mN/m De Soria et al.26 give an experimental value of 24.24 mN/m for this mixture. Surface tensions for aqueous solutions are more difficult to predict than those for nonaqueous mixtures because of the nonlinear dependence on mole fraction. Small concentrations of the organic material may significantly affect the mixture surface tension value. For many binary organic-water mixtures, the method of Tamura, Kurata, and Odani110 may be used: where

(2-172)

ψw is defined by the relation:



Chloroacetic acid

V(chloroacetic acid) q = 2  V(acetic acid) Expected errors are less than 10 percent when q is less than 5 and within 20 percent when q is greater than 5.

Example 43 Estimate surface tension of a water-methanol mixture. Equation (2-171) can be used with a water-methanol mixture at 303.15 K when the methanol mole fraction is 0.122. From Jasper,43 σw = 71.40 mN/m, and σo = 21.73 mN/m. The density of water (per Ref. 24) is 55.16 kmol/m3; Vw = 0.01813 m3/kmol. The density of methanol is 24.49 kmol/m3 (Ref. 24); Vo = 0.04083 m3/kmol. For methanol, q = 1. Using Eq. (2-173) to obtain ψw: ψw (0.878)(0.01813) log10  = log10  1 − ψw (0.122)(0.04083)



(ψw)q (xwVw)q log10  = log10  (xwVw + xoVo)1 − q (1 − ψw) xoVo





q σoV 2/3 o + 44.1   − σwV w2/3 T q where

Example Acetic acid: q = 2 Acetone: q = 2

(2-171)

σm = mixture surface tension, mN/m σw = surface tension of pure water, mN/m σo = surface tension of pure organic component, mN/m ψo = 1 − ψw



q Number of carbon atoms One less than the number of carbon atoms Number of carbon atoms times the ratio of the molar volume of the halogen derivative to the parent fatty acid

Halogen derivatives of fatty acids

2

2

1/4 σ m1/4 = ψwσ 1/4 w + ψoσ o

Organic component Fatty acids, alcohols Ketones

(2-173)

xw = bulk mole fraction of pure water xo = bulk mole fraction of pure organic component Vw = molar volume of pure water, m3/kmol Vo = molar volume of pure organic component, m3/kmol T = temperature, K q = constant depending upon the size and type of the organic component; see table:



44.1 +  [(21.73)(0.04083)2/3 − (71.40)(0.01813)2/3] 303.15 = 0.505 − 0.342 = 0.163 ψw  = 100.163 = 1.455 1 − ψw 1.455 ψw =  = 0.593 2.455

2-374

PHYSICAL AND CHEMICAL DATA

Using Eq. (2-172): ψo = 1 − 0.593 = 0.407. Substituting into Eq. (2-171): σ m1/4 = (0.593)(71.40)1/4 + (0.407)(21.73)1/4 σ m1/4 = 2.603 σm = 45.91 mN/m The reported experimental value is 46.1 mN/m.110

FLAMMABILITY PROPERTIES Flash points, lower and upper flammability limits, and autoignition temperatures are the three properties used to indicate safe operating limits of temperature when processing organic materials. Prediction methods are somewhat erratic, but, together with comparisons with reliable experimental values for families or similar compounds, they are valuable in setting a conservative value for each of the properties. The DIPPR compilation includes evaluated values for over 1000 common organics. Detailed examples of most of the methods discussed are available in Danner and Daubert.22 The flash point is the lowest temperature at which a liquid gives off sufficient vapor to form an ignitable mixture with air near the surface of the liquid or within the vessel used. ASTM test methods include procedures using a closed cup (ASTM D56, ASTM D93, and ASTM D3828), which is preferred, and an open cup (ASTM D92 and ASTM D1310). When several values are available, the lowest temperature is usually taken in order to assure safe operation of the process. The method of Shebeko et al.96 is the preferred flash point prediction method. The formula of the compound, the system pressure, and vapor pressure data for the compound must be available or estimable. Equation (2-174) is the basic equation. P Psat =  = 0 1 + 4.76(2β − 1)

(2-174)

(NH − NX) NO β = NC + NS +  −  4 2 N’s are the numbers of atoms of carbon (C), sulfur (S), hydrogen (H), halogens (X), and oxygen (O) in the molecule. P is the total system pressure. Psat is the vapor pressure of the compound at the flash point temperature. If Psat is available as a function of temperature, Eq. (2-174) can be solved directly for the flash point temperature. Otherwise, trial and error with a table of Psat vs. T is required. Errors average about 5°C but may be as much as 15°C. An alternate method for flash point prediction is the method of Gmehling and Rasmussen33 and depends on the lower flammability limit (discussed later). Vapor pressure as a function of temperature is also required. The method is generally not as accurate as the preceding method as flammability limit errors are propagated. The authors have also extended the method to defined mixtures of organics. The upper and lower flammability limits are the boundary-line mixtures of vapor or gas with air, which, if ignited, will just propagate flame and are given in terms of percent by volume of gas or vapor in the air. Each of these limits also has a temperature at which the flammability limits are reached. The temperature corresponding to the lower-limit partial vapor pressure should equal the flash point. The

temperature corresponding to the upper-limit partial vapor pressure is somewhat above the lower limit and is usually considerably below the autoignition temperature. Flammability limits are calculated at one atmosphere total pressure and normally are considered synonymous with explosive limits. Limits in oxygen rather than air are sometimes measured and available. Limits are generally reported at 298 K and 1 atm. If temperature or pressure are increased, the lower limit will decrease while the upper limit will increase, giving a wider range of compositions over which flame will propagate. The most generally applicable method for prediction of the property is the method of Seaton,95 which depends only on the molecular structure of the molecule and utilizes second order (Benson-type) groups to construct the molecule. Equation (2-175) sums the groups’ number of each type group (ni) to get both the upper and lower limits. zu or

(ni fi ) zl =  nf

ii gi

 

(2-175)

Two sets of fi and gi are given in the article for each second-order group to cover both upper (u) and lower (l) limits (z) in volume percent units. A study of this method for about 80 organic compounds in 14 families shows absolute errors of 0.15 percent and 2.3 percent for the lower and upper limits, respectively. The upper limit prediction should not be used for ethers. Alternate group contribution methods dependent only on molecular structure are the method of Shebeko et al.97 modified by High and Siegel39 for lower flammability limit and the method of High and Danner38 for upper flammability limit. Both methods are detailed by Danner and Daubert.22 A study comparing these methods with the Seaton method shows slightly higher absolute errors of 0.23 percent and 2.9 percent for the lower and upper limits, respectively. The upper limit prediction should not be used for ethers. Both methods are recommended to be used only for qualitative guidance. Lower flammability limits can also be back-calculated from a known flash point by the method of Gmehling and Rasmussen33 discussed earlier. The autoignition temperature is the minimum temperature for a substance to initiate self-combustion in air in the absence of a spark or flame. The temperature is no lower than and is generally considerably higher than the temperature corresponding to the upper flammability limit. Large differences can occur in reported values determined by different procedures. The lowest reasonable value should be accepted in order to assure safety. Values are also sometimes given in oxygen rather than in air. Values for hydrocarbons other than alkynes and alkadienes can be predicted by the method of Suzuki et al.108 The best model includes the descriptors Tc, Pc, the parachor, the molecular surface area (which can be approximated by the van der Waals area), and the zero-order connectivity index. Excluding alkynes and alkadienes, a study for 58 alkanes, aromatics, and cycloalkanes showed an average deviation from experimental values of about 30 K. Another method of estimating autoignition temperatures is to compare values for a compound with other members of its homologous series on a plot vs. carbon number as the temperature decreases and carbon number increases. Affens1 gives a formal procedure for such estimation.

Section 3

Mathematics*

Bruce A. Finlayson, Ph.D., Rehnberg Professor and Chair, Department of Chemical Engineering, University of Washington: Member, National Academy of Engineering (Numerical methods and all general material; section editor) James F. Davis, Ph.D., Professor of Chemical Engineering, Ohio State University (Intelligent Systems) Arthur W. Westerberg, Ph.D., Swearingen University Professor of Chemical Engineering, Carnegie Mellon University: Member, National Academy of Engineering (Optimization) Yoshiyuki Yamashita, Ph.D., Associate Professor of Chemical Engineering, Tohoku University, Sendai, Japan (Intelligent Systems)

MATHEMATICS General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miscellaneous Mathematical Constants. . . . . . . . . . . . . . . . . . . . . . . . . . The Real-Number System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Progressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algebraic Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-7 3-8 3-8 3-9 3-9

MENSURATION FORMULAS Plane Geometric Figures with Straight Boundaries . . . . . . . . . . . . . . . . Plane Geometric Figures with Curved Boundaries . . . . . . . . . . . . . . . . Solid Geometric Figures with Plane Boundaries . . . . . . . . . . . . . . . . . . Solids Bounded by Curved Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miscellaneous Formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Irregular Areas and Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-10 3-10 3-11 3-11 3-12 3-12

ELEMENTARY ALGEBRA Operations on Algebraic Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . The Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Progressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Permutations, Combinations, and Probability. . . . . . . . . . . . . . . . . . . . . Theory of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-12 3-13 3-13 3-14 3-14

ANALYTIC GEOMETRY Plane Analytic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solid Analytic Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-16 3-18

PLANE TRIGONOMETRY Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functions of Circular Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inverse Trigonometric Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relations between Angles and Sides of Triangles . . . . . . . . . . . . . . . . . . Hyperbolic Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Approximations for Trigonometric Functions . . . . . . . . . . . . . . . . . . . . .

3-20 3-20 3-21 3-22 3-22 3-23

DIFFERENTIAL AND INTEGRAL CALCULUS Differential Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multivariable Calculus Applied to Thermodynamics . . . . . . . . . . . . . . . Integral Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-23 3-26 3-27

INFINITE SERIES Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Operations with Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tests for Convergence and Divergence. . . . . . . . . . . . . . . . . . . . . . . . . . Series Summation and Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-30 3-31 3-31 3-32

COMPLEX VARIABLES Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Special Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trigonometric Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Powers and Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elementary Complex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex Functions (Analytic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-33 3-33 3-33 3-33 3-33 3-34

* The contributions of William F. Ames (retired), Georgia Institute of Technology; Arthur E. Hoerl (deceased), University of Delaware; and M. Zuhair Nashed, University of Delaware, to material that was used from the sixth edition is gratefully acknowledged. 3-1

3-2

MATHEMATICS

DIFFERENTIAL EQUATIONS Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ordinary Differential Equations of the First Order . . . . . . . . . . . . . . . . Ordinary Differential Equations of Higher Order . . . . . . . . . . . . . . . . . Special Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-35 3-36 3-36 3-37 3-38

DIFFERENCE EQUATIONS Elements of the Calculus of Finite Differences . . . . . . . . . . . . . . . . . . . Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-41 3-41

INTEGRAL EQUATIONS Classification of Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relation to Differential Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Methods of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-42 3-43 3-43

INTEGRAL TRANSFORMS (OPERATIONAL METHODS) Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convolution Integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fourier Cosine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-44 3-45 3-45 3-46 3-46

MATRIX ALGEBRA AND MATRIX COMPUTATIONS Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Matrix Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-47 3-48

NUMERICAL APPROXIMATIONS TO SOME EXPRESSIONS Approximation Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

NUMERICAL ANALYSIS AND APPROXIMATE METHODS Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Solution of Linear Equations. . . . . . . . . . . . . . . . . . . . . . . . . Numerical Solution of Nonlinear Equations in One Variable . . . . . . . . Interpolation and Finite Differences. . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Integration (Quadrature) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Solution of Ordinary Differential Equations as Initial Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ordinary Differential Equations-Boundary Value Problems . . . . . . . . . Numerical Solution of Integral Equations. . . . . . . . . . . . . . . . . . . . . . . . Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Solution of Partial Differential Equations. . . . . . . . . . . . . . . Spline Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fast Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-54 3-57 3-60 3-60 3-61 3-64 3-64

OPTIMIZATION Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conditions for Optimality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strategies of Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-65 3-66 3-67

STATISTICS Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enumeration Data and Probability Distributions . . . . . . . . . . . . . . . . . . Measurement Data and Sampling Densities. . . . . . . . . . . . . . . . . . . . . . Tests of Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Error Analysis of Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Factorial Design of Experiments and Analysis of Variance . . . . . . . . . .

3-69 3-71 3-72 3-76 3-83 3-87 3-87

DIMENSIONAL ANALYSIS PROCESS SIMULATION

3-49

3-49 3-50 3-50 3-51 3-53 3-53

INTELLIGENT SYSTEMS IN PROCESS ENGINEERING

GENERAL REFERENCES: The list of references for this section is selected to provide a broad perspective on classical and modern mathematical methods that are useful in chemical engineering. The references supplement and extend the treatment given in this section. Also included are selected references to important areas of mathematics that are not covered in the Handbook but that may be useful for certain areas of chemical engineering, e.g., additional topics in numerical analysis and software, optimal control and system theory, linear operators, and functional-analysis methods. Readers interested in brief summaries of theory, together with many detailed examples and solved problems on various topics of college mathematics and mathematical methods for engineers, are referred to the Schaum’s Outline Series in Mathematics, published by the McGraw-Hill Book Comapny.

32. 33. 34. 35. 36. 37. 38. 39.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.

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237. Rall, L. B. Computational Solution of Nonlinear Operator Equations, Wiley, New York (1969) and Dover, New York (1981). 238. Ralston, A., and A. Rabinowitz. A First Course in Numerical Analysis, 2d ed., McGraw-Hill, New York (1978). 239. Ramirez, W. F. Computational Methods for Process Simulations, Butterworths, Boston (1989). 240. Rauch, J. Partial Differential Equations, Springer-Verlag (1991). 241. Reddy, J. N., and D. K. Gartling. The Finite Element Method in Heat Transfer and Fluid Dynamics, CRC Press (1994). 242. Reklaitis, G. V. Introduction to Material and Energy Balances, Wiley (1983). 243. Rekalitis, G. V., and H. D. Spriggs. Proceedings of the First International Conference on Foundations of Computer-Aided Operations, Elsevier Science Publishers, Inc., New York (1987). 244. Reklaitis, G. V., A. Ravindran, and K. M. Ragsdell. Engineering Optimization Methods and Applications, Wiley, New York (1983). 245. Rhee, H.-K., R. Aris, and N. R. Amundson. First-Order Partial Differential Equations, vol. I., Prentice Hall, Englewood Cliffs, NJ (1986). 246. ———. Matrix Computations and Mathematical Software, McGrawHill, New York (1981). 247. ———. Numerical Methods, Software, and Analysis, 2d ed., Academic, New York (1993). 248. Rich, E., and K. Kevin. Artificial Intelligence, 2d ed., McGraw-Hill, New York (1991). 249. Riggs, J. B. An Introduction to Numerical Methods for Chemical Engineers, Texas Tech Univ. Press, Lubbock, TX (1994). 250. Rippin, D. W. T., J. C. Hale, and J. F. Davis (ed.). Proceedings of the Second International Conference on Foundations of Computer-Aided Operations, CACHE Corporation, Austin, TX (1994). 251. Ritchmyer, R., and K. Morton. Difference Methods for Initial-Value Problems, 2d ed., Interscience, New York (1967). 252. Saaty, T. L., and J. Bram. Nonlinear Mathematics, McGraw-Hill, New York (1964) and Dover, New York (1981). 253. Schiesser, W. E. The Numerical Method of Lines, Academic Press (1991). 254. Schittkowski, K. Num. Math. 38 (1982): 83. 255. Seader, J. D. Computer Modeling of Chemical Processes, AIChE Monog. Ser. No. 15 (1985). 256. Seborg, D. E., T. F. Edgar, and D. A. Mellichamp. Process Dynamics and Control, Wiley, New York (1989). 257. Shampine, L. Numerical Solution of Ordinary Differential Equations, Chapman & Hall (1994). 258. Shapiro, S. C., D. Eckroth et. al (ed.). Encyclopedia of Artificial Intelligence, Wiley, New York (1987). 259. Shanno, D. F. Math. Comp. 24 (1970): 647. 260. Shenk, A. Calculus and Analytic Geometry, Goodyear Publishing Company, Santa Monica, CA (1977). 261. Shockley, J. E. Calculus and Analytic Geometry, Saunders, Philadelphia (1982). 262. Siirola, J. J., I. E. Grossmann, and G. Stephanopoulos. Proceedings of the Second International Conference on Foundations of Computer-Aided Design, Elsevier Science Publishers, Inc., New York (1990). 263. Simmons, G. F. Differential Equations, McGraw-Hill, New York (1972). 264. Simmonds, J. G. A Brief on Tensor Analysis, Springer-Verlag (1994). 265. Sincich, T., and Mendenhall, W. Statistics for Engineering and the Sciences, 4th ed., Prentice Hall, Englewood Cliffs, NJ (1995). 266. Sincovec, R. F. Math. Comp. 26 (1972): 893–895. 267. Smith, I. M., J. L. Siemienivich, and I. Gladwell. “A Comparison of Old and New Methods for Large Systems of Ordinary Differential Equations Arising from Parabolic Partial Differential Equations,” Num. Anal. Rep. Department of Engineering, no. 13, University of Manchester, England (1975). 268. Smith, W. K. Analytic Geometry, Macmillan (1972). 269. Sobel, M. A., and N. Lerner. College Algebra, Prentice Hall, Englewood Cliffs, NJ (1983). 270. Sod, G. A. Numerical Methods in Fluid Dynamics, Cambridge Univ. Press (1985). 271. Sokolnikoff, I. S., and Sokolnikoff, E. S. Higher Mathematics for Engineers and Physicists, McGraw-Hill, New York (1941). 272. Spiegel, M. R. Applied Differential Equations, 3d ed., Prentice Hall, Englewood Cliffs, NJ (1981). 273. Stakgold, I. Green’s Functions and Boundary Value Problems, Interscience, New York (1979). 274. Stein, S. K. Calculus and Analytic Geometry, 3d ed., McGraw-Hill, New York (1982). 275. Stephanopoulos, G., and J. F. Davis (eds.). Artificial Intelligence in Process Engineering, CACHE Monograph Series, CACHE, Austin (1990–1992).

276. Stephanopoulos, G., and H. Chonghun. “Intelligent Systems in Process Engineering: A Review,” Proceedings of PSE ’94, Korea (1994). 277. Stillwell, J. C. Elements of Algebra, CRC Press, New York (1994). 278. Stoer, J., and R. Bulirsch. Introduction to Numerical Analysis, Springer, New York (1993). 279. Strang, G. Linear Algebra and Its Applications, 2d ed., Academic, New York (1980). 280. Strang, G. Introduction to Linear Algebra, Wellesley-Cambridge, Cambridge, MA (1993). 281. ——— and G. Fix. An Analysis of the Finite Element Method, Prentice Hall, Englewood Cliffs, NJ (1973). 282. Swokowski, E. W. Calculus with Analytic Geometry, 2d ed., Prindle, Weber & Schmidt, Boston (1981). 283. Taylor, A. E., and D. C. Lay. Introduction to Functional Analysis, 2d ed., Wiley, New York (1980). 284. Umeda, T., and A. Ichikawa. I&EC Proc. Design Develop. 10 (1971): 229. 285. Vasantharajan, S., and L. T. Biegler. Computers and Chemical Engineering 12 (1988): 1087. 286. Vemuri, V., and W. Karplus. Digital Computer Treatment of Partial Differential Equations, Prentice Hall, Englewood Cliffs, NJ (1981). 287. Vichnevetsky, R. Computer Methods for Partial Differential Equations, vols. 1 and 2, Prentice Hall, Englewood Cliffs, NJ (1981, 1982). 288. Villadsen, J. V., and M. L. Michelsen. Solution of Differential Equation Models by Polynomial Approximation, Prentice Hall, Englewood Cliffs, NJ (1978). 289. Villadsen, J., and W. E. Stewart. Chem. Eng. Sci. 22 (1967): 1483–1501. 290. Walas, S. M. Modeling with Differential Equations in Chemical Engineering, Butterworth-Heinemann, Stoneham, MA (1991). 291. Weisberg, S. Applied Linear Regression, 2d ed., Wiley, New York (1985). 292. Weld, D. S., and J. de Kleer (ed.). Readings in Qualitative Reasoning About Physical Systems, Morgan Kaufman, San Mateo, CA (1990). 293. Westerberg, A. W., H. P. Hutchison, R. L. Motard, and P. Winter. Process Flowsheeting, Cambridge University Press, London (1979). 294. Westerberg, A. W., and H. H. Chien (ed.). Proceedings of the Second International Conference on Foundations of Computer-Aided Design, CACHE Corporation, Austin, TX (1984). 295. Westerberg, A. W. “Optimization” in A. K. Sunol, D. W. T. Rippin, G. V. Reklaitis, O. Hortacsu (eds.), Batch Processing Systems Engineering: Current Status and Future Directions, vol. 143, NATO ASI Series F, Springer, Berlin (1995). 296. Whipkey, K. L., and M. N. Whipkey. The Power of Calculus, 3d ed., Wiley, New York (1979). 297. Wilkinson, J. H. The Algebraic Eigenvalue Problem, Clarendon Press, Oxford (1988). 298. Williams, G. Computational Linear Algebra with Models, 2d ed., Allyn and Bacon, Boston (1981). 299. Wolfram, S. Mathematica, Addison-Wesley, New York (1988). 300. Wouk, A. A Course of Applied Functional Analysis, Interscience, New York (1979). 301. Wylie, C. R. Advanced Engineering Mathematics, 5th ed., McGraw-Hill, New York (1982). 302. Young, D. M. Iterative Solution for Large Linear Systems, Academic, New York (1971). 303. Zienkiewicz, O. C., and R. L. Taylor. The Finite Element Method, McGraw-Hill, London (1989). 304. ——— and K. Morgan. Finite Elements and Approximations, Wiley, New York (1983). REFERENCES FOR GENERAL AND SPECIFIC TOPICS Advanced engineering mathematics: Upper undergraduate level, 19, 80, 127, 144, 156, 173, 194, 203, 209, 301. Graduate level, 79, 127, 212, 273. Mathematical tables, mathematical dictionaries, and handbooks of mathematical functions and formulas, 1, 28, 48, 57, 118, 126, 188, 208, 217, 222, 234. Mathematical modeling of physical phenomena, 17, 19, 31, 44, 98, 132, 194, 273. Mathematical theory of reaction, diffusion, and transport processes, 10, 16, 19, 88, 223. Mathematical methods in chemical engineering, 13, 15, 61, 85, 104, 106, 138, 239, 249, 288. Inequalities, 28, 126, 139, 290. Vector and tensor analysis, 18, 163, 173, 264. Special functions in physics and engineering, 190, 197, 235. Green’s functions and applications, 75, 127, 273. Perturbation and asymptotic methods in applied mathematics, 170, 215, 216. Approximation theory and interpolation, 86, 87. Functional analysis; linear operators, 129, 147, 162, 174, 196, 216, 226, 283, 300. Generalized inverses and least-squares problems, 33, 128, 189.

MATHEMATICS

3-7

MATHEMATICS GENERAL The basic problems of the sciences and engineering fall broadly into three categories: 1. Steady state problems. In such problems the configuration of the system is to be determined. This solution does not change with time but continues indefinitely in the same pattern, hence the name “steady state.” Typical chemical engineering examples include steady temperature distributions in heat conduction, equilibrium in chemical reactions, and steady diffusion problems. 2. Eigenvalue problems. These are extensions of equilibrium problems in which critical values of certain parameters are to be determined in addition to the corresponding steady-state configurations. The determination of eigenvalues may also arise in propagation problems. Typical chemical engineering problems include those in heat transfer and resonance in which certain boundary conditions are prescribed. 3. Propagation problems. These problems are concerned with predicting the subsequent behavior of a system from a knowledge of the initial state. For this reason they are often called the transient (time-varying) or unsteady-state phenomena. Chemical engineering examples include the transient state of chemical reactions (kinetics), the propagation of pressure waves in a fluid, transient behavior of an adsorption column, and the rate of approach to equilibrium of a packed distillation column. The mathematical treatment of engineering problems involves four basic steps: 1. Formulation. The expression of the problem in mathematical language. That translation is based on the appropriate physical laws governing the process. 2. Solution. Appropriate mathematical operations are accomplished so that logical deductions may be drawn from the mathematical model. 3. Interpretation. Development of relations between the mathematical results and their meaning in the physical world. 4. Refinement. The recycling of the procedure to obtain better predictions as indicated by experimental checks. Steps 1 and 2 are of primary interest here. The actual details are left to the various subsections, and only general approaches will be discussed. The formulation step may result in algebraic equations, difference equations, differential equations, integral equations, or combinations of these. In any event these mathematical models usually arise from statements of physical laws such as the laws of mass and energy conservation in the form. Input of conserved quantity − output of conserved quantity + conserved quantity produced = accumulation of conserved quantity Rate of input of conserved quantity − rate of output of conserved quantity + rate of conserved quantity produced = rate of accumulation of conserved quantity These statements may be abbreviated by the statement Input − output + production = accumulation

FIG. 3-1

Boundary conditions.

is called a boundary-value problem (Fig. 3-1). Schematically, the problem is characterized by a differential equation plus an open region in which the equation holds and, on the boundaries of the region, by certain conditions (boundary conditions) that are dictated by the physical problem. The solution of the equation must satisfy the differential equation inside the region and the prescribed conditions on the boundary. In mathematical language, the propagation problem is known as an initial-value problem (Fig. 3-2). Schematically, the problem is characterized by a differential equation plus an open region in which the equation holds. The solution of the differential equation must satisfy the initial conditions plus any “side” boundary conditions. The description of phenomena in a “continuous” medium such as a gas or a fluid often leads to partial differential equations. In particular, phenomena of “wave” propagation are described by a class of partial differential equations called “hyperbolic,” and these are essentially different in their properties from other classes such as those that describe equilibrium (“elliptic”) or diffusion and heat transfer (“parabolic”). Prototypes are: 1. Elliptic. Laplace’s equation ∂2u ∂2u
2 +
2 = 0 ∂x ∂y Poisson’s equation

∂2u ∂2u
2 +
2 = g(x,y) ∂x ∂y These do not contain the variable t (time) explicitly; accordingly, their solutions represent equilibrium configurations. Laplace’s equation corresponds to a “natural” equilibrium, while Poisson’s equation corresponds to an equilibrium under the influence of an external force of density proportional to g(x, y). 2. Parabolic. The heat equation ∂u ∂2u ∂2u
=
2 +
2 ∂t ∂x ∂y describes nonequilibrium or propagation states of diffusion as well as heat transfer. 3. Hyperbolic. The wave equation ∂2u ∂2u ∂2u =
2 +
2
∂t2 ∂x ∂y describes wave propagation of all types when the assumption is made that the wave amplitude is small and that interactions are linear. The solution phase has been characterized in the past by a concentration on methods to obtain analytic solutions to the mathematical

When the basic physical laws are expressed in this form, the formulation is greatly facilitated. These expressions are quite often given the names, “material balance,” “energy balance,” and so forth. To be a little more specific, one could write the law of conservation of energy in the steady state as Rate of energy in − rate of energy out + rate of energy produced = 0 Many general laws of the physical universe are expressible by differential equations. Specific phenomena are then singled out from the infinity of solutions of these equations by assigning the individual initial or boundary conditions which characterize the given problem. In mathematical language one such problem, the equilibrium problem,

FIG. 3-2

Propagation problem.

3-8

MATHEMATICS

equations. These efforts have been most fruitful in the area of the linear equations such as those just given. However, many natural phenomena are nonlinear. While there are a few nonlinear problems that can be solved analytically, most cannot. In those cases, numerical methods are used. Due to the widespread availability of software for computers, the engineer has quite good tools available. Numerical methods almost never fail to provide an answer to any particular situation, but they can never furnish a general solution of any problem. The mathematical details outlined here include both analytic and numerical techniques useful in obtaining solutions to problems. Our discussion to this point has been confined to those areas in which the governing laws are well known. However, in many areas, information on the governing laws is lacking. Interest in the application of statistical methods to all types of problems has grown rapidly since World War II. Broadly speaking, statistical methods may be of use whenever conclusions are to be drawn or decisions made on the basis of experimental evidence. Since statistics could be defined as the technology of the scientific method, it is primarily concerned with the first two aspects of the method, namely, the performance of experiments and the drawing of conclusions from experiments. Traditionally the field is divided into two areas: 1. Design of experiments. When conclusions are to be drawn or decisions made on the basis of experimental evidence, statistical techniques are most useful when experimental data are subject to errors. The design of experiments may then often be carried out in such a fashion as to avoid some of the sources of experimental error and make the necessary allowances for that portion which is unavoidable. Second, the results can be presented in terms of probability statements which express the reliability of the results. Third, a statistical approach frequently forces a more thorough evaluation of the experimental aims and leads to a more definitive experiment than would otherwise have been performed. 2. Statistical inference. The broad problem of statistical inference is to provide measures of the uncertainty of conclusions drawn from experimental data. This area uses the theory of probability, enabling scientists to assess the reliability of their conclusions in terms of probability statements. Both of these areas, the mathematical and the statistical, are intimately intertwined when applied to any given situation. The methods of one are often combined with the other. And both in order to be successfully used must result in the numerical answer to a problem—that is, they constitute the means to an end. Increasingly the numerical answer is being obtained from the mathematics with the aid of computers. MISCELLANEOUS MATHEMATICAL CONSTANTS Numerical values of the constants that follow are approximate to the number of significant digits given. π = 3.1415926536 e = 2.7182818285 γ = 0.5772156649 ln π = 1.1447298858 log π = 0.4971498727 Radian = 57.2957795131° Degree = 0.0174532925 rad Minute = 0.0002908882 rad Second = 0.0000048481 rad

Pi Napierian (natural) logarithm base Euler’s constant Napierian (natural) logarithm of pi, base e Briggsian (common logarithm of pi, base 10

THE REAL-NUMBER SYSTEM The natural numbers, or counting numbers, are the positive integers: 1, 2, 3, 4, 5, . . . . The negative integers are −1, −2, −3, . . . . A number in the form a/b, where a and b are integers, b ≠ 0, is a rational number. A real number that cannot be written as the quotient of two integers is called an irrational number, e.g.,
2,
3,
5, π, 3 e,
2. There is a one-to-one correspondence between the set of real numbers and the set of points on an infinite line (coordinate line).

Order among Real Numbers; Inequalities a > b means that a − b is a positive real number. If a < b and b < c, then a < c. If a < b, then a  c < b  c for any real number c. If a < b and c > 0, then ac < bc. If a < b and c < 0, then ac > bc. If a < b and c < d, then a + c < b + d. If 0 < a < b and 0 < c < d, then ac < bd. If a < b and ab > 0, then 1/a > 1/b. If a < b and ab < 0, then 1/a < 1/b. Absolute Value For any real number x, |x| = x −x



if x ≥ 0 if x < 0

Properties If |x| = a, where a > 0, then x = a or x = −a. |x| = |−x|; −|x| ≤ x ≤ |x|; |xy| = |x| |y|. If |x| < c, then −c < x < c, where c > 0. ||x| − |y|| ≤ |x + y| ≤ |x| + |y|.
x2 = |x|. a = c , then a + b = c + d , a − b = c − d , Proportions If





b d d b d b a−b c−d = .

a+b c+d Indeterminants Form

Example

(∞)(0) 00 ∞0 1∞

xe−x xx (tan x)cos x (1 + x)1/x

x→∞ x → 0+ − x→aπ x → 0+

∞−∞ 0
0 ∞
∞

1 −
x− 1
x+ sin x
x ex
x

x→∞ x→0 x→∞

Limits of the type 0/∞, ∞/0, 0∞, ∞ ⋅ ∞, (+∞) + (+∞), and (−∞) + (−∞) are not indeterminate forms. Integral Exponents (Powers and Roots) If m and n are positive integers and a, b are numbers or functions, then the following properties hold: a−n = 1/an a≠0 (ab)n = anbn (an)m = anm, n


a  = a1/n

anam = an + m if

a>0

mn

 =
a, a > 0
a 
 m

n

n

am a > 0 am/n = (am)1/n =
, a0 = 1 (a ≠ 0) 0a = 0 (a ≠ 0) Infinity (∞) is not a real number. It is possible to extend the realnumber system by adjointing to it “∞” and “−∞,” and within the extended system, certain operations involving +∞ or −∞ are possible. For example, if 0 < a < 1, then a∞ = lim x→∞ ax = 0, whereas if a > 1, then a∞ = ∞, ∞a = ∞ (a > 0), ∞a = 0 (a < 0). Care should be taken in the case of roots and fractional powers of a product; e.g.,
 xy ≠
x
y if x and y are negative. This rule applies if one is careful about the domain of the functions involved; so
 xy =
 x
y if x > 0 and y > 0. Given any number b > 0, there is a unique function f(x) defined for all real numbers x such that (1) f(x) = bx for all rational x; (2) f is increasing if b > 1, constant if b = 1, and decreasing if 0 < b < 1. This function is called the exponential function bx. For any b > 0, f(x) = bx is

MATHEMATICS a continuous function. Also with a,b > 0 and x,y any real numbers, we have (ab)x = axbx bx by = bx + y (bx)y = bxy The exponential function with base b can also be defined as the inverse of the logarithmic function. The most common exponential function in applications corresponds to choosing b the transcendental number e. Logarithms log ab = log a + log b, a > 0, b > 0 log an = n log a log (a/b) = log a − log b n log
a = (1/n) log a The common logarithm (base 10) is denoted log a or log10a. The natural logarithm (base e) is denoted ln a (or in some texts loge a). Roots If a is a real number, n is a positive integer, then x is called the nth root of a if xn = a. The number of nth roots is n, but not all of them are necessarily real. The principal nth root means the following: (1) if a > 0 the principal nth root is the unique positive root, (2) if a < 0, and n odd, it is the unique negative root, and (3) if a < 0 and n even, it is any of the complex roots. In cases (1) and (2), the root can be found on a calculator by taking y = ln a/n and then x = e y. In case (3), see the section on complex variables. PROGRESSIONS Arithmetic Progression n−1

1

n

 (a + kd) = na +
2 n(n − 1)d =
2 (a + ᐉ)

k=0

where ᐉ is the last term, ᐉ = a + (n − 1)d. Geometric Progression n a(rn − 1) ar k − 1 =
(r ≠ 1)  r−1 k=1 Arithmetic-Geometric Progression n−1 a − [a + (n − 1)d]r n dr(1 − r n − 1) (a + kd)r k =

+

 1−r (1 − r)2 k=0 n 1 2 n (n + 1)2(2n2 + 2n − 1)  k5 =
12 k=1 2

k=1 n

 (2k − 1)

3



2

n

n

n

k k

k

k=1

2

k

k=1

2

k=1

The equality holds if, and only if, the vectors a, b are linearly dependent (i.e., one vector is scalar times the other vector). Minkowski’s Inequality Let a1, a2, . . . , an and b1, b2, . . . , bn be any two sets of complex numbers. Then for any real number p > 1,

  |a + b |  ≤   |a |  +   |b |  1/p

n

k

k

1/p

n

p

k

k=1

1/p

n

p

k

k=1

p

k=1

Hölder’s Inequality Let a1, a2, . . . , an and b1, b2, . . . , bn be any two sets of complex numbers, and let p and q be positive numbers with 1/p + 1/q = 1. Then

  a b  ≤   |a |    |b |  n

1/p

n

k k

k

k=1

1/q

n

p

q

k

k=1

k=1

The equality holds if, and only if, the sequences |a1|p, |a2|p, . . . , |an|p and |b1|q, |b2|q, . . . , |bn|q are proportional and the argument (angle) of the complex numbers akb k is independent of k. This last condition is of course automatically satisfied if a1, . . . , an and b1, . . . , bn are positive numbers. Lagrange’s Inequality Let a1, a2, . . . , an and b1, b2, . . . , bn be real numbers. Then 2

n

n

2 k

k=1



2 k

k=1

(akbj − aj bk)2

1≤k≤j≤n

Example Two chemical engineers, John and Mary, purchase stock in the same company at times t1, t2, . . . , tn, when the price per share is respectively p1, p2, . . . , pn. Their methods of investment are different, however: John purchases x shares each time, whereas Mary invests P dollars each time (fractional shares can be purchased). Who is doing better? While one can argue intuitively that the average cost per share for Mary does not exceed that for John, we illustrate a mathematical proof using inequalities. The average cost per share for John is equal to n

x  pi 1 n Total money invested i=1



=
=
 pi Number of shares purchased n i=1 nx

nP n
=
n n P 1



i i = 1 pi = 1 pi

= n2(2n2 − 1)



1 γ = lim 
− ln n = 0.577215 n→∞ m m=1 ALGEBRAIC INEQUALITIES Arithmetic-Geometric Inequality Let An and Gn denote respectively the arithmetic and the geometric means of a set of positive numbers a1, a2, . . . , an. The An ≥ Gn, i.e., a1 + a2 + ⋅⋅⋅ + an

≥ (a1a2 ⋅⋅⋅ an)1/n n The equality holds only if all of the numbers ai are equal. Carleman’s Inequality The arithmetic and geometric means just defined satisfy the inequality n

G

r

r=1

  a b  ≤   |a |   |b | 

The average cost per share for Mary is

1 =
n(4n2 − 1) 3

k=1

n

⋅⋅⋅ ar)1/r ≤ neAn

where e is the best possible constant in this inequality. Cauchy-Schwarz Inequality Let a = (a1, a2, . . . , an), b = (b1, b2, . . . , bn), where the ai’s and bi’s are real or complex numbers. Then

k=1

k=1

2

1 2

r=1

k k

n

n

n

 (a a

n

 (2k − 1) = n  (2k − 1)

or, equivalently,

  a b  =   a   b  −

(r ≠ 1)

3-9

≤ neAn

Thus the average cost per share for John is the arithmetic mean of p1, p2, . . . , pn, whereas that for Mary is the harmonic mean of these n numbers. Since the harmonic mean is less than or equal to the arithmetic mean for any set of positive numbers and the two means are equal only if p1 = p2 = ⋅⋅⋅ = pn, we conclude that the average cost per share for Mary is less than that for John if two of the prices pi are distinct. One can also give a proof based on the Cauchy-Schwarz inequality. To this end, define the vectors a = (p1−1/2, p2−1/2, . . . , pn−1/2)

b = (p11/2, p21/2, . . . , pn1/2)

Then a ⋅ b = 1 + ⋅⋅⋅ + 1 = n, and so by the Cauchy-Schwarz inequality n 1 (a ⋅ b)2 = n2 ≤ 
i = 1 pi

n

p

i

i=1

with the equality holding only if p1 = p2 = ⋅⋅⋅ = pn. Therefore n

 pi n i=1

n ≤ 1 

n i = 1 pi

3-10

MATHEMATICS

MENSURATION FORMULAS Let A denote areas and V, volumes, in the following.

Radius R of Circumscribed Circle abc R =


4
 s( s −a )( s − b )( s −c)

PLANE GEOMETRIC FIGURES WITH STRAIGHT BOUNDARIES Triangles (see also “Plane Trigonometry”) A = a bh where b = base, h = altitude. Rectangle A = ab where a and b are the lengths of the sides. Parallelogram (opposite sides parallel) A = ah = ab sin α where a, b are the lengths of the sides, h the height, and α the angle between the sides. See Fig. 3-3. Rhombus (equilateral parallelogram) A = aab where a, b are the lengths of the diagonals. Trapezoid (four sides, two parallel) A = a(a + b)h where the lengths of the parallel sides are a and b, and h = height. Quadrilateral (four-sided) A = aab sin θ where a, b are the lengths of the diagonals and the acute angle between them is θ. Regular Polygon of n Sides See Fig. 3-4. 1 180° A =
nl 2 cot
where l = length of each side 4 n l 180° R =
csc
where R is the radius of the circumscribed circle 2 n l 180° r =
cot
where r is the radius of the inscribed circle 2 n 360° β=
n (n − 2)180° θ =

n β β l = 2r tan
= 2R sin
2 2 Inscribed and Circumscribed Circles with Regular Polygon of n Sides Let l = length of one side.

Figure

n

Area

Radius of circumscribed circle

Equilateral triangle Square Pentagon Hexagon Octagon Decagon Dodecagon

3 4 5 6 8 10 12

0.4330 l2 1.0000 l2 1.7205 l2 2.5981 l2 4.8284 l2 7.6942 l2 11.1962 l2

0.5774 l 0.7071 l 0.8507 l 1.0000 l 1.3065 l 1.6180 l 1.8660 l

Radius of inscribed circle 0.2887 l 0.5000 l 0.6882 l 0.8660 l 1.2071 l 1.5388 l 1.9318 l

Radius r of Circle Inscribed in Triangle with Sides a, b, c r=

FIG. 3-3

 (s − a)(s − b)(s − c)

s

Parallelogram.

where s = a(a + b + c)

FIG. 3-4

Regular polygon.

Area of Regular Polygon of n Sides Inscribed in a Circle of Radius r A = (nr 2/2) sin (360°/n) Perimeter of Inscribed Regular Polygon P = 2nr sin (180°/n) Area of Regular Polygon Circumscribed about a Circle of Radius r A = nr 2 tan (180°/n) Perimeter of Circumscribed Regular Polygon 180° P = 2nr tan
n PLANE GEOMETRIC FIGURES WITH CURVED BOUNDARIES Circle (Fig. 3-5) Let C = circumference r = radius D = diameter A = area S = arc length subtended by θ l = chord length subtended by θ H = maximum rise of arc above chord, r − H = d θ = central angle (rad) subtended by arc S C = 2πr = πD (π = 3.14159 . . .) S = rθ = aDθ l = 2
 r2 − d2 = 2r sin (θ/2) = 2d tan (θ/2) 1 1 θ d =

4 r2 −l2 =
l cot
2 2 2 S −1 d −1 l θ =
= 2 cos
= 2 sin
r r D A (circle) = πr2 = dπD2 A (sector) = arS = ar 2θ A (segment) = A (sector) − A (triangle) = ar 2(θ − sin θ) r−H  = r2 cos−1
− (r − H)
2rH − H2 r Ring (area between two circles of radii r1 and r2 ) The circles need not be concentric, but one of the circles must enclose the other. A = π(r1 + r2)(r1 − r2)

FIG. 3-5

Circle.

r1 > r2

MENSURATION FORMULAS

3-11

Volume and Surface Area of Regular Polyhedra with Edge l

FIG. 3-6

Ellipse.

Ellipse (Fig. 3-6)

FIG. 3-7

Parabola.

Let the semiaxes of the ellipse be a and b A = πab C = 4aE(k)

where e2 = 1 − b2/a2 and E(e) is the complete elliptic integral of the second kind, π 1 2 E(e) =
1 −
e2 + ⋅⋅⋅ 2 2





2 2+ b ]. [an approximation for the circumference C = 2π
(a )/2 Parabola (Fig. 3-7) y2 x2 +y2 2x +
4 Length of arc EFG =
4 x2 +y2 +
ln

2x y

4 Area of section EFG =
xy 3 Catenary (the curve formed by a cord of uniform weight suspended freely between two points A, B; Fig. 3-8) y = a cosh (x/a) Length of arc between points A and B is equal to 2a sinh (L/a). Sag of the cord is D = a cosh (L/a) − 1. SOLID GEOMETRIC FIGURES WITH PLANE BOUNDARIES Cube Volume = a3; total surface area = 6a2; diagonal = a
3, where a = length of one side of the cube. Rectangular Parallelepiped Volume = abc; surface area = 2  2(ab + ac + bc); diagonal =
 a2 + b +c2, where a, b, c are the lengths of the sides. Prism Volume = (area of base) × (altitude); lateral surface area = (perimeter of right section) × (lateral edge). Pyramid Volume = s (area of base) × (altitude); lateral area of regular pyramid = a (perimeter of base) × (slant height) = a (number of sides) (length of one side) (slant height). Frustum of Pyramid (formed from the pyramid by cutting off the top with a plane

1⋅A 2)h V = s (A1 + A2 +
A where h = altitude and A1, A2 are the areas of the base; lateral area of a regular figure = a (sum of the perimeters of base) × (slant height).

FIG. 3-8

Catenary.

Type of surface

Name

Volume

Surface area

4 equilateral triangles 6 squares 8 equilateral triangles 12 pentagons 20 equilateral triangles

Tetrahedron Hexahedron (cube) Octahedron Dodecahedron Icosahedron

0.1179 l3 1.0000 l3 0.4714 l3 7.6631 l3 2.1817 l3

1.7321 l2 6.0000 l2 3.4641 l2 20.6458 l2 8.6603 l2

SOLIDS BOUNDED BY CURVED SURFACES Cylinders (Fig. 3-9) V = (area of base) × (altitude); lateral surface area = (perimeter of right section) × (lateral edge). Right Circular Cylinder V = π (radius)2 × (altitude); lateral surface area = 2π (radius) × (altitude). Truncated Right Circular Cylinder V = πr 2h; lateral area = 2πrh h = a (h1 + h2) Hollow Cylinders Volume = πh(R2 − r 2), where r and R are the internal and external radii and h is the height of the cylinder. Sphere (Fig. 3-10) V (sphere) = 4⁄ 3πR3, jπD3 V (spherical sector) = wπR2h = jπh1(3r 22 + h12) V (spherical segment of one base) = jπh1(3r 22 + h12) V (spherical segment of two bases) = jπh 2(3r 12 + 3r 22 + h 22 ) A (sphere) = 4πR2 = πD2 A (zone) = 2πRh = πDh A (lune on the surface included between two great circles, the inclination of which is θ radians) = 2R2θ. Cone V = s (area of base) × (altitude). Right Circular Cone V = (π/3) r 2h, where h is the altitude and r is the radius of the base; curved surface area = πr
 r2 + h2, curved sur2 face of the frustum of a right cone = π(r1 + r2)
 h2 +( r1 −r 2), where r1, r2 are the radii of the base and top, respectively, and h is the altitude; volume of the frustum of a right cone = π(h/3)(r 21 + r1r2 + r 22) = h/3(A1 + A2 +
 A 1A2), where A1 = area of base and A2 = area of top. Ellipsoid V = (4 ⁄3) πabc, where a, b, c are the lengths of the semiaxes. Torus (obtained by rotating a circle of radius r about a line whose distance is R > r from the center of the circle) V = 2π2Rr 2

Surface area = 4π2Rr

Prolate Spheroid (formed by rotating an ellipse about its major axis [2a]) Surface area = 2πb2 + 2π(ab/e) sin−1 e

V = 4 ⁄3πab2

where a, b are the major and minor axes and e = eccentricity (e < 1). Oblate Spheroid (formed by the rotation of an ellipse about its minor axis [2b]) Data as given previously.

FIG. 3-9

Cylinder.

FIG. 3-10

Sphere.

3-12

MATHEMATICS b2 1+e Surface area = 2πa2 + π
ln
e 1−e

V = 4 ⁄3πa2b

MISCELLANEOUS FORMULAS See also “Differential and Integral Calculus.” Volume of a Solid Revolution (the solid generated by rotating a plane area about the x axis) V=π

[ f(x)] dx b

2

a

where y = f(x) is the equation of the plane curve and a ≤ x ≤ b. Area of a Surface of Revolution

y ds

IRREGULAR AREAS AND VOLUMES

b

S = 2π

a

y/d x) 2 dx and y = f(x) is the equation of the plane where ds =
1 +(d curve rotated about the x axis to generate the surface. Area Bounded by f(x), the x Axis, and the Lines x = a, x = b A=

1. If a plane area is revolved about a line which lies in its plane but does not intersect the area, then the volume generated is equal to the product of the area and the distance traveled by the area’s center of gravity. 2. If an arc of a plane curve is revolved about a line that lies in its plane but does not intersect the arc, then the surface area generated by the arc is equal to the product of the length of the arc and the distance traveled by its center of gravity. These theorems are useful for determining volumes V and surface areas S of solids of revolution if the centers of gravity are known. If S and V are known, the centers of gravity may be determined.

f(x) dx b

[ f(x) ≥ 0]

Irregular Areas Let y0, y1, . . . , yn be the lengths of a series of equally spaced parallel chords and h be their distance apart. The area of the figure is given approximately by any of the following: AT = (h/2)[(y0 + yn) + 2(y1 + y2 + ⋅⋅⋅ + yn − 1)] As = (h/3)[(y0 + yn) + 4(y1 + y3 + y5 + ⋅⋅⋅ + yn − 1) + 2(y2 + y4 + ⋅⋅⋅ + yn − 2)]

a

Length of Arc of a Plane Curve If y = f(x), Length of arc s =

(n even, Simpson’s rule)

The greater the value of n, the greater the accuracy of approximation. Irregular Volumes To find the volume, replace the y’s by crosssectional areas Aj and use the results in the preceding equations.

dy 1 + 
 dx dx b

(trapezoidal rule)

2

a

If x = g(y), Length of arc s =

dx 1 + 
 dy dy 2

d

c

If x = f(t), y = g(t), Length of arc s =

dy dx  + 
dt
dt  dt  t1

2

2

t0

In general, (ds)2 = (dx)2 + (dy)2. Theorems of Pappus (for volumes and areas of surfaces of revolution)

FIG. 3-11

Irregular area.

ELEMENTARY ALGEBRA REFERENCES:

20, 102, 108, 191, 269, 277.

Dividend Divisor ex + 1 | 3e2x + ex + 1 3ex − 2 quotient 3e2x + 3ex

OPERATIONS ON ALGEBRAIC EXPRESSIONS

−2ex + 1 −2ex − 2

An algebraic expression will here be denoted as a combination of letters and numbers such as 3ax − 3xy + 7x2 + 7x 3/ 2 − 2.8xy Addition and Subtraction Only like terms can be added or subtracted in two algebraic expressions. Example (3x + 4xy − x ) + (3x + 2x − 8xy) = 5x − 4xy + 2x . 2

2

2

Example (2x + 3xy − 4x1/2) + (3x + 6x − 8xy) = 2x + 3x + 6x − 5xy − 4x1/2. Multiplication Multiplication of algebraic expressions is term by term, and corresponding terms are combined. Example (2x + 3y − 2xy)(3 + 3y) = 6x + 9y + 9y − 6xy . 2

2

Division This operation is analogous to that in arithmetic. Example Divide 3e2x + ex + 1 by ex + 1.

+ 3 (remainder) Therefore, 3e + e + 1 = (e + 1)(3e − 2) + 3. 2x

x

x

x

Operations with Zero All numerical computations (except division) can be done with zero: a + 0 = 0 + a = a; a − 0 = a; 0 − a = −a; (a)(0) = 0; a0 = 1 if a ≠ 0; 0/a = 0, a ≠ 0. a/0 and 0/0 have no meaning. Fractional Operations x −x x −x x −x x ax −
= −
=
=
;
=
;
=
, if a ≠ 0. y −y −y y y −y y ay

 

x z xz

=
; y y y


y 
t  =
yt ; x

z

xz

  

xt x/y x t
=

=
z/t y z yz

Factoring That process of analysis consisting of reducing a given expression into the product of two or more simpler expressions called factors. Some of the more common expressions are factored here: (1) (x2 − y2) = (x − y)(x + y)

ELEMENTARY ALGEBRA s (n − 1)d 2s a = l − (n − 1)d =
−
=
− l n 2 n

(2) x2 + 2xy + y2 = (x + y)2 (3) x + ax + b = (x + c)(x + d) where c + d = a, cd = b 2

l−a 2(s − an) 2(nl − s) d=
=
=
n−1 n(n − 1) n(n − 1)

(4) by2 + cy + d = (ey + f)(gy + h) where eg = b, fg + eh = c, fh = d (5) x2 + y2 + z2 + 2yz + 2xz + 2xy = (x + y + z)2

2l + d +
 (2 l + d )2 −8 ds 2s l−a n =
+ 1 =
=


2d d l+a

(6) x2 − y2 − z2 − 2yz = (x − y − z)(x + y + z) (7) x2 + y2 + z2 − 2xy − 2xz + 2yz = (x − y − z)2 (8) x3 − y3 = (x − y)(x2 + xy + y2) (9) (x + y ) = (x + y)(x − xy + y ) 3

3

2

2

(10) (x4 − y4) = (x − y)(x + y)(x2 + y2) (11) x5 + y5 = (x + y)(x4 − x3y + x2y2 − xy3 + y4) (12) xn − yn = (x − y)(xn − 1 + xn − 2y + xn − 3y2 + ⋅⋅⋅ + yn − 1) Laws of Exponents (an)m = anm; an + m = an ⋅ am; an/m = (an)1/m; an − m = an/am; a1/m = m
a; a1/2 =
a;
 x2 = |x| (absolute value of x). For x > 0, y > 0,
xy  =
x n
 y; for x > 0
 xm = xm/n;
1 /x = 1/
x

The arithmetic mean or average of two numbers a, b is (a + b)/2; of n numbers a1, . . . , an is (a1 + a2 + ⋅⋅⋅ + an)/n. A geometric progression is a succession of terms such that each term, except the first, is derivable from the preceding by the multiplication of a quantity r called the common ratio. All such progressions have the form a, ar, ar 2, . . . , ar n − 1. With a = first term, l = last term, r = ratio, n = number of terms, s = sum of the terms, the following relations hold: [a + (r − 1)s] (r − 1)sr n − 1 l = ar n − 1 =

=

r rn − 1 a(r n − 1) a(1 − r n) rl − a lr n − l s=
=
=
=
r−1 1−r r − 1 rn − rn − 1

THE BINOMIAL THEOREM

l (r − 1)s s−a log l − log a a=
=
r =
log r =

rn − l rn − 1 s−l n−1

If n is a positive integer, n(n − 1) (a + b)n = an + nan − 1b +
an − 2 b2 2! n n(n − 1)(n − 2) n n−j j +

an − 3b3 + ⋅⋅⋅ + bn =  a b j 3! j=0



where

n! = number of combinations of n things taken j at  nj =
j!(n − j)!

log l − log a log[a + (r − 1)s] − log a n =

+ 1 =


log r log r ; of n The geometric mean of two nonnegative numbers a, b is
ab numbers is (a1a2 . . . an)1/n. Example Find the sum of 1 + a + d + ⋅⋅⋅ + 1⁄64. Here a = 1, r = a, n = 7. Thus a(1⁄64) − 1 s =

= 127/64 a−1

a time. n! = 1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅⋅⋅ n, 0! = 1. Example Find the sixth term of (x + 2y) . The sixth term is obtained by setting j = 5. It is 12 12 − 5 x (2y)5 = 792x7(2y)5 5

a ar n s = a + ar + ar 2 + ⋅⋅⋅ + ar n − 1 =
−
1−r 1−r

12

 

  j  = (1 + 1) 14

Example

n

14

3-13

a lim s =
1−r which is called the sum of the infinite geometric progression. If |r| < 1,

then

n→∞

Example The present worth (PW) of a series of cash flows Ck at the end

= 214.

of year k is

j=0

PROGRESSIONS

n Ck PW = 
k k = 1 (1 + i) where i is an assumed interest rate. (Thus the present worth always requires specification of an interest rate.) If all the payments are the same, Ck = R, the present worth is n 1 PW = R 
k k = 1 (1 + i) This can be rewritten as n R 1 R n−1 1 =

PW =

1 + i k = 1 (1 + i)k − 1 1 + i j = 0 (1 + i) j

An arithmetic progression is a succession of terms such that each term, except the first, is derivable from the preceding by the addition of a quantity d called the common difference. All arithmetic progressions have the form a, a + d, a + 2d, a + 3d, . . . . With a = first term, l = last term, d = common difference, n = number of terms, and s = sum of the terms, the following relations hold:

This is a geometric series with r = 1/(1 + i) and a = R/(1 + i). The formulas above give R (1 + i)n − 1 PW (=s) =


(1 + i)n i The same formula applies to the value of an annuity (PW) now, to provide for equal payments R at the end of each of n years, with interest rate i.

If n is not a positive integer, the sum formula no longer applies and an infinite series results for (a + b)n. The coefficients are obtained from the first formulas in this case. Example (1 + x)1/2 = 1 + ax − a ⋅ dx2 + a ⋅ d ⋅ 3⁄ 6 x3 ⋅⋅⋅ (convergent for

x2 < 1).

Additional discussion is under “Infinite Series.”

d l = a + (n − 1)d = −
+ 2

2 d s + a −
d2  2

s (n − 1) =
+
d n 2 n n n s =
[2a + (n − 1)d] =
(a + l) =
[2l − (n − 1)d] 2 2 2

A progression of the form a, (a + d )r, (a + 2d )r 2, (a + 3d )r 3, etc., is a combined arithmetic and geometric progression. The sum of n such terms is a − [a + (n − 1)d]r n rd(1 − r n − 1) s =

+

I−r (1 − r)2 a If |r| < 1, lim s =
+ rd/(1 − r)2. n→∞ 1−r

3-14

MATHEMATICS

The non-zero numbers a, b, c, etc., form a harmonic progression if their reciprocals 1/a, 1/b, 1/c, etc., form an arithmetic progression. Example The progression 1, s, 1⁄5, 1⁄7, . . . , 1⁄31 is harmonic since 1, 3, 5, 7, . . . , 31 form an arithmetic progression. The harmonic mean of two numbers a, b is 2ab/(a + b). PERMUTATIONS, COMBINATIONS, AND PROBABILITY Each separate arrangement of all or a part of a set of things is called a permutation. The number of permutations of n things taken r at a time, written n! P(n, r) =
= n(n − 1)(n − 2) ⋅⋅⋅ (n − r + 1) (n − r)! Example The permutations of a, b, c two at a time are ab, ac, ba, ca, cb, and bc. The formula is P(3,2) = 3!/1! = 6. The permutations of a, b, c three at a time are abc, bac, cab, acb, bca, and cba. Each separate selection of objects that is possible irrespective of the order in which they are arranged is called a combination. The number of combinations of n things taken r at a time, written C(n, r) = n!/ [r!(n − r)!].

has three complex roots. If the coefficients are real numbers, then at least one of the roots must be real. The cubic equation x3 + bx2 + cx + d = 0 may be reduced by the substitution x = y − (b/3) to the form y3 + py + q = 0, where p = s(3c − b2), q = 1⁄27(27d − 9bc + 2b3). This equation has the solutions y1 = A + B, y2 = −a(A + B) + (i
3/2)(A − B), 3 y3 = −a(A + B) − (i
3/2)(A − B), where i2 = −1, A =
 − q /2 + R,
 3 3 2 B =
 − q /2 − R, and R = (p/3) + (q/2) . If b, c, d are all real and if
 R > 0, there are one real root and two conjugate complex roots; if R = 0, there are three real roots, of which at least two are equal; if R < 0, there are three real unequal roots. If R < 0, these formulas are impractical. In this case, the roots are given by yk =  2
 − p /3 cos [(φ/3) + 120k], k = 0, 1, 2 where φ = cos−1

Example Two dice may be thrown in 36 separate ways. What is the probability of throwing such that their sum is 7? Seven may arise in 6 ways: 1 and 6, 2 and 5, 3 and 4, 4 and 3, 5 and 2, 6 and 1. The probability of shooting 7 is j.

2

3

and the upper sign applies if q > 0, the lower if q < 0. Example x3 + 3x2 + 9x + 9 = 0 reduces to y3 + 6y + 2 = 0 under x = y − 1. 3 3 4. The desired roots in y are Here p = 6, q = 2, R = 9. Hence A =
2, B =
− 3 3 3 3 3 3 4 and −a(
2 −
4)  (i
3/2)(
2 +
4). The roots in x are x =
2 −
− y − 1. Example y3 − 7y + 7 = 0. p = −7, q = 7, R < 0. Hence 28 φ
cos 
+ 120k  3 3 27 φ φ = 
,
= 3°37′52″. 28 3

xk = −

Example The combinations of a, b, c taken 2 at a time are ab, ac, bc; taken 3 at a time is abc. An important relation is r! C(n, r) = P(n, r). If an event can occur in p ways and fail to occur in q ways, all ways being equally likely, the probability of its occurrence is p/(p + q), and that of its failure q/(p + q).

q /4
 −p /27

where

The roots are approximately −3.048916, 1.692020, and 1.356897.

Example Many equations of state involve solving cubic equations for the compressibility factor Z. For example, the Redlich-Kwong-Soave equation of state requires solving Z 3 − Z 2 + cZ + d = 0,

d<0

THEORY OF EQUATIONS

where c and d depend on critical constants of the chemical species. In this case, only positive solutions, Z > 0, are desired.

Linear Equations A linear equation is one of the first degree (i.e., only the first powers of the variables are involved), and the process of obtaining definite values for the unknown is called solving the equation. Every linear equation in one variable is written Ax + B = 0 or x = −B/A. Linear equations in n variables have the form

Quartic Equations See Ref. 118. General Polynomials of the nth Degree Denote the general polynomial equation of degree n by

a11 x1 + a12 x2 + ⋅⋅⋅ + a1n xn = b1 a21 x1 + a22 x2 + ⋅⋅⋅ + a2n xn = b2 ⯗ am1 x1 + am2 x2 + ⋅⋅⋅ + amn xn = bm The solution of the system may then be found by elimination or matrix methods if a solution exists (see “Matrix Algebra and Matrix Computations”). Quadratic Equations Every quadratic equation in one variable is expressible in the form ax 2 + bx + c = 0. a ≠ 0. This equation has two solutions, say, x1, x2, given by

4ac  −b 
 x1 b2− =

x2 2a

If a, b, c are real, the discriminant b − 4ac gives the character of the roots. If b2 − 4ac > 0, the roots are real and unequal. If b2 − 4ac < 0, the roots are complex conjugates. If b2 − 4ac = 0 the roots are real and equal. Two quadratic equations in two variables can in general be solved only by numerical methods (see “Numerical Analysis and Approximate Methods”). If one equation is of the first degree, the other of the second degree, a solution may be obtained by solving the first for one unknown. This result is substituted in the second equation and the resulting quadratic equation solved. Cubic Equations A cubic equation, in one variable, has the form x3 + bx2 + cx + d = 0. Every cubic equation having complex coefficients 2

P(x) = a0 x n + a1 x n − 1 + ⋅⋅⋅ + an − 1 x + an = 0 If n > 4, there is no formula which gives the roots of the general equation. For fourth and higher order (even third order), the roots can be found numerically (see “Numerical Analysis and Approximate Methods”). However, there are some general theorems that may prove useful. Remainder Theorems When P(x) is a polynomial and P(x) is divided by x − a until a remainder independent of x is obtained, this remainder is equal to P(a). Example P(x) = 2x4 − 3x2 + 7x − 2 when divided by x + 1 (here a = −1) results in P(x) = (x + 1)(2x3 − 2x2 − x + 8) − 10 where −10 is the remainder. It is easy to see that P(−1) = −10. Factor Theorem If P(a) is zero, the polynomial P(x) has the factor x − a. In other words, if a is a root of P(x) = 0, then x − a is a factor of P(x). If a number a is found to be a root of P(x) = 0, the division of P(x) by (x − a) leaves a polynomial of degree one less than that of the original equation, i.e., P(x) = Q(x)(x − a). Roots of Q(x) = 0 are clearly roots of P(x) = 0. Example P(x) = x3 − 6x2 + 11x − 6 = 0 has the root + 3. Then P(x) = (x − 3)(x2 − 3x + 2). The roots of x2 − 3x + 2 = 0 are 1 and 2. The roots of P(x) are therefore 1, 2, 3. Fundamental Theorem of Algebra Every polynomial of degree n has exactly n real or complex roots, counting multiplicities.

ELEMENTARY ALGEBRA Every polynomial equation a0 x n + a1 x n − 1 + ⋅⋅⋅ + an = 0 with rational coefficients may be rewritten as a polynomial, of the same degree, with integral coefficients by multiplying each coefficient by the least common multiple of the denominators of the coefficients. Example The coefficients of 3⁄2 x4 + 7⁄3 x3 − 5⁄6 x2 + 2x − j = 0 are rational

numbers. The least common multiple of the denominators is 2 × 3 = 6. Therefore, the equation is equivalent to 9x4 + 14x3 − 5x2 + 12x − 1 = 0.

Upper Bound for the Real Roots Any number that exceeds all the roots is called an upper bound to the real roots. If the coefficients of a polynomial equation are all of like sign, there is no positive root. Such equations are excluded here since zero is the upper bound to the real roots. If the coefficient of the highest power of P(x) = 0 is negative, replace the equation by −P(x) = 0. If in a polynomial P(x) = c0 x n + c1 x n − 1 + ⋅⋅⋅ + cn − 1 x + cn = 0, with c0 > 0, the first negative coefficient is preceded by k coefficients which are positive or zero, and if G denotes the greatest of the numerical values of the negative coefficients, then each real root is less than k 1 +
 G /c0. A lower bound to the negative roots of P(x) = 0 may be found by applying the rule to P(−x) = 0. Example P(x) = x + 2x + 4x − 8x − 32 = 0. Here3 k = 5 (since 2 coefficients are zero), G = 32, c0 = 1. The upper bound is 1 +
32 = 3. P(−x) = −x7 − 5 2x + 4x4 − 8x2 − 32 = 0. −P(−x) = x73+ 2x5 − 4x4 + 8x2 + 32 = 0. Here k = 3, G = 4, c0 = 1. The lower bound is −(1 +
4) ≈ −2.587. Thus all real roots r lie in the range −2.587 < r < 3. 7

5

4

2

Descartes Rule of Signs The number of positive real roots of a polynomial equation with real coefficients either is equal to the number v of its variations in sign or is less than v by a positive even integer. The number of negative roots of P(x) = 0 either is equal to the number of variations of sign of P(−x) or is less than that number by a positive even integer. Example P(x) = x4 + 3x3 + x − 1 = 0. v = 1; so P(x) has one positive root. P(−x) = x4 − 3x3 − x − 1. Here v = 1; so P(x) has one negative root. The other two roots are complex conjugates. Example P(x) = x4 − x2 + 10x − 4 = 0. v = 3; so P(x) has three or one positive roots. P(−x) = x4 − x2 − 10x − 4. v = 1; so P(x) has exactly one negative root. Numerical methods are often used to find the roots of polynomials. A detailed discussion of these techniques is given under “Numerical Analysis and Approximate Methods.” Determinants Consider the system of two linear equations a11x1 + a12x2 = b1 a21x1 + a22x2 = b2 If the first equation is multiplied by a22 and the second by −a12 and the results added, we obtain (a11a22 − a21a12)x1 = b1a22 − b2a12 The expression a11a22 − a21a12 may be represented by the symbol a11





a11 a12 a a22 |A| = 21 ⯗ an1 an2

a13 ⋅⋅⋅ a1n ⋅⋅⋅⋅⋅ a2n



an3 ⋅⋅⋅ ann

is called a determinant. The n2 quantities aij are called the elements of the determinant. In the determinant |A| let the ith row and jth column be deleted and a new determinant be formed having n − 1 rows and columns. This new determinant is called the minor of aij denoted Mij.



Example





a11 a12 a13 a21 a22 a23 = a31A31 + a32A32 + a33A33 a31 a32 a33 = a31

aa









a13 a a a a − a32 11 13 + a33 11 12 a23 a21 a23 a21 a22

12 22



In general, Aij will be determinants of order n − 1, but they may in turn be expanded by the rule. Also, n

a

j=1

n

ji



A jk =  a ij A jk = |A| 0 j=1

i=k i≠k

Fundamental Properties of Determinants 1. The value of a determinant |A| is not changed if the rows and columns are interchanged. 2. If the elements of one row (or one column) of a determinant are all zero, the value of |A| is zero. 3. If the elements of one row (or column) of a determinant are multiplied by the same constant factor, the value of the determinant is multiplied by this factor. 4. If one determinant is obtained from another by interchanging any two rows (or columns), the value of either is the negative of the value of the other. 5. If two rows (or columns) of a determinant are identical, the value of the determinant is zero. 6. If two determinants are identical except for one row (or column), the sum of their values is given by a single determinant obtained by adding corresponding elements of dissimilar rows (or columns) and leaving unchanged the remaining elements. Example

 31 25  +  47 25  = 13 + 6 = 19

Directly

 78 25  = 35 − 16 = 19

By rule 6

7. The value of a determinant is not changed if to the elements of any row (or column) are added a constant multiple of the corresponding elements of any other row (or column). 8. If all elements but one in a row (or column) are zero, the value of the determinant is the product of that element times its cofactor. The evaluation of determinants using the definition is quite laborious. The labor can be reduced by applying the fundamental properties just outlined. The solution of n linear equations (not all bi zero) a11 x1 + a12x2 + ⋅⋅⋅ + a1n xn = b1



21



The cofactor Aij of the element aij is the signed minor of aij determined by the rule Aij = (−1) i + jMij. The value of |A| is obtained by forming any of the n n equivalent expressions  j = 1 aij Aij ,  t = 1 aij Aij, where the elements aij must be taken from a single row or a single column of A.

a12 = a11a22 − a21a12 a22 This symbol is called a determinant of second order. The value of the square array of n2 quantities aij, where i = 1, . . . , n is the row index, j = 1, . . . , n the column index, written in the form

a



a11 a12 a13 a a a21 a22 a23 The minor of a23 is M23 = 11 12 a31 a32 a31 a32 a33

Example

3-15

a21 x1 + a22x2 + ⋅⋅⋅ + a2n xn = b2 ⯗⯗⯗ ⯗ an1x1 + an2x2 + ⋅⋅⋅ + annxn = bn a11 ⋅⋅⋅ a1n a21 ⋅⋅⋅ a2n where |A| = ⯗ ≠0 an1 ⋅⋅⋅ ann





has a unique solution given by x1 = |B1|/ |A|, x2 = |B2|/ |A|, . . . , xn = |Bn|/ |A|, where Bk is the determinant obtained from A by replacing its kth column by b1, b2, . . . , bn. This technique is called Cramer’s rule. It requires more labor than the method of elimination and should not be used for computations.

3-16

MATHEMATICS

ANALYTIC GEOMETRY REFERENCES: 108, 188, 193, 260, 261, 268, 274, 282. Analytic geometry uses algebraic equations and methods to study geometric problems. It also permits one to visualize algebraic equations in terms of geometric curves, which frequently clarifies abstract concepts.

PLANE ANALYTIC GEOMETRY Coordinate Systems The basic concept of analytic geometry is the establishment of a one-to-one correspondence between the points of the plane and number pairs (x, y). This correspondence may be done in a number of ways. The rectangular or cartesian coordinate system consists of two straight lines intersecting at right angles (Fig. 3-12). A point is designated by (x, y), where x (the abscissa) is the distance of the point from the y axis measured parallel to the x axis, positive if to the right, negative to the left. y (ordinate) is the distance of the point from the x axis, measured parallel to the y axis, positive if above, negative if below the x axis. The quadrants are labeled 1, 2, 3, 4 in the drawing, the coordinates of points in the various quadrants having the depicted signs. Another common coordinate system is the polar coordinate system (Fig. 3-13). In this system the position of a point is designated by the pair (r, θ), r =
 x2 +y2 being the distance to the origin 0(0,0) and θ being the angle the line r makes with the positive x axis (polar axis). To change from polar to rectangular coordinates, use x = r cos θ and y = r sin θ. To change from rectangular to polar coordinates, use r =
 x2 +y2 and θ = tan−1 (y/x) if x ≠ 0; θ = π/2 if x = 0. The distance between two points (x1, y1), (x2, y2) is defined 2 2 by d =
 (x1 −x +(y1 −y 2) 2) in rectangular coordinates or by d = 2 2 r 1 +r −2 r1 r2cos (θ1 − θ
 2 2) in polar coordinates. Other coordinate systems are sometimes used. For example, on the surface of a sphere latitude and longitude prove useful. The Straight Line (Fig. 3-14) The slope m of a straight line is the tangent of the inclination angle θ made with the positive x axis. If (x1, y1) and (x2, y2) are any two points on the line, slope = m = (y2 − y1)/ (x2 − x1). The slope of a line parallel to the x axis is zero; parallel to the y axis, it is undefined. Two lines are parallel if and only if they have the same slope. Two lines are perpendicular if and only if the product of their slopes is −1 (the exception being that case when the lines are parallel to the coordinate axes). Every equation of the type Ax + By + C = 0 represents a straight line, and every straight line has an equation of this form. A straight line is determined by a variety of conditions:

origin are given (see Fig. 3-15). Let p = length of the perpendicular and α the angle that the perpendicular makes with the positive x axis. The equation of the line is x cos  + y sin  = p. The equation of a line perpendicular to a given line of slope m and passing through a point (x1, y1) is y − y1 = −(1/m) (x − x1). The distance from a point (x1, y1) to a line with equation Ax + by + C = 0 is |Ax1 + By1 + C| d =

A2 + B2
 Example If it is known that centigrade C and Fahrenheit F are linearly related and when C = 0°, F = 32°; C = 100°, F = 212°, find the equation relating C and F and that point where C = F. By using the two-point form, the equation is 212 − 32 F − 32 =
(C − 0) 100 − 0 or F = 9⁄ 5C + 32. Equivalently 100 − 0 C − 0 =
(F − 32) 212 − 32 or C = 5⁄ 9(F − 32). Letting C = F, we have from either equation F = C = −40.

Occasionally some nonlinear algebraic equations can be reduced to linear equations under suitable substitutions or changes of variables. In other words, certain curves become the graphs of lines if the scales or coordinate axes are appropriately transformed. Example Consider y = bxn. B = log b. Taking logarithms log y = n log x + log b. Let Y = log y, X = log x, B = log b. The equation then has the form Y = nX + B, which is a linear equation. Consider k = k0 exp (−E/RT), taking logarithms loge k = loge k0 − E/(RT). Let Y = loge k, B = loge k0, and m = −E/R, X = 1/T, and the result is Y = mX + B. Next consider y = a + bxn. If the substitution t = x n is made, then the graph of y is a straight line versus t. Asymptotes The limiting position of the tangent to a curve as the point of contact tends to an infinite distance from the origin is called an asymptote. If the equation of a given curve can be expanded in a Laurent power series such that n n b f(x) =  ak x k + 
kk k=0 k=0 x n

lim f(x) =  akxk

and Given conditions (1) (2) (3) (4) (5)

Parallel to x axis Parallel y axis Point (x1, y1) and slope m Intercept on y axis (0, b), m Intercept on x axis (a, 0), m

(6)

Two points (x1, y1), (x2, y2)

(7)

Two intercepts (a, 0), (0, b)

Equation of line y = constant x = constant y − y1 = m(x − x1) y = mx + b y = m(x − a) y2 − y1 y − y1 =
(x − x1) x2 − x1 x/a + y/b = 1

The angle β a line with slope m1 makes with a line having slope m2 is given by tan β = (m2 − m1)/(m1m2 + 1). A line is determined if the length and direction of the perpendicular to it (the normal) from the

FIG. 3-12

Rectangular coordinates.

x→∞

FIG. 3-13

Polar coordinates.

k=0

then the equation of the asymptote is y =  k = 0 ak x k. If n = 1, then the asymptote is (in general oblique) a line. In this case, the equation of the asymptote may be written as n

y = mx + b

m = lim f ′(x) x→∞

b = lim [ f(x) − xf ′(x)] x→∞

Geometric Properties of a Curve When the Equation Is Given The analysis of the properties of an equation is facilitated by the investigation of the equation by using the following techniques: 1. Points of maximum, minimum, and inflection. These may be investigated by means of the calculus.

FIG. 3-14

Straight line.

FIG. 3-15

Determination of line.

ANALYTIC GEOMETRY 2.

Symmetry. Let F(x, y) = 0 be the equation of the curve.

The table characterizes the curve represented by the equation.

Condition on F(x, y)

Symmetry

B2 − 4AC < 0

F(x, y) = F(−x, y) F(x, y) = F(x, −y) F(x, y) = F(−x, −y) F(x, y) = F(y, x)

With respect to y axis With respect to x axis With respect to origin With respect to the line y = x

A < 0 A ≠ C, an ellipse A < 0 A = C, a circle A > 0, no locus

3. Extent. Only real values of x and y are considered in obtaining the points (x, y) whose coordinates satisfy the equation. The extent of them may be limited by the condition that negative numbers do not have real square roots. 4. Intercepts. Find those points where the curves of the function cross the coordinate axes. 5. Asymptotes. See preceding discussion. 6. Direction at a point. This may be found from the derivative of the function at a point. This concept is useful for distinguishing among a family of similar curves. Example y2 = (x2 + 1)/(x2 − 1) is symmetric with respect to the x and y axis, the origin, and the line y = x. It has the vertical asymptotes x = 1. When x = 0, y2 = −1; so there are no y intercepts. If y = 0, (x2 + 1)/(x2 − 1) = 0; so there are no x intercepts. If |x| < 1, y2 is negative; so |x| > 1. From x2 = ( y2 + 1)/(y2 − 1), y = 1 are horizontal asymptotes and |y| > 1. As x → 1+, y → + ∞; as x → + ∞, y → + 1. The graph is given in Fig. 3-16. Conic Sections The curves included in this group are obtained from plane sections of the cone. They include the circle, ellipse, parabola, hyperbola, and degeneratively the point and straight line. A conic is the locus of a point whose distance from a fixed point called the focus is in a constant ratio to its distance from a fixed line, called the directrix. This ratio is the eccentricity e. If e = 0, the conic is a circle; if 0 < e < 1, the conic is an ellipse; if e = 1, the conic is a parabola; if e > 1, the conic is a hyperbola. Every conic section is representable by an equation of second degree. Conversely, every equation of second degree in two variables represents a conic. The general equation of the second degree is Ax2 + Bxy + Cy2 + Dx + Ey + F = 0. Let  be defined as the determinant



2A = B D

FIG. 3-16

B 2C E

D E 2F



Graph of y2 = (x2 + 1)/(x2 − 1)

≠0

=0

B2 − 4AC = 0

B2 − 4AC > 0

Parabola

Hyperbola

2 parallel lines if Q = D2 + E2 − 4(A + C)F > 0 1 straight line if Q = 0, no locus if Q < 0

Point

2 intersecting straight lines

Example 3x2 + 4xy − 2y2 + 3x − 2y + 7 = 0.



6 = 4 3

4 −4 −2



3 −2 = −596 ≠ 0, B2 − 4AC = 40 > 0 14

The curve is therefore a hyperbola.

To translate the axes to a new origin at (h, k), substitute for x and y in the original equation x + h and y + k. Translation of the axes can always be accomplished to eliminate the linear terms in the seconddegree equation in two variables having no xy term. Example x2 + y2 + 2x − 4y + 2 = 0. Rewrite this as x2 + 2x + 1 + y2 − 4y + 4 − 5 + 2 = 0 or (x + 1)2 + (y − 2)2 = 3. Let u = x + 1, v = y − 2. Then u2 + v2 = 3. The axis has been translated to the new origin (−1, 2). The type of curve determined by a specific equation of the second degree can also be easily determined by reducing it to a standard form by translation and/or rotation. In the case in which the equation has no xy term, the procedure is merely to complete the squares of the terms in x and y separately. To rotate the axes through an angle α, substitute for x the quantity x cos α − y sin α and for y the quantity x sin α + y cos α. A rotation of the axes through α = a cot−1 (A − C)/B will eliminate the crossproduct term in the general second-degree equation. Example Consider 3x2 + 2xy + y2 − 2x + 3y = 7. A rotation of axes through α = a cot−1 1 = 22a° eliminates the cross-product term. The following tabulation gives the form of the more common equations. Polar equation

Type of curve

(1) r = a (2) r = 2a cos θ (3) r = 2a sin θ (4) r2 − 2br cos (θ − β) + b2 − a2 = 0

Circle Circle Circle Circle at (b, β), radius a

ke (5) r =

1 − e cos θ

e = 1 parabola 0 < e < 1 ellipse e > 1 hyperbola

Some common equations in parametric form are given below. (1) (x − h)2 + (y − k)2 = a2 (x − h)2 (y − k)2 (2)
+
=1 a2 b2

(3) z2 + y2 = a2 x (4) y = a cosh
a (5) Cycloid

3-17

x = h + a cos θ y = k + a sin θ x = h + a cos φ y = k + a sin φ −at x=
t2 +1
 a y=
t2 + 1


Circle (Fig. 3-23) Parameter is angle θ.

s x = a sinh−1
a y2 = a2 + s2 x = a(φ − sin φ) y = a(1 − cos φ)

Catenary (Fig. 3-24; such as hanging cable under gravity) Parameter s = arc length from (0, a) to (x, y). See Fig. 3-24.

Ellipse (Fig. 3-20) Parameter is angle φ. dy Circle Parameter is t =
= slope of tangent at (x, y). dx

3-18

MATHEMATICS

FIG. 3-17

Circle center (0,0) r = a.

FIG. 3-18

Circle center (a,0) r = 2a cos θ.

FIG. 3-19

Circle center (0,a) r = 2a sin θ.

FIG. 3-20

Ellipse, 0 < e < 1.

FIG. 3-21

Hyperbola, e > 1, r = ke/(1 − e cos θ).

FIG. 3-22

Parabola, e = 1.

Circle at (b, β), radius a: r2 − 2br cos (θ − β)  b2 − a2 = 0.

Graphs of Polar Equations The equation r = 0 corresponds to x = 0, y = 0 regardless of θ. The same point may be represented in several different ways; thus the point (2, π/3) or (2, 60°) has the following representations: (2, 60°), (2, −300°). These are summarized in (2, 60° + n 360°), n = 0, 1, 2, or in radian measure [2, (π/3) + 2nπ], n = 0, 1, 2. Plotting of polar equations can be facilitated by the following steps: 1. Find those points where r is a maximum or minimum. 2. Find those values of θ where r = 0, if any. 3. Symmetry: The curve is symmetric about the origin if the equation is unchanged when θ is replaced by θ  π, symmetric about the x axis if the equation is unchanged when θ is replaced by −θ, and symmetric about the y axis if the equation is unchanged when θ is replaced by π − θ. Parametric Equations It is frequently useful to write the equations of a curve in terms of an auxiliary variable called a parameter. For example, a circle of radius a, center at (0, 0), can be written in the equivalent form x = a cos θ, y = a sin φ where θ is the parameter. Similarly, x = a cos φ, y = b sin φ are the parametric equations of the ellipse x2/a2 + y2/b2 = 1 with parameter φ.

FIG. 3-23

Circle.

FIG. 3-24

Cycloid.

SOLID ANALYTIC GEOMETRY Coordinate Systems The commonly used coordinate systems are three in number. Others may be used in specific problems (see Ref. 212). The rectangular (cartesian) system (Fig. 3-25) consists of mutually orthogonal axes x, y, z. A triple of numbers (x, y, z) is used to represent each point. The cylindrical coordinate system (r, θ, z; Fig. 3-26) is frequently used to locate a point in space. These are essentially the polar coordinates (r, θ) coupled with the z coordinate. As

FIG. 3-25

Cartesian coordinates.

FIG. 3-26

Cylindrical coordinates.

ANALYTIC GEOMETRY before, x = r cos θ, y = r sin θ, z = z and r2 = x2 + y2, y/x = tan θ. If r is held constant and θ and z are allowed to vary, the locus of (r, θ, z) is a right circular cylinder of radius r along the z axis. The locus of r = C is a circle, and θ = constant is a plane containing the z axis and making an angle θ with the xz plane. Cylindrical coordinates are convenient to use when the problem has an axis of symmetry. The spherical coordinate system is convenient if there is a point of symmetry in the system. This point is taken as the origin and the coordinates (ρ, φ, θ) illustrated in Fig. 3-27. The relations are x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ, and r = ρ sin φ. θ = constant is a plane containing the z axis and making an angle θ with the xz plane. φ = constant is a cone with vertex at 0. ρ = constant is the surface of a sphere of radius ρ, center at the origin 0. Every point in the space may be given spherical coordinates restricted to the ranges 0 ≤ φ ≤ π, ρ ≥ 0, 0 ≤ θ < 2π. Lines and Planes The distance between two points (x1, y1, z1), 2 2 2 (x2, y2, z2) is d =
 (x −x +(y −y +( z1 −z 1 2) 1 2) 2). There is nothing in the geometry of three dimensions quite analogous to the slope of a line in the plane case. Instead of specifying the direction of a line by a trigonometric function evaluated for one angle, a trigonometric function evaluated for three angles is used. The angles α, β, γ that a line segment makes with the positive x, y, and z axes, respectively, are called the direction angles of the line, and cos α, cos β, cos γ are called the direction cosines. Let (x1, y1, z1), (x2, y2, z2) be on the line. Then cos α = (x2 − x1)/d, cos β = (y2 − y1)/d, cos γ = (z2 − z1)/d, where d = the distance between the two points. Clearly cos2 α + cos2 β + cos2 γ = 1. If two lines are specified by the direction cosines (cos α1, cos β1, cos γ1), (cos α2, cos β2, cos γ2), then the angle θ between the lines is cos θ = cos α1 cos α2 + cos β1 cos β2 + cos γ1 cos γ2. Thus the lines are perpendicular if and only if θ = 90° or cos α1 cos α2 + cos β1 cos β2 + cos γ1 cos γ2 = 0. The equation of a line with direction cosines (cos α, cos β, cos γ) passing through (x1, y1, z1) is (x − x1)/cos α = (y − y1)/cos β = (z − z1)/cos γ. The equation of every plane is of the form Ax + By + Cz + D = 0. The numbers A B C

,

,

2 2 2 2 2 2       A + B + C2
 A + B + C2
 A + B + C2


FIG. 3-27

Spherical coordinates.

FIG. 3-29

y2 z2 x2 Ellipsoid.
2 +
2 +
2 = 1 (sphere if a = b = c) b c a

FIG. 3-30

y2 z2 x2 Hyperboloid of one sheet.
2 +
2 −
2 = 1 b c a

FIG. 3-31

y2 z2 x2 Hyperboloid of two sheets.
2 +
2 −
2 = −1 b c a

FIG. 3-28

Parabolic cylinder.

are direction cosines of the normal lines to the plane. The plane through the point (x1, y1, z1) whose normals have these as direction cosines is A(x − x1) + B(y − y1) + C(z − z1) = 0. Example Find the equation of the plane through (1, 5, −2) perpendicular to the line (x + 9)/7 = (y − 3)/−1 = z/8. The numbers (7, −1, 8) are called direction numbers. They are a constant multiple of the direction cosines. cos α = 7/114, cos β = −1/114, cos γ = 8/114. The plane has the equation 7(x − 1) − 1(y − 5) + 8(z + 2) = 0 or 7x − y + 8z + 14 = 0. The distance from the point (x1, y1, z1) to the plane Ax + By + Cz + D = 0 is |Ax1 + By1 + Cz1 + D| d =


2+   B2+ C2
A

Space Curves Space curves are usually specified as the set of points whose coordinates are given parametrically by a system of equations x = f(t), y = g(t), z = h(t) in the parameter t. Example The equation of a straight line in space is (x − x1)/a = (y − y1)/b = (z − z1)/c. Since all these quantities must be equal (say, to t), we may write x = x1 + at, y = y1 + bt, z = z1 + ct, which represent the parametric equations of the line. Example The equations z = a cos βt, y = a sin βt, z = bt, a, β, b positive constants, represent a circular helix. Surfaces The locus of points (x, y, z) satisfying f(x, y, z) = 0, broadly speaking, may be interpreted as a surface. The simplest surface is the plane. The next simplest is a cylinder, which is a surface generated by a straight line moving parallel to a given line and passing through a given curve. Example The parabolic cylinder y = x2 (Fig. 3-28) is generated by a straight line parallel to the z axis passing through y = x2 in the plane z = 0.

3-19

3-20

MATHEMATICS

A surface whose equation is a quadratic in the variables x, y, and z is called a quadric surface. Some of the more common such surfaces are tabulated and pictured in Figs. 3-29 to 3-37.

Elliptic paraboloid.

FIG. 3-33

y2 z2 x2 FIG. 3-32 Cone.
2 +
2 +
2 = 0 b c a

y2 x2
2 +
2 + 2z = 0 b a

FIG. 3-35

y2 x2 Elliptic cylinder.
2 +
2 = 1 b a

FIG. 3-36

Hyperbolic cylinder. y2 x2
2 −
2 = 1 b a

y2 x2 FIG. 3-34 Hyperbolic paraboloid.
2 −
2 + 2z = 0 b a

FIG. 3-37

Parabolic cylinder. y2 + 2ax = 0

PLANE TRIGONOMETRY REFERENCES:

20, 108, 131, 158, 166, 202.

ANGLES An angle is generated by the rotation of a line about a fixed center from some initial position to some terminal position. If the rotation is clockwise, the angle is negative; if it is counterclockwise, the angle is positive. Angle size is unlimited. If α, β are two angles such that α + β = 90°, they are complementary; they are supplementary if α + β = 180°. Angles are most commonly measured in the sexagesimal system or by radian measure. In the first system there are 360 degrees in one complete revolution; one degree = 1⁄90 of a right angle. The degree is subdivided into 60 minutes; the minute is subdivided into 60 seconds. In the radian system one radian is the angle at the center of a circle subtended by an arc whose length is equal to the radius of the circle. Thus 2 rad = 360°; 1 rad = 57.29578°; 1° = 0.01745 rad; 1 min = 0.00029089 rad. The advantage of radian measure is that it is dimensionless. The quadrants are conventionally labeled as Fig. 3-38 shows.

FIG. 3-38

Quadrants.

FUNCTIONS OF CIRCULAR TRIGONOMETRY The trigonometric functions of angles are the ratios between the various sides of the reference triangles shown in Fig. 3-39 for the various quadrants. Clearly r =
 x2 +y2 ≥ 0. The fundamental functions (see Figs. 3-40, 3-41, 3-42) are

FIG. 3-39

Triangles.

FIG. 3-40

Graph of y = sin x.

PLANE TRIGONOMETRY

3-21

Relations between Functions of a Single Angle sec θ = 1/ cos θ; csc θ = 1/sin θ, tan θ = sin θ/cos θ = sec θ/csc θ = 1/cot θ; sin2 θ + cos2 θ = 1; 1 + tan2 θ = sec2 θ; 1 + cot2 θ = csc2 θ. For 0 ≤ θ ≤ 90° the following results hold: sin θ = cos θ/cot θ =
1 −co s2 θ = cos θ tan θ

FIG. 3-41

tan θ 1 θ θ =

=

= 2 sin
cos
+t an2 θ
1 +co t2 θ
1 2 2

 

Graph of y = cos x.

and

1 2θ =

cos θ =
1 −sin 2 θ +t an
1 cot θ sin θ θ θ =

=
= cos2
− sin2
2 2 +co t2 θ tan θ
1





The cofunction property is very important. cos θ = sin (90° − θ), sin θ = cos (90° − θ), tan θ = cot (90° − θ), cot θ = tan (90° − θ), etc. Functions of Negative Angles sin (−θ) = −sin θ, cos (−θ) = cos θ, tan (−θ) = −tan θ, sec (−θ) = sec θ, csc (−θ) = −csc θ, cot (−θ) = −cot θ. FIG. 3-42

Graph of y = tan x.

Plane Trigonometry Sine of θ = sin θ = y/r Cosine of θ = cos θ = x/r Tangent of θ = tan θ = y/x

Secant of θ = sec θ = r/x Cosecant of θ = csc θ = r/y Cotangent of θ = cot θ = x/y

Magnitude and Sign of Trigonometric Functions 0 ≤ θ ≤ 360° Function

0° to 90°

90° to 180°

180° to 270°

270° to 360°

sin θ csc θ cos θ sec θ tan θ cot θ

+0 to +1 +∞ to +1 +1 to 0 +1 to +∞ +0 to +∞ +∞ to +0

+1 to +0 +1 to +∞ −0 to −1 −∞ to −1 −∞ to −0 −0 to −∞

−0 to −1 −∞ to −1 −1 to −0 −1 to −∞ +0 to +∞ +∞ to +0

−1 to −0 −1 to −∞ +0 to +1 +∞ to +1 −∞ to −0 −0 to −∞

Identities Sum and Difference Formulas Let x, y be two angles. sin (x  y) = sin x cos y  cos x sin y; cos (x  y) = cos x cos y  sin x sin y; tan (x  y) = (tan x  tan y)/(1  tan x tan y); sin x  sin y = 2 sin a(x  y) cos a(x  y); cos x + cos y = 2 cos a(x + y) cos a(x − y); cos x − cos y = −2 sin a(x + y) sin a(x − y); tan x  tan y = [sin (x  y)]/(cos x cos y); sin2 x − sin2 y = cos2 y − cos2 x = sin (x + y) sin (x − y); cos2 x − sin2 y = cos2 y − sin2 x = cos (x + y) cos (x − y); sin (45° + x) = cos (45° − x); sin (45° − x) = cos (45° + x); tan (45°  x) = cot (45°  x). A cos x + 2 2   B sin x =
 A + B2 sin (α + x) =
 A + B2 cos (β − x) where tan α = A/B, tan β = B/A; both α and β are positive acute angles. Multiple and Half Angle Identities Let x = angle, sin 2x = 2 sin x cos x; sin x = 2 sin ax cos ax; cos 2x = cos2 x − sin2x = 1 − 2 sin2x = 2 cos2x − 1. tan 2x = (2 tan x)/(1 − tan2 x); sin 3x = 3 sin x − 4 sin3x; cos 3x = 4 cos3 x − 3 cos x. tan 3x = (3 tan x − tan3 x)/(1 − 3 tan2 x); sin 4x = 4 sin x cos x − 8 sin3 x cos x; cos 4x = 8 cos4 x − 8 cos2 x + 1. x sin
=
 a (1 −co s x) 2

 

x (1   cos
=
a +o csx) 2

Values of the Trigonometric Functions for Common Angles θ°

θ, rad

sin θ

cos θ

tan θ

0 30 45 60 90

0 π/6 π/4 π/3 π/2

0 1/2
2/2
3/2 1

1
3/2
2/2 1/2 0

0
3/3 1
3 +∞

If 90° ≤ θ ≤ 180°, sin θ = sin (180° − θ); cos θ = −cos (180° − θ); tan θ = −tan (180° − θ). If 180° ≤ θ ≤ 270°, sin θ = −sin (270° − θ); cos θ = −cos (270° − θ); tan θ = tan (270° − θ). If 270° ≤ θ ≤ 360°, sin θ = −sin (360° − θ); cos θ = cos (360° − θ); tan θ = −tan (360° − θ). The reciprocal properties may be used to find the values of the other functions. If it is desired to find the angle when a function of it is given, the procedure is as follows: There will in general be two angles between 0° and 360° corresponding to the given value of the function.

1 − cos x sin x 1 − cos x
=
=
   1 + cos x 1 + cos x sin x

x tan
= 2

Relations between Three Angles Whose Sum Is 180° Let x, y, z be the angles. x y z six x + sin y + sin z = 4 cos
cos
cos
2 2 2

  

  

x y z cos x + cos y + cos z = 4 sin
sin
sin
+ 1 2 2 2

  

x y z sin x + sin y − sin z = 4 sin
sin
cos
2 2 2

sin x + sin y + sin z = 2 cos x cos y cos z + 2; tan x + tan y + tan z = tan x tan y tan z; sin 2x + sin 2y + sin 2z = 4 sin x sin y sin z. 2

2

2

INVERSE TRIGONOMETRIC FUNCTIONS Given (a > 0)

Find an acute angle θ0 such that

Required angles are

sin θ = +a cos θ = +a tan θ = +a sin θ = −a cos θ = −a tan θ = −a

sin θ0 = a cos θ0 = a tan θ0 = a sin θ0 = a cos θ0 = a tan θ0 = a

θ0 and (180° − θ0) θ0 and (360° − θ0) θ0 and (180° + θ0) 180° + θ0 and 360° − θ0 180° − θ0 and 180° + θ0 180° − θ0 and 360° − θ0

y = sin −1 x = arcsin x is the angle y whose sine is x. Example y = sin−1 a, y is 30°. The complete solution of the equation x = sin y is y = (−1)n sin−1 x + n(180°), −π/2 ≤ sin−1 x ≤ π/2 where sin−1 x is the principal value of the angle whose sine is x. The range of principal values of the cos−1 x is 0 ≤ cos−1 x ≤ π and −π/2 ≤ tan−1 x ≤ π/2. If these restrictions are allowed to hold, the following formulas result:

3-22

MATHEMATICS

x −x2
1 sin−1 x = cos−1
1 −x2 = tan−1
2 = cot−1
x −x
1 1 1 π = sec−1
2 = csc−1
=
− cos−1 x x 2 −x
1 −x2
1 −x2 = tan−1
cos−1 x = sin−1
1 x FIG. 3-44

x 1 = cot−1
2 = sec−1
x −x
1

Right triangle.

a =
(c  + b)( c−  b) = c sin α = b tan α

x 1 tan−1 x = sin−1
2 = cos−1
2 −x +x
1
1 +x2 1
1 = cot−1
= sec−1
1 +x2 = csc−1
x x RELATIONS BETWEEN ANGLES AND SIDES OF TRIANGLES Solutions of Triangles (Fig. 3-43) Let a, b, c denote the sides and α, β, γ the angles opposite the sides in the triangle. Let 2s = a + b + c, A = area, r = radius of the inscribed circle, R = radius of the circumscribed circle, and h = altitude. In any triangle α + β + γ = 180°.

 +a)( c−  a) = c cos α = a cot α b =
(c a b a 2 2 a  b, sin α =
, cos α =
, tan α =
, β = 90° − α c =
 c c b

1 a2 b2 tan α c2 sin 2α A =
ab =
=
=
2 2 tan α 2 4 Oblique Triangles (Fig. 3-45) There are four possible cases. 1. Given b, c and the included angles α, b−c 1 1 1 1 tan
(β + γ)
(β + γ) = 90° −
α; tan
(β − γ) =
b +c 2 2 2 2 1 1 1 1 b sin α β =
(β + γ) +
(β − γ); γ =
(β + γ) −
(β − γ); a =
2 2 2 2 sin β 2. Given the three sides a, b, c, s = a (a + b + c); r=

Triangle.

Law of Sines sin α/a = sin β/ b = sin γ/ c. Law of Tangents a + b tan a(α + β) b + c tan a(β + γ) a + c tan a(α + γ)
=

;
=

;
=

a − b tan a(α − β) b − c tan a(β − γ) a − c tan a(α − γ) Law of Cosines a2 = b2 + c2 − 2bc cos α; b2 = a2 + c2 − 2ac cos β; c = a2 + b2 − 2ab cos γ. Other Relations In this subsection, where appropriate, two more formulas can be generated by replacing a by b, b by c, c by a, α by β, β by γ, and γ by α. cos α = (b2 + c2 − a2)/2bc; a = b cos γ + c = cos β; sin α = (2/bc)
 s( s −a )( s − b )( s −c) ; 2

     

α sin
= 2

(s − b)(s − c) α

; cos
= bc 2

s(s − a) 1
; A =
bh bc 2

1 a2 sin β sin γ =
ab sin γ =

=
s( s−  a)( s−  b)( s− c) = rs 2 2 sin α where r =

Oblique triangle.

Right Triangle (Fig. 3-44) Given one side and any acute angle α or any two sides, the remaining parts can be obtained from the following formulas:

1 π = csc−1
2 =
− sin−1 x  −x 2
1

FIG. 3-43

FIG. 3-45

(s − a)(s − b)(s − c)

 s

R = a/(2 sin α) = abc/4A; h = c sin a = a sin γ = 2rs/b. Example a = 5, b = 4, α = 30°. Use the law of sines. 0.5/5 = sin β/4, sin β = 2⁄ 5, β = 23°35′, γ = 126°25′. So c = sin 126°25′/ 1⁄10 = 10(.8047) = 8.05. The relations given here suffice to solve any triangle. One method for each triangle is given.

(s − a)(s − b)(s − c)

 s

1 r 1 r 1 r tan
α =
; tan
β =
; tan
γ =
2 s−a 2 s−b 2 s−c 3. Given any two sides a, c and an angle opposite one of them α, sin γ = (c sin α)/a; β = 180° − a − γ; b = (a sin β)/(sin α). There may be two solutions here. γ may have two values γ1, γ2; γ1 < 90°, γ2 = 180° − γ1 > 90°. If α + γ2 > 180°, use only γ1. This case may be impossible if sin γ > 1. 4. Given any side c and two angles α and β, γ = 180° − α − β; a = (c sin α)/(sin γ); b = (c sin β)/(sin γ). HYPERBOLIC TRIGONOMETRY The hyperbolic functions are certain combinations of exponentials ex and e−x. ex + e−x ex − e−x sinh x ex − e−x cosh x =
; sinh x =
; tanh x =
=
2 2 cosh x ex + e−x ex + e−x 1 cosh x 1 2 coth x =
=
=
; sech x =
=
; ex − e−x tanh x sinh x cosh x ex + e−x 2 1 csch x =
=
sinh x ex − e−x Fundamental Relationships sinh x + cosh x = ex; cosh x − sinh x = e−x; cosh2 x − sinh2 x = 1; sech2 x + tanh2 x = 1; coth2 x − csch2 x = 1; sinh 2x = 2 sinh x cosh x; cosh 2x = cosh2 x + sinh2 x = 1 + 2 sinh2 x = 2 cosh2 x − 1. tanh 2x = (2 tanh x)/(1 + tanh2 x); sinh (x  y) = sinh x cosh y  cosh x sinh y; cosh (x  y) = cosh x cosh y  sinh x sinh y; 2 sinh2 x/2 = cosh x − 1; 2 cosh2 x/2 = cosh x + 1; sinh (−x) = −sinh x; cosh (−x) = cosh x; tanh (−x) = −tanh x. When u = a cosh x, v = a sinh x, then u2 − v2 = a2; which is the equation for a hyperbola. In other words, the hyperbolic functions in the parametric equations u = a cosh x, v = a sinh x have the same relation to the hyperbola u2 − v2 = a2 that the equations u = a cos θ, v = a sin θ have to the circle u2 + v2 = a2.

DIFFERENTIAL AND INTEGRAL CALCULUS Inverse Hyperbolic Functions If x = sinh y, then y is the inverse hyperbolic sine of x written y = sinh−1 x or arcsinh x. sinh−1 x = loge (x +
 x2 +1) 1 1+x −1 x2 +1 ); tanh −1 x =
loge
; cosh x = loge (x +
 2 1−x 1 x+1 −x2 1 +
1 coth−1 x =
loge
; sech−1 x = loge

; 2 x−1 x





1 +
1 +x2 csch−1 = loge

x Magnitude of the Hyperbolic Functions cosh x ≥ 1 with equality only for x = 0; −∞ < sinh x < ∞; −1 < tanh x < 1. cosh x ∼ ex/2 as x → ∞; sinh x → ex/2 as x → ∞.





3-23

APPROXIMATIONS FOR TRIGONOMETRIC FUNCTIONS For small values of θ (θ measured in radians) sin θ ≈ θ, tan θ ≈ θ; cos θ ≈ 1 − (θ2/2). The following relations actually hold: sin θ < θ < tan θ; cos θ < sin θ/θ < 1; θ
1 −θ2 < sin θ < θ; cos θ < θ/tan θ < 1; θ θ2 θ 1 −
< sin θ < θ and θ < tan θ <
2 − θ
1 2





The behavior ratio of the functions as θ → 0 is given by the following: lim sin θ/θ = 1; sin θ/tan θ = 1. θ→0

DIFFERENTIAL AND INTEGRAL CALCULUS REFERENCES: 114, 158, 260, 261, 274, 282, 296. See also “General References: References for General and Specific Topics—Advanced Calculus.” For computer evaluations of the calculus described here, see Refs. 68, 299.

Limits The limit of function f(x) as x approaches a (a is finite or else x is said to increase without bound) is the number N. lim f(x) = N x→a

DIFFERENTIAL CALCULUS An Example of Functional Notation Suppose that a storage warehouse of 16,000 ft3 is required. The construction costs per square foot are $10, $3, and $2 for walls, roof, and floor respectively. What are the minimum cost dimensions? Thus, with h = height, x = width, and y = length, the respective costs are Walls = 2 × 10hy + 2 × 10hx = 20h(y + x) Roof = 3xy Floor = 2xy Total cost = 2xy + 3xy + 20h(x + y) = 5xy + 20h(x + y)

This states that f(x) can be calculated as close to N as desirable by making x sufficiently close to a. This does not put any restriction on f(x) when x = a. Alternatively, for any given positive number ε, a number δ can be found such that 0 < |a − x| < δ implies that |N − f(x)| < ε. The following operations with limits (when they exist) are valid: lim bf(x) = b lim f(x) x→a

x→a

(3-1)

x→a

x→a

lim f(x) x→a f(x) lim
=
lim g(x) x→a g(x)

(3-2)

x→a

Solving for h from Eq. (3-2), h = volume/xy = 16,000/xy 320,000 Cost = 5xy +
(y + x) = 5xy + 320,000 xy In this form it can be shown that the minimum x = y; therefore Cost = 5x2 + 640,000 (1/x)

x→a

(3-3)


x +
y  1

1

(3-4)

cost will occur for

By evaluation, the smallest cost will occur when x = 40. Cost = 5(1600) + 640,000/40 = $24,000 The dimensions are then x = 40 ft, y = 40 ft, h = 16,000/(40 × 40) = 10 ft. Symbolically, the original cost relationship is written Cost = f(x, y, h) = 5xy + 20h(y + x) and the volume relation Volume = g(x, y, h) = xyh = 16,000 In terms of the derived general relationships (3-1) and (3-2), x, y, and h are independent variables—cost and volume, dependent variables. That is, the cost and volume become fixed with the specification of dimensions. However, corresponding to the given restriction of the problem, relative to volume, the function g(x, y, z) = xyh becomes a constraint function. In place of three independent and two dependent variables the problem reduces to two independent (volume has been constrained) and two dependent as in functions (3-3) and (3-4). Further, the requirement of minimum cost reduces the problem to three dependent variables (x, y, h) and no degrees of freedom, that is, freedom of independent selection.

x→a

lim [ f(x)g(x)] = lim f(x) ⋅ lim g(x)

and the restriction Total volume = xyh

x→a

lim [ f(x) + g(x)] = lim f(x) + lim g(x) x→a

if

lim g(x) ≠ 0 x→a

Continuity A function f(x) is continuous at the point x = a if lim [ f(a + h) − f(a)] = 0 h→0

Rigorously, it is stated f(x) is continuous at x = a if for any positive ε there exists a δ > 0 such that |f(a + h) − f(a)| < ε for all x with |x − a| < δ. For example, the function (sin x)/x is not continuous at x = 0 and therefore is said to be discontinuous. Discontinuities are classified into three types: 1. Removable 2. Infinite 3. Jump

y = sin x/x at x = 0 y = 1/x at x = 0 y = 10/(1 + e1/x) at x = 0+ y = 0+ x=0 y=0 x = 0− y = 10

Derivative The function f(x) has a derivative at x = a, which can be denoted as f ′(a), if f(a + h) − f(a) lim

h→0 h exists. This implies continuity at x = a. Conversely, a function may be continuous but not have a derivative. The derivative function is df f(x + h) − f(x) f ′(x) =
= lim

dx h→0 h Differentiation Define ∆y = f(x + ∆x) − f(x). Then dividing by ∆x ∆y f(x + ∆x) − f(x)
=

∆x ∆x

3-24

MATHEMATICS ∆y dy lim
=
∆x dx

Call

∆x→0

Example Find the derivative of y = sin x. dy sin (x + ∆x) − sin(x)
= lim

dx ∆x→0 ∆x sin x cos ∆x + sin ∆x cos x − sin x = lim



∆x→0 ∆x sin ∆x cos x sin x(cos ∆x − 1) = lim

+ lim

∆x→0 ∆x→0 ∆x ∆x sin ∆x = cos x since lim
= 1 ∆x→0 ∆x

Differential Operations The following differential operations are valid: f, g, . . . are differentiable functions of x, c is a constant; e is the base of the natural logarithms. dc (3-5)
=0 dx dx
=1 dx

(3-6)

d df dg
(f + g) =
+
dx dx dx

(3-7)

d dg df
(f × g) = f
+ g
dx dx dx dy 1
=
dx dx/dy

if

(3-8) dx
≠0 dy

d df
f n = nf n − 1
dx dx d f g(df/dx) − f(dg/dx)

=

dx g g2



df df dv
=
×
dx dv dx

d tan x = sec x dx

(3-20)

d cot x = −csc2 x dx

(3-21)

d sec x = tan x sec x dx

(3-22)

d csc x = −cot x csc x dx

(3-23)

d sin−1 x = (1 − x2)−1/2 dx

(3-24)

d cos−1x = −(1 − x2)−1/2 dx

(3-25)

d tan−1 x = (1 + x2)−1 dx

(3-26)

d cot−1 x = −(1 + x2)−1 dx

(3-27)

d sec−1 x = x−1(x2 − 1)−1/2 dx

(3-28)

d csc−1 x = −x−1(x2 − 1)−1/2 dx

(3-29)

d sinh x = cosh x dx

(3-30)

d cosh x = sinh x dx

(3-31)

d tanh x = sech2 x dx

(3-32)

d coth x = −csch2 x dx

(3-33)

d sech x = −sech x tanh x dx

(3-34)

d csch x = −csch x coth x dx

(3-35)

d sinh x = (x + 1)

(3-36)

−1

(3-9)

−1/2

2

dx

d cosh−1 = (x2 − 1)−1/2 dx

(3-37)

d tanh−1 x = (1 − x2)−1 dx

(3-38)

d coth−1 x = −(x2 − 1)−1 dx

(3-39)

d sech−1 x = −(1/x)(1 − x2)−1/2 dx

(3-40)

d csch x = −x (x + 1)

(3-41)

−1

−1

−1/2

2

dx

Example Find dy/dx for y =
x cos (1 − x ). 2

(3-10)

dy d d
=
x
cos (1 − x2) + cos (1 − x2)

x dx dx dx

(3-11)

d d
cos (1 − x2) = −sin (1 − x2)
(1 − x2) dx dx

Using (3-8) (3-19)

= −sin (1 − x )(0 − 2x) 2

(chain rule)

(3-12)

g

df df dg
= gf g − 1
+ f g ln f
dx dx dx

(3-13)

dax
= (ln a) ax dx

(3-14)

(3-5), (3-10)

d
x 1
=
x−1/2 dx 2

(3-10)

1 dy
= 2x3/2 sin (1 − x2) +
x−1/2 cos (1 − x2) dx 2

Example Find the derivative of tan x with respect to sin x. v = sin x

Example Derive dy/dx for x2 + y3 = x + xy + A. Here

(3-19)

2

f(x + ∆x) − f(x) dy lim


= ∆x→0 dx ∆x

then

d cos x = −sin x dx

y = tan x

d d d d d
x2 +
y3 =
x +
xy +
A dx dx dx dx dx

d tan x dy dy dx
=
=

d sin x dv dx dv 1 d tan x =

d sin x dx

dx = sec2 x/cos x

dy dy 2x + 3y2
= 1 + y + x
+ 0 dx dx by rules (3-10), (3-10), (3-6), (3-8), and (3-5) respectively. dy 2x − 1 − y Thus
=

dx x − 3y2

Differentials dex = ex dx

(3-15a)

d(ax) = ax log a dx

(3-15b)

d ln x = (1/x) dx

(3-16)

d log x = (log e/x)dx

(3-17)

d sin x = cos x dx

(3-18)

Using (3-12) (3-9) (3-18), (3-20)

Very often in experimental sciences and engineering functions and their derivatives are available only through their numerical values. In particular, through measurements we may know the values of a function and its derivative only at certain points. In such cases the preceding operational rules for derivatives, including the chain rule, can be applied numerically. Example Given the following table of values for differentiable functions f and g; evaluate the following quantities:

DIFFERENTIAL AND INTEGRAL CALCULUS x

f(x)

f′(x)

g(x)

g′(x)

1 3 4

3 0 −2

1 2 10

4 4 3

−4 7 6

d
[ f(x) + g(x)]| x = 4 = f ′(4) + g′(4) = 10 + 6 = 16 dx f ′

f ′(1)g(1) − f(1)g′(1)

1 ⋅ 4 − 3(−4)

=

=
= 1 
g  (1) =


[g(1)] (−4) 16 2

16

Example Given f(x) = 3x3 + 2x + 1, calculate all derivative values at x = 3. df(x)
= 9x2 + 2 dx

x = 3, f′(3) = 9(9) + 2 = 83

d2f(x) = 18x
dx2

x = 3, f″(3) = 18(3) = 54

d3f(x) = 18
dx3

x = 3, f ″(3) = 18

dnf(x) =0
dxn

for n ≥ 4

If f ′(x) > 0 on (a, b), then f is increasing on (a, b). If f ′(x) < 0 on (a, b), then f is decreasing on (a, b). The graph of a function y = f(x) is concave up if f ′ is increasing on (a, b); it is concave down if f ′ is decreasing on (a, b). If f ″(x) exists on (a, b) and if f ″(x) > 0, then f is concave up on (a, b). If f ″(x) < 0, then f is concave down on (a, b). An inflection point is a point at which a function changes the direction of its concavity. Indeterminate Forms: L’Hospital’s Theorem Forms of the type 0/0, ∞/∞, 0 × ∞, etc., are called indeterminates. To find the limiting values that the corresponding functions approach, L’Hospital’s theorem is useful: If two functions f(x) and g(x) both become zero at x = a, then the limit of their quotient is equal to the limit of the quotient of their separate derivatives, if the limit exists or is + ∞ or − ∞. sin x x sin x d sin x cos x lim
= lim
= lim
= 1 x→0 x→0 x→0 x dx 1

Example Find lim
. n→0 Here

(1.1) x

Example Find lim .
1000 x→∞ (1.1)x d(1.1)x (ln 1.1)(1.1)x = lim
= lim

lim
x→∞ x1000 x→∞ dx1000 x→∞ 1000x999 1.1x Obviously lim
= ∞ since repeated application of the rule will reduce the x→∞ x1000 denominator to a finite number 1000! while the numerator remains infinitely large.

x3 6 lim x3 e−x = lim
x = lim
x = 0 x→∞ x→∞ e x→∞ e

∂e ∂z ∂x
= y
+ ey
∂x ∂x ∂x

∂ey ∂z 2 ∂y
= ex
+ x
∂y ∂y ∂y

= 2xyex + e y

= ex + xey

2

2

Order of Differentiation It is generally true that the order of differentiation is immaterial for any number of differentiations or variables provided the function and the appropriate derivatives are continuous. For z = f(x, y) it follows: ∂3f ∂3f ∂3f =
=

∂y2 ∂x ∂y ∂x ∂y ∂x ∂y2 General Form for Partial Differentiation 1. Given f(x, y) = 0 and x = g(t), y = h(t). df ∂f dx ∂f dy Then
=

+

dt ∂x dt ∂y dt d 2f ∂2f dx
2 =
2
dt ∂x dt

∂2

∂2

f dy f ∂f d x

+

+

  +2
∂x ∂y dt dt ∂y  dt  ∂x dt 2

dx

dy

y = (1 − x)

1/x

ln y = (1/x) ln (1 − x) ln(1 − x) lim (ln y) = lim
= −1 x→0 x→0 x lim y = e−1

2

2

2

2

∂f d 2y +

∂y dt2 Example Find df/dt for f = xy, x = ρ sin t, y = ρ cos t. df ∂(xy) d ρ sin t ∂(xy) d ρ cos t
=

+

dt ∂x dt ∂y dt = y(ρ cos t) + x(−ρ sin t) = ρ2 cos2 t − ρ2 sin2 t









Given f(x, y) = 0 and x = g(t, s), y = h(t, s). ∂f ∂f ∂x ∂f ∂y Then
=

+

∂t ∂x ∂t ∂y ∂t 2.

∂f ∂f ∂x ∂f ∂y
=

+

∂s ∂x ∂s ∂y ∂x Differentiation of Composite Function dy ∂f/∂x ∂f Rule 1. Given f(x, y) = 0, then
= −

≠ 0 . dx ∂f/∂y ∂y Rule 2. Given f(u) = 0 where u = g(x), then df du
= f ′(u)
dx dx du   + f ′(u)
dx

d 2f du
2 = f ″(u)
dx dx

2

2

2

df d sin2 u d
1 −x2
=


dx du dx



1 = 2 sin u cos u
(−2x)(1 − x2)−1/2 2

Example Find lim (1 − x)1/x. x→0

x→0

2

x2

Example Find df/dx for f = sin2 u and u =
1−x2

Example Find lim x3 e−x. x→∞

Therefore,

Example Find ∂z/∂x and ∂z/∂y for z = ye x + xey.



x

Let

Partial Derivative The abbreviation z = f(x, y) means that z is a function of the two variables x and y. The derivative of z with respect to x, treating y as a constant, is called the partial derivative with respect to x and is usually denoted as ∂z/∂x or ∂f(x, y)/∂x or simply fx. Partial differentiation, like full differentiation, is quite simple to apply. Conversely, the solution of partial differential equations is appreciably more difficult than that of differential equations.

2

Higher Differentials The first derivative of f(x) with respect to x is denoted by f′ or df/dx. The derivative of the first derivative is called the second derivative of f(x) with respect to x and is denoted by f ″, f (2), or d 2 f/dx 2; and similarly for the higher-order derivatives.

3-25

− u
1 = −2
sin u cos u u 2

Rule 3. Given f(u) = 0 where u = g(x,y), then ∂f ∂u ∂f ∂u
= f ′(u)
+
= f ′(u)
∂x ∂x ∂y ∂y



3-26

MATHEMATICS ∂2f ∂u
2 = f ″
∂x ∂x

∂   + f′
∂x 2

2

u

V is specified once T and p are specified; it is therefore a state function. All applications are for closed systems with constant mass. If a process is reversible and only p-V work is done, the first law and differentials can be expressed as follows.

2

∂2f ∂u ∂u ∂2u
= f ″

+ f′
∂x ∂y ∂x ∂y ∂x ∂y ∂2f ∂u
2 = f ″
∂y ∂y

dU = T dS − p dV dH = T dS + V dp dA = −S dT − p dV dG = −S dT + V dp

∂2u

  + f′
∂y 2

2

MULTIVARIABLE CALCULUS APPLIED TO THERMODYNAMICS Many of the functional relationships needed in thermodynamics are direct applications of the rules of multivariable calculus. This section reviews those rules in the context of the needs of themodynamics. These ideas were expounded in one of the classic books on chemical engineering thermodynamics.151 State Functions State functions depend only on the state of the system, not on past history or how one got there. If z is a function of two variables, x and y, then z(x,y) is a state function, since z is known once x and y are specified. The differential of z is dz = M dx + N dy The line integral



(M dx + N dy)

is independent of the path in x-y space if and only if ∂M ∂N
=
∂y ∂x The total differential can be written as ∂z ∂z dz =
dx +
dy ∂x y ∂y x and the following condition guarantees path independence. ∂ ∂z ∂ ∂z

=

∂y ∂x y ∂x ∂y x

 

 

 

∂2z ∂2z
=
∂y ∂x ∂x ∂y

or

 

(3-42)

(3-43)

x

z

∂z ∂z ∂y (∂y/∂x)z (3-45)
=−

=−
∂x y ∂x z ∂y x (∂y/∂z)x Alternatively, divide Eq. (3-43) by dy when holding some other variable w constant to obtain ∂z ∂z ∂x ∂z (3-46)
=

+
∂y w ∂x v ∂y w ∂y x Also divide both numerator and denominator of a partial derivative by dw while holding a variable y constant to get

    

      

(∂z/∂w) ∂z ∂w = 
 
 
∂z∂x  =
(∂x/∂w) ∂w ∂x y y

y

 

∂T ∂p = −
 
∂V  ∂S

(3-47)

y

Themodynamic State Functions In thermodynamics, the state functions include the internal energy, U; enthalpy, H; and Helmholtz and Gibbs free energies, A and G, respectively, defined as follows: H=U+pV A = U − TS G = H − TS = U + pV − TS = A + pV S is the entropy, T the absolute temperature, p the pressure, and V the volume. These are also state functions, in that the entropy is specified once two variables (like T and p) are specified, for example. Likewise,

V

This is one of the Maxwell relations, and the other Maxwell relations can be derived in a similar fashion by applying Eq. (3-44). ∂T ∂V
=
∂p S ∂S p ∂S ∂p
=
∂V T ∂T V

        ∂S ∂V = −
 
∂p  ∂T

p

In process simulation it is necessary to calculate enthalpy as a function of state variables. This is done using the following formulas, derived from the above relations by considering S and H as functions of T and p. ∂V dH = Cp dT + V − T
dp ∂T p Enthalpy differences are then given by the following formula. T2 p 2 ∂V H(T2, p2) − H(T1, p1) = Cp(T, p1) dT + V−T
dp T1 p1 ∂T p T2,p The same manipulations can be done for internal energy as a function of T and V. (∂V/∂T) dU = CV dT − p + T
p dV (∂V/∂p)T



Rearrangement gives

y

 

T

(3-44)

  dy

 

S

Example Suppose z is constant and apply Eq. (3-43). ∂z ∂z 0 =
dx +
∂x y ∂y

 

This is

C

 

Alternatively, if the internal energy is considered a function of S and V, then the differential is: ∂U ∂U dU =
dS +
dV ∂S V ∂V S This is the equivalent of Eq. (3-43) and gives the following definitions. ∂U ∂U T=
, p=−
∂S V ∂V S Since the internal energy is a state function, then Eq. (3-44) must be satisfied. ∂2U ∂2U
=
∂V ∂S ∂S ∂V



 



  



Partial Derivatives of All Thermodynamic Functions The various partial derivatives of the thermodynamic functions can be classified into six groups. In the general formulas below, the variables U, H, A, G or S are denoted by Greek letters, while the variables V, T, or p are denoted by Latin letters. Type I (3 possibilities plus reciprocals) ∂a ∂p General:
; Specific:
∂b c ∂T V Eq. (3-45) gives ∂p ∂V ∂p (∂V/∂T)p
=−

=−
∂T V ∂T p ∂V T (∂V/∂p)T

 

 

    

Type II

(30 possibilities) ∂α ∂G General:
; Specific:
∂b c ∂T

 

 

V

DIFFERENTIAL AND INTEGRAL CALCULUS The differential for G gives ∂G ∂p
= −S + V
∂T V ∂T V Using the other equations for U, H, A, or S gives the other possibilities. Type III (15 possibilities plus reciprocals) ∂a ∂V General:
; Specific:
∂b α ∂T S First expand the derivative using Eq. (3-45). (∂S/∂T)V ∂V ∂S ∂V
=−

=−
∂T S ∂T V ∂S T (∂S/∂V)T Then evaluate the numerator and denominator as type II derivatives. CV ∂V



T ∂V CV ∂p T



= − ∂V ∂p =
∂T S T ∂V −





∂T p ∂V T ∂T p

 

 

 

f ′(x) dx = f(x) + c where c is an arbitrary constant to be determined by the problem. By virtue of the known formulas for differentiation the following relationships hold (a is a constant):

(du + dv + dw) = du + dv + dw a dv = a dv v v dv =
+ c (n ≠ −1) n+1

 

    

 

n

dv
= ln |v| + c v



∂G

∂G

v

 

∂T

=

=

(∂A/∂T) ∂A   ∂T   ∂A  p

p

(∂G/∂T)p

p

p

This operation has created two type II derivatives; by substitution we obtain ∂G S
=

∂A p S + p (∂V/∂T)p Type V (60 possibilities) ∂α ∂G General:
; Specific:
∂b β ∂p A Start from the differential for dG. Then we get ∂G ∂T
= −S
+ V ∂p A ∂p A The derivative is type III and can be evaluated by using Eq. (3-45). ∂G (∂A/∂p)T
= S
+ V ∂p A (∂A/∂T)p The two type II derivatives are then evaluated. ∂G Sp (∂V/∂p)T
=

+ V ∂p A S + p (∂V/∂T)p These derivatives are also of interest for free expansions or isentropic changes. Type VI (30 possibilities plus reciprocals) ∂α ∂G General:
; Specific:
∂β γ ∂A H We use Eq. (3-47) to obtain two type V derivatives. (∂G/∂T)H ∂G
=
∂A H (∂A/∂T)H These can then be evaluated using the procedures for Type V derivatives.

 

 

(3-54) (3-55) (3-56)

2

(3-57) (3-58) (3-59) (3-60)

2

−1

2

2

v
+c a

(3-61)

dv 1 v−a
=
ln 
 + c v −a 2a v+a 2

(3-62)

2

dv

= ln |v +
 v a| + c v a
 sec v dv = ln (sec v + tan v) + c csc v dv = ln (csc v − cot v) + c

 

 

(3-53)

2

dv
= sin a −v


2

2

 

 

v

2

 

 

(3-52)

−1

 

 

(3-50)

v

v

 

 

(3-49)

(3-51)

a a dv =
+c ln a e dv = e + c sin v dv = −cos v + c cos v dv = sin v + c sec v dv = tan v + c csc v dv = −cot v + c sec v tan v dv = sec v + c csc v cot v dv = −csc v + c dv 1 v
=
tan
+ c v +a a a

These derivatives are of importance for reversible, adiabatic processes (such as in an ideal turbine or compressor), since then the entropy is constant. An example is the Joule-Thomson coefficient. ∂T 1 ∂V
=
−V + T
∂p H Cp ∂T p Type IV (30 possibilities plus reciprocals) ∂α ∂G General:
; Specific:
∂β c ∂A p Use Eq. (3-47) to introduce a new variable.

 

(3-48)

n+1

   

  

3-27

2

2

(3-63) (3-64) (3-65)

Example Derive ∫ av dv = (av/ln a) + c. By reference to the differentiation formula dav/dv = av ln a, or in the more usable form d(av/ln a) = av dv, let f ′ = av dv; then f = av/ln a and hence ∫ av dv = (av/ln a) + c. Example Find ∫ (3x2 + ex − 10) dx using Eq. (3-48). ∫ (3x2 + ex − 10) dx = 3 ∫ x2 dx + ∫ ex dx − 10 ∫ dx = x3 + ex − 10x + c (by Eqs. 3-50, 3-53). 7x dx
. Let v = 2 − 3x ; dv = −6x dx 2 − 3x 7x dx x dx 7 −6x dx
=7
=−

2 − 3x 2 − 3x 6 2 − 3x 7 dv =−

6 v

Example Find Thus

2

2

2

2

2

INTEGRAL CALCULUS

7 = −
ln |v| + c 6

Indefinite Integral If f ′(x) is the derivative of f(x), an antiderivative of f ′(x) is f(x). Symbolically, the indefinite integral of f ′(x) is

7 = −
ln |2 − 3x2| + c 6

3-28

MATHEMATICS

Example—Constant of Integration By definition the derivative of x3 is 3x2, and x3 is therefore the integral of 3x2. However, if f = x3 + 10, it follows that f′ = 3x2, and x3 + 10 is therefore also the integral of 3x2. For this reason the constant c in ∫ 3x2 dx = x3 + c must be determined by the problem conditions, i.e., the value of f for a specified x. Methods of Integration In practice it is rare when generally encountered functions can be directly integrated. For example, the integrand in ∫
 si nx dx which appears quite simple has no elementary function whose derivative is
 si nx. In general, there is no explicit way of determining whether a particular function can be integrated into an elementary form. As a whole, integration is a trial-and-error proposition which depends on the effort and ingenuity of the practitioner. The following are general procedures which can be used to find the elementary forms of the integral when they exist. When they do not exist or cannot be found either from tabled integration formulas or directly, the only recourse is series expansion as illustrated later. Indefinite integrals cannot be solved numerically unless they are redefined as definite integrals (see “Definite Integral”), i.e., F(x) = ∫ f(x) dx, indefinite, whereas F(x) = ∫ ax f(t) dt, definite. Direct Formula Many integrals can be solved by transformation in the integrand to one of the forms given previously. Example Find ∫ x
3 x + 10 dx. Let v = 3x + 10 for which dv = 9x dx. 2

Thus

3

3

x
3x +10 dx = (3x + 10) 2

3

3

1/2

(x2 dx)

(3x + 10)

1 =
9

v

1/2

x dx

= (3 + 4x) 1/4

1 =
4

1/2

(9x2 dx)

dv

3/2

1 v =

+c 9 3⁄ 2

General The number of possible transformations one might use are unlimited. No specific overall rules can be given. Success in handling integration problems depends primarily upon experience and ingenuity. The following example illustrates the extent to which alternative approaches are possible.

e dx− 1

Example Find
. Let ex = y; then ex dx = dy or dx = 1/y dy. x dx (1/y) dy e −2 dy y−1
=
=
= ln
= ln
e −1 y−1 y −y e y x

x

A B C =
+
+
x+2 x−2 x−1 A(x − 2)(x − 1) + B(x + 2)(x − 1) + C(x + 2)(x − 2) =





(x + 2)(x − 2)(x − 1) x2(A + B + C) + x(−3A + B) + (2A − 2B − 4C) =




(x + 2)(x − 2)(x − 1) Equate coefficients and solve for A, B, and C. A+B+C=0 −3A + B = 0 2A − 2B − 4C = 1 A = 1⁄12, B = d, C = −s 1 1 1 1 =
+
−


x3 − x2 − 4x + 4 12(x + 2) 4(x − 2) 3(x − 1)

x +a Let x = a tan θ
 2

a2 −x2 Let x = a sin θ
  − 9 x
4 2 2

dx. Let x =
sin θ; then dx =
cos θ dθ. x 3 3 2

2

 (2 /3 ) −x θ 2 2/3
1 −sin dx = 3


cos θ dθ


x (2/3) sin θ  3 2

2

2

2

2

=3

cos θ
dθ sin θ

=3

cot θ dθ

Hence

dx dx dx dx

=
+
−
x − x − 4x + 4 12(x + 2) 4(x − 2) 3(x − 1)

2

3

2

d(uv) = u dv + v du

2

= −3 cot θ − 3θ + c by trigonometric transform − 9 x2 3
4 = −

− 3 sin−1
x + c in terms of x x 2

Algebraic Substitution Functions containing elements of the type (a + bx)1/n are best handled by the algebraic transformation yn = a + bx. Example Find

. Let 3 + 4x = y4; then 4dx = 4y3 dy and 1/4

2

Parts An extremely useful formula for integration is the relation

2

(3 +x dx4x)

x

2

Partial Fractions Rational functions are of the type f(x)/g(x) where f(x) and g(x) are polynomial expressions of degrees m and n respectively. If the degree of f is higher than g, perform the algebraic division—the remainder will then be at least one degree less than the denominator. Consider the following types: Type 1 Reducible denominator to linear unequal factors. For example, 1 1 =




x3 − x2 − 4x + 4 (x + 2)(x − 2)(x − 1)

x2 −a2 Let x = a sec θ


3

4

1 1 =
(3 + 4x)7/4 −
(3 + 4x)3/4 + c 28 4

[by Eq. (3-50)]

Trigonometric Substitution This technique is particularly well adapted to integrands in the form of radicals. For these the function is transformed into a trigonometric form. In the latter form they may be more easily recognizable relative to the identity formulas. These functions and their transformations are

Example Find

2

1 y7 3 y3 =

−

+c 4 7 4 3

2 =
(3x3 + 10)3/2 + c 27

2

y (y − 3) dy

2

1 =
9

3

y4 − 3

y3 dy 4

y

and or

u dv + v du u dv = uv − v du uv =

No general rule for breaking an integrand can be given. Experience alone limits the use of this technique. It is particularly useful for trigonometric and exponential functions. Example Find xex dx. Let u=x du = dx

and

dv = ex dx v = ex

DIFFERENTIAL AND INTEGRAL CALCULUS

xe dx = xe − e dx x

Therefore

x

Techniques for determining when integration is valid under these conditions are available in the references. However, the following simplified rules will, in general, serve as a guide for most practical applications. Rule 1 For the integral ∞ φ(x) dx
0 xn

x

= xex − ex + c

Example Find ex sin x dx. Let u = ex



dv = sin x dx v = −cos x

du = ex dx

e sin x dx = −e cos x + e cos x dx x

x

x

u = ex

Again

dv = cos x dx v = sin x

du = ex dx

e sin x dx = −e cos x + e sin x − e sin x dx + c x

x

x

x

c = (ex/2)(sin x − cos x) +
2

Series Expansion When an explicit function cannot be found, the integration can sometimes be carried out by a series expansion. Example Find e− x dx. Since 2

4

6

x x 2 e− x = 1 − x2 +
−
+ ⋅⋅⋅ 2! 3!

e

−x2

dx =

dx − x dx +
2!x dx −
3!x dx + ⋅⋅⋅ 4



if φ(x) is bounded, the integral will converge for n < 1 and diverge for n ≥ 1. Thus 1 1
dx x 0




will converge (exist) since a = n < 1. Properties The fundamental theorem of calculus states

f(x) dx = F(b) − F(a)

6

for all x

b

a

dF(x)/dx = f(x)

where

Definite Integral The concept and derivation of the definite integral are completely different from those for the indefinite integral. These are by definition different types of operations. However, the formal operation ∫ as it turns out treats the integrand in the same way for both. Consider the function f(x) = 10 − 10e−2x. Define x1 = a and xn = b, and suppose it is desirable to compute the area between the curve and the coordinate axis y = 0 and bounded by x1 = a, xn = b. Obviously, by a sufficiently large number of rectangles this area could be approximated as closely as desired by the formula



if φ(x) is bounded, the integral will converge for n > 1 and not converge for n ≤ 1. ∞ It is easily seen that ∫ 0 e−x dx converges by noting 1/x2 > 1/ex > 0 for large x. Rule 2 For the integral b φ(x)
n dx, a (a − x)

2

x3 x5 x7 = x −
+
−
+ ⋅⋅⋅ 3 5.2! 7.3!

n−1

Other properties of the definite integral are

c[ f(x) dx] = c f(x) dx b

[ f (x) + f (x)] dx = f (x) dx + f (x) dx

a

i

2

a

a

f(x) dx = − f(x) dx b

a

a

b

f(x) dx = b

c

a

f(x) dx +

a

f(x) dx b

c

f(x) dx = (b − a) f(ξ) for some ξ in (a, b)

xi − 1 ≤ ξi − 1 ≤ xi

b

i+1

− xi)

i=1

f(x) dx indefinite integral where dF/dx = f(x) F(a, b) = f(x) dx definite integral F(α) = f(x, α) dx

∂
∂b

f(x) dx = f(b)

∂
∂a

f(x) dx = −f(a)

dF(α)
= dα

F(x) =

a b

a

b

a

∂f(x, α)
dx if a and b are constant ∂α b

a

b

d

a

when F(x) =



d

c

c

b

a

b(x)

f(x, y) dy

a(x)

a

There are certain restrictions of the integration definition, “The function f(x) must be continuous in the finite interval (a, b) with at most a finite number of finite discontinuities,” which must be observed before integration formulas can be generally applied. Two of these restrictions give rise to so-called improper integrals and require special handling. These occur when ∞ 1. The limits of integration are not both finite, i.e., ∫ 0 e−x dx. 2. The function becomes infinite within the interval of integration, i.e., 1 1
dx x 0


b

dx f(x, α) dα = dα f(x, α) dx

b



b

1

a

where the points x1, x2, . . . , xn are equally spaced. For a rigorous definition of the definite integral the references should be consulted. Thus, the value of a definite integral depends on the limits a, b, and any selected variable coefficients in the function but not on the dummy variable of integration x. Symbolically

or

2

a

n

n→∞

b

1

The definite integral of f(x) is defined as

f(x) dx = lim  f(ξ )(x

a

b

i=1

+ ⋅⋅⋅ + f(ξn − 1)(b − xn − 1)

b

a

f(ξi)(xi + 1 − xi) = f(ξ1)(x2 − a) + f(ξ2)(x3 − x2)

b

3-29

the Leibniz rule gives dF db da
=
f [x, b(x)] −
f [x, a(x)] + dx dx dx





b(x)

a(x)

∂f
dy ∂x

π/2

Example Find

sin x dx.

0



π/2

0

since

π sin x dx = [−cos x]0π/2 = − cos
− cos 0 = 1 2



−d cos x/dx = sin x



(3-66)

3-30

MATHEMATICS dx
. Direct application of the formula would yield (x − 1) 2

Example Find the incorrect value

1 dx
= −
 (x − 1) x−1 2

2

0

c = π/2 φ(α) = −tan−1 α + π/2

2

= −2

0

It should be noted that f(x) = 1/(x − 1) becomes unbounded as x → 1 and by Rule 2 the integral diverges and hence is said not to exist. Methods of Integration All the methods of integration available for the indefinite integral can be used for definite integrals. In addition, several others are available for the latter integrals and are indicated below. Change of Variable This substitution is basically the same as previously indicated for indefinite integrals. However, for definite integrals, the limits of integration must also be changed: i.e., for x = φ(t), 2

f (x) dx = b

Integration It is sometimes useful to generate a double integral to solve a problem. By this approach, the fundamental theorem indicated by Eq. (3-66) can be used.

0

x 1

α

Consider

0

dα x b

α

dα x a

2

π/2

0

cos2 θ dθ = 16[aθ + d sin 2θ]0π/2 = 4π



∞

e−αx sin x
dx (α > 0) x Since this is a continuous function of α, it may be differentiated under the integral sign ∞ dφ
= − e−αx sin x dx 0 dα φ(α) =

b

dα b+1
= ln
α+1 a+1





= −1/(1 + α ) φ(α) = −tan−1 α + c 2

1

b

α



1

dα =

a

0

b+1 xb − xα
dx = ln
ln x a+1



xb − xα
dx ln x



Complex Variable Certain definite integrals can be evaluated by the technique of complex variable integration. This is described in the references for “Complex Variables.” Numerical Because of the property of definite integrals another method for obtaining their solution is available which cannot be applied to indefinite integrals. This involves a numerical approximation based on the previously outlined summation definition:

0



dx x 0



Therefore

Differentiation Here the application of the general rules for differentiating under the integral sign may be useful. Example Find



a

dx =

0

1

x = 4 sin θ (x = 0, θ = 0) dx = 4 cos θ dθ (x = 4, θ = π/2) 0

1

α


16−x dx. Let 2

0

dx =

0

But also b


16−x dx = 16

1

a

t0

4

Then

1 dx =
(α > −1) α+1

Then multiplying both sides by dα and integrating between a and b,

4

0

xb − xα
dx ln x

1

f [φ(t)]φ′(t) dt

t = t0 when x = a t = t1 when x = b

Example Find



Example Find

t1

a

where

and since φ(α) → 0 as α → ∞,

2

0

n−1

lim  f(ξi)(xi + 1 − xi) =

n→∞

where x1 = a

and

1

f(x) dx b

a

xn = b

Examples of this procedure are given in the subsection “Numerical Analysis and Approximate Methods.”

INFINITE SERIES REFERENCES: 53, 126, 127, 163. For asymptotic series and asymptotic methods, see Refs. 51, 127.

DEFINITIONS

while valid for any finite value of r and n now takes on a different interpretation. In this sense it is necessary to consider the limit of Sn as n increases indefinitely: S = lim Sn

A succession of numbers or terms that are formed according to some definite rule is called a sequence. The indicated sum of the terms of a sequence is called a series. A series of the form a0 + a1(x − c) + a2(x − c)2 + ⋅⋅⋅ + an(x − c)n + ⋅⋅⋅ is called a power series. Consider the sum of a finite number of terms in the geometric series (a special case of a power series). Sn = a + ar + ar 2 + ar 3 + ⋅⋅⋅ + ar n − 1

(3-67)

For any number of terms n, the sum equals 1 − rn Sn = a
1−r In this form, the geometric series is assumed finite. In the form of Eq. (3-67), it can further be defined that the terms in the series be nonending and therefore an infinite series. S = a + ar + ar 2 + ⋅⋅⋅ + ar n + ⋅⋅⋅ However, the defined sum of the terms [Eq. (3-67)] 1 − rn Sn = a
r≠1 1−r

(3-68)

n→∞

1 − rn = a lim
n→∞ 1 − r For this, it is stated the infinite series converges if the limit of Sn approaches a fixed finite value as n approaches infinity. Otherwise, the series is divergent. On this basis an analysis of 1 − rn S = a lim
n→∞ 1 − r shows that if r is less than 1 but greater than −1, the infinite series is convergent. For values outside of the range −1 < r < 1, the series is divergent because the sum is not defined. The range −1 < r < 1 is called the region of convergence. (We assume a ≠ 0.) Consider the divergence of Eq. (3-68) when r = −1 and +1. For the former case r = −1, S = a + a(−1) + a(−1)2 + a(−1)3 + ⋅⋅⋅ + a(−1)n + ⋅⋅⋅ = a − a + a − a + a − ⋅⋅⋅

INFINITE SERIES and for which 1 − rn S = a lim
n→∞ 1 − r 1 − (−1)n = a lim
undefined limit (if a ≠ 0) n→∞ 1+1 Since the limit sum does not exist, the series is divergent. This is defined as a bounded or oscillating divergent series. Similarly for the value r = +1, S = a + a(1) + a(1)2 + a(1)3 + ⋅⋅⋅ + a(1)n + ⋅⋅⋅ S = a + a + a + a + ⋅⋅⋅ + a + ⋅⋅⋅

The series is also divergent but defined as an unbounded divergent series. There are also two types of convergent series. Consider the new series 1 1 1 1 S = 1 −
+
−
+ ⋅⋅⋅ + (−1)n + 1
+ ⋅⋅⋅ (3-69) 2 3 4 n It can be shown that the series (3-69) does converge to the value S = log 2. However, if each term is replaced by its absolute value, the series becomes unbounded and therefore divergent (unbounded divergent): 1 1 1 1 S = 1 +
+
+
+
+ ⋅⋅⋅ (3-70) 2 3 4 5 In this case the series (3-69) is defined as a conditionally convergent series. If the replacement series of absolute values also converges, the series is defined to converge absolutely. Series (3-69) is further defined as an alternating series, while series (3-70) is referred to as a positive series. OPERATIONS WITH INFINITE SERIES 1. The convergence or divergence of an infinite series is unaffected by the removal of a finite number of finite terms. This is a trivial theorem but useful to remember, especially when using the comparison test to be described in the subsection “Tests for Convergence and Divergence.” 2. If a series is conditionally convergent, its sums can be made to have any arbitrary value by a suitable rearrangement of the series; it can in fact be made divergent or oscillatory (Riemann’s theorem). This seemingly paradoxical theorem can be illustrated by the following example. 1 1 1 1 1 2 3 4 5 6 The series is rearranged so that each positive term is followed by two negative terms: 1 1 1 1 1 1 1 1 t = 1 −
−
+
−
−
+
−
−
+ ⋅⋅⋅ 2 4 3 6 8 5 10 12 Define t3n for the first 3n terms in the series

Example S = 1 −
+
−
+
−
+ ⋅⋅⋅













1 1 1 1 1 1 1 1 t3n = 1 −
−
+
−
−
+ ⋅⋅⋅ +
−
−
2 4 3 6 8 2n − 1 4n − 2 4n 1 1 1 1 1 1 =
−
+
−
+ ⋅⋅⋅ +
−
2 4 6 8 4n − 2 4n



1 1 1 1 1 1 =
1 −
+
−
+ ⋅⋅⋅ +
−
2 2 3 4 2n − 1 2n

3. A series of positive terms, if convergent, has a sum independent of the order of its terms; but if divergent, it remains divergent however its terms are rearranged. 4. An oscillatory series can always be made to converge by grouping the terms in brackets. Example Consider the series 1 2 3 4 5 1 −
+
−
+
−
+ ⋅⋅⋅ 2 3 4 5 6 which oscillates between the values 0.306 and 1.306. However, the series 1 2 3 4 5 1 1 1 1 1 −
+
−
+
−
+ ⋅⋅⋅ =
−
−
−
− ⋅⋅⋅  0.306 ⋅⋅⋅ 2 3 4 5 6 2 12 30 56



(a ≠ 0)

3-31

 

 



and



 

 



1 1 2 3 4 5 6 1 1 1 −
−
−
−
−
−
+ ⋅⋅⋅ = 1 +
+
+
+ ⋅⋅⋅ = 1.306 ⋅⋅⋅ 2 3 4 5 6 7 6 20 42

5. A power series can be inverted, provided the first-degree term is not zero. Given y = b1 x + b2 x2 + b3 x3 + b4 x4 + b5 x5 + b6 x6 + b7 x7 + ⋅⋅⋅ then

x = B1 y + B2 y2 + B3 y3 + B4 y4 + B5 y5 + B6 y6 + B7 y7 + ⋅⋅⋅

where B1 = 1/b1 B2 = −b2 /b13 B3 = (1/b15 ) (2b22 − b1b3 ) B4 = (1/b17 )(5b1b2 b3 − b12 b4 − 5b23 ) Additional coefficients are available in the references. 6. Two series may be added or subtracted term by term provided each is a convergent series. The joint sum is equal to the sum (or difference) of the individuals. 7. The sum of two divergent series can be convergent. Similarly, the sum of a convergent series and a divergent series must be divergent. Example Given 1+n

∞

=
+
+
+
+ ⋅⋅⋅  
n  1 4 9 16

n=1

2

3

1−n

∞

5

4

(a divergent series)

2

= −
−
−
+ ⋅⋅⋅  
n  4 9 16

n=1

1

2

3

(a divergent series)

2

1+n

1−n

1+n+1−n

+  
 =  

  
n  n n

However,

2

2

2

1 = 2 
2 n

(convergent)

8. A power series may be integrated term by term to represent the integral of the function within an interval of the region of convergence. If f(x) = a0 + a1x + a2x2 + ⋅⋅⋅, then



x2

x1

f(x) dx =



x2

x1

a0 dx +



x2

x1

a1x dx +



x2

a2 x2 dx + ⋅⋅⋅

x1

9. A power series may be differentiated term by term and represents the function df(x)/dx within the same region of convergence as f(x). TESTS FOR CONVERGENCE AND DIVERGENCE



1 =
S2n 2 where S2n is the sum of the first 2n terms of the original series. Thus 1 lim t3n = lim
S2n n→∞ n→∞ 2 1 t=
S 2 and since lim t3n + 2 = lim t3n + 1 = lim t3n, it follows the sum of the series t is (a) S. Hence a rearrangement of the terms of an alternating series alters the sum of the series.

In general, the problem of determining whether a given series will converge or not can require a great deal of ingenuity and resourcefulness. There is no all-inclusive test which can be applied to all series. As the only alternative, it is necessary to apply one or more of the developed theorems in an attempt to ascertain the convergence or divergence of the series under study. The following defined tests are given in relative order of effectiveness. For examples, see references on advanced calculus. 1. Comparison Test. A series will converge if the absolute value of each term (with or without a finite number of terms) is less than the corresponding term of a known convergent series. Similarly, a positive series is divergent if it is termwise larger than a known divergent series of positive terms.

3-32

MATHEMATICS

2. nth-Term Test. A series is divergent if the nth term of the series does not approach zero as n becomes increasingly large. 3. Ratio Test. If the absolute ratio of the (n + 1) term divided by the nth term as n becomes unbounded approaches a. A number less than 1, the series is absolutely convergent b. A number greater than 1, the series is divergent c. A number equal to 1, the test is inconclusive 4. Alternating-Series Leibniz Test. If the terms of a series are alternately positive and negative and never increase in value, the absolute series will converge, provided that the terms tend to zero as a limit. 5. Cauchy’s Root Test. If the nth root of the absolute value of the nth term, as n becomes unbounded, approaches a. A number less than 1, the series is absolutely convergent b. A number greater than 1, the series is divergent c. A number equal to 1, the test is inconclusive 6. Maclaurin’s Integral Test. Suppose  an is a series of positive terms and f is a continuous decreasing function such that f(x) ≥ 0 for 1∞ ≤ x < ∞ and f(n) = an. Then the series and the improper integral ∫ 1 f(x) dx either both converge or both diverge. SERIES SUMMATION AND IDENTITIES Sums for the First n Numbers to Integer Powers n



j=1 n



j=1 n



j=1 n



j=1

n(n + 1) j =
= 1 + 2 + 3 + 4 + ⋅⋅⋅ + n 2 n(n + 1)(2n + 1) j 2 =

= 12 + 22 + 32 + 42 + ⋅⋅⋅ + n2 6

Taylor’s Series x2 x3 f(x + h) = f(h) + xf ′(h) +
f ″(h) +
f ′′′(h) + ⋅⋅⋅ 2! 3! f ″(x0) f ′′′(x0) or f(x) = f(x0) + f ′(x0) (x − x0) +
(x − x0)2 +
(x − x0)3 + ⋅⋅⋅ 2! 3! Example Find a series expansion for f(x) = ln (1 + x) about x0 = 0. f′(x) = (1 + x)−1, thus

f(0) = 0,

f″(x) = −(1 + x)−2,

f′(0) = 1,

f ″(0) = −1,

which converges for −1 < x ≤ 1.

Maclaurin’s Series x2 x3 f(x) = f(0) + xf ′(0) +
f ″(0) +
f ′′′(0) + ⋅⋅⋅ 2! 3! This is simply a special case of Taylor’s series when h is set to zero. Exponential Series x2 x3 xn ex = 1 + x +
+
+ ⋅⋅⋅ +
+ ⋅⋅⋅ − ∞ < x < ∞ 2! 3! n! Logarithmic Series

n2(n + 1)2 j =

= 13 + 23 + 33 + ⋅⋅⋅ + n3 4

x−1 1 x−1 ln x =
+

x 2 x

n(n + 1)(2n + 1)(3n2 + 3n − 1) j 4 =



= 14 + 24 + 34 + ⋅⋅⋅ + n4 30

ln x = 2

n

 [a + (k − 1)d] = a + (a + d ) + (a + 2d )

+ (a + 3d ) + ⋅⋅⋅ + [a + (n − 1)]d

k=1



 ar

j−1

3

1

x−1 +

+ ⋅⋅⋅


x + 1 3 x + 1 3

1

(x > a)

(x > 0)

−∞ < x < ∞ −∞ < x < ∞

x3 1 3 x5 1 3 5 x7 sin−1 x = x +
+
⋅
⋅
+
⋅
⋅
⋅
+ ⋅⋅⋅ 6 2 4 5 2 4 6 7

= a + ar + ar 2 + ar 3 + ⋅⋅⋅ + ar n − 1 1 − rn =a
1−r

2

x2 x4 x 6 cos x = 1 −
+
−
+ ⋅⋅⋅ 2! 4! 6!

Geometric Progression j=1

x−1

x−1 + ⋅⋅⋅  +
3 
x 

Trigonometric Series* x3 x5 x7 sin x = x −
+
−
+ ⋅⋅⋅ 3! 5! 7!

1 = na +
n(n − 1)d 2

n

f ′′′(1) = 2, etc.

x2 x3 x4 xn ln (x + 1) = x −
+
−
+ ⋅⋅⋅ + (−1)n + 1
+ ⋅⋅⋅ 2 3 4 n

3

Arithmetic Progression

f′′′(x) = 2(1 + x)−3, etc.

r≠1

1 1 1 tan−1 x = x −
x3 +
x5 −
x7 + ⋅⋅⋅ 3 5 7

∂f f(x, y) = f(x0, y0) +
∂x

n

1 1 1 1 1 1 1 +
+
+
+ ⋅⋅⋅ +

=
a +
a + d a + 2d a + 3d a + 4d a + nd k = 0 a + kd

1 ∂2f +

2 2! ∂x



The reciprocals of the terms of the arithmetic-progression series are called harmonic progression. No general summation formulas are available for this series. n(n − 1) (x + y)n = xn + nx n − 1y +
x n − 2 y2 2! n(n − 1)(n − 2) n − 3 3 n! +

x y + ⋅⋅⋅ +
x n − ryr + ⋅⋅⋅ + yn 3! (n − r)!r! n(n − 1) n(n − 1)(n − 2) (1  x)n = 1  nx +
x2 

x3 + ⋅⋅⋅ (x2 < 1) 2! 3!

(x2 < 1)

Taylor Series The Taylor series for a function of two variables, expanded about the point (x0, y0), is

Harmonic Progression

Binomial Series

(x2 < 1)



x 0, y0

∂f (x − x0) +
∂y

∂2f (x − x0)2 + 2
x 0, y0 ∂x∂y





(y − y0) x 0, y0

(x − x0)(y − y0) x 0 , y0

∂2f +
2 ∂y





(y − y0)2 + ⋅⋅⋅ x 0, y0

Partial Sums of Infinite Series, and How They Grow Calculus textbooks devote much space to tests for convergence and divergence of series that are of little practical value, since a convergent

* tan x series has awkward coefficients and should be computed as sin x (sign)

. 2  −sin x
1





COMPLEX VARIABLES series either converges rapidly, in which case almost any test (among those presented in the preceding subsections) will do; or it converges slowly, in which case it is not going to be of much use unless there is

3-33

some way to get at its sum without adding up an unreasonable number of terms. To find out, as accurately as possible, how fast a convergent series converges and how fast a divergent series diverges, see Ref. 34.

COMPLEX VARIABLES REFERENCES: General. 73, 163, 172, 179. Applied and computational complex analysis. 141, 146, 179.

Numbers of the form z = x + iy, where x and y are real, i = −1, are called complex numbers. The numbers z = x + iy are representable in the plane as shown in Fig. 3-46. The following definitions and terminology are used: 1. Distance OP = r = modulus of z written |z|. |z| =
 x2 + y2. 2. x is the real part of z. 3. y is the imaginary part of z. 4. The angle θ, 0 ≤ θ < 2π, measured counterclockwise from the positive x axis to OP is the argument of z. θ = arctan y/x = arcsin y/r = arccos x/r if x ≠ 0, θ = π/2 if x = 0 and y > 0. 5. The numbers r, θ are the polar coordinates of z. 6. z = x − iy is the complex conjugate of z. 2

ALGEBRA Let z1 = x1 + iy1, z2 = x2 + iy2. Equality z1 = z2 if and only if x1 = x2 and y1 = y2. Addition z1 + z2 = (x1 + x2) + i(y1 + y2). Subtraction z1 − z2 = (x1 − x2) + i(y1 − y2). Multiplication z1 ⋅ z2 = (x1x2 − y1y2) + i(x1y2 + x2y1). x2y1 − x1y2 x1x2 + y1y2 Division z1 /z2 =

+ i

, z2 ≠ 0. x 22 + y22 x 22 + y22 SPECIAL OPERATIONS zz = x2 + y2 = |z|2; z1z2 = z1  z2; z1 = z1; z1z2 = z1z2; |z1 ⋅ z2| = |z1| ⋅ |z2|; arg (z1 ⋅ z2) = arg z1 + arg z2; arg (z1 /z2) = arg z1 − arg z2; i4n = 1 for n any integer; i2n = −1 where n is any odd integer; z + z = 2x; z − z = 2iy. Every complex quantity can be expressed in the form x + iy. TRIGONOMETRIC REPRESENTATION By referring to Fig. 3-46, there results x = r cos θ, y = r sin θ so that z = x + iy = r (cos θ + i sin θ), which is called the polar form of the complex number. cos θ + i sin θ = e iθ. Hence z = x + iy = re iθ. z = x − iy = re−iθ. Two important results from this are cos θ = (eiθ + e−iθ)/2 and sin θ = (eiθ − e−iθ)/2i. Let z1 = r1e iθ1, z2 = r2e iθ2. This form is convenient for multiplication for z1z2 = r1 r2 e i(θ1 + θ2) and for division for z1 /z2 = (r1 /r2)ei(θ1 − θ2), z2 ≠ 0. POWERS AND ROOTS If n is a positive integer, z = (re ) = r e If n is a positive integer, n

FIG. 3-46

Complex plane.

iθ n

n inθ

= r (cos nθ + i sin nθ). n

θ + 2kπ θ + 2kπ z1/ n = r 1/ nei[(θ + 2kπ)/n] = r 1/n cos
+ i sin
n n









and selecting values of k = 0, 1, 2, 3, . . . , n − 1 give the n distinct values of z1/n. The n roots of a complex quantity are uniformly spaced around a circle, with radius r 1/n, in the complex plane in a symmetric fashion. Example Find the three cube roots of −8. Here r = 8, θ = π. The roots are z0 = 2(cos π/3 + i sin π/3) = 1 + i
3, z1 = 2(cos π + i sin π) = −2, z2 = 2(cos 5π/3 + i sin 5π/3) = 1 − i
3. ELEMENTARY COMPLEX FUNCTIONS Polynomials A polynomial in z, anzn + an − 1zn − 1 + ⋅⋅⋅ + a0, where n is a positive integer, is simply a sum of complex numbers times integral powers of z which have already been defined. Every polynomial of degree n has precisely n complex roots provided each multiple root of multiplicity m is counted m times. Exponential Functions The exponential function ez is defined by the equation ez = ex + iy = ex ⋅ eiy = ez(cos y + i sin y). Properties: e0 = 1; ez1 ⋅ ez2 = ez1 + z2; ez1/ez2 = ez1 − z2; ez + 2kπi = ez. Trigonometric Functions sin z = (eiz − e−iz)/2i; cos z = (eiz + e−iz)/2; tan z = sin z/cos z; cot z = cos z/sin z; sec z = 1/cos z; csc z = 1/sin z. Fundamental identities for these functions are the same as their real counterparts. Thus cos2 z + sin2 z = 1, cos (z1  z2) = cos z1 cos z2  sin z1 sin z2, sin (z1  z2) = sin z1 cos z2  cos z1 sin z2. The sine and cosine of z are periodic functions of period 2π; thus sin (z + 2π) = sin z. For computation purposes sin z = sin (x + iy) = sin x cosh y + i cos x sinh y, where sin x, cosh y, etc., are the real trigonometric and hyperbolic functions. Similarly, cos z = cos x cosh y − i sin x sinh y. If x = 0 in the results given, cos iy = cosh y, sin iy = i sinh y. Example Find all solutions of sin z = 3. From previous data sin z = sin x cosh y + i cos x sinh y = 3. Equating real and imaginary parts sin x cosh y = 3, cos x sinh y = 0. The second equation can hold for y = 0 or for x = π/2, 3π/2, . . . . If y = 0, cosh 0 = 1 and sin x = 3 is impossible for real x. Therefore, x = π/2, 3π/2, . . . (2n + 1)π/2, n = 0, 1, 2, . . . . However, sin 3π/2 = −1 and cosh y ≥ 1. Hence x = π/2, 5π/2, . . . . The solution is z = [(4n + 1)π]/2 + i cosh−13, n = 0, 1, 2, 3, . . . . Example Find all solutions of ez = −i. ez = ex(cos y + i sin y) = −i. Equating real and imaginary parts gives ex cos y = 0, ex sin y = −1. From the first y = π/2, 3π/2, . . . . But ex > 0. Therefore, y = 3π/2, 7π/2, −π/2, . . . . Then x = 0. The solution is z = i[(4n + 3)π]/2. Two important facets of these functions should be recognized. First, the sin z is unbounded; and, second, ez takes all complex values except 0. Hyperbolic Functions sinh z = (ez − e−z)/2; cosh z = (ez + e−z)/2; tanh z = sinh z/cosh z; coth z = cosh z/sinh z; csch z = 1/sinh z; sech z = 1/cosh z. Identities are: cosh2 z − sinh2 z = 1; sinh (z1 + z2) = sinh z1 cosh z2 + cosh z1 sinh z2; cosh (z1 + z2) = cosh z1 cosh z2 + sinh z1 sinh z2; cosh z + sinh z = ez; cosh z − sinh z = e−z. The hyperbolic sine and hyperbolic cosine are periodic functions with the imaginary period 2πi. That is, sinh (z + 2πi) = sinh z. Logarithms The logarithm of z, log z = log |z| + i(θ + 2nπ), where log |z| is taken to the base e and θ is the principal argument of z, that is, the particular argument lying in the interval 0 ≤ θ < 2π. The logarithm of z is infinitely many valued. If n = 0, the resulting logarithm is called the principal value. The familiar laws log z1 z2 = log z1 + log z2, log z1 /z2 = log z1 − log z2, log zn = n log z hold for the principal value.

3-34

MATHEMATICS

 π4



Example log (1 + i) = log
2 + i
+ 2nπ . General powers of z are defined by zα = eα log z. Since log z is infinitely many valued, so too is zα unless α is a rational number. DeMoivre’s formula can be derived from properties of ez. zn = rn (cos θ + i sin θ)n = rn (cos nθ + i sin nθ) (cos θ + i sin θ)n = cos nθ + i sin nθ

Thus

Example ii = eilog i = ei[log|i| + i(π/2 + 2nπ)] = e−(π/2 + 2nπ). Thus i i is real with principal value (n = 0) = e−π/2. Example (
2)1 + i = e(1 + i) log
2 = elog
2. ei log
2 =
2. (cos log
2 + i sin log
2) =
2[cos (0.3466) + i sin (0.3466)]. Inverse Trigonmetric Functions cos−1 z = −i log (z 
 z2 −1); i + z i −1 2 −1 sin z = −i log (iz 
1 −z); tan z =
log
. These functions 2 i−z are infinitely many valued. Inverse Hyperbolic Functions cosh−1 z = log (z 
 z2−1); 1+z 1 z2 +1); tanh−1 z =
log
. sinh−1 z = log (z 
 2 1−z

  



COMPLEX FUNCTIONS (ANALYTIC) In the real-number system a greater than b(a > b) and b less than c(b < c) define an order relation. These relations have no meaning for complex numbers. The absolute value is used for ordering. Some important relations follow: |z| ≥ x; |z| ≥ y; |z1  z2| ≤ |z1| + |z2|; |z1 − z2| ≥ ||z1| − |z2||; |z| ≥ (|x| + |y|)/
2. Parts of the complex plane, commonly called regions or domains, are described by using inequalities.  Example |z − 3| ≤ 5. This is equivalent to
(x −3 )2 +y2 ≤ 5, which is the set of all points within and on the circle, centered at x = 3, y = 0 of radius 5.

Example |z − 1| ≤ x represents the set of all points inside and on the parabola 2x = y2 + 1 or, equivalently, 2x ≥ y2 + 1. Functions of a Complex Variable If z = x + iy, w = u + iv and if for each value of z in some region of the complex plane one or more values of w are defined, then w is said to be a function of z, w = f(z). Some of these functions have already been discussed, e.g., sin z, log z. All functions are reducible to the form w = u(x, y) + iv(x, y), where u, v are real functions of the real variables x and y.

∂v/∂x = −∂u/∂y. The last two equations are called the CauchyRiemann equations. The derivative dw ∂u ∂v ∂v ∂u
=
+i
=
−i
dz ∂x ∂x ∂y ∂y If f(z) possesses a derivative at zo and at every point in some neighborhood of z0, then f(z) is said to be analytic at z0. If the Cauchy-Riemann equations are satisfied and ∂u ∂u ∂v ∂v u, v,
,
,
,
∂x ∂y ∂x ∂y are continuous in a region of the complex plane, then f(z) is analytic in that region. Example w = zz = x2 + y2. Here u = x2 + y2, v = 0. ∂u/∂x = 2x, ∂u/∂y = 2y, ∂v/∂x = ∂v/∂y = 0. These are continuous everywhere, but the Cauchy-Riemann equations hold only at the origin. Therefore, w is nowhere analytic, but it is differentiable at z = 0 only. Example w = ez = ex cos y + iex sin y. u = ex cos y, v = ex sin y. ∂u/∂x = ex cos y, ∂u/∂y = −ex sin y, ∂v/∂x = ex sin y, ∂v/∂y = ex cos y. The continuity and CauchyRiemann requirements are satisfied for all finite z. Hence ez is analytic (except at ∞) and dw/∂z = ∂u/∂x + i(∂v/∂x) = ez. 1 z

x − iy x +y

x x +y

y x +y

Example w =
=
=
−i
2 2 2 2 2 2 It is easy to see that dw/dz exists except at z = 0. Thus 1/z is analytic except at z = 0.

Singular Points If f(z) is analytic in a region except at certain points, those points are called singular points. Example 1/z has a singular point at zero. Example tan z has singular points at z = (2n + 1)(π/2), n = 0, 1, 2, . . . . The derivatives of the common functions, given earlier, are the same as their real counterparts. Example (d/dz)(log z) = 1/z, (d/dz)(sin z) = cos z. Harmonic Functions Both the real and the imaginary parts of any analytic function f = u + iv satisfy Laplace’s equation ∂2φ/∂x2 + ∂2φ/∂y2 = 0. A function which possesses continuous second partial derivatives and satisfies Laplace’s equation is called a harmonic function.

Example z3 = (x + iy)3 = x3 + 3x2(iy) + 3x(iy)2 + (iy)3 = (x3 − 3xy2) +

Example ez = ex cos y + iex sin y. u = ex cos y, ∂u/∂x = ex cos y, ∂2u/∂x2 = ex cos y, ∂u/∂y = −ex sin y, ∂2u/∂y2 = −ex cos y. Clearly ∂2u/∂x2 + ∂2u/∂y2 = 0. Similarly, v = ex sin y is also harmonic.

Example cos z = cos x cosh y − i sin x sinh y.

If w = u + iv is analytic, the curves u(x, y) = c and v(x, y) = k intersect at right angles, if wi(z) ≠ 0.

i(3x2y − y3).

Differentiation The derivative of w = f(z) is dw f(z + ∆z) − f(z) lim


= ∆z→0 dz ∆z and for the derivative to exist the limit must be the same no matter how ∆z approaches zero. If w1, w2 are differentiable functions of z, the following rules apply: dw2 d(w1w2) dw1 dw2 d(w1  w2) dw1
= w2
+ w1


=

dz dz dz dz dz dz d(w1/w2) w2(dw1/dz) − w1(dw2/dz)
=


dz w22 dw1n dw1
= nw1n − 1
dz dz For w = f(z) to be differentiable, it is necessary that ∂u/∂x = ∂v/∂y and and

Example z3 = (x3 − 3xy2) + i(3x2y − y3). Set u = x3 − 3xy2 = c, v = 3x2y − y3 = k. By implicit differentiation there results, respectively, dy/dx = (x2 − y2)/2xy, dy/dx = 2xy/(y2 − x2), which are clearly negative reciprocals, the condition for perpendicularity. Integration In much of the work with complex variables a simple extension of integration called line or curvilinear integration is of fundamental importance. Since any complex line integral can be expressed in terms of real line integrals, we define only real line integrals. Let F(x,y) be a real, continuous function of x and y and c be any continuous curve of finite length joining the points A and B (Fig. 3-47). F(x,y) is not related to the curve c. Divide c up into n segments, ∆si, whose projection on the x axis is ∆xi and on the y axis is ∆yi. Let (εi, ηi) be the coordinates of an arbitrary point on ∆si. The limits of the sums

 F(ε , η ) ∆s = F(x, y) ds n

lim

∆si→0

i

i=1

i

i

c

DIFFERENTIAL EQUATIONS

Conformal transformation.

FIG. 3-48

Line integral.

FIG. 3-47

grals are the same as those for ordinary integrals. That is, ∫c [ f(z)  g(z)] dz = ∫c f(z) dz  ∫c g(z) dz; ∫c kf(z) dz = k ∫c f(z) dz for any constant k, etc.

 F(ε , η ) ∆x = F(x, y) dx n

lim

∆si→0

i

i

i

c

i=1

 F(ε , η ) ∆ y = F(x, y) dy

lim

∆si→0

i

i

c

2

are known as line integrals. Much of the initial strangeness of these integrals will vanish if it be observed that the ordinary definite integral b ∫a f(x) dx is just a line integral in which the curve c is a line segment on the x axis and F(x, y) is a function of x alone. The evaluation of line integrals can be reduced to evaluation of ordinary integrals. Example ∫c y(1 + x) dy, where c: y = 1 − x21 from (−1, 0) to (1, 0). Clearly y = 1 − x2, dy = −2x dx. Thus ∫c y(1 + x) dy = −2 ∫ −1 (1 − x2)(1 + x)x dx = −8⁄15. Example ∫c x2y ds, c is the square whose vertices are (0, 0), (1, 0), (1, 1),  x2 + dy2. When dx = 0, ds = dy. From (0, 0) to (1, 0), y = 0, dy = (0, 1). ds =
d 0. Similar arguments for the other sides give

x y ds = 0.x dx + y dy + x dx + 0.y dy = a − s = ⁄ 1

2

1

0

2

0

0

2

0

16

1

2

c

i

i=1

(x + iy) dz along c: y = x, 0 to 1 + i. This becomes (x + iy) dz = (x dx − y dy) + i (y dx + x dy) = x dx − x dx + i x dx + i x dx = − ⁄

Example

n

c

3-35

1

Let f(z) be any function of z, analytic or not, and c any curve as above. The complex integral is calculated as ∫c f(z) dz = ∫c (u dx − v dy) + i ∫c (v dx + u dy), where f(z) = u(x, y) + iv(x, y). Properties of line inte-

2

c

c

1

2

c

1

1

1

2

0

2

0

0

16

+ 5i/6

0

Conformal Mapping Every function of a complex variable w = f(z) = u(x, y) + iv(x, y) transforms the x, y plane into the u, v plane in some manner. A conformal transformation is one in which angles between curves are preserved in magnitude and sense. Every analytic function, except at those points where f ′(z) = 0, is a conformal transformation. See Fig. 3-48. Example w = z2. u + iv = (x2 − y2) + 2ixy or u = x2 − y2, v = 2xy. These are the transformation equations between the (x, y) and (u, v) planes. Lines parallel to the x axis, y = c1 map into curves in the u, v plane with parametric equations u = x2 − c12, v = 2c1x. Eliminating x, u = (v2/4c12) − c12, which represents a family of parabolas with the origin of the w plane as focus, the line v = 0 as axis and opening to the right. Similar arguments apply to x = c2. The principles of complex variables are useful in the solution of a variety of applied problems. See the references for additional information.

DIFFERENTIAL EQUATIONS REFERENCES: Ordinary Differential Equations: Elementary level, 41, 44, 62, 81, 204, 236, 263. Intermediate level, 30, 43, 144. Theory and Advanced topics, 252. Applications, 9, 263. Partial Differential Equations: Elementary level and solution methods, 9, 41, 61, 72, 144, 156, 229. Theory and advanced level, 79, 220, 240. See also “Numerical Analysis and Approximate Methods” and “General References: References for General and Specific Topics—Advanced Engineering Mathematics” for additional references on topics in ordinary and partial differential equations.

The natural laws in any scientific or technological field are not regarded as precise and definitive until they have been expressed in mathematical form. Such a form, often an equation, is a relation between the quantity of interest, say, product yield, and independent variables such as time and temperature upon which yield depends. When it happens that this equation involves, besides the function itself, one or more of its derivatives it is called a differential equation. Example The homogeneous bimolecular reaction A + B → C is characterized by the differential equation dx/dt = k(a − x)(b − x), where a = initial concentration of A, b = initial concentration of B, and x = x(t) = concentration of C as a function of time t. k

Example The differential equation of heat conduction in a moving fluid with velocity components vx, vy is ∂u ∂u ∂u K ∂2u ∂2u
+ vx
+ vy
=

2 +
2 ∂t ∂x ∂y ρcp ∂x ∂y





where u = u(x, y, t) = temperature, K = thermal conductivity, ρ = density, and cp = specific heat at constant pressure.

ORDINARY DIFFERENTIAL EQUATIONS When the function involved in the equation depends upon only one variable, its derivatives are ordinary derivatives and the differential equation is called an ordinary differential equation. When the function depends upon several independent variables, then the equation is called a partial differential equation. The theories of ordinary and partial differential equations are quite different. In almost every respect the latter is more difficult. Whichever the type, a differential equation is said to be of nth order if it involves derivatives of order n but no higher. The equation in the first example is of first order and that in the second example of second order. The degree of a differential equation is the power to which the derivative of the highest order is raised after the equation has been cleared of fractions and radicals in the dependent variable and its derivatives. A relation between the variables, involving no derivatives, is called a solution of the differential equation if this relation, when substituted in the equation, satisfies the equation. A solution of an ordinary differential equation which includes the maximum possible number of “arbitrary” constants is called the general solution. The maximum number of “arbitrary” constants is exactly equal to the order of the dif-

3-36

MATHEMATICS

ferential equation. If any set of specific values of the constants is chosen, the result is called a particular solution.

are distinct real roots, r1 and r2, say, the solution is y = Aer1x + Ber2x, where A and B are arbitrary constants.

Example The general solution of (d2x/dt2) + k2x = 0 is x = A cos kt + B sin kt, where A, B are arbitrary constants. A particular solution is x = a cos kt + 3 sin kt.

Example y″ + 4y′ + 3 = 0. The characteristic equation is m2 + 4m + 3 = 0. The roots are −3 and −1, and the general solution is y = Ae−3x + Be−x.

In the case of some equations still other solutions exist called singular solutions. A singular solution is any solution of the differential equation which is not included in the general solution. Example y = x(dy/dx) − d(dy/dx) has the general solution y = cx − dc , where c is an arbitrary constant; y = x2 is a singular solution, as is easily verified. 2

2

ORDINARY DIFFERENTIAL EQUATIONS OF THE FIRST ORDER Equations with Separable Variables Every differential equation of the first order and of the first degree can be written in the form M(x, y) dx + N(x, y) dy = 0. If the equation can be transformed so that M does not involve y and N does not involve x, then the variables are said to be separated. The solution can then be obtained by quadrature, which means that y = ∫ f(x) dx + c, which may or may not be expressible in simpler form. Example Two liquids A and B are boiling together in a vessel. Experimentally it is found that the ratio of the rates at which A and B are evaporating at any time is proportional to the ratio of the amount of A (say, x) to the amount of B (say, y) still in the liquid state. This physical law is expressible as (dy/dt)/(dx/dt) = ky/x or dy/dx = ky/x, where k is a proportionality constant. This equation may be written dy/y = k(dx/x), in which the variables are separated. The solution is ln y = k ln x + ln c or y = cxk. Exact Equations The equation M(x, y) dx + N(x, y) dy = 0 is exact if and only if ∂M/∂y = ∂N/∂x. In this case there exists a function w = f(x, y) such that ∂f /∂x = M, ∂f /∂y = N, and f(x, y) = C is the required solution. f(x, y) is found as follows: treat y as though it were constant and evaluate ∫ M(x, y) dx. Then treat x as though it were constant and evaluate ∫ N(x, y) dy. The sum of all unlike terms in these two integrals (including no repetitions) is f(x, y). Example (2xy − cos x) dx + (x2 − 1) dy = 0 is exact for ∂M/∂y = 2x, ∂N/∂x = 2x. ∫ M dx = ∫ (2xy − cos x) dx = x2y − sin x, ∫ N dy = ∫ (x2 − 1) dy = x2y − y. The solution is x2y − sin x − y = C, as may easily be verified. Linear Equations A differential equation is said to be linear when it is of first degree in the dependent variable and its derivatives. The general linear first-order differential equation has the form dy/dx + P(x)y = Q(x). Its general solution is y = e−∫ P dx

Qe

∫ P dx

dx + C



Example A tank initially holds 200 gal of a salt solution in which 100 lb is dissolved. Six gallons of brine containing 4 lb of salt run into the tank per minute. If mixing is perfect and the output rate is 4 gal/min, what is the amount A of salt in the tank at time t? The differential equation of A is dA/dt + [1/(100 + t)]A = 4. Its general solution is A = 2(100 + t) + C/(100 + t). At t = 0, ′A = 100; so the particular solution is A = 2(100 + t) − 104/(100 + t). ORDINARY DIFFERENTIAL EQUATIONS OF HIGHER ORDER The higher-order differential equations, especially those of order 2, are of great importance because of physical situations describable by them. Equation y(n) = f(x) Such a differential equation can be solved by n integrations. The solution will contain n arbitrary constants. Linear Differential Equations with Constant Coefficients and Right-Hand Member Zero (Homogeneous) The solution of y″ + ay′ + by = 0 depends upon the nature of the roots of the characteristic equation m2 + am + b = 0 obtained by substituting the trial solution y = emx in the equation. Distinct Real Roots If the roots of the characteristic equation

Multiple Real Roots If r1 = r2, the solution of the differential equation is y = e r1x(A + Bx). Example y″ + 4y + 4 = 0. The characteristic equation is m2 + 4m + 4 = 0 with roots −2 and −2. The solution is y = e−2x(A + Bx). Complex Roots If the characteristic roots are p  iq, then the solution is y = epx(A cos qx + B sin qx). Example The differential equation My″ + Ay′ + ky = 0 represents the vibration of a linear system of mass M, spring constant k, and damping constant , the roots of the characteristic equation A. If A < 2
kM A Mm2 + Am + k = 0 are complex −
 i 2M and the solution is y = e−(At/2M)

k A − 

 M 2M 

2

A − 
t + c sin  − 
t

c cos  M 2M   M 2M   k

A

1

2

k

2

2

This solution is oscillatory, representing undercritical damping.

All these results generalize to homogeneous linear differential equations with constant coefficients of order higher than 2. These equations (especially of order 2) have been much used because of the ease of solution. Oscillations, electric circuits, diffusion processes, and heat-flow problems are a few examples for which such equations are useful. Second-Order Equations: Dependent Variable Missing Such an equation is of the form dy d 2y F x,
,
2 = 0 dx dx It can be reduced to a first-order equation by substituting p = dy/dx and dp/dx = d 2y/dx2. Second-Order Equations: Independent Variable Missing Such an equation is of the form dy d 2y F y,
,
2 = 0 dx dx









dy d 2y dp
= p,
2 = p
dx dx dy The result is a first-order equation in p, dp F y, p, p
= 0 dy Set





Example The capillary curve for one vertical plate is given by


dx  

d 2y 4y 1+
2 =
dx c2 Its solution by this technique is c 2 c2 −y2 −
 c2 − h x +
 0 =
2 where c, h0 are physical constants.

dy

2 3/2

cosh

−1

c c
− cosh−1
y h0



Example The equation governing chemical reaction in a porous catalyst in plane geometry of thickness L is dc d 2c D
2 = k f(c),
(0) = 0, c(L) = c0 dx dx where D is a diffusion coefficient, k is a reaction rate parameter, c is the concentration, k f(c) is the rate of reaction, and c0 is the concentration at the boundary. Making the substitution gives k dp p
=
f(c) dc D

DIFFERENTIAL EQUATIONS



c p2 k f(c) dc
=
2 D c(0) If the reaction is very fast, c(0) ≈ 0 and the average reaction rate is related to p(L). See Ref. 106. This variable is given by 1/2 2k c0 p(L) =
f(c) dc D 0 Thus, the average reaction rate can be calculated without solving the complete problem.

Integrating gives





Linear Nonhomogeneous Differential Equations Linear Differential Equations Right-Hand Member f(x) ≠ 0 Again the specific remarks for y″ + ay′ + by = f(x) apply to differential equations of similar type but higher order. We shall discuss two general methods. Method of Undetermined Coefficients Use of this method is limited to equations exhibiting both constant coefficients and particular forms of the function f(x). In most cases f(x) will be a sum or product of functions of the type constant, xn (n a positive integer), emx, cos kx, sin kx. When this is the case, the solution of the equation is y = H(x) + P(x), where H(x) is a solution of the homogeneous equations found by the method of the preceding subsection and P(x) is a particular integral found by using the following table subject to these conditions: (1) When f(x) consists of the sum of several terms, the appropriate form of P(x) is the sum of the particular integrals corresponding to these terms individually. (2) When a term in any of the trial integrals listed is already a part of the homogeneous solution, the indicated form of the particular integral is multiplied by x. Form of Particular Integral

If f(x) is

Then P(x) is

a (constant) axn aerx c cos kx d sin kx

A (constant) An xn + An − 1 xn − 1 + ⋅⋅⋅A1 x + A0 Berx

gxnerx cos kx hxnerx sin kx

A cos kx + B sin kx

(Anxn + ⋅⋅⋅ + A0)erx cos kx + (Bn xn + ⋅⋅⋅ + B0)erx sin kx

d 2y

1 dy

dx

x dx

Example (1 − x2)
2 −

= x. The homogeneous equation d 2y 1 dy (1 − x2)
2 −

= 0 x dx dx dp dx
=
p x(1 − x2) when we set dy/dx = p. Upon integrating twice, y = c1
 x2 − 1 + c2 is the homogeneous solution. Now assume that the particular solution has the form y = u
 x2 −1 + v. The equations for u and v become reduces to

u′ = du/dx =
 x2 −1 dv 2 v′ =
= 1 − x dx so that 1 x2 −1 − ln (x +
 x2 −1)] and v = x − x3/3. u =
[x
 2 The complete solution is x x3 1 x2 −1 + c2 +
−
−

 x2 −1 ln (x +
 x2 −1). y = c1
 2 6 2

Perturbation Methods If the ordinary differential equation has a parameter that is small and is not multiplying the highest derivative, perturbation methods can give solutions for small values of the parameter. Example Consider the differential equation for reaction and diffusion in a catalyst; the reaction is second order: c″ = ac2, c′(0) = 0, c(1) = 1. The solution is expanded in the following Taylor series in a. c(x, a) = c0(x) + ac1(x) + a2c2(x) + … The goal is to find equations governing the functions {ci(x)} and solve them. Substitution into the equations gives the following equations: c0″(x) + a c″1(x) + a2c″2(x) + … = a[c0(x) + ac1(x) + a2c2(x) + …]2 c′0(0) + ac′1(0) + a2c′2(0) + … = 0 c0(1) + ac1(1) + a2c2(1) + … = 1 Like terms in powers of a are collected to form the individual problems. c″0 = 0,

Since the form of the particular integral is known, the constants may be evaluated by substitution in the differential equation. Example y″ + 2y′ + y = 3e − cos x + x . The characteristic equation is (m + 1)2 = 0 so that the homogeneous solution is y = (c1 + c2x)e−x. To find a particular solution we use the trial solution from the table, y = a1e2x + a2 cos x + a3 sin x + a4x3 + a5x2 + a6x + a7. Substituting this in the differential equation collecting and equating like terms, there results a1 = s, a2 = 0, a3 = −a, a4 = 1, a5 = −6, a6 = 18, and a7 = −24. The solution is y = (c1 + c2x)e−x + se2x − a sin x + x3 − 6x2 + 18x − 24. 2x

3

Method of Variation of Parameters This method is applicable to any linear equation. The technique is developed for a second-order equation but immediately extends to higher order. Let the equation be y″ + a(x)y′ + b(x)y = R(x) and let the solution of the homogeneous equation, found by some method, be y = c1 f1(x) + c2 f2(x). It is now assumed that a particular integral of the differential equation is of the form P(x) = uf1 + vf2 where u, v are functions of x to be determined by two equations. One equation results from the requirement that uf1 + vf2 satisfy the differential equation, and the other is a degree of freedom open to the analyst. The best choice proves to be u′f1 + v′f2 = 0

and

u′f 1′ + v′f′2 = R(x)

f2 du Then u′ =
= −

R(x) dx f1 f 2′ − f2 f 1′ f1 dv v′ =
=

R(x) dx f1 f 2′ − f2 f 1′ and since f1, f2, and R are known u, v may be found by direct integration.

3-37

c″1 = c 02,

c′0(0) = 0, c′1(0) = 0,

c″2 = 2c0 c1,

c0(1) = 1 c1(1) = 0

c′2(0) = 0,

c2(1) = 0

The solution proceeds in turn. c0(x) = 1,

(x2 − 1) c1(x) =
, 2

5 − 6x2 + x4 c2(x) =

12

SPECIAL DIFFERENTIAL EQUATIONS (SEE REF. 1) Euler’s Equation The linear equation x ny(n) + a1x n − 1y(n − 1) + ⋅⋅⋅ + an − 1xy′ + any = R(x) can be reduced to a linear equation with constant coefficients by the change of variable x = et. To solve the homogeneous equation substitute y = x r into it, cancel the powers of x, which are the same for all terms, and solve the resulting polynomial for r. In case of multiple or complex roots there results the form y = x r(log x)r and y = x α[cos (β log x) + i sin (β log x)]. Example Solve x2y″ − 2y = 0. By setting y = xr, xr[r(r − 1) − 2] = 0. The roots of r 2 − r − 2 = 0 are r = 2, −1. The general solution is y = Ax2 + B/x. The equation (ax + b)ny(n) + a1(ax + b)n − 1y(n − 1) + ⋅⋅⋅ + an y = R(x) can be reduced to the Euler form by the substitution ax + b = z. It may be treated without change of variable, the homogeneous equation having solutions of the form y = (ax + b)r. Bessel’s Equation The linear equation x2(d 2y/dx2) + (1 − 2α) x(dy/dx) + [β 2 γ 2 x 2γ + (α2 − p2γ 2)]y = 0 is the general Bessel equation. By series methods, not to be discussed here, this equation can be shown to have the solution y = AxαJp(βxγ) + BxαJ−p(βxγ)

p not an integer or zero

y = AxαJp(βxγ) + BxαYp(βxγ)

p an integer

3-38 where

MATHEMATICS ∞

PARTIAL DIFFERENTIAL EQUATIONS

  

k!Γ(p + k + 1)

x Jp(x) =
2

p

The analysis of situations involving two or more independent variables frequently results in a partial differential equation.

k=0

 Γ(n) = x

x J−p(x) =
2

(−1)k(x/2)2k

–p

∞



k=0

(−1)k(x/2)2k

k!Γ(k + 1 − p)

∞

n − 1 −x

e dx

p not an integer

n>0

0

is the gamma function. For p an integer x p ∞ (−1)k(x/2)2k Jp(x) =


2 k = 0 k!(p + k)!



(Bessel function of the first kind of order p) [ Jp(x) cos (pπ) − J−p(x)] Yp(x) =


sin (pπ) (replace right-hand side by limiting value if P is an integer or zero). The series converge for all x. Much of the importance of Bessel’s equation and Bessel functions lies in the fact that the solutions of numerous linear differential equations can be expressed in terms of them. Example d 2y/dx2 + [9x − (63/4x2)]y = 0. In general form this is x2(d 2y/dx2) + (9x3 − 63⁄ 4)y = 0. Thus α = a, γ = 3⁄ 2; β = 2, p = 8⁄ 3. The solution is (since p integer) y = Ax1/2J8/3(2x3/2) + Bx1/2J–8/3(2x3/2). Tables are available for the evaluation of many of these functions. Example The heat flow through a wedge-shaped fin is characterized by the equation x2(d 2y/dx2) + x(dy/dx) − a2xy = 0, where y = T − Tair, α is a combination of physical constants, and x = distance from fin end. By comparing this with the standard equation, there results α = 0, p = 0, γ = a, β 2 = −4a2 or β = 2ai. The solution is y = AJ0(2ai
x) + BY0(2ai
x). Legendre’s Equation The Legendre equation (1 − x2)y″ − 2xy′ + n(n + 1)y = 0, n ≥ 0, has the solution y = Aun(x) + Bvn(x) for n not an integer where n(n + 1) n(n − 2)(n + 1)(n + 3) un(x) = 1 −
x2 +


x4 2! 4! n(n − 2)(n − 4)(n + 1)(n + 3)(n + 5) 6 −



x + ⋅⋅⋅ 6! (n − 1)(n + 2) 3 (n − 1)(n − 3)(n + 2)(n + 4) 5 vn(x) = x −

x +


x − ⋅⋅⋅⋅ 3! 5! If n is an even integer or zero, un is a polynomial in x. If n is an odd integer, then vn is a polynomial. The interval of convergence for the series is −1 < x < 1. If n is an integer, set un(x) vn(x) Pn(x) =
(n even or zero), Pn =
(n odd) un(1) vn(1) The polynomials Pn are the so-called Legendre polynomials, P0(x) = 1, P1(x) = x, P2(x) = a(3x2 − 1), P3(x) = a(5x3 − 3x), . . . . Laguerre’s Equation The Laguerre equation x(d 2y/dx2) + (c − x) (dy/dx) − ay = 0 is satisfied by the confluent hypergeometric function. See Refs. 1 and 173. Hermite’s Equation The Hermite equation y″ − 2xy′ + 2ny = 0 is satisfied by the Hermite polynomial of degree n, y = AHn(x) if n is a positive integer or zero. H0(x) = 1, H1(x) = 2x, H2(x) = 4x2 − 2, H3(x) = 8x3 − 12x, H4(x) = 16x4 − 48x2 + 12, Hr + 1(x) = 2xHr(x) − 2rHr − 1(x). Example y″ − 2xy′ + 6y = 0. Here n = 3; so y = AH3 = A(8x3 − 12x) is a solution.

Chebyshev’s Equation The equation (1 − x2)y″ − xy′ + n2y = 0 for n a positive integer or zero is satisfied by the nth Chebyshev polynomial y = ATn(x). T0(x) = 1, T1(x) = x, T2(x) = 2x2 − 1, T3(x) = 4x3 − 3x, T4(x) = 8x4 − 8x2 + 1; Tr + 1(x) = 2xTr(x) − Tr − 1(x). Example (1 − x2)y″ − xy′ + 36y = 0. Here n = 6. A solution is y = T6(x) = 2xT5(x) − T4(x) = 2x(2xT4 − T3) − T4 = 32x6 − 48x4 + 18x2 − 1. Further details on these special equations and others can be found in the literature.

Example The equation ∂T/∂t = K(∂2T/∂x2) represents the unsteady onedimensional conduction of heat. Example The equation for the unsteady transverse motion of a uniform beam clamped at the ends is ∂4y ρ ∂2y =0
4 +

∂x EI ∂t2 Example The expansion of a gas behind a piston is characterized by the simultaneous equations ∂u ∂u c2 ∂ρ ∂ρ ∂ρ ∂u
+ u
+

= 0 and
+ u
+ ρ
= 0 ∂t ∂x ρ ∂x ∂t ∂x ∂x Example The heating of a diathermanous solid is characterized by the equation α(∂2θ/∂x2) + βe−γz = ∂θ/∂t. The partial differential equation ∂2f/∂x ∂y = 0 can be solved by two integrations yielding the solution f = g(x) + h(y), where g(x) and h(y) are arbitrary differentiable functions. This result is an example of the fact that the general solution of partial differential equations involves arbitrary functions in contrast to the solution of ordinary differential equations, which involve only arbitrary constants. A number of methods are available for finding the general solution of a partial differential equation. In most applications of partial differential equations the general solution is of limited use. In such applications the solution of a partial differential equation must satisfy both the equation and certain auxiliary conditions called initial and/or boundary conditions, which are dictated by the problem. Examples of these include those in which the wall temperature is a fixed constant T(x0) = T0, there is no diffusion across a nonpermeable wall, and the like. In ordinary differential equations these auxiliary conditions allow definite numbers to be assigned to the constants of integration. In partial differential equations the boundary conditions demand that the arbitrary functions resulting from integration assume specific forms. Except for a few cases (some first-order equations, D’Alembert’s solution of the wave equation, and others) a procedure which first determines the arbitrary functions and then specializes them to fit the boundary conditions is usually not feasible. A more fruitful attack is to determine directly a set of particular solutions and then combine them so that the boundary conditions are satisfied. The only area in which much analysis has been accomplished is for linear homogeneous partial differential equations. Such equations have the property that if f1, f2, . . . , ∞ fn, . . . are individually solutions, then the function f =  i=1 fi is also a solution, provided the series converges and is differentiable up to the order (termwise) of the equation. Partial Differential Equations of Second and Higher Order Many of the applications to scientific problems fall naturally into partial differential equations of second order, although there are important exceptions in elasticity, vibration theory, and elsewhere. A second-order differential equation can be written as ∂2u ∂2u ∂2u a
2 + b
+ c
2 = f ∂x ∂x∂y ∂y where a, b, c, and f depend upon x, y, u, ∂u/∂x, and ∂u/∂y. This equation is hyperbolic, parabolic, or elliptic, depending on whether the discriminant b2 − 4ac is >0, =0, or <0, respectively. Since a, b, c, and f depend on the solution, the type of equation can be different at different x and y locations. If the equation is hyperbolic, discontinuities can be propagated. See Refs. 11, 79, 105, 159, and 192. Phenomena of propagation such as vibrations are characterized by equations of “hyperbolic” type which are essentially different in their properties from other classes such as those which describe equilibrium (elliptic) or unsteady diffusion and heat transfer (parabolic). Prototypes are as follows: Elliptic Laplace’s equation ∂2u/∂x2 + ∂2u/∂y2 = 0 and Poisson’s equation ∂2u/∂x2 + ∂2u/∂y2 = g(x, y) do not contain the variable time

DIFFERENTIAL EQUATIONS explicitly and consequently represent equilibrium configurations. Laplace’s equation is satisfied by static electric or magnetic potential at points free from electric charges or magnetic poles. Other important functions satisfying Laplace’s equation are the velocity potential of the irrotational motion of an incompressible fluid, used in hydrodynamics; the steady temperature at points in a homogeneous solid, and the steady state of diffusion through a homogeneous body. The gravitational potential V at points occupied by mass of density d satisfies Poisson’s equation ∂2 V/∂x2 + ∂2 V/∂y2 + ∂2 V/∂z2 = −4πd. Parabolic The heat equation ∂T/∂t = ∂ 2T/∂x2 + ∂2T/∂y2 represents nonequilibrium or unsteady states of heat conduction and diffusion. Hyperbolic The wave equation ∂2u/∂t2 = c2(∂2u/∂x2 + ∂2u/∂y2) represents wave propagation of many varied types. Quasilinear first-order differential equations are like ∂u ∂u a
+b
=f ∂x ∂y where a, b, and f depend on x, y, and u, with a2 + b2 ≠ 0. This equation can be solved using the method of characteristics, which writes the solution in terms of a parameter s, which defines a path for the characteristic. dx dy du
= a,
= b,
= f ds ds ds These equations are integrated from some initial conditions. For a specified value of s, the value of x and y shows the location where the solution is u. The equation is semilinear if a and b depend just on x and y (and not u), and the equation is linear if a, b, and f all depend on x and y, but not u. Such equations give rise to shock propagation, and conditions have been derived to deduce the presence of shocks, Ref. 245. For further information, see Refs. 79, 159, 192, and 245. An example of a linear hyperbolic equation is the advection equation for flow of contaminants when the x and y velocity components are u and v, respectively. ∂c ∂c ∂c
+u
+v
=0 ∂t ∂x ∂y The equations for flow and adsorption in a packed bed or chromatography column give a quasilinear equation. ∂c ∂c df ∂c φ
+ φ u
+ (1 − φ)

= 0 ∂t ∂x dc ∂t Here n = f(c) is the relation between concentration on the adsorbent and fluid concentration. The solution of problems involving partial differential equations often revolves about an attempt to reduce the partial differential equation to one or more ordinary differential equations. The solutions of the ordinary differential equations are then combined (if possible) so that the boundary conditions as well as the original partial differential equation are simultaneously satisfied. Three of these techniques are illustrated. Similarity Variables The physical meaning of the term “similarity” relates to internal similitude, or self-similitude. Thus, similar solutions in boundary-layer flow over a horizontal flat plate are those for which the horizontal component of velocity u has the property that two velocity profiles located at different coordinates x differ only by a scale factor. The mathematical interpretation of the term similarity is a transformation of variables carried out so that a reduction in the number of independent variables is achieved. There are essentially two methods for finding similarity variables, “separation of variables” (not the classical concept) and the use of “continuous transformation groups.” The basic theory is available in Ames (see the references). Example The equation ∂θ/∂x = (A/y)(∂2θ/∂y2) with the boundary conditions θ = 0 at x = 0, y > 0; θ = 0 at y = ∞, x > 0; θ = 1 at y = 0, x > 0 represents the nondimensional temperature θ of a fluid moving past an infinitely wide flat plate immersed in the fluid. Turbulent transfer is neglected, as is molecular transport except in the y direction. It is now assumed that the equation and the boundary conditions can be satisfied by a solution of the form θ = f(y/xn) = f(u), where θ =

3-39

0 at u = ∞ and θ = 1 at u = 0. The purpose here is to replace the independent variables x and y by the single variable u when it is hoped that a value of n exists which will allow x and y to be completely eliminated in the equation. In this case since u = y/xn, there results after some calculation ∂θ/∂x = −(nu/x)(dθ/du), ∂2θ/∂y2 = (1/x2n)(d2θ/du2), and when these are substituted in the equation, −(1/x)nu(dθ/du) = (1/x3n)(A/u)(d2θ/du2). For this to be a function of u only, choose n = s. There results (d2θ/du2) + (u2/3A)(dθ/du) = 0. Two integrations and use of the boundary conditions for this ordinary differential equation give the solution θ=



∞

u



exp (−u3/9A) du 

∞

exp (−u3/9A) du

0

Group Method The type of transformation can be deduced using group theory. For a complete exposition, see Refs. 9, 12, and 145; a shortened version is in Ref. 106. Basically, a similarity transformation should be considered when one of the independent variables has no physical scale (perhaps it goes to infinity). The boundary conditions must also simplify (and combine) since each transformation leads to a differential equation with one fewer independent variable. Example A similarity variable is found for the problem ∂c ∂ ∂c
=
D(c)
, c(0,t) = 1, c(∞,t) = 0, c(x,0) = 0 ∂t ∂x ∂x Note that the length dimension goes to infinity, so that there is no length scale in the problem statement; this is a clue to try a similarity transformation. The transformation examined here is





t = a α t,

x = a β x, c = a γ c With this substitution, the equation becomes ∂c ∂ ∂c aα − γ
= a2β − γ
D(a−γ c)
∂t ∂x ∂x Group theory says a system is conformally invariant if it has the same form in the new variables; here, that is





γ = 0, α − γ = 2β − γ,

or α = 2β

The invariants are β δ=
α

x η =
δ , t and the solution is

c(x, t) = f(η)tγ/α We can take γ = 0 and δ = β/α = a. Note that the boundary conditions combine because the point x = ∞ and t = 0 give the same value of η and the conditions on c at x = ∞ and t = 0 are the same. We thus make the transformation x η =
, c(x, t) = f(η) 4 D
 0t The use of the 4 and D0 makes the analysis below simpler. The result is





df d df
D(c)
+ 2η
= 0, dη dη dη

f(0) = 1,

f(∞) = 0

Thus, we solve a two-point boundary value problem instead of a partial differential equation. When the diffusivity is constant, the solution is the error function, a tabulated function. c(x,t) = 1 − erf η = erfc η erf η =



η

0



e−ξ dξ  2

∞

2

e−ξ dξ

0

Separation of Variables This is a powerful, well-utilized method which is applicable in certain circumstances. It consists of assuming that the solution for a partial differential equation has the form U = f(x)g(y). If it is then possible to obtain an ordinary differential equation on one side of the equation depending only on x and on the other side only on y, the partial differential equation is said to be separable in the variables x, y. If this is the case, one side of the equation is a function of x alone and the other of y alone. The two can be equal only if each is a constant, say λ. Thus the problem has again been reduced to the solution of ordinary differential equations. Example Laplace’s equation ∂2V/∂x2 + ∂2V/∂y2 = 0 plus the boundary conditions V(0, y) = 0, V(l, y) = 0, V(x, ∞) = 0, V(x, 0) = f(x) represents the steadystate potential in a thin plate (in z direction) of infinite extent in the y direction

3-40

MATHEMATICS

and of width l in the x direction. A potential f(x) is impressed (at y = 0) from x = 0 to x = 1, and the sides are grounded. To obtain a solution of this boundaryvalue problem assume V(x, y) = f(x)g(y). Substitution in the differential equation yields f ″(x)g(y) + f(x)g″(y) = 0, or g″(y)/g(y) = −f″(x)/f(x) = λ2 (say). This system becomes g″(y) − λ2g(y) = 0 and f ″(x) + λ2f(x) = 0. The solutions of these ordinary differential equations are respectively g(y) = Aeλy + Be−λy, f(x) = C sin λx + D cos λx. Then f(x)g(y) = (Aeλy + Be−λy) (C sin λx + D cos λx). Now V(0, y) = 0 so that f(0)g(y) = (Aeλy + Be−λy) D  0 for all y. Hence D = 0. The solution then has the form sin λx (Aeλy + Be−λy) where the multiplicative constant C has been eliminated. Since V(l, y) = 0, sin λl(Aeλy + Be−λy)  0. Clearly the bracketed function of y is not zero, for the solution would then be the identically zero solution. Hence sin λl = 0 or λn = nπ/l, n = 1, 2, . . . where λn = nth eigenvalue. The solution now has the form sin (nπx/l)(Aenπy/l + Be−nπy/l). Since V(x, ∞) = 0, A must be taken to be zero because ey becomes arbitrarily large as y → ∞. The solution then reads Bn sin (nπx/l)e−nπy/l, where Bn is the multiplicative constant. The differential equation is linear and homogeneous so that n∞= 1 Bne−nπy/l sin (nπx/l) is also a solution. Satisfaction of the last boundary condition is ensured by taking 2 Bn =
l

f(x) sin (nπx/l) dx = Fourier sine coefficients of f(x) l

0

Further, convergence and differentiability of this series are established quite easily. Thus the solution is ∞ nπx V(x, y) =  Bne−nπy/l sin
l n=1

Example The diffusion problem ∂c ∂ ∂c
=
D(c)
, c(0, t) = 1, c(∞, t) = 0, c(x, 0) = 0 ∂t ∂x ∂x can be solved by separation of variables. First transform the problem so that the boundary conditions are homogeneous (having zeroes on the right-hand side). Let c(x, t) = 1 − x + u(x, t) Then u(x, t) satisfies





∂u ∂ 2u
= D
2 , ∂t ∂x

u(x, 0) = x − 1,

u(0, t) = 0,

u(1, t) = 0

Assume a solution of the form u(x, t) = X(x) T(t), which gives dT d 2X X
=DT
dt dx2 Since both sides are constant, this gives the following ordinary differential equations to solve. 1 d 2X 1 dT = −λ

= −λ,

D T dt X dx2 The solution of these is T=Ae

−λDt

,

 x + E sin
λ x X = B cos
λ

The combined solution for u(x,t) is

 x + E sin
λ  x) e−λDt u = A (B cos
λ Apply the boundary condition that u(0,t) = 0 to give B = 0. Then the solution is

 x)e−λDt u = A (sin
λ where the multiplicative constant E has been eliminated. Apply the boundary condition at x = L.

 L)e−λDt 0 = A (sin
λ This can be satisfied by choosing A = 0, which gives no solution. However, it can also be satisfied by choosing λ such that

 L = 0,
λ L=nπ sin
λ n2π2 Thus λ=
L2 The combined solution can now be written as





sin nπx 2 2 2 u = A
e−n π Dt/L L

Since the initial condition must be satisfied, we use an infinite series of these functions. ∞ sin nπx 2 2 2 u =  An
e−n π Dt/L L n=1 At t = 0, we satisfy the initial condition. ∞ sin nπx x − 1 =  An
L n=1 This is done by multiplying the equation by sin mπx
L and integrating over x: 0 → L. (This is the same as minimizing the mean-square error of the initial condition.) This gives L AmL
= (x − 1) sin mπx dx 0 2 which completes the solution.











Integral-Transform Method A number of integral transforms are used in the solution of differential equations. Only one, the Laplace transform, will be discussed here [for others, see “Integral Transforms (Operational Methods)”]. The one-sided Laplace transform indicated ∞ by L[ f(t)] is defined by the equation L[ f(t)] = ∫0 f(t)e−st dt. It has numerous important properties. The ones of interest here are L[ f′(t)] = sL[f(t)] − f(0); L[ f″(t)] = s2L[f(t)] − sf(0) − f′(0); L[ f (n)(t)] = snL[ f(t)] − sn − 1f(0) − sn − 2f′(0) − ⋅⋅⋅ − f (n − 1)(0) for ordinary derivatives. For partial derivatives an indication of which variable is being transformed avoids confusion. Thus, if ∂y y = y(x, t), Lt
= sL[y(x, t)] − y(x, 0) ∂t



∂y dLt[y(x, t)] Lt
=

∂x dx since L[y(x, t)] is “really” only a function of x. Otherwise the results are similar. These facts coupled with the linearity of the transform, i.e., L[af(t) + bg(t)] = aL[ f(t)] + bL[g(t)], make it a useful device in solving some linear differential equations. Its use reduces the solution of ordinary differential equations to the solution of algebraic equations for L[y]. The solution of partial differential equations is reduced to the solution of ordinary differential equations. In both situations the inverse transform must be obtained either from tables, of which there are several, or by use of complex inversion methods.



whereas

Example The equation ∂c/∂t = D(∂2c/∂x2) represents the diffusion in a semi-infinite medium, x ≥ 0. Under the boundary conditions c(0, t) = c0, c(x, 0) = 0 find a solution of the diffusion equation. By taking the Laplace transform of both sides with respect to t, ∞ ∂2c 1 ∞ −st ∂c e−st
2 dt =
e
dt 0 ∂x D 0 ∂t





sF d2F = (1/D)sF − c(x, 0) =

dx2 D where F(x, s) = Lt[c(x, t)]. Hence d 2F s −
F=0
dx2 D The other boundary condition transforms into F(0, s) = c0 /s. Finally the solution of the ordinary differential equation for F subject to F(0, s) = c0 /s and F remains x. Reference to a table shows that the funcfinite as x → ∞ is F(x, s) = (c0 /s)e−
s/D tion having this as its Laplace transform is or

 



2 c(x, t) = c0 1 −

π



t x/2
D

0



e−u du 2

Matched-Asymptotic Expansions Sometimes the coefficient in front of the highest derivative is a small number. Special perturbation techniques can then be used, provided the proper scaling laws are found. See Refs. 32, 170, and 180.

DIFFERENCE EQUATIONS

3-41

DIFFERENCE EQUATIONS REFERENCES:

30, 43.

Certain situations are such that the independent variable does not vary continuously but has meaning only for discrete values. Typical illustrations occur in the stagewise processes found in chemical engineering such as distillation, staged extraction systems, and absorption columns. In each of these the operation is characterized by a finite between-stage change of the dependent variable in which the independent variable is the integral number of the stage. The importance of difference equations is twofold: (1) to analyze problems of the type described and (2) to obtain approximate solutions of problems which lead, in their formulation, to differential equations. In this subsection only problems of analysis are considered; the application to approximate solutions is considered under “Numerical Analysis and Approximate Methods.” ELEMENTS OF THE CALCULUS OF FINITE DIFFERENCES Let y = f(x) be defined for discrete equidistant values of x, which will be denoted by xn. The corresponding value of y will be written yn = f(xn). The first forward difference of f(x) denoted by ∆f(x) = f(x + h) − f(x) where h = xn − xn − 1 = interval length. Example Let f(x) = x2. Then ∆f(x) = (x + h)2 − x2 = 2hx + h2. The second forward difference is obtained by taking the difference of the first; thus ∆∆f(x) = ∆2f(x) = ∆f(x + h) − ∆f(x) = f(x + 2h) − 2f(x + h) + f(x). Example f(x) = x2, ∆2f(x) = ∆[∆f(x)] = ∆2hx + ∆h2 = 2h(x + h) − 2hx + h2 −

h2 = 2h2.

Similarly the nth forward difference is defined by the relation ∆nf(x) = ∆[∆n − 1f(x)]. Other difference relations are also quite useful. Some of these are ∇f(x) = f(x) − f(x − h), which is called the backward difference, and δf(x) = f [x + (h/2)] − f [x − (h/2)], called the central difference. Some properties of the operator ∆ are quite important. If C is any constant, ∆C = 0; if f(x) is any function of period h, ∆f(x) = 0 (in fact, periodic functions of period h play the same role here as constants do in the differential calculus); ∆[ f(x) + g(x)] = ∆f(x) + ∆g(x); ∆m[∆nf(x)] = ∆m + nf(x); ∆[f(x)g(x)] = f(x) ∆g(x) + g(x + h) ∆f(x) g(x) ∆f(x) − f(x) ∆g(x) f(x) ∆
=


g(x) g(x)g(x + h)



Example ∆(x sin x) = x∆ sin x + sin (x + h) ∆x = 2x sin (h/2) cos [x + (h/2)] +

h sin (x + h).

DIFFERENCE EQUATIONS A difference equation is a relation between the differences and the independent variable, φ(∆ny, ∆n − 1y, . . . , ∆y, y, x) = 0, where φ is some given function. The general case in which the interval between the successive points is any real number h, instead of 1, can be reduced to that with interval size 1 by the substitution x = hx′. Hence all further difference-equation work will assume the interval size between successive points is 1. Example f(x + 1) − (α + 1)f(x) + αf(x − 1) = 0. Common notation usually is yx = f(x). This equation is then written yx + 1 − (α + 1)yx + αyx − 1 = 0. Example yx + 2 + 2yx yx + 1 + yx = x2. Example yx + 1 − yx = 2x. The order of the difference equation is the difference between the largest and smallest arguments when written in the form of the second example. The first and second examples are both of order 2, while the third example is of order 1. A linear difference equation involves no

products or other nonlinear functions of the dependent variable and its differences. The first and third examples are linear, while the second example is nonlinear. A solution of a difference equation is a relation between the variables which satisfies the equation. If the difference equation is of order n, the general solution involves n arbitrary constants. The techniques for solving difference equations resemble techniques used for differential equations. Equation Dny = a The solution of ∆ny = a, where a is a constant, is a polynomial of degree n plus an arbitrary periodic function of period 1. That is, y = (axn/n!) + c1xn − 1 + c2xn − 2 + ⋅⋅⋅ + cn + f(x), where f(x + 1) = f(x). Example ∆3y = 6. The solution is y = x3 + c1x2 + c2x + c3 + f(x); c1, c2, c3 are arbitrary constants, and f(x) is an arbitrary periodic function of period 1. Equation yx + 1 - yx = φ(x) This equation states that the first difference of the unknown function is equal to the given function φ(x). The solution by analogy with solving the differential equation dy/dx = φ(x) by integration is obtained by “finite integration” or summation. When there are only a finite number of data points, this is easily accomplished by writing yx = y0 +  tx = 1 φ(t − 1), where the data points are numbered from 1 to x. This is the only situation considered here. Examples If φ(x) = 1, yx = x. If φ(x) = x, yx = [x(x − 1)]/2. If φ(x) = ax, a ≠ 0, yx = ax/(a − 1). In all cases y0 = 0. Other examples may be evaluated by using summation, that is, y2 = y1 + φ(1), y3 = y2 + φ(2) = y1 + φ(1) + φ(2), y4 = y3 + φ(3) = y1 + φ(1) + φ(2) + φ(3), . . . , yx = y1 +  tx =− 11 φ(t). Example yx + 1 − ryx = 1, r constant, x > 0 and y0 = 1. y1 = 1 + r, y2 = 1 + r + r2, . . . , yx = 1 + r + ⋅⋅⋅ + rx = (1 − rx + 1)/(1 − r) for r ≠ 1 and yx = 1 + x for r = 1. Linear Difference Equations The linear difference equation of order n has the form Pnyx + n + Pn − 1yx + n − 1 + ⋅⋅⋅ + P1yx + 1 + P0yx = Q(x) with Pn ≠ 0 and P0 ≠ 0 and Pj ; j = 0, . . . , n are functions of x. Constant Coefficient and Q(x) = 0 (Homogeneous) The solution is obtained by trying a solution of the form yx = cβ x. When this trial solution is substituted in the difference equation, a polynomial of degree n results for β. If the solutions of this polynomial are denoted by β1, β2, . . . , βn then the following cases result: (1) if all the βj’s are n real and unequal, the solution is yx =  j = 1 cj β jx, where the c1, . . . , cn are arbitrary constants; (2) if the roots are real and repeated, say, βj has multiplicity m, then the partial solution corresponding to βj is β jx(c1 + c2 x + ⋅⋅⋅ + cm xm − 1); (3) if the roots are complex conjugates, say, a + ib = peiθ and a − ib = pe−iθ, the partial solution corresponding to this pair is px(c1 cos θx + c2 sin θx); and (4) if the roots are multiple complex conjugates, say, a + ib = peiθ and a − ib = pe−iθ are m-fold, then the partial solution corresponding to these is px[(c1 + c2 x + ⋅⋅⋅ + cm x m − 1) cos θx + (d1 + d2 x + ⋅⋅⋅ + dm x m − 1) sin θx]. Example The equation yx + 1 − (α + 1)yx + αyx − 1 = 0, y0 = c0 and ym + 1 = xm + 1/k represents the steady-state composition of transferable material in the raffinate stream of a staged countercurrent liquid-liquid extraction system. Clearly y is a function of the stage number x. α is a combination of system constants. By using the trial solution yx = cβx, there results β2 − (α + 1)β + α = 0, so that β1 = 1, β2 = α. The general solution is yx = c1 + c2αx. By using the side conditions, c1 = c0 − c2, c2 = (ym + 1 − c0)/(αm + 1 − 1). The desired solution is (yx − c0)/(ym + 1 − c0) = (αx − 1)/(αm + 1 − 1). Example yx + 3 − 3yx + 2 + 4yx = 0. By setting yx = cβ x, there results β3 − 3β2 + 4 = 0 or β1 = −1, β2 = 2, β3 = 2. The general solution is yx = c1(−1)x + 2x(c2 + c3x). Example yx + 1 − 2yx + 2yx − 1 = 0. β1 = 1 + i, β2 = 1 − i. p =
1 + 1 =
2, θ = π/4. The solution is yx = 2 x/2 [c1 cos (xπ/4) + c2 sin (xπ/4)]. Constant Coefficients and Q(x) ≠ 0 (Nonhomogeneous) In this case the general solution is found by first obtaining the homoge-

3-42

MATHEMATICS

neous solution, say, yHx and adding to it any particular solution with Q(x) ≠ 0, say, y xP. There are several means of obtaining the particular solution. Method of Undetermined Coefficients If Q(x) is a product or linear combination of products of the functions ebx, a x, x p (p a positive integer or zero) cos cx and sin cx, this method may be used. The “families” [ax], [ebx], [sin cx, cos cx] and [x p, x p − 1, . . . , x, 1] are defined for each of the above functions in the following way: The family of a term fx is the set of all functions of which fx and all operations of the form a x + y, cos c(x + y), sin c(x + y), (x + y)p on fx and their linear combinations result in. The technique involves the following steps: (1) Solve the homogeneous system. (2) Construct the family of each term. (3) If the family has no representative in the homogeneous solution, assume y Px is a linear combination of the families of each term and determine the constants so that the equation is satisfied. (4) If a family has a representative in the homogeneous solution, multiply each member of the family by the smallest integral power of x for which all such representatives are removed and revert to step 3. Example yx + 1 − 3yx + 2yx − 1 = 1 + ax. a ≠ 0. The homogeneous solution is yxH = c1 + c2 2x. The family of 1 is 1 and of ax is ax. However, 1 is a solution of the homogeneous system. Therefore, try yxP = Ax + Bax. Substituting in the equation there results a yx = c1 + c2 2x − x +

axa ≠ 1, a ≠ 2 (a − 1)(a − 2) If a = 1, yx = c1 + c22x − 2x. If a = 2, yx = c1 + c22x − x + x2x.

Example The family of x23x is [x23x, x3x, 3x]. Method of Variation of Parameters This technique is applicable to general linear difference equations. It is illustrated for the second-order system yx + 2 + Ayx + 1 + Byx = φ(x). Assume that the homogeneous solution has been found by some technique and write yxH = c1ux + c2vx. Assume that a particular solution yxP = Dxux + Exvx. Ex and Dx can be found by solving the equations: ux + 1φ(x) Ex + 1 − Ex =


ux + 1vx + 2 − ux + 2vx + 1 vx + 1φ(x) Dx + 1 − Dx =


vx + 1ux + 2 − vx + 2ux + 1 by summation. The general solution is then yx = yxP + yxH. Variable Coefficients The method of variation of parameters applies equally well to the linear difference equation with variable coefficients. Techniques are therefore needed to solve the homogeneous system with variable coefficients. Equation yx + 1 - axyx = 0 By assuming that this equation is valid for x ≥ 0 and y0 = c, the solution is yx = c  xn = 1 an − 1. x+2 x+1

Example yx + 1 +
yx = 0. The solution is x x+1 n+1 2 3 yx = c  −
= c(−1)x
⋅
⋅⋅⋅
= (−1)xc(x + 1) n 1 2 x n=1





Example yx + 1 − xyx = 0. The solution is yx = c(x − 1)!

Reduction of Order If one homogeneous solution, say, ux, can be found by inspection or otherwise, an equation of lower order can be obtained by the substitution vx = yx /ux. The resultant equation must be satisfied by vx = constant or ∆vx = 0. Thus the equation will be of reduced order if the new variable Ux = ∆(yx /ux) is introduced. Example (x + 2)yx + 2 − (x + 3)yx + 1 + yx = 0. By observation ux = 1 is a solution. Set Ux = ∆yx = yx + 1 − yx. There results (x + 2)Ux + 1 − Ux = 0, which is of degree one lower than the original equation. The complete solution for yx is finally x 1 yx = c0 
+ c1 n = 0 n! Factorization If the difference equation can be factored, then the general solution can be obtained by solving two or more successive equations of lower order. Consider yx + 2 + Axyx + 1 + Bxyx = φ(x). If there exists ax, bx such that ax + bx = −Ax and axbx = Bx, then the difference equation may be written yx + 2 − (ax + bx) yx + 1 + axbxyx = φ(x). First solve Ux + 1 − bxUx = φ(x) and then yx + 1 − axyx = Ux. Example yx + 2 − (2x + 1)yx + 1 + (x2 + x)yx = 0. Set ax = x, bx = x + 1. Solve ux + 1 − (x + 1)ux = 0 and then yx + 1 − xyx = ux. Substitution If it is possible to rearrange a difference equation so that it takes the form afx + 2yx + 2 + bfx + 1yx + 1 + cfxyx = φ(x) with a, b, c constants, then the substitution ux = fxyx reduces the equation to one with constant coefficients. Example (x + 2)2yx + 2 − 3(x + 1)2yx + 1 + 2x2yx = 0. Set ux = x2yx. The equation becomes ux + 2 − 3ux + 1 + 2ux = 0, which is linear and easily solved by previous methods. The substitution ux = yx / fx reduces afx fx + 1yx + z + bfx fx + 2yx + 1 + cfx + 1 fx + 2yx = φ(x) to an equation with constant coefficients. Example x(x + 1)yx + 2 + 3x(x + 2)yx + 1 − 4(x + 1)(x + 2)yx = x. Set ux = yx /fx = yx /x. Then yx = xux, yx + 1 = (x + 1)ux + 1 and yx + 2 = (x + 2)ux + 2. Substitution in the equation yields x(x + 1)(x + 2)ux + 2 + 3x(x + 2)(x + 1)uu + 1 − 4x(x + 1)(x + 2) ux = x or ux + 2 + 3ux + 1 − 4ux = 1/(x + 1)(x + 2), which is a linear equation with constant coefficients. Nonlinear Difference Equations: Riccati Difference Equation The Riccati equation yx + 1yx + ayx + 1 + byx + c = 0 is a nonlinear difference equation which can be solved by reduction to linear form. Set y = z + h. The equation becomes zx + 1zx + (h + a)zx + 1 + (h + b)zx + h2 + (a + b)h + c = 0. If h is selected as a root of h2 + (a + b)h + c = 0 and the equation is divided by zx + 1zx there results [(h + b)/zx + 1] + [(h + a)/zx] + 1 = 0. This is a linear equation with constant coefficients. The solution is 1 yx = h +



a+h x 1 c −

−

b+h (a + h) + (b + h)





Example This equation is obtained in distillation problems, among others, in which the number of theoretical plates is required. If the relative volatility is assumed to be constant, the plates are theoretically perfect, and the molal liquid and vapor rates are constant, then a material balance around the nth plate of the enriching section yields a Riccati difference equation.

INTEGRAL EQUATIONS REFERENCES: 75, 79, 105, 195, 273. See also “Numerical Analysis and Approximate Methods.”

An integral equation is any equation in which the unknown function appears under the sign of integration and possibly outside the sign of integration. If derivatives of the dependent variable appear elsewhere in the equation, the equation is said to be integrodifferential.

CLASSIFICATION OF INTEGRAL EQUATIONS Volterra integral equations have an integral with a variable limit. The Volterra equation of the second kind is u(x) = f(x) + λ

K(x, t)u(t) dt x

a

INTEGRAL EQUATIONS whereas a Volterra equation of the first kind is u(x) = λ

METHODS OF SOLUTION

K(x, t)u(t) dt x

a

Equations of the first kind are very sensitive to solution errors so that they present severe numerical problems. Volterra equations are similar to initial value problems. A Fredholm equation of the second kind is u(x) = f(x) + λ

K(x, t)u(t) dt b

a

whereas a Fredholm equation of the first kind is u(x) =

b

a

The limits of integration are fixed, and these problems are analogous to boundary value problems. An eigenvalue problem is a homogeneous equation of the second kind, and solutions exist only for certain λ.

K(x, t)u(t) dt b

a

See Refs. 105 and 195 for further information and existence proofs. If the unknown function u appears in the equation in any way except to the first power, the integral equation is said to be nonlinear. b The equation u(x) = f(x) + ∫a K(x, t)[u(t)]3/2 dt is nonlinear. The differential equation du/dx = g(x, u) is equivalent to the nonlinear integral x equation u(x) = c + ∫a g[t, u(t)] dt. An integral equation is said to be singular when either one or both of the limits of integration become infinite or if K(x, t) becomes infinite for one or more points of the interval under discussion.



∞

cos (xt)u(t) dt and f(x) =

The Leibniz rule (see “Integral Calculus”) can be used to show the equivalence of the initial-value problem consisting of the secondorder differential equation d 2y/dx2 + A(x)(dy/dx) + B(x)y = f(x) together with the prescribed initial conditions y(a) = y0, y′(a) = y′0 to the integral equation. y(x) =

K(x, t)y(t) dt + F(x) x

and

(x − t)f(t) dt + [A(a)y + y′](x − a) + y x

0





0

0

a

This integral equation is a Volterra equation of the second kind. Thus the initial-value problem is equivalent to a Volterra integral equation of the second kind. Example d 2y/dx2 + x2(dy/dx) + xy = x, y(0) = 1, y′(0) x= 0. Here A(x) = x2, B(x) = x, f(x) = x. The equivalent integral equation is y(x) = ∫ 0 K(x, t)y(t) dt + F(x) x where K(x, t) = t(x − t) − t2 and F(x) = ∫ 0 (x − t)t dt + 1 = x3/6 + 1. Combining these x y(x) = ∫ 0 t[x − 2t]y(t) dt + x3/6 + 1. Eigenvalue problems can also be related. For example, the problem (d y/dx2) + λy = 0 with y(0) = 0, y(a) = 0 is equivalent to the integral a equation y(x) = λ ∫ 0 K(x, t)y(t) dt, where K(x, t) = (t/a)(a − x) when t < x and K(x, t) = (x/a)(a − t) when t > x. The differential equation may be recovered from the integral equation by differentiating the integral equation by using the Leibniz rule.







Method of Successive Approximations Consider the equation b y(x) = f(x) + λ ∫ a K(x, t)y(t) dt. In this method a unique solution is obtained in sequence form as follows: Substitute in the right-hand member of the equation y0(t) for y(t). Upon integration there results b y1(t) = f(x) + λ ∫ a K(x, t)y0(t) dt. Continue in like manner by replacing y0 by y1, y1 by y2, etc. A series of functions y0(x), y1(x), y2(x), . . . are obtained which satisfy the equations yn(x) = f(x) + λ

K(x, t)y b

(t) dt

n−1

a

Then yn(x) = f(x) + λ K(x, t)f(t) dt + λ K(x, t) ∫ a K(t, t1)f(t1) dt1 dt + b λ3 ∫ ab K(x, t) ∫ a K(t, t ) K(t1, t2)f(t2) dt2 dt1 dt + ⋅⋅⋅ + Rn, where Rn is the remainder, and max. y0 |Rn| ≤ |λn| Mn(b − a)n a≤x≤b 2 b ∫a

∫ ab b 1 ∫a



b



where M = maximum value of |K| in the rectangle a ≤ t ≤ b, a ≤ x ≤ b. If |λ|M(b − a) < 1, lim Rn = 0. Then yn(x) → y(x), which is the unique n→∞ solution. 1

Example Consider the equation y(x) = 1 + λ ∫ 0 (1 − 3xt)y(t) dt.

(1 − 3xt) dt + λ (1 − 3xt) (1 − 3tt ) dt dt + ⋅⋅⋅ 1

1

1

2

1

0

K(x, t) = (t − x)[B(t) − A′(t)] − A(t) F(x) =



L[1] 1/p p E(t) = L−1

= L−1
2 = L−1
= cosh
 λt 1 − λL[K(t)] 1 − λ/p p2 − λ

y(x) = 1 + λ

a

where

Example In a certain linear system, the effect E(t) due to a cause C = λE at time τ is a function only of the elapsed time t − τ. If the system has the activity level 1 at time t < 0, thet cause λE and effect (E) relation is given by the integralt equation E(t) = 1 + λ ∫ 0 K(t − τ)E(τ) dτ. Let K(t − τ) = t − τ. Then E(t) = 1 + λ ∫ 0 (t − τ)E(τ) dτ. By using the transform method

x

RELATION TO DIFFERENTIAL EQUATIONS



L[ f(x)] L[ f(x)] L[u(x)] =

, u(x) = L−1

1 − λL[K(x)] 1 − λL[K(x)] Equations of the type considered here occur quite frequently in practice in what can be called “cause-and-effect” systems.



u(t)
dt are both 0 0 x − t singular. The kernel of the first equation is cos (xt), and that of the second is (x − t)−1.

Example u(x) = x +

In general, the solution of integral equations is not easy, and a few exact and approximate methods are given here. Often numerical methods must be employed, as discussed in “Numerical Solution of Integral Equations.” Equations of Convolution Type The equation u(x) = f(x) + x λ ∫ 0 K(x − t)u(t) dt is a special case of the linear integral equation of the second kind of Volterra type. The integral part is the convolution integral discussed under “Integral Transforms (Operational Methods)”; so the solution can be accomplished by Laplace transforms; L[u(x)] = L[ f(x)] + λL[u(x)]L[K(x)] or



K(x, t)u(t) dt

u(x) = λ

3-43

0

1

0

3 1 1 3 λ4 1 3 = 1 + λ 1 −
x + λ2
+
λ3 1 −
x +
+
λ5 1 −
x + ⋅⋅⋅ 2 4 4 2 16 16 2







λ λ 3 = 1 +
+
+ ⋅⋅⋅ 1 + λ 1 −
x 4 16 2



2

 

4

1 + λ(1 − ⁄ x) =

, 1 − dλ2 32









|λ|<2

Example dy/dx = x2 x+ y. x0 = 0, y0 = 1. This problem is equivalent to the integral equation y = 1 + ∫ 0 (x2 + y) dx. Let the initial approximation for y be 1. Then x x3 y(1) = 1 + (x2 + 1) dx = 1 + x +
0 3



2

y(2) = 1 +

[x + y x

2

0

] dx = 1 +

(1)

x + 1 + x +
x3  dx x

3

2

0

x4 x2 x3 = 1 + x +
+
+
, etc. 2 3 12

3-44

MATHEMATICS

INTEGRAL TRANSFORMS (OPERATIONAL METHODS) REFERENCES:

63, 64, 71, 72, 97, 137, 217.

The term “operational method” implies a procedure of solving differential and difference equations by which the boundary or initial conditions are automatically satisfied in the course of the solution. The technique offers a very powerful tool in the applications of mathematics, but it is limited to linear problems. Most integral transforms are special cases of the equation g(s) = b ∫ a f(t)K(s, t) dt in which g(s) is said to be the transform of f(t) and K(s, t) is called the kernel of the transform. A tabulation of the more important kernels and the interval (a, b) of applicability follows. The first three transforms are considered here.

Properties of the Laplace Transform 1. The Laplace transform is a linear operator: L{af(t) + bg(t)} = aL{ f(t)} + bL{g(t)} for any constants a, b and any two functions f and g whose Laplace transforms exist. 2. The Laplace transform of a real-valued function is real for real s. If f(t) is a complex-valued function, f(t) = u(t) + iv(t), where u and v are real, then L{ f(t)} = L{u(t)} + iL{v(t)}. Thus L{u(t)} is the real part of L{ f(t)}, and L{v(t)} is the imaginary part of L{ f(t)}. 3. The Laplace transform of a function in the class Λ has derivatives of all orders, and L{tkf(t)} = (−1)kd kF(s)/ds k, k = 1, 2, 3, . . . .

Name of transform

(a, b)

Laplace

(0, ∞)

Fourier

(−∞, ∞)

Fourier cosine

(0, ∞)

Fourier sine

(0, ∞)

Mellin

(0, ∞)

Hankel

(0, ∞)

K(s, t) e−st 1
e−ist 
2π


π2 cos st 
π2 sin st



2as a e−st sin at dt =
, s > 0. By property 3,
= (s2 + a2)2 s2 + a2 e−st t sin at dt = L{t sin at}.

Example

0

0

Example By applying property 3 with f(t) = 1 and using the preceding results, we obtain k! dk 1 L{tk} = (−1)k
k
=
ds s sk + 1



provided Re s > 0; k = 1, 2, . . . . Similarly, we obtain



t

tJν (st), ν ≥ −a

The Laplace transform of a function f(t) is defined by F(s) = L{f(t)} = ∞ ∫ 0 e−stf(t) dt, where s is a complex variable. Note that the transform is an improper integral and therefore may not exist for all continuous functions and all values of s. We restrict consideration to those values of s and those functions f for which this improper integral converges. The function L[f(t)] = g(s) is called the direct transform, and L−1[g(s)] = f(t) is called the inverse transform. Both the direct and the inverse transforms are tabulated for many often-occurring functions. In general, 1  + i∞ st L−1[g(s)] =
e g(s) ds 2πi  − i∞ and to evaluate this integral requires a knowledge of complex variables, the theory of residues, and contour integration. A function is said to be piecewise continuous on an interval if it has only a finite number of finite (or jump) discontinuities. A function f on 0 < t < ∞ is said to be of exponential growth at infinity if there exist constants M and α such that |f(t)| ≤ Meαt for sufficiently large t. Sufficient Conditions for the Existence of Laplace Transform Suppose f is a function which is (1) piecewise continuous on every finite interval 0 < t < T, (2) of exponential growth at infinity, and (3) ∫ 0δ |f(t)| dt exist (finite) for every finite δ > 0. Then the Laplace transform of f exists for all complex numbers s with sufficiently large real part. Note that condition 3 is automatically satisfied if f is assumed to be piecewise continuous on every finite interval 0 ≤ t < T. The function f(t) = t−1/2 is not piecewise continuous on 0 ≤ t ≤ T but satisfies conditions 1 to 3. Let Λ denote the class of all functions on 0 < t < ∞ which satisfy conditions 1 to 3.



Example Let f(t) be the Heaviside step function at t = t0; i.e., f(t) = 0 for t ≤ t0, and f(t) = 1 for t > t0. Then ∞ T 1 e−st0 e−st dt = lim e−st dt = lim
(e−st0 − e−sT) =
L{ f(t)} = T→∞ t0 T→∞ s t0 s provided s > 0.



Example Let f(t) = eat, t ≥ 0, where a is a real number. Then L{eat} = ∞ ∫ 0 e−(s − a) dt = 1/(s − a), provided Re s > a.



dk 1 k! L{tkeat} = (−1)k
k
=

ds s − a (s − a)k + 1

s−1

LAPLACE TRANSFORM





∞

∞

4. Frequency-shift property (or, equivalently, the transform of an exponentially modulated function). If F(s) is the Laplace transform of a function f(t) in the class Λ, then for any constant a, L{eat f(t)} = F(s − a). 1 (s + a)

Example L{te−at} =
2 , s > 0. 5. Time-shift property. Let u(t − a) be the unit step function at t = a. Then L{f (t − a)u(t − a)} = e−asF(s). 6. Transform of a derivative. Let f be a differentiable function such that both f and f ′ belong to the class Λ. Then L{ f ′(t)} = sF(s) − f(0). 7. Transform of a higher-order derivative. Let f be a function which has continuous derivatives up to order n on (0, ∞), and suppose that f and its derivatives up to order n belong to the class Λ. Then L{ f ( j)(t)} = s jF(s) − s j − 1f(0) − s j − 2 f ′(0) − ⋅ ⋅ ⋅ − sf ( j − 2)(0) − f ( j − 1)(0) for j = 1, 2, . . . , k. Example L{ f″(t)} = s2L{ f(t)} − sf(0) − f′(0) L{ f″′(t)} = s3L{ f(t)} − s2f(0) − sf ′(0) − f″(0)

Example Solve y″ + y = 2et, y(0) = y′(0) = 2. L[y″] = −y′(0) − sy(0) + s2L[y] = −2 − 2s + s2L[y]. Thus 2 −2 − 2s + s2L[y] + L[y] = 2L[et] =
s−1 1 1 s 2s2 L[y] =

=
+
+
(s − 1)(s2 + 1) s − 1 s2 + 1 s2 + 1 Hence y = et + cos t + sin t.

A short table (Table 3-1) of very common Laplace transforms and inverse transforms follows. The references include more detailed ∞ tables. NOTE: Γ(n + 1) = ∫ 0 xne−x dx (gamma function); Jn(t) = Bessel function of the first kind of order n. t 1 1 0 8. L f(t) dt =
L[ f(t)] +
f(t) dt a s s a







1 s

s −1 a 

Example Find f(t) if L[ f(t)] =
2
2 2 Therefore f(t) =

sinh at
1a sinh at dt dt =
a1
− t a t

t

0

0

2





1 1 L
sinh at =
a s2 − a2

INTEGRAL TRANSFORMS (OPERATIONAL METHODS) TABLE 3-1

Laplace Transforms

f(t)

g(s)

f(t)

g(s)

1

1/s

e−at(1 − at)

s
2 (s + a)

tn, (n a + integer)

n!
sn + 1

t sin at
2a

s
(s2 + a2)2

tn, n ≠ + integer

Γ(n + 1)
sn + 1

1
2 sin at sinh at 2a

s
s4 + 4a4

cos at

s
s2 + a2

cos at cosh at

s2
s4 + 4a4

a
s2 + a2

sin at

2

s
s4 − a4

1
(sinh at + sin at) 2a

cosh at

s
s2 − a2

a (cosh at + cos at)

s2
s4 − a4

sinh at

a
s2 − a2

sin at
t

a tan−1
s

e−at

1
s+a

J0(at)

1
s2 +a2


e−bt cos at

s+b

(s + b)2 + a2

Jn(at) nan
t

1

s2 +a2 − sn


e−bt sin a

a

(s + b)2 + a2

) J0 (2
at

1
e−a/s s

3-45

I0 = zero-order Bessel function of an imaginary argument. For large u, In(u) ∼ eu/
2π u . Hence for large n, exp [−(
θ  −
n )2] y(n, θ) ∼


2π1/2(nθ)1/4 or for sufficiently large n, the peak concentration occurs near θ = n.

Other applications of Laplace transforms are given under “Differential Equations.” CONVOLUTION INTEGRAL The convolution integral (faltung) of two functions f(t), r(t) is x(t) = f(t)°r(t) = ∫ 0t f(τ)r(t − τ) dτ. Example t° sin t =

τ sin (t − τ) dτ = t − sin t. t

0

L[ f(t)]L[h(t)] = L[ f(t)°h(t)] Z-TRANSFORM

See Refs. 198, 218, and 256. The z-transform is useful when data is available at only discrete points. Let f *(t) = f(tk) be the value of f at the sample points tk = k ∆t,

k = 0, 1, 2, . . .

Then the function f *(t) is ∞

9.



f(t) L
= t

 ⋅⋅⋅

∞

f(t) L
= tk

g(s) ds

s

∞

s

f *(t) =  f(tk) δ(t − tk) k=0

∞

k

g(s)(ds)

Take the Laplace transform of this.

s



sin at Example L
= t



10.



∞

L[sin at] ds =

s

s

a ds s
= cot−1
s2 + a2 a

01

ta

∞0

k=0

∞

k=0

L[u(t − a)] = e−as/s

at t = a elsewhere

L[u′(t − a)] = e−as

12. L−1[e−asg(s)] = f(t − a)u(t − a) (second shift theorem). 13. If f(t) is periodic of period b, i.e., f(t + b) = f(t), then b 1 L[f(t)] =
e−stf(t) dt 1 − e−bs 0



k=0

g*(z) =  f(tk) z−k

The unit impulse function is δ(a) = u′(t − a) =

∞

For convenience, replace e−s∆t by z and call g*(z) the z-transform of f *(t).

The unit step function u(t − a) =

11.

∞

∞

g*(s) = L[ f *(t)] =  f(tk) e−stk =  f(tk) e−s∆tk

k integrals



Example The partial differential equations relating gas composition to position and time in a gas chromatograph are ∂y/∂n + ∂x/∂θ = 0, ∂y/∂n = x − y, where x = mx′, n = (kGaP/Gm)h, θ = (mkGaP/ρB)t and GM = molar velocity, y = mole fraction of the component in the gas phase, ρB = bulk density, h = distance from the entrance, P = pressure, kG = mass-transfer coefficient, and m = slope of the equilibrium line. These equations are equivalent to ∂2y/∂n ∂θ + ∂y/∂n + ∂y/∂θ = 0, where the boundary conditions considered here are y(0, θ) = 0 and x(n, 0) = y(n, 0) + (∂y/∂n) (n, 0) = δ(0) (see property 11). The problem is conveniently solved by using the Laplace transform of y with respect to n; write ∞ g(s, θ) = ∫ 0 e−nsy(n, θ) dn. Operating on the partial differential equation gives s(dg/dθ) − (∂y/∂θ) (0, θ) + sg − y(0, θ) + dg/dθ = 0 or (s + 1) (dg/dθ) + sg = (∂y/∂θ) (0, θ) + y(0, θ) = 0. The second boundary condition gives g(s, 0) + sg(s, 0) − y(0, 0) = 1 or g(s, 0) + sg(s, 0) = 1 (L[δ(0)] = 1). A solution of the ordinary differential equation for g consistent with this second condition is 1 g(s, θ) =
e−sθ /(s + 1) s+1

θ) where Inversion of this transform gives the solution y(n, θ) = e−(n + θ) I0(2
n

The z-transform is used in process control when the signals are at intervals of ∆t. A brief table (Table 3-2) is provided here. The z-transform can also be used to solve difference equations, just like the Laplace transform can be used to solve differential equations. TABLE 3-2

z-Transforms

f(k)

g*(z)

1(k)

1
1 − z−1

k ∆t

∆t z−1
(1 − z−1)2

(k ∆t)n − 1

∂n − 1 1 lim (−1)n − 1


a→0 ∂an − 1 1 − e−a∆tz−1

sin a k ∆t

z−1 sin a ∆t


(1 − 2 z−1 cos a ∆t + z−2)

cos a k ∆t

1 − z−1 cos a ∆t


(1 − 2 z−1 cos a ∆t + z−2)

e−ak∆t

1

1 − e−a∆tz−1

e−bk∆t cos a k ∆t

1 − z−1 e−b∆t cos a ∆t



1 − 2 z−1 e−b∆t cos a ∆t + z−2 e−2b∆t

1
e−bk∆t sin a k ∆t b

1 z−1 e−b∆t sin a ∆t




b 1 − 2 z−1 e−b∆t cos a ∆t + z−2 e−2b∆t





3-46

MATHEMATICS

Example The difference equation for y(k) is

and its inverse by

y(k) + a1 y(k − 1) + a2 y(k − 2) = b1u(k)

F c−1[g(s)] = f(t) =

Take the z-transform −2

The inverse transform must be found, usually from a table of inverse transforms.



The Fourier transform is given by ∞ 1 F[ f(t)] =
f(t)e−ist dt = g(s)   2 π
−∞ and its inverse by ∞ 1 g(s)eist dt = f(t) F−1[g(s)] =
π −∞
2





1−a≤t≤a has F[ f(t)] = 0 elsewhere a a a 2 sin sa eist dt + e−ist dt = 2 cos st dt =
0 0 0 s





a

−a



Example Find F[e−a|t|] by property 7. e−a|t| is even. So L[e−at] = 1/(s + a). Therefore, F[e−a|t|] = 1/(is + a) + 1/(−is + a) = 2a/(s2 + a2). Tables of this transform may be found in Higher Transcendental Functions, vols. I, II, and III, A. Erdelyi, et al., McGraw-Hill, New York, 1953–1955. FOURIER COSINE TRANSFORM The Fourier cosine transform is given by


π2

a

cos st dt =

0



 



4. Fc[t−2nf(t)] = (−1)n (d 2ng/ds2n). 5. Fc[t2n + 1f(t)] = (−1)n (d 2n + 1/ds2n + 1) Fs[ f(t)]. A short table (Table 3-3) of Fourier cosine transforms follows.

e−ist dt =

Properties of the Fourier Transform Let F [ f (t)] = g(s); F−1[g(s)] = f(t). 1. F[ f (n)(t)] = (is)nF[ f(t)]. 2. F[af(t) + bh(t)] = aF[ f(t)] + bF [h(t)]. 3. F[ f(−t)] = g(−s). 1 s 4. F[ f(at)] =
g
, a > 0. a a 5. F[e−iwt f(t)] = g(s + w). 6. F[ f(t + t1)] = eist1g(s). 7. F [ f(t)] = G(is) + G(−is) if f(t) = f(−t) ( f even) F[ f(t)] = G(is) − G(−is) if f(t) = −f(−t) ( f odd) where G(s) = L[f(t)]. This result allows the use of the Laplacetransform tables to obtain the Fourier transforms.

Fc[f(t)] = g(s) =


π2

1 s+b s−b Fc[ f(at) cos bt] =
g
+ g
, a, b > 0 2a a a

3.

In brief, the condition for the Fourier transform to exist is that ∞ ∫ -∞ |f(t)| dt < ∞, although certain functions may have a Fourier transform even if this is violated.



0
10

Properties of the Fourier Cosine Transform Fc[ f(t)] = g(s). 1. Fc[af(t) + bh(t)] = aFc[ f(t)] + bFc[h(t)]. 2. Fc[ f(at)] = (1/a)g(s/a).





Example Fc[ f(t)], f(t) = 2 sin as

π s

FOURIER TRANSFORM

Example The function f(t) =

g(s) cos st ds

The Fourier sine transform Fs is obtainable by replacing the cosine by the sine in these integrals.

u*(z) y*(z) =

1 + a1z−1 + a2z−2

Then

∞

0

(1 + a1z + a2z ) y*(z) = u*(z) −1


π2

∞

f(t) cos st dt

0

TABLE 3-3

Fourier Transforms g(s)

2/π

f(t) t 2−t 0 t−1/2 0 (t − a)−1/2 (t2 + a2)−1 e−at

0
1
2 [2 cos s − 1 − cos 2s] s π1/2(2s)−1/2

0
π1/2(2s)−1/2 [cos as − sin as]

a>0

a πa−1e−as

e−at

2

a>0

a
s2 + a2 2 aπ1/2a−1/2e−s /4a

sin at
t

a>0



π/2 π/4 0

sa

Example The temperature in the semi-infinite rod 0 ≤ x < ∞ is determined by the differential equation ∂ ∂t = k(∂2 ∂x2) and the condition = 0 when t = 0, x ≥ 0; ∂ ∂x = −µ = constant when x = 0, t > 0. By using the Fourier cosine transform a solution may be found as 2µ

(x, t) =
π



∞

0

cos px
(1 − e−kp2t ) dp. p

MATRIX ALGEBRA AND MATRIX COMPUTATIONS REFERENCES: General (textbooks that cover at an introductory level a variety of topics that constitute a core of numerical methods for practicing engineers), 2, 3, 4, 22, 56, 59, 70, 77, 133, 135, 143, 150, 155, 219. Numerical solution of nonlinear equations, 153, 171, 237, 302. Numerical solution of ordinary differential equations, 76, 117, 127, 185, 257. Numerical solution of integral equa-

tions, 23, 26, 129, 162. Numerical solution of partial differential equations, 11, 76, 127, 133, 155, 210, 251, 286, 287, 213, 233, 253. Spline functions and applications, 38, 56, 70, 230. Finite elements and applications, 5, 29, 83, 130, 164, 210, 241, 281, 287, 303, 304. Fast Fourier transforms, 47, 56, 135, 238. Software, 187, 231.

MATRIX ALGEBRA AND MATRIX COMPUTATIONS MATRIX ALGEBRA Matrices A rectangular array of rows and n columns a11 a A = (aij) = 21 ⯗ am1



mn quantities, arranged in m



… …

a1n a2n

…

amn

is called a matrix. The elements aij may be real or complex. The notation aij means the element in the ith row and jth column, i is called the row index, j the column index. If m = n the matrix is said to be square and of order n. A matrix, even if it is square, does not have a numerical value, as a determinant does. However, if the matrix A is square, a determinant can be formed which has the same elements as the matrix A. This is called the determinant of the matrix and is written det (A) or |A|. If A is square and det (A) ≠ 0, A is said to be nonsingular; if det (A) = 0, A is said to be singular. A matrix A has rank r if and only if it has a nonvanishing determinant of order r and no nonvanishing determinant of order > r. Equality of Matrices Let A = (aij), B = (bij). Two matrices A and B are equal (=) if and only if they are identical; that is, they have the same number of rows and the same number of columns and equal corresponding elements (aij = bij for all i and j). Addition and Subtraction The operations of addition (+) and subtraction (−) of two or more matrices are possible if and only if they have the same number of rows and columns. Thus A  B = (aij  bij); i.e., addition and subtraction are of corresponding elements. Transposition The matrix obtained from A by interchanging the rows and columns of A is called the transpose of A, written A′ or AT.



1 2 Example A = 12 31 46 , AT = 3 1 4 6





Note that (AT )T = A.

Multiplication Let A = (aij), i = 1, . . . , m1; j = 1, . . . , m2. B = (bij), i = 1, . . . , n1, j = 1, . . . , n2. The product AB is defined if and only if the number of columns of A (m2) equals the number of rows of B(n1), i.e., n1 = m2. For two such matrices the product P = AB is defined by summing the element by element products of a row of A by a column of B. This is the row by column rule. Thus n1

pij =  aikbkj k=1

The resulting matrix has m1 rows and n2 columns. Example



3 1 5

2 1 4

0 −2

1 0

5 1



6 = 3



−4 −2 −8

3 1 5

17 6 29

24 9 42



It is helpful to remember that the element pij is formed from the ith row of the first matrix and the jth column of the second matrix. The matrix product is not commutative. That is, AB ≠ BA in general. Inverse of a Matrix A square matrix A is said to have an inverse if there exists a matrix B such that AB = BA = I, where I is the identity matrix of order n. 1 0.... 0 0 1.. ⯗ 1 0 0 ....0 1





−1

The inverse B is a square matrix of the order of A, designated by A . Thus AA−1 = A−1A = I. A square matrix A has an inverse if and only if A is nonsingular. Certain relations are important: (1)

(AB)−1 = B−1A−1

(2)

(AB)T = BTAT

(3)

(A−1)T = (AT )−1

(4)

(ABC)−1 = C−1B−1A−1

3-47

Scalar Multiplication Let c be any real or complex number. Then cA = (caij). Adjugate Matrix of a Matrix Let Aij denote the cofactor of the element aij in the determinant of the matrix A. The matrix BT where B = (Aij) is called the adjugate matrix of A written adj A = BT. The elements bij are calculated by taking the matrix A, deleting the ith row and jth column, and calculating the determinant of the remaining matrix times (−1)i + j. Then A−1 = adj A/ |A|. This definition may be used to calculate A−1. However, it is very laborious and the inversion is usually accomplished by numerical techniques shown under “Numerical Analysis and Approximate Methods.”



3

0

3

6

Example Let A = −1 2

adj A = BT =



0 6 −12



−1 1 3

Form B = (A i j), B =



0 −6 2

6 12 −2

−12 −18 6





−6 2 12 −2 ; |A| = 12 −18 6





−a 1 −3 ⁄ 2

0 adj A A−1 =
= a |A| −1

1 ⁄6 −1⁄6 a

Linear Equations in Matrix Form Every set of n nonhomogeneous linear equations in n unknowns a11 x1 + a12 x2 + ⋅⋅⋅ + a1n xn = b1 a21 x1 + a22 x2 + ⋅⋅⋅ + a2n xn = b2   an1 x1 + an2 x2 + ⋅⋅⋅ + ann xn = bn can be written in matrix form as AX = B, where A = (aij), X T = [x1 ⋅⋅⋅ xn], and BT = [b1 ⋅⋅⋅ bn]. The solution for the unknowns is X = A−1B. Special Square Matrices 1. A triangular matrix is a matrix all of whose elements above or below the main diagonal (set of elements a11, . . . , ann) are zero. If A is triangular, det (A) = a11. a22 . . . ann. 2. A diagonal matrix is one such that all elements both above and below the main diagonal are zero (i.e., aij = 0 for all i ≠ j). If all diagonal elements are equal, the matrix is called scalar. If A is diagonal, A = (aij), A−1 = (1/aij). 3. If aij = aji for all i and j (i.e., A = AT ), the matrix is symmetric. 4. If aij = −aji for i ≠ j but the aij are not all zero, the matrix is skew. 5. If aij = −aji for all i and j (i.e., aii = 0), the matrix is skew symmetric. 6. If AT = A−1, the matrix A is orthogonal. 7. If the matrix A* = (aij)T, aij = complex conjugate of aij, A* is the hermitian conjugate of A. 8. If A = A−1, A is involutory. 9. If A = A*, A is hermitian. 10. If A = −A*, A is skew hermitian. 11. If A−1 = A*, A is unitary. If A is any matrix, then AAT and ATA are square symmetric matrices, usually of different order.



5

Example Let A = 3



35 AAT = 22 8

2

1 3 4 1 −2 0



22 51 3

8 3 , 9



3 4 1 5

2 −2 0 1

38 13 18 17

13 21 7 18

18 7 10 5

5 0 1 5 , AT = 3 1 0



ATA =



 

17 18 5 26

Using a program such as MATLAB, these are easily calculated.

Matrix Calculus Differentiation Let the elements of A = [aij(t)] be differentiable dA daij(t) functions of t. Then
=
. dt dt





3-48

MATHEMATICS







interchange two rows of A to produce a (nonzero) pivot, such as if we start with 0 2 A= 1 6



sin t cos t dA cos t −sin t Example A = −cos , = . t sin t
sin t cos t dt



Integration The integral ∫ A dt = [∫ aij(t) dt].









2 t 2t Example A = t2 e2t , ∫ A dt = tt3/2 . /3 et

The matrix B = A − λI is called the characteristic (eigen) matrix of A. Here A is square of order n, λ is a scalar parameter, and I is the n × n identity. det B = det (A − λI) = 0 is the characteristic (eigen) equation for A. The characteristic equation is always of the same degree as the order of A. The roots of the characteristic equation are called the eigenvalues of A.



 

Example A = 13 28 , B = 13 28 − λ0 λ0 = 1−λ3 8−λ2



is the characteristic matrix and f(λ) = det (B) = det (A − λI) = (1 − λ)(8 − λ) − 6 = 2 − 9λ + λ2 = 0 is the characteristic equation. The eigenvalues of A are the roots of λ2 − 9λ + 2 = 0, which are (9 
73)/2.

A nonzero matrix Xi, which has one column and n rows, called a column vector satisfying the equation (A − λI)Xi = 0 and associated with the ith characteristic root λi is called an eigenvector. Vector and Matrix Norms To carry out error analysis for approximate and iterative methods for the solutions of linear systems, one needs notions for vectors in Rn and for matrices that are analogous to the notion of length of a geometric vector. Let Rn denote the set of all vectors with n components, x = (x1, . . . , xn). In dealing with matrices it is convenient to treat vectors in Rn as columns, and so x = (x1, . . . , xn)T; however, we shall here write them simply as row vectors. A norm on Rn is a real-valued function f defined on Rn with the following properties: 1. f(x) ≥ 0 for all x
Rn. 2. f(x) = 0 if and only if x = (0, 0, . . . , 0). 3. f(ax) = |a| f(x) for all real numbers a and x
Rn. 4. f(x + y) f(x) + f(y) for all x, y
Rn. The usual notation for a norm is f(x) =  x. The norm of a matrix is κ(A)  A  A−1 where

If A is a square matrix and if principal submatrices of A are all nonsingular, then we may choose P as the identity in the preceding factorization and obtain A = LU. This factorization is unique if L is normalized (as assumed previously), so that it has unit elements on the main diagonal. Solution of Ax = b by Using LU Factorization Suppose that the indicated system is compatible and that A = LU (the case PA = LU is similarly handled and amounts to rearranging the equations). Let z = Ux. Then Ax = LUx = b implies that Lz = b. Thus to solve Ax = b we first solve Lz = b for z and then solve Ux = z for x. This procedure does not require that A be invertible and can be used to determine all solutions of a compatible system Ax = b. Note that the systems Lz = b and Ux = z are both in triangular forms and thus can be easily solved. The LU decomposition is essentially a Gaussian elimination, arranged for maximum efficiency (Ref. 112). The chief reason for doing an LU decomposition is that it takes fewer multiplications than would be needed to find an inverse. Also, once the LU decomposition has been found, it is possible to solve for multiple right-hand sides with little increase in work. The multiplication count for an n × n matrix and m right-hand sides is 1 1 operation count =
n3 −
n + mn2 3 3 If an inverse is desired, it can be calculated by solving for the LU decomposition and then solving n problems with right-hand sides consisting of all zeroes except one entry. Thus 4n2/3 − n/3 multiplications are required for the inverse. The determinant is given by n

Det A =  aii(i) i=1

where aii(i) are the diagonal elements obtained in the LU decomposition. A tridiagonal matrix is one in which the only nonzero entries lie on the main diagonal and the diagonal just above and just below the main diagonal. The set of equations can be written as ai xi − 1 + bi xi + ci xi + 1 = di The LU decomposition is b1 = b1 for k=2,n do ak ak a′k =
, b′k = bk −
ck − 1 b′k − 1 b′k − 1 enddo d′1 = d1 for k=2,n do d′k = dk − a′k d′k − 1 enddo xn = d n′ / b′n for k=n−1,1 do d k′ − ck xk + 1 xk =

b′k enddo

n  A x A = supx ≠ 0
= maxk  |ajk| j=1  x

The norm is useful when doing numerical calculations. If the computer’s floating-point precision is 10−6, then κ = 106 indicates an illconditioned matrix. If the floating-point precision is 10−12 (double precision), then a matrix with κ = 1012 may be ill-conditioned. Two other measures are useful and are more easily calculated: (k) |det A| maxk |akk | Ratio =

V =

, αi = (a 2i1 + a 2i2 + . . . a 2in)1/2 (k) , mink |akk | α1 α2 . . . αn where akk(k) are the diagonal elements of the LU decomposition. MATRIX COMPUTATIONS The principal topics in linear algebra involve systems of linear equations, matrices, vector spaces, linear transformations, eigenvalues and eigenvectors, and least-squares problems. The calculations are routinely done on a computer. LU Factorization of a Matrix To every m × n matrix A there exists a permutation matrix P, a lower triangular matrix L with unit diagonal elements, and an m × n (upper triangular) echelon matrix U such that PA = LU. The Gauss elimination is in essence an algorithm to determine U, P, and L. The permutation matrix P may be needed since it may be necessary in carrying out the Gauss elimination to

The operation count for an n × n matrix with m right-hand sides is 2(n − 1) + m(3n − 2) If

|bi| > |ai| + |ci |

no pivoting is necessary, and this is true for many boundary-value problems and partial-differential equations. Sparse matrices are ones in which the majority of the elements are

NUMERICAL ANALYSIS AND APPROXIMATE METHODS zero. If the structure of the matrix is exploited, the solution time on a computer is greatly reduced. See Refs. 27, 55, 95, 96, 101, and 246. The conjugate gradient method is one method for solving sparse matrix problems, since it only involves multiplication of a matrix times a vector. Thus the sparseness of the matrix is easy to exploit. The conjugate gradient method is an iterative method that converges for sure in n iterations where the matrix is an n × n matrix. See Refs. 142 and 206. The singular value decomposition is useful when the matrix is singular or nearly singular (see Ref. 231). Matrix methods, in particular finding the rank of the matrix, can be used to find the number of independent reactions in a reaction set. If the stoichiometric numbers for the reactions and molecules are put in the form of a matrix, the rank of the matrix gives the number of independent reactions (see Ref. 13). Pivoting in Gauss Elimination It might seem that the Gauss elimination completely disposes of the problem of finding solutions of linear systems, and theoretically it does. In practice, however, things are not so simple. Example Assume three-decimal floating arithmetic (i.e., only the three most significant digits of any number are retained), and solve the following system by Gauss elimination: 0.000100x1 + 1.00x2 = 1.00 1.00x1 + 1.00x2 = 2.00 We obtain 0.100 × 10−3x1 + 0.100 × 101x2 = 0.100 × 101 −0.100 × 105x2 = −0.100 × 105 so that x2 = 1.00 and x1 = 0.00.

3-49

We check our solution by computing the residual vector r = b − Ax: r1 = 0.100 × 101 − 0.100 × 10−3x1 − 0.100 × 101x2 = 0.00 r2 = 0.200 × 101 − 0.100 × 101x1 − 0.100 × 101x2 = 0.100 × 101 The fact that r2 = 1 indicates that our “solution” is not very good. Indeed the exact solution of the system is x1 = 1.00010 and x2 = 0.99990, so the result computed by Gauss elimination is pretty bad. Now reverse the order of the equations (that is, pivot) and solve 0.100 × 101x1 + 0.100 × 101x2 = 0.200 × 101 0.100 × 101x2 = 0.100 × 101 so that x2 = 1.00 and x1 = 1.00. In this case the residual vector is r1 = 0.00 and r2 = 0.100 × 10−3, a considerable improvement over the previous result. In fact, the solution is as good as one could hope for by using three-digit arithmetic.

The moral of the preceding example is that the order of equations can make a large difference in how good an answer is obtained. It should be clear that the poor results in the first case are caused by having the large multiplier (0.100 × 101)/(0.100 × 10−3), which resulted from dividing by a relatively small a11. It is not enough just to avoid zero “pivots”; one must also avoid using pivots that are relatively small. This magnification of errors can be reduced if we arrange that the pivot at any stage is larger in magnitude than any remaining element in the column. If this is done, the multipliers will then be less than or equal to 1 in magnitude. Gauss elimination modified in this manner is called pivotal condensation or partial pivoting. This is routinely done by computer programs.

NUMERICAL APPROXIMATIONS TO SOME EXPRESSIONS APPROXIMATION IDENTITIES For the following relationships the sign  means approximately equal to, when X is small: Approximation 1
1X 1X 1+Y
1+YX 1X

Approximation X  X1

1 2 −n

(1  X)  1  nX

Approximation

Approximation

(1  X)n  1  nX

X (1  X)−1/2  1 
2

(a  X)2  a2  2aX

ez  1 + X

sin X  X(X rad)

tan X  X

2Y + X (Y  + X) 

Y 2

X2 X 2 +  X2  Y +

small
Y 2Y Y





NUMERICAL ANALYSIS AND APPROXIMATE METHODS REFERENCES: General (textbooks that cover at an introductory level a variety of topics that constitute a core of numerical methods for practicing engineers), 2, 3, 4, 22, 56, 59, 70, 77, 133, 135, 143, 150, 155, 219. Numerical solution of nonlinear equations, 153, 171, 237, 302. Numerical solution of ordinary differential equations, 76, 117, 127, 185, 257. Numerical solution of integral equations, 23, 26, 129, 162. Numerical solution of partial differential equations, 11, 76, 127, 133, 155, 210, 251, 286, 287, 213, 233, 253. Spline functions and applications, 38, 56, 70, 230. Finite elements and applications, 5, 29, 83, 130, 164, 210, 241, 281, 287, 303, 304. Fast Fourier transforms, 47, 56, 135, 238. Software, 187, 231.

INTRODUCTION The goal of approximate and numerical methods is to provide convenient techniques for obtaining useful information from mathematical formulations of physical problems. Often this mathematical statement is not solvable by analytical means. Or perhaps analytic solutions are available but in a form that is inconvenient for direct interpretation

numerically. In the first case it is necessary either to attempt to approximate the problem satisfactorily by one which will be amenable to analysis, to obtain an approximate solution to the original problem by numerical means, or to use the two techniques in combination. Numerical techniques therefore do not yield exact results in the sense of the mathematician. Since most numerical calculations are inexact, the concept of error is an important feature. The error associated with an approximate value is defined as True value = approximate value + error The four sources of error are as follows: 1. Gross errors. These result from unpredictable human, mechanical, or electrical mistakes. 2. Round-off errors. These are the consequence of using a number specified by m correct digits to approximate a number which requires more than m digits for its exact specification. For example, approximate the irrational number
2 by 1.414. Such errors are often

3-50

MATHEMATICS

present in experimental data, in which case they may be called inherent errors, due either to empiricism or to the fact that the computer dictates the number of digits. Such errors may be especially damaging in areas such as matrix inversion or the numerical solution of partial differential equations when the number of algebraic operations is extremely large. 3. Truncation errors. These errors arise from the substitution of a finite number of steps for an infinite sequence of steps which would yield the exact result. To illustrate this error consider the infinite series for e−x ⋅ e−x = 1 − x + x2/2 − x3/6 + ET(x), where ET is the truncation error, ET = (1/24)e−εx4, 0 < ε < x. If x is positive, ε is also positive. Hence e−ε < 1. The approximation e−x ≈ 1 − x + x2/2 − x3/6 is in error by a positive amount smaller than (1/24)x4. 4. Inherited errors. These arise as a result of errors occurring in the previous steps of the computational algorithm. The study of errors in a computation is related to the theory of probability. In what follows a relation for the error will be given in certain instances. NUMERICAL SOLUTION OF LINEAR EQUATIONS See the section entitled “Matrix Algebra and Matrix Computation.” NUMERICAL SOLUTION OF NONLINEAR EQUATIONS IN ONE VARIABLE Special Methods for Polynomials Consider a polynomial equation of degree n: P(x) = a0 xn + a1xn − 1 + a2 xn − 2 + ⋅⋅⋅ + an − 1x + an = 0

(3-71)

with real coefficients. P(x) has exactly n roots, which may be real or complex. If all the coefficients of P(x) are integers, then any rational root, say, r/s (r, s integers, having no common divisors) of P(x), must be such that r is an integral divisor of an and s is an integral divisor of a0. Further, any polynomial with rational coefficients may be converted into one with integral coefficients by multiplying by the lowest common multiple of the denominators of the coefficients. Example 3x4 − 5⁄ 3 x2 + 1⁄5 x − 2 = 0. The lowest common multiple of the denominator is 15. Thus multiplying by 15 (which does not change the roots) gives 45x4 − 25x2 + 3x − 30 = 0. The only possible rational roots r/s are such that r may have the values 30, 15, 10, 6, 5, 3, 2, 1. s may have the values 45, 15, 9, 5, 3, 1. The possible rational roots may then be formed from all possible quotients, having no common factor. In addition to these results, one can obtain an upper and lower bound for the real roots by the following device: If a0 > 0 in Eq. (3-71) and if in Eq. (3-71) the first negative coefficient is preceded by k coefficients which are positive or zero, and if G is the greatest of the absolute values of the negative coefficients, then each real root is less k than 1 +
 G/a 0. Example P(x) = x5 + 3x4 − 7x2 − 40x + 2 = 0. Here a03 = 1, G = 40, and k = 3

Example f(x) = x4 − 13x2 + 4x − 2 = 0 has three changes in sign; therefore, there are either three or one positive roots. f(−x) = x4 − 13x2 − 4x − 2 has one change in sign. Therefore, there is one negative root. General Methods for Nonlinear Equations in One Variable Successive Substitutions Let f(x) = 0 be the nonlinear equation to be solved. If this is rewritten as x = F(x), then an iterative scheme can be set up in the form xk + 1 = F(xk). To start the iteration an initial guess must be obtained graphically or otherwise. The convergence or divergence of the procedure depends upon the method of writing x = F(x), of which there will usually be several forms. However, if a is a root of f(x) = 0, and if |F′(a)| < 1, then for any initial approximation sufficiently close to a, the method converges to a. This process is called first order because the error in xk + 1 is proportional to the first power of the error in xk for large k. Example f(x) = x3 − x − 1 = 0. A rough plot shows a real root of approximately 1.3. The equation can be written in the form x = F(x) in several ways such as x = x3 − 1, x = 1/(x2 − 1), and x = (1 + x)1/3. In the first case F′(x) = 3x2 = 5.07 at x = 1.3, in the second F(1.3) = −5.46, and only in the third case is F′(1.3) < 1. Hence only the third iterative process has a chance to converge. This is illustrated in the following table. Step k

x = x3 − 1

x = 1/(x2 − 1)

x = (1 + x)1/3

0 1 2 3 4

1.3 1.197 0.7150 −0.6345

1.3 1.4493 0.9088 −5.742

1.3 1.32 1.3238 1.3245 1.3247

Another way of writing the equation is xk + 1 = xk + β f(xk). The choice of β is made such that |1 + β df/dx(a)| < 1. Convergence is guaranteed by the theorem given for simultaneous equations.

Methods of Perturbation Let f(x) = 0 be the equation. In general, the iterative relation is xk + 1 = xk − [ f(xk)/ak] where the iteration begins with x0 as an initial approximation and αk as some functional. Newton-Raphson Procedure This variant chooses αk = f ′(xk) where f ′ = df/dx and geometrically consists of replacing the graph of f(x) by the tangent line at x = xk in each successive step. If f′(x) and f ″(x) have the same sign throughout an interval a ≤ x ≤ b containing the solution, with f(a), f(b) of opposite signs, then the process converges starting from any x0 in the interval a ≤ x ≤ b. The process is second order. (0.5)x − 0.5 0.3

Example f(x) = x − 1 +

f′ (x) = 1 − 2.3105[0.5]x An approximate root (obtained graphically) is 2.

since we must supply 0 as the coefficient for x . Thus 1 +
40 ≈ 4.42 is an upper bound for the real roots.

Step k

xk

f(xk)

f′(xk)

A lower bound to the real roots may be found by applying the criterion to the equation P(−x).

0 1 2

2 1.6054 1.4632

0.1667 0.0342 0.0055

0.4224 0.2407 0.1620

3

Example P(−x) = −x5 + 3x4 − 7x2 + 40x + 2 = 0, which is equivalent to x5 − 3x4 + 7x2 − 40x − 2 = 0 since a0 must be +. Then a0 = 1, G = 40, and k = 1. Hence −(1 + 40) = −41 is a lower bound. Thus all real roots −41 < r < 4.42. One last result is helpful in getting an estimate of how many positive and negative real roots there are. Descartes Rule The number of positive real roots of a polynomial with real coefficients is either equal to the number of changes in sign v or is less than v by a positive even integer. The number of negative roots of f(x) is either equal to the number of variations of sign of f(−x) or is less than this by a positive even integer.

Method of False Position This variant is commenced by finding x0 and x1 such that f(x0), f(x1) are of opposite signs. Then α1 = slope of secant line joining [x0, f(x0)] and [x1, f(x1)] so that x1 − x0 x2 = x1 −

f(x1) f(x1) − f(x0) In each following step αk is the slope of the line joining [xk, f(xk)] to the most recently determined point where f(xj) has the opposite sign from that of f(xk). This method is of first order. If one uses the most recently determined point (regardless of sign), the method is a secant method.

NUMERICAL ANALYSIS AND APPROXIMATE METHODS Method of Wegstein This is a variant of the method of successive substitutions which forces and/or accelerates convergence. The iterative procedure xk + 1 = F(xk) is revised by setting ˆxk + 1 = F(xk) and then taking xk + 1 = qxk + (1 − q)ˆxk + 1, where q is a suitably chosen number which may be taken as constant throughout or may be adjusted at each step. Wegstein found that suitable q’s are: Behavior of successive substitution process

Range of optimum q

Oscillatory convergence Oscillatory divergence Monotonic convergence Monotonic divergence

0
At each step q may be calculated to give a locally optimum value by setting xˆ k + 1 − xˆ k q =

xˆ k + 1 − 2 xˆ k + 1 + xˆ k− 1 The Wegstein method is a secant method applied to g(x)  x − F(x). Numerical Solution of Simultaneous Nonlinear Equations The techniques illustrated here will be demonstrated for two simultaneous equations f(x, y) = 0, g(x, y) = 0. They immediately generalize to more than two simultaneous equations. Method of Successive Substitutions Write a system of equations as αi = fi(α),

or α = f(α)

The following theorem guarantees convergence. Let α be the solution to αi = fi(α). Assume that given h > 0, there exists a number 0 < µ < 1 such that n ∂fi
≤ µ for |xi − αi| < h, i = 1, . . . , n  ∂x j=1 j

 

x Then

k+1 i

= fi(x ik)

x ik → αi

Step k

xk

yk

f(xk, yk)

g(xk, yk)

0 1 2 3

0.4 0.43673 0.42672 0.42925

0.3 0.24184 0.25573 0.24943

−0.26 0.078 −0.0170 0.0077

0.07 0.0175 −0.007 0.0010

Method of Continuity (Homotopy) In the case of n equations in n unknowns, when n is large, determining the approximate solution may involve considerable effort. In such a case the method of continuity is admirably suited for use on digital computers. It consists basically of the introduction of an extra variable into the n equations fi(x1, x2, . . . , xn) = 0

i = 1, . . . , n

(3-72)

and replacing them by fi(x1, x2, . . . , xn, λ) = 0

i = 1, . . . , n

(3-73)

where λ is introduced in such a way that the functions (3-73) depend in a simple way upon λ and reduce to an easily solvable system for λ = 0 and to the original equations (3-72) for λ = 1. A system of ordinary differential equations, with independent variable λ, is then constructed by differentiating Eqs. (3-73) with respect to λ. There results n ∂fi dxf ∂fi (3-74)

+
=0  ∂λ j = 1 ∂xj dλ where x1, . . . , xn are considered as functions of λ. Equations (3-74) are integrated, with initial conditions obtained from Eqs. (3-73) with λ = 0, from λ = 0 to λ = 1. If the solution can be continued to λ = 1, the values of x1, . . . , xn for λ = 1 will be a solution of the original equations. If the integration becomes infinite, the parameter λ must be introduced in a different fashion. Integration of the differential equations (which are usually nonlinear in λ) may be accomplished by using techniques described under “Numerical Solution of Ordinary Differential Equations.” Other Methods Other methods can be found in the literature. See Ref. 66. INTERPOLATION AND FINITE DIFFERENCES

as k increases (see Ref. 106). Newton-Raphson Method To solve the set of equations Fi(x1, x2, . . . , xn) = 0,

or Fi({xj}) = 0,

or Fi(x) = 0

one uses a truncated Taylor series to give n ∂F 0 = Fi({x k}) + 
i j = 1 ∂xj



xk

(x jk + 1 − x jk)

Linear Interpolation If a function f(x) is approximately linear in a certain range, then the ratio f(x1) − f(x0)

= f [x0, x1] x1 − x0 is approximately independent of x0, x1 in the range. The linear approximation to the function f(x), x0 < x < x1 then leads to the interpolation formula f(x) ≈ f(x0) + (x − x0)f [x0, x1] x − x0 ≈ f(x0) +
[ f(x1) − f(x0)] x1 − x0 1 ≈
[(x1 − x) f(x0) − (x0 − x)f(x1)] x1 − x0

Thus one solves iteratively from one point to another. n

A

k ij

(x

k+1 j

− x ) = −Fi({x }) k

k j

j=1

where

3-51

∂F A kij =
i ∂xj



xk

This method requires solution of sets of linear equations until the functions are zero to some tolerance or the changes of the solution between iterations is small enough. Convergence is guaranteed provided the norm of the matrix A is bounded, F(x) is bounded for the initial guess, and the second derivative of F(x) with respect to all variables is bounded. See Refs. 106 and 155. Example f(x, y) = 4x2 + 6x − 4xy + 2y2 − 3 g(x, y) = 2x2 − 4xy + y2

Example Find cosh 0.83 by linear interpolation given cosh 0.8 and cosh 0.9. xi

f(xi)

xi − 0.83

0.8 0.9

1.33743 1.43309

−0.03 +0.07

f(0.83) ≈ 1/0.10[(0.07)(1.33743) − (−0.03)(1.43309)] f(0.83) ≈ 1.36613

By plotting one of the approximate roots is found to be x0 = 0.4, y0 = 0.3. At this point there results ∂f/∂x = 8, ∂f/∂y = −0.4, ∂g/∂x = 0.4, and ∂g/∂y = −1.

Since the true five-place value is 1.36468, it is seen that here linear interpolation gives three significant figures.

8(xk + 1 − xk) − 0.4( yk + 1 − yk) = +0.26

Divided Differences of Higher Order and Higher-Order Interpolation The first-order divided difference f [x0, x1] was defined previously. Divided differences of second and higher order are defined iteratively by

0.4(xk + 1 − xk) − 1( yk + 1 − yk) = −0.07 The first few iteration steps are as follows:

3-52

MATHEMATICS f [x1, x2] − f[x0, x1] f [x0, x1, x2] =

x2 − x0 ⯗ f [x1, . . . , xk] − f[x0, x1, . . . , xk − 1] f [x0, x1, . . . , xk] =



xk − x0

and a convenient form for computational purposes is k′ f(xj) f [x0, x1, . . . , xk] = 


j = 0 (xj − x0)(xj − x1) ⋅⋅⋅ (xj − xk) for any k ≥ 0, where the ′ means that the term (xj − xj) is omitted in the denominator. For example, f(x0) f(x1) f(x2) f [x0, x1, x2] =

+

+

(x0 − x1)(x0 − x2) (x1 − x0)(x1 − x2) (x2 − x0)(x2 − x1) If the accuracy afforded by a linear approximation is inadequate, a generally more accurate result may be based upon the assumption that f(x) may be approximated by a polynomial of degree 2 or higher over certain ranges. This assumption leads to Newton’s fundamental interpolation formula with divided differences f(x) ≈ f(x0) + (x − x0) f[x0, x1] + (x − x0)(x − x1) f [x0, x1, x2] + ⋅⋅⋅ + (x − x0)(x − x1) ⋅⋅⋅ (x − xn − 1) f [x0, x1, . . . , xn] + En(x) 1 En(x) = error =
f n + 1(ε)π(x) (n + 1)! where minimum (x0, . . . , x) < ε < maximum (x0, x1, . . . , xn, x) and π(x) = (x − x0)(x − x1) ⋅⋅⋅ (x − xn). In order to use the previous equation most effectively one may first form a divided-difference table. For example, for third-order interpolation the difference table is

where

and the finite interpolation formula through the points y0, y1, . . . , yn is written as follows: α (α − 1) α (α − 1) ⋅⋅⋅ (α − n + 1) yα = y0 + α ∆y0 +
∆2y0 + ⋅⋅⋅ +


∆ny0 2! n! Keeping only the first two terms gives a straight line through (x0, y0)–(x1, y1); keeping the first three terms gives a quadratic function of position going through those points plus (x2, y2). The value α = 0 gives x = x0; α = 1 gives x = x1, and so on. Equally Spaced Backward Differences Backward differences are defined by ∇yn = yn − yn − 1 ∇ 2yn = ∇yn − ∇yn − 1 = yn − 2 yn − 1 + yn − 2 The interpolation polynomial of order n through the points y0, y−1, . . . , y−n is α (α + 1) α (α + 1) ⋅⋅⋅ (α + n − 1) yα = y0 + α ∇y0 +
∇2y0 + ⋅⋅⋅ +


∇ny0 2! n! The value of α = 0 gives x = x0; α = −1 gives x = x−1, and so on. Alternatively, the interpolation polynomial of order n through the points y1, y0, y−1, . . . , y−n is α (α − 1) yα = y1 + (α − 1) ∇y1 +
∇ 2y1 2! (α − 1) α (α + 1) ⋅⋅⋅ (α + n − 2) + ⋅⋅⋅ +



∇ny1 n! Now α = 1 gives x = x1; α = 0 gives x = x0. Central Differences The central difference denoted by h h δf(x) = f x +
− f x −
2 2



 











h h δ2f(x) = δn − 1 f x +
− δn − 1 f x −
2 2 is useful for calculating at the interior points of tabulated data. Lagrange Interpolation Formulas A global polynomial is defined over the entire region of space m

where each entry is given by taking the difference between diagonally adjacent entries to the left, divided by the abscissas corresponding to the ordinates intercepted by the diagonals passing through the calculated entry. Equally Spaced Forward Differences If the ordinates are equally spaced, i.e., xj − xj − 1 = ∆x for all j, then the first differences are denoted by ∆f(x0) = f(x1) − f(x0) or ∆y0 = y1 − y0, where y = f(x). The differences of these first differences, called second differences, are denoted by ∆2y0, ∆2y1, . . . , ∆2yn. Thus ∆2y0 = ∆y1 − ∆y0 = y2 − y1 − y1 + y0 = y2 − 2y1 + y0 And in general j

∆ y0 =  (−1) j

n

n=0



j y n j−n



j! j where =
= binomial coefficients. n n!( j − n)! If the ordinates are equally spaced, xn + 1 − xn = ∆x yn = y(xn) then the first and second differences are denoted by ∆yn = yn + 1 − yn ∆2yn = ∆yn + 1 − ∆yn = yn + 2 − 2yn + 1 + yn A new variable is defined xα − x0 α=
∆x

Pm(x) =  cj x j j=0

This polynomial is of degree m (highest power is x m) and order m + 1 (m + 1 parameters {cj}). If we are given a set of m + 1 points y1 = f(x1), y2 = f(x2), . . . , ym + 1 = f(xm + 1) then Lagrange’s formula gives a polynomial of degree m that goes through the m + 1 points: (x − x2) (x − x3) ⋅⋅⋅ (x − xm + 1) Pm(x) =



y1 (x1 − x2) (x1 − x3) ⋅⋅⋅ (x1 − xm + 1) (x − x1) (x − x3) ⋅⋅⋅ (x − xm + 1) +



y2 + ⋅⋅⋅ (x2 − x1) (x2 − x3) ⋅⋅⋅ (x2 − xm + 1) (x − x1) (x − x2) ⋅⋅⋅ (x − xm + 1) +



ym + 1 (xm + 1 − x1) (xm + 1 − x2) ⋅⋅⋅ (xm + 1 − xm) Note that each coefficient of yj is a polynomial of degree m that vanishes at the points {xj} (except for one value of j) and takes the value of 1.0 at that point: Pm(xj) = yj,

j = 1, 2, . . . , m + 1

If the function f(x) is known, the error in the approximation is, per Ref. 14, |xm + 1 − x1|m + 1 |error(x)| ≤

maxx1 ≤ x ≤ xm + 1 | f (m + 2)(x)| (m + 2)! The evaluation of Pm(x) at a point other than at the defining points can be made with Neville’s algorithm (Ref. 231). Let P1 be the value at x of the unique function passing through the point (x1, y1); or P1 = y1. Let

NUMERICAL ANALYSIS AND APPROXIMATE METHODS P12 be the value at x of the unique polynomial passing through the points x1 and x2. Likewise, Pijk . . . r is the unique polynomial passing through the points xi, xj, xk, . . . xr. Then use the table x1 y1 = P1 x2 y2 = P2 x3 y3 = P3 x4 y4 = P4

3-53

f 0′ = 1/10h [−2f−2 − f−1 + f1 + 2f2] If derivatives are required at end points, with all points and ordinates to one side, the derivatives are f0′ = 1/20h [−21 f0 + 13 f1 + 17 f2 − 9 f3]

P12

P123

P23

P234

P34

f0′ = 1/20h [21 f0 − 13 f−1 − 17 f−2 + 9 f−1] f−1 ′ = 1/20h [ 11 f0 − 3 f−1 − 7 f−2 − f−1]

These entries are defined using (x − xi + m) Pi(i + 1) . . . (i + m − 1) + (xi − x) P(i + 1)(i + 2) . . . (i + m) Pi(i + 1) . . . (i + m) =





xi − xi + m For example, consider P1234. The terms on the right-hand side involve P123 and P234. The “parents,” P123 and P234, already agree at points 2 and 3. Here i = 1, m = 3; thus, the parents agree at xi + 1, . . . , xi + m − 1 already. The formula makes Pi(i + 1) . . . (i + m) agree with the function at the additional points xi + m and xi. Thus, Pi(i + 1) . . . (i + m) agrees with the function at all the points {xi, xi + 1, . . . xi + m}. NUMERICAL DIFFERENTIATION Numerical differentiation should be avoided whenever possible, particularly when data are empirical and subject to appreciable observation errors. Errors in data can affect numerical derivatives quite strongly; i.e., differentiation is a roughening process. When such a calculation must be made, it is usually desirable first to smooth the data to a certain extent. Use of Interpolation Formula If the data are given over equidistant values of the independent variable x, an interpolation formula such as the Newton formula (see Refs. 143 and 185) may be used and the resulting formula differentiated analytically. If the independent variable is not at equidistant values, then Lagrange’s formulas must be used. By differentiating three- and five-point Lagrange interpolation formulas the following differentiation formulas result for equally spaced tabular points: Three-Point Formulas Let x0, x1, x2 be the three points. 1 h2 f ′(x0) =
[−3f(x0) + 4f(x1) − f(x2)] +
f ′″(ε) 2h 3 1 h2 f ′(x1) =
[−f(x0) + f(x2)] −
f ′″(ε) 2h 6 h2 1 f ′(x2) =
[ f(x0) − 4f(x1) + 3f(x2)] +
f ′″(ε) 2h 3 where the last term is an error term min. xj < ε < max. xj. j

f1′ = 1/20h [−11 f0 + 3 f1 + 7 f2 + f3]

P1234

j

Smoothing Techniques These techniques involve the approximation of the tabular data by a least-squares fit of the data by using some known functional form, usually a polynomial (for the concept of least squares see “Statistics”). In place of approximating f(x) by a single least-squares polynomial of degree n over the entire range of the tabulation, it is often desirable to replace each tabulated value by the value taken on by a least-squares polynomial of degree n relevant to a subrange of 2M + 1 points centered, when possible, at the point for which the entry is to be modified. Thus each smoothed value replaces a tabulated value. Let fj = f(xj) be the tabular points and yj = smoothed values. First-Degree Least Squares with Three Points y0 = j[5f0 + 2 f1 − f2] y1 = s[ f0 + f1 + f2] y2 = j[−f0 + 2 f1 + 5f2] Second-Degree Least Squares with Five Points For five evenly spaced points x−2, x−1, x0, x1, and x2 (separated by distance h) and their ordinates f−2, f−1, f0, f1, and f2, assume a parabola is fit by least squares. Then the derivative at the center point is

Numerical Derivatives The results given above can be used to obtain numerical derivatives when solving problems on the computer, in particular for the Newton-Raphson method and homotopy methods. Suppose one has a program, subroutine, or other function evaluation device that will calculate f given x. One can estimate the value of the first derivative at x0 using f [x0(1 + ε)] − f [x0] df
≈

dx x0 ε (a first-order formula) or f [x0(1 + ε)] − f [x0(1 − ε)] df
≈


dx x0 2ε





(a second-order formula). The value of ε is important; a value of 10−6 is typical, but smaller or larger values may be necessary depending on the computer precision and the application. One must also be sure that the value of x0 is not zero and use a different increment in that case. NUMERICAL INTEGRATION (QUADRATURE) A multitude of formulas have been developed to accomplish numerical integration, which consists of computing the value of a definite integral from a set of numerical values of the integrand. Newton-Cotes Integration Formulas (Equally Spaced Ordinates) for Functions of One Variable The definite integral b ∫ a f(x) dx is to be evaluated. Trapezoidal Rule This formula consists of subdividing the interval a ≤ x ≤ b into n subintervals a to a + h, a + h to a + 2h, . . . and replacing the graph of f(x) by the result of joining the ends of adjacent ordinates by line segments. If fj = f(xj) = f(a + jh), f0 = f(a), fn = f(b), the integration formula is b h f(x) dx =
[ f0 + 2f1 + 2f2 + ⋅⋅⋅ + 2fn − 1 + fn] + En a 2



nh3 (b − a)3 |En| =
|f″ (ε)| =
| f ″(ε)| a<ε


nh5 (b − a)5 (IV) |En| =
| f (IV)(ε)| =
| f (ε)| a<ε
x = a + (b − a)u,

dx = (b − a)du

f(x) dx = (b − a) f(u) du b

a

1

0

3-54

MATHEMATICS

f(u) du =  W f(u ) 1

m

i

0

i

i=1

The quadrature is exact when f is a polynomial of degree 2m − 1 in x. Because there are m weights and m Gauss points, we have 2m parameters that are chosen to exactly represent a polynomial of degree 2m − 1, which has 2m parameters. The Gauss points and weights are given in the table. Gaussian Quadrature Points and Weights m

ui

Wi

2

0.21132 48654 0.78867 51346 0.11270 16654 0.50000 00000 0.88729 83346 0.06943 18442 0.33000 94783 0.66999 05218 0.93056 81558 0.04691 00771 0.23076 53450 0.50000 00000 0.76923 46551 0.95308 99230

0.50000 00000 0.50000 00000 0.27777 77778 0.44444 44445 0.27777 77778 0.17392 74226 0.32607 25774 0.32607 25774 0.17392 74226 0.11846 34425 0.23931 43353 0.28444 44444 0.23931 43353 0.11846 34425

3

4

5

Example Evaluate the same integral (3-75) using the trapezoid rule and then apply the Romberg method. To achieve four-digit accuracy, any result from J2 through L1 are suitable, even though the base results (I1–I4) are not that close. I1 = 0.967058363

I2 = 0.923704741 J1 = 0.909253534

f(x) dx = b

1/a

a

1/b

b

g2(x)

sin x dx

f(x, y) dx dy =

g1(x)

(3-75)

Using the Gaussian quadrature formulas gives the following values for various values of m. Clearly, three internal points, requiring evaluation of the integrand at only three points, gives excellent results.

1 2 3 4 5

0.908185 0.910089 0.909336367 0.909330666 0.909330674

Romberg’s Method Romberg’s method uses extrapolation techniques to improve the answer (Ref. 231). If we let I1 be the value of the integral obtained using interval size h = ∆x, and I2 be the value of I obtained when using interval size h/2, and I0 the true value of I, then the error in a method is approximately hm, or I1 ≈ I0 + chm h I2 ≈ I0 + c
2

 

m

Replacing the ≈ by an equality (an approximation) and solving for c and I0 gives 2mI2 − I1 I0 =
2m − 1 This process can also be used to obtain I1, I2, . . . , by halving h each time, and then calculating new estimates from each pair, calling them J1, J2, . . . ; that is, in the formula above, replace I0 with J1. The formulas are reapplied for each pair of J to obtain K1, K2, . . . The process continues until the required tolerance is obtained. I1

I2 I3 I4 J1 J2 J3 K1 K2 L1

Romberg’s method is most useful for a low-order method (small m) because significant improvement is then possible.

b



g2(x)

f(x, y) dy

Gaussian quadrature can also be used in two dimensions, provided the integration is on a square or can be transformed to one. (Domain transformations might be used to convert the domain to a square.)

f(x, y) dx dy =  W 1

I

G(x) dx g1(x)

0

m

ab > 0

a

G(x) =

1

−x

 

1 1
2 f
du u u

Two-Dimensional Formula Two-dimensional integrals can be calculated by breaking down the integral into one-dimensional integrals. a

e

I4 = 0.910227902 J3 = 0.909330366 K2 = 0.909331898 L1 = 0.909325916

Singularities When the integrand has singularities, a variety of techniques can be tried. The integral may be divided into one part that can be integrated analytically near the singularity and another part that is integrated numerically. Sometimes a change of argument allows analytical integration. Series expansion might be helpful, too. When the domain is infinite, it is possible to use Gauss-Legendre or Gauss-Hermite quadrature. Also a transformation can be made (Ref. 26). For example, let u = 1/x and then

Example Calculate the value of the following integral. I=

I3 = 0.912920511 J2 = 0.909325768 K1 = 0.909349846

1

mx

xi

0

0

i=1

my

W

yj

f(xi, yj)

j=1

NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS AS INITIAL VALUE PROBLEMS A differential equation for a function that depends on only one variable, often time, is called an ordinary differential equation. The general solution to the differential equation includes many possibilities; the boundary or initial conditions are needed to specify which of those are desired. If all conditions are at one point, then the problem is an initial value problem and can be integrated from that point on. If some of the conditions are available at one point and others at another point, then the ordinary differential equations become two-point boundary value problems, which are treated in the next section. Initial value problems as ordinary differential equations arise in control of lumped parameter models, transient models of stirred tank reactors, and in all models where there are no spatial gradients in the unknowns. A higher-order differential equation y(n) + F(y(n − 1), y(n − 2), . . . , y′, y) = 0 with initial conditions Gi(y(n − 1)(0), y(n − 2)(0), . . . , y(0), y(0)) = 0,

i = 1, . . . , n

can be converted into a set of first-order equations using d(i − 1)y d dyi − 1 =
y(i − 2) =
yi  y(i − 1) =
dt(i − 1) dt dt The higher-order equation can be written as a set of first-order equations. dy1
= y2 dt dy2
= y3 dt

NUMERICAL ANALYSIS AND APPROXIMATE METHODS dy3
= y4 dt ... dyn
= −F(yn − 1, yn − 2, . . . , y2, y1) dt The initial conditions would have to be specified for variables y1(0), . . . , yn(0), or equivalently y(0), . . . , y(n − 1)(0). The set of equations is then written as dy
= f(y, t) dt All the methods in this section are described for a single equation; the methods apply to multiple equations. See Refs. 106 and 185 for more details. Euler’s method is first-order. yn + 1 = yn + ∆t f(yn) and errors are proportional to ∆t. The second-order Adams-Bashforth method is ∆t yn + 1 = yn +
[3 f(yn) − f(yn − 1)] 2 Errors are proportional to ∆ t2, and high-order methods are available. Notice that the higher-order explicit methods require knowing the solution (or the right-hand side) evaluated at times in the past. Since these were calculated to get to the current time, this presents no problem except for starting the problem. Then it may be necessary to use Euler’s method with a very small step size for several steps in order to generate starting values at a succession of time points. The error terms, order of the method, function evaluations per step, and stability limitations are listed in Ref. 106. The advantage of the high-order Adams-Bashforth method is that it uses only one function evaluation per step yet achieves high-order accuracy. The disadvantage is the necessity of using another method to start. Runge-Kutta methods are explicit methods that use several function evaluations for each time step. Runge-Kutta methods are traditionally written for f(t, y). The first-order Runge-Kutta method is Euler’s method. A second-order Runge-Kutta method is ∆t yn + 1 = yn +
[ f n + f(tn + ∆t, yn + ∆t f n)] 2 while the midpoint scheme is also a second-order Runge-Kutta method. ∆t ∆t yn + 1 = yn + ∆t f tn +
, yn +
f n 2 2 A popular fourth-order Runge-Kutta method is the Runge-KuttaFeldberg formulas (Ref. 111), which have the property that the method is fourth-order but achieves fifth-order accuracy. The popular integration package RKF45 is based on this method.



k1 = ∆t f(tn, yn) ∆t k1 k2 = ∆t f tn +
, yn +
4 4

 



 









12 1932 7200 7296 k4 = ∆t f tn +
∆t, yn +
k1 −
k2 +
k3 13 2197 2197 2197

439 3680 845 k5 = ∆t f tn + ∆t, yn +
k1 − 8k2 +
k3 −
k4 216 513 4104

∆t 8 3544 1859 11 k6 = ∆t f tn +
, yn −
k1 + 2k2 −
k3 +
k4 −
k5 2 27 2565 4104 40



1 25 1408 2197 yn + 1 = yn +
k1 +
k3 +
k4 −
k5 216 2565 4104 5

2 16 6656 28561 9 zn + 1 = yn +
k1 +
k3 +
k4 −
k5 +
k6 135 12825 56430 50 55 The value of yn + 1 − zn + 1 is an estimate of the error in yn + 1 and can be used in step-size control schemes. Usually one would use a high-order method to achieve high accuracy. The Runge-Kutta-Feldberg method is popular because it is high order and does not require a starting method (as does an AdamsBashforth method). However, it does require four function evaluations per time step, or four times as many as a fourth-order Adams-Bashforth method. For problems in which the function evaluations are a significant portion of the calculation time, this might be important. Given the speed of present-day computers and the widespread availability of microcomputers (which can be run while you are doing something else, if need be), the efficiency of the methods is most important only for very large problems that are going to be solved many times. For other problems, the most important criterion for choosing a method is probably the time the user spends setting up the problem. The stability limits for the explicit methods are based on the largest eigenvalue of the linearized system of equations (see Ref. 106). For linear problems, the eigenvalues do not change, so that the stability and oscillation limits must be satisfied for every eigenvalue of the matrix A. When solving nonlinear problems, the equations are linearized about the solution at the local time, and the analysis applies for small changes in time, after which a new analysis about the new solution must be made. Thus, for nonlinear problems, the eigenvalues keep changing, and the largest stable time step changes, too. The stability limits are: Euler method, λ ∆t ≤ 2 Runge-Kutta, 2nd order, λ ∆t < 2 Runge-Kutta-Feldberg, λ ∆t < 3.0 Richardson extrapolation can be used to improve the accuracy of a method. Suppose we step forward one step ∆t with a pth-order method. Then redo the problem, this time stepping forward from the same initial point, but in two steps of length ∆t/2, thus ending at the same point. Call the solution of the one-step calculation y1 and the solution of the two-step calculation y2. Then an improved solution at the new time is given by 2 py2 − y1 y=
2p − 1 This gives a good estimate provided ∆t is small enough that the method is truly convergent with order p. This process can also be repeated in the same way Romberg’s method was used for quadrature. The error term in the various methods can be used to deduce a step size that will give a user-specified accuracy. Most packages today are based on a user-specified tolerance; the step-size is changed during the calculation to achieve that accuracy. The accuracy itself is not guaranteed, but it improves as the tolerance is decreased. Implicit Methods By using different interpolation formulas involving yn + 1, it is possible to derive implicit integration methods. Implicit methods result in a nonlinear equation to be solved for yn + 1 so that iterative methods must be used. The backward Euler method is a first-order method. yn + 1 = yn + ∆t f(yn + 1)

3 3 9 k3 = ∆t f t +
∆t, yn +
k1 +
k2 8 32 32 n

3-55



Errors are proportional to ∆t for small ∆t. The trapezoid rule is a second-order method. ∆t yn + 1 = yn +
[ f(yn) + f(yn + 1)] 2 Errors are proportional to ∆ t2 for small ∆ t. When the trapezoid rule is used with the finite difference method for solving partial differential equations, it is called the Crank-Nicolson method. The implicit methods are stable for any step size but do require the solution of a set of nonlinear equations, which must be solved iteratively. The set of equations can be solved using the successive substitution method or Newton-Raphson method. See Ref. 36 for an application to dynamic distillation problems. The best packages for stiff equations (see below) use Gear’s back-

3-56

MATHEMATICS

ward difference formulas. The formulas of various orders are, per Refs. 59 and 117, (1) yn + 1 = yn + ∆t f(yn + 1) 4 1 2 (2) yn + 1 =
yn −
yn − 1 +
∆t f(yn + 1) 3 3 3 18 9 2 6 (3) yn + 1 =
yn −
yn − 1 +
yn − 2 +
∆t f(yn + 1) 11 11 11 11 48 36 16 3 12 (4) yn + 1 =
yn −
yn − 1 +
yn − 2 −
yn − 3 +
∆t f(yn + 1) 25 25 25 25 25 300 300 200 75 12 (5) yn + 1 =
yn −
yn − 1 +
yn − 2 −
yn − 3 +
yn − 4 137 137 137 137 137 60 +
∆t f(yn + 1) 137 Stiffness The concept of stiffness is described for a system of linear equations. dy
=Ay dt Let λi be the eigenvalues of the matrix A (Ref. 267). Then, per Ref. 181, the stiffness ratio is defined as maxi |Re (λi)| SR =

mini |Re (λi)| SR = 20 is not stiff, SR = 103 is stiff, and SR = 106 is very stiff. If the problem is nonlinear, then the solution is expanded about the current state. n dyi ∂fi
= fi [y(tn)] + 
[yj − yj(tn)] dt j = 1 ∂yj The question of stiffness then depends on the solution at the current time. Consequently nonlinear problems can be stiff during one time period and not stiff during another. While the chemical engineer may not actually calculate the eigenvalues, it is useful to know that they determine the stability and accuracy of the numerical scheme and the step size used. Problems are stiff when the time constants for different phenomena have very different magnitudes. Consider flow through a packed bed reactor. The time constants for different phenomena are: 1. Time for device flow-through L φAL tflow =
=
u Q where Q is the volumetric flow rate, A is the cross sectional area, L is the length of the packed bed, and φ is the void fraction; 2. Time for reaction 1 tr × n =
k where k is a rate constant (time−1); 3. Time for diffusion inside the catalyst εR2 tinternal diffusion =
De where ε is the porosity of the catalyst, R is the catalyst radius, and De is the effective diffusion coefficient inside the catalyst; 4. Time for heat transfer is R2 ρsCsR2 tinternal heat transfer =
=
α ke where ρs is the catalyst density, Cs is the catalyst heat capacity per unit mass, ke is the effective thermal conductivity of the catalyst, and α is the thermal diffusivity. For example, in the model of a catalytic converter for an automobile (Ref. 103), the time constants for internal diffusion was 0.3 seconds; internal heat transfer, 21 seconds; and device flow-through, 0.003 seconds. The device flow-through is so fast that it might as well be instantaneous. The stiffness is approximately 7000.

Implicit methods must be used to integrate the equations. Alternatively, a quasistate model can be developed (Ref. 239). Differential-Algebraic Systems Sometimes models involve ordinary differential equations subject to some algebraic constraints. For example, the equations governing one equilibrium stage (as in a distillation column) are dxn M
= Vn + 1yn + 1 − Lnxn − V nyn + Ln − 1xn − 1 dt xn − 1 − xn = En(xn − 1 − x*,n) N

x =1 i

i=1

where x and y are the mole fraction in the liquid and vapor, respectively; L and V are liquid and vapor flow rates, respectively; M is the holdup; and the superscript is the stage number. The efficiency is E, and the concentration in equilibrium with the vapor is x*. The first equation is an ordinary differential equation for the mass of one component on the stage, while the third equation represents a constraint that the mass fractions add to one. This is a differential-algebraic system of equations. Differential-algebraic equations can be written in the general notation dy F t, y,
= 0 dt





To solve the general problem using the backward Euler method, replace the nonlinear differential equation with the nonlinear algebraic equation for one step. yn + 1 − yn F t, yn + 1,
= 0 ∆t This equation must be solved for yn + 1. The Newton-Raphson method can be used, and if convergence is not achieved within a few iterations, the time step can be reduced and the step repeated. In actuality, the higher-order backward-difference Gear methods are used in DASSL (Ref. 224). Differential-algebraic systems are more complicated than differential systems because the solution may not always be defined. Pontelides et al. (Ref. 226) introduced the term index to identify the possible problems. The index is defined as the minimum number of times the equations need to be differentiated with respect to time to convert the system to a set of ordinary differential equations. These higher derivatives may not exist, and the process places limits on which variables can be given initial values. Sometimes the initial values must be constrained by the algebraic equations (Ref. 226). For a differential-algebraic system modeling a distillation tower, Ref. 226 shows that the index depends on the specification of pressure for the column. Byrne and Ponzi (Ref. 58) also list several chemical engineering examples of differential-algebraic systems and solve one involving two-phase flow. Computer Software Efficient computer packages are available for solving ordinary differential equations as initial value problems. The packages are widely available and good enough that most chemical engineers use them and do not write their own. Here we discuss three of them: RKF45, LSODE, and EPISODE. In each of the packages, the user specifies the differential equation to be solved and a desired error criterion. The package then integrates in time and adjusts the step size to achieve the error criterion within the limitations imposed by stability. A popular explicit, Runge-Kutta package is RKF45, developed by Forsythe et al. (Ref. 111). The method is based on the Runge-KuttaFeldberg formulas. Notice there that an estimate of the truncation error at each step is available. Then the step size can be reduced until this estimate is below the user-specified tolerance. The method is thus automatic, and the user is assured of the results. Note, however, that the tolerance is set on the local truncation error, namely from one step to another, whereas the user is usually interested in the global truncation error, or the error after several steps. The global error is generally made smaller by making the tolerance smaller, but the absolute accuracy is not the same as the tolerance. If the problem is stiff, then very





NUMERICAL ANALYSIS AND APPROXIMATE METHODS small step sizes are used; the computation becomes very lengthy. The RKF45 code discovers this and returns control to the user with a message indicating the problem is too hard to solve with RKF45. A popular implicit package is LSODE, a version of Gear’s method (Ref. 117) written by Alan Hindmarsh at Lawrence Livermore Laboratory (Ref. 148). In this package, the user specifies the differential equation to be solved and the tolerance desired. Now the method is implicit and therefore stable for any step size. The accuracy may not be acceptable, however, and sets of nonlinear equations must be solved. Thus, in practice the step size is limited but not nearly so much as in the Runge-Kutta methods. In these packages, both the step size and order of the method are adjusted by the package. Suppose we are calculating with a kth order method. The truncation error is determined by the (k + 1)th order derivative. This is estimated using difference formulas and the values of the right-hand sides at previous times. An estimate is also made for the kth and (k + 2)th derivative. Then it is possible to estimate the error in a (k − 1)th order method, a kth order method, and a (k + 1)th order method. Furthermore, the step size needed to satisfy the tolerance with each of these methods can be determined. Then we can choose the method and step size for the next step that achieves the biggest step, with appropriate adjustments due to the different work required for each order. The package generally starts with a very small step size and a first-order method, the backward Euler method. Then it integrates along, adjusting the order up (and later down) depending on the error estimates. The user is thus assured that the local truncation error meets the tolerance. There is a further difficulty, since the set of nonlinear equations must be solved. Usually a good guess of the solution is available, since the solution is evolving in time and past history can be extrapolated. Thus, the Newton-Raphson method will usually converge. The package protects itself, though, by only doing three iterations. If convergence is not reached within this many iterations, then the step size is reduced and the calculation is redone for that time step. The convergence theorem for the Newton-Raphson method (p. 3-50) indicates that the method will converge if the step size is small enough. Thus the method is guaranteed to work. Further economies are possible. The Jacobian needed in the Newton-Raphson method can be fixed over several time steps. Then, if the iteration does not converge, the Jacobian can be reevaluated at the current time-step. If the iteration still does not converge, then the step-size is reduced and a new Jacobian is evaluated. Also the successive substitution method can be used, which is even faster, except that it may not converge. However, it, too, will converge if the time step is small enough. Comparisons of the methods and additional details are provided for chemical engineering problems by Refs. 59 and 106. Generally, the Runge-Kutta methods give extremely good accuracy, especially when the step size is kept small for stability reasons. When the computation time is comparable for LSODE and RKF45, the RKF45 package generally gives much more accurate results. The RKF45 package is unsuitable, however, for many chemical reactor problems because they are so stiff. Generally, though, standard packages must have a highorder explicit method (usually a version of Runge-Kutta) and a multistep, implicit method (usually a version of GEAR, EPISODE, or LSODE). The package DASSL (Ref. 224) uses similar principles to solve the differential-algebraic systems. The software described here is available by electronic mail over the Internet. Sending the message send index to [email protected]

will retrieve an index and descriptions of how to obtain the software. Stability, Bifurcations, Limit Cycles Some aspects of this subject involve the solution of nonlinear equations; other aspects involve the integration of ordinary differential equations; applications include chaos and fractals as well as unusual operation of some chemical engineering equipment. Ref. 176 gives an excellent introduction to the subject and the details needed to apply the methods. Ref. 66 gives more details of the algorithms. A concise survey with some chemical engineering examples is given in Ref. 91. Bifurcation results are closely connected with stability of the steady states, which is essentially a transient phenomenon.

3-57

Sensitivity Analysis When solving differential equations, it is frequently necessary to know the solution as well as the sensitivity of the solution to the value of a parameter. Such information is useful when doing parameter estimation (to find the best set of parameters for a model) and for deciding if a parameter needs to be measured accurately. See Ref. 105. ORDINARY DIFFERENTIAL EQUATIONS-BOUNDARY VALUE PROBLEMS Diffusion problems in one dimension lead to boundary value problems. The boundary conditions are applied at two different spatial locations: at one side the concentration may be fixed and at the other side the flux may be fixed. Because the conditions are specified at two different locations, the problems are not initial value in character. It is not possible to begin at one position and integrate directly because at least one of the conditions is specified somewhere else and there are not enough conditions to begin the calculation. Thus, methods have been developed especially for boundary value problems. Shooting Methods The first method is one that utilizes the techniques for initial value problems but allows for an iterative calculation to satisfy all the boundary conditions. Consider the nonlinear boundary value problem d 2y dy
2 = f x, y,
, y(0) = α, y(1) = β dx dx Convert this second-order equation into two first-order equations along with the boundary conditions written to include a parameter s to represent the unknown value of v(0) = dy/dx(0). dy dv
= v,
= f(x, y, v), y(0) = α, v(0) = s dx dx The parameter s is chosen so that the last boundary condition is satisfied: y(1) = β. Define the function





χ(s) = y(1, s) − β and iterate on s to make χ(s) = 0. Note that the condition at x = 0 is satisfied for any s, the differential equation is satisfied by the integration routine, and only the last boundary condition is yet to be satisfied. Both successive substitution and the Newton-Raphson methods can be used. The technique can be used when the boundary conditions are more general and convergence can be proved (see Refs. 106 and 167). Computer software exists: the IMSL program DTPTB uses DVERK, which employs Runge-Kutta integration to integrate the ordinary differential equations (Ref. 55). Finite Difference Method To apply the finite difference method, we first spread grid points through the domain. Figure 3-49 shows a uniform mesh of n points (nonuniform meshes are possible, too). The unknown, here c(x), at a grid point xi is assigned the symbol ci = c(xi). The finite difference method can be derived easily by using a Taylor expansion of the solution about this point. Expressions for the derivatives are: dc dc ci + 1 − ci d 2c ∆x ci − ci − 1 d 2c ∆x
=
−
2
+ ⋅⋅⋅ ,
=
+
2
+ ⋅⋅⋅ dx i ∆x dx i 2 dx i ∆x dx i 2









dc ci + 1 − ci − 1 d c ∆x
=

−
3
dx i 2∆x dx i 3! The truncation error in the first two expressions is proportional to ∆x, and the methods are said to be first-order. The truncation error in the third expression is proportional to ∆x2, and the method is said to be second-order. Usually the last equation is used to insure the best accuracy. The finite difference representation of the second derivative is:



FIG. 3-49

3

Finite difference mesh; ∆x uniform.



2

3-58

MATHEMATICS

d 2c ci + 1 − 2ci + ci − 1 d4c 2∆x2 −
4
+ ⋅⋅⋅
2 =

dx i ∆x2 dx i 4! The truncation error is proportional to ∆x2. To solve a differential equation, it is evaluated at a point i and then these expressions are inserted for the derivatives.





FIG. 3-50

Example Consider the equation for convection, diffusion, and reaction in

Finite difference grid with variable spacing.

a tubular reactor. 1 d 2c dc

2 −
= Da R(c) Pe dx dx The finite difference representation is 1 ci + 1 − 2ci + ci − 1 ci + 1 − ci − 1 −

= Da R(ci)


Pe ∆x2 2∆x This equation is written for i = 2 to n − 1, or the internal points. The equations would then be coupled but would also involve the values of c1 and cn, as well. These are determined from the boundary conditions. If the boundary condition involves a derivative, it is important that the derivatives be evaluated using points that exist. Three possibilities exist: dc c2 − c1
=
dx 1 ∆x



dc −3c1 + 4c2 − c3
=

dx 1 2∆x The third alternative is to add a false point, outside the domain, as c0 = c(x = −∆x). c2 − c0 dc
=
dx 1 2∆x Since this equation introduces a new variable, c0, another equation is needed and is obtained by writing the finite difference equation for i = 1, too. The sets of equations can be solved using the Newton-Raphson method. The first form of the derivative gives a tridiagonal system of equations, and the standard routines for solving tridiagonal equations suffice. For the other two options, some manipulation is necessary to put them into a tridiagonal form (see Ref. 105). Frequently, the transport coefficients, such as diffusion coefficient or thermal conductivity, depend on the dependent variable, concentration, or temperature, respectively. Then the differential equation might look like dc d
D(c)
= 0 dx dx This could be written as two equations. dJ dc −
=0 J = −D(c)
dx dx Because the coefficient depends on c, the equations are more complicated. A finite difference method can be written in terms of the fluxes at the midpoints, i + 1/2. ci + 1 − ci Ji + 1/2 − Ji − 1/2 −

=0 Ji +1/2 = −D(ci + 1/2)
∆x ∆x These are combined to give the complete equation. D(ci + 1/2) (ci + 1 − ci) − D(ci − 1/2) (ci − ci − 1) =0




∆x2 This represents a set of nonlinear algebraic equations that can be solved with the Newton-Raphson method. However, in this case, a viable iterative strategy is to evaluate the transport coefficients at the last value and then solve D(cki + 1/2) (cik++11 − c ik + 1) − D(cik− 1/2) (cik + 1 − c ik−+11) =0





∆x2 The advantage of this approach is that it is easier to program than a full NewtonRaphson method. If the transport coefficients do not vary radically, then the method converges. If the method does not converge, then it may be necessary to use the full Newton-Raphson method. There are three common ways to evaluate the transport coefficient at the midpoint. The first one uses the transport coefficient evaluated at the average value of the solutions on either side. 1 D (ci + 1/2) ≈ D
(ci + 1 + ci) 2 The truncation error of this approach is ∆x2 (Ref. 106). The second approach uses the average of the transport coefficients on either side. 1 D (ci + 1/2) ≈
[D(ci + 1) + D(ci)] 2













The truncation error of this approach is also ∆x2 (Ref. 106). The third approach uses an “upstream” transport coefficient. D (ci + 1/2) ≈ D(ci + 1),

when D (ci + 1) > D(ci)

D (ci + 1/2) ≈ D(ci),

when D (ci + 1) < D(ci)

This approach is used when the transport coefficients vary over several orders of magnitude, and the “upstream” direction is defined as the one in which the transport coefficient is larger. The truncation error of this approach is only ∆x (Refs. 106 and 107), but this approach is useful if the numerical solutions show unrealistic oscillations. If the grid spacing is not uniform, the formulas must be revised. The notation is shown in Fig. 3-50. The finite-difference form of the equations is then Ji + 1/2 − Ji − 1/2 ci + 1 − ci ci − ci − 1 −

= 0 Ji + 1/2 = −Di + 1/2
, Ji − 1/2 = −Di − 1/2
a(∆xi + ∆xi + 1) ∆xi + 1 ∆xi If average diffusion coefficients are used, then the finite difference equation is as follows.





1 1 1


(Di + 1 + Di) (ci + 1 − ci) −
(Di + Di − 1) (ci − ci − 1) = 0 ∆xi + 1 + ∆xi ∆xi + 1 ∆xi Rigorous error bounds are discussed for linear ordinary differential equations solved with the finite difference method by Isaacson and Keller (Ref. 107). Computer software exists to solve two-point boundary value problems. The IMSL routine DVCPR uses the finite difference method with a variable step size (Ref. 247). Finlayson (Ref. 106) gives FDRXN for reaction problems.

Example A reaction diffusion problem is solved with the finite difference method. d 2c
2 = φ2c, dx

dc
(0) = 0, dx

c(1) = 1

The solution is derived for φ = 2. It is solved several times, first with two intervals and three points (at x = 0, 0.5, 1), then with four intervals, then with eight intervals. The reason is that when an exact solution is not known, one must use several ∆x and see that the solution converges as ∆x approaches zero. With two intervals, the equations are as follows. The points are x1 = 0, x2 = 0.5, and x3 = 1.0; and the solution at those points are c1, c2, and c3, respectively. A false boundary is used at x0 = −0.5. c0 − c2
= 0, 2∆x

c0 − 2c1 + c2 − φ2c1 = 0,

∆x2

c1 − 2c2 + c3 − φ2c2 = 0,

∆x2

c3 = 1

The solution is c1 = 0.2857, c2 = 0.4286, and c3 = 1.0. Since the solution is only an approximation and approaches the exact solution only as ∆x approaches zero, it is necessary to find out if ∆x is small enough to be considered zero. This is done by solving the problem again with more grid points. The value of concentration at x = 0 takes the following values for different ∆x. These values are extrapolated using the Richardson extrapolation technique to give c(0) = 0.265826. Using this value as the best estimate of the exact solution, the errors in the solution are tabulated versus ∆x. Clearly the errors go as ∆x2 (decreasing by a factor of 4 when ∆x decreases by a factor of 2), thus validating the solution. The exact solution (given below) is 0.265802. n−1

∆x

c(0)

2 4 8

0.5 0.25 0.125

0.285714 0.271043 0.267131

n−1

∆x

Error in c(0)

2 4 8

0.5 0.25 0.125

0.01989 0.00521 0.00130

Finite Difference Methods Solved with Spreadsheets A convenient way to solve the finite difference equations for simple

NUMERICAL ANALYSIS AND APPROXIMATE METHODS

3-59

The residual is then made orthogonal to the set of basis functions. db b (x)  a
− φ R  a b (x) dx = 0 dx 1

2

NT

j

i

0

FIG. 3-51

Finite difference method using spreadsheets.

problems is to use a computer spreadsheet. The equations for the problem solved in the example can be cast into the following form 2c2 c1 =

2 + φ2∆x2

i=1

NT

i

2

i i

2

j = 1, . . . , NT

i=1

This is the process that makes the method a Galerkin method. The basis for the orthogonality condition is that a function that is made orthogonal to each member of a complete set is then zero. The residual is being made orthogonal, and if the basis functions are complete and you use infinitely many of them, then the residual is zero. Once the residual is zero, the problem is solved. This equation is integrated by parts to give the following equation NT

− i=1



1

0

b (x)R  a b (x) dx

dbj dbi

dxai = φ2 dx dx

1

NT

j

i i

0

i=1

j = 1, . . . , NT − 1

ci + 1 + ci − 1 ci =

2 + φ2∆x2 cn + 1 = 1 Let us solve the problem using 6 nodes, or 5 intervals. Then the connection between the cell in the spreadsheet and the nodal value is shown in Fig. 3-51. The following equations are placed into the various cells. A1: = 2*B1/(2.+(phi*dx)**2) B1: = (A1 + C1)/(2.+(phi*dx)**2) F1: = 1.

(3-76)

This equation defines the Galerkin method and a solution that satisfies this equation (for all j = 1, . . . , ∞) is called a weak solution. For an approximate solution, the equation is written once for each member of the trial function, j = 1, . . . , NT − 1, and the boundary condition is applied. NT

 a b (1) = c i

i

B

i=1

The equation in cell B1 is copied into cells C1 though E1. Then turn on the iteration scheme in the spreadsheet and watch the solution converge. Whether or not convergence is achieved can depend on how you write the equations, so some experimentation may be necessary. Theorems for convergence of the successive substitution method are useful in this regard. Orthogonal Collocation The orthogonal collocation method has found widespread application in chemical engineering, particularly for chemical reaction engineering. In the collocation method, the dependent variable is expanded in a series of orthogonal polynomials, and the differential equation is evaluated at certain collocation points. The collocation points are the roots to an orthogonal polynomial, as first used by Lanczos (Refs. 182 and 183). A major improvement was proposed by Villadsen and Stewart (Refs. 288 and 289), who proposed that the entire solution process be done in terms of the solution at the collocation points rather than the coefficients in the expansion. This method is especially useful for reaction-diffusion problems that frequently arise when modeling chemical reactors. It is highly efficient when the solution is smooth, but the finite difference method is preferred when the solution changes steeply in some region of space. See Ref. 105 for comparisons. Galerkin Finite Element Method In the finite element method, the domain is divided into elements and an expansion is made for the solution on each finite element. In the Galerkin finite element method an additional idea is introduced: the Galerkin method is used to solve the equation. The Galerkin method is explained before the finite element basis set is introduced, using the equations for reaction and diffusion in a porous catalyst pellet. d 2c
2 = φ2R(c) dx dc
(0) = 0, c(1) = 1 dx The unknown solution is expanded in a series of known functions {bi(x)} with unknown coefficients {ai}. NT

c(x) =  ai bi(x) i=1

The trial solution is substituted into the differential equation to obtain the residual. NT NT d 2bi Residual =  ai
− φ2R  aibi(x) 2 dx i=1 i=1





The Galerkin finite element method results when the Galerkin method is combined with a finite element trial function. The domain is divided into elements separated by nodes, as in the finite difference method. The solution is approximated by a linear (or sometimes quadratic) function of position within the element. These approximations are substituted into Eq. (3-76) to provide the Galerkin finite element equations. The element integrals are defined as 1 e BJI = −
∆xe



1

0

dNJ dNI

du, du du

F Je = φ2∆xe

N (u)R  c N (u) du 1

NP

e I

J

0

I

I=1

and the entire method can be written in the following compact notation:

B

c =  FJe

e e JI I

e

e

The matrices for various terms are given in the table. This equation can also be written in the form AAc = f where the matrix AA is sparse; if linear elements are used, the matrix is tridiagonal. Once the solution is found, the solution at any point can be recovered from ce(u) = c eI = 1(1 − u) + c eI = 2u for linear elements. Element Matrices for Galerkin Method with Linear Shape Functions

dN dN N1 = 1 − u, N2 = u,
1 = −1,
2 = 1 du du



−a a du =   N
−a a du N N du = sj js, N du = aa, N u du = js 1

0



dNJ dNI 1 −1 ,

du = −1 1 du du

1

dNI

J

0

1

J

0

1

1

I

J

0

j

0

Example Solve the specified problem when φ = 2, the rate expression is linear, R(c) = c, and the boundary condition is 1.0. The Galerkin finite element method is used with ∆x = 0.33333. The element nodes are at x = 0, 0.3333, 0.6667, and 1.0. The solution at x = 1.0 is c4 = 1.0. The Galerkin equations for one element are obtained from the table.

−11 −11 = φ ∆x  s⁄ s⁄ 

−

2

16

2

16

3-60

MATHEMATICS

When these are summed over all elements the result is



−1 1 0 0

1 −1 − 1 1 0

0 1 −1 − 1 0

 

0 0 1 1

1

c1 ⁄6 s 0 1 4 1⁄6 s + s c2 ⁄6 =
1 c3 0 ⁄ 6 s + s 9 0 0 c4 0

0 0 ⁄6 0

1

   c1 0 0 c2 + 0 c3 1 c4

u(x) = f(x) + λ(b − a)

After rearrangement this is



31⁄27

−25⁄27 0

−25⁄27 62⁄27 −25⁄27

0

−25⁄27 62⁄27

   c1 c2 = c3

25

⁄27

Cubic B-Splines Cubic B-splines can also be used to solve differential equations (Refs. 105 and 266). Adaptive Meshes In many two-point boundary value problems, the difficulty in the problem is the formation of a boundary layer region, or a region in which the solution changes very dramatically. In such cases, it is prudent to use small mesh spacing there, either with the finite difference method or the finite element method. If the region is known a priori, small mesh spacings can be assumed at the boundary layer. If the region is not known, though, other techniques must be used. These techniques are known as adaptive mesh techniques. The mesh size is made small where some property of the solution is large. For example, if the truncation error of the method is nth order, then the nth-order derivative of the solution is evaluated and a small mesh is used where it is large. Alternatively, the residual (the differential equation with the numerical solution substituted into it) can be used as a criterion. See Refs. 21 and 107. It is also possible to define the error that is expected from a method one order higher and one order lower. Then a decision about whether to increase or decrease the order of the method can be made, taking into account the relative work of the different orders. This provides a method of adjusting both the mesh spacing (∆x, or sometimes called h) and the degree of polynomial (p). Such methods are called h-p methods. Singular Problems and Infinite Domains If the solution being sought has a singularity, it may be difficult to find a good numerical solution. Sometimes even the location of the singularity may not be known (Ref. 11). One method of solving such problems is to refine the mesh near the singularity, relying on the better approximation due to a smaller ∆x. Another approach is to incorporate the singular trial function into the approximation. Thus, if the solution approaches f(x) as x goes to zero and f(x) becomes infinite, one may define a new variable u(x) = y(x) − f(x) and derive an equation for u. The differential equation is more complicated, but the solution is better near the singularity. See Refs. 39 and 231. Sometimes the domain is semi-infinite, as in boundary layer flow. The domain can be transformed from the x domain (0–∞) to the η domain (1–0) using the transformation η = exp (−x). Another approach is to use a variable mesh, perhaps with the same transformation. For example, use η = exp (−βx) and a constant mesh size in η; the value of β is found experimentally. Still another approach is to solve on a finite mesh in which the last point is far enough away that its location does not influence the solution (Ref. 59). A location that is far enough away must be found by trial and error. NUMERICAL SOLUTION OF INTEGRAL EQUATIONS

b

a

for u(x)

n

i

i

(3-78)

i

where t1, . . . , tn are points of subdivision of the t axis, a ≤ t ≤ b, and the c’s are coefficients whose values depend upon the type of numerical integration formula used. Now Eq. (3-78) must hold for all values of x, a ≤ x ≤ b; so it must hold for x = t1, x = t2, . . . , x = tn. Substituting for x successively t1, t2, . . . , tn and setting u(ti) = ui, f(ti) = fi, we get n linear algebraic equations for the n unknowns u1, . . . , un. That is, ui = fi + (b − a)[c1k(ti, t1)u1 + c2 k(ti, t2)u2 + ⋅⋅⋅ + cnk(ti, tn)un]

i = 1, 2, . . . , n

These uj may be solved for by the methods under “Numerical Solution of Linear Equations and Associated Problems” and substituted into Eq. (3-78) to yield an approximate solution for Eq. (3-77). Example Solve numerically u(x) = x + s ∫ 01 (t + x)u(t) dt. In this example a = 0, b = 1. Take n = 3, t1 = 0, t2 = a, t3 = 1. Then Eq. (3-78) takes the form (for which we have used the parabolic rule) a u(x) = x + (s)
[(t1 + x)u(t1) + 4(t2 + x)u(t2) + (t3 + x)u(t3)] 3 = x + (1/18)[(t1 + x)u(t1) + 4(t2 + x)u(t2) + (t3 + x)u(t3)] This must hold for all x, 0 ≤ x ≤ 1. Here t1 = 0, t2 = a, and t3 = 1. Evaluate at x = ti. u(t1) = t1 + 1⁄18[2t1u(t1) + 4(t2 + t1)u(t2) + (t3 + t1)u(t3)] u(t2) = t2 + 1⁄18[(t1 + t2)u(t1) + 4(2t2)u(t2) + (t3 + t2)u(t3)] u(t3) = t3 + 1⁄18[(t1 + t3)u(t1) + 4(t2 + t3)u(t2) + 2t3u(t3)] By setting in the values of t1, t2, t3 and u(ti) = ui, 18u1 − 2u2 − u3 = 0 −u1 + 28u2 − 3u3 = 18 −u1 − 6u2 + 16u3 = 18 with the solution u1 = 12⁄ 71, u2 = 57⁄71, u3 = 102⁄71. Thus u(x) = x + 1⁄18[x12⁄71 + 4(a + x)57⁄71 + (1 + x)102⁄71] = 90⁄ 71x + 12⁄71

Because of the work involved in solving large systems of simultaneous linear equations it is desirable that only a small number of u’s be computed. Thus the gaussian integration formulas are useful because of the economy they offer. See references on numerical solutions of integral equations. Solutions for Volterra equations are done in a similar fashion, except that the solution can proceed point by point, or in small groups of points depending on the quadrature scheme. See Refs. 105 and 195. There are methods that are analogous to the usual methods for integrating differential equations (Runge-Kutta, predictor-corrector, Adams methods, etc.). Explicit methods are fast and efficient until the time step is very small to meet the stability requirements. Then implicit methods are used, even though sets of simultaneous algebraic equations must be solved. The major part of the calculation is the evaluation of integrals, however, so that the added time to solve the algebraic equations is not excessive. Thus, implicit methods tend to be preferred (Ref. 195). Volterra equations of the first kind are not well posed, and small errors in the solution can have disastrous consequences. The boundary element method uses Green’s functions and integral equations to solve differential equations (Refs. 45 and 200). MONTE CARLO SIMULATIONS

In this subsection is considered a method of solving numerically the Fredholm integral equation of the second kind:

k(x, t)u(t) dt

 c k(x, t )u(t ) i=1

0 0

The solution is c1 = 0.2560, c2 = 0.3174, c3 = 0.5312, and c4 = 1. The exact solution is derived using the section entitled “Ordinary Differential Equations: Linear Differential Equations with Constant Coefficients.” e2x + e−2x cosh (2x) c=
=
e2 + e−2 cosh (2) The values of the exact solution at the same finite element nodes are c1 = 0.2658, c2 = 0.3271, c3 = 0.5392, and c4 = 1, indicating that the three-element finite element solution is accurate within 3 percent. When the exact solution is not known, the problem must be solved several times, each with a different number of elements, so that convergence is seen as the number of elements increases.

u(x) = f(x) + λ

The method discussed arises because a definite integral can be closely approximated by any of several numerical integration formulas (each of which arises by approximating the function by some polynomial over an interval). Thus the definite integral in Eq. (3-77) can be replaced by an integration formula, and Eq. (3-77) may be written

(3-77)

Some physical problems, such as those involving interaction of molecules, are usually formulated as integral equations. Monte Carlo methods are especially well-suited to their solution. This section cannot give a comprehensive treatment of such methods, but their use in

NUMERICAL ANALYSIS AND APPROXIMATE METHODS

3-61

calculating the value of an integral will be illustrated. Suppose we wish to calculate the integral G=



Ω0

g(x)f(x) dx

where the distribution function f(x) satisfies: f(x) ≥ 0,



Ω0

f(x) dx = 1

The distribution function f(x) can be taken as constant; for example, 1/Ω0. We choose variables x1, x2, . . . , xN randomly from f(x) and form the arithmetic mean 1 GN =
 g(xi) N i The quantity GN is an estimation of G, and the fundamental theorem of Monte Carlo guarantees that the expected value of GN is G, if G exists (Ref. 161). The error in the calculation is given by σ1 ε=
N1/2 where σ 21 is calculated from σ 21 =



Ω0

g2(x) f(x) dx − G2

Thus the number of terms needed to achieve a specified accuracy can be calculated once an estimate of σ 21 is known. σ12 N=
ε Various methods, such as influence sampling, can be used to reduce the number of calculations needed (Ref. 161). NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS Parabolic Equations in One Dimension By combining the techniques applied to initial value problems and boundary value problems it is possible to easily solve parabolic equations in one dimension. The method is often called the method of lines. It is illustrated here using the finite difference method, but the Galerkin finite element method and the orthogonal collocation method can also be combined with initial value methods in similar ways. The analysis is done by example. Example Consider the diffusion equation, with boundary and initial conditions.

∂c ∂2c
= D
2 ∂t ∂x c(x, 0) = 0 c(0, t) = 1, c(1, t) = 0

We denote by ci the value of c(xi, t) at any time. Thus, ci is a function of time, and differential equations in ci are ordinary differential equations. By evaluating the diffusion equation at the ith node and replacing the derivative with a finite difference equation, the following working equation is derived for each node i, i = 2, . . . , n (see Fig. 3-52). dci ci + 1 − 2ci + ci − 1
= D

dt ∆x2 This can be written in the general form of a set of ordinary differential equations by defining the matrix AA. dc
= AAc dt This set of ordinary differential equations can be solved using any of the standard methods, and the stability of the integration of these equations is governed by the largest eigenvalue of AA. If Euler’s method is used for integration, the time step is limited by 2 ∆t ≤
|λ|max

FIG. 3-52

Computational molecules. h = ∆x = ∆y.

whereas, if the Runge-Kutta-Feldberg method is used, the 2 in the numerator is replaced by 3.0. The largest eigenvalue of AA is bounded by Gerschgorin’s Theorem (Ref. 155, p. 135). n 4D |λ|max ≤ max 2 < j < n  |AA ji | =
2 ∆x i=2 This gives the well-known stability limit D 1 ∆t
2 ≤
∆x 2

3-62

MATHEMATICS

The smallest eigenvalue is independent of ∆x (it is Dπ 2/L2) so that the ratio of largest to smallest eigenvalue is proportional to 1/∆x2. Thus, the problem becomes stiff as ∆x approaches zero (Ref. 106). Another way to study the stability of explicit equations is to use the positivity theorem. For Euler’s method, the equations can be written in the form c in + 1 − cin cni+ 1 − 2c ni + cni− 1

= D

∆t ∆x2 n n where c i = c(xi, t ). Then the new value is given by





D∆t n D∆t n D∆t n c in + 1 =
ci + 1 + 1 − 2
ci +
ci − 1 ∆x2 ∆x2 ∆x2 Theorem. If c ni + 1 = Acni+ 1 + Bc in + Ccni− 1 and A, B, and C are positive and A + B + C ≤ 1, then the scheme is stable and the errors die out. Here the theorem requires D∆t 1−2
>0 ∆x2 which gives the same stability condition (Ref. 106). Implicit methods can also be used. Write a finite difference form for the time derivative and average the right-hand sides, evaluated at the old and new time. c in + 1 − cin cni+ 1 − 2cni + cni− 1 cin++11 − 2c in + 1 + cni −+11 + Dθ




= D(1 − θ)

2 ∆t ∆x ∆x2 Now the equations are of the form









D∆tθ n + 1 D∆tθ n + 1 D∆tθ n + 1 −
ci + 1 + 1 + 2
ci −
ci − 1 ∆x2 ∆x2 ∆x2 D∆t(1 − θ) n = c in +

(ci + 1 − 2c in + cni− 1) ∆x2 and require solving a set of simultaneous equations, which have a tridiagonal structure. Using θ = 0 gives the Euler method (as above), θ = 0.5 gives the Crank-Nicolson method, and θ = 1 gives the backward Euler method. The Crank-Nicolson method is also the same as applying the trapezoid rule to do the integration. The stability limit is given by D∆t 0.5 ≤

∆x2 1 − 2θ If the ∆t satisfies the following equation, then the solution will not oscillate from node to node (a numerical artifact). See Ref. 106. 0.25 D∆t ≤

∆x2 1−θ Other methods can be used in space, such as the finite element method, the orthogonal collocation method, or the method of orthogonal collocation on finite elements (see Ref. 106). Spectral methods employ Chebyshev polynomials and the Fast Fourier Transform and are quite useful for hyperbolic or parabolic problems on rectangular domains (Ref. 125). Packages exist that use various discretizations in the spatial direction and an integration routine in the time variable. PDECOL uses B-splines for the spatial direction and various GEAR methods in time (Ref. 247). PDEPACK and DSS (Ref. 247) use finite differences in the spatial direction and GEARB in time (Ref. 66). REACOL (Ref. 106) uses orthogonal collocation in the radial direction and LSODE in the axial direction, while REACFD uses finite difference in the radial direction; both codes are restricted to modeling chemical reactors. Elliptic Equations Elliptic equations can be solved with both finite difference and finite element methods. One-dimensional elliptic problems are two-point boundary value problems. Two- and threedimensional elliptic problems are often solved with iterative methods when the finite difference method is used and direct methods when the finite element method is used. So there are two aspects to consider: how the equations are discretized to form sets of algebraic equations and how the algebraic equations are then solved.

The prototype elliptic problem is steady-state heat conduction or diffusion, ∂2T ∂2T +
=Q k
∂x2 ∂y2





possibly with a heat generation term per unit volume, Q. The boundary conditions taken here are T = f(x, y) on the boundary (S) with f a known function. Illustrations are given for constant thermal conductivity k while Q is a known function of position. The finite difference formulation is given using the following nomenclature: Ti, j = T(i∆x, j∆y) The finite difference formulation is then (see Fig. 3-52) Ti + 1, j − 2Ti, j + Ti − 1, j Ti, j + 1 − 2Ti, j + Ti, j − 1 +


= Qi, j


∆x2 ∆y2

(3-79)

Ti, j = f(xi, yj) on S If the boundary is parallel to a coordinate axis any derivative is evaluated as in the section on boundary value problems, using either a onesided, centered difference or a false boundary. If the boundary is more irregular and not parallel to a coordinate line then more complicated expressions are needed and the finite element method may be the better method. Equation (3-79) is rewritten in the form ∆ x2 ∆x2 Qi, j 2 1 +
2 Ti, j = Ti + 1, j + Ti − 1, j +
2 (Ti, j + 1 + Ti, j − 1) − ∆x2
∆y ∆y k





The relaxation method solves this equation iteratively. Qi, j ∆ x2 ∆ x2 2 1 +
2 T*i, j = Tis+ 1, j + T si −+ 1,1 j +
2 (T i,s j + 1 + T i,s +j −1 1) − ∆ x2
∆y ∆y k





T si,+j 1 = Ti,s j + β(T*i, j − Ti,s j) If β = 1, this is the Gauss-Seidel method. If β > 1, it is overrelaxation; if β < 1 it is underrelaxation. The value of β may be chosen empirically, 0 < β < 2, but it can be selected theoretically for simple problems like this (Refs. 106 and 221). In particular, these equations can be programmed in a spreadsheet and solved using the iteration feature, provided the boundaries are all rectangular. The alternating direction method can be used for elliptic problems by using sequences of iteration parameters (Refs. 106 and 221). The method is well suited to transient problems as well. These are the classical iterative techniques. Recently preconditioned conjugate gradient methods have been developed (see Ref. 100). In these methods, a series of matrix multiplications are done iteration by iteration; and the steps lend themselves to the efficiency available in parallel computers. In the multigrid method, the problem is solved on several grids, each more refined than the previous one. As one iterates between the solutions on the different grids, one converges to the solution of the algebraic equations. See Juncu and Mihail (Ref. 68) for a chemical engineering application. The Galerkin finite element method (FEM) is useful for solving elliptic problems and is particularly effective when the domain or geometry is irregular. As an example, cover the domain with triangles and define a trial function on each triangle. The trial function takes the value 1.0 at one corner and 0.0 at the other corners and is linear in between. See Fig. 3-53. These trial functions on each triangle are pieced together to give a trial function on the whole domain. General treatments of the finite element method are available (see references). The steps in the solution method are similar to those described for boundary value problems, except now the problems are much bigger so that the numerical analysis must be done very carefully to be efficient. Most engineers, though, just use a finite element program without generating it. There are three major caveats that must be addressed. The first one is that the solution is dependent on the mesh laid down, and the only way to assess the accuracy of the solution is to solve the problem with a more refined mesh. The second concern is that the solution obeys the shape of the trial function inside

NUMERICAL ANALYSIS AND APPROXIMATE METHODS the element. Thus, if linear functions are used on triangles, a threedimensional view of the solution, plotting the solution versus x and y, consists of a series of triangular planes joined together at the edges, as in a geodesic dome. The third caveat is that the Galerkin finite element method is applied to both the differential equations and the boundary conditions. Computer programs are usually quite general and may allow the user to specify boundary conditions that are not realistic. Also, natural boundary conditions are satisfied if no other boundary condition (ones involving derivatives) is set at a node. Thus, the user of finite element codes must be very clear what boundary conditions and differential equations are built into the computer code. When the problem is nonlinear, the Newton-Raphson method is used to iterate from an initial guess. Nonlinear problems lead to complicated integrals to evaluate, and they are usually evaluated using Gaussian quadrature. One nice feature of the finite element method is the use of natural boundary conditions. It may be possible to solve the problem on a domain that is shorter than needed to reach some limiting condition (such as at an outflow boundary). The externally applied flux is still applied at the shorter domain, and the solution inside the truncated domain is still valid. Examples are given in Refs. 67 and 107. The effect of this is to allow solutions in domains that are smaller, thus saving computation time and permitting the solution in semi-infinite domains. A general purpose package for general two-dimensional domains and rectangular three-dimensional rectangular domains is ELLPACK (Ref. 247). This package allows choice of a variety of methods: finite difference, Hermite collocation, spline Galerkin, collocation, as well as others. Comparisons of the various methods are available (Ref. 154). The program FISHPAK solves the Helmholtz equation in multiple dimensions when the domain is separable (since fast methods like FFT are used). See Ref. 247. Hyperbolic Equations The most common situation yielding hyperbolic equations involves unsteady phenomena with convection. Two typical equations are the convective diffusive equation ∂c ∂2c ∂c
+ u
= D
2 ∂t ∂x ∂x and the chromatography equation (Ref. 245) ∂c ∂c df ∂c φ
+ φu
+ (1 − φ)

= 0 ∂t ∂x dc ∂t where φ is the void fraction and f(c) gives the equilibrium relation between the concentration in the fluid phase and the concentration in the solid phase. If the diffusion coefficient is zero, the convective diffusion equation is hyperbolic. If D is small, the phenomenon may be essentially hyperbolic, even though the equations are parabolic. Thus the numerical methods for hyperbolic equations may be useful even for parabolic equations. Equations for several methods are given here, as taken from the book by Finlayson (Ref. 107). If the convective term is treated with a centered difference expression, the solution exhibits oscillations from node to node, and these only go away if a very fine grid is used. The

FIG. 3-53 Trial functions for Galerkin finite element method: linear polynomial on triangle.

3-63

simplest way to avoid the oscillations with a hyperbolic equation is to use upstream derivatives. If the flow is from left to right, this would give dci ci − ci − 1 ci + 1 − 2ci + ci − 1
+ u
= D

dt ∆x ∆x2 d ci − ci − 1
[ φci + (1 − φ)f(ci)] + φui
= 0 dt ∆x (See Ref. 227 for the reason the equation is written in this form.) The effect of using upstream derivatives is to add artificial or numerical diffusion to the model. This can be ascertained by rearranging the finite difference form of the convective diffusion equation dci ci + 1 − ci − 1 u∆x ci + 1 − 2ci + ci − 1
+ u

= D +


dt 2∆x 2 ∆x2 Thus the diffusion coefficient has been changed from u∆x D to D +
2 Another method often used for hyperbolic equations is the MacCormack method. This method has two steps, and it is written here for the convective diffusion equation. u∆t D∆t n c*i n + 1 = cin −
(cni+ 1 − c ni ) +
(c i + 1 − 2c ni + cni − 1) ∆x ∆x2





1 u∆t c in + 1 =
(cin + c*i n + 1) −
(c*i n + 1 − c*i −n 1+ 1) 2 2∆x D∆t +
2 (c*i +n 1+ 1 − 2c*i n + 1 + c*i −n 1+ 1) 2∆x The concentration profile is steeper for the MacCormack method than for the upstream derivatives, but oscillations can still be present. The flux-corrected transport method can be added to the MacCormack method. A solution is obtained both with the upstream algorithm and the MacCormack method and then they are combined to add just enough diffusion to eliminate the oscillations without smoothing the solution too much. The algorithm is complicated and lengthy but well worth the effort (Refs. 37, 107, and 270). Stability conditions can be constructed in terms of Co = u∆t/∆x and r = D∆t/∆x2 by using Fourier analysis (Ref. 107). All the methods require u∆t Co =
≤ 1 ∆x where Co is the Courant number. How much Co should be less than one depends on the method and on r = D∆t/∆x2. For example, the upstream method requires Co ≤ 1 − 2r. The MacCormack method depends less on r and is stable for most Co as long as r ≤ 0.5. Each of these methods is trying to avoid oscillations that would disappear if the mesh were fine enough. For the steady convective diffusion equation, these oscillations do not occur provided u∆x
≤1 2D For large velocity u, the ∆x must be small to meet this condition. An alternative is to use a small ∆x in regions where the solution changes drastically. Since these regions change in time, it is necessary that the elements or grid points move. The criteria to move the grid points can be quite complicated, and typical methods are reviewed in Ref. 107. Similar considerations apply to the nonlinear chromatography problem (Ref. 227). See especially Ref. 192. Parabolic Equations in Two or Three Dimensions Computations become much more lengthy when there are two or more spatial dimensions. For example, we may have the unsteady heat conduction equation ∂T ∂2T ∂2T ρCp
= k
+
−Q ∂t ∂x2 ∂y2





3-64

MATHEMATICS

In the finite difference method an explicit technique would evaluate the right-hand side at the nth time level. T i,n j+ 1 − T i,nj k n ρCp

=
2 (T in+ 1, j − 2T i, j + T in− 1, j) ∆t ∆x k +
2 (T ni, j + 1 − 2T i,n j + Ti,nj − 1) − Q ∆y When Q = 0 and ∆x = ∆y, the time step limit can be found using the positivity rule. ∆x2ρC ∆x2 ∆t ≤
p or
4k 4D These time steps are smaller than for one-dimensional problems. For three dimensions, the limit is ∆x2 ∆t ≤
6D To avoid such small time steps, which become smaller as ∆x decreases, an implicit method could be used. This leads to large, sparse matrices rather than convenient tridiagonal matrices. These can be solved, but the alternating direction method is also useful (Ref. 221). This reduces a problem on an n n grid to a series of 2n one-dimensional problems on an n grid. SPLINE FUNCTIONS Splines are functions that match given values at the points x1, . . . , xNT and have continuous derivatives up to some order at the knots, or the points x2, . . . , xNT − 1. Cubic splines are most common; see Ref. 38. The function is represented by a cubic polynomial within each interval (xi, xi + 1) and has continuous first and second derivatives at the knots. Two more conditions can be specified arbitrarily. These are usually the second derivatives at the two end points, which are commonly taken as zero; this gives the natural cubic splines. Take yi = y(xi) at each of the points xi, and let ∆xi = xi + 1 − xi. Then, in the interval (xi, xi + 1), the function is represented as a cubic polynomial.

FAST FOURIER TRANSFORM (REF. 231) Suppose a signal y(t) is sampled at equal intervals

n = . . . , −2, −1, 0, 1, 2, . . .

∆ = sampling rate (e.g., number of samples per second) The Fourier transform and inverse transform are Y(ω) =



∞

−∞

y(t)eiωt dt



1 ∞ Y(ω)e−iωt dt y(t) =
2π −∞ The Nyquist critical frequency or critical angular frequency is 1 π fc =
, ωc =
2∆ ∆ If a function y(t) is bandwidth-limited to frequencies smaller than fc, such as Y(ω) = 0

for ω > ωc

then the function is completely determined by its samples yn. Thus, the entire information content of a signal can be recorded by sampling at a rate ∆−1 = 2fc. If the function is not bandwidth-limited, then aliasing occurs. Once a sample rate ∆ is chosen, information corresponding to frequencies greater than fc is simply aliased into that range. The way to detect this in a Fourier transform is to see if the transform approaches zero at fc; if not, aliasing has occurred, and a higher sampling rate is needed. Next, suppose we have N samples, where N is even yk = y(tk)

tk = k∆

k = 0,1,2, . . . , N − 1

and the sample rate is ∆. With only N values {yk}, it is not possible to determine the complete Fourier transform Y(ω). We calculate the value Y(ωn) at the discrete points 2πn N N ωn =
, n = −
, . . . , 0, . . . ,
N∆ 2 2 Yn =

N−1

 ye k

2πikn/N

k=0

Y(ωn) = ∆Yn

Ci(x) = a0i + a1i x + a2i x2 + a3i x3 The interpolating function takes on specified values at the knots and has continuous first and second derivatives at the knots. Within the ith interval, the function is (x − xi) Ci(x) = Ci(xi) + C′i (xi)(x − xi) + C″i (xi)
2 (x − xi)3 + [C″i (xi + 1) − C″i (xi)]
6∆xi where Ci(xi) = yi. The second derivative C″i (xi) = y″i is found by solving the following tridiagonal system of equations: yi − yi − 1 yi + 1 − yi y″i − 1∆xi − 1 + y″i 2(∆xi − 1 + ∆xi) + y″i + 1∆xi = 6
−
∆xi − 1 ∆xi Since the continuity conditions apply only for i = 2, . . . , NT − 1, we have only NT − 2 conditions for the NT values of y″i . Two additional conditions are needed, and these are usually taken as the value of the second derivative at each end of the domain, y″1 , y″NT. If these values are zero, we get the natural cubic splines; they can also be set to achieve some other purpose, such as making the first derivative match some desired condition at the two ends. With these values taken as zero in the natural cubic spline, we have a NT − 2 system of tridiagonal equations, which is easily solved. Once the second derivatives are known at each of the knots, the first derivatives are given by ∆x yi + 1 − yi ∆x y′i =
− y″i
i − y″i + 1
i ∆xi 3 6 The function itself is then known within each element.



yn = y(n∆),



The discrete inverse Fourier transform is 1 N−1 yk =
 Yne−2πikn/N N n=0 The fast Fourier transform (FFT) is used to calculate the Fourier transform as well as the inverse Fourier transform. A discrete Fourier transform of length N can be written as the sum of two discrete Fourier transforms, each of length N/2. Yk = Yke + W k Yko Here Yk is the kth component of the Fourier transform of y, and Yke is the kth component of the Fourier transform of the even components of {yj} and is of length N/2. Similarly, Yko is the kth component of the Fourier transform of the odd components of {yj} and is of length N/2. W is a constant, which is taken to the kth power. W = e2πi/N Since Yk has N components, while Y ek and Yko have N/2 components, Y ek and Yko are repeated once to give N components in the calculation of Yk. This decomposition can be used recursively. Thus, Y ek is split into even and odd terms of length N/4. Yke = Ykee + W k Ykeo Yko = Ykoe + W k Ykoo This process is continued until there is only one component. For this reason, the number N is taken as a power of 2. The vector {yj } is filled with zeroes, if need be, to make N = 2 p for some p. For the computer program, see Ref. 26. The standard Fourier transform takes N 2 operations to calculation, whereas the fast Fourier transform takes only N log2 N. For large N, the difference is significant; at N = 100 it is a factor of 15, but for N = 1000 it is a factor of 100.

OPTIMIZATION The discrete Fourier transform can also be used for differentiating a function, and this is used in the spectral method for solving differential equations. Suppose we have a grid of equidistant points L xn = n∆x, n = 0, 1, 2, . . . , 2N − 1, ∆x =
2N The solution is known at each of these grid points {Y(xn)}. First the Fourier transform is taken. 1 2N − 1 yk =
 Y(xn)e−2ikπx n /L 2N n = 0 The inverse transformation is 1 N Y(x) =
 yke2ikπx/L L k = −N Differentiate this to get dY 1 N 2πik
=
 yk
e2ikπx/L dx L k = −N L

3-65

Thus at the grid points



dY 1 N 2πik
=
 yk
e2ikπx n /L dx n L k = −N L The process works as follows. From the solution at all grid points the Fourier transform is obtained using FFT, {yk}. Then this is multiplied by 2πik/L to obtain the Fourier transform of the derivative. 2πik yk = yk
L Then the inverse Fourier transform is taken using FFT, giving the value of the derivative at each of the grid points.



dY 1 N
=
 yke2ikπx n /L dx n L k = −N

OPTIMIZATION INTRODUCTION Optimization should be viewed as a tool to aid in decision making. Its purpose is to aid in the selection of better values for the decisions that can be made by a person in solving a problem. To formulate an optimization problem, one must resolve three issues. First, one must have a representation of the artifact that can be used to determine how the artifact performs in response to the decisions one makes. This representation may be a mathematical model or the artifact itself. Second, one must have a way to evaluate the performance—an objective function—which is used to compare alternative solutions. Third, one must have a method to search for the improvement. This section concentrates on the third issue, the methods one might use. The first two items are difficult ones, but discussing them at length is outside the scope of this section. Example optimization problems are: (1) determining the optimal thickness of pipe insulation; (2) finding the best equipment sizes and operating schedules for the design of a new batch process to make a given slate of products; (3) choosing the best set of operating conditions for a set of experiments to determine the constants in a kinetic model for a given reaction; (4) finding the amounts of a given set of ingredients one should use for making a carbon rod to be used as an electrode in an arc welder. For the first problem, one will usually write a mathematical model of how insulation of varying thicknesses restricts the loss of heat from a pipe. Evaluation requires that one develop a cost model for the insulation (a capital cost in dollars) and the heat that is lost (an operating cost in dollars/year). Some method is required to permit these two costs to be compared, such as a present worth analysis. Finally, if the model is simple enough, the method one can use is to set the derivative of the evaluation function to zero with respect to wall thickness to find candidate points for its optimal thickness. For the second problem, selecting a best operating schedule involves discrete decisions, which will generally require models that have integer variables. It may not be possible to develop a mathematical model for the fourth problem if not enough is known to characterize the performance of a rod versus the amounts of the various ingredients used in its manufacture. The rods may have to be manufactured and judged by ranking the rods relative to each other, perhaps based partially or totally on opinions. Pattern search methods have been devised to attack problems in this class. In this section assume a mathematical model is possible for the problem to be solved. The model may be encoded in a subroutine and be known only implicitly, or the equations may be known explicitly. A general form for such an optimization problem is min F = F(z), such that h(z) = 0 and g(z) ≤ 0

where F represents a specified objective function that is to be minimized. Functions h and g represent equality and inequality constraints that must be satisfied at the final problem solution. Variables z are used to model such things as flows, mole fractions, physical properties, temperatures, and sizes. The objective function F is generally assumed to be a scalar function, one which represents such things as cost, net present value, safety, or flexibility. Sometimes several objective functions are specified (e.g., minimizing cost while maximizing reliability); these are commonly combined into one function, or else one is selected for the optimization while the others are specified as constraints. Equations h(z) = 0 are typically algebraic equations, linear or nonlinear, when modeling steady-state processes, or algebraic coupled with ordinary and/or partial differential equations when optimizing time-varying processes. Inequalities g(z) ≤ 0 put limits on the values variables can take, such as a minimum and maximum temperature, or they restrict one pressure to be greater than another. An important issue is how to solve large problems that occur in distributed systems. The optimization of distributed systems is discussed in Refs. 52, 120, 244, and 285. For further reading on optimization, readers are directed to Refs. 120 and 244 as well as introductory texts on optimization applied to chemical engineering (Refs. 99 and 225). The material in this section is part of a more advanced treatment (Ref. 295). Packages There are a number of packages available for optimization, some of which are listed here. 1. Frameworks • GAMS. This framework is commercially available. It provides a uniform language to access several different optimization packages, many of them listed below. It will convert the model as expressed in “GAMS” into the form needed to run the package chosen. • AMPL. This framework is by Fourier and coworkers (Ref. 113) at Northwestern University. It is well suited for constructing complex models. • ASCEND. This framework is by Westerberg and coworkers (Ref. 295) at Carnegie-Mellon University. It features an objectoriented modeling language and is well suited for constructing complex models. 2. Algebraic optimization with equality and inequality constraints • SQP. A package by Biegler at Carnegie-Mellon University. • MINOS5.4. A package available from Stanford Research Institute (affiliated with Stanford University). This package is the state of the art for mildly nonlinear programming problems. • GRG. A package from Lasdon at the University of Texas, Dept. of Management Science. 3. Linear programming. Most current commercial codes for lin-

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MATHEMATICS

ear programming extend the Simplex algorithm, and they can typically handle problems with up to 15,000 constraints. • MPSX. From IBM • SCICONIC. From the company of that name • MINOS5.4 • Cplex. A package by R. Bixby at Rice University and Cplx, Inc. CONDITIONS FOR OPTIMALITY


du dF

i

∆h = 0

= 0,

i = 1, 2, . . . , r

Equality Constrained Problems—Lagrange Multipliers Form a scalar function, called the Lagrange function, by adding each of the equality constraints multiplied by an arbitrary multiplier to the objective function. m

Local Minimum Point for Unconstrained Problems Consider the following unconstrained optimization problem: Min {F(u) | u
Rr} u

If F is continuous and has continuous first and second derivatives, it is necessary that F is stationary with respect to all variations in the independent variables u at a point û, which is proposed as a minimum to F; that is, ∂F or ∇u F = 0 at u = û (3-80)
= 0, i = 1, 2, . . . , r ∂ui These are only necessary conditions, as point û may be a minimum, maximum, or saddle point. Sufficient conditions are that any local move away from the optimal point û gives rise to an increase in the objective function. Expand F in a Taylor series locally around the candidate point û up to second-order terms: 1 2 F(u) = F(û) + ∇u F T  û (u − û) +
(u − û)T∇uu F  û (u − û) + … 2 If û satisfies necessary conditions [Eq. (3-80)], the second term disappears in this last line. Sufficient conditions for the point to be a local 2 minimum are that the matrix of second partial derivatives ∇uu F is positive definite. This matrix is symmetric, so all of its eigenvalues are real; to be positive definite, they must all be greater than zero. Constrained Derivatives—Equality Constrained Problems Consider minimizing the objective function F written in terms of n variables z and subject to m equality constraints h(z) = 0, or Mzin {F(z) | h(z) = 0, z
Rn, h:Rn → Rm}

(3-81)

The point zˆ is tested to see if it could be a minimum point. It is necessary that F be stationary for all infinitesimal moves for z that satisfy the equality constraints. Linearize the m equality constraints around zˆ, getting h(ˆz + ∆z) = h(ˆz) + ∇z hT  zˆ ∆z

(3-82)

where ∆z = z − zˆ . There are m constraints here, so m of the variables are dependent, leaving r = n − m independent variables. Partition the variables ∆z into a set of m dependent variables ∆x and r = n − m independent variables ∆u. Equation (3-82), rearranged and then rewritten in terms of these variables, becomes

L(x, u, λ) = F(x, u) +  λi hi(x, u) = F(x, u) + λT h(x, u) i=1

At any point where the functions h(z) are zero, the Lagrange function equals the objective function. Next differentiate L with respect to variables x, u, and λ. ∇x LT  zˆ = ∇x F T  zˆ + λT∇h xT  zˆ

= 0T

(3-83)

∇u L  zˆ = ∇u F  zˆ + λ ∇h 

=0

(3-84)

∇λ LT  zˆ = hT(x, u)

=0

T

T

T

T u zˆ

T T

Solve Eq. (3-83) for the Lagrange multipliers λT = −∇x F T[∇h Tx ]−1

(3-85)

and then eliminate these multipliers from Eq. (3-84). ∇u LT = ∇u F T − ∇x F T[∇hTx ]∇h Tu = 0 T ∇uL is equal to the constrained derivatives for the problem, which should be zero at the solution to the problem. Also, these stationarity conditions very neatly provide the necessary conditions for optimality of an equality-constrained problem. Lagrange multipliers are often referred to as shadow prices, adjoint variables, or dual variables, depending on the context. Suppose the variables are at an optimum point for the problem. Perturb the variables such that only constraint hi changes. We can write ∆L = ∆F + λi ∆hi = 0 which is zero because, as just shown, the Lagrange function is at a stationary point at the optimum. Solving for the change in the objective function: ∆F = −λi∆hi The multiplier tells how the optimal value of the objective function changes for this small change in the value of a constraint while holding all the other constraints at zero. It is for this reason that they are often called shadow prices. Equality- and Inequality-Constrained Problems—KuhnTucker Multipliers Next a point is tested to see if it is an optimum one when there are inequality constraints. The problem is Mzin {F(z) | h(z) = 0, g(z) ≤ 0, z
Rn, F:Rn → R1, h:Rn → Rm, g:Rn → Rp}

The Lagrange function here is similar to that used above.

∆h = ∇x hT  zˆ ∆x + ∇uhT  zˆ ∆u = 0

L(z, λ, µ)  F(z) + λTh(z) + µTg(z)

This enables the solution for ∆x. Linearize the objective function F(z) in terms of the partitioned variables

Each of the inequality constraints gi(z) multiplied by what is called a Kuhn-Tucker multiplier µi is added to form the Lagrange function. The necessary conditions for optimality, called the Karush-KuhnTucker conditions for inequality-constrained optimization problems, are

∆F = ∇x F T  zˆ ∆x + ∇u F T  zˆ ∆u and substitute for ∆x. ∆F = {∇x F T − ∇u F T[∇x hT]−1 ∇u hT}zˆ ∆u



dF =
du

T ∆h = 0

r



dF ∆u = 
dui i=1

∆h = 0

∆ui

There is one term for each ∆ui in the row vector which is in the curly braces {}. These terms are called constrained derivatives, which tells how the object function changes when the independent variables ui are changed while keeping the constraints satisfied (by varying the dependent variables xi). Necessary conditions for optimality are that these constrained derivatives are zero; that is,

∇z L zˆ = ∇z F zˆ + ∇z h zˆ λ + ∇z g zˆ µ = 0 ∇λL = h(z) = 0 g(z) ≤ 0 µi gi(z) = 0 ,

i = 1, 2, . . . , p

µi ≥ 0 ,

i = 1, 2, . . . , p

(3-86)

Conditions in Eq. (3-86), called complementary slackness conditions, state that either the constraint gi(z) = 0 and/or its corresponding multiplier µi is zero. If constraint gi(z) is zero, it is behaving like an equality constraint, and its multiplier µi is exactly the same as a Lagrange multiplier for an equality constraint. If the constraint is

OPTIMIZATION away from zero, it is not a part of the problem and should not affect it. Setting its multiplier to zero removes it from the problem. As the goal is to minimize the objective function, releasing the constraint into the feasible region must not decrease the objective function. Using the shadow price argument above, it is evident that the multiplier must be nonnegative (Ref. 177). Sufficiency conditions to assure that a Kuhn-Tucker point is a local minimum point require one to prove that the objective function will increase for any feasible move away from such a point. To carry out such a test, one has to generate the matrix of second derivatives of the Lagrange function with respect to all the variables z evaluated at zˆ . The test is seldom done, as it requires too much work. STRATEGIES OF OPTIMIZATION The theory just covered tells if a candidate point is or is not the optimum point, but how is the candidate point found? The simplest strategy is to place a grid of points throughout the feasible space, evaluating the objective function at every grid point. If the grid is fine enough, then the point yielding the highest value for the objective function can be selected as the optimum. Twenty variables gridded over only ten points would take place over 1020 points in our grid, and, at one nanosecond per evaluation, it would take in excess of four thousand years to carry out these evaluations. Most strategies limit themselves to finding a local minimum point in the vicinity of the starting point for the search. Such a strategy will find the global optimum only if the problem has a single minimum point or a set of “connected” minimum points. A “convex” problem has only a global optimum. Pattern Search Suppose the optimization problem is to find the right mix of a given set of ingredients and the proper baking temperature and time to make the best cake possible. A panel of judges can be formed to judge the cakes; assume they are only asked to rank the cakes and that they can do that task in a consistent manner. Our approach will be to bake several cakes and ask the judges to rank them. For this type of problem, pattern-search methods can be used to find the better conditions for manufacturing the product. We shall only describe the ideas behind this approach. Details on implementing it can be found in Ref. 284. The complex method is one such pattern search method (see Fig. 3-54). First, form a “complex” of at least r + 1 (r = 2 and 4 points are used in Fig. 3-54) different points at which to bake the cakes by picking a range of suitable values for the r independent variables for the baking process. Bake the cakes and then ask the judges to identify the worst cake. For each independent variable, form the average value at which it was run in the complex. Draw a line from the coordinates of the worst cake through the average point—called the centroid—and continue on that line a distance that is twice that between these two points. This point will be the next test point. First decide if it is feasible. If so, bake the cake and discover if it leads to a cake that is better than the worst cake from the last set of cakes. If it is not feasible or it is not better, then return half the distance toward the average values from the last

FIG. 3-54

Complex method, a pattern search optimization method.

3-67

test and try again. If it is better, toss out the worst point of the last test and replace it with this new one. Again, ask the judges to find the worst cake. Continue as above until the cakes are all the same quality in the most recent test. It might pay to restart at this point, stopping finally if the restart leads to no improvement. The method takes large steps if the steps are being successful in improving the recipe. It collapses onto a set of points quite close to each other otherwise. The method works reasonably well, but it requires one to bake lots of cakes. The following strategies are all examples of Generalized Reduced Gradient (GRG) methods. Optimization of Unconstrained Objective Assume the objective function F is a function of independent variables ui, i = 1 ⋅⋅⋅ r. A computer program, given the values for the independent variables, can calculate F and its derivatives with respect to each ui. Assume that F is well approximated as an as-yet-unknown quadratic function in u. 1 F ≈ a + bTu +
uTQu 2 where a is a scalar; b, a vector; and Q, an r × r symmetric positive definite matrix. The gradient of the approximate function is ∇u F = b + Qu Setting the gradient to zero allows an estimate for its minimum. u = −Q−1b

(3-87)

Initially, Q and b are not known and the calculation proceeds as follows: b contains r unknown coefficients and Q another r(r + 1)/2. To estimate b and Q, the computer code is used repeatedly, getting r equations each time—namely (∇u F)(1) = b + Qu(1) (∇u F)(2) = b + Qu(2) ... (∇u F )(t) = b + Qu(t)

(3-88)

As soon as there are as many independent equations as there are unknown coefficients, these linear equations are solved for b and Q. A proper choice of the points u(i) guarantees getting independent equations to solve here. Given b and Q, Eq. (3-87) provides a new estimate for u as a candidate minimum point. The subroutine is used again to obtain the gradient of F at this point. If the gradient is essentially zero, the calculations stop, since a point has been found that satisfies the necessary conditions for optimality. If not, the equations are written in the form of Eq. (3-88) for this new point, adding them to the set while removing the oldest set of equations. The new set of equations for b and Q are solved, and the calculations continue until a minimum point is found. If removal of the oldest equations from the set in Eq. (3-88) leads to a singular set of equations, then different equations have to be selected for removal. Alternatively, all the older equations can be kept, with the new ones added to the top of the list. Pivoting can be done by proceeding down the list until a nonsingular set of equations is found. Then the older equations are used only if necessary. Also, since only one set of equations is being replaced, clever methods are available to find the solution to the equations with much less work than is required to solve the set of equations the first time (Refs. 89 and 259). Quadratic Fit for the Equality Constrained Case Next consider solving a problem of the form of Eq. (3-82). For each iteration k: 1. Enter with values provided for variables u(k). 2. Given values for u(k), solve equations h(x, u) = 0 for x(k). These will be m equations in m unknowns. If the equations are nonlinear, solving can be done using a variant of the Newton-Raphson method. 3. Use Eq. (3-85) to solve for the Lagrange multipliers λ(k). If the Newton-Raphson method (or any or several variants to it) is used to solve the equations, the jacobian matrix ∇ xT h| z (k) and its LU factors are already known so solving Eq. (3-85) requires very little effort. 4. Substitute λ(k) into Eq. (3-84), which in general will not be zero. The gradient ∇u L(k) computed will be the constrained derivatives of F with respect to the independent variables u(k). 5. Return.

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MATHEMATICS

The calculations begin with given values for the independent variables u and exit with the (constrained) derivatives of the objective function with respect to them. Use the routine described above for the unconstrained problem where a succession of quadratic fits is used to move toward the optimal point for an unconstrained problem. This approach is a form of the generalized reduced gradient (GRG) approach to optimizing, one of the better ways to carry out optimization numerically. Inequality Constrained Problems To solve inequality constrained problems, a strategy is needed that can decide which of the inequality constraints should be treated as equalities. Once that question is decided, a GRG type of approach can be used to solve the resulting equality constrained problem. Solving can be split into two phases: phase 1, where the goal is to find a point that is feasible with respect to the inequality constraints; and phase 2, where one seeks the optimum while maintaining feasibility. Phase 1 is often accomplished by ignoring the objective function and using instead p

F=

i=1



g 2i (z) if gi(z) > 0) 0 otherwise

until all the inequality constraints are satisfied. Then at each point, check which of the inequality constraints are active, or exactly equal to zero. These can be placed into the active set and treated as equalities. The remaining can be put aside to be used only for testing. A step can then be proposed using the GRG algorithm. If it does not cause one to violate any of the inactive inequality constraints, the step is taken. Otherwise one can add the closest inactive inequality constraint to the active set. Finding the closet inactive equality will almost certainly require a line search in the direction proposed by the GRG algorithm. When one comes to a stationary point, one has to test the active inequality constraints at that point to see if they should remain active. This test is done by examining the sign (they should be nonnegative if they are to remain active) of their respective Kuhn-Tucker multipliers. If any should be released, it has to be done carefully as the release of a constraint changes the multipliers for all the constraints. One can find oneself cycling through the testing to decide whether to release the constraints. A correct approach is to add slack variables s to the problem to convert the inequality constraints to equalities and then require the slack variables to remain positive. The multipliers associated with the inequalities s ≥ 0 all behave independently, and their sign tells one directly to keep or release the constraints. In other words, simultaneously release all the slack variables that have multipliers strictly less than zero. If released, the slack variables must be treated as a part of the set of independent variables until one is well away from the associated constraints for this approach to work. Successive Quadratic Programming (SQP) The above approach to finding the optimum is called a feasible path method, as it attempts at all times to remain feasible with respect to the equality and inequality constraints as it moves to the optimum. A quite different method exists called the Successive Quadratic Programming (SQP) method, which only requires one be feasible at the final solution. Tests that compare the GRG and SQP methods generally favor the SQP method so it has the reputation of being one of the best methods known for nonlinear optimization for the type of problems considered here. Assume certain inequality constraints will be active at the final solution. The necessary conditions for optimality are ∇z L(z, µ, λ) = ∇F + ∇g A µ + ∇hλ = 0,

gA(z) = 0,

h(z) = 0

Then one can apply Newton’s method to the necessary conditions for optimality, which are a set of simultaneous (non)linear equations. The Newton equations one would write are



∇zzL[z(i), u(i), λ(i)]

∇gA[z(i)]

∇h[z(i)]

∇gA[z(i)]T

0

0

∇h[z(i)]T

0

0

 

∆z(i)

∇zL[z(i), µ(i), λ(i)]

∆µ(i) = −

gA[z(i)]

∆λ(i)

h[z(i)]



A sufficient condition for a unique Newton direction is that the matrix of constraint derivatives is of full rank (linear independence

of constraints) and the Hessian matrix of the Lagrange function [∇zz L(z, µ, λ)] projected into the space of the linearized constraints is positive definite. The linearized system actually represents the solution of the following quadratic programming problem: 1 Min ∇F [z(i)]T∆z +
∆ zT ∇zz L[z(i), µ(i), λ(i)]∆z ∆z 2 subject to gA[z(i)] + ∇g A[z(i)]T∆z = 0

and

h[z(i)] + ∇h[z(i)]T∆z = 0

Reformulating the necessary conditions as a linear quadratic program has an interesting side effect. We can simply add linearizations of the inactive inequalities to the problem and let the active set be selected by the algorithm used to solve the linear quadratic program. Problems with calculating second derivatives as well as maintaining positive definiteness of the Hessian matrix can be avoided by approximating this matrix by B(i) using a quasi-Newton formula such as BFGS (Refs. 50, 84, 109, 110, 122, and 259). One maintains positive definiteness by skipping the update if it causes the matrix to lose this property. Here gradients of the Lagrange function are used to calculate the update formula (Refs. 136 and 228). The resulting quadratic program, which generates the search direction at each iteration i, becomes: 1 Min ∇F[z(i)]T∆z +
∆zTB(i)∆z ∆z 2 subject to

g[z(i)] + ∇g[z(i)]T∆z ≤ 0 h[z(i)] + ∇h[z(i)]T∆z = 0

This linear quadratic program will have a unique solution if B(i) is kept positive definite. Efficient solution methods exist for solving it (Refs. 119 and 123). Finally, to ensure convergence of this algorithm from poor starting points, a step size α is chosen along the search direction so that the point at the next iteration (zi + 1 = zi + αd) is closer to the solution of the NLP (Refs. 65, 136, and 254). These problems get very large as the Lagrange function involves all the variables in the problem. If one has a problem with 5000 variables z and the problem has only 10 degrees of freedom (i.e., the partitioning will select 4990 variables x and only 10 variables u), one is still faced with maintaining a matrix B that is 5000 × 5000. See Westerberg (Ref. 40) for references to this case. Interior Point Algorithms for Linear Programming Problems There has been considerable excitement in the popular press about so-called interior point algorithms (Ref. 23) for solving extremely large linear programming problems. Computational demands for these algorithms grow less rapidly than for the Simplex algorithm, with a breakeven point being a few thousand constraints. A key idea for an interior method is that one heads across the feasible region to locate the solution rather than around its edges as one does for the Simplex algorithm. This move is found by computing the direction of steepest descent for the objective function with respect to changing the slack variables. Variables u are computed in terms of the slack variables by using the inequality constraints. The direction of steepest descent is a function of the scaling of the variables used for the problem. See Refs. 6, 124, 199, and 295. Linear Programming The combined term linear programming is given to any method for finding where a given linear function of several variables takes on an extreme value, and what that value is, when the variables are nonnegative and are constrained by linear equalities or inequalities. A very general problem consists of maximizing f = nf = 1 cj zj subject to the constraints zj ≥ 0 ( j = 1, 2, . . . , n) n and j = 1 aij zj ≤ bi (i = 1, 2, . . . , m). With S the set of all points whose coordinates zj satisfy all the constraints, we must ask three questions: (1) Are the constraints consistent? If not, S is empty and there is no solution. (2) If S is not empty, does the function f become unbounded on S? If so, the problem has no solution. If not, then there is a point P of S that is optimal in the sense that if Q is any point of S then f(Q) ≤ f(P). (3) How can we find P? The simplex algorithm, in a sense, prepares the problem before cal-

STATISTICS culation in such a way that favorable answers to these questions are tentatively assumed for the given problem and can be guaranteed for the prepared problem. The calculations then reveal whether or not those assumptions are justified for the given problem. The simplex algorithm terminates automatically, yielding full information on the given problem and so-called dual problem. The dual of the general problem of linear programming is to minimizem d(µ1, . . . , µm) = m  i = 1 µi bi subject to µi ≥ 0 (i = 1, 2, . . . , m) and  i = 1 µ i aif ≥ cj( j = 1, 2, . . . , n). Let A be the matrix [aij], c = [cj], U = [µi] be row vectors, and B = [bi]T, Z = [zj]T be column vectors. In matrix form the original (primal) problem is to maximize f(Z) = CZ subject to Z ≥ 0, AZ ≤ B. The dual is to minimize d(U) = UB subject to U ≥ 0, UA ≥ C. Example Maximize 3z1 + 4z2 subject to the constraints z1 ≥ 0, z2 ≥ 0, 2z1 + 4z2 ≤ 8, and 4z1 + 3z2 ≤ 10. The dual problem is to minimize 8µ1 + 10µ2 subject to the constraints µ1 ≥ 0, µ2 ≥ 0, 2µ1 + 4µ2 ≥ 3, and 5µ1 + 3µ2 ≥ 4. Simplex Method 1. Original problem. Let the column vector [zj]T = z ( j = 1, 2, n . . . , n) and the row vector [cj] = c. To maximize f(z) =  j = 1 cj zj = cTz subject to then n constraints zj ≥ 0 ( j = 1, . . . , n) and m further constraints hi:  j = 1 aij zj° ibi (i = 1, 2, . . . , m) where °i can be ≥ = or ≤. If any bi ≤ 0, multiply hi by − 1; thus we may assume bi ≥ 0. We suppose the m constraints have been arranged so that °i is ≥ for i = 1, . . . , g; °i is = for i = g + 1, . . . , g + e; °i is ≤ for i = g + e + 1, . . . , g + e + l = m. 2. Adjusted original problem. Introduce m + g further variables

3-69

with associated constraints and coefficients for use in f. Thus, replacing j by j + m, f becomes f(x) =  mj =+mn + 1 cj zj and constraints zj ≥ 0 and hi:  mj =+mn + 1 aij zj° i bi, i = 1, . . . , m. 3. Prepared problem. For i = 1, . . . , g replace hi by Hi: zi +  mj =+mn + 1 aij zj zm + n + i = bi, define ci = −M(M > 0 and “large”) and Cm + n + i = 0, and add the constraints zi ≥ 0, zm + n + i ≥ 0. For i = g + e + 1, . . . , m replace hi by Hi. xi +  mj =+mn + 1 aij zj = bi, define ci = 0, and adjoin zi ≥ 0. Let J run from 1 to N = n + m + g; put Z = [zj]T and j = m + 1, . . . , m + n. The new function to be maximized is f(Z) = N g=e vm + n  J = 1 cj z j. Actually this is f(Z) = −M  i = 1 zi +  j = m + 1 cjzj, for all other coefficients are zero. Thus for J = g + e + 1, . . . , m and m + n + 1, . . . , N the variables z j make no contribution to f. They are called slack variables, since they take up the slack permitted by the inequalities (≤ and ≥) in hi. Any variable zj, i = 1, . . . , g + e whose value is not zero gives rise to a large negative term −Mzi. Such a term will keep f(Z) less than it would be with that zi = 0. The effect of ci = −M(i = 1, . . . , g + e) is to make it likely that the optimal solution will have the artificial variables zi = 0(i = 1, . . . , g + e). N The prepared problem now has the form—maximize f(Z) =  j = 1 cjzJ N subject to zJ ≥ 0 and Hi:  j = 1 aijzj = (i = 1, . . . , m), where bi ≥ 0, aiβ = δiβ =

i≠β (β = 1, . . . , m), ai,m + n + β = −δiβ (β = 1, . . . , g) i≠β

l

0

and aij came from hi. The set of feasible points SP (points satisfying all constraints) for the prepared problem is not empty, and f(Z) is bounded above on SP.

STATISTICS GENERAL REFERENCES:

69, 93, 134, 169, 186, 211, 265, 291.

INTRODUCTION Statistics represents a body of knowledge which enables one to deal with quantitative data reflecting any degree of uncertainty. There are six basic aspects of applied statistics. These are: 1. Type of data 2. Random variables 3. Models 4. Parameters 5. Sample statistics 6. Characterization of chance occurrences From these can be developed strategies and procedures for dealing with (1) estimation and (2) inferential statistics. The following has been directed more toward inferential statistics because of its broader utility. Detailed illustrations and examples are used throughout to develop basic statistical methodology for dealing with a broad area of applications. However, in addition to this material, there are many specialized topics as well as some very subtle areas which have not been discussed. The references should be used for more detailed information. Type of Data In general, statistics deals with two types of data: counts and measurements. Counts represent the number of discrete outcomes, such as the number of defective parts in a shipment, the number of lost-time accidents, and so forth. Measurement data are treated as a continuum. For example, the tensile strength of a synthetic yarn theoretically could be measured to any degree of precision. A subtle aspect associated with count and measurement data is that some types of count data can be dealt with through the application of techniques which have been developed for measurement data alone. This ability is due to the fact that some simplified measurement statistics serve as an excellent approximation for the more tedious count statistics. Random Variables Applied statistics deals with quantitative data. In tossing a fair coin the successive outcomes would tend to be

different, with heads and tails occurring randomly over a period of time. Given a long strand of synthetic fiber, the tensile strength of successive samples would tend to vary significantly from sample to sample. Counts and measurements are characterized as random variables, that is, observations which are susceptible to chance. Virtually all quantitative data are susceptible to chance in one way or another. Models Part of the foundation of statistics consists of the mathematical models which characterize an experiment. The models themselves are mathematical ways of describing the probability, or relative likelihood, of observing specified values of random variables. For example, in tossing a coin once, a random variable x could be defined by assigning to x the value 1 for a head and 0 for a tail. Given a fair coin, the probability of observing a head on a toss would be a .5, and similarly for a tail. Therefore, the mathematical model governing this experiment can be written as x

P(x)

0 1

.5 .5

where P(x) stands for what is called a probability function. This term is reserved for count data, in that probabilities can be defined for particular outcomes. The probability function that has been displayed is a very special case of the more general case, which is called the binomial probability distribution. For measurement data which are considered continuous, the term probability density is used. For example, consider a spinner wheel which conceptually can be thought of as being marked off on the circumference infinitely precisely from 0 up to, but not including, 1. In spinning the wheel, the probability of the wheel’s stopping at a specified marking point at any particular x value, where 0 ≤ x < 1, is zero, for example, stopping at the value x =
 .5. For the spinning wheel, the probability density function would be defined by f(x) = 1 for 0 ≤ x < 1. Graphically, this is shown in Fig. 3-55. The relative-probability

3-70

MATHEMATICS 1 n x =
 xi with n = 6 n i=1 = (1/6)(.1009 + .3754 + ⋅⋅⋅ + .6606) = .3899 The sample standard deviation s is defined by the computation s= Density function.

FIG. 3-55

=

concept refers to the fact that density reflects the relative likelihood of occurrence; in this case, each number between 0 and 1 is equally likely. For measurement data, probability is defined by the area under the curve between specified limits. A density function always must have a total area of 1. P [0 ≤ x ≤ .4] = .4 P [.2 ≤ x ≤ .9] = .7 P[.6 ≤ x < 1] = .4 and so forth. Since the probability associated with any particular point value is zero, it makes no difference whether the limit point is defined by a closed interval (≤ or ≥) or an open interval (< or >).

Many different types of models are used as the foundation for statistical analysis. These models are also referred to as populations. Parameters As a way of characterizing probability functions and densities, certain types of quantities called parameters can be defined. For example, the center of gravity of the distribution is defined to be the population mean, which is designated as µ. For the coin toss µ = .5, which corresponds to the average value of x; i.e., for half of the time x will take on a value 0 and for the other half a value 1. The average would be .5. For the spinning wheel, the average value would also be .5. Another parameter is called the standard deviation, which is designated as σ. The square of the standard deviation is used frequently and is called the popular variance, σ2. Basically, the standard deviation is a quantity which measures the spread or dispersion of the distribution from its mean µ. If the spread is broad, then the standard deviation will be larger than if it were more constrained. For specified probability and density functions, the respective means and variances are defined by the following: Probability functions

Probability density functions E(x) = µ =

Var(x) = σ =  (x − µ) P(x) 2

x f(x) dx x

x

Var(x) = σ =

2

2

(x − µ) f(x) dx 2

x

x

(xi − x)2

n−1

  

n x 2i − ( xi)2

n(n − 1)

In effect, this represents the root of a statistical average of the squares. The divisor quantity (n − 1) will be referred to as the degrees of freedom. The value of n − 1 is used in the denominator because the deviations from the sample average must total zero, or

 (x − x) = 0 i

Thus knowing n − 1 values of xi − x permits calculation of the nth value of xi − x. The sample value of the standard deviation for the data given is .3686. The following is a tabulation of the deviations (xi − xj) for the data:

Example For the density of Fig. 3-55 the

E(x) = µ =  x P(x)

 

where E(x) is defined to be the expected or average value of x. Sample Statistics Many types of sample statistics will be defined. Two very special types are the sample mean, designated as x, and the sample standard deviation, designated as s. These are, by definition, random variables. Parameters like µ and σ are not random variables; they are fixed constants. Example In an experiment, six random numbers (rounded to four decimal places) were observed from the uniform distribution f(x) = 1 for 0 ≤ x < 1: .1009 .3754 .0842 .9901 .1280 .6606 The sample mean corresponds to the arithmetic average of the observations, which will be designated as x1 through x6, where

x .1009 .3754 .0842 .9901 .1280 .6606

x − x −.2890 −.0145 −.3057 .6002 −.2619 .2707

x = .3899

s = .3686

In effect, the standard deviation quantifies the relative magnitude of the deviation numbers, i.e., a special type of “average” of the distance of points from their center. In statistical theory, it turns out that the corresponding variance quantities s2 have remarkable properties which make possible broad generalities for sample statistics and therefore also their counterparts, the standard deviations. For the corresponding population, the parameter values are µ = .50 and σ = .2887. If, instead of using individual observations only, averages of six were reported, then the corresponding population parameter values would be µ = .50 and σx = σ/
6 = .1179. The corresponding variance for an average will be written occasionally as Var (x) = var (x)/n. In effect, the variance of an average is inversely proportional to the sample size n, which reflects the fact that sample averages will tend to cluster about µ much more closely than individual observations. This is illustrated in greater detail under “Measurement Data and Sampling Densities.” Characterization of Chance Occurrences To deal with a broad area of statistical applications, it is necessary to characterize the way in which random variables will vary by chance alone. The basic foundation for this characteristic is laid through a density called the gaussian, or normal, distribution. Determining the area under the normal curve is a very tedious procedure. However, by standardizing a random variable that is normally distributed, it is possible to relate all normally distributed random variables to one table. The standardization is defined by the identity z = (x − µ)/σ, where z is called the unit normal. Further, it is possible to standardize the sampling distribution of averages x by the identity z = (x − µ)/(σ/
 n). A remarkable property of the normal distribution is that, almost regardless of the distribution of x, sample averages x will approach the gaussian distribution as n gets large. Even for relatively small values of n, of about 10, the approximation in most cases is quite close. For example, sample averages of size 10 from the uniform distribution will have essentially a gaussian distribution. Also, in many applications involving count data, the normal distribution can be used as a close approximation. In particular, the approximation is quite close for the binomial distribution within certain guidelines.

STATISTICS ENUMERATION DATA AND PROBABILITY DISTRIBUTIONS Introduction Many types of statistical applications are characterized by enumeration data in the form of counts. Examples are the number of lost-time accidents in a plant, the number of defective items in a sample, and the number of items in a sample that fall within several specified categories. The sampling distribution of count data can be characterized through probability distributions. In many cases, count data are appropriately interpreted through their corresponding distributions. However, in other situations analysis is greatly facilitated through distributions which have been developed for measurement data. Examples of each will be illustrated in the following subsections. Binomial Probability Distribution Nature Consider an experiment in which each outcome is classified into one of two categories, one of which will be defined as a success and the other as a failure. Given that the probability of success p is constant from trial to trial, then the probability of observing a specified number of successes x in n trials is defined by the binomial distribution. The sequence of outcomes is called a Bernoulli process, Nomenclature n = total number of trials x = number of successes in n trials p = probability of observing a success on any one trial pˆ = x/n, the proportion of successes in n trials Probability Law x n x P(x) = P
= p (1 − p)n − x x = 0, 1, 2, . . . , n x n

  

where

 x =
x!(n − x)! n

n!

Properties E(x) = np E( p ˆ)=p

5. The number of spinning machines that break down during 1 day at a large plant. Nomenclature x = total number of arrivals in a total length L or total period T a = average rate of arrivals for a unit length or unit time λ = aL = expected or average number of arrivals for the total length L λ = aT = expected or average number of arrivals for the total time T Probability Law Given that a is constant for the total length L or period T, the probability of observing x arrivals in some period L or T is given by λx P(x) =
e−λ x = 0, 1, 2, . . . x! Properties E(x) = λ

Var (x) = λ

Example The number of broken filaments in a threadline has been averaging .015 per yard. What is the probability of observing exactly two broken filaments in the next 100 yd? In this example, a = .015/yd and L = 100 yd; therefore λ = (.015)(100) = 1.5: (1.5)2 P(x = 2) =
e−1.5 = .2510 2! Example A commercial item is sold in a retail outlet as a unit product. In the past, sales have averaged 10 units per month with no seasonal variation. The retail outlet must order replacement items 2 months in advance. If the outlet starts the next 2-month period with 25 items on hand, what is the probability that it will stock out before the end of the second month? Given a = 10/month, then λ = 10 × 2 = 20 for the total period of 2 months: ∞

25

26

0

P(x ≥ 26) =  P(x) = 1 −  P(x)



2025 20x −20 −20 20 202 e =e 1 +
+
+ ⋅⋅⋅ +
0
x! 1 2! 25! 25



= .887815

Var(x) = np(1 − p) Var( p ˆ ) = p(1 − p)/n

Geometric Probability Distribution Nature Consider an experiment in which each outcome is classified into one of two categories, one of which will be defined as a success and the other as a failure. Given that the probability of success p is constant from trial to trial, then the probability of observing the first success on the xth trial is defined by the geometric distribution. Nomenclature p = probability of observing a success on any one trial x = the number of trials to obtain the first success Probability Law P(x) = p(1 − p)x − 1

x = 1, 2, 3, . . .

Properties E(x) = 1/p

3-71

Var (x) = (1 − p)/p2

Poisson Probability Distribution Nature In monitoring a moving threadline, one criterion of quality would be the frequency of broken filaments. These can be identified as they occur through the threadline by a broken-filament detector mounted adjacent to the threadline. In this context, the random occurrences of broken filaments can be modeled by the Poisson distribution. This is called a Poisson process and corresponds to a probabilistic description of the frequency of defects or, in general, what are called arrivals at points on a continuous line or in time. Other examples include: 1. The number of cars (arrivals) that pass a point on a high-speed highway between 10:00 and 11:00 A.M. on Wednesdays 2. The number of customers arriving at a bank between 10:00 and 10:10 A.M. 3. The number of telephone calls received through a switchboard between 9:00 and 10:00 A.M. 4. The number of insurance claims that are filed each week

Therefore P(x ≥ 26) = .112185 or roughly an 11 percent chance of a stockout.

Hypergeometric Probability Distribution Nature In an experiment in which one samples from a relatively small group of items, each of which is classified in one of two categories, A or B, the hypergeometric distribution can be defined. One example is the probability of drawing two red and two black cards from a deck of cards. The hypergeometric distribution is the analog of the binomial distribution when successive trials are not independent, i.e., when the total group of items is not infinite. This happens when the drawn items are not replaced. Nomenclature N = total group size n = sample group size X = number of items in the total group with a specified attribute A N − X = number of items in the total group with the other attribute B x = number of items in the sample with a specified attribute A n − x = number of items in the sample with the other attribute B

Category A Category B Total

Probability Law P(x) =

Population

Sample

X N−X N

x n−x n

Nn −− Xx   XxNn

nX E(x) =
N N−n var (x) = nP(1 − P)
N−1

3-72

MATHEMATICS

Example What is the probability that an appointed special committee of 4 has no female members when the members are randomly selected from a candidate group of 10 males and 7 females? 10 7 4 0 P(x = 0) = = .0882 17 4

    

Example A bin contains 300 items, of which 240 are good and 60 are defective. In a sample of 6 what is the probability of selecting 4 good and 2 defective items by chance?

 4  2  = .2478 300 6 240

P(x) =

60

Multinomial Distribution Nature For an experiment in which successive outcomes can be classified into two or more categories and the probabilities associated with the respective outcomes remain constant, then the experiment can be characterized through the multinomial distribution. Nomenclature n = total number of trials k = total number of distinct categories pj = probability of observing category j on any one trial, j = 1, 2, . . . , k xj = total number of occurrences in category j in n trials Probability Law n! P(x1, x2, . . . , xk) =

p1x1 p2x2 ⋅⋅⋅ pkx k x1! x2! . . . xk! Example In tossing a die 12 times, what is the probability that each face value will occur exactly twice? 12! 1 2 1 2 1 2 1 2 1 2 1 2 p(2, 2, 2, 2, 2, 2) =







= .003438 2!2!2!2!2!2! 6 6 6 6 6 6

Monday 16

Tuesday 17

Wednesday 18

Thursday 19

Friday 20

Groups of 25 pooled

37.68 36.38 38.43 39.07 33.06 x = 36.92 s = 2.38

35.97 35.92 36.51 33.89 36.01 35.66 1.02

33.71 32.34 33.29 32.81 37.13 33.86 1.90

35.61 37.13 31.37 35.89 36.33 35.27 2.25

36.65 37.91 42.18 39.25 33.32 37.86 3.27

35.91 2.52

Pooled sample of 100: x = 35.16

Even if the process were at steady state, tensile strength, a key property would still reflect some variation. Steady state, or stable operation of any process, has associated with it a characteristic variation. Superimposed on this is the testing method, which is itself a process with its own characteristic variation. The observed variation is a composite of these two variations. Assume that the table represents “typical” production-line performance. The numbers themselves have been generated on a computer and represent random observations from a population with µ = 35 and a population standard deviation σ = 2.45. The sample values reflect the way in which tensile strength can vary by chance alone. In practice, a production supervisor unschooled in statistics but interested in high tensile performance would be despondent on the eighth day and exuberant on the twentieth day. If the supervisor were more concerned with uniformity, the lowest and highest points would have been on the eleventh and seventeenth days. An objective of statistical analysis is to serve as a guide in decision making in the context of normal variation. In the case of the production supervisor, it is to make a decision, with a high probability of being correct, that something has in fact changed the operation. Suppose that an engineering change has been made in the process and five new tensile samples have been tested with the results: 36.81 38.34 34.87 39.58 36.12

      

MEASUREMENT DATA AND SAMPLING DENSITIES Introduction The following example data are used throughout this subsection to illustrate concepts. Consider, for the purpose of illustration, that five synthetic-yarn samples have been selected randomly from a production line and tested for tensile strength on each of 20 production days. For this, assume that each group of five corresponds to a day, Monday through Friday, for a period of 4 weeks: Monday 1

Tuesday 2

Wednesday 3

Thursday 4

Friday 5

36.48 35.33 35.92 32.28 31.61 x = 34.32 s = 2.22

38.06 31.86 33.81 30.30 35.27 33.86 3.01

35.28 36.58 38.81 33.31 33.88 35.57 2.22

36.34 36.25 30.46 37.37 37.52 35.59 2.92

36.73 37.17 33.07 34.27 36.94 35.64 1.85

6

7

8

9

10

38.67 32.08 33.79 32.85 35.22 x = 34.52 s = 2.60

36.62 33.05 35.43 36.63 31.46 34.64 2.30

35.03 36.22 32.71 32.52 27.23 32.74 3.46

35.80 33.16 35.19 32.91 35.44 34.50 1.36

36.82 36.49 32.83 32.43 34.16 34.54 2.03

11

12

13

14

15

39.63 34.38 36.51 30.00 39.64 x = 36.03 s = 4.04

34.52 37.39 34.16 35.76 37.63 35.89 1.59

36.05 35.36 35.00 33.61 36.98 35.40 1.25

36.64 31.18 36.13 37.51 39.05 36.10 2.96

31.57 36.21 33.84 35.01 34.95 34.32 1.75

Groups of 25 pooled

34.19 2.35

x = 37.14 s = 1.85

In this situation, management would inquire whether the product has been improved by increased tensile strength. To answer this question, in addition to a variety of analogous questions, it is necessary to have some type of scientific basis upon which to draw a conclusion. A scientific basis for the evaluation and interpretation of data is contained in the accompanying table descriptions. These tables characterize the way in which sample values will vary by chance alone in the context of individual observations, averages, and variances.

Table number

35.00 2.40

s = 2.47

Designated symbol

Variable

Sampling distribution of

3-4

z

x−µ
σ

3-4

z

x − µ
σ/
 n

Averages

3-5

t

x − µ
s/
 n

Averages when σ is unknown*

3-6

χ2

(s2/σ2)(df)

3-7

F

s12 /s 22

Observations*

Variances* Ratio of two independent sample variances*

*When sampling from a gaussian distribution.

35.55 2.42

Normal Distribution of Observations Many types of data follow what is called the gaussian, or bell-shaped, curve; this is especially true of averages. Basically, the gaussian curve is a purely mathematical function which has very special properties. However, owing to some mathematically intractable aspects primary use of the function is restricted to tabulated values.

STATISTICS

3-73

0 and 2 is .4772; therefore, the area between 1 and 2 is .4772 − .3413 = .1359. Also, since the normal curve is symmetric, areas to the left can be determined in exactly the same way. For example, the area between −2 and +1 would include the area between −2 and 0, .4772 (the same as 0 to 2), plus the area between 0 and 1, .3413, or a total area of .8185. Any types of observation which are applicable to the normal curve can be transformed to Z values by the relationship z = (x − µ)/σ and, conversely, Z values to x values by x = µ + σz, as shown in Fig. 3-56. For example, for tensile strength, with µ = 35 and σ = 2.45, this would dictate z = (x − 35)/2.45 and x = 35 + 2.45z. Example What proportion of tensile values will fall between 34 and 36? FIG. 3-56

z1 = (34 − 35)/2.45 = −.41

Transformation of z values.

z2 = (36 − 35)/2.45 = .41

P[−.41 ≤ z ≤ .41] = .3182, or roughly 32 percent

Basically, the tabled values represent area (proportions or probability) associated with a scaling variable designated by Z in Fig. 3-56. The normal curve is centered at 0, and for particular values of Z, designated as z, the tabulated numbers represent the corresponding area under the curve between 0 and z. For example, between 0 and 1 the area is .3413. (Get this number from Table 3-4. The value of A includes the area on both sides of zero. Thus we want A/2. For z = 1, A = 0.6827, A/2 = 0.3413. For z = 2, A/2 = 0.4772.) The area between

TABLE 3-4

The value 0.3182 is interpolated from Table 3-4 using z = 0.40, A = 0.3108, and z = 0.45, A = 0.3473.

Example What midrange of tensile values will include 95 percent of the sample values? Since P[−1.96 ≤ z ≤ 1.96] = .95, the corresponding values of x are x1 = 35 − 1.96(2.45) = 30.2 x2 = 35 + 1.96(2.45) = 39.8 P[30.2 ≤ x ≤ 39.8] = .95

or

Ordinates and Areas between Abscissa Values -z and +z of the Normal Distribution Curve

z

X

Y

A

1−A

z

X

0 0.05  .10  .15  .20

µ µ  0.05σ µ  .10σ µ  .15σ µ  .20σ

0.399 .398 .397 .394 .391

0.0000 .0399 .0797 .1192 .1585

1.0000 0.9601 .9203 .8808 .8415

1.50 1.55 1.60 1.65 1.70

µ  1.50σ µ  1.55σ µ  1.60σ µ  1.65σ µ  1.70σ

0.1295 .1200 .1109 .1023 .0940

0.8664 .8789 .8904 .9011 .9109

0.1336 .1211 .1096 .0989 .0891

    

.25 .30 .35 .40 .45

µ µ µ µ µ

.25σ .30σ .35σ .40σ .45σ

.387 .381 .375 .368 .361

.1974 .2358 .2737 .3108 .3473

.8026 .7642 .7263 .6892 .6527

1.75 1.80 1.85 1.90 1.95

µ  1.75σ µ  1.80σ µ  1.85σ µ  1.90σ µ  1.95σ

.0863 .0790 .0721 .0656 .0596

.9199 .9281 .9357 .9446 .9488

.0801 .0719 .0643 .0574 .0512

    

.50 .55 .60 .65 .70

µ µ+ µ µ µ

.50σ .55σ .60σ .65σ .70σ

.352 .343 .333 .323 .312

.3829 .4177 .4515 .4843 .5161

.6171 .5823 .5485 .5157 .4839

2.00 2.05 2.10 2.15 2.20

µ  2.00σ µ  2.05σ µ  2.10σ µ  2.15σ µ  2.20σ

.0540 .0488 .0440 .0396 .0335

.9545 .9596 .9643 .9684 .9722

.0455 .0404 .0357 .0316 .0278

    

.75 .80 .85 .90 .95

µ µ µ µ µ

.75σ .80σ .85σ .90σ .95σ

.301 .290 .278 .266 .254

.5467 .5763 .6047 .6319 .6579

.4533 .4237 .3953 .3681 .3421

2.25 2.30 2.35 2.40 2.45

µ  2.25σ µ  2.30σ µ  2.35σ µ  2.40σ µ  2.45σ

.0317 .0283 .0252 .0224 .0198

.9756 .9786 .9812 .9836 .9857

.0244 .0214 .0188 .0164 .0143

1.00 1.05 1.10 1.15 1.20

µ  1.00σ µ  1.05σ µ  1.10σ µ  1.15σ µ  1.20σ

.242 .230 .218 .206 .194

.6827 .7063 .7287 .7499 .7699

.3173 .2937 .2713 .2501 .2301

2.50 2.55 2.60 2.65 2.70

µ  2.50σ µ  2.55σ µ  2.60σ µ  2.65σ µ  2.70σ

.0175 .0154 .0136 .0119 .0104

.9876 .9892 .9907 .9920 .9931

.0124 .0108 .0093 .0080 .0069

1.25 1.30 1.35 1.40 1.45

µ  1.25σ µ  1.30σ µ  1.35σ µ  1.40σ µ  1.45σ

.183 .171 .160 .150 .139

.7887 .8064 .8230 .8385 .8529

.2113 .1936 .1770 .1615 .1471

2.75 2.80 2.85 2.90 2.95

µ  2.75σ µ  2.80σ µ  2.85σ µ  2.90σ µ  2.95σ

.0091 .0079 .0069 .0060 .0051

.9940 .9949 .9956 .9963 .9968

.0060 .0051 .0044 .0037 .0032

1.50

µ  1.50σ

.130

.8664

.1336

3.00 4.00 5.00

µ  3.00σ µ  4.00σ µ  5.00s

.0044 .0001 .000001

.9973 .99994 .9999994

.0027 .00006 .0000006

0.000  .126  .253  .385  .524  .674  .842

µ µ  0.126σ µ  .253σ µ  .385σ µ  .524σ µ  .674σ µ  .842σ

0.3989 .3958 .3863 .3704 .3477 .3178 .2800

.0000 .1000 .2000 .3000 .4000 .5000 .6000

1.0000 0.9000 .8000 .7000 .6000 .5000 .4000

1.036 1.282 1.645 1.960 2.576 3.291 3.891

µ  1.036σ µ  1.282σ µ  1.645σ µ  1.960σ µ  2.576σ µ  3.291σ µ  3.891σ

Y

0.2331 .1755 .1031 .0584 .0145 .0018 .0002

A

0.7000 .8000 .9000 .9500 .9900 .9990 .9999

1−A

0.3000 .2000 .1000 .0500 .0100 .0010 .0001

3-74

MATHEMATICS

Normal Distribution of Averages An examination of the tensile-strength data previously tabulated would show that the range (largest minus the smallest) of tensile strength within days averages 5.72. The average range in x values within each week is 2.37, while the range in the four weekly averages is 1.72. This reflects the fact that averages tend to be less variable in a predictable way. Given that the variance of x is var (x) = σ2, then the variance of x based on n observations is var (x) = σ2/n. For averages of n observations, the corresponding relationship for the Z-scale relationship is σ z = (x − µ/σ/
 n) or z x = µ +

 n Example What proportion of daily tensile averages will fall between 34 and 36? z1 = (34 − 35)/(2.45/
5) = −.91

z2 = (36 − 35)/(2.45/
5) = .91

P[−.91 ≤ z ≤ .91] = .6372, or roughly 64 percent

Example What midrange of daily tensile averages will include 95 percent of the sample values? x1 = 35 − 1.96(2.45/
5) = 32.85 x2 = 35 + 1.96(2.45/
5) = 37.15 P[32.85 ≤ x ≤ 37.15] = .95

or

Example What proportion of weekly tensile averages will fall between 34 and 36? z1 = (34 − 35)/(2.45/
25) = −2.04 z2 = (36 − 35)/(2.45/
25) = 2.04 P[−2.04 ≤ z ≤ 2.04] = .9586, or roughly 96 percent

Distribution of Averages The normal curve relies on a knowledge of σ, or in special cases, when it is unknown, s can be used with the normal curve as an approximation when n > 30. For example, with n > 30 the intervals x  s and x  2s will include roughly 68 and 95 percent of the sample values respectively when the distribution is normal. In applications sample sizes are usually small and σ unknown. In these cases, the t distribution can be used where t = (x − µ)/(s/
n ) or  x = µ + ts/
n The t distribution is also symmetric and centered at zero. It is said to be robust in the sense that even when the individual observations x are not normally distributed, sample averages of x have distributions which tend toward normality as n gets large. Even for small n of 5 through 10, the approximation is usually relatively accurate. In reference to the tensile-strength table, consider the summary statistics x and s by days. For each day, the t statistic could be computed. If this were repeated over an extensive simulation and the resultant t quantities plotted in a frequency distribution, they would match the corresponding distribution of t values summarized in Table 3-5. Since the t distribution relies on the sample standard deviation s, the resultant distribution will differ according to the sample size n. To designate this difference, the respective distributions are classified according to what are called the degrees of freedom and abbreviated as df. In simple problems, the df are just the sample size minus 1. In more complicated applications the df can be different. In general, degrees of freedom are the number of quantities minus the number of constraints. For example, four numbers in a square which must have row and column sums equal to zero have only one df, i.e., four numbers minus three constraints (the fourth constraint is redundant).

TABLE 3-5 df

t.30

t.20

t.10

t.05

t.025

t.01

t.005

1 2 3 4 5

0.325 .289 .277 .271 .267

0.727 .617 .584 .569 .559

1.376 1.061 0.978 .941 .920

3.078 1.886 1.638 1.533 1.476

6.314 2.920 2.353 2.132 2.015

12.706 4.303 3.182 2.776 2.571

31.821 6.965 4.541 3.747 3.365

63.657 9.925 5.841 4.604 4.032

6 7 8 9 10

.265 .263 .262 .261 .260

.553 .549 .546 .543 .542

.906 .896 .889 .883 .879

1.440 1.415 1.397 1.383 1.372

1.943 1.895 1.860 1.833 1.812

2.447 2.365 2.306 2.262 2.228

3.143 2.998 2.896 2.821 2.764

3.707 3.499 3.355 3.250 3.169

11 12 13 14 15

.260 .259 .259 .258 .258

.540 .539 .538 .537 .536

.876 .873 .870 .868 .866

1.363 1.356 1.350 1.345 1.341

1.796 1.782 1.771 1.761 1.753

2.201 2.179 2.160 2.145 2.131

2.718 2.681 2.650 2.624 2.602

3.106 3.055 3.012 2.977 2.947

16 17 18 19 20

.258 .257 .257 .257 .257

.535 .534 .534 .533 .533

.865 .863 .862 .861 .860

1.337 1.333 1.330 1.328 1.325

1.746 1.740 1.734 1.729 1.725

2.120 2.110 2.101 2.093 2.086

2.583 2.567 2.552 2.539 2.528

2.921 2.898 2.878 2.861 2.845

21 22 23 24 25

.257 .256 .256 .256 .256

.532 .532 .532 .531 .531

.859 .858 .858 .857 .856

1.323 1.321 1.319 1.318 1.316

1.721 1.717 1.714 1.711 1.708

2.080 2.074 2.069 2.064 2.060

2.518 2.508 2.500 2.492 2.485

2.831 2.819 2.807 2.797 2.787

26 27 28 29 30

.256 .256 .256 .256 .256

.531 .531 .530 .530 .530

.856 .855 .855 .854 .854

1.315 1.314 1.313 1.311 1.310

1.706 1.703 1.701 1.699 1.697

2.056 2.052 2.048 2.045 2.042

2.479 2.473 2.467 2.462 2.457

2.779 2.771 2.763 2.756 2.750

40 60 120 ∞

.255 .254 .254 .253

.529 .527 .526 .524

.851 .848 .845 .842

1.303 1.296 1.289 1.282

1.684 1.671 1.658 1.645

2.021 2.000 1.980 1.960

2.423 2.390 2.358 2.326

2.704 2.660 2.617 2.576

Above values refer to a single tail outside the indicated limit of t. For example, for 95 percent of the area to be between −t and +t in a two-tailed t distribution, use the values for t0.025 or 2.5 percent for each tail.

Example What is the sample value of t for the first day of tensile data? Sample t = (34.32 − 35)/(2.22/
5) = −.68 Note that on the average 90 percent of all such sample values would be expected to fall within the interval 2.132.

t Distribution for the Difference in Two Sample Means Population Variances Are Equal The t distribution can be readily extended to the difference in two sample means when the respective populations have the same variance σ2: (x1 − x2) − (µ1 − µ2) t =

sp
1 /n +1 /n2 1  where s 2p is a pooled variance defined by (n1 − 1)s12 + (n2 − 1)s22 s 2p =


(n1 − 1) + (n2 − 1) In this application, the t distribution has (n1 + n2 − 2) df. Population Variances Are Unequal When population variances are unequal, an approximate t quantity can be used:

Example For a sample size n = 5, what values of t define a midarea of 90 percent. For 4 df the tabled value of t corresponding to a midarea of 90 percent is 2.132; i.e., P[−2.132 ≤ t ≤ 2.132] = .90. Example For a sample size n = 25, what values of t define a midarea of 95 percent? For 24 df the tabled value of t corresponding to a midarea of 95 percent is 2.064; i.e., P[−2.064 ≤ t ≤ 2.064] = .95.

Values of t

t.40

with and

(x1 − x2) − (µ1 − µ2) t =

a + b
 a = s12 / n1 b = s22 / n2 (a + b)2 df =


a2/(n1 − 1) + b2/(n2 − 1)

STATISTICS Chi-Square Distribution For some industrial applications, product uniformity is of primary importance. The sample standard deviation s is most often used to characterize uniformity. In dealing with this problem, the chi-square distribution can be used where χ 2 = (s2/σ2) (df). The chi-square distribution is a family of distributions which are defined by the degrees of freedom associated with the sample variance. For most applications, df is equal to the sample size minus 1. The probability distribution function is −(df )2 p(y) = y0ydf − 2 exp
2 where y0 is chosen such that the integral of p(y) over all y is one. In terms of the tensile-strength table previously given, the respective chi-square sample values for the daily, weekly, and monthly figures could be computed. The corresponding df would be 4, 24, and 99 respectively. These numbers would represent sample values from the respective distributions which are summarized in Table 3-6. In a manner similar to the use of the t distribution, chi square can be interpreted in a direct probabilistic sense corresponding to a midarea of (1 − α):





P[χ 21 ≤ (s2/σ2)(df) ≤ χ 22 ] = 1 − α where χ 21 corresponds to a lower-tail area of α/2 and χ 22 an upper-tail area of α/2. The basic underlying assumption for the mathematical derivation of chi square is that a random sample was selected from a normal distribution with variance σ2. When the population is not normal but skewed, square probabilities could be substantially in error. Example On the basis of a sample size n = 5, what midrange of values will include the sample ratio s/σ with a probability of 90 percent? Use Table 3-6 for 4 df and read χ 21 = 0.484 for a lower tail area of 0.05/2, 2.5 percent, and read χ 22 = 11.1 for an upper tail area of 97.5 percent.

TABLE 3-6

3-75

P[.484 ≤ (s2/σ2)(4) ≤ 11.1] = .90 P[.35 ≤ s/σ ≤ 1.66] = .90

or

Example On the basis of a sample size n = 25, what midrange of values will include the sample ratio s/σ with a probability of 90 percent? P[12.4 ≤ (s2/σ2)(24) ≤ 39.4] = .90 P[.72 ≤ s/σ ≤ 1.28] = .90

or

This states that the sample standard deviation will be at least 72 percent and not more than 128 percent of the population variance 90 percent of the time. Conversely, 10 percent of the time the standard deviation will underestimate or overestimate the population standard deviation by the corresponding amount. Even for samples as large as 25, the relative reliability of a sample standard deviation is poor.

The chi-square distribution can be applied to other types of application which are of an entirely different nature. These include applications which are discussed under “Goodness-of-Fit Test” and “Two-Way Test for Independence of Count Data.” In these applications, the mathematical formulation and context are entirely different, but they do result in the same table of values. F Distribution In reference to the tensile-strength table, the successive pairs of daily standard deviations could be ratioed and squared. These ratios of variance would represent a sample from a distribution called the F distribution or F ratio. In general, the F ratio is defined by the identity F(γ1, γ 2) = s12 / s22 where γ1 and γ2 correspond to the respective df’s for the sample variances. In statistical applications, it turns out that the primary area of interest is found when the ratios are greater than 1. For this reason, most tabled values are defined for an upper-tail area. However, defining F2 to be that value corresponding to an upper-tail area of α/2, then F1 for a lower-tail area of α/2 can be determined through the identity

Percentiles of the c2 Distribution Percent

df

10

90

95

97.5

0.000039 .0100 .0717 .207 .412

0.00016 .0201 .115 .297 .554

0.00098 .0506 .216 .484 .831

0.0039 .1026 .352 .711 1.15

0.0158 .2107 .584 1.064 1.61

2.71 4.61 6.25 7.78 9.24

3.84 5.99 7.81 9.49 11.07

5.02 7.38 9.35 11.14 12.83

6.63 9.21 11.34 13.28 15.09

7.88 10.60 12.84 14.86 16.75

6 7 8 9 10

.676 .989 1.34 1.73 2.16

.872 1.24 1.65 2.09 2.56

1.24 1.69 2.18 2.70 3.25

1.64 2.17 2.73 3.33 3.94

2.20 2.83 3.49 4.17 4.87

10.64 12.02 13.36 14.68 15.99

12.59 14.07 15.51 16.92 18.31

14.45 16.01 17.53 19.02 20.48

16.81 18.48 20.09 21.67 23.21

18.55 20.28 21.96 23.59 25.19

11 12 13 14 15

2.60 3.07 3.57 4.07 4.60

3.05 3.57 4.11 4.66 5.23

3.82 4.40 5.01 5.63 6.26

4.57 5.23 5.89 6.57 7.26

5.58 6.30 7.04 7.79 8.55

17.28 18.55 19.81 21.06 22.31

19.68 21.03 22.36 23.68 25.00

21.92 23.34 24.74 26.12 27.49

24.73 26.22 27.69 29.14 30.58

26.76 28.30 29.82 31.32 32.80

16 18 20 24 30

5.14 6.26 7.43 9.89 13.79

5.81 7.01 8.26 10.86 14.95

6.91 8.23 9.59 12.40 16.79

7.96 9.39 10.85 13.85 18.49

9.31 10.86 12.44 15.66 20.60

23.54 25.99 28.41 33.20 40.26

26.30 28.87 31.41 36.42 43.77

28.85 31.53 34.17 39.36 46.98

32.00 34.81 37.57 42.98 50.89

34.27 37.16 40.00 45.56 53.67

40 60 120

20.71 35.53 83.85

22.16 37.48 86.92

24.43 40.48 91.58

26.51 43.19 95.70

29.05 46.46 100.62

51.81 74.40 140.23

55.76 79.08 146.57

59.34 83.30 152.21

63.69 88.38 158.95

66.77 91.95 163.64

1 2 3 4 5

0.5

1

2.5

5

99

99.5

For large values of degrees of freedom the approximate formula



2 χ 2a = n 1 −
+ za 9n

2
  9n

3

where za is the normal deviate and n is the number of degrees of freedom, may be used. For example, χ.299 = 60[1 − 0.00370 + 2.326(0.06086)]3 = 60(1.1379)3 = 88.4 for the 99th percentile for 60 degrees of freedom.

3-76

MATHEMATICS F1(γ1, γ 2) = 1/F2(γ 2, γ1)

With 103 df and α = .05, t = 1.96 using t.025 in Table 3-5. Therefore

The F distribution, similar to the chi square, is sensitive to the basic assumption that sample values were selected randomly from a normal distribution. Example For two sample variances with 4 df each, what limits will bracket their ratio with a midarea probability of 90 percent? Use Table 3-7 with 4 df in the numerator and denominator and upper 5 percent points (to get both sides totaling 10 percent). The entry is 6.39. Thus: P[1/6.39 ≤ s 12 /s 22 ≤ 6.39] = .90 P[.40 ≤ s1 /s2 ≤ 2.53] = .90

or

Confidence Interval for a Mean For the daily sample tensilestrength data with 4 df it is known that P[−2.132 ≤ t ≤ 2.132] = .90. This states that 90 percent of all samples will have sample t values which fall within the specified limits. In fact, for the 20 daily samples exactly 16 do fall within the specified limits (note that the binomial with n = 20 and p = .90 would describe the likelihood of exactly none through 20 falling within the prescribed limits—the sample of 20 is only a sample). Consider the new daily sample (with n = 5, x = 37.14, and s = 1.85) which was observed after a process change. In this case, the same probability holds. However, in this instance the sample value of t cannot be computed, since the new µ, under the process change, is not known. Therefore P[−2.132 ≤ (37.14 − µ)/(1.85/
5) ≤ 2.132] = .90. In effect, this identity limits the magnitude of possible values for µ. The magnitude of µ can be only large enough to retain the t quantity above −2.132 and small enough to retain the t quantity below +2.132. This can be found by rearranging the quantities within the bracket; i.e., P[35.78 ≤ µ ≤ 38.90] = .90. This states that we are 90 percent sure that the interval from 35.78 to 38.90 includes the unknown parameter µ. In general, s s P x − t
≤ µ ≤ x + t
= 1 − α
 n
 n





where t is defined for an upper-tail area of α/2 with (n − 1) df. In this application, the interval limits (x + t s/
n ) are random variables which will cover the unknown parameter µ with probability (1 − α). The converse, that we are 100 (1 − α) percent sure that the parameter value is within the interval, is not correct. This statement defines a probability for the parameter rather than the probability for the interval. Example What values of t define the midarea of 95 percent for weekly samples of size 25, and what is the sample value of t for the second week? P[−2.064 ≤ t ≤ 2.064] = .95 (34.19 − 35)/(2.35/
25) = 1.72.

and

Example For the composite sample of 100 tensile strengths, what is the 90 percent confidence interval for µ? Use Table 3-5 for t.05 with df ≈ ∞.





2.47 2.47 P 35.16 − 1.645
< µ < 35.16 + 1.645
= .90
 100
 100 or

P[34.75 ≤ µ ≤ 35.57] = .90

Confidence Interval for the Difference in Two Population Means The confidence interval for a mean can be extended to include the difference between two population means. This interval is based on the assumption that the respective populations have the same variance σ2:

 1/n 2 ≤ µ1 − µ2 ≤ (x1 − x2) + tsp
1/n   2 (x1 − x2) − tsp
1/n 1/n 1 + 1 + Example Compute the 95 percent confidence interval based on the original 100-point sample and the subsequent 5-point sample: 99(2.47)2 + 4(1.85)2 s p2 =


= 5.997 103 or

sp = 2.45

00  = −1.98  2.20 +1/5 (35.16 − 37.14)  1.96(2.45)
1/1 or

−4.18 ≤ (µ1 − µ2) ≤ .22

Note that if the respective samples had been based on 52 observations each rather than 100 and 5, the uncertainty factor would have been .94 rather than the observed 2.20. The interval width tends to be minimum when n1 = n2.

Confidence Interval for a Variance The chi-square distribution can be used to derive a confidence interval for a population variance σ2 when the parent population is normally distributed. For a 100(1 − α) percent confidence interval (df )s2 (df )s2 ≤ σ2 ≤

2 χ2 χ12 where χ 12 corresponds to a lower-tail area of α/2 and χ 22 to an upper-tail area of α/2. Example For the first week of tensile-strength samples compute the 90 percent confidence interval for σ2 (df = 24, corresponding to n = 25, using 5 percent and 95 percent in Table 3-6): 24(2.40)2 24(2.40)2
≤ σ2 ≤
36.4 13.8 3.80 ≤ σ2 ≤ 10.02 or

1.95 ≤ σ ≤ 3.17

TESTS OF HYPOTHESIS General Nature of Tests The general nature of tests can be illustrated with a simple example. In a court of law, when a defendant is charged with a crime, the judge instructs the jury initially to presume that the defendant is innocent of the crime. The jurors are then presented with evidence and counterargument as to the defendant’s guilt or innocence. If the evidence suggests beyond a reasonable doubt that the defendant did, in fact, commit the crime, they have been instructed to find the defendant guilty; otherwise, not guilty. The burden of proof is on the prosecution. Jury trials represent a form of decision making. In statistics, an analogous procedure for making decisions falls into an area of statistical inference called hypothesis testing. Suppose that a company has been using a certain supplier of raw materials in one of its chemical processes. A new supplier approaches the company and states that its material, at the same cost, will increase the process yield. If the new supplier has a good reputation, the company might be willing to run a limited test. On the basis of the test results it would then make a decision to change suppliers or not. Good management would dictate that an improvement must be demonstrated (beyond a reasonable doubt) for the new material. That is, the burden of proof is tied to the new material. In setting up a test of hypothesis for this application, the initial assumption would be defined as a null hypothesis and symbolized as H0. The null hypothesis would state that yield for the new material is no greater than for the conventional material. The symbol µ0 would be used to designate the known current level of yield for the standard material and µ for the unknown population yield for the new material. Thus, the null hypothesis can be symbolized as H0: µ ≤ µ0. The alternative to H0 is called the alternative hypothesis and is symbolized as H1: µ > µ0. Given a series of tests with the new material, the average yield x would be compared with µ0. If x < µ0, the new supplier would be dismissed. If x > µ0, the question would be: Is it sufficiently greater in the light of its corresponding reliability, i.e., beyond a reasonable doubt? If the confidence interval for µ included µ0, the answer would be no, but if it did not include µ0, the answer would be yes. In this simple application, the formal test of hypothesis would result in the same conclusion as that derived from the confidence interval. However, the utility of tests of hypothesis lies in their generality, whereas confidence intervals are restricted to a few special cases.

TABLE 3-7 F Distribution Upper 5% Points (F.95)

Degrees of freedom for denominator

Degrees of freedom for numerator 1

2

3

4

5

6

7

8

9

10

12

15

20

24

30

40

60

120

∞

1 2 3 4 5

161 18.5 10.1 7.71 6.61

200 19.0 9.55 6.94 5.79

216 19.2 9.28 6.59 5.41

225 19.2 9.12 6.39 5.19

230 19.3 9.01 6.26 5.05

234 19.3 8.94 6.16 4.95

237 19.4 8.89 6.09 4.88

239 19.4 8.85 6.04 4.82

241 19.4 8.81 6.00 4.77

242 19.4 8.79 5.96 4.74

244 19.4 8.74 5.91 4.68

246 19.4 8.70 5.86 4.62

248 19.4 8.66 5.80 4.56

249 19.5 8.64 5.77 4.53

250 19.5 8.62 5.75 4.50

251 19.5 8.59 5.72 4.46

252 19.5 8.57 5.69 4.43

253 19.5 8.55 5.66 4.40

254 19.5 8.53 5.63 4.37

6 7 8 9 10

5.99 5.59 5.32 5.12 4.96

5.14 4.74 4.46 4.26 4.10

4.76 4.35 4.07 3.86 3.71

4.53 4.12 3.84 3.63 3.48

4.39 3.97 3.69 3.48 3.33

4.28 3.87 3.58 3.37 3.22

4.21 3.79 3.50 3.29 3.14

4.15 3.73 3.44 3.23 3.07

4.10 3.68 3.39 3.18 3.02

4.06 3.64 3.35 3.14 2.98

4.00 3.57 3.28 3.07 2.91

3.94 3.51 3.22 3.01 2.85

3.87 3.44 3.15 2.94 2.77

3.84 3.41 3.12 2.90 2.74

3.81 3.38 3.08 2.86 2.70

3.77 3.34 3.04 2.83 2.66

3.74 3.30 3.01 2.79 2.62

3.70 3.27 2.97 2.75 2.58

3.67 3.23 2.93 2.71 2.54

11 12 13 14 15

4.84 4.75 4.67 4.60 4.54

3.98 3.89 3.81 3.74 3.68

3.59 3.49 3.41 3.34 3.29

3.36 3.26 3.18 3.11 3.06

3.20 3.11 3.03 2.96 2.90

3.09 3.00 2.92 2.85 2.79

3.01 2.91 2.83 2.76 2.71

2.95 2.85 2.77 2.70 2.64

2.90 2.80 2.71 2.65 2.59

2.85 2.75 2.67 2.60 2.54

2.79 2.69 2.60 2.53 2.48

2.72 2.62 2.53 2.46 2.40

2.65 2.54 2.46 2.39 2.33

2.61 2.51 2.42 2.35 2.29

2.57 2.47 2.38 2.31 2.25

2.53 2.43 2.34 2.27 2.20

2.49 2.38 2.30 2.22 2.16

2.45 2.34 2.25 2.18 2.11

2.40 2.30 2.21 2.13 2.07

16 17 18 19 20

4.49 4.45 4.41 4.38 4.35

3.63 3.59 3.55 3.52 3.49

3.24 3.20 3.16 3.13 3.10

3.01 2.96 2.93 2.90 2.87

2.85 2.81 2.77 2.74 2.71

2.74 2.70 2.66 2.63 2.60

2.66 2.61 2.58 2.54 2.51

2.59 2.55 2.51 2.48 2.45

2.54 2.49 2.46 2.42 2.39

2.49 2.45 2.41 2.38 2.35

2.42 2.38 2.34 2.31 2.28

2.35 2.31 2.27 2.23 2.20

2.28 2.23 2.19 2.16 2.12

2.24 2.19 2.15 2.11 2.08

2.19 2.15 2.11 2.07 2.04

2.15 2.10 2.06 2.03 1.99

2.11 2.06 2.02 1.98 1.95

2.06 2.01 1.97 1.93 1.90

2.01 1.96 1.92 1.88 1.84

21 22 23 24 25

4.32 4.30 4.28 4.26 4.24

3.47 3.44 3.42 3.40 3.39

3.07 3.05 3.03 3.01 2.99

2.84 2.82 2.80 2.78 2.76

2.68 2.66 2.64 2.62 2.60

2.57 2.55 2.53 2.51 2.49

2.49 2.46 2.44 2.42 2.40

2.42 2.40 2.37 2.36 2.34

2.37 2.34 2.32 2.30 2.28

2.32 2.30 2.27 2.25 2.24

2.25 2.23 2.20 2.18 2.16

2.18 2.15 2.13 2.11 2.09

2.10 2.07 2.05 2.03 2.01

2.05 2.03 2.01 1.98 1.96

2.01 1.98 1.96 1.94 1.92

1.96 1.94 1.91 1.89 1.87

1.92 1.89 1.86 1.84 1.82

1.87 1.84 1.81 1.79 1.77

1.81 1.78 1.76 1.73 1.71

30 40 60 120 ∞

4.17 4.08 4.00 3.92 3.84

3.32 3.23 3.15 3.07 3.00

2.92 2.84 2.76 2.68 2.60

2.69 2.61 2.53 2.45 2.37

2.53 2.45 2.37 2.29 2.21

2.42 2.34 2.25 2.18 2.10

2.33 2.25 2.17 2.09 2.01

2.27 2.18 2.10 2.02 1.94

2.21 2.12 2.04 1.96 1.88

2.16 2.08 1.99 1.91 1.83

2.09 2.00 1.92 1.83 1.75

2.01 1.92 1.84 1.75 1.67

1.93 1.84 1.75 1.66 1.57

1.89 1.79 1.70 1.61 1.52

1.84 1.74 1.65 1.55 1.46

1.79 1.69 1.59 1.50 1.39

1.74 1.64 1.53 1.43 1.32

1.68 1.58 1.47 1.35 1.22

1.62 1.51 1.39 1.25 1.00

Upper 1% Points (F.99)

Degrees of freedom for denominator

Degrees of freedom for numerator 1

2

3

4

5

6

7

8

9

10

12

15

20

24

30

40

60

120

∞

1 2 3 4 5

4052 98.5 34.1 21.2 16.3

5000 99.0 30.8 18.0 13.3

5403 99.2 29.5 16.7 12.1

5625 99.2 28.7 16.0 11.4

5764 99.3 28.2 15.5 11.0

5859 99.3 27.9 15.2 10.7

5928 99.4 27.7 15.0 10.5

5982 99.4 27.5 14.8 10.3

6023 99.4 27.3 14.7 10.2

6056 99.4 27.2 14.5 10.1

6106 99.4 27.1 14.4 9.89

6157 99.4 26.9 14.2 9.72

6209 99.4 26.7 14.0 9.55

6235 99.5 26.6 13.9 9.47

6261 99.5 26.5 13.8 9.38

6287 99.5 26.4 13.7 9.29

6313 99.5 26.3 13.7 9.20

6339 99.5 26.2 13.6 9.11

6366 99.5 26.1 13.5 9.02

6 7 8 9 10

13.7 12.2 11.3 10.6 10.0

10.9 9.55 8.65 8.02 7.56

9.78 8.45 7.59 6.99 6.55

9.15 7.85 7.01 6.42 5.99

8.75 7.46 6.63 6.06 5.64

8.47 7.19 6.37 5.80 5.39

8.26 6.99 6.18 5.61 5.20

8.10 6.84 6.03 5.47 5.06

7.98 6.72 5.91 5.35 4.94

7.87 6.62 5.81 5.26 4.85

7.72 6.47 5.67 5.11 4.71

7.56 6.31 5.52 4.96 4.56

7.40 6.16 5.36 4.81 4.41

7.31 6.07 5.28 4.73 4.33

7.23 5.99 5.20 4.65 4.25

7.14 5.91 5.12 4.57 4.17

7.06 5.82 5.03 4.48 4.08

6.97 5.74 4.95 4.40 4.00

6.88 5.65 4.86 4.31 3.91

11 12 13 14 15

9.65 9.33 9.07 8.86 8.68

7.21 6.93 6.70 6.51 6.36

6.22 5.95 5.74 5.56 5.42

5.67 5.41 5.21 5.04 4.89

5.32 5.06 4.86 4.70 4.56

5.07 4.82 4.62 4.46 4.32

4.89 4.64 4.44 4.28 4.14

4.74 4.50 4.30 4.14 4.00

4.63 4.39 4.19 4.03 3.89

4.54 4.30 4.10 3.94 3.80

4.40 4.16 3.96 3.80 3.67

4.25 4.01 3.82 3.66 3.52

4.10 3.86 3.66 3.51 3.37

4.02 3.78 3.59 3.43 3.29

3.94 3.70 3.51 3.35 3.21

3.86 3.62 3.43 3.27 3.13

3.78 3.54 3.34 3.18 3.05

3.69 3.45 3.25 3.09 2.96

3.60 3.36 3.17 3.00 2.87

16 17 18 19 20

8.53 8.40 8.29 8.19 8.10

6.23 6.11 6.01 5.93 5.85

5.29 5.19 5.09 5.01 4.94

4.77 4.67 4.58 4.50 4.43

4.44 4.34 4.25 4.17 4.10

4.20 4.10 4.01 3.94 3.87

4.03 3.93 3.84 3.77 3.70

3.89 3.79 3.71 3.63 3.56

3.78 3.68 3.60 3.52 3.46

3.69 3.59 3.51 3.43 3.37

3.55 3.46 3.37 3.30 3.23

3.41 3.31 3.23 3.15 3.09

3.26 3.16 3.08 3.00 2.94

3.18 3.08 3.00 2.92 2.86

3.10 3.00 2.92 2.84 2.78

3.02 2.92 2.84 2.76 2.69

2.93 2.83 2.75 2.67 2.61

2.84 2.75 2.66 2.58 2.52

2.75 2.65 2.57 2.49 2.42

21 22 23 24 25

8.02 7.95 7.88 7.82 7.77

5.78 5.72 5.66 5.61 5.57

4.87 4.82 4.76 4.72 4.68

4.37 4.31 4.26 4.22 4.18

4.04 3.99 3.94 3.90 3.86

3.81 3.76 3.71 3.67 3.63

3.64 3.59 3.54 3.50 3.46

3.51 3.45 3.41 3.36 3.32

3.40 3.35 3.30 3.26 3.22

3.31 3.26 3.21 3.17 3.13

3.17 3.12 3.07 3.03 2.99

3.03 2.98 2.93 2.89 2.85

2.88 2.83 2.78 2.74 2.70

2.80 2.75 2.70 2.66 2.62

2.72 2.67 2.62 2.58 2.53

2.64 2.58 2.54 2.49 2.45

2.55 2.50 2.45 2.40 2.36

2.46 2.40 2.35 2.31 2.27

2.36 2.31 2.26 2.21 2.17

30 40 60 120 ∞

7.56 7.31 7.08 6.85 6.63

5.39 5.18 4.98 4.79 4.61

4.51 4.31 4.13 3.95 3.78

4.02 3.83 3.65 3.48 3.32

3.70 3.51 3.34 3.17 3.02

3.47 3.29 3.12 2.96 2.80

3.30 3.12 2.95 2.79 2.64

3.17 2.99 2.82 2.66 2.51

3.07 2.89 2.72 2.56 2.41

2.98 2.80 2.63 2.47 2.32

2.84 2.66 2.50 2.34 2.18

2.70 2.52 2.35 2.19 2.04

2.55 2.37 2.20 2.03 1.88

2.47 2.29 2.12 1.95 1.79

2.39 2.20 2.03 1.86 1.70

2.30 2.11 1.94 1.76 1.59

2.21 2.02 1.84 1.66 1.47

2.11 1.92 1.73 1.53 1.32

2.01 1.80 1.60 1.38 1.00

Interpolation should be performed using reciprocals of the degrees of freedom. 3-77

3-78

MATHEMATICS

Test of Hypothesis for a Mean Procedure Nomenclature µ = mean of the population from which the sample has been drawn σ = standard deviation of the population from which the sample has been drawn µ0 = base or reference level H0 = null hypothesis H1 = alternative hypothesis α = significance level, usually set at .10, .05, or .01 t = tabled t value corresponding to the significance level α. For a two-tailed test, each corresponding tail would have an area of α/2, and for a one-tailed test, one tail area would be equal to α. If σ2 is known, then z would be used rather than the t. t = (x − µ0)/(s/
 n) = sample value of the test statistic. Assumptions 1. The n observations x1, x2, . . . , xn have been selected randomly. 2. The population from which the observations were obtained is normally distributed with an unknown mean µ and standard deviation σ. In actual practice, this is a robust test, in the sense that in most types of problems it is not sensitive to the normality assumption when the sample size is 10 or greater. Test of Hypothesis 1. Under the null hypothesis, it is assumed that the sample came from a population whose mean µ is equivalent to some base or reference designated by µ0. This can take one of three forms: Form 1

Form 2

Form 3

H0: µ = µ0 H1: µ ≠ µ0 Two-tailed test

H0: µ ≤ µ0 H1: µ > µ0 Upper-tailed test

H0: µ ≥ µ0 H1: µ < µ0 Lower-tailed test

2. If the null hypothesis is assumed to be true, say, in the case of a two-sided test, form 1, then the distribution of the test statistic t is known. Given a random sample, one can predict how far its sample value of t might be expected to deviate from zero (the midvalue of t) by chance alone. If the sample value of t does, in fact, deviate too far from zero, then this is defined to be sufficient evidence to refute the assumption of the null hypothesis. It is consequently rejected, and the converse or alternative hypothesis is accepted. 3. The rule for accepting H0 is specified by selection of the α level as indicated in Fig. 3-57. For forms 2 and 3 the α area is defined to be in the upper or the lower tail respectively. 4. The decision rules for each of the three forms are defined as follows: If the sample t falls within the acceptance region, accept H0 for lack of contrary evidence. If the sample t falls in the critical region, reject H0 at a significance level of 100α percent. Example Application. In the past, the yield for a chemical process has been established at 89.6 percent with a standard deviation of 3.4 percent. A new supplier of raw materials will be used and tested for 7 days. Procedure 1. The standard of reference is µ0 = 89.6 with a known σ = 3.4. 2. It is of interest to demonstrate whether an increase in yield is achieved with the new material; H0 says it has not; therefore, H0: µ ≤ 89.6

H1: µ > 89.6

3. Select α = .05, and since σ is known (the new material would not affect the day-to-day variability in yield), the test statistic would be z with a corresponding critical value cv(z) = 1.645 (Table 3-5, df = ∞). 4. The decision rule: Accept H0 if sample z < 1.645 Reject H0 if sample z > 1.645 5. A 7-day test was carried out, and daily yields averaged 91.6 percent with a sample standard deviation s = 3.6 (this is not needed for the test of hypothesis). 6. For the data sample z = (91.6 − 89.6)/(3.4/
7) = 1.56. 7. Since the sample z < cv(z), accept the null hypothesis for lack of contrary evidence; i.e., an improvement has not been demonstrated beyond a reasonable doubt.

Example Application. In the past, the break strength of a synthetic yarn has averaged 34.6 lb. The first-stage draw ratio of the spinning machines has been increased. Production management wants to determine whether the break strength has changed under the new condition. Procedure 1. The standard of reference is µ0 = 34.6. 2. It is of interest to demonstrate whether a change has occurred; therefore, H0: µ = 34.6

Accept H0 if −2.086 < sample t < 2.086 Reject H0 if sample t < −2.086 or > 2.086 6. For the data sample t = (35.55 − 34.6)/(2.041/
21) = 2.133. 7. Since 2.133 > 2.086, reject H0 and accept H1. It has been demonstrated that an improvement in break strength has been achieved.

Two-Population Test of Hypothesis for Means Nature Two samples were selected from different locations in a plastic-film sheet and measured for thickness. The thickness of the respective samples was measured at 10 close but equally spaced points in each of the samples. It was of interest to compare the average thickness of the respective samples to detect whether they were significantly different. That is, was there a significant variation in thickness between locations? From a modeling standpoint statisticians would define this problem as a two-population test of hypothesis. They would define the respective sample sheets as two populations from which 10 sample thickness determinations were measured for each. In order to compare populations based on their respective samples, it is necessary to have some basis of comparison. This basis is predicated on the distribution of the t statistic. In effect, the t statistic characterizes the way in which two sample means from two separate populations will tend to vary by chance alone when the population means and variances are equal. Consider the following: Population 1

Acceptance region.

Population 2

Normal

Sample 1

Normal

Sample 2

µ1

n1 x1 s12

µ2

n2 x2 s22

σ

FIG. 3-57

H1: µ ≠ 34.6

3. Select α = .05, and since with the change in draw ratio the uniformity might change, the sample standard deviation would be used, and therefore t would be the appropriate test statistic. 4. A sample of 21 ends was selected randomly and tested on an Instron with the results x = 35.55 and s = 2.041. 5. For 20 df and a two-tailed α level of 5 percent, the critical values of t are given by 2.086 with a decision rule (Table 3-5, t.025, df = 20):

2 1

σ

2 2

Consider the hypothesis µ1 = µ2. If, in fact, the hypothesis is correct, i.e., µ1 = µ2 (under the condition σ 12 = σ 22), then the sampling distribution of (x1 − x2) is predictable through the t distribution. The observed sample values then can be compared with the corresponding t distribution. If the sample values are reasonably close (as reflected through the α level), that is, x1 and x2 are not “too different” from each other on the basis of the t distribution, the null hypothesis would be accepted. Conversely, if they deviate from each other “too much” and the deviation is therefore not ascribable to chance, the conjecture would be questioned and the null hypothesis rejected.

STATISTICS Example Application. Two samples were selected from different locations in a plastic-film sheet. The thickness of the respective samples was measured at 10 close but equally spaced points. Procedure 1. Demonstrate whether the thicknesses of the respective sample locations are significantly different from each other; therefore, H0: µ1 = µ2 2. 3.

H1: µ1 ≠ µ2

Select α = .05. Summarize the statistics for the respective samples: Sample 1

Sample 2

1.473 1.484 1.484 1.425 1.448

1.367 1.276 1.485 1.462 1.439

1.474 1.501 1.485 1.435 1.348

1.417 1.448 1.469 1.474 1.452

x1 = 1.434

s1 = .0664

x2 = 1.450

s2 = .0435

4. As a first step, the assumption for the standard t test, that σ 12 = σ 22, can be tested through the F distribution. For this hypothesis, H0: σ 12 = σ 22 would be tested against H1: σ 12 ≠ σ 22. Since this is a two-tailed test and conventionally only the upper tail for F is published, the procedure is to use the largest ratio and the corresponding ordered degrees of freedom. This achieves the same end result through one table. However, since the largest ratio is arbitrary, it is necessary to define the true α level as twice the value of the tabled value. Therefore, by using Table 3-7 with α = .05 the corresponding critical value for F(9,9) = 3.18 would be for a true α = .10. For the sample,

Test of Hypothesis for Matched Pairs: Procedure Nomenclature di = sample difference between the ith pair of observations s = sample standard deviation of differences µ = population mean of differences σ = population standard deviation of differences µ0 = base or reference level of comparison H0 = null hypothesis H1 = alternative hypothesis α = significance level t = tabled value with (n − 1) df t = (d n), the sample value of t  − µ0)/(s/
 Assumptions 1. The n pairs of samples have been selected and assigned for testing in a random way. 2. The population of differences is normally distributed with a mean µ and variance σ2. As in the previous application of the t distribution, this is a robust procedure, i.e., not sensitive to the normality assumption if the sample size is 10 or greater in most situations. Test of Hypothesis 1. Under the null hypothesis, it is assumed that the sample came from a population whose mean µ is equivalent to some base or reference level designated by µ0. For most applications of this type, the value of µ0 is defined to be zero; that is, it is of interest generally to demonstrate a difference not equal to zero. The hypothesis can take one of three forms:

Sample F = (.0664/.0435)2 = 2.33

Form 1

Form 2

Therefore, the ratio of sample variances is no larger than one might expect to observe when in fact σ 12 = σ 22. There is not sufficient evidence to reject the null hypothesis that σ 12 = σ 22. 5. For 18 df and a two-tailed α level of 5 percent the critical values of t are given by 2.101 (Table 3-5, t0.025, df = 18). 6. The decision rule:

H0: µ = µ0 H1: µ ≠ µ0 Two-tailed test

H0: µ ≤ µ0 H1: µ > µ0 Upper-tailed test

Accept H0 if −2.101 ≤ sample t ≤ 2.101 Reject H0 otherwise 7.

For the sample the pooled variance estimate is given by 9(.0664)2 + 9(.0435)2 (.0664)2 + (.0435)2 s p2 =


=

= .00315 9+9 2 sp = .056

or 8.

The sample statistic value of t is 1.434 − 1.450 Sample t =

= −.64 0 0 .056
1/1 + 1/1

9. Since the sample value of t falls within the acceptance region, accept H0 for lack of contrary evidence; i.e., there is insufficient evidence to demonstrate that thickness differs between the two selected locations.

Test of Hypothesis for Paired Observations Nature In some types of applications, associated pairs of observations are defined. For example, (1) pairs of samples from two populations are treated in the same way, or (2) two types of measurements are made on the same unit. For applications of this type, it is not only more effective but necessary to define the random variable as the difference between the pairs of observations. The difference numbers can then be tested by the standard t distribution. Examples of the two types of applications are as follows: 1. Sample treatment a. Two types of metal specimens buried in the ground together in a variety of soil types to determine corrosion resistance b. Wear-rate test with two different types of tractor tires mounted in pairs on n tractors for a defined period of time 2. Same unit a. Blood-pressure measurements made on the same individual before and after the administration of a stimulus b. Smoothness determinations on the same film samples at two different testing laboratories

3-79

Form 3 H0: µ ≥ µ0 H1: µ < µ0 Lower-tailed test

2. If the null hypothesis is assumed to be true, say, in the case of a lower-tailed test, form 3, then the distribution of the test statistic t is known under the null hypothesis that limits µ = µ0. Given a random sample, one can predict how far its sample value of t might be expected to deviate from zero by chance alone when µ = µ0. If the sample value of t is too small, as in the case of a negative value, then this would be defined as sufficient evidence to reject the null hypothesis. 3. Select α. 4. The critical values or value of t would be defined by the tabled value of t with (n − 1) df corresponding to a tail area of α. For a twotailed test, each tail area would be α/2, and for a one-tailed test there would be an upper-tail or a lower-tail area of α corresponding to forms 2 and 3 respectively. 5. The decision rule for each of the three forms would be to reject the null hypothesis if the sample value of t fell in that area of the t distribution defined by α, which is called the critical region. Otherwise, the alternative hypothesis would be accepted for lack of contrary evidence. Example Application. Pairs of pipes have been buried in 11 different locations to determine corrosion on nonbituminous pipe coatings for underground use. One type includes a lead-coated steel pipe and the other a bare steel pipe. Procedure 1. The standard of reference is taken as µ0 = 0, corresponding to no difference in the two types. 2. It is of interest to demonstrate whether either type of pipe has a greater corrosion resistance than the other. Therefore, H0: µ = 0

H1: µ ≠ 0

3. Select α = .05. Therefore, with n = 11 the critical values of t with 10 df are defined by t = 2.228 (Table 3.5, t.025). 4. The decision rule: Accept H0 if −2.228 ≤ sample t ≤ 2.228 Reject H0 otherwise 5. The sample of 11 pairs of corrosion determinations and their differences are as follows:

3-80

MATHEMATICS

Soil type

Lead-coated steel pipe

Bare steel pipe

d = difference

A B C D E F

27.3 18.4 11.9 11.3 14.8 20.8

41.4 18.9 21.7 16.8 9.0 19.3

−14.1 −0.5 −9.8 −5.5 5.8 1.5

G H I J K

17.9 7.8 14.7 19.0 65.3

32.1 7.4 20.7 34.4 76.2

−14.2 0.4 −6.0 −15.4 −10.9

6.

The sample statistics: 11  d 2 − ( d)2 s2 =

= 52.56 11 × 10

d = −6.245 

s = 7.25

or

Sample t = (−6.245 − 0)/(7.25/
11) = −2.86 7. Since the sample t of −2.86 < tabled t of −2.228, reject H0 and accept H1; that is, it has been demonstrated that, on the basis of the evidence, lead-coated steel pipe has a greater corrosion resistance than bare steel pipe.

Example Application. A stimulus was tested for its effect on blood pressure. Ten men were selected randomly, and their blood pressure was measured before and after the stimulus was administered. It was of interest to determine whether the stimulus had caused a significant increase in the blood pressure. Procedure 1. The standard of reference was taken as µ0 ≤ 0, corresponding to no increase. 2. It was of interest to demonstrate an increase in blood pressure if in fact an increase did occur. Therefore, H0: µ0 ≤ 0

H1: µ0 > 0

3. Select α = .05. Therefore, with n = 10 the critical value of t with 9 df is defined by t = 1.833 (Table 3-5, t.05, one-sided). 4. The decision rule: Accept H0 if sample t < 1.833 Reject H0 if sample t > 1.833 5. The sample of 10 pairs of blood pressure and their differences were as follows:

proportion of workers in a plant who are out sick, (2) lost-time worker accidents per month, (3) defective items in a shipment lot, and (4) preference in consumer surveys. The procedure for testing the significance of a sample proportion follows that for a sample mean. In this case, however, owing to the nature of the problem the appropriate test statistic is Z. This follows from the fact that the null hypothesis requires the specification of the goal or reference quantity p0, and since the distribution is a binomial proportion, the associated variance is [p0(1 − p0)]n under the null hypothesis. The primary requirement is that the sample size n satisfy normal approximation criteria for a binomial proportion, roughly np > 5 and n(1 − p) > 5. Test of Hypothesis for a Proportion: Procedure Nomenclature p = mean proportion of the population from which the sample has been drawn p0 = base or reference proportion [p0(1 − p0)]/n = base or reference variance p ˆ = x/n = sample proportion, where x refers to the number of observations out of n which have the specified attribute H0 = assumption or null hypothesis regarding the population proportion H1 = alternative hypothesis α = significance level, usually set at .10, .05, or .01 z = Tabled Z value corresponding to the significance level α. The sample sizes required for the z approximation according to the magnitude of p0 are given in Table 3-5. z = (p ˆ − p0)/
 p 1 − p /n, the sample value of the test 0( 0) statistic Assumptions 1. The n observations have been selected randomly. 2. The sample size n is sufficiently large to meet the requirement for the Z approximation. Test of Hypothesis 1. Under the null hypothesis, it is assumed that the sample came from a population with a proportion p0 of items having the specified attribute. For example, in tossing a coin the population could be thought of as having an unbounded number of potential tosses. If it is assumed that the coin is fair, this would dictate p0 = 1/2 for the proportional number of heads in the population. The null hypothesis can take one of three forms:

Individual

Before

After

d = difference

Form 1

Form 2

Form 3

1 2 3 4 5

138 116 124 128 155

146 118 120 136 174

8 2 −4 8 19

H0: p = p0 H1: p ≠ p0 Two-tailed test

H0: p ≤ p0 H1: p > p0 Upper-tailed test

H0: p ≥ p0 H1: p < p0 Lower-tailed test

6 7 8 9 10

129 130 148 143 159

133 129 155 148 155

4 −1 7 5 −4

2. If the null hypothesis is assumed to be true, then the sampling distribution of the test statistic Z is known. Given a random sample, it is possible to predict how far the sample proportion x/n might deviate from its assumed population proportion p0 through the Z distribution. When the sample proportion deviates too far, as defined by the significance level α, this serves as the justification for rejecting the assumption, that is, rejecting the null hypothesis. 3. The decision rule is given by Form 1: Accept H0 if lower critical z < sample z < upper critical z Reject H0 otherwise Form 2: Accept H0 if sample z < upper critical z Reject H0 otherwise Form 3: Accept H0 if lower critical z < sample z Reject H0 otherwise

6.

The sample statistics: d  = 4.4

s = 6.85

Sample t = (4.4 − 0)/(6.85/
10) = 2.03 7. Since the sample t = 2.03 > critical t = 1.833, reject the null hypothesis. It has been demonstrated that the population of men from which the sample was drawn tend, as a whole, to have an increase in blood pressure after the stimulus has been given. The distribution of differences d seems to indicate that the degree of response varies by individuals.

Example Test of Hypothesis for a Proportion Nature Some types of statistical applications deal with counts and proportions rather than measurements. Examples are (1) the

Application. A company has received a very large shipment of rivets. One product specification required that no more than 2 percent of the rivets have diameters greater than 14.28 mm. Any rivet with a diameter greater than this would be classified as defective. A random sample of 600 was selected and

STATISTICS tested with a go–no-go gauge. Of these, 16 rivets were found to be defective. Is this sufficient evidence to conclude that the shipment contains more than 2 percent defective rivets? Procedure 1. The quality goal is p ≤ .02. It would be assumed initially that the shipment meets this standard; i.e., H0: p ≤ .02. 2. The assumption in step 1 would first be tested by obtaining a random sample. Under the assumption that p ≤ .02, the distribution for a sample proportion would be defined by the z distribution. This distribution would define an upper bound corresponding to the upper critical value for the sample proportion. It would be unlikely that the sample proportion would rise above that value if, in fact, p ≤ .02. If the observed sample proportion exceeds that limit, corresponding to what would be a very unlikely chance outcome, this would lead one to question the assumption that p ≤ .02. That is, one would conclude that the null hypothesis is false. To test, set H0: p ≤ .02

H1: p > .02

3. Select α = .05. 4. With α = .05, the upper critical value of Z = 1.645 (Table 3-5, t.05, df = ∞, one-sided). 5. The decision rule:

Test of Hypothesis 1. Under the null hypothesis, it is assumed that the respective two samples have come from populations with equal proportions p1 = p2. Under this hypothesis, the sampling distribution of the corresponding Z statistic is known. On the basis of the observed data, if the resultant sample value of Z represents an unusual outcome, that is, if it falls within the critical region, this would cast doubt on the assumption of equal proportions. Therefore, it will have been demonstrated statistically that the population proportions are in fact not equal. The various hypotheses can be stated: Form 1

Form 2

Form 3

H0: p1 = p2 H1: p1 ≠ p2 Two-tailed test

H0: p1 ≤ p2 H1: p1 > p2 Upper-tailed test

H0: p1 ≥ p2 H1: p1 < p2 Lower-tailed test

2.

Accept H0 if sample z < 1.645 Reject H0 if sample z > 1.645 6.

The sample z is given by (16/600) − .02 Sample z =

02)( .9 8)/ 600
(. = 1.17

7. Since the sample z < 1.645, accept H0 for lack of contrary evidence; there is not sufficient evidence to demonstrate that the defect proportion in the shipment is greater than 2 percent.

Test of Hypothesis for Two Proportions Nature In some types of engineering and management-science problems, we may be concerned with a random variable which represents a proportion, for example, the proportional number of defective items per day. The method described previously relates to a single proportion. In this subsection two proportions will be considered. A certain change in a manufacturing procedure for producing component parts is being considered. Samples are taken by using both the existing and the new procedures in order to determine whether the new procedure results in an improvement. In this application, it is of interest to demonstrate statistically whether the population proportion p2 for the new procedure is less than the population proportion p1 for the old procedure on the basis of a sample of data. Test of Hypothesis for Two Proportions: Procedure Nomenclature p1 = population 1 proportion p2 = population 2 proportion n1 = sample size from population 1 n2 = sample size from population 2 x1 = number of observations out of n1 that have the designated attribute x2 = number of observations out of n2 that have the designated attribute p ˆ 1 = x1/n1, the sample proportion from population 1 p ˆ 2 = x2/n2, the sample proportion from population 2 α = significance level H0 = null hypothesis H1 = alternative hypothesis z = tabled Z value corresponding to the stated significance level α ˆ2 p ˆ1−p z =



, the sample value of Z
pˆ 1(1 − pˆ 1)/n1 + pˆ 2(1 − pˆ 2)/n2 Assumptions 1. The respective two samples of n1 and n2 observations have been selected randomly. 2. The sample sizes n1 and n2 are sufficiently large to meet the requirement for the Z approximation; i.e., x1 > 5, x2 > 5.

3-81

The decision rule for form 1 is given by Accept H0 if lower critical z < sample z < upper critical z Reject H0 otherwise

Example Application. A change was made in a manufacturing procedure for component parts. Samples were taken during the last week of operations with the old procedure and during the first week of operations with the new procedure. Determine whether the proportional numbers of defects for the respective populations differ on the basis of the sample information. Procedure 1. The hypotheses are H0: p1 = p2

H1: p1 ≠ p2

2. Select α = .05. Therefore, the critical values of z are 1.96 (Table 3-4, A = 0.9500). 3. For the samples, 75 out of 1720 parts from the previous procedure and 80 out of 2780 parts under the new procedure were found to be defective; therefore, p ˆ 1 = 75/1720 = .0436 4.

p ˆ 2 = 80/2780 = .0288

The decision rule: Accept H0 if −1.96 ≤ sample Z ≤ 1.96 Reject H0 otherwise

5.

The sample statistic: .0436 − .0288 Sample z =




0436)( .9 564)/ 1720 288)( .9 712)/ 2780 +(.0
(. = 2.53

6. Since the sample z of 2.53 > tabled z of 1.96, reject H0 and conclude that the new procedure has resulted in a reduced defect rate.

Goodness-of-Fit Test Nature A standard die has six sides numbered from 1 to 6. If one were really interested in determining whether a particular die was well balanced, one would have to carry out an experiment. To do this, it might be decided to count the frequencies of outcomes, 1 through 6, in tossing the die N times. On the assumption that the die is perfectly balanced, one would expect to observe N/6 occurrences each for 1, 2, 3, 4, 5, and 6. However, chance dictates that exactly N/6 occurrences each will not be observed. For example, given a perfectly balanced die, the probability is only 1 chance in 65 that one will observe 1 outcome each, for 1 through 6, in tossing the die 6 times. Therefore, an outcome different from 1 occurrence each can be expected. Conversely, an outcome of six 3s would seem to be too unusual to have occurred by chance alone. Some industrial applications involve the concept outlined here. The basic idea is to test whether or not a group of observations follows a preconceived distribution. In the case cited, the distribution is uniform; i.e., each face value should tend to occur with the same frequency. Goodness-of-Fit Test: Procedure Nomenclature Each experimental observation can be classified into one of r possible categories or cells.

3-82

MATHEMATICS

r = total number of cells Oj = number of observations occurring in cell j Ej = expected number of observations for cell j based on the preconceived distribution N = total number of observations f = degrees of freedom for the test. In general, this will be equal to (r − 1) minus the number of statistical quantities on which the Ej’s are based (see the examples which follow for details). Assumptions 1. The observations represent a sample selected randomly from a population which has been specified. 2. The number of expectation counts Ej within each category should be roughly 5 or more. If an Ej count is significantly less than 5, that cell should be pooled with an adjacent cell. Computation for Ej On the basis of the specified population, the probability of observing a count in cell j is defined by pj. For a sample of size N, corresponding to N total counts, the expected frequency is given by Ej = Npj. Test Statistics: Chi Square r (Oj − Ej)2 χ2 = 
with f df Ej j=1 Test of Hypothesis 1. H0: The sample came from the specified theoretical distribution H1: The sample did not come from the specified theoretical distribution 2. For a stated level of α, Reject H0 if sample χ2 > tabled χ2 Accept H0 if sample χ2 < tabled χ2 Example Application A production-line product is rejected if one of its characteristics does not fall within specified limits. The standard goal is that no more than 2 percent of the production should be rejected. Computation 1. Of 950 units produced during the day, 28 units were rejected. 2. The hypotheses: H0: the process is in control H1: the process is not in control 3. Assume that α = .05; therefore, the critical value of χ2(1) = 3.84 (Table 3-6, 95 percent, df = 1). One degree of freedom is defined since (r − 1) = 1, and no statistical quantities have been computed for the data. 4. The decision rule: Reject H0 if sample χ2 > 3.84 Accept H0 otherwise 5. Since it is assumed that p = .02, this would dictate that in a sample of 950 there would be on the average (.02)(950) = 19 defective items and 931 acceptable items:

Category

Observed Oj

Expectation Ej = 950pj

Acceptable Not acceptable Total

922 28 950

931 19 950

(922 − 931)2 (28 − 19)2 Sample χ =

+

931 19 = 4.35 with critical χ2 = 3.84 2

6. Conclusion. Since the sample value exceeds the critical value, it would be concluded that the process is not in control.

Example Application A frequency count of workers was tabulated according to the number of defective items that they produced. An unresolved question is whether the observed distribution is a Poisson distribution. That is, do observed and expected frequencies agree within chance variation? Computation 1. The hypotheses: H0: there are no significant differences, in number of defective units, between workers H1: there are significant differences

2. 3.

Assume that α = .05. Test statistic: No. of defective units 0 1 2 3 4 5 6 7 8 9 ≥10 Sum

Oj 8 7 9 12 9 6 3 2 0 1 0 52

Ej

10

2.06 6.64 10.73 11.55 9.33 6.03

8.70 pool

3.24 1.50 .60 .22 .10 52

6

5.66 pool

The expectation numbers Ej were computed as follows: For the Poisson distribution, λ = E(x); therefore, an estimate of λ is the average number of defective units per worker, i.e., λ = (1/52)(0 × 3 + 1 × 7 + ⋅⋅⋅ + 9 × 1) = 3.23. Given this approximation, the probability of no defective units for a worker would be (3.23)0/0!)e−3.23 = .0396. For the 52 workers, the number of workers producing no defective units would have an expectation E = 52(0.0396) = 2.06, and so forth. The sample chi-square value is computed from (10 − 8.70)2 (9 − 10.73)2 (6 − 5.66)2 χ2 =

+

+ ⋅⋅⋅ +

8.70 10.73 5.66 = .53 4. The critical value of χ2 would be based on four degrees of freedom. This corresponds to (r − 1) − 1, since one statistical quantity λ was computed from the sample and used to derive the expectation numbers. 5. The critical value of χ2(4) = 9.49 (Table 3-6) with α = .05; therefore, accept H0.

Two-Way Test for Independence for Count Data Nature When individuals or items are observed and classified according to two different criteria, the resultant counts can be statistically analyzed. For example, a market survey may examine whether a new product is preferred and if it is preferred due to a particular characteristic. Count data, based on a random selection of individuals or items which are classified according to two different criteria, can be statistically analyzed through the χ2 distribution. The purpose of this analysis is to determine whether the respective criteria are dependent. That is, is the product preferred because of a particular characteristic? Two-Way Test for Independence for Count Data: Procedure Nomenclature 1. Each observation is classified into each of two categories: a. The first one into 2, 3, . . . , or r categories b. The second one into 2, 3, . . . , or c categories 2. Oij = number of observations (observed counts) in cell (i, j) with i = 1, 2, . . . , r j = 1, 2, . . . , c 3. N = total number of observations 4. Eij = computed number for cell (i,j) which is an expectation based on the assumption that the two characteristics are independent 5. Ri = subtotal of counts in row i 6. Cj = subtotal of counts in column j 7. α = significance level 8. H0 = null hypothesis 9. H1 = alternative hypothesis 10. χ2 = critical value of χ2 corresponding to the significance level α and (r − 1)(c − 1) df c,r (Oij − Eij)2 11. Sample χ2 = 

Eij i, j Assumptions 1. The observations represent a sample selected randomly from a large total population.

STATISTICS 2. The number of expectation counts Eij within each cell should be approximately 2 or more for arrays 3 × 3 or larger. If any cell contains a number smaller than 2, appropriate rows or columns should be combined to increase the magnitude of the expectation count. For arrays 2 × 2, approximately 4 or more are required. If the number is less than 4, the exact Fisher test should be used. Test of Hypothesis Under the null hypothesis, the classification criteria are assumed to be independent, i.e., H0: the criteria are independent H1: the criteria are not independent For the stated level of α, Reject H0 if sample χ2 > tabled χ2 Accept H0 otherwise Computation for Eij Compute Eij across rows or down columns by using either of the following identities: R Eij = Cj
i across rows N

 

 

Cj Eij = Ri
down columns N Sample c Value 2

(Oij − Eij)2 χ2 = 

Eij i, j

In the special case of r = 2 and c = 2, a more accurate and simplified formula which does not require the direct computation of Eij can be used: [|O11O22 − O12O21| − aN]2N χ2 =


R1R2C1C2 Example Application A market research study was carried out to relate the subjective “feel” of a consumer product to consumer preference. In other words, is the consumer’s preference for the product associated with the feel of the product, or is the preference independent of the product feel? Procedure 1. It was of interest to demonstrate whether an association exists between feel and preference; therefore, assume H0: feel and preference are independent H1: they are not independent 2. A sample of 200 people was asked to classify the product according to two criteria: a. Liking for this product b. Liking for the feel of the product

127/200 = 63.5 percent suggest further that there are other attributes of the product which tend to nullify the beneficial feel of the product.

LEAST SQUARES When experimental data is to be fit with a mathematical model, it is necessary to allow for the fact that the data has errors. The engineer is interested in finding the parameters in the model as well as the uncertainty in their determination. In the simplest case, the model is a linear equation with only two parameters, and they are found by a least-squares minimization of the errors in fitting the data. Multiple regression is just linear least squares applied with more terms. Nonlinear regression allows the parameters of the model to enter in a nonlinear fashion. The following description of maximum likelihood applies to both linear and nonlinear least squares (Ref. 231). If each measurement point yi has a measurement error ∆yi that is independently random and distributed with a normal distribution about the true model y(x) with standard deviation σi, then the probability of a data set is N 1 yi − y(xi) 2 P =  exp −

∆y 2 σi i=1

 











Yes

No

Ri

114 55 169

13 18 31

= 127 = 73 200

3. Select α = .05; therefore, with (r − 1)(c − 1) = 1 df, the critical value of χ2 is 3.84 (Table 3-6, 95 percent). 4. The decision rule: Accept H0 if sample χ2 < 3.84 Reject H0 otherwise 5.







∞

ta − 1e−t dt

0

Like feel Yes No Cj



Here, yi is the measured value, σi is the standard deviation of the ith measurement, and ∆y is needed to say a measured value ∆y has a certain probability. Given a set of parameters (maximizing this function), the probability that this data set plus or minus ∆y could have occurred is P. This probability is maximized (giving the maximum likelihood) if the negative of the logarithm is minimized. N yi − y(xi) 2
− N log ∆y 
2 σi i=1 Since N, σi, and ∆y are constants, this is the same as minimizing χ2. N yi − y(xi; a1, . . . , aM) 2 χ2 = 


σi i=1 with respect to the parameters {aj}. Note that the standard deviations {σi} of the measurements are expected to be known. The goodness of fit is related to the number of degrees of freedom, ν = N − M. The probability that χ2 would exceed a particular value (χ0)2 is ν 1 P = 1 − Q
,
χ 20 2 2 where Q(a, x) is the incomplete gamma function x 1 Q(a, x) =
e−t ta − 1 dt (a > 0) Γ(a) 0 and Γ(a) is the gamma function Γ(a) =

Like product

3-83

The sample value of χ2 by using the special formula is [|114 × 18 − 13 × 55| − 100]2200 Sample χ2 =



(169)(31)(127)(73) = 6.30

6. Since the sample χ2 of 6.30 > tabled χ2 of 3.84, reject H0 and accept H1. The relative proportionality of E11 = 169(127/200) = 107.3 to the observed 114 compared with E22 = 31(73/200) = 11.3 to the observed 18 suggests that when the consumer likes the feel, the consumer tends to like the product, and conversely for not liking the feel. The proportions 169/200 = 84.5 percent and

Both functions are tabulated in mathematical handbooks (Ref. 1). The function P gives the goodness of fit. Call χ02 the value of χ2 at the minimum. Then P > 0.1 represents a believable fit; if Q > 0.001, it might be an acceptable fit; smaller values of Q indicate the model may be in error (or the σi are really larger.) A “typical” value of χ2 for a moderately good fit is χ2 ∼ ν. Asymptotically for large ν, the statistic χ2 becomes normally distributed with a mean ν and a standard deviation (2 ν) (Ref. 231).
 If values σi are not known in advance, assume σi = σ (so that its value does not affect the minimization of χ2). Find the parameters by minimizing χ2 and compute: N [yi − y(xi)]2 σ2 = 

N i=1 This gives some information about the errors (i.e., the variance and standard deviation of each data point), although the goodness of fit, P, cannot be calculated. The minimization of χ2 requires yi − y(xi) ∂y(xi; a1, . . . , aM)

= 0, 


σ ∂a N

i=1

2 i

k

k = 1, . . . , M

3-84

MATHEMATICS

Linear Least Squares When the model is a straight line yi − a − bxi χ2(a, b) = 

σi i=1 N

Define

N 1 S = 
2 , i = 1 σi N



N xi Sx = 
, 2 i = 1 σi

2 i



y = a + bx + ε where x = strip-method determination y = magnetic-method determination

N y Sy = 
i2 i = 1 σi



N

x Sxx = 
2 , i = 1 σi

Nomenclature. The calibration between the magnetic and the stripping methods can be determined through the model

2

xiyi Sxy = 
, 2 i = 1 σi

Sample data



Thickness, 10−5 ln

N

1 S ti =
xi −
x , σi S

Stt =  t 2i

Stripping method, x

Magnetic method, y

1 σ b2 =
Stt

104 114 116 129 132 139

85 115 105 127 120 121

We thus get the values of a and b with maximum likelihood as well as the variances of a and b. Using the value of χ2 for this a and b, we can also calculate the goodness of fit, P. In addition, the linear correlation coefficient r is related by

174 312 338 465 720

155 250 310 443 630

Then

1 N tiyi b=

, Stt i = 1 σi

Sy − Sxb a=
, S

Sx Cov (a, b) = −
, SStt



i=1



1 S 2x σ a2 =
1 +
, S SStt Cov (a, b) rab =
σaσb

N

χ2 = (1 − r2)  (yi − y)2 i=1

(xi − x)(yi − y)

σ2i i=1 r =


N N (xi − x )2 (yi − y)2



  2 σi σ 2i i=1 i=1

Computations. The normal equations are defined by na + ( x)b =  y

N

Here

 

Values of r near 1 indicate a positive correlation; r near −1 means a negative correlation and r near zero means no correlation. The form of the equations here is given to provide good accuracy when many terms are used and to provide the variances of the parameters. Another form of the equations for a and b is simpler, but is sometimes inaccurate unless many significant digits are kept in the calculations. The minimization of χ2 when σi is the same for all i gives the following equations for a and b. N

N

i=1

i=1

aN + b  xi =  yi N

N

N

a  xi + b  x =  yi xi 2 i

i=1

i=1

i=1

N

The solution is

N

N

N  yi xi −  xi  yi i=1 i=1 i=1 b =


N N 2 2 N  x i −  xi



i=1

N



N

y =  yi N, i=1

i=1





( x)a + ( x2)b =  xy

For the sample

11a + 2743b = 2461 2743a + 1,067,143b = 952,517 with  y2 = 852,419. The solution to the normal equations is given by a = 3.19960

χ2 =  y2 − a  y − b  xy if a sufficient number of significant digits is retained (usually six or seven digits are sufficient). Here χ2 = 2175.14 If the normalized method is used in addition, the value of Stt is 3.8314 × 105/σ2, where σ2 is the variance of the measurement of y. The values of a and b are, of course, the same. The variances of a and b are σ a2 = 0.2532σ2, σ b2 = 2.610 × 10−6σ2. The correlation coefficient is 0.996390, which indicates that there is a positive correlation between x and y. The small value of the variance for b indicates that this parameter is determined very well by the data. The residuals show no particular pattern, and the predictions are plotted along with the data in Fig. 3-58. If the variance of the measurements of y is known through repeated measurements, then the variance of the parameters can be made absolute.

Multiple Regression A general linear model is one expressed as M

x =  xi N i=1

a = y − bx

The value of χ2 can be calculated from the formula N

N

N

i=1

i=1

i=1

χ2 =  y 2i − a  yi − b  yi xi It is usually advisable to plot the observed pairs of yi versus xi to support the linearity assumption and to detect potential outliers. Suspected outliers can be omitted from the least-squares “fit” and then subsequently tested on the basis of the least-squares fit.

y(x) =  akXk(x) k=1

where the parameters are {ak}, and the expression is linear with respect to them, and Xk(x) can be any (nonlinear) functions of x, not depending on the parameters {ak}. Then: N M 1
2 yi −  aj Xj(xi) Xk(xi) = 0, k = 1, . . . , M  i = 1 σi j=1 This is rewritten as M N N 1 yi
2 Xj(xi)Xk(xi) aj = 
2 Xk(xi)   j = 1 i = 1 σi i = 1 σi









M

Example Application. Brenner (Magnetic Method for Measuring the Thickness of Non-magnetic Coatings on Iron and Steel, National Bureau of Standards, RP1081, March 1938) suggests an alternative way of measuring the thickness of nonmagnetic coatings of galvanized zinc on iron and steel. This procedure is based on a nondestructive magnetic method as a substitute for the standard destructive stripping method. A random sample of 11 pieces was selected and measured by both methods.

b = .884362

The error sum of squares can be computed from the formula

or as

α

a = βk

kj j

j=1

Solving this set of equations gives the parameters {aj}, which maximize the likelihood. The variance of aj is σ2(aj) = Cj j where Cj k = α−1 j k, or C is the inverse of α. The covariance of aj and ak

STATISTICS

FIG. 3-58

3-85

Plot of data and correlating line. 4.5βˆ 0 + 2.85βˆ 1 + 2.025βˆ 2 = 14.08957

is given by Cj k. If rounding errors affect the result, then we try to make the functions orthogonal. For example, using Xk(x) = xk − 1

2.85βˆ 0 + 2.025βˆ 1 + 1.5333βˆ 2 = 8.828813 The algebraic solution to the simultaneous equations is βˆ = 3.19513 βˆ = .4425 βˆ = −.7653

will cause rounding errors for a smaller M than

0

Xk(x) = Pk − 1(x)

1

2

The inverse of the product matrix

where Pk − 1 are orthogonal polynomials. If necessary, a singular value decomposition can be used. Various global and piecewise polynomials can be used to fit the data. Most approximations are to be used with M < N. One can sometimes use more and more terms, and calculating the value of χ2 for each solution. Then stop increasing M when the value of χ2 no longer increases with increasing M. Example Application. Merriman (“The Method of Least Squares Applied to a Hydraulic Problem,” J. Franklin Inst., 233–241, October 1877) reported on a study of stream velocity as a function of relative depth of the stream. Sample data

α=



α−1 =

is

10 4.5 2.85



4.5 2.85 2.025

.6182 −2.5909 2.2727

2.85 2.025 1.5333



−2.5909 16.5530 −17.0455



2.2727 −17.0455 18.9394

The variances are then the diagonal elements of the inverse of matrix α (0.6182, 16.5530, 18.9394) times the variance of the measurement of y, σ y2. The value of χ2 is 5.751 × 10−5, the correlation coefficient r = 0.99964, and σ = 0.002398. t values. A sample t value can be computed for each regression coefficient j through the identity tj = βˆ j /(ˆσ
cjj), where cjj is the ( j,j) element in the inverse. For the two variables x1 and x2,

Velocity, y, ft/s

Coefficient

cjj

Sample t value

.1 .2 .3 .4

3.1950 3.2299 3.2532 3.2611 3.2516

.4425 −.7653

16.55 18.94

45.3 −73.3

.5 .6 .7 .8 .9

3.2282 3.1807 3.1266 3.0594 2.9759

Depth* 0

*As a fraction of total depth. Model. Owing to the curvature of velocity with depth, a quadratic model was specified:

Computational note. From a computational standpoint, it is usually advisable to define the variables in deviation units. For example, in the problem presented, let x1 = depth −  dep th  = depth − .45 For expansion terms such as a square, define x2 = x21 − x¯21 ( x¯21 = .0825) For the previous sample data, Deviation units

Velocity = β0 + β1x1 + β2x2 where x2 = x 12 . Normal equations. The three normal equations are defined by (n)βˆ + ( x )βˆ + ( x )βˆ =  y 0



1

1



2

2

( x1)βˆ 0 + ( x12)βˆ 1 + ( x1x2)βˆ 2 =  x1 y ( x2)βˆ 0 + ( x1x2)βˆ 1 + ( x22)βˆ 2 =  x2 y For the sample data, the normal equations are 10βˆ + 4.5βˆ + 2.85 βˆ = 31.7616 0

1

2

x1

x2

x1

x2

−.45 −.35 −.25 −.25 −.15 −.05

.12 .04 −.02 −.02 −.06 −.08

.05 .15 .25 .35 .45

−.08 −.06 −.02 .04 .12

The resultant analysis-of-variance tables will remain exactly the same. However,

3-86

MATHEMATICS

the corresponding coefficient t value for the linear coefficient will usually be improved. This is an idiosyncrasy of regression modeling. With the coded data presented, the least-squares solution is given by Yˆ = 3.17616 − .2462x1 − .7653x2 with a corresponding t value for βˆ 1 = −.2462 of t = −63.63. When expansion terms are used but not expanded about the mean, the corresponding t values for the generating terms should not be used. For example, if x3 = x1x2 is used rather than the correct expansion (x1 − x1)(x2 − x2), then the corresponding t values for x1 and x2 should not be used.

Nonlinear Least Squares There are no analytic methods for determining the most appropriate model for a particular set of data. In many cases, however, the engineer has some basis for a model. If the parameters occur in a nonlinear fashion, then the analysis becomes more difficult. For example, in relating the temperature to the elapsed time of a fluid cooling in the atmosphere, a model that has an asymptotic property would be the appropriate model (temp = a + b exp(−c time), where a represents the asymptotic temperature corresponding to t → ∞. In this case, the parameter c appears nonlinearly. The usual practice is to concentrate on model development and computation rather than on statistical aspects. In general, nonlinear regression should be applied only to problems in which there is a welldefined, clear association between the two variables; therefore, a test of hypothesis on the significance of the fit would be somewhat ludicrous. In addition, the generalization of the theory for the associate confidence intervals for nonlinear coefficients is not well developed. The Levenberg-Marquardt method is used when the parameters of the model appear nonlinearly (Ref. 231). We still define N yi − y(xi; a) χ2(a) = 

σ 2i i=1





2

1. Choose a and calculate χ2(a). 2. Choose λ, say λ = 0.001. 3. Solve Eq. (3-91) for ak + 1 and evaluate χ2(ak + 1). 4. If χ2(ak + 1) ≥ χ2(ak) then increase λ by a factor of, say, 10 and go back to step 3. This makes the step more like a steepest descent. 5. If χ2(ak + 1) < χ2(ak) then update a, i.e., use a = ak + 1, decrease λ by a factor of 10, and go back to step 3. 6. Stop the iteration when the decrease in χ2 from one step to another is not statistically meaningful, i.e., less than 0.1 or 0.01 or 0.001. 7. Set λ = 0 and compute the estimated covariance matrix: C = α−1. This gives the standard errors in the fitted parameters a. For normally distributed errors the parameter region in which χ2 = constant can give boundaries of the confidence limits. The value of a obtained in the Marquardt method gives the minimum χ2min. If we set χ2 = χ2min + ∆χ for some ∆χ and then look at contours in parameter space where χ12 = constant then we have confidence boundaries at the probability associated with χ 12. For example, in a chemical reactor with radial dispersion the heat transfer coefficient and radial effective heat conductivity are closely connected: decreasing one and increasing the other can still give a good fit. Thus, the confidence boundaries may look something like Fig. 3-59. The ellipse defined by ∆χ2 = 2.3 contains 68.3 percent of the normally distributed data. The curve defined by ∆χ2 = 6.17 contains 95.4 percent of the data. Example Application. Data were collected on the cooling of water in the atmosphere as a function of time. Sample data

and near the optimum represent χ2 by 1 χ2(a) = χ02 − dT ⋅ a +
aT ⋅ D ⋅ a 2 where d is an M × 1 vector and D is an M × M matrix. We then calculate iteratively D ⋅ (ak + 1 − ak) = −∇χ2(ak)

(3-89)

The notation alk means the lth component of a evaluated on the kth iteration. If ak is a poor approximation to the optimum, we might use steepest descent instead. ak + 1 − ak = −constant × ∇χ2(ak)

(3-90)

and choose the constant somehow to decrease χ2 as much as possible. The gradient of χ2 is N ∂χ2 yi − y(xi; a) ∂y(xi; a)
= −2 


∂ak σ 2i ∂ak i=1

k = 1, 2, . . . , M

The second derivative (in D) is N ∂2χ2 1 ∂y(xi; a) ∂y(xi; a) ∂2y(xi; a)
= 2 
2

− [yi − y(xi; a)]
∂ak∂al ∂ak ∂al ∂ak∂al i=1 σi



Time x

Temperature y

0 1 2 3 5

92.0 85.5 79.5 74.5 67.0

7 10 15 20

60.5 53.5 45.0 39.5

Model form. On the basis of the nature of the data, an exponential model was selected initially to represent the trend y = a + becx. In this example, the resultant temperature would approach as an asymptotic (a with c negative) the wet-bulb temperature of the surrounding atmosphere. Unfortunately, this temperature was not reported. Using a computer package in MATLAB gives the following results: a = 33.54, b = 57.89, c = 0.11. The value of χ2 is 1.83. An alternative form of model is y = a + b/(c + x). For this model the results were a = 9.872, b = 925.7, c = 11.27, and the value of χ2 is 0.19. Since this model had a smaller value of χ2, it might be the chosen one, but it is only a fit of the specified data and may not be generalized beyond that. Both curve fits give an equivalent plot. The second form is shown in Fig. 3-60.

Both Eq. (3-89) and Eq. (3-90) are included if we write M

 α′

kl

(a kl + 1 − a lk ) = βk

(3-91)

l=1

where

N 1 ∂y(xi; a) ∂y(xi; a) α′kl = 
2

∂ak ∂al i=1 σi N 1 ∂y(xi; a) α′kk = 
2
∂ak i=1 σi



k≠1

 (1 + λ) 2

N yi − y(xi; a) ∂y(xi; a) β k = 


σ 2i ∂ak i=1

The second term in the second derivative is dropped because it is usually small [remember that yi will be close to y(xi, a)]. The LevenbergMarquardt method then iterates as follows

FIG. 3-59

Parameter estimation for heat transfer.

STATISTICS

FIG. 3-60

Data in nonlinear regression example.

ERROR ANALYSIS OF EXPERIMENTS Consider the problem of assessing the accuracy of a series of measurements. If measurements are for independent, identically distributed observations, then the errors are independent and uncorrelated. Then y, the experimentally determined mean, varies about E(y), the true mean, with variance σ2/n, where n is the number of observations in y. Thus, if one measures something several times today, and each day, and the measurements have the same distribution, then the variance of the means decreases with the number of samples in each day’s measurement, n. Of course, other factors (weather, weekends) may make the observations on different days not distributed identically. Consider next the problem of estimating the error in a variable that cannot be measured directly but must be calculated based on results of other measurements. Suppose the computed value Y is a linear combination of the measured variables {yi}, Y = α1y1 + α2y2 + … . Let the random variables y1, y2, . . . have means E(y1), E(y2), . . . and variances σ2(y1), σ2(y2), . . . . The variable Y has mean E(Y) = α1E(y1) + α2 E(y2) + … and variance (Ref. 82) n

n

σ2(Y) =  α2i σ2(yi) + 2  i=1

n



αi αj Cov (yi, yj)

i=1 j=i+1

If the variables are uncorrelated and have the same variance, then σ2(Y) =

3-87

  α σ n

2 i

2

i=1

Next suppose the model relating Y to {yi} is nonlinear, but the errors are small and independent of one another. Then a change in Y is related to changes in yi by ∂Y ∂Y dY =
dy1 +
dy2 + … ∂y1 ∂y2 If the changes are indeed small, then the partial derivatives are constant among all the samples. Then the expected value of the change, E(dY), is zero. The variances are given by the following equation (Refs. 25 and 40): N ∂Y 2 σ2(dY) = 
σ i2 ∂yi i=1 Thus, the variance of the desired quantity Y can be found. This gives an independent estimate of the errors in measuring the quantity Y from the errors in measuring each variable it depends upon.

 

Example Suppose one wants to measure the thermal conductivity of a solid (k). To do this, one needs to measure the heat flux (q), the thickness of the

sample (d), and the temperature difference across the sample (∆T). Each measurement has some error. The heat flux (q) may be the rate of electrical heat ˙ divided by the area (A), and both quantities are measured to some tolinput (Q) erance. The thickness of the sample is measured with some accuracy, and the temperatures are probably measured with a thermocouple to some accuracy. These measurements are combined, however, to obtain the thermal conductivity, and it is desired to know the error in the thermal conductivity. The formula is d ˙ k=
Q A∆T The variance in the thermal conductivity is then k 2 k 2 k 2 2 k 2 σ k2 =
σ d2 +
σ Q˙ +
σ A2 +
σ 2∆T ˙ Q d A ∆T

 

 

 

 

FACTORIAL DESIGN OF EXPERIMENTS AND ANALYSIS OF VARIANCE Statistically designed experiments consider, of course, the effect of primary variables, but they also consider the effect of extraneous variables and the interactions between variables, and they include a measure of the random error. Primary variables are those whose effect you wish to determine. These variables can be quantitative or qualitative. The quantitative variables are ones you may fit to a model in order to determine the model parameters (see the section “Least Squares”). Qualitative variables are ones you wish to know the effect of, but you do not try to quantify that effect other than to assign possible errors or magnitudes. Qualitative variables can be further subdivided into Type I variables, whose effect you wish to determine directly, and Type II variables, which contribute to the performance variability and whose effect you wish to average out. For example, if you are studying the effect of several catalysts on yield in a chemical reactor, each different type of catalyst would be a Type I variable because you would like to know the effect of each. However, each time the catalyst is prepared, the results are slightly different due to random variations; thus, you may have several batches of what purports to be the same catalyst. The variability between batches is a Type II variable. Since the ultimate use will require using different batches, you would like to know the overall effect including that variation, since knowing precisely the results from one batch of one catalyst might not be representative of the results obtained from all batches of the same catalyst. A randomized block design, incomplete block design, or Latin square design (Ref. 40), for example, all keep the effect of experimental error in the blocked variables from influencing the effect of the primary variables. Other uncontrolled variables are accounted for by introducing randomization in parts of the experimental design. To study all variables and their interaction requires a factorial design, involving all possible

3-88

MATHEMATICS

combinations of each variable, or a fractional factorial design, involving only a selected set. Statistical techniques are then used to determine which are the important variables, what are the important interactions, and what the error is in estimating these effects. The discussion here is only a brief overview of the excellent Ref. 40. Suppose we have two methods of preparing some product and we wish to see which treatment is best. When there are only two treatments, then the sampling analysis discussed in the section “TwoPopulation Test of Hypothesis for Means” can be used to deduce if the means of the two treatments differ significantly. When there are more treatments, the analysis is more detailed. Suppose the experimental results are arranged as shown in the table: several measurements for each treatment. The goal is to see if the treatments differ significantly from each other; that is, whether their means are different when the samples have the same variance. The hypothesis is that the treatments are all the same, and the null hypothesis is that they are different. The statistical validity of the hypothesis is determined by an analysis of variance. Estimating the Effect of Four Treatments Treatment

Treatment average Grand average

1

2

3

4

— — —

— — — —

— — — — —

— — — — — — —

—

— —

—

The data for k = 4 treatments is arranged in the table. For each treatment, there are nt experiments and the outcome of the ith experiment with treatment t is called yti. Compute the treatment average nt

y

Basically the test for whether the hypothesis is true or not hinges on a comparison of the within-treatment estimate sR2 (with νR = N − k degrees of freedom) with the between-treatment estimate s2T (with νT = k − 1 degrees of freedom). The test is made based on the F distribution for νR and νT degrees of freedom (Table 3-7). Next consider the case that uses randomized blocking to eliminate the effect of some variable whose effect is of no interest, such as the batch-to-batch variation of the catalysts in the chemical reactor example. Suppose there are k treatments and n experiments in each treatment. The results from nk experiments can be arranged as shown in the block design table; within each block, the various treatments are applied in a random order. Compute the block average, the treatment average, as well as the grand average as before. Block Design with Four Treatments and Five Blocks Treatment

1

2

3

4

Block average

Block 1 Block 2 Block 3 Block 4 Block 5

— — — — —

— — — — —

— — — — —

— — — — —

— — — — —

The following quantities are needed for the analysis of variance table. Name

Formula

dof

average

2 SA = nky 

blocks

2 SB = k  i = 1 (y i − y)

n−1

treatments

ST = n  t = 1 (y t − y)

k−1

residuals

SR =  t = 1  i = 1 (yti − yi − yt + y)2

(n − 1)(k − 1)

total

S = t = 1  i = 1 y

N = nk

1

n

k

2

k

k

n

n

2 ti

ti

i=1

The key test is again a statistical one, based on the value of

yt =
nt

sT2 ,
sR2

Also compute the grand average k

 n y

t t

k

N =  nt

t=1

y =

, N

t=1

Next compute the sum of squares of deviations from the average within the tth treatment nt

St =  (yti − yt)2 i=1

Since each treatment has nt experiments, the number of degrees of freedom is nt − 1. Then the sample variances are St s t2 =
nt − 1 The within-treatment sum of squares is k

SR =  St t=1

and the within-treatment sample variance is SR sR2 =
N−k Now, if there is no difference between treatments, a second estimate of σ2 could be obtained by calculating the variation of the treatment averages about the grand average. Thus compute the betweentreatment mean square k ST 2 sT2 =
, ST =  nt(y t − y) k−1 t=1

ST sT2 =
, k−1

SR s R2 =

(n − 1)(k − 1)

and the F distribution for νR and νT degrees of freedom (Table 3-7). The assumption behind the analysis is that the variations are linear (Ref. 40). There are ways to test this assumption as well as transformations to make if it is not true. Reference 40 also gives an excellent example of how the observations are broken down into a grand average, a block deviation, a treatment deviation, and a residual. For twoway factorial design in which the second variable is a real one rather than one you would like to block out, see Ref. 40. To measure the effects of variables on a single outcome a factorial design is appropriate. In a two-level factorial design, each variable is considered at two levels only, a high and low value, often designated as a + and −. The two-level factorial design is useful for indicating trends, showing interactions, and it is also the basis for a fractional factorial design. As an example, consider a 23 factorial design with 3 variables and 2 levels for each. The experiments are indicated in the factorial design table. Two-Level Factorial Design with Three Variables Variable Run

1

2

3

1 2 3 4 5 6 7 8

− + − + − + − +

− − + + − − + +

− − − − + + + +

DIMENSIONAL ANALYSIS The main effects are calculated by calculating the difference between results from all high values of a variable and all low values of a variable; the result is divided by the number of experiments at each level. For example, for the first variable: [(y2 + y4 + y6 + y8) − (y1 + y3 + y5 + y7)] Effect of variable 1 =



4 Note that all observations are being used to supply information on each of the main effects and each effect is determined with the precision of a fourfold replicated difference. The advantage of a one-at-atime experiment is the gain in precision if the variables are additive and the measure of nonadditivity if it occurs (Ref. 40). Interaction effects between variables 1 and 2 are obtained by calculating the difference between the results obtained with the high and low value of 1 at the low value of 2 compared with the results obtained with the high and low value 1 at the high value of 2. The 12-interaction is [(y4 − y3 + y8 − y7) − (y2 − y1 + y6 − y5)] 12-interaction =



2

3-89

The key step is to determine the errors associated with the effect of each variable and each interaction so that the significance can be determined. Thus, standard errors need to be assigned. This can be done by repeating the experiments, but it can also be done by using higher-order interactions (such as 123 interactions in a 24 factorial design). These are assumed negligible in their effect on the mean but can be used to estimate the standard error (see Ref. 40). Then, calculated effects that are large compared with the standard error are considered important, while those that are small compared with the standard error are considered to be due to random variations and are unimportant. In a fractional factorial design one does only part of the possible experiments. When there are k variables, a factorial design requires 2k experiments. When k is large, the number of experiments can be large; for k = 5, 25 = 32. For a k this large, Box et al. (Ref. 82, p. 376) do a fractional factorial design. In the fractional factorial design with k = 5, only 16 experiments are done. Cropley (Ref. 82) gives an example of how to combine heuristics and statistical arguments in application to kinetics mechanisms in chemical engineering.

DIMENSIONAL ANALYSIS Dimensional analysis allows the engineer to reduce the number of variables that must be considered to model experiments or correlate data. Consider a simple example in which two variables F1 and F2 have the units of force and two additional variables L1 and L2 have the units of length. Rather than having to deduce the relation of one variable on the other three, F1 = fn (F2, L1, L2), dimensional analysis can be used to show that the relation must be of the form F1 /F2 = fn (L1 /L2). Thus considerable experimentation is saved. Historically, dimensional analysis can be done using the Rayleigh method or the Buckingham pi method. This brief discussion is equivalent to the Buckingham pi method but uses concepts from linear algebra; see Ref. 13 for further information. The general problem is posed as finding the minimum number of variables necessary to define the relationship between n variables. Let {Qi} represent a set of fundamental units, like length, time, force, and so on. Let [Pi] represent the dimensions of a physical quantity Pi; there are n physical quantities. Then form the matrix αij

Q1 Q2 … Qm

[P1]

[P2]

…

[Pn]

α11 α21

α12 α22

… …

α1n α2n

αm1

αm2

…

αmn

in which the entries are the number of times each fundamental unit appears in the dimensions [Pi]. The dimensions can then be expressed as follows.

Fluid thermal conductivity = k = (F/θT) Fluid specific heat = cp = (FL/MT) Dimensional constant = gc = (ML/Fθ2) The matrix α in this case is as follows. [Pi]

Qj

F M L θ T

Example: Buckingham Pi Method—Heat-Transfer Film Coefficient It is desired to determine a complete set of dimensionless groups with which to correlate experimental data on the film coefficient of heat transfer between the walls of a straight conduit with circular cross section and a fluid flowing in that conduit. The variables and the dimensional constant believed to be involved and their dimensions in the engineering system are given below: Film coefficient = h = (F/LθT) Conduit internal diameter = D = (L) Fluid linear velocity = V = (L/θ) Fluid density = ρ = (M/L3) Fluid absolute viscosity = µ = (M/Lθ)

D

V

ρ

µ

k

Cp

gc

1 0 −1 −1 −1

0 0 1 0 0

0 0 1 −1 0

0 1 −3 0 0

0 1 −1 −1 0

1 0 0 −1 −1

1 −1 1 0 −1

−1 1 1 −2 0

Here m ≤ 5, n = 8, p ≥ 3. Choose D, V, µ, k, and gc as the primary variables. By examining the 5 × 5 matrix associated with those variables, we can see that its determinant is not zero, so the rank of the matrix is m = 5; thus, p = 3. These variables are thus a possible basis set. The dimensions of the other three variables h, ρ, and Cp must be defined in terms of the primary variables. This can be done by inspection, although linear algebra can be used, too. hD h [h] = D−1k+1; thus
=
is a dimensionless group D−1 k k ρVD ρ [ρ] = µ1V −1D −1; thus

=
is a dimensionless group µ1V−1D−1 µ Cpµ Cp [Cp] = k+1µ−1; thus
=
is a dimensionless group k+1 µ−1 k Thus, the dimensionless groups are [Pi] hD ρVD Cp µ :
,
,


Q1α1i Q2α2i⋅⋅⋅Qmαmi k µ k

[Pi] = Q1α1i Q2α2i⋅⋅⋅Qmαmi Let m be the rank of the α matrix. Then p = n − m is the number of dimensionless groups that can be formed. One can choose m variables {Pi} to be the basis and express the other p variables in terms of them, giving p dimensionless quantities.

h

The dimensionless group hD/k is called the Nusselt number, NNu, and the group Cp µ /k is the Prandtl number, NPr. The group DVρ/µ is the familiar Reynolds number, NRe , encountered in fluid-friction problems. These three dimensionless groups are frequently used in heat-transfer-film-coefficient correlations. Functionally, their relation may be expressed as φ(NNu, NPr, NRe) = 0 or as

(3-91)

NNu = φ1(NPr, NRe)

It has been found that these dimensionless groups may be correlated well by an equation of the type hD/k = K(cpµ/k)a(DVρ/µ)b in which K, a, and b are experimentally determined dimensionless constants. However, any other type of algebraic expression or perhaps simply a graphical relation among these three groups that accurately fits the experimental data would be an equally valid manner of expressing Eq. (3-91).

3-90

MATHEMATICS

Naturally, other dimensionless groups might have been obtained in the example by employing a different set of five repeating quantities that would not form a dimensionless group among themselves. Some of these groups may be found among those presented in Table 3-8. Such a complete set of three dimensionless groups might consist of Stanton, Reynolds, and Prandtl numbers or of Stanton, Peclet, and Prandtl numbers. Also, such a complete set different from that obtained in the preceding example will result from a multiplication of appropriate powers of the Nusselt, Prandtl, and Reynolds numbers. For such a set to be complete, however, it must satisfy the condition that each of the three dimensionless groups be independent of the other two.

TABLE 3-8 Dimensionless Groups in the Engineering System of Dimensions Biot number Condensation number Number used in condensation of vapors Euler number Fourier number Froude number Graetz number Grashof number Mach number Nusselt number Peclet number Prandtl number Reynolds number Schmidt number Stanton number Weber number

NBi NCo NCv NEu NFo NFr NGz NGr NMa NNu NPe NPr NRe NSc NSt NWe

hL/k (h/k)(µ2/ρ2g)1/3 L3ρ2gλ/kµ∆t gc(−dp)/ρV2 kθ/ρcL2 V2/Lg wc/kL L3ρ2βg∆t/µ2 V/Va hD/k DVρc/k cµ/k DVρ/µ µ/ρDυ h/cVρ LV2ρ/σgc

PROCESS SIMULATION Classification Process simulation refers to the activity in which mathematical models of chemical processes and refineries are modeled with equations, usually on the computer. The usual distinction must be made between steady-state models and transient models, following the ideas presented in the introduction to this section. In a chemical process, of course, the process is nearly always in a transient mode, at some level of precision, but when the time-dependent fluctuations are below some value, a steady-state model can be formulated. This subsection presents briefly the ideas behind steady-state process simulation (also called flowsheeting), which are embodied in commercial codes. The transient simulations are important for designing startup of plants and are especially useful for the operating of chemical plants. Process Modules The usual first step in process simulation is to perform a mass and energy balance for a chosen process. The most important aspect of the simulation is that the thermodynamic data of the chemicals be modeled correctly. The computer results of vaporliquid equilibria, for example, must be checked against experimental data to insure their validity before using the data in more complicated computer calculations. At this first level of detail, it is not necessary to know the internal parameters for all the units, since what is desired is just the overall performance. For example, in a heat exchanger design, it suffices to know the heat duty, the total area, and the temperatures of the output streams; the details like the percentage baffle cut, tube layout, or baffle spacing can be specified later when the details of the proposed plant are better defined. Each unit operation is modeled by a subroutine, which is governed by equations (presented throughout this book). Some of the inputs to the units are known, some are specified by the user as design variables, and some are to be found using the simulation. It is important to know the number of degrees of freedom for each option of the unit operation, because at least that many parameters must be specified in order for the simulation to be able to calculate unit outputs. Sometimes the quantities the user would like to specify are targets, and parameters in the unit operation are to be changed to meet that target. This is not always possible, and the designer will have to adjust the parameters of the unit operation to achieve the desired target, possibly using the convergence tools discussed below. For example, in a reaction/separation system, if there is an impurity that must be purged, a common objective is to set the purge fraction so that the impurity concentration into the reactor is kept at some moderate value. Yet the solution techniques do not readily lend themselves to this connection, so convergence strategies must be employed.

Solution Strategies Consider a chemical process consisting of a series of units, such as distillation towers, reactors, and so forth. If the feed to the process is known and the operating parameters of the unit operations are specified by the user, then one can begin with the first unit, take the process input, calculate the unit output, carry that output to the input of the next unit, and continue the process. In this way, one can simulate the entire process. However, if the process involves a recycle stream, as nearly all chemical processes do, then when the calculation is begun, it is discovered that the recycle stream is unknown. Thus the calculation cannot begin. This situation leads to the need for an iterative process: the flow rates, temperature, and pressure of the unknown recycle stream are guessed and the calculations proceed as before. When one reaches the end of the process, where the recycle stream is formed to return to the inlet, it is necessary to check to see if the recycle stream is the same as assumed. If not, an iterative procedure must be used to cause convergence. The techniques like Wegstein (see “Numerical Solution of Nonlinear Equations in One Variable”) can be used to accelerate the convergence. When doing these iterations, it is useful to analyze the process using precedence ordering and tearing to minimize the number of recycle loops (Refs. 201, 242, 255, and 293). When the recycle loops interact with one another the iterations may not lead to a convergent solution. The designer usually wants to specify stream flow rates or parameters in the process, but these may not be directly accessible. For example, the desired separation may be known for a distillation tower, but the simulation program requires the specification of the number of trays. It is left up to the designer to choose the number of trays that lead to the desired separation. In the example of the purge stream/ reactor impurity, a controller module may be used to adjust the purge rate to achieve the desired reactor impurity. This further complicates the iteration process. An alternative method of solving the equations is to solve them as simultaneous equations. In that case, one can specify the design variables and the desired specifications and let the computer figure out the process parameters that will achieve those objectives. It is possible to overspecify the system or give impossible conditions. However, the biggest drawback to this method of simulation is that large sets (10,000s) of algebraic equations must be solved simultaneously. As computers become faster, this is less of an impediment. For further information, see Refs. 90, 175, 255, and 293. For information on computer software, see the Annual CEP Software Directory (Ref. 8) and other articles (Refs. 7 and 175).

INTELLIGENT SYSTEMS IN PROCESS ENGINEERING

3-91

INTELLIGENT SYSTEMS IN PROCESS ENGINEERING REFERENCES: General, 232, 248, 258, 275, 276. Knowledge-Based Systems, 49, 232, 275. Neural Networks, 54, 140. Qualitative Simulation, 178, 292. Fuzzy Logic, 94. Genetic Algorithms, 121. Applications, 15, 24, 205, 232, 250, 262, 294.

Intelligent system is a term that refers to computer-based systems that include knowledge-based systems, neural networks, fuzzy logic and fuzzy control, qualitative simulation, genetic algorithms, natural language understanding, and others. The term is often associated with a variety of computer programming languages and/or features that are used as implementation media, although this is an imprecise use. Examples include object-oriented languages, rule-based languages, prolog, and lisp. The term intelligent system is preferred over the term artificial intelligence. The three intelligent-system technologies currently seeing the greatest amount of industrial application are knowledge-based systems, fuzzy logic, and artificial neural networks. These technologies are components of distributed systems. Mathematical models, conventional numeric and statistical approaches, neural networks, knowledge-based systems, and the like, all have their place in practical implementation and allow automation of tasks not welltreated by numerical algorithms. Fundamentally, intelligent-system techniques are modeling techniques. They allow the encoding of qualitative models that draw upon experience and expertise, thereby extending modeling capacity beyond mathematical description. An important capability of intelligent system techniques is that they can be used not only to model physical behaviors but also decision-making processes. Decision processes reflect the selection, application, and interpretation of highly relevant pieces of information to draw conclusions about complex situations. Activity-specific decision processes can be expressed at a functional level, such as diagnosis, design, planning, and scheduling, or as their generic components, such as classification, abduction, and simulation. Decision process models address how information is organized and structured and then assimilated into active decisions. Knowledge-Based Systems Knowledge-based system (KBS) approaches capture the structural and information processing features of qualitative problem solving associated with sequential consideration, selection, and search. These technologies not only provide the means of capturing decision-making knowledge but also offer a medium for exploiting efficient strategies used by experts. KBSs, then, are computer programs that model specific ways of organizing problem-specific fragments of knowledge and then searching through them by establishing appropriate relationships to reach correct conclusions. Deliberation is a general label for the algorithmic process for sorting through the knowledge fragments. The basic components of KBSs are knowledge representation (structure) and search. They are the programming mechanisms that facilitate the use and application of the problem-specific knowledge appropriate to solving the problem. Together they are used to form conclusions, decisions, or interpretations in a symbolic form. See Refs. 49, 232, and 275. Qualitative simulation is a specific KBS model of physical processes that are not understood well enough to develop a physics-based numeric model. Corrosion, fouling, mechanical wear, equipment failure, and fatigue are not easily modeled, but decisions about them can be based on qualitative reasoning. See Refs. 178 and 292. Qualitative description of physical behaviors require that each continuous variable space be quantized. Quantization is typically based on landmark values that are boundary points separating qualitatively distinct regions of continuous values. By using these qualitative quantity descriptions, dynamic relations between variables can be modeled as qualitative equations that represent the structure of the system. The

solution to the equations represents the possible sequences of qualitative states as well as the explanations for changes in behaviors. Building and explaining a complex model requires a unified view called an ontology. Methods of qualitative reasoning can be based on different viewpoints; the dominant viewpoints are device, process, and constraints. Behavior generation is handled with two approaches: (1) simulating successive states from one or more initial states, and (2) determining all possible state-to-state transitions once all possible states are determined. Fuzzy Logic Fuzzy logic is a formalism for mapping between numerical values and qualitative or linguistic interpretations. This is useful when it is difficult to define precisely such terms as “high” and “low,” since there may be no fixed threshold. Fuzzy sets use the concept of degree of membership to overcome this problem. Degree of membership allows a descriptor to be associated with a range of numeric values but in varying degrees. A fuzzy set is explicitly defined by a degree of membership for each linguistic variable that is applicable, mA(x) where mA is the degree of membership for linguistic variable A. For fuzzy sets, logical operators, such as complement (NOT), intersection (AND), and union (OR) are defined. The following are typical definitions. NOT: mNOT A(x) = 1 − mA(x) AND: mA AND B(x) = min [mA(x), mB(x)] OR: mA OR B(x) = max [mA(x), mB(x)] Using these operators, fuzzy inference mechanisms are then developed to manipulate rules that include fuzzy values. The largest difference between fuzzy inference and ordinary inference is that fuzzy inference allows “partial match” of input and produces an “interpolated” output. This technology is useful in control also. See Ref. 94. Artificial Neural Networks An artificial neural network (ANN) is a collection of computational units that are interconnected in a network. Knowledge is captured in the form of weights, and input-output mappings are produced by the interactions of the weights and the computational units. Each computational unit combines weighted inputs and generates an output base on an activation function. Typical activation functions are (1) specified limit, (2) sigmoid, and (3) gaussian. ANNs can be feedforward, with multiple layers of intermediate units, or feedback (sometimes called recurrent networks). The ability to generalize on given data is one of the most important performance characteristics. With appropriate selection of training examples, an optimal network architecture, and appropriate training, the network can map a relationship between input and output that is complete but bounded by the coverage of the training data. Applications of neural networks can be broadly classified into three categories: 1. Numeric-to-numeric transformations are used as empirical mathematical models where the adaptive characteristics of neural networks learn to map between numeric sets of input-output data. In these modeling applications, neural networks are used as an alternative to traditional data regression schemes based on regression of plant data. Backpropagation networks have been widely used for this purpose. 2. Numeric-to-symbolic transformations are used in patternrecognition problems where the network is used to classify input data vectors into specific labeled classes. Pattern recognition problems include data interpretation, feature identification, and diagnosis. 3. Symbolic-to-symbolic transformations are used in various symbolic manipulations, including natural language processing and rulebased system implementation. See Refs. 54 and 140.

Section 4

Thermodynamics

Hendrick C. Van Ness, D.Eng., Howard P. Isermann Department of Chemical Engineering, Rensselaer Polytechnic Institute; Fellow, American Institute of Chemical Engineers; Member, American Chemical Society Michael M. Abbott, Ph.D., Howard P. Isermann Department of Chemical Engineering, Rensselaer Polytechnic Institute; Member, American Institute of Chemical Engineers

INTRODUCTION Postulate 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Postulate 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Postulate 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Postulate 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Postulate 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4-3 4-3 4-4 4-4 4-4

VARIABLES, DEFINITIONS, AND RELATIONSHIPS Constant-Composition Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy and Entropy as Functions of T and P. . . . . . . . . . . . . . . . . . Internal Energy and Entropy as Functions of T and V. . . . . . . . . . . . Heat-Capacity Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Systems of Variable Composition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Molar Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gibbs/Duhem Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Molar Gibbs Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Ideal Gas State and the Compressibility Factor . . . . . . . . . . . . . Residual Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4-5 4-5 4-6 4-6 4-6 4-7 4-7 4-7 4-8 4-8 4-8

SOLUTION THERMODYNAMICS Ideal Gas Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fugacity and Fugacity Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fundamental Residual-Property Relation. . . . . . . . . . . . . . . . . . . . . . . . The Ideal Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fundamental Excess-Property Relation . . . . . . . . . . . . . . . . . . . . . . . . . Summary of Fundamental Property Relations . . . . . . . . . . . . . . . . . . . . Property Changes of Mixing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Behavior of Binary Liquid Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4-8 4-9 4-9 4-10 4-10 4-11 4-11 4-12

EVALUATION OF PROPERTIES Residual-Property Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid/Vapor Phase Transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4-14 4-15

Liquid-Phase Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties from PVT Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pitzer’s Corresponding-States Correlation . . . . . . . . . . . . . . . . . . . . . Alternative Property Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . Virial Equations of State. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generalized Correlation for the Second Virial Coefficient. . . . . . . . . Cubic Equations of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Benedict/Webb/Rubin Equation of State . . . . . . . . . . . . . . . . . . . . . . Expressions for the Excess Gibbs Energy . . . . . . . . . . . . . . . . . . . . . . . .

4-15 4-15 4-16 4-16 4-19 4-20 4-20 4-21 4-22

EQUILIBRIUM Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Phase Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 1: Application of the Phase Rule . . . . . . . . . . . . . . . . . . . . . Vapor/Liquid Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gamma/Phi Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data Reduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solute/Solvent Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K-Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equation-of-State Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid/Liquid and Vapor/Liquid/Liquid Equilibria . . . . . . . . . . . . . . . . Chemical-Reaction Stoichiometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chemical-Reaction Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Standard Property Changes of Reaction . . . . . . . . . . . . . . . . . . . . . . . Equilibrium Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 2: Single-Reaction Equilibrium . . . . . . . . . . . . . . . . . . . . . . Complex Chemical-Reaction Equilibria . . . . . . . . . . . . . . . . . . . . . . . Example 3: Minimization of Gibbs Energy . . . . . . . . . . . . . . . . . . . . .

4-24 4-24 4-25 4-25 4-25 4-26 4-27 4-28 4-28 4-30 4-31 4-31 4-31 4-32 4-32 4-33 4-34

THERMODYNAMIC ANALYSIS OF PROCESSES Calculation of Ideal Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lost Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analysis of Steady-State, Steady-Flow Processes . . . . . . . . . . . . . . . . . . Example 4: Lost-Work Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4-34 4-35 4-35 4-36

4-1

4-2

THERMODYNAMICS

Nomenclature and Units Symbols are omitted that are correlation- or application-specific. Symbol A âi B C D B′ C′ D′ Bij Cijk CP CV EK EP fi fˆi G g g H Ki Kj ki M

Mi Mi  ∆M ∆M°j m m ˙ n ni P Pc Pisat pi Q Q˙ R

U.S. customary units

Definition

SI units

Helmholtz energy Activity of species i in solution 2d virial coefficient, density expansion 3d virial coefficient, density expansion 4th virial coefficient, density expansion 2d virial coefficient, pressure expansion 3d virial coefficient, pressure expansion 4th virial coefficient, pressure expansion Interaction 2d virial coefficient Interaction 3d virial coefficient Heat capacity at constant pressure Heat capacity at constant volume Kinetic energy Gravitational potential energy Fugacity of pure species i Fugacity of species i in solution Molar or unit-mass Gibbs energy Acceleration of gravity
GE/RT Molar or unit-mass enthalpy

J Dimensionless

Equilibrium K-value, yi/xi Equilibrium constant for chemical reaction j Henry’s constant Molar or unit-mass value of any extensive thermodynamic property of a solution Molar or unit-mass value of any extensive property of pure species i Partial molar property of species i in solution Property change of mixing Standard property change of reaction j Mass Mass flow rate Number of moles Number of moles of species i Absolute pressure Critical pressure Saturation or vapor pressure of species i Partial pressure of species i in gas mixture (
yiP) Heat Rate of heat transfer Universal gas constant

Dimensionless Dimensionless

Btu/lb mol or Btu/lbm Dimensionless Dimensionless

kPa

psi

cm3/mol 6

Btu Dimensionless cm3/mol

2

cm /mol

cm6/mol2

cm9/mol3

cm9/mol3

kPa−1

kPa−1

kPa−2

kPa−2

−3

kPa

kPa−3

cm3/mol

cm3/mol

cm6/mol2

cm6/mol2

J/(mol⭈K)

Btu/(lb mol⭈R)

J/(mol⭈K)

Btu/(lb mol⭈R)

J J kPa kPa

Btu Btu psi psi

J/mol or J/kg

Btu/lb mol or Btu/lbm ft/s2

m/s2 J/mol or J/kg

Symbol

Definition

SI units

S

Molar or unit-mass entropy

T Tc U

Absolute temperature Critical temperature, Molar or unit-mass internal energy Velocity Molar or unit-mass volume

u V W Ws ˙s W xi

yi Z z

Work Shaft work for flow process Shaft power for flow process Mole fraction in general or liquid-phase mole fraction of species i in solution Vapor-phase mole fraction of species i in solution Compressibility factor Elevation above a datum level

J/(mol⭈K) or J/(kg⭈K) K K J/mol or J/kg

U.S. customary units

m/s m3/mol or m3/kg J J J/s Dimensionless

Btu/(lb mol⭈R) or Btu/(lb⭈R) R R Btu/lb mol or Btu/lbm ft/s ft3/lb mol or ft3/lbm Btu Btu Btu/s Dimensionless

Dimensionless

Dimensionless

Dimensionless m

Dimensionless ft

Superscripts E id ig l lv R t v ∞

Denotes excess thermodynamic property Denotes value for an ideal solution Denotes value for an ideal gas Denotes liquid phase Denotes phase transition from liquid to vapor Denotes residual thermodynamic property Denotes a total value of a thermodynamic property Denotes vapor phase Denotes a value at infinite dilution

C c H r rev

Denotes a value for a colder heat reservoir Denotes a value for the critical state Denotes a value for a hotter heat reservoir Denotes a reduced value Denotes a reversible process

α,β

As superscripts, identify phases Volume expansivity, species i Reaction coordinate for reaction j Defined by Eq. (4-72) Heat-capacity ratio, CP/CV Activity coefficient of species i in solution Chemical potential of species i Stoichiometric number of species i in reaction j Molar density As a subscript, denotes a heat reservoir Defined by Eq. (4-283) Fugacity coefficient of pure species i Fugacity coefficient of species i in solution Acentric factor

Subscripts

Greek letters

βi εj Γi(T) γ γi

kg kg/s

lbm lbm/s

kPa kPa kPa

psi psi psi

ρ σ

kPa

psi

Φi φi

J J/s J/(mol⭈K)

Btu Btu/s Btu/(lb mol⭈R)

µi νi, j

φˆ i ω

K−1 mol

R−1 lb mol

J/mol Dimensionless Dimensionless

Btu/lb mol Dimensionless Dimensionless

J/mol Dimensionless

Btu/lb mol Dimensionless

mols/m3

lb moles/ft3

Dimensionless Dimensionless

Dimensionless Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

GENERAL REFERENCES: Abbott, M.M., and H.C. Van Ness, Schaum’s Outline of Theory and Problems of Thermodynamics, 2d ed., McGraw-Hill, New York, 1989. Tester, J.W. and M. Modell, Thermodynamics and its Applications, 3d ed., Prentice-Hall, Englewood Cliffs, N.J., 1996. Prausnitz, J.M., R.N. Lichtenthaler, and E.G. de Azevedo, Molecular Thermodynamics of Fluid-Phase Equilibria, 2d ed., Prentice-Hall, Englewood Cliffs, N.J., 1986. Reid, R.C., J.M. Prausnitz, and B.E. Poling, The Properties of Gases and Liquids, 4th ed.,

McGraw-Hill, New York, 1987. Sandler, S.I., Chemical and Engineering Thermodynamics, 2d ed., Wiley, New York, 1989. Smith, J.M., H.C. Van Ness, and M.M. Abbott, Introduction to Chemical Engineering Thermodynamics, 5th ed., McGraw-Hill, New York, 1996. Van Ness, H.C., and M.M. Abbott, Classical Thermodynamics of Nonelectrolyte Solutions: With Applications to Phase Equilibria, McGraw-Hill, New York, 1982.

INTRODUCTION Thermodynamics is the branch of science that embodies the principles of energy transformation in macroscopic systems. The general restrictions which experience has shown to apply to all such transformations are known as the laws of thermodynamics. These laws are primitive; they cannot be derived from anything more basic. The first law of thermodynamics states that energy is conserved; that, although it can be altered in form and transferred from one place to another, the total quantity remains constant. Thus, the first law of thermodynamics depends on the concept of energy; but, conversely, energy is an essential thermodynamic function because it allows the first law to be formulated. This coupling is characteristic of the primitive concepts of thermodynamics. The words system and surroundings are similarly coupled. A system is taken to be any object, any quantity of matter, any region, and so on, selected for study and set apart (mentally) from everything else, which is called the surroundings. The imaginary envelope which encloses the system and separates it from its surroundings is called the boundary of the system. Attributed to this boundary are special properties which may serve either (1) to isolate the system from its surroundings, or (2) to provide for interaction in specific ways between system and surroundings. An isolated system exchanges neither matter nor energy with its surroundings. If a system is not isolated, its boundaries may permit exchange of matter or energy or both with its surroundings. If the exchange of matter is allowed, the system is said to be open; if only energy and not matter may be exchanged, the system is closed (but not isolated), and its mass is constant. When a system is isolated, it cannot be affected by its surroundings. Nevertheless, changes may occur within the system that are detectable with such measuring instruments as thermometers, pressure gauges, and so on. However, such changes cannot continue indefinitely, and the system must eventually reach a final static condition of internal equilibrium. For a closed system which interacts with its surroundings, a final static condition may likewise be reached such that the system is not only internally at equilibrium but also in external equilibrium with its surroundings. The concept of equilibrium is central in thermodynamics, for associated with the condition of internal equilibrium is the concept of state. A system has an identifiable, reproducible state when all its properties, such as temperature T, pressure P, and molar volume V, are fixed. The concepts of state and property are again coupled. One can equally well say that the properties of a system are fixed by its state. Although the properties T, P, and V may be detected with measuring instruments, the existence of the primitive thermodynamic properties (see Postulates 1 and 3 following) is recognized much more indirectly. The number of properties for which values must be specified in order to fix the state of a system depends on the nature of the system and is ultimately determined from experience. When a system is displaced from an equilibrium state, it undergoes a process, a change of state, which continues until its properties attain new equilibrium values. During such a process the system may be

caused to interact with its surroundings so as to interchange energy in the forms of heat and work and so to produce in the system changes considered desirable for one reason or another. A process that proceeds so that the system is never displaced more than differentially from an equilibrium state is said to be reversible, because such a process can be reversed at any point by an infinitesimal change in external conditions, causing it to retrace the initial path in the opposite direction. Thermodynamics finds its origin in experience and experiment, from which are formulated a few postulates that form the foundation of the subject. The first two deal with energy: POSTULATE 1 There exists a form of energy, known as internal energy, which for systems at internal equilibrium is an intrinsic property of the system, functionally related to its characteristic coordinates. POSTULATE 2 (FIRST LAW OF THERMODYNAMICS) The total energy of any system and its surroundings is conserved. Internal energy is quite distinct from such external forms as the kinetic and potential energies of macroscopic bodies. Although a macroscopic property characterized by the macroscopic coordinates T and P, internal energy finds its origin in the kinetic and potential energies of molecules and submolecular particles. In applications of the first law of thermodynamics, all forms of energy must be considered, including the internal energy. It is therefore clear that Postulate 2 depends on Postulate 1. For an isolated system, the first law requires that its energy be constant. For a closed (but not isolated) system, the first law requires that energy changes of the system be exactly compensated by energy changes in the surroundings. Energy is exchanged between such a system and its surroundings in two forms: heat and work. Heat is energy crossing the system boundary under the influence of a temperature difference or gradient. A quantity of heat Q represents an amount of energy in transit between a system and its surroundings, and is not a property of the system. The convention with respect to sign makes numerical values of Q positive when heat is added to the system and negative when heat leaves the system. Work is again energy in transit between a system and its surroundings, but resulting from the displacement of an external force acting on the system. Like heat, a quantity of work W represents an amount of energy, and is not a property of the system. The sign convention, analogous to that for heat, makes numerical values of W positive when work is done on the system by the surroundings and negative when work is done on the surroundings by the system. When applied to closed (constant-mass) systems for which the only form of energy that changes is the internal energy, the first law of thermodynamics is expressed mathematically as dU t = dQ + dW (4-1) 4-3

4-4

THERMODYNAMICS

where U t is the total internal energy of the system. Note that dQ and dW, differential quantities representing energy exchanges between the system and its surroundings, serve to account for the energy change of the surroundings. On the other hand, dU t is directly the differential change in internal energy of the system. Integration of Eq. (4-1) gives for a finite process ∆Ut = Q + W

(4-2)

where ∆U is the finite change given by the difference between the final and initial values of U t. The heat Q and work W are finite quantities of heat and work; they are not properties of the system nor functions of the thermodynamic coordinates that characterize the system.

encountered in chemical technology is one for which the primary characteristic variables are temperature T, pressure P, molar volume V, and composition, not all of which are necessarily independent. Such systems are usually made up of fluids (liquid or gas) and are called PVT systems. For closed systems of this kind, the work of a reversible process may always be calculated from dWrev = −P dV t

t

POSTULATE 3 There exists a property called entropy, which for systems at internal equilibrium is an intrinsic property of the system, functionally related to the measurable coordinates which characterize the system. For reversible processes, changes in this property may be calculated by the equation: dS t = dQrev /T

(4-3)

where St is the total entropy of the system and T is the absolute temperature of the system. POSTULATE 4 (SECOND LAW OF THERMODYNAMICS) The entropy change of any system and its surroundings, considered together, resulting from any real process is positive, approaching zero when the process approaches reversibility. In the same way that the first law of thermodynamics cannot be formulated without the prior recognition of internal energy as a property, so also the second law can have no complete and quantitative expression without a prior assertion of the existence of entropy as a property. The second law requires that the entropy of an isolated system either increase or, in the limit, where the system has reached an equilibrium state, remain constant. For a closed (but not isolated) system it requires that any entropy decrease in either the system or its surroundings be more than compensated by an entropy increase in the other part or that in the limit, where the process is reversible, the total entropy of the system plus its surroundings be constant. The fundamental thermodynamic properties that arise in connection with the first and second laws of thermodynamics are internal energy and entropy. These properties, together with the two laws for which they are essential, apply to all types of systems. However, different types of systems are characterized by different sets of measurable coordinates or variables. The type of system most commonly

(4-4)

t

where P is the absolute pressure and V is the total volume of the system. This equation follows directly from the definition of mechanical work. POSTULATE 5 The macroscopic properties of homogeneous PVT systems at internal equilibrium can be expressed as functions of temperature, pressure, and composition only. This postulate imposes an idealization, and is the basis for all subsequent property relations for PVT systems. The PVT system serves as a satisfactory model in an enormous number of practical applications. In accepting this model one assumes that the effects of fields (e.g., electric, magnetic, or gravitational) are negligible and that surface and viscous-shear effects are unimportant. Temperature, pressure, and composition are thermodynamic coordinates representing conditions imposed upon or exhibited by the system, and the functional dependence of the thermodynamic properties on these conditions is determined by experiment. This is quite direct for molar or specific volume V, which can be measured, and leads immediately to the conclusion that there exists an equation of state relating molar volume to temperature, pressure, and composition for any particular homogeneous PVT system. The equation of state is a primary tool in applications of thermodynamics. Postulate 5 affirms that the other molar or specific thermodynamic properties of PVT systems, such as internal energy U and entropy S, are also functions of temperature, pressure, and composition. These molar or unit-mass properties, represented by the plain symbols V, U, and S, are independent of system size and are called intensive. Temperature, pressure, and the composition variables, such as mole fraction, are also intensive. Total-system properties (V t, U t, St ) do depend on system size, and are extensive. For a system containing n moles of fluid, M t = nM, where M is a molar property. Applications of the thermodynamic postulates necessarily involve the abstract quantities internal energy and entropy. The solution of any problem in applied thermodynamics is therefore found through these quantities.

VARIABLES, DEFINITIONS, AND RELATIONSHIPS Consider a single-phase closed system in which there are no chemical reactions. Under these restrictions the composition is fixed. If such a system undergoes a differential, reversible process, then by Eq. (4-1) dU t = dQrev + dWrev

where i is an index identifying the chemical species present. When U, S, and V represent specific (unit-mass) properties, n is replaced by m. Equation (4-5) shows that for the single-phase, nonreacting, closed system specified, nU = u(nS, nV)

Substitution for dQrev and dWrev by Eqs. (4-3) and (4-4) gives dU t = T dS t − P dV t

Then

Although derived for a reversible process, this equation relates properties only and is valid for any change between equilibrium states in a closed system. It may equally well be written d(nU) = T d(nS) − P d(nV)

(4-5)

where n is the number of moles of fluid in the system and is constant for the special case of a closed, nonreacting system. Note that n
n1 + n2 + n3 + ⋅⋅⋅ =  ni i

∂(nU) d(nU) = ᎏ ∂(nS)





nV,n

∂(nU) d(nS) + ᎏ ∂(nV)





d(nV) nS,n

where the subscript n indicates that all mole numbers ni (and hence n) are held constant. Comparison with Eq. (4-5) shows that ∂(nU) =T (4-6) ᎏ ∂(nS) nV,n





∂(nU) ᎏ ∂(nV) 

= −P nS,n

(4-7)

VARIABLES, DEFINITIONS, AND RELATIONSHIPS Consider now an open system consisting of a single phase and assume that nU =
(nS, nV, n1, n2, n3, . . .) Then ∂(nU) ∂(nU) ∂(nU) d(nS) + ᎏ d(nV) +  ᎏ dni d(nU) = ᎏ ∂(nS) nV,n ∂(nV) nS,n ∂ni nS,nV,n i where the summation is over all species present in the system and subscript nj indicates that all mole numbers are held constant except the ith. Let ∂(nU) µi
ᎏ ∂ni nS,nV,nj















j



Together with Eqs. (4-6) and (4-7), this definition allows elimination of all the partial differential coefficients from the preceding equation: d(nU) = T d(nS) − P d(nV) +  µi dni

(4-8)

i

Equation (4-8) is the fundamental property relation for singlephase PVT systems, from which all other equations connecting properties of such systems are derived. The quantity µi is called the chemical potential of species i, and it plays a vital role in the thermodynamics of phase and chemical equilibria. Additional property relations follow directly from Eq. (4-8). Since ni = xin, where xi is the mole fraction of species i, this equation may be rewritten: d(nU) − T d(nS) + P d(nV) −  µi d(xin) = 0 i

Upon expansion of the differentials and collection of like terms, this becomes



 



dU − T dS + P dV −  µi dxi n + U − TS + PV −  xiµi dn = 0 i

i

Since n and dn are independent and arbitrary, the terms in brackets must separately be zero. Then dU = T dS − P dV +  µi dxi

(4-9)

i

U = TS − PV +  xiµi

(4-10)

i

Equations (4-8) and (4-9) are similar, but there is an important difference. Equation (4-8) applies to a system of n moles where n may vary; whereas Eq. (4-9) applies to a system in which n is unity and invariant. Thus Eq. (4-9) is subject to the constraint that i xi = 1 or that i dxi = 0. In this equation the xi are not independent variables, whereas the ni in Eq. (4-8) are. Equation (4-10) dictates the possible combinations of terms that may be defined as additional primary functions. Those in common use are: Enthalpy Helmholtz energy Gibbs energy

H
U + PV A
U − TS G
U + PV − TS = H − TS

(4-11) (4-12) (4-13)

Additional thermodynamic properties are related to these and arise by arbitrary definition. Multiplication of Eq. (4-11) by n and differentiation yields the general expression: d(nH) = d(nU) + P d(nV) + nV dP d(nH) = T d(nS) + nV dP +  µi dni

and so on, as a function of a particular set of independent variables; these are the canonical variables for the property. The choice of which equation to use in a particular application is dictated by convenience. However, the Gibbs energy G is special, because of its unique functional relation to T, P, and the ni, which are the variables of primary interest in chemical processing. A similar set of equations is developed from Eq. (4-9). This set also follows from the preceding set when n = 1 and ni = xi. The two sets are related exactly as Eq. (4-8) is related to Eq. (4-9). The equations written for n = 1 are, of course, less general. Furthermore, the interdependence of the xi precludes those mathematical operations which depend on independence of these variables. CONSTANT-COMPOSITION SYSTEMS For 1 mole of a homogeneous fluid of constant composition Eqs. (4-8) and (4-14) through (4-16) simplify to: dU = T dS − P dV dH = T dS + V dP dA = −S dT − P dV dG = −S dT + V dP Implicit in these are the following: ∂U ∂H T= ᎏ = ᎏ ∂S V ∂S

    ∂U ∂A −P =  ᎏ  =  ᎏ  ∂V ∂V ∂H ∂G V = ᎏ = ᎏ ∂P ∂P

(4-14)

i

The total differentials of nA and nG are obtained similarly: d(nA) = −nS dT − P d(nV) +  µi dni

(4-15)

d(nG) = −nS dT + nV dP +  µi dni

(4-16)

i

i

Equations (4-8) and (4-14) through (4-16) are equivalent forms of the fundamental property relation. Each expresses a property nU, nH,

(4-17) (4-18) (4-19) (4-20)

(4-21)

P

S

T

S

T

(4-22) (4-23)

∂A ∂G (4-24) −S = ᎏ = ᎏ ∂T V ∂T P In addition, the common Maxwell equations result from application of the reciprocity relation for exact differentials: ∂T ∂P (4-25) ᎏ =− ᎏ ∂V S ∂S V

   

    ∂T ∂V = ᎏ ᎏ ∂P   ∂S  S

∂P ∂S = ᎏ ᎏ ∂T   ∂V  V

(4-26) P

(4-27) T

∂V ∂S = − ᎏ  ᎏ ∂T  ∂P P

(4-28)

T

In all these equations the partial derivatives are taken with composition held constant. Enthalpy and Entropy as Functions of T and P At constant composition the molar thermodynamic properties are functions of temperature and pressure (Postulate 5). Thus ∂H ∂H dH = ᎏ dT + ᎏ dP (4-29) ∂T P ∂P T

 

 

∂S ∂S (4-30) dS = ᎏ dT + ᎏ dP ∂T P ∂P T The obvious next step is to eliminate the partial-differential coefficients in favor of measurable quantities. The heat capacity at constant pressure is defined for this purpose: ∂H CP
ᎏ (4-31) ∂T P It is a property of the material and a function of temperature, pressure, and composition. Equation (4-18) may first be divided by dT and restricted to constant pressure, and then be divided by dP and restricted to constant temperature, yielding the two equations:

 

Substitution for d(nU) by Eq. (4-8) reduces this result to:

4-5

 

 

4-6

THERMODYNAMICS ∂H

∂S = T ᎏ ᎏ ∂T  ∂T P

P

∂H ∂S ᎏ =T ᎏ +V ∂P T ∂P T In view of Eq. (4-31), the first of these becomes ∂S CP (4-32) ᎏ =ᎏ ∂T P T and in view of Eq. (4-28), the second becomes ∂H ∂V (4-33) ᎏ =V−T ᎏ ∂P T ∂T P Combination of Eqs. (4-29), (4-31), and (4-33) gives ∂V dH = CP dT + V − T ᎏ dP (4-34) ∂T P and in combination Eqs. (4-30), (4-32), and (4-28) yield CP ∂V dS = ᎏ dT − ᎏ dP (4-35) T ∂T P Equations (4-34) and (4-35) are general expressions for the enthalpy and entropy of homogeneous fluids at constant composition as functions of T and P. The coefficients of dT and dP are expressed in terms of measurable quantities. Internal Energy and Entropy as Functions of T and V Because V is related to T and P through an equation of state, V rather than P can serve as an independent variable. In this case the internal energy and entropy are the properties of choice; whence ∂U ∂U dU = ᎏ dT + ᎏ dV (4-36) ∂T V ∂V T

 

 

 

 

 



 

 

 

 

∂S ∂S (4-37) dS = ᎏ dT + ᎏ dV ∂T V ∂V T The procedure now is analogous to that of the preceding section. Define the heat capacity at constant volume by ∂U CV
ᎏ (4-38) ∂T V It is a property of the material and a function of temperature, pressure, and composition. Two relations follow immediately from Eq. (4-17): ∂U ∂S ᎏ =T ᎏ ∂T V ∂T V

 

 

 

    ∂U ∂S = T ᎏ − P ᎏ ∂V  ∂V T

T

As a result of Eq. (4-38) the first of these becomes CV ∂S (4-39) ᎏ =ᎏ ∂T V T and as a result of Eq. (4-27), the second becomes ∂U ∂P (4-40) ᎏ =T ᎏ −P ∂V T ∂T V Combination of Eqs. (4-36), (4-38), and (4-40) gives ∂P dU = CV dT + T ᎏ − P dV (4-41) ∂T V and Eqs. (4-37), (4-39), and (4-27) together yield ∂P CV dS = ᎏ dT + ᎏ dV (4-42) T ∂T V Equations (4-41) and (4-42) are general expressions for the internal energy and entropy of homogeneous fluids at constant composition as functions of temperature and molar volume. The coefficients of dT and dV are expressed in terms of measurable quantities.

 

 

 

     

Heat-Capacity Relations In Eqs. (4-34) and (4-41) both dH and dU are exact differentials, and application of the reciprocity relation leads to ∂CP ∂2V (4-43) ᎏ = −T ᎏ2 ∂P T ∂T P

 

 

∂CV

∂2P

= T ᎏ ᎏ ∂V  ∂T 2

T

(4-44)

V

Thus, the pressure or volume dependence of the heat capacities may be determined from PVT data. The temperature dependence of the heat capacities is, however, determined empirically and is often given by equations such as CP = α + βT + γ T 2 Equations (4-35) and (4-42) both provide expressions for dS, which must be equal for the same change of state. Equating them and solving for dT gives T ∂V T ∂P dT = ᎏ ᎏ dP + ᎏ ᎏ dV CP − CV ∂T P CP − CV ∂T V However, at constant composition T = T(P,V), and ∂T ∂T dT = ᎏ dP + ᎏ dV ∂P V ∂V P Equating coefficients of either dP or dV in these two expressions for dT gives ∂V ∂P CP − CV = T ᎏ ᎏ (4-45) ∂T P ∂T V Thus the difference between the two heat capacities may be determined from PVT data. Division of Eq. (4-32) by Eq. (4-39) yields the ratio of these heat capacities: CP (∂S/∂T)P (∂S/∂V)P(∂V/∂T)P ᎏ = ᎏ = ᎏᎏ CV (∂S/∂T)V (∂S/∂P)V(∂P/∂T)V

 

 

 

 

  

Replacement of each of the four partial derivatives through the appropriate Maxwell relation gives finally C ∂V ∂P γ
ᎏP = ᎏ (4-46) ᎏ CV ∂P T ∂V S where γ is the symbol conventionally used to represent the heatcapacity ratio. The Ideal Gas The simplest equation of state is the ideal gas equation: PV = RT

  

where R is a universal constant, values of which are given in Table 1-9. The following partial derivatives are obtained from the ideal gas equation: ∂P R P ∂2P ᎏ =ᎏ=ᎏ ᎏ2 = 0 ∂T V V T ∂T V

 

 

∂V R V ∂V =ᎏ=ᎏ =0 ᎏ ᎏ ∂T  P T ∂T  2

2

P

P

∂P P ᎏ =−ᎏ ∂V T V The general equations for constant-composition fluids derived in the preceding subsections reduce to very simple forms when the relations for an ideal gas are substituted into them:

 

∂U ∂H = ᎏ =0 ᎏ ∂V   ∂P  T

T

∂S R ∂S R =−ᎏ =ᎏ ᎏ ᎏ ∂P  P ∂V  V T

dU = CV dT

T

VARIABLES, DEFINITIONS, AND RELATIONSHIPS dH = CP dT CV R dS = ᎏ dT + ᎏ dV T V

 

 

 

 

Combining these expressions with Eq. (4-48) and collecting like terms gives ∂M ∂M dM − ᎏ dT − ᎏ dP −   Mi dxi n + M −   Mi xi dn = 0 ∂T P,x ∂P T,x i i



CP R dS = ᎏ dT − ᎏ dP T P ∂CV

∂CP

∂ ln P C γ
ᎏP = − ᎏ CV ∂ ln V





and

SYSTEMS OF VARIABLE COMPOSITION The composition of a system may vary because the system is open or because of chemical reactions even in a closed system. The equations developed here apply regardless of the cause of composition changes. Partial Molar Properties Consider a homogeneous fluid solution comprised of any number of chemical species. For such a PVT system let the symbol M represent the molar (or unit-mass) value of any extensive thermodynamic property of the solution, where M may stand in turn for U, H, S, and so on. A total-system property is then nM, where n =  ini and i is the index identifying chemical species. One might expect the solution property M to be related solely to the properties Mi of the pure chemical species which comprise the solution. However, no such generally valid relation is known, and the connection must be established experimentally for every specific system. Although the chemical species which make up a solution do not in fact have separate properties of their own, a solution property may be arbitrarily apportioned among the individual species. Once an apportioning recipe is adopted, then the assigned property values are quite logically treated as though they were indeed properties of the species in solution, and reasoning on this basis leads to valid conclusions. For a homogeneous PVT system, Postulate 5 requires that nM = (T, P, n1, n2, n3 , . . .) The total differential of nM is therefore





P,n

∂(nM) dT + ᎏ ∂P





T,n

∂(nM) dP +  ᎏ ∂ni i





dni T,P,nj

where subscript n indicates that all mole numbers ni are held constant, and subscript nj signifies that all mole numbers are held constant except the ith. This equation may also be written ∂M d(nM) = n ᎏ ∂T

 

P,x

∂M dT + n ᎏ ∂P

 

T,x

∂(nM) dP +  ᎏ ∂ni i





dni T,P,nj

where subscript x indicates that all mole fractions are held constant. The derivatives in the summation are called partial molar properties M i; by definition, ∂(nM) M (4-47) i
ᎏ ∂ni T,P,nj





The basis for calculation of partial properties from solution properties is provided by this equation. Moreover, the preceding equation becomes ∂M ∂M d(nM) = n ᎏ dT + n ᎏ dP +  M (4-48) i dni ∂T P,x ∂P T,x i

 

 

 

Important equations follow from this result through the relations: d(nM) = n dM + M dn dni = d(xin) = xi dn + n dxi



and arbitrary, the terms in brackets ∂M

ᎏ ∂P 

T,x

dP +  M i dxi

(4-49)

i

M =  xiM i

(4-50)

i

S

These equations clearly show that for an ideal gas U, H, CP, and CV are functions of temperature only and are independent of P and V. The entropy of an ideal gas, however, is a function of both T and P or of both T and V.

∂(nM) d(nM) = ᎏ ∂T

 

 

T

CP − CV = R

 

Since n and dn are independent must separately be zero; whence ∂M dM = ᎏ dT + ∂T P,x

= ᎏ =0 ᎏ ∂V   ∂P  T

4-7

Equation (4-49) is merely a special case of Eq. (4-48); however, Eq. (4-50) is a vital new relation. Known as the summability equation, it provides for the calculation of solution properties from partial properties. Thus, a solution property apportioned according to the recipe of Eq. (4-47) may be recovered simply by adding the properties attributed to the individual species, each weighted by its mole fraction in solution. The equations for partial molar properties are also valid for partial specific properties, in which case m replaces n and the xi are mass fractions. Equation (4-47) applied to the definitions of Eqs. (4-11) through (4-13) yields the partial-property relations: Hi =  Ui + PV  i A Ui − TS i i =  G Hi − TS i i =  Pertinent examples on partial molar properties are presented in Smith, Van Ness, and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed., Sec. 10.3, McGraw-Hill, New York, 1996). Gibbs/Duhem Equation Differentiation of Eq. (4-50) yields dM =  xi dM i +  M i dxi i

i

Since this equation and Eq. (4-49) are both valid in general, their right-hand sides can be equated, yielding ∂M ∂M (4-51) ᎏ dT + ᎏ dP −  xi dM i = 0 ∂T P,x ∂P T,x i

 

 

This general result, the Gibbs/Duhem equation, imposes a constraint on how the partial molar properties of any phase may vary with temperature, pressure, and composition. For the special case where T and P are constant:  =0  x dM i

(constant T, P)

i

(4-52)

i

Symbol M may represent the molar value of any extensive thermodynamic property; for example, V, U, H, S, or G. When M
H, the derivatives (∂H/∂T)P and (∂H/∂P)T are given by Eqs. (4-31) and (4-33). Equations (4-49), (4-50), and (4-51) then become ∂V dH = CP dT + V − T ᎏ dP +  H (4-53) i dxi ∂T P,x i



 

H =  xiH i

(4-54)

i

∂V CP dT + V − T ᎏ dP −  xi dH (4-55) i = 0 ∂T P,x i Similar equations are readily derived when M takes on other identities. Equation (4-47), which defines a partial molar property, provides a general means by which partial property values may be determined. However, for a binary solution an alternative method is useful. Equation (4-50) for a binary solution is



 

M = x1 M 1 + x2 M 2

(4-56)

Moreover, the Gibbs/Duhem equation for a solution at given T and P, Eq. (4-52), becomes x1 dM 1 + x2 dM 2 = 0

(4-57)

4-8

THERMODYNAMICS

These two equations can be combined to give dM M 1 = M + x2 ᎏ dx1

The following equation is a mathematical identity:

dM M2 = M − x1 ᎏ  dx1

nG 1 nG d ᎏ
ᎏ d(nG) − ᎏ2 dT RT RT RT

(4-58b)

Substitution for d(nG) by Eq. (4-16) and for G by H − TS (Eq. [4-13]) gives, after algebraic reduction,

Thus for a binary solution, the partial properties are given directly as functions of composition for given T and P. For multicomponent solutions such calculations are complex, and direct use of Eq. (4-47) is appropriate. Partial Molar Gibbs Energy Implicit in Eq. (4-16) is the relation ∂(nG) µi = ᎏ ∂ni T,P,nj





In view of Eq. (4-47), the chemical potential and the partial molar Gibbs energy are therefore identical: µi =  Gi

(4-59)

The reciprocity relation for an exact differential applied to Eq. (416) produces not only the Maxwell relation, Eq. (4-28), but also two other useful equations: ∂µi ∂(nV) = Vi (4-60) ᎏ = ᎏ ∂P T,n ∂ni T,P,nj

    ∂µ ∂(nS) = − ᎏ  ᎏ ∂T  ∂n i

P,n

i

T,P,nj

= −S i

(4-61)

In a solution of constant composition, µi = µ(T,P); whence ∂µi ∂µi dµi
dG i = ᎏ dT + ᎏ dP ∂T P,n ∂P T,n

 

or

 

dG i = −S i dT + V i dP

(4-62)

Comparison with Eq. (4-20) provides an example of the parallelism that exists between the equations for a constant-composition solution and those for the corresponding partial properties. This parallelism exists whenever the solution properties in the parent equation are related linearly (in the algebraic sense). Thus, in view of Eqs. (4-17), (4-18), and (4-19): dU i = T dS i − P dV i Vi dP dH i = T dS i +  dA i = −S i dT − P dV i

 

(4-58a)

(4-63) (4-64) (4-65)

nG nV nH µi d ᎏ = ᎏ dP − ᎏ2 dT +  ᎏ dni RT RT RT RT i

 

(4-66)

Equation (4-66) is a useful alternative to the fundamental property relation given by Eq. (4-16). All terms in this equation have the units of moles; moreover, the enthalpy rather than the entropy appears on the right-hand side. The Ideal Gas State and the Compressibility Factor The simplest equation of state for a PVT system is the ideal gas equation: PV ig = RT ig

where V is the ideal-gas–state molar volume. Similarly, H ig, S ig, and G ig are ideal gas–state values; that is, the molar enthalpy, entropy, and Gibbs energy values that a PVT system would have were the ideal gas equation the correct equation of state. These quantities provide reference values to which actual values may be compared. For example, the compressibility factor Z compares the true molar volume to the ideal gas molar volume as a ratio: V V PV Z=ᎏ =ᎏ=ᎏ V ig RT/P RT Generalized correlations for the compressibility factor are treated in Sec. 2. Residual Properties These quantities compare true and ideal gas properties through differences: MR
M − M ig

(4-67)

where M is the molar value of an extensive thermodynamic property of a fluid in its actual state and Mig is the corresponding value for the ideal gas state of the fluid at the same T, P, and composition. Residual properties depend on interactions between molecules and not on characteristics of individual molecules. Since the ideal gas state presumes the absence of molecular interactions, residual properties reflect deviations from ideality. Most commonly used of the residual properties are: V R
V − V ig HR
H − H ig SR
S − S ig GR
G − G ig

Residual volume Residual enthalpy Residual entropy Residual Gibbs energy

Note that these equations hold only for species in a constantcomposition solution.

SOLUTION THERMODYNAMICS M iig(T, P) = Miig(T, pi)

IDEAL GAS MIXTURES An ideal gas is a model gas comprising imaginary molecules of zero volume that do not interact. Each chemical species in an ideal gas mixture therefore has its own private properties, uninfluenced by the presence of other species. The partial pressure of species i in a gas mixture is defined as pi = xiP

The partial molar property, other than the volume, of a constituent species in an ideal gas mixture is equal to the corresponding molar property of the species as a pure ideal gas at the mixture temperature but at a pressure equal to its partial pressure in the mixture.

This is expressed mathematically for generic partial property M iig by the equation

(4-68)

Miig = Miig  ig

where Mi is evaluated at the mixture T and P. Thus, for the enthalpy,

(i = 1, 2, . . . , N)

where xi is the mole fraction of species i. The sum of the partial pressures clearly equals the total pressure. Gibbs’ theorem for a mixture of ideal gases may be stated as follows:

(M ≠ V)

For those properties of an ideal gas that are independent of P, for example, U, H, and CP, this becomes simply

Hiig = Hiig  The entropy of an ideal gas does depend on pressure: dSiig = −R d ln P

(constant T )

Integration from pi to P gives P P Siig(T, P) − Siig(T, pi) = −R ln ᎏ = −R ln ᎏ = R ln xi pi xi P Whence

Siig(T, pi) = Siig(T, P) − R ln xi

(4-69)

SOLUTION THERMODYNAMICS Substituting this result into Eq. (4-68) written for the entropy gives Sl = S − R ln xi  ig

ig i

(4-70)

where Siig is evaluated at the mixture T and P. For the Gibbs energy of an ideal gas mixture, Gig = Hig − TSig; the parallel relation for partial properties is Giig =  Hiig − TSiig In combination with Eqs. (4-69) and (4-70), this becomes G Hiig − TSiig + RT ln xi iig =  or

µiig
 Giig = Giig + RT ln xi

(4-72)

where Γi(T), the integration constant for a given temperature, is a function of temperature only. Equation (4-71) now becomes µigi = Γi(T) + RT ln xi P

(4-73)

FUGACITY AND FUGACITY COEFFICIENT The chemical potential µi plays a vital role in both phase and chemicalreaction equilibria. However, the chemical potential exhibits certain unfortunate characteristics which discourage its use in the solution of practical problems. The Gibbs energy, and hence µi, is defined in relation to the internal energy and entropy, both primitive quantities for which absolute values are unknown. Moreover, µi approaches negative infinity when either P or xi approaches zero. While these characteristics do not preclude the use of chemical potentials, the application of equilibrium criteria is facilitated by introduction of the fugacity, a quantity that takes the place of µi but which does not exhibit its less desirable characteristics. The origin of the fugacity concept resides in Eq. (4-72), an equation valid only for pure species i in the ideal gas state. For a real fluid, an analogous equation is written: Gi
Γi(T) + RT ln f i

(4-74)

in which a new property f i replaces the pressure P. This equation serves as a partial definition of the fugacity f i. Subtraction of Eq. (4-72) from Eq. (4-74), both written for the same temperature and pressure, gives f Gi − Giig = RT ln ᎏi P According to the definition of Eq. (4-67), Gi − Giig is the residual Gibbs energy, GiR. The dimensionless ratio fi /P is another new property called the fugacity coefficient φi. Thus, (4-75) G = RT ln φi fi where φi
ᎏ (4-76) P The definition of fugacity is completed by setting the ideal-gas–state fugacity of pure species i equal to its pressure: R i

f iig = P Thus, for the special case of an ideal gas, GiR = 0, φi = 1, and Eq. (4-72) is recovered from Eq. (4-74). The definition of the fugacity of a species in solution is parallel to the definition of the pure-species fugacity. An equation analogous to the ideal gas expression, Eq. (4-73), is written for species i in a fluid mixture: µ
Γ (T) + RT ln fˆ (4-77) i

i

i in solution. Since it is not a partial molar property, it is identified by a circumflex rather than an overbar. Subtracting Eq. (4-73) from Eq. (4-77), both written for the same temperature, pressure, and composition, yields fˆi µi − µiig = RT ln ᎏ xiP Analogous to the defining equation for the residual Gibbs energy of a mixture, GR
G − Gig, is the definition of a partial molar residual Gibbs energy: G Gi −  Gigi = µi − µigi iR


(4-71)

Elimination of Giig from this equation is accomplished by Eq. (4-20), written for pure species i as: RT dGiig = Viig dP = ᎏ dP = RT d ln P (constant T) P Integration gives Giig = Γi(T) + RT ln P

4-9

i

where the partial pressure xi P is replaced by fˆi , the fugacity of species

GiR = RT ln φˆ i (4-78)  ˆ f i where by definition φˆ i
ᎏ (4-79) xiP The dimensionless ratio φˆ i is called the fugacity coefficient of species i in solution. Eq. (4-78) is the analog of Eq. (4-75), which relates φi to GiR. For an ideal gas,  GiR is necessarily 0; therefore φˆ iig = 1, and fˆ ig = x P Therefore

i

i

Thus, the fugacity of species i in an ideal gas mixture is equal to its partial pressure. Pertinent examples are given in Smith, Van Ness, and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed., Secs. 10.5–10.7, McGraw-Hill, New York, 1996). FUNDAMENTAL RESIDUAL-PROPERTY RELATION In view of Eq. (4-59), the fundamental property relation given by Eq. (4-66) may be written Gi  nH nG nV d ᎏ = ᎏ dP − ᎏ2 dT +  ᎏ dni (4-80) RT RT RT RT i

 

This equation is general, and may be written for the special case of an ideal gas: G nGig nVig nHig iig d ᎏ = ᎏ dP − ᎏ2 dT +  ᎏ dni RT RT RT RT i

 

Subtraction of this equation from Eq. (4-80) gives GiR nGR nVR nHR  d ᎏ = ᎏ dP − ᎏ2 dT +  ᎏ dni RT RT RT RT i

 

(4-81)

where the definitions GR
G − Gig and G iR
G i − G iig have been imposed. Equation (4-81) is the fundamental residual-property relation. An alternative form follows by introduction of the fugacity coefficient as given by Eq. (4-78): nGR nVR nHR d ᎏ = ᎏ dP − ᎏ2 dT +  ln φˆ i dni (4-82) RT RT RT i

 

These equations are of such generality that for practical application they are used only in restricted forms. Division of Eq. (4-82) by dP and restriction to constant T and composition leads to: VR ∂(GR/RT) ᎏ = ᎏᎏ RT ∂P





(4-83)

T,x

Similarly, division by dT and restriction to constant P and composition gives HR ∂(GR/RT) ᎏ = −T ᎏᎏ RT ∂T





(4-84)

P,x

Also implicit in Eq. (4-82) is the relation ∂(nGR/RT) ln φˆ i = ᎏᎏ (4-85) T,P,nj ∂ni This equation demonstrates that ln φˆ i is a partial property with respect to GR/RT. The partial-property analogs of Eqs. (4-83) and (4-84) are therefore:





4-10

THERMODYNAMICS ∂ ln φ ᎏ ∂P  ∂ ln φˆ ᎏ ∂T  ˆi

ViR  =ᎏ T,x RT

(4-86)

H iR = − ᎏ2 (4-87) P,x RT The partial-property relationship of ln φˆ i to GR/RT also means that the summability relation applies; thus GR (4-88) ᎏ =  xi ln φˆ i RT i i

THE IDEAL SOLUTION The ideal gas is a useful model of the behavior of gases and serves as a standard to which real gas behavior can be compared. This is formalized by the introduction of residual properties. Another useful model is the ideal solution, which serves as a standard to which real solution behavior can be compared. This is formalized by introduction of excess properties. The partial molar Gibbs energy of species i in an ideal gas mixture is given by Eq. (4-71). This equation takes on new meaning when Giig, the Gibbs energy of pure species i in the ideal gas state, is replaced by Gi, the Gibbs energy of pure species i as it actually exists at the mixture T and P and in the same physical state (real gas, liquid, or solid) as the mixture. It then becomes applicable to species in real solutions; indeed, to liquids and solids as well as to gases. The ideal solution is therefore defined as one for which G iid
Gi + RT ln xi

(4-89)

where superscript id denotes an ideal-solution property. This equation is the basis for development of expressions for all other thermodynamic properties of an ideal solution. Equations (4-60) and (4-61), applied to an ideal solution with µi replaced by  Gi, can be written id id ∂G ∂G i i V and Siid = − ᎏ iid = ᎏ  ∂P T,x ∂T P,x Appropriate differentiation of Eq. (4-89) in combination with these relations and Eqs. (4-23) and (4-24) yields









id i

Thus, the fugacity coefficient of species i in an ideal solution is equal to the fugacity coefficient of pure species i in the same physical state as the solution and at the same T and P. Ideal solution behavior is often approximated by solutions comprised of molecules not too different in size and of the same chemical nature. Thus, a mixture of isomers conforms very closely to ideal solution behavior. So do mixtures of adjacent members of a homologous series. FUNDAMENTAL EXCESS-PROPERTY RELATION The residual Gibbs energy and the fugacity coefficient are useful where experimental PVT data can be adequately correlated by equations of state. Indeed, if convenient treatment of all fluids by means of equations of state were possible, the thermodynamic-property relations already presented would suffice. However, liquid solutions are often more easily dealt with through properties that measure their deviations from ideal solution behavior, not from ideal gas behavior. Thus, the mathematical formalism of excess properties is analogous to that of the residual properties. If M represents the molar (or unit-mass) value of any extensive thermodynamic property (e.g., V, U, H, S, G, and so on), then an excess property M E is defined as the difference between the actual property value of a solution and the value it would have as an ideal solution at the same temperature, pressure, and composition. Thus, M E
M − Mid

(4-99)

This definition is analogous to the definition of a residual property as given by Eq. (4-67). However, excess properties have no meaning for pure species, whereas residual properties exist for pure species as well as for mixtures. In addition, analogous to Eq. (4-99) is the partialproperty relation, MiE =  Mi −  Miid (4-100) 

Gid =  xiGi + RT  xi ln xi

(4-93)

Vid =  xiVi

(4-94)

where  M iE is a partial excess property. The fundamental excessproperty relation is derived in exactly the same way as the fundamental residual-property relation and leads to analogous results. Equation (4-80), written for the special case of an ideal solution, is subtracted from Eq. (4-80) itself, yielding: GiE nGE nV E nH E  d ᎏ = ᎏ dP − ᎏ2 dT +  ᎏ dni (4-101) RT RT RT RT i This is the fundamental excess-property relation, analogous to Eq. (4-81), the fundamental residual-property relation. The excess Gibbs energy is of particular interest. Equation (4-77) may be written: G  = Γ (T ) + RT ln fˆ

S =  xiSi − R  xi ln xi

(4-95)

In accord with Eq. (4-97) for an ideal solution, this becomes

Hid =  xiHi

(4-96)

Viid = Vi 

(4-90)

S iid = Si − R ln xi

(4-91)

G + TS Since H i =   , substitutions by Eqs. (4-89) and (4-91) yield id

same T and P. Division of both sides of Eq. (4-97) by xiP and substitution of φˆ iid for fˆiid/xiP (Eq. [4-79]) and of φi for fi /P (Eq. [4-76]) gives an alternative form: φˆ iid = φi (4-98)

id i

Hiid = Hi 

(4-92)

The summability relation, Eq. (4-50), written for the special case of an ideal solution, may be applied to Eqs. (4-89) through (4-92): i

i

i

id

i

i

i

i

i

i

When this equation and Eq. (4-74) are combined with Eq. (4-89), Γi(T) is eliminated, and the resulting expression reduces to fˆiid = xi fi (4-97) This equation, known as the Lewis/Randall rule, applies to each species in an ideal solution at all conditions of T, P, and composition. It shows that the fugacity of each species in an ideal solution is proportional to its mole fraction; the proportionality constant is the fugacity of pure species i in the same physical state as the solution and at the

i

i

G iid = Γi (T ) + RT ln xi fi

i

A simple equation for the fugacity of a species in an ideal solution follows from Eq. (4-89). Written for the special case of species i in an ideal solution, Eq. (4-77) becomes µ id
 G id = Γ (T) + RT ln fˆid i

 

By difference

fˆi G Giid = RT ln ᎏ i −  xi fi The left-hand side is the partial excess Gibbs energy  GiE; the dimensionless ratio fˆi/xifi appearing on the right is called the activity coefficient of species i in solution, and is given the symbol γi. Thus, by definition, fˆi γi
ᎏ (4-102) xi fi and

G iE = RT ln γi

(4-103)

Comparison with Eq. (4-78) shows that Eq. (4-103) relates γi to G iE exactly as Eq. (4-78) relates φˆ i to  GiR. For an ideal solution,G iE =0, and therefore γi = 1.

SOLUTION THERMODYNAMICS VE ∂(GE/RT) ᎏ = ᎏᎏ RT ∂P

An alternative form of Eq. (4-101) follows by introduction of the activity coefficient through Eq. (4-103):

  E

E



E

nH nG nV d ᎏ = ᎏ dP − ᎏ2 dT +  ln γi dni RT RT RT i



(4-117)

T,x

HE ∂(GE/RT) ᎏ = −T ᎏᎏ RT ∂T



(4-104)

4-11



(4-118)

P,x

and from Eq. (4-104) SUMMARY OF FUNDAMENTAL PROPERTY RELATIONS For convenience, the three other fundamental property relations, Eqs. (4-16), (4-80), and (4-82), expressing the Gibbs energy and related properties as functions of T, P, and the ni, are collected here: d(nG) = nV dP − nS dT +  µi dni

(4-16)

 

(4-80)

i

G nG nV nH i d ᎏ = ᎏ dP − ᎏ2 dT +  ᎏ dni RT RT RT RT i

  R

R

R

nG nV nH d ᎏ = ᎏ dP − ᎏ2 dT +  ln φˆ i dni RT RT RT i

(4-82)

These equations and Eq. (4-104) may also be written for the special case of 1 mole of solution by setting n = 1 and ni = xi. The xi are then subject to the constraint that i xi = 1. If written for 1 mole of a constant-composition solution, they become: dG = V dP − S dT

(4-105)

 

(4-106)

 

(4-107)

 

(4-108)

G V H d ᎏ = ᎏ dP − ᎏ2 dT RT RT RT GR VR HR d ᎏ = ᎏ dP − ᎏ2 dT RT RT RT GE VE HE d ᎏ = ᎏ dP − ᎏ2 dT RT RT RT

These equations are, of course, valid as a special case for a pure species; in this event they are written with subscript i affixed to the appropriate symbols. The partial-property analogs of these equations are: dG Vi dP −  Si dT i = dµi =  µi G V H i i i d ᎏ = d ᎏ = ᎏ dP − ᎏ2 dT RT RT RT RT

   

 

G Vi H    d ᎏ = d ln φˆ i = ᎏ dP − ᎏ2 dT RT RT RT R i

R

R i

 

H G ViE iE  iE d ᎏ = d ln γi = ᎏ dP − ᎏ2 dT RT RT RT

(4-110)







∂ ln γi

ᎏ ∂T 

H iE = − ᎏ2 (4-121) P,x RT Finally, an especially useful form of the Gibbs/Duhem equation follows from Eq. (4-116):

 x d ln γ = 0 i

(constant T,P)

i

Since ln γi is a partial property with respect to GE/RT, the following form of the summability equation is valid: GE (4-123) ᎏ =  xi ln γi RT i The analogy between equations derived from the fundamental residual- and excess-property relations is apparent. Whereas the fundamental residual-property relation derives its usefulness from its direct relation to equations of state, the excess-property formulation is useful because VE, HE, and γi are all experimentally accessible. Activity coefficients are found from vapor/liquid equilibrium data, and VE and HE values come from mixing experiments. PROPERTY CHANGES OF MIXING If M represents a molar thermodynamic property of a homogeneous fluid solution, then by definition, ∆M
M −  xi Mi

(4-124)

i

(4-111) (4-112)

where by definition

(4-113)

G V H i ᎏ dP − ᎏ2 dT =  xi d ᎏ RT RT RT i

(4-114)

VR HR ᎏ dP − ᎏ2 dT =  xi d ln φˆ i RT RT i

(4-115)

∆M =  xi ∆ M i

(4-125)

Mi − Mi ∆ M i


(4-126)

i

All three quantities are for the same T, P, and physical state. Eq. (4-126) defines a partial molar property change of mixing, and Eq. (4-125) is the summability relation for these properties. Each of Eqs. (4-93) through (4-96) is an expression for an ideal solution property, and each may be combined with the defining equation for an excess property (Eq. [4-99]), yielding GE = G −  xiGi − RT  xi ln xi

(4-127)

V = V −  xiVi

(4-128)

i

i

E

HE VE ᎏ dP − ᎏ2 dT =  xi d ln γi RT RT i

(4-122)

i

where ∆M is the property change of mixing, and Mi is the molar property of pure species i at the T and P of the solution and in the same physical state (gas or liquid). The summability relation, Eq. (4-50), may be combined with Eq. (4-124) to give

i

 



(4-109)

Finally, a Gibbs/Duhem equation is associated with each fundamental property relation: V dP − S dT =  xi dµi

∂(nGE/RT) ln γi = ᎏᎏ (4-119) T,P,nj ∂ni The last relation demonstrates that ln γi is a partial property with respect to GE/RT. The partial-property analogs of Eqs. (4-117) and (4-118) follow from Eq. (4-112): V ∂ ln γi iE (4-120) ᎏ =ᎏ ∂P T,x RT

(4-116)

This depository of equations stores an enormous amount of information. The equations themselves are so general that their direct application is seldom appropriate. However, by inspection one can write a vast array of relations valid for particular applications. For example, Eqs. (4-83) and (4-84) come directly from Eq. (4-107); Eqs. (4-86) and (4-87), from (4-111). Similarly, from Eq. (4-108),

i

SE = S −  xiSi + R  xi ln xi

(4-129)

HE = H −  xiHi

(4-130)

i

i

i

In view of Eq. (4-124), these may be written G E = ∆G − RT  xi ln xi i

(4-131)

4-12

THERMODYNAMICS V E = ∆V

(4-132)

S = ∆S + R  xi ln xi E

H E = ∆H

i

(4-133) (4-134)

where ∆G, ∆V, ∆S, and ∆H are the Gibbs energy change of mixing, the volume change of mixing, the entropy change of mixing, and the enthalpy change of mixing. For an ideal solution, each excess property is zero, and for this special case ∆Gid = RT  xi ln xi ∆Vid = 0

i

(4-135) (4-136)

∆S = −R  xi ln xi id

(4-137)

i

(4-138) ∆Hid = 0 Property changes of mixing and excess properties are easily calculated one from the other. The most commonly encountered property changes of mixing are the volume change of mixing ∆V and the enthalpy change of mixing ∆H, commonly called the heat of mixing. These properties are directly measurable and are identical to the corresponding excess properties. Pertinent examples are given in Smith, Van Ness, and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed., Sec. 11.4, McGraw-Hill, New York, 1996). BEHAVIOR OF BINARY LIQUID SOLUTIONS Property changes of mixing and excess properties find greatest application in the description of liquid mixtures at low reduced tempera-

tures, that is, at temperatures well below the critical temperature of each constituent species. The properties of interest to the chemical engineer are V E (
∆V), H E (
∆H), S E, ∆S, G E, and ∆G. The activity coefficient is also of special importance because of its application in phase-equilibrium calculations. The behavior of binary liquid solutions is clearly displayed by plots of M E, ∆M, and ln γi vs. x1 at constant T and P. The volume change of mixing (or excess volume) is the most easily measured of these quantities and is normally small. However, as illustrated by Fig. 4-1, it is subject to individualistic behavior, being sensitive to the effects of molecular size and shape and to differences in the nature and magnitude of intermolecular forces. The heat of mixing (excess enthalpy) and the excess Gibbs energy are also experimentally accessible, the heat of mixing by direct measurement and GE (or ln γi) indirectly as a product of the reduction of vapor/liquid equilibrium data. Knowledge of HE and GE allows calculation of SE by Eq. (4-13) written for excess properties, HE − GE SE = ᎏ T

(4-139)

with ∆S then given by Eq. (4-133). Figure 4-2 displays plots of ∆H, ∆S, and ∆G as functions of composition for 6 binary solutions at 50°C. The corresponding excess properties are shown in Fig. 4-3; the activity coefficients, derived from Eq. (4-119), appear in Fig. 4-4. The properties shown here are insensitive to pressure, and for practical purposes represent solution properties at 50°C (122°F) and low pressure (P ≈ 1 bar [14.5 psi]).

FIG. 4-1 Excess volumes at 25°C for liquid mixtures of cyclohexane(1) with some other C6 hydrocarbons.

SOLUTION THERMODYNAMICS

(a)

(b)

(c)

(d)

(e)

(f)

Property changes of mixing at 50°C for 6 binary liquid systems: (a) chloroform(1)/n-heptane(2); (b) acetone(1)/ methanol(2); (c) acetone(1)/chloroform(2); (d) ethanol(1)/n-heptane(2); (e) ethanol(1)/chloroform(2); ( f) ethanol(1)/water(2). FIG. 4-2

(a)

(b)

(c)

(d)

(e)

(f)

Excess properties at 50°C for 6 binary liquid systems: (a) chloroform(1)/n-heptane(2); (b) acetone(1)/methanol(2); (c) acetone(1)/chloroform(2); (d) ethanol(1)/n-heptane(2); (e) ethanol(1)/chloroform(2); ( f) ethanol(1)/water(2). FIG. 4-3

4-13

4-14

THERMODYNAMICS

(a)

(b)

(c)

(d)

(e)

(f)

Activity coefficients at 50°C for 6 binary liquid systems: (a) chloroform(1)/n-heptane(2); (b) acetone(1)/ methanol(2); (c) acetone(1)/chloroform(2); (d) ethanol(1)/n-heptane(2); (e) ethanol(1)/chloroform(2); ( f) ethanol(1)/ water(2).

FIG. 4-4

EVALUATION OF PROPERTIES RESIDUAL-PROPERTY FORMULATIONS The most satisfactory calculational procedure for thermodynamic properties of gases and vapors requires PVT data and ideal gas heat capacities. The primary equations are based on the concept of the ideal gas state and the definitions of residual enthalpy and residual entropy: H = Hig + HR

and

S = Sig + SR

The enthalpy and entropy are simple sums of the ideal gas and residual properties, which are evaluated separately. For the ideal gas state at constant composition, dH = C dT ig P

ig

dT dP dSig = CPig ᎏ − R ᎏ T P Integration from an initial ideal gas reference state at conditions T0 and P0 to the ideal gas state at T and P gives: Hig = H0ig +

C

ig P

dT

C

ig P

dT P ᎏ − R ln ᎏ T P0

T

T0

Sig = S0ig +

T

T0

H = H0ig +

C T

T0

ig P

dT + HR

(4-140)

T

T0

ig P

CPig = A + BT + CT 2 + DT −2

(4-142)

where A, B, C, and D are constants characteristic of the particular gas, ig and either C or D is 0. Evaluation of the integrals ∫ CP dT and ig ig ∫ (CP /T)dT is accomplished by substitution for CP , followed by formal integration. For temperature limits of T0 and T the results are conveniently expressed as follows: T B C D τ−1 ig CP dT = AT0(τ − 1) + ᎏ T02(τ2 − 1) + ᎏ T03 (τ3 − 1) + ᎏ ᎏ T0 2 3 T0 τ (4-143) ig T CP D τ+1 2 and ᎏ dT = A ln τ + BT0 + CT0 + ᎏ ᎏ (τ − 1) T0 T τ 2T02 2 (4-144) T where τ
ᎏ T0







Substitution into the equations for H and S yields

C

dT P (4-141) ᎏ − R ln ᎏ + SR T P0 The reference state at T0 and P0 is arbitrarily selected, and the values assigned to H0ig and S0ig are also arbitrary. In practice, only changes in H and S are of interest, and the reference-state values ultimately cancel in their calculation. ig The ideal-gas–state heat capacity CP is a function of T but not of P. ig For a mixture, the heat capacity is simply the molar average i xi CP . ig Empirical equations giving the temperature dependence of CP are available for many pure gases, often taking the form S = S0ig +











EVALUATION OF PROPERTIES Equations (4-140) and (4-141) may sometimes be advantageously expressed in alternative form through use of mean heat capacities: H = H0ig + 〈CPig〉H(T − T0) + HR (4-145) T P ig R ig S = S0 + 〈CP 〉S ln ᎏ − R ln ᎏ + S (4-146) T0 P0 where 〈CPig〉H and 〈CPig〉S are mean heat capacities specific respectively to enthalpy and entropy calculations. They are given by the following equations: B C D 〈CPig〉H = A + ᎏ T0 (τ + 1) + ᎏ T02(τ 2 + τ + 1) + ᎏ2 (4-147) 2 3 τ T0





D 〈CPig〉S = A + BT0 + CT02 + ᎏ τ 2 T02

τ+1 τ−1 ᎏ  ᎏ 2   ln τ 

(4-148)

LIQUID/VAPOR PHASE TRANSITION When a differential amount of a pure liquid in equilibrium with its vapor in a piston-and-cylinder arrangement evaporates at constant temperature T and vapor pressure Pisat, Eq. (4-16) applied to the process reduces to d(niGi) = 0, whence ni dGi + Gi dni = 0

B ln P sat = A − ᎏ (4-152) T+C A principal advantage of this equation is that values of the constants A, B, and C are readily available for a large number of species. The accurate representation of vapor-pressure data over a wide temperature range requires an equation of greater complexity. The Wagner equation, one of the best, expresses the reduced vapor pressure as a function of reduced temperature: Aτ + Bτ 1.5 + Cτ 3 + Dτ 6 ln Prsat = ᎏᎏᎏ (4-153) 1−τ where here τ
1 − Tr and A, B, C, and D are constants. Values of the constants either for this equation or the Antoine equation are given for many species by Reid, Prausnitz, and Poling (The Properties of Gases and Liquids, 4th ed., App. A, McGraw-Hill, New York, 1987). LIQUID-PHASE PROPERTIES Given saturated-liquid enthalpies and entropies, the calculation of these properties for pure compressed liquids is accomplished by integration at constant temperature of Eqs. (4-34) and (4-35):

(4-149)

where Gil and Giv are the molar Gibbs energies of the individual phases. If the temperature of a two-phase system is changed and if the two phases continue to coexist in equilibrium, then the vapor pressure must also change in accord with its temperature dependence. Since Eq. (4-149) holds throughout this change, dGil = dGiv Substituting the expressions for dGil and dGiv given by Eq. (4-16) yields Vil dPisat − Sil dT = Viv dPisat − Siv dT which upon rearrangement becomes dPisat Siv − Sil ∆Silv =ᎏ ᎏ=ᎏ v l dT Vi − Vi ∆Vilv The entropy change ∆Silv and the volume change ∆Vilv are the changes which occur when a unit amount of a pure chemical species is transferred from phase l to phase v at constant temperature and pressure. Integration of Eq. (4-18) for this change yields the latent heat of phase transition: ∆ Hilv = T∆Silv Thus, ∆Si = ∆Hi /T, and substitution in the preceding equation gives lv

lv

dPisat ∆ Hilv (4-150) ᎏ=ᎏ dT T∆Vilv Known as the Clapeyron equation, this is an exact thermodynamic relation, providing a vital connection between the properties of the liquid and vapor phases. Its use presupposes knowledge of a suitable vapor pressure vs. temperature relation. Empirical in nature, such relations are approximated by the equation B ln P sat = A − ᎏ (4-151) T where A and B are constants for a given species. This equation gives a rough approximation of the vapor-pressure relation for its entire temperature range. Moreover, it is an excellent basis for interpolation between values that are reasonably spaced. The Antoine equation, which is more satisfactory for general use, has the form

 −

P

Hi = Hisat +

Since the system is closed, dni = 0 and, therefore, dGi = 0; this requires the molar (or specific) Gibbs energy of the vapor to be identical with that of the liquid: Gil = Giv

4-15

sat

Vi(1 − βiT)dP

(4-154)

βiVi dP

(4-155)

Pi

Si = Sisat

P sat

Pi

where the volume expansivity of species i at temperature T is 1 ∂Vi βi
ᎏ ᎏ (4-156) Vi ∂T P Since βi and Vi are weak functions of pressure for liquids, they are usually assumed constant at the values for the saturated liquid at temperature T.

 

PROPERTIES FROM PVT CORRELATIONS The empirical representation of the PVT surface for pure materials is treated later in this section. We first present general equations for evaluation of reduced properties from such representations. Equation (4-83), applied to a pure material, may be written GR VR d ᎏ = ᎏ dP (constant T) RT RT Integration from zero pressure to arbitrary pressure P gives P GR VR (constant T) ᎏ = ᎏ dP 0 RT RT where at the lower limit GR/RT is set equal to zero on the basis that the zero-pressure state is an ideal gas state. The residual volume is related directly to the compressibility factor: ZRT RT RT V R
V − V ig = ᎏ − ᎏ = (Z − 1) ᎏ P P P

 



VR Z − 1 ᎏ=ᎏ RT P

whence



(4-157)

P GR dP (constant T ) (4-158) ᎏ = (Z − 1) ᎏ 0 RT P Differentiation of Eq. (4-158) with respect to temperature in accord with Eq. (4-84), gives P HR ∂Z dP (constant T ) (4-159) ᎏ = −T ᎏ ᎏ 0 RT ∂T P P Equation (4-13) written for residual properties becomes SR HR GR (4-160) ᎏ=ᎏ−ᎏ R RT RT

Therefore

 

4-16

THERMODYNAMICS

In view of Eq. (4-75), Eqs. (4-158) and (4-160) may be expressed alternatively as P dP ln φ = (Z − 1) ᎏ (constant T ) (4-161) 0 P

Again, in alternative form, HR (HR)0 (HR)1 ᎏ=ᎏ+ωᎏ RTc RTc RTc

SR HR and (4-162) ᎏ = ᎏ − ln φ R RT Values of Z and of (∂Z/∂T)P come from experimental PVT data, and the integrals in Eqs. (4-158), (4-159), and (4-161) may be evaluated by numerical or graphical methods. Alternatively, the integrals are expressed analytically when Z is given by an equation of state. Residual properties are therefore evaluated from PVT data or from an appropriate equation of state. Pitzer’s Corresponding-States Correlation A three-parameter corresponding-states correlation of the type developed by Pitzer, K.S. (Thermodynamics, 3d ed., App. 3, McGraw-Hill, New York, 1995) is described in Sec. 2. It has as its basis an equation for the compressibility factor:

where



Z = Z0 + ωZ1 0

(4-163)

1

where Z and Z are each functions of reduced temperature Tr and reduced pressure Pr. The acentric factor ω is defined by Eq. (2-23). The Tr and Pr dependencies of functions Z0 and Z1 are shown by Figs. 2-1 and 2-2. Generalized correlations are developed here for the residual enthalpy, residual entropy, and the fugacity coefficient. Equations (4-161) and (4-159) are put into generalized form by substitution of the relationships P = Pc Pr

T = Tc Tr

dP = Pc dPr

dT = Tc dTr

The resulting equations are: ln φ =



Pr

0

dP (Z − 1) ᎏr Pr

HR ᎏ = −T r2 RTc

and

∂Z  ᎏ ∂T 

(4-164)

Pr

0

r

Pr

dPr ᎏ Pr

(4-165)

(HR)0 ᎏ = −T r2 RTc

∂Z  ᎏ ∂T  ∂Z  ᎏ ∂T  Pr

0

0 r

Pr

(4-168)

dPr ᎏ Pr

Pr 1 (HR)1 dPr ᎏ = −T r2 ᎏ 0 RTc r Pr Pr The residual entropy is given by Eq. (4-162), here written SR 1 HR (4-169) ᎏ = ᎏ ᎏ − ln φ R Tr RTc Pitzer’s original correlations for Z and the derived quantities were determined graphically and presented in tabular form. Since then, analytical refinements to the tables have been developed, with extended range and accuracy. The most popular Pitzer-type correlation is that of Lee and Kesler (AIChE J., 21, pp. 510–527 [1975]). These tables cover both the liquid and gas phases, and span the ranges 0.3 ≤ Tr ≤ 4.0 and 0.01 ≤ Pr ≤ 10.0. Shown by Figs. 4-5 and 4-6 are isobars of −(HR)0/RTc and −(HR)1/RTc with Tr as independent variable drawn from these tables. Figures 4-7 and 4-8 are the corresponding plots for −ln φ0 and −ln φ1. Figures 4-9 and 4-10 are isotherms of φ0 and φ1 with Pr as independent variable. Although the Pitzer correlations are based on data for pure materials, they may also be used for the calculation of mixture properties. A set of recipes is required relating the parameters Tc, Pc, and ω for a mixture to the pure-species values and to composition. One such set is given by Eqs. (2-80) through (2-82) in Sec. 2, which define pseudoparameters, so called because the defined values of Tc, Pc, and ω have no physical significance for the mixture. Alternative Property Formulations Direct application of Eqs. (4-159) and (4-161) can be made only to equations of state that are solvable for volume, that is, that are volume explicit. Most equations of state are in fact pressure explicit, and alternative equations are required.

 

The terms on the right-hand sides of these equations depend only on the upper limit Pr of the integrals and on the reduced temperature at which they are evaluated. Thus, values of ln φ and HR/RTc may be determined once and for all at any reduced temperature and pressure from generalized compressibility factor data. Substitution for Z in Eq. (4-164) by Eq. (4-163) yields ln φ =



Pr

0

dP (Z0 − 1) ᎏr + ω Pr



Pr

0

dP Z1 ᎏr Pr

This equation may be written in alternative form as ln φ = ln φ0 + ω ln φ1 where

ln φ0




ln φ1




Pr

0 Pr

0

(4-166)

dP (Z0 − 1) ᎏr Pr dP Z1 ᎏr Pr

Since Eq. (4-166) may also be written φ = (φ0)(φ1)ω

(4-167)

correlations may be presented for φ0 and φ1 as well as for their logarithms. Differentiation of Eq. (4-163) yields ∂Z ∂Z = ᎏ ᎏ ∂T  ∂T 0

r

Pr

r

Pr

∂Z1 +ω ᎏ ∂Tr

 

Pr

Substitution for (∂Z/∂Tr)Pr in Eq. (4-165) gives: HR ᎏ = −T r2 RTc

∂Z  ᎏ ∂T  Pr

0

0 r

dPr ᎏ − ωT r2 Pr Pr

∂Z  ᎏ ∂T  Pr

0

1 r

Pr

dPr ᎏ Pr

Correlation of −(HR)0/RTc, drawn from the tables of Lee and Kesler (AIChE J., 21, pp. 510–527 [1975]).

FIG. 4-5

EVALUATION OF PROPERTIES

4-17

Correlation of −(HR)1/RTc, drawn from the tables of Lee and Kesler (AIChE J., 21, pp. 510–527 [1975]).

FIG. 4-6

Equation (4-158) is converted through application of the general relation PV = ZRT. Differentiation at constant T gives P dV + V dP = RT dZ which is readily transformed to dP dZ dV ᎏ=ᎏ−ᎏ P Z V

(constant T )

(constant T )

FIG. 4-8 Correlation of [−ln φ1] vs. Tr, drawn from the tables of Lee and Kesler (AIChE J., 21, pp. 510–527 [1975]).

Substitution into Eq. (4-158) leads to V GR dV (4-170) ᎏ = Z − 1 − ln Z − (Z − 1) ᎏ ∞ RT V The molar volume may be eliminated in favor of the molar density, ρ = V−1, to give ρ GR dρ (4-171) ᎏ = Z − 1 − ln Z + (Z − 1) ᎏ 0 RT ρ For a pure material, Eq. (4-75) shows that GR/RT = ln φ, in which case Eqs. (4-170) and (4-171) directly yield values of ln φ: V dV ln φ = Z − 1 − ln Z − (Z − 1) ᎏ (4-172) ∞ V









ρ

dρ (Z − 1) ᎏ (4-173) ρ where subscript i is omitted for simplicity. The corresponding equations for HR are most readily found from Eq. (4-107) applied to a pure material. In view of Eqs. (4-75) and (4-157), this equation may be written HR dP d ln φ = (Z − 1) ᎏ − ᎏ2 dT RT P Division by dT and restriction to constant V gives, upon rearrangement, HR Z − 1 ∂P ∂ ln φ ᎏ2 = ᎏ ᎏ − ᎏ RT P ∂T V ∂T V Differentiation of P = ZRT/V provides the first derivative on the right and differentiation of Eq. (4-172) provides the second. Substitution then leads to V HR ∂Z dV (4-174) ᎏ=Z−1+T ᎏ ᎏ ∞ RT ∂T V V ln φ = Z − 1 − ln Z +

0

  

FIG. 4-7 Correlation of [−ln φ0] vs. Tr, drawn from the tables of Lee and Kesler (AIChE J., 21, pp. 510–527 [1975]).

 



4-18

THERMODYNAMICS

FIG. 4-10 Correlation of φ1 vs. Pr, drawn from the tables of Lee and Kesler (AIChE J., 21, pp. 510–527 [1975]).

Direct application of these results is possible only to equations of state explicit in volume. For pressure-explicit equations of state, alternative recipes are required. The basis is Eq. (4-82), which in view of Eq. (4-157) may be written nGR n(Z − 1) nHR d ᎏ = ᎏ dP − ᎏ2 dT +  ln φˆ i dni RT P RT i Division by dni and restriction to constant T, nV, and nj ( j ≠ i) leads to ∂(nGR/RT) n(Z − 1) ∂P ln φˆ i = ᎏᎏ −ᎏ ᎏ T,nV,nj ∂ni P ∂ni T,nV,nj But P = (nZ)RT/nV, and therefore ∂P P ∂(nZ) =ᎏ ᎏ ᎏ ∂ni T,nV,nj nZ ∂ni T,nV,nj Combination of the last two equations gives ∂(nGR/RT) Z − 1 ∂(nZ) ln φˆ i = ᎏᎏ − ᎏ ᎏ (4-178) T,nV,nj ∂ni Z ∂ni T,nV,nj Alternatively, ∂(nGR/RT) Z − 1 ∂(nZ) ln φˆ i = ᎏᎏ − ᎏ ᎏ (4-179) T,ρ/n,nj ∂ni Z ∂ni T,ρ/n,nj These equations may either be applied to the results of integrations of Eqs. (4-170) and (4-171) or directly to Eqs. (4-170) and (4-171) as written for a mixture. In the latter case the following analogs of Eq. (4-176) are obtained: V ∂(nZ) dV ln φˆ i = − − 1 ᎏ − ln Z (4-180) ᎏ ∞ ∂ni T,nV,nj V

 

FIG. 4-9 Correlation of φ0 vs. Pr, drawn from the tables of Lee and Kesler (AIChE J., 21, pp. 510–527 [1975]).

  ρ

H ∂Z dρ (4-175) ᎏ=Z−1−T ᎏ ᎏ 0 RT ∂T ρ ρ As before, the residual entropy is found by Eq. (4-162). In applications to equilibrium calculations, the fugacity coefficients of species in a mixture φˆ i are required. Given an expression for GR/RT as determined from Eq. (4-158) for a constant-composition mixture, the corresponding recipe for ln φˆ i is found through the partialproperty relation ∂(nGR/RT) ln φˆ i = ᎏᎏ (4-85) T,P,nj ∂ni There are two ways to proceed: operate on the result of the integration of Eq. (4-158) in accord with Eq. (4-85) or apply Eq. (4-85) directly to Eq. (4-158), obtaining P dP ln φˆ i = (Z (4-176) i − 1) ᎏ 0 P where  Zi is the partial compressibility factor, defined as ∂(nZ) Z (4-177) i
ᎏ ∂ni T,P,nj R

Alternatively,















 

 

























   ∂(nZ) ln φˆ = −   ᎏ  ∂n ρ

i

0

i

T,ρ/n,nj





dρ − 1 ᎏ − ln Z ρ

(4-181)

EVALUATION OF PROPERTIES Virial Equations of State The virial equation in density is an infinite-series representation of the compressibility factor Z in powers of molar density ρ (or reciprocal molar volume V−1) about the real-gas state at zero density (zero pressure): Z = 1 + Bρ + Cρ2 + Dρ3 + ⋅ ⋅ ⋅

(4-182)

The density-series virial coefficients B, C, D, . . . , depend on temperature and composition only. The composition dependencies are given by the exact recipes B =  yiyjBij

(4-183)

C =    yiyjykCijk

(4-184)

i

i

j

j

k

where yi, yj, and yk are mole fractions for a gas mixture, with indices i, j, and k identifying species. The coefficient Bij characterizes a bimolecular interaction between molecules i and j, and therefore Bij = Bji. Two kinds of second virial coefficient arise: Bii and Bjj, wherein the subscripts are the same (i = j); and Bij, wherein they are different (i ≠ j). The first is a virial coefficient for a pure species; the second is a mixture property, called a cross coefficient. Similarly for the third virial coefficients: Ciii, Cjjj, and Ckkk are for the pure species; and Ciij = Ciji = Cjii, and so on, are cross coefficients. Although the virial equation itself is easily rationalized on empirical grounds, the “mixing rules” of Eqs. (4-183) and (4-184) follow rigorously from the methods of statistical mechanics. The temperature derivatives of B and C are given exactly by dB dBij (4-185) ᎏ =   yiyj ᎏ dT dT i j dC dCijk (4-186) ᎏ =    yiyjyk ᎏ dT dT i j k An alternative form of the virial equation expresses Z as an expansion in powers of pressure about the real-gas state at zero pressure (zero density): Z = 1 + B′P + C′P 2 + D′P 3 + ⋅ ⋅ ⋅ (4-187) Equation (4-187) is the virial equation in pressure, and B′, C′, D′, . . . , are the pressure-series virial coefficients. Like the density-series coefficients, they depend on temperature and composition only. Moreover, the two sets of coefficients are related: B B′ = ᎏ (4-188) RT

  





















P (4-196) ln φˆ i = 2  ykBki − B ᎏ RT k Equation (4-191) is explicit in pressure, and Eqs. (4-173), (4-175), and (4-181) are therefore applicable. Direct substitution of Eq. (4-191) into Eq. (4-173) yields 3 ln φ = 2Bρ + ᎏ Cρ2 − ln Z (4-197) 2

Whence

(4-189)

∂Z

dC =ᎏρ+ᎏρ ᎏ ∂T  dT dT

Moreover,

dB

2

ρ

and so on Application of an infinite series to practical calculations is, of course, impossible, and truncations of the virial equations are in fact employed. The degree of truncation is conditioned not only by the temperature and pressure but also by the availability of correlations or data for the virial coefficients. Values can usually be found for B (see Sec. 2), and often for C (see, e.g., De Santis and Grande, AIChE J., 25, pp. 931–938 [1979]), but rarely for higher-order coefficients. Application of the virial equations is therefore usually restricted to two- or three-term truncations. For pressures up to several bars, the two-term expansion in pressure, with B′ given by Eq. (4-188), is usually preferred: BP Z=1+ᎏ (4-190) RT For supercritical temperatures, it is satisfactory to ever-higher pressures as the temperature increases. For pressures above the range where Eq. (4-190) is useful, but below the critical pressure, the virial expansion in density truncated to three terms is usually suitable:

(4-191)

Equations for derived properties may be developed from each of these expressions. Consider first Eq. (4-190), which is explicit in volume. Equations (4-159), (4-161), and (4-176) are therefore applicable. Direct substitution for Z in Eq. (4-161) gives BP ln φ = ᎏ (4-192) RT Differentiation of Eq. (4-190) yields ∂Z dB B P ᎏ = ᎏ−ᎏ ᎏ ∂T P dT T RT Whence, by Eq. (4-159) HR P B dB (4-193) ᎏ=ᎏ ᎏ−ᎏ RT R T dT and by Eq. (4-162), SR P dB (4-194) ᎏ=−ᎏᎏ R R dT Multiplication of Eq. (4-190) by n gives P nZ = n + (nB) ᎏ RT Differentiation in accord with Eq. (4-177) yields ∂(nB) P Z i = 1 + ᎏ ᎏ ∂ni T,nj RT Whence, by Eq. (4-176), ∂(nB) P ln φˆ i = ᎏ ᎏ ∂ni T,nj RT Equation (4-183) can be written 1 nB = ᎏ   nknlBkl n k l from which, by differentiation, ∂(nB) = 2  ykBki − B (4-195) ᎏ ∂ni T,nj k



and so on

C − B2 C′ = ᎏ (RT)2

Z = 1 + Bρ + Cρ2

4-19



 



HR dB T dC (4-198) ᎏ = B − T ᎏ ρ + C − ᎏ ᎏ ρ2 RT dT 2 dT The residual entropy is given by Eq. (4-162). Application of Eq. (4-181) provides an expression for ln φˆ i. First, from Eq. (4-191), whence

∂(nZ) ᎏ ∂n 

∂(nB) ∂(nC) =1+ B+ ᎏ ρ + 2C + ᎏ ∂ni T,nj ∂ni Substitution into Eq. (4-181) gives, on integration, i



T,ρ/n,nj





 ρ

2

T,nj

∂(nB) 1 ∂(nC) ln φˆ i = B + ᎏ ρ + ᎏ 2C + ᎏ ρ2 − ln Z ∂ni T,nj 2 ∂ni T,nj The mole-number derivative of nB is given by Eq. (4-195); the corresponding derivative of nC, similarly found from Eq. (4-184), is ∂(nC) = 3   ykylCkli − 2C (4-199) ᎏ ∂ni T,nj k l















4-20

THERMODYNAMICS

Finally, 3 ln φˆ i = 2ρ  ykBki + ᎏ ρ2   ykylCkli − ln Z (4-200) 2 k k l In a process calculation, T and P, rather than T and ρ (or T and V), are usually the favored independent variables. Application of Eqs. (4-197), (4-198), and (4-200) therefore requires prior solution of Eq. (4-191) for Z or ρ. Since Z = P/ρRT, Eq. (4-191) may be written in two equivalent forms: BP CP 2 Z3 − Z2 − ᎏ Z − ᎏ2 = 0 (4-201) RT (RT)

 

 

 

B 1 P (4-202) ρ3 + ᎏ ρ2 + ᎏ ρ − ᎏ = 0 C C CRT In the event that three real roots obtain for these equations, only the largest Z (smallest ρ) appropriate for the vapor phase has physical significance, because the virial equations are suitable only for vapors and gases. Generalized Correlation for the Second Virial Coefficient Perhaps the most useful of all Pitzer-type correlations is the one for the second virial coeffieient. The basic equation (see Eq. [2-68]) is BPc (4-203) ᎏ = B0 + ωB1 RTc where for a pure material B0 and B1 are functions of reduced temperture only. Substitution for B by this expression in Eq. (4-190) yields P Z = 1 + (B0 + ωB1) ᎏr (4-204) Tr By differentiation, ∂Z dB0/dTr B0 dB1/dT B1 + ωPr ᎏr − ᎏ ᎏ = Pr ᎏ − ᎏ 2 ∂Tr Pr Tr Tr Tr Tr2 Substitution of these equations into Eqs. (4-164) and (4-165) and integration gives P ln φ = (B0 + ωB1) ᎏr (4-205) Tr or

 















HR dB0 dB1 (4-206) ᎏ = Pr B0 − Tr ᎏ + ω B1 − Tr ᎏ RTc dTr dTr The residual entropy follows from Eq. (4-162): SR dB0 dB1 (4-207) ᎏ = −Pr ᎏ + ω ᎏ R dTr dTr In these equations, B0 and B1 and their derivatives are well represented by 0.422 B0 = 0.083 − ᎏ (4-208) Tr1.6 and





0.172 B1 = 0.139 − ᎏ Tr4.2 dB0 0.675 ᎏ=ᎏ dTr Tr2.6

(4-209) (4-210)

dB1 0.722 (4-211) ᎏ=ᎏ dTr Tr5.2 Though limited to pressures where the two-term virial equation in pressure has approximate validity, this correlation is applicable to most chemical-processing conditions. As with all generalized correlations, it is least accurate for polar and associating molecules. Although developed for pure materials, this correlation can be extended to gas or vapor mixtures. Basic to this extension is the mixing rule for second virial coefficients and its temperature derivative: B =   yiyjBij

(4-183)

dB dBij ᎏ =   yiyj ᎏ dT dT i j

(4-185)

i

j

Values for the cross coefficients and their derivatives in these equations are provided by writing Eq. (4-203) in extended form: RTcij 0 Bij = ᎏ (B + ωij B1) (4-212) Pcij where B0, B1, dB0/dTr, and dB1/dTr are the same functions of Tr as given by Eqs. (4-208) through (4-211). Differentiation produces dBij RTcij dB0 dB1 ᎏ = ᎏ ᎏ + ωij ᎏ dT Pcij dT dT







0

1



dBij R dB dB (4-213) ᎏ = ᎏ ᎏ + ωij ᎏ dT Pcij dTrij dTrij where Trij = T/Tcij. The following are combining rules for calculation of ωij, Tcij, and Pcij as given by Prausnitz, Lichtenthaler, and de Azevedo (Molecular Thermodynamics of Fluid-Phase Equilibria, 2d ed., pp. 132 and 162, Prentice-Hall, Englewood Cliffs, N.J., 1986): ωi + ωj ωij = ᎏ (4-214) 2 Tcij = (Tci Tcj)1/2(1 − kij) (4-215) Zcij RTcij Pcij = ᎏ (4-216) Vcij Zci + Zcj (4-217) with Zcij = ᎏ 2 1/3 1/3 3 Vci + Vc j (4-218) Vcij = ᎏᎏ 2 In Eq. (4-215), kij is an empirical interaction parameter specific to an i-j molecular pair. When i = j and for chemically similar species, kij = 0. Otherwise, it is a small (usually) positive number evaluated from minimal PVT data or in the absence of data set equal to zero. When i = j, all equations reduce to the appropriate values for a pure species. When i ≠ j, these equations define a set of interaction parameters having no physical significance. For a mixture, values of Bij and dBij /dT from Eqs. (4-212) and (4-213) are substituted into Eqs. (4183) and (4-185) to provide values of the mixture second virial coefficient B and its temperature derivative. Values of HR and SR for the mixture are then given by Eqs. (4-193) and (4-194), and values of ln φˆ i for the component fugacity coefficients are given by Eq. (4-196). Cubic Equations of State The simplest expressions that can (in principle) represent both the vapor- and liquid-phase volumetric behavior of pure fluids are equations cubic in molar volume. All such expressions are encompassed by the generic equation or





RT a(V − η) P = ᎏ − ᎏᎏᎏ (4-219) V − b (V − b)(V 2 + δV + ε) where parameters b, θ, δ, ε, and η can each depend on temperature and composition. Special cases are obtained by specification of values or expressions for the various parameters. The modern development of cubic equations of state started in 1949 with publication of the Redlich/Kwong equation (Redlich and Kwong, Chem. Rev., 44, pp. 233–244 [1949]): RT a(T) P=ᎏ−ᎏ (4-220) V − b V(V + b) a a(T) = ᎏ T1/2 and a and b are functions of composition only. This equation, like other cubic equations of state, has three volume roots, of which two may be complex. Physically meaningful values of V are always real, positive, and greater than the constant b. When T > Tc, solution for V at any positive value of P yields only one real positive root. When T = Tc, this is also true, except at the critical pressure, where there are three roots, all equal to Vc. For T < Tc, only one real positive root exists at high pressures, but for a range of lower pressures there are three real positive roots. Here, the middle root is of no significance; the smallest root is a liquid or liquidlike volume, and the largest root is a vapor or vaporlike volume. The volumes of saturated liquid and satuwhere

EVALUATION OF PROPERTIES rated vapor are given by the smallest and largest roots when P is the saturation vapor pressure Psat. The application of cubic equations of state to mixtures requires expression of the equation-of-state parameters as functions of composition. No exact theory like that for the virial coefficients prescribes this composition dependence, and empirical mixing rules provide approximate relationships. The mixing rules that have found general favor for the Redlich/Kwong equation are: with aij = aji, and

a =   yi yj aij

(4-221)

b =  yi bi

(4-222)

i

j

i

The aij are of two types: pure-species parameters (like subscripts) and interaction parameters (unlike subscripts). The bi are parameters for the pure species. Parameter evaluation may be accomplished with the equations 0.42748R2 Tcij2.5 aij = ᎏᎏ (4-223) Pcij 0.08664RTci (4-224) bi = ᎏᎏ Pci where Eqs. (4-215) through (4-218) provide for the calculation of the Tcij and Pcij. Multiplication of the Redlich/Kwong equation (Eq. [4-220]) by V/RT leads to its expression in alternative form: a 1 h Z=ᎏ−ᎏ (4-225) ᎏ 1 − h bRT1.5 1 + h









a h h Z−1=ᎏ−ᎏ ᎏ 1 − h bRT1.5 1 + h

Whence

(4-226)

bP h=ᎏ (4-227) ZRT Equations (4-170) and (4-174) in combination with Eq. (4-226) give GR a ln (1 + h) (4-228) ᎏ = Z − 1 − ln (1 − h)Z − ᎏ RT bRT1.5 where



R







H 3a ln (1 + h) (4-229) ᎏ=Z−1− ᎏ RT 2 bRT 1.5 Once a and b are determined by Eqs. (4-221) through (4-224), then for given T and P values of Z, GR/RT, and HR/RT are found by Eqs. (4-225), (4-228), and (4-229) and SR/R by Eq. (4-160). The procedure requires initial solution of Eqs. (4-225) and (4-227) for Z and h. The original Redlich/Kwong equation is rarely satisfactory for vapor/liquid equilibrium calculations, and equations have been developed specific to this purpose. The two most popular are the Soave/ Redlich/Kwong (SRK) equation, a modification of the Redlich/Kwong equation (Soave, Chem. Eng. Sci., 27, pp. 1197–1203 [1972]), and the Peng/Robinson (PR) equation (Peng and Robinson, Ind. Eng. Chem. Fundam., 15, pp. 59–64 [1976]). Both equations are designed specifically to yield reasonable vapor pressures for pure fluids. Thus, there is no assurance that molar volumes calculated by these equations are more accurate than values given by the original Redlich/Kwong equation. Written for pure species i the SRK and PR equations are special cases of the following: RT ai(T ) P = ᎏ − ᎏᎏ (4-230) Vi − bi (Vi + εbi)(Vi + σbi) and

where

Ωaα(Tri; ωi)R2 Tci2 ai(T) = ᎏᎏ Pci

Ωb RTci bi = ᎏ Pci and ε σ, Ωa, and Ωb are equation-specific constants. For the Soave/ Redlich/Kwong equation: α(Tri; ωi) = [1 + (0.480 + 1.574ωi − 0.176ω2i ) (1 − Tri1/2)]2

4-21

For the Peng/Robinson equation: α(Tri; ωi) = [1 + (0.37464 + 1.54226ωi − 0.26992ω i2) (1 − Tri1/2)]2 Written for a mixture, Eq. (4-230) becomes RT a(T) P = ᎏ − ᎏᎏ (4-231) V − b (V + εb)(V + σb) where a and b are mixture values, related to the ai and bi by mixing rules. Equation (4-170) applied to Eq. (4-231) leads to ai  bi bi (V − b)Z a/bRT V + σb  ln φˆ i = ᎏ (Z − 1) − ln ᎏ + ᎏ 1 + ᎏ − ᎏ ln ᎏ b a b V ε−σ V + εb (4-232)





where ai and bi are partial parameters for species i, defined by ∂(na) (4-233) ai = ᎏ ∂ni T,nj

  ∂(nb) b = ᎏ  ∂n

and

(4-234)

i

i

T,nj

These are general equations that do not depend on the particular mixing rules adopted for the composition dependence of a and b. The mixing rules given by Eqs. (4-221) and (4-222) can certainly be employed with these equations. However, for purposes of vapor/liquid equilibrium calculations, a special pair of mixing rules is far more appropriate, and will be introduced when these calculations are treated. Solution of Eq. (4-232) for fugacity coefficient φˆ i at given T and P requires prior solution of Eq. (4-231) for V, from which is found Z = PV/RT. Benedict/Webb/Rubin Equation of State The BWR equation of state with Z as the dependent variable is written A0 C0 a Z = 1 + B0 − ᎏ −ᎏ ρ + b − ᎏ ρ2 RT RT 3 RT



 



c aα + ᎏ ρ5 + ᎏ3 ρ2 (1 + γρ2) exp (−γρ2) (4-235) RT RT All eight parameters depend on composition; moreover, parameters C0, b, and γ are for some applications treated as functions of T. By Eq. (4-171), the residual Gibbs energy is GR C0 3 a A0 6aα ᎏ = 2 B0 − ᎏ − ᎏ3 ρ + ᎏ b − ᎏ ρ2 + ᎏ ρ5 RT RT RT 2 RT 5RT













c + ᎏ3 (2γ 2ρ4 + γρ2 − 2) exp (−γ ρ2) + 2 − ln Z (4-236) 2γRT With allowance for T dependence of C0, b, and γ, Eq. (4-175) yields HR 1 dC0 2A0 4C0 ᎏ = B0 − ᎏ − ᎏ3 + ᎏ2 ᎏ ρ RT RT RT RT dT









1 db 3a 6aα − ᎏ T ᎏ − 2b + ᎏ ρ2 + ᎏ ρ5 2 dT RT 5RT





c + ᎏ3 (2γ 2ρ4 − γρ2 − 6) exp (−γρ2) + 6 2γRT





c dγ −ᎏ (4-237) ᎏ (γ 2ρ4 + 2γρ2 + 2) exp (−γρ2) − 2 2γ 2RT 2 dT The residual entropy is given by Eq. (4-160). Computation of ln φˆ i is done via Eq. (4-181). The result is A0 + A 0 C0 + C  0i ln φˆ i = B0 +  B0i − ᎏi − ᎏ ρ RT RT 3





4aα + aα 2a + ai 2 1 i + αai ρ5 bi − ᎏ ρ + ᎏᎏ + ᎏ 2b +  5RT RT 2



c + ᎏ3 2γRT



2 γi



1 + ᎏγ  γ ρ + ᎏγ − ᎏc  ρ

ci γi −2 1+ᎏ −ᎏ c γ



γi



2 4

ci

2

γi

 exp (−γρ ) + 2 1 + ᎏc − ᎏγ  − ln Z 2

ci

(4-238)

4-22

THERMODYNAMICS

Here the quantities with overbars are partial parameters for species i, defined for arbitrary parameter π by ∂(nπ) πi
ᎏ (4-239)  ∂ni T,nj Application of these equations requires specific mixing rules. For example, if



π=



 y π 

r

1/r k k

(4-240)

The symmetrical nature of these relations is evident. The infinitedilution values of the activity coefficients are ln γ 1∞ = ln γ 2∞ = B. If D = ⋅ ⋅ ⋅ = 0, then GE ᎏ = B + C(x1 − x2) = B + C(2x1 − 1) x1 x2 RT and in this case GE/x1 x2 RT is linear in x1. The substitutions, B + C = A21 and B − C = A12 transform this expression into the Margules equation: GE/x1 x2 RT = A21 x1 + A12 x2

k

where r is a small integer, the recipe for  πi is πi 1/r π (4-241) i = π r ᎏ − (r − 1) π Specifically, if r = 3 for π
c; then ci 1/3 −2 ci = c 3 ᎏ c where ci is the parameter for pure i and c is the parameter for the mixture, given by

 



 

c=



 y c 

3

1/3 k k

k

Equation-of-state examples are given in Smith, Van Ness, and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed., Secs. 3.4–3.7 and 6.2–6.6, McGraw-Hill, New York, 1996). EXPRESSIONS FOR THE EXCESS GIBBS ENERGY In principle, equation-of-state procedures can be used for the calculation of liquid-phase as well as gas-phase properties, and much has been accomplished in the development of PVT equations of state suitable for both phases. However, a widely used alternative for the liquid phase is application of excess properties. The excess property of primary importance for engineering calculations is the excess Gibbs energy GE, because its canonical variables are T, P, and composition, the variables usually specified or sought in a design calculation. Knowing GE as a function of T, P, and composition, one can in principle compute from it all other excess properties (see, for example, Eqs. [4-117] through [4-119]). As noted with respect to Fig. 4-1, the excess volume for liquid mixtures is usually small; the pressure dependence of GE may then be safely ignored. Thus, the engineering efforts at describing GE center on representing its composition and temperature dependence. For binary systems at constant T, GE is a function of just x1, and the quantity most conveniently represented by an equation is GE/x1x2RT. The simplest procedure is to express this quantity as a power series in x1: GE ᎏ = a + bx1 + cx 12 + ⋅ ⋅ ⋅ (constant T) x1 x2 RT An equivalent power series with certain advantages is known as the Redlich/Kister expansion (Redlich, Kister, and Turnquist, Chem. Eng. Progr. Symp. Ser. No. 2, 48, pp. 49–61 [1952]): GE ᎏ = B + C(x1 − x2) + D(x1 − x2)2 + ⋅ ⋅ ⋅ x1x2 RT In application, different truncations of this series are appropriate. For each particular expression representing GE/x1x2 RT, specific expressions for ln γ1 and ln γ2 result from application of Eq. (4-119). When all parameters are zero, GE/RT = 0, and the solution is ideal. If C = D = ⋅ ⋅ ⋅ = 0, then GE ᎏ=B x1 x2 RT where B is a constant for a given temperature. The corresponding equations for ln γ1 and ln γ2 are ln γ1 = Bx22

(4-242)

ln γ2 = Bx 21

(4-243)

(4-244)

Application of Eq. (4-119) yields ln γ1 = x 22 [A12 + 2(A21 − A12)x1]

(4-245)

ln γ2 = x 21 [A21 + 2(A12 − A21)x2]

(4-246)

An alternative equation is obtained when the reciprocal quantity x1x2RT/GE is expressed as a linear function of x1: x1x2 ᎏ = B′ + C′ (x1 − x2) = B′ + C′ (2x1 − 1) GE/RT This may also be written: x1x2 = B′ (x1 + x2) + C′ (x1 − x2) = (B′ + C′)x1 + (B′ − C′)x2 ᎏ GE/RT The substitutions B′ + C′ = 1/A′21 and B′ − C′ = 1/A′12 produce x1x2 x1 x2 A′12 x1 + A′21 x2 = ᎏ + ᎏ = ᎏᎏ ᎏ GE/RT A′21 A′12 A′12 A′21 or

GE A′12 A′21 ᎏ = ᎏᎏ x1x2 RT A′12 x1 + A′21 x2

(4-247)

The activity coefficients implied by this equation are given by A′12 x1 −2 (4-248) ln γ1 = A′12 1 + ᎏ A′21 x2









A′21 x2 ln γ2 = A′21 1 + ᎏ A′12 x1

−2

(4-249)

These are known as the van Laar equations. When x1 = 0, ln γ 1∞ = A′12; when x2 = 0, ln γ 2∞ = A′21. The Redlich/Kister expansion, the Margules equations, and the van Laar equations are all special cases of a very general treatment based on rational functions, that is, on equations for GE given by ratios of polynomials (Van Ness and Abbott, Classical Thermodynamics of Nonelectrolyte Solutions: With Applications to Phase Equilibria, Sec. 5-7, McGraw-Hill, New York, 1982). Although providing great flexibility in the fitting of VLE data for binary systems, they are without theoretical foundation, with no rational basis for their extension to multicomponent systems. Nor do they incorporate an explicit temperature dependence for the parameters. Modern theoretical developments in the molecular thermodynamics of liquid-solution behavior are often based on the concept of local compositon, presumed to account for the short-range order and nonrandom molecular orientations that result from differences in molecular size and intermolecular forces. Introduced with the publication of a model of GE behavior known as the Wilson equation ( J. Am. Chem. Soc., 86, pp. 127–130 [1964]), it prompted the development of alternative local-composition models, most notably the NRTL (Non-Random-Two-Liquid) equation of Renon and Prausnitz (AIChE J., 14, pp. 135–144 [1968]) and the UNIQUAC (UNIversal QUAsi-Chemical) equation of Abrams and Prausnitz (AIChE J., 21, pp. 116–128 [1975]). A further significant development, based on the UNIQUAC equation, is the UNIFAC method (UNIQUAC Functional-group Activity Coefficients). Proposed by Fredenslund, Jones, and Prausnitz (AIChE J., 21, pp. 1086–1099 [1975]) and given detailed treatment by Fredenslund, Gmehling, and Rasmussen (Vapor-Liquid Equilibrium Using UNIFAC, Elsevier, Amsterdam, 1977), it provides for the calculation of activity coefficients from contributions of the various groups making up the molecules of a solution.

EVALUATION OF PROPERTIES The Wilson equation, like the Margules and van Laar equations, contains just two parameters for a binary system (Λ12 and Λ21), and is written: GE (4-250) ᎏ = −x1 ln (x1 + x2Λ12) − x2 ln (x2 + x1Λ21) RT Λ12 Λ21 ln γ1 = −ln (x1 + x2Λ12) + x2 ᎏᎏ − ᎏᎏ x1 + x2Λ12 x2 + x1Λ21



Λ21 Λ12 ln γ2 = −ln (x2 + x1Λ21) − x1 ᎏᎏ − ᎏᎏ x1 + x2Λ12 x2 + x1Λ21



whence





(4-252)

Both Λ12 and Λ21 must be positive numbers. The NRTL equation contains three parameters for a binary system and is written: GE G21τ21 G12τ12 (4-253) ᎏ = ᎏᎏ + ᎏᎏ x1x2 RT x1 + x2 G21 x2 + x1 G12 G21 ln γ1 = x 22 τ21 ᎏᎏ x1 + x2 G21

 

G τ  + ᎏᎏ (x + x G ) 

(4-254)

 

G τ  + ᎏᎏ (x + x G ) 

(4-255)

G12 ln γ2 = x 21 τ12 ᎏᎏ x2 + x1 G12 Here

G12 = exp (−ατ12)

12 12

2

1

12

2

21 21

1

2

21

2

b12 b21 τ21 = ᎏ τ12 = ᎏ RT RT where α, b12, and b21, parameters specific to a particular pair of species, are independent of composition and temperature. The infinite-dilution values of the activity coefficients are given by the equations:

and

ln γ 1∞ = τ21 + τ12 exp (−ατ12) ln γ 2∞ = τ12 + τ 21 exp (−ατ 21)

j

Function g contains pure-species parameters only, whereas function g R incorporates two binary parameters for each pair of molecules. For a multicomponent system, Φ θ gC =  xi ln ᎏi + 5  qi xi ln ᎏi (4-260) xi Φi i i gR = − qi xi ln i

where

 θ τ 

(4-261)

j ji

j

xi ri Φi
ᎏ  xj rj

(4-262)

xi qi θi
ᎏ  xj qj

(4-263)

j

and

Subscript i identifies species, and j is a dummy index; all summations are over all species. Note that τji ≠ τij; however, when i = j, then τii = τjj = 1. In these equations ri (a relative molecular volume) and qi (a relative molecular surface area) are pure-species parameters. The influence of temperature on g enters through the interaction parameters τji of Eq. (4-261), which are temperature dependent: −(uji − uii) τji = exp ᎏᎏ (4-264) RT Parameters for the UNIQUAC equation are therefore values of (uji − uii). An expression for ln γi is found by application of Eq. (4-119) to the UNIQUAC equation for g (Eqs. [4-259] through [4-261]). The result is given by the following equations: ln γ i = ln γ iC + ln γ iR

The local-composition models have limited flexibility in the fitting of data, but they are adequate for most engineering purposes. Moreover, they are implicitly generalizable to multicomponent systems without the introduction of any parameters beyond those required to describe the constituent binary systems. For example, the Wilson equation for multicomponent systems is written: GE (4-256) ᎏ = − xi ln  xj Λij RT i j and

(4-259)

j

G21 = exp (−ατ21)

xkΛki ln γi = 1 − ln  xj Λij −  ᎏ  xj Λkj j k

g = gC + g R C

ln γ 2∞ = −ln Λ21 + 1 − Λ12

2

binary (in contrast to multicomponent) systems. This makes parameter determination for the local-composition models a task of manageable proportions. The UNIQUAC equation treats g
GE/RT as comprised of two additive parts, a combinatorial term g C, accounting for molecular size and shape differences, and a residual term g R (not a residual property), accounting for molecular interactions:

(4-251)

ln γ 1∞ = −ln Λ12 + 1 − Λ21

2

4-23

(4-257)

where Λij = 1 for i = j, and so on. All indices in these equations refer to the same species, and all summations are over all species. For each ij pair there are two parameters, because Λij ≠ Λji. For example, in a ternary system the three possible ij pairs are associated with the parameters Λ12, Λ21; Λ13, Λ31; and Λ23, Λ32. The temperature dependence of the parameters is given by: −a V Λij = ᎏj exp ᎏij (i ≠ j) (4-258) Vi RT where Vj and Vi are the molar volumes at temperature T of pure liquids j and i, and aij is a constant independent of composition and temperature. Thus the Wilson equation, like all other local-composition models, has built into it an approximate temperature dependence for the parameters. Moreover, all parameters are found from data for

(4-265)



J J ln γ i = 1 − Ji + ln Ji − 5qi 1 − ᎏi + ln ᎏi Li Li C

τij ln γ iR = qi 1 − ln si −  θj ᎏ sj j





where in addition to Eqs. (4-263) and (4-264) ri Ji = ᎏ  rj xj



(4-266) (4-267)

(4-268)

j

qi Li = ᎏ  qj xj

(4-269)

si =  θl τ li

(4-270)

j

l

Again subscript i identifies species, and j and l are dummy indicies. Values for the parameters ri, qi, and (uij − ujj) are given by Gmehling, Onken, and Arlt (Vapor-Liquid Equilibrium Data Collection, Chemistry Data Series, vol. I, parts 1–8, DECHEMA, Frankfurt/Main, 1974–1990). The Wilson parameters Λij, NRTL parameters Gij, and UNIQUAC parameters τij all inherit a Boltzmann-type T dependence from the origins of the expressions for GE, but it is only approximate. Computations of properties sensitive to this dependence (e.g., heats of mixing and liquid/liquid solubility) are in general only qualitatively correct.

4-24

THERMODYNAMICS

EQUILIBRIUM CRITERIA The equations developed in preceding sections are for PVT systems in states of internal equilibrium. The criteria for internal thermal and mechanical equilibrium are well known, and need not be discussed in detail. They simply require uniformity of temperature and pressure throughout the system. The criteria for phase and chemical-reaction equilibria are less obvious. Consider a closed PVT system, either homogeneous or heterogeneous, of uniform T and P, which is in thermal and mechanical equilibrium with its surroundings, but which is not initially at internal equilibrium with respect to mass transfer or with respect to chemical reaction. Changes occurring in the system are then irreversible, and must necessarily bring the system closer to an equilibrium state. The first and second laws written for the entire system are dU t = dQ + dW dQ dSt ≥ ᎏ T

dW = −P dV t dU t + P dV t − T dSt ≤ 0

The inequality applies to all incremental changes toward the equilibrium state, whereas the equality holds at the equilibrium state where any change is reversible. Various constraints may be put on this expression to produce alternative criteria for the directions of irreversible processes and for the condition of equilibrium. For example, it follows immediately that ≤0

Alternatively, other pairs of properties may be held constant. The most useful result comes from fixing T and P, in which case d(Ut + PVt − TSt)T,P ≤ 0 t ≤0 dGT,P

or

i

dn″i = −dn′i

 (µ′ − µ″) dn′ = 0

Therefore

i

i

i

i

Since the dn′i are independent and arbitrary, it follows that µ′i = µ″i This is the criterion of two-phase equilibrium. It is readily generalized to multiple phases by successive application to pairs of phases. The general result is

i

Since mechanical equilibrium is assumed,

dU

i

Since the system is closed and without chemical reaction, material balances require that

(4-272)

Substitution for each µi by Eq. (4-77) produces the equivalent result (4-273) fˆ ′ = fˆ ″ = fˆ ′′′ = ⋅ ⋅ ⋅

dU t − dW − T dSt ≤ 0

t St,V t

t dG T,P
d(nG)T,P =  µ′i dn′i +  µ″i dn″i = 0

µ′i = µ″i = µ′′′ i = ⋅ ⋅ ⋅

Combination gives

Whence

If the two-phase system is at equilibrium, then application of Eq. (4-271) yields

This expression shows that all irreversible processes occurring at constant T and P proceed in a direction such that the total Gibbs energy of the system decreases. Thus the equilibrium state of a closed system is the state with the minimum total Gibbs energy attainable at the given T and P. At the equilibrium state, differential variations may occur in the system at constant T and P without producing a change in Gt. This is the meaning of the equilibrium criterion t =0 dGT,P

(4-271)

This equation may be applied to a closed, nonreactive, two-phase system. Each phase taken separately is an open system, capable of exchanging mass with the other, and Eq. (4-16) may be written for each phase: d(nG)′ = −(nS)′ dT + (nV)′ dP +  µ′i dn′i i

d(nG)″ = −(nS)″ dT + (nV)″ dP +  µ″i dn″i i

where the primes and double primes denote the two phases and the presumption is that T and P are uniform throughout the two phases. The change in the Gibbs energy of the two-phase system is the sum of these equations. When each total-system property is expressed by an equation of the form nM = (nM)′ + (nM)″

i

i

These are the criteria of phase equilibrium applied in the solution of practical problems. For the case of equilibrium with respect to chemical reaction within a single-phase closed system, combination of Eqs. (4-16) and (4-271) leads immediately to

 µ dn = 0 i

i

(4-274)

i

For a system in which both phase and chemical-reaction equilibrium prevail, the criteria of Eqs. (4-272) and (4-274) are superimposed. THE PHASE RULE The intensive state of a PVT system is established when its temperature and pressure and the compositions of all phases are fixed. However, for equilibrium states these variables are not all independent, and fixing a limited number of them automatically establishes the others. This number of independent variables is given by the phase rule, and is called the number of degrees of freedom of the system. It is the number of variables which may be arbitrarily specified and which must be so specified in order to fix the intensive state of a system at equilibrium. This number is the difference between the number of variables needed to characterize the system and the number of equations that may be written connecting these variables. For a system containing N chemical species distributed at equilibrium among π phases, the phase-rule variables are temperature and pressure, presumed uniform throughout the system, and N − 1 mole fractions in each phase. The number of these variables is 2 + (N − 1)π. The masses of the phases are not phase-rule variables, because they have nothing to do with the intensive state of the system. The equations that may be written connecting the phase-rule variables are: 1. Equation (4-272) for each species, giving (π − 1)N phaseequilibrium equations. 2. Equation (4-274) for each independent chemical reaction, giving r equations. The total number of independent equations is therefore (π − 1)N + r. In their fundamental forms these equations relate chemical potentials, which are functions of temperature, pressure, and composition, the phase-rule variables. Since the degrees of freedom of the system F is the difference between the number of variables and the number of equations,

this sum is given by

F = 2 + (N − 1)π − (π − 1)N − r

d(nG) = (nV) dP − (nS) dT +  µ′i dn′i +  µ″i dn″i i

i

or

F=2−π+N−r

(4-275)

EQUILIBRIUM The number of independent chemical reactions r can be determined as follows: 1. Write formation reactions from the elements for each chemical compound present in the system. 2. Combine these reaction equations so as to eliminate from the set all elements not present as elements in the system. A systematic procedure is to select one equation and combine it with each of the other equations of the set so as to eliminate a particular element. This usually reduces the set by one equation for each element eliminated, though two or more elements may be simultaneously eliminated. The resulting set of r equations is a complete set of independent reactions. More than one such set is often possible, but all sets number r and are equivalent. Example 1: Application of the Phase Rule a. For a system of two miscible nonreacting species in vapor/liquid equilibrium, F=2−π+N−r=2−2+2−0=2 The two degrees of freedom for this system may be satisfied by setting T and P, or T and y1, or P and x1, or x1 and y1, and so on, at fixed values. Thus, for equilibrium at a particular T and P, this state (if possible at all) exists only at one liquid and one vapor composition. Once the two degrees of freedom are used up, no further specification is possible that would restrict the phase-rule variables. For example, one cannot in addition require that the system form an azeotrope (assuming this possible), for this requires x1 = y1, an equation not taken into account in the derivation of the phase rule. Thus, the requirement that the system form an azeotrope imposes a special constraint and reduces the number of degrees of freedom to one. b. For a gaseous system consisting of CO, CO2, H2, H2O, and CH4 in chemical-reaction equilibrium, F=2−π+N−r=2−1+5−2=4 The value of r = 2 is found from the formation reactions: C + a O2 → CO C + O2 → CO2 H2 + a O2 → H2O C + 2H2 → CH4 Systematic elimination of C and O2 from this set of chemical equations reduces the set to two. Three possible pairs of equations may result, depending on how the combination of equations is effected. Any pair of the following three equations represents a complete set of independent reactions, and all pairs are equivalent. CH4 + H2O → CO + 3H2 CO + H2O → CO2 + H2 CH4 + 2H2O → CO2 + 4H2 The result, F = 4, means that one is free to specify, for example, T, P, and two mole fractions in an equilibrium mixture of these five chemical species, provided nothing else is arbitrarily set. Thus, it cannot simultaneously be required that the system be prepared from specified amounts of particular constituent species.

Since the phase rule treats only the intensive state of a system, it applies to both closed and open systems. Duhem’s theorem, on the other hand, is a rule relating to closed systems only: For any closed system formed initially from given masses of prescribed chemical species, the equilibrium state is completely determined by any two properties of the system, provided only that the two properties are independently variable at the equilibrium state. The meaning of completely determined is that both the intensive and extensive states of the system are fixed; not only are T, P, and the phase compositions established, but so also are the masses of the phases. VAPOR/LIQUID EQUILIBRIUM Vapor/liquid equilibrium (VLE) relationships (as well as other interphase equilibrium relationships) are needed in the solution of many engineering problems. The required data can be found by experiment, but such measurements are seldom easy, even for binary systems, and they become rapidly more difficult as the number of constituent species increases. This is the incentive for application of thermodynamics to the calculation of phase-equilibrium relationships.

4-25

The general VLE problem involves a multicomponent system of N constituent species for which the independent variables are T, P, N − 1 liquid-phase mole fractions, and N − 1 vapor-phase mole fractions. (Note that i xi = 1 and i yi = 1, where xi and yi represent liquid and vapor mole fractions respectively.) Thus there are 2N independent variables, and application of the phase rule shows that exactly N of these variables must be fixed to establish the intensive state of the system. This means that once N variables have been specified, the remaining N variables can be determined by simultaneous solution of the N equilibrium relations: fˆ l = fˆ v (i = 1, 2, . . . , N) (4-276) i

i

where superscripts l and v denote the liquid and vapor phases, respectively. In practice, either T or P and either the liquid-phase or vapor-phase composition are specified, thus fixing 1 + (N − 1) = N independent variables. The remaining N variables are then subject to calculation, provided that sufficient information is available to allow determination of all necessary thermodynamic properties. Gamma/Phi Approach For many VLE systems of interest the pressure is low enough that a relatively simple equation of state, such as the two-term virial equation, is satisfactory for the vapor phase. Liquid-phase behavior, on the other hand, may be conveniently described by an equation for the excess Gibbs energy, from which activity coefficients are derived. The fugacity of species i in the liquid phase is then given by Eq. (4-102), written fˆ l = γ x f i

i i i

while the vapor-phase fugacity is given by Eq. (4-79), written fˆiv = φˆ iv yi P Equation (4-276) is now expressed as (i = 1, 2, . . . , N ) γi xi fi = φˆ i yiP

(4-277)

The identifying superscripts l and v are omitted here with the understanding that γi and fi are liquid-phase properties, whereas φˆ i is a vapor-phase property. Applications of Eq. (4-277) represent what is known as the gamma/phi approach to VLE calculations. Evaluation of φˆ i is usually by Eq. (4-196), based on the two-term virial equation of state, but other equations, such as Eq. (4-200), are also applicable. The activity coefficient γi is evaluated by Eq. (4-119), which relates ln γi to GE/RT as a partial property. Thus, what is required for the liquid phase is a relation between GE/RT and composition. Equations in common use for this purpose have already been described. The fugacity fi of pure compressed liquid i must be evaluated at the T and P of the equilibrium mixture. This is done in two steps. First, one calculates the fugacity coefficient of saturated vapor φiv = φisat by an integrated form of Eq. (4-161), written for pure species i and evaluated at temperature T and the corresponding vapor pressure P = Pisat. Equation (4-276) written for pure species i becomes f iv = f il = f isat

(4-278)

sat

where fi indicates the value both for saturated liquid and for saturated vapor. The corresponding fugacity coefficient is fisat φisat = ᎏ (4-279) Pisat This fugacity coefficient applies equally to saturated vapor and to saturated liquid at given temperature T. Equation (4-278) can therefore equally well be written φ iv = φ il

(4-280)

The second step is the evaluation of the change in fugacity of the liquid with a change in pressure to a value above or below Pisat. For this isothermal change of state from saturated liquid at Pisat to liquid at pressure P, Eq. (4-105) is integrated to give Gi − Gisat =



P

sat Pi

Vi dP

Equation (4-74) is then written twice: for Gi and for Gisat. Subtraction provides another expression for Gi − Gisat:

4-26

THERMODYNAMICS

fi Gi − Gisat = RT ln ᎏ fisat Equating the two expressions for Gi − Gisat yields P 1 fi ln ᎏ = ᎏ sat Vi dP sat RT Pi fi Since Vi, the liquid-phase molar volume, is a very weak function of P at temperatures well below Tc, an excellent approximation is often obtained when evaluation of the integral is based on the assumption that Vi is constant at the value for saturated liquid, Vil: fi Vil(P − Pisat) ln ᎏ = ᎏᎏ sat fi RT Substituting fisat = φ isatPisat (Eq. [4-279]), and solving for fi gives Vil(P − Pisat) fi = φ isat Pisat exp ᎏᎏ (4-281) RT The exponential is known as the Poynting factor. Equation (4-277) may now be written



yiPΦi = xiγi Pisat

(i = 1, 2, . . . , N)

(4-282)

φˆ i −Vi (P − Pi ) Φi = ᎏ exp ᎏᎏ (4-283) φisat RT If evaluation of φisat and φˆ i is by Eqs. (4-192) and (4-196), this reduces to PB i − PisatBii − Vil(P − Pisat) Φi = exp ᎏᎏᎏ (4-284) RT where  Bi is given by Eq. (4-195): ∂(nB) Bi
ᎏ = 2  ykBki − B (4-285)  ∂ni T,nj k with B evaluated by Eq. (4-183). The N equations represented by Eq. (4-282) in conjunction with Eq. (4-284) may be used to solve for N unspecified phase-equilibrium variables. For a multicomponent system the calculation is formidable, but well suited to computer solution. The types of problems encountered for nonelectrolyte systems at low to moderate pressures (well below the critical pressure) are discussed by Smith, Van Ness, and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed., McGraw-Hill, New York, 1996). When Eq. (4-282) is applied to VLE for which the vapor phase is an ideal gas and the liquid phase is an ideal solution, it reduces to a very simple expression. For ideal gases, fugacity coefficients φˆ i and φisat are unity, and the right-hand side of Eq. (4-283) reduces to the Poynting factor. For the systems of interest here this factor is always very close to unity, and for practical purposes Φi = 1. For ideal solutions, the activity coefficients γi are also unity. Equation (4-282) therefore reduces to

 

where

l

sat





yiP = xiPisat





(i = 1, 2, . . . , N)

an equation which expresses Raoult’s law. It is the simplest possible equation for VLE, and as such fails to provide a realistic representation of real behavior for most systems. Nevertheless, it is useful as a standard of comparison. When an appropriate correlating equation for G E is not available, reliable estimates of activity coefficients may often be obtained from a group-contribution correlation. The Analytical Solution of Groups (ASOG) method (Kojima and Tochigi, Prediction of Vapor-Liquid Equilibrium by the ASOG Method, Elsevier, Amsterdam, 1979) and the UNIFAC method are both well developed. Additional references of interest include Hansen et al. (Ind. Eng. Chem. Res., 30, pp. 2352– 2355 [1991]), Gmehling and Schiller (Ibid., 32, pp. 178–193 [1993]); Larsen et al. (Ibid., 26, pp. 2274–2286 [1987]); and Tochigi et al. ( J. Chem. Eng. Japan, 23, pp. 453–463 [1990]). Data Reduction Correlations for GE and the activity coefficients are based on VLE data taken at low to moderate pressures. The ASOG and UNIFAC group-contribution methods depend for validity on parameters evaluated from a large base of such data. The process

of finding a suitable analytic relation for g (
GE/RT) as a function of its independent variables T and x1, thus producing a correlation of VLE data, is known as data reduction. Although g is in principle also a function of P, the dependence is so weak as to be universally and properly neglected. Given here is a brief description of the treatment of data taken for binary systems under isothermal conditions. A more comprehensive development is given by Van Ness ( J. Chem. Thermodyn., 27, pp. 113–134 [1995]; Pure & Appl. Chem., 67, pp. 859–872 [1995]). Presumed in all that follows is the existence of an equation inherently capable of representing correct values of GE for the liquid phase as a function of x1: g
GE/RT =  (x1; α, β, . . .)

(4-286)

where α, β, and so on, represent adjustable parameters. The measured variables of binary VLE are x1, y1, T, and P. Experimental values of the activity coefficient of species i in the liquid are related to these variables by Eq. (4-282), written: y*i P* γ*i = ᎏ Φi (i = 1, 2) (4-287) xiPisat where Φi is given by Eq. (4-283), and the asterisks denote experimental values. A simple summability relation analogous to Eq. (4-123) defines an experimental value of g*: g*
x1 ln γ 1* + x2 ln γ 2*

(4-288)

Moreover, Eq. (4-122), the Gibbs/Duhem equation, may be written for experimental values in a binary system as d ln γ1* d ln γ 2* x1 ᎏ + x2 ᎏ = 0 (4-289) dx1 dx1 Because experimental measurements are subject to systematic error, sets of values of ln γ 1* and ln γ 2* determined by experiment may not satisfy, that is, may not be consistent with, the Gibbs/Duhem equation. Thus, Eq. (4-289) applied to sets of experimental values becomes a test of the thermodynamic consistency of the data, rather than a valid general relationship. Values of g given by the correlating equation, Eq. (4-286), are called derived values, and associated derived values of the activity coefficients are given by specialization of Eqs. (4-58): dg γ1 = exp g + x2 ᎏ (4-290) dx1









dg (4-291) γ2 = exp g − x1 ᎏ dx1 These two equations may be combined to yield dg γ1 (4-292) ᎏ = ln ᎏ dx1 γ2 This equation applies to derived property values. The corresponding experimental values are given by differentiation of Eq. (4-288): dg* d ln γ 1* d ln γ 2* ᎏ = x1 ᎏ + ln γ 1* + x2 ᎏ − ln γ 2* dx1 dx1 dx1 dg* γ 1* d ln γ 1* d ln γ 2* ᎏ = ln ᎏ + x1 ᎏ + x2 ᎏ dx1 γ 2* dx1 dx1 Subtraction of Eq. (4-293) from Eq. (4-292) gives

or

(4-293)

γ1 dg dg* γ 1* d ln γ 1* d ln γ 2* ᎏ − ᎏ = ln ᎏ − ln ᎏ − x1 ᎏ + x2 ᎏ dx1 dx1 γ2 γ 2* dx1 dx1 The differences between like terms represent residuals between derived and experimental values. Defining these residuals as γ γ γ 1* δ g
g − g* and δ ln ᎏ1
ln ᎏ1 − ln ᎏ γ2 γ2 γ 2* puts this equation into the form dδg γ1 d ln γ 1* d ln γ 2* ᎏ = δ ln ᎏ − x1 ᎏ + x2 ᎏ dx1 γ2 dx1 dx1









EQUILIBRIUM If a data set is reduced so as to make the δg residuals scatter about zero, then the derivative on the left is effectively zero, and the preceding equation becomes γ d ln γ1* d ln γ 2* δ ln ᎏ1 = x1 ᎏ + x2 ᎏ (4-294) γ2 dx1 dx1 The right-hand side of this equation is exactly the quantity that Eq. (4-289), the Gibbs/Duhem equation, requires to be zero for consistent data. The residual on the left is therefore a direct measure of deviations from the Gibbs/Duhem equation. The extent to which values of this residual fail to scatter about zero measures the departure of the data from consistency with respect to this equation. The data-reduction procedure just described provides parameters in the correlating equation for g that make the δg residuals scatter about zero. This is usually accomplished by finding the parameters that minimize the sum of squares of the residuals. Once these parameters are found, they can be used for the calculation of derived values of both the pressure P and the vapor composition y1. Equation (4-282) is solved for yiP and written for species 1 and for species 2. Adding the two equations gives x1γ1P1sat x2γ2P2sat P=ᎏ +ᎏ (4-295) Φ1 Φ2 whence by Eq. (4-282), x1γ1P1sat y1 = ᎏ (4-296) Φ1P These equations allow calculation of the primary residuals: δP
P − P*

and

δy1
y1 − y1*

If the experimental values P* and y1* are closely reproduced by the correlating equation for g, then these residuals, evaluated at the experimental values of x1, scatter about zero. This is the result obtained when the data are thermodynamically consistent. When they are not, these residuals do not scatter about zero, and the correlation for g does not properly reproduce the experimental values P* and y1*. Such a correlation is, in fact, unnecessarily divergent. An alternative is to process just the P-x1 data; this is possible because the P-x1-y1 data set includes more information than necessary. Assuming that the correlating equation is appropriate to the data, one merely searches for values of the parameters α, β, and so on, that yield pressures by Eq. (4-295) that are as close as possible to the measured values. The usual procedure is to minimize the sum of squares of the residuals δP. Known as Barker’s method (Austral. J. Chem., 6, pp. 207–210 [1953]), it provides the best possible fit of the experimental pressures. When the experimental data do not satisfy the Gibbs/Duhem equation, it cannot precisely represent the experimental y1 values; however, it provides a better fit than does the procedure that minimizes the sum of the squares of the δg residuals. Worth noting is the fact that Barker’s method does not require experimental y1* values. Thus the correlating parameters α, β, and so on, can be evaluated from a P-x1 data subset. Common practice now is, in fact, to measure just such data. They are, of course, not subject to a test for consistency by the Gibbs/Duhem equation. The world’s store of VLE data has been compiled by Gmehling et al. (VaporLiquid Equilibrium Data Collection, Chemistry Data Series, vol. I, parts 1–8, DECHEMA, Frankfurt am Main, 1979–1990). Solute/Solvent Systems The gamma/phi approach to VLE calculations presumes knowledge of the vapor pressure of each species at the temperature of interest. For certain binary systems species 1, designated the solute, is either unstable at the system temperature or is supercritical (T > Tc ). Its vapor pressure cannot be measured, and its fugacity as a pure liquid at the system temperature f1 cannot be calculated by Eq. (4-281). Equations (4-282) and (4-283) are applicable to species 2, designated the solvent, but not to the solute, for which an alternative approach is required. Figure 4-11 shows a typical plot of the liquidphase fugacity of the solute fˆ1 vs. its mole fraction x1 at constant temperature. Since the curve representing fˆ1 does not extend all the way to x1 = 1, the location of f1, the liquid-phase fugacity of pure species 1, is not established. The tangent line at the origin, representing Henry’s

FIG. 4-11

4-27

Plot of solute fugacity fˆ1 vs. solute mole fraction.

law, provides alternative information. The slope of the tangent line is Henry’s constant, defined as fˆ1 k1
lim ᎏ (4-297) x1→0 x 1 This is the definition of k1 for temperature T and for a pressure equal to the vapor pressure of the pure solvent P2sat. The activity coefficient of the solute at infinite dilution is fˆ1 1 fˆ1 lim γ1 = lim ᎏ = ᎏ lim ᎏ x1→0 x1→0 x f x1→0 x f 1 1 1 1 In view of Eq. (4-297), this becomes γ1∞ = k1/f1, or k1 f1 = ᎏ (4-298) γ ∞1 where γ 1∞ represents the infinite-dilution value of the activity coefficient of the solute. Since both k1 and γ 1∞ are evaluated at P2sat, this pressure also applies to f1. However, the effect of P on a liquid-phase fugacity, given by a Poynting factor, is very small, and for practical purposes may usually be neglected. The activity coefficient of the solute, given by fˆ1 y1P φˆ 1 γ1
ᎏ =ᎏ x1 f1 x1 f1 then becomes y1P φˆ 1γ1∞ γ1 = ᎏ x1k1 For the solute, this equation takes the place of Eqs. (4-282) and (4-283). Solution for y1 gives x1(γ1/γ1∞)k1 y1 = ᎏᎏ (4-299) φˆ 1P For the solvent, species 2, the analog of Eq. (4-296) is x2γ 2 P2sat y2 = ᎏ (4-300) Φ2P Since y1 + y2 = 1, x1(γ1/γ1∞)k1 x2γ2 P2sat P = ᎏᎏ +ᎏ (4-301) φˆ 1 Φ2 Note that the same correlation that provides for the evaluation of γ1 also allows evaluation of γ ∞1 . There remains the problem of finding Henry’s constant from the available VLE data. For equilibrium fˆ1
fˆ1l = fˆ1v = y1Pφˆ 1 Division by x1 gives fˆ1 y1 ᎏ = Pφˆ 1 ᎏ x1 x1

4-28

THERMODYNAMICS

Henry’s constant is defined as the limit as x1 → 0 of the ratio on the left; therefore y k1 = P2sat φˆ 1∞ lim ᎏ1 x1→0 x 1 The limiting value of y1/x1 can be found by plotting y1/x1 vs. x1 and extrapolating to zero. K-Values A measure of how a given chemical species distributes itself between liquid and vapor phases is the equilibrium ratio: y Ki
ᎏi (4-302) xi

with the theoretically based Wong/Sandler mixing rules for parameters a and b these equations provide the means for accurate correlation and prediction of VLE data. The first of the Wong/Sandler mixing rules relates the difference in mixture quantities b and a/RT to the corresponding differences (identified by subscripts) for the pure species: a b − ᎏ =   xp xq E pq (4-305) RT p q where





1 ap aq E pq
ᎏ bp − ᎏ + bq − ᎏ (1 − k pq) 2 RT RT

(4-306)

Usually called simply a K-value, it adds nothing to thermodynamic knowledge of VLE. However, its use may make for computational convenience, allowing formal elimination of one set of mole fractions {yi} or {xi} in favor of the other. Moreover, it characterizes lightness of a constituent species. For a light species, tending to concentrate in the vapor phase, K > 1; for a heavy species, tending to concentrate in the liquid phase, K < 1. Empirical correlations for K-values found in the older literature have little relation to thermodynamics. Their proper evaluation comes directly from Eq. (4-277): γifi y Ki
ᎏi = ᎏ (4-303) xi φˆ i P

Binary interaction parameters kpq are determined for each pq pair (p ≠ q) from experimental data. Note that kpq = kqp and kpp = kqq = 0. Since the quantity on the left-hand side of Eq. (4-305) represents the second virial coefficient as predicted by Eq. (4-231), the basis for Eq. (4-305) lies in Eq. (4-183), which expresses the quadratic dependence of the mixture second virial coefficient on mole fraction. The second Wong/Sandler mixing rule relates ratios of a/RT to b: a (4-307) ᎏ=1−D bRT

When Raoult’s law applies, this becomes Ki = Pisat/P. In general, Kvalues are functions of T, P, liquid composition, and vapor composition, making their direct and accurate correlation impossible. Those correlations that do exist are approximate and severely limited in application. The DePriester correlation, for example, gives K-values for light hydrocarbons (Chem. Eng. Prog. Symp. Ser. No. 7, 49, pp. 1–43 [1953]). Equation-of-State Approach Although the gamma/phi approach to VLE is in principle generally applicable to systems comprised of subcritical species, in practice it has found use primarily where pressures are no more than a few bars. Moreover, it is most satisfactory for correlation of constant-temperature data. A temperature dependence for the parameters in expressions for GE is included only for the localcomposition equations, and it is at best only approximate. A generally applicable alternative to the gamma/phi approach results when both the liquid and vapor phases are described by the same equation of state. The defining equation for the fugacity coefficient, Eq. (4-79), may be applied to each phase:

The quantity GE/RT is given by an appropriate correlation for the excess Gibbs energy of the liquid phase, and is evaluated at the mixture composition, regardless of whether the mixture is liquid or vapor. The constant c is specific to the equation of state. The theoretical basis for these equations can be found in the literature (Wong and Sandler, op. cit.; Ind. Eng. Chem. Res., 31, pp. 2033–2039 [1992]; Eubank, et al., Ind. Eng. Chem. Res., 34, pp. 314–323 [1995]). Elimination of a from Eq. (4-305) by Eq. (4-307) provides an expression for b: 1 b = ᎏ   xp xq Epq (4-309) D p q

Liquid: Vapor:

fˆil = φˆ il xiP fˆiv = φˆ iv yiP (i = 1, 2, . . . , N)

(4-304)

This introduces the compositions xi and yi into the equilibrium equations, but neither is explicit, because the φˆ i are functions, not only of T and P, but of composition. Thus Eq. (4-304) represents N complex relationships connecting T, P, the xi, and the yi, suitable for computer solution. Given an appropriate equation of state, one or another of Eqs. (4-178) through (4-181) provides for expression of the φˆ i as functions of T, P, and composition. Because of inadequacies in empirical mixing rules, such as those given by Eqs. (4-221) and (4-222), the equation-of-state approach was long limited to systems exhibiting modest and well-behaved deviations from ideal solution behavior in the liquid phase; for example, to systems containing hydrocarbons and cryogenic fluids. However, the introduction by Wong and Sandler (AIChE J., 38, pp. 671–680 [1992]) of a new class of mixing rules for cubic equations of state has greatly expanded their useful application to VLE. The Soave/Redlich/Kwong (SRK) and the Peng/Robinson (PR) equations of state, both expressed by Eqs. (4-230) and (4-231), were developed specifically for VLE calculations. The fugacity coefficients implicit in these equations are given by Eq. (4-232). When combined

GE ap D
1 + ᎏ −  xp ᎏ cRT bp RT p

(4-308)

Mixture parameter a then follows from Eq. (4-307): a = bRT(1 − D)

(4-310)

Equations (4-233) and (4-234) may now be applied for the evaluation of partial parameters ai and  bi: 1 ln γ ai bi = ᎏ 2 xj Eij − b 1 + ᎏi − ᎏ  D c biRT j



and

Equation (4-276) now becomes xi φˆ il = yi φˆ iv

where





ai ln γi bi  ai = bRT ᎏ − ᎏ + a ᎏ − 1 bi RT c b



 



(4-311) (4-312)

For pure species i, Eq. (4-232) reduces to (Vi − bi)Zi ai /biRT Vi + σbi ln φi = Zi − 1 − ln ᎏᎏ + ᎏ ln ᎏ Vi ε−σ Vi + εbi

(4-313)

This equation may be applied separately to the liquid phase and to the vapor phase to yield the pure-species values φil and φiv. For vapor/ liquid equilibrium (Eq. [4-280]), these two quantities are equal. Given parameters ai and bi, the pressure P in Eq. (4-230) that makes these two values equal is Pisat, the equilibrium vapor pressure of pure species i as predicted by the equation of state. The correlations for α(Tri; ωi) that follow Eq. (4-230) are designed to provide values of ai that yield pure-species vapor pressures which, on average, are in reasonable agreement with experiment. However, reliable correlations for Pisat as a function of temperature are available for many pure species. Thus when Pisat is known for a particular temperature, ai should be evaluated so that the equation of state correctly predicts this known value. The procedure is to write Eq. (4-313) for each of the phases, combining the two equations in accord with Eq. (4-280), written ln φil = ln φiv

EQUILIBRIUM The resulting expression may be solved for ai: Vi − bi bi RT(ε − σ) ln ᎏ ᎏ + Ziv − Zil Viv − bi ai = (Vil + σbi)(Viv + εbi) ln ᎏᎏᎏ (Vil + εbi)(Viv + σbi)





l

(4-314)

where ln γi∞ comes from the GE correlation and ln φi is given by Eq. (4-313) written for the liquid phase. Equation (4-315) supplies a value for ln φˆ i∞ which, when used with Eq. (4-232), ultimately (see following) leads to values for kpq. For a binary system comprised of species p and q, Eqs. (4-232), (4-312), and (4-315) may be written for species p at infinite dilution. The three resulting equations are then combined to yield b p∞ ln γ p∞ + ln φp − Mp ᎏ = ᎏᎏ bq Zq − 1

(4-316)

where (Vq − bq)Zq 1 ap ln γ p∞ Vq + σbq Mp
−ln ᎏᎏ +ᎏ ᎏ −ᎏ ln ᎏ (4-317) Vq ε − σ bp RT c Vq + εbq By Eq. (4-311) written for species p at infinite dilution in a pq binary, 2Epq ln γ p∞ ap ᎏᎏ − 1 − ᎏᎏ + ᎏᎏ ∞ b b c RT q p bp  ᎏ = ᎏᎏᎏ (4-318) aq 1 − ᎏᎏ bq bqRT





Equations (4-316) and (4-318) are set equal, Epq is eliminated by Eq. (4-306), and kpq is replaced by kp, its infinite-dilution value at xp → 0. Solution for kp then yields ln γ + ln φ − M ln γ a ᎏ + b 1 + ᎏᎏ − ᎏᎏ b − ᎏRaᎏT  ᎏ Z −1 c b RT q

q

∞ p

p

p

∞ p

q

p

kp = 1 − ᎏᎏᎏᎏᎏᎏ ap aq bp − ᎏᎏ + bq − ᎏᎏ RT RT (4-319) q

p

where ln φp comes from Eq. (4-313). All values in Eq. (4-319) are for the liquid phase at P = Pqsat. The analogous equation for kq, the infinitedilution value of k pq at xq → 0 is written ln γ + ln φ − M ln γ a ᎏ + b 1 + ᎏᎏ − ᎏᎏ b − ᎏRaᎏT  ᎏ c Z −1 b RT p

p

∞

q

q

q

∞

p

q

q

kq = 1 − ᎏᎏᎏᎏᎏᎏ ap aq bp − ᎏᎏ + bq − ᎏᎏ RT RT (4-320) p

A second advantage is that the procedure, applied for infinite dilution of each species, yields two values of kpq from which a composition-dependent function can be generated, a simple linear relation proving fully satisfactory: kpq = kp xq + kq xp

where Ziv = Pisat Viv/RT and Zil = Pisat Vil/RT. Values of Viv and Vil come from solution of Eq. (4-230) for each phase with P = Pisat at temperature T. Since a value of ai is required for these calculations, an iterative procedure is implemented with an initial value for ai from the appropriate correlation for α(Tri; ωi). The binary interaction parameters kpq are evaluated from liquidphase GE correlations for binary systems. The most satisfactory procedure is to apply at infinite dilution the relation between a liquid-phase activity coefficient and its underlying fugacity coefficients, γ i∞ = φˆ i∞/φi. Rearrangement of the logarithmic form yields ln φˆ i∞ = ln γ i∞ + ln φi (4-315)

q

where Mq is given by an equation analogous to Eq. (4-317) but with subscripts reversed. All values in Eq. (4-320) are for the liquid phase at P = Ppsat. One advantage of this procedure is that kp and kq are found directly from the pure-species parameters ap, aq, bp, and bq. In addition, the required values of ln γp∞ and ln γq∞ can be found from experimental data for the pq binary system, independent of the correlating expression used for GE.

4-29

(4-321)

The two values kp and kq are usually not very different, and kpq is not strongly composition dependent. Nevertheless, the quadratic dependence of b − (a/RT) on composition indicated by Eq. (4-305) is not exactly preserved. Since this quantity is not a true second virial coefficient, only a value predicted by a cubic equation of state, a strict quadratic dependence is not required. Moreover, the compositiondependent kpq leads to better results than does use of a constant value. The equation-specific constants for the SRK and PR equations are given by the following table:

ε σ Ωa Ωb c

SRK equation

PR equation

0 1 0.42748 0.08664 0.69315

−0.414214 2.414214 0.457235 0.077796 0.62323

Outlined below are the steps required for of a VLE calculation of vapor-phase composition and pressure, given the liquid-phase composition and temperature. A choice must be made of an equation of state. Only the Soave/Redlich/Kwong and Peng/Robinson equations, as represented by Eqs. (4-230) and (4-231), are considered here. These two equations usually give comparable results. A choice must also be made of a two-parameter correlating expression to represent the liquid-phase composition dependence of GE for each pq binary. The Wilson, NRTL (with α fixed), and UNIQUAC equations are of general applicability; for binary systems, the Margules and van Laar equations may also be used. The equation selected depends on evidence of its suitability to the particular system treated. Reasonable estimates of the parameters in the equation must also be known at the temperature of interest. These parameters are directly related to infinite-dilution values of the activity coefficients for each pq binary. Input information includes the known values of T and {xi}, as well as the equation-of-state and GE-expression parameters. Estimates are also needed of P and {yi}, the quantities to be evaluated, and these require some preliminary calculations: 1. For the chosen equation of state (with appropriate values of ε, σ, and c), find values of bi and preliminary values of ai for each species from the information following Eq. (4-230). 2. If the vapor pressure Pisat for species i at temperature T is known, determine a new value for ai by Eqs. (4-314) and (4-230). 3. Evaluate kp and kq by Eqs. (4-319) and (4-320) for each pq binary. 4. Although pressure P is to be determined, an estimate is required to permit any VLE calculations at all. A reasonable initial value is the sum of the pure-species vapor pressures, each weighted by its known liquid-phase mole fraction. 5. The vapor-phase composition is also to be determined, and it, too, is required to initiate calculations. Assuming both the liquid and vapor phases to be ideal solutions, Eqs. (4-98) and (4-304) combine to give φil yi = xi ᎏ φiv Evaluation of the pure-species values φil and φiv by Eq. (4-313) then provides values for yi. Since these are not constrained to sum to unity, they should be normalized to yield an initial vapor-phase composition. Given estimates for P and {yi} an iterative procedure can be initiated: 1. At the known liquid-phase composition, evaluate D by Eq. (4-308), b and a by Eqs. (4-309) and (4-310), and {b i} and {ai} by Eqs. (4-311) and (4-312).

4-30

THERMODYNAMICS

2. Evaluate {φˆ il }. The mixture volume V is determined from the equation of state, Eq. (4-231), applied to the liquid phase at the given composition, T, and P. 3. Repeat the two preceding items for the vapor-phase composition, thus evaluating {φˆ vi }. 4. Eq. (4-304) is now written φˆ l yi = xi ᎏvi φˆ i The values of yi so calculated are normalized by division by  i yi. 5. Recalculate the φˆ iv, and continue this iterative procedure until it converges to a fixed value for  i yi. This sum is appropriate to the pressure P for which the calculations have been made. Unless the sum is unity, the pressure is adjusted and the iteration process is repeated. Systematic adjustment of pressure P continues until  i yi = 1. The pressure and vapor compositions so found are the equilibrium values for the given temperature and liquid-phase composition as predicted by the equation of state. A vast store of liquid-phase excess-property data for binary systems at temperatures near 30°C and somewhat higher is available in the literature. Effective use of these data to extend G E correlations to higher temperatures is critical to the procedure considered here. The key relations are Eq. (4-118),

 

GE HE d ᎏ = − ᎏ2 dT RT RT

(constant P,x)

and the excess-property analog of Eq. (4-31), dH E = CPE dT

Integration of the first of these equations from T0 to T gives

  

T

GE GE HE (4-322) ᎏ= ᎏ − ᎏ dT RT RT T0 T0 RT 2 Similarly, the second equation may be integrated from T1 to T:



T

E

CP dT

(4-323)

T1

In addition, we may write ∂CPE dC PE = ᎏ ∂T Integration from T2 to T yields



CPE = CPE2 +



dT

P,x

∂C  ᎏ ∂T  T

E P

dT

P,x

T2

Combining this equation with Eqs. (4-322) and (4-323) leads to H T T ᎏ − 1 ᎏ   − ᎏ RT   T T

GE GE ᎏ= ᎏ RT RT

E

T0

1

T1

0

  ∂C 1 I
 ᎏ   ᎏ RT ∂T

 

CPE2 T T T −ᎏ ln ᎏ − ᎏ − 1 ᎏ1 − I R T0 T0 T

T

where

T0

2

T

T

T1

T2

E P



 

(4-324)

dT dT dT

P,x

This general equation makes use of excess Gibbs-energy data at temperature T0, excess enthalpy (heat-of-mixing) data at T1, and excess heat-capacity data at T2. Evaluation of the integral I requires informaE tion with respect to the temperature dependence of CP . Because of the relative paucity of excess heat-capacity data, the most reasonable assumption is that this quantity is constant, independent of T. In this event, the integral is zero, and the closer T0 and T1 are to T, the less the influence of this assumption. When no information is available with E respect to CP , and excess enthalpy data are available at only a single temperature, the excess heat capacity must be assumed zero. In this case only the first two terms on the right-hand side of Eq. (4-324) are retained, and it more rapidly becomes imprecise as T increases. Our primary interest in Eq. (4-324) is its application to binary systems at infinite dilution of one of the constituent species. For this pur-





E P







 

 

C T T T − ᎏ ln ᎏ − ᎏ − 1 ᎏ1 x1x2R T0 T0 T As shown by Smith, Van Ness and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed., Chap. 11, McGraw-Hill, New York, 1996), GE
ln γi∞ ᎏ x1x2 RT xi = 0 The preceding equation may therefore be written HE T T1 ln γi∞ = (ln γi∞)T0 − ᎏ ᎏ−1 ᎏ x1x2 RT T1,xi = 0 T0 T











E P





 

C T T T − ᎏ ln ᎏ − ᎏ − 1 ᎏ1 (4-325) x1x2 R xi = 0 T0 T0 T The methanol(1)/acetone(2) system serves as a specific example in conjunction with the Peng/Robinson equation of state. At a base temperature T0 of 323.15 K (50°C), both VLE data (Van Ness and Abbott, Int. DATA Ser., Ser. A, Sel. Data Mixtures, 1978, p. 67 [1978]) and excess enthalpy data (Morris, et al., J. Chem. Eng. Data, 20, pp. 403– 405 [1975]) are available. From the former, (ln γ 1∞)T0 = 0.6281

(constant P,x)

HE = H1E +

E

pose, we divide Eq. (4-324) by the product x1x2. For CP independent of T (and thus with I = 0), it then becomes GE GE HE T T1 ᎏ= ᎏ − ᎏ ᎏ−1 ᎏ x1x2RT x1x2RT T0 x1x2RT T1 T0 T

and

(ln γ 2∞)T0 = 0.6557

and from the latter HE HE = 1.3636 and = 1.0362 ᎏ ᎏ x1x2 RT T0,x 1 = 0 x1x2 RT T0 ,x 2 = 0 The Margules equations (Eqs. [4-244], [4-245], and [4-246]) are well suited to this system, and the parameters for this equation are given as







A12 = ln γ 1∞

and



A21 = ln γ 2∞

This information allows prediction of VLE at 323.15 K and at the higher temperatures, 372.8, 397.7, and 422.6 K, for which measured VLE values are given by Wilsak, et al. (Fluid Phase Equilibria, 28, pp. 13–37 [1986]). Values of ln γi∞ and hence of the Margules parameters at the higher temperatures are given by Eq. (4-325) with CPE = 0. The pure-species vapor pressures in all cases are the measured values reported with the data sets. Results of these calculations are displayed in Table 4-1, where the parentheses enclose values from the gamma/ phi approach as reported in the papers cited. The results at 323.15 K (581.67 R) show both the suitability of the Margules equation for correlation of data for this system and the capability of the equation-of-state method to reproduce the data. Results for the three higher temperatures indicate the quality of predictions based only on vapor-pressure data for the pure species and on mixture data at 323.15 K (581.67 R). Extrapolations based on the same data to still higher temperatures can be expected to become progressively less accurate. When Eq. (4-325) can no longer be expected to produce reasonable values, better results are obtained for higher temperatures by assuming that the parameters, A12, A21, k1, and k2, do not change further at still-higher temperatures. This is also the course to be followed for extrapolation to supercritical temperatures. Only the Wilson, NRTL, and UNIQUAC equations are suited to the treatment of multicomponent systems. For such systems, the parameters are determined for pairs of species exactly as for binary systems. Examples treating the calculation of VLE are given in Smith, Van Ness, and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed., Chap. 12, McGraw-Hill, New York, 1996). LIQUID/LIQUID AND VAPOR/LIQUID/LIQUID EQUILIBRIA Equation (4-273) is the basis for both liquid/liquid equilibria (LLE) and vapor/liquid/liquid equilibria (VLLE). Thus, for LLE with superscripts α and β denoting the two phases, Eq. (4-273) is written

EQUILIBRIUM TABLE 4-1

VLE Results for Methanol(1)/Acetone(2)

T, K

ln γ1∞

ln γ 2∞

k1

k2

RMS δP, kPa

RMS % δP

323.15

0.6281 (0.6281) 0.4465 (0.4607) 0.3725 (0.3764) 0.3072 (0.3079)

0.6557 (0.6557) 0.5177 (0.5271) 0.4615 (0.4640) 0.4119 (0.3966)

0.1395

0.0955

0.12

0.1432

0.1056

0.1454

0.1118

0.1480

0.1192

0.08 (0.06) 0.85 (0.83) 2.46 (1.39) 7.51 (2.38)

372.8 397.7 422.6

fˆiα = fˆiβ

(i = 1, 2, . . . , N)

(4-326)

Eliminating fugacities in favor of activity coefficients gives x αi γ iα = xiβ γ iβ

(i = 1, 2, . . . , N)

(4-327)

For most LLE applications, the effect of pressure on the γ i can be ignored, and thus Eq. (4-327) constitutes a set of N equations relating equilibrium compositions to each other and to temperature. For a given temperature, solution of these equations requires a single expression for the composition dependence of GE suitable for both liquid phases. Not all expressions for GE suffice, even in principle, because some cannot represent liquid/liquid phase splitting. The UNIQUAC equation is suitable, and therefore prediction is possible by the UNIFAC method. A special table of parameters for LLE calculations is given by Magnussen, et al. (Ind. Eng. Chem. Process Des. Dev., 20, pp. 331–339 [1981]). A comprehensive treatment of LLE is given by Sorensen, et al. (Fluid Phase Equilibria, 2, pp. 297–309 [1979]; 3, pp. 47–82 [1979]; 4, pp. 151–163 [1980]). Data for LLE are collected in a three-part set compiled by Sorensen and Arlt (Liquid-Liquid Equilibrium Data Collection, Chemistry Data Series, vol. V, parts 1–3, DECHEMA, Frankfurt am Main, 1979–1980). For vapor/liquid/liquid equilibria, Eq. (4-273) gives fˆiα = fˆiβ = fˆiv (i = 1, 2, . . . , N) (4-328) where α and β designate the two liquid phases. With activity coefficients applied to the liquid phases and fugacity coefficients to the vapor phase, the 2N equilibrium equations for subcritical VLLE are xiα γiα fiα = yi φˆ iP (all i) (4-329) xiβ γiβ fiβ = yi φˆ iP



E

As for LLE, an expression for G capable of representing liquid/liquid phase splitting is required; as for VLE, a vapor-phase equation of state for computing the φˆ i is also needed.

0.32 0.55

∆nN, j ∆n1, j ∆n2, j (4-331) ᎏ=ᎏ=⋅⋅⋅=ᎏ ν1, j ν2, j νN, j Since all of these terms are equal, they can be equated to the change in a single quantity εj, called the reaction coordinate for reaction j, thereby giving i = 1, 2, . . . , N ∆ni, j = νi, j ∆εj (4-332) j = I, II, . . . , r



Since the total change in mole number ∆ni is just the sum of the changes ∆ni,j resulting from the various reactions, ∆ni =  ∆ni, j =  νi, j ∆εj j

sign (νi,j ) =

(4-333)

If the initial number of moles of species i is ni0 and if the convention is adopted that εj = 0 for each reaction in this initial state, then ni = ni0 +  νi, j εj

(i = 1, 2, . . . , N)

(4-334)

j

Equation (4-334) is the basic expression of material balance for a closed system in which r chemical reactions occur. It shows for a reacting system that at most r mole number–related quantities εj are capable of independent variation. Note the absence of implied restrictions with respect to chemical-reaction equilibria; the reactioncoordinate formalism is merely an accounting scheme, valid for tracking the progress of each reaction to any arbitrary level of conversion. The reaction coordinate has units of moles. A change in εj of 1 mole signifies a mole of reaction, meaning that reaction j has proceeded to such an extent that the change in mole number of each reactant and product is equal to its stoichiometric number. CHEMICAL-REACTION EQUILIBRIA The general criterion of chemical-reaction equilibria is given by Eq. (4-274). For a system in which just a single reaction occurs, Eq. (4-334) becomes dni = νi dε

whence

Substitution for dni in Eq. (4-274) leads to

 νµ = 0 i i

where the |νi| are stoichiometric coefficients and the Ai stand for chemical formulas. The νi themselves are called stoichiometric numbers, and associated with them is a sign convention such that the value is positive for a product and negative for a reactant. More generally, for a system containing N chemical species, any or all of which can participate in r chemical reactions, the reactions can be represented by the equations:

where

(i = 1, 2, . . . , N)

j

|ν1|A1 + |ν2|A2 + ⋅ ⋅ ⋅ → |ν3|A3 + |ν4|A4 + ⋅ ⋅ ⋅

( j = I, II, . . . , r)

0.004 (0.006) 0.014 (0.013) 0.009 (0.006)

ni = ni0 + νiε

Consider a phase in which a chemical reaction occurs according to the equation

0 = νi, j Ai

RMS δy1

0.22

CHEMICAL-REACTION STOICHIOMETRY

i

4-31

(4-330)

− for a reactant species

+ for a product species

If species i does not participate in reaction j, then νi,j = 0. The stoichiometric numbers provide relations among the changes in mole numbers of chemical species which occur as the result of chemical reaction. Thus, for reaction j:

(4-335)

i

Generalization of this result to multiple reactions produces

ν

i, j

µi = 0

( j = I, II, . . . , r)

(4-336)

i

Standard Property Changes of Reaction A standard property change for the reaction aA + bB → lL + mM is defined as the property change that occurs when a moles of A and b moles of B in their standard states at temperature T react to form l moles of L and m moles of M in their standard states also at temperature T. A standard state of species i is its real or hypothetical state as a pure species at temperature T and at a standard-state pressure P°. The standard property change of reaction j is given the symbol ∆Mj°, and its general mathematical definition is ∆Mj°
 νi, j Mi° i

(4-337)

4-32

THERMODYNAMICS

For species present as gases in the actual reactive system, the standard state is the pure ideal gas at pressure P°. For liquids and solids, it is usually the state of pure real liquid or solid at P°. The standard-state pressure P° is fixed at 100 kPa. Note that the standard states may represent different physical states for different species; any or all of the species may be gases, liquids, or solids. The most commonly used standard property changes of reaction are ∆Gj°
 νi, jGi° =  νi,j µi°

(4-338)

∆Hj°
 νi, j Hi°

(4-339)

∆C°Pi
 νi, j C°Pi

(4-340)

i

i

i

i

The standard Gibbs-energy change of reaction ∆Gj° is used in the calculation of equilibrium compositions. The standard heat of reaction ∆Hj° is used in the calculation of the heat effects of chemical reaction, and the standard heat-capacity change of reaction is used for extrapolating ∆Hj° and ∆Gj° with T. Numerical values for ∆Hj° and ∆Gj° are computed from tabulated formation data, and ∆C°Pi is determined from empirical expressions for the T dependence of the C°Pi (see, e.g., Eq. [4-142]). Equilibrium Constants For practical application, Eq. (4-336) must be reformulated. The initial step is elimination of the µi in favor of fugacities. Equation (4-74) for species i in its standard state is subtracted from Eq. (4-77) for species i in the equilibrium mixture, giving µi = Gi° + RT ln âi

(4-341)

where, by definition, âi
fˆi /fi° and is called an activity. Substitution of this equation into Eq. (4-341) yields, upon rearrangement,

 [ν

(Gi° + RT ln âi)] = 0

i, j

i

or

 (ν

Gi°) + RT  ln âiνi,j = 0

i, j

i

i

− (νi,jGi°)

ln  âiνi, j = ᎏᎏ RT i The right-hand side of this equation is a function of temperature only for given reactions and given standard states. Convenience suggests setting it equal to ln Kj; whence i

or

 (y φˆ ) i i

νi, j

i

νj

P ᎏ P° 

= Kj

(all j)

(4-344)

where νj
i νi,j and P° is the standard-state pressure of 100 kPa, expressed in the same units used for P. The yi may be eliminated in favor of equilibrium values of the reaction coordinates εj. Then, for fixed temperature Eqs. (4-344) relate the εj to P. In principle, specification of the pressure allows solution for the εj. However, the problem may be complicated by the dependence of the φˆ i on composition, that is, on the εj. If the equilibrium mixture is assumed an ideal solution, then each φˆ i becomes φi, the fugacity coefficient of pure species i at the mixture T and P. This quantity does not depend on composition and may be determined from experimental data, from a generalized correlation, or from an equation of state. An important special case of Eq. (4-344) is obtained for gas-phase reactions when the phase can be assumed an ideal gas. In this event φˆ i = 1, and P νj = Kj (all j) (4-345) i (yi)νi, j ᎏ P° In the general case the evaluation of the φˆ i requires an iterative process. An initial step is to set the φˆ i equal to unity and to solve the problem by Eq. (4-345). This provides a set of yi values, allowing evaluation of the φˆ i by, for example, Eq. (4-196), (4-200), or (4-231). Equation (4-344) can then be solved for a new set of yi values, and the process continues to convergence. For liquid-phase reactions, Eq. (4-342) is modified by introduction of the activity coefficient, γi = fˆi /xi fi, where xi is the liquid-phase mole fraction. The activity is then f fˆi âi
ᎏ = γi xi ᎏi fi° fi° Both fi and fi° represent fugacity of pure liquid i at temperature T, but at pressures P and P°, respectively. Except in the critical region, pressure has little effect on the properties of liquids, and the ratio fi/fi° is often taken as unity. When this is not acceptable, this ratio is evaluated by the equation f 1 P Vi(P − P°) ln ᎏi = ᎏ Vi dP ᎏᎏ fi° RT P° RT When the ratio fi /fi° is taken as unity, âi = γi xi, and Eq. (4-342) becomes

 



 (γ x ) i i

νi,j

= Kj

(all j)

(4-346)

i

â

i

νi, j

= Kj

(all j)

(4-342)

i

−∆Gj° (4-343) Kj
exp ᎏ RT Quantity Kj is the chemical-reaction equilibrium constant for reaction j, and ∆Gj° is the corresponding standard Gibbs-energy change of reaction (see Eq. [4-338]). Although called a “constant,” Kj is a function of T, but only of T. The activities in Eq. (4-342) provide the connection between the equilibrium states of interest and the standard states of the constituent species, for which data are presumed available. The standard states are always at the equilibrium temperature. Although the standard state need not be the same for all species, for a particular species it must be the state represented by both G°i and the fi° upon which the activity âi is based. The application of Eq. (4-342) requires explicit introduction of composition variables. For gas-phase reactions this is accomplished through the fugacity coefficient: âi
fˆi/fi° = yi φˆ iP/fi° where

and Eq. (4-342) becomes





However, the standard state for gases is the ideal gas state at the standard-state pressure, for which fi° = P°. Therefore yi φˆ iP âi = ᎏ P°

Here the difficulty is to determine the γi , which depend on the xi . This problem has not been solved for the general case. Two courses are open: the first is experiment; the second, assumption of solution ideality. In the latter case, γi = 1, and Eq. (4-346) reduces to

 (x ) i

νi,j

= Kj

(all j)

(4-347)

i

the law of mass action. The significant feature of Eqs. (4-345) and (4-347), the simplest expressions for gas- and liquid-phase reaction equilibrium, is that the temperature-, pressure-, and compositiondependent terms are distinct and separate. Example 2: Single-Reaction Equilibrium Consider the equilibrium state at 1,000 K and atmospheric pressure for the reaction CO + H2O → CO2 + H2 Let the feed stream contain 3 mol CO, 1 mol H2O, and 2 mol CO2 for every mole of H2 present. This initial constitution forms the basis for calculation, and for this single reaction, Eq. (4-334) becomes ni = ni0 + νiε. Whence nCO = 3 − ε nH2O = 1 − ε nCO2 = 2 + ε nH2 = 1 + ε

n =7 i

i

Each mole fraction is therefore given by yi = ni/7.

EQUILIBRIUM At 1,000 K, ∆G° = −2680 J per mole of reaction; whence by Eq. (4-343) 2680 K = exp ᎏᎏ = 1.38 (8.314)(1000) For the given conditions, the assumption of ideal gases is appropriate; Eq. (4-345) written for a single reaction (subscript j omitted) with ν = 0 becomes

ᎏ2 7+ᎏε  ᎏ1 7+ᎏε  = K = 1.38  y = ᎏᎏ ᎏ3 7−ᎏε  ᎏ1 7−ᎏε  i

or

νi i

limited to gas-phase reactions for which the problem is to find the equilibrium composition for given T and P and for a given initial feed. 1. Formulate the constraining material-balance equations, based on conservation of the total number of atoms of each element in a system comprised of w elements. Let subscript k identify a particular atom, and define Ak as the total number of atomic masses of the kth element in the feed. Further, let aik be the number of atoms of the kth element present in each molecule of chemical species i. The material balance for element k is then

 na

i ik

na

or

ε = 0.258

i ik

2.

nCO = 2.74 mol nH2O = 0.74 mol nCO2 = 2.26 mol nH2 = 1.26 mol

yCO = 0.391 yH2O = 0.106 yCO2 = 0.323 yH2 = 0.180

 i ni = 7.00 mol

 i yi = 1.000

(4-353)

− Ak = 0

(k = 1, 2, . . . , w)



− Ak = 0

(k = 1, 2, . . . , w)

 λ  n a k

i ik

k

∆CP° ᎏ dT R

(4-351)

where for simplicity subscript j has been suppressed. A convenient integrated form of Eq. (4-349) is −∆G° ∆H0° − ∆G0° ∆H° 1 ln K = ᎏ = ᎏᎏ −ᎏ+ᎏ RT RT RT T

i ik

Summed over k, these equations give

Integration of Eq. (4-350) from reference temperature T0 (usually 298.15 K) to temperature T gives T0

 n a i

For an endothermic reaction ∆Hj° is positive; for an exothermic reaction it is negative. The temperature dependence of ∆ Hj° is given by d∆Hj° (4-350) ᎏ = ∆C°Pj dT

T

Multiply each element balance by λk, a Lagrange multiplier: λk

The effect of temperature on the equilibrium constant follows from Eq. (4-106): d(∆Gj°/RT) −∆Hj° (4-348) ᎏᎏ = ᎏ dT RT 2 The total derivative is appropriate here because property changes of reaction are functions of temperature only. In combination with Eq. (4-343) this gives d ln Kj ∆ Hj° (4-349) ᎏ=ᎏ dT RT 2



(k = 1, 2, . . . , w)

i

Thus, for the equilibrium mixture,

∆H° = ∆H0° + R

= Ak

i

(2 + ε)(1 + ε) ᎏᎏ = 1.38 (3 − ε)(1 − ε)

whence

4-33



T

T0

CP° ᎏ dT R

(4-352)

where ∆H°/RT is given by Eq. (4-351). In the more extensive compilations of data, values of ∆G° and ∆H° for formation reactions are given for a wide range of temperatures, rather than just at the reference temperature of 298.15 K. (See in particular TRC Thermodynamic Tables—Hydrocarbons and TRC Thermodynamic Tables—Non-hydrocarbons, serial publications of the Thermodynamics Research Center, Texas A & M University System, College Station, Tex.; “The NBS Tables of Chemical Thermodynamic Properties,” J. Physical and Chemical Reference Data, 11, supp. 2 [1982]. Where data are lacking, methods of estimation are available; these are reviewed by Reid, Prausnitz, and Poling, The Properties of Gases and Liquids, 4th ed., Chap. 6, McGraw-Hill, New York, 1987. For an estimation procedure based on molecular structure, see Constantinou and Gani, Fluid Phase Equilibria, 103, pp. 11–22 [1995]. (See also Sec. 2.) Complex Chemical-Reaction Equilibria When the composition of an equilibrium mixture is determined by a number of simultaneous reactions, calculations based on equilibrium constants become complex and tedious. A more direct procedure (and one suitable for general computer solution) is based on minimization of the total Gibbs energy Gt in accord with Eq. (4-271). The treatment here is

3.

i



− Ak = 0

Form a function F by addition of this sum to Gt: F = Gt +  λk k

 n a

i ik

i

− Ak



t

Function F is identical with G , because the summation term is zero. However, the partial derivatives of F and Gt with respect to ni are different, because function F incorporates the constraints of the material balances. 4. The minimum value of both F and Gt is found when the partial derivatives of F with respect to ni are set equal to zero: ∂F ∂Gt = ᎏ +  λkaik = 0 ᎏ ∂ni T,P,nj ∂ni T,P,nj k The first term on the right is the definition of the chemical potential; whence

 

 

µi +  λk aik = 0

(i = 1, 2, . . . , N)

(4-354)

k

However, the chemical potential is given by Eq. (4-341); for gas-phase reactions and standard states as the pure ideal gases at P°, this equation becomes fˆi µi = G°i + RT ln ᎏ P° If Gi° is arbitrarily set equal to zero for all elements in their standard states, then for compounds G°i = ∆Gf°, the standard Gibbs-energy i change of formation for species i. In addition, the fugacity is eliminated in favor of the fugacity coefficient by Eq. (4-79), fˆi = yiφˆ iP. With these substitutions, the equation for µi becomes yi φˆ iP µi = ∆Gf°i + RT ln ᎏ P° Combination with Eq. (4-354) gives yi φˆ iP ∆Gf°i + RT ln ᎏ +  λkaik = 0 P° k

(i = 1, 2, . . . , N)

(4-355)

If species i is an element, ∆Gf°i is zero. There are N equilibrium equations (Eqs. [4-355]), one for each chemical species, and there are w material-balance equations (Eqs. [4-353]), one for each element—a total of N + w equations. The unknowns in these equations are the ni (note that yi = ni / i ni), of which there are N, and the λk, of which there are w—a total of N + w unknowns. Thus, the number of equations is sufficient for the determination of all unknowns. Equation (4-355) is derived on the presumption that the φˆ i are known. If the phase is an ideal gas, then each φˆ i is unity. If the phase is an ideal solution, each φˆ i becomes φi, and can at least be estimated. For real gases, each φˆ i is a function of the yi , the quantities being calculated. Thus an iterative procedure is indicated, initiated with each φˆ i

4-34

THERMODYNAMICS

set equal to unity. Solution of the equations then provides a preliminary set of yi. For low pressures or high temperatures this result is usually adequate. Where it is not satisfactory, an equation of state with the preliminary yi gives a new and more nearly correct set of φˆ i for use in Eq. (4-355). Then a new set of yi is determined. The process is repeated to convergence. All calculations are well suited to computer solution. In this procedure, the question of what chemical reactions are involved never enters directly into any of the equations. However, the choice of a set of species is entirely equivalent to the choice of a set of independent reactions among the species. In any event, a set of species or an equivalent set of independent reactions must always be assumed, and different assumptions produce different results. Example 3: Minimization of Gibbs Energy Calculate the equilibrium compositions at 1,000 K and 1 bar of a gas-phase system containing the species CH4, H2O, CO, CO2, and H2. In the initial unreacted state there are present 2 mol of CH4 and 3 mol of H2O. Values of ∆Gf° at 1,000 K are ∆G°f CH = 19,720 J/mol

The five equations for the five species then become: CH4:

19,720 nCH λC 4λH ᎏ + ln ᎏ4 + ᎏ + ᎏ = 0 i ni RT RT RT

H2O:

−192,420 nH2O 2λH λO +ᎏ+ᎏ=0 ᎏᎏ + ln ᎏ i ni RT RT RT

CO:

−200,240 nCO λC λO ᎏᎏ + ln ᎏ + ᎏ + ᎏ = 0 i ni RT RT RT

CO2:

−395,790 nCO λC 2λO ᎏᎏ + ln ᎏ2 + ᎏ + ᎏ = 0 i ni RT RT RT

H2:

nH2 2λH ln ᎏ +ᎏ=0 i ni RT

The three material-balance equations (Eq. [4-353]) are:

∆G°f H O = −192,420 J/mol

C:

nCH4 + nCO + nCO2 = 2

∆G°f CO = −200,240 J/mol

H:

4nCH4 + 2nH2O + 2nH2 = 14

∆G°f CO = −395,790 J/mol

O:

nH2O + nCO + 2nCO2 = 3

4

2

2

The required values of Ak are determined from the initial numbers of moles, and the values of aik come directly from the chemical formulas of the species. These are shown in the accompanying table.

Simultaneous computer solution of these eight equations, with RT = 8,314 J/mol and

n = n i

Carbon

Oxygen

Hydrogen

Ak = no. of atomic masses of k in the system AC = 2 Species i CH4 H2O CO CO2 H2

AO = 3

AH = 14

aik = no. of atoms of k per molecule of i aCH4,C = 1 aH2O,C = 0 aCO,C = 1 aCO2,C = 1 aH2,C = 0

aCH4,O = 0 aH2O,O = 1 aCO,O = 1 aCO2,O = 2 aH2,O = 0

CH4

+ nH2O + nCO + nCO2 + nH2

i

Element k

aCH4,H = 4 aH2O,H = 2 aCO,H = 0 aCO2,H = 0 aH2,H = 2

At 1 bar and 1,000 K the assumption of ideal gases is justified, and the φˆ i are all unity. Since P = 1 bar, Eq. (4-355) is written: ∆Gf° ni λk ᎏi + ln ᎏ +  ᎏ aik = 0 RT Σi ni RT k

produces the following results (yi = ni/i ni): yCH4 = 0.0196

λC ᎏ = 0.7635 RT

yH2O = 0.0980 yCO = 0.1743

λO ᎏ = 25.068 RT

yCO2 = 0.0371 yH2 = 0.6711

λH ᎏ = 0.1994 RT

 y = 1.000 i

i

The values of λk/RT are of no significance, but are included to make the results complete.

THERMODYNAMIC ANALYSIS OF PROCESSES Real irreversible processes can be subjected to thermodynamic analysis. The goal is to calculate the efficiency of energy use or production and to show how energy loss is apportioned among the steps of a process. The treatment here is limited to steady-state, steady-flow processes, because of their predominance in chemical technology. CALCULATION OF IDEAL WORK In any steady-state, steady-flow process requiring work, a minimum amount must be expended to bring about a specific change of state in the flowing fluid. In a process producing work, a maximum amount is attainable for a specific change of state in the flowing fluid. In either case, the limiting value obtains when the specific change of state is accomplished completely reversibly. The implications of this requirement are: 1. The process is internally reversible within the control volume. 2. Heat transfer external to the control volume is reversible. The second item means that heat exchange between system and surroundings must occur at the temperature of the surroundings, presumed to constitute a heat reservoir at a constant and uniform temperature

Tσ. This may require Carnot engines or heat pumps internal to the system that provide for the reversible transfer of heat from the temperature of the flowing fluid to that of the surroundings. Since Carnot engines and heat pumps are cyclic, they undergo no net change of state. The entropy change of the surroundings, found by integration of Eq. (4-3), is ∆Sσ = Qσ/Tσ; whence Qσ = Tσ ∆Sσ (4-356) Since heat transfer with respect to the surroundings and with respect to the system are equal but of opposite sign, Qσ = −Q. Moreover, the second law requires for a reversible process that the entropy changes of system and surroundings be equal but of opposite sign: ∆Sσ = −∆St. Equation (4-356) can therefore be written Q = Tσ∆St. In terms of rates this becomes ˙ = Tσ ∆(Sm) Q ˙ fs (4-357) ˙ where Q = rate of heat transfer with respect to the system m ˙ = mass rate of flow of fluid In addition, ∆ denotes the difference between exit and entrance streams, and fs indicates that the term applies to all flowing streams.

THERMODYNAMIC ANALYSIS OF PROCESSES The energy balance for a steady-state steady-flow process resulting from the first law of thermodynamics is 1 ˙ +W ˙s ∆ H + ᎏ u2 + zg m ˙ =Q (4-358) fs 2 where H = specific enthalpy of flowing fluid u = velocity of flowing fluid z = elevation of flowing fluid above datum level g = local acceleration of gravity Ws = shaft work ˙ in Eq. (4-358) by Eq. (4-357) gives Eliminating Q



 



 

1 ˙ s(rev) ∆ H + ᎏ u2 + zg m ˙ = Tσ ∆(Sm) ˙ fs + W fs 2 ˙ s(rev) indicates that the shaft work is for a completely where W ˙ ideal. Thus reversible process. This work is called the ideal work W 1 2 ˙ ideal = ∆ H + ᎏ W u + zg m ˙ − Tσ ∆(Sm) ˙ fs (4-359) fs 2 In most applications to chemical processes, the kinetic- and potential-energy terms are negligible compared with the others; in this event Eq. (4-359) is written ˙ ideal = ∆(Hm) W ˙ fs − Tσ ∆(Sm) ˙ fs (4-360)



 

For the special case of a single stream flowing through the system, Eq. (4-360) becomes ˙ ideal = m(∆H W ˙ − Tσ ∆S) (4-361) Division by m ˙ puts this equation on a unit-mass basis Wideal = ∆H − Tσ ∆S (4-362) A completely reversible processes is hypothetical, devised solely to find the ideal work associated with a given change of state. Its only connection with an actual process is that it brings about the same change of state as the actual process, allowing comparison of the actual work of a process with the work of the hypothetical reversible process. Equations (4-359) through (4-362) give the work of a completely reversible process associated with given property changes in the flowing streams. When the same property changes occur in an actual ˙ s (or Ws) is given by an energy balance, and process, the actual work W comparison can be made of the actual work with the ideal work. When ˙ ideal (or Wideal) is positive, it is the minimum work required to bring W about a given change in the properties of the flowing streams, and is ˙ s. In this case a thermodynamic efficiency ηt is defined smaller than W as the ratio of the ideal work to the actual work: ˙ ideal W ηt (work required) = ᎏ (4-363) ˙s W ˙ ideal| is the maximum work ˙ ideal (or Wideal) is negative, | W When W obtainable from a given change in the properties of the flowing ˙ s|. In this case, the thermodynamic effistreams, and is larger than |W ciency is defined as the ratio of the actual work to the ideal work: ˙s W ηt(work produced) = ᎏ (4-364) ˙ ideal W LOST WORK Work that is wasted as the result of irreversibilities in a process is ˙ lost, and is defined as the difference between the called lost work W actual work of a process and the ideal work for the process. Thus, by definition, Wlost
Ws − Wideal (4-365) In terms of rates this is written ˙ lost
W ˙ s −W ˙ ideal W (4-366) The actual work rate comes from Eq. (4-358) 1 2 ˙ s=∆ H+ᎏ ˙ W u + zg m ˙ −Q fs 2 Subtracting the ideal work rate as given by Eq. (4-359) yields ˙ ˙ lost = Tσ∆(Sm) ˙ fs − Q (4-367) W



 

4-35

For the special case of a single stream flowing through the control volume, ˙ lost = mT ˙ W ˙ σ∆S − Q (4-368) Division of this equation by m ˙ gives Wlost = Tσ∆S − Q

(4-369)

where the basis is now a unit amount of fluid flowing through the control volume. The total rate of entropy increase (in both system and surroundings) as a result of a process is ˙ Q S˙ total = ∆(Sm) ˙ fs − ᎏ (4-370) Tσ For a single stream, division by m ˙ provides an equation based on a unit amount of fluid flowing through the control volume: Q Stotal = ∆S − ᎏ (4-371) Tσ Multiplication of Eq. (4-370) by Tσ gives ˙ ˙ fs − Q TσS˙ total = Tσ∆(Sm) Since the right-hand sides of this equation and of Eq. (4-367) are identical, it follows that ˙ lost = TσS˙ total W (4-372) For flow of a single stream on the basis of a unit amount of fluid, this becomes Wlost = TσStotal

(4-373)

Since the second law of thermodynamics requires that S˙ total ≥ 0 and Stotal ≥ 0 it follows that ˙ lost ≥ 0 W

and

Wlost ≥ 0

When a process is completely reversible, the equality holds, and the lost work is zero. For irreversible processes the inequality holds, and the lost work, that is, the energy that becomes unavailable for work, is positive. The engineering significance of this result is clear: The greater the irreversibility of a process, the greater the rate of entropy production and the greater the amount of energy that becomes unavailable for work. Thus, every irreversibility carries with it a price. ANALYSIS OF STEADY-STATE, STEADY-FLOW PROCESSES Many processes consist of a number of steps, and lost-work calculations are then made for each step separately. Writing Eq. (4-372) for each step of the process and summing gives

 W˙ lost = Tσ  S˙ total Dividing Eq. (4-372) by this result yields ˙ lost S˙ total W =ᎏ ᎏ ˙ lost  S˙ total W Thus, an analysis of the lost work, made by calculation of the fraction that each individual lost-work term represents of the total lost work, is the same as an analysis of the rate of entropy generation, made by expressing each individual entropy-generation term as a fraction of the sum of all entropy-generation terms. An alternative to the lost-work or entropy-generation analysis is a work analysis. This is based on Eq. (4-366), written ˙ lost = W ˙ s −W ˙ ideal W (4-374) For a work-requiring process, all of these work quantities are positive ˙ s >W ˙ ideal. The preceding equation is then expresed as and W ˙ ideal +  W ˙ lost ˙ s =W (4-375) W A work analysis gives each of the individual work terms on the right as ˙ s. a fraction of W

4-36

THERMODYNAMICS TABLE 4-2

States and Values of Properties for the Process of Fig. 4-12*

Point

P, bar

T, K

Composition

State

H, J/mol

S, J/(mol ⋅ K)

1 2 3 4 5 6 7

55.22 1.01 1.01 55.22 1.01 1.01 1.01

300 295 295 147.2 79.4 90 300

Air Pure O2 91.48% N2 Air 91.48% N2 pure O2 Air

Superheated Superheated Superheated Superheated Saturated vapor Saturated vapor Superheated

12,046 13,460 12,074 5,850 5,773 7,485 12,407

82.98 118.48 114.34 52.08 75.82 83.69 117.35

*Properties on the basis of Miller and Sullivan, U.S. Bur. Mines Tech. Pap. 424 (1928).

˙ s and W ˙ ideal are negative, and For a work-producing process, W ˙ ideal | > |W ˙ s|. Equation (4-374) in this case is best written: |W ˙ s| + ˙ lost ˙ ideal| = |W W (4-376) |W



A work analysis here expresses each of the individual work terms on ˙ ideal|. A work analysis cannot be carried out the right as a fraction of |W ˙ ideal is negative, indiin the case where a process is so inefficient that W ˙ s is positive, indicating that the process should produce work, but W cating that the process in fact requires work. A lost-work or entropy-generation analysis is always possible. Example 4: Lost-Work Analysis Make a work analysis of a simple Linde system for the separation of air into gaseous oxygen and nitrogen, as depicted in Fig. 4-12. Table 4-2 lists a set of operating conditions for the numbered points of the diagram. Heat leaks into the column of 147 J/mol of entering air and into the exchanger of 70 J/mol of entering air have been assumed. Take Tσ = 300 K. The basis for analysis is 1 mol of entering air, assumed to contain 79 mol % N2 and 21 mol % O2. By a material balance on the nitrogen, 0.79 = 0.9148 x; whence x = 0.8636 mol of nitrogen product 1 − x = 0.1364 mol of oxygen product Calculation of Ideal Work If changes in kinetic and potential energies are neglected, Eq. (4-360) is applicable. From the tabulated data, ∆(Hm) ˙ fs = (13,460)(0.1364) + (12,074)(0.8636) − (12,407)(1) = −144 J ∆(Sm) ˙ fs = (118.48)(0.1364) + (114.34)(0.8636) − (117.35)(1) = −2.4453 J/K

Thus, by Eq. (4-360), ˙ ideal = −144 − (300)(−2.4453) = 589.6 J W Calculation of Actual Work of Compression For simplicity, the work of compression is calculated by the equation for an ideal gas in a three-stage reciprocating machine with complete intercooling and with isentropic compression in each stage. The work so calculated is assumed to represent 80 percent of the actual work. The following equation may be found in any number of textbooks on thermodynamics: nγRT1 P2 (γ − 1)/nγ ˙ s = ᎏᎏ W −1 ᎏ (0.8)(γ − 1) P1

 

where



n = number of stages, here taken as 3 γ = ratio of heat capacities, here taken as 1.4 T1 = initial absolute temperature, 300 K P2 /P1 = overall pressure ratio, 54.5 R = universal gas constant, 8.314 J/(mol⋅K)

The efficiency factor of 0.8 is already included in the equation. Substitution of the remaining values gives (3)(1.4)(8.314)(300) ˙ s = ᎏᎏᎏ W (54.5)0.4/(3)(1.4) − 1 = 15,171 J (0.8)(0.4) The heat transferred to the surroundings during compression as a result of intercooling and aftercooling to 300 K is found from the first law:





Diagram of simple Linde system for air separation.

FIG. 4-12

˙ s = (12,046 − 12,407) − 15,171 = −15,532 J Q˙ = m(∆H) ˙ −W Calculation of Lost Work Equation (4-367) may be applied to each of the major units of the process. For the compressor/cooler, ˙ lost = (300)[(82.98)(1) − (117.35)(1)] − (−15,532) W = 5,221.0 J For the exchanger, ˙ lost = (300)[(118.48)(0.1364) + (114.34)(0.8636) + (52.08)(1) W − (75.82)(0.8636) − (83.69)(0.1364) − (82.98)(1)] − 70 = 2,063.4 J Finally, for the rectifier, ˙ lost = (300)[(75.82)(0.8636) + (83.69)(0.1364) − (52.08)(1)] − 147 W = 7,297.0 J Work Analysis Since the process requires work, Eq. (4-375) is appropriate for a work analysis. The various terms of this equation appear as entries in the following table, and are on the basis of 1 mol of entering air. ˙s % of W ˙ ideal W ˙ lost: W ˙ lost: W ˙ lost: W ˙s W

Compressor/cooler Exchanger Rectifier

589.6 J 5,221.0 J 2,063.4 J 7,297.0 J

3.9 34.4 13.6 48.1

15,171.0 J

100.0

The thermodynamic efficiency of this process as given by Eq. (4-363) is only 3.9 percent. Significant inefficiencies reside with each of the primary units of the process.

Section 5

Heat and Mass Transfer*

James G. Knudsen, Ph.D., Professor Emeritus of Chemical Engineering, Oregon State University; Member, American Institute of Chemical Engineers, American Chemical Society; Registered Professional Engineer (Oregon). (Conduction and Convection; Condensation, Boiling; Section Coeditor) Hoyt C. Hottel, S.M., Professor Emeritus of Chemical Engineering, Massachusetts Institute of Technology; Member, National Academy of Sciences, American Academy of Arts and Sciences, American Institute of Chemical Engineers, American Chemical Society, Combustion Institute. (Radiation) Adel F. Sarofim, Sc.D., Lammot du Pont Professor of Chemical Engineering and Assistant Director, Fuels Research Laboratory, Massachusetts Institute of Technology; Member, American Institute of Chemical Engineers, American Chemical Society, Combustion Institute. (Radiation) Phillip C. Wankat, Ph.D., Professor of Chemical Engineering, Purdue University; Member, American Institute of Chemical Engineers, American Chemical Society, International Adsorption Society. (Mass Transfer Section Coeditor) Kent S. Knaebel, Ph.D., President, Adsorption Research, Inc.; Member, American Institute of Chemical Engineers, American Chemical Society, International Adsorption Society. Professional Engineer (Ohio). (Mass Transfer Section Coeditor)

HEAT TRANSFER Modes of Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5-8

HEAT TRANSFER BY CONVECTION

HEAT TRANSFER BY CONDUCTION Fourier’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Three-Dimensional Conduction Equation . . . . . . . . . . . . . . . . . . . . . . . Thermal Conductivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Steady-State Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One-Dimensional Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conduction through Several Bodies in Series . . . . . . . . . . . . . . . . . . . Conduction through Several Bodies in Parallel. . . . . . . . . . . . . . . . . . Several Bodies in Series with Heat Generation . . . . . . . . . . . . . . . . . Example 1. Steady-State Conduction with Heat Generation . . . . . . . Two-Dimensional Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Unsteady-State Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One-Dimensional Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two-Dimensional Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conduction with Change of Phase. . . . . . . . . . . . . . . . . . . . . . . . . . . .

5-8 5-8 5-9 5-9 5-9 5-9 5-10 5-10 5-10 5-10 5-10 5-10 5-11 5-11

Coefficient of Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Individual Coefficient of Heat Transfer. . . . . . . . . . . . . . . . . . . . . . . . Overall Coefficient of Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . Representation of Heat-Transfer Film Coefficients . . . . . . . . . . . . . . Natural Convection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nusselt Equation for Various Geometries . . . . . . . . . . . . . . . . . . . . . . Simplified Dimensional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . Simultaneous Loss by Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enclosed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Forced Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analogy between Momentum and Heat Transfer . . . . . . . . . . . . . . . . Laminar Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transition Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 2. Calculation of j Factors in an Annulus . . . . . . . . . . . . . . . Jackets and Coils of Agitated Vessels . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5-12 5-12 5-12 5-12 5-12 5-12 5-12 5-12 5-12 5-12 5-14 5-14 5-15 5-16 5-16 5-17 5-19

* The contribution to the section on Interphase Mass Transfer of Mr. William M. Edwards (editor of Sec. 14), who was an author for the sixth edition, is acknowledged. 5-1

5-2

HEAT AND MASS TRANSFER

Nonnewtonian Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5-19 5-19

HEAT TRANSFER WITH CHANGE OF PHASE Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Condensation Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Condensation Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boiling (Vaporization) of Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boiling Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boiling Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5-20 5-20 5-20 5-22 5-22 5-22

HEAT TRANSFER BY RADIATION General References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature for Radiative Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . Nature of Thermal Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Blackbody Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radiative Exchange between Surfaces and Solids . . . . . . . . . . . . . . . . . Emittance and Absorptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Black-Surface Enclosures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 3: Calculation of View Factor . . . . . . . . . . . . . . . . . . . . . . . . Example 4: Calculation of Exchange Area. . . . . . . . . . . . . . . . . . . . . . Non-Black Surface Enclosures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 5: Radiation in a Furnace Chamber . . . . . . . . . . . . . . . . . . . Emissivities of Combustion Products . . . . . . . . . . . . . . . . . . . . . . . . . . . Gaseous Combustion Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 6: Calculation of Gas Emissivity and Absorptivity . . . . . . . . Flames and Particle Clouds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radiative Exchange between Gases or Suspended Matter and a Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 7: Radiation in Gases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single-Gas-Zone/Two-Surface-Zone Systems . . . . . . . . . . . . . . . . . . . The Effect of Nongrayness of Gas on Total-Exchange Area . . . . . . . Example 8: Effective Gas Emissivity . . . . . . . . . . . . . . . . . . . . . . . . . . Treatment of Refractory Walls Partially Enclosing a Radiating Gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combustion Chamber Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 9: Radiation in a Furnace . . . . . . . . . . . . . . . . . . . . . . . . . . . MASS TRANSFER General References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fick’s First Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mutual Diffusivity, Mass Diffusivity, Interdiffusion Coefficient . . . .

5-23 5-23 5-24 5-24 5-25 5-25 5-27 5-29 5-29 5-29 5-31 5-32 5-33 5-34 5-35 5-36 5-36 5-37 5-37 5-38 5-39 5-40 5-41

5-42 5-42 5-42 5-46

Self Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tracer Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass-Transfer Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem Solving Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuity and Flux Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Material Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flux Expressions: Simple Integrated Forms of Fick’s First Law . . . . Stefan-Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diffusivity Estimation—Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Binary Mixtures—Low Pressure—Nonpolar Components . . . . . . . . Binary Mixtures—Low Pressure—Polar Components . . . . . . . . . . . . Self-Diffusivity—High Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Supercritical Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low-Pressure/Multicomponent Mixtures . . . . . . . . . . . . . . . . . . . . . . Diffusivity Estimation—Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stokes-Einstein and Free-Volume Theories . . . . . . . . . . . . . . . . . . . . Dilute Binary Nonelectrolytes: General Mixtures . . . . . . . . . . . . . . . Binary Mixtures of Gases in Low-Viscosity, Nonelectrolyte Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dilute Binary Mixtures of a Nonelectrolyte in Water . . . . . . . . . . . . . Dilute Binary Hydrocarbon Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . Dilute Binary Mixtures of Nonelectrolytes with Water as the Solute. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dilute Dispersions of Macromolecules in Nonelectrolytes . . . . . . . . Concentrated, Binary Mixtures of Nonelectrolytes . . . . . . . . . . . . . . Binary Electrolyte Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multicomponent Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diffusion of Fluids in Porous Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interphase Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass-Transfer Principles: Dilute Systems . . . . . . . . . . . . . . . . . . . . . . Mass-Transfer Principles: Concentrated Systems . . . . . . . . . . . . . . . . HTU (Height Equivalent to One Transfer Unit) . . . . . . . . . . . . . . . . NTU (Number of Transfer Units) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions of Mass-Transfer Coefficients ˆkG and ˆkL . . . . . . . . . . . . . Simplified Mass-Transfer Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass-Transfer Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effects of Total Pressure on ˆkG and ˆkL . . . . . . . . . . . . . . . . . . . . . . . . . Effects of Temperature on ˆkG and ˆkL . . . . . . . . . . . . . . . . . . . . . . . . . . Effects of System Physical Properties on ˆkG and ˆkL . . . . . . . . . . . . . . Effects of High Solute Concentrations on ˆkG and ˆkL . . . . . . . . . . . . . Influence of Chemical Reactions on ˆkG and ˆkL . . . . . . . . . . . . . . . . . . Effective Interfacial Mass-Transfer Area a . . . . . . . . . . . . . . . . . . . . . Volumetric Mass-Transfer Coefficients Kˆ Ga and Kˆ La . . . . . . . . . . . . . Chilton-Colburn Analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5-46 5-46 5-46 5-46 5-46 5-46 5-47 5-47 5-48 5-48 5-49 5-49 5-49 5-50 5-50 5-50 5-50 5-51 5-52 5-52 5-52 5-52 5-52 5-53 5-54 5-54 5-54 5-54 5-56 5-57 5-57 5-57 5-57 5-58 5-61 5-64 5-66 5-69 5-69 5-74 5-78 5-79

HEAT AND MASS TRANSFER

5-3

Nomenclature and Units Specialized heat transfer nomenclature used for radiative heat transfer is defined in the subsection “Heat Transmission by Radiation.” Nomenclature for mass transfer is defined in the subsection “Mass Transfer.” Symbol a ax a′ A

b b′ bf B c1, c2, etc. c, cp C Cr dm D Dc De Dj Dotl Dp Ds Dt D1, D2 EH EM f F Fa Fg Fc Ft Fs Fw FD FT g, gL gc G Gmax Gmf h ham, hlm hb hcg hc + hr

Definition Proportionality coefficient Cross-sectional area of a fin Proportionality factor Area of heat transfer surface; Ai for inside; Ao for outside; Am for mean; Aavg for average; A1, A2, and A3 for points 1, 2, and 3 respectively; AB for bare surface of finned tube; Af for finned portion of tube; Auf for external area of unfinned portion of finned tube; Aof for external area of finned tube before fins are attached, equals Ao; Aoe for effective area of finned surface; AT for total external area of finned tube; Ad for surface area of dirt (scale) deposit Proportionality coefficient Proportionality factor Height of fin Material constant = 5D−0.5 Constants of integration Specific heat at constant pressure; cs for specific heat of solid; cg for specific heat of gas Thermal conductance, equals kA/x, hA, or UA; C1, C2, C3, Cn, thermal conductance of sections 1, 2, 3, and n respectively of a composite body Correlating constant; proportionality coefficient Depth of divided solids bed Diameter; Do for outside; Di for inside; Dr for root diameter of finned tube Diameter of a coil or helix Equivalent diameter of a cross section, usually 4 times free area divided by wetted perimeter; Dw for equivalent diameter of window Diameter of a jacketed cylindrical vessel Outside diameter of tube bundle Diameter of packing in a packed tube Inside diameter of heat-exchanger shell Solids-processing vessel diameter Diameter at points 1 and 2 respectively; inner and outer diameter of annulus respectively Eddy conductivity of heat Eddy viscosity Fanning friction factor; f1 for inner wall and f2 for outer wall of annulus; fk for ideal tube bank; skin friction drag coefficient Entrance factors Dry solids feed rate Gas volumetric flow rate Fraction of total tubes in cross-flow; Fbp for fraction of cross-flow area available for bypass flow Factor, ratio of temperature difference across tube-side film to overall mean temperature difference Factor, ratio of temperature difference across shell-side film to overall mean temperature difference Factor, ratio of temperature difference across retaining wall to overall mean temperature difference between bulk fluids Factor, ratio of temperature difference across combined dirt or scale films to overall mean temperature difference between bulk fluids Temperature-difference correction factor Acceleration due to gravity Conversion factor Mass velocity, equals V␳ or W/S; Gv for vapor mass velocity Mass velocity through minimum free area between rows of tubes normal to the fluid stream Minimum fluidizing mass velocity Local individual coefficient of heat transfer, equals dq/(dA)(∆T) Film coefficient based on arithmetic-mean temperature difference and logarithmic-mean temperature difference respectively Film coefficient delivered at base of fin Effective combined coefficient for simultaneous gas-vapor cooling and vapor condensation Combined coefficient for conduction, convection, and radiation between surface and surroundings

SI units

U.S. customary units

Dimensionless m2

Dimensionless ft2

m2

ft2

m

ft

J/(kg⋅K)

Btu/(lb⋅°F)

J/(s⋅K)

Btu/(h⋅°F)

Dimensionless m m

Dimensionless ft ft

m m

ft ft

m m m m m m

ft ft ft ft ft ft

J/(s⋅m⋅K) Pa⋅s Dimensionless

Btu/(h⋅ft⋅°F) lb/(ft⋅h) Dimensionless

kg/(s⋅m2) m3 (s⋅m2 of bed area)

lb/(h⋅ft2) ft3/(h⋅ft2 of bed area)

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

981 m/s2 1.0 (kg⋅m)/(N⋅s2) kg/(m2⋅s) kg/(m2⋅s)

(4.18)(108) ft/h2 (4.17)(108)(lb⋅ft)/(lbf⋅h2) lb/(h⋅ft2) lb/(h⋅ft2)

kg/(m2⋅s) J/(m2⋅s⋅K) J/(m2⋅s⋅K)

lb/(h⋅ft2) Btu/(h⋅ft2⋅°F) Btu/(h⋅ft2⋅°F)

J/(m2⋅s⋅K) J/(m2⋅s⋅K)

Btu/(h⋅ft2⋅°F) Btu/(h⋅ft2⋅°F)

J/(m2⋅s⋅K)

Btu/(h⋅ft2⋅°F)

5-4

HEAT AND MASS TRANSFER

Nomenclature and Units (Continued ) Symbol hdo, hdi hf hfi hfo hF, hs hi, ho hk h1 h′ hlm hr hT hw j J Jb, Jc, Jl, Jr k kv kavg, km kf kw K′ lc L Lo Lu LF LH Lp m M M n nt nr n′ nb Nr N

NB Nd NGr NNu NPe NPr NRe NSt Nss p pf p, p′

Definition Film coefficient for dirt or scale on outside or inside respectively of a surface Film coefficient for finned-tube exchangers based on total external surface Effective outside film coefficient of a finned tube based on inside area Film coefficient for air film of an air-cooled finned-tube exchanger based on external bare surface Effective film coefficient for dirt or scale on heat-transfer surface Film coefficient for heat transfer for inside and outside surface respectively Film coefficient for ideal tube bank; hs for shell side of baffled exchanger; hsv for coefficient at liquid-vapor interface Condensing coefficient on top tube; hN coefficient for N tubes in a vertical row Film coefficient for enclosed spaces Film coefficient based on log-mean temperature difference Heat-transfer coefficient for radiation Coefficient of total heat transfer by conduction, convection, and radiation between the surroundings and the surface of a body subject to unsteady-state heat transfer Equivalent coefficient of retaining wall, equals k/x Ordinate, Colburn j factor, equals f/2; jH for heat transfer; jH1 for inner wall of annulus; jH2 for outer wall of annulus; jk for heat transfer for ideal tube bank Mechanical equivalent of heat Correction factors for baffle bypassing, baffle configuration, baffle leakage, and adverse temperature gradient respectively Thermal conductivity; k1, k2, k3, thermal conductivities of bodies 1, 2, and 3 Thermal conductivity of vapor; k1 for liquid thermal conductivity; ks for thermal conductivity of solid Mean thermal conductivity Thermal conductivity of fluid at film temperature Thermal conductivity of retaining-wall material Property of non-Newtonian fluid Baffle cut; ls for baffle spacing Length of heat-transfer surface Flow rate Undisturbed length of path of fluid flow Thickness of dirt or scale deposit Depth of fluidized bed Diameter of agitator blade Ratio, term, or exponent as defined where used Molecular weight Weight of fluid Position ratio or number Number of tubes in parallel in a heat exchanger Number of rows in a vertical plane Flow-behavior index for nonnewtonian fluids Number of baffle-type coils Speed of agitator Number of tubes in a vertical row; or number of tubes in a bundle; Nb for number of baffles; NT for total number of tubes in exchanger; Nc for number of tubes in one cross-flow section; Ncw for number of cross-flow rows in each window Biot number, hT ∆x/k Proportionality coefficient, dimensionless group Grashof number, L3ρ2gβ ∆t/µ2 Nusselt number, hD/k or hL/k Peclet number, DGc/k Prandtl number, cµ/k Reynolds number, DG/µ Stanton number, NNu/NRe NPr Number of sealing strips Pressure Perimeter of a fin Center-to-center spacing of tubes in tube bundle (tube pitch); pn for tube pitch normal to flow; pp for tube pitch parallel to flow

SI units

U.S. customary units

J/(m2⋅s⋅K)

Btu/(h⋅ft2⋅°F)

J/(m2⋅s⋅K)

Btu/(h⋅ft2⋅°F)

J/(m2⋅s⋅K) J/(m2⋅s⋅K)

Btu/(h⋅ft2⋅°F) Btu/(h⋅ft2⋅°F)

J/(m2⋅s⋅K) J/(m2⋅s⋅K)

Btu/(h⋅ft2⋅°F) Btu/(h⋅ft2⋅°F)

J/(m2⋅s⋅K)

Btu/(h⋅ft2⋅°F)

J/(m2⋅s⋅K)

Btu/(h⋅ft2⋅°F)

J/(m2⋅s⋅K) J/(m2⋅s⋅K) J/(m2⋅s⋅K) J/(m2⋅s⋅K)

Btu/(h⋅ft2⋅°F) Btu/(h⋅ft2⋅°F) Btu/(h⋅ft2⋅°F) Btu/(h⋅ft2⋅°F)

J/(m2⋅s⋅K) Dimensionless

Btu/(h⋅ft2⋅°F) Dimensionless

1.0(N⋅m)/J

778(ft⋅lbf)/Btu

J/(m⋅s⋅K)

(Btu⋅ft)/(h⋅ft2⋅°F)

J/(m⋅s⋅K)

(Btu⋅ft)/(h⋅ft2⋅°F)

J/(m⋅s⋅K) J/(m⋅s⋅K) J/(m⋅s⋅K)

(Btu⋅ft)/(h⋅ft2⋅°F) (Btu⋅ft)/(h⋅ft2⋅°F) (Btu⋅ft)/(h⋅ft2⋅°F)

m m kg/s m m m m

ft ft lb/h ft ft ft ft

kg/mol kg Dimensionless

lb/mol lb Dimensionless

rad/s

r/h

Dimensionless

Dimensionless

kPa m m

lbf/ft2 abs ft ft

HEAT AND MASS TRANSFER

5-5

Nomenclature and Units (Continued ) Symbol ∆p P P′ ∆Pbk, ∆Pwk q q′ (q/A)max Q Q r

rj R

Rj S

Sr s t t1, t2, tn t′ t′1, t′2 t″1, t″2 tb tH, tL ts tsv tw t∞ TH, TL T ∆T, ∆t

∆tam, ∆tlm ∆tom ∆TH, ∆tH ∆TL, ∆tL ∆Tm, ∆tm u u° U U1, U2 Uco, Ucv, Uct, Ura Um v Vr

Definition Pressure of the vapor in a bubble minus saturation pressure of a flat liquid surface Absolute pressure; Pc for critical pressure Spacing between adjacent baffles on shell side of a heat exchanger (baffle pitch) Pressure drop for ideal-tube-bank cross-flow and ideal window respectively; ∆Ps for shell side of baffled exchanger Rate of heat flow, equals Q/θ Rate of heat generation Maximum heat flux in nucleate boiling Quantity of heat; rate of heat transfer Quantity of heat; QT for total quantity Radius; cylindrical and spherical coordinate; distance from midplane to a point in a body; r1 for inner wall of annulus; r2 for outer wall of annulus; ri for inside radius of tube; rm for distance from midplane or center of a body to the exterior surface of the body Inside radius Thermal resistance, equals x/kA, 1/UA, 1/hA; R1, R2, R3, Rn for thermal resistance of sections 1, 2, 3, and n of a composite body; RT for sum of individual resistances of several resistances in series or parallel; Rdi and Rdo for dirt or scale resistance on inner and outer surface respectively Ratio of total outside surface of finned tube to area of tube having same root diameter Cross-sectional area; Sm for minimum cross-sectional area between rows of tubes, flow normal to tubes; Stb for tube-to-baffle leakage area for one baffle; Ssb for shell-to-baffle area for one baffle; Sw for area for flow through window; Swg for gross window area; Swt for window area occupied by tubes Slope of rotary shell Specific gravity of fluid referred to liquid water Bulk temperature; temperature at a given point in a body at time θ Temperature at points 1, 2, and n in a system through which heat is being transferred Temperature of surroundings Inlet and outlet temperature respectively of hotter fluid Inlet and outlet temperature respectively of colder fluid Initial uniform bulk temperature of a body; bulk temperature of a flowing fluid High and low temperature respectively on tube side of a heat exchanger Surface temperature Saturated-vapor temperature Wall temperature Temperature of undisturbed flowing stream High and low temperature respectively on shell side of a heat exchanger Absolute temperature; Tb for bulk temperature; Tw for wall temperature; Tv for vapor temperature; Tc for coolant temperature; Te for temperature of emitter; Tr for temperature of receiver Temperature difference; ∆t1, ∆t2, and ∆t3 temperature difference across bodies 1, 2, and 3 or at points 1, 2, and 3; ∆To, ∆to for overall temperature difference; ∆tb for temperature difference between surface and boiling liquid Arithmetic- and logarithmic-mean temperature difference respectively Mean effective overall temperature difference Greater terminal temperature difference Lesser terminal temperature difference Mean temperature difference Velocity in x direction Friction velocity Overall coefficient of heat transfer; Uo for outside surface basis; U′ for overall coefficient between liquid-vapor interface and coolant Overall coefficient of heat transfer at points 1 and 2 respectively Overall coefficients for divided solids processing by conduction, convection, contact, and radiation mechanism respectively Mean overall coefficient of heat transfer Velocity in y direction Volume of rotating shell

SI units

U.S. customary units

kPa

lbf/ft2 abs

kPa m

lbf/ft2 ft

kPa

lbf/ft2

W, J/s J/(s⋅m3) J/(s⋅m2) J/s J m

Btu/h Btu/(h⋅ft3) Btu/(h⋅ft2) Btu/h Btu ft

Dimensionless (s⋅K)/J

Dimensionless (h⋅°F)/Btu

m2

ft2

K K

°F °F

K K K K

°F °F °F °F

K

°F

K K K K K

°F °F °F °F °F

K

°R

K

°F, °R

K K K K K m/s m/s J/(s⋅m2⋅K)

°F °F °F °F °F ft/h ft/h Btu/(h⋅ft2⋅°F)

J/(s⋅m2⋅K) J/(s⋅m2⋅K)

Btu/(h⋅ft2⋅°F) Btu/(h⋅ft2⋅°F)

J/(s⋅m2⋅K) m/s m3

Btu/(h⋅ft2⋅°F) ft/h ft3

5-6

HEAT AND MASS TRANSFER

Nomenclature and Units (Concluded ) Symbol V V′, Vs VF Vg, Vl V′max w w W Wr W1, Wo xq x X y y+ Y z zp ZH

Definition

SI units

Velocity Velocity Face velocity of a fluid approaching a bank of finned tubes Specific volume of gas, liquid Maximum velocity through minimum free area between rows of tubes normal to the fluid stream Velocity in z direction Flow rate Total mass rate of flow; mass rate of vapor generated; WF for total rate of vapor condensation in one tube Weight rate of flow Total mass rate of flow on tube side and shell side respectively of a heat exchanger Vapor quality, xi for inlet quality, xo for outlet quality Coordinate direction; length of conduction path; xs for thickness of scale; x1, x2, and x3 at positions 1, 2, and 3 in a body through which heat is being transferred Factor Coordinate direction Wall distance Factor Coordinate direction Distance (perimeter) traveled by fluid across fin Ratio of sensible heat removed from vapor to total heat transferred

U.S. customary units

m/s m/s m/s m3/kg m/s

ft/h ft/s ft/h ft3/lb ft/h

m/s kg/s kg/s

ft/h lb/h lb/h

kg/(s⋅tube) kg/s

lb/(h⋅tube) lb/h

kg/s m

lb/h ft

Dimensionless m Dimensionless Dimensionless m m Dimensionless

Dimensionless ft Dimensionless Dimensionless ft ft Dimensionless

m2/s

ft2/h

K−1 ° kg/(s2 − n′⋅m) kg/(s⋅m)

°F−1 ° lb/(ft⋅s2 − n′) lb/(h⋅ft)

m m m2/s

ft ft ft2/h

s

h

J/kg m Pa⋅s

Btu/lb ft lb/(h⋅ft)

m2/s kg/m3

ft2/h lb/ft3

N/m

lbf/ft

N/m2

lbf/ft2

rad Dimensionless

rad Dimensionless

Greek symbols α β β′ γ Γ δs δ δsb ε εv η θ θb λ λm µ

ν ρ σ
τ φ φp Φ ω Ω

Thermal diffusivity, equals k/ρc; αe for effective thermal diffusivity of powdered solids Volumetric coefficient of thermal expansion Contact angle of a bubble Fluid consistency Mass rate of flow of a falling film from a tube or surface per unit perimeter, equals w/πD for vertical tube, w/2L for horizontal tube Correction factor, ratio of nonnewtonian to newtonian shear rates Cell width Diametral shell-to-baffle clearance Eddy diffusivity; εM for eddy diffusivity of momentum; εH for eddy diffusivity of heat Fraction of voids in porous bed Fluidization efficiency Time Baffle cut Latent heat (enthalpy) of vaporization (condensation) Radius of maximum velocity Viscosity; µw for viscosity at wall temperature; µb for viscosity at bulk temperature; µf for viscosity at film temperature; µG, µg, and µv for viscosity of gas or vapor; µL, µl for viscosity of liquid; µw for viscosity at wall; µl for viscosity of fluid at inner wall of annulus Kinematic viscosity Density; ρL, ρl for density of liquid; ρG, ρv for density of gas or vapor; ρs for density of solid Surface tension between a liquid and its vapor Term indicating summation of variables Shear stress τw for shear stress at the wall Velocity-potential function Particle sphericity Viscous-dissipation function Angle of repose of powdered solid Fin efficiency

GENERAL REFERENCES: Becker, Heat Transfer, Plenum, New York, 1986. Bejan, Convection Heat Transfer, Wiley, New York, 1984. Bird, Stewart, and Lightfoot, Transport Phenomena, Wiley, New York, 1960. Carslaw and Jaeger, Conduction of Heat in Solids, Clarendon Press, Oxford, 1959. Chapman, Heat Transfer, 2d ed., Macmillan, New York, 1967. Drew and Hoopes, Advances in Chemical Engineering, Academic, New York, vol. 1, 1956; vol. 2, 1958; vol. 5, 1964; vol. 6, 1966; vol. 7, 1968. Dusinberre, Heat Transfer Calculations by Finite Differences, International Textbook, Scranton, Pa., 1961. Eckert and Drake, Heat and Mass Transfer, 2d ed., McGraw-Hill, New York, 1959. Gebhart, Heat Transfer, McGraw-Hill, New York, 1961. Irvine and Hartnett, Advances in Heat Transfer, Academic, New York, vol. 1, 1964; vol. 2, 1965; vol. 3, 1966. Grigull and Sandner, Heat Conduction, Hemisphere Publishing, 1984. Jakob, Heat Transfer, Wiley, New York, vol. 1, 1949; vol. 2, 1957. Jakob and Hawkins, Elements of Heat Transfer, 3d ed., Wiley, New York, 1957. Kakac, Bergles, and Mayinger, Heat Exchangers: Thermal Hydraulic Fundamentals and Design, Hemisphere Publishing, Washington, 1981. Kakac and Yener, Convective Heat Transfer, Hemisphere Publishing, Washington, 1980. Kay, An Introduction to Fluid Mechanics and Heat Transfer, 2d ed., Cambridge University Press, Cambridge, England, 1963. Kays, Convective Heat and Mass Transfer, McGraw-Hill, New York, 1966. Kays and London, Compact Heat Exchangers, 3d ed., McGraw-Hill, New York, 1984. Kern, Process Heat Transfer, McGrawHill, New York, 1950. Knudsen and Katz, Fluid Dynamics and Heat Transfer, McGraw-Hill, New York, 1958. Kraus, Analysis and Evaluation of Extended Surface Thermal Systems, Hemisphere Publishing, Washington, 1982. Kutatladze, A Concise Encyclopedia of Heat Transfer, 1st English ed., Pergamon, New York, 1966. Lykov, Heat and Mass Transfer in Capillary Porous Bodies, translated from Russian, Pergamon, New York, 1966. McAdams, Heat Transmission, 3d ed., McGraw-Hill, New York, 1954. Mickley, Sherwood, and Reed, Applied Mathematics in Chemical Engineering, 2d ed., McGraw-Hill, New York, 1957. Rohsenow and Choi, Heat, Mass, and Momentum Transfer, Prentice-Hall, Englewood Cliffs, N.J., 1961. Schlünder (ed.), Heat Exchanger Design Handbook, Hemisphere Publishing, Washington, 1983 (Book 2; Chapter 2.4 [Conduction], Chapter 2.5 [Convection], Chapter 2.6 [Condensation], Chapter 2.7 [Boiling]). Skelland, Non-Newtonian Flow and Heat Transfer, Wiley, New York, 1967. Taborek and Bell, Process Heat Exchanger Design, Hemisphere Publishing, Washington, 1984. Taborek, Hewitt, and Afghan, Heat Exchangers: Theory and Practice, Hemisphere Publishing, Washington, 1983. TSederberg, Thermal Conductivity of Liquids and Gases, M.I.T., Cambridge, Mass., 1965. Welty, Wicks, and Wilson, Fundamentals of Momentum, Heat and Mass Transfer, 3d ed., Wiley, New York, 1984. Zenz and Othmer, Fluidization and Fluid Particle Systems, Reinhold, New York, 1960.

31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65.

Lee and Thodos, Ind. Eng. Chem. Fundam., 22, 17–26 (1983). Lee and Thodos, Ind. Eng. Chem. Res., 27, 992–997 (1988). Lees and Sarram, J. Chem. Eng. Data, 16, 1, 41 (1971). Leffler and Cullinan, Ind. Eng. Chem. Fundam., 9, 84, 88 (1970). Lugg, Anal. Chem., 40, 1072 (1968). Marrero and Mason, AIChE J., 19, 498 (1973). Mathur and Thodos, AIChE J., 11, 613 (1965). Matthews and Akgerman, AIChE J., 33, 881 (1987). Matthews, Rodden and Akgerman, J. Chem. Eng. Data, 32, 317 (1987). Olander, AIChE J., 7, 175 (1961). Passut and Danner, Chem. Eng. Prog. Symp. Ser., 140, 30 (1974). Perkins and Geankoplis, Chem. Eng. Sci., 24, 1035–1042 (1969). Pinto and Graham, AIChE J., 32, 291 (1986). Pinto and Graham, AIChE J., 33, 436 (1987). Quale, Chem. Rev., 53, 439 (1953). Rathbun and Babb, Ind. Eng. Chem. Proc. Des. Dev., 5, 273 (1966). Reddy and Doraiswamy, Ind. Eng. Chem. Fundam., 6, 77 (1967). Riazi and Whitson, Ind. Eng. Chem. Res., 32, 3081 (1993). Robinson, Edmister, and Dullien, Ind. Eng. Chem. Fundam., 5, 75 (1966). Rollins and Knaebel, AIChE J., 37, 470 (1991). Siddiqi, Krahn, and Lucas, J. Chem. Eng. Data, 32, 48 (1987). Siddiqi and Lucas, Can. J. Chem. Eng., 64, 839 (1986). Smith and Taylor, Ind. Eng. Chem. Fundam., 22, 97 (1983). Sridhar and Potter, AIChE J., 23, 4, 590 (1977). Stiel and Thodos, AIChE J., 7, 234 (1961). Sun and Chen, Ind. Eng. Chem. Res., 26, 815 (1987). Tanford, Phys. Chem. of Macromolecule, Wiley, New York, NY (1961). Taylor and Webb, Comput. Chem. Eng., 5, 61 (1981). Tyn and Calus, J. Chem. Eng. Data, 20, 310 (1975). Umesi and Danner, Ind. Eng. Chem. Process Des. Dev., 20, 662 (1981). Van Geet and Adamson, J. Phys. Chem., 68, 2, 238 (1964). Vignes, Ind. Eng. Chem. Fundam., 5, 184 (1966). Wilke, Chem. Eng. Prog., 46, 2, 95 (1950). Wilke and Chang, AIChE J., 1, 164 (1955). Wilke and Lee, Ind. Eng. Chem., 47, 1253 (1955).

REFERENCES FOR DIFFUSIVITIES IN POROUS SOLIDS, TABLE 5-20 66. Ruthven, Principles of Adsorption & Adsorption Processes, Wiley, 1984. 67. Satterfield, Mass Transfer in Heterogeneous Catalysis, MIT Press, 1970. 68. Suzuki, Adsorption Engineering, Kodansha—Elsevier, 1990. 69. Yang, Gas Separation by Adsorption Processes, Butterworths, 1987.

REFERENCES FOR DIFFUSIVITIES 1. Akita, Ind. Eng. Chem. Fundam., 10, 89 (1981). 2. Asfour and Dullien, Chem. Eng. Sci., 41, 1891 (1986). 3. Blanc, J. Phys., 7, 825 (1908). 4. Brokaw, Ind. Eng. Chem. Process Des. and Dev., 8, 2, 240 (1969). 5. Caldwell and Babb, J. Phys. Chem., 60, 51 (1956). 6. Catchpole and King, Ind. Eng. Chem. Res., 33, 1828 (1994). 7. Chen and Chen, Chem. Eng. Sci., 40, 1735 (1985). 8. Chung, Ajlan, Lee and Starling, Ind. Eng. Chem. Res., 27, 671 (1988). 9. Condon and Craven, Aust. J. Chem., 25, 695 (1972). 10. Cullinan, AIChE J., 31, 1740–1741 (1985). 11. Cullinan, Can. J. Chem. Eng., 45, 377–381 (1967). 12. Cussler, AIChE J., 26, 1 (1980). 13. Darken, Trans. Am. Inst. Mining Met. Eng., 175, 184 (1948). 14. Debenedetti and Reid, AIChE J., 32, 2034 (1986); see errata: AIChE J., 33, 496 (1987). 15. Elliott, R. W. and H. Watts, Can. J. Chem., 50, 31 (1972). 16. Erkey and Akgerman, AIChE J., 35, 443 (1989). 17. Ertl, Ghai, and Dullien, AIChE J., 20, 1, 1 (1974). 18. Fairbanks and Wilke, Ind. Eng. Chem., 42, 471 (1950). 19. Fuller, Schettler and Giddings, Ind. Eng. Chem., 58, 18 (1966). 20. Ghai, Ertl, and Dullien, AIChE J., 19, 5, 881 (1973). 21. Gordon, J. Chem. Phys., 5, 522 (1937). 22. Graham and Dranoff, Ind. Eng. Chem. Fundam., 21, 360–365 (1982). 23. Graham and Dranoff, Ind. Eng. Chem. Fundam., 21, 365–369 (1982). 24. Gurkan, AIChE J., 33, 175–176 (1987). 25. Hayduk and Laudie, AIChE J., 20, 3, 611 (1974). 26. Hayduk and Minhas, Can. J. Chem. Eng., 60, 195 (1982). 27. Hildebrand, Science, 174, 490 (1971). 28. Hiss and Cussler, AIChE J., 19, 4, 698 (1973). 29. Jossi, Stiel, and Thodos, AIChE J., 8, 59 (1962). 30. Krishnamurthy and Taylor, Chem. Eng. J., 25, 47 (1982).

REFERENCES FOR TABLES 5-21 TO 5-28 70. Bahmanyar, Chang-Kakoti, Garro, Liang, and Slater, Chem. Engr. Rsch. Des., 68, 74 (1990). 71. Beenackers and van Swaaij, Chem. Engr. Sci., 48, 3109 (1993). 72. Bird, Stewart and Lightfoot, Transport Phenomena, Wiley, 1960. 73. Blatt, Dravid, Michaels, and Nelson in Flinn (ed.), Membrane Science and Technology, 47, Plenum, 1970. 74. Bolles and Fair, Institution Chem. Eng. Symp. Ser., 56, 3/35 (1979). 75. Bolles and Fair, Chem. Eng., 89(14), 109 (July 12, 1982). 76. Bravo and Fair, Ind. Eng. Chem. Process Des. Dev., 21, 162 (1982). 77. Bravo, Rocha and Fair, Hydrocarbon Processing, 91 (Jan. 1985). 78. Brian and Hales, AIChE J., 15, 419 (1969). 79. Calderbank and Moo-Young, Chem. Eng. Sci., 16, 39 (1961). 80. Chilton and Colburn, Ind. Eng. Chem., 26, 1183 (1934). 81. Cornell, Knapp, and Fair, Chem. Engr. Prog., 56(7), 68 (1960). 82. Cornet and Kaloo, Proc. 3rd Int’l. Congr. Metallic Corrosion—Moscow, 3, 83 (1966). 83. Cussler, Diffusion: Mass Transfer in Fluid Systems, Cambridge, 1984. 84. Dwivedi and Upadhyay, Ind. Eng. Chem. Process Des. Develop, 16, 1657 (1977). 85. Eisenberg, Tobias, and Wilke, Chem. Engr. Prog. Symp. Sec., 51(16), 1 (1955). 86. Elzinga and Banchero, Chem. Engr. Progr. Symp. Ser., 55(29), 149 (1959). 87. Fair, “Distillation” in Rousseau (ed.), Handbook of Separation Process Technology, Wiley, 1987. 88. Faust, Wenzel, Clump, Maus, and Andersen, Principles of Unit Operations, 2d ed., Wiley, 1980. 89. Frossling, Gerlands Beitr. Geophys., 52, 170 (1938). 90. Garner and Suckling, AIChE J., 4, 114 (1958). 91. Geankoplis, Transport Processes and Unit Operations, 3d ed., Prentice Hall, 1993. 5-7

5-8 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120. 121. 122. 123. 124. 125. 126. 127. 128.

HEAT AND MASS TRANSFER Gibilaro, Davies, Cooke, Lynch, and Middleton, Chem. Engr. Sci., 40, 1811 (1985). Gilliland and Sherwood, Ind. Engr. Chem., 26, 516 (1934). Griffith, Chem. Engr. Sci., 12, 198 (1960). Gupta and Thodos, AIChE J., 9, 751 (1963). Gupta and Thodos, Ind. Eng. Chem. Fundam., 3, 218 (1964). Harriott, AIChE J., 8, 93 (1962). Hausen, Verfahrenstech. Beih. Z. Ver. Dtsch. Ing., 4, 91 (1943). Heertjes, Holve, and Talsma, Chem. Engr. Sci., 3, 122 (1954). Hines and Maddox, Mass Transfer: Fundamentals and Applications, Prentice Hall, 1985. Hsiung and Thodos, Int. J. Heat Mass Transfer, 20, 331 (1977). Hsu, Sato, and Sage, Ind. Engr. Chem., 46, 870 (1954). Hughmark, Ind. Eng. Chem. Fundam., 6, 408 (1967). Johnson, Besic, and Hamielec, Can. J. Chem. Engr., 47, 559 (1969). Johnstone and Pigford, Trans. AIChE, 38, 25 (1942). Kafesjian, Plank, and Gerhard, AIChE J., 7, 463 (1961). Kelly and Swenson, Chem. Eng. Prog., 52, 263 (1956). King, Separation Processes, 2d ed., McGraw-Hill (1980). Kirwan, “Mass Transfer Principles” in Rousseau, Handbook of Separation Process Technology, Wiley, 1987. Klein, Ward, and Lacey, “Membrane Processes—Dialysis and ElectroDialysis” in Rousseau, Handbook of Separation Process Technology, Wiley, 1987. Kohl, “Absorption and Stripping” in Rousseau, Handbook of Separation Process Technology, Wiley, 1987. Kojima, Uchida, Ohsawa, and Iguchi, J. Chem. Engng. Japan, 20, 104 (1987). Koloini, Sopcic, and Zumer, Chem. Engr. Sci., 32, 637 (1977). Lee, Biochemical Engineering, Prentice Hall, 1992. Lee and Foster, Appl. Catal., 63, 1 (1990). Lee and Holder, Ind. Engr. Chem. Res., 34, 906 (1995). Levich, Physicochemical Hydrodynamics, Prentice Hall, 1962. Levins and Gastonbury, Trans. Inst. Chem. Engr., 50, 32, 132 (1972). Lim, Holder, and Shah, J. Supercrit. Fluids, 3, 186 (1990). Linton and Sherwood, Chem. Engr. Prog., 46, 258 (1950). Ludwig, Applied Process Design for Chemical and Petrochemical Plants, 2d ed., vol. 2, Gulf Pub. Co., 1977. McCabe, Smith, and Harriott, Unit Operations of Chemical Engineering, 5th ed., McGraw-Hill, 1993. Nelson and Galloway, Chem. Engr. Sci., 30, 7 (1975). Notter and Sleicher, Chem. Eng. Sci., 26, 161 (1971). Ohashi, Sugawara, Kikuchi, and Konno, J. Chem. Engr. Japan, 14, 433 (1981). Onda, Takeuchi, and Okumoto, J. Chem. Engr. Japan, 1, 56 (1968). Pasternak and Gauvin, AIChE J., 7, 254 (1961). Pasternak and Gauvin, Can. J. Chem. Engr., 38, 35 (April 1960).

129. 130. 131. 132. 133. 134. 135. 136. 137. 138. 139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150. 151. 152. 153. 154. 155. 156. 157. 158. 159. 160. 161. 162. 163. 164. 165. 166. 167. 168. 169.

Perez and Sandall, AIChE J., 20, 770 (1974). Petrovic and Thodos, Ind. Eng. Chem. Fundam., 7, 274 (1968). Pinczewski and Sideman, Chem. Engr. Sci., 29, 1969 (1974). Prandtl, Phys. Zeit., 29, 487 (1928). Prasad and Sirkar, AIChE J., 34, 177 (1988). Rahman and Streat, Chem. Engr. Sci., 36, 293 (1981). Ranz and Marshall, Chem. Engr. Prog., 48, 141, 173 (1952). Reiss, Ind. Eng. Chem. Process Des. Develop, 6, 486 (1967). Riet, Ind. Eng. Chem. Process Des. Dev., 18, 357 (1979). Rowe, Chem. Engr. Sci., 30, 7 (1975). Rowe, Claxton, and Lewis, Trans. Inst. Chem. Engr. London, 43, 14 (1965). Ruckenstein and Rajagopolan, Chem. Engr. Commun., 4, 15 (1980). Ruthven, Principles of Adsorption & Adsorption Processes, Wiley, 1984. Satterfield, AIChE J., 21, 209 (1975). Schluter and Deckwer, Chem. Engr. Sci., 47, 2357 (1992). Schmitz, Steiff, and Weinspach, Chem. Engng. Technol., 10, 204 (1987). Sherwood, Brian, Fisher, and Dresner, Ind. Eng. Chem. Fundam., 4, 113 (1965). Sherwood, Pigford, and Wilke, Mass Transfer, McGraw-Hill, 1975. Shulman, Ullrich, Proulx, and Zimmerman, AIChE J., 1, 253 (1955). Shulman and Margolis, AIChE J., 3, 157 (1957). Siegel, Sparrow, and Hallman, Appl. Sci. Res. Sec. A., 7, 386 (1958). Sissom and Pitts, Elements of Transport Phenomena, McGraw-Hill, 1972. Skelland, Diffusional Mass Transfer, Wiley (1974). Skelland and Cornish, AIChE J., 9, 73 (1963). Skelland and Moeti, Ind. Eng. Chem. Res., 29, 2258 (1990). Skelland and Tedder, “Extraction—Organic Chemicals Processing” in Rousseau, Handbook of Separation Process Technology, Wiley, 1987, pp. 405–466. Skelland and Wellek, AIChE J., 10, 491, 789 (1964). Slater, “Rate Coefficients in Liquid-Liquid Extraction Systems” in Godfrey and Slater, Liquid-Liquid Extraction Equipment, Wiley, 1994, pp. 45–94. Steinberger and Treybal, AIChE J., 6, 227 (1960). Steiner, L., Chem. Eng. Sci., 41, 1979 (1986). Taylor and Krishna, Multicomponent Mass Transfer, Wiley, 1993. Tournie, Laguerie, and Couderc, Chem. Engr. Sci., 34, 1247 (1979). Treybal, Mass Transfer Operations, 3d ed., McGraw-Hill, 1980. Von Karman, Trans. ASME, 61, 705 (1939). Wakao and Funazkri, Chem. Engr. Sci., 33, 1375 (1978). Wankat, Equilibrium-Staged Separations, Prentice Hall, 1988. Wankat, Rate-Controlled Separations, Chapman-Hall, 1990. Wilson and Geankoplis, Ind. Eng. Chem. Fundam., 5, 9 (1966). Yagi and Yoshida, Ind. Eng. Chem. Process Des. Dev., 14, 488 (1975). Yang, Gas Separation by Adsorption Processes, Butterworths, 1987. Yoshida, Ramaswami, and Hougen, AIChE J., 8, 5 (1962).

HEAT TRANSFER MODES OF HEAT TRANSFER There are three fundamental types of heat transfer: conduction, convection, and radiation. All three types may occur at the same time, and it is advisable to consider the heat transfer by each type in any particular case. Conduction is the transfer of heat from one part of a body to another part of the same body, or from one body to another in physical contact with it, without appreciable displacement of the particles of the body.

Convection is the transfer of heat from one point to another within a fluid, gas, or liquid by the mixing of one portion of the fluid with another. In natural convection, the motion of the fluid is entirely the result of differences in density resulting from temperature differences; in forced convection, the motion is produced by mechanical means. When the forced velocity is relatively low, it should be realized that “free-convection” factors, such as density and temperature difference, may have an important influence. Radiation is the transfer of heat from one body to another, not in contact with it, by means of wave motion through space.

HEAT TRANSFER BY CONDUCTION FOURIER’S LAW Fourier’s law is the fundamental differential equation for heat transfer by conduction: dQ/dθ = −kA(dt/dx)

direction of the flow of heat, i.e., the temperature gradient. The factor k is called the thermal conductivity; it is a characteristic property of the material through which the heat is flowing and varies with temperature.

(5-1)

where dQ/dθ (quantity per unit time) is the rate of flow of heat, A is the area at right angles to the direction in which the heat flows, and −dt/dx is the rate of change of temperature with the distance in the

THREE-DIMENSIONAL CONDUCTION EQUATION Equation (5-1) is used as a basis for derivation of the unsteady-state three-dimensional energy equation for solids or static fluids:

HEAT TRANSFER BY CONDUCTION ∂t ∂t ∂ ∂t ∂ ∂t ∂ cρ ᎏ = ᎏ k ᎏ + ᎏ k ᎏ + ᎏ k ᎏ + q′ ∂θ ∂x ∂x ∂y ∂y ∂z ∂z

 

 

 

(5-2)

where x, y, z are distances in the rectangular coordinate system and q′ is the rate of heat generation (by chemical reaction, nuclear reaction, or electric current) in the solid per unit of volume. Solution of Eq. (5-2) with appropriate boundary and initial conditions will give the temperature as a function of time and location in the material. Equation (5-2) may be transformed into spherical or cylindrical coordinates to conform more closely to the physical shape of the system. THERMAL CONDUCTIVITY Thermal conductivity varies with temperature but not always in the same direction. The thermal conductivities for many materials, as a function of temperature, are given in Sec. 2. Additional and more comprehensive information may often be obtained from suppliers of the materials. Impurities, especially in metals, can give rise to variations in thermal conductivity of from 50 to 75 percent. In using thermal conductivities, engineers should remember that conduction is not the sole method of transferring heat and that, particularly with liquids and gases, radiation and convection may be much more important. The thermal conductivity at a given temperature is a function of the apparent, or bulk, density. Thus, at 0°C (32°F), k for asbestos wool is 0.09 J/(m⭈s⭈K) [0.052 Btu/(hr⭈ft⭈°F)] when the bulk density is 400 kg/m3 (24.9 lb/ft3) and is 0.19 (0.111) for a density of 700 (43.6). In determining the apparent thermal conductivities of granular solids, such as granulated cork or charcoal grains, Griffiths (Spec. Rep. 5, Food Investigation Board, H. M. Stationery Office, 1921) found that air circulates within the mass of granular solid. Under a certain set of conditions, the apparent thermal conductivity of a charcoal was 9 percent greater when the test section was vertical than when it was horizontal. When the apparent conductivity of a mixture of cellular or porous nonhomogeneous solid is determined, the observed temperature coefficient may be much larger than for the homogeneous solid alone, because heat is transferred not only by the mechanism of conduction but also by convection in the gas pockets and by radiation from surface to surface of the individual particles. If internal radiation is an important factor, a plot of the apparent conductivity as ordinate versus temperature should show a curve concave upward, since radiation increases with the fourth power of the absolute temperature. Griffiths noted that cork, slag, wool, charcoal, and wood fibers, when of good quality and dry, have thermal conductivities about 2.2 times that of still air, whereas a highly cellular form of rubber, 112 kg/m3 (7 lb/ft3), had a thermal conductivity only 1.6 times that of still air. In measuring the apparent thermal conductivity of diathermanous substances such as quartz (especially when exposed to radiation emitted at high temperatures), it should be remembered that a part of the heat is transmitted by radiation. Bridgman [Proc. Am. Acad. Arts Sci., 59, 141 (1923)] showed that the thermal conductivity of liquids is increased by only a few percent under a pressure of 100,330 kPa (1000 atm). The thermal conductivity of some liquids varies with temperature through a maximum. It is often necessary for the engineer to estimate thermal conductivities; methods are indicated in Sec. 2. Equation (5-2) considers the thermal conductivity to be variable. If k is expressed as a function of temperature, Eq. (5-2) is nonlinear and difficult to solve analytically except for certain special cases. Usually in complicated systems numerical solution by means of computer is possible. A complete review of heat conduction has been given by Davis and Akers [Chem. Eng., 67(4), 187, (5), 151 (1960)] and by Davis [Chem. Eng., 67(6), 213, (7), 135 (8), 137 (1960)]. STEADY-STATE CONDUCTION For steady flow of heat, the term dQ/dθ in Eq. (5-1) is constant and may be replaced by Q/θ or q. Likewise, in Eq. (5-2) the term ∂ t/∂θ is zero. Hence, for constant thermal conductivity, Eq. (5-2) may be expressed as ∇ 2 t = (q′/k)

(5-3)

5-9

One-Dimensional Conduction Many heat-conduction problems may be formulated into a one-dimensional or pseudo-onedimensional form in which only one space variable is involved. Forms of the conduction equation for rectangular, cylindrical, and spherical coordinates are, respectively, ∂2 t q′ (5-4a) ᎏ2 = − ᎏ ∂x k

 

1 d dt q′ ᎏᎏ rᎏ =−ᎏ r dr dr k



(5-4b)



1 d dt q′ (5-4c) ᎏ2 ᎏ r 2 ᎏ = − ᎏ r dr dr k These are second-order differential equations which upon integration become, respectively, t = −(q′x2/2k) + c1x + c2

(5-5a)

t = −(q′r /4k) + c1 ln r + c2

(5-5b)

t = −(q′r 2/6k) − (c1/r) + c2

(5-5c)

2

Constants of integration c1 and c2 are determined by the boundary conditions, i.e., temperatures and temperature gradients at known locations in the system. For the case of a solid surface exposed to surroundings at a different temperature and for a finite surface coefficient, the boundary condition is expressed as hT (ts − t′ ) = −k(dt/dx)surf

(5-6)

Inspection of Eqs. (5-5a), (5-5b), and (5-5c) indicates the form of temperature profile for various conditions and geometries and also reveals the effect of the heat-generation term q′ upon the temperature distributions. In the absence of heat generation, one-dimensional steady-state conduction may be expressed by integrating Eq. (5-1): x2 t2 dx q (5-7) ᎏ = − k dt x1 t1 A Area A must be known as a function of x. If k is constant, Eq. (5-7) is expressed in the integrated form





q = kAavg (t1 − t2)/(x2 − x1)

(5-8)



x2

1 dx (5-9) Aavg = ᎏ ᎏ x2 − x1 x1 A Examples of values of Aavg for various functions of x are shown in the following table. where

Area proportional to

Aavg

Constant x

A1 = A2 A2 − A1 ln (A2/A1) 2 A1 A

x2

Usually, thermal conductivity k is not constant but is a function of temperature. In most cases, over the ranges of values used the relation is linear. Integration of Eq. (5-7), with k linear in t, gives x2 dx q (5-10) ᎏ = kavg(t1 − t2) x1 A where kavg is the arithmetic-average thermal conductivity between temperatures t1 and t2. This average probably gives results which are correct within the precision of the data in the majority of cases, though a special integration can be made whenever k is known to be greatly different from linear in temperature. Conduction through Several Bodies in Series Figure 5-1 illustrates diagrammatically the temperature gradients accompanying the steady conduction of heat in series through three solids.



5-10

HEAT AND MASS TRANSFER For fuel, at x = 0, t = 570°C (1050°F), dt/dx = 0 (this follows if the temperature is finite at the midplane). For fuel and cladding, at x = x1, tf = tc, kf (dt/dx) = kc (dt/dx) For cladding, at x = x2, tc − 400 = −(kc /42,600)(dt/dx) For the fuel, the first integration of Eq. (10-4a) gives dtf /dx = −(q′/kf )x + c1 which gives c1 = 0 when the boundary condition is applied. Thus the second integration gives tf = −(q′/2kf )x2 + c2 from which c2 is determined to be 570 (1050) upon application of the boundary condition. Thus the temperature profile in the fuel is tf = −(q′/2kf )x2 + 570

FIG. 5-1

Temperature gradients for steady heat conduction in series through three solids.

The temperature profile in the cladding is obtained by integrating Eq. (10-4a) twice with q′ = 0. Hence

Since the heat flow through each of the three walls must be the same,

There are now three unknowns, c1, c2, and q′, and three boundary conditions by which they can be determined. At x = x1,

(dtc /dx) = c1

q = (k1 A1 ∆t1/x1) = (k2 A2 ∆ t2 /x2) = (k3 A3 ∆ t3 /x3)

(5-11)

then

∆t1 = qR1

∆ t2 = qR2

(5-12) ∆ t3 = qR3

(5-13)

Adding the individual temperature drops, noting that q is uniform, q(R1 + R2 + R3) = ∆t1 + ∆ t2 + ∆ t3 =
∆ t

(5-14)

q =
∆ t/RT = (t1 − t4)/RT

(5-15)

or

where RT is the overall resistance and is the sum of the individual resistances in series, then RT = R1 + R2 + ⋅ ⋅ ⋅ + Rn

(5-16)

When a wall is constructed of several layers of solids, the joints at adjacent layers may not perfectly exclude air spaces, and these additional resistances should not be overlooked. Conduction through Several Bodies in Parallel For n resistances in parallel, the rates of heat flow are additive: q = ∆t/R1 + ∆ t/R2 + ⋅ ⋅ ⋅ + ∆ t/Rn



1 1 1 q= ᎏ+ᎏ+⋅⋅⋅+ᎏ R1 R2 Rn

 ∆t

q = (C1 + C2 + ⋅ ⋅ ⋅ + Cn)∆ t =
C ∆ t

tc = c1 x + c2

q′x12 /2kf + 570 = c1x1 + c2 − kf q′x1/kf = kc c1 At x = x2,

Since, by definition, individual thermal resistance R = x/ kA

and

(5-17a) (5-17b)

c1 x 2 + c2 − 200 = −(kc /42,600)c1 From which q′ = (2.53)(109) J/(m3⋅s)[(2.38)(108)Btu/(h⋅ft3)] c1 = −(1.92)(105) c2 = 724

Two-Dimensional Conduction If the temperature of a material is a function of two space variables, the two-dimensional conduction equation is (assuming constant k) ∂ 2t/∂x2 + ∂ 2 t/∂y2 = −q′/k

(5-18)

When q′ is zero, Eq. (5-18) reduces to the familiar Laplace equation. The analytical solution of Eq. (10-18) as well as of Laplace’s equation is possible for only a few boundary conditions and geometric shapes. Carslaw and Jaeger (Conduction of Heat in Solids, Clarendon Press, Oxford, 1959) have presented a large number of analytical solutions of differential equations applicable to heat-conduction problems. Generally, graphical or numerical finite-difference methods are most frequently used. Other numerical and relaxation methods may be found in the general references in the “Introduction.” The methods may also be extended to three-dimensional problems. UNSTEADY-STATE CONDUCTION

(5-17c)

where R1 to Rn are the individual resistances and C1 to Cn are the individual conductances; C = kA/x. Several Bodies in Series with Heat Generation The simple Fourier type of equation indicated by Eq. (5-15) may not be used when heat generation occurs in one of the bodies in the series. In this case, Eq. (5-5a), (5-5b), or (5-5c) must be solved with appropriate boundary conditions. Example 1: Steady-State Conduction with Heat Generation A plate-type nuclear fuel element, consisting of a uranium-zirconium alloy (3.2)(10−3) m (0.125 in) thick clad on each side with a (6.4)(10−4)-m- (0.025-in-) thick layer of zirconium, is cooled by water under pressure at 200°C (400°F), the heat-transfer coefficient being 42,600 J/(m2⋅s⋅K) [7500 Btu/(h⋅ft2⋅°F)]. If the temperature at the center of the fuel must not exceed 570°C (1050°F), determine the maximum rate of heat generation in the fuel. The zirconium and zirconium alloy have a thermal conductivity of 21 J/(m⋅s⋅K) [12 Btu/(h⋅ft2)(°F/ft)]. Solution. Equation (5-4a) may be integrated for each material. The heat generation is zero in the cladding, and its value for the fuel may be determined from the integrated equations. Let x = 0 at the midplane of the fuel. Then x1 = (1.6)(10−3) m (0.0625 in) at the cladding-fuel interface and x2 = (2.2)(10−3) m (0.0875 in) at the cladding-water interface. Let the subscripts c, f refer to cladding and fuel respectively. The boundary conditions are:

When temperatures of materials are a function of both time and space variables, more complicated equations result. Equation (5-2) is the three-dimensional unsteady-state conduction equation. It involves the rate of change of temperature with respect to time ∂ t/ ∂θ. Solutions to most practical problems must be obtained through the use of digital computers. Numerous articles have been published on a wide variety of transient conduction problems involving various geometrical shapes and boundary conditions. One-Dimensional Conduction The one-dimensional transient conduction equations are (for constant physical properties) ∂ t/∂θ = α(∂2t/∂x2) + q′/cρ ∂t α ∂ ∂t q′ ᎏ=ᎏᎏ rᎏ +ᎏ ∂θ r dr ∂r cρ

 

(rectangular coordinates)

(5-19a)

(cylindrical coordinates)

(5-19b)

q′ ∂t α ∂ ∂t (spherical coordinates) (5-19c) ᎏ = ᎏ2 ᎏ r2 ᎏ + ᎏ ∂θ r ∂r ∂r cρ These equations have been solved analytically for solid slabs, cylinders, and spheres. The solutions are in the form of infinite series, and usually the results are plotted as curves involving four ratios [Gurney and Lurie, Ind. Eng. Chem., 15, 1170 (1923)] defined as follows with q′ = 0:

 

HEAT TRANSFER BY CONDUCTION Y = (t′ − t)/(t′ − tb)

X = kθ/ρcrm2

5-11

Since each ratio is dimensionless, any consistent units may be employed in any ratio. The significance of the symbols is as follows: t′ = temperature of the surroundings; tb = initial uniform temperature of the body; t = temperature at a given point in the body at the time θ measured from the start of the heating or cooling operations; k = uniform thermal conductivity of the body; ρ = uniform density of the body; c = specific heat of the body; hT = coefficient of total heat transfer between the surroundings and the surface of the body expressed as heat transferred per unit time per unit area of the surface per unit difference in temperature between surroundings and surface; r = distance, in the direction of heat conduction, from the midpoint or midplane of the body to the point under consideration; rm = radius of a sphere or cylinder, one-half of the thickness of a slab heated from both faces, the total thickness of a slab heated from one face and insulated perfectly at the other; and x = distance, in the direction of heat conduction, from the surface of a semi-infinite body (such as the surface of the earth) to the point under consideration. In making the integrations which lead to the curves shown, the following factors were assumed constant: c, hT, k, r, rm, t′, x, and ρ. The working curves are shown in Figs. 5-2 to 5-5 for cylinders of infinite length, spheres, slabs of infinite faces, and semi-infinite solids respectively, with Y plotted as ordinates on a logarithmic scale versus X as abscissas to an arithmetic scale, for various values of the ratios m and n. To facilitate calculations involving instantaneous rates of cooling or heating of the semi-infinite body, Fig. 5-5 shows also a curve of dY/dX versus X. Similar plots to a larger scale are given in McAdams,

Brown and Marco, Schack, and Stoever (see “Introduction: General References”). For a solid of infinite thickness (Fig. 5-5) and with m = 0, 2 z (5-21) Y = ᎏ exp (−z2) dz π 0  where z = 1/ 2X and the “error integral” may be evaluated from standard mathematical tables. Various numerical and graphical methods are used for unsteadystate conduction problems, in particular the Schmidt graphical method (Foppls Festschrift, Springer-Verlag, Berlin, 1924). These methods are very useful because any form of initial temperature distribution may be used. Two-Dimensional Conduction The governing differential equation for two-dimensional transient conduction is q′ ∂t ∂2t ∂2t (5-22) ᎏ = α ᎏ2 + ᎏ2 + ᎏ ∂θ ∂x ∂y cρ McAdams (Heat Transmission, 3d ed., McGraw-Hill, New York, 1954) gives various forms of transient difference equations and methods of solving transient conduction problems. The availability of computers and a wide variety of computer programs permits virtually routine solution of complicated conduction problems. Conduction with Change of Phase A special type of transient problem (the Stefan problem) involves conduction of heat in a material when freezing or melting occurs. The liquid-solid interface moves with time, and in addition to conduction, latent heat is either generated or absorbed at the interface. Various problems of this type are discussed by Bankoff [in Drew et al. (eds.), Advances in Chemical Engineering, vol. 5, Academic, New York, 1964].

Heating and cooling of a solid cylinder having an infinite ratio of length to diameter.

FIG. 5-4 Heating and cooling of a solid slab having a large face area relative to the area of the edges.

m = k/hTrm

n = r/rm

FIG. 5-2

(5-20a,b) (5-20c,d)





Heating and cooling of a solid of infinite thickness, neglecting edge effects. (This may be used as an approximation in the zone near the surface of a body of finite thickness.)

FIG. 5-5 FIG. 5-3

Heating and cooling of a solid sphere.



5-12

HEAT AND MASS TRANSFER

HEAT TRANSFER BY CONVECTION COEFFICIENT OF HEAT TRANSFER In many cases of heat transfer involving either a liquid or a gas, convection is an important factor. In the majority of heat-transfer cases met in industrial practice, heat is being transferred from one fluid through a solid wall to another fluid. Assume a hot fluid at a temperature t1 flowing past one side of a metal wall and a cold fluid at t7 flowing past the other side to which a scale of thickness xs adheres. In such a case, the conditions obtaining at a given section are illustrated diagrammatically in Fig. 5-6. For turbulent flow of a fluid past a solid, it has long been known that, in the immediate neighborhood of the surface, there exists a relatively quiet zone of fluid, commonly called the film. As one approaches the wall from the body of the flowing fluid, the flow tends to become less turbulent and develops into laminar flow immediately adjacent to the wall. The film consists of that portion of the flow which is essentially in laminar motion (the laminar sublayer) and through which heat is transferred by molecular conduction. The resistance of the laminar layer to heat flow will vary according to its thickness and can range from 95 percent of the total resistance for some fluids to about 1 percent for other fluids (liquid metals). The turbulent core and the buffer layer between the laminar sublayer and turbulent core each offer a resistance to heat transfer which is a function of the turbulence and the thermal properties of the flowing fluid. The relative temperature difference across each of the layers is dependent upon their resistance to heat flow. The Energy Equation A complete energy balance on a flowing fluid through which heat is being transferred results in the energy equation (assuming constant physical properties): ∂t ∂t ∂t ∂t cρ ᎏ + u ᎏ + v ᎏ + w ᎏ ∂θ ∂x ∂y ∂z





∂2 t ∂ 2 t ∂ 2 t = k ᎏ2 + ᎏ2 + ᎏ2 + q′ + Φ (5-23) ∂x ∂y ∂z where Φ is the term accounting for energy dissipation due to fluid viscosity and is significant in high-speed gas flow and in the flow of highly viscous liquids. Except for the time term, the left-hand terms of Eq. (523) are the so-called convective terms involving the energy carried by the fluid by virtue of its velocity. Therefore, the solution of the equation is dependent upon the solution of the momentum equations of flow. Solutions of Eq. (5-23) exist only for several simple flow cases and geometries and mainly for laminar flow. For turbulent flow the difficulties of expressing the fluid velocity as a function of space and time coordinates and of obtaining reliable values of the effective thermal conductivity of the flowing fluid have prevented solution of the equation unless simplifying assumptions and approximations are made.





Individual Coefficient of Heat Transfer Because of the complicated structure of a turbulent flowing stream and the impracticability of measuring thicknesses of the several layers and their temperatures, the local rate of heat transfer between fluid and solid is defined by the equations dq = hi d Ai (t1 − t3) = ho d Ao (t5 − t7)

(5-24)

where hi and ho are the local heat-transfer coefficients inside and outside the wall, respectively, and temperatures are defined by Fig. 5-6. The definition of the heat-transfer coefficient is arbitrary, depending on whether bulk-fluid temperature, centerline temperature, or some other reference temperature is used for t1 or t7. Equation (5-24) is an expression of Newton’s law of cooling and incorporates all the complexities involved in the solution of Eq. (5-23). The temperature gradients in both the fluid and the adjacent solid at the fluid-solid interface may also be related to the heat-transfer coefficient: dt dt dq = h i dAi (t1 − t3) = −k ᎏ = −k ᎏ (5-25) dx fluid dx solid Equation (5-25) holds for the liquid only if laminar flow exists immediately adjacent to the solid surface. The integration of Eq. (5-24) will give out out dq dq Ai = or Ao = (5-26) ᎏ ᎏ in in hi ∆ ti ho ∆ to which may be evaluated only if the quantities under the integral can be expressed in terms of a single variable. If q is a linear function of ∆t and h is constant, then Eq. (5-26) gives hA(∆tin − ∆ tout) q = ᎏᎏ (5-27) ln (∆ tin/∆ tout) where the ∆t factor is the logarithmic-mean temperature difference between the wall and the fluid. Frequently experimental data report average heat-transfer coefficients based upon an arbitrarily defined temperature difference, the two most common being hlm A(∆tin − ∆ tout) q = ᎏᎏ (5-28a) ln(∆ tin/∆ tout) ham A(∆tin + ∆ tout) q = ᎏᎏ (5-28b) 2 where hlm and ham are average heat-transfer coefficients based upon the logarithmic-mean temperature difference and the arithmeticaverage temperature difference, respectively. Overall Coefficient of Heat Transfer In testing commercial heat-transfer equipment, it is not convenient to measure tube temperatures (t3 or t4 in Fig. 5-6), and hence the overall performance is expressed as an overall coefficient of heat transfer U based on a convenient area dA, which may be dAi, dAo, or an average of dAi and dAo; whence, by definition,













dq = U dA (t1 − t7)

(5-29)

U is called the “overall coefficient of heat transfer,” or merely the “overall coefficient.” The rate of conduction through the tube wall and scale deposit is given by kdAavg(t3 − t4) dq = ᎏᎏ = hd dAd(t4 − t5) (5-30) x Upon eliminating t3, t4, t5 from Eqs. (5-24), (5-29), and (5-30), the complete expression for the steady rate of heat flow from one fluid through the wall and scale to a second fluid, as illustrated in Fig. 5-6, is t1 − t7 dq = ᎏᎏᎏᎏ = U dA (t1 − t7) (5-31)* 1 x 1 1 ᎏᎏ + ᎏᎏ + ᎏᎏ + ᎏᎏ hi dAi k dAavg hd dAd ho dAo Temperature gradients for a steady flow of heat by conduction and convection from a warmer to a colder fluid separated by a solid wall.

FIG. 5-6

* Normally, dirt and scale resistance must be considered on both sides of the tube wall. The area dA is any convenient reference area.

HEAT TRANSFER BY CONVECTION

tally and are given in Table 5-1. Fluid properties are evaluated at tf = (ts + t′)/2. For vertical plates and cylinders and 1 < NPr < 40, Kato, Nishiwaki, and Hirata [Int. J. Heat Mass Transfer, 11, 1117 (1968)] recommend the relations 0.36 0.175 NNu = 0.138NGr (NPr − 0.55) (5-33a)

Representation of Heat-Transfer Film Coefficients There are two general methods of expressing film coefficients: (1) dimensionless relations and (2) dimensional equations. The dimensionless relations are usually indicated in either of two forms, each yielding identical results. The preferred form is that suggested by Colburn [Trans. Am. Inst. Chem. Eng., 29, 174–210 (1933)]. It relates, primarily, three dimensionless groups: the Stanton number h/cG, the Prandtl number cµ/k, and the Reynolds number DG/µ. For more accurate correlation of data (at Reynolds number <10,000), two additional dimensionless groups are used: ratio of length to diameter L/D and ratio of viscosity at wall (or surface) temperature to viscosity at bulk temperature. Colburn showed that the product of the Stanton number and the two-thirds power of the Prandtl number (and, in addition, power functions of L/D and µw /µ for Reynolds number <10,000) is approximately equal to half of the Fanning friction factor f/2. This product is called the Colburn j factor. Since the Colburn type of equation relates heat transfer and fluid friction, it has greater utility than other expressions for the heat-transfer coefficient. The classical (and perhaps more familiar) form of dimensionless expressions relates, primarily, the Nusselt number hD/k, the Prandtl number cµ/k, and the Reynolds number DG/µ. The L/D and viscosityratio modifications (for Reynolds number <10,000) also apply. The dimensional equations are usually expansions of the dimensionless expressions in which the terms are in more convenient units and in which all numerical factors are grouped together into a single numerical constant. In some instances, the combined physical properties are represented as a linear function of temperature, and the dimensional equation resolves into an equation containing only one or two variables.

for NGr > 109, and 0.25 0.25 NNu = 0.683NGr NPr [NPr /(0.861 + NPr)]0.25

h = b(∆t)mL3m − 1 (5-34) Values of b in SI and U.S. customary units are given in Table 5-1 for air, water, and organic liquids. Simultaneous Loss by Radiation The heat transferred by radiation is often of significant magnitude in the loss of heat from surfaces to the surroundings because of the diathermanous nature of atmospheric gases (air). It is convenient to represent radiant-heat transfer, for this case, as a radiation film coefficient which is added to the film coefficient for convection, giving the combined coefficient for convection and radiation (hc + hr). In Fig. 5-7 values of the film coefficient for radiation hr are plotted against the two surface temperatures for emissivity = 1.0. Table 5-2 shows values of (hc + hr) from single horizontal oxidized pipe surfaces. Enclosed Spaces The rate of heat transfer across an enclosed space is calculated from a special coefficient h′ based upon the temperature difference between the two surfaces, where h′ = (q/A)/ (ts1 − ts2). The value of h′L/k may be predicted from Eq. (5-32) by using the values of a and m given in Table 5-3. For vertical enclosed cells 10 in high and up to 2-in gap width, Landis and Yanowitz (Proc. Third Int. Heat Transfer Conf., Chicago, 1966, vol. II, p. 139) give

Natural convection occurs when a solid surface is in contact with a fluid of different temperature from the surface. Density differences provide the body force required to move the fluid. Theoretical analyses of natural convection require the simultaneous solution of the coupled equations of motion and energy. Details of theoretical studies are available in several general references (Brown and Marco, Introduction to Heat Transfer, 3d ed., McGraw-Hill, New York, 1958; and Jakob, Heat Transfer, Wiley, New York, vol. 1, 1949; vol. 2, 1957) but have generally been applied successfully to the simple case of a vertical plate. Solution of the motion and energy equations gives temperature and velocity fields from which heat-transfer coefficients may be derived. The general type of equation obtained is the so-called Nusselt equation: L3ρ2gβ ∆t cµ m hL (5-32a) ᎏ = a ᎏᎏ ᎏ k µ2 k

q

0.84

(NGr NPr)0.28



3

7

hδ δ ᎏ = 0.2 + 0.145 ᎏ NGr k D1





0.25

δ exp −0.02 ᎏ D1



b, air at Configuration

Y = NGrNPr

a

m

Vertical surfaces L = vertical dimension < 3 ft

<104 104 < Y < 109 >109 <10−5 −5 10 < Y < 10−3 10−3 < Y < 1 1 < Y < 104 104 < Y < 109 >109 105 < Y < 2 × 107(FU) 2 × 107 < Y < 3 × 1010(FU) 3 × 105 < Y < 3 × 1010(FD)

1.36 0.59 0.13 0.49 0.71 1.09 1.09 0.53 0.13 0.54 0.14 0.27

1⁄ 5

Horizontal flat surface



(5-36)

for 0.55 < δ/D1 < 2.65, where NGr is based upon gap width δ and D1 is the core diameter of the annulus.

Values of a, m, and b for Eqs. (5-32) and (5-34)

Horizontal cylinder L = diameter < 8 in

(5-35)

for 2 × 10 < NGr NPr (δ/L) < 10 , where q/A is the uniform heat flux and ∆ t is the temperature difference at L/2. Equation (5-35) is applicable for air, water, and silicone oils. For horizontal annuli Grugal and Hauf (Proc. Third Int. Heat Transfer Conf., Chicago, 1966, vol. II, p 182) report 3

(5-32b) NNu = a(NGr NPr) Nusselt Equation for Various Geometries Natural-convection coefficients for various bodies may be predicted from Eq. (5-32). The various numerical values of a and m have been determined experimen-

NOTE:

δ

= 0.123(δ/L) ᎏA  ᎏ k ∆t

m

TABLE 5-1

(5-33b)

for NGr < 109. Simplified Dimensional Equations Equation (5-32) is a dimensionless equation, and any consistent set of units may be used. Simplified dimensional equations have been derived for air, water, and organic liquids by rearranging Eq. (5-32) into the following form by collecting the fluid properties into a single factor:

NATURAL CONVECTION



5-13

d s 0 1⁄25 1⁄ 10 1⁄ 5 d s d s d

b, water at

b, organic liquid at

21°C

70°F

21°C

70°F

21°C

70°F

1.37 1.24

0.28 0.18

127

26

59

12

1.32 1.24 1.86

0.27 0.18 0.38

0.88

0.18

FU = facing upward; FD = facing downward. b in SI units is given in °C column; b in U.S. customary units, in °F column.

5-14

HEAT AND MASS TRANSFER siderable effort has been directed toward deriving some simple relationship between momentum and heat transfer. The methodology has been to use easily observed velocity profiles to obtain a measure of the diffusivity of momentum in the flowing stream. The analogy between heat and momentum is invoked by assuming that diffusion of heat and diffusion of momentum occur by essentially the same mechanism so that a relatively simple relationship exists between the diffusion coefficients. Thus, the diffusivity of momentum is used to predict temperature profiles and thence by Eq. (10-25) to predict the heat-transfer coefficient. The analogy has been reasonably successful for simple geometries and for fluids of very low Prandtl number (liquid metals). For highPrandtl-number fluids the empirical analogy of Colburn [Trans. Am. Inst. Chem. Eng., 29, 174 (1933)] has been very successful. A j factor for momentum transfer is defined as j = f/2, where f is the friction factor for the flow. The j factor for heat transfer is assumed to be equal to the j factor for momentum transfer j = h/cG(cµ/k)2/3

(5-37)

More involved analyses for circular tubes reduce the equations of motion and energy to the form (ν + εM)du τgc (5-38a) ᎏ = − ᎏᎏ dy ρ (α + εH)dt q/A (5-38b) ᎏ = − ᎏᎏ dy cρ where εH is the eddy diffusivity of heat and εM is the eddy diffusivity of momentum. The units of diffusivity are L2/θ. The eddy viscosity is EM = ρεM, and the eddy conductivity of heat is EH = εHcρ. Values of εM are determined via Eq. (5-38a) from experimental velocitydistribution data. By assuming εH/εM = constant (usually unity), Eq. (10-38b) is solved to give the temperature distribution from which the heat-transfer coefficient may be determined. The major difficulties in solving Eq. (5-38b) are in accurately defining the thickness of the various flow layers (laminar sublayer and buffer layer) and in obtaining a suitable relationship for prediction of the eddy diffusivities. For assistance in predicting eddy diffusivities, see Reichardt (NACA Tech. Memo 1408, 1957) and Strunk and Chao [Am. Inst. Chem. Eng. J., 10, 269 (1964)]. Internal and External Flow Two main types of flow are considered in this subsection: internal or conduit flow, in which the fluid completely fills a closed stationary duct, and external or immersed flow, in which the fluid flows past a stationary immersed solid. With internal flow, the heat-transfer coefficient is theoretically infinite at the location where heat transfer begins. The local heat-transfer coefficient rapidly decreases and becomes constant, so that after a certain length the average coefficient in the conduit is independent of the length. The local coefficient may follow an irregular pattern, however, if obstructions or turbulence promoters are present in the duct. For immersed flow, the local coefficient is again infinite at the point where heating begins, after which it decreases and may show various irregularities depending upon the configuration of the body. Usually in this instance the local coefficient never becomes constant as flow proceeds downstream over the body. When heat transfer occurs during immersed flow, the rate is dependent upon the configuration of the body, the position of the body, the proximity of other bodies, and the flow rate and turbulence of the

FIG. 5-7 Radiation coefficients of heat transfer hr. To convert British thermal units per hour-square foot-degrees Fahrenheit to joules per square meter– second–kelvins, multiply by 5.6783; °C = (°F − 32)/1.8.

FORCED CONVECTION Forced-convection heat transfer is the most frequently employed mode of heat transfer in the process industries. Hot and cold fluids, separated by a solid boundary, are pumped through the heat-transfer equipment, the rate of heat transfer being a function of the physical properties of the fluids, the flow rates, and the geometry of the system. Flow is generally turbulent, and the flow duct varies in complexity from circular tubes to baffled and extended-surface heat exchangers. Theoretical analyses of forced-convection heat transfer have been limited to relatively simple geometries and laminar flow. Analyses of turbulent-flow heat transfer have been based upon some mechanistic model and have not generally yielded relationships which were suitable for design purposes. Usually for complicated geometries only empirical relationships are available, and frequently these are based upon limited data and special operating conditions. Heattransfer coefficients are strongly influenced by the mechanics of flow occurring during forced-convection heat transfer. Intensity of turbulence, entrance conditions, and wall conditions are some of the factors which must be considered in detail as greater accuracy in prediction of coefficients is required. Analogy between Momentum and Heat Transfer The interrelationship of momentum transfer and heat transfer is obvious from examining the equations of motion and energy. For constant fluid properties, the equations of motion must be solved before the energy equation is solved. If fluid properties are not constant, the equations are coupled, and their solutions must proceed simultaneously. ConTABLE 5-2

Nominal pipe diameter, in 1 3 5 10

Values of (hc + hr)*

Btu/(h⋅ft2⋅°F from pipe to room) For horizontal bare standard steel pipe of various sizes in a room at 80°F Temperature difference, °F

30

50

100

150

200

250

300

350

400

450

500

550

600

650

700

2.16 1.97

2.26 2.05 1.95 1.87

2.50 2.25 2.15 2.07

2.73 2.47 2.36 2.29

3.00 2.73 2.61 2.54

3.29 3.00 2.90 2.82

3.60 3.31 3.20 3.12

3.95 3.69 3.54 3.47

4.34 4.03 3.90 3.84

4.73 4.43

5.16 4.85

5.60 5.26

6.05 5.71

6.51 6.19

6.98 6.66

1.80

*Bailey and Lyell [Engineering, 147, 60 (1939)] give values for (hc + hr) up to ∆ts of 1000°F. °C = (°F − 32)/1.8; 5.6783 Btu/(h⋅ft2⋅°F) = J/(m2⋅s⋅/K).

HEAT TRANSFER BY CONVECTION TABLE 5-3

Values of a and m for Eq. (5-32)

Configuration Vertical spaces Horizontal spaces

NGr NPr(δ/L)3

a

m

2 × 104 to 2 × 105 2 × 105 to 107 104 to 3 × 105 3 × 105 to 107

0.20 (δ/L)−5/36 0.071 (δ/L)1/9 0.21 (δ/L)−1/4 0.075

d s d s

stream. The heat-transfer coefficient varies over the immersed body, since both the thermal and the momentum boundary layers vary in thickness. Relatively simple relationships are available for simple configurations immersed in an infinite flowing fluid. For complicated configurations and assemblages of bodies such as are found on the shell side of a heat exchanger, little is known about the local heattransfer coefficient; empirical relationships giving average coefficients are all that are usually available. Research that has been conducted on local coefficients in complicated geometries has not been extensive enough to extrapolate into useful design relationships. Laminar Flow Normally, laminar flow occurs in closed ducts when NRe < 2100 (based on equivalent diameter De = 4 × free area ÷ perimeter). Laminar-flow heat transfer has been subjected to extensive theoretical study. The energy equation has been solved for a variety of boundary conditions and geometrical configurations. However, true laminar-flow heat transfer very rarely occurs. Natural-convection effects are almost always present, so that the assumption that molecular conduction alone occurs is not valid. Therefore, empirically derived equations are most reliable. Data are most frequently correlated by the Nusselt number (NNu)lm or (NNU)am, the Graetz number NGz = (NRe NPr D/L), and the Grashof (natural-convection effects) number NGr. Some correlations consider only the variation of viscosity with temperature, while others also consider density variation. Theoretical analyses indicate that for very long tubes (NNu)lm approaches a limiting value. Limiting Nusselt numbers for various closed ducts are shown in Table 5-4. Circular Tubes For horizontal tubes and constant wall temperature, several relationships are available, depending on the Graetz number. For 0.1 < NGz < 104, Hausen’s [Allg. Waermetech., 9, 75 (1959)], the following equation is recommended. 0.8 µb 0.14 0.19NGz (NNu)lm = 3.66 + ᎏᎏ (5-39) ᎏ 0.467 1 + 0.117NGz µw

 

For NGz > 100, the Sieder-Tate relationship [Ind. Eng. Chem., 28, 1429 (1936)] is satisfactory for small diameters and ∆t’s: 1/3 (NNu)am = 1.86NGz (µb /µw)0.14

(5-40)

A more general expression covering all diameters and ∆ t’s is 1/3 obtained by including an additional factor 0.87(1 + 0.015NGr ) on the right side of Eq. (5-40). The diameter should be used in evaluating NGr. An equation published by Oliver [Chem. Eng. Sci., 17, 335 (1962)] is also recommended.

TABLE 5-4 Values of Limiting Nusselt Number in Laminar Flow in Closed Ducts Limiting Nusselt number NGr < 4.0 Configuration Circular tube Concentric annulus Equilateral triangle Rectangles Aspect ratio: 1.0 (square) 0.713 0.500 0.333 0.25 0 (parallel planes)

For laminar flow in vertical tubes a series of charts developed by Pigford [Chem. Eng. Prog. Symp. Ser. 17, 51, 79 (1955)] may be used to predict values of ham. Annuli Approximate heat-transfer coefficients for laminar flow in annuli may be predicted by the equation of Chen, Hawkins, and Solberg [Trans. Am. Soc. Mech. Eng., 68, 99 (1946)]:

Constant wall temperature

Constant heat flux

3.66

4.36 Eq. (10-42) 3.00

2.89

3.63 3.78 4.11 4.77 5.35 8.24

3.39 7.60

µb

ᎏ   ᎏ D  µ 

0.45 0.5 De (NNu)am = 1.02N Re N Pr ᎏ L

δ = cell width, L = cell length.

5-15

0.4

D2

0.8

1

0.14 0.05 NGr

(5-41)

1

Limiting Nusselt numbers for slug-flow annuli may be predicted (for constant heat flux) from Trefethen (General Discussions on Heat Transfer, London, ASME, New York, 1951, p. 436): 8(m − 1)(m2 − 1)2 (NNu)lm = ᎏᎏᎏ 4m4 ln m − 3m4 + 4m2 − 1

(5-42)

where m = D2/D1. The Nusselt and Reynolds numbers are based on the equivalent diameter, D2 − D1. Limiting Nusselt numbers for laminar flow in annuli have been calculated by Dwyer [Nucl. Sci. Eng., 17, 336 (1963)]. In addition, theoretical analyses of laminar-flow heat transfer in concentric and eccentric annuli have been published by Reynolds, Lundberg, and McCuen [Int. J. Heat Mass Transfer, 6, 483, 495 (1963)]. Lee [Int. J. Heat Mass Transfer, 11, 509 (1968)] presented an analysis of turbulent heat transfer in entrance regions of concentric annuli. Fully developed local Nusselt numbers were generally attained within a region of 30 equivalent diameters for 0.1 < NPr < 30, 104 < NRe < 2 × 105, 1.01 < D2 /D1 < 5.0. Parallel Plates and Rectangular Ducts The limiting Nusselt number for parallel plates and flat rectangular ducts is given in Table 5-4. Norris and Streid [Trans. Am. Soc. Mech. Eng., 62, 525 (1940)] report for constant wall temperature 1/3 (NNu)lm = 1.85NGz

(5-43)

for NGz > 70. Both Nusselt number and Graetz numbers are based on equivalent diameter. For large temperature differences it is advisable to apply the correction factor (µb/µw)0.14 to the right side of Eq. (5-43). For rectangular ducts Kays and Clark (Stanford Univ., Dept. Mech. Eng. Tech. Rep. 14, Aug. 6, 1953) published relationships for heating and cooling of air in rectangular ducts of various aspect ratios. For most noncircular ducts Eqs. (5-39) and (5-40) may be used if the equivalent diameter (= 4 × free area/wetted perimeter) is used as the characteristic length. See also Kays and London, Compact Heat Exchangers, 3d ed., McGraw-Hill, New York, 1984. Immersed Bodies When flow occurs over immersed bodies such that the boundary layer is completely laminar over the whole body, laminar flow is said to exist even though the flow in the mainstream is turbulent. The following relationships are applicable to single bodies immersed in an infinite fluid and are not valid for assemblages of bodies. In general, the average heat-transfer coefficient on immersed bodies is predicted by NNu = Cr(NRe)m(NPr)1/3

(5-44)

Values of Cr and m for various configurations are listed in Table 5-5. The characteristic length is used in both the Nusselt and the Reynolds numbers, and the properties are evaluated at the film temperature = (tw + t∞)/2. The velocity in the Reynolds number is the undisturbed free-stream velocity. Heat transfer from immersed bodies is discussed in detail by Eckert and Drake, Jakob, and Knudsen and Katz (see “Introduction: General References”), where equations for local coefficients and the effects of unheated starting length are presented. Equation (5-44) may also be expressed as m−1 NSt NPr2/3 = Cr NRe = f/2

(5-45)

where f is the skin-friction drag coefficient (not the form drag coefficient). Falling Films When a liquid is distributed uniformly around the periphery at the top of a vertical tube (either inside or outside) and allowed to fall down the tube wall by the influence of gravity, the fluid

5-16

HEAT AND MASS TRANSFER

TABLE 5-5

Laminar-Flow Heat Transfer over Immersed Bodies [Eq. (5-44)] Configuration

Characteristic length

Flat plate parallel to flow Circular cylinder axes perpendicular to flow

Plate length Cylinder diameter

Non-circular cylinder, axis Perpendicular to flow, characteristic Length perpendicular to flow

Square, short diameter Square, long diameter Hexagon, short diameter Hexagon, long diameter

Sphere*

Diameter

NRe

NPr

Cr

m

103 to 3 × 105 1–4 4 – 40 40 – 4000 4 × 103 – 4 × 104 4 × 104 – 2.5 × 105 5 × 103 – 105 5 × 103 – 105 5 × 103 – 105 5 × 103 – 2 × 104 2 × 104 – 105 1 – 7 × 104

>0.6

0.648 0.989 0.911 0.683 0.193 0.0266 0.104 0.250 0.155 0.162 0.0391 0.6

0.50 0.330 0.385 0.466 0.618 0.805 0.675 0.588 0.638 0.638 0.782 0.50

>0.6

>0.6 0.6 – 400

*Replace NNu by NNu − 2.0 in Eq. (5-44).

does not fill the tube but rather flows as a thin layer. Similarly, when a liquid is applied uniformly to the outside and top of a horizontal tube, it flows in layer form around the periphery and falls off the bottom. In both these cases the mechanism is called gravity flow of liquid layers or falling films. For the turbulent flow of water in layer form down the walls of vertical tubes the dimensional equation of McAdams, Drew, and Bays [Trans. Am. Soc. Mech. Eng., 62, 627 (1940)] is recommended: hlm = bΓ1/3

(5-46)

where b = 9150 (SI) or 120 (U.S. customary) and is based on values of Γ = WF/πD ranging from 0.25 to 6.2 kg/(m⭈s) [600 to 15,000 lb/(h⭈ft)] of wetted perimeter. This type of water flow is used in vertical vaporin-shell ammonia condensers, acid coolers, cycle water coolers, and other process-fluid coolers. The following dimensional equations may be used for any liquid flowing in layer form down vertical surfaces: For

k3ρ2g 4Γ ᎏ > 2100 hlm = 0.01 ᎏ µ µ2

For

4Γ k2ρ4/3cg2/3 ᎏ < 2100 ham = 0.50 ᎏᎏ µ Lµ1/3





4Γ  ᎏk  ᎏ µ  1/3

cµ

1/3

µ

1/3

(5-47a)

ᎏ  ᎏ µ   µ  1/3

1/4

4Γ

1/9

(5-47b)

w

Equation (5-47b) is based on the work of Bays and McAdams [Ind. Eng. Chem., 29, 1240 (1937)]. The significance of the term L is not clear. When L = 0, the coefficient is definitely not infinite. When L is large and the fluid temperature has not yet closely approached the wall temperature, it does not appear that the coefficient should necessarily decrease. Within the finite limits of 0.12 to 1.8 m (0.4 to 6 ft), this equation should give results of the proper order of magnitude. For falling films applied to the outside of horizontal tubes, the Reynolds number rarely exceeds 2100. Equations may be used for falling films on the outside of the tubes by substituting πD/2 for L. For water flowing over a horizontal tube, data for several sizes of pipe are roughly correlated by the dimensional equation of McAdams, Drew, and Bays [Trans. Am. Soc. Mech. Eng., 62, 627 (1940)]. ham = b (Γ/D0)1/3

4, 91 (1934)] fits both the laminar extreme and the fully turbulent extreme quite well. D 2/3 µb 0.14 2/3 (NNu)am = 0.116(NRe − 125)NPr1/3 1 + ᎏ (5-49) ᎏ L µw between 2100 and 10,000. It is customary to represent the probable magnitude of coefficients in this region by hand-drawn curves (Fig. 5-8). Equation (5-40) is plotted as a series of curves ( j factor versus Reynolds number with L/D as parameters) terminating at Reynolds number = 2100. Continuous curves for various values of L/D are then hand-drawn from these terminal points to coincide tangentially with the curve for forced-convection, fully turbulent flow [Eq. (5-50c)].

    

Turbulent Flow Circular Tubes Numerous relationships have been proposed for predicting turbulent flow in tubes. For high-Prandtl-number fluids, relationships derived from the equations of motion and energy through the momentum-heat-transfer analogy are more complicated and no more accurate than many of the empirical relationships that have been developed. For NRe > 10,000, 0.7 < NPr < 170, for properties based on the bulk temperature and for heating, the Dittus-Boelter equation [Boelter, Cherry, Johnson and Martinelli, Heat Transfer Notes, McGrawHill, New York (1965)] may be used: 0.8 0.4 NNu = 0.0243 NRe NPr (µ b /µw)0.14

(5-50a)

For cooling, the relationship is 0.14 0.3 NNu = 0.0265 N 0.8 Re N Pr (µb /µw)

(5.50b)

The Colburn correlation is 2/3 −0.2 (µw /µb)0.14 = 0.023NRe jH = NSt NPr

(5-50c)

In Eq. (5-50c), the viscosity-ratio factor may be neglected if properties are evaluated at the film temperature (tb + tw)/2.

(5-48)

where b = 3360 (SI) or 65.6 (U.S. customary) and Γ ranges from 0.94 to 4 kg/m⋅s) [100 to 1000 lb/(h⋅ft)]. Falling films are also used for evaporation in which the film is both entirely or partially evaporated (juice concentration). This principle is also used in crystallization (freezing). The advantage of high coefficient in falling-film exchangers is partially offset by the difficulties involved in distribution of the film, maintaining complete wettability of the tube, and pumping costs required to lift the liquid to the top of the exchanger. Transition Region Turbulent-flow equations for predicting heat transfer coefficients are usually valid only at Reynolds numbers greater than 10,000. The transition region lies in the range 2000 < NRe < 10,000. No simple equation exists for accomplishing a smooth mathematical transition from laminar flow to turbulent flow. Of the relationships proposed, Hausen’s equation [Z. Ver. Dtsch. Ing. Beih. Verfahrenstech., No.

Graphical representation of the Colburn j factor for the heating and cooling of fluids inside tubes. The curves for NRe below 2100 are based on Eq. (5-40). L is the length of each pass in feet. The curve for NRe above 10,000 is represented by Eq. (5-50c).

FIG. 5-8

HEAT TRANSFER BY CONVECTION For the transition and turbulent regions, including diameter to length effects, Gnielinski [Int. Chem. Eng., 16, 359 (1976)] recommends a modification of an equation suggested by Petukhov and Popov [High Temp., 1, 69 (1963)]. This equation applies in the ranges 0 < D/L < 1, 0.6 < NPr < 2000, 2300 < NRe < 106. ( f/2)(NRe − 1000) NPr D 1+ ᎏ NNu = ᎏᎏᎏ 2/3 1 + 12.7( f/2)0.5(NPr − 1) L

µb

   ᎏ µ  2/3

0.14

(5-51a)

w

The Fanning friction f is determined by an equation recommended by Filonenko [Teploenergetika, 1, 40 (1954)] f = 0.25 (1.82 log10 NRe − 1.64)−2

(5-51b)

Any other appropriate friction factor equation for smooth tubes may be used. Approximate predictions for rough pipes may be obtained from Eq. (5-50c) if the right-hand term is replaced by f / 2 for the rough pipe. For air, Nunner (Z. Ver. Dtsch. Ing. Forsch., 1956, p. 455) obtains frough (NNu)rough (5-52) ᎏᎏ = ᎏ (NNu)smooth fsmooth Dippery and Sabersky [Int. J. Heat Mass Transfer, 6, 329 (1963)] present a complete discussion of the influence of roughness on heat transfer in tubes. Dimensional Equations for Various Conditions For gases at ordinary pressures and temperatures based on cµ/k = 0.78 and µ = (1.76)(10−5) Pa⋅s [0.0426 lb/(ft⋅h)] h = bcρ0.8(V 0.8/D0.2)

(5-53)

where b = (3.04)(10−3) (SI) or (1.44)(10−2) (U.S. customary). For air at atmospheric pressure h = b(V 0.8/D0.2)

(5-54)

where b = 3.52 (SI) or (4.35)(10−4) (U.S. customary). For water [based on a temperature range of 5 to 104°C (40 to 220°F)] h = 1057 (1.352 + 0.02t) (V 0.8/D0.2)

(5-55a)

in SI units with t = °C, or h = 0.13(1 + 0.011t)(V 0.8/D0.2)

(5-55b)

in U.S. customary units with t = °F. For organic liquids, based on c = 2.092 J/kg⋅K)[0.5 Btu/(lb⋅°F)], k = 0.14 J/(m⋅s⋅K) [0.08 Btu/(h⋅ft⋅°F)], µb = (1)(10−3) Pa⋅s (1.0 cP), and ρ = 810 kg/m3 (50 lb/ft3), h = b(V 0.8/D0.2)

(5-56)

where b = 423 (SI) or (5.22)(10 ) (U.S. customary). Within reasonable limits, coefficients for organic liquids are about one-third of the values obtained for water. Entrance effects are usually not significant industrially if L/D > 60. Below this limit Nusselt recommended the conservative equation for 10 < L/D < 400 and properties evaluated at bulk temperature −2

0.8 NNu = 0.036NRe NPr1/3(L/D)−0.054

(5-57)

It is common to correlate entrance effects by the equation hm /h = 1 + F(D/L)

(5-58)

where h is predicted by Eq. (5-50a) or (5-50b), and hm is the mean coefficient for the pipe in question. Values of F are reported by Boelter, Young, and Iverson [NACA Tech. Note 1451, 1948] and tabulated by Kays and Knudsen and Katz (see “Introduction: General References”). Selected values of F are as follows: Fully developed velocity profile 1.4 Abrupt contraction entrance 6 90° right-angle bend 7 180° round bend 6

5-17

For large temperature differences different equations are necessary and usually are specifically applicable to either gases or liquids. Gambill (Chem. Eng., Aug. 28, 1967, p. 147) provides a detailed review of high-flux heat transfers to gases. He recommends 0.8 0.4 0.021NRe NPr NNu = ᎏᎏ (5-59) (Tw /Tb)0.29 + 0.0019 (L/D) for 10 < L/D < 240, 110 < Tb < 1560 K (200 < Tb < 2800°R), 1.1 < (Tw /Tb) < 8.0, and properties evaluated at Tb. For liquids, Eq. (5-50c) is generally satisfactory. Annuli For diameter ratios D1/D2 > 0.2, Monrad and Pelton’s equation [Trans. Am. Inst. Chem. Eng., 38, 593 (1942)] is recommended for either or both the inner and outer tube: 0.8 1/3 NNu = 0.020NRe NPr (D2/D1)0.53 (5-60a) Equation (5-51a) may also be used for smooth annuli as follows: (NNu)ann D1 (5-60b) ᎏ=φ ᎏ (NNu)tube D2 The hydraulic diameter D2 − D1 is used in NNu, NRe, and D/L is used for the annulus. The function on the right of Eq. (5-60b) is given by Petukhov and Roizen [High Temp., 2, 65 (1964)] as follows: Inner tube heated 0.86 (D1 /D2)−0.16 Outer tube heated 1 − 0.14 (D1 /D2)0.6 If both tubes are heated, the function is the sum of the above two functions divided by 1 + D1/D2 [Stephan, Chem. Ing. Tech., 34, 207 (1962)]. The Colburn form of relationship may be employed for the individual walls of the annulus by using the individual friction factor for each wall [see Knudsen, Am. Inst. Chem. Eng. J., 8, 566 (1962)]: jH1 = (NSt)1NPr2/3 = f1 /2 (5-61a)

 

jH2 = (NSt)2 NPr2/3 = f2 /2 (5-61b) Rothfus, Monrad, Sikchi, and Heideger [Ind. Eng. Chem., 47, 913 (1955)] report that the friction factor f2 for the outer wall bears the same relation to the Reynolds number for the outer portion of the annular stream 2(r22 − λm)Vρ /r2 µ as the friction factor for circular tubes does to the Reynolds number for circular tubes, where r2 is the radius of the outer tube and λm is the position of maximum velocity in the annulus, estimated from r22 − r12 λm = ᎏ (5-62)* ln (r2/r1)2 To calculate the friction factor f1 for the inner tube use the relation f2 r2(λm − r12) f1 = ᎏᎏ (5-63) r1(r22 − λm) There have been several analyses of turbulent heat transfer in annuli: for example, Deissler and Taylor (NACA Tech. Note 3451, 1955), Kays and Leung [Int. J. Heat Mass Transfer, 6, 537 (1963)], Lee [Int. J. Heat Transfer, 11, 509 (1968)], Sparrow, Hallman and Siegel [Appl. Sci. Res., 7A, 37 (1958)], and Johnson and Sparrow [Am. Soc. Mech. Eng. J. Heat Transfer, 88, 502 (1966)]. The reader is referred to these for details of the analyses. For annuli containing externally finned tubes the heat-transfer coefficients are a function of the fin configurations. Knudsen and Katz (Fluid Dynamics and Heat Transfer, McGraw-Hill, New York, 1958) present relationships for transverse finned tubes, spined tubes, and longitudinal finned tubes in annuli. Noncircular Ducts Equations (5-50a) and (5-50b) may be employed for noncircular ducts by using the equivalent diameter De = 4 × free area per wetted perimeter. Kays and London (Compact Heat Exchangers, 3rd ed., McGraw-Hill, New York, 1984) give charts for various noncircular ducts encountered in compact heat exchangers. Vibrations and pulsations generally tend to increase heat-transfer coefficients. Example 2: Calculation of j Factors in an Annulus Calculate the heat-transfer j factors for both walls of an annulus for the following condi-

* Equation (5-62) predicts the point of maximum velocity for laminar flow in annuli and is only an approximate equation for turbulent flow. Brighton and Jones [Am. Soc. Mech. Eng. Basic Eng., 86, 835 (1964)] and Macagno and McDougall [Am. Inst. Chem. Eng. J., 12, 437 (1966)] give more accurate equations for predicting the point of maximum velocity for turbulent flow.

5-18

HEAT AND MASS TRANSFER

tions: D1 = 0.0254 m (1.0 in); D2 = 0.0635 m (2.5 in); water at 15.6°C (60°F); µ/ρ = (1.124)(10−6) m2/s [(1.21)(10−5) ft2/s]; velocity = 1.22 m/s (4 ft/s). 0.06352 − 0.02542 λm = ᎏᎏᎏ2 = (4.621)(10−4) m2 (0.716 in2) 4 ln (0.0635/0.0254) 2(r 22 − λm)Vρ 2[0.03182 − (4.621)(10−4)(1.22)] Re2 = ᎏᎏ = ᎏᎏᎏᎏ = (3.74)(104) r2µ (0.0318)(1.124)(10−6) From Eq. (5-51b), f2 = 0.0055. Hence 2/3 jH2 = (NSt)2 N Pr = 0.00275 From Eq. (5-63), (0.0055)(0.0318)[(4.621)(10−4) − 0.01272] = 0.00754 f1 = ᎏᎏᎏᎏᎏ (0.0127)[0.03182 − (4.621)(10−4)] 2/3 from which jH1 = (NSt)1N Pr = 0.00377. These results indicate that for this system the heat-transfer coefficient on the inner tube is about 40 percent greater than on the outer tube.

Coils For flow inside helical coils, Reynolds number above 10,000, multiply the value of the film coefficient obtained from the applicable equation for straight tubes by the term (1 + 3.5 Di /Dc). For flow inside helical coils, Reynolds number less than 10,000, substitute the term (Dc /Di)1/2 for (L/Di) where the latter appears in the applicable equation for straight tubes (frequently as part of the Graetz number). For flat spiral (pancake) coils, in which the ratio Dc /Di varies for each turn, a different value of coefficient will be obtained for each turn; a weighted average based on length per turn is used. For flow outside helical coils use the equation for flow normal to a bank of tubes, in-line flow. Finned Tubes (Extended Surface) When the film coefficient on the outside of a metal tube is much lower than that on the inside, as when steam condensing in a pipe is being used to heat air, externally finned (or extended) heating surfaces are of value in increasing substantially the rate of heat transfer per unit length of tube. The data on extended heating surfaces, for the case of air flowing outside and at right angles to the axes of a bank of finned pipes, can be represented approximately by the dimensional equation derived from 0.6 VF0.6 p′ hf = b ᎏ (5-64) ᎏ 0.4 D0 p′ − D0 −3 where b = 5.29 (SI) or (5.39)(10 ) (U.S. customary); hf is the film coefficient of heat transfer on the air side; VF is the face velocity of the air; p′ is the center-to-center spacing, m, of the tubes in a row; and D0 is the outside diameter, m, of the bare tube (diameter at the root of the fins). In atmospheric air-cooled finned tube exchangers, the air-film coefficient from Eq. (5-64) is sometimes converted to a value based on outside bare surface as follows: Af + Auf A hfo = hf ᎏ = hf ᎏT (5-65) Aof Ao in which hfo is the air-film coefficient based on external bare surface; hf is the air-film coefficient based on total external surface; AT is total external surface, and Ao is external bare surface of the unfinned tube; Af is the area of the fins; Auf is the external area of the unfinned portion of the tube; and Aof is area of tube before fins are attached. Fin efficiency is defined as the ratio of the mean temperature difference from surface to fluid divided by the temperature difference from fin to fluid at the base or root of the fin. Graphs of fin efficiency for extended surfaces of various types are given by Gardner [Trans. Am. Soc. Mech. Eng., 67, 621 (1945)]. Heat-transfer coefficients for finned tubes of various types are given in a series of papers [Trans. Am. Soc. Mech. Eng., 67, 601 (1945)]. For flow of air normal to fins in the form of short strips or pins, Norris and Spofford [Trans. Am. Soc. Mech. Eng., 64, 489 (1942)] correlate their results for air by the dimensionless equation of Pohlhausen: hm cpµ 2/3 zpGmax −0.5 = 1.0 ᎏ (5-66) ᎏ ᎏ cpGmax k µ for values of zpGmax/µ ranging from 2700 to 10,000.



 







For the general case, the treatment suggested by Kern (Process Heat Transfer, McGraw-Hill, New York, 1950, p. 512) is recommended. Because of the wide variations in fin-tube construction, it is convenient to convert all film coefficients to values based on the inside bare surface of the tube. Thus to convert the film coefficient based on outside area (finned side) to a value based on inside area Kern gives the following relationship: hfi = (ΩAf + Ao)(hf /Ai)

(5-67)

in which hfi is the effective outside film coefficient based on the inside area, hf is the outside film coefficient calculated from the applicable equation for bare tubes, Af is the surface area of the fins, Ao is the surface area on the outside of the tube which is not finned, Ai is the inside area of the tube, and Ω is the fin efficiency defined as Ω = (tanh mbf)/mbf

(5-68)

m = (hf pf / kax)1/2 m−1 (ft−1)

(5-69)

in which and bf = height of fin. The other symbols are defined as follows: pf is the perimeter of the fin, ax is the cross-sectional area of the fin, and k is the thermal conductivity of the material from which the fin is made. Fin efficiencies and fin dimensions are available from manufacturers. Ratios of finned to inside surface are usually available so that the terms A f, Ao, and Ai may be obtained from these ratios rather than from the total surface areas of the heat exchangers. Banks of Tubes For heating and cooling of fluids flowing normal to a bank of circular tubes at least 10 rows deep the following equations are applicable: Colburn type:

 

h cµ ᎏ ᎏ cGmax k

2/3

a = ᎏᎏ =j (DoGmax/µ)0.4

(5-70)

cµ  ᎏ k 

(5-71)

Nusselt type:



hD DoGmax ᎏ=a ᎏ k µ

0.6

1/3

The dimensionless constant a in these equations varies depending upon conditions. Conditions, Reynolds number > 3000

Value of a

Flow normal to apex of diamond, staggered arrangement No leakage Normal leakage in baffled exchanger Flow normal to flat side of diamond, not staggered (in-line) arrangement No leakage Normal leakage in baffled exchanger

0.330 0.198 0.260 0.156

For Reynolds number less than 3000, Eq. (5-70) would give conservative results, but greater accuracy (if desired) may be obtained by using the following equation.

 

h cµ ᎏ ᎏ cGmax k

2/3

a = ᎏᎏ =j (DoGmax /µ)m

(5-72)

in which the constant a and exponent m are as follows: Reynolds number

m

Tube pitch

100–300

0.492

Staggered In-line

1–100

0.590

Staggered In-line

Leakage

a

None Normal None Normal None Normal None Normal

0.695 0.416 0.548 0.329 1.086 0.650 0.855 0.513

HEAT TRANSFER BY CONVECTION The following dimensional equations (5-73 to 5-77) are based on flow normal to a bank of staggered tubes without leakage. Multiply the values obtained for h by 0.6 for normal leakage and, in addition, by 0.79 for in-line (not staggered) tube arrangement. c1/3k2/3ρ0.6 V 0.6 max h = b ᎏᎏ (5-73) µ0.267 D00.4 where b = 0.33 (SI) or 0.261 (U.S. customary). For gases at ordinary pressures and temperatures, based on cµ/k = 0.78; µ = (1.76)(10−5) Pa⭈s [0.0426 lb/(ft⭈h)], 0.6 Gmax h = bc ᎏ (5-74) 0.4 D0 where b = (4.82)(10−3) (SI) or 0.109 (U.S. customary). For air at atmospheric pressure 0.6 Vmax h=bᎏ (5-75) D00.4 where b = 5.33 (SI) or (5.44)(10−3) (U.S. customary). For water based on a temperature range 7 to 104°C (40 to 220°F) 0.6 Vmax h = 986(1.21 + 0.0121t) ᎏ (5-76a) D00.4 in SI units and t in °C. 0.6 Vmax h = 1.01(1 + 0.0067t) ᎏ (5-76b) D00.4 in U.S. customary units and t in °F. For organic liquids, based on c = 2.22 J/(kg⭈K) [0.53 Btu/(lb⭈°F)], k = 0.14 J/(m⭈s⭈K) [0.08 Btu/ (h⭈ft⭈°F)], µb = (1)(10−3) Pa⭈s (1.0 cP), ρ = 810 kg/m3 (50 lb/ft3), V 0.6 max h=bᎏ (5-77) D00.4 where b = 400 (SI) or 0.408 (U.S. customary).

5-19

LIQUID METALS Liquid metals constitute a class of heat-transfer media having Prandtl numbers generally below 0.01. Heat-transfer coefficients for liquid metals cannot be predicted by the usual design equations applicable to gases, water, and more viscous fluids with Prandtl numbers greater than 0.6. Relationships for predicting heat-transfer coefficients for liquid metals have been derived from solution of Eqs. (5-38a) and (5-38b). By the momentum-transfer-heat-transfer analogy, the eddy conductivity of heat is kNPr(EM /µ) ≈ k for small NPr. Thus in the solution of Eqs. (5-38a) and (5-38b) the knowledge of the thickness of various layers of flow is not critical. In fact, assumption of slug flow and constant conductivity (=k) across the duct gives reasonable values of heat-transfer coefficients for liquid metals. For constant heat flux: NNu = 5 + 0.025(NRe NPr)0.8

(5-81)

For constant wall temperature: NNu = 7 + 0.025(NRe NPr)0.8

(5-82)

For 0.003 < NPr < 0.05 and constant heat flux, Sleicher and Rouse [Int. J. Heat Mass Transfer, 18, 677 (1975)] obtained the correlation 0.85 NNu = 6.3 + 0.0167 NRe NPr0.93

(5-83)

For parallel plates and annuli with D2 /D1 < 1.4 and uniform heat flux, Seban [Trans. Am. Soc. Mech. Eng., 72, 789 (1950)] obtained the equation NNu = 5.8 + 0.020(NRe NPr)0.8

(5-84)

JACKETS AND COILS OF AGITATED VESSELS

For annuli only, application of a factor of 0.70(D2 /D1)0.53 is recommended for Eqs. (5-81) and (5-82). For more accurate semiempirical relationships for tubes, annuli, and rod bundles, refer to Dwyer [Am. Inst. Chem. Eng. J., 9, 261 (1963)]. Hsu [Int. J. Heat Mass Transfer, 7, 431 (1964)] and Kalish and Dwyer [Int. J. Heat Mass Transfer, 10, 1533 (1967)] discuss heat transfer to liquid metals flowing across banks of tubes. Hsu recommends the equations

See Sec. 18.

NNu = 0.81NRe NPr (φ/D)1/2

(for uniform heat flux)

NONNEWTONIAN FLUIDS

NNu = 0.096NRe NPr (φ/D)

(for cosine surface temperature)

1/2

A wide variety of nonnewtonian fluids are encountered industrially. They may exhibit Bingham-plastic, pseudoplastic, or dilatant behavior and may or may not be thixotropic. For design of equipment to handle or process nonnewtonian fluids, the properties must usually be measured experimentally, since no generalized relationships exist to predict the properties or behavior of the fluids. Details of handling nonnewtonian fluids are described completely by Skelland (NonNewtonian Flow and Heat Transfer, Wiley, New York, 1967). The generalized shear-stress rate-of-strain relationship for nonnewtonian fluids is given as d ln (D ∆P/4L) n′ = ᎏᎏ (5-78) d ln (8V/D) as determined from a plot of shear stress versus velocity gradient. For circular tubes, NGz > 100, n′ > 0.1, and laminar flow 1/3 (NNu)lm = 1.75 δ1/3 s NGz

(5-79)

where δs = (3n′ + 1)/4n′. When natural-convection effects are considered, Metzer and Gluck [Chem. Eng. Sci., 12, 185 (1960)] obtained the following for horizontal tubes: NPr NGr D 0.4 1/3 γb 0.14 (NNu)lm = 1.75 δ 1/3 NGz + 12.6 ᎏ (5-80) ᎏ s L γw where properties are evaluated at the wall temperature, i.e., γ = gc K′8n′ − 1 and τw = K′(8V/D)n′. Metzner and Friend [Ind. Eng. Chem., 51, 879 (1959)] present relationships for turbulent heat transfer with nonnewtonian fluids. Relationships for heat transfer by natural convection and through laminar boundary layers are available in Skelland’s book (op. cit.).





  

(5-85) (5-86)

where the heat-transfer coefficient is based on the average circumferential temperature around the tubes, the Reynolds number is based on the superficial velocity through the tube bank, D is the tube outside diameter, and φ is a velocity potential function having the following values:

D/p′

φ/D square pitch

φ/D equilateral triangular pitch

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

2.00 2.02 2.07 2.16 2.30 2.52 2.84 3.34 4.23

2.00 2.02 2.06 2.15 2.27 2.45 2.71 3.11 3.80

Equations (5-85) and (5-86) are useful in calculating tube-surface temperatures. Further information on liquid-metal heat transfer in tube banks is given by Hsu for spheres and elliptical rod bundles [Int. J. Heat Mass Transfer, 8, 303 (1965)] and by Kalish and Dwyer for oblique flow across tube banks [Int. J. Heat Mass Transfer, 10, 1533 (1967)]. For additional details of heat transfer with liquid metals for various systems see Dwyer (1968 ed., Na and Nak supplement to Liquid Metals Handbook) and Stein (“Liquid Metal Heat Transfer,” in Advances in Heat Transfer, vol. 3, Academic, New York, 1966).

5-20

HEAT AND MASS TRANSFER

HEAT TRANSFER WITH CHANGE OF PHASE In any operation in which a material undergoes a change of phase, provision must be made for the addition or removal of heat to provide for the latent heat of the change of phase plus any other sensible heating or cooling that occurs in the process. Heat may be transferred by any one or a combination of the three modes—conduction, convection, and radiation. The process involving change of phase involves mass transfer simultaneous with heat transfer. CONDENSATION Condensation Mechanisms Condensation occurs when a saturated vapor comes in contact with a surface whose temperature is below the saturation temperature. Normally a film of condensate is formed on the surface, and the thickness of this film, per unit of breadth, increases with increase in extent of the surface. This is called film-type condensation. Another type of condensation, called dropwise, occurs when the wall is not uniformly wetted by the condensate, with the result that the condensate appears in many small droplets at various points on the surface. There is a growth of individual droplets, a coalescence of adjacent droplets, and finally a formation of a rivulet. Adhesional force is overcome by gravitational force, and the rivulet flows quickly to the bottom of the surface, capturing and absorbing all droplets in its path and leaving dry surface in its wake. Film-type condensation is more common and more dependable. Dropwise condensation normally needs to be promoted by introducing an impurity into the vapor stream. Substantially higher (6 to 18 times) coefficients are obtained for dropwise condensation of steam, but design methods are not available. Therefore, the development of equations for condensation will be for the film type only. The physical properties of the liquid, rather than those of the vapor, are used for determining the film coefficient for condensation. Nusselt [Z. Ver. Dtsch. Ing., 60, 541, 569 (1916)] derived theoretical relationships for predicting the film coefficient of heat transfer for condensation of a pure saturated vapor. A number of simplifying assumptions were used in the derivation. The Reynolds number of the condensate film (falling film) is 4Γ/µ, where Γ is the weight rate of flow (loading rate) of condensate per unit perimeter kg/(s⭈m) [lb/(h⭈ft)]. The thickness of the condensate film for Reynolds number less than 2100 is (3µΓ/ρ2g)1/3. Condensation Coefficients Vertical Tubes For the following cases Reynolds number < 2100 and is calculated by using Γ = WF /πD. The Nusselt equation for the heat-transfer coefficient for condensate films may be written in the following ways (using liquid physical properties and where L is the cooled length and ∆t is tsv − ts): Colburn type: h cµ 5.35 ᎏ ᎏ=ᎏ cG k 4Γ/µ Γ WF2 ρ2g where G = ᎏᎏ = ᎏ (3µΓ/ρ2g)1/3 29.6D2µ





(5-87)

1/3

kg/(s⭈m2) [lb/(h⭈ft2)]





1/4

L3ρ2g = 0.925 ᎏ µΓ





1/3

(5-88)

Dimensional: h = b(k3ρ2D/µbWF)1/3

(5-89)

where b = 127 (SI) or 756 (U.S. customary). For steam at atmospheric pressure, k = 0.682 J/(m⭈s⭈K) [0.394 Btu/(h⭈ft⭈°F)], ρ = 960 kg/m3 (60 lb/ft3), µb = (0.28)(10−3) Pa⭈s (0.28 cP), h = b(D/WF)1/3

h = b(D/WF)1/3

(5-91)

where b = 457 (SI) or 1080 (U.S. customary). Horizontal Tubes For the following cases Reynolds number < 2100 and is calculated by using Γ = WF /2L. Colburn type: h cµ 4.4 (5-92) ᎏᎏ=ᎏ cG k 4Γ/µ Γ WF2 ρ2g 1/3 kg/(s⭈m2) [lb/(h⭈ft2)] G = ᎏᎏ = ᎏ (3µΓ/ρ2g)1/3 12L2µ Nusselt type: D3ρ2g 1/3 hD D3ρ2gλ 1/4 = 0.76 ᎏ (5-93)* ᎏ = 0.73 ᎏ k kµ ∆t µΓ Dimensional:













h = b(k3ρ2L/µbWF)1/3

(5-90)

(5-94)

where b = 205.4 (SI) or 534 (U.S. customary). For steam at atmospheric pressure h = b(L/WF)1/3

(5-95)

where b = 2080 (SI) or 4920 (U.S. customary). For organic vapors at normal boiling point h = b(L/WF)1/3

(5-96)

where b = 324 (SI) or 766 (U.S. customary). Figure 5-9 is a nomograph for determining coefficients of heat transfer for condensation of pure vapors. Banks of Horizontal Tubes (NRe < 2100) In the idealized case of N tubes in a vertical row where the total condensate flows smoothly from one tube to the one beneath it, without splashing, and still in laminar flow on the tube, the mean condensing coefficient hN for the entire row of N tubes is related to the condensing coefficient for the top tube h1 by hN = h1N−1/4

(5-97)

Dukler Theory The preceding expressions for condensation are based on the classical Nusselt theory. It is generally known and conceded that the film coefficients for steam and organic vapors calculated by the Nusselt theory are conservatively low. Dukler [Chem. Eng. Prog., 55, 62 (1959)] developed equations for velocity and temperature distribution in thin films on vertical walls based on expressions of Deissler (NACA Tech. Notes 2129, 1950; 2138, 1952; 3145, 1959) for the eddy viscosity and thermal conductivity near the solid boundary. According to the Dukler theory, three fixed factors must be known to establish the value of the average film coefficient: the terminal Reynolds number, the Prandtl number of the condensed phase, and a dimensionless group Nd defined as follows: 2/3 2 0.553 0.78 0.16 Nd = (0.250µ1.173 ρG ) L µG /(g D ρL

Nusselt type: hL L3ρ2gλ ᎏ = 0.943 ᎏ k kµ ∆t

where b = 2954 (SI) or 6978 (U.S. customary). For organic vapors at normal boiling point, k = 0.138 J/(m⭈s⭈K) [0.08 Btu/(h⭈ft⭈°F)], ρ = 720 kg/m3 (45 lb/ft3), µb = (0.35)(10−3) Pa⭈s (0.35 cP),

(5-98)

Graphical relationships of these variables are available in Document 6058, ADI Auxiliary Publications Project, Library of Congress, Washington. If rigorous values for condensing-film coefficients are desired, especially if the value of Nd in Eq. (5-98) exceeds (1)(10−5), it is suggested that these graphs be used. For the case in which interfacial shear is zero, Fig. 5-10 may be used. It is interesting to note that, according to the Dukler development, there is no definite transition Reynolds number; deviation from Nusselt theory is less at low Reynolds numbers; and when the Prandtl number of a fluid is less * If the vapor density is significant, replace ρ2 with ρl(ρl − ρv).

HEAT TRANSFER WITH CHANGE OF PHASE

5-21

Chart for determining film coefficient hm for film-type condensation of pure vapor, based on Eqs. 5-88 and 5-93. For 4 ρ 2k3/µ  is in U.S. customary units; to convert feet to vertical tubes multiply hm by 1.2. If 4Γ/µf exceeds 2100, use Fig. 5-10. λ meters, multiply by 0.3048; to convert inches to centimeters, multiply by 2.54; and to convert British thermal units per hour– square foot–degrees Fahrenheit to watts per square meter–kelvins, multiply by 5.6780.

FIG. 5-9

than 0.4 (at Reynolds number above 1000), the predicted values for film coefficient are lower than those predicted by the Nusselt theory. The Dukler theory is applicable for condensate films on horizontal tubes and also for falling films, in general, i.e., those not associated with condensation or vaporization processes.

Vapor Shear Controlling For vertical in-tube condensation with vapor and liquid flowing cocurrently downward, if gravity controls, Figs. 5-9 and 5-10 may be used. If vapor shear controls, the Carpenter-Colburn correlation (General Discussion on Heat Transfer, London, 1951, ASME, New York, p. 20) is applicable:

5-22

HEAT AND MASS TRANSFER

Dukler plot showing average condensing-film coefficient as a function of physical properties of the condensate film and the terminal Reynolds number. (Dotted line indicates Nusselt theory for Reynolds number < 2100.) [Reproduced by permission from Chem. Eng. Prog., 55, 64 (1959).] FIG. 5-10

1/2 hµl / kl ρl1/2 = 0.065(NPr)1/2 l Fvc

(5-99a)

Fvc = fG2vm /2ρv 2 1/2 Gvi2 + GviGvo + Gvo Gvm = ᎏᎏᎏ 3 and f is the Fanning friction factor evaluated at

where



(5-99b)



(5-99c)

(NRe)vm = DiGvm /µv

(5-99d)

and the subscripts vi and vo refer to the vapor inlet and outlet, respectively. An alternative formulation, directly in terms of the friction factor, is h = 0.065 (cρkf/2µρv)1/2Gvm

(5-99e)

expressed in consistent units. Another correlation for vapor-shear-controlled condensation is the Boyko-Kruzhilin correlation [Int. J. Heat Mass Transfer, 10, 361 (1967)], which gives the mean condensing coefficient for a stream between inlet quality xi and outlet quality xo: hDi DiGT 0.8 (ρ /ρ (ρ /ρ  m)i +  m)o (5-100a) ᎏ = 0.024 ᎏ (NPr)l0.43 ᎏᎏᎏ kl µl 2 where GT = total mass velocity in consistent units ρ ρl − ρv (5-100b) ᎏ = 1 + ᎏ xi ρm i ρv





 

ρ ρ −ρ =1+ᎏx ᎏ ρ  ρ l

and

v

(5-100c)

o

m

o

v

For horizontal in-tube condensation at low flow rates Kern’s modification (Process Heat Transfer, McGraw-Hill, New York, 1950) of the Nusselt equation is valid: k 3l ρl (ρl − ρv)gλ 1/4 Lk 3l ρl(ρl − ρv)g 1/3 hm = 0.761 ᎏᎏ = 0.815 ᎏᎏ (5-101) WF µl π µl Di ∆t where WF is the total vapor condensed in one tube and ∆ t is tsv − ts . A more rigorous correlation has been proposed by Chaddock [Refrig. Eng., 65(4), 36 (1957)]. Use consistent units. At high condensing loads, with vapor shear dominating, tube orientation has no effect, and Eq. (5-100a) may also be used for horizontal tubes. Condensation of pure vapors under laminar conditions in the presence of noncondensable gases, interfacial resistance, superheating, variable properties, and diffusion has been analyzed by Minkowycz and Sparrow [Int. J. Heat Mass Transfer, 9, 1125 (1966)].









BOILING (VAPORIZATION) OF LIQUIDS Boiling Mechanisms Vaporization of liquids may result from various mechanisms of heat transfer, singly or combinations thereof.

For example, vaporization may occur as a result of heat absorbed, by radiation and convection, at the surface of a pool of liquid; or as a result of heat absorbed by natural convection from a hot wall beneath the disengaging surface, in which case the vaporization takes place when the superheated liquid reaches the pool surface. Vaporization also occurs from falling films (the reverse of condensation) or from the flashing of liquids superheated by forced convection under pressure. Pool boiling refers to the type of boiling experienced when the heating surface is surrounded by a relatively large body of fluid which is not flowing at any appreciable velocity and is agitated only by the motion of the bubbles and by natural-convection currents. Two types of pool boiling are possible: subcooled pool boiling, in which the bulk fluid temperature is below the saturation temperature, resulting in collapse of the bubbles before they reach the surface, and saturated pool boiling, with bulk temperature equal to saturation temperature, resulting in net vapor generation. The general shape of the curve relating the heat-transfer coefficient to ∆tb, the temperature driving force (difference between the wall temperature and the bulk fluid temperature) is one of the few parametric relations that are reasonably well understood. The familiar boiling curve was originally demonstrated experimentally by Nukiyama [ J. Soc. Mech. Eng. ( Japan), 37, 367 (1934)]. This curve points out one of the great dilemmas for boiling-equipment designers. They are faced with at least six heat-transfer regimes in pool boiling: natural convection (+), incipient nucleate boiling (+), nucleate boiling (+), transition to film boiling (−), stable film boiling (+), and film boiling with increasing radiation (+). The signs indicate the sign of the derivative d(q/A)/d ∆ tb. In the transition to film boiling, heat-transfer rate decreases with driving force. The regimes of greatest commercial interest are the nucleate-boiling and stable-film-boiling regimes. Heat transfer by nucleate boiling is an important mechanism in the vaporization of liquids. It occurs in the vaporization of liquids in kettle-type and natural-circulation reboilers commonly used in the process industries. High rates of heat transfer per unit of area (heat flux) are obtained as a result of bubble formation at the liquid-solid interface rather than from mechanical devices external to the heat exchanger. There are available several expressions from which reasonable values of the film coefficients may be obtained. The boiling curve, particularly in the nucleate-boiling region, is significantly affected by the temperature driving force, the total system pressure, the nature of the boiling surface, the geometry of the system, and the properties of the boiling material. In the nucleate-boiling regime, heat flux is approximately proportional to the cube of the temperature driving force. Designers in addition must know the minimum ∆t (the point at which nucleate boiling begins), the critical ∆t (the ∆t above which transition boiling begins), and the maximum heat flux (the heat flux corresponding to the critical ∆t). For designers who do not have experimental data available, the following equations may be used. Boiling Coefficients For the nucleate-boiling coefficient the Mostinski equation [Teplenergetika, 4, 66 (1963)] may be used: q 0.7 P 0.17 P 1.2 P 10 h = bPc0.69 ᎏ 1.8 ᎏ + 4 ᎏ + 10 ᎏ (5-102) A Pc Pc Pc where b = (3.75)(10−5)(SI) or (2.13)(10−4) (U.S. customary), Pc is the critical pressure and P the system pressure, q/A is the heat flux, and h is the nucleate-boiling coefficient. The McNelly equation [J. Imp. Coll. Chem. Eng. Soc., 7(18), (1953)] may also be used: 0.33 qc 0.69 Pkl 0.31 ρl h = 0.225 ᎏl (5-103) ᎏ ᎏ−1 Aλ σ ρv where cl is the liquid heat capacity, λ is the latent heat, P is the system pressure, kl is the thermal conductivity of the liquid, and σ is the surface tension. An equation of the Nusselt type has been suggested by Rohsenow [Trans. Am. Soc. Mech. Eng., 74, 969 (1952)].

    

 

 

    



hD/k = Cr(DG/µ)2/3(cµ/k)−0.7 in which the variables assume the following form: 1/2 1/2 hβ′ gcσ β′ gcσ W 2/3 cµ = Cr ᎏ ᎏᎏ ᎏ ᎏ ᎏ ᎏ k g(ρL − ρv) µ g(ρL − ρv) A k





 



 

(5-104a) −0.7

(5-104b)

The coefficient Cr is not truly constant but varies from 0.006 to 0.015.* It is possible that the nature of the surface is partly responsible for the variation in the constant. The only factor in Eq. (5-104b) not readily available is the value of the contact angle β′. Another Nusselt-type equation has been proposed by Forster and Zuber:† 0.62 1/3 NNu = 0.0015NRe NPr (5-105) which takes the following form:

α  W 2σ cρLπ ᎏᎏ ᎏ kρv A ∆p

ρL

  ᎏ ∆pg  α  ρ cρ ∆T π cµ = 0.0015  ᎏ  ᎏᎏ  ᎏ  µ λρ k 1/2

1/4

c

L

2 0.62

L

1/ 2

(5-106)

v

where α = k/ρc (all liquid properties) ∆p = pressure of the vapor in a bubble minus saturation pressure of a flat liquid surface Equations (5-104b) and (5-106) have been arranged in dimensional form by Westwater. The numerical constant may be adjusted to suit any particular set of data if one desires to use a certain criterion. However, surface conditions vary so greatly that deviations may be as large as ⫾25 percent from results obtained. The maximum heat flux may be predicted by the KutateladseZuber [Trans. Am. Soc. Mech. Eng., 80, 711 (1958)] relationship, using consistent units: (ρl − ρv)σg 1/4 q = 0.18gc1/4ρv λ ᎏᎏ (5-107) ᎏ A max ρ2v Alternatively, Mostinski presented an equation which approximately represents the Cichelli-Bonilla [Trans. Am. Inst. Chem. Eng., 41, 755 (1945)] correlation:

 





HEAT TRANSFER BY RADIATION

5-23

  1 − ᎏPP 

(5-108)

(q/A)max P ᎏ=b ᎏ Pc Pc

0.35

0.9

c

where b = 0.368(SI) or 5.58 (U.S. customary); Pc is the critical pressure, Pa absolute; P is the system pressure; and (q/A)max is the maximum heat flux. The lower limit of applicability of the nucleate-boiling equations is from 0.1 to 0.2 of the maximum limit and depends upon the magnitude of natural-convection heat transfer for the liquid. The best method of determining the lower limit is to plot two curves: one of h versus ∆ t for natural convection, the other of h versus ∆ t for nucleate boiling. The intersection of these two curves may be considered the lower limit of applicability of the equations. These equations apply to single tubes or to flat surfaces in a large pool. In tube bundles the equations are only approximate, and designers must rely upon experiment. Palen and Small [Hydrocarbon Process., 43(11), 199 (1964)] have shown the effect of tube-bundle size on maximum heat flux. p q gσ(ρl − ρv) 1/4 = b ᎏ ρv λ ᎏᎏ (5-109) ᎏ Do NT A max ρ2v

 





where b = 0.43 (SI) or 61.6 (U.S. customary), p is the tube pitch, Do is the tube outside diameter, and NT is the number of tubes (twice the number of complete tubes for U-tube bundles). For film boiling, Bromley’s [Chem. Eng. Prog., 46, 221 (1950)] correlation may be used: kv3(ρl − ρv)ρv g h = b ᎏᎏ µv Do ∆ tb





1/4

(5-110)

where b = 4.306 (SI) or 0.620 (U.S. customary). Katz, Myers, and Balekjian [Pet. Refiner, 34(2), 113 (1955)] report boiling heat-transfer coefficients on finned tubes.

HEAT TRANSFER BY RADIATION GENERAL REFERENCES: Much of the pertinent literature on radiative heat transfer has been surveyed in the following texts: Goody, Atmospheric Radiation, Clarendon Press, Oxford, 1964. Sparrow and Cess, Radiation Heat Transfer, Brooks/Cole Publishing Company, Belmont, Calif., 1966. Hottel and Sarofim, Radiative Transfer, McGraw-Hill, New York, 1967. Love, Radiative Heat Transfer, Merrill, Columbus, 1968. Siegel and Howell, Thermal Radiation Heat Transfer, NASA SP-164, GPO, Washington, 1968. Edwards, Radiation Heat Transfer Notes, Hemisphere Publishing Corp., 1981; Howell and Siegel, Thermal Radiative Heat Transfer, McGraw-Hill, 3d ed., 1992; Brewster, Thermal Radiative Transfer and Properties, Wiley, 1992; Modest, Radiative Heat Transfer, McGraw-Hill, 1993. Additional sources are the Journal of Applied Optics and the Journal of the Optical Society of America, particularly for surface properties; the Journal of Quantitative Spectroscopy and Radiative Transfer for gas properties; the Journal of Heat Transfer and the International Journal of Heat and Mass Transfer for broad coverage; and the Journal of the Institute of Energy for applications to industrial furnaces.

Thermal radiation—electromagnetic energy in transport—is emitted within matter excited by temperature; it is absorbed in other matter at distances from the source which depend on the mean free path of the photons emitted. The ratio of the mean free path involved in an energy-transport process to a characteristic dimension of the system of interest determines the mathematical structure of the formulation. In molecular conduction this ratio is minute (unless the system or the density of matter is minute, which is the case of free molecular flow), and a differential equation of energy diffusion is involved. In gas radiation the ratio is generally large enough to give rise to an integral equation, with an unknown function inside the integral. Solids gener-

ally have small enough photon mean free paths (high enough absorption coefficients) for the radiation escaping through the surface to have originated close to the surface; radiative loss is then identifiable with its surface temperature, but an integral equation is still involved if all the surfaces of an enclosure filled with a diathermanous medium like air are not specified as to temperature or are not black. Radiation differs from conduction and convection not only in mathematical structure but in its much higher sensitivity to temperature. It is of dominating importance in furnaces because of their temperature, and in cryogenic insulation because of the vacuum existing between particles. The temperature at which it accounts for roughly half of the total heat loss from a surface in air depends on such factors as surface emissivity and the convection coefficient. For pipes in free convection, this is room temperature; for fine wires of low emissivity it is above red heat. Gases at combustion-chamber temperatures lose more than 90 percent of their energy by radiation from the carbon dioxide, water vapor, and particulate matter. NOMENCLATURE FOR RADIATIVE TRANSFER Terms that are defined at specific places in the text are excluded. a = effective energy fraction of blackbody spectrum in which a nongray gas absorbs. A = area. c = number concentration of particles in a cloud. c1, c2 = first and second Planck-law constants.

* Reported by Westwater in Drew and Hoopes, Advances in Chemical Engineering, vol. I, Academic, New York, 1956, p. 15. † Forster, J. Appl. Phys., 25, 1067 (1954); Forster and Zuber, J. Appl. Phys., 25, 474 (1954); Forster and Zuber, Conference on Nuclear Engineering, University of California, Los Angeles, 1955; excellent treatise on boiling of liquids by Westwater in Drew and Hoopes, Advances in Chemical Engineering, vol. I, Academic, New York, 1956.

5-24

HEAT AND MASS TRANSFER

C = axis-to-axis distance of separation of tubes. Cb = mean specific heat of combustion products from base temperature To to leaving-gas temperature TE. C = cold-surface fraction of a furnace enclosure. CW = correction factor for pressure broadening of radiation from water vapor. d = particle diameter. D = tube diameter; characteristic dimension; dimensionless firing density. D′ = reduced firing density. E = hemispherical emissive power of a blackbody. f = fraction of blackbody radiation lying below λ. fv = volume fraction of space occupied by particles. F = direct view factor; Fij, fraction of isotropic radiation from Ai intercepted directly by Aj. F = total view factor from black source to black sink, with  allowance for refractory surfaces (subscripts identify source and sink). Ᏺij = total view factor, radiation from i to j both directly and indirectly, expressed as fraction of blackbody radiation from Ai. gs = direct-exchange area between gas volume and surface. G S  = total-exchange area between gas and surface; subscript R indicates allowance for radiatively adiabatic surfaces. h = coefficient of convective heat transfer. H˙ = enthalpy of fuel plus air entering combustion chamber. I = intensity, radiant-energy-flux density per unit solid angle of divergence. ij = shorthand for sisj. k = absorption or emission coefficient; or thermal conductivity. K = constant defined in connection with Eq. (5-147). L = mean beam length; L0, at vanishingly small optical thickness; Lm, average value. L = wall-loss group. Lc = dimensionless convective loss. Lo = dimensionless wall-opening loss. Lr = dimensionless refractory-wall loss. ˙ = mass flow rate. m n = refractive index. p = partial pressure, atm.; subscript c, CO2; subscript w, water vapor. P = total pressure atm. q = heat-flux density, energy per time-area. ˙ = heat flux, energy per time. Q r = separating distance; or electrical resistivity; or refractory (radiatively adiabatic) surface. ss AF, direct-exchange area (subscripts identify surface zones). SS   AᏲ, total-exchange area. T = absolute temperature. Subscript 1 (or G), radiating surface (or gas) temperature; subscript E, exit-gas; subscript o, base temperature; subscript F, pseudoadiabatic flame temperature based on C p averaged from To to TE. U = overall coefficient of heat transfer, gas convection to refractory wall to ambient air. W = total leaving-flux density (also radiosity). α = absorptivity or absorptance; α12, absorptance of surface 1 for radiation from surface 2. ∆ = difference between radiating temperature and leaving-gas temperature divided by the pseudoadiabatic flame temperature TF. ε = emissivity or emittance. η = thermal efficiency. Subscript G, gas-side; subscript 1, sinkside. θ = polar angle. λ = wavelength. µm = micrometer (m−6). ρ = reflectance; ρs, specular reflectance. σ = Stefan-Boltzmann constant. τ = ratio of temperature to TF. Subscript G, gas; subscript 1 sink; subscript o, base. τ = transmittance. Ω = solid angle. ω = albedo of a surface.

NATURE OF THERMAL RADIATION Consider a pencil of radiation, defined as all the rays passing through each of two small widely separated areas dA1 and dA2. The rays at dA1 will have a solid angle of divergence dΩ1 equal to the apparent area of dA2 viewed from dA1, divided by the square of the separating distance. Let the normal to dA1 make the angle θ1 with the pencil. The flux density q (energy per time-area) normal to the beam and per unit solid ˙1 angle of its divergence is called the intensity I, and the flux dQ (energy per time) through the area dA1 (of apparent area dA1 cos θ1 normal to the beam) is therefore given by ˙ 1 = dA1(cos θ1)q = I dA1(cos θ1) dΩ1 dQ (5-111) The intensity I along a pencil, in the absence of absorption or scatter, is constant (unless the beam passes into a medium of different refractive index n; then I1/n21 = I2 /n22). The emissive power* of a surface is the flux density (energy per time-surface area) due to emission from it throughout a hemisphere. If the intensity I of emission from a surface is independent of the angle of emission, Eq. (5-111) may be integrated to show that the surface emissive power is πI, though the emission is throughout 2π sr. Blackbody Radiation Engineering calculations of thermal radiation from surfaces are best keyed to the radiation characteristics of the blackbody, or ideal radiator. The characteristic properties of a blackbody are that it absorbs all the radiation incident on its surface and that the quality and intensity of the radiation it emits are completely determined by its temperature. The total radiative flux throughout a hemisphere from a black surface of area A and absolute temperature T is given by the Stefan-Boltzmann law: ˙ = AσT 4 Q or q = σT 4 (5-112) The Stefan-Boltzmann constant σ has the value (0.1713)(10−8) Btu/ (ft2 ⭈h⭈°R4); (1.00)(10−8) CHU/(ft2 ⭈h⭈K4); (4.88)(10−8) kcal/(m2 ⭈h⭈K4); (1.356)(10−12) cal (cm2 ⭈s⭈K4); (5.67)(10−12) W/(cm2 ⭈K4); (5.67)(10−8) W/(m2 ⭈K4); or in terms of Planck constants, c1(π/c2)4/15. From the definition of emissive power, σT 4 is the total emissive power of a blackbody, called E; the intensity IB of blackbody emission is E/π or σT 4/π. The spectral distribution of energy flux from a black body is expressed by Planck’s law: Eλ dλ = (2πhc2 n2 λ−5)/(ehc/kλT − 1) dλ

(n2c1λ−5)/(ec / λT − 1) dλ 2

(5-113) (5-114)

where Eλ dλ is the hemispherical flux density lying in the wavelength range λ to λ + dλ; h is Planck’s constant, (6.6256)(10−27) erg⭈s; c is the velocity of light in vacuo, (2.9979)(1010) cm/s; k is the Boltzmann constant, (1.3805)(10−16) erg/K; λ is the wavelength measured in vacuo; and n is the refractive index of the emitter (λ = nλm, where λm is the wavelength measured in the medium; Eλ d λ = Eλm d λm, where Eλ and Eλm are both measured in the medium; engineers commonly use Eλ). Equation (10-191) may be written Eλ c1(λT)−5 =ᎏ (5-115) ᎏ 2 5 nT ec /λT − 1 The first and second Planck-law constants c1 and c2 are respectively (3.740)(10−16) (J⭈m2)/s and (1.4388)(10−2) m⭈K. The term Eλ/n2T 5, clearly a function only of the product λT, is given in Fig. 5-11 which may be visualized as the monochromatic emissive power versus wavelength measured in vacuo of a black surface at 1 K discharging in vacuo. The wavelength of maximum intensity is seen to be inversely proportional to the absolute temperature. The relation is known as Wien’s displacement law: λmaxT = (2.898)(10−3) m⭈K. This can be misleading, however, since the wavelength of maximum intensity depends on whether intensity is defined in terms of wavelength interval or frequency interval. More useful displacement laws refer to the value of λT corresponding to maximum energy per unit fractional change in wavelength or frequency [(3.67)(10−3) m⭈K] or to the value of λT corresponding to half of the energy [(4.11)(10−3) m⭈K]. Figure 2

* Variously called, in the literature, emittance, total hemispherical intensity, or radiant flux density.

HEAT TRANSFER BY RADIATION

5-25

Distribution of energy in the spectrum of a blackbody. To convert microns to micrometers, multiply by unity. To convert ergs per square centimeter–second–micron–K5 to watts per square meter, per meter, per K5, multiply by 10−3. FIG. 5-11

5-11 carries, at the top, a scale giving the fraction f of the total energy in the spectrum that lies below λT. A generalization useful for identifying the spectral range of greatest interest in evaluations of radiative transfer is that roughly half of the energy from a black surface lies within the twofold range of λT geometrically centered on 3.67 × 10−3, i.e., from λT = (3.67/2)(10−3) to (3.67 × 2)(10−3) m⭈K. One limiting form of the Planck equation, approached as λT → 0, is the Wien equation [Eqs. (5-113) and (5-114)] with the 1 missing in the denominator. The error is less than 1 percent when λT < (3)(10−3) m⭈K or when T < 4800 K if an optical pyrometer with red screen (λ = 0.65µm) is used. RADIATIVE EXCHANGE BETWEEN SURFACES OF SOLIDS Emittance and Absorptance The ratio of the total radiating power of a real surface to that of a black surface at the same temperature is called the emittance of the surface (for a perfectly plane surface, the emissivity), designated by ε. Subscripts λ, θ, and n may be assigned to differentiate monochromatic, directional, and surfacenormal values respectively from the total hemispherical value. If radi-

ation is incident on a surface, the fraction absorbed is called the absorptance (absorptivity), a term to which two subscripts may be appended, the first to identify the temperature of the surface and the second to identify the spectral energy distribution of the surface. According to Kirchhoff’s law, the emissivity and absorptivity of a surface in surroundings at its own temperature are the same for both monochromatic and total radiation. When the temperatures of the surface and its surroundings differ, the total emissivity and absorptivity of the surface often are found to be different, but, because absorptivity is substantially independent of irradiation density, the monochromatic emissivity and absorptivity of surfaces are for all practical purposes the same. The difference between total emissivity and absorptivity depends on the variation, with wavelength, of ελ and on the difference between the emitter temperature and the effective source temperature. Consider radiative exchange between a body of area A1 and temperature T1 and black surroundings at T2. The net interchange is given by ˙ 1 = 2 = A1 Q



∞

[ελEλ(T1) − αλEλ(T2)] dλ

0

= A1(ε1σT 41 − α12σT 42 )

(5-116)

5-26 where

HEAT AND MASS TRANSFER

 ε df =  ε df

ε1 =

1

λ

λT 1

(5-117)

λ

λT 2

(5-118)

0

and

α12

1

0

The value of ε1 (or α12, the absorptivity of surface A1 for blackbody radiation at T2) is the area under a curve of ελ versus f, the latter read as a function of λT1 (or λT2) from the top ordinate of Fig. 5-11. For a gray surface, ε1 = α12 = ελ. A selective surface is one whose ελ changes dramatically with wavelength. If this change is unidirectional, ε1 and α12 are, according to Eqs. (5-116) to (5-118), markedly different when the absolute-temperature ratio is far from 1; e.g., when T1 = 294 K (530°R; ambient temperature), and T2 = 6000 K (10,800°R; effective solar temperature), ε1 = 0.9 and α12 = 0.1 to 0.2 for a white paint, but ε1 can be as low as 0.12 and α12 above 0.9 for a thin layer of copper oxide on bright aluminum. The effect of radiation-source temperature on the low-temperature absorptivity of a number of additional materials is presented in Fig. 5-12. It will be noted that polished aluminum (curve 15) and anodized (surface-oxidized) aluminum (curve 13), representative of metals and nonmetals respectively, respond oppositely to a change in the temperature of the radiation source. The absorptance of surfaces for solar

FIG. 5-12 Variation of absorptivity with temperature of radiation source. (1) Slate composition roofing. (2) Linoleum, red brown. (3) Asbestos slate. (4) Soft rubber, gray. (5) Concrete. (6) Porcelain. (7) Vitreous enamel, white. (8) Red brick. (9) Cork. (10) White dutch tile. (11) White chamotte. (12) MgO, evaporated. (13) Anodized aluminum. (14) Aluminum paint. (15) Polished aluminum. (16) Graphite. The two dashed lines bound the limits of data on gray paving brick, asbestos paper, wood, various cloths, plaster of paris, lithopone, and paper. To convert degrees Rankine to kelvins, multiply by (5.556)(10−1).

radiation may be read from the right of Fig. 10-45, if solar radiation is assumed to consist of blackbody radiation from a source at 5800 K (10,440°R). Although values of emittance and absorptance depend in very complex ways on the real and imaginary components of the refractive index and on the geometrical structure of the surface layer, the generalizations that follow are possible. Polished Metals 1. ε λ in the infrared is governed by free-electron contributions, is quite low, and is a function of the resistivity-wavelength quotient r/ λ (Fig. 5-13). For λ > 8µm, ε λ,n is approximately 0.0365 r/ λ , where r is in ohm-meters and λ in micrometers (the Drude or HagenRubens relation). At shorter wavelengths, bound-electron contributions become significant and ελ increases, sometimes exhibiting maxima; values of 0.4 to 0.8 are common in the visible spectrum (0.4 to 0.7 µm). ελ is approximately proportional to the square root of the absolute temperature (ελ ∝ r, and r ∝ T) in the far infrared (λ > 8µm), is temperature-insensitive in the near infrared (0.7 to 1.5 µm), and decreases slightly as temperature increases in the visible. 2. Total emittance is substantially proportional to absolute temperature; at moderate temperature, εn = 0.058T rT, where T is in kelvin. 3. The total absorptance of a metal at T1 for radiation from a black or gray source at T2 is equal to the emissivity evaluated at the geometric mean of T1 and T2. Figure 5-13 gives values of ελ, ελ,n, and their ratio as a function of r/λ (dashed lines); and total emissivities ε, εn and their ratio as a function of rT (solid lines). Although the figure is based on free-electron contributions to emissivity in the far infrared, the relations for total emissivity are remarkably good even at high temperatures. Unless extraordinary pains are taken to prevent oxidation, however, a metallic surface may exhibit several times the emittance or absorptance of a polished specimen. The emittance of iron and steel, for example, varies widely with degree of oxidation and roughness; clean metallic surfaces have an emittance of from 0.05 to 0.45 at ambient temperatures to 0.4 to 0.7 at high temperatures; oxidized and/or rough surfaces range from 0.6 to 0.95 at low temperatures to 0.9 to 0.95 at high temperatures. Refractory Materials Grain size and concentration of trace impurities are important. 1. Most refractory materials have an ελ of 0.8 to 1.0 at wavelengths beyond 2 to 4 µm; ελ decreases rapidly toward shorter wavelengths for materials that are white in the visible but retains its high value for black materials such as FeO and Cr2O3. Small concentrations of FeO

FIG. 5-13 Hemispherical and normal emissivities of metals and their ratio. Dashed lines: monochromatic (spectral) values versus r/λ. Solid lines: total values versus rT. To convert ohm-centimeter-kelvins to ohm-meter-kelvins, multiply by 10−2.

HEAT TRANSFER BY RADIATION and Cr2O3 or other colored oxides can cause marked increases in the emittance of materials that normally are white. The sensitivity of the emittance of refractory oxides to small additions of absorbing materials is demonstrated by the results of calculations, shown in Fig. 5-14, of the emittance of a semi-infinite absorbing-scattering medium as a function of its albedo: the ratio of the scatter coefficient to the sum of scatter and absorption coefficients. The results, pertinent to the radiative properties of fibrous materials, paints, oxide coatings, and refractories, show that when absorption accounts for only 0.5 percent (10 percent) of the total attenuation within the medium, the emittance is greater than 0.15 (0.5). ελ for refractory materials varies little with temperature, with the exception of some white oxides which at high temperatures become good emitters in the visible spectrum as a consequence of the induced electronic transitions. 2. Refractory materials generally have a total emittance which is high (0.7 to 1.0) at ambient temperatures and decreases with increase in temperature; a change from 1000 to 1570°C (1850 to 2850°F) may cause a decrease in ε of one-fourth to one-third. 3. The emittance and absorptance increase with increase in grain size over a grain-size range of 1 to 200 µm. 4. The ratio ε/εn of hemispherical to normal emissivity of polished surfaces varies with refractive index n from 1 at n = 1.0 to 0.93 at n = 1.5 (common glass) and back to 0.96 at n = 3. 5. The ratio ε/εn for a surface composed of particulate matter which scatters isotropically varies with ε from 1 when ε = 1 to 0.8 when ε = 0.07 (see Fig. 5-14). 6. The total absorptance shows a decrease with increase in temperature of the radiation source similar to the decrease in emittance with increase in the specimen temperature. Figure 5-12 shows a regular variation of α12 with T2. When T2 is not very different from T1, α12 may be expressed as ε1(T2 /T1)m. It may be shown that Eq. (5-116) is then approximated by m ˙ 1,net = σA1εav 1 + ᎏ Q (T 14 − T 42) (5-119) 4 where εav is evaluated at the arithmetic mean of T1 and T2. For metals m is about 0.5; for nonmetals it is small and negative. Table 5-6, based on a critical evaluation of early data, is illustrative of the emittance of materials encountered in engineering practice; it shows the wide variation possible in the emissivity of a particular material due to variations in surface roughness and thermal pretreatment. (With few exceptions the values refer to emission normal to the surface; see above for conversion to hemispherical values.) More recent data support the range of emittance values given in Table 5-6 and their dependence on surface conditions. Extensive compilations of data are provided by Schmidt and Furthmann (Mitt. KaiserWilhelm-Inst. Eisenforsch., 109, 225), covering data to 1928; by Gubareff, Jansen, and Torborg [Thermal Radiation Properties Sur-





5-27

vey, Honeywell Research Center, Minneapolis, 1960), covering data to 1940; and by Goldsmith, Waterman, and Hirschhorn [Thermophysical Properties of Matter, Purdue University (Touloukian, Ed.), Plenum, 1970]. For opaque materials, the reflectance ρ is the complement of the absorptance. The directional distribution of the reflected radiation depends on the material, its degree of roughness or grain size, and, if a metal, its state of oxidation. Polished surfaces of homogeneous materials reflect specularly. In contrast, the intensity of the radiation reflected from a perfectly diffuse, or Lambert, surface is independent of direction. The directional distribution of reflectance of many oxidized metals, refractory materials, and natural products approximates that of a perfectly diffuse reflector. A better model, adequate for many calculational purposes, is achieved by assuming that the total reflectance ρ is the sum of diffuse and specular components ρD and ρ S . Black-Surface Enclosures View Factor and Direct-Exchange Area When several surfaces are present, the need arises for evaluating a geometrical factor F, called the direct view factor. In the following discussion, restriction is to black surfaces, the intensity from which is independent of angle of emission. Define F12 as the fraction of the radiation leaving surface A1 in all directions which is intercepted by surface A2. Since the net interchange between A1 and A2 must be zero when their temperatures are alike, it follows that A1F12 = A2F21. This product, having the dimensions of area, is called the direct-exchange area and is designated for brevity by 12( 2 1). It is sometimes designated s1s2. Clearly, 11 + 12 + 13 + ⭈⭈⭈ = A1; and when A1 cannot “see” itself, 11 = 0. From Eq. (5-111) and the definition of F: Q1 − 2 dA1(cos θ1) dΩ1 A1F12 s1s2 ᎏ = ᎏᎏ A1 A2 E1 π





dA1(cos θ1) dA2 (cos θ2) (5-120) ᎏᎏᎏ πr 2 where A(cos θ) is the projection of A normal to r, the line connecting dA1 and dA2. Values of s1s2 (or of F12) may be obtained by integrating either Eq. (5-120) or an equivalent contour integral (see Hottel and Sarofim, Radiative Transfer, McGraw-Hill, New York, 1967, chap. 2). Such values are given for opposed parallel disks or rectangles in Fig. 5-15. For rectangles of dimensions L1 and L2, F   F 1F2, where F1 and F2 are for squares of sides L1 and L2. The view factor for rectangles in perpendicular planes and having a common edge length x and ratios Y and Z of widths to common length is given by =

A1

1 FYZ = ᎏ πY







1 Y2 ᎏ ln (1 + Y 2) ᎏ2 4 1+Y

A2

Z 1+Z ᎏᎏ  ᎏ Y + Z  1 + Y + Z  Y2

Z2

2

2

2

1 − Y 2 − Z2

2

2

2

1 1 1 2+  + Y tan−1 ᎏ + Z tan−1 ᎏ − Y Z2 tan−1 ᎏᎏ Y2 + Z2  Y Z



(5-121) The direct-exchange area is given by x2 sysz = ᎏ (5-122) π When the maximum dimensions of each of two plane surfaces is small relative to their center-to-center separating distance r, Eq. (5-120) gives A1(cos θ1)A2(cos θ2) 12 = ᎏᎏᎏ (5-123) πr2 and when, in addition, the normals to A1 and A2 are in a common plane, 12 = A1A2n1n2/πr2

Hemispherical emittance εh and the ratio of hemispherical to normal emittance εh/εn for a semi-infinite absorbing-scattering medium.

FIG. 5-14

(5-124)

where n1 is the normal-to-A1 component of the distance to A2. Equation (5-124) is, for example, in error only by +7 percent for the case of opposed squares separated by 3 times their side dimension. The view factors are given for finite coaxial coextensive cylinders in Fig. 5-16,

5-28

HEAT AND MASS TRANSFER

TABLE 5-6

Normal Total Emissivity of Various Surfaces A. Metals and Their Oxides Surface

Aluminum Highly polished plate, 98.3% pure Polished plate Rough plate Oxidized at 1110°F Aluminum-surfaced roofing Calorized surfaces, heated at 1110°F. Copper Steel Brass Highly polished: 73.2% Cu, 26.7% Zn 62.4% Cu, 36.8% Zn, 0.4% Pb, 0.3% Al 82.9% Cu, 17.0% Zn Hard rolled, polished: But direction of polishing visible But somewhat attacked But traces of stearin from polish left on Polished Rolled plate, natural surface Rubbed with coarse emery Dull plate Oxidized by heating at 1110°F Chromium; see Nickel Alloys for Ni-Cr steels Copper Carefully polished electrolytic copper Commercial, emeried, polished, but pits remaining Commercial, scraped shiny but not mirrorlike Polished Plate, heated long time, covered with thick oxide layer Plate heated at 1110°F Cuprous oxide Molten copper Gold Pure, highly polished Iron and steel Metallic surfaces (or very thin oxide layer): Electrolytic iron, highly polished Polished iron Iron freshly emeried Cast iron, polished Wrought iron, highly polished Cast iron, newly turned Polished steel casting Ground sheet steel Smooth sheet iron Cast iron, turned on lathe Oxidized surfaces: Iron plate, pickled, then rusted red Completely rusted Rolled sheet steel Oxidized iron Cast iron, oxidized at 1100°F Steel, oxidized at 1100°F Smooth oxidized electrolytic iron Iron oxide Rough ingot iron

t, °F*

Emissivity*

440–1070 73 78 390–1110 100

0.039–0.057 0.040 0.055 0.11–0.19 0.216

390–1110 390–1110

0.18–0.19 0.52–0.57

476–674 494–710 530

0.028–0.031 0.033–0.037 0.030

70 73 75 100–600 72 72 120–660 390–1110 100–1000

0.038 0.043 0.053 0.096 0.06 0.20 0.22 0.61–0.59 0.08–0.26

176

0.018

66

0.030

72 242

0.072 0.023

77 390–1110 1470–2010 1970–2330

0.78 0.57 0.66–0.54 0.16–0.13

440–1160

0.018–0.035

350–440 800–1880 68 392 100–480 72 1420–1900 1720–2010 1650–1900 1620–1810

0.052–0.064 0.144–0.377 0.242 0.21 0.28 0.435 0.52–0.56 0.55–0.61 0.55–0.60 0.60–0.70

68 67 70 212 390–1110 390–1110 260–980 930–2190 1700–2040

0.612 0.685 0.657 0.736 0.64–0.78 0.79 0.78–0.82 0.85–0.89 0.87–0.95

Surface Sheet steel, strong rough oxide layer Dense shiny oxide layer Cast plate: Smooth Rough Cast iron, rough, strongly oxidized Wrought iron, dull oxidized Steel plate, rough High temperature alloy steels (see Nickel Alloys). Molten metal Cast iron Mild steel Lead Pure (99.96%), unoxidized Gray oxidized Oxidized at 390°F. Mercury Molybdenum filament Monel metal, oxidized at 1110°F Nickel Electroplated on polished iron, then polished Technically pure (98.9% Ni, + Mn), polished Electroplated on pickled iron, not polished Wire Plate, oxidized by heating at 1110°F Nickel oxide Nickel alloys Chromnickel Nickelin (18–32 Ni; 55–68 Cu; 20 Zn), gray oxidized KA-2S alloy steel (8% Ni; 18% Cr), light silvery, rough, brown, after heating After 42 hr. heating at 980°F. NCT-3 alloy (20% Ni; 25% Cr.), brown, splotched, oxidized from service NCT-6 alloy (60% Ni; 12% Cr), smooth, black, firm adhesive oxide coat from service Platinum Pure, polished plate Strip Filament Wire Silver Polished, pure Polished Steel, see Iron. Tantalum filament Tin—bright tinned iron sheet Tungsten Filament, aged Filament Zinc Commercial, 99.1% pure, polished Oxidized by heating at 750°F. Galvanized sheet iron, fairly bright Galvanized sheet iron, gray oxidized

t, °F* 75 75 73 73 100–480 70–680 100–700

2370–2550 2910–3270 260–440 75 390 32–212 1340–4700 390–1110 74

Emissivity* 0.80 0.82 0.80 0.82 0.95 0.94 0.94–0.97

0.29 0.28 0.057–0.075 0.281 0.63 0.09–0.12 0.096–0.292 0.41–0.46 0.045

440–710

0.07–0.087

68 368–1844 390–1110 1200–2290

0.11 0.096–0.186 0.37–0.48 0.59–0.86

125–1894

0.64–0.76

70

0.262

420–914 420–980

0.44–0.36 0.62–0.73

420–980

0.90–0.97

520–1045

0.89–0.82

440–1160 1700–2960 80–2240 440–2510

0.054–0.104 0.12–0.17 0.036–0.192 0.073–0.182

440–1160 100–700

0.0198–0.0324 0.0221–0.0312

2420–5430 76

0.194–0.31 0.043 and 0.064

80–6000 6000

0.032–0.35 0.39

440–620 750 82 75

0.045–0.053 0.11 0.228 0.276

260–1160

0.81–0.79

1900–2560 206–520 209–362

0.526 0.952 0.959–0.947

B. Refractories, Building Materials, Paints, and Miscellaneous Asbestos Board Paper Brick Red, rough, but no gross irregularities Silica, unglazed, rough Silica, glazed, rough Grog brick, glazed See Refractory Materials below.

74 100–700 70 1832 2012 2012

0.96 0.93–0.945 0.93 0.80 0.85 0.75

Carbon T-carbon (Gebr. Siemens) 0.9% ash (this started with emissivity at 260°F. of 0.72, but on heating changed to values given) Carbon filament Candle soot Lampblack-waterglass coating

HEAT TRANSFER BY RADIATION TABLE 5-6

5-29

Normal Total Emissivity of Various Surfaces (Concluded ) B. Refractories, Building Materials, Paints, and Miscellaneous Surface

Same Thin layer on iron plate Thick coat Lampblack, 0.003 in. or thicker Enamel, white fused, on iron Glass, smooth Gypsum, 0.02 in. thick on smooth or blackened plate Marble, light gray, polished Oak, planed Oil layers on polished nickel (lube oil) Polished surface, alone +0.001-in. oil +0.002-in. oil +0.005-in. oil Infinitely thick oil layer Oil layers on aluminum foil (linseed oil) Al foil +1 coat oil +2 coats oil Paints, lacquers, varnishes Snowhite enamel varnish or rough iron plate Black shiny lacquer, sprayed on iron Black shiny shellac on tinned iron sheet Black matte shellac Black lacquer Flat black lacquer White lacquer

t, °F*

Emissivity*

260–440 69 68 100–700 66 72

0.957–0.952 0.927 0.967 0.945 0.897 0.937

70 72 70 68

0.903 0.931 0.895 0.045 0.27 0.46 0.72 0.82

212 212 212

0.087† 0.561 0.574

Surface Oil paints, sixteen different, all colors Aluminum paints and lacquers 10% Al, 22% lacquer body, on rough or smooth surface 26% Al, 27% lacquer body, on rough or smooth surface Other Al paints, varying age and Al content Al lacquer, varnish binder, on rough plate Al paint, after heating to 620°F. Paper, thin Pasted on tinned iron plate On rough iron plate On black lacquered plate Plaster, rough lime Porcelain, glazed Quartz, rough, fused Refractory materials, 40 different poor radiators

t, °F*

Emissivity*

212

0.92–0.96

212

0.52

212

0.3

212 70 300–600

0.27–0.67 0.39 0.35

66 66 66 50–190 72 70 1110–1830

0.924 0.929 0.944 0.91 0.924 0.932

good radiators 73 76 70 170–295 100–200 100–200 100–200

0.906 0.875 0.821 0.91 0.80–0.95 0.96–0.98 0.80–0.95

Roofing paper Rubber Hard, glossy plate Soft, gray, rough (reclaimed) Serpentine, polished Water

69 74 76 74 32–212



0.65 – 0.75 0.70 0.80 – 0.85 0.85 – 0.90 0.91

} }{



0.945 0.859 0.900 0.95–0.963

*When two temperatures and two emissivities are given, they correspond, first to first and second to second, and linear interpolation is permissible. °C = (°F − 32)/1.8. †Although this value is probably high, it is given for comparison with the data by the same investigator to show the effect of oil layers. See Aluminum, Part A of this table.

and for an infinite plane parallel to a system of rows of parallel tubes as curves 1 and 3 of Fig. 5-17. The exchange area between any two area elements of a sphere is independent of their relative shape and position and is simply the product of the areas divided by the area of the whole sphere; i.e., any spot on a sphere has equal views of all other spots. For surfaces in two-dimensional systems (with third dimension infinite), A1F12 per unit length in the third dimension may be obtained simply by evaluating, in a cross-sectional view, the sum of lengths of crossed strings from the ends of A1 to the ends of A2 less the sum of uncrossed strings from and to the same points, all divided by 2. The strings must be so drawn that all the flux from one surface to the other must cross each of a pair of crossed strings and neither of a pair of uncrossed ones. If one surface can see the other around both sides of an obstruction, two more pairs of strings are involved.

Example 3: Calculation of View Factor Evaluate the view factor between two parallel circular tubes long enough compared with their diameter D or their axis-to-axis separating distance C to make the problem twodimensional. With reference to Fig. 5-18, the crossed-strings method yields, per unit of axial length, 1/2 2(EFGH − HJ) D C 2 C A1F12 = ᎏᎏ = D sin−1 ᎏ + ᎏ − 1 − ᎏ 2 C D D



 

Results for a large number of other cases are given by Hottel and Sarofim (op. cit., chap. 2) and Hamilton and Morgan (NACA-TN2836, December 1952). A comprehensive bibliography is provided by Siegel and Howell (Thermal Radiation Heat Transfer, McGraw-Hill, 1992). The view factor F may often be evaluated from that for simpler configurations by the application of three principles: that of reciprocity, AiFij = AjFji; that of conservation,
Fij = 1; and that due to Yamauti [Res. Electrotech. Lab. (Tokyo), 148, 1924; 194, 1927; 250, 1929], showing that the exchange areas AF between two pairs of surfaces are equal when there is a one-to-one correspondence for all sets of symmetrically placed pairs of elements in the two surface combinations. Example 4: Calculation of Exchange Area The exchange area between the two squares 1 and 4 of Fig. 5-19 is to be evaluated. The following exchange areas may be obtained from Eq. 5-121. F for common-side rectangles: 13 = 0.24, 24 = 2 × 0.29 = 0.5S, (1 + 2)(3 + 4) = 3 × 0.32 = 0.96. Expression of (1 + 2)(3 + 4) in terms of its components yields (1 + 2)(3 3+ 4) = 13 + 14 + 23  + 24. And by the Yamauti principle 14 = 23, since for every pair of elements in 1 and 4 there is a corresponding pair in 2 and 3. Therefore, (1 1+ 2)(3 3+ 4) − 13 − 24 14 = ᎏᎏᎏ = 0.07 2 Figure 5-16 may be used in the same way.

FIG. 5-15

Radiation between parallel planes, directly opposed.

Non-Black-Surface Enclosures In the following discussion we are concerned with enclosures containing gray sources and sinks, radiatively adiabatic surfaces, and no absorbing gas. The calculation of interchange between a source and a sink under conditions involving successive multiple reflections from other source-sink surfaces in the

5-30

HEAT AND MASS TRANSFER

(a) FIG. 5-16

(b)

View factors for a system of two concentric coaxial c to inner cylinder. (b) Inner surface of outer cylinder to itself.

FIG. 5-18

Direct exchange between parallel circular tubes.

FIG. 5-19

Illustration of the Yamauti principle.

Distribution of radiation to rows of tubes irradiated from one side. Dashed lines: direct view factor F from plane to tubes. Solid lines: total view factor  F for black tubes backed by a refractory surface. FIG. 5-17

enclosure, as well as reradiation from refractory surfaces which are in radiative equilibrium, can become complicated. Zone Method Let a zone of a furnace enclosure be an area small enough to make all elements of itself have substantially equivalent “views” of the rest of the enclosure. (In a furnace containing a symmetry plane, parts of a single zone would lie on either side of the plane.) Zones are of two classes: source-sink surfaces, designated by numerical subscripts and having areas A1, A2, . . . , and emissivities ε1, ε2, . . . ; and surfaces at which the net radiant-heat flux is zero (fulfilled by the average refractory wall in which difference between internal convection and external loss is minute compared with incident radiation), designated by letter subscripts starting with r, and having areas Ar, As,. . . . It may be shown (see, for example, Hottel and Sarofim, op. cit., chap. 3) that the net radiation interchange between source-sink zones i and j is given by ˙ i = j = AiᏲijσ T i4 − AjᏲjiσ T 4j Q (5-125)

Ᏺij is called the total view factor from i to j, and the term AiᏲij, sometimes designated S iS j, is called the total interchange area shared by areas Ai and Aj and depends on the shape of the enclosure and the emissivity and absorptivity of the source and sink zones. Restriction here is to gray source-sink zones, for which AiᏲij = AjᏲji; the more general case is treated elsewhere (Hottel and Sarofim, op. cit., chap. 5). Evaluation of the AᏲ’s that characterize an enclosure involves solution of a system of radiation balances on the surfaces. If the assumption is made that all the zones of the enclosure are gray and emit and reflect diffusely,* then the direct-exchange area ij, as evaluated for the black-surface pair Ai and Aj, applies to emission and reflections between them. If at a surface the total leaving-flux density, emitted plus reflected, is denoted by W (and called by some the radiosity and by others the exitance), radiation balances take the form: * So-called Lambert surfaces, which emit or reflect with an intensity independent of angle; approximately satisfied by most nonmetallic, tarnished, oxidized, or rough surfaces.

HEAT TRANSFER BY RADIATION For source-sink j, AjεjEj + ρj
(ij)Wi = AjWj

(5-126)

i

For adiabatic surface r,


(ir)W = A W i

r

(5-127)

r

i

where ρ is reflectance and the summation is over all surfaces in the enclosure. In matrix notation, Eq. (10-196) becomes, with source or sink zones represented by 1, 2, 3 . . . and adiabatic zones by r, s, t . . . ,



A 11  − ᎏᎏ1 ρ1

12

1r

1s

12

A 22 − ᎏᎏ2 ρ2

2r

2s

1r 1s

2r 2s

rr − Ar rs

rs ss − As

W2

=

Refractory temperature is obtained from Wr = Er = σTr4. The more general use of Eq. (5-128) is to obtain the set of total interchange areas AᏲ⋅ which constitute a complete description of the effect of shape, size, and emissivity on radiative flux, independent of the presence or absence of other transfer mechanisms. It may be shown that Aiεi Ajεj D′ij AiᏲij AjᏲji  SiS j = ᎏ ᎏ − ᎏ − δijεj ρi ρj D





(5-130)

where D is the determinant of the square coefficient matrix in Eq. 5-128) and D′ij is the cofactor of its ith row and jth column, or −1)i + j times the minor of D formed by crossing out the ith row and ith column, and δij is the Kronecker delta, 1 when i = j, otherwise 0. As an example, consider radiation between two surfaces A1 and A2, which together form a complete enclosure. Equation (5-130) takes the form



12 A 11 − ᎏᎏ1 ρ1 12

12 A 22 − ᎏᎏ2 ρ2



(5-131)

Only one direct-view factor F12 or direct-exchange area 12 is needed because F11 equals 1 − F12 and F22 equals 1 − F21 equals 1 − F12 A1 /A2. Then 11 equals A1 − 12 and 22 equals A2 − 21. With these substitutions, Eq. (5-131) becomes A1 A1Ᏺ12 = ᎏᎏᎏ 1 1 A1 1 ᎏᎏ + ᎏᎏ − 1 + ᎏᎏ ᎏᎏ − 1 F12 ε1 A2 ε2





Ᏺ12 = ε1

(5-135)

F12( A2 F21) represents the total interchange where the expression A1 area for the limiting case of a black source and black sink (the refractory emissivity is of no moment). The factor F  is known exactly for a few geometrically simple cases and may be approximated for others. If A1 and A2 are equal parallel disks, squares, or rectangles, connected by nonconducting but reradiating refractory walls, then  F is given by Fig. 5-15, curves 5 to 8. If A1 represents an infinite plane and A2 is one or two rows of infinite parallel tubes in a parallel plane and if the only other surface is a refractory surface behind the tubes,  F12 is given by curve 5 or 6 of Fig. 5-17. If an enclosure may be divided into several radiant-heat sources or sinks A1, A2, etc., and the rest of the enclosure (reradiating refractory surface) may be lumped together as Ar at a uniform temperature Tr, then the total interchange area for zone pairs in the black system is given by (1r)(r2) A1F (5-137) 12( A2F 21) = 12 + ᎏ Ar − rr For the two-source-sink-zone system to which Eq. (5-136) applies, Eq. (5-137) simplifies to

(5-128)

  



0 0

This represents a system of simultaneous equations equal in number to the number of rows of the square matrix. Each equation consists, on the left, of the sum of the products of the members of a row of the square matrix and the corresponding members of the W-column matrix and, on the right, of the member of that row in the third matrix. With this set of equations solved for Wi, the net flux at any surface Ai is given by Q˙ i,net = (Aiεi /ρi)(Ei − Wi) (5-129)

A1ε1 A2ε2 A1Ᏺ12 = ᎏ ᎏ ρ1 ρ2

 

Aε − ᎏ1ᎏ1 E1 ρ1 A2ε2 − ᎏᎏE2 ρ2

Wr Ws

 

2. Sphere of area A1 concentric with surrounding sphere of area A2. F12 = 1. Then A1 A1Ᏺ12 = ᎏᎏ (5-134) 1 A1 1 ᎏᎏ + ᎏᎏ ᎏᎏ ε1 A2 ε2 − 1 3. Body of surface A1 having no negative curvature, surrounded by very much larger surface A2. F12 = 1 and A1/A2 → 0. Then Many furnace problems are adequately handled by dividing the enclosure into but two source-sink zones A1 and A2 and any number of no-flux zones Ar, As,. . . . For this case Eq. (5-130) yields 1 1 1 1 1 1 1 ᎏ ᎏ = ᎏ ᎏ − 1 + ᎏ ᎏ − 1 + ᎏ (5-136) A1Ᏺ12 A2Ᏺ21 A1 ε1 A2 ε2 A1F 12

  W1

5-31

(5-132)

Special cases include 1. Parallel plates, large compared to clearance. Substitution of F12 = 1 and A1 = A2 gives A1 (5-133) A1Ᏺ12 = ᎏᎏ 1 1 ᎏᎏ + ᎏᎏ − 1 ε1 ε2













A 1F 12 = 12 + 1/(1/1 r + 1/2 r)

(5-138)

and if A1 and A2 each can see none of itself, there is further simplification to 1 A1F 12 = 12 + ᎏᎏᎏ 1/(A1 − 12) + 1/(A2 − 12) A1A2 − (1 2)2 = ᎏᎏ (5-139) A1 + A2 − 2(1 2) which necessitates the evaluation of but one geometrical factor F. Equation (5-136) covers many of the problems of radiant-heat interchange between source and sink in a furnace enclosure. The error due to single zoning of source and sink is small even if the views of the enclosure from different parts of each zone are quite different, provided the emissivity is fairly high; the error in F  is zero if it is obtainable from Fig. 5-15 or 5-17, small if Eq. (5-137) is used and the variation in temperature over the refractory is small. An approach to any desired accuracy can be made by use of Eqs. (5-126) and (5-130) with division of the surfaces into more zones. From the definitions of F,  F, and Ᏺ it is to be noted that F11 + F12 + F13 + ⋅⋅⋅ + F1r + F1s + ⋅⋅⋅ = 1 F F12 +  F13 + ⋅⋅⋅ = 1 11 +  Ᏺ11 + Ᏺ12 + Ᏺ13 + ⋅⋅⋅ = ε1 Example 5: Radiation in a Furnace Chamber A furnace chamber of rectangular parallelepipedal form is heated by the combustion of gas inside vertical radiant tubes lining the sidewalls. The tubes are of 0.127-m (5-in) outside diameter on 0.305-m (12-in) centers. The stock forms a continuous plane on the hearth. Roof and end walls are refractory. Dimensions are shown in Fig. 5-20. The radiant tubes and stock are gray bodies having emissivities of 0.8 and 0.9 respectively. What is the net rate of heat transmission to the stock by radiation when the mean temperature of the tube surface is 816°C (1500°F) and that of the stock is 649°C (1200°F)?

5-32

FIG. 5-20

HEAT AND MASS TRANSFER

Furnace-chamber cross section. To convert feet to meters, multiply

by 0.3048.

This problem must be broken up into two parts, first considering the walls with their refractory-backed tubes. To imaginary planes A2 of area 1.83 by 3.05 m (6 by 10 ft) and located parallel to and inside the rows of radiant tubes, the tubes emit radiation σ T 14A1Ᏺ12, which equals σT 14A2 Ᏺ21. To find Ᏺ21, use Fig. 5-17, curve 5, from which F 21 = 0.81. Then from Eq. (10-200) 1 ᎏᎏᎏᎏ Ᏺ21 = 1 1 = 0.702 12 1 ᎏᎏ − 1 + ᎏᎏ ᎏᎏ − 1 + ᎏᎏ 1 5π 0.8 0.81 This amounts to saying that the system of refractory-backed tubes is equal in radiating power to a continuous plane A2 replacing the tubes and refractory back of them, having a temperature equal to that of the tubes and an equivalent or effective emissivity of 0.702. The new simplified furnace now consists of an enclosure formed by two 1.83by 3.05-m (6- by 10-ft) radiating sidewalls (area A2, emissivity 0.702), a 1.52- by 3.05-m (5- by 10-ft) receiving plane on the floor A3, and refractory surfaces Ar to complete the enclosure (ends, roof, and floor side strips). The desired heat transfer is Q˙ 2 ⭵ 3 = σ (T 14 − T 34)A2 Ᏺ23



  



To evaluate Ᏺ23, start with the direct interchange factor F23. F23 = F from A2 to (A3 + a strip of Ar alongside A3, which has a common edge with A2) minus F from A2 to the strip only. These two F’s may be evaluated from Fig. 5-20 and Eq. (5-121). For the first F, Y = 1.83/3.05, Z = 1.98/3.05, and F = 0.239; for the second F, Y = 1.83/3.05, Z = 0.46/3.05, and F = 0.100. Then F23 = 0.239 − 0.10 = 0.139. Now F  may be evaluated. From Eq. 5-137 et seq., 1 1 A2F23 = 23 ÷ ᎏᎏ F 23 = F23 + ᎏᎏᎏ 1/2r + 1/3r (1/F2r) + (A2/A3)(1/F3r) Since A2 “sees” Ar, A3, and some of itself (the plane opposite), F2r = 1 − F22 − F23. F22, the direct interchange factor between parallel 1.83- by 3.05-m (6- by 10-ft) rectangles separated by 2.44 m (8 ft), may be taken as the geometric mean of the factors for 1.83-m (6-ft) squares separated by 2.44 m (8 ft) and for 3.05-m (10-ft) squares separated by 2.44 m (8 ft). These come from Fig. 5-15, curve 2, according to which F22 = 0 .13 ×0 .255 = 0.182. Then F2r = 1 − 0.182 − 0.139 = 0.679. The other required direct factor is F3r = 1 − F32 = 1 − F23 A2/A3 = 1 − (0.139)(11.14)/4.65 = 0.666. Then 1 F 23 = 0.139 + ᎏᎏᎏᎏ = 0.336 (1/0.679) + (11.14/4.65)(1/0.666) Having  F23, we may now evaluate the factor Ᏺ23: 1 Ᏺ23 = ᎏᎏᎏᎏᎏᎏ = 0.273 (1/0.336) + [(1/0.702) − 1] + (11.14/4.65)[(1/0.9) − 1] ˙ net = σ(T 14 − T 43)A2 Ᏺ23 = 5.67(10.894 − 9.224)(11.15)(0.273) Q = 118,000 J/s (402,000 Btu/h) A result of interest is obtained by dividing the term A2 Ᏺ23(11.15 × 0.273, or 3.04) by the actual area A1 of the radiating tubes (0.127π)(18.3)(2) = 14.6 m2 (157 ft2). Thus 3.04/14.6 = 0.208, which means that the net radiation from a tube to the stock is 20.8 percent as much as if the tube were black and completely surrounded by black stock.

Integral Formulation The zone method has the purpose of dodging the solution of an integral equation. If in Eq. (5-126) the zone on which the radiation balance is formulated is decreased to a differential element, that equation becomes dAi dAf (cos θi)(cos θj)Wi dA jεj Ej + ρj ᎏᎏᎏ = dA jWj (5-140) r2



which is an integral equation with the unknown function W inside the integral. Integration is over the entire surface area. Exact solutions have been carried out for only a few simple cases. One of these is the evaluation of emittance of an isothermal spherical cavity, for which dAi dAf (cos θi)(cos θj)/r2 in the integral of Eq. (5-140) becomes dAi dAj /4πR2, where R is the sphere radius. For this special case W is, from Eq. (10-202), constant over the inner surface of the cavity and given by εE W = ᎏᎏ (5-141) 1 − ρ(1 − A1/4πR2) where A1 is the curved area of a hole in the sphere’s surface. The ratio W/E is the effective emittance of the hole as sensed by a narrow-angle receiver viewing the cavity interior. If the material of construction of the cavity is a diffuse emitter and reflector and has an emissivity of 0.5 and the cavity is to appear at least 98 percent black, the curved area A1 of the hole must be smaller than 2 percent of the total surface area of the sphere. Enclosures of Surfaces That Are Not Diffuse Reflectors If no restriction that the surfaces be diffuse emitters and reflectors is imposed, Eq. (5-140) becomes much more complex. The W’s are replaced by πI’s and εj, Ii, and Ij all become functions of the angle of the leaving beam, and ρj goes inside the integral and becomes a function of angles of incidence and reflection. Seldom are such details of reflectance known. When they are and a solution is needed, the Monte Carlo method of tracing the history of a large number of beams emitted from random positions and in random initial directions is probably the best method of obtaining a solution. Another approach is possible, however, because of the tendency of most surfaces to fit a simpler reflection model. The total reflectance ρ( 1 − ε) can be represented by the sum of a diffuse component ρD and a specular component ρS. For applications see Hottel and Sarofim (op. cit., chap. 5). The method yields the following relation for exchange between concentric spheres or infinite cylinders: A1Ᏺ12  S1S 2 1 = ᎏᎏᎏᎏᎏᎏ (5-142) 1/A1ε1 + (1/A2)(1/ε2 − 1) + [ρs2 /(1 − ρs2)](1/A1 − 1/A2) When there is no specular reflectance, the third term in the denominator drops out, in agreement with Eqs. (5-134) and (5-135). When the reflectance is exclusively specular, the denominator becomes 1/A1ε1 + ρs2 /A1(1 − ρs2), easily derivable from first principles. EMISSIVITIES OF COMBUSTION PRODUCTS The radiation from a flame is due to radiation from burning soot particles of microscopic and submicroscopic dimensions, from suspended larger particles of coal, coke, or ash, and from the water vapor and carbon dioxide in the hot gaseous combustion products. The contribution of radiation emitted by the combustion process itself, so-called chemiluminescence, is relatively negligible. Common to these problems is the effect of the shape of the emitting volume on the radiative flux; this is considered first. Mean Beam Lengths Evaluation of radiation from a nonisothermal volume is beyond the scope of this section (see Hottel and Sarofim, Radiative Transfer, McGraw-Hill, New York, 1967, chap. 11). Consider an isothermal gas confined within the volume bounded by the solid angle dΩ with vertex at dA and making the angle θ with the normal to dA. The ratio of the emission to dA from the gas to that from a blackbody at the gas temperature and filling the field of view dΩ is called the gas emissivity ε. Clearly, ε depends on the path length L through the volume to dA. A hemispherical volume radiating to a spot on the center of its base represents the only case in which L is independent of direction. Flux at that spot relative to hemispherical blackbody flux is thus an alternative way to visualize emissivity. The flux density to a small area of interest on the envelope of an emitter volume of any shape can be matched by that at the base of a hemispherical volume of some radius L, which is called the mean beam length. It is found that although the ratio of L to a characteristic dimension D of the shape varies with opacity, the variation is small enough for most engineering purposes to permit use of a constant ratio LM/D,

HEAT TRANSFER BY RADIATION where LM is the average mean beam length. LM can be defined to apply either to a spot on the envelope or to any finite portion of its area. An important limiting case is that of opacity approaching zero (pD → 0, where p = partial pressure of the emitter constituent). For this case, L (called L0) equals 4V/A (V = gas volume; A = bounding area) when interest is in radiation to the entire envelope. For the range of pD encountered in practice, the optimum value of L (now LM) lies between 0.8 to 0.95 times L0. For shapes not reported in Table 5-7, a factor of 0.88 (or LM = 0.88L0 = 3.5V/A) is recommended. Instead of using the average-mean-beam-length concept to approximate A1[ε(Lm)] (the flux per unit black emissive power from a gas volume partially bounded by a surface of area A1), one may calculate the flux rigorously by integration, over the gas volume and over A1, of the expression 4k dv τ(r) dA cos θ/πr 2. Here k is the emission coefficient of the gas, and τ(r) is the transmittance through the distance r between dv and dA. The result has the dimensions of area and, by analogy to ss, is called gs1, the direct-exchange area between the gas zone and the surface zone (Hottel and Sarofim, op. cit., chap. 7). The use of A1[ε(Lm)] instead of gs1 is adequate when the problem is such that all the gas can be treated as a single zone in contact with A1, and having a mean radiating temperature, but the gs concept is clearly useful if allowance is to be made for temperature variations within the gas. Gaseous Combustion Products Radiation from water vapor and carbon dioxide occurs in spectral bands in the infrared. In magnitude it overshadows convection at furnace temperatures. Carbon Dioxide The contribution εc to the emissivity of a gas containing CO2 depends on gas temperature TG, on the CO2 partial pressure-beam length product pc L and, to a much lesser extent, on the total pressure P. Constants for use in evaluating εc at a total pressure of 101.3 kPa (1 atm) are given in Table 5-8 (more on this later). The gas absorptivity αc equals the emissivity when the absorbing gas and the emitter are at the same temperature. When the emitter surface temperature is T1, αc is (TG /T1)0.65 times εc, evaluated using Table 5-8 at T1 instead of TG and at pc LT1 /TG instead of pc L. Line broadening, due to TABLE 5-7

increases either in total pressure or in partial pressure of CO2, makes a correction necessary. However, at a total pressure of 101.3 kPa (1 atm) the correction factor may be ignored, since it decreases with increase in temperature and is never more than 4 percent at temperatures above 1111 K(2000°R). Estimations of the correction in systems up to 1013.3 kPa (10 atm) are given by Hottel and Sarofim (op. cit., p. 228), and by Edwards [J. Opt. Soc. Am., 50, 617 (1960)] who in addition presents data on CO2-band emission for use in calculations involving spectrally selective surfaces. The principal emission bands of CO2 are at about 2.64 to 2.84, 4.13 to 4.5, and 13 to 17 µm. Water Vapor The contribution εw to the emissivity of a gas containing H2O depends on TG and pw L and on total pressure P and partial pressure pw. Table 5-8 gives constants for use in evaluating εw. Allowance for departure from the special pressure conditions is made by multiplying εw by a correction factor Cw read from Fig. 5-21 as a function of (pw + P) and pw L. The absorptivity αw of water vapor for blackbody radiation is εw evaluated from Table 5-8 but at T1 instead of TG and at pw LT1 /TG instead of pw L. Multiply by (TG /T1)0.45. The correction factor Cw still applies. Spectral data for water vapor, tabulated for 371 wavelength intervals from 1 to 40 µm, are also available [Ferriso, Ludwig, and Thompson, J. Quant. Spectros. Radiat. Transfer, 6, 241–273 (1966)]. The principal emission is in bands at about 2.55 to 2.84, 5.6 to 7.6, and 12 to 25 µm. Carbon Dioxide–Water-Vapor Mixtures When these gases are present together, the total radiation due to both is somewhat less than the sum of the separately calculated effects, because each gas is somewhat opaque to radiation from the other in the wavelength regions 2.7 and 15 µm. Allowance for spectral overlap, the effect of pressure, and the effect of soot luminosity would make computation tedious. Table 5-8 gives constants for use in direct calculation, for H2O/CO2 mixtures, of the product εGT . The product term is used because it varies much less with T than does εG alone. Constants are given for mixtures, in nonradiating gases, of water vapor alone, CO2 alone, and four pw /pc mixtures.

Mean Beam Lengths for Volume Radiation Shape

Sphere Infinite cylinder Semi-infinite cylinder, radiating to: Center of base Entire base Right-circle cylinder, ht. = diam. radiating to: Center of base Whole surface Right-circle cylinder, ht. = 0.5 diam. radiating to: End Side Total surface Right-circle cylinder, ht. = 2 × diam. radiating to: End Side Total surface Infinite cylinder, half-circle cross section radiating to middle of flats Rectangular parallelepipeds: 1:1:1 (cube) 1:1:4, radiating to: 1 × 4 face 1 × 1 face Whole surface 1:2:6, radiating to: 2 × 6 face 1 × 6 face 1 × 2 face Whole surface Infinite parallel planes Space outside bank of parallel tubes on equilateral triangular centers Tube diam. = clearance Tube diam. = a clearance Tube centers on squares, diam. = clearance

5-33

Characteristic dimension, D

L 0/D

L M /D

Diameter Diameter

0.67 1.0

0.63 0.94

Diameter Diameter

1.0 0.81

0.90 0.65

Diameter Diameter

0.76 0.67

0.71 0.60

Diameter Diameter Diameter

0.47 0.52 0.50

0.43 0.46 0.45

Diameter Diameter Diameter

0.73 0.82 0.80

0.60 0.76 0.73

Radius

1.26

Edge

0.67

0.60

Shortest edge Shortest edge Shortest edge

0.90 0.86 0.89

0.82 0.71 0.81

Shortest edge Shortest edge Shortest edge Shortest edge Clearance

1.18 1.24 1.18 1.2 2.00

1.76

Clearance 3.4 Clearance 4.45 Clearance 4.1

2.8 3.8 3.5

0.82 0.85 0.85

5-34

HEAT AND MASS TRANSFER

TABLE 5-8

Emissivity eG of H2O:CO2 Mixtures Limited range for furnaces, valid over 25-fold range of pw + cL, 0.046–1.15 m atm (0.15–3.75 ft. atm)

pw /pc pw ᎏ pw + pc

0

a

1

2

3

∞

0

s(0.3–0.42)

a(0.42–0.5)

w(0.6–0.7)

e(0.7–0.8)

1

corresponding to (CH6)x, covering future high H2 fuels

H2O only

CO2 only

corresponding to (CH)x, covering coal, heavy oils, pitch

corresponding to (CH2)x, covering distillate oils, paraffins, olefines

corresponding to CH4, covering natural gas and refinery gas

Constants b and n of Eq., εGT = b(pL − 0.015)n, pL = m atm, T = K T, K

b

n

b

n

b

n

b

n

b

n

b

n

1000 1500 2000

188 252 267

0.209 0.256 0.316

384 448 451

0.33 0.38 0.45

416 495 509

0.34 0.40 0.48

444 540 572

0.34 0.42 0.51

455 548 594

0.35 0.42 0.52

416 548 632

0.400 0.523 0.640

T, °R

b

n

b

n

b

n

b

n

b

n

b

n

1800 2700 3600

264 335 330

0.209 0.256 0.316

467 514 476

0.33 0.38 0.45

501 555 519

0.34 0.40 0.48

534 591 563

0.34 0.42 0.51

541 600 577

0.35 0.42 0.52

466 530 532

0.400 0.523 0.640

Constants b and n of Eq., εGT = b(pL − 0.05)n, pL = ft. atm, T = °R

Full range, valid over 2000-fold range of pw + cL, 0.005–10.0 m atm (0.016–32.0 ft. atm) Constants of Eq., log10 εGTG = a0 + a1 log pL + a2 log2 pL + a3 log3 pL pL = m atm, T = K pw ᎏ pc

pw ᎏ pw + pc

0

0

a

s

1

a

2

w

3

e

∞

1

pL = ft. atm, T = °R

T, K

a0

a1

a2

a3

T, °R

a0

a1

a2

a3

1000 1500 2000 1000 1500 2000 1000 1500 2000 1000 1500 2000 1000 1500 2000 1000 1500 2000

2.2661 2.3954 2.4104 2.5754 2.6451 2.6504 2.6090 2.6862 2.7029 2.6367 2.7178 2.7482 2.6432 2.7257 2.7592 2.5995 2.7083 2.7709

0.1742 0.2203 0.2602 0.2792 0.3418 0.4279 0.2799 0.3450 0.4440 0.2723 0.3386 0.4464 0.2715 0.3355 0.4372 0.3015 0.3969 0.5099

−0.0390 −0.0433 −0.0651 −0.0648 −0.0685 −0.0674 −0.0745 −0.0816 −0.0859 −0.0804 −0.0990 −0.1086 −0.0816 −0.0981 −0.1122 −0.0961 −0.1309 −0.1646

0.0040 0.00562 −0.00155 0.0017 −0.0043 −0.0120 −0.0006 −0.0039 −0.0135 0.0030 −0.0030 −0.0139 0.0052 0.0045 −0.0065 0.0119 0.00123 −0.0165

1800 2700 3600 1800 2700 3600 1800 2700 3600 1800 2700 3600 1800 2700 3600 1800 2700 3600

2.4206 2.5248 2.5143 2.6691 2.7074 2.6686 2.7001 2.7423 2.7081 2.7296 2.7724 2.7461 2.7359 2.7811 2.7599 2.6720 2.7238 2.7215

0.2176 0.2695 0.3621 0.3474 0.4091 0.4879 0.3563 0.4561 0.5210 0.3577 0.4384 0.5474 0.3599 0.4403 0.5478 0.4102 0.5330 0.6666

−0.0452 −0.0521 −0.0627 −0.0674 −0.0618 −0.0489 −0.0736 −0.0756 −0.0650 −0.0850 −0.0944 −0.0871 −0.0896 −0.1051 −0.1021 −0.1145 −0.1328 −0.1391

0.0040 0.00562 −0.00155 0.0017 −0.0043 −0.0120 −0.0006 −0.0039 −0.0135 0.0030 −0.0030 −0.0139 0.0052 0.0045 −0.0065 0.0119 0.00123 −0.0165

NOTE: pw /(pw + pc) of s, a, w, and e may be used to cover the ranges 0.2–0.4, 0.4–0.6, 0.6–0.7, and 0.7–0.8, respectively, with a maximum error in εG of 5 percent at pL = 6.5 m atm, less at lower pLs. Linear interpolation reduces the error generally to less than 1 percent. Linear interpolation or extrapolation on T introduces an error generally below 2 percent, less than the accuracy of the original data.

Four suffice, since a change halfway from one mixture ratio to the adjacent one changes the emissivity by a maximum of but 5 percent; linear interpolation may be used if considered necessary. The constants are given for three temperatures, adequate for linear interpolation since εGT  changes a maximum of one-sixth due to a change from one temperature base halfway to the adjacent one. The interpolation relation, with TH and TL representing the higher and lower base temperatures bracketing T, and with the brackets in the term [A(x)] indicating that the parentheses refer not to a multiplier but to an argument, is [εGT H(pL)](TG − TL) + [εGT L(pL)](TH − TG) εGT G = ᎏᎏᎏᎏᎏ (5-143) 500 Extrapolation to a temperature that is above the highest or below the lowest of the three base temperatures in Table 5-8 uses the same formulation, but one of its terms becomes negative. The gas absorptivity may also be obtained from the constants for emissivities. The product αG1T1 (gas absorptivity for black surface radiation), x (surface temperature), is εGT 1 evaluated at T1 instead of TG and at pLT1/TG instead of pL, then multiplied by (TG /T1)0.5, or pLT1 TG 0.5 αG1T1 = εGT (5-144) 1 ᎏ ᎏ TG T1

 

  

The exponent 0.5 is an adequate average of the exponents for the pure components. The interpolation relation for absorptivity is TG 0.5 T1 − TL pLTH αG1T1 = εGT ᎏ H ᎏ ᎏ TG TH 500

 

     

pLTL + εGT L ᎏ TG



T T −T ᎏ  ᎏ T   500  G

0.5

H

1

(5-145)

L

The base temperature pair TH and TL can be different for evaluating εG and αG1 if TG and T1 are far enough apart. Example 6: Calculation of Gas Emissivity and Absorptivity This example will use only SI units, except that pressure will be in atm, not kPa. Flue gas containing 6 percent CO2 and 11 percent H2O vapor, wet basis, flows through a bank of tubes of 0.1016 in (4-in) outside diameter on equilateral 0.2032 m (8-in) triangular centers. In a section in which the gas and tube surface temperatures are 691°C (964 K) and 413°C (686 K), what are the emissivity and absorptivity of the gas? From Table 5-8, Lm = (2.8)(0.01016) = 0.2845 m (only SI units will be used in this example). p = pw + pc = 0.17 atm; pL = 0.0484 m atm, barely large enough to justify the short method, the top part of Table 5-8. pw /pc = 11/6, near enough to 2 to use col. 5. Since both TG and T1 are below the lowest T in the top part of the table, use the nearest pair, TH = 1500 K and TL = 1000 K. At TH, b = 540, n = 0.42. εGT H = 540(0.0484 − 0.015)0.42 = 129.5. At TL,

HEAT TRANSFER BY RADIATION

5-35

cles are small relative to the wavelength of the radiation of interest [diameters (2)(10−8) to (1.4)(10−7) m (200 to 1400 Å)], the monochromatic emissivity ελ depends on the total particle volume per unit volume of space fv regardless of particle size. It is given by ελ = 1 − e−Kf v L/λ

(5-146)

where L is the path length. Use of the perfect gas law and a material balance allows the restatement of the above to ελ = 1 − e−KPSL/λT

Correction factor for converting emissivity of water vapor to values of Pw and PT other than 0 to 1 atm respectively. To convert atmosphere-feet to kilopascal-meters, multiply by 30.89; to convert atmospheres to kilo-pascals, multiply by (1.0133)(102).

FIG. 5-21

(5-147)

where P is the total pressure (atm) and S is the mole fraction of soot in the gas. S depends on the fractional conversion fc of the fuel carbon to soot and is the mole fraction, wet basis, of carbon in gaseous form (CO2, CO, CH4, etc.) times fc /(1 − fc) or, with negligible error, times fc, which is a very small number. Evaluation of K is complex, and its numerical value depends somewhat on the age of the soot, the temperature at which it is formed, and its hydrogen content. It is recommended that K = 0.526 [K/atm]. The total emissivity of soot εs is obtained by integration over the wavelength spectrum, giving KPSL 15 εs = 1 − ᎏ ψ(3) 1 + ᎏ , (5-148) 4 c2 where ψ(3)(x) is the pentagamma function of x. It may be shown that an excellent approximation to Eq. (5-148) is

 



εs = 1 − [1 + 34.9SPL]−4 b = 444, n = 0.34, εGT L = 444(0.0484 − 0.015)0.34 = 139.8. From interpolation Eq. (5-143) (here extrapolation), εGT G = [129.5 (964 − 1000) + 139.5 (1500 − 964)]/500 = 140.2. For αG1T1 with T1 = TH = 1500, pLTH /TG = 0.0753. From Eq. (5-144),  αGHT αGLT H = 540 (0.0753 − 0.015)0.42(964/1500)0.5 = 133.1. Similarly,  L = 444 (0.0502 − 0.015)0.34 (964/1000)0.5 = 139.7. From Eq. (5-145) 133.1(686 − 1000) + 139.7(1500 − 686) αG1T1 = ᎏᎏᎏᎏᎏ = 143.8 500 Then εG = 140.2/964 = 0.145 and αG1 = 143.8/686 = 0.210. If the longer method (the bottom part of Table 5-8) were used, εG = 0.141 and αG1 = 0.206.

Other Gases Because of their practical importance, the emissivities of CO2 and H2O have been studied much more extensively than those of other gases, and the values summarized in the preceding paragraphs are based on extensive measurement of both total and integrated spectral values. Correction for pressure has reduced the disagreement among experimenters. A summary of the less adequate information on other gases appears in Table 5-9. Flames and Particle Clouds Luminous Flames Luminosity conventionally refers to soot radiation; it is important when combustion occurs under such conditions that the hydrocarbons in the flame are subject to heat in the absence of sufficient air well mixed on a molecular scale. Because soot parti-

TABLE 5-9

(5-149)

where PL is in atm m. The error is less the lower εs, and is only 0.5 percent at εs = 0.5 and 0.8 percent at 0.67. There is at present no method of predicting soot concentration of a luminous flame analytically; reliance must be placed on experimental measurement on flames similar to that of interest. Visual observation is misleading; a flame so bright as to hide the wall behind it may be far from a “black” radiator. The chemical kinetics and fluid mechanics of soot burnout have not progressed far enough to evaluate the soot fraction fc for relatively complex systems. Additionally, the soot in a combustion chamber is highly localized, and a mean value is needed for calculation of the radiative heat transfer performance of the chamber. On the basis of limited experience with fitting data to a model, the following procedure is recommended when total combustion chamber performance is being estimated: (1) When pitch, or a highly aromatic fuel, is burned, one percent of the fuel carbon appears as soot. This produces values of εs of 0.4–0.5 and εG + s of 0.6–0.7. These values are lower than some measurements on pitch flames, but the measurements are usually taken through the flame at points of high luminosity. (2) When No. 2 fuel oil is burned, 33 percent of the fuel carbon appears as soot (but that number varies greatly with burner design). (3) When natural gas is burned, any soot contribution to emissivity may be ignored. Admittedly, the numbers given should be functions of burner design and excess air, and they should be considered tentative, subject to change when good data show they are off target. The Inter-

Total Emissivities of Some Gases 1000°R

1600°R

2200°R

2800°R

Temperature Px L, (atm)(ft)

0.01

0.1

1.0

0.01

0.1

1.0

0.01

0.1

1.0

0.01

0.1

1.0

NH3a SO2b CH4c COd NOe HCl f

0.047 0.020 0.020 0.011 0.0046 0.00022

0.20 0.13 0.060 0.031 0.018 0.00079

0.61 0.28 0.15 0.061 0.060 0.0020

0.020 0.013 0.023 0.022 0.0046 0.00036

0.120 0.090 0.072 0.057 0.021 0.0013

0.44 0.32 0.194 0.10 0.070 0.0033

0.0057 0.0085 0.022 0.022 0.0019 0.00037

0.051 0.051 0.070 0.050 0.010 0.0014

0.25 0.27 0.185 0.080 0.040 0.0036

(0.001) 0.0058 0.019 (0.012) 0.00078 0.00029

(0.015) 0.043 0.059 (0.035) 0.004 0.0010

(0.14) 0.20 0.17 (0.050) 0.025 0.0027

NOTE: Figures in this table are taken from plots in Hottel and Sarofim, Radiative Transfer, McGraw-Hill, New York, 1967, chap. 6. Values in parentheses are extrapolated. To convert degrees Rankine to kelvins, multiply by (5.556)(10−1). To convert atmosphere-feet to kilopascal-meters, multiply by 30.89. a Total-radiation measurements of Port (Sc.D. thesis in chemical engineering, MIT, 1940) at 1-atm total pressure, L = 1.68 ft, T to 2000°R. b Calculations of Guerrieri (S.M. thesis in chemical engineering, MIT, 1932) from room-temperature absorption measurements of Coblentz (Investigations of Infrared Spectra, Carnegie Institution, Washington, 1905) with poor allowance for temperature. c Band measurements of Lee and Happel [Ind. Eng. Chem. Fundam., 3, 167 (1964)] at T up to 2050°R plus calculations to extrapolate temperature to 3800°R. d Total-radiation measurements of Ullrich (Sc.D. thesis in chemical engineering, MIT, 1953) at 1-atm total pressure, L = 1.68 ft, T to 2200°R. e Calculations of Malkmus and Thompson [J. Quant. Spectros. Radiat. Transfer, 2, 16 (1962)], to T = 5400°R and PL = 30 atm ⋅ ft. f Calculations of Malkmus and Thompson [J. Quant. Spectros. Radiat. Transfer, 2, 16 (1962)], to T = 5400°R and PL = 300 atm ⋅ ft.

5-36

HEAT AND MASS TRANSFER

where a/v is the projected area of the particles per unit volume of space. If the particles have no negative curvature (a particle can see none of itself) and are randomly oriented, a is a′/4, where a′ is the actual surface area; and if the particles are uniform, a/v = cA = cA′/4 where A and A′ are the projected and total areas of each particle and c is the number concentration of particles. For spherical particles, this gives εM = 1 − e−(π/4)cd L = 1 − e−1.5fvL/d (5-153)

Evaluation of αG1 is unnecessary when T1 is less than one-half TG; αG1 may then be assumed equal to εG. If the bounding surface is gray rather than black, multiplication of Eq. (5-154) by surface emissivity ε1 allows properly for reduction of the primary beams, gas-to-surface or surface-to-gas, but secondary reflections are ignored. The correction then lies between ε1 and 1, and for most industrially important surfaces with ε1 > 0.8 a value of (1 + ε1)/ 2 is adequate. Rigorous allowance for this and other factors is presented later, e.g., Eq. (5-163). If the bounding walls are mostly sink-type surfaces of area A1 and temperature T1, but in small part refractory surfaces of area Ar in radiative equilibrium at unknown temperature Tr, an energy balance on Ar is in principle necessary to determine Tr and the effect on energy flux. However, the total heat transfer to the sink may be visualized as corresponding to its having an effective area equal to its own plus a fraction x of that of the refractory, with the only temperatures involved being those of the gas and the heat sink. The fraction x varies from zero when the ratio of refractory to heat-sink surface is very high to unity when the ratio is very low and the value of εG is low. If Ar is small compared with A1, a value for x of 0.7 may be used in the approximate method. Long Exchanger This case, in which axial radiative flux is ignored, includes most radiatively modified heat exchangers of interest to chemical engineers. When the gas temperature transverse to the flow direction is reasonably uniform and the chamber is long compared with its mean hydraulic radius, the opposed upstream and downstream fluxes through the flow cross section will substantially cancel (hot combustion products through tubes or across tube banks, tunnel kilns, billet-reheating furnaces, Example 7). Under these conditions, the radiative contribution to local flux density q may be formulated in terms of local temperatures and beam lengths or exchange areas evaluated for a two-dimensional system infinite in the flow direction. The local flux density at the sink A1 is then

As an example, consider heavy fuel oil (CH1.5, specific gravity, 0.95) atomized to a surface mean particle diameter of d, burned with 20 percent excess air to produce coke-residue particles having the original drop diameter and suspended in combustion products at 1204°C (2200°F). The flame emissivity due to the particles along a path of L m will be, with d in micrometers, εM = 1 − e−24.3L/d (5-154)

where h is the local convective heat-transfer coefficient and qr(TG, T1) the radiation contribution calculated from TG, T1, εG, and ε1 by using the approximate treatment in the preceding subsection or the more rigorous treatment in the following subsection. If m ˙ Cp is the hourly heat capacity of the gas stream, the temperature of which changes by dTG over the sink-area increment dA1, then

national Flame Foundation has recorded data on many luminous flames from gas, oil, and coal (see J. Inst. Energy, formerly J. Inst. Fuel, 1956 to present). Combined Soot, H2O, and CO2 Radiation The spectral overlap of H2O and CO2 radiation has been taken into account by the constants for obtaining εG. Additional overlap occurs when soot emissivity εs is added. If the emission bands of water vapor and CO2 were randomly placed in the spectrum and soot radiation were gray, the combined emissivity would be εG plus εs minus an overlap correction εGεs. But monochromatic soot emissivity is higher the shorter the wavelength, and in a highly sooted flame at 1500 K half the soot emission lies below 2.5 µm where H2O and CO2 emission is negligible. Then the correction εGεs must be reduced, and the following is recommended: εG + s = εG + εs − MεGεs (5-150) where M depends mostly on TG and to a much less extent on optical density SPL. Values that have been calculated from this simple model can be represented with acceptable error by T M = 1.07 + 18 SPL − 0.27 ᎏ (5-151) 1000 Clouds of Large Black Particles The emissivity εM of a cloud of particles with a perimeter large compared with wavelength λ is εM = 1 − e−(a/v)L (5-152)





2

With 200-µm particles and an L of 3.05 m (10 ft), the particle contribution to emissivity will be 0.31. Soot luminosity will increase this; particle burnout will decrease it. Clouds of Nonblack Particles The correction for nonblackness of the particles is complicated by multiple scatter of the radiation reflected by each particle. The emissivity εM of a cloud of gray particles of individual surface emissivity ε1 can be estimated by the use of Eq. (5-151), with its exponent multiplied by ε1, if the optical thickness (a/v)L does not exceed about 2. Modified Eq. (5-151) would predict an approach of εM to 1 as L → ∞, an impossibility in a scattering system; the asymptotic value of εM can be read from Fig. 5-14 as εh, with albedo ω given by particle-surface reflectance 1 − ε1. Particles with a perimeter lying between 0.5 and 5 times the wavelength of interest can be handled with difficulty by use of the Mie equations (see Hottel and Sarofim, op. cit., chaps. 12 and 13). Summation of Separate Contributions to Gas or Flame Emissivity Flame emissivity εG + s due to joint emission from gas and soot has already been treated. If massive-particle emissivity εM, such as from fly ash, coal char, or carbonaceous cenospheres from heavy fuel oil, are present, it is recommended that the total emissivity be approximated by εG + s + εM − (εG + s)(εM) RADIATIVE EXCHANGE BETWEEN GASES OR SUSPENDED MATTER AND A BOUNDARY ˙ between an Local Radiative Exchange The interchange rate Q isothermal gas mass at TG and its isothermal black bounding surface of area A1 is given by ˙ = A1σ(TG4 εG − T 41 αG1) Q (5-155)

q(TG, T1) = qr(TG, T1) + h(TG − T1)

˙ Cp dTG [q(TG, T1)]dA1 = −m from which

(5-156)

(5-157)



TG,inlet

Cp dTG (5-158) ᎏ q(TG, T1) The area under a curve of Cp /q versus TG or 1/q versus the specific enthalpy i may be used to solve for the area A1 required to obtain a given outlet temperature or to obtain the outlet temperature given A1. Three points generally suffice to determine the area under the curve within 10 percent. Instead of using graphical integration, which can handle any complexity of variation of flux density q with TG and T1 along an interchanger flow path, one may evaluate a mean flux density based on mean gas and sink temperatures, based in turn on terminal temperatures. It has been found empirically that fair results are obtained by the use of a mean surface temperature equal to the arithmetic mean of the terminal surface temperatures and by the use of a mean gas temperature equal to the mean surface temperature plus the logarithmic mean of the temperature difference, gas to surface, at the two ends of the exchanger. When radiation dominates the transfer process, however, graphical integration is safer. ˙ A1 = m

TG,outlet

Example 7: Radiation in Gases Flue gas containing 6 percent carbon dioxide and 11 percent water vapor by volume (wet basis) flows through the convection bank of an oil tube still consisting of rows of 0.102-m (4-in) tubes on 0.203-m (8-in) centers, nine 7.62-m (25-ft) tubes in a row, the rows staggered to put the tubes on equilateral triangular centers. The flue gas enters at 871°C (1144 K, 1600°F) and leaves at 538°C (811 K, 1000°F). The oil flows in a countercurrent direction to the gas and rises from 316 to 427°C (600 to 800°F). Tube surface emissivity is 0.8. What is the average heat-input rate, due to gas radiation alone, per square meter of external tube area?

HEAT TRANSFER BY RADIATION With each row of tubes there is associated (0.203)(3/2) = 0.176 m (0.577 ft) of wall height, of area [(0.203)(9)(2) + (7.62)(2)]0.176 − (9)(2)(π)(0.0508)2 = 3.18 m2 (34.2 ft2). One row of tubes has an area of (π)(0.102)(7.62)(9) = 22.0 m2 (236 ft2). If the recommended factor of 0.7 on the refractory area is used, the effective area of the tubes is [22.0 + (0.7)(3.18)]/22.0 = 1.10 m2/m2 of actual area. The exact evaluation of the outside tube temperature from the known oil temperature would involve a knowledge of the oil-film coefficient, tube-wall resistance, and rate of heat flow into the tube, the evaluation usually involving trial and error. However, for the present purpose the temperature drop through the tube wall and oil film will be assumed to be 41.7°C (75°F), making the tube surface temperatures 357°C (675°F) and 468°C (875°F); the average is 412°C (775°F). The radiating gas temperature is (871 − 468) − (538 − 357) tg = 412 + ᎏᎏᎏᎏ 2.3 log [(871 − 468)/(538 − 357)] = 412 + 278 = 690°C (1274°F) These temperatures, partial pressures, and dimensions were used in Example 6 to determine gas emissivity and absorptivity, εG = 0.145; αG1 = 0.210. The approximate effective emissivity of the boundary is (0.8 + 1)/2 = 0.9. Then from Eq. (5-155), modified to allow for sink emissivity and for the presence of a small amount of refractory boundary, ˙ 1 = q = (0.9)(1.10)σ(TG4 εG − T41αG) Q/A = (0.9)(1.10)(5.67)[(9.63)4(0.145) − (6.85)4(0.210)] = 4405 J/(m2 tube area⋅s) [1396, Btu/(ft2 tube area⋅h)] This is equivalent to a convection coefficient of 4405/278, or 15.85 W/(m2)(K) which is of the order of magnitude expected of the convection coefficient itself. Radiation rapidly becomes dominant as the system temperature rises.

Total-Exchange Areas  SS GS  and   The arguments leading to the development of the interchange factor AiᏲij ( S iS j) between surfaces apply to the case of absorption within the gas volume if in the evaluation of the direct-exchange areas allowance is made for attenuation of the radiant beam through the gas. This necessitates nothing more than redefinition, in Eqs. (5-126) to (5-130), of every term ij( Si S j AiFij) to represent, per unit black emissive power, flux from Ai through an absorbing gas to Aj. This may be visualized as multiplication of AiFij by the mean gas transmittance Tij(= 1 − εG for a gray gas). In a system containing an isothermal gas and source-sink boundaries of areas A1 . . . An, the total emission from A1 per unit of its black emissive power is ε1A1, of which  S1S 1 + S 1S 2 + ⋅⋅⋅ + S 1S n is absorbed in the various source-sink surfaces by multiple reflections. The difference has been absorbed in the gas and is called the gas-surface totalexchange area  GS 1 G S1 = A1ε1 −
S  1S i

(5-159)

i

Note that though S 1S i is never used in calculating radiative exchange, its value is necessary for use of Eq. (5-159) to calculate  GS . If the gas volume is not isothermal and is zoned, an additional magnitude, the gas-to-gas total-exchange area  GiG j, arises (see Hottel and Sarofim. Radiative Transfer, McGraw-Hill, New York, 1967, chap. 11). Space does not permit derivations of special cases; only the single-gas-zone system is treated here. Single-Gas-Zone/Two-Surface-Zone Systems An enclosure consisting of but one isothermal gas zone and two gray surface zones can, properly specified, model so many industrially important radiation problems as to merit detailed presentation. One can evaluate the total radiation flux between any two of the three zones, including multiple reflection at all surfaces. ˙ G⇔1 =  Q GS 1σ(TG4 − T 41) ˙ 1⇔2 =  Q S1S 2σ(T14 − T 42 ) The total-exchange area takes a relatively simple closed form, even when important allowance is made for gas radiation not being gray and when a reduction of the number of system parameters is introduced by assuming that one of the surface zones, if refractory, is radiatively adiabatic. Before allowance is made for these factors, the case of a gray gas enclosed by two source-sink surface zones will be presented. Modification of Eq. 5-130, as discussed in the first paragraph of this subsection, combined with the assumption that a single mean beam length applies to all transfers; that is, that there is but one gas transmittance τ(= 1 − εG), gives

5-37

A1ε1ε2F12 S (5-160) 1S 2 = ᎏᎏᎏᎏᎏ 1/τ + τρ1ρ2(1 − F12 /C2) − ρ1(1 − F12) − ρ2(1 − F21) A1ε12(F11 + ρ2τ(F12 /C2 − 1) (5-161) S 1S 1 = ᎏᎏᎏᎏᎏ 1/τ + τρ1ρ2(1 − F12 /C2) − ρ1(1 − F12) − ρ2(1 − F21) A1ε1εG(1/τ + ρ2(F12 /C2 − 1)) (5-162) G S 1 = ᎏᎏᎏᎏᎏ 1/τ + τρ1ρ2(1 − F12 /C2) − ρ1(1 − F12) − ρ2(1 − F21) These three expressions suffice to formulate total-exchange areas for gas-enclosing arrangements which include, for example, the four cases illustrated in Table 5-10. An additional surface arrangement of importance is a single-zone surface enclosing gas. With the gas assumed gray, the simplest derivation of G S 1 is to note that the emission from surface A1 per unit of its blackbody emissive power is A1ε1, of which the fractions εG and (1 − εG)ε1 are absorbed by the gas and the surface, respectively, and the surface-reflected residue always repeats this distribution. Therefore, εG A1 G GS (5-163) S single surface  1 = A1ε1 ᎏᎏ = ᎏᎏ zone surround1 1 εG + (1 − εG)ε1 ing gray gas ᎏᎏ + ᎏᎏ − 1 εG ε1 Alternatively, G S1 could be obtained from Case 1 of Table 5-10 by  letting plane area A1 approach 0, leaving A2 as the sole surface zone. Departure of gas from grayness has so marked an effect on radiative transfer that the subject will be presented prior to discussion of the systems covered by Table 5-10. The Effect of Nongrayness of Gas on Total-Exchange Area A radiating gas departs from grayness in two ways: (1) Its transmittance τ through successive path lengths Lm due to surface reflection, instead of being constant, keeps increasing because at the wavelengths of high absorption the incremental absorption decreases with increasing pathlength; (2) Gas emissivity εG and absorptivity αG1 are not the same unless T1 equals TG. The total emissivity of a real gas, the spectral emissivity, and absorptivity ελ that varies in any way with λ can be expressed as the a-weighted mean of a suitable number of gray-gas emissivity or absorptivity terms εG,i or αG,i, representing the gray-gas emissivity or absorptivity in the energy fractions ai of the blackbody spectrum. Then n

n

0

0

εG =
aiεG,i =
ai(1 − e−k i pL)

(5-164)

For simplicity, n should be as low as is consistent with small error. The retention of but two terms is feasible when one considers that if αG1 is so fitted that the first absorption and the second following surface reflection are correct, then further attenuation of the beam by successive surface reflections makes the errors in those absorptions decrease in importance. Let the gas be modeled as the sum of one gray gas plus a clear gas, with the gray gas occupying the energy fraction a of the blackbody spectrum and the clear gas the fraction (1 − a). Then [εG(pL)] = a(1 − e−kpL) + (1 − a)0 [εG(2pL)] = a(1 − e−2kpL) + (1 − a)0

(5-165)

Solution of these gives εG(pL) a = ᎏᎏ εG(2pL) 2−ᎏ ᎏ εG(pL) εG(pL) kpL = −ln 1 − ᎏ a





(5-166)

Note that these values are specific to the subject problem in which the mean beam length is Lm, with εGs evaluated from basic data, such as Table 5-8. (1 − e−kpL) in Eq. (5-165) represents the emissivity of a gray gas, which will be called εG,i. For later use, note that, εG(pL) εG,i = ᎏ (5-167) a To allow for the difference between emissivity and absorptivity and combine them into a single emissivity-absorptivity term called effec-

5-38

HEAT AND MASS TRANSFER

TABLE 5-10

Total-Exchange Areas for Four Arrangements of Two-Zone-Surface Enclosures of a Gray Gas

A plane surface A1 and a surface A2 completing the enclosure

Infinite parallel planes

Concentric spherical or infinite cylindrical surface zones, A1 inside

Two-surface-zone spherical surface, each zone one or more parts; or speckled enclosure, any shape

F12 = 1

F12 = F21 = 1

F12 = 1; F21 = A1/A2

F12 = F22 = C2; F21 = F11 = C1

S 1S 2 ε1ε2 ᎏ=ᎏ A1 D1

S 1S 2 ε1ε2 ᎏ=ᎏ A1 D2

S 1S 2 ε1ε2 ᎏ=ᎏ A1 D3

S 1S 2 ε1ε2C2 ᎏ=ᎏ A1 D4

G S 1 ε1εG(1/τ + ρ2A1/A2) ᎏ = ᎏᎏ A1 D1

G S 1 ε1εG(1/τ + ρ2) ᎏ = ᎏᎏ A1 D2

G S 1 ε1εG(1/τ + ρ2A1/A2) ᎏ = ᎏᎏ A1 D3

G S 1 ε1εG/τ ᎏ=ᎏ A1 D4

G S 2 ε2εG(1/τ + ρ1A1/A2) ᎏ = ᎏᎏ A2 D1

S 1S 1 ε 12 ρ2τ ᎏ=ᎏ A1 D2

G S 2 ε2εG(1/τ + ρ1A1/A2) ᎏ = ᎏᎏ A2 D3

S 1S 1 ε 21 C1 ᎏ=ᎏ A1 D4

S 1S 1 ε 12 τρ2A1/A2 ᎏ = ᎏᎏ A1 D1

D2 1/τ − τρ1ρ2

S 1S 1 ε 12 ρ2τA1/A2 ᎏ = ᎏᎏ A1 D3

D4 1/τ − ρ1C1 − ρ2C2









1 A1 D1 ᎏ − ρ2 1 − ᎏ (1 − τρ1) τ A2

1 A1 D3 ᎏ − ρ2 1 − ᎏ (1 − τρ1) τ A2

tive emissivity εG,e, one must first evaluate absorptivity αG1 using Eq. (5-161). Formulation of the net direct exchange can then be used to define εG,e: σ(εGTG4 − αG1T 41) σεG,e(T G4 − T 41) εG,e − αG1(T41/ TG4 ) (5-168) or εG,e = ᎏᎏ 1 − (T1 / TG)4 The emissivity and absorptivity of use in converting gray-gas totalexchange areas to real-gas values are εG,e and ae, the latter obtained by using Eq. (5-166), except that εG,e (pL) replaces εG(pL); the same for εG,e (2pL). This means that, for the conversion, four terms will have to be formulated: εG(pL), εG(2pL), αG1(pL), and αG1(2pL). The gray emissivity term εG,i of Eq. (5-167) now becomes εG,e /ae. Conversion of gray-gas total exchange areas  GS SS  and   to their nongray form depends on the fact that the relation between radiative transfer and blackbody emissive power σT 4 is linear and proportional. The gray-gas-equivalent emissivity εG,e /ae is applicable only to the energy fraction ae of σT 4. In consequence, to convert  GS SS  or   to its nongray form, wherever εG or τ appears in G S  it must be replaced by εG,e /ae or (1 − εG,e /ae), respectively; the overall result is then multiplied by ae. The converted S S  is in two parts: The gray-gas contribution involves, as above, replacement of εG by εG,e /ae and τ by (1 − εG,e /ae), and the result multiplied by ae; for the clear-gas contribution, εG is replaced by 0 and τ by 1, and the result is multiplied by (1 − ae) and added to the gray-gas contribution. The simplest application of this simple gray-plus-clear model of gas radiation is the case of a single gas zone surrounded by a single surface zone, Eq. (5-163) for a gray gas. The gray-plus-clear model gives ae GS  1 (5-169) ᎏ = ᎏᎏ ae 1 A1 ᎏᎏ + ᎏᎏ − 1 εG,e ε1

at 1000 K. The surrounding gas is at 1500 K and is well stirred. Find the effective gas emissivity εG,e, the weighting factor ae, and the surface radiative flux density. Solution: Combustion is 1CH4 + 2 × 1.2 O2 + 2 × 1.2 × 79/21 N2 + 2 × 1.2 × 100/21 × 0.0088 H2O, going to 1 CO2 + [2 + 2 × 1.2 × (100/21) × 0.0088] H2O + 0.4 O2 + 9.03 N2 = 12.53 moles per mole of CH4; pc + pw = (1 + 2.1)/12.53 = 0.2474 atm. The mean beam length, Lm = 0.88 × 4V/AT = 0.88 × 4(10 × 3 × 5)/ [2 × (10 × 3 + 10 × 5 + 3 × 5)] = 2.779 m. pLm = 0.2474 × 2.779 = 0. 6875 m atm. From emissivity Table 5-8, b(1500) = 540; n(1500) = 0.42; b(1000) = 444; n(1000) = 0.34. εG(pL) = 540(0.6875 − 0.015)0.42/1500 = 0.3047; αG1(pL) = 444(0.6875 × 1000/1500 − 0.015)0.34 (1500/1000)0.5/1000 = 0.4124. Then εG,e(pL) = [0.3047 − 0.4124(1000/1500)4]/[1 − (1000/1500)4] = 0.2782. Repeat all 3 computations for pL = 2 × 0.6875 to give εG(2pL) = 0.4096, αG1 (2pL) = 0.5250, εG,e(2pL) = 0.3812. Then ae = 0.2782/(2 − 0.3812/0.2782) = 0.4418 and the emissivity substitute = 0.2782/0.4418 = 0.6297. For a single enveloping surface zone, the total-exchange area comes from Eq. (5-169); G S 1/A1 = ae/(ae/εG,e + 1/ε1 − 1) = ˙ 0.4418/(0.4418/0.2782 + 1/0.8 − 1) = 0.2404. The flux density is Q/A = q = 4 4 (G S 1/A)σ(TG − T 1) = 0.2404 × 56.7 × [(1500/1000)4 − (1000/1000)4] = 55.37 2 kW/m (17,550 Btu/sq ft hr). (Note that allowing for average humidity in air adds 5 percent to H2O and approximately 2 percent to gas emissivity.)

Example 8: Effective Gas Emissivity Methane is burned to completion with 20 percent excess air (air half-saturated with water vapor at 298 K (60°F), 0.0088 mols H2O/mol dry air) in a furnace chamber of floor dimensions 3 × 10 m and height 5 m. The whole surface is a gray-energy sink of emissivity 0.8

Total-exchange areas for the basic one-gas two-surface model [Eqs. (5-160) to (5-162)], used to evaluate the cases in Table 5-10, take the following form when converted by the above described procedure to their nongray form: ae A1F12ε1ε2 S 1S 2 = ᎏᎏᎏᎏᎏ 1/(1 − εG,e /ae) + (1 − εG,e /ae)ρ1ρ2(1 − F12C2) − ρ1(1 − F12) − ρ2(1 − F21) (1 − ae)A1F12ε1ε2 + ᎏᎏᎏᎏᎏ 1 + ρ1ρ2(1 − F12C2) − ρ1(1 − F12) − ρ2(1 − F21) A1ε1εG,e(1/(1 − εG,e /ae) + ρ2(F12 /C2 − 1) G S 1 = ᎏᎏᎏᎏᎏ 1/(1 − εG,e /ae) + (1 − εG,e /ae)ρ1ρ2(1 − F12C2) − ρ1(1 − F12) − ρ2(1 − F21)

(5-170) (5-171)

Modification of Table 5-10 to make the total-exchange areas conform to the gray-plus-clear gas model is straightforward, following the instructions presented above. The results are given in Table 5-11.

HEAT TRANSFER BY RADIATION TABLE 5-11

5-39

Conversion of Some Total-Exchange Areas to Their Gray-Plus-Clear Values

1. Plane slab A1 and surface A2, completing an enclosure of gas (F12 = 1) 1S S 2 aε1ε2 (1 − a)ε1ε2 ᎏ=ᎏ + ᎏᎏ A1 D1 1 − ρ2(1 − ε1A1/A2) ρ1εG

− ρ 1 −  ᎏ ε + ᎏ  D = 1ᎏ − ε /a A a 1

A1

1

2

1

G

2

S G 1 ε1εG(1/(1 − εG /a) + ρ2A1 /A2) ᎏ = ᎏᎏᎏ A1 D1 S G 2 ε2εG(1/(1 − εG /a) + ρ1A1/A2) ᎏ = ᎏᎏᎏ A2 D1 2. Infinite parallel planes, gas between (F12 = F21 = 1) S S aε1ε2 (1 − a)ε1ε2 1 2 ᎏ=ᎏ + ᎏᎏ A1 D2 1 − ρ1ρ2 εG

− 1 − ᎏ ρ ρ D = ᎏ 1 − ε /a  a 1

2

1 2

G

S G 1 ε1εG(1/(1 − εG /a) + ρ2) ᎏ = ᎏᎏᎏ A1 D2 3. Concentric spherical or infinite cylindrical surface zones, A1 inside (F12 = 1; F21 = A1/A2) 1S S 2 aε1ε2 (1 − a)ε1ε2 ᎏ = ᎏ + ᎏᎏ A1 D3 1 − ρ2(1 − ε1A1/A2) 1 ρ1εG A D3 = ᎏᎏ − ρ2 1 − ᎏ1 ε1 + ᎏ (1 − εG /a) A2 a

  



S G 1 ε1εG(1/(1 − εG /a) + ρ2A1/A2) ᎏ = ᎏᎏᎏ A1 D3 S G 2 ε2εG(1/(1 − εG /a) + ρ1A1/A2) ᎏ = ᎏᎏᎏ A2 D3 4. Spherical enclosure of two surface zones or “speckled” A1:A2 enclosure (F12 = F22 = C2; F21 = F11 = C1) S S aε1ε2C2 (1 − a)ε1ε2C2 1 2 ᎏ = ᎏ + ᎏᎏ A1 D4 1 − ρ1C1 − ρ2C2 −ρC −ρC D = ᎏᎏ (1 − ε /a) 1

4

1

1

2

2

G

C = Aᎏ +A A1

1

1

2

GS  1 ε1εG /(1 − εG /a) ᎏ = ᎏᎏ A1 D4

Treatment of Refractory Walls Partially Enclosing a Radiating Gas Another modification of the results in Table 5-10 becomes important when one of the surface zones is radiatively adiabatic; the need to find its temperature can be eliminated. If surface A2, now called Ar, is radiatively adiabatic, its net radiative exchange with A1 must equal its net exchange with the gas.

or

4 4 G SS S r(TG4 − T 4r ) =  r(T 1 r − T 1) TG4 − T 4r T 4r − T 41 T G4 − T 41 ᎏ = ᎏ = ᎏᎏ 1/G 1/S 1/G S r rS 1 S r + 1/S rS 1

(5-172)

GS The net flux from gas G is  GS 1σ(TG4 − T14) +  rσ(TG4 − Tr4) which, with replacement of the last term using Eq. (5-172), gives the single term



1 σ(TG4 − T14) G S1 + ᎏᎏ  1 1 ᎏᎏ + ᎏᎏ G Sr S  rS 1



The bracketed term is called (G S 1)R, the total exchange area from G to A1 with assistance from a refractory surface. In summary,





1 ˙ G ⇔ 1 = (G S 1 + ᎏᎏ σ(TG4 − T14) Q S 1)Rσ(TG4 − T 14) = G 1 1 ᎏᎏ + ᎏᎏ G Sr S  rS 1

(5-173)

Table 5-10 supplies the forms for the three terms needed to formulate (G S 1)R, with Ar substituted for A2. If, in addition, allowance is to be made for the gas not being gray, εG,e and ae are evaluated using values of the emissivity and absorptivity calculated using Table 5-8, and the procedure described in the previous subsection is followed with εG,e /ae replacing εG together with the addition of a clear-gas contribution, when S S  is at issue. It is tempting to say that a surface A2 (or Ar) could be made radiatively adiabatic simply by assigning its reflectance ρ a value of 1, making the terms in the brackets of Eq. (5-173) much easier to evaluate and the result much simpler. This is valid only if the gas is gray. If it is not, Ar is a net absorber of radiation

5-40

HEAT AND MASS TRANSFER

in the spectral energy fraction a (or ae) and a net emitter in the cleargas fraction (1 − a). Conversion of  GS  to (G S 1)R will be carried out for two of the four cases of Table 5-10. Case 1 is an idealization of a metal-heating slab furnace or glass furnace, with its plane sink A1 combining with refractory surface Ar to complete the enclosure. With insertion into Eq. (5173) of G SrS S 1, G S r, and  1 after converting each to its gray plus clear form, one obtains



(G C1 1 S 1)R εG ᎏ = ᎏ ε1 ρ2 ᎏ + ᎏ A1 D1 Cr 1 − ᎏεGᎏ a

+



εr εr + ρrε1(C1/Cr) 1 ᎏᎏᎏ + ρ1 + (Cr /C1)[1 − (εG/a)] ε1 C1 ᎏ a εr + ρrε1 ᎏ + (1 − a)D1 εG Cr







(5-174) where D1 = 1/(1 − εG/a) − ρr{1 − (C1/Cr)[1 − ρ1(1 − εG/a)]}. Conversion of (G S ) 1 R to applicability to a gray gas comes by making a equal 1, producing the enormous simplification to  ᎏ A (GS1)R 1

gray gas

1 = ᎏᎏᎏ 1 + C1(1/εG − 2) ρ1 ᎏᎏ + ᎏᎏ 1 − C1εG ε1

(5-175)

Note that the emissivity and reflectance of the refractory are without effect on (G S 1)R if the gas is gray. The second conversion of G S ) S  to (G 1 R will be Case 4 of Table 5-10, the two-surface-zone enclosure with computation simplified by assuming that the direct-view factor from any spot to a surface equals the fraction of the whole enclosure that the surface occupies (the speckled-furnace model). This case can be considered an idealization of many processing furnaces such as distilling and cracking coil furnaces, with parts of the enclosure tube-covered and part left refractory. (But the refractory under the tubes is not to be classified as part of the refractory zone.) Again, one starts with substitution into Eq. (5-173) of the terms G SrS S 1, G S r, and  1 from Table 5-10, Case 4, with all terms first converted to their gray-plus-clear form. To indicate the procedure, one of the components,  SrS 1, wil be formulated. Sr S1  Crε1εr Crε1εr ᎏ = a ᎏ + (1 − a) ᎏᎏ A1 D4 1 − ρ1C1 − ρrCr Crε1εr εG(1 − a)(a − εG) = ᎏ 1 + ᎏᎏ D4 1 − ρ1C1 − ρrCr With D4 = 1/(1 − εG/a) − ρ1C1 − ρrCr, the result of the full substitution simplifies to





1 (G S 1)R ᎏ = ᎏᎏᎏᎏ A1 1 1 1 1/a − 1 C1 ᎏᎏ − ᎏᎏ + ᎏᎏ + ᎏ ᎏ εG a ε1 ε1 + εr(Cr /C1)





For a gray gas (a = 1), the above becomes 1 (G S 1)R ᎏ = ᎏᎏ 1 1 A1 C1 ᎏᎏ − 1 + ᎏᎏ εG ε1





(5-176)

(5-177)

Eq. (5-176) has wide applicability. COMBUSTION CHAMBER HEAT TRANSFER Treatment of radiative transfer in combustion chambers is available at varying levels of complexity, including allowance for temperature variation in both gas and refractory walls (Hottel and Sarofim,

Radiative Transfer, McGraw-Hill, New York, 1967, chap. 14). A less rigorous treatment suffices, however, for handling many problems. There are two limiting cases: the long chamber with gas temperature varying only in the direction of gas flow (already treated) and the compact chamber containing a gas or a flame to which can be assigned an effective or average radiating temperature. The latter will be considered. Stirred-Chamber Model; Refractory Wall Loss Negligible What furnace engineers most need is a closed-form solution of the problem, theoretically sound in structure and therefore containing a minimum number of parameters and no empirical constants and, preferably, physically visualizable. They can then (1) correlate data on existing furnaces, (2) develop a performance equation for standard design, or (3) estimate performance of a new furnace type on which no data are available. An equation representing an energy balance on a combustion chamber of two surface zones, a heat sink A1 at temperature T1, and a refractory surface Ar assumed radiatively adiabatic at Tr, is most simply solved if the total enthalpy input H is expressed as mC ˙ p(TF − To); m ˙ is the mass rate of fuel plus air; and TF is a pseudoadiabatic flame temperature based on a mean specific heat from base temperature To up to the gas exit temperature TE rather than up to TF. The heat transfer rate Q˙ out of the gas is then H˙ − mC ˙ p(TE − To) or mC ˙ p(TF − TE). The energy balance, with ambient temperature taken as conventional base To, is ˙ − mC (Q˙ =)H ˙ p(TE − To) = (G S 1)R σ(TG4 − T 14 ) + h1A1(TG − T1) + AoFoσ(TG4 − To4) + UAr(TG − To)

(5-178)

S 1)RσTF4 To make the relation dimensionless, divide through by (G and let all temperatures, expressed as ratios to TF, be called τs. For clarity, the terms are tabulated: mC ˙ p/(G S 1)RσTF3 = D, dimensionless firing density l.h.s. term = D(1 − τE) 1st r.h.s. term = τ G4 − τ 14 (h1A1/(G GS  1)R σTF3) = Lc, convection number (dimensionless) AoFo /(G S 1)R = Lo, wall-openings loss number, (dimensionless) UAr /(G S 1)RσTF3 = Lr, refractory-wall loss number (dimensionless) The equation then becomes D(1 − τE) = τG4 − τ14 + Lc(τG − τ1) + Lo(τG4 − τo4) + Lr(τG − τo)

(5-179)

This equation has two unknowns (τG and τE), and an empirical relation between them is needed. Many have been tried, and one of the best is to assume that the excess of TG over TE expressed as a ratio to TF (zero for a perfectly stirred chamber) is a constant ∆ [ (TG − TE)/ TF]. Although ∆ should vary with burner type, the effects of firing rate and percent excess air are small. In the absence of performance data on the kind of furnace under study, assume ∆ = 300/TF, °R or 170/TF, K. The left side of Eq. (5-178) then becomes D(1 − τG + ∆), and with coefficients of τG and τ G4 collected, the equation becomes D + Lc + Lr τ14 + Lcτ1 + Loτ 4o + Lrτo + D(1 + ∆) τG4 + ᎏᎏ τG − ᎏᎏᎏᎏ = 0 1 + Lo 1 + Lo (5-180)



 



Though this is a quartic equation, it is capable of explicit solution because of the absence of second and third degree terms. Trial-anderror enters, however, because (G S 1)R and C p are mild functions of TG and related TE, respectively, and a preliminary guess of TG is necessary. An ambiguity can exist in interpretation of terms. If part of the enclosure surface consists of screen tubes over the chamber-gas exit to a convection section, radiative transfer to those tubes is included in the chamber energy balance, but convection is not, because it has no effect on chamber gas temperature. With Eq. (5-180) solved, the gas-side efficiency ηG is (1 − τG + ∆)/ (1 − τo). The sink-side efficiency η1 is less by the amount (Lo(τG4 − τL4) + Lr(τG − τo))/D(1 − τo) and is also given by [(G GS  1)Rσ(TG4 − T 41) + h1A1(TG − T1)]/H˙. It must be remembered that the efficiency η1

HEAT TRANSFER BY RADIATION includes the losses through the wall from the backside of any wall–mounted heat sinks. Though the results must be considered approximations, depending as they do on the empirical ∆, the equation may be used to find the effect of firing rate, excess air, and air preheat on efficiency. With some performance data available, the small effect of various factors on ∆ may be found. The first term on the right side of Eq. (5-179) is so nearly dominant for most furnaces that consideration of the main features of chamber performance is clarified by ignoring the loss terms Lo and Lr or by assuming that they and Lc have a constant mean value. The relation of a modified chamber efficiency ηG(1 − τo) to a modified firing density D/(1 − τo) and to the normalized sink temperature τ = T1 /TF is shown in Fig. 5-23, which is based on Eq. (5-178), with the radiative and convective transfer terms (G S 1)Rσ(TG4 − T 41) + h1A1(TG − T1) replaced by a combined radiation/conduction term (G GS  1)R,cσ(TG4 − T 41), where 3 (G GS GS ; TG1 is adequately approximated by the  1)R,c = (G  1)R + h1A1/4σTG1 arithmetic mean of TG and T1. Example 9: Radiation in a Furnace Consider a furnace 3 m × 10 m × 5 m fired with methane and 20 percent excess air, at a methane firing rate of 2500 kg/hr. Two rows of 5-inch (0.127 m) tubes (outer diameter) are mounted on equilateral triangular centers, with center-to-center distance twice the tube diameter, on 60 percent of the interior surface of the chamber. The radiative properties of the gases for an enclosure of these dimensions, containing the same combustion products, have been estimated in Example 8 for a gas temperature of 1500 K and a sink temperature of 1000 K: 12.53 moles of combustion products are generated per mole of fuel, with a mean molar heat capacity between a base temperature of 298 K and the exit gas exit temperature TE, adequately represented for this example by M C p = 7.01 + 0.875 (T/1000) over a TE range of 800 to 1600 K. The lower heating value of CH4 is 191,760 cal/g mole. The air is preheated to 600°C, and has a mean  MC p of 7.31 cal/g mole. The alloy tube emissivity ε1 is 0.7 and may be assumed gray; the mean tube surface temperature is 700°C. The convection coefficient, gas-to-tube plane and to refractory surface is 0.0170 kW/m2 °C; hc + r on the outside surface is 0.0114 kW/m2 °C. The 0.343-m-thick refractory walls and roof have a k of 0.00050 kW/m °C and an assumed εr of 0.6; the walls are pierced by four 0.10-m × 0.23-m peepholes. The gas exit area, 1 m × 10 m, is tube-screen-covered. What is the sink-side efficiency η1, the gas exit temperature TE, and the mean flux density through the tube surface? Solution: Temporary basis—1 mole entering CH4. Since no molal change occurs when CH4 burns completely with half-saturated entering air 20 percent in excess of stoichiometric, the total number of moles produced equal 11.53 moles. Entering enthalpy = 191,760 + 11.53 × 7.31(600 − 25) = 240,220 Kcal/kg mol CH4. H˙ [of Eq. (5-178)] = 240,220 × (2500/16.04) = 37.44E6 Kcal/hr × 4.186/3600 = 43.54E3 Kw. m ˙ = [16.04 + 2 × 1.2(100/21)(29 + 0.0088 × 18.016)](2500/16.04) = 54,440 kg/hr. Trial and error solution necessitates several sets of computations of TG to check assumed TE; only the last of these will be given. The first, to save time by using results attained elsewhere, assumes TG = 1500 K; the resulting TG is 369 K higher. The second set assumes TG = 2017 K; the resulting TG is 80 K lower. Linear interpolation indicates the third set should assume TG = 1934 K. That set is presented: TE = 1934 − 170 = 1764.  M Cp = 7.01 + 0.875 × (1764/1000) = 8.554 cal/(gmol)(K). TF = 240,220/(12.53 × 8.554) +

5-41

298 = 2539 K.  Cp = (8.554/16.04)(4.186/3600) = 0.6201E-3 kw-hr/(kg)(K). For εG, pLm = 0.247 × 2.779 = 0.6875 m atm. Use of Table 5-8, with TH = 2000 K (b = 572, n = 0.51) and TL = 1500 (b = 540, n = 0.42) gives εG(pL) = [572(0.6875 − 0.015)0.51(1934 − 1500) + 540(0.6725)0.42(2000 − 1934)]/(500 × 1934) = 0.2409. For αG1, with T1 = 1000, Table 5-8 gives b = 444, n = 0.34. αG1(pL) = 444(0.6875 × 1000/1934 − 0.015)0.34(1934/1000)0.5/1000 = 0.4281. εG,e(pL) = [0.2409 − 0.4281(1000/1934)4]/(1 − 1/1.9344) = 0.2265. 2pLm = 2 × 0.6875 = 1.375 m atm. εG(2pL) = [572(1.375 − 0.015)0.51 × 434 + 540(1.36)0.42 × 66]/(500 × 1934) = 0.3422. αG1(2pL) = 444 × (1.375 × 1000/1934 − 0.015)0.34(1.934)0.5/1000 = 0.5459. εG,e(2pL) = [0.3422 − 0.5459(1/1.934)4/(1 − 1/1.9344) = 0.3265. ae = 0.2265/(2 − 0.3265/0.2265) = 0.4056. Of all these, only εG,e(pL) and ae will be used from here on. From Eq. (5-176), (G S1)R / A1 = 1/[0.6(1/0.2265 − 1/0.4056) +  1/0.87 + (1/0.4056 − 1)/(0.87 + 0.6 (0.4/0.6))] = 0.2879. A1 = (3 × 10 + 3 × 5 + 2 10 × 5) × 2 = 190 × 0.6 = 114 m . Ar (with floor area omitted for loss) = 190 × 0.4 − 10 × 3 = 46 m2. (G S 1)R = 0.2879 × 114 = 32.82 m2. (G S 1)RσTF3 = 32.82 × (56.7E − 12) × 25393 = 30.46 kw/K. D = 54,440 × (0.6201E − 3)/30.46 = 1.1083. ∆ = 170/2539 = 0.06696. Lc = 0.017 × 114/30.46 = 0.0636. Lo = 4 × 0.1 × 0.23 × 0.335/32.82 = 9.4E − 4. Lr = 0.0012 × 46/30.46 = 0.001812. In Eq. (5-180), the coefficient of τG equals (1.1083 + 0.0636 + 0.0018)/1.00094 = 1.1726. The constant in the equation equals [(100/2539)4 + 0.0636(1/2.539) + 0.00094 × 0.11744 + 0.00181(298/2539) + 1.1083(1 + 170/2539)]/1.00094 = 1.2307. The equation to solve is: τG4 + 1.1726τG − 1.2307 = 0. Solution gives τG = 0.7620; TG = 0.762 × 2539 = 1935 K. Te = 1765 K, only 1 K above value assumed for obtaining C p and εG. ηG = (1 − τG)/(1 − τo) = (1 − 1765/2539)/(1 − 298/2539) = 0.3454. Sink-side efficiency η1 0.3454 − [0.00094(0.7624 − 0.117444)] + 0.001812(0.762 − 0.1174)/1.1083(1 − 0.1174) = 0.344, not including convection to screen tubes ˙ η1)/A1 = 43,540 × 0.3439/114 = 131.3 kw/m2. covering gas exit. qplane of sink = (H qtube surf = 131.3 (2D/2πD) = 41.8 kw/m2 × 3412 × 0.30482 = 13,300 Btu/(ft2)(hr). TE = 1765 K = 1492°C = 2717°F.

In Fig. 5-22, the shaded areas indicate the operating regimes of a wide range of furnace types. Note the significant properties of the function presented. (1) As firing rate D′ goes down, the efficiency rises and approaches 1 − τ1 in the limit. (This conclusion is modified if wall losses are significant.) (2) Changes in sink temperature have little effect if τ1 < 0.3. (3) As the furnace walls approach complete coverage by a black sink [Cε1 → 1 in Eqs. (5-176) and (5-177)] and as convection becomes unimportant, the effect of flame emissivity on D becomes one of inverse proportionality; thus at very high firing rates at which efficiency approaches inverse proportionality to D, the efficiency of heat transfer varies directly as εG (gas-turbine chambers), but at low firing rates εG has relatively little effect. (4) When Cε1 << 1 because of a nonblack sink or much refractory surface, the effect of changing flame emissivity is to produce a much less than proportional effect on heat flux. The factor ∆, the allowance for imperfect stirring, must be estimated. Values in the range of 93 to 149°C (200 to 300°F) have been found to produce data correlation for a series of tests on marine boilers. Equation (5-179) and Fig. 5-22 serve as a framework for correlating the performance of furnaces with flow patterns—plug flow, parabolic

FIG. 5-22 Thermal performance of well-stirred furnace chambers; reduced efficiency as a function of reduced firing density D and reduced sink temperature τ1. (a) Radiant section, oil tube stills, cracking coils. (b) Domestic boiler combustion chambers. (c) Open-hearth furnaces. (d) Soaking pits.

5-42

HEAT AND MASS TRANSFER

FIG. 5-23

Effect of wall-loss factor L on combustion-chamber performance; L′ = 0.02, and τ = 0.5 and 0.7.

profile, and recirculatory flow—which differ from the well-stirred model [Hottel and Sarofim, Int. J. Mass Heat Transfer, 8, 1153 (1965)]. As expected, plug-flow furnaces show somewhat higher efficiency, mild-recirculation types somewhat lower efficiency, and strong-recirculation furnaces a performance closely similar to that of the well-stirred model.

If data on several furnaces of a single class are available, a similar treatment can lead to a partially empirical equation based on simplified rules for obtaining (G S1)R,c or an effective ∆. Because Eq. (5-178) has a  structure which covers a wide range of furnace types and has a sound theoretical basis, it provides safer structures of empirical design equations than many such equations available in the engineering literature.

MASS TRANSFER GENERAL REFERENCES: Bird, Stewart, and Lightfoot, Transport Phenomena, Wiley, New York, 1960. Cussler, Diffusion: Mass Transfer in Fluid Systems, Cambridge University Press, Cambridge, 1984. Danner and Daubert, Manual for Predicting Chemical Process Design Data, AIChE, New York, 1983. Daubert and Danner, Physical and Thermodynamic Properties of Pure Chemicals, Taylor and Francis, Bristol, PA, 1989–1995. Fahien, Fundamentals of Transport Phenomena, McGraw-Hill, New York, 1983. Foust, Wenzel, Clump, Maus, and Andersen, Principles of Unit Operations, 2d ed., Wiley, New York, 1980. Gammon, Marsh, and Dewan, Transport Properties and Related Thermodynamic Data of Binary Mixtures, AIChE, New York, Part 1, 1993; Part 2, 1994. Geankoplis, Transport Processes and Unit Operations, 3d ed., Prentice Hall, Englewood Cliffs, NJ, 1993. Hines and Maddox, Mass Transfer: Fundamentals and Applications, Prentice Hall, Englewood Cliffs, NJ, 1985. Kirwan, “Mass Transfer Principles,” Chap. 2 in Rousseau, R. W. (ed.), Handbook of Separation Process Technology, Wiley, New York, 1987. McCabe, Smith, and Harriott, Unit Operations of Chemical Engineering, 5th ed., McGraw-Hill, New York, 1993. Reid, Prausnitz, and Poling, The Properties of Gases and Liquids, 4th ed., McGrawHill, New York, 1987. Schwartzberg and Chao, Food Technol., 36(2), 73 (1982). Sherwood, Pigford, and Wilke, Mass Transfer, McGraw-Hill, New York, 1975. Skelland, Diffusional Mass Transfer, Wiley, New York, 1974. Taylor and Krishna, Multicomponent Mass Transfer, Wiley, New York, 1993. Treybal, Mass-Transfer Operations, 3d ed., McGraw-Hill, New York, 1980.

INTRODUCTION This part of Sec. 5 provides a concise guide to solving problems in situations commonly encountered by chemical engineers. It deals with

diffusivity and mass-transfer coefficient estimation and common flux equations, although material balances are also presented in typical coordinate systems to permit a wide range of problems to be formulated and solved. Mass-transfer calculations involve transport properties, such as diffusivities, and other empirical factors that have been found to relate mass-transfer rates to measured “driving forces” in myriad geometries and conditions. The context of the problem dictates whether the fundamental or more applied coefficient should be used. One key distinction is that, whenever there is flow parallel to an interface through which mass transfer occurs, the relevant coefficient is an empirical combination of properties and conditions. Conversely, when diffusion occurs in stagnant media or in creeping flow without transverse velocity gradients, ordinary diffusivities may be suitable for solving the problem. In either case, it is strongly suggested to employ data, whenever available, instead of relying on correlations. Units employed in diffusivity correlations commonly followed the cgs system. Similarly, correlations for mass transfer correlations used the cgs or English system. In both cases, only the most recent correlations employ SI units. Since most correlations involve other properties and physical parameters, often with mixed units, they are repeated here as originally stated. Common conversion factors are listed in Table 1-4. Fick’s First Law This law relates flux of a component to its composition gradient, employing a constant of proportionality called a

MASS TRANSFER

5-43

Nomenclature and Units—Mass Transfer Symbols a

Definition

Acs A′ ap c ci cP d db dc d drop dimp dpore DA′A DAB D°AB Deff DK DL DS E ES f F g gc G Ga GM h′ hT H H

Effective interfacial mass transfer area per unit volume Cross-sectional area of vessel Constant (see Table 5-28-I) See a Concentration = P/RT for an ideal gas Concentration of component i = xi c at gas-liquid interface Specific heat Characteristic length Bubble diameter Column diameter Sauter mean diameter Impeller diameter Pore diameter Self-diffusivity (= DA at xA = 1) Mutual diffusivity Mutual diffusivity at infinite dilution of A in B Effective diffusivity within a porous solid = εpD/τ Knudson diffusivity for gases in small pores Liquid phase diffusion coefficient Surface diffusivity Energy dissipation rate/mass Activation energy for surface diffusion Friction factor for fluid flow Faraday’s constant Acceleration due to gravity Conversion factor Gas-phase mass flux Dry air flux Molar gas-phase mass flux Heat transfer coefficient Total height of tower packing Compartment height Henry’s law constant

H′

Henry’s law constant

HG HOG HL HOL HTU jD jH jM m JA

J JSi k k k k′

Height of one transfer unit based on gas-phase resistance Height of one overall gas-phase mass-transfer unit Height of one transfer unit based on liquid-phase resistance Height of one overall liquid-phase mass-transfer unit Height of one transfer unit (general) Chilton-Colburn factor for mass transfer, Eq. (5-291) Chilton-Colburn factor for heat transfer See jD Mass flux of A by diffusion with respect to the mean mass velocity Molar flux of A by diffusion with respect to mean molar velocity Molar flux of A with respect to mean volume velocity Molar flux by surface diffusion Boltzmann’s constant Film mass transfer coefficient Thermal conductivity Mass-transfer coefficient for dilute systems

kG

Gas-phase mass-transfer coefficient for dilute systems

k′G

Gas-phase mass-transfer coefficient for dilute systems

kGa kˆ Ga

Volumetric gas-phase mass-transfer Overall volumetric gas-phase mass-transfer coefficient for concentrated systems Liquid phase mass transfer coefficient for pure absorption (no reaction) Liquid-phase mass-transfer coefficient for dilute systems

J

M A

V A

ˆL k° kL k′L kˆ L kLa K K KG

Liquid-phase mass-transfer coefficient for dilute systems Liquid-phase mass-transfer coefficient for concentrated systems Volumetric liquid-phase mass-transfer coefficient for dilute systems Overall mass transfer coefficient α/R = specific conductance Overall gas-phase mass-transfer coefficient for dilute systems

SI units

U.S. customary units

m2/m3

ft2/ft3

m2 or cm2

ft2

mol/m3 or mol/l or gequiv/l mol/m3 or mol/l or gequiv/l kJ/(kg⋅K) m or cm m m or cm m m m or cm m2/s or cm2/s m2/s or cm2/s m2/s or cm2/s m2/s m2/s or cm2/s m2/s m2/s or cm2/s

lbmol/ft3 lbmol/ft3 Btu/(lb⋅°F) ft ft ft ft ft ft ft2/h ft2/h ft2/h ft2/h ft2/h ft2/h ft2/h

J/mol or cal/mol Dimensionless 96,487 Coulomb/gequiv m/s2 1.0 kg/(s⋅m2) kg/(s⋅m2) kmol/(s⋅m2) W/(m2⋅K) = J/(s⋅m2⋅K) m m kPa/(mole-fraction solute in liquid phase) kPa/[kmol/(m3 solute in liquid phase)]

Dimensionless

m m m m m Dimensionless Dimensionless

ft/h2 4.17 × 108 lb ft/[lbf⋅h2] lb/(h⋅ft2) lb/(h⋅ft2) (lbmol)/(h⋅ft2) Btu/(h⋅ft2⋅°F) ft ft (lbf/in2)/(mole-fraction solute in liquid phase) (lbf/in2)/[(lbmol)/(ft3 solute in liquid phase)] or atm/[(lbmole)/ (ft3 solute in liquid phase)] ft ft ft ft ft Dimensionless Dimensionless

kmol/(m2⋅s) or mol/(cm2⋅s)

lbmol/(ft2⋅h)

kmol/(m2⋅s) or mol/(cm2⋅s)

lbmol/(ft2⋅h)

kmol/(m2⋅s) kmol/(m2⋅s) or gmol/(cm2⋅s) 8.9308 × 10−10 gequiv ohm/s m/s or cm/s (J⋅m)/(s⋅m2⋅K) kmol/[(s⋅m2)(kmol/m3)] or m/s kmol/[(s⋅m2)(kPa solute partial pressure)] kmol/[(s⋅m2)(mole fraction in gas)] kmol/[(s⋅m3)(mole fraction)] kmol/(s⋅m3)

lbmol/(ft2 ⋅h) lbmol/(ft2⋅h) ft/hr Btu/(h⋅ft⋅°F) lbmol/[(h⋅ft2)(lbmol/ft3)] or ft/hr lbmol/[(h⋅ft2)lbf/in2 solute partial pressure)] lbmol/[(h⋅ft2)(mole fraction in gas)] (lbmol)/[(h⋅ft3)(mole fraction)] lbmol/(h⋅ft3)

kmol/(s⋅m2)

lbmol/(h⋅ft2)

kmol/[(s⋅m2)(mole-fraction solution in liquid)] kmol/[(s⋅m2)(kmol/m3)] or m/s kmol/(s⋅m2)

(lbmol)/[(h⋅ft2)(mole-fraction solute in liquid)] (lbmol)/[(h⋅ft2)(lbmol/ft3)] or ft/h lbmol/(h⋅ft2)

kmol/[(s⋅m3)(mole fraction)]

(lbmol)/[(h⋅ft3)(mole fraction)]

m/s or cm/s ohm/cm kmol/[(s⋅m2)(mole fraction)]

ft/h (lbmol)/[(h⋅ft2)(mole fraction)]

5-44

HEAT AND MASS TRANSFER

Nomenclature and Units—Mass Transfer (Continued ) Symbols Kˆ G

Definition

KGa

Overall gas-phase mass-transfer coefficient for concentrated systems Overall volumetric gas-phase mass-transfer dilute systems

K′Ga

Overall volumetric gas-phase mass-transfer dilute systems

(Ka)H KL Kˆ L

Overall enthalpy mass-transfer coefficient Overall liquid-phase mass-transfer coefficient Liquid-phase mass-transfer coefficient for concentrated systems Overall volumetric liquid-phase mass-transfer coefficient for dilute systems Overall volumetric liquid-phase mass-transfer coefficient for concentrated systems Liquid-phase mass flux Molar liquid-phase mass flux Slope of equilibrium curve = dy/dx (mole-fraction solute in gas)/(mole-fraction solute in liquid) Molality of solute Molecular weight of species i Mass in a control volume V Valences of cationic and anionic species See Table 5-28-I Mass flux of A with respect to fixed coordinates Impeller speed Number deck levels Interphase mass-transfer rate of solute A per interfacial area with respect to fixed coordinates Number of components Froude Number (dimp N2/g)

KLa Kˆ La L LM m m Mi M |n+ n −| n′ nA N N′ NA Nc NFr

NOG NOL NTU NKn NPr NRe NSc NSh NSt NW p pB,M pi pT P P Pc Per Q rA R

ρ∞ gx3 Grashof number ᎏ2 ᎏ −1 (µ/ρ) ρo Number of overall gas-phase mass-transfer units Number of overall liquid-phase mass-transfer units Number of transfer units (general) Knudson number = l/dpore Prandtl number (cpµ/k) Reynolds number (Gd/µG) Schmidt number (µG/ρGDAB) or (µL/ρLDL) Sherwood number (ˆkGRTd/DABpT) Stanton number (ˆkG/GM) or (ˆkL/LM) Weber number (ρcN2d3imp/σ) Solute partial pressure in bulk gas Log mean partial pressure difference of stagnant gas B Solute partial pressure at gas-liquid interface Total system pressure Pressure Power Critical pressure Perimeter/area Volumetric flow rate Radius of dilute spherical solute Gas constant

R Ri s S t tf T Tb Tc Tr u, v uo u∞ uL vs vT vTS V V Vb Vi vi

Solution electrical resistance Radius of gyration of the component i molecule Fractional surface-renewal rate Tower cross-sectional area = πd2/4 Contact time Formation time of drop Temperature Normal boiling point Critical temperature Reduced temperature = T/Tc Fluid velocity Blowing or suction velocity Velocity away from object Superficial liquid velocity in vertical direction Slip velocity Terminal velocity Stokes law terminal velocity Packed volume in tower Control volume Volume at normal boiling point Molar volume of i at its normal boiling point Partial molar volume of i

NGr







SI units

U.S. customary units

kmol/(s⋅m2)

lbmol/(h⋅ft2)

kmol/[(s⋅m3)(mole-fraction solute in gas)] kmol/[(s⋅m3)(kPa solute partial pressure)] kmol/[(s⋅m2)(mole fraction)] kmol/[(s⋅m2)(mole fraction)] kmol/(s⋅m2)

(lbmol)/[(h⋅ft3(mole-fraction solute in gas)] (lbmol)/[(h⋅ft3)(lbf/in2 solute partial pressure)] lb/[(h⋅ft3)(lb water/lb dry air)] (lbmol)/[(h⋅ft2)(mole fraction)] (lbmol)/(h⋅ft2)

kmol/[(s⋅m3)(mole-fraction solute in liquid)] kmol/(s⋅m3)

(lbmol)/[(h⋅ft3)(mole-fraction solute in liquid)] (lbmol)/(h⋅ft3)

kg/(s⋅m2) kmol/(s⋅m2) Dimensionless

lb/(h⋅ft2) (lbmol)/(h⋅ft2) Dimensionless

mol/1000 g solvent kg/kmol or g/mol kg or g Dimensionless Dimensionless kg/(s⋅m2) Revolution/s Dimensionless kmol/(s⋅m2)

lb/lbmol lb Dimensionless Dimensionless lb/(h⋅ft2) Revolution/min Dimensionless (lbmol)/(h⋅ft2)

Dimensionless Dimensionless

Dimensionless Dimensionless

Dimensionless

Dimensionless

Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless kPa Dimensionless kPa kPa Pa Watts Pa m−1 m3/s Å 8.314 J/mol K = 8.314 Pa m3/(mol K) = 82.057 atm cm3/mol K ohm Å s−1 m2 s s K K K Dimensionless m/s or cm/s m/s m/s m/s m/s m/s m/s m3 m3 or cm3 m3/kmol or cm3/mol m3/kmol or cm3/mol m3/kmol or cm3/mol

Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless lbf/in2 Dimensionless lbf/in2 lbf/in2 lbf/in2 or atm lbf/in2 or atm ft−1 ft3/h 10.73 ft3 psia/lbmol⋅h

h−1 ft2 h h °R °R °R Dimensionless ft/h ft/h ft/h ft/h ft/h ft/h ft/h ft3 ft3 ft3/lbmol ft3/lbmol ft3/lbmol

MASS TRANSFER

5-45

Nomenclature and Units—Mass Transfer (Concluded ) Symbols Vmli Vtower w x x xA xo xBM o BM

x

xi y yBM o yBM yi yio z

Definition

SI units

Molar volume of the liquid-phase component i at the melting point Tower volume per area Width of film Length along plate Mole-fraction solute in bulk-liquid phase Mole fraction of component A Mole-fraction solute in bulk liquid in equilibrium with bulk-gas solute concentration y Logarithmic-mean solvent concentration between bulk liquid and interface values Logarithmic-mean inert-solvent concentration between bulk-liquid value and value in equilibrium with bulk gas Mole-fraction solute in liquid at gas-liquid interface Mole-fraction solute in bulk-gas phase Logarithmic-mean inert-gas concentration (5-262) Logarithmic-mean inert-gas concentration Mole fraction solute in gas at interface Mole-fraction solute in gas at interface in equilibrium with the liquid-phase interfacial solute concentration xi Direction of unidimensional diffusion

U.S. customary units

m3/kmol or cm3/mol

ft3/lbmol

m3/m2 m m (kmol solute)/(kmol liquid) kmole A/kmole fluid (kmol solute)/(kmol liquid)

ft3/ft2 ft ft (lbmol solute)/(lb mol liquid) lbmole A/lb mole fluid (lbmol solute)/(lbmol liquid)

(kmol solvent)/(kmol liquid)

(lbmol solvent)/(lbmol liquid)

(kmol solvent)/(kmol liquid)

(lbmol solvent)/(lbmol liquid)

(kmol solute)/(kmol liquid) (kmol solute)/(kmol gas) (kmol inert gas)/(kmol gas) (kmol inert gas)/(kmol gas) (kmole solute)/(kmol gas) (kmol solute)/(kmol gas)

(lbmol solute)/(lbmol liquid) (lbmol solute)/(lbmol gas) (lbmol inert gas)/(lbmol gas) (lbmol inert gas)/(lbmol gas) (lbmol solute)/(lbmol gas) (lbmol solute)/(lbmol gas)

m

ft

Dimensionless cm−1 Dimensionless m Dimensionless

Dimensionless

m3/m3 Dimensionless Dimensionless Dimensionless Dimensionless cm2/(gequiv⋅ohm) cm2/ohm gequiv cm2/gequiv ohm Debeyes cP or Pa s kg/(s⋅m) kg/(s⋅m) m2/s kg/m3 or g/cm3 kg/m3 or g/cm3 kg/m3 kg/m3 kmol/m3 kg/m3 or g/cm3 Dimensionless

ft3/ft3 Dimensionless Dimensionless Dimensionless Dimensionless

Dimensionless Dimensionless dyn/cm Å dyn/cm Å Dimensionless Dimensionless Radians/s Dimensionless

Dimensionless Dimensionless lbf/ft

Greek symbols α α β δ ε ε εA εAB γi γ⫾ λ+ λ− Λ Λo µi µi µG µL ν ρ ρc ρc ρG ρ L ρp ρr ψi ψ Ψ σ σi σi σAB τ ω ω Ω

1 + NB/NA Conductance cell constant (measured) 5/6 MA1/2 P1/3 c /T c Effective thickness of stagnant-film layer Fraction of discontinuous phase in continuous phase for two-phase flow Void fraction available for gas flow or fractional gas holdup Characteristic Lennard-Jones energy (εAεB)1/2 Activity coefficient of solute i Mean ionic activity coefficient of solute Infinite dilution conductance of cation and anion 1000 K/C = λ+ + λ− = Λo + f(C) Infinite dilution conductance Dipole moment of i Viscosity of pure i Gas-phase viscosity Liquid-phase viscosity Kinematic viscosity = ρ/µ Density of A Critical density of A Density continuous phase Gas-phase density Average molar density of liquid phase Particle density Reduced density = ρ/ρc Parachor of component i = Viσ1/4 Parameter, Table 5-28-G Shape factor, Table 5-27-A Interfacial tension Characteristic length Surface tension of component i Binary pair characteristic length = (σA + σB)/2 Intraparticle tortuosity Pitzer’s acentric factor = −[1.0 + log10(P*/Pc)] Rotational velocity Diffusion collision integral = f(kT/εAB) Subscript

A B G m L super

Solute component in liquid or gas phase Inert-gas or inert-solvent component Gas phase Mean value Liquid phase Superficial velocity

*

At equilibrium

Superscript

Dimensionless ft Dimensionless

lb/(h⋅ft) lb/(h⋅ft) lb/(h⋅ft) ft2/h lb/ft3 lb/ft3 lb/ft3 lb/ft3 (lbmol)/ft3 lb/ft3 Dimensionless

Dimensionless Dimensionless

5-46

HEAT AND MASS TRANSFER

diffusivity. It can be written in several forms, depending on the units and frame of reference. Three that are related but not identical are dcA dxA dwA (5-181) V JA = −DAB ᎏ ≈ M JA = −cDAB ᎏ ∝ m JA = −ρDAB ᎏ dz dz dz The first equality (on the left-hand side) corresponds to the molar flux with respect to the volume average velocity, while the equality in the center represents the molar flux with respect to the molar average velocity and the one on the right is the mass flux with respect to the mass average velocity. These must be used with consistent flux expressions for fixed coordinates and for NC components, such as: NC

NC

NC

J + wA
ni

m A

i=1

NA = V JA + cA
NiV i = M JA + xA
Ni = ᎏᎏ MA i=1 i=1

(5-182)

In each case, the term containing the summation accounts for conveyance, which is the amount of component A carried by the net flow in the direction of diffusion. Its impact on the total flux can be as much as 10 percent. In most cases it is much less, and it is frequently ignored. Some people refer to this as the “convective” term, but that conflicts with the other sense of convection which is promoted by flow perpendicular to the direction of flux. Mutual Diffusivity, Mass Diffusivity, Interdiffusion Coefficient Diffusivity is denoted by DAB and is defined by Fick’s first law as the ratio of the flux to the concentration gradient, as in Eq. (5-181). It is analogous to the thermal diffusivity in Fourier’s law and to the kinematic viscosity in Newton’s law. These analogies are flawed because both heat and momentum are conveniently defined with respect to fixed coordinates, irrespective of the direction of transfer or its magnitude, while mass diffusivity most commonly requires information about bulk motion of the medium in which diffusion occurs. For liquids, it is common to refer to the limit of infinite dilution of A in B using the symbol, D°AB. When the flux expressions are consistent, as in Eq. (5-182), the diffusivities in Eq. (5-181) are identical. As a result, experimental diffusivities are often measured under constant volume conditions but may be used for applications involving open systems. It turns out that the two versions are very nearly equivalent for gas-phase systems because there is negligible volume change on mixing. That is not usually true for liquids, however. Self Diffusivity Self-diffusivity is denoted by DA′A and is the measure of mobility of a species in itself; for instance, using a small concentration of molecules tagged with a radioactive isotope so they can be detected. Tagged and untagged molecules presumably do not have significantly different properties. Hence, the solution is ideal, and there are practically no gradients to “force” or “drive” diffusion. This kind of diffusion is presumed to be purely statistical in nature. In the special case that A and B are similar in molecular weight, polarity, and so on, the self-diffusion coefficients of pure A and B will be approximately equal to the mutual diffusivity, DAB. Second, when A and B are the less mobile and more mobile components, respectively, their self-diffusion coefficients can be used as rough lower and upper bounds of the mutual diffusion coefficient. That is, DA′A ≤ DAB ≤ DB′B. Third, it is a common means for evaluating diffusion for gases at high pressure. Self-diffusion in liquids has been studied by many [Easteal AIChE J. 30, 641 (1984), Ertl and Dullien, AIChE J. 19, 1215 (1973), and Vadovic and Colver, AIChE J. 18, 1264 (1972)]. Tracer Diffusivity Tracer diffusivity, denoted by DA′B is related to both mutual and self-diffusivity. It is evaluated in the presence of a second component B, again using a tagged isotope of the first component. In the dilute range, tagging A merely provides a convenient method for indirect composition analysis. As concentration varies, tracer diffusivities approach mutual diffusivities at the dilute limit, and they approach self-diffusivities at the pure component limit. That is, at the limit of dilute A in B, DA′ B → D°AB and DB ′A → DB ′ B; likewise at the limit of dilute B in A, DB ′A → D°BA and DA ′B → DA′A. Neither the tracer diffusivity nor the self-diffusivity has much practical value except as a means to understand ordinary diffusion and as

order-of-magnitude estimates of mutual diffusivities. Darken’s equation [Eq. (5-220)] was derived for tracer diffusivities but is often used to relate mutual diffusivities at moderate concentrations as opposed to infinite dilution. Mass-Transfer Coefficient Denoted by kc, kx, Kx, and so on, the mass-transfer coefficient is the ratio of the flux to a concentration (or composition) difference. These coefficients generally represent rates of transfer that are much greater than those that occur by diffusion alone, as a result of convection or turbulence at the interface where mass transfer occurs. There exist several principles that relate that coefficient to the diffusivity and other fluid properties and to the intensity of motion and geometry. Examples that are outlined later are the film theory, the surface renewal theory, and the penetration theory, all of which pertain to idealized cases. For many situations of practical interest like investigating the flow inside tubes and over flat surfaces as well as measuring external flow through banks of tubes, in fixed beds of particles, and the like, correlations have been developed that follow the same forms as the above theories. Examples of these are provided in the subsequent section on mass-transfer coefficient correlations. Problem Solving Methods Most, if not all, problems or applications that involve mass transfer can be approached by a systematic course of action. In the simplest cases, the unknown quantities are obvious. In more complex (e.g., multicomponent, multiphase, multidimensional, nonisothermal, and/or transient) systems, it is more subtle to resolve the known and unknown quantities. For example, in multicomponent systems, one must know the fluxes of the components before predicting their effective diffusivities and vice versa. More will be said about that dilemma later. Once the known and unknown quantities are resolved, however, a combination of conservation equations, definitions, empirical relations, and properties are applied to arrive at an answer. Figure 5-24 is a flowchart that illustrates the primary types of information and their relationships, and it applies to many mass-transfer problems. CONTINUITY AND FLUX EXPRESSIONS Material Balances Whenever mass-transfer applications involve equipment of specific dimensions, flux equations alone are inadequate to assess results. A material balance or continuity equation must also be used. When the geometry is simple, macroscopic balances suffice. The following equation is an overall mass balance for such a unit having Nm bulk-flow ports and Nn ports or interfaces through which diffusive flux can occur: Nm Nn dM (5-183) ᎏ =
mi +
ni Acsi dt i = 1 i=1 where M represents the mass in the unit volume V at any time t; mi is the mass flow rate through the ith port; and ni is the mass flux through the ith port, which has a cross-sectional area of Acsi. The corresponding balance equation for individual components includes a reaction term: Nm Nn dMj (5-184) ᎏ =
mij +
nij Acsi + rj V dt i=1 i=1 For the jth component, mij = miwij is the component mass flow rate in stream i; wij is the mass fraction of component j in stream i; and rj is the net reaction rate (mass generation minus consumption) per unit volume V that contains mass M. If it is inconvenient to measure mass flow rates, the product of density and volumetric flow rate is used instead. In addition, most situations that involve mass transfer require material balances, but the pertinent area is ambiguous. Examples are packed columns for absorption, distillation, or extraction. In such cases, flow rates through the discrete ports (nozzles) must be related to the mass-transfer rate in the packing. As a result, the mass-transfer rate is determined via flux equations, and the overall material balance incorporates the stream flow rates mi and integrated fluxes. In such instances, it is common to begin with the most general, differential material balance equations. Then, by eliminating terms that are negligible, the simplest applicable set of equations remains to be solved.

MASS TRANSFER

5-47

Flowchart illustrating problem solving approach using mass-transfer rate expressions in the context of mass conservation.

FIG. 5-24

Table 5-12 provides material balances for Cartesian, cylindrical, and spherical coordinates. The generic form applies over a unit crosssectional area and constant volume: ∂ρ j (5-185a) ᎏ = −∇ ⋅ nj + rj ∂t where nj = ρvj. Applying Fick’s law and expressing composition as concentration gives ∂cj (5-185b) ᎏ = −v ⋅ ∇cj + Dj∇2cj + rj ∂t Flux Expressions: Simple Integrated Forms of Fick’s First Law Simplified flux equations that arise from Eqs. (5-181) and (5-182) can be used for unidimensional, steady-state problems with binary mixtures. The boundary conditions represent the compositions xAL and xAR at the left-hand and right-hand sides of a hypothetical layer having thickness ∆z. The principal restriction of the following equations is that the concentration and diffusivity are assumed to be constant. As written, the flux is positive from left to right, as depicted in Fig. 5-25. 1. Equimolar counterdiffusion (NA = −NB) dxA DAB NA = M JA = −DAB c ᎏ = ᎏ c (xAL − xAR) (5-189) dz ∆z 2. Unimolar diffusion (NA ≠ 0, NB = 0) 1 − xA DAB NA = M JA + xANA = ᎏ c ln ᎏR (5-190) ∆z 1 − xAL 3. Steady state diffusion (NA ≠ −NB ≠ 0) NA ᎏᎏ − xAR NA + NB NA DAB ᎏᎏ (5-191) NA = M JA + xA(NA + NB) = ᎏ ᎏ c ln NA NA + NB ∆z ᎏᎏ − xAL NA + NB TABLE 5-12

Nc

Ni = −cDim ∇xi + xi
Nj

In the late 1800s, the development of the kinetic theory of gases led to a method for calculating multicomponent gas diffusion (e.g., the flux of each species in a mixture). The methods were developed simultaneously by Stefan and Maxwell. The problem is to determine the diffusion coefficient Dim. The Stefan-Maxwell equations are simpler in principle since they employ binary diffusivities: Nc

1 ∇xi =
ᎏ (xiNj − xj Ni) j = 1 cDij





(5-186)

∂nzj 1 ∂rnrj 1 ∂nθj ∂ρj ᎏ = − ᎏ ᎏ + ᎏ ᎏ + ᎏ + rj ∂t ∂z r ∂r r ∂θ



Cylindrical



(5-187)

2 ∂nθj sin θ ∂nφj ∂ρj 1 ∂r nrj 1 1 ᎏ = − ᎏ2 ᎏ + ᎏ ᎏᎏ + ᎏ ᎏ + rj ∂t ∂r r sin θ ∂θ r sin θ ∂φ r



(5-193)

If Eqs. (5-192) and (5-193) are combined, the multicomponent diffusion coefficient may be assessed in terms of binary diffusion coefficients [see Eq. (5-204)]. For gases, the values Dij of this equation are approximately equal to the binary diffusivities for the ij pairs. The Stefan-Maxwell diffusion coefficients may be negative, and the method may be applied to liquids, even for electrolyte diffusion [Kraaijeveld, Wesselingh, and Kuiken, Ind. Eng. Chem. Res., 33, 750 (1994)]. Approximate solutions have been developed by linearization [Toor, H.L., AIChE J., 10, 448 and 460 (1964); Stewart and Prober, Ind. Eng. Chem. Fundam., 3, 224 (1964)]. Those differ in details but yield about the same accuracy. More recently, efficient algorithms for solving the equations exactly have been developed (see Taylor and Krishna, Krishnamurthy and Taylor, and Taylor and Webb).

Equation ∂nyj ∂nzj ∂nxj ∂ρj ᎏ = − ᎏ + ᎏ + ᎏ + rj ∂t ∂x ∂y ∂z

Cartesian

(5-192)

j=1

Continuity Equation in Various Coordinate Systems

Coordinate System

Spherical

The unfortunate aspect of the last relationship is that one must know a priori the ratio of the fluxes to determine the magnitudes. It is not possible to solve simultaneously the pair of equations that apply for components A and B because the equations are not independent. Stefan-Maxwell Equations Following Eq. (5-182), a simple and intuitively appealing flux equation for applications involving Nc components is



(5-188)

5-48

FIG. 5-25

HEAT AND MASS TRANSFER Before using diffusivities from either data or correlations, it is a good idea to check their reasonableness with respect to values that have been commonly observed in similar situations. Table 5-13 is a compilation of several rules of thumb. These values are not authoritative; they simply represent guidelines based on experience. Diffusivity correlations for gases are outlined in Table 5-14. Specific parameters for individual equations are defined in the specific text regarding each equation. References are given after Table 5-19. The errors reported for Eq. (5-194) through (5-197) were compiled by Reid et al., who compared the predictions with 68 experimental values of DAB. Errors cited for Eqs. (5-198) to (5-202) were reported by the authors. Binary Mixtures—Low Pressure—Nonpolar Components Many evaluations of correlations are available (Elliott and Watts; Lugg; Marrero and Mason). The differences in accuracy of the correlations are minor, and thus the major concern is ease of calculation. The Fuller-Schettler-Giddings equation is usually the simplest correlation to use and is recommended by Reid et al. Chapman-Enskog (Bird et al.) and Wilke-Lee The inherent assumptions of these equations are quite restrictive (i.e., low density,

Hypothetical film and boundary conditions.

DIFFUSIVITY ESTIMATION—GASES Whenever measured values of diffusivities are available, they should be used. Typically, measurement errors are less than those associated with predictions by empirical or even semitheoretical equations. A few general sources of data are Sec. 2 of this handbook, Schwartzberg and Chao; Reid et al.; Gammon et al.; and Daubert and Danner. Many other more restricted sources are listed under specific topics later in this subsection. TABLE 5-13

Rules of Thumb for Diffusivities (See Cussler, Reid et al., Schwartzberg and Chao) Di magnitude

Di range

m2/s

cm2/s

m2/s

Gas at atmospheric pressure

10−5

0.1

10−4–10−6

Liquid Liquid occluded in solid matrix Polymers and glasses

10−9 10−10 10−12

10−5 10−6 10−8

10−8–10−10 10−8–10−12 10−10–10−14

10−4–10−6 10−4–10−8 10−6–10−10

Solid

10−14

10−10

10−10–10−34

10−6–10−30

Continuous phase

TABLE 5-14

cm2/s 1–10−2

Comments Accurate theories exist, generally within ⫾10%; DiP  constant; Di ∝ T1.66 to 2.0 Approximate correlations exist, generally within ⫾25% Hard cell walls: Deff /Di = 0.1 to 0.2. Soft cell walls: Deff /Di = 0.3 to 0.9 Approximate theories exist for dilute and concentrated limits; strong composition dependence Approximate theories exist; strong temperature dependence

Correlations of Diffusivities for Gases

Authors*

Equation

Error

1. Binary Mixtures—Low Pressure—Nonpolar Chapman-Enskog

1/2 0.001858T 3/2 MAB DAB = ᎏᎏ Pσ 2AB ΩD

(5-194)

7.3%

Wilke-Lee [65]

1/2 1/2 (0.0027 − 0.0005 MAB ) T 3/2MAB DAB = ᎏᎏᎏᎏ 2 PσAB ΩD

(5-195)

7.0%

Fuller-Schettler-Giddings [19]

1/2 0.001T1.75MAB DAB = ᎏᎏᎏ 2 P [(
v)A1/3 + (
v)1/3 B ]

(5-196)

5.4%

(5-197)

9.0%

2. Binary Mixtures—Low Pressure—Polar Brokaw [4]

1/2 0.001858T 3/2MAB DAB = ᎏᎏ 2 Pσ ABΩD

3. Self-Diffusivity—High Pressure Mathur-Thodos [37]

10.7 × 10−5T DAA = ᎏᎏr {ρr ≤ 1.5} βρr

(5-198)

5%

Lee-Thodos [31]

0.77 × 10−5Tr DAA = ᎏᎏ {ρr ≤ 1} ρrδ

(5-199)

0.5%

Lee-Thodos [32]

(0.007094G + 0.001916)2.5Tr DAA = ᎏᎏᎏ , [ρr > 1, G < 1] δ

(5-200)

17%

4. Supercritical Mixtures Sun and Chen [56]

1.23 × 10−10T DAB = ᎏᎏ µ0.799VC0.49 A

(5-201)

5%

Catchpole and King [6]

(ρ−0.667 − 0.4510) (1 + MA/MB) R DAB = 5.152 DcTr ᎏᎏᎏᎏ (1 + (VcB /VcA)0.333)2

(5-202)

10%

*References are listed on pages 5-7 and 5-8.

MASS TRANSFER TABLE 5-15

5-49

Estimates for εi and σi (K, Å, atm, cm3, mol)

Critical point

ε/k = 0.75 Tc

1/3 σ = 0.841 V1/3 c or 2.44 (Tc/Pc)

Critical point

ε/k = 65.3 Tc zc3.6

1.866 V1/3 c σ = ᎏᎏ z1.2 c

Normal boiling point Melting point

ε/k = 1.15 Tb ε/k = 1.92 Tm

σ = 1.18 V1/3 b σ = 1.222 V1/3 m

Acentric factor

ε/k = (0.7915 + 0.1693 ω) Tc

T σ = (2.3551 − 0.087 ω) ᎏc Pc

 

1/3

NOTE: These values may not agree closely, so usage of a consistent basis is suggested (e.g., data at the normal boiling point).

spherical atoms), and the intrinsic potential function is empirical. Despite that, they provide good estimates of DAB for many polyatomic gases and gas mixtures, up to about 1000 K and a maximum of 70 atm. The latter constraint is because observations for many gases indicate that DABP is constant up to 70 atm. The characteristic length is σAB = (σA + σB)/2 in A˚. In order to estimate ΩD for Eqs. (5-194) or (5-195), two empirical equations are available. The first is: ΩD = (44.54T *−4.909 + 1.911T *−1.575)0.10 (5-203a)

the gas pairs [e.g., (CH3)2O & CH3Cl]. Despite that, Reid, op. cit., found the average error was 9.0 percent for combinations of mixtures (including several polar-nonpolar gas pairs), temperatures and pressures. In this equation, ΩD is calculated as described previously, and other terms are:

where T* = kT/εAB and εAB = (εA εB)1/2. Estimates for σi and εi are given in Table 5-15. This expression shows that ΩD is proportional to temperature roughly to the −0.49 power at low temperatures and to the −0.16 power at high temperature. Thus, gas diffusivities are proportional to temperatures to the 2.0 power and 1.66 power, respectively, at low and high temperatures. The second is: G A C E ΩD = ᎏ + ᎏᎏ + ᎏᎏ + ᎏᎏ (5-203b) T*B exp (DT*) exp (FT*) exp (HT*)

Self-Diffusivity—High Pressure The criterion of high pressure is vague at best. For most “permanent” gases, such as the major constituents of air, it would mean P > 70 atm. For less volatile components, the criterion would be lower. At present, accurate prediction of mutual diffusion coefficients for dense gas mixtures is not possible. One major reason for this is the scarcity of data. Most high-pressure diffusion experiments have measured the self-diffusion coefficient. The general observation is that the product DP is near constant at low pressure, is not constant at high pressure, but rather decreases as pressure increases. In addition, although there are usually negligible composition effects on diffusivity of gases at low pressures, the effects are not negligible at high pressures. Mathur-Thodos showed that for reduced densities less than unity, the product DAAρ is approximately constant at a given temperature. Thus, by knowing the value of the product at low pressure, it is possible to estimate its value at a higher pressure. They found at higher pressures the density increases, but the product DAAρ decreases rapidly. In their correlation, β = MA1/2PC1/3/T C5/6. Lee-Thodos presented a generalized treatment of self-diffusivity for gases (and liquids). These correlations have been tested for more than 500 data points each. The average deviation of the first is 0.51 percent, and that of the second is 17.2 percent. δ = MA1/2/ 2 5/6 0.1 P1/2 c V c , s/cm , and where G = (X* − X)/(X* − 1), X = ρr /T r , and X* = ρr /T r0.1 evaluated at the solid melting point. Lee and Thodos expanded their earlier treatment of self-diffusivity to cover 58 substances and 975 data points, with an average absolute deviation of 5.26 percent. Their correlation is too involved to repeat here, but those interested should refer to the original paper. Supercritical Mixtures Debenedetti-Reid showed that conventional correlations based on the Stokes-Einstein relation (for liquid phase) tend to overpredict diffusivities in the supercritical state. Nevertheless, they observed that the Stokes-Einstein group DABµ/T was constant. Thus, although no general correlation applies, only one data point is necessary to examine variations of fluid viscosity and/or temperature effects. They explored certain combinations of aromatic solids in SF6 and CO2. Sun-Chen examined tracer diffusion data of aromatic solutes in alcohols up to the supercritical range and found their data correlated with average deviations of 5 percent and a maximum deviation of 17 percent for their rather limited set of data. Catchpole-King examined binary diffusion data of near-critical fluids in the reduced density range of 1 to 2.5 and found that their data correlated with average deviations of 10 percent and a maximum deviation of 60 percent. They observed two classes of behavior. For the first, no correction factor was required (R = 1). That class was comprised of alcohols as solvents with aromatic or aliphatic solutes, or carbon dioxide as a solvent with aliphatics except ketones as solutes, or

where A = 1.06036, B = 0.15610, C = 0.1930, D = 0.47635, E = 1.03587, F = 1.52996, G = 1.76474, and H = 3.89411. Fuller-Schettler-Giddings The parameters and constants for this correlation were determined by regression analysis of 340 experimental diffusion coefficient values of 153 binary systems. Values of
vi used in this equation are in Table 5-16. Binary Mixtures—Low Pressure—Polar Components The Brokaw correlation was based on the Chapman-Enskog equation, but σAB* and ΩD* were evaluated with a modified Stockmayer potential for polar molecules. Hence, slightly different symbols are used. That potential model reduces to the Lennard-Jones 6-12 potential for interactions between nonpolar molecules. As a result, the method should yield accurate predictions for polar as well as nonpolar gas mixtures. Brokaw presented data for 9 relatively polar pairs along with the prediction. The agreement was good: an average absolute error of 6.4 percent, considering the complexity of some of TABLE 5-16 Atomic Diffusion Volumes for Use in Estimating DAB by the Method of Fuller, Schettler, and Giddings Atomic and Structural Diffusion–Volume Increments, vi (cm3/mol) C H O (N)

16.5 1.98 5.48 5.69

(Cl) (S) Aromatic ring Heterocyclic ring

19.5 17.0 −20.2 −20.2

Diffusion Volumes for Simple Molecules, Σvi (cm3/mol) H2 D2 He N2 O2 Air Ar Kr (Xe) Ne

7.07 6.70 2.88 17.9 16.6 20.1 16.1 22.8 37.9 5.59

CO CO2 N2O NH3 H2O (CCl2F2) (SF5) (Cl2) (Br2) (SO2)

18.9 26.9 35.9 14.9 12.7 114.8 69.7 37.7 67.2 41.1

Parentheses indicate that the value listed is based on only a few data points.

ΩD* = ΩD + 0.19 δ 2AB/T* σAB* = (σA* σB*)1/2 δAB = (δA δB)1/2 εAB* = (εA*εB*)1/2

T* = kT/εAB* σi* = [1.585 Vbi /(1 + 1.3 δ2i )]1/3 δi = 1.94 × 103 µ2i /VbiTbi εi* /k = 1.18 (1 + 1.3 δ 2i )Tbi

5-50

HEAT AND MASS TRANSFER

ethylene as a solvent with aliphatics except ketones and naphthalene as solutes. For the second class, the correction factor was R = X 0.17. The class was comprised of carbon dioxide with aromatics; ketones and carbon tetrachloride as solutes; and aliphatics (propane, hexane, dimethyl butane), sulfur hexafluoride, and chlorotrifluoromethane as solvents with aromatics as solutes. In addition, sulfur hexafluoride combined with carbon tetrachloride, and chlorotrifluoromethane combined with 2-propanone were included in that class. In all cases, X = (1 + (VCB /VCA)1/3)2/(1 + MA /MB) was in the range of 1 to 10. Low-Pressure/Multicomponent Mixtures These methods are outlined in Table 5-17. Stefan-Maxwell equations were discussed earlier. Smith-Taylor compared various methods for predicting multicomponent diffusion rates and found that Eq. (5-204) was superior among the effective diffusivity approaches, though none is very good. They also found that linearized and exact solutions are roughly equivalent and accurate. Blanc provided a simple limiting case for dilute component i diffusing in a stagnant medium (i.e., N ≈ 0), and the result, Eq. (5-205), is known as Blanc’s law. The restriction basically means that the compositions of all the components, besides component i, are relatively large and uniform. Wilke obtained solutions to the Stefan-Maxwell equations. The first, Eq. (5-206), is simple and reliable under the same conditions as Blanc’s law. This equation applies when component i diffuses through a stagnant mixture. It has been tested and verified for diffusion of toluene in hydrogen + air + argon mixtures and for diffusion of ethyl propionate in hydrogen + air mixtures (Fairbanks and Wilke). When the compositions vary from one boundary to the other, Wilke recommends that the arithmetic average mole fractions be used. Wilke also suggested using the Stefan-Maxwell equation, which applies when the fluxes of two or more components are significant. In this situation, the mole fractions are arithmetic averages of the boundary conditions, and the solution requires iteration because the ratio of fluxes is not known a priori. DIFFUSIVITY ESTIMATION—LIQUIDS Many more correlations are available for diffusion coefficients in the liquid phase than for the gas phase. Most, however, are restricted to binary diffusion at infinite dilution D°AB or to self-diffusivity DA′A. This reflects the much greater complexity of liquids on a molecular level. For example, gas-phase diffusion exhibits negligible composition effects and deviations from thermodynamic ideality. Conversely, liquid-phase diffusion almost always involves volumetric and thermodynamic effects due to composition variations. For concentrations greater than a few mole percent of A and B, corrections are needed to obtain the true diffusivity. Furthermore, there are many conditions that do not fit any of the correlations presented here. Thus, careful consideration is needed to produce a reasonable estimate. Again, if diffusivity data are available at the conditions of interest, then they are strongly preferred over the predictions of any correlations. Stokes-Einstein and Free-Volume Theories The starting point for many correlations is the Stokes-Einstein equation. This equation is derived from continuum fluid mechanics and classical thermodynamics for the motion of large spherical particles in a liquid.

For this case, the need for a molecular theory is cleverly avoided. The Stokes-Einstein equation is (Bird et al.) kT (5-207) DAB = ᎏ 6πrAµB where A refers to the solute and B refers to the solvent. This equation is applicable to very large unhydrated molecules (M > 1000) in lowmolecular-weight solvents or where the molar volume of the solute is greater than 500 cm3/mol (Reddy and Doraiswamy; Wilke and Chang). Despite its intellectual appeal, this equation is seldom used “as is.” Rather, the following principles have been identified: (1) The diffusion coefficient is inversely proportional to the size rA  VA1/3 of the solute molecules. Experimental observations, however, generally indicate that the exponent of the solute molar volume is larger than one-third. (2) The term DABµB /T is approximately constant only over a 10-to-15 K interval. Thus, the dependence of liquid diffusivity on properties and conditions does not generally obey the interactions implied by that grouping. For example, Robinson et al. found that: ln DAB ∝ −1/T. (3) Finally, pressure does not affect liquid-phase diffusivity much, since µB and VA are only weakly pressure-dependent. Pressure does have an impact at very high levels. Another advance in the concepts of liquid-phase diffusion was provided by Hildebrand, who adapted a theory of viscosity to selfdiffusivity. He postulated that DA′A = B(V − Vms)/Vms, where DA′A is the self-diffusion coefficient, V is the molar volume, and Vms is the molar volume at which fluidity is zero (i.e., the molar volume of the solid phase at the melting temperature). The difference (V − Vms) can be thought of as the free volume, which increases with temperature; and B is a proportionality constant. Ertl and Dullien [ibid.] found that Hildebrand’s equation could not fit their data with B as a constant. They modified it by applying an empirical exponent n (a constant greater than unity) to the volumetric ratio. The new equation is not generally useful, however, since there is no means for predicting n. The theory does identify the free volume as an important physical variable, since n > 1 for most liquids implies that diffusion is more strongly dependent on free volume than is viscosity. Dilute Binary Nonelectrolytes: General Mixtures These correlations are outlined in Table 5-18. Wilke-Chang This correlation for D°AB is one of the most widely used, and it is an empirical modification of the Stokes-Einstein equation. It is not very accurate, however, for water as the solute. Otherwise, it applies to diffusion of very dilute A in B. The average absolute error for 251 different systems is about 10 percent. φB is an association factor of solvent B that accounts for hydrogen bonding. Component B

φB

Water Methanol Ethanol Propanol Others

2.26 1.9 1.5 1.2 1.0

The value of φB for water was originally stated as 2.6, although when the original data were reanalyzed, the empirical best fit was 2.26.

TABLE 5-17 Relationships for Diffusivities of Multicomponent Gas Mixtures at Low Pressure Authors* Stefan-Maxwell, Smith and Taylor [53]

Blanc [13]

Wilke [63]

Equation



Dim = 1 − xi Dim = Dim =

NC

NC

j=1

xj

NC

*References are listed at the beginning of this subsection.

i

j=1

xiNi

j

ij

(5-204)

i

(5-205)

ij


 j=1 j≠i

j

−1


ᎏ D  j=1

NC

D 
N /N /
x − ᎏ N /

xj ᎏ Dij

−1

(5-206)

MASS TRANSFER TABLE 5-18

5-51

Correlations for Diffusivities of Dilute, Binary Mixtures of Nonelectrolytes in Liquids

Authors*

Equation

Error

1. General Mixtures Wilke-Chang [64]

7.4 × 10−8 (φBMB)1/2 T D°AB = ᎏᎏᎏ µB VA0.6

(5-208)

20%

Tyn-Calus [59]

8.93 × 10−8 (VA/VB2 )1/6 (ψB/ψA)0.6 T D°AB = ᎏᎏᎏᎏ µB

(5-209)

10%

Umesi-Danner [60]

2.75 × 10−8 (RB/RA2/3) T D°AB = ᎏᎏᎏ µB

(5-210)

16%

Siddiqi-Lucas [52]

9.89 × 10−8 VB0.265 T D°AB = ᎏᎏᎏ V A0.45 µB0.907

(5-211)

13%

  ᎏ V 

(5-212)

18%

(5-213)

6%

2. Gases in Low Viscosity Liquids Sridhar-Potter [54]

Chen-Chen [7]

Vc D°AB = DBB ᎏB VcA

2/3

VB

mlB

 

(βVcB)2/3(RTcB)1/2 T (Vr − 1) ᎏ D°AB = 2.018 × 10−9 ᎏᎏ MA1/6 (MBVcA)1/3 TcB

1/2

3. Aqueous Solutions Hayduk-Laudie [25]

13.16 × 10−5 D°AW = ᎏᎏ µ w1.14 VA0.589

(5-214)

18%

Siddiqi-Lucas [52]

D°AW = 2.98 × 10−7 VA−0.5473 µ w−1.026 T

(5-215)

13%

Hayduk-Minhas [26]

A − 0.791) D°AB = 13.3 × 10−8 T1.47 µ(10.2/V VA−0.71 B

(5-216)

5%

Matthews-Akgerman [38]

D°AB = 32.88 MA−0.61VD−1.04 T 0.5 (VB − VD)

(5-217)

5%

Riazi-Whitson [48]

(ρDAB)° µ DAB = 1.07 ᎏ ᎏ ρ µ°

(5-218)

15%

4. Hydrocarbon Mixtures

−0.27 − 0.38 ω + (−0.05 + 0.1 ω)P r

 

*References are listed on pages 5-7 and 5-8.

Random comparisons of predictions with 2.26 versus 2.6 show no consistent advantage for either value, however. It has been suggested to replace the exponent of 0.6 with 0.7 and to use an association factor of 0.7 for systems containing aromatic hydrocarbons. These modifications, however, are not recommended by Umesi and Danner. Lees and Sarram present a comparison of the association parameters. The average absolute error for 87 different solutes in water is 5.9 percent. Tyn-Calus This correlation requires data in the form of molar volumes and parachors ψi = Viσ1/4 i (a property which, over moderate temperature ranges, is nearly constant), measured at the same temperature (not necessarily the temperature of interest). The parachors for the components may also be evaluated at different temperatures from each other. Quale has compiled values of ψi for many chemicals. Group contribution methods are available for estimation purposes (Reid et al.). The following suggestions were made by Reid et al.: The correlation is constrained to cases in which µB < 30 cP. If the solute is water or if the solute is an organic acid and the solvent is not water or a short-chain alcohol, dimerization of the solute A should be assumed for purposes of estimating its volume and parachor. For example, the appropriate values for water as solute at 25°C are VW = 37.4 cm3/mol and ψW = 105.2 cm3g1/4/s1/2mol. Finally, if the solute is nonpolar, the solvent volume and parachor should be multiplied by 8µB. Umesi-Danner They developed an equation for nonaqueous solvents with nonpolar and polar solutes. In all, 258 points were involved ˚ of the component in the regression. Ri is the radius of gyration in A molecule, which has been tabulated by Passut and Danner for 250 compounds. The average absolute deviation was 16 percent, compared with 26 percent for the Wilke-Chang equation.

Siddiqi-Lucas In an impressive empirical study, these authors examined 1275 organic liquid mixtures. Their equation yielded an average absolute deviation of 13.1 percent, which was less than that for the Wilke-Chang equation (17.8 percent). Note that this correlation does not encompass aqueous solutions; those were examined and a separate correlation was proposed, which is discussed later. Binary Mixtures of Gases in Low-Viscosity, Nonelectrolyte Liquids Sridhar-Potter derived an equation for predicting gas diffusion through liquid by combining existing correlations. Hildebrand had postulated the following dependence of the diffusivity for a gas in a liquid: D°AB = DB′B(VcB /VcA)2/3, where DB′B is the solvent self-diffusion coefficient and Vci is the critical volume of component i, respectively. To correct for minor changes in volumetric expansion, Sridhar and Potter multiplied the resulting equation by VB / VmlB, where VmlB is the molar volume of the liquid B at its melting point and DB′B can be estimated by the equation of Ertl and Dullien (see p. 5-50). Sridhar and Potter compared experimentally measured diffusion coefficients for twenty-seven data points of eleven binary mixtures. Their average absolute error was 13.5 percent, but Chen and Chen analyzed about 50 combinations of conditions and 3 to 4 replicates each and found an average error of 18 percent. This correlation does not apply to hydrogen and helium as solutes. However, it demonstrates the usefulness of self-diffusion as a means to assess mutual diffusivities and the value of observable physical property changes, such as molar expansion, to account for changes in conditions. Chen-Chen Their correlation was based on diffusion measurements of 50 combinations of conditions with 3 to 4 replicates each and exhibited an average error of 6 percent. In this correlation, Vr = VB / [0.9724 (VmlB + 0.04765)] and VmlB = the liquid molar volume at the

5-52

HEAT AND MASS TRANSFER

melting point, as discussed previously. Their association parameter β [which is different from the definition of that symbol in Eq. (5-219)] accounts for hydrogen bonding of the solvent. Values for acetonitrile and methanol are: β = 1.58 and 2.31, respectively. Dilute Binary Mixtures of a Nonelectrolyte in Water The correlations that were suggested previously for general mixtures, unless specified otherwise, may also be applied to diffusion of miscellaneous solutes in water. The following correlations are restricted to the present case, however. Hayduk-Laudie They presented a simple correlation for the infinite dilution diffusion coefficients of nonelectrolytes in water. It has about the same accuracy as the Wilke-Chang equation (about 5.9 percent). There is no explicit temperature dependence, but the 1.14 exponent on µw compensates for the absence of T in the numerator. That exponent was misprinted (as 1.4) in the original article and has been reproduced elsewhere erroneously. Siddiqi-Lucas These authors examined 658 aqueous liquid mixtures in an empirical study. They found an average absolute deviation of 19.7 percent. In contrast, the Wilke-Chang equation gave 35.0 percent and the Hayduk-Laudie correlation gave 30.4 percent. Dilute Binary Hydrocarbon Mixtures Hayduk-Minhas presented an accurate correlation for normal paraffin mixtures that was developed from 58 data points consisting of solutes from C5 to C32 and solvents from C5 to C16. The average error was 3.4 percent for the 58 mixtures. Matthews-Akgerman The free-volume approach of Hildebrand was shown to be valid for binary, dilute liquid paraffin mixtures (as well as self-diffusion), consisting of solutes from C8 to C16 and solvents of C6 and C12. The term they referred to as the “diffusion volume” was simply correlated with the critical volume, as VD = 0.308 Vc. We can infer from Table 5-15 that this is approximately related to the volume at the melting point as VD = 0.945 Vm. Their correlation was valid for diffusion of linear alkanes at temperatures up to 300°C and pressures up to 3.45 MPa. Matthews et al. and Erkey and Akgerman completed similar studies of diffusion of alkanes, restricted to n-hexadecane and n-octane, respectively, as the solvents. Riazi-Whitson They presented a generalized correlation in terms of viscosity and molar density that was applicable to both gases and liquids. The average absolute deviation for gases was only about 8 percent, while for liquids it was 15 percent. Their expression relies on the Chapman-Enskog correlation [Eq. (5-194)] for the low-pressure diffusivity and the Stiel-Thodos correlation for low-pressure viscosity: xAµ°A MA1/2 + xBµ°B MB1/2 µ° = ᎏᎏᎏ xAMA1/2 + xBMB1/2 i where µ°i ξi = 3.4 × 10−4 Tr0.94 for Tr i < 1.5 or µ°i ξi = 1.778 × 10−4 (4.58 2/3 i /Pc i Tr i − 1.67)5/8 for Tr i > 1.5. In these equations, ξi = Tc1/6 M1/2 i , and units are in cP, atm, K, and mol. For dense gases or liquids, the Chung et al. or Jossi-Stiel-Thodos correlation may be used to estimate viscosity. The latter is: (µ − µ°) ξ + 10−4 = (0.1023 + 0.023364 ρr + 0.058533 ρ 2r − 0.040758 ρ3r + 0.093324 ρ4r)4 where

(xA TcA + xBTcB)1/6 ξ = ᎏᎏᎏᎏ (xAMA + xBMB)1/2 (xA PcA + xB PcB)

and

ρr = (xA VcA + xB VcB)ρ.

Dilute Binary Mixtures of Nonelectrolytes with Water as the Solute Olander modified the Wilke-Chang equation to adapt it to the infinite dilution diffusivity of water as the solute. The modification he recommended is simply the division of the right-hand side of the Wilke-Chang equation by 2.3. Unfortunately, neither the WilkeChang equation nor that equation divided by 2.3 fit the data very well. A reasonably valid generalization is that the Wilke-Chang equation is accurate if water is very insoluble in the solvent, such as pure hydrocarbons, halogenated hydrocarbons, and nitro-hydrocarbons. On the other hand, the Wilke-Chang equation divided by 2.3 is accurate for solvents in which water is very soluble, as well as those that have low viscosities. Such solvents include alcohols, ketones, carboxylic acids,

and aldehydes. Neither equation is accurate for higher-viscosity liquids, especially diols. Dilute Dispersions of Macromolecules in Nonelectrolytes The Stokes-Einstein equation has already been presented. It was noted that its validity was restricted to large solutes, such as spherical macromolecules and particles in a continuum solvent. The equation has also been found to predict accurately the diffusion coefficient of spherical latex particles and globular proteins. Corrections to StokesEinstein for molecules approximating spheroids is given by Tanford. Since solute-solute interactions are ignored in this theory, it applies in the dilute range only. Hiss-Cussler Their basis is the diffusion of a small solute in a fairly viscous solvent of relatively large molecules, which is the opposite of the Stokes-Einstein assumptions. The large solvent molecules investigated were not polymers or gels but were of moderate molecular weight so that the macroscopic and microscopic viscosities were the same. The major conclusion is that D°AB µ2/3 = constant at a given temperature and for a solvent viscosity from 5 × 10−3 to 5 Pa s or greater (5 to 5 × 103 cP). This observation is useful if D°AB is known in a given high-viscosity liquid (oils, tars, etc.). Use of the usual relation of D°AB ∝ 1/µ for such an estimate could lead to large errors. Concentrated, Binary Mixtures of Nonelectrolytes Several correlations that predict the composition dependence of DAB are summarized in Table 5-19. Most are based on known values of D°AB and D°BA. In fact, a rule of thumb states that, for many binary systems, D°AB and D°BA bound the DAB vs. xA curve. Cullinan’s equation predicts diffusivities even in lieu of values at infinite dilution, but requires accurate density, viscosity, and activity coefficient data. Since the infinite dilution values D°AB and D°BA are generally unequal, even a thermodynamically ideal solution like γA = γB = 1 will exhibit concentration dependence of the diffusivity. In addition, nonideal solutions require a thermodynamic correction factor to retain the true “driving force” for molecular diffusion, or the gradient of the chemical potential rather than the composition gradient. That correction factor is: ∂ ln γA βA = 1 + ᎏ (5-219) ∂ ln xA Caldwell-Babb Darken observed that solid-state diffusion in metallurgical applications followed a simple relation. His equation related the tracer diffusivities and mole fractions to the mutual diffusivity: DAB = (xA DB + xB DA) βA

(5-220)

Caldwell and Babb used virtually the same equation to evaluate the mutual diffusivity for concentrated mixtures of common liquids. Van Geet and Adamson tested that equation for the n-dodecane (A) and n-octane (B) system and found the average deviation of DAB from experimental values to be −0.68 percent. In addition, that equation was tested for benzene + bromobenzene, n-hexane + n-dodecane, benzene + CCl 4, octane + decane, heptane + cetane, benzene + diphenyl, and benzene + nitromethane with success. For systems that depart significantly from thermodynamic ideality, it breaks down, sometimes by a factor of eight. For example, in the binary systems acetone + CCl 4, acetone + chloroform, and ethanol + CCl 4, it is not accurate. Thus, it can be expected to be fairly accurate for nonpolar hydrocarbons of similar molecular weight but not for polar-polar mixtures. Siddiqi et al. found that this relation was superior to those of Vignes and Leffler and Cullinan for a variety of mixtures. Umesi and Danner found an average absolute deviation of 13.9 percent for 198 data points. Rathbun-Babb suggested that Darken’s equation could be improved by raising the thermodynamic correction factor βA to a power, n, less than unity. They looked at systems exhibiting negative deviations from Raoult’s law and found n = 0.3. Furthermore, for polarnonpolar mixtures, they found n = 0.6. In a separate study, Siddiqi and Lucas followed those suggestions and found an average absolute error of 3.3 percent for nonpolar-nonpolar mixtures, 11.0 percent for polarnonpolar mixtures, and 14.6 percent for polar-polar mixtures. Siddiqi et al. examined a few other mixtures and found that n = 1 was probably best. Thus, this approach is, at best, highly dependent on the type of components being considered.

MASS TRANSFER

5-53

TABLE 5-19 Correlations of Diffusivities for Concentrated, Binary Mixtures of Nonelectrolyte Liquids Authors*

Equation

Caldwell-Babb [5] Rathbun-Babb [46] Vignes [62] Leffler-Cullinan [34]

DAB = (xA DBA ° + xB DAB ° )βA DAB = (xA DBA ° + xB DAB ° )βAn DAB = DAB ° xB DBA ° xA βA DAB µ mix = (DAB ° µ B)xB (DBA ° µA) xA βA

Cussler [12]

K ∂ ln xA DAB = D0 1 + ᎏ ᎏ −1 xA xB ∂ ln aA

Cullinan [10]

kT 2πxA xB βA DAB = ᎏᎏ ᎏᎏ 2πµ mix (V/A)1/3 1 + βA (2πxA xB − 1)

Asfour-Dullien [2]

DAB ° DAB = ᎏ µB





−1/2







(5-221) (5-222) (5-223) (5-224) (5-225)



1/2

ζµ β  ᎏ µ  xB

DBA °

xA

(5-227)

A

A

DAB = (CB V B D AB ° + CA V A D BA ° )βA

Siddiqi-Lucas [52]

(5-226)

(5-228)

Relative errors for the correlations in this table are very dependent on the components of interest and are cited in the text. *See pages 5-7 and 5-8 for references.

Vignes empirically correlated mixture diffusivity data for 12 binary mixtures. Later Ertl et al. evaluated 122 binary systems, which showed an average absolute deviation of only 7 percent. None of the latter systems, however, was very nonideal. Leffler-Cullinan modified Vignes’ equation using some theoretical arguments to arrive at Eq. (5-224), which the authors compared to Eq. (5-223) for the 12 systems mentioned above. The average absolute maximum deviation was only 6 percent. Umesi and Danner, however, found an average absolute deviation of 11.4 percent for 198 data points. For normal paraffins, it is not very accurate. In general, the accuracies of Eqs. (5-223) and (5-224) are not much different, and, since Vignes’ is simpler to use, it is suggested. The application of either should be limited to nonassociating systems that do not deviate much from ideality (0.95 < βA < 1.05). Cussler studied diffusion in concentrated associating systems and has shown that, in associating systems, it is the size of diffusing clusters rather than diffusing solutes that controls diffusion. Do is a reference diffusion coefficient discussed hereafter; aA is the activity of component A; and K is a constant. By assuming that Do could be predicted by Eq. (5-223) with β = 1, K was found to be equal to 0.5 based on five binary systems and validated with a sixth binary mixture. The limitations of Eq. (5-225) using Do and K defined previously have not been explored, so caution is warranted. Gurkan showed that K should actually be closer to 0.3 (rather than 0.5) and discussed the overall results. Cullinan presented an extension of Cussler’s cluster diffusion theory. His method accurately accounts for composition and temperature dependence of diffusivity. It is novel in that it contains no adjustable constants, and it relates transport properties and solution thermodynamics. This equation has been tested for six very different mixtures by Rollins and Knaebel, and it was found to agree remarkably well with data for most conditions, considering the absence of adjustable parameters. In the dilute region (of either A or B), there are systematic errors probably caused by the breakdown of certain implicit assumptions (that nevertheless appear to be generally valid at higher concentrations). Asfour-Dullien developed a relation for predicting alkane diffusivities at moderate concentrations that employs: Vfm 2/3 MxAMxB ᎏ ζ= ᎏ (5-229) Vf xAVf xB Mm





where Vfxi = V ; the fluid free volume is Vf i = Vi − Vml i for i = A, B, and m, in which Vml i is the molar volume of the liquid at the melting point and xA2 2 xA xB xB2 −1 Vmlm = ᎏ +ᎏ +ᎏ VmlA VmlAB VmlB xi fi



and

1/3 V1/3 mlA + V mlB VmlAB = ᎏᎏ 2







3

and µ is the mixture viscosity; Mm is the mixture mean molecular weight; and βA is defined by Eq. (5-219). The average absolute error of this equation is 1.4 percent, while the Vignes equation and the Leffler-Cullinan equation give 3.3 percent and 6.2 percent, respectively. Siddiqi-Lucas suggested that component volume fractions might be used to correlate the effects of concentration dependence. They found an average absolute deviation of 4.5 percent for nonpolarnonpolar mixtures, 16.5 percent for polar-nonpolar mixtures, and 10.8 percent for polar-polar mixtures. Binary Electrolyte Mixtures When electrolytes are added to a solvent, they dissociate to a certain degree. It would appear that the solution contains at least three components: solvent, anions, and cations. If the solution is to remain neutral in charge at each point (assuming the absence of any applied electric potential field), the anions and cations diffuse effectively as a single component, as for molecular diffusion. The diffusion of the anionic and cationic species in the solvent can thus be treated as a binary mixture. Nernst-Haskell The theory of dilute diffusion of salts is well developed and has been experimentally verified. For dilute solutions of a single salt, the well-known Nernst-Haskell equation (Reid et al.) is applicable: 1 1 1 1 ᎏᎏ + ᎏᎏ ᎏᎏ + ᎏᎏ n− n n RT n+ + − ᎏᎏ = 8.9304 × 10−10 T ᎏᎏ (5-230) D°AB = ᎏ 1 1 F 2 ᎏ1ᎏ + ᎏ1ᎏ ᎏᎏ0 + ᎏᎏ0 λ0+ λ0− λ+ λ−

   

   

where D°AB = diffusivity based on molarity rather than normality of dilute salt A in solvent B, cm2/s. The previous definitions can be interpreted in terms of ionicspecies diffusivities and conductivities. The latter are easily measured and depend on temperature and composition. For example, the equivalent conductance Λ is commonly tabulated in chemistry handbooks as the limiting (infinite dilution) conductance Λo and at standard concentrations, typically at 25°C. Λ = 1000 K/C = λ+ + λ− = Λo + f(C), (cm2/ohm gequiv); K = α/R = specific conductance, (ohm cm)−1; C = solution concentration, (gequiv/ᐉ); α = conductance cell constant (measured), (cm−1); R = solution electrical resistance, which is measured (ohm); and f(C) = a complicated function of concentration. The resulting equation of the electrolyte diffusivity is |z+| + |z−| DAB = ᎏᎏᎏ (5-231) (|z−| / D+) + (|z+| / D−) where |z⫾| represents the magnitude of the ionic charge and where the cationic or anionic diffusivities are D⫾ = 8.9304 × 10−10 T λ⫾ / |z⫾| cm2/s. The coefficient is kN0 /F 2 = R/F 2. In practice, the equivalent conductance of the ion pair of interest would be obtained and supplemented with conductances of permutations of those ions and one independent cation and anion. This would allow determination of all the ionic conductances and hence the diffusivity of the electrolyte solution.

5-54

HEAT AND MASS TRANSFER

Gordon Typically, as the concentration of a salt increases from infinite dilution, the diffusion coefficient decreases rapidly from D°AB. As concentration is increased further, however, DAB rises steadily, often becoming greater than D°AB. Gordon proposed the following empirical equation, which is applicable up to concentrations of 2N: ln γ⫾ 1 µB 1+ᎏ (5-232) DAB = D°AB ᎏ ᎏ CBVB µ ln m where D°AB is given by the Nernst-Haskell equation. References that tabulate γ⫾ as a function of m, as well as other equations for DAB, are given by Reid et al. Multicomponent Mixtures No simple, practical estimation methods have been developed for predicting multicomponent liquiddiffusion coefficients. Several theories have been developed, but the necessity for extensive activity data, pure component and mixture volumes, mixture viscosity data, and tracer and binary diffusion coefficients have significantly limited the utility of the theories (see Reid et al.). The generalized Stefan-Maxwell equations using binary diffusion coefficients are not easily applicable to liquids since the coefficients are so dependent on conditions. That is, in liquids, each Dij can be strongly composition dependent in binary mixtures and, moreover, the binary Dij is strongly affected in a multicomponent mixture. Thus, the convenience of writing multicomponent flux equations in terms of binary coefficients is lost. Conversely, they apply to gas mixtures because each Dij is practically independent of composition by itself and in a multicomponent mixture (see Taylor and Krishna for details). One particular case of multicomponent diffusion that has been examined is the dilute diffusion of a solute in a homogeneous mixture (e.g., of A in B + C). Umesi and Danner compared the three equations given below for 49 ternary systems. All three equations were equivalent, giving average absolute deviations of 25 percent. Perkins-Geankoplis





n

Dam µ m0.8 =
xj D°A j µ0.8 j

(5-233)

j=1 j≠A

Cullinan This is an extension of Vignes’ equation to multicomponent systems:

applications of interest are outlined in Table 5-20. Applications of these equations are found in Secs. 16, 22, and 23. Diffusion within the largest cavities of a porous medium is assumed to be similar to ordinary or bulk diffusion except that it is hindered by the pore walls (see Eq. 5-236). The tortuosity τ that expresses this hindrance has been estimated from geometric arguments. Unfortunately, measured values are often an order of magnitude greater than those estimates. Thus, the effective diffusivity Deff (and hence τ) is normally determined by comparing a diffusion model to experimental measurements. The normal range of tortuosities for silica gel, alumina, and other porous solids is 2 ≤ τ ≤ 6, but for activated carbon, 5 ≤ τ ≤ 65. In small pores and at low pressures, the mean free path ᐉ of the gas molecule (or atom) is significantly greater than the pore diameter dpore. Its magnitude may be estimated from 3.2 µ RT 1/2 ᐉ = ᎏ ᎏ , in m P 2πM As a result, collisions with the wall occur more frequently than with other molecules. This is referred to as the Knudsen mode of diffusion and is contrasted with ordinary or bulk diffusion, which occurs by intermolecular collisions. At intermediate pressures, both ordinary diffusion and Knudsen diffusion may be important [see Eqs. (5-239) and (5-240)]. For gases and vapors that adsorb on the porous solid, surface diffusion may be important, particularly at high surface coverage [see Eqs. (5-241) and (5-244)]. The mechanism of surface diffusion may be viewed as molecules hopping from one surface site to another. Thus, if adsorption is too strong, surface diffusion is impeded, while if adsorption is too weak, surface diffusion contributes insignificantly to the overall rate. Surface diffusion and bulk diffusion usually occur in parallel [see Eqs. (5-245) and (5-246)]. Although Ds is expected to be less than Deff, the solute flux due to surface diffusion may be larger than that due to bulk diffusion if ∂qi / ∂z >> ∂Ci / ∂z. This can occur when a component is strongly adsorbed and the surface coverage is high. For all that, surface diffusion is not well understood. The references in Table 5-20 should be consulted for further details.





INTERPHASE MASS TRANSFER

DIFFUSION OF FLUIDS IN POROUS SOLIDS

Transfer of material between phases is important in most separation processes in which two phases are involved. When one phase is pure, mass transfer in the pure phase is not involved. For example, when a pure liquid is being evaporated into a gas, only the gas-phase mass transfer need be calculated. Occasionally, mass transfer in one of the two phases may be neglected even though pure components are not involved. This will be the case when the resistance to mass transfer is much larger in one phase than in the other. Understanding the nature and magnitudes of these resistances is one of the keys to performing reliable mass transfer. In this section, mass transfer between gas and liquid phases will be discussed. The principles are easily applied to the other phases. Mass-Transfer Principles: Dilute Systems When material is transferred from one phase to another across an interface that separates the two, the resistance to mass transfer in each phase causes a concentration gradient in each, as shown in Fig. 5-26 for a gas-liquid interface. The concentrations of the diffusing material in the two phases immediately adjacent to the interface generally are unequal, even if expressed in the same units, but usually are assumed to be related to each other by the laws of thermodynamic equilibrium. Thus, it is assumed that the thermodynamic equilibrium is reached at the gas-liquid interface almost immediately when a gas and a liquid are brought into contact. For systems in which the solute concentrations in the gas and liquid phases are dilute, the rate of transfer may be expressed by equations which predict that the rate of mass transfer is proportional to the difference between the bulk concentration and the concentration at the gas-liquid interface. Thus

Diffusion in porous solids is usually the most important factor controlling mass transfer in adsorption, ion exchange, drying, heterogeneous catalysis, leaching, and many other applications. Some of the

where NA = mass-transfer rate, k′G = gas-phase mass-transfer coefficient, k′L = liquid-phase mass-transfer coefficient, p = solute partial pressure in

Dam =

n

 (D° ) Aj

xj

(5-234)

j=1 j≠A

Leffler-Cullinan They extended their binary relation to an arbitrary multicomponent mixture, as follows: Dam µm =

n

 (D° µ ) Aj

j

xj

(5-235)

j=1 j≠A

where DAj is the dilute binary diffusion coefficient of A in j; DAm is the dilute diffusion of A through m; xj is the mole fraction; µj is the viscosity of component j; and µm is the mixture viscosity. Akita Another case of multicomponent dilute diffusion of significant practical interest is that of gases in aqueous electrolyte solutions. Many gas-absorption processes use electrolyte solutions. Akita presents experimentally tested equations for this case. Graham-Dranoff They studied multicomponent diffusion of electrolytes in ion exchangers. They found that the Stefan-Maxwell interaction coefficients reduce to limiting ion tracer diffusivities of each ion. Pinto-Graham Pinto and Graham studied multicomponent diffusion in electrolyte solutions. They focused on the Stefan-Maxwell equations and corrected for solvation effects. They achieved excellent results for 1-1 electrolytes in water at 25°C up to concentrations of 4M.

NA = k′G(p − pi) = k′L(ci − c)

(5-248)

MASS TRANSFER TABLE 5-20

5-55

Relations for Diffusion in Porous Solids

Mechanism

Equation

Bulk diffusion in pores

ε pD Deff = ᎏ τ

Knudsen diffusion

T DK = 48.5 dpore ᎏ M

 

Applies to

1/2

in m2/s

References*

(5-236)

Gases or liquids in large pores. NK n = ᐉ/d pore < 0.01

[67]

(5-237)

Dilute (low pressure) gases in small pores. NK n = ᐉ/d pore > 10

Geankoplis, [68, 69]

(5-238)

"

"

"

"

(5-239)

"

"

"

"

ε pDK DKeff = ᎏ τ dC Ni = −DK ᎏi dz Combined bulk and Knudsen diffusion

1 − α xA 1 Deff = ᎏ + ᎏ Deff DKeff



−1



NB α=1+ᎏ NA



1 1 Deff = ᎏ + ᎏ Deff DKeff Surface diffusion

−1



(5-240)

NA = NB

dq JSi = −DSeff ρp ᎏi dz

(5-241)

Adsorbed gases or vapors

ε p DS DSeff = ᎏ τ

(5-242)

"

DSθ = 0 DSθ = ᎏ (1 − θ)

(5-243)

θ = fractional surface coverage ≤ 0.6

(5-244)

"

"

"

"

(5-245)

"

"

"

"

(5-246)

"

"

"

"

(5-247)

"

"

"

"

 

−E DS = DS′ (q) exp ᎏS RT

 

Parallel bulk and surface diffusion

Geankoplis, [66, 69]

NA ≠ NB

  

 

dp dq J = − Deff ᎏi + DSeff ρp ᎏi dz dz

 

dp J = −Dapp ᎏi dz

 

dqi Dapp = Deff + DSeff ρp ᎏ dpi

"

"

[66, 68, 69]

"

[68]

*See pages 5-7 and 5-8 for references.

bulk gas, pi = solute partial pressure at interface, c = solute concentration in bulk liquid, and ci = solute concentration in liquid at interface. The mass-transfer coefficients k′G and k′L by definition are equal to the ratios of the molal mass flux NA to the concentration driving forces (p − pi) and (ci − c) respectively. An alternative expression for the rate of transfer in dilute systems is given by NA = kG(y − yi) = kL(xi − x) (5-249)

where NA = mass-transfer rate, kG = gas-phase mass-transfer coefficient, kL = liquid-phase mass-transfer coefficient, y = mole-fraction solute in bulk-gas phase, yi = mole-fraction solute in gas at interface, x = mole-fraction solute in bulk-liquid phase, and xi = mole-fraction solute in liquid at interface. The mass-transfer coefficients defined by Eqs. (5-248) and (5-249) are related to each other as follows: kG = k′G pT

(5-250)

kL = k′L ρ (5-251) L where pT = total system pressure employed during the experimental determinations of k′G values and ρ L = average molar density of the liquid phase. The coefficient kG is relatively independent of the total system pressure and therefore is more convenient to use than k′G, which is inversely proportional to the total system pressure. The above equations may be used for finding the interfacial concentrations corresponding to any set of values of x and y provided the ratio of the individual coefficients is known. Thus (y − yi)/(xi − x) = kL /kG = k′Lρ L/k′G pT = LMHG /GMHL

FIG. 5-26

Concentration gradients near a gas-liquid interface.

(5-252)

where LM = molar liquid mass velocity, GM = molar gas mass velocity, HL = height of one transfer unit based on liquid-phase resistance, and HG = height of one transfer unit based on gas-phase resistance. The last term in Eq. (5-252) is derived from Eqs. (5-271) and (5-273). Equation (5-252) may be solved graphically if a plot is made of the equilibrium vapor and liquid compositions and a point representing

5-56

HEAT AND MASS TRANSFER

the bulk concentrations x and y is located on this diagram. A construction of this type is shown in Fig. 5-27, which represents a gasabsorption situation. The interfacial mole fractions yi and xi can be determined by solving Eq. (5-252) simultaneously with the equilibrium relation y°i = F(xi) to obtain yi and xi. The rate of transfer may then be calculated from Eq. (5-249). If the equilibrium relation y°i = F(xi) is sufficiently simple, e.g., if a plot of y°i versus xi is a straight line, not necessarily through the origin, the rate of transfer is proportional to the difference between the bulk concentration in one phase and the concentration (in that same phase) which would be in equilibrium with the bulk concentration in the second phase. One such difference is y − y°, and another is x° − x. In this case, there is no need to solve for the interfacial compositions, as may be seen from the following derivation. The rate of mass transfer may be defined by the equation NA = KG(y − y°) = kG(y − yi) = kL(xi − x) = KL(x° − x) (5-253) where KG = overall gas-phase mass-transfer coefficient, KL = overall liquid-phase mass-transfer coefficient, y° = vapor composition in equilibrium with x, and x° = liquid composition in equilibrium with vapor of composition y. This equation can be rearranged to the formula 1 1 y − y° 1 1 yi − y° 1 1 yi − y° ᎏ=ᎏ ᎏ =ᎏ+ᎏ ᎏ =ᎏ+ᎏ ᎏ KG kG y − yi kG kG y − yi kG kL xi − x













(5-254) in view of Eq. (5-252). Comparison of the last term in parentheses with the diagram of Fig. 5-27 shows that it is equal to the slope of the chord connecting the points (x,y°) and (xi,yi). If the equilibrium curve is a straight line, then this term is the slope m. Thus 1/KG = (1/ kG + m/ kL) (5-255) When Henry’s law is valid (pA = HxA or pA = H′CA), the slope m can be computed according to the relationship m = H/ pT = H′ρ (5-256) L/ pT where m is defined in terms of mole-fraction driving forces compatible with Eqs. (5-249) through (5-255), i.e., with the definitions of kL, kG, and KG. If it is desired to calculate the rate of transfer from the overall concentration difference based on bulk-liquid compositions (x° − x), the appropriate overall coefficient KL is related to the individual coefficients by the equation 1/KL = (1/kL + 1/mkG) (5-257) Conversion of these equations to a k′G, k′L basis can be accomplished readily by direct substitution of Eqs. (5-250) and (5-251). Occasionally one will find k′L or K′L values reported in units (SI) of meters per second. The correct units for these values are kmol/

[(s⋅m2)(kmol/m3)], and Eq. (5-251) is the correct equation for converting them to a mole-fraction basis. When k′G and K′G values are reported in units (SI) of kmol/[(s⋅m2) (kPa)], one must be careful in converting them to a mole-fraction basis to multiply by the total pressure actually employed in the original experiments and not by the total pressure of the system to be designed. This conversion is valid for systems in which Dalton’s law of partial pressures (p = ypT) is valid. Comparison of Eqs. (5-255) and (5-257) shows that for systems in which the equilibrium line is straight, the overall mass transfer coefficients are related to each other by the equation KL = mKG

(5-258)

When the equilibrium curve is not straight, there is no strictly logical basis for the use of an overall transfer coefficient, since the value of m will be a function of position in the apparatus, as can be seen from Fig. 5-27. In such cases the rate of transfer must be calculated by solving for the interfacial compositions as described above. Experimentally observed rates of mass transfer often are expressed in terms of overall transfer coefficients even when the equilibrium lines are curved. This procedure is empirical, since the theory indicates that in such cases the rates of transfer may not vary in direct proportion to the overall bulk concentration differences (y − y°) and (x° − x) at all concentration levels even though the rates may be proportional to the concentration difference in each phase taken separately, i.e., (xi − x) and (y − yi). In most types of separation equipment such as packed or spray towers, the interfacial area that is effective for mass transfer cannot be accurately determined. For this reason it is customary to report experimentally observed rates of transfer in terms of transfer coefficients based on a unit volume of the apparatus rather than on a unit of interfacial area. Such volumetric coefficients are designated as KGa, kLa, etc., where a represents the interfacial area per unit volume of the apparatus. Experimentally observed variations in the values of these volumetric coefficients with variations in flow rates, type of packing, etc., may be due as much to changes in the effective value of a as to changes in k. Calculation of the overall coefficients from the individual volumetric coefficients is made by means of the equations 1/KGa = (1/kGa + m/kLa)

(5-259)

1/KLa = (1/kLa + 1/mkGa)

(5-260)

Because of the wide variation in equilibrium, the variation in the values of m from one system to another can have an important effect on the overall coefficient and on the selection of the type of equipment to use. For example, if m is large, the liquid-phase part of the overall resistance might be extremely large where kL might be relatively small. This kind of reasoning must be applied with caution, however, since species with different equilibrium characteristics are separated under different operating conditions. Thus, the effect of changes in m on the overall resistance to mass transfer may partly be counterbalanced by changes in the individual specific resistances as the flow rates are changed. Mass-Transfer Principles: Concentrated Systems When solute concentrations in the gas and/or liquid phases are large, the equations derived above for dilute systems no longer are applicable. The correct equations to use for concentrated systems are as follows: NA = kˆ G(y − yi)/yBM = kˆ L(xi − x)/xBM = Kˆ G(y − y°)/y°BM = Kˆ L(x° − x)/x°BM

(5-261)

where (NB = 0)

Identification of concentrations at a point in a countercurrent absorption tower.

FIG. 5-27

(1 − y) − (1 − yi) yBM = ᎏᎏ ln [(1 − y)/(1 − yi)] (1 − y) − (1 − y°) y°BM = ᎏᎏᎏ ln [(1 − y)/(1 − y°)] (1 − x) − (1 − xi) xBM = ᎏᎏ ln [(1 − x)/(1 − xi)] (1 − x) − (1 − x°) x°BM = ᎏᎏ ln [(1 − x)/(1 − x°)]

(5-262) (5-263) (5-264) (5-265)

MASS TRANSFER and where kˆ G and kˆ L are the gas-phase and liquid-phase mass-transfer coefficients for concentrated systems and Kˆ G and Kˆ L are the overall gas-phase and liquid-phase mass-transfer coefficients for concentrated systems. These coefficients are defined later in Eqs. (5-268) to (5-270). The factors yBM and xBM arise from the fact that, in the diffusion of a solute through a second stationary layer of insoluble fluid, the resistance to diffusion varies in proportion to the concentration of the insoluble stationary fluid, approaching zero as the concentration of the insoluble fluid approaches zero. See Eq. (5-190). The factors y°BM and x°BM cannot be justified on the basis of masstransfer theory since they are based on overall resistances. These factors therefore are included in the equations by analogy with the corresponding film equations. In dilute systems the logarithmic-mean insoluble-gas and nonvolatileliquid concentrations approach unity, and Eq. (5-261) reduces to the dilute-system formula. For equimolar counter diffusion (e.g., binary distillation), these log-mean factors should be omitted. See Eq. (5-189). Substitution of Eqs. (5-262) through (5-265) into Eq. (5-261) results in the following simplified formula: NA = kˆ G ln [(1 − yi)/(1 − y)] = Kˆ G ln [(1 − y°)/(1 − y)] = kˆ L ln [(1 − x)/(1 − xi)] (5-266) = Kˆ L ln [(1 − x)/(1 − x°)] ˆ ˆ ˆ ˆ Note that the units of kG, KG, kL, and KL are all identical to each other, i.e., kmol/(s⋅m2) in SI units. The equation for computing the interfacial gas and liquid compositions in concentrated systems is (y − yi)/(xi − x) = kˆ LyBM / kˆ GxBM = LMHGyBM /GMHL x BM = kL /kG

(5-267) This equation is identical to the one for dilute systems since kˆ G = kGyBM and kˆ L = kLxBM. Note, however, that when kˆ G and kˆ L are given, the equation must be solved by trial and error, since xBM contains xi and yBM contains yi. The overall gas-phase and liquid-phase mass-transfer coefficients for concentrated systems are computed according to the following equations: 1 yBM 1 xBM 1 yi − y° =ᎏ (5-268) ᎏ ᎏ+ᎏᎏ ᎏ Kˆ G y°BM kˆ G y°BM kˆ L xi − x





1 xBM 1 yBM 1 x° − xi (5-269) =ᎏᎏ+ᎏᎏ ᎏ ᎏ Kˆ L x°BM kˆ L x°BM kˆ G y − yi When the equilibrium curve is a straight line, the terms in parentheses can be replaced by the slope m as before. In this case the overall mass-transfer coefficients for concentrated systems are related to each other by the equation Kˆ L = m Kˆ G(x°BM /y°BM) (5-270)





All these equations reduce to their dilute-system equivalents as the inert concentrations approach unity in terms of mole fractions of inert concentrations in the fluids. HTU (Height Equivalent to One Transfer Unit) Frequently the values of the individual coefficients of mass transfer are so strongly dependent on flow rates that the quantity obtained by dividing each coefficient by the flow rate of the phase to which it applies is more nearly constant than the coefficient itself. The quantity obtained by this procedure is called the height equivalent to one transfer unit, since it expresses in terms of a single length dimension the height of apparatus required to accomplish a separation of standard difficulty. The following relations between the transfer coefficients and the values of HTU apply: H = G /k ay = G /kˆ a (5-271) G

M

G

BM

M

G

HOG = GM /KGay°BM = GM/Kˆ Ga H = L /k ax = L /kˆ a

(5-272)

HOL = LM /KLax°BM = LM/Kˆ La

(5-274)

L

M

L

BM

M

L

(5-273)

5-57

The equations that express the addition of individual resistances in terms of HTUs, applicable to either dilute or concentrated systems, are yBM mGM xBM HOG = ᎏ HG + ᎏ (5-275) ᎏ HL y°BM LM y°BM xBM LM yBM (5-276) HOL = ᎏ HL + ᎏ ᎏ HG x°BM mGM x°BM These equations are strictly valid only when m, the slope of the equilibrium curve, is constant, as noted previously. NTU (Number of Transfer Units) The NTU required for a given separation is closely related to the number of theoretical stages or plates required to carry out the same separation in a stagewise or plate-type apparatus. For equimolal counterdiffusion, such as in a binary distillation, the number of overall gas-phase transfer units NOG required for changing the composition of the vapor stream from y1 to y2 is y1 dy (5-277) NOG = ᎏ y2 y − y° When diffusion is in one direction only, as in the absorption of a soluble component from an insoluble gas, y1 y°BM dy (5-278) NOG = ᎏᎏ y2 (1 − y)(y − y°) The total height of packing required is then





hT = HOGNOG

(5-279)

When it is known that HOG varies appreciably within the tower, this term must be placed inside the integral in Eqs. (5-277) and (5-278) for accurate calculations of hT. For example, the packed-tower design equation in terms of the overall gas-phase mass-transfer coefficient for absorption would be expressed as follows: y1 y°BM dy GM hT = (5-280) ᎏ ᎏᎏ y2 KGay°BM (1 − y)(y − y°) where the first term under the integral can be recognized as the HTU term. Convenient solutions of these equations for special cases are discussed later. Definitions of Mass-Transfer Coefficients kˆ G and kˆ L The mass-transfer coefficient is defined as the ratio of the molal mass flux NA to the concentration driving force. This leads to many different ways of defining these coefficients. For example, gas-phase masstransfer rates may be defined as N = k (y − y ) = k′ (p − p ) = kˆ (y − y )/y (5-281)

 

A

G

i



G

i

G

i

BM

where the units (SI) of kG are kmol/[(s⋅m2)(mole fraction)], the units of k′G are kmol/[(s⋅m2)(kPa)], and the units of kˆ G are kmol/(s⋅m2). These coefficients are related to each other as follows: kG = kGyBM = k′G pT yBM

(5-282)

where pT is the total system pressure (it is assumed here that Dalton’s law of partial pressures is valid). In a similar way, liquid-phase mass-transfer rates may be defined by the relations N = k (x − x) = k′ (c − c) = kˆ (x − x)/x (5-283) A

L

i

L

i

L

i

BM

where the units (SI) of kL are kmol/[(s⋅m2)(mole fraction)], the units of k′L are kmol/[(s⋅m2)(kmol/m3)] or meters per second, and the units of kˆ L are kmol/(s⋅m2). These coefficients are related as follows: (5-284) kˆ L = kLxBM = k′Lρ LxBM is the molar density of the liquid phase in units (SI) of kilowhere ρ L moles per cubic meter. Note that, for dilute solutions where xBM ⬟ 1, kL and kˆ L will have identical numerical values. Similarly, for dilute gases kˆ G ⬟ kG. Simplified Mass-Transfer Theories In certain simple situations, the mass-transfer coefficients can be calculated from first principles. The film, penetration, and surface-renewal theories are attempts to extend these theoretical calculations to more complex sit-

5-58

HEAT AND MASS TRANSFER

uations. Although these theories are often not accurate, they are useful to provide a physical picture for variations in the mass-transfer coefficient. For the special case of steady-state unidirectional diffusion of a component through an inert-gas film in an ideal-gas system, the rate of mass transfer is derived as 1 − yi DABpT (y − yi) DABpT NA = ᎏ (5-285) ᎏ = ᎏ ln ᎏ RT δG yBM RT δG 1−y where DAB = the diffusion coefficient or “diffusivity,” δG = the “effective” thickness of a stagnant-gas layer which would offer a resistance to molecular diffusion equal to the experimentally observed resistance, and R = the gas constant. [Nernst, Z. Phys. Chem., 47, 52 (1904); Whitman, Chem. Mat. Eng., 29, 149 (1923), and Lewis and Whitman, Ind. Eng. Chem., 16, 1215 (1924)]. The film thickness δG depends primarily on the hydrodynamics of the system and hence on the Reynolds number and the Schmidt number. Thus, various correlations have been developed for different geometries in terms of the following dimensionless variables: NSh = kˆ GRTd/DABpT = f(NRe,NSc) (5-286) where NSh is the Sherwood number, NRe (= Gd/µG) is the Reynolds number based on the characteristic length d appropriate to the geometry of the particular system; and NSc (= µG /ρGDAB) is the Schmidt number. According to this analysis one can see that for gas-absorption problems, which often exhibit unidirectional diffusion, the most appropriate driving-force expression is of the form (y − yi)/yBM, and the most appropriate mass-transfer coefficient is therefore kˆ G. This concept is to be found in all the key equations for the design of mass-transfer equipment. The Sherwood-number relation for gas-phase mass-transfer coefficients as represented by the film diffusion model in Eq. (5-286) can be rearranged as follows: NSh = (kˆ G /GM)NReNSc = NStNReNSc = f (NRe,NSc) (5-287) where NSt = kˆ G /GM = k′G pBM /GM is known as the Stanton number. This equation can now be stated in the alternative functional forms NSt = kˆ G /GM = g(NRe,NSc) (5-288) 2 /3 (5-289) jD = NSt ⋅ NSc where j is the Chilton-Colburn “j factor” for mass transfer (discussed later). The important point to note here is that the gas-phase masstransfer coefficient kˆ G depends principally upon the transport properties of the fluid (NSc) and the hydrodynamics of the particular system involved (NRe). It also is important to recognize that specific masstransfer correlations can be derived only in conjunction with the investigator’s particular assumptions concerning the numerical values of the effective interfacial area a of the packing. The stagnant-film model discussed previously assumes a steady state in which the local flux across each element of area is constant; i.e., there is no accumulation of the diffusing species within the film. Higbie [Trans. Am. Inst. Chem. Eng., 31, 365 (1935)] pointed out that industrial contactors often operate with repeated brief contacts between phases in which the contact times are too short for the steady state to be achieved. For example, Higbie advanced the theory that in a packed tower the liquid flows across each packing piece in laminar flow and is remixed at the points of discontinuity between the packing elements. Thus, a fresh liquid surface is formed at the top of each piece, and as it moves downward, it absorbs gas at a decreasing rate until it is mixed at the next discontinuity. This is the basis of penetration theory. If the velocity of the flowing stream is uniform over a very deep region of liquid (total thickness, δT >>  Dt), the time-averaged masstransfer coefficient according to penetration theory is given by k′L = 2 D πt (5-290) L/ where k′L = liquid-phase mass-transfer coefficient, DL = liquid-phase diffusion coefficient, and t = contact time. In practice, the contact time t is not known except in special cases in which the hydrodynamics are clearly defined. This is somewhat

similar to the case of the stagnant-film theory in which the unknown quantity is the thickness of the stagnant layer δ (in film theory, the liquid-phase mass-transfer coefficient is given by k′L = DL /δ). The penetration theory predicts that k′L should vary by the square root of the molecular diffusivity, as compared with film theory, which predicts a first-power dependency on D. Various investigators have reported experimental powers of D ranging from 0.5 to 0.75, and the Chilton-Colburn analogy suggests a w power. Penetration theory often is used in analyzing absorption with chemical reaction because it makes no assumption about the depths of penetration of the various reacting species, and it gives a more accurate result when the diffusion coefficients of the reacting species are not equal. When the reaction process is very complex, however, penetration theory is more difficult to use than film theory, and the latter method normally is preferred. Danckwerts [Ind. Eng. Chem., 42, 1460 (1951)] proposed an extension of the penetration theory, called the surface renewal theory, which allows for the eddy motion in the liquid to bring masses of fresh liquid continually from the interior to the surface, where they are exposed to the gas for finite lengths of time before being replaced. In his development, Danckwerts assumed that every element of fluid has an equal chance of being replaced regardless of its age. The Danckwerts model gives

s k′L = D

(5-291)

where s = fractional rate of surface renewal. Note that both the penetration and the surface-renewal theories predict a square-root dependency on D. Also, it should be recognized that values of the surface-renewal rate s generally are not available, which presents the same problems as do δ and t in the film and penetration models. The predictions of correlations based on the film model often are nearly identical to predictions based on the penetration and surfacerenewal models. Thus, in view of its relative simplicity, the film model normally is preferred for purposes of discussion or calculation. It should be noted that none of these theoretical models has proved adequate for making a priori predictions of mass-transfer rates in packed towers, and therefore empirical correlations such as those outlined later in Table 5-28. must be employed. Mass-Transfer Correlations Because of the tremendous importance of mass transfer in chemical engineering, a very large number of studies have determined mass-transfer coefficients both empirically and theoretically. Some of these studies are summarized in Tables 5-21 to 5-28. Each table is for a specific geometry or type of contactor, starting with flat plates, which have the simplest geometry (Table 5-21); then wetted wall columns (Table 5-22); flow in pipes and ducts (Table 5-23); submerged objects (Table 5-24); drops and bubbles (Table 5-25); agitated systems (Table 5-26); packed beds of particles for adsorption, ion exchange, and chemical reaction (Table 5-27); and finishing with packed bed two-phase contactors for distillation, absorption and other unit operations (Table 5-28). Graphical correlations for the Bolles and Fair correlation (Table 5-28-G) are in Figs. 5-28 to 5-30. Although extensive, these tables are not meant to be encyclopedic. For simple geometries, one may be able to determine a theoretical (T) form of the mass-transfer correlation. For very complex geometries, only an empirical (E) form can be found. In systems of intermediate complexity, semiempirical (S) correlations where the form is determined from theory and the coefficients from experiment are often useful. Although the major limitations and constraints in use are usually included in the tables, obviously many details cannot be included in this summary form. Readers are strongly encouraged to check the references before using the correlations in important situations. Note that even authoritative sources occasionally have typographical errors in the fairly complex correlation equations. Thus, it is a good idea to check several sources, including the original paper. The references will often include figures comparing the correlations with data. These figures are very useful since they provide a visual picture of the scatter in the data. Since there are often several correlations that are applicable, how does one choose the correlation to use? First, the engineer must determine which correlations are closest to the current situation. This

MASS TRANSFER TABLE 5-21

5-59

Mass Transfer Correlations for a Single Flat Plate or Disk—Transfer to or from Plate to Fluid Situation

A. Laminar, local, flat plate, forced flow

Laminar, average, flat plate, forced flow

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlation k′x NSh,x = ᎏ = 0.323(NRe,x)1/2(NSc)1/3 D Coefficient 0.332 is a better fit.

B. Laminar, local, flat plate, blowing or suction and forced flow

C. Laminar, local, flat plate, natural convection vertical plate

[T] Low M.T. rates. Low mass-flux, constant property systems. NSh,x is local k. Use with arithmetic difference in concentration. Coefficient 0.323 is Blasius’ approximate solution.

[100] p. 183 [108] p. 526 [146] p. 79 [150] p. 518

xu ∞ ρ NRe,x = ᎏ , x = length along plate µ

[151] p. 110

Lu ∞ ρ NRe,L = ᎏ , 0.664 (Polhausen) µ

k′m L NSh,avg = ᎏ = 0.646(NRe,L)1/2(NSc)1/3 D k′m is mean mass-transfer coefficient for dilute systems.

j-factors

References*

f jD = jH = ᎏ = 0.664(NRe,L)−1/2 2 k′x NSh,x = ᎏ = (Slope)y = 0 (NRe,x)1/2(NSc)1/3 D

k′x −1/4 1/4 NSh,x = ᎏ = 0.508 N1/2 N Gr Sc (0.952 + NSc) D

is a better fit for NSc > 0.6, NRe,x < 3 × 105. [S] Analogy. Nsc = 1.0, f = drag coefficient. jD is defined in terms of k′m.

[151] p. 271

[T] Blowing is positive. Other conditions as above. uo Re ,x ᎏᎏ N u∞ 0.6 0.5 0.25 0.0 −2.5 ᎏᎏ ᎏᎏᎏ (Slope)y = 0 0.01 0.06 0.17 0.332 1.64

[100] p. 185

[T] Low MT rates. Dilute systems, ∆ρ/ρ << 1. NGr NSc < 108. Use with arithmetic concentration difference. x = length from plate bottom.

[151] p. 120

[150] p. 271

gx 3 ρ∞ NGr = ᎏ2 ᎏ −1 (µ/ρ) ρ0



D. Laminar, stationary disk

Laminar, spinning disk

E. Laminar, inclined, plate

k′d disk 8 NSh = ᎏ = ᎏ D π k′d disk 1/3 NSh = ᎏ = 0.879N 1/2 Re N Sc D

3 2 1/3 x ρ g sinα NSh,avg = 0.783 N1/9 ᎏᎏ Re,film N Sc µ2







[T] Stagnant fluid. Use arithmetic concentration difference.

[146] p. 240

[T] Asymptotic solution for large NSc. NRe < ∼ 104 u = ωddisk/2, ω = rotational speed, rad/s. Rotating disks are often used in electrochemical research.

[117] p. 60

[T] Constant-property liquid film with low masstransfer rates. Use arithmetic concentration difference.

[151] p. 130

[146] p. 240

2/9

4Qρ NRe,film = ᎏ < 2000 µ2 k′m x NSh,avg = ᎏ D



3µQ δ film = ᎏᎏ wρg sinα

F. Turbulent, local flat plate, forced flow

Turbulent, average, flat plate, forced flow

w = width of plate, δ f = film thickness, α = angle of inclination, x = distance from start soluble surface.



1/3

= film thickness

k′x 0.8 NSh,x = ᎏ = 0.0292N Re,x , D k′L 0.8 , average coefficient NSh,avg = ᎏ = 0.0365N Re,L D

Newtonian fluid. Solute does not penetrate past region of linear velocity profile. Differences between theory and experiment.

[146] p. 209

[S] Low mass-flux with constant property system. Use with arithmetic concentration difference. NSc = 1.0, NRe,x > 105

[100] p. 191 [146] p. 201 [151] p. 221

Based on Prandtl’s 1/7-power velocity law,



y u ᎏ= ᎏ u∞ δ G. Laminar and turbulent, flat plate, forced flow

f −0.2 jD = jH = ᎏ = 0.037 N Re,L 2

1/7

Chilton-Colburn analogies, NSc = 1.0, (gases), f = drag coefficient. Corresponds to item 5-21-F and refers to same conditions. 8000 < NRe < 300,000. Can apply analogy, jD = f/2, to entire plate (including laminar portion) if average values are used.

[100] p. 193 [109] p. 112 [146] p. 201 [151] p. 271

5-60

HEAT AND MASS TRANSFER

TABLE 5-21

Mass Transfer Correlations for a Single Flat Plate or Disk—Transfer to or from Plate to Fluid (Concluded ) Situation

H. Laminar and turbulent, flat plate, forced flow

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlation 0.8 NSh,avg = 0.037N 1/3 Sc (N Re,L − 15,500) to NRe,L = 320,000

k′m L NSh,avg = ᎏ , NSc > 0.5 D Entrance effects are ignored. NRe,Cr is transition laminar to turbulent.

NSh,avg = 0.037N 1/3 Sc



References*

[E] Use arithmetic concentration difference.



0.664 1/2 0.8 0.8 × N Re,L − N Re,Cr + ᎏ N Re,Cr 0.037

[109] p. 112 [146] p. 201

in range 3 × 105 to 3 × 106. I. Turbulent, local flat plate, natural convection, vertical plate Turbulent, average, flat plate, natural convection, vertical plate

k′x 2/5 7/15 NSh,x = ᎏ = 0.0299N Gr N Sc D 2/3 −2/5 × (1 + 0.494N Sc ) 2/5 7/15 2/3 −2/5 NSh,avg = 0.0249N Gr N Sc × (1 + 0.494N Sc )

[S] Low solute concentration and low transfer rates. Use arithmetic concentration difference. gx3 ρ∞ NGr = ᎏ2 ᎏ −1 (µ/ρ) ρ0



[151] p. 225



NGr > 1010 Assumes laminar boundary layer is small fraction of total. k′m L NSh,avg = ᎏ D

J. Turbulent, vertical plate

k′m x x 3ρ 2g 2/9 NSh,avg = ᎏ = 0.327N Re,film N 1/3 Sc ᎏ µ2 D



 

Q2 δ film = 0.172 ᎏ w 2g K. Turbulent, spinning disk

L. Mass transfer to a flat plate membrane in a stirred vessel



2/9

[E] See 5-21-E for terms.

[151] p. 229

4Qρ NRe,film = ᎏ > 2360 wµ2 Solute remains in laminar sublayer.

1/3

k ′ddisk 1.1 1/3 NSh = ᎏ = 5.6N Re N Sc D

k′dtank b c NSh = ᎏ = aN Re N Sc D a depends on system. a = 0.0443 [73, 165]; b is often 0.65–0.70 [110]. If

[E] Use arithmetic concentration difference. 6 × 105 < NRe < 2 × 106 120 < NSc < 1200 u = ωddisk /2 where ω = rotational speed, radians/s. NRe = ρωd 2/2µ.

[82] [146] p. 241

[E] Use arithmetic concentration difference. ω = stirrer speed, radians/s. Useful for laboratory dialysis, R.O., U.F., and microfiltration systems.

[73] [110] p. 965 [165] p. 738

2 ωd tank ρ NRe = ᎏ µ b = 0.785 [73]. c is often 0.33 but other values have been reported [110].

*See pages 5-7 and 5-8 for references.

involves recognizing the similarity of geometries, which is often challenging, and checking that the range of parameters in the correlation is appropriate. For example, the Bravo, Rocha, and Fair correlation for distillation with structured packings with triangular cross-sectional channels (Table 5-28-H) uses the Johnstone and Pigford correlation for rectification in vertical wetted wall columns (Table 5-22-D). Recognizing that this latter correlation pertains to a rather different application and geometry was a nontrivial step in the process of developing a correlation. If several correlations appear to be applicable, check to see if the correlations have been compared to each other and to the data. When a detailed comparison of correlations is not available, the following heuristics may be useful: 1. Mass-transfer coefficients are derived from models. They must be employed in a similar model. For example, if an arithmetic concentration difference was used to determine k, that k should only be used in a mass-transfer expression with an arithmetic concentration difference. 2. Semiempirical correlations are often preferred to purely empirical or purely theoretical correlations. Purely empirical correlations are dangerous to use for extrapolation. Purely theoretical correlations may predict trends accurately, but they can be several orders of magnitude off in the value of k. 3. Correlations with broader data bases are often preferred.

4. The analogy between heat and mass transfer holds over wider ranges than the analogy between mass and momentum transfer. Good heat transfer data (without radiation) can often be used to predict mass-transfer coefficients. 5. More recent data is often preferred to older data, since end effects are better understood, the new correlation often builds on earlier data and analysis, and better measurement techniques are often available. 6. With complicated geometries, the product of the interfacial area per volume and the mass-transfer coefficient is required. Correlations of kap or of HTU are more accurate than individual correlations of k and ap since the measurements are simpler to determine the product kap or HTU. 7. Finally, if a mass-transfer coefficient looks too good to be true, it probably is incorrect. To determine the mass-transfer rate, one needs the interfacial area in addition to the mass-transfer coefficient. For the simpler geometries, determining the interfacial area is straightforward. For packed beds of particles a, the interfacial area per volume can be estimated as shown in Table 5-27-A. For packed beds in distillation, absorption, and so on in Table 5-28, the interfacial area per volume is included with the mass-transfer coefficient in the correlations for HTU. For agitated liquid-liquid systems, the interfacial area can be estimated

MASS TRANSFER

5-61

TABLE 5-22 Mass Transfer Correlations for Falling Films with a Free Surface in Wetted Wall Columns—Transfer between Gas and Liquid Situation A. Laminar, vertical wetted wall column

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlation x k′m x NSh,avg = ᎏ ≈ 3.41 ᎏ D δfilm (first term of infinite series)



3µQ δfilm = ᎏ wρg



1/3

[T] Low rates M.T. Use with log mean concentration difference. Parabolic velocity distribution in films. 4Qρ NRe,film = ᎏ < 20 wµ Derived for flat plates, used for tubes if

= film thickness

C. Turbulent, vertical wetted wall column with ripples





[E] Use with log mean concentration difference for correlations in B and C. NRe is for gas. NSc for vapor in gas. 2000 < NRe ≤ 35,000, 0.6 ≤ NSc ≤ 2.5. Use for gases, dt = tube diameter.

0.15

[E] For gas systems with rippling.





4Qρ Fits B for ᎏ = 1000 wµ





4Qρ 30 ≤ ᎏ < 1200 wµ k′m d t 0.8 1/3 NSh,avg = ᎏ = 0.023N Re N Sc D

D. Rectification in vertical wetted wall column with turbulent vapor flow, Johnstone and Pigford correlation

k′G dcol pBM 0.33 NSh,avg = ᎏᎏ = 0.0328(N R′ e) 0.77 N Sc Dv p 3000 < N R′ e < 40,000, 0.5 < NSc < 3 d col vrel ρv N′Re = ᎏ , v rel = gas velocity relative to µv 3 liquid film = ᎏ uavg in film 2

[151] p. 137 [161] p. 50

 

k′m dt 0.83 0.44 N Sc NSh,avg = ᎏ = 0.023N Re D A coefficient 0.0163 has also been reported using NRe′, where v = v of gas relative to liquid film. k′m d t 0.83 0.44 4Qρ NSh,avg = ᎏ = 0.00814N Re N Sc ᎏ D wµ

[146] p. 78

ρg 1/2 rtube ᎏ > 3.0. σ = surface tension 2σ If NRe,film > 20, surface waves and rates increase. An approximate solution Dapparent can be used. Ripples are suppressed with a wetting agent good to NRe = 1200.

w = film width (circumference in column)

B. Turbulent, vertical wetted wall column

References*

[88] p. 266 [93] [100] p.181 [146] p. 211 [151] p. 265 [159] p. 212 [161] p. 71 [88] p. 266 [106] [146] p. 213

[E] “Rounded” approximation to include ripples. Includes solid-liquid mass-transfer data to find s 0.83 coefficient on NSc. May use N Re . Use for liquids. See also Table 5-23. [E] Use logarithmic mean driving force at two ends of column. Based on four systems with gas-side resistance only. pBM = logarithmic mean partial pressure of nondiffusing species B in binary mixture. p = total pressure Modified form is used for structured packings (See Table 5-28-H).

[105] [146] p. 214

*See pages 5-7 and 5-8 for references.

from the dispersed phase holdup and mean drop size correlations. Godfrey, Obi, and Reeve [Chem. Engr. Prog. 85, 61 (Dec. 1989)] summarize these correlations. For many systems,  ddrop/dimp = −0.6 (const)NWe where NWe = ρc N 2d 3imp /σ. Effects of Total Pressure on kˆ G and kˆ L The influence of total system pressure on the rate of mass transfer from a gas to a liquid or to a solid has been shown to be the same as would be predicted from stagnant-film theory as defined in Eq. (5-285), where kˆ G = DABpT /RT δG (5-292) Since the quantity DAB pT is known to be relatively independent of the pressure, it follows that the rate coefficients kˆ G, kGyBM, and k′G pTyBM (= k′G pBM) do not depend on the total pressure of the system, subject to the limitations discussed later. Investigators of tower packings normally report k′Ga values measured at very low inlet-gas concentrations, so that yBM = 1, and at total pressures close to 100 kPa (1 atm). Thus, the correct rate coefficient for use in packed-tower designs involving the use of the driving force (y − yi)/yBM is obtained by multiplying the reported k′Ga values by the value of pT employed in the actual test unit (e.g., 100 kPa) and not the total pressure of the system to be designed.

From another point of view one can correct the reported values of k′Ga in kmol/[(s⋅m3)(kPa)], valid for a pressure of 101.3 kPa (1 atm), to some other pressure by dividing the quoted values of k′Ga by the design pressure and multiplying by 101.3 kPa, i.e., (k′Ga at design pressure pT) = (k′Ga at 1 atm) × 101.3/pT. One way to avoid a lot of confusion on this point is to convert the experimentally measured k′Ga values to values of kˆ Ga straightaway, before beginning the design calculations. A design based on the rate coefficient kˆ Ga and the driving force (y − yi)/yBM will be independent of the total system pressure with the following limitations: caution should be employed in assuming that kˆ Ga is independent of total pressure for systems having significant vapor-phase nonidealities, for systems that operate in the vicinity of the critical point, or for total pressures higher than about 3040 to 4050 kPa (30 to 40 atm). Experimental confirmations of the relative independence of kˆ G with respect to total pressure have been widely reported. Deviations do occur at extreme conditions. For example, Bretsznajder (Prediction of Transport and Other Physical Properties of Fluids, Pergamon Press, Oxford, 1971, p. 343) discusses the effects of pressure on the DABpT product and presents experimental data on the self-diffusion of CO2 which show that the D-p product begins to decrease at a pressure of

5-62

HEAT AND MASS TRANSFER

TABLE 5-23

Mass-Transfer Correlations for Flow in Pipes and Ducts—Transfer is from Wall to Fluid

Situation A. Tubes, laminar, fully developed parabolic velocity profile, developing concentration profile, constant wall concentration

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlation 0.0668(d t /x)NRe NSc k′d t NSh = ᎏ = 3.66 + ᎏᎏᎏ 1 + 0.04[(d t /x)NRe NSc]2/3 D

[T] Use log mean concentration difference. For x/d t ᎏ < 0.10, NRe < 2100. NRe NSc x = distance from tube entrance. Good agreement with experiment at values

References* [98] [100] p. 176 [108] p. 525 [151] p. 159

π d 104 > ᎏ ᎏt NReNSc > 10 4 x B. Tubes, fully developed concentration profile

k′d NSh = ᎏt = 3.66 D

[T] Subset of 5-23-A for fully developed concentration profile. x/d t ᎏ > 0.1 NRe NSc

C. Tubes, approximate solution

 

k′d d NSh,x = ᎏt = 1.077 ᎏt D x

 

E. Laminar, fully developed parabolic velocity profile, constant mass flux at wall

F. Laminar, alternate

(NRe NSc)1/3

[T] For arithmetic concentration difference.

(NRe NSc)1/3



∞

 

 

Graetz solution for heat transfer written for M.T.

11 1 ∞ exp [−λ 2j (x /rt)/(NRe NSc)] NSh, x = ᎏ − ᎏ
ᎏᎏᎏ 48 2 j = 1 Cj λ 4j



λ 2j 25.68 83.86 174.2 296.5 450.9



[T] Use arithmetic concentration difference. Fits W gas data well, for ᎏ < 50 (fit is fortuitous). Dρx NSh,avg = (k′m d t)/D. a1 = 2.405, a 2 = 5.520, a 3 = 8.654, a4 = 11.792, a 5 = 14.931. Graphical solutions are in references.

H. Vertical tubes, laminar flow, forced and natural convection

cj 7.630 × 10−3 2.058 × 10−3 0.901 × 10−3 0.487 × 10−3 0.297 × 10−3

0.023(dt /L)NRe NSc NSh = 4.36 + ᎏᎏᎏ 1 + 0.0012(dt/L)NRe NSc

[T] Use log mean concentration difference. NRe < 2100





[149] [151] p. 167

k′d NSh,x = ᎏt D vd t ρ NRe = ᎏ µ [T]

k′dt 48 NSh = ᎏ = ᎏ = 4.3636 D 11 (NGr NSc d/L)3/4 1/3 1 ⫾ 0.0742 ᎏᎏ NSh,avg = 1.62N Gz NGz

[91] p. 443 [120] [151] p. 150

−1



k′d Nsh = ᎏt D Use log mean concentration difference. NRe < 2100 G. Laminar, fully developed concentration and velocity profile

[151] p. 166

W ᎏ > 400 ρDx Leveque’s approximation: Concentration BL is thin. Assume velocity profile is linear. High mass velocity. Fits liquid data well.

1/3

−2 a 2j (x/rt) 1 − 4
a −2 j exp ᎏᎏ 1 dt NRe NSc j=1 NSh,avg = ᎏ ᎏ NRe NSc ᎏᎏᎏᎏ ∞ 2 L −2 a 2j (x/rt) −2 1 + 4
a j exp ᎏᎏ NRe NSc j=1

j 1 2 3 4 5

[151] p. 165

1/3

k′d d NSh,avg = ᎏt = 1.615 ᎏt D L

D. Tubes, laminar, uniform plug velocity, developing concentration profile, constant wall concentration

[98]

[98]

[100] p. 176

[T] Use log mean concentration difference. NRe < 2100

[98] [151] p. 167

[T] Approximate solution. Use minus sign if forced and natural convection oppose each other. NRe NSc d NGz = ᎏ L

[140]

1/3

g∆ρd 3 NGr = ᎏ ρν 2 Good agreement with experiment. I. Tubes, laminar, RO systems

J. Tubes and parallel plates, laminar RO

 

k′m d t ud 2 NSh,avg = ᎏ = 1.632 ᎏt D DL

1/3

Graphical solutions for concentration polarization. Uniform velocity through walls.

Use arithmetic concentration difference. Thin concentration polarization layer, not fully developed. NRe < 2000, L = length tube.

[73] [165] p. 738

[T]

[145] [165] p. 762

MASS TRANSFER TABLE 5-23

5-63

Mass-Transfer Correlations for Flow in Pipes and Ducts—Transfer is from Wall to Fluid (Continued ) Comments E = Empirical, S = Semiempirical, T = Theoretical

References*

Graphical solution

[T] Low transfer rates.

[151] p. 176

k′(2h) NSh = ᎏ = 7.6 D

[T] h = distance between plates. Use log mean concentration difference.

[151] p. 177

Situation K. Parallel plates, laminar, parabolic velocity, developing concentration profile, constant wall concentration L. 5-23-K, fully developed

Correlation

NRe NSc ᎏ < 20 x/(2h) M. Parallel plates, laminar, parabolic velocity, developing concentration profile, constant mass flux at wall N. 5-23-M, fully developed

O. Laminar flow, vertical parallel plates, forced and natural convection

Graphical solution

[T] Low transfer rates.

[151] p. 176

k′(2h) NSh = ᎏ = 8.23 D

[T] Use log mean concentration difference.

[151] p. 177

NRe NSc ᎏ < 20 x/(2h)



(NGr NSc h/L)3/4 NSh,avg = 1.47N 1/3 Gz 1 ⫾ 0.0989 ᎏᎏ NGz

1/3

[T] Approximate solution. Use minus sign if forced and natural convection oppose each other.

 

uH 2p k′(2Hp) NSh,avg = ᎏ = 2.354 ᎏ D DL

Q. Tubes, turbulent

k′m d t 1/3 NSh,avg = ᎏ = 0.023N 0.83 Re N Sc D

R. Tubes, turbulent

k′m d t 0.44 NSh,avg = ᎏ = 0.023N 0.83 Re N Sc D

S. Tubes, turbulent

k′d t 0.346 NSh = ᎏ = 0.0096 N 0.913 Re N Sc D

T. Tubes, turbulent, smooth tubes, Reynolds analogy

f k′d t NSh = ᎏ = ᎏ NRe NSc D 2 f = Fanning friction faction

1/3



f jD = jH = ᎏ 2 f NSh −0.2 = 0.023N Re If ᎏ = 0.023N −0.2 Re , jD = ᎏ 2 NRe N 1/3 Sc

Thin concentration polarization layer. Short tubes, concentration profile not fully developed. Use arithmetic concentration difference.

[73] [165] p. 738

[E] Use with log mean concentration difference at two ends of tube. 2100 < NRe < 35,000 0.6 < NSc < 3000 From wetted wall column and dissolution data— see Table 5-22-B. Good fit for liquids.

[88] p. 266

[E] Evaporation of liquids. Use with log mean concentration difference. See item above. Better fit for gases. 2000 < NRe < 35,000 0.6 < NSc < 2.5.

[93][100] p. 181 [109] p. 112 [146] p. 211

[E] 430 < NSc < 100,000. Dissolution data. Use for high NSc.

[122] p. 668

[T] Use arithmetic concentration difference. NSc near 1.0 Turbulent core extends to wall. Of limited utility.

[91] p. 438 [100] p. 171 [151] p. 239 [159] p. 250

[T] Use log-mean concentration difference. Relating jD to f/2 approximate. NPr and NSc near 1.0. Low concentration. Results about 20% lower than experiment. 3 × 104 < NRe < 106

[72] p. 400, 647 [80][88] p. 269 [151] p. 264 [159] p. 251

k′d t NSh = ᎏ D jD = jH = f (NRe, geometry and B.C.)

V. Tubes, turbulent, smooth tubes, constant surface concentration, Prandtl analogy *See pages 5-7 and 5-8 for references.

[140]

NRe NSc h NGz = ᎏ L g∆ρh3 NGr = ᎏ ρν 2 Good agreement with experiment.

P. Parallel plates, laminar, RO systems

U. Tubes, turbulent, smooth tubes, Chilton-Colburn analogy



k′d ( f /2)NRe NSc NSh = ᎏt = ᎏᎏ D 1 + 5f2 /(NSc − 1) f −0.25 ᎏ = 0.04 NRe 2

[100] p. 181 [120] [161] p. 72

[E] Good over wide ranges. [72] p. 647 [80] [T] Use arithmetic concentration difference. Improvement over Reynolds analogy. Best for NSc near 1.0.

[100] p. 173 [132] [151] p. 241

5-64

HEAT AND MASS TRANSFER

TABLE 5-23

Mass-Transfer Correlations for Flow in Pipes and Ducts—Transfer is from Wall to Fluid (Concluded )

Situation

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlation

References*

W. Tubes, turbulent, smooth tubes, Constant surface concentration, Von Karman analogy

( f/2)NReNSc ᎏᎏᎏᎏᎏ NSh = 5 2 (NSc − 1) + ln 1 + ᎏᎏ (NSc − 1) 1 + 5f/ 6 f −0.25 ᎏ = 0.04 N Re 2

[T] Use arithmetic concentration difference. NSh = k′dt /D. Improvement over Prandtl, NSc < 25.

[100] p. 173 [151] p. 243 [159] p. 250 [162]

X. Tubes, turbulent, smooth tubes, constant surface concentration

For 0.5 < NSc < 10:

[S] Use arithmetic concentration difference. Based on partial fluid renewal and an infrequently replenished thin fluid layer for high Nsc. Good fit to available data.

[100] p. 179 [131]







9/10 1/2 NSh,avg = 0.0097 N Re N Sc −1/3 −1/6 × (1.10 + 0.44N Sc − 0.70N Sc )

For 10 < NSc < 1000:

u bulk d t NRe = ᎏ ν

NSh,avg

k′avg d t NSh,avg = ᎏ D

9/10 1/2 −1/3 −1/6 0.0097N Re N Sc (1.10 + 0.44N Sc − 0.70N Sc ) = ᎏᎏᎏᎏᎏ 1/2 −1/3 −1/6 1 + 0.064 N Sc (1.10 + 0.44N Sc − 0.70N Sc )

For NSc > 1000: 9/10 1/3 N Sh,avg = 0.0102 N Re N Sc

Y. Turbulent flow, tubes

NSh NSh −0.12 −2/3 NSt = ᎏ =ᎏ = 0.0149N Re N Sc NPe NRe NSc

Z. Turbulent flow, noncircular ducts

Use correlations with

[E] Smooth pipe data. Data fits within 4% except at NSc > 20,000, where experimental data is underpredicted.

[124]

NSc > 100, 105 > NRe > 2100 Can be suspect for systems with sharp corners.

4 cross-sectional area d eq = ᎏᎏᎏ wetted perimeter Parallel plates:

[151] p. 289 [165] p. 738

2hw d eq = 4 ᎏ 2w + 2h *See pages 5-7 and 5-8 for references.

approximately 8100 kPa (80 atm). For reduced temperatures higher than about 1.5, the deviations are relatively modest for pressures up to the critical pressure. However, deviations are large near the critical point (see also p. 5-49). The effect of pressure on the gas-phase viscosity also is negligible for pressures below about 5060 kPa (50 atm). For the liquid-phase mass-transfer coefficient kˆ L, the effects of total system pressure can be ignored for all practical purposes. Thus, when using kˆ G and kˆ L for the design of gas absorbers or strippers, the primary pressure effects to consider will be those which affect the equilibrium curves and the values of m. If the pressure changes affect the hydrodynamics, then kˆ G, kˆ L, and a can all change significantly. Effects of Temperature on kˆ G and kˆ L The Stanton-number relationship for gas-phase mass transfer in packed beds, NSt = kˆ G /GM = g(NRe,NSc) (5-293)

With regard to the liquid-phase mass-transfer coefficient, Whitney and Vivian found that the effect of temperature upon kLa could be explained entirely by variations in the liquid-phase viscosity and diffusion coefficient with temperature. Similarly, the oxygen-desorption data of Sherwood and Holloway [Trans. Am. Inst. Chem. Eng., 36, 39 (1940)] show that the influence of temperature upon HL can be explained by the effects of temperature upon the liquid-phase viscosity and diffusion coefficients. It is important to recognize that the effects of temperature on the liquid-phase diffusion coefficients and viscosities can be very large and therefore must be carefully accounted for when using kˆ L or HL data. For liquids the mass-transfer coefficient kˆ L is correlated in terms of design variables by relations of the form NSt = kˆ L/LM = f(NRe,NSc) (5-294)

indicates that for a given system geometry the rate coefficient kˆ G depends only on the Reynolds number and the Schmidt number. Since the Schmidt number for a gas is independent of temperature, the principal effect of temperature upon kˆ G arises from changes in the gas viscosity with changes in temperature. For normally encountered temperature ranges, these effects will be small owing to the fractional powers involved in Reynolds-number terms (see Tables 5-21 to 5-28). It thus can be concluded that for all practical purposes kˆ G is independent of temperature and pressure in the normal ranges of these variables. For modest changes in temperature the influence of temperature upon the interfacial area a may be neglected. For example, in experiments on the absorption of SO2 in water, Whitney and Vivian [Chem. Eng. Prog., 45, 323 (1949)] found no appreciable effect of temperature upon k′Ga over the range from 10 to 50°C.

A general relation for HL which may be used as the basis for applying temperature corrections is as follows: a HL = bNRe N1/2 Sc

(5-295)

where b is a proportionality constant and the exponent a may range from about 0.2 to 0.5 for different packings and systems. The liquidphase diffusion coefficients may be corrected from a base temperature T1 to another temperature T2 by using the Einstein relation as recommended by Wilke [Chem. Eng. Prog., 45, 218 (1949)]: D2 = D1(T2 /T1)(µ1/µ2)

(5-296)

The Einstein relation can be rearranged to the following equation for relating Schmidt numbers at two temperatures: NSc2 = NSc1(T1 /T2)(ρ1 /ρ2)(µ2 /µ1)2

(5-297)

MASS TRANSFER TABLE 5-24

5-65

Mass Transfer Correlations for Flow Past Submerged Objects

Situation A. Single sphere

B. Single sphere, creeping flow with forced convection

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlation k′G pBLM RTd s 2r NSh = ᎏᎏ =ᎏ PD r − rs r/rs 2 5 10 50 ∞ (asymptotic limit) NSh 4.0 2.5 2.22 2.04 2.0 k′d NSh = ᎏ = [4.0 + 1.21(NRe NSc)2/3]1/2 D

k′d NSh = ᎏ = a(NRe NSc)1/3 D a = 1.01, 1.0, or 0.991 C. Single spheres, molecular diffusion, and forced convection, low flow rates

1/2 1/3 N Sh = 2.0 + AN Re N Sc A = 0.5 to 0.62

A = 0.60.

[T] Use with log mean concentration difference. r = distance from sphere, rs, ds = radius and diameter of sphere. No convection.

[151] p. 18

[T] Use with log mean concentration difference. Average over sphere. Numerical calculations. (NRe NSc) < 10,000 NRe < 1.0. Constant sphere diameter. Low mass-transfer rates.

[78][109] p. 114 [122] p. 671 [146] p. 214

[T] Fit to above ignoring molecular diffusion.

[117] p. 80

1000 < (NReNSc) < 10,000.

146 p. 215

[E] Use with log mean concentration difference. Average over sphere. Frössling Eq. (A = 0.552), 2 ≤ NRe ≤ 800, 0.6 ≤ Nsc ≤ 2.7. NSh lower than experimental at high NRe. [E] Ranz and Marshall 2 ≤ NRe ≤ 200, 0.6 ≤ Nsc ≤ 2.5. See also Table 5-27-L.

[72] [89] p. 409, 647 [100], p. 194 109 p. 114 151 p. 276 [72] p. 409, 647 [135] [146] p. 217 [151] p. 276 [90][91] p. 446 [146] p. 217 [139][151] p. 276 [102][151] p. 276

[E] Liquids 2 ≤ NRe ≤ 2,000. Graph in Ref. 146, p. 217–218. [E] 100 ≤ NRe ≤ 700; 1,200 ≤ NSc ≤ 1525.

A = 0.95. A = 0.95. A = 0.544.

References*

[E] Use with arithmetic concentration difference. NSc = 1; 50 ≤ NRe ≤ 350.

D. 5-24-C

k′d 0.35 NSh = ᎏs = 2.0 + 0.575N 1/2 Re N Sc D

[E] Use with log mean concentration difference. NSc ≤ 1, NRe < 1.

[94][151] p. 276

E. 5-24-C

k′d 0.53 1/3 NSh = ᎏs = 2.0 + 0.552N Re N Sc D

[E] Use with log mean concentration difference. 1.0 < NRe ≤ 48,000 Gases: 0.6 ≤ NSc ≤ 2.7.

[91] p. 446

[S] Correlates large amount of data and compares to published data. vr = relative velocity between fluid and sphere, m/s. CDr = drag coefficient for single particle fixed in fluid at velocity vr. See 5-27-G for calculation details and other applications.

[125]

[E] Use with arithmetic concentration difference. Liquids, 2000 < NRe < 17,000. High NSc, graph in Ref. 146, p. 217–218.

[91] p. 446 [157] [146] p. 217

k′d 0.6 1/3 NSh = ᎏs = 0.33N Re N Sc D

[E] 1500 ≤ NRe ≤ 12,000.

[151] p. 276

k′d 0.56 1/3 NSh = ᎏs = 0.43N Re N Sc D

[E] 200 ≤ NRe ≤ 4 × 104, “air” ≤ NSc ≤ “water.”

[151] p. 276

k′d 0.514 1/3 NSh = ᎏs = 0.692N Re N Sc D

[E] 500 ≤ NRe ≤ 5000.

[128] [151] p. 276

k′d 1/3 NSh = ᎏs = AN 1/2 Re N Sc , A = 0.82 D

[E] 100 < NRe ≤ 3500, NSc = 1560.

[151] p. 276

A = 0.74

[E] 120 ≤ NRe ≤ 6000, NSc = 2.44.

A = 0.582

[E] 300 ≤ NRe ≤ 7600, NSc = 1200.

[151] p. 276 [152]

jD = 0.600(N Re)−0.487

[E] Use with arithmetic concentration difference.

[151] p. 276

k′d cyl NSh = ᎏ D

50 ≤ NRe ≤ 50,000; gases, 0.6 ≤ NSc ≤ 2.6; liquids; 1000 ≤ NSc ≤ 3000. Data scatter ⫾ 30%.

[91] p. 450

F. Single spheres, forced concentration, any flow rate

E1/3d p4/3ρ k′L d s NSh = ᎏ = 2.0 + 0.59 ᎏ µ D





0.57 1/3 N Sc

Energy dissipation rate per unit mass of fluid (ranges 570 < NSc < 1420): ᎏ   ᎏ d  s m2

v 3r

CDr E= ᎏ 2

3

p

E d ρ 2 < ᎏᎏ < 63,000 µ



G. Single spheres, forced convection, high flow rates, ignoring molecular diffusion

H. Single cylinders, perpendicular flow

1/3

4/3 p



k′d 0.62 1/3 NSh = ᎏs = 0.347N Re N Sc D

5-66

HEAT AND MASS TRANSFER

TABLE 5-24

Mass Transfer Correlations for Flow Past Submerged Objects (Concluded )

Situation

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlation Can use jD = jH. Graphical correlation.

I. 5-24-H

J. Rotating cylinder in an infinite liquid, no forced flow

[E] Used with linear concentration difference.

[72] p. 408, 647

k′d s NSh = ᎏ D

[146] p. 236 [151] p. 273

k′ 0.644 −0.30 j′D = ᎏ N Sc = 0.0791N Re v

[E] Used with arithmetic concentration difference.

[85]

Results presented graphically to NRe = 241,000.

112 < NRe ≤ 100,000. 835 < NSc < 11490

[146] p. 238

vdcyl µ ω dcyl NRe = ᎏ where v = ᎏ = peripheral velocity ρ 2

K. Oblate spheroid, forced convection

References*

NSh −0.5 = 0.74 N Re jD = ᎏ NRe N 1/3 Sc

k′ = mass-transfer coefficient, cm/s; ω = rotational speed, radian/s. Useful geometry in electrochemical studies. [E] Used with arithmetic concentration difference. 120 ≤ NRe ≤ 6000; standard deviation 2.1%. Eccentricities between 1:1 (spheres) and 3:1.

total surface area dch vρ NRe = ᎏ , dch = ᎏᎏᎏ µ perimeter normal to flow

[151] p. 284 [152]

e.g., for cube with side length a, dch = 1.27a. k′d ch NSh = ᎏ D L. Other objects, including prisms, cubes, hemispheres, spheres, and cylinders; forced convection

M. Other objects, molecular diffusion limits

jD = 0.692N

Shape is often approximated by drops. [109] p. 115

v d ch ρ , N Re,p = ᎏ µ

−0.486 Re,p

[E] Used with arithmetic concentration difference.

Terms same as in 5-24-J.

k′d ch NSh = ᎏ =A D

500 ≤ N Re, p ≤ 5000. Turbulent. Agrees with cylinder and oblate spheroid results, ⫾15%. Assumes molecular diffusion and natural convection are negligible.

[127, 128] [151] p. 285

[T] Use with arithmetic concentration difference. Hard to reach limits in experiments.

[109] p. 114

[E] Use with logarithmic mean concentration difference.

[133]

 Spheres and cubes A = 2, tetrahedrons A = 26 . octahedrons 22 N. Shell side of microporous hollow fiber module for solvent extraction

0.33 NSh = β[d h(1 − ϕ)/L]N 0.6 Re N Sc

Kdh  NSh = ᎏ D β = 5.8 for hydrophobic membrane.

d h vρ N Re = ᎏ ,K  = overall mass-transfer coefficient µ

β = 6.1 for hydrophilic membrane.

dh = hydraulic diameter 4 × cross-sectional area of flow = ᎏᎏᎏᎏ wetted perimeter ϕ = packing fraction of shell side. L = module length. Based on area of contact according to inside or outside diameter of tubes depending on location of interface between aqueous and organic phases. Can also be applied to gas-liquid systems with liquid on shell side.

See Table 5-27 for flow in packed beds. *See pp. 5-7 and 5-8 for references.

Substitution of this relation into Eq. (5-295) shows that for a given geometry the effect of temperature on HL can be estimated as HL2 = HL1(T1 /T2)1/2(ρ1 /ρ2)1/2(µ2 /µ1)1 − a

(5-298)

In using these relations it should be noted that for equal liquid flow rates HL2 /HL1 = (kˆ La)1/(kˆ La)2 (5-299) ˆ L When ˆ G and k Effects of System Physical Properties on k designing packed towers for nonreacting gas-absorption systems for

which no experimental data are available, it is necessary to make corrections for differences in composition between the existing test data and the system in question. For example, the test data of Fellinger for ammonia-water absorption on various packings are frequently used as a base (see Table 5-28-B). In these tests it is estimated that HG = 0.9HOG, so that one may wish to use these data as the basis for estimating HG or kˆ Ga values for other systems. This may be done by tak0.5 −0.5 ing HG proportional to NSc and kˆ Ga proportional to NSc , based on a value of NSc for NH3-air of 0.66 at 25°C. The coefficient kG varies as the diffusivity DAB to the 0.5 power. It should be noted, however, that

MASS TRANSFER TABLE 5-25

5-67

Mass-Transfer Correlations for Drops and Bubbles

Conditions A. Single liquid drop in immiscible liquid, drop formation, discontinuous (drop) phase coefficient

Correlations ρd kˆ d,f = A ᎏ Md

   Dd ᎏ π tf

av



24 A = ᎏ (0.8624) (extension by fresh surface 7 elements) kˆ df = 0.0432 d ρd × ᎏp ᎏ tf Md

−0.334

µd ᎏᎏ d gc d pσ ρ

     

C. Single liquid drop in immiscible liquid, drop formation, continuous phase coefficient D. 5-25-C

av

0.089

uo ᎏ dp g

G. 5-25-E, continuous phase coefficient, stagnant drops, spherical

H. 5-25-E, oblate spheroid



Dc

av

f

   

F. 5-25-E

−0.601

kL,c = 0.386 ρc × ᎏ Mc

E. Single liquid drop in immiscible liquid, free rise or fall, discontinuous phase coefficient, stagnant drops

d p2 ᎏ t f Dd

ᎏ    πt

ρc kˆ cf = 4.6 ᎏ Mc

Dc ᎏ tf

av

−dp ρd kL,d,m = ᎏ ᎏ 6t M d

0.5

ρcσgc ᎏ ∆ ρgt fµ c

   0.407

gt f2 ᎏ dp

0.148

−Dd j 2π 2 t

∞

  ln ᎏπ6
ᎏj1 exp ᎏ (d /2) 

−d p ρd kˆ L,d,m = ᎏ ᎏ 6t Md

[T,S] Use arithmetic mole fraction difference.

[151] p. 399

Fits some, but not all, data. Low mass transfer rate. Md = mean molecular weight of dispersed phase; tf = formation time of drop. kL,d = mean dispersed liquid phase M.T. coefficient kmole/[s ⋅ m2 (mole fraction)].

A = 1.31 (semiempirical value)

B. 5-25-A

References*

1/2

24 A = ᎏ (penetration theory) 7



Comments E = Empirical, S = Semiempirical, T = Theoretical

2

av

2

j=1

p

2

πD t   ln 1 − ᎏ d /2 1/2 1/2 d

av

p

kL,c,m dc ρc NSh = ᎏ = 0.74 ᎏ Dc Mc

 N

(NSc)1/3

1/2 Re

av

ρc kL,c,m d3 NSh = ᎏ = 0.74 ᎏ Dc Mc

  (N

)1/2(NSc,c)1/3

Re,3

av

[E] Use arithmetic mole fraction difference. Based on 23 data points for 3 systems. Average absolute deviation 26%. Use with surface area of drop after detachment occurs. uo = velocity through nozzle; σ = interfacial tension.

[151] p. 401

[T] Use arithmetic mole fraction difference. Based on rate of bubble growth away from fixed orifice. Approximately three times too high compared to experiments.

[151] p. 402

[E] Average absolute deviation 11% for 20 data points for 3 systems.

[151] p. 402 [154] p. 434

[T] Use with log mean mole fraction differences based on ends of column. t = rise time. No continuous phase resistance. Stagnant drops are likely if drop is very viscous, quite small, or is coated with surface active agent. kL,d,m = mean dispersed liquid M.T. coefficient.

[151] p. 404 [154] p. 435

[S] See 5-25-E. Approximation for fractional extractions less than 50%.

[151] p. 404 [154] p. 435

vs d p ρc [E] NRe = ᎏ , special case Eq. (5-254). µc

[151] p. 407

vs = slip velocity between drop and continuous phase.

[152][154] p. 436

[E] Used with log mean mole fraction. Differences based on ends of extraction column; 100 measured values ⫾2% deviation. Based on area oblate spheroid.

[151] p. 285, 406, 407

[154] p. 434

vsd3ρc NRe,3 = ᎏ µc total drop surface area vs = slip velocity, d3 = ᎏᎏᎏ perimeter normal to flow I. Single liquid drop in immiscible liquid, Free rise or fall, discontinuous phase coefficient, circulating drops

J. 5-25-I



dp 3 kdr,circ = − ᎏ ln ᎏ 6θ 8

∞


B

j=1

2 j

λ j64Ddθ exp − ᎏ d p2





Eigenvalues for Circulating Drop k d d p /Dd

λ1

λ2

3.20 10.7 26.7 107 320 ∞

0.262 0.680 1.082 1.484 1.60 1.656

0.424 4.92 5.90 7.88 8.62 9.08

d p ρd kˆ L,d,circ = − ᎏ ᎏ 6θ Md

λ3

B1

B2

B3

15.7 19.5 21.3 22.2

1.49 1.49 1.49 1.39 1.31 1.29

0.107 0.300 0.495 0.603 0.583 0.596

0.205 0.384 0.391 0.386

R πD θ   ln 1 − ᎏᎏ d /2 1/2

av

1/2 1/2 d

p

[T] Use with arithmetic concentration difference.

[86][99][151] p. 405

θ = drop residence time. A more complete listing of eigenvalues is given by Refs. 86 and 99.

[161] p. 523

k′L,d,circ is m/s.

[E] Used with mole fractions for extraction less than 50%, R ≈ 2.25.

[151] p. 405

5-68

HEAT AND MASS TRANSFER

TABLE 5-25

Mass-Transfer Correlations for Drops and Bubbles (Continued )

Conditions K. 5-25-I

ˆk L,d,circ d p NSh = ᎏ Dd ρd = 31.4 ᎏ Mf

2 −0.125 d p v s ρc N Sc,d ᎏ σg c

−0.34

  ᎏ d 

L. Liquid drop in immiscible liquid, free rise or fall, continuous phase coefficient, circulating single drops

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlations

4Dd t 2 p

av



−0.37



k′L,c d p NSh,c = ᎏ Dd





d p g1/3 0.484 = 2 + 0.463N Re,drop N 0.339 ᎏ Sc,c D c2/3

1/8 K = N Re,drop

M. 5-25-L, circulating, single drop

µcvs ᎏ σg c

[E] Used with log mean mole fraction difference. dp = diameter of sphere with same volume as drop. 856 ≤ NSc ≤ 79,800, 2.34 ≤ σ ≤ 4.8 dynes/cm.

[154] p. 435 [155]

[E] Used as an arithmetic concentration difference.

[103]

d pv sρc NRe,drop = ᎏ µc

 F 0.072

F = 0.281 + 1.615K + 3.73K 2 − 1.874K µc ᎏ µd

References*

Solid sphere form with correction factor F.

   1/4

k L,c d p ρc NSh = ᎏ = 0.6 ᎏ Dc Mc

1/6

 N

1/2 Re,drop

1/2 N Sc,c

[E] Used as an arithmetic concentration difference. Low σ.

av

[151] p. 407

d pv sρc NRe,drop = ᎏ µc N. 5-25-L, circulating swarm of drops

O. Liquid drops in immiscible liquid, free rise or fall, discontinuous phase coefficient, oscillating drops

P. 5-25-O

Q. Single liquid drop in immiscible liquid, range rigid to fully circulating

ρc k L,c = 0.725 ᎏ Mc

 N

−0.43 Re,drop

−0.58 N Sc,c v s (1 − φ d)

av

k L,d,osc d p NSh = ᎏ Dd ρd = 0.32 ᎏ Md

−0.14

t   ᎏ d  4Dd 2 p

av

σ 3g c3 ρ2c 0.68 N Re,drop ᎏ gµ 4c∆ρ





0.10

0.00375v k L,d,osc = ᎏᎏs 1 + µ d /µ c

Rigid drops: 104 < NPe,c < 106

[E] Used as an arithmetic concentration difference. Low σ, disperse-phase holdup of drop swarm. φ d = volume fraction dispersed phase.

[151] p. 407 [154] p. 436

[E] Used with a log mean mole fraction difference. Based on ends of extraction column.

[151] p. 406

d pvsρc NRe,drop = ᎏ µc d p = diameter of sphere with volume of drop. Average absolute deviation from data, 10.5%. 411 ≤ NRe ≤ 3114 Low interfacial tension (3.5–5.8 dynes), µc < 1.35 centipoise.

[154] p. 435 [155]

[T] Use with log mean concentration difference. Based on end of extraction column. No continuous phase resistance. kL,d,osc in cm/s, vs = drop velocity relative to continuous phase.

[146] p. 228 [151] p. 405

[E] Allows for slight effect of wake.

[156] p. 58 [158]

[E] Used with log mean mole fraction difference. 23 data points. Average absolute deviation 25%. t f = formation time.

[151] p. 408

[E] Used with log mean mole fraction difference. 20 data points. Average absolute deviation 22%.

[151] p. 409

kcdp 0.5 0.33 NSh,c,rigid = ᎏ = 2.43 + 0.774N Re N Sc Dc 0.33 + 0.0103NReN Sc

Circulating drops: 10 < NRe < 1200, 190 < NSc < 241,000, 103 < NPe,c < 106



2 NSh,c,fully circular = ᎏ N 0.5 Pe,c π 0.5 Drops in intermediate range: NSh,c − NSh,c,rigid ᎏᎏᎏ = 1 − exp [−(4.18 × 10−3)N 0.42 Pe,c] NSh,c,fully circular − NSh,c,rigid R. Coalescing drops in immiscible liquid, discontinuous phase coefficient

S. 5-25-R, continuous phase coefficient

d p ρd kˆ d,coal = 0.173 ᎏ ᎏ t f Md

µ   ᎏ ρ D 

∆ ρgd p2 × ᎏ σg c



−1.115

d

av

d

d

v  ᎏ D  1.302

2 s f

t

0.146

d

ρ kˆ c,coal = 5.959 × 10−4 ᎏ M

 

ρdu3s

av

d p2 ρc ρd v3s

ᎏᎏ   ᎏ g µ   µ σg 

D × ᎏc tf

0.5

c

0.332

d

c

0.525

MASS TRANSFER TABLE 5-25

5-69

Mass-Transfer Correlations for Drops and Bubbles (Concluded )

Conditions T. Single liquid drops in gas, gas side coefficient

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlations kˆg Mg d p P 1/2 N 1/3 ᎏ = 2 + AN Re,g Sc,g Dgas ρg A = 0.552 or 0.60.

References*

[E] Used for spray drying (arithmetic partial pressure difference).

[88] p. 489

d pρgvs NRe,g = ᎏ , vs = slip velocity between drop µg

[111] p. 388 [135]

and gas stream. Sometimes written with: MgP ᎏ = RT ρg U. Single water drop in air, liquid side coefficient

 

DL kL = 2 ᎏ πt

1/2

, short contact times

DL k L = 10 ᎏ , long contact times dp V. Single bubbles of gas in liquid, continuous phase coefficient, very small bubbles

k′c d b NSh = ᎏ = 1.0(NReNSc)1/3 Dc

W. 5-25-V, medium to large bubbles

k′c d b NSh = ᎏ = 1.13(NReNSc)1/2 Dc

X. 5-25-W

k′c d b db NSh = ᎏ = 1.13(NReNSc)1/2 ᎏᎏ Dc 0.45 + 0.2d b

Y. Rising small bubbles of gas in liquid, continuous phase





k′c d b 1/3 NSh = ᎏ = 2 + 0.31(NGr)1/3N Sc , d b < 0.25 cm Dc

[T] Use arithmetic concentration difference. Penetration theory. t = contact time of drop. Gives plot for k G a also. Air-water system.

[111] p. 389

[T] Solid-sphere Eq. (see Table 5-24-B). d b < 0.1 cm, k′c is average over entire surface of bubble.

[122] p. 673 [146] p. 214

[T] Use arithmetic concentration difference. Droplet equation: d b > 0.5 cm.

[146] p. 231

[S] Use arithmetic concentration difference. Modification of above (W), db > 0.5 cm. 500 ≤ NRe ≤ 8000. No effect SAA for dp > 0.6 cm.

[104][146] p. 231

[E] Use with arithmetic concentration difference.

[79][91] p. 451

d |ρG − ρL|g NRa = ᎏᎏ = Raleigh number µ L DL Note that NRa = NGr NSc. Valid for single bubbles or swarms. Independent of agitation as long as bubble size is constant.

[109] p. 119 [161] p. 156

[E] Use with arithmetic concentration difference. Hg = fractional gas holdup, volume gas/total volume. For large bubbles, k′c is independent of bubble size and independent of agitation or liquid velocity. Resistance is entirely in liquid phase for most gas-liquid mass transfer.

[79][91] p. 452 [109] p. 119

3 b

Z. 5-25-Y, large bubbles

k′c d b NSh = ᎏ = 0.42 (NGr)1/3N 1/2 Sc , d b > 0.25 cm Dc 6 Hg Interfacial area ᎏᎏ = a = ᎏ volume db

[114] p. 249

See Table 5-26 for agitated systems. *See pages 5-7 and 5-8 for references.

there is conflicting evidence concerning this exponent (w versus a) as discussed by Yadav and Sharma [Chem. Eng. Sci., 34, 1423 (1979)]. The existing data indicate that kˆ La is proportional to the square root of the solute-diffusion coefficient, and since the interfacial area a does not depend on DL, it follows that kˆ L is proportional to DL0.5. An analysis of the design variables involved indicates that kˆ L should be −0.5 proportional to NSc when the Reynolds number is held constant. It should be noted that the influence of substituting solvents of widely differing viscosities upon the interfacial area a can be very large. One therefore should be cautious about extrapolating kˆ La data to account for viscosity effects between different solvent systems. ˆ G and k ˆ L As disEffects of High Solute Concentrations on k cussed previously, the stagnant-film model indicates that kˆ G should be independent of yBM and kG should be inversely proportional to yBM. The data of Vivian and Behrman [Am. Inst. Chem. Eng. J., 11, 656 (1965)] for the absorption of ammonia from an inert gas strongly suggest that the film model’s predicted trend is correct. This is another indication that the most appropriate rate coefficient to use is kˆ G and the proper driving-force term is of the form (y − yi)/yBM.

The use of the rate coefficient kˆ L and the driving force (xi − x)/xBM is believed to be appropriate. For many practical situations the liquidphase solute concentrations are low, thus making this assumption unimportant. ˆ G and k ˆ L When a chemInfluence of Chemical Reactions on k ical reaction occurs, the transfer rate may be influenced by the chemical reaction as well as by the purely physical processes of diffusion and convection within the two phases. Since this situation is common in gas absorption, gas absorption will be the focus of this discussion. One must consider the impacts of chemical equilibrium and reaction kinetics on the absorption rate in addition to accounting for the effects of gas solubility, diffusivity, and system hydrodynamics. There is no sharp dividing line between pure physical absorption and absorption controlled by the rate of a chemical reaction. Most cases fall in an intermediate range in which the rate of absorption is limited both by the resistance to diffusion and by the finite velocity of the reaction. Even in these intermediate cases the equilibria between the various diffusing species involved in the reaction may affect the rate of absorption.

5-70

HEAT AND MASS TRANSFER

TABLE 5-26

Mass-Transfer Correlations for Particles, Drops, and Bubbles in Agitated Systems Comments E = Empirical, S = Semiempirical, T = Theoretical

References*

k′LT d p 1/3 ᎏ = 2 + 0.6N 1/2 Re,T N Sc D

[S] Use log mean concentration difference. Modified Frossling equation:

[97][146] p. 220

Replace vslip with v T = terminal velocity. Calculate Stokes’ law terminal velocity

v Ts d p ρc NRe,Ts = ᎏ µc

Situation A. Solid particles suspended in agitated vessel containing vertical baffles, continuous phase coefficient

Correlation

d p2|ρ p − ρc|g v Ts = ᎏᎏ 18µ c

(Reynolds number based on Stokes’ law.) v T d p ρc NRe,T = ᎏ µc

and correct: NRe,Ts 1 v T /v Ts 0.9

10 0.65

100 0.37

1,000 0.17

10,000 0.07

100,000 0.023

Approximate: k′L = 2k′LT B. 5-26-A C. Solid, neutrally buoyant particles, continuous phase coefficient

Graphical comparisons experiments and correlations.

 

k′Ld p d imp 0.36 NSh = ᎏ = 2 + 0.47N 0.62 ᎏ Re,p N Sc D d tank

(terminal velocity Reynolds number.) k′L almost independent of d p. Harriott suggests different correction procedures. Range k′L /k′LT is 1.5 to 8.0.

[87]

[E,S] For spheres. Includes transpiration effects and changing diameters.

[78][146] p. 222

[E] Use log mean concentration difference. Density unimportant if particles are close to neutrally buoyant. [E] E = energy dissipation rate per unit mass fluid

[109] p. 115 [118] p. 132

0.17

Graphical comparisons are in Ref. 109, p. 116.

[161] p. 523

Pgc =ᎏ , P = power Vtank ρc E1/3d p4/3 NRe,p = ᎏ ν Also used for drops. Geometric effect (d imp/d tank) is usually unimportant. Ref. 118 gives a variety of references on correlations. D. 5-26-C, small particles

0.52 NSh = 2 + 0.52N Re,p N 1/3 Sc , NRe,p < 1.0





E. Solid particles with significant density difference

F. Small solid particles, gas bubbles or liquid drops, dp < 2.5 mm

k′L d p d 3p |ρp − ρ c| NSh = ᎏ = 2 + 0.31 ᎏᎏ µ cD D

G. Highly agitated systems; solid particles, drops, and bubbles; continuous phase coefficient

H. Liquid drops in baffled tank with flat six-blade turbine







 

[E] Use log mean concentration difference. NSh standard deviation 11.1%. vslip calculated by methods given in reference.

[118]

[E] Use log mean concentration difference. g = 9.80665 m/s 2. Second term RHS is free-fall or rise term. For large bubbles, see Table 5-25-Z.

[79][91] p. 451 [114] p. 249

[E] Use arithmetic concentration difference. Use when gravitational forces overcome by agitation. Up to 60% deviation. Correlation prediction is low (Ref. 118). (P/Vtank) = power dissipated by agitator per unit volume liquid.

[79][83] p. 231 [91] p. 452

[E] Use arithmetic concentration difference. Studied for five systems.

[154] p. 437

1/4

1.582

2 NRe = d imp Nρc /µ c , NOh = µ c /(ρc d impσ)1/2

1.025 N 1.929 Re N Oh

φ = volume fraction dispersed phase. N = impeller speed (revolutions/time). For dtank = htank, average absolute deviation 23.8%.

k′c d p 1/3 NSh = ᎏ = 1.237 × 10 −5 N Sc N 2/3 D ρd d p2

ᎏ  ᎏ D   σ 

5/12 d imp × N Fr ᎏ dp

[109] p. 116

1/3

(ND)1/2 k′c a = 2.621 × 10−3 ᎏ d imp d imp × φ 0.304 ᎏ d tank

I. Liquid drops in baffled tank, low volume fraction dispersed phase

0.38 N Sc



(P/Vtank)µ cg c 2/3 k′LN Sc = 0.13 ᎏᎏ ρc2

[E] Terms same as above.

1/2

d pv slip k′L d p NSh = ᎏ = 2 + 0.44 ᎏ D ν

dp

1/2

5/4

φ−1/2

tank

Stainless steel flat six-blade turbine. Tank had four baffles. Correlation recommended for φ ≤ 0.06 [Ref. 156] a = 6φ /dˆ 32, where dˆ 32 is Sauter mean diameter when 33% mass transfer has occurred.

[E] 180 runs, 9 systems, φ = 0.01. kc is timeaveraged. Use arithmetic concentration difference.







2 d impN 2 d imp NSc NRe = ᎏ , NFr = ᎏ µc g



d p = particle or drop diameter; σ = interfacial tension, N/m; φ = volume fraction dispersed phase; a = interfacial volume, 1/m; and kcαD c2/3 implies rigid drops. Negligible drop coalescence. Average absolute deviation—19.71%. Graphical comparison given by Ref. 153.

[153, 156] p. 78

MASS TRANSFER TABLE 5-26

5-71

Mass-Transfer Correlations for Particles, Drops, and Bubbles in Agitated Systems (Concluded )

Situation J. Gas bubble swarms in sparged tank reactors

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlation ν 1/3 P/VL a qG ν 1/3 b k′L a ᎏ2 =C ᎏ ᎏ ᎏ2 g ρ(νg 4)1/3 VL g Rushton turbines: C = 7.94 × 10−4, a = 0.62, b = 0.23. Intermig impellers: C = 5.89 × 10−4, a = 0.62, b = 0.19.

 



  

 

K. 5-26-J

P k′L a = 2.6 × 10−2 ᎏ VL

L. 5-26-J

P k′L a = 2.0 × 10−3 ᎏ VL

M. 5-26-J, baffled tank with standard blade Rushton impeller

P k′L a = 93.37 ᎏ VL

N. 5-26-M

d 2imp µ eff k′L a ᎏ = 7.57 ᎏ D ρD

 

 

References*

[E] Use arithmetic concentration difference. Done for biological system, O2 transfer. htank /Dtank = 2.1; P = power, kW. VL = liquid volume, m3. qG = gassing rate, m3/s. k′L a = s −1. Since a = m2/m3, ν = kinematic viscosity, m2/s. Low viscosity system. Better fit claimed with qG /VL than with uG (see 5-26-K to O).

[143]

u 0.5 G

[E] Use arithmetic concentration difference. Ion free water VL < 2.6, uG = superficial gas velocity in m/s. 500 < P/VL < 10,000. P/VL = watts/m3, VL = liquid volume, m3.

[115, 137]

u 0.2 G

[E] Use arithmetic concentration difference. Water with ions. 0.002 < VL < 4.4, 500 < P/VL < 10,000. Same definitions as 5-26-J.

[115, 117]

[E] Air-water. Same definitions as 5-26-J. 0.005 < uG < 0.025, 3.83 < N < 8.33, 400 < P/VL < 7000 h = Dtank = 0.305 or 0.610 m. VG = gas volume, m3, N = stirrer speed, rpm. Method assumes perfect liquid mixing.

[92, 115]

[E] Use arithmetic concentration difference. CO2 into aqueous carboxyl polymethylene. Same definitions as 5-26-M. µeff = effective viscosity from power law model, Pa⋅s. σ = surface tension liquid, N/m.

[115, 129]

[E] Use arithmetic concentration difference. O2 into aqueous glycerol solutions. O2 into aqueous millet jelly solutions. Same definitions as 5-26-M.

[115, 167]

[E] Use arithmetic concentration difference. Solids are glass beads, d p = 320 µm. ε s = solids holdup m3/m3 liquid, (k′L a)o = mass transfer in absence of solids. Ionic salt solution— noncoalescing.

[71, 144]

[E] Use arithmetic concentration difference. Variety of solids, d p > 150 µm (glass, amberlite, polypropylene). Tap water. Slope very different than item P. Coalescence may have occurred.

[71, 112]

0.4

0.7

0.76

u G0.45

µ  ᎏ µ



0.5

d 2impNρL × ᎏ µ eff

G

0.694

eff

u d ᎏ σ  1.11

G

0.447

d imp = impeller diameter, m; D = diffusivity, m2/s O. 5-26-M, bubbles

P. Gas bubble swarm in sparged stirred tank reactor with solids present

Q. 5-26-P



k′Lad 2imp d 2impNρ ᎏ = 0.060 ᎏ µ eff D

d N µ u ᎏ ᎏ g   σ  2 imp

2

0.19

eff

k′La ᎏ = 1 − 3.54(εs − 0.03) (k′La)o 300 ≤ P/Vrx < 10,000 W/m3, 0.03 ≤ εs ≤ 0.12 0.34 ≤ uG ≤ 4.2 cm/s, 5 < µ L < 75 Pa⋅s k′La ᎏ = 1 − εs (k′La)o

G

0.6

See also Table 5-25. *See pages 5-7 and 5-8 for references.

The gas-phase rate coefficient kˆ G is not affected by the fact that a chemical reaction is taking place in the liquid phase. If the liquidphase chemical reaction is extremely fast and irreversible, the rate of absorption may be governed completely by the resistance to diffusion in the gas phase. In this case the absorption rate may be estimated by knowing only the gas-phase rate coefficient kˆ G or else the height of one gas-phase transfer unit HG = GM /(kˆ Ga). It should be noted that the highest possible absorption rates will occur under conditions in which the liquid-phase resistance is negligible and the equilibrium back pressure of the gas over the solvent is zero. Such situations would exist, for instance, for NH3 absorption into an acid solution, for SO2 absorption into an alkali solution, for vaporization of water into air, and for H2S absorption from a dilute-gas stream into a strong alkali solution, provided there is a large excess of reagent in solution to consume all the dissolved gas. This is known as the gas-phase mass-transfer limited condition, when both the liquidphase resistance and the back pressure of the gas equal zero. Even when the reaction is sufficiently reversible to allow a small back pres-

sure, the absorption may be gas-phase-controlled, and the values of kˆ G and HG that would apply to a physical-absorption process will govern the rate. The liquid-phase rate coefficient kˆ L is strongly affected by fast chemical reactions and generally increases with increasing reaction rate. Indeed, the condition for zero liquid-phase resistance (m/kˆ L) implies that either the equilibrium back pressure is negligible, or that kˆ L is very large, or both. Frequently, even though reaction consumes the solute as it is dissolving, thereby enhancing both the mass-transfer coefficient and the driving force for absorption, the reaction rate is slow enough that the liquid-phase resistance must be taken into account. This may be due either to an insufficient supply of a second reagent or to an inherently slow chemical reaction. In any event the value of kˆ L in the presence of a chemical reaction normally is larger than the value found when only physical absorption occurs, kˆ L0 . This has led to the presentation of data on the effects of chemical reaction in terms of the “reaction factor” or “enhancement factor” defined as

5-72

HEAT AND MASS TRANSFER

TABLE 5-27

Mass Transfer Correlations for Fixed and Fluidized Beds Transfer is to or from particles.

Situation A. Heat or mass transfer in packed bed for gases and liquids

(shape factor, Ψ)

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlation −0.51 jD = jH = 0.91ΨN Re , 0.01 < NRe < 50 1/3 Equivalent NSh = 0.91ΨN 0.49 Re N Sc . −0.41 jD = jH = 0.61ΨN Re , 50 < NRe < 1000 1/3 Equivalent NSh = 0.61ΨN 0.59 Re N Sc .

particle 1.00

sphere 0.91

[E] Different constants and shape factors reported in other references. Evaluate terms at film temperature or composition. cylinder 0.81

References* [100] p. 194 [169]

k′d NSh NSh = ᎏs , jD = ᎏ NRe N 1/3 D Sc vsuper ρ NRe = ᎏ , v super = superficial velocity µΨa surface area a = ᎏᎏ = 6(1 − ε)/d p volume For spheres, dp = diameter. ar t. u .A re a For nonspherical: d p = 0.567 P Srf Results are from too-short beds—use with caution.

B. For gases, fixed and fluidized beds, Gupta and Thodos correlation

2.06 jH = jD = ᎏ , 90 ≤ NRe ≤ A εN 0.575 Re

v superd p ρ [E] For spheres. NRe = ᎏ µ

[95, 96]

Equivalent:

A = 2453 [Ref. 151], A = 4000 [Ref. 100]. For NRe > 1900, j H = 1.05j D. Heat transfer result is in absence of radiation.

[100] p. 195 [151]

2.06 0.425 1/3 N Sc NSh = ᎏ N Re ε For other shapes:

k′d NSh = ᎏs D

ε jD ᎏ = 0.79 (cylinder) or 0.71 (cube) (ε j D)sphere Graphical results are available for NRe from 1900 to 10,300. C. For gases, for fixed beds, Petrovic and Thodos correlation

0.357 0.641 1/3 NSh = ᎏ N Re N Sc ε

D. For gases and liquids, fixed and fluidized beds

0.4548 jD = ᎏ , 10 ≤ NRe ≤ 2000 εN 0.4069 Re NSh k′d , NSh = ᎏs jD = ᎏ NReN 1/3 D Sc

E. For gases, fixed beds

F. For liquids, fixed bed, Wilson and Geankoplis correlation

0.499 jD = ᎏ 0.382 εN Re 1.09 jD = ᎏ , 0.0016 < NRe < 55 2/3 εN Re 165 ≤ NSc ≤ 70,600, 0.35 < ε < 0.75 Equivalent: 1.09 1/3 NSh = ᎏ N 1/3 Re N Sc ε 0.25 jD = ᎏ , 55 < NRe < 1500, 165 ≤ NSc ≤ 10,690 εN 0.31 Re Equivalent: 0.25 0.69 1/3 N Sc NSh = ᎏ N Re ε

[E] Packed spheres, deep beds, 3 < NRe < 900 can be extrapolated to NRe < 2000. Corrected for axial dispersion with axial Peclet number = 2.0. Prediction is low at low NRe. NRe defined as in 5-27-A and B.

[130][141] p. 214 [163]

[E] Packed spheres, deep bed. Average deviation ⫾20%, NRe = dpvsuperρ/µ. Can use for fluidized beds. 10 ≤ NRe ≤ 4000.

[85][91] p. 447

[E] Data on sublimination of naphthalene spheres dispersed in inert beads. 0.1 < NRe < 100, NSc = 2.57. Correlation coefficient = 0.978.

[101]

[E] Beds of spheres,

[91] p. 448

d pVsuperρ NRe = ᎏ µ Deep beds. k′d s NSh = ᎏ D

[100] p. 195 [151] p. 287 [166]

MASS TRANSFER TABLE 5-27

5-73

Mass Transfer Correlations for Fixed and Fluidized Beds (Continued )

Situation G. For liquids, fixed beds, Ohashi et al. correlation

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlation E 1/3d 4/3 k′d p ρ NSh = ᎏs = 2 + 0.51 ᎏ µ D





References*

0.60

N 1/3 Sc

E = Energy dissipation rate per unit mass of fluid

[S] Correlates large amount of published data. Compares number of correlations, v r = relative velocity, m/s. In packed bed, v r = v super /ε.

[125]

 

v 3r = 50(1 − ε)ε 2 CDo ᎏ , m2/s 3 dp 50(1 − ε)CD = ᎏᎏ ε



ᎏ d  v 3super p

General form: α E 1/3 D 4/3 p ρ β NSh = 2 + K ᎏᎏ N Sc µ





applies to single particles, packed beds, two-phase tube flow, suspended bubble columns, and stirred tanks with different definitions of E.

CDo = single particle drag coefficient at v super cal−m culated from CDo = AN Re . i NRe 0 to 5.8 5.8 to 500 >500

A 24 10 0.44

m 1.0 0.5 0

Ranges for packed bed: 0.001 < NRe < 1000 505 < NSc < 70600 E 1/3d p4/3ρ 0.2 < ᎏᎏ < 4600 µ Compares different situations versus general correlation. See also 5-24-F.

H. For liquids, fixed and fluidized beds

1.1068 ε jD = ᎏ , 1.0 < N Re ≤ 10 N 0.72 Re NSh k′d ε jD = ᎏ , NSh = ᎏs NRe N 1/3 D Sc

I. For gases and liquids, fixed and fluidized beds, Dwivedi and Upadhyay correlation

0.765 0.365 +ᎏ ε jD = ᎏ N 0.82 N 0.386 Re Re Gases: 10 ≤ N Re ≤ 15,000. Liquids: 0.01 ≤ N Re ≤ 15,000.

[E] Spheres: d pv superρ NRe = ᎏ µ [E] Deep beds of spheres, NSh jD = ᎏ N Re N 1/3 Sc

[84][91] p. 448

[84][100] p. 196

d pv superρ k′d s NRe = ᎏ , NSh = ᎏ µ D Based on 20 gas studies and 17 liquid studies. Recommended instead of 5-27-D or F. J. For gases and liquids, fixed bed

K. For liquids, fixed and fluidized beds, Rahman and Streat correlation L. For liquids and gases, Ranz and Marshall correlation

−0.415 , 10 ≤ NRe ≤ 2500 jD = 1.17N Re

[E] Spheres:

k′ pBM 2/3 jD = ᎏ ᎏ NSc vav P

d pv superρ NRe = ᎏ µ

Comparison with other results are shown.

Variation in packing that changes ε not allowed for. Extensive data referenced. 0.5 < NSc < 15,000.

0.86 NSh = ᎏ NReN1/3 Sc , 2 ≤ NRe ≤ 25 ε

[E] Can be extrapolated to NRe = 2000. NRe = dpvsuperρ/µ. Done for neutralization of ion exchange resin.

[134]

[E] Based on freely falling, evaporating spheres (see 5-24-C). Has been applied to packed beds. Prediction is low compared to experimental data for packed beds. Limit of 2.0 at low NRe is too high. Not corrected for axial dispersion.

[135][141] p. 214 [163][168] p. 106

k′d 1/2 NSh = ᎏ = 2.0 + 0.6N 1/3 Sc N Re D d pv superρ NRe = ᎏ µ

M. For liquids and gases, Wakao and Funazkri correlation

0.6 NSh = 2.0 + 1.1N 1/3 Sc N Re , 3 < NRe < 10,000

k′film d p NSh = ᎏ t D Graphical comparison with data shown by Refs. 141, p. 215, and 163.

[146] p. 241

ρf vsuperρ [E] NRe = ᎏ µ

[141] p. 214 [163]

Correlate 20 gas studies and 16 liquid studies. Corrected for axial dispersion with:

[165] p. 376 [168] p. 106

εDaxial ᎏ = 10 + 0.5NScNRe D Daxial is axial dispersion coefficient.

5-74

HEAT AND MASS TRANSFER

TABLE 5-27

Mass Transfer Correlations for Fixed and Fluidized Beds (Concluded )

Situation

N. Liquid fluidized beds

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlation (2ξ/εm)(1 − ε)1/2 2ξ/ε m + ᎏᎏ − 2 tan h (ξ/ε m) [1 − (1 − ε)1/3]2 NSh = ᎏᎏᎏᎏᎏ ξ/ε m ᎏᎏ − tan h (ξ/ε m) 1 − (1 − ε1/2)





where α 1 1/2 − 1 ᎏ N 1/3 ξ= ᎏ Sc N Re (1 − ε)1/3 2





This simplifies to: α2

ε 1 − 2m 1 NSh = ᎏ ᎏ−1 (1 − ε)1/3 (1 − ε)1/3



O. Liquid fluidized beds

ᎏ2 N

0.306 ρ s − ρ NSh = 0.250N 0.023 ᎏ Re N Ga ρ











m = 1 for NRe > 2; m = 0.5 for NRe < 1.0; ε = voidage; α = const. Best fit data is α = 0.7. Comparison of theory and experimental ion exchange results in Ref. 113.

0.410 N Sc

(ε < 0.85)

[E] Correlate amount of data from literature. Compare large number of published correlations.

(ε > 0.85)

k′L d p d p ρvsuper NSh = ᎏ , NRe = ᎏ D µ



0.297 0.404 N Sc

0.300 0.400 N Sc

[113, 123, 138]

Vsuper d pξ NRe = ᎏ µ

2/3 N Sc (NRe < 0.1)

This can be simplified (with slight loss in accuracy at high ε) to 0.323 ρ s − ρ NSh = 0.245N Ga ᎏ ρ

k′L d p NSh = ᎏ D

Re

0.282

−0.057 0.332 ρs − ρ NSh = 0.304N Re N Ga ᎏ ρ

[S] Modification of theory to fit experimental data. For spheres, m = 1, NRe > 2.

References*

[160]

µ d p3 ρ 2g , NSc = ᎏ NGa = ᎏ µ2 ρD 1.6 < NRe < 1320, 2470 < NGa < 4.42 × 106 ρs − ρ 0.27 < ᎏ < 1.114, 305 < NSc < 1595 ρ Predicts very little dependence of NSh on velocity.

P. Liquid film flowing over solid particles with air present, trickle bed reactors, fixed bed

Q. Supercritical fluids in packed bed

kL 1/3 NSh = ᎏ = 1.8N 1/2 Re N Sc , 0.013 < NRe < 12.6 aD

L [E] NRe = ᎏ aµ

two-phases, liquid trickle, no forced flow of gas. 1/2 1/3 NSh = 0.8N Re N Sc , one-phase, liquid only.

L = superficial liquid flow rate, kg/m2s. a = surface area/col. volume, m2/m3. Irregular granules of benzoic acid, 0.29 ≤ dp ≤ 1.45 cm.



2 NSh N Re N 1/3 Sc = 0.1813 ᎏ ᎏᎏ NGr (NSc NGr)1/4



1/4



1/3 3/4 (N 1/2 Re N Sc )

2 1/3 N Re N Sc + 1.2149 ᎏ NGr

R. Supercritical fluids in packed bed





3/4

1/3 (N 1/2 NSh Re N Sc ) = 0.5265 ᎏᎏ ᎏᎏ (NSc NGr)1/4 (NSc NGr)1/4



2 N Re N 1/3 Sc + 2.48 ᎏ NGr





0.6439



− 0.01649



[E] Natural and forced convection, 4 < NRe < 135.

[119]

[E] Natural and forced convection. 0.3 < NRe < 135. Improvement of correlation in Q.

[116]

1/3

1.6808

− 0.8768

[142]

1.553

NOTE: For NRe < 3 convective contributions which are not included may become important. Use with logarithmic concentration difference (integrated form) or with arithmetic concentration difference (differential form). *See pages 5-7 and 5-8 for references.

φ = kˆ L / kˆ L0 ≥ 1 (5-300) 0 ˆ ˆ where kL = mass-transfer coefficient with reaction and kL = masstransfer coefficient for pure physical absorption. It is important to understand that when chemical reactions are involved, this definition of kˆ L is based on the driving force defined as the difference between the concentration of unreacted solute gas at the interface and in the bulk of the liquid. A coefficient based on the total of both unreacted and reacted gas could have values smaller than the physical-absorption mass-transfer coefficient kˆ L0 . When liquid-phase resistance is important, particular care should be taken in employing any given set of experimental data to ensure that the equilibrium data used conform with those employed by the original author in calculating values of kˆ L or HL. Extrapolation to widely different concentration ranges or operating conditions should

be made with caution, since the mass-transfer coefficient kˆ L may vary in an unexpected fashion, owing to changes in the apparent chemicalreaction mechanism. Generalized prediction methods for kˆ L and HL do not apply when chemical reaction occurs in the liquid phase, and therefore one must use actual operating data for the particular system in question. A discussion of the various factors to consider in designing gas absorbers and strippers when chemical reactions are involved is presented by Astarita, Savage, and Bisio, Gas Treating with Chemical Solvents, Wiley (1983) and by Kohl and Ricsenfeld, Gas Purification, 4th ed., Gulf (1985). Effective Interfacial Mass-Transfer Area a In a packed tower of constant cross-sectional area S the differential change in solute flow per unit time is given by −d(GMSy) = NAa dV = NAaS dh

(5-301)

MASS TRANSFER

5-75

TABLE 5-28 Mass Transfer Correlations for Packed Two-Phase Contactors—Absorption, Distillation, Cooling Towers, and Extractors (Packing Is Inert) Situation A. Absorption, counter-current, liquid-phase coefficient HL, Sherwood and Holloway correlation for random packings

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlations

 

L n 0.5 HL = a L ᎏ N Sc,L , L = lb/hr ft 2 µL LM HL = ᎏ kˆ L a L M = lbmoles/hr ft 2, kˆ L = lbmoles/hr ft 2, a = ft 2/ft 3, µ L in lb/(hr ft). Ranges for 5-28-B (G and L) Packing

aG

b

c

G

L

aL

References*

[E] From experiments on desorption of sparingly soluble gases from water. Graphs [Ref. 146], p. 606. Equation is dimensional. A typical value of n is 0.3 [Ref. 91] p. 633 has constants in kg, m, and s units for use in 5-28-A and B with kˆ G in kgmole/s m2 and kˆ L in kgmole/s m2 (kgmole/m3). Constants for other packings are given by Refs. 121 p. 187 and 161, p. 239.

[121] p. 187 [122] p. 714 [146] p. 606 [164] p. 660

[E] Based on ammonia-water-air data in Fellinger’s 1941 MIT thesis. Curves: Refs. 121, p. 186 and 146 p. 607. Constants given in 5-28-A. The equation is dimensional.

[91] p. 633 [121] p. 189 [146] p. 607 [164] p. 660

[E] Compared napthalene sublimination to aqueous absorption to obtain kˆ G , a, and kˆ L separately. Raschig rings and Berl saddles. d p = diameter of sphere with same surface area as packing piece. ε Lo = operating void space = ε − φ Li , where ε = void fraction w/o liquid, and φ Li = liquid holdup. Same definition as 5-28-A and B. Onda et al. correlation (5-28-D) is preferred. G = ρGv super,gas

[76][123] p. 174, 186 [151, 158] [161] p. 203

n

Raschig rings 3/8 inch 1 1 2

2.32 7.00 6.41 3.82

0.45 0.39 0.32 0.41

0.47 0.58 0.51 0.45

32.4 0.30 0.811 0.30 1.97 0.36 5.05 0.32

0.74 0.24 0.40 0.45

200–500 200–800 200–600 200–800

500–1500 0.00182 0.46 400–500 0.010 0.22 500–4500 — — 500–4500 0.0125 0.22

Berl saddles 1/2 inch 1/2 1 1.5

200–700 200–800 200–800 200–1000

500–1500 0.0067 0.28 400–4500 — — 400–4500 0.0059 0.28 400–4500 0.0062 0.28

Range for 5-28-A is 400 < L < 15,000 lb/hr ft2 B. Absorption counter-current, gasphase coefficient HG, for random packing

0.5 a G(G) bN Sc,v HG = ᎏᎏ , G = lb/hr ft 2 (L) c

GM HG = ᎏ kˆ G a G M = lbmoles/hr ft 2, kˆ G = lbmoles/hr ft 2.

C. Absorption, counter-current, gasliquid individual coefficients and interfacial area, Shulman data for random packings



2/3 dpG k GN Sc,v ᎏ = 1.195 ᎏᎏ GM µ G(1 − ε Lo)

dp L kˆ L d p ᎏ = 25.1 ᎏ DL µL

 



−0.36

0.45

N 0.5 Sc,L

Interfacial area a per volume given for Racshig rings and Berl saddles in graphical form by Refs. 78 and 121 p. 178, and in equation form by Ref. 161, p. 205. Liquid holdups are given by Refs. 161 (p. 206), 148, or 121, p. 174. D. Absorption and and distillation, counter-current, gas and liquid individual coefficients and wetted surface area, Onda et al. correlation for random packings



G k′G RT ᎏ=A ᎏ a pDG a pµ G



0.7

−2.0 N 1/3 Sc,G (a pd′p)

A = 5.23 for packing ≥ 1/2 inch (0.012 m) A = 2.0 for packing < 1/2 inch (0.012 m) k′G = lbmoles/hr ft 2 atm [kg mol/s m2 (N/m2)] ρL k L′ ᎏ µ Lg

 

1/3



L = 0.0051 ᎏ aw µL



2/3

−1/2 N Sc,L (a p d′p)0.4

[E] Gas absorption and desorption from water and organics plus vaporization of pure liquids for Raschig rings, saddles, spheres, and rods. d′p = nominal packing size, a p = dry packing surface area/volume, a w = wetted packing surface area/volume. Equations are dimensionally consistent, so any set of consistent units can be used. σ = surface tension, dynes/cm.

k′L = lbmoles/hr ft 2 (lbmoles/ft 3) [kgmoles/s m2 (kgmoles/m3)] aw ᎏ = 1 − exp ap



σ −1.45 ᎏc σ

  ᎏ aµ 

L2a p × ᎏ ρ L2 g

0.75

0.1

L

p

L

 ᎏ ρ σa  −0.05

L

L

p

0.2

Critical surface tensions, σ C = 61 (ceramic), 75 (steel), 33 (polyethylene), 40 (PVC), 56 (carbon) dynes/cm. Graphical comparison with data in Ref. 126.

L 4 < ᎏ < 400 aw µ L G 5 < ᎏ < 1000 ap µG Most data ± 20% of correlation, some ± 50%.

[76][88] p. 399 [111] p. 380 [126][159] p. 355

5-76

HEAT AND MASS TRANSFER

TABLE 5-28 Mass Transfer Correlations for Packed Two-Phase Contactors—Absorption, Distillation, Cooling Towers, and Extractors (Packing Is Inert) (Continued ) Situation E. Distillation and absorption, counter-current, random packings, modification of Onda correlation, Bravo and Fair correlation

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlations Use Onda’s correlations (5-28-D) for k′G and k′L. Calculate: G L HG = ᎏ , HL = ᎏ k G′ aePMG k′LaeρL HOG = HG + λHL where m λ=ᎏ LM/GM Using

References*

[E] Use’s Bolles & Fair (Ref. 75) data base to determine new effective area ae to use with Onda et al. (Ref. 126) correlation. Same definitions as 5-28-D. P = total pressure, atm; MG = gas, molecular weight; m = local slope of equilibrium curve; LM/GM = slope operating line; Z = height of packing in feet. Equation for ae is dimensional. Fit to data for effective area quite good for distillation. Good for absorption at low values of (Nca,L × NRe,G), but correlation is too high at higher values of (NCa,L × NRe,G).

[76]

[E] Based on oxygen transfer from water to air 77°F. Liquid film resistance controls. (Dwater @ 77°F = 2.4 × 10−5). Equation is dimensional. Data was for thin-walled polyethylene Raschig rings. Correlation also fit data for spheres. Fit ⫾25%. See Reiss for graph.

[136] [142] p. 217

σ0.5 ae = 0.498ap ᎏ (NCa,LNRe,G)0.392 Z0.4 where

 

6G NRe,G = ᎏ , apµG LµL NCa,L = ᎏ (dimensionless) ρLσgc F. Absorption, co-current downward flow, random packings

Air-oxygen-water results correlated by k′La = 0.12EL0.5. Extended to other systems.



DL k′La = 0.12EL0.5 ᎏ 2.4 × 105



0.5

∆p EL = ᎏ vL ∆L 2-phase −1 k′La = s DL = cm/s EL = ft, lbf/s ft3 vL = superficial liquid velocity, ft/s

 

∆p ᎏ = pressure loss in two-phase flow = lbf/ft2 ft ∆L k′G a = 2.0 + 0.91EG2/3 for NH3 ∆p Eg = ᎏ vg ∆L 2-phase vg = superficial gas velocity, ft/s.

 

G. Absorption, stripping, distillation, counter-current, HL, and HG, random packings, Cornell et al. correlation, and Bolles and Fair correlation

For Raschig rings, Berl saddles, and spiral tile: Z 0.15 φCflood 0.5 HL = ᎏ N Sc,L ᎏ 3.28 3.05 Cflood = 1.0 if below 40% flood—otherwise, use Fig. 5-28. φ shown in Fig. 5-29 for different packings and sizes. Range 0.02 < φ < 0.300. Aψ(d′col)mZ0.33N0.5 Sc,G ᎏᎏᎏᎏ HG = µL 0.16 ρwater 1.25 σwater 0.8 n L ᎏᎏ ᎏᎏ ᎏᎏ µwater ρL σL

 



 

 

[E] Ammonia absorption into water from air at 70°F. Gas-film resistance controls. Thin-walled polyethylene Raschig rings and 1-inch Intalox saddles. Fit ⫾25%. See Reiss for fit. Terms defined as above. [E] Z = packed height, m of each section with its own liquid distribution. The original work is reported in English units. Cornell et al. (Ref. 81) review early literature. Improved fit of Cornell’s φ values given by Bolles and Fair (Refs. 74 and 75) and in Fig. 5-29.



A = 0.017 (rings) or 0.029 (saddles) d′col = column diameter in m (if diameter > 0.6 m, use d′col = 0.6) m = 1.24 (rings) or 1.11 (saddles) n = 0.6 (rings) or 0.5 (saddles) ψ is given in Fig. 5-30. Range: 25 < ψ < 190 m.

L = liquid rate, kg/(sm2), µwater = 1.0 Pa ⋅ s, ρwater = 1000 kg/m3, σwater = 72.8 mN/m (72.8 dynes/cm). HG and HL will vary from location to location. Design each section of packing separately.

[136]

[74, 75, 81] [100] p. 428 [111] p. 381 [151] p. 353 [164] p. 651

MASS TRANSFER

5-77

TABLE 5-28 Mass Transfer Correlations for Packed Two-Phase Contactors—Absorption, Distillation, Cooling Towers, and Extractors (Packing Is Inert) (Concluded ) Situation H. Distillation and absorption. Counter-current flow. Structured packings. Gauze-type with triangular flow channels, Bravo, Rocha, and Fair correlation

Correlations

[T] Check of 132 data points showed average deviation 14.6% from theory. Johnstone and Pigford [Ref. 105] correlation (5-22-D) has exponent on NRe rounded to 0.8. Assume gauze packing is completely wet. Thus, aeff = ap to calculate HG and HL. Same approach may be used generally applicable to sheet-metal packings, but they will not be completely wet and need to estimate transfer area. L = liquid flux, kg/s m2. G = vapor flux, kg/s m2. Fit to data shown in Ref. 77.

Equivalent channel:



1 1 deq = Bh ᎏ + ᎏ B + 2S 2S

Comments E = Empirical, S = Semiempirical, T = Theoretical



Use modified correlation for wetted wall column (See 5-22-D) k′vdeq 0.333 NSh,v = ᎏ = 0.0338N0.8 Re,vNSc,v Dv

References* [77] [87] p. 310, 326 [159] p. 356, 362

G L HG = ᎏ , HL = ᎏ k′vapρv k′LapρL

deqρv(Uv,eff + UL,eff) NRe,v = ᎏᎏ µv where effective velocities Uv,super Uv,eff = ᎏ ε sin θ 3Γ ρL2 g UL,eff = ᎏ ᎏ 2ρL 3µLΓ





0.333

L ,Γ=ᎏ Per

Perimeter 4S + 2B Per = ᎏᎏ = ᎏ Area Bh Calculate k′L from penetration model (use time for liquid to flow distance s). k′L = 2(DLUL,eff /πS)1/2. I. High-voidage packings, cooling towers, splash-grid packings

J. Liquid-liquid extraction, packed towers

K. Liquid-liquid extraction in Rotating-disc contactor (RDC)

(Ka)HVtower L −n′ ᎏᎏ = 0.07 + A′N′ ᎏ L Ga A′ and n′ depend on deck type (Ref. 107), 0.060 ≤ A′ ≤ 0.135, 0.46 ≤ n′ ≤ 0.62. General form fits the graphical comparisons (Refs. 146 and 164).

 



lb water lb/(h)(ft3) ᎏ lb dry air Vtower = tower volume, ft3/ft2. If normal packings are used, use absorption masstransfer correlations or Ref. 88, p. 452. [156] p. 79

[70][156] p. 79

kd,RDC N H ᎏ = 1.0 + 1.825 ᎏ ᎏ kd NCr Dtank

kc, kd are for drops (Table 5-25) N = impeller speed Breakage occurs when N > NCr. Maximum enhancement before breakage was factor of 2.0. H = compartment height, Dtank = tank diameter, σ = interfacial tension, N/m. Done in 0.152 and 0.600 m RDC.

See Table 5-26-F, G, H, and I.

[E]

 

kc,RDC N ᎏ = 1.0 + 2.44 ᎏ kc NCr



2.5

ᎏ D  H

tank

 

*See pages 5-7 and 5-8 for references.



[107][121] p. 220 [146] p. 286 [164] p. 681

[E]

Use k values for drops (Table 5-25). Enhancement due to packing is at most 20%. Packing decreases drop size and increases interfacial area.

σ NCr = 7.6 × 10−4 ᎏ ddrop µc

L. Liquid-liquid extraction, stirred tanks

[E] General form. Ga = lb dry air/hr ft2. L = lb/h ft2, N′ = number of deck levels. (Ka)H = overall enthalpy transfer coefficient =

5-78

HEAT AND MASS TRANSFER

Liquid-film correction factor (Table 5-28-G) for operation at high percent of flood. [Cornell et al., Chem. Eng. Prog., 56(8), 68 (1960).]

FIG. 5-28

where a = interfacial area effective for mass transfer per unit of packed volume and V = packed volume. Owing to incomplete wetting of the packing surfaces and to the formation of areas of stagnation in the liquid film, the effective area normally is significantly less than the total external area of the packing pieces. The effective interfacial area depends on a number of factors, as discussed in a review by Charpentier [Chem. Eng. J., 11, 161 (1976)]. Among these factors are (1) the shape and size of packing, (2) the packing material (for example, plastic generally gives smaller interfacial areas than either metal or ceramic), (3) the liquid mass velocity, and (4), for small-diameter towers, the column diameter. Whereas the interfacial area generally increases with increasing liquid rate, it apparently is relatively independent of the superficial gas mass velocity below the flooding point. According to Charpentier’s review, it appears valid to assume that the interfacial area is independent of the column height when specified in terms of unit packed volume (i.e., as a). Also, the existing data for chemically reacting gas-liquid systems (mostly aqueous electrolyte solutions) indicate that the interfacial area is independent of the chemical system. However, this situation may not hold true for systems involving large heats of reaction. Rizzuti et al. [Chem. Eng. Sci., 36, 973 (1981)] examined the influence of solvent viscosity upon the effective interfacial area in packed columns and concluded that for the systems studied the effective interfacial area a was proportional to the kinematic viscosity raised to the 0.7 power. Thus, the hydrodynamic behavior of a packed absorber is strongly affected by viscosity effects. Surface-tension effects also are important, as expressed in the work of Onda et al. (see Table 5-28-D). In developing correlations for the mass-transfer coefficients kˆ G and kˆ L, the various authors have assumed different but internally compatible correlations for the effective interfacial area a. It therefore would be inappropriate to mix the correlations of different authors unless it has been demonstrated that there is a valid area of overlap between them. Volumetric Mass-Transfer Coefficients Kˆ Ga and Kˆ La Experimental determinations of the individual mass-transfer coefficients kˆ G and kˆ L and of the effective interfacial area a involve the use of extremely difficult techniques, and therefore such data are not plentiful. More often, column experimental data are reported in terms of overall volumetric coefficients, which normally are defined as follows: and

K′Ga = nA /(hTSpT ∆y°1m)

(5-302)

KLa = nA /(hTS ∆x°1m)

(5-303)

where K′Ga = overall volumetric gas-phase mass-transfer coefficient, KLa = overall volumetric liquid-phase mass-transfer coefficient, nA =

Hl correlation for various packings (Table 5-28-G). To convert meters to feet, multiply by 3.281; to convert pounds per hour-square foot to kilograms per second-square meter, multiply by 0.001356; and to convert millimeters to inches, multiply by 0.0394. [Bolles and Fair, Inst. Chem. Eng. Symp. Ser., no. 56, 3.3/35 (1969).]

FIG. 5-29

overall rate of transfer of solute A, hT = total packed depth in tower, S = tower cross-sectional area, pT = total system pressure employed during the experiment, and ∆x°1m and ∆y°1m are defined as (y − y°)1 − (y − y°)2 ∆y°1m = ᎏᎏᎏ ln [(y − y°)1/(y − y°)2]

(5-304)

MASS TRANSFER

5-79

vary with the total system pressure except when the liquid-phase resistance is negligible (i.e., when either m = 0, or kˆ La is very large, or both). Extrapolation of KGa data for absorption and stripping to conditions other than those for which the original measurements were made can be extremely risky, especially in systems involving chemical reactions in the liquid phase. One therefore would be wise to restrict the use of overall volumetric mass-transfer-coefficient data to conditions not too far removed from those employed in the actual tests. The most reliable data for this purpose would be those obtained from an operating commercial unit of similar design. Experimental values of HOG and HOL for a number of distillation systems of commercial interest are also readily available. Extrapolation of the data or the correlations to conditions that differ significantly from those used for the original experiments is risky. For example, pressure has a major effect on vapor density and thus can affect the hydrodynamics significantly. Changes in flow patterns affect both masstransfer coefficients and interfacial area. Chilton-Colburn Analogy When a fluid moves over either a liquid or a solid surface, the eddy motion that causes mass transfer also causes heat transfer and fluid friction owing to the transfer of thermal energy and momentum respectively. This close similarity among the mechanisms for the transfer of mass, heat, and momentum was brought out in the Reynolds analogy (see Table 5-23-T), which stated that the following dimensionless ratios are equal: kˆ G/GM = h′/cpG = f/2 (5-306) where h′ = heat-transfer coefficient, cp = specific heat, G = mass flux, and f = friction factor. Experimental data for mass transfer into gas streams agree approximately with Eq. (5-306) when the Schmidt number is close to unity and in smooth, straight tubes or along flat plates when the pressure drop is due entirely to skin friction against the surface. It does not, however, agree for cases involving “form” drag as well as skin friction. Also, it does not account for the mass-transfer resistance of the region of fluid near the liquid or solid boundary in which mass transfer occurs principally by molecular (as opposed to turbulent) motion. Colburn [Trans. Am. Inst. Chem. Eng., 29, 174 (1933)] and Chilton and Colburn [Ind. Eng. Chem., 26, 1183 (1934)] showed empirically that the resistance of the laminar sublayer can be expressed by the following modification of the Reynolds analogy: ( kˆ G/GM)NSc2/3 = jM = (h′/cpG)NPr2/3 = jH = f/2 (5-307) Hg correlation for various packings (Table 5-28-G). To convert meters to feet, multiply by 3.281; to convert millimeters to inches, multiply by 0.03937. [Bolles and Fair, Inst. Chem. Eng. Symp. Ser., no. 56, 3.3/35 (1979).] FIG. 5-30

(x° − x)2 − (x° − x)1 ∆x°1m = ᎏᎏᎏ (5-305) ln [(x° − x)2/(x° − x)1] where subscripts 1 and 2 refer to the bottom and top of the tower respectively. Experimental K′Ga and KLa data are available for most absorption and stripping operations of commercial interest (see Sec. 15). The solute concentrations employed in these experiments normally are very low, so that KLa ⬟ Kˆ La and K′GapT ⬟ Kˆ Ga, where pT is the total pressure employed in the actual experimental-test system. Unlike the individual gas-film coefficient kˆ Ga, the overall coefficient Kˆ Ga will and

for turbulent flow through straight tubes (see Table 5-23-U) and across plane surfaces (see Table 5-21-G), and jM = jH ≤ f/2

(5-308)

for turbulent flow around cylinders (see Table 5-24-I), where jM = mass-transfer factor, jH = heat-transfer factor, NPr = cp µ/k = Prandtl number, and k = thermal conductivity; other symbols are as defined earlier. On occasion one will find that heat-transfer-rate data are available for a system in which mass-transfer-rate data are not readily available. The Chilton-Colburn analogy provides a procedure for developing estimates of the mass-transfer rates based on heat-transfer data. Extrapolation of experimental jM or jH data obtained with gases to predict liquid systems (and vice versa) should be approached with caution, however. When pressure-drop or friction-factor data are available, one may be able to place an upper bound on the rates of heat and mass transfer, according to Eq. (5-308).

Section 6

Fluid and Particle Dynamics*

James N. Tilton, Ph.D., P.E., Senior Consultant, Process Engineering, E. I. du Pont de Nemours & Co.; Member, American Institute of Chemical Engineers; Registered Professional Engineer (Delaware)

FLUID DYNAMICS Nature of Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deformation and Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Viscosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kinematics of Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compressible and Incompressible Flow . . . . . . . . . . . . . . . . . . . . . . . Streamlines, Pathlines, and Streaklines . . . . . . . . . . . . . . . . . . . . . . . . One-dimensional Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rate of Deformation Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laminar and Turbulent Flow, Reynolds Number . . . . . . . . . . . . . . . . Conservation Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Macroscopic and Microscopic Balances . . . . . . . . . . . . . . . . . . . . . . . Macroscopic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass Balance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Momentum Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total Energy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanical Energy Balance, Bernoulli Equation . . . . . . . . . . . . . . . . Microscopic Balance Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass Balance, Continuity Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cauchy Momentum and Navier-Stokes Equations . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 1: Force Exerted on a Reducing Bend. . . . . . . . . . . . . . . . . Example 2: Simplified Ejector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 3: Venturi Flowmeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 4: Plane Poiseuille Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . Incompressible Flow in Pipes and Channels. . . . . . . . . . . . . . . . . . . . . . Mechanical Energy Balance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Friction Factor and Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . Laminar and Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Velocity Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Entrance and Exit Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Residence Time Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Noncircular Channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonisothermal Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Open Channel Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-Newtonian Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Economic Pipe Diameter, Turbulent Flow . . . . . . . . . . . . . . . . . . . . .

6-4 6-4 6-4 6-4 6-5 6-5 6-5 6-5 6-5 6-5 6-5 6-6 6-6 6-6 6-6 6-6 6-6 6-7 6-7 6-7 6-7 6-7 6-8 6-8 6-8 6-8 6-9 6-9 6-9 6-9 6-9 6-10 6-11 6-11 6-11 6-12 6-12 6-12 6-13 6-14

Economic Pipe Diameter, Laminar Flow . . . . . . . . . . . . . . . . . . . . . . Vacuum Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Molecular Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Slip Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frictional Losses in Pipeline Elements . . . . . . . . . . . . . . . . . . . . . . . . . . Equivalent Length and Velocity Head Methods . . . . . . . . . . . . . . . . . Contraction and Entrance Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 5: Entrance Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expansion and Exit Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fittings and Valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 6: Losses with Fittings and Valves . . . . . . . . . . . . . . . . . . . . Curved Pipes and Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Screens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jet Behavior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow through Orifices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compressible Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mach Number and Speed of Sound . . . . . . . . . . . . . . . . . . . . . . . . . . Isothermal Gas Flow in Pipes and Channels. . . . . . . . . . . . . . . . . . . . Adiabatic Frictionless Nozzle Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 7: Flow through Frictionless Nozzle . . . . . . . . . . . . . . . . . . Adiabatic Flow with Friction in a Duct of Constant Cross Section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 8: Compressible Flow with Friction Losses . . . . . . . . . . . . . Convergent/Divergent Nozzles (De Laval Nozzles) . . . . . . . . . . . . . . Multiphase Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquids and Gases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gases and Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluid Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Perforated-Pipe Distributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 9: Pipe Distributor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Slot Distributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turning Vanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Perforated Plates and Screens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Beds of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Flow Straightening Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluid Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stirred Tank Agitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pipeline Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tube Banks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transition Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6-14 6-14 6-15 6-15 6-16 6-16 6-16 6-16 6-17 6-17 6-17 6-18 6-19 6-20 6-21 6-22 6-22 6-22 6-22 6-23 6-23 6-25 6-25 6-26 6-26 6-29 6-32 6-32 6-33 6-33 6-33 6-33 6-34 6-34 6-34 6-34 6-35 6-36 6-36 6-38

* The author acknowledges the contribution of B. C. Sakiadis, editor of this section in the sixth edition of the Handbook. 6-1

6-2

FLUID AND PARTICLE DYNAMICS

Laminar Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Beds of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fixed Beds of Granular Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tower Packings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluidized Beds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boundary Layer Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flat Plate, Zero Angle of Incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . Cylindrical Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous Flat Surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous Cylindrical Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vortex Shedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coating Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Falling Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minimum Wetting Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laminar Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of Surface Traction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flooding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hydraulic Transients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Water Hammer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6-38 6-38 6-38 6-39 6-40 6-40 6-40 6-40 6-40 6-40 6-41 6-41 6-42 6-42 6-42 6-42 6-43 6-43 6-43 6-44 6-44

Example 10: Response to Instantaneous Valve Closing . . . . . . . . . . . Pulsating Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cavitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Closure Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eddy Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computational Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dimensionless Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6-44 6-44 6-44 6-45 6-45 6-46 6-46 6-47 6-48

PARTICLE DYNAMICS Drag Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Terminal Settling Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spherical Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonspherical Rigid Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hindered Settling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time-dependent Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gas Bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid Drops in Liquids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid Drops in Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wall Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6-50 6-50 6-50 6-51 6-52 6-52 6-53 6-53 6-54 6-54

Nomenclature and Units* In this listing, symbols used in this section are defined in a general way and appropriate SI units are given. Specific definitions, as denoted by subscripts, are stated at the place of application in the section. Some specialized symbols used in the section are defined only at the place of application. Some symbols have more than one definition; the appropriate one is identified at the place of application. Symbol

Definition

SI units

U.S. customary units

a A b b c cf C Ca C0 CD d D De Dij

Pressure wave velocity Area Wall thickness Channel width Acoustic velocity Friction coefficient Conductance Capillary number Discharge coefficient Drag coefficient Diameter Diameter Dean number Deformation rate tensor components Elastic modulus Energy dissipation rate Eotvos number Fanning friction factor Vortex shedding frequency Force Cumulative residence time distribution Froude number Acceleration of gravity Mass flux Enthalpy per unit mass Liquid depth Ratio of specific heats Kinetic energy of turbulence Power law coefficient Viscous losses per unit mass Length Mass flow rate Mass Mach number Morton number Molecular weight Power law exponent Blend time number Best number Power number Pumping number Pressure Entrained flow rate Volumetric flow rate Throughput (vacuum flow) Heat input per unit mass Radial coordinate Radius Ideal gas universal constant Volume fraction of phase i Reynolds number Density ratio

m/s m2 m m m/s Dimensionless m3/s Dimensionless Dimensionless Dimensionless m m Dimensionless 1/s

ft/s ft2 in ft ft/s Dimensionless ft3/s Dimensionless Dimensionless Dimensionless ft ft Dimensionless 1/s

Pa J/s Dimensionless Dimensionless 1/s N Dimensionless

lbf/in2 ft ⋅ lbf/s Dimensionless Dimensionless 1/s lbf Dimensionless

Dimensionless m/s2 kg/(m2 ⋅ s) J/kg m Dimensionless J/kg kg/(m ⋅ s2 − n) J/kg m kg/s kg Dimensionless Dimensionless kg/kgmole Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Pa m3/s m3/s Pa ⋅ m3/s J/kg m m J/(kgmole ⋅ K) Dimensionless Dimensionless Dimensionless

Dimensionless ft/s2 lbm/(ft2 ⋅ s) Btu/lbm ft Dimensionless ft ⋅ lbf/lbm lbm/(ft ⋅ s2 − n) ft ⋅ lbf/lbm ft lbm/s lbm Dimensionless Dimensionless lbm/lbmole Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless lbf/in2 ft3/s ft3/s lbf ⋅ ft3/s Btu/lbm ft ft Btu/(lbmole ⋅ R) Dimensionless Dimensionless Dimensionless

E E˙ v Eo f f F F Fr g G h h k k K lv L m ˙ M M M Mw n Nb ND NP NQ p q Q Q δQ r R R Ri Re s

Symbol

Definition

SI units

U.S. customary units

s S S S St t t T u u U v V V We ˙s W δWs x y z z

Entropy per unit mass Slope Pumping speed Surface area per unit volume Strouhal number Time Force per unit area Absolute temperature Internal energy per unit mass Velocity Velocity Velocity Velocity Volume Weber number Rate of shaft work Shaft work per unit mass Cartesian coordinate Cartesian coordinate Cartesian coordinate Elevation

J/(kg ⋅ K) Dimensionless m3/s l/m Dimensionless s Pa K J/kg m/s m/s m/s m/s m3 Dimensionless J/s J/kg m m m m

Btu/(lbm ⋅ R) Dimensionless ft3/s l/ft Dimensionless s lbf/in2 R Btu/lbm ft/s ft/s ft/s ft/s ft3 Dimensionless Btu/s Btu/lbm ft ft ft ft

α α β β γ˙ Γ

Velocity profile factor Included angle Velocity profile factor Bulk modulus of elasticity Shear rate Mass flow rate per unit width Boundary layer or film thickness Kronecker delta Pipe roughness Void fraction Turbulent dissipation rate Residence time Angle Mean free path Viscosity Kinematic viscosity Density Surface tension Cavitation number Components of total stress tensor Shear stress Time period Components of deviatoric stress tensor Energy dissipation rate per unit volume Angle of inclination Vorticity

Greek symbols

δ δij


θ θ λ µ ν ρ σ σ σij τ τ τij Φ φ ω

Dimensionless Radians Dimensionless Pa l/s kg/(m ⋅ s)

Dimensionless Radians Dimensionless lbf/in2 l/s lbm/(ft ⋅ s)

m

ft

Dimensionless m Dimensionless J/(kg ⋅ s) s Radians m Pa ⋅ s m2/s kg/m3 N/m Dimensionless Pa

Dimensionless ft Dimensionless ft ⋅ lbf/(lbm ⋅ s) s Radians ft lbm/(ft ⋅ s) ft2/s lbm/ft3 lbf/ft Dimensionless lbf/in2

Pa s Pa

lbf/in2 s lbf/in2

J/(m3 ⋅ s)

ft ⋅ lbf/(ft3 ⋅ s)

Radians 1/s

Radians 1/s

* Note that with U.S. Customary units, the conversion factor gc may be required to make equations in this section dimensionally consistent; gc = 32.17 (lbm⋅ft)/lbf⋅s2). 6-3

6-4

FLUID AND PARTICLE DYNAMICS

FLUID DYNAMICS GENERAL REFERENCES: Batchelor, An Introduction to Fluid Dynamics, Cambridge University, Cambridge, 1967; Bird, Stewart, and Lightfoot, Transport Phenomena, Wiley, New York, 1960; Brodkey, The Phenomena of Fluid Motions, Addison-Wesley, Reading, Mass., 1967; Denn, Process Fluid Mechanics, Prentice-Hall, Englewood Cliffs, N.J., 1980; Landau and Lifshitz, Fluid Mechanics, 2d ed., Pergamon, 1987; Govier and Aziz, The Flow of Complex Mixtures in Pipes, Van Nostrand Reinhold, New York, 1972, Krieger, Huntington, N.Y., 1977; Panton, Incompressible Flow, Wiley, New York, 1984; Schlichting, Boundary Layer Theory, 8th ed., McGraw-Hill, New York, 1987; Shames, Mechanics of Fluids, 3d ed., McGraw-Hill, New York, 1992; Streeter, Handbook of Fluid Dynamics, McGraw-Hill, New York, 1971; Streeter and Wylie, Fluid Mechanics, 8th ed., McGraw-Hill, New York, 1985; Vennard and Street, Elementary Fluid Mechanics, 5th ed., Wiley, New York, 1975; Whitaker, Introduction to Fluid Mechanics, Prentice-Hall, Englewood Cliffs, N.J., 1968, Krieger, Huntington, N.Y., 1981.

NATURE OF FLUIDS Deformation and Stress A fluid is a substance which undergoes continuous deformation when subjected to a shear stress. Figure 6-1 illustrates this concept. A fluid is bounded by two large parallel plates, of area A, separated by a small distance H. The bottom plate is held fixed. Application of a force F to the upper plate causes it to move at a velocity U. The fluid continues to deform as long as the force is applied, unlike a solid, which would undergo only a finite deformation.

A

F V

y

H

Fluids without any solidlike elastic behavior do not undergo any reverse deformation when shear stress is removed, and are called purely viscous fluids. The shear stress depends only on the rate of deformation, and not on the extent of deformation (strain). Those which exhibit both viscous and elastic properties are called viscoelastic fluids. Purely viscous fluids are further classified into time-independent and time-dependent fluids. For time-independent fluids, the shear stress depends only on the instantaneous shear rate. The shear stress for time-dependent fluids depends on the past history of the rate of deformation, as a result of structure or orientation buildup or breakdown during deformation. A rheogram is a plot of shear stress versus shear rate for a fluid in simple shear flow, such as that in Fig. 6-1. Rheograms for several types of time-independent fluids are shown in Fig. 6-2. The Newtonian fluid rheogram is a straight line passing through the origin. The slope of the line is the viscosity. For a Newtonian fluid, the viscosity is independent of shear rate, and may depend only on temperature and perhaps pressure. By far, the Newtonian fluid is the largest class of fluid of engineering importance. Gases and low molecular weight liquids are generally Newtonian. Newton’s law of viscosity is a rearrangement of Eq. (6-1) in which the viscosity is a constant: du τ = µγ˙ = µ  (6-2) dy All fluids for which the viscosity varies with shear rate are nonNewtonian fluids. For non-Newtonian fluids the viscosity, defined as the ratio of shear stress to shear rate, is often called the apparent viscosity to emphasize the distinction from Newtonian behavior. Purely viscous, time-independent fluids, for which the apparent viscosity may be expressed as a function of shear rate, are called generalized Newtonian fluids.

x Deformation of a fluid subjected to a shear stress.

The SI units of viscosity are kg/(m ⋅ s) or Pa ⋅ s (pascal second). The cgs unit for viscosity is the poise; 1 Pa ⋅ s equals 10 poise or 1000 centipoise (cP) or 0.672 lbm/(ft ⋅ s). The terms absolute viscosity and shear viscosity are synonymous with the viscosity as used in Eq. (6-1). Kinematic viscosity ν ⬅ µ/ρ is the ratio of viscosity to density. The SI units of kinematic viscosity are m2/s. The cgs stoke is 1 cm2/s. Rheology In general, fluid flow patterns are more complex than the one shown in Fig. 6-1, as is the relationship between fluid deformation and stress. Rheology is the discipline of fluid mechanics which studies this relationship. One goal of rheology is to obtain constitutive equations by which stresses may be computed from deformation rates. For simplicity, fluids may be classified into rheological types in reference to the simple shear flow of Fig. 6-1. Complete definitions require extension to multidimensional flow. For more information, several good references are available, including Bird, Armstrong, and Hassager, (Dynamics of Polymeric Liquids, vol. 1: Fluid Mechanics, Wiley, New York, 1977); Metzner, (“Flow of Non-Newtonian Fluids” in Streeter, Handbook of Fluid Dynamics, McGraw-Hill, New York, 1971); and Skelland (Non-Newtonian Flow and Heat Transfer, Wiley, New York, 1967).

ti c as m tic ha las g p n o Bi pl

τy

Ps eu d

The force is directly proportional to the area of the plate; the shear stress is τ = F/A. Within the fluid, a linear velocity profile u = Uy/H is established; due to the no-slip condition, the fluid bounding the lower plate has zero velocity and the fluid bounding the upper plate moves at the plate velocity U. The velocity gradient γ˙ = du/dy is called the shear rate for this flow. Shear rates are usually reported in units of reciprocal seconds. The flow in Fig. 6-1 is a simple shear flow. Viscosity The ratio of shear stress to shear rate is the viscosity, µ. τ µ= (6-1) γ˙

Shear stress τ

FIG. 6-1

lat Di

t an

an toni Ne w Shear rate |du/dy| FIG. 6-2

Shear diagrams.

Non-Newtonian fluids include those for which a finite stress τy is required before continuous deformation occurs; these are called yield-stress materials. The Bingham plastic fluid is the simplest yield-stress material; its rheogram has a constant slope µ∞, called the infinite shear viscosity. τ = τy + µ∞γ˙ (6-3) Highly concentrated suspensions of fine solid particles frequently exhibit Bingham plastic behavior. Shear-thinning fluids are those for which the slope of the rheogram decreases with increasing shear rate. These fluids have also been called pseudoplastic, but this terminology is outdated and discouraged. Many polymer melts and solutions, as well as some solids suspensions, are shear-thinning. Shear-thinning fluids without yield stresses typically obey a power law model over a range of shear rates. τ = Kγ˙ n (6-4) The apparent viscosity is µ = Kγ˙ n − 1

(6-5)

FLUID DYNAMICS The factor K is the consistency index or power law coefficient, and n is the power law exponent. The exponent n is dimensionless, while K is in units of kg/(m ⋅ s2 − n). For shear-thinning fluids, n < 1. The power law model typically provides a good fit to data over a range of one to two orders of magnitude in shear rate; behavior at very low and very high shear rates is often Newtonian. Shear-thinning power law fluids with yield stresses are sometimes called Herschel-Bulkley fluids. Numerous other rheological model equations for shear-thinning fluids are in common use. Dilatant, or shear-thickening, fluids show increasing viscosity with increasing shear rate. Over a limited range of shear rate, they may be described by the power law model with n > 1. Dilatancy is rare, observed only in certain concentration ranges in some particle suspensions (Govier and Aziz, pp. 33–34). Extensive discussions of dilatant suspensions, together with a listing of dilatant systems, are given by Green and Griskey (Trans. Soc. Rheol, 12[1], 13–25 [1968]); Griskey and Green (AIChE J., 17, 725–728 [1971]); and Bauer and Collins (“Thixotropy and Dilatancy,” in Eirich, Rheology, vol. 4, Academic, New York, 1967). Time-dependent fluids are those for which structural rearrangements occur during deformation at a rate too slow to maintain equilibrium configurations. As a result, shear stress changes with duration of shear. Thixotropic fluids, such as mayonnaise, clay suspensions used as drilling muds, and some paints and inks, show decreasing shear stress with time at constant shear rate. A detailed description of thixotropic behavior and a list of thixotropic systems is found in Bauer and Collins (ibid.). Rheopectic behavior is the opposite of thixotropy. Shear stress increases with time at constant shear rate. Rheopectic behavior has been observed in bentonite sols, vanadium pentoxide sols, and gypsum suspensions in water (Bauer and Collins, ibid.) as well as in some polyester solutions (Steg and Katz, J. Appl. Polym. Sci., 9, 3, 177 [1965]). Viscoelastic fluids exhibit elastic recovery from deformation when stress is removed. Polymeric liquids comprise the largest group of fluids in this class. A property of viscoelastic fluids is the relaxation time, which is a measure of the time required for elastic effects to decay. Viscoelastic effects may be important with sudden changes in rates of deformation, as in flow startup and stop, rapidly oscillating flows, or as a fluid passes through sudden expansions or contractions where accelerations occur. In many fully developed flows where such effects are absent, viscoelastic fluids behave as if they were purely viscous. In viscoelastic flows, normal stresses perpendicular to the direction of shear are different from those in the parallel direction. These give rise to such behaviors as the Weissenberg effect, in which fluid climbs up a shaft rotating in the fluid, and die swell, where a stream of fluid issuing from a tube may expand to two or more times the tube diameter. A parameter indicating whether viscoelastic effects are important is the Deborah number, which is the ratio of the characteristic relaxation time of the fluid to the characteristic time scale of the flow. For small Deborah numbers, the relaxation is fast compared to the characteristic time of the flow, and the fluid behavior is purely viscous. For very large Deborah numbers, the behavior closely resembles that of an elastic solid. Analysis of viscoelastic flows is very difficult. Simple constitutive equations are unable to describe all the material behavior exhibited by viscoelastic fluids even in geometrically simple flows. More complex constitutive equations may be more accurate, but become exceedingly difficult to apply, especially for complex geometries, even with advanced numerical methods. For good discussions of viscoelastic fluid behavior, including various types of constitutive equations, see Bird, Armstrong, and Hassager (Dynamics of Polymeric Liquids, vol. 1: Fluid Mechanics, vol. 2: Kinetic Theory, Wiley, New York, 1977); Middleman (The Flow of High Polymers, Interscience (Wiley) New York, 1968); or Astarita and Marrucci (Principles of Non-Newtonian Fluid Mechanics, McGraw-Hill, New York, 1974). Polymer processing is the field which depends most on the flow of non-Newtonian fluids. Several excellent texts are available, including Middleman (Fundamentals of Polymer Processing, McGraw-Hill, New York, 1977) and Tadmor and Gogos (Principles of Polymer Processing, Wiley, New York, 1979).

6-5

There is a wide variety of instruments for measurement of Newtonian viscosity, as well as rheological properties of non-Newtonian fluids. They are described in Van Wazer, Lyons, Kim, and Colwell, (Viscosity and Flow Measurement, Interscience, New York, 1963); Coleman, Markowitz, and Noll (Viscometric Flows of Non-Newtonian Fluids, Springer-Verlag, Berlin, 1966); Dealy and Wissbrun (Melt Rheology and its Role in Plastics Processing, Van Nostrand Reinhold, 1990). Measurement of rheological behavior requires wellcharacterized flows. Such rheometric flows are thoroughly discussed by Astarita and Marrucci (Principles of Non-Newtonian Fluid Mechanics, McGraw-Hill, New York, 1974). KINEMATICS OF FLUID FLOW Velocity The term kinematics refers to the quantitative description of fluid motion or deformation. The rate of deformation depends on the distribution of velocity within the fluid. Fluid velocity v is a vector quantity, with three cartesian components vx, vy, and vz. The velocity vector is a function of spatial position and time. A steady flow is one in which the velocity is independent of time, while in unsteady flow v varies with time. Compressible and Incompressible Flow An incompressible flow is one in which the density of the fluid is constant or nearly constant. Liquid flows are normally treated as incompressible, except in the context of hydraulic transients (see following). Compressible fluids, such as gases, may undergo incompressible flow if pressure and/or temperature changes are small enough to render density changes insignificant. Frequently, compressible flows are regarded as flows in which the density varies by more than 5 to 10 percent. Streamlines, Pathlines, and Streaklines These are curves in a flow field which provide insight into the flow pattern. Streamlines are tangent at every point to the local instantaneous velocity vector. A pathline is the path followed by a material element of fluid; it coincides with a streamline if the flow is steady. In unsteady flow the pathlines generally do not coincide with streamlines. Streaklines are curves on which are found all the material particles which passed through a particular point in space at some earlier time. For example, a streakline is revealed by releasing smoke or dye at a point in a flow field. For steady flows, streamlines, pathlines, and streaklines are indistinguishable. In two-dimensional incompressible flows, streamlines are contours of the stream function. One-dimensional Flow Many flows of great practical importance, such as those in pipes and channels, are treated as onedimensional flows. There is a single direction called the flow direction; velocity components perpendicular to this direction are either zero or considered unimportant. Variations of quantities such as velocity, pressure, density, and temperature are considered only in the flow direction. The fundamental conservation equations of fluid mechanics are greatly simplified for one-dimensional flows. A broader category of one-dimensional flow is one where there is only one nonzero velocity component, which depends on only one coordinate direction, and this coordinate direction may or may not be the same as the flow direction. Rate of Deformation Tensor For general three-dimensional flows, where all three velocity components may be important and may vary in all three coordinate directions, the concept of deformation previously introduced must be generalized. The rate of deformation tensor Dij has nine components. In Cartesian coordinates, ∂v ∂v Dij = i + j ∂xj ∂xi

冢

冣

(6-6)

where the subscripts i and j refer to the three coordinate directions. Some authors define the deformation rate tensor as one-half of that given by Eq. (6-6). Vorticity The relative motion between two points in a fluid can be decomposed into three components: rotation, dilatation, and deformation. The rate of deformation tensor has been defined. Dilatation refers to the volumetric expansion or compression of the fluid, and vanishes for incompressible flow. Rotation is described by a tensor ωij = ∂vi / ∂xj − ∂vj / ∂xi. The vector of vorticity given by one-half the

6-6

FLUID AND PARTICLE DYNAMICS

curl of the velocity vector is another measure of rotation. In twodimensional flow in the x-y plane, the vorticity ω is given by ∂v 1 ∂v (6-7) ω =  y − x 2 ∂x ∂y Here ω is the magnitude of the vorticity vector, which is directed along the z axis. An irrotational flow is one with zero vorticity. Irrotational flows have been widely studied because of their useful mathematical properties and applicability to flow regions where viscous effects may be neglected. Such flows without viscous effects are called inviscid flows. Laminar and Turbulent Flow, Reynolds Number These terms refer to two distinct types of flow. In laminar flow, there are smooth streamlines and the fluid velocity components vary smoothly with position, and with time if the flow is unsteady. The flow described in reference to Fig. 6-1 is laminar. In turbulent flow, there are no smooth streamlines, and the velocity shows chaotic fluctuations in time and space. Velocities in turbulent flow may be reported as the sum of a time-averaged velocity and a velocity fluctuation from the average. For any given flow geometry, a dimensionless Reynolds number may be defined for a Newtonian fluid as Re = LU ρ /µ where L is a characteristic length. Below a critical value of Re the flow is laminar, while above the critical value a transition to turbulent flow occurs. The geometry-dependent critical Reynolds number is determined experimentally.

冢

冣

CONSERVATION EQUATIONS Macroscopic and Microscopic Balances Three postulates, regarded as laws of physics, are fundamental in fluid mechanics. These are conservation of mass, conservation of momentum, and conservation of energy. In addition, two other postulates, conservation of moment of momentum (angular momentum) and the entropy inequality (second law of thermodynamics) have occasional use. The conservation principles may be applied either to material systems or to control volumes in space. Most often, control volumes are used. The control volumes may be either of finite or differential size, resulting in either algebraic or differential conservation equations, respectively. These are often called macroscopic and microscopic balance equations. Macroscopic Equations An arbitrary control volume of finite size Va is bounded by a surface of area Aa with an outwardly directed unit normal vector n. The control volume is not necessarily fixed in space. Its boundary moves with velocity w. The fluid velocity is v. Figure 6-3 shows the arbitrary control volume. Mass balance Applied to the control volume, the principle of conservation of mass may be written as (Whitaker, Introduction to Fluid Mechanics, Prentice-Hall, Englewood Cliffs, N.J., 1968, Krieger, Huntington, N.Y., 1981) d ρ dV + ρ(v − w) ⋅ n dA = 0 (6-8)  Aa dt Va This equation is also known as the continuity equation.

冕

冕

Area Aa

Volume Va

Simplified forms of Eq. (6-8) apply to special cases frequently found in practice. For a control volume fixed in space with one inlet of area A1 through which an incompressible fluid enters the control volume at an average velocity V1, and one outlet of area A2 through which fluid leaves at an average velocity V2, as shown in Fig. 6-4, the continuity equation becomes V1 A1 = V2 A2 (6-9) The average velocity across a surface is given by V = (1/A)

冕 v dA A

where v is the local velocity component perpendicular to the inlet surface. The volumetric flow rate Q is the product of average velocity and the cross-sectional area, Q = VA. The average mass velocity is G = ρV. For steady flows through fixed control volumes with multiple inlets and/or outlets, conservation of mass requires that the sum of inlet mass flow rates equals the sum of outlet mass flow rates. For incompressible flows through fixed control volumes, the sum of inlet flow rates (mass or volumetric) equals the sum of exit flow rates, whether the flow is steady or unsteady. Momentum Balance Since momentum is a vector quantity, the momentum balance is a vector equation. Where gravity is the only body force acting on the fluid, the linear momentum principle, applied to the arbitrary control volume of Fig. 6-3, results in the following expression (Whitaker, ibid.). d ρv dV + ρv(v − w) ⋅ n dA = ρg dV + tn dA (6-10)  Aa Va Aa dt Va Here g is the gravity vector and tn is the force per unit area exerted by the surroundings on the fluid in the control volume. The integrand of the area integral on the left-hand side of Eq. (6-10) is nonzero only on the entrance and exit portions of the control volume boundary. For the special case of steady flow at a mass flow rate m ˙ through a control volume fixed in space with one inlet and one outlet, (Fig. 6-4) with the inlet and outlet velocity vectors perpendicular to planar inlet and outlet surfaces, giving average velocity vectors V1 and V2, the momentum equation becomes

冕

冕

冕

冕

m(β ˙ 2V2 − β1V1) = −p1A1 − p2A2 + F + Mg

n outwardly directed unit normal vector

V2 2

w boundary velocity

V1

v fluid velocity FIG. 6-3

Arbitrary control volume for application of conservation equations.

(6-11)

where M is the total mass of fluid in the control volume. The factor β arises from the averaging of the velocity across the area of the inlet or outlet surface. It is the ratio of the area average of the square of velocity magnitude to the square of the area average velocity magnitude. For a uniform velocity, β = 1. For turbulent flow, β is nearly unity, while for laminar pipe flow with a parabolic velocity profile, β = 4/3. The vectors A1 and A 2 have magnitude equal to the areas of the inlet and outlet surfaces, respectively, and are outwardly directed normal to the surfaces. The vector F is the force exerted on the fluid by the nonflow boundaries of the control volume. It is also assumed that the stress vector tn is normal to the inlet and outlet surfaces, and that its magnitude may be approximated by the pressure p. Equation (6-11) may be generalized to multiple inlets and/or outlets. In such cases, the mass flow rates for all the inlets and outlets are not equal. A distinct flow rate m ˙ i applies to each inlet or outlet i. To generalize the equation, ⴚpA terms for each inlet and outlet, − mβV ˙ terms for each inlet, and mβV ˙ terms for each outlet are included. Balance equations for angular momentum, or moment of momentum, may also be written. They are used less frequently than the lin-

FIG. 6-4

Fixed control volume with one inlet and one outlet.

FLUID DYNAMICS ear momentum equations. See Whitaker (Introduction to Fluid Mechanics, Prentice-Hall, Englewood Cliffs, N.J., 1968, Krieger, Huntington, N.Y., 1981; or Shames (Mechanics of Fluids, 3d ed., McGraw-Hill, New York, 1992). Total Energy Balance The total energy balance derives from the first law of thermodynamics. Applied to the arbitrary control volume of Fig. 6-3, it leads to an equation for the rate of change of the sum of internal, kinetic, and gravitational potential energy. In this equation, u is the internal energy per unit mass, v is the magnitude of the velocity vector v, z is elevation, g is the gravitational acceleration, and q is the heat flux vector: d  dt

冕 ρ冢u + v2 + gz冣 dV + 冕 ρ冢u + v2 + gz冣(v − w) ⋅ n dA = 冕 (v ⋅ t ) dA − 冕 (q ⋅ n) dA (6-12) 2

2

Va

Aa

n

Aa

Aa

The first integral on the right-hand side is the rate of work done on the fluid in the control volume by forces at the boundary. It includes both work done by moving solid boundaries and work done at flow entrances and exits. The work done by moving solid boundaries also includes that by such surfaces as pump impellers; this work is called ˙ S. shaft work; its rate is W A useful simplification of the total energy equation applies to a particular set of assumptions. These are a control volume with fixed solid boundaries, except for those producing shaft work, steady state conditions, and mass flow at a rate m ˙ through a single planar entrance and a single planar exit (Fig. 6-4), to which the velocity vectors are perpendicular. As with Eq. (6-11), it is assumed that the stress vector tn is normal to the entrance and exit surfaces and may be approximated by the pressure p. The equivalent pressure, p + ρgz, is assumed to be uniform across the entrance and exit. The average velocity at the entrance and exit surfaces is denoted by V. Subscripts 1 and 2 denote the entrance and exit, respectively. V 12 V 22 h1 + α1  + gz1 = h2 + α2  + gz2 − δQ − δWS 2 2

(6-13)

Here, h is the enthalpy per unit mass, h = u + p/ρ. The shaft work per ˙ s /m. unit of mass flowing through the control volume is δWS = W ˙ Similarly, δQ is the heat input rate per unit of mass. The factor α is the ratio of the cross-sectional area average of the cube of the velocity to the cube of the average velocity. For a uniform velocity profile, α = 1. In turbulent flow, α is usually assumed to equal unity; in turbulent pipe flow, it is typically about 1.07. For laminar flow in a circular pipe with a parabolic velocity profile, α = 2. Mechanical Energy Balance, Bernoulli Equation A balance equation for the sum of kinetic and potential energy may be obtained from the momentum balance by forming the scalar product with the velocity vector. The resulting equation, called the mechanical energy balance, contains a term accounting for the dissipation of mechanical energy into thermal energy by viscous forces. The mechanical energy equation is also derivable from the total energy equation in a way that reveals the relationship between the dissipation and entropy generation. The macroscopic mechanical energy balance for the arbitrary control volume of Fig. 6-3 may be written, with p = thermodynamic pressure, as d  dt

冕 ρ冢v2 + gz冣 dV + 冕 ρ冢v2 + gz冣(v − w) ⋅ n dA = 冕 p ⵱ ⋅ v dV + 冕 (v ⋅ t ) dA − 冕 Φ dV 2

2

Va

Aa

n

Va

Aa

(6-14)

Va

The last term is the rate of viscous energy dissipation to internal energy, E˙ v = 冕Va Φ dV, also called the rate of viscous losses. These losses are the origin of frictional pressure drop in fluid flow. Whitaker and Bird, Stewart, and Lightfoot provide expressions for the dissipation function Φ for Newtonian fluids in terms of the local velocity gradients. However, when using macroscopic balance equations the local velocity field within the control volume is usually unknown. For such cases additional information, which may come from empirical correlations, is needed.

6-7

For the same special conditions as for Eq. (6-13), the mechanical energy equation is reduced to p 2 dp V12 V 22 (6-15) α1  + gz1 + δWS = α2  + gz2 +  + lv p1 2 2 ρ ˙ is the energy dissipation per unit mass. This equation Here lv = E˙ v /m has been called the engineering Bernoulli equation. For an incompressible flow, Eq. (6-15) becomes p2 p1 V12 V22 (6-16)  + α1  + gz1 + δWS =  + α2  + gz2 + lv ρ ρ 2 2 The Bernoulli equation can be written for incompressible, inviscid flow along a streamline, where no shaft work is done. p1 V12 p2 V22 (6-17)  +  + gz1 =  +  + gz2 ρ 2 ρ 2 Unlike the momentum equation (Eq. [6-11]), the Bernoulli equation is not easily generalized to multiple inlets or outlets. Microscopic Balance Equations Partial differential balance equations express the conservation principles at a point in space. Equations for mass, momentum, total energy, and mechanical energy may be found in Whitaker (ibid.), Bird, Stewart, and Lightfoot (Transport Phenomena, Wiley, New York, 1960), and Slattery (Momentum, Heat and Mass Transfer in Continua, 2d ed., Krieger, Huntington, N.Y., 1981), for example. These references also present the equations in other useful coordinate systems besides the cartesian system. The coordinate systems are fixed in inertial reference frames. The two most used equations, for mass and momentum, are presented here. Mass Balance, Continuity Equation The continuity equation, expressing conservation of mass, is written in cartesian coordinates as ∂ρ ∂ρvx ∂ρvy ∂ρvz (6-18) +++=0 ∂t ∂x ∂y ∂z In terms of the substantial derivative, D/Dt, Dρ ∂ρ ∂ρ ∂ρ ∂ρ ∂vx ∂vy ∂vz  ⬅  + vx  + vy  + vz  = −ρ  +  +  (6-19) Dt ∂t ∂x ∂y ∂z ∂x ∂y ∂z The substantial derivative, also called the material derivative, is the rate of change in a Lagrangian reference frame, that is, following a material particle. In vector notation the continuity equation may be expressed as Dρ (6-20)  = −ρ∇ ⋅ v Dt For incompressible flow, ∂v ∂v ∂v ∇ ⋅ v = x + y + z = 0 (6-21) ∂x ∂y ∂z Stress Tensor The stress tensor is needed to completely describe the stress state for microscopic momentum balances in multidimensional flows. The components of the stress tensor σij give the force in the j direction on a plane perpendicular to the i direction, using a sign convention defining a positive stress as one where the fluid with the greater i coordinate value exerts a force in the positive i direction on the fluid with the lesser i coordinate. Several references in fluid mechanics and continuum mechanics provide discussions, to various levels of detail, of stress in a fluid (Denn; Bird, Stewart, and Lightfoot; Schlichting; Fung [A First Course in Continuum Mechanics, 2d. ed., Prentice-Hall, Englewood Cliffs, N.J., 1977]; Truesdell and Toupin [in Flügge, Handbuch der Physik, vol. 3/1, Springer-Verlag, Berlin, 1960]; Slattery [Momentum, Energy and Mass Transfer in Continua, 2d ed., Krieger, Huntington, N.Y., 1981]). The stress has an isotropic contribution due to fluid pressure and dilatation, and a deviatoric contribution due to viscous deformation effects. The deviatoric contribution for a Newtonian fluid is the threedimensional generalization of Eq. (6-2):

冕

冢

冣

τij = µDij

(6-22)

σij = (−p + λ∇ ⋅ v)δij + τij

(6-23)

The total stress is

6-8

FLUID AND PARTICLE DYNAMICS

The identity tensor δij is zero for i ≠ j and unity for i = j. The coefficient λ is a material property related to the bulk viscosity, κ = λ + 2µ/3. There is considerable uncertainty about the value of κ. Traditionally, Stokes’ hypothesis, κ = 0, has been invoked, but the validity of this hypothesis is doubtful (Slattery, ibid.). For incompressible flow, the value of bulk viscosity is immaterial as Eq. (6-23) reduces to σij = −pδij + τij

(6-24)

Similar generalizations to multidimensional flow are necessary for non-Newtonian constitutive equations. Cauchy Momentum and Navier-Stokes Equations The differential equations for conservation of momentum are called the Cauchy momentum equations. These may be found in general form in most fluid mechanics texts (e.g., Slattery [ibid.]; Denn; Whitaker; and Schlichting). For the important special case of an incompressible Newtonian fluid with constant viscosity, substitution of Eqs. (6-22) and (6-24) lead to the Navier-Stokes equations, whose three Cartesian components are ∂v ∂v ∂v ∂v ρ x + vx x + vy x + vz x ∂t ∂x ∂y ∂z

冢

冣

∂p ∂2vx ∂2vx ∂2vx =−+µ  + + + ρgx (6-25) ∂x ∂x2 ∂y2 ∂z2

冢

∂vy ∂vy ∂vy ∂vy ρ  + vx  + vy  + vz  ∂t ∂x ∂y ∂z

冢

冣

冣

∂p ∂2vy ∂2vy ∂2vy =−+µ  + + + ρgy ∂y ∂x2 ∂y2 ∂z2

冢

∂vz ∂vz ∂vz ∂vz ρ  + vx  + vy  + vz  ∂t ∂x ∂y ∂z

冢

冣

(6-26)

冣

∂p ∂2vz ∂2vz ∂2vz =−+µ  + + + ρgz (6-27) ∂z ∂x2 ∂y2 ∂z2

冢

冣

Stokes equations, Dirichlet and Neumann, or essential and natural, boundary conditions may be satisfied by different means. Fluid statics, discussed in Sec. 10 of the Handbook in reference to pressure measurement, is the branch of fluid mechanics in which the fluid velocity is either zero or is uniform and constant relative to an inertial reference frame. With velocity gradients equal to zero, the momentum equation reduces to a simple expression for the pressure field, ∇p = ρg. Letting z be directed vertically upward, so that gz = −g where g is the gravitational acceleration (9.806 m2/s), the pressure field is given by dp/dz = −ρg

Example 1: Force Exerted on a Reducing Bend An incompressible fluid flows through a reducing elbow (Fig. 6-5) situated in a horizontal plane. The inlet velocity V1 is given and the pressures p1 and p2 are measured. Selecting the inlet and outlet surfaces 1 and 2 as shown, the continuity equation Eq. (6-9) can be used to find the exit velocity V2 = V1A1/A2. The mass flow rate is obtained by m ˙ = ρV1A1. Assume that the velocity profile is nearly uniform so that β is approximately unity. The force exerted on the fluid by the bend has x and y components; these can be found from Eq. (6-11). The x component gives Fx = m(V ˙ 2x − V1x) + p1A1x + p2 A2x while the y component gives Fy = m(V ˙ 2y − V1y) + p1 A1y + p2 A2y

In vector notation, Dv ∂v (6-28) ρ  =  + (v ⋅ ∇)v = −∇p + µ∇2v + ρg Dt ∂t The pressure and gravity terms may be combined by replacing the pressure p by the equivalent pressure P = p + ρgz. The left-hand side terms of the Navier-Stokes equations are the inertial terms, while the terms including viscosity µ are the viscous terms. Limiting cases under which the Navier-Stokes equations may be simplified include creeping flows in which the inertial terms are neglected, potential flows (inviscid or irrotational flows) in which the viscous terms are neglected, and boundary layer and lubrication flows in which certain terms are neglected based on scaling arguments. Creeping flows are described by Happel and Brenner (Low Reynolds Number Hydrodynamics, Prentice-Hall, Englewood Cliffs, N.J., 1965); potential flows by Lamb (Hydrodynamics, 6th ed., Dover, New York, 1945) and Milne-Thompson (Theoretical Hydrodynamics, 5th ed., Macmillan, New York, 1968); boundary layer theory by Schlichting (Boundary Layer Theory, 8th ed., McGraw-Hill, New York, 1987), and lubrication theory by Batchelor (An Introduction to Fluid Dynamics, Cambridge University, Cambridge, 1967) and Denn (Process Fluid Mechanics, Prentice-Hall, Englewood Cliffs, N.J., 1980). Because the Navier-Stokes equations are first-order in pressure and second-order in velocity, their solution requires one pressure boundary condition and two velocity boundary conditions (for each velocity component) to completely specify the solution. The no slip condition, which requires that the fluid velocity equal the velocity of any bounding solid surface, occurs in most problems. Specification of velocity is a type of boundary condition sometimes called a Dirichlet condition. Often boundary conditions involve stresses, and thus velocity gradients, rather than the velocities themselves. Specification of velocity derivatives is a Neumann boundary condition. For example, at the boundary between a viscous liquid and a gas, it is often assumed that the liquid shear stresses are zero. In numerical solution of the Navier-

(6-29)

This equation applies to any incompressible or compressible static fluid. For an incompressible liquid, pressure varies linearly with depth. For compressible gases, p is obtained by integration accounting for the variation of ρ with z. The force exerted on a submerged planar surface of area A is given by F = pc A where pc is the pressure at the geometrical centroid of the surface. The center of pressure, the point of application of the net force, is always lower than the centroid. For details see, for example, Shames, where may also be found discussion of forces on curved surfaces, buoyancy, and stability of floating bodies. Examples Four examples follow, illustrating the application of the conservation equations to obtain useful information about fluid flows.

The velocity components are V1x = V1, V1y = 0, V2x = V2 cos θ, and V2y = V2 sin θ. The area vector components are A1x = −A1, A1y = 0, A 2x = A 2 cos θ, and A 2y = A 2 sin θ. Therefore, the force components may be calculated from Fx = m(V ˙ 2 cos θ − V1) − p1A1 + p2A2 cos θ Fy = mV ˙ 2 sin θ + p2A2 sin θ The force acting on the fluid is F; the equal and opposite force exerted by the fluid on the bend is ⴚF.

Example 2: Simplified Ejector Figure 6-6 shows a very simplified sketch of an ejector, a device that uses a high velocity primary fluid to pump another (secondary) fluid. The continuity and momentum equations may be

V2 θ

V1 F

y x Force at a reducing bend. F is the force exerted by the bend on the fluid. The force exerted by the fluid on the bend is ⴚF. FIG. 6-5

FLUID DYNAMICS

6-9

y H

x

FIG. 6-8

FIG. 6-6

Draft-tube ejector.

applied on the control volume with inlet and outlet surfaces 1 and 2 as indicated in the figure. The cross-sectional area is uniform, A1 = A2 = A. Let the mass flow rates and velocities of the primary and secondary fluids be m ˙ p, m ˙ s, Vp and Vs. Assume for simplicity that the density is uniform. Conservation of mass gives m˙2 = m ˙p +m ˙ s. The exit velocity is V2 = m ˙ 2 /(ρA). The principle momentum exchange in the ejector occurs between the two fluids. Relative to this exchange, the force exerted by the walls of the device are found to be small. Therefore, the force term F is neglected from the momentum equation. Written in the flow direction, assuming uniform velocity profiles, and using the extension of Eq. (6-11) for multiple inlets, it gives the pressure rise developed by the device: (p2 − p1)A = (m˙p + m ˙ s)V2 − m ˙ pVp − m ˙ sVs Application of the momentum equation to ejectors of other types is discussed in Lapple (Fluid and Particle Dynamics, University of Delaware, Newark, 1951) and in Sec. 10 of the Handbook.

Example 3: Venturi Flowmeter An incompressible fluid flows

Plane Poiseuille flow.

ish in the x direction. Since the flow field is infinite in the z direction, all velocity derivatives should be zero in the z direction. Therefore, velocity components are a function of y alone. It is also assumed that there is no flow in the z direction, so vz = 0. The continuity equation Eq. (6-21), with vz = 0 and ∂vx / ∂x = 0, reduces to dvy  = 0. dy Since vy = 0 at y = H/2, the continuity equation integrates to vy = 0. This is a direct result of the assumption of fully developed flow. The Navier-Stokes equations are greatly simplified when it is noted that vy = vz = 0 and ∂vx /∂x = ∂vx /∂z = ∂vx /∂t = 0. The three components are written in terms of the equivalent pressure P: ∂P ∂2vx 0=−+µ ∂x ∂y2 ∂P 0=− ∂y ∂P 0=− ∂z

through the venturi flowmeter in Fig. 6-7. An equation is needed to relate the flow rate Q to the pressure drop measured by the manometer. This problem can be solved using the mechanical energy balance. In a well-made venturi, viscous losses are negligible, the pressure drop is entirely the result of acceleration into the throat, and the flow rate predicted neglecting losses is quite accurate. The inlet area is A and the throat area is a. With control surfaces at 1 and 2 as shown in the figure, Eq. (6-17) in the absence of losses and shaft work gives

The latter two equations require that P is a function only of x, and therefore ∂P/∂x = dP/dx. Inspection of the first equation shows one term which is a function only of x and one which is only a function of y. This requires that both terms are constant. The pressure gradient −dP/dx is constant. The x-component equation becomes d 2vx 1 dP =  dy2 µ dx

p1 V 12 p2 V 22 +=+ ρ 2 ρ 2 The continuity equation gives V2 = V1A/a, and V1 = Q/A. The pressure drop measured by the manometer is p1 − p2 = (ρm − ρ)g∆z. Substituting these relations into the energy balance and rearranging, the desired expression for the flow rate is found. 2(ρm − ρ)g∆z 1 Q =   A ρ[(A/a)2 − 1]

Two integrations of the x-component equation give

冪莦

Example 4: Plane Poiseuille Flow An incompressible Newtonian fluid flows at a steady rate in the x direction between two very large flat plates, as shown in Fig. 6-8. The flow is laminar. The velocity profile is to be found. This example is found in most fluid mechanics textbooks; the solution presented here closely follows Denn. This problem requires use of the microscopic balance equations because the velocity is to be determined as a function of position. The boundary conditions for this flow result from the no-slip condition. All three velocity components must be zero at the plate surfaces, y = H/2 and y = −H/2. Assume that the flow is fully developed, that is, all velocity derivatives van-

1

2

1 dP vx =   y2 + C1y + C2 2µ dx where the constants of integration C1 and C2 are evaluated from the boundary conditions vx = 0 at y = H/2. The result is

冢

冣冤 冢 冣 冥

H2 dP 2y 2 vx =  −  1 −  8µ dx H This is a parabolic velocity distribution. The average velocity V = H/2 (1/H) 冕 −H/2 vx dy is dP H2 V= − 12µ dx

冢

This flow is one-dimensional, as there is only one nonzero velocity component, vx, which, along with the pressure, varies in only one coordinate direction.

INCOMPRESSIBLE FLOW IN PIPES AND CHANNELS Mechanical Energy Balance The mechanical energy balance, Eq. (6-16), for fully developed incompressible flow in a straight circular pipe of constant diameter D reduces to p2 p1 (6-30)  + gz1 =  + gz 2 + lv ρ ρ In terms of the equivalent pressure, P ⬅ p + ρgz, P1 − P2 = ρ lv

∆z

FIG. 6-7

Venturi flowmeter.

冣

(6-31)

The pressure drop due to frictional losses lv is proportional to pipe length L for fully developed flow and may be denoted as the (positive) quantity ∆P ⬅ P1 − P2. Friction Factor and Reynolds Number For a Newtonian fluid in a smooth pipe, dimensional analysis relates the frictional pressure drop per unit length ∆P/L to the pipe diameter D, density ρ, and average velocity V through two dimensionless groups, the Fanning friction factor f and the Reynolds number Re.

6-10

FLUID AND PARTICLE DYNAMICS D∆P f⬅ 2ρV 2L

(6-32)

DVρ Re ⬅  (6-33) µ For smooth pipe, the friction factor is a function only of the Reynolds number. In rough pipe, the relative roughness
/D also affects the friction factor. Figure 6-9 plots f as a function of Re and
/D. Values of
for various materials are given in Table 6-1. The Fanning friction factor should not be confused with the Darcy friction factor used by Moody (Trans. ASME, 66, 671 [1944]), which is four times greater. Using the momentum equation, the stress at the wall of the pipe may be expressed in terms of the friction factor: ρV 2 τw = f  2

(6-34)

Laminar and Turbulent Flow Below a critical Reynolds number of about 2,100, the flow is laminar; over the range 2,100 < Re < 5,000 there is a transition to turbulent flow. For laminar flow, the Hagen-Poiseuille equation 16 f = , Re

Re ≤ 2,100

(6-35)

may be derived from the Navier-Stokes equation and is in excellent agreement with experimental data. It may be rewritten in terms of volumetric flow rate, Q = VπD2/4, as π∆PD4 Q = , Re ≤ 2,100 (6-36) 128µL

TABLE 6-1 Values of Surface Roughness for Various Materials* Surface roughness ε, mm

Material Drawn tubing (brass, lead, glass, and the like) Commercial steel or wrought iron Asphalted cast iron Galvanized iron Cast iron Wood stove Concrete Riveted steel

0.00152 0.0457 0.122 0.152 0.259 0.183–0.914 0.305–3.05 0.914–9.14

* From Moody, Trans. Am. Soc. Mech. Eng., 66, 671–684 (1944); Mech. Eng., 69, 1005–1006 (1947). Additional values of ε for various types or conditions of concrete wrought-iron, welded steel, riveted steel, and corrugated-metal pipes are given in Brater and King, Handbook of Hydraulics, 6th ed., McGraw-Hill, New York, 1976, pp. 6-12–6-13. To convert millimeters to feet, multiply by 3.281 × 10−3.

For turbulent flow in smooth tubes, the Blasius equation gives the friction factor accurately for a wide range of Reynolds numbers. 0.079 f= , 4,000 < Re < 105 (6-37) Re0.25 The Colebrook formula (Colebrook, J. Inst. Civ. Eng. [London], 11, 133–156 [1938–39]) gives a good approximation for the f-Re-(
/D) data for rough pipes over the entire turbulent flow range: 1.256 1
 = −4 log  +  3.7D Re兹f苶 兹苶f

冤

冥

Re > 4,000

Fanning Friction Factors. Reynolds number Re = DVρ/µ, where D = pipe diameter, V = velocity, ρ = fluid density, and µ = fluid viscosity. (Based on Moody, Trans. ASME, 66, 671 [1944].)

FIG. 6-9

(6-38)

FLUID DYNAMICS An equation by Churchill (Chem. Eng., 84[24], 91–92 [Nov. 7, 1977]) for both smooth and rough tubes offers the advantage of being explicit in f: 1 0.27
 = −4 log  + (7/Re)0.9 Re > 4,000 (6-39) D 兹苶f In laminar flow, f is independent of
/D. In turbulent flow, the friction factor for rough pipe follows the smooth tube curve for a range of Reynolds numbers (hydraulically smooth flow). For greater Reynolds numbers, f deviates from the smooth pipe curve, eventually becoming independent of Re. This region, often called complete turbulence, is frequently encountered in commercial pipe flows. The Reynolds number above which f becomes essentially independent of Re is (Davies, Turbulence Phenomena, Academic, New York, 1972, p. 37) 20[3.2 − 2.46 ln (
/D)] Re =  (6-40) (
/D) Roughness may also affect the transition from laminar to turbulent flow (Schlichting). Common pipe flow problems include calculation of pressure drop given the flow rate (or velocity) and calculation of flow rate (or velocity) given pressure drop. When flow rate is given, the Reynolds number is first calculated to determine the flow regime, so that the appropriate relations between f and Re (or pressure drop and velocity or flow rate) are used. When pressure drop is given and the velocity is unknown, the Reynolds number and flow regime cannot be immediately determined. It is necessary to assume the flow regime and then verify by checking Re afterward. With experience, the initial guess for the flow regime will usually prove correct. When solving Eq. (6-38) for velocity when pressure drop is given, it is useful to note that the right-hand side is independent of velocity since Re兹苶f = (D3/2/µ)兹苶 ρ苶 ∆苶 P苶 /(苶2苶 L苶). As Fig. 6-9 suggests, the friction factor is uncertain in the transition range 2,100 < Re < 4,000 and a conservative choice should be made for design purposes. Velocity Profiles In laminar flow, the solution of the NavierStokes equation, corresponding to the Hagen-Poiseuille equation, gives the velocity v as a function of radial position r in a circular pipe of radius R in terms of the average velocity V = Q/A. The parabolic profile, with centerline velocity twice the average velocity, is shown in Fig. 6-10. r2 v = 2V 1 − 2 (6-41) R In turbulent flow, the velocity profile is much more blunt, with most of the velocity gradient being in a region near the wall, described by a universal velocity profile. It is characterized by a viscous sublayer, a turbulent core, and a buffer zone in between.

冤

冥

冢

冣

Viscous sublayer u+ = y+

for

y+ < 5

(6-42)

u+ = 5.00 ln y+ − 3.05

for

5 < y+ < 30

(6-43)

for

y+ > 30

(6-44)

Buffer zone Turbulent core u+ = 2.5 ln y+ + 5.5

Here, u+ = v/u* is the dimensionless, time-averaged axial velocity, u* =

r z

R

(

2 v = 2V 1 – r 2 R

v max = 2V FIG. 6-10

velocity V.

Parabolic velocity profile for laminar flow in a pipe, with average

6-11

兹苶 τw/ρ 苶 is the friction velocity and τw = fρV 2/2 is the wall stress. The

friction velocity is of the order of the root mean square velocity fluctuation perpendicular to the wall in the turbulent core. The dimensionless distance from the wall is y+ = yu ρ/µ. The universal velocity * profile is valid in the wall region for any cross-sectional channel shape. For incompressible flow in constant diameter circular pipes, τw = ∆P/4L where ∆P is the pressure drop in length L. In circular pipes, Eq. (6-44) gives a surprisingly good fit to experimental results over the entire cross section of the pipe, even though it is based on assumptions which are valid only near the pipe wall. For rough pipes, the velocity profile in the turbulent core is given by u+ = 2.5 ln y/
+ 8.5

for

y+ > 30

(6-45)

when the dimensionless roughness
+ =
u*ρ/µ is greater than 5 to 10; for smaller
+, the velocity profile in the turbulent core is unaffected by roughness. For velocity profiles in the transition region, see Patel and Head ( J. Fluid Mech., 38, part 1, 181–201 [1969]) where profiles over the range 1,500 < Re < 10,000 are reported. Entrance and Exit Effects In the entrance region of a pipe, some distance is required for the flow to adjust from upstream conditions to the fully developed flow pattern. This distance depends on the Reynolds number and on the flow conditions upstream. For a uniform velocity profile at the pipe entrance, the computed length in laminar flow required for the centerline velocity to reach 99 percent of its fully developed value is (Dombrowski, Foumeny, Ookawara and Riza, Can. J. Chem. Engr., 71, 472–476 [1993]) Lent /D = 0.370 exp (−0.148Re) + 0.0550Re + 0.260

(6-46)

In turbulent flow, the entrance length is about Lent /D = 40

(6-47)

The frictional losses in the entrance region are larger than those for the same length of fully developed flow. (See the subsection, “Frictional Losses in Pipeline Elements,” following.) At the pipe exit, the velocity profile also undergoes rearrangement, but the exit length is much shorter than the entrance length. At low Re, it is about one pipe radius. At Re > 100, the exit length is essentially 0. Residence Time Distribution For laminar Newtonian pipe flow, the cumulative residence time distribution F(θ) is given by θavg F(θ) = 0 for θ< 2 1 θavg 2 θavg F(θ) = 1 −   for θ≥ (6-48) 4 θ 2 where F(θ) is the fraction of material which resides in the pipe for less than time θ and θavg is the average residence time, θ = V/L. The residence time distribution in long transfer lines may be made narrower (more uniform) with the use of flow inverters or static mixing elements. These devices exchange fluid between the wall and central regions. Variations on the concept may be used to provide effective mixing of the fluid. See Godfrey (“Static Mixers,” in Harnby, Edwards, and Nienow, Mixing in the Process Industries, 2d ed., Butterworth Heinemann, Oxford, 1992); Gretta and Smith (Trans. ASME J. Fluids Eng., 115, 255–263 [1993]); Kemblowski and Pustelnik (Chem. Eng. Sci., 43, 473–478 [1988]). A theoretically derived equation for flow in helical pipe coils by Ruthven (Chem. Eng. Sci., 26, 1113–1121 [1971]; 33, 628–629 [1978]) is given by θavg 1 θavg 2.81 F(θ) = 1 −   for 0.5 <  < 1.63 (6-49) 4 θ θ and was substantially confirmed by Trivedi and Vasudeva (Chem. Eng. Sci., 29, 2291–2295 [1974]) for 0.6 < De < 6 and 0.0036 < D/Dc < 0.097 where De = Re兹苶 D苶 /Dc苶 is the Dean number and Dc is the diameter of curvature of the coil. Measurements by Saxena and Nigam (Chem. Eng. Sci., 34, 425–426 [1979]) indicate that such a distribution will hold for De > 1. The residence time distribution for helical coils is narrower than for straight circular pipes, due to the secondary flow which exchanges fluid between the wall and center regions.

冢 冣

冢 冣冤 冥

6-12

FLUID AND PARTICLE DYNAMICS

In turbulent flow, axial mixing is usually described in terms of turbulent diffusion or dispersion coefficients, from which cumulative residence time distribution functions can be computed. Davies (Turbulence Phenomena, Academic, New York, 1972, p. 93), gives DL = 1.01νRe0.875 for the longitudinal dispersion coefficient. Levenspiel (Chemical Reaction Engineering, 2d ed., Wiley, New York, 1972, pp. 253–278) discusses the relations among various residence time distribution functions, and the relation between dispersion coefficient and residence time distribution. Noncircular Channels Calculation of frictional pressure drop in noncircular channels depends on whether the flow is laminar or turbulent, and on whether the channel is full or open. For turbulent flow in ducts running full, the hydraulic diameter DH should be substituted for D in the friction factor and Reynolds number definitions, Eqs. (6-32) and (6-33). The hydraulic diameter is defined as four times the channel cross-sectional area divided by the wetted perimeter. For example, the hydraulic diameter for a circular pipe is DH = D, for an annulus of inner diameter d and outer diameter D, DH = D − d, for a rectangular duct of sides a, b, DH = ab/[2(a + b)]. The hydraulic radius RH is defined as one-fourth of the hydraulic diameter. With the hydraulic diameter subsititued for D in f and Re, Eqs. (6-37) through (6-40) are good approximations. Note that V appearing in f and Re is the actual average velocity V = Q/A; for noncircular pipes; it is not Q/(πDH2 /4). The pressure drop should be calculated from the friction factor for noncircular pipes. Equations relating Q to ∆P and D for circular pipes may not be used for noncircular pipes with D replaced by DH because V ≠ Q/(πDH2 /4). Turbulent flow in noncircular channels is generally accompanied by secondary flows perpendicular to the axial flow direction (Schlichting). These flows may cause the pressure drop to be slightly greater than that computed using the hydraulic diameter method. For data on pressure drop in annuli, see Brighton and Jones (J. Basic Eng., 86, 835–842 [1964]); Okiishi and Serovy (J. Basic Eng., 89, 823–836 [1967]); and Lawn and Elliot (J. Mech. Eng. Sci., 14, 195–204 [1972]). For rectangular ducts of large aspect ratio, Dean (J. Fluids Eng., 100, 215–233 [1978]) found that the numerator of the exponent in the Blasius equation (6-37) should be increased to 0.0868. Jones (J. Fluids Eng., 98, 173–181 [1976]) presents a method to improve the estimation of friction factors for rectangular ducts using a modification of the hydraulic diameter–based Reynolds number. The hydraulic diameter method does not work well for laminar flow because the shape affects the flow resistance in a way that cannot be expressed as a function only of the ratio of cross-sectional area to wetted perimeter. For some shapes, the Navier-Stokes equations have been integrated to yield relations between flow rate and pressure drop. These relations may be expressed in terms of equivalent diameters DE defined to make the relations reduce to the second form of the Hagen-Poiseulle equation, Eq. (6-36); that is, DE ⬅ (128QµL/π∆P)1/4. Equivalent diameters are not the same as hydraulic diameters. Equivalent diameters yield the correct relation between flow rate and pressure drop when substituted into Eq. (6-36), but not Eq. (6-35) because V ≠ Q/(πDE/4). Equivalent diameter DE is not to be used in the friction factor and Reynolds number; f ≠ 16/Re using the equivalent diameters defined in the following. This situation is, by arbitrary definition, opposite to that for the hydraulic diameter DH used for turbulent flow. Ellipse, semiaxes a and b (Lamb, Hydrodynamics, 6th ed., Dover, New York, 1945, p. 587): 32a3b3 1/4 (6-50) DE =  a2 + b2 Rectangle, width a, height b (Owen, Trans. Am. Soc. Civ. Eng., 119, 1157–1175 [1954]): 128ab3 1/4 DE =  (6-51) πK a/b = 1 1.5 2 3 4 5 10 ∞ K = 28.45 20.43 17.49 15.19 14.24 13.73 12.81 12

冢

冣

冢

冣

Annulus, inner diameter D1 outer diameter D2 (Lamb, op. cit., p. 587):

D22 − D12 DE = (D22 − D12) D22 + D12 −  ln (D2 /D1)

冦

冤

冥冧

1/4

(6-52)

For isosceles triangles and regular polygons, see Sparrow (AIChE J., 8, 599–605 [1962]), Carlson and Irvine (J. Heat Transfer, 83, 441–444 [1961]), Cheng (Proc. Third Int. Heat Transfer Conf., New York, 1, 64–76 [1966]), and Shih (Can. J. Chem. Eng., 45, 285–294 [1967]). The critical Reynolds number for transition from laminar to turbulent flow in noncircular channels varies with channel shape. In rectangular ducts, 1,900 < Rec < 2,800 (Hanks and Ruo, Ind. Eng. Chem. Fundam., 5, 558–561 [1966]). In triangular ducts, 1,600 < Rec < 1,800 (Cope and Hanks, Ind. Eng. Chem. Fundam., 11, 106–117 [1972]; Bandopadhayay and Hinwood, J. Fluid Mech., 59, 775–783 [1973]). Nonisothermal Flow For nonisothermal flow of liquids, the friction factor may be increased if the liquid is being cooled or decreased if the liquid is being heated, because of the effect of temperature on viscosity near the wall. In shell and tube heat-exchanger design, the recommended practice is to first estimate f using the bulk mean liquid temperature over the tube length. Then, in laminar flow, the result is divided by (µa /µw)0.23 in the case of cooling or (µa /µw)0.38 in the case of heating. For turbulent flow, f is divided by (µa /µw)0.11 in the case of cooling or (µa /µw)0.17 in case of heating. Here, µa is the viscosity at the average bulk temperature and µw is the viscosity at the average wall temperature (Seider and Tate, Ind. Eng. Chem., 28, 1429–1435 [1936]). In the case of rough commercial pipes, rather than heat-exchanger tubing, it is common for flow to be in the “complete” turbulence regime where f is independent of Re. In such cases, the friction factor should not be corrected for wall temperature. If the liquid density varies with temperature, the average bulk density should be used to calculate the pressure drop from the friction factor. In addition, a (usually small) correction may be applied for acceleration effects by adding the term G2[(1/ρ2) − (1/ρ1)] from the mechanical energy balance to the pressure drop ∆P = P1 − P2, where G is the mass velocity. This acceleration results from small compressibility effects associated with temperature-dependent density. Christiansen and Gordon (AIChE J., 15, 504–507 [1969]) present equations and charts for frictional loss in laminar nonisothermal flow of Newtonian and non-Newtonian liquids heated or cooled with constant wall temperature. Frictional dissipation of mechanical energy can result in significant heating of fluids, particularly for very viscous liquids in small channels. Under adiabatic conditions, the bulk liquid temperature rise is given by ∆T = ∆P/Cv ρ for incompressible flow through a channel of constant cross-sectional area. For flow of polymers, this amounts to about 4°C per 10 MPa pressure drop, while for hydrocarbon liquids it is about 6°C per 10 MPa. The temperature rise in laminar flow is highly nonuniform, being concentrated near the pipe wall where most of the dissipation occurs. This may result in significant viscosity reduction near the wall, and greatly increased flow or reduced pressure drop, and a flattened velocity profile. Compensation should generally be made for the heat effect when ∆P exceeds 1.4 MPa (203 psi) for adiabatic walls or 3.5 MPa (508 psi) for isothermal walls (Gerard, Steidler, and Appeldoorn, Ind. Eng. Chem. Fundam., 4, 332–339 [1969]). Open Channel Flow For flow in open channels, the data are largely based on experiments with water in turbulent flow, in channels of sufficient roughness that there is no Reynolds number effect. The hydraulic radius approach may be used to estimate a friction factor with which to compute friction losses. Under conditions of uniform flow where liquid depth and cross-sectional area do not vary significantly with position in the flow direction, there is a balance between gravitational forces and wall stress, or equivalently between frictional losses and potential energy change. The mechanical energy balance reduces to lv = g(z1 − z2). In terms of the friction factor and hydraulic diameter or hydraulic radius, 2 f V 2L f V 2L lv =  =  = g(z1 − z2) (6-53) DH 2RH The hydraulic radius is the cross-sectional area divided by the wetted perimeter, where the wetted perimeter does not include the free sur-

FLUID DYNAMICS face. Letting S = sin θ = channel slope (elevation loss per unit length of channel, θ = angle between channel and horizontal), Eq. (6-53) reduces to 2gSRH V=  (6-54) f The most often used friction correlation for open channel flows is due to Manning (Trans. Inst. Civ. Engrs. Ireland, 20, 161 [1891]) and is equivalent to 29n2 f= (6-55) RH1/3 where n is the channel roughness, with dimensions of (length)1/6. Table 6-2 gives roughness values for several channel types. For gradual changes in channel cross section and liquid depth, and for slopes less than 10°, the momentum equation for a rectangular channel of width b and liquid depth h may be written as a differential equation in the flow direction x.

冪莦

dh h db f V 2(b + 2h)  (1 − Fr) − Fr   = S −  dx b dx 2gbh

冢 冣

(6-56)

For a given fixed flow rate Q = Vbh, and channel width profile b(x), Eq. (6-56) may be integrated to determine the liquid depth profile h(x). The dimensionless Froude number is Fr = V 2/gh. When Fr = 1, the flow is critical, when Fr < 1, the flow is subcritical, and when Fr > 1, the flow is supercritical. Surface disturbances move at a wave velocity c = 兹苶 gh; they cannot propagate upstream in supercritical flows. The specific energy Esp is nearly constant. 2

V Esp = h +  2g

(6-57)

This equation is cubic in liquid depth. Below a minimum value of Esp there are no real positive roots; above the minimum value there are two positive real roots. At this minimum value of Esp the flow is critical; that is, Fr = 1, V = 兹苶 gh, and Esp = (3/2)h. Near critical flow conditions, wave motion and sudden depth changes called hydraulic jumps are likely. Chow (Open Channel Hydraulics, McGraw-Hill, New York, 1959), discusses the numerous surface profile shapes which may exist in nonuniform open channel flows. For flow over a sharp-crested weir of width b and height L, from a liquid depth H, the flow rate is given approximately by 2 Q =  Cd b兹2苶g苶(H − L)3/2 3

(6-58)

where Cd ≈ 0.6 is a discharge coefficient. Flow through notched weirs is described under flow meters in Sec. 10 of the Handbook.

TABLE 6-2 Eq. (6-55)

Average Values of n for Manning Formula, Surface

n, m1/6

n, ft1/6

Cast-iron pipe, fair condition Riveted steel pipe Vitrified sewer pipe Concrete pipe Wood-stave pipe Planed-plank flume Semicircular metal flumes, smooth Semicircular metal flumes, corrugated Canals and ditches Earth, straight and uniform Winding sluggish canals Dredged earth channels Natural-stream channels Clean, straight bank, full stage Winding, some pools and shoals Same, but with stony sections Sluggish reaches, very deep pools, rather weedy

0.014 0.017 0.013 0.015 0.012 0.012 0.013 0.028

0.011 0.014 0.011 0.012 0.010 0.010 0.011 0.023

0.023 0.025 0.028

0.019 0.021 0.023

0.030 0.040 0.055 0.070

0.025 0.033 0.045 0.057

SOURCE: Brater and King, Handbook of Hydraulics, 6th ed., McGraw-Hill, New York, 1976, p. 7–22. For detailed information, see Chow, Open-Channel Hydraulics, McGraw-Hill, New York, 1959, pp. 110–123.

6-13

Non-Newtonian Flow For isothermal laminar flow of timeindependent non-Newtonian liquids, integration of the Cauchy momentum equations yields the fully developed velocity profile and flow rate–pressure drop relations. For the Bingham plastic fluid described by Eq. (6-3), in a pipe of diameter D and a pressure drop per unit length ∆P/L, the flow rate is given by πD3τw 4τy τ y4 Q= 1−+ 32µ∞ 3τw 3τ w4

冤

冥

(6-59)

where the wall stress is τw = D∆P/(4L). The velocity profile consists of a central nondeforming plug of radius rP = 2τy /(∆P/L) and an annular deforming region. The velocity profile in the annular region is given by 1 ∆P vz =   (R2 − r2) − τy(R − r) , rP ≤ r ≤ R (6-60) µ∞ 4L where r is the radial coordinate and R is the pipe radius. The velocity of the central, nondeforming plug is obtained by setting r = rP in Eq. (6-60). When Q is given and Eq. (6-59) is to be solved for τw and the pressure drop, multiple positive roots for the pressure drop may be found. The root corresponding to τw < τy is physically unrealizable, as it corresponds to rp > R and the pressure drop is insufficient to overcome the yield stress. For a power law fluid, Eq. (6-4), with constant properties K and n, the flow rate is given by ∆P 1/n n Q=π  (6-61)  R(1 + 3n)/n 2KL 1 + 3n and the velocity profile by ∆P 1/n n (6-62) vz =   [R(1 + n)/n − r (1 + n)/n] 2KL 1+n Similar relations for other non-Newtonian fluids may be found in Govier and Aziz and in Bird, Armstrong, and Hassager (Dynamics of Polymeric Liquids, vol. 1: Fluid Mechanics, Wiley, New York, 1977). For steady-state laminar flow of any time-independent viscous fluid, at average velocity V in a pipe of diameter D, the RabinowitschMooney relations give a general relationship for the shear rate at the pipe wall. 8V 1 + 3n′ γ˙ w =   (6-63) D 4n′ where n′ is the slope of a plot of D∆P/(4L) versus 8V/D on logarithmic coordinates, d ln [D∆P/(4L)] n′ =  (6-64) d ln (8V/D) By plotting capillary viscometry data this way, they can be used directly for pressure drop design calculations, or to construct the rheogram for the fluid. For pressure drop calculation, the flow rate and diameter determine the velocity, from which 8V/D is calculated and D∆P/(4L) read from the plot. For a Newtonian fluid, n′ = 1 and the shear rate at the wall is γ˙ = 8V/D. For a power law fluid, n′ = n. To construct a rheogram, n′ is obtained from the slope of the experimental plot at a given value of 8V/D. The shear rate at the wall is given by Eq. (6-63) and the corresponding shear stress at the wall is τw = D∆P/(4L) read from the plot. By varying the value of 8V/D, the shear rate versus shear stress plot can be constructed. The generalized approach of Metzner and Reed (AIChE J., 1, 434 [1955]) for time-independent non-Newtonian fluids defines a modified Reynolds number as Dn′V 2 − n′ρ ReMR ⬅  (6-65) K′8n′ − 1 where K′ satisfies 8V n′ D∆P (6-66)  = K′  4L D With this definition, f = 16/ReMR is automatically satisfied at the value of 8V/D where K′ and n′ are evaluated. Equation (6-66) may be obtained by integration of Eq. (6-64) only when n′ is a constant, as, for

冤

冥

冢

冢

冣 冢

冣 冢

冣

冣

冢

冣

冢 冣

6-14

FLUID AND PARTICLE DYNAMICS

example, the cases of Newtonian and power law fluids. For Newtonian fluids, K′ = µ and n′ = 1; for power law fluids, K′ = K[(1 + 3n)/ (4n)]n and n′ = n. For Bingham plastics, K′ and n′ are variable, given as a function of τw (Metzner, Ind. Eng. Chem., 49, 1429–1432 [1957]). K = τ 1w− n′

冤

µ∞  1 − 4τy /3τw + (τy /τw)4/3

冥

3n + 1 µeff = K  4n

冢

n′

(6-67)

1 − 4τy /(3τw) + (τy /τw)4/3 n′ =  1 − (τy /τw)4

(6-68)

For laminar flow of power law fluids in channels of noncircular cross section, see Schecter (AIChE J., 7, 445–448 [1961]), Wheeler and Wissler (AIChE J., 11, 207–212 [1965]), Bird, Armstrong, and Hassager (Dynamics of Polymeric Liquids, vol. 1: Fluid Mechanics, Wiley, New York, 1977), and Skelland (Non-Newtonian Flow and Heat Transfer, Wiley, New York, 1967). Steady state, fully developed laminar flows of viscoelastic fluids in straight, constant-diameter pipes show no effects of viscoelasticity. The viscous component of the constitutive equation may be used to develop the flow rate–pressure drop relations, which apply downstream of the entrance region after viscoelastic effects have disappeared. A similar situation exists for time-dependent fluids. The transition to turbulent flow begins at ReMR in the range of 2,000 to 2,500 (Metzner and Reed, AIChE J., 1, 434 [1955]). For Bingham plastic materials, K′ and n′ must be evaluated for the τw condition in question in order to determine ReMR and establish whether the flow is laminar. An alternative method for Bingham plastics is by Hanks (Hanks, AIChE J., 9, 306 [1963]; 14, 691 [1968]; Hanks and Pratt, Soc. Petrol. Engrs. J., 7, 342 [1967]; and Govier and Aziz, pp. 213–215). The transition from laminar to turbulent flow is influenced by viscoelastic properties (Metzner and Park, J. Fluid Mech., 20, 291 [1964]) with the critical value of ReMR increased to beyond 10,000 for some materials. For turbulent flow of non-Newtonian fluids, the design chart of Dodge and Metzner (AIChE J., 5, 189 [1959]), Fig. 6-11, is most widely used. For Bingham plastic materials in turbulent flow, it is generally assumed that stresses greatly exceed the yield stress, so that the friction factor–Reynolds number relationship for Newtonian fluids applies, with µ∞ substituted for µ. This is equivalent to setting n′ = 1 and τy /τw = 0 in the Dodge-Metzner method, so that ReMR = DVρ/µ∞. Wilson and Thomas (Can. J. Chem. Eng., 63, 539–546 [1985]) give friction factor equations for turbulent flow of power law fluids and Bingham plastic fluids. Power law fluids: 1−n 1+n 1 1  =  + 8.2  + 1.77 ln  1 + n 2 fN 兹苶f 兹苶

冢

where fN is the friction factor for Newtonian fluid evaluated at Re = DVρ/µeff where the effective viscosity is

冣

(6-69)

FIG. 6-11 Fanning friction factor for non-Newtonian flow. (From Dodge and Metzner, Am. Inst. Chem. Eng. J., 5, 189 [1959]).

n−1

冣 冢 D冣 8V

n−1

(6-70)

Bingham fluids: (1 − ξ)2 1 1  =  + 1.77 ln  + ξ(10 + 0.884ξ) 1+ξ fN 兹苶f 兹苶

冢

冣

(6-71)

where fN is evaluated at Re = DVρ/µ∞ and ξ = τy /τw. Iteration is required to use this equation since τw = fρV 2/2. Drag reduction in turbulent flow can be achieved by adding soluble high molecular weight polymers in extremely low concentration to Newtonian liquids. The reduction in friction is generally believed to be associated with the viscoelastic nature of the solutions effective in the wall region. For a given polymer, there is a minimum molecular weight necessary to initiate drag reduction at a given flow rate, and a critical concentration above which drag reduction will not occur (Kim, Little and Ting, J. Colloid Interface Sci., 47, 530–535 [1974]). Drag reduction is reviewed by Hoyt (J. Basic Eng., 94, 258–285 [1972]); Little, et al. (Ind. Eng. Chem. Fundam., 14, 283–296 [1975]) and Virk (AIChE J., 21, 625–656 [1975]). At maximum possible drag reduction in smooth pipes, 1 50.73  = −19 log  (6-72) Re兹苶f 兹苶f

冢

or, approximately,

冣

0.58 f= Re0.58

(6-73)

for 4,000 < Re < 40,000. The actual drag reduction depends on the polymer system. For further details, see Virk (ibid.). Economic Pipe Diameter, Turbulent Flow The economic optimum pipe diameter may be computed so that the last increment of investment reduces the operating cost enough to produce the required minimum return on investment. For long cross-country pipelines, alloy pipes of appreciable length and complexity, or pipelines with control valves, detailed analyses of investment and operating costs should be made. Peters and Timmerhaus (Plant Design and Economics for Chemical Engineers, 4th ed., McGraw-Hill, New York, 1991) provide a detailed method for determining the economic optimum size. For pipelines of the lengths usually encountered in chemical plants and petroleum refineries, simplified selection charts are often adequate. In many cases there is an economic optimum velocity that is nearly independent of diameter, which may be used to estimate the economic diameter from the flow rate. For low-viscosity liquids in schedule 40 steel pipe, economic optimum velocity is typically in the range of 1.8 to 2.4 m/s (5.9 to 7.9 ft/s). For gases with density ranging from 0.2 to 20 kg/m3 (0.013 to 1.25 lbm/ft3), the economic optimum velocity is about 40 m/s to 9 m/s (131 to 30 ft/s). Charts and rough guidelines for economic optimum size do not apply to multiphase flows. Economic Pipe Diameter, Laminar Flow Pipelines for the transport of high-viscosity liquids are seldom designed purely on the basis of economics. More often, the size is dictated by operability considerations such as available pressure drop, shear rate, or residence time distribution. Peters and Timmerhaus (ibid., Chap. 10) provide an economic pipe diameter chart for laminar flow. For non-Newtonian fluids, see Skelland (Non-Newtonian Flow and Heat Transfer, Chap. 7, Wiley, New York, 1967). Vacuum Flow When gas flows under high vacuum conditions or through very small openings, the continuum hypothesis is no longer appropriate if the channel dimension is not very large compared to the mean free path of the gas. When the mean free path is comparable to the channel dimension, flow is dominated by collisions of molecules with the wall, rather than by collisions between molecules. An approximate expression based on Brown, et al. (J. Appl. Phys., 17, 802–813 [1946]) for the mean free path is 8RT  冢 冣 冪莦 πM

2µ λ=  p

w

(6-74)

FLUID DYNAMICS The Knudsen number Kn is the ratio of the mean free path to the channel dimension. For pipe flow, Kn = λ/D. Molecular flow is characterized by Kn > 1.0; continuum viscous (laminar or turbulent) flow is characterized by Kn < 0.01. Transition or slip flow applies over the range 0.01 < Kn < 1.0. Vacuum flow is usually described with flow variables different from those used for normal pressures, which often leads to confusion. Pumping speed S is the actual volumetric flow rate of gas through a flow cross section. Throughput Q is the product of pumping speed and absolute pressure. In the SI system, Q has units of Pa ⋅ m3/s. Q = Sp

(6-75)

The mass flow rate w is related to the throughput using the ideal gas law. Mw w=Q (6-76) RT Throughput is therefore proportional to mass flow rate. For a given mass flow rate, throughput is independent of pressure. The relation between throughput and pressure drop ∆ p = p1 − p2 across a flow element is written in terms of the conductance C. Resistance is the reciprocal of conductance. Conductance has dimensions of volume per time. Q = C∆p (6-77) The conductance of a series of flow elements is given by 1 1 1 1  =  +  +  + ⋅⋅⋅ C C1 C2 C3 while for elements in parallel,

(6-78)

C = C1 + C2 + C3 + ⋅⋅⋅

(6-79)

For a vacuum pump of speed Sp withdrawing from a vacuum vessel through a connecting line of conductance C, the pumping speed at the vessel is SpC S= (6-80) Sp + C Molecular Flow Under molecular flow conditions, conductance is independent of pressure. It is proportional to 兹苶 T苶 /Mw苶, with the proportionality constant a function of geometry. For fully developed pipe flow, πD3 RT C=  (6-81) 8L Mw For an orifice of area A, RT (6-82) C = 0.40A  Mw Conductance equations for several other geometries are given by Ryans and Roper (Process Vacuum System Design and Operation, Chap. 2, McGraw-Hill, New York, 1986). For a circular annulus of outer and inner diameters D1 and D2 and length L, the method of Guthrie and Wakerling (Vacuum Equipment and Techniques, McGraw-Hill, New York, 1949) may be written

冪莦

1 K =  1 + (L/D)

for

0 ≤ L/D ≤ 0.75

6-15 (6-84)

1 + 0.8(L/D) K = 2 for L/D > 0.75 (6-85) 1 + 1.90(L/D) + 0.6(L/D) For L/D > 100, the error in neglecting the end correction by using the fully developed pipe flow equation (6-81) is less than 2 percent. For rectangular channels, see Normand (Ind. Eng. Chem., 40, 783–787 [1948]). Yu and Sparrow ( J. Basic Eng., 70, 405–410 [1970]) give a theoretically derived chart for slot seals with or without a sheet located in or passing through the seal, giving mass flow rate as a function of the ratio of seal plate thickness to gap opening. Slip Flow In the transition region between molecular flow and continuum viscous flow, the conductance for fully developed pipe flow is most easily obtained by the method of Brown, et al. (J. Appl. Phys., 17, 802–813 [1946]), which uses the parameter X=

2µ RT  冪莦π8 冢Dλ 冣 = 冢 M p D 冣 冪莦

(6-86)

m

where pm is the arithmetic mean absolute pressure. A correction factor F, read from Fig. 6-12 as a function of X, is applied to the conductance for viscous flow. πD4pm C=F (6-87) 128µL For slip flow through square channels, see Milligan and Wilkerson (J. Eng. Ind., 95, 370–372 [1973]). For slip flow through annuli, see Maegley and Berman (Phys. Fluids, 15, 780–785 [1972]). The pump-down time θ for evacuating a vessel in the absence of air in-leakage is given approximately by p1 − p0 V θ = t ln  (6-88) S0 p2 − p0 where Vt = volume of vessel plus volume of piping between vessel and pump; S0 = system speed as given by Eq. (6-80), assumed independent of pressure; p1 = initial vessel pressure; p2 = final vessel pressure; and p0 = lowest pump intake pressure attainable with the pump in question. See Dushman and Lafferty (Scientific Foundations of Vacuum Technique, 2d ed., Wiley, New York, 1962). The amount of inerts which has to be removed by a pumping system after the pump-down stage depends on the in-leakage of air at the various fittings, connections, and so on. Air leakage is often correlated with system volume and pressure, but this approach introduces uncer-

冢 冣 冢

冣

冪莦

(D1 − D2)2(D1 + D2) C = 0.42K  L

RT  冪莦 M

(6-83)

w

where K is a dimensionless constant with values given in Table 6-3. For a short pipe of circular cross section, the conductance as calculated for an orifice from Eq. (6-82) is multiplied by a correction factor K which may be approximated as (Kennard, Kinetic Theory of Gases, McGraw-Hill, New York, 1938, pp. 306–308) TABLE 6-3

Constants for Circular Annuli

D2 /D1

K

D2 /D1

K

0 0.259 0.500

1.00 1.072 1.154

0.707 0.866 0.966

1.254 1.430 1.675

FIG. 6-12 Correction factor for Poiseuille’s equation at low pressures. Curve A: experimental curve for glass capillaries and smooth metal tubes. (From Brown, et al., J. Appl. Phys., 17, 802 [1946].) Curve B: experimental curve for iron pipe (From Riggle, Courtesy of E. I. du Pont de Nemours & Co.)

6-16

FLUID AND PARTICLE DYNAMICS

tainty because the number and size of leaks does not necessily correlate with system volume, and leakage is sensitive to maintenance quality. Ryans and Roper (Process Vacuum System Design and Operation, McGraw-Hill, New York, 1986) present a thorough discussion of air leakage.

Contraction and Entrance Losses For a sudden contraction at a sharp-edged entrance to a pipe or sudden reduction in crosssectional area of a channel, as shown in Fig. 6-13a, the loss coefficient based on the downstream velocity V2 is given for turbulent flow in Crane Co. Tech Paper 410 (1980) approximately by

冢

A K = 0.5 1 − 2 A1

FRICTIONAL LOSSES IN PIPELINE ELEMENTS The viscous or frictional loss term in the mechanical energy balance for most cases is obtained experimentally. For many common fittings found in piping systems, such as expansions, contractions, elbows and valves, data are available to estimate the losses. Substitution into the energy balance then allows calculation of pressure drop. A common error is to assume that pressure drop and frictional losses are equivalent. Equation (6-16) shows that in addition to frictional losses, other factors such as shaft work and velocity or elevation change influence pressure drop. Losses lv for incompressible flow in sections of straight pipe of constant diameter may be calculated as previously described using the Fanning friction factor: ∆P 2 f V 2L lv =  =  (6-89) ρ D where ∆P = drop in equivalent pressure, P = p + ρgz, with p = pressure, ρ = fluid density, g = acceleration of gravity, and z = elevation. Losses in the fittings of a piping network are frequently termed minor losses or miscellaneous losses. These descriptions are misleading because in process piping fitting losses are often much greater than the losses in straight piping sections. Equivalent Length and Velocity Head Methods Two methods are in common use for estimating fitting loss. One, the equivalent length method, reports the losses in a piping element as the length of straight pipe which would have the same loss. For turbulent flows, the equivalent length is usually reported as a number of diameters of pipe of the same size as the fitting connection; Le /D is given as a fixed quantity, independent of D. This approach tends to be most accurate for a single fitting size and loses accuracy with deviation from this size. For laminar flows, Le /D correlations normally have a size dependence through a Reynolds number term. The other method is the velocity head method. The term V 2/2g has dimensions of length and is commonly called a velocity head. Application of the Bernoulli equation to the problem of frictionless discharge at velocity V through a nozzle at the bottom of a column of liquid of height H shows that H = V 2/2g. Thus H is the liquid head corresponding to the velocity V. Use of the velocity head to scale pressure drops has wide application in fluid mechanics. Examination of the Navier-Stokes equations suggests that when the inertial terms dominate the viscous terms, pressure gradients are expected to be proportional to ρV 2 where V is a characteristic velocity of the flow. In the velocity head method, the losses are reported as a number of velocity heads K. Then, the engineering Bernoulli equation for an incompressible fluid can be written ρV22 ρV12 ρV 2 p1 − p2 = α2  − α1  + ρg(z2 − z1) + K  (6-90) 2 2 2 where V is the reference velocity upon which the velocity head loss coefficient K is based. For a section of straight pipe, K = 4 f L/D.

(a) FIG. 6-13

(b)

冣

(6-91)

Example 5: Entrance Loss Water, ρ = 1000 kg/m3, flows from a large vessel through a sharp-edged entrance into a pipe at a velocity in the pipe of 2 m/s. The flow is turbulent. Estimate the pressure drop from the vessel into the pipe. With A2 /A1 ∼ 0, the viscous loss coefficient is K = 0.5 from Eq. (6-91). The mechanical energy balance, Eq. (6-16) with V1 = 0 and z2 − z1 = 0 and assuming uniform flow (α2 = 1) becomes ρV22 ρV22 p1 − p2 =  + 0.5  = 4,000 + 2,000 = 6,000 Pa 2 2 Note that the total pressure drop consists of 0.5 velocity heads of frictional loss contribution, and 1 velocity head of velocity change contribution. The frictional contribution is a permanent loss of mechanical energy by viscous dissipation. The acceleration contribution is reversible; if the fluid were subsequently decelerated in a frictionless diffuser, a 4,000 Pa pressure rise would occur.

For a trumpet-shaped rounded entrance, with a radius of rounding greater than about 15 percent of the pipe diameter (Fig. 6-13b), the turbulent flow loss coefficient K is only about 0.1 (Vennard and Street, Elementary Fluid Mechanics, 5th ed., Wiley, New York, 1975, pp. 420–421). Rounding of the inlet prevents formation of the vena contracta, thereby reducing the resistance to flow. For laminar flow the losses in sudden contraction may be estimated for area ratios A2 /A1 < 0.2 by an equivalent additional pipe length Le given by Le /D = 0.3 + 0.04Re

(6-92)

where D is the diameter of the smaller pipe and Re is the Reynolds number in the smaller pipe. For laminar flow in the entrance to rectangular ducts, see Shah (J. Fluids Eng., 100, 177–179 [1978]) and Roscoe (Philos. Mag., 40, 338–351 [1949]). For creeping flow, Re < 1, of power law fluids, the entrance loss is approximately Le/D = 0.3/n (Boger, Gupta, and Tanner, J. Non-Newtonian Fluid Mech., 4, 239–248 [1978]). For viscoelastic fluid flow in circular channels with sudden contraction, a toroidal vortex forms upstream of the contraction plane. Such flows are reviewed by Boger (Ann. Review Fluid Mech., 19, 157–182 [1987]). For creeping flow through conical converging channels, inertial acceleration terms are negligible and the viscous pressure drop ∆p = ρlv may be computed by integration of the differential form of the Hagen-Poiseuille equation Eq. (6-36), provided the angle of convergence is small. The result for a power law fluid is 3n + 1 ∆p = K  4n

冢

8V 1  1 − 冢  冣 冥冧 冣 冢 D 冣 冦 6n tan (α/2) 冤 D n

2

n

D2

2

3n

1

where D1 = inlet diameter D2 = exit diameter V2 = velocity at the exit α = total included angle

(c)

(d)

Contractions and enlargements: (a) sudden contraction, (b) rounded contraction, (c) sudden enlargement, and (d) uniformly diverging duct.

(6-93)

FLUID DYNAMICS Equation (6-93) agrees with experimental data (Kemblowski and Kiljanski, Chem. Eng. J. (Lausanne), 9, 141–151 [1975]) for α < 11°. For Newtonian liquids, Eq. (6-93) simplifies to 8V 1 D 3 ∆p = µ 2  1 − 2 (6-94) D2 6 tan (α/2) D1 For creeping flow through rectangular or two-dimensional converging channels, the differential form of the Hagen-Poiseulle equation with equivalent diameter given by Eq. (6-49) may be used, provided the convergence is gradual. Expansion and Exit Losses For ducts of any cross section, the frictional loss for a sudden enlargement (Fig. 6-13c) with turbulent flow is given by the Borda-Carnot equation: A 2 V12 − V22 V12 lv =  = 1 − 1 (6-95) 2 2 A2 where V1 = velocity in the smaller duct V2 = velocity in the larger duct A1 = cross-sectional area of the smaller duct A2 = cross-sectional area of the larger duct

冢 冣冦

冤 冢 冣 冥冧

冢

冣

Equation (6-95) is valid for incompressible flow. For compressible flows, see Benedict, Wyler, Dudek, and Gleed ( J. Eng. Power, 98, 327–334 [1976]). For an infinite expansion, A1/A2 = 0, Eq. (6-95) shows that the exit loss from a pipe is 1 velocity head. This result is easily deduced from the mechanical energy balance Eq. (6-90), noting that p1 = p2. This exit loss is due to the dissipation of the discharged jet; there is no pressure drop at the exit. For creeping Newtonian flow (Re < 1), the frictional loss due to a sudden enlargement should be obtained from the same equation for a sudden contraction (Eq. [6-92]). Note, however, that Boger, Gupta, and Tanner (ibid.) give an exit friction equivalent length of 0.12 diameter, increasing for power law fluids as the exponent decreases. For laminar flows at higher Reynolds numbers, the pressure drop is twice that given by Eq. (6-95). This results from the velocity profile factor α in the mechanical energy balance being 2.0 for the parabolic laminar velocity profile. If the transition from a small to a large duct of any cross-sectional shape is accomplished by a uniformly diverging duct (see Fig. 6-13d) with a straight axis, the total frictional pressure drop can be computed by integrating the differential form of Eq. (6-89), dlv /dx = 2 f V 2/D over the length of the expansion, provided the total angle α between the diverging walls is less than 7°. For angles between 7 and 45°, the loss coefficient may be estimated as 2.6 sin (α/2) times the loss coefficient for a sudden expansion; see Hooper (Chem. Eng., Nov. 7, 1988). Gibson (Hydraulics and Its Applications, 5th ed., Constable, London 1952, p. 93) recommends multiplying the sudden enlargement loss by 0.13 for 5° < α < 7.5° and by 0.0110α1.22 for 7.5° < α < 35°. For angles greater than 35 to 45°, the losses are normally considered equal to those for a sudden expansion, although in some cases the losses may be greater. Expanding flow through standard pipe reducers should be treated as sudden expansions. Trumpet-shaped enlargements for turbulent flow designed for constant decrease in velocity head per unit length were found by Gibson (ibid., p. 95) to give 20 to 60 percent less frictional loss than straight taper pipes of the same length. A special feature of expansion flows occurs when viscoelastic liquids are extruded through a die at a low Reynolds number. The extrudate may expand to a diameter several times greater than the die diameter, whereas for a Newtonian fluid the diameter expands only 10 percent. This phenomenon, called die swell, is most pronounced with short dies (Graessley, Glasscock, and Crawley, Trans. Soc. Rheol., 14, 519–544 [1970]). For velocity distribution measurements near the die exit, see Goulden and MacSporran (J. Non-Newtonian Fluid Mech., 1, 183–198 [1976]) and Whipple and Hill (AIChE J., 24, 664–671 [1978]). At high flow rates, the extrudate becomes distorted, suffering melt fracture at wall shear stresses greater than 105 N/m2. This phenomenon is reviewed by Denn (Ann. Review Fluid Mech., 22, 13–34 [1990]). Ramamurthy (J. Rheol., 30, 337–357 [1986]) has found a dependence of apparent stick-slip behavior in melt fracture to be dependent on the material of construction of the die.

6-17

Fittings and Valves For turbulent flow, the frictional loss for fittings and valves can be expressed by the equivalent length or velocity head methods. As fitting size is varied, K values are relatively more constant than Le /D values, but since fittings generally do not achieve geometric similarity between sizes, K values tend to decrease with increasing fitting size. Table 6-4 gives K values for many types of fittings and valves. Manufacturers of valves, especially control valves, express valve capacity in terms of a flow coefficient Cv, which gives the flow rate through the valve in gal/min of water at 60°F under a pressure drop of 1 lbf/in2. It is related to K by C1d 2 Cv =  (6-96) K 兹苶 where C1 is a dimensional constant equal to 29.9 and d is the diameter of the valve connections in inches. For laminar flow, data for the frictional loss of valves and fittings are meager. (Beck and Miller, J. Am. Soc. Nav. Eng., 56, 62–83 [1944]; Beck, ibid., 56, 235–271, 366–388, 389–395 [1944]; De Craene, Heat. Piping Air Cond., 27[10], 90–95 [1955]; Karr and Schutz, J. Am. Soc. Nav. Eng., 52, 239–256 [1940]; and Kittredge and Rowley, Trans. ASME, 79, 1759–1766 [1957]). The data of Kittredge and Rowley indicate that K is constant for Reynolds numbers above 500 to 2,000, but increases rapidly as Re decreases below 500. Typical values for K for laminar flow Reynolds numbers are shown in Table 6-5. Methods to calculate losses for tee and wye junctions for dividing and combining flow are given by Miller (Internal Flow Systems, 2d ed., Chap. 13, BHRA, Cranfield, 1990), including effects of Reynolds number, angle between legs, area ratio, and radius. Junctions with more than three legs are also discussed. The sources of data for the loss coefficient charts are Blaisdell and Manson (U.S. Dept. Agric. Res. Serv. Tech. Bull. 1283 [August 1963]) for combining flow and Gardel (Bull. Tech. Suisses Romande, 85[9], 123–130 [1957]; 85[10], 143–148 [1957]) together with additional unpublished data for dividing flow. Miller (Internal Flow Systems, 2d ed., Chap. 13, BHRA, Cranfield, 1990) gives the most complete information on losses in bends and curved pipes. For turbulent flow in circular cross-section bends of constant area, as shown in Fig. 6-14a, a more accurate estimate of the loss coefficient K than that given in Table 6-4 is K = K*CReCoCf

(6-97)

where K*, given in Fig. 6-14b, is the loss coefficient for a smoothwalled bend at a Reynolds number of 106. The Reynolds number correction factor CRe is given in Fig. 6-14c. For 0.7 < r/D < 1 or for K* < 0.4, use the CRe value for r/D = 1. Otherwise, if r/D < 1, obtain CRe from K* CRe =  K* + 0.2(1 − CRe, r/D = 1)

(6-98)

The correction Co (Fig. 6-14d) accounts for the extra losses due to developing flow in the outlet tangent of the pipe, of length Lo. The total loss for the bend plus outlet pipe includes the bend loss K plus the straight pipe frictional loss in the outlet pipe 4f Lo /D. Note that Co = 1 for Lo /D greater than the termination of the curves on Fig. 6-14d, which indicate the distance at which fully developed flow in the outlet pipe is reached. Finally, the roughness correction is frough Cf =  fsmooth

(6-99)

where frough is the friction factor for a pipe of diameter D with the roughness of the bend, at the bend inlet Reynolds number. Similarly, fsmooth is the friction factor for smooth pipe. For Re > 106 and r/D ≥ 1, use the value of Cf for Re = 106. Example 6: Losses with Fittings and Valves It is desired to calculate the liquid level in the vessel shown in Fig. 6-15 required to produce a discharge velocity of 2 m/s. The fluid is water at 20°C with ρ = 1,000 kg/m3 and µ = 0.001 Pa ⋅ s, and the butterfly valve is at θ = 10°. The pipe is 2-in Schedule 40, with an inner diameter of 0.0525 m. The pipe roughness is 0.046 mm. Assuming the flow is turbulent and taking the velocity profile factor α = 1, the engineering Bernoulli equation Eq. (6-16), written between surfaces 1 and 2, where the

6-18

FLUID AND PARTICLE DYNAMICS

TABLE 6-4 Additional Frictional Loss for Turbulent Flow through Fittings and Valvesa

Type of fitting or valve 45° ell, standardb,c,d,e,f 45° ell, long radiusc 90° ell, standardb,c,e,f,g,h Long radius b,c,d,e Square or miter h 180° bend, close returnb,c,e Tee, standard, along run, branch blanked off e Used as ell, entering rung,i Used as ell, entering branchc,g,i Branching flowi,j,k Coupling c,e Unione Gate valve,b,e,m open e openn a openn d openn Diaphragm valve,o open e openn a openn d openn Globe valve,e,m Bevel seat, open a openn Composition seat, open a openn Plug disk, open e open n a open n d open n Angle valve,b,e open Y or blowoff valve,b,m open Plug cockp θ = 5° θ = 10° θ = 20° θ = 40° θ = 60° Butterfly valve p θ = 5° θ = 10° θ = 20° θ = 40° θ = 60° Check valve,b,e,m swing Disk Ball Foot valvee Water meter,h disk Piston Rotary (star-shaped disk) Turbine-wheel a

Additional friction loss, equivalent no. of velocity heads, K 0.35 0.2 0.75 0.45 1.3 1.5 0.4 1.0 1.0 1l 0.04 0.04 0.17 0.9 4.5 24.0 2.3 2.6 4.3 21.0 6.0 9.5 6.0 8.5 9.0 13.0 36.0 112.0 2.0 3.0 0.05 0.29 1.56 17.3 206.0 0.24 0.52 1.54 10.8 118.0 2.0q 10.0q 70.0q 15.0 7.0r 15.0r 10.0r 6.0r

Lapple, Chem. Eng., 56(5), 96–104 (1949), general survey reference. “Flow of Fluids through Valves, Fittings, and Pipe,” Tech. Pap. 410, Crane Co., 1969. c Freeman, Experiments upon the Flow of Water in Pipes and Pipe Fittings, American Society of Mechanical Engineers, New York, 1941. d Giesecke, J. Am. Soc. Heat. Vent. Eng., 32, 461 (1926). e Pipe Friction Manual, 3d ed., Hydraulic Institute, New York, 1961. f Ito, J. Basic Eng., 82, 131–143 (1960). g Giesecke and Badgett, Heat. Piping Air Cond., 4(6), 443–447 (1932). h Schoder and Dawson, Hydraulics, 2d ed., McGraw-Hill, New York, 1934, p. 213. i Hoopes, Isakoff, Clarke, and Drew, Chem. Eng. Prog., 44, 691–696 (1948). j Gilman, Heat. Piping Air Cond., 27(4), 141–147 (1955). k McNown, Proc. Am. Soc. Civ. Eng., 79, Separate 258, 1–22 (1953); discussion, ibid., 80, Separate 396, 19–45 (1954). For the effect of branch spacing on junction losses in dividing flow, see Hecker, Nystrom, and Qureshi, Proc. Am. Soc. Civ. Eng., J. Hydraul. Div., 103(HY3), 265–279 (1977). l This is pressure drop (including friction loss) between run and branch, based on velocity in the mainstream before branching. Actual value depends on the flow split, ranging from 0.5 to 1.3 if mainstream enters run and from 0.7 to 1.5 if mainstream enters branch. m Lansford, Loss of Head in Flow of Fluids through Various Types of 1a-in. Valves, Univ. Eng. Exp. Sta. Bull. Ser. 340, 1943. b

pressures are both atmospheric and the fluid velocities are 0 and V = 2 m/s, respectively, and there is no shaft work, simplifies to V2 gZ =  + lv 2 Contributing to lv are losses for the entrance to the pipe, the three sections of straight pipe, the butterfly valve, and the 90° bend. Note that no exit loss is used because the discharged jet is outside the control volume. Instead, the V 2/2 term accounts for the kinetic energy of the discharging stream. The Reynolds number in the pipe is DVρ 0.0525 × 2 × 1000 Re =  =  = 1.05 × 105 µ 0.001 From Fig. 6-9 or Eq. (6-38), at
/D = 0.046 × 10−3/0.0525 = 0.00088, the friction factor is about 0.0054. The straight pipe losses are then 4fL V 2 lv(sp) =   D 2

冢 冣

4 × 0.0054 × (1 + 1 + 1) V 2 =   0.0525 2

冢

冣

V2 = 1.23  2 The losses from Table 6-4 in terms of velocity heads K are K = 0.5 for the sudden contraction and K = 0.52 for the butterfly valve. For the 90° standard radius (r/D = 1), the table gives K = 0.75. The method of Eq. (6-94), using Fig. 6-14, gives K = K*CReCoCf

冢

0.0054 = 0.24 × 1.24 × 1.0 ×  0.0044 = 0.37

冣

This value is more accurate than the value in Table 6-4. The value fsmooth = 0.0044 is obtainable either from Eq. (6-37) or Fig. 6-9. The total losses are then V2 V2 lv = (1.23 + 0.5 + 0.52 + 0.37)  = 2.62  2 2 and the liquid level Z is 1 V2 V2 V2 Z =   + 2.62  = 3.62  g 2 2 2g

冢

冣

3.62 × 2 =  = 0.73 m 2 × 9.81 2

Curved Pipes and Coils For flow through curved pipe or coil, a secondary circulation perpendicular to the main flow called the Dean effect occurs. This circulation increases the friction relative to straight pipe flow and stabilizes laminar flow, delaying the transition Reynolds number to about D Recrit = 2,100 1 + 12  (6-100) Dc where Dc is the coil diameter. Equation (6-100) is valid for 10 < Dc / D < 250. The Dean number is defined as Re De =  (6-101) (Dc /D)1/2 In laminar flow, the friction factor for curved pipe fc may be expressed in terms of the straight pipe friction factor f = 16/Re as (Hart, Chem. Eng. Sci., 43, 775–783 [1988])

冢

冪莦冣

TABLE 6-5 Additional Frictional Loss for Laminar Flow through Fittings and Valves Additional frictional loss expressed as K Type of fitting or valve

Re = 1,000

500

100

50

90° ell, short radius Gate valve Globe valve, composition disk Plug Angle valve Check valve, swing

0.9 1.2 11 12 8 4

1.0 1.7 12 14 8.5 4.5

7.5 9.9 20 19 11 17

16 24 30 27 19 55

SOURCE: From curves by Kittredge and Rowley, Trans. Am. Soc. Mech. Eng., 79, 1759–1766 (1957).

FLUID DYNAMICS

V2 = 2 m/s 2

1

m

1

Z

90° horizontal bend 1m FIG. 6-15

1m

6-19

10, 328–332 [1971]). For friction loss in laminar flow through semicircular ducts, see Masliyah and Nandakumar (AIChE J., 25, 478– 487 [1979]); for curved channels of square cross section, see Cheng, Lin, and Ou (J. Fluids Eng., 98, 41–48 [1976]). For non-Newtonian (power law) fluids in coiled tubes, Mashelkar and Devarajan (Trans. Inst. Chem. Eng. (London), 54, 108–114 [1976]) propose the correlation fc = (9.07 − 9.44n + 4.37n2)(D/Dc)0.5(De′)−0.768 + 0.122n (6-104) where De′ is a modified Dean number given by 1 6n + 2 De′ =   8 n

冢

Tank discharge example.

D  冣 Re 冪莦 D n

MR

(6-105)

c

De1.5 fc /f = 1 + 0.090  (6-102) 70 + De For turbulent flow, equations by Ito (J. Basic Eng, 81, 123 [1959]) and Srinivasan, Nandapurkar, and Holland (Chem. Eng. [London] no. 218, CE113-CE119 [May 1968]) may be used, with probable accuracy of 15 percent. Their equations are similar to 0.0073 0.079 fc =  + (6-103) Re0.25 兹(D 苶c苶/D 苶)苶 The pressure drop for flow in spirals is discussed by Srinivasan, et al. (loc. cit.) and Ali and Seshadri (Ind. Eng. Chem. Process Des. Dev.,

where ReMR is the Metzner-Reed Reynolds number, Eq. (6-65). This correlation was tested for the range De′ = 70 to 400, D/Dc = 0.01 to 0.135, and n = 0.35 to 1. See also Oliver and Asghar (Trans. Inst. Chem. Eng. [London], 53, 181–186 [1975]). Screens The pressure drop for incompressible flow across a screen of fractional free area α may be computed from

(a)

(b)

(c)

(d)

冢

冣

Loss coefficients for flow in bends and curved pipes: (a) flow geometry, (b) loss coefficient for a smooth-walled bend at Re = 106, (c) Re correction factor, (d) outlet pipe correction factor (From D. S. Miller, Internal Flow Systems, 2d. ed., BHRA, Cranfield, U.K., 1990.)

FIG. 6-14

6-20

FLUID AND PARTICLE DYNAMICS

FIG. 6-16

Screen discharge coefficients, plain square-mesh screens. (Courtesy of E. I. du Pont de Nemours

& Co.)

ρV 2 ∆p = K  2

(6-106)

where ρ = fluid density V = superficial velocity based upon the gross area of the screen K = velocity head loss 1 − α2

冢 冣 冢 α 冣

1 K = 2 C

2

(6-107)

The discharge coefficient for the screen C with aperture Ds is given as a function of screen Reynolds number Re = Ds(V/α)ρ/µ in Fig. 6-16 for plain square-mesh screens, α = 0.14 to 0.79. This curve fits most of the data within 20 percent. In the laminar flow region, Re < 20, the discharge coefficient can be computed from

苶e苶 C = 0.1兹R

(6-108)

Coefficients greater than 1.0 in Fig. 6-16 probably indicate partial pressure recovery downstream of the minimum aperture, due to rounding of the wires. Grootenhuis (Proc. Inst. Mech. Eng. [London], A168, 837–846 [1954]) presents data which indicate that for a series of screens, the total pressure drop equals the number of screens times the pressure drop for one screen, and is not affected by the spacing between screens or their orientation with respect to one another, and presents a correlation for frictional losses across plain square-mesh screens and sintered gauzes. Armour and Cannon (AIChE J., 14, 415–420 [1968]) give a correlation based on a packed bed model for plain, twill, and “dutch” weaves. For losses through monofilament fabrics see Pedersen (Filtr. Sep., 11, 586–589 [1975]). For screens inclined at an angle θ, use the normal velocity component V ′ V′ = V cos θ

(6-109)

(Carothers and Baines, J. Fluids Eng., 97, 116–117 [1975]) in place of V in Eq. (6-106). This applies for Re > 500, C = 1.26, α ≤ 0.97 and 0 < θ < 45°, for square-mesh screens and diamond-mesh netting. Screens inclined at an angle to the flow direction also experience a tangential stress. For non-Newtonian fluids in slow flow, friction loss across a square-woven or full-twill-woven screen can be estimated by considering the screen as a set of parallel tubes, each of diameter equal to the average minimal opening between adjacent wires, and length twice the diameter, without entrance effects (Carley and Smith, Polym. Eng. Sci., 18, 408–415 [1978]). For screen stacks, the losses of individual screens should be summed.

JET BEHAVIOR A free jet, upon leaving an outlet, will entrain the surrounding fluid, expand, and decelerate. To a first approximation, total momentum is conserved as jet momentum is transferred to the entrained fluid. For practical purposes, a jet is considered free when its cross-sectional area is less than one-fifth of the total cross-sectional flow area of the region through which the jet is flowing (Elrod, Heat. Piping Air Cond., 26(3), 149–155 [1954]), and the surrounding fluid is the same as the jet fluid. A turbulent jet in this discussion is considered to be a free jet with Reynolds number greater than 2,000. Additional discussion on the relation between Reynolds number and turbulence in jets is given by Elrod (ibid.). Abramowicz (The Theory of Turbulent Jets, MIT Press, Cambridge, 1963) provides a thorough discourse on the theory of turbulent jets. Hussein, et al. (J. Fluid Mech., 258, 31–75 [1994]) give extensive velocity data for a free jet, as well as an extensive discussion of free jet experimentation and comparison of data with momentum conservation equations. A turbulent free jet is normally considered to consist of four flow regions (Tuve, Heat. Piping Air Cond., 25(1), 181–191 [1953]; Davies, Turbulence Phenomena, Academic, New York, 1972) as shown in Fig. 6-17: 1. Region of flow establishment—a short region whose length is about 6.4 nozzle diameters. The fluid in the conical core of the same length has a velocity about the same as the initial discharge velocity. The termination of this potential core occurs when the growing mixing or boundary layer between the jet and the surroundings reaches the centerline of the jet. 2. A transition region that extends to about 8 nozzle diameters. 3. Region of established flow—the principal region of the jet. In this region, the velocity profile transverse to the jet is self-preserving when normalized by the centerline velocity.

FIG. 6-17

Configuration of a turbulent free jet.

FLUID DYNAMICS TABLE 6-6

Turbulent Free-Jet Characteristics Where Both Jet Fluid and Entrained Fluid Are Air Rounded-inlet circular jet

Longitudinal distribution of velocity along jet center line*† x Vc D0 for 7 <  < 100  = K V0 D0 x K=5 for V0 = 2.5 to 5.0 m/s K = 6.2 for V0 = 10 to 50 m/s Radial distribution of longitudinal velocity†

冢 冣

冢冣

V r log c = 40  Vr x

2

x for 7 <  < 100 D0

Jet angle°† x for  < 100 D0

α ⯝ 20° Entrainment of surrounding fluid‡

x for 7 <  < 100 D0

q x  = 0.32  q0 D0

Rounded-inlet, infinitely wide slot jet Longitudinal distribution of velocity along jet centerline‡

冢 冣

x Vc B0 0.5 for 5 <  < 2,000 and V0 = 12 to 55 m/s  = 2.28  V0 B0 x Transverse distribution of longitudinal velocity‡

冢 冣

冢冣

y V log c = 18.4  Vx x

2

x for 5 <  < 2,000 B0

Jet angle‡ α is slightly larger than that for a circular jet Entrainment of surrounding fluid‡

冢 冣

q x  = 0.62  q0 B0

0.5

x for 5 <  < 2,000 B0

*Nottage, Slaby, and Gojsza, Heat, Piping Air Cond., 24(1), 165–176 (1952). †Tuve, Heat, Piping Air Cond., 25(1), 181–191 (1953). ‡Albertson, Dai, Jensen, and Rouse, Trans. Am. Soc. Civ. Eng., 115, 639–664 (1950), and Discussion, ibid., 115, 665–697 (1950).

4. A terminal region where the residual centerline velocity reduces rapidly within a short distance. For air jets, the residual velocity will reduce to less than 0.3 m/s, (1.0 ft/s) usually considered still air. Several references quote a length of 100 nozzle diameters for the length of the established flow region. However, this length is dependent on initial velocity and Reynolds number. Table 6-6 gives characteristics of rounded-inlet circular jets and rounded-inlet infinitely wide slot jets (aspect ratio > 15). The information in the table is for a homogeneous, incompressible air system under isothermal conditions. The table uses the following nomenclature: B0 = slot height D0 = circular nozzle opening q = total jet flow at distance x q0 = initial jet flow rate r = radius from circular jet centerline y = transverse distance from slot jet centerline Vc = centerline velocity Vr = circular jet velocity at r Vy = velocity at y Witze (Am. Inst. Aeronaut. Astronaut. J., 12, 417–418 [1974]) gives equations for the centerline velocity decay of different types of subsonic and supersonic circular free jets. Entrainment of surrounding fluid in the region of flow establishment is lower than in the region of established flow (see Hill, J. Fluid Mech., 51, 773–779 [1972]). Data of Donald and Singer (Trans. Inst. Chem. Eng. [London], 37, 255–267

6-21

[1959]) indicate that jet angle and the coefficients given in Table 6-6 depend upon the fluids; for a water system, the jet angle for a circular jet is 14° and the entrainment ratio is about 70 percent of that for an air system. Most likely these variations are due to Reynolds number effects which are not taken into account in Table 6-6. Rushton (AIChE J., 26, 1038–1041 [1980]) examined available published results for circular jets and found that the centerline velocity decay is given by D0 Vc (6-110)  = 1.41Re0.135  V0 x where Re = D0V0ρ/µ is the initial jet Reynolds number. This result corresponds to a jet angle tan α/2 proportional to Re−0.135. Characteristics of rectangular jets of various aspect ratios are given by Elrod (Heat., Piping, Air Cond., 26[3], 149–155 [1954]). For slot jets discharging into a moving fluid, see Weinstein, Osterle, and Forstall (J. Appl. Mech., 23, 437–443 [1967]). Coaxial jets are discussed by Forstall and Shapiro (J. Appl. Mech., 17, 399–408 [1950]), and double concentric jets by Chigier and Beer (J. Basic Eng., 86, 797–804 [1964]). Axisymmetric confined jets are described by Barchilon and Curtet (J. Basic Eng., 777–787 [1964]). Restrained turbulent jets of liquid discharging into air are described by Davies (Turbulence Phenomena, Academic, New York, 1972). These jets are inherently unstable and break up into drops after some distance. Lienhard and Day (J. Basic Eng. Trans. AIME, p. 515 [September 1970]) discuss the breakup of superheated liquid jets which flash upon discharge. Density gradients affect the spread of a single-phase jet. A jet of lower density than the surroundings spreads more rapidly than a jet of the same density as the surroundings, and, conversely, a denser jet spreads less rapidly. Additional details are given by Keagy and Weller (Proc. Heat Transfer Fluid Mech. Inst., ASME, pp. 89–98, June 22–24 [1949]) and Cleeves and Boelter (Chem. Eng. Prog., 43, 123–134 [1947]). Few experimental data exist on laminar jets (see Gutfinger and Shinnar, AIChE J., 10, 631–639 [1964]). Theoretical analysis for velocity distributions and entrainment ratios are available in Schlichting and in Morton (Phys. Fluids, 10, 2120–2127 [1967]). Theoretical analyses of jet flows for power law non-Newtonian fluids are given by Vlachopoulos and Stournaras (AIChE J., 21, 385–388 [1975]), Mitwally (J. Fluids Eng., 100, 363 [1978]), and Sridhar and Rankin (J. Fluids Eng., 100, 500 [1978]).

冢 冣

FLOW THROUGH ORIFICES Section 10 of this Handbook describes the use of orifice meters for flow measurement. In addition, orifices are commonly found within pipelines as flow-restricting devices, in perforated pipe distributing and return manifolds, and in perforated plates. Incompressible flow through an orifice in a pipeline as shown in Fig. 6-18, is commonly described by the following equation for flow rate Q in terms of pressure drop across the orifice ∆p, the orifice area Ao, the pipe crosssectional area A, and the density ρ. Q = Co Ao

2∆p/ρ  冪莦 [1 − (A /A) ] o

2

Vena contracta Pipe area A Orifice area A o FIG. 6-18

Flow through an orifice.

(6-111)

6-22

FLUID AND PARTICLE DYNAMICS Mach Number and Speed of Sound The Mach number M = V/c is the ratio of fluid velocity, V, to the speed of sound or acoustic velocity, c. The speed of sound is the propagation velocity of infinitesimal pressure disturbances and is derived from a momentum balance. The compression caused by the pressure wave is adiabatic and frictionless, and therefore isentropic.

.90

Co, orifice number

.85 .80

Data scatter ±2%

c=

∂p 莦 冪冢莦莦 ∂ρ 冣

(6-112)

s

.75

The derivative of pressure p with respect to density ρ is taken at constant entropy s. For an ideal gas,

.70

∂p

= 冢 ∂ρ 冣 M kRT

s

.65 0

50

100

150

200

ρg∆p , Froude number Do Orifice coefficient vs. Froude number. (Courtesy E. I. duPont de Nemours & Co.)

FIG. 6-19

where

w

k = ratio of specific heats, Cp /Cv R = universal gas constant (8,314 J/kgmol K) T = absolute temperature Mw = molecular weight

Hence for an ideal gas, c=

kRT  冪莦 M

(6-113)

w

The velocity of approach term [1 − (Ao /A)2] accounts for the kinetic energy approaching the orifice, while the orifice coefficient or discharge coefficient Co accounts for the vena contracta effect which causes the fluid to accelerate to velocity greater than Q/Ao. The downstream pressure measurement corresponding to ∆p in Eq. (6-111) is taken at the vena contracta. Downstream of the vena contracta, the velocity decelerates and some pressure recovery may be expected. Any pressure recovery is completed about 4 to 8 pipe diameters downstream of the orifice. As an approximation, the pressure recovery, expressed as a fraction of the orifice pressure drop, is approximately equal to the area ratio Ao /A. When the orifice discharges into a large chamber, instead of being installed within a pipe, there is negligible pressure recovery. Equation (6-111) may also be used for flow across a perforated plate with open area Ao and total area A. The orifice coefficient has a value of about 0.62 at large Reynolds numbers (Re = DoVoρ/µ > 20,000), although values ranging from 0.60 to 0.70 are frequently used. At lower Reynolds numbers, the orifice coefficient varies with both Re and with the area or diameter ratio. See Sec. 10 for more details. When liquids discharge vertically downward from orifices into gas, gravity increases the discharge coefficient. Figure 6-19 shows this effect, giving the discharge coefficient in terms of a modified Froude number, Fr = ρg∆p/Do. The orifice coefficient deviates from its value for sharp-edged orifices when the orifice wall thickness exceeds about 75 percent of the orifice diameter. Some pressure recovery occurs within the orifice and the orifice coefficient increases. Pressure drop across segmental orifices is roughly 10 percent greater than that for concentric circular orifices of the same open area. COMPRESSIBLE FLOW Flows are typically considered compressible when the density varies by more than 5 to 10 percent. In practice compressible flows are normally limited to gases, supercritical fluids, and multiphase flows containing gases. Liquid flows are normally considered incompressible, except for certain calculations involved in hydraulic transient analysis (see following) where compressibility effects are important even for nearly incompressible liquids with extremely small density variations. Textbooks on compressible gas flow include Shapiro (Dynamics and Thermodynamics of Compressible Fluid Flow, vol. I and II, Ronald Press, New York [1953]) and Zucrow and Hofmann (Gas Dynamics, vol. I and II, Wiley, New York [1976]). In chemical process applications, one-dimensional gas flows through nozzles or orifices and in pipelines are the most important applications of compressible flow. Multidimensional external flows are of interest mainly in aerodynamic applications.

Most often, the Mach number is calculated using the speed of sound evaluated at the local pressure and temperature. When M = 1, the flow is critical or sonic and the velocity equals the local speed of sound. For subsonic flow M < 1 while supersonic flows have M > 1. Compressibility effects are important when the Mach number exceeds 0.1 to 0.2. A common error is to assume that compressibility effects are always negligible when the Mach number is small. The proper assessment of whether compressibility is important should be based on relative density changes, not on Mach number. Isothermal Gas Flow in Pipes and Channels Isothermal compressible flow is often encountered in long transport lines, where there is sufficient heat transfer to maintain constant temperature. Velocities and Mach numbers are usually small, yet compressibility effects are important when the total pressure drop is a large fraction of the absolute pressure. For an ideal gas with ρ = pMw /RT, integration of the differential form of the momentum or mechanical energy balance equations, assuming a constant friction factor f over a length L of a channel of constant cross section and hydraulic diameter DH, yields,

冤

冢 冣冥

p1 RT 4 f L p21 − p22 = G2   + 2 ln  Mw DH p2

(6-114)

where the mass velocity G = w/A = ρV is the mass flow rate per unit cross-sectional area of the channel. The logarithmic term on the righthand side accounts for the pressure change caused by acceleration of gas as its density decreases, while the first term is equivalent to the calculation of frictional losses using the density evaluated at the average pressure (p1 + p2)/2. Solution of Eq. (6-114) for G and differentiation with respect to p2 reveals a maximum mass flux Gmax = p2兹苶 Mw苶/苶 (R苶 T苶) and a corresponding exit velocity V2,max = 兹苶 R苶 T苶 /M苶w and exit Mach number M2 = 1/兹苶k. This apparent choking condition, though often cited, is not physically meaningful for isothermal flow because at such high velocities, and high rates of expansion, isothermal conditions are not maintained. Adiabatic Frictionless Nozzle Flow In process plant pipelines, compressible flows are usually more nearly adiabatic than isothermal. Solutions for adiabatic flows through frictionless nozzles and in channels with constant cross section and constant friction factor are readily available. Figure 6-20 illustrates adiabatic discharge of a perfect gas through a frictionless nozzle from a large chamber where velocity is effectively zero. A perfect gas obeys the ideal gas law ρ = pMw /RT and also has constant specific heat. The subscript 0 refers to the stagnation conditions in the chamber. More generally, stagnation conditions refer to the conditions which would obtained by isentropically decelerating a gas flow to zero velocity. The minimum area section, or throat, of the nozzle is at the nozzle exit. The flow through the nozzle is isentropic

FLUID DYNAMICS

p0

FIG. 6-20

p1

from upstream pressure p0 to external pressure p2, Equations (6-115) through (6-122) are best used as follows. The critical pressure is first determined from Eq. (6-119). If p2 > p*, then the flow is subsonic (subcritical, unchoked). Then p1 = p2 and M1 may be obtained from Eq. (6-115). Substitution of M1 into Eq. (6-118) then gives the desired mass velocity G. Eqs. (6-116) and (6-117) may be used to find the exit temperature and density. On the other hand, if p2 ≤ p*, then the flow is choked and M1 = 1. Then p1 = p*, and the mass velocity is G* obtained from Eq. (6-122). The exit temperature and density may be obtained from Eqs. (6-120) and (6-121). When the flow is choked, G = G* is independent of external downstream pressure. Reducing the downstream pressure will not increase the flow. The mass flow rate under choking conditions is directly proportional to the upstream pressure.

p2

Isentropic flow through a nozzle.

because it is frictionless (reversible) and adiabatic. In terms of the exit Mach number M1 and the upstream stagnation conditions, the flow conditions at the nozzle exit are given by p0 k − 1 2 k / (k − 1) (6-115)  = 1 +  M1 p1 2

冢

冣

T0 k−1 2  = 1 +  M1 T1 2

(6-116)

ρ0 k−1 2 (6-117)  = 1 +  M1 ρ1 2 The mass velocity G = w/A, where w is the mass flow rate and A is the nozzle exit area, at the nozzle exit is given by 1 / (k − 1)

冢

G = p0

冣

kM M   冪莦 RT k−1 w 0

1

(k + 1) / 2(k − 1)

冢1 + 2 M 冣

(6-118)

2 1

冣

T* 2 = T0 k + 1 ρ* 2 =  ρ0 k+1

冢

G* = p0

(6-120) 1/(k − 1)

冣

(6-121)

kM 2 莦  冣莦 冪冢莦莦 k + 1 冣 莦莦莦莦莦冢莦莦 RT (k + 1)/(k − 1)

Example 7: Flow through Frictionless Nozzle Air at p0 and temperature T0 = 293 K discharges through a frictionless nozzle to atmospheric pressure. Compute the discharge mass flux G, the pressure, temperature, Mach number, and velocity at the exit. Consider two cases: (1) p0 = 7 × 105 Pa absolute, and (2) p0 = 1.5 × 105 Pa absolute. 1. p0 = 7.0 × 105 Pa. For air with k = 1.4, the critical pressure ratio from Eq. (6-119) is p*/p0 = 0.5285 and p* = 0.5285 × 7.0 × 105 = 3.70 × 105 Pa. Since this is greater than the external atmospheric pressure p2 = 1.01 × 105 Pa, the flow is choked and the exit pressure is p1 = 3.70 × 105 Pa. The exit Mach number is 1.0, and the mass flux is equal to G* given by Eq. (6-118). G* = 7.0 × 105 ×

2 1.4 × 29 = 1,650 kg/m ⋅ s 莦 冪冢莦莦 冢 1.4 + 1 冣 莦莦莦莦莦莦莦莦 8314 × 293 冣莦 (1.4 + 1)/(1.4 − 1)

2

The exit temperature, since the flow is choked, is 2 T* T* =  T0 =  × 293 = 244 K T0 1.4 + 1

冢 冣 冢

冣

苶T 苶*/ 苶M 苶w苶 = 313 m/s. The exit velocity is V = Mc = c* = 兹kR 2. p0 = 1.5 × 105 Pa. In this case p* = 0.79 × 105 Pa, which is less than p2. 5 Hence, p1 = p2 = 1.01 × 10 Pa. The flow is unchoked (subsonic). Equation (6-115) is solved for the Mach number. 1.4/(1.4 − 1) 1.5 × 105 1.4 − 1 5 = 1 +  M 21 1.01 × 10 2

冢

冣

M1 = 0.773

These equations are consistent with the isentropic relations for a perfect gas p/p0 = (ρ/ρ0)k, T/T0 = (p/p0)(k − 1)/k. Equation (6-116) is valid for adiabatic flows with or without friction; it does not require isentropic flow. However, Eqs. (6-115) and (6-117) do require isentropic flow. The exit Mach number M1 may not exceed unity. At M1 = 1, the flow is said to be choked, sonic, or critical. When the flow is choked, the pressure at the exit is greater than the pressure of the surroundings into which the gas flow discharges. The pressure drops from the exit pressure to the pressure of the surroundings in a series of shocks which are highly nonisentropic. Sonic flow conditions are denoted by *; sonic exit conditions are found by substituting M1 = M*1 = 1 into Eqs. (6-115) to (6-118). p* 2 k/(k − 1) (6-119) =  p0 k+1

冢

6-23

w

(6-122)

0

Note that under choked conditions, the exit velocity is V = V* = c* = 兹苶 kR苶 T苶 */苶 M苶w, not 兹kR 苶T 苶苶 苶苶w. Sonic velocity must be evaluated at the 0/M exit temperature. For air, with k = 1.4, the critical pressure ratio p*/p0 is 0.5285 and the critical temperature ratio T*/T0 = 0.8333. Thus, for air discharging from 300 K, the temperature drops by 50 K (90 R). This large temperature decrease results from the conversion of internal energy into kinetic energy and is reversible. As the discharged jet decelerates in the external stagant gas, it recovers its initial enthalpy. When it is desired to determine the discharge rate through a nozzle

Substitution into Eq. (6-118) gives G. G = 1.5 × 105 ×

1.4 × 29

 冪莦 8,314 × 293

0.773 ×  = 337 kg/m2 ⋅ s (1.4 + 1)/2(1.4 − 1) 1.4 − 1 1 +  × 0.7732 2 The exit temperature is found from Eq. (6-116) to be 261.6 K or −11.5°C. The exit velocity is

冢 冢

V = Mc = 0.773 ×

冣

冣

1.4 × 8314 × 261.6  = 250 m/s 冪莦莦 29

Adiabatic Flow with Friction in a Duct of Constant Cross Section Integration of the differential forms of the continuity, momentum, and total energy equations for a perfect gas, assuming a constant friction factor, leads to a tedious set of simultaneous algebraic equations. These may be found in Shapiro (Dynamics and Thermodynamics of Compressible Fluid Flow, vol. I, Ronald Press, New York, 1953) or Zucrow and Hofmann (Gas Dynamics, vol. I, Wiley, New York, 1976). Lapple’s (Trans. AIChE., 39, 395–432 [1943]) widely cited graphical presentation of the solution of these equations contained a subtle error, which was corrected by Levenspiel (AIChE J., 23, 402–403 [1977]). Levenspiel’s graphical solutions are presented in Fig. 6-21. These charts refer to the physical situation illustrated in Fig. 6-22, where a perfect gas discharges from stagnation conditions in a large chamber through an isentropic nozzle followed by a duct of length L. The resistance parameter is N = 4fL/DH, where f = Fanning friction factor and DH = hydraulic diameter. The exit Mach number M2 may not exceed unity. M2 = 1 corresponds to choked flow; sonic conditions may exist only at the pipe exit. The mass velocity G* in the charts is the choked mass flux for an isentropic nozzle given by Eq. (6-118). For a pipe of finite length,

6-24

FLUID AND PARTICLE DYNAMICS

(a)

(b) Design charts for adiabatic flow of gases; (a) useful for finding the allowable pipe length for given flow rate; (b) useful for finding the discharge rate in a given piping system. (From Levenspiel, Am. Inst. Chem. Eng. J., 23, 402 [1977].) FIG. 6-21

the mass flux is less than G* under choking conditions. The curves in Fig. 6-21 become vertical at the choking point, where flow becomes independent of downstream pressure. The equations for nozzle flow, Eqs. (6-114) through (6-118), remain valid for the nozzle section even in the presence of the discharge pipe. Equations (6-116) and (6-120), for the temperature variation, may also be used for the pipe, with M2, p2 replacing M1, p1 since they are valid for adiabatic flow, with or without friction. The graphs in Fig. 6-21 are based on accurate calculations, but are

L p0

p1

p2

p3

D FIG. 6-22

entrance.

Adiabatic compressible flow in a pipe with a well-rounded

FLUID DYNAMICS difficult to interpolate precisely. While they are quite useful for rough estimates, precise calculations are best done using the equations for one-dimensional adiabatic flow with friction, which are suitable for computer programming. Let subscripts 1 and 2 denote two points along a pipe of diameter D, point 2 being downstream of point 1. From a given point in the pipe, where the Mach number is M, the additional length of pipe required to accelerate the flow to sonic velocity (M = 1) is denoted Lmax and may be computed from k+1  M 2 2  4f Lmax 1 − M 2 k + 1 k−1 + ln (6-123) =  1 +  M 2 D kM 2 2k 2

冢

冣

With L = length of pipe between points 1 and 2, the change in Mach number may be computed from

冢

4fL 4f Lmax =  D D

冣 − 冢 D 冣 4f Lmax

1

(6-124) 2

Eqs. (6-116) and (6-113), which are valid for adiabatic flow with friction, may be used to determine the temperature and speed of sound at points 1 and 2. Since the mass flux G = ρv = ρcM is constant, and ρ = PMw /RT, the pressure at point 2 (or 1) can be found from G and the pressure at point 1 (or 2). The additional frictional losses due to pipeline fittings such as elbows may be added to the velocity head loss N = 4 f L/DH using the same velocity head loss values as for incompressible flow. This works well for fittings which do not significantly reduce the channel crosssectional area, but may cause large errors when the flow area is greatly reduced, as, for example, by restricting orifices. Compressible flow across restricting orifices is discussed in Sec. 10 of this Handbook. Similarly, elbows near the exit of a pipeline may choke the flow even though the Mach number is less than unity due to the nonuniform velocity profile in the elbow. For an abrupt contraction rather than rounded nozzle inlet, an additional 0.5 velocity head should be added to N. This is a reasonable approximation for G, but note that it allocates the additional losses to the pipeline, even though they are actually incurred in the entrance. It is an error to include one velocity head exit loss in N. The kinetic energy at the exit is already accounted for in the integration of the balance equations. Example 8: Compressible Flow with Friction Losses Calculate the discharge rate of air to the atmosphere from a reservoir at 106 Pa gauge and 20°C through 10 m of straight 2-in Schedule 40 steel pipe (inside diameter = 0.0525 m), and 3 standard radius, flanged 90° elbows. Assume 0.5 velocity heads lost for the elbows. For commercial steel pipe, with a roughness of 0.046 mm, the friction factor for fully rough flow is about 0.0047, from Eq. (6-38) or Fig. 6-9. It remains to be verified that the Reynolds number is sufficiently large to assume fully rough flow. Assuming an abrupt entrance with 0.5 velocity heads lost, 10 N = 4 × 0.0047 ×  + 0.5 + 3 × 0.5 = 5.6 0.0525 The pressure ratio p3 /p0 is 1.01 × 105 = 0.092  (1 × 106 + 1.01 × 105) From Fig. 6-21b at N = 5.6, p3 /p0 = 0.092 and k = 1.4 for air, the flow is seen to be choked. At the choke point with N = 5.6 the critical pressure ratio p2 /p0 is about 0.25 and G/G* is about 0.48. Equation (6-122) gives G* = 1.101 × 106 ×

2 1.4 × 29 = 2,600 kg/m ⋅ s 莦 冪冢莦莦 冢 1.4 + 1 冣 莦莦莦莦莦莦莦莦 8,314 × 293.15 冣莦 (1.4 + 1)/(1.4 − 1)

and a Reynolds number of 3.6 × 106. Over the entire pipe length the Reynolds number is very large and the fully rough flow friction factor choice was indeed valid.

Once the mass flux G has been determined, Fig. 6-21a or 6-21b can be used to determine the pressure at any point along the pipe, simply by reducing 4fL/DH and computing p2 from the figures, given G, instead of the reverse. Charts for calculation between two points in a pipe with known flow and known pressure at either upstream or downstream locations have been presented by Loeb (Chem. Eng., 76[5], 179–184 [1969]) and for known downstream conditions by Powley (Can. J. Chem. Eng., 36, 241–245 [1958]). Convergent/Divergent Nozzles (De Laval Nozzles) During frictionless adiabatic one-dimensional flow with changing crosssectional area A the following relations are obeyed: dA dp 1 − M 2 dρ dV (6-125)  = 2 (1 − M 2) =   = −(1 − M 2)  M2 A ρV ρ V Equation (6-125) implies that in converging channels, subsonic flows are accelerated and the pressure and density decrease. In diverging channels, subsonic flows are decelerated as the pressure and density increase. In subsonic flow, the converging channels act as nozzles and diverging channels as diffusers. In supersonic flows, the opposite is true. Diverging channels act as nozzles accelerating the flow, while converging channels act as diffusers decelerating the flow. Figure 6-23 shows a converging/diverging nozzle. When p2 /p0 is less than the critical pressure ratio (p*/p0), the flow will be subsonic in the converging portion of the nozzle, sonic at the throat, and supersonic in the diverging portion. At the throat, where the flow is critical and the velocity is sonic, the area is denoted A*. The cross-sectional area and pressure vary with Mach number along the converging/ diverging flow path according to the following equations for isentropic flow of a perfect gas: (k + 1) / 2(k − 1) A 1 2 k−1 (6-126)  =   1 +  M2 A* M k + 1 2

冤

冢

冣冥

k / (k − 1) p0 k−1 (6-127)  = 1 +  M2 p 2 The temperature obeys the adiabatic flow equation for a perfect gas, k−1 T0 (6-128)  = 1 +  M2 T 2 Equation (6-128) does not require frictionless (isentropic) flow. The sonic mass flux through the throat is given by Eq. (6-122). With A set equal to the nozzle exit area, the exit Mach number, pressure, and temperature may be calculated. Only if the exit pressure equals the ambient discharge pressure is the ultimate expansion velocity reached in the nozzle. Expansion will be incomplete if the exit pressure exceeds the ambient discharge pressure; shocks will occur outside the nozzle. If the calculated exit pressure is less than the ambient discharge pressure, the nozzle is overexpanded and compression shocks within the expanding portion will result. The shape of the converging section is a smooth trumpet shape similar to the simple converging nozzle. However, special shapes of the diverging section are required to produce the maximum supersonic exit velocity. Shocks result if the divergence is too rapid and excessive boundary layer friction occurs if the divergence is too shallow. See Liepmann and Roshko (Elements of Gas Dynamics, Wiley, New York, 1957, p. 284). If the nozzle is to be used as a thrust device, the diverg-

冢

冣

2

Multiplying by G/G* = 0.48 yields G = 1,250 kg/m2 ⋅ s. The discharge rate is w = GA = 1,250 × π × 0.05252/4 = 2.7 kg/s. Before accepting this solution, the Reynolds number should be checked. At the pipe exit, the temperature is given by Eq. (6-120) since the flow is choked. Thus, T2 = T* = 244.6 K. The viscosity of air at this temperature is about 1.6 × 10−5 Pa ⋅ s. Then DVρ DG 0.0525 × 1,250 Re =  =  =  = 4.1 × 106 µ µ 1.6 × 10−5 At the beginning of the pipe, the temperature is greater, giving greater viscosity

6-25

FIG. 6-23

Converging/diverging nozzle.

6-26

FLUID AND PARTICLE DYNAMICS

ing section can be conical with a total included angle of 30° (Sutton, Rocket Propulsion Elements, 2d ed., Wiley, New York, 1956). To obtain large exit Mach numbers, slot-shaped rather than axisymmetric nozzles are used. MULTIPHASE FLOW Multiphase flows, even when restricted to simple pipeline geometry, are in general quite complex, and several features may be identified which make them more complicated than single-phase flow. Flow pattern description is not merely an identification of laminar or turbulent flow. The relative quantities of the phases and the topology of the interfaces must be described. Because of phase density differences, vertical flow patterns are different from horizontal flow patterns, and horizontal flows are not generally axisymmetric. Even when phase equilibrium is achieved by good mixing in two-phase flow, the changing equilibrium state as pressure drops with distance, or as heat is added or lost, may require that interphase mass transfer, and changes in the relative amounts of the phases, be considered. Wallis (One-dimensional Two-phase Flow, McGraw-Hill, New York, 1969) and Govier and Aziz present mass, momentum, mechanical energy, and total energy balance equations for two-phase flows. These equations are based on one-dimensional behavior for each phase. Such equations, for the most part, are used as a framework in which to interpret experimental data. Reliable prediction of multiphase flow behavior generally requires use of data or correlations. Two-fluid modeling, in which the full three-dimensional microscopic (partial differential) equations of motion are written for each phase, treating each as a continuum, occupying a volume fraction which is a continuous function of position, is a rapidly developing technique made possible by improved computational methods. For some relatively simple examples not requiring numerical computation, see Pearson (Chem. Engr. Sci., 49, 727–732 [1994]). Constitutive equations for two-fluid models are not yet sufficiently robust for accurate general-purpose two-phase flow computation, but may be quite good for particular classes of flows. Liquids and Gases For cocurrent flow of liquids and gases in vertical (upflow), horizontal, and inclined pipes, a very large literature of experimental and theoretical work has been published, with less work on countercurrent and cocurrent vertical downflow. Much of the effort has been devoted to predicting flow patterns, pressure drop, and volume fractions of the phases, with emphasis on fully developed flow. In practice, many two-phase flows in process plants are not fully developed. The most reliable methods for fully developed gas/liquid flows use mechanistic models to predict flow pattern, and use different pressure drop and void fraction estimation procedures for each flow pattern. Such methods are too lengthy to include here, and are well suited to incorporation into computer programs; commercial codes for gas/liquid pipeline flows are available. Some key references for mechanistic methods for flow pattern transitions and flow regime– specific pressure drop and void fraction methods include Taitel and Dukler (AIChE J., 22, 47–55 [1976]), Barnea, et al. (Int. J. Multiphase Flow, 6, 217–225 [1980]), Barnea (Int. J. Multiphase Flow, 12, 733–744 [1986]), Taitel, Barnea, and Dukler (AIChE J., 26, 345–354 [1980]), Wallis (One-dimensional Two-phase Flow, McGraw-Hill, New York, 1969), and Dukler and Hubbard (Ind. Eng. Chem. Fundam., 14, 337–347 [1975]). For preliminary or approximate calculations, flow pattern maps and flow regime–independent empirical correlations, are simpler and faster to use. Such methods for horizontal and vertical flows are provided in the following. In horizontal pipe, flow patterns for fully developed flow have been reported in numerous studies. Transitions between flow patterns are gradual, and subjective owing to the visual interpretation of individual investigators. In some cases, statistical analysis of pressure fluctuations has been used to distinguish flow patterns. Figure 6-24 (Alves, Chem. Eng. Progr., 50, 449–456 [1954]) shows seven flow patterns for horizontal gas/liquid flow. Bubble flow is prevalent at high ratios of liquid to gas flow rates. The gas is dispersed as bubbles which move at velocity similar to the liquid and tend to concentrate near the top of the pipe at lower liquid velocities. Plug flow describes a pattern in which alternate plugs of gas and liquid move along the upper

Gas/liquid flow patterns in horizontal pipes. (From Alves, Chem. Eng. Progr., 50, 449–456 [1954].)

FIG. 6-24

part of the pipe. In stratified flow, the liquid flows along the bottom of the pipe and the gas flows over a smooth liquid/gas interface. Similar to stratified flow, wavy flow occurs at greater gas velocities and has waves moving in the flow direction. When wave crests are sufficiently high to bridge the pipe, they form frothy slugs which move at much greater than the average liquid velocity. Slug flow can cause severe and sometimes dangerous vibrations in equipment because of impact of the high-velocity slugs against bends or other fittings. Slugs may also flood gas/liquid separation equipment. In annular flow, liquid flows as a thin film along the pipe wall and gas flows in the core. Some liquid is entrained as droplets in the gas core. At very high gas velocities, nearly all the liquid is entrained as small droplets. This pattern is called spray, dispersed, or mist flow. Approximate prediction of flow pattern may be quickly done using flow pattern maps, an example of which is shown in Fig. 6-25 (Baker, Oil Gas J., 53[12], 185–190, 192–195 [1954]). The Baker chart remains widely used; however, for critical calculations the mechanistic model methods referenced previously are generally preferred for their greater accuracy, especially for large pipe diameters and fluids with physical properties different from air/water at atmospheric pressure. In the chart, λ = (ρ′G ρ′L)1/2 1 µ′L ψ =  2 σ′ (ρ′L )

冤

(6-129)

冥

1/3

(6-130)

Flow-pattern regions in cocurrent liquid/gas flow through horizontal pipes. To convert lbm/(ft2 ⋅ s) to kg/(m2 ⋅ s), multiply by 4.8824. (From Baker, Oil Gas J., 53[12], 185–190, 192, 195 [1954].)

FIG. 6-25

FLUID DYNAMICS GL and GG are the liquid and gas mass velocities, µ′L is the ratio of liquid viscosity to water viscosity, ρ′G is the ratio of gas density to air density, ρ′L is the ratio of liquid density to water density, and σ′ is the ratio of liquid surface tension to water surface tension. The reference properties are at 20°C (68°F) and atmospheric pressure, water density 1,000 kg/m3 (62.4 lbm/ft3), air density 1.20 kg/m3 (0.075 lbm/ft3), water viscosity 0.001 Pa ⋅ s, (1.0 cp) and surface tension 0.073 N/m (0.0050 lbf/ft). The empirical parameters λ and ψ provide a crude accounting for physical properties. The Baker chart is dimensionally inconsistent since the dimensional quantity GG / λ is plotted against a dimensionless one, GLλψ/GG, and so must be used with GG in lbm/(ft2 ⋅ s) units on the ordinate. To convert to kg/(m2 ⋅ s), multiply by 4.8824. Rapid approximate predictions of pressure drop for fully developed, incompressible horizontal gas/liquid flow may be made using the method of Lockhart and Martinelli (Chem. Eng. Prog., 45, 39–48 [1949]). First, the pressure drops that would be expected for each of the two phases as if flowing alone in single-phase flow are calculated. The Lockhart-Martinelli parameter X is defined in terms of the ratio of these pressure drops: (∆ p/L) 1/2 X = L (6-131) (∆ p/L)G The two-phase pressure drop may be then be estimated from either of the single-phase pressure drops, using ∆p ∆p (6-132)  = YL  L TP L L

冤

冥

冢 冣

冢 冣

∆p 冢 L 冣

冢 冣

∆p = YG  (6-133) L G where YL and YG are read from Fig. 6-26 as functions of X. The curve labels refer to the flow regime (laminar or turbulent) found for each of the phases flowing alone. The common turbulent-turbulent case is approximated well by 1 20 YL = 1 +  + 2 (6-134) X X Lockhart and Martinelli (ibid.) correlated pressure drop data from pipes 25 mm (1 in) in diameter or less within about 50 percent. In general, the predictions are high for stratified, wavy, and slug flows and low for annular flow. The correlation can be applied to pipe diameters up to about 0.1 m (4 in) with about the same accuracy. or

TP

Parameters for pressure drop in liquid/gas flow through horizontal pipes. (Based on Lockhart and Martinelli, Chem. Engr. Prog., 45, 39 [1949].)

FIG. 6-26

6-27

The volume fraction, sometimes called holdup, of each phase in two-phase flow is generally not equal to its volumetric flow rate fraction, because of velocity differences, or slip, between the phases. For each phase, denoted by subscript i, the relations among superficial velocity Vi, in situ velocity vi, volume fraction Ri, total volumetric flow rate Qi, and pipe area A are (6-135) Qi = Vi A = vi Ri A Vi (6-136) vi =  Ri The slip velocity between gas and liquid is vs = vG − vL . For two-phase gas/liquid flow, RL + RG = 1. A very common mistake in practice is to assume that in situ phase volume fractions are equal to input volume fractions. For fully developed incompressible horizontal gas/liquid flow, a quick estimate for RL may be obtained from Fig. 6-27, as a function of the Lockhart-Martinelli parameter X defined by Eq. (6-131). Indications are that liquid volume fractions may be overpredicted for liquids more viscous than water (Alves, Chem. Eng. Prog., 50, 449–456 [1954]), and underpredicted for pipes larger than 25 mm diameter (Baker, Oil Gas J., 53[12], 185–190, 192–195 [1954]). A method for predicting pressure drop and volume fraction for non-Newtonian fluids in annular flow has been proposed by Eisenberg and Weinberger (AIChE J., 25, 240–245 [1979]). Das, Biswas, and Matra (Can. J. Chem. Eng., 70, 431–437 [1993]) studied holdup in both horizontal and vertical gas/liquid flow with non-Newtonian liquids. Farooqi and Richardson (Trans Inst. Chem. Engrs., 60, 292–305, 323–333 [1982]) developed correlations for holdup and pressure drop for gas/non-Newtonian liquid horizontal flow. They used a modified Lockhart-Martinelli parameter for non-Newtonian liquid holdup. They found that two-phase pressure drop may actually be less than the single-phase liquid pressure drop with shear thinning liquids in laminar flow. Pressure drop data for a 1-in feed tee with the liquid entering the run and gas entering the branch are given by Alves (Chem. Eng. Progr., 50, 449–456 [1954]). Pressure drop and division of two-phase annular flow in a tee are discussed by Fouda and Rhodes (Trans. Inst. Chem. Eng. [London], 52, 354–360 [1974]). Flow through tees can result in unexpected flow splitting. Further reading on gas/liquid flow through tees may be found in Mudde, Groen, and van den Akker (Int. J. Multiphase Flow, 19, 563–573 [1993]); Issa and Oliveira (Computers and Fluids, 23, 347–372 [1993]) and Azzopardi and Smith (Int. J. Multiphase Flow, 18, 861–875 [1992]). Results by Chenoweth and Martin (Pet. Refiner, 34[10], 151–155 [1955]) indicate that single-phase data for fittings and valves can be used in their correlation for two-phase pressure drop. Smith, Murdock, and Applebaum (J. Eng. Power, 99, 343–347 [1977]) evaluated existing correlations for two-phase flow of steam/water and other gas/liquid mixtures through sharp-edged orifices meeting ASTM standards for flow measurement. The correlation of Murdock (J. Basic Eng., 84, 419–433 [1962]) may be used for these orifices. See also Collins and Gacesa (J. Basic Eng., 93, 11–21 [1971]), for measurements with steam and water beyond the limits of this correlation.

Liquid volume fraction in liquid/gas flow through horizontal pipes. (From Lockhart and Martinelli, Eng. Prog., 45, 39 [1949].) FIG. 6-27

6-28

FLUID AND PARTICLE DYNAMICS

For pressure drop and holdup in inclined pipe with upward or downward flow, see Beggs and Brill (J. Pet. Technol., 25, 607–617 [1973]); the mechanistic model methods referenced above may also be applied to inclined pipes. Up to 10° from horizontal, upward pipe inclination has little effect on holdup (Gregory, Can. J. Chem. Eng., 53, 384–388 [1975]). For fully developed incompressible cocurrent upflow of gases and liquids in vertical pipes, a variety of flow pattern terminologies and descriptions have appeared in the literature; some of these have been summarized and compared by Govier, Radford, and Dunn (Can. J. Chem. Eng., 35, 58–70 [1957]). One reasonable classification of patterns is illustrated in Fig. 6-28. In bubble flow, gas is dispersed as bubbles throughout the liquid, but with some tendency to concentrate toward the center of the pipe. In slug flow, the gas forms large Taylor bubbles of diameter nearly equal to the pipe diameter. A thin film of liquid surrounds the Taylor bubble. Between the Taylor bubbles are liquid slugs containing some bubbles. Froth or churn flow is characterized by strong intermittency and intense mixing, with neither phase easily described as continuous or dispersed. There remains disagreement in the literature as to whether churn flow is a real fully developed flow pattern or is an indication of large entry length for developing slug flow (Zao and Dukler, Int. J. Multiphase Flow, 19, 377–383 [1993]; Hewitt and Jayanti, Int. J. Multiphase Flow, 19, 527–529 [1993]). Ripple flow has an upward-moving wavy layer of liquid on the pipe wall; it may be thought of as a transition region to annular, annular mist, or film flow, in which gas flows in the core of the pipe while an annulus of liquid flows up the pipe wall. Some of the liquid is entrained as droplets in the gas core. Mist flow occurs when all the liquid is carried as fine drops in the gas phase; this pattern occurs at high gas velocities, typically 20 to 30 m/s (66 to 98 ft/s). The correlation by Govier, et al. (Can. J. Chem. Eng., 35, 58–70 [1957]), Fig. 6-29, may be used for quick estimate of flow pattern. Slip, or relative velocity between phases, occurs for vertical flow as well as for horizontal. No completely satisfactory, flow regime–independent correlation for volume fraction or holdup exists for vertical flow. Two frequently used flow regime–independent methods are those by Hughmark and Pressburg (AIChE J., 7, 677 [1961]) and Hughmark (Chem. Eng. Prog., 58[4], 62 [April 1962]). Pressure drop in upflow may be calculated by the procedure described in Hughmark (Ind. Eng. Chem. Fundam., 2, 315–321 [1963]). The mechanistic, flow regime–based methods are advisable for critical applications. For upflow in helically coiled tubes, the flow pattern, pressure drop, and holdup can be predicted by the correlations of Banerjee,

FIG. 6-29 Flow-pattern regions in cocurrent liquid/gas flow in upflow through vertical pipes. To convert ft/s to m/s, multiply by 0.3048. (From Govier, Radford, and Dunn, Can. J. Chem. Eng., 35, 58–70 [1957].)

Rhodes, and Scott (Can. J. Chem. Eng., 47, 445–453 [1969]) and Akagawa, Sakaguchi, and Ueda (Bull JSME, 14, 564–571 [1971]). Correlations for flow patterns in downflow in vertical pipe are given by Oshinowo and Charles (Can. J. Chem. Eng., 52, 25–35 [1974]) and Barnea, Shoham, and Taitel (Chem. Eng. Sci., 37, 741–744 [1982]). Use of drift flux theory for void fraction modeling in downflow is presented by Clark and Flemmer (Chem. Eng. Sci., 39, 170–173 [1984]). Downward inclined two-phase flow data and modeling are given by Barnea, Shoham, and Taitel (Chem. Eng. Sci., 37, 735–740 [1982]). Data for downflow in helically coiled tubes are presented by Casper (Chem. Ing. Tech., 42, 349–354 [1970]). The entrance to a drain is flush with a horizontal surface, while the entrance to an overflow pipe is above the horizontal surface. When such pipes do not run full, considerable amounts of gas can be drawn down by the liquid. The amount of gas entrained is a function of pipe diameter, pipe length, and liquid flow rate, as well as the drainpipe outlet boundary condition. Extensive data on air entrainment and liquid head above the entrance as a function of water flow rate for pipe diameters from 43.9 to 148.3 mm (1.7 to 5.8 in) and lengths from about 1.22 to 5.18 m (4.0 to 17.0 ft) are reported by Kalinske (Univ. Iowa Stud. Eng., Bull. 26, pp. 26–40 [1939–1940]). For heads greater than the critical, the pipes will run full with no entrainment. The critical head h for flow of water in drains and overflow pipes is given in Fig. 6-30. Kalinske’s results show little effect of the height of protrusion of overflow pipes when the protrusion height is greater than about one pipe diameter. For conservative design, McDuffie (AIChE J., 23, 37–40 [1977]) recommends the following relation for minimum liquid height to prevent entrainment. h 2 Fr ≤ 1.6  (6-137) D

冢 冣

FIG. 6-28 Flow patterns in cocurrent upward vertical gas/liquid flow. (From Taitel, Barnea, and Dukler, AIChE J., 26, 345–354 [1980]. Reproduced by permission of the American Institute of Chemical Engineers © 1980 AIChE. All rights reserved.)

FIG. 6-30 Critical head for drain and overflow pipes. (From Kalinske, Univ. Iowa Stud. Eng., Bull. 26 [1939–1940].)

FLUID DYNAMICS where the Froude number is defined by VL Fr ⬅  g(苶 ρL苶苶 −苶 ρ苶 D苶 /ρL苶 兹苶 G)苶

(6-138)

where g = acceleration due to gravity VL = liquid velocity in the drain pipe ρL = liquid density ρG = gas density D = pipe inside diameter h = liquid height For additional information, see Simpson (Chem. Eng., 75(6), 192–214 [1968]). A critical Froude number of 0.31 to ensure vented flow is widely cited. Recent results (Thorpe, 3d Int. Conf. Multi-phase Flow, The Hague, Netherlands, 18–20 May 1987, paper K2, and 4th Int. Conf. Multi-phase Flow, Nice, France, 19–21 June 1989, paper K4) show hysteresis, with different critical Froude numbers for flooding and unflooding of drain pipes, and the influence of end effects. Wallis, Crowley, and Hagi (Trans. ASME J. Fluids Eng., 405–413 [ June 1977]) examine the conditions for horizontal discharge pipes to run full. Flashing flow and condensing flow are two examples of multiphase flow with phase change. Flashing flow occurs when pressure drops below the bubble point pressure of a flowing liquid. A frequently used one-dimensional model for flashing flow through nozzles and pipes is the homogeneous equilibrium model which assumes that both phases move at the same in situ velocity, and maintain vapor/ liquid equilibrium. It may be shown that a critical flow condition, analogous to sonic or critical flow during compressible gas flow, is given by the following expression for the mass flux G in terms of the derivative of pressure p with respect to mixture density ρm at constant entropy: ∂p Gcrit = ρm (6-139)  ∂ρm s

冪冢莦莦莦冣

苶p 苶/∂ 苶ρ 苶苶 The corresponding acoustic velocity 兹(∂ m)s苶 is normally much less than the acoustic velocity for gas flow. The mixture density is given in terms of the individual phase densities and the quality (mass flow fraction vapor) x by 1 x 1−x (6-140) =+ ρm ρG ρL Choked and unchoked flow situations arise in pipes and nozzles in the same fashion for homogeneous equilibrium flashing flow as for gas flow. For nozzle flow from stagnation pressure p0 to exit pressure p1, the mass flux is given by p1 dp G2 = −2ρ2m1 (6-141)  p0 ρm The integration is carried out over an isentropic flash path: flashes at constant entropy must be carried out to evaluate ρm as a function of p. Experience shows that isenthalpic flashes provide good approximations unless the liquid mass fraction is very small. Choking occurs when G obtained by Eq. (6-141) goes through a maximum at a value of p1 greater than the external discharge pressure. Equation (6-139) will also be satisfied at that point. In such a case the pressure at the nozzle exit equals the choking pressure and flashing shocks occur outside the nozzle exit. For homogeneous flow in a pipe of diameter D, the differential form of the Bernoulli equation (6-15) rearranges to dp G2 1 dx′ G2 =0 (6-142)  + g dz +  d  + 2f   ρm ρm ρm D ρm2 where x′ is distance along the pipe. Integration over a length L of pipe assuming constant friction factor f yields

冕

冕

−

p

2

p1

ρm dp − g

冕

z2

z1

ρm2 dz

G =  ln (ρm1 /ρm2) + 2f L/D 2

(6-143)

Frictional pipe flow is not isentropic. Strictly speaking, the flashes must be carried out at constant h + V 2/2 + gz, where h is the enthalpy

6-29

per unit mass of the two-phase flashing mixture. The flash calculations are fully coupled with the integration of the Bernoulli equation; the velocity V must be known at every pressure p to evaluate ρm. Computational routines, employing the thermodynamic and material balance features of flowsheet simulators, are the most practical way to carry out such flashing flow calculations, particularly when multicompent systems are involved. Significant simplification arises when the mass fraction liquid is large, for then the effect of the V 2/2 term on the flash splits may be neglected. If elevation effects are also negligible, the flash computations are decoupled from the Bernoulli equation integration. For many horizontal flashing flow calculations, this is satisfactory and the flash computatations may be carried out first, to find ρm as a function of p from p1 to p2, which may then be substituted into Eq. (6-143). With flashes carried out along the appropriate thermodynamic paths, the formalism of Eqs. (6-139) through (6-143) applies to all homogeneous equilibrium compressible flows, including, for example, flashing flow, ideal gas flow, and nonideal gas flow. Equation (6-118), for example, is a special case of Eq. (6-141) where the quality x = 1 and the vapor phase is a perfect gas. Various nonequilibrium and slip flow models have been proposed as improvements on the homogeneous equilibrium flow model. See, for example, Henry and Fauske (Trans. ASME J. Heat Transfer, 179–187 [May 1971]). Nonequilibrium and slip effects both increase computed mass flux for fixed pressure drop, compared to homogeneous equilibrium flow. For flow paths greater than about 100 mm, homogeneous equilibrium behavior appears to be the best assumption (Fischer, et al., Emergency Relief System Design Using DIERS Technology, AIChE, New York [1992]). For shorter flow paths, the best estimate may sometimes be given by linearly interpolating (as a function of length) between frozen flow (constant quality, no flashing) at 0 length and equilibrium flow at 100 mm. In a series of papers by Leung and coworkers (AIChE J., 32, 1743–1746 [1986]; 33, 524–527 [1987]; 34, 688–691 [1988]; J. Loss Prevention Proc. Ind., 2[2], 78–86 [April 1989]; 3(1), 27–32 [January 1990]; Trans. ASME J. Heat Transfer, 112, 524–528, 528–530 [1990]; 113, 269–272 [1991]) approximate techniques have been developed for homogeneous equilibrium calculations based on pseudo–equation of state methods for flashing mixtures. Relatively less work has been done on condensing flows. Slip effects are more important for condensing than for flashing flows. Soliman, Schuster, and Berenson (J. Heat Transfer, 90, 267–276 [1968]) give a model for condensing vapor in horizontal pipe. They assume the condensate flows as an annular ring. The LockhartMartinelli correlation is used for the frictional pressure drop. To this pressure drop is added an acceleration term based on homogeneous flow, equivalent to the G2d(1/ρm) term in Eq. (6-142). Pressure drop is computed by integration of the incremental pressure changes along the length of pipe. For condensing vapor in vertical downflow, in which the liquid flows as a thin annular film, the frictional contribution to the pressure drop may be estimated based on the gas flow alone, using the friction factor plotted in Fig. 6-31, where ReG is the Reynolds number for the gas flowing alone (Bergelin, et al., Proc. Heat Transfer Fluid Mech. Inst., ASME, June 22–24, 1949, pp. 19–28). 2fG′ ρGVG2 dp −  =  (6-144) dz D To this should be added the GG2 d(1/ρG)/dx term to account for velocity change effects. Gases and Solids The flow of gases and solids in horizontal pipe is usually classified as either dilute phase or dense phase flow. Unfortunately, there is no clear dilineation between the two types of flow, and the dense phase description may take on more than one meaning, creating some confusion (Knowlton, et al., Chem. Eng. Progr., 90(4), 44–54 [April 1994]). For dilute phase flow, achieved at low solids-to-gas weight ratios (loadings), and high gas velocities, the solids may be fully suspended and fairly uniformly dispersed over the pipe cross section (homogeneous flow), particularly for low-density or small particle size solids. At lower gas velocities, the solids may bounce along the bottom of the pipe. With higher loadings and lower gas velocities, the particles may settle to the bottom of the pipe, form-

6-30

FLUID AND PARTICLE DYNAMICS

Friction factors for condensing liquid/gas flow downward in vertical pipe. In this correlation Γ/ρL is in ft2/h. To convert ft2/h to m2/s, multiply by 0.00155. (From Bergelin, et al., Proc. Heat Transfer Fluid Mech. Inst., ASME, 1949, p. 19.) FIG. 6-31

ing dunes, with the particles moving from dune to dune. In dense phase conveying, solids tend to concentrate in the lower portion of the pipe at high gas velocity. As gas velocity decreases, the solids may first form dense moving strands, followed by slugs. Discrete plugs of solids may be created intentionally by timed injection of solids, or the plugs may form spontaneously. Eventually the pipe may become blocked. For more information on flow patterns, see Coulson and Richardson (Chemical Engineering, vol. 2, 2d ed., Pergamon, New York, 1968, p. 583); Korn (Chem. Eng., 57[3], 108–111 [1950]); Patterson ( J. Eng. Power, 81, 43–54 [1959]); Wen and Simons (AIChE J., 5, 263–267 [1959]); and Knowlton, et al. (Chem. Eng. Progr., 90[4], 44–54 [April 1994]). For the minimum velocity required to prevent formation of dunes or settled beds in horizontal flow, some data are given by Zenz (Ind. Eng. Chem. Fundam., 3, 65–75 [1964]), who presented a correlation for the minimum velocity required to keep particles from depositing on the bottom of the pipe. This rather tedious estimation procedure may also be found in Govier and Aziz, who provide additional references and discussion on transition velocities. In practice, the actual conveying velocities used in systems with loadings less than 10 are generally over 15 m/s, (49 ft/s) while for high loadings (>20) they are generally less than 7.5 m/s (24.6 ft/s) and are roughly twice the actual solids velocity (Wen and Simons, AIChE J., 5, 263–267 [1959]). Total pressure drop for horizontal gas/solid flow includes acceleration effects at the entrance to the pipe and frictional effects beyond the entrance region. A great number of correlations for pressure gradient are available, none of which is applicable to all flow regimes. Govier and Aziz review many of these and provide recommendations on when to use them. For upflow of gases and solids in vertical pipes, the minimum conveying velocity for low loadings may be estimated as twice the terminal settling velocity of the largest particles. Equations for terminal settling velocity are found in the “Particle Dynamics” subsection, following. Choking occurs as the velocity is dropped below the minimum conveying velocity and the solids are no longer transported, collapsing into solid plugs (Knowlton, et al., Chem. Eng. Progr., 90[4], 44–54 [April 1994]). See Smith (Chem. Eng. Sci., 33, 745–749 [1978]) for an equation to predict the onset of choking. Total pressure drop for vertical upflow of gases and solids includes acceleration and frictional affects also found in horizontal flow, plus potential energy or hydrostatic effects. Govier and Aziz review many of the pressure drop calculation methods and provide recommendations for their use. See also Yang (AIChE J., 24, 548–552 [1978]). Drag reduction has been reported for low loadings of small diameter particles (<60 µm diameter), ascribed to damping of turbulence near the wall (Rossettia and Pfeffer, AIChE J., 18, 31–39 [1972]). For dense phase transport in vertical pipes of small diameter, see

Sandy, Daubert, and Jones (Chem. Eng. Prog., 66, Symp. Ser., 105, 133–142 [1970]). The flow of bulk solids through restrictions and bins is discussed in symposium articles (J. Eng. Ind., 91[2] [1969]) and by Stepanoff (Gravity Flow of Bulk Solids and Transportation of Solids in Suspension, Wiley, New York, 1969). Some problems encountered in discharge from bins include (Knowlton, et al., Chem. Eng. Progr., 90[4], 44–54 [April 1994]) flow stoppage due to ratholing or arching, segregation of fine and coarse particles, flooding upon collapse of ratholes, and poor residence time distribution when funnel flow occurs. Solid/liquid or slurry flow may be divided roughly into two categories based on settling behavior (see Etchells in Shamlou, Processing of Solid-Liquid Suspensions, Chap. 12, Butterworth-Heinemann, Oxford, 1993). Nonsettling slurries are made up of very fine, highly concentrated, or neutrally buoyant particles. These slurries are normally treated as pseudohomogeneous fluids. They may be quite viscous and are frequently non-Newtonian. Slurries of particles that tend to settle out rapidly are called settling slurries or fast-settling slurries. While in some cases positively buoyant solids are encountered, the present discussion will focus on solids which are more dense than the liquid. For horizontal flow of fast-settling slurries, the following rough description may be made (Govier and Aziz). Ultrafine particles, 10 µm or smaller, are generally fully syspended and the particle distributions are not influenced by gravity. Fine particles 10 to 100 µm (3.3 × 10−5 to 33 × 10−5 ft) are usually fully suspended, but gravity causes concentration gradients. Medium-size particles, 100 to 1000 µm, may be fully suspended at high velocity, but often form a moving deposit on the bottom of the pipe. Coarse particles, 1,000 to 10,000 µm, (0.0033 to .033 ft), are seldom fully suspended and are usually conveyed as a moving deposit. Ultracoarse particles larger than 10,000 µm (0.033 ft) are not suspended at normal velocities unless they are unusually light. Figure 6-32, taken from Govier and Aziz, schematically indicates four flow pattern regions superimposed on a plot of pressure gradient vs. mixture velocity VM = VL + VS = (QL + QS)/A where VL and VS are the superficial liquid and solid velocities, QL and QS are liquid and solid volumetric flow rates, and A is the pipe cross-sectional area. VM4 is the transition velocity above which a bed exists in the bottom of the pipe, part of which is stationary and part of which moves by saltation, with the upper particles tumbling and bouncing over one another, often with formation of dunes. With a broad particle-size distribution, the finer particles may be fully suspended. Near VM4, the pressure gra-

FIG. 6-32 Flow pattern regimes and pressure gradients in horizontal slurry flow. (From Govier and Aziz, The Flow of Complex Mixtures in Pipes, Van Nostrand Reinhold, New York, 1972.)

FLUID DYNAMICS dient rapidly increases as VM decreases. Above VM3, the entire bed moves. Above VM2, the solids are fully suspended; that is, there is no deposit, moving or stationary, on the bottom of the pipe. However, the concentration distribution of solids is asymmetric. This flow pattern is the most frequently used for fast-settling slurry transport. Typical mixture velocities are in the range of 1 to 3 m/s (3.3 to 9.8 ft/s). The minimum in the pressure gradient is found to be near VM2. Above VM1, the particles are symmetrically distributed, and the pressure gradient curve is nearly parallel to that for the liquid by itself. The most important transition velocity, often regarded as the minimum transport or conveying velocity for settling slurries, is VM2. The Durand equation (Durand, Minnesota Int. Hydraulics Conf., Proc., 89, Int. Assoc. for Hydraulic Research [1953]; Durand and Condolios, Proc. Colloq. On the Hyd. Transport of Solids in Pipes, Nat. Coal Board [UK], Paper IV, 39–35 [1952]) gives the minimum transport velocity as VM2 = FL[2gD(s − 1)]0.5 where

(6-145)

g = acceleration of gravity D = pipe diameter s = ρS /ρL = ratio of solid to liquid density FL = a factor influenced by particle size and concentration

Probably FL is a function of particle Reynolds number and concentration, but Fig. 6-33 gives Durand’s empirical correlation for FL as a function of particle diameter and the input, feed volume fraction solids, CS = QS /(QS + QL). The form of Eq. (6-145) may be derived from turbulence theory, as shown by Davies (Chem. Eng. Sci., 42, 1667–1670 [1987]). No single correlation for pressure drop in horizontal solid/liquid flow has been found satisfactory for all particle sizes, densities, concentrations, and pipe sizes. However, with reference to Fig. 6-32, the following simplifications may be considered. The minimum pressure gradient occurs near VM2 and for conservative purposes it is generally desirable to exceed VM2. When VM2 is exceeded, a rough guide for pressure drop is 25 percent greater than that calculated assuming that the slurry behaves as a psuedohomogeneous fluid with the density of the mixture and the viscosity of the liquid. Above the transition velocity to symmetric suspension, VM1, the pressure drop closely approaches the pseuodohomogeneous pressure drop. The following

Durand factor for minimum suspension velocity. (From Govier and Aziz, The Flow of Complex Mixtures in Pipes, Van Nostrand Reinhold, New York, 1972.)

FIG. 6-33

6-31

correlation by Spells (Trans. Inst. Chem. Eng. [London], 33, 79–84 [1955]) may be used for VM1. DVM1ρM 0.775 2 = 0.075  gDS(s − 1) (6-146) VM1 µ where D = pipe diameter DS = particle diameter (such that 85 percent by weight of particles are smaller than DS ρM = the slurry mixture density µ = liquid viscosity s = ρS /ρL = ratio of solid to liquid density

冢

冣

Between VM2 and VM1 the concentration of solids gradually becomes more uniform in the vertical direction. This transition has been modeled by several authors as a concentration gradient where turbulent diffusion balances gravitational settling. See, for example, Karabelas (AIChE J., 23, 426–434 [1977]). Published correlations for pressure drop are frequently very complicated and tedious to use, may not offer significant accuracy advantages over the simple guide given here, and many of them are applicable only for velocities above VM2. One which does include the effect of sliding beds is due to Gaessler (Doctoral Dissertation, Technische Hochshule, Karlsruhe, Germany [1967]; reproduced by Govier and Aziz, pp. 668–669). Turian and Yuan (AIChE J., 23, 232–243 [1977]; see also Turian and Oroskar, AIChE J., 24, 1144 [1978]) segregated a large body of data into four flow regime groups and developed empirical correlations for predicting pressure drop in each flow regime. Pressure drop data for the flow of paper stock in pipes are given in the data section of Standards of the Hydraulic Institute (Hydraulic Institute, 1965). The flow behavior of fiber suspensions is discussed by Bobkowicz and Gauvin (Chem. Eng. Sci., 22, 229–241 [1967]), Bugliarello and Daily (TAPPI, 44, 881–893 [1961]), and Daily and Bugliarello (TAPPI, 44, 497–512 [1961]). In vertical flow of fast-settling slurries, the in situ concentration of solids with density greater than the liquid will exceed the feed concentration C = QS /(QS + QL) for upflow and will be smaller than C for downflow. This results from slip between the phases. The slip velocity, the difference between the in situ average velocities of the two phases, is roughly equal to the terminal settling velocity of the solids in the liquid. Specification of the slip velocity for a pipe of a given diameter, along with the phase flow rates, allows calculation of in situ volume fractions, average velocities, and holdup ratios by simple material balances. Slip velocity may be affected by particle concentration and by turbulence conditions in the liquid. Drift-flux theory, a framework incorporating certain functional forms for empirical expressions for slip velocity, is described by Wallis (One-Dimensional Two-Phase Flow, McGraw-Hill, New York, 1969). Minimum transport velocity for upflow for design purposes is usually taken as twice the particle settling velocity. Pressure drop in vertical pipe flow includes the effects of kinetic and potential energy (elevation) changes and friction. Rose and Duckworth (The Engineer, 227[5,903], 392 [1969]; 227[5,904], 430 [1969]; 227[5,905], 478 [1969]; see also Govier and Aziz, pp. 487–493) have developed a calculation procedure including all these effects, which may be applied not only to vertical solid/liquid flow, but also to gas/solid flow and to horizontal flow. For fast-settling slurries, ensuring conveyance is usually the key design issue while pressure drop is somewhat less important. For nonsettling slurries conveyance is not an issue, because the particles do not separate from the liquid. Here, viscous and rheological behavior, which control pressure drop, take on critical importance. Fine particles, often at high concentration, form nonsettling slurries for which useful design equations can be developed by treating them as homogeneous fluids. These fluids are usually very viscous and often non-Newtonian. Shear-thinning and Bingham plastic behavior are common; dilatancy is sometimes observed. Rheology of such fluids must in general be empirically determined, although theoretical results are available for some very limited circumstances. Further discussion of both fast-settling and nonsettling slurries may be found in Shook (in Shamlou, Processing of Solid-Liquid Suspensions, Chap. 11, Butterworth-Heinemann, Oxford, 1993).

6-32

FLUID AND PARTICLE DYNAMICS

FLUID DISTRIBUTION Uniform fluid distribution is essential for efficient operation of chemical-processing equipment such as contactors, reactors, mixers, burners, heat exchangers, extrusion dies, and textile-spinning chimneys. To obtain optimum distribution, proper consideration must be given to flow behavior in the distributor, flow conditions upstream and downstream of the distributor, and the distribution requirements of the equipment. Even though the principles of fluid distribution have been well developed for more than three decades, they are frequently overlooked by equipment designers, and a significant fraction of process equipment needlessly suffers from maldistribution. In this subsection, guides for the design of various types of fluid distributors, taking into account only the flow behavior within the distributor, are given. Perforated-Pipe Distributors The simple perforated pipe or sparger (Fig. 6-34) is a common type of distributor. As shown, the flow distribution is uniform; this is the case in which pressure recovery due to kinetic energy or momentum changes, frictional pressure drop along the length of the pipe, and pressure drop across the outlet holes have been properly considered. In typical turbulent flow applications, inertial effects associated with velocity changes may dominate frictional losses in determining the pressure distribution along the pipe, unless the length between orifices is large. Application of the momentum or mechanical energy equations in such a case shows that the pressure inside the pipe increases with distance from the entrance of the pipe. If the outlet holes are uniform in size and spacing, the discharge flow will be biased toward the closed end. Disturbances upstream of the distributor, such as pipe bends, may increase or decrease the flow to the holes at the beginning of the distributor. When frictional pressure drop dominates the inertial pressure recovery, the distribution is biased toward the feed end of the distributor.

FIG. 6-34

Perforated-pipe distributor.

For turbulent flow, with roughly uniform distribution, assuming a constant friction factor, the combined effect of friction and inertial (momentum) pressure recovery is given by 4fL ρVi2 ∆p =  − 2K  (discharge manifolds) (6-147) 3D 2 where ∆p = net pressure drop over the length of the distributor L = pipe length D = pipe diameter f = Fanning friction factor Vi = distributor inlet velocity

冢

冣

The factor K would be 1 in the case of full momentum recovery, or 0.5 in the case of negligible viscous losses in the portion of flow which remains in the pipe after the flow divides at a takeoff point (Denn, pp. 126–127). Experimental data (Van der Hegge Zijnen, Appl. Sci. Res., A3, 144–162 [1951–1953]; and Bailey, J. Mech. Eng. Sci., 17, 338–347 [1975]), while scattered, show that K is probably close to 0.5 for discharge manifolds. For inertially dominated flows, ∆p will be negative. For return manifolds the recovery factor K is close to 1.0, and the pressure drop between the first hole and the exit is given by ρVe2 4fL ∆p =  + 2K  (return manifolds) (6-148) 3D 2 where Ve is the pipe exit velocity. One means to obtain a desired uniform distribution is to make the average pressure drop across the holes ∆ po large compared to the pressure variation over the length of pipe ∆ p. Then, the relative variation in pressure drop across the various holes will be small, and so will be the variation in flow. When the area of an individual hole is

冢

冣

small compared to the cross-sectional area of the pipe, hole pressure drop may be expressed in terms of the discharge coefficient Co and the velocity across the hole Vo as 1 ρV o2 (6-149) ∆po =   C o2 2 Provided Co is the same for all the holes, the percent maldistribution, defined as the percentage variation in flow between the first and last holes, may be estimated reasonably well for small maldistribution by (Senecal, Ind. Eng. Chem., 49, 993–997 [1957]) ∆p − |∆p|  冣 冢 冪莦 ∆p

Percent maldistribution = 100 1 −

o

(6-150)

o

This equation shows that for 5 percent maldistribution, the pressure drop across the holes should be about 10 times the pressure drop over the length of the pipe. For discharge manifolds with K = 0.5 in Eq. (6-147), and with 4f L/3D << 1, the pressure drop across the holes should be 10 times the inlet velocity head, ρVi2/2 for 5 percent maldistribution. This leads to a simple design equation. Discharge manifolds, 4f L/3D << 1, 5% maldistribution: Vo Ap  =  = 兹1苶0苶Co Vi Ao

(6-151)

Here Ap = pipe cross-sectional area and Ao is the total hole area of the distributor. Use of large hole velocity to pipe velocity ratios promotes perpendicular discharge streams. In practice, there are many cases where the 4f L/3D term will be less than unity but not close to zero. In such cases, Eq. (6-151) will be conservative, while Eqs. (6-147), (6-149), and (6-150) will give more accurate design calculations. In cases where 4f L/(3D) > 2, friction effects are large enough to render Eq. (6-151) nonconservative. When significant variations in f along the length of the distributor occur, calculations should be made by dividing the distributor into small enough sections that constant f may be assumed over each section. For return manifolds with K = 1.0 and 4f L/(3D) << 1, 5 percent maldistribution is achieved when hole pressure drop is 20 times the pipe exit velocity head. Return manifolds, 4f L/3D << 1, 5% maldistribution: Vo Ap (6-152)  =  = 兹苶2苶0Co Ve Ao When 4f L/3D is not negligible, Eq. (6-152) is not conservative and Eqs. (6-148), (6-149), and (6-150) should be used. One common misconception is that good distribution is always provided by high pressure drop, so that increasing flow rate improves distribution by increasing pressure drop. Conversely, it is mistakenly believed that turndown of flow through a perforated pipe designed using Eqs. (6-151) and (6-152) will cause maldistribution. However, when the distribution is nearly uniform, decreasing the flow rate decreases ∆p and ∆po in the same proportion, and Eqs. (6-151) and (6-152) are still satisfied, preserving good distribution independent of flow rate, as long as friction losses remain small compared to inertial (velocity head change) effects. Conversely, increasing the flow rate through a distributor with severe maldistribution will not generally produce good distribution. Often, the pressure drop required for design flow rate is unacceptably large for a distributor pipe designed for uniform velocity through uniformly sized and spaced orifices. Several measures may be taken in such situations. These include the following: 1. Taper the diameter of the distributor pipe so that the pipe velocity and velocity head remain constant along the pipe, thus substantially reducing pressure variation in the pipe. 2. Vary the hole size and/or the spacing between holes to compensate for the pressure variation along the pipe. This method may be sensitive to flow rate and a distributor optimized for one flow rate may suffer increased maldistribution as flow rate deviates from design rate. 3. Feed or withdraw from both ends, reducing the pipe flow velocity head and required hole pressure drop by a factor of 4.

FLUID DYNAMICS

6-33

The orifice discharge coefficient Co is usually taken to be about 0.62. However, Co is dependent on the ratio of hole diameter to pipe diameter, pipe wall thickness to hole diameter ratio, and pipe velocity to hole velocity ratio. As long as all these are small, the coefficient 0.62 is generally adequate. Example 9: Pipe Distributor A 3-in schedule 40 (inside diameter 7.793 cm) pipe is to be used as a distributor for a flow of 0.010 m3/s of water (ρ = 1,000 kg/m3, µ = 0.001 Pa ⋅ s). The pipe is 0.7 m long and is to have 10 holes of uniform diameter and spacing along the length of the pipe. The distributor pipe is submerged. Calculate the required hole size to limit maldistribution to 5 percent, and estimate the pressure drop across the distributor. The inlet velocity computed from Vi = Q/A = 4Q/(πD2) is 2.10 m/s, and the inlet Reynolds number is DVi ρ 0.07793 × 2.10 × 1000 Re =  =  = 1.64 × 105 µ 0.001 For commercial pipe with roughness
= 0.046 mm, the friction factor is about 0.0043. Approaching the last hole, the flow rate, velocity and Reynolds number are about one-tenth their inlet values. At Re = 16,400 the friction factor f is about 0.0070. Using an average value of f = 0.0057 over the length of the pipe, 4 fL/3D is 0.068 and may reasonably be neglected so that Eq. (6-151) may be used. With Co = 0.62, Vo A p  =  = 兹1苶0苶Co = 兹1苶0苶 × 0.62 = 1.96 Vi Ao With pipe cross-sectional area Ap = 0.00477 m2, the total hole area is 0.00477/1.96 = 0.00243 m2. The area and diameter of each hole are then 0.00243/10 = 0.000243 m2 and 1.76 cm. With Vo /Vi = 1.96, the hole velocity is 1.96 × 2.10 = 4.12 m/s and the pressure drop across the holes is obtained from Eq. (6-149). 1000(4.12)2 1 ρVo2 1 ∆ po =   = 2 ×  = 22,100 Pa Co2 2 0.62 2 Since the hole pressure drop is 10 times the pressure variation in the pipe, the total pressure drop from the inlet of the distributor may be taken as approximately 22,100 Pa. Further detailed information on pipe distributors may be found in Senecal (Ind. Eng. Chem., 49, 993–997 [1957]). Much of the information on tapered manifold design has appeared in the pulp and paper literature (Spengos and Kaiser, TAPPI, 46[3], 195–200 [1963]; Madeley, Paper Technology, 9[1], 35–39 [1968]; Mardon, et al., TAPPI, 46[3], 172–187 [1963]; Mardon, et al., Pulp and Paper Magazine of Canada, 72[11], 76–81 [November 1971]; Trufitt, TAPPI, 58[11], 144–145 [1975]). Slot Distributors These are generally used in sheeting dies for extrusion of films and coatings and in air knives for control of thickness of a material applied to a moving sheet. A simple slotted pipe for turbulent flow conditions may give severe maldistribution because of nonuniform discharge velocity, but also because this type of design does not readily give perpendicular discharge (Koestel and Tuve, Heat. Piping Air Cond., 20[1], 153–157 [1948]; Senecal, Ind. Eng. Chem., 49, 993–997 [1957]; Koestel and Young, Heat. Piping Air Cond., 23[7], 111–115 [1951]). For slots in tapered ducts where the duct cross-sectional area decreases linearly to zero at the far end, the discharge angle will be constant along the length of the duct (Koestel and Young, ibid.). One way to ensure an almost perpendicular discharge is to have the ratio of the area of the slot to the cross-sectional area of the pipe equal to or less than 0.1. As in the case of perforatedpipe distributors, pressure variation within the slot manifold and pressure drop across the slot must be carefully considered. In practice, the following methods may be used to keep the diameter of the pipe to a minimum consistent with good performance (Senecal, Ind. Eng. Chem., 49, 993–997 [1957]): 1. Feed from both ends. 2. Modify the cross-sectional design (Fig. 6-35); the slot is thus farther away from the influence of feed-stream velocity. 3. Increase pressure drop across the slot; this can be accomplished by lengthening the lips (Fig. 6-35). 4. Use screens (Fig. 6-35) to increase overall pressure drop across the slot. Design considerations for air knives are discussed by Senecal (ibid.). Design procedures for extrusion dies when the flow is laminar,

FIG. 6-35

Modified slot distributor.

as with highly viscous fluids, are presented by Bernhardt (Processing of Thermoplastic Materials, Rheinhold, New York, 1959, pp. 248– 281). Turning Vanes In applications such as ventilation, the discharge profile from slots can be improved by turning vanes. The tapered duct is the most amenable for turning vanes because the discharge angle remains constant. One way of installing the vanes is shown in Fig. 6-36. The vanes should have a depth twice the spacing (Heating, Ventilating, Air Conditioning Guide, vol. 38, American Society of Heating, Refrigerating and Air-Conditioning Engineers, 1960, pp. 282–283) and a curvature at the upstream end of the vanes of a circular arc which is tangent to the discharge angle θ of a slot without vanes and perpendicular at the downstream or discharge end of the vanes (Koestel and Young, Heat. Piping Air Cond., 23[7], 111–115 [1951]). Angle θ can be estimated from Cd As cot θ =  (6-153) Ad where A s = slot area A d = duct cross-sectional area at upstream end Cd = discharge coefficient of slot Vanes may be used to improve velocity distribution and reduce frictional loss in bends, when the ratio of bend turning radius to pipe diameter is less than 1.0. For a miter bend with low-velocity flows, simple circular arcs (Fig. 6-37) can be used, and with high-velocity flows, vanes of special airfoil shapes are required. For additional details and references, see Ower and Pankhurst (The Measurement of Air Flow, Pergamon, New York, 1977, p. 102); Pankhurst and Holder (Wind-Tunnel Technique, Pitman, London, 1952, pp. 92–93); Rouse (Engineering Hydraulics, Wiley, New York, 1950, pp. 399–401); and Jorgensen (Fan Engineering, 7th ed., Buffalo Forge Co., Buffalo, 1970, pp. 111, 117, 118). Perforated Plates and Screens A nonuniform velocity profile in turbulent flow through channels or process equipment can be smoothed out to any desired degree by adding sufficient uniform resistance, such as perforated plates or screens across the flow channel, as shown in Fig. 6-38. Stoker (Ind. Eng. Chem., 38, 622–624 [1946]) provides the following equation for the effect of a uniform resistance on velocity profile:

冪莦莦

(V1,max /V)2 + α2 − α1 + α2K V2,max (6-154)  =  V 1+K Here, V is the area average velocity, K is the number of velocity heads of pressure drop provided by the uniform resistance, ∆ p = KρV 2/2, and α is the velocity profile factor used in the mechanical energy bal-

FIG. 6-36 Turning vanes in a slot distributor.

6-34

FLUID AND PARTICLE DYNAMICS entrance to a vessel, it is sometimes necessary to diffuse the momentum of the inlet jet discharging from the feed pipe in order to produce a more uniform velocity profile within the vessel. Methods for this application exist, but remain largely in the domain of proprietary, commercial design. FLUID MIXING

FIG. 6-37

Miter bend with vanes.

ance, Eq. (6-13). It is ratio of the area average of the cube of the velocity, to the cube of the area average velocity V. The shape of the exit velocity profile appears twice in Eq. (6-154), in V2,max / V and α2. Typically, K is on the order of 10, and the desired exit velocity profile is fairly uniform so that α2 ∼ 1.0 may be appropriate. Downstream of the resistance, the velocity profile will gradually reestablish the fully developed profile characteristic of the Reynolds number and channel shape. The screen or perforated plate open area required to produce the resistance K may be computed from Eqs. (6-107) or (6-111). Screens and other flow restrictions may also be used to suppress stream swirl and turbulence (Loehrke and Nagib, J. Fluids Eng., 98, 342–353 [1976]). Contraction of the channel, as in a venturi, provides further reduction in turbulence level and flow nonuniformity. Beds of Solids A suitable depth of solids can be used as a fluid distributor. As for other types of distribution devices, a pressure drop of 10 velocity heads is typically used, here based on the superficial velocity through the bed. There are several substantial disadvantages to use of particle beds for flow distribution. Heterogeneity of the bed may actually worsen rather than improve distribution. In general, uniform flow may be found only downstream of the point in the bed where sufficient pressure drop has occurred to produce uniform flow. Therefore, inefficiency results when the bed also serves reaction or mass transfer functions, as in catalysts, adsorbents, or tower packings for gas/liquid contacting, since portions of the bed are bypassed. In the case of trickle flow of liquid downward through column packings, inlet distribution is critical since the bed itself is relatively ineffective in distributing the liquid. Maldistribution of flow through packed beds also arises when the ratio of bed diameter to particle size is less than 10 to 30. Other Flow Straightening Devices Other devices designed to produce uniform velocity or reduce swirl, sometimes with reduced pressure drop, are available. These include both commercial devices of proprietary design and devices discussed in the literature. For pipeline flows, see the references under flow inverters and static mixing elements previously discussed in the “Incompressible Flow in Pipes and Channels” subsection. For large area changes, as at the

FIG. 6-38

Smoothing out a nonuniform profile in a channel.

Mixing of fluids is a discipline of fluid mechanics. Fluid motion is used to accelerate the otherwise slow processes of diffusion and conduction to bring about uniformity of concentration and temperature, blend materials, facilitate chemical reactions, bring about intimate contact of multiple phases, and so on. As the subject is too broad to cover fully, only a brief introduction and some references for further information are given here. Several texts are available. These include Harnby, Edwards, and Nienow (Mixing in the Process Industries, 2d ed., Butterworths, London, 1992), Oldshue (Fluid Mixing Technology, McGraw-Hill, New York, 1983), Tatterson (Fluid Mixing and Gas Dispersion in Agitated Tanks, McGraw-Hill, New York, 1991), Uhl and Gray (Mixing, vols. I–III, Academic, New York, 1966, 1967, 1986), and Nagata (Mixing: Principles and Applications, Wiley, New York, 1975). A good overview of stirred tank agitation is given in the series of articles from Chemical Engineering (110–114, Dec. 8, 1975; 139–145, Jan. 5, 1976; 93–100, Feb. 2, 1976; 102–110, Apr. 26, 1976; 144–150, May 24, 1976; 141– 148, July 19, 1976; 89–94, Aug. 2, 1976; 101–108, Aug. 30, 1976; 109– 112, Sept. 27, 1976; 119–126, Oct. 25, 1976; 127–133, Nov. 8, 1976). Process mixing is commonly carried out in pipeline and vessel geometries. The terms radial mixing and axial mixing are commonly used. Axial mixing refers to mixing of materials which pass a given point at different times, and thus leads to backmixing. For example, backmixing or axial mixing occurs in stirred tanks where fluid elements entering the tank at different times are intermingled. Mixing of elements initially at different axial positions in a pipeline is axial mixing. Radial mixing occurs between fluid elements passing a given point at the same time, as, for example, between fluids mixing in a pipeline tee. Turbulent flow, by means of the chaotic eddy motion associated with velocity fluctuation, is conducive to rapid mixing and, therefore, is the preferred flow regime for mixing. Laminar mixing is carried out when high viscosity makes turbulent flow impractical. Stirred Tank Agitation Turbine impeller agitators, of a variety of shapes, are used for stirred tanks, predominantly in turbulent flow. Figure 6-39 shows typical stirred tank configurations and timeaveraged flow patterns for axial flow and radial flow impellers. In order to prevent formation of a vortex, four vertical baffles are normally installed. These cause top-to-bottom mixing and prevent mixing-ineffective swirling motion. For a given impeller and tank geometry, the impeller Reynolds number determines the flow pattern in the tank: D2Nρ ReI =  (6-155) µ where D = impeller diameter, N = rotational speed, and ρ and µ are the liquid density and viscosity. Rotational speed N is typically reported in revolutions per minute, or revolutions per second in SI units. Radians per second are almost never used. Typically, ReI > 104 is required for fully turbulent conditions throughout the tank. A wide transition region between laminar and turbulent flow occurs over the range 10 < ReI < 104. The power P drawn by the impeller is made dimensionless in a group called the power number: P NP =  (6-156) ρN 3D5 Figure 6-40 shows power number vs. impeller Reynolds number for a typical configuration. The similarity to the friction factor vs. Reynolds number behavior for pipe flow is significant. In laminar flow, the power number is inversely proportional to Reynolds number, reflecting the dominance of viscous forces over inertial forces. In turbulent flow, where inertial forces dominate, the power number is nearly constant.

FLUID DYNAMICS Nb = tb N

FIG. 6-39 Typical stirred tank configurations, showing time-averaged flow patterns for axial flow and radial flow impellers. (From Oldshue, Fluid Mixing Technology, McGraw-Hill, New York, 1983.)

Impellers are sometimes viewed as pumping devices; the total volumetric flow rate Q discharged by an impeller is made dimensionless in a pumping number: Q NQ = 3 (6-157) ND Blend time tb, the time required to achieve a specified maximum standard deviation of concentration after injection of a tracer into a stirred tank, is made dimensionless by multiplying by the impeller rotational speed:

6-35 (6-158)

Dimensionless pumping number and blend time are independent of Reynolds number under fully turbulent conditions. The magnitude of concentration fluctuations from the final well-mixed value in batch mixing decays exponentially with time. The design of mixing equipment depends on the desired process result. There is often a tradeoff between operating cost, which depends mainly on power, and capital cost, which depends on agitator size and torque. For some applications bulk flow throughout the vessel is desired, while for others high local turbulence intensity is required. Multiphase systems introduce such design criteria as solids suspension and gas dispersion. In very viscous systems, helical ribbons, extruders, and other specialized equipment types are favored over turbine agitators. Pipeline Mixing Mixing may be carried out with mixing tees, inline or motionless mixing elements, or in empty pipe. In the latter case, large pipe lengths may be required to obtain adequate mixing. Coaxially injected streams require lengths on the order of 100 pipe diameters. Coaxial mixing in turbulent single-phase flow is characterized by the turbulent diffusivity (eddy diffusivity) DE which determines the rate of radial mixing. Davies (Turbulence Phenomena, Academic, New York, 1972) provides an equation for DE which may be rewritten as DE ∼ 0.015DVRe−0.125 where

(6-159)

D = pipe diameter V = average velocity Re = pipe Reynolds number, DVρ/µ ρ = density µ = viscosity

Properly designed tee mixers, with due consideration given to main stream and injected stream momentum, are capable of producing high degrees of uniformity in just a few diameters. Forney (Jet Injection for Optimum Pipeline Mixing, in “Encyclopedia of Fluid Mechanics,” vol. 2., Chap. 25, Gulf Publishing, 1986) provides a thorough discussion of tee mixing. Inline or motionless mixers are generally of

FIG. 6-40 Dimensionless power number in stirred tanks. (Reprinted with permission from Bates, Fondy, and Corpstein, Ind. Eng. Chem. Process Design Develop., 2, 310 [1963].)

6-36

FLUID AND PARTICLE DYNAMICS TUBE BANKS Pressure drop across tube banks may not be correlated by means of a single, simple friction factor—Reynolds number curve, owing to the variety of tube configurations and spacings encountered, two of which are shown in Fig. 6-41. Several investigators have allowed for configuration and spacing by incorporating spacing factors in their friction factor expressions or by using multiple friction factor plots. Commercial computer codes for heat-exchanger design are available which include features for estimating pressure drop across tube banks. Turbulent Flow The correlation by Grimison (Trans. ASME, 59, 583–594 [1937]) is recommended for predicting pressure drop for turbulent flow (Re ≥ 2,000) across staggered or in-line tube banks for tube spacings [(a/Dt), (b/Dt)] ranging from 1.25 to 3.0. The pressure drop is given by

FIG. 6-41

2 4 f NrρVmax ∆p =  2

Tube-bank configurations.

where proprietary commercial design, and may be selected for viscous or turbulent, single or multiphase mixing applications. They substantially reduce required pipe length for mixing.

(6-160)

f = friction factor Nr = number of rows of tubes in the direction of flow ρ = fluid density Vmax = fluid velocity through the minimum area available for flow.

Upper chart: Friction factors for staggered tube banks with minimum fluid flow area in transverse openings. Lower chart: Friction factors for staggered tube banks with minimum fluid flow area in diagonal openings. (From Grimison, Trans. ASME, 59, 583 [1937].) FIG. 6-42

FLUID DYNAMICS

FIG. 6-43

6-37

Friction factors for in-line tube banks. (From Grimison, Trans. ASME, 59, 583 [1937].)

For banks of staggered tubes, the friction factor for isothermal flow is obtained from Fig. (6-42). Each “fence” (group of parametric curves) represents a particular Reynolds number defined as DtVmaxρ Re =  (6-161) µ where Dt = tube outside diameter and µ = fluid viscosity. The numbers along each fence represent the transverse and inflow-direction spacings. The upper chart is for the case in which the minimum area for flow is in the transverse openings, while the lower chart is for the case in which the minimum area is in the diagonal openings. In the latter case, Vmax is based on the area of the diagonal openings and Nr is the number of rows in the direction of flow minus 1. A critical comparison of this method with all the data available at the time showed an average deviation of the order of 15 percent. (Boucher and Lapple, Chem. Eng. Prog., 44, 117–134 [1948]). For tube spacings greater than 3 tube diameters, the correlation by Gunter and Shaw (Trans. ASME, 67, 643–660 [1945]) can be used as an approximation. As an approximation, the pressure drop can be taken as 0.72 velocity head (based on Vmax per row of tubes for tube spacings commonly encountered in practice (Lapple, et al., Fluid and Particle Mechanics, University of Delaware, Newark, 1954). For banks of in-line tubes, f for isothermal flow is obtained from Fig. 6-43. Average deviation from available data is on the order of 15 percent. For tube spacings greater than 3Dt, the charts of Gram, Mackey, and Monroe (Trans. ASME, 80, 25–35 [1958]) can be used. As an approximation, the pressure drop can be taken as 0.32 velocity head (based on Vmax) per row of tubes (Lapple, et al., Fluid and Particle Mechanics, University of Delaware, Newark, 1954). For turbulent flow through shallow tube banks, the average friction factor per row will be somewhat greater than indicated by Figs. 6-42 and 6-43, which are based on 10 or more rows depth. A 30 percent increase per row for 2 rows, 15 percent per row for 3 rows and 7 percent per row for 4 rows can be taken as the maximum likely to be encountered (Boucher and Lapple, Chem. Eng. Prog., 44, 117–134 [1948]). For a single row of tubes, the friction factor is given by Curve B in Fig. 6-44 as a function of tube spacing. This curve is based on the

the data of several experimenters, all adjusted to a Reynolds number of 10,000. The values should be substantially independent of Re for 1,000 < Re < 100,000. For extended surfaces, which include fins mounted perpendicularly to the tubes or spiral-wound fins, pin fins, plate fins, and so on, friction data for the specific surface involved should be used. For details, see Kays and London (Compact Heat Exchangers, 2d ed., McGraw-Hill, New York, 1964). If specific data are unavailable, the correlation by Gunter and Shaw (Trans. ASME, 67, 643–660 [1945]) may be used as an approximation. When a large temperature change occurs in a gas flowing across a

FIG. 6-44 Friction factors vs. transverse spacing for single row of tubes. (From Boucher and Lapple, Chem. Eng. Prog., 44, 117 [1948].)

6-38

FLUID AND PARTICLE DYNAMICS

tube bundle, gas properties should be evaluated at the mean temperature Tm = Tt + K ∆Tlm

(6-162)

where Tt = average tube-wall temperature K = constant ∆Tlm = log-mean temperature difference between the gas and the tubes. Values of K averaged from the recommendations of Chilton and Genereaux (Trans. AIChE, 29, 151–173 [1933]) and Grimison (Trans. ASME, 59, 583–594 [1937]) are as follows: for in-line tubes, 0.9 for cooling and −0.9 for heating; for staggered tubes, 0.75 for cooling and −0.8 for heating. For nonisothermal flow of liquids across tube bundles, the friction factor is increased if the liquid is being cooled and decreased if the liquid is being heated. The factors previously given for nonisothermal flow of liquids in pipes (“Incompressible Flow in Pipes and Channels”) should be used. For two-phase gas/liquid horizontal cross flow through tube banks, the method of Diehl and Unruh (Pet. Refiner, 37[10], 124–128 [1958]) is available. Transition Region This region extends roughly over the range 200 < Re < 2,000. Figure 6-45 taken from Bergelin, Brown, and Doberstein (Trans. ASME, 74, 953–960 [1952]) gives curves for friction factor fT for five different configurations. Pressure drop for liquid flow is given by 2 µs 0.14 4 fT Nr ρVmax ∆p =  (6-163)  2 µb where Nr = number of major restrictions encountered in flow through the bank (equal to number of rows when minimum flow area occurs in transverse openings, and to number of rows minus 1 when it occurs in the diagonal openings); ρ = fluid density; Vmax = velocity through minimum flow area; µs = fluid viscosity at tube-surface temperature and µb = fluid viscosity at average bulk temperature. This method gives the friction factor within about 25 percent. Laminar Region Bergelin, Colburn, and Hull (Univ. Delaware

冢 冣

Eng. Exp. Sta. Bull., 2 [1950]) recommend the following equations for pressure drop with laminar flow (Rev < 100) of liquids across banks of plain tubes with pitch ratios P/Dt of 1.25 and 1.50: 2 280Nr Dt 1.6 µs m ρV max (6-164) ∆p =     Rev P µb 2

冢 冣 冢 冣冢

冣

0.57 m= (6-165) (Rev)0.25 where Rev = DvVmaxρ/µb; Dv = volumetric hydraulic diameter [(4 × free-bundle volume)/(exposed surface area of tubes)]; P = pitch (= a for in-line arrangements, = a or c [whichever is smaller] for staggered arrangements), and other quantities are as defined following Eq. (6-163). Bergelin, et al. (ibid.) show that pressure drop per row is independent of the number of rows in the bank with laminar flow. The pressure drop is predicted within about 25 percent. The validity of extrapolating Eq. (6-164) to pitch ratios larger than 1.50 is unknown. The correlation of Gunter and Shaw (Trans. ASME, 67, 643–660 [1945]) may be used as an approximation in such cases. For laminar flow of non-Newtonian fluids across tube banks, see Adams and Bell (Chem. Eng. Prog., 64, Symp. Ser., 82, 133–145 [1968]). Flow-induced tube vibration occurs at critical fluid velocities through tube banks, and is to be avoided because of the severe damage that can result. Methods to predict and correct vibration problems may be found in Eisinger (Trans. ASME J. Pressure Vessel Tech., 102, 138–145 [May 1980]) and Chen (J. Sound Vibration, 93, 439–455 [1984]). BEDS OF SOLIDS Fixed Beds of Granular Solids Pressure-drop prediction is complicated by the variety of granular materials and of their packing arrangement. For flow of a single incompressible fluid through an incompressible bed of granular solids, the pressure drop may be estimated by the correlation given in Fig. 6-46 (Leva, Chem. Eng., 56[5], 115–117 [1949]), or Fluidization, McGraw-Hill, New York, 1959). The modified friction factor and Reynolds number are defined by Dpρφ3s − n
3|∆p| fm ⬅  2G2L(1 −
)3 − n

(6-166)

DpG Re′ ⬅  (6-167) µ where −∆ p = pressure drop L = depth of bed Dp = average particle diameter, defined as the diameter of a sphere of the same volume as the particle
= void fraction n = exponent given in Fig. 6-46 as a function of Re′ φs = shape factor defined as the area of sphere of diameter Dp divided by the actual surface area of the particle G = fluid superficial mass velocity based on the empty chamber cross section ρ = fluid density µ = fluid viscosity As for any incompressible single-phase flow, the equivalent pressure P = p + ρgz where g = acceleration of gravity z = elevation, may be used in place of p to account for gravitational effects in flows with vertical components. In creeping flow (Re′ < 10), 100 fm =  (6-168) Re′ At high Reynolds numbers the friction factor becomes nearly constant, approaching a value of the order of unity for most packed beds. In terms of S, particle surface area per unit volume of bed, Friction factors for transition region flow across tube banks. (Pitch is the minimum center-to-center tube spacing.) (From Bergelin, Brown, and Doberstein, Trans. ASME, 74, 953 [1952].)

FIG. 6-45

6(1 −
) Dp =  φsS

(6-169)

FLUID DYNAMICS

FIG. 6-46

6-39

Friction factor for beds of solids. (From Leva, Fluidization, McGraw-Hill, New York, 1959, p. 49.)

Porous Media Packed beds of granular solids are one type of the general class referred to as porous media, which include geological formations such as petroleum reservoirs and aquifers, manufactured materials such as sintered metals and porous catalysts, burning coal or char particles, and textile fabrics, to name a few. Pressure drop for incompressible flow across a porous medium has the same qualitative behavior as that given by Leva’s correlation in the preceding. At low Reynolds numbers, viscous forces dominate and pressure drop is proportional to fluid viscosity and superficial velocity, and at high Reynolds numbers, pressure drop is proportional to fluid density and to the square of superficial velocity. Creeping flow (Re′ <∼ 1) through porous media is often described in terms of the permeability k and Darcy’s Law: −∆P µ (6-170) =V L k where V = superficial velocity. The SI units for permeability are m2. Creeping flow conditions generally prevail in geological porous media. For multidimensional flows through isotropic porous media, the superficial velocity V and pressure gradient ∇P vectors replace the corresponding one-dimensional variables in Eq. (6-170). µ ∇P = −  V (6-171) k For isotropic homogeneous porous media (uniform permeability and porosity), the pressure for creeping incompressible single phase-flow may be shown to satisfy the LaPlace equation: ∇ 2P = 0

(6-172)

For anisotropic or oriented porous media, as are frequently found in geological media, permeability varies with direction and a permeability tensor K, with nine components Kij giving the velocity compenent in the i direction due to a pressure gradient in the j direction, may be introduced. For further information, see Slattery (Momentum, Energy and Mass Transfer in Continua, Krieger, Huntington, New York, 1981, p. 193–218). See also Dullien (Chem. Eng. J. [Laussanne], 10, 1,034 [1975]) for a review of pressure-drop methods in singlephase flow. Solutions for Darcy’s law for several geometries of interest in petroleum reservoirs and aquifers, for both incompressible and compressible flows, are given in Craft and Hawkins (Applied Petro-

leum Reservoir Engineering, Prentice-Hall, Englewood Cliffs, N.J., 1959). See also Todd (Groundwater Hydrology, 2nd ed., Wiley, New York, 1980). For granular solids of mixed sizes the average particle diameter may be calculated as 1 xi (6-173) =冱 Dp Dp,i i where xi = weight fraction of particles of size Dp,i. For isothermal compressible flow of a gas with constant compressibility factor Z through a packed bed of granular solids, an equation similar to Eq. (6-114) for pipe flow may be derived: 2ZRG2T v 2fmL(1 −
)3 − n p21 − p22 =  ln 2 +  Mw v1 φ 3s − n
3Dp

冤

冥

(6-174)

where p1 = upstream absolute pressure p2 = downstream absolute pressure R = gas constant T = absolute temperature Mw = molecular weight v1 = upstream specific volume of gas v2 = downstream specific volume of gas For creeping flow of power law non-Newtonian fluids, the method of Christopher and Middleton (Ind. Eng. Chem. Fundam., 4, 422–426 [1965]) may be used: 150HLV n(1 −
)2 −∆p =  (6-175) Dp2 φ 2s
3 K 3 n Dp2 φ2s
4 (1 − n)/2 H= 9+ (6-176)  12 n (1 −
)2 where V = G/ρ = superficial velocity, K, n = power law material constants, and all other variables are as defined in Eq. (6-166). This correlation is supported by data from Christopher and Middleton (ibid.), Gregory and Griskey (AIChE J., 13, 122–125 [1967]), Yu, Wen, and Bailie (Can. J. Chem. Eng., 46, 149–154 [1968]), Siskovic, Gregory, and Griskey (AIChE J., 17, 176–187 [1978]), Kemblowski and Mertl (Chem. Eng. Sci., 29, 213–223 [1974]), and Kemblowski and Dziu-

冢

冣冤

冥

6-40

FLUID AND PARTICLE DYNAMICS

minski (Rheol. Acta, 17, 176–187 [1978]). The measurements cover the range n = 0.50 to 1.60, and modified Reynolds number Re′ = 10−8 to 10, where DpV 2 − nρ Re′ =  (6-177) H For the case n = 1 (Newtonian fluid), Eqs. (6-175) and (6-176) give a pressure drop 25 percent less than that given by Eqs. (6-166) through (6-168). For viscoelastic fluids see Marshall and Metzner (Ind. Eng. Chem. Fundam., 6, 393–400 [1967]), Siskovic, Gregory, and Griskey (AIChE J., 13, 122–125 [1967]) and Kemblowski and Dziubinski (Rheol. Acta, 17, 176–187 [1978]). For gas flow through porous media with small pore diameters, the slip flow and molecular flow equations previously given (see the “Vacuum Flow” subsection) may be applied when the pore is of the same or smaller order as the mean free path, as described by Monet and Vermeulen (Chem. Eng. Prog., 55, Symp. Ser., 25 [1959]). Tower Packings For the flow of a single fluid through a bed of tower packing, pressure drop may be estimated using the preceding methods. See also Sec. 14 of this Handbook. For countercurrent gas/liquid flow in commercial tower packings, both structured and unstructured, several sources of data and correlations for pressure drop and flooding are available. See, for example, Strigle (Random Packings and Packed Towers, Design and Applications, Gulf Publishing, Houston, 1989; Chem. Eng. Prog., 89[8], 79–83 [August 1993]), Hughmark (Ind. Eng. Chem. Fundam., 25, 405–409 [1986]), Chen (Chem. Eng. Sci., 40, 2139–2140 [1985]), Billet and Mackowiak (Chem. Eng. Technol., 11, 213–217 [1988]), Krehenwinkel and Knapp (Chem. Eng. Technol., 10, 231–242 [1987]), Mersmann and Deixler (Ger. Chem. Eng., 9, 265–276 [1986]) and Robbins (Chem. Eng. Progr., 87[5], 87–91 [May 1991]). Data and correlations for flooding and pressure drop for structured packings are given by Fair and Bravo (Chem. Eng. Progr., 86[1], 19–29 [January 1990]). Fluidized Beds When gas or liquid flows upward through a vertically unconstrained bed of particles, there is a minimum fluid velocity at which the particles will begin to move. Above this minimum velocity, the bed is said to be fluidized. Fluidized beds are widely used, in part because of their excellent mixing and heat and mass transfer characteristics. See Sec. 17 of this Handbook for detailed information. BOUNDARY LAYER FLOWS Boundary layer flows are a special class of flows in which the flow far from the surface of an object is inviscid, and the effects of viscosity are manifest only in a thin region near the surface where steep velocity gradients occur to satisfy the no-slip condition at the solid surface. The thin layer where the velocity decreases from the inviscid, potential flow velocity to zero (relative velocity) at the solid surface is called the boundary layer. The thickness of the boundary layer is indefinite because the velocity asymptotically approaches the free-stream velocity at the outer edge. The boundary layer thickness is conventionally taken to be the distance for which the velocity equals 0.99 times the free-stream velocity. The boundary layer may be either laminar or turbulent. Particularly in the former case, the equations of motion may be simplified by scaling arguments. Schlichting (Boundary Layer Theory, 8th ed., McGraw-Hill, New York, 1987) is the most comprehensive source for information on boundary layer flows. Flat Plate, Zero Angle of Incidence For flow over a wide, thin flat plate at zero angle of incidence with a uniform free-stream velocity, as shown in Fig. 6-47, the critical Reynolds number at which the boundary layer becomes turbulent is normally taken to be

where

xVρ Rex =  = 500,000 µ V = free-stream velocity ρ = fluid density µ = fluid viscosity x = distance from leading edge of the plate

(6-178)

Uniform free-stream velocity V

y

δ(x )

x L FIG. 6-47

Boundary layer on a flat plate at zero angle of incidence.

However, the transition Reynolds number depends on free-stream turbulence and may range from 3 × 105 to 3 × 106. The laminar boundary layer thickness δ is a function of distance from the leading edge: δ ≈ 5.0xRe −0.5 x

(6-179)

The total drag on the plate of length L and width b for a laminar boundary layer, including the drag on both surfaces, is: FD = 1.328bLρV 2ReL−0.5

(6-180)

For non-Newtonian power law fluids (Acrivos, Shah, and Peterson, AIChE J., 6, 312–317 [1960]; Hsu, AIChE J., 15, 367–370 [1969]), FD = CbLρV 2ReL′− 1/(1 + n)

(6-181)

n = 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 C = 2.075 1.958 1.838 1.727 1.627 1.538 1.460 1.390 1.328 where Re′L = ρV 2 − nLn/K and K and n are the power law material constants (see Eq. [6-4]). For a turbulent boundary layer, the thickness may be estimated as δ ≈ 0.37xRe −0.2 x

(6-182)

and the total drag force on both sides of the plate of length L is 0.455 1,700 −  ρbLV 2 5 × 105 < ReL < 109 (6-183) FD =  (log ReL)2.58 ReL Here the second term accounts for the laminar leading edge of the boundary layer and assumes that the critical Reynolds number is 500,000. Cylindrical Boundary Layer Laminar boundary layers on cylindrical surfaces, with flow parallel to the cylinder axis, are described by Glauert and Lighthill (Proc. R. Soc. [London], 230A, 188–203 [1955]), Jaffe and Okamura (Z. Angew. Math. Phys., 19, 564–574 [1968]) and Stewartson (Q. Appl. Math., 13, 113–122 [1955]). For a turbulent boundary layer, the total drag may be estimated as

冤

冥

FD = 苶cj苶πrLρV (6-184) where r = cylinder radius, L = cylinder length, and the average friction coefficient is given by (White, J. Basic Eng., 94, 200–206 [1972]) L 0.4 ReL−1/3 (6-185) 苶cj苶 = 0.0015 + 0.30 + 0.015  r 6 9 6 for ReL = 10 to 10 and L/r < 10 . Continuous Flat Surface Boundary layers on continuous surfaces drawn through a stagnant fluid are shown in Fig. 6-48. Figure 6-48a shows the continuous flat surface (Sakiadis, AIChE J., 7, 26–28, 221–225, 467–472 [1961]). The critical Reynolds number for transition to turbulent flow may be greater than the 500,000 value for the finite flat-plate case discussed previously (Tsou, Sparrow, and Kurtz, J. Fluid Mech., 26, 145–161 [1966]). For a laminar boundary layer, the thickness is given by 2

冤

冢 冣冥

δ = 6.37xRe x−0.5

(6-186)

FLUID DYNAMICS

6-41

(a)

FIG. 6-49 Boundary layer parameters for continuous cylindrical surfaces. (From Sakiadis, Am. Inst. Chem. J., 7, 467 [1961].)

(b) Continuous surface: (a) continuous flat surface, (b) continuous cylindrical surface. (From Sakiadis, Am. Inst. Chem. Eng. J., 7, 221, 467 [1961].)

FIG. 6-48

(Chem. Eng. Sci., 32, 1517–1525 [1977] and Crane (Z. Angew. Math. Phys., 26, 619–622 [1975]). VORTEX SHEDDING

and the total drag exerted on the two surfaces is FD = 1.776bLρV 2ReL−0.5

(6-187)

The total flow rate of fluid entrained by the surface is q = 3.232bLVReL−0.5

(6-188)

The theoretical velocity field was experimentally verified by Tsou, Sparrow, and Goldstein (Int. J. Heat Mass Transfer, 10, 219–235 [1967]) and Szeri, Yates, and Hai ( J. Lubr. Technol., 98, 145–156 [1976]). For non-Newtonian power law fluids see Fox, Erickson, and Fan (AIChE J., 15, 327–333 [1969]). For a turbulent boundary layer, the thickness is given by δ = 1.01xRe −0.2 x

(6-189)

and the total drag on both sides by FD = 0.056bLρV 2ReL−0.2

(6-190)

and the total entrainment by q = 0.252bLVReL−0.2

(6-191)

When the laminar boundary layer is a significant part of the total length of the object, the total drag should be corrected by subtracting a calculated turbulent drag for the length of the laminar section and then adding the laminar drag for the laminar section. Tsou, Sparrow, and Goldstein (Int. J. Heat Mass Transfer, 10, 219–235 [1967]) give an improved analysis of the turbulent boundary layer; their data indicate that Eq. (6-190) underestimates the drag by about 15 percent. Continuous Cylindrical Surface The continuous surface shown in Fig. 6-48b is applicable, for example, for a wire drawn through a stagnant fluid (Sakiadis, AIChE J., 7, 26–28, 221–225, 467–472 [1961]). The critical-length Reynolds number for transition is Rex = 200,000. The laminar boundary layer thickness, total drag, and entrainment flow rate may be obtained from Fig. 6-49; the drag and entrainment rate are obtained from the momentum area Θ and displacement area ∆ evaluated at x = L. FD = ρV 2Θ

(6-192)

q = V∆

(6-193)

Further laminar boundary layer analysis is given by Crane (Z. Angew. Math. Phys., 23, 201–212 [1972]). For a turbulent boundary layer, the total drag may be roughly estimated using Eqs. (6-184) and (6-185) for finite cylinders. Measured forces by Kwon and Prevorsek (J. Eng. Ind., 101, 73–79 [1979]) are greater than predicted this way. The laminar boundary layer on deforming continuous surfaces with velocity varying with axial position is discussed by Vleggaar

When fluid flows past objects or through orifices or similar restrictions, vortices may periodically be shed downstream. Objects such as smokestacks, chemical-processing columns, suspended pipelines, and electrical transmission lines can be subjected to damaging vibrations and forces due to the vortices, especially if the shedding frequency is close to a natural vibration frequency of the object. The shedding can also produce sound. See Krzywoblocki (Appl. Mech. Rev., 6, 393–397 [1953]) and Marris (J. Basic Eng., 86, 185–196 [1964]). Development of a vortex street, or von Kármán vortex street is shown in Fig. 6-50. Discussions of the vortex street may be found in Panton (pp. 387–393). The Reynolds number is DVρ Re =  (6-194) µ where D = diameter of cylinder or effective width of object V = free-stream velocity ρ = fluid density µ = fluid viscosity For flow past a cylinder, the vortex street forms at Reynolds numbers above about 40. The vortices initially form in the wake, the point of formation moving closer to the cylinder as Re is increased. At a Reynolds number of 60 to 100, the vortices are formed from eddies attached to the cylinder surface. The vortices move at a velocity slightly less than V. The frequency of vortex shedding f is given in terms of the Strouhal number, which is approximately constant over a wide range of Reynolds numbers. fD St ⬅  (6-195) V For 40 < Re < 200 the vortices are laminar and the Strouhal number has a nearly constant value of 0.2 for flow past a cylinder. Between Re = 200 and 400 the Strouhal number is no longer constant and the wake becomes irregular. Above about Re = 400 the vortices become turbulent, the wake is once again stable, and the Strouhal number remains constant at about 0.2 up to a Reynolds number of about 10 5.

FIG. 6-50

Vortex street behind a cylinder.

6-42

FLUID AND PARTICLE DYNAMICS

Above Re = 105 the vortex shedding is difficult to see in flow visualization experiments, but velocity measurements still show a strong spectral component at St = 0.2 (Panton, p. 392). Experimental data suggest that the vortex street disappears over the range 5 × 105 < Re < 3.5 × 106, but is reestablished at above 3.5 × 106 (Schlichting). Vortex shedding exerts alternating lateral forces on a cylinder, perpendicular to the flow direction. Such forces may lead to severe vibration or mechanical failure of cylindrical elements such as heatexchanger tubes, transmission lines, stacks, and columns when the vortex shedding frequency is close to resonant bending frequency. According to Den Hartog (Proc. Nat. Acad. Sci., 40, 155–157 [1954]), the vortex shedding and cylinder vibration frequency will shift to the resonant frequency when the calculated shedding frequency is within 20 percent of the resonant frequency. The well-known Tacoma Narrows bridge collapse resulted from resonance between a torsional oscillation and vortex shedding (Panton, p. 392). Spiral strakes are sometimes installed on tall stacks so that vortices at different axial positions are not shed simultaneously. The alternating lateral force FK, sometimes called the von Kármán force, is given by (Den Hartog, Mechanical Vibrations, 4th ed., McGraw-Hill, New York, 1956, pp. 305–309): ρV 2 FK = CK A  (6-196) 2 where CK = von Kármán coefficient A = projected area perpendicular to the flow ρ = fluid density V = free-stream fluid velocity For a cylinder, CK = 1.7. For a vibrating cylinder, the effective projected area exceeds, but is always less than twice, the actual cylinder projected area (Rouse, Engineering Hydraulics, Wiley, New York, 1950). The following references pertain to discussions of vortex shedding in specific structures: steel stacks (Osker and Smith, Trans. ASME, 78, 1381–1391 [1956]; Smith and McCarthy, Mech. Eng., 87, 38–41 [1965]); chemical-processing columns (Freese, J. Eng. Ind., 81, 77–91, [1959]); heat exchangers (Eisinger, Trans. ASME J. Pressure Vessel Tech., 102, 138–145 [May 1980]; Chen, J. Sound Vibration, 93, 439–455 [1984]; Gainsboro, Chem. Eng. Prog., 64[3], 85–88 [1968]; “Flow-Induced Vibration in Heat Exchangers,” Symp. Proc., ASME, New York, 1970); suspended pipe lines (Baird, Trans. ASME, 77, 797–804 [1955]); and suspended cable (Steidel, J. Appl. Mech., 23, 649–650 [1956]). COATING FLOWS In coating flows, liquid films are entrained on moving solid surfaces. For general discussions, see Ruschak (Ann. Rev. Fluid Mech., 17, 65–89 [1985]), Cohen and Gutoff (Modern Coating and Drying Technology, VCH Publishers, New York, 1992), and Middleman (Fundamentals of Polymer Processing, McGraw-Hill, New York, 1977). It is generally important to control the thickness and uniformity of the coatings. In dip coating, or free withdrawal coating, a solid surface is withdrawn from a liquid pool, as shown in Fig. 6-51. It illustrates many of the features found in other coating flows, as well. Tallmadge and Gutfinger (Ind. Eng. Chem., 59[11], 19–34 [1967]) provide an early review of the theory of dip coating. The coating flow rate and film thickness are controlled by the withdrawal rate and the flow behavior in the meniscus region. For a withdrawal velocity V and an angle of inclination from the horizontal φ, the film thickness h may be estimated for low withdrawal velocities by ρg h  σ

冢 冣

where

1/2

0.944 =  Ca2/3 (1 − cos φ)1/2

g = acceleration of gravity Ca = µV/σ = capillary number µ = viscosity σ = surface tension

(6-197)

FIG. 6-51

Dip coating.

Equation (6-197) is asymptotically valid as Ca → 0 and agrees with experimental data up to capillary numbers in the range of 0.01 to 0.03. In practice, where high production rates require high withdrawal speeds, capillary numbers are usually too large for Eq. (6-197) to apply. Approximate analytical methods for larger capillary numbers have been obtained by numerous investigators, but none appears wholly satisfactory, and some are based on questionable assumptions (Ruschak, Ann. Rev. Fluid Mech., 17, 65–89 [1985]). With the availability of high-speed computers and the development of the field of computational fluid dynamics, numerical solutions accounting for two-dimensional flow aspects, along with gravitational, viscous, inertial, and surface tension forces are now the most effective means to analyze coating flow problems. Other common coating flows include premetered flows, such as slide and curtain coating, where the film thickness is an independent parameter that may be controlled within limits, and the curvature of the mensiscus adjusts accordingly; the closely related blade coating; and roll coating and extrusion coating. See Ruschak (ibid.), Cohen and Gutoff (Modern Coating and Drying Technology, VCH Publishers, New York, 1992) and Middleman (Fundamentals of Polymer Processing, McGraw-Hill, New York, 1977). For dip coating of wires, see Taughy (Int. J. Numerical Meth. Fluids, 4, 441–475 [1984]). Many coating flows are subject to instabilities that lead to unacceptable coating defects. Three-dimensional flow instabilities lead to such problems as ribbing. Air entrainment is another common defect. FALLING FILMS Minimum Wetting Rate The minimum liquid rate required for complete wetting of a vertical surface is about 0.03 to 0.3 kg/m ⋅ s for water at room temperature. The minimum rate depends on the geometry and nature of the vertical surface, liquid surface tension, and mass transfer between surrounding gas and the liquid. See Ponter, et al. (Int. J. Heat Mass Transfer, 10, 349–359 [1967]; Trans. Inst. Chem. Eng. [London], 45, 345–352 [1967]), Stainthorp and Allen (Trans. Inst. Chem. Eng. [London], 43, 85–91 [1967]) and Watanabe, et al. (J. Chem. Eng. [Japan], 8[1], 75 [1975]). Laminar Flow For films falling down vertical flat surfaces, as shown in Fig. 6-52, or vertical tubes with small film thickness compared to tube radius, laminar flow conditions prevail for Reynolds numbers less than about 2,000, where the Reynolds number is given by 4Γ Re =  (6-198) µ where Γ = liquid mass flow rate per unit width of surface and µ = liq-



,,  V

Γ = mass flow rate per unit width of surface

δ

FIG. 6-52

Falling film.

uid viscosity. For a flat film surface, the following equations may be derived. The film thickness δ is 3Γµ 1/3 δ=  (6-199) ρ2g The average film velocity is Γ gρδ2 V== (6-200) ρδ 3µ The downward velocity profile u(x) where x = 0 at the solid surface and x = δ at the liquid/gas interface is given by 2x x 2 (6-201) u = 1.5V  −  δ δ These equations assume that there is no drag force at the gas/liquid interface, such as would be produced by gas flow. For a flat surface inclined at an angle θ with the horizontal, the preceding equations may be modified by replacing g by g sin θ. For films falling inside vertical tubes with film thickness up to and including the full pipe radius, see Jackson (AIChE J., 1, 231–240 [1955]). These equations have generally given good agreement with experimental results for low-viscosity liquids (<0.005 Pa ⋅ s) (< 5 cp) whereas Jackson (ibid.) found film thicknesses for higher-viscosity liquids (0.01 to 0.02 Pa⋅s (10 to 20 cp) were significantly less than predicted by Eq. (6-197). At Reynolds numbers of 25 or greater, surface waves will be present on the liquid film. West and Cole (Chem. Eng. Sci., 22, 1388– 1389 [1967]) found that the surface velocity u(x = δ) is still within 7 percent of that given by Eq. (6-201) even in wavy flow. For laminar non-Newtonian film flow, see Bird, Armstrong, and Hassager (Dynamics of Polymeric Liquids, vol. 1: Fluid Mechanics, Wiley, New York, 1977, p. 215, 217), Astarita, Marrucci, and Palumbo (Ind. Eng. Chem. Fundam., 3, 333–339 [1964]) and Cheng (Ind. Eng. Chem. Fundam., 13, 394–395 [1974]). Turbulent Flow In turbulent flow, Re > 2,000, for vertical surfaces, the film thickness may be estimated to within 25 percent using Γ1.75µ0.25 1/3 δ = 0.304  (6-202) ρ2g Replace g by g sin θ for a surface inclined at angle θ to the horizontal. The average film velocity is V = Γ/ρδ. Tallmadge and Gutfinger (Ind. Eng. Chem., 59[11], 19–34 [1967]) discuss prediction of drainage rates from liquid films on flat and cylindrical surfaces. Effect of Surface Traction If a drag is exerted on the surface of the film because of motion of the surrounding fluid, the film thickness will be reduced or increased, depending upon whether the drag acts with or against gravity. Thomas and Portalski (Ind. Eng. Chem., 50, 1081–1088 [1958]), Dukler (Chem. Eng. Prog., 55[10], 62–67 [1959]) and Kosky (Int. J. Heat Mass Transfer, 14, 1220–1224 [1971]) have presented calculations of film thickness and film velocity. Film thickness data for falling water films with cocurrent and countercurrent air flow in pipes are given by Zhivaikin (Int. Chem. Eng., 2, 337–341 [1962]). Zabaras, Dukler, and Moalem-Maron (AIChE J., 32, 829–843

冢 冣

冤

冢

冢 冣冥

FLUID DYNAMICS

6-43

[1986]) and Zabaras and Dukler (AIChE J., 34, 389–396 [1988]) present studies of film flow in vertical tubes with both cocurrent and countercurrent gas flow, including measurements of film thickness, wall shear stress, wave velocity, wave amplitude, pressure drop, and flooding point for countercurrent flow. Flooding With countercurrent gas flow, a condition is reached with increasing gas rate for which flow reversal occurs and liquid is carried upward. The mechanism for this flooding condition has been most often attributed to waves either bridging the pipe or reversing direction to flow upward at flooding. However, the results of Zabaras and Dukler (ibid.) suggest that flooding may be controlled by flow conditions at the liquid inlet and that wave bridging or upward wave motion does not occur, at least for the 50.8-mm diameter pipe used for their study. Flooding mechanisms are still incompletely understood. Under some circumstances, as when the gas is allowed to develop its normal velocity profile in a “calming length” of pipe beneath the liquid draw-off, the gas superficial velocity at flooding will be increased, and increases with decreasing length of wetted pipe (Hewitt, Lacy, and Nicholls, Proc. Two-Phase Flow Symp., University of Exeter, paper 4H, AERE-4 4614 [1965]). A bevel cut at the bottom of the pipe with an angle 30° from the vertical will increase the flooding velocity in small-diameter tubes at moderate liquid flow rates. If the gas approaches the tube from the side, the taper should be oriented with the point facing the gas entrance. Figures 6-53 and 6-54 give correlations for flooding in tubes with square and slant bottoms (courtesy Holmes, DuPont Co.) The superficial mass velocities of gas and liquid GG and GL, and the physical property parameters λ and ψ are the same as those defined for the Baker chart (“Multiphase Flow” subsection, Fig. 6-25). For tubes larger than 50 mm (2 in), flooding velocity appears to be relatively insensitive to diameter and the flooding curves for 1.98-in diameter may be used.

Flooding in vertical tubes with square top and square bottom. To convert lbm/(ft2 ⋅s) to kg/(m2 ⋅s), multiply by 4.8824; to convert in to mm, multiply by 25.4. (Courtesy of E. I. du Pont de Nemours & Co.) FIG. 6-53

冣

Flooding in vertical tubes with square top and slant bottom. To convert lbm/(ft2 ⋅s) to kg/(m2 ⋅s), multiply by 4.8824; to convert in to mm, multiply by 25.4. (Courtesy of E. I. du Pont de Nemours & Co.) FIG. 6-54

6-44

FLUID AND PARTICLE DYNAMICS

HYDRAULIC TRANSIENTS Many transient flows of liquids may be analyzed by using the full timedependent equations of motion for incompressible flow. However, there are some phenomena that are controlled by the small compressibility of liquids. These phenomena are generally called hydraulic transients. Water Hammer When liquid flowing in a pipe is suddenly decelerated to zero velocity by a fast-closing valve, a pressure wave propagates upstream to the pipe inlet, where it is reflected; a pounding of the line commonly known as water hammer is often produced. For an instantaneous flow stoppage of a truly incompressible fluid in an inelastic pipe, the pressure rise would be infinite. Finite compressibility of the fluid and elasticity of the pipe limit the pressure rise to a finite value. The Joukowski formula gives the maximum pressure rise as ∆ p = ρa∆V where

(6-203)

ρ = liquid density ∆V = change in liquid velocity a = pressure wave velocity

The wave velocity is given by β苶 /ρ 兹苶 a =  +苶(苶 β苶 /E苶 )(苶 D苶 /b苶) 兹苶1苶 where

(6-204)

β = liquid bulk modulus of elasticity E = elastic modulus of pipe wall D = pipe inside diameter b = pipe wall thickness

The numerator gives the wave velocity for perfectly rigid pipe, and the denominator corrects for wall elasticity. This formula is for thin-walled pipes; for thick-walled pipes, the factor D/ b is replaced by (Do2 + Di2 ) 2  (Do2 − Di2 ) where Do = pipe outside diameter Di = pipe inside diameter Example 10: Response to Instantaneous Valve Closing Compute the wave speed and maximum pressure rise for instantaneous valve closing, with an initial velocity of 2.0 m/s, in a 4-in Schedule 40 steel pipe with elastic modulus 207 × 109 Pa. Repeat for a plastic pipe of the same dimensions, with E = 1.4 × 109 Pa. The liquid is water with β = 2.2 × 109 Pa and ρ = 1,000 kg/m3. For the steel pipe, D = 102.3 mm, b = 6.02 mm, and the wave speed is 苶/ρ 苶 兹β a =  苶)( 苶D 苶/b 苶)苶 +苶β (苶/E 兹1苶苶 苶苶 ×苶10苶9苶/1苶0苶0苶0苶 兹2苶.2 =  9 9 苶苶 苶0苶2苶.3 苶/6 苶.0 苶2苶)苶 +苶(2苶.2 ×苶10苶苶 /2苶0苶7苶苶 ×苶10苶苶 )(1 兹1苶苶 = 1365 m/s The maximum pressure rise is ∆ p = ρa∆V = 1,000 × 1,365 × 2.0 = 2.73 × 106 Pa For the plastic pipe,

苶苶 ×苶10苶9苶/1苶0苶0苶0苶 兹2苶.2 a =  9 9 苶苶 苶苶 苶0苶2苶.3 苶/6 苶.0 苶2苶)苶 +苶(2苶.2 ×苶10苶苶 /1苶.4 ×苶 10苶苶 )(1 兹1苶苶 = 282 m/s ∆p = ρa∆V = 1,000 × 282 × 2.0 = 5.64 × 105 Pa

The maximum pressure surge is obtained when the valve closes in less time than the period τ required for the pressure wave to travel from the valve to the pipe inlet and back, a total distance of 2L. 2L τ= (6-205) a The pressure surge will be reduced when the time of flow stoppage exceeds the pipe period τ, due to cancellation between direct and

reflected waves. Wood and Jones (Proc. Am. Soc. Civ. Eng., J. Hydraul. Div., 99, (HY1), 167–178 [1973]) present charts for reliable estimates of water-hammer pressure for different valve closure modes. Wylie and Streeter (Hydraulic Transients, McGraw-Hill, New York, 1978) describe several solution methods for hydraulic transients, including the method of characteristics, which is well suited to computer methods for accurate solutions. A rough approximation for the peak pressure for cases where the valve closure time tc exceeds the pipe period τ is (Daugherty and Franzini, Fluid Mechanics with Engineering Applications, McGraw-Hill, New York, 1985): τ ∆p ≈  ρa∆V (6-206) tc Successive reflections of the pressure wave between the pipe inlet and the closed valve result in alternating pressure increases and decreases, which are gradually attenuated by fluid friction and imperfect elasticity of the pipe. Periods of reduced pressure occur while the reflected pressure wave is traveling from inlet to valve. Degassing of the liquid may occur, as may vaporization if the pressure drops below the vapor pressure of the liquid. Gas and vapor bubbles decrease the wave velocity. Vaporization may lead to what is often called liquid column separation; subsequent collapse of the vapor pocket can result in pipe rupture. In addition to water hammer induced by changes in valve setting, including closure, numerous other hydraulic transient flows are of interest, as, for example (Wylie and Streeter, Hydraulic Transients, McGraw-Hill, New York, 1978), those arising from starting or stopping of pumps; changes in power demand from turbines; reciprocating pumps; changing elevation of a reservoir; waves on a reservoir; turbine governor hunting; vibration of impellers or guide vanes in pumps, fans, or turbines; vibration of deformable parts such as valves; draft-tube instabilities due to vortexing; and unstable pump or fan characteristics. Tube failure in heat exhangers may be added to this list. Pulsating Flow Reciprocating machinery (pumps and compressors) produces flow pulsations, which adversely affect flow meters and process control elements and can cause vibration and equipment failure, in addition to undesirable process results. Vibration and damage can result not only from the fundamental frequency of the pulse producer but also from higher harmonics. Multipiston double-acting units reduce vibrations. Pulsation dampeners are often added. Damping methods are described by M. W. Kellogg Co. (Design of Piping Systems, rev. 2d ed., Wiley, New York, 1965). For liquid phase pulsation damping, gas-filled surge chambers, also known as accumulators, are commonly used; see Wylie and Streeter (Hydraulic Transients, McGraw-Hill, New York, 1978). Software packages are commercially available for simulation of hydraulic transients. These may be used to analyze piping systems to reveal unsatisfactory behavior, and they allow the assessment of design changes such as increases in pipe-wall thickness, changes in valve actuation, and addition of check valves, surge tanks, and pulsation dampeners. Cavitation Loosely regarded as related to water hammer and hydraulic transients because it may cause similar vibration and equipment damage, cavitation is the phenomenon of collapse of vapor bubbles in flowing liquid. These bubbles may be formed anywhere the local liquid pressure drops below the vapor pressure, or they may be injected into the liquid, as when steam is sparged into water. Local low-pressure zones may be produced by local velocity increases (in accordance with the Bernoulli equation; see the preceding “Conservation Equations” subsection) as in eddies or vortices, or near boundary contours; by rapid vibration of a boundary; by separation of liquid during water hammer; or by an overall reduction in static pressure, as due to pressure drop in the suction line of a pump. Collapse of vapor bubbles once they reach zones where the pressure exceeds the vapor pressure can cause objectionable noise and vibration and extensive erosion or pitting of the boundary materials. The critical cavitation number at inception of cavitation, denoted σi, is useful in correlating equipment performance data: (p − pv) σi =  (6-207) ρV 2/2

冢 冣

FLUID DYNAMICS where

p = static pressure in undisturbed flow pv = vapor pressure ρ = liquid density V = free-stream velocity of the liquid

The value of the cavitation number for incipient cavitation for a specific piece of equipment is a characteristic of that equipment. Cavitation numbers for various head forms of cylinders, for disks, and for various hydrofoils are given by Holl and Wislicenus (J. Basic Eng., 83, 385–398 [1961]) and for various surface irregularities by Arndt and Ippen (J. Basic Eng., 90, 249–261 [1968]), Ball (Proc. ASCE J. Constr. Div., 89(C02), 91–110 [1963]), and Holl (J. Basic Eng., 82, 169–183 [1960]). As a guide only, for blunt forms the cavitation number is generally in the range of 1 to 2.5, and for somewhat streamlined forms the cavitation number is in the range of 0.2 to 0.5. Critical cavitation numbers generally depend on a characteristic length dimension of the equipment in a way that has not been explained. This renders scale-up of cavitation data questionable. For cavitation in flow through orifices, Fig. 6-55 (Thorpe, Int. J. Multiphase Flow, 16, 1023–1045 [1990]) gives the critical cavitation number for inception of cavitation. To use this cavitation number in Eq. (6-207), the pressure p is the orifice backpressure downstream of the vena contracta after full pressure recovery, and V is the average velocity through the orifice. Fig. 6-55 includes data from Tullis and Govindarajan (ASCE J. Hydraul. Div., HY13, 417–430 [1973]) modified to use the same cavitation number definition; their data also include critical cavitation numbers for 30.50- and 59.70-cm pipes (12.00- to 23.50-in). Very roughly, compared with the 15.40-cm pipe, the cavitation number is about 20 percent greater for the 30.50-cm (12.01-in) pipe and about 40 percent greater for the 59.70-cm (23.50in) diameter pipe. Inception of cavitation appears to be related to release of dissolved gas and not merely vaporization of the liquid. For further discussion of cavitation, see Eisenberg and Tulin (Streeter, Handbook of Fluid Dynamics, Sec. 12, McGraw-Hill, New York, 1961). TURBULENCE Turbulent flow occurs when the Reynolds number exceeds a critical value above which laminar flow is unstable; the critical Reynolds number depends on the flow geometry. There is generally a transition regime between the critical Reynolds number and the Reynolds number at which the flow may be considered fully turbulent. The transition regime is very wide for some geometries. In turbulent flow, variables such as velocity and pressure fluctuate chaotically; statistical methods are used to quantify turbulence. Time Averaging In turbulent flows it is useful to define timeaveraged and fluctuation values of flow variables such as velocity com-

6-45

ponents. For example, the x-component velocity fluctuation v′x is the difference between the actual instantaneous velocity vx and the timeaveraged velocity v苶苶x:

(6-208) v′x(x, y, z, t) = vx(x, y, z, t) − v苶苶x(x, y, z) The actual and fluctuating velocity components are, in general, functions of the three spatial coordinates x, y, and z and of time t. The time-averaged velocity v苶苶x is independent of time for a stationary flow. Nonstationary processes may be considered where averages are defined over time scales long compared to the time scale of the turbulent fluctuations, but short compared to longer time scales over which the time-averaged flow variables change due, for example, to time-varying boundary conditions. The time average over a time interval 2T centered at time t of a turbulently fluctuating variable ζ(t) is defined as 1 t+T ζ(τ) dτ (6-209) 苶ζ(苶t苶)苶 =  2T t − T where τ = dummy integration variable. For stationary turbulence, ζ苶 does not vary with time. 1 t+T ζ苶 = lim  ζ(τ) dτ (6-210) T→∞ 2T t − T

冕

冕

The time average of a fluctuation ζ苶′苶 = ζ  苶ζ = 0. Fluctuation magnitudes are quantified by root mean squares. 苶x′苶 v˜ ′x = 冪(v ) 2苶

(6-211)

In isotropic turbulence, statistical measures of fluctuations are equal in all directions. v˜ ′x = v˜ ′y = v˜ ′z

(6-212)

In homogeneous turbulence, turbulence properties are independent of spatial position. The kinetic energy of turbulence k is given by 1 (6-213) k =  (˜v′x2 + v˜ ′y2 + v˜ ′z2) 2 Turbulent velocity fluctuations ultimately dissipate their kinetic energy through viscous effects. Macroscopically, this energy dissipation requires pressure drop, or velocity decrease. The energy dissipation rate per unit mass is usually denoted
. For steady flow in a pipe, the average energy dissipation rate per unit mass is given by 2f V3
= (6-214) D where ρ = fluid density f = Fanning friction factor D = pipe inside diameter When the continuity equation and the Navier-Stokes equations for incompressible flow are time averaged, equations for the timeaveraged velocities and pressures are obtained which appear identical to the original equations (6-18 through 6-28), except for the appearance of additional terms in the Navier-Stokes equations. Called Reynolds stress terms, they result from the nonlinear effects of momentum transport by the velocity fluctuations. In each i-component (i = x, y, z) Navier-Stokes equation, the following additional terms appear on the right-hand side: 3 ∂τ(t) ji  冱 ∂xj j=1 with j components also being x, y, z. The Reynolds stresses are given by τ(t) vi′苶 v′j (6-215) ij = −ρ苶

FIG. 6-55 Critical cavitation number vs. diameter ratio β. (Reprinted from Thorpe, “Flow regime transitions due to cavitation in the flow through an orifice,” Int. J. Multiphase Flow, 16, 1023–1045. Copyright © 1990, with kind permission from Elsevier Science, Ltd., The Boulevard, Langford Lane, Kidlington OX5 1GB, United Kingdom.)

The Reynolds stresses are nonzero because the velocity fluctuations vi′苶 v′j in general in different coordinate directions are correlated so that 苶 is nonzero. Although direct numerical simulations under limited circumstances have been carried out to determine (unaveraged) fluctuating velocity fields, in general the solution of the equations of motion for turbulent flow is based on the time-averaged equations. This requires semi-

6-46

FLUID AND PARTICLE DYNAMICS

empirical models to express the Reynolds stresses in terms of timeaveraged velocities. This is the closure problem of turbulence. In all but the simplest geometries, numerical methods are required. Closure Models Many closure models have been proposed. A few of the more important ones are introduced here. Many employ the Boussinesq approximation, simplified here for incompressible flow, which treats the Reynolds stresses as analogous to viscous stresses, introducing a scalar quantity called the turbulent or eddy viscosity µt . ∂v苶苶i ∂v苶j苶 vi′苶 v′j = µt  + (6-216) −ρ 苶 ∂xj ∂xi An additional turbulence pressure term equal to −wkδij, where k = turbulent kinetic energy and δij = 1 if i = j and δij = 0 if i ≠ j, is sometimes included in the right-hand side. To solve the equations of motion using the Boussinesq approximation, it is necessary to provide equations for the single scalar unknown µt (and k, if used) rather than the nine unknown tensor components τ (t) ij . With this approximation, and using the effective viscosity µeff = µ + µt, the time-averaged momentum equation is similar to the original Navier-Stokes equation, with time-averaged variables and µeff replacing the instantaneous variables and molecular viscosity. However, solutions to the time-averaged equations for turbulent flow are not identical to those for laminar flow because µeff is not a constant. The universal turbulent velocity profile near the pipe wall presented in the preceding subsection “Incompressible Flow in Pipes and Channels” may be developed using the Prandtl mixing length approximation for the eddy viscosity, d 苶vx苶 µt = ρ lP2  (6-217) dy where lP is the Prandtl mixing length. The turbulent core of the universal velocity profile is obtained by assuming that the mixing length is proportional to the distance from the wall. The proportionality constant is one of two constants adjusted to fit experimental data. The Prandtl mixing length concept is useful for shear flows parallel to walls, but is inadequate for more general three-dimensional flows. A more complicated semiempirical model commonly used in numerical computations, and found in most commercial software for computational fluid dynamics (CFD; see the following subsection), is the k–
model described by Launder and Spaulding (Lectures in Mathematical Models of Turbulence, Academic, London, 1972). In this model the eddy viscosity is assumed proportional to the ratio k2/
. k2 (6-218) µt = ρCµ 
where the value Cµ = 0.09 is normally used. Semiempirical partial differential conservation equations for k and
derived from the NavierStokes equations with simplifying closure assumptions are coupled with the equations of continuity and momentum: ∂ ∂  (ρk) +  (ρv苶苶ik) ∂t ∂xi

冢

冣

冨 冨

∂ µt ∂k ∂v苶苶i ∂v苶j苶 ∂v苶苶i +   − ρ
=    + µt  ∂xi σk ∂xi ∂xj ∂xi ∂xj

冢

冣

冢

冣

(6-219)

∂ ∂  (ρ
) +  (ρv苶苶i
) ∂t ∂xi ρ
2 ∂ µt ∂

µt ∂v苶苶i ∂v苶j苶 ∂v苶苶i +   − C2
 (6-220) =    + C1
  ∂xi σ
∂xi k ∂xj ∂xi ∂xj k In these equations summations over repeated indices are implied. The values for the empirical constants C1
= 1.44, C2
= 1.92, σk = 1.0, and σ
= 1.3 are widely accepted (Launder and Spaulding, The Numerical Computation of Turbulent Flows, Imperial Coll. Sci. Tech. London, NTIS N74-12066 [1973]). The k–
model has proved reasonably accurate for many flows without highly curved streamlines or significant swirl. It usually underestimates flow separation and overestimates turbulence production by normal straining. The k–
model is suitable for high Reynolds number flows. See Virendra, Patel, Rodi,

冢

冣

冢

冣

and Scheuerer (AIAA J., 23, 1308–1319 [1984]) for a review of low Reynolds number k–
models. More advanced models, more complex and computationally intensive, are being developed. For example, the renormalization group theory (Yakhot and Orszag, J. Scientific Computing, 1, 1–51 [1986]; Yakhot, Orszag, Thangam, Gatski, and Speziale, Phys. Fluids A, 4, 1510–1520 [1992]) modification of the k–
model provides theoretical values of the model constants and provides substantial improvement in predictions of flows with stagnation, separation, normal straining, transient behavior such as vortex shedding, and relaminarization. Stress transport models provide equations for all nine Reynolds stress components, rather than introducing eddy viscosity. Algebraic closure equations for the Reynolds stresses are available, but are no longer in common use. Differential Reynolds stress models (e.g., Launder, Reece, and Rodi, J. Fluid Mech., 68, 537–566 [1975]) use differential conservation equations for all nine Reynolds stress components. In direct numerical simulation of turbulent flows, the solution of the unaveraged equations of motion is sought. Due to the extreme computational intensity, solutions to date have been limited to relatively low Reynolds numbers (Re < about 10,000 to 20,000) in simple geometries such as channel flow. See, for example, Kim, Moin, and Moser (J. Fluid Mech., 177, 133 [1987]). Since computational grids must be sufficiently fine to resolve even the smallest eddies, the computational difficulty rapidly becomes prohibitive as Reynolds number increases. Large eddy simulations use models for subgrid turbulence while solving for larger-scale fluctuations. Eddy Spectrum The energy that produces and sustains turbulence is extracted from velocity gradients in the mean flow, principally through vortex stretching. At Reynolds numbers well above the critical value there is a wide spectrum of eddy sizes, often described as a cascade of energy from the largest down to the smallest eddies. The largest eddies are of the order of the equipment size. The smallest are those for which viscous forces associated with the eddy velocity fluctuations are of the same order as inertial forces, so that turbulent fluctuations are rapidly damped out by viscous effects at smaller length scales. Most of the turbulent kinetic energy is contained in the larger eddies, while most of the dissipation occurs in the smaller eddies. Large eddies, which extract energy from the mean flow velocity gradients, are generally anisotropic. At smaller length scales, the directionality of the mean flow exerts less influence, and local isotropy is approached. The range of eddy scales for which local isotropy holds is called the equilibrium range. Davies (Turbulence Phenomena, Academic, New York, 1972) presents a good discussion of the spectrum of eddy lengths for welldeveloped isotropic turbulence. The smallest eddies, usually called Kolmogorov eddies (Kolmogorov, Compt. Rend. Acad. Sci. URSS, 30, 301; 32, 16 [1941]), have a characteristic velocity fluctuation v˜ K′ given by v˜ K′ = (ν
)1/4

(6-221)

where ν = kinematic viscosity and
= energy dissipation per unit mass. The size of the Kolmogorov eddy scale is lK = (ν3/
)1/4

(6-222)

The Reynolds number for the Kolmogorov eddy, ReK = lK v˜ ′k /ν, is equal to unity by definition. In the equilibrium range, which exists for welldeveloped turbulence and extends from the medium eddy sizes down to the smallest, the energy dissipation at the smaller length scales is supplied by turbulent energy drawn from the bulk flow and passed down the spectrum of eddy lengths according to the scaling rule (˜v′)3
= (6-223) l which is consistent with Eqs. (6-221) and (6-222). For the medium, or energy-containing, eddy size, (˜ve′)3
= (6-224) le For turbulent pipe flow, the friction velocity u* = 兹苶 τw 苶 /ρ used earlier in describing the universal turbulent velocity profile may be used as an estimate for v˜ e′. Together with the Blasius equation for the friction fac-

FLUID DYNAMICS tor from which
may be obtained (Eq. 6-214), this provides an estimate for the energy-containing eddy size in turbulent pipe flow: le = 0.05DRe−1/8

(6-225)

where D = pipe diameter and Re = pipe Reynolds number. Similarly, the Kolmogorov eddy size is lK = 4DRe−0.78

(6-226)

Most of the energy dissipation occurs on a length scale about 5 times the Kolmogorov eddy size. The characteristic fluctuating velocity for these energy-dissipating eddies is about 1.7 times the Kolmogorov velocity. The eddy spectrum is normally described using Fourier transform methods; see, for example, Hinze (Turbulence, McGraw-Hill, New York, 1975), and Tennekes and Lumley (A First Course in Turbulence, MIT Press, Cambridge, 1972). The spectrum E(κ) gives the fraction of turbulent kinetic energy contained in eddies of wavenumber ∞ between κ and κ + dκ, so that k = 冕0 E(κ) dκ. The portion of the equilibrium range excluding the smallest eddies, those which are affected by dissipation, is the inertial subrange. The Kolmogorov law gives E(κ) ∝ κ−5/3 in the inertial subrange. Several texts are available for further reading on turbulent flow, including Tennekus and Lumley (ibid.), Hinze (Turbulence, McGrawHill, New York, 1975), Landau and Lifshitz (Fluid Mechanics, 2d ed., Chap. 3, Pergamon, Oxford, 1987) and Panton (Incompressible Flow, Wiley, New York, 1984). COMPUTATIONAL FLUID DYNAMICS Computational fluid dynamics (CFD) emerged in the 1980s as a significant tool for fluid dynamics both in research and in practice, enabled by rapid development in computer hardware and software. Commercial CFD software is widely available. Computational fluid dynamics is the numerical solution of the equations of continuity and momentum (Navier-Stokes equations for incompressible Newtonian fluids) along with additional conservation equations for energy and material species in order to solve problems of nonisothermal flow, mixing, and chemical reaction. Textbooks include Fletcher (Computational Techniques for Fluid Dynamics, vol. 1: Fundamental and General Techniques, and vol. 2: Specific Techniques for Different Flow Categories, Springer-Verlag, Berlin, 1988), Hirsch (Numerical Computation of Internal and External Flows, vol. 1: Fundamentals of Numerical Discretization, and vol. 2: Computational Methods for Inviscid and Viscous Flows, Wiley, New York, 1988), Peyret and Taylor (Computational Methods for Fluid Flow, Springer-Verlag, Berlin, 1990), Canuto, Hussaini, Quarteroni, and Zang (Spectral Methods in Fluid Dynamics, Springer-Verlag, Berlin, 1988), Anderson, Tannehill, and Pletcher (Computational Fluid Mechanics and Heat Transfer, Hemisphere, New York, 1984), and Patankar (Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, D.C., 1980). A wide variety of numerical methods has been employed, but three basic steps are common to all CFD methods: 1. Subdivision or discretization of the flow domain into cells or elements. There are methods, called boundary element methods, in which the surface of the flow domain, rather than the volume, is discretized, but the vast majority of CFD work uses volume discretization. Discretization produces a set of grid lines or curves which define a mesh and a set of nodes at which the flow variables are to be calculated. The equations of motion are solved approximately on a domain defined by the grid. Curvilinear or body-fitted coordinate system grids may be used to ensure that the discretized domain accurately represents the true problem domain. 2. Discretization of the governing equations. In this step, the exact partial differential equations to be solved are replaced by approximate algebraic equations written in terms of the nodal values of the dependent variables. Among the numerous discretization methods, finite difference, finite volume, and finite element methods are the most common. The finite difference method estimates spatial derivatives in terms of the nodal values and spacing between nodes. The governing equations are then written in terms of

6-47

the nodal unknowns at each interior node. Finite volume methods, related to finite difference methods, may be derived by a volume integration of the equations of motion, with application of the divergence theorem, reducing by one the order of the differential equations. Equivalently, macroscopic balance equations are written on each cell. Finite element methods are weighted residual techniques in which the unknown dependent variables are expressed in terms of basis functions interpolating among the nodal values. The basis functions are substituted into the equations of motion, resulting in error residuals which are multiplied by the weighting functions, integrated over the control volume, and set to zero to produce algebraic equations in terms of the nodal unknowns. Selection of the weighting functions defines the various finite element methods. For example, Galerkin’s method uses the nodal interpolation basis functions as weighting functions. Each method also has its own method for implementing boundary conditions. The end result after discretization of the equations and application of the boundary conditions is a set of algebraic equations for the nodal unknown variables. Discretization in time is also required for the ∂/∂t time derivative terms in unsteady flow; finite differencing in time is often used. The descretized equations represent an approximation of the exact equations, and their solution gives an approximation for the flow variables. The accuracy of the solution improves as the grid is refined; that is, as the number of nodal points is increased. 3. Solution of the algebraic equations. For creeping flows, the algebraic equations are linear and a linear matrix equation is to be solved. Both direct and iterative solvers have been used. For most flows, the nonlinear inertial terms in the momentum equation are important and the algebraic discretized equations are therefore nonlinear. Solution yields the nodal values of the unknowns. CFD solutions, especially for complex three-dimensional flows, generate very large quantities of solution data. Computer graphics have greatly improved the ability to examine CFD solutions and visualize flow. CFD methods are used for incompressible and compressible, creeping, laminar and turbulent, Newtonian and non-Newtonian, and isothermal and nonisothermal flows. Chemically reacting flows, particularly in the field of combustion, have been simulated. Solution accuracy must be considered from several perspectives. These include convergence of the algorithms for solving the nonlinear discretized equations and convergence with respect to refinement of the mesh so that the discretized equations better approximate the exact equations and, in some cases, so that the mesh more accurately fits the true geometry. The possibility that steady-state solutions are unstable must always be considered. In addition to numerical sources of error, modeling errors are introduced in turbulent flow, where semiempirical closure models are used to solve time-averaged equations of motion, as discussed previously. Most commercial CFD codes include the k–
turbulence model, which has been by far the most widely used. More accurate models, such as differential Reynolds stress and renormalization group theory models, are also becoming available. Significant solution error is known to result in some problems from inadequacy of the turbulence model. Closure models for nonlinear chemical reaction source terms may also contribute to inaccuracy. Direct numerical simulation and large eddy simulation, which involve solutions for velocity fluctuations, under limited conditions or with certain modeling assumptions, remain primarily research areas. In its general sense, multiphase flow is not currently solvable by computational fluid dynamics. However, in certain cases reasonable solutions are possible. These include well-separated flows where the phases are confined to relatively well-defined regions separated by one or a few interfaces and flows in which a second phase appears as discrete particles of known size and shape whose motion may be approximately computed with drag coefficient formulations, or rigorously computed with refined meshes applying boundary conditions at the particle surface. Two-fluid modeling, in which the phases are treated as overlapping continua, with each phase occupying a volume fraction that is a continuous function of position (and time) is a useful approximation which is becoming available in commercial software. See Elghobashi and Abou-Arab ( J. Physics Fluids, 26, 931–938 [1983]) for a k–
model for two-fluid systems.

6-48

FLUID AND PARTICLE DYNAMICS

Figure 6-56 gives an example CFD calculation for time-dependent flow past a square cylinder at a Reynolds number of 22,000 (Choudhury, et al., Trans. ASME Fluids Div., Lake Tahoe, Nev. [1994]). The computation was done with an implementation of the renormalization group theory k–
model. The series of contour plots of stream function shows a sequence in time over about 1 vortex-shedding period. The calculated Strouhal number (Eq. [6-195]) is 0.146, in excellent agreement with experiment, as is the time-averaged drag coefficient, CD = 2.24. Similar computations for a circular cylinder at Re = 14,500 have given excellent agreement with experimental measurements for St and CD (Introduction to the Renormalization Group Method and Turbulence Modeling, Fluent, Inc., 1993).

DIMENSIONLESS GROUPS For purposes of data correlation, model studies, and scale-up, it is useful to arrange variables into dimensionless groups. Table 6-7 lists many of the dimensionless groups commonly found in fluid mechanics problems, along with their physical interpretations and areas of application. More extensive tabulations may be found in Catchpole and Fulford (Ind. Eng. Chem., 58[3], 46–60 [1966]) and Fulford and Catchpole (Ind. Eng. Chem., 60[3], 71–78 [1968]).

FIG. 6-56 Computational fluid dynamic simulation of flow over a square cylinder, showing one vortex shedding period. (From Choudhury, et al., Trans. ASME Fluids Div., TN-076 [1994].)

FLUID DYNAMICS TABLE 6-7

6-49

Dimensionless Groups and Their Significance

Name

Symbol

Formula

Archimedes number

Ar

gL (ρp − ρ)ρ  µ2

inertial forces × buoyancy forces  (viscous forces)2

Particle settling

Bingham number

Bm

τ yL  µ ∞V

yield stress  viscous stress

Flow of Bingham plastics = yield number, Y

Bingham Reynolds number

ReB

LVρ  µ∞

inertial force  viscous force

Flow of Bingham plastics

Blake number

B

Vρ  µ(1 −
)s

inertial force  viscous force

Beds of solids

Bond number

Bo

(ρL − ρG)L2g  σ

gravitational force  surface-tension force

Atomization = Eotvos number, Eo

Capillary number

Ca

µV  σ

viscous force  surface-tension force

Two-phase flows, free surface flows

Cauchy number

C

ρV 2  β

inertial force  compressibility force

Compressible flow, hydraulic transients

Cavitation number

σ

p − pv  ρV 2/2

excess pressure above vapor pressure  velocity head

Cavitation

Dean number

De

Re  (Dc/D)1/2

inertial force Reynolds number ×  centrifugal force

Flow in curved channels

Deborah number

De

λ

fluid relaxation time  flow characteristic time

Viscoelastic flow

Drag coefficient

CD

FD  AρV 2/2

drag force  projected area × velocity head

Flow around objects, particle settling

Elasticity number

El

λµ 2 ρL

elastic force  inertial force

Viscoelastic flow

Euler number

Eu

∆p 2 ρV

frictional pressure loss  2 × velocity head

Fluid friction in conduits

Fanning friction factor

f

D∆ p 2τ w =  2ρV 2L ρV 2

wall shear stress  velocity head

Fluid friction in conduits Darcy friction factor = 4f

Froude number

Fr

V2  gL

inertial force  gravity force

苶 Often defined as Fr = V/兹gL

Densometric Froude number

Fr′

ρV 2  (ρd − ρ)gL

inertial force  gravity force

or

Hedstrom number

He

L2 τYρ  µ ∞2

Bingham Reynolds number × Bingham number

Flow of Bingham plastics

Hodgson number

H

V′ω∆p  q 苶p 苶

time constant of system  period of pulsation

Pulsating gas flow

Mach number

M

V  c

fluid velocity  sonic velocity

Compressible flow

Ohnesorge number

Z

µ  (ρLσ)1/2

viscous force  (inertial force × surface tension force)1/2

Weber number Atomization =  Reynolds number

Peclet number

Pe

LV  D

convective transport  diffusive transport

Heat, mass transfer, mixing

Pipeline parameter

Pn

aVo  2gH

maximum water-hammer pressure rise  2 × static pressure

Water hammer

3

Physical interpretation

Comments

V Fr′ =  苶d苶− 苶苶 苶L 苶/ρ 苶 ρ)g 兹(ρ

Newton number

6-50

FLUID AND PARTICLE DYNAMICS

TABLE 6-7

Dimensionless Groups and Their Significance (Concluded )

Name

Symbol

Formula

Physical interpretation

Comments

Power number

Po

P  ρN 3L5

impeller drag force  inertial force

Prandtl velocity ratio

v+

v  (τ w /ρ)1/2

velocity normalized by friction velocity

Reynolds number

Re

LVρ  µ

inertial force  viscous force

Strouhal number

St

f′ L  V

vortex shedding frequency × characteristic flow time scale

Weber number

We

ρV 2L  σ

Nomenclature a A c D Dc D′ f′ FD g H L N p pv p 苶 ∆p

Wave speed Projected area Sonic velocity Diameter of pipe Diameter of curvature Diffusivity Vortex shedding frequency Drag force Acceleration of gravity Static head Characteristic length Rotational speed Pressure Vapor pressure Average static pressure Frictional pressure drop

Agitation Turbulent flow near a wall, friction velocity = 兹苶 τw苶 /ρ

inertial force  surface tension force SI Units Nomenclature m/s m m/s m m m2/s 1/s N m/s m m 1/s Pa Pa Pa Pa

P q 苶 s v V V′ 
λ µ µ∞ ρ ρG, ρL ρd σ ω

Vortex shedding, von Karman vortex streets Bubble, drop formation SI Units

Power Average volumetric flow rate Particle area/particle volume Local fluid velocity Characteristic or average fluid velocity System volume Bulk modulus Void fraction Fluid relaxation time Fluid viscosity Infinite shear viscosity (Bingham plastics) Fluid density Gas, liquid densities Dispersed phase density Surface tension Characteristic frequency or reciprocal time scale of flow

Watts m3/s 1/m m/s m/s m3 Pa m3 s Pa ⋅ s Pa ⋅ s kg/m3 kg/m3 kg/m3 N/m 1/s

PARTICLE DYNAMICS GENERAL REFERENCES: Brodkey, The Phenomena of Fluid Motions, AddisonWesley, Reading, Mass., 1967; Clift, Grace, and Weber, Bubbles, Drops and Particles, Academic, New York, 1978; Govier and Aziz, The Flow of Complex Mixtures in Pipes, Van Nostrand Reinhold, New York, 1972, Krieger, Huntington, N.Y., 1977; Lapple, et al., Fluid and Particle Mechanics, University of Delaware, Newark, 1951; Levich, Physicochemical Hydrodynamics, PrenticeHall, Englewood Cliffs, N.J., 1962; Orr, Particulate Technology, Macmillan, New York, 1966; Shook and Roco, Slurry Flow, Butterworth-Heinemann, Boston, 1991; Wallis, One-dimensional Two-phase Flow, McGraw-Hill, New York, 1969.

DRAG COEFFICIENT Whenever relative motion exists between a particle and a surrounding fluid, the fluid will exert a drag upon the particle. In steady flow, the drag force on the particle is CD APρu2 FD =  (6-227) 2 where FD = drag force CD = drag coefficient AP = projected particle area in direction of motion ρ = density of surrounding fluid u = relative velocity between particle and fluid The drag force is exerted in a direction parallel to the fluid velocity. Equation (6-227) defines the drag coefficient. For some solid bodies, such as aerofoils, a lift force component perpendicular to the liquid velocity is also exerted. For free-falling particles, lift forces are generally not important. However, even spherical particles experience lift forces in shear flows near solid surfaces.

TERMINAL SETTLING VELOCITY A particle falling under the action of gravity will accelerate until the drag force balances gravitational force, after which it falls at a constant terminal or free-settling velocity ut, given by ut = where

2gm (ρ − ρ)  冪莦 ρρ A C p

p

p

P

(6-228)

D

g = acceleration of gravity m p = particle mass ρp = particle density

and the remaining symbols are as previously defined. Settling particles may undergo fluctuating motions owing to vortex shedding, among other factors. Oscillation is enhanced with increasing separation between the mass and geometric centers of the particle. Variations in mean velocity are usually less than 10 percent. The drag force on a particle fixed in space with fluid moving is somewhat lower than the drag force on a particle freely settling in a stationary fluid at the same relative velocity. Spherical Particles For spherical particles of diameter dp, Eq. (6-228) becomes 4gdp(ρp − ρ) ut =  (6-229) 3ρCD The drag coefficient for rigid spherical particles is a function of particle Reynolds number, Rep = dpρu/µ where µ = fluid viscosity, as shown in Fig. 6-57. At low Reynolds number, Stokes’ Law gives 24 CD =  Rep < 0.1 (6-230) Rep

冪莦

PARTICLE DYNAMICS

6-51

FIG. 6-57 Drag coefficients for spheres, disks, and cylinders: Ap = area of particle projected on a plane normal to direction of motion; C = overall drag coefficient, dimensionless; Dp = diameter of particle; Fd = drag or resistance to motion of body in fluid; Re = Reynolds number, dimensionless; u = relative velocity between particle and main body of fluid; µ = fluid viscosity; and ρ = fluid density. (From Lapple and Shepherd, Ind. Eng. Chem., 32, 605 [1940].)

which may also be written FD = 3π µudp

Rep < 0.1

(6-231)

and gives for the terminal settling velocity gdp2 (ρp − ρ) Rep < 0.1 (6-232) ut =  18µ In the intermediate regime (0.1 < Rep < 1,000), the drag coefficient may be estimated within 6 percent by 24 CD =  1 + 0.14Re 0.70 0.1 < Rep < 1,000 (6-233) p Rep In the Newton’s Law regime, which covers the range 1,000 < Rep < 350,000, CD = 0.445, within 13 percent. In this region, Eq. (6-227) becomes gdp(ρp − ρ) ut = 1.73  1,000 < Rep < 350,000 (6-234) ρ 6 Between about Rep = 350,000 and 1 × 10 , the drag coefficient drops dramatically in a drag crisis owing to the transition to turbulent flow in the boundary layer around the particle, which delays aft separation, resulting in a smaller wake and less drag. Beyond Re = 1 × 106, the drag coefficient may be estimated from (Clift, Grace, and Weber): 8 × 104 CD = 0.19 −  Rep > 1 × 106 (6-235) Rep Drag coefficients may be affected by turbulence in the free-stream flow; the drag crisis occurs at lower Reynolds numbers when the free stream is turbulent. Torobin and Guvin (AIChE J., 7, 615–619 [1961]) found that the drag crisis Reynolds number decreases with increasing free-stream turbulence, reaching a value of 400 when the relative turbulence intensity, defined as 兹苶 u苶苶′/U u苶′ is the rms 苶R is 0.4. Here 兹苶 fluctuating velocity and 苶 UR is the relative velocity between the particle and the fluid. For particles settling in non-Newtonian fluids, correlations are

冢 冣冢

冪莦

冣

given by Dallon and Christiansen (Preprint 24C, Symposium on Selected Papers, part III, 61st Ann. Mtg. AIChE, Los Angeles, Dec. 1–5, 1968) for spheres settling in shear-thinning liquids, and by Ito and Kajiuchi (J. Chem. Eng. Japan, 2[1], 19–24 [1969]) and Pazwash and Robertson (J. Hydraul. Res., 13, 35–55 [1975]) for spheres settling in Bingham plastics. Beris, Tsamopoulos, Armstrong, and Brown ( J. Fluid Mech., 158 [1985]) present a finite element calculation for creeping motion of a sphere through a Bingham plastic. Nonspherical Rigid Particles The drag on a nonspherical particle depends upon its shape and orientation with respect to the direction of motion. The orientation in free fall as a function of Reynolds number is given in Table 6-8. The drag coefficients for disks (flat side perpendicular to the direction of motion) and for cylinders (infinite length with axis perpendicular to the direction of motion) are given in Fig. 6-57 as a function of Reynolds number. The effect of length-to-diameter ratio for cylinders in the Newton’s law region is reported by Knudsen and Katz (Fluid Mechanics and Heat Transfer, McGraw-Hill, New York, 1958). Pettyjohn and Christiansen (Chem. Eng. Prog., 44, 157–172 [1948]) present correlations for the effect of particle shape on freesettling velocities of isometric particles. For Re < 0.05, the terminal or free-settling velocity is given by TABLE 6-8

Free-Fall Orientation of Particles

Reynolds number*

Orientation

0.1–5.5

All orientations are stable when there are three or more perpendicular axes of symmetry. Stable in position of maximum drag. Unpredictable. Disks and plates tend to wobble, while fuller bluff bodies tend to rotate. Rotation about axis of least inertia, frequently coupled with spiral translation.

5.5–200 200–500 500–200,000 SOURCE:

From Becker, Can. J. Chem. Eng., 37, 85–91 (1959). *Based on diameter of a sphere having the same surface area as the particle.

6-52

FLUID AND PARTICLE DYNAMICS gd s2(ρp − ρ) ut = K1  18µ

(6-236)

ψ (6-237) K1 = 0.843 log  0.065 where ψ = sphericity, the surface area of a sphere having the same volume as the particle, divided by the actual surface area of the particle; ds = equivalent diameter, equal to the diameter of the equivalent sphere having the same volume as the particle; and other variables are as previously defined. In the Newton’s law region, the terminal velocity is given by

冢

ut =

冣

4d (ρ − ρ)g  冪莦 3K ρ s

p

(6-238)

3

K3 = 5.31 − 4.88ψ

(6-239)

Equations (6-236) to (6-239) are based on experiments on cubeoctahedrons, octahedrons, cubes, and tetrahedrons for which the sphericity ψ ranges from 0.906 to 0.670, respectively. See also Clift, Grace, and Weber. A graph of drag coefficient vs. Reynolds number with ψ as a parameter may be found in Brown, et al. (Unit Operations, Wiley, New York, 1950) and in Govier and Aziz. For particles with ψ < 0.67, the correlations of Becker (Can. J. Chem. Eng., 37, 85–91 [1959]) should be used. Reference to this paper is also recommended for intermediate region flow. Settling characteristics of nonspherical particles are discussed by Clift, Grace, and Weber, Chaps. 4 and 6. The terminal velocity of axisymmetric particles in axial motion can be computed from Bowen and Masliyah (Can. J. Chem. Eng., 51, 8–15 [1973]) for low–Reynolds number motion: V′ gDs2(ρp − ρ) ut =   (6-240) 18µ K2 K2 = 0.244 + 1.035 − 0.7122 + 0.4413

(6-241)

where Ds = diameter of sphere with perimeter equal to maximum particle projected perimeter V′ = ratio of particle volume to volume of sphere with diameter Ds  = ratio of surface area of particle to surface area of a sphere with diameter Ds and other variables are as defined previously. Hindered Settling When particle concentration increases, particle settling velocities decrease because of hydrodynamic interaction between particles and the upward motion of displaced liquid. The suspension viscosity increases. Hindered settling is normally encountered in sedimentation and transport of concentrated slurries. Below 0.1 percent volumetric particle concentration, there is less than a 1 percent reduction in settling velocity. Several expressions have been given to estimate the effect of particle volume fraction on settling velocity. Maude and Whitmore (Br. J. Appl. Phys., 9, 477–482 [1958]) give, for uniformly sized spheres, ut = ut0 (1 − c)n

(6-242)

where ut = terminal settling velocity ut0 = terminal velocity of a single sphere (infinite dilution) c = volume fraction solid in the suspension n = function of Reynolds number Rep = dput0 ρ/µ as given Fig. 6-58 In the Stokes’ law region (Rep < 0.3), n = 4.65 and in the Newton’s law region (Rep > 1,000), n = 2.33. Equation (6-242) may be applied to particles of any size in a polydisperse system, provided the volume fraction corresponding to all the particles is used in computing terminal velocity (Richardson and Shabi, Trans. Inst. Chem. Eng. [London], 38, 33–42 [1960]). The concentration effect is greater for nonspherical and angular particles than for spherical particles (Steinour, Ind. Eng. Chem., 36, 840–847 [1944]). Theoretical developments for low–Reynolds number flow assemblages of spheres are given by Happel and Brenner (Low Reynolds Number Hydrodynamics, Prentice-

Values of exponent n for use in Eq. (6-240). (From Maude and Whitmore, Br. J. Appl. Phys., 9, 481 [1958]. Courtesy of the Institute of Physics and the Physical Society.)

FIG. 6-58

Hall, Englewood Cliffs, N.J., 1965) and Famularo and Happel (AIChE J., 11, 981 [1965]) leading to an equation of the form ut0 ut =  1 + γc1/3

(6-243)

where γ is about 1.3. As particle concentration increases, resulting in interparticle contact, hindered settling velocities are difficult to predict. Thomas (AIChE J., 9, 310 [1963]) provides an empirical expression reported to be valid over the range 0.08 < ut /ut0 < 1: ut (6-244) ln  = −5.9c ut0

冢 冣

Time-Dependent Motion The time-dependent motion of particles is computed by application of Newton’s second law, equating the rate of change of particle motion to the net force acting on the particle. Rotation of particles may also be computed from the net torque. For large particles moving through low-density gases, it is usually sufficient to compute the force due to fluid drag from the relative velocity and the drag coefficient computed for steady flow conditions. For two- and three-dimensional problems, the velocity appearing in the particle Reynolds number and the drag coefficient is the amplitude of the relative velocity. The drag force, not the relative velocity, is to be resolved into vector components to compute the particle acceleration components. Clift, Grace, and Weber (Bubbles, Drops and Particles, Academic, London, 1978) discuss the complexities that arise in the computation of transient drag forces on particles when the transient nature of the flow is important. Analytical solutions for the case of a single particle in creeping flow (Rep = 0) are available. For example, the creeping motion of a sphericial particle released from rest in a stagnant fluid is described by dU ρ dU ρpV  = g(ρp − ρ)V − 3πµdpU −  V  dt 2 dt

冢冣

3 苶ρ 苶µ 苶 −  d 2p 兹π 2

(dU/dt) ds 冕  苶苶s 兹t苶− t

t=s

(6-245)

0

Here, U = particle velocity, positive in the direction of gravity, and V = particle volume. The first term on the right-hand side is the net gravitational force on the particle, accounting for buoyancy. The second is the steady-state Stokes drag (Eq. 6-231). The third is the added mass or virtual mass term, which may be interpreted as the inertial effect of the fluid which is accelerated along with the particle. The volume of the added mass of fluid is half the particle volume. The last term, the Basset force, depends on the entire history of the transient motion, with past motions weighted inversely with the square root of elapsed time. Clift, et al. provide integrated solutions. In turbulent flows, particle velocity will closely follow fluid eddy velocities when (Clift et al.)

PARTICLE DYNAMICS

6-53

Terminal velocity of air bubbles in water at 20°C. (From Clift, Grace, and Weber, Bubbles, Drops and Particles, Academic, New York, 1978). FIG. 6-59

d p2 [(2ρp /ρ) + 1] τ0 >>  (6-246) 36ν where τ0 = oscillation period or eddy time scale, the right-hand side expression is the particle relaxation time, and ν = kinematic viscosity. Gas Bubbles Fluid particles, unlike rigid solid particles, may undergo deformation and internal circulation. Figure 6-59 shows rise velocity data for air bubbles in stagnant water. In the figure, Eo = Eotvos number, g(ρL − ρG)d b2 /σ, where ρL = liquid density, ρG = gas density, db = bubble diameter, and σ = surface tension. Small bubbles (<1-mm [0.04-in] diameter) remain spherical and rise in straight lines. The presence of surface active materials generally renders small bubbles rigid, and they rise roughly according to the drag coefficient and terminal velocity equations for spherical solid particles. Bubbles roughly in the range 2- to 8-mm (0.079- to 0.32-in) diameter assume flattened, ellipsoidal shape, and rise in a zig-zag or spiral pattern. This motion increases dissipation and drag, and the rise velocity may actually decrease with increasing bubble diameter in this region, characterized by rise velocities in the range of 20 to 30 cm/s (0.7 to 1.0 ft/s). Large bubbles, >8-mm (0.32-in) diameter, are greatly deformed, assuming a mushroomlike, spherical cap shape. These bubbles are unstable and may break into smaller bubbles. Carefully purified water, free of surface active materials, allows bubbles to freely circulate even when they are quite small. Under creeping flow conditions Reb = dburρL /µL < 1, where ur = bubble rise velocity and µL = liquid viscosity, the bubble rise velocity may be computed analytically from the Hadamard-Rybczynski formula (Levich, Physicochemical Hydrodynamics, Prentice-Hall, Englewood Cliffs, N.J., 1962, p. 402). When µG /µL << 1, which is normally the case, the rise velocity is 1.5 times the rigid sphere Stokes law velocity. However, in practice, most liquids, including ordinary distilled water, contain sufficient surface active materials to render small bubbles rigid. Larger bubbles undergo deformation in both purified and ordinary liquids; however, the variation in rise velocity for large bubbles with degree of purity is quite evident in Fig. 6-59. For additional discussion, see Clift, et al., Chap. 7. Figure 6-60 gives the drag coefficient as a function of bubble or drop Reynolds number for air bubbles in water and water drops in air, compared with the standard drag curve for rigid spheres. Information on bubble motion in non-Newtonian liquids may be found in Astarita and Apuzzo (AIChE J., 11, 815–820 [1965]); Calderbank, Johnson, and Loudon (Chem. Eng. Sci., 25, 235–256 [1970]); and Acharya, Mashelkar, and Ulbrecht (Chem. Eng. Sci., 32, 863–872 [1977]).

Liquid Drops in Liquids Very small liquid drops in immisicibile liquids behave like rigid spheres, and the terminal velocity can be approximated by use of the drag coefficient for solid spheres up to a Reynolds number of about 10 (Warshay, Bogusz, Johnson, and Kintner, Can. J. Chem. Eng., 37, 29–36 [1959]). Between Reynolds numbers of 10 and 500, the terminal velocity exceeds that for rigid spheres owing to internal circulation. In normal practice, the effect of drop phase viscosity is neglected. Grace, Wairegi, and Nguyen (Trans. Inst. Chem. Eng., 54, 167–173 [1976]; Clift, et al., op. cit., pp. 175–177) present a correlation for terminal velocity valid in the range M < 10−3 Eo < 40 Re > 0.1 (6-247) where M = Morton number = gµ4∆ρ/ρ2σ3 Eo = Eotvos number = g∆ρd 2/σ Re = Reynolds number = duρ/µ ∆ρ = density difference between the phases ρ = density of continuous liquid phase d = drop diameter µ = continuous liquid viscosity σ = surface tension u = relative velocity

FIG. 6-60 Drag coefficient for water drops in air and air bubbles in water. Standard drag curve is for rigid spheres. (From Clift, Grace, and Weber, Bubbles, Drops and Particles, Academic, New York, 1978.)

6-54

FLUID AND PARTICLE DYNAMICS

The correlation is represented by J = 0.94H0.757

(2 < H ≤ 59.3)

(6-248)

J = 3.42H0.441

(H > 59.3)

(6-249)

µ 4 H =  EoM−0.149  3 µw

冢 冣

where

−0.14

(6-250)

J = ReM 0.149 + 0.857

(6-251)

Note that the terminal velocity may be evaluated explicitly from µ u =  M −0.149( J − 0.857) ρd

(6-252)

In Eq. (6-250), µ = viscosity of continuous liquid and µw = viscosity of water, taken as 0.9 cP (0.0009 Pa ⋅ s). For drop velocities in non-Newtonian liquids, see Mhatre and Kinter (Ind. Eng. Chem., 51, 865–867 [1959]); Marrucci, Apuzzo, and Astarita (AIChE J., 16, 538–541 [1970]); and Mohan, et al. (Can. J. Chem. Eng., 50, 37–40 [1972]). Liquid Drops in Gases Liquid drops falling in stagnant gases appear to remain spherical and follow the rigid sphere drag relationships up to a Reynolds number of about 100. Large drops will deform, with a resulting increase in drag, and in some cases will shatter. The largest water drop which will fall in air at its terminal velocity is about 8 mm (0.32 in) in diameter, with a corresponding velocity of about 9 m/s (30 ft/s). Drops shatter when the Weber number defined as ρGu2d We =  σ

(6-253)

exceeds a critical value. Here, ρG = gas density, u = drop velocity, d = drop diameter, and σ = surface tension. A value of Wec = 13 is often cited for the critical Weber number. Terminal velocities for water drops in air have been correlated by Berry and Prnager (J. Appl. Meteorol., 13, 108–113 [1974]) as Re = exp [−3.126 + 1.013 ln ND − 0.01912(ln ND)2]

(6-254)

for 2.4 < ND < 107 and 0.1 < Re < 3550. The dimensionless group ND (often called the Best number [Clift, et al.]) is given by 4ρ∆ρgd 3 ND =  3µ2

(6-255)

and is proportional to the similar Archimedes and Galileo numbers. Figure 6-61 gives calculated settling velocities for solid spherical particles settling in air or water using the standard drag coefficient curve for spherical particles. For fine particles settling in air, the Stokes-Cunningham correction has been applied to account for particle size comparable to the mean free path of the gas. The correction is less than 1 percent for particles larger than 16 µm settling in air. Smaller particles are also subject to Brownian motion. Motion of particles smaller than 0.1 µm is dominated by Brownian forces and gravitational effects are small. Wall Effects When the diameter of a settling particle is significant compared to the diameter of the container, the settling velocity is reduced. For rigid spherical particles settling with Re < 1, the correction given in Table 6-9 may be used. The factor kw is multiplied by the settling velocity obtained from Stokes’ law to obtain the corrected setTABLE 6-9 Wall Correction Factor for Rigid Spheres in Stokes’ Law Region β*

kw

β

kw

0.0 0.05 0.1 0.2 0.3

1.000 0.885 0.792 0.596 0.422

0.4 0.5 0.6 0.7 0.8

0.279 0.170 0.0945 0.0468 0.0205

SOURCE:

From Haberman and Sayre, David W. Taylor Model Basin Report

1143, 1958. *β = particle diameter divided by vessel diameter.

FIG. 6-61 Terminal velocities of spherical particles of different densities settling in air and water at 70°F under the action of gravity. To convert ft/s to m/s, multiply by 0.3048. (From Lapple, et al., Fluid and Particle Mechanics, University of Delaware, Newark, 1951, p. 292.)

tling rate. For values of diameter ratio β = particle diameter/vessel diameter less than 0.05, kw = 1/(1 + 2.1β) (Zenz and Othmer, Fluidization and Fluid-Particle Systems, Reinhold, New York, 1960, pp. 208–209). In the range 100 < Re < 10,000, the computed terminal velocity for rigid spheres may be multiplied by k′w to account for wall effects, where k′w is given by (Harmathy, AIChE J., 6, 281 [1960]) 1 − β2 k′w = 4 (6-256) +苶 β苶 兹苶1苶 For gas bubbles in liquids, there is little wall effect for β < 0.1. For β > 0.1, see Uto and Kintner (AIChE J., 2, 420–424 [1956]), Maneri and Mendelson (Chem. Eng. Prog., 64, Symp. Ser., 82, 72–80 [1968]), and Collins (J. Fluid Mech., 28, part 1, 97–112 [1967]).

Section 7

Reaction Kinetics

Stanley M. Walas, Ph.D., Professor Emeritus, Department of Chemical and Petroleum Engineering, University of Kansas; Fellow, American Institute of Chemical Engineers

REACTION KINETICS Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Primary Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7-3 7-3 7-3

IDEAL REACTORS

RATE EQUATIONS Rate of Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Law of Mass Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concentration, Moles, Partial Pressure, and Mole Fraction . . . . . . . . . Typical Units of Specific Rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 1: Rates of Change at Constant V or Constant P . . . . . . . . . Reaction Time in Flow Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constants of the Rate Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . From the Differential Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . From the Integrated Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . From Half-Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex Rate Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple Reactions and Stoichiometric Balances . . . . . . . . . . . . . . . . . . Single Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stoichiometric Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 2: Analysis of Three Simultaneous Reactions . . . . . . . . . . . Mechanisms of Some Complex Reactions. . . . . . . . . . . . . . . . . . . . . . . . Phosgene Synthesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ozone and Chlorine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hydrogen Bromide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enzyme Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chain Polymerization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solid Catalyzed Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . With Diffusion between Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Catalysis by Solids: Langmuir-Hinshelwood Mechanism . . . . . . . . . . . . Adsorptive Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dissociation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Different Sites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dual Sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reactant in the Gas Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chemical Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 3: Phosgene Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chemical Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 4: Reaction between Methane and Steam . . . . . . . . . . . . . . Approach to Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 5: Percent Approach to Equilibrium . . . . . . . . . . . . . . . . . . Integration of Rate Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 6: Laplace Transform Application . . . . . . . . . . . . . . . . . . . .

7-5 7-5 7-5 7-5 7-7 7-7 7-7 7-8 7-8 7-8 7-8 7-8 7-8 7-8 7-8 7-10 7-10 7-10 7-10 7-10 7-10 7-10 7-11 7-11 7-11 7-11 7-11 7-11 7-11 7-12 7-12 7-12 7-12 7-13 7-13 7-14 7-14 7-14 7-15

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Material and Energy Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Batch Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Daily Yield. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Filling and Emptying Periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimum Operation of Reversible Reactions . . . . . . . . . . . . . . . . . . . Continuous Stirred Tank Reactors (CSTR) . . . . . . . . . . . . . . . . . . . . . . . Example 7: A Four-Stage Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 8: Consecutive Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 9: Comparison of Batch and CSTR Volumes . . . . . . . . . . . . Different Sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tubular and Packed Bed Flow Reactors . . . . . . . . . . . . . . . . . . . . . . . . . Frictional Pressure Drop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recycle and Separation Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 10: Reactor Size with Recycle. . . . . . . . . . . . . . . . . . . . . . . . Heat Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Batch Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CSTR Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plug Flow Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Packed Bed Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Unsteady Conditions with Accumulation Terms . . . . . . . . . . . . . . . . . . . Example 11: Balances of a Semibatch Process . . . . . . . . . . . . . . . . . .

7-15 7-15 7-15 7-16 7-16 7-16 7-17 7-17 7-17 7-17 7-19 7-19 7-19 7-19 7-20 7-20 7-21 7-21 7-22 7-22 7-22 7-22 7-23

LARGE SCALE OPERATIONS Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple Steady States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonideal Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laminar Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Residence Time Distribution (RTD) . . . . . . . . . . . . . . . . . . . . . . . . . . Segregated Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 12: Segregated Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maximum Mixedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimum Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heterogeneous Reactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reaction and Separation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7-23 7-23 7-23 7-23 7-24 7-25 7-25 7-25 7-25 7-25 7-25 7-25 7-25 7-26 7-26 7-27 7-1

7-2

REACTION KINETICS

ACQUISITION OF DATA Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Batch Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple Phases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solid Catalysts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References for Laboratory Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7-27 7-27 7-27 7-27 7-28 7-28 7-28 7-28

SOLVED PROBLEMS Equilibrium of Formation of Ethylbenzene . . . . . . . . . . . . . . . . . Optimum Cycle Period with Downtime . . . . . . . . . . . . . . . . . . . .

7-28 7-29

P1. P2.

P3. P4. P5. P6. P7. P8. P9. P10. P11. P12. P13. P14. P15. P16.

Parallel Reactions of Butadiene . . . . . . . . . . . . . . . . . . . . . . . . . . . Batch Reaction with Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . A Semibatch Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimum Reaction Temperature with Downtime. . . . . . . . . . . . . Rate Equations from CSTR Data . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of Batch and CSTR Operations . . . . . . . . . . . . . . . . . Instantaneous and Gradual Feed Rates . . . . . . . . . . . . . . . . . . . . . Filling and Unsteady Operating Period of a CSTR . . . . . . . . . . . Second-Order Reaction in Two Stages . . . . . . . . . . . . . . . . . . . . . Butadiene Dimerization in a TFR . . . . . . . . . . . . . . . . . . . . . . . . Autocatalytic Reaction with Recycle . . . . . . . . . . . . . . . . . . . . . . Minimum Residence Time in a PFR . . . . . . . . . . . . . . . . . . . . . . Heat Transfer in a Cylindrical Reactor. . . . . . . . . . . . . . . . . . . . . Pressure Drop and Conversion in a PFR . . . . . . . . . . . . . . . . . . .

7-29 7-29 7-30 7-30 7-30 7-31 7-31 7-31 7-32 7-32 7-32 7-32 7-33 7-33

Nomenclature and Units Following is a listing of typical nomenclature expressed in SI and U.S. customary units. Specific definitions and units are stated at the place of application in this section.

Symbol A, B, C, . . . •

A Ca C0 Cp CSTR D, De, Dx Deff DK E(t) E(tr) fa F(t) ∆G Ha ∆Hr K, Ke, Ky, Kφ k, kc, kp L n na n′a nt pa Pe PFR Q r ra R Re Sc

Definition Names of substances, or their concentrations Free radical, as CH •3 Concentration of substance A Initial mean concentration in vessel Heat capacity Continuous stirred tank reactor Dispersion coefficient Effective diffusivity Knudsen diffusivity Residence time distribution Normalized residence time distribution Ca /Ca 0 or na /na 0, fraction of A remaining unconverted Age function of tracer Gibbs energy change Hatta number Heat of reaction Chemical equilibrium constant Specific rate of reaction Length of path in reactor Parameter of Erlang or Gamma distribution, or number of stages in a CSTR battery Number of mols of A present Number of mols flowing per unit time; the prime (′) may be omitted when context is clear Total number of mols Partial pressure of substance A Peclet number for dispersion Plug flow reactor Heat transfer Radial position Rate of reaction of A per unit volume Radius of cylindrical vessel Reynolds number Schmidt number

SI units

U.S. customary units

kg mol/m3 lb mol/ft3 kg mol/m3 lb mol/ft3 kJ/(kg⋅K)

Btu/(lbm⋅°F)

m2/s m2/s m2/s

ft2/s ft2/s ft2/s

Symbol t t
tr TFR u u(t) V V′ Vr x xa z

Definition Time Mean residence time t/t
, reduced time Tubular flow reactor Linear velocity Unit step input Volume of reactor contents Volumetric flow rate Volume of reactor Axial position in a reactor 1 − fa = 1 − Ca /Ca0 or 1 − na /na0, fraction of A converted x/L, normalized axial position

U.S. customary units

SI units s s

s s

m/s

ft/s

m3 m3/s m3 m

ft3 ft3/s ft3 ft

Pa⋅s m2/s Pa kg/m3

lbm/(ft⋅s) ft2/s psi lbm/ft3

Greek letters

kJ

Btu

β γ3(t) δ(t)

kJ/kg mol

Btu/lb mol

ε

Variable m

Variable ft

␪ η Λ(t) µ ν π ρ ρ

kPa

psi

kJ m Variable

Btu ft Variable

m

ft

σ2(t) σ2(tr) τ τ φ φm

r/R, normalized radial position Skewness of distribution Unit impulse input, Dirac function Fraction void space in a packed bed t/t
, reduced time, fraction of surface covered by adsorbed species Effectiveness of porous catalyst Intensity function Viscosity υ/ρ, kinematic viscosity Total pressure Density r/R, normalized radial position in a pore Variance Normalized variance t/t
, reduced time Tortuosity Thiele modulus Modified Thiele modulus Subscripts

0

Subscript designating initial or inlet conditions, as in Ca0, na0, V′0 , . . .

REACTION KINETICS GENERAL REFERENCES 1. Aris, Elementary Chemical Reactor Analysis, Prentice-Hall, 1969. 2. Bamford and Tipper (eds.), Comprehensive Chemical Kinetics, Elsevier, 1969–date. 3. Boudart, Kinetics of Chemical Processes, Prentice-Hall, 1968. 4. Brotz, Fundamentals of Chemical Reaction Engineering, Addison-Wesley, 1965. 5. Butt, Reaction Kinetics and Reactor Design, Prentice-Hall, 1980. 6. Capello and Bielski, Kinetic Systems: Mathematical Description of Kinetics in Solution, Wiley, 1972. 7. Carberry, Chemical and Catalytic Reaction Engineering, McGraw-Hill, 1976. 8. Carberry and Varma (eds.), Chemical Reaction and Reactor Engineering, Dekker, 1987. 9. Chen, Process Reactor Design, Allyn & Bacon, 1983. 10. Cooper and Jeffreys, Chemical Kinetics and Reactor Design, PrenticeHall, 1971. 11. Cremer and Watkins (eds.), Chemical Engineering Practice, vol. 8: Chemical Kinetics, Butterworths, 1965. 12. Denbigh and Turner, Chemical Reactor Theory, Cambridge, 1971. 13. Fogler, Elements of Chemical Reaction Engineering, Prentice-Hall, 1992. 14. Froment and Bischoff, Chemical Reactor Analysis and Design, Wiley, 1990. 15. Hill, An Introduction to Chemical Engineering Kinetics and Reactor Design, Wiley, 1977. 16. Holland and Anthony, Fundamentals of Chemical Reaction Engineering, Prentice-Hall, 1989.

17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.

7-3

Horak and Pasek, Design of Industrial Chemical Reactors from Laboratory Data, Heyden, 1978. Kafarov, Cybernetic Methods in Chemistry and Chemical Engineering, Mir Publishers, 1976. Laidler, Chemical Kinetics, Harper & Row, 1987. Levenspiel, Chemical Reaction Engineering, Wiley, 1972. Lewis (ed.), Techniques of Chemistry, vol. 4: Investigation of Rates and Mechanisms of Reactions, Wiley, 1974. Naumann, Chemical Reactor Design, Wiley, 1987. Panchenkov and Lebedev, Chemical Kinetics and Catalysis, Mir Publishers, 1976. Petersen, Chemical Reaction Analysis, Prentice-Hall, 1965. Rase, Chemical Reactor Design for Process Plants: Principles and Case Studies, Wiley, 1977. Rose, Chemical Reactor Design in Practice, Elsevier, 1981. Smith, Chemical Engineering Kinetics, McGraw-Hill, 1981. Steinfeld, Francisco, and Hasse, Chemical Kinetics and Dynamics, Prentice-Hall, 1989. Ulrich, Guide to Chemical Engineering Reactor Design and Kinetics, Ulrich, 1993. Walas, Reaction Kinetics for Chemical Engineers, McGraw-Hill, 1959; reprint, Butterworths, 1989. Walas, Chemical Reaction Engineering Handbook of Solved Problems, Gordon & Breach Publishers, 1995. Westerterp, van Swaaij, and Beenackers, Chemical Reactor Design and Operation, Wiley, 1984.

REACTION KINETICS INTRODUCTION From an engineering viewpoint, reaction kinetics has these principal functions: Establishing the chemical mechanism of a reaction Obtaining experimental rate data Correlating rate data by equations or other means Designing suitable reactors Specifying operating conditions, control methods, and auxiliary equipment to meet the technological and economic needs of the reaction process Reactions can be classified in several ways. On the basis of mechanism they may be: 1. Irreversible 2. Reversible 3. Simultaneous 4. Consecutive A further classification from the point of view of mechanism is with respect to the number of molecules participating in the reaction, the molecularity: 1. Unimolecular 2. Bimolecular and higher Related to the preceding is the classification with respect to order. In the power law rate equation r = kCap Cbq, the exponent to which any particular reactant concentration is raised is called the order p or q with respect to that substance, and the sum of the exponents p + q is the order of the reaction. At times the order is identical with the molecularity, but there are many reactions with experimental orders of zero or fractions or negative numbers. Complex reactions may not conform to any power law. Thus, there are reactions of: 1. Integral order 2. Nonintegral order 3. Non–power law; for instance, hyperbolic With respect to thermal conditions, the principal types are: 1. Isothermal at constant volume 2. Isothermal at constant pressure 3. Adiabatic 4. Temperature regulated by heat transfer According to the phases involved, reactions are: 1. Homogeneous, gaseous, liquid or solid 2. Heterogeneous:

Controlled by diffusive mass transfer Controlled by chemical factors A major distinction is between reactions that are: 1. Uncatalyzed 2. Catalyzed with homogeneous or solid catalysts Equipment is also a basis for differentiation, namely: 1. Stirred tanks, single or in series 2. Tubular reactors, single or in parallel 3. Reactors filled with solid particles, inert or catalytic: Fixed bed Moving bed Fluidized bed, stable or entrained Finally, there are the operating modes: 1. Batch 2. Continuous flow 3. Semibatch or semiflow Clearly, these groupings are not mutually exclusive. The chief distinctions are between homogeneous and heterogeneous reactions and between batch and flow reactions. These distinctions most influence the choice of equipment, operating conditions, and methods of design. PRIMARY NOMENCLATURE The participant A is identified by the subscript a. Thus, the concentration is Ca; the number of mols is na; the fractional conversion is xa; the partial pressure is pa; and the rate of decomposition is ra. Capital letters are also used to represent concentration on occasion; thus, A instead of Ca. The flow rate in mol is n′a but the prime (′) is left off when the meaning is clear from the context. The volumetric flow rate is V′; reactor volume is Vr or simply V of batch reactors; the total pressure is π; and the temperature is T. The concentration is Ca = na /V or n′a / V′. Throughout this section, equations are presented without specification of units. Use of any consistent unit set is appropriate. SUMMARY Basic kinetic relations of this section are summarized in Table 7-1.

TABLE 7-1

Basic Rate Equations The relation between ra and Ca must be established (numerically if need be) from the second line before the integration can be completed.

1. The reference reaction is νa A + νbB + ⋅ ⋅ ⋅ → νrR + νsS + ⋅ ⋅ ⋅ ∆ν = νr + νs + ⋅ ⋅ ⋅ − (νa + νb + ⋅ ⋅ ⋅) 2. Stoichiometric balance for any component i: ν ni = ni0 ⫾ ᎏi (na0 − na) νa + for product (right-hand side, RHS) − for reactant (left-hand side, LHS) ν Ci = Ci0 ⫾ ᎏi (Ca0 − Ca), at constant T and V only νa

 

{

 

∆ν nt = nt0 + ᎏ (na0 − na) νa

 

 

ν ra = kCαa [Cb0 − ᎏb (Ca0 − Ca)]β⋅ ⋅ ⋅ νa where it is not necessarily true that α = νa′, β = νb′, ⋅ ⋅ ⋅

 

4. At constant volume, Ca = na/Vr Ca0

Ca



na0

na

5. Ideal gases at constant pressure:



  

α − 1 na0 na



[nt0 + (∆ν/νa)(na0 − na)]α − 1 dna ᎏᎏᎏ nαa

b′ −E k = k∞ exp ᎏ = exp a′ − ᎏ RT T E = energy of activation





k1

A+B→C+D (1) k2 (2) C+D→A+B k3 (3) A+C→E The rates are related by: ra = ra1 + ra2 + ra3 = k1CaCb − k2CcCd + k3CaCc rb = −rd = k1CaCb − k2CcCd rc = k1CaCb + k2CcCd + k3CaCc re = −k3CaCc The number of independent rate equations is the same as the number of independent stoichiometric relations. In the present example, Reactions (1) and (2) are reversible reactions and are not independent. Accordingly, Cc and Cd, for example, can be eliminated from the equations for ra and rb which then become an integrable system. Usually only systems of linear differential equations with constant coefficients are solvable analytically. 8. Mass transfer resistance: Cai = interfacial concentration of reactant A





Ca0

Ca

7-4

1 dCa ᎏᎏ (Ca − ra/kd)α

ra = kPaPbθ3v 1 θv = ᎏᎏᎏ (1 + K
a
Pa
+ KbPb + ⋅ ⋅ ⋅)

(3)

n Pa = ᎏa P nt ni ni0 ⫾ (νi/νa)(na0 − na) Pi = ᎏ P = ᎏᎏᎏ P nt nt0 + (∆ν/νa)(na0 − na)

ntRT P

kt =



na0

na

dna ᎏ2 , VPaPbθv

⯗

dC r ra = − ᎏa = kd(Ca − Cai) = kCαai = k Ca − ᎏa kd dt

(2)

• Reaction A2 + B → R + S, with A2 dissociated upon adsorption and with surface reaction rate controlling:

for a Case (2) batch reaction

10. A continuously stirred tank reactor (CSTR) battery Material balances: n′a0 = n′a + ra1Vr1

7. Simultaneous reactions. The overall rate is the algebraic sum of the rates of the individual reactions. For example, take the three reactions:

kt =

summation over all substances absorbed

V=ᎏ

6. Temperature effect on the specific rate:

 

Ke = PrPs /PaPb (equilibrium constant)

products, RHS +− for for reactants, LHS

∆ν ntRT RT Vr = ᎏ = ᎏ nt0 + ᎏ (na0 − na) P P νa ra = kCαa RT kt = ᎏ P

(1)

• At constant P and T the Pi are eliminated in favor of ni and the total pressure by:

1 dCa ᎏᎏᎏᎏ Caα[Cb0 − (νb /νa)(Ca0 − Ca)]β ⋅ ⋅ ⋅

Vr−1 + α + β dna ᎏᎏᎏ nαa [nb0 + (νb /νa)(na0 − na)]β⋅ ⋅ ⋅ Completed integrals for some values of α and β are in Table 7-4. kt =



1 θv = ᎏᎏ , 1 + KjPj

ν = kC [Cb0 − ᎏb (Ca0 − Ca)]νb⋅ ⋅ ⋅ νa



/

Ka PrPs θv = 1 1 + ᎏ ᎏ + KbPb + KrPr + KsPs + KlPl Ke Pb

r = kPaPbθv2

1 dna ra = − ᎏ ᎏ = kC νaaCbνb⋅ ⋅ ⋅ Vr dt

kt =

1 dn ra = − ᎏ ᎏa = kPaθv V dt

l is an adsorbed substance that is chemically inert. • Surface reaction rate controlling:

3. Law of mass action:

νa a

9. Solid-catalyzed reactions. Some Langmuir-Hinshelwood mechanisms for the reference reaction A + B → R + S (see also Tables 7.2, 7.3): • Adsorption rate of A controlling:

n′a,j − 1 = n′aj + rajVrj,

for the jth stage

For a first-order reaction, with ra = kCa: Caj 1 ᎏ = ᎏᎏᎏᎏ Ca0 (1 + k1t
1)(1 + k2 t
2)⋅ ⋅ ⋅(1 + kj t
j) 1 = ᎏj (1 + kt
i) for j tanks in series with the same temperatures and residence times t
i = Vri /V′i, where V′ is the volumetric flow rate.

11. Plug flow reactor (PFR): α



dn′a rs = − ᎏ = kCαa Cbβ⋅ ⋅ ⋅ dVr α

β

n′ ⋅⋅⋅   ᎏ V′ 

n′a =k ᎏ V′

b

(4)

RATE EQUATIONS TABLE 7-1

Basic Rate Equations (Concluded )

12. Material and energy balances for batch, CSTR, and PFR in Tables 7-5, 7-6, and 7-7. SOURCE:

7-5

13. Notation. A, B, R, S are participants in the reaction; the letters also are used to represent concentrations. Ci = ni/Vr or n′i /V′, concentration ni = mol of component i in the reactor n′i = molal flow rate of component i Vr = volume of reactor V′ = volumetric flow rate νi = stoichiometric coefficient ri = rate of reaction of substance i [mol/(unit time)(unit volume)] α,β = empirical exponents in a rate equation

Adapted from Walas, Chemical Process Equipment Selection and Design, Butterworth-Heinemann, 1990.

RATE EQUATIONS RATE OF REACTION The term rate of reaction means the rate of decomposition per unit volume, 1 dn ra = − ᎏ ᎏa , mol/(unit time) (unit volume) (7-1) V dt na0 dxa (7-2) = ᎏ ᎏ, n0 = na0(1 − xa) V dt where xa is the fractional conversion of substance A. A rate of formation will have the opposite sign. The negative sign is required for the rate of decomposition to be a positive number. When the volume is constant, dC ra = − ᎏa only at constant volume (7-3) dt Law of Mass Action The effect of concentration on the rate is isolated as ra = kf(Ca, Cb, . . .)

the rate equation is (7-6)

dCa ⇒−ᎏ at constant volume (7-7) dt The exponents (p, q, r, . . .) are empirical, but they are identical with the stoichiometric coefficients (νa, νb, νc, . . .) when the stoichiometric equation truly represents the mechanism of reaction. The first group of exponents identifies the order of the reaction, the stoichiometric coefficients the molecularity. Effect of Temperature The Arrhenius equation relates the specific rate to the absolute temperature, −E k = k0 exp ᎏ (7-8) RT

  B = exp A − ᎏ  T

CONCENTRATION, MOLES, PARTIAL PRESSURE, AND MOLE FRACTION Any property of a reacting system that changes regularly as the reaction proceeds can be formulated as a rate equation which should be convertible to the fundamental form in terms of concentration, Eq. (7-4). Examples are the rates of change of electrical conductivity, of pH, or of optical rotation. The most common other variables are partial pressure pi and mole fraction Ni. The relations between these units are nt pi ni = VCi = nt Ni = ᎏ (7-11) π where the subscript t denotes the total mol and π the total pressure. For ideal gases, nt RT V=ᎏ π

(7-4)

where the specific rate k is independent of concentration but does depend on temperature, catalysts, and other factors. The law of mass action states that the rate is proportional to the concentrations of the reactants. For the reaction νa A + νb B + νcC + . . . ⇒ νr R + νsS + . . . (7-5) 1 dn ra = − ᎏ ᎏa = kCap Cbq Ccr . . . V dt

last equation, the reaction is believed to have a complex mechanism (Fig. 7-1g).

(7-9)

B ln k = A − ᎏ (7-10) T E is called the activation energy and k0 the preexponential factor. When presumably accurate data deviate from linearity as stated by the

nt RT ni V πV (7-12) ni = ᎏ Ci = ᎏ pi = ᎏ pi = ᎏ Ni π π RT RT Other volume-explicit equations of state are sometimes required, such as the compressibility equation V = zRT/P or the truncated virial equation V = (1 + B′P)RT/P. The quantities z and B′ are not constants, so some kind of averaging will be required. More accurate equations of state are even more difficult to use but are not often justified for kinetic work. Designate δa as the increase in the total mol per mol decrease of substance A according to the stoichiometric equation Eq. (7-5): (νr + νs + . . .) − (νa + νb + νc + . . .) δa = ᎏᎏᎏᎏ (7-13) νa The total number of mols present is nt = nt0 + δa(na0 − na) = nt0 + δa xa = nt0 + δb xb = . . . (7-14) Accordingly, dn dn dn dn − ᎏt = δa ᎏa = δ b ᎏb = δc ᎏc = . . . dt dt dt dt The various differentials are 1 nt dni = d(VCi) = ᎏ d(Vpi) = d(nt Ni) = ᎏ dNi RT 1 + δi Ni The rate equation 1 dCa ra = − ᎏ ᎏ = kcCaα V dt

(7-15)

(7-16)

(7-17)

ln C

l/C q – 1

REACTION KINETICS

t r = kCq, slope = k(q – 1) (a)

t r = kC, slope = –k (b)

( b)

ln r

ln t1/2

(a)

ln C

ln C0 r = kCq, slope = 1 – q (d)

r = kCq, slope = q (c)

(c)

( d)

108

ln k

ln k4

107 106 105 104

l/T k = exp(A – E/RT), slope = –E/R (e) (e)

Range 800–3000 K 103 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1000/T Variable activation energy (f) ( f)

106 k

7-6

220 Oxidation of acetylene on manganese dioxide 200 catalyst 180 External diffusion 160 140 Pore diffusion 120 100 80 60 Chemical rate 40 20 10 12 14 16 18 20 22 24 26 28 30 10,000/T (g)

Constants of the power law and Arrhenius equations by linearization: (a) integrated equation, (b) integrated first order, (c) differential equation, (d) half-time method, (e) Arrhenius equation, ( f) variable activation energy, and (g) change of mechanism with temperature (T in K).

FIG. 7-1

RATE EQUATIONS can be expressed in terms of pressure and mole fraction, α

 p

1 d(Vpa ) 1 − ᎏ ᎏ = kc ᎏ RTV dt RT or at constant volume,

α a

 

(7-19)

1 α−1 (7-20) kp = kc ᎏ RT Typical Units of Specific Rates For order α, typical units are:

 

kc kp

(L/g mol)α − 1 ⋅s−1, and s−1 when first order (g mol)/L⋅ s⋅atmα

α−1

1 N   ᎏ 1+δ N 

α a

π α−1 1 = kc ᎏ ᎏ N aα RT 1 + δa Na Various derivatives are evaluated in numerical Example 1.



(7-21)

Consider the ideal gas reaction 2A ⇒ B + 2C occurring at 800°R, starting with 5 lb mol of pure A at 10 atm. The rate equation is

1 dn ra = − ᎏ ᎏa = 700 Ca2 lb mol/(ft3 ⋅h) V dt Evaluate the various rates of change at the time when the rate of reaction is ra = 0.1 lb mol/(ft3⋅h) and the reaction proceeds at (1) constant volume, and (2) constant pressure. 1 dn ra = − ᎏ ᎏa = 700 Ca2 = 0.1 lb mol/(ft3⋅h) V dt 0.1 ᎏ = 0.01195 lb mol/ft

 700

3

na 0 RT 5(0.729)(800) V0 = ᎏ = ᎏᎏ = 291.6 ft3 π0 10 5 na 0 Ca0 = ᎏ = ᎏ = 0.01715 lb mol/ft3 V0 291.6 nt = 0.5(3na 0 − na) lb mol (3na 0 − na) RT V = ᎏᎏ = 29.16(15 − na) ft3 2π0 n π = ᎏt π0 = 3na 0 − na atm nt 0 At constant volume, na = V0Ca = 291.6(0.01195) = 3.4853 lb mol dna dCa ᎏ = V0 ᎏ = −291.6(0.1) = −29.16 lb mol/h dt dt n 2na Na = ᎏa = ᎏ nt 3na 0 − na

 

dNa 6 dna 6 ᎏ = ᎏᎏ2 ᎏ = ᎏᎏ2 (−29.16) dt na 0(3 − na /na 0) dt 5(3 − 3.4853/5) = −6.598 h−1 na RT pa = ᎏ atm V0

 

V = 29.16 (15 − 3.8768) = 324.4 ft3 n na Ca = ᎏa = ᎏᎏ lb mol/ft3 V 29.16(15 − na )







2π0na pa = Naπ0 = ᎏ atm 3na 0 − na

Example 1: Rates of Change at Constant V or Constant P

Ca =

since

dNa 6na 0 dna 30 ᎏ = ᎏᎏ2 ᎏ = ᎏᎏ2 (−32.44) = −7.8658 h−1 dt (3na 0 − na) dt (15 − 3.8768)

a

  

dna ᎏ = −Vra = −324.4(0.1) = −32.44 lb mol/h dt

= 0.1349 lb mol/(ft3 ⋅h)

α a

or a



dπ 5 dna ᎏ = − ᎏ ᎏ = 29.16 atm/h dt na 0 dt At constant pressure, na = VCa = 29.16(15 − na) (0.01195) = 3.8768 lb mol



α

 N

dN n − ᎏa = kc ᎏt dt V



dCa 1 15 dna 1 15 ᎏ = ᎏ ᎏᎏ2 ᎏ = ᎏ ᎏᎏ2 (−32.44) dt 29.16 (15 − na) dt 29.16 (15 − 3.8768)

Furthermore, nt dNa nt − ᎏᎏ ᎏ = kc ᎏ V(1 + δaNa) dt V

n 3na 0 − na na π = ᎏt π0 = ᎏᎏ π0 = 5 3 − ᎏ atm nt 0 2na0 na 0

(7-18)

dpa 1 α−1 α pa = k p paα − ᎏ = kc ᎏ dt RT where the specific rate in terms of partial pressure is

7-7

dpa RT dna 0.729(800) ᎏ = ᎏ ᎏ = ᎏᎏ (−29.16) = −58.32 atm/h dt V0 dt 291.6

dpa 6π0 dna (6)(10)(5) ᎏ = ᎏᎏ2 ᎏ = ᎏᎏ2 (−32.44) = −78.66 atm/h dt (3na 0 − na) dt (15 − 3.8768) (3na 0 − na)RT 3 V = ᎏᎏ ft 2π0





dV RT dna 0.729(800) ᎏ = ᎏ − ᎏ = ᎏᎏ (32.44) = 945.95 ft3/h dt 2π0 dt 20 d na 0 − na 1 dxa 1 dna ᎏ = ᎏ ᎏ = − ᎏ ᎏ = − ᎏ (−32.44) = 6.488 h−1 dt dt na0 na0 dt 5





SUMMARY Rate

At constant V

At constant P

dna dt, lb mol/h dNa /dt, h−1 dpa /dt, atm/h dπ/dt, atm/h dV/dt, ft3/h dxa /dt, h−1

−29.16 −6.598 −58.32 29.16 0 5.832

−32.44 −7.866 −78.66 0 946.0 6.488

REACTION TIME IN FLOW REACTORS Flow reactors usually operate at nearly constant pressure, and thus at variable density when there is a change of moles of gas or of temperature. An apparent residence time is the ratio of reactor volume and the inlet volumetric flow rate, Vr t
app = ᎏ (7-22) V′0 The true residence time is obtained by integration of the rate equation, t
=

dn dV dn ᎏ =  ᎏ =  ᎏᎏ V′ V′r kV′(n/V′) r

q

(7-23)

The apparent time is readily evaluated and is popularly used to indicate the loading of a flow reactor. A related concept is that of space velocity, which is a ratio of a flow rate at STP (usually 60°F, 1 atm) to the size of the reactor. The most common versions in typical units are: GHSV (gas hourly space velocity) = (volumes of feed as gas at STP/h)/(volume of reactor or its content of catalyst) = SCFH gas feed/ft3. LHSV (liquid hourly space velocity) = (volume of liquid feed at 60°F/h)/(ft3 of reactor) = SCFH liquid feed/ft3. WHSV (weight hourly space velocity) = (lb feed/h)/(lb catalyst). It is usually advisable to spell out the units when the acronym is used, since the units are arbitrary.

7-8

REACTION KINETICS

CONSTANTS OF THE RATE EQUATION The problem is to apply experimental data to find the constants of assumed rate equations, of which some of the simpler examples are: dC r = − ᎏ = kCq dt

(7-24)





(7-25)

dCa q (7-26) ra = − ᎏ = kCap Cb dt Experimental data that are most easily obtained are of (C, t), (p, t), (r, t), or (C, T, t). Values of the rate are obtainable directly from measurements on a continuous stirred tank reactor (CSTR), or they may be obtained from (C, t) data by numerical means, usually by first curve fitting and then differentiating. When other properties are measured to follow the course of reaction—say, conductivity—those measurements are best converted to concentrations before kinetic analysis is started. The most common ways of evaluating the constants are from linear rearrangements of the rate equations or their integrals. Figure 7-1 examines power law and Arrhenius equations, and Fig. 7-2 has some more complex cases. From the Differential Equation Linear regression can be applied with the differential equation to obtain constants. Taking logarithms of Eq. (7-25), or

b ln r = a − ᎏ + q ln C (7-27) T The variables that are combined linearly are ln r, 1/T, and ln C. Multilinear regression software can be used to find the constants, or only three sets of the data suitably spaced can be used and the constants found by simultaneous solution of three linear equations. For a linearized Eq. (7-26) the variables are logarithms of r, Ca, and Cb. The logarithmic form of Eq. (7-24) has only two constants, so the data can be plotted and the constants read off the slope and intercept of the best straight line. From the Integrated Equation The integral of Eq. (7-24) is 1 C k = ᎏ ln ᎏ0 , t − t0 C

when q = 1

(7-28)

C0 − 1 C0 q − 1 − 1 , when q ≠ 1 (7-29) ᎏᎏ ᎏ (t − t0) (q − 1) C A value of q is assumed and values of k are calculated for each data point. The correct value of q has been chosen when the values of k are nearly constant or show no drift. This procedure is applicable for a rate equation of any complexity if it can be integrated. Eqs. (7-28) and (7-29) can also be put into linear form: q

 

 

C ln ᎏ0 = k(t − t0), C q−1



when q = 1

(7-30)

1 q−1 = ᎏ + k(q − 1) (t − t0), when q ≠ 1 (7-31) C0 When the plots are collinear, the correct value of k is found from the slope of the best straight line. From Half-Times The time by which one-half of the reactant has been converted is called the half-time. From Eq. (7-24),

ᎏC  1

 

kt1/2 = ln 2,

[r − f (C , a, b, . . .)] i

dC b r = − ᎏ = exp a − ᎏ Cq dt t

or

Complex Rate Equations Complex rate equations may require individual treatment, although the examples in Fig. 7-2 are all linearizable. A perfectly general procedure is nonlinear regression. For instance, when r = f(C, a, b, . . .) where (a, b, . . .) are the constants to be found, the condition is

q=1

2q − 1 − 1 q≠1 (7-32) ᎏᎏ q (q − 1)C0 − 1 When several sets of (C0, t1/2) are known, values of q are tried until one is found that makes all k values substantially the same. Alternatively, the constants may be found from a linearized plot, 2q − 1 − 1 ln t1/2 = ln ᎏ + (1 − q) ln C0 (q − 1)k

(7-33)

2

i

⇒ Minimum

(7-34)

∂Σ ∂Σ (7-35) ᎏ=ᎏ=...=0 ∂a ∂b Much professional software is devoted to this problem. A diskette for sets of differential and algebraic equations with parameters to be found by this method is by Constantinides (Applied Numerical Methods with Personal Computers, McGraw-Hill, 1987). The acquisition of kinetic data and parameter estimation can be at quite a sophisticated level, particularly for solid catalytic reactions: statistical design of experiments, refined equipment, computer monitoring of data acquisition, and statistical evaluation of the data. Two papers are devoted to this topic by Hofmann (in Chemical Reaction Engineering, ACS Advances in Chemistry, 109, 519–534 [1972]; in de Lasa, ed., Chemical Reactor Design and Technology, Martinus Nijhoff, 1985, pp. 69–105).

and

MULTIPLE REACTIONS AND STOICHIOMETRIC BALANCES Single Reaction For the stoichiometric equation, Eq. (7-5), the relations between the conversions of the several participants are x na0 − na nb0 − nb nr0 − nr ns0 − ns ᎏ=ᎏ=ᎏ=...=−ᎏ=−ᎏ=... νa νa νb νr νs n na0 − x Ca = ᎏa = ᎏ , V V

nb0 − νb x/νa Cb = ᎏᎏ , V

(7-36)

Cc0 − νc x/νa Cc = ᎏᎏ , and so on V (7-37)

ν Cb = Cb0 − ᎏb (Ca0 − Ca) νa

Also,

νc (7-38) Cc = Cc0 − ᎏ (Ca0 − Ca), and so on νa Accordingly, the rate equation can be written in terms of the single dependent variable x; thus, 1 dn 1 dx ra = − ᎏ ᎏa = ᎏ ᎏ V dt V dt na0 − x p nb0 − νb x/νa =k ᎏ ᎏᎏ V V and in terms of concentrations,





n − ν x /ν . . .  ᎏᎏ  V q

c0

c

a

r

(7-39)

dC q ra = − ᎏa = kCap Cb Ccr . . . dt q r νb ν Cc0 − ᎏc (Ca0 − Ca) . . . (7-40) = kCap Cb0 − ᎏ (Ca0 − Ca) νa νa Eq. (7-39) becomes integrable when V is properly expressed in terms of the composition of the system, and Eq. (7-40) can be integrated as it stands. Multiple Reactions When a substance participates in several reactions at the same time, its net rate of decomposition is the algebraic sum of its rates in the individual reactions. Identify the rates of the individual steps with subscripts, (dC/dt)1, (dC/dt)2, . . . . Take this case of three reactions,





1

A+B⇒C 2

A+C⇒D+E 3

D+E⇒A+C



RATE EQUATIONS 12

6

11

5 4

9

C/r

t/(C0 – C)

10

3

8 2

7 6

1

y = 5.03 + 0.99×

C/r = 0.72 + 5.73C 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 C (b)

5 1

2

3 4 5 In (C0/C)/(C0 – C)

6

7

(a) (a )

(b)

20 19

110

18

100

17 y2

90

y2

15

80

y1

y1

16 14

70

13 12 11 10

60 50 0

1

2

3

4

5

x1, x2

(c)) (c 9.00

–3.25

Regression data Calculated value

7.80

–3.75 –4.25

5.40

–4.75

4.20

–5.25

y

6.60

3.00 0

2

4

6 8 Point no. (d)

10

–5.75 0

12

2

4

6 8 Point no. (e)

10

12

FIG. 7-2 Linear analysis of catalytic rate equations. (a), (b) Sucrose hydrolysis with an enzyme, r = kM/(M + C). Data are (C, t) curve-fitted with a fourth-degree polynomial and differentiated for r − (−dC/dt). Integrated equation, t 1 M ln (C0 /C) k = 0.199, M = 4.98 ᎏ = ᎏ + ᎏ ᎏ, C0 − C k k C0 − C Linearized rate equation, C M C M = 4.13 ᎏ = ᎏ + ᎏ, r k k poor agreement. (c) For a solid catalyzed reaction, two possible equations in linear form are y1 = Pa /r = a + bpa and y2 = Pa /r
= a + bpa, of which the second appears to fit. (d), (e) Hydrogenation of octenes, Hougen and Watson (Chemical Process Principles, Wiley, 1947, p. 943). The hyperbolic and power law fits are of about equal quality. Pressure in atm; r in lb mol/(ft3 ⋅h).

y=

PP ᎏ = a + bp + cp + dp

 r u h

u

s

h

= 2.7655 + 1.5247pu + 1.0092ps + 1.1291ph ln r = ln k + a ln Pu + b ln ps + c ln ph = −4.059 + 0.469 ln pu − 0.2356 ln ps + 0.5997 ln ph r = 0.0173 P u0.469 p −0.2356 P h0.5997 s

7-9

7-10

REACTION KINETICS

The overall rates of the several participants are

dC ᎏ = 3k1AB2 − k2AC − k3CD dt

ra = ra1 + ra2 + ra3 = −k1CaCb − k2CaCc + k3CdCe

= 3k1AB2 − C{k2A + k3[D0 + 3(A0 − A) − 3(B0 − B)]}

rb = −k1CaCb rc = k1CaCb − k2CaCc + k3CdCe rd = re = k2CaCe − k3CdCe

(7-41)

The number of independent rate equations is the same as the number of independent stoichiometric relations. In this example, reactions 2 and 3 are reversible and are not independent, so there are only two independent rate equations. Some reactions apparently represented by single stoichiometric equations are in reality the result of several reactions, often involving short-lived intermediates. After a set of such elementary reactions is postulated by experience, intuition, and exercise of judgment, a rate equation is deduced and checked against experimental rate data. Several examples are given under “Mechanisms of Some Complex Reactions,” following. Stoichiometric Balances The amounts of all participants in a group of reactions can be expressed in terms of a number of key components equal to the number of independent stoichiometric relations. The independent rate equations will then involve only those key components and will be, in principle, integrable. For a single equation, Eqs. (7-36) and (7-37) relate the amounts of the several participants. For multiple reactions, the procedure for finding the concentrations of all participants starts by assuming that the reactions proceed consecutively. Key components are identified. Intermediate concentrations are identified by subscripts. The resulting concentration from a particular reaction is the starting concentration for the next reaction in the series. The final value carries no subscript. After the intermediate concentrations are eliminated algebraically, the compositions of the excess components will be expressible in terms of the key components.

MECHANISMS OF SOME COMPLEX REACTIONS The rates of many reactions are not represented by application of the law of mass action on the basis of their overall stoichiometric relations. They appear, rather, to proceed by a sequence of first- and secondorder processes involving short-lived intermediates which may be new species or even unstable combinations of the reactants; for 2A + B ⇒ C, the sequence could be A + B ⇒ AB followed by A + AB ⇒ C. Free radicals are molecular fragments having one or more unpaired electrons, usually short-lived (milliseconds) and highly reactive. They are detectable spectroscopically and some have been isolated. They occur as initiators and intermediates in such basic phenomena as oxidation, combustion, photolysis, and polymerization. The rate equation of a process in which they are involved is developed on the postulate that each free radical is at equilibrium or its net rate of formation is zero. Several examples of free radical and catalytic mechanisms will be cited, all possessing nonintegral power law or hyperbolic rate equations. Phosgene Synthesis CO + Cl2 ⇒ COCl2, but with the sequence: Cl2 ⇔ 2Cl• Cl• + CO ⇔ COCl• COCl• + Cl2 ⇒ COCl2 + Cl•

1

Assuming the first two reactions to be in equilibrium, an expression is found for the concentration of COCl• and when this is substituted into the third equation the rate becomes

2

rCOCl 2 = k(CO)(Cl2)3/2

Example 2: Analysis of Three Simultaneous Reactions Consider the three reactions A + 2B ⇒ 3C A + C ⇒ 2D C + D ⇒ 2E

Cl2 + O3 ⇒ ClO• + ClO•2

with A, B, and C the key components. Apply Eq. (7-37), B0 − B C1 − C0 A0 − A1 = ᎏ =ᎏ 2 3

ClO•2 + O3 ⇒ ClO•3 + O2 ClO•3 + O3 ⇒ ClO•2 + 2O2

D2 − D0 A1 − A = C1 − C2 = ᎏ 2

ClO•3 + ClO•3 ⇒ Cl2 + 3O2

E − E0 (7-42) C2 − C = D2 − D = ᎏ 2 Elimination of the concentrations with subscripts 1 and 2 will find D and E in terms of A, B, and C, with the same results that are achieved by the following method. This alternative procedure is called the xyz method. The amount of change by the first reaction is x, by the second y, and by the third z. For the same example, A = A0 − x − y

(7-47)

Hydrogen Bromide H2 + Br2 ⇒ 2HBr (Bodenstein, 1906). The chain of reactions is: 1

Br2 ⇒ 2Br• 2

Br + H2 ⇒ HBr •

H• + Br2 ⇒ HBR + Br•

C = C0 + 3x − y − z

Br• + Br• ⇒ Br2 (7-43)

Elimination of x, y, and z gives for the excess components: D0 − D = −3(A0 − A) + 3(B0 − B) E0 − E = 2(A0 − A) − 4(B0 − B) − 2(C0 − C)

rO3 = k(Cl 2)1/ 2 (O3)3/ 2

H• + HBr ⇒ H2 + Br•

D = D0 + 2y − z

The differential equations for the three key components become: dA ᎏ = −k1AB2 − k2AC dt

The chain carriers ClO•2 and ClO•3 are assumed to attain steady state. Then,

3

B = B0 − 2x

E = E0 + 2z

(7-46)

Ozone and Chlorine The assumed sequence is:

3

dB ᎏ = −2k1AB2 dt

(7-45)

These equations will have to be solved numerically for A, B, and C as functions of time; then D and E can be found by algebra. Alternatively, five differential equations can be written and solved directly for the five participants as functions of time, thus avoiding the use of stoichiometric balances, although these are really involved in the formulation of the differential equations.

(7-44)

Assuming equilibrium for the concentrations of the free radicals, the rate equation becomes d(HBr) ᎏ = k1(Br•)(H2) + k2(H•)(Br2) − k3(H•)(HBr) dt k1(H2)(Br2)3/ 2 = ᎏᎏ (7-48) k2(Br2) + k3(HBr) Enzyme Kinetics The enzyme E and the reactant S are assumed to form a complex ES that then dissociates into product P and uncombined enzyme.

RATE EQUATIONS 1

S+E⇔ ES 2 3

ES ⇒ E + P If equilibrium holds, (S)(E) (S)[(E0) − (ES)] ᎏ = ᎏᎏ = K m (ES) (ES) where (E0) is the total of the free and combined enzyme and K m is a dissociation constant. Solve for (ES) and substitute into the rate equation, d(P) k(E0)(S) rp = ᎏ = k(ES) = ᎏ (7-49) dt K m + (S) This hyperbolic equation is named after Michaelis and Menten (Biochem. Zeit., 49, 333 [1913]). Chain Polymerization The growth process of a polymer postulates a three-step mechanism: 1. An initiator I generates a free radical R• 2. The free radical reacts repeatedly with monomer by a process called propagation. 3. The free radical eventually disappears by some reaction, called termination. The stoichiometric equations are 1

I ⇒ 2R• 2

R + M ⇒ RM•, •

initiation

kp

RM• + M ⇒ RM2• or

kp

RMn• + M ⇒ RM•n + 1,

propagation

kt

RM + RM ⇒ R2Mn + m • n

or

• m

RMn + RMm,

termination

The rates of formation of the free radicals R• and M• reach steady states, dR•n ᎏ = 2k1(I) − k2(R•)(M) = 0 dt dM• ᎏ = k2(R•)(M) − 2kt(M•)2 = 0 dt These equations are solved for (R•) and (M•) and substituted into the propagation equation. The rate of polymerization becomes dM k 1/2 rp = − ᎏ = kp(M•)(M) = kp ᎏ1 (M)(I)1/2 (7-50) dt kt Thus, the process of chain polymerization is first-order with respect to monomer and half-order with respect to initiator. Solid Catalyzed Reaction The pioneers were Langmuir ( J. Am. Chem. Soc., 40, 1361 [1918]) and Hinshelwood (Kinetics of Chemical Change, Oxford, 1940). For a gas phase reaction A + B ⇒ Products, catalyzed by a solid, the postulated mechanism consists of the following: 1. The reactants are first adsorbed on the surface, where they subsequently react and the product is desorbed. 2. The rate of adsorption is proportional to the partial pressure and to the fraction of uncovered surface ϑv. 3. The rate of desorption of A is proportional to the fraction ϑa of the surface covered by A. 4. Adsorptive equilibrium is maintained. 5. The rate of reaction between adsorbed species is proportional to their amounts on the surface. The net rates of adsorption are:

 

ra = ka paϑv − k−aϑa ⇒ 0 Substitute ϑv = 1 − ϑa − ϑb and solve for the coverages: ka ϑa = ᎏ paϑv = Ka paϑv k−a

  k ϑ = ᎏ p ϑ = K p ϑ k b

b

−b

v

b

b

y=

(7-51)

pp 1+KP +K p ᎏ = ᎏᎏ

 r

K
kK a

b

a a

b

a

b

b

More about this topic is presented later. WITH DIFFUSION BETWEEN PHASES When reactants are distributed between several phases, migration between phases ordinarily will occur: with gas/liquid, from the gas to the liquid; with fluid/solid, from the fluid to the solid; between liquids, possibly both ways because reactions can occur in either or both phases. The case of interest is at steady state, where the rate of mass transfer equals the rate of reaction in the destined phase. Take a hyperbolic rate equation for the reaction on a surface. Then, r = rd = rs k2(C − r/k1) k2Cs = k1(C − Cs) = ᎏ = ᎏᎏ (7-52) 1 + k3Cs 1 + k3(C − r/k1) The unknown intermediate concentration Cs has been mathematically eliminated from the last term. In this case, r can be solved for explicitly, but that is not always possible with surface rate equations of greater complexity. The mass transfer coefficient k1 is usually obtainable from correlations. When the experimental data are of (C, r) the other constants can be found by linear plotting. CATALYSIS BY SOLIDS: LANGMUIR-HINSHELWOOD MECHANISM A plausible mechanism of solid catalytic reactions is that the participants chemisorb on the surface and react while in the adsorbed state. The process of adsorption of A on an active site of the surface σ is represented by A + σ ⇒ Aσ and the reaction between adsorbed molecules, for instance, by Aσ + Bσ ⇒ Cσ + Dσ Adsorptive Equilibrium The fraction of the surface covered by A at equilibrium is ϑa = Ka paϑv

(7-53)

1 ϑv = ᎏᎏᎏᎏ 1 + Ka pa + Kb pb + KcPc + Kd pd + . . .

(7-54)

where terms may be added for adsorbed inerts that may be present, and analogous expressions for the other participants. The rate of reaction between species in adsorptive equilibrium is then r = kpa pbϑv2

(7-55)

Dissociation A diatomic molecule A2 may adsorb as atoms, A2 + 2σ ⇒ 2Aσ with the result,
a
pa
K ϑa = ᎏᎏᎏ = K

apa
ϑv
a
1 + K pa
+ Kbpb + . . .

rb = kb pbϑv − k−bϑb ⇒ 0

b

1 ϑv = ᎏᎏ 1 + Ka pa + K b pb The rate of surface reaction is: kKaKb papb r = kϑaϑb = ᎏᎏ2 (1 + Ka pa + K b pb) The linearized form can be used to find the constants,

7-11

v

and the rate of the reaction is 2Aσ + Bσ ⇒ Products r = k′ϑa2 ϑb = kpa pbϑv3

(7-56)

Different Sites When A and B adsorb on chemically different sites σ1 and σ2, the rate of the reaction A + B ⇒ Unadsorbed products

7-12

REACTION KINETICS

is

Walas (Reaction Kinetics for Chemical Engineers, McGraw-Hill, 1959; Butterworths, 1989, pp. 153–164), and Rase (Chemical Reactor Design for Process Plants, vol. 1, Wiley, 1977, pp. 178–191). All of the relations developed here assume that only one step is controlling. A more general case is that of the reaction A ⇒ B with five steps controlling, namely

kpa pb r = ᎏᎏᎏ (7-57) (1 + Kapa)(1 + Kbpb) Dual Sites When the numbers of moles of reactants and products are unequal, A ⇔ M + N, the mechanism is assumed to be

r = k1(pag − pai)

Aσ + σ ⇔ Mσ + Nσ

θa r = k2 paiθv − ᎏ k3



and the rate ϑmϑn pmpn r = k ϑaϑv − ᎏ = k pa − ᎏ ϑv2 K K



 





Adsorption of A

r = k4θa

k(pa − pmpn /K) = ᎏᎏᎏ (7-58) (1 + Kapa + Kmpm + Knpn)2 Reactant in the Gas Phase When A in the gas phase reacts with adsorbed B,

Surface reaction

θb r = k5 pbiθv − ᎏ k6





Desorption of B

r = k7(pbi − pbg)

Diffusion of B from the surface

(7-62)

where θv = 1 − θa − θb . At steady state these rates are all the same. Upon elimination of the unmeasurable quantities pai, pbi, ϑa, ϑb, and ϑv, the relation becomes k5(k7 pbg + r) r r = ᎏᎏ 1−ᎏ k7 k4

A + Bσ ⇒ Products kpa pb r = kpaϑb = kpapbϑv = ᎏᎏ (7-59) (1 + ΣKi pi) Chemical Equilibrium When A is not in adsorptive equilibrium, it is assumed to be in chemical equilibrium, with p*a = pmpn /Ke pb. This expression is substituted for pa wherever it appears in the rate equation. Then













r 1 r k1r 1 1 − k5 pbg + ᎏ + ᎏ 1 − ᎏ − ᎏ (7-63) ᎏ+ᎏ k7 k6 k4 k1 pag − r k2 k3 k4 Combinations of several adsorption and surface reaction steps are usually not felt to be necessary, since so many alternatives are available individually. Single steps in combination with diffusion to the surface are usually adequate, as in the case leading to Eq. (7-52). Over the usual limited range of conditions, a power law rate equation often appears to be as satisfactory a fit of the data as a more complex Langmuir-Hinshelwood equation. The example of the hydrogenation of octenes is shown in Fig. 7-2d and 7-2e, and another case follows.

kpm pn/Ke r = kp*a pb ϑv2 = ᎏᎏᎏᎏᎏ (7-60) (1 + K a pmpn /K e pb + K b pb + K m pm + K n pn)2 All of these relations are brought together in the fundamental form (kinetic term)(driving force) r = ᎏᎏᎏ (7-61) adsorption term Table 7-2 summarizes the cases when all substances are in adsorptive equilibrium and the surface reaction controls. In Table 7-3, substance A is not in adsorptive equilibrium, so its adsorption rate is controlling. Details of the derivations of these and some other equations are presented by Yang and Hougen (Chem. Eng. Prog., 46, 146 [1950]),

TABLE 7-2

Diffusion of A to the surface

Example 3: Phosgene Synthesis Rate data were obtained by Potter and Baron (Chem. Eng. Prog., 47, 478 [1951]) for the reaction CO (A) + Cl2 (B2) ⇒ COCl2 (C) at 30.6. Three correlations of approximately equal statis-

Surface-reaction Controlling (Adsorptive Equilibrium Maintained of All Participants)

Reaction 1. A → M + N A→M+N A→M+N

Special condition General case Sparsely covered surface Fully covered surface

2. A A M

Basic rate equation

Driving force

Adsorption term

r = kθa r = kθa r = kθa

pa pa 1

1 + Kapa + Kmpm + Knpn 1 1

r = k1θa − k−1θm

pm pa − ᎏ K

1 + Kapa + Kmpm (1 + Kapa + Kmpm + Knpn)2

3. A A M + N

Adsorbed A reacts with vacant site

r = k1θaθv − k−1θmθn

pmpn pa − ᎏ K

4. A2 A M

Dissociation of A2 upon adsorption

r = k1θa2 − k−1θmθv

5. A + B → M + N A+B→M+N

Adsorbed B reacts with A in gas but not with adsorbed A

r = kθaθb r = kpaθb

pm pa − ᎏ K papb papb

2

(1 + K ap
a + Kmpm)

(1 + Kapa + Kbpb + Kmpm + Knpn)2 1 + Kapa + Kbpb + Kmpm + Knpn

6. A + B A M

r = k1θaθb − k−1θmθv

pm papb − ᎏ K

(1 + Kapa + Kbpb + Kmpm)2

7. A + B A M + N

r = k1θaθb − k−1θmθn

pmpn papb − ᎏ K

(1 + Kapa + Kbpb + Kmpm + Knpn)2

r = k1θa2θb − k−1θmθnθv

pmpn papb − ᎏ K

3

(1 + K apa
+ Kbpb + Kmpm + Knpn)

8. A2 + B A M + N NOTE:

Dissociation of A2 upon adsorption

The rate equation is: k (driving force) r = ᎏᎏ adsorption term When an inert substance I is adsorbed, the term Kipi is to be added to the adsorption term. SOURCE: From Walas, Reaction Kinetics for Chemical Engineers, McGraw-Hill, 1959; Butterworths, 1989.

RATE EQUATIONS TABLE 7-3

7-13

Adsorption-rate Controlling (Rapid Surface Reaction)

Reaction

Special condition

Basic rate equation

1. A → M + N

r = kpaθv

2. A A M

θa r = k paθv − ᎏ Ka

3. A A M + N

θa r = k paθv − ᎏ Ka

 

Driving force

Adsorption term

pa

Kapmpn 1+ᎏ + Kmpm + Knpn K



pm pa − ᎏ K

Kapm 1+ᎏ + Kmpm K



pmpn pa − ᎏ K

Kapmpn 1+ᎏ + Kmpm + Knpn K

pm pa − ᎏ K

ᎏ+K p  1 +  K

pa

Kapmpn 1 + ᎏ + Kbpb + Kmpm + Knpn Kpb

4. A2 A M

Dissociation of A2 upon adsorption

θa2 r = k paθv2 − ᎏ Ka

5. A + B → M + N

Unadsorbed A reacts with adsorbed B

r = kpaθv





2

Kapm

m m

6. A + B A M

θa r = k paθv − ᎏ Ka



pm pa − ᎏ Kpb

Kn pm 1 + ᎏ + Kbpb + Kmpm Kpb

7. A + B A M + N

θa r = k paθv − ᎏ Ka



pmpn pa − ᎏ Kpb

Kapmpn 1 + ᎏ + Kbpb + Kmpm + Knpn Kpb

θa2 r = k paθv2 − ᎏ Ka

pmpn pa − ᎏ Kpb

ᎏ+Kp +K p 1 +  Kp

8. A2 + B A M + N

 

Dissociation of A2 upon adsorption





Kapmpn

b b

m m



+ Knpn

b

2

NOTES:

The rate equation is: k (driving force) r = ᎏᎏ adsorption term Adsorption rate of substance A is controlling in each case. When an inert substance I is adsorbed, the term Kipi is to be added to the adsorption term. SOURCE: From Walas, Reaction Kinetics for Chemical Engineers, McGraw Hill, 1959; Butterworths, 1989.

tical validity are:

Aσ + 2Bσ ⇒ C + 3σ papb 1/3
b
− 0.00046pc) y= ᎏ = 0.34(1 − 0.061pa + 0.0032p r 2. Aσ + B2σ ⇒ C + 2σ papb 1/2 y= ᎏ = 2.38(1 + 1.98pb + 0.59pc) r 0.58 −0.68 3. r = 0.02p1.33 a pb pc The data are partial pressures, atm and the rate r, g mol phosgene made/(h⋅g catalyst). The first is ruled out because the constants physically cannot be negative. Although the other correlations are equally valid statistically, the Langmuir-Hinshelwood may be preferred to the power law form because it is more likely to be amenable to extrapolation. 1.

   

CHEMICAL EQUILIBRIUM

d ln K ∆Hr ᎏ = ᎏ2 dT RT This is integrable to 1 ln K = ln K298 + ᎏ R



T

∆Hr298 + 298 ∆Cp dT dT ᎏᎏᎏ 298 T2



T

(7-65)

where ∆Hr is the enthalpy change of reaction. Over a moderate temperature range, an adequate form of relation is





b K = exp a + ᎏ (7-66) t Gaseous equilibria are expressed in terms of fugacities or fugacity coefficients. In terms of partial pressures, pi = yiπ,

The rate of a reversible reaction k1

aA + bB ⇔ cC + dD k2

may be written



found by simultaneous solution of the several equations. The equilibrium composition of a mixture of known chemical species also can be found by a process of Gibbs energy minimization without the formulation of stoichiometric equations. Examples of the calculation of equilibria are in books on thermodynamics and in Walas (Phase Equilibria in Chemical Engineering, Butterworths, 1985). The equilibrium constant depends on temperature according to



Ccc Cdd r = k1 Caa Cbb − ᎏ (7-64) Ke In terms of the compositions at equilibrium, the equilibrium constant is d Ccec Cde Ke = ᎏ a b Cae Cbe With the aid of the stoichiometric “degree of advancement,” Ca0 − Ca Cb0 − Cb CC0 − Cc Cd0 − Cd ε=ᎏ=ᎏ=−ᎏ=−ᎏ a b c d the equilibrium constant can be written in terms of a single variable. When several reactions occur simultaneously, each reaction is characterized by its own εi. When the Kes are known, the composition can be

pcc pdd Kp = ᎏ = K yπ c + d − a − b (7-67) paa pbb Pressure affects the composition of an equilibrium mixture, but not the equilibrium constant itself. Although the equilibrium constant can be evaluated in terms of kinetic data, it is usually found independently so as to simplify finding the other constants of the rate equation. With K e known, the correct exponents of Eq. (7-64) can be found by choosing trial sets until k1 comes out approximately constant. When the exponents are small integers or simple fractions, this process is not overly laborious. Example 4: Reaction between Methane and Steam At 600°C the principal reactions between methane and steam are CH4 + H2O ⇔ CO + 3H2

CO +

1−x 5−x

x−y 5−x−y

x

3x

H2O ⇔ CO2 + H2 y

3x + y

Σ = 6 + 2x

7-14

REACTION KINETICS

where K1 = 0.574, K2 = 2.21. Starting with 1 mol methane and 5 mol steam, (x − y)(3x + y)3 y(3x + y) ᎏᎏᎏ2 = 0.574, ᎏᎏ = 2.21 (1 − x)(5 − x − y)(6 + 2x) (x − y)(5 − x − y) Simultaneous solution by the Newton-Raphson method yields x = 0.9121, y = 0.6328. Accordingly, the fractional compositions are: (1 − x) CH4 = ᎏ = 0.0112 (6 + 2x) (x − y) CO = ᎏ = 0.0357 (6 + 2x) y CO2 = ᎏ = 0.0809 (6 + 2x) (5 − x − y) H2O = ᎏᎏ = 0.4416 (6 + 2x)

Example 5: Percent Approach to Equilibrium For a reversible reaction with rate equation r = k[A2 − (1 − A)2/16], the size function kVr /V′ of a plug flow reactor will be found in terms of percent approach to equilibrium: 1 dA kVr , Aequilib = 0.2000 ᎏ = ᎏᎏ 2 2 A A − (1 − A) /16 V′



Percent approach

70

90

95

98

99

99.5

100

A kVr /V′

0.440 1.319

0.280 3.053

0.240 4.309

0.216 6.090

0.208 7.600

0.204 9.315

0.2 ∞

The volume escalates rapidly at high percent approaches.

(3x + y) H2 = ᎏ = 0.4306 (6 + 2x)

INTEGRATION OF RATE EQUATIONS

Approach to Equilibrium As equilibrium is approached the rate of reaction falls off, and the reactor size required to achieve a specified conversion goes up. At some point, the cost of increased reactor size will outweigh the cost of discarded or recycled unconverted material. No simple rule for an economic appraisal is really possible, but sometimes a basis of 95 percent of equilibrium converTABLE 7-4

sion is taken. For adiabatic operation, a certain approach to equilibrium temperature is common practice, say within 10 to 20°C (18 to 36°F), a number possibly based on experience with a particular process.

In either batch or flow systems, many single-rate equations lead to integrands that are ratios of low-degree polynomials that can be integrated by inspection or with the briefest of integral tables. Some of the cases of frequent occurrence are summarized in Table 7-4. When the problem is to relate C and t, the constants are known, and the polynomials are of second degree or higher, numerical integration may save

Some Isothermal Rate Equations and Their Integrals

1. A → Products: dA − ᎏ kAq dt A q=1 ᎏ = exp [−k(t − t0)], A0 1 ᎏᎏᎏ q−1 1 + (q − 1)kA0 (t − t0)



RT RT ∆ν V = nt ᎏ = nt0 + ᎏ (na0 − na) ᎏ P νa P



1/(q − 1)

,

k(t − t0) =

q≠1

2. A + B → Products: dA − ᎏ = kAB = kA(A + B0 − A0) dt 1 A0(A + B0 − A0) k(t − t0) = ᎏ ln ᎏᎏ B0 − A0 AB0

k1

k1

4. Reversible reaction, second order, A + B A R + S: k2

SOURCE:

1 ᎏn1 − ᎏ n   , when α = 2 a

a0

k2

These linear equations are solved for the transforms as D = s2 + (k1 + k2 + k3)s + k1k2

q=0

2αA + β + q ᎏᎏ , ᎏᎏ 2αA + β + q  2αA + β − q 

dA − ᎏ = k1A − k2B dt dB − ᎏ = −k1A + (k2 + k3)B dt dC − ᎏ = −k2B dt (s + k1)A
+ k3B
= A0 −k1A
+ (s + k2 + k3)B
= B0 −k2B
+ sC
= C0

q≠0

0

5. The reaction νaA → νrR + νsS between ideal gases at constant T and P: knαa dn − ᎏa = ᎏ dt Vα − 1



Laplace transformations are made and rearranged to

dA − ᎏ = k1AB − k2RS = k1A(A + B0 − A0) dt − k2(A0 + R0 − A)(A0 + S0 − A) = αA2 + βA − γ α = k1 − k2 β = k1(B0 − A0) + k2(2A0 + R0 + S0) γ = k2(A0 + R0)(A0 + S0)
2
+

α q = β 4
γ


2αA0 + β − q

RT ∆ν ᎏ nb0 + ᎏ P νa na0 ∆ν − ᎏ ln ᎏ νa na

in general

Rate equations are

k2

1 ᎏ ln q

V ᎏ dna, nαa

k3

dA − ᎏ = k1A − k2(A0 + B0 − A) = (k1 + k2)A − k2(A0 + B0) dt k1A0 − k2B0 (k1 + k2)(t − t0) = ln ᎏᎏᎏ (k1 + k2)A − k2(A0 + B0)

{

na

AA B →C

k1

k(t − t0) =





α−1

6. Equations readily solvable by Laplace transforms. For example:

3. Reversible reaction A AB:

2αA0 + β ᎏ, 2αA + β





na0

A0s + (k2 + k3)A0 + K3B0 A
= ᎏᎏᎏ D B0s + k1(A0 + B0) B
= ᎏᎏ D k2
B + C0 C
=ᎏ s Inversion of the transforms can be made to find the concentrations A, B, and C as functions of the time t.

Adapted from Walas, Chemical Process Equipment Selection and Design, Butterworth-Heinemann, 1990.

IDEAL REACTORS time and preserve reliability. Some 40 cases of integrations at constant volume are developed by Capellos and Bielski (Kinetic Systems Mathematical Descriptions of Chemical Kinetics, Wiley, 1972). Sets of first-order rate equations are solvable by Laplace transform (Rodiguin and Rodiguina, Consecutive Chemical Reactions, Van Nostrand, 1964). The methods of linear algebra are applied to large sets of coupled first-order reactions by Wei and Prater (Adv. Catal., 13, 203 [1962]). Reactions of petroleum fractions are examples of this type. Example 6: Laplace Transform Application For the reaction A ⇒B ⇔C 1

2

3

B0 = C0 = 0 dA ᎏ = k1A dt

with the rate equations are

sA
− A0 = −k1A


The transforms are

k3A0 sB A − (k2 + k3)
B
= ᎏ + (k1 − k3)
s Explicitly,

A0 A
=ᎏ s + k1

The reactions are 2A ⇒ B ⇒ C. The partial solutions are A0 A = ᎏᎏ (1 + 2k1A0 t) dB k1A20 ᎏ + k2B = ᎏᎏ2 dt (1 + 2k1 A0t) Although the differential equation is first-order linear, its integration requires evaluation of an infinite series of integrals of increasing difficulty. 1 2 2. The reactions are A ⇒ B and A + B ⇒ C. After A is expressed in terms of B by elimination of t, dB B−k k k = ᎏ1 ᎏ = (k1 − k2B) A0 − B0 + B − 2k ln ᎏ , dt B+k k2 but this cannot be integrated analytically. 1 2 1 2 1 3. For the reactions A ⇒ B, 2B ⇒ C; 2A ⇒ B ⇒ C; 2A ⇒ B, 2 2B ⇒ C; the rate equations are solved in terms of higher transcendental functions by Chien ( J. Am. Chem. Soc., 76, 2256 [1948]). For the first case, with B0 = 0: A = exp (−k1t) 1

1.

2



dB ᎏ = k1A − k2B + k3(A0 − A − B) dt

k3 k1 − k3 A0 B
= ᎏᎏ ᎏ + ᎏ s + k2 + k3 s s + k1









where

A and B as functions of t are found by inversion with a table of L-T pairs.

When even second-order reactions are included in a group to be analyzed, individual integration methods may be needed. Three cases of coupled first- and second-order reactions will be touched on. All of them are amenable only with difficulty to the evaluation of specific rates from kinetic data. Numerical integrations are often necessary.

7-15

τ iJ1(γ) − βH1 (γ) B = A0 ᎏ ᎏᎏ K J0(γ) + β i H0(i) τ = exp (−k1t) K = k1k2 A0
τ
γ = 2iK β = iJ1(γ)/H1(i)(γ) (i)

The notation of the Bessel functions is that of Jahnke and Emde (Tables of Functions with Formulas and Curves, Dover, 1945; Teubner, 1960).

IDEAL REACTORS INTRODUCTION A useful classification of kinds of reactors is in terms of their concentration distributions. The concentration profiles of certain limiting cases are illustrated in Fig. 7-3; namely, of batch reactors, continuously stirred tanks, and tubular flow reactors. Basic types of flow reactors are illustrated in Fig. 7-4. Many others, employing granular catalysts and for multiphase reactions, are illustrated throughout Sec. 23. The present material deals with the sizes, performances and heat effects of these ideal types. They afford standards of comparison. In a batch reactor, all the reactants are loaded at once; the concentration then varies with time, but at any one time it is uniform throughout. Agitation serves to mix separate feeds initially and to enhance heat transfer. In a semibatch operation, some of the reactants are charged at once and the others are then charged gradually. In an ideal continuously stirred tank reactor (CSTR), the conditions are uniform throughout and the condition of the effluent is the same as the condition in the tank. When a battery of such vessels is employed in series, the concentration profile is step-shaped if the abscissa is the total residence time or the stage number. The residence time of individual molecules varies exponentially from zero to infinity, as illustrated in Fig. 7-3e. In another kind of ideal flow reactor, all portions of the feed stream have the same residence time; that is, there is no mixing in the axial direction but complete mixing radially. It is called a plug flow reactor (PFR), or a tubular flow reactor (TFR), because this flow pattern is characteristic of tubes and pipes. As the reaction proceeds, the concentration falls off with distance. Often, complete mixing cannot be approached for economic reasons. Inactive or dead zones, bypassing, and limitations of energy input are common causes. Packed beds are usually predominantly used in plug flow reactors, but they may also have small mixing zones

superimposed in series or in parallel. In tubular reactors for viscous fluids, laminar or non-Newtonian behavior gives rise to variations of residence time. Deviations from ideal behavior are analyzed at length in Sec. 23. MATERIAL AND ENERGY BALANCES These balances are based on the general conservation law, Input + Sources = Outputs + Sinks + Accumulation

(7-68)

The terms may be quantities or rates of flow of material or enthalpy. Inputs and outputs are streams that cross the vessel boundaries. A heat of reaction within the vessel is a source. A depletion of reactant in the vessel is a sink. Accumulation is the time derivative of the content of the reference quantity in the vessel; of the volume times the concentration, ∂VrCa /∂t; or of the total enthalpy of the vessel contents, ∂[WCp(T − Tref )] /∂t. BATCH REACTORS Batch reactors are tanks, usually provided with agitation and some mode of heat transfer to maintain temperature within a desirable range. They are primarily employed for relatively slow reactions of several hours duration, since the downtime for filling and emptying large equipment may be an hour or so. Agitation maintains uniformity and improves heat transfer. Modes of heat transfer are illustrated in Figs. 23-1 and 23-2. Except in the laboratory, batch reactors are mostly liquid phase. In semibatch operation, a gas of limited solubility may be fed in gradually as it is used up. Batch reactors are popular in practice because of their flexibility with respect to reaction time and to the kinds and quantities of reactions that they can process.

Concentration

REACTION KINETICS

Concentration

7-16

(b) (b)

(a)

Concentration

Time

Concentration

Time (a) (a)

1

2 3 4 5 Stage number

(b)

(c)

Distance along reactor

(c)

(d)

1.0 0.9 0.8

(d)

n=1

0.7

n=5 PFR

C/C0

0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0

0.5

1.0 1.5 Reduced time, t / t

2.0

(e)

2.5

(f)

Types of flow reactors: (a) stirred tank battery, (b) vertically staged, (c) compartmented, (d) single-jacketed tube, (e) shell and tube, ( f) semiflow stirred tank.

FIG. 7-4

(e) Concentration profiles in batch and continuous flow: (a) batch time profile, (b) semibatch time profile, (c) five-stage distance profile, (d) tubular flow distance profile, (e) residence time distributions in single, five-stage, and PFR; the shaded area represents the fraction of the feed that has a residence time between the indicated abscissas.

FIG. 7-3

Material and energy balances of a nonflow reactor are summarized in Table 7-5. Several batch operations are summarized in Fig. 7-5. Daily Yield Say the downtime for filling and emptying a reactor is td and no reaction occurs during these periods. The reaction time tr of a first-order reaction, for instance, is given by ktr = −ln (1 − x). The daily yield with n batches per day will be 24Vr kC0 x 24VrC0 x y = nVr(C0 − C) = ᎏ = ᎏᎏ (7-69) tr + td −ln (1 − x) + ktd Some conditions at which the daily yield is a maximum are ktd x

0.01 0.13

0.1 0.45

0.5 0.68

5.0 0.88

Thus, the required conversion goes up as the downtime increases. Details are in Problem P2 of the “Solved Problems” subsection. Filling and Emptying Periods Say the pumping rate is V′, the full tank volume is Vr1, and the rate of reaction is r = kCaq. For t ≤ Vr1/V ′, the material balances with Eq. (7-68) are as follows.

Filling: Vr = V′t,

Ca = Ca0 , when t = 0, d(VrCa) dC V′Ca0 = 0 + rVr + ᎏ = kVrCaq + V′Ca + V′t ᎏa dt dt

 

Vr dCa (7-70) Ca0 = Ca + k ᎏ Caq + t ᎏ V′ dt where the variables are separable. Emptying: Vr = Vr1 − V′t, Ca = Ca1, when t = 0, d(VrCa) dC q 0 = V′Ca + kVrCa + ᎏ = V′Ca + kVrCaq − V′Ca + Vr ᎏa dt dt or

dCa (7-71) ᎏ = −kCaq dt This is the same equation as for the full tank, but applies only for t ≤ Vr1/V′. Figure 7-5e shows a complete batch cycle. Optimum Operation of Reversible Reactions Often, equilibrium composition becomes less favorable and the rate of reaction becomes more favorable as the temperature increases, so a best condition may exist. If the temperature is adjusted at each composition to

IDEAL REACTORS TABLE 7-5 Material and Energy Balances of a Nonflow Reactor

Changes in density because of reaction or temperature changes are often small enough to be ignored. Then the volumetric flow rate is uniform and the balance becomes

Rate equations: α

 

1 dn n ra = − ᎏ ᎏa = kC αa = k ᎏa Vr dθ Vr



b′ k = exp a′ − ᎏ T′

(1)





T

∆Cp dT

Rate of heat transfer: Q′ = UA(Ts − T)

(5)

d(VrCa) V′0 Ca0 = V′Ca + Vr ra + ᎏ dt where the reactor volume Vr is a known function of time. For a power law rate equation at steady state,

Enthalpy balance: dT 1 UA(Ts − T) ᎏ = ᎏ ∆Hr + ᎏᎏα dna ρVrC Vrk(na /Vr)
p



dT 1 UA(Ts − T) ᎏ = ᎏ ∆Hr + ᎏᎏ dCa ρC VrkCaα
p



when





Ca = Ca0

(7) (8)

Solve Eq. (6) to find T = f(Ca); combine Eqs. (1) and (2) and integrate as 1 ᎏᎏᎏ dCa Caαexp [a′ − b′/f(Ca)]

(9)

SOURCE: Adapted from Walas, Chemical Process Equipment Selection and Design, Butterworth-Heinemann, 1990.

Ca, n − 1 = Can +
tnran

Example 7: A Four-Stage Unit When the material balances are Cn − 1 = Cn + 1.5[Cn /(0.2 + Cn)]2 and C0 = 2, the successive outlet concentrations are found by RootSolver to be 0.985, 0.580, 0.389, and 0.281. The simplest problem is when all of the stages have the same kt
; then one of the three variables (kt
, n, or Can) can be found when the others are specified. For first-order reactions, 1 Can ᎏ = ᎏᎏᎏᎏ Ca0 (1 + k1t
1)(1 + k2t
2) . . . (1 + knt
n)

(7-80)

1 ⇒ ᎏᎏn (1 + kt
total /n)

(7-81)

for identical stages. For multiple reactions, material balances are required for each stoichiometry.

k1 = A1 exp (−B1 /T) and k2 = A2 exp (−B2 /T). The condition (∂r/∂T)x = 0

Example 8: Consecutive Reactions Take the reaction A ⇒ B ⇒ C, with B0 = C0 = 0. Define ϑ = k1t
A0, α = 1/(1 + k1 t
), β = 1/(1 + k 2 t
). Then by setting up successive material balances, equations for the effluent from the nth stage are derived as 1

(7-72)

which tells what the temperature must be at each fractional conversion for the minimum reactor size. Practically, it may be difficult to vary the temperature of a batch reactor in this way, but the operation may be more nearly feasible with a CSTR battery or a PFR. Figure 7-5f shows an example of such a temperature profile for a batch reactor. CONTINUOUS STIRRED TANK REACTORS (CSTR) Flow reactors are used for greater production rates when the reaction time is comparatively short, when uniform temperature is desired, when labor costs are high. CSTRs are used singly or in multiple units in series, in either separate vessels or single, compartmented shells. Material and energy balances are based on the conservation law, Eq. (7-69). In the operation of liquid phase reactions at steady state, the input and output flow rates are constant so the holdup is fixed. The usual control of the discharge is on the liquid level in the tank. When the mixing is adequate, concentration and temperature are uniform, and the effluent has these same properties. The steady state material balance on a reactant A is V 0′ Ca0 = V′Ca + Vr ra

(7-79)

The set of equations for all stages can be solved in succession, starting with the inlet to the first stage as Ca0.

r = k1(1 − x) − k2 x

leads to

(7-78)

A summary of material and energy balances is in Table 7-6. For each vessel of a series,

make the rate a maximum, then a minimum reactor size or maximum conversion will result. Take the first-order reversible process,

B1 − B2 T = ᎏᎏ A1B1(1 − x) ln ᎏᎏ A2B2x

(7-77)

Ca0 = Ca + kt
Caq (6)

1 C
p = ᎏ niCpi ρVr

Ca

(7-75)

(4)

(the simplest case is when UA and Ts are constant)

Ca0

Vr
t = ᎏ V′

Ca0 − Ca (7-76) ra = ᎏ t
emphasizes how CSTR measurements can provide data for the development of rate equations without integrating them. During startup or discharge the material balance becomes

(3)

298



(7-74)

A useful rearrangement,

∆Hr = ∆Hr298 +

θ=

Ca0 = Ca +
tra where the residence time is

(2)

Heat of reaction:

T = T0

7-17

(7-73)

αβϑ Bn = ᎏ (α n − β n) α−β When n ⇒ ∞, this equation reduces to An = A0αn

Cn = A0 − An − Bn

2

(7-82)

B k1 (7-83) ᎏ = ᎏ [exp (−k1t
) − exp (−k2t
)] A0 k2 − k1 This is the equation for a plug flow reactor. It can be derived directly from the rate equations with the aid of Laplace transforms. The sequences of secondorder reactions of Figs. 7-5a and 7-5c required numerical integrations.

When CSTRs are operated in series, the sum of the reactor volumes drops off sharply with the number of stages. An economical number often is only 3 to 6, since the benefit of reduced volume may be outweighed by the increased cost of multiple agitators, pumps, and controls. When all stages are in a single shell, the economics are more favorable to large numbers of stages, but the single-shell arrangements lose some of the flexibility of the multiple-tank designs. Example 9: Comparison of Batch and CSTR Volumes For a first-order reaction, the ratio of n-stage CSTR and batch volumes is n[(C0 /C)1/n − 1] Ratio = ᎏᎏ ln(C0 /C)

REACTION KINETICS 1

1.0 0.9 0.8 0.7 0.6

c0 c3

.5 c2

c1 0 0

A/A0

ci /c00

c4

C/A0

A/A0

Batch

0.5 0.4

One-stage B/A0

0.3 0.2

.8

1.6

2.4

0.1 0.0 0

3.2

Two-stage

2

1

3

b/c00

Time, t

(a) (a)

(b) (b)

2

4

5

Concentration

2.0 1.8 1.6

1

C B A

0 0

k1 = 2, k2 = 1

1

2

3

4

5

1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0

C B A 2

4

t

6 8 10 12 14 CSTR residence time

16

18

20

(d) (d)

(c) (c)

0.50

2000 1800 (0.7494, 20)

.6 .5 .4 .3

(0.1375, 174)

Temperature

Concentration, A

2200 1 .9 .8 .7

(0.1498, 154) 0

40

80 120 Time, min (e)

160

200

0.40

Temperature

1600

0.35

1400

0.30

1200

0.25

Equilibrium at 600

1000

0.20 0.15

800 600

.2 .1 0

0.45

Time below T = 600

400 200 0.0

Time

7-18

0.10

Time at 600 0.1

0.2

0.3 0.4 0.5 0.6 0.7 Fraction converted, f

0.8

0.9

0.05 0.00 1.0

(f)

FIG. 7-5 Some batch operations, the P-code refers to detailed solutions in Walas (Chemical Reaction Engineering Handbook of Solved Problems, Gordon & Breach, 1995). (a) Methane chlorination in batch reactor or PFR; abscissa is ratio of chlorine to methane; P4.03.20. (b) Product yields of A ⇒ B ⇒ C with k1 = 2 and k2 = 1 in batch reactor and CSTR; P4.04.60. (c) The reactions 2A ⇒ B and 2B + 2C ⇒ D with k1 = 1.0 in batch reactor; P4.04.46. (d) Same as c but in CSTR. (e) Fill for 20 min, react, and discharge for 20 min with ra = 0.03[Ca(0.2 + Ca) − 0.04(1 − Ca)2]; P4.09.18. ( f ) Best temperature profile for a reversible reaction, r = k1(1 − x) − k2x; when t = 0.1, x = 0.37 at 600 R, and x = 0.51 with optimum temperature profile; x = 0.81 when t = 0.5 and final temperature is 250 R; the full range is impractical; P4.11.02.

IDEAL REACTORS TABLE 7-6

Material and Energy Balance of a CSTR

when

Mean residence time: Vr t
= ᎏ V′

(1)

Temperature dependence: b′ k = exp a′ − ᎏ T





(2)

Rate equation: (Ca0 − Ca) x = ᎏᎏ Ca0

ra = kCαa = kCαa0(1 − x)α,

C1q + 1 + C2q [(q − 1)C1 − qC0] = 0

and

The sketch identifies the nomenclature.

(3)

Material balance: Ca0 = Ca + kt
Ca

(4) (5)

x = ktCαa0− 1(1 − x)α

7-19

q = 1,



C1 = C 0C
, 2

and
t1 =
t2

For higher orders
t1 ≠ t
2 and the sum is less than twice the sum of equal stages, although usually not much different from that sum. As an example, when the second-order reaction between benzoquinone and cyclopentadiene is done in a three-stage unit, the reactor sizes are 3.25, 4.68, and 6.27, totaling 14.20, as compared to 14.56 with three equal stages (where consistent units are used). Details are in Walas (Chemical Reaction Engineering Handbook of Solved Problems, p. 4.11.15, Gordon & Breach, 1995). Selectivity A significant respect in which CSTRs may differ from batch (or PFR) reactors is in the product distribution of complex reactions. However, each particular set of reactions must be treated individually to find the superiority. For the consecutive reactions A ⇒ B ⇒ C, Fig. 7-5b shows that a higher peak value of B is reached in batch reactors than in CSTRs; as the number of stages increases the batch performance is approached. TUBULAR AND PACKED BED FLOW REACTORS

Enthalpy balance:

n′H − n′ H i

i

i0

i0

= Q′ − ∆Hr(n′a0 − n′a)

Hi =



(6)

T

Cpi dT

(7)

298

∆Hr = ∆Hr298 +



T

∆Cp dT

(8)

298

For the reaction aA + bB → rR + sS: ∆Cp = rCpr + sCps − aCpa − bCpb

(9)

When the heat capacities are equal and constant, the heat balance is: C
pρV′(T − T0) = Q′ − ∆Hr298V′(Ca0 − Ca)

(10)

SOURCE: Adapted from Walas, Chemical Process Equipment Selection and Design, Butterworth-Heinemann, 1990.

Some values are C0 /C n

2

10

20

1 5

2.89 1.07

3.91 1.27

6.34 1.37

The ratio goes up sharply as the conversion increases and down sharply as the number of stages increases. For higher-order reactions the numbers are of comparable magnitudes.

Different Sizes Ordinarily, it is most economical to make all stages of a CSTR battery the same size. For a first-order reaction the resulting total volume is a minimum for a specified performance, but not so for other orders. Take a two-stage battery: C0 − C1 C1 − C2 +ᎏ
t1 +
t2 = ᎏ kC1q kC2q With C0 and C2 specified, the condition for a minimum is ∂(t
1 +
t2) ᎏ=0 ∂C1

Tubular reactors are made up of one or more tubes in parallel, each of less than approximately 100-mm (3.94-in) diameter. With fluids of normal viscosity, plug flow exists in tubes of this size, with all molecules having essentially the same residence time. In packed beds of larger diameters, large-scale convection may be inhibited to such an extent that plug flow is also approached. Continuous gas phase reactions are predominantly done in such units, as are many liquid phase processes. Immiscible liquids are best handled in stirred tanks, although in-line mixers can facilitate such reactions in pipes. Reaction times are mostly short, made feasible by elevated temperatures. In such large-scale operations as oil cracking, the tubes may be several hundred meters long in a trombonelike arrangement. Temperature control is by heat transfer through the walls or by cold-shot injection. Shell-and-tube arrangements can provide large amounts of heat transfer. Product distribution of complex reactions is like that of batch reactors, but different from that of CSTRs. Material and energy balances of a plug flow reactor are summarized in Table 7-7. For convenience, the loading on a flow reactor is expressed as a size of reactor per unit of flow rate, say Vr /V′, and is labeled the space velocity. Some of the units in practical use are stated in the Introduction. How the actual residence time is calculated when the density of flow varies is illustrated in Table 7-8. Tubular flow reactors operate at nearly constant pressure. How the differential material balance is integrated for a number of secondorder reactions will be explained. When na is the molal flow rate of reactant A the flow reactor equation is −dna = na0 dx = −V′dCa = ra dVr (7-84) na0 dna (7-85) or Vr = ᎏ na ra The equation is rendered integrable by application of the stoichiometry of the reaction, the ideal gas law, and, for instance, the power law for rate of reaction. Some details are shown in Table 7-9. Frictional Pressure Drop Usually this does not have a significant effect on the reactor size, except perhaps when the flow is twophase. Some approximate relations will be cited that are adequate for pressure-drop calculations of homogeneous flow reactions in pipelines. The pressure drop is given by f ρu2 −dP = ᎏ dL (7-86) 2gD A good approximation to the friction factor in the turbulent flow range is µD 0.2 f = 0.046(Re)−0.2 = 0.044 ᎏ (7-87) W The mass flow rate is



 

W = 0.7854D2ρu

7-20

REACTION KINETICS

TABLE 7-7

Material and Energy Balances of a Plug Flow Reactor (PFR)

The balances are made over a differential volume dVr of the reactor. Rate equation:

At constant Ts, Eq. (7) may be integrated numerically to yield the temperature as a function of the number of moles T = φ(n′a)

−dn′a dVr = ᎏ ra

(8)

(1) Then the reactor volume is found by integration

1 V′ α = − ᎏ ᎏ dn′a k n′a

  

b′ = −exp −a′ + ᎏ T

(2) α

dn′ ᎏ Pn′  n′tRT

a

(3)

a



n′a0

Vr =

n′a

1 dn′a ᎏᎏᎏᎏ exp [a′ − b′/φ(n′a)][Pn′a /n′tRφ(n′a)]α

(9)

Adiabatic process: dQ = 0

(10)

Enthalpy balance: ∆Hr = ∆Hr298 +



T

∆Cp dT

(4)

298

4U dQ = U(Ts − T) dAp = ᎏ (Ts − T) dVr D 4U(Ts − T) = − ᎏᎏ dn′a Dra dQ + ∆Hr dn′a = ni dHi = niCpi dT dT ᎏ = ∆Hr − 4U(Ts − T)/Dra = f(T, Ts, n′a) dn′a ᎏᎏᎏ nicpi

(5) The balance around one end of the reactor is (6) (7)

n

Hi0 − Hr0(n′a0 − n′a) = niHi = ni

i0

C

pi

dT

(11)

With reference temperature at T0, enthalpies Hi0 = 0 ∆Hr0 = ∆Hr298 +



T0

∆Cp dT

(12)

298

Substituting Eq. (12) into Eq. (10)

−∆H

r298

+



T0

298



∆Cp dT (n′a0 − n′a) = ni



T

Cpi dT

(13)

T0

Adiabatic process with ∆Cp = 0 and with constant heat capacities ∆Hr298(n′a0 − n′a) T = T0 − ᎏᎏ niCpi

(14)

This expression is substituted instead of Eq. (8) to find the volume with Eq. (9).

The density in terms of the molecular weight M is M PM0nt0 ρ=ᎏ=ᎏ V RTnt Also, in terms of the tube length dL, dVr = 0.7854D2dL Combining,

0.046W1.8µ0.2RT[nt0 + δa(na0 − na)] −dP = ᎏᎏᎏᎏ dVr gD6.8M0nt0P

(7-88)

This is to be solved simultaneously with the flow reactor equation, Eq. (7-84). Alternatively, dVr can be eliminated from Eq. (7-88) for a direct relation between P and na. More accurate relations than Eqs. (7-86) and (7-87) are described in Sec. 11 of this Handbook. RECYCLE AND SEPARATION MODES All reactor modes can sometimes be advantageously operated with recycling of part of the product or intermediate streams. Heated or cooled recycle streams serve to moderate undesirable temperature travels, and they can be processed for changes in composition before being returned. Say the recycle flow rate in a PFR is V r′ and the fresh feed rate is V0′, with the ratio R = Vr′ /V′0. With a fresh feed concentration of C0 and a product of C2 the composite feed concentration is C0 + RC2 C1 = ᎏ 1+R

(7-89)

The change in concentration across the reactor becomes C2 − C0 ∆C = C1 − C2 = ᎏ (7-90) 1+R Accordingly, the change in concentration (or in temperature) across the reactor can be made as small as desired by upping the recycle ratio. Eventually, the reactor can become a differential unit with substantially constant temperature, while substantial differences will concurrently arise between the fresh feed inlet and the product withdrawal outlet. Such an operation is useful for obtaining experimental data for analysis of rate equations. In the simplest case, where the product is recycled without change, the flow reactor equation at constant density with a power law is −V′0 (1 + R)dC = kCqdVr C1 dC and Vr = V′0 (1 + R) (7-91) ᎏq C2 kC Recycling increases the size of the reactor and degrades the plug flow characteristics, so there must be practical compensation by adjustment of the temperature or composition.



Example 10: Reactor Size with Recycle For first-order reaction with C0 /C2 = 10 and R = 5, C1 /C2 = 2.5. The relative reactor sizes with recycle and without are (1 + 5) ln (C1 /C2) 5.497 Ratio = ᎏᎏ = ᎏ = 1.193 ln (C0 /C2) 4.605 With reversible reactions, recycling is warranted when improvement in conversion can be realized by removing some of the product in a separator and returning only unconverted material. In some CSTR operations, the product is removed continuously by extraction or azeotropic distillation. The gasoline addi-

IDEAL REACTORS TABLE 7-8

True Contact Time in A PFR

The ratio Vr /V′0 is of the volume of the reactor to the incoming volumetric rate and has the dimensions of time. It will be compared with the true residence time when the number of mols changes as reaction goes on, or P and T also change.

     π =k ᎏ RT

q

n rα = k ᎏα V′

q

nα ᎏ nt

q

t

q

(1)

α

The definition of the rate of reaction and the law of mass action is:

 

1 dn n rα = − ᎏ ᎏα = k ᎏα V′ dt V′

q

Rearrange to

 

n dn   ᎏ n 

1 dnα 1 V′ q 1 RT dt = − ᎏ ᎏ = − ᎏ ᎏ dnα = − ᎏ ᎏ V′ rα kV′ nα k π

q−1

q−1

t

q α

Example: Take q = 1, nt0 = nα0. 1 (δα + 1)nα0 − δαnα 1 ᎏᎏ dnα = ᎏ (δα + 1) ln ᎏ − δαx nα 1−x k

nα0

nα

α0

t

t0

α

The rate equations will be stated in these terms for a number of reactions. In all these cases, the integrands are ratios of second-degree equations. The moderately complex integrations are accomplished with the aid of a table of integrals, or by MATHEMATICA, or numerically when the constants are known. 2A⇒M (1) A+B⇒M (2) 2A ⇔ M (3) A+B⇔M (4) Part (1):

nt = nt0 + δα(nα0 − nα)



α

(2)

α

Eqs. (1) and (2) are the desired comparison. Before integrating, substitute



reactor volume ᎏᎏ molal input rate

nα π Cα = ᎏ = ᎏ V′ RT

α

t0

1 k

dx ᎏ, rα

π n −x ᎏ  ᎏnn  = ᎏ RT  n + δ x 

nt q ᎏ dnα nα q−1

r

t=ᎏ

x

x0

nt = nt0 + δα(nα0 − nα) = nt0 + δαx

   dV 1 RT n ᎏ = − ᎏ  ᎏ   ᎏ  dn V′ kn π n

nα



V′ = ntRT/π

1 RT dVr = − ᎏ ᎏ k π



−dnα = nα0dx = −V′dCα = V′Cα0dx = rαdVr

Vr ᎏ= nα0

−dnα = rαdVr

Vr ᎏ= V′0

Tubular flow reactors usually operate at nearly constant pressure. For a reactant A, the differential material balance is:

One form of the integration is:

The differential balance on the reactant is:

nα0

TABLE 7-9 Integration of Rate Equations of a PFR at Constant Pressure

q

ntRT V′ = ᎏ π

0

7-21



1 1 dnα ᎏ = ᎏ ln ᎏ nα 1−x k

(1 − 2) δα = ᎏ = −0.5 2

 

RT V′ = ᎏ (nt0 − 0.5x) π Vr ᎏ= nα0

dx 1 n − 0.5x = ᎏ (R
T
)   ᎏᎏ  dx ᎏ r n −x k 2

x

2

t0

x0

α

α0

Part (2): (1 − 2) δα = ᎏ = −1 1

The ratio t δαx y = ᎏ = δα + 1 − ᎏᎏ ln [1/(1 − x)] Vr /V′0 >1 when δα < 0 <1 when δα > 0

  (n − x)(n − x) V 1 π ᎏ = ᎏ  ᎏ   ᎏᎏ dx n (n − x) k RT RT V′ = ᎏ (nt0 − x) π 2 x

r

α0

x0

α0

b0

2

t0

Part (3): tive methyl-tert-butyl ether is made in a distillation column where reaction and simultaneous separation occur.

HEAT EFFECTS The heat balance of a reactor is made up of three terms: Heat of reaction + Heat transfer = Gain of sensible and latent heats by the mixture. This establishes the temperature as a function of the composition T = f(na)

δa = − 0.5

  RT n −x RT n + 0.5n x r = k  ᎏ   ᎏᎏ  − k  ᎏ  ᎏᎏ  π n − 0.5x π n − 0.5x

RT V′ = ᎏ (nt0 − 0.5x) π 2

α

α0

1

2

α0

m0

2

t0

t0

Part (4): δα = −0.5

which may be substituted into the equations of the specific rate and the equilibrium constant B k = exp A + ᎏ f(na)









D K e = exp C + ᎏ f(na) With these substitutions the rate equation remains a function of the composition alone. Heat balances of several kinds of reactors are summarized in Tables 7-5, 7-6, 7-7 and 7-10. Enthalpy changes of processes depend only on the end states. Normally the enthalpy change of reaction is known at some standard tem-

RT 2 (nα0 − x)(nb0 − x) RT rα = k1 ᎏ ᎏᎏ − k2 ᎏ π (nt0 − 0.5x)2 π

 

nm0 + 0.5nα0x

  ᎏᎏ n − 0.5x  t0

perature, Tb = 298 K (536 R), for instance. The simplest formulation of the heat balance, accordingly, is to consider the reaction to occur at this temperature, transfer whatever heat is required, and raise the enthalpy of the reaction products to their final values. Batch Reactions For a batch reaction, the heat balance is −(∆Hr)Tb(na0 − na) + Q = ni(HiT − HiTb)

(7-92)

7-22

REACTION KINETICS −(∆Hr)Tb(na0 − na) + UA(Tm − T) = VrρCp(T − Tb) (7-95) UA or −(∆Hr)Tb(Ca0 − Ca) + ᎏ (Tm − T) = ρCp(T − Tb) (7-96) Vr CSTR Reactions For a CSTR reaction, the quantities ni are molal flow rates. Per unit of time,

TABLE 7-10 Material and Energy Balances of a Packed Bed Reactor

 

Diffusivity and thermal conductivity are taken appreciable only in the radial direction. Material balance equation: ρ ∂x D ∂2x 1 ∂x ᎏ − ᎏ ᎏ2 + ᎏ ᎏ − ᎏ rc = 0 ∂z u ∂r u0C0 r ∂r Energy balance equation:





⇒ ni

∆Hrρ ∂T k ∂2T 1 ∂T + ᎏ ᎏ + ᎏ rc = 0 ᎏ−ᎏ ᎏ ∂z GCp ∂r2 GCp r ∂r



−(∆Hr)TbVrra + Q = ni(HiT − HiTb)

(1)



(2)

At the inlet: x(0, r) = x0

(3) (4)

T(0, r) = T0

(5)

∂x ᎏ=0 ∂r

(6)

∂T U ᎏ = ᎏ (T′ − T) ∂r k

(7)

At the wall:

When the temperature T′ of the heat-transfer medium is not constant, another enthalpy balance must be formulated to relate T′ with the process temperature T. A numerical solution of these equations may be obtained in terms of finite difference equivalents, taking m radial increments and n axial ones. With the following equivalents for the derivatives, the solution may be carried out by direct iteration: r = m(∆r)

(7-98)

 

⇒ ntCpt dT

∂x ∂T ᎏ=ᎏ=0 ∂r ∂r

r = R,

Cpi dT

Tb

The last equation applies in the absence of phase change. Plug Flow Reactions The differential relations in a cylindrical vessel are 4 4UA dA = ᎏ dVr −∆HrT dna + ᎏ dVr = niCpi dT (7-99) D D

At the center: r = 0,



(7-97)

T

(7-100)

Note that the enthalpy change of reaction is a function of temperature, but a mean value often is adequate. The various heat balances are to be solved simultaneously with the appropriate material balances, but when the temperatures can be solved for explicitly their equivalents are simply substituted into the equations for k and Ke and the material balance is solved alone. Packed Bed Reactors The commonest vessels are cylindrical. They will have gradients of composition and temperature in the radial and axial directions. The partial differential equations of the material and energy balances are summarized in Table 7-10. Example 4 of “Modeling of Chemical Reactions” in Sec. 23 is an application of such equations. A variety of provisions for heat transfer are illustrated in Figs. 23-1 to 23-3 and elsewhere in Sec. 23. UNSTEADY CONDITIONS WITH ACCUMULATION TERMS

z = n(∆z)

(8)

∂T Tm,n + 1 − Tm,n ᎏ = ᎏᎏ ∂z ∆z

(9)

∂T Tm + 1,n − Tm,n ᎏ = ᎏᎏ ∂r ∆r

(10)

∂2T Tm + 1,n − 2Tm,n + Tm − 1,n = ᎏᎏᎏ ᎏ ∂r2 (∆r)2

(11)

Expressions for the x derivatives are of the same form: rc = rate of reaction, a function of s and T G = mass flow rate, mass/(time)(superficial cross section) u = linear velocity D = diffusivity k = thermal conductivity SOURCE: Adapted from Walas, Chemical Process Equipment Selection αnd Design, Butterworth-Heinemann, 1990.

The solvent, as well as any other inerts, and the mass of the vessel are included in this summation. The heat exchange through a jacket or coils at temperature Tm is Q = UA(Tm − T)

(7-93)

When phase changes are absent, −(∆Hr)Tb(na0 − na) + UA(Tm − T) = ni



T

Cpi dT

(7-94)

Tb

When the mixture can be characterized by an overall heat capacity,

Unsteady material and energy balances are formulated with the conservation law, Eq. (7-68). The sink term of a material balance is Vr ca and the accumulation term is the time derivative of the content of reactant in the vessel, or ∂(VrCa)/∂t, where both Vr and Ca depend on the time. An unsteady condition in the sense used in this section always has an accumulation term. This sense of unsteadiness excludes the batch reactor where conditions do change with time but are taken account of in the sink term. Startup and shutdown periods of batch reactors, however, are classified as unsteady; their equations are developed in the “Batch Reactors” subsection. For a semibatch operation in which some of the reactants are preloaded and the others are fed in gradually, equations are developed in Example 11, following. For a CSTR the unsteady material balance is d(VrCa) V′Ca0 = V′Ca + Vrra + ᎏ (7-101) dt Enthalpy balances also will have accumulation terms. Conditions that give rise to unsteadiness are changes in feed rate, composition, or temperature. In the case of Fig. 7-6, a sinusoidal input of feed rate is introduced. The output concentration also appears to vary sinusoidally. The amplitude of the response is lower as the specific rate is increased. If a sinusoidal variation of the temperature of the heat transfer medium in the jacket or coil occurs, say Tm = Tm0(1 + α sin β t)

(7-102)

the balances will be dT −∆HrVrra + UA(Tm − T) = V′ρCp(T − T0) + ρVrCp ᎏ dt

(7-103)

dCa (7-104) V′Ca0 = V′Ca + Vrra + Vr ᎏ dt Since each input of mass to a perfect plug flow unit is independent of what has been input previously, its condition as it moves along the

LARGE SCALE OPERATIONS

Example 11: Balances of a Semibatch Process The reaction A + B ⇒ Products is carried out by first charging B into the vessel to a concentration Cb0 and a volume Vr0, then feeding a solution of concentration Ca0 at volumetric rate V′ for a time t. Volume of solution in the tank:

1.50 Feed rate 1.20

Vr = Vr0 + V′t

1.00 0.90

(7-105)

Stoichiometric balance:

Output C, k = 1

0.618

0.60

V′tCa 0 − VrCa = Vr 0Cb 0 − Vr Cb Vr 0Cb0 − V′tCa0 Cb = Ca + ᎏᎏ Vr0 + V′t

Output C, k = 5 0.358

0.30 0.00 0

7-23

(7-106)

Material balance on A: Input = Output + Sink + Accumulation

3

6

9

d(CaVr) V′Ca0 = 0 + kVrCaCb + ᎏ dt

12

t

dCa = kVrCaCb + Vr ᎏ + CaV′ dt

FIG. 7-6 Sinusoidal input of feed rate to a CSTR. Input: F = 1 + 0.2 sin (t); output: dC/dt = (1 − C)[1 + 0.2 sin (t)] − kc2. Straight lines are for constant feed rate.

reactor will be determined solely by its initial condition and its residence time, independently of what comes before and after. Practically, of course, some interaction will occur at the boundaries of successive inputs of different compositions or temperatures. This is governed by diffusional behaviors that are beyond the scope of the present work.

dCa V′Ca V′Ca0 ᎏ + kCaCb + ᎏ = ᎏ dt Vr Vr

(7-107)

Eqs. (7-105), (7-106), and (7-107) are combined into dCa V′ Vr0Cb0 − V′tCa0 ᎏ = ᎏ (Ca0 − Ca) − kCa Ca + ᎏᎏ dt Vr0 + V′t Vr0 + V′t





(7-108)

A numerical integration is required.

LARGE SCALE OPERATIONS INTRODUCTION In this category are included a number of topics that become especially significant on the industrial scale. Some of this material is covered at length in Sec. 23, so only an outline is provided here. MULTIPLE STEADY STATES Phenomena of multiple steady states and instabilities occur particularly with nonisothermal CSTRs. Some isothermal processes with hyperbolic rate equations and processes with porous catalysts also can have such behavior. Mathematically, multiplicities become evident when heat and material balances are combined. Both are functions of temperature, the latter through the rate equation which depends on temperature by way of the Arrhenius law. The curves representing these balances may intersect in several points. For first order in a CSTR, the material balance in terms of the fraction converted can be written kt
b x = ᎏ, k = exp a − ᎏ (7-109) 1 + kt
T and the energy balance Heat generation = Sensible heat gain −∆HrVrCf (1 − x) ᎏᎏ + UA(Tm − T) = ρCpVr (T − Tf ) (7-110) t
These balances can be plotted two ways, as shown in Fig. 7-7: 1. x from both equations can be plotted against T, with the intersections at the steady state values of T and corresponding values of x. 2. The LHS (heat generation) and RHS (heat removal) of Eq. (7-110) are plotted against T after x has been eliminated between the two balances; the intersections identify the same steady state temperatures as the plot in Fig. 7-7a. Conditions at which the slope of the heat generation line is greater than that of the heat removal line are unstable, and where it is less the condition is stable (see Fig. 7-7b). At an unstable point, any fluctua-





tion in conditions will move the temperature to a neighboring point. Control systems always produce small fluctuations of the process variables, as in the sinusoidal case of Fig. 7-6. If the fluctuations occur while the system is at an unstable point, the steadiness will disappear. In the case of Fig. 7-7c, as the unstable position is approached (T = 280, C = 2.4) the profiles of T and C become erratic and eventually degenerate to the condition at the stable point on the right (Figs. 7-7d and 7-7e). Either of the two stable operating conditions can be selected by adjusting the positions of the curves so that only one intersection is obtained. In a plant, long-time unstable operation is unlikely because of imprecise temperature control. Plug flow reactors with recycle exhibit some of the characteristics of CSTRs, including the possibility of multiple steady states. This topic is explored by Perlmutter (Stability of Chemical Reactors, PrenticeHall, 1972). Endothermic reactions possess only one steady state. For complex reactions and with multistage CSTRs, more than three steady states can exist (as in Fig. 23-17c). Most of the work on multiplicities and instabilities has been done only on paper. No plant studies and a very few laboratory studies are mentioned in the comprehensive reviews of Razon and Schmitz (Chem. Eng. Sci., 42, 1,005–1,047 [1987]) and Morbidelli et al. (in Carberry and Varma, Chemical Reaction and Reactor Engineering, Dekker, 1987, pp. 973–1,054). NONIDEAL BEHAVIOR Reactors that are nominally CSTRs or PFRs may in practice deviate substantially from ideal mixing or nonmixing. This topic is developed at length in Sec. 23, so only a few summary statements are made here. More information about this topic also may be found in Nauman and Buffham (Mixing in Continuous Flow Systems, Wiley, 1983). Laminar Flow With highly viscous fluids the linear velocity along a streamline varies with the radial position. Laminar flow is characteristic of some polymeric systems. Figure 23-21 shows how the conversion is poorer in laminar flow than with uniform flow over the

7-24

REACTION KINETICS

1.0

100

0.9

90 Mass balance

Heat generation

80

0.7

70 Heat balance

0.6 0.5 0.4

Qg and Qr

Fraction converted

0.8

Heat removal

60 50 40

0.3

30

0.2

20

0.1 0.0 350 360 370 380 390 400 410 420 430 440 450 Temperature

10 0 350 360 370 380 390 400 410 420 430 440 450 Temperature

(a) (a)

(b) (b)

3.0 2.7

Concentration, C

2.4 2.1

Tf = 250

Heat balance Tf = 300

1.8 1.5 1.2 0.9 0.6

Material balance

0.3 0.0 250 270 290 310 330 350 370 390 410 430 450 Temperature, T (c) (c)

450

Tf = 250

1.5

0 0

T

C

3

200

400

600

800

1000

Tf = 250

350

250 0

200

400

600

t

t

(d)

(e)

800

1000

Multiplicity and instability of first-order reactions in CSTRs. For (a) and (b), k = exp (25 − 10,000/T), x = k/(1 + k), 200x + (350 − T) = T − 350. For curves (c) to (e), k = exp (25 − 7,550/T), C = 3/(1 + 300k), C = 3 − 0.2(T − Tf). Unsteady state, 300dC/dt = 3 − (1 + 300k)C, C0 = 0; 300dT/dt = Tf − T + 15,000kC, T0 = 300. Temperature in °C. FIG. 7-7

cross section for first- and second-order reactions. Another adverse effect with viscous solutions is poor heat transfer. Accordingly, stirred tanks are often preferred to tubular units for such applications. The equations for radial and axial distributions of composition and temperature in laminar flow are studied by Nauman (Chemical Reactor Design, Wiley, 1987, pp. 165–203). Residence Time Distribution (RTD) This is established by injecting a known amount of tracer into the feed stream and monitor-

ing its concentration in the effluent. At present there are no correlations of this kind of behavior that could be used for design of a new process, but such information about existing units is of value for diagnostic purposes. The RTD is a distinctive characteristic of mixing behavior. In Fig. 7-3e, the CSTR has an RTD that varies as the negative exponential of the time and the PFR is represented by a vertical line at tr = 1. Multistage units and many packed beds have bell-shaped RTDs, like that of

LARGE SCALE OPERATIONS the five-stage unit of Fig. 7-3e and the large-scale units of Figs. 23-10 and 23-11. An equation that represents such shapes is called the Erlang, nn RTD = ᎏ trn − 1 exp (−ntr) (7-111) Γ(n) where n is interpreted as a number of CSTRs in series. When n is integral, Γ(n) is replaced by (n − 1)!. Plots of these curves are shown in Fig. 23-9. When the RTD of a vessel is known, its performance as a reactor for a first-order reaction, and the range within which its performance will fall for other orders, can be predicted. Segregated Flow A real example is bead polymerization of styrene and some other materials. The reactant is in the form of individual small beads suspended in a fluid and retarded from agglomeration by colloids on their surfaces. Accordingly, they go through the reactor as independent bodies and attain conversions under batch conditions with their individual residence times. This is called segregated flow. With a particular RTD, conversion is a maximum with this flow pattern. The mean conversion of all the segregated elements then is given by ∞ C C
(RTD) ᎏ dtr (7-112) ᎏ= 0 C0 C0 batch



For first order

 

C ᎏ C 

batch

ᎏ C 

batch

0

For second order

C

0

= exp (−kt
tr)

(7-113)

1 = ᎏᎏ (1 + kC0t
tr)

(7-114)

Example 12: Segregated Flow The pilot unit of Fig. 23-11 with

n = 9.3 has

RTD = 13188tr8.3 exp (−9.3tr) Some values of mean concentration ratio
C/C0 of first- and second-order reactions obtained with Eq. (7-112) are: kt
or kC0t


1

2

5

10

First order C
/C0 Second order
C/C0 Second order PFR

0.386 0.513 0.500

0.163 0.349 0.333

0.018 0.180 0.167

0.100 0.091

Maximum Mixedness With a particular RTD, this pattern provides a lower limit to the attainable conversion. It is explained in Sec. 23. Some comparisons of conversions with different flow patterns are made in Fig. 23-14. Segregated conversion is easier to calculate and is often regarded as a somewhat plausible mechanism, so it is often the only one taken into account. Dispersion In tubes, and particularly in packed beds, the flow pattern is disturbed by eddies whose effect is taken into account by a dispersion coefficient in Fick’s diffusion law. A PFR has a dispersion coefficient of 0 and a CSTR of ∞. Some rough correlations of the Peclet number uL/D in terms of Reynolds and Schmidt numbers are Eqs. (23-47) to (23-49). There is also a relation between the Peclet number and the value of n of the RTD equation, Eq. (7-111). The dispersion model is sometimes said to be an adequate representation of a reactor with a “small” deviation from plug flow, without specifying the magnitude of small. As a point of superiority to the RTD model, the dispersion model does have the empirical correlations that have been cited and can therefore be used for design purposes within the limits of those correlations. OPTIMUM CONDITIONS Optimization of a process is an activity whereby the best conditions are found for attainment of a maximum or minimum of some desired objective. In the broadest sense, an industrial process has maximum profit as its goal, but there are also problems with less-ambitious goals that do not involve money or the whole plant.

7-25

The best quality to be found may be a temperature, a temperature program, a concentration, a conversion, a yield of preferred product, a cycle period for a batch reaction, a daily production level, a kind of reactor, a size for a reactor, an arrangement of reactor elements, provisions for heat transfer, profit or cost, and so on—a maximum or minimum of some of these factors. Among the constraints that may be imposed on the process are temperature range, pressure range, corrosiveness, waste disposal, and others. Once the objective and the constraints have been set, a mathematical model of the process can be subjected to a search strategy to find the optimum. Simple calculus is adequate for some problems, or Lagrange multipliers can be used for constrained extrema. When a full plant simulation can be made, various alternatives can be put through the computer. Such an operation is called flowsheeting. A chapter is devoted to this topic by Edgar and Himmelblau (Optimization of Chemical Processes, McGraw-Hill, 1988) where they list a number of commercially available software packages for this purpose, one of the first of which was Flowtran. With many variables and constraints, linear and nonlinear programming may be applicable, as well as various numerical gradient search methods. Maximum principle and dynamic programming are laborious and have had only limited applications in this area. The various mathematical techniques are explained and illustrated, for instance, by Edgar and Himmelblau (Optimization of Chemical Processes, McGraw-Hill, 1988). A few specific conclusions about optimum performance can be stated: 1. The minimum total volume of a CSTR battery for first-order reaction, and near-minimum for second-order, is obtained when all vessels are the same size. 2. An economical optimum number of CSTRs and their auxiliaries in series is 4 to 5. 3. In a sequence of PFR and CSTR, better performance is obtained with the PFR last. Performance of reversible reactions is improved with the CSTR at a higher temperature. 4. For the consecutive reactions A ⇒ B ⇒ C, a higher yield of intermediate B is obtained in batch reactors or PFRs than in CSTRs. 5. When the desirable product of a complex reaction is favored by a high concentration of some reactant, batch or semibatch reactors can be made superior to CSTRs. 6. Conversion by a reversible reaction is enhanced by starting out at high temperature and ending at low temperature if equilibrium conversion drops off at high temperature. 7. For a reversible reaction, the minimum size or maximum conversion is obtained when the rate of reaction is kept at a maximum at each conversion by adjustment of the temperature. Variables It is possible to identify a large number of variables that influence the design and performance of a chemical reactor with heat transfer, from the vessel size and type; catalyst distribution among the beds; catalyst type, size, and porosity; to the geometry of the heattransfer surface, such as tube diameter, length, pitch, and so on. Experience has shown, however, that the reactor temperature, and often also the pressure, are the primary variables; feed compositions and velocities are of secondary importance; and the geometric characteristics of the catalyst and heat-exchange provisions are tertiary factors. Tertiary factors are usually set by standard plant practice. Many of the major optimization studies cited by Westerterp et al. (1984), for instance, are devoted to reactor temperature as a means of optimization. The complexity of temperature regulation of three major commercial reversible processes are represented in Figs. 23-3a, 23-3e, and 23-3f. Presumably, these profiles have been established by fine-tuning the operations over a period of time. Objective Function This is the quantity for which a minimax is sought. For a complete manufacturing plant, it is related closely to the economy of the plant. Subsidiary problems may be to optimize conversion, production, selectivity, energy consumption, and so on in terms of temperature, pressure, catalyst, or other pertinent variables. Case Studies Several collections of more or less detailed solutions of optimization problems are cited, as follows. 1. Of the 23 studies listed under “Modeling of Chemical Reactors” in Sec. 23, a number are optimization oriented. Added to them

7-26

REACTION KINETICS

may be a detailed study of an existing sulfuric acid plant by Crowe et al. (Chemical Plant Simulation, Prentice-Hall, 1971). 2. Chen (Process Reactor Design, Allyn & Bacon, 1983) does the following examples mostly with simple calculus: Batch reactors—optimum residence time for series and complex reactions, minimum cost, optimal operating temperature, and maximum rate of reaction CSTRs—minimum volume of battery, maximum yield, optimal temperature for reversible reaction, minimum total cost, reactor volume with recycle, maximum profit for reversible reaction with recycle, and heat loss Tubular flow reactors—minimum volume for second-order reversible reactions, maximum yield of consecutive reactions, minimum cost with and without recycle, and maximum profit with recycle Packed bed reactor optimization Size comparison for first- and second-order and reversible reactions Selectivity of parallel and consecutive reactions and of reactions in a porous catalyst 3. Edgar and Himmelblau (Optimization of Chemical Processes, McGraw-Hill, 524–550, 1988) supply many references to other problems in the literature: Optimal residence time for the reactions A ⇔ B followed by B ⇒ P or X Optimal time for a biochemical CSTR Selection of feedstock for thermal cracking to ethylene by linear programming Maximum yield from a four-stage CSTR by nonlinear programming Optimal design of ammonia synthesis by differential equation solution and a numerical gradient search A C4 alkylation process by sequential quadratic programming 4. Westerterp, van Swaaij, and Beenackers (Chemical Reactor Design and Operation, Wiley, 1984, pp. 674–746) also supply many references to other problems in the literature: Optimized costs for several gas phase reactions: (1) A + B ⇒ P; (2) A + B ⇔ P; and (3) A + B ⇒ P, A + 2B ⇒ X, P + B ⇒ X Ammonia cold-shot converter Maximum yield of first-order consecutive reactions in CSTR by application of Lagrange multipliers Autothermal reactor for methanol synthesis using a numerical search technique Minimum reactor volumes of isothermal and nonisothermal cascades by dynamic programming Optimum temperature profiles of 2A ⇒ B ⇒ P by the maximum principle Optimizing the temperature for A ⇒ P and A ⇒ X by the maximum principle Westerterp et al. (1984; see Case Study 4, preceding) conclude, “Thanks to mathematical techniques and computing aids now available, any optimization problem can be solved, provided it is realistic and properly stated. The difficulties of optimization lie mainly in providing the pertinent data and in an adequate construction of the objective function.” HETEROGENEOUS REACTIONS Heterogeneous reactions of industrial significance occur between all combinations of gas, liquid, and solid phases. The solids may be inert or reactive or catalysts in granular form. Some noncatalytic examples are listed in Table 7-11, and processes with solid catalysts are listed under “Catalysis” in Sec. 23. Equipment and operating conditions of heterogeneous processes are covered at some length in Sec. 23; only some highlights will be pointed out here. Reactants migrate between phases in order to react: from gas phase to liquid, from fluid to solid, and between liquids when the reaction occurs in both phases. One of the liquids usually is aqueous. Resistance to mass transfer may have a strong effect on the overall rate of reaction. A principal factor is the interfacial area. Its magnitude is enhanced by agitation, spraying, sparging, use of trays or packing, and by size reduction or increase of the porosity of solids. These are the same operations that are used to effect physical mass transfer between

TABLE 7-11

Industrial Noncatalytic Heterogeneous Reaction

Gas/solid Action of chlorine on uranium oxide to recover volatile uranium chloride Removal of iron oxide impurity from titanium oxide by volatilization by action of chlorine Combustion and gasification of coal Manufacture of hydrogen by action of steam on iron Manufacture of blue gas by action of steam on carbon Calcium cyanamide by action of atmospheric nitrogen on calcium carbide Burning of iron sulfide ores with air Nitriding of steel Liquid/solid Ion exchange Acetylene by action of water on calcium carbide Cyaniding of steel Hydration of lime Action of liquid sulfuric acid on solid sodium chloride or on phosphate rock or on sodium nitrate Leaching of uranium ores with sulfuric acid Gas/liquid Sodium thiosulfate by action of sulfur dioxide on aqueous sodium carbonate and sodium sulfide Sodium nitrite by action of nitric oxide and oxygen on aqueous sodium carbonate Sodium hypochlorite by action of chlorine on aqueous sodium hydroxide Ammonium nitrate by action of ammonia on aqueous nitric acid Nitric acid by absorption of nitric oxide in water Recovery of iodine by action of sulfur dioxide on aqueous sodium iodate Hydrogenation of vegetable oils with gaseous hydrogen Desulfurization of gases by scrubbing with aqueous ethanolamines Liquid/liquid Caustic soda by reaction of sodium amalgam and water Nitration of organic compounds with aqueous nitric acid Formation of soaps by action of aqueous alkalies on fats or fatty acids Sulfur removal from petroleum fractions by aqueous ethanolamines Treating of petroleum products with sulfuric acid Solid/solid Manufacture of cement Boron carbide from boron oxide and carbon Calcium silicate from lime and silica Calcium carbide by reaction of lime and carbon Leblanc soda ash Gas/liquid/solid Hydrogenation or liquefaction of coal in oil slurry SOURCE: Adapted from Walas, Reaction Kinetics for Chemical Engineers, McGraw-Hill, 1959; Butterworths, 1989.

phases and the equipment can be similar, except that more heat transfer may be needed because of substantial heats of reaction. Chemical reaction always enhances the rate of mass transfer between phases. The possible magnitudes of such enhancements are indicated in Tables 23-6 and 23-7. They are no more predictable than are specific rates of chemical reactions and must be found experimentally for each case, or in the relatively sparse literature on the subject. Mechanisms The most widely investigated heterogeneous reactions have been gas/liquid and fluid/solid catalyst. The Hatta theory or Langmuir-Hinshelwood mechanisms can suggest the forms of rate equations but they always involve parameters to be found empirically. Because liquid diffusivities are low, most liquid/liquid reactions are believed to be mass-transfer controlled. In some cases the phase in which reaction occurs has been identified, but there are cases where both phases are active. Phase-transfer catalysts enhance the transfer of reactant from an aqueous to an organic phase and thus speed up reactions. Mass transfer responds less strongly to change of temperature than does chemical rate, so this feature can be used to discriminate between possible controlling mechanisms. A sensitivity to stirring rate or to a change in linear velocity also will indicate the presence of major resistance to mass transfer. No single pattern appears to hold for reactions of solids, but much is known about the behavior of important operations like cement manufacturing, ore roasting, and lime burning.

ACQUISITION OF DATA Reaction and Separation Some multiphase operations combine simultaneous reaction and separation. A few examples follow. 1. The yield of furfural from xylose is improved by countercurrent extraction with tetralin (Schoenemann, Proc. 2d Europ. Symp. Chem. React. Eng., Pergamon, 1961, p. 30). 2. The reaction of vinyl acetate and stearic acid makes vinyl stearate and acetic acid but also some unwanted ethylidene acetate. A high selectivity is obtained by reaction in a distillation column with acetic acid overhead and vinyl stearate to the bottom (Geelen and Wiffels, Proc. 3d Europ. Symp. Chem. React. Eng., Pergamon, 1964, p. 125).

7-27

3. The hydrolysis of fats is improved by running in a countercurrent extraction column (Donders et al., Proc. 4th Europ. Symp. Chem. React. Eng., Pergamon, 1968, pp. 159–168). 4. In the production of KNO3 from KCl and HNO3, the product HCl is removed continuously from the aqueous phase by contact with amyl alcohol, thus forcing the reaction to completion (Baniel and Blumberg, Chim. Ind., 4, 27 [1957]). 5. Methyl-tert-butyl ether, a gasoline additive, is made from isobutene and methanol with distillation in a bed of acidic ionexchange resin catalyst. The MTBE goes to the bottom with purity above 99 percent and unreacted materials overhead.

ACQUISITION OF DATA INTRODUCTION Kinetic data are acquired in the laboratory as a basis for design of large-scale equipment or for an understanding of its performance, or for the interpretation of possible reaction mechanisms. All levels of sophistication of equipment, statistical design of experiments, execution, and statistical analysis of the data are reported in the literature. Before serious work is undertaken, the appropriate literature should be consulted. The bibliography of Shah (Gas-Liquid-Solid Reactor Design, McGraw-Hill, 1979), for instance, has 145 items classified into 22 categories of reactor types. The criteria for selection of laboratory reactors include equipment cost, ease of operation, ease of data analysis, accuracy, versatility, temperature uniformity, and controllability, suitability for mixed phases, and scale-up feasibility. A number of factors limit the accuracy with which parameters for the design of commercial equipment can be determined. The parameters may depend on transport properties for heat and mass transfer that have been determined under nonreacting conditions. Inevitably, subtle differences exist between large and small scale. Experimental uncertainty is also a factor, so that under good conditions with modern equipment kinetic parameters can never be determined more precisely than ⫾5 to 10 percent (Hofmann, in de Lasa, Chemical Reactor Design and Technology, Martinus Nijhoff, 1986, p. 72). Composition The law of mass action is expressed as a rate in terms of chemical compositions of the participants, so ultimately the variation of composition with time must be found. The composition is determined in terms of a property that is measured by some instrument and calibrated in terms of composition. Among the measures that have been used are titration, pressure, refractive index, density, chromatography, spectrometry, polarimetry, conductimetry, absorbance, and magnetic resonance. In some cases the composition may vary linearly with the observed property, but in every case a calibration is needed. Before kinetic analysis is undertaken, the data are converted to composition as a function of time (C, t), or to composition and temperature as functions of time (C, T, t). In a steady CSTR the rate is observed as a function of residence time. When a reaction has many participants, which may be the case even of apparently simple processes like pyrolysis of ethane or synthesis of methanol, a factorial or other experimental design can be made and the data subjected to a response surface analysis (Davies, Design and Analysis of Industrial Experiments, Oliver & Boyd, 1954). A quadratic of this type for the variables x1, x2, and x3 is r = k1 x1 + k2 x2 + k3 x3 + k11 x12 + k22 x22 + k33 x32 + k12x1x2 + k13x1x3 + k23 x2 x3 (7-115) Analysis of such a correlation may reveal the significant variables and interactions, and may suggest some model, say of the L-H type, that could be analyzed in more detail by a regression process. The variables xi could be various parameters of heterogeneous processes as well as concentrations. An application of this method to isomerization of n-pentane is given by Kittrel and Erjavec (Ind. Eng. Chem. Proc. Des. Dev., 7, 321 [1968]).

The constants of rate equations of single reactions often can be found by one of the linearization schemes of Fig. 7-1. Nonlinear regression methods can treat any kind of rate equation, even models made up of differential and algebraic equations together, for instance dA ᎏ = −k1A dt dB ᎏ = k1A − k2B2 + k3C dt C = A0 + B0 + C0 − A − B Software for these procedures is supplied, for example, by Constantinides (Applied Numerical Methods with Personal Computers, McGraw-Hill, 1987, pp. 577–614, with diskette) and by the commercial product SimuSolv (Mitchell and Gauthier Associates, 200 Baker Street, Concord, MA 01742). These do the integration, find the constants and their statistical criteria, and make the plots. SimuSolv is claimed “to provide maximum efficiency in problem solving with minimum involvement in computational procedures.” Since the computer does the work, many possibilities may be considered. For the reaction cyclohexanol to cyclohexanone, 36 experiments at 6 temperature levels were made and more than 50 rate equations were tested (Hofmann, in de Lasa, Chemical Reactor Design and Technology, Martinus Nijhoff, 1986, p. 72). A rate equation for methanol from CO2 and H2 was selected from 44 possibilities by Beenackers and Graaf (in Cheremisinoff, Handbook of Heat and Mass Transfer, vol. 3, Gulf Publishing, 1989, pp. 671–699). They used a spinning basket reactor like the item shown in Fig. 23-29c. EQUIPMENT Many configurations of laboratory reactors have been employed. Rase (Chemical Reactor Design for Process Plants, Wiley, 1977) and Shah (Gas-Liquid-Solid Reactor Design, McGraw-Hill, 1979) each have about 25 sketches, and Shah’s bibliography has 145 items classified into 22 categories of reactor types. Jankowski et al. (Chemische Technik, 30, 441–446 [1978]) illustrate 25 different kinds of gradientless laboratory reactors for use with solid catalysts. Laboratory reactors are of two main types: 1. Designed to obtain such fundamental data as chemical rates free of mass transfer resistances or other complications. Some of the heterogeneous reactors of Fig. 23-29, for instance, employ known interfacial areas, thus avoiding one uncertainty. 2. Simulations of the kinds of reactor intended for the pilot or plant scale. How to do the scale-up to the plant size, however, is a sizable problem in itself. Batch Reactors In the simplest kind of investigation, reactants can be loaded into a number of ampules, kept in a thermostatic bath for various periods, and analyzed. In terms of cost and versatility, the stirred batch reactor is the unit of choice for homogeneous or slurry reactions and even gas/liquid reactions when provision is made for recirculation of the gas. They are especially suited to reactions with half-lives in excess of 10 min. Sam-

7-28

REACTION KINETICS

ples are taken at intervals and the reaction is stopped by cooling, usually by at least 50°C (122°F), by dilution, or by destroying a residual reactant such as an acid or base; analysis can then be made at leisure. Analytic methods that do not necessitate termination of reaction include measurements of (1) the amount of gas produced, (2) the gas pressure in a constant volume vessel, (3) absorption of light, (4) electrical or thermal conductivity, (5) polarography, (6) viscosity of polymerization, and so on. The readings of any instrument should be calibrated to chemical composition or concentration. Operation may be isothermal, with the important effect of temperature determined from several isothermal runs, or the composition and temperature may be recorded simultaneously and the data regressed simultaneously. Finding the parameters of the nonisothermal equation r = exp (a + b/T) Cq is only a little more difficult than for r = kCq. Rates, dC/dt, are found by numerical differentiation of (C, t) data. On the laboratory scale, it is usually safe to assume that a batch reactor is stirred to uniform composition, but for critical cases such as high viscosities this could be checked with tracer tests. CSTRs and other devices that require flow control are more expensive and difficult to operate. Particularly in steady operation, however, the great merit of CSTRs is their isothermicity and the fact that their mathematical representation is algebraic, involving no differential equations, thus making data analysis simpler. For laboratory research purposes, CSTRs are considered feasible for holding times of 1 to 4,000 s, reactor volumes of 2 to 400 cm3 (0.122 to 24.4 in3) and flow rates of 0.1 to 2.0 cm3/s. Flow Reactors Fast reactions and those in the gas phase are generally done in tubular flow reactors, just as they are often done on the commercial scale. Some heterogeneous reactors are shown in Fig. 23-29; the item in Fig. 23-29g is suited to liquid/liquid as well as gas/liquid. Stirred tanks, bubble and packed towers, and other commercial types are also used. The operation of such units can sometimes be predicted from independent data of chemical and mass transfer rates, correlations of interfacial areas, droplet sizes, and other data. Usually it is not possible to measure compositions along a TFR, although temperatures can sometimes be measured. Mostly TFRs are kept at nearly constant temperatures. Small-diameter tubes immersed in a fluidized sand bed or molten lead or salt can hold quite constant temperatures of a few hundred degrees. A recycle unit like that shown in Fig. 23-29a can be operated as a differential reactor with arbitrarily small conversion and temperature change. This and the CSTR are the preferred laboratory devices nowadays, unless the budget allows for only a batch stirred flask. Test work in a tubular flow unit may be desirable if the commercial unit is to be of that type, although rate data from any kind of laboratory equipment are adaptable to the design of most kinds of large-scale equipment. Larger TFRs may be used in pilot plants to test predictions by data from gradientless reactors. Multiple Phases Reactions between gas/liquid, liquid/liquid, and fluid/solid phases are often tested in CSTRs. Other laboratory types are suggested by the commercial units depicted in appropriate sketches in Sec. 23. Liquids can be reacted with gases of low solubili-

ties in stirred vessels, with the liquid charged first and the gas fed continuously at the rate of reaction or dissolution, sometimes with recirculation in larger units. The reactors of Fig. 23-29 are designed to have known interfacial areas. Most equipment for gas absorption without reaction is adaptable to absorption with reaction. The many types of equipment for liquid/liquid extraction also are adaptable to reactions of immiscible phases. Solid Catalysts Processes with solid catalysts are affected by diffusion of heat and mass (1) within the pores of the pellet, (2) between the fluid and the particle, and (3) axially and radially within the packed bed. Criteria in terms of various dimensionless groups have been developed to tell when these effects are appreciable. They are discussed by Mears (Ind. Eng. Chem. Proc. Des. Devel., 10, 541–547 [1971]; Ind. Eng. Chem. Fund., 15, 20–23 [1976]) and Satterfield (Heterogeneous Catalysis in Practice, McGraw-Hill, 1991, p. 491). For catalytic investigations, the rotating basket or fixed basket with internal recirculation are the standard devices nowadays, usually more convenient and less expensive than equipment with external recirculation. In the fixed basket type, an internal recirculation rate of 10 to 15 or so times the feed rate effectively eliminates external diffusional resistance, and temperature gradients. A unit holding 50 cm3 (3.05 in3) of catalyst can operate up to 800 K (1440 R) and 50 bar (725 psi). When deactivation occurs rapidly (in a few seconds during catalytic cracking, for instance), the fresh activity can be found with a transport reactor through which both reactants and fresh catalyst flow without slip and with short contact time. Since catalysts often are sensitive to traces of impurities, the time-deactivation of the catalyst usually can be evaluated only with commercial feedstock, preferably in a pilot plant. Physical properties of catalysts also may need to be checked periodically, including pellet size, specific surface, porosity, pore size and size distribution, and effective diffusivity. The effectiveness of a porous catalyst is found by measuring conversions with successively smaller pellets until no further change occurs. These topics are touched on by Satterfield (Heterogeneous Catalysis in Industrial Practice, McGraw-Hill, 1991). REFERENCES FOR LABORATORY REACTORS Berty, “Laboratory reactors for catalytic studies”, in Leach, ed., Applied Industrial Catalysis, vol. 1, Academic, 1983, pp. 41–57. Danckwerts, Gas-Liquid Reactions, McGraw-Hill, 1970. Hofmann, “Industrial process kinetics and parameter estimation”, in ACS Advances in Chemistry, 109, 519–534 (1972); “Kinetic data analysis and parameter estimation”, in de Lasa, ed., Chemical Reactor Design and Technology, Martinus Nijhoff, 1986, pp. 69–105. Horak and Pasek, Design of Industrial Chemical Reactors from Laboratory Data, Heyden, 1978. Rase, Chemical Reactor Design for Process Plants, Wiley, 1977, pp. 195–259. 124 references. Shah, Gas-Liquid-Solid Reactor Design, McGraw-Hill, 1979, pp. 149–179. 145 references.

SOLVED PROBLEMS These numerical problems deal with ideal types of batch, continuously stirred, and plug flow reactors, for which the formulas are summarized in Tables 7-5 to 7-7. They find parameters of rate equations, conversions, vessel sizes, or operating conditions. Numerical methods are adopted for most integrations and differential equations. Several ODE softwares are readily available, including POLYMATH, which is obtainable through the AIChE. A larger and broader collection of solutions is provided by Walas (Chemical Reaction Engineering Handbook of Solved Problems, Gordon & Breach, 1995). P1.

EQUILIBRIUM OF FORMATION OF ETHYLBENZENE

Ethylbenzene is made from benzene and ethylene in the gas phase at 260°C and 40 atm.

C6H6 + C2H4 ⇔ C6H5C2H5 Equimolal proportions of the reactants are used. Thermodynamic data at 298 K are tabulated. The specific heats are averages. Find: (1) the enthalpy change of reaction at 298 and 573 K; (2) equilibrium constant at 298 and 573 K; (3) fractional conversion at 573 K.

C6H6 C2H4 C6H5C2H5 ∆

Cp

∆Hf

∆Gf

28 5 38 −5

19,820 12,496 7,120 −25,196

30,989 16,282 31,208 −16,063

SOLVED PROBLEMS ∆HT = ∆H298 +



T

∆Cp dT = −25,196 −5(T − 298)

298

= −26,576 at 573 K

(1)

−∆G298 16,063 ln K298 = ᎏ = ᎏᎏ = 26.90 298R (1.987)(298)

dA − ᎏ = k1AB = 5.9AB dt

(1)

dB − ᎏ = k1AB + k2B2 = 5.9AB + 1.443B2 dt

(2)

Dividing these equations,

d ln K ∆H 23,706 5 −ᎏ ᎏ = ᎏ2 = − ᎏ dT RT RT2 RT

dB k2B ᎏ−ᎏ=1 dA k1A This is a linear equation whose solution is

11,854 2.50  ᎏ + ᎏ  dT = 6.17 T T 573

(2)

2

298

K = 485

k1 B = ᎏ A + IAk 2 /k 1 = 1.32A − 0.010A0.245 k1 − k2

(4)

The integration constant was evaluated with A0 = B0 = 0.010. Substituting (4) into (1),

x(2 − x) = ᎏᎏ2 40(1 − x) x = 0.9929, fraction converted P2.

(3)

dA − ᎏ = 5.9A(1.32A − 0.010A0.245) dt

(3)

OPTIMUM CYCLE PERIOD WITH DOWNTIME

Find the optimum cycle period for a first-order batch reaction with a downtime of ϑd h per batch.

The variables are separable, but an integration in closed form is not possible because of the odd exponent. Numerical integration followed by substitution into (4) will provide both A and B as functions of t. The plots, however, are of solutions of the original differential equations with ODE.

dC − ᎏ = kC dϑ

.01

.01

 

Butadiene

1 C0 ϑ = ᎏ ln ᎏ k C

(5)

gmol/liter

ln K298 = 26.9 −

7-29

A, Acrolein

Number of daily batches: B, Butadiene

24 n=ᎏ ϑ + ϑd

0 0

0 0

.01

10

20

30

40

50

t

Acrolein

Daily yield: 24Vr(C0 − C) 24kVrC0(1 − C/C0) y = Vr(C0 − C)n = ᎏᎏ = ᎏᎏ 1 k ln (C0 /C) + ϑd ln (C0 /C) + kϑd The ordinate of the plot is y/24kVrC0 which is proportional to the daily yield. The peaks in this curve are at these values of the parameters: kϑd C/C0

0.01 0.87

0.10 0.65

1 0.32

5 0.12

BATCH REACTION WITH HEAT TRANSFER

A second-order reaction proceeds in a batch reactor provided with heat transfer. Initial conditions are T0 = 350 and C0 = 1. Other data are:



5,000 k = exp 16 − ᎏ T



∆Hr = −(5000 + 5T )

ft3/(lb mol⋅h)

(1)

Btu/lb mol

(2)

Btu/h

(3)

ρCp = 50

1.0 0.9

l – C/C0 ln(C0 /C) + kθd

P4.

The rate of heat transfer is

kθd = 0.01

0.8 0.7

Q = UA(300 − T )

0.1

0.6

The temperature T and the time t will be found in terms of fractional conversion x when UA/Vr = 0 or 150. The rate equation may be written:

0.5 0.4 0.3 0.2

1.0

0.1 0.0 0.0

5.0 0.1

0.2 0.3

0.4

0.5 0.6 C/C0

0.7

0.8 0.9

1.0

dt 1 ᎏ = ᎏᎏ2 dx kC0(1 − x)

(4)

The differential heat balance is ρCpVr dT = Qdt − ∆HrVr C0dx

P3.

PARALLEL REACTIONS OF BUTADIENE

Butadiene (B) reacts with acrolein (A) and also forms a dimer according to the reactions C4H6 + C3H4O ⇒ C7H10O, 2 C4H6 ⇒ C8H12 1

2

The reaction is carried out in a closed vessel at 330°C, starting at 1 atm with equal concentrations of A and B, 0.010 g mol /L each. Specific rates are k1 = 5.900 and k2 = 1.443 L /(g mol⋅min). Find (1) B as a function of A; (2) A and B as functions of t.

Substituting for dt from Eq. (4) and rearranging,



dT 1 Q ᎏ = ᎏ ᎏᎏ2 − ∆HrC0 dx ρCp Vr kC0(1 − x)



UA(300 − T) = 0.02 ᎏᎏ + 5000 + 5T Vr k(1 − x)2





(5)

Equations (1), (4), and (5) are solved simultaneously with UA/Vr = 0 or 150. In the adiabatic case, the temperature tends to run away.

7-30

REACTION KINETICS 0.2

500 UA/Vr = 0



0.1

150

t

T

150 400

0 350

300 0

.2

.4

.6

.8

0 1

X

P5.

The downtime is 1 h per batch. Find the temperature at which the daily production is a maximum. The reaction time of one batch is 1 f df tb = ᎏ ᎏᎏ (1) k 0 (1 − f )2 − f 2/Ke 24 Batches/day = ᎏ tb + 1 24 Daily production = ᎏ VrCa0 f tb + 1 Maximize P = f/(tb + 1) as a function of temperature. Eq. (1) is integrated with POLYMATH for several temperatures and the results are plotted. The tabulation gives the integration at 550 K. The peak value of P = f/(tb + 1) = 0.1941 at 550 K, tb = 0.6, and f = 0.3105.

A SEMIBATCH PROCESS

A tank is charged initially with vr0 = 100 L of a solution of concentration Cb0 = 2 g mol/L. Another solution is then pumped in at V′ = 5 L/min with concentration Ca0 = 0.8 until a stoichiometric amount has been added. The rate equation is

Maximum daily production = 0.1941(24)VrCa0 = 4.66Vr kg mol/d 2.2

r = 0.015CaCb g mol/(L⋅min)

2.0

Find the concentration during the filling period and for 50 min afterward.

1.8 1.6 tb, 10P

Vr = 100 + 5t 100(2) − 5(0.8)t 40 − 0.8t Cb = Ca + ᎏᎏ = Ca + ᎏ 100 + 5t 20 + t

(1)

1.4 1.2

(2)

1.0

dCa 0.8 − Ca 40 − 0.8t (3) ᎏ = ᎏ − 0.015Ca Ca + ᎏ dt 20 + t 20 + t The input is continued until 200 lb mol of A have been added, which is for 50 min. Eq. (3) is integrated for this time interval. After input is discontinued the rate equation is dC − ᎏa = kCa2 (4) dt At t = 50, Ca = Ca1 = 0.4467. 1 1 (5) Ca = ᎏᎏ = ᎏᎏᎏ 1/Ca1 + k(t − 50) 2.2386 + 0.015(t − 50) Plots are shown for several specific rates, including k = 0 when no reaction takes place.

0.6





0.4 0.2 400 425 450 475 500 525 550 575 600 625 650 Temperature, T

The equations:







p = f /(t + 1)

k = 0.015 Ca



d( f ) 2 ᎏ = k∗ (1 − f )∗∗2 − f ∗∗ ᎏ d(t) ke x = 550 2,500 k = exp 4.5 − ᎏ x 5,178 ke = exp 28.8 − ᎏᎏ x − .037 ∗ x

 

k=0

Initial values:

t0 = 0.0

Final value:

tf = 2.0000

0.3 k = 0.050

100

50 t

tb

0.8

0.6

0 0

10P

P6. OPTIMUM REACTION TEMPERATURE WITH DOWNTIME A liquid phase reaction 2A ⇔ B + C has the rate equation 2 C C f2 b c 2 (1 − f )2 − ᎏ , kg mol/(m3⋅h) ra = k Ca2 − ᎏ = kCa0 Ke Ke where f = fractional conversion Ca0 = 1 2,500 k = exp 4.5 − ᎏ T 5,178 Ke = exp 28.8 − 0.037T − ᎏ T

f0 = 0.0

t

f

p

0.0 0.2000 0.4000 0.6000 0.8000 1.0000 1.2000 1.4000 1.6000 1.8000 2.0000

0.0 0.1562 0.2535 0.3105 0.3427 0.3605 0.3702 0.3755 0.3784 0.3799 0.3807

0.0 0.1302 0.1811 0.1941 0.1904 0.1802 0.1683 0.1565 0.1455 0.1357 0.1269

1





 









P7.

RATE EQUATIONS FROM CSTR DATA

For the consecutive reactions 2A ⇒ B and 2B ⇒ C, concentrations were measured as functions of residence time in a CSTR. In all experiments, Ca0 = 1 lb mol/ft3. Volumetric flow rate was constant. The data are tabulated in the first three columns. Check the proposed rate equations, ra = k1Caα β

rb = −0.5k1Caα + k2Cb

SOLVED PROBLEMS Write and rearrange the material balances on the CSTR.

7-31

The solution is,

Ca0 = Ca +
tra Ca0 − Ca ra = ᎏ = k1Caα t


(1)

Cb0 − Cb β β (2) rb = ᎏ = −0.5k1Caα + k2Cb = −0.5ra + k2Cb t
Numerical values of ra, rb, and rb + 0.5ra are tabulated. The constants of the rate equations are evaluated from the plots of the linearized equations, ln ra = ln k1 + α ln Ca = −2.30 + 2.001 ln Ca ln (rb + 0.5ra ) = ln k2 + β ln Cb = −4.606 + 0.9979 ln Cb which make the rate equations ra = 0.1003Ca2.00

(3)

rb = −0.0502Ca2 + 0.01Cb0.998

(4)


t

Ca

Cb

ra

−rb

rb + 0.5ra

10 20 40 100 450

1.000 0.780 0.592 0.400 0.200

0.4545 0.5083 0.5028 0.400 0.1636

0.100 0.061 0.0352 0.0160 0.0040

0.04545 0.02542 0.01257 0.0040 0.000364

0.00455 0.00508 0.00503 0.0040 0.00164

2

P9.

C1 = 0.1994,

= 60.1% conversion

C2 = 0.1025,

= 79.5% conversion

INSTANTANEOUS AND GRADUAL FEED RATES

Initially a reactor contains 2 m3 of a solvent. A solution containing 2 kg mol/m3 of reactant A is pumped in at the rate of 0.06 m3/min until the volume becomes 4 m3. The rate equation is ra = 0.25Ca, 1/min. Compare the time-composition profile of this operation with charging all of the feed instantaneously. During the filling period, Vr = 2 + 0.06t d(VrCa) dC dV V′Ca0 = kVrCa + ᎏ = kVrCa + Vr ᎏa + Ca ᎏr dt dt dt dCa 0.06(2) = 0.25(2 + 0.06t)Ca + (2 + 0.06t) ᎏ + 0.06Ca dt dCa 0.12 −(0.56 + 0.015t)Ca Ca0 = 0 (1) ᎏ = ᎏᎏᎏ , dt 2 + 0.06t When all of A is charged at the beginning, dCa Ca0 = 0.5 (2) ᎏ = −0.25Ca, dt The integrals of these two equations are plotted. A peak value, Ca = 0.1695, is reached in the first operation at t = 10. .5

10–1

.4 3 2

(Ca, ra) .3 Ca

10–2

Charged at once

.2 Charged gradually

3 2 10–3 10–1

(Cb, 0.5ra + rb) 2

3 4 Ca, Cb

5

6

.1 7 8 9 100

0 0

8

16

24

32

40

t

P8.

COMPARISON OF BATCH AND CSTR OPERATIONS

A solution containing 0.5 lb mol/ft3 of reactive component is to be treated at 25 ft3/h. The rate equation is −dC r = ᎏ = 2.33C1.7 lb mol/(ft3⋅h) dt 1. If the downtime is 45 min per batch, what size reactor is needed for 90% conversion? 2. What percentage conversion is attained with a two-stage CSTR, each vessel being 50 ft3? Part 1: The integral of the rate equation is solved for the time, 1 1 1 1 t = ᎏ (C−0.7 −C−0.7 −ᎏ 0 ) = ᎏ ᎏ 0.7k 0.7(2.33) 0.050.7 0.50.7



= 4.00 h 24 Number of batches = ᎏᎏ = 5.053/d (4 + 0.75) 24(25) Reactor volume Vr = ᎏ = 118.7 ft3 5.053 Part 2: 50 τ=ᎏ=2 25 0.5 = C1 + τr = C1 + 2(2.33)C1.7 1 C1 = C2 + 2(2.33)C1.7 2



P10. FILLING AND UNSTEADY OPERATING PERIOD OF A CSTR A stirred reactor is being charged at 5 ft3/min with a concentration of 2 mol/ft3. The reactor has a capacity of 150 ft3 but is initially empty. The rate of reaction is r = 0.02C2 lb mol/(ft3⋅min). After the tank is filled, pumping is continued and overflow is permitted at the same flow rate. Find the concentration in the tank when it first becomes full, and find how long it takes for the effluent concentration to get within 95% of the steady state value. Filling period: Vr = V′t





dVrC dC V′C0 = kVrC2 + ᎏ = kVrC2 + C + t ᎏ V′ dt dt dC C0 − ktC2 − C 2 − 0.02tC2 − C C = 2 when t = 0 ᎏ = ᎏᎏ = ᎏᎏ , dt t t The numerical solution is C = 1.3269 when t = 30. Unsteady period: dC V′C0 = V′C + kVrC2 + Vr ᎏ dt dC C0 − C − kτC2 2 − C − 0.02(30)C2 ᎏ = ᎏᎏ = ᎏᎏ , dt τ 30 C = 1.3269 when t = 30.

REACTION KINETICS nt = ns + na + nb = 0.5na0 + na + 0.5(na0 − na) = na0 + 0.5na na Pa = ᎏᎏ (1) na0 + 0.5na

The variables are separable, but the plot is of a numerical solution. The steady state concentration is 1.1736. At 95% approach to steady state from the condition at t = 30, C = 0.05(1.3269) + 0.95(1.1736) = 1.1813

0.5(na0 − na) Pb = ᎏᎏ na0 + 0.5na

From a printout of the solution, t = 67.4 min at this value. 2

Pb 118 na2 na0 − na (1) ra = k Pa2 − ᎏ = ᎏᎏ ᎏᎏ − ᎏ Ke na0 + 0.5na na0 + 0.5na 2(1.27) Put na0 = 6, substitute Eq. (1) into the flow reactor equation, and integrate numerically. 6 dna Vr = ᎏ = 0.0905 m3 3.6 ra

C



1.5

(67.4, 1.1813)

P11.

50 t

100

P13.

A second order reaction is conducted in two equal CSTR stages. The residence time per stage is τ = 1 and the specific rate is kC0 = 0.5. Feed concentration is C0. Two cases are to be examined: (1) with pure solvent initially in the tanks; and (2) with concentrations C0 initially in both tanks, that is, with C10 = C20 = C0. The unsteady balances on the two reactors are C fi = ᎏi C0 dC1 FC0 = FC1 + VrkC12 + Vr ᎏ dt df 1 = f1 + 0.5f 12 + ᎏ1 dt

AUTOCATALYTIC REACTION WITH RECYCLE

Part of the effluent from a PFR is returned to the inlet. The recycle ratio is R, fresh feed rate is F0 F R = ᎏr F0 Ft = Fr + F0 = F0(R + 1) The concentration of the mixed feed is Ca0 + RCaf Cat = ᎏᎏ 1+R where Caf is the outlet concentration. For the autocatalytic reaction A ⇒ B, the rate equation is ra = kCaCb = kCa(Ca0 − Ca)

(1)

The flow reactor equation is

dC2 FC1 = FC2 + Vr kC22 + Vr ᎏ dt

−Ft dCa = −F0(R + 1)dCa = ra dVr = kCa(Ca0 − Ca)dVr

df f1 = f2 + 0.5f 22 + ᎏ2 (2) dt The steady state values are the same for both starting conditions, obtained by zeroing the derivatives in Eqs. (1) and (2). Then

kVr ᎏ = (R + 1) F0

kVr /Fo

5

P14. .8

1.6

2.4

3.2

4

t

P12.

BUTADIENE DIMERIZATION IN A TFR

A mixture of 0.5 mol of steam per mol of butadiene is dimerized in a tubular reactor at 640°C and 1 atm. The forward specific rate is k = 118 g mol/(L⋅h⋅atm2) and the equilibrium constant is 1.27. Find the length of 10-cm ID tube for 40% conversion when the total feed rate is 9 kg mol/h. 2A ⇔ B na0 = 6 kg mol/h

1.6 1.4 1.2

Left

1

4

0.8

3

0.6

2

0.4

1 0 10–3 2 3

.2 .1 0 0

1.8 Right

6

Stage 2

f20 = 0

.3

2.0

7

f10 = 0

.5 .4

dCa ᎏᎏ Ca(Ca0 − Ca)

8

Stage 1

f20 = 1

caf

9

1 .9

.6

cat

10

f2 = 0.7321

f10 = 1



The plot is for Ca0 = 2 and Caf = 0.04. The minimum reactor size is at a recycle ratio R = 0.23 and mixed feed Cat = 1.57.

The plots are of numerical solutions.

.8 .7



0.0905(106) L = ᎏᎏ = 1,153 cm 78.5

SECOND-ORDER REACTION IN TWO STAGES

f1 = 0.5702,





(30, 1.3269)

1 0



Cat

7-32

0.2 0.0 10–2

2 3 10–1 2 3 Recycle ratio, R

100

2 3

101

MINIMUM RESIDENCE TIME IN A PFR

A reversible reaction A ⇔ B is conducted in a plug flow reactor. The rate equation is x r = kCa0 1 − x − ᎏ Ke where Ca0 = 4 5,800 k = exp 17.2 − ᎏ T



  9,000 K = exp −24.7 + ᎏ  T e



SOLVED PROBLEMS = na0[20(1 − x) + 15(2x)dT + 24U(T − Tw)dVr

Find the conditions for minimum Vr /V′ when conversion is 80%. The flow reactor equation is

dT −∆Hrra − 24U(T − Tw) 8,000ra −120(T − 630) ᎏ = ᎏᎏᎏ = ᎏᎏᎏ d(Vr /na0) 20 − 10x 20 − 10x

1−x−x −dna = V′Ca0dx = kCa0 ᎏ dVr Ke







Vr 1 0.8 dx 1 1 (1) ᎏ=ᎏ ᎏᎏ = ᎏᎏ ln ᎏᎏ V′ k 0 1 − x − x/Ke k(1 + 1/Ke) 0.2 − 0.8/Ke The plot of this equation shows the minimum to be Vr /V′ = 2.04 at T = 340 K.

(4)

Differential Eqs. (3) and (4) are solved simultaneously with auxiliary Eqs. (1) and (2) by ODE. The solutions with U = 5 and U = 0 are shown. 1000

1

4 2

4.00

1. 2. 3. 4.

3.50 T

3.25

800

3.00

T, U = 5 T, U = 0 x, U = 5 x, U = 0

0.5

x

3

3.75

Vr /V′

7-33

1

2.75 2.50 2.25 600

2.00

Minimum 2.04 at 340 K 1.75 1.50 330 332 334 336 338 340 342 344 346 348 350

0

T

P15.

P16.

HEAT TRANSFER IN A CYLINDRICAL REACTOR

A reaction A ⇒ 2B runs in a tube provided with a cooling jacket that keeps the wall at 630 R. Inlet is pure A at 650 R and 50 atm. Other data are stated in the following. Find the profiles of temperature and conversion along the reactor, both with heat transfer and adiabatically. 1 Tube diameter D = ᎏ ft 6 Cpa = 20, Cpb = 15 Btu/(lb mol R)

(1)

Heat transfer area dA = (4/D)dVr = 24dVr Rate equation: na na π 50k na ra = k ᎏ = k ᎏ ᎏ = ᎏ ᎏ V RT nt 0.729T 2na0 − na





 









dx P(1 − x) (1) ᎏ = 0.02 ᎏ dL 1 + 2x where several factors have been combined into the numerical coefficient. The pressure gradient due to friction is proportional to the flowing mole rate, 1 + 2x, and inversely to the density or the pressure. Here again, several factors are incorporated into a numerical coefficient, making



 (2)

dP 1 + 2x − ᎏ = 0.6 ᎏ (2) dL P The numbered equations are integrated and plotted. They show the typical fall in pressure as conversion with an increase in the number of moles proceeds at constant pressure, x = 0.48 when L = 10.

na x=1−ᎏ na0 Flow reactor:

.5 P

−dna = na0dx = radVr

∆Hr dna = −∆Hrra dVr = niCpi dT + U(T − Tw)dA

(3)

x and P/10

.4

dx ᎏ = ra d(Vr /na0) Heat balance over a differential volume dVr:



or



68.6k 1 − x =ᎏ ᎏ T 1+x

n na kP 1 − x kP ra = k ᎏa = ᎏ ᎏᎏ =ᎏ ᎏ V′ RT 3na0 − na RT 1 + 2x When put into the plug flow equation,



3,000 k = exp 7.82 − ᎏ T Heat transfer coefficient U = 5 Btu/(ft3⋅h R)

    

PRESSURE DROP AND CONVERSION IN A PFR

A reaction A ⇒ 3B takes place in a tubular flow reactor at constant temperature and an inlet pressure of 5 atm. The rate equation is

kP 1 − x na0 dx = ᎏ ᎏ AdL RT 1 + 2x

∆Hr = −8,000 Btu/(lb mol A)



0 1

0.5 Vr /na0

.3

x

.2 .1 0 0

2

4

6 L

8

10

Section 8

Process Control

Thomas F. Edgar, Ph.D., Professor of Chemical Engineering, University of Texas, Austin, TX. (Advanced Control Systems, Process Measurements, Section Editor) Cecil L. Smith, Ph.D., Principal, Cecil L. Smith Inc., Baton Rouge, LA. (Batch Process Control, Telemetering and Transmission, Digital Technology for Process Control, Process Control and Plant Safety) F. Greg Shinskey, B.S.Ch.E., Consultant (retired from Foxboro Co.), North Sandwich, NH. (Fundamentals of Process Dynamics and Control, Unit Operations Control) George W. Gassman, B.S.M.E., Senior Research Specialist, Final Control Systems, Fisher Controls International, Inc., Marshalltown, IA. (Controllers, Final Control Elements, and Regulators) Paul J. Schafbuch, Ph.D., Senior Research Specialist, Final Control Systems, Fisher Controls International, Inc., Marshalltown, IA. (Controllers, Final Control Elements, and Regulators) Thomas J. McAvoy, Ph.D., Professor of Chemical Engineering, University of Maryland, College Park, MD. (Fundamentals of Process Dynamics and Control) Dale E. Seborg, Ph.D., Professor of Chemical Engineering, University of California, Santa Barbara, CA. (Advanced Control Systems)

FUNDAMENTALS OF PROCESS DYNAMICS AND CONTROL The General Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-4 Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-4 Feedforward Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-4 Computer Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-4 Process Dynamics and Mathematical Models . . . . . . . . . . . . . . . . . . . . . 8-4 Open-Loop versus Closed-Loop Dynamics . . . . . . . . . . . . . . . . . . . . 8-4 Physical Models versus Empirical Models . . . . . . . . . . . . . . . . . . . . . 8-5 Nonlinear versus Linear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-6 Simulation of Dynamic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-6 Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-6 Transfer Functions and Block Diagrams . . . . . . . . . . . . . . . . . . . . . . . 8-6 Continuous versus Discrete Models . . . . . . . . . . . . . . . . . . . . . . . . . . 8-7 Process Characteristics in Transfer Functions. . . . . . . . . . . . . . . . . . . 8-7 Fitting Dynamic Models to Experimental Data . . . . . . . . . . . . . . . . . 8-10 Feedback Control System Characteristics. . . . . . . . . . . . . . . . . . . . . . . . 8-11 Closing the Loop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-11 On/Off Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-12 Proportional Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-12 Proportional-plus-Integral (PI) Control . . . . . . . . . . . . . . . . . . . . . . . 8-12 Proportional-plus-Integral-plus-Derivative (PID) Control . . . . . . . . 8-12 Controller Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-13 Controller Performance Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-13 Tuning Methods Based on Known Process Models . . . . . . . . . . . . . . 8-14 Tuning Methods When Process Model Is Unknown . . . . . . . . . . . . . 8-15

ADVANCED CONTROL SYSTEMS Benefits of Advanced Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Advanced Control Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Feedforward Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cascade Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time-Delay Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selective and Override Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adaptive Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fuzzy Logic Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Statistical Process Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multivariable Control Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Control Strategies for Multivariable Control Problems . . . . . . . . . . . Decoupling Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pairing of Controlled and Manipulated Variables . . . . . . . . . . . . . . . . RGA Method for 2 × 2 Control Problems . . . . . . . . . . . . . . . . . . . . . . RGA Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model Predictive Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Advantages and Disadvantages of MPC . . . . . . . . . . . . . . . . . . . . . . . Economic Incentives for Automation Projects . . . . . . . . . . . . . . . . . . Basic Features of MPC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Implementation Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integration of MPC and On-Line Optimization . . . . . . . . . . . . . . . . . Real-Time Process Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Essential Features of Optimization Problems. . . . . . . . . . . . . . . . . . .

8-16 8-16 8-16 8-18 8-19 8-19 8-20 8-21 8-21 8-22 8-22 8-23 8-23 8-24 8-24 8-25 8-25 8-25 8-25 8-25 8-27 8-27 8-27 8-28 8-1

8-2

PROCESS CONTROL

Development of Process (Mathematical) Models . . . . . . . . . . . . . . . . Formulation of the Objective Function. . . . . . . . . . . . . . . . . . . . . . . . Unconstrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single Variable Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multivariable Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonlinear Programming. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expert Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8-28 8-29 8-30 8-30 8-30 8-30 8-31 8-31

UNIT OPERATIONS CONTROL Process and Instrumentation Diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . Control of Heat Exchangers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Steam-Heated Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exchange of Sensible Heat. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distillation Column Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Controlling Quality of a Single Product. . . . . . . . . . . . . . . . . . . . . . . . Controlling Quality of Two Products . . . . . . . . . . . . . . . . . . . . . . . . . . Chemical Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Composition Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Controlling Evaporators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Drying Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8-31 8-32 8-32 8-32 8-33 8-33 8-33 8-33 8-33 8-35 8-35 8-36

BATCH PROCESS CONTROL Batch versus Continuous Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Batches and Recipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Routing and Production Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . Production Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Batch Automation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interlocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discrete Device States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Process States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Regulatory Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sequence Logic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Batch Production Facilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equipment Suite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Process Unit or Batch Unit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Item of Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structured Batch Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Product Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Process Technology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8-37 8-38 8-39 8-39 8-39 8-39 8-39 8-40 8-40 8-41 8-41 8-42 8-42 8-42 8-42 8-42 8-42 8-42 8-42

PROCESS MEASUREMENTS General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Accuracy and Repeatability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamics of Process Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . Selection Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermocouples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resistance Thermometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Filled-System Thermometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bimetal Thermometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pyrometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Accuracy of Pyrometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid-Column Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elastic-Element Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electrical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orifice Meter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Venturi Meter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbine Meter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vortex-Shedding Flowmeters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ultrasonic Flowmeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetic Flowmeters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coriolis Mass Flowmeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Level Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Float-Actuated Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Head Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electrical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sonic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physical Property Measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Density and Specific Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Viscosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8-43 8-43 8-43 8-44 8-44 8-45 8-45 8-45 8-45 8-45 8-46 8-46 8-46 8-47 8-47 8-47 8-47 8-48 8-48 8-48 8-48 8-48 8-48 8-48 8-49 8-49 8-49 8-49 8-49 8-49 8-49 8-50 8-50 8-50 8-50 8-50

Refractive-Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dielectric Constant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Conductivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chemical Composition Analyzers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chromatographic Analyzers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Infrared Analyzers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ultraviolet and Visible-Radiation Analyzers . . . . . . . . . . . . . . . . . . . . Paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electroanalytical Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conductometric Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement of pH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specific-Ion Electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moisture Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dew-Point Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Piezoelectric Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Capacitance Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Oxide Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Photometric Moisture Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gear Train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Differential Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hall-Effect Sensors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sampling Systems for Process Analyzers . . . . . . . . . . . . . . . . . . . . . . . . . Selecting the Sampling Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sample Withdrawal from Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sample Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sample Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8-50 8-50 8-50 8-50 8-51 8-51 8-51 8-51 8-51 8-51 8-51 8-51 8-51 8-51 8-52 8-52 8-52 8-52 8-52 8-52 8-52 8-52 8-52 8-52 8-52 8-53 8-53

TELEMETERING AND TRANSMISSION Analog Signal Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Digital Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analog Input and Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pulse Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Serial Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Microprocessor-Based Transmitters . . . . . . . . . . . . . . . . . . . . . . . . . . Transmitter/Actuator Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Filtering and Smoothing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alarms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8-53 8-53 8-53 8-54 8-54 8-54 8-54 8-55 8-55

DIGITAL TECHNOLOGY FOR PROCESS CONTROL Hierarchy of Information Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement Devices and Actuators . . . . . . . . . . . . . . . . . . . . . . . . . Regulatory Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Supervisory Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Production Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Corporate Information Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distributed Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distributed Database and the Database Manager . . . . . . . . . . . . . . . . . Historical Database Subsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Process Control Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single-Loop Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Programmable Logic Controllers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intercomputer Communications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8-56 8-56 8-57 8-57 8-57 8-57 8-57 8-58 8-59 8-59 8-59 8-60 8-60

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS Electronic and Pneumatic Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . Electronic Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pneumatic Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Control Valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Valves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Special Application Valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Valve-Control Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Valve Application Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Process Valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Valves for On/Off Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure Relief Valves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Check Valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adjustable Speed Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Regulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Self-Operated Regulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pilot-Operated Regulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Over-Pressure Protection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8-61 8-61 8-62 8-64 8-64 8-65 8-66 8-67 8-73 8-77 8-77 8-78 8-78 8-79 8-79 8-80 8-81 8-81

PROCESS CONTROL AND PLANT SAFETY Role of Automation in Plant Safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integrity of Process Control Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . Considerations in Implementation of Safety Interlock Systems. . . . . . . Interlocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8-81 8-82 8-82 8-83 8-84

Nomenclature Symbol

Definition

A Aa Ac Am Av A1 b B B i* c,C cA Cd Ci C i* CL Co Cr Cv D D i* D(s) e E f F,f FL gc gi G Gc Gf GL Gm Gp Gt Gv hi h1 H i Ii j J k kf kr K Kc KL Km Kp Ku L Lp m,M mc Mo Mr Mw n N p1 pa pv P Pu q qb Qa rc R

Area Actuator area Amplitude of controlled variable Output amplitude limits Cross sectional area of valve Cross sectional area of tank Controller output bias Bottoms flow rate Limit on control Controlled variable Concentration of A Discharge coefficient Inlet concentration Limit on control move Specific heat of liquid Integration constant Heat capacity of reactants Valve flow coefficient Distillate flow rate Limit on output Decoupler transfer function Error Economy of evaporator Function of time Feed flow rate Pressure recovery factor Unit conversion constant Algebraic inequality constraint Transfer function Controller transfer function Feedforward controller transfer function Load transfer function Sensor transfer function Process transfer function Transmitter transfer function Valve transfer function Algebraic equality constraints Liquid head in tank Latent heat of vaporization Summation index Impulse response coefficient Time index Objective function or performance index Time index Flow coefficient Kinetic rate constant Gain Controller gain Load transfer function gain Measurement gain Process gain Ultimate controller gain (stability) Disturbance or load variable Sound pressure level Manipulated variable Number of constraints Mass flow Mass of reactants Molecular weight Number of data points, number of stages or effects Number of inputs/outputs, model horizon Pressure Actuator pressure Vapor pressure Proportional band (%) Proportional band (ultimate) Radiated energy flux Energy flux to a black body Flow rate Number of constraints Equal percentage valve characteristic

Symbol

Definition

R,r RT R1 s s Si t T Tb Tf TR u U V Vs w wi W x xi xT X y Y z zi Z

Set point Resistance in temperature sensor Valve resistance Laplace transform variable Search direction Step response coefficient Time Temperature Base temperature Exhaust temperature Reset time Controller output Heat transfer coefficient Volume Product value Mass flow rate Weighting factor Steam flow rate Mass fraction Optimization variable Pressure drop ratio factor Transform of deviation variable Process output, controlled variable, valve travel Controller tuning law, expansion factor z-transform variable Feed mole fraction (distillation) Compressibility factor

α αT β γ δ ∆q ∆t ∆T ∆u ε ζ θ λ Λ ξ ρ σ Στ τ τD τF τI τL τn τp τo φPI

Digital filter coefficient Temperature coefficient of resistance Resistance thermometer parameter Ratio of specific heats Move suppression factor Load step change Time step Temperature change Control move Spectral emissivity, step size Damping factor (second order system) Time delay Relative gain array parameter, wavelength Relative gain array Deviation variable Density Stefan-Boltzmann constant Total response time Time constant Derivative time (PID controller) Filter time constant Integral time (PID controller) Load time constant Natural period of closed loop Process time constant Period of oscillation Phase lag

A b c eff F i L m p s set t u v

Species A Best Controller Effective Feedforward Initial, inlet Load, disturbance Measurement or sensor Process Steady state Set point value Transmitter Ultimate Valve

Greek symbols

Subscripts

8-3

FUNDAMENTALS OF PROCESS DYNAMICS AND CONTROL THE GENERAL CONTROL SYSTEM A process is shown in Fig. 8-1 with a manipulated input M, a load input L, and a controlled output C, which could be flow, pressure, liquid level, temperature, composition, or any other inventory, environmental, or quality variable that is to be held at a desired value identified as the set point R. The load may be a single variable or aggregate of variables acting either independently or manipulated for other purposes, affecting the controlled variable much as the manipulated variable does. Changes in load may occur randomly as caused by changes in weather, diurnally with ambient temperature, manually when operators change production rate, stepwise when equipment is switched in or out of service, or cyclically as the result of oscillations in other control loops. Variations in load will drive the controlled variable away from set point, requiring a corresponding change in the manipulated variable to bring it back. The manipulated variable must also change to move the controlled variable from one set point to another. An open-loop system positions the manipulated variable either manually or on a programmed basis, without using any process measurements. This operation is acceptable for well-defined processes without disturbances. An automanual transfer switch is provided to allow manual adjustment of the manipulated variable in case the process or the control system is not performing satisfactorily. A closed-loop system uses the measurement of one or more process variables to move the manipulated variable to achieve control. Closedloop systems may include feedforward, feedback, or both. Feedback Control In a feedback control loop, the controlled variable is compared to the set point R, with the difference, deviation, or error e acted upon by the controller to move m in such a way as to minimize the error. This action is specifically negative feedback, in that an increase in deviation moves m so as to decrease the deviation. (Positive feedback would cause the deviation to expand rather than diminish and therefore does not regulate.) The action of the controller is selectable to allow use on process gains of both signs. The controller has tuning parameters related to proportional, integral, derivative, lag, deadtime, and sampling functions. A negative feedback loop will oscillate if the controller gain is too high, but if it is too low, control will be ineffective. The controller parameters must be properly related to the process parameters to ensure closed-loop stability while still providing effective control. This is accomplished first by the proper selection of control modes to satisfy the requirements of the process, and second by the appropriate tuning of those modes. Feedforward Control A feedforward system uses measurements of disturbance variables to position the manipulated variable in such a way as to minimize any resulting deviation. The disturbance

FIG. 8-1

8-4

variables could be either measured loads or the set point, the former being more common. The feedforward gain must be set precisely to offset the deviation of the controlled variable from the set point. Feedforward control is usually combined with feedback control to eliminate any offset resulting from inaccurate measurements and calculations and unmeasured load components. The feedback controller can either bias or multiply the feedforward calculation. Computer Control Computers have been used to replace analog PID controllers, either by setting set points of lower level controllers in supervisory control, or by driving valves directly in direct digital control. Single-station digital controllers perform PID control in one or two loops, including computing functions such as mathematical operations, characterization, lags, and deadtime, with digital logic and alarms. Distributed control systems provide all these functions, with the digital processor shared among many control loops; separate processors may be used for displays, communications, file servers, and the like. A host computer may be added to perform highlevel operations such as scheduling, optimization, and multivariable control. More details on computer control are provided later in this section. PROCESS DYNAMICS AND MATHEMATICAL MODELS GENERAL REFERENCES: Seborg, Edgar, and Mellichamp, Process Dynamics and Control, Wiley, New York, 1989; Marlin, Process Control, McGraw-Hill, New York, 1995; Ogunnaike and Ray, Process Dynamics Modeling and Control, Oxford University Press, New York, 1994; Smith and Corripio, Principles and Practices of Automatic Process Control, Wiley, New York, 1985

Open-Loop versus Closed-Loop Dynamics It is common in industry to manipulate coolant in a jacketed reactor in order to control conditions in the reactor itself. A simplified schematic diagram of such a reactor control system is shown in Fig. 8-2. Assume that the reactor temperature is adjusted by a controller that increases the coolant flow in proportion to the difference between the desired reactor temperature and the temperature that is measured. The proportionality constant is Kc. If a small change in the temperature of the inlet stream occurs, then depending on the value of Kc, one might observe the reactor temperature responses shown in Fig. 8-3. The top plot shows the case for no control (Kc = 0), which is called the open loop, or the normal dynamic response of the process by itself. As Kc increases, several effects can be noted. First, the reactor temperature responds faster and faster. Second, for the initial increases in Kc, the maximum deviation in the reactor temperature becomes smaller. Both of these effects are desirable so that disturbances from normal operation have

Block diagram for feedforward and feedback control.

FUNDAMENTALS OF PROCESS DYNAMICS AND CONTROL

FIG. 8-2

Reactor control system.

as small an effect as possible on the process under study. As the gain is increased further, eventually a point is reached where the reactor temperature oscillates indefinitely, which is undesirable. This point is called the stability limit, where Kc = Ku, the ultimate controller gain. Increasing Kc further causes the magnitude of the oscillations to increase, with the result that the control valve will cycle between full open and closed. The responses shown in Fig. 8-3 are typical of the vast majority of regulatory loops encountered in the process industries. Figure 8-3 shows that there is an optimal choice for Kc, somewhere between 0 (no control) and Ku (stability limit). If one has a dynamic model of a process, then this model can be used to calculate controller settings. In Fig. 8-3, no time scale is given, but rather the figure shows relative responses. A well-designed controller might be able to speed up the response of a process by a factor of roughly two to four. Exactly how fast the control system responds is determined by the dynamics of the process itself. Physical Models versus Empirical Models In developing a dynamic process model, there are two distinct approaches that can be

FIG. 8-3

Typical control system responses.

8-5

taken. The first involves models based on first principles, called physical models, and the second involves empirical models. The conservation laws of mass, energy, and momentum form the basis for developing physical models. The resulting models typically involve sets of differential and algebraic equations that must be solved simultaneously. Empirical models, by contrast, involve postulating the form of a dynamic model, usually as a transfer function, which is discussed below. This transfer function contains a number of parameters that need to be estimated. For the development of both physical and empirical models, the most expensive step normally involves verification of their accuracy in predicting plant behavior. To illustrate the development of a physical model, a simplified treatment of the reactor, shown in Fig. 8-2 is used. It is assumed that the reactor is operating isothermally and that the inlet and exit volumetric flows and densities are the same. There are two components, A and B, in the reactor, and a single first order reaction of A → B takes place. The inlet concentration of A, which we shall call ci, varies with time. A dynamic mass balance for the concentration of A (cA) can be written as follows: dcA Vᎏ = Fci − FcA − krVc (8-1) dt In Eq. (8-1), the flow in of A is Fci, the flow out is FcA, and the loss via reaction is krVcA, where V = reactor volume and kr = kinetic rate constant. In this example, ci is the input, or forcing variable, and cA is the output variable. If V, F, and kr are constant, Eq. (8-1) can be rearranged by dividing by (F + krV) so that it only contains two groups of parameters. The result is: dc τ ᎏ = Kci − cA (8-2) dt where τ = V/(F + krV) and K = F/(F + krV). For this example, the resulting model is a first-order differential equation in which τ is called the time constant and K the process gain.

8-6

PROCESS CONTROL

As an alternative to deriving Eq. (8-2) from a dynamic mass balance, one could simply postulate a first-order differential equation to be valid (empirical modeling). Then it would be necessary to estimate values for τ and K so that the postulated model described the reactor’s dynamic response. The advantage of the physical model over the empirical model is that the physical model gives insight into how reactor parameters affect the values of τ, and K, which in turn affects the dynamic response of the reactor. Nonlinear versus Linear Models If V, F, and k are constant, then Eq. (8-1) is an example of a linear differential equation model. In a linear equation, the output and input variables and their derivatives only appear to the first power. If the rate of reaction were second order, then the resulting dynamic mass balance would be: dc V ᎏ = Fci − FcA − kr VcA2 (8-3) dt Since cA appears in this equation to the second power, the equation is nonlinear. The difference between linear systems and nonlinear systems can be seen by considering the steady state behavior of Eq. (8-1) compared to Eq. (8-3) (the left-hand side is zero; i.e., dcA /dt = 0). For a given change in ci, ∆ci, the change in cA calculated from Eq. (8-1), or ∆c, is always proportional to ∆ci, and the proportionality constant is K [see Eq. (8-2)]. The change in the output of a system divided by a change in the input to the system is called the process gain. Linear systems have constant process gains for all changes in the input. By contrast, Eq. (8-3) gives a ∆c that varies in proportion to ∆ ci but with the proportionality factor being a function of the concentration levels in the reactor. Thus, depending on where the reactor operates, a change in ci produces different changes in cA. In this case, the process has a nonlinear gain. Systems with nonlinear gains are more difficult to control than linear systems that have constant gains. Simulation of Dynamic Models Linear dynamic models are particularly useful for analyzing control-system behavior. The insight gained through linear analysis is invaluable. However, accurate dynamic process models can involve large sets of nonlinear equations. Analytical solution of these models is not possible. Thus, in these cases, one must turn to simulation approaches to study process dynamics and the effect of process control. Equation (8-3) will be used to illustrate the simulation of nonlinear processes. If dcA /dt on the left-hand side of Eq. (8-3) is replaced with its finite difference approximation, one gets: cA(t) + ∆ t ⋅ [Fci(t) − FcA(t) − krVcA(t)2] cA(t + ∆ t) = ᎏᎏᎏᎏ V

(8-4)

Starting with an initial value of cA and knowing ci(t), Eq. (8-4) can be solved for cA(t + ∆ t). Once cA(t + ∆ t) is known, the solution process can be repeated to calculate cA(t + 2∆ t), and so on. This approach is called the Euler integration method; while it is simple, it is not necessarily the best approach to numerically integrating nonlinear differential equations. To achieve accurate solutions with an Euler approach, one often needs to take small steps in time, ∆t. A number of more sophisticated approaches are available that allow much larger step sizes to be taken but require additional calculations. One widely used approach is the fourth-order Runge Kutta method, which involves the following calculations: Fci(t) − FcA − krVcA2 define f(cA,t) = ᎏᎏᎏ (8-5) V then with

cA(t + ∆t) = cA(t) + ∆t(m1 + 2m2 + 2m3 + m4) m1 = f [cA(t), t]

(8-6) (8-7)

m1∆t ∆t m2 = f cA(t) + ᎏ , t + ᎏ 2 2



(8-8)

∆t m2∆t m3 = f cA(t) + ᎏ , t + ᎏ 2 2



(8-9)





m4 = f [cA(t) + m3∆t, t + ∆t]

(8-10)

In this method, the mi’s are calculated sequentially in order to take a step in time. Even though this method requires calculation of the four additional mi values, for equivalent accuracy the fourth-order Runge Kutta method can result in a faster numerical solution, since a larger step, ∆t, can be taken with it. Increasingly sophisticated simulation packages are being used to calculate the dynamic behavior of processes and test control system behavior. These packages have good user interfaces, and they can handle stiff systems where some variables respond on a time scale that is much much faster or slower than other variables. A simple Euler approach cannot effectively handle stiff systems, which frequently occur in chemical-process models. Laplace Transforms When mathematical models are used to describe process dynamics in conjunction with control-system analysis, the models generally involve linear differential equations. Laplace transforms are very effective for solving linear differential equations. The key advantage of using Laplace transforms is that they convert differential equations into algebraic equations. The resulting algebraic equations are easier to solve than the original differential equations. When the Laplace transform is applied to a linear differential equation in time, the result is an algebraic equation in a new variable, s, called the Laplace variable. To get the solution to the original differential equation, one needs to invert the Laplace transform. Table 8-1 gives a number of useful Laplace transform pairs, and more extensive tables are available (Seborg, Edgar, and Mellichamp, Process Dynamics and Control, Wiley, New York, 1989). To illustrate how Laplace transforms work, consider the problem of solving Eq. (8-2), subject to the initial condition that cA = 0 at t = 0, and ci is constant. If cA were not initially zero, one would define a deviation variable between cA and its initial value (cA − c0). Then the transfer function would be developed using this deviation variable. Taking the Laplace transform of both sides of Eq. (8-2) gives: τdcA £ ᎏ = £(Kci) − £(cA) (8-11) dt Denoting the £(c) as CA(s) and using the relationships in Table 8-1 gives: Kc τsCA(s) = ᎏi − CA(s) (8-12) s Equation (8-12) can be solved for CA to give: Kci /s CA(s) = ᎏ (8-13) τs + 1 Using the entries in Table 8-1, Eq. (8-13) can be inverted to give the transient response of cA as:





cA(t) = (Kci)(1 − e−t/τ)

(8-14)

Equation (8-14) shows that cA starts from 0 and builds up exponentially to a final concentration of Kci. Note that to get Eq. (8-14), it was only necessary to solve the algebraic Eq. (8-12) and then find the inverse of CA(s) in Table 8-1. The original differential equation was not solved directly. In general, techniques such as partial fraction expansion must be used to solve higher order differential equations with Laplace transforms. Transfer Functions and Block Diagrams A very convenient and compact method of representing the process dynamics of linear systems involves the use of transfer functions and block diagrams. A transfer function can be obtained by starting with a physical model as TABLE 8-1

Frequently Used Laplace Transforms

Time function, f(t)

Transform, F(s)

A At Ae−at A(1 − e−t/τ) A sin (ωt) f(t − θ) df/dt ∫ f(t) dt

A/s2 A/s A/(s + a) A/[s(τs + 1)] Aω/(s2 + ω2) e−θsF(s) sF(s) − f(0) F(s)/s

FUNDAMENTALS OF PROCESS DYNAMICS AND CONTROL

8-7

discussed previously. If the physical model is nonlinear, then it first needs to be linearized around an operating point. The resulting linearized model is then approximately valid in a region around this operating point. To illustrate how transfer functions are developed, Eq. (8-2) will again be used. First, one defines deviation variables, which are the process variables minus their steady state values at the operating point. For Eq. (8-2), there would be deviation variables for both cA and ci, and these are defined as: ξ = cA − cs

(8-15)

ξi = ci − cis

(8-16)

where the subscript s stands for steady state. Substitution of Eq. (8-15) and (8-16) into Eq. (8-2) gives: dξ τ ᎏ = K ξi − ξ + (Kcis − cs) (8-17) dt The term in parentheses in Eq. (8-17) is zero at steady state and thus it can be dropped. Next the Laplace transform is taken, and the resulting algebraic equation solved. Denoting X(s) as the Laplace transform of ξ and X i(s) as the transform of ξ i, the final transfer function can be written as: K X (8-18) ᎏ=ᎏ Xi τs + 1 Equation (8-18) is an example of a first-order transfer function. As mentioned above, an alternative to formally deriving Eq. (8-18) involves simply postulating its form and then identifying its two parameters, the process gain K and time constant τ, to fit the process under study. In fitting the parameters, data can be generated by forcing the process. If step forcing is used, then the resulting response is called the process reaction curve. Often transfer functions are placed in block diagrams, as shown in Fig. 8-4. Block diagrams show how changes in an input variable affect an output variable. Block diagrams are a means of concisely representing the dynamics of a process under study. Since linearity is assumed in developing a block diagram, if more than one variable affects an output, the contributions from each can be added together. Continuous versus Discrete Models The preceding discussion has focused on systems where variables change continuously with time. Most real processes have variables that are continuous in nature, such as temperature, pressure, and flow. However, some processes involve discrete events, such as the starting or stopping of a pump. In addition, modern plants are controlled by digital computers, which are discrete by nature. In controlling a process, a digital system samples variables at a fixed rate, and the resulting system is a sampled data system. From one sampling instant until the next, variables are assumed to remain fixed at their sampled values. Similarly, in controlling a process, a digital computer sends out signals to control elements, usually valves, at discrete instants of time. These signals remain fixed until the next sampling instant. Figure 8-5 illustrates the concept of sampling a continuous function. At integer values of the sampling rate, ∆ t, the value of the variable to be sampled is measured and held until the next sampling instant. To deal with sampled data systems, the z transform has been developed. The z transform of the function given in Fig. 8-5 is defined as

FIG. 8-5

Sampled data example.

In an analogous manner to Laplace transforms, one can develop transfer functions in the z domain as well as block diagrams. Tables of z transform pairs have been published (Seborg, Edgar, and Mellichamp, Process Dynamics and Control, Wiley, New York, 1989) so that the discrete transfer functions can be inverted back to the time domain. The inverse gives the value of the function at the discrete sampling instants. Sampling a continuous variable results in a loss of information. However, in practical applications, sampling is fast enough that the loss is typically insignificant and the difference between continuous and discrete modeling is small in terms of its effect on control. Increasingly, model predictive controllers that make use of discrete dynamic models are being used in the process industries. The purpose of these controllers is to guide a process to optimum operating points. These model predictive control algorithms are typically run at much slower sampling rates than are used for basic control loops such as flow control or pressure control. The discrete dynamic models used are normally developed from data generated from plant testing as discussed hereafter. For a detailed discussion of modeling sampled data systems, the interested reader is referred to textbooks on digital control (Astrom and Wittenmark, Computer Controlled Systems, Prentice Hall, Englewood Cliffs, NJ, 1984). Process Characteristics in Transfer Functions In many cases, process characteristics are expressed in the form of transfer functions. In the previous discussion, a reactor example was used to illustrate how a transfer function could be derived. Here, another system involving flow out of a tank, shown in Fig. 8-6, is considered. Proportional Element First, consider the outflow through the exit valve on the tank. If the flow through the line is turbulent, then Bernoulli’s equation can be used to relate the flow rate through the valve to the pressure drop across the valve as: f1 = kf Av2(h gc− h 1  0)

where f1 = flow rate, kf = flow coefficient, Av = cross sectional area of the restriction, gc = constant, h1 = liquid head in tank, and h0 = atmo-

∞

Z( f ) =  f(n ∆t)z−n

(8-19)

n=0

FIG. 8-4

First-order transfer function.

(8-20)

FIG. 8-6

Single tank with exit valve.

8-8

PROCESS CONTROL

spheric pressure. This relationship between flow and head is nonlinear, and it can be linearized around a particular operating point to give: 1 f1 − f1s = ᎏ (h1 − h1s) (8-21) R1 2 2 where R1 = f1s /(gc kf A ) is called the resistance of the valve in analogy with an electrical resistance. The transfer function relating changes in flow to changes in head is shown in Fig. 8-7, and it is an example of a pure gain system with no dynamics. In this case, the process gain is K = 1/R1. Such a system has an instantaneous dynamic response, and for a step change in head, there is an immediate step change in flow, as shown in Fig. 8-8. The exact magnitude of the step in flow depends on the operating flow, f1s, as the definition of R1 shows. First-Order Lag (Time Constant Element) Next consider the system to be the tank itself. A dynamic mass balance on the tank gives: dh A1 ᎏ1 = fi − f1 (8-22) dt where A1 is the cross sectional area of the tank and fi is the inlet flow. By substituting Eq. (8-21) into Eq. (8-22), and following the approach discussed above for deriving transfer functions, one can develop the transfer function relating changes in h1 to changes in fi. The resulting transfer function is another example of a first-order system, shown in Fig. 8-4, and it has a gain, K = R1, and a time constant, τ1 = R1A1. For a step change in fi, h1 follows a decaying exponential response from its initial value, h1s, to a final value of h1s + R1∆fi (Fig. 8-9). At a time equal to τ1, the transient in h1 is 63 percent finished; and at 3τ1, the response is 95 percent finished. These percentages are the same for all firstorder processes. Thus, knowledge of the time constant of a first-order process gives insight into how fast the process responds to sudden input changes. Capacity Element Now consider the case where the valve in Fig. 8-7 is replaced with a pump. In this case, it is reasonable to assume that the exit flow from the tank is independent of the level in the tank. For such a case, Eq. (8-22) still holds, except that f1 no longer depends on h1. For changes in fi, the transfer function relating changes in h1 to changes in fi is shown in Fig. 8-10. This is an example of a pure capacity process, also called an integrating system. The cross sectional area of the tank is the chemical process equivalent of an electrical capacitor. If the inlet flow is step forced while the outlet is held

 

FIG. 8-7

FIG. 8-8

FIG. 8-9

Response of first-order system.

constant, then the level builds up linearly as shown in Fig. 8-11. Eventually the liquid would overflow the tank. Second-Order Element Because of their linear nature, transfer functions can be combined in a straightforward manner. Consider the two tank system shown in Fig. 8-12. For tank 1, the transfer function relating changes in f1 to changes in fi can be obtained by combining two first order transfer functions to give: F1(s) 1 (8-23) ᎏ = ᎏᎏ Fi(s) R1A1s + 1 Since f1 is the inlet flow to tank 2, the transfer function relating changes in h2 to changes in f1 has the same form as that given in Fig. 8-4: R2 H2(s) (8-24) ᎏ = ᎏᎏ F1(s) A2 R2 s + 1 Equations (8-23) and (8-24) can be multiplied together to give the final transfer function relating changes in h2 to changes in fi as shown in Fig. 8-13. This is an example of a second-order transfer function. This transfer function has a gain R1R2 and two time constants, R1A1 and R2 A2. For two equal tanks, a step change in fi produces the S-shaped response in level in the second tank shown in Fig. 8-14. General Second-Order Element Figure 8-3 illustrates the fact that closed loop systems often exhibit oscillatory behavior. A general

Proportional element transfer function.

Response of proportional element.

FIG. 8-10

Pure capacity transfer function.

FIG. 8-11

Response of pure capacity system.

FUNDAMENTALS OF PROCESS DYNAMICS AND CONTROL

FIG. 8-12

Two tanks in series.

FIG. 8-13

Second-order transfer function.

FIG. 8-15

General second-order transfer function.

FIG. 8-16

Response of general second-order system.

8-9

Higher-Order Lags If a process is described by a series of n firstorder lags, the overall system response becomes proportionally slower with each lag added. The special case of a series of n first-order lags with equal time constants has a transfer function given by: K G(s) = ᎏn (8-25) (τs + 1) The step response of this transfer function is shown in Fig. 8-19. Note that all curves reach about 60 percent of their final value at t = nτ.

FIG. 8-14

Response of second-order system.

second-order transfer function that can exhibit oscillatory behavior is important for the study of automatic control systems. Such a transfer function is given in Fig. 8-15. For a step input, the transient responses shown in Fig. 8-16 result. As can be seen when ζ < 1, the response oscillates and when ζ > 1, the response is S-shaped. Few open-loop chemical processes exhibit an oscillating response; most exhibit an S-shaped step response. Distance-Velocity Lag (Dead-Time Element) The dead-time element, commonly called a distance-velocity lag, is often encountered in process systems. For example, if a temperature-measuring element is located downstream from a heat exchanger, a time delay occurs before the heated fluid leaving the exchanger arrives at the temperature measurement point. If some element of a system produces a dead-time of θ time units, then an input to that unit, f(t), will be reproduced at the output as f(t − θ). The transfer function for a pure dead-time element is shown in Fig. 8-17, and the transient response of the element is shown in Fig. 8-18.

FIG. 8-17

Dead-time transfer function.

FIG. 8-18

Response of dead-time system.

8-10

FIG. 8-19

PROCESS CONTROL

FIG. 8-20

Example of 2 × 2 transfer function.

FIG. 8-21

Plot of experimental data.

Response of nth order lags.

Higher-order systems can be approximated by a first or second-order plus dead-time system for control system design. Multiinput, Multioutput Systems The dynamic systems considered up to this point have been examples of single-input, singleoutput (SISO) systems. In chemical processes, one often encounters systems where one input can affect more than one output. For example, assume that one is studying a distillation tower in which both reflux and boilup are manipulated for control purposes. If the output variables are the top and bottom product compositions, then each input affects both outputs. For this distillation example, the process is referred to as a 2 × 2 system to indicate the number of inputs and outputs. In general, multiinput, multioutput (MIMO) systems can have n inputs and m outputs with n ≠ m, and they can be nonlinear. Such a system would be called an n × m system. An example of a transfer function for a 2 × 2 linear system is given in Fig. 8-20. Note that since linear systems are involved, the effects of the two inputs on each output are additive. In many process-control systems, one input is selected to control one output in a MIMO system. For m output there would be m such selections. For this type of control strategy, one needs to consider which inputs and outputs to couple together, and this problem is referred to as loop pairing. Another important issue that arises involves interaction between control loops. When one loop makes a change in its manipulated variable, the change affects the other loops in the system. These changes are the direct result of the multivariable nature of the process. In some cases, the interaction can be so severe that overall control-system performance is drastically reduced. Finally, some of the modern approaches to process control tackle the MIMO problem directly, and they simultaneously use all manipulated variables to control all output variables rather than pairing one input to one output (see later section on multivariable control). Fitting Dynamic Models to Experimental Data In developing empirical transfer functions, it is necessary to identify model parameters from experimental data. There are a number of approaches to process identification that have been published. The simplest approach involves introducing a step test into the process and recording the response of the process, as illustrated in Fig. 8-21. The x’s in the figure represent the recorded data. For purposes of illustration, the process under study will be assumed to be first order with deadtime and have the transfer function:

C(s) G(s) = ᎏ = K exp(−θs)/(τs + 1) (8-26) M(s) The response produced by Eq. (8-26), c(t), can be found by inverting the transfer function, and it is also shown in Fig. 8-21 for a set of model parameters, K, τ, and θ, fitted to the data. These parameters are calculated using optimization to minimize the squared difference between the model predictions and the data, i.e., a least squares approach. Let each measured data point be represented by cj (measured response), tj (time of measured response), j = 1 to n. Then the least squares problem can be formulated as: min τ,θ,K

n

ˆ )]  [c − c(t j

j

2

(8-27)

j=0

which can be solved to calculate the optimal values of K, τ, and θ. A number of software packages are available for minimizing Eq. (8-27). One operational problem that step forcing causes is the fact that the process under study is moved away from its steady state operating point. Plant managers may be reluctant to allow large steady state changes, since normal production will be disturbed by the changes. As a result, alternative methods of forcing actual processes have been developed, and these included pulse testing and pseudo random binary signal (PRBS) forcing, both of which are illustrated in Fig. 8-22. With pulse forcing, one introduces a step, and then after a period of time the input is returned to its original value. The result is that the process dynamics are excited, but after the forcing, the process returns to its original steady state. PRBS forcing involves a series of pulses of fixed height and random duration, as shown in Fig. 8-22. The advantage of PRBS is that forcing can be concentrated on particular frequency ranges that are important for control-system design.

FUNDAMENTALS OF PROCESS DYNAMICS AND CONTROL

8-11

FIG. 8-23 Both load regulation and setpoint response require high gains for the feedback controller.

FIG. 8-22

Pulse and PRBS testing.

Transfer function models are linear in nature, but chemical processes are known to exhibit nonlinear behavior. One could use the same type of optimization objective as given in Eq. (8-26) to determine parameters in nonlinear first-principle models, such as Eq. (8-3) presented earlier. Also, nonlinear empirical models, such as neural network models, have recently been proposed for process applications. The key to the use of these nonlinear empirical models is having high-quality process data, which allows the important nonlinearities to be identified. FEEDBACK CONTROL SYSTEM CHARACTERISTICS GENERAL REFERENCES: Shinskey, Feedback Controllers for the Process Industries, McGraw-Hill, New York, 1994; Seborg, Edgar, and Mellichamp, Process Dynamics and Control, Wiley, New York, 1989.

FIG. 8-24

There are two objectives in applying feedback control: regulating the controlled variable at set point following changes in load, and responding to set-point changes; the latter called servo operation. In fluid processes, almost all control loops must contend with variations in load; therefore, regulation is of primary importance. While most loops will operate continuously at fixed set points, frequent changes in set points can occur in flow loops and in batch production. The most common mechanism for achieving both objectives is feedback control, because it is the simplest and most universally applicable approach to the problem. Closing the Loop The simplest representation of the closed feedback loop is shown in Fig. 8-23. The load is shown entering the process at the same point as the manipulated variable because that is the most common point of entry, and also because, lacking better information, the transfer function gains in the path of the manipulated variable are the best estimates of those in the load path. In general, the load never impacts directly on the controlled variable without passing through the dominant lag in the process. Where the load is unmeasured, its current value can be observed to be the controller output required to keep the controlled variable C at set point R. If the loop is opened, either by placing the controller in manual operation or by setting its gains to zero, the load will have complete influence over the controlled variable, and the set point will have none. Only by closing the loop with controller gains as high as possible will the influence of the load be minimized and that of the set point be maximized. There is a practical limit to the controller gains, however, at the point where the controlled variable develops a uniform oscillation (see Fig. 8-24). This is defined as the limit of stability, and it is reached when the product of gains in the loop GcGvGp for that frequency of oscillation is equal to 1.0. If a change in a parameter in the loop causes an increase from this condition, oscillations will expand, creating a dangerous situation where safe limits of operation could be exceeded. Consequently, control loops should be left in a condition

Transition to instability as controller gain increases.

8-12

PROCESS CONTROL

where the loop gain is less than 1.0 by a safe margin that allows for possible variations in process parameters. In controller design, a choice must be made between performance and robustness. Performance is a measure of how well a given controller with certain parameter settings regulates a variable relative to the best loop performance with optimal controller settings. Robustness is a measure of how small a change in a process parameter is required to bring the loop from its current state to the limit of stability. Increasing controller performance by raising its gains can be expected to decrease robustness. Both performance and robustness are functions of the process being controlled, the selection of the controller, and the tuning of the controller parameters. On/Off Control An on/off controller is used for manipulated variables having only two states. They commonly control temperatures in homes, electric water-heaters and refrigerators, and pressure and liquid level in pumped storage systems. On/off control is satisfactory where slow cycling is acceptable because it always leads to cycling when the load lies between the two states of the manipulated variable. The cycle will be positioned symmetrically about the set point only if the normal value of the load is equidistant between the two states of the manipulated variable. The period of the symmetrical cycle will be approximately 4θ, where θ is the deadtime in the loop. If the load is not centered between the states of the manipulated variable, the period will tend to increase, and the cycle follows a sawtooth pattern. Every on/off controller has some degree of deadband, also known as lockup, or differential gap. Its function is to prevent erratic switching between states, thereby extending the life of contacts and motors. Instead of changing states precisely when the controlled variable crosses set point, the controller will change states at two different points for increasing and decreasing signals. The difference between these two switching points is the deadband (see Fig. 8-25); it increases the amplitude and period of the cycle, similar to the effect of dead time. A three-state controller is used to drive either a pair of independent on/off actuators such as heating and cooling valves, or a bidirectional motorized actuator. The controller is actually two on/off controllers, each with deadband, separated by a dead zone. When the controlled variable lies within the dead zone, neither output is energized. This controller can drive a motorized valve to the point where the manipulated variable matches the load, thereby avoiding cycling. Proportional Control A proportional controller moves its output proportional to the deviation in the controlled variable from set point: 100 u = Kce + b = ᎏ e + b (8-28) P where e = ⫾(r − c), the sign selected to produce negative feedback. In

FIG. 8-25

On/off controller characteristics.

some controllers, proportional gain Kc is expressed as a pure number; in others, it is set as 100/P, where P is the proportional band in percent. The output bias b of the controller is also known as manual reset. The proportional controller is not a good regulator, because any change in output to a change in load results in a corresponding change in the controlled variable. To minimize the resulting offset, the bias should be set at the best estimate of the load and the proportional band set as low as possible. Processes requiring a proportional band of more than a few percent will control with unacceptable values of offset. Proportional control is most often used for liquid level where variations in the controlled variable carry no economic penalty, and where other control modes can easily destabilize the loop. It is actually recommended for controlling the level in a surge tank when manipulating the flow of feed to a critical downstream process. By setting the proportional band just under 100 percent, the level is allowed to vary over the full range of the tank capacity as inflow fluctuates, thereby minimizing the resulting rate of change of manipulated outflow. This technique is called averaging level control. Proportional-plus-Integral (PI) Control Integral action eliminates the offset described above by moving the controller output at a rate proportional to the deviation from set point. Although available alone in an integral controller, it is most often combined with proportional action in a PI controller: 100 1 u = ᎏ e + ᎏ e dt + C0 (8-29) P τI where τI is the integral time constant in minutes; in some controllers, it is introduced as integral gain or reset rate 1/τI in repeats per minute. The last term in the equation is the constant of integration, the value the controller output has when integration begins. The PI controller is by far the most commonly used controller in the process industries. The summation of the deviation with its integral in the above equation can be interpreted in terms of frequency response of the controller (Seborg, Edgar, and Mellichamp, Process Dynamics and Control, Wiley, New York, 1989). The PI controller produces a phase lag between zero and 90 degrees: τ0 φPI = −tan−1 ᎏ (8-30) 2πτI where τ0 is the period of oscillation of the loop. The phase angle should be kept between 15 degrees for lag-dominant processes and 45 degrees for dead-time-dominant processes for optimum results. Proportional-plus-Integral-plus-Derivative (PID) Control The derivative mode moves the controller output as a function of the rate-of-change of the controlled variable, which adds phase lead to the controller, increasing its speed of response. It is normally combined





FUNDAMENTALS OF PROCESS DYNAMICS AND CONTROL with proportional and integral modes. The noninteracting form of the PID controller appears functionally as:



100 1 u=ᎏ e+ᎏ P τI

dc e dt + τ ᎏ +C dt  D

0

(8-31)

where τD is the derivative time constant. Note that derivative action is applied to the controlled variable rather than to the deviation, as it should not be applied to the set point; the selection of the sign for the derivative term must be consistent with the action of the controller. Figure 8-26 compares typical loop responses for P, PI, and PID controllers, along with the uncontrolled case. In some analog PID controllers, the integral and derivative terms are combined serially rather than in parallel as done in the last equation. This results in interaction between these modes, such that the effective values of the controller parameters differ from their set values as follows: τIeff = τI + τD 1 τDeff = ᎏᎏ 1/τD + 1/τI τD 100 Kc = ᎏ 1 + ᎏ P τI





(8-32)

The performance of the interacting controller is almost as good as the noninteracting controller on most processes, but the tuning rules differ because of the above relationships. With digital PID controllers, the noninteracting version is commonly used. There is always a gain limit placed upon the derivative term—a value of 10 is typical. However, interaction decreases the derivative gain below this value by the factor 1 + τD /τI, which is the reason for the decreased performance of the interacting PID controller. Sampling in a digital controller has a similar effect, limiting derivative gain to the ratio of derivative time to the sample interval of the controller. Noise on the controlled variable is amplified by derivative action, preventing its use in controlling flow and liquid level. Derivative action is recommended for control of temperature and composition, reducing the integrated error (IE) by a factor of two over PI control with no loss in robustness (Shinskey, Feedback Controllers for the Process Industries, McGraw-Hill, New York, 1994). CONTROLLER TUNING The performance of a controller depends as much on its tuning as its design. Tuning must be applied by the end user to fit the controller to the controlled process. There are many different approaches to controller tuning based on the particular performance criteria selected,

whether load or set-point changes are most important, whether the process is lag- or deadtime-dominant, and the availability of information about the process dynamics. The earliest definitive work in this field was done at the Taylor Instrument Company by Ziegler and Nichols (Trans. ASME, 759, 1942), tuning PI and interacting PID controllers for optimum response to step load changes applied to lag-dominant processes. While these tuning rules are still in use, they are approximate and do not apply to set-point changes, dead-timedominant processes, or noninteracting PID controllers (Seborg, Edgar, and Mellichamp, Process Dynamics and Control, Wiley, New York, 1989). Controller Performance Criteria The most useful measures of controller performance in an industrial setting are the maximum deviation in the controlled variable resulting from a disturbance and its integral. The disturbance could be to the set point or to the load, depending on the variable being controlled and its context in the process. The size of the deviation and its integral are proportional to the size of the disturbance (if the loop is linear at the operating point). While actual disturbances in a plant setting may appear to be random, the controller needs a reliable test to determine how well it is tuned. The disturbance of choice for test purposes is the step, because it can be applied manually, and by containing all frequencies including zero, it exercises all modes of the controller. When tuned optimally for step disturbances, the controller should be well-tuned for most other disturbances as well. Figure 8-27 shows the optimum response of a controlled variable to a step change in load. A step change in load may be simulated by stepping the controller output while it is in the manual mode followed immediately by transfer to automatic. The maximum deviation is the most important criterion for variables that could exceed safe operating levels such as steam pressure, drum level, and steam temperature in a boiler. The same rule applies to product quality, which could violate specifications and therefore be rejected. If the product can be accumulated in a downstream storage tank, however, its average quality is more important, and this is a function of the deviation integrated over the residence time in the tank. Deviation in the other direction, where the product is better than specification, is safe, but it increases production costs in proportion to the integrated deviation because quality is given away. For a PI or PID controller, the integrated deviation—better known as the integrated error IE—is related to the controller settings: ∆ uPτ IE = ᎏI (8-33) 100 where ∆u is the difference in controller output between two steady states, as required by a change in load or set point. The proportional band P and integral time τI are the indicated settings of the controller

Response for a step change in disturbance with tuned P, PI, and PID controllers and with no control. FIG. 8-26

8-13

8-14

PROCESS CONTROL

FIG. 8-27

The minimum-IAE response to a step load change has little overshoot and is well-damped.

for both interacting and noninteracting PID controllers. Although the derivative term does not appear in the relationship, its use typically allows a 50 percent reduction in integral time and therefore in IE. The integral time in the IE expression should be augmented by the sample interval if the controller is digital, the time constant of any filter used, and the value of any deadtime compensator. It would appear from the above that minimizing IE is simply a matter of minimizing the P and τI settings of the controller. However, settings will be reached that produce uniform oscillations—an unacceptable situation. It is preferable, instead, to find a combination of controller settings that minimize integrated absolute error IAE, which for both load and set-point changes is a well-damped response with minimal overshoot. Figure 8-27 is an example of a minimum-IAE response to a step change in load for a lag-dominant process. Because of the very small overshoot, the IAE will be only slightly larger than the IE. Loops that are tuned to minimize IAE tend to be close to minimum IE and also minimum peak deviation. The performance of a controller (and its tuning) must be based on what is achievable for a given process. The concept of best practical IE (IEb) for a step change in load ∆q can be estimated (Shinskey, Feedback Controllers for the Process Industries, McGraw-Hill, New York, 1994): IEb = ∆qKLτL(1 − e−θ/τ L )

(8-34)

where KL is the gain and τL the primary time constant in the load path, and θ the dead time in the manipulated path to the controlled variable. If the load or its gain is unknown, ∆ u and K(= Kv Kp) may be substituted. If the process is non-self-regulating (i.e., it is an integrator), the relationship is ∆qθ2 IEb = ᎏ (8-35) τ1 where τ1 is the time constant of the process integrator. The peak deviation with the best practical response curve is: IEb eb = ᎏ (8-36) θ + τ2 where τ2 is the time constant of a common secondary lag (e.g., in the measuring device). The performance for any controller can be measured against this standard by comparing the IE it achieves in responding to a step load change with the best practical IE. Potential performance improvements by tuning PI controllers on lag-dominant processes lie in the 20–30 percent range, while for PID controllers they fall between 40–60 percent, varying with secondary lags.

Tuning Methods Based on Known Process Models The most accurate tuning rules for controllers have been based on simulation, where the process parameters can be specified and IAE and IE can be integrated during the simulation as an indication of performance. Controller settings are then iterated until a minimum IAE is reached for a given disturbance. These optimum settings are then related to the parameters of the simulated process in tables, graphs, or equations, as a guide to tuning controllers for processes whose parameters are known (Seborg, Edgar, and Mellichamp, Process Dynamics and Control, Wiley, New York, 1989). This is a multidimensional problem, however, in that the relationships change as a function of process type, controller type, and source of disturbance. Table 8-2 summarizes these rules for minimum-IAE load response for the most common controllers. The process gain K and time constant τm are obtained from the product of Gv and Gp in Fig. 8-23. Derivative action is not effective for dead-time-dominant processes. For non-self-regulating processes, τ is the time constant of the integrator. The last category of distributed lag includes all heat-transfer processes, backmixed vessels, and processes having multiple interacting lags such as distillation columns;  τ represents the total response time of these processes (i.e., the time required for 63 percent complete response to a step input). Any secondary lag, sampling interval, or filter time constant should be added to deadtime θ. The principal limitation to using these rules is that the true process parameters are often unknown. Steady-state gain K can be calculated from a process model or determined from the steady-state results of a step test as ∆c/∆u, as shown in Fig. 8-28. The test will not be viable, however, if the time constant of the process τm is longer than a few TABLE 8-2

Tuning Rules Using Known Process Parameters

Process

Controller

P

τI

Dead-time-dominant Lag-dominant

PI PI PIDn PIDi PI PIDn PIDi PI PIDn PIDi

250K 106K θ/τm 77K θ/τm 106K θ/τm 106 θ/τ1 78 θ/τ1 108 θ/τ1 20K 10K 15K

0.5 θ 4.0 θ 1.8 θ 1.5 θ 4.0 θ 1.9 θ 1.6 θ 0.50  τ 0.30  τ 0.25  τ

Non-self-regulating Distributed lags

NOTE:

n = noninteracting; i = interacting controller modes

τD

0.45 θ 0.55 θ 0.48 θ 0.58 θ 0.09  τ 0.10  τ

FUNDAMENTALS OF PROCESS DYNAMICS AND CONTROL

8-15

If a steady state can be reached, gain K and time constant τ can be estimated from a step response; if not, use τ1 instead.

FIG. 8-28

minutes, since five time constants must elapse to approach a steady state within one percent, and unrequested disturbances may intervene. Estimated dead-time θ is the time from the step to the intercept of a straight line tangent to the steepest part of the response curve. The estimated time constant τ is the time from that point to 63 percent of the complete response. In the presence of a secondary lag, these results will not be completely accurate, however. The time for 63 percent response may be more accurately calculated as the residence time of the process: its volume divided by current volumetric flow rate. Tuning Methods When Process Model Is Unknown Ziegler and Nichols developed two tuning methods for processes with unknown parameters. The open-loop method uses a step test without waiting for a steady state to be reached and is therefore applicable to very slow processes. Deadtime is estimated from the intercept of the tangent in Fig. 8-28, whose slope is also used. If the process is non-self-regulating, the controlled variable will continue to follow this slope, changing an amount equal to ∆u in a time equal to its time constant. This time estimate τ1 is used along with θ to tune controllers according to Table 8-3, applicable to lag-dominant processes. A more recent tuning approach uses integral criteria such as the integral of the squared error (ISE), integral of the absolute error (IAE), and the time-weighted IAE (ITAE) of Seborg, Edgar, and Mellichamp (Process Dynamics and Control, Wiley, New York, 1989). The controller parameters are selected to minimize various integrals. Power-law correlations for PID controller settings have been tabulated for a range of first-order model parameters. The best tuning parameters have been fitted using a general equation, Y = A(θ/τ)B, where Y depends on the particular controller mode to be evaluated (KC, τI, τD). There are several features of the correlations that should be noted:

TABLE 8-3

Tuning Rules Using Slope and Intercept

Controller

P

τI

τD

PI PIDn PIDi

150 θ/τ 75 θ/τ 113 θ/τ

3.5 θ 2.1 θ 1.8 θ

— 0.63 θ 0.70 θ

NOTE:

1. The controller gain is inversely proportional to the process gain for constant dead time and time constant. 2. The allowable controller gain is higher when the ratio of dead time to time constant becomes smaller. This is because dead time has a destabilizing effect on the control system, limiting the controller gain, while a larger time constant generally demands a higher controller gain. A recent addition to the model-based tuning correlations is Internal Model Control (Rivera, Morari, and Skogestad, “Internal Model Control 4: PID Controller Design,” IEC Proc. Des. Dev., 25, 252, 1986), which offers some advantages over the other methods described here. However, the correlations are similar to the ones discussed above. Other plant testing and controller design approaches such as frequency response can be used for more complicated models. The Ziegler and Nichols closed-loop method requires forcing the loop to cycle uniformly under proportional control. The natural period τn of the cycle—the proportional controller contributes no phase shift to alter it—is used to set the optimum integral and derivative time constants. The optimum proportional band is set relative to the undamped proportional band Pu, which produced the uniform oscillation. Table 8-4 lists the tuning rules for a lag-dominant process. A uniform cycle can also be forced using on/off control to cycle the manipulated variable between two limits. The period of the cycle will be close to τn if the cycle is symmetrical; the peak-to-peak amplitude of the controlled variable Ac divided by the difference between the output limits Am is a measure of process gain at that period and is therefore related to Pu for the proportional cycle: π A Pu = 100 ᎏ ᎏc (8-37) 4 Am The factor π/4 compensates for the square wave in the output. Tuning rules are given in Table 8-4.

n = noninteracting, i = interacting controller modes

TABLE 8-4

Tuning Rules Using Proportional Cycle

Controller

P

τI

τD

PI PIDn PIDi

1.70 Pu 1.30 Pu 1.80 Pu

0.81 τn 0.48 τn 0.39 τn

— 0.11 τn 0.14 τn

8-16

PROCESS CONTROL

ADVANCED CONTROL SYSTEMS BENEFITS OF ADVANCED CONTROL The economics of most processes are determined by the steady-state operating conditions. Excursions from these steady-state conditions generally average out and have an insignificant effect on the economics of the process, except when the excursions lead to off-specification products. In order to enhance the economic performance of a process, the steady-state operating conditions must be altered in a manner that leads to more efficient process operation. The following hierarchy is used for process control: Level 0: Measurement devices and actuators Level 1: Regulatory control Level 2: Supervisory control Level 3: Production control Level 4: Information technology Levels 2, 3, and 4 clearly affect the process economics, as all three levels are directed to optimizing the process in some manner. However, level 0 (measurement devices and actuators) and level 1 (regulatory control) would appear to have no effect on process economics. Their direct effect is indeed minimal, but indirectly, they have a major effect. Basically, these levels provide the foundation for all higher levels. A process cannot be optimized until it can be operated consistently at the prescribed targets. Thus, a high degree of regulatory control must be the first goal of any automation effort. In turn, the measurements and actuators provide the process interface for regulatory control. For most processes, the optimum operating point is determined by a constraint. The constraint might be a product specification (a product stream can contain no more than 2 percent ethane); violation of this constraint causes off-specification product. The constraint might be an equipment limit (vessel pressure rating is 300 psig); violation of this constraint causes the equipment protection mechanism (pressure relief device) to activate. As the penalties are serious, violation of such constraints must be very infrequent. If the regulatory control system were perfect, the target could be set exactly equal to the constraint (that is, the target for the pressure controller could be set at the vessel relief pressure). However, no regulatory control system is perfect. Therefore, the value specified for the target must be on the safe side of the constraint, thus giving the control system some “elbow room.” How much depends on the following: 1. The performance of the control system (i.e., how effectively it responds to disturbances). The faster the control system reacts to a disturbance, the closer the process can be operated to the constraint. 2. The magnitude of the disturbances to which the control system must respond. If the magnitude of the major disturbances can be reduced, the process can be operated closer to the constraint. One measure of the performance of a control system is the variance of the controlled variable from the target. Both improving the control system and reducing the disturbances will lead to a lower variance in the controlled variable. In a few applications, improving the control system leads to a reduction in off-specification product and thus improved process economics. However, in most situations, the process is operated sufficiently far from the constraint that very little, if any, off-specification product results from control system deficiencies. Management often places considerable emphasis on avoiding off-spec production, so consequently the target is actually set far more conservatively than it should be. In most applications, simply improving the control system does not directly lead to improved process economics. Instead, the control system improvement must be accompanied by shifting the target closer to the constraint. There is always a cost of operating a process in a conservative manner. The cost may be a lower production rate, a lower process efficiency, a product giveaway, or otherwise. When management places extreme emphasis on avoiding off-spec production, the natural reaction is to operate very conservatively, thus incurring other costs.

The immediate objective of an advanced control effort is to reduce the variance in an important controlled variable. However, this effort must be coupled with a commitment to adjust the target for this controlled variable so that the process is operated closer to the constraint. In large throughput (commodity) processes, very small shifts in operating targets can lead to large economic returns. ADVANCED CONTROL TECHNIQUES GENERAL REFERENCES: Seborg, Edgar, and Mellichamp, Process Dynamics and Control, John Wiley and Sons, New York, 1989. Stephanopoulos, Chemical Process Control: An Introduction to Theory and Practice, Prentice Hall, Englewood Cliffs, New Jersey, 1984. Shinskey, Process Control Systems, 3d ed., McGraw-Hill, New York, 1988. Ogunnaike and Ray, Process Dynamics, Modeling, and Control, Oxford University Press, New York, 1994.

While the single-loop PID controller is satisfactory in many process applications, it does not perform well for processes with slow dynamics, time delays, frequent disturbances, or multivariable interactions. We discuss several advanced control methods hereafter that can be implemented via computer control, namely feedforward control, cascade control, time-delay compensation, selective and override control, adaptive control, fuzzy logic control, and statistical process control. Feedforward Control If the process exhibits slow dynamic response and disturbances are frequent, then the application of feedforward control may be advantageous. Feedforward (FF) control differs from feedback (FB) control in that the primary disturbance or load (L) is measured via a sensor and the manipulated variable (m) is adjusted so that deviations in the controlled variable from the set point are minimized or eliminated (see Fig. 8-29). By taking control action based on measured disturbances rather than controlled variable error, the controller can reject disturbances before they affect the controlled variable c. In order to determine the appropriate settings for the manipulated variable, one must develop mathematical models that relate: 1. The effect of the manipulated variable on the controlled variable 2. The effect of the disturbance on the controlled variable These models can be based on steady-state or dynamic analysis. The performance of the feedforward controller depends on the accuracy of both models. If the models are exact, then feedforward control offers the potential of perfect control (i.e., holding the controlled variable precisely at the set point at all times because of the ability to predict the appropriate control action). However, since most mathematical models are only approximate and since not all disturbances are measurable, it is standard practice to utilize feedforward control in conjunction with feedback control. Table 8-5 lists the relative advantages and disadvantages of feedforward and feedback control. By combining the two control methods, the strengths of both schemes can be utilized. FF control therefore attempts to eliminate the effects of measurable disturbances, while FB control would correct for unmeasurable

FIG. 8-29

Block diagram for feedforward control configuration.

ADVANCED CONTROL SYSTEMS

8-17

TABLE 8-5 Relative Advantages and Disadvantages of Feedback and Feedforward Advantages

Disadvantages Feedforward

• Acts before the effect of a disturbance has been felt by the system

• Requires measurement of all possible disturbances and their direct measurement

• Good for systems with large time constant or deadtime

• Cannot cope with unmeasured disturbances

• Does not introduce instability in the closed-loop response

• Sensitive to process/model error

Feedback • Does not require identification and measurement of any disturbance for corrective action

• Control action not taken until the effect of the disturbance has been felt by the system

• Does not require an explicit process model

• Unsatisfactory for processes with large time constants and frequent disturbances

• Controller can be robust to process/ model errors

• May cause instability in the closed-loop response

disturbances and modeling errors. This is often referred to as feedback trim. These controllers have become widely accepted in the chemical process industries since the 1960s. Design Based on Material and Energy Balances Consider a heat exchanger example (see Fig. 8-30) to illustrate the use of FF and FB control. The control objective is to maintain T2, the exit liquid temperature, at the desired value (or set point) Tset despite variations in inlet liquid flow rate F and inlet liquid temperature T1. This is done by manipulating W, the steam flow rate. A feedback control scheme would entail measuring T2, comparing T2 to Tset, and then adjusting W. A feedforward control scheme requires measuring F and T1, and adjusting W (knowing Tset), in order to control exit temperature, T2. Figure 8-31 shows the control system diagrams for FB and FF control. A feedforward control algorithm can be designed for the heat exchanger in the following manner. Using a steady-state energy balance and assuming no heat loss from the heat exchanger, WH = FC(T2 − T1)

(8-38)

(a)

(b) (a) Feedback control of a heat exchanger. (b) Feedforward control of a heat exchanger.

FIG. 8-31

where H = latent heat of vaporization CL = specific heat of liquid. Rearranging Eq. (8-38), CL W=ᎏ F(T2 − T1) H

(8-39)

or

W = K1F(T2 − T1)

(8-40)

with

CL K1 = ᎏ H

(8-41)

W = K1F(Tset − T1)

(8-42)

Replace T2 by Tset:

FIG. 8-30

A heat exchanger diagram.

Equation (8-42) can be used in the FF calculation, assuming one knows the physical properties CL and H. Of course, it is probable that the model will contain errors (e.g., unmeasured heat losses, incorrect CL or H). Therefore, K1 can be designated as an adjustable parameter that can be tuned. The use of a physical model for FF control is desirable since it provides a physical basis for the control law and gives an a priori estimate of what the tuning parameters are. Note that such a model could be nonlinear [e.g., in Eq. (8-42), F and Tset are multiplied]. Block Diagram Analysis One shortcoming of this feedforward design procedure is that it is based on the steady-state characteristics of the process and as such, neglects process dynamics (i.e., how fast the controlled variable responds to changes in the load and manipulated variables). Thus, it is often necessary to include “dynamic compensation” in the feedforward controller. The most direct method of designing the FF dynamic compensator is to use a block diagram of a general process, as shown in Fig. 8-32. Gt represents the disturbance transmitter, Gf is the feedforward controller, GL relates the load to the controlled variable, Gv is the valve, and Gp is the process. Gm is the output transmitter and Gc is the feedback controller. All blocks correspond to transfer functions (via Laplace transforms). Using block diagram algebra and Laplace transform variables, the controlled variable C(s) is given by GtGf L(s) + GLL(s) C(s) = ᎏᎏ 1 + GmGcGvGp

(8-43)

8-18

PROCESS CONTROL

FIG. 8-32

Block diagram for feedback-feedforward control.

For disturbance rejection [L(s) ≠ 0] we require that C(s) = 0, or zero error. Solving Eq. (8-43) for Gf, −GL Gf = ᎏ (8-44) GtGvGp Suppose the dynamics of GL and Gp are first order; in addition, assume that Gv = Kv and Gt = Kt (constant gains for simplicity). KL C(s) GL(s) = ᎏ =ᎏ (8-45) τLs + 1 L(s)

GcGvGpKm C(s) ᎏ = ᎏᎏ R(s) 1 + GcGvGpGm

(8-48)

−K(τps + 1) KL τps + 1 (8-47) Gf(s) = − ᎏ ⋅ ᎏ = ᎏᎏ τLs + 1 KpKvKt τLs + 1 The above FF controller can be implemented using analog elements or more commonly by a digital computer. Figure 8-33 compares typical responses for PID FB control, steady-state FF control (s = 0), dynamic FF control, and combined FF/FB control. In practice, the engineer can tune K, τp and τL in the field to improve the performance of the FF controller. The feedforward controller can also be simplified to provide steady-state feedforward control. This is done by setting s = 0 in Gf(s). This might be appropriate if there is uncertainty in the dynamic models for GL and Gp. Other Considerations in Feedforward Control The tuning of feedforward and feedback control systems can be performed independently. In analyzing the block diagram in Fig. 8-32, note that Gf is chosen to cancel out the effects of the disturbance L(s) as long as there are no model errors. For the feedback loop, therefore, the effects of L(s) can also be ignored, which for the servo case is:

Note that the characteristic equation will be unchanged for the FF + FB system, hence system stability will be unaffected by the presence of the FF controller. In general, the tuning of the FB controller can be less conservative than for the case of FB alone, since smaller excursions from the set point will result. This in turn would make the dynamic model Gp(s) more accurate. The tuning of the controller in the feedback loop can be theoretically performed independent of the feedforward loop (i.e., the feedforward loop does not introduce instability in the closed-loop response). For more information on feedforward/feedback control applications and design of such controllers, refer to the general references. Cascade Control One of the disadvantages of using conventional feedback control for processes with large time lags or delays is that disturbances are not recognized until after the controlled variable deviates from its set point. In these processes, correction by feedback control is generally slow and results in long-term deviation from set point. One way to improve the dynamic response to load changes is by using a secondary measurement point and a secondary controller; the secondary measurement point is located so that it recognizes the upset condition before the primary controlled variable is affected. One such approach is called cascade control, which is routinely used in most modern computer control systems. Consider a chemical reactor, where reactor temperature is to be controlled by coolant flow to the jacket of the reactor (Fig. 8-34). The reactor temperature can be influenced by changes in disturbance variables such as feed rate or feed temperature; a feedback controller could be employed to compensate for such disturbances by adjusting a valve on the coolant flow to the reactor jacket. However, suppose an increase occurs in the

(a)

(b)

Kp C(s) Gp(s) = ᎏ = ᎏ τps + 1 U(s) Using Eq. (8-44),

FIG. 8-33

control.

(8-46)

(a) Comparison of FF (steady state model) and PID FB control for load change; (b) comparison of FF (dynamic model) and combined FF/FB

ADVANCED CONTROL SYSTEMS coolant temperature as a result of changes in the plant coolant system. This will cause a change in the reactor temperature measurement, although such a change will not occur quickly, and the corrective action taken by the controller will be delayed. Cascade control is one solution to this problem (see Fig. 8-35). Here the jacket temperature is measured, and an error signal is sent from this point to the coolant control valve; this reduces coolant flow, maintaining the heat transfer rate to the reactor at a constant level and rejecting the disturbance. The cascade control configuration will also adjust the setting of the coolant control valve when an error occurs in reactor temperature. The cascade control scheme shown in Fig. 8-35 contains two controllers. The primary controller is the reactor temperature coolant temperature controller. It measures the reactor temperature, compares it to the set point, and computes an output, which is the set point for the coolant flow rate controller. This secondary controller compares the set point to the coolant temperature measurement and adjusts the valve. The principal advantage of cascade control is that the secondary measurement (jacket temperature) is located closer to a potential disturbance in order to improve the closed-loop response. Figure 8-36 shows the block diagram for a general cascade control system. In tuning of a cascade control system, the secondary controller (in the inner loop) is tuned first with the primary controller in manual. Often only a proportional controller is needed for the secondary loop, since offset in the secondary loop can be treated by using proportional plus integral action in the primary loop. When the primary controller is transferred to automatic, it can be tuned using the techniques described earlier in this section. For more information on theoretical analysis of cascade control systems, see the general references for a discussion of applications of cascade control. Time-Delay Compensation Time delays are a common occurrence in the process industries because of the presence of recycle loops, fluid-flow distance lags, and “dead time” in composition measurements resulting from use of chromatographic analysis. The presence of a time delay in a process severely limits the performance of a conventional PID control system, reducing the stability margin of the closed-loop control system. Consequently, the controller gain must be reduced below that which could be used for a process without delay. Thus, the response of the closed-loop system will be sluggish compared to that of the system with no time delay. In order to improve the performance of time-delay systems, special control algorithms have been developed to provide time-delay com-

FIG. 8-34

Conventional control of an exothermic chemical reactor.

FIG. 8-35

8-19

Cascade control of an exothermic chemical reactor.

pensation. The Smith predictor technique is the best known algorithm; a related method is called the analytical predictor. Various investigators have found that based on integral squared error, the performance of the Smith predictor can be as much as 30 percent better than for a conventional controller. The Smith predictor is a model-based control strategy that involves a more complicated block diagram than that for a conventional feedback controller, although a PID controller is still central to the control strategy (see Fig. 8-37). The key concept is based on better coordination of the timing of manipulated variable action. The loop configuration takes into account the fact that the current controlled variable measurement is not a result of the current manipulated variable action, but the value taken 0 time units earlier. Time-delay compensation can yield excellent performance; however, if the process model parameters change (especially the time delay), the Smith predictor performance will deteriorate and is not recommended unless other precautions are taken. Selective and Override Control When there are more controlled variables than manipulated variables, a common solution to this problem is to use a selector to choose the appropriate process variable from among a number of available measurements. Selectors can be based on either multiple measurement points, multiple final control elements, or multiple controllers, as discussed below. Selectors are used to improve the control system performance as well as to protect equipment from unsafe operating conditions. One type of selector device chooses as its output signal the highest (or lowest) of two or more input signals. This approach is often referred to as auctioneering. On instrumentation diagrams, the symbol HS denotes high selector and LS a low selector. For example, a high selector can be used to determine the hot-spot temperature in a fixed-bed chemical reactor. In this case, the output from the high selector is the input to the temperature controller. In an exothermic catalytic reaction, the process may run away due to disturbances or changes in the reactor. Immediate action should be taken to prevent a dangerous rise in temperature. Because a hot spot may potentially develop at one of several possible locations in the reactor, multiple (redundant) measurement points should be employed. This approach minimizes the time required to identify when a temperature has risen too high at some point in the bed. The use of high or low limits for process variables is another type of selective control, called an override. The feature of anti-reset windup in feedback controllers is a type of override. Another example is a distillation column with lower and upper limits on the heat input to the column reboiler. The minimum level ensures that liquid will remain

8-20

PROCESS CONTROL

FIG. 8-36 Block diagram of the cascade control system. For a chemical reactor, L1 would correspond to a feed temperature or composition disturbance, while L2 would be a change in the cooling water temperature.

on the trays, while the upper limit is determined by the onset of flooding. Overrides are also used in forced-draft combustion-control systems to prevent an imbalance between air flow and fuel flow, which could result in unsafe operating conditions. Other types of selective systems employ multiple final control elements or multiple controllers. In some applications, several manipulated variables are used to control a single process variable (also called split-range control). Typical examples include the adjustment of both inflow and outflow from a chemical reactor in order to control reactor pressure or the use of both acid and base to control pH in waste-water treatment. In this approach, the selector chooses from several controller outputs which final control element should be adjusted (Marlin, Process Control, McGraw-Hill, New York, 1995).

Adaptive Control Process control problems inevitably require on-line tuning of the controller constants to achieve a satisfactory degree of control. If the process operating conditions or the environment changes significantly, the controller may have to be retuned. If these changes occur quite frequently, then adaptive control techniques should be considered. An adaptive control system is one in which the controller parameters are adjusted automatically to compensate for changing process conditions. During the 1980s, several adaptive controllers were field-tested and commercialized in the U.S. and abroad, including products by ASEA (Sweden), Leeds and Northrup, Foxboro, and Sattcontrol. At the present time, some form of adaptive tuning is available on almost all PID controllers. The ASEA adaptive controller, Novatune, was

˜

˜ = G*e ˜ − θs (G* ˜ = model withFIG. 8-37 Block diagram of the Smith predictor. The process model used in the controller is G ˜ out delay; e−θs = time delay element).

ADVANCED CONTROL SYSTEMS announced in 1983 and is generally based on minimum-variancecontrol algorithms. Both feedforward and feedback control capabilities reside in the hardware. The unit has been tested successfully in reactor and paper machine control applications in Europe and in pH control of wastewater in the United States. Foxboro developed a self-tuning PID controller that is based on a so-called “expert system” approach for adjustment of the controller parameters. The on-line tuning of Kc, τI, and τD is based on the closedloop transient response to a step change in set point. By evaluating the salient characteristics of the response (e.g., the decay ratio, overshoot, and closed-loop period), the controller parameters can be updated without actually finding a new process model. The details of the algorithm, however, are proprietary. The Sattcontroller (also marketed by Fisher-Rosemount) has an autotuning function that is based on placing the process in a controlled oscillation at very low amplitude, comparable with that of the noise level of the process. This is done via a relay-type step function with hysteresis. The autotuner identifies the dynamic parameters of the process (the ultimate gain and period) and automatically calculates Kc, τI, and τD using empirical tuning rules. Gain scheduling can also be implemented with this controller, using up to three sets of PID controller parameters. The subject of adaptive control is one of current interest. New algorithms are presently under development, but these need to be fieldtested before industrial acceptance can be expected. It is clear, however, that digital computers will be required for implementation of self-adaptive controllers due to their complexity. An adaptive controller is inherently nonlinear and therefore more complicated than the conventional PID controller. Fuzzy Logic Control The application of fuzzy logic to process control requires the concepts of fuzzy rules and fuzzy inference. A fuzzy rule, also known as a fuzzy IF-THEN statement, has the form: If x then y where x specifies a vector of input variables and corresponding membership values and y specifies an output variable and its corresponding membership value. For example, if input1 = high and input2 = low, then output = medium. Three functions are required to perform logical inferencing with the fuzzy rules. The fuzzy AND is the product of a rule’s input membership values, generating a weight for the rule’s output. The fuzzy OR is a normalized sum of the weights assigned to each rule that contributes to a particular decision. The third function used is defuzzification, which generates a crisp final output. In one approach, the crisp output is the weighted average of the peak element values:  [w(i) p(i)]/ [w(i)]. With a single feedback control architecture, information that is readily available to the algorithm includes the error signal, the difference between the process variable and the set point variable, change in error from previous cycles to the current cycle, changes to the set point variable, change of the manipulated variable from cycle to cycle, and the change in the process variable from past to present. In addition, multiple combinations of the system response data are available. As long as the irregularity lies in that dimension wherein fuzzy decisions are being based or associated, the result should be enhanced performance. This enhanced performance should be demonstrated in both the transient and steady-state response. If the system tends to have changing dynamic characteristics or exhibits nonlinearities, fuzzy logic control should offer a better alternative to using constant PID settings. Most fuzzy logic software begins building its information base during the autotune function. In fact, the majority of the information used in the early stages of system startup comes from the autotune solutions. In addition to single-loop process controllers, products that have benefited from the implementation of fuzzy logic are: • Camcorders with automatic compensation for operator-injected noise such as shaking and moving • Elevators with decreased wait time, making intelligent floor decisions and minimizing travel and power consumption

8-21

• Antilock braking systems with quickly reacting independent wheel decisions based on current and acquired knowledge • Television with automatic color, brightness, and acoustic control based on signal and environmental conditions Sometimes fuzzy logic controllers are combined with pattern recognition software such as artificial neural networks (Kosko, Neural Networks and Fuzzy Systems, Prentice Hall, Englewood Cliffs, New Jersey, 1992). Statistical Process Control Statistical process control (SPC), also called statistical quality control (SQC), involves the application of statistical concepts to determine whether a process is operating satisfactorily. The ideas involved in statistical quality control are over fifty years old, but only recently with the growing worldwide focus on increased productivity have applications of SPC become widespread. If a process is operating satisfactorily (or “in control”), then the variation of product quality falls within acceptable bounds, usually the minimum and maximum values of a specified composition or property (product specification). Figure 8-38 illustrates the typical spread of values of the controlled variable that might be expected to occur under steady-state operating conditions. The mean and root mean square (RMS) deviation are identified in Fig. 8-38 and can be computed from a series of n observations c1, c2, . . . cn as follows: 1 n mean: c = ᎏ  ci (8-49) n i=1 RMS deviation: 1/ 2 1 n (8-50) σ = ᎏ  (ci − c)2 n i=1 The RMS deviation is a measure of the spread of values for c around the mean. A large value of σ indicates that wide variations in c occur. The probability that the controlled variable lies between the values of c1 and c2 is given by the area under the distribution between c1 and c2 (histogram). If the histogram follows a normal probability distribution, then 99.7 percent of all observations should lie with ⫾3σ of the mean (between the lower and upper control limits). These limits are used to determine the quality of control. If all data from a process lie within the ⫾3σ limits, then we conclude that nothing unusual has happened during the recorded time period. The process environment is relatively unchanged, and the product quality lies within specification. On the other hand, if repeated violations of the ⫾3σ limits occur, then the process environment has changed and the process is out of control. One way to codify abnormal behavior is the so-called Western Electric rules, which identify cases where a process is out of control: 1. One point that occurs outside the upper or lower control limits 2. Any seven consecutive points lying on the same side of the center line (mean) 3. Any seven consecutive points that increase or decrease 4. Any nonrandom pattern In the above list, one assumes that sample values are independent (i.e., not correlated). There are important economic consequences of a process being out of control; for example, product waste and customer dissatisfaction. Hence, statistical process control does provide a way to continuously monitor process performance and improve product quality. A typical process may go out of control due to several reasons, including • Persistent disturbances from the weather • An undetected grade change in raw materials • A malfunctioning instrument or control system Statistical quality control is a diagnostic tool—that is, an indicator of quality problems—but it does not identify the source of the problem or the corrective action to be taken. The Shewhart chart provides a way to analyze variability of a single measurement, as discussed in the following example. The data in Fig. 8-39 were obtained from the monitoring of pH in a yarn-soaking kettle used in textile manufacturing (Seborg, Edgar, and Mellichamp, Process Dynamics and Control, Wiley, New York, 1989). Because pH has a crucial influence on color and durability of the yarn, it is important to maintain pH within a range that gives the best results for both characteristics. The pH is considered to be in control between values of 4.25 and 4.64. At the






8-22

PROCESS CONTROL

Histogram plotting frequency of occurrence. c = mean, σ = rms deviation. Also shown is fit by normal probability distribution.

FIG. 8-38

25th day, the data show that pH is out of control; this might imply that a property change in the raw material has occurred and must be corrected with the supplier. However, a real-time correction would be preferable. In Fig. 8-39, the pH was adjusted by slowly adding more acid to the vats until it came back into control (on day 29). In continuous processes where automatic feedback control has been implemented, the feedback mechanism theoretically ensures that product quality is at or near the set point regardless of process disturbances. This, of course, requires that an appropriate manipulated variable has been identified for adjusting the product quality. However, even under feedback control, there may be daily variations of product quality because of disturbances or equipment or instrument malfunctions. These occurrences can be analyzed using the concepts of statistical quality control. More details on statistical process control are available in several textbooks (Grant and Leavenworth, Statistical Quality Control, McGraw-Hill, New York, 1980; Montgomery, Introduction to Statistical Quality Control, Wiley, New York, 1985). MULTIVARIABLE CONTROL PROBLEMS GENERAL REFERENCES: Shinskey, F. G., Process Control Systems, 3d ed., McGraw-Hill, New York, 1988. Seborg, D. E., T. F. Edgar, and D. A. Mellichamp, Process Dynamics and Control, Wiley, New York, 1989. McAvoy, T. J., Interaction Analysis, ISA, Research Triangle Park, North Carolina, 1983.

Process control books and journal articles tend to emphasize problems with a single controlled variable. In contrast, most practical

FIG. 8-39

problems are multivariable control problems because many process variables must be controlled. In fact, for virtually any important industrial process, at least two variables must be controlled: product quality and throughput. In this section, strategies for multivariable control problems are considered. Three examples of simple multivariable control problems are shown in Fig. 8-40. The in-line blending system blends pure components A and B to produce a product stream with flow rate w and mass fraction of A, x. Adjusting either inlet flow rate wA or wB affects both of the controlled variables w and x. For the pH neutralization process in Figure 8-40(b), liquid level h and the pH of the exit stream are to be controlled by adjusting the acid and base flow rates wa and wb. Each of the manipulated variables affects both of the controlled variables. Thus, both the blending system and the pH neutralization process are said to exhibit strong process interactions. In contrast, the process interactions for the gas-liquid separator in Fig. 8-40(c) are not as strong because one manipulated variable, liquid flow rate L, has only a small and indirect effect on one controlled variable, pressure P. Strong process interactions can cause serious problems if a conventional multiloop feedback control scheme (e.g., PI or PID controllers) is employed. The process interactions can produce undesirable control loop interactions where the controllers fight each other. Also, it may be difficult to determine the best pairing of controlled and manipulated variables. For example, in the in-line blending process in Fig. 8-40(a), should w be controlled with wA and x with wB, or vice versa? Control Strategies for Multivariable Control Problems If a conventional multiloop control strategy performs poorly due to control loop interactions, a number of solutions are available:

Process control chart for the average daily pH readings.

ADVANCED CONTROL SYSTEMS

(a)

(b)

(c) FIG. 8-40

Physical examples of multivariable control problems.

a. Detune one or more of the control loops b. Choose different controlled or manipulated variables (or pairings) c. Use a decoupling control system d. Use a multivariable control scheme (e.g., model predictive control) Detuning a controller (e.g., using a smaller controller gain or a larger reset time) tends to reduce control loop interactions by sacrificing the performance for the detuned loops. This approach may be acceptable if some of the controlled variables are faster or less important than others. The selection of controlled and manipulated variables is of crucial importance in designing a control system. In particular, a judicious choice may significantly reduce control loop interactions. For the blending process in Fig. 8-40(a), a straightforward control strategy would be to control x by adjusting wA, and w by adjusting wB. But

8-23

physical intuition suggests that it would be better to control x by adjusting the ratio wA /(wA + wB) and to control product flow rate w by the sum wA + wB. Thus, the new manipulated variables would be: M1 = wA /(wA + wB) and M2 = wA + wB. In this control scheme, M1 only affects x and M2 only affects w. Thus, the control loop interactions have been eliminated. Similarly, for the pH neutralization process in Fig. 8-40(b), the control loop interactions would be greatly reduced if pH is controlled by M1 = wa /(wa + wb) and liquid level h is controlled by M2 = wa + wb. Decoupling Control Systems Decoupling control systems provide an alternative approach for reducing control loop interactions. The basic idea is to use additional controllers called “decouplers” to compensate for undesirable process interactions. As an illustrative example, consider the simplified block diagram for a representative decoupling control system shown in Fig. 8-41. The two controlled variables C1 and C2 and two manipulated variables M1 and M2 are related by four process transfer functions, Gp11, Gp12, and so on. For example, Gp11 denotes the transfer function between M1 and C1: C1(s) (8-51) ᎏ = Gp11(s) M1(s) Figure 8-41 includes two conventional feedback controllers: Gc1 controls C1 by manipulating M1, and Gc2 controls C2 by manipulating M2. The output signals from the feedback controllers serve as input signals to the two decouplers D12 and D21. The block diagram is in a simplified form because the load variables and transfer functions for the final control elements and sensors have been omitted. The function of the decouplers is to compensate for the undesirable process interactions represented by Gp12 and Gp21. Suppose that the process transfer functions are all known. Then the ideal design equations are: Gp12(s) D12(s) = − ᎏ (8-52) Gp11(s) Gp21(s) (8-53) D21(s) = − ᎏ Gp22(s) These decoupler design equations are very similar to the ones for feedforward control in an earlier section. In fact, decoupling can be interpreted as a type of feedforward control where the input signal is the output of a feedback controller rather than a measured load variable. In principle, ideal decoupling eliminates control loop interactions and allows the closed-loop system to behave as a set of independent control loops. But in practice, this ideal behavior is not attained for a variety of reasons, including imperfect process models and the presence of saturation constraints on controller outputs and manipulated variables. Furthermore, the ideal decoupler design equations in (8-52) and (8-53) may not be physically realizable and thus would have to be approximated. In practice, other types of decouplers and decoupling control configurations have been employed. For example, in partial decoupling, only a single decoupler is employed (i.e., either D12 or D21 in Fig. 8-41 is set equal to zero). This approach tends to be more robust than complete decoupling and is preferred when one of the controlled variables is more important than the other. Static decouplers can be used to reduce the steady-state interactions between control loops. They can be designed by replacing the transfer functions in Eqs. (8-52) and (8-53) with the corresponding steady-state gains, Kp12 D12(s) = − ᎏ (8-54) Kp11 Kp21 (8-55) D21(s) = − ᎏ Kp22 The advantage of static decoupling is that less process information is required: namely, only steady-state gains. Nonlinear decouplers can be used when the process behavior is nonlinear. Pairing of Controlled and Manipulated Variables A key decision in multiloop-control-system design is the pairing of manipu-

8-24

PROCESS CONTROL

FIG. 8-41

Decoupling control system.

lated and controlled variables. This is referred to as the controllerpairing problem. Suppose there are N controlled variables and N manipulated variables. Then N! distinct control configurations exist. For example, if N = 5, then there are 120 different multiloop control schemes. In practice, many of them would be rejected based on physical insight or previous experience. But a smaller number (e.g., 5–15) may appear to be feasible and further analysis would be warranted. Thus, it is very useful to have a simple method for choosing the most promising control configuration. The most popular and widely used technique for determining the best controller pairing is the relative gain array (RGA) method (Bristol, “On a New Measure of Process Interaction,” IEEE Trans. Auto. Control, AC-11, 133, 1966). The RGA method provides two important items of information: 1. A measure of the degree of process interactions between the manipulated and controlled variables 2. A recommended controller pairing An important advantage of the RGA method is that it requires minimal process information: namely, steady-state gains. Another advantage is that the results are independent of both the physical units used and the scaling of the process variables. The chief disadvantage of the RGA method is that it neglects process dynamics, which can be an important factor in the pairing decision. Thus, the RGA analysis should be supplemented with an evaluation of process dynamics. Although extensions of the RGA method that incorporate process dynamics have been reported, these extensions have not been widely applied. RGA Method for 2 ⴛ 2 Control Problems To illustrate the use of the RGA method, consider a control problem with two inputs and two outputs. The more general case of N × N control problems is considered elsewhere (McAvoy, Interaction Analysis, ISA, Research Triangle Park, North Carolina, 1983). As a starting point, it is assumed that a linear, steady-state process model is available, C1 = K11 M1 + K12 M2

(8-56)

C2 = K21 M1 + K22 M2

(8-57)

where M1 and M2 are steady-state values of the manipulated inputs; C1 and C2 are steady-state values of the controlled outputs; and the values K are steady-state gains. The C and M variables are deviation variables from nominal steady-state values. This process model could be obtained in a variety of ways, such as by linearizing a theoretical model or by calculating steady-state gains from experimental data or a steady-state simulation.

By definition, the relative gain λij between the ith manipulated variable and the jth controlled variable is defined as: open-loop gain between Ci and Mj λij = ᎏᎏᎏᎏ (8-58) closed-loop gain between Ci and Mj where the open-loop gain is simply Kij from Eqs. (8-56) and (8-57). The closed-loop gain is defined to be the steady-state gain between Mj and Ci when the other control loop is closed and no offset occurs due to the presence of integral control action. The RGA for the 2 × 2 process is denoted by Λ=

λλ

11 21



λ12 λ22

(8-59)

The RGA has the important normalization property that the sum of the elements in each row and each column is exactly one. Consequently, the RGA in Eq. (8-59) can be written as Λ=

1 −λ λ



1−λ λ

(8-60)

where λ can be calculated from the following formula: 1 λ = ᎏᎏ K12 K21 1 − ᎏᎏ K11 K22

(8-61)

Ideally, the relative gains that correspond to the proposed controller pairing should have a value of one since Eq. (8-58) implies that the open and closed-loop gains are then identical. If a relative gain equals one, the steady-state operation of this loop will not be affected when the other control loop is changed from manual to automatic, or vice versa. Consequently, the recommendation for the best controller pairing is to pair the controlled and manipulated variables so that the corresponding relative gains are positive and close to one. RGA Example In order to illustrate use of the RGA method, consider the following steady-state version of a transfer function model for a pilot-scale, methanol-water distillation column (Wood and Berry, “Terminal Composition Control of a Binary Distillation Column,” Chem. Eng. Sci., 28, 1707, 1973): K11 = 12.8, K12 = −18.9, K21 = 6.6, and K22 = −19.4. It follows that λ = 2 and Λ=

−12 −12

(8-62)

Thus it is concluded that the column is fairly interacting and the recommended controller pairing is to pair C1 with M1 and C2 with M2.

ADVANCED CONTROL SYSTEMS MODEL PREDICTIVE CONTROL Introduction The model-based control strategy that has been most widely applied in the process industries is model predictive control (MPC). It is a general method that is especially well-suited for difficult multiinput, multioutput (MIMO) control problems where there are significant interactions between the manipulated inputs and the controlled outputs. Unlike other model-based control strategies, MPC can easily accommodate inequality constraints on input and output variables such as upper and lower limits or rate-of-change limits. A key feature of MPC is that future process behavior is predicted using a dynamic model and available measurements. The controller outputs are calculated so as to minimize the difference between the predicted process response and the desired response. At each sampling instant, the control calculations are repeated and the predictions updated based on current measurements. In typical industrial applications, the set point and target values for the MPC calculations are updated using on-line optimization based on a steady-state model of the process. Constraints on the controlled and manipulated variables can be routinely included in both the MPC and optimization calculations. The extensive MPC literature includes survey articles (Garcia, Prett, and Morari, Automatica, 25, 335, 1989; Richalet, Automatica, 29, 1251, 1993) and books (Prett and Garcia, Fundamental Process Control, Butterworths, Stoneham, Massachusetts, 1988; Soeterboek, Predictive Control—A Unified Approach, Prentice Hall, Englewood Cliffs, New Jersey, 1991). The current widespread interest in MPC techniques was initiated by pioneering research performed by two industrial groups in the 1970s. Shell Oil (Houston, TX) reported their Dynamic Matrix Control (DMC) approach in 1979, while a similar technique, marketed as IDCOM, was published by a small French company, ADERSA, in 1978. Since then, there have been over one thousand applications of these and related MPC techniques in oil refineries and petrochemical plants around the world. Thus, MPC has had a substantial impact and is currently the method of choice for difficult multivariable control problems in these industries. However, relatively few applications have been reported in other process industries, even though MPC is a very general approach that is not limited to a particular industry. Advantages and Disadvantages of MPC Model Predictive Control offers a number of important advantages: 1. It is a general control strategy for MIMO processes with inequality constraints on input and output variables. 2. It can easily accommodate difficult or unusual dynamic behavior such as large time delays and inverse responses. 3. Since the control calculations are based on optimizing control system performance, MPC can be readily integrated with on-line optimization strategies to optimize plant performance. 4. The control strategy can be easily updated on-line to compensate for changes in process conditions, constraints, or performance criteria. But current versions of MPC have significant disadvantages: 1. The MPC strategy is very different from conventional multiloop control strategies and thus initially unfamiliar to plant personnel. 2. The MPC calculations can be relatively complicated (e.g., solving an LP or QP problem at each sampling instant) and thus require a significant amount of computer resources and effort. 3. The development of a dynamic model from plant data is time consuming, typically requiring one to three weeks of around-the-clock plant tests. 4. Since empirical models are generally used, they are only valid over the range of conditions that were considered during the plant tests. 5. Theoretical studies have demonstrated that MPC can perform poorly for some types of process disturbances, especially when output constraints are employed (Lundstrom, Lee, Morari, and Skogestad, Computers Chem. Eng., 19, 409, 1995). Since MPC has been widely used and has had considerable impact, there is a broad consensus that its advantages far outweigh its disadvantages. Economic Incentives for Automation Projects Industrial applications of advanced process control strategies such as MPC are

8-25

motivated by the need for improvements regarding safety, product quality, environmental standards, and economic operation of the process. One view of the economics incentives for advanced automation techniques is illustrated in Fig. 8-42. Distributed control systems (DCS) are widely used for data acquisition and conventional singleloop (PID) control. Usually, they are the most expensive part of the entire control system. The addition of advanced regulatory control systems such as decouplers, selective controls, and time-delay compensation can provide additional benefits for a modest incremental cost. But experience has indicated that the major benefits can be obtained for relatively small incremental costs through a combination of MPC and on-line optimization. The results in Fig. 8-42 are shown qualitatively, rather than quantitatively, because the actual costs and benefits are application-dependent. A key reason why MPC has become a major commercial and technical success is that there are numerous vendors who are licensed to market MPC products and install them on a turnkey basis. Consequently, even medium-sized companies are able to take advantage of this new technology. Payout times of 3–12 months have been reported. Basic Features of MPC Model predictive control strategies have a number of distinguishing features: 1. A dynamic model of the process is used to predict the future outputs over a prediction horizon consisting of the next p sampling periods. 2. A reference trajectory is used to represent the desired output response over the prediction horizon. 3. Inequality constraints on the input and output variables can be included as an option. 4. At each sampling instant, a control policy consisting of the next m control moves is calculated. The control calculations are based on minimizing a quadratic or linear performance index over the prediction horizon while satisfying the constraints. 5. The performance index is expressed in terms of future control moves and the predicted deviations from the reference trajectory. 6. A receding horizon approach is employed. At each sampling instant, only the first control move (of the m moves that were calculated) is actually implemented. Then the predictions and control calculations are repeated at the next sampling instant. These distinguishing features of MPC will now be described in more detail.

FIG. 8-42

industries.

Economic incentives for automation projects in the process

8-26

PROCESS CONTROL

A key feature of MPC is that a dynamic model of the process is used to predict future values of the controlled outputs. There is considerable flexibility concerning the choice of the dynamic model. For example, a physical model based on first principles (e.g., mass and energy balances) or an empirical model could be selected. Also, the empirical model could be a linear model (e.g., transfer function, step response model, or state space model) or a nonlinear model (e.g., neural net model). However, most industrial applications of MPC have relied on linear empirical models, which may include simple nonlinear transformations of process variables. The original formulations of MPC (i.e., DMC and IDCOM) were based on empirical linear models expressed in either step-response or impulse-response form. For simplicity, we will consider only a singleinput, single-output (SISO) model. However, the SISO model can be easily generalized to the MIMO models that are used in industrial applications. The step response model relating a single controlled variable y and a single manipulated variable u can be expressed as

sion of MPC is based on step-response models, while IDCOM utilizes impulse response models. The receding horizon feature of MPC is shown in Fig. 8-44 with the current sampling instant denoted by k. Past input signals [u(i) for i < k] are used to predict the output at the next p sampling instants [y(k ˆ + i) for i = 1, 2, . . . , p]. The control calculations are performed to generate an m-step control policy [u(k), u(k + 1), . . . , u(k + m)], which optimizes the performance index. The first control action, u(k), is implemented. Then at the next sampling instant (k + 1), the prediction and control calculations are repeated in order to determine u(k + 1). In Fig. 8-44, the reference trajectory (or target) is considered to be constant. Other possibilities include a gradual or step set point change that can be generated by on-line optimization. The performance index for MPC applications is usually a linear or quadratic function of the predicted errors and calculated future control moves. For example, the following quadratic performance index has been widely used:

N

y(k) ˆ =  Si ∆u(k − i) + y(0)

∆u(k)

i=1

where y(k) ˆ is the predicted value of y at the k-sampling instant; u(k) is the value of the manipulated input at time k; and the model parameters Si are referred to as the step-response coefficients. The initial value y(0) is assumed to be known. The change in the manipulated input from one sampling instant to the next is denoted by ∆u(k) = u(k) − u(k − 1) 䉭

(8-64)

The step-response model is also referred to as a finite impulse response (FIR) model or a discrete convolution model. In principle, the step-response coefficients can be determined from the output response to a step change in the input. A typical response to a unit step change in input u is shown in Fig. 8-43. The step response coefficients Si are simply the values of the output variable at the sampling instants, after the initial value y(0) has been subtracted. Theoretically, they can be determined from a single-step response, but, in practice, a number of “bump tests” are required to compensate for unanticipated disturbances, process nonlinearities, and noisy measurements. The step-response model in Eq. (8-63) is equivalent to the following impulse response model:

where the impulse response coefficients Ii are related to the stepresponse coefficients by Ii = Si − Si − 1. Step- and impulse-response models typically contain a large number of parameters because the model horizon N is usually quite large (30 < N < 70). In fact, these models are often referred to as nonparametric models. The DMC ver-

FIG. 8-43

Step response for u, a unit step change in the input.

i=1

i=1

ˆ + i) e(k + i) = r(k + i) − y(k 䉭

(8-66)

(8-67)

where r(k + i) is the reference value at time k + i, and ∆u(k) denotes the vector of current and future control moves over the next m sampling instants: ∆u(k) = [∆u(k), ∆u(k + 1), . . . , ∆u(k + m − 1)]T

(8-68)

Equation (8-66) contains two types of design parameters that can also be used for tuning purposes. The move suppression factor δ penalizes large control moves, while the weighting factors wi allow the predicted errors to be weighed differently at each time step, if desired. Inequality constraints on the future inputs or their rates of change are widely used in the MPC calculations. For example, if both upper and lower limits are required, the constraints could be expressed as: Bi∗ ≤ u(k + i) ≤ B*i

for i = 1, 2, . . . , m

(8-69)

Ci∗ ≤ ∆u(k + i) ≤ C*i

for i = 1, 2, . . . , m

(8-70)

where Bi and Ci are constants. Constraints on the predicted outputs are sometimes included as well: Di∗ ≤ y(k ˆ + i) ≤ D *i

(8-65)

i=1

m

The value e(k + i) denotes the predicted error at time (k + i),

N

y(k) ˆ =  Ii u(k − i) + y(0)

p

min J =  wie2(k + i) + δ  ∆u2(k + i − 1)

(8-63)

for i = 1, 2, . . . , p

(8-71)

The minimization of the quadratic performance index in Eq. (8-66), subject to the constraints in Eq. (8-69) to (8-71) and the step-response model in Eq. (8-63), can be formulated as a standard quadratic pro-

FIG. 8-44

The “moving horizon” approach of model predictive control.

ADVANCED CONTROL SYSTEMS gramming (QP) problem. Consequently, efficient QP solution techniques can be employed. When the inequality constraints in Eqs. (8-69) to (8-71) are omitted, the optimization problem has an analytical solution (Prett and Garcia, Fundamental Process Control, Butterworths, Stoneham, Massachusetts, 1988; Soeterboek, Predictive Control—A Unified Approach, Prentice Hall, Englewood Cliffs, New Jersey, 1991). If the quadratic terms in Eq. (8-66) are replaced by linear terms, a linear programming program (LP) problem results that can also be solved using standard methods. This MPC formulation for SISO control problems can easily be extended to MIMO problems. Implementation Issues A critical factor in the successful application of any model-based technique is the availability of a suitable dynamic model. In typical MPC applications, an empirical model is identified from data acquired during extensive plant tests. The experiments generally consist of a series of bump tests in the manipulated variables. Typically, the manipulated variables are adjusted one at a time and the plant tests require a period of one to three weeks. The step or impulse response coefficients are then calculated using linearregression techniques such as least-squares methods. However, details concerning the procedures utilized in the plant tests and subsequent model identification are considered to be proprietary information. The scaling and conditioning of plant data for use in model identification and control calculations can be key factors in the success of the application. The MPC control problem illustrated in Eqs. (8-66) to (8-71) contains a variety of design parameters: model horizon N, prediction horizon p, control horizon m, weighting factors wi, move suppression factor δ, the constraint limits Bi, Ci, and Di, and the sampling period ∆t. Some of these parameters can be used to tune the MPC strategy, notably the move suppression factor δ, but details remain largely proprietary. One commercial controller, Honeywell’s RMPCT ® (Robust Multivariable Predictive Control Technology), provides default tuning parameters based on the dynamic process model and the model uncertainty. Integration of MPC and On-Line Optimization As indicated in Fig. 8-42, significant potential benefits can be realized by using a combination of MPC and on-line optimization. At the present time, most commercial MPC packages integrate the two methodologies in a hierarchical configuration such as the one shown in Fig. 8-45. The MPC calculations are performed quite often (e.g., every 1–10 min) and implemented as set points for PID control loops at the DCS level. The targets and constraints for the MPC calculations are generated by solving a steady-state optimization problem (LP or QP) based on a linear process model. These calculations may be performed as often as the MPC calculations. As an option, the targets and constraints for the LP or QP optimization can be generated from a nonlinear process model using a nonlinear optimization technique. These calculations tend to be performed less frequently (e.g., every 1–24 hours) due to the complexity of the calculations and the process models. The combination of MPC and frequent on-line optimization has been successfully applied in oil refineries and petrochemical plants around the world. REAL-TIME PROCESS OPTIMIZATION GENERAL REFERENCES: Biles and Swain, Optimization and Industrial Experimentation, Wiley—Interscience, New York, 1980. Dantzig, Linear Programming and Extensions, Princeton University Press, Princeton, New Jersey, 1963. Edgar and Himmelblau, Optimization of Chemical Processes, McGraw-Hill, New York, 1987. Fletcher, Practical Methods of Optimization. Wiley, New York, 1980. Gill, Murray, and Wright, Practical Optimization, Academic Press, New York, 1981. Murtagh, Advanced Linear Programming, McGraw-Hill, New York, 1983. Murty, Linear Programming, Wiley, New York, 1983. Reklaitis, Ravindran, and Ragsdell, Engineering Optimization, Wiley—Interscience, New York, 1984.

The chemical industry has undergone significant changes during the past 20 years due to the increased cost of energy and raw materials, more stringent environmental regulations, and intense worldwide competition. Modifications of both plant-design procedures and plant operating conditions have been implemented in order to reduce costs

FIG. 8-45

8-27

Hierarchical control configuration for MPC and on-line optimiza-

tion.

and meet constraints. One of the most important engineering tools that can be employed in such activities is optimization. As plant computers have become more powerful, the size and complexity of problems that can be solved by optimization techniques have correspondingly expanded. A wide variety of problems in the operation and analysis of chemical plants (as well as many other industrial processes) can be solved by optimization. Real-time optimization means that the process-operating conditions (set points) are evaluated on a regular basis and optimized. Sometimes this is called steady-state optimization or supervisory control. This section examines the basic characteristics of optimization problems and their solution techniques and describes some representative benefits and applications in the chemical and petroleum industries. Typical problems in chemical engineering process design or plant operation have many possible solutions. Optimization is concerned with selecting the best among the entire set by efficient quantitative methods. Computers and associated software make the computations involved in the selection feasible and cost-effective. Engineers work to improve the initial design of equipment and strive for enhancements in the operation of the equipment once it is installed in order to realize the most production, the greatest profit, the maximum cost, the least energy usage, and so on. In plant operations, benefits arise from improved plant performance, such as improved yields of valuable products (or reduced yields of contaminants), reduced energy consumption, higher processing rates, and longer times between shutdowns. Optimization can also lead to reduced maintenance costs, less equipment wear, and better staff utilization. It is helpful to systematically identify the objective, constraints, and degrees of freedom in a process or a plant if such benefits as improved quality of designs, faster and more reliable troubleshooting, and faster decision making are to be achieved. Optimization can take place at many levels in a company, ranging

8-28

PROCESS CONTROL

from a complex combination of plants and distribution facilities down through individual plants, combinations of units, individual pieces of equipment, subsystems in a piece of equipment, or even smaller entities. Problems that can be solved by optimization can be found at all these levels. While process design and equipment specification are usually performed prior to the implementation of the process, optimization of operating conditions is carried out monthly, weekly, daily, hourly, or even every minute. Optimization of plant operations determines the set points for each unit at the temperatures, pressures, and flow rates that are the best in some sense. For example, the selection of the percentage of excess air in a process heater is quite critical and involves a balance on the fuel-air ratio to assure complete combustion and at the same time make the maximum use of the heating potential of the fuel. Typical day-to-day optimization in a plant minimizes steam consumption or cooling water consumption, optimizes the reflux ratio in a distillation column, or allocates raw materials on an economic basis [Latour, Hydro Proc., 58(6), 73, 1979, and Hydro. Proc., 58(7), 219, 1979]. A real-time optimization (RTO) system determines set point changes and implements them via the computer control system without intervention from unit operators. The RTO system completes all data transfer, optimization calculations, and set point implementation before unit conditions change and invalidate the computed optimum. In addition, the RTO system should perform all tasks without upsetting plant operations. Several steps are necessary for implementation of RTO, including determination of the plant steady state, data gathering and validation, updating of model parameters (if necessary) to match current operations, calculation of the new (optimized) set points, and the implementation of these set points. To determine if a process unit is at steady state, a program monitors key plant measurements (e.g., compositions, product rates, feed rates, and so on) and determines if the plant is steady enough to start the sequence. Only when all of the key measurements are within the allowable tolerances is the plant considered steady and the optimization sequence started. Tolerances for each measurement can be tuned separately. Measured data are then collected by the optimization computer. The optimization system runs a program to screen the measurements for unreasonable data (gross error detection). This validity checking automatically modifies the model updating calculation to reflect any bad data or when equipment is taken out of service. Data validation and reconciliation (on-line or off-line) is an extremely critical part of any optimization system. The optimization system then may run a parameter-fitting case that updates model parameters to match current plant operation. The integrated process model calculates such items as exchanger heat transfer coefficients, reactor performance parameters, furnace efficiencies, and heat and material balances for the entire plant. Parameter fitting allows for continual updating of the model to account for plant deviations and degradation of process equipment. After completion of the parameter fitting, the information regarding the current plant constraints, the control status data, and the economic values for feed products, utilities, and other operating costs are collected. The economic values are updated by the planning and scheduling department on a regular basis. The optimization system then calculates the optimized set points. The steady-state condition of the plant is rechecked after the optimization case is successfully completed. If the plant is still steady, then the values of the optimization targets are transferred to the process-control system for implementation. After a line-out period, the process-control computer resumes the steadystate detection calculations, restarting the cycle. Essential Features of Optimization Problems The solution of optimization problems involves the use of various tools of mathematics. Consequently, the formulation of an optimization problem requires the use of mathematical expressions. From a practical viewpoint, it is important to mesh properly the problem statement with the anticipated solution technique. Every optimization problem contains three essential categories: 1. An objective function to be optimized (revenue function, cost function, etc.) 2. Equality constraints (equations)

3. Inequality constraints (inequalities) Categories 2 and 3 comprise the model of the process or equipment; category 1 is sometimes called the economic model. No single method or algorithm of optimization exists that can be applied efficiently to all problems. The method chosen for any particular case will depend primarily on (1) the character of the objective function, (2) the nature of the constraints, and (3) the number of independent and dependent variables. Table 8-6 summarizes the six general steps for the analysis and solution of optimization problems (Edgar and Himmelblau, Optimization of Chemical Processes, McGraw-Hill, New York, 1988). You do not have to follow the cited order exactly, but you should cover all of the steps eventually. Shortcuts in the procedure are allowable, and the easy steps can be performed first. Steps 1, 2, and 3 deal with the mathematical definition of the problem: identification of variables and specification of the objective function and statement of the constraints. If the process to be optimized is very complex, it may be necessary to reformulate the problem so that it can be solved with reasonable effort. Later in this section, we discuss the development of mathematical models for the process and the objective function (the economic model). Step 5 in Table 8-6 involves the computation of the optimum point. Quite a few techniques exist to obtain the optimal solution for a problem. We describe several classes of methods below; Fig. 8-46 is a diagram for selection of individual optimization techniques. In general, the solution of most optimization problems involves the use of a digital computer to obtain numerical answers. Over the past 15 years, substantial progress has been made in developing efficient and robust digital methods for optimization calculations. Much is known about which methods are most successful. Virtually all numerical optimization methods involve iteration, and the effectiveness of a given technique often depends on a good first guess for the values of the variables at the optimal solution. After the optimum is computed, a sensitivity analysis for the objective function value should be performed to determine the effects of errors or uncertainty in the objective function, mathematical model, or other constraints. Development of Process (Mathematical) Models Constraints in optimization problems arise from physical bounds on the variables, empirical relations, physical laws, and so on. The mathematical relations describing the process also comprise constraints. Two general categories of models exist: 1. Those based on physical theory 2. Those based on strictly empirical descriptions Mathematical models based on physical and chemical laws (e.g., mass and energy balances, thermodynamics, chemical reaction kinetics) are frequently employed in optimization applications. These models are conceptually attractive because a general model for any system size can be developed before the system is constructed. On the other hand, an empirical model can be devised that simply correlates inputoutput data without any physiochemical analysis of the process. For TABLE 8-6 Problems

The Six Steps Used to Solve Optimization

1. Analyze the process itself so that the process variables and specific characteristics of interest are defined (i.e., make a list of all of the variables). 2. Determine the criterion for optimization and specify the objective function in terms of the above variables together with coefficients. This step provides the performance model (sometimes called the economic model when appropriate). 3. Develop via mathematical expressions a valid process or equipment model that relates the input-output variables of the process and associated coefficients. Include both equality and inequality constraints. Use well-known physical principles (mass balances, energy balances), empirical relations, implicit concepts, and external restrictions. Identify the independent and dependent variables (number of degrees of freedom). 4. If the problem formulation is too large in scope, (a) break it up into manageable parts and/or (b) simplify the objective function and model. 5. Apply a suitable optimization technique to the mathematical statement of the problem. 6. Check the answers and examine the sensitivity of the result to changes in the coefficients in the problem and the assumptions.

ADVANCED CONTROL SYSTEMS

FIG. 8-46

8-29

Diagram for selection of optimization techniques with algebraic constraints and objective function.

these models, optimization is often used to fit a model to process data, using a procedure called parameter estimation. The well-known least squares curve-fitting procedure is based on optimization theory, assuming that the model parameters are contained linearly in the model. One example is the yield matrix, where the percentage yield of each product in a unit operation is estimated for each feed component using process data rather than employing a mechanistic set of chemical reactions.

Formulation of the Objective Function The formulation of objective functions is one of the crucial steps in the application of optimization to a practical problem. You must be able to translate the desired objective into mathematical terms. In the chemical process industries, the objective function often is expressed in units of currency (e.g., U.S. dollars) because the normal industrial goal is to minimize costs or maximize profits subject to a variety of constraints. A typical economic model involves the costs of raw materials, values

8-30

PROCESS CONTROL

of products, costs of production, as functions of operating conditions, projected sales figures, and the like. An objective function can be expressed in terms of these quantities; for example, annual operating profit ($/yr) might be expressed as: J =  FsVs −  FrCr − OC s

where

(8-72)

r

J = profit/time

s FsVs = sum of product flow rates times respective product values (income) F C r

r

r

= sum of feed flows times respective unit costs

OC = operating costs/time

Unconstrained Optimization Unconstrained optimization refers to the case where no inequality constraints are present and all equality constraints can be eliminated by solving for selected dependent variables followed by substitution for them in the objective function. Very few realistic problems in process optimization are unconstrained. However, it is desirable to have efficient unconstrained optimization techniques available since these techniques must be applied in real time and iterative calculations cost computer time. The two classes of unconstrained techniques are single-variable optimization and multivariable optimization. Single Variable Optimization Many process optimization problems can be reduced to the variation of a single variable so as to maximize profit or some other overall process objective function. Some examples of single variable optimization include optimizing the reflux ratio in a distillation column or the air/fuel ratio in a furnace (Martin, Latour, and Richard, Chem. Engr. Prog., 77, September, 1981). While most processes actually are multivariable processes with several operating degrees of freedom, often we choose to optimize only the most important variable in order to keep the strategy uncomplicated. One characteristic implicitly required in a single variable optimization problem is that the objective function J be unimodal in the variable x. The selection of a method for one-dimensional search is based on the trade-off between number of function evaluations versus computer time. We can find the optimum by evaluating the objective function at many values of x using a small grid spacing (∆x) over the allowable range of x values, but this method is generally inefficient. There are three classes of techniques that can be used efficiently for one-dimensional search: 1. Indirect 2. Region elimination 3. Interpolation Indirect methods seek to solve the necessary condition dJ/dx = 0 by iteration, but these methods are not as popular as the second two classes. Region elimination methods include equal interval search, dichotomous search (or bisecting), Fibonacci search, and golden section. These methods do not use information on the shape of the function (other than being unimodal) and thus tend to be rather conservative. The third class of techniques uses repeated polynomial fitting to predict the optimum. These interpolation methods tend to converge rapidly to the optimum without being very complicated. Two interpolation methods, quadratic and cubic interpolation, have been used in many optimization packages. Multivariable Optimization In multivariable optimization problems, there is no guarantee that the optimum can be reached in a reasonable amount of computer time. The numerical optimization of general nonlinear multivariable objective functions requires that efficient and robust techniques be employed. Efficiency is important since iteration is employed. For example, in multivariable “grid” search for a problem with four independent variables, an equally spaced grid for each variable is prescribed. For ten values of each of the four variables, there would be 104 total function evaluations required to find the best answer for the grid intersections. However, this computational effort still may not yield a result close enough to the true optimum, requiring further search. Therefore, grid search is a very inefficient method for most problems involving many variables. In multivariable optimization, the difficulty of dealing with multi-

variable functions is usually resolved by treating the problem as a series of one-dimensional searches. For a given starting point, a search direction s is specified, and the optimum is found by searching along that direction. The step size ε is the distance moved along s. Then a new search direction is determined, followed by another onedimensional search. The algorithm used to specify the search direction depends on the optimization method. There are two basic types of unconstrained optimization algorithms: (1) those requiring function derivatives and (2) those that do not. The nonderivative methods are of interest in optimization applications because these methods can be readily adapted to the case in which experiments are carried out directly on the process. In such cases, an actual process measurement (such as yield) can be the objective function, and no mathematical model for the process is required. Methods that do not require derivatives are called direct methods and include sequential simplex (Nelder-Meade) and Powell’s method. The sequential simplex method is quite satisfactory for optimization with two or three independent variables, is simple to understand, and is fairly easy to execute. Powell’s method is more efficient than the simplex method and is based on the concept of conjugate search directions. The second class of multivariable optimization techniques in principle requires the use of partial derivatives, although finite difference formulas can be substituted for derivatives; such techniques are called indirect methods and include the following classes: 1. Steepest descent (gradient) method 2. Conjugate gradient (Fletcher-Reeves) method 3. Newton’s method 4. Quasi-Newton methods The steepest descent method is quite old and utilizes the intuitive concept of moving in the direction where the objective function changes the most. However, it is clearly not as efficient as the other three. Conjugate gradient utilizes only first-derivative information, as does steepest descent, but generates improved search directions. Newton’s method requires second derivative information but is very efficient, while quasi-Newton retains most of the benefits of Newton’s method but utilizes only first derivative information. All of these techniques are also used with constrained optimization. Constrained Optimization When constraints exist and cannot be eliminated in an optimization problem, more general methods must be employed than those described above, since the unconstrained optimum may correspond to unrealistic values of the operating variables. The general form of a nonlinear programming problem allows for a nonlinear objective function and nonlinear constraints, or Minimize

J(x1, x2, . . . , xn)

Subject to

hi(x1, x2, . . . , xn) = 0

(i = 1, rc)

gi(x1, x2, . . . , xn) ≥ 0

(i = 1, mc)

(8-73)

In this case, there are n process variables with rc equality constraints and mc inequality constraints. Such problems pose a serious challenge to performing optimization calculations in a reasonable amount of time. Typical constraints in chemical process optimization include operating conditions (temperatures, pressures, and flows have limits), storage capacities, and product purity specifications. An important class of constrained optimization problems is one in which both the objective function and constraints are linear. The solution of these problems is highly structured and can be obtained rapidly. The accepted procedure, linear programming (LP), has become quite popular in the past twenty years, solving a wide range of industrial problems. It is increasingly being used for on-line optimization. For processing plants, there are several different kinds of linear constraints that may arise, making the LP method of great utility. 1. Production limitation due to equipment throughput restrictions, storage limits, or market constraints. 2. Raw material (feedstock) limitation. 3. Safety restrictions on allowable operating temperatures and pressures. 4. Physical property specifications placed on the composition of the final product. For blends of various products, we usually assume that a composite property can be calculated through the averaging of pure component physical properties.

UNIT OPERATIONS CONTROL 5. Material and energy balances of the steady-state model. The optimum in linear programming lies at the constraint intersections, which was generalized to any number of variables and constraints by George Dantzig. The Simplex algorithm is a matrix-based numerical procedure for which many digital computer codes exist, both for mainframe and microcomputers (Edgar and Himmelblau, Optimization of Chemical Processes, McGraw-Hill, New York, 1987; Schrage, Linear, Integer, and Quadratic Programming with LINDO, Scientific Press, Palo Alto, California, 1983). The algorithm can handle virtually any number of inequality constraints and any number of variables in the objective function and utilizes the observation that only the constraint boundaries need to be examined to find the optimum. In some instances, nonlinear optimization problems even with nonlinear constraints can be linearized so that the LP algorithm can be employed to solve them (called successive linear programming or SLP). In the process industries, the Simplex algorithm has been applied to a wide range of problems, including refinery scheduling, olefins production, the optimal allocation of boiler fuel, and the optimization of a total plant. Nonlinear Programming The most general case for optimization occurs when both the objective function and constraints are nonlinear, a case referred to as nonlinear programming. While the idea behind the search methods used for unconstrained multivariable problems are applicable, the presence of constraints complicates the solution procedure. In practice, one of the best current general algorithms (best on the basis of many tests) using iterative linearization is the Generalized Reduced Gradient algorithm (GRG). The GRG algorithm employs linear or linearized constraints, defines new variables that are normal to the constraints, and expresses the gradient (or other search direction) in terms of this normal basis (Liebman, Lasdon, Schrage, and Waren, GINO, Scientific Press, Palo Alto, California, 1986). Other established types of constrained optimization methods include the following types of algorithms: 1. Penalty functions with augmented Lagrangian method (an enhancement of the classical Lagrange multiplier method) 2. Successive quadratic programming All of these methods have been utilized to solve nonlinear programming problems in the field of chemical engineering design and operations (Lasdon and Waren, Oper. Res., 5, 34, 1980). Nonlinear programming is receiving increased usage in the area of real-time optimization. One important class of nonlinear programming techniques is called quadratic programming (QP), where the objective function is quadratic and the constraints are linear. While the solution is iterative, it can be obtained quickly as in linear programming. This is the basis for the newest type of constrained multivariable control algorithms called model predictive control. The dominant method used in the refining industry utilizes the solution of a QP and is called dynamic matrix con-

8-31

trol or DMC. See the earlier subsection on model predictive control for more details. EXPERT SYSTEMS An expert system is a computer program that uses an expert’s knowledge in a particular domain to solve a narrowly focused, complex problem. An off-line system uses information entered manually and produces results in visual form to guide the user in solving the problem at hand. An on-line system uses information taken directly from process measurements to perform tasks automatically or instruct or alert operating personnel to the status of the plant. Each expert system has a rule base created by the expert to respond the way the expert would to sets of input information. Expert systems used for plant diagnostics and management usually have an open rule base, which can be changed and augmented as more experience accumulates and more tasks are to be automated. The system begins as an empty shell with an assortment of functions such as equation-solving, logic, and simulation, as well as input and display tools to allow an expert to construct a proprietary rule base. The “expert” in this case would be the person or persons having the deepest knowledge about the process, its problems, its symptoms, and remedies. Converting these inputs into meaningful outputs is the principal task in constructing a rule base. Skill at computer programming is especially helpful, although most shells allow rules to be entered in the vernacular. Firstprinciples models (deep knowledge) produce the most accurate results, although heuristics are always required to establish limits. A closed expert system is one designed by an expert to be sold in quantity for use by others (where open systems tend to be unique). It is closed to keep users from altering the rule base and thereby changing the product. Common examples in process control are autotuning and self-tuning controllers whose rule base is designed by one or more experts in that field. Once packaged and sold, its rule base cannot be changed in the field no matter how poorly it performs the task; revisions must be made by the manufacturer in later releases as for any software product. The development vehicle used to create and test the rule base must be as flexible as possible, allowing easy alterations and expansion of the rule base with whatever displays can convey the most information. The delivery vehicle, however, should be virtually transparent to the user, conveying only as much information as needed to solve the problem at hand. Self-tuning controllers can perform their task without explicitly informing users, but their output and status is available on demand, and their operation may be easily limited or interrupted. To be successful, the scope of an expert system must be limited to a narrow group of common problems that are readily solved by conventional means, and where the return on investment is greatest. Widening the scope usually requires more complex methods and treats less common problems having lower return.

UNIT OPERATIONS CONTROL PROCESS AND INSTRUMENTATION DIAGRAMS GENERAL REFERENCES: Shinskey, Process Control Systems, 3d ed., McGrawHill, New York, 1988. Luyben, Practical Distillation Control, Van Nostrand Reinhold, New York, 1992.

The process and instrumentation (P&I) diagram provides a graphical representation of the control configuration for the process. The P&I diagrams illustrate the measurement devices that provide inputs to the control strategy, the actuators that will implement the results of the control calculations, and the function blocks that provide the control logic. The symbology for drawing P&I diagrams generally follows standards developed by one of the following organizations: 1. International Society for Measurement and Control (ISA). The chemicals, refining, and foods industries generally follow this standard.

2. Scientific Apparatus Manufacturers Association (SAMA). The fossil-fuel electric utility industry generally follows this standard. Both organizations update their standards from time to time, primarily because the continuing evolutions in control-system hardware provide additional possibilities for implementing control schemes. Although arguments can be made for the advantages of each symbology, the practices within an industry seem to be mainly the result of historical practice with no indication of any significant shift. Most companies adopt one of the standards but then tailor or extend the symbology to best suit their internal practices. Such companies maintain an internal document and/or drawing that specifies the symbology used on their P&I diagrams. Their internal personnel and all contractors are instructed to adhere to this symbology when developing P&I diagrams. Figure 8-47 presents a P&I diagram for a simple temperature control loop that adheres to the ISA symbology. The measurement

8-32

PROCESS CONTROL

Temperature leaving a heat exchanger responds as a distributed lag, the gain and time constant of which vary inversely with flow.

FIG. 8-48

FIG. 8-47

Example of a process and instrument diagram.

devices and most elements of the control logic are represented by circles. In Figure 8-47, circles are used to designate the following: 1. TT102 is the temperature measurement device. 2. TC102 is the temperature controller. 3. TY102 is the current-to-pneumatic (I/P) transducer. The symbol for the control valve in Fig. 8-47 is for a pneumatic positioning valve without a valve positioner. Electronic signals (that is, 4–20 milliamp current loops) are represented by dashed lines. In Fig. 8-47, these include the following: 1. The signal from the measurement device to the controller. 2. The signal from the controller to the I/P transducer. Pneumatic signals are represented by solid lines that are crosshatched intermittently. The signal from the I/P transducer to the pneumatic positioning valve is pneumatic. The ISA symbology provides different symbols for different types of actuators. Furthermore, variations for the controller symbol distinguish control algorithms implemented in DCS technology from panel-mounted single-loop controllers. CONTROL OF HEAT EXCHANGERS Steam-Heated Exchangers Steam, the most common heating medium, transfers its latent heat in condensing, causing heat flow to be proportional to steam flow. Thus, a measurement of steam flow is essentially a measure of heat transfer. Consider raising a liquid from temperature T1 to T2 by condensing steam: Q = WH = FCL(T2 − T1)

(8-74)

where W and H are the mass flow of steam and its latent heat, F and CL are the mass flow and specific heat of the liquid, and Q is the rate of heat transfer. The response of controlled temperature to steam flow is linear: H dT2 (8-75) ᎏ=ᎏ dW FCL However, the steady-state process gain described by this derivative varies inversely with liquid flow: Adding a given increment of heat flow to a smaller flow of liquid produces a greater temperature rise. Dynamically, the response of liquid temperature to a step in steam flow is that of a distributed lag, shown in Fig. 8-48. The time required to reach 63 percent complete response,  τ, is essentially the residence time of the fluid in the exchanger, which is its volume divided

by its flow. The residence time then varies inversely with flow. Table 8-2 gives optimum settings for PI and PID controllers for distributed lags, the proportional band varying directly with steady-state gain, and integral and derivative settings directly with  τ. Since both these parameters vary inversely with liquid flow, fixed settings for the temperature controller are optimal at only one flow rate. Undamped oscillations will be produced when the flow decreases by one-third from the value at which the controller was optimally tuned, whereas increasing flow rates produces an overdamped response. The stable operating range can be broadened to one-half the original flow by using an equal-percentage steam valve whose gain varies directly with flow. The best solution is to adapt the PID settings to change inversely with measured flow, thereby keeping the controller optimally tuned for all flow rates. Feedforward control can also be applied by multiplying the liquid flow measurement—after dynamic compensation—by the output of the temperature controller, the result used to set steam flow in cascade. Feedforward is capable of a reduction in integrated error as much as a hundredfold but requires the use of a steam-flow loop and dynamic compensator to approach this. Steam flow is sometimes controlled by manipulating a valve in the condensate line rather than the steam line, because it is smaller and hence less costly. Heat transfer, then, is changed by raising or lowering the level of condensate flooding the heat-transfer surface, an operation that is slower than manipulating a steam valve. Protection also needs to be provided against an open condensate valve blowing steam into the condensate system. Exchange of Sensible Heat When there is no change in phase, heat transfer is no longer linear with flow of the manipulated stream, as illustrated by Fig. 8-49. Here again, an equal-percentage valve should be used on that stream to linearize the loop. The variable dynamics of the distributed lag apply, limiting the stable operating range in the same way as for the steam-heated exchanger. These heat exchangers are also sensitive to variations in the temperature of the manipulated stream, an increasingly common problem where heat is being recovered at variable temperatures for reuse in heat transfer. Figure 8-50 shows a temperature controller (TC) setting a heatflow controller (QC) in cascade. A measurement of the manipulated flow is multiplied by its temperature difference across the heat exchanger to calculate the current heat-transfer rate, using the right side of Eq. (8-74). Variations in supply temperature, then, appear as variations in calculated heat transfer, which the QC can quickly correct by adjusting the manipulated flow. An equal-percentage valve is still required to linearize the secondary loop, but the primary loop of temperature-setting heat flow is linear. Feedforward can be added by multiplying the dynamically compensated flow measurement of the other fluid by the output of the temperature controller.

UNIT OPERATIONS CONTROL F=D+B

8-33 (8-76)

as well as a balance on each component: Fzi = Dyi + Bxi

(8-77)

where z, y, and x are mol fractions of component i in the respective streams. Combining these equations gives a relationship between the composition of the products and their relative portion of the feed: D B zi − xi (8-78) ᎏ=1−ᎏ=ᎏ F F yi − xi

Heat-transfer rate in sensible-heat exchange varies nonlinearly with flow of the manipulated fluid. FIG. 8-49

Manipulating heat flow linearizes the loop and protects against variations in supply temperature.

FIG. 8-50

When manipulating a stream whose flow is independently determined, such as flow of a product or a heat-transfer fluid from a fired heater, a three-way valve is used to divert the required flow to the heat exchanger. This does not alter the linearity of the process or its sensitivity to supply variations and even adds the possibility of independent flow variations. The three-way valve should have equal-percentage characteristics, and heat-flow control may be even more beneficial. DISTILLATION COLUMN CONTROL Distillation columns have four or more closed loops—increasing with the number of product streams and their specifications—all of which interact with each other to some extent. Because of this interaction, there are many possible ways to pair manipulated and controlled variables through controllers and other mathematical functions with widely differing degrees of effectiveness. Columns also differ from each other, so that no single rule of configuring control loops can be applied successfully to all. The following rules apply to the most common separations. Controlling Quality of a Single Product If one of the products of a column is far more valuable than the others, its quality should be controlled to satisfy given specifications, and its recovery should be maximized by minimizing losses of its principal component in other streams. This is achieved by maximizing reflux ratio consistent with flooding limits on trays, which means maximizing the flow of reflux or vapor, whichever is limiting. The same rule should be followed when heating and cooling have little value. A typical example is the separation of high-purity propylene from much lower-valued propane, usually achieved with waste heat from quench water from the cracking reactors. The most important factor affecting product quality is the material balance. In separating a feed stream F into distillate D and bottom B products, an overall mole-flow balance must be maintained:

From the above, it can be seen that control of either xi or yi requires both product flow rates to change with feed rate and feed composition. Figure 8-51 shows a propylene-propane fractionator controlled at maximum boilup by the differential pressure controller (DPC) across the trays. This loop is fast enough to reject upsets in the temperature of the quench water quite easily. Pressure is controlled by manipulating the heat-transfer surface in the condenser through flooding. If the condenser should become overloaded, pressure will rise above set point, but this has no significant effect on the other control loops. Temperature measurements on this column are not helpful, as the difference between the component boiling points is too small. Propane content in the propylene distillate is measured by a chromatographic analyzer sampling the overhead vapor for fast response and is controlled by the analyzer controller (AC) manipulating the ratio of distillate to feed rates. The feedforward signal from feed rate is dynamically compensated by f(t) and nonlinearly characterized by f(x) to account for variations in propylene recovery as feed rate changes. Distillate flow can be measured and controlled more accurately than reflux flow by a factor equal to the reflux ratio—in this column, typically between 10 and 20. Therefore, reflux flow is placed under accumulator level control (LC). Yet composition responds to the difference between boilup and reflux. To eliminate the lag inherent in the response of the level controller, reflux flow is driven by the subtractor in the direction opposite to distillate flow—this is essential to fast response of the composition loop. Controlling Quality of Two Products Where the two products have similar value, or where heating and cooling costs are comparable to product losses, the compositions of both products should be controlled. This introduces the possibility of strong interaction between the two composition loops, as they tend to have the same speed of response. To minimize interaction, most columns should have distillate composition controlled by reflux ratio and bottom composition by boilup or preferably boilup-to-bottom ratio. These loops are insensitive to variations in feed rate, eliminating the need for feedforward control, and they also reject heat-balance upsets quite effectively. Figure 8-52 shows a depropanizer controlled by reflux and boilup ratios. The actual mechanism through which these ratios are manipulated is as D/(L + D) and B/(V + B), where L is reflux flow and V is vapor boilup, which decouples the temperature loops from the liquid-level loops. Column pressure here is controlled by flooding both condenser and accumulator; however, there is no LC on the accumulator, so this arrangement will not function with an overloaded condenser. Temperatures are used as indications of composition in this column because of the substantial difference in boiling points between propane and butanes. However, off-key components such as ethane do effect the accuracy of the relationship so that an analyzer controller is used to set the top temperature controller (TC) in cascade. If the products from a column are especially pure, even this configuration may produce excessive interaction between the composition loops. Then the composition of the less pure product should be controlled by manipulating its own flow; the composition of the remaining product should be controlled by manipulating reflux ratio if it is the distillate or boilup ratio if it is the bottom product. CHEMICAL REACTORS Composition Control The first requirement for successful control of a chemical reactor is to establish the proper stoichiometry, that is, to control the flow rates of the reactants in the proportions needed

8-34

PROCESS CONTROL

FIG. 8-51

FIG. 8-52

The quality of high-purity propylene should be controlled by manipulating the material balance.

Depropanizers require control of both products, here using reflux-ratio and boilup-ratio manipulation.

UNIT OPERATIONS CONTROL to satisfy the reaction chemistry. In a continuous reactor, this begins by setting ingredient flow rates in ratio to one another. However, because of variations in the purity of the feed streams and inaccuracy in flow metering, some indication of excess reactant such as pH or a composition measurement should be used to trim the ratios. Many reactions are incomplete, leaving one or more reactants unconverted. They are separated from the products of the reaction and recycled to the reactor, usually contaminated with inert components. While reactants can be recycled to complete conversion (extinction), inerts can accumulate to the point of impeding the reaction and must be purged from the system. Inerts include noncondensible gases that must be vented and nonvolatiles from which volatile products must be stripped. If one of the reactants differs in phase from the others and the products, it may be manipulated to close the material balance on that phase. For example, a gas reacting with liquids to produce a liquid product may be added, as it is consumed to control reactor pressure; a gaseous purge would be necessary. Similarly, a liquid reacting with a gas to produce a gaseous product could be added, as it is consumed to control liquid level in the reactor; a liquid purge would be required. Where a large excess of one reactant A is used to minimize side reactions, the unreacted excess is typically sent to a storage tank for recycling. Its flow from the recycle storage tank is set in the desired ratio to the flow of reactant B, with the flow of fresh A manipulated to control recycle tank level if the feed is a liquid or tank pressure if it is a gas. Some catalysts travel with the reactants and must be recycled in the same way. With batch reactors, it may be possible to add all reactants in their proper quantities initially if the reaction rate can be controlled by injection of initiator or adjustment of temperature. In semibatch operation, one key ingredient is flow-controlled into the batch at a rate that sets the production. This ingredient should not be manipulated for temperature control of an exothermic reactor, as the loop includes two dominant lags—concentration of the reactant and heat capacity of the reaction mass—and can easily go unstable. Temperature Control Reactor temperature should always be controlled by heat transfer. Endothermic reactions require heat and therefore are eminently self-regulating. Exothermic reactions produce heat, which tends to raise reaction temperature, thereby increasing reaction rate and producing more heat. This positive feedback is countered by negative feedback in the cooling system, which removes more heat as reactor temperature rises. Most continuous reactors have enough heat-transfer surface relative to reaction mass so that negative feedback dominates and they are self-regulating. But most batch reactors do not and are therefore steady-state unstable. Unstable reactors are controllable, but the temperature controller requires a high gain, and the cooling system must have enough margin to accommodate the largest expected disturbance in heat load. Figure 8-53 shows the recommended system for controlling the temperature of an exothermic reactor, either continuous or batch. The circulating pump on the coolant loop is absolutely essential to effective temperature control in keeping dead time minimum and constant—without it, dead time varies inversely with cooling load, causing limit cycling at low loads. Heating is usually required to raise the temperature to reaction conditions, although it is often locked out in a batch reactor once initiator is introduced. The valves are operated in split range, the heating valve opening from 50–100 percent of controller output, and the cooling valve opening from 0–50 percent. The cascade system linearizes the reactor temperature loop, speeds its response, and protects it from disturbances in the cooling system. The flow of heat removed per unit of coolant flow is directly proportional to the temperature rise of the coolant, which varies with both the temperature of the reactor and the rate of heat transfer from it. Using an equal-percentage cooling valve helps compensate for this nonlinearity, although it is incomplete—a preferred arrangement would be to manipulate coolant flow using a heat-flow controller as described in Fig. 8-50. The flow of heat across the heat-transfer surface is linear with both temperatures, leaving the primary loop with a constant gain. Using the coolant exit rather than inlet temperature as the secondary controlled variable moves the jacket dynamics from the primary to the secondary

8-35

The reactor temperature controller sets coolant outlet temperature in cascade, with primary integral feedback taken from the secondary temperature measurement.

FIG. 8-53

loop, reducing the period of the primary loop. Performance and robustness are both improved by using the secondary temperature measurement as the feedback signal to the integral mode of the primary controller. (This feature may only be available with controllers that integrate by positive feedback.) This places the entire secondary loop in the integral path of the primary controller, effectively pacing its integral time to the rate at which the secondary temperature is able to respond. The primary controller may also be left in the automatic mode at all times without integral windup. The primary time constant of the reactor is MrCr τ1 = ᎏ (8-79) UA where Mr and Cr are the mass and heat capacity of the reactants, and U and A are the overall heat-transfer coefficient and area respectively. This system was tested on a pilot reactor where the heat-transfer area and mass could both be changed by a factor of two, changing τ1 by a factor of four as confirmed by observations of rates of temperature rise. Yet the controllers configured as described in Fig. 8-53 did not require retuning as τ1 varied. The primary controller should be PID, and the secondary controller at least PI in this system; if the secondary controller has no integral mode, the primary controller will control with offset. Set point overshoot in batch reactor control can be avoided by setting derivative time of the primary controller higher than its integral time, but this is only effective with interacting PID controllers. CONTROLLING EVAPORATORS The most important consideration in controlling the quality of concentrate from an evaporator is forcing the vapor rate to match the flow of excess solvent entering in the feed. The mass flow of solid material entering and leaving are equal in the steady state: M0 x0 = Mn xn

(8-80)

where M0 and x0 are the mass flow and solid fraction of the feed, and Mn and xn are their values in the product after n effects of evaporation. The total solvent evaporated from all the effects must then be

W=M

0



x − Mn = M0 1 − ᎏ0 xn



(8-81)

8-36

PROCESS CONTROL

For a steam-heated evaporator, each unit of steam W0 applied produces a known amount of evaporation based on the number of effects and their fractional economy E:

 W = nEW0

(8-82)

(A comparable statement can be made with regard to the power applied to a mechanical recompression evaporator.) In summary, the steam flow required to increase the solid content of the feed from x0 to xn is M0(1 − x0 / xn) W0 = ᎏᎏ (8-83) nE The usual measuring device for feed flow is a magnetic flowmeter, which is a volumetric device whose output F must be multiplied by density ρ to produce mass flow M0. For most aqueous solutions which are fed to evaporators, the product of density and the function of solid content appearing above is linear with density: x Fρ 1 − ᎏ0 ≈ F [1 − m(ρ − 1)] (8-84) xn where slope m is determined by the desired product concentration, and density is in g/ml. The required steam flow in lb/h for feed measured in gal/min is then 500F [1 − m(ρ − 1)] W0 = ᎏᎏᎏ (8-85) nE where the factor of 500 converts gal/min of water to lb/h. The factor nE is about 1.74 for a double-effect evaporator and 2.74 for a tripleeffect. Using a thermocompressor (ejector) driven with 150-lb/in2 steam on a single-effect evaporator gives an nE of 2.05; it essentially adds the equivalent of one effect to the evaporator train. A cocurrent evaporator train with its controls is illustrated in Fig. 8-54. The control system applies equally well to countercurrent or mixed-feed evaporators, the principal difference being the tuning of the dynamic compensator f(t), which must be done in the field to minimize the short-term effects of changes in feed flow on product quality. Solid concentration in the product is usually measured as density; feedback trim is applied by the AC adjusting slope m of the density function, which is the only term related to xn. This recalibrates the system whenever xn must move to a new set point. The accuracy of the system depends on controlling heat flow; therefore, if steam pressure varies, compensation must be applied to cor-



FIG. 8-54



rect for both steam density and enthalpy as a function of pressure. Some evaporators must use unreliable sources of low-pressure steam. In this case, the measurement of pressure-compensated steam flow can be used to set feed flow by solving the last equation for F using W0 as a variable. The steam-flow controller would be set for a given production rate, but the dynamically compensated steam-flow measurement would be the input signal to calculate the feed-flow set point. Both of these configurations are widely used in controlling corn-syrup concentrators. DRYING OPERATIONS Controlling dryers is much different than controlling evaporators because on-line measurements of feed rate and composition and product composition are rarely available. Most dryers transfer moisture from wet feed into hot dry air in a single pass. The process is generally very self-regulating, in that moisture becomes progressively harder to remove from the product as it dries: This is known as fallingrate drying. Controlling the temperature of the air leaving a cocurrent dryer tends to regulate the moisture in the product, as long as feed rate and the moisture in the feed and air are reasonably constant. At constant outlet air temperature, product moisture tends to rise with all three of these variables. In the absence of moisture analyzers, regulation of product quality can be improved by raising the temperature of the exhaust air in proportion to the evaporative load. The evaporative load can be estimated by the loss in temperature of the air passing through the dryer in the steady state. Changes in load are first observed in upsets in exhaust temperature at a given inlet temperature; the controller then responds by returning the exhaust air to its original temperature by changing that of the inlet air. Figure 8-55 illustrates the simplest application of this principal as the linear relationship T0 = Tb + K∆T

(8-86)

where T0 is the set point for exhaust temperature elevated above a base temperature Tb corresponding to zero-load operation, and ∆T is the drop in air temperature from inlet to outlet. Coefficient K must be set to regulate product moisture over the expected range of evaporative load. If set too low, product moisture will increase with increasing load; if set too high, it will decrease with increasing load. While K can be estimated from the model of a dryer, it does depend on the rate-of-

Controlling evaporators requires matching steam flow and evaporative load, here using feedforward control.

BATCH PROCESS CONTROL

FIG. 8-55

8-37

Product moisture from a cocurrent dryer can be regulated through temperature control indexed to heat load.

drying curve for the product, its particle size, and whether the load variations are due primarily to changes in feed rate or feed moisture. It is important to have the most accurate measurement of exhaust temperature attainable. Note that Fig. 8-55 shows the sensor inserted into the dryer upstream of the rotating seal, because leakage there could cause the temperature in the exhaust duct to read low—even lower than the wet-bulb temperature, an impossibility without leakage of either heat or outside air. The calculation of exhaust-temperature set point forms a positivefeedback loop capable of destabilizing the dryer. For example, an increase in load causes the controller to raise inlet temperature, which will in turn raise the calculated set point calling for a further increase in inlet temperature. The gain in the set point loop, K, typically is well below the gain of the exhaust temperature measurement responding to the same change in inlet temperature. Negative feedback then dominates in the steady state, but the response of the exhaust temperature measurement is delayed by the dryer. A similar lag f(t) is shown inserted in the set point loop to prevent positive feedback from dominating in the short term, which could cause cycling. If product moisture is measured off-line, analytical results can be used to adjust K and Tb manually. If an on-line analyzer is used, the analyzer controller would be most effective in adjusting the bias Tb, as shown in the figure. While the rotary dryer shown is commonly used for grains and minerals, this system has been successfully applied to fluid-bed drying of plastic pellets, air-lift drying of wood fibers, and spray drying of milk solids. The air may be steam-heated as shown or heated by direct combustion of fuel, provided that a representative measurement of inlet air temperature can be made. If it cannot, then evaporative load can be inferred from a measurement of fuel flow, replacing ∆T in the set point calculation.

If the feed flows countercurrent to the air, as is the case when drying granulated sugar, exhaust temperature does not respond to variations in product moisture. For these dryers, product moisture can better be regulated by controlling its temperature at the point of discharge. Conveyor-type dryers are usually divided into a number of zones, each separately heated with recirculation of air which raises its wet-bulb temperature. Only the last two zones may require indexing of exhaust-air temperature as a function of ∆T. Batch drying, used on small lots like pharmaceuticals, begins operation by blowing air at constant inlet temperature through saturated product in constant-rate drying, where ∆T is constant at its maximum value ∆Tc. When product moisture reaches the point where fallingrate drying begins, the exhaust temperature begins to rise. The desired product moisture will be reached at a corresponding exhaust temperature Tf, which is related to the temperature Tc observed during constant-rate drying, as well as ∆Tc: Tf = Tc + K∆Tc

(8-87)

The control system requires the values of Tc and ∆Tc observed during the first minutes of operation to be stored as the basis for the above calculation of end point. When the exhaust temperature then reaches the value calculated, drying is terminated. Coefficient K can be estimated from models but requires adjustment on-line to reach product specifications repeatedly. Products having different moisture specifications or particle size will require different settings of K, but the system does compensate for variations in feed moisture, batch size, air moisture, and inlet temperature. Some exhaust air may be recirculated to control the dewpoint of the inlet air, thereby conserving energy toward the end of the batch and when the ambient air is especially dry.

BATCH PROCESS CONTROL BATCH VERSUS CONTINUOUS PROCESSES GENERAL REFERENCES: Fisher, Batch Control Systems: Design, Application, and Implementation, ISA, Research Triangle Park, North Carolina, 1990; Rosenof and Ghosh, Batch Process Automation, Van Nostrand Reinhold, New York, 1987.

When categorizing process plants, the following two extremes can be identified:

1. Commodity plants. These plants are custom-designed to produce large amounts of a single product (or a primary product plus one or more secondary products). An example is a chlorine plant, where the primary product is chlorine and the secondary products are hydrogen and sodium hydroxide. Usually the margins (product value less manufacturing costs) for the products from commodity plants are small, so the plants must be designed and operated for best possible efficiencies. Although a few are batch, most commodity plants are

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PROCESS CONTROL

continuous. Factors such as energy costs are life-and-death issues for such plants. 2. Specialty plants. These plants are capable of producing small amounts of a variety of products. Such plants are common in fine chemicals, pharmaceuticals, foods, and so on. In specialty plants, the margins are usually high, so factors such as energy costs are important but not life-and-death issues. As the production amounts are relatively small, it is not economically feasible to dedicate processing equipment to the manufacture of only one product. Instead, batch processing is utilized so that several products (perhaps hundreds) can be manufactured with the same process equipment. The key issue in such plants is to manufacture consistently each product in accordance with its specifications. The above two categories represent the extremes in process configurations. The term semibatch designates plants in which some processing is continuous but other processing is batch. Even processes that are considered to be continuous can have a modest amount of batch processing. For example, the reformer unit within a refinery is thought of as a continuous process, but the catalyst regeneration is normally a batch process. In a continuous process, the conditions within the process are largely the same from one day to the next. Variations in feed composition, plant utilities (e.g., cooling water temperature), catalyst activities, and other variables occur, but normally these changes are either about an average (e.g., feed compositions) or exhibit a gradual change over an extended period of time (e.g., catalyst activities). Summary data such as hourly averages, daily averages, and the like are meaningful in a continuous process. In a batch process, the conditions within the process are continually changing. The technology for making a given product is contained in the product recipe that is specific to that product. Such recipes normally state the following: 1. Raw material amounts. This is the stuff needed to make the product. 2. Processing instructions. This is what must be done with the stuff in order to make the desired product. This concept of a recipe is quite consistent with the recipes found in cookbooks.

FIG. 8-56

Batch control overview.

Sometimes the term recipe is used to designate only the raw material amounts and other parameters to be used in manufacturing a batch. Although appropriate for some batch processes, this concept is far too restrictive for others. For some products, the differences from one product to the next are largely physical as opposed to chemical. For such products, the processing instructions are especially important. The term formula is more appropriate for the raw material amounts and other parameters, with recipe designating the formula and the processing instructions. The above concept of a recipe permits the following three different categories of batch processes to be identified: 1. Cyclical batch. Both the formula and the processing instructions are the same from batch to batch. Batch operations within processes that are primarily continuous often fall into this category. The catalyst regenerator within a reformer unit is a cyclical batch process. 2. Multigrade. The processing instructions are the same from batch to batch, but the formula can be changed to produce modest variations in the product. In a batch PVC plant, the different grades of PVC are manufactured by changing the formula. In a batch pulp digester, the processing of each batch or cook is the same, but at the start of each cook, the process operator is permitted to change the formula values for chemical-to-wood ratios, cook time, cook temperature, and so on. 3. Flexible batch. Both the formula and the processing instructions can change from batch to batch. Emulsion polymerization reactors are a good example of a flexible batch facility. The recipe for each product must detail both the raw materials required and how conditions within the reactor must be sequenced in order to make the desired product. Of these, the flexible batch is by far the most difficult to automate and requires a far more sophisticated control system than either the cyclical batch or the multigrade batch facility. Batches and Recipes Each batch of product is manufactured in accordance with a product recipe, which contains all information (formula and processing instructions) required to make a batch of the product (see Fig. 8-56). For each batch of product, there will be one and only one product recipe. However, a given product recipe is nor-

BATCH PROCESS CONTROL mally used to make several batches of product. To uniquely identify a batch of product, each batch is assigned a unique identifier called the batch ID. Most companies adopt a convention for generating the batch ID, but this convention varies from one company to the next. In most batch facilities, more than one batch of product will be in some stage of production at any given time. The batches in progress may or may not be using the same recipe. The maximum number of batches that can be in progress at any given time is a function of the equipment configuration for the plant. The existence of multiple batches in progress at a given time presents numerous opportunities for the process operator to make errors, such as charging a material to the wrong batch. Charging a material to the wrong batch is almost always detrimental to the batch to which the material is incorrectly charged. Unless this error is recognized quickly so that the proper charge can be made, the error is also detrimental to the batch to which the charge was supposed to be made. Such errors usually lead to an off-specification batch, but the consequences could be more serious and result in a hazardous condition. Recipe management refers to the assumption of such duties by the control system. Each batch of product is tracked throughout its production, which may involve multiple processing operations on various pieces of processing equipment. Recipe management assures that all actions specified in the product recipe are performed on each batch of product made in accordance with that recipe. As the batch proceeds from one piece of processing equipment to the next, recipe management is also responsible for assuring that the proper type of process equipment is used and that this processing equipment is not currently in use by another batch. By assuming such responsibilities, the control system greatly reduces the incidences where operator error results in off-specification batches. Such a reduction in error is essential to implement just-intime production practices, where each batch of product is manufactured at the last possible moment. When a batch (or batches) are made today for shipment by overnight truck, there is insufficient time for producing another batch to make up for an off-specification batch. Routing and Production Monitoring In some facilities, batches are individually scheduled. However, in most facilities, production is scheduled by product runs, where a run is the production of a stated quantity of a given product. From the stated quantity and the standard yield of each batch, the number of batches can be determined. As this is normally more than one batch of product, a production run is normally a sequence of some number of batches of the same product. In executing a production run, the following issues must be addressed (see Fig. 8-56): 1. Processing equipment must be dedicated to making the run. More than one run is normally in progress at a given time. The maximum number of runs simultaneously in progress depends on the equipment configuration of the plant. Routing involves determining which processing equipment will be used for each production run. 2. Raw material must be utilized. When a production run is scheduled, the necessary raw materials must be allocated to the production run. As the individual batches proceed, the consumption of raw materials must be monitored for consistency with the allocation of raw materials to the production run. 3. The production quantity for the run must be achieved by executing the appropriate number of batches. The number of batches is determined from a standard yield for each batch. However, some batches may achieve yields higher than the standard yield, but other batches may achieve yields lower than the standard yield. The actual yields from each batch must be monitored and significant deviations from the expected yields must be communicated to those responsible for scheduling production. The last two activities are key components of production monitoring, although production monitoring may also involve other activities such as tracking equipment utilization. Production Scheduling In this regard, it is important to distinguish between scheduling runs (sometimes called long-term scheduling) and assigning equipment to runs (sometimes called routing or short-term scheduling). As used herein, production scheduling refers to scheduling runs and is usually a corporate-level as opposed to a

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plant-level function. Short-term scheduling or routing was previously discussed and is implemented at the plant level. The long-term scheduling is basically a material resources planning (MRP) activity involving the following: 1. Forecasting. Orders for long-delivery raw materials are issued at the corporate level based on the forecast for the demand for products. The current inventory of such raw materials is also maintained at the corporate level. This constitutes the resources from which products can be manufactured. 2. The orders for products. Orders are normally received at the corporate level and then assigned to individual plants for production and shipment. Although the scheduling of some products is based on required product inventory levels, scheduling based on orders and shipping directly to the customer (usually referred to as just-in-time) avoids the costs associated with maintaining product inventories. 3. Plant locations and capacities. While producing a product at the nearest plant usually lowers transportation costs, plant capacity limitations sometimes dictate otherwise. Any company competing in the world economy needs the flexibility to accept orders on a worldwide basis and then assign them to individual plants to be filled. Such a function is logically implemented within the corporate-level information technology framework. BATCH AUTOMATION FUNCTIONS Automating a batch facility requires a spectrum of functions. Interlocks Some of these are provided for safety and are properly called safety interlocks. However, others are provided to avoid mistakes in processing the batch and are properly called process interlocks. Discrete Device States Discrete devices such as two-position valves can be driven to either of two possible states. Such devices can be optionally outfitted with limit switches that indicate the state of the device. For two-position valves, the following combinations are possible: 1. No limit switches 2. One limit switch on the closed position 3. One limit switch on the open position 4. Two limit switches In process-control terminology, the discrete device driver is the software routine that generates the output to a discrete device such as a valve and also monitors the state feedback information to ascertain that the discrete device actually attains the desired state. Given the variety of discrete devices used in batch facilities, this logic must include a variety of capabilities. For example, valves do not instantly change states, but instead each valve exhibits a travel time for the change from one state to another. To accommodate this characteristic of the field device, the processing logic within the discrete device driver must provide for a user-specified transition time for each field device. When equipped with limit switches, the potential states for a valve are as follows: 1. Open. The valve has been commanded to open, and the limit switch inputs are consistent with the open state. 2. Closed. The valve has been commanded to close, and the limit switch inputs are consistent with the closed state. 3. Transition. This is a temporary state that is only possible after the valve has been commanded to change state. The limit switch inputs are not consistent with the commanded state, but the transition time has not expired. 4. Invalid. The transition time has expired, and the limit switch inputs are not consistent with the commanded state for the valve. The invalid state is an abnormal condition that is generally handled in a manner similar to process alarms. The transition state is not considered to be an abnormal state but may be implemented in either of the following ways: 1. Drive and wait. Further actions are delayed until the device attains its commanded state. 2. Drive and proceed. Further actions are initiated while the device is in the transition state. The latter is generally necessary for devices with long travel times, such as flush-fitting reactor discharge valves that are motor-driven.

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Closing such valves is normally via drive and wait; however, drive and proceed is usually appropriate when opening the valve. Although two-state devices are most common, the need occasionally arises for devices with three or more states. For example, an agitator may be on high speed, on slow speed, or off. Process States Batch processing usually involves imposing the proper sequence of states on the process. For example, a simple blending sequence might be as follows: 1. Transfer specified amount of material from tank A to tank R. The process state is “Transfer from A.” 2. Transfer specified amount of material from tank B to tank R. The process state is “Transfer from B.” 3. Agitate for specified period of time. The process state is “Agitate without cooling.” 4. Cool (with agitation) to specified target temperature. The process state is “Agitate with cooling.” For each process state, the various discrete devices are expected to be in a specified device state. For process state “Transfer from A,” the device states might be as follows: 1. Tank A discharge valve: open. 2. Tank R inlet valve: open. 3. Tank A transfer pump: running. 4. Tank R agitator: off. 5. Tank R cooling valve: closed. For many batch processes, process state representations are a very convenient mechanism for representing the batch logic. A grid or table can be constructed, with the process states as rows and the discrete device states as columns (or vice versa). For each process state, the state of every discrete device is specified to be one of the following: 1. Device state 0, which may be valve closed, agitator off, and so on 2. Device state 1, which may be valve open, agitator on, and so on 3. No change or don’t care This representation is easily understandable by those knowledgeable about the process technology and is a convenient mechanism for conveying the process requirements to the control engineers responsible for implementing the batch logic. Many batch software packages also recognize process states. A configuration tool is provided to define a process state. With such a mechanism, the batch logic does not need to drive individual devices but can simply command that the desired process state be achieved. The system software then drives the discrete devices to the device states required for the target process state. This normally includes the following: 1. Generating the necessary commands to drive each device to its proper state 2. Monitoring the transition status of each device to determine when all devices have attained their proper states 3. Continuing to monitor the state of each device to assure that the devices remain in their proper states Should any discrete device not remain in its target state, failure logic must be initiated. We will use the control of a simple mixing process (Fig. 8-57) to demonstrate various batch control strategies found in commercial systems. To start the operation sequence, a solenoid valve (VN7) is opened to introduce liquid A. When the liquid level in the tank reaches an intermediate level (LH2), flow B is started to turn on the mixer. When the liquid level is high (LXH2), flow B is stopped and the discharge valve is opened (VN9). The discharge valve is closed and the motor stopped when the tank level reaches the low limit (LL2). The operator may start another mixing cycle by depressing the start button again. It should be noted that this simplified control strategy does not deal with emergency process conditions. Timing of equipment sequencing, such as making sure valve 8 is closed before opening the discharge valve, is not considered. However, this example fully demonstrates the device interlocking and signal latching often encountered in sequential process control. This process is event triggered and can be easily programmed using sequential logic [Figure 8-58a]. Many PLC implementations start the programming phase with sequential logic design. Gate 1 ensures that

FIG. 8-57

Process schematics of a mixing tank.

the process will not start, when requested, if the tank level is not low. Gate 3 opens valve 7 for flow A only if valve 8 is not opened. Gate 2 latches the operator request once valve 7 is opened such that the operator may release the push button. Gate 4 starts flow B and the mixer motor when the intermediate level is reached. The start signal is fed into gate 3 to terminate flow A. At the high tank level, gate 6 opens the discharge valve. This signal is fed into gate 4 to stop flow B and the mixer motor. Gate 5 latches in the discharge signal until the tank is drained. Note that for a DCS, this sequential logic can be entered entirely as Boolean functional blocks. Figure 8-58b is the ladder logic diagram for the same mixing process. It involves rungs of parallel circuits containing relays (the circles) and contacts. Parallel bars on the rungs represent contacts. A slashed pair of bars depict a normally closed contact. A normally open momentary contact is shown on rung 1 in Fig. 8-58b. The ladder logic and diagram builder in PLCs can be programmed easily because there are only a limited number of symbols required in ladder logic diagrams. The translation from sequential logic to ladder logic is straightforward. In general, two or more contacts on the same rung forms an AND gate. Contacts on branches of a rung form an OR gate. For example, contact C1 on rung 1 is normally open, unless the tank level is low. Contact CR8 is normally closed unless relay CR8 on rung 2 is energized. An operator-actuated push button, HS4, and the contact C1 forms an AND gate equivalent to gate 1 in Fig. 8-58a. Therefore, when the operator depresses the push button when the tank level is low, relay CR7 is energized, which closes contact CR7 on branch rung 1A. Once contact CR7 is latched in, the operator may release the button. The junction connecting rungs 1 and 1A is equivalent to the output of the OR gate 2 in Fig. 8-58a. Regulatory Control For most batch processes, the discrete logic requirements overshadow the continuous control requirements. For many batch processes, the continuous control can be provided by simple loops for flow, pressure, level, and temperature. However, very sophisticated advanced control techniques are occasionally applied. As temperature control is especially critical in reactors, the simple feedback approach is replaced by model-based strategies that rival if not exceed the sophistication of advanced control loops in continuous plants. In some installations, alternative approaches for regulatory control

BATCH PROCESS CONTROL

(a)

(b) FIG. 8-58

Logic diagrams for the control of the mixing tank.

may be required. Where a variety of products are manufactured, the reactor may be equipped with alternative heat-removal capabilities, including the following: 1. Jacket filled with cooling water. Most such jackets are oncethrough, but some are recirculating. 2. Heat exchanger in a pump-around loop. 3. Reflux condenser. The heat removal capability to be used usually depends on the product being manufactured. Therefore, regulatory loops must be configured for each possible option, and sometimes for certain combinations of the possible options. These loops are enabled and disabled depending on the product being manufactured. The interface between continuous controls and discrete controls is also important. For example, a feed might be metered into a reactor at a variable rate, depending on another feed or possibly on reactor temperature. However, the product recipe calls for a specified quantity of this feed. The flow must be totalized (i.e., integrated), and when the flow total attains a specified value, the feed must be terminated. The discrete logic must have access to operational parameters such as controller modes. That is, the discrete logic must be able to switch a controller to manual, auto, or cascade. Furthermore, the discrete logic must be able to force the controller output to a specified value. Sequence Logic Sequence logic must not be confused with discrete logic. Discrete logic is especially suitable for interlocks or per-

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missives, e.g., the reactor discharge valve must be closed in order for the feed valve to be opened. Sequence logic is used to force the process to attain the proper sequence of states. For example, a feed preparation might be to first charge A, then charge B, then mix, and finally cool. Although discrete logic can be used to implement sequence logic, other alternatives are often more attractive. Sequence logic is often, but not necessarily, coupled with the concept of a process state. Basically, the sequence logic determines when the process should proceed from the current state to the next, and sometimes what the next state should be. Sequence logic must encompass both normal and abnormal process operations. Thus, sequence logic is often viewed as consisting of two distinct but related parts: 1. Normal logic. This sequence logic provides for the normal or expected progression from one process state to another. 2. Failure logic. This logic provides for responding to abnormal conditions, such as equipment failures. Of these, the failure logic can easily be the most demanding. The simplest approach is to stop or hold on any abnormal condition, and let the process operator sort things out. However, this is not always acceptable. Some failures lead to hazardous conditions that require immediate action; waiting for the operator to decide what to do is not acceptable. The appropriate response to such situations is best determined in conjunction with the process hazards analysis. No single approach has evolved as the preferred way to implement sequence logic. The approaches utilized include the following: 1. Discrete logic. Sequence logic can be implemented via ladder logic, and this approach is common when sequence logic is implemented in programmable logic controllers (PLCs). 2. Programming languages. Traditional procedural languages do not provide the necessary constructs for implementing sequence logic. This necessitates one of the following: a. Special languages. The necessary extensions for sequence logic are provided by extending the syntax of the programming language. This is the most common approach within distributed control systems (DCSs). The early implementations used BASIC as the starting point for the extensions; the later implementations used C as the starting point. A major problem with this approach is portability, especially from one manufacturer to the next but sometimes from one product version to the next within the same manufacturer’s product line. b. Subroutine or function libraries. The facilities for sequence logic are provided via subroutines or functions that can be referenced from programs written in FORTRAN or C. This requires a generalpurpose program development environment and excellent facilities to trap the inevitable errors in such programs. Operating systems with such capabilities have long been available on the larger computers, but not for the microprocessors utilized within DCS systems. However, such operating systems are becoming more common within DCS systems. 3. State machines. This technology is commonly applied within the discrete manufacturing industries. However, its migration to process batch applications has been limited. 4. Graphical implementations. For sequence logic, the flowchart traditionally used to represent the logic of computer programs must be extended to provide parallel execution paths. Such extensions have been implemented in a graphical representation called Grafcet. As process engineers have demonstrated a strong dislike for ladder logic, PLC manufacturers are considering providing Grafcet either in addition to or as an alternative to ladder logic. As none of the above have been able to dominate the industry, it is quite possible that future developments will provide a superior approach for implementing sequence logic. BATCH PRODUCTION FACILITIES Especially for flexible batch applications, the batch logic must be properly structured in order to be implemented and maintained in a reasonable manner. An underlying requirement is that the batch process equipment be properly structured. The following structure is appropriate for most batch production facilities.

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Plant A plant is the collection of production facilities at a geographical site. The production facilities at a site normally share warehousing, utilities, and the like. Equipment Suite An equipment suite is the collection of equipment available for producing a group of products. Normally, this group of products is similar in certain respects. For example, they might all be manufactured from the same major raw materials. Within the equipment suite, material transfer and metering capabilities are available for these raw materials. The equipment suite contains all of the necessary types of processing equipment (reactors, separators, and so on) required to convert the raw materials into salable products. A plant may consist of only one suite of equipment, but large plants usually contain multiple equipment suites. Process Unit or Batch Unit A process unit is a collection of processing equipment that can, at least at certain times, be operated in a manner completely independent from the remainder of the plant. A process unit normally provides a specific function in the production of a batch of product. For example, a process unit might be a reactor complete with all associated equipment (jacket, recirculation pump, reflux condenser, and so on). However, each feed preparation tank is usually a separate process unit. With this separation, preparation of the feed for the next batch can be started as soon as the feed tank is emptied for the current batch. All but the very simplest equipment suites contain multiple process units. The minimum number of process units is one for each type of processing equipment required to make a batch of product. However, many equipment suites contain multiple process units of each type. In such equipment suites, multiple batches and multiple production runs can be in progress at a given time. Item of Equipment An item of equipment is a hardware item that performs a specific purpose. Examples are pumps, heat exchangers, agitators, and the like. A process unit could consist of a single item of equipment, but most process units consist of several items of equipment that must be operated in harmony in order to achieve the function expected of the process unit. Device A device is the smallest element of interest to batch logic. Examples of devices include measurement devices and actuators. STRUCTURED BATCH LOGIC

product must be heated to a specified temperature. Whether heating is undertaken with steam or hot oil is irrelevant to the product technology. By restricting the product recipe to a given product technology, the same product recipe can be used to make products at different sites. Timing diagrams (such as Fig. 8-59) are one way to represent a recipe. At a given site, the specific approach to be used to heat a vessel is important. The traditional approach is for an engineer at each site to expand the product recipe into a document that explains in detail how the product is to be made at the site. This document goes by various names, although standard operating procedure or SOP is a common one. Depending on the level of detail to which it is written, the SOP could specify exactly which valves must be opened in order to heat the contents of a vessel. Thus, the SOP is site-dependent, and contains both product technology and process technology. In structuring the logic for a flexible batch application, the following organization permits product technology to be cleanly separated from process technology: • A recipe consists of a formula and one or more processing operations. Ideally, only product technology is contained in a recipe. • A processing operation consists of one or more phases. Ideally, only product technology is contained in a processing operation. • A phase consists of one or more actions. Ideally, only process technology is contained in a phase. In this structure, the recipe and processing operations would be the same at each site that manufactures the product. However, the logic that comprises each phase would be specific to a given site. Using the heating example from above, each site would require a phase to heat the contents of the vessel. However, the logic within the phase at one site would accomplish the heating by opening the appropriate steam valves, while the logic at the other site would accomplish the heating by opening the appropriate hot oil valves. Usually the critical part of structuring batch logic is the definition of the phases. There are two ways to approach this: 1. Examine the recipes for the current products for commonality, and structure the phases to reflect this commonality. 2. Examine the processing equipment to determine what processing capabilities are possible, and write phases to accomplish each possible processing capability.

Flexible batch applications must be pursued using a structured approach to batch logic. In such applications, the same processing equipment is used to make a variety of products. In most facilities, little or no proprietary technology is associated with the equipment itself; the proprietary technology is how this equipment is used to produce each of the products. The primary objective of the structured approach is to separate cleanly the following two aspects of the batch logic: Product Technology Basically, this encompasses the product technology, such as how to mix certain molecules to make other molecules. This technology ultimately determines the chemical and physical properties of the final product. The product recipe is the principal source for the product technology. Process Technology The process equipment permits certain processing operations (e.g., heat to a specified temperature) to be undertaken. Each processing operation will involve certain actions (e.g., opening appropriate valves). The need to keep these two aspects separated is best illustrated by a situation where the same product is to be made at different plants. While it is possible that the processing equipment at the two plants is identical, this is rarely the case. Suppose one plant uses steam for heating its vessels, but the other uses a hot oil system as the source of heat. When a product recipe requires that material is to be heated to a specified temperature, each plant can accomplish this objective, but will go about it in quite different ways. The ideal case for a product recipe is as follows: 1. Contains all of the product technology required to make a product 2. Contains no equipment-dependent information, that is, no process technology In the previous example, such a recipe would simply state that the

FIG. 8-59

Sample process timing diagram.

PROCESS MEASUREMENTS There is the additional philosophical issue of whether to have a large number of simple phases with few options each, or a small number of complex phases with numerous options. The issues are a little different from structuring a complex computer program into subprograms. Each possible alternative will have advantages and disadvantages. As the phase contains no product technology, the implementation of a phase must be undertaken by those familiar with the process equipment. Furthermore, they should undertake this on the basis that the result will be used to make a variety of products, not just those that are initially contemplated. The development of the phase logic must also encompass all equipment-related safety issues. The phase should accomplish a clearly defined objective, so the implementers should be able to thoroughly consider all relevant issues in accomplishing this objective. The phase logic is defined in detail, implemented in the control system, and then thoroughly tested. Except when the processing equipment is modified, future modifications to the phase should be infrequent. The result should be a very dependable module that can serve as a building block for batch logic.

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Even for flexible batch applications, a comprehensive menu of phases should permit most new products to be implemented using currently existing phases. By reusing exising phases, numerous advantages accrue: 1. The engineering effort to introduce a new recipe at a site is reduced. 2. The product is more likely to be on-spec the first time, thus avoiding the need to dispose of off-spec product. 3. The new product can be supplied to customers sooner, hopefully before competitors can supply the product. There is also a distinct advantage in maintenance. When a problem with a phase is discovered and the phase logic is corrected, the correction is effectively implemented in all recipes that use the phase. If a change is implemented in the processing equipment, the affected phases must be modified accordingly and then thoroughly tested. These modifications are also effectively implemented in all recipes that use these phases.

PROCESS MEASUREMENTS GENERAL REFERENCES: Benedict, Fundamentals of Temperature, Pressure, and Flow Measurements, Wiley, New York, 1969. Considine, Process Instruments and Control Handbook, McGraw-Hill, New York, 1993. Considine and Ross, Handbook of Applied Instrumentation, McGraw-Hill, New York, 1964. Doebelin, Measurement Systems: Application and Design, 4th ed. McGrawHill, New York, 1990. Ginesi and Annarummo, “User Tips for Mass, Volume Flowmeters,” Tech, 41, April, 1994. ISA Transducer Compendium, 2d ed., Plenum, New York, 1969. Liptak, Instrument Engineers Handbook, Chilton, Philadelphia, 1995. Michalski, Eckersdorf, and McGhee, Temperature Measurement, Wiley, Chichester, 1991. Nichols, G. D., On-Line Process Analyzers, Wiley, New York, 1988.

GENERAL CONSIDERATIONS Process measurements encompass the application of the principles of metrology to the process in question. The objective is to obtain values for the current conditions within the process and make this information available in a form usable by either the control system, process operators, or any other entity that needs to know. The term “measured variable” or “process variable” designates the process condition that is being determined. Process measurements fall into two categories: 1. Continuous measurements. An example of a continuous measurement is a level measurement device that determines the liquid level in a tank (in meters). 2. Discrete measurements. An example of a discrete measurement is a level switch that indicates the presence or absence of liquid at the location at which the level switch is installed. In continuous processes, most process control applications rely on continuous measurements. In batch processes, many of the process control applications will utilize discrete as well as continuous measurements. In both types of processes, the safety interlocks and process interlocks rely largely on discrete measurements. Continuous Measurements In most applications, continuous measurements are considerably more ambitious than discrete measurements. Basically, discrete measurements involve a yes/no decision, whereas continuous measurements may entail considerable signal processing. The components of a typical continuous measurement device are as follows: • Sensor. This component produces a signal that is related in a known manner to the process variable of interest. The sensors in use today are primarily of the electrical analog variety, and the signal is in the form of a voltage, a resistance, a capacitance, or some other directly measurable electrical quantity. Prior to the mid 1970s, instruments tended to use sensors whose signal was mechanical in nature, and thus compatible with pneumatic technology. Since that time the

fraction of sensors that are digital in nature has grown considerably, often eliminating the need for analog-to-digital conversion. • Signal processing. The signal from most sensors is related in a nonlinear fashion to the process variable of interest. In order for the output of the measurement device to be linear with respect to the process variable of interest, linearization is required. Furthermore, the signal from the sensor might be affected by variables other than the process variable. In this case, additional variables must be sensed and the signal from the sensor compensated to account for the other variables. For example, reference junction compensation is required for thermocouples (except when used for differential temperature measurements). • Transmitter. The measurement device output must be a signal that can be transmitted over some distance. Where electronic analog transmission is used, the low range on the transmitter output is 4 milliamps, and the upper range is 20 milliamps. Microprocessor-based transmitters (often referred to as smart transmitters) are usually capable of transmitting the measured variable digitally in engineering units. Accuracy and Repeatability Definitions of terminology pertaining to process measurements can be obtained from standard S51.1 from the International Society of Measurment and Control (ISA) and standard RC20-11 from the Scientific Apparatus Manufacturers Association (SAMA), both of which are updated periodically. An appreciation of accuracy and repeatability is especially important. Some applications depend on the accuracy of the instrument, but other applications depend on repeatability. Excellent accuracy implies excellent repeatability; however, an instrument can have poor accuracy but excellent repeatability. In some applications, this is acceptable, as discussed below. Range and Span A continuous measurement device is expected to provide credible values of the measured value between a lower range and an upper range. The difference between the upper range and the lower range is the span of the measurement device. The maximum value for the upper range and the minimum value for the lower range depend on the principles on which the measurement device is based and on the design chosen by the manufacturer of the measurement device. If the measured variable is greater than the upper range or less than the lower range, the measured variable is said to be out-of-range or the measurement device is said to be overranged. Accuracy Accuracy refers to the difference between the measured value and the true value of the measured variable. Unfortunately, the true value is never known, so in practice accuracy refers to the difference between the measured value and an accepted standard value for the measured variable.

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Accuracy can be expressed in a number of ways: 1. As an absolute difference in the units of the measured variable 2. As a percent of the current reading 3. As a percent of the span of the measured variable 4. As a percent of the upper range of the span For process measurements, accuracy as a percent of span is the most common. Manufacturers of measurement devices always state the accuracy of the instrument. However, these statements always specify specific or reference conditions at which the measurement device will perform with the stated accuracy, with temperature and pressure most often appearing in the reference conditions. When the measurement device is applied at other conditions, the accuracy is affected. Manufacturers usually also provide some statements on how accuracy is affected when the conditions of use deviate from the referenced conditions in the statement of accuracy. Although appropriate calibration procedures can minimize some of these effects, rarely can they be totally eliminated. It is easily possible for such effects to cause a measurement device with a stated accuracy of 0.25 percent of span at reference conditions to ultimately provide measured values with accuracies of 1 percent or less. Microprocessor-based measurement devices usually provide better accuracy than the traditional electronic measurement devices. In practice, most attention is given to accuracy when the measured variable is the basis for billing, such as in custody transfer applications. However, whenever a measurement device provides data to any type of optimization strategy, accuracy is very important. Repeatability Repeatability refers to the difference between the measurements when the process conditions are the same. This can also be viewed from the opposite perspective. If the measured values are the same, repeatability refers to the difference between the process conditions. For regulatory control, repeatability is of major interest. The basic objective of regulatory control is to maintain uniform process operation. Suppose that on two different occasions, it is desired that the temperature in a vessel be 80°C. The regulatory control system takes appropriate actions to bring the measured variable to 80°C. The difference between the process conditions at these two times is determined by the repeatability of the measurement device. In the use of temperature measurement for control of the separation in a distillation column, repeatability is crucial but accuracy is not. Composition control for the overhead product would be based on a measurement of the temperature on one of the trays in the rectifying section. A target would be provided for this temperature. However, at periodic intervals, a sample of the overhead product is analyzed in the laboratory and the information provided to the process operator. Should this analysis be outside acceptable limits, the operator would adjust the set point for the temperature. This procedure effectively compensates for an inaccurate temperature measurement; however, the success of this approach requires good repeatability from the temperature measurement. Dynamics of Process Measurements Especially where the measurement device is incorporated into a closed loop control configuration, dynamics are important. The dynamic characteristics depend on the nature of the measurement device, and also on the nature of components associated with the measurement device (for example, thermowells and sample conditioning equipment). The term measurement system designates the measurement device and its associated components. The following dynamics are commonly exhibited by measurement systems: • Time constants. Where there is a capacity and a throughput, the measurement device will exhibit a time constant. For example, any temperature measurement device has a thermal capacity (mass times heat capacity) and a heat flow term (heat transfer coefficient and area). Both the temperature measurement device and its associated thermowell will exhibit behavior typical of time constants. • Dead time. Probably the best example of a measurement device that exhibits pure dead time is the chromatograph, because the analysis is not available for some time after a sample is injected. Additional dead time results from the transportation lag within the sample

system. Even continuous analyzer installations are plagued by dead time from the sample system. • Underdamped behavior. Measurement devices with mechanical components often have a natural harmonic and can exhibit underdamped behavior. The displacer type of level measurement device is capable of such behavior. While the manufacturers of measurement devices can supply some information on the dynamic characteristics of their devices, interpretation is often difficult. Measurement device dynamics are quoted on varying bases, such as rise time, time to 63 percent response, settling time, and so on. Even where the time to 63 percent response is quoted, it might not be safe to assume that the measurement device exhibits first-order behavior. Where the manufacturer of the measurement device does not supply the associated equipment (thermowells, sample conditioning equipment, and the like), the user must incorporate the characteristics of these components to obtain the dynamics of the measurement system. An additional complication is that most dynamic data are stated for configurations involving reference materials such as water, air, and so on. The nature of the process material will affect the dynamic characteristics. For example, a thermowell will exhibit different characteristics when immersed in a viscous organic emulsion than when immersed in water. It is often difficult to extrapolate the available data to process conditions of interest. Similarly, it is often impossible, or at least very difficult, to experimentally determine the characteristics of a measurement system under the conditions where it is used. It is certainly possible to fill an emulsion polymerization reactor with water and determine the dynamic characteristics of the temperature measurement system. However, it is not possible to determine these characteristics when the reactor is filled with the emulsion under polymerization conditions. The primary impact of unfavorable measurement dynamics is on the performance of closed loop control systems. This explains why most control engineers are very concerned with measurement dynamics. The goal to improve the dynamic characteristics of measurement devices is made difficult because the discussion regarding measurement dynamics is often subjective. Selection Criteria The selection of a measurement device entails a number of considerations given below, some of which are almost entirely subjective. 1. Measurement span. The measurement span required for the measured variable must lie entirely within the instrument’s envelope of performance. 2. Performance. Depending on the application, accuracy, repeatability, or perhaps some other measure of performance is appropriate. Where closed loop control is contemplated, speed of response must be included. 3. Reliability. Data available from the manufacturers can be expressed in various ways and at various reference conditions. Often, previous experience with the measurement device within the purchaser’s organization is weighted most heavily. 4. Materials of construction. The instrument must withstand the process conditions to which it is exposed. This encompasses considerations such as operating temperatures, operating pressures, corrosion, and abrasion. For some applications, seals or purges may be necessary. 5. Prior use. For the first installation of a specific measurement device at a site, training of maintenance personnel and purchases of spare parts might be necessary. 6. Potential for releasing process materials to the environment. Fugitive emissions are receiving ever increasing attention. Exposure considerations, both immediate and long term, for maintenance personnel are especially important when the process fluid is either corrosive or toxic. 7. Electrical classification. Article 500 of the National Electric Code provides for the classification of the hazardous nature of the process area in which the measurement device will be installed. If the measurement device is not inherently compatible with this classification, suitable enclosures must be purchased and included in the installation costs.

PROCESS MEASUREMENTS 8. Physical access. Subsequent to installation, maintenance personnel must have physical access to the measurement device for maintenance and calibration. If additional structural facilities are required, they must be included in the installation costs. 9. Cost. There are two aspects of the cost: a. Initial purchase and installation (capital costs). b. Recurring costs (operational expense). This encompasses instrument maintenance, instrument calibration, consumables (for example, titrating solutions must be purchased for automatic titrators), and any other costs entailed in keeping the measurement device in service. Calibration Calibration entails the adjustment of a measurement device so that the value from the measurement device agrees with the value from a standard. The International Standards Organization (ISO) has developed a number of standards specifically directed to calibration of measurement devices. Furthermore, compliance with the ISO 9000 standards requires that the working standard used to calibrate a measurement device must be traceable to an internationally recognized standard such as those maintained by the National Institute of Standards and Technology (NIST). Within most companies, the responsibility for calibrating measurement devices is delegated to a specific department. Often, this department may also be responsible for maintaining the measurement device. The specific calibration procedures depend on the type of measurement device. The frequency of calibration is normally predetermined, but earlier action may be dictated if the values from the measurement device become suspect. Calibration of some measurement devices involves comparing the measured value with the value from the working standard. Pressure and differential pressure transmitters are calibrated in this manner. Calibration of analyzers normally involves using the measurement device to analyze a specially prepared sample whose composition is known. These and similar approaches can be applied to most measurement devices. Flow is an important measurement whose calibration presents some challenges. When a flow measurement device is used in applications such as custody transfer, provision is made to pass a known flow through the meter. However, such a provision is costly and is not available for most in-process flowmeters. Without such a provision, a true calibration of the flow element itself is not possible. For orifice meters, calibration of the flowmeter normally involves calibration of the differential pressure transmitter, and the orifice plate is usually only inspected for deformation, abrasion, and so on. Similarly, calibration of a magnetic flowmeter normally involves calibration of the voltage measurement circuitry, which is analogous to calibration of the differential pressure transmitter for an orifice meter. TEMPERATURE MEASUREMENTS Measurement of the hotness or coldness of a body or fluid is commonplace in the process industries. Temperature-measuring devices utilize systems with properties that vary with temperature in a simple, reproducible manner and thus can be calibrated against known references (sometimes called secondary thermometers). The three dominant measurement devices used in automatic control are thermocouples, resistance thermometers, and pyrometers and are applicable over different temperature regimes. Thermocouples Temperature measurements using thermocouples are based on the discovery by Seebeck in 1821 that an electric current flows in a continuous circuit of two different metallic wires if the two junctions are at different temperatures. The thermocouple may be represented diagrammatically as shown in Fig. 8-60. A and B are the two metals, and T1 and T2 are the temperatures of the junctions. Let T1 and T2 be the reference junction (cold junction) and the measuring junction, respectively. If the thermoelectric current i flows in the direction indicated in Fig. 8-60, metal A is customarily referred to as thermoelectrically positive to metal B. Metal pairs used for thermocouples include platinum-rhodium (the most popular and accurate), chromel-alumel, copper-constantan, and iron-constantan. The thermal emf is a measure of the difference in temperature between T2 and T1. In control systems the reference junction is usually located at

FIG. 8-60

8-45

Basic circuit of Seebeck effect.

the emf-measuring device. The reference junction may be held at constant temperature such as in an ice bath or a thermostated oven, or it may be at ambient temperature but electrically compensated (coldjunction-compensated circuit) so that it appears to be held at a constant temperature. Resistance Thermometers The resistance thermometer depends upon the inherent characteristics of materials to change in electrical resistance when they undergo a change in temperature. Industrial resistance thermometers are usually constructed of platinum, copper, or nickel, and more recently semiconducting materials such as thermistors are being used. Basically, a resistance thermometer is an instrument for measuring electrical resistance that is calibrated in units of temperature instead of in units of resistance (typically ohms). Several common forms of bridge circuits are employed in industrial resistance thermometry, the most common being the Wheatstone bridge. A resistance thermometer detector (RTD) consists of a resistance conductor (metal), which generally shows an increase in resistance with temperature. The following equation represents the variation of resistance with temperature (°C): RT = R0(1 + a1T + a2T 2 + . . . + anT n) R0 = resistance at 0°C

(8-88)

The temperature coefficient of resistance αT is expressed as: 1 dR αT = ᎏ ᎏT (8-89) RT dT For most metals, αT is positive. For many pure metals, the coefficient is essentially constant and stable over large portions of their useful range. Typical resistance versus temperature curves for platinum, copper, and nickel are given in Fig. 8-61, with platinum usually the metal of choice. Platinum has a useful range of −200°C to 800°C, while Nickel (−80°C to 320°C) and copper (−100°C to 100°C) are more limited. Detailed resistance versus temperature tables are available from the National Bureau of Standards and suppliers of resistance thermometers. Table 8-7 gives recommended temperature measurement ranges for thermocouples and RTDs. Resistance thermometers are receiving increased usage because they are about ten times more accurate than thermocouples. Thermistors Thermistors are nonlinear temperature-dependent resistors, and normally only the materials with negative temperature TABLE 8-7 Recommended Temperature Measurement Ranges for RTDs and Thermocouples Resistance thermometer detectors (RTDs) −200°C–+850°C −80°C–+320°C

100V Pt 120V Ni Thermocouples Type B Type E Type J Type K Type N Type R Type S Type T

700°C–+1820°C −175°C–+1000°C −185°C–+1200°C −175°C–+1372°C 0°C–+1300°C 125°C–+1768°C 150°C–+1768°C −170°C–+400°C

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Typical resistance-thermometer curves for platinum, copper, and nickel wire, where RT = resistance at temperature T and R0 = resistance at 0°C.

FIG. 8-61

coefficient of resistance (NTC type) are used. The resistance is related to temperature as: 1 1 RT = RTr exp β ᎏ − ᎏ (8-90) T Tr






where Tr is a reference temperature, which is generally 298 K. Thus 1 dR αT = ᎏ ᎏT (8-91) RT dT The value of β is of the order of 4000, so at room temperature (298 K), αT = −0.045 for thermistor and 0.0035 for 100 Ω Platinum RTD. Compared with RTDs, NTC type thermistors are advantageous in that the detector dimension can be made small, the resistance value is higher (less affected by the resistances of the connecting leads), the temperature sensitivity is higher, and the thermal inertia of the sensor is low. Disadvantages of thermistors to RTDs include nonlinear characteristics and low measuring temperature range. Filled-System Thermometers The filled-system thermometer is designed to provide an indication of temperature some distance removed from the point of measurement. The measuring element (bulb) contains a gas or liquid that changes in volume, pressure, or vapor pressure with temperature. This change is communicated through a capillary tube to a Bourdon tube or other pressure- or volume-sensitive device. The Bourdon tube responds so as to provide a motion related to the bulb temperature. Those systems that respond to volume changes are completely filled with a liquid. Systems that respond to pressure changes either are filled with a gas or are partially filled with a volatile liquid. Changes in gas or vapor pressure with changes in bulb temperatures are carried through the capillary to the Bourdon. The latter bulbs are sometimes constructed so that the capillary is filled with a nonvolatile liquid. Fluid-filled bulbs deliver enough power to drive controller mechanisms and even directly actuate control valves. These devices are characterized by large thermal capacity, which sometimes leads to slow response, particularly when they are enclosed in a thermal well for process measurements. Filled-system thermometers are used extensively in industrial processes for a number of reasons. The simplicity

of these devices allows rugged construction, minimizing the possibility of failure with a low level of maintenance, and inexpensive overall design of control equipment. In case of system failure, the entire unit must be replaced or repaired. As normally used in the process industries, the sensitivity and percentage of span accuracy of these thermometers are generally the equal of those of other temperature-measuring instruments. Sensitivity and absolute accuracy are not the equal of those of short-span electrical instruments used in connection with resistance-thermometer bulbs. Also, the maximum temperature is somewhat limited. Bimetal Thermometers Thermostatic bimetal can be defined as a composite material made up of strips of two or more metals fastened together. This composite, because of the different expansion rates of its components, tends to change curvature when subjected to a change in temperature. With one end of a straight strip fixed, the other end deflects in proportion to the temperature change, the square of the length, and inversely as the thickness, throughout the linear portion of the deflection characteristic curve. If a bimetallic strip is wound into a helix or a spiral and one end is fixed, the other end will rotate when heat is applied. For a thermometer with uniform scale divisions, a bimetal must be designed to have linear deflection over the desired temperature range. Bimetal thermometers are used at temperatures ranging from 580°C down to −180°C and lower. However, at the low temperatures the rate of deflection drops off quite rapidly. Bimetal thermometers do not have long-time stability at temperatures above 430°C. Pyrometers Planck’s distribution law gives the radiated energy flux qb(λ, T)d λ in the wavelength range λ to λ + dλ from a black surface: 1 C1 qb(λ, T) = ᎏ (8-92) ᎏ λ5 ec2 /λT − 1 where C1 = 3.7418 × 1010 µW µm4 cm−2, and C2 = 14,388 µm K. If the target object is a black body and if the pyrometer has a detector that measures the specific wavelength signal from the object, the temperature of the object can be exactly estimated from Eq. (8-92). While it is possible to construct a physical body that closely approxi-

PROCESS MEASUREMENTS

8-47

mates black body behavior, most real-world objects are not black bodies. The deviation from a black body can be described by the spectral emissivity q(T) εT = ᎏ (8-93) qb(T)

the ratio of the radiant intensities of the nonblack body, whose temperature is to be measured at the same wavelength, then Wien’s law gives ελ1 exp−C2/λ1T exp−C2/λ1Tc =ᎏ (8-99) ᎏᎏ ελ2 exp−C2/λ2T exp−C2/λ2Tc

where q(λ, T) is the radiated energy flux from a real body in the wavelength range λ to λ + dλ and 0 < ελ,T < 1. Integrating Eq. (8-92) over all wavelengths gives the Stefan-Boltzmann equation

where T is the true temperature of the body. Rearranging Eq. (8-99) gives

qb(T) =



∞

qb(λ, T) dλ

0

= σT 4

(8-94)

where σ is the Stefan-Boltzmann constant. Similar to Eq. (8-93), the emissivity εT for the total radiation is q(T) εT = ᎏ (8-95) qb(T) where q(T) is the radiated energy flux from a real body with emissivity εT. Total Radiation Pyrometers In total radiation pyrometers, the thermal radiation is detected over a large range of wavelengths from the object at high temperature. The detector is normally a thermopile, which is built by connecting several thermocouples in series to increase the temperature measurement range. The pyrometer is calibrated for black bodies, so the indicated temperature Tp should be converted for non-black body temperature. Photoelectric Pyrometers Photoelectric pyrometers belong to the class of band radiation pyrometers. The thermal inertia of thermal radiation detectors does not permit the measurement of rapidly changing temperatures. For example, the smallest time constant of a thermal detector is about 1 msec, while the smallest time constant of a photoelectric detector can be about 1 or 2 sec. Photoelectric pyrometers may use photoconductors, photodiodes, photovoltaic cells, or vacuum photocells. Photoconductors are built from glass plates with thin film coatings of 1 µm thickness, using PbS, CdS, PbSe or PbTe. When the incident radiation has the same wavelength as the materials are able to absorb, the captured incident photons free photoelectrons, which form an electric current. Photodiodes in germanium or silicon are operated with a reverse bias voltage applied. Under the influence of the incident radiation their conductivity as well as their reverse saturation current is proportional to the intensity of the radiation within the spectral response band from 0.4 to 1.7 µm for Ge and 0.6 to 1.1 µm for Si. Because of the above characteristics, the operating range of a photoelectric pyrometer can be either spectral or in a specific band. Photoelectric pyrometers can be applied for a specific choice of the wavelength. Disappearing Filament Pyrometers Disappearing filament pyrometers can be classified as spectral pyrometers. The brightness of a lamp filament is changed by adjusting the lamp current until the filament disappears against the background of the target, at which point the temperature is measured. Since the detector is the human eye, it is difficult to calibrate for on-line measurements. Ratio Pyrometers The ratio pyrometer is also called the twocolor pyrometer. Two different wavelengths are utilized for detecting the radiated signal. If one uses Wien’s law for small values of λT, the detected signals from spectral radiant energy flux emitted at the wavelengths λ1 and λ2 with emissivities ελ1 and ελ2 are −C2/λ1T Sλ1 = KC1ελ1λ−5 1 exp

(8-96)

−C2/λ2T Sλ2 = KC1ελ2λ−5 2 exp

(8-97)

The ratio of the signals Sλ1 and Sλ2 is ελ1 λ2 Sλ ᎏ1 = ᎏ ᎏ Sλ2 ελ2 λ1

C 1 ᎏ − ᎏ    exp
ᎏ T λ λ 5

1

2

2

(8-98)

1

Nonblack or nongrey bodies are characterized by wavelength dependence of their spectral emissivity. Let Tc be defined as the temperature of the body corresponding to the temperature of a black body. If the ratio of its radiant intensities at the wavelengths λ1, and λ2 equals




ln ελ1/ελ2 1 T = ᎏᎏ +ᎏ 1 1 Tc C2 ᎏᎏ − ᎏᎏ λ1 λ2







−1

(8-100)

For black or grey bodies, Eq. (8-98) reduces to 1 Sλ λ2 5 C2 1 (8-101) ᎏ1 = ᎏ exp ᎏ ᎏ − ᎏ Sλ2 λ1 T λ2 λ1 Thus, by measuring Sλ1 and Sλ2, the temperature T can be estimated. Accuracy of Pyrometers Most of the temperature estimation methods for pyrometers assume that the object is either a grey body or has known emissivity values. The emissivity of the nonblack body depends on the internal state or the surface geometry of the objects. Also, the medium through which the thermal radiation passes is not always transparent. These inherent uncertainties of the emissivity values make the accurate estimation of the temperature of the target objects difficult. Proper selection of the pyrometer and accurate emissivity values can provide a high level of accuracy.

 






PRESSURE MEASUREMENTS Pressure defined as force per unit area is usually expressed in terms of familiar units of weight-force and area or the height of a column of liquid that produces a like pressure at its base. Process pressuremeasuring devices may be divided into three groups: (1) those that are based on the measurement of the height of a liquid column, (2) those that are based on the measurement of the distortion of an elastic pressure chamber, and (3) electrical sensing devices. Liquid-Column Methods Liquid-column pressure-measuring devices are those in which the pressure being measured is balanced against the pressure exerted by a column of liquid. If the density of the liquid is known, the height of the liquid column is a measure of the pressure. Most forms of liquid-column pressure-measuring devices are commonly called manometers. When the height of the liquid is observed visually, the liquid columns are contained in glass or other transparent tubes. The height of the liquid column may be measured in length units or be calibrated in pressure units. Depending on the pressure range, water and mercury are the liquids most frequently used. Since the density of the liquid used varies with temperature, the temperature must be taken into account for accurate pressure measurements. Elastic-Element Methods Elastic-element pressure-measuring devices are those in which the measured pressure deforms some elastic material (usually metallic) within its elastic limit, the magnitude of the deformation being approximately proportional to the applied pressure. These devices may be loosely classified into three types: Bourdon tube, bellows, and diaphragm. Bourdon-Tube Elements Probably the most frequently used process pressure-indicating device is the C-spring Bourdon-tube pressure gauge. Gauges of this general type are available in a wide variety of pressure ranges and materials of construction. Materials are selected on the basis of pressure range, resistance to corrosion by the process materials, and effect of temperature on calibration. Gauges calibrated with pressure, vacuum, compound (combination pressure and vacuum), and suppressed-zero ranges are available. Bellows Element The bellows element is an axially elastic cylinder with deep folds or convolutions. The bellows may be used unopposed, or it may be restrained by an opposing spring. The pressure to be measured may be applied either to the inside or to the space outside the bellows, with the other side exposed to atmospheric pressure. For measurement of absolute pressure either the inside or the space outside of the bellows can be evacuated and sealed. Differential pres-

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sures may be measured by applying the pressures to opposite sides of a single bellows or to two opposing bellows. Diaphragm Elements Diaphragm elements may be classified into two principal types: those that utilize the elastic characteristics of the diaphragm and those that are opposed by a spring or other separate elastic element. The first type usually consists of one or more capsules, each composed of two diaphragms bonded together by soldering, brazing, or welding. The diaphragms are flat or corrugated circular metallic disks. Metals commonly used in diaphragm elements include brass, phosphor bronze, beryllium copper, and stainless steel. Ranges are available from fractions of an inch of water to about 206.8 kPa gauge. The second type of diaphragm is used for containing the pressure and exerting a force on the opposing elastic element. The diaphragm is a flexible or slack diaphragm of rubber, leather, impregnated fabric, or plastic. Movement of the diaphragm is opposed by a spring that determines the deflection for a given pressure. This type of diaphragm is used for the measurement of extremely low pressure, vacuum, or differential pressure. Electrical Methods Strain Gauges When a wire or other electrical conductor is stretched elastically, its length is increased and its diameter is decreased. Both of these dimensional changes result in an increase in the electrical resistance of the conductor. Devices utilizing resistancewire grids for measuring small distortions in elastically stressed materials are commonly called strain gauges. Pressure-measuring elements utilizing strain gauges are available in a wide variety of forms. They usually consist of one of the elastic elements described earlier to which one or more strain gauges have been attached to measure the deformation. There are two basic strain-gauge forms: bonded and unbonded. Bonded strain gauges are those which are bonded directly to the surface of the elastic element whose strain is to be measured. The unbonded-strain-gauge transducer consists of a fixed frame and an armature which moves with respect to the frame in response to the measured pressure. The strain-gauge wire filaments are stretched between the armature and frame. The strain gauges are usually connected electrically in a Wheatstone-bridge configuration. Strain-gauge pressure transducers are manufactured in many forms for measuring gauge, absolute, and differential pressures and vacuum. Full-scale ranges from 25.4 mm of water to 10,134 MPa are available. Strain gauges bonded directly to a diaphragm pressure-sensitive element usually have an extremely fast response time and are suitable for high-frequency dynamic-pressure measurements. Piezoresistive Transducers A variation of the conventional strain-gauge pressure transducer uses bonded single-crystal semiconductor wafers, usually silicon, whose resistance varies with strain or distortion. Transducer construction and electrical configurations are similar to those using conventional strain gauges. A permanent magnetic field is applied perpendicular to the resonating sensor. An AC current causes the resonator to vibrate, and the resonant frequency is a function of the pressure (tension) of the resonator. The principal advantages of piezoresistive transducers are a much higher bridge voltage output and smaller size. Full-scale output voltages of 50 to 100 mV/V of excitation are typical. Some newer devices provide digital rather than analog output. Piezoelectric Transducers Certain crystals produce a potential difference between their surfaces when stressed in appropriate directions. Piezoelectric pressure transducers generate a potential difference proportional to a pressure-generated stress. Because of the extremely high electrical impedance of piezoelectric crystals at low frequency, these transducers are usually not suitable for measurement of static process pressures. FLOW MEASUREMENTS Flow, defined as volume per unit of time at specified temperature and pressure conditions, is generally measured by positive-displacement or rate meters. The term “positive-displacement meter” applies to a device in which the flow is divided into isolated measured volumes when the number of fillings of these volumes is counted in some man-

ner. The term “rate meter” applies to all types of flowmeters through which the material passes without being divided into isolated quantities. Movement of the material is usually sensed by a primary measuring element that activates a secondary device. The flow rate is then inferred from the response of the secondary device by means of known physical laws or from empirical relationships. The principal classes of flow-measuring instruments used in the process industries are variable-head, variable-area, positive-displacement, and turbine instruments, mass flowmeters, vortex-shedding and ultrasonic flowmeters, magnetic flowmeters, and more recently, Coriolis mass flowmeters. Head meters are covered in more detail in Sec. 5. Orifice Meter The most widely used flowmeter involves placing a fixed-area flow restriction (an orifice) in the pipe carrying the fluid. This flow restriction causes a pressure drop that can be related to flow rate. The sharp-edge orifice is popular because of its simplicity, low cost, and the large amount of research data on its behavior. For the orifice meter, the flow rate Qa for a liquid is given by CdA2 Qa = ᎏᎏ ⋅ −A (2A /1)2 1

2(p − p ) ᎏᎏ  ρ 1

2

(8-102)

where p1 − p2 is the pressure drop, ρ is the density, A1 is the pipe crosssectional area, A2 is the orifice cross-sectional area, and Cd is the discharge coefficient. The discharge coefficient Cd varies with the Reynolds number at the orifice and can be calibrated with a single fluid, such as water (typically Cd ≈ 0.6). If the orifice and pressure taps are constructed according to certain standard dimensions, quite accurate (about 0.4 to 0.8 percent error) values of Cd may be obtained. It should also be noted that the standard calibration data assume no significant flow disturbances such as elbows, valves, and so on, for a certain minimum distance upstream of the orifice. The presence of such disturbances close to the orifice can cause errors of as much as 15 percent. Accuracy in measurements limits the meter to a range of 3:1. The orifice has a relatively large permanent pressure loss that must be made up by the pumping machinery. Venturi Meter The venturi tube operates on exactly the same principle as the orifice [see Eq. (8-102)]. Discharge coefficients of venturis are larger than those for orifices and vary from about 0.94 to 0.99. A venturi gives a definite improvement in power losses over an orifice and is often indicated for measuring very large flow rates, where power losses can become economically significant. The initial higher cost of a venturi over an orifice may thus be offset by reduced operating costs. Rotameter A rotameter consists of a vertical tube with a tapered bore in which a float changes position with the flow rate through the tube. For a given flow rate the float remains stationary since the vertical forces of differential pressure, gravity, viscosity, and buoyancy are balanced. The float position is the output of the meter and can be made essentially linear with flow rate by making the tube area vary linearly with the vertical distance. Turbine Meter If a turbine wheel is placed in a pipe containing a flowing fluid, its rotary speed depends on the flow rate of the fluid. A turbine can be designed whose speed varies linearly with flow rate. The speed can be measured accurately by counting the rate at which turbine blades pass a given point, using magnetic pickup to produce voltage pulses. By feeding these pulses to an electronic pulse-rate meter, one can measure flow rate by summing the pulses during a timed interval. Turbine meters are available with full-scale flow rates ranging from about 0.1 to 30,000 gpm for liquids and 0.1 to 15,000 ft3/min for air. Nonlinearity can be less than 0.05 percent in the larger sizes. Pressure drop across the meter varies with the square of flow rate and is about 3 to 10 psi at full flow. Turbine meters can follow flow transients quite accurately since their fluid/mechanical time constant is of the order of 2 to 10 msec. Vortex-Shedding Flowmeters These flowmeters take advantage of vortex shedding, which occurs when a fluid flows past a nonstreamlined object (a blunt body). The flow cannot follow the shape of the object and separates from it, forming turbulent vortices or eddies at the object’s side surfaces. As the vortices move downstream, they grow in size and are eventually shed or detached from the object.

PROCESS MEASUREMENTS Shedding takes place alternately at either side of the object, and the rate of vortex formation and shedding is directly proportional to the volumetric flow rate. The vortices are counted and used to develop a signal linearly proportional to the flow rate. The digital signals can easily be totaled over an interval of time to yield the flow rate. Accuracy can be maintained regardless of density, viscosity, temperature, or pressure when the Reynolds number is greater than 10,000. There is usually a low flow cutoff point below which the meter output is clamped at zero. This flowmeter is recommended for use with relatively clean, low viscosity liquids, gases, and vapors, and rangeability of 10:1 to 20:1 is typical. A sufficient length of straight-run pipe is necessary to prevent distortion in the fluid velocity profile. Ultrasonic Flowmeters All ultrasonic flowmeters are based upon the variable time delays of received sound waves that arise when a flowing liquid’s rate of flow is varied. Two fundamental measurement techniques, depending upon liquid cleanliness, are generally used. In the first technique, two opposing transducers are inserted in a pipe so that one transducer is downstream from the other. These transducers are then used to measure the difference between the velocity at which the sound travels with the direction of flow and the velocity at which it travels against the direction of flow. The differential velocity is measured either by (1) direct time delays using sound wave burst or (2) frequency shifts derived from beat-together, continuous signals. The frequency-measurement technique is usually preferred because of its simplicity and independence of the liquid static velocity. A relatively clean liquid is required to preserve the uniqueness of the measurement path. In the second technique, the flowing liquid must contain scatters in the form of particles or bubbles that will reflect the sound waves. These scatters should be traveling at the velocity of the liquid. A Doppler method is applied by transmitting sound waves along the flow path and measuring the frequency shift in the returned signal from the scatters in the process fluid. This frequency shift is proportional to liquid velocity. Magnetic Flowmeters The principle behind these flowmeters is Faraday’s law of electromagnetic inductance. The magnitude of the voltage induced in a conductive medium moving at right angles through a magnetic field is directly proportional to the product of the magnetic flux density, the velocity of the medium, and the path length between the probes. A minimum value of fluid conductivity is required to make this approach viable. The pressure of multiple phases or undissolved solids can affect the accuracy of the measurement if the velocities of the phases are different than that for straightrun pipe. Magmeters are very accurate over wide flow ranges and are especially accurate at low flow rates. Typical applications include metering viscous fluids, slurries, or highly corrosive chemicals. Because magmeters should be filled with fluid, the preferred installation is in vertical lines with flow going upwards. However, magmeters can be used in tight piping schemes where it is impractical to have long pipe runs, typically requiring lengths equivalent to five or more pipe diameters. Coriolis Mass Flowmeters Coriolis mass flowmeters utilize a vibrating tube in which Coriolis acceleration of a fluid in a flow loop can be created and measured. They can be used with virtually any liquid and are extremely insensitive to operating conditions, with high pressure over ranges of 100:1. These meters are more expensive than volumetric meters and range in size from g to 6 inches. Due to the circuitous path of flow through the meter, Coriolis flowmeters exhibit higher than average pressure changes. The meter should be installed so that it will remain full of fluid, with the best installation in a vertical pipe with flow going upward. There is no Reynolds number limitation with this meter, and it is quite insensitive to velocity profile distortions and swirl, hence there is no requirement for straight piping upstream. LEVEL MEASUREMENTS The measurement of level can be defined as the determination of the location of the interface between two fluids, separable by gravity, with respect to a fixed datum plane. The most common level measurement is that of the interface between a liquid and a gas. Other level mea-

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surements frequently encountered are the interface between two liquids, between a granular or fluidized solid and a gas, and between a liquid and its vapor. A commonly used basis for classification of level devices is as follows: float-actuated, displacer, and head devices, and a miscellaneous group that depends mainly on fluid characteristics. Float-Actuated Devices Float-actuated devices are characterized by a buoyant member that floats at the interface between two fluids. Since a significant force is usually required to move the indicating mechanism, float-actuated devices are generally limited to liquid-gas interfaces. By properly weighting the float, they can be used to measure liquid-liquid interfaces. Float-actuated devices may be classified on the basis of the method used to couple the float motion to the indicating system as discussed below. Chain or Tape Float Gauge In these types of gauges, the float is connected to the indicating mechanism by means of a flexible chain or tape. These gauges are commonly used in large atmospheric storage tanks. The gauge-board type is provided with a counterweight to keep the tape or chain taut. The tape is stored in the gauge head on a spring-loaded reel. The float is usually a pancake-shaped hollow metal float with guide wires from top to bottom of the tank to constrain it. Lever and Shaft Mechanisms In pressurized vessels, floatactuated lever and shaft mechanisms are frequently used for level measurement. This type of mechanism consists of a hollow metal float and lever attached to a rotary shaft, which transmits the float motion to the outside of the vessel through a rotary seal. Magnetically Coupled Devices A variety of float-actuated level devices that transmit the float motion by means of magnetic coupling have been developed. Typical of this class of devices are magnetically operated level switches and magnetic-bond float gauges. A typical magnetic-bond float gauge consists of a hollow magnet-carrying float that rides along a vertical nonmagnetic guide tube. The follower magnet is connected and drives an indicating dial similar to that on a conventional tape float gauge. The float and guide tube are in contact with the measured fluid and come in a variety of materials for resistance to corrosion and to withstand high pressures or vacuum. Weighted floats for liquid-liquid interfaces are available. Head Devices A variety of devices utilize hydrostatic head as a measure of level. As in the case of displacer devices, accurate level measurement by hydrostatic head requires an accurate knowledge of the densities of both heavier-phase and lighter-phase fluids. The majority of this class of systems utilize standard-pressure and differential-pressure measuring devices. Bubble-Tube Systems The commonly used bubble-tube system sharply reduces restrictions on the location of the measuring element. In order to eliminate or reduce variations in pressure drop due to the gas flow rate, a constant differential regulator is commonly employed to maintain a constant gas flow rate. Since the flow of gas through the bubble tube prevents entry of the process liquid into the measuring system, this technique is particularly useful with corrosive or viscous liquids, liquids subject to freezing, and liquids containing entrained solids. Electrical Methods Two electrical characteristics of fluids— conductivity and dielectric constant—are frequently used to distinguish between two phases for level-measurement purposes. An application of electrical conductivity is the fixed-point level detection of a conductive liquid such as high and low water levels. A voltage is applied between two electrodes inserted into the vessel at different levels. When both electrodes are immersed in the liquid, a current flows. Capacitance-type level measurements are based on the fact that the electrical capacitance between two electrodes varies with the dielectric constant of the material between them. A typical continuous level-measurement system consists of a rod electrode positioned vertically in a vessel, the other electrode usually being the metallic vessel wall. The electrical capacitance between the electrodes is a measure of the height of the interface along the rod electrode. The rod is usually conductively insulated from process fluids by a coating of plastic. The dielectric constant of most liquids and solids is markedly higher than that of gases and vapors. The dielectric constant of water and other polar liquids is also higher than that of hydrocarbons and other nonpolar liquids.

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Thermal Methods Level-measuring systems may be based on the difference in thermal characteristics between the fluids, such as temperature or thermal conductivity. A fixed-point level sensor based on the difference in thermal conductivity between two fluids consists of an electrically heated thermistor inserted into the vessel. The temperature of the thermistor and consequently its electrical resistance increase as the thermal conductivity of the fluid in which it is immersed decreases. Since the thermal conductivity of liquids is markedly higher than that of vapors, such a device can be used as a point level detector for liquid-vapor interface. Sonic Methods A fixed-point level detector based on sonicpropagation characteristics is available for detection of a liquid-vapor interface. This device uses a piezoelectric transmitter and receiver, separated by a short gap. When the gap is filled with liquid, ultrasonic energy is transmitted across the gap, and the receiver actuates a relay. With a vapor filling the gap, the transmission of ultrasonic energy is insufficient to actuate the receiver. PHYSICAL PROPERTY MEASUREMENTS Physical-property measurements are sometimes equivalent to composition analyzers, because the composition can frequently be inferred from the measurement of a selected physical property. Density and Specific Gravity For binary or pseudobinary mixtures of liquids or gases or a solution of a solid or gas in a solvent, the density is a function of the composition at a given temperature and pressure. Specific gravity is the ratio of the density of a noncompressible substance to the density of water at the same physical conditions. For nonideal solutions, empirical calibration will give the relationship between density and composition. Several types of measuring devices are described below. Liquid Column Density may be determined by measuring the gauge pressure at the base of a fixed-height liquid column open to the atmosphere. If the process system is closed, then a differential pressure measurement is made between the bottom of the fixed height liquid column and the vapor over the column. If vapor space is not always present, the differential-pressure measurement is made between the bottom and top of a fixed-height column with the top measurement being made at a point below the liquid surface. Displacement There are a variety of density-measurement devices based on displacement techniques. A hydrometer is a constant-weight, variable-immersion device. The degree of immersion, when the weight of the hydrometer equals the weight of the displaced liquid, is a measure of the density. The hydrometer is adaptable to manual or automatic usage. Another modification includes a magnetic float suspended below a solenoid, the varying magnetic field maintaining the float at a constant distance from the solenoid. Change in position of the float, resulting from a density change, excites an electrical system which increases or decreases the current through the solenoid. Direct Mass Measurement One type of densitometer measures the natural vibration frequency and relates the amplitude to changes in density. The density sensor is a U-shaped tube held stationary at its node points and allowed to vibrate at its natural frequency. At the curved end of the U is an electrochemical device that periodically strikes the tube. At the other end of the U, the fluid is continuously passed through the tube. Between strikes, the tube vibrates at its natural frequency. The frequency changes directly in proportion to changes in density. A pickup device at the curved end of the U measures the frequency and electronically determines the fluid density. This technique is useful because it is not affected by the optical properties of the fluid. However, particulate matter in the process fluid can affect the accuracy. Radiation-Density Gauges Gamma radiation may be used to measure the density of material inside a pipe or process vessel. The equipment is basically the same as for level measurement, except that here the pipe or vessel must be filled over the effective, irradiated sample volume. The source is mounted on one side of the pipe or vessel and the detector on the other side with appropriate safety radiation shielding surrounding the installation. Cesium 137 is used as the radi-

ation source for path lengths under 610 mm (24 in) and cobalt 60 above 610 mm. The detector is usually an ionization gauge. The absorption of the gamma radiation is a function of density. Since the absorption path includes the pipe or vessel walls, an empirical calibration is used. Appropriate corrections must be made for the source intensity decay with time. Viscosity Continuous viscometers generally measure either the resistance to flow or the drag or torque produced by movement of an element (moving surface) through the fluid. Each installation is normally applied over a narrow range of viscosities. Empirical calibration over this range allows use on both newtonian and nonnewtonian fluids. One such device uses a piston inside a cylinder. The hydrodynamic pressure of the process fluid raises the piston to a preset height. Then the inlet valve closes and the piston is allowed to free-fall, and the time of travel (typically a few seconds) is a measure of viscosity. Other geometries include the rotation of a spindle inside a sample chamber and a vibrating probe immersed in the fluid. Because viscosity depends on temperature, the viscosity measurement must be thermostated with a heater or cooler. Refractive-Index When light travels from one medium (e.g., air or glass) into another (e.g., a liquid), it undergoes a change of velocity and, if the angle of incidence is not 90°, a change of direction. For a given interface, angle, temperature, and wavelength of light the amount of deviation or refraction will depend on the composition of the liquid. If the sample is transparent, the normal method is to measure the refraction of light transmitted through the glass-sample interface. If the sample is opaque, the reflectance near the critical angle at a glass-sample interface is measured. In an on-line refractometer, the process fluid is separated from the optics by a prism material. A beam of light is focused on a point in the fluid which creates a conic section of light at the prism, striking the fluid at different angles (greater than or less than the critical angle). The critical angle depends on the species concentrations; as the critical angle changes, the proportions of reflected and refracted light change. A photodetector produces a voltage signal proportional to the light refracted, when compared to a reference signal. Refractometers can be used with opaque fluids and in streams that contain particulates. Dielectric Constant The dielectric constant of material represents its ability to reduce the electric force between two charges separated in space. This property is useful in process control for polymers, ceramic materials, and semiconductors. Dielectric constants are measured with respect to vacuum (1.0); typical values range from 2 (benzene) to 33 (methanol) to 80 (water). The value for water is higher than for most plastics. A measuring cell is made of glass or some other insulating material and is usually doughnut-shaped, with the cylinders coated with metal, which constitute the plates of the capacitor. Thermal Conductivity All gases and vapor have the ability to conduct heat from a heat source. At a given temperature and physical environment, radiation, and convection heat losses will be stabilized and the temperature of the heat source will be mainly dependent on the thermal conductivity and thus the composition of the surrounding gases. Thermal-conductivity analyzers normally consist of a sample cell and a reference cell, each containing a combined heat source and detector. These cells are normally contained in a metal block with two small cavities in which the detectors are mounted. The sample flows through the sample-cell cavity past the detector. The reference cell is an identical cavity with a detector through which a known gas flows. The combined heat source and detectors are normally either wire filaments or thermistors heated by a constant current. Since their resistance is a function of temperature, the sample-detector resistance will vary with sample composition while the reference-detector resistance will remain constant. The output from the detector bridge will be a function of sample composition. CHEMICAL COMPOSITION ANALYZERS A number of composition analyzers used for process monitoring and control require chemical conversion of one or more sample components preceding quantitative measurement. These reactions include

PROCESS MEASUREMENTS formation of suspended solids for turbidimetric measurement, formation of colored materials for colorimetric detection, selective oxidation or reduction for electrochemical measurement, and formation of electrolytes for measurement by electrical conductance. Some nonvolatile materials may be separated and measured by gas chromatography after conversion into volatile derivatives. Chromatographic Analyzers Chromatographic analyzers are widely used for the separation and measurement of volatile compounds and of compounds that can be quantitatively converted into volatile derivatives. These materials are separated by placing a portion of the sample in a chromatographic column and carrying the compounds through the column with a gas stream. As a result of the different affinities of the sample components for the column packing, the compounds emerge successively as binary mixtures with the carrier gas. A detector at the column outlet measures some physical property which can be related to the concentrations of the compounds in the carrier gas. Both the concentration peak height and the peak height-time integral, (i.e., peak area) can be related to the concentration of the compound in the original sample. The two detectors most commonly used for process chromatographs are the thermalconductivity detector and the hydrogen-flame ionization detector. Thermal-conductivity detectors, discussed earlier, require calibration for the thermal response of each compound. Hydrogen-flame ionization detectors are more complicated than thermal-conductivity detectors but are capable of 100 to 10,000 times greater sensitivity for hydrocarbons and organic compounds. For ultrasensitive detection of trace impurities, carrier gases must be specially purified. Infrared Analyzers Many gaseous and liquid compounds absorb infrared radiation to some degree. The degree of absorption at specific wavelengths depends on molecular structure and concentration. There are two common detector types for nondispersive infrared analyzers. These analyzers normally have two beams of radiation, an analyzing and a reference beam. One type of detector consists of two gas-filled cells separated by a diaphragm. As the amount of infrared energy absorbed by the detector gas in one cell changes, the cell pressure changes. This causes movement in the diaphragm, which in turn causes a change in capacitance between the diaphragm and a reference electrode. This change in electrical capacitance is measured as the output. The second type of detector consists of two thermopiles or two bolometers, one in each of the two radiation beams. The infrared radiation absorbed by the detector is measured by a differential thermocouple output or a resistance-thermometer (bolometer) bridge circuit. With gas-filled detectors, a chopped light system is normally used in which one side of the detector sees the source through the analyzing beam and the other side the reference beam, alternating at a frequency of a few hertz. Ultraviolet and Visible-Radiation Analyzers Many gas and liquid compounds absorb radiation in the near-ultraviolet or visible region. For example, organic compounds containing aromatic and carbonyl structural groups are good absorbers in the ultraviolet region. Also many inorganic salts and gases absorb in the ultraviolet or visible region. In contrast, straight-chain and saturated hydrocarbons, inert gases, air, and water vapor are essentially transparent. Process analyzers are designed to measure the absorbance in a particular wavelength band. The desired band is normally isolated by means of optical filters. When the absorbance is in the visible region, the term “colorimetry” is used. A phototube is the normal detector. Appropriate optical filters are used to limit the energy reaching the detector to the desired level and the desired wavelength region. Since absorption by the sample is logarithmic if a sufficiently narrow wavelength region is used, an exponential amplifier is sometimes used to compensate and produce a linear output. Paramagnetism A few gases including O2, NO, and NO2 exhibit paramagnetic properties as a result of unpaired electrons. In a nonuniform magnetic field, paramagnetic gases, because of their magnetic susceptibility, tend to move toward the strongest part of the field, thus displacing diamagnetic gases. Paramagnetic susceptibility of these gases decreases with temperature. These effects permit measurement of the concentration of the strongest paramagnetic gas, oxy-

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gen. This analyzer used a dumbbell suspended in the magnetic field which is repelled or attracted toward the magnetic field depending on the magnetic susceptibility of the gas. ELECTROANALYTICAL INSTRUMENTS Conductometric Analysis Solutions of electrolytes in ionizing solvents (e.g., water) conduct current when an electrical potential is applied across electrodes immersed in the solution. Conductance is a function of ion concentration, ionic charge, and ion mobility. Conductance measurements are ideally suited for measurement of the concentration of a single strong electrolyte in dilute solutions. At higher concentrations, conductance becomes a complex, nonlinear function of concentration requiring suitable calibration for quantitative measurements. Measurement of pH The primary detecting element in pH measurement is the glass electrode. A potential is developed at the pH-sensitive glass membrane as a result of differences in hydrogen ion activity in the sample and a standard solution contained within the electrode. This potential measured relative to the potential of the reference electrode gives a voltage that is expressed as pH. Instrumentation for pH measurement is among the most widely used process-measurement devices. Rugged electrode systems and highly reliable electronic circuits have been developed for this use. After installation, the majority of pH measurement problems are sensor-related, mostly on the reference side, including junction plugging, poisoning, and depletion of electrolyte. For the glass (measuring electrode), common difficulties are broken or cracked glass, coating, and etching or abrasion. Symptoms such as drift, sluggish response, unstable readings, and inability to calibrate are indications of measurement problems. On-line diagnostics such as impedance measurements, wiring checks, and electrode temperature are now available in most instruments. Other characteristics that can be measured off-line include efficiency or slope and asymmetry potential (offset), which indicate whether the unit should be cleaned or changed [Nichols, Chem. Engr. Prog., 90(12), 64, 1994; McMillan, Chem. Engr. Prog., 87(12), 30, 1991]. Specific-Ion Electrodes In addition to the pH glass electrode specific for hydrogen ions, a number of electrodes that are selective for the measurement of other ions have been developed. This selectivity is obtained through the composition of the electrode membrane (glass, polymer, or liquid-liquid) and the composition of the electrode. These electrodes are subject to interference from other ions, and the response is a function of the total ionic strength of the solution. However, electrodes have been designed to be highly selective for specific ions, and when properly used, these provide valuable process measurements. MOISTURE MEASUREMENT Moisture measurements are important in the process industries because moisture can foul products, poison reactions, damage equipment, or cause explosions. Moisture measurements include both absolute-moisture methods and relative-humidity methods. The absolute methods are those that provide a primary output that can be directly calibrated in terms of dew-point temperature, molar concentration, or weight concentration. Loss of weight on heating is the most familiar of these methods. The relative-humidity methods are those that provide a primary output that can be more directly calibrated in terms of percentage of saturation of moisture. Dew-Point Method For many applications, the dew point is the desired moisture measurement. When concentration is desired, the relation between water content and dew point is well-known and available. The dew-point method requires an inert surface whose temperature can be adjusted and measured, a sample gas stream flowing past the surface, a manipulated variable for adjusting the surface temperature to the dew point, and a means of detecting the onset of condensation. Although the presence of condensate can be detected electrically, the original and most often used method is the optical detection of

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change in light reflection from an inert metallic-surface mirror. Some instruments measure the attenuation of reflected light at the onset of condensation. Others measure the increase of light dispersed and scattered by the condensate instead of, or in addition to, the reflectedlight measurement. Surface cooling is obtained with an expendable refrigerant liquid, conventional mechanical refrigeration, or thermoelectric cooling. Surface-temperature measurement is usually made with a thermocouple or a thermistor. Piezoelectric Method A piezoelectric crystal in a suitable oscillator circuit will oscillate at a frequency dependent on its mass. If the crystal has a stable hygroscopic film on its surface, the equivalent mass of the crystal varies with the mass of water sorbed in the film. Thus, the frequency of oscillation depends on the water in the film. The analyzer contains two such crystals in matched oscillator circuits. Typically, valves alternately direct the sample to one crystal and a dry gas to the other on a 30-s cycle. The oscillator frequencies of the two circuits are compared electronically, and the output is the difference between the two frequencies. This output is then representative of the moisture content of the sample. The output frequency is usually converted to a variable DC voltage for meter readout and recording. Multiple ranges are provided for measurement from about 1 ppm to near saturation. The dry reference gas is preferably the same as the sample except for the moisture content of the sample. Other reference gases which are adsorbed in a manner similar to the dried sample gas may be used. The dry gas is usually supplied by an automatic dryer. The method requires a vapor sample to the detector. Mist striking the detector destroys the accuracy of measurement until it vaporizes or is washed off the crystals. Water droplets or mist may destroy the hygroscopic film, thus requiring crystal replacement. Vaporization or gasliquid strippers may sometimes be used for the analysis of moisture in liquids. Capacitance Method Several analyzers utilize the high dielectric constant of water for its detection in solutions. The alternating electric current through a capacitor containing all or part of the sample between the capacitor plates is measured. Selectivity and sensitivity are enhanced by increasing the concentration of moisture in the cell by filling the capacitor sample cell with a moisture-specific sorbent as part of the dielectric. This both increases the moisture content and reduces the amount of other interfering sample components. Granulated alumina is the most frequently used sorbent. These detectors may be cleaned and recharged easily and with satisfactory reproducibility if the sorbent itself is uniform. Oxide Sensors Aluminum oxide can be used as a sensor for moisture analysis. A conductivity cell has one electrode node of aluminum, which is anodized to form a thin film of aluminum oxide, followed by coating with a thin layer of gold (the opposite electrode). Moisture is selectively adsorbed through the gold layer and into the hygroscopic aluminum oxide layer, which in turn determines the electrical conductivity between gold and aluminum oxide. This value can be related to ppm water in the sample. This sensor can operate between near vacuum to several hundred atmospheres, and it is independent of flow rate (including static conditions). Temperature, however, must be carefully monitored. A similar device is based on phosphorous pentoxide. Moisture content influences the electrical current between two inert metal electrodes, which are fabricated as a helix on the inner wall of a tubular nonconductive sample cell. For a constant DC voltage applied to the electrodes, a current flows which is proportional to moisture. The moisture is absorbed into the hygroscopic phosphorous pentoxide, where the current electrolyzes the water molecules into hydrogen and oxygen. This sensor will handle moisture up to 1000 ppm and 6 atm pressure. Similar to the aluminum oxide ion, temperature control is very important. Photometric Moisture Analysis This analyzer requires a light source, a filter wheel rotated by a synchronous motor, a sample cell, a detector to measure the light transmitted, and associated electronics. Water has two absorption bands in the near infrared region at 1400 and 1900 nm. This analyzer can measure moisture in liquid or gaseous samples at levels from 5 ppm up to 100 percent, depending on other chemical species in the sample. Response time is less than 1 s, and samples can be run up to 300°C and 400 psig.

OTHER TRANSDUCERS Gear Train Rotary motion and angular position are easily transduced by various types of gear arrangements. A gear train in conjunction with a mechanical counter is a direct and effective way to obtain a digital readout of shaft rotations. The numbers on the counter can mean anything desired, depending on the gear ratio and the actuating device used to turn the shaft. A pointer attached to a gear train can be used to indicate a number of revolutions or a small fraction of a revolution for any specified pointer rotation. Differential Transformer These devices produce an AC electrical output from linear movement of an armature. They are very versatile in that they can be designed for a full range of output with any range of armature travel up to several inches. The transformers have one or two primaries and two secondaries connected to oppose each other. With an AC voltage applied to the primary, the output voltage depends on the position of the armature and the coupling. Such devices produce accuracies of 0.5 to 1.0 percent of full scale and are used to transmit forces, pressures, differential pressures, or weights up to 1500 m. They can also be designed to transmit rotary motion. Hall-Effect Sensors Some semiconductor materials exhibit a phenomenon in the presence of a magnetic field which is adaptable to sensing devices. When a current is passed through one pair of wires attached to a semiconductor, such as germanium, another pair of wires properly attached and oriented with respect to the semiconductor will develop a voltage proportional to the magnetic field present and the current in the other pair of wires. Holding the exciting current constant and moving a permanent magnet near the semiconductor produce a voltage output proportional to the movement of the magnet. The magnet may be attached to a process-variable measurement device which moves the magnet as the variable changes. Hall-effect devices provide high speed of response, excellent temperature stability, and no physical contact. SAMPLING SYSTEMS FOR PROCESS ANALYZERS The sampling system consists of all the equipment required to present a process analyzer with a clean representative sample of a process stream and to dispose of that sample. When the analyzer is part of an automatic control loop, the reliability of the sampling system is as important as the reliability of the analyzer or the control equipment. Sampling systems have several functions. The sample must be withdrawn from the process, transported, conditioned, introduced into the analyzer, and disposed. Probably the most common problem in sample-system design is the lack of realistic information concerning the properties of the process material at the sampling point. Another common problem is the lack of information regarding the conditioning required so that the analyzer may utilize the sample without malfunction for long periods of time. Some samples require enough conditioning and treating that the sampling systems become equivalent to miniature online processing plants. These systems possess many of the same fabrication, reliability, and operating problems as small-scale pilot plants except that the sampling system must generally operate reliably for much longer periods of time. Selecting the Sampling Point The selection of the sampling point is based primarily on supplying the analyzer with a sample whose composition or physical properties are pertinent to the control function to be performed. Other considerations include selecting locations that provide representative homogeneous samples with minimum transport delay, locations that collect a minimum of contaminating material, and locations that are accessible for test and maintenance procedures. Sample Withdrawal from Process A number of considerations are involved in the design of sample-withdrawal devices that will provide representative samples. For example, in a horizontal pipe that conveys process fluid, a sample point on the bottom of the pipe will collect a maximum amount of rust, scale, or other solid materials being carried along by the process fluid. In a gas stream, such a location will also collect a maximum amount of liquid contaminants. A sample point on the top side of a pipe will, for liquid streams, collect a

TELEMETERING AND TRANSMISSION maximum amount of vapor contaminants being carried along. Bends in the piping that produce swirls or cause centrifugal concentration of the denser phase may cause maximum contamination to be at unexpected locations. Two-phase process materials are difficult to sample for a total-composition representative sample. A typical method for obtaining a sample of process fluid well away from vessel or pipe walls is an eduction tube inserted through a packing gland. This sampling method withdraws liquid sample and vaporizes it for transporting to the analyzer location. The transport lag time from the end of the probe to the vaporizer is minimized by using tubing having a small internal volume compared with pipe and valve volumes. This sample probe may be removed for maintenance and reinstalled without shutting down the process. The eduction tube is made of material that will not corrode so that it will slide through the packing gland even after long periods of service. There may be a small amount of process-fluid leakage until the tubing is withdrawn sufficiently to close the gate valve. A swaged ferrule on the end of the tube prevents accidental ejection of the eduction tube prior to removal of the packing gland. The section of pipe surrounding the eduction tube and extending into the process vessel provides mechanical protection for the eduction tube. Sample Transport Transport time, the time elapsed between sample withdrawal from the process and its introduction into the analyzer, should be minimized, particularly if the analyzer is an automatic analyzer-controller. Any sample-transport time in the analyzercontroller loop must be treated as equivalent to process dead time in determining conventional feedback controller settings or in evaluating controller performance. Reduction in transport time usually means transporting the sample in the vapor state.

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Design considerations for sample-lines are as follows: 1. The structural strength or protection must be compatible with the area through which the sample line runs. 2. Line size and length must be small enough to meet transporttime requirements without excessive pressure drop or excessive bypass of sample at the analyzer input. 3. Line size and internal-surface quality must be adequate to prevent clogging by the contaminants in the sample. 4. The prevention of a change of state of the sample may require insulation, refrigeration, or heating of the sample line. 5. Sample-line material must be such as to minimize corrosion due to sample or environment. Sample Conditioning Sample conditioning usually involves the removal of contaminants or some deleterious component from the sample mixture and/or the adjustment of temperature, pressure, and flow rate of the sample to values acceptable to the analyzer. Some of the more common contaminants that must be removed are rust, scale, corrosion products, deposits due to chemical reactions, and tar. In sampling some process streams, the material to be removed may include the primary-process product such as polymer or the main constituent of the stream such as oil. In other cases, the material to be removed is present in trace quantities. For example, water in an online chromatograph sample can damage the chromatographic column packing. When contaminants or other materials that will hinder analysis represent a large percentage of the stream composition, their removal may significantly alter the integrity of the sample. In some cases, removal must be done as part of the analysis function so that removed material can be accounted for. In other cases, proper calibration of the analyzer output will suffice.

TELEMETERING AND TRANSMISSION ANALOG SIGNAL TRANSMISSION Modern control systems permit the measurement device, the control unit, and the final actuator to be physically separated by several hundred meters, if necessary. This requires the transmission of the measured variable from the measurement device to the control unit, and the transmission of the controller output from the control unit to the final actuator. In each case, transmission of a single value in only one direction is required. Such requirements can be met by analog signal transmission. A span is defined for the value to be transmitted, and the value is basically transmitted as a percent of this span. For the measured variable, the logical span is the measurement span. For the controller output, the logical span is the range of the final actuator (e.g., valve fully closed to valve fully open). For pneumatic transmission systems, the signal range used for the transmission is 3 to 15 psig. In each pneumatic transmission system, there can be only one transmitter, but there can be any number of receivers. When most measurement devices were pneumatic, pneumatic transmission was the logical choice. However, with the displacement of pneumatic measurement devices by electronic devices, pneumatic transmission is becoming less common but is unlikely to totally disappear. In order for electronic transmission systems to be less susceptible to interference from magnetic fields, current is used for the transmission signal instead of voltage. The signal range is 4 to 20 milliamps. In each circuit or “current loop,” there can be only one transmitter. There can be more than one receiver, but not an unlimited number. For each receiver, a 250 ohm “range resistor” is inserted into the current loop, which provides a 1- to 5-volt input to the receiving device. The number of receivers is limited by the power available from the transmitter. Both pneumatic and electronic transmission use a “live zero.” This

enables the receiver to distinguish a transmitted value of zero percent of span from a transmitter or transmission system failure. Transmission of zero percent of span provides a signal of 4 milliamps in electronic transmission. Should the transmitter or the transmission system fail (i.e., an open circuit in a current loop), the signal level would be zero milliamps. For most measurement variable transmissions, the lower range of the measurement span corresponds to 4 milliamps and the upper range of the measurement span corresponds to 20 milliamps. On an open circuit, the measured variable would fail to its lower range. In some applications, this is undesirable. For example, in a fired heater that is heating material to a target temperature, failure of the temperature measurement to its lower span value would drive the output of the combustion control logic to the maximum possible firing rate. In such applications, the analog transmission signal is normally inverted, with the upper range of the measurement span corresponding to 4 milliamps and the lower range of the measurement span corresponding to 20 milliamps. On an open circuit, the measured variable would fail to its upper range. For the fired heater, failure of the measured variable to its upper span would drive the output of the combustion control logic to the minimum firing rate. DIGITAL SYSTEMS With the advent of the microprocessor, digital technology began to be used for data collection, feedback control, and all other information processing requirements in production facilities. Such systems must acquire data from a variety of measurement devices, and control systems must drive final actuators. Analog Input and Outputs Analog inputs are generally divided into two categories:

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1. High level. Where the source is a process transmitter, the range resistor in the current loop converts the 4–20 milliamp signal into a 1–5 volt signal. The conversion equipment can be unipolar (i.e., capable of processing only positive voltages). 2. Low level. The most common low level signals are inputs from thermocouples. These inputs rarely exceed 30 millivolts, and could be zero or even negative. The conversion equipment must be bipolar (i.e., capable of processing positive and negative voltages). Ultimately, such signals are converted to digital values via an analog-to-digital (A/D) converter. However, the A/D converter is normally preceded by two other components: 1. Multiplexer. This permits one A/D converter to service multiple analog inputs. The number of inputs to a multiplexer is usually between 8 and 256. 2. Amplifier. As A/D converters require high level signals, a high gain amplifier is required to convert low-level signals into high-level signals. One of the important parameters for the A/D converter is its resolution. The resolution is stated in terms of the number of significant binary digits (bits) in the digital value. As the repeatability of most process transmitters is around 0.1 percent, the minimum acceptable resolution for a bipolar A/D converter is 12 bits, which translates to 11 data bits plus one bit for the sign. With this resolution, the analog input values can be represented to 1 part in 211, or one part in 2048. Normally, a 5-volt input is converted to a digital value of 2000, which effectively gives a resolution of 1 part in 2000 or 0.05 percent. Very few process control systems utilize resolutions higher than 14 bits, which translates to a resolution of 1 part in 8000 or 0.0125 percent. For 4–20 milliamp inputs, the resolution is not quite as good as stated above. For a 12-bit bipolar A/D converter, 1-volt converts to a digital value of 400. Thus, the range for the digital value is 400 to 2000, making the effective input resolution 1 part in 1600, or 0.0625 percent. On the output side, dedicated digital-to-analog converters are provided for each analog output. Outputs are normally unipolar, and require a lower resolution than inputs. A 10-bit resolution is normally sufficient, giving a resolution of 1 part in 1000 or 0.1%. Pulse Inputs Where the sensor within the measurement device is digital in nature, analog-to-digital conversion can be avoided. For rotational devices, the rotational element can be outfitted with a shaft encoder that generates a known number of pulses per revolution. The digital system can process such inputs in either of the following ways: 1. Count the number of pulses over a fixed interval of time. 2. Determine the time for a specified number of pulses. 3. Determine the duration of time between the leading (or trailing) edges of successive pulses. Of these, the first option is the most commonly used in process applications. Turbine flowmeters are probably the most common example where pulse inputs are used. Another example is a watt-hour meter. Basically any measurement device that involves a rotational element can be interfaced via pulses. Occasionally, a nonrotational measurement device can generate pulse outputs. One example is the vortex shedding meter, where a pulse can be generated when each vortex passes over the detector. Serial Interfaces Some very important measurement devices cannot be reasonably interfaced via either analog or pulse inputs. Two examples are the following: 1. Chromatographs can perform a total composition analysis for a sample. It is possible but inconvenient to provide an analog input for each component. Furthermore, it is often desirable to capture other information, such as the time that the analysis was made (normally the time the sample was injected). 2. Load cells are capable of resolutions of 1 part in 100,000. A/D converters for analog inputs cannot even approach such resolutions. One approach to interfacing with such devices is serial interfaces. This involves two aspects: 1. Hardware interface. The RS-232 interface standard is the basis for most serial interfaces. 2. Protocol. This is interpreting the sequence of characters

transmitted by the measurement device. There are no standards for protocols, which means that custom software is required. One advantage of serial interfaces is that two-way communication is possible. For example, a “tare” command can be issued to a load cell. Microprocessor-Based Transmitters The cost of microprocessor technology has declined to the point where it is economically feasible to incorporate a microprocessor into each transmitter. Such microprocessor-based transmitters are often referred to as “smart” transmitters. As opposed to conventional or “dumb” transmitters, the smart transmitters offer the following capabilities: 1. Checks on the internal electronics, such as verifying that the voltage levels of internal power supplies are within specifications. 2. Checks on environmental conditions within the instruments, such as verifying that the case temperature is within specifications. 3. Compensation of the measured value for conditions within the instrument, such as compensating the output of a pressure transmitter for the temperature within the transmitter. Smart transmitters are much less affected by temperature and pressure variations than conventional transmitters. 4. Compensation of the measured value for other process conditions, such as compensating the output of a capacitance level transmitter for variations in process temperature. 5. Linearizing the output of the transmitter. Functions such as square root extraction of the differential pressure for a head-type flowmeter can be done within the instrument instead of within the control system. 6. Configuring the transmitter from a remote location, such as changing the span of the transmitter output. 7. Automatic recalibration of the transmitter. Although this is highly desired by users, the capabilities, if any, in this respect depend on the type of measurement. Due to these capabilities, smart transmitters offer improved performance over conventional transmitters. Transmitter/Actuator Networks With the advent of smart transmitters and smart actuators, the limitations of the 4–20 milliamp analog signal transmission retard the full utilization of the capabilities of the smart devices. For smart transmitters, the following capabilities are required: 1. Transmission of more than one value from a transmitter. Information beyond the measured variable is available from the smart transmitter. For example, a smart pressure transmitter can also report the temperature within its housing. Knowing that this temperature is above normal values permits corrective action to be taken before the device fails. Such information is especially important during the initial commissioning of a plant. 2. Bidirectional transmission. Configuration parameters such as span, engineering units, resolution, and so on, must be communicated to the smart transmitter. Similar capabilities are required for smart actuators. In order to meet their initial requirements, several manufacturers have developed digital communications capabilities for communicating with smart transmitters. These can be used either in addition to or in lieu of the 4–20 milliamp signal. Although most manufacturers release enough information on their communications features to permit another manufacturer to provide compatible instruments (and in some cases provide an open communication standard), the communications capability provided by a manufacturer may be proprietary. Users purchase their transmitters from a variety of manufacturers, so this situation limits the full utilization of the capabilities of smart transmitters and valves. Efforts to develop a standard for a communications network have not proceeded smoothly. The International Society for Measurement and Control (ISA) has attempted to develop a standard generally referred to as fieldbus. The standards effort attempted to develop a world standard, encompassing European, Japanese, and American products. This effort focused on developing a single standard with which all manufacturers would comply. Currently, efforts are mostly being directed to providing the capability for interoperability between the products of the manufacturers with competing communications networks. Meanwhile, users are reluctant to make major commitments, and are continuing to rely primarily on the traditional 4–20 milliamp transmission.

TELEMETERING AND TRANSMISSION FILTERING AND SMOOTHING A signal received from a process transmitter generally contains the following features: 1. Low-frequency process disturbances. The control system is expected to react to these disturbances. 2. High-frequency process disturbances. The frequency of these disturbances is beyond the capability of the control system to effectively react. 3. Measurement noise. 4. Stray electrical pickup, primarily 50- or 60-cycle AC. Frequencies are measured in Hertz (Hz), with 60-cycle AC being a 60-Hz frequency. The objective of filtering and smoothing is to remove the last three components, leaving only the low frequency process disturbances. Normally this has to be accomplished using the proper combination of analog and digital filters. Sampling a continuous signal results in a phenomenon often referred to as aliasing or foldover. In order to represent a sinusoidal signal, a minimum of four samples are required during each cycle. That is, the sampling interval must be at least 1/4th the period of the sinusoid. Consequently, when a signal is sampled at a frequency ωs, all frequencies higher than (π/2)ωs cannot be represented at their original frequency. Instead, they are present in the sampled signal with their original amplitude but at a lower frequency harmonic. Because of the aliasing or foldover issues, a combination of analog and digital filtering is usually required. The sampler (i.e., the A/D converter) must be preceded by an analog filter that rejects those highfrequency components such as stray electrical pickup that would result in foldover when sampled. In commercial products, analog filters are normally incorporated into the input processing hardware by the manufacturer. The software then permits the user to specify digital filtering to remove any undesirable low-frequency components. On the analog side, the filter is often the conventional resistorcapacitor or RC filter. However, other possibilities exist. For example, one type of A/D converter is called an “integrating A/D” because the converter basically integrates the input signal over a fixed interval of time. By making the interval 1/60th second, this approach provides excellent rejection of any 60-Hz electrical noise. On the digital side, the input processing software generally provides for smoothing via the exponentially weighted moving average, which is the digital counterpart to the RC network analog filter. The smoothing equation is as follows: yi = αxi + (1 − α)yi − 1 where

(8-103)

xi = current value of input yi = current output from filter yi − 1 = previous output from filter α = filter coefficient

The degree of smoothing is determined by the filter coefficient α, with α = 1 being no smoothing and α = 0 being infinite smoothing (no effect of new measurements). The filter coefficient α is related to the filter time constant τF and the sampling interval ∆t by the following equation: −∆t α = 1 − exp ᎏ (8-104) τF or by the approximation ∆t α=ᎏ (8-105) ∆t + τF Another approach to smoothing is to use the arithmetic moving average, which is represented by the following equation:

 


 x  n

i+1−j

j=1

yi = ᎏᎏ n

(8-106)

The term “moving” is applied because the filter software maintains a storage array with the previous n values of the input. When a new value is received, the oldest value in the storage array is replaced with

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the new value, and the arithmetic average recomputed. This permits the filtered value to be updated each time a new input value is received. In process applications, determining τF (or α) for the exponential filter and n for the moving average filter is often done merely by observing the behavior of the filtered value. If the filtered value is “bouncing,” the degree of smoothing (that is, τF or n) is increased. This can easily lead to an excessive degree of filtering, which will limit the performance of any control system that uses the filtered value. The degree of filtering is best determined from the frequency spectrum of the measured input, but such information is rarely available for process measurements. ALARMS The purpose of an alarm is to alert the process operator to a process condition that requires immediate attention. An alarm is said to occur whenever the abnormal condition is detected and the alert is issued. An alarm is said to return to normal when the abnormal condition no longer exists. Analog alarms can be defined on measured variables, calculated variables, controller outputs, and the like. For analog alarms, the following possibilities exist: 1. High/low alarms. A high alarm is generated when the value is greater than or equal to the value specified for the high-alarm limit. A low alarm is generated when the value is less than or equal to the value specified for the low-alarm limit. 2. Deviation alarms. An alarm limit and a target are specified. A high deviation alarm is generated when the value is greater than or equal to the target plus the deviation alarm limit. A low deviation alarm is generated when the value is less than or equal to the target minus the deviation alarm limit. 3. Trend or rate-of-change alarms. A limit is specified for the maximum rate of change, usually specified as a change in the measured value per minute. A high trend alarm is generated when the rate of change of the variable is greater than or equal to the value specified for the trend alarm limit. A low trend alarm is generated when the rate of change of the variable is less than or equal to the negative of the value specified for the trend alarm limit. Most systems permit multiple alarms of a given type to be configured for a given value. For example, configuring three high alarms provides a high alarm, a high-high alarm, and a high-high-high alarm. One operational problem with analog alarms is that noise in the variable can cause multiple alarms whenever its value approaches a limit. This can be avoided by defining a deadband on the alarm. For example, a high alarm would be processed as follows: 1. Occurrence. The high alarm is generated when the value is greater than or equal to the value specified for the high-alarm limit. 2. Return to normal. The high-alarm return to normal is generated when the value is less than or equal to the high alarm limit less the deadband. As the degree of noise varies from one input to the next, the deadband must be individually configurable for each alarm. Discrete alarms can be defined on discrete inputs, limit switch inputs from on/off actuators and so on. For discrete alarms, the following possibilities exist: 1. Status alarms. An expected or normal state is specified for the discrete value. A status alarm is generated when the discrete value is other than its expected or normal state. 2. Change-of-state alarm. A change-of-state alarm is generated on any change of the discrete value. The expected sequence of events on an alarm is basically as follows: 1. The alarm occurs. This usually activates an audible annunciator. 2. The alarm occurrence is acknowledged by the process operator. When all alarms have been acknowledged, the audible annunciator is silenced. 3. Corrective action is initiated by the process operator. 4. The alarm condition returns to normal. However, additional requirements are imposed at some plants. Sometimes the process operator must acknowledge the alarm’s return to normal. Some plants require that the alarm occurrence be reissued

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if the alarm remains in the occurred state longer than a specified period of time. Consequently, some “personalization” of the alarming facilities is done. When alarms were largely hardware-based (i.e., the panel alarm systems), the purchase and installation of the alarm hardware imposed a certain discipline on the configuration of alarms. With digital systems, the suppliers have made it extremely easy to configure alarms. In fact, it is sometimes easier to configure alarms on a measured value than not to configure the alarms. Furthermore, the engineer assigned the responsibility for defining alarms should ensure that an abnormal process condition will not go undetected because an alarm has not been configured. When alarms are defined on every measured and calculated variable, the result is an excessive number of alarms, most of which are duplicative and unnecessary. The accident at the Three Mile Island nuclear plant clearly demonstrated that an alarm system can be counterproductive. An excessive number of alarms can distract the operator’s attention from the real problem that needs to be addressed. Alarms that merely tell the operator something that is already known do the same. In fact, a very good definition of a nuisance alarm is one that informs the operator of a situation of which the operator is already aware. The only problem with applying this definition is determining what the operator already knows. Unless some discipline is imposed, engineering personnel, especially where contractors are involved, will define far more alarms than plant operations require. This situation may be addressed by simply setting the alarm limits to values such that the alarms never occur. However, changes in alarms and alarm limits are changes from the perspective of the Process Safety Management regulations. It is prudent to impose the necessary discipline to avoid an excessive number of alarms. Potential guidelines are as follows: 1. For each alarm, a specific action is expected from the process operator. Operator actions such as “call maintenance” are inappropriate with modern systems. If maintenance needs to know, modern systems can inform maintenance directly. 2. Alarms should be restricted to abnormal situations for which the process operator is responsible. A high alarm on the temperature in one of the control system cabinets should not be issued to the process operator. Correcting this situation is the responsibility of maintenance, not the process operator. 3. Process operators are expected to be exercising normal surveillance of the process. Therefore, alarms are not appropriate for situations known to the operator either through previous alarms or through normal process surveillance. The “sleeping operator” problem can be addressed by far more effective means than the alarm system.

4. When the process is operating normally, no alarms should be triggered. Within the electric utility industry, this design objective is known as “darkboard.” Application of darkboard is especially important in batch plants, where much of the process equipment is operated intermittently. Ultimately, guidelines such as those above will be taken seriously only if production management carefully configures the alarms. The consequences of excessive and redundant alarms will be felt primarily by those responsible for production operations. Therefore, production management must make adequate resources available for reviewing and analyzing the proposed alarm configurations. Another serious distraction to a process operator is the multiple alarm event, where a single event within the process results in multiple alarms. When the operator must individually acknowledge each alarm, considerable time can be lost in silencing the obnoxious annunciator before the real problem is addressed. Air-handling systems are especially vulnerable to this, where any fluctuation in pressure (for example, resulting from a blower trip) can cause a number of pressure alarms to occur. Point alarms (high alarms, low alarms, status alarms, etc.) are especially vulnerable to the multiple alarm event. This can be addressed in one of two ways: 1. Ganging alarms. Instead of individually issuing the point alarms, all alarms associated with a certain aspect of the process are simply wired to give a single trouble alarm. The responsibility rests entirely with the operator to determine the nature of the problem. 2. Intelligent alarms. Logic is incorporated into the alarm system to determine the nature of the problem and then issue a single alarm to the process operator. Sometimes this is called an expert system. While the intelligent alarm approach is clearly preferable, substantial process analysis is required to support intelligent alarming. Meeting the following two objectives is quite challenging: 1. The alarm logic must consistently detect abnormal conditions within the process. 2. The alarm logic must not issue an alert to an abnormal condition when in fact none exists. Often the latter case is more challenging than the former. Logically, the intelligent alarm effort must be linked to the process hazards analysis. Developing an effective intelligent alarming system requires substantial commitments of effort, involving both process engineers, control systems engineers, and production personnel. Methodologies such as expert systems can facilitate the implementation of an intelligent alarming system, but they must still be based on a sound analysis of the potential process hazards.

DIGITAL TECHNOLOGY FOR PROCESS CONTROL GENERAL REFERENCES: Fortier, Design and Analysis of Distributed RealTime Systems, McGraw-Hill, New York, 1985; Hawryszkiewycs, Database Analysis and Design, Science Research Associates Inc., Chicago, 1984; Khambata, Microprocessors/Microcomputers: Architecture, Software, and Systems, 2d ed., Wiley, New York, 1987; Liptak, Instrument Engineers Handbook, Chilton Book Company, Philadelphia, 1995; Mellichamp (ed.), Real-Time Computing with Applications to Data Acquisition and Control, Van Nostrand Reinhold, New York, 1983.

Since the 1970s, process controls have evolved from pneumatic analog technology to electronic analog technology to microprocessorbased controls. Electronic analog technology has virtually disappeared from process controls. Pneumatic controls continue to be manufactured, but they are relegated to special situations where pneumatics can offer a unique advantage. Process controls are dominated by programmable electronic systems (PES), most of which are based on microprocessor technology.

HIERARCHY OF INFORMATION SYSTEMS Coupling digital controls with networking technology permits information to be passed from level-to-level within a corporation at high rates of speed. This technology is capable of presenting the measured variable from a flow transmitter installed in a plant in a remote location anywhere in the world to the company headquarters in less than a second. A hierarchical representation of the information flow within a company leads to a better understanding of how information is passed from one layer to the next. Such representations can be developed in varying degrees of detail, and most companies have developed one that describes their specific practices. The following hierarchy consists of five levels. Measurement Devices and Actuators Often referred to as level 0, this layer couples the control and information systems to the process. The measurement devices provide information on the cur-

DIGITAL TECHNOLOGY FOR PROCESS CONTROL rent conditions within the process. The actuators permit control decisions to be imposed on the process. Although traditionally analog, smart transmitters and smart valves based on microprocessor technology will eventually dominate this layer. Regulatory Controls The objective of this layer is to operate the process at or near the targets supplied by others, be it the process operator or a higher layer in the hierarchy. In order to achieve consistent process operations, a high degree of automatic control is required from the regulatory layer. The direct result is a reduction in variance in the key process variables. More uniform product quality is an obvious benefit. However, consistent process operation is a prerequisite for optimizing the process operations. To ensure success for the upper level functions, the first objective of any automation effort must be to achieve a high degree of regulatory control. Supervisory Controls The regulatory layer blindly attempts to operate the process at the specified targets, regardless of the appropriateness of these targets. Determining the most appropriate targets is the responsibility of the supervisory layer. Given the current production targets for a unit, supervisory control determines how the process can be best operated to meet the production targets. Usually this optimization has a limited scope, being confined to a single production unit or possibly even a single unit operation within a production unit. Supervisory control translates changes in factors such as current process efficiencies, current energy costs, cooling medium temperatures, and so on, to changes in process operating targets so as to optimize process operations. Production Controls The nature of the production control logic differs greatly between continuous and batch plants. A good example of production control in a continuous process is refinery optimization. From the assay of the incoming crude oil, the values of the various possible refined products, the contractual commitments to deliver certain products, the performance measures of the various units within a refinery, and the like, it is possible to determine the mix of products that optimizes the economic return from processing this crude. The solution of this problem involves many relationships and constraints and is solved with techniques such as linear programming. In a batch plant, production control often takes the form of routing or short-term scheduling. For a multiproduct batch plant, determining the long term schedule is basically a manufacturing resource planning (MRP) problem, where the specific products to be manufactured and the amounts to be manufactured are determined from the outstanding orders, the raw materials available for production, the production capacities of the process equipment, and other factors. The goal of the MRP effort is the long-term schedule, which is a list of the products to be manufactured over a specified period of time (often one week). For each product on the list, a target amount is also specified. To manufacture this amount usually involves several batches. The term “production run” often refers to the sequence of batches required to make the target amount of product, so in effect the long term schedule is a list of production runs. Most multiproduct batch plants have more than one piece of equipment of each type. Routing refers to determining the specific pieces of equipment that will be used to manufacture each run on the long term production schedule. For example, the plant might have five reactors, eight neutralization tanks, three grinders, and four packing machines. For a given run, a rather large number of possible routes are possible. Furthermore, rarely is only one run in progress at a given time. The objective of routing is to determine the specific pieces of production equipment to be used for each run on the long-term production schedule. Given the dynamic nature of the production process (equipment failures, insertion/deletion of runs into the longterm schedule, etc.), the solution of the routing problem continues to be quite challenging. Corporate Information Systems Terms such as management information systems (MIS) and information technology (IT) are frequently used to designate the upper levels of computer systems within a corporation. From a control perspective, the functions performed at this level are normally long-term and/or strategic. For example, in a processing plant, long-term contracts are required with the providers of the feedstocks. A forecast must be developed for the demand for possible products from the plant. This demand must be translated into

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needed raw materials, and then contracts executed with the suppliers to deliver these materials on a relatively uniform schedule. While most companies within the process industries recognize the importance of information technology in managing their businesses, this technology has been a source of considerable frustration and disappointment. Schedule delays, cost overruns, and failure of the final product to perform as expected have often eroded the credibility of information technology. However, immense potential remains for the technology, and process companies have no choice but to seek continuous improvement. DISTRIBUTED CONTROL SYSTEMS Although digital control technology was first applied to process control in 1959, the total dependence of the early centralized architectures on a single computer for all control and operator interface functions resulted in complex systems with dubious reliability. Adding a second processor increased both the complexity and the cost. Consequently, many installations provided analog backup systems to protect against a computer malfunction. Microprocessor technology permitted these technical issues to be addressed in a cost-effective manner. In the mid-1970s, a process control architecture referred to as a distributed control system (DCS) was introduced and almost instantly became a commercial success. A DCS consists of some number of microprocessor-based nodes that are interconnected by a digital communications network, often called a data highway. The key features of this architecture are as follows: 1. The process control functions and the operator interface, also referred to as man-machine interface (MMI) or human-machine interface (HMI), is provided by separate nodes. This approach is referred to as split-architecture, and it permits considerable flexibility in choosing a configuration that most appropriately meets the needs of the application. 2. The process control functions can be distributed functionally and/or geographically. Functional distribution permits related control functions to be grouped and implemented in a single node. Geographical distribution permits the process control nodes to be physically located near the equipment being controlled. As the digital communications network is based on local area network (LAN) technology, the nodes within the DCS can be physically separated by thousands of meters. 3. Redundancy can be provided where appropriate, the following being typical: a. Multiple operator interface nodes can be provided to reduce the impact of an operator interface node failure. b. The digital communications network is normally redundant to the extent that at least two independent paths are available between any two nodes of the DCS. c. Consisting basically of processor and memory, the process control nodes are highly reliable, with mean-times-between-failures approaching 100 years. Redundant configurations are available for especially critical applications. 4. As the data within the DCS are digital in nature, interfaces to upper level computers are technically easier to implement. Unfortunately, the proprietary nature of the communications networks within commercial DCS products complicate the implementation of such interfaces. Truly open DCS architectures, at least as the term “open” is used in the mainstream of computing, are not yet available. Figure 8-62 depicts a hypothetical distributed control system. A number of different unit configurations are illustrated. This system consists of many commonly used DCS components, including multiplexers (MUXs), single/multiple-loop controllers, programmable logic controllers (PLCs), and smart devices. A typical system includes the following elements as well: • Host computers. These are the most powerful computers in the system, capable of performing functions not normally available in other units. They act as the arbitrator unit to route internodal communications. An operator interface is supported and various peripheral devices are coordinated. Computationally intensive tasks, such as optimization or advanced control strategies, are processed here. • Data highway. This is the communication link between com-

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FIG. 8-62

A typical DDC system.

ponents of a network. Coaxial cable is often used. A redundant pair is normally supplied to reduce possibility of link failure. • Real-time clocks (RTCs). Real-time systems are required to respond to events, as they occur, in a timely manner. This is especially crucial in process control systems where control actions applied at the wrong time may amplify process deviations or destabilize the processes. The nodes in the systems are interrupted periodically by the real-time clocks to maintain the actual elapsed times. • Operator control stations. These typically consist of color graphics monitors with special keyboards, in addition to a conventional alphanumeric keyboard, containing keys to perform dedicated functions. Operators may supervise and control processes from these stations. A control station may contain a number of printers for alarm logging, report printing, or hard-copying process graphics. • Remote control units. These units are used to control unit processes. Basic control functions such as the PID algorithm are implemented here. Depending on other hardware components used, data acquisition capability may be required to perform digital control. They may be configured to supply process set points to single-loop controllers. Radio telemetry may be installed to communicate with MUX units located at great distances. • Programmer consoles. These are programming terminals. Developing system software on the host machines is a common prac-

tice by many system suppliers. This eliminates compatibility problems between development and target environments. Programming capability is normally retained when the system is delivered such that system users may develop their own application programs. • Mass storage device. Typically, fixed-head hard disk drives are used to store active data, including on-line and historical databases and non-memory-resident programs. Memory-resident programs are stored to allow loading at system startups. The tape drives are used for archives and backups. DISTRIBUTED DATABASE AND THE DATABASE MANAGER A database is a centralized location for data storage. The use of databases enhances system performance by maintaining complex relations between data elements while reducing data redundancy. A database may be built based on the relational model, the entity relationship model, or some other model. The database manager is a system utility program or programs acting as the gatekeeper to the databases. All functions retrieving or modifying data must submit a request to the manager. Information required to access the database include the tag name of the database entity, often referred to as a point, the attributes to be accessed, and the values if modifying. The database manager maintains the integrity of the databases by executing a request only

DIGITAL TECHNOLOGY FOR PROCESS CONTROL when not processing other conflicting requests. Although a number of functions may read the same data item at the same time, writing by a number of functions or simultaneous read and write of the same data item is not permitted. To allow flexibility, the database manager must also perform point addition or deletion. However, the ability to create a point type or to add or delete attributes of a point type is not normally required because, unlike other data processing systems, a process control system normally involves a fixed number of point types and related attributes. For example, analog and binary input and output types are required for process I/O points. Related attributes for these point types include tag names, values, and hardware addresses. Different system manufacturers may define different point types using different data structures. We will discuss other commonly used point types and attributes as they appear. Historical Database Subsystem We have discussed the use of on-line databases. An historical database is built similar to an on-line database. Unlike their on-line counterparts, the information stored in a historical database is not normally accessed directly by other subsystems for process control and monitoring. Periodic reports and longterm trends are generated based on the archived data. The reports are often used for long-term planning and system performance evaluations such as statistical process (quality) control. The trends may be used to detect process drifts or to compare process variations at different times. The historical data is sampled at user-specified intervals. A typical process plant contains a large number of data points, but it is not feasible to store data for all points at all times. The user determines if a data point should be included in the list of archive points. Most systems provide archive-point menu displays. The operators are able to add or delete data points to the archive point lists. The sampling periods are normally some multiples of their base scan frequencies. However, some systems allow historical data sampling of arbitrary intervals. This is necessary when intermediate virtual data points that do not have the scan frequency attribute are involved. The archive point lists are continuously scanned by the historical database software. On-line databases are polled for data. The times of data retrieval are recorded with the data obtained. To conserve storage space, different data compression techniques are employed by various manufacturers. The historical data may be used for long-term trending. The live trends data are displayed but not stored. Therefore, these trends cannot be recalled once cleared off the screens. The historical trend of any archive point may be displayed at any time because the values used are extracted from the archived data. Zooming, that is, axis scaling, is allowed by most systems. As a result, the displayed data point intervals may not be multiples of stored data intervals. Many systems provide data interpolation and smoothing functions to process the stored data when they are displayed. The live and historical trend displays are superior to strip charts in many ways. In addition to conventional trend recording, the zoom-in capability allows close examination of recorded data, whereas zoom-out compresses long-term data within a screen. Exact data sampled at any time point can be extracted by cursor positioning. Strip-chart recorders have disappeared from many modern plants. Periodic reports, including shift, daily, weekly, monthly, and quarterly reports, are printed based on archived data. Some reports may contain simply the stored data in certain specific arrangements. More often, quantities such as mean values, standard deviations, or other calculated values are included. Instead of hard-coding reports to user specifications, many system suppliers provide report generation packages in the form of metalanguages. These packages allow users to configure report formats suitable for their particular requirements. The report generator interprets the configuration files prepared by the users to create reports. Due to the infrequent execution, the report generator is normally operated in the batch mode. DCS manufacturers have devoted considerable efforts to make it easy to implement and enhance process control configurations within their products. Although programming in the traditional sense is possible within most products, the majority of the functions required for a process control application can be implemented by configuring as opposed to programming.

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PROCESS CONTROL LANGUAGE A digital control system involves software development. The introduction of high-level programming languages such as FORTRAN and BASIC in the 1960s was considered a major breakthrough. For process control applications, some companies have incorporated libraries of software routines for these languages, but others have developed speciality pseudolanguages. These implementations are characterized by their statement-oriented language structure. Although substantial savings in time and efforts can be realized, software development costs continue to be significant. The most successful and user-friendly approach, which is now adopted by virtually all commercial systems, is the fill-in-the-forms or table-driven process control languages (PCLs). The core of these languages is a number of basic functional blocks or software modules. All modules are defined as database points. Using a module is analogous to calling a subroutine in conventional programs. In general, each module contains some inputs and an output. The programming involves soft-wiring outputs of blocks to inputs of other blocks. Some modules may require additional parameters to direct module execution. The users are required to fill in the sources of input values, the destinations of output values, and the parameters in blanks of forms or tables prepared for the modules. The source and destination blanks may be filled with process I/Os when appropriate. To connect modules, some systems require filling the tag names of modules originating or receiving data. Additional programs are often required to resolve ambiguities when connecting multiple input-output modules. Another method involves the use of intermediate data points. The blanks in a pair of interconnecting modules are filled with the tag name of the same data point. Batch jobs and/or interactive data entry may be performed to fill the databases. A completed control strategy resembles a data flow diagram. The soft-wiring of modules is similar to hard-wiring analog-electronic circuits in analog computers. Additional database space must be allocated when intermediate data points are used. A system can be designed to use process I/O points as intermediates. However, the data acquisition software must be programmed to bypass these points when scanned. All system builders provide virtual data point types if the intermediate data storage scheme is adopted. These points are not scanned by the data acquisition software. Memory space requirements are reduced by eliminating unnecessary attributes such as hardware addresses and scan frequencies. It should be noted that the fill-in-the-forms technique is applicable to all data point types. All process control languages contain PID controller blocks. The digital PID controller is normally programmed to execute in velocity form. A pulse duration output may be used to receive the velocity output directly. Where positional signal is expected, an operating mode bit is used to enable an internal integrator. This flexibility is not normally available in analog controllers. Unlike an analog controller, the three modes in a digital PID controller do not interact. This simplifies the tuning effort. In addition to the tuning constants, a typical digital PID controller contains some entries not normally found in an analog controller: • When a process error is below certain tolerable deadband, the controller ceases modifying output. This is referred to as gap action. • The magnitude of change in a velocity output is limited by a change clamp. • A pair of output clamps is used to restrict a positional output value from exceeding specified limits. • The controller action can be disabled by triggering a binary deactivate input signal, during process startup, shutdown, or when some abnormal conditions exist. Although modules are supplied and their internal configurations are different from system to system, their basic functionalities are the same. SINGLE-LOOP CONTROLS With the exception of pneumatic controllers for special applications, commercial single-loop controllers are almost entirely microprocessorbased. The most basic products provide only the PID control algo-

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rithm, but the more powerful versions provide a set of general-purpose algorithms comparable to those in a DCS. For applications such as cascade control and multivariable control, the manufacturers of singleloop controllers provide multiloop versions of their products. These multiloop controllers have much in common with the process control node in a DCS. Single-loop controllers provide both the process control functions and the operator interface function. This makes them ideally suited to very small applications, where only two or three loops are required. However, it is possible to couple single-loop controllers to a personal computer (PC) to provide the operator interface function. Such installations are extremely cost effective, and with the keen competition in PC-based products, the capabilities are comparable and sometimes even better than that provided by a DCS. However, this approach makes sense only up to about 25 loops. Initially, the microprocessor-based single-loop controllers made the power of digital control affordable to those with small processes. To compete with these products in small applications, the DCS suppliers have introduced micro-DCS versions of their products. As a PC-based operator interface is usually a component of the micro-DCS, there is sometimes little distinction between a micro-DCS and a system consisting of single-loop controllers coupled to a PC-based operator interface. PROGRAMMABLE LOGIC CONTROLLERS The programmable logic controller (PLC) was the first digital technology to successfully compete with conventional technology in industrial control applications. Initially developed in the early 1970s for applications within the manufacturing industries (principally automotive), the PLC proved to be superb for implementing discrete logic. The earliest PLCs were limited to discrete I/O, basic Boolean logic functions (AND, OR, NOT), timers, and counters. However, versions soon appeared with analog I/O, math functions, PID control algorithms, and other functions required for process control applications. Developed to replace hard-wired relay logic, the early PLCs were “programmed” using the same ladder logic diagrams used to represent logic implemented with hard-wired relays. As the initial target market was electrical, programming in ladder logic was a definite

advantage, and some union contracts specifically required that such discrete logic be presented as ladder diagrams. However, ladder logic is not the programming medium preferred by instrument engineers, which hampers the acceptance of PLCs for process control. Alternatives to ladder logic are available for programming PLCs, but established perceptions are slow to change. Developed specifically for implementing discrete logic, PLCs continue to provide the best route to implementing such logic. The manufacturers of PLCs provide robust, cost-effective discrete I/O modules. Regardless of its acceptability, ladder logic is the most efficient means for implementing discrete logic. Because PLCs scan the discrete logic very rapidly, a 100-millisecond scan rate is considered very slow for a PLC. The process control modules of a DCS often implement discrete logic using function blocks, which is less efficient than ladder logic and normally results in a slower scan rate. A few DCS process control modules have used ladder logic to implement discrete logic, but their discrete I/O capabilities and slow scan rates rarely match that of a PLC. Consequently, for applications heavy with discrete logic, most DCS suppliers will incorporate one or more PLCs into their system. Being excellent at discrete logic, PLCs are a potential candidate for implementing interlocks. Process interlocks are clearly acceptable for implementation within a PLC. Implementation of safety interlocks in programmable electronic systems (such as a PLC) is not universally accepted. Many organizations continue to require that all safety interlocks be hard-wired, but implementing safety interlocks in a PLC that is dedicated to safety functions is accepted by some as being equivalent to the hard-wired approach. INTERCOMPUTER COMMUNICATIONS A group of computers becomes a network when intercomputer communication is established. Prior to the 1980s, all system suppliers used proprietary protocols to network their systems. Ad hoc approaches were sometimes used to connect third-party equipment, which was not cost-effective in system maintenance, upgrade, and expansion. The recent introduction of standardized protocols has led to a decrease in initial capital cost. Most current DCS network protocol designs are based on the ISO-OSI* seven-layer model. The most notable effort in standardizing plant automation protocols

* Abbreviated from International Standards Organization-Open System Interconnection. They are the physical, data link, network, transports session, presentation, and application layers. Only the physical, data link, and application layers are present in the mini-MAP.

FIG. 8-63

A DCS using broadband data highway and fieldbus.

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS was initiated by General Motors in the early 1980s. This culminated in the Manufacturing Automation Protocol (MAP), which adopted the ISO-OSI standards as its basis. MAP specifies a broadband backbone local area network (LAN), which incorporates a selection of existing standard protocols suitable for manufacturing automation while defining additional protocols where no standard previously existed. Although intended for discrete component systems, MAP has evolved to address the integration of DCSs used in process control as well. Due to various technical reasons, MAP has gained limited acceptance by the process industries as of 1990. Engineering societies, including ISA and IEEE, and many operating companies are collaborating to refine MAP for wider support. More microprocessor-based process equipment, such as smart instruments and single-loop controllers, with digital communications capability are now becoming available and are used extensively in process plants. A fieldbus, which is a low-cost protocol, is necessary to perform efficient communication between the DCS and these devices. So-called mini-MAP architecture was developed to satisfy process control and instrumentation requirements while incorporating existing ISA standards. It is intended to improve access time while

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allowing communications to a large number of microprocessor-based devices. The mini-MAP contains only three of the seven layers specified by the ISO-OSI model; therefore, a mini-MAP device cannot communicate with devices on the MAP bus directly. The development of MAP/EPA (Enhanced Performance Architecture) is parallel to that of the mini-MAP. This scheme adopts the full seven-layer model with a reduced set of MAP protocols. The MAP/EPA is compatible to both the complete MAP and the mini-MAP. Another benefit of standardizing the fieldbus is that it encourages third-party traditional equipment manufacturers to enter the smart equipment market, resulting in increased competition and improved equipment quality. Figure 8-63 illustrates a LAN-based DCS. Irrespective of the protocol used, communication programs act as servers to the database manager. When some functions request data from a remote node, the database manager will transfer the request to the remote node database manager via the communication programs. The remote node communication programs will relay the request to the resident database manager and return the obtained data. The remote database access and the existence of communications equipment and software are transparent to plant operators.

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS GENERAL REFERENCES: Baumann, Control Valve Primer, 2d ed., ISA, 1994; Considine, Process/Industrial Instruments & Controls Handbook, 4th ed., McGraw-Hill, 1993; Driskell, Control-Valve Selection and Sizing, ISA, 1983; Hammitt, Cavitation and Multiphase Flow Phenomena, McGraw-Hill, 1980; Norton, Fundamentals of Noise and Vibration Analysis for Engineers, Cambridge University Press, 1989; Ulanski, Valve and Actuator Technology, McGraw-Hill, 1991.

External control of the process is achieved by devices that are specially designed, selected and configured for the intended processcontrol application. The text below covers three very common function classifications of process-control devices: controllers, final control elements, and regulators. The process controller is the “master” of the process-control system. It accepts a set point and other inputs and generates an output or outputs that it computes from a rule or set of rules that are part of its internal configuration. The controller output serves as an input to another controller or, more often, as an input to a final control element. The final control element is the device that affects the flow in the piping system of the process. The final control element serves as an interface between the process controller and the process. Control valves and adjustable speed pumps are the principal types discussed. Regulators, though not controllers or final control elements, perform the combined function of these two devices (controller and final control element) along with the measurement function commonly associated with the process variable transmitter. The uniqueness, control performance, and widespread usage of the regulator make it deserving of a functional grouping of its own. ELECTRONIC AND PNEUMATIC CONTROLLERS Electronic Controllers Almost all of the electronic process controllers used today are microprocessor-based devices. These processor-based controllers contain, or have access to, input/output (I/O) interface electronics that allow various types of signals to enter and leave the controller’s processor. The controller, depending on its type, uses sufficient read-only-memory (ROM) and read/write-accessiblememory (RAM) to perform the controller function. The resolution of the analog I/O channels of the controller vary somewhat, with 12-bit and 14-bit conversions quite common. Sample rates for the majority of the constant sample rate controllers range from 1 to 10 samples/second. Hard-wired single-pole, low-pass filters are installed on the analog inputs to the controller to protect the sampler from aliasing errors.

Distributed Control Systems Some knowledge of the distributed control system (DCS) is useful in understanding electronic controllers. A DCS is a process control system with sufficient performance to support large-scale real-time process applications. The DCS has (1) an operations workstation with a cathode ray tube (CRT) for display; (2) a controller subsystem that supports various types of controllers and controller functions; (3) an I/O subsystem for transducing data; (4) a higher-level computing platform for performing process supervision, information processing, and analysis; and (5) communication networks to tie the DCS subsystems, plant areas, and other plant systems together. The component controllers used in the controller subsystem portion of the DCS can be of various types and include multiloop controllers, programmable logic controllers, personal computer controllers, singleloop controllers, and fieldbus controllers. The type of electronic controller utilized depends on the size and functional characteristic of the process application being controlled. See the earlier section on distributed control systems. Multiloop Controllers The multiloop controller is a DCS network device that uses a single 32-bit microprocessor to provide control functions to many process loops. The controller operates independent of the other devices on the DCS network and can support from 20 to 500 loops. Data acquisition capability for up to 1000 analog and discrete I/O channels or more can also be provided by this controller. The multiloop controller contains a variety of function blocks (for example, PID, totalizer, lead/lag compensator, ratio control, alarm, sequencer, and Boolean) that can be “soft-wired” together to form complex control strategies. The multiloop controller, as part of a DCS, communicates with other controllers and man/machine interface (MMI) devices also on the DCS network. Programmable Logic Controllers The programmable logic controller (PLC) originated as a solid-state replacement for the hardwired relay control panel and was first used in the automotive industry for discrete manufacturing control. Today, PLCs are used to implement Boolean logic functions, timers, counters, and some math functions and PID control. PLCs are often used with on/off process control valves. PLCs are classified by the number of the I/O functions supported. There are several sizes available, with the smallest PLCs supporting less than 128 I/O channels and the largest supporting over 1023 I/O channels. I/O modules are available that support high-current motor loads, general-purpose voltage and current loads, discrete inputs, ana-

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log I/O and special-purpose I/O for servomotors, stepping motors, high-speed pulse counting, resolvers, decoders, multiplexed displays, and keyboards. PLCs often come with lights or other discrete indicators to determine the status of key I/O channels. When used as an alternative to a DCS, the PLC is programmed with a handheld or computer-based loader. The PLC is typically programmed with basic ladder logic or a high-level computer language such as BASIC, FORTRAN, or C. Programmable logic controllers use 16- or 32-bit microprocessors and offer some form of point-to-point communications such as RS-232C, RS-422, or RS-485. Personal Computer Controller Because of its high performance at low cost and its unexcelled ease of use, application of the personal computer (PC) as a platform for process controllers is growing. When configured to perform scan, control, alarm, and data acquisition (SCADA) functions and combined with a spreadsheet or database management application, the PC controller can be a lowcost, basic alternative to the DCS or PLC. Using the PC for control requires installation of a board into the expansion slot in the computer, or the PC can be connected to an external I/O module using a standard communication port on the PC (RS-232, RS-422, or IEEE-488). The controller card/module supports 16- or 32-bit microprocessors. Standardization and high volume in the PC market has produced a large selection of hardware and software tools for PC controllers. Single-Loop Controller The single-loop controller (SLC) is a process controller that produces a single output. SLCs can be pneumatic, analog electronic, or microprocessor-based. Pneumatic SLCs are discussed in the pneumatic controller section, and analog electronic SLC is not discussed because it has been virtually replaced by the microprocessor-based design. The microprocessor-based SLC uses an 8- or 16-bit microprocessor with a small number of digital and analog process input channels with control logic for the I/O incorporated within the controller. Analog inputs and outputs are available in the standard ranges (1–5 volts DC and 4–20 mA DC). Direct process inputs for temperature sensors (thermistor RTD and thermocouple types) are available. Binary outputs are also available. The face of the SLC has some form of visible display and pushbuttons that are used to view or adjust control values and configuration. SLCs are available for mounting in panel openings as small as 48 mm by 48 mm (1.9 by 1.9 inches). The processor based SLC allows the operator to select a control strategy from a predefined set of control functions. Control functions include PID, on/off, lead/lag, adder/subtractor, multiply/divider, filter functions, signal selector, peak detector, and analog track. SLCs feature auto/manual transfer switching, multi-setpoint, self diagnostics, gain scheduling, and perhaps also time sequencing. Most processorbased SLCs have self-tuning PID control algorithms. Sample times for the microprocessor-based SLCs vary from 0.1 to 0.4 seconds. Low-pass analog electronic filters are installed on the process inputs to stop aliasing errors caused by fast changes in the process signal. Input filter time constants are typically in the range from 0.1 to 1 s. Microprocessor-based SLCs may be made part of a DCS by using the communication port (RS-488 is common) on the controller or may be operated in a standalone mode independent of the DCS. Fieldbus Controller The benefits of eliminating all analog communication links to and from the devices in the process loop (including final control elements and measurement transmitters) have stimulated considerable interest in standardizing a suitable digital fieldbus communication network. Although a universal network standard is not currently complete (see “Digital Field Communications” in this section), several manufacturers have made available field devices that feature basic process-controller functionality. These controllers, known as fieldbus controllers, reside in the final control element or measurement transmitter and are considered to be an option available with these control devices. A suitable communications modem is present in the device to interface with a proprietary, PCbased, or hybrid analog/digital bus network. Presently, fieldbus controllers are single-loop controllers with 8- and 16-bit microprocessors and are options to digital field-control devices. These controllers support the basic PID control algorithm

and are projected to increase in functionality as the controller market develops. Parameters relating to intrinsic safety, communication type and bit rate, level of DCS support, and ultimate controller performance differentiate currently available fieldbus controllers. Controller Reliability and Application Trends Critical process-control applications demand a high level of reliability from the electronic controller. Some methods that improve the reliability of electronic controllers include: (1) focusing on robust circuit design using quality components; (2) using redundant circuits, modules, or subsystems where necessary; (3) using small-sized backup systems when needed; (4) reducing repair time and using more powerful diagnostics; and (5) distributing functionality to more independent modules to limit the impact of a failed module. Currently, the trend in process control is away from centralized process control and toward an increased number of small distributedcontrol or PLC systems. This trend will put emphasis on the evolution of the fieldbus controller and continued growth of the PC-based controller. Also, as hardware and software improves, the functionality of the controller will increase, and the supporting hardware will be physically smaller. Hence, the traditional lines between the DCS and the PLC will become less distinct as systems will be capable of supporting either function set. Pneumatic Controllers The pneumatic controller is an automatic controller that uses pneumatic pressure as a power source and generates a single pneumatic output pressure. The pneumatic controller is used in single-loop control applications and is often installed on the control valve or on an adjacent pipestand or wall in close proximity to the control valve and/or measurement transmitter. Pneumatic controllers are used in areas where it would be hazardous to use electronic equipment, in locations without power, in situations where maintenance personnel are more familiar with pneumatic controllers, or in applications where replacement with modern electronic controls has not been justified. Process-variable feedback for the controller is achieved by one of two methods. The process variable can (1) be measured and transmitted to the controller by using a separate measurement transmitter with a 0.2–1.0-bar (3–15-psig) pneumatic output, or (2) be sensed directly by the controller, which contains the measurement sensor within its enclosure. Controllers with integral sensing elements are available that sense pressure, differential pressure, temperature, and level. Some controller designs have the set point adjustment knob in the controller, making set point adjustment a local and manual operation. Other types receive a set point from a remotely located pneumatic source, such as a manual air set regulator or another controller, to achieve set point adjustment. There are versions of the pneumatic controller that support the useful one-, two-, and three-mode combinations of proportional, integral, and derivative actions. Other options include auto/manual transfer stations, antireset windup circuitry, on/off control, and process-variable and set point indicators. Pneumatic controllers are made of Bourdon tubes, bellows, diaphragms, springs, levers, cams, and other fundamental transducers to accomplish the control function. If operated on clean, dry plant air, they offer good performance and are extremely reliable. Pneumatic controllers are available with one or two stages of pneumatic amplification, with the two-stage designs having faster dynamic response characteristics. An example of a pneumatic PI controller is shown in Fig. 8-64a. This controller has two stages of pneumatic amplification and a Bourdon tube input element that measures process pressure. The Bourdon tube element is a flattened tube that has been formed into a curve so that changes in pressure inside the tube cause vertical motions to occur at the ungrounded end. This motion is transferred to the left end of the beam, as shown. The resulting motion of the beam is detected by the pneumatic nozzle amplifier, which, by proper sizing of the nozzle and fixed orifice diameters, causes the pressure internal to the nozzle to rise and fall with vertical beam motion. The internal nozzle pressure is routed to the pneumatic relay. The relay, which is constructed like the booster relay described in the “Valve Control Devices” subsection, has a direct linear input-to-output pressure characteristic. The output of the relay is the controller’s output and is piped away to the final control element.

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS

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(a)

(b) FIG. 8-64

Pneumatic controller: (a) example; (b) frequency response characteristic.

To generate the P and I control modes, a feedback circuit consisting of two bellows and two small metering valves has been added to the pneumatic amplifier system described above. The first valve is the proportional gain valve and is adjusted to provide an output pressure that is ratioed to the output pressure from the relay. The ratio used is set by manual adjustment of this valve. The output from the proportional gain valve is connected to a proportional feedback bellows, which provides negative feedback around the pneumatic amplifiers. The output pressure from the proportional gain valve is also connected to another valve, called the reset valve. This valve meters flow to and from a second bellows, known as the reset bellows. The resistance provided by the valve and the capacitance relating to the volume of the bellows forms a time constant that becomes the reset time constant for the controller. The frequency response characteristics of a pneumatic PI controller and an ideal PI controller are shown in Fig. 8-64b. Notice that the gain of the pneumatic controller reaches a limit at the lowest frequencies. This limit is due to a less-than-infinite amount of forward amplifier gain. The manufacturer of the controller designs the forward gain term to be as high as possible to better approximate ideal reset action in the controller but never to reach the ideal. Reset gain for available pneumatic controllers runs between 20 and 100 times the gain implied by a proportional gain of unity. Unity proportional gain implies that a 100 percent change in process input (i.e., a full range change) will generate a 100 percent change in controller output.

The reset time, which is user-adjustable, can range from 0.05 seconds to 80 minutes or more, depending on controller design. The reset time constant, when converted to frequency 1/2(TR) Hz (where TR is the reset time in seconds), determines the frequency where the reset and proportional response characteristics of the controller merge (see Fig. 8-64b). Tuning the reset adjustment on the controller moves the reset frequency to the left or right along the frequency axis and thereby affects the reset action of the controller. The response limit for the controller is a function of the design of the relay, the size of the load volume to which the controller is attached, the setting of the proportional band valve, and the forward gain designed into the pneumatic amplifiers. The frequency response limit (sometimes called the controller bandwidth) for a pneumatic controller into a small instrument load volume is in the 5- to 8-Hz range for two-stage pneumatic designs like the one shown. The dynamic response of the controller to set point changes is essentially the same as that indicated for process-variable changes. The set point adjustment mechanism affects the vertical motion of the nozzle over the beam and results in actions in the controller similar to those produced by changes in the process pressure. The main shortcomings of the pneumatic controller is its lack of flexibility when compared to modern electronic controller designs. Increased range of adjustability, choice of alternate control algorithms, the communication link to the control system, and other features and services provided by the electronic controller make it a superior choice in most of today’s applications.

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CONTROL VALVES A control valve consists of a valve, an actuator, and possibly one or more valve-control devices. The valves discussed in this section are applicable to throttling control (i.e., where flow through the valve is regulated to any desired amount between maximum and minimum limits). Other valves such as check, isolation, and relief valves are addressed in the next subsection. As defined, control valves are automatic control devices that modify the fluid flow rate as specified by the controller. Valves Valves are categorized according to their design style. These styles can be grouped into type of stem motion—linear or rotary. The valve stem is the rod, shaft, or spindle that connects the actuator with the closure member (i.e., a movable part of the valve that is positioned in the flow path to modify the rate of flow). Motion of either type is known as travel. The major categories are described briefly below. Globe and Angle The most common linear stem-motion control valve is the globe valve. The name comes from the globular shaped cavities around the port. In general, a port is any fluid passageway, but often the reference is to the passage that is blocked off by the closure member when the valve is closed. In globe valves, the closure member is called a plug. The plug in the valve shown in Fig. 8-65 is guided by a large-diameter port and moves within the port to provide the flow control orifice of the valve. A very popular alternate construction is a cage-guided plug as illustrated in Fig. 8-66. In many such designs, openings in the cage provide the flow control orifices. The valve seat is the zone of contact between the moving closure member and the stationary valve body, which shuts off the flow when the valve is closed. Often the seat in the body is on a replaceable part known as a seat ring. This stationary seat can also be designed as an integral part of the cage. Plugs may also be port-guided by wings or a skirt that fits snugly into the seat-ring bore. One distinct advantage of cage guiding is the use of balanced plugs in single-port designs. The unbalanced plug depicted in Fig. 8-65 is subjected to a static pressure force equal to the port area times the valve pressure differential (plus the stem area times the downstream pressure) when the valve is closed. In the balanced design (Fig. 8-66),

Actuator yoke mounting boss

note that both the top and bottom of the plug are subjected to the same downstream pressure when the valve is closed. Leakage via the plug-to-cage clearance is prevented by a plug seal. Both plug types are subjected to hydrostatic force due to internal pressure acting on the stem area and to dynamic flow forces when the valve is flowing. The plug, cage, seat ring, and associated seals are known as the trim. A key feature of globe valves is that they allow maintenance of the trim via a removable bonnet without removing the valve body from the line. Bonnets are typically bolted on but may be threaded in smaller sizes. Angle valves are an alternate form of the globe valve. They often share the same trim options and have the top-entry bonnet style. Angle valves can eliminate the need for an elbow but are especially useful when direct impingement of the process fluid on the body wall is to be avoided. Sometimes it is not practical to package a long trim within a globe body, so an angle body is used. Some angle bodies are self draining, which is an important feature for dangerous fluids. Butterfly The classic design of butterfly valves is shown in Fig. 8-67. Its chief advantage is high capacity in a small package and a very low initial cost. Much of the size and cost advantage is due to the wafer body design, which is clamped between two pipeline flanges. In the simplest design, there is no seal as such, merely a small clearance gap between the disc OD and the body ID. Often a true seal is provided by a resilient material in the body that is engaged via an interference fit with the disc. In a lined butterfly valve, this material covers the entire body ID and extends around the body ends to eliminate the need for pipeline joint gaskets. In a fully lined valve, the disc is also coated to minimize corrosion or erosion. A high-performance butterfly valve has a disc that is offset from the shaft center line. This eccentricity causes the seating surface to move away from the seal once the disc is out of the closed position, reducing friction and seal wear. Also known as an eccentric disc valve, its advantage is improved shutoff while maintaining high ultimate capacity at a

Stem

Graphite packing

Spring-loaded PTFE V-ring packing

Bonnet

Bonnet gasket

Backup ring

Guide bushing Plug

Cage

Body

Seat ring

Gasket

Post-guided contour-plug globe valve with metal seat and raisedface flange end connections. (Courtesy Fisher-Rosemount.)

FIG. 8-65

Seal ring Valve plug

Seat

PTFE disk

Metal disk retainer

Cage-guided balanced-plug globe valve with polymer seat and plug seal. (Courtesy Fisher-Rosemount.)

FIG. 8-66

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS

8-65

Partial cutaway of wafer-style lined butterfly valve. (Courtesy Fisher-Rosemount.)

Bodies are usually in two or three pieces or have a removable retainer to facilitate installing seals. End connections are usually flanged or threaded in small sizes, although segmented-ball valves are offered in wafer style also. Plug There are two substantially different rotary-valve design categories referred to as plug valves. The first consists of a cylindrical or slightly conical plug with a port through it. The plug rotates to vary the flow much like a ball valve. The body is top-entry but is geometrically simpler than a globe valve and thus can be lined with fluorocarbon polymer to protect against corrosion. These plug valves have excellent shutoff but are generally not for modulating service due to high friction. A variation of the basic design (similar to the eccentric butterfly disc) only makes sealing contact in the closed position and is used for control. The other rotary plug design is portrayed in Fig. 8-69. The seating surface is substantially offset from the shaft, producing a ball-valve-like motion with the additional cam action of the plug into the seat when closing. In reverse flow, high-velocity fluid motion is directed inward— impinging on itself and only contacting the plug and seat ring. Multi-Port This term refers to any valve or manifold of valves with more than one inlet or outlet. For throttling control, the threeway body is used for blending (two inlets, one outlet) or as a divertor (one inlet, two outlets). A three-way valve is most commonly a special globelike body with special trim that allows flow both over and under the plug. Two rotary valves and a pipe tee can also be used. Special three-, four-, and five-way ball-valve designs are used for switching applications.

reasonable cost. This cost advantage relative to other design styles is particularly true in sizes above 6-inch nominal pipe size (NPS). Improved shutoff is due to advances in seal technologies, including polymer, flexing metal, combination metal with polymer inserts, and so on, many utilizing pressure assist. Ball Ball valves get their name from the shape of the closure member. One version uses a full spherical member with a cylindrical bore through it. The ball is rotated d turn from the full-closed to the full-open position. If the bore is the same diameter as the mating-pipe fitting ID, the valve is referred to as full-bore. If the hole is undersized, the ball valve is considered to be a venturi style. A segmented ball is a portion of a hollow sphere—large enough to block the port when closed. Segmented balls often have a V-shaped contour along one edge, which provides a desirable flow characteristic (see Fig. 8-68). Both full-ball and segmented-ball valves are known for their low resistance to flow when full open. Shutoff leakage is minimized through the use of flexing or spring-loaded elastomeric or metal seals.

Special Application Valves Digital Valves True digital valves consist of discrete solenoidoperated flow ports that are sized according to binary weighing. The valve can be designed with sharp-edged orifices or with streamlined nozzles that can be used for flow metering. Precise control of the throttling-control orifice is the strength of the digital valve. Digital valves are mechanically complicated and expensive, and they have considerably reduced maximum flow capacities over the globe and rotary valve styles. Cryogenic Service Valves designed to minimize heat absorption for throttling liquids and gases below 80 K are called cryogenic service valves. These valves are designed with small valve bodies to minimize heat absorption and long bonnets between the valve and actuator to allow for extra layers of insulation around the valve. For extreme cases, vacuum jacketing can be constructed around the entire valve to minimize heat influx. High Pressure Valves used for pressures nominally above 760

Stem PTFE molded to inside diameter and sides of backup ring Bearing Shaft

Body

Special disc design to reduce dynamic torque

FIG. 8-67

V-notch ball Ball seal

Seal protector ring Gasket

Actuator mounting Follower shaft

Groove pin Body

Drive shaft Bearing Taper key

FIG. 8-68

Packing follower

Segmented ball valve. Partial view of actuator mounting shown 90° out of position. (Courtesy Fisher-Rosemount.)

8-66

PROCESS CONTROL

Eccentric plug valve shown in erosion-resistant reverse flow direction. Shaded components can be made of hard metal or ceramic materials. (Courtesy Fisher-Rosemount.)

FIG. 8-69

bar (11,000 psi, pressures above ANSI Class 4500) are often customdesigned for specific applications. Normally, these valves are of the plug type and use specially hardened plug and seat assemblies. Internal surfaces are polished, and internal corners and intersecting bores are smoothed to reduce high localized stresses in the valve body. Steam loops in the valve body are available to raise the body temperature to increase ductility and impact strength of the body material. High-Viscous Process Used most extensively by the polymer industry, the valve for high-viscous fluids is designed with smooth finished internal passages to prevent stagnation and polymer degradation. These valves are available with integral body passages through which a heat-transfer fluid is pumped to keep the valve and process fluid heated. Pinch The industrial equivalent of controlling flow by pinching a soda straw is the pinch valve. Valves of this type use fabric-reinforced elastomer sleeves that completely isolate the process fluid from the metal parts in the valve. The valve is actuated by applying air pressure directly to the outside of the sleeve, causing it to contract or pinch. Another method is to pinch the sleeve with a linear actuator with a specially attached foot. Pinch valves are used extensively for corrosive material service and erosive slurry service. This type of valve is used in applications with pressure drops up to 10 bar (145 psi). Fire-Safe Valves that handle flammable fluids may have additional safety-related requirements for minimal external leakage, minimal internal (downstream) leakage, and operability during and after a fire. Being fire-safe does not mean being totally impervious to fire, but a sample valve must meet particular specifications such as American Petroleum Institute (API) 607, Factory Mutual Research Corp. (FM) 7440, or the British Standard 5146 under a simulated fire test. Due to very high flame temperature, metal seating (either primary or as a backup to a burned-out elastomer) is mandatory. Solids Metering The control valves described earlier are primarily used for the control of fluid (liquid or gas) flow. Sometimes these valves, particularly the ball, butterfly, or sliding gate valves, are used to throttle dry or slurry solids. More often, special throttling mechanisms like venturi ejectors, conveyers, knife-type gate valves, or rotating vane valves are used. The particular solids-metering valve hardware

depends on the volume, density, particle shape, and coarseness of the solids to be handled. Actuators An actuator is a device that applies the force (torque) necessary to cause a valve’s closure member to move. Actuators must overcome pressure and flow forces; friction from packing, bearings or guide surfaces, and seals; and provide the seating force. In rotary valves, maximum friction occurs in the closed position and the moment necessary to overcome it is referred to as breakout torque. The rotary valve shaft torque generated by steady-state flow and pressure forces is called dynamic torque. It may tend to open or close the valve depending on valve design and travel. Dynamic torque per unit pressure differential is largest in butterfly valves at roughly 70° open. In linear stem-motion valves, the flow forces should not exceed available actuator force, but this is usually accounted for by default when the seating force is provided. Actuators often provide a failsafe function. In the event of an interruption in the power source, the actuator will place the valve in a predetermined safe position, usually either full open or full closed. Safety systems are often designed to trigger local failsafe action at specific valves to cause a needed action to occur, which may not be a complete process or plant shutdown. Actuators are classified according to their power source. The nature of these sources leads naturally to design features that make their performance characteristics distinct. Pneumatic Despite the availability of more sophisticated alternatives, the pneumatically driven actuator is still by far the most popular type. Historically the most common has been the spring and diaphragm design (Fig. 8-70). The compressed air input signal fills a chamber sealed by an elastomeric diaphragm. The pressure force on the diaphragm plate causes a spring to be compressed and the actuator stem to move. This spring provides the failsafe function and contributes to the dynamic stiffness of the actuator. If the accompanying valve is “push-down-to-close,” the actuator depicted in Fig. 8-70 would be described as “air-to-close” or synonymously as fail-open. A Air connection Diaphragm casing Diaphragm and stem shown in up position Diaphragm plate Lower diaphragm casing Actuator spring Actuator stem Spring seat

Spring adjustor Stem connector

Yoke Travel indicator disk Indicator scale FIG. 8-70 Spring and diaphragm actuator with an “up” fail-safe mode. Spring adjuster allows slight alteration of bench set. (Courtesy Fisher-Rosemount.)

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS slightly different design yields “air-to-open” or fail-closed action. The spring is typically precompressed to provide a significant available force in the failed position (e.g., to provide seating load). The spring also provides a proportional relationship between the force generated by air pressure and stem position. The pressure range over which a spring and diaphragm actuator strokes in the absence of valve forces is known as the bench set. The chief advantages of spring and diaphragm actuators are their high reliability, low cost, adequate dynamic response, and failsafe action—all of which are inherent in their simple design. Alternately, the pressurized chamber can be formed by a circular piston with a seal on its outer edge sliding within a cylindrical bore. Higher operating pressure (6 bar [∼90 psig] is typical) and longer strokes are possible. Piston actuators can be spring-opposed but many times are in a dual-acting configuration (i.e., compressed air is applied to both sides of the piston with the net force determined from the pressure difference—see Fig. 8-71). Dynamic stiffness is usually higher with piston designs than with spring and diaphragm actuators; see “Positioner/Actuator Stiffness.” Failsafe action, if necessary, is achieved without a spring through the use of additional solenoid valves, trip valves, or relays. See “Valve Control Devices.” Motion Conversion Actuator power units with translational output can be adapted to rotary valves that generally need 90° or less rotation. A lever is attached to the rotating shaft and a link with pivoting means on the end connects to the linear output of the power unit, an arrangement similar to an internal combustion engine crankshaft, connecting rod, and piston. When the actuator piston, or more commonly the diaphragm plate, is designed to tilt, one pivot can be eliminated (see Fig. 8-71). Scotch yoke and rack and pinion arrangements are also commonly used, especially with piston power units. Friction and changing mechanical advantage of these motion-conversion mechanisms means the available torque may vary greatly with travel. One notable exception is vane-style rotary actuators whose offset “piston” pivots, giving direct rotary output. Hydraulic The design of typical hydraulic actuators is similar to double-acting piston pneumatic types. One key advantage is the high pressure (typically 35 to 70 bar [500 to 1000 psi]), which leads to high thrust in a smaller package. The incompressible nature of the

O-ring seal Piston

Piston rod

Cylinder Sliding seal

Lever

Double-acting piston rotary actuator with lever and tilting piston for motion conversion. (Courtesy Fisher-Rosemount.)

FIG. 8-71

8-67

hydraulic oil means these actuators have very high dynamic stiffness. The incompressibility and small chamber size connote fast stroking speed and good frequency response. The disadvantages include high initial cost, especially when considering the hydraulic supply. Maintenance is much more difficult than with pneumatics, especially on the hydraulic positioner. Electrohydraulic actuators have similar performance characteristics and cost/maintenance ramifications. The main difference is that they contain their own electric-powered hydraulic pump. The pump may run continuously or be switched on when a change in position is required. Their main application is remote sites without an air supply when a failsafe spring return is needed. Electric The most common electric actuators use a typical motor—three-phase AC induction, capacitor-start split-phase induction, or DC. Normally the motor output passes through a large gear reduction and, if linear motion output is required, a ball screw or thread. These devices can provide large thrust, especially given their size. Lost motion in the gearing system does create backlash, but if not operating across a thrust reversal, this type of actuator has very high stiffness. Usually the gearing system is self-locking, which means that forces on the closure member cannot move it by spinning a nonenergized motor. This behavior is called a lock-in-last-position failsafe mode. Some gear systems (e.g., low-reduction spur gears) can be backdriven. A solenoid-activated mechanical brake or locking current to motor field coils is added to provide lock-in-last-position fail mode. A battery backup system for a DC motor can guard against power failures. Otherwise, an electric actuator is not acceptable if fail-open/ closed action is mandatory. Using electrical power requires environmental enclosures and explosion protection, especially in hydrocarbon-processing facilities; see the full discussion in “Valve Control Devices.” Unless sophisticated speed-control power electronics is used, position modulation is achieved via bang-bang control. Mechanical inertia causes overshoot, which is (1) minimized by braking and/or (2) hidden by adding dead band to the position control. Without these provisions, high starting currents would cause motors to overheat from constant “hunting” within the position loop. Travel is limited with power interruption switches or with force (torque) electromechanical cutouts when the closed position is against a mechanical stop (e.g., a globe valve). Electric actuators are often used for on/off service. Stepper motors can be used instead, and they, as their name implies, move in fixed incremental steps. Through gear reduction, the typical number of increments for 90° rotation range from 5000 to 10,000; hence positioning resolution at the actuator is excellent. Position overshoot is not an issue, and added dead band need only be a few steps away. An electromagnetic solenoid can be used to directly actuate the plug on very small linear stem-motion valves. A solenoid is usually designed as a two-position device, so this valve control is on/off. Special solenoids with position feedback can provide proportional action for modulating control. Force requirements of medium-sized valves can be met with piloted plug designs, which use process pressure to assist the solenoid force. Piloted plugs are also used to minimize the size of common pneumatic actuators, especially when there is need for high seating load. Manual A manually positioned valve is by definition not an automatic control valve, but it may be involved with process control. For rotary valves, the manual operator can be as simple as a lever, but a wheel driving a gear reduction is necessary in larger-size valves. Linear motion is normally created with a wheel turning a screw-type device. A manual override is usually available as an option for the powered actuators listed above. For spring-opposed designs, an adjustable travel stop will work as a one-way manual input. In more complex designs the handwheel can provide loop control override via an engagement means. Some gear-reduction systems of electric actuators allow the manual positioning to be independent from the automatic positioning without declutching. Valve-Control Devices Devices mounted on the control valve that interface various forms of input signals, monitor and transmit valve position, or modify valve response are valve-control devices. In some applications, several auxiliary devices are used together on the

8-68

PROCESS CONTROL

same control valve. For example, mounted on the control valve, one may find a current-to-pressure transducer, a valve positioner, a volume booster relay, a solenoid valve, a trip valve, a limit switch, a process controller, and/or a stem-position transmitter. Figure 8-72 shows a valve positioner mounted on the yoke leg of a spring and diaphragm actuator. As most throttling control valves are still operated by pneumatic actuators, the control-valve device descriptions that follow relate primarily to devices that are used with pneumatic actuators. The function of hydraulic and electrical counterparts are very similar. Specific details on a particular valve-control device are available from the vendor of the device. Transducers The current-to-pressure transducer (I/P transducer) is a conversion interface that accepts a standard 4–20 mA input current from the process controller and converts it to a pneumatic output in a standard pneumatic pressure range (normally 0.2–1.0 bar [3–15 psig] or, less frequently, 0.4–2.0 bar [6–30 psig]). The output pressure generated by the transducer is connected directly to the pressure connection on a spring-opposed diaphragm actuator or to the input of a pneumatic valve positioner. Figure 8-73a is the schematic of a basic I/P transducer. The transducer shown is characterized by (1) an input conversion that generates an angular displacement of the beam proportional to the input current, (2) a pneumatic amplifier stage that converts the resulting angu-

(a)

(b)

(c) Current to pressure transducer components parts: (a) direct current to pressure conversion; (b) pneumatic booster amplifier (relay); (c) block diagram of a modern I/P transducer. FIG. 8-73

FIG. 8-72 Valve and actuator with valve positioner attached. (Courtesy FisherRosemount.)

lar displacement to pneumatic pressure, and (3) a pressure area that serves as a means to return the beam back to very near its original position when the new output pressure is achieved. The result is a device that generates a pressure output that tracks the input current signal. The transducer shown in Fig. 8-73a is used to provide pressure to small load volumes (normally 4.0 in3 or less), such as a positioner or booster input. With only one stage of pneumatic amplification, the flow capacity of this transducer is limited and not sufficient to provide responsive load pressure directly to a pneumatic actuator. The flow capacity of the transducer can be increased by adding a booster relay like the one shown in Fig. 8-73b. The flow capacity of the booster relay is nominally fifty to one hundred times that of the nozzle amplifier shown in Fig. 8-73a and makes the combined transducer/booster suitably responsive to operate pneumatic actuators. This type of transducer is stable into all sizes of load volumes and produces measured accuracy (see Instrument Society of America [ISA]S51.1-1979, “Process Instrumentation Terminology” for the definition of measured accuracy) of 0.5 percent to 1.0 percent of span. Better measured accuracy results from the transducer design shown in Fig. 8-73c. In this design, pressure feedback is taken at the output of the booster-relay stage and fed back to the main summer. This allows the transducer to correct for errors generated in the pneumatic booster as well as errors in the I/P-conversion stage. Also, particularly with the new analog electric and digital versions of this design, PID control is used in the transducer-control network to give extremely good static accuracy, fast dynamic response, and reasonable

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS stability into a wide range of load volumes (small instrument bellows to large actuators). Also, environmental factors such as temperature change, vibration, and supply pressure fluctuation affect this type of transducer the least. Even a perfectly accurate I/P transducer cannot compensate for stem-position errors generated by friction, backlash, and varying force loads coming from the actuator and valve. To do this compensation, a different control-valve device, known as a valve positioner, is required. Valve Positioners The valve positioner, when combined with an appropriate actuator, forms a complete closed-loop valve-position control system. This system makes the valve stem conform to the input signal coming from the process controller in spite of force loads that the actuator may encounter while moving the control valve. Usually, the valve positioner is contained in its own enclosure and is mounted on the control valve. The key parts of the positioner/actuator system, shown in Fig. 8-74a, are (1) an input-conversion network, (2) a stem-position feedback network, (3) a summing junction, (4) an amplifier network, and (5) an actuator. The input-conversion network shown is the interface between the input signal and the summer. This block converts the input current or pressure (from an I/P transducer or a pneumatic process controller) to a voltage, an electric current, a force, torque, displacement or other particular variable that can be directly used by the summer. The input conversion usually contains a means to adjust the slope and offset of the block to provide for a means of spanning and zeroing the positioner during calibration. In addition, means for changing the sense (known as “action”) of the input/output characteristic are oftentimes addressed in this block. Also, exponential, logarithmic or other predetermined characterization can be put in this block to provide a characteristic that is useful in offsetting or reinforcing a nonlinear valve or process characteristic. The stem-position feedback network converts stem travel to a useful form for the summer. This block includes the feedback linkage,

(a)

(b) Positioner/actuators: (a) generic block diagram; (b) an example of a pneumatic positioner/actuator.

FIG. 8-74

8-69

which varies with actuator type. Depending on positioner design, the stem-position feedback network can provide span and zero and characterization functions similar to that described for the inputconversion block. The amplifier network provides signal conversion and suitable static and dynamic compensation for good positioner performance. Control from this block usually reduces down to a form of proportional or proportional plus derivative control. The output from this block in the case of a pneumatic positioner is a single connection to the spring and diaphragm actuator or two connections for push-pull operation of a springless piston actuator. The action of the amplifier network and the action of the stem-position feedback can be reversed together to provide for reversed positioner action. By design, the gain of the amplifier network shown in Fig. 8-74a is made very large. Large gain in the amplifier network means that only a small proportional deviation will be required to position the actuator through its active range of travels. This means that the signals into the summer track very closely and that the gain of the input-conversion block and the stem-position feedback block determine the closedloop relationship between the input signal and the stem travel. Large amplifier gain also means that only a small amount of additional stem-travel deviation will result when large external force loads are applied to the actuator stem. For example, if the positioner’s amplifier network has a gain of 50 and assuming that high packing-box friction loads require 25 percent of the actuator’s range of thrust to move the actuator, then only 25 percent/50 or 0.5 percent deviation between input signal and output travel will result due to valve friction. Figure 8-74b is an example of a pneumatic positioner/actuator. The input signal is a pneumatic pressure that (1) moves the summing beam, which (2) operates the spool valve amplifier, which (3) provides flow to and from the piston actuator, which (4) causes the actuator to move and continue moving until (5) the feedback force returns the beam to its original position and stops valve travel at a new position. Typical positioner operation is thereby achieved. Static performance measurements related to positioner/actuator operation are: conformity, measured accuracy, hysteresis, dead band, repeatability, and locked stem-pressure gain. Definitions and standardized test procedures for determining these measurements can be found in ISA-S75.13-1989, “Method of Evaluating the Performance of Positioners with Analog Input Signals and Pneumatic Output”. Dynamics of Pneumatic Positioners Dynamically, the pneumatic positioner is characterized by the combined effects of gain and capacitance and nonlinear effects such as valve friction and flowcapacity saturation. Generally, there is a threshold level of input signal below which the positioner output will not respond at all. This band is on the order of 0.1 percent of input span for pneumatic positioners but can be larger if significant valve friction is present. Above this threshold level, but below the level that causes velocity saturation, the positioner is approximately linear and likened to a second-order lowpass filter (see Fig. 8-75a). Natural frequencies range from 0.3 to 3.0 Hz and damping ratios of 0.6 to 2.0 are common and dependent on positioner design and the physical size of the actuator volume. At higher drive levels, the flow capacity of the positioner is reached and attenuation of the resulting travel begins. Positioner/Actuator Stiffness Minimizing the effect of dynamic loads on valve-stem travel is an important characteristic of the positioner/actuator. Stem position must be maintained in spite of changing reaction forces caused by valve throttling. These forces can be random in nature (buffeting force) or result from a negative sloped force/stem travel characteristic (negative gradient); either could result in valvestem instability and loss of control. To reduce and eliminate the effect of these forces, the effective stiffness of the positioner/actuator must be made sufficiently high to maintain adequate stability of the valve stem. The stiffness characteristic of the positioner/actuator varies with frequency. Figure 8-75b indicates the stiffness of the positioner/actuator is high at low frequencies and is directly related to the lockedstem pressure gain provided by the positioner. As frequency increases, a dip in the stiffness curve results from dynamic gain attenuation in the pneumatic amplifiers in the positioner. The value at the bottom of the dip is the sum of the mechanical stiffness of the spring in the actu-

8-70

PROCESS CONTROL

(a)

(b) Frequency response curves for a pneumatic positioner/actuator: (a) input signal to stem travel for a 69-inch2 spring and diaphragm actuator with a 1.5-inch total travel and 3–15 psig input pressure; (b) dynamic stiffness for the same positioner/actuator. FIG. 8-75

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS ator and the air spring effect produced by air enclosed in the actuator casing. The air spring effect results from adiabatic expansion and compression of the air in the actuator casing. Numerically, the small perturbation value for air spring stiffness in Newtons/meter is given by Eq. (8-107). γPaA2a Air spring rate = ᎏ (8-107) V where γ is ratio of specific heats (1.4 for air), Pa is the actuator pressure in Pascal absolute, Aa is the actuator pressure area in m2, and V is the internal actuator volume in m3. Notice in the figure that the minimum stiffness value (mechanical spring stiffness + air spring stiffness) is several times larger than the stiffness produced by the spring in the actuator (shown as a dotted line) by itself. This indicates that the air spring stiffness is quite significant and worth considering in actuator design and actuator sizing. To the right of the dip, the inertia effects of the mass of the moving parts of the valve and actuator cause the overall system stiffness to rise with increasing frequency. Positioner Application Positioners are widely used on pneumatic valve actuators. More often than not, they provide improved process-loop control because they reduce valve-related nonlinearity. Dynamically, positioners maintain their ability to improve controlvalve performance for sinusoidal input frequencies up to about one half of the positioner bandwidth. At input frequencies greater than this, the attenuation in the positioner amplifier network gets large, and valve nonlinearity begins to affect final control-element performance more significantly. Because of this, the most successful use of the positioner occurs when the positioner-response bandwidth is greater than twice that of the most dominant time lag in the process loop. Some typical examples of where the dynamics of the positioner are sufficiently fast to improve process control are the following: 1. In a distributed control system (DCS) process loop with an electronic transmitter. The DCS controller and the electronic transmitter have time constants that are dominant over the positioner response. Positioner operation is therefore beneficial in reducing valve-related nonlinearity. 2. In a process loop with a pneumatic controller and a large process time constant. Here the process time constant is dominant, and the positioner will improve the linearity of the final control element. Some common processes with large time constants that benefit from positioner application are liquid level, temperature, large volume gas pressure, and mixing. 3. Additional situations where valve positioners are used are as follows: • On springless actuators where the actuator is not usable for throttling control without position feedback. • When split ranging is required to control two or more valves sequentially. In the case of two valves, the smaller control valve is calibrated to open in the lower half of the input signal range and a larger valve is calibrated to open in the upper half of the input signal range. Calibrating the input command signal range in this way is known as split-range operation and increases the practical range of throttling process flows over that of a single valve. • In open-loop control applications where best static accuracy is needed. On occasion, positioner use can degrade process control. Such is the case when the process controller, the process, and the process transmitter have time constants that are similar or smaller than that of the positioner/actuator. This situation is characterized by low processcontroller P gain (P gain < 0.5), and hunting or limit cycling of the process variable is observed. Improvements here can be made by doing one of the following: • Install a dominant first-order low-pass filter in the loop ahead of the positioner and retune the process loop. This should allow increased proportional gain in the process loop and reduce hunting. Possible means for adding the filter include adding it to the firmware of the DCS controller, by adding an external RC network on the output of the process controller or by enabling the filter function in the

8-71

input of the positioner if it is available. Also, some transducers, when connected directly to the actuator, form a dominant first-order lag that can be used to stabilize the process loop. • Select a positioner with a faster response characteristic. Booster Relays The booster relay is a single-stage power amplifier having a fixed gain relationship between the input and output pressures. The device is packaged as a complete standalone unit with pipe-thread connections for input, output, and supply pressure. The booster amplifier shown in Fig. 8-73b shows the basic construction of the booster relay. Enhanced versions are available that provide specific features such as: (1) variable gain to split the output range of a pneumatic controller to operate more than one valve or to provide additional actuator force; (2) low hysteresis for relaying measurement and control signals; (3) high flow capacity for increased actuatorstroking speed; and (4) arithmetic, logic, or other compensation functions for control system design. A particular type of booster relay, called a dead-band booster, is shown in Fig. 8-76. This booster is designed to be used exclusively between the output of a valve positioner and the input to a pneumatic actuator. It is designed to provide extra flow capacity to stroke the actuator faster than with the positioner alone. The dead-band booster is designed intentionally with a large dead band (approximately 5 percent of the input span), elastomer seats for tight shutoff, and an adjustable bypass valve connected between the input and the output of the booster. The bypass valve is tuned to provide the best compromise between increased actuator stroking speed and positioner/actuator stability. With the exception of the dead-band booster, the application of booster relays has diminished somewhat by the increased use of current-to-pressure transducers, electropneumatic positioners, and electronic control systems. Transducers and valve positioners serve much the same functionality as the booster relay in addition to interfacing with the electronic process controller. Solenoid Valves The electric solenoid valve has two output states. When sufficient electric current is supplied to the coil, an internal armature moves against a spring to an extreme position. This motion causes an attached pneumatic or hydraulic valve to operate. When current is removed, the spring returns the armature and the attached solenoid valve to the deenergized position. An intermediate pilot stage is sometimes used when additional force is required to operate the main solenoid valve. Generally, solenoid valves are used to pressurize or vent the actuator casing for on/off control-valve application and safety shutdown applications. Input signal Diaphragms

Exhaust port

Bypass valve adjusting screw Adjustable restriction

Exhaust

Supply port

Supply

FIG. 8-76

Output to actuator

Dead-band booster relay. (Courtesy Fisher-Rosemount.)

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PROCESS CONTROL

Trip Valves The trip valve is part of a system that is used where a specific valve action (i.e., fail up, fail down, or lock in last position) is required when pneumatic supply pressure to the control valve falls below a preset level. Trip systems are used primarily on springless piston actuators requiring fail-open or fail-closed action. An air storage or “volume” tank and a check valve are used with the trip valve to provide power to stroke the valve when supply pressure is lost. Trip valves are designed with hysteresis around the trip point to avoid instability when the trip pressure and the reset pressure settings are too close to the same value. Limit Switches and Stem-Position Transmitters Travel-limit switches, position switches, and valve-position transmitters are devices that, when mounted on the valve, actuator, damper, louver, or other throttling element, detect the component’s relative position. The switches are used to operate alarms, signal lights, relays, solenoid valves, or discrete inputs into the control system. The valve-position transmitter generates a 4–20-mA output that is proportional to the position of the valve. Fire and Explosion Protection Electrical equipment can be a source of ignition in environments with combustible concentrations of gas, liquid, dust, fibers, or flyings. Most of the time it is possible to locate the electronic equipment away from these hazardous areas. However, where electric or electronic valve-mounted instruments must be used in areas where there is a hazard of fire or explosion, the equipment must be designed to meet requirements for safety. Articles 500 through 504 of the National Electrical Code address definitions of hazardous locations and the requirements for electrical devices in locations where fire or explosion hazard exists. NFPA (National Fire Protection Agency) 497M addresses the properties and group classification of gas, vapor, and dust for electrical devices in hazardous locations. With valve-mounted accessories, the approved protection concepts most often used for safety protection are explosion-proof, intrinsically safe, nonincendive, and dust-ignition-proof. The explosion-proof enclosure is designed such that an explosion in the interior of the enclosure containing the electronic circuits will be contained. The enclosure will not allow sufficient flame to escape to the exterior to cause an ignition. Also, a surface temperature rating is given to the device. This rating must indicate a lower surface temperature than the ignition temperature of the gas in the hazardous area. Explosion-proof enclosures are characterized by strong metal enclosures with special close-fitting access covers and breathers that contain an ignition to the inside of the enclosure. Field wiring in the hazardous environment is enclosed in a metal conduit of the mineralinsulated-cable type. All conduit and cable connections or cable terminations are threaded and explosion-proof. Conduit seals are put into the conduit or cable system at locations defined by the National Electric Code (Article 501) to prevent gas and vapor leakage and to prevent flames from passing from one part of the conduit system to the other. The intrinsically safe (I.S.) control-valve device contains circuits that are incapable of releasing sufficient electrical or thermal energy to cause ignition of a specified hazardous mixture under normal or fault operating conditions of the circuit. I.S. circuits are designed with voltage- and current-limiting networks added where necessary to achieve approved levels of safety. I.S. field wiring need not be enclosed in metal conduit but must be kept separate from wiring for nonintrinsically safe circuits. Intrinsically safe field wiring must be energy limited, usually by a Zener diode barrier circuit located in the control room. The manufacture of the intrinsically safe control-valve devices must list the identification number of the control drawing on the nameplate attached to the approved device. The control drawing contains information showing approved combinations of accessories and other connected apparatus such as Zener diode energy barriers. ANSI/ISA S12.12, “Nonincendive Electrical Equipment for Use in Class I and Class II, Division 2 and Class III, Division 1 and 2 Hazardous (Classified) Locations,” addresses requirements for nonincendive electrical equipment and wiring. Nonincendive apparatus and/or field wiring refers to approved equipment or wiring that is incapable of imparting sufficient energy to ignite the specified hazardous atmosphere under normal circuit operating conditions. Nonincendive protection is considered for applications where hazardous concentrations

of flammable gas and vapors or combustible dusts are only present under abnormal operating conditions or in those applications where easily ignited fibers or flyings are present in sufficient quantities to cause ignitable mixtures. For applications where hazardous concentrations of flammable gas and vapors or combustible dust are present continuously, intermittently, or periodically, more stringent protection (see Division 1, National Electrical Code, Article 500) offered by explosion-proof or intrinsically safe concepts is required. The dust-ignition-proof protection concept excludes dust from entering the device enclosure and will not permit arcs, sparks, or heat generated by the device to cause ignition of external suspensions or accumulations of the dust. Enclosure requirements can be found in ANSI/UL 1203-1994, “Explosion-Proof and Dust-Ignition-Proof Electrical Equipment for Use in Hazardous Locations.” Certified testing and approval for control-valve devices used in hazardous locations is normally procured by the manufacturer of the device. The manufacturer often goes to a third party laboratory for testing and certification. Applicable approval standards are available from CSA, CENELEC, FM, SAA, and UL. Environmental Enclosures Enclosures for valve accessories are sometimes required to provide protection from specific environmental conditions. The National Electrical Manufacturers Association (NEMA) provides descriptions and test methods for equipment used in specific environmental conditions in NEMA 250. Protection against rain, windblown dust, hose-directed water, and external ice formation are examples of environmental conditions that are covered by NEMA standards. Also, the electronic control-valve device’s level of immunity to, and emission of, electromagnetic interference (EMI) can be an issue in the chemical-valve environment. EMI requirements for the controlvalve devices are presently mandatory in the European Community but voluntary in the United States, Japan, and the rest of the world. International Electrotechnical Commission (IEC) 801, Parts 1 through 4, “Electromagnetic Compatibility for Industrial Process Measurement and Control Equipment,” defines tests and requirements for control-device immunity. Immunity and emission standards are addressed in CENELEC (European Committee for Electrotechnical Standardization) EN 50 081-1:1992, EN 50 081-2:1993, EN 50 082-1:1992, and prEN 50 082-2:1994. Digital Field Communications An increasing number of valvemounted devices are available that support digital communications in addition to, or in place of, the traditional 4–20 mA current signal. These control-valve devices have increased functionality, resulting in reduced setup time, improved control, combined functionality of traditionally separate devices, and control-valve diagnostic capability. Digital communications also allow the control system to become completely distributed where, for example, the process PID controller could reside in the valve positioner or in the process transmitter. The high-performance, all-digital, multidrop communication protocol for use in the process-control industry is known as fieldbus. Presently there are several regional and industry-based fieldbus standards including the French standard, FIP (NFC 4660x approved by UTE), the German standard, Profibus (DIN 19245 approved by DKE), and proprietary standards by DCS vendors. As of 1997, none of these fieldbus standards have been adopted by international standards organizations. The International Electrotechnical Commission (IEC) Standards Committee 65C Working Group 6 (IEC SC65C WG6) and the ISA Standards and Practices Committee 50 (ISA SP50) are presently working on a fieldbus standard, but at the time of this writing, the standard is unfinished. One interim solution supported by some valvedevice products is the hybrid communication method, where both analog and digital communication capabilities are present in the same device. This scheme has the advantage of allowing the communicating valve-control device to be retrofit into a traditional 4–20 mA current loop and still support digital communications between the final control element and the control room. Here the current signal is used to communicate the primary signal value, and the digital communication channel carries secondary variable information, configuration information, calibration information, and alert and diagnostic information. An example of a hybrid protocol that is open (not proprietary) and

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS

FIG. 8-77 Hybrid point-to-point communications between the control room and the control valve device.

in use by several manufacturers of control-valve devices is known as HART ®* (Highway Addressable Remote Transducer) protocol (see Fig. 8-77). With this protocol, the digital communications occur over the same two wires that provide the 4–20 mA process control signal without disrupting the process signal. The protocol uses the frequency-shift keying (FSK) technique where two individual frequencies, one representing the mark and the other representing the space, are superimposed on the 4–20 mA current signal. As the average value of the signals used is zero, there is no DC offset value added to the 4–20 mA signal. Valve Application Technology Functional requirements and the properties of the controlled fluid determine which valve and actuator types are best for a specific application. If demands are modest and no unique valve features are required, the valve-design style selection may be determined solely by cost. If so, general-purpose globe or angle valves provide exceptional value, especially in sizes less than 3-inch NPS and hence are very popular. Beyond type selection, there are many other valve specifications that must be determined properly in order to ultimately yield-improved process control. Materials and Pressure Ratings Valves must be constructed from materials that are sufficiently immune to corrosive or erosive action by the process fluid. Common body materials are cast iron, steel, stainless steel, high-nickel alloys, and copper alloys such as bronze. Trim materials usually need a greater immunity due to the higher fluid velocity in the throttling region. High hardness is desirable in erosive and cavitating applications. Heat-treated and precipitation-hardened stainless steels are common. High hardness is also good for guiding, bearing, and seating surfaces; cobalt-chromium alloys are utilized in cast or wrought form and frequently as welded overlays called hard facing. In less stringent situations, chrome plating, heat-treated nickel coatings, and ion nitriding are used. Tungsten carbide and ceramic trim are warranted in extremely erosive services. See Sec. 28, “Materials of Construction,” for specific material properties. Since the valve body is a pressurized vessel, it is usually designed to comply with a standardized system of pressure ratings. Two common systems are described in the standards ANSI B16.34 and DIN 2401. * HART is a trademark owned by Rosemount, Inc.

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Internal pressure limits under these standards are divided into broad classes, with specific limits being a function of material and temperature. Manufacturers also assign their own pressure ratings based on internal design rules. A common insignia is “250 WOG,” which means a pressure rating of 250 psig (∼17 bar) in water, oil, or gas at ambient temperature. “Storage and Process Vessels” in Sec. 10 provides introductory information on compliance of pressure-vessel design to industry codes (e.g., ASME Boiler and Pressure Vessel Code—Section VIII, ASME B31.3 Chemical Plant and Petroleum Refinery Piping). Valve bodies are also standardized to mate with common piping connections: flanged, butt-weld end, socket-weld end, and screwed end. Dimensional information for some of these joints and class pressure-temperature ratings are included in Sec. 10, “Process Plant Piping.” Control valves have their own standardized face-to-face dimensions that are governed by ISA Standards S75.03, 04, 12, 14, 15, 16, 20, and 22. Butterfly valves are also governed by API 609 and Manufacturers Standardization Society (MSS) SP-67 and 68. Sizing Throttling control valves must be selected to pass the required flow rate given expected pressure conditions. Sizing is not merely matching the end connection size with surrounding piping; it is a key step in ensuring that the process can be properly controlled. Sizing methods range from simple models based on elementary fluid mechanics to very complex models when unusual thermodynamics or nonideal behaviors occur. Basic sizing practices have been standardized upon (e.g., ISA S75.01) and are implemented as PC-based programs by manufacturers. The following is a discussion of very basic sizing equations and the associated physics. Regardless of the particular process variable being controlled (e.g., temperature, level, pH), the output of a control valve is flow rate. The throttling valve performs its function of manipulating flow rate by virtue of being an adjustable resistance to flow. Flow rate and pressure conditions are normally known when a process is designed and the valve resistance range must be matched accordingly. In the tradition of orifice and nozzle discharge coefficients, this resistance is embodied in the valve flow coefficient Cv. The mass flow rate (w) in kg/h is given for a liquid by (p 1−  w = 27.3Cvρ p 2)

(8-108)

where p1 and p2 are upstream and downstream static pressure in bar, respectively. The density of the fluid ρ is expressed in kg/m3. This equation is valid for nonvaporizing, turbulent-flow conditions for a valve with no attached fittings. The relationship can be derived from the principles of conservation of mass and energy. A more complete presentation of sizing relationships is given in ISA S75.01, including provisions for pipe reducers, vaporizing liquids, and Reynolds number effects. While the above equation gives the relationship between pressure and flow from a macroscopic point of view, it does not explain what is going on inside the valve. Valves create a resistance to flow by restricting the cross sectional area of the flow passage and also by forcing the fluid to change direction as it passes through the body and trim. The conservation of mass principle dictates that, for steady flow, density × average velocity × cross sectional area equals a constant. The average velocity of the fluid stream at the minimum restriction in the valve is therefore much higher than at the inlet. Note that due to the abrupt nature of the flow contraction that forms the minimum passage, the main fluid stream may separate from the passage walls and form a jet that has an even smaller cross section, the so-called vena contracta. The ratio of minimum stream area to the corresponding passage area is called the contraction coefficient. As the fluid expands from the minimum cross sectional area to the full passage area in the downstream piping, large amounts of turbulence are generated. Direction changes can also induce significant amounts of turbulence. Some of the potential energy that was stored in the fluid by pressurizing it (e.g., the work done by a pump) is first converted into the kinetic energy of the fast-moving fluid at the vena contracta. Some of that kinetic energy turns into the kinetic energy of turbulence. As the turbulent eddies break down into smaller and smaller structures, viscous effects ultimately convert all of the turbulent energy into heat. Therefore, a valve converts fluid energy from one form to another. For many valve constructions, it is reasonable to approximate the

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PROCESS CONTROL

fluid transition from the valve inlet to the minimum cross section of the flow stream as an isentropic or lossless process. This minimum pressure, pvc, can be estimated from the Bernoulli relationship. See Sec. 6 (“Fluid and Particle Mechanics”) for more background information. Downstream of the vena contracta, the flow is definitely not lossless due to all the turbulence that is generated. As the flow passage area increases and the fluid slows down, some of the kinetic energy of the fluid is converted back to potential energy as the pressure recovers. The remaining energy that is permanently lost via turbulence accounts for the permanent pressure or head loss of the valve. The relative amount of pressure that is recouped determines whether the valve is considered to be high or low recovery. See Fig. 8-78 for an illustration of how the mean pressure changes as fluid moves through a valve. The flow-passage geometry at and downstream of the vena contracta primarily determines the amount of recovery. The amount of recovery is quantified by the liquid pressure recovery factor FL where FL =

p −p ᎏ  p −p 1

2

1

vc

(8-109)

A key limitation of sizing Eq. (8-109) is the limitation to incompressible fluids. For gases and vapors, density is dependent on pressure. For convenience, compressible fluids are often assumed to follow the ideal-gas-law model. Deviations from ideal behavior are corrected for, to first order, with nonunity values of compressibility factor Z. (See Sec. 2, “Physical and Chemical Data,” for definitions and data for common fluids.) For compressible fluids

w = 94.8Cv P1Y

xM ᎏ  TZ w

where P1 is in bar absolute, T1 is inlet temperature in K, Mw is the molecular weight, and x is the dimensionless pressure-drop ratio (p1 − p2)/p1. The expansion factor Y accounts for changes in the fluid density as the fluid passes through the valve and for variation in the contraction coefficient with pressure drop. For convenience the experimental data is approximated by a simple relationship: 1.4x xTγ Y=1−ᎏ for x≤ᎏ (8-111) 3xTγ 1.4 where γ is the ratio of specific heats and xT is the pressure drop ratio factor. Even though a fluid may be compressible, if the value of x is small, the flow will behave as though it is incompressible. In the limit as x goes to zero, Eq. (8-110) reduces to the incompressible form Eq. (8-108) with ρ expressed via the ideal-gas equation of state. Compressible fluids exhibit a phenomenon known as choking. Given a nozzle geometry with fixed inlet conditions, the mass flow rate will increase as P2 is decreased up to a maximum amount at the critical pressure drop. The velocity at the vena contracta has reached sonic and a standing shock has formed. This shock causes a step change in pressure as flow passes through it, and further reduction in P2 does not increase mass flow. xT is a parameter of the flow model that relates to the critical pressure-drop ratio but also accounts for valve geometry effects. The value of xT varies with flow-path geometry; a rough estimate for conventional valves is one-half. In the choked case, xTγ x>ᎏ and Y = 0.67 (8-112) 1.4 Noise Control Sound is a fluctuation of air pressure that can be detected by the human ear. Sound travels through any fluid (e.g., the air) as a compression/expansion wave. This wave travels radially outward in all directions from the sound source. The pressure wave induces an oscillating motion in the transmitting medium that is superimposed on any other net motion it may have. These waves are reflected, refracted, scattered, and absorbed as they encounter solid objects. Sound is transmitted through solids in a complex array of types of elastic waves. Sound is characterized by its amplitude, frequency, phase, and direction of propagation. Sound strength is therefore location-dependent and is often quantified as a sound pressure level (Lp) in dB based on the root-meansquare (rms) sound-pressure (pS) value, where



pS Lp = 10 log10 ᎏ preference

FIG. 8-78 Generic depictions of average pressure at subsequent cross sections throughout a control valve. FLs selected for illustration are 0.9 and 0.63 for low and high recovery, respectively. Internal pressure in the high-recovery valve is shown as a dashed line for flashing conditions (p2 < pv) with pv = B.

(8-110)

1



2

(8-113)

For airborne sound, the reference pressure is 2 × 10−5 Pa (29 × 10−10 psi), which is nominally the human threshold of hearing at 1000 Hz. The corresponding sound pressure level is 0 dB. Conversation is about 50 dB, and a jackhammer operator is subject to 100 dB. Extreme levels such as a jet engine at takeoff might produce 140 dB at a distance of 3 m, which is a pressure amplitude of 200 Pa (29 × 10−3 psi). These examples demonstrate both the sensitivity and wide dynamic range of the human ear. Traveling sound waves carry energy. Sound intensity I is a measure of the power passing through a unit area in a specified direction and is related to pS. Measuring sound intensity in a process plant gives clues as to the location of the source. As one moves away from the source, the fact that the energy is spread over a larger area requires that sound pressure level decrease. For example, doubling one’s distance from a point source reduces the Lp by 6 dB. Viscous action from the induced fluid motion absorbs additional acoustic energy. However, in free air, this viscous damping is negligible over short distances (on the order of a meter). Noise is a group of sounds with many nonharmonic frequency components of varying amplitudes and random phase. The turbulence generated by a throttling valve creates noise. As a valve converts potential energy to heat, some of it becomes acoustic energy as an intermediate step. Valves handling large amounts of compressible fluid through a large pressure change create the most noise because more total power is being transformed. Liquid flows are noisy only

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS under special circumstances as will be seen in the next subsection. Due to the random nature of turbulence and the broad distribution of length and velocity scales of turbulent eddies, valve-generated sound is usually random, broad-spectrum noise. Total sound pressure level from two such statistically uncorrelated sources is (in dB): (pS1)2 + (pS2)2 Lp = 10 log10 ᎏᎏ (8-114) (preference)2 For example, two sources of equal strength combine to create an Lp that is 3 dB higher. While noise is annoying to listen to, the real reasons for being concerned about noise are its impact on people and equipment. Hearing loss can occur due to long-term exposure to moderately high or even short exposure to very high noise levels. The U.S. Occupational Safety and Health Act (OSHA) has specific guidelines for permissible levels and exposure times. The human ear has a frequency-dependent sensitivity to sound. When the effect on humans is the criteria, Lp measurements are weighted to account for the ear’s response. This socalled A-weighted scale is defined in ANSI S1.4 and is commonly reported as LpA. Figure 8-79 illustrates the difference between actual and perceived airborne sound pressure level. At sufficiently high levels, noise and the associated vibration can damage equipment. There are two approaches to fluid-generated noise control—source or path treatment. Path treatment means absorbing or blocking the transmission of noise after it has been created. The pipe itself is a barrier. The sound pressure level inside a standard schedule pipe is roughly 40–60 dB higher than on the outside. Thicker walled pipe reduces levels somewhat more, and adding acoustical insulation on the outside of the pipe reduces ambient levels up to 10 dB per inch of thickness. Since noise propagates relatively unimpeded inside the




FIG. 8-79



Valve-generated sound pressure level spectrums.

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pipe, barrier approaches require the entire downstream piping system to be treated in order to be totally effective. In-line silencers place absorbent material inside the flow stream, thus reducing the level of the internally propagating noise. Noise reductions up to 25 dB can be achieved economically with silencers. The other approach to valve noise problems is the use of quiet trim. Two basic strategies are used to reduce the initial production of noise—dividing the flow stream into multiple paths and using several flow resistances in series. Lp is proportional to mass flow and is dependent on vena contracta velocity. If each path is an independent source, it is easy to show from Eq. (8-114) that p2s is inversely proportional to the number of passages; additionally, smaller passage size shifts the predominate spectral content to higher frequencies, where structural resonance may be less of a problem. Series resistances or multiple stages can reduce maximum velocity and/or produce back pressure to keep jets issuing from multiple passages acting independently. While some of the basic principles are understood, predicting noise for a particle-flow passage requires some empirical data as a basis. Valve manufacturers have developed noise-prediction methods for the valves they build. ISA S75.17 is a public-domain methodology for standard (nonlow noise) valve types, although treatment of some multistage, multipath types is underway. Low-noise hardware consists of special cages in linear stem valves, perforated domes or plates and multichannel inserts in rotary valves, and separate devices that use multiple fixed restrictions. Cavitation and Flashing From the discussion on pressure recovery it was seen that the pressure at the vena contracta can be much lower than the downstream pressure. If the pressure on a liquid falls below its vapor pressure (pv ), the liquid will vaporize. Due to the effect of surface tension, this vapor phase will first appear as bubbles. These bubbles are carried downstream with the flow, where they collapse if the pressure recovers to a value above pv. This pressure-driven process of vapor-bubble formation and collapse is known as cavitation. Cavitation has three negative side effects in valves—noise and vibration, material removal, and reduced flow. The bubble-collapse process is a violent asymmetrical implosion that forms a high-speed microjet and induces pressure waves in the fluid. This hydrodynamic noise and the mechanical vibration that it can produce are far stronger than other noise-generation sources in liquid flows. If implosions occur adjacent to a solid component, minute pieces of material can be removed, which, over time, will leave a rough, cinderlike surface. The presence of vapor in the vena contracta region puts an upper limit on the amount of liquid that will pass through a valve. A mixture of vapor and liquid has a lower density than the liquid alone. While Eq. (8-108) is not applicable to two-phase flows because pressure changes are redistributed due to varying density and the two phases do not necessarily have the same average velocity, it does suggest that lower density reduces total mass flow rate. Figure 8-80 illustrates a typical flow-rate-to-pressure-drop relationship. As with compressible gas flow at a given p1, flow increases as p2 is decreased until the flow chokes (i.e., no additional fluid will pass). The transition between incompressible and choked flow is gradual because, within the convoluted flow passages of valves, the pressure is actually an uneven distribution at each cross section, and, consequently, vapor-formation zones increase gradually. In fact, isolated zones of bubble formation or incipient cavitation often occur at pressure drops well below that at which a reduction in flow is noticeable. The similarity between liquid and gas choking is not serendipitous; it is surmised that the two-phase fluid is traveling at the mixture’s sonic velocity in the throat when choked. Complex fluids with components having varying vapor pressures and/or entrained noncondensable gases (e.g., crude oil) will exhibit soft vaporization/implosion transitions. There are several methods to reduce cavitation or at least its negative side effects. Material damage is slowed by using harder materials and by directing the cavitating stream away from passage walls (e.g., with an angle body flowing down). Sometimes the system can be designed to place the valve in a higher p2 location or add downstream resistance, which creates back pressure. A low recovery valve has a higher minimum pressure for a given p2 and so is a means to eliminate the cavitation itself, not just its side effects. In Fig. 8-78, if pv < “B” neither valve will cavitate substantially. For pv > “B” but < “A,” the

8-76

PROCESS CONTROL and seating surfaces become nicked or worn. Leak passages across the seat-contact line, known as wire drawing, may form and become worse over time—even in hard metal seats under sufficiently high pressure differentials. Polymers used for seat and plug seals and internal static seals include: PTFE (polytetrafluoroethylene) and other fluorocarbons, polyethylene, nylon, polyether-ether-ketone, and acetal. Fluorocarbons are often carbon or glass-filled to improve mechanical properties and heat resistance. Temperature and chemical compatibility with the process fluid are the key selection criteria. Polymer-lined bearings and guides are used to decrease friction, which lessens dead band and reduces actuator force requirements. See Sec. 28, “Materials of Construction,” for properties. Packing forms the pressure-tight seal, where the stem protrudes through the pressure boundary. Packing is typically made from PTFE or, for high temperature, a bonded graphite. If the process fluid is toxic, more sophisticated systems such as dual packing, live-loaded, or a flexible metal bellows may be warranted. Packing friction can significantly degrade control performance. Pipe, bonnet, and internal-trim joint gaskets are typically a flat sheet composite. Gaskets intended to absorb dimensional mismatch are typically made from filled spiralwound flat stainless steel wire with PTFE or graphite filler. The use of asbestos in packing and gaskets has largely been eliminated. Flow Characteristics The relationship between valve flow and valve travel is called the valve-flow characteristic. The purpose of flow characterization is to make loop dynamics independent of load, so that a single controller tuning remains optimal for all loads. Valve gain is one factor affecting loop dynamics. In general, gain is the ratio of change in output to change in input. The input of a valve is travel (y) and the output is flow (w). Since pressure conditions at the valve can depend on flow (hence travel), valve gain is dw ∂w dCv ∂w dp1 ∂w dp2 ᎏ=ᎏᎏ+ᎏᎏ+ᎏᎏ dy ∂Cv dy ∂p1 dy ∂p2 dy

FIG. 8-80

Liquid flow rate versus pressure drop (assuming constant P1 and Pv).

high recovery valve will cavitate substantially, but the low recovery valve will not. Special anticavitation trims are available for globe/angle valves and more recently for some rotary valves. These trims use multiple contraction/expansion stages or other distributed resistances to boost FL to values sometimes near unity. If p2 is below pv, the two-phase mixture will continue to vaporize in the body outlet and/or downstream pipe until all liquid phase is gone, a condition known as flashing. The resulting huge increase in specific volume leads to high velocities, and any remaining liquid droplets acquire much of the higher vapor-phase velocity. Impingement of these droplets can produce material damage, but it differs from cavitation damage because it exhibits a smooth surface. Hard materials and directing the two-phase jets away from solid surfaces are means to avoid this damage. Seals, Bearings, and Packing Systems In addition to their control function, valves often need to provide shutoff. ANSI B16.104, FCI 70-2 (1991), and IEC 534-4 all recognize six standard classifications and define their as-shipped qualification tests. Class I is an amount agreed to by user and supplier with no test needed. Classes II, III, and IV are based on an air test with maximum leakage of 0.5 percent, 0.1 percent, and 0.01 percent of rated capacity, respectively. Class V restricts leakage to 5 × 10−6 ml of water per second per mm of port diameter per bar differential. Class VI allows 0.15 to 6.75 ml per minute of air to escape depending on port size; this class implies the need for interference-fit elastomeric seals. With the exception of Class V, all classes are based on standardized pressure conditions that may not represent actual conditions. Therefore, it is difficult to estimate leakage in service. Leakage normally increases over time as seals

(8-115)

An inherent valve flow characteristic is defined as the relationship between flow rate and travel, under constant pressure conditions. Since the last two terms in Eq. (8-115) are zero in this case, the inherent characteristic is necessarily also the relationship between flow coefficient and travel. Figure 8-81 shows three common inherent characteristics. A linear characteristic has a constant slope, meaning the inherent valve gain is a constant. The most popular characteristic is equal-percentage, which gets its name from the fact that equal changes in travel produce equal-percentage changes in the existing flow coefficient. In other words, the slope of the curve is proportional to Cv or equivalently that inherent valve gain is proportional to flow. The equal-percentage characteristic can be expressed mathematically by




 

y Cv(y) = (rated Cv) exp ᎏ − 1 ln R rated y

(8-116)

This expression represents a set of curves parameterized by R. Note that Cv (y = 0) equals (rated Cv)/R rather than zero; real equalpercentage characteristics deviate from theory at some small travel to meet shutoff requirements. An equal-percentage characteristic provides perfect compensation for a process that has gain inversely proportional to flow (e.g., liquid pressure). Quick opening does not have a standardized mathematical definition. Its shape arises naturally from high-capacity plug designs used in on/off service globe valves. Frequently, pressure conditions at the valve will change with flow rate. This so-called process influence [the last two terms on the right hand side of Eq. (8-115)] combine with inherent gain to express the installed valve gain. The flow-versus-travel relationship for a specific set of conditions is called the installed flow characteristic. Typically, valve ∆p decreases with load, since pressure losses in the piping system increase with flow. Figure 8-82 illustrates how allocation of total system head to the valve influences the installed flow characteristics. For a linear or quick-opening characteristic, this transition toward a concave down shape would be more extreme. This effect of typical process pressure variation, which causes equal-percentage character-

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS

FIG. 8-81

Typical inherent flow characteristics.

istics to have fairly constant installed gain, is one reason the equalpercentage characteristic is the most popular. Due to clearance flow, flow force gradients, seal friction, and the like, flow cannot be throttled to an arbitrarily small value. Installed rangeability is the ratio of maximum to minimum controllable flow. The actuator and positioner, as well as the valve, influence the installed rangeability. Inherent rangeability is defined as the ratio of the largest to the smallest Cv within which the characteristic meets specified criteria (see ISA S75.11). The R value in the equalpercentage definition is a theoretical rangeability only. While high installed rangeability is desirable, it is also important not to oversize a valve; otherwise, turndown (ratio of maximum normal to minimum controllable flow) will be limited. Sliding stem valves are characterized by altering the contour of the plug when the port and plug determine the minimum (controlling) flow area. Passage area versus travel is also easily manipulated in characterized cage designs. Inherent rangeability varies widely, but typical values are 30 for contoured plugs and 20–50 for characterized cages. While these types of valves can be characterized, the degree to which manufacturers conform to the mathematical ideal is revealed by plotting measured Cv versus travel. Note that ideal equal percentage will plot as a straight line on a semilog graph. Custom characteristics that compensate for a specific process are possible. Rotary stem-valve designs are normally offered only in their naturally occurring characteristic, since it is difficult to appreciably alter this. If additional characterization is required, the positioner or controller may be characterized. However, these approaches are less direct, since it is possible for device nonlinearity and dynamics to distort the compensation.

8-77

Installed flow characteristic as a function of percent of total system head allocated to the control valve (assuming constant head pump, no elevation head loss, and an R equal 30 equal-percentage inherent characteristic).

FIG. 8-82

OTHER PROCESS VALVES In addition to the throttling control valve, other types of process valves are used to manipulate the process. Valves for On/Off Applications Valves are often required for service that is primarily nonthrottling in nature. Valves in this category, depending on the service requirements, may be of the same design as the types used for throttling control or, as in the case of gate valves, different in design. Valves in this category usually have tight shutoff when they are closed and low pressure drops when they are wide open. The on/off valve can be operated manually, such as by handwheel or lever; or automatically, with pneumatic or electric actuators. Batch Batch process operation is an application requiring on/off valve service. Here the valve is opened and closed to provide reactant, catalyst, or product to and from the batch reactor. Like the throttling control valve, the valve used in this service must be designed to open and close thousands of times. For this reason, valves used in this application are often the same valves used in continuous throttling applications. Ball valves are especially useful in batch operations. The ball valve has a straight-through flow passage that reduces pressure drop in the wide-open state and provides tight shutoff capability when closed. In addition, the segmented ball valve provides for shearing action between the ball and the ball seat that promotes closure in slurry service. Isolation A means for pressure-isolating control valves, pumps, and other piping hardware for installation and maintenance is another

8-78

PROCESS CONTROL

common application for an on/off valve. In this application, the valve is required to have tight shutoff so that leakage is stopped when the piping system is under repair. As the need to cycle the valve in this application is far less than that of a throttling control valve, the wear characteristics of the valve are less important. Also, because there are many required in a plant, the isolation valve needs to be reliable, simple in design and simple in operation. The gate valve, shown in Figure 8-83, is the most widely used valve in this application. The gate valve is composed of a gate-like disc that moves perpendicular to the flow stream. The disc is moved up and down by a threaded screw that is rotated to effect disc movement. Because the disc is large and at right angles to the process pressure, large seat loading for tight shutoff is possible. Wear produced by high seat loading during the movement of the disk prohibits the use of the gate valve for throttling applications. Pressure Relief Valves Definitions for pressure relief valves, relief valves, pilot-operated pressure relief valves and safety valves, are found in the ASME Boiler and Pressure Vessel Code, Section VIII, Division 1, “Rules for Construction of Pressure Vessels,” Paragraphs UG-125 and UG-126. The pressure-relief valve is an automatic pressure relieving device designed to open when normal conditions are exceeded and to close again when normal conditions are restored. Within this class there are relief valves, pilot operated pressure relief valves, and safety valves. Relief valves (see Fig. 8-84) have spring-loaded disks that close a main orifice against a pressure source. As pressure rises, the disk begins to rise off the orifice and a small amount of fluid passes through the valve. Continued rise in pressure above the opening pressure causes the disk to open the orifice in a proportional fashion. The main orifice reduces and closes when the pressure returns to the set pres-

FIG. 8-84

FIG. 8-83

Gate valve. (Courtesy Crane Valves.)

Relief valve. (Courtesy Teledyne Fluid Systems, Farris Engineering.)

sure. Additional sensitivity to over-pressure conditions can be improved by adding an auxiliary pressure relief valve (pilot) to the basic pressure relief valve. This combination is known as a pilot-operated pressure relief valve. The safety valve is similar to the relief valve except it is designed to open fully, or pop, with only a small amount of pressure over the rated limit. Conventional safety valves are sensitive to downstream pressure and may have unsatisfactory operating characteristics in variable back pressure applications. The balanced safety relief valve is available and minimizes the effect of downstream pressure on performance. Check Valves The purpose of a check valve is to allow relatively unimpeded flow in the desired direction but to prevent flow in the reverse direction. Two common designs are swing-type and lift-type check valves—the names of which denote the motion of the closure member. In the forward direction, flow forces overcome the weight of

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS the member or a spring to open the flow passage. With reverse pressure conditions, flow forces drive the closure member into the valve seat, thus providing shutoff. ADJUSTABLE SPEED PUMPS An alternative to throttling a process with a process-control valve and a fixed speed pump is by adjusting the speed of the process pump and not using a throttling control valve at all. Pump speed can be varied by using variable-speed prime movers such as turbines, motors with magnetic or hydraulic couplings, and electric motors. Each of these methods of modulating pump speed has its own strengths and weaknesses but all offer energy savings and dynamic performance advantages over throttling with a control valve. The centrifugal pump directly driven by a variable-speed electric motor is the most commonly used hardware combination for adjustable speed pumping. The motor is operated by an electronicmotor speed controller whose function is to generate the voltage or current waveform required by the motor to make the speed of the motor track the input command input signal from the process controller. The most popular form of motor speed control for adjustable-speed pumping is the voltage-controlled pulse-width-modulated (PWM) frequency synthesizer and AC squirrel-cage induction motor combination. The flexibility of application of the PWM motor drive and its 90 percent+ electrical efficiency along with the proven ruggedness of the traditional AC induction motor makes this combination popular. From an energy-consumption standpoint, the power required to maintain steady process flow with an adjustable-speed pump system (three-phase PWM drive and a squirrel-cage induction motor driving a centrifugal pump on water) is less than that required with a conventional control valve and a fixed speed pump. Figure 8-85 shows this to be the case for a system where 100 percent of the pressure loss is due to flow velocity losses. At 75 percent flow, Fig. 8-85 shows the constant speed-pump/control-valve use at a 10.1-kW rate where throttling with the adjustable speed pump and no control valve used at a 4.1-kW rate. This trend of reduced energy consumption is true for the entire range of flows, although amounts vary. From a dynamic-response standpoint, the adjustable speed pump has a dynamic characteristic that is more suitable in process-control applications than those characteristics of control valves. The small amplitude response of an adjustable speed pump does not contain the dead band or the dead time commonly found in the small amplitude response of the control valve. Nonlinearities associated with frictions in the valve and discontinuities in the pneumatic portion of the control-valve instrumentation are not present with electronic

8-79

variable-speed drive technology. As a result, process control with the adjustable speed pump does not exhibit limit cycles, problems related to low controller gain and generally degraded process loop performance caused by control valve nonlinearities. Unlike the control valve, the centrifugal pump has poor or nonexistent shutoff capability. A flow check valve or an automated on/off valve may be required to achieve shutoff requirements. This requirement may be met by automating an existing isolation valve in retrofit applications. REGULATORS A regulator is a compact device that maintains the process variable at a specific value in spite of disturbances in load flow. It combines the functions of the measurement sensor, controller, and final control element into one self-contained device. Regulators are available to control pressure, differential pressure, temperature, flow, liquid level, and other basic process variables. They are used to control the differential across a filter press, heat exchanger, or orifice plate. Regulators are used for monitoring pressure variables for redundancy, flow check, and liquid surge relief. Regulators may be used in gas blanketing systems to maintain a protective environment above any liquid stored in a tank or vessel as the liquid is pumped out. When the temperature of the vessel is suddenly cooled, the regulator maintains the tank pressure and protects the walls of the tank from possible collapse. Regulators are known for their fast dynamic response. The absence of time delay that often comes with more sophisticated control systems makes the regulator useful in applications requiring fast corrective action. Regulators are designed to operate on the process pressures in the pipeline without any other sources of energy. Upstream and downstream pressures are used to supply and exhaust the regulator. Exhausting is back to the downstream piping so that no contamination or leakage to the external environment occurs. This makes regulators useful in remote locations where power is not available or where external venting is not allowed. The regulator is limited to operating on processes with clean, nonslurry process fluids. The small orifice and valve assemblies contained in the regulator can plug and malfunction if the process fluid that operates the regulator is not sufficiently clean. Regulators are normally not suited to systems that require constant set point adjustment. Although regulators are available with capability to respond to remote set point adjustment, this feature adds complexity to the regulator and may be better addressed by a control-valvebased system. In the simplest of regulators, tuning of the regulator for best control is accomplished by changing a spring, an orifice, or a nozzle.

FIG. 8-85 Pressure, flow, and power for a throttling process using (1) a control valve and a constant speed pump and (2) an adjustable speed pump.

8-80

PROCESS CONTROL

Spring Main throttling valve Diaphragm

(b)

(a) FIG. 8-86

Regulators: (a) self-operated; (b) pilot-operated. (Courtesy Fisher-Rosemount.)

Self-Operated Regulators Self-operated regulators are the simplest form of regulator. This regulator (see Fig. 8-86a) is composed of a main throttling valve, a diaphragm or piston to sense pressure, and a spring. The self-contained regulator is completely operated by the process fluid, and no outside control lines or pilot stage is used. In general, self-operated regulators are simple in construction, easy to operate and maintain, and are usually stable devices. Except for some of the pitot tube types, self-operated regulators have very good dynamic response characteristics. This is because any change in the

FIG. 8-87

controlled variable registers directly and immediately upon the main diaphragm to produce a quick response to the disturbance. The disadvantage of the self-operated regulator is that it is not generally capable of maintaining a set point as load flow is increased. Because of the proportional nature of the spring and diaphragmthrottling effect, offset from set point occurs in the controlled variable as flow increases. Figure 8-87 shows a typical regulation curve for the self-contained regulator. Reduced set point offset with increasing load flow can be achieved

Pressure regulation curves for three regulator types.

PROCESS CONTROL AND PLANT SAFETY by adding a pitot tube to the self-operated regulator. The tube is positioned somewhere near the vena contracta of the main regulator valve. As flow though the valve increases, the measured feedback pressure from the pitot tube drops below the control pressure. This causes the main valve to open or boost more than it would if the static value of control pressure were acting on the diaphragm. The resultant effect keeps the control pressure closer to set point and thus prevents a large drop in process pressure during high-load-flow conditions. Figure 8-87 shows the improvement that the pitot-tube regulator provides over the regulator without the tube. A side effect of adding a pitot-tube method is that the response of the regulator can be slowed due to the restriction provided by the pitot tube. Pilot-Operated Regulators Another category of regulators uses a pilot stage to provide the load pressure on the main diaphragm. This pilot is a regulator itself that has the ability to multiply a small change in downstream pressure into a large change in pressure applied to the regulator diaphragm. Due to this high-gain feature, pilot-operated regulators can achieve a dramatic improvement in steady-state accuracy over that achieved with a self-operated regulator. Figure 8-87 shows for regulation at high flows the pilot-operated regulator is best of the three regulators shown.

8-81

The main limitation of the pilot-operated regulator is stability. When the gain in the pilot amplifier is raised too much, the loop can become unstable and oscillate or hunt. The two-path pilot regulator (see b) is also available. This regulator combines the effects of selfoperated and the pilot-operated styles and mathematically produces the equivalent of proportional plus reset control of the process pressure. Over-Pressure Protection Figure 8-87 shows a characteristic rise in control pressure that occurs at low or zero flow. This lockup tail is due to the effects of imperfect plug and seat alignment and the elastomeric effects of the main throttle valve. If for some reason the main throttle valve fails to completely shut off, or if the valve shuts off but the control pressure continues to rise for other reasons, the lockup tail could get very large, and the control pressure could rise to extremely high valves. Damage to the regulator or the downstream pressure volume could occur. To avoid this situation, some regulators are designed with a built-in over-pressure relief mechanism. Over-pressure relief circuits usually are composed of a spring-opposed diaphragm and valve assembly that vents the downstream piping when the control pressure rises above the set point pressure.

PROCESS CONTROL AND PLANT SAFETY GENERAL REFERENCES: Guidelines for Safe Automation of Chemical Processes, AIChE Center for Chemical Process Safety, New York, 1993.

Accidents in chemical plants make headline news, especially when there is loss of life or the general public is affected in even the slightest way. This increases the public’s concern and may lead to government action. The terms hazard and risk are defined as follows: • Hazard. A potential source of harm to people, property, or the environment • Risk. Possibility of injury, loss, or an environmental accident created by a hazard Safety is the freedom from hazards and thus the absence of any associated risks. Unfortunately, absolute safety cannot be realized. The design and implementation of safety systems must be undertaken with a view of two issues: • Regulatory. The safety system must be consistent with all applicable codes and standards as well as “generally accepted good engineering practices.” • Technical. Just meeting all applicable regulations and “following the crowd” does not relieve a company of its responsibilities. The safety system must work. The regulatory environment will continue to change. As of this writing, the key regulatory instrument is OSHA 29 CFR 1910.119 that pertains to process safety management within plants in which certain chemicals are present. In addition to government regulation, industry groups and professional societies are producing documents ranging from standards to guidelines. Instrument Society of America Standard S84.01, “Application of Safety Instrumented Systems for the Process Industries,” is in draft form at the date of this writing. The Guidelines for Safe Automation of Chemical Processes from the American Institute of Chemical Engineers’ Center for Chemical Process Safety (1993) provides a comprehensive coverage of the various aspects of safety, and, although short on specifics, it is very useful to operating companies developing their own specific safety practices (that is, it does not tell you what to do, but it helps you decide what is proper for your plant). The ultimate responsibility for safety rests with the operating company; OSHA 1910.119 is clear on this. Each company is expected to develop (and enforce) its own practices in the design, installation, testing, and maintenance of safety systems. Fortunately, some companies make these documents public. Monsanto’s Safety System Design Practices was published in its entirety in the proceedings of the International Symposium and Workshop on Safe Chemical Process Automation, Houston, Texas, September 27–29, 1994 (available from

the American Institute of Chemical Engineers’ Center for Chemical Process Safety). ROLE OF AUTOMATION IN PLANT SAFETY As microprocessor-based controls displaced hardwired electronic and pneumatic controls, the impact on plant safety has definitely been positive. When automated procedures replace manual procedures for routine operations, the probability of human errors leading to hazardous situations is lowered. The enhanced capability for presenting information to the process operators in a timely manner and in the most meaningful form increases the operator’s awareness of the current conditions in the process. Process operators are expected to exercise due diligence in the supervision of the process, and timely recognition of an abnormal situation reduces the likelihood that the situation will progress to the hazardous state. Figure 8-88 depicts the layers of safety protection in a typical chemical plant. Although microprocessor-based process controls enhance plant safety, their primary objective is efficient process operation. Manual

FIG. 8-88

Layers of safety protection in chemical plants.

8-82

PROCESS CONTROL

operations are automated to reduce variability, to minimize the time required, to increase productivity, and so on. Remaining competitive in the world market demands that the plant be operated in the best manner possible, and microprocessor-based process controls provide numerous functions that make this possible. Safety is never compromised in the effort to increase competitiveness, but enhanced safety is a by-product of the process-control function and is not a primary objective. By attempting to maintain process conditions at or near their design values, the process controls also attempt to prevent abnormal conditions from developing within the process. Although process controls can be viewed as a protective layer, this is really a by-product and not the primary function. Where the objective of a function is specifically to reduce risk, the implementation is normally not within the process controls. Instead, the implementation is within a separate system specifically provided to reduce risk. This system is generally referred to as the safety interlock system. As safety begins with the process design, an inherently safe process is the objective of modern plant designs. When this cannot be achieved, process hazards of varying severity will exist. Where these hazards put plant workers and/or the general public at risk, some form of protective system is required. Process safety management addresses the various issues, ranging from assessment of the process hazard to assuring the integrity of the protective equipment installed to cope with the hazard. When the protective system is an automatic action, it is incorporated into the safety interlock system, not within the process controls. INTEGRITY OF PROCESS CONTROL SYSTEMS Ensuring the integrity of process controls involves both hardware issues, software issues, and human issues. Of these, the hardware issues are usually the easiest to assess and the software issues the most difficult. The hardware issues are addressed by providing various degrees of redundancy, by providing multiple sources of power and/or an uninterruptible power supply, and the like. The manufacturers of process controls provide a variety of configuration options. Where the process is inherently safe and infrequent shutdowns can be tolerated, nonredundant configurations are acceptable. For more demanding situations, an appropriate requirement might be that no single component failure can render the process-control system inoperable. For the very critical situations, triple-redundant controls with voting logic might be appropriate. The difficulty is assessing what is required for a given process. Another difficulty is assessing the potential for human errors. If redundancy is accompanied with increased complexity, the resulting increased potential for human errors must be taken into consideration. Redundant systems require maintenance procedures that can correct problems in one part of the system while the remainder of the system is in full operation. When conducting maintenance in such situations, the consequences of human errors can be rather unpleasant. The use of programmable systems for process control present some possibilities for failures that do not exist in hard-wired electromechanical implementations. Probably the one of most concern is latent defects or “bugs” in the software, either the software provided by the supplier or the software developed by the user. The source of this problem is very simple. There is no methodology available that can be applied to obtain absolute assurance that a given set of software is completely free of defects. Increased confidence in a set of software is achieved via extensive testing, but no amount of testing results in absolute assurance that there are no defects. This is especially true of real-time systems, where the software can easily be exposed to a sequence of events that was not anticipated. Just because the software performs correctly for each event individually does not mean that it will perform correctly when two (or more) events occur at nearly the same time. This is further complicated by the fact that the defect may not be in the programming; it may be in how the software was designed to respond to the events.

The testing of any collection of software is made more difficult as the complexity of the software increases. Software for process control has progressively become more complex, mainly because the requirements have progressively become more demanding. To remain competitive in the world market, processes must be operated at higher production rates, within narrower operating ranges, closer to equipment limits, and so on. Demanding applications require sophisticated control strategies, which translate into more complex software. Even with the best efforts of both supplier and user, complex software systems are unlikely to be completely free of defects. CONSIDERATIONS IN IMPLEMENTATION OF SAFETY INTERLOCK SYSTEMS Where hazardous conditions can develop within a process, a protective system of some type must be provided. Sometimes these are in the form of process hardware such as pressure relief devices. However, sometimes logic must be provided for the specific purpose of taking the process to a state where the hazardous condition cannot exist. The term safety interlock system is normally used to designate such logic. The purpose of the logic within the safety interlock system is very different from the logic within the process controls. Fortunately, the logic within the safety interlock system is normally much simpler than the logic within the process controls. This simplicity means that a hardwired implementation of the safety interlock system is usually an option. Should a programmable implementation be chosen, this simplicity means that latent defects in the software are less likely to be present. Most safety systems only have to do simple things, but they must do them very, very well. The difference in the nature of process controls and safety interlock systems leads to the conclusion that these two should be physically separated (see Fig. 8-89). That is, safety interlocks should not be piggy-backed onto a process-control system. Instead, the safety interlocks should be provided by equipment, either hard-wired or programmable, that is dedicated to the safety functions. As the process controls become more complex, faults are more likely. Separation means that faults within the process controls have no consequences in the safety interlock system. Modifications to the process controls are more frequent than modifications to the safety interlock system. Therefore, physically separating the safety interlock system from the process controls provides the following benefits: 1. The possibility of a change to the process controls leading to an unintentional change to the safety interlock system is eliminated. 2. The possibility of a human error in the maintenance of the process controls having consequences for the safety interlock system is eliminated. 3. Management of change is simplified. 4. Administrative procedures for software-version control are more manageable. Separation also applies to the measurement devices and actuators. Although the traditional point of reference for safety interlock systems is a hard-wired implementation, a programmed implementation is an alternative. The potential for latent defects in software implementation is a definite concern. Another concern is that solid-state components are not guaranteed to fail to the safe state. The former is addressed by extensive testing; the latter is addressed by manufacturer-supplied and/or user-supplied diagnostics that are routinely executed by the processor within the safety interlock system. Although issues must be addressed in programmable implementations, the hard-wired implementations are not perfect either. Where a programmed implementation is deemed to be acceptable, the choice is usually a programmable logic controller (PLC) that is dedicated to the safety function. PLCs are programmed using the traditional relay ladder diagrams used for hard-wired implementations. The facilities for developing, testing, and troubleshooting PLCs are excellent. However, for PLCs used in safety interlock systems, administrative procedures must be developed and implemented to address the following issues:

PROCESS CONTROL AND PLANT SAFETY

FIG. 8-89

8-83

Total control system with parallel tasks.

1. Version controls for the PLC program must be implemented and rigidly enforced. Revisions to the program must be reviewed in detail and thoroughly tested before implementing in the PLC. The various versions must be clearly identified so that there can be no doubt as to what logic is provided by each version of the program. 2. The version of the program that is currently being executed by the PLC must be known with absolute certainty. It must simply not be possible for a revised version of the program undergoing testing to be downloaded to the PLC. Constant vigilance is required to prevent lapses in such administrative procedures. INTERLOCKS An interlock is a protective response initiated on the detection of a process hazard. The interlock system consists of the measurement devices, logic solvers, and final control elements that recognize the hazard and initiate an appropriate response. Most interlocks consist of one or more logic conditions that detect out-of-limit process conditions and respond by driving the final control elements to the safe states. For example, one must specify that a valve fails open or fails closed. The potential that the logic within the interlock could contain a defect or bug is a strong incentive to keep it simple. Within process plants, most interlocks are implemented with discrete logic, which means either hard-wired electromechanical devices or programmable logic controllers. Interlocks within process plants can be broadly classified as follows: 1. Safety interlocks. These are designed to protect the public, the plant personnel, and possibly the plant equipment from process hazards. 2. Process interlocks. These are designed to prevent process conditions that would unduly stress equipment (perhaps leading to minor damage), lead to off-specification product, and so on. Basically, the process interlocks address hazards whose consequences essentially lead to a monetary loss, possibly even a short plant shut-

down. The more serious hazards are addressed by the safety interlocks. Implementation of process interlocks within process control systems is perfectly acceptable. Furthermore, it is also permissible (and probably advisable) that responsible operations personnel be authorized to bypass or ignore a process. Safety interlocks must be implemented within the separate safety interlock system. Bypassing or ignoring safety interlocks by operations personnel is simply not permitted. When this is necessary for actions such as verifying that the interlock continues to be functional, such situations must be infrequent and incorporated into the design of the interlock. Safety interlocks are assigned to categories that reflect the severity of the consequences should the interlock fail to perform as intended. The specific categories used within a company is completely at the discretion of the company. However, most companies use categories that distinguish among the following: 1. Hazards that pose a risk to the public. Complete redundancy is normally required. 2. Hazards that could lead to injury of company personnel. Partial redundancy is often required (for example, redundant measurements but not redundant logic). 3. Hazards that could result in major equipment damage and consequently lengthy plant downtime. No redundancy is normally required for these, although redundancy is always an option. Situations that result in minor equipment damage that can be quickly repaired do not generally require a safety interlock; however, a process interlock might be appropriate. A process hazards analysis is intended to identify the safety interlocks required for a process and to provide the following for each: 1. The hazard that is to be addressed by the safety interlock. 2. The classification of the safety interlock. 3. The logic for the safety interlock, including inputs from measurement devices and outputs to actuators. The process hazards analysis is conducted by an experienced, multidisciplinary team that examines the process design, the plant equipment, operating procedures, and so on, using techniques such as

8-84

PROCESS CONTROL

hazard and operability studies (HAZOP), failure mode and effect analysis (FEMA), and others. The process hazards analysis recommends appropriate measures to reduce the risk, including (but not limited to) the safety interlocks to be implemented in the safety interlock system. Diversity is recognized as a useful approach to reduce the number of defects. The team that conducts the process hazards analysis does not implement the safety interlocks but provides the specifications for the safety interlocks to another organization for implementation. This organization reviews the specifications for each safety interlock, seeking clarifications as necessary from the process hazards analysis team and bringing any perceived deficiencies to the attention of the process hazards analysis team. Diversity can be used to further advantage in redundant configurations. Where redundant measurement devices are required, different technology can be used for each. Where redundant logic is required, one can be programmed and one hard-wired. Reliability of the interlock systems has two aspects: 1. It must react should the hazard arise. 2. It must not react when there is no hazard. Emergency shutdowns often pose risks in themselves, and therefore they should be undertaken only when truly appropriate. The need to avoid extraneous shutdowns is not just to avoid disruption in production operations. Although safety interlocks can inappropriately initiate shutdowns, the process interlocks are usually the major source of problems. It is possible to configure so many process interlocks that it is not possible to operate the plant.

TESTING As part of the detailed design of each safety interlock, written test procedures must be developed for the following purposes: 1. Assure that the initial implementation complies with the requirements defined by the process hazards analysis team. 2. Assure that the interlock (hardware, software, and I/O) continues to function as designed. The design must also determine the time interval on which this must be done. Often these tests must be done with the plant in full operation. The former is the responsibility of the implementation team and is required for the initial implementation and following any modification to the interlock. The latter is the responsibility of plant maintenance, with plant management responsible for seeing that it is done on the specified interval of time. Execution of each test must be documented, showing when it was done, by whom, and the results. Failures must be analyzed for possible changes in the design or implementation of the interlock. These tests must encompass the complete interlock system, from the measurement devices through the final control elements. Merely simulating inputs and checking the outputs is not sufficient. The tests must duplicate the process conditions and operating environments as closely as possible. The measurement devices and final control elements are exposed to process and ambient conditions and thus are usually the most likely to fail. Valves that remain in the same position for extended periods of time may stick in that position and not operate when needed. The easiest component to test is the logic; however, this is the least likely to fail.

Section 9

Process Economics*

F. A. Holland, D.Sc., Ph.D., Consultant in Heat Energy Recycling; Research Professor, University of Salford, England; Fellow, Institution of Chemical Engineers, London. (Section Editor) J. K. Wilkinson, M.Sc., Consultant Chemical Engineer; Fellow, Institution of Chemical Engineers, London.

INVESTMENT AND PROFITABILITY Annual Costs, Profits, and Cash Flows . . . . . . . . . . . . . . . . . . . . . . . . . . Contribution and Breakeven Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . Capital Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Depreciation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Traditional Measures of Profitability . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time Value of Money . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 1: Capitalized Cost of Equipment . . . . . . . . . . . . . . . . . . . . Modern Measures of Profitability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 2: Net Present Value for Different Depreciation Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 3: Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Learning Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 4: Estimation of Average Cost of Incremental Units. . . . . . Risk and Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 5: Probability Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 6: Calculation of Probability of Meeting a Sales Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 7: Calculation of Probability of Sales . . . . . . . . . . . . . . . . . . Example 8: Calculation of Probability of Equipment Breakdowns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 9: Calculation of Probability of Machine Failures . . . . . . . . Example 10: Logistics Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 11: Parameter Method of Risk Analysis . . . . . . . . . . . . . . . . Example 12: Expected Value of Net Profit . . . . . . . . . . . . . . . . . . . . . Example 13: Evaluation of Investment Priorities Using Probability Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 14: Estimation of Probability of a Research and Development Program Breaking Even . . . . . . . . . . . . . Example 15: Utility Function Curve . . . . . . . . . . . . . . . . . . . . . . . . . . Inflation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 16: Effect of Inflation on Net Present Value . . . . . . . . . . . . Example 17: Effect of Fuel Cost on Project Economics . . . . . . . . . .

9-5 9-7 9-7 9-7 9-8 9-10 9-13 9-13 9-16 9-19 9-20 9-20 9-22 9-23 9-24 9-24 9-25 9-25 9-25 9-27 9-28 9-30 9-30 9-32 9-33 9-34 9-34 9-38

ACCOUNTING AND COST CONTROL Principles of Accounting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Financing Assets by Equity and Debt . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparative Company Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cost of Capital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 18: Risk-Free Cost of Capital . . . . . . . . . . . . . . . . . . . . . . . . Management and Cost Accounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Allocation of Overheads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 19: Overhead in Two Different Products . . . . . . . . . . . . . . . Inventory Evaluation and Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 20: Inventory Computations . . . . . . . . . . . . . . . . . . . . . . . . . Working Capital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Budgets and Cost Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MANUFACTURING-COST ESTIMATION General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One Main Product Plus By-Products. . . . . . . . . . . . . . . . . . . . . . . . . . . . Two Main Products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 21: Calculation of Contributions to Income for Multiple Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Direct Manufacturing Costs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Indirect Manufacturing Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rapid Manufacturing-Cost Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . Manufacturing Cost as a Basis for Product Pricing. . . . . . . . . . . . . . . . . Standard Costs for Budgetary Control. . . . . . . . . . . . . . . . . . . . . . . . . . . Example 22: Direct-Material-Mixture Variance . . . . . . . . . . . . . . . . . Contribution Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Valuation of Recycled Heat Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FIXED-CAPITAL-COST ESTIMATION Total Capital Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cost Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9-39 9-42 9-44 9-47 9-47 9-48 9-48 9-49 9-49 9-50 9-52 9-54

9-55 9-55 9-56 9-56 9-57 9-57 9-57 9-58 9-59 9-60 9-61 9-62

9-63 9-63

* The contribution of the late Mr. F. A. Watson, who was an author for the Sixth edition, is acknowledged. 9-1

9-2

PROCESS ECONOMICS

Types and Accuracy of Estimates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rapid Estimations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 23: Estimation of Total Installed Cost of a Plant . . . . . . . . . Equipment Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Piping Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electrical and Instrumentation Estimation . . . . . . . . . . . . . . . . . . . . . . .

9-63 9-64 9-68 9-72 9-73 9-73

Auxiliaries Estimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Use of Computers in Cost Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . Startup Costs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Construction Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Project Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overseas Construction Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9-74 9-75 9-76 9-76 9-77 9-78

Nomenclature and Units Symbol a A AA AD (ATR) b bc B c c cB cD cI cL c° C CCT CDS (CEQ)DEL CK CL CRS CRW CWS (CI) (COP)A (CR) (CRR) (CSR) d d (DR) (DCFRR) e e exp (a) (EMIP) fAF fAP

Definition Empirical constant in general equations Annual income or expenditure particularized by the subscript Annual allowances against tax other than for depreciation of fixed assets Annual writing down (depreciation) of fixed assets, allowable against tax Asset-turnover ratio defined by Eq. (9-131) Empirical constant in general equations Deviation from budgeted capacity Parametric constant in Eq. (9-204) Empirical constant in general equations Cost (or income) per unit of sales or production particularized by the subscript Cost of base heat supply Cost of heat energy delivered by a heat pump defined by Eq. (9-240) Cost of high-grade energy supplied to the compressor of a vapor compression heat pump Cost of labor per unit of production Standard cost particularized by the subscript Cost particularized by the subscript Installed cost of a cooling tower Installed cost of a demineralized-water system Delivered-equipment cost Capitalized cost of a fixed asset defined by Eq. (9-47) Cost of land and other nondepreciable assets Installed cost of a refrigeration system Installed cost of a river-water supply system Installed cost of a water-softening system Cost index as used in Eq. (9-246) Actual coefficient of performance of a heat pump Capital ratio defined by Eq. (9-134) Capital-rate-of-return ratio defined by Eq. (9-56) Contribution-sales ratio defined by Eq.(9-236) Empirical constant in general equations Symbol indicating differentiation Debt ratio defined by Eq. (9-139) Discounted-cash-flow rate of return Empirical constant in general equations Base of natural logarithms, 2.71828 Exponential function of a, ea Equivalent maximum investment period defined by Eq. (9-55) Annuity future-worth factor, i[(1 + i)n − 1]−1 Annuity present-worth factor, fAF(1 + i)n

Units Various $/year $/year $/year Dimensionless Various

Symbol fd fi fk fp f(x) F Fn i

Dimensionless Dimensionless Various

ie

$/unit

ir i′ I

$/unit $/GJ $/GJ $/hour $/hour

im

kn K ln (a) log (a) m

$ $ $

m

$ $

n N

$

N (NPV) p(x)

$ $ $ Dimensionless Dimensionless

(MSF)

P Pa Pb Pe

Year Year

P1

Dimensionless

Ps

Various Dimensionless Dimensionless Year−1 Various Dimensionless Dimensionless Year Dimensionless Dimensionless

Ps′ Pw (PBP) (PM) (PSR) q QD r R R° RB

Definition

Units

Discount factor, (1 + i)−n Compound-interest factor, (1 + i)n Capitalized-cost factor, fAP/i Piping-cost factor defined by Eq. (9-249) Distribution function of x variously defined Future value of a sum of money Sum of fd for Years 1 to n Interest rate per period, usually annual, often the cost of capital Effective interest rate defined by Eq. (9-111) Minimum acceptable interest rate defined by Eq. (9-107) Entrepreneurial-risk interest rate Nominal annual interest rate Value of inventory particularized by the subscript Constants in Eq. (9-81) Effective value of the first unit of production Logarithm to the base e of a Logarithm to the base 10 of a Number of interest periods due per year Number of units removed from inventory Measured-survival function defined by Eq. (9-106) Number of years, units, etc. Slope of the learning curve defined by Eq. (9-64) Number of inventory orders per year Net present value Probability of the variable having the value x Present value of a sum of money Production time worked Budgeted production

Dimensionless Dimensionless Dimensionless Dimensionless

Production efficiency defined by Eq. (9-216) Level of productive activity defined by Eq. (9-217) Actual production rate Book value of asset at the end of year s′ Budgeted working time Payback period defined by Eq. (9-30) Profit margin defined by Eq. (9-127) Profit-sales ratio defined by Eq. (9-235) Quantity defining the scale of operation Process-heat-rate requirement Fraction of range of the independent variable Production rate Standard production rate Breakeven production rate

Dimensionless $ Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless $ Various $/unit, time/ unit, etc. Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless $ Dimensionless $ Hour Standard hour Dimensionless Dimensionless Standard hour $ Hour Year Dimensionless Dimensionless Various GJ/hour Dimensionless Units/year Units/year Units/year

9-3

9-4

PROCESS ECONOMICS

Nomenclature and Units (Concluded ) Symbol R0 RS (ROA) (ROE) (ROI) s s′ s° S t tC tSU T U V W x x
X y y Y Y Y
z

Definition

Units

Scheduled production rate Sales rate Return on assets defined by Eq. (9-129) Return on equity defined by Eq. Eq. (9-130) Return on investment defined by Eq. (9-128) Scheduled number of productive years Number of productive years to date Sample standard deviation Scrap value of a depreciable asset Fractional tax rate payable on adjusted income Time taken to construct plant Time taken to start up plant Auxiliary variable defined by Eq. (9-92) Size of inventory order Variable cost of inventory order Power supplied at shaft of a heat pump General variable Mean value of x Cumulative production from startup Cumulative probability Operating time of a heat pump Cumulative average cost, production time, etc. Operating-labor rate in Eq. (9-204) Cumulative-average batch cost, etc. Standard score defined by Eq. (9-73)

Units/year Units/year Dimensionless Dimensionless Dimensionless

Various $ Dimensionless Years Years Various Units $/unit GJ/hour Various Units Dimensionless Hours/year $/unit, hour/ unit, etc. labor-hour/ton $/unit, etc. Dimensionless

Greek symbols α β β δ ∆ η η θ σ  φ φP ψ χ χ

Proportionality factor in Eq. (9-168) Proportionality factor in Eq. (9-171) Exponent in Eqs. (9-106) and (9-117) Symbol indicating partial differentiation Symbol indicating a difference of like quantities Contribution efficiency defined by Eq. (9-119) Margin of safety defined by Eq. (9-229) Time taken to produce a given amount of product Population standard deviation Symbol indicating a sum of like quantities Fractional increase in production rate Parameter defined with Eq. (9-254) Parameter defined with Eq. (9-241) Plant capacity in Eq. (9-204) Weight of product per unit of raw material Subscripts

A BD BL

Allowance against tax other than for capital depreciation Depreciation allowance shown in company balance sheet Within project boundary limits

Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Hour Various Dimensionless Dimensionless Dimensionless Dimensionless Tons/day Dimensionless

Symbol BOH CF CI DCF DME FC FE FGE FIFO FIN FME FOH GE GP IME INV IO IT IW L L LIFO max M ME N NCI NOH NNP NP OH P RM s′ S SAV ST SVOH TC TE TFE TVE U U VE VGE VME VOH W WC WAV 1, 2, j, n

Definition Budgeted overhead Cash flow after payment of tax and expenses Cash income after payment of expenses Discounted cash flow Direct manufacturing expense Fixed capital Fixed expense Fixed general expense On a first-in–first-out basis Financial-resources inventory Fixed manufacturing expense Fixed overhead General expense Gross profit Indirect manufacturing expense Inventory Inventory-orders cost Income tax payable Inventory working cost Labor-earnings index Lower-quartile value of the variable Last-in–first-out basis Maximum value Median value of the variable Manufacturing expense At agreed normal production rate Net cash income after payment of tax Overhead cost at agreed normal production rate Net profit after payment of tax Net profit before payment of tax Overhead cost Profit Raw material In the s′th productive year From sales and other income On a simple-average basis Steel-price index Semivariable overhead Total capital Total expense Total fixed expense Total variable expense Utilities Upper-quartile value of the variable Variable expense Variable general expense Variable manufacturing expense Variable overhead expense Weighted value Working capital On a weighted-average basis 1st, 2d, jth, nth item, year, etc.

GENERAL REFERENCES: Allen, D. H., Economic Evaluation of Projects, 3d ed., Institution of Chemical Engineers, Rugby, England, 1991. Aries, R. S. and R. D. Newton, Chemical Engineering Cost Estimation, McGraw-Hill, New York, 1955. Baasel, W. D., Preliminary Chemical Engineering Plant Design, 2d ed., Van Nostrand Reinold, New York, 1989. Barish, N. N. and S. Kaplan, Economic Analysis for Engineering and Management Decision Making, 2d ed., McGraw-Hill, New York, 1978. Bierman, H., Jr. and S. Smidt, The Capital Budgeting Decision, Economic Analysis and Financing of Investment Projects, 7th ed., Macmillan, London, 1988. Canada, J. R. and J. A. White, Capital Investment Decision: Analysis for Management and Engineering, 2d ed., Prentice Hall, Englewood Cliffs, NJ, 1980. Carsberg, B. and A. Hope, Business Investment Decisions under Inflation, Macdonald & Evans, London, 1976. Chemical Engineering (ed.), Modern Cost Engineering, McGraw-Hill, New York, 1979. Garvin, W. W., Introduction to Linear Programming, McGraw-Hill, New York, 1960. Gass, S. I., Linear Programming, McGraw-Hill, New York, 1985. Granger, C. W. J., Forecasting in Business and Economics, Academic Press, New York, 1980. Hackney, J. W. and K. K. Humphreys (ed.), Control and Management of Capital Projects, 2d ed., McGraw-Hill, New York, 1991. Happel, J., W. H. Kapfer, B. J. Blewitt, P. T. Shannon, and D. G. Jordan, Process Economics, American Institute of Chemical Engineers, New York, 1974. Hill, D. A. and L. E. Rockley, Secrets of Successful Financial Management, Heinemann, London, 1990. Holland, F. A., F. A. Watson, and J. K. Wilkinson, Introduction to Process Economics, 2d ed., Wiley, London, 1983. Humphreys, K. K. (ed.), Jelen’s Cost and Optimization Engineering, 3d ed., McGraw-Hill, New York, 1991. Institution of Chemical Engineers (ed.), A Guide to Capital Cost Estimation, Institution of Chemical Engineers, Rugby, England, 1988. Jordan, R. B., How to Use the Learning Curve, Materials Management Institute, Boston, 1965. Kharbanda, O. P. and E. A. Stallworthy, Capital Cost Estimating in the Process Indus-

tries, 2d ed., Butterworth-Heinemann, London, 1988. Kirkman, P. R., Accounting under Inflationary Conditions, 2d ed., Routledge, Chapman & Hall, London, 1978. Liddle, C. J. and A. M. Gerrard, The Application of Computers to Capital Cost Estimation, Institution of Chemical Engineers, Rugby, England, 1975. Loomba, N. P., Linear Programming, McGraw-Hill, New York, 1964. Merrett, A. J. and A. Sykes, The Finance and Analysis of Capital Projects, Longman, London, 1963. Merrett, A. J. and A. Sykes, Capital Budgeting and Company Finance, Longman, London, 1966. Ostwald, P. F., Engineering Cost Estimating, 3d ed., Prentice Hall, Englewood Cliffs, NJ, 1991. Park, W. R. and D. E. Jackson, Cost Engineering Analysis, 2d ed., Wiley, New York, 1984. Peters, M. S. and K. D. Timmerhaus, Plant Design and Economics for Chemical Engineers, 4th ed., McGraw-Hill, New York, 1991. Pilcher, R., Principles of Construction Management, 3d ed., McGraw-Hill, New York, 1992. Popper, H. (ed.), Modern Cost Estimating Techniques, McGraw-Hill, New York, 1970. Raiffa, H. and R. Schlaifer, Applied Statistical Decision Theory, Harper & Row (Harvard Business), New York, 1984. Ridge, W. J., Value Analysis for Better Management, American Management Association, New York, 1969. Rose, L. M., Engineering Investment Decisions: Planning under Uncertainty, Elsevier, Amsterdam, 1976. Rudd, D. F. and C. C. Watson, The Strategy of Process Engineering, Wiley, New York, 1968. Thorne, H. C. and J. B. Weaver (ed.), Investment Appraisal for Chemical Engineers, American Institute of Chemical Engineers, New York, 1991. Weaver, J. B., ‘Project Selection in the 1980’s’, Chem. Eng. News 37–46 (Nov. 2, 1981). Wells, G. L., Process Engineering with Economic Objectives, Wiley, New York, 1973. Wilkes, F. M., Capital Budgeting Techniques, 2d ed., Wiley, London, 1983. Wood, E. G., Costing Matters for Managers, Beekman Publications, London, 1977. Woods, D. R., Process Design and Engineering, Prentice Hall, Englewood Cliffs, NJ, 1993. Wright, M. G., Financial Management, McGraw-Hill, London, 1970.

NOMENCLATURE An attempt has been made to bring together most of the methods currently available for project evaluation and to present them in such a way as to make the methods amenable to modern computational techniques. To this end the practices of accountants and others have been reduced, where possible, to mathematical equations which are usually solvable with an electronic hand calculator equipped with scientific function keys. To make the equations suitable for use on high-speed computers an attempt has been made to devise a nomenclature which is suitable for machines using ALGOL, COBOL, or FORTRAN compilers. The number of letters and numbers used to define a variable has usually been limited to five. The letters are mnemonic in English wherever possible and are derived in two ways. First, when a standard accountancy phrase exists for a term, this has been abbreviated in capital letters and enclosed in parentheses, e.g., (ATR), for assets-toturnover ratio; (DCFRR), for discounted-cash-flow rate of return. Clearly, the parentheses are omitted when the letter group is used to define the variable name for the computer. Second, a general symbol is defined for a type of variable and is modified by a mnemonic subscript, e.g., an annual cash quantity ATC, annual total capital outlay, $/year. Clearly, the symbols are written on one line when the letter group is used to define a variable name for the computer. In other cases, when well-known standard symbols exist, they have been

adopted, e.g., z for the standard score as used in the normal distribution. Also, a, b, c, d, and e have been used to denote empirical constants and x and y to denote general variables where their use does not clash with other meanings of the same symbols. The coverage in this section is so wide that nomenclature has sometimes proved a problem which has required the use of primes, asterisks, and other symbols not universally acceptable in the naming of computer variables. However, it is realized that each individual will program only his or her preferred methods, which will release some symbols for other uses. Also, it is not difficult to replace a forbidden symbol by an acceptable one; e.g., cRM might be rendered CARM and PS′ as PSP by using A for asterisk and P for prime. For compilers which recognize only one alphabetical case, an extra prefix can be used to distinguish between uppercase and lowercase letters, for which purpose the letters U and L have been used only in a restricted way in the nomenclature. It is, of course, impossible to allow for all possible variations of equation requirements and machine capability, but it is hoped that the nomenclature in the table presented at the beginning of the section will prove adequate for most purposes and will be capable of logical extension to other more specialized requirements.

INVESTMENT AND PROFITABILITY In order to assess the profitability of projects and processes it is necessary to define precisely the various parameters. Annual Costs, Profits, and Cash Flows To a large extent, accountancy is concerned with annual costs. To avoid confusion with other costs, annual costs will be referred to by the letter A.

The revenue from the annual sales of product AS, minus the total annual cost or expense required to produce and sell the product ATE, excluding any annual provision for plant depreciation, is the annual cash income ACI: A CI = AS − A TE (9-1) 9-5

9-6

PROCESS ECONOMICS

Net annual cash income ANCI is the annual cash income ACI, minus the annual amount of tax AIT: A NCI = ACI − AIT

(9-2)

Taxable income is (ACI − AD − AA), where AD is the annual writingdown allowance and AA is the annual amount of any other allowances. A distinction is made between the writing-down allowance permissible for the computation of tax due, the actual depreciation in value of an asset, and the book depreciation in value of that asset as shown in the company position statement. There is no necessary connection between these values unless specified by law, although the first two or all three are often assigned the same value in practice. Some governments give cash incentives to encourage companies to build plants in otherwise unattractive areas. Neither AD nor AA involves any expenditure of cash, since they are merely book transactions. The annual amount of tax AIT is given by AIT = (A CI − AD − AA) t

(9-3)

where t is the fractional tax rate. The value of t is determined by the appropriate tax authority and is subject to change. For most developed countries the value of t is about 0.35 or 35 percent. The annual amount of tax AIT included in Eq. (9-2) does not necessarily correspond to the annual cash income ACI in the same year. The tax payments in Eq. (9-2) should be those actually paid in that year. In the United States, companies pay about 80 percent of the tax on estimated current-year earnings in the same year. In the United Kingdom, companies do not pay tax until at least 9 months after the end of the accounting period, which, for the most part, amounts to paying tax on the previous year’s earnings. When assessing projects for different countries, engineers should acquaint themselves with the tax situation in those countries. In modern methods of profitability assessment, cash flows are more meaningful than profits, which tend to be rather loosely defined. The net annual cash flow after tax is given by A CF = A NCI − ATC

(9-4)

where ATC is the annual expenditure of capital, which is not necessarily zero after the plant has been built. For example, working capital, plant additions, or modifications may be required in future years. The total annual expense ATE required to produce and sell a product can be written as the sum of the annual general expense AGE and the annual manufacturing cost or expense A ME: A TE = A GE + A ME

(9-5)

Annual general expense A GE arises from the following items: adminis-

tration, sales, shipping of product, advertising and marketing, technical service, research and development, and finance. The terms gross annual profit A GP and net annual profit ANP are commonly used by accountants and misused by others. Normally, both A GP and A NP are calculated before tax is deducted. Gross annual profit A GP is given by A GP = AS − A ME − A BD

A NP = AGP − AGE

(9-7)

Equation (9-7) can also be written as A NP = ACI − A BD

(9-8)

Net annual profit after tax ANNP can be written as A NNP = A NCI − ABD

(9-9)

The relationships among the various annual costs given by Eqs. (9-1) through (9-9) are illustrated diagrammatically in Fig. 9-1. The top half of the diagram shows the tools of the accountant; the bottom half, those of the engineer. The net annual cash flow ACF, which excludes any provision for balance-sheet depreciation ABD, is used in two of the more modern methods of profitability assessment: the net-presentvalue (NPV) method and the discounted-cash-flow-rate-of-return (DCFRR) method. In both methods, depreciation is inherently taken care of by calculations which include capital recovery. Annual general expense AGE can be written as the sum of the fixed and variable general expenses: AGE = AFGE + AVGE

(9-10)

Similarly, annual manufacturing expense AME can be written as the sum of the fixed and variable manufacturing expenses: AME = AFME + A VME

(9-11)

A variable expense is considered to be one which is directly proportional to the rate of production RP or of sales RS as is most appropriate to the case under consideration. Unless the variation in finishedproduct inventory is large when compared with the total production over the period in question, it is usually sufficiently accurate to consider RP and RS to be represented by the same-numerical-value R units of sale or production per year. A fixed expense is then considered to be one which is not directly proportional to R, such as overhead charges. Fixed expenses are not necessarily constant but may be sub-

Relationship between annual costs, annual profits, and cash flows for a project. ABD = annual depreciation allowance; ACF = annual net cash flow after tax; ACI = annual cash income; AGE = annual general expense; AGP = annual gross profit; AIT = annual tax; AME = annual manufacturing cost; ANCI = annual net cash income; ANNP = annual net profit after taxes; ANP = annual net profit; AS = annual sales; ATC = annual total cost; (DCFRR) = discounted-cash-flow rate of return; (NPV) = net present value. FIG. 9-1

(9-6)

where ABD is the balance-sheet annual depreciation charge, which is not necessarily the same as AD used in Eq. (9-3) for tax purposes. Net annual profit ANP is simply

INVESTMENT AND PROFITABILITY

9-7

Breakeven chart showing relationship between contribution and fixed expense. FIG. 9-4

FIG. 9-2

Conventional breakeven chart.

ject to stepwise variation at different levels of production. Some authors consider such steps as included in a semivariable expense, which is less amenable to mathematical analysis than the above division of expenses. Contribution and Breakeven Charts These can be used to give valuable preliminary information prior to the use of the more sophisticated and time-consuming methods based on discounted cash flow. If the sales price per unit of sales is cS and the variable expense is cVE per unit of production, Eq. (9-7) can be rewritten as ANP = R(cS − cVE) − AFE

(9-12)

where R(cS − cVE) is known as the annual contribution. The net annual profit is zero at an annual production rate RB = AFE/(cS − cVE)

(9-13)

where RB is the breakeven production rate. Breakeven charts can be plotted in any of the three forms shown in Figs. 9-2, 9-3, and 9-4. The abscissa shown as annual sales volume R is also frequently plotted as a percentage of the designed production or sales capacity R0. In the case of ships, aircraft, etc., it is then called the percentage utilization. The percentage margin of safety is defined as 100(R0 − RB)/R0. A decrease in selling price cS will decrease the slope of the lines in Figs. 9-2, 9-3, and 9-4 and increase the required breakeven value RB for a given level of fixed expense AFE. Capital Costs The total capital cost CTC of a project consists of the fixed-capital cost CFC plus the working-capital cost CWC, plus the cost of land and other nondepreciable costs CL:

CTC = CFC + CWC + CL

TABLE 9-1

FIG. 9-3

Breakeven chart showing fixed expense as a burden cost.

(9-14)

The project may be a complete plant, an addition to an existing plant, or a plant modification. The working-capital cost of a process or a business normally includes the items shown in Table 9-1. Since working capital is completely recoverable at any time, in theory if not in practice, no tax allowance is made for its depreciation. Changes in working capital arising from varying trade credits or payroll or inventory levels are usually treated as a necessary business expense except when they exceed the tax debt due. If the annual income is negative, additional working capital must be provided and included in the ATC for that year. The value of land and other nondepreciables often increases over the working life of the project. These are therefore not treated in the same way as other capital investments but are shown to have made a (taxable) profit or loss only when the capital is finally recovered. Working capital may vary from a very small fraction of the total capital cost to almost the whole of the invested capital, depending on the process and the industry. For example, in jewelry-store operations, the fixed capital is very small in comparison with the working capital. On the other hand, in the chemical-process industries, the working capital is likely to be in the region of 10 to 20 percent of the value of the fixed-capital investment. Depreciation The term “depreciation” is used in a number of different contexts. The most common are: 1. A tax allowance 2. A cost of operation 3. A means of building up a fund to finance plant replacement 4. A measure of falling value In the first case, the annual taxable income is reduced by an annual depreciation charge or allowance which has the effect of reducing the annual amount of tax payable. The annual depreciation charge is merely a book transaction and does not involve any expenditure of cash. The method of determining the annual depreciation charge must be agreed to by the appropriate tax authority. In the second case, depreciation is considered to be a manufacturing cost in the same way as labor cost or raw-materials cost. However, Working-Capital Costs

Raw materials for plant startup Raw-materials, intermediate, and finished-product inventories Cost of handling and transportation of materials to and from stores Cost of inventory control, warehouse, associated insurance, security arrangements, etc. Money to carry accounts receivable (i.e., credit extended to customers) less accounts payable (i.e., credit extended by suppliers) Money to meet payrolls when starting up Readily available cash for emergencies Any additional cash required to operate the process or business

9-8

PROCESS ECONOMICS

it is more difficult to estimate a depreciation cost per unit of product than it is to do so for labor or raw-materials costs. In the net-presentvalue (NPV) and discounted-cash-flow-rate-of-return (DCFRR) methods of measuring profitability, depreciation, as a cost of operation, is implicitly accounted for. (NPV) and (DCFRR) give measures of return after a project has generated sufficient income to repay, among other things, the original investment and any interest charges that the invested money would otherwise have brought into the company. In the third case, depreciation is considered as a means of providing for plant replacement. In the rapidly changing modern chemicalprocess industries, many plants will never be replaced because the processes or products have become obsolete during their working life. Management should be free to invest in the most profitable projects available, and the creation of special-purpose funds may hinder this. However, it is desirable to designate a proportion of the retained income as a fund from which to finance new capital projects. These are likely to differ substantially from the projects that originally generated the income. In the fourth case, a plant or a piece of equipment has a limited useful life. The primary reason for the decrease in value is the decrease in future life and the consequent decrease in the number of years for which income will be earned. At the end of its life, the equipment may be worth nothing, or it may have a salvage or scrap value S. Thus a fixed-capital cost CFC depreciates in value during its useful life of s years by an amount that is equal to (CFC − S). The useful life is taken from the startup of the plant. On the basis of straight-line depreciation, the average annual amount of depreciation AD over a service life of s years is given by AD = (CFC − S)/s

(9-15)

The book value after the first year P1 is given by P1 = CFC − AD

(9-16)

The book value at the end of a specified number of years s′ is given by Ps′ = CFC − s′AD

(9-17)

The principal use of a particular depreciation rate is for tax purposes. The permitted annual depreciation is subtracted from the annual income before the latter is taxed. The basis for depreciation in a particular case is a matter of agreement between the taxation authority and the company, in conformity with tax laws. Other commonly used methods of computing depreciation are the declining-balance method (also known as the fixed-percentage method) and the sum-of-years-digits method. On the basis of declining-balance (fixed-percentage) depreciation, the book value at the end of the first year is given by P1 = CFC(1 − r)

(9-18)

where r is a fraction to be agreed with the taxation authority. The book value at the end of specified number of years s′ is given by Ps′ = CFC(1 − r)s′

(9-19)

When the fraction r is chosen to be 2/s, i.e., twice the reciprocal of the service life s, the method is called the double-declining-balance method. The declining-balance method of depreciation allows equipment or plant to be depreciated by a greater amount during the earlier years than during the later years. This method does not allow equipment or plant to be depreciated to a zero value at the end of the service life. On the basis of sum-of-years-digits depreciation, the annual amount of depreciation for a specified number of years s′ for a plant of fixed-capital cost CFC, scrap value S, and service life s is given by s − s′ + 1 ADs′ = ᎏᎏ (CFC − S) 1+2+3+⋅⋅⋅+s





(9-20)













AD CFC − S = ᎏ fAF where fAF is the annuity future-worth factor given by

or

(9-24)

fAF = i / [(1 + i)s − 1] In the sinking-fund method of depreciation, the effect of interest is to make the annual decrease of the book value of the equipment or plant less in the early than in the later years with consequent higher tax due in the earlier years when recovery of the capital is most important. It is preferable not to think of annual depreciation as a contribution to a fund to replace equipment at the end of its life but as part of the difference between the revenue and the expenditure, which difference is tax-free. Some of the preceding methods of computing depreciation are not allowed by taxation authorities in certain countries. When calculating depreciation, it is necessary to obtain details of the methods and rates permitted by the appropriate authority and to use the information provided. Figure 9-5 shows the fall in book value with time for a piece of equipment having a fixed-capital cost of $120,000, a useful life of 10 years, and a scrap value of $20,000. This fall in value is calculated by using (1) straight-line depreciation, (2) double-declining depreciation, and (3) sum-of-years-digits depreciation. Traditional Measures of Profitability Rate-of-Return Methods Although traditional rate-of-return methods have the advantage of simplicity, they can yield very misleading results. They are based on the relation Percent rate of return = [(annual profit)/(invested capital)]100

(9-25)

Since different meanings are ascribed to both annual profit and invested capital in Eq. (9-25), it is important to define the terms precisely. The invested capital may refer to the original total capital investment, the depreciated investment, the average investment, the current value of the investment, or something else. The annual profit may refer to the net annual profit before tax A NP, the net annual profit after tax A NNP, the annual cash income before tax A CI, or the annual cash income after tax A NCI. The fractional interest rate of return based on the net annual profit after tax and the original investment is i = ANNP /CTC

(9-26)

which can be written in terms of Eq. (9-9) as

Equation (9-20) can also be rewritten in the form 2(s − s′ + 1) ADs′ = ᎏᎏ (CFC − S) s(s + 1)

It can be shown that the book value at the end of a particular year s′ is 1 + 2 + ⋅ ⋅ ⋅ + (s − s′) Ps′ = 2 ᎏᎏᎏ (CFC − S) + S (9-22) s(s + 1) The sum-of-years-digits depreciation allows equipment or plant to be depreciated by a greater amount during the early years than during the later years. A fourth method of computing depreciation (now seldom used) is the sinking-fund method. In this method, the annual depreciation AD is the same for each year of the life of the equipment or plant. The series of equal amounts of depreciation AD, invested at a fractional interest rate i and made at the end of each year over the life of the equipment or plant of s years, is used to build up a future sum of money equal to (CFC − S). This last is the fixed-capital cost of the equipment or plant minus its salvage or scrap value and is the total amount of depreciation during its useful life. The equation relating (CFC − S) and AD is simply the annual cost or payment equation, written either as (1 + i)s − 1 CFC − S = AD ᎏᎏ (9-23) i

i = (ANCI /CTC) − (ABD / CTC) (9-21)

(9-27)

where ABD is the balance-sheet annual depreciation. The main disadvantage of using Eq. (9-27) is that the fractional depreciation rate

INVESTMENT AND PROFITABILITY

FIG. 9-7 FIG. 9-5

Book value against time for various depreciation methods.

9-9

Effect of double-declining depreciation on rate of return for a proj-

ect.

Estimated salvage value of plant items S = $20,000 ABD /CTC is arbitrarily assessed. Its value will affect the fractional rate of return considerably and may lead to erroneous conclusions when making comparisons between different companies. This is particularly true when making international comparisons. Figures 9-6, 9-7, and 9-8 show the effect of the depreciation method on profit for a project described by the following data: Net annual cash income after tax ANCI = $25,500 in each of 10 years Fixed-capital cost CFC = $120,000

Working capital CWC = $10,000 Cost of land CL = $20,000 In Eq. (9-27), i can be taken either on the basis of the net annual cash income for a particular year or on the basis of an average net annual cash income over the length of the life of the project. The equations corresponding to Eq. (9-26) based on depreciated and average investment are given respectively as follows: and

Effect of straight-line depreciation on rate of return for a project. ABD = annual depreciation allowance; ANCI = annual net cash income after tax; ANNP = annual net profit after payment of tax; CTC = total capital cost.

i = A NNP /(Ps′ + CWC + CL)

(9-28)

i = 2A NNP /(CFC + S + 2CWC + 2CL)

(9-29)

FIG. 9-6

FIG. 9-8

project.

Effect of sum-of-years-digits depreciation on rate of return for a

9-10

PROCESS ECONOMICS

where Ps′ is the book value of the fixed-capital investment at the end of a particular year s′. If i is taken on the basis of average values for ANNP over the length of the project, an average value for the working capital CWC must be used. In Eqs. (9-28) and (9-29), the computations are based on unchanging values of the cost of land and other nondepreciable costs CL. This is unrealistic, since the value of land has a tendency to rise. In such circumstances, the accountancy principle of conservatism requires that the lowest valuation be adopted. Payback Period Another traditional method of measuring profitability is the payback period or fixed-capital-return period. Actually, this is really a measure not of profitability but of the time it takes for cash flows to recoup the original fixed-capital expenditure. The net annual cash flow after tax is given by A CF = A NCI − A TC (9-4) where ATC is the annual expenditure of capital, which is not necessarily zero after the plant has been built. The payback period (PBP) is the time required for the cumulative net cash flow taken from the startup of the plant to equal the depreciable fixed-capital investment (CFC − S). It is the value of s′ that satisfies

Annual Compound Interest It is more common to use compound interest, in which F and P are related by F = P(1 + i)n F = Pfi

or

where the compound-interest factor fi = (1 + i)n. Values for compound-interest factors are readily available in tables. The present value P of a future sum of money F is P = F/(1 + i)n F = P/fd

or



A CF = CFC − S

(9-30)

The payback-period method takes no account of cash flows or profits received after the breakeven point has been reached. The method is based on the premise that the earlier the fixed capital is recovered, the better the project. However, this approach can be misleading. Let us consider projects A and B, having net annual cash flows as listed in Table 9-2. Both projects have initial fixed-capital expenditures of $100,000. On the basis of payback period, project A is the more desirable since the fixed-capital expenditure is recovered in 3 years, compared with 5 years for project B. However, project B runs for 7 years with a cumulative net cash flow of $110,000. This is obviously more profitable than project A, which runs for only 4 years with a cumulative net cash flow of only $10,000. Time Value of Money A large part of business activity is based on money that can be loaned or borrowed. When money is loaned, there is always a risk that it may not be returned. A sum of money called interest is the inducement offered to make the risk acceptable. When money is borrowed, interest is paid for the use of the money over a period of time. Conversely, when money is loaned, interest is received. The amount of a loan is known as the principal. The longer the period of time for which the principal is loaned, the greater the total amount of interest paid. Thus, the future worth of the money F is greater than its present worth P. The relationship between F and P depends on the type of interest used. Table 9-3 gives examples of compound-interest factors and example compound-interest calculations. Simple Interest When simple interest is used, F and P are related by F = P(1 + ni) (9-31) where i is the fractional interest rate per period and n is the number of interest periods. Normally, the interest period is 1 year, in which case i is known as the effective interest rate. TABLE 9-2

Cash Flows for Two Projects

(9-34) (9-35)

where the discount factor fd is fd = 1 / fi = 1/[(1 + i)n] Values for the discount factors are readily available in tables which show that it will take 7.3 years for the principal to double in amount if compounded annually at 10 percent per year and 14.2 years if compounded annually at 5 percent per year. For the case of different annual fractional interest rates (i1,i2, . . . ,in in successive years), Eq. (9-32) should be written in the form F = P(1 + i1)(1 + i2)(1 + i3) ⋅ ⋅ ⋅ (1 + in)

s′ = (PBP)

s′ = 0

(9-32) (9-33)

(9-36)

Short-Interval Compound Interest If interest payments become due m times per year at compound interest, mn payments are required in n years. The nominal annual interest rate i′ is divided by m to give the effective interest rate per period. Hence, F = P[1 + (i′/m)]mn

(9-37)

It follows that the effective annual interest i is given by i = [1 + (i′/m)]m − 1

(9-38)

The annual interest rate equivalent to a compound-interest rate of 5 percent per month (i.e., i′/m = 0.05) is calculated from Eq. (9-38) to be i = (1 + 0.05)12 − 1 = 0.796, or 79.6 percent/year Continuous Compound Interest As m approaches infinity, the time interval between payments becomes infinitesimally small, and in the limit Eq. (9-37) reduces to F = P exp (i′n)

(9-39)

A comparison of Eqs. (9-32) and (9-39) shows that the nominal interest rate i′ on a continuous basis is related to the effective interest rate i on an annual basis by exp (i′n) = (1 + i)n

(9-40)

Numerically, the difference between continuous and annual compounding is small. In practice, it is probably far smaller than the errors in the estimated cash-flow data. Annual compound interest conforms more closely to current acceptable accounting practice. However, the small difference between continuous and annual compounding may be significant when applied to very large sums of money. Let us suppose that $100 is invested at a nominal interest rate of 5 percent. We then compute the future worth of the investment after 2 years and also compute the effective annual interest rate for the following kinds of interest: (1) simple, (2) annual compound, (3) monthly compound, (4) daily compound, and (5) continuous compound. The following tabulation shows the results of the calculations, along with the appropriate equation to be used:

Cash flows ACF Year

Project A

Project B

0 1 2 3 4 5 6 7  A CF

$100,000 50,000 30,000 20,000 10,000 0 0 0 $ 10,000

$100,000 0 10,000 20,000 30,000 40,000 50,000 60,000 $110,000

3 years

5 years

Payback period (PBP)

Interest type

Equation

Future worth F

Effective rate i, %

Equation

1 2 3 4 5

(9-31) (9-32) (9-37) (9-37) (9-39)

$110.000 $110.250 $110.495 $110.516 $110.517

5 5 5.117 5.1267 5.1271

(9-31) (9-38) (9-38) (9-38) (9-38)

When computing the effective annual rate for continuous compounding, the first term of Eq. (9-38), [1 + (i′/m)]m, approaches ei′ as m approaches infinity.

TABLE 9-3

Compound Interest Factors* (For examples demonstrating use see end of table.) Single payment

Uniform annual series

Single payment

Uniform annual series

Compoundamount factor

Presentworth factor

Sinkingfund factor

Capitalrecovery factor

Compoundamount factor

Presentworth factor

Compoundamount factor

Presentworth factor

Sinkingfund factor

Capitalrecovery factor

Compoundamount factor

Presentworth factor

n

Given P, to find F (1 + i)n

Given F, to find P 1 ᎏn (1 + i)

Given F, to find A i ᎏᎏ (1 + i)n − 1

Given P, to find A i(1 + i)n ᎏᎏ (1 + i)n − 1

Given A, to find F (1 + i)n − 1 ᎏᎏ i

Given A, to find P (1 + i)n − 1 ᎏᎏ i(1 + i)n

Given P, to find F (1 + i)n

Given F, to find P 1 ᎏn (1 + i)

Given F, to find A i ᎏᎏ (1 + i)n − 1

Given P, to find A i(1 + i)n ᎏᎏ (1 + i)n − 1

Given A, to find F (1 + i)n − 1 ᎏᎏ i

Given A, to find P (1 + i)n − 1 ᎏᎏ i(1 + i)n

1 2 3 4 5

1.050 1.103 1.158 1.216 1.276

0.9524 .9070 .8638 .8227 .7835

1.00000 0.48780 .31721 .23201 .18097

1.05000 0.53780 .36721 .28201 .23097

1.000 2.050 3.153 4.310 5.526

0.952 1.859 2.723 3.546 4.329

1.060 1.124 1.191 1.262 1.338

0.9434 .8900 .8396 .7921 .7473

1.00000 0.48544 .31411 .22859 .17740

1.06000 0.54544 .37411 .28859 .23740

1.000 2.060 3.184 4.375 5.637

0.943 1.833 2.673 3.465 4.212

1 2 3 4 5

6 7 8 9 10

1.340 1.407 1.477 1.551 1.629

.7462 .7107 .6768 .6446 .6139

.14702 .12282 .10472 .09069 .07940

.19702 .17282 .15472 .14069 .12950

6.802 8.142 9.549 11.027 12.578

5.076 5.786 6.463 7.108 7.722

1.419 1.504 1.594 1.689 1.791

.7050 .6651 .6274 .5919 .5584

.14336 .11914 .10104 .08702 .07587

.20336 .17914 .16104 .14702 .13587

6.975 8.394 9.897 11.491 13.181

4.917 5.582 6.210 6.802 7.360

6 7 8 9 10

11 12 13 14 15

1.710 1.796 1.886 1.980 2.079

.5847 .5568 .5303 .5051 .4810

.07039 .06283 .05646 .05102 .04634

.12039 .11283 .10646 .10102 .09634

14.207 15.917 17.713 19.599 21.579

8.306 8.863 9.394 9.899 10.380

1.898 2.012 2.133 2.261 2.397

.5268 .4970 .4688 .4423 .4173

.06679 .05928 .05296 .04758 .04296

.12679 .11928 .11296 .10758 .10296

14.972 16.870 18.882 21.015 23.276

7.887 8.384 8.853 9.295 9.712

11 12 13 14 15

16 17 18 19 20

2.183 2.292 2.407 2.527 2.653

.4581 .4363 .4155 .3957 .3769

.04227 .03870 .03555 .03275 .03024

.09227 .08870 .08555 .08275 .08024

23.657 25.840 28.132 30.539 33.066

10.838 11.274 11.690 12.085 12.462

2.540 2.693 2.854 3.026 3.207

.3936 .3714 .3503 .3305 .3118

.03895 .03544 .03236 .02962 .02718

.09895 .09544 .09236 .08962 .08718

25.673 28.213 30.906 33.760 36.786

10.106 10.477 10.828 11.158 11.470

16 17 18 19 20

21 22 23 24 25

2.786 2.925 3.072 3.225 3.386

.3589 .3418 .3256 .3101 .2953

.02800 .02597 .02414 .02247 .02095

.07800 .07597 .07414 .07247 .07095

35.719 38.505 41.430 44.502 47.727

12.821 13.163 13.489 13.799 14.094

3.400 3.604 3.820 4.049 4.292

.2942 .2775 .2618 .2470 .2330

.02500 .02305 .02128 .01968 .01823

.08500 .08305 .08128 .07968 .07823

39.993 43.392 46.996 50.816 54.865

11.764 12.042 12.303 12.550 12.783

21 22 23 24 25

26 27 28 29 30

3.556 3.733 3.920 4.116 4.322

.2812 .2678 .2551 .2429 .2314

.01956 .01829 .01712 .01605 .01505

.06956 .06829 .06712 .06605 .06505

51.113 54.669 58.403 62.323 66.489

14.375 14.643 14.898 15.141 15.372

4.549 4.822 5.112 5.418 5.743

.2198 .2074 .1956 .1846 .1741

.01690 .01570 .01459 .01358 .01265

.07690 .07570 .07459 .07358 .07265

59.156 63.706 68.528 73.640 79.058

13.003 13.211 13.406 13.591 13.765

26 27 28 29 30

31 32 33 34 35

4.538 4.765 5.003 5.253 5.516

.2204 .2099 .1999 .1904 .1813

.01413 .01328 .01249 .01176 .01107

.06413 .06328 .06249 .06176 .06107

70.761 75.299 80.064 85.067 90.320

15.593 15.803 16.003 16.193 16.374

6.088 6.453 6.841 7.251 7.686

.1643 .1550 .1462 .1379 .1301

.01179 .01100 .01027 .00960 .00897

.07179 .07100 .07027 .06960 .06897

84.802 90.890 97.343 104.184 111.435

13.929 14.084 14.230 14.368 14.498

31 32 33 34 35

40 45 50

7.040 8.985 11.467

.1420 .1113 .0872

.00828 .00626 .00478

.05828 .05626 .05478

120.800 159.700 209.348

17.159 17.774 18.256

10.286 13.765 18.420

.0972 .0727 .0543

.00646 .00470 .00344

.06646 .06470 .06344

154.762 212.744 290.336

15.046 15.456 15.762

40 45 50

55 60 65 70 75

14.636 18.679 23.840 30.426 38.833

.0683 .0535 .0419 .0329 .0258

.00367 .00283 .00219 .00170 .00132

.05367 .05283 .05219 .05170 .05132

272.713 353.584 456.798 588.529 756.654

18.633 18.929 19.161 19.343 19.485

24.650 32.988 44.145 59.076 79.057

.0406 .0303 .0227 .0169 .0126

.00254 .00188 .00139 .00103 .00077

.06254 .06188 .06139 .06103 .06077

394.172 533.128 719.083 967.932 1,300.949

15.991 16.161 16.289 16.385 16.456

55 60 65 70 75

80 85 90 95 100

49.561 63.254 80.730 103.035 131.501

.0202 .0158 .0124 .0097 .0076

.00103 .00080 .00063 .00049 .00038

.05103 .05080 .05063

971.229 1,245.087 1,594.607 2,040.694 2,610.025

19.596 19.684 19.752 19.806 19.848

105.796 141.579 189.465 253.546 339.302

.0095 .0071 .0053 .0039 .0029

.00057 .00043 .00032 .00024 .00018

.06057 .06043 .06032 .06024 .06018

1,746.600 2,342.982 3,141.075 4,209.104 5,638.368

16.509 16.549 16.579 16.601 16.618

80 85 90 95 100

5% Compound Interest Factors

.05038

n

6% Compound Interest Factors

9-11

9-12

PROCESS ECONOMICS TABLE 9-3

Compound Interest Factors (Concluded )

Examples of Use of Table and Factors Given: $2500 is invested now at 5 percent. Required: Accumulated value in 10 years (i.e., the amount of a given principal). Solution:

F = P(1 + i)n = $2500 × 1.0510 Compound-amount factor = (1 + i)n = 1.0510 = 1.629 F = $2500 × 1.629 = $4062.50

Given: $19,500 will be required in 5 years to replace equipment now in use. Required: With interest available at 3 percent, what sum must be deposited in the bank at present to provide the required capital (i.e., the principal which will amount to a given sum)? 1 1 Solution: P = F ᎏn = $19,500 ᎏ5 (1 + i) 1.03 Present-worth factor = 1/(1 + i)n = 1/1.035 = 0.8626 P = $19,500 × 0.8626 = $16,821 Given: $50,000 will be required in 10 years to purchase equipment. Required: With interest available at 4 percent, what sum must be deposited each year to provide the required capital (i.e., the annuity which will amount to a given fund)? i 0.04 Solution: A = F ᎏᎏ = $50,000 ᎏᎏ (1 + i)n − 1 1.0410 − 1 i 0.04 Sinking-fund factor = ᎏᎏ = ᎏᎏ = 0.08329 (1 + i)n − 1 1.0410 − 1 A = $50,000 × 0.08329 = $4,164 Given: $20,000 is invested at 10 percent interest. Required: Annual sum that can be withdrawn over a 20-year period (i.e., the annuity provided by a given capital). i(1 + i)n 0.10 × 1.1020 Solution: A = P ᎏᎏ = $20,000 ᎏᎏ (1 + i)n − 1 1.1020 − 1 i(1 + i)n 0.10 × 1.1020 Capital-recovery factor = ᎏᎏ = ᎏᎏ = 0.11746 (1 + i)n − 1 1.1020 − 1 A = $20,000 × 0.11746 = $2349.20 Given: $500 is invested each year at 8 percent interest. Required: Accumulated value in 15 years (i.e., amount of an annuity). (1 + i)n − 1 1.0815 − 1 Solution: F = A ᎏᎏ = $500 ᎏᎏ i 0.08 (1 − i)n − 1 1.0815 − 1 Compound-amount factor = ᎏᎏ = ᎏᎏ = 27.152 i 0.08 F = $500 × 27.152 = $13,576 Given: $8000 is required annually for 25 years. Required: Sum that must be deposited now at 6 percent interest. (1 + i)n − 1 1.0625 − 1 Solution: P = A ᎏᎏ = $8000 ᎏᎏ i(1 + i)n 0.06 × 1.0625 (1 + i)n − 1 1.0625 − 1 Present-worth factor = ᎏᎏ = ᎏᎏ = 12.783 i(1 + i)n 0.06 × 1.0625 P = $8000 × 12.783 = $102,264 *Factors presented for two interest rates only. By using the appropriate formulas, values for other interest rates may be calculated.

INVESTMENT AND PROFITABILITY Annual Cost or Payment A series of equal annual payments A invested at a fractional interest rate i at the end of each year over a period of n years may be used to build up a future sum of money F. These relations are given by (1 + i)n − 1 F = A ᎏᎏ (9-41) i





F = A/fAF

or

(9-42)

where the annuity future-worth factor is fAF = i / [(1 + i)n − 1] Values for fAF are readily available in tables. Equation (9-41) can be combined with Eq. (9-34) to yield (1 + i)n − 1 P = A ᎏᎏ i(1 + i)n





P = A/ fAP

(9-43)

fAP = [i(1 + i)n]/[(1 + i)n − 1] Values for fAP are also available in tables. Alternatively, the annual payment A required to build up a future sum of money F with a present value of P is given by A = FfAF

(9-45) (9-46)

A = PfAP

Equation (9-41) represents the future sum of a series of uniform annual payments that are invested at a stated interest rate over a period of years. This procedure defines an ordinary annuity. Other forms of annuities include the annuity due, in which payments are made at the beginning of the year instead of at the end; and the deferred annuity, in which the first payment is deferred for a definite number of years. Capitalized Cost A piece of equipment of fixed-capital cost CFC will have a finite life of n years. The capitalized cost of the equipment CK is defined by (CK − CFC)(1 + i)n = CK − S (9-47) CK is in excess of CFC by an amount which, when compounded at an annual interest rate i for n years, will have a future worth of CK less the salvage or scrap value S. If the renewal cost of the equipment remains constant at (CFC − S) and the interest rate remains constant at i, then CK is the amount of capital required to replace the equipment in perpetuity. Equation (9-47) may be rewritten as S (1 + i)n CK = CFC − ᎏn ᎏᎏ (9-48) (1 + i) (1 + i)n − 1



or





CK = (CFC − Sfd) f k

(9-49)

where fd is the discount factor and fk, the capitalized-cost factor, is fk = [(1 + i)n]/[(1 + i)n − 1] Values for each factor are available in tables. Example 1: Capitalized Cost of Equipment A piece of equipment has been installed at a cost of $100,000 and is expected to have a working life of 10 years with a scrap value of $20,000. Let us calculate the capitalized cost of the equipment based on an annual compound-interest rate of 5 percent. Therefore, we substitute values into Eq. (9-48) to give $20,000 (1 + 0.05)10 CK = $100,000 − ᎏᎏ ᎏᎏ (1 + 0.05)10 (1 + 0.05)10 − 1



the venture and the greater the justification for putting the capital at risk. A profitability estimate is an attempt to quantify the desirability of taking this risk. The ways of assessing profitability to be considered in this section are (1) discounted-cash-flow rate of return (DCFRR), (2) net present value (NPV) based on a particular discount rate, (3) equivalent maximum investment period (EMIP), (4) interest-recovery period (IRP), and (5) discounted breakeven point (DBEP). Cash Flow Let us consider a project in which CFC = $1,000,000, CWC = $90,000, and CL = $10,000. Hence, CTC = $1,100,000 from Eq. (9-14). If all this capital expenditure occurs in Year 0 of the project, then ATC = $1,100,000 in Year 0 and −ATC = −$1,100,000. From Eq. (9-4), it is seen that any capital expenditure makes a negative contribution to the net annual cash flow ACF. Let us consider another project in which the fixed-capital expenditure is spread over 2 years, according to the following pattern: CFC = CFC0 + CFC1

(9-44)

where P is the present worth of the series of future equal annual payments A and the annuity present-worth factor is





CK = [$100,000 − ($20,000/1.62889)](2.59009) CK = $227,207

Modern Measures of Profitability An investment in a manufacturing process must earn more than the cost of capital for it to be worthwhile. The larger the additional earnings, the more profitable

9-13

Year 0

Year 1

CFC0 = $400,000 CL = 10,000 ATC = 410,000

CFC1 = $600,000 CWC = 90,000 ATC = 690,000

In the final year of the project, the working capital and the land are recovered, which in this case cost a total of $100,000. Thus, in the final year of the project, ATC = −$100,000 and −ATC = +$100,000. From Eq. (9-4), it is seen that any capital recovery makes a positive contribution to the net annual cash flow. During the development and construction stages of a project, ACI and AIT are both zero in Eqs. (9-2) and (9-4). For this period, the cash flow for the project is negative and is given by ACF = −ATC

(9-50)

Figure 9-9 shows the cash-flow stages in a project. The expenditure during the research and development stage is normally relatively small. It will usually include some preliminary process design and a market survey. Once the decision to go ahead with the project has been taken, detailed process-engineering design will commence, and the rate of expenditure starts to increase. The rate is increased still further when equipment is purchased and construction gets under way. There is no return on the investment until the plant is started up. Even during startup, there is some additional expenditure. Once the plant is operating smoothly, an inflow of cash is established. During the early stages of a project, there may be a tax credit because of the existence of expenses without corresponding income. Discounted Cash Flow The present value P of a future sum of money F is given by P = Ffd (9-51) where fd = 1/(1 + i)n, the discount factor. Values for this factor are readily available in tables. For example, $90,909 invested at an annual interest rate of 10 percent becomes $100,000 after 1 year. Similarly, $38,554 invested at 10 percent becomes $100,000 after 10 years. Thus, cash flow in the early years of a project has a greater value than the same amount in the later years of a project. Therefore, it pays to receive money as soon as possible and to delay paying out money for as long as possible. Time is taken into account by using the annual discounted cash flow ADCF, which is related to the annual cash flow ACF and the discount factor fd by (9-52) ADCF = A CF fd Thus, at the end of any year n, (ADCF)n = (ACF)n / (1 + i)n The sum of the annual discounted cash flows over n years,  ADCF, is known as the net present value (NPV) of the project: n

(NPV) =  (ADCF)n 0

(9-53)

9-14

PROCESS ECONOMICS

FIG. 9-9

Effect of discount rate on cash flows.

The value of (NPV) is directly dependent on the choice of the fractional interest rate i. An interest rate can be selected to make (NPV) = 0 after a chosen number of years. This value of i is found from n (ACF)n (ACF)0 (ACF)1 +ᎏ +⋅⋅⋅+ᎏ =0 (9-54) 0 (ADCF)n = ᎏ (1 + i)0 (1 + i)1 (1 + i)n Equation (9-54) may be solved for i either graphically or by an iterative trial-and-error procedure. The value of i given by Eq. (9-54) is known as the discounted-cash-flow rate of return (DCFRR). It is also known as the profitability index, true rate of return, investor’s rate of return, and interest rate of return. Cash-Flow Curves Figure 9-9 shows the cash-flow stages in a project together with their discounted-cash-flow values for the data given in Table 9-4. In addition to cash-flow and discounted-cash-flow curves, it is also instructive to plot cumulative-cash-flow and cumulative-discounted-cash-flow curves. These are shown in Fig. 9-10 for the data in Table 9-4. The cost of capital may also be considered as the interest rate at which money can be invested instead of putting it at risk in a manufacturing process. Let us consider the process data listed in Table 9-4 and plotted in Fig. 9-10. If the cost of capital is 10 percent, then the appropriate discounted-cash-flow curve in Fig. 9-10 is abcdef. Up to point e, or 8.49 years, the capital is at risk. Point e is the discounted breakeven point (DBEP). At this point, the manufacturing process TABLE 9-4

has paid back its capital and produced the same return as an equivalent amount of capital invested at a compound-interest rate of 10 percent. Beyond the breakeven point, the capital is no longer at risk and any cash flow above the horizontal baseline, ADCF = 0, is in excess of the return on an equivalent amount of capital invested at a compoundinterest rate of 10 percent. Thus, the greater the area above the baseline, the more profitable the process. When (NPV) and (DCFRR) are computed, depreciation is not considered as a separate expense. It is simply used as a permitted writingdown allowance to reduce the annual amount of tax in accordance with the rules applying in the country of earning. The tax payable is deducted in accordance with Eq. (9-2) in the year in which it is paid, which may differ from the year in which the corresponding income was earned. A (DCFRR) of, say, 15 percent implies that 15 percent per year will be earned on the investment, in addition to which the project generates sufficient money to repay the original investment plus any interest payable on borrowed capital plus all taxes and expenses. It is not normally possible to make a comprehensive assessment of profitability with a single number. The shape of the cumulative-cashflow and cumulative-discounted-cash-flow curves both before and after the breakeven point is an important factor. D. H. Allen [Chem. Eng., 74, 75–78 (July 3, 1967)] accounted for the shape of the cumulative-undiscounted-cash-flow curve up to the

Annual Cash Flows and Discounted Cash Flows for a Project Discounted at 10%

Discounted at 20%

Discounted at 25%

Year

ACF, $

 ACF, $

fd

ADCF, $

 ADCF, $

fd

A DCF, $

 ADCF, $

fd

A DCF, $

 ADCF, $

0 1 2 3 4

−10,000 −30,000 −60,000 −750,000 −150,000

−10,000 −40,000 −100,000 −850,000 −1,000,000

1.00000 0.90909 0.82645 0.75131 0.68301

−10,000 −27,273 −49,587 −563,483 −102,452

−10,000 −37,273 −86,860 −650,343 −752,795

1.00000 0.83333 0.69444 0.57870 0.48225

−10,000 −25,000 −41,666 −434,025 −72,338

−10,000 −35,000 −76,666 −510,691 −583,029

1.00000 0.80000 0.64000 0.51200 0.40960

−10,000 −24,000 −38,400 −384,000 −61,440

−10,000 −34,000 −72,400 −456,400 −517,840

5 6 7 8 9

+200,000 +300,000 +400,000 +400,000 +360,000

−800,000 −500,000 −100,000 +300,000 +660,000

0.62092 0.56447 0.51316 0.46651 0.42410

+124,184 +169,341 +205,264 +186,604 +152,676

−628,611 −459,270 −254,006 −67,402 +85,274

0.40188 0.33490 0.27908 0.23257 0.19381

+80,376 +100,470 +111,632 +93,028 +69,772

−502,653 −402,183 −290,551 −197,523 −127,751

0.32768 0.26214 0.20972 0.16777 0.13422

+65,536 +78,642 +83,888 +67,108 +48,319

−452,304 −373,662 −289,774 −222,666 −174,347

10 11 12 13 14

+320,000 +280,000 +240,000 +240,000 +400,000

+980,000 +1,260,000 +1,500,000 +1,740,000 +2,140,000

0.38554 0.35049 0.31863 0.28966 0.26333

+123,373 +98,137 +76,471 +69,518 +105,332

+208,647 +306,784 +383,255 +452,773 +558,105

0.16151 0.13459 0.11216 0.09346 0.07789

+51,683 +37,685 +26,918 +22,430 +31,156

−76,068 −38,383 −11,465 +10,965 +42,121

0.10737 0.08590 0.06872 0.05498 0.04398

+34,358 +24,052 +16,493 +13,195 +17,592

−139,989 −115,937 −99,444 −86,249 −68,657

NOTE: ACF is net annual cash flow, ADCF is net annual discounted cash flow, fd is discount factor at stated interest,  ACF is cumulative cash flow, and  ADCF is cumulative discounted cash flow.

INVESTMENT AND PROFITABILITY

FIG. 9-10

Effect of discount rate on cumulative cash flows.

breakeven point e0 in Fig. 9-10 by using a parameter known as the equivalent maximum investment period (EMIP), which is defined as area (a0 to e0) (EMIP) = ᎏᎏ for ACF ≤ 0 (9-55) ( ACF)max where the area (a0 to e0) refers to the area below the horizontal baseline ( ACF = 0) on the cumulative-cash-flow curve in Fig. 9-10. The sum ( ACF)max is the maximum cumulative expenditure on the project, which is given by point d0 in Fig. 9-10. (EMIP) is a time in years. It is the equivalent period during which the total project debt would be outstanding if it were all incurred at one instant and all repaid at one instant. Clearly, the shorter the (EMIP), the more attractive the project. Allen accounted for the shape of the cumulative-cash-flow curve

FIG. 9-11

9-15

beyond the breakeven point by using a parameter known as the interest-recovery period (IRP). This is the time period (illustrated in Fig. 9-11) that makes the area (e0 to f0) above the horizontal baseline equal to the area (a0 to e0) below the horizontal baseline on the cumulativecash-flow curve. C. G. Sinclair [Chem. Process. Eng., 47, 147 (1966)] has considered similar parameters to the (EMIP) and (IRP) based on a cumulativediscounted-cash-flow curve. Consideration of the cash-flow stages in Fig. 9-10 shows the factors that can affect the (EMIP) and (IRP). If the required capital investment is increased, it is necessary to increase the rate of income after startup for the (EMIP) to remain the same. In order to have the (EMIP) small, it is necessary to keep the research and development, design, and construction stages short.

Cumulative cash flow against time, showing interest recovery period.

9-16

PROCESS ECONOMICS ACF = $110,000/year − (−$100,000/year)

Example 2: Net Present Value for Different Depreciation Methods The following data describe a project. Revenue from annual sales and the total annual expense over a 10-year period are given in the first three columns of Table 9-5. The fixed-capital investment CFC is $1,000,000. Plant items have a zero salvage value. Working capital CWC is $90,000, and cost of land CL is $10,000. There are no tax allowances other than depreciation; i.e., AA is zero. The fractional tax rate t is 0.50. We shall calculate for these data the net present value (NPV) for the following depreciation methods and discount factors: a. Straight-line, 10 percent b. Straight-line, 20 percent c. Double-declining, 10 percent d. Sum-of-years-digits, 10 percent e. Straight-line, 10 percent; income tax delayed for 1 year In addition, we shall calculate the discounted-cash-flow rate of return (DCFRR) with straight-line depreciation. a. We begin the calculations for this example by finding the total capital cost CTC for the project from Eq. (9-14). Here, CTC = $1,100,000. In Year 0, this amount is the same as the net annual capital expenditure ATC and is listed in Table 9-5. The annual rate of straight-line depreciation of the fixed-capital investment CFC, from $1,000,000 at startup to a salvage value S, of zero at the end of a productive life s of 10 years, is given by AD = (CFC − S)/s AD = ($1,000,000 − $0)/10 years = $100,000/year The annual cash income ACI for Year 1, when AS = $400,000 per year and ATE = $100,000 per year, is, from Eq. (9-1), $300,000 per year. Values for subsequent years are calculated in the same way and listed in Table 9-4. Annual amount of tax AIT for Year 1, when ACI = $300,000 per year, AD = $100,000 per year, AA = $0 per year, and t = 0.5, is found from Eq. (9-3) to be AIT = [($300,000 − $100,000 − $0)/year](0.5) = $100,000/year Values for subsequent years are calculated in the same way and listed in Table 9-4. Net annual cash flow (after tax) ACF for Year 0, when ACI = $0 per year, AIT = $0 per year, and ATC = $1,100,000 per year, is found from Eq. (9-4) to be ACF = $0/year − $1,100,000/year = −$1,100,000/year Net annual cash flow (after tax) ACF for Year 1, when ACI = $300,000 per year, AIT = $100,000 per year, and ATC = $0 per year, is found from Eqs. (9-2) and (9-4) to be ACF = $200,000/year − $0/year = $200,000/year Values for the years up to and including Year 9 are calculated in the same way and listed in Table 9-5. At the end of Year 10, the working capital (CWC = $90,000) and the cost of land (CL = $10,000) are recovered, so that the annual expenditure of capital ATC in Year 10 is −$100,000 per year. Hence, the net annual cash flow (after tax) for Year 10 must reflect this recovery. By using Eq. (9-4),

TABLE 9-5

= $210,000/year The net annual discounted cash flow ADCF for Year 1, when ACF = $200,000 per year and fd = 0.90909 (for i = 10 percent), is found from Eq. (9-52) to be ADCF = ($200,000/year)(0.90909) = $181,820/year Values for subsequent years are calculated in the same way and listed in Table 9-5. The net present value (NPV) is found by summing the values of ADCF for each year, as in Eq. (9-53). The net present value is found to be $276,210, as given by the final entry in Table 9-5. b. The same procedure is used for i = 20 percent. The discount factors to be used in a table similar to Table 9-5 must be those for 20 percent. The (NPV) is found to be −$151,020. c. The calculations are similar to those for subexample a except that depreciation is computed by using the double-declining method of Eq. (9-19). The net present value is found to be $288,530. d. Again, the calculations are similar to those for subexample a except that depreciation is computed by using the sum-of-years-digits method of Eq. (9-20). The net present value is found to be $316,610. e. The calculations follow the same procedure as for subexample a, but the annual amount of tax AIT is calculated for a particular year and then deducted from the annual cash income ACI for the following year. The net present value for Year 11 is found to be $341,980. The discounted-cash-flow rate of return (DCFRR) can readily be obtained approximately by interpolation of the (NPV) for i = 10 percent and i = 20 percent: (DCFRR) = 0.100 + [($276,210)(0.20 − 0.10)]/[$276,210 − (−$151,020)] (DCFRR) = 0.164, or 16.4 percent The calculation of (DCFRR) usually requires a trial-and-error solution of Eq. (9-57), but rapidly convergent methods are available [N. H. Wild, Chem. Eng., 83, 153–154 (Apr. 12, 1976)]. For simplicity linear interpolation is often used. A comparison of the (NPV) values for a 10 percent discount factor shows clearly that double-declining depreciation is more advantageous than straightline depreciation and that sum-of-years-digits depreciation is more advantageous than the double-declining method. However, a significant advantage is obtained by delaying the payment of tax for 1 year even with straight-line depreciation.

This example is a simplified one. The cost of the working capital is assumed to be paid for in Year 0 and returned in Year 10. In practice, working capital increases with the production rate. Thus there may be an annual expenditure on working capital in a number of years subsequent to Year 0. Except in loss-making years, this is usually treated as an expense of the process. In loss-making years the cash injection for working capital is included in the ATC for that year. Analysis of Techniques Both the (NPV) and the (DCFRR) methods are based on discounted cash flows and in that sense are vari-

Annual Cash Flows, Straight-Line Depreciation, and 10 Percent Discount Factor Before tax

After tax

Year

AS, $

ATE, $

ACI, $

AD + AA, $

AIT, $

ATC, $

ACF, $

fd

ADCF, $

(NPV), $

0 1 2 3

0 400,000 500,000 500,000

0 100,000 100,000 110,000

0 300,000 400,000 390,000

0 100,000 100,000 100,000

0 200,000 300,000 290,000

0 100,000 150,000 145,000

+1,100,000 0 0 0

−1,100,000 200,000 250,000 245,000

1.0000 0.90909 0.82645 0.75131

−1,100,000 181,820 206,610 184,070

−1,100,000 −918,180 −711,570 −527,500

4 5 6 7

500,000 520,000 520,000 520,000

120,000 130,000 130,000 140,000

380,000 390,000 390,000 380,000

100,000 100,000 100,000 100,000

280,000 290,000 290,000 280,000

140,000 145,000 145,000 140,000

0 0 0 0

240,000 245,000 245,000 240,000

0.68301 0.62092 0.56447 0.51316

163,920 152,120 138,300 123,160

−363,580 −211,460 −73,160 +50,000

8 9 10

390,000 350,000 280,000

140,000 150,000 160,000

250,000 200,000 120,000

100,000 100,000 100,000

150,000 100,000 20,000

75,000 50,000 10,000

0 0 −100,000

175,000 150,000 210,000

0.46651 0.42410 0.38554

81,640 63,610 80,960

+131,640 +195,250 +276,210

AS = revenue from annual sales. ATE = total annual expense. ACI = annual cash income. AD + AA = annual depreciation and other tax allowances. ACI − AD − AA = taxable income. AIT = (ACI − AD − AA)t = amount of tax at t = 0.5.

ACI − AD − AA, $

ATC = total annual capital expenditure. ACF = ACI − AIT − ATC = net annual cash flow. fd = discount factor at 10%. ADCF = net annual discounted cash flow. (NPV) =  ADCF = net present value.

INVESTMENT AND PROFITABILITY ations of the same basic method. However, when ranking different projects on the basis of profitability, they can produce different results. Discounted-cash-flow rate of return (DCFRR) has the advantage of being unique and readily understood. However, when used alone, it gives no indication of the scale of the operation. The (NPV) indicates the monetary return, but unlike that of the (DCFRR) its value depends on the base year chosen for the calculation. Additional information is needed before its significance can be appreciated. However, when a company is considering investment in a portfolio of projects, individual (NPV)s have the advantage of being additive. This is not true of (DCFRR)s. Increasing use is being made of the capital-rate-of-return ratio (CRR), which is the net present value (NPV) divided by the maximum cumulative expenditure or maximum net outlay, −( ACF)max (CRR) = (NPV)/(ACF)max

for

ACF ≤ 0

(9-56)

The maximum net outlay is very important, since no matter how profitable a project is, the matter is academic if the company is unable to raise the money to undertake the project. An (NPV) or (DCFRR) estimation will be no better than the accuracy of the projected cash flows over the life of the project. Clearly, one is likely to predict cash flows more accurately for 2 or 3 years ahead than, say, for 9 or 10 years ahead. However, since the cash flows for the later years are discounted to a greater extent than the cash flows for the earlier years, the latter have less effect on the overall estimation. Nevertheless, the difficulty of predicting cash flows in later years and the inherent lack of confidence in these predictions are serious disadvantages of the (DCFRR) method. In this respect (NPV)s are more useful since they are calculated for each year of a project. Thus, a project with a favorable (NPV) in the early years is a promising one. One way of overcoming these disadvantages of the (DCFRR) method is to make estimates of the times required to reach certain values of (DCFRR). For example, how many years will it take to reach (DCFRR)s of 10 percent, 15 percent, 20 percent per year, etc.? Although (DCFRR) trial-and-error calculations and (NPV) calculations are tedious if done manually, computer programs which are suitable for programmable pocket calculators can readily be written to make calculations easier. It is possible for some projects to reach a stage at which repairs, replacements, etc., can exceed net earnings in a particular year. In this case the cumulative-discounted-cash-flow or net-present-value curve plotted against time has a genuine maximum. It is important when appraising by (NPV) and (DCFRR) not to consider the past in profitability estimations. Good money should never follow bad. It is unwise to continue to put money into a project if a more profitable project exists, even though this course may involve scrapping an expensive plant. Other considerations may, however, outweigh purely financial criteria in a particular case. No single value for a profitability estimate should be accepted without further consideration. An intelligent consideration of the cumulative-cash-flow and cumulative-discounted-cash-flow curves such as those shown in Fig. 9-10, together with experience and good judgment, is the best way of assessing the financial merit of a project. When considering future projects, top management will most likely require the discounted-cash-flow rate of return and the payback period. However, the estimators should also supply management with the following: Cumulative discounted-cash-flow or (NPV) curve for a discount rate of 10 percent per year or other agreed aftertax cost of capital Maximum net outlay, ( ACF)max, for ACF ≤ 0 Discounted breakeven point (DBEP) Plot of capital-return ratio (CRR) against time over the life of the project for a discount rate at the cost of capital Number of years to reach discounted-cash-flow rates of return of, say, 15 and 25 percent per year respectively Comparisons on the basis of time can be summarized by the following: Duration of the project Breakeven point (BEP)

9-17

Discounted breakeven point (DBEP) Equivalent maximum investment period (EMIP) Interest-recovery period (IRP) Payback period (PBP) Comparisons on the basis of cash can be summarized by the following: Maximum cumulative expenditure on the project, ( ACF)max, for ACF ≤ 0 Maximum discounted cumulative expenditure on the project Cumulative net annual cash flow  ACF Cumulative net annual discounted cash flow  ADCF or net present value (NPV) Capitalized cost CK Comparisons on the basis of interest can be summarized as (1) the net present value (NPV) and (2) the discounted-cash-flow rate of return (DCFRR), which from Eqs. (9-53) and (9-54) is given formally as the fractional interest rate i which satisfies the relationship n

(NPV) =  (ADCF)n = 0

(9-57)

0

When comparing project profitability, the ranking on the basis of net present value (NPV) may differ from that on the basis of discounted-cash-flow rate of return (DCFRR). Let us consider the data for two projects: Cost of capital

Project C

Project D

i, % 4 8 12 16

(NPV), $ +100,000 +41,000 −2,000 −32,000

(NPV), $ +62,000 +28,000 +10,000 −4,000

These (NPV) data are plotted against the cost of capital, as shown in Fig. 9-12. The discounted-cash-flow rate of return is the value of i that satisfies Eq. (9-5). From Fig. 9-12, (NPV) = 0 at a (DCFRR) of 11.8 percent for project C and 14.7 percent for project D. Thus, on the basis of (DCFRR), project D is more profitable than project C. The (NPV) of project C is equal to that of project D at a cost of capital i = 9.8 percent. If the cost of capital is greater than 9.8 percent, project D has the higher (NPV) and is, therefore, the more profitable. If the cost of capital is less than 9.8 percent, project C has the higher (NPV) and is the more profitable. Benefit of Early Cash Flows It pays to receive cash inflows as early as possible and to delay cash outflows as long as possible. Let us consider the net annual cash flows (after tax) ACF for projects E, F, and G, listed in Table 9-6. The cumulative annual cash flows  ACF and cumulative discounted annual cash flows  ADCF, using a discount of 10 percent for these projects, are also listed in Table 9-6. We notice that the cumulative annual cash flow for each project is +$1000.

FIG. 9-12

Effect of cost of capital on net present value.

9-18

PROCESS ECONOMICS

TABLE 9-6

Cash-Flow Data for Projects E, F, and G

TABLE 9-7

Cash-Flow Data for Projects H and I

Discounted at 10% Year

ACF, $

 ACF, $

0 1 2 3

−5000 +3000 +2000 +1000

−5000 −2000 0 +1000

0 1 2 3

−5000 +1000 +2000 +3000

−5000 −4000 −2000 +1000

0 1 2 3

−5000 +2000 +2000 +2000

−5000 −3000 −1000 +1000

Discounted at 10%

fd

ADCF, $

 ADCF = (NPV), $

1.0000 0.90909 0.82645 0.75131

−5000 +2727 +1653 +751

ADCF, $

 ADCF = (NPV), $

1.0000 0.90909 0.82645 0.75131 0.68301

−50,000 9,091 8,265 7,513 6,830

−50,000 −40,909 −32,644 −25,131 −18,301

+10,000 +10,000 +10,000 +10,000 +10,000

0.62092 0.56447 0.51316 0.46651 0.42410

6,209 5,645 5,132 4,665 4,241

−12,092 −6,447 −1,315 +3,350 +7,591

−40,000 +20,000 +20,000 +20,000 +20,000 +20,000 +20,000

0.75131 0.68301 0.62092 0.56447 0.51316 0.46651 0.42410

−30,052 +13,660 +12,418 +11,289 +10,263 +9,330 +8,482

−30,052 −16,392 −3,974 +7,315 +17,578 +26,908 +35,390

Year

ACF, $

fd

−5000 −2273 −620 +131

0 1 2 3 4

−50,000 +10,000 +10,000 +10,000 +10,000

−5000 +909 +1653 +2254

−5000 −4091 −2438 −184

5 6 7 8 9

−5000 +1818 +1653 +1503

−5000 −3182 −1529 −26

3 4 5 6 7 8 9

Project E

Project H

Project F 1.0000 0.90909 0.82645 0.75131

Project G 1.0000 0.90909 0.82645 0.75131

The (DCFRR) is the discount rate that satisfies Eq. (9-57) in the final year of the project. We can approximate the (DCFRR) for each project as follows: For project E,  ACF = +$1000 in Year 3 for i = 0 percent  ADCF = +$131 in Year 3 for i = 10 percent  ADCF = $0 in Year 3 for i = (DCFRR) Therefore, 1000/(1000 − 131)  (DCFRR)/10 (DCFRR)  11.5 percent Similarly for project F,

Project I

land have been neglected since the latter is the same for each project and the former would also favor project I. Incremental Comparisons A company may have the choice of, say, investing $10,000 in project J, which will give a (DCFRR) of 16 percent, or $7000 in project K, which will give a (DCFRR) of 18 percent. Should it spend $10,000 on project J or spend only $7000 on project K and invest the difference of $3000 elsewhere? Both projects have lives of 10 years and constant positive net annual cash flows ACF of $2069 and $1558 for projects J and K respectively. The corresponding (NPV)s at a discount factor of 10 percent are +$2710 and +$2560 respectively. These data are summarized as follows:

1000/(1000 + 184)  (DCFRR)/10 (DCFRR)  8.4 percent Similarly for project G, 1000/(1000 + 26)  (DCFRR)/10 (DCFRR)  9.7 percent In terms of net present value (NPV), the projects in order of merit are E, G, and F, with (NPV)s of +$131, −$26, and −$184 respectively. In terms of (DCFRR), the projects in order of merit are also E, G, and F, with (DCFRR) values of 11.5 percent, 9.7 percent, and 8.4 percent respectively. When to Scrap an Existing Process Let us suppose that a company invests $50,000 in a manufacturing process that has positive net annual flows (after tax) ACF of $10,000 in each year. During the third year of operation, an alternative process becomes available. The new process would require an investment of $40,000 but would have positive net annual cash flows (after tax) of $20,000 in each year. The cost of capital is 10 percent, and it is estimated that a market will exist for the product for at least 6 more years. Should the company continue with the existing process (project H), or should it scrap project H and adopt the new process (project I)? The net annual cash flows ACF and cumulative discounted annual cash flow  ADCF for a discount factor of 10 percent are listed in Table 9-7 for the two projects. At the end of Year 9, the net present values are (NPV) = +$35,390 for project I (NPV) = +$7591 for project H The difference is +$27,779, which is numerically greater than the money lost by the end of Year 3 for project H. Thus project H should be scrapped, and the new project I adopted if only economic reasons need to be considered. Recovery of working capital and the cost of

ACF, $, in Year 0 ACF, $, in each of Years 1–10 (NPV), i = 10 percent, $ (DCFRR), percent

Project J

Project K

−10,000 +2,069 +2,710 16

−7,000 +1,558 +2,560 18

Project ( J − K) −3,000 +511 +150 12.4

From the difference in cash flows between the projects, the discounted-cash-flow rate of return (DCFRR) for project (J-K) can be shown as 12.4 percent. This is significantly lower than for either project J or project K. Thus, if the $3000 can be invested to give a return greater than 12.4 percent, project K should be chosen in preference to project J. Comparisons on the Basis of Capitalized Cost A machine in a process generates a positive net cash flow of $1000. Two alternatives are available: machine L, costing $2000, requires replacement every 4 years, and machine M, costing $3000, requires replacement every 6 years. Neither machine has any scrap value. The cost of capital is 10 percent. Which machine is the more profitable to operate? In this case, the lives of the machines are unequal, and the comparison is conveniently made on the basis of capitalized cost. This puts lives on the same basis, which is an infinite number of years. The net annual cash flows generated by each machine are equal. The capitalized cost CK of a piece of fixed-capital cost CFC is the amount of capital required to ensure that the equipment may be renewed in perpetuity. For a piece of equipment with no scrap value, CK is given by (1 + i)n CK = CFC ᎏᎏ (9-58) (1 + i)n − 1 For machine L, CK = ($2000)(3.15471) = $6309.42





INVESTMENT AND PROFITABILITY For machine M, CK = ($3000)(2.29607) = $6888.21 Thus, machine L with the lower capitalized cost is the more profitable to operate. Relationship between (PBP) and (DCFRR) For the case of a single lump-sum capital expenditure CFC which generates a constant annual cash flow ACF in each subsequent year, the payback period is given by the equation (PBP) = CFC / ACF

(9-59)

if the scrap value of the capital outlay may be taken as zero. For this simplified case the net present value (NPV) after n years with money invested at a required aftertax compound annual fractional interest rate i is given by the equation (NPV) = CFC − ACF Fn

(9-60)

n

where

1 Fn =  ᎏn (1 + i) 1

When (NPV) = 0, the value of i given by Eq. (9-60) is the discountedcash-flow rate of return (DCFRR), and in this case Eqs. (9-59) and (9-60) can be combined to give: (PBP) = Fn

(9-61)

Figure 9-13 is a plot of Eq. (9-61) in the form of the number of years n required to reach a certain discounted-cash-flow rate of return (DCFRR) for a given payback period (PBP). The figure is a modification of plots previously published by A. G. Bates [Hydrocarbon Process., 45, 181–186 (March 1966)], C. Estrup [Br. Chem. Eng., 16, 171 (February–March 1971)], and F. A. Holland and F. A. Watson [Process Eng. Econ., 1, 293–299 (December 1976)]. In the limiting case when n approaches infinity, Eq. (9-61) can be written as (DCFRR)max = 1/(PBP)

(9-62)

which means, for example, that if the payback period is 4 years, the maximum possible discounted-cash-flow rate of return which can be reached is 25 percent. The corresponding (DCFRR) for (PBP) = 10 years is 10 percent.

FIG. 9-13

9-19

Equations (9-59), (9-60), (9-61), and (9-62) may be used as they stand to assess expenditure on energy-conservation measures since a constant amount of energy is saved in each year subsequent to the capital outlay. However, the annual cash flows ACF corresponding to the energy savings remain constant only if there is no inflation or if the money values are corrected to their purchasing power at the time of the capital expenditure. Sensitivity Analysis An economic study should pinpoint the areas most susceptible to change. It is easier to predict expenses than either sales or profits. Fairly accurate estimates of capital costs and processing costs can be made. However, for the most part, errors in these estimates have a correspondingly smaller effect than changes in sales price, sales volume, and the costs of raw materials and distribution. Sales and raw-materials prices may be affected by any of the following: discounts and allowances, availability of substitutes, contract pricing, government regulations, quality and form of the materials, and competition. Sales volume may be affected by any of the following: new uses for the product, new markets, advertising, quality, overcapacity, replacement by another product, competition, and timing of entry into the market. Distribution costs depend on plant location, physical state of the material (whether liquid, gas, or solid), nature of the material (whether corrosive, explosive, flammable, perishable, or toxic), freight rates, and labor costs. Distribution costs may be affected by any of the following: new methods of materials handling, safety regulations, productivity agreements, wage rates, transportation systems, storage systems, quality, losses, and seasonal effects. It is worthwhile to make tables or plot curves that show the effect of variations in costs and prices on profitability. This procedure is called sensitivity analysis. Its purpose is to determine to which factors the profitability of a project is most sensitive. Sensitivity analysis should always be carried out to observe the effect of departures from expected values. For many years, companies and countries have lived with the problem of inflation, or the falling value of money. Costs—in particular, labor costs—tend to rise each year. Failure to account for this trend in predicting future cash flows can lead to serious errors and misleading profitability estimates. Another important factor is the tendency of product prices to fall as the total national or international volume of production increases. Sales prices may fall by 20 percent for a doubling in volume or production.

Relationship between payback period and discounted-cash-flow rate of return.

9-20

PROCESS ECONOMICS

FIG. 9-14

Net present value against time, showing effect of adverse changes in cash flows.

No profitability estimate is better than the inherent accuracy of the data. Example 3: Sensitivity Analysis The following data describe a project. Revenue from annual sales and total annual expense over a 10-year period are given in the first three columns of Table 9-5. The fixed-capital investment CFC is $1 million. Plant items have a zero salvage value. Working capital CWC is $90,000, and the cost of land CL is $10,000. There are no tax allowances other than depreciation; i.e., AA is zero. The fractional tax rate t is 0.50. For this project, the net present value for a 10 percent discount factor and straight-line depreciation was shown to be $276,210 and the discounted-cash-flow rate of return to be 16.4 percent per year. We shall use these data and the accompanying information of Table 9-5 as the base case and calculate for straight-line depreciation the net present value (NPV) with a 10 percent discount factor and the discounted-cash-flow rate of return (DCFRR) for the project with the following situations.

Case a b c d e

Modification Revenue AS reduced by 10 percent per year Revenue AS reduced by 20 percent per year Total expense ATE increased by 10 percent per year Fixed-capital investment increased by 10 percent AS reduced by 10 percent per year, ATE increased by 10 percent per year, and CFC increased by 10 percent

The results are shown in Figs. 9-14 and 9-15 and Tables 9-8 and 9-9.

Learning Curves It is usual to learn from experience. Consequently, the time taken to produce an article, the number of spoiled batches, the cost per unit of production, etc., tend to decrease with the number of units produced. The relationships are expressed for the ideal case by

TABLE 9-8 Annual Cash Flows, Straight-Line Depreciation, and 10 Percent Discount Factor When Revenue Is Reduced by 10 Percent per Year Base case Year

fd

∆ADCF, $

∆(NPV), $

Reduced (NPV), $

0 40,000 50,000 50,000

1.0000 0.90909 0.82645 0.75131

0 18,180 20,660 18,780

0 18,180 38,840 57,620

−1,100,000 −936,360 −750,410 −585,120

−363,580 −211,460 −73,160 +50,000

50,000 52,000 52,000 52,000

0.68301 0.62092 0.56447 0.51316

17,070 16,140 14,680 13,340

74,690 90,830 105,510 118,850

−438,270 −302,290 −178,670 −68,850

+131,640 +195,250 +276,210

39,000 35,000 28,000

0.46651 0.42410 0.38554

9,100 7,420 5,390

127,950 135,370 140,760

+3,690 +59,880 +135,450

AS, $

ATE, $

(NPV), $

0 1 2 3

0 400,000 500,000 500,000

0 100,000 100,000 110,000

−1,100,000 −918,180 −711,570 −527,500

4 5 6 7

500,000 520,000 520,000 520,000

120,000 130,000 130,000 140,000

8 9 10

390,000 350,000 280,000

140,000 150,000 160,000

AS = base revenue from annual sales before tax. ATE = base total annual expense before tax. (NPV) = base net present value after tax. ∆AS = decrease in annual revenue.

∆AS, $

fd = discount factor at 10%. ∆ADCF = decrease in net discounted cash flow at income tax rate = 0.5. ∆(NPV) =  ∆ADCF = decrease in net present value. Reduced (NPV) =  ADCF = reduced net present value after tax.

INVESTMENT AND PROFITABILITY

FIG. 9-15

where Y = X= K= N=

Decrease in net present value against time resulting from adverse changes in cash flows.

Y = KX N (9-63) cumulative-average cost, production time, etc., per unit cumulative production, units effective value of first unit produced slope of straight-line plot of Y versus X on log-log paper

The particular learning curve is usually characterized by the percentage reduction in the cumulative average value Y when the number of units X is doubled. From this definition it follows that N = log (characteristic/100)/log 2

(9-64)

The cost cME of the last unit of a block bringing the cumulative production to X units is, from Eq. (9-63), cME = K[X N + 1 − (X − 1) N + 1]

(9-65)

These unit costs, or the time taken to produce the last unit, etc., may be plotted on cartesian coordinates against the number of units produced to provide a standard against which the performance of a new employee, a new machine, etc., can be judged. Figure 9-16 shows such a plot for the subsequent example. In general, cost data will be available for multiple units. Typically, the cost of production for 1 week or of a specific order is computed and an average cost per unit obtained. This average value Y
for the batch should be plotted against the corresponding learning-curve value
X calculated by Eq. (9-66): XN = (X2N + 1 − X1N + 1)/(X2 − X1)


(9-66)

where X1 and X2 are the cumulative production before and after the batch. This form of the equation is useful when only the previous production history of the process is known, from the serial numbers or otherwise. TABLE 9-9

Summary of Results of Sensitivity Analysis

Case Base case A S reduced 10% per year A S reduced 20% per year A TE increased 10% per year CFC increased 10% Combined: A S reduced 10% per year A TE increased 10% per year CFC increased 10%

}

9-21

(NPV), $ i = 10%

(DCFRR), %

276,210 135,450 −5,330 238,430 206,890

16.4 12.9 9.8 15.0 14.0

28,420

10.6

A straight line may be fitted to the (X,Y) or (X
,Y
) pairs of data when plotted on log-log graph paper from which the slope N and the intercept log K with X = 1 may be read. Alternatively, the method of least squares may be used to estimate the values of K and N, giving the best fit to the available data. It will be noted that a value of N = 0, corresponding to a characteristic of 100 percent for the learning curve, implies that the value of Y is independent of X. This would imply that learning by experience was not possible and thus corresponds to an optimally designed process or one for which the costs are determined by external factors. Similarly, a value of N = −1, corresponding to the 50 percent learning curve, implies that the cost of production is inversely proportional to the number produced, which is absurd. Projects having characteristics less than 70 percent are impractical. Low characteristics are typical of hasty entry into a market in an attempt to preempt it. Characteristics tend to increase with experience, so that established and mature projects are likely to have characteristics around 95 percent. Characteristics close to 100 percent are unlikely to be achieved because of random factors such as changes in personnel, accidents, supply delays, etc. Figure 9-17 represents a typical practical case, from which it can be seen that the curve has a point of inflexion but eventually settles down to an approximately straight line of lower slope than that of the conventionally defined learning curve. At some point it is useful to change to the equation of this mature project line. Significant changes in working, such as the introduction of new equipment, the influx of a large number of inexperienced workers, or a temporary reduction in skills after a long shutdown, may produce a sudden increase in all the cumulative-average curves. The simplest way to handle this, when the next accurate costing is available, is to deduct the value of X obtained from the curve from that actually achieved and to use this value as a constant correction to X until the next break in the curve is reached. If the causes of such steps recur, the size of the step can often be related to a particular cause. In such cases the estimated step change can be used for predictions until the next accurately determined values are obtained. Applications for the learning curve are already extensive, and new uses can often be found. Care is needed in applying the techniques to ensure that it is possible for learning to take place. In projecting prices, etc., unusual items, such as the cost of the special setting up of tools or factory rearrangements, should be excluded from the production costs used to establish the learning curve. In times of inflation, costs should be corrected for the effects of inflation in the manner to be shown subsequently. Production times or spoilage rates are not affected by cost allocations or inflation and may prove to be better

9-22

PROCESS ECONOMICS

FIG. 9-16

FIG. 9-17

Cartesian plot of learning curve.

Logarithmic plot of learning curve.

standards of performance where appropriate. However, the learning curve is often required when preparing quotations for batch production runs, particularly when competition is likely to be keen. In such cases the average cost of the production run Y
between cumulative production totals of X1 and X2 may be estimated by Eq. (9-67) when the previous cumulative-average cost Y1 is known:

extensively and provides many tables of factors. The uses considered include estimating starting costs, determining labor requirements, establishing factory cost targets, checking employee-training progress, the make-or-buy decision, aid in purchasing negotiations, and aid in establishing a selling price.

Y
= Y1[(X2 / X1)N + 1 − 1] / [(X2 / X1) − 1]

The cost of an initial batch of 21 units, exclusive of special tools and setting-up costs, averaged $120 per unit. The average cost of the next batch of 80 units was $75.81. Let us establish the learning curve implied by these data and hence estimate the probable average cost of the next 50 units. We shall establish also the unit-cost curve to be used as a control during follow-up orders. If the batch units are capable of continuous subdivision, we proceed as follows. We substitute the given values of the cumulative-average cost Y and cumulative production X for the first batch into Eq. (9-63) to give, by taking logarithms of each side,

(9-67)

In process engineering, fractional units can often be produced so that the learning curve can be treated as being continuous. When only discrete numbers of units can be produced, the learning curve is strictly a histogram. In order to allow for this it is sufficient to increase the value of X by half a unit before applying the above equations. The difference is significant at small values of X, such as may be used for the initial estimates of K and N. As the project matures, it is better to use the equations as presented, as the cost of the first unit K is an entirely notional one. Major technological changes should, of course, be treated as the start of a new project. R. B. Jordan (How to Use the Learning Curve, Materials Management Institute, Boston, 1965) discusses the uses of the learning curve

Example 4: Estimation of Average Cost of Incremental Units

log 120 = log K + N log 21 The cost of the first batch is 120 × 21 = $2520, and that of the second batch is 75.81 × 80 = $6065. The total cost of the first 101 units is therefore $8585, with a cumulative-average unit cost of $85. We substitute as before to give log 85 = log K + N log 101

INVESTMENT AND PROFITABILITY

FIG. 9-18

9-23

Effect of learning on the average cost of a product.

From these equations it follows that K = $234.15 and N = −0.2196. This line is plotted in Fig. 9-18. From Eq. (9-64) it follows that the value of the characteristic of this learning curve = 100 antilog (−0.2196 log 2) = 85.0 percent. From Eq. (9-65) the production cost of the third unit is

tion. In many cases, not all the x values will be different. In such circumstances, Eq. (9-68) can be written as

cME = ($234.15)(30.7804 − 20.7804) = $149.70

where f(xi) is the frequency with which a particular value xi occurs. It is often convenient to divide the frequency of occurrence by the total number of items. In this case, f(xi) becomes the relative frequency of occurrence of the value xi, and  f(xi) = 1. The values of x may be either discrete or continuous. The number of sales of, say, automobiles in any one day must be an integer. If a business sells 4 automobiles, this represents all possible values of x in the range of 3.5 to 4.5. When x represents a continuous variable quantity, it is sometimes convenient to take the total or relative frequency of occurrences within a given range of x values. These frequencies can then be plotted against the midvalues of x to form a histogram. In this case, the ordinate should be the frequency per unit of width x. This makes the area under any bar proportional to the probability that the value of x will lie in the given range. If the relative frequency is plotted as ordinate, the sum of the areas under the bars is unity. If x is a continuous variable and the interval ranges are made smaller and smaller, a smooth curve will eventually result. The area under such a curve between x1 and x2 represents the probability that a randomly selected item will have a value of x lying in the range x1 to x2. This is the information that is desired. Data available from past experience can be used to generate frequency distribution curves. It is essential for a company to have an efficient commercial-intelligence system to assess market conditions. Accuracy of sales forecasting can also be increased by a careful study of past sales records, price trends, etc. However, the uncertainty of an estimate increases the farther into the future that the estimate is projected. Estimates of sales income and other types of forecasts are usually based on the opinions of experts. Experts should be able to estimate maximum, minimum, and most likely, or modal, values for a quantity. The modal value is not necessarily midway between the minimum and maximum values, since many distributions are skewed. An expert may be asked to estimate the probability of the occurrence of certain values on each side of the mode. When experts are questioned separately, the procedure is known as the Delphic method. Strictly speaking, this method requires that the opinion of each expert be assessed by a coordinator, who then feeds the results back to see if the opinions of one expert are modified by those of others. The process is repeated until agreement is reached. In practice, the procedure is too tedious to be repeated more than once. It is useful to compare the past predictions of each expert with the results obtained in practice. This information enables the opinions to be weighted by the coordinator. When the experts work in close collaboration, it is not possible to avoid some collusion. In this case, it is

Values calculated in this way are plotted in Fig. 9-16 and also in Fig. 9-18. It will be noted that after about 10 units this latter curve becomes parallel to the cumulative-average-cost curve and that the Y values are (N + 1) times those obtained from the latter curve. Since the cumulative-average cost Y2 of the first 101 units was $85, it follows from Eq. (9-67) that the average cost of the third batch of 50 units, bringing the cumulative total to 151, is given by Y
3 = ($85)[(151/101)0.7804 − 1]/[(151/101) − 1] = $63.30 per unit This may be used as a cost guide when quoting the order. If the units of production may not be subdivided, the procedure is similar except that all X values are increased by 0.5 unit in establishing the curves. The results are not sufficiently different to be significant for estimation purposes. To the above costs must be added back any unit costs omitted from those to which learning might bring improvement. These will normally include overheads and specific charges on the project such as the unit cost of special tools, jigs, etc.

Risk and Uncertainty Discounted-cash-flow rates of return (DCFRR) and net present values (NPV) for future projects can never be predicted absolutely because the cash-flow data for such projects are subject to uncertainty. Therefore, when stating predicted values of (DCFRR) and (NPV) for projects, it is also desirable to give a measure of confidence in the predictions. For example, for a particular project it may be estimated that there is a 90 percent chance of the (DCFRR) being greater than 10 percent, a 50 percent chance of its being greater than 16 percent, and only a 10 percent chance of its being greater than 20 percent. Management retains the power of decision to proceed with the project or not, but the probability data provide desirable information for the decision. The estimation of probabilities requires the use of statistics. Thus statistical methods play an increasing role in decision making. Predictions from Limited Data Predictions of future sales price, sales volume, etc., are normally based on a very limited amount of data about past events. Furthermore, it would not be convenient to use the entire population of past events even if it were available. A statistic is a measure, based on limited information from a sample, that allows the corresponding parameter of the population to be estimated. The mean value x
of a property x is a statistic based on a sample of n items defined by x
= (x1 + x2 + x3 + ⋅ ⋅ ⋅ + xn)/n (9-68) The mean
x is the statistic corresponding to the population parameters µ, which is the arithmetic average of all the items in the popula-

x
= [ xi f(xi)] / [ f(xi)]

(9-69)

9-24

PROCESS ECONOMICS

better to arrive at a single consensus opinion by a free and open discussion. This is the think-tank method. Its main disadvantage is that rank or aggressiveness might unduly weight one or more opinions. The opinions of the experts, however obtained, provide a basis for plotting a frequency or probability distribution curve. If the relative frequency is plotted as ordinate, the total area under the curve is unity. The area under the curve between two values of the quantity is the probability that a randomly selected value will fall in the range between the two values of the quantity. These probabilities are mere estimates, and their reliability depends on the skill of the forecasters. The estimated (DCFRR) and the estimated (NPV) are both functions of the estimated cumulative revenue from annual sales  AS, the estimated cumulative total annual cost or expense  ATE, and the estimated fixed capital cost CFC of the plant. The revenue from annual sales for each year is in turn the product of the sales price and sales volume. Initially it is desirable to select those values from the distribution curves of  AS,  ATE, and CFC which enable the maximum and minimum (DCFRR) and (NPV) to be calculated. If the maximum values of (DCFRR) and (NPV) are not acceptable to the company, the project should promptly be rejected. If the minimum values of (DCFRR) and (NPV) are acceptable, a detailed assessment should be made. If the maximum values of (DCFRR) and (NPV) are acceptable but the minimum values are not, the feasibility study should be continued. Mathematical Models for Distribution Curves Mathematical models have been developed to fit the various distribution curves. It is most unlikely that any frequency distribution curve obtained in practice will exactly fit a curve plotted from any of these mathematical models. Nevertheless, the approximations are extremely useful, particularly in view of the inherent inaccuracies of practical data. The most common are the binomial, Poisson, and normal, or gaussian, distributions. A normal distribution curve is bell-shaped (see Sec. 3). The curve obeys the relationship exp − [(x − µ)2/2σ2] f(x) = ᎏᎏᎏ (9-70) σ(2π)0.5 where σ is known as the true standard deviation. The standard deviation s° from a sample is given by (xi −
x)2f(xi) s° = ᎏᎏ  f(xi) − 1





0.5

(9-71)

The standard deviation s° for the sample corresponds to the true standard deviation σ for the whole population in the same way that the mean
x of the sample corresponds to the arithmetic average µ for the whole population. Equation (9-70) can be written more compactly as f(z) = [exp (−z2 / 2) ]/ [(2π)0.5]

(9-72)

where the standard score z is z = (x − µ)/σ

(9-73)

The area under the curve of f(z) is unity if the abscissa extends from minus infinity to plus infinity. The area under the curve between z1 and z2 is the probability that a randomly selected value of x will lie in the range z1 and z2, since this is the relative frequency with which that range of values would be represented in an infinite number of trials. An event that will definitely occur has a probability of unity. An event that will definitely not occur has a probability of zero. Equation (9-72) can be integrated between limits to determine the probability that a random value lies between the selected limits. Extensive tables of f(z) and the associated integral are available (see Sec. 3). A frequency distribution curve can be used to plot a cumulativefrequency curve. This is the curve of most importance in business decisions and can be plotted from a normal frequency distribution curve (see Sec. 3). The cumulative curve represents the probability of a random value z having a value of, say, z1 or less. If a property or variable c is a function of several other variables x1, x2, etc., it can be written in the form c = φ(x1, x2, . . . xn)

(9-74)

If each x is a normally distributed independent variable, then ∂c 2 ∂c 2 ∂c 2 σc2 = ᎏ σ12 + ᎏ σ22 + ⋅ ⋅ ⋅ + ᎏ σn2 (9-75) ∂x1 ∂x2 ∂xn where σc is the standard deviation of the variable c and σ1, σ2, etc., are the standard deviations of the variables x1, x2, etc. Many distributions occurring in business situations are not symmetrical but skewed, and the normal distribution curve is not a good fit. However, when data are based on estimates of future trends, the accuracy of the normal approximation is usually acceptable. This is particularly the case as the number of component variables x1, x2, etc., in Eq. (9-74) increases. Although distributions of the individual variables (x1, x2, etc.) may be skewed, the distribution of the property or variable c tends to approach the normal distribution. Let us consider an event that must have one of two outcomes. It must either occur with probability p1 or fail to occur with probability p2. Since these are exclusive events and the probability that something will happen is unity, it follows that

 

 

 

p1 + p2 = 1

(9-76)

Provided that no learning process is involved (so that the value of p1 is not influenced by previous results), the probability of x successes in n trials is given by the term containing p1x in the expansion of the binomial: n! (p1 + p2)n = p1n + ⋅ ⋅ ⋅ ᎏ p1x p2(n − x) ⋅ ⋅ ⋅ + p2n (9-77) x!(n − x)! where x and n are integers and x! (read as x factorial) is the product of all integers from unity to x. Example 5: Probability Calculation If a six-sided die marked with the numbers 1, 2, 3, 4, 5, and 6 is thrown, the probability that any given number will be uppermost is 1/6. If the die is thrown twice in succession, then the probability of a given sequence of numbers occurring, say, 5 followed by 6, is (1/6)(1/6) = 1/36. The chance of any particular number occurring 0, 1, 2, 3, or 4 times in four throws of the die (or in a simultaneous throw of four dice) is given by the successive terms of Eq. (9-77), expanded as (5⁄6 + j)4 = (1)(5⁄6)4(j)0 + (4)(5⁄6)3(j)1 + (6)(5⁄6)2(j)2 + (4)(5⁄6)1(j)3 + (1)(5⁄6)0(j)4 = 0.4823 + 0.3858 + 0.1157 + 0.0154 + 0.0008 = 1 The distribution of the number of successes is skewed toward the low numbers. In particular, there is only a slightly better than even-money chance that any given number will occur even once in four throws. Such highly unsymmetrical distributions cannot be approximated by the normal distribution curve. However, an increasing number of throws will result in totals that are close to the normal distribution. This fact can be used to approximate such a distribution without the enormous labor of the calculations required by the use of Eq. (9-77). Possible values of the total of four throws of a die are integers from 4 to 24 and hence represent values in the range from 3.5 to 24.5. The mean value
x of this range is given by Eq. (9-68) as
x = (3.5 + 24.5)/2 = 14.0. The cumulative probability of a normally distributed variable lying within 4 standard deviations of the mean is 0.49997. Therefore, it is more than 99.99 percent (0.49997/0.50000) certain that a random value will be within ⫾ 4σ from the mean. For practical purposes, σ may be taken as one-eighth of the range of certainty, and the standard deviation can be obtained: s °  (24.5 − 3.5)/8 = 2.625 From Eq. (9-73) the standard score becomes z = (x − µ)/σ  (x −
x)/s° For a total score of 4 (i.e., x = 4), the standard score is approximately z = (4 − 14)/2.625 = −3.81. Since the normal curve is symmetrical about z = 0, the height of the ordinate at z = −3.81 is the same as that at z = +3.81. From tables of values of cumulative probabilities of the normal distribution, the height of the ordinate is 0.0003 in units of 1/σ. The relative frequency of 4 occurring is thus approximately 0.0003/2.625 = 0.0001. This concept can be used to translate Delphic or other opinions into probability distributions and hence into useful decision-making tools.

Example 6: Calculation of Probability of Meeting a Sales Demand A store that is open 5 days a week is to promote a new product. The manager believes that not more than 5 units will be sold in any one day, but he cannot be more precise about the probable sales pattern. Stocks are delivered once per week. What size should the first order be to give a 95 percent certainty of meeting demand?

INVESTMENT AND PROFITABILITY Since the product is sold in units, the possible range of weekly sales is from −0.5 to +25.5 units. Therefore, the mean of the sales distribution will be
x  [25.5 − (−0.5)]/2 = 13 The standard deviation for this example will be

9-25

Since something must happen, the probability of 4 or more breakdowns is 1 − 0.0498 − 0.1496 − 0.2240 − 0.2240 = 0.3526 A simple trial will show how much more easily the preceding calculation is carried out than direct use of Eq. (9-77).

s°  [25.5 − (−0.5)]/8 = 3.25 From this, the approximate frequency distribution of daily sales can be derived by using Eqs. (9-70), (9-72), or (9-73). The desired area to the right of z = 0 for the normal probability distribution curve is 0.95 − 0.50 = 0.45. For this value the standard score z = 1.645. x  13 + (1.645)(3.25) = 18.35 Hence, to be 95 percent certain of meeting demand, 19 units should be purchased.

If the value of n in Eq. (9-77) is large and neither p1 nor p2 is too close to zero, the binomial distribution can be approximated by x − np1 z=ᎏ (9-78)
n
p

1p2 The approximation of Eq. (9-78) is good enough for most purposes if np1 and np2 are each greater than 5.

The necessary value of λ may often be established as in the following example. Example 9: Calculation of Probability of Machine Failures In a production period of 100 days, 0, 1, 2, 3, and 4 machine failures occurred in a single day on 41, 37, 15, 6, and 1 occasions respectively. Let us fit a Poisson distribution to the data and estimate the maximum number of machine failures likely to occur in 1 day of a 300-day year. The mean number of failures is found from Eq. (9-69) to be 0(41) + 1(37) + 2(15) + 3(6) + 4(1)
x = ᎏᎏᎏᎏ = 0.89 41 + 37 + 15 + 6 + 1 The standard deviation is found from Eq. (9-71):



Example 7: Calculation of Probability of Sales The records of a business show that never more than 1 item is sold in a day and that 2 sales per week can be expected. What is the probability of selling between 90 and 120 items in a 300-day year? In a year consisting of 50 weeks of 6 days, the mean or expected value of the distribution is 100 items. The probability of a sale of an item on a given day is p1 = 100/300 = 1/3, and of no sale is p2 = 2/3. From Eq. (9-78),

(xi −
x) (0 − 0.89)2 (1 − 0.89)2 (2 − 0.89)2 (3 − 0.89)2 (4 − 0.89)2

There are times when the frequency measurement is an integral number of events in a given segment of a continuum, for example, the number of automobiles passing a given point in 1 h or the number of leaks in a given length of hosepipe. In such cases, the correct frequency distribution is the Poisson distribution, in which the probability of x events per unit of a continuum occurring is given by f(x) = λxe−λ/x!

(9-79)

where x is an integer, e is the base of natural logarithms, and λ is a parameter of the system λ = µ = σ2. As λ increases, the Poisson distribution approaches the normal distribution, with the relationship


z = (x − λ)/ λ

(9-80)

When the value of p1 is very close to zero in Eq. (9-77), so that the occurrence of the event is rare, the binomial distribution can be approximated by the Poisson distribution with λ = np1 when n > 50 while np1 < 5. Example 8: Calculation of Probability of Equipment Breakdown The daily chance of a breakdown in a production line operated continuously for 300 days per year is estimated at 1 percent from past performance. Let us estimate the probability of 4 or more breakdowns in the coming year. For n = 300 and p1 = 0.01, λ  np1 = 3 < 5, the probability of no breakdown is found from Eq. (9-79) to be f(0) = (3)0e−3/0! = 1/e3 = 0.0498 Similarly, Breakdowns

Probability

1 2 3

f(1) = (3)1e−3/1! = 0.1496 f(2) = (3)2e−3/2! = 0.2240 f(3) = (3)3e−3/3! = 0.2240

f(xi)

(xi −
x)2f(xi)

41 37 15 6 1  (xi −
x)−2 f(xi)  f(xi) − 1

32.48 0.45 18.48 26.71 9.67 = 87.79 = 99

2

x − (300)(1/3) x − 100 z = ᎏᎏ = ᎏ 8.165
0
0
)(
1
/3
)(
2
/3
)
(3 The integral range of 90 to 120 items contains all possible values of x from 89.5 to 120.5. For x = 89.5, z = −1.286; and for x = 120.5, z = 2.511. The cumulative probability of a standard score of 1.286 is 0.11, while that of a standard score of 2.511 is 0.99. Therefore, the probability of annual sales in the range of 90 to 120 items is (0.99 − 0.11) = 0.88, or 88 percent.



 (xi −
x)2f(xi) 0.5 s° = ᎏᎏ  f(xi) − 1 The steps for calculating the numerator and denominator for this equation are tabulated as follows:

Therefore, s° = (87.79/99)0.5 = 0.9417. The Poisson distribution is a good fit since λ = µ  x
= 0.8900 λ = σ2  (s°)2  0.8868

and

The Poisson distribution is found from Eq. (9-79) to be f(x) = [(0.89)xe−0.89]/x! By substituting the appropriate values of x for this example into the preceding equation, we find f(0) = 0.4107, f(1) = 0.3655, f(2) = 0.1627, f(3) = 0.0483, f(4) = 0.0107, f(5) = 0.0019, and f(6) = 0.0003. Hence, in 300 days the expected maximum number of breakdowns in 1 day is 5 since (300)f(6) = 0.09 occurrence.

In many business applications, Eq. (9-74) can be reduced to the linear relationship c = k1x1 + k2x2 + ⋅ ⋅ ⋅ + knxn

(9-81)

where the k’s are constants. Equation (9-75) then becomes σc2 = k12σ12 + k22σ22 + ⋅ ⋅ ⋅ + kn2σn2

(9-82)

On the other hand, for a product function such as c = x1x2

(9-83)

Eq. (9-75) can be written in the form σc2/c2 = σ12/x12 + σ22 /x12

(9-84)

The discounted-cash-flow rate of return (DCFRR) and net present value (NPV) are functions of the cumulative revenue from annual sales  ATE and the fixed-capital cost of the plant CFC, among other factors. Equation (9-75) can be written for (DCFRR) and for (NPV) as



∂(DCFRR) 2 σ (DCFRR) = ᎏᎏ ∂ AS

σ 2

2

 AS



∂(DCFRR) + ᎏᎏ ∂ ATE

σ 2

2

 ATE

∂(DCFRR) + ᎏᎏ ∂CFC



σ 2

2 CFC

(9-85)

9-26

PROCESS ECONOMICS

∂(NPV)2 2 ∂(NPV)2 2 ∂(NPV)2 2 σ2(NPV) = ᎏ σ  AS + ᎏ σ  ATE + ᎏ σ CFC ∂ AS ∂ ATE ∂CFC

(9-86)

The revenue from annual sales AS of a product at an annual production rate R and sales price of cs per unit of production is AS = RcS

(9-87)

Equation (9-84) can be written as: σ AS = (AS/R)2σR + (AS/cs)2σ c2s 2

2

(9-88)

An extensive example illustrating the use of Eqs. (9-81) through (9-86) in establishing the probability of attaining a given value of the net present value or less in a particular year of a project was presented by Holland et al. [F. A. Holland, F. A. Watson, and J. K. Wilkinson, Chem. Eng., 81, 105–110 (Jan. 7, 1974)]. The result is shown in Fig. 9-19. Decision makers often prefer to have graphs showing the probability of attaining a value greater than a given value. Such curves are easily obtained by subtracting the probability of achieving a given value or less from 100 percent. Figure 9-20 was obtained in this way and shows the probability of attaining a (DCFRR) greater than a given value. Monte Carlo Method The Monte Carlo method makes use of random numbers. A digital computer can be used to generate pseudorandom numbers in the range from 0 to 1. To describe the use of random numbers, let us consider the frequency distribution curve of a particular factor, e.g., sales volume. Each value of the sales volume has a certain probability of occurrence. The cumulative probability of that value (or less) being realized is a number in the range from 0 to 1. Thus, a random number in the same range can be used to select a random value of the sales volume. In the same way, random values of the other factors can be obtained. These can then be combined to give random values of (DCFRR) and (NPV) and, in turn, used to plot cumulative-probability curves for (DCFRR) and (NPV). The computer may be required to perform some 10,000 to 50,000 calculations. The use of the Monte Carlo method in project appraisal was illustrated by Holland et al. [F. A. Holland, F. A. Watson, and J. K. Wilkinson, Chem. Eng., 81, 76–79 (Feb. 4, 1974)]. The cumulativeprobability curves of (DCFRR) and (NPV) can never be more accurate than the opinions on which they are based, and comparable accuracy can be obtained by the use of S-shaped curves with relatively small computational effort. S-Shaped Curves K. D. Tocher (The Art of Simulation, rev. ed., English Universities Press, London, 1967) presented a comprehen-

FIG. 9-20 Probability of a given discounted-cash-flow rate of return or more for a project.

sive treatment of the generation of random and pseudorandom numbers and their use in a wide range of simulated processes. He also considered sampling techniques from the various statistical distributions and the design of simulated processes. It will be noted that the cumulative distribution curves are S-shaped, and Tocher (op. cit., p. 16) recommended as a general equation for such curves x = a + by + cy2 + d(1 − y)2 ln y + ey2 ln (1 − y)

(9-89)

in which x varies from −∞ to +∞ as y varies from 0 to 1. The underlying frequency curve corresponding to Eq. (9-89) is 1 dx 1−y ᎏ = ᎏ = b + 2cy + d(1 − y) ᎏ − 2 ln y p(x) dy y







y + ey 2 ln (1 − y) − ᎏ 1−y



(9-90)

If necessary, the fit can be improved by increasing the order of the polynomial part of Eq. (9-89), so that this approach provides a very flexible method of simulation of a cumulative-frequency distribution. The method can even be extended to J-shaped curves, which are characterized by a maximum frequency at x = 0 and decreasing frequency for increasing values of x, by considering the reflexion of the curve in the y axis to exist. The resulting single maximum curve can then be sampled correctly by Monte Carlo methods if the vertical scale is halved and only absolute values of x are considered. When the data do not warrant the accuracy of Eq. (9-89) or Eq. (9-90), simpler curves will usually suffice if the frequency distribution may be assumed to have a single maximum value. Let us consider a product which is sold entirely on the basis of personal recommendation. The rate of sale will depend on the number of people who have already bought the product. Thus initially sales will increase exponentially. Eventually the market will be saturated, and only replacement purchases will be made. If the frequency curve may be assumed to be symmetrical about a single maximum value, the cumulative distribution curve is known as the logistics curve and is defined by Eq. (9-91): y = c / [1 + a exp (−bx)] FIG. 9-19

Probability of a given net present value or less for a project.

(9-91)

where y varies between zero and c as x ranges from −∞ to +∞.

INVESTMENT AND PROFITABILITY

9-27

Although only three constants appear explicitly in Eq. (9-91), two further constants are implied by the choice of zero as the lower bound of y and the point of inflexion at y = c/2. The usual use of Eq. (9-91) is in sales forecasting, in which case y is sales demand and x is time. If such a curve already exists, the value of c can be read as the upper asymptote and a and b obtained by the use of an auxiliary variable T where

When a cumulative-frequency curve can be satisfactorily represented by a logistics curve, the underlying frequency curve can be obtained by differentiation of Eq. (9-91) as

The probability-density function for the normal distribution curve calculated from Eq. (9-95) by using the values of a, b, and c obtained in Example 10 is also compared with precise values in Table 9-10. In such symmetrical cases the best fit is to be expected when the median or 50 percentile xM is used in conjunction with the lower quartile or 25 percentile xL or with the upper quartile or 75 percentile xU. These statistics are frequently quoted, and determination of values of a, b, and c by using xM with xL and with xU is an indication of the symmetry of the curve. When the agreement is reasonable, the mean values of b so determined should be used to calculate the corresponding value of a. In practice most distribution curves are not symmetrical about the median but are inherently skewed. The effect of an advertising campaign is usually to increase the rate of sales in the early years. It may also increase the level of mature demand for the product, but this mature demand must be asymptotic to a finite upper limit of sales c. Such a curve is positively skewed since (xM − xL) < (xU − xM). This situation can often be approximated by the Gompertz curve defined by Eq. (9-96):

T = x2 (at y = r2c) − x1 (at y = r1c)

(9-92)

b = [ln (1/r1 − 1) − ln (1/r2 − 1)]/T

(9-93)

a = (1/r1 − 1) exp (bx1)

(9-94)

a = (1/r2 − 1) exp (bx2)

or

If the values of a obtained from Eq. (9-94) differ significantly, the logistics curve is not a suitable representation of the data. Example 10: Logistics Curve We shall derive the logistics curve representing the cumulative-frequency distributions of the normal distribution curve defined by Eqs. (9-72) and (9-73). In this case, y varies between a cumulative probability of zero and unity as z varies from −∞ to +∞. Since the upper bound is unity, c = 1. From Table 9-10 the area under the right-hand side of the curve between z = 0 and z = z may be read. Since the frequency curve is symmetrical about the mean, this is also the area between z = 0 and z = z. Hence, the area under the frequency curve, which represents the cumulative probability, is 0.50000 at z = 0 and the 80 percentile, for which the area is 0.80000, corresponds to the value z = 0.842. We substitute these values into Eqs. (9-92) through (9-94) to give T = 0.842 − 0.000 = 0.842

abc exp (−bx) dy p(x) = ᎏ = ᎏᎏ2 dx [1 + a exp (−bx)]

ln y = ln c − a exp (−bx)

a = 1.0000 or 1.00000

b = 0.8794/(xU − xM)

From Eq. (9-91) the corresponding logistic curve is y = [1 + exp (−1.6464z)]

−1

The cumulative-frequency function calculated from this simple expression is compared with the precise value in Table 9-10.

Data for Normal Distribution Curve Area under normal curve, cumulative probability, Ordinate of normal distribution curve, p(z)

p(z) dz z

y=

(9-96)

which has its point of inflexion at 0.3679 c. In terms of the upper and lower quartiles and the median,

b = [ln (1/0.50 − 1) − ln (1/0.80 − 1)]/0.842 = 1.6464

TABLE 9-10

(9-95)

Precise

Estimated

Precise

Estimated

0.000 0.100 0.200 0.253 0.300

0.3989 0.3970 0.3910 0.3864 0.3814

0.4116 0.4088 0.4006 0.3943 0.3875

0.0000 0.03983 0.07926 0.10000 0.11791

0.00000 0.04017 0.08158 0.10265 0.12103

0.400 0.500 0.524 0.600 0.700

0.3683 0.3521 0.3478 0.3332 0.3123

0.3700 0.3491 0.3436 0.3255 0.3003

0.15542 0.19146 0.20000 0.22575 0.25804

0.15894 0.19492 0.20323 0.22866 0.25996

0.800 0.842 0.900 1.000 1.200

0.2897 0.2798 0.2661 0.2420 0.1942

0.2744 0.2634 0.2484 0.2231 0.1761

0.28814 0.30000 0.31594 0.34134 0.38493

0.28870 0.30000 0.31484 0.33840 0.37822

1.282 1.400 1.600 1.645 1.800

0.1953 0.1497 0.1109 0.1032 0.0790

0.1587 0.1358 0.1029 0.0964 0.0769

0.40000 0.41234 0.44520 0.45000 0.46407

0.39194 0.40929 0.43303 0.43752 0.45092

2.000 2.500 3.000 4.000 ∞

0.0540 0.0175 0.0044 0.0001 0.0000

0.0569 0.0260 0.0116 0.0023 0.0000

0.47725 0.49379 0.49865 0.49997 0.50000

0.46418 0.48395 0.49289 0.49862 0.50000

(9-98)

a = 0.6931 exp (bxM)

(9-99)

The suitability of the Gompertz fit to the curve can be assessed by comparing the values of b calculated from Eqs. (9-97) and (9-98), and, if suitable, the average value of b may be used in Eq. (9-99) to calculate the corresponding value of a to ensure a fit at the median and reasonable accuracy over the more important practical range within a couple of standard deviations on either side of the median. The underlying frequency distribution curve of the Gompertz curve may be obtained by differentiation of Eq. (9-96) to give p(x) = dy/dx = yab exp (−bx)

0

Standard score, z

(9-97)

b = 0.6931/(xM − xL)

(9-100)

The logistic and Gompertz curves are of the general shape illustrated by Fig. 9-19. They may be adapted to fit curves of the general shape illustrated by Fig. 9-20 by a little mathematical manipulation. As an example, let us consider the current ratio, the ratio of current assets to current debts, as is quoted in Dun & Bradstreet statistics. A typical value for United States industrial chemical companies might be listed as xL = 1.82, xM = 2.59, and xU = 3.25. First, we notice that (xM − xL) > (xU − xM). This curve is, therefore, negatively skewed, or reversed S-shaped, and the logistics curve is not suitable. Nor can the Gompertz equation be used directly. However, it is clear that if the curve is drawn upside down and backward, the transformed curve will be positively skewed. Mathematically, this is equivalent to interchanging the upper and lower bounds and considering the dependent variable to be (c − y). In the present case the quoted values represent the cumulative probabilities that the current ratio will be less than the quoted value and hence the value of y ranges between zero and unity. Hence, c = 1. In the transformed curve xL = 3.25, xM = 2.59, and xU = 1.82. Hence, from Eq. (9-97) b = 0.8794/(1.82 − 2.59) = −1.1421 and from Eq. (9-98) b = 0.6931/(2.59 − 3.25) = −1.0502 The variation is within 5 percent of the mean value of b = −1.0961, and the transformed curve should be sufficiently accurate for many purposes. From Eq. (9-99) a = 0.6931 exp [(−1.0961)(2.59)] = 0.04054

9-28

PROCESS ECONOMICS In all such S-shaped curves the range of x is from −∞ to +∞, so that there is always a finite possibility of negative values of x occurring. In the present case the definition of the current ratio makes values of x below zero meaningless. The error of some 4 percent in the cumulative-probability curve implied by this factor may be tolerable in a given case. It can be shown [J. J. Molder and E. G. Rogers, Manage. Sci., 15, B-76 (1968)] that for continuous events it is possible to estimate the mean and standard deviation of a skewed distribution from estimates of a low value, a most likely or modal value, and a high value. It is suggested, since it is difficult to make very fine subjective judgments as to probabilities, that the range most likely to be accurate is that for which there is a 10 percent chance of a value less than the low value and a 10 percent chance of a value greater than the high value. These values will usually imply a skewed distribution. For the suggested 80 percent confidence level the best available estimates are x
= (low value) + [(2)(modal value)] + (high value) s° = [(high value) − (low value)] / 2.65

(9-101) (9-102)

On this basis an alternative approach to risk analysis is the parameter method [D. O. Cooper and L. B. Davidson, Chem. Eng. Prog., 72, 73–78 (November 1976)].

FIG. 9-21

Cumulative probability of a given current ratio.

Hence, from Eq. (9-96) the equation of the transformed curve is ln (1 − y) = −0.04054 exp (1.0961x) Since d(1 − y) = −dy, the corresponding underlying frequency distribution curve is from Eq. (9-100): p(x) = +0.0444(1 − y) exp (1.0961x) Values of y and p(x) calculated from last two equations are plotted in Figs. 9-21 and 9-22 respectively.

Example 11: Parameter Method of Risk Analysis Let us consider the project outlined in Table 9-5. It is estimated that the basic data represent the most likely values and that there is a 10 percent chance that AS will be reduced by more than 20 percent or will be increased by more than 5 percent. In the same way the low and high levels at 10 percent probability for ATE are considered to be 5 percent below and 25 percent above the base figures respectively. The low and high values for CFC are considered to be 5 percent below and 30 percent above the base figure, while changes in other parameters are considered to be immaterial. With a cost of capital i of 10 percent the various cash flows can be discounted and summed. Thus for the base cases  As fd = $2,815,600,  ATE fd = $754,716,  AD fd = $614,457, and  CWC fd = $61,446. With corporate taxes payable at 50 percent the aftertax cash flows of the first three items are (1 − 0.50) of the sums calculated above. The discounted working capital and the fixed-capital outlay are not subject to tax. These most probable values are listed and summed in Table 9-11 and, after adjustment for tax, give the modal value of the (NPV) as $276,224. A reduction of 20 percent in As for each year will result in a 20 percent reduction in  As fd below the modal value, i.e., a reduction of (0.20)($2,815,600) = $563,120. The aftertax effect of this reduction on the contribution to (NPV) is (1 − 0.50)($563,120) = $281,560, making the low value $1,407,800 − $281,560 = $1,126,240 or, more directly, (0.8)($1,407,800). Other values in Table 9-11 are calculated in a similar manner. The mean value of each of the distributions is obtained from these high, modal, and low values by the use of Eq. (9-101). If the distribution is skewed, the mean and the mode will not coincide. However, the mean values may be summed to give the mean value of the (NPV) as $161,266. The standard deviation of each of the distributions is calculated by the use of Eq. (9-75). The fact that the (NPV) of the mean or the mode is the sum of the individual mean or modal values implies that Eq. (9-81) is appropriate with all the k’s equal to unity. Hence, by Eq. (9-81) the standard deviation of the (NPV) is the root mean square of the individual standard deviations. In the present case s° = $166,840 for the (NPV). If the resulting distribution is assumed to be normal, then the cumulative distribution curve can immediately be generated. From Table 9-10, a standard score of 4 corresponds to a probability of 0.5 + 0.49997 = 0.99997 and one of −4 to a probability of 0.5 − 0.49997 = 0.00003, virtually unity and zero respectively. From Eq. (9-73) a standard score of 4 corresponds to an (NPV) of $161,266 + (4)($166,840) = $828,626 and one of −4 to an (NPV) of $506,094. Values of (NPV) corresponding to other confidence limits may be calculated in the same way and plotted to give the curve of Fig. 9-23. TABLE 9-11

Data for Risk Analysis (NPV), $/year

FIG. 9-22

Probability of a given current ratio.

Parameter

Low

Modal

High

Mean

Standard deviation

AS ATE CFC CWC

1,126,240 −471,698 −900,603 −61,446

1,407,800 −377,358 −692,772 −61,446

1,478,190 −358,490 −658,133 −61,446

1,355,008 −396,226 −736,070 −61,446

132,811 −42,720 −91,498 0

INVESTMENT AND PROFITABILITY

9-29

used in place of Eq. (9-86) to estimate the overall variance of the main variable. When the estimates are well founded, the skewness may be preserved by using a distribution such as the Gompertz. The median of that curve occurs as y = 0.5 c, while the point of inflexion corresponds to the mode at y = c/exp (1) = 0.3679 c. The statistician Karl Pearson suggested as a simple measure of skewness Skewness = 3 (mean − median)/σ

(9-103)

with an empirical approximation in terms of the mode given by (Mean − mode) = 3 (mean − median)

(9-104)

Applying these equations to the present problem, (Mean − mode) = $161,266 − $276,224 = −$114,958 Skewness = −$114,958/$166,840 = −0.6890 For symmetrical distributions, such as the logistic or normal, the skewness should be zero. The Gompertz distribution requires the distribution to be positively skewed, which can be achieved by treating −(NPV) as the independent variable and (c − y) as the dependent variable. From Eq. (9-104) the median of the distribution is given approximately as Median = [$161,266 − (−$114,958)]/3 = $199,585 Substituting values into Eq. (9-96) with −(NPV) as the independent variable to give, since the range of y is zero to unity, ln (1 − 0.5) = ln (1) − a exp [(−b)(−$199,585)] ln (1 − 1/e) = ln (1) − a exp [(−b)(−$276,224)]

Cumulative probability of a given net present value or less for a project showing normal and Gompertz approximations.

FIG. 9-23

As has been stated, with the uncertainties attached to many business assessments of the range of various factors, the central-limit theorem implies that the assumption of a normal distribution of the main variable is sufficiently accurate provided that there are several factors contributing to that main variable. The results are as informative as most Monte Carlo estimates and have the advantage that they can be rapidly obtained without recourse to a digital computer, although a good desk calculator speeds the work. Strictly, the variables should be independent and additive. Thus it is better, for example, to treat (AS − ATVE) as a single variable since both sales income and total variable expense are related to the annual rate of sales R. In such cases the standard deviation of ATVE would be added to or subtracted from that of AS before squaring to obtain the variance according as the uncertainty of the group was greater or less than that of the individual factors. Also, when a product such as AS = RcS is involved, Eq. (9-84) should be used to estimate the variance rather than Eq. (9-82). When the predominant uncertainties are multiplied together, a log-normal distribution may provide a better final distribution. A similar technique may be applied to the (DCFRR) provided that Eq. (9-85) is

FIG. 9-24

−ln [ln (0.5)/ln (1 − 1/e)] whence b = ᎏᎏᎏ = −5.388 × 10−6/$ [(−$199,585) − (−$276,224)] −ln 0.5 a = ᎏᎏᎏᎏ = 2.0315 exp [(5.388 × 10−6/$)(−$199,585)] The Gompertz curve of the distribution is then, in terms of (NPV), ln y = −2.0315 exp [−5.388 × 10−6(NPV)] For the same degree of certainty as before, the minimum value of the (NPV) is likely to be ln (0.00003) ln ᎏᎏ − 5.388 × 10−6 = −$303,365 −2.0315 and the maximum of $2,064,569 calculated in the same way for y = 0.99997. Other values are calculated in the same way and are plotted as in Fig. 9-23. Decision Trees In a typical decision tree, illustrated in a very simplified form by Fig. 9-24, each node represents a decision point (DP) at which one or more alternatives are available. Some quantifiable result of each alternative is chosen as a basis for comparison: for example, the net present value (NPV). A value is assigned to the probability of attaining each result, either cumulative or not as required. These may be obtained by the methods just described or otherwise. The estimates are subject to the restriction that the sum of the proba-

Effect of decision-tree options on net present value.





9-30

PROCESS ECONOMICS

bilities for all branches leaving each node shall be unity since some decision must be taken there. In considering two investments, we shall let option B be a safe investment having a base net present value (NPV)B that is independent of any competition. We shall let option A yield a net present value (NPV)A1 if no competition exists and (NPV)A2 if competition exists. We shall then let the probabilities of no competition and competition be p1 and p2 respectively. Then p2 must equal (1 − p1). The expected (NPV) for option A can be written, from Eq. (9-105), which follows, as (NPV)WA = p1(NPV)A1 + (1 − p1)(NPV)A2 where (NPV)WA is the weighted net present value for option A based on the probabilities of encountering no competition p1 and of encountering competition (1 − p1). In the same way the expected (NPV) for option B is given by (NPV)WB = p1(NPV)B + (1 − p1)(NPV)B = (NPV)B The gain in the expected value of option A over option B is thus ∆(NPV)W = (NPV)WA − (NPV)WB Let us suppose that the options represented in Fig. 9-24 were such that (NPV)B = (0.5)(NPV)A1 = 2(NPV)A2. Then substitution leads to ∆(NPV)W = [2p1 + 0.5(1 − p1) − 1](NPV)B = (1.5p1 − 0.5)(NPV)B The choice is immaterial when ∆(NPV)W = 0, i.e., when p1 = 1/3. If the probability of no competition is greater than 1/3, option A should be chosen; otherwise option B should be chosen. The technique is based on the methods of linear algebra and the theory of games. When the problem contains many multibranched decision points, a computer may be needed to follow all possible paths and list them in order of desirability in terms of the quantitative criterion chosen. The decision maker may then concentrate on the routes at the top of the list and choose from among them by using other, possibly subjective criteria. The technique has many uses which are well covered in an extensive literature and will not be further considered here. Numerical Measures of Risk Without risk and the reward for successfully accepting risk, there would be no business activity. In estimating the probabilities of attaining various levels of net present value (NPV) and discounted-cash-flow rate of return (DCFRR), there was a spread in the possible values of (NPV) and (DCFRR). A number of methods have been suggested for assessing risks and rewards to be expected from projects. Let us consider a proposed project in which there is a probability p1 that a net present value (NPV)1 will result, a probability p2 that (NPV)2 will result, etc. A weighted average (NPV)w, known as the expected value, can then be calculated from (NPV)w = p1(NPV)1 + p2(NPV)2 + ⋅ ⋅ ⋅

(9-105)

where p1 + p2 + ⋅ ⋅ ⋅ = 1.0. Analogous equations may be written for other additive measures of profitability such as net profit. Example 12: Expected Value of Net Profit Let us consider a contractor who stands to make a net profit of $100,000 on a contract. The cost of preparing the bid on the contract is $10,000. There are four competing contractors, each with a probability p1 = 0.25 of obtaining the contract. Thus, each contractor has a probability p2 = 0.75 of not obtaining the contract. Therefore, the expected value of the project is 0.25($100,000) + 0.75(−$10,000) = $17,500 In this case, the potential gain is 10 times greater than the potential loss.

If the potential loss can bankrupt the company, then decisions are not necessarily made on the basis of expected value even though the potential gain may be very high. Also, decisions are not necessarily made on the basis of expected value if the potential loss represents a relatively small amount of money to the company. Between these two extremes, expected value can be a very useful criterion, particularly for a company with a large number of projects.

A company may be considering a project with a very high potential rate of return and a low risk, but it may prove impossible to raise the money to start the project. Conversely, the company may be prepared to undertake an extremely risky project if the investment is trivial. Thus, the attitude of a company to risk depends on the circumstances. Money does not hold the same value for each company or each individual. A dollar may keep a pauper from starvation while being a trivial amount to the person who gave it. Attempts have been made to quantify a company’s attitude to money, risk, and uncertainty by asking business executives a number of questions such as the following: “Your company has signed a business contract with potential aftertax proceeds of $P. The probability of achieving the net gain of $P is, say, p1 = 0.75, and the probability of a net loss of $P is p2 = 0.25. If you would rather keep the contract, how much cash would you accept for your interest in it? If you would rather be released from the contract, how much cash would you pay to be released from it?” The same questions may then be asked for different values of the probabilities p1 and p2. The answers to these questions can give an indication of the importance to the company of $P at various levels of risk and are used to plot the utility curve in Fig. 9-25. Positive values are the amounts of money that the company would accept in order to forgo participation. Negative values are the amounts the company would pay in order to avoid participation. Only when the utility value and the expected value (i.e., the straight line in Fig. 9-25) are the same can net present value (NPV) and discounted-cash-flow rate of return (DCFRR) be justified as investment criteria. Since the utility curve has such a subjective basis, most companies prefer the objectivity of (NPV) and (DCFRR) over the range of the normal income and expenditure budget. Subjective methods tend to be reserved for exceptionally high risk projects. A utility curve such as that in Fig. 9-25 is specific to a certain sum of money. The curve is likely to be different for, say, P = $10,000. Figure 9-25 can only be used to consider projects that fall within the range of −$100,000 to +$100,000. Other utility curves must be used to cover projects that lie outside this range. R. O. Swalm [“Utility Theory—Insight into Risk Taking,” Harv. Bus. Rev., 44, 123–136 (November–December 1966)] found that many business executives had difficulty in appreciating fine shades of odds and confined his considerations to even-money bets. He asked various executives to state what guaranteed sum of money they considered equivalent to a gamble related to the toss of a coin. If the coin fell on one side, they would win a given sum of money; if the coin fell on the other side, they would get nothing. Swalm started by considering a sum of money equivalent to twice the maximum expenditure that the executive could authorize in 1 year. This was used to obtain a further utility. In this way, a utility curve could be sketched. Swalm chose an arbitrary utility scale based on a range of −120 utiles to +120 utiles. (NOTE: It is as incorrect to compare utiles by ratio as it is to imply that an object at 30°C is twice as hot as an object at 15°C.) Swalm found that most executives are conservative in their expenditure and that the patterns of utility curves are very similar if plotted with an ordinate range of ⫾1 unit. The unit, in this case, is the maximum authorized annual expenditure of the executive. Such curves may appear to differ quite widely when plotted in terms of absolute money values. The curves also show that executives tend to be more conservative when considering a loss than they do when considering a reduced gain. Example 13: Evaluation of Investment Priorities Using Prob-

ability Calculations A company is considering investment in one or more of three projects, A, B, and C. We wish to evaluate the investment priorities if the probabilities of attaining various net present values (NPV) are as listed in the third column of Table 9-11. Equation (9-105) gives the expected value for (NPV)w. Hence for project A, (NPV)w is computed from the data in Table 9-12 and found to be (NPV) w = 0.1($95,000) + 0.8($45,000) + 0.1(−$75,000) (NPV) w = $9,500 + $36,000 − $7,500 (NPV) w = $38,000 Corresponding values for projects B and C are calculated in the same way and are listed in Table 9-12.

INVESTMENT AND PROFITABILITY TABLE 9-12 Comparison of Projects in Terms of Expected Value and Expected Utility

Project

(NPV), Probability, $ p

Equivalent probability of winning Probable Expected Expected $100,000 utility value, $ utility, $

A

95,000 45,000 −75,000

0.1 0.8 0.1 1.0

0.80 0.34 0.03

0.080 0.272 0.003 0.355

9,500 36,000 −7,500 38,000

8,000 27,200 300 35,500

B

50,000 20,000 −60,000

0.2 0.6 0.2 1.0

0.37 0.23 0.04

0.074 0.138 0.008 0.220

10,000 12,000 −12,000 10,000

7,400 13,800 800 22,000

C

45,000 10,000 −60,000

0.1 0.6 0.3 1.0

0.35 0.20 0.04

0.035 0.120 0.012 0.167

4,500 6,000 −18,000 −7,500

3,500 12,000 1,200 16,700

In project A, the probability p = 0.1 for (NPV) = $95,000. Figure 9-25 shows that $95,000 is the amount of money that this company would pay for a 0.8 probability of gaining $100,000. There is, therefore, a 0.2 probability of losing $100,000. In this case, a probability of p = 0.1 of attaining $95,000 is equivalent to a probable utility of (0.1)(0.8) = 0.08 of gaining $100,000. Equation (9-105) can be used to calculate the expected utility if the probabilities p1, p2, etc., are replaced by the probable utilities and if the net present values (NPV)1, (NPV)2, etc., are each replaced by $100,000. For project A, the expected utility Uw is U w = [0.1(0.8) + 0.8(0.34) + 0.1(0.03)]$100,000 U w = $8,000 + $27,200 + $300 U w = $35,500 Corresponding values for projects B and C are calculated in the same way and are listed in Table 9-12. The straight line in Fig. 9-25 represents the situation in which the expected value and the expected utility are equal over the range of −$100,000 to

FIG. 9-25

Utility-function plot for $100,000.

9-31

+$100,000. In this case, decisions can be taken on the basis of the highest expected value as a routine matter. In other cases, decisions should be made on the basis of the highest expected utility. The utility curve in Fig. 9-25 represents the present attitude of management to $100,000. This curve should be updated as the company’s business position changes. In this example, the utility curve is above the straight line. This represents a tendency on the part of the company’s decision makers to gamble. When it is below the straight line, the utility curve implies conservatism. The investment priorities should be to implement project A and then, if finance is available, project B. It might appear that project C should also be considered in view of the expected utility of $16,700. However, it is better to do nothing than to implement project C. The utility of doing nothing, which is equivalent to paying $0, is read from Fig. 9-25 to be 0.17. This gives a corresponding probable utility of (1.0)(0.17)($100,000), or $17,000. This is a better result than investing in project C. In this example, the order of priorities based on expected utilities is the same as that based on expected values. However, the order of priorities is clear-cut on the basis of expected value but much less so on the basis of expected utility.

Capital is at risk until the breakeven point has been reached. It is common practice to give consideration to the discounted breakeven point (DBEP), the time at which the (NPV) is zero when discounting at the cost of capital. At any time after the (DBEP), the project will have recovered its cost and provided a greater return on the capital than the cost of capital. It is customary for management to spread risk by diversifying the activities of a company among a portfolio of projects. R. L. Reul [Chem. Eng. (London), 238, CE 120–125 (May 1970)] has defined a parameter, which he calls the measured-survival function (MSF), given by (MSF) = 1 − (1 − p)β

(9-106)

where (MSF) is the probability that a portfolio of bets with a similar strategy will at least break even, and β is the amount of one win divided by the amount of each bet. Reul has applied Eq. (9-106) to the research and development activities of a company. Equation (9-106) is based on the simplified assumption that a project either succeeds with probability p and achieves the expected reward or fails completely with probability (1 − p). Therefore, (MSF) is the probability of at least

9-32

PROCESS ECONOMICS

one success when β similar projects are undertaken and represents a conservative measure of risk. It follows that β > 1 and hence that (MSF) > p. Many projects may result in greater returns or have an increased probability of attaining a given return if more money is spent. Each alternative derivable result from a given project is treated as a separate risk in the portfolio. Research and development activities do not, in themselves, produce a salable product. Thus, they cannot directly generate a return on capital outlay. A successful research and development project is one that results in an activity that earns revenue for the company. The life cycle of the revenue from an individual product may be as shown in Fig. 9-26. This revenue has to pay not only for the successful project but for all the unsuccessful research and development activities. It is common practice to consider all R&D as a portfolio. Disbursements for R&D are relatively flexible and can be switched from less favorable to more favorable projects at short notice. When considering individual projects, β should be taken as the lesser of expected proceeds if project is successful β = ᎏᎏᎏᎏᎏ disbursement on project total expenditure on all projects over budget period β = ᎏᎏᎏᎏᎏᎏ expenditure on project over budget period Because the projects in a portfolio will usually have different probabilities of success and different rewards for success, β and p in Eq. (9-106) are conservatively estimated as follows: total annual proceeds if all projects are successful β = ᎏᎏᎏᎏᎏᎏ total annual disbursements on all projects or

total expected value of all projects p = ᎏᎏᎏᎏᎏ total proceeds if all projects are successful The expected value can be calculated from Eq. (9-105). The relationship between (MSF), p, and 1/β in Eq. (9-106) is shown graphically in Fig. 9-27. It is the responsibility of management to decide on an acceptable value of the (MSF) for its company. The value chosen will depend on the company’s attitude to risk that can be quantified in the form of a utility curve such as the one shown in Fig. 9-25, from which a value of equivalent (MSF) can be obtained. It is also the responsibility of management to estimate the probabilities for the success of individual projects after due consideration of all the data provided by the various departments. The rate of return on investment that is acceptable to management is a function of these responsibilities. Each industry has a reasonably well defined return on investment that reflects the degree of risk inherent in that industry. If management decisions are faulty, the company either will overspend or will miss opportunities. With a disbursement of $1000 in Year 0, the discounted breakeven point (DBEP) will be reached in 3 years at a compound-interest rate of 30 percent if the annual net profit ANP = $550.63 per year. Thus, a

FIG. 9-27

Measured-survival-function plot.

discounted-cash-flow rate of return (DCFRR) of 30 percent corresponds to 1 $1000 ᎏ = ᎏᎏᎏ = 0.61 β (3 years)($550.63/year) For 1/β = 0.61 and an (MSF) = 0.999, the probability of individual success is read from Fig. 9-27 to be p = 0.985. Similarly, it can be deduced that if (MSF) = 0.999 and p = 0.95, a (DCFRR) of 45 percent is required; if breakeven in 20 years is acceptable, then a (DCFRR) of only 10 percent is needed. Example 14: Estimation of Probability of a Research and Development Program Breaking Even Details of the estimates for the current research and development program of a company are given in Table 9-13. We shall estimate the probability that this portfolio will at least break even. The total annual proposed disbursement for R&D is $500,000. The effective total annual income if all projects reached their anticipated income would be $1,300,000. Therefore, β = $1,300,000/$500,000 = 2.600 Project A has an expected value of (0.95)($500,000/year) = $475,000/year; project B has an expected value of (0.90)($400,000/year) = $360,000/year; and so on. We sum these values to obtain the total expected value of the portfolio as $1,109,500 per year. Hence, p = ($1,109,500/year)/($1,300,000/year) = 0.8535 TABLE 9-13 Example of a Portfolio of Projects for a Research and Development Program

FIG. 9-26

Life cycle of products.

Project

Proposed disbursement for coming year

Annual aftertax income if successful

A B C D E F G H Totals

$125,000 100,000 100,000 80,000 50,000 20,000 20,000 5,000 $500,000

$ 500,000 400,000 125,000 100,000 60,000 30,000 70,000 15,000 $1,400,000

Probability of success 0.95 0.90 0.80 0.75 0.70 0.65 0.50 0.20

INVESTMENT AND PROFITABILITY From Fig. 9-27, for a probability of success p = 0.8535 and a value 1/β = 1/2.600 = 0.3846, the (MSF) is 99.3 percent. This is the probability that this portfolio will at least break even. Alternatively, we can substitute the values for p and β into Eq. (9-106) to get (MSF) = 1 − (1 − 0.8535)2.6 = 0.9932, or 99.32 percent The (MSF) and utility curves can be related.

Example 15: Utility-Function Curve Let us sketch a utilityfunction curve that is equivalent to the following pattern of measured-survival functions (MSF), which expresses the observed strategy of a particular manager when spending an authorized annual budget of $1,000,000: Case

Potential proceeds annually, $

(MSF), %

a b c d

Above 600,000 300,000–600,000 0–300,000 Losses

99.9 95.0 65.0 75.0

We shall plot the resultant curve on a utility scale of ⫾120 utiles against a potential gain of ⫾$1,000,000. The required axes range from −$1,000,000 per year to +$1,000,000 per year, and from −120 utiles to +120 utiles. Utiles can be compared by ratio on an absolute scale only. Hence, for purposes of calculation the axes are moved to provide a working range of $0 per year to $2,000,000 per year and 0 to 240 utiles as in Fig. 9-28. On these axes, a potential gain of $600,000 per year corresponds to an absolute amount of (600,000 + 1,000,000) = $1,600,000 per year, and a potential loss of $200,000 per year to an absolute amount of (−200,000 + 1,000,000) = $800,000 per year. a. For annual proceeds above $600,000 per year, (MSF) is 99.9 percent. If the certainty of an annual gain of $600,000 has to be abandoned in an effort to obtain an annual gain of $1,000,000, then on an absolute scale β = ($2,000,000/year)/($1,600,000) = 1.2500 With 1/β = 0.8000 and (MSF) = 99.9 percent, we find the required probability of success by solving Eq. (9-106) for p: p = 1 − (1 − 0.999)0.800 = 0.996 The utility of an amount of money is its utility when it is certain to be obtained, multiplied by its probability of being attained. On a scale in which an absolute annual income of $2,000,000 per year has a utility of 240 utiles, the utility of $1,600,000 is (0.996)(240), or 239 utiles. b. For annual proceeds between $300,000 and $600,000, (MSF) = 95 percent. If the certainty of an annual gain of $300,000 has to be abandoned to obtain an annual gain of $600,000, then, as before, 1/β = $1,300,000/year/$1,600,000/year = 0.8125 p = 1 − (1 − 0.95)0.8125 = 0.912 Since, to this manager, the utility of an absolute income of $1,600,000 is 239 utiles, the value of $1,300,000 is (0.912)(239) = 218 utiles. On the original scales, potential annual proceeds of $300,000 have a utility of (218 − 120), or 98 utiles.

9-33

c and d. Values of utility at other potential annual gains are calculated in the same way and shown graphically in Fig. 9-28.

This strategy is extremely conservative when high gains are possible but becomes less so for smaller potential gains. If potential losses are involved, the strategy is a fair one for which (NPV) would be an accurate guide for choosing alternatives. Insurance and Risk In the venture-premium method of assessment, risky investments are required to yield a rate of return that adds a premium to the cost of finance. D. F. Rudd and C. C. Watson (The Strategy of Process Engineering, Wiley, New York, 1968, p. 91) consider this relationship: im = i + ir (9-107) where i is the cost of capital, i m is the minimum acceptable interest rate of return on the investment, and i r is known as the risk rate. They suggested that each project should pay an insurance premium ir to guarantee the expected profits. The magnitude of ir is proportional to the amount of capital to be risked. It is also a function of the degree of risk involved. Working capital and capital for auxiliary facilities are assumed to be risk-free. Thus, the risk rate is applied only to the fraction of the capital investment likely to be lost if the project is unexpectedly terminated. The main objection to the venture-premium method is that the assessment of the riskiness of a project may be too subjective. This could lead to the rejection of potentially attractive proposals and the acceptance of projects that merely appear to be risk-free. Insurance is protection against risk. Commercial insurance companies minimize their own risks by covering a large number of individuals against a given risk and also by offering coverage on a wide variety of different types of risk. It is frequently quite difficult to assess the probability of success of a particular research and development project. It is much easier for an insurance company to assess its probabilities from its casualty tables. Businesses tend to provide their own insurance cover when individual claims are likely to be a small fraction of the available capital. The cost of commercial insurance is about 30 percent higher than would be necessary to cover the same risk in one’s own company. However, for low-probability, high-cost risks, most businesses prefer to insure with a commercial insurance company. Such risks include loss of plant or buildings due to fire and losses of revenue due to delays in startup or strikes. It is also becoming necessary to insure against factors not normally considered until recently. These include possible lawsuits for polluting the environment. The cost of insurance increases the annual total expense ATE. Thus, overinsurance can lead to an unnecessary decrease in profitability. The management of a company must ultimately judge its own risks. As an example, let us calculate the required risk rate for a project that is described by the following: (1) risk strategy is equivalent to an (MSF) of 99 percent, (2) payback of risk capital is 3 years, (3) cost of capital i is 10 percent, and (4) probability of complete success of the project is estimated as 95 percent. First, we calculate the value of β in Eq. (9-106). For this project, with (MSF) = 0.99 and p = 0.95, log [1 − (MSF)] log (0.01) β = ᎏᎏ = ᎏ = 1.537 log (1 − p) log (0.05) To recover this amount of capital and interest in 3 years, the average net annual cash flow ACF required is ACF = 1.537/3 = 0.5124 ($/year)/($ invested) In effect, in computing the average net annual cash flow per dollar invested, the value of fAP of Eq. (9-46) has been obtained for this example. From tables of the annuity present-worth factor fAP the value of the interest rate is found to be im = 0.25 when fAP = 0.5124 with n = 3 years. Hence, by substituting appropriate values into Eq. (9-107) and solving for the required risk rate,

FIG. 9-28

Utility-function plot illustrating managerial strategy.

ir = im − i = 25 − 10 = 15 percent

9-34

PROCESS ECONOMICS

based on the payback period of the risk capital. All capital CWC is completely recoverable without risk and requires interest only at 10 percent. The unrecovered part of the risk capital CFC attracts the additional risk interest rate of 15 percent, which should be reduced as the risk capital is written down. A different view of risk is expressed in Eq. (9-108): [1 + (DCFRR)] = (1 + i)(1 + i′r)

(9-108)

The (DCFRR) represents the return on all capital invested after such capital has been paid back, together with any interest incurred by borrowing it, and after payment of all expenses, including taxes, associated with the project. It thus represents the entrepreneurial return to the company for managing the total capital employed. If the cost of capital i is set at the best risk-free use of that capital, such as the interest rate on a bank deposit or on government bonds, etc., i′r represents the increased entrepreneurial return on the capital for taking the risks involved. This is a useful concept since the probability of achieving a given (DCFRR), and hence of a particular value of i′r, may be estimated by the methods detailed previously. We notice that i, as so defined, implies that all taxes and interest have been paid. Thus, $100 deposited in a bank at a rate of 10 percent with half of the money borrowed at 15 percent and corporation tax at 40 percent would result in a risk-free income after tax of [($100)(0.10/year) − (0.5)($100)(0.15/year)](1 − 0.40) = $1.5/year The same money invested in a project with a (DCFRR) of 10 percent would, by Eq. (9-108), obtain an entrepreneurial return i′r = 8.37 percent on the whole investment, i.e., $8.37/$100. Investment of the entrepreneur’s own money would only achieve an aftertax return of (0.1)(1 − 0.40) = 6 percent on $50, or $3/$100 of total investment. The incentive to the entrepreneur to manage the project thus corresponds to a tax-free income of $5.37/$100 of total investment. In practice, money is borrowed from more than one source at different interest rates and at different tax liabilities. The effective cost of capital in such cases can be obtained by an extension of the above reasoning and is treated in detail by A. J. Merrett and A. Sykes (Capital Budgeting and Company Finance, Longmans, London, 1966, pp. 30–48). Inflation It is currently necessary to evaluate the profitability of proposed investments whose future earnings are virtually certain to be eroded by inflation. It has been common practice to ignore the effects of inflation. This is done on the reasonable grounds that predicting the market rate of interest, and thus the appropriate discount rate for future cash flows, is difficult enough without having to worry about inflation as well. But failure at least to try to predict inflation rates and take them into account can greatly distort a project’s economics, especially at the double-digit rates that have been found throughout the world. It is the common experience that a given amount of money buys less and less of goods and services as time goes by. The problem is to express this experience quantitatively. Published figures for inflation rates are based on some particular mixture of goods and services that is chosen to represent the material wants of the average citizen. If a given quantity of this specific mixture cost $100 last year and now costs $120, then the mix has suffered a 20 percent rate of inflation. The purchasing power of the currency (i.e., of the $120) in respect of these goods and services has consequently fallen by a factor of ($120 − 100)/$120, or 16.7 percent. Two kinds of inflation can be considered: general, or open, inflation and repressed, or differential, inflation. In the first case, all costs and TABLE 9-14

prices increase at a uniform rate. Thus, the same rate of inflation will be calculated regardless of the particular mixture of goods and services chosen. In the second case, the rate of inflation will depend on the spending spectrum of the individual or company. For instance, a given company’s labor costs and material costs may inflate at different rates. To quite a large extent, inflation becomes repressed, or differential, in such fields as taxation, import control, and price restriction. The effect of inflation on the real value of future earnings from a project should not be confused with the effect of the market rates of interest on those earnings. Strictly speaking, the market interest rate and the inflation rate are not fully independent, at least according to some economic theorists. However, they are here treated as being separate. Because of each effect, a dollar of project income next year has a smaller true value than does a dollar in hand today. The interestrate effect could be offset because a dollar could be financially invested at the prevailing interest rate and the dollar plus interest earnings recouped in a year. By contrast, the inflation effect comes about simply because a dollar can buy more now than a year hence because of an irreversible rise in prices. The distinction is clarified in the following subsections. Effect of Inflation on (NPV) When computing the (NPV) for a proposed project, error arises if the actual cash flows are simply added together instead of adjusting all the values to their purchasing power in a particular year. The reason lies in the basis of (NPV) calculations. We shall rewrite Eq. (9-57) to give n ACFn (NPV) = ACF0 +  ᎏ (9-109) (1 + i)n 1 Equation (9-109) is valid for the case of no inflation. In the case of general inflation at a fractional rate ii, this equation can be written in the modified form n A CF0 (NPV) = ACF0 +  ᎏᎏ (9-110) n (1 + i) (1 + ii)n 1 Equation (9-110) enables all the net annual cash flows to be corrected to their purchasing power in Year 0. If the inflation rate is zero, Eq. (9-110) becomes identical with Eq. (9-109). The following example illustrates the effect of inflation on (NPV) as well as on the taxes the company pays. Example 16: Effect of Inflation on Net Present Value Let us consider a simplified project in which $1,100,000 of capital is spent in Year 0, $1,000,000 for fixed-capital items and $100,000 for working capital. The fixed capital is depreciated on a straight-line basis to a book value of zero at the end of Year 5. The annual sales revenue in Years 1 through 5 is $500,000. There is no inflation. The $100,000 of working capital is recovered at the end of Year 5. The taxation rate is 50 percent, and the market interest rate is 10 percent. Table 9-14 lists the cash-flow data for this project, showing that the (NPV) at the end of Year 5 is $99,326 by using Eq. (9-109). Let us modify this example by assuming that there is a general inflation rate of 20 percent per year and that the project analyst ignores the inflation and (inappropriately) applies Eq. (9-109). The revenue and expense data for this case are shown in Table 9-15, yielding an (NPV) of $431,269. When Eq. (9-109) is (inappropriately) used for the same example with various other rates of inflation, the resulting (NPV)s can be plotted as the upper line in Fig. 9-29. If the inflation is correctly taken into account by applying Eq. (9-110), the results are strikingly different. By further discounting the discounted cash flows ADCF of Table 9-15 by the fd factors corresponding to an inflation rate of 20 percent before summing, it can be seen that the project actually incurs a negative (NPV) of $208,733 in uninflated-money terms. The lower line in Fig. 9-29

(NPV) Calculations with No Inflation Revenue from sales, AS

Total Cash income, Depreciation Taxable Amount of tax at charge, income, t = 0.5, AIT expenses, ACI (= AS − ATE ATE) AD (ACI − AD) [= (ACI − AD)t]

Net cash flow Discount factor at i = 10%, fd after tax, ACF 1 (= ACI − = ᎏᎏn AIT − ATC) (1 + 0.1)

Discounted net cash flow, ADCF [= ACF( fd)]

Net present value (NPV),

Year, n

Net capital expenditure, ATC

0 1 2 3 4 5

$1,100,000 0 0 0 0 −100,000

0 $500,000 500,000 500,000 500,000 500,000

0 $100,000 100,000 100,000 100,000 100,000

−$1,100,000 300,000 300,000 300,000 300,000 400,000

−$1,100,000 272,727 247,935 225,393 204,903 248,368

−$1,100,000 −827,273 −579,338 −353,945 −149,042 +99,326

0 $400,000 400,000 400,000 400,000 400,000

0 $200,000 200,000 200,000 200,000 200,000

0 $200,000 200,000 200,000 200,000 200,000

0 $100,000 100,000 100,000 100,000 100,000





1.00000 0.90909 0.82645 0.75131 0.68301 0.62092

n

=  A  DCF

0

INVESTMENT AND PROFITABILITY TABLE 9-15

(NPV) Calculations with Inflation Present But Not Allowed For

Year, n

Net capital expenditure, ATC

Revenue from sales, AS

Total expenses, ATE

Cash income, ACI

0 1 2 3 4 5

$1,100,000 0 0 0 0 −100,000

0 $500,000 600,000 720,000 864,000 1,036,800

0 $100,000 120,000 144,000 172,800 207,360

0 $400,000 480,000 576,000 691,200 829,440

Depreciation charge, AD

Taxable income, (ACI − AD)

Amount of tax at t = 0.5, AIT

0 $200,000 200,000 200,000 200,000 200,000

0 $200,000 280,000 376,000 491,200 629,440

0 $100,000 140,000 188,000 245,600 314,720

extends the example by assuming other rates of inflation. Figure 9-29 shows that the effect of inflation, if not taken into account, is to make a project seem more profitable than it actually is. Table 9-15 shows that the total amount of tax actually paid over the 5-year period was $988,320. This becomes $534,272 in uninflated-money terms when the tax for each year is corrected to its purchasing power in Year 0, using fd factors for the 20 percent inflation rate employed for the example. Calculations for other rates of inflation can also be made, and the results plotted as in Fig. 9-30.

This confirms that although the tax paid will increase with inflation, the gain to the government is more apparent than real. It is interesting to note that although the tax paid corrected to its purchasing power in Year 0 is almost constant irrespective of the inflation rate, it does go through a maximum at an inflation rate of about 17 percent in this example. Effect of Inflation on (DCFRR) A net annual cash flow ACF will have a cash value of ACF(1 + i) 1 year later if invested at a fractional interest rate i. If there is inflation at an annual rate ii, then an effective rate of return or interest rate ie can be defined by the equation ACF(1 + ie) = [ACF(1 + i)]/(1 + ii)

FIG. 9-29

9-35

Effect of inflation rate on net present value for a project.

(9-111)

Net cash flow, ACF

Discount factor at i = 10%, fd

Discounted net cash flow, ADCF

Net present value (NPV)

−$1,100,000 300,000 340,000 388,000 445,600 614,720

1.00000 0.90909 0.82645 0.75131 0.68301 0.62092

−$1,100,000 272,727 280,993 291,508 304,349 381,692

−$1,100,000 −827,273 −546,280 −254,772 +49,577 +431,269

which can be simplified and rewritten to give ie = i − ii − ie ii

(9-112)

In the context of the discounted-cash-flow rate of return, Eq. (9-112) becomes ie = (DCFRR) − ii − ie ii

(9-113)

In this equation, (DCFRR) can be viewed as the nominal discountedcash-flow rate of return uncorrected for inflation and ie can be thought of as the true or real discounted-cash-flow rate of return. Instead of using Eq. (9-113), it is unfortunately common practice to try to obtain the true or effective rate of return by calculating the nominal (DCFRR), based on actual net annual cash flows uncorrected for inflation, and then subtracting the inflation rate from it as if ie = (DCFRR) − ii

(9-114)

Equation (9-113) shows that Eq. (9-114) is only approximately true and should be used, if at all, solely for low interest rates. Let us consider the case of a nominal (DCFRR) of 5 percent and an inflation rate of 3 percent. Equation (9-14) yields an approximate effective return rate of 2 percent, compared with the real effective rate of 1.94 percent given by Eq. (9-113); i.e., there is an error of 3.1 percent. Now let us consider the case of a nominal (DCFRR) of 25 percent and an inflation rate of 23 percent. Equation (9-114) yields an approximate effective return rate of 2 percent, compared with 1.63 percent from Eq. (9-113); in this case, the error that results is 22.7 percent. Inflation, (DCFRR), and Payback Period More insight into the effect of inflation on (DCFRR) calculations can be gained by considering the payback period (PBP), which is defined as the elapsed time necessary for the positive aftertax cash flows from the project to

FIG. 9-30

Effect of inflation rate on taxes paid for a project.

9-36

PROCESS ECONOMICS

recoup the original fixed-capital expenditure. In this definition, the cash flows are not discounted to allow for the market rate of interest or for the inflation rate, so that a project with a given (PBP) could show various values for its (DCFRR) and a given (DCFRR) could pertain to projects with various payback periods. We shall consider the simple case of (1) a single capital expenditure made immediately before the start of production and (2) equal positive net annual cash flows ACF in all the productive years of the project. For this case, Eq. (9-109) can be rewritten in terms of the payback period and the (DCFRR) as follows: n 1 (PBP) =  ᎏᎏn (9-115) [(1 + (DCFRR)] 1

ignores it and mistakenly takes a (DCFRR) of the project at its nominal value instead of converting it to an ie. Equation (9-115) rearranged into the form ie = [1 + (DCFRR)]/(1 + ii) − 1

(9-116)

shows that in order to achieve an ie of 20 percent when the general inflation rate is likewise 20 percent, a project must generate a nominal (DCFRR) of 44 percent. This is the basis for the uppermost line in Fig. 9-31. Other lines pertaining to other rates of inflation could be plotted in the same way. Let us assume that 20 percent inflation prevails but that the analyst

shows that with a nominal (DCFRR) of 20 percent and a general inflation rate of 20 percent, the true or effective rate of interest is zero. This is the basis for the lowest line in Fig. 9-31. Points for lines corresponding to other rates of inflation could be plotted onto that figure. Plots similar to Fig. 9-31 can be drawn for other (DCFRR) values. Figure 9-31 shows that the elapsed time necessary to reach a nominal (DCFRR) for a given project decreases sharply with inflation. This figure, like Fig. 9-29, shows that the effect of inflation is to make a project seem more profitable than it actually is. The magnitude of the effect comes through even more clearly in Fig. 9-32, a plot of the time needed to reach a nominal (DCFRR) of 20 percent against the inflation rate for various values of (PBP). This plot also shows that the longer the payback period, the greater the increase in apparent profitability of the project. The true rates of return ie can be calculated from Eq. (9-116) to be 20, 9.09, 0, and −7.69 percent respectively for general inflation rates of 0, 10, 20, and 30 percent. Thus, although the time required for a project with a payback period of 4 years to reach a nominal (DCFRR) of 20 percent is reduced from almost 9 years under conditions of no inflation to less than 3a years for 30 percent inflation, the true rate of return that prevails for the latter condition is −7.69 percent, implying that the project loses money in real terms. It is interesting to note that, in order to reach a real (DCFRR) or ie of 20 percent within a reasonable project lifetime when the general inflation rate is 20 percent, it follows from Fig. 9-31 that the payback period for the project must not be much in excess of 2 years. Although it is difficult to carry out economic-feasibility studies on projects in a time of high inflation, it is important to try to predict inflation rates and allow for them in such studies. When different people talk about inflation, they often adopt different concepts without realizing it. The area of conceptual uncertainty

FIG. 9-31 Effect of inflation rate on the relationship between the payback period and the discounted-cash-flow rate of return.

FIG. 9-32

The relationship set out in Eq. (9-115) can also be viewed via a different chain of causality with (DCFRR) as a given parameter, (PBP) as the independent variable, and n as the variable whose value is being sought. Such an approach is the basis for the lines in Fig. 9-31, each of which shows the number of years of project life required to achieve an effective interest rate or a (DCFRR) of 20 percent by projects having various payback periods. The three lines differ from each other with respect to the matter of inflation. If there is no inflation, then the middle line pertains. Because there is no inflation, the nominal (DCFRR) is equal to or identical with ie, the real discounted-cash-flow rate of return, as can be seen from the relationship expressed in Eq. (9-113). When inflation does exist, the relevant parameter is ie, which is different from the nominal (DCFRR). Equation (9-113), manipulated into equivalent form, (DCFRR) = (1 + ie)(1 + ii) − 1

Adverse effect of inflation for higher payback periods.

INVESTMENT AND PROFITABILITY

9-37

Relationship between measured-survival function, number of payback periods, and contribution efficiency.

FIG. 9-33

can be said to lie somewhere between the upper and lower lines shown on Fig. 9-31 in most cases. Inflation and the (MSF) By applying the measured-survivalfunction concept to manufacturing projects rather than to research and development, we can define a modified (MSF) for a given project as (MSF) = 1 − (1 − η)β

(9-117)

Here, β is the number of payback periods that have elapsed since the project started to generate positive net annual cash flows ACF up to any given year n since project startup. It is given by n=n

 (A

)

CF n

n=0

β = ᎏᎏ CFC − S

(9-118)

If all the net annual cash flows in Eq. (9-118) are based on their purchasing power in Year 0, then β is independent of inflation. As for the contribution efficiency η, it is the ratio of (1) the annual profit that can actually be achieved in a given year for a given sales volume to (2) the profit that could be obtained if no repayment of capital or interest were required and all fixed-expense items were credited free to the project. It is defined by η = [R(cs − cVE) − AFE]/[R(cs − cVE)]

(9-119)

where R is the annual production rate or sales volume in physical units, cS is the sales price per unit, cVE is the variable production and selling cost per unit, and AFE is the annual fixed cost. If the project gets a “free ride,” i.e., if AFE is zero, then η takes on its maximum possible value of unity. Conversely, if the project and its production rate are only at the breakeven point, then η becomes zero. Therefore, contribution efficiency can be regarded as a measure of the probability of success for the project. The relationship between the number of payback periods, the contribution efficiency, and the measured-survival function as set out in Eq. (9-117) is plotted in Fig. 9-33. The contribution efficiency defined by Eq. (9-119) may vary from year to year. In that case, Eq. (9-117) can be written in the modified form (MSF) = 1 − [(1 − η1)(1 − η2) ⋅ ⋅ ⋅ (1 − ηn)]β/n

manufacturing projects. That decision reflects and helps quantify the company’s attitude toward risk. Thus, (MSF) should in practice be regarded as a given or predetermined variable, and Eq. (9-117) accordingly becomes more useful if it is rearranged. For instance, the values of contribution efficiency for a given value of (MSF) are related to the number of elapsed payback periods by η = 1 − [1 − (MSF)]1/β

(9-121)

If the acceptable (MSF) is 0.9, this can be satisfied by a project having η = 0.9 and β = 1, or a project having η = 0.684 and β = 2, and so on. Once Eq. (9-121) has been used to calculate a required contribution efficiency [given the (MSF) and the expected number of payback periods of project life], Eq. (9-119) can be applied to determine the necessary selling price if R, cVE, and AFE are known. Similarly, Eq. (9-119) can be used to find the required production rate if cS is known. It is also possible to combine (MSF) considerations with evaluation of the true discounted-cash-flow rate of return (DCFRR) by using Eq. (9-62). The relationship of Eq. (9-59) is independent of inflation if all money values are based on those prevailing in the startup year. For this case, Fig. 9-34 shows the true (DCFRR) reached in a given time, expressed as the number of elapsed payback periods β for various values of the payback period. Let us consider a project having a contribution efficiency of 0.684 and a payback period of 3 years. Figure 9-33 shows that when two payback periods have elapsed, a measured-survival function of 0.9 has

(9-120)

where η1, η2, . . . ηn are the contribution efficiencies in Years 1, 2, . . . n respectively. As in the case of the (MSF) defined by Reul for research and development projects, it is the responsibility of management in a particular manufacturing company to decide on an acceptable level of (MSF) for

FIG. 9-34 Real discounted-cash-flow rate of return against number of payback periods for various payback periods.

9-38

PROCESS ECONOMICS

been attained. In addition, Fig. 9-34 shows that the discounted-cashflow rate of return reached at that time is 24 percent. Effects of Differential Inflation Inflation can be general or differential. In the first case, all costs and prices increase at a uniform rate. In the second, government controls and other factors cause the various costs and prices to inflate at different rates. The onset of general inflation does not change the value of the contribution efficiency η, as can be seen from Eq. (9-119), and it does not affect the value of β if the cash flows in Eq. (9-118) are converted to their purchasing power in Year 0. Thus, general inflation does not cause the measured-survival function to change. Differential inflation, on the other hand, can affect the measuredsurvival function. We shall assume, for instance, that the sales price per unit product cS in Eq. (9-119) is frozen at a constant level while some or all of the production costs are allowed to rise. This causes the value of η to decrease; therefore, (MSF) likewise decreases, as can be seen from Eq. (9-117). Let us consider the effect of differential inflation on the overall profitability of the project of the last example. The effect of general inflation on this project showed that the apparent profitability rises sharply, to an (NPV) of $431,269 at a general inflation rate of 20 percent. However, when the cash flows of the (NPV) are properly corrected to their purchasing power in Year 0, the (NPV) instead becomes $208,733. The effect of differential inflation on this project emerges in Fig. 9-35, with all (NPV)s corrected to their purchasing power in Year 0. The top line shows (NPV) for various rates of general inflation. The bottom line shows (NPV) for the differential-inflation case in which only the costs are allowed to increase while product selling price and thus cash income remain constant from year to year. The middle line shows the effect of general inflation when the price rises are delayed by 1 year. The figure confirms that both of these situations take away from the attractiveness of the project.

The effect upon total taxes paid, when they are corrected to their purchasing power in Year 0, is shown in Fig. 9-36. Differential inflation not only decreases the profitability of the project to its owner but also decreases the revenue received by the taxing authority. The method of calculation is identical to that of the earlier example. Another instance of differential inflation occurs when the prices of goods and services rise uniformly but the cost of borrowing money, the interest rate charged on a loan, does not rise. If the fractional inflation rate is ii, a fractional interest rate iL on a loan can be corrected to an effective rate of interest by Eq. (9-116) with iL substituted for (DCFRR). The effect of various amounts of loan, borrowed at various interest rates iL, on the net present value of a particular, fairly simple project is shown in Fig. 9-37. Thus, if $25,000 were borrowed at an interest rate of 15 percent for the project, the (NPV) would be about $43,000 at a zero inflation rate. But if the inflation for goods and services ii is 10 percent, the effective interest rate for that loan can be calculated from Eq. (9-116) to be only 4.55 percent. It is seen from Fig. 9-37 that this increases the (NPV) of the project to $48,000. This confirms the economic advantage of borrowing at a fixed interest rate in a time of general inflation. A topical aspect of differential inflation is the question of energy costs. Will the cost of a particular fuel rise or fall in relation to prices in general, and if so, what effect will this have on the economics of a project? Example 17: Effect of Fuel Cost on Project Economics A process unit is heated by gas. We assume that $100 spent on energyconservation measures for this particular unit at the end of 1980 would save 200 therms (21.1 GJ) of gas energy in each subsequent year. If the cost of gas in 1980 is $x per therm, the annual dollar savings at 1980 prices is $200x. The (NPV) at the end of year n for this project is n (200)x (NPV) = −100 +  ᎏn (1 + i) 1

if the appropriate discount factor is i. This is independent of inflation provided that the cost of gas rises in line with any general rate of inflation. However, if the real cost of gas rises at a fractional annual rate r over and above the general inflation rate, it should be modified into the form n (200x)(1 + r)n (NPV) = −100 +  ᎏᎏ (1 + i)n 1 This equation confirms that as the gas price rises because of inflation, the attractiveness of the conservation project also rises.

FIG. 9-35

Effect of differential inflation on inflation-corrected net present

value.

FIG. 9-36

Effect of differential inflation on inflation-corrected tax revenue.

FIG. 9-37

Effect of loan interest rate on the net present value of a project.

ACCOUNTING AND COST CONTROL

9-39

ACCOUNTING AND COST CONTROL Principles of Accounting Accounting is the art of recording business transactions in a systematic manner. Financial statements are both the basis for and the result of management decisions. Such statements can tell managers or engineers a great deal about their company, provided that they can interpret the information correctly. Since a fair allocation of costs requires considerable technical knowledge of operations in the chemical-process industries, a close liaison between the senior process engineers and the accountants in a company is desirable. Indeed, the success of a company depends on a combination of financial, technical, and managerial skills. Accounting is also the language of business, and the different departments of management use it to communicate within a broad context of financial and cost terms. Engineers involved in feasibility studies and detailed process evaluations are dependent for financial information on the company accountants, especially for information on the way in which the company intends to allocate its overhead costs. It is vital that engineers correctly interpret such information and that they can, if necessary, make the accountants understand the effect of the chosen method of allocation. The method of allocating overheads can seriously affect the assigned costs of a project and hence the apparent cash flows for that project. Since these cash flows are used to assess profitability by the net-present-value (NPV) and discounted-cash-flow-rate-of-return (DCFRR) methods, unfair allocation of overhead costs can result in a wrong choice between alternative projects. In addition to understanding the principles of accountancy and obtaining a working knowledge of its practical techniques, engineers should be aware of possible inaccuracies of accounting information in the same way that they allow for errors in any technical data. At first acquaintance, the language of accountancy appears illogical to most engineers. Although accountants normally express themselves in tabular form, the basis of all their practice can be simply expressed by Capital = assets − liabilities (9-122) Equation (9-122) can alternatively be written as Assets = capital + liabilities (9-123) Capital, often referred to as net worth, is the money value of the business, since assets are the money values of things the business owns while liabilities are the money values of the things the business owes. Most engineers have great difficulty in thinking of capital (also known as ownership) as a liability. This is easily overcome once it is realized that a business is a legal entity in its own right, owing money to the individuals who own it. This realization is absolutely essential when considering large companies with stockholders and is used for consistency even for sole ownerships and partnerships. If an individual puts up $10,000 capital to start a business, then that business has a liability to repay $10,000 to the individual. It is even more difficult to think of profit as being a liability. Profit is the increase in money value available for distribution to the owners and effectively represents the interest obtained on the capital. If the profit is not distributed, it represents an increase in capital by the normal concept of compound interest. Thus, if the individual’s business makes a profit of $5000, the liability to the individual is increased to $15,000. With this concept in mind, Eq. (9-123) can be expanded to Assets = capital + liabilities + profit (9-124) where the capital is considered as the cash investment in the business and is distinguished from the resultant profit in the same way that principal and interest are separated. Profit (as referred to above) is the difference between the total cash revenue from sales and the total of all costs and other expenses incurred in making those sales. With this definition, Eq. (9-124) can be further expanded to Assets + expenses = capital + liabilities + revenue from sales (9-125) Engineers usually have the greatest difficulty in regarding an expense as being equivalent to an asset, as is implied by Eq. (9-125). Let us consider a one-person business. We assume for a given period

a profit of $5000 and total expenses excluding the individual’s earnings of $8000. Also we assume that the individual’s labor to the business in this period is worth $12,000. The revenue required from sales would be $25,000. Effectively, the individual has made a personal income of $17,000 in the period but has apportioned it to the business as $12,000 expense for the individual’s labor and $5000 return on capital. In larger businesses, there will also be those who receive salaries but do not hold stock and, therefore, receive no profits and stockholders who receive profits but no salaries. Thus, the difference between expenses and profits is very practical. The period covered by the published accounts of a company is usually 1 year, but the details from which these accounts are compiled are entered daily in a journal. The journal is a chronological listing of every transaction of the business, with details of the corresponding income or expenditure. For the smallest businesses, this may provide sufficient documentation, but in most cases the unsystematic nature of the journal can lead to computational errors. Therefore, the usual practice is to keep accounts that are listings of transactions related to a specific topic such as “purchase-of-oil account.” This account would list the cost of each purchase of oil, together with the date of purchase, as extracted from the journal. Principles of Double-Entry Accounting Many of the accounts involve both income and expenditure. The general practice is to keep accounts by the double-entry system, which may be summarized by Debits = credits

(9-126)

The principle of double entry dates from the fifteenth century and is based on the premise that every transaction involves a giver and a receiver of value. Double entry requires that each transaction be entered into two accounts, the convention being that the account of the giver is credited and the account of the receiver is debited with the same amount of money, as noted in the journal. For convenience, each account is divided centrally, and the debit items are entered on the right-hand side. It is also usual to provide a cross-reference to the journal entry so that errors and omissions can be checked. Let us consider the purchase of $50,000 worth of plant equipment by company A, paid for by check. The accounting entries are: debit the plant-equipment account $50,000, and credit the bank account $50,000. The plant-equipment account is then said to have a debit balance of $50,000, and the bank account a credit balance of $50,000, if these happen to be the only entries. If company A then sells $100,000 worth of product that is paid for by check, the accounting entries are: credit the sales account $100,000, and debit the bank account $100,000. The bank account will now have a debit balance of ($100,000 − $50,000) = $50,000, and the sales account a credit balance of $100,000, if this happens to be the only sale to date in the accounting period. In principle, the debiting and crediting of accounts are relatively straightforward. However, a great deal of practice is essential in order to achieve proficiency. Although it is not at all necessary for engineers to compete with professional accountants in this field, engineers should appreciate what accountants do and why they do it. Of the accounts considered in the preceding illustrations, the plantequipment and bank accounts are asset accounts, and the sales account is a liability account. To increase an asset, debit the asset account; to increase a liability, credit the liability account. Conversely, to decrease an asset, credit the asset account; to decrease a liability, debit the liability account. Closing the Books At the end of the accounting period, the individual accounts are closed by balancing each in accordance with Eq. (9-126). The balances are transferred either to the balance sheet in the case of capital expenditure or to the income statement in the case of revenue expenditure. An alternative name for the balance sheet is the position statement; the income statement is also called the trading and profit-and-loss account. The purpose of capital expenditure, such as the purchase of a piece of plant equipment for $50,000, is to earn future revenue. In contrast, the purpose of revenue expenditure is to maintain existing business.

9-40

PROCESS ECONOMICS

TABLE 9-16

Income Statement for ABC Company

Revenue Sales revenue Other revenue

$1,900,000 100,000 $2,000,000

Expenses Raw materials Wages Utilities Depreciation Other expenses Income taxes

953,000 185,000 44,000 68,000 376,000 194,000 1,820,000

Net profit (after tax)

$ 180,000

Revenue expenditure includes the direct material costs and direct labor costs incurred in the manufacture of a product, together with the associated overheads that include maintenance of the plant. Since these expenses are debits, the debit balance for a given accounting period is obtained by adding up the debit balances from each individual expenditure account. Similarly, since revenues from sales and other income are credits, the credit balance for a given accounting period is obtained by adding up the credit balances from each individual income or revenue account. To ascertain profit or loss (calculated as income minus expenditure for a given accounting period), income and expenditure must be matched. For example, any rent paid in advance beyond the current accounting period should not be included in the profit or loss calculation. Similarly, goods sold but not yet paid for in a given accounting period should not be included in the revenue total for that period. An income statement such as the one shown in Table 9-16 is used to obtain the profit or loss for a given period. The debit and credit balances of all the accounts that do not represent expenditure or income for a given accounting period are entered as assets and liabilities in a balance sheet such as that shown in Table 9-17. There is no rigid format for either the income statement or the balance sheet. Tables 9-16 and 9-17 show common layouts for the income statement and balance sheet respectively, but these are not the only forms. For example, vertical balance sheets, with the assets listed above the liabilities and equity, are also popular. Some expenditures are partly capital and partly revenue. For example, repair and improvement work may be done on a plant simultaneously. In this case, the repair work should be classified as revenue expenditure and the plant-improvement work as capital expenditure. TABLE 9-17

Accounting Concepts and Conventions Accounting is based on the following concepts: (1) money measurement, (2) business entity, (3) going concern, (4) cost, and (5) matching. Concept 1. “Money measurement” means that only those facts that can be represented in monetary terms are recorded. The balance sheet and income statement for a company give no indication as to what might happen in the future. The company may be about to be successfully sued for a large sum of money, or a competitor may be launching a new product that will seriously reduce future sales of the company’s products. Concept 2. “Business entity” means that accounts are kept for the company quite independently of the people who may own the company. For example, if an individual puts an additional $10,000 into a one-person business, the accounts show that the business is $10,000 richer. They do not show that the individual’s personal wealth has been depleted by $10,000. Concept 3. “Going concern” means that the accounting is based on the premise that the business will continue indefinitely. It is most unlikely that the values of the assets shown in the balance sheet are what the assets would realize if sold. No attempt is made in normal accounting to measure the value of the business to a potential buyer. Concept 4. “Cost” means that the assets are normally shown in the balance sheet at cost price together with their subsequent depreciation. Some assets such as land may be considerably more valuable than when originally purchased, but no indication of this is given in the balance sheet. However, some governments now require a note giving the current estimated value of the land. Concept 5. “Matching” means that the revenue in a given accounting period should correspond to the expenses for that accounting period. Accounting is also based on the following conventions: (1) materiality, (2) conservatism, or prudence, and (3) consistency. Materiality deals with determining whether certain expenditures will have a significant effect on a company’s accounting procedures. This is a matter of judgment that is to be made by each company. Obviously, the purchase of a vehicle is a material item, but writing paper or tools for maintenance are less obvious. Although such items may last well beyond the current accounting period, it may not be worth the accounting effort to treat them as material items. Some companies will treat a particular item as capital; other companies, as expenditure. Clearly, the purchase of a piece of equipment costing, say, $1000, will be regarded as less material by a giant company than by a small one. Conservatism, or prudence, means monetary values that tend to understate rather than overstate the profit are taken. Consistency means that accounting items are normally treated in

Balance Sheet for XYZ Company Liabilities and stockholders’ equity (thousands of dollars)

Assets (thousands of dollars) Current assets Cash Notes and accounts receivable Inventories: Finished products Work in process Raw materials and supplies (at cost) Total inventories Total current assets Investments and long-term receivables (at cost) Property, plant, and equipment (at cost) Land Buildings Machinery and equipment Less accumulated depreciation Net property, plant and equipment Prepaid and deferred, charges Total assets

$ 38,893 110,740 17,396 56,690 35,790 109,876 259,509 94,009 6,110 63,848 106,185 176,143 75,163 100,980 6,094 $460,592

Current liabilities Notes payable Accounts payable and accrued liabilities Accrued taxes Total current liabilities Long-term liabilities Deferred income taxes Other deferred credits Stockholders’ equity Common stock, $20 par value Shares authorized, 7,750,000 Shares issued, 4,794,450 Capital in excess of par value of common stock Retained earnings Total stockholders’ equity

Total liabilities and stockholders’ equity

$ 34,507 106,433 7,264 148,204 67,677 13,225 2,307

95,889 31,798 101,492 229,179

$460,592

ACCOUNTING AND COST CONTROL the same way over an indefinite number of years. For example, an individual item would not be treated as an expenditure during one year and as a capital item during the next year without good reason being given. Balance Sheet The balance sheet, also called the position statement, presents an accounting view of the financial status of a company at a particular point in time. A typical balance sheet is shown in Table 9-17. Although a balance sheet has two sides that balance, it is not part of the double-entry system. In fact, it is not an account but rather a statement listing all the assets of a company and the various claims against these assets on the last day of the accounting period. The assets must be equal to the claims against them at all times. Those who have claims against the assets are the owners (stockholders in a business corporation) and the people to whom the company owes money. In the case of the latter, the company is said to have liabilities to its creditors. The total claim against the assets is often labeled “liabilities and owners’ equity.” Assets are classified as current or fixed, and liabilities as current or long-term. Fixed assets are material items that have a relatively long life and normally include land, buildings, plant, vehicles, etc. They are held for the specific purpose of earning revenue and are not for sale in the normal course of business. Current assets include cash and those items that can be fairly easily converted into cash, such as rawmaterials inventories, etc. In contrast to fixed assets, current assets are acquired for the specific purpose of conversion into cash in the normal course of business. However, what is regarded as a fixed asset by one type of company might be regarded as a current asset by another. For example, a chemical company would normally classify its vehicles as a fixed asset. However, a company whose primary business was to sell vehicles would classify them as a current asset. Similarly, the distinction between current and long-term liabilities is also not clear-cut. Current liabilities include accounts payable (money owed to creditors), taxes payable, dividends payable, etc., if due within a year. Long-term liabilities include deferred income taxes, bonds, notes, etc., that do not have to be paid within a year. The owners’ equity includes the par, or face, value of the capital received from stockholders and any retained earnings. The balance sheet shows only the nominal value and not the current or real value of this capital. A balance sheet includes items that are not regarded as assets or liabilities in normal language, such as expenditures carried forward and accumulated profits. Accountants regard assets as resources that have not yet been used up. Assets are normally shown on the balance sheet at cost minus accumulated depreciation. In this sense, the depreciation charge for an accounting period is the means of converting a part of an asset into a current expenditure that is then listed as an expense in the income statement. Let us consider plant equipment costing $1 million and purchased on Jan. 1, 1988. Table 9-18 shows the provision for the depreciation account for 1988, 1989, and 1990 for straight-line depreciation, assuming a service life of 10 years and zero scrap value. The credit entries of $100,000 for the depreciation in each year are balanced by the depreciation charge of $100,000 debited to the income statement (or trading and profit-and-loss account) in each year. Table 9-19 shows the correTABLE 9-18 Account

Provision for Depreciation of Plant-Equipment

1988 Dec. 31 Balance carried down

$100,000

Jan. 1 Balance brought down Dec. 31 Debited to income statement

0 $100,000

1989 Dec. 31 Balance carried down

$200,000 $200,000

Jan. 1 Balance brought down Dec. 31 Debited to income statement

$100,000 100,000 $200,000

1990 Dec. 31 Balance carried down

$300,000 $300,000

Jan. 1 Balance brought down Dec. 31 Debited to income statement

$200,000 100,000 $300,000

TABLE 9-19

9-41

Balance-Sheet Entries

As of Dec. 31, 1988 Plant equipment at cost Less depreciation to date

$1,000,000 100,000 $900,000

As of Dec. 31, 1989 Plant equipment at cost Less depreciation to date

$1,000,000 200,000 $800,000

As of Dec. 31, 1990 Plant equipment at cost Less depreciation to date

$1,000,000 300,000 $700,000

sponding entries in the balance sheets for the years 1988, 1989, and 1990. Entries for subsequent years are made in the same way. A balance sheet is true only for one particular point in time; it tells nothing about the trends in a company. However, by comparing balance sheets for successive years, management can follow changes in the various items. If the observed trend is undesirable, management can take corrective action. Since the accounting period of 1 year is long for most businesses, it is usual to draw up balance sheets at more frequent intervals for control purposes. These may be less formal than those issued annually to the stockholders. In general, balance sheets are less useful to management than are income statements. Income Statement Income statements range from the very simple presentation shown in Table 9-16 to the more informative and more complex presentation shown in Table 9-20. The income statement shows the revenue and the corresponding expenses that were incurred to earn that revenue over a period of time. It is the most obvious measure of the efficiency of a business. Although published income statements are normally for 1-year periods, many companies use monthly income statements for internal purposes. Income statements are very useful tools to assist management in controlling a business and planning for the future. Since management needs to follow the trends of the normal expenses, extraordinary expenses such as those incurred as a result of a major fire or flood should be shown separately. If revenue and expenses are not properly matched, an understatement or an overstatement of profit may occur. If raw materials were previously purchased at a lower cost than their current cost, profit will be overstated. Any overstatement of profit will mean that more tax will be paid. One of the most important items in an income statement is depreciation expense. Although depreciation should not be thought of as a means to build up a fund to replace plant, it nevertheless does enable money to be retained in the business by reducing the profit available for distribution to stockholders. It is of course a duty of both accountants and management to see that sufficient money is retained in the business to replace assets and to invest such money in other processes or outside investment. A further duty of accountants and management is to ensure that the company always has sufficient working capital to enable it to carry on its business. Types of Accountancy The traditional work of accountants has been to prepare balance sheets and income statements. Nowadays, accountants are becoming increasingly concerned with forward planning. Modern accountancy can roughly be divided into two branches, financial accountancy and management or cost accountancy. Financial accountancy is concerned with stewardship. This involves the preparation of balance sheets and income statements that represent the interest of stockholders and are consistent with existing legal requirements. Taxation is an important element of financial accounting. Management accounting is concerned with decision making and control. This is the branch of accountancy closest to the interest of most process engineers. Management accounting is concerned with standard costing, budgetary control, and investment decisions. Accounting statements present only facts that can be expressed in financial terms. They do not indicate whether a company is developing new products that will ensure a sound business future. A company

9-42

PROCESS ECONOMICS

TABLE 9-20 Income Statement for a Mature Year for a New Chemical Product, Produced at 10 Million lb/Year

Revenue from annual sales AS Direct manufacturing expense ADME Raw materials Catalysts and solvents Operating labor Operating supervision Utilities Operating maintenance Operating supplies Royalties and patents Total ADME Indirect manufacturing expense AIME Payroll overhead Central laboratory General plant overhead Packaging and storage Property taxes Insurance Total AIME

Unit values, cents/lb

% sales revenue, %

20.00

100.0

$ 884,000 69,000 102,000 20,000 22,000

8.84 0.69 1.02 0.20 0.22

44.2 3.4 5.1 1.0 1.1

21,000 4,000 10,000 $1,132,000

$1,132,000

0.21 0.04 0.10 11.32

1.1 0.2 0.5 56.6

$132,000

0.28 0.10 0.52 0.22 0.14 0.06 1.32

1.4 0.5 2.6 1.1 0.7 0.3 6.6

$1,264,000

12.64

63.2

68,000

0.68

3.4

74,000 124,000

0.74 1.24

3.7 6.2

40,000 10,000

0.40 0.10

2.0 0.5

308,000

0.60 3.08

3.0 15.4

$1,640,000 $360,000

16.40 3.60

82.0 18.0

$2,000,000

28,000 10,000 52,000 22,000 14,000 6,000 $132,000

Total manufacturing expense (excluding depreciation) AME Depreciation ABD Other expenses Administration Sales and shipping Advertising and marketing Technical service Research and development Total other expenses Total expense ATE Net annual profit ANP

60,000 308,000

net annual profit (ROA) = ᎏᎏ 100 (9-129) total assets where (ROA) is called the return on assets. In the second case, Eq. (9-128) can be written as net annual profit (ROE) = ᎏᎏᎏ 100 (9-130) stockholders’ equity where (ROE) is the return on equity. Asset-turnover ratio (ATR) is a commonly used measure of company performance, defined as revenue from annual sales (ATR) = ᎏᎏᎏ 100 (9-131) total assets A comparison between Eqs. (9-127), (9-129), and (9-131) shows that (ROA) = (ATR)(PM) (9-132) Thus (ROA) can be improved by increasing either (ATR) or (PM). A variation of Eq. (9-131) is the fixed-asset turnover ratio (FATR), defined as revenue from annual sales (FATR) = ᎏᎏᎏ 100 (9-133) fixed assets Clearly, (FATR) is of less value than (ATR) when applied to companies that use relatively large amounts of working capital. The (FATR) is the inverse of the capital ratio (CR) for single projects. (CR) is defined as (CR) = CFC / AS (9-134) where CFC is the fixed-capital cost for a green-fields (grass-roots) site and AS is the revenue from annual sales. The fixed assets in Eq. (9-133) and those included in the total assets in Eqs. (9-129) and (9-131) are usually taken at their written-down, or book, value, which may differ significantly from their market value. This is one disadvantage in using Eqs. (9-129), (9-131), and (9-133). The revenue from annual sales referred to in Eqs. (9-127), (9-131), and (9-132) is normally taken to be the gross turnover, which includes intergroup sales. However, intergroup sales are eliminated in consolidated or group accounts. Again, revenue from annual sales must be clearly defined before comparisons are made with other companies. Let us consider the simplified balance-sheet or position statement shown in Table 9-21. Essentially, total assets are related to liabilities and stockholders’ equity by Total assets = stockholders’ equity + total debt

(9-135)

Equation (9-135) can also be written as may have impressive current financial statements and yet be heading for bankruptcy in a few years’ time if provision is not being made for the introduction of sufficient new products or services. Financing Assets by Equity and Debt Financial Ratios Probably the most commonly mentioned ratio is the profit margin (PM), defined as net annual profit (PM) = ᎏᎏᎏ 100 (9-127) revenue from annual sales Another common ratio is the return on investment (ROI), defined as net annual profit (ROI) = ᎏᎏ 100 (9-128) investment In both Eq. (9-127) and Eq. (9-128), the net annual profit can be either before or after tax. It can also include interest and dividends receivable, etc. Obviously, the net annual profit must be clearly defined before comparisons are made with other companies. Similarly the term “investment” in Eq. (9-128) can have a variety of meanings. The two most common ones (used when assessing the profitability of companies as opposed to projects) are total assets and owners’ equity or capital employed. In the first case, Eq. (9-128) can be written as

Stockholders’ equity = total assets − total debt

(9-136)

Equations (9-130) and (9-136) can be combined to give net annual profit (ROE) = ᎏᎏᎏ 100 (9-137) total assets − total debt Equation (9-137) can also be written to include a quantity called the debt ratio (DR), which gives net annual profit 100 (ROE) = ᎏᎏ ᎏ (9-138) total assets 1 − (DR)





TABLE 9-21 Simplified Balance Sheets for Companies X and Y X Company balance sheet

Total assets $100,000

Total debt Stockholders’ equity

0 $100,000

Total liabilities and stockholders’ equity

$100,000

Y Company balance sheet

Total assets $100,000

Total debt Stockholders’ equity

$ 50,000 $ 50,000

Total liabilities and stockholders’ equity

$100,000

ACCOUNTING AND COST CONTROL

9-43

where (DR) is the debt ratio as given by total debt (DR) = ᎏᎏ (9-139) total assets Return on assets (ROA) can be related to the return on equity (ROE) by combining Eqs. (9-129) and (9-138): (ROA) = (ROE)/[1 − (DR)]

(9-140)

(ROE) can also be related to the asset-turnover ratio (ATR) and the profit margin (PM) by combining Eqs. (9-132) and (9-140): (ROE) = [(ATR)(PM)]/[1 − (DR)]

(9-141)

Financing by Debt, or Leverage The debt ratio (DR) is also known as the leverage, or gearing, ratio. Highly levered companies have a high proportion of debt to total assets. At first glance, it may appear that the use of leverage is a simple way of increasing the return on equity (ROE). However, interest charges have to be paid on the debt. Whether leverage is a good thing or not will depend on exactly what the interest charges are in relation to the return on assets and the return on equity. Let us consider the simplified balance sheets of two companies, X and Y, shown in Table 9-21. Companies X and Y have a debt, or leverage, ratio of zero and 0.5 respectively. Let us assume that the debt is of the debenture type for tax purposes and that the interest rate is 10 percent per annum. The return on equity (ROE) after tax is given in Table 9-22 for companies X and Y for various values of net annual profit ANP before tax. ANNP is the net annual profit after tax. The data for Table 9-21 are plotted in Fig. 9-38. This figure shows that leverage has no effect on the (ROE) when the interest rate charged for the borrowed money is equal to the return on assets (ROA) before tax. Leverage provides increased (ROE) values when the (ROA) is greater than the interest rate charged for the borrowed money and decreased (ROE) values when it is less. The greater the debt, or leverage, ratio (DR), the more sensitive the (ROE) is to a change in (ROA) and the steeper the slope of the line in Fig. 9-38. Dividends to stockholders are paid out of the net annual profit after tax ANNP, from which the (ROE) after tax in Fig. 9-38 is calculated. Thus, the higher the leverage, the greater the financial risk to the stockholder. This risk is not the same as the business risk of the company, which is a function of its overall prospects in its particular industry. Leverage increases the return to the stockholders when the (ROA) is higher than the interest rate on debt and decreases the return when the (ROA) is lower than the interest rate. Whether the assets of a company are financed largely by stockholders’ equity (also called net worth), or largely by debt, or by some combination of the two depends on a number of factors. If sales do not fluctuate, a company is in a good position to pay the fixed interest charges on debt. This is also the case if the revenue from sales is steadily increasing. In this case, any new common stock issued by the company is likely to command a good price, and it also increases the attractiveness of equity financing. The attitude of management is also an important factor in deterTABLE 9-22 Return on Equity after Tax for Companies X and Y (ROA) before tax

5%

10%

15%

20%

X company ANP Less tax at 50%

$5,000 ($2,500)

$10,000 ($ 5,000)

$15,000 ($ 7,500)

$20,000 ($10,000)

ANNP (ROE) after tax

$2,500 2a %

$ 5,000 5%

$ 7,500 7a %

$10,000 10%

Y company ANP before interest Less interest

$5,000 ($5,000)

$10,000 ($ 5,000)

$15,000 ($ 5,000)

$20,000 ($ 5,000)

ANP after interest Less tax at 50%

0 0

$ 5,000 ($ 2,500)

$10,000 ($ 5,000)

$15,000 ($ 7,500)

ANNP (ROE) after tax

0 0

$ 2,500 5%

$ 5,000 10%

$ 7,500 15%

FIG. 9-38

Effect of leverage on the return on equity.

mining how much debt financing is used. In a small firm in which management owns most of the equity, management may be very reluctant to issue further amounts of common stock that would lead to a dilution of its control. Furthermore, if management has great confidence in future prospects, it will wish to ensure the maximum return for itself. In contrast, the equity in a large company is widely distributed, and the issue of further amounts of common stock has little effect on the control of the company. The difference between equity financing and debt financing is not always clear-cut. For example, preferred stock can be classified as stockholders’ equity or debt, depending on who is doing the financial analysis. Equity Financing Typically, the company balance sheet will show the stockholders’ equity and list the preferred stock, common stock, and retained earnings as in Table 9-23. The issue of common stock is the basic method of financing a company. Common stockholders take the ultimate risk in a business because they have no right to a return on their investment. However, they have the right to elect the directors of the company, who in turn are responsible for the management of the business. Stockholders are likely to vote the board of directors out if adequate dividends are not paid. Usually the liability of stockholders is limited to the nominal, or par, value of their stock, and hence they can lose only what they have already paid for the stock. If the liability is not limited by law, the personal assets of the stockholders are at risk in the event of company bankruptcy, in proportion to the amount of stock held. Preferred stock is often used as an alternative to debt when companies do not wish to issue additional common stock or to incur the fixed interest charges required to finance debt. Preferred stockholders are not normally allowed to vote for the board of directors. They have the right to receive fixed amounts of dividends before common stockholders are paid any dividends. However, a company does not have to pay dividends. The board of directors may decide to pay small or no dividends in a particular year. Holders of cumulative preferred stock TABLE 9-23 Stockholders’ Equity as Shown in Section of a Company’s Balance Sheet Preferred stock, par value $100 per share Authorized 2000 issued and outstanding Less discount on preferred stock Preferred-stock equity Common stock, par value $10 per share Authorized and issued 100,000 shares Amount paid in excess of par Retained earnings Common-stock equity Total stockholders’ equity

1,500

$ 150,000 (10,000) $ 140,000

$1,000,000 100,000 $1,100,000 100,000 $1,200,000 $1,340,000

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PROCESS ECONOMICS

are entitled to receive compensation for the previous underpayment of dividends when the company again pays dividends. Common stockholders have a right to the residual assets of a company in the event of dissolution or liquidation but only after all the creditors and then any liabilities to the preferred stockholders have been paid. The larger the proportion of debt financing in a company, the smaller the amount the common stockholders are likely to receive if the company is liquidated. Common stockholders normally have a preemptive right to the first option to purchase any additional issues of common stock. This prevents management from using an additional issue of common stock to override the control exercised by existing stockholders. Preemptive rights also protect existing stockholders from having the value of their shares decreased by such dilution, since the same net earnings would be spread over more units of stock. Let us consider the very simplified case of a company with 100,000 shares of common stock, each with a market value of $10, giving a total market value of $1,000,000. If a further 50,000 shares are sold at $4 each, the total market value of the 150,000 shares is $1,200,000, or $8 each. This means that the new stockholders have gained at the expense of the original ones. The preemptive right is designed to prevent this. In practice, the situation is rather more complex than is indicated here. Both common and preferred stocks normally have a par, or nominal, value. In the case of common stock, the market value at the time of issue usually differs from the par value. Stock can be issued either at a premium or at a discount, depending on prevailing economic conditions and the strength of the company. The difference between the actual amount paid and the par value is listed in the stockholders’equity section of the balance sheet, as shown in Table 9-23. The issuance of stock at a premium or a discount is done to protect existing stockholders. In the case of preferred stock, the par value has more meaning than with common stock, since it is the amount due preferred stockholders if the company goes into liquidation, provided that this is a condition of issue. The advantage of using common stock to finance assets is that it does not incur fixed interest charges. Furthermore, there is no maturity date, as there is with all loans and most preference issues. Common stock can often be issued more easily than debt can be financed. However, the flotation costs of common stock can be quite high, especially when stock values are depressed, so that large discounts for the stock are needed to induce purchase. Stockholders’ equity in a company is made up of the capital contributed by the stockholders and the capital generated from retained earnings. The presence of retained earnings on a balance sheet, as shown in Table 9-23, does not necessarily mean that they are matched by an equal amount of cash. In fact, there may be little or no cash available. The retained earnings shown on a balance sheet may be largely fictitious. For example, the assets on a balance sheet may be worth less than shown by at least the value of the retained earnings. Purchase and Sale of Equities Stockholders usually require an adequate return on their investment, and the quoted price of the stock reflects the consensus opinion of investors as to the current health of the company. Purchases or sales are normally made through stockbrokers. Most stock transactions are completed through organized security exchanges on which the stock is listed. Such exchanges have physical existence in the form of buildings located in different regions of the country. Each exchange has members who are often the nominated representatives of large brokerage firms having offices in various cities. These offices are in constant telephone and telegraph communication with the members at the exchange, passing on requests to buy or sell specified stocks. Since brokers live by commissions and charges on transactions, they attempt to match such requests either directly or by dealings with other brokers. In the United Kingdom, brokers must deal through an independent “jobber,” similar in function to a specialist broker, who quotes a low price for sales and a higher one for purchases before the jobber knows whether the broker is buying or selling. The difference represents the jobber’s margin, or “turn.” If requests to buy exceed offers for sale, the price of the stock rises until someone is tempted to sell. Conversely, if an excess of stock is offered for sale, the price is likely to fall.

It is an advantage to a company to be listed on a stock exchange since its investors can more easily sell their stock if they decide to do so. This increased liquidity makes investors more willing to accept a lower rate of return, which effectively lowers the cost of capital to the company. Because dealings in the stock of a listed company are published, a healthy company engenders confidence that makes it easier to obtain other forms of finance. In the absence of a regular market, stock transactions are necessarily infrequent, and prices are liable to wide fluctuation, which may make creditors wary and possibly lead to bankruptcy proceedings. Such dealings are usually referred to as “over-thecounter” and are confined to the relatively few specialist brokers who hold inventories of such stock and are prepared to “make a market” in them or are limited to private transactions. Retained Earnings Much confusion is caused by the practice of dividing retained earnings under various headings such as reserve for replacement of plant, reserve for contingencies, etc. This procedure also restricts the flexibility of management in expenditure decisions. The amount of retained earnings shown on a balance sheet should not be taken as a measure of the amount of future dividends that the company is likely to pay. A contract may exist that specifies a minimum balance of retained earnings, which is then not available for dividends until bonds issued by the company have been retired. Dividends can be paid either as cash or in the form of an additional issue of stock. A stock split is really a stock dividend, and both are used to reduce the price of stock when management considers that it is too high. A stock dividend is essentially a transfer of retained earnings to the common-stock account and makes the amount transferred unavailable for future dividends. A stock dividend may be used in place of a cash dividend when a company is short of cash. Debt Financing In practice, debt financing covers a variety of fixed-income securities, both long-term and short-term. The most common forms of long-term debt are bonds, mortgages, and debentures. A bond is simply a long-term promissory note. It is a contract established between borrower and lender in a document called an indenture. A bond indenture includes a detailed description of assets that are pledged, together with any protective clauses and provisions for redemption. A trustee is appointed to look after the interest of the bondholders. The trustee is normally a commercial bank. Bonds may be issued with a call provision that enables a company to redeem its bonds at any date earlier than scheduled. Obviously, this would be an advantage to a company in times of falling interest rates. However, a company has to pay more than the par value of the bond for this privilege. The additional amount is called the bond premium. Sometimes a company uses a sinking fund to retire a bond. A series of equal annual payments A, invested at a fractional interest rate i and made at the end of each year over a period of n years, is equivalent to a sum of money of present value P, given by A = PfAP

(9-46)

where fAP is the annuity present-worth factor, which is fAP = [i(1 + i)n]/[(1 + i)n − 1] A company may use a sinking fund in a variety of ways, but the simplest is to pay a fixed amount A at the end of each year to buy and retire bonds until after n years all the bonds have been retired. This annual payment may prove a significant strain on the resources of a company. Failure to make the payment could result in bankruptcy. In the case of income bonds, a company is required only to pay interest when it earns it. A mortgage is a bond in which specific real assets are pledged as security. A senior mortgage has a prior claim on assets. A junior mortgage is normally a second mortgage on the residual value of the assets. A blanket mortgage is a pledge on all real property owned by a company. A debenture is an unsecured bond. Strong companies are in a better position to issue debentures than weak companies since they have less need to pledge specific assets. Debenture holders are really general creditors. Subordinated debenture holders have claims on assets only after the claims of certain other claimants have been met. The issue of subordinated debentures provides a tax advantage for a com-

ACCOUNTING AND COST CONTROL TABLE 9-24

9-45

Comparative Ratios for Selected United States Industry Groups for 1992*

Industry Number of companies ᎏᎏᎏ Ratio

Agricultural chemicals 2391

Paints and allied products 238

Petroleum refining 98

Plastic materials and resins

Soap and other detergents

234

66

Current assets ᎏᎏ Current debt

2.0

2.8

1.3

1.8

2.2

Net profit ⫻ 100, % ᎏᎏᎏ net sales

1.9

1.8

1.5

1.5

3.8

Net profit ⫻ 100, % ᎏᎏᎏ net worth

9.3

6.8

6.7

14.3

16.3

Net sales ᎏᎏᎏ Net working capital

9.3

5.2

15.8

9.9

5.9

Collection period, days

27.8

39.6

34.8

39.3

44.7

Net sales ᎏᎏ Inventory

11.4

8.1

14.6

11.9

8.9

Fixed assets ⫻ 100, % ᎏᎏᎏ Net worth

43.0

30.8

131.5

20.0

42.3

Current debt ⫻ 100, % ᎏᎏᎏ Net worth

52.1

42.0

68.4

96.8

46.9

Total debt ⫻ 100, % ᎏᎏᎏ Net worth

71.9

66.2

147.7

111.0

61.4

116.1

89.0

193.7

143.6

119.4

Current debt ⫻ 100, % ᎏᎏᎏ Inventory

*Reprinted with the special permission of Dun & Bradstreet International. Numbers above are median values.

pany compared with the issue of preferred stock because the interest payable is a tax-deductible expense. A financial analyst looking at a company from a potential common stockholder’s point of view is likely to classify preferred stock as debt. In contrast, bondholders and general creditors are likely to regard preferred stock as additional equity. Since preferred stock is a hybrid type of security, it may be issued by a company whose management is divided over the question of whether to use equity or debt to finance additional assets. However, preferred stock does have the disadvantage that the dividends are not allowed as a tax-deductible expense. Comparative Company Data Table 9-24 gives comparative company data that have been compiled by Dun & Bradstreet for various types of processing industries. The median value for each ratio is given. Row 1 in Table 9-24 is the current assets Current ratio = ᎏᎏ (9-142) current liabilities liquid assets Compare Quick ratio = ᎏᎏ (9-143) current liabilities Row 2 in Table 9-24 is the profit margin (PM) of Eq. (9-127). In this case, the net profit referred to is the net annual profit after tax and depreciation ANNP. The net sales is the revenue from annual sales AS after deductions for returns, allowances, and discounts for gross sales. Row 3 in Table 9-24 is the return on equity (ROE) of Eq. (9-130). In this case, the net worth is the tangible net worth representing the sum of the preferred and common stocks and the surplus and undistributed profits or retained earnings, less any intangible items such as goodwill, etc. Row 7 in Table (9-24) is the Average collection period average value of accounts receivable = ᎏᎏᎏᎏ (9-144) revenue from sales per day The funded debt (referred to in row 14) consists of mortgages,

bonds, debentures, serial notes, or other obligations with maturity of more than 1 year from the statement date. Robert Morris Associates also compiles extensive comparative company data for various industries. In addition to ratios similar to the Dun & Bradstreet ratios shown in Table 9-24, Robert Morris Associates gives very useful breakdowns of assets and liabilities for various industries. Table 9-25 shows a breakdown of assets and liabilities for United States manufacturers of industrial inorganic chemicals. Application of Overall Company Ratios The various ratios for a hypothetical company are listed in Table 9-26. The balance sheet shown in Table 9-27 has been built up from the ratios in Table 9-26 in terms of the revenue from net annual sales AS. Let us calculate the following values for the right-hand side of the balance sheet as follows: From ratio 5 Net worth = AS/2.50 = 0.4 AS From ratio 11 Total debt = (0.4 AS)(0.65) = 0.26 AS From ratio 10 Current debt = (0.4 AS)(0.35) = 0.14 AS Long-term debt = total debt − current debt Long-term debt = 0.26 AS − 0.14 AS = 0.12 AS We calculate the following values for the left-hand side of the balance sheet: From ratio 99-45 Fixed assets = (0.4 AS)(0.74) = 0.29 AS From ratio 1 Current assets = (0.14 AS)(2.60) = 0.36 AS From ratio 8 Inventory = AS /7.14 = 0.14 AS From ratio 7 Accounts receivable = (AS /365)(61) = 0.167 AS

9-46

PROCESS ECONOMICS TABLE 9-25 Typical Balance Sheet for a United States Manufacturer of Industrial Inorganic Chemicals Assets

%

Cash Marketable securities Net receivables Net inventory All other current assets Total current assets Fixed assets All other noncurrent assets

4.3 2.2 26.2 24.6 0.9 58.3 36.3 5.4

Total assets

100.0

Liabilities Short-term due to banks Due to trade Income taxes Current maturities long-term debt All other current liabilities Total current debt Noncurrent debt unsubordinated Total unsubordinated debt Subordinated debt Tangible net worth Total liabilities and stockholders’ equity

% 5.6 14.5 3.1 2.1 5.0 30.3 18.3 48.6 3.1 48.3 100.0

Abridged from Annual Statement Studies, 1973 ed., copyright 1973 by Robert Morris Associates, Philadelphia.

Cash and short-term investments = total current assets − (inventory + accounts receivable) Cash and short-term investments = 0.364 AS − 0.307 AS = 0.057 AS In addition to the data for the balance sheet, we calculate the net annual profit (after tax), i.e., ratio 2, to be ANNP = 0.04 AS. TABLE 9-26

Ratios for a Typical Industrial Chemical Company

No.

Ratio

1

Current assets ᎏᎏ = 2.60 Current debt

2

100 = 4.00 ᎏᎏ Net sales 

3

100 = 10.0 ᎏᎏ Net worth 

4

100 = 18.18 ᎏᎏᎏ Net working capital 

5

Net sales ᎏᎏ = 2.50 Net worth

6

Net sales ᎏᎏᎏ = 4.50 Net working capital

7

accounts receivable Collection period = ᎏᎏᎏ = 61 days sales per day

8

Net sales ᎏᎏ = 7.14 Inventory

9



Net profit

Net profit

Net profit



Fixed assets ᎏᎏ 100 = 74.00 Net worth

10

100 = 35.00 ᎏᎏ Net worth 

11

100 = 65.00 ᎏᎏ Net worth 

12

100 = 63.00 ᎏᎏᎏ Net working capital 

13



Current debt ᎏᎏ 100 = 100.00 Inventory

14



Funded debt ᎏᎏᎏ 100 = 76.50 Net working capital

Current debt

Total debt

Inventory





In practice, the ratios are obtained from the information published in the balance sheet. The advantage of the above presentation is that it relates everything to the revenue from net annual sales and hence underlines the importance of sales. Careful study of the ratios can produce many inferences as to the health of the company. For example, the leverage, or debt, ratio (DR) for this example is total debt 0.260 A (DR) = ᎏᎏ = ᎏS = 0.40 total assets 0.660 AS This value is quite low and does not present any problems of control by debtors, such as can arise when (DR) is greater than 1. From Table 9-27 we calculate the ratio for 0.140 AS Current debt ᎏᎏᎏᎏ = ᎏ = 2.45 Cash + short-term investments 0.057 AS Therefore, requests for early repayment by more than 40 percent of the debtors could be met. Hence, no liquidity problems are likely to arise, and advantage can be taken of discounts for early payment. Also, the current debt could be met by sale of the inventory, which takes (0.140 AS /AS)(365), or 51 days. The quick ratio is 1/2.45 = 0.407. If it is assumed that current debtors are due for payment within 61 days, the same time as that allowed to creditors, no bankruptcy petitions are likely. The profit of 10 percent, indicated by ratio 3 in Table 9-26, will be reduced by any dividend due to preferred stockholders, because such payments are not part of fixed-debt expenses; the residue is shared among the ordinary stockholders. If all the long-term debts were in redeemable 6 percent preferred shares, then (from ratio 3) the net annual profit (after tax) is ANNP = 0.10(0.40 AS), or 0.04 AS. Interest due on preferred shares is 0.06(0.12 AS), or 0.0072 AS. Therefore, the earnings for the ordinary shares are (0.04 AS − 0.0072 AS)/(0.4 AS − 0.12 AS) = 0.1171 This value corresponds to 11.71 cents per dollar of common stockholders’ equity. If it is assumed that available interest rates offered by banks, government, etc., for no-risk investment of capital are 10 percent, then the maximum economic market price of $100 stock units in this hypothetical company is about $117. If all the debt is in bonds, etc., earnings on ordinary stock would be 10 cents per dollar of net worth, and the maximum economic price of the stock would be about $100 unless stock prices were expected to rise. Other ratios can easily be deduced from those listed. For example, the return on assets (ROA) and the asset-turnover ratio (ATR) are net annual profit 0.04 A (ROA) = ᎏᎏ 100 = ᎏS = 6.06 total assets 0.66 AS revenue from annual sales (ATR) = ᎏᎏᎏ 100 total assets (ATR) = (AS /0.66 AS)100 = 151.5

ACCOUNTING AND COST CONTROL

9-47

TABLE 9-27 Balance Sheet for a Typical Industrial Chemical Company, Dec. 31, 1991 Assets Current assets Cash Accounts receivable Inventory Total current assets Fixed assets Total assets

Liabilities and stockholders’ equity 0.057 AS 0.167 AS 0.140 AS 0.364 AS 0.296 AS 0.660 AS

Liabilities Current debt Long-term debt

0.140 AS 0.120 AS

Total debt Net worth Total liabilities and stockholders’ equity

0.260 AS 0.400 AS 0.660 AS

Cost of Capital The value of the interest rate of return used in calculating the net present value (NPV) of a project is usually referred to as the cost of capital. It is not a constant value since it depends on the financial structure of the company, the policy of the company toward a particular project, the local method of assessing taxation, and, in some cases, the measure of risk associated with the particular project. The last-named factor is best dealt with by calculating the entrepreneur’s risk allowance inherent in the project i′r from Eq. (9-108), written in the form i′r = [1 + (DCFRR)]/(1 + i) − 1 where i is the cost of capital exclusive of the risk allowance. The value of i′r should be compared with the probability of exceeding or of failing to achieve an (NPV) of zero when using that value of i. The decision to proceed can then be made with a full knowledge of the odds against success. The decision can be related to the company attitude to budgets of the relevant size by the use of probable utilities, as has already been discussed. Cash flows used in calculating (NPV) and (DCFRR) should, of course, be corrected for the anticipated rates of inflation, preferably to the time when the utility curve was obtained. This is important since inflation is likely to have a distorting effect on utility curves obtained at different times. This may be due to an unconscious wish to protect against inflation by achieving higher rewards while assigning less importance to any losses incurred, thus tending toward a gambling outlook. In the absence of a risk allowance the cost of capital becomes a technical financial computation based on sources of funds and company policy. As such it will usually be presented as a figure specified for use in a particular appraisal and is therefore of little concern to the project assessor. However, the following résumé indicates the kinds of factors to be considered. In most companies the objective of company policy is to maximize the financial return to the equity stockholders. This is not invariably the case, since a young company will often plow back an unusually large proportion of its profits to encourage growth. Also, it is increasingly the case that projects are undertaken to restore or preserve an environmental amenity or to bring work into a particular locality. In such circumstances a low value of the cost of capital might be assigned to the project. In many government projects a limited loss is acceptable, in which case the value of i would be negative. When the objective is to maximize the aftertax return to the stockholders, a balance must be struck between the proportion of aftertax company profits which are retained to permit growth of the company assets and the proportion which are distributed to provide an income for the stockholders in the form of dividends. The latter will usually be subject to personal income taxes, sometimes at higher-than-normal rates. The growth potential should be reflected in an increased value of the stock as quoted on the stock exchange. Such growth may result in the imposition of capital gains or inheritance taxes. The selection of the right proportion of earnings to be retained is crucial since this affects the appeal of the company to investors and hence its credit worthiness in the eyes of creditors. The optimum split is influenced by the type of investor since institutional tax rates and exemptions often differ from those applied to private investors. It is for this reason that the optimum split is sensitive to local taxation policy. Most companies can maintain a given level of business only by continuous reinvestment in plant and equipment. If company growth is

required, additional investment is essential. In general, a company has only three sources of new money, namely, cash received from the sale of newly issued shares, retained earnings, and debt capital of all kinds including deferred taxes. In certain circumstances cash grants may be forthcoming from government sources. Each of these sources has its own effective rate of interest, and it is the weighted average of these rates which constitutes the cost of capital exclusive of risk allowance. There is no interest payable directly on equity stocks, but there is a concealed rate expected by investors. Without the expectation of a certain return on their investment they would not invest in a new issue, nor would they retain existing holdings of stock. The sale of stocks on the stock exchange does not affect the cash holding of the company, but new issues must be at prices lower than existing values quoted on the exchange unless great confidence exists that the new money will produce an increased income greatly in excess of the reduction in earnings per share caused by the new issue. Stock carrying a fixed interest rate normally has the interest treated as an allowable expense before tax in the same way as a bank overdraft, which is a relatively short-term source of debt. Deferred taxes carry an interest rate which, like an overdraft, is normally compounded daily at a nominal annual rate but naturally is not an allowable expense for tax purposes. Cash owing on outstanding bills carries a notional rate of interest since in many cases prompt settlement of bills would attract a cash discount. Example 18: Risk-Free Cost of Capital A company requires an investment of $100,000 in new plant to maintain its present sales. Let us determine the current cost of capital to the company and the risk-free cost of capital that it should assign to the plant-replacement project, given the following data. Company assets: from stock sales $ 300,000 from retained earnings 200,000 as bills due 100,000 as deferred taxes 200,000 as bank overdraft 200,000 Total assets Current annual income

$1,000,000 $ 200,000

Bills are due on monthly account with a 2 percent discount for cash. Overdraft and deferred-tax interest are compounded daily at nominal annual interest rates of 15 and 9 percent respectively. Corporation tax, capital gains tax, and personal income tax rates are 50, 40, and 30 percent respectively. The current rate of inflation is at 8 percent per year. The traditional return expected by investors is 7 percent per year net of all taxes in real terms. The interest-rate equivalent of the cash discounts is 2 percent per month, since this discount could be obtained every month if payment were to be made at the beginning of the month rather than, as at present, at its end. Since the bills are settled monthly, the notional interest is paid monthly and should not be compounded. The discount is equivalent to 12 monthly simple-interest payments per year. Hence, from Eq. (9-31) the effective annual interest rate on discounts = (12)(0.02) = 0.24 = 24 percent. It would, therefore, be a good use of surplus cash to reduce this debt as quickly as possible. This would require cash equivalent to one-sixth of the annual bills due, or $16,700, to be available. It can, therefore, be assumed that this level of liquidity is not available for capital projects, either as working capital to reduce the debt or for fixed-capital projects. Further, since the new project will not increase sales, it cannot generate further debt of this kind. Hence, this source is not available to capitalize the new project. Since the overdraft is payable daily at a nominal annual interest rate of 15 percent, it follows from Eq. (9-38) that the effective annual interest rate on overdraft = (1 + 0.15/365)365 − 1 = 16.18 percent. Similarly, the effective annual interest rate on deferred tax = (1 + 0.09/365)365 − 1 = 9.42 percent.

9-48

PROCESS ECONOMICS

The new plant will not increase sales and will therefore not increase the tax debt, so that this source is not available to capitalize the project. An increase in overdraft may be available, subject to a maximum imposed by the acceptable gearing of the company. Since the liquidity of the company is so low, it is possible that it is already extended to its maximum debt, in which case the gearing $300,000 + $200,000 Total equity ᎏᎏ = ᎏᎏᎏᎏ = 1.00 Total debt $100,000 + $200,000 + $200,000 Since neither increased bills due nor increased tax debt is available to finance the new project, this implies that the required $100,000 of new capital will be available as $50,000 from increased overdraft and $50,000 from increased equity. The effective interest rate on the equity involved must therefore be calculated. Equity is available from two sources. First, the company can sell new stock which, if in the form of ordinary shares, carries no interest payment. Although this course appears cheap, its use for projects which do not increase earnings, at least to a compensatory level, is usually inadvisable. This leaves retained earnings as the most likely source of equity for the present project. Equity holders require a real return on their outlay, which they assume to be at the stock-market price if this differs from the face value of the stock, of 7 percent net of all taxes. Retained earnings attract a 40 percent capital gains tax; hence the actual interest rate required on distribution forgone is 7/(1 − 0.40) = 11.67 percent. This is in real terms and at a time of 8 percent inflation rate must be increased in cash terms to (1 + 0.1167)(1.08) − 1 = 20.60 percent. In the same way the effective interest rate on distributed earnings, on which an income tax of 30 percent is payable, may be calculated to be [0.07/(1 − 0.30) + 1](1.08) − 1 = 18.80 percent. If the shares are currently valued at par by the stock exchange, this would require distribution, on the $300,000 of issued equity, of ($300,000)(0.188) = $56,400. This amount is required after corporation tax has been paid on the earnings of $200,000. Thus the earnings which can be retained are ($200,000)(1 − 0.50) − $56,400 = $43,600. This is close to the $50,000 required. If the company were well regarded on the stock exchange, a slight reduction in distributed dividends might not reduce share values since the purpose is to maintain future earnings. However, it would be possible for share values to be reduced by up to ($56,400 − $50,000)/$56,400 = 0.1135 = 11.35 percent. If this happened, the effective interest rate on the retained earnings should be increased to 20.60/(1 − 0.1135) = 23.24 percent. These rates are net of corporation tax, which is the correct form for use in (NPV) calculations. The interest rate due on deferred tax is also net of corporation tax at 9.42 percent. The interest payable on overdrafts is an expense fully allowable against tax, so that the effective aftertax rate is reduced to (16.18)(1 − 0.50) = 8.09 percent. Similarly, as the advantage forgone on the discounts would tend to increase company profits and hence tax due, the effective aftertax gain is reduced to (24)(1 − 0.50) = 12.00 percent. The present cost of capital to the company is the weighted-average interest payable on the various sources of funds on an after-corporation-tax basis. This is readily calculable to be at least [($300,000)(0.00) + ($200,000)(20.60) + ($100,000)(12.00) + ($200,000)(9.42) + ($200,000)(8.09)]/($1,000,000) = 8.82 percent. The cost of capital to the new project, with only two sources, should be [($50,000)(20.60) + ($50,000)(8.09)]/($100,000) = 14.35 percent. Since this project is essential if current production is to be maintained, many companies would assess the cost of capital at somewhere near the lower value. Values of cost of capital in the region of 10 percent are to be expected in developed countries at the present time. We notice in particular that inflation does not affect quoted interest rates when assessing present values of cost of capital. It must, however, be taken into account in assessing the interest rate on the dividend which will be expected by investors. As has been stated, it is alternatively possible to assign to the cost of capital the best risk-free return available on the money. The assessment then proceeds as discussed in connection with Eq. (9-108).

Management and Cost Accounting In any given time period, cost may be divided into expired and unexpired cost. An expired cost is an expense; an unexpired cost is an asset. This division is the basis for income statements and balance sheets. Cost accounting is the name traditionally given to accounting for manufacturing costs. The manufacturing cost of a product is traditionally taken as the sum of the costs for (1) direct materials, (2) direct labor, (3) manufacturing overheads, and (4) administration, selling, and finance. Two methods are in general use in accounting for manufacturing costs, absorption costing and marginal costing. In absorption costing, which is the traditional method, all manufacturing overhead costs are included in the cost of sales. In marginal costing only variable manufacturing overhead costs are included in the cost of sales. Marginal costing is more valuable than absorption costing in decision making.

However, it is sometimes quite difficult to separate costs and particularly manufacturing overhead costs into fixed and variable components. In the long term virtually all costs are variable. The difference between the two methods assumes great importance in inventory evaluation. In cost accounting, costs are identified with cost centers. These are accounting devices which may or may not have a physical existence. In the simplest case of a plant manufacturing a single product, the entire plant may be the cost center. In practice there are two major classifications of cost accounting systems, job costing and process costing. In the former, costs are collected for each job or batch irrespective of the accounting period. This system is normally used in construction work. Process costing is normally used in continuous and semicontinuous processes. Costs are collected for a specific accounting period. Allocation of Overheads How overheads are allocated can affect the total cost of a product and, hence, the estimated future cash flows for a project. Since these cash flows are used in the net-presentvalue (NPV) and discounted-cash-flow-rate-of-return (DCFRR) methods for estimating profitability, erroneous allocations could result in the wrong choice of project. The modern trend is for overhead costs to become an increasing proportion of total product costs. This results from the ever-greater sophistication of process plants. Therefore, it is highly desirable that chemical engineers should have some say in the allocation of overheads and that this should not be left entirely to accountants. Direct costs are those that can be directly charged to a single product. The most obvious direct cost is for raw materials, of which the quantity consumed is directly proportional to the amount of product manufactured. Direct process labor is also considered to be a direct cost. However, many costs cannot be directly charged to an individual product. These so-called indirect, burden, or overhead costs range from the lighting and heating required for the plant and offices to the cafeteria and medical facilities provided. When several products are made in a plant, it becomes increasingly difficult to allocate overheads correctly among the various products. A number of different methods are commonly used to estimate the amount of overhead to be allocated to an individual product. These methods are necessary because accountancy costs become prohibitive for charging all costs directly to an individual product. Unfortunately, there is always an arbitrary element inherent in the process of allocation. Overheads in the chemical-process industries are commonly calculated as a percentage of (1) direct materials cost, (2) direct labor cost, or (3) prime or direct costs. Other methods of allocating overheads are on the basis of (1) plant area, (2) number of employees, (3) capital value, and (4) electric power. These listings do not include all the methods in use. The validity of a particular method depends on the process and the industry. An inappropriate method can lead to misleading and even absurd results. Let us consider the manufacture of metal ornaments. The processing cost, exclusive of material, may vary very little for a wide range of materials. However, the direct materials cost will be much greater for precious than for base metals. In this case, an overhead allocation on the basis of direct material costs could be very misleading, while one based on direct labor cost could be quite accurate. Problems can also arise when allocating overheads on the basis of direct labor cost. Let us consider a company that evaluates overheads at 125 percent of direct labor cost. A process plant employs seven operators, each with a direct cost of $10,000 per budget period. As a result of a works-study exercise, it is found that the plant can operate satisfactorily with six operators. The actual cost saving is likely to be far nearer to the direct labor savings of $10,000 per period than to the calculated saving of $10,000 + $10,000(125/100) = $22,500 per period. The $22,500 calculated saving is the direct labor cost plus overheads taken as 125 percent of the direct labor cost. A thorough analysis should be made before production is stopped on a product that is losing money. Although direct costs of the discontinued product will be saved, overheads are not eliminated, as might be inferred from taking overheads as a percentage of direct material, direct labor, or prime costs. The plant is still there, together with its

ACCOUNTING AND COST CONTROL associated costs for interest charges, insurance, painting, some maintenance, etc. Continued production can still make a useful contribution to such overheads. This contribution is lost if production is stopped and must then be borne by other products. Problems can arise with each of the methods used for allocating overheads. Two process plants may occupy similar areas yet have vastly different material or labor costs. Problems can also arise with an individual plant that can be used to make different products, as Example 19 will show. Example 19: Overhead in Two Different Projects Let us consider a plant that can make either product A or product B. At normal capacity, the overhead cost is known to be $2.50 per unit. Product A has a direct materials cost of $8 per unit and a direct labor cost of $2 per unit. For simplicity, the prime cost is here taken as the sum of these two costs, i.e., $10 per unit. In Table 9-28, a correct overhead cost of $2.50 per unit at normal capacity is calculated by taking either 31.25 percent of the direct materials cost, 125 percent of the direct labor cost, or 25 percent of the prime cost. All these methods give a total cost of $12.50 per unit and a profit of $1.50 per unit for a selling price of $14 per unit. The alternative, product B, has a direct materials cost of $6 per unit and a direct labor cost of $4 per unit. In Table 9-28 overhead costs of $1.875 per unit, $5 per unit, and $2.50 per unit are calculated by taking 31.25 percent of direct materials cost, 125 percent of direct labor cost, and 25 percent of prime cost respectively. Total costs are $11.875 per unit, $15 per unit, and $12.50 per unit respectively and profits of $2.125 per unit, −$1 per unit, and $1.50 per unit respectively, for a selling price of $14 per unit. An alternative to allocating overheads by using a single method is to classify the various overheads into groups and to use the most appropriate allocation for each group. For example, depreciation would be allocated on the basis of capital cost, while indirect labor might be allocated either on the basis of direct labor cost or on the number of employees. Clearly, this alternative method is more complex, increases the associated accountancy costs, and is prone to misinterpretation and possibly abuse. Inventory Evaluation and Control Inventory Effect on Cash Income and Profit When the annual production rate is equal to the annual sales volume, the revenue from annual sales AS is AS = Rcs

(9-87)

where R is the production rate, units per year, and cs is the sales price per unit. In this case, the annual cash income ACI and the net annual profit ANP before tax are given respectively from Eqs. (9-1), (9-8), and (9-12) by ACI = R(cs − cTVE) − ATFE

(9-145)

ANP = R(cs − cTVE) − ATFE − ABD

(9-146)

where cTVE is the total variable expense per unit of production, ATFE is the total annual fixed expense required to produce and sell a product but excluding any annual provision for plant depreciation, and ABD is the balance-sheet annual depreciation charge. However, in a given accounting period the sales volume may differ from the volume of production. In this case, the inventory of finished product I1 at the beginning of the accounting period will differ from

TABLE 9-28

that at the end, I2, and Eqs. (9-145) and (9-146) need to be written in modified form as ACI = R(cS − cTVE) − ATFE + (I1 − I2)

(9-147)

ANP = R(cS − cTVE) − (ATFE + ABD) + (I1 − I2)

(9-148)

If the annual sales volume exceeds the annual production rate R by an amount ∆R, then Eqs. (9-147) and (9-148) can be written as ACI = R(cS − cTVE) − ATFE + ∆R(cS − cINV)

(9-149)

ANP = R(cS − cTVE) − (ATFE + ABD) + ∆R(cS − cINV)

(9-150)

where cINV is the value per unit of inventory of the finished product. Clearly, the value of cINV affects both the annual cash income and the net annual profit. Since annual cash incomes are the basic data for (NPV) and (DCFRR) methods of estimating profitability, the actual value per unit of inventory is of direct importance for chemical engineers engaged in economic assessments. Let us divide Eq. (9-150) by Eq. (9-87) to give ANP cTVE (ATFE + ABD) ∆R cINV ᎏ = 1 − ᎏ − ᎏᎏ + ᎏ 1 − ᎏ AS cS AS R cS





   c A (A + A ) ∆R ᎏ = (CSR) − ᎏᎏ +  ᎏ 1 − ᎏ  A A R c NP

TFE

S

BD

INV

S

(9-151) (9-152)

S

where (CSR) = 1 − (cTVE /cS) is the contribution to the sales-price ratio. When the ratio of net annual profit to revenue from annual sales ANP /AS is expressed as a percentage it is known as the profit margin. The terms (CSR) and (ATFE + ABD)/AS have a similar order of magnitude in chemical processing. For example, (CSR) values of 0.1 to 0.4 are typical, and these are quite close to a value of 0.2 that is common in general chemical processing for (ATFE + ABD)/AS. Thus, the profit margin is very sensitive to the value cINV per unit of inventory. For example, let us consider that both (CSR) and (ATFE + ABD)/AS are equal to 0.3. In this case, Eq. (9-152) can be written as ANP /AS = (∆R/R)(1 − cINV/cS)

(9-153)

If the sales volume exceeds the annual production rate by 10 percent and the inventory is valued at the sales price, then Eq. (9-153) shows that the profit margin is (ANP /AS)100 = 0 percent. If the inventory is valued at the total variable cost, then the profit margin (ANP /AS)100 = (0.1)(1 − 0.7) (100) = 3 percent. Hence, the value of the inventory is of vital importance. Unfortunately, there is no universally accepted method for valuing inventory. The value of cINV per unit can be taken on any of the following bases: 1. Direct material plus direct labor cost. 2. Direct material plus direct labor plus other direct production expenses. (This is the total variable production cost.) 3. Total variable production cost plus fixed production overhead cost. 4. Total variable cost. (This includes both production and general expenses.) 5. Total cost. (This includes variable and fixed production and general expenses.) Methods 1, 2, and 4 are termed direct costing, variable costing, and marginal costing respectively. Although direct costing is being increas-

Profits of Products with Different Methods of Overhead Allocation Product A—basis of overhead allocation

Direct materials cost Direct labor cost Prime or direct cost Overhead cost Total cost Selling price Profit/(loss)

9-49

Product B—basis of overhead allocation

31.25% of direct materials cost, $/unit

125% of direct labor cost, $/unit

25% of prime cost, $/unit

31.25% of direct materials cost, $/unit

125% of direct labor cost, $/unit

25% of prime cost, $/unit

8.00 2.00 10.00 2.50 12.50 14.00 1.50

8.00 2.00 10.00 2.50 12.50 14.00 1.50

8.00 2.00 10.00 2.50 12.50 14.00 1.50

6.000 4.000 10.000 1.875 11.875 14.000 2.125

6.000 4.000 10.000 5.000 15.000 14.000 (1.000)

6.000 4.000 10.000 2.500 12.500 14.000 1.500

9-50

PROCESS ECONOMICS

ingly used for internal accounting and control purposes, it is not acceptable to tax authorities as a basis for calculating profit. Tax authorities and most accountants favor method 3, which is known as absorption costing. We have already seen that this method has the disadvantage that the fixed-overhead cost per unit is determined for a particular normal production rate. If the production rate exceeds the normal, there is overabsorption of fixed overheads. Conversely, if the production rate falls below the normal, there is underabsorption of fixed overheads. Method 5 is known as full-absorption costing. Most people agree that general expenses incurred in administration, selling, distribution, etc., should not be included in the cost of inventory. In fact, many feel that no costs should be absorbed before they have been incurred. In general, method 2 is favored by engineers and method 3 by most accountants. However, the accountancy convention is to value at either cost or market value, whichever is the lower. In the methods considered, either actual or standard costs can be used. Note that method 3 shows a higher profit than method 2 when sales volume exceeds the production rate and a lower profit when the production rate exceeds sales volume. The total profits (before tax) over the life of a project are independent of the method used to value inventory. Over a project life of n years, Eq. (9-148) can be written as n

A

n

NP

0

n

p

χ <m 1

Hence, the value of the inventory, IFIFO at any given time is IFIFO = CI − CFIFO

0

(9-154)

0

where the last term in Eq. (9-154) becomes  n0 (Im − Im + 1) = I0 − In + 1. Since there will be no material in inventory in the year before the project starts or in the year after it terminates, I0 = In + 1 = 0. Hence, total profits do not depend on individual values for I. However, the annual profit ANP (before tax) does depend on the value of the inventory. Since the tax payable in any individual year is based on ANP, the net annual profit ANNP (after tax) is also dependent on the method chosen for valuing inventory. Frequently, a particular method for valuing inventory is chosen to delay payment of tax as long as it is legally possible to do so. So far, only the inventory of finished product has been considered. There are also inventories of raw materials and work in process, i.e., partially processed materials or intermediate products, to be considered. It is necessary to modify Eqs. (9-147) through (9-154) accordingly to take these inventories into account. Effect of Raw-Materials Prices Raw materials for the chemicalprocess industries are subject to relatively wide variations in price. These effects on profits will now be considered. When the price of raw materials varies from week to week, not all the units in storage will have been purchased at the same price. Let us consider χ units in storage at the start of the inventory period, purchased at a price c1 per unit. Additional quantities χ2, χ3, etc., are purchased at prices c2, c3, etc., per unit respectively until finally χn units are purchased at the latest price of cn per unit at the end of the inventory period. The total value of the inventory CINV at the end of the inventory period (in the absence of any withdrawal) is given by n

CINV =  χ j cj

(9-155)

p



p



CFIFO =  χ j cj + m −  χ j cp + 1 1

n

p

1

1

providing that the value of m satisfies n

m <  χj 0

1

(9-156)

p





IFIFO =  χ j cj −  χ jcj − m −  χ j cp + 1 1

(9-158)

In the LIFO method, the m units taken out of storage are valued at their purchase price, beginning with the latest item purchased. In a similar manner, the value of the material CLIFO taken out of inventory is given by



n



n

CLIFO =  χ j cj + m −  χ j cp − 1 p

p

(9-159)

where p is the smallest integer, such that n

χ

j

≤m

p

Hence, the value of the inventory ILIFO at any given time is ILIFO = CI − CLIFO

(9-160)

In terms of Eqs. (9-155) and (9-159), Eq. (9-160) can be written n

n

1

p





n

ILIFO =  χ j cj −  χ j cj − m −  χ j cp − 1 p

(9-161)

Example 20: Inventory Computation Let us consider 10 successive batches of raw materials, of 1000 units, purchased in a time of rising prices in which c1 = $0.10 per unit, c2 = $0.11 per unit, etc., as listed in Table 9-29. The total cost of the purchases in the inventory is found from Eq. (9-155) to be $1450. Let us calculate the value of the raw-materials inventory after, say, 5500 units have been withdrawn from inventory, first by using FIFO and then by using LIFO. We substitute the appropriate quantities into Eq. (9-158) for FIFO, keeping in mind that p = 5, n = 10, and m = 5500, to get IFIFO = $1450 − $600 − (5500 − 5000)($0.15) IFIFO = $775 In a similar manner, we substitute values into Eq. (9-161) for LIFO (except that p is now 6) to get ILIFO = $1450 − $850 − (5500 − 5000)($0.14) ILIFO = $530 Thus, in a time of rising raw-materials prices, the FIFO method gives a higher value for the remaining inventory than will LIFO. In a time of falling prices, the FIFO method will give a lower value for the remaining inventory than will LIFO.

Average-Cost Basis for Inventory Either a simple average or a weighted average can be used to value inventory cost. Using the simple-average method, the value CSAV of the material taken out of inventory is given by CSAV = m(c1 + cn)/2

1

The value of the inventory I at any given time depends on the values ascribed to the units withdrawn from inventory. There are five methods for valuing inventory: (1) FIFO (first-in–first-out), (2) LIFO (last-in–first-out), (3) average cost, (4) standard cost, and (5) market value. In the FIFO method, the units taken out of storage are valued at their purchase price beginning with the earliest item purchased. If a number of units m are removed from inventory during the period, the total cost of these items on a FIFO basis is given by:

(9-157)

In terms of Eqs. (9-155) and (9-156), Eq. (9-157) can be written as

n

=  R(cS − cTVE) −  (ATFE + ABD) +  (Im − Im + 1) 0

and where p is the largest integer, such that

TABLE 9-29

(9-162)

Costs of Inventory with Rising Prices p

p

n

n

x,

xc,

x,

xc,

units

$

units

$

j

j j

j

j j

p

xj, units

cj , $/unit

xj cj , $

1 2 3 4 5

1,000 1,000 1,000 1,000 1,000

0.10 0.11 0.12 0.13 0.14

100 110 120 130 140

1,000 2,000 3,000 4,000 5,000

100 210 330 460 600

10,000 9,000 8,000 7,000 6,000

1,450 1,350 1,240 1,120 990

6 7 8 9 10

1,000 1,000 1,000 1,000 1,000

0.15 0.16 0.17 0.18 0.19

150 160 170 180 190

6,000 7,000 8,000 9,000 10,000

750 910 1,080 1,260 1,450

5,000 4,000 3,000 2,000 1,000

850 700 540 370 190

1

1

p

p

ACCOUNTING AND COST CONTROL The value of the inventory ISAV at any given time is ISAV = CI − CSAV

(9-163)

In terms of Eqs. (9-155) and (9-162), Eq. (9-163) can be written as n

ISAV =  χ j cj − [m(c1 + cn)/2]

(9-164)

1

We shall consider the value of the inventory after 5500 units have been withdrawn, using the data listed in Table 9-29. On the basis of a simple average, the materials withdrawn are priced at ($0.10 + $0.19)/2 = $0.145 per unit. Since the total cost of the purchases in the raw-materials inventory is found from Eq. (9-155) to be $1450, the value of the inventory after 5500 units have been withdrawn is calculated from Eq. (9-164) to be ISAV = $1450 − (5500)($0.145) = $652.50 For the weighted-average method, the value of the material CWAV taken out of inventory is given by n

n

1

1

CWAV = m  χ j cj  χ j

(9-165)

IWAV = CI − CWAV

(9-166)

In terms of Eqs. (9-153) and (9-164), Eq. (9-166) can be written n

 

n

IWAV =  χ j cj 1 − m χ j 1

1



(9-167)

Let us use Eq. (9-167) to value the inventory, after 5500 units have been withdrawn, by employing the data of Table 9-29. IWAV = $1450 [1 − (5500/10,000)] = $652.50 For this example, the values of ISAV and IWAV are the same because the batches purchased are of equal size and the prices are linearly progressive. This is a combination rarely found in practice. A more realistic example, in which the buyer seeks to purchase at the lowest price, is provided by the data of Table 9-30. The quantities bought vary according to price, but some may have been made at high prices to maintain production. On the basis of Table 9-30, when 5500 items have been removed from inventory, the value of the inventory by using the FIFO, LIFO, simple-average, and weighted average methods respectively is

p

n

xc,

x,

xc,

units

$

units

$

100 165 65 200 30

1,000 2,500 3,000 5,000 5,200

100 265 330 530 560

10,000 9,000 7,500 7,000 5,000

1,175 1,075 910 845 645

140 275 45 20 135

6,200 8,700 9,000 9,100 10,000

700 975 1,020 1,040 1,175

4,800 3,800 1,300 1,000 900

615 475 200 155 135

xj , units

cj , $/unit

xj cj , $

1 2 3 4 5

1,000 1,500 500 2,000 200

0.10 0.11 0.13 0.10 0.15

6 7 8 9 10

1,000 2,500 300 100 900

0.14 0.11 0.15 0.20 0.15

j

1

j j

1

(9-169)

j

p

j j

p

(9-170)

For a given number of orders per year N, the value of the inventory is proportional to the magnitude of the average individual order U. Hence, Eq. (9-170) can be written as AI = βU + (FR/U) + VR

n

x,

p

(9-168)

where α is the proportionality factor which is of the order of 0.25 for many industries. In contrast, some costs can be reduced as a result of larger inventories. For example, larger discounts can be obtained on bulk purchases and deliveries. In addition, larger inventories reduce the risk of losing sales and goodwill through interruptions to production and consequently running out of stocks of finished goods. The cost of placing an order for materials is partly fixed and partly variable. The annual cost of ordering AIO is given by

AI = αI + FN + VR

Costs of Inventory with Fluctuating Prices p

AIW = αI

where F is the fixed cost per order, N is the number of orders per year, V is the variable cost of ordering per unit of production, and R is the annual production rate. Although the administrative cost of placing an order will be more or less fixed, the shipping costs are proportional to the size of the order and, hence, for the year are proportional to the annual production rate. The total annual cost of inventory AI is the sum of Eqs. (9-168) and (9-169):

Pros and Cons of Inventory Valuation In the standard-price method of inventory valuation, all materials are taken out at the same TABLE 9-30

price. In addition to simplicity, the method has the advantage that the efficiency of raw-materials purchase is constantly checked. In both the average-cost and the standard-cost methods of valuing inventory, materials are not charged out at actual cost. Thus, the amount of profit or loss for the period may be varied by the method chosen to value the inventory. For this reason, accountants usually insist that the method of inventory valuation be consistent from period to period. This causes inertia but does not prevent a change of method when it can be justified. In such cases, it is usual to inform stockholders of the change because the influence on declared profits can be large. Unfortunately, there is no right or wrong way to value inventory, although certain methods are not allowed in certain countries for taxassessment purposes. For example, LIFO is not allowed in the United Kingdom. As a general rule, the method used should be the one that gives the lowest tax liability. However, it is generally accepted that consistency is also a virtue in inventory valuation. It is important to realize that the method used to value inventory for cost accounting purposes is not necessarily the one used to draw up the balance sheet and financial accounts. In this case, inventory is valued either at the cost given by another method or at the market value, whichever is lower. Inventory Control The optimum size of inventory depends on the type of industry and on the skills available to the individual company. Inventories are high in the tobacco industry and low in perishable-foods businesses. The larger the inventory, the larger the warehousing and associated costs. These costs include insurance, taxes, depreciation, handling and security charges, etc., and can be taken as roughly proportional to the value of the inventory I. The annual cost AIW of maintaining an inventory of value I is given by

AIO = FN + VR

IFIFO = $1175 − $560 − (5500 − 5200)($0.10) IFIFO = $573.00 ILIFO = $1175 − $645 − (5500 − 5000)($0.10) ILIFO = $480.00 ISAV = $1175 − 5500 [($0.10 + $0.15)/2] ISAV = $487.50 IWAV = $1175[1 − (5000/10,000)] IWAV = $528.75

9-51

(9-171)

where β is a proportionality factor and the number of orders per year N has been written as R/U. By differentiating Eq. (9-171) with respect to U, the optimum size of order can be estimated. The differential is: dAI /dU = β − (FR/U 2)

(9-172)

Setting the right-hand side of Eq. (9-172) equal to zero yields


R
/β
U = F

(9-173)

where U is now the optimum size of order. A number of models have been developed to enable managers to handle inventories in the most profitable manner. These models can be applied to other elements of working capital, such as cash.

9-52

PROCESS ECONOMICS

Working Capital The amount and disposition of working capital and the efficiency of its use determine the immediate prospects for future growth in a company. The bulk of managerial effort in a company is directly or indirectly concerned with the manipulation of working capital. Insufficient or misused working capital is the commonest cause of business failure. Engineers concerned with cost estimations tend to make estimates of the fixed-capital cost of a project, leaving considerations of working capital to the accountants. Although the estimation of fixed-capital cost is more straightforward from an engineering point of view, the estimation of working capital is of vital importance both for an individual project and for the company as a whole. Working capital can range from about 10 percent to almost 100 percent of the invested capital, depending on the industry, and is an important factor in the profitability index of a business. For this reason, it is best to compare the performance of an individual company with that of others that are as similar as possible. Gross working capital is normally defined as total current assets, while net working capital is current assets minus current liabilities. Current assets normally amount to more than half of the total assets of a company. When accountants wish to emphasize working capital in a company, they present the balance sheet in the vertical form, as shown in Table 9-31. In this table, the stockholders’ equity of $91,650 funds the sum of $38,650 (net working capital) and the sum of $53,000 (fixed assets less the long-term loan). The flow of working capital is diagrammatically illustrated in Fig. 9-39. The necessary working capital varies with sales volume or production rate. For example, sales and hence accounts receivable will double for a doubling in sales volume. In addition, an increase in sales volume normally requires increased inventories of raw materials, work in progress, and finished goods, all of which tie up capital. An increase in sales volume may lead to a relative shortage of working capital. In turn, this may mean that accounts payable cannot be paid in time and that valuable cash discounts may be lost or interest and penalty charges incurred. Creditors may take legal action to obtain payments and thereby put an additional strain on the current assets of the company to provide legal fees. Often, such action leads other creditors to take similar steps, which may lead a fundamentally sound company into bankruptcy. A shortage of cash may prevent a company from taking advantage of large discounts available for bulk purchase of raw materials. The importance of the availability of adequate cash or near cash can be seen by considering an account payable within 28 days, with a 2 percent discount allowed if paid within 7 days. If cash is not available to pay the account within 7 days, this is then equivalent to paying 2 percent interest on the money for the remaining 21-day period, or an annual compound-interest rate of more than 41 percent. Adequate cash and a history of prompt payment of accounts strengthen the credit standing of a company and make it easier to obtain bank loans, etc. A company needs additional cash as a contingency against fires, floods, strikes, etc., as well as for additional adver-

FIG. 9-39

TABLE 9-31

Balance Sheet for BCD Company, Dec. 31, 1991

Current assets Cash Accounts receivable Inventories Raw materials Work in process Finished product Gross working capital Current liabilities Accounts payable Notes payable Bank loans Accruals payable

$14,575 35,575 6,000 3,750 10,000 $69,900 25,000 3,000 3,000 250 31,250 $38,650

Net working capital Fixed assets Long-term loan

75,000 22,000 $53,000

Stockholders’ equity

$91,650

tising required to counteract the activities of competitors. This additional money is normally held as interest-bearing investments that can be turned into cash on short notice. Working-Capital Ratios Financial analysts make extensive use of ratios in assessing the economic health of a company. For evaluating the ability of a company to successfully maintain and develop its immediate business activities, analysts apply a current ratio and a quick (or acid-test) ratio, as given by current assets Current ratio = ᎏᎏ (9-142) current liabilities liquid assets Quick ratio = ᎏᎏ (9-143) current liabilities Liquid assets are those that can be realized almost immediately, such as cash, accounts receivable, and marketable securities. Although inventories are current assets, they must not be regarded as liquid assets because they cannot usually be converted into cash without winding up the business. Although a high current ratio is desirable, this may be achieved by having unnecessarily high inventories that bring no profit except when commodity prices are rising rapidly. The quick ratio is less misleading in this respect. Good management practice will hold inventories at the lowest possible levels consistent with customer satisfaction and efficient plant operation. Excessive inventories are unproductive and are an investment having little or no rate of return. Excessive inventories should be maintained only when supplies are erratic or rising in price. Management should normally aim for a high inventory-turnover ratio, as given by:

Flow of working capital showing relationship between current liabilities and current assets.

ACCOUNTING AND COST CONTROL revenue from annual sales Inventory-turnover ratio = ᎏᎏᎏ average value of inventory

(9-174)

Similarly, good management practice is to hold accounts receivable at a low level and to have a high accounts-receivable-turnover ratio, as given by Accounts-receivable-turnover ratio revenue from annual sales = ᎏᎏᎏᎏ (9-175) average value of accounts receivable Alternatively, it is good practice to have a low average collection period, as given by Average collection period average value of accounts receivable = ᎏᎏᎏᎏ (9-144) revenue from sales per day Equation (9-144) gives the number of days during which sales are tied up in receivables. An accounts-receivable-turnover ratio of 12 is considered fairly good for a manufacturing company. This implies an average collection period of about 1 month. The price obtained for the goods should include an allowance for interest (otherwise obtainable) on the money tied up for such a period. Since ratios, like balance sheets, refer to a particular point in time, they have a limited use unless they are compared with previous values. A study of ratio trends indicates whether or not a company is approaching a working-capital or a liquidity crisis and may enable management to compare the performance of the company with that of competitors. Funds Statement A typical funds statement is shown in Table 9-32. It displays the change in net working capital and can be obtained from a statement of changes in working capital, such as the one shown in Table 9-33. A funds statement shows where the cash came from and how it was used. Changes in working capital ∆CWC in an annual accounting period can be represented by ∆CWC = AS − ATE −  ∆CFC −  ∆CL +  ∆CFIN

(9-176)

where AS is the revenue from annual sales of a product, ATE is the total cost or expense required to produce and sell the product but excluding any annual provision for plant depreciation,  ∆CFC is the sum of the changes in depreciable fixed assets,  ∆CL is the sum of the changes in nondepreciable fixed assets, and  ∆CFIN is the sum of the

TABLE 9-32

Funds Statement for Year Ending Dec. 31, 1991

Sources of funds Cash income from operations New finance Sale of plant equipment Total sources of funds Applications of funds Purchase of land Cash dividend on stock Total applications of funds Increase in net working capital

TABLE 9-33

$15,000 10,000 2,000 $27,000 $3,000 9,000 $12,000 $15,000

Statement of Changes in Working Capital

Cash Accounts receivable Inventories Total current assets Current liabilities Net working capital Current ratio Quick ratio

Jan. 1, 1991

Jan. 1, 1992

Change

$ 80,000 20,000 30,000 $130,000 ($ 60,000) $ 70,000 2.17 1.67

$ 94,000 35,000 25,000 $154,000 ($ 69,000) $ 85,000 2.23 1.87

+$14,000 +15,000 −5,000 +$24,000 (+$ 9,000) +$15,000

9-53

changes in financial resources such as loans, bonds, preferred stock, common stock, etc. Equation (9-176) can also be written as ∆CWC = ACI −  ∆CFC −  ∆CL +  ∆CFIN

(9-177)

where ACI is the annual cash income, which is the main source of funds for most companies. In this case, the annual cash income excludes all noncash expenses such as the balance-sheet annual depreciation charge ABD, which is purely a book transaction. A positive value of any term in Eq. (9-177) implies an increase in working capital, and a negative value a decrease. For example, the sale of fixed assets such as plant, buildings, land, etc., is a source of cash, and the purchase of fixed assets uses up cash. Similarly, an increase in financial resources in the form of loans and stock and bond issues is a source of cash, and a decrease in financial resources in the form of repayment of loans, retirement of stocks and bonds, and the payment of cash dividends uses up cash. (Note that a stock dividend as opposed to a cash dividend does not use up cash.) The relation between (1) net annual cash flow ACF after tax for individual projects, (2) the annual amount of tax AIT, and (3) the annual expenditure of capital A⌼C is ACF = ACI − AIT − A⌼C

(9-178)

Equation (9-178) for a single project is really analogous to Eq. (9-177) for a company. An income statement or profit-and-loss account gives the net annual profit A NP before tax. In order to assess the annual cash income A CI as a source of funds from the value of the net annual profit A NP given in the income statement, it is necessary to add back all noncash expenses such as the balance-sheet annual depreciation charge A BD. This practice sometimes erroneously suggests that depreciation is a source of funds, whereas cash income is the only source of funds. Although ABD does not affect working capital in any way, the annual depreciation charge AD does affect the annual amount of tax AIT given by A IT = (A CI − AD − AA)t

(9-3)

where t is the fractional tax rate and AA is the annual amount of any other allowances. Thus, the net annual cash income is affected by depreciation allowances, as follows: A NCI = A CI − A IT

(9-2)

In this sense, depreciation makes working capital available by reducing the cash outflow for taxes. When a fixed asset is sold at a price that differs from its book value, an accounting gain or loss is recorded. This gain or loss does not affect working capital, which has simply been increased by the amount of cash received from the sale. For example, if an item of plant is sold for $40,000 (whatever its book value), the increase in working capital resulting from the sale is $40,000. If the book value of the plant had been $50,000, the accounting loss of $10,000 on the sale would be included as an expense when calculating the cash income. This $10,000 must be added back to the cash income in order to get the cash income excluding noncash expenses as required for ACI. Conversely, any accounting gain on the sale of a fixed asset must be subtracted if it has been included in the cash income. Book values of fixed assets are determined by the balance-sheet annual depreciation charges ABD, which do not affect working capital. Although the accounting gain or loss on the sale of a fixed asset is based on its book value, working capital is not affected by depreciation assessments. Transactions that change the character of the net working capital but do not affect its value occur in a company. For example, a cash payment of $10,000 for accounts payable reduces both the current asset of cash by $10,000 and the current liability of accounts payable by $10,000, leaving the net working capital unchanged. However, this transaction affects both the current and the quick ratios. After the balance sheet and the income statement or profit-and-loss account, the funds statement is generally regarded as the most important financial document. However, many financial managers regard a statement showing changes in cash as being of equal importance. As with balance sheets and income statements, there is no rigid format for funds statements. These vary in the amount of detail given and

9-54

PROCESS ECONOMICS

TABLE 9-34 Statement of Changes in Working Capital and Sources and Applications of Funds for BCD Company Balance sheets Cash Accounts receivable Inventories Total current assets Accounts payable Notes payable Bank loans Accruals payable Total current liabilities Net working capital Fixed assets Net assets Long-term loan Stockholders’ equity

Change in funds

Dec. 31, 1991

Dec. 31, 1992

$ 14,575 35,575 19,750 $ 69,900 $ 25,000 3,000 3,000 250 $ 31,250 $ 38,650 75,000 $113,650 $ 22,000 $ 91,650

$ 13,000 36,000 20,750 $ 69,750 $ 27,000 2,500 2,000 300 $ 31,800 $ 37,950 85,000 $122,950 $ 22,000 $100,950

Applications Sources $ 1,575 $

425 1,000 2,000 500 1,000 50

10,000

$12,925

9,300 $12,925

also in their layout. Funds statements help management in planning for the future, for example, in timing and making financial provision for future expenditures. A typical management accountant’s statement for changes in working capital and sources and applications of funds is shown in Table 9-34. This is based on the following relation: an increase in application of funds equals an increase in sources of funds. The relation can also be expressed as follows: an increase in assets plus a decrease in liabilities equals an increase in liabilities plus a decrease in assets. In Table 9-34, the cash income for the year has been absorbed in the increase in stockholders’ equity, which consists of issued stock plus retained profits or income. General Considerations Many people regard the management of working capital as essentially a cash-flow problem. Certainly, expansion involves increased investment in fixed assets and in the various items that comprise the current assets, all of which require cash. If this leads to a shortage of cash so that a company cannot pay its liabilities, then a situation called overtrading results, and the company may ultimately be forced into liquidation. Therefore, it is essential that a company have access to readily available cash or sources of short-term financing, preferably without having to specify a particular asset as collateral, since such collateral can only be pledged to an agreed realizable value that is much less than its true value. Accounts payable, also called trade credit, are the major source of short-term financing. Accounts payable normally amount to about 40 percent of the current liabilities of a manufacturing company. Such short-term financing is relatively expensive when available discounts are lost. The second most important source of short-term financing is notes payable from commercial banks. Banks normally require a borrower to maintain a compensating balance. For example, if a company requires a loan of $100,000, it must borrow more than this, say, $120,000 (on which it pays interest), in order to maintain a minimum checking-account balance of $20,000. Commercial banks also provide a wide variety of other services that can be of great help to companies in temporary financial difficulties. The third most important source of short-term financing is the commercial paper, or promissory notes, of large companies. This is the

cheapest form of finance. However, the amount of money available is a function of the excess liquidity of the large companies at a given time, and the money may have to be repaid at relatively short notice. Sometimes, fixed assets are purchased via short-term loans, which can lead to liquidity problems. For the most part, fixed assets should be financed from long-term or permanent capital such as stocks or bonds. The proven ability of management to handle working capital efficiently will put a company in a better position to obtain such longterm capital when required, because the confidence of bankers and stockholders will have been obtained. Budgets and Cost Control R. J. Bull (Accounting in Business, 2d ed., Butterworth, London, 1972, p. 163) defined a budget as a comprehensive and coordinated plan, expressed in monetary terms, directing and controlling the resources and trading activities of an enterprise for some specified period in the future. A budget is not a forecast. A forecast is an estimate of the future which may or may not be attained. A budget is an overall objective based on a forecast. In addition, a budget defines the detailed objectives to be achieved by various levels of management in an organization. Since the achievement of these objectives requires the complete cooperation of management, the targets set must be realistic in terms of resources and past performance. A comparison of actual with budgeted results can be used as the basis for control at the company, departmental, plant, or project level. In addition, a continuing record of performance should be maintained to provide the data for the preparation of further budgets. Since company accounts are normally published annually, 1 year is commonly taken as a budget period. However, budget periods can vary widely depending on the nature of the operation. For example, a sales budget may be for a period of, say, 3 or 6 months, while the budget period for the installation of, say, a nuclear power station would extend over many years. The basic objectives of budgets are planning and control. The first step is to determine the limiting factor. For example, budgeted sales cannot exceed the maximum productive capacity of the available plant. Since all the activities in the plan are interrelated, the extent of the plan is determined by the limiting factor. After the plan is put into operation, actual progress is monitored against established standards. These data may subsequently lead to the plan’s being modified in order to achieve the objectives more effectively. R. Pilcher (Appraisal and Control of Project Costs, McGraw-Hill, Maidenhead, England, 1973, p. 233) stated the main purposes of a cost control system to be: 1. To provide immediate warning of uneconomic operations in both the long and the short terms 2. To provide the relevant feedback, carefully qualified in detail by all the conditions under which the work has been carried out, to the estimator who is responsible for establishing the standards in the past and in the future 3. To provide data to assist in the valuation of those variations that will arise during the course of the work 4. To promote cost consciousness 5. To summarize progress Budgeted income statements are identical in form to actual income statements. However, the budgeted numbers are objectives rather than achievements. Budgetary models based on mathematical equations are increasingly being used. These may be used to determine rapidly the effect of changes in variables. Variance analysis is discussed in the treatment of manufacturing-cost estimation.

MANUFACTURING-COST ESTIMATION The annual manufacturing cost or expense AME can be written as the sum of the direct manufacturing or prime cost ADME and the indirect manufacturing or overhead cost AIME: AME = ADME + AIME

(9-179)

The determination of direct or prime costs is more straightforward than the determination of indirect or overhead costs. When more than one product is involved, the question arises as to the correct distribution of overhead costs between the various products. In addition to fixed and variable costs there are mixed or semivari-

MANUFACTURING-COST ESTIMATION able costs. These have a fixed and a variable element. The variable element can vary with production linearly, stepwise, or in a curvilinear manner. However, it is a convenient simplification to divide all manufacturing costs into fixed and linearly variable costs. General Considerations Manufacturing costs are best considered in the context of the manufacturing, trading, and profit-and-loss accounts. Typical examples of these are shown in Tables 9-35, 9-36, and 9-37, respectively. These are based on the conventional accountancy period of 1 year. The gross annual profit A GP in Table 9-36 is dependent on the balance-sheet annual depreciation charge A BD, which is not necessarily the same as the depreciation allowance used for tax purposes. Since A BD is arbitrarily chosen, it can be used to make the gross annual profit A GP high or low according to the company policy. The gross annual profit A GP is also dependent on the method of valuing the inventory. For example, raw materials may have been purchased at the beginning of the accounting year at, say, 9 cents per kilogram. The purchase price may have risen to, say, 12 cents per kilogram at the end of the accounting year. If valuation of the inventory is made at the higher purchase price, the production cost is lower and the gross profit higher than if the valuation is made at the lower purchase price. Although in this case the profit looks better, a higher tax is payable. TABLE 9-35

$

Less inventory of raw materials Dec. 31, 1991 Cost of raw materials consumed Direct wages Direct utilities Other direct expenses Prime cost or direct manufacturing expense, ADME Payroll overhead General plant overhead Other indirect expenses Indirect manufacturing expense, AIME Depreciation, ABD

35,000 870,000 25,000 $ 930,000 51,000 $ 879,000 122,000 22,000 104,000 $1,127,000 28,000 52,000 52,000 $ 132,000 68,000 $ 200,000

Add work in progress, Jan. 1, 1991 Less work in progress, Dec. 31, 1991 Production cost of products

TABLE 9-36

Less inventory of finished products, Dec. 31, 1991 Gross profit Sales

TABLE 9-37

$1,127,000

Net profit before taxes Gross profit

$ 74,000 124,000 40,000 10,000 60,000 $308,000

AS = RcS

(9-180)

(9-181)

AVGE = RcVGE

(9-182)

AVME = RcVME

(9-183)

In practice, annual direct variable costs such as raw materials, utilities, etc., are not always proportional to the production rate. Substituting Eqs. (9-182) and (9-183) into the first term on the right of Eq. (9-180) yields AS − (AVGE + AVME) = R[cS − (cVGE + cVME)]

200,000 $1,327,000 30,000 $1,357,000 35,000 $1,322,000

$ 200,000 1,322,000 $1,522,000 190,000 $1,332,000 668,000 $2,000,000

Profit-and-Loss Account

Administration Sales and shipping Advertising and marketing Technical service Research and development

ACI = [AS − (AVGE + AVME)] − (AFGE + AFME)

where AVGE and AVME are the annual variable general and variable manufacturing expenses respectively and AFGE and AFME are the annual fixed general and fixed manufacturing expenses respectively. Revenue from annual sales is

Likewise, for AVME,

Trading Account

Inventory of finished products, Jan. 1, 1991 Add production cost of products

The profit is also dependent on the method of valuing the work in progress. We shall consider the manufacture of 100,000 kg of product with a prime or direct manufacturing cost of 10 cents per kilogram and an additional indirect manufacturing expense of 5 cents per kilogram. We assume that 90,000 kg of the product is sold and 10,000 kg is stored in inventory. The value of the inventory is $1000 on the basis of prime or direct cost and $1500 when the indirect manufacturing expense is included. It is still a controversial question as to whether manufacturing overheads should be absorbed in the cost. The latter is known as absorption costing and is the traditional accounting method. Direct costing is being increasingly used and is particularly favored by engineers. The annual cash income ACI before tax, in terms of the revenue from annual sales AS of a product and the components of the total annual cost or expense required to produce and sell the product (excluding any allowance for plant depreciation), is expressed by

where R is the production rate and cS the sales price per unit of production. Similarly, AVGE can be taken as proportional to the annual production rate R and the variable general expense per unit of production cVGE:

Manufacturing Account

Inventory of raw materials, Jan. 1, 1991 Add purchases Add carriage inward

9-55

$308,000 360,000 $668,000

(9-184)

In Eq. (9-184), the term [cS − (cVGE + cVME)] is the contribution to cash income. For simultaneous production of more than one product, Eq. (9-184) can be written as AS − (AVGE + AVME) = R1 {(cS)1 − [(cVGE)1 + (cVME)1]} + R2 {(cS)2 − [(cVGE)2 + (cVME)2]} + . . .

(9-185)

where the subscripts 1, 2, etc., refer to the various coproducts. The contribution to cash income made by a particular product depends on the method of accounting. Widely different values for the contribution can be calculated by using different methods of assigning manufacturing expenses. The cost of each item in a cost estimate should be presented in such a way that the estimate can be modified and updated at any time in the future when revised data become available. Published data and shortcut estimating methods can be used to calculate the approximate manufacturing cost of a new product. However, most companies have extensive data on various items of cost such as overheads, property taxes, etc. These data should be used whenever possible to give the estimate that is most valid for a particular company. For a new product, the ratio of manufacturing expense to sales price cME/cS should be compared with the ratio of total manufacturing cost or expense to sales revenue for the company as a whole. If the ratio cME/cS is less than or equal to the ratio for the company, then the proposed sales price appears to be reasonable and the product is probably commercially viable. This comparison is, of course, used only as an approximate guide in preliminary assessments. One Main Product Plus By-Products We shall let one unit of raw material yield χ1, χ2, etc., weights of products 1, 2, etc., respectively. The variable general expense per unit of raw material will be

9-56

PROCESS ECONOMICS

cVGE and the variable manufacturing expense per unit of raw material will be cVME. The unit of raw material may be either a single material or a mixture of several components. I. Leibson and G. A. Trischman [Chem. Eng., 78, 69–74 (May 31, 1971)] showed the effects on manufacturing expenses of alternate feedstocks having different compositions and costs for producing the same products. If product 1 is the main product, then this carries all the variable expenses, so that (cVGE)1 = cVGE / χ1

(9-186)

(cVME)1 = cVME / χ1

(9-187)

By Value We shall let one unit of raw material yield χ1 and χ2 weights of products 1 and 2 respectively, with values of χ1(cS)1 and χ2(cS)2 respectively; (cS)1 and (cS)2 are the sales prices of products 1 and 2 per unit of production. We shall let the variable general expense per unit of raw material be cVGE and the variable manufacturing expense per unit of raw material be cVME. If the variable expenses are shared by value then χ1(cS)1cVGE (9-198) (cVGE)1 = ᎏᎏ [χ1(cS)1 + χ2(cS)2]χ1

where (cVGE)1 and (cVME)1 are the variable general and variable manufacturing expenses respectively per unit of product 1. For by-product 2, etc., (cVGE)2 = 0 and (cVME)2 = 0, etc. For this case, Eq. (9-184) becomes AS − (AVGE + AVME) = R1 [(cS)1 − (cVGE + cVME)/χ1] + R2(cS)2 + . . .

(9-188)

If R is the annual rate at which the raw material is consumed, then R1 = χ1 R

(9-189)

R2 = χ2 R

(9-190)

In terms of one unit of raw material, Eq. (9-185) can be combined with Eq. (9-189) and (9-190) and written as [AS − (AVGE + AVME)]/R = χ1(cS)1 − (cVGE + cVME) + χ2(cS)2 + . . .

(9-191)

Equation (9-191) gives the contribution of products 1, 2, etc., per unit of raw material. In this particular case, the contribution of the main product per unit weight of raw material is χ1(cS)1 − (cVGE + cVME). The contribution of product 2 is χ2(cS)2, i.e., its selling price per unit weight of raw material.

χ1(cS)1cVME (cVME)1 = ᎏᎏ [χ1(cS)1 + χ2(cS)2]χ1

(9-199)

χ2(cS)2cVGE (cVGE)2 = ᎏᎏ [χ1(cS)1 + χ2(cS)2]χ2

(9-200)

χ2(cS)2cVME (9-201) (cVME)2 = ᎏᎏ [χ1(cS)1 + χ2(cS)2]χ2 where (cVGE)1 and (cVME)1 are the variable general and variable manufacturing expenses per unit of product 1 respectively and (cVGE)2 and (cVME)2 are the comparable expenses for product 2. For this case, Eq. (9-185) becomes (cS)1(cVGE + cVME) AS − (AVGE + AVME) = R1 (cS)1 − ᎏᎏ [χ1(cS)1 + χ2(cS)2]



(cS)2(cVGE + cVME) + R2 (cS)2 − ᎏᎏ (9-202) [χ1(cS)1 + χ2(cS)2] This analysis can be extended for any number of coproducts. In terms of one unit of raw material, Eq. (9-202) can be combined with Eqs. (9-189) and (9-190) and written as AS − (AVGE + AVME) (cS)1(cVGE + cVME) ᎏᎏ = χ1 (cS)1 − ᎏᎏ R [χ1(cS)1 + χ2(cS)2]





Two Main Products By Weight We shall let one unit of raw material yield χ1 and χ2 weights of products 1 and 2 respectively. The variable general expense per unit of raw material will be cVGE and the variable manufacturing expense per unit of raw material cVME. In practice, it is rare for χ1 + χ2 to be exactly unity. If the variable expenses are shared by weight, then

(cS)2(cVGE + cVME) + χ2 (cS)2 − ᎏᎏ (9-203) [χ1(cS)1 + χ2(cS)2] Equation (9-203) gives the contribution of products 1 and 2 per unit weight of raw material.

(cVGE)1 = χ1 cVGE / [(χ1 + χ2)χ1]

0.32 kg of product 1 and χ2 = 0.64 kg of product 2. The balance of the raw material goes to waste. Product 1 sells at (cS)1 = 40 cents per kilogram, and product 2 sells at (cS)2 = 12 cents per kilogram. The variable general and variable manufacturing expenses, including raw materials, cVGE + cVME, total 10 cents per kilogram. Let us calculate (a) the total contribution to cash income per kilogram of raw material and (b) the individual contribution of each product to cash income per kilogram of raw material for the following:

(9-192)

(cVME)1 = χ1 cVME / [(χ1 + χ2)χ1]

(9-193)

(cVGE)2 = χ2 cVGE / [(χ1 + χ2)χ2]

(9-194)

(cVME)2 = χ2 cVME / [(χ1 + χ2)χ2]

(9-195)

where (cVGE)1 and (cVME)1 are the variable general and variable manufacturing expenses per unit of product 1 respectively and (cVGE)2 and (cVME)2 are the comparable expenses for product 2. For this example, Eq. (9-185) becomes (cVGE + cVME) AS − (AVGE + AVME) = R1 (cS)1 − ᎏᎏ (χ1 + χ2)





(cVGE + cVME) + R2 (cS)2 − ᎏᎏ (9-196) (χ1 + χ2) In Eqs. (9-192) through (9-196) the sum of χ1 and χ2 may be equal to or less than 1. The above analysis can be extended for any number of coproducts. In terms of one unit of raw material, Eq. (9-196) can be combined with Eqs. (9-189) and (9-190) and written as AS − (AVGE + AVME) (cVGE + cVME) ᎏᎏ = χ1 (cS)1 − ᎏᎏ R (χ1 + χ2)









(cVGE + cVME) + χ2 (cS)2 − ᎏᎏ (9-197) (χ1 + χ2) Equation (9-197) gives the contribution of products 1 and 2 per unit weight of raw material.







Example 21: Calculation of Contributions to Income for Multiple Products One kilogram of raw material is used to manufacture χ1 =

Case

Condition

1 2 3 4

Product 1 as the main product charged for all the variable expenses Product 2 as the main product charged for all the variable expenses Products 1 and 2 sharing the variable expenses on the basis of weight Products 1 and 2 sharing the variable expenses on the basis of value

Case 1. The total contribution per kilogram of raw material is obtained by substituting the appropriate values into Eq. (9-191). For these conditions, χ1(cS)1 + χ2(cS)2 − (cVGE + cVME) = 0.32(40 cents/kg) + 0.64 (12 cents/kg) − 10 cents/kg = 12.8 cents/kg + 7.68 cents/kg − 10 cents/kg = 10.48 cents/kg The individual contribution of product 1 as the main product is 12.8 cents − 10 cents = 2.8 cents per kilogram of raw material. The individual contribution of product 2 as the by-product is 7.68 cents per kilogram of raw material. Case 2. The total contribution when product 2 is the main product is also obtained from Eq. (9-191) and as in Case 1 is found to be 10.48 cents per kilogram.

MANUFACTURING-COST ESTIMATION The individual contribution of product 2 as the main product is 7.68 cents − 10 cents = −2.32 cents per kilogram of raw material. The individual contribution of product 1 as the by-product is 12.8 cents per kilogram of raw material. Case 3. When products 1 and 2 share the variable expenses on the basis of weight, the total contribution per kilogram of raw material is found by substituting the unit costs into Eq. (9-197). Values for each term are χ1(cS)1 = 12.8 cents/kg

(same as Case 1)

χ2(cS)2 = 7.68 cents/kg

(same as Case 1)

[χ1 /(χ1 + χ2)](cVGE + cVME) = (0.32/0.96)(10 cents/kg) = 3.34 cents/kg [χ2 /(χ1 + χ2)](cVGE + cVME) = (0.64/0.96)(10 cents/kg) = 6.66 cents/kg Therefore, the total contribution becomes 12.8 + 7.68 − 3.34 − 6.66 = 10.48 cents per kilogram. The individual contribution of product 1 is 12.8 cents − 3.34 cents = 9.46 cents per kilogram of raw material. The individual contribution of product 2 is 7.68 cents − 6.66 cents = 1.02 cents per kilogram of raw material. Case 4. When products 1 and 2 share the variable expenses on the basis of value, the total contribution per kilogram of raw material is found by substituting the unit costs into Eq. (9-203). Values of each term are χ1(cs)1 = 12.8 cents/kg

(same as Case 1)

χ2(cs)2 = 7.68 cents/kg

(same as Case 1)

χ1(cs)1(cVGE + cVME) (12.8)(10) ᎏᎏ = ᎏᎏ = 6.24 cents/kg χ1(cs)1 + χ2(cs)2 12.8 + 7.68 χ2(cs)2(cVGE + cVME) (7.68)(10) ᎏᎏ = ᎏᎏ = 3.76 cents/kg χ1(cs)1 + χ2(cs)2 12.8 + 7.68 Therefore, the total contribution becomes 12.8 + 7.68 − 6.24 − 3.76 = 10.48 cents per kilogram. The individual contribution of product 1 is found as 12.8 cents − 6.24 cents = 6.56 cents per kilogram of raw material. The individual contribution of product 2 then becomes 7.68 cents − 3.76 cents = 3.92 cents per kilogram of raw material.

Direct Manufacturing Costs Direct manufacturing costs include raw materials, operating labor, utilities, and some miscellaneous items. A summary of the characteristics of each follows. Raw Materials The cost of raw materials is normally the largest item of expense in the manufacturing cost of a product. The quantities of raw materials consumed can be calculated from material balances. Material costs are conveniently presented in tables that give the following: name of material, form and grade, method of delivery, unit of measure, cost per unit, source of cost, annual consumption, annual cost, fractional consumption per unit of production, and cost per unit of production. Net consumption of materials should be used for catalysts, solvents, filter aids, etc., that may have a recovery value. Current prices of chemicals are published in various trade journals. However, quotations from suppliers should be used whenever possible. It may be possible for a company to negotiate the purchase of a material at a cost per unit that is significantly lower than the current published price. This is particularly true if large quantities are involved. Thus, estimates should be presented for both minimum and maximum costs. Price trends, availability, and quality are other factors that should be considered. A knowledge of price trends is particularly important for a product that a company may not manufacture for several years. The yield in a chemical reaction determines the quantities of materials in the material balance. Assumed yields are used to obtain approximate exploratory estimates. In this case, possible ranges should be given. Firmer estimates require yields based on laboratory or, preferably, pilot-plant work. Operating Labor The cost of operating labor is the second largest item of expense in the manufacturing cost. Labor requirements for a process can be estimated from an intelligent study of the equipment flow sheet, paying careful attention to the various primary process steps such as fractionation, filtration, etc. The hourly wage rate should be that currently paid in the company. Once the number of persons required per shift has been estimated for a particular production rate, the annual labor cost and the labor cost per unit of production can be estimated. H. E. Wessel [Chem. Eng., 59, 209–210 (July 1952)] made a study of the operating-labor requirements in the United States chemical

9-57

industry and presented the data as a plot of labor-hours per ton per processing step versus plant capacity in tons per day. These data can be represented by: log10 Y = 0.783 log10 χ + 1.252 + B

(9-204)

where Y is the operating-labor-hours per ton per processing step; χ is plant capacity, tons per day; and B is a constant having values of 0.132 when multiple units are used to increase capacity or when the process is completely batch, of 0 for the average chemical-processing plant, and of −0.167 for large, highly automated plants or plants concerned with fluid processing. Wessel’s data for the United States chemical industry refer to the short ton equal to 2000 lb, or 907.2 kg. Labor requirements are higher in countries with lower productivities. The approximate cost of supervision for operating labor is equivalent to 10 percent of the labor cost for simple operations and 25 percent for complex operations. Utilities These include steam, cooling water, process water, electricity, fuel, compressed air, and refrigeration. The consumption of utilities can be estimated from the material and energy balances for the process, together with the equipment flow sheet. Let us consider a cooler in the equipment flow sheet. The required rate of heat removal is known from the balances, and the rate of cooling water can be calculated once the inlet and outlet temperatures of the water have been specified. The calculation of the consumption of other utilities is also straightforward. Allowances should be made for wastage. The current cost per unit for each utility is usually well known in a company. Thus, the annual cost for utilities and the utilities cost per unit of production can be estimated. The latter is normally much smaller than the raw-materials and labor costs. However, a great deal more work is involved in calculating the utilities cost than for any other item in the manufacturing cost. Unfortunately, there are no satisfactory shortcut methods for doing this. When the utilities cost is relatively small, it may be possible to make an intelligent guess on the basis of known costs for similar processes in the company. Alternatively, published data for the consumption of utilities per unit of production for various processes may be used. Miscellaneous Direct Costs Estimates for the cost of maintenance and repairs, operating supplies, royalties, and patents are best based on company records for similar processes. A rough average value for the annual cost of maintenance is 6 percent of the capital cost of the plant. This percentage can vary from 2 to 10 percent, depending on the severity of plant operation. Approximately half of the maintenance costs are for materials and half for labor. Royalty and patents costs are in the order of 1 to 5 percent of the sales price of the product. Indirect Manufacturing Costs Estimates for the cost of payroll overhead, control laboratory, general plant overhead, packaging, and storage facilities are best based on company records for similar processes. Payroll overhead includes the cost of pensions, holidays, sick pay, etc., and is normally between 15 and 20 percent of the operating-labor cost. Laboratory work is required for product quality control, and its cost is approximately 10 to 20 percent of the operating-labor cost. Plant overhead includes the cost of medical, safety, recreational, effluent-disposal, and warehousing facilities, etc. In general, the larger the plant, the lower the overhead per unit of production. Plantoverhead costs can vary between 15 and 150 percent of the operatinglabor cost. Packaging costs depend on the physical and chemical nature of the product as well as on its use and value. The cost of packaging is as high as one-third of the selling price for soaps and pharmaceuticals. Rapid Manufacturing-Cost Estimates Fixed manufacturing costs are a function of the fixed-capital investment and are independent of the production rate of the plant. Property taxes or rates depend on location. They may be taken as 2 percent of the fixedcapital cost of the plant in the absence of specific data. The cost of insurance depends on both location and the hazardous nature of the materials handled. This cost is normally of the order of 1 percent of the fixed-capital cost of the plant.

9-58

PROCESS ECONOMICS

The manufacturing cost of a product is the sum of the processing or conversion cost and the cost of raw materials. The processing cost can be roughly broken down into three parts: investment-related cost, labor-related cost, and utility cost. Companies usually include in the charge for overhead the following items: operating supplies, supervision, indirect payroll expenses, plant protection, plant office, general plant overhead, and control laboratory. This overhead charge is frequently taken as an equivalent percentage of the direct labor cost. The percentage is best obtained from company records. Although it can vary over a wide range, a reasonable value is 125 percent. In this case, the labor-related cost would be 2.25 times the direct labor cost. For a 6000-h year and N persons per shift earning cL$ per hour, the annual labor-related cost would be 13,500 (cL N). Let us consider a plant of fixed-capital cost CFC. If the annual property taxes are taken as 0.02 CFC, insurance as 0.01 CFC, and maintenance as 0.06 CFC, the annual investment-related cost would be 0.09 CFC. Annual utilities cost is AU. The annual processing cost Ap can be represented by A p = η1CFC + η2cL N + AU

(9-205)

where the factors η1 and η2 can be obtained from the data available in a particular company and have the dimensions of year −1 and hours per year respectively. Substituting the information previously given into Eq. (9-205) yields the relationship A p = 0.09CFC + 13,500cL N + AU

(9-206)

Equation (9-206) represents very closely the manufacturing costs of a particular company and is typical of the coefficients to be expected. The annual processing cost AP2 for a similar plant of a different size designed for an annual production rate R2 can be approximately calculated from an equation of the form AP2 = η1CFC1(R2/R1)0.7 + η2cLN1(R2/R1)0.25 + AU1(R2/R1)

(9-207)

(F. A. Holland, F. A. Watson, and J. K. Wilkinson, Introduction to Process Economics, 2d ed. Wiley, London and New York, 1983, p. 158). The processing cost per unit of production for a plant with an annual production R2 can be approximately calculated from AP2 100 R R2 0.7 R2 0.25 + AU1 ᎏ2 (9-208) ᎏ = ᎏ η1 CFC ᎏ + η2cL N1 ᎏ R2 R2 R1 R1 R1 where AP2 /R2 is in cents per kilogram. Equation (9-208) can be used to compute data for plots such as Fig. 9-40, which shows the decrease in processing cost per unit of production AP2 /R2 with increasing plant size. Manufacturing Cost as a Basis for Product Pricing Pricing on the basis of cost plus a fair profit has the disadvantage of ignoring demand. The modern approach is to price on the basis of market research. However, the classic cost-plus-fair-profit approach can still give useful complementary information. This can be done by any of the following three methods:



 

 

 

1. Absorption pricing 2. Rate-of-return pricing 3. Marginal pricing The gross annual profit A GP for a product is given by AGP = AS − AME − ABD

(9-6)

where AS is the revenue from annual sales, AME the annual manufacturing cost, and ABD the balance-sheet annual depreciation charge. Equation (9-6) can also be rewritten in the form AS = A GP + AVME + AFME + ABD

(9-209)

where AVME and AFME are the annual variable and fixed manufacturing costs or expenses respectively. Equation (9-209) can also be rewritten as cS = cVME + (AGP /R) + (AFME + ABD)/R

(9-210)

where R is the annual sales volume taken as equal to the annual production rate, cS is the sales price per unit of production, and cVME is the variable manufacturing expense per unit of production. Absorption pricing is based on a normal annual production rate R. The gross profit per unit AGP/R is taken as a fixed percentage χ of the fixed plus variable manufacturing costs given by the equation AGP χ AFME + ABD (9-211) ᎏ = ᎏ cVME + ᎏᎏ R 100 R We combine Eqs. (9-210) and (9-211) to give 100 + χ AFME + ABD (9-212) cS = ᎏ cVME + ᎏᎏ 100 R Equations (9-210), (9-211), and (9-212) are based on a fixed normal annual production rate R. Let us consider a change in annual production rate to R + ∆R. In order to maintain the gross profit per unit as AGP /R, the sales price per unit of production would need to be cS − ∆cS. For this case Eq. (9-210) can be written in the modified form



 











cS − ∆ cS = cVME + (AGP/R) + [(AFME + ABD)/(R + ∆R)]

(9-213)

We subtract Eq. (9-213) from Eq. (9-212) to give ∆R AFME + ABD (9-214) ∆cS = ᎏ ᎏᎏ R R + ∆R Equation (9-214) gives the overpricing ∆ cS per unit of production for an increase in annual production rate ∆ R. Equation (9-214) also gives the underpricing ∆ cS per unit of production for a decrease in annual production rate ∆ R. In the first case the fixed costs or overheads are said to be overabsorbed and in the second case underabsorbed. Absorption pricing is rigid and arbitrary and may result in business being turned away if the fixed sales price cS cannot be obtained even though the business may give a useful contribution to fixed costs. Rate-of-return pricing is a modified form of absorption pricing. It is based on the equation AGP AGP CTC (9-215) ᎏ= ᎏ ᎏ AME AME CTC where CTC is the total capital employed. Equation (9-215) can also be written as





  

Percentage markup on cost = (capital-turnover ratio)(projected rate of return on capital)

FIG. 9-40

Decrease in processing cost against increase in plant size.

The percentage markup on cost is calculated for a known capitalturnover ratio and a desired rate of return on capital. As with absorption pricing, the percentage markup on manufacturing cost per unit of production is calculated for a normal annual production rate. If this production rate is exceeded, the rate of return on capital will be higher than projected because of the decrease in unit cost. Conversely, if the production rate is lower than normal, the rate of return on capital will be lower than projected because of the increase in unit cost. For production rates both higher and lower than the normal production rate, the percentage markup is based on the normal unit cost. Thus the method is strictly valid only for the normal production rate.

MANUFACTURING-COST ESTIMATION With marginal pricing a company chooses its selling prices so as to maximize the total contribution  R(cS − cVME) from its various products. The method is particularly useful for large multiproduct companies with extensive existing facilities since it is the marginal cost that must be considered as the base case when entering into competition with another company. The lowest acceptable price for a product is that which gives the lowest worthwhile contribution to fixed costs. Marginal pricing enables a company to develop a more aggressive pricing policy than when using absorption or rate-of-return pricing. Absorption, rate-of-return, and marginal pricing have been considered here on the basis of manufacturing cost. Total cost, which is the sum of manufacturing and general costs, can also be considered as the basis. In this case the appropriate profit to consider is the net annual profit rather than the gross annual profit. Standard Costs for Budgetary Control For convenience and simplicity, we shall consider the total cost of a manufactured product to be the sum of the material, labor, and overhead costs. Standard costs are those that have been predetermined and budgeted for the manufacture of a given amount of product in a given time. The deviation of the actual cost from the standard cost is called the variance. It is far easier to make comparisons between periods by using variances than by using actual production data. The different variances for material, labor, and overhead costs are listed in Table 9-38. Standard costing is extensively used in budgetary-control systems. Criteria for the establishment of standards range from the maximum possible under ideal conditions to those expected under normal conditions. Past or historical costs are not always the best basis for setting up standards because past performance may have been unnecessarily inefficient. Static and Flexible Budgets Overhead cost can significantly affect the profitability of a project and is the only cost outside the control of the project manager. The project is expected to contribute a definite amount toward the expenses of the company and will be charged this amount even if the production rate is zero. This is the fixed component of the overhead cost and will include directly allocable costs such as depreciation and a proportion of general costs such as office salaries and heating. Other nonproduction costs such as indirect labor may vary linearly with the production rate and represent the variable component of the overhead. Costs that are neither fixed nor variable but occur in disTABLE 9-38

Variances

Revenue from sales RcS − R*c*S =

Sales price R(cS − c*S) +

Sales volume c*S(R − R*)

Profit RcNP − R*c*NP

=

Unit profit R(cNP − c*NP) +

Quantity c*NP(R − R*)

Total cost RcTE − R*c*TE

=

Unit cost R(cTE − c*TE) +

Production rate c*TE(R − R*)

Direct material cost Material price RcRM − R*c*RM = R(cRM − c*RM) +

Material usage c*RM(R − R*)

Direct labor cost θcL = θ*c*L

=

Overhead cost COH − θ*c*NOH

=

Wage rate Labor efficiency θ(cL − c*L) + c*L(θ − θ*) Budgeted cost (COH − θcBOH) +

Volume Efficiency θ(cBOH − c*NOH) + c*NOH(θ − θ*)

cBOH = Flexible budgeted overhead cost, $/h. cL = Actual labor cost, $/h. c*NOH = Standard overhead cost based on normal production rate, $/h. cNP = Actual net profit before tax, $/unit. COH = Actual overhead cost, $/period. cRM = Actual raw-materials cost, $/unit. cS = Actual selling price, $/unit. cTE = Actual total cost, $/unit. R = Actual quantity, units/period. R* = Standard quantity, units/period. θ = Actual time to produce a given quantity, h. θ* = Standard time to produce a given quantity, h. cNP = cS − cTE. NOTE: The asterisk on all items not otherwise defined indicates the standard cost for that item.

TABLE 9-39 Overhead cost, $/month Fixed Variable Semivariable Total

9-59

Flexible Budget for Overhead Costs Production, 1 million lb/month 8

9

10

11

12

40,000 40,000 40,000 120,000

40,000 45,000 40,000 125,000

40,000 50,000 60,000 150,000

40,000 55,000 60,000 155,000

40,000 60,000 90,000 190,000

crete steps at various production levels (such as supervisory labor) are the semivariable component of the overhead cost. It is an easy matter to determine these various components for various production rates and list them as shown in Table 9-39. Two types of overhead budget are currently in use. The static (often referred to as the fixed) budgeted overhead cost is related to the standard budgeted production rate. The flexible budgeted overhead cost is that shown as the total cost in Table 9-39. Values for intermediate production rates are often obtained by interpolation. This is justifiable only when semivariable costs are a negligible part of overhead costs. Flexible budgeting is more widely used than static budgeting despite certain logical difficulties. This is so because production in many cases is seasonal and the use of a static production norm might distort evaluation of performance. Variances are the difference between the actual costs expended and the budgeted costs expected. Variances are unfavorable if positive and favorable if negative. Any variance should be explained and, if necessary, controlled; the largest variance should be considered first. Let us consider the overhead-cost data for Table 9-39 with 10 million kg per month as the standard production rate. The static budgeted overhead is then $150,000 per month, or 1.5 cents per kilogram. We assume that the actual overhead is $186,000 for a month in which 12 million kg was produced. Then, the static budgeted overhead cost would be 12 million(1.5), or $180,000 per month. Therefore, the variance is $186,000 − $180,000 = +$6000, which is unfavorable because $6000 more was spent than was anticipated. From Table 9-39 we find that the flexible budgeted overhead cost for a production rate of 12 million kg per month is $190,000. The corresponding variance is $186,000 minus $190,000, or −$4,000, which is favorable because $4,000 less was spent than was anticipated. Thus, the use of flexible budgeting makes this particular performance look better without changing either the production rate or a single cost of the planned budget. The Standard Hour The standard hour can be defined as the number of units of output expected to be produced in 1 h. It is often used as a measure of output rate by cost accountants. Let us consider a batch processing unit that can produce either 1000 kg of product A in a cycle time of 5 h or 900 kg of product B in a cycle time of 3 h. Thus, for this processing unit a standard hour is 200 kg of product A or 300 kg of product B. In a budget period of, say, 1000 h, it is possible to produce 200,000 kg of product A, or 300,000 kg of product B, or any appropriate combination of the two products. For example, let us assume that production requirements are twice as great for product A as for product B, i.e., a ratio of 600 kg, or 3 standard hours, of product A to 300 kg, or 1 standard hour, of product B. On this basis, for a budget period of 1000 h, 750 standard hours [(e)1000] would be used to produce 750(200) = 150,000 kg of product A, and 250 standard hours [(1/4)1,000] would be used to produce 250(300) = 75,000 kg of product B. Production efficiency Pe can be calculated from Pe = (Ps /Pa )100 (9-216) where Ps is the actual production rate in standard hours and Pa is the actual hours worked. The level of production activity Pᐉ can be calculated from Pᐉ = (Ps /Pb )100 (9-217) where Pb is the budgeted production in standard hours. The deviation from budgeted capacity bc can be calculated from bc = (Pa /Pw)100 (9-218) where Pw is the budgeted number of working hours.

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PROCESS ECONOMICS

TABLE 9-40

Sales, Profits, and Manufacturing Costs

Component

Actual, $/period

Standard, $/period

Variance, $/period

Direct materials cost Direct labor cost Overhead cost Manufacturing cost Revenue from sales Gross profit

325,080 55,001 75,000 455,081 588,240 133,159

350,000 51,600 68,370 469,970 570,000 100,030

−24,920 +3,401 +6,630 −14,889 +18,240 +33,129

Actual Costs versus Standard Costs Let us consider the sales, profits, and manufacturing-cost data in Table 9-40. The gross profit is $33,129 per period better than expected. Clearly, there is less incentive to investigate overall costs when the profit variance is favorable than if the profit were less than expected. However, standard costing enables an objective analysis of the data, whether good or bad, to be made. The individual variances in Table 9-40 show that the increased profit is due to reduced material costs, which affect manufacturing costs to a greater extent than increased labor and overhead costs. The individual variances also show that to an even greater extent the increased profit is due to the increase in sales revenue. Clearly, management will wish to investigate both labor and overhead costs for any inefficiencies and to ascertain the reasons for the improved sales revenue. If necessary, the standard values can be revised. We notice that profit is obtained as the difference between two large cash sums and that variances of some 3 percent in manufacturing costs and sales revenue have resulted in a variance of some 33 percent in gross profit. Table 9-40 is a very simplified presentation. In a full standard costing system, the direct material, direct labor, and overhead variances are broken down into component parts to enable an even closer look at the operation. Standard costing is an invaluable aid to management for controlling a business. Variances: Direct Material Cost Since the variance in direct material cost ∆ (RcRM) is the difference between actual cost and standard cost, ∆ (RcRM) = RcRM − R*c*RM (9-219) where R is the actual quantity and R* is the standard quantity, units per period, cRM is the actual price, and c*RM is the standard price, $ per unit. Equation (9-219) can be written in an expanded form: ∆ (RcRM) = R(cRM − c*RM) + c*RM(R − R*)

(9-220)

where R(cRM − c*RM), known as the direct-material-price variance, is the actual quantity multiplied by the deviation in unit price, and c*RM(R − R*), known as the direct-material-usage variance, is the standard unit price multiplied by the deviation in quantity. By using the data of Table 9-41, let us calculate the direct materials cost and the standard direct materials cost as RcRM = 1,806,000(0.18) = $325,080/period R*c*RM = 1,750,000(0.20) = $350,000/period

of (1,806,000)(0.18 − 0.20) = −$36,120 per period. This variance is favorable. Likewise, we calculate a direct-material-usage variance of (0.20)(1,806,000 − 1,750,000) = $11,200 per period. This variance is unfavorable. In this case, the favorable direct-material-cost variance was achieved because of a lower unit price despite an inefficient material usage, which needs to be investigated. (There is no room for complacency since the lower unit price may well be temporary.) In the case of mixtures of raw materials, the direct-material-usage variance can be further subdivided into (1) a direct-material-mixture variance and (2) a direct-material-yield variance. The former is due to the difference between the actual and standard mixture compositions, and the latter to the difference between the actual and standard yields. Here, the standard yield is the output expected from the standard input of material. The yield variance denotes the extent of loss of material. The direct-material-mixture variance can be illustrated by Example 22. Example 22: Direct-Material-Mixture Variance A standard mixture of 100 units of material contains 70 percent of material A at $0.08 per unit and 30 percent of material B at $0.12 per unit. The standard mixture cost is 70(0.08) + 30(0.12) = $9.20. Now let us consider a mixture of 100 units containing 75 percent of material A and 25 percent of material B. The cost of this mixture at standard prices is 75(0.08) + 25(0.12) = $9.00. The direct-material-mixture variance is $9.00 − $9.20 = −$0.20 and is favorable. The favorable variance has been brought about by using more of the lower-priced material A and less of the higher-priced material B. The direct-material-yield variance is illustrated as follows. Let us assume that the standard mixture (cost $9.20 for 100 units) has a standard loss of 20 percent, making the cost $9.20 for 80 units, or $0.115 per unit of output. Now let us consider the actual loss to be 30 percent, leaving 70 units of output for each 100 units of input. The directmaterial-yield variance is 0.115(80 − 70) = $1.15 and is unfavorable. Variances: Direct Labor Cost Since the variance in direct labor cost ∆ (θcL) is the difference between actual cost and standard cost, ∆ (θcL ) = θcL − θ*c*L

(9-221)

where cL is the actual pay or wage rate, $ per hour; c*L is the standard pay or wage rate, $ per hour; θ is the actual time taken to produce a given quantity of product in a given period, hours; and θ* is the standard time taken to produce a given quantity of product in a given period, hours. Equation (9-221) can also be written in expanded form: ∆ (θcL ) = θ(cL − c*L ) + c*L(θ − θ*)

(9-222)

where θ(cL − c*L), known as the direct pay, or wage-rate, variance, is the actual time taken to produce a given output multiplied by the deviation in wage rate, and c*L(θ − θ*), known as the direct-laborefficiency variance, is the standard wage rate multiplied by the deviation in time taken to produce a given output. By using the data of Table 9-41, we calculate that 1 standard hour corresponds to (1,750,000/6250) = 280 units/standard hour Standard time to actual production is

From Eq. (9-219) we calculate the direct-material-cost variance as −$24,920 per period. This variance is favorable. However, by using the relations of Eq. (9-220) we calculate a direct-material-price variance

θ* = (1,806,000/280) = 6450 standard hours/period Direct labor cost is θcL = 6509(8.45) = $55,001/period

TABLE 9-41

Cost Data for Problems

Standard direct labor cost is

Factor

Actual value

Standard value

Raw-materials cost, $/unit Direct labor cost, $/unit Production rate, units/period Production time h/period Fixed and semivariable overhead cost, $/period Variable overhead cost, $/h Overhead cost, $/period

cRM = 0.18 cL = 8.45 R = 1,806,000 θ = 6509

c *RM = 0.20 c *L = 8.00 R* = 1,750,000 θ *N = 6250

COH = 75,000

C *FOH = 60,000 C *VOH = 1.00

θ*c*L = 6450(8.00) = $51,600/period From Eq. (9-219) we calculate the direct-labor-cost variance as $3401 per period. This variance is unfavorable. However, by using the relations of Eq. (9-220), we calculate a direct pay, or wage-rate, variance of θ(cL − c*L) = 6509(8.45 − 8.00) = $2929 per period. This direct pay variance is unfavorable. Likewise, we calculate the direct-laborefficiency variance as c*L(θ − θ*) = 8.00(6509 − 6450) = $472 per period. This variance is also unfavorable.

MANUFACTURING-COST ESTIMATION For this example, the adverse direct-labor cost variance of $3401 is due to both a higher wage rate per hour and a higher number of laborhours. The direct-labor-cost variance can, if necessary, be broken down into a direct-labor-idle-time variance in addition to the direct-wagerate and direct-labor-efficiency variances. The direct-labor-idle-time variance is simply the number of idle labor-hours in the period multiplied by the standard wage rate. This is rarely relevant to the conditions existing in process plants except when maintenance is involved. Variances: Overhead Cost The variance in overhead cost ∆COH is the difference between actual overhead cost and static standard overhead cost, ∆COH = COH − θ*c*NOH

(9-223)

where COH is the actual overhead cost incurred in a given period, $ per period; c*NOH is the static standard overhead cost based on the normal production rate, $ per hour; θ is the actual time taken to produce a given quantity of product in a given period; and θ* is the standard time taken to produce a given quantity of product in a given period, hours. Equation (9-223) can also be written in expanded form ∆COH = (COH − θcBOH) + θ(cBOH − c*NOH) + c*NOH (θ − θ*)

(9-224)

where cBOH is the flexible budgeted overhead cost at the actual production rate or operating capacity. In Eq. (9-223), (COH − θcBOH) is known as the budgeted overheadcost variance, θ(cBOH − c*NOH) as the overhead-volume variance, and c*NOH(θ − θ*) as the overhead-efficiency variance. The last is analogous to the labor-efficiency variance and is the standard overhead rate multiplied by the deviation in time taken to produce a given output. Also in Eq. (9-224), cBOH is simply the flexible budgeted overhead cost in dollars per hour for the actual production rate, and the overhead-volume variance θ(cBOH − c*NOH) is the actual time taken to produce a given output multiplied by the difference between the flexible budgeted overhead cost and the standard overhead cost in dollars per hour. The budgeted overhead-cost variance (COH − θcBOH) is the difference between the actual overhead cost and the actual time (in hours) required to produce the given output multiplied by the flexible budgeted overhead cost (in dollars per hour). We shall write the fixed overhead cost for the budget period as C* FOH, the semivariable overhead cost as C* SVOH, and the standard hours to produce the agreed normal production as θ*N. The standard overhead cost at the agreed normal production rate can then be calculated from c*NOH = [(C*FOH + C*SVOH)/θ*N] + c*VOH

(9-225)

where c*VOH is the standard variable overhead cost, $ per hour. For production rates that differ from the agreed normal rate, the flexible budgeted overhead cost is given by c BOH = [(C*FOH + C*SVOH)/θ] + c*VOH

(9-226)

where θ is the actual hours taken to produce a given amount of product. For production rates lower than normal, the fixed overheads are underused, and the flexible budgeted overhead cost cBOH is greater than the standard overhead cost c*NOH. For production rates higher than normal, cBOH is less than c*NOH. It is common practice in cost accountancy to treat the standard semivariable cost C*SVOH at the normal production rate as part of the standard fixed cost. In this case, Eqs. (9-225) and (9-226) can be written respectively as c*NOH = (C*FOH/θ*N) + c*VOH

(9-227)

cBOH = (C*FOH /θ) + c*VOH

(9-228)

By using the data of Table 9-40 in Eq. (9-227), we calculate the standard overhead cost to be c*NOH = (60,000/6250) + 1.00 = $10.60/h From Eq. (9-228) and the data in Table 9-40 we calculate the flexible budgeted overhead cost to be cBOH = (60,000/6509) + 1.00 = $10.22/h

9-61

By substituting into Eq. (9-223), we calculate the overhead cost variance: ∆cOH = 75,000 − 6450(10.60) = $6630 This variance is unfavorable. (Note that standard time for actual production was previously calculated to be 6450 h per period.) The overhead cost variance comprises (1) budgeted overhead-cost variance, (2) overhead-volume variance, and (3) overhead-efficiency variance. The calculations for each follow: cOH − θcBOH = 75,000 − 6509(10.22) = $8478 Budgeted overhead-cost variance is positive and, therefore, unfavorable. θ(cBOH − c*NOH) = 6509(10.22 − 10.60) = −$2473 Overhead volume variance is negative and favorable. c*NOH(θ − θ*) = 10.60(6509 − 6450) = $625 Overhead efficiency variance is positive and unfavorable. The total variances in each category are listed in Table 9-40. Chemical engineers usually make detailed evaluations of costs rather than evaluations for profits or sales. However, the latter can be analyzed in a similar manner to costs by using the equations shown in Table 9-38. For this purpose, the sign convention will be reversed because an increase in sales or profits would be considered favorable, whereas an increase in cost would be considered unfavorable. The equations can be applied to both batch and continuous processes. Budgets can be used for both forward planning and control. Variances show managers what their costs should have been and how near they came to meeting budgeted values. Managers will be able to assess, over a number of budget periods, the rate of improvement in performance in their areas of responsibility. A good budgetary system not only should provide detailed information and an appraisal of performance but also should motivate people to improve performance. Contribution Analysis Contribution analysis can be used to make rapid assessments of the effect of changes in manufacturing costs on profitability. A dimensionless contribution efficiency η can be defined by rewriting Eq. (9-12) in the form R(cS − cVE) − AFE η = ᎏᎏ (9-229) R(cS − cVE) [F. A. Holland and F. A. Watson, Eng. Process Econ., 1, 135–143 (1976)]. This represents the ratio of the net annual profit ANP actually achieved divided by the profit which could be obtained if no repayment of capital or interest were required and all fixed-expense items were credited free to the project. The contribution efficiency η is also the profit per unit of contribution. A value for η of unity would be obtained for a very high production rate R whether cS is greater or less than cVE. For the unusual case of cS being equal to c VE, the value of η would become negatively infinite for a finite annual fixed expense A FE or positively infinite if A FE became negative because of excessive subsidy of expenses. However, for most projects which are intended to pay their own expenses and taxes, ANP must be positive, and hence cS is usually greater than cVE, so that η will normally have values in the range of zero to unity. For projects which are not intended to make a profit but are provided for their social or amenity value, the aim should be to bring the value of η as near to zero as possible. The breakeven production rate RB is defined by Eq. (9-13) as the production rate at which the project makes neither a profit nor a loss. Equation (9-13) and (9-229) can be combined to give η = (R − R B)/R

(9-230)

which shows that the contribution efficiency η is a function of the production rate R and that η has the value zero when the production rate is the breakeven production rate RB. For all real projects R B will be positively finite while R cannot be less than zero, and hence the practical range of η is from negative infinity to unity. At first glance it might appear that it is desirable to have a value of η as near to unity as possible. However, this is not necessarily so. Reference to Eq. (9-229) will show that if the unit contribution (cS − cVE)

9-62

PROCESS ECONOMICS

is positive, as it must be if the project is ever to make a profit, a value of unity for the contribution efficiency η implies a negligible value of the annual fixed expense AFE when compared with the annual contribution R(cS − cVE). In such cases either R(cS − cVE) is very large, thus attracting competition in spite of high capital charges, or the fixed expenses are very low, so that it is easy for many small competitors to enter the market. The result of such competition often leads to a rapid reduction in sales price cS. Ball-point pens and electronic calculators were both drastically reduced in price as a result of competitors entering the market. Since the variable expense per unit of production cVE is by definition independent of production rate, the unit contribution (cS − cVE) and hence the value of η will be reduced. On the other hand, a very large company may be in a stronger and more stable position with a modest contribution efficiency and relatively high fixed costs which will deter competitors from entering the market and thereby depressing the sales price. In this argument it is implicit that cS is also independent of output from the project under consideration. In many cases this will be the case since if many small buyers can choose from many alternative producers, the individual producer cannot adjust the price to suit its output, while at the opposite extreme a group of producers that are in a position to make such adjustments are also likely to attract the attention of antitrust legislation. It is of interest to be able to examine the effect of changes in productivity on the profitability of projects. Historically, labor costs have been regarded as variable costs, implying that if workers doubled their output their net wages also doubled. This may have been the case for some piecework rates, but it is generally not true today. It is not normally possible to reduce the work force in step with falling demand or to recruit and train labor in step with increasing demand. In general it is better to consider labor as a fixed cost, with any part of a production bonus which is truly proportional to output included in the variable expense cVE. If the annual fixed expense AFE varies significantly with production rate R owing to this factor, then the breakeven chart will consist of curves, the simplicity of the method is lost, and it will be assumed that a particular change in productivity agreements implies a step change in AFE and/or in cVE. Let us consider an increase in productivity for the same fixed labor cost, with other fixed costs remaining the same. We shall let the original production rate R be increased by an increase in productivity by a fraction φ. Therefore, ∆R = Rφ

(9-231)

where ∆R is the increase in sales volume or production rate. By substitution into Eq. (9-229), the resulting increase in profit ∆A NP is given by A NP + ∆ANP = AS + ∆ AS − (AVE + ∆ AVE) − (AFE + 0)

(9-232)

We shall subtract ANP from Eq. (9-232) to give ∆ A NP = ∆ AS − ∆ AVE = ∆ R(cS − cVE)

(9-233)

It follows from Eqs. (9-12), (9-229), and (9-233) that ∆ ANP /ANP = φ/η

(9-234)

Thus a change of productivity φ of 10 percent will result in a 10 percent increase in profit when η = 1 and a very large increase in profit when η is close to zero. If η is negative, increased productivity reduces the profit (or increases the loss). Equation (9-234) illustrates the enormous influence that go-slow tactics can have on the profitability of companies and processes which have low contribution efficiencies, since a slowdown has little effect on AFE. It is sometimes the case that in different countries productivity per worker varies considerably in similar industries. When poor productivity is not the result of technical or capital inadequacy, it should be possible to increase profitability without a proportionate increase in AFE. Breakeven charts present a snapshot of the present situation by means of graphs which are generally drawn in the manner shown in Figs. 9-2, 9-3, and 9-4. Since the lines are straight, this implies that cS, cVE, and AFE will remain constant over the range of variation of R, which is of interest. The values would be based on the production rate currently achieved (or scheduled), since all the data are available from

the financial analysis of current production, so that cS would be (AS)0 /R0, and so on. Two well-known ratios used by financial analysts are the profit-tosales ratio (PSR) and the contribution-to-sales ratio (CSR). The profitto-sales ratio is simply Eq. (9-127) for profit margin (PM) rewritten as the ratio (PSR) = ANP /RcS

(9-235)

The contribution-to-sales ratio (CSR) = (AS − AVE)/AS = (cS − cVE)/cS

(9-236)

We substitute these values into Eq. (9-229) to give η = (PSR)/(CSR)

(9-237)

The contribution efficiency at the scheduled output η0 is given by substituting the value of the scheduled output into Eq. (9-229) to give R0 (cS − cVE) − AFE η0 = ᎏᎏ R0 (cS − cVE) The characteristic shape of a given breakeven chart of the type represented by Figs. (9-2), (9-3), and (9-4) can be defined by the two ratios (CSR) and η0, while the scale of the project can be defined by a single annual cost such as AFE. This information may be used for the rapid investigation of the likely effect on current profits obtainable by changes in various factors such as prices, expenses, and throughput. It should be noted that this technique is not intended to replace discounted methods of investment appraisal but to provide a rapid assessment of the probable effect of changes in current conditions. If the current profitability is always maximized, then the discountedcash-flow present value will always be made as great as conditions in a changing world will permit. Valuation of Recycled Heat Energy The rising cost of energy is having an inflationary effect on manufacturing costs. One obvious way to reduce energy costs is to recycle heat energy whenever possible [S. A. K. El-Meniawy, F. A. Watson, and F. A. Holland, Indian Chem. Eng., 22 (July–September 1980)]. Heat pumps are particularly suitable for recycling heat energy in the chemical-process industries. For the outlay of an additional fixedcapital expenditure CFC on a heat-pump system, a considerable reduction in the annual heating cost can be effected. Let us consider a process unit requiring heat at the rate of QD GJ/h operating for y h in a year. We shall let the unit cost of this base heating requirement be cB $ per gigajoule. Therefore the annual heating cost for this unit is QDycB $ per year. We then consider the use of a heat pump to supply this heat so that QD = W(COP)A

(9-238)

where W is the rate of energy input to the compressor in gigajoules per hour and (COP)A is the actual coefficient of performance of the heat pump. When interest charges are involved, the fixed-capital expenditure on a heat-pump system CFC can be related to an annual cost AFC for the estimated life of the heat pump in years by the equation AFC = CFC fAP

(9-239)

where fAP = [i(1 + i)n]/[(1 + i)n − 1], the annuity present-worth factor, and i is the fractional interest rate per year payable on the borrowed money. A given value for AFC enables a cost in, say, dollars per gigajoule, to be assigned to the heat energy made available by a heat pump. We shall consider a heat-pump system which operates for y h/year and consumes W GJ/h of high-grade energy to drive the compressor. We shall let the unit cost of the input energy to the compressor be cI $ per gigajoule. The annual amount of heat delivered by the heat pump is QDy, which in terms of Eq. (9-238) can be written as W(COP)Ay in gigajoules per year. The annual cost of this delivered heat, neglecting any maintenance cost, is (WcI y + AFC) in dollars per year. Therefore the unit cost cD of the heat energy delivered by a compressor-driven pump is cD = (WcI y + AFC)/[W(COP)Ay] $/GJ

(9-240)

FIXED-CAPITAL-COST ESTIMATION The ratio of the unit costs of the delivered heat energy and the input energy can be obtained by combining Eqs. (9-239) and (9-240) to give (cD /cI) = [(1 + ψ)/(COP)A] (9-241) where ψ = (CFC /W)( fAP /ycI). ψ is a dimensionless parameter which contains cost and usage data for a particular heat pump; it should have as low a value as possible in order to minimize the unit cost of the delivered heat cD. The cost CFC/W is the fixed-capital cost per unit of input energy in $ (GJh−1)−1, and this should also be as low as possible consistent with a long life and good reliability for the heat pump. Clearly y, the number of operating hours per year, should approach as closely as possible to the maximum value of 8760 h for a 365-day year. Since all costs refer to a given year, Eqs. (9-239), (9-240), and (9-241) are independent of inflation. The annual cost of heat delivered by the heat pump is QDycD, where the unit cost cD is given by Eq. (9-240). Therefore, the annual saving on heating costs in dollars per year is QDy(cB − cD), which can also be written in terms of Eq. (9-238) as W(COP)Ay(cB − cD). The payback period in years for a heat-pump system is the additional fixed-capital cost CFC divided by the annual saving on heating costs. This can be written as CFC 1 (PBP) = ᎏ (9-242) ᎏᎏ W (COP)Ay(cB − cD) which can also be written in terms of Eq. (9-241) as CFC 1 (9-243) (PBP) = ᎏ ᎏᎏᎏ W y[(COP)AcB − cI(1 + ψ)] For the special case of the unit cost of the input energy to the compressor being the same as the unit cost of the base heat supply, i.e., cB = cI, Eq. (9-243) simplifies to CFC 1 (PBP) = ᎏ (9-244) ᎏᎏᎏ W ycI[(COP)A − (1 + ψ)]

 

 



 

9-63

Equation (9-244) can be used to calculate the payback period when electricity, oil, or gas, etc., is used to drive the compressor and also to provide the base heating. Equation (9-244) shows that, to have a low payback period (PBP), CFC/W and ψ should be small and y, (COP)A, and cI large. Clearly as the unit cost of input energy cI increases, the economics of heat pumps becomes more favorable. The value of ψ will for most cases be less than 0.2 and with the right application may well be less than 0.1. Values for the annuity presentworth factor will in most cases be less than 0.15. Since 1 bbl (0.159 m3) of oil is normally quoted as having a thermalenergy value of 6.12 GJ, a world oil price of, say, US$40 per barrel is equivalent to US$6.54 per gigajoule. For simplicity, we substitute cI = US$6 w per gigajoule and a conservative value of [(COP)A − (1 + ψ)] = 3 into Eq. (9-244) to give (PBP) = (CFC / W)(1/20y)

(9-245)

where (PBP) is the payback period in years, y is the operating hours per year, and (CFC/W ) is the fixed-capital cost in US$ (GJh−1)−1 of primary-energy input. Equation (9-245) shows that in this particular case the fixed-capital cost per unit of input energy (CFC/W) must not exceed $160,000 (GJh−1)−1, or $576 per kilowatt, to have a 1-year payback period if the heat pump is operational for 8000 h/year. For this case the corresponding value of ψ is about 0.12 for a heat pump with an operating life of 10 years purchased with money borrowed at a 10 percent rate of interest. Equation (9-245) also shows that the fixed-capital cost per unit of energy input (CFC/W) must not exceed $40,000 (GJh−1)−1, or $144 per kilowatt, to have a 1-year payback period if the heat pump is operational for only 2000 h/year. For this case the corresponding value of ψ is also about 0.12 for a heat pump with an operating life of 10 years purchased with money borrowed at a 10 percent rate of interest.

FIXED-CAPITAL-COST ESTIMATION Total Capital Cost The installed cost of the fixed-capital investment CFC is obviously an essential item which must be forecast before an investment decision can be made. It forms part of the total capital investment CTC, defined by Eq. (9-14). The fixed-capital investment is usually regarded as the capital needed to provide all the depreciable facilities. It is sometimes divided into two classes by defining battery limits and auxiliary facilities for the project. The boundary for battery limits includes all manufacturing equipment but excludes administrative offices, storage areas, utilities, and other essential and nonessential auxiliary facilities. Cost Indices The value of money will change because of inflation and deflation. Hence cost data can be accurate only at the time when they are obtained and soon go out of date. Data from cost records of equipment and projects purchased in the past may be converted to present-day values by means of a cost index. The present cost of the item is found by multiplying the historical cost by the ratio of the present cost index divided by the index applicable at the previous date. Ideally each cost item affected by inflation should be forecast separately. Labor costs, construction costs, raw-materials and energy prices, and product prices all change at different rates. Composite indices are derived by adding weighted fractions of the component indices. Most cost indices represent national averages, and local values may differ considerably. Table 9-42 presents information on some cost indices for the United States. Engineering News-Record updates its constructioncost index in March, June, September, and December. The Oil and Gas Journal gives the Nelson-Farrar refinery indices in the first issue of each quarter. The Chemical Engineering plant-cost index and Marshall and Swift equipment-cost index are given in each issue of the publication Chemical Engineering. Derivation of the base values is referred to in the respective publications.

Table 9-43 is based on the method suggested by J. Cran [Eng. Process Econ., 2, 89–90 (1977)]. He showed that reasonably accurate plant-cost indices for various countries could be derived by using two component indices in the equation (CI)P = 0.327(CI)ST + 0.673(CI)L

(9-246)

where (CI)ST is the steel-price index and (CI)L the earnings index for labor in the particular country. Most of the data required can be obtained from the United Nations Monthly Bulletin of Statistics or the Organization for Economic Cooperation and Development (OECD) annual review of the iron and steel industry. In Table 9-43, the plantcost indices have been brought to a common base of 1980 = 100. The values given do not relate costs in one country to those in another country, as this involves many complex and difficult problems. However, the table indicates the inflationary trends in plant costs since 1980 for each of the countries listed. Types and Accuracy of Estimates Capital-cost estimates may be required for a variety of reasons, among others to enable feasibility studies to be carried out, to enable a manufacturing company to select from alternative investments, to assist in selection from alternative designs, to provide information for planning the appropriation of capital, and to enable a contractor to bid on a new project. It is therefore essential to achieve the greatest accuracy of estimation with a minimum expenditure of time and money. Two simple rules are invaluable in aiding the production of consistently accurate estimates: 1. Check the completeness of the project scope. 2. Reduce the effect of bias by using statistically proven methods of estimation based on experience. Estimates which are lower than actual project costs are often the result of sizable omissions of equipment, services, or auxiliary facilities

9-64

PROCESS ECONOMICS

TABLE 9-42

Cost Indices for the United States Price indices*

Construction cost indices

Year

Consumer

Producer

Industrial chemicals

1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993

100 110 117 121 126 131 133 138 144 151 159 165 170 175

100 108 113 114 115 115 114 117 122 127 132 132 133 136

100 113 106 103 102 100 97 102 106 108 111 111 110 111

Chemical equipment†

Process plants‡

Petroleum refinery§

General construction¶

660 721 746 761 780 790 798 814 852 895 915 931 943 967

261 297 314 317 323 325 318 324 343 355 358 361 358 359

286 315 340 357 370 374 380 391 406 417 427 436 445 453

303 330 357 380 387 392 401 412 422 429 441 452 470 489

*Bureau of Labor Statistics, 1980 = 100. †Marshall & Swift, Chem. Eng., 1926 = 100. ‡Chem. Eng., plant index, 1957–1959 = 100. §Nelson-Farrar refinery index, Oil Gas J., 1967 = 100. ¶Eng. News Rec., 1967 = 100.

rather than of errors in pricing or estimation methods. To avoid this, the use of a checklist of items involved in a new project as given in Table 9-44 can be invaluable. The first stage toward producing an accurate estimate is to use a standard cost code for all construction projects. Table 9-45 shows a suitable numerical cost code, and Table 9-46 shows a typical alphabetical-numerical code. The cost-code system can be used throughout the estimating and construction stages for the collection of cost data by manual or computer methods. There are numerous types of fixedcapital-cost estimates, but in 1958 the American Association of Cost Engineers defined five types as follows: 1. Order-of-magnitude estimate (ratio estimate). Rule-of-thumb method based on cost data for previous similar types of plant; probable error within 10 to 50 percent. 2. Study estimate ( factored estimate). Better than order-ofmagnitude; requires knowledge of major items of equipment; used for feasibility surveys; probable error up to 30 percent. 3. Preliminary estimate (budget-authorization estimate). Requires more detailed information than study estimate; probable error up to 20 percent. 4. Definitive estimate (project-control estimate). Based on considerable data prior to preparation of completed drawings and specifications; probable error within 10 percent. 5. Detailed estimate ( firm or contractor’s estimate). Requires completed drawings, specifications, and site surveys; probable error within 5 percent. Greater accuracy of estimation may be achieved, within limits, by the expenditure of more time and money. The greater the accuracy required, the greater the time and effort needed to obtain the design and cost data prior to making the estimate. W. R. Park investigated the cost and accuracy of estimates for a project with a total cost of $1 million as shown in Fig. 9-41 (Cost EngiTABLE 9-43

neering Analysis, Wiley, New York, 1973, p. 133). Table 9-47 shows typical average costs for producing estimates [adapted from A. Pikulik and H. E. Diaz, Chem. Eng., 84, 106–122 (Oct. 10, 1977)]. Rapid Estimations Ratio Methods J. E. Haselbarth [Chem. Eng., 74, 214–215 (Dec. 4, 1967)] published data giving the total capital investment per unit of annual production capacity CTC/R. Table 9-48 lists data for many processes involving production units constructed on a previously developed site. Plants built on a green-field site would cost about 30 to 40 percent more, but enlargements of an existing plant would cost about 20 to 30 percent less than the values given in Table 9-48. Total fixed-capital investments for installations within the battery limits are given in Table 9-48. These refer to North American values corresponding to a Marshall and Swift index of 1000 and are adapted from the data of D. R. Woods (Financial Decision Making in the Process Industry, Prentice Hall, Englewood Cliffs, NJ, 1975, pp. 288–290). L. Lynn and R. F. Howland [Chem. Eng., 67, 131–136 (Feb. 8, 1960)] studied the capital ratios for 17 process industries and summarized data for more than 1000 processes. The capital ratio (CR) for a plant erected on a green-field site is defined as the ratio of the fixed-capital investment CFC to the annual sales revenue AS: (CR) = CFC/AS

(9-247)

However, Lynn and Howland included in the fixed-capital cost not only money invested in production and storage facilities but also that invested in land, research and development costs, and any auxiliary facilities necessary to support the process. Typical values of capital ratios for the year 1958 are listed in Table 9-49. Both of the preceding methods are relatively inaccurate and can be used only for rough screening. They have the advantage that an esti-

International Plant-Cost Indices*

Year

Australia

Belgium

Canada

Denmark

France

Germany

Italy

Japan

Netherlands

Spain

Sweden

United Kingdom

United States

1985 1986 1987 1988 1989 1990 1991 1992 1993

151 160 170 180 193 203 208 215 218

134 135 138 141 147 152 161 167 171

137 142 146 153 160 166 173 174 180

143 148 159 163 177 184 193 199 198

171 178 186 191 200 204 210 217 223

121 124 128 133 137 142 150 160 162

201 206 217 231 253 260 269 286 292

112 111 110 115 122 126 127 127 129

119 121 123 124 126 130 133 140 142

190 207 222 233 249 267 283 303 315

151 162 171 183 199 215 227 232 244

148 156 166 182 200 216 232 241 240

124 122 123 131 137 136 138 136 136

*From Process Eng., London, U.K. (monthly). Annual averages for each year (1980 = 100).

FIXED-CAPITAL-COST ESTIMATION TABLE 9-44

9-65

Checklist of Items for Fixed-Capital-Cost Estimates

Land: Surveys Fees Property cost Site development: Site clearing Grading Roads, access and on-site Walkways Railroads Fence Parking areas Other paved areas Wharves and piers Recreational facilities Landscaping Process buildings: (List as required) Include in each as required substructure, superstructures, platforms, supports, stairways, ladders, access ways, cranes, monorails, hoists, elevators Auxiliary buildings: Administration and office Medical or dispensary Cafeteria Garage Product warehouse(s) Parts or stores warehouse Maintenance shops—electric, piping, sheet metal, machine, welding, carpenters, instrument Guard and safety Hose houses Change houses Smoking stations (in hazardous plants) Personnel building Shipping office and platforms Research laboratory Control laboratories Building services: Plumbing Heating Ventilation Dust collection Air conditioning Sprinkler systems Elevators, escalators Building lighting Telephones Fire alarm Paging Intercommunication systems Painting Process equipment: (List carefully from checked flow sheets) Nonprocess equipment: Office furniture and equipment Cafeteria equipment Safety and medical equipment Shop equipment Automotive heavy maintenance and yard material-handling equipment Laboratory equipment Lockers and locker-room benches Garage equipment Shelves, bins, pallets, hand trucks Housekeeping equipment Fire extinguishers, hoses, fire engines Process appurtenances: Piping—carbon steel, alloy, cast iron, lead-lined, aluminum, copper, asbestos-cement, ceramic, plastic, rubber, reinforced concrete Pipe hangers, fittings, valves Insulation—piping, equipment Instruments Instrument panels Electrical—panels, switches, motors, conduit, wire, fittings, feeders, grounding, instrument and control wiring

Utilities: Boiler plant Incinerator Ash disposal Boiler feed-water treatment Electric generation Electrical substations Refrigeration plant Air plant Wells River intake Primary water treatment—filtration, coagulation, aeration Secondary water treatment—deionization, demineralization, pH and hardness control Cooling towers Water storage Effluent outfall Process-waste sewers Process-waste pumping stations Sanitary-waste sewers Sanitary-waste pumping stations Impounders, collection basins Waste treatment, including gases Storm sewers Yard distribution and facilities (outside battery limits): Process pipe lines—steam, condensate, water, gas, fuel oil, air, fire, instrument, and electric lines Raw-material and finished-product handling equipment—elevators, hoists, conveyors, airveyors, cranes Raw-material and finished-product storage—tanks, spheres, drums, bins, silos Fuel receiving, blending, and storage Product loading stations Track and truck scales Miscellaneous: Demolition and alteration work Catalysts Chemicals (initial charge only) Spare parts and noninstalled equipment spares Surplus equipment, supplies and equipment allowance Equipment rentals (for construction) Premium time (for construction) Inflation cost allowance Freight charges Taxes and insurance Duties Allowance for modifications and extra construction work during startup Engineering costs: Administrative Process, project, and general engineering Drafting Cost engineering Procurement, expediting, and inspection Travel and living expense Reproductions Communications Scale model Outside architect and engineering fees Construction expense: Construction, operation, and maintenance of temporary sheds, offices, roads, parking lots, railroads, electrical, piping, communication, and fencing Construction tools and equipment Warehouse personnel and expense Construction supervision Accounting and timekeeping Purchasing, expediting, and traffic Safety and medical Guards and watchmen Travel and transportation allowance for craft labor Fringe benefits Housekeeping Weather protection Permits, special licenses, field tests Rental of off-site space Contractor’s home office expense and fees Taxes and insurance, interest

9-66

PROCESS ECONOMICS

TABLE 9-45 Code Group Direct costs: 0–349 350–399 400–449 450–489 490–499 500–599 600–699

700–729 730–749 750 760–769 770–799 800–819 820–830 Indirect costs: 850–874 875–889 900–969 970–994 995 996

Standard Cost Code (Numerical) Summary Subdivision Process section (process equipment listed in flow-sheet sequence 1, 2, 3, 4, 5, . . . , N, preceded by process identification number) Site development Process buildings Auxiliary buildings Nonprocess equipment Process appurtenances (piping, insulation, instrumentation, electrical, etc.) Utilities and yard services (boiler plant, refrigeration, compressed air, water supply and treatment, effluents, fire protection, yard piping, yard electrical, yard materials handling, raw and finished-product storage) Substructures Superstructures Painting Building services Demolition and alteration to existing structure Surplus equipment, supplies and materials, royalty payments Design modifications, construction modifications, and extra work during startup Home office engineering Architect’s and engineer’s charges and fees Construction expenses Taxes and insurance Contractor’s home office expenses Contractor’s fees

mate can be made in a few minutes, and they do not require design work or process flow sheets. Step Count Methods These methods, used for order-ofmagnitude estimates, are based on the definition of the functional units required to carry out the process. A functional unit is a signifiTABLE 9-46 A B C D E F G H J K L M N O P Q R S T U V W X 100 200 300 320 340 360 380 400 500 600 700 800 900 950

Standard Cost Code (Alphabetical-Numerical)

Site work and foundations Buildings (less foundations) Steel structures and platforms (other than buildings) Heat exchangers Fractionating towers Tanks and drums Pumps and pump drivers Compressors and blowers Reactors and converters Grinding, crushing, and classifying equipment Materials-handling equipment Fired heaters Catalysts and chemicals Laboratory equipment Piping Instruments and controls Electrical Insulation and painting Utility equipment (boilers, generators, refrigeration) Plant and building accessories (railroads, fence, etc.) Laboratory equipment Safety equipment Warehouse spares Equipment and materials delivered and installed (A through X) Sales and use taxes Temporary facilities Construction equipment, tools, and supplies Construction supervision Field office expense Warehousing expense Payroll taxes and insurance Home office engineering costs Procurement costs Resident engineering Royalty payments Engineering administrative overhead and profit Constructor’s administrative overhead and profit

FIG. 9-41

Relationship between cost and accuracy of cost estimations.

cant process step, including all the equipment and ancillaries to operate. The sum of the costs of the functional units gives an estimate of the total capital cost of the plant. Usually a functional unit is a unit operation, a unit process, or a separation involving energy transfer, moving parts, and possibly a high level of internals. Pumping and heat exchange form part of a functional unit. “Inprocess” storage is ignored, but large storage requirements for raw materials, intermediates, or products are usually estimated separately. Sometimes there are difficulties in the identification and definition of the functional units involved in a process. For a particular process, the capital cost per functional unit is given by CF = f (q, T, p, Mc, CI) (9-248) where q is capacity, T is operating temperature, p is pressure, Mc is a materials of construction factor, and CI is a relevant cost index. Typical methods are those of F. C. Zevnik and R. L. Buchanan [Chem. Eng. Progr., 59, 70–77 (Feb. 1963)] and J. H. Taylor (Eng. & Proc. Econ., 2, 259–267, 1977). The former is mainly a graphical method of estimating the cost per functional unit (CF) based on the capacity, the maximum pressure, the maximum temperature, and the materials of construction. The Taylor method requires the determination of the costliness index, which is dependent on the complexity of the process. A simpler method was suggested by S. R. Timms (M.Phil. thesis, Aston University, England, 1980) to give the battery limits cost for gas phase processes only in U.S. dollars with a Marshall and Swift index of 1000. The simple equation is CBL = 11,800Nq0.615

(9-249a)

Taking into account materials of construction, temperature, and pressure effects, this becomes CBL = 5500Nq0.639Mc(Tmax)0.066(pmax)−0.016

(9-249b)

where N is the number of functional units; q is the capacity (tons/y); Mc is the materials of construction factor: 1.0 for carbon steel, 1.15 for TABLE 9-47 (1990)*

Typical Average Costs for Making Estimates Cost of project

Type of estimate Order of magnitude estimate Study estimate Preliminary estimate Definitive estimate Detailed estimate

Less than $2 million

$2 to $10 million

$10 to $100 million

$

$

$

3,000 20,000 50,000 80,000 200,000

6,000 40,000 80,000 160,000 520,000

13,000 80,000 130,000 320,000 1,000,000

*Adapted from A. Pikulik and H. E. Diaz, “Cost Estimating for Major Process Equipment.” Chem. Eng., 84, 106–122 (Oct. 10, 1977).

FIXED-CAPITAL-COST ESTIMATION TABLE 9-48

9-67

Capital-Cost Data for Processing Plants*

Product

Process route

Size, 1000 metric tons/year

Acetaldehyde Acetic acid Acetone Ammonia Ammonium nitrate Butanol Chlorine Ethylene Ethylene oxide Formaldehyde (37%) Glycol Hydrofluoric acid Methanol Nitric acid (conc.) Phosphoric acid Polyethylene (high density) Propylene Sulfuric acid Urea

Ethylene Methanol/CO Propylene Steam reforming Ammonia/nitric acid Propylene/CO/H2O Electrolysis of NaCl Refinery gases Ethylene/O2 Methanol Ethylene/Cl2 Hydrogen fluoride/H2O CO2/natural gas/steam Ammonia oxidation Calcium phosphate/H2SO4 Ethylene

50 10 100 100 100 50 50 50 50 10 5 10 60 100 5 5

Refinery gases Sulfur Ammonia/CO2

10 100 60

Alkylation (H2SO4) Coking (delayed) Coking (fluid) Cracking (fluid) Cracking Distillation (atm) Distillation (vac.) Hydrotreating Reforming Polymerization

Catalytic Thermal Thermal Catalytic Thermal 65% vaporized 65% vaporized Catalytic desulfurization Catalytic Catalytic

10 10 10 10 10 100 100 10 10 10

Total capital cost ($) ᎏᎏᎏ metric ton/yr of product

Size range 1000 metric tons/year

Exponent n‡

13.7 6.6 35.0 26.0 6.0 44.0 31.0 14.4 55.0 17.7 16.6 8.8 14.4 6.6 3.3 17.7

274 660 350 260 60 880 620 288 1100 1770 3320 880 240 66 660 3540

20–150 3–30 30–300 30–300 30–300 20–150 20–150 20–150 20–150 3–30 2–20 3–30 20–200 30–300 2–20 2–20

0.70 0.68 0.45 0.70 0.65 0.40 0.45 0.83 0.78 0.55 0.75 0.68 0.60 0.60 0.60 0.65

3.3 3.3 8.8

330 33 147

3–30 30–300 20–200

0.70 0.65 0.70

2100 2880 1770 1770 550 354 210 330 3200 550

3–30 3–30 3–30 3–30 3–30 30–300 30–300 3–30 3–30 3–30

0.60 0.38 0.42 0.70 0.70 0.90 0.70 0.65 0.60 0.58

Approximate cost† $ × 106

Chemical plants

Refinery units 21.0 28.8 17.7 17.7 5.5 35.4 21.0 3.3 32.0 5.5

*Adapted from M. S. Peters and K. D. Timmerhaus, “Plant Design and Economics for Chemical Engineers”, 4th ed., McGraw-Hill, New York, 1991. †All costs are approximate U.S.A. values with M & S = 1000, assuming 330 operating days per year. ‡Exponents apply roughly for threefold capacity ratio extending either way from the plant size given.

low-grade stainless steel, 1.2 for medium-grade stainless steel, 1.3 for high-grade stainless steel; Tmax is the maximum process temperature (K); and pmax is the maximum process pressure (bars). For liquid and/or solid handling, A. V. Bridgwater and F. D. G. Bossom (Proceedings of the 6th International Cost Engineering Conference, Mexico, October, 1980) suggested the following equations: CBL = 2160N(q/s)0.675

(9-250)

for plant capacities (q) above 60,000 tons/y, where CBL is the capital cost for battery limits in U.S. dollars (M & S = 1,000); N is the number of functional units; and s is the reactor conversion (weight of desired TABLE 9-49

Capital Ratios for Process Industries Industry

Capital ratio,* 1958

Chemicals, general Carbon black Explosives Glass Fibers, synthetic Foodstuffs, processed Inorganics, heavy Nonferrous metals Petroleum Pharmaceuticals Pigments, paints, and inks Pulp and paper Resins and plastics Rubber Soap and detergents Steel Sulfur Average

2.02 3.98 1.64 1.46 3.44 0.66 2.24 3.31 3.08 0.92 1.04 2.01 1.90 1.04 0.69 2.78 1.97 2.01

*Capital ratio = (fixed-capital investment)/(annual sales revenue).

reactor product/weight of reactor input). Thus, q/s represents the feed throughput in tons per year. For plant capacities below 60,000 tons per year, the equation becomes: CBL = 189,300N(q/s)0.30

(9-251)

In general, the step count method of estimation can be applied to any special situation to derive a model equation for that particular industry or group of processes. Exponential Methods Rapid capital-cost estimates can be made by using capacity-ratio exponents based on existing cost data of a company or drawn from published correlations. If the cost of a piece of equipment or plant of size or capacity q1 is C1, then the cost of a similar piece of equipment or plant of size or capacity q2 can be calculated from C2 = C1(q2 /q1)n

(9-252)

where the value of the exponent n depends on the type of equipment or plant. Cost indices should be used to bring the cost data to a common year. Table 9-48 gives typical values of n for various processes along with the cost of a plant of given capacity at a particular time and the capacity range of applicability. For process plants, capacity is expressed in terms of annual production capacity in metric tons per year. Exponential cost correlations have been developed for individual items of equipment. Care must be taken in determining whether the cost of the equipment has been expressed as free on board (FOB), delivered (DEL), or installed (INST), as this is not always clearly stated. In many cases the cost must be correlated in terms of parameters related to capacity such as surface area for heat exchangers or power for grinding equipment. There are four main sources of error in such cost correlations: 1. Oversimplification by correlating the cost of equipment in terms of a single variable

9-68

PROCESS ECONOMICS

2. Representation of data by using a simple exponential relationship 3. Failure to include the effects of technological improvements 4. Errors incurred because of special circumstances Table 9-50 gives typical values of the exponent n for many types of equipment. Prices are North American with a Marshall and Swift index of 1000, mainly for carbon steel equipment. Factor Estimations Most factor methods for estimating the total installed cost of a process plant are based on a combination of materials, labor, and overhead cost components. These can be conveniently grouped as 1. Cost of major items of equipment 2. Cost of complete installation of equipment 3. Auxiliary equipment to make the process work 4. Engineering and field expenses 5. Contractor’s fees and contingencies A great variety of factors are in use, depending on the time available and the accuracy expected. Normally the input information required is the base cost. Determination of this cost usually requires a knowledge of equipment sizes, probably using mass and energy balances for the proposed process. Equipment or Base Cost The total cost of the main-plant items is generally used as the base cost. Again, care must be taken with equipment costs which may be quoted as installed (INST), delivered to site (DEL), or free on board the delivery vehicle at the place of manufacture or other specified location (FOB). Base equipment includes all equipment within the battery limits whose cost is as significant as the cost of a pump. For example, storage tanks, knockout drums, accumulators, heat exchangers, and pumps are classed as main-plant items (MPI). Early in the development of the process-flow diagram, it is advisable to increase the estimated (MPI) cost by 10 to 20 percent to allow for later additions. When the scope of the process has been well defined, (MPI) costs should be increased by 1 to 10 percent. For order-of-magnitude estimates the cost of equipment delivered (CEQ)DEL varies approximately from 1.1 to 1.25 times the FOB cost (CEQ)FOB. The factor would be at the lower end of the range for domestic purchases and at the higher end for imports. Installation costs include unpacking, mounting, and connecting up to existing auxiliaries or utilities. The cost of equipment installed (CEQ)INST varies with type and size but generally ranges from 1.4 to 2.2 times the deliveredequipment cost (CEQ)DEL. Single-factor methods collect the various items of expenditure into one factor, which is usually used to multiply the total cost of delivered equipment  (CEQ)DEL to give the fixed-capital cost for plant within the battery limits: (CFC)BL = f

 (CEQ)DEL

(9-253)

Typical values for single factors f for battery-limit-plant costs (for a carbon steel plant including auxiliaries but not land) are as follows: Solids processing (S) 3.8 Solids-fluid processing (S-F) 4.1 Fluid processing (F) 4.8 Thus the factors vary with the type of processing, although the boundaries between the classifications are not clear-cut and considerable judgment is required in selection of the correct factor. Multiple-factor methods include the cost contributions for each given activity, which can be added together to give an overall factor. This factor can be used to multiply the total cost of delivered equipment  (CEQ)DEL to produce an estimate of the total fixed-capital investment either for grass-roots or for battery-limit plants. The costs may be divided into four groups: 1. Cost of plant within battery limits 2. Cost of auxiliaries 3. Cost of engineering and field expenses 4. Cost of contractor’s fees plus contingency allowance Table 9-51 gives typical values of such factors for carbon steel installations taken from the data of D. R. Woods (Financial Decision Making in the Process Industry, Prentice Hall, Englewood Cliffs, NJ, 1975, p. 184). Auxiliaries and site preparation are given as factors of the delivered-equipment cost in Table 9-51, whereas C. A. Miller [Chem.

Eng., 72, 226–236 (Sept. 13. 1965)] expresses auxiliary costs as factors of the battery-limit (BL) cost. Table 9-52 gives the factors from the breakdown of Miller, which is more detailed than that of Woods. Example 23: Estimation of Total Installed Cost of a Plant Let us estimate the total installed cost for a grass-roots plant producing an organic chemical (S-F process) on a continuous basis. We assume that the total cost of delivered equipment  (CEQ)DEL is $1 million and use suitable factors from Table 9-51. The estimated values for the various contributions are given in Table 9-53, resulting in an estimate of $4,280,000 for the total fixed-capital investment, including a contingency factor. A multiple-factor method for predesign cost estimating has been put forward by D. H. Allen and R. C. Page [Chem. Eng., 82, 142–150 (Mar. 3, 1975)] for fluid-type plants (F) that include some vapor processing. The method requires the following input information: 1. Plant flow sheet giving main-plant items and process streams 2. Total process-stream input per year 3. Extreme temperature and pressure conditions, if any 4. Materials of construction for main-plant items 5. Operating phases for each main-plant item 6. Expectation of any unusually high or low, direct or indirect initial costs By means of 12 procedural steps involving the input information, several equations, graphs, and tables, the total cost of delivered equipment  (CEQ)DEL is estimated. This is then converted into a grass-roots investment estimate by dividing  (CEQ)DEL by a single factor ranging from 15 to 30 percent (average value, 21 percent). The method is rapid and is claimed to be accurate within −20 to +25 percent, but it has only been tested by using data published in the literature for eight plants. Multiple-Factor Methods That Separate Materials and Labor These methods have become increasingly popular. While they are similar to the preceding methods, labor and materials costs are considered separately. Hence it is possible to allow for variations in efficiency and labor costs in different localities or countries. H. C. Bauman (Fundamentals of Cost Engineering in the Chemical Industry, Van Nostrand Reinhold, New York, 1964, p. 295) divides most of the components of Table 9-51 into material and labor components, quoting the data as ranges and medians of the percentage of the total fixed-capital investment. In Table 9-54, Bauman’s data have been converted to factors of the delivered-equipment cost for a grass-roots installation. A study has been made by A. V. Bridgwater [Chem. Eng., 86, 119–121 (Nov. 5, 1979)] of the geographical variations in capital costs. He concluded that because of trade and competition basic equipment costs do not vary significantly in the industrialized countries of the western world. The main differences in construction costs at various international locations are due to variations in labor costs and productivity, the use of specialized equipment, and sundry local factors. Table 9-55 gives location factors for the construction of chemical plants of similar function in various countries (1993 values). The factors have been corrected by Bridgwater for location variations in labor costs and efficiency and converted at the average value of the exchange rate. Factor Methods Using the Modular Approach These are methods used for estimating the cost of major-equipment units and have been proposed by several authors. Perhaps the most comprehensive is the method suggested by K. M. Guthrie [Chem. Eng., 76, 114–142 (Mar. 24, 1969)]. Table 9-56 gives average factors for majorequipment items based on a (CEQ)FOB cost for carbon steel units. To the FOB cost of the item is added, by means of factors, the total materials cost to complete the module M. Erection and setting costs L are added as a factor or calculated from the L/M cost ratio to give M + L = X, the direct module cost. Indirect costs, such as freight, taxes, insurance, engineering, and field expense, are added to (M + L) to give the total module cost. This excludes contingency allowances, contractor’s fees, auxiliaries, site development, land, and industrial buildings, which may have to be added when applicable. The factors in Table 9-56 were based on mid-1968 prices for a United States Gulf Coast location.

FIXED-CAPITAL-COST ESTIMATION TABLE 9-50

Typical Exponents for Equipment Cost versus Capacity Equipment

Agitator, turbine, top entry, open, FOB Agitator, turbine, top entry, closed, FOB Blower, centrifugal, 4 lbf/in2 (27.6 kN/m2), DEL, excluding motor Cone crusher, FOB, crusher only Jaw crusher, FOB, excluding motor Jaw crusher, FOB, excluding motor Centrifugal pump, C/S, FOB, excluding motor Centrifugal pump, C/S, FOB, excluding motor Conveyor, belt, C/S, FOB, excluding motor Conveyor, screw, C/S, DEL, excluding motor Centrifuge, automatic batch, horizontal, C/S, FOB Compressor, reciprocating, <1000 lbf/in2, FOB, including motor Crystallizer, forced circulation, C/S, FOB Dryer, drum, C/S, FOB, excluding motor Dryer, vacuum, shelf, C/S, FOB, excluding trays, vacuum equipment Dust collector, cloth, shaker type, FOB, including motors Dust collector, multicyclones, FOB Electrostatic precipitator, FOB Ejector, single-stage, 100 psig, steam, FOB Ejector, two-stage, FOB, including condenser, piping Ejector, multistage, FOB, including condenser, piping Filter, vertical-pressure leaf, C/S, DEL Filter, plate and frame, C/S, DEL Filter, vacuum rotary drum, C/S, FOB, including motor Heat exchanger, shell-tube, floating head, C/S, DEL; fixed tube × 0.85; U tube × 0.87; kettle × 1.35 Heat exchanger, thermal screw, C/S, FOB, excluding motor Kettle, jacketed, glass-lined, FOB Motors, ac induction, wound rotor, TEFC, FOB Motors, ac induction, wound rotor, TEFC, FOB Piping, typical straight run, C/S, FOB, $/ft Installed: $/ft × 6 to 7 Complex network: FOB $/ft × 2 Installed: $/ft × 13 Pressure vessel horizontal drum (150 psig), C/S Jacketed reactors, including mixer, FOB Refrigeration, packaged mechanical, INST Screen, vibrating, single-deck, DEL, including motor Stack, carbon steel Tanks: atm, horizontal cylinder, C/S, FOB Vertical cylinder, C/S, FOB Vertical jacketed, C/S, FOB Vertical agitated, C/S, FOB, including motor Towers, distillation including internals, INST NOTE:

9-69

Size

Approximate cost, $000

Unit

10 (7.5) 10 (7.5)

hp (kW) hp (kW)

7.0 10.7

10 (4.72)

67

100 (74.6) 10 (7.5) 100 (74.6) 10 (7.5)

103 sft3/min (sm3/s) hp (kW) hp (kW) hp (kW) hp (kW)

100 (74.6)

hp (kW)

100 (9.3) 70 (540) 20 (1.86) 300 (224)

2

2

ft (m ) ft × m diameter (m × mm diameter) Filter area, ft2 (m2) hp (kW)

100 (91)

ton/day (Mg/day) ft2 (m2)

100 (9.3)

ft2 (m2)

10 (4.7)

3

3

3

3

sft /min (m /s)

Exponent 0.45 0.56

0.5–150 (0.24–71)

0.60

30–300 (22.4–224) 1–60 (0.75–44.7) 60–400 (44.7–300) 0.5–40 (0.37–30)

0.92 0.65 0.81 0.30

4.4

40–400 (30–300)

0.67

6.7

60–200 (5.6–18.6)

0.50

50–100 (390–780)

0.46

130 34 284 1.6

100 (9.3)

4

Size range 2–30 (1.5–22.4) 2–200 (1.5–150)

10 100

7–80 (0.65–7.43)

0.65

133

1–20000 (0.75–1490)

0.84

283

10–1000 (9.1–970)

0.59

73

10–400 (0.9–37)

0.52

17

15–1000 (1.4–93)

0.56

17

4

10 –5 × 10 (0.47–23.6)

0.79 0.66 0.39 0.81 0.50

3

1 (3.4 × 10−3)

sft /min (m /s) ft3/min at 40°C (m3/s) lb/h (air/mmHg absolute) [kg/h/(N/m2)]

10 (3.4 × 10−2)

[kg/h/(N/m2)]

16.7

100 (9.3) 100 (9.3) 100 (9.3)

ft2 (m2) ft2 (m2) ft2 (m2)

17 5.7 63.3

10 –1.5 × 10 (0.47–70.8) 103–8 × 104 (0.47–73.8) 8 × 104–106 (37.8–472) 0.2–30 (6.8 × 10−4–0.1) 0.2–10 (6.8 × 10−4–3.4 × 10−2) 0.2–100 (6.8 × 10−4–0.34) 30–1500 (2.8–140) 10–1000 (0.9–93) 10–1500 (0.9–140)

1000 (93)

ft2 (m2)

21.7

20–20000 (1.9–1860)

0.59

100 (9.3)

ft2 (m2)

33

10–400 (0.9–37)

0.78

100 (0.38) 10 (7.5)

U.S. gal (m3) hp (kW)

53 12.3

50–1000 (0.2–3.8) 10–25 (7.5–18.6)

0.48 0.56

hp (kW)

19.3

25–200 (18.6–149)

0.77

1–24 (25–610)

1.33

4

10 (4.7) 104 (4.7) 2 × 105 (94) 3 (10−2)

70 (52)

6 (152)

1000 (3.8)

7 77 383 2.7 6.3

Nominal diameter in (mm)

0.0093

U.S. gal (m3)

6.3

3

100 (0.38)

U.S. gal (m )

100 (351.7)

U.S. tons (kW)

500 (46)

ft2 (m2)

10

ft (m) U.S. gal (m3) U.S. gal (m3) U.S. gal (m3) U.S. gal (m3) feed, lb/year ᎏᎏ 106

1000 (3.8) 1000 (3.8) 1000 (3.8) 1000 (3.8) 4000 (trays)

All costs are North American values with M & S = 1000.



9.3



3

5

100–80000 (0.4–302)

0.43 0.26 0.57 0.55 0.48

0.62

10–4000 (0.04–15.1)

0.53

10–1000 (35.2–3520)

0.73

150–700 (14–65)

0.62

— 4.7

20–150 (6.1–45.7) 100–40000 (0.4–151)

1.00 0.57

3.3 15 12.3

100–20000 (0.4–76) 70–1500 (0.26–5.7) 100–20000 (0.4–76)

0.30 0.57 0.50

300–30000

1.00

133

0.65

3300

9-70

PROCESS ECONOMICS

TABLE 9-51

Factors to Convert Delivered-Equipment Costs into Fixed-Capital Investment Grass-roots plants Details

Equipment, delivered Installed Piping Structural steel foundations, reinforced concrete Electrical Instruments Battery-limits building and service Excavation and site preparation Auxiliaries Total physical plant Field expense Engineering Direct plant costs Contractor’s fees, overhead, profit Contingency CFC: total fixed-capital investment

TABLE 9-52

Battery-limit installations

Solids processing

Solids-fluid processing

Fluid processing

Solids processing

Solids-fluid processing

Fluid processing

1.00 0.19–0.23 0.07–0.23

1.00 0.39–0.43 0.30–0.39

1.00 0.45 0.16

1.00 0.39 0.31

0.13–0.25 0.03–0.12 0.33–0.50 0.03–0.18 0.14–0.30 2.37 0.10–0.12

0.08–0.17 0.13 0.26–0.35 0.08–0.22 0.48–0.55 2.97 0.35–0.43 0.35–0.43 3.73 0.09–0.17 0.39 4.27

1.00 0.76 0.33 0.28 0.09 0.13 0.45

0.10 0.09 0.25 0.13 0.40 2.58 0.39 0.33 3.30 0.17 0.34 3.81

0.10 0.13 0.39 0.10 0.55 2.97 0.34 0.32 3.63 0.18 0.36 4.17

1.00 0.27–0.47 0.66–1.20 0–0.13 0.09–0.11

2.48 0.30–0.33 0.26 3.06

Included above 3.04 0.41 3.45 0.17 0.36 3.98

0.18–0.34 0.10 0.70 3.50 0.41 0.33 4.24 0.21 0.42 4.87

Factor Method of Miller (Based on Delivered-Equipment Costs = 100)* Battery-limit costs (range of factors in percent of basic equipment); average unit cost of main-plant item (MPI)

Field erection of basic equipment

Equipment foundations and structural supports

Piping, including ductwork but excluding insulation

Insulation of equipment only

Insulation of piping only

High percentage of equipment involving high field labor Average (mild steel) equipment High percentage of corrosion materials and other highunit-cost equipment involving little field erection High: predominance of compressors or mild steel equipment requiring heavy foundations Average: for mild steel fabricated-equipment solids Average: for predominance of alloy and other high-unitprice fabricated equipment Low: equipment more or less sitting on floor Piling or rock excavation High: gases and liquids, petrochemicals, plants with substantial ductwork Average for chemical plants: liquids, electrolytic plants Liquids and solids Low: solids Very high: substantial mild steel equipment requiring lagging and very low temperatures High: substantial equipment requiring lagging and high temperatures (petrochemicals) Average for chemical plants Low Very high: substantial mild steel piping requiring lagging and very low temperatures High: substantial piping requiring lagging and high temperatures (petrochemicals) Average for chemical plants Low

Under $9000

$9000 to $15,000

$15,000 to $21,000

$21,000 to $30,000

$30,000 to $39,000

$39,000 to $51,000

$51,000

23/18

21/17

19.5/16

18.5/15

17.5/14.2

16.5/13.5

15.5/13

18/12.5

17/11.5

16/10.8

15/10

14.2/9.2

13.5/8.5

13/8

12.5/7.5

11.5/6.7

10.8/6

10/5.5

9.2/5.2

8.5/5

8/4.8

17/12

15/10

14/9

12/8

10.5/6

12.5/7

11/6

9.5/5

8/4

7/3

7/3

8/3

8.5/3

7.5/3

6.5/2.5

5.5/2

4.5/1.5

5/0

4/0

3/0

2.5/0

2/0

1.5/0

1/0

105/65

90/58

Increase above values by 25 to 100% 80/48 70/40 58/34

50/30

42/25

65/33

58/27

48/22

40/16

34/12

30/10

25/9

33/13 13/5 13/10

27/10 10/4 11.5/8.5

22/8 8/3 10/7.4

16/6 6/2 9/6.2

12/5 5/1 7.8/5.3

10/4 4/0 6.8/4.5

9/3 3/0 5.8/3.5

10.3/7.5

9/6.3

7.8/5.2

6.7/4.2

5.7/3.4

4.7/3.8

4.8/2.5

7.8/3.4 3.5/0 22/16

6.5/2.6 2.7/0 19/13

5.5/2.1 2.2/0 16/11

4.5/1.7 1.8/0 14/9

3.6/1.4 1.5/0 12/7

2.9/1.1 1.2/0 9/5

2.2/.8 1/0 6/3.5

18/14

15/12

13/10

11/8

9/6

7/4

4.5/2.5

16/12 14/8

14/10 12/6

12/8 10/5

10/6 8/4

8/4 6/3

6/2 4/2

4/2 2/1

FIXED-CAPITAL-COST ESTIMATION TABLE 9-52

9-71

Factor Method of Miller (Based on Delivered-Equipment Costs = 100) (Concluded ) Battery-limit costs (range of factors in percent of basic equipment); average unit cost of main-plant item (MPI) Under $9000

All electrical except building, lighting, and instrumentation

Instrumentation*

Miscellaneous, including site preparation, painting, and other items not accounted for above

Electrolytic plants, including rectification equipment Plants with mild steel equipment, heavy drives, solids Plants with alloy or high-unitcost equipment, chemical and petrochemical plants Substantial instrumentation, central control panels, petrochemicals Miscellaneous chemical plants Little instrumentation, solids Top of range—large complicated processes; bottom of range—smaller, simple processes

$9000 to $15,000

$15,000 to $21,000

$21,000 to $30,000

$30,000 to $39,000

$39,000 to $51,000

55/42

50/38

45/33

40/30

35/26

26/17

22.5/15

19.5/12.5

17/10

14/8.5

12/7

10/6

18/9.5

15.5/8.5

13/6.5

11/5.5

9/4.5

7.3/3.5

6/2.5

58/31

46/24

37/18

29/13

23/10

18/7

32/13 21/9

26/10 17/7

20/7 13/5

15/5 10/3

11/3 7/2

8/2 5/1

$51,000

Range for all values of basic equipment is 6 to 1%

Building evaluation when most process units are located inside buildings

Buildings—architectural and structural, excludes building services†

High, brick and steel

Medium

Economical

+4

+2

0

Very high unit cost equipment

Mostly alloy steel

Mixed materials

−3

−2

−1

Very high

Intermediate

Atmosphere

−1

0

Quality of construction

Type of equipment

−2

Operations pressures cost equipment, chemical

Evaluation

Costly carbon steel 0

Building class = algebraic sum = Average unit cost of MPI

Cost of process Units inside buildings

Building class

Under $9000

$9000 to $15,000

$15,000 to $21,000

$21,000 to $30,000

$30,000 to $39,000

$39,000 to $51,000

$51,000

+2 +1 to −1 −2

92/68 72/49 50/37

82/61 62/43 44/33

74/56 56/38 40/29

67/49 51/33 35/25

59/44 45/29 30/21

52/39 41/26 27/18

46/33 36/21 23/15

37/16

32/13

28/11

24/8

20/6

17/4

14/2

Open-air plants with minor buildings Building services‡

Compressed air for general service only Electric lighting Sprinklers Plumbing Heating Ventilation: Without air conditioning With air conditioning Total overall average§

High

Normal

Low

4 18 10 20 25

1a 9 6 12 16

0.5 5 3 3 8

18 45 85

8 35 55

0 25 20

The above factors apply to those items normally classified as building services. They do not include (1) services located outside the building such as substations, outside sewers, and outside water lines, all of which are considered to be outside the battery limit as well as outside the building; and (2) process services. *Courtesy C. A. Miller of Canadian Industries Ltd. and the American Association of Cost Engineers. NOTE: The average unit cost of the main-plant items is the total cost of the MPI divided by the total number of items. Figures include up to 3 percent for BL outside lighting, which is not covered in building services. *Total instrumentation cost does not vary a great deal with size and hence is not readily calculated as a percentage of basic equipment. This is particularly true for distillation systems. If in doubt, detailed estimates should be made. †When building specifications and dimensions are known, a high-speed building-cost estimate is recommended, especially if buildings are a significant item of cost. If a separate estimate is not possible, evaluate the buildings as shown before selecting the factors. ‡The following factors are for battery-limit (process) buildings only and are expressed in percentage of the building architectural and structural cost. They are not related to the basic equipment cost. §The totals provide the ranges for the type of building involved and are useful when individual service requirements are not known. Note that the overall averages are not the sum of the individual columns.

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TABLE 9-53

Estimate Using Factors from Table 9-51

Details (solids-fluid, grass-roots plant)

Factor assumed

Cost, $

Percentage of total

1.00 0.41 0.34 0.13 0.13 0.30 0.15 0.52 2.98 0.39 0.39 3.76 0.13 0.39 4.28

1,000,000 410,000 340,000 130,000 130,000 300,000 150,000 520,000 2,980,000 390,000 390,000 3,760,000 130,000 390,000 4,280,000

23.4 9.6 8.0 3.0 3.0 7.0 3.5 12.2 69.7 9.1 9.1 87.9 3.0 9.1 100.0

Equipment, delivered Installed Piping Electrical Instruments Battery-limit building and service Excavation and site preparation Auxiliaries Total physical plant Field expense Engineering Direct plant costs Contractor’s fees, overhead, profit Contingency Total fixed-capital investment

TABLE 9-55 Location Factors for Chemical Plants of Similar Functions (1993 Values)

Australia Austria Belgium Canada Central Africa Central America Denmark Finland France Germany Greece India imported element indigenous element Ireland Italy Japan Malaysia Middle East New Zealand North Africa imported element indigenous element Norway Portugal South Africa South America Spain Sweden Switzerland Turkey United Kingdom United States

TABLE 9-54 Typical Factors with Separation of Materials and Labor* Total factor Equipment delivered Installation Instruments installed Piping Foundations and steel Insulation painting Electrical Battery-limit building Site preparation Auxiliaries Physical-plant cost

1.00 0.09 0.13 0.29 0.18 0.11 0.18 0.21 0.08 0.55 2.82

Engineering and home office Field expense Direct plant cost

0.31 0.43 3.56

Contractor’s fees Contingency Fixed-capital cost

0.17 0.39 4.12

Materials factor

Labor factor

0.09 0.155 0.08 0.025 0.06 0.13

0.09 0.04 0.135 0.10 0.085 0.12 0.08

0.01 0.30

0.30 0.13

1.04 0.85 0.70 1.14 1.51 1.20 0.85 0.88 0.73 0.76 0.80 0.80 0.25 0.70 0.79 1.46 0.42 0.84 1.27 0.65 0.44 0.92 1.00 0.90 1.36 0.83 0.75 0.94 0.80 0.76 1.00

NOTE: 1. Increase a factor by 10 percent for each 1000 mi or part of 1000 mi that the new plant is distant from a major manufacturing or import center, or both. 2. When materials or labor, or both, are obtained from more than a single source, prorate the appropriate factors. 3. Investment incentives have been ignored.

*Based on the data of H. C. Bauman, Fundamentals of Cost Engineering in the Chemical Industry, Van Nostrand Reinhold, New York, 1964, p. 295, for essentially carbon steel equipment.

A. Pikulik and H. E. Diaz [Chem. Eng., 84, 106–122 (Oct. 10, 1977)] presented a graphical method for estimating the fabricated cost of distillation columns and pressure vessels, storage tanks, fired heaters, pumps and drivers, compressors and drivers, and vacuum equipment. TABLE 9-56

Factor (United States = 1.0)

Location

Equipment Costs The cost of delivered equipment forms the basis of most methods of estimating the fixed-capital cost. The equipment required can usually be divided into (1) processing equipment, (2) equipment for handling and storage of raw materials, and (3) finished products handling and storage equipment. Quotations for equipment costs from fabricators or suppliers are the most accurate. Therefore most companies base their costs on (1) quotations from fabricators, (2) past purchase records updated with

Factors for Individual Items* Exchangers Details

Furnaces

FOB equipment Piping Concrete Steel Instruments Electrical Insulation Paint Total materials = M Erection and setting (L) X, excluding site preparation and auxiliaries (M + L) Freight, insurance, taxes, engineering, home office, construction Overhead or field expense Total module factor

1.00 0.18 0.10 0.04 0.02 1.34 0.30 1.64 0.60 2.24

Shell and tube 1.00 0.46 0.05 0.03 0.10 0.02 0.05 1.71 0.63 2.34 0.08 0.95 3.37

Vessels Air-cooled

Vertical

Horizontal

Pump and driver

Compressor and driver

1.00 0.18 0.02

1.00 0.61 0.10 0.08 0.12 0.05 0.08 0.01 2.05 0.95 3.00 0.08 1.12 4.20

1.00 0.42 0.06

1.00 0.30 0.04

1.00 0.21 0.12

0.06 0.05 0.05 0.01 1.65 0.59 2.24 0.08 0.92 3.24

0.03 0.31 0.03 0.01 1.72 0.70 2.42 0.08 0.97 3.47

0.08 0.16 0.03 0.01 1.61 0.58 2.19 0.08 0.97 3.24

0.05 0.12 0.01 1.38 0.38 1.76 0.70 2.46

*From K. M. Guthrie, Chem. Eng., 76, 114–142 (Mar. 24, 1969). Based on FOB equipment cost = 100 (carbon steel).

Tanks 1.00

1.20 0.13 1.33 0.08 1.41

FIXED-CAPITAL-COST ESTIMATION appropriate cost indices, and (3) exponential methods of adjusting prices for capacity changes. A large number of graphs for estimating the costs of various types of process equipment and auxiliaries are presented in the excellent text by M. S. Peters and K. D. Timmerhaus (Plant Design and Economics for Chemical Engineers, 4th ed., McGraw-Hill, New York, 1991, Chaps. 14–16) at January 1990 prices. Information on process equipment costs also appears, from time to time, in various journals (see Table 9-50). Although these published cost data are extremely useful for making rapid estimates, no published data can compete with the detailed and usually confidential cost records of large companies. Piping Estimation The cost of fabrication and installation of process-plant piping appears to range from 18 to 61 percent of the FOB equipment cost as indicated in Table 9-56. This would normally represent about 7 to 15 percent of the installed plant cost and is obviously a significant item. The various available piping-estimation methods are as follows: 1. Detailed pricing from piping drawings 2. Guthrie method 3. Dickson N method 4. Pricing by weight of specific types of pipe 5. Price estimation by cost per joint 6. Pricing as a factor of equipment cost 7. Pricing as a factor of total plant installed cost The first five methods are applicable only after rigorous circuit analysis and when piping layouts and isometric drawings or scale models are available for quantity takeoff (e.g., pipe size, length, and specification, flanges and valve count, etc.). Guthrie’s method [K. M. Guthrie, Chem. Eng., 76, 201–216 (Apr. 14, 1969)] is mainly graphical, using average mid-1968 costs for a United States Gulf Coast location. The Dickson N method [R. A. Dickson, Chem. Eng., 54, 121–123 (November 1947)] is a variation of the detailed price takeoff. Various circuits for each type of pipe are completely priced for a base size. Another chart gives an N factor for all other pipe sizes. Multiplying the cost of the circuit for the base size by the appropriate N factor yields the estimated cost of the new circuit of the desired pipe size. The method depends for its accuracy on periodic repricing of the base-size circuits in order to keep the base charts up to date. Estimating by weight requires virtually complete takeoff, including weight calculations and a full record of past costs on this basis. Its only advantage lies in the time saved in the detailed estimates of the cost of piping components. Estimating by cost per joint depends on the accumulation of past data, analyzed and conveniently correlated for use. The main advantage of the method lies in the fact that good engineering flow sheets can be used for the estimation. Figure 9-42 is a plot of the number of labor-hours of field erection time per joint against the nominal pipe size of shop-fabricated carbon steel and low-alloy pipe. The unit of work measurement used in this

Labor-hours required to erect large quantities of shop-fabricated steel and low-alloy piping.

FIG. 9-42

TABLE 9-57

9-73

Components of Total Installed Piping Cost

Material Labor Indirect costs Factors applicable to labor-hours Crafts involved in piping erection

Pipe, valves, fittings, nuts, bolts, gaskets, and hangers Cut, erect, align, fit, bolt, thread or weld, and test Handle and haul, store, scaffold, lost time, tools and rentals, contractor’s overhead and profit Craft rate, productivity, height, and complexity Pipefitters, laborers, carpenters, warehouse workers, teamsters, and operating engineers

method is the pipe joint, requiring two joints for couplings and valves, three for tees, etc., as most of the labor-hours involved in pipe erection are expended in making connections. The additional costs of handling, suspending, and placing lengths of pipe in position are included in the chart. It should be noted that Fig. 9-42 gives labor-hours only. Material costs must be obtained by price takeoff from drawings on which all valves and instrument connections are shown. Pipe lengths and fittings are taken off by referring to the equipment-layout plan and elevation drawing. The graph of Fig. 9-42 can be updated by using actual costs for a specific job, in which case the labor cost per joint represents a total labor cost including all the factors applicable to labor shown in Table 9-57. It should be possible to analyze statistically uniform data from a number of complete jobs to determine the value of each factor for various project locations. Methods 6 and 7 are simpler procedures, using factors for estimating piping costs when neither flow sheets nor detailed piping drawings are available. Tables 9-51, 9-52, 9-54, and 9-56 include typical values of piping factors based on total equipment cost, delivered or FOB, as indicated in the particular table. These methods require some degree of judgment in selecting the appropriate factor, based on experience gained by comparing piping costs for similar previously installed process plants. A rough method of estimating the piping factor as a percentage of the total delivered cost of major process equipment (excluding instruments and electrical items) was presented by E. S. Sokullu in the form fP = 11φ1.6 P

(9-254)

where φP = (number of actual pipes on flow diagram)/(number of major process equipment units) [Chem. Eng., 76, 148–150 (Feb. 10, 1969)]. The equipment-unit method would appear to give more accuracy than the preceding methods, particularly for unfamiliar process arrangements. It requires the accumulation of piping costs for various sizes of main-plant items such as pumps, heat exchangers, evaporators, tanks, and columns. Basically it is assumed that piping designs for specific items are similar for most projects. Statistical analysis of such data shows good agreement with the more detailed takeoff pricing methods. Since for most processes the length of pipe used is a small proportion of the total piping cost, the assumption of an average length of piping per main-plant item, based on actual costs for several previous jobs, should give sufficient accuracy. Correction for escalation of costs can be carried out by using a single cost index, unlike methods 1 to 5. Most of the factorial methods of estimation given previously, with the exception of the method of Allen and Page (loc. cit.), tend to estimate costs which are based on carbon steel equipment or installations. Table 9-58 gives typical multiplying factors for converting carbon steel costs to equivalent-alloy costs for a few items of equipment. (Adapted from A Guide to Capital Cost Estimating, 3d ed., Institution of Chemical Engineers, Rugby, England, 1988, p. 70.) Electrical and Instrumentation Estimation These costs usually range from 4 to 10 percent of the total installed plant cost, with a median value of about 7.5 percent. As with piping estimation, the process design must be almost completed before detailed drawings and specifications can be prepared for estimating purposes. However, actual electrical costs can be up to 100 percent higher than estimated costs, and so it is important to attempt to maintain the accuracy range within reasonable limits.

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PROCESS ECONOMICS

TABLE 9-58 Typical Factors for Converting Carbon Steel Cost to Equivalent-Alloy Costs Material

Pumps, etc.

Other equipment

All carbon steel Stainless steel, Type 410 Stainless steel, Type 304 Stainless steel, Type 316 Stainless steel, Type 310 Rubber-lined steel Bronze Monel

1.00 1.43 1.70 1.80 2.00 1.43 1.54 3.33

1.00 2.00 2.80 2.90 3.33 1.25

Material Carbon steel shell and tubes Carbon steel shell, aluminum tubes Carbon steel shell, monel tubes Carbon steel shell, 304 stainless tubes 304 stainless steel shell and tubes

Heat exchangers 1.00 1.25 2.08 1.67 2.86

During the design stages, frequent changes in the type and sizes of equipment lead to delays in establishing electrical requirements. Hence it is very difficult to obtain a detailed estimate of the cost of the electrical part of the project. For order-of-magnitude and study estimates, an appropriate factor in the range 4 to 10 percent of the total installed plant cost can be used. However, for budget-authorization or preliminary estimates requiring an accuracy within 5 percent more accurate methods are necessary. The methods available for electrical estimates are as follows: 1. Detailed takeoff 2. Factored electrical cost as a percentage of total installed plant cost for specific types of plant 3. Unit pricing The detailed-takeoff method can rarely if ever be used. When detailed drawings are available, costs may be estimated by pricing materials and components from suppliers’ catalogs or, for special items, from quotations. Handbooks are available which give typical values of the labor-hours required to perform units of installation work, such as installation of switches, starters, motors, conduit wiring, and push buttons of various sizes, for both hazardous and nonhazardous areas. Labor rates can be obtained from various government statistical sources or elsewhere. For the United States the National Electrical Contractors Association publishes an excellent manual of electrical costs. From the complete plans and specifications, the estimator can take off materials, estimate the labor cost, apply appropriate factors for labor efficiency, productivity, and local conditions, and achieve good results. The factor estimate, if based on tested actual data, gives good results in the study estimate and often proves adequate at the preliminary estimate stage. It is essential to accumulate from past experience data showing actual electrical costs (1) as a percentage of total installed plant cost and (2) as a percentage of installed equipment costs. Studies of electrical installations for more than 100 plants (H. C. Bauman, Fundamentals of Cost Engineering in the Chemical Industry, Van Nostrand Reinhold, New York, 1964, p. 134) showed electrical costs ranging from about 4 to 11 percent of total plant cost, with a median for battery-limit process plants of 7.5 percent. The corresponding range based on installed equipment costs was 15 to 40 percent, with a median of 26 percent. Thus, it appears that there is a better correlation between electrical costs and total installed plant cost than with installed equipment costs. Table 9-59, taken from Bauman’s data, gives typical values of electrical costs as a percentage of total installed plant cost. Cost ranges for installed instrumentation costs are also included in Table 9-59, as these would form part of electrical costs. The ranges of values are rather wide, depending on the degree of automatic control required. Electrical costs involve four main components: (1) power wiring, (2) lighting, (3) transformation and service, and (4) instrument and control wiring. A breakdown of these component costs as a percentage of total electrical cost is given in Table 9-60.

TABLE 9-59 Electrical and Instrumentation as Percentage of Total Installed Plant Cost Electrical (process and service)

Instrumentation

Type of plant

Range, %

Median, %

Range, %

Median, %

Solids plants Grass-roots process Battery-limit process

3.7–10.7 4.0–7.9 4.3–10.1

5.4 % 5.9 % 7.5 %

0.3–6.0 1.9–4.3 0.1–7.9

0.8 3.2 3.7

TABLE 9-60 Component Electrical Costs as Percentage of Total Electrical Cost Component

Range, %

Median, %

Power wiring Lighting Transformation and service Instrument-control wiring

25–50 7–25 9–65 3–8

40 12 40 5

The unit-cost method can give a quick and accurate estimation, provided it is based on accumulated data from many jobs on various types of plant. The actual data are analyzed to provide unit-cost information for electrical components as follows: 1. Total installed cost per motor 2. Total installed cost per lighting outlet by type 3. Total installed cost of receptacles by type (incandescent, fluorescent, etc.) 4. Total installed cost for each wired instrumentation point 5. Total installed cost for each unit of transformation 6. Total installed cost per lineal foot of distribution by type (overhead bare and insulated, underground) 7. Total installed cost of each interlock point Each unit cost contains all the costs involved in the installation of that unit. For motors installed costs include the starter, conduit, wire, and a proportionate share of the service panelboard and busbars. The motor cost is not included since this will be part of the equipment cost. In the case of lighting, the installed cost includes the lighting fixtures, the conduit and wire, and a proportional share of the lighting panelboard and service switching costs. Auxiliaries Estimation Chemical-plant auxiliaries normally include all structures, equipment, and services which are not directly involved in the process. Within this broad range there are two major classifications, utilities and service facilities. The typical cost range for auxiliaries is from 20 to 40 percent of the total installed plant cost. For a small continuous-process plant making a single product, the cost of auxiliaries would lie in the lower part of the range, while for large multiprocess grass-roots plants the factor would tend to be near the upper limit of the range. Auxiliary Buildings Typical variations in the cost of auxiliaries for a variety of process plants are given in Table 9-61. The widest variation is shown for auxiliary buildings, which is not surprising in view of the many types and quality of materials and the wide variation in methods of construction. For example, amenities buildings such as offices, cafeteria, first-aid rooms, gatehouses, and control rooms would necessitate fairly expensive brick and plaster-wall construction. On the other hand, services buildings such as substations, switch rooms, and pump or compressor houses would cost about 5 to 10 percent less. Provision of air conditioning, furniture, and equipment for cafeteria, laboratory, and office buildings would add about 50 percent to the basic cost of the building. Steam-Generating Facilities These form the second largest investment item for chemical-plant auxiliary equipment. Variations in capacity, location indoors or outdoors, the type of fuel used, pressure and temperature levels, and the type of process served have an important effect on actual cost as well as on cost relative to other auxiliary items. Package boiler installations can be purchased as shop-built units which are assembled, piped, and wired ready to be erected on the owner’s foundations. They are available in units up to about 136,000 kg/h (300,000 lb/h), although units larger than about 45,360

FIXED-CAPITAL-COST ESTIMATION TABLE 9-61 Typical Ranges of Auxiliary Facilities as Percentage of Total Installed Plant Cost Grass roots and large additions Auxiliary buildings Steam generation Refrigeration, including distribution Water supply, cooling, and pumping Finished-product storage Process-waste systems Raw-materials storage Steam distribution Electrical distribution Air compressor and distribution Water distribution Fire protection system Water treatment Railroads Roads and walks Gas supply and distribution Sanitary-waste disposal Communications Yard and fence lighting

River-water pumping and filtering installations can be approximately correlated by

Range, %

Median, %

3–9 2.6–6 1–3 0.4–3.7 0.7–2.4 0.4–1.8 0.3–3.2 0.2–2 0.4–2.1 0.2–3.0 0.1–2 0.3–1.0 0.2–1.1 0.3–0.9 0.2–1.2 0.2–0.4 0.1–0.4 0.1–0.3 0.1–0.3

5.0 3.0 2.0 1.8 1.5 1.1 1.1 1.0 1.0 1.0 0.9 0.7 0.6 0.6 0.6 0.3 0.3 0.2 0.2

kg/h (100,000 lb/h) may be available only on a semierected basis. It is usually necessary to obtain firm price quotations that take into account all the factors involved. Housing the boiler installations in buildings will generally increase the cost by about 7 to 9 percent per kilogram-hour (15 to 20 percent per pound-hour) of steam-generating capacity over the cost for outdoor installations or for installations in existing buildings. For most chemical plants, process steam is used at pressures of 1.825 MN/m2 (250 psig), saturated or lower. When combined heat and power generation is economically justified, the steam may be generated at about 5.96 MN/m2 (850 psig) appropriately superheated and used to drive back-pressure steam turbines passing out process steam at the required pressure level. Electricity A reliable and adequate electricity supply is usually available through government or private enterprises. Owing to the increasing cost of purchased electricity, many companies have installed combined heating and power (CHP) generation systems. A cogeneration plant may (1) be owned and operated by the industrial user or the utility, (2) serve or be isolated from one or more industrial users, or (3) form an integral part of the local utility grid. Typical costs for generating steam range from $7.90 to $9.50 per 1000 kg ($3.60 to $4.30 per 1000 lb) at 3550 kPa (500 psig); $3.70 to $7.70 per 1000 kg ($1.70 to $3.50 per 1000 lb) at 790 kPa (100 psig); and $2.00 to $3.70 per 1000 kg ($0.90 to $1.70 per 1000 lb) for exhaust steam. The above costs apply for a Marshall and Swift index of 1000. For most plants, electric distribution systems start at the power company’s service point on the plant’s property. The choice of an electrical system will depend on many factors relating to the particular project. A wide range of items, from switchgear, transformers, and motor control centers to cabling, earthing, and lighting, are required. Normally, competitive quotations would be obtained to take an estimate beyond the study stage. Various types of overhead and underground distribution systems may be used depending on local conditions. Generally, an overhead system will incur only about 30 percent of the cost of an underground distribution system. Water Systems These systems usually form the third highest cost item in chemical-plant auxiliaries, with cooling towers representing the largest part of the investment. Although the installed cost increases with the terminal temperature range, an approximate cost correlation is given by CCT = 100q0.87

9-75

(9-255)

where CCT is the installed cost of the cooling tower in United States dollars for a Marshall and Swift (M & S) index of 1000 and q is the capacity in United States gallons per minute over the range from (1)(103) to (1)(105) U.S. gal/min.

CRW = 0.65q0.81

(9-256)

where CRW is the installed cost of the river-water system in United States dollars for an M & S index of 1000 and q is the capacity over the range from (4)(105) to (1)(107) U.S. gal/day. Similarly, installed costs of water-softening systems can be correlated in United States dollars (M & S = 1000) as follows: CWS = 1380q0.44

(9-257)

over the range of capacity from (3)(107) to (1)(109) U.S. gal/day and of demineralizing systems by CDS = 0.17q1.9 (9-258) over the range of q values from (1)(104) to (4)(105) U.S. gal/day. Actual water-treatment costs may vary widely from the above, depending on the quality of the water, the percentage of dissolved solids, and the total hardness. Refrigeration Systems These systems are being used increasingly in chemical processing. Installed costs of packaged mechanical units in United States dollars (M & S = 1000) can be approximately correlated by CRS = 4630q0.73 (9-259) where q is the capacity in tons of refrigeration over the range from 10 to 1000 tons. One ton of refrigeration is equivalent to a rate of heat removal of 3.517 kW (12,000 Btu/h). Roads and Walkways The cost of roads and walkways in chemical plants is difficult to estimate, since these vary with type of construction and thickness of applied cover. Some typical unit costs for roads are as follows: For 305-mm (12-in) gravel base covered with 76-mm (3-in) asphalt, the cost is $17.10 per square meter ($14.30 per square yard); for a reinforced concrete slab with a 152-mm (6-in) subbase, the cost is from $28.40 to $35.10 per square meter ($23.80 to $29.30 per square yard), depending on the thickness of concrete (for M & S = 1000). Installed costs for railroads, including switches and frogs, can be roughly estimated as follows (for M & S = 1000): Linear meters

Linear feet

$/meter

$/foot

152–305 305–915 915–3050 Above 3050

500–1000 1000–3000 3000–10000 Above 10000

230.00 210.00 200.00 187.00

70.00 64.00 61.00 57.00

Usually the cost of roads and walkways amounts to 0.2 to 1.2 percent of the fixed capital cost with a typical value of 0.6 percent. Similarly, railroads cost 0.3 to 0.9 percent of the fixed capital cost, having an average value of 0.6 percent. Use of Computers in Cost Estimation A large part of estimation consists of the collection and storage of data obtained from records of actual plant costs. The data then must be correlated and updated and the required information rapidly retrieved for use in further cost estimations. A comprehensive survey (C. J. Liddle and A. M. Gerrard, The Application of Computers to Capital Cost Estimation, Institution of Chemical Engineers, London, 1975, pp. 6–17) suggests that large chemical manufacturers, equipment vendors, and some contractors are using the computer increasingly for data retrieval, followed by simple correlation and the application of factorial methods to cost estimation. In the case of equipment vendors, the computer’s contribution appears to be particularly worthwhile owing to the elimination of estimating errors in producing price quotations. Several companies have developed an automated quotation system to overcome delay and inaccuracy in estimating and bidding. Such systems appear to have been developed by firms already possessing significant computing facilities, since the cost of computer time is small compared with the cost of the computer. Qualitatively, operating costs for an automated quotation system appear to be about half of those of a manual system

9-76

PROCESS ECONOMICS

of price quotation. The methods of estimation used are based on the manual methods described previously. Several of the larger chemical manufacturers, particularly those in the petrochemicals field, have developed computer packages based on manual methods of factorial estimating. Usually the input data consists of the cost of each main-plant item (MPI) obtained from quotations or historical records. The program then estimates the costs of erection, piping, instrumentation, electricals, civil engineering, and lagging for each (MPI) in turn by adding a series of factors. These account for the complexity of the process and the constructional difficulties for each (MPI) to produce an estimate of the overall plant cost. It is obviously necessary to introduce appropriate inflation indices to bring the estimated costs up to date. For process plants, it is often possible to use these cost-estimation programs with a design or flow-sheet program to optimize on a particular component or even over the whole plant (A Guide to Capital Cost Estimating, 3d ed. Institution of Chemical Engineers, Rugby, England, 1988, pp. 48–49). However, it must be remembered that optimization is expensive on computer time, although there appear to be no data available on the cost effectiveness of the computer in this area. It is also possible to incorporate the capital-cost estimate in an investment evaluation involving forecasts of expenses and revenue from sales. Thus, by means of the computer design and costing can be brought together. There is an immediate feedback of information, resulting in improved design and lower costs. In some types of plants, costing data can be fed in as a subroutine to the design programs. All these possibilities assume that the total cost of using the computer is not unreasonable. Startup Costs Startup problems can reduce aftertax earnings during the early years, the most serious effect being to delay the startup of production, causing a loss of earnings. An accurate estimate of startup time and cost can help in (1) predicting the availability of new products, (2) planning market entry, and (3) estimating the overall profitability because of more accurate cash-flow forecasts and (NPV) calculations [R. P. Feldman, Chem. Eng., 76, 87–90 (Nov. 3, 1969)]. Startup costs are defined as the total of those costs directly related to bringing a new production facility into operation. They should not include the costs of entering or expanding a business. Hence startup costs include the following: 1. All expenses due to changes in process and equipment after completion of construction but excluding those due to changes in project scope 2. All labor costs after completing construction, especially those incurred in checking the functioning of equipment 3. All costs incurred during the startup period but excluding normal operating expenses 4. Expenses for training plant personnel even if incurred before startup has officially begun 5. All research and development costs incurred during startup The following expenses should not be included: 1. Marketing costs 2. Expense of training sales representatives 3. Penalties for shipping outside optimum freight areas 4. Costs associated with starting a new company 5. Lost sales unless there is a contract with a penalty 6. Profit lost due to timing Startup time may be defined as the time span between the end of construction and the beginning of normal operation. Hence it should start when the contractor finishes the whole plant or a specified section of it to enable comparisons to be made with other startup times. It is usual to define “normal” operation as (1) operations at a certain percentage of design capacity, (2) a specified number of days of continuous operation, or (3) the capability of making products of a specified purity. It is essential for project and production management to agree beforehand on the definition to be applied. Obtaining agreement on the definition of “normal” operation is important since (1) it sets a target for field personnel, (2) it ensures that everyone is striving for the same target, (3) it permits comparisons with other plants, and (4) it determines a cutoff point for completion of startup. It may be neces-

sary to wait until the plant is running well to obtain the actual total cost of startup. For control purposes it is advisable to estimate startup cost and time beforehand and then try to stay within the estimates. The general parameters which can be used to estimate startup cost CSU, which are usually between 2 and 20 percent of the battery-limit fixed-capital cost, are as follows: 1. Direct fixed-capital cost for plant (battery-limit capital), (CFC)BL 2. Newness of process and technology, b 3. Newness of type and size of equipment, c 4. Labor quality and quantity, d 5. Interplant dependency, e Hence startup cost may be expressed as CSU = (CFC)BL [0.10 + b + c + d + ne]

(9-260)

When applied to large air-separation and ammonia plants (1000 to 1400 metric tons/day), the following values for the parameters can be used: b = 0.05 for a radically new process = 0.02 for a relatively new process = −0.02 for an old process c = 0.07 if radically new = 0.04 if very new = 0.02 if relatively new = −0.03 if old d = 0.04 if labor is in very short supply = 0.02 if labor is in short supply = −0.1 if labor is in surplus supply e = 0.04 if plant is very dependent on another = 0.02 if moderately dependent on another = −0.02 if independent and n = number of plants or sections making up the process chain Startup time tSU for these plants may be estimated from construction time tC by developing an equation similar to Eq. (9-260): tSU = tC(0.15 + b + c + d + ne)

(9-261)

For the same type of plant the values of the parameters are b = 0.15 for a radically new process = 0.05 for a relatively new process = −0.01 for an old process c = 0.15 if radically new = 0.08 if very new = 0.05 if relatively new = −0.01 if old d = 0.15 if labor is in very short supply = 0.05 if labor is in short supply = −0.01 if labor is in surplus supply e = 0.25 if plant is very dependent on another = 0.10 if moderately dependent on another = −0.02 if independent and n = number of plants or sections making up the process chain It should be noted that these values are based on previous experience with certain types of plants, but appropriate values which apply to other processes and locations could be selected. Construction Time The duration of construction is difficult to estimate owing to the large number of variables involved. In general, estimates of construction time tend to be overoptimistic, especially for

FIXED-CAPITAL-COST ESTIMATION larger projects. Usually projects costing less than $5 million at 1993 prices can be completed in 10 to 18 months, while those costing more than $10 million may take from 18 to 42 months to complete. Delays of up to 12 months behind schedule are quite possible, particularly when there are labor problems. As mentioned previously, such delays will usually result in increased construction costs. Often, a more serious effect is loss of earnings resulting from a delayed startup. Both of these factors increase the payback period and reduce the attainable net present value and discounted-cash-flow rate of return of the project. Project Control Having made a good estimate of the capital cost and the expected construction time, it is essential to introduce an effective system for controlling expenditure of time and money during construction. Good capital-cost control can cut down expenditures even when the definitive estimate is not very accurate. It is most important for management to receive early warnings if overruns in expenditure or time are likely to occur. Effective cost control should start from the beginning of the project at the research and development stage and continue through the design and estimating stages to initial operation of the plant [J. W. Hackney and K. K. Humphreys (eds.), Control and Management of Capital Projects, 2d ed., McGraw-Hill, New York, 1991]. The stages discussed here are the later steps after authorization of funds and during project construction. After the purchasing department has placed the orders for equipment and materials, speed and efficiency during the construction stage is most important in ensuring the financial success of the project. Field expenditure during construction can amount to 30 to 60 percent of the fixed-capital cost and includes the costs of all labor, installed equipment, and materials together with associated process piping, electrical instrumentation, and insulation. Construction therefore requires efficient execution and prompt feedback of progress information, necessitating a good cost-control system. Figure 9-43 shows the flow of information needed for cost control. The chart assumes a definitive estimate which has been linked to a standard code of accounts. As construction proceeds, up-to-date costcontrol reports are supplied to the field cost engineer. From the home office the engineer receives monthly reports of engineering and drafting labor-hours used and money expended, together with a list of drawings and specifications completed up to that time. Monthly expenditures and current commitments come in coded detail from the job ledgers of the accountants. Timekeepers’ records give details of craft and nonmanual labor-hour expenditures. Quantities of equipment and material held on site are reported daily by quantity surveyors to the construction superintendent. All purchase orders are posted in the ledgers as current commitments, whether they are placed at the home office or in the field, and an up-to-date warehouse inventory is maintained.

FIG. 9-43

Information flow for cost control.

9-77

At the end of each month, the field cost engineer collects all current information on a detailed cost report form. As these are actual costs, they can be used to estimate future job costs to completion. Daily reports of unit-cost progress for concrete, excavation, masonry, steel, piping, and electrical work, etc., are then used to predict possible overruns or underruns for the various items. Analysis and comparison with the original estimate point out trouble spots for early attention. If an item is running into difficulty, it is red-flagged to the resident and project engineers for remedial action. In practice, the existence of a tight cost-control system tends to spread a cost consciousness among the personnel involved in the project. Such an awareness, even in construction-equipment maintenance and job housekeeping, can lead to efficient cost control throughout. Cost reports should be brief but informative, preferably in summary form. They should report expenditures and commitments, estimated costs to complete, and expected overruns or underruns of the authorized budget for each important item of cost. Brief notes should emphasize significant deviations from predicted cost. Any large, persistent overrun should have already been investigated and reported to the project and construction managers for immediate attention. If an expected overrun cannot be avoided, the current summary cost report should serve as justification for a request for additional funds. When organized efficiently, the cost-control system should require no more paperwork than for the normal construction procedure. The cost of cost control appears to vary between 0.2 and 0.5 percent of the total project value. Proper use of the normal records available for craft-labor time, warehouse-inventory control, and the usual accounting purposes should be adequate. The savings achieved by good cost control should far exceed the additional costs of operating the system. Additional details on the technique are given by H. C. Bauman (Fundamentals of Cost Engineering in the Chemical Industry, Van Nostrand Reinhold, New York, 1964, pp. 190–196). Scheduling construction to ensure that the project is completed in the shortest possible time is an essential part of project control. The project-control estimate defines to a large extent the constructiontime schedule. It is then possible to prepare a master schedule from the control estimate by carefully sequencing and synchronizing the installation work according to past experience. Drawings are usually completed in predictable order owing to the dependence of certain designs on preceding work. The normal order of completion of drawings and specifications is (1) site work, (2) substructures, (3) equipment and building superstructures, (4) equipment layouts, (5) piping, (6) insulation, (7) instrumentation, and (8) electrical work. Detailed planning and scheduling then involve establishing the items of work required and determining the correct sequence of work and the number of persons required to perform each item of work.

9-78

PROCESS ECONOMICS

From this information it is possible to prepare bar charts by using a 4-week month and noting on the chart the interdependence of the various functions. The starting time for each class of work is fixed on the chart, and the duration is calculated from the labor-hours allocated to that work from the control estimate. Work should progress smoothly as time elapses, but the operations must be linked by the order of necessary precedence. Starting times for the various items of work will be staggered as drawings are released and also to smooth out labor requirements. Sketching the bar chart is commenced by inserting the arrival dates of key items, observing any necessary precedence. Estimates of the duration of erection time can be made to obtain the starting date for process piping. Since the supporting structure must be in place when the key item arrives, it is possible to work back along the bar chart to the preparation of the foundations. From the complete bar chart, built up in a similar manner, a tentative startup date can be set after allowing a few weeks for tidying up bits and pieces. Some activities can be speeded up, but it is necessary to estimate the increased cost of so doing. Actual progress made with construction work can be indicated on the bar chart by filling in the open bars according to the percentage toward completion. Comparison of the actual progress bar for the whole project with the cumulative labor-hour curve indicates whether the job is ahead of schedule or not. If corrective action is required, effort should be concentrated on the key or critical items. Large projects will usually require network analysis using the critical-path method (CPM) or program evaluation and review technique (PERT) in the planning, scheduling, and progress-control stages. Examples of bar charts and a fuller description of network analysis are given by J. W. Hackney and K. K. Humphreys (eds.), Control and Management of Capital Projects, 2d ed., McGraw-Hill, New York, 1991. A detailed treatment of the use of PERT and CPM techniques as applied to contract bidding strategy and to project control is given by L. A. Swanson and H. L. Pazer (Pertsim: Text and Simulation, International Textbook, Scranton, Pa., 1969), who present a hand simulation technique based on probabilistic methods. Overseas Construction Costs Although Table 9-55 gives location factors for the construction of chemical plants of similar function in various countries at 1993 values, these may vary differentially over a period of time owing to local changes in labor costs and productivity. Hence, it is often necessary to estimate the various components of overseas construction costs separately. Equipment and material prices will depend on local labor costs and the availability of raw materials. If the basic materials have to be imported, costs in the source area become important and import duties and freight charges must be added. Equipment and material normally amount to about 40 to 45 percent of the costs of a typical chemical plant. In general, equipment and material costs are slightly cheaper in European countries and Japan, whereas in Mexico and Canada they are nearer the United States average. Construction labor makes up about 20 to 35 percent of total costs for a chemical plant. Table 9-62 compares average 1994 hourly rates TABLE 9-62

for various types of construction labor in several countries with those for the United States. Fringe benefits are known in countries other than the United States as “social charges”; they vary considerably in degree of coverage from country to country. Typical allowances in these countries include family benefits based on number of children, health service, maternity benefits, disability allowances, grants for funeral expenses, old-age and war pensions, unemployment benefits, and pension schemes. Additional fringe benefits may include paid holidays, starting allowances for new workers, relocation grants, severance pay, profit sharing, production bonuses, special gratuities, and sometimes housing allowances. It is essential to investigate the local situation thoroughly to determine the benefits payable and the additional cost on the basic hourly wage rate. Labor productivity is very much dependent on the health and well-being of the workers and also on the availability of laborsaving tools and construction equipment. The frequency of strikes, holidays, slowdowns, and political unrest will also depress productivity. Closedshop practices or demarcation disputes will also affect the productivity of labor. The use of standard equipment, parts, and methods tends to improve productivity. In a particular country, productivity will depend largely on the number of hours worked per week. Production will increase with the number of hours worked during the week, but as more overtime is worked, fatigue will produce a falloff in productivity. In the United States, construction craft labor usually work a normal 40 h/week. The United Kingdom operates a 40-h schedule, although there is strong pressure to reduce this to 38 h/week. European countries tend to work a normal 40 h/week, and some Far Eastern countries may work up to 45 h/week. Productivity of local craft labor also depends on the use and availability of modern mechanical tools and construction equipment. Normally, the low cost of labor in certain countries tends to cut out the purchase or hire of sophisticated laborsaving equipment and to encourage the employment of large pools of labor, particularly in developing countries such as India, Pakistan, southeast Asian countries, and many African countries. In turn, this usually leads to higher construction costs. The use of laborsaving equipment is prevalent in Canada, western Europe, Japan, and, to an increasing extent, the Middle East. Complete Plant Costs It is difficult to compare costs of domestic and overseas plants owing to the wide variation in types of plants and sizes and the rapid changes in technology. Useful data are scarce, and the following comparisons must be used with caution and then only for order-of-magnitude estimates of fixed-capital costs. The method uses a breakdown of costs for a typical chemical plant installed in the United States, as shown in Fig. 9-44. Costs of equipment, appurtenances, construction, and engineering with material and labor separate are given as a percentage of total installed United States costs. The four components of cost are defined as follows: Equipment includes all prefabricated machines, appliances, or systems such as tanks, heat exchangers, pumps, motors, switchgear, and boilers. Appurtenances are auxiliary items which cover materials, such as pipes, valves, fittings, conduit, wire, tubing, and insulation.

Comparative National Labor Costs

Hourly labor costs Germany Switzerland Belgium Netherlands Japan Austria United States France Italy Britain Ireland Spain Greece Portugal

$

Relative to U.S.

25.35 22.49 20.84 19.82 19.05 19.02 16.58 16.05 15.69 12.90 11.94 11.36 6.29 4.70

1.53 1.36 1.25 1.20 1.15 1.15 1.00 0.97 0.95 0.78 0.72 0.69 0.38 0.28

NOTE: Figures include fringe benefits. SOURCE: Adapted from Economist Intelligence

Unit January 1994.

FIG. 9-44

Typical breakdown of chemical-plant costs by major component.

FIXED-CAPITAL-COST ESTIMATION Construction expense includes the cost of construction equipment, tools, sheds, railroad trackage, road materials, welding machines, scaffolding, and timber, which are all used in construction but do not form a permanent part of the plant. Engineering is mainly labor but has a small component cost which can be classified with equipment and materials, such as tools, paper, pencils, and reproduction costs. In total, labor amounts to 34 percent and material to 66 percent of total installed costs. Table 9-63 uses the data of Fig. 9-44 to compare the relative fixedcapital costs for plant construction in other countries with those for the United States. The relative cost ratios were developed from data similar to those in Table 9-62. Labor ratios were corrected for the different local rates and hours per working week, job duration, and degree of mechanization available in other countries. Some of these factors are difficult to estimate, and the final “total” ratios give a reasonable order-of-magnitude value for relative construction costs for equivalent plants in the countries indicated. TABLE 9-64

9-79

TABLE 9-63 Relative Plant Construction Costs in Various Countries Compared with the United States Country United States England Italy Mexico Australia Canada France Germany Japan

Equipment

Material

Labor

Engineering

Total

0.28 0.26 0.22 0.26 0.38 0.32 0.27 0.26 0.29

0.38 0.41 0.32 0.35 0.54 0.44 0.40 0.36 0.35

0.26 0.18 0.31 0.35 0.29 0.31 0.22 0.32 0.22

0.08 0.05 0.05 0.04 0.09 0.08 0.06 0.06 0.04

1.00 0.90 0.90 1.00 1.30 1.15 0.95 1.00 0.90

The choice of an overseas manufacturing site involves the consideration of many political and economic factors in addition to costs. Table 9-64 gives a list of 92 items which should be taken into account when choosing a plant location for manufacturing abroad.

Factors in Choosing a Foreign Manufacturing Site

Economic factors Size of GNP and rate of growth Is there a working development plan? Resistance to recession Relative dependence on imports and exports Foreign-exchange position Balance-of-payments outlook Stability of currency; convertibility Remittance and repatriation regulations Balance of economy (industry-agriculture-trade) Size of market for your products; rate of growth Size of population; rate of growth Per capita income; rate of growth Income distribution Current or prospective membership in a customs union Price levels; rate of inflation Political factors Stability of government; its form Presence or absence of class antagonism Special political, ethnic, and social problems Attitude toward private and foreign investment Acceptability of United States investment by government Acceptability of United States investment by customers and competitors Presence or absence of nationalization threat Presence or absence of state industries Do state industries receive favored treatment? Concentration of influence in small groups Treaty of friendship or establishment with United States? Government factors Are fiscal and monetary policies sound? Freedom from bureaucratic red tape Fairness and honesty of administrative procedures Degree of antiforeign or anti–United States discrimination Fairness of courts Clear and modern corporate investment laws Patentability of your products Presence or absence of price controls Restrictions on 100 percent United States or foreign ownership Geographic factors Efficiency of transport (railways, waterways, highways) Port facilities Free ports, free zones, bonded warehouses Proximity of site to export markets Proximity of site to suppliers, customers Proximity to raw-material sources Existing supporting industry Availability of local raw materials Availability of power, water, gas Reliability of utilities Waste-disposal facilities Can exports be easily made? Can imports be easily made?

Geographic factors (Continued) Are plant sites readily available? Cost of suitable land Labor factors Availability of English-speaking managerial, technical, office personnel Availability of skilled labor Availability of semiskilled and unskilled labor Level of worker productivity Training facilities Outlook for increase in labor supply Degree of skill and discipline at all levels Tranquillity of labor relations Presence or absence of militant or Communist-dominated unions Degree of labor voice in management Freedom to hire and fire Compulsory and voluntary fringe benefits Social security taxes Total cost, including fringes, compared with alternative sites Compulsory or customary profit sharing Tax factors Tax rates (corporate and personal income, capital, withholding, turnover, excise, payroll, capital gains, customs, other indirect and local taxes) General tax morality Fairness and incorruptibility of tax authorities Long-term trend for taxes Taxation of export income and income earned abroad Tax incentives for new businesses Depreciation rates Tax-loss carry-forward and carry-back Joint tax treaties Duty and tax drawbacks when imported goods are exported Availability of tariff protection Capital-sources factors Availability of local capital Costs of local borrowing Normal terms for local borrowing Availability of convertible currencies locally Modern banking system Government credit aids to new businesses Availability and cost of export financing, insurance Do United States or European capital sources favor loans here? Business factors Availability of United States government investment insurance General business morality State of marketing and distribution system Are administrative procedures simple and effective? Normal profit margins in general, in your industry Competitive situation in your industry; is it cartelized? What are antitrust and restrictive practices laws, and do they conflict with United States laws? Availability of amenities for United States expatriate executives and families

Section 10

Transport and Storage of Fluids

Meherwan P. Boyce, Ph.D., President & CEO, Boyce Engineering International, Inc.; Fellow, American Society of Mechanical Engineers; Registered Professional Engineer.

INTRODUCTION MEASUREMENT OF FLOW Static Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Local Static Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Average Static Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specifications for Piezometer Taps . . . . . . . . . . . . . . . . . . . . . . . . . . . Special Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermocouples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resistive Thermal Detectors (RTD) . . . . . . . . . . . . . . . . . . . . . . . . . . Static Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Velocity Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pitot Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Traversing for Mean Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anemometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow Visualization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Head Meters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid-Column Manometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tube Size for Manometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiplying Gauges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanical Pressure Gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conditions of Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calibration of Gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Venturi Meters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow Nozzles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Critical Flow Nozzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orifice Meters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elbow Meters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Area Meters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass Flowmeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Axial-Flow Transverse-Momentum Mass Flowmeter . . . . . . . . . . . . . Inferential Mass Flowmeter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weirs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two-Phase Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gas-Solid Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gas-Liquid Mixtures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid-Solid Mixtures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10-6 10-6 10-6 10-6 10-7 10-7 10-7 10-8 10-8 10-8 10-8 10-9 10-10 10-11 10-11 10-11 10-11 10-12 10-12 10-13 10-13 10-13 10-13 10-14 10-14 10-15 10-17 10-17 10-18 10-18 10-18 10-19 10-19 10-19 10-19 10-19 10-20 10-20 10-20 10-20

PUMPING OF LIQUIDS AND GASES Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Centrifugal Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electromagnetic Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transfer of Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanical Impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement of Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Capacity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total Dynamic Head. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total Suction Head . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Static Suction Head . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total Discharge Head. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Static Discharge Head . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Velocity Head . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Viscosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Friction Head . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Work Performed in Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Suction Limitations of a Pump. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NPSH Requirements for Other Liquids . . . . . . . . . . . . . . . . . . . . . . . Example 1: NPSH Calculation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pump Selection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Range of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pump Materials of Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Presence of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Centrifugal Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Casings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Action of a Centrifugal Pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Centrifugal-Pump Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . System Curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pump Selection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Process Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sealing the Centrifugal Chemical Pump . . . . . . . . . . . . . . . . . . . . . . . Double-Suction Single-Stage Pumps. . . . . . . . . . . . . . . . . . . . . . . . . . Close-Coupled Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Canned-Motor Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vertical Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sump Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multistage Centrifugal Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Propeller and Turbine Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Axial-Flow (Propeller) Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbine Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Regenerative Pumps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10-20 10-21 10-21 10-22 10-22 10-22 10-22 10-22 10-22 10-22 10-22 10-22 10-22 10-22 10-22 10-22 10-22 10-22 10-23 10-23 10-23 10-23 10-23 10-23 10-24 10-24 10-24 10-24 10-24 10-24 10-25 10-25 10-28 10-28 10-28 10-29 10-29 10-29 10-29 10-29 10-29 10-29 10-31 10-31 10-1

10-2

TRANSPORT AND STORAGE OF FLUIDS

Positive-Displacement Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reciprocating Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Piston Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diaphragm Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotary Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gear Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Screw Pumps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluid-Displacement Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jet Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electromagnetic Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vibration Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 2: Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pump Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pump Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compression of Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theory of Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adiabatic Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compressor Selection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fans and Blowers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Centrifugal Blowers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Forward-Curved Blade Blowers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Backward-Curved Blade Blowers . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fan Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compressors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous-Flow Compressors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Centrifugal Compressors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compressor Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Impeller Fabrication. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Axial Flow Compressors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Positive Displacement Compressors . . . . . . . . . . . . . . . . . . . . . . . . . . Rotary Compressors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reciprocating Compressors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ejectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ejector Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Uses of Ejectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vacuum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vacuum Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sealing of Rotating Shafts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Noncontact Seals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Labyrinth Seals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ring Seals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fixed Seal Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Floating Seal Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Packing Seal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanical Face Seals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanical Seal Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Internal and External Seals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bearings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Types of Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thrust Bearings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thrust-Bearing Power Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10-31 10-32 10-32 10-33 10-34 10-34 10-34 10-35 10-35 10-35 10-36 10-36 10-37 10-37 10-37 10-37 10-37 10-45 10-45 10-45 10-46 10-46 10-46 10-47 10-47 10-47 10-48 10-49 10-49 10-50 10-50 10-51 10-56 10-56 10-57 10-57 10-57 10-58 10-59 10-59 10-61 10-61 10-61 10-61 10-62 10-62 10-63 10-64 10-64 10-65 10-66 10-67

PROCESS-PLANT PIPING Codes and Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Units: Pipe and Tubing Sizes and Ratings . . . . . . . . . . . . . . . . . . . . . . Pressure-Piping Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . National Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Governmental Regulations: OSHA . . . . . . . . . . . . . . . . . . . . . . . . . . . Code Contents and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pipe-System Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specific Material Precautions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Metallic Pipe Systems: Carbon Steel and Stainless Steel . . . . . . . . . . . . Pipe and Tubing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flanged Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bends and Fittings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Valves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cast Iron, Ductile Iron, and High-Silicon Iron. . . . . . . . . . . . . . . . . . . . Cast Iron and Ductile Iron. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . High-Silicon Iron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonferrous-Metal Piping Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aluminum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Copper and Copper Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lead and Lead-Lined Steel Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnesium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nickel and Nickel Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10-67 10-67 10-67 10-67 10-67 10-68 10-68 10-68 10-68 10-70 10-70 10-71 10-75 10-83 10-87 10-92 10-92 10-92 10-93 10-93 10-93 10-95 10-95 10-95

Titanium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zirconium (Tin 1.2 to 1.7 Percent) . . . . . . . . . . . . . . . . . . . . . . . . . . . Flexible Metal Hose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonmetallic Pipe and Lined Pipe Systems . . . . . . . . . . . . . . . . . . . . . . . Asbestos Cement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gravity Sewer Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Impervious Graphite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cement-Lined Steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chemical Ware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vitrified-Clay Sewer Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Glass Pipe and Tubing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Glass-Lined Steel Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chemical-Porcelain Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fused Silica or Fused Quartz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wood and Wood-Lined Steel Pipe. . . . . . . . . . . . . . . . . . . . . . . . . . . . Plastic-Lined and Rubber-Lined Steel Pipe . . . . . . . . . . . . . . . . . . . . Rubber-Lined Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plastic Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Piping-System Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Safeguarding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classification of Fluid Services . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design Criteria: Metallic Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure Design of Metallic Components: Wall Thickness . . . . . . . . Thermal Expansion and Flexibility: Metallic Piping . . . . . . . . . . . . . . Reactions: Metallic Piping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pipe Supports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design Criteria: Nonmetallic Pipe. . . . . . . . . . . . . . . . . . . . . . . . . . . . Fabrication, Assembly, and Erection . . . . . . . . . . . . . . . . . . . . . . . . . . . . Welding, Brazing, or Soldering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bending and Forming. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preheating and Heat Treatment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Joining Nonmetallic Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Assembly and Erection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examination, Inspection, and Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . Examination and Inspection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examination Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Type and Extent of Required Examination . . . . . . . . . . . . . . . . . . . . . Comparison of Piping-System Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . Forces of Piping on Process Machinery and Piping Vibration . . . . . . . . Heat Tracing of Piping Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Types of Heat-Tracing Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Choosing the Best Tracing System. . . . . . . . . . . . . . . . . . . . . . . . . . . .

10-95 10-96 10-96 10-96 10-96 10-97 10-97 10-97 10-98 10-98 10-98 10-98 10-99 10-99 10-99 10-100 10-100 10-100 10-100 10-100 10-102 10-102 10-102 10-102 10-103 10-103 10-109 10-122 10-124 10-126 10-126 10-126 10-127 10-127 10-127 10-127 10-127 10-127 10-129 10-130 10-133 10-133 10-133 10-133 10-137

STORAGE AND PROCESS VESSELS Storage of Liquids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Atmospheric Tanks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure Tanks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculation of Tank Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Container Materials and Safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pond and Underground Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Storage of Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gas Holders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution of Gases in Liquids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Storage in Pressure Vessels, Bottles, and Pipe Lines . . . . . . . . . . . . . Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cost of Storage Facilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bulk Transport of Fluids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pipe Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tanks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tank Cars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tank Trucks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marine Transportation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Materials of Construction for Bulk Transport . . . . . . . . . . . . . . . . . . . Pressure Vessels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Code Administration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ASME Code Section VIII, Division 1 . . . . . . . . . . . . . . . . . . . . . . . . . ASME Code Section VIII, Division 2 . . . . . . . . . . . . . . . . . . . . . . . . . Additional ASME Code Considerations . . . . . . . . . . . . . . . . . . . . . . . Other Regulations and Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vessels with Unusual Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . Vessel Codes Other Than ASME. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vessel Design and Construction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Care of Pressure Vessels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure-Vessel Cost and Weight. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10-138 10-138 10-138 10-139 10-140 10-141 10-141 10-141 10-141 10-141 10-141 10-142 10-142 10-142 10-142 10-142 10-143 10-143 10-144 10-144 10-144 10-144 10-147 10-148 10-149 10-149 10-150 10-151 10-151 10-151

Nomenclature and Units In this listing, symbols used in the section are defined in a general way and appropriate SI and U.S. customary units are given. Specific definitions, as denoted by subscripts, are stated at the place of application in the section. Some specialized symbols used in the section are defined only at the place of application. Symbol A A A∞ a a a B b b C C C

C C Ca C1 cp cv D D, D0 d E E Ea Ec Ej Em F F F f f f G g gc H H, h Had h h i i ii io I J K

Definition Area Factor for determining minimum value of R1 Free-stream speed of sound Area Duct or channel width Coefficient, general Height Duct or channel height Coefficient, general Coefficient, general Conductance Sum of mechanical allowances (thread or groove depth) plus corrosion or erosion allowances Cold-spring factor Constant Capillary number Estimated self-spring or relaxation factor Constant-pressure specific heat Constant-volume specific heat Diameter Outside diameter of pipe Diameter Modulus of elasticity Quality factor As-installed Young’s modulus Casting quality factor Joint quality factor Minimum value of Young’s modulus Force Friction loss Correction factor Frequency Friction factor Stress-range reduction factor Mass velocity Local acceleration due to gravity Dimensional constant Depth of liquid Head of fluid, height Adiabatic head Flexibility characteristic Height of truncated cone; depth of head Specific enthalpy Stress-intensification factor In-plane stressintensification factor Out-plane stress intensification factor Electric current Mechanical equivalent of heat Index, constant or flow parameter

SI units m2

U.S. customary units

Symbol

ft2

K K1

m2 m

ft2 ft

m m

ft ft

m3/s mm

ft3/s in

k k k L L L M Mi, mi Mo

Dimensionless

Dimensionless

Mt M∞ m m N

J/(kg⋅K)

Btu/(lb⋅°R)

N N

J/(kg⋅K)

Btu/(lb⋅°R)

m mm m N/m2

ft in ft lbf/ft2

MPa

kip/in2 (ksi)

MPa

kip/in2 (ksi)

N (N⋅m)/kg Dimensionless Hz Dimensionless

lbf (ft⋅lbf)/lb Dimensionless l/s Dimensionless

NS NDe NFr NRe NWe NPSH n n n n P Pad p p Q Q Q Q

kg/(s⋅m2) m/s2

lb/(s⋅ft2) ft/s2

q R

1.0 (kg⋅m)/(N⋅s2) m m N⋅m/kg

32.2 (lb⋅ft)/ (lbf⋅s2) ft ft lbf⋅ft/lbm

R R R R

m

in

J/kg

Btu/lb

R R Ra

Rm A 1.0 (N⋅m)/J

A 778 (ft⋅lbf)/ Btu R1

Definition Fluid bulk modulus of elasticity Constant in empirical flexibility equation Ratio of specific heats Flexibility factor Adiabatic exponent cp /cv Length Developed length of piping between anchors Dish radius Molecular weight In-plane bending moment Out-plane bending moment Torsional moment Free stream Mach number Mass Thickness Number of data points or items Frictional resistance Equivalent full temperature cycles Strouhal number Dean number Froude number Reynolds number Weber number Net positive suction head Polytropic exponent Pulsation frequency Constant, general Number of items Design gauge pressure Adiabatic power Pressure Power Heat Volume Volume rate of flow (liquids) Volume rate of flow (gases) Volume flow rate Gas constant Radius Electrical resistance Head reading Range of reaction forces or moments in flexibility analysis Cylinder radius Universal gas constant Estimated instantaneous reaction force or moment at installation temperature Estimated instantaneous maximum reaction force or moment at maximum or minimum metal temperature Effective radius of miter bend

SI units

U.S. customary units

N/m2

lbf/ft2

Dimensionless

Dimensionless

m m

ft ft

m kg/mol N⋅mm N⋅mm

in lb/mol in⋅lbf in⋅lbf

N⋅mm

in⋅lbf

kg m Dimensionless

lb ft Dimensionless

Dimensionless

Dimensionless

Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless m

Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless ft

Hz

1/s

Dimensionless kPa kW Pa kW J m3 m3/h

Dimensionless lbf/in2 hp lbf/ft2 hp Btu ft3 gal/min

m3/h

ft3/min (cfm)

m3/s 8314 J/ (K⋅mol) m Ω m N or N⋅mm

ft3/s 1545 (ft⋅lbf)/ (mol⋅°R) ft Ω ft lbf or in⋅lbf

m J/(kg⋅K) N or N⋅mm

ft (ft⋅lbf)/(lbm⋅°R) lbf or in⋅lbf

N or N⋅mm

lbf or in⋅lbf

mm

in

10-3

10-4

TRANSPORT AND STORAGE OF FLUIDS

Nomenclature and Units (Concluded ) Symbol r r rc rk r2 S S S S S SA SE SL ST Sb Sc Sh St s s T Ts T 苶 T 苶b T 苶h t t t tm

tr U u u V V

Definition Radius Pressure ratio Critical pressure ratio Knuckle radius Mean radius of pipe using nominal wall thickness 苶 T Specific surface area Fluid head loss Specific energy loss Speed Basic allowable stress for metals, excluding factor E, or bolt design stress Allowable stress range for displacement stress Computed displacementstress range Sum of longitudinal stresses Allowable stress at test temperature Resultant bending stress Basic allowable stress at minimum metal temperature expected Basic allowable stress at maximum metal temperature expected Torsional stress Specific gravity Specific entropy Temperature Effective branch-wall thickness Nominal wall thickness of pipe Nominal branch-pipe wall thickness Nominal header-pipe wall thickness Head or shell radius Pressure design thickness Time Minimum required thickness, including mechanical, corrosion, and erosion allowances Pad or saddle thickness Straight-line distance between anchors Specific internal energy Velocity Velocity Volume

SI units

U.S. customary units

m Dimensionless

ft Dimensionless

m mm

in in

m2/m3 Dimensionless m/s2 m3/s MPa

ft2/ft3 Dimensionless lbf/lb ft3/s kip/in2 (ksi)

MPa

kip/in2 (ksi)

MPa

kip/in2 (ksi)

MPa

kip/in2 (ksi)

MPa

kip/in2 (ksi)

MPa MPa

kip/in2 (ksi) kip/in2 (ksi)

MPa

kip/in2 (ksi)

MPa

kip/in2 (ksi)

J/(kg⋅K) K (°C) mm

Btu/(lb⋅°R) °R (°F) in

mm

in

mm

in

mm

in

mm mm s mm

in in s in

mm m

in ft

J/kg m/s m/s m3

Btu/lb ft/s ft/s ft3

Symbol v W W w x x x Y y y Z Z Ze z z

Definition

SI units

Specific volume Work Weight Weight flow rate Weight fraction Distance or length Value of expression [(p2 /p1)(k − 1/k) − 1] Expansion factor Distance or length Resultant of total displacement strains Section modulus of pipe Vertical distance Effective section modulus for branch Gas-compressibility factor Vertical distance

U.S. customary units

m3/kg N⋅m kg kg/s Dimensionless m

ft3/lb lbf⋅ft lb lb/s Dimensionless ft

Dimensionless m mm

Dimensionless ft in

mm3 m mm3

in3 ft in3

Dimensionless

Dimensionless

m

ft

Greek symbols α α σ α, β, θ β β Γ Γ δ ε ε η ηad ηp θ λ µ ν ρ σ σc τ φ φ φ ψ ψ

Viscous-resistance coefficient Angle Half-included angle Angles Inertial-resistance coefficient Ratio of diameters Liquid loading Pulsation intensity Thickness Wall roughness Voidage—fractional free volume Viscosity, nonnewtonian fluids Adiabatic efficiency Polytropic efficiency Angle Molecular mean free-path length Viscosity Kinematic viscosity Density Surface tension Cavitation number Shear stress Shape factor Angle Flow coefficient Pressure coefficient Sphericity

1/m2

1/ft2

° ° ° 1/m

° ° ° 1/ft

Dimensionless kg/(s⋅m) Dimensionless m m Dimensionless

Dimensionless lb/(s⋅ft) Dimensionless ft ft Dimensionless

Pa⋅s

lb/(ft⋅s)

° m

° ft

Pa⋅s m2/s kg/m3 N/m Dimensionless N/m2 Dimensionless °

lb/(ft⋅s) ft2/s lb/ft3 lbf/ft Dimensionless lbf/ft2 Dimensionless °

Dimensionless

Dimensionless

INTRODUCTION

10-5

INTRODUCTION Transportation and the storage of fluids (gases and liquids) involves the understanding of the properties and behavior of fluids. The study of fluid dynamics is the study of fluids and their motion in a force field. Flows can be classified into two major categories: (a) incompressible and (b) compressible flow. Most liquids fall into the incompressible-flow category, while most gases are compressible in nature. A perfect fluid can be defined as a fluid that is nonviscous and nonconducting. Fluid flow, compressible or incompressible, can be classified by the ratio of the inertial forces to the viscous forces. This ratio is represented by the Reynolds number (NRe). At a low Reynolds number, the flow is considered to be laminar, and at high Reynolds numbers, the flow is considered to be turbulent. The limiting types of flow are the inertialess flow, sometimes called Stokes flow, and the inviscid flow that occurs at an infinitely large Reynolds number. Reynolds numbers (dimensionless) for flow in a pipe is given as: ρVD NRe = ᎏ (10-1) µ where ρ is the density of the fluid, V the velocity, D the diameter, and µ the viscosity of the fluid. In fluid motion where the frictional forces interact with the inertia forces, it is important to consider the ratio of the viscosity µ to the density ρ. This ratio is known as the kinematic viscosity (ν). Tables 10-1 and 10-2 give the kinematic viscosity for several fluids. A flow is considered to be adiabatic when there is no transfer of heat between the fluid and its surroundings. An isentropic flow is one in which the entropy of each fluid element remains constant. To fully understand the mechanics of flow, the following definitions explain the behavior of various types of fluids in both their static and flowing states. A perfect fluid is a nonviscous, nonconducting fluid. An example of this type of fluid would be a fluid that has a very small viscosity and conductivity and is at a high Reynolds number. An ideal gas is one that obeys the equation of state: P (10-2) ᎏ = RT ρ where P = pressure, ρ = density, R is the gas constant per unit mass, and T = temperature. A flowing fluid is acted upon by many forces that result in changes in pressure, temperature, stress, and strain. A fluid is said to be isotropic when the relations between the components of stress and those of the rate of strain are the same in all directions. The fluid is said to be Newtonian when this relationship is linear. These pressures and temperatures must be fully understood so that the entire flow picture can be described. The static pressure in a fluid has the same value in all directions and can be considered as a scalar point function. It is the pressure of a flowing fluid. It is normal to the surface on which it acts and at any TABLE 10.1

TABLE 10.2

Kinematic Viscosity Temperature

Liquid

°C

°F

ν × 106 (ft2/sec)

Glycerine Mercury Mercury Lubricating oil Lubricating oil Lubricating oil

20 0 100 20 40 60

68 32 212 68 104 140

7319 1.35 0.980 4306 1076 323

given point has the same magnitude irrespective of the orientation of the surface. The static pressure arises because of the random motion in the fluid of the molecules that make up the fluid. In a diffuser or nozzle, there is an increase or decrease in the static pressure due to the change in velocity of the moving fluid. Total Pressure is the pressure that would occur if the fluid were brought to rest in a reversible adiabatic process. Many texts and engineers use the words total and stagnation to describe the flow characteristics interchangeably. To be accurate, the stagnation pressure is the pressure that would occur if the fluid were brought to rest adiabatically or diabatically. Total pressure will only change in a fluid if shaft work or work of extraneous forces are introduced. Therefore, total pressure would increase in the impeller of a compressor or pump; it would remain constant in the diffuser. Similarly, total pressure would decrease in the turbine impeller but would remain constant in the nozzles. Static temperature is the temperature of the flowing fluid. Like static pressure, it arises because of the random motion of the fluid molecules. Static temperature is in most practical installations impossible to measure since it can be measured only by a thermometer or thermocouple at rest relative to the flowing fluid that is moving with the fluid. Static temperature will increase in a diffuser and decrease in a nozzle. Total temperature is the temperature that would occur when the fluid is brought to rest in a reversible adiabatic manner. Just like its counterpart total pressure, total and stagnation temperatures are used interchangeably by many test engineers. Dynamic temperature and pressure are the difference between the total and static conditions. Pd = PT − Ps (10-3) Td = TT − Ts (10-4) where subscript d refers to dynamic, T to total, and s to static. Another helpful formula is: 1 PK = ᎏ ρV 2 (10-5) 2 For incompressible fluids, PK = Pd .

Density, Viscosity, and Kinematic Viscosity of Water and Air in Terms of Temperature Air at a pressure of 760 mm Hg (14.696 lbf/in2)

Water

(°C)

(°F)

Density ␳ (lbf sec2/ft4)

Viscosity µ × 106 (lbf sec/ft2)

−20 −10 0 10 20 40 60 80 100

−4 14 32 50 68 104 140 176 212

— — 1.939 1.939 1.935 1.924 1.907 1.886 1.861

— — 37.5 27.2 21.1 13.68 9.89 7.45 5.92

Temperature

Conversion factors: 1 kp sec2/m4 = 0⋅01903 lbf sec2/ft4 (= slug/ft3) 1 lbf sec2/ft4 = 32.1719 lb/ft3 (lb = lb mass; lbf = lb force) 1 kp sec2/m4 = 9.80665 kg/m3 (kg = kg mass; kp = kg force) 1 kg/m3 = 16.02 lb/ft3

Kinematic viscosity ν × 106 (ft2/sec) — — 19.4 14.0 10.9 7.11 5.19 3.96 3.19

Density ␳ (lbf sec2/ft4)

Viscosity µ × 106 (lbf sec/ft2)

Kinematic viscosity ν × 106 (ft2/sec)

0.00270 0.00261 0.00251 0.00242 0.00234 0.00217 0.00205 0.00192 0.00183

0.326 0.338 0.350 0.362 0.375 0.399 0.424 0.449 0.477

122 130 140 150 160 183 207 234 264

10-6

TRANSPORT AND STORAGE OF FLUIDS

MEASUREMENT OF FLOW This subsection deals with the techniques of measuring pressures, temperatures, velocities, and flow rates of flowing fluids. STATIC PRESSURE Local Static Pressure In a moving fluid, the local static pressure is equal to the pressure on a surface which moves with the fluid or to the normal pressure (for newtonian fluids) on a stationary surface which parallels the flow. The pressure on such a surface is measured by making a small hole perpendicular to the surface and connecting the opening to a pressure-sensing element (Fig. 10-1a). The hole is known as a piezometer opening or pressure tap. Measurement of local static pressure is frequently difficult or impractical. If the channel is so small that introduction of any solid object disturbs the flow pattern and increases the velocity, there will be a reduction and redistribution of the static pressure. If the flow is in straight parallel lines, aside from the fluctuations of normal turbulence, the flat disk (Fig. 10-1b) and the bent tube (Fig. 10-1c) give satisfactory results when properly aligned with the stream. Slight misalignments can cause serious errors. Diameter of the disk should be 20 times its thickness and 40 times the static opening; the face must be flat and smooth, with the knife edges made by beveling the underside. The piezometer tube, such as that in Fig. 10-1c, should have openings with size and spacing as specified for a pitot-static tube (Fig. 10-6). Readings given by open straight tubes (Fig. 10-1d and 1e are too low due to flow separation. Readings of closed tubes oriented perpendicularly to the axis of the stream and provided with side openings (Fig. 10-1e) may be low by as much as two velocity heads. Average Static Pressure In most cases, the object of a staticpressure measurement is to obtain a suitable average value for substitution in Bernoulli’s theorem or in an equivalent flow formula. This can be done simply only when the flow is in straight lines parallel to the confining walls, such as in straight ducts at sufficient distance downstream from bends (2 diameters) or other disturbances. For such streams, the sum of static head and gravitational potential head is the same at all points in a cross section taken perpendicularly to the axis of flow. Thus the exact location of a piezometer opening about the periphery of such a cross section is immaterial provided its elevation is known. However, in stating the static pressure, the custom is to give the value at the elevation corresponding to the centerline of the stream. With flow in curved passages or with swirling flow, determination of a true average static pressure is, in general, impractical. In metering, straightening vanes are often placed upstream of the pressure tap to eliminate swirl. Fig. 10-2 shows various flow equalizers and straighteners. Specifications for Piezometer Taps The size of a static opening should be small compared with the diameter of the pipe and yet large compared with the scale of surface irregularities. For reliable results, it is essential that (1) the surface in which the hole is made be substantially smooth and parallel to the flow for some distance on either side of the opening, and (2) the opening be flush with the surface and possess no “burr” or other irregularity around its edge.

Rounding of the edge is often employed to ensure absence of a burr. Pressure readings will be high if the tap is inclined upstream, is rounded excessively on the upstream side, has a burr on the downstream side, or has an excessive countersink or recess. Pressure readings will be low if the tap is inclined downstream, is rounded excessively on the downstream side, has a burr on the upstream side, or protrudes into the flow stream. Errors resulting from these faults can be large. Recommendations for pressure-tap dimensions are summarized in Table 10-3. Data from several references were used in arriving at these composite values. The length of a pressure-tap opening prior to

L D Simple vane flow straightener (a) (a)

L Multitube flow straightener

D1

(b) (b)

Equalizer (perforated plate or screen) (c) (c)

D 3 min. 2 Combination equalizer and straightener (d) (d)

b

Q 2D Multiplate type equalizer and straightener (e) (e) Flow equalizers and straighteners [Power Test Code 10, Compressors and Exhausters, Amer. Soc. of Mechanical Engineers, 1965].

FIG. 10-2 FIG. 10-1

Measurement of static pressure.

MEASUREMENT OF FLOW TABLE 10-3

Pressure-Tap Holes

Nominal inside pipe diameter, in

Maximum diameter of pressure tap, mm (in)

Radius of hole-edge rounding, mm (in)

1 2 3 4 8 16

3.18 (f) 6.35 (d) 9.53 (r) 12.7 (a) 12.7 (a) 19.1 (e)

<0.40 (1⁄64) 0.40 (1⁄64) 0.40–0.79 (1⁄64–h) 0.79 (h) 0.79–1.59 (h–g) 0.79–1.59 (h–g)

any enlargement in the tap channel should be at least two tap diameters, preferably three or more. A piezometer ring is a toroidal manifold into which are connected several sidewall static taps located around the perimeter of a common cross section. Its intent is to give an average pressure if differences in pressure other than those due to static head exist around the perimeter. However, there is generally no assurance that a true average is provided thereby. The principal advantage of the ring is that use of several holes in place of a single hole reduces the possibility of completely plugging the static openings. For information on prediction of static-hole error, see Shaw, J. Fluid Mech., 7, 550–564 (1960); Livesey, Jackson, and Southern, Aircr. Eng., 34, 43–47 (February 1962). For nonnewtonian fluids, pressure readings with taps may also be low because of fluid-elasticity effects. This error can be largely eliminated by using flush-mounted diaphragms. For information on the pressure-hole error for nonnewtonian fluids, see Han and Kim, Trans. Soc. Rheol., 17, 151–174 (1973); Novotny and Eckert, Trans. Soc. Rheol., 17, 227–241 (1973); and Higashitani and Lodge, Trans. Soc. Rheol., 19, 307–336 (1975). Dynamic pressure may be measured by use of a pitot tube that is a simple impact tube. These tubes measure the pressure at a point where the velocity of the fluid is brought to zero. Pitot tubes must be parallel to the flow. The pitot tube is sensitive to yaw or angle attack. In general angles of attack over 10° should be avoided. In cases where the flow direction is unknown, it is recommended to use a Kiel probe. Figure 10-3 shows a Kiel probe. This probe will read accurately to an angle of about 22° with the flow. Special Tubes A variety of special forms of the pitot tube have been evolved. Folsom (loc. cit.) gives a description of many of these special types together with a comprehensive bibliography. Included

FIG. 10-3

10-7

are the impact tube for boundary-layer measurements and shielded total-pressure tubes. The latter are insensitive to angle of attack up to 40°. Chue [Prog. Aerosp. Sci., 16, 147–223 (1975)] reviews the use of the pitot tube and allied pressure probes for impact pressure, static pressure, dynamic pressure, flow direction and local velocity, skin friction, and flow measurements. A reversed pitot tube, also known as a pitometer, has one pressure opening facing upstream and the other facing downstream. Coefficient C for this type is on the order of 0.85. This gives about a 40 percent increase in pressure differential as compared with standard pitot tubes and is an advantage at low velocities. There are commercially available very compact types of pitometers which require relatively small openings for their insertion into a duct. The pitot-venturi flow element is capable of developing a pressure differential 5 to 10 times that of a standard pitot tube. This is accomplished by employing a pair of concentric venturi elements in place of the pitot probe. The low-pressure tap is connected to the throat of the inner venturi, which in turn discharges into the throat of the outer venturi. For a discussion of performance and application of this flow element, see Stoll, Trans. Am. Soc. Mech. Eng., 73, 963–969 (1951). TOTAL TEMPERATURE For most points requiring temperature monitoring, either thermocouples or resistive thermal detectors (RTD’s) can be used. Each type of temperature transducer has its own advantages and disadvantages, and both should be considered when temperature is to be measured. Since there is considerable confusion in this area, a short discussion of the two types of transducers is necessary. Thermocouples The various types of thermocouples provide transducers suitable for measuring temperatures from −330 to 5000°F (−201 to 2760°C). The useful ranges for the various types are shown in Fig. 10-4. Thermocouples function by producing a voltage proportional to the temperature differences between two junctions of dissimilar metals. By measuring this voltage, the temperature difference can be determined. It is assumed that the temperature is known at one of the junctions; therefore, the temperature at the other junction can be determined. Since the thermocouples produce a voltage, no external power supply is required to the test junction; however, for accurate measurement, a reference junction is required. For a temperature monitoring system, reference junctions must be placed at

Kiel probe. Accurate measurements can be made at angles up to 22.5° with the flow stream.

10-8

TRANSPORT AND STORAGE OF FLUIDS

Copper/constantan

300

Iron/constantan

300

Chromel/alumel

300

750

1600

32

Platinum, 10% rhodium platinum

32

Platinum, 13% rhodium platinum

32

Platinel 1813 platinel 1503

Type J

2300

Chromel/constantan

Platinum, 30% rhodium Platinum, 6% rhodium

Type T

1800

Type K

Type E

2800

2900

100

Type R

3270

32

2372

Indium/indium 60% rhodium 40% FIG. 10-4

Type S

2552

3326

Ranges of various thermocouples.

each thermocouple or similar thermocouple wire installed from the thermocouple to the monitor where there is a reference junction. Properly designed thermocouple systems can be accurate to approximately ⫾2°F (⫾1°C). Resistive Thermal Detectors (RTD) RTDs determine temperature by measuring the change in resistance of an element due to temperature. Platinum is generally utilized in RTDs because it remains mechanically and electrically stable, resists contaminations, and can be highly refined. The useful range of platinum RTDs is −454–1832°F (−270–1000°C). Since the temperature is determined by the resistance in the element, any type of electrical conductor can be utilized to connect the RTD to the indicator; however, an electrical current must be provided to the RTD. A properly designed temperature monitoring system utilizing RTDs can be accurate ⫾0.02°F (⫾0.01°C).

or more sidewall taps to measure local static pressure. The combined pitot-static tube shown in Fig. 10-6 consists of a jacketed impact tube with one or more rows of holes, 0.51 to 1.02 mm (0.02 to 0.04 in) in diameter, in the jacket to measure the static pressure. Velocity V0 m/s (ft/s) at the point where the tip is located is given by 苶苶 苶苶 苶苶 苶 V0 = C 兹2 gc苶h ∆苶 = C 兹2 gc苶P (苶T苶− P苶/P S)苶S

V0 = C

STATIC TEMPERATURE Since this temperature requires the thermometer or thermocouple to be at rest relative to the flowing fluid, it is impractical to measure. It can be, however, calculated from the measurement of total temperature and total and static pressure. TO TS = ᎏᎏ PO (k − 1)/k ᎏᎏ PS

冢 冣

2g k p p −1 ᎏ 莦莦 ᎏ 莦莦 ᎏ莦 冪莦 k − 1 冢 ρ 冣莦冤冢 p 冣 莦莦莦莦莦冥莦 c

0

i

0

0

(10-6)

VELOCITY MEASUREMENTS Pitot Tubes The combination of pitot tubes in conjunction with sidewall static taps measures local or point velocities by measuring the difference between the total pressure and the static pressure. The pitot tube shown in Fig. 10-5 consists of an impact tube whose opening faces directly into the stream to measure impact pressure, plus one

(10-7)

where C = coefficient, dimensionless; gc = dimensional constant; ∆h = dynamic pressure (∆hs g /gc ), expressed in (N⋅m)/kg [(ft⋅lbf )/lb or ft of fluid flowing]; ∆hs = differential height of static liquid column corresponding to ∆h; g = local acceleration due to gravity; gc = dimensional constant; pi = impact pressure; p0 = local static pressure; and ρ0 = fluid density measured at pressure p0 and the local temperature. With gases at velocities above 60 m/s (about 200 ft/s), compressibility becomes important, and the following equation should be used:

FIG. 10-5

Pitot tube with sidewall static tap.

(k − 1)/k

(10-8)

MEASUREMENT OF FLOW

10-9

The relationships between total and static temperature and pressure are given by the following relationship: TT k−1 ᎏ = 1 + ᎏ M2 TS 2

(10-12)

k−1 PT ᎏ = 1 + ᎏ M2 PS 2

冢

FIG. 10-6

Pitot-static tube.

where k is the ratio of specific heat at constant pressure to that at constant volume. (See ASME Research Committee on Fluid Meters Report, op. cit., p. 105.) Coefficient C is usually close to 1.00 (⫾0.01) for simple pitot tubes (Fig. 10-5) and generally ranges between 0.98 and 1.00 for pitot-static tubes (Fig. 10-6). There are certain limitations on the range of usefulness of pitot tubes. With gases, the differential is very small at low velocities; e.g., at 4.6 m/s (15.1 ft/s) the differential is only about 1.30 mm (0.051 in) of water (20°C) for air at 1 atm (20°C), which represents a lower limit for 1 percent error even when one uses a micromanometer with a precision of 0.0254 mm (0.001 in) of water. Equation does not apply for Mach numbers greater than 0.7 because of the interference of shock waves. For supersonic flow, local Mach numbers can be calculated from a knowledge of the dynamic and true static pressures. The free stream Mach number (M∞) is defined as the ratio of the speed of the stream (V∞) to the speed of sound in the free stream: A∞ =

∂P ᎏ 冣莦 冪莦冢莦 ∂␳

s=c

V∞ M∞ = ᎏ ∂P ᎏᎏ ∂ρ s = c

冪冢莦莦莦冣莦

(10-9) (10-10)

where S is the entropy. For isentropic flow, this relationship and pressure can be written as: V∞ M∞ = ᎏ (10-11) 苶T 苶苶s 兹kR

(10-13)

With liquids at low velocities, the effect of the Reynolds number upon the coefficient is important. The coefficients are appreciably less than unity for Reynolds numbers less than 500 for pitot tubes and for Reynolds numbers less than 2300 for pitot-static tubes [see Folsom, Trans. Am. Soc. Mech. Eng., 78, 1447–1460 (1956)]. Reynolds numbers here are based on the probe outside diameter. Operation at low Reynolds numbers requires prior calibration of the probe. The pitot-static tube is also sensitive to yaw or angle of attack than is the simple pitot tube because of the sensitivity of the static taps to orientation. The error involved is strongly dependent upon the exact probe dimensions. In general, angles greater than 10° should be avoided if the velocity error is to be 1 percent or less. Disturbances upstream of the probe can cause large errors, in part because of the turbulence generated and its effect on the staticpressure measurement. A calming section of at least 50 pipe diameters is desirable. If this is not possible, the use of straightening vanes or a honeycomb is advisable. The effect of pulsating flow on pitot-tube accuracy is treated by Ower et al., op. cit., pp. 310–312. For sinusoidal velocity fluctuations, the ratio of indicated velocity to actual mean velocity is given by the factor 兹1苶苶 +苶 λ2苶2 /苶, where λ is the velocity excursion as a fraction of the mean velocity. Thus, the indicated velocity would be about 6 percent high for velocity fluctuations of ⫾50 percent, and pulsations greater than ⫾20 percent should be damped to avoid errors greater than 1 percent. The error increases as the frequency of flow oscillations approaches the natural frequency of the pitot tube and the density of the measuring fluid approaches the density of the process fluid [see Horlock and Daneshyar, J. Mech. Eng. Sci., 15, 144–152 (1973)]. Pressures substantially lower than true impact pressures are obtained with pitot tubes in turbulent flow of dilute polymer solutions [see Halliwell and Lewkowicz, Phys. Fluids, 18, 1617–1625 (1975)]. Traversing for Mean Velocity Mean velocity in a duct can be obtained by dividing the cross section into a number of equal areas, finding the local velocity at a representative point in each, and averaging the results. In the case of rectangular passages, the cross section is usually divided into small squares or rectangles and the velocity is found at the center of each. In circular pipes, the cross section is divided into several equal annular areas as shown in Fig. 10-7. Read-

Velocity ration versus Reynolds number for smooth circular pipes. [Based on data from Rothfus, Archer, Klimas, and Sikchi, Am. Inst. Chem. Eng. J., 3,208 (1957).] FIG. 10-7

(k − 1)/k

冣

10-10

TRANSPORT AND STORAGE OF FLUIDS

ings of velocity are made at the intersections of a diameter and the set of circles which bisect the annuli and the central circle. For an N-point traverse on a circular cross section, make readings on each side of the cross section at 苶n 苶苶 苶)苶 percent 100 × 兹(2 −苶)/ 1苶N

(n = 1, 2, 3 to N/2)

of the pipe radius from the center. Traversing several diameters spaced at equal angles about the pipe is required if the velocity distribution is unsymmetrical. With a normal velocity distribution in a circular pipe, a 10-point traverse theoretically gives a mean velocity 0.3 percent high; a 20-point traverse, 0.1 percent high. For normal velocity distribution in straight circular pipes at locations preceded by runs of at least 50 diameters without pipe fittings or other obstructions, the graph in Fig. 10-7 shows the ratio of mean velocity V to velocity at the center umax plotted against the Reynolds number, where D = inside pipe diameter, ρ = fluid density, and µ = fluid viscosity, all in consistent units. Mean velocity is readily determined from this graph and a pitot reading at the center of the pipe if the quantity Dumax ρ/µ is less than 2000 or greater than 5000. The method is unreliable at intermediate values of the Reynolds number. Methods for determining mean flow rate from probe measurements under nonideal conditions are described by Mandersloot, Hicks, and Langejan [Chem. Eng. (London), no. 232, CE370-CE380 (1969)]. Anemometers An anemometer may be any instrument for measurement of gas velocity, e.g., a pitot tube, but usually the term refers to one of the following types. The vane anemometer is a delicate revolution counter with jeweled bearings, actuated by a small windmill, usually 75 to 100 mm (about 3 to 4 in) in diameter, constructed of flat or slightly curved radially disposed vanes. Gas velocity is determined by using a stopwatch to find the time interval required to pass a given number of meters (feet) of gas as indicated by the counter. The velocity so obtained is inversely proportional to gas density. If the original calibration was carried out in a gas of density ρ0 and the density of the gas stream being metered is ρ1, the true gas velocity can be found as follows: From the calibration curve for the instrument, find Vt,0 corresponding to the quantity Vm兹苶 ρ苶 ρ苶0, where Vm = measured velocity. Then the actual velocity Vt,1 1/苶 is equal to Vt,0兹苶 ρ苶 ρ苶1. In general, when working with air, the effects of 0/苶 atmospheric-density changes can be neglected for all velocities above 1.5 m/s (about 5 ft/s). In all cases, care must be taken to hold the anemometer well away from one’s body or from any object not normally present in the stream. Vane anemometers can be used for gas-velocity measurements in the range of 0.3 to 45 m/s (about 1 to 150 ft/s), although a given instrument generally has about a twentyfold velocity range. Bearing friction has to be minimized in instruments designed for accuracy at the low end of the range, while ample rotor and vane rigidity must be provided for measurements at the higher velocities. Vane anemometers are sensitive to shock and cannot be used in corrosive atmospheres. Therefore, accuracy is questionable unless a recent calibration has been made and the history of the instrument subsequent to calibration is known. For additional information, see Ower et al., op. cit., chap. VIII. A turbine flowmeter consists of a straight flow tube containing a turbine which is free to rotate on a shaft supported by one or more bearings and located on the centerline of the tube. Means are provided for magnetic detection of the rotational speed, which is proportional to the volumetric flow rate. Its use is generally restricted to clean, noncorrosive fluids. Additional information on construction, operation, range, and accuracy can be obtained from Holzbock (Instruments for Measurement and Control, 2d ed., Reinhold, New York, 1962, pp. 155–162). For performance characteristics of these meters with liquids, see Shafer, J. Basic Eng., 84, 471–485 (December 1962); or May, Chem. Eng., 78(5), 105–108 (1971); and for the effect of density and Reynolds number when used in gas flowmetering, see Lee and Evans, J. Basic Eng., 82, 1043–1057 (December 1965). The current meter is generally used for measuring velocities in open channels such as rivers and irrigation channels. There are two types, the cup meter and the propeller meter. The former is more widely used. It consists of six conical cups mounted on a vertical axis pivoted at the ends and free to rotate between the rigid arms of a

U-shaped clevis to which a vaned tailpiece is attached. The wheel rotates because of the difference in drag for the two sides of the cup, and a signal proportional to the revolutions of the wheel is generated. The velocity is determined from the count over a period of time. The current meter is generally useful in the range of 0.15 to 4.5 m/s (about 0.5 to 15 ft/s) with an accuracy of ⫾2 percent. For additional information see Creager and Justin, Hydroelectric Handbook, 2d ed., Wiley, New York, 1950, pp. 42–46. The hot-wire anemometer consists essentially of an electrically heated fine wire (generally platinum) exposed to the gas stream whose velocity is being measured. An increase in fluid velocity, other things being equal, increases the rate of heat flow from the wire to the gas, thereby tending to cool the wire and alter its electrical resistance. In a constant-current anemometer, gas velocity is determined by measuring the resulting wire resistance; in the constant-resistance type, gas velocity is determined from the current required to maintain the wire temperature, and thus the resistance, constant. The difference in the two types is primarily in the electric circuits and instruments employed. The hot-wire anemometer can, with suitable calibration, accurately measure velocities from about 0.15 m/s (0.5 ft/s) to supersonic velocities and detect velocity fluctuations with frequencies up to 200,000 Hz. Fairly rugged, inexpensive units can be built for the measurement of mean velocities in the range of 0.15 to 30 m/s (about 0.5 to 100 ft/s). More elaborate, compensated units are commercially available for use in unsteady flow and turbulence measurements. In calibrating a hot-wire anemometer, it is preferable to use the same gas, temperature, and pressure as will be encountered in the intended application. In this case the quantity I 2R w /∆t can be plotted against 兹苶 V, where I = hot-wire current, Rw = hot-wire resistance, ∆t = difference between the wire temperature and the gas bulk temperature, and V = mean local velocity. A procedure is given by Wasan and Baid [Am. Inst. Chem. Eng. J., 17, 729–731 (1971)] for use when it is impractical to calibrate with the same gas composition or conditions of temperature and pressure. Andrews, Bradley, and Hundy [Int. J. Heat Mass Transfer, 15, 1765–1786 (1972)] give a calibration correlation for measurement of small gas velocities. The hot-wire anemometer is treated in considerable detail in Dean, op. cit., chap. VI; in Ladenburg et al., op. cit., art. F-2; by Grant and Kronauer, Symposium on Measurement in Unsteady Flow, American Society of Mechanical Engineers, New York, 1962, pp. 44–53; ASME Research Committee on Fluid Meters Report, op. cit., pp. 105–107; and by Compte-Bellot, Ann. Rev. Fluid Mech., 8, pp. 209–231 (1976). The hot-wire anemometer can be modified for liquid measurements, although difficulties are encountered because of bubbles and dirt adhering to the wire. See Stevens, Borden, and Strausser, David Taylor Model Basin Rep. 953, December 1956; Middlebrook and Piret, Ind. Eng. Chem., 42, 1511–1513 (1950); and Piret et al., Ind. Eng. Chem., 39, 1098–1103 (1947). The hot-film anemometer has been developed for applications in which use of the hot-wire anemometer presents problems. It consists of a platinum-film sensing element deposited on a glass substrate. Various geometries can be used. The most common involves a wedge with a 30° included angle at the end of a tapered rod. The wedge is commonly 1 mm (0.039 in) long and 0.2 mm (0.0079 in) wide on each face. Compared with the hot wire, it is less susceptible to fouling by bubbles or dirt when used in liquids, has greater mechanical strength when used with gases at high velocities and high temperatures, and can give a higher signal-to-noise ratio. For additional information see Ling and Hubbard, J. Aeronaut. Sci., 23, 890–891 (1956); and Ling, J. Basic Eng., 82, 629–634 (1960). The heated-thermocouple anemometer measures gas velocity from the cooling effect of the gas stream flowing across the hot junctions of a thermopile supplied with constant electrical power input. Alternate junctions are maintained at ambient temperature, thus compensating for the effect of ambient temperature. For details see Bunker, Proc. Instrum. Soc. Am., 9, pap. 54-43-2 (1954). A glass-coated bead thermistor anemometer can be used for the measurement of low fluid velocities, down to 0.001 m/s (0.003 ft/s) in air and 0.0002 m/s (0.0007 ft/s) in water [see Murphy and Sparks, Ind. Eng. Chem. Fundam., 7, 642–645 (1968)].

MEASUREMENT OF FLOW The laser-Doppler anemometer measures local fluid velocity from the change in frequency of radiation, between a stationary source and a receiver, due to scattering by particles along the wave path. A laser is commonly used as the source of incident illumination. The measurements are essentially independent of local temperature and pressure. This technique can be used in many different flow systems with transparent fluids containing particles whose velocity is actually measured. For a brief review of the laser-Doppler technique see Goldstein, Appl. Mech. Rev., 27, 753–760 (1974). For additional details see Durst, Melling, and Whitelaw, Principles and Practice of Laser-Doppler Anemometry, Academic, New York, 1976. Flow Visualization A great many techniques have been developed for the visualization of velocity patterns, particularly for use in water-tunnel and wind-tunnel studies. In the case of liquids, the more common methods of revealing flow lines involve the use of dye traces, the addition of aluminum flake, plastic particles, globules of equal density liquid (dibutyl phatalate and kerosene) and glass spheres, and the use of polarized light with a doubly refractive liquid or suspension. For the last-named techniques, called flow birefringence, see Prados and Peebles, Am. Inst. Chem. Eng. J., 5, 225–234 (1959). The velocity pattern for laminar flow in a two-dimensional system can be quantitatively mapped by using an electrolytic-tank analog or a conductivepaper analog with a suitable combination of resistances, sources, and sinks. The hydrogen-bubble technique has been proposed for flow visualization and velocity field mapping in liquids. A fine wire, usually of the order of 0.013 to 0.05 mm (0.0005 to 0.002 in) in diameter, is employed as the negative electrode of a direct-current circuit in a water channel. Hydrogen bubbles, formed at the wire by periodic electrical pulses, are swept off by hydrodynamic forces and follow the flow. The bubbles are made visible by lighting at an oblique angle to the direction of view. For details see Schraub, Kline, Henry, Runstadler, and Little, J. Basic. Eng., 87, 429–444 (1965); or Davis and Fox, J. Basic Eng., 89, 771–781 (1967). Thomas and Rice [ J. Appl. Mech., 40, 321–325 (1973)] applied the hydrogen-bubble technique for velocity measurements in thin liquid films. Durelli and Norgard [Exp. Mech., 12, 169–177 (1972)] compare the flow birefringence and hydrogen-bubble techniques. In the case of gases, flow lines can be revealed through the use of smoke traces or the addition of a lightweight powder such as balsa dust to the stream. One of the best smoke generators is the reaction of titanium tetrachloride with moisture in the air. A woodsmokegeneration system is described by Yu, Sparrow, and Eckert [Int. J. Heat Mass Transfer, 15, 557–558 (1972)]. Tufts of wool or nylon attached at one end to a solid surface can be used to reveal flow phenomena in the vicinity of the surface. Optical methods commonly employed depend upon changes in the refractive index resulting from the presence of heated wires or secondary streams in the flow field or upon changes in density in the primary gas as a result of compressibility effects. The three common techniques are the shadowgraph, the schlieren, and the interferometer. All three theoretically can give quantitative information on the velocity profiles in a two-dimensional system, but in practice only the interferometer is commonly so used. The optical methods are described by Ladenburg et al. (op. cit., pp. 3–108). For additional information on other methods, see Goldstein, Modern Developments in Fluid Dynamics, vol. I, London, 1938, pp. 280–296. The water table is frequently used to simulate two-dimensional compressible-flow phenomena in gases. It provides an effective, lowcost means for velocity and pressure-distribution studies or for flow visualization using either shadowgraph or schlieren techniques. In the water table, the wave velocity corresponds to the velocity of sound in the gas, streaming water flow corresponds to subsonic flow, shooting water flow corresponds to supersonic flow, and a hydraulic jump corresponds to a shock wave. From precise measurements of water depth, it is possible to calculate corresponding gas temperatures, pressures, and densities. For information on water-table design and operation see Orlin, Lindner, and Bitterly, Application of the Analogy between Water Flow with a Free Surface and Two-Dimensional Compressible Gas Flow, NACA Rep. 875, 1947, or Mathews, The Design, Operation, and Uses of the Water Channel as an Instrument for the Investigation of Compressible-Flow Phenomena, NACA Tech. Note

10-11

2008, 1950. Additional theoretical background can be obtained from Preiswerk, Application of the Methods of Gas Dynamics to Water Flows with Free Surface, part I: Flows with No Energy Dissipation, NACA Tech. Mem. 934, 1940; part II: Flows with Momentum Discontinuities (Hydraulic Jumps), NACA Tech. Mem. 935, 1940. HEAD METERS General Principles If a constriction is placed in a closed channel carrying a stream of fluid, there will be an increase in velocity, and hence an increase in kinetic energy, at the point of constriction. From an energy balance, as given by Bernoulli’s theorem (see subsection “Energy Balance”), there must be a corresponding reduction in pressure. Rate of discharge from the constriction can be calculated by knowing this pressure reduction, the area available for flow at the constriction, the density of the fluid, and the coefficient of discharge C. The last-named is defined as the ratio of actual flow to the theoretical flow and makes allowance for stream contraction and frictional effects. The metering characteristics of commonly used head meters are reviewed and grouped by Halmi [ J. Fluids Eng., 95, 127–141 (1973)]. The term static head generally denotes the pressure in a fluid due to the head of fluid above the point in question. Its magnitude is given by the application of Newton’s law (force = mass × acceleration). In the case of liquids (constant density), the static head ph Pa (lbf/ft2) is given by ph = hρg/ gc

(10-14a)

where h = head of liquid above the point, m (ft); ρ = liquid density; g = local acceleration due to gravity; and gc = dimensional constant. The head developed in a compressor or pump is the energy force per unit mass. In the measuring systems it is often misnamed as (ft) while the units are really ft-lb/lbm or kilojoules. For a compressor or turbine, it is represented by the following relationship: E = U1Vθ1 − U2Vθ2

(10-14b)

where U is the blade speed and Vθ is the tangential velocity component of absolute velocity. This equation is known as the Euler equation. Liquid-Column Manometers The height, or head [Eq. (10-14a)], to which a fluid rises in an open vertical tube attached to an apparatus containing a liquid is a direct measure of the pressure at the point of attachment and is frequently used to show the level of liquids in tanks and vessels. This same principle can be applied with U-tube gauges (Fig. 10-8a) and equivalent devices (such as that shown in Fig. 10-8b) to measure pressure in terms of the head of a fluid other than the one under test. Most of these gauges may be used either as open or as differential manometers. The manometric fluid that constitutes the measured liquid column of these gauges may be any liquid immiscible with the fluid under pressure. For high vacuums or for high pressures and large pressure differences, the gauge liquid is a high-density liquid, generally mercury; for low pressures and small pressure differences, a low-density liquid (e.g., alcohol, water, or carbon tetrachloride) is used. The open U tube (Fig. 10-8a) and the open gauge (Fig. 10-8b) each show a reading hM m (ft) of manometric fluid. If the interface of the manometric fluid and the fluid of which the pressure is wanted is K m (ft) below the point of attachment, A, ρA is the density of the lat-

(a) FIG. 10-8

Open manometers.

(b)

10-12

TRANSPORT AND STORAGE OF FLUIDS

ter fluid at A, and ρM is that of the manometric fluid, then gauge pressure pA Pa (lbf/ft2) at A is pA = (hMρM − KρA)(g/gc )

(10-15)*

where g = local acceleration due to gravity and gc = dimensional constant. The head HA at A as meters (feet) of the fluid at that point is hA = hM(ρM /ρA) − K

(10-16)*

When a gas pressure is measured, unless it is very high, ρA is so much smaller than ρM that the terms involving K in these formulas are negligible. The differential U tube (Fig. 10-9) shows the pressure difference between taps A and B to be pA − pB = [hM (ρM − ρA) + KAρA − KBρB](g/gc)

(10-17)*

where hM is the difference in height of the manometric fluid in the U tube; KA and KB are the vertical distances of the upper surface of the manometric fluid above A and B respectively; ρA and ρB are the densities of the fluids at A and B respectively; and ρM is the density of the manometric fluid. If either pressure tap is above the higher level of manometric fluid, the corresponding K is taken to be negative. Valve D, which is kept closed when the gauge is in use, is used to vent off gas which may accumulate at these high points. The inverted differential U tube, in which the manometric fluid may be a gas or a light liquid, can be used to measure liquid pressure differentials, especially for the flow of slurries where solids tend to settle out. Additional details on the use of this manometer can be obtained from Doolittle (op. cit., p. 18). Closed U tubes (Fig. 10-10) using mercury as the manometric fluid serve to measure directly the absolute pressure p of a fluid, provided that the space between the closed end and the mercury is substantially a perfect vacuum. The mercury barometer (Fig. 10-11) indicates directly the absolute pressure of the atmosphere in terms of height of the mercury column. Normal (standard) barometric pressure is 101.325 kPa by definition. Equivalents of this pressure in other units are 760 mm mercury (at 0°C), 29.921 inHg (at 0°C), 14.696 lbf/in2, and 1 atm. For cases in which barometer readings, when expressed by the height of a mercury column, must be corrected to standard temperature (usually 0°C), appropriate temperature correction factors are given in ASME PTC, op. cit., pp. 23–26; and Weast, Handbook of Chemistry and Physics, 59th ed., Chemical Rubber, Cleveland, 1978–1979, pp. E39–E41. Tube Size for Manometers To avoid capillary error, tube diameter should be sufficiently large and the manometric fluids of such densities that the effect of capillarity is negligible in comparison with the gauge reading. The effect of capillarity is practically negligible for tubes with inside diameters 12.7 mm (a in) or larger (see ASME PTC, op. cit., p. 15). Small diameters are generally permissible for U tubes because the capillary displacement in one leg tends to cancel that in the other. The capillary rise in a small vertical open tube of circular cross section dipping into a pool of liquid is given by 4σgc cos θ h = ᎏᎏ (10-18) gD(ρ1 − ρ2) Here σ = surface tension, D = inside diameter, ρ1 and ρ2 are the densities of the liquid and gas (or light liquid) respectively, g = local acceleration due to gravity, gc = dimensional constant, and θ is the contact angle subtended by the heavier fluid. For most organic liquids and water, the contact angle θ is zero against glass, provided the glass is wet with a film of the liquid; for mercury against glass, θ = 140° (International Critical Tables, vol. IV, McGraw-Hill, New York, 1928, pp. 434–435). For further discussion of capillarity, see Schwartz, Ind. Eng. Chem., 61(1), 10–21 (1969). * The line leading from the pressure tap to the gauge is assumed to be filled with fluid of the same density as that in the apparatus at the location of the pressure tap; if this is not the case, ρA is the density of the fluid actually filling the gauge line, and the value given for hA must be multiplied by ρA/ρ, where ρ is the density of the fluid whose head is being measured.

FIG. 10-9

Differential U tube.

FIG. 10-10

Closed U tube.

Multiplying Gauges To attain the requisite precision in measurement of small pressure differences by liquid-column manometers, means must often be devised to magnify the readings. Of the schemes that follow, the second and third may give tenfold multiplication; the fourth, as much as thirtyfold. In general, the greater the multiplication, the more elaborate must be the precautions in the use of the gauge if the gain in precision is not to be illusory. 1. Change of manometric fluid. In open manometers, choose a fluid of lower density. In differential manometers, choose a fluid such that the difference between its density and that of the fluid being measured is as small as possible. 2. Inclined U tube (Fig. 10-12). If the reading R m (ft) is taken as shown and R0 m (ft) is the zero reading, by making the substitution hM = (R − R0) sin θ, the formulas of preceding paragraphs give (pA − pB) when the corresponding upright U tube is replaced by one inclined. For precise work, the gauge should be calibrated because of possible variations in tube diameter and slope. 3. The draft gauge (Fig. 10-13). Commonly used for low gas heads, this gauge has for one leg of the U a reservoir of much larger bore than the tubing that forms the inclined leg. Hence variations of level in the inclined tube produce little change in level in the reservoir. Although hM may be readily computed in terms of reading R and the dimensions of the tube, calibration of the gauge is preferable; often the changes of level in the reservoir are not negligible, and also variations in tube diameter may introduce serious error into the computation. Commercial gauges are often provided with a scale giving hM directly in height of water column, provided a particular liquid (often not water) fills the tube; failure to appreciate that the scale is incorrect unless the gauge is filled with the specified liquid is a frequent source of error. If the scale reads correctly when the density of the gauge liquid is ρ0, then the reading must be multiplied by ρ/ρ0 if the density of the fluid actually in use is ρ. 4. Two-fluid U tube (Fig. 10-14). This is a highly sensitive device for measuring small gas heads. Let A be the cross-sectional area of each of the reservoirs and a that of the tube forming the U; let ρ1 be

FIG. 10-11

Mercury barometer.

FIG. 10-12

Inclined U tube.

MEASUREMENT OF FLOW

FIG. 10-13

Draft gauge.

FIG. 10-14

Two-fluid U tube.

the density of the lighter fluid and ρ2 that of the heavier fluid; and if R is the reading and R0 its value with zero pressure difference, then the pressure difference is g a pA − pB = (R − R0) ρ2 − ρ1 + ᎏ ρ1 ᎏ (10-19) A gc where g = local acceleration due to gravity and gc = dimensional constant. When A/a is sufficiently large, the term (a/A) ρ1 in Eq. (10-19) becomes negligible in comparison with the difference (ρ2 − ρ1). However, this term should not be omitted without due consideration. In applying Eq. (10-19), the densities of the gauge liquids may not be taken from tables without the possibility of introducing serious error, for each liquid may dissolve appreciable quantities of the other. Before the gauge is filled, the liquids should be shaken together, and the actual densities of the two layers should be measured for the temperature at which the gauge is to be used. When high magnification is being sought, the U tube may have to be enclosed in a constanttemperature bath so that (ρ2 − ρ1) may be accurately known. In general, if highest accuracy is desired, the gauge should be calibrated. Several micromanometers, based on the liquid-column principle and possessing extreme precision and sensitivity, have been developed for measuring minute gas-pressure differences and for calibrating low-range gauges. Some of these micromanometers are available commercially. These micromanometers are free from errors due to capillarity and, aside from checking the micrometer scale, require no calibration. See Doolittle, op. cit., p. 21. Mechanical Pressure Gauges The Bourdon-tube gauge indicates pressure by the amount of flection under internal pressure of an oval tube bent in an arc of a circle and closed at one end. These gauges are commercially available for all pressures below atmospheric and for pressures up to 700 MPa (about 100,000 lbf/in2) above atmospheric. Details on Bourdon-type gauges are given by Harland [Mach. Des., 40(22), 69–74 (Sept. 19, 1968)]. A diaphragm gauge depends for its indication on the deflection of a diaphragm, usually metallic, when subjected to a difference of pressure between the two faces. These gauges are available for the same general purposes as Bourdon gauges but are not usually employed for high pressures. The aneroid barometer is a type of diaphragm gauge. Small pressure transducers with flush-mounted diaphragms are commercially available for the measurement of either steady or fluctuating pressures up to 100 MPa (about 15,000 lbf/in2). The metallic diaphragms are as small as 4.8 mm (t in) in diameter. The transducer is mounted on the apparatus containing the fluid whose pressure is to be measured so that the diaphragm is flush with the inner surface of the apparatus. Deflection of the diaphragm is measured by unbonded strain gauges and recorded electrically. With nonnewtonian fluids the pressure measured at the wall with non-flush-mounted pressure gauges may be in error (see subsection “Static Pressure”). Bourdon and diaphragm gauges that show both pressure and vacuum indications on the same dial are called compound gauges. Conditions of Use Bourdon tubes should not be exposed to temperatures over about 65°C (about 150°F) unless the tubes are specifically designed for such operation. When the pressure of a hotter fluid is to be measured, some type of liquid seal should be used to keep the hot fluid from the tube. In using either a Bourdon or a diaphragm gauge to measure gas pressure, if the gauge is below the

冢

冣

10-13

pressure tap of the apparatus so that liquid can collect in the lead, the gauge reading will be too high by an amount equal to the hydrostatic head of the accumulated liquid. For measuring pressures of corrosive fluids, slurries, and similar process fluids which may foul Bourdon tubes, a chemical gauge, consisting of a Bourdon gauge equipped with an appropriate flexible diaphragm to seal off the process fluid, may be used. The combined volume of the tube and the connection between the diaphragm and the tube is filled with an inert liquid. These gauges are available commercially. Further details on pressure-measuring devices are found in Sec. 22. Calibration of Gauges Simple liquid-column manometers do not require calibration if they are so constructed as to minimize errors due to capillarity (see subsection “Liquid-Column Manometers”). If the scales used to measure the readings have been checked against a standard, the accuracy of the gauges depends solely upon the precision of determining the position of the liquid surfaces. Hence liquid-column manometers are primary standards used to calibrate other gauges. For high pressures and, with commercial mechanical gauges, even for quite moderate pressures, a deadweight gauge (see ASME PTC, op. cit., pp. 36–41; Doolittle, op. cit., p. 33; Jones, op. cit., p. 43; Sweeney, op. cit., p. 104; and Tongue, op. cit., p. 29) is commonly used as the primary standard because it is safer and more convenient than use of manometers. When manometers are used as high-pressure standards, an extremely high mercury column may be avoided by connecting a number of the usual U tubes in series. Multiplying gauges are standardized by comparing them with a micromanometer. Procedure in the calibration of a gauge consists merely of connecting it, in parallel with a standard gauge, to a reservoir wherein constant pressure may be maintained. Readings of the unknown gauge are then made for various reservoir pressures as determined by the standard. Calibration of high-vacuum gauges is described by Sellenger [Vacuum, 18(12), 645–650 (1968)]. Venturi Meters The standard Herschel-type venturi meter consists of a short length of straight tubing connected at either end to the pipe line by conical sections (see Fig. 10-15). Recommended proportions (ASME PTC, op. cit., p. 17) are entrance cone angle α1 = 21 ⫾ 2°, exit cone angle α2 = 5 to 15°, throat length = one throat diameter, and upstream tap located 0.25 to 0.5 pipe diameter upstream of the entrance cone. The straight and conical sections should be joined by smooth curved surfaces for best results. The practical working equation for weight rate of discharge, adopted by the ASME Research Committee on Fluid Meters for use with either gases or liquids, is 2gc(p1 − p2)ρ1 w = q1ρ1 = CYA2 ᎏᎏ 1 − β4

冪莦

苶1苶− 苶苶 苶1苶 = KYA2 兹2苶苶 gc(p p苶 2)ρ

(10-20)

where A2 = cross-sectional area of throat; C = coefficient of discharge, 苶苶 dimensionless; gc = dimensional constant; K = C/兹1 −苶 β4苶, dimensionless; p1, p2 = pressure at upstream and downstream static pressure taps respectively; q1 = volumetric rate of discharge measured at upstream

FIG. 10-15

Herschel-type venturi tube.

10-14

TRANSPORT AND STORAGE OF FLUIDS

pressure and temperature; w = weight rate of discharge; Y = expansion factor, dimensionless; β = ratio of throat diameter to pipe diameter, dimensionless; and ρ1 = density at upstream pressure and temperature. For the flow of gases, expansion factor Y, which allows for the change in gas density as it expands adiabatically from p1 to p2, is given by Y=

k 1−r 1−β ᎏ莦 ᎏᎏ 莦 ᎏ 莦冣 冪莦r 莦莦冢莦 k − 1 冣 莦冢莦 1 − r 冣 莦冢莦 1−β r (k − 1)/k

4

2/k

4 2/k

(10-21)

for venturi meters and flow nozzles, where r = p2 /p1 and k = specific heat ratio cp /cv. Values of Y computed from Eq. (10-21) are given in Fig. 10-16 as a function of r, k, and β. For the flow of liquids, expansion factor Y is unity. The change in potential energy in the case of an inclined or vertical venturi meter must be allowed for. Equation (10-20) is accordingly modified to give w = q1ρ = CA2

[2g (p − p ) + 2gρ(Z − Z )]ρ ᎏᎏᎏᎏ 冪莦莦 1−β c

1

2

1

4

2

(10-22)

where g = local acceleration due to gravity and Z1, Z2 = vertical heights above an arbitrary datum plane corresponding to the centerline pressure-reading locations for p1 and p2 respectively. Value of the discharge coefficient C for a Herschel-type venturi meter depends upon the Reynolds number and to a minor extent upon the size of the venturi, increasing with diameter. A plot of C versus pipe Reynolds number is given in ASME PTC, op. cit., p. 19. A value of 0.984 can be used for pipe Reynolds numbers larger than 200,000. Permanent pressure loss for a Herschel-type venturi tube depends upon diameter ratio β and discharge cone angle α2. It ranges from 10 to 15 percent of the pressure differential (p1 − p2) for small angles (5 to 7°) and from 10 to 30 percent for large angles (15°), with the larger losses occurring at low values of β (see ASME PTC, op. cit., p. 12). See Benedict, J. Fluids Eng., 99, 245–248 (1977), for a general equation for pressure loss for venturis installed in pipes or with plenum inlets. For flow measurement of steam and water mixtures with a Herschel-type venturi in 2a-in- and 3-in-diameter pipes, see Collins and Gacesa, J. Basic Eng., 93, 11–21 (1971). A variety of short-tube venturi meters are available commercially. They require less space for installation and are generally (although not always) characterized by a greater pressure loss than the corresponding Herschel-type venturi meter. Discharge coefficients vary widely for different types, and individual calibration is recommended if the manufacturer’s calibration is not available. Results of tests on the Dall flow tube are given by Miner [Trans. Am. Soc. Mech. Eng., 78, 475–479 (1956)] and Dowdell [Instrum. Control Syst., 33, 1006–1009 (1960)]; and on the Gentile flow tube (also called Beth flow tube or Foster flow tube) by Hooper [Trans. Am. Soc. Mech. Eng., 72, 1099–1110 (1950)]. The use of a multiventuri system (in which an inner venturi discharges into the throat of an outer venturi) to increase both the dif-

ferential pressure for a given flow rate and the signal-to-loss ratio is described by Klomp and Sovran [J. Basic Eng., 94, 39–45 (1972)]. Flow Nozzles A simple form of flow nozzle is shown in Fig. 10-17. It consists essentially of a short cylinder with a flared approach section. The approach cross section is preferably elliptical in shape but may be conical. Recommended contours for long-radius flow nozzles are given in ASME PTC, op. cit., p. 13. In general, the length of the straight portion of the throat is about one-half throat diameter, the upstream pressure tap is located about one pipe diameter from the nozzle inlet face, and the downstream pressure tap about one-half pipe diameter from the inlet face. For subsonic flow, the pressures at points 2 and 3 will be practically identical. If a conical inlet is preferred, the inlet and throat geometry specified for a Herschel-type venturi meter can be used, omitting the expansion section. Rate of discharge through a flow nozzle for subcritical flow can be determined by the equations given for venturi meters, Eq. (10-20) for gases and Eq. (10-22) for liquids. The expansion factor Y for nozzles is the same as that for venturi meters [Eq. (10-21), Fig. 10-16]. The value of the discharge coefficient C depends primarily upon the pipe Reynolds number and to a lesser extent upon the diameter ratio β. Curves of recommended coefficients for long-radius flow nozzles with pressure taps located one pipe diameter upstream and one-half pipe diameter downstream of the inlet face of the nozzle are given in ASME PTC, op. cit., p. 15. In general, coefficients range from 0.95 at a pipe Reynolds number of 10,000 to 0.99 at 1,000,000. The performance characteristics of pipe-wall-tap nozzles (Fig. 10-17) and throat-tap nozzles are reviewed by Wyler and Benedict [ J. Eng. Power, 97, 569–575 (1975)]. Permanent pressure loss across a subsonic flow nozzle is approximated by 1 − β2 p1 − p4 = ᎏ2 (p1 − p2) (10-23) 1+β where p1, p2, p4 = static pressures measured at the locations shown in Fig. 10-17; and β = ratio of nozzle throat diameter to pipe diameter, dimensionless. Equation (10-23) is based on a momentum balance assuming constant fluid density (see Lapple et al., Fluid and Particle Mechanics, University of Delaware, Newark, 1951, p. 13). See Benedict, loc. cit., for a general equation for pressure loss for nozzles installed in pipes or with plenum inlets. Nozzles show higher loss than venturis. Permanent pressure loss for laminar flow depends on the Reynolds number in addition to β. For details, see Alvi, Sridharan, and Lakshamana Rao, J. Fluids Eng., 100, 299–307 (1978). Critical Flow Nozzle For a given set of upstream conditions, the rate of discharge of a gas from a nozzle will increase for a decrease in the absolute pressure ratio p2/p1 until the linear velocity in the throat reaches that of sound in the gas at that location. The value of p2/p1 for which the acoustic velocity is just attained is called the critical pressure ratio rc. The actual pressure in the throat will not fall below p1rc even if a much lower pressure exists downstream. The critical pressure ratio rc can be obtained from the following theoretical equation, which assumes a perfect gas and a frictionless nozzle: k−1 k+1 rc(1 − k)/k + ᎏ β4 rc2/k = ᎏ (10-24) 2 2

冢

FIG. 10-16

Values of expansion factor Y for orifices, nozzles, and venturis.

FIG. 10-17

Flow-nozzle assembly.

冣

MEASUREMENT OF FLOW This reduces, for β ≤ 0.2, to 2 k/(k − 1) rc = ᎏ (10-25) k+1 where k = ratio of specific heats cp /cv and β = diameter ratio. A table of values of rc as a function of k and β is given in the ASME Research Committee on Fluid Meters Report, op. cit., p. 68. For small values of β, rc = 0.487 for k = 1.667, 0.528 for k = 1.40, 0.546 for k = 1.30, and 0.574 for k = 1.15. Under critical flow conditions, only the upstream conditions p1, v1, and T1 need be known to determine flow rate, which, for β ≤ 0.2, is given by

冢

wmax = CA2

冣

p 2 ᎏ 冣莦冢莦莦 ᎏ莦 冪g莦k莦冢莦莦 v k + 1 冣 莦莦莦莦莦 (k + 1)/(k − 1)

1

c

(10-26)

1

For a perfect gas, this corresponds to wmax = CA2p1

2 M ᎏ 冣莦冢莦莦 ᎏ莦 冪g莦k莦冢莦莦 RT k + 1 冣 莦莦莦莦莦 (k + 1)/(k − 1)

c

(10-27)

1

For air, Eq. (10-26) reduces to 苶苶1 wmax = C1CA2 p1 /兹T

(10-28)

where A2 = cross-sectional area of throat; C = coefficient of discharge, dimensionless; gc = dimensional constant; k = ratio of specific heats, cp /cv; M = molecular weight; p1 = pressure on upstream side of nozzle; R = gas constant; T1 = absolute temperature on upstream side of nozzle; v1 = specific volume on upstream side of nozzle; C1 = dimensional constant, 0.0405 SI units (0.533 U.S. customary units); and wmax = maximum-weight flow rate. Discharge coefficients for critical flow nozzles are, in general, the same as those for subsonic nozzles. See Grace and Lapple, Trans. Am. Soc. Mech. Eng., 73, 639–647 (1951); and Szaniszlo, J. Eng. Power, 97, 521–526 (1975). Arnberg, Britton, and Seidl [ J. Fluids Eng., 96, 111–123 (1974)] present discharge-coefficient correlations for circular-arc venturi meters at critical flow. For the calculation of the flow of natural gas through nozzles under critical-flow conditions, see Johnson, J. Basic Eng., 92, 580–589 (1970). Orifice Meters A square-edged or sharp-edged orifice, as shown in Fig. 10-18, is a clean-cut square-edged hole with straight walls perpendicular to the flat upstream face of a thin plate placed crosswise of the channel. The stream issuing from such an orifice

10-15

attains its minimum cross section (vena contracta) at a distance downstream of the orifice which varies with the ratio β of orifice to pipe diameter (see Fig. 10-19). For a centered circular orifice in a pipe, the pressure differential is customarily measured between one of the following pressure-tap pairs. Except in the case of flange taps, all measurements of distance from the orifice are made from the upstream face of the plate. 1. Corner taps. Static holes drilled one in the upstream and one in the downstream flange, with the openings as close as possible to the orifice plate. 2. Radius taps. Static holes located one pipe diameter upstream and one-half pipe diameter downstream from the plate. 3. Pipe taps. Static holes located 2a pipe diameters upstream and eight pipe diameters downstream from the plate. 4. Flange taps. Static holes located 25.4 mm (1 in) upstream and 25.4 mm (1 in) downstream from the plate. 5. Vena-contracta taps. The upstream static hole is one-half to two pipe diameters from the plate. The downstream tap is located at the position of minimum pressure (see Fig. 10-19). Radius taps are best from a practical standpoint; the downstream pressure tap is located at about the mean position of the vena contracta, and the upstream tap is sufficiently far upstream to be unaffected by distortion of the flow in the immediate vicinity of the orifice (in practice, the upstream tap can be as much as two pipe diameters from the plate without affecting the results). Vena-contracta taps give the largest differential head for a given rate of flow but are inconvenient if the orifice size is changed from time to time. Corner taps offer the sometimes great advantage that the pressure taps can be built into the plate carrying the orifice. Thus the entire apparatus can be quickly inserted in a pipe line at any convenient flanged joint without having to drill holes in the pipe. Flange taps are similarly convenient, since by merely replacing standard flanges with special orifice flanges, suitable pressure taps are made available. Pipe taps give the lowest differential pressure, the value obtained being close to the permanent pressure loss.

(a)

(b) FIG. 10-18 Square-edged or sharp-edged orifices. The plate at the orifice opening must not be thicker than one-thirtieth of the pipe diameter, one-eighth of the orifice diameter, or one-fourth of the distance from the pipe wall to the edge of the opening. (a) Pipe-line orifice. (b) Types of plates.

FIG. 10-19 Coefficient of discharge for square-edged circular orifices for NRe > 30,000 with the upstream tap located between one and two pipe diameters from the orifice plate [Spitzglass, Trans. Am. Soc. Mech. Eng., 44, 919 (1922).]

10-16

TRANSPORT AND STORAGE OF FLUIDS

Rate of discharge through an orifice meter is given by Eq. (10-8) for either liquids or gases. For the case of subsonic flow of a gas (rc < r < 1.0), the expansion factor Y for orifices is approximated by Y = 1 − [(1 − r)/k](0.41 + 0.35β4)

(10-29)

where r = ratio of downstream to upstream static pressure (p2 /p1), k = ratio of specific heats (cp /cv), and β = diameter ratio. (See also Fig. 10-16.) Values of Y for supercritical flow of a gas (r < rc) through orifices are given by Benedict [J. Basic Eng., 93, 121–137 (1971)]. For the case of liquids, expansion factor Y is unity, and Eq. (10-22) should be used, since it allows for any difference in elevation between the upstream and downstream taps. Coefficient of discharge C for a given orifice type is a function of the Reynolds number NRe (based on orifice diameter and velocity) and diameter ratio β. At Reynolds numbers greater than about 30,000, the coefficients are substantially constant. For square-edged or sharpedged concentric circular orifices, the value will fall between 0.595 and 0.620 for vena-contracta or radius taps for β up to 0.8 and for flange taps for β up to 0.5. Figure 10-19 gives the coefficient of dis苶苶 charge K, including the velocity-of-approach factor (1/兹1 −苶 β4苶), as a function of β and the location of the downstream tap. Precise values of K are given in ASME PTC, op. cit., pp. 20–39, for flange taps, radius taps, vena-contracta taps, and corner taps. Precise values of C are given in the ASME Research Committee on Fluid Meters Report, op. cit., pp. 202–207, for the first three types of taps. The discharge coefficient of sharp-edged orifices was shown by Benedict, Wyler, and Brandt [ J. Eng. Power, 97, 576–582 (1975)] to increase with edge roundness. Typical as-purchased orifice plates may exhibit deviations on the order of 1 to 2 percent from ASME values of the discharge coefficient. In the transition region (NRe between 50 and 30,000), the coefficients are generally higher than the above values. Although calibration is generally advisable in this region, the curves given in Fig. 10-20 for corner and vena-contracta taps can be used as a guide. In the laminarflow region (NRe < 50), the coefficient C is proportional to 兹苶 NRe 苶. For 1 < NRe < 100, Johansen [Proc. R. Soc. (London), A121, 231–245 (1930)] presents discharge-coefficient data for sharp-edged orifices with corner taps. For NRe < 1, Miller and Nemecek [ASME Paper 58-A-106 (1958)] present correlations giving coefficients for sharpedged orifices and short-pipe orifices (L/D from 2 to 10). For shortpipe orifices (L/D from 1 to 4), Dickerson and Rice [ J. Basic Eng., 91,

546–548 (1969)] give coefficients for the intermediate range (27 < NRe < 7000). See also subsection “Contraction and Entrance Losses.” Permanent pressure loss across a concentric circular orifice with radius or vena-contracta taps can be approximated for turbulent flow by (p1 − p4)/(p1 − p2) = 1 − β2

(10-30)

where p1, p2 = upstream and downstream pressure-tap readings respectively, p4 = fully recovered pressure (four to eight pipe diameters downstream of the orifice), and β = diameter ratio. See ASME PTC, op. cit., Fig. 5. See Benedict, J. Fluids Eng., 99, 245–248 (1977), for a general equation for pressure loss for orifices installed in pipes or with plenum inlets. Orifices show higher loss than nozzles or venturis. Permanent pressure loss for laminar flow depends on the Reynolds number in addition to β. See Alvi, Sridharan, and Lakshmana Rao, loc. cit., for details. For the case of critical flow through a square- or sharp-edged concentric circular orifice (where r ≤ rc, as discussed earlier in this subsection), use Eqs. (10-26), (10-27), and (10-28) as given for criticalflow nozzles. However, unlike nozzles, the flow through a sharp-edged orifice continues to increase as the downstream pressure drops below that corresponding to the critical pressure ratio rc. This is due to an increase in the cross section of the vena contracta as the downstream pressure is reduced, giving a corresponding increase in the coefficient of discharge. At r = rc, C is about 0.75, while at r ⬵ 0, C has increased to about 0.84. See Grace and Lapple, loc. cit.; and Benedict, J. Basic Eng., 93, 99–120 (1971). Measurements by Harris and Magnall [Trans. Inst. Chem. Eng. (London), 50, 61–68 (1972)] with a venturi (β = 0.62) and orifices with radius taps (β = 0.60–0.75) indicate that the discharge coefficient for nonnewtonian fluids, in the range N′Re (generalized Reynolds number) 3500 to 100,000, is approximately the same as for newtonian fluids at the same Reynolds number. Quadrant-edge orifices have holes with rounded edges on the upstream side of the plate. The quadrant-edge radius is equal to the thickness of the plate at the orifice location. The advantages claimed for this type versus the square- or sharp-edged orifice are constantdischarge coefficients extending to lower Reynolds numbers and less possibility of significant changes in coefficient because of erosion or other damage to the inlet shape. Values of discharge coefficient C and Reynolds numbers limit for constant C are presented in Table 10-4, based on Ramamoorthy and Seetharamiah [J. Basic Eng., 88, 9–13 (1966)] and Bogema and Monkmeyer (J. Basic Eng., 82, 729–734 (1960)]. At Reynolds numbers above those listed for the upper limits, the coefficients rise abruptly. As Reynolds numbers decrease below those listed for the lower limits, the coefficients pass through a hump and then drop off. According to Bogema, Spring, and Ramamoorthy [J. Basic Eng., 84, 415–418 (1962), the hump can be eliminated by placing a fine-mesh screen about three pipe diameters upstream of the orifice. This reduces the lower NRe limit to about 500. Permanent pressure loss across quadrant-edge orifices for turbulent flow is somewhat lower than given by Eq. (10-30). See Alvi, Sridharan, and Lakshmana Rao, loc. cit., for values of discharge coefficient and permanent pressure loss in laminar flow. TABLE 10-4 Orifices

Discharge Coefficients for Quadrant-Edge * for Limiting N Re constant coefficient†

Coefficient of discharge for square-edged circular orifices with corner taps. [Tuve and Sprenkle, Instruments, 6, 201 (1933).] FIG. 10-20

β

C‡

K‡

Lower

Upper

0.225 0.400 0.500 0.600 0.630

0.770 0.780 0.824 0.856 0.885

0.771 0.790 0.851 0.918 0.964

5,000 5,000 4,000 3,000 3,000

60,000 150,000 200,000 120,000 105,000

*Based on pipe diameter and velocity. †For a precision of about ⫾0.5 percent. ‡Can be used with corner taps, flange taps, or radius taps.

MEASUREMENT OF FLOW Segmental and eccentric orifices are frequently used for gas metering when there is a possibility that entrained liquids or solids would otherwise accumulate in front of a concentric circular orifice. This can be avoided if the opening is placed on the lower side of the pipe. For liquid flow with entrained gas, the opening is placed on the upper side. The pressure taps should be located on the opposite side of the pipe from the opening. Coefficient C for a square-edged eccentric circular orifice (with opening tangent to pipe wall) varies from about 0.61 to 0.63 for β’s from 0.3 to 0.5, respectively, and pipe Reynolds numbers > 10,000 for either vena-contracta or flange taps (where β = diameter ratio). For square-edged segmental orifices, the coefficient C falls generally between 0.63 and 0.64 for 0.3 ≤ β ≤ 0.5 and pipe Reynolds numbers > 10,000, for vena-contracta or flange taps, where β = diameter ratio for an equivalent circular orifice = 兹苶 α (α = ratio of orifice to pipe cross-sectional areas). Values of expansion factor Y are slightly higher than for concentric circular orifices, and the location of the vena contracta is moved farther downstream as compared with concentric circular orifices. For further details, see ASME Research Committee on Fluid Meters Report, op. cit., pp. 210–213. For permanent pressure loss with segmental and eccentric orifices with laminar pipe flow see Lakshmana Rao and Sridharan, Proc. Am. Soc. Civ. Eng., J. Hydraul. Div., 98 (HY 11), 2015–2034 (1972). Annular orifices can also be used to advantage for gas metering when there is a possibility of entrained liquids or solids and for liquid metering with entrained gas present in small concentrations. Coefficient K was found by Bell and Bergelin [Trans. Am. Soc. Mech. Eng., 79, 593–601 (1957)] to range from about 0.63 to 0.67 for annulus Reynolds numbers in the range of 100 to 20,000 respectively for values of 2L/(D − d) less than 1 where L = thickness of orifice at outer edge, D = inside pipe diameter, and d = diameter of orifice disk. The annulus Reynolds number is defined as NRe = (D − d)(G/ µ)

(10-31)

where G = mass velocity pV through orifice opening and µ = fluid viscosity. The above coefficients were determined for β’s (= d/D) in the range of 0.95 to 0.996 and with pressure taps located 19 mm (e in) upstream of the disk and 230 mm (9 in) downstream in a 5.25-indiameter pipe. Elbow Meters A pipe elbow can be used as a flowmeter for liquids if the differential centrifugal head generated between the inner and outer radii of the bend is measured by means of pressure taps located midway around the bend. Equation (10-22) can be used, except that the pressure-difference term (p1 − p2) is now taken to be the differential centrifugal pressure and β is taken as zero if one assumes no change in cross section between the pipe and the bend. The discharge coefficient should preferably be determined by calibration, but as a guide it can be estimated within ⫾6 percent for circular pipe for Reynolds numbers greater than 105 from C = 0.98 兹苶 R苶 2苶 D, c /苶 where Rc = radius of curvature of the centerline and D = inside pipe diameter in consistent units. See Murdock, Foltz, and Gregory, J. Basic Eng., 86, 498–506 (1964); or the ASME Research Committee on Fluid Meters Report, op. cit., pp. 75–77. Accuracy Square-edged orifices and venturi tubes have been so extensively studied and standardized that reproducibilities within 1 to 2 percent can be expected between standard meters when new and clean. This is therefore the order of reliability to be had, if one assumes (1) accurate measurement of meter differential, (2) selection of the coefficient of discharge from recommended published literature, (3) accurate knowledge of fluid density, (4) accurate measurement of critical meter dimensions, (5) smooth upstream face of orifice, and (6) proper location of the meter with respect to other flow-disturbing elements in the system. Care must also be taken to avoid even slight corrosion or fouling during use. Presence of swirling flow or an abnormal velocity distribution upstream of the metering element can cause serious metering error unless calibration in place is employed or sufficient straight pipe is inserted between the meter and the source of disturbance. Table 10-5 gives the minimum lengths of straight pipe required to avoid appreciable error due to the presence of certain fittings and valves either upstream or downstream of an orifice or nozzle. These values were

10-17

extracted from plots presented by Sprenkle [Trans. Am. Soc. Mech. Eng., 67, 345–360 (1945)]. Table 10-5 also shows the reduction in spacing made possible by the use of straightening vanes between the fittings and the meter. Entirely adequate straightening vanes can be provided by fitting a bundle of thin-wall tubes within the pipe. The center-to-center distance between tubes should not exceed onefourth of the pipe diameter, and the bundle length should be at least 8 times this distance. The distances specified in Table 10-5 will be conservative if applied to venturi meters. For specific information on requirements for venturi meters, see a discussion by Pardoe appended to Sprenkle (op. cit.). Extensive data on the effect of installation on the coefficients of venturi meters are given elsewhere by Pardoe [Trans. Am. Soc. Mech. Eng., 65, 337–349 (1943)]. In the presence of flow pulsations, the indications of head meters such as orifices, nozzles, and venturis will often be undependable for several reasons. First, the measured pressure differential will tend to be high, since the pressure differential is proportional to the square of flow rate for a head meter, and the square root of the mean differential pressure is always greater than the mean of the square roots of the differential pressures. Second, there is a phase shift as the wave passes through the metering restriction which can affect the differential. Third, pulsations can be set up in the manometer leads themselves. Frequency of the pulsation also plays a part. At low frequencies, the meter reading can generally faithfully follow the flow pulsations, but at high frequencies it cannot. This is due to inertia of the fluid in the manometer leads or of the manometric fluid, whereupon the meter would give a reading intermediate between the maximum and minimum flows but having no readily predictable relation to the mean flow. Pressure transducers with flush-mounted diaphragms can be used together with high-speed recording equipment to provide accurate records of the pressure profiles at the upstream and downstream pressure taps, which can then be analyzed and translated into a mean flow rate. TABLE 10-5 Locations of Orifices and Nozzles Relative to Pipe Fittings Distances in pipe diameters, D1 Distance, upstream fitting to orifice

Type of fitting upstream

D2 D1

Without straightening vanes

Single 90° ell, tee, or cross used as ell

0.2 0.4 0.6 0.8

6 6 9 20

9 12

8

2 short-radius 90° ells in form of S

0.2 0.4 0.6 0.8

7 8 13 25

8 10 15

6 11

2 long- or short-radius 90° ells in perpendicular planes

0.2 0.4 0.6 0.8

15 18 25 40

9 10 11 13

5 6 7 9

Contraction or enlargement

0.2 0.4 0.6 0.8

8 9 10 15

Vanes have no advantage

Globe valve or stop check

0.2 0.4 0.6 0.8

9 10 13 21

9 10 10 13

Gate valve, wide open, or plug cocks

0.2 0.4 0.6 0.8

6 6 8 14

With straightening vanes

Distance, vanes to orifice

Distance, nearest downstream fitting from orifice 2 4 2 4 2 4 2 4

5 6 6 9

Same as globe valve

2 4 2 4

10-18

TRANSPORT AND STORAGE OF FLUIDS

The rather general practice of producing a steady differential reading by placing restrictions in the manometer leads can result in a reading which, under a fixed set of conditions, may be useful in control of an operation but which has no readily predictable relation to the actual average flow. If calibration is employed to compensate for the presence of pulsations, complete reproduction of operating conditions, including source of pulsations and waveform, is necessary to ensure reasonable accuracy. According to Head [Trans. Am. Soc. Mech. Eng., 78, 1471–1479 (1956)], a pulsation-intensity limit of Γ = 0.1 is recommended as a practical pulsation threshold below which the performance of all types of flowmeters will differ negligibly from steady-flow performance (an error of less than 1 percent in flow due to pulsation). Γ is the peak-totrough flow variation expressed as a fraction of the average flow rate. According to the ASME Research Committee on Fluid Meters Report (op. cit., pp. 34–35), the fractional metering error E for liquid flow through a head meter is given by (1 + E)2 = 1 + Γ 2/8

(10-32)

When the pulsation amplitude is such as to result in a greater-thanpermissible metering error, consideration should be given to installation of a pulsation damper between the source of pulsations and the flowmeter. References to methods of pulsation-damper design are given in the subsection “Unsteady-State Behavior.” Pulsations are most likely to be encountered in discharge lines from reciprocating pumps or compressors and in lines supplying steam to reciprocating machinery. For gas flow, a combination involving a surge chamber and a constriction in the line can be used to damp out the pulsations to an acceptable level. The surge chamber is generally located as close to the pulsation source as possible, with the constriction between the surge chamber and the metering element. This arrangement can be used for either a suction or a discharge line. For such an arrangement, the metering error has been found to be a function of the Hodgson number NH, which is defined as NH = Qn ∆ps /qps

(10-33)

where Q = volume of surge chamber and pipe between metering element and pulsation source; n = pulsation frequency; ∆ps = permanent pressure drop between metering element and surge chamber; q = average volume flow rate, based on gas density in the surge chamber; and ps = pressure in surge chamber. Herning and Schmid [Z. Ver. Dtsch. Ing., 82, 1107–1114 (1938)] presented charts for a simplex double-acting compressor for the prediction of metering error as a function of the Hodgson number and s, the ratio of piston discharge time to total time per stroke. Table 10-6a gives the minimum Hodgson numbers required to reduce the metering error to 1 percent as given by the charts (for specific heat ratios between 1.28 and 1.37). Schmid [Z. Ver. Dtsch. Ing., 84, 596–598 (1940)] presented similar charts for a duplex double-acting compressor and a triplex double-acting compressor for a specific heat ratio of 1.37. Table 10-6b gives the minimum Hodgson numbers correspond-

ing to a 1 percent metering error for these cases. The value of Q ∆ps can be calculated from the appropriate Hodgson number, and appropriate values of Q and ∆ps selected so as to satisfy this minimum requirement. AREA METERS General Principles The underlying principle of an ideal area meter is the same as that of a head meter of the orifice type (see subsection “Orifice Meters”). The stream to be measured is throttled by a constriction, but instead of observing the variation with flow of the differential head across an orifice of fixed size, the constriction of an area meter is so arranged that its size is varied to accommodate the flow while the differential head is held constant. A simple example of an area meter is a gate valve of the rising-stem type provided with static-pressure taps before and after the gate and a means for measuring the stem position. In most common types of area meters, the variation of the opening is automatically brought about by the motion of a weighted piston or float supported by the fluid. Two different cylinder- and piston-type area meters are described in the ASME Research Committee on Fluid Meters Report, op. cit., pp. 82–83. Rotameters The rotameter, an example of which is shown in Fig. 10-21, has become one of the most popular flowmeters in the chemical-process industries. It consists essentially of a plummet, or “float,” which is free to move up or down in a vertical, slightly tapered tube having its small end down. The fluid enters the lower end of the tube and causes the float to rise until the annular area between the float and the wall of the tube is such that the pressure drop across this constriction is just sufficient to support the float. Typically, the tapered tube is of glass and carries etched upon it a nearly linear scale on which the position of the float may be visually noted as an indication of the flow. Interchangeable precision-bore glass tubes and metal metering tubes are available. Rotameters have proved satisfactory both for gases and for liquids at high and at low pressures. A single instrument can readily cover a tenfold range of flow, and by providing floats of different densities a two-hundredfold range is practicable. Rotameters are available with pneumatic, electric, and electronic transmitters for actuating remote recorders, integrators, and automatic flow con-

TABLE 10-6a Minimum Hodgson Numbers Simplex double-acting compressor s

NH

s

NH

0.167 0.333 0.50

1.31 1.00 0.80

0.667 0.833 1.00

0.60 0.43 0.34

TABLE 10-6b Minimum Hodgson Numbers Duplex double-acting compressor

Triplex double-acting compressor

s

NH

s

NH

0.167 0.333 0.50 0.667 0.833 1.00

1.00 0.70 0.30 0.10 0.05 0.00

0.167 0.333 0.50 0.667 0.833 1.00

0.85 0.30 0.15 0.06 0.00 0.00

FIG. 10-21

Rotameter.

MEASUREMENT OF FLOW trollers (see Considine, op. cit., pp. 4-35–4-36, and Sec. 22 of this Handbook). Rotameters require no straight runs of pipe before or after the point of installation. Pressure losses are substantially constant over the whole flow range. In experimental work, for greatest precision, a rotameter should be calibrated with the fluid which is to be metered. However, most modern rotameters are precision-made so that their performance closely corresponds to a master calibration plot for the type in question. Such a plot is supplied with the meter upon purchase. According to Head [Trans. Am. Soc. Mech. Eng., 76, 851–862 (1954)], flow rate through a rotameter can be obtained from w = qρ = KDf

W (ρ − ρ)ρ ᎏᎏ 冪莦 ρ f

f

(10-34)

f

and

冤

冪莦莦莦莦莦莦莦 冥

Dt µ K = φ ᎏ , ᎏᎏ Df Wf(ρf − ρ)ρ ᎏᎏ ρf

(10-35)

where w = weight flow rate; q = volume flow rate; ρ = fluid density; K = flow parameter, m1/2/s (ft1/2/s); Df = float diameter at constriction; Wf = float weight; ρf = float density; Dt = tube diameter at point of constriction; and µ = fluid viscosity. The appropriate value of K is obtained from a composite correlation of K versus the parameters shown in Eq. (10-35) corresponding to the float shape being used. The relation of Dt to the rotameter reading is also required for the tube taper and size being used. The ratio of flow rates for two different fluids A and B at the same rotameter reading is given by wA KA ᎏ=ᎏ wB KB

(ρ − ρ )ρ ᎏᎏ 冪莦 (ρ − ρ )ρ f

A

A

f

B

B

(10-36)

A measure of self-compensation, with respect to weight rate of flow, for fluid-density changes can be introduced through the use of a float with a density twice that of the fluid being metered, in which case an increase of 10 percent in ρ will produce a decrease of only 0.5 percent in w for the same reading. The extent of immunity to changes in fluid viscosity depends upon the shape of the float. According to Baird and Cheema [Can. J. Chem. Eng., 47, 226–232 (1969)], the presence of square-wave pulsations can cause a rotameter to overread by as much as 100 percent. The higher the pulsation frequency, the less the float oscillation, although the error can still be appreciable even when the frequency is high enough so that the float is virtually stationary. Use of a damping chamber between the pulsation source and the rotameter will reduce the error. Additional information on rotameter theory is presented by Fischer [Chem. Eng., 59(6), 180–184 (1952)], Coleman [Trans. Inst. Chem. Eng., 34, 339–350 (1956)], and McCabe and Smith (Unit Operations of Chemical Engineering, 3d ed., McGraw-Hill, New York, 1976, pp. 215–218). MASS FLOWMETERS General Principles There are two main types of mass flowmeters: (1) the so-called true mass flowmeter, which responds directly to mass flow rate, and (2) the inferential mass flowmeter, which commonly measures volume flow rate and fluid density separately. A variety of types of true mass flowmeters have been developed, including the following: (a) the Magnus-effect mass flowmeter, (b) the axial-flow, transverse-momentum mass flowmeter, (c) the radial-flow, transversemomentum mass flowmeter, (d) the gyroscopic transverse-momentum mass flowmeter, and (e) the thermal mass flowmeter. Type b is the basis for several commercial mass flowmeters, one version of which is briefly described here. Axial-Flow Transverse-Momentum Mass Flowmeter This type is also referred to as an angular-momentum mass flowmeter. One embodiment of its principle involves the use of axial flow through a driven impeller and a turbine in series. The impeller imparts angular momentum to the fluid, which in turn causes a torque to be imparted

10-19

to the turbine, which is restrained from rotating by a spring. The torque, which can be measured, is proportional to the rotational speed of the impeller and the mass flow rate. Inferential Mass Flowmeter There are several types in this category, including the following: 1. Head meters with density compensation. Head meters such as orifices, venturis, or nozzles can be used with one of a variety of densitometers [e.g., based on (a) buoyant force on a float, (b) hydraulic coupling, (c) voltage output from a piezoelectric crystal, or (d) radiation absorption]. The signal from the head meter, which is proportional to ρV 2 (where ρ = fluid density and V = fluid velocity), is multiplied by ρ given by the densitometer. The square root of the product is proportional to the mass flow rate. 2. Head meters with velocity compensation. The signal from the head meter, which is proportional to ρV 2, is divided by the signal from a velocity meter to give a signal proportional to the mass flow rate. 3. Velocity meters with density compensation. The signal from the velocity meter (e.g., turbine meter, electromagnetic meter, or sonic velocity meter) is multiplied by the signal from a densitometer to give a signal proportional to the mass flow rate. Additional information on mass-flowmeter principles can be obtained from Yeaple (Hydraulic and Pneumatic Power and Control, McGraw-Hill, New York, 1966, pp. 125–128), Halsell [Instrum. Soc. Am. J., 7, 49–62 (June 1960)], and Flanagan and Colman [Control, 7, 242–245 (1963)]. Information on commercially available mass flowmeters is given in the latter two references. WEIRS Liquid flow in an open channel may be metered by means of a weir, which consists of a dam over which, or through a notch in which, the liquid flows. The terms “rectangular weir,” “triangular weir,” etc., generally refer to the shape of the notch in a notched weir. All weirs considered here have flat upstream faces that are perpendicular to the bed and walls of the channel. Sharp-edged weirs have edges like those of square or sharpedged orifices (see subsection “Orifice Meters”). Notched weirs are ordinarily sharp-edged. Weirs not in the sharp-edged class are, for the most part, those described as broad-crested weirs. The head h0 on a weir is the liquid-level height above the crest or base of the notch. The head must be measured sufficiently far upstream to avoid the drop in level occasioned by the overfall which begins at a distance about 2h0 upstream from the weir. Surface-level measurements should be made a distance of 3h0 or more upstream, preferably by using a stilling box equipped with a high-precision level gauge, e.g., a hook gauge or float gauge. With sharp-edged weirs, the sheet of discharging liquid, called the “nappe,” contracts as it leaves the opening and free discharge occurs. Rounding the upstream edge will reduce the contraction and increase the flow rate for a given head. A clinging nappe may result if the head is very small, if the edge is well rounded, or if air cannot flow in beneath the nappe. This, in turn, results in an increase in the discharge rate for a given head as compared with that for a free nappe. For further information on the effect of the nappe, see Gibson, Hydraulics and Its Applications, 5th ed., Constable, London, 1952; and Chow, Open-Channel Hydraulics, McGraw-Hill, New York, 1959. Flow through a rectangular weir (Fig. 10-22) is given by 苶g苶 q = 0.415(L − 0.2h0)h1.5 0 兹2

(10-37)

where q = volume flow rate, L = crest length, h0 = weir head, and g =

FIG. 10-22

Rectangular weir.

10-20

TRANSPORT AND STORAGE OF FLUIDS

local acceleration due to gravity. This is known as the modified Francis formula for a rectangular sharp-edged weir with two end corrections; it applies when the velocity-of-approach correction is small. The Francis formula agrees with experiments within 3 percent if (1) L is greater than 2h0, (2) velocity of approach is 0.6 m/s (2 ft/s) or less, (3) height of crest above bottom of channel is at least 3h0, and (4) h0 is not less than 0.09 m (0.3 ft). Narrow rectangular notches (h0 > L) have been found to give about 93 percent of the discharge predicted by the Francis formula. Thus 苶g苶 q = 0.386Lh1.5 (10-38) 0 兹2 In this case, no end corrections are applied even though the formula applies only for sharp-edged weirs. See Schoder and Dawson, Hydraulics, McGraw-Hill, New York, 1934, p. 175, for further details. The triangular-notch weir has the advantage that a single notch can accommodate a wide range of flow rates, although this in turn reduces its accuracy. The discharge for sharp- or square-edged weirs is given by 苶g苶)/tan φ q = (0.31h2.5 (10-39) 0 兹2 See Eq. (10-37) for nomenclature. Angle φ is illustrated in Fig. 10-23. Equations (10-37), (10-38), and (10-39) are applicable only to the flow of water. However, for the case of triangular-notch weirs Lenz [Trans. Am. Soc. Civ. Eng., 108, 759–802 (1943)] has presented correlations predicting the effect of viscosity over the range of 0.001 to 0.15 Pa⋅s (1 to 150 cP) and surface tension over the range of 0.03 to 0.07 N/m (30 to 70 dyn/cm). His equation predicts about an 8 percent increase in flow for a liquid of 0.1-Pa⋅s (100-cP) viscosity compared with water at 0.001 Pa⋅s (1 cP) and about a 1 percent increase for a liquid with one-half of the surface tension of water. For fluids of moderate viscosity, Ranga Raju and Asawa [Proc. Am. Soc. Civ. Eng., J. Hydraul. Div., 103 (HY 10), 1227–1231 (1977)] find that the effect of viscosity and surface tension on the discharge flow rate for rectangular and triangular-notch (φ = 45°) weirs can be neglected when (NRe)0.2(NWe)0.6 > 900

(10-40)

gh 苶03 /ν, g = local acceleration due to where NRe (Reynolds number) = 兹苶 gravity, h0 = weir head, ν = kinematic viscosity; NWe (Weber number) = ρgh02 /gcσ, ρ = density, gc = dimensional constant, and σ = surface tension. For the flow of high-viscosity liquids over rectangular weirs, see Slocum, Can. J. Chem. Eng., 42, 196–200 (1964). His correlation is based on data for liquids with viscosities in the range of 2.5 to 500 Pa⋅s (25 to 5000 cP), in which range the discharge decreases markedly for a given head as viscosity is increased.

FIG. 10-23

Triangular weir.

Information on other types of weirs can be obtained from Addison, op. cit.; Gibson, Hydraulics and Its Applications, 5th ed., Constable, London, 1952; Henderson, Open Channel Flow, Macmillan, New York, 1966; Linford, Flow Measurement and Meters, Spon, London, 1949; Lakshmana Rao, “Theory of Weirs,” in Advances in Hydroscience, vol. 10, Academic, New York, 1975; and Urquhart, Civil Engineering Handbook, 4th ed., McGraw-Hill, New York, 1959. TWO-PHASE SYSTEMS It is generally preferable to meter each of the individual components of a two-phase mixture separately prior to mixing, since it is difficult to meter such mixtures accurately. Problems arise because of fluctuations in composition with time and variations in composition over the cross section of the channel. Information on metering of such mixtures can be obtained from the following sources. Gas-Solid Mixtures Carlson, Frazier, and Engdahl [Trans. Am. Soc. Mech. Eng., 70, 65–79 (1948)] describe the use of a flow nozzle and a square-edged orifice in series for the measurement of both the gas rate and the solids rate in the flow of a finely divided solid-ingas mixture. The nozzle differential is sensitive to the flow of both phases, whereas the orifice differential is not influenced by the solids flow. Farbar [Trans. Am. Soc. Mech. Eng., 75, 943–951 (1953)] describes how a venturi meter can be used to measure solids flow rate in a gassolids mixture when the gas rate is held constant. Separate calibration curves (solids flow versus differential) are required for each gas rate of interest. Cheng, Tung, and Soo [J. Eng. Power, 92, 135–149 (1970)] describe the use of an electrostatic probe for measurement of solids flow in a gas-solids mixture. Goldberg and Boothroyd [Br. Chem. Eng., 14, 1705–1708 (1969)] describe several types of solids-in-gas flowmeters and give an extensive bibliography. Gas-Liquid Mixtures An empirical equation was developed by Murdock [J. Basic Eng., 84, 419–433 (1962)] for the measurement of gas-liquid mixtures using sharp-edged orifice plates with either radius, flange, or pipe taps. An equation for use with venturi meters was given by Chisholm [Br. Chem. Eng., 12, 454–457 (1967)]. A procedure for determining steam quality via pressure-drop measurement with upflow through either venturi meters or sharp-edged orifice plates was given by Collins and Gacesa [J. Basic Eng., 93, 11–21 (1971)]. Liquid-Solid Mixtures Liptak [Chem. Eng., 74(4), 151–158 (1967)] discusses a variety of techniques that can be used for the measurement of solids-in-liquid suspensions or slurries. These include metering pumps, weigh tanks, magnetic flowmeter, ultrasonic flowmeter, gyroscope flowmeter, etc. Shirato, Gotoh, Osasa, and Usami [J. Chem. Eng. Japan, 1, 164– 167 (January 1968)] present a method for determining the mass flow rate of suspended solids in a liquid stream wherein the liquid velocity is measured by an electromagnetic flowmeter and the flow of solids is calculated from the pressure drops across each of two vertical sections of pipe of different diameter through which the suspension flows in series.

PUMPING OF LIQUIDS AND GASES GENERAL REFERENCES: Paul N. Garay, P. E., Pump Application Desk Book, Fairmont Press, 1993. John W. Dufor and William E. Nelson, Centrifugal Pump Sourcebook, McGraw-Hill, 1992. Process Pumps, ITT Fluid Technology Corporation, 1992. James Corley, “The Vibration Analysis of Pumps: A Tutorial,” Fourth International Pump Symposium, Texas A & M University, Houston, Texas, May 1987.

INTRODUCTION A pump is a physical contrivance that is used to deliver fluids from one location to another through conduits. Over the years, numerous pump

designs have evolved to meet differing requirements. The basic requirements to define the application are suction and delivery pressures, pressure loss in transmission, and the flow rate. Special requirements may exist in food, pharmaceutical, nuclear, and other industries that impose material selection requirements of the pump. The primary means of transfer of energy to the fluid that causes flow are gravity, displacement, centrifugal force, electromagnetic force, transfer of momentum, mechanical impulse, and a combination of these energy-transfer mechanisms. Gravity and centrifugal force are the most common energy-transfer mechanisms in use. Pump designs have largely been standardized. Based on application

PUMPING OF LIQUIDS AND GASES experience, numerous standards have come into existence. As special projects and new application situations for pumps develop, these standards will be updated and revised. Common pump standards are: 1. American Petroleum Institute (API) Standard 610, Centrifugal Pumps for Refinery Service 2. American Waterworks Association (AWWA) E101, Deep Well Vertical Turbine Pumps 3. Underwriters Laboratories (UL) UL 51, UL343, UL1081, UL448, UL1247 4. National Fire Protection Agency (NFPA) NFPA-20 Centrifugal Fire Pumps 5. American Society of Mechanical Engineers (ASME) 6. American National Standards Institute 7. Hydraulic Institute Standards (Application) These standards specify design, construction, and testing details such as material selection, shop inspection and tests, drawings and other uses required, clearances, construction procedures, and so on.

Piston Plunger Reciprocating

There are four (4) major types of pumps: (1) positive displacement, (2) dynamic (kinetic), (3) lift, and (4) electromagnetic. Piston pumps are positive displacement pumps. The most common centrifugal pumps are of dynamic type; ancient bucket-type pumps are lift pumps; and electromagnetic pumps use electromagnetic force and are common in modern reactors. Canned pumps are also becoming popular in the petrochemical industry because of the drive to minimize fugitive emissions. Figure 10-24 shows pump classification: TERMINOLOGY Displacement Discharge of a fluid from a vessel by partially or completely displacing its internal volume with a second fluid or by mechanical means is the principle upon which a great many fluidtransport devices operate. Included in this group are reciprocatingpiston and diaphragm machines, rotary-vane and gear types, fluid piston compressors, acid eggs, and air lifts.

Double Acting

Simplex Duplex

Steam

Single Acting Double Acting

Simplex Duplex Triplex Multiplex

Power

Simplex Multiplex

Diaphragm

Fluid Operated Mech. Operated

Positive Displacement Single Rotor

Vane Piston Flexible Member Screw

Multiple Rotor

Gear Lobe Circumferential Piston Screw

Blow Cover

Rotary

Fluid Ring Radial Flow Mixed Flow

Single Suction Double Suction Double Valves

Self-priming Non-self-priming Open Impeller Single Stage Semi-open Impeller Multistage Closed Impeller

Axial Flow

Single Suction

Single Stage Multistage

Peripheral

Single Stage Multistage

Self-priming Non-self-priming

Regenerative

Special

Viscous Drag Jet (Ejector-boosted) Gas Lift Hydraulic Ram Electromagnetic Screw Centrifugal Rotating Casing (Pitot)

Pumps Centrifugal

Kinetic

Open Screw FIG. 10-24

Classification of pumps (Courtesty of Hydraulic Institute).

10-21

Open Impeller Closed Impeller

10-22

TRANSPORT AND STORAGE OF FLUIDS

The large variety of displacement-type fluid-transport devices makes it difficult to list characteristics common to each. However, for most types it is correct to state that (1) they are adaptable to highpressure operation, (2) the flow rate through the pump is variable (auxiliary damping systems may be employed to reduce the magnitude of pressure pulsation and flow variation), (3) mechanical considerations limit maximum throughputs, and (4) the devices are capable of efficient performance at extremely low-volume throughput rates. Centrifugal Force Centrifugal force is applied by means of the centrifugal pump or compressor. Though the physical appearance of the many types of centrifugal pumps and compressors varies greatly, the basic function of each is the same, i.e., to produce kinetic energy by the action of centrifugal force and then to convert this energy into pressure by efficiently reducing the velocity of the flowing fluid. In general, centrifugal fluid-transport devices have these characteristics: (1) discharge is relatively free of pulsation; (2) mechanical design lends itself to high throughputs, capacity limitations are rarely a problem; (3) the devices are capable of efficient performance over a wide range of pressures and capacities even at constant-speed operation; (4) discharge pressure is a function of fluid density; and (5) these are relatively small high-speed devices and less costly. A device which combines the use of centrifugal force with mechanical impulse to produce an increase in pressure is the axial-flow compressor or pump. In this device the fluid travels roughly parallel to the shaft through a series of alternately rotating and stationary radial blades having airfoil cross sections. The fluid is accelerated in the axial direction by mechanical impulses from the rotating blades; concurrently, a positive-pressure gradient in the radial direction is established in each stage by centrifugal force. The net pressure rise per stage results from both effects. Electromagnetic Force When the fluid is an electrical conductor, as is the case with molten metals, it is possible to impress an electromagnetic field around the fluid conduit in such a way that a driving force that will cause flow is created. Such pumps have been developed for the handling of heat-transfer liquids, especially for nuclear reactors. Transfer of Momentum Deceleration of one fluid (motivating fluid) in order to transfer its momentum to a second fluid (pumped fluid) is a principle commonly used in the handling of corrosive materials, in pumping from inaccessible depths, or for evacuation. Jets and eductors are in this category. Absence of moving parts and simplicity of construction have frequently justified the use of jets and eductors. However, they are relatively inefficient devices. When air or steam is the motivating fluid, operating costs may be several times the cost of alternative types of fluid-transport equipment. In addition, environmental considerations in today’s chemical plants often inhibit their use. Mechanical Impulse The principle of mechanical impulse when applied to fluids is usually combined with one of the other means of imparting motion. As mentioned earlier, this is the case in axial-flow compressors and pumps. The turbine or regenerative-type pump is another device which functions partially by mechanical impulse. Measurement of Performance The amount of useful work that any fluid-transport device performs is the product of (1) the mass rate of fluid flow through it and (2) the total pressure differential measured immediately before and after the device, usually expressed in the height of column of fluid equivalent under adiabatic conditions. The first of these quantities is normally referred to as capacity, and the second is known as head. Capacity This quantity is expressed in the following units. In SI units capacity is expressed in cubic meters per hour (m3/h) for both liquids and gases. In U.S. customary units it is expressed in U.S. gallons per minute (gal/min) for liquids and in cubic feet per minute (ft3/min) for gases. Since all these are volume units, the density or specific gravity must be used for conversion to mass rate of flow. When gases are being handled, capacity must be related to a pressure and a temperature, usually the conditions prevailing at the machine inlet. It is important to note that all heads and other terms in the following equations are expressed in height of column of liquid.

Total Dynamic Head The total dynamic head H of a pump is the total discharge head hd minus the total suction head hs. Total Suction Head This is the reading hgs of a gauge at the suction flange of a pump (corrected to the pump centerline°), plus the barometer reading and the velocity head hvs at the point of gauge attachment: hs = hgs + atm + hvs

(10-41)

If the gauge pressure at the suction flange is less than atmospheric, requiring use of a vacuum gauge, this reading is used for hgs in Eq. (10-41) with a negative sign. Before installation it is possible to estimate the total suction head as follows: hs = hss − hfs

(10-42)

where hss = static suction head and hfs = suction friction head. Static Suction Head The static suction head hss is the vertical distance measured from the free surface of the liquid source to the pump centerline plus the absolute pressure at the liquid surface. Total Discharge Head The total discharge head hd is the reading hgd of a gauge at the discharge flange of a pump (corrected to the pump centerline*), plus the barometer reading and the velocity head hvd at the point of gauge attachment: hd = hgd + atm + hvd

(10-43)

Again, if the discharge gauge pressure is below atmospheric, the vacuum-gauge reading is used for hgd in Eq. (10-43) with a negative sign. Before installation it is possible to estimate the total discharge head from the static discharge head hsd and the discharge friction head hfd as follows: hd = hsd + hfd

(10-44)

Static Discharge Head The static discharge head hsd is the vertical distance measured from the free surface of the liquid in the receiver to the pump centerline,* plus the absolute pressure at the liquid surface. Total static head hts is the difference between discharge and suction static heads. Velocity Since most liquids are practically incompressible, the relation between the quantity flowing past a given point in a given time and the velocity of flow is expressed as follows: Q = Av

(10-45)

This relationship in SI units is as follows: v (for circular conduits) = 3.54 Q/d 2

(10-46)

where v = average velocity of flow, m/s; Q = quantity of flow, m3/h; and d = inside diameter of conduit, cm. This same relationship in U.S. customary units is v (for circular conduits) = 0.409 Q/d 2

(10-47)

where v = average velocity of flow, ft/s; Q = quantity of flow, gal/min; and d = inside diameter of conduit, in. Velocity Head This is the vertical distance by which a body must fall to acquire the velocity v. hv = v2/2g

(10-48)

Viscosity (See Sec. 5 for further information.) In flowing liquids the existence of internal friction or the internal resistance to relative motion of the fluid particles must be considered. This resistance is called viscosity. The viscosity of liquids usually decreases with rising temperature. Viscous liquids tend to increase the power required by a pump, to reduce pump efficiency, head, and capacity, and to increase friction in pipe lines. Friction Head This is the pressure required to overcome the resistance to flow in pipe and fittings. It is dealt with in detail in Sec. 5. * On vertical pumps, the correction should be made to the eye of the suction impeller.

PUMPING OF LIQUIDS AND GASES

where kW is the pump power output, kW; H = total dynamic head, N⋅m/kg (column of liquid); Q = capacity, m3/h; and ρ = liquid density, kg/m3. When the total dynamic head H is expressed in pascals, then kW = HQ/3.599 × 106

hp = HQ/1.714 × 103

(10-52)

Pump efficiency = (power output)/(power input)

(10-53)

Suction Limitations of a Pump Whenever the pressure in a liquid drops below the vapor pressure corresponding to its temperature, the liquid will vaporize. When this happens within an operating pump, the vapor bubbles will be carried along to a point of higher pressure, where they suddenly collapse. This phenomenon is known as cavitation. Cavitation in a pump should be avoided, as it is accompanied by metal removal, vibration, reduced flow, loss in efficiency, and noise. When the absolute suction pressure is low, cavitation may occur in the pump inlet and damage result in the pump suction and on the impeller vanes near the inlet edges. To avoid this phenomenon, it is necessary to maintain a required net positive suction head (NPSH)R, which is the equivalent total head of liquid at the pump centerline less the vapor pressure p. Each pump manufacturer publishes curves relating (NPSH)R to capacity and speed for each pump. When a pump installation is being designed, the available net positive suction head (NPSH)A must be equal to or greater than the (NPSH)R for the desired capacity. The (NPSH)A can be calculated as follows: (NPSH)A = hss − hfs − p

(10-54)

If (NPSH)A is to be checked on an existing installation, it can be determined as follows: (NPSH)A = atm + hgs − p + hvs

(10-55)

Practically, the NPSH required for operation without cavitation and vibration in the pump is somewhat greater than the theoretical. The actual (NPSH)R depends on the characteristics of the liquid, the total head, the pump speed, the capacity, and impeller design. Any suction condition which reduces (NPSH)A below that required to prevent cavitation at the desired capacity will produce an unsatisfactory installation and can lead to mechanical difficulty. NPSH Requirements for Other Liquids NPSH values depend on the fluid being pumped. Since water is considered a standard fluid

Is

50 40 30 20 15 10

1.0

(10-51)

The power input to a pump is greater than the power output because of internal losses resulting from friction, leakage, etc. The efficiency of a pump is therefore defined as

ta

u ob

ne

ta

Bu

5 4 3 2 1.5

(10-50)

where hp is the pump-power output, hp; H = total dynamic head, lbf⋅ft/lbm (column of liquid); Q = capacity, U.S. gal/min; and s = liquid specific gravity. When the total dynamic head H is expressed in pounds-force per square inch, then

ne

100

In U.S. customary units, hp = HQs/3.960 × 103

e

an

op

Pr

0

50

100

10 8 6 5 4 3 2 1.5 1.0 0.5

NPSH reductions, feet

(10-49)

500 400 300 200 150

Re fri ge Me thy ran t la lco R-1 ho 1 l Wa ter

kW = HQρ/3.670 × 105

1000

Vapor pressure, PSIA

Work Performed in Pumping To cause liquid to flow, work must be expended. A pump may raise the liquid to a higher elevation, force it into a vessel at higher pressure, provide the head to overcome pipe friction, or perform any combination of these. Regardless of the service required of a pump, all energy imparted to the liquid in performing this service must be accounted for; consistent units for all quantities must be employed in arriving at the work or power performed. When arriving at the performance of a pump, it is customary to calculate its power output, which is the product of (1) the total dynamic head and (2) the mass of liquid pumped in a given time. In SI units power is expressed in kilowatts; horsepower is the conventional unit used in the United States. In SI units,

10-23

150 200 250 300 400

Temperature, F NPSH reductions for pumps handling hydrocarbon liquids and high-temperature water. This chart has been constructed from test data obtained using the liquids shown (Hydraulic Institute Standards).

FIG. 10-25

for pumping, various correction methods have been developed to evaluate NPSH when pumping other fluids. The most recent of these corrective methods has been developed by Hydraulic Institute and is shown in Fig. 10-25. The chart shown in Fig. 10-25 is for pure liquids. Extrapolation of data beyond the ranges indicated in the graph may not produce accurate results. Figure 10-25 shows the variation of vapor pressure and NPSH reductions for various hydrocarbons and hot water as a function of temperature. Certain rules apply while using this chart. When using the chart for hot water, if the NPSH reduction is greater than one-half of the NPSH required for cold water, deduct one-half of cold water NPSH to obtain the corrected NPSH required. On the other hand, if the value read on the chart is less than one-half of cold water NPSH, deduct this chart value from the cold water NPSH to obtain the corrected NPSH. Example 1: NPSH Calculation Suppose a selected pump requires a minimum NPSH of 16 ft (4.9 m) when pumping cold water; What will be the NPSH limitation to pump propane at 55°F (12.8°C) with a vapor pressure of 100 psi? Using the chart in Fig. 10-25, NPSH reduction for propane gives 9.5 ft (2.9 m). This is greater than one-half of cold water NPSH of 16 ft (4.9 m). The corrected NPSH is therefore 8 ft (2.2 m) or one-half of cold water NPSH. PUMP SELECTION When selecting pumps for any service, it is necessary to know the liquid to be handled, the total dynamic head, the suction and discharge heads, and, in most cases, the temperature, viscosity, vapor pressure, and specific gravity. In the chemical industry, the task of pump selection is frequently further complicated by the presence of solids in the liquid and liquid corrosion characteristics requiring special materials of construction. Solids may accelerate erosion and corrosion, have a tendency to agglomerate, or require delicate handling to prevent undesirable degradation. Range of Operation Because of the wide variety of pump types and the number of factors which determine the selection of any one type for a specific installation, the designer must first eliminate all but those types of reasonable possibility. Since range of operation is always an important consideration, Fig. 10-26 should be of assistance. The boundaries shown for each pump type are at best approximate, as unusual applications for which the best selection contradicts the chart

10-24

TRANSPORT AND STORAGE OF FLUIDS

Pump coverage chart based on normal ranges of operation of commercially available types. Solid lines: use left ordinate, head scale. Broken lines: use right ordinate, pressure scale. To convert gallons per minute to cubic meters per hour, multiply by 0.2271; to convert feet to meters, multiply by 0.3048; and to convert pounds-force per square inch to kilopascals, multiply by 6.895.

FIG. 10-26

will arise. In most cases, however, Fig. 10-26 will prove useful in limiting consideration to two or three types of pumps. Pump Materials of Construction In the chemical industry, the selection of pump materials of construction is dictated by considerations of corrosion, erosion, personnel safety, and liquid contamination. The experience of pump manufacturers is often valuable in selecting materials. See section on materials. Presence of Solids When a pump is required to pump a liquid containing suspended solids, there are unique requirements which must be considered. Adequate clear-liquid hydraulic performance and the use of carefully selected materials of construction may not be all that is required for satisfactory pump selection. Dimensions of all internal passages are critical. Pockets and dead spots, areas where solids can accumulate, must be avoided. Close internal clearances are undesirable because of abrasion. Flushing connections for continuous or intermittent use should be provided. For installations in which suspended solids must be handled with a minimum of solids breakage or degradation, such as pumps feeding filter presses, special attention is required; either a low-shear positivedisplacement pump or a recessed-impeller centrifugal pump may be called for. Ease of maintenance is of increasing importance in today’s economy. Chemical pump installations that require annual maintenance costing 2 or 3 times the original investment are not uncommon. In most cases this expense is the result of improper selection. CENTRIFUGAL PUMPS The centrifugal pump is the type most widely used in the chemical industry for transferring liquids of all types—raw materials, materials in manufacture, and finished products—as well as for general services of water supply, boiler feed, condenser circulation, condensate return, etc. These pumps are available through a vast range of sizes, in capacities from 0.5 m3/h to 2 × 104 m3/h (2 gal/min to 105 gal/min), and for discharge heads (pressures) from a few meters to approximately 48 MPa (7000 lbf/in2). The size and type best suited to a particular application can be determined only by an engineering study of the problem. The primary advantages of a centrifugal pump are simplicity, low first cost, uniform (nonpulsating) flow, small floor space, low maintenance expense, quiet operation, and adaptability for use with a motor or a turbine drive. A centrifugal pump, in its simplest form, consists of an impeller rotating within a casing. The impeller consists of a number of blades,

either open or shrouded, mounted on a shaft that projects outside the casing. Its axis of rotation may be either horizontal or vertical, to suit the work to be done. Closed-type, or shrouded, impellers are generally the most efficient. Open- or semiopen-type impellers are used for viscous liquids or for liquids containing solid materials and on many small pumps for general service. Impellers may be of the single-suction or the double-suction type—single if the liquid enters from one side, double if it enters from both sides. Casings There are three general types of casings, but each consists of a chamber in which the impeller rotates, provided with inlet and exit for the liquid being pumped. The simplest form is the circular casing, consisting of an annular chamber around the impeller; no attempt is made to overcome the losses that will arise from eddies and shock when the liquid leaving the impeller at relatively high velocities enters this chamber. Such casings are seldom used. Volute casings take the form of a spiral increasing uniformly in cross-sectional area as the outlet is approached. The volute efficiently converts the velocity energy imparted to the liquid by the impeller into pressure energy. A third type of casing is used in diffuser-type or turbine pumps. In this type, guide vanes or diffusers are interposed between the impeller discharge and the casing chamber. Losses are kept to a minimum in a well-designed pump of this type, and improved efficiency is obtained over a wider range of capacities. This construction is often used in multistage high-head pumps. Action of a Centrifugal Pump Briefly, the action of a centrifugal pump may be shown by Fig. 10-27. Power from an outside source is applied to shaft A, rotating the impeller B within the stationary casing C. The blades of the impeller in revolving produce a reduction in pressure at the entrance or eye of the impeller. This causes liquid to flow into the impeller from the suction pipe D. This liquid is forced outward along the blades at increasing tangential velocity. The velocity head it has acquired when it leaves the blade tips is changed to pressure head as the liquid passes into the volute chamber and thence out the discharge E. Centrifugal-Pump Characteristics Figure 10-28 shows a typical characteristic curve of a centrifugal pump. It is important to note that at any fixed speed the pump will operate along this curve and at no other points. For instance, on the curve shown, at 45.5 m3/h (200 gal/min) the pump will generate 26.5-m (87-ft) head. If the head is increased to 30.48 m (100 ft), 27.25 m3/h (120 gal/min) will be delivered. It is not possible to reduce the capacity to 27.25 m3/h (120 gal/ min) at 26.5-m (87-ft) head unless the discharge is throttled so that 30.48 m (100 ft) is actually generated within the pump. On pumps with variable-speed drivers such as steam turbines, it is possible to change the characteristic curve, as shown by Fig. 10-29. As shown in Eq. (10-48), the head depends upon the velocity of the fluid, which in turn depends upon the capability of the impeller to transfer energy to the fluid. This is a function of the fluid viscosity and the impeller design. It is important to remember that the head produced will be the same for any liquid of the same viscosity. The pressure rise, however, will vary in proportion to the specific gravity. For quick pump selection, manufacturers often give the most essential performance details for a whole range of pump sizes. Figure 10-30 shows typical performance data for a range of process pumps based on suction and discharge pipes and impeller diameters. The performance data consists of pump flow rate and head. Once a pump

FIG. 10-27

A simple centrifugal pump.

PUMPING OF LIQUIDS AND GASES TABLE 10-7

The Affinity Laws Constant impeller diameter

Characteristic curve of a centrifugal pump operating at a constant speed of 3450 r/min. To convert gallons per minute to cubic meters per hour, multiply by 0.2271; to convert feet to meters, multiply by 0.3048; to convert horsepower to kilowatts, multiply by 0.746; and to convert inches to centimeters, multiply by 2.54.

FIG. 10-28

Characteristic curve of a centrifugal pump at various speeds. To convert gallons per minute to cubic meters per hour, multiply by 0.2271; to convert feet to meters, multiply by 0.3048; to convert horsepower to kilowatts, multiply by 0.746; and to convert inches to centimeters, multiply by 2.54.

FIG. 10-29

meets a required specification, then a more detailed performance data for the particular pump can be easily found based on the curve reference number. Figure 10-31 shows a more detailed pump performance curve that includes, in addition to pump head and flow, the break horsepower required, NPSH required, number of vanes, and pump efficiency for a range of impeller diameters. If detailed manufacturer-specified performance curves are not available for a different size of the pump or operating condition, a best estimate of the off-design performance of pumps can be obtained through similarity relationship or the affinity laws. These are: 1. Capacity (Q) is proportional to impeller rotational speed (N). 2. Head (h) varies as square of the impeller rotational speed. 3. Break horsepower (BHP) varies as the cube of the impeller rotational speed. These equations can be expressed mathematically and appear in Table 10-7.

10-25

Constant impeller speed

Capacity

Q1 N1 ᎏ=ᎏ Q2 N2

Q1 D1 ᎏ=ᎏ Q2 D2

Head

H1 (N1)2 ᎏ = ᎏ2 H2 (N2)

h1 (D1)2 ᎏ = ᎏ2 h2 (D2)

Break horsepower

BHP1 (N1)3 ᎏ = ᎏ3 BHP2 (N2)

BHP1 (P1)3 ᎏ = ᎏ3 BHP2 (P2)

System Curves In addition to the pump design, the operational performance of a pump depends upon factors such as the downstream load characteristics, pipe friction, and valve performance. Typically, head and flow follow the following relationship: (Q2 )2 h2 (10-56) ᎏ2 = ᎏ (Q1) h1 where the subscript 1 refers to the design condition and 2 to the actual conditions. The above equation indicates that head will change as a square of the water flow rate. Figure 10-32 shows the schematic of a pump, moving a fluid from tank A to tank B, both of which are at the same level. The only force that the pump has to overcome in this case is the pipe function, variation of which with fluid flow rate is also shown in the figure. On the other for the use shown in Figure 10-33, the pump in addition to pipe friction should overcome head due to difference in elevation between tanks A and B. In this case, elevation head is constant, whereas the head required to overcome friction depends on the flow rate. Figure 10-34 shows the pump performance requirement of a valve opening and closing. Pump Selection One of the parameters that is extremely useful in selecting a pump for a particular application is specific speed Ns. Specific speed of a pump can be evaluated based on its design speed, flow, and head. NQ0.5 NQ1/2 Ns = ᎏ or Ns = ᎏ (10-57) 0.75 H H3/4 where N = rpm, Q is flow rate in gpm, and H is head in ft⋅lbf /lbm. Specific speed is a parameter that defines the speed at which impellers of geometrically similar design have to be run to discharge one gallon per minute against a one-foot head. In general, pumps with a low specific speed have a low capacity and high specific speed, high capacity. Specific speeds of different types of pumps are shown in Table 10-8 for comparison. Another parameter that helps in evaluating the pump suction limitations, such as cavitation, is suction specific speed. NQ1/2 S = ᎏᎏ (10-58) (NPSH)3/4 Typically, for single-suction pumps, suction-specific speed above 11,000 is considered excellent. Below 7000 is poor and 7000–9000 is of an average design. Similarly, for double-suction pumps, suctionspecific speed above 14,000 is considered excellent, below 7000 is poor, and 9000–11,000 is average. Figure 10-35 shows the schematic of specific-speed variation for different types of pumps. The figure clearly indicates that, as the specific speed increases, the ratio of the impeller outer diameter D1 to inlet or eye diameter D2 decreases, tending to become unity for pumps of axial-flow type. TABLE 10-8

Specific Speeds of Different Types of Pumps

Pump type

Specific speed range

Below 2,000 2,000–5,000 4,000–10,000 9,000–15,000

Process pumps and feed pumps Turbine pumps Mixed-flow pumps Axial-flow pumps

TRANSPORT AND STORAGE OF FLUIDS

800

3500 16

15

600

12

400

13

11

14 5 8

7

Selection and curves

6

200

10

Feet

9 100 80

1

2

4 3

60 40

20

20

200

60 80 100

40

400

600 800 1000

2000

GPM Range Pump No. 1 1.5 × 6 E 731 Plus 2 3 × 1.5 × 6 731 Plus 3 3×2×6 731 Plus 4 4×3×6 731 Plus 5 1.5 × 1 × 8 731 Plus 6 3 × 1.5 × 8 731 Plus 7 3 × 1.5 × 8.5 E 731 Plus 8 3 × 2 × 8.5 E 731 Plus FIG. 10-30

Range Pump No. 9 4 × 3 × 8.5 731 Plus 10 6 × 4 × 8.5 731 Plus 11 2 × 1 × 10 E 731 Plus 12 3 × 1.5 × 11 E 731 Plus 13 3 × 2 × 11 731 Plus 14 4 × 3 × 11 731 Plus 15 3 × 1.5 × 13 E 731 Plus 16 3 × 2 × 13 731 Plus

Curve A-8475 A-6982 A-8159 A-8551 A-8153 A-8155 A-8529 A-8506

Pump 1.5 × 1 × 6 E 731 Plus

Speed 3525 RPM

Impeller data Imp. no.

175

6.0′′

Total head in feet

150

45

5.0′′

100

4.5′′

75

4.0′′

50

55

58

61

63

No. of vanes

Max. Inlet sphere 0.28′′ area

63

6

2.7σ

65 68

55 50

3.5′′

50

P-3708

Max. dia. 6.0′′ Min. dia. 3.5′′

5.5′′

125

Curve A-8969 A-8547 A-8496 A-8543 A-8456 A-7342 A-8492 A-7338

Performance curves for a range of open impeller pumps.

Curve A-8475-1

B

A

C

25

E

F

3.0 4.0

10

6.0

8.0

10.0

D

NPSI Req’d (ft)

A BHP

10-26

B

5

C F

0

0

20

40

60

E

D

100 120 80 U.S. gallons per minute

140

160

FIG. 10-31 Typical pump performance curve. The curve is shown for water at 85°F. If the specific gravity of the fluid is other than unity, BHP must be corrected.

PUMPING OF LIQUIDS AND GASES A

B

10-27 B

Flow

A

Variation of total head versus flow rate to overcome friction.

he

ad

cu

Total head, feet

Capacity, GPM FIG. 10-32

rv e

Sy

ste

m

fric

ti o

n

Total head, feet

Flow

S

te ys

System friction

m

Static difference

Variation of total head as a function of flow rate to overcome both friction and static head.

FIG. 10-33

Valve partially closed Valve open

Head

Pump head capacity curve

Sy

2

ead ste m h

c u rv

1

e

Frictional resistance

S1

Total static head

Q2

Q1 Capacity FIG. 10-34

Typical steady-state response of a pump system with a valve fully and partially open.

20000

15000

9000 10000

8000

7000

6000

5000

4000

3000

2000

1500

1000

900

800

700

600

500

Values of specific speed, H2

D2

D2 Radial-vane area D2 >2 D1 FIG. 10-35

Francis-flow area

Mixed-flow area

Axial-flow area

D2 = 1.5 to 2 D1

D2 < 1.5 D1

D2 <1 D1

Specific speed variations of different types of pump.

Axis of rotation

10-28

TRANSPORT AND STORAGE OF FLUIDS

Typically, axial flow pumps are of high flow and low head type and have a high specific speed. On the other hand, purely radial pumps are of high head and low flow rate capability and have a low specific speed. Obviously, a pump with a moderate flow and head has an average specific speed. A typical pump selection chart such as shown in Fig. 10-36 calculates the specific speed for a given flow, head, and speed requirements. Based on the calculated specific speed, the optimal pump design is indicated. Process Pumps This term is usually applied to single-stage pedestal-mounted units with single-suction overhung impellers and with a single packing box. These pumps are ruggedly designed for ease in dismantling and accessibility, with mechanical seals or packing arrangements, and are built especially to handle corrosive or otherwise difficult-to-handle liquids. Specifically but not exclusively for the chemical industry, most pump manufacturers now build to national standards horizontal and vertical process pumps. American National Standards Institute (ANSI) Standards B73.1—1977 and B73.2—1975 apply to the horizontal (Fig. 10-37a) and vertical in-line (Fig. 10-37b) pumps respectively. The horizontal pumps are available for capacities up to 900 m3/h (4000 gal/min); the vertical in-line pumps, for capacities up to 320 m3/h (1400 gal/min). Both horizontal and vertical in-line pumps are available for heads up to 120 m (400 ft). The intent of each ANSI specification is that pumps from all vendors for a given nominal capacity and total dynamic head at a given rotative speed shall be dimensionally interchangeable with respect to mounting, size, and location of suction and discharge nozzles, input shaft, base plate, and foundation bolts. The vertical in-line pumps, although relatively new additions, are finding considerable use in chemical and petrochemical plants in the United States. An inspection of the two designs will make clear the relative advantages and disadvantages of each. Chemical pumps are available in a variety of materials. Metal pumps are the most widely used. Although they may be obtained in iron, bronze, and iron with bronze fittings, an increasing number of pumps of ductile-iron, steel, and nickel alloys are being used. Pumps are also available in glass, glass-lined iron, carbon, rubber, rubberlined metal, ceramics, and a variety of plastics, such units usually being employed for special purposes. Sealing the Centrifugal Chemical Pump Although detailed treatment of shaft seals is presented in the subsection “Sealing of Rotating Shafts,” it is appropriate to mention here the special problems

50 60 80 100 400

Capacity, GPM 150 200 300 400 600 800 1000

Head, ft of liquid

300 200

97

150

17

14

11

50

50

rp pm m

0r

2000 4000 35 5 29 0 50 rpm rp m

70 rp rp m m

100 80 60 50 40 30 Specific speed 400

600 800 1000

2000

4000

Typical impeller proportion as a function of specific speed above FIG. 10-36 Relationships between specific speed, rotative speed, and impeller

proportions (Worthington Pump Inc., Pump World, vol. 4, no. 2, 1978).

Horizontal process pump conforming to American National Standards Institute (ANSI) Standard B73.1-1977.

FIG. 10-37a

FIG. 10-37b Vertical in-line process pump conforming to ANSI Standard B73.2-1975. The pump shown is driven by a motor through flexible coupling. Not shown but also conforming to ANSI Standard B73.2 are vertical in-line pumps with rigid couplings and with no coupling (impeller-mounted on an extended motor shaft).

of sealing centrifugal chemical pumps. Current practice demands that packing boxes be designed to accommodate both packing and mechanical seals. With either type of seal, one consideration is of paramount importance in chemical service: the liquid present at the sealing surfaces must be free of solids. Consequently, it is necessary to provide a secondary compatible liquid to flush the seal or packing whenever the process liquid is not absolutely clean. The use of packing requires the continuous escape of liquid past the seal to minimize and to carry away the frictional heat developed. If the effluent is toxic or corrosive, quench glands or catch pans are usually employed. Although packing can be adjusted with the pump operating, leaking mechanical seals require shutting down the pump to correct the leak. Properly applied and maintained mechanical seals usually show no visible leakage. In general, owing to the more effective performance of mechanical seals, they have gained almost universal acceptance. Double-Suction Single-Stage Pumps These pumps are used for general water-supply and circulating service and for chemical service when liquids that are noncorrosive to iron or bronze are being handled. They are available for capacities from about 5.7 m3/h (25 gal/ min) up to as high as 1.136 × 104 m3/h (50,000 gal/min) and heads up to 304 m (1000 ft). Such units are available in iron, bronze, and iron with bronze fittings. Other materials increase the cost; when they are required, a standard chemical pump is usually more economical.

PUMPING OF LIQUIDS AND GASES

FIG. 10-38

10-29

Close-coupled pump.

Close-Coupled Pumps (Fig. 10-38) Pumps equipped with a built-in electric motor or sometimes steam-turbine-driven (i.e., with pump impeller and driver on the same shaft) are known as closecoupled pumps. Such units are extremely compact and are suitable for a variety of services for which standard iron and bronze materials are satisfactory. They are available in capacities up to about 450 m3/h (2000 gal/min) for heads up to about 73 m (240 ft). Two-stage units in the smaller sizes are available for heads to around 150 m (500 ft). Canned-Motor Pumps (Fig. 10-39) These pumps command considerable attention in the chemical industry. They are closecoupled units in which the cavity housing the motor rotor and the pump casing are interconnected. As a result, the motor bearings run in the process liquid and all seals are eliminated. Because the process liquid is the bearing lubricant, abrasive solids cannot be tolerated. Standard single-stage canned-motor pumps are available for flows up to 160 m3/h (700 gal/min) and heads up to 76 m (250 ft). Two-stage units are available for heads up to 183 m (600 ft). Canned-motor pumps are being widely used for handling organic solvents, organic heat-transfer liquids, and light oils as well as many clean toxic or hazardous liquids or for installations in which leakage is an economic problem. Vertical Pumps In the chemical industry, the term vertical process pump (Fig. 10-40) generally applies to a pump with a vertical shaft having a length from drive end to impeller of approximately 1 m (3.1 ft) minimum to 20 m (66 ft) or more. Vertical pumps are used as either wet-pit pumps (immersed) or dry-pit pumps (externally mounted) in conjunction with stationary or mobile tanks containing difficult-to-handle liquids. They have the following advantages: the liquid level is above the impeller, and the pump is thus self-priming; and the shaft seal is above the liquid level and is not wetted by the pumped liquid, which simplifies the sealing task. When no bottom connections are permitted on the tank (a safety consideration for highly corrosive or toxic liquid), the vertical wet-pit pump may be the only logical choice. These pumps have the following disadvantages: intermediate or line bearings are generally required when the shaft length exceeds about 3 m (10 ft) in order to avoid shaft resonance problems; these

FIG. 10-40 Vertical process pump for dry-pit mounting. (Courtesy of Lawrence Pumps, Inc.)

bearings must be lubricated whenever the shaft is rotating. Since all wetted parts must be corrosion-resistant, low-cost materials may not be suitable for the shaft, column, etc. Maintenance is more costly since the pumps are larger and more difficult to handle. For abrasive service, vertical cantilever designs requiring no line or foot bearings are available. Generally, these pumps are limited to about a 1-m (3.1-ft) maximum shaft length. Vertical pumps are also used to pump waters to reservoirs. One such application in the Los Angeles water basin has fourteen 4-stage pumps, each pump requiring 80,000 Hp to drive them. Sump Pumps These are small single-stage vertical pumps used to drain shallow pits or sumps. They are of the same general construction as vertical process pumps but are not designed for severe operating conditions. Multistage Centrifugal Pumps These pumps are used for services requiring heads (pressures) higher than can be generated by a single impeller. All impellers are in series, the liquid passing from one impeller to the next and finally to the pump discharge. The total head then is the summation of the heads of the individual impellers. Deepwell pumps, high-pressure water-supply pumps, boiler-feed pumps, fire pumps, and charge pumps for refinery processes are examples of multistage pumps required for various services. Multistage pumps may be of the volute type (Fig. 10-41), with single- or double-suction impellers (Fig. 10-42), or of the diffuser type (Fig. 10-43). They may have horizontally split casings or, for extremely high pressures, 20 to 40 MPa (3000 to 6000 lbf/in2), vertically split barrel-type exterior casings with inner casings containing diffusers, interstage passages, etc. PROPELLER AND TURBINE PUMPS

FIG. 10-39

Co.)

Canned-motor pump (Courtesy of Chempump Division, Crane

Axial-Flow (Propeller) Pumps (Fig. 10-44) These pumps are essentially very-high-capacity low-head units. Normally they are designed for flows in excess of 450 m3/h (2000 gal/min) against heads of 15 m (50 ft) or less. They are used to great advantage in closed-loop circulation systems in which the pump casing becomes merely an elbow in the line. A common installation is for calandria circulation. A characteristic curve of an axial-flow pump is given in Fig. 10-45.

10-30

TRANSPORT AND STORAGE OF FLUIDS

FIG. 10-41

Six-stage volute-type pump.

FIG. 10-42

Two-stage pump having double-suction impellers.

FIG. 10-43

Seven-stage diffuser-type pump.

PUMPING OF LIQUIDS AND GASES

FIG. 10-44

10-31

Axial-flow elbow-type propeller pump. (Courtesy of Lawrence

Pumps, Inc.)

Characteristic curve of an axial-flow pump. To convert gallons per minute to cubic meters per hour, multiply by 0.2271; to convert feet to meters, multiply by 0.3048; and to convert horsepower to kilowatts, multiply by 0.746.

FIG. 10-45

FIG. 10-46

Vertical multistage turbine, or mixed-flow, pump.

POSITIVE-DISPLACEMENT PUMPS Turbine Pumps The term “turbine pump” is applied to units with mixed-flow (part axial and part centrifugal) impellers. Such units are available in capacities from 20 m3/h (100 gal/min) upward for heads up to about 30 m (100 ft) per stage. Turbine pumps are usually vertical. A common form of turbine pump is the vertical pump, which has the pump element mounted at the bottom of a column that serves as the discharge pipe (see Fig. 10-46). Such units are immersed in the liquid to be pumped and are commonly used for wells, condenser circulating water, large-volume drainage, etc. Another form of the pump has a shell surrounding the pumping element which is connected to the intake pipe. In this form, the pump is used on condensate service in power plants and for process work in oil refineries. Regenerative Pumps Also referred to as turbine pumps because of the shape of the impeller, regenerative pumps employ a combination of mechanical impulse and centrifugal force to produce heads of several hundred meters (feet) at low volumes, usually less than 20 m3/h (100 gal/min). The impeller, which rotates at high speed with small clearances, has many short radial passages milled on each side at the periphery. Similar channels are milled in the mating surfaces of the casing. Upon entering, the liquid is directed into the impeller passages and proceeds in a spiral pattern around the periphery, passing alternately from the impeller to the casing and receiving successive impulses as it does so. Figure 10-47 illustrates a typical performance-characteristic curve. These pumps are particularly useful when low volumes of lowviscosity liquids must be handled at higher pressures than are normally available with centrifugal pumps. Close clearances limit their use to clean liquids. For very high heads, multistage units are available.

Whereas the total dynamic head developed by a centrifugal, mixedflow, or axial-flow pump is uniquely determined for any given flow by the speed at which it rotates, positive-displacement pumps and those which approach positive displacement will ideally produce whatever head is impressed upon them by the system restrictions to flow. Actually, with slippage neglected, the maximum head attainable is determined by the power available in the drive and the strength of the pump parts. An automatic relief valve set to open at a safe pressure

FIG. 10-47 Characteristic curves of a regenerative pump. To convert gallons per minute to cubic meters per hour, multiply by 0.2271; to convert feet to meters, multiply by 0.3048; and to convert horsepower to kilowatts, multiply by 0.746.

10-32

TRANSPORT AND STORAGE OF FLUIDS

higher than the normal or maximum discharge pressure is generally required on the discharge side of all positive-displacement pumps. In general, overall efficiencies of positive-displacement pumps are higher than those of centrifugal equipment because internal losses are minimized. On the other hand, the flexibility of each piece of equipment in handling a wide range of capacities is somewhat limited. Positive-displacement pumps may be of either the reciprocating or the rotary type. In all positive-displacement pumps, a cavity or cavities are alternately filled and emptied of the pumped fluid by the action of the pump. Reciprocating Pumps There are three classes of reciprocating pumps: piston pumps, plunger pumps, and diaphragm pumps. Basically, the action of the liquid-transferring parts of these pumps is the same, a cylindrical piston, plunger, or bucket or a round diaphragm being caused to pass or flex back and forth in a chamber. The device is equipped with valves for the inlet and discharge of the liquid being pumped, and the operation of these valves is related in a definite manner to the motions of the piston. In all modern-design reciprocating pumps, the suction and discharge valves are operated by pressure difference. That is, when the pump is on its suction stroke and the pump cavity is increasing in volume, the pressure is lowered within the pump cavity, permitting the higher suction pressure to open the suction valve and allowing liquid to flow into the pump. At the same time, the higher discharge-line pressure holds the discharge valve closed. Likewise on the discharge stroke, as the pump cavity is decreasing in volume, the higher pressure developed in the pump cavity holds the suction valve closed and opens the discharge valve to expel liquid from the pump into the discharge line. The overall efficiency of these pumps varies from about 50 percent for the small pumps to about 90 percent or more for the larger sizes. As shown in Fig. 10-48, reciprocating pumps, except when used for metering service, are frequently provided on the discharge side with gas-charged chambers, the purpose of which is to limit pressure pulsation and to provide a more uniform flow in the discharge line. In many installations, surge chambers are required on the suction side as well. Piping layouts should be studied to determine the most effective size and location. If surge chambers are used, provision should be made to keep the chamber charged with gas. A surge chamber filled with liquid is of no value. A liquid-level gauge is desirable to permit checking the amount of gas in the chamber. Reciprocating pumps may be of single-cylinder or multicylinder design. Multicylinder pumps have all cylinders in parallel for increased capacity. Piston-type pumps may be single-acting or doubleacting; i.e., pumping may be accomplished from one or both ends of the piston. Plunger pumps are always single-acting. The following tabulation (Table 10-9) provides data on the flow variation of reciprocating pumps of various designs. Piston Pumps There are two ordinary types of piston pumps, simplex double-acting pumps and duplex double-acting pumps. Simplex Double-Acting Pumps These pumps may be directacting (i.e., direct-connected to a steam cylinder) or power-driven (through a crank and flywheel from the crosshead of a steam engine).

TABLE 10-9

Flow Variation of Reciprocating Pumps

Number of cylinders

Single- or doubleacting

Flow variation per stroke from mean, percent

Single Single Duplex Duplex Triplex Quintuplex

Single Double Single Double Single and double Single

+220 to −100 +60 to −100 +24.1 to −100 +6.1 to −21.5 +1.8 to −16.9 +1.8 to −5.2

Figure 10-48 is a direct-acting pump, designed for use at pressures up to 0.690 MPa (100 lbf/in2). In this figure, the piston consists of disks A and B, with packing rings C between them. A bronze liner for the water cylinder is shown at D. Suction valves are E1 and E2. Discharge valves are F1 and F2. Duplex Double-Acting Pumps These pumps differ primarily from those of the simplex type in having two cylinders whose operation is coordinated. They may be direct-acting, steam-driven, or power-driven with crank and flywheel. A duplex outside-end-packed plunger pump with pot valves, of the type used with hydraulic presses and for similar service, is shown in Fig. 10-49. In this drawing, plunger A is direct-connected to rod B, while plunger C is operated from the rod by means of yoke D and tie rods. Plunger pumps differ from piston pumps in that they have one or more constant-diameter plungers reciprocating through packing glands and displacing liquid from cylinders in which there is considerable radial clearance. They are always single-acting, in the sense that only one end of the plunger is used in pumping the liquid. Plunger pumps are available with one, two, three, four, five, or even more cylinders. Simplex and duplex units are often built in a horizontal design. Those with three or more cylinders are usually of vertical

(a)

(b) FIG. 10-48

Double-acting steam-driven reciprocating pump.

FIG. 10-49

Duplex single-acting plunger pump.

PUMPING OF LIQUIDS AND GASES

FIG. 10-52

FIG. 10-50

Adrich-Groff variable-stroke power pump. (Courtesy of Ingersoll-

Rand.)

design. The driver may be an electric motor, a steam or gas engine, or a steam turbine. This is the common type of power pump. An example, arranged for belt drive, is shown in Fig. 10-50 from which the action may be readily traced. Occasionally plunger pumps are constructed with opposed cylinders and plungers connected by yokes and tie rods; this arrangement, in effect, constitutes a double-acting unit. Simplex plunger pumps mounted singly or in gangs with a common drive are quite commonly used as metering or proportioning pumps (Fig. 10-51). Frequently a variable-speed drive or a strokeadjusting mechanism is provided to vary the flow as desired. These pumps are designed to measure or control the flow of liquid within a deviation of ⫾2 percent with capacities up to 11.35 m3/h (50 gal/min) and pressures as high as 68.9 MPa (10,000 lbf/in2). Diaphragm Pumps These pumps perform similarly to piston and plunger pumps, but the reciprocating driving member is a flexible diaphragm fabricated of metal, rubber, or plastic. The chief advantage of this arrangement is the elimination of all packing and seals exposed to the liquid being pumped. This, of course, is an important asset for equipment required to handle hazardous or toxic liquids.

FIG. 10-51

Plunger-type metering pump. (Courtesy of Milton Roy Co.)

10-33

Mechanically actuated diaphragm pump.

A common type of low-capacity diaphragm pump designed for metering service employs a plunger working in oil to actuate a metallic or plastic diaphragm. Built for pressures in excess of 6.895 MPa (1000 lbf/in2) with flow rates up to about 1.135 m3/h (5 gal/min) per cylinder, such pumps possess all the characteristics of plunger-type metering pumps with the added advantage that the pumping head can be mounted in a remote (even a submerged) location entirely separate from the drive. Figure 10-52 shows a high-capacity 22.7-m3/h (100-gal/min) pump with actuation provided by a mechanical linkage. Pneumatically Actuated Diaphragm Pumps (Fig. 10-53) These pumps require no power source other than plant compressed air. They must have a flooded suction, and the pressure is, of course, limited to the available air pressure. Because of their slow speed and large valves, they are well suited to the gentle handling of liquids for which degradation of suspended solids should be avoided. A major consideration in the application of diaphragm pumps is the realization that diaphragm failure will probably occur eventually. The consequences of such failure should be realistically appraised before selection, and maintenance procedures should be established accordingly.

FIG. 10-53 Pneumatically actuated diaphragm pump for slurry service. (Courtesy of Dorr-Olvier Inc.)

10-34

TRANSPORT AND STORAGE OF FLUIDS

Rotary Pumps In rotary pumps the liquid is displaced by rotation of one or more members within a stationary housing. Because internal clearances, although minute, are a necessity in all but a few special types, capacity decreases somewhat with increasing pump differential pressure. Therefore, these pumps are not truly positivedisplacement pumps. However, for many other reasons they are considered as such. The selection of materials of construction for rotary pumps is critical. The materials must be corrosion-resistant, compatible when one part is running against another, and capable of some abrasion resistance. Gear Pumps When two or more impellers are used in a rotarypump casing, the impellers will take the form of toothed-gear wheels as in Fig. 10-54, of helical gears, or of lobed cams. In each case, these impellers rotate with extremely small clearance between them and between the surfaces of the impellers and the casing. In Fig. 10-54, the two toothed impellers rotate as indicated by the arrows; the suction connection is at the bottom. The pumped liquid flows into the spaces between the impeller teeth as these cavities pass the suction opening. The liquid is then carried around the casing to the discharge opening, where it is forced out of the impeller teeth mesh. The arrows indicate this flow of liquid. Rotary pumps are available in two general classes, interior-bearing and exterior-bearing. The interior-bearing type is used for handling liquids of a lubricating nature, and the exterior-bearing type is used with nonlubricating liquids. The interior-bearing pump is lubricated by the liquid being pumped, and the exterior-bearing type is oillubricated. The use of spur gears in gear pumps will produce in the discharge pulsations having a frequency equivalent to the number of teeth on both gears multiplied by the speed of rotation. The amplitude of these disturbances is a function of tooth design. The pulsations can be reduced markedly by the use of rotors with helical teeth. This in turn introduces end thrust, which can be eliminated by the use of doublehelical or herringbone teeth. Screw Pumps A modification of the helical gear pump is the screw pump. Both gear and screw pumps are positive displacement

FIG. 10-55

FIG. 10-56

FIG. 10-54

Positive-displacement gear-type rotary pump.

pumps. Figure 10-55 illustrates a two-rotor version in which the liquid is fed to either the center or the ends, depending upon the direction of rotation, and progresses axially in the cavities formed by the meshing threads or teeth. In three-rotor versions, the center rotor is the driving member while the other two are driven. Figure 10-56 shows still another arrangement, in which a metal rotor of unique design rotates without clearance in an elastomeric stationary sleeve. Screw pumps, because of multiple dams that reduce slip, are well adapted for producing higher pressure rises, for example, 6.895 MPa (1000 lbf/in2), especially when handling viscous liquids such as heavy oils. The all-metal pumps are generally subject to the same limitations on handling abrasive solids as conventional gear pumps. In addition, the wide bearing spans usually demand that the liquid have considerable lubricity to prevent metal-to-metal contact.

Two-rotor screw pump. (Courtesy of Warren Quimby Pump Co.)

Single-rotor screw pump with an elastomeric lining. (Courtesy of Moyno Pump Division, Robbins & Myers, Inc.)

PUMPING OF LIQUIDS AND GASES Among the liquids handled by rotary pumps are mineral oils, vegetable oils, animal oils, greases, glucose, viscose, molasses, paints, varnish, shellac, lacquers, alcohols, catsup, brine, mayonnaise, sizing, soap, tanning liquors, vinegar, and ink. Some screw-type units are specially designed for the gentle handling of large solids suspended in the liquid. Fluid-Displacement Pumps In addition to pumps that depend on the mechanical action of pistons, plungers, or impellers to move the liquid, other devices for this purpose employ displacement by a secondary fluid. This group includes air lifts and acid eggs. The air lift is a device for raising liquid by means of compressed air. In the past it was widely used for pumping wells, but it has been less widely used since the development of efficient centrifugal pumps. It operates by introducing compressed air into the liquid near the bottom of the well. The air-and-liquid mixture, being lighter than liquid alone, rises in the well casing. The advantage of this system of pumping lies in the fact that there are no moving parts in the well. The pumping equipment is an air compressor, which can be located on the surface. A simplified sketch of an air lift for this purpose is shown in Fig. 10-57. Ingersoll-Rand has developed empirical information on air-lift performance which is available upon request. An important application of the gas-lift principle involves the extraction of oil from wells. There are several references to both practical and theoretical work involving gas lift performance and related problems. Recommended sources are American Petroleum Institute, Drilling and Production Practices, 1952, pp. 257–317, and 1939, p. 266; Trans. Am. Soc. Mining Metall. Eng., 92, 296–313 (1931), 103, 170–186 (1933), 118, 56–70 (1936), 192, 317–326 (1951), 189, 73–82 (1950), and 198, 271–278 (1953); Trans Am. Soc. Mining Metall., and Pet. Eng., 213 (1958), and 207, 17–24 (1956); and Univ. Wisconsin Bull., Eng. Ser., 6, no. 7 (1911, reprinted 1914). An acid egg, or blowcase, consists of an egg-shaped container which can be filled with a charge of liquid that is to be pumped. This container is fitted with an inlet pipe for the charge, an outlet pipe for the discharge, and a pipe for the admission of compressed air or gas, as illustrated in Fig. 10-58. Pressure of air or gas on the surface of the liquid forces it out of the discharge pipe. Such pumps can be handoperated or arranged for semiautomatic or automatic operation. JET PUMPS Jet pumps are a class of liquid-handling device that makes use of the momentum of one fluid to move another. Ejectors and injectors are the two types of jet pumps of interest to chemical engineers. The ejector, also called the siphon, exhauster, or eductor, is designed for use in operations in which the head pumped against is low and is less than the head of the fluid used for pumping.

FIG. 10-57

head.

10-35

FIG. 10-58 A form of acid egg. External controls required for automatic operation are not shown.

The injector is a special type of jet pump, operated by steam and used for boiler feed and similar services, in which the fluid being pumped is discharged into a space under the same pressure as that of the steam being used to operate the injector. Figure 10-59 shows a simple design for a jet pump of the ejector type. The pumping fluid enters through the nozzle at the left and passes through the venturi nozzle at the center and out of the discharge opening at the right. As it passes into the venturi nozzle, it develops a suction that causes some of the fluid in the suction chamber to be entrained with the stream and delivered through this discharge. The efficiency of an ejector or jet pump is low, being only a few percent. The head developed by the ejector is also low except in special types. The device has the disadvantage of diluting the fluid pumped by mixing it with the pumping fluid. In steam injectors for boiler feed and similar services in which the heat of the steam is recovered, efficiency is close to 100 percent. The simple ejector or siphon is widely used, in spite of its low efficiency, for transferring liquids from one tank to another, for lifting acids, alkalies, or solid-containing liquids of an abrasive nature, and for emptying sumps. ELECTROMAGNETIC PUMPS The necessity of circulating liquid-metal heat-transfer media in nuclear-reactor systems has led to development of electromagnetic pumps. All electromagnetic pumps utilize the motor principle: a conductor in a magnetic field, carrying a current which flows at right angles to the direction of the field, has a force exerted on it, the force being mutually perpendicular to both the field and the current. In all electromagnetic pumps, the fluid is the conductor. This force, suitably directed in the fluid, manifests itself as a pressure if the fluid is suitably contained. The field and current can be produced in a number of different ways and the force utilized variously.

Simplified sketch of an air lift, showing submergence and total FIG. 10-59

Simple ejector using a liquid-motivating fluid.

10-36

TRANSPORT AND STORAGE OF FLUIDS

,,

Vibration

Log frequency

Frequency range Fault to be detected FIG. 10-60

Simplified diagram of a direct-current-operated electromagnetic

pump.

VIBRATION MONITORING One of the major factors that causes pump failure is vibration, which usually causes seal damage and oil leakage. Vibration in pumps is caused by numerous factors such as cavitation, impeller unbalance, loose bearings, and pipe pulsations. Typically, large-amplitude vibration occurs when the frequency of vibration coincides with that of the natural frequency of the pump system. This results in a catastrophic operating condition that should be avoided. If the natural frequency is close to the upper end of the operating speed range, then the pump system should be stiffened to reduce vibration. On the other hand, if the natural frequency is close to the lower end of the operating range, the unit should be made more flexible. During startup, the pump system may go through its system natural frequency, and vibration can occur. Continuous operation at this operating point should be avoided. ASME recommends periodic monitoring of all pumps. Pump vibration level should fall within the prescribed limits. The reference vibration level is measured during acceptance testing. This level is specified by the manufacturer. During periodic maintenance, the vibration level should not exceed alert level (see Table 10-10). If the measured level exceeds the alert level then preventive maintenance should be performed, by diagnosing the cause of vibration and reducing the vibration level prior to continued operation. TABLE 10-10

Alert mils., microns

Vr < 0.5 0.5 < Vr < 2.0 2.0 < Vr < 5.0 5.0 < Vr

1.0 2Vr 2+Vr 1.4Vr

Medium

Unbalance Misalignment Bent shaft Oil whirl Eccentricity

Wear Faults in gears

High Faults in rolling element bearings

Frequency range of typical machinery faults.

Typical problems and their vibration frequency ranges are shown in Fig. 10-61. Collection and analysis of vibration signatures is a complex procedure. By looking at a vibration spectrum, one can identify which components of the pump system are responsible for a particular frequency component. Comparison of vibration signatures at periodic intervals reveals if a particular component is deteriorating. The following example illustrates evaluation of the frequency composition of an electric motor gear pump system. Example 2: Vibration Consider an electric motor rotating at 1800 rpm driving an 8-vane centrifugal pump rotating at 600 rpm. For this 3:1 speed reduction, assume a gear box having two gears of 100 and 300 tooth. Since 60 Hz is 1 rpm, Motor frequency = 1800/60 = 30 Hz Pump frequency = 600/60 = 10 Hz Gear mesh frequency = 300 teeth × 600 rpm = 3000 Hz Vane frequency = 8 × 600 rpm = 80 Hz An ideal vibration spectra for this motor-gear pump assembly would appear as shown in Fig. 10-62. Figure 10-63 shows an actual pump vibration spectra. In the figure, several amplitude peaks occur at several frequencies.

2 Pump speed Motor speed 0.2

Gear mesh freq.

Vane freq.

0.02

Alert Levels

Reference value mils.

FIG. 10-61

Accel. level G

Both alternating- and direct-current units are available. While dc pumps (Fig. 10-60) are simpler, their high-current requirement is a definite limitation; ac pumps can readily obtain high currents by making use of transformers. Multipole induction ac pumps have been built in helical and linear configurations. Helical units are effective for relatively high heads and low flows, while linear induction pumps are best suited to large flows at moderate heads. Electromagnetic pumps are available for flow rates up to 2.271 × 103 m3/h (10,000 gal/min), and pressures up to 2 MPa (300 lbf/in2) are practical. Performance characteristics resemble those of centrifugal pumps.

Low

Action required mils., microns 1.5 3Vr 4+Vr 1.8Vr

10 30

FIG. 10-62

80

1000 Frequency, Hz

3000 5000

An ideal vibration spectra from an electric motor pump assembly.

1. 0.

0

0.5

BRG system

1.0

Supersynchronous

Radial gap B

2 × N = Misaligned

Hydraulic unbal. misalignment

2.0 3.0 Rotor vibration frequency/machine speed (CPS/RPS)

Hydraulic FIG. 10-63

10-37

,,,, ,,,, Vane passing

Rotational synchronous 1 × N = Dyn. unbal.

, , , ,,, , ,,,,, Oil whip

2.

Suction recirculation

3.

Oil-whirl; rubbing; resonance

Shaft vibration amplitudes, mils

0.46 × N Subsynchronous 0.54 × N

PUMPING OF LIQUIDS AND GASES

Hydraulic

An actual pump vibration spectra.

PUMP DIAGNOSTICS

COMPRESSION OF GASES

As the mechanical integrity of the pump system changes, the amplitude of vibration levels change. In some cases, in order to identify the source of vibration, pump speed may have to be varied, as these problems are frequency- or resonance-dependent. Pump impeller imbalance and cavitation are related to this category. Table 10-11 classifies different types of pump-related problems, their possible causes and corrective actions. Typical pump-related problems are classified under 1. Cavitation-type problems 2. Capacity-type problems 3. Motor overload problems It is advisable in most of these cases to use accelerometers. Displacement probes will not give the high-frequency signals and velocity probes because their mechanical design is very directional and prone to deterioration. Figure 10-64 shows the signal from the various types of probes.

Theory of Compression In any continuous compression process the relation of absolute pressure p to volume V is expressed by the formula

PUMP SPECIFICATIONS Pump specifications depend upon numerous factors but mostly on application. Typically, the following factors should be considered while preparing a specification. 1. Application, scope and type 2. Service conditions 3. Operating conditions 4. Consternation-application-specific details and special considerations • Casing connection • Impeller details • Shaft • Shifting box details—lubrications, sealing, etc. • Bearing frame and bearings • Base plate and couplings • Materials • Special operating conditions and miscellaneous items Table 10-12 is based on the API and ASME codes.

pV n = C = constant

(10-59)

The plot of pressure versus volume for each value of exponent n is known as the polytropic curve. Since the work W performed in proceeding from p1 to p2 along any polytropic curve (Fig. 10-65) is W=

冕 p dV 2

(10-60)

1

it follows that the amount of work required is dependent upon the polytropic curve involved and increases with increasing values of n. The path requiring the least amount of input work is n = 1, which is equivalent to isothermal compression. For adiabatic compression (i.e., no heat is being added or taken away during the process), n = k = ratio of specific heat at constant pressure to that at constant volume. Since most compressors operate along a polytropic path approaching the adiabatic, compressor calculations are generally based on the adiabatic curve. Some formulas based upon the adiabatic equation and useful in compressor work are as follows: Pressure, volume, and temperature relations for perfect gases: p2 /p1 = (V1 / V2)k

(10-61)

T2 /T1 = (V1 / V2)k − 1

(10-62)

p2 /p1 = (T2 / T1)k / (k − 1)

(10-63)

Adiabatic Calculations Adiabatic head is expressed as follows: In SI units, k × RT p (k − 1) / k Had = ᎏ1 ᎏ2 −1 (10-64a) k−1 p1 where Had = adiabatic head, N⋅m/kg; R = gas constant, J/(kg⋅K) = 8314/ molecular weight; T1 = inlet gas temperature, K; p1 = absolute inlet pressure, kPa; and p2 = absolute discharge pressure, kPa.

冤冢 冣

冥

TABLE 10-11

Pump Problems Possible causes

Corrective action Cavitating-type problems

Plugged suction screen. Piping gaskets with undersized IDs installed, a very common problem in small pumps. Column tray parts or ceramic packing lodged in the impeller eye. Deteriorated impeller eye due to corrosion. Flow rate is high enough above design that NPSH for flow rate has increased above NPSH. Lined pipe collapsed at gasket area or ID due to buildup of corrosion products between liner and carbon-steel pipe. Poor suction piping layout, too many elbows in too many planes, a tee branch almost directly feeding the suction of the other pump, or not enough straight run before the suction flange of the pump. Vertical pumps experience a vortex formation due to loss of submergence required by the pump. Observe the suction surface while the pump is in operation, if possible. Spare pump begins to cavitate when attempt is made to switch it with the running pump. The spare is “backed off” by the running pump because its shutoff head is less than the head produced by the running pump. This is a frequent problem when one pump is turbine-driven and one is motor-driven. Suction piping configuration causes adverse fluid rotation when approaching impeller. Velocity of the liquid is too high as it approaches the impeller eye. Pump is operating at a low-flow-producing suction recirculation in the impeller eye. This results in a cavitationlike sound.

Check for indications of the presence of screen. Remove and clean screen. Install proper-sized gaskets. Remove suction piping and debris. Replace impeller and overhaul pump. Reduce flow rate to that of design. Replace deteriorated piping. Redesign piping layout, using fewer elbows and laterals for tees, and have five or more straight pipe diameters before suction flange. Review causes of vortexing. Consider installation of a vortex breaker such as a bell mouth umbrella or changes to sump design. Throttle discharge of running pump until spare can get in system. Slow down running pump if it is a turbine or variable-speed motor.

Install sufficient straight run of suction piping, or install vanes in piping to break up prerotation. Install larger suction piping or reduce flow through pump. Install bypass piping back to suction vessel to increase flow through pump. Remember bypass flow may have to be as high as 50 percent of design flow.

Capacity-type problems Check the discharge block valve opening first. It may be partially closed and thus the problem. Wear-ring clearances are excessive (closed impeller design). Impeller-to-case or head clearances are excessive (open impeller design). Air leaks into the system if the pump suction is below atmospheric pressure. Increase in piping friction to the discharge vessel due to the following: 1. Gate has fallen off the discharge valve stem. 2. Spring is broken in the spring-type check valve. 3. Check valve flapper pin is worn, and flapper will not swing open. 4. Lined pipe collapsing. 5. Control valve stroke improperly set, causing too much pressure drop. Suction and/or discharge vessel levels are not correct, a problem mostly seen in lower-speed pumps. Motor running backward or impeller of double suction design is mounted backward. Discharge pressure developed in both cases is about one-half design value. Entrained gas from the process lowering NPSH available. Polymer or scale buildup in discharge nozzle areas. Mechanical seal in suction system under vacuum is leaking air into system, causing pump curve to drop. The pump may have formed a vortex at high flow rates or low liquid level. Does the vessel have a vortex breaker? Does the incoming flow cause the surface to swirl or be agitated? Variable-speed motor running too slowly. Bypassing is occurring between volute channels in a double-volute pump casing due to a casting defect or extreme erosion. The positions of impellers are not centered with diffuser vanes. Several impellers will cause vibration and lower head output. When the suction system is under vacuum, the spare pump has difficulty getting into system. Certain pump designs use an internal bypass orifice port to alter head-flow curve. High liquid velocities often erode the orifice, causing the pump to go farther out on the pump curve. The system head curve increase corrects the flow back up the curve. Replacement impeller is not correct casting pattern; therefore NPSH required is different. Volute and cutwater area of casing is severely eroded.

10-38

Open block valve completely. Overhaul pump. Renew wear rings if clearance is about twice design value for energy and performance reasons. Reposition impeller to obtain correct clearance. Take actions as needed to eliminate air leaks. Take the following actions: 1. Repair or replace gate valve. 2. Repair valve by replacing spring. 3. Overhaul check valve; restore proper clearance to pin and flapper bore. 4. Replace damaged pipe. 5. Adjust control valve stroke as necessary. Calibrate level controllers as necessary. Check for proper rotation and mounting of impeller. Reverse motor leads if necessary. Reduce entrained gas in liquid by process changes as needed. Shut down pump and remove scale or deposits. Change percentage balance of seal faces or increase spring tension. Reduce flow to design rates. Raise liquid level in suction vessel. Install vortex breaker in suction vessel. Adjust motor speed as needed. Overhaul pump; repair eroded area. Overhaul pump; reposition individual impellers as needed. Reposition whole rotor by changing thrust collar locator spacer. Install a positive-pressure steam (from running pump) to fill the suction line from the block valve through the check valve. Overhaul pump, restore orifice to correct size.

Overhaul pump, replace impeller with correct pattern. Overhaul pump; replace casing or repair by welding. Stress-relieve after welding as needed.

PUMPING OF LIQUIDS AND GASES TABLE 10-11

10-39

Pump Problems (Concluded ) Possible causes

Corrective action Overload problems

Polymer buildup between wear surfaces (rings or vanes). Excessive wear ring (closed impeller) or cover-case clearance (open impeller).

Remove buildup to restore clearances. Replace wear rings or adjust axial clearance of open impeller. In severe cases, cover or case must be replaced. Overhaul pump, replacing parts as needed.

Pump circulating excessive liquid back to suction through a breakdown bushing or a diffuser gasket area. Minimum-flow loop left open at normal rates, or bypass around control valve is open. Discharge piping leaking under liquid level in sump-type design. Electrical switch gear problems cause one phase to have low amperage. Specific gravity is higher than design specification.

Close minimum-flow loop or control valve bypass valve. Inspect piping for leakage. Replace as needed. Check out switch gear and repair as necessary. Change process to adjust specific gravity to design value, or throttle pump to reduce horsepower requirements. This will not correct problem with some vertical turbine pumps that have a flat horsepower-required curve. Replace motor with one of larger size, or reduce flow rate. Increase clearance of impeller to casing.

Pump motor not sized for end of curve operation. Open impeller has slight rub on casing. Most often occurs in operations from 250 to 400°F due to piping strain and differential growth in the pump. A replacement impeller was not trimmed to the correct diameter.

In U.S. customary units, k Had = ᎏ RT1 k−1

(k − 1) / k

冤冢 冣 p2 ᎏ p1

Remove impeller from pump and turn to correct diameter.

冥

−1

where kWad = power, kW; W = mass flow, kg/s × 9.806 N/kg; and Q1 = volume rate of gas flow, m3/h, at compressor inlet conditions. In U.S. customary units, WHad k WRT p (k − 1) / k hpad = ᎏ = ᎏ ᎏ1 ᎏ2 −1 (10-65b) 550 k − 1 550 p1

(10-64b)

冤冢 冣

where Had = adiabatic head, ft⋅lbf/lbm; R = gas constant, (ft⋅lbf )/ (lbm⋅°R) = 1545/molecular weight; T1 = inlet gas temperature, °R; p1 = absolute inlet pressure, lbf/in2; and p2 = absolute discharge pressure, lbf/in2. The work expended on the gas during compression is equal to the product of the adiabatic head and the mass flow of gas handled. Therefore, the adiabatic power is as follows: In SI units, WHad k × WRT1 p (k − 1) / k kWad = ᎏ = ᎏᎏ ᎏ2 −1 (10-65a) 3 10 k−1 p1

冤冢 冣

k kWad = 2.78 × 10−4 ᎏ Q1p1 k−1

Vibration amplitude, mil, in/sec, or G

or

(k − 1) / k

2

冥

−1

1

冤冢ᎏp 冣 p2 1

T2 = T1(p2 /p1)(k − 1) / k

(10-66a)

sp

lac

a ler

en

t, m

il

1

n,

G

tio

em

10

ce Ac Upper frequency limit of velocity measurement, 3 kHz

Constant velocity, 0.3 in/s Lower amplitude limit of displacement measurement, 0.1 mil

0.1

Lower amplitude limit of acceleration measurement, 0.056 G

0.01 1

10

102 103 Frequency, Hz

104

Range of displacement measurement Range of velocity measurement Range of acceleration measurement FIG. 10-64

(10-66b)

(10-67)

The work in a compressor under ideal conditions as previously shown occurs at constant entropy. The actual process is a polytropic process as shown in Fig. 10-65 and given by the equation of state Pv n = constant.

102 Di

冥

−1

where hpad = power, hp; W = mass flow, lb/s; and Q1 = volume rate of gas flow, ft3/min. Adiabatic discharge temperature is

冥

冤冢ᎏpp 冣

k hpad = 4.36 × 10−3 ᎏ Q1p1 k−1

or

冥

(k − 1) / k

Limitations on machinery vibrations analysis systems and transducers.

105

TABLE 10-12

API and ASME Codes

Specification

Specification

Description

1.0

Scope: This specification covers horizontal, end suction, vertically split, single-stage centrifugal pumps with top centerline discharge and “back pullout” feature.

2.0

Service Conditions: Pump shall be designed to operate satisfactorily with a reasonable service life when operated either intermittently or continuously in typical process applications.

3.0

Operating Conditions: Capacity U.S. gallons per minute Head ( ft total head) ( psig). Speed Suction Pressure ( ft head) (positive) (lift) ( Liquid to be handled Specific gravity Viscosity ( Temperature of liquid at inlet °F Solids content % Max. size

4.0

*Omit if not applicable.

4.8

Bearing Frame and Bearings: 4.8.1 Bearing Frame. Frames shall be equipped with axial radiating fins extending the length of the frame to aid in heat dissipation. Frame shall be provided with ductile iron outboard bearing housing. Both ends of the frame shall be provided with lip-type oil seals and labyrinth-type deflectors of metallic reinforced synthetic rubber to prevent the entrance of contaminants. 4.8.2 Bearings. Pump bearings shall be heavy-duty, antifriction balltype on both ends. The single row inboard bearing, nearest the impeller, shall be free to float within the frame and shall carry only radial load. The double row outboard bearing (F4-G1 and F4-I1) or duplex angular contact bearing (F4-H1), coupling end, shall be locked in place to carry radial and axial thrust loads. Bearings shall be designed for a minimum life of 20,000 hours in any normal pump operating range.

4.9

Bearing Lubrication. Ball bearings shall be oil-mist—lubricated by means of a slinger. The oil slinger shall be mounted on the shaft between the bearings to provide equal lubrication to both bearings. Bulls-eye oil-sight glasses shall be provided on both sides of the frame to provide a positive means of checking the proper oil level from either side of the pump. A tapped and plugged hole shall also be provided in both sides of the frame to mount bottle-type constant-level oilers where desired. A tapped and plugged hole shall be provided on both sides for optional straight-through oil cooling device.

5.0

Baseplate and Coupling: 5.1 Baseplate. Baseplates shall be rigid and suitable for mounting pump and motor. Baseplates shall be of channel steel construction. 5.2 Coupling. Coupling shall be flexible-spacer type. Coupling shall have at least three–and–one-half–inch spacer length for ease of rotating element removal. Both coupling hubs shall be provided with flats 180° apart to facilitate removal of impeller. Coupling shall not require lubrication.*

6.0

Mechanical Modifications Required for High Temperature: 6.1 Modifications Required, Temperature Range 250–350°F. Pumps for operation in this range shall be provided with a water-jacketed stuffing box. 6.2 Modifications Required, Temperature Range 351–550°F (Maximum). Pumps for operation in this range shall be provided with a water-jacketed stuffing box and a water-cooled bearing frame.

7.0

Materials: Pump materials shall be selected to suit the particular service requirements. 7.1 Cast Iron—316 SS Fitted. 15″ only; pump shall have cast iron casing and stuffing box cover. 316 SS metal impeller; shaft shall be 1045 steel with 316 SS sleeve. 7.2 All Duron Metal. All pump materials shall be Duron metal. Shaft shall be 1045 steel, with 316 SS sleeve. 316 SS metal impeller optional. 7.3 All AISI 316 Stainless Steel. All pump materials shall be AISI 316 stainless steel. Shaft should be 1045 steel, with 316 SS sleeve. 7.4 All #20 Stainless Steel. All pump materials shall be #20 SS stainless steel. Shaft shall be 316 SS, with #20 SS sleeve. 7.5 All CD-4MCu. All pump materials shall be CD-4MCu. Shaft shall be 316 SS, with #20 SS sleeve.

8.0

Miscellaneous: 8.1 Nameplates. All nameplates and other data plates shall be stainless steel, suitably secured to the pump. 8.2 Hardware. All machine bolts, stud nuts, and capscrews shall be of the hex-head type. 8.3 Rotation. Pump shall have clockwise rotation viewed from its driven end. 8.4 Parts Numbering. Parts shall be completely identified with a numerical system (no alphabetical letters) to facilitate parts inventory control and stocking. Each part shall be properly identified by a separate number, and those parts that are identical shall have the same number to effect minimum spare parts inventory.

rpm psig) )

Pump Construction: 4.1 Casing. Casing shall be vertically split with self-venting top centerline discharge, with an integral foot located directly under the casing for added support. All casings shall be of the “back pullout” design with suction and discharge nozzles cast integrally. Casings shall be provided with bosses in suction and discharge nozzles, and in bottom of casing for gauge taps and drain tap. (Threaded taps with plugs shall be provided for these features.) 4.2 Casing Connections. Connections shall be A.N.S.I. flat-faced flanges. [Cast iron (125) (250) psig rated] [Duron metal, steel, alloy steel (150) (300) psig rated] 4.3 Casing Joint Gasket. A confined-type nonasbestos gasket suitable for corrosive service shall be provided at the casing joint. 4.4 Impeller. Fully-open impeller with front edge having contoured vanes curving into the suction for minimum NPSH requirements and maximum efficiency shall be provided. A hex head shall be cast in the eye of the impeller to facilitate removal, and eliminate need for special impeller removing tool. All impellers shall have radial “pump-out” vanes on the back side to reduce stuffing box pressure and aid in eliminating collection of solids at stuffing box throat. Impellers shall be balanced within A.N.S.I. guidelines to ISO tolerances. 4.4.1 Impeller Clearance Adjustment. All pumps shall have provisions for adjustment of axial clearance between the leading edge of the impeller and casing. This adjustment shall be made by a precision microdial adjustment at the outboard bearing housing, which moves the impeller forward toward the suction wall of the casing. 4.5 Shafts. Shafts shall be suitable for hook-type sleeve. Shaft material shall be (SAE 1045 steel on Duron and 316 stainless steel pumps) or (AISI 316 stainless steel on CD-4MCu pumps and #20 stainless steel pumps). Shaft deflection shall not exceed .005 at the vertical centerline of the impeller. 4.6 Shaft Sleeve. Renewable hook-type shaft sleeve that extends through the stuffing box and gland shall be provided. Shaft sleeve shall be (316 stainless steel), (#20 stainless steel) or (XH-800 Ni-chromeboron coated 316 stainless steel with coated surface hardness of approximately 800 Brinnell). 4.7 Stuffing Box. Stuffing box shall be suitable for packing, single (inside or outside) or double-inside mechanical seal without modifications. Stuffing box shall be accurately centered by machined rabbit fits on case and frame adapter. 4.7.1 Packed Stuffing Box. The standard packed stuffing box shall consist of five rings of graphited nonasbestos packing; a stainless steel packing base ring in the bottom of the box to prevent extrusion of the packing past the throat; a teflon seal cage, and a two-piece 316 stainless steel packing gland to insure even pressure on the packing. Ample space shall be provided for repacking the stuffing box. 4.7.1.1 Lubrication-Packed Stuffing Box. A tapped hole shall be provided in the stuffing box directly over the seal cage for lubrication and cooling of the packing. Lubrication liquid shall be supplied (from an external source) (through a by-pass line from the pump discharge nozzle). 4.7.2 Stuffing Box with Mechanical Seal. Mechanical seal shall be of the (single inside) (single outside) (double inside) (cartridge) type and (balanced) (unbalanced). Stuffing box is to be (standard) (oversize) (oversize tapered).

Description Suitable space shall be provided in the standard and oversized stuffing box for supplying a (throttle bushing) (dilution control bushing) with single seals. Throttle bushings and dilution control bushings shall be made of (glass-filled teflon) (a suitable metal material). 4.7.2.1 Lubrication—Stuffing Box with Mechanical Seals. Suitable tapped connections shall be provided to effectively lubricate, cool, flush, quench, etc., as required by the application or recommendations of the mechanical seal manufacturer.

PUMPING OF LIQUIDS AND GASES

FIG. 10-65

10-41

Polytropic compression curves.

Adiabatic efficiency is given by the following relationship Ideal work ηad = ᎏᎏ Actual work

(10-68)

In terms of the change total temperatures the relationship can be written as: T2 − T1 (10-69) ηad = ᎏ T2a − T1 where T2a is the total actual discharge temperature of the gas. The adiabatic efficiency can be represented in terms of total pressure change:

ηad

(k − 1 )/ k

冢ᎏPPᎏ冣 − 1 = ᎏᎏ 冢ᎏPPᎏ冣 − 1 2

In U.S. customary units,

1

2

FIG. 10-66 Factors for use in adiabatic formula. Values of X to be used in finding XG may be obtained from Table 10-3. (By permission of Compressed Air Data.)

(10-70)

(n − 1 )/ n

hpad = 0.0154Q1p1X

1

T2 = T1(X + 1)

Polytropic head can be expressed by the following relationship. n Had = ᎏ ZRT1 n−1

(n − 1 )/ n

冦冤冢ᎏPP 冣 2

冥冧

−1

1

(10-71)

Likewise for polytropic efficiency, which is often considered as the small stage efficiency, or the hydraulic efficiency: (k − 1)/k η pc = ᎏ (10-72) (n − 1)/n Polytropic efficiency is the limited value of the isentropic efficiency as the pressure ratio approaches 1.0, and the value of the polytropic efficiency is higher than the corresponding adiabatic efficiency as seen in Fig. 10-66. A characteristic of polytropic efficiency is that the polytropic efficiency of a multistage unit is equal to the stage efficiency if each stage has the same efficiency. Air and a number of other gases have a value of k = 1.39 to 1.41. To simplify calculations for these gases, tables have been made of the bracketed expression [(p2 / p1)(k − 1) / k − 1] in these equations for a value of k = 1.395. These are known as X factors, and they are given in Table 10-13. By using X factors, the adiabatic formulas for k = 1.395 read as follows: Adiabatic temperature, pressure, and volume relations: V1 / V2 = p2 / [(X + 1)p1]

(10-73)

T2 /T1 = X + 1

(10-74)

T2 − T1 = T1X = T2[X/(X + 1)]

(10-75)

Adiabatic power: In SI units, kWad = 9.81 × 10−4Q1p1X

(10-76b)

Adiabatic discharge temperature:

(10-76a)

(10-77)

To find the X factor XG for a gas of any k value refer to Fig. 6-34. This figure gives values of XG /X for gases having specific-heat ratios between 1.0 and 1.4. The factor XG is then the product of XG /X from Fig. 10-66 and the value X from Table 10-13 for desired compression ratio. Adiabatic power for gases other than air: In SI units, kWad = 6.37 × 10−4Q1 p1X/d

(10-78a)

In U.S. customary units, hpad = 1 × 10−2Q1 p1X/d

(10-78b)

where d = 2.922 (k − 1)/k. If the compression cycle approaches the isothermal condition, pV = constant, as is the case when several stages with intercoolers are used, a simple approximation of the power is obtained from the following formula: In SI units, kW = 2.78 × 10−4Q1p1 ln p2 /p1

(10-79a)

In U.S. customary units, hp = 4.4 × 10−3Q1p1 ln p2 /p1

(10-79b)

For multistage compressors of Ns number of stages with adiabatic compression in each stage, equal division of work between stages, and intercooling to the intake temperature, the following formulas are helpful: In SI units, 6.37 × 10−4NsQ1p1 Ns 苶G苶+ 苶苶1 − 1) kWad = ᎏᎏ (兹X (10-80a) d

10-42

TRANSPORT AND STORAGE OF FLUIDS

TABLE 10-13

Values of X for Normal Air and Perfect Diatomic Gases X = r 0.283 − 1

r

0

1

2

3

4

5

6

7

8

9

r

0

1

2

3

4

5

6

7

8

9

1.00 1.01 1.02 1.03 1.04

0.00 000 .00 282 .00 562 .00 840 .01 116

028 310 590 868 144

057 338 618 895 171

085 366 646 923 199

113 394 673 951 226

141 422 701 978 253

169 450 729 006 281

198 478 757 034 308

226 506 785 061 336

254 534 812 089 363

1.75 1.76 1.77 1.78 1.79

0.17 160 .17 349 .17 538 .17 725 .17 912

179 368 556 744 930

198 387 575 762 949

217 406 594 781 968

236 425 613 800 986

255 443 631 818 005

274 462 650 837 023

292 481 669 856 042

311 500 688 874 061

330 519 706 893 079

1.05 1.06 1.07 1.08 1.09

.01 390 .01 663 .01 933 .02 202 .02 469

418 690 960 229 495

445 717 987 255 522

472 744 014 282 549

500 771 041 309 575

527 798 068 336 602

554 825 095 362 628

581 852 122 389 655

608 879 148 416 681

636 906 175 442 708

1.80 1.81 1.82 1.83 1.84

.18 098 .18 283 .18 468 .18 652 .18 835

116 302 486 670 853

135 320 505 688 871

153 339 523 707 890

172 357 541 725 908

191 376 560 743 926

209 394 578 762 944

228 412 596 780 962

246 431 615 798 981

265 449 633 816 999

1.10 1.11 1.12 1.13 1.14

.02 734 .02 997 .03 259 .03 519 .03 778

760 024 285 545 804

787 050 311 571 829

813 076 337 597 855

840 102 363 623 881

866 129 389 649 906

892 155 415 675 932

919 181 441 700 958

945 207 467 726 983

971 233 493 752 009

1.85 1.86 1.87 1.88 1.89

.19 017 .19 199 .19 380 .19 560 .19 740

035 217 398 578 758

054 235 416 596 776

072 253 434 614 794

090 271 452 632 811

108 289 470 650 829

126 308 488 668 847

144 326 506 686 865

163 344 524 704 883

181 362 542 722 901

1.15 1.16 1.17 1.18 1.19

.04 035 .04 290 .04 543 .04 796 .05 046

060 315 569 821 071

086 341 594 846 096

111 366 619 871 121

137 391 644 896 146

162 417 670 921 171

188 442 695 946 196

213 467 720 971 221

239 493 745 996 245

264 518 770 021 270

1.90 1.91 1.92 1.93 1.94

.19 919 .20 097 .20 275 .20 452 .20 628

937 115 292 469 645

954 133 310 487 663

972 150 328 504 681

990 168 345 522 698

008 186 363 540 716

026 204 381 557 733

044 221 399 575 751

061 239 416 593 768

079 257 434 610 786

1.20 1.21 1.22 1.23 1.24

.05 295 .05 543 .05 789 .06 034 .06 277

320 567 813 058 301

345 592 838 082 325

370 617 862 107 350

394 641 887 131 374

419 666 911 155 398

444 691 936 180 422

469 715 960 204 446

493 740 985 228 470

518 764 009 253 495

1.95 1.96 1.97 1.98 1.99

.20 804 .20 979 .21 153 .21 327 .21 500

821 996 170 344 517

839 013 188 361 534

856 031 205 379 552

874 048 222 396 569

891 066 240 413 586

909 083 257 431 603

926 101 275 448 620

944 118 292 465 638

961 135 309 482 655

1.25 1.26 1.27 1.28 1.29

.06 519 .06 759 .06 998 .07 236 .07 472

543 783 022 260 496

567 807 046 283 520

591 831 070 307 543

615 855 094 331 567

639 879 117 354 590

663 903 141 378 614

687 927 165 402 637

711 951 189 425 661

735 974 212 449 684

2.00 2.01 2.02 2.03 2.04

.21 672 .21 844 .22 015 .22 186 .22 356

689 861 032 203 373

707 878 049 220 390

724 895 066 237 407

741 913 084 254 424

758 930 101 271 441

775 947 118 288 458

792 964 135 305 474

810 981 152 322 491

827 998 169 339 508

1.30 1.31 1.32 1.33 1.34

.07 708 .07 941 .08 174 .08 405 .08 635

731 965 197 428 658

754 988 220 451 681

778 011 243 474 704

801 035 267 497 727

825 058 290 520 750

848 081 313 543 773

871 104 336 566 795

895 128 359 589 818

918 151 382 612 841

2.05 2.06 2.07 2.08 2.09

.22 525 .22 694 .22 863 .23 030 .23 197

542 711 879 047 214

559 728 896 064 231

576 745 913 080 247

593 762 930 097 264

610 778 946 114 281

627 795 963 130 297

644 812 980 147 314

660 829 997 164 331

677 846 013 181 347

1.35 1.36 1.37 1.38 1.39

.08 864 .09 092 .09 318 .09 543 .09 767

887 114 341 566 790

910 137 363 588 812

932 160 386 611 834

955 182 408 633 857

978 205 431 655 879

001 228 453 678 901

023 250 476 700 923

046 273 498 723 946

069 295 521 745 968

2.10 2.11 2.12 2.13 2.14

.23 364 .23 530 .23 695 .23 860 .24 024

380 546 712 877 041

397 563 728 893 057

414 579 745 909 074

430 596 761 926 090

447 613 778 942 106

463 629 794 959 123

480 646 811 975 139

497 662 827 992 155

513 679 844 008 172

1.40 1.41 1.42 1.43 1.44

.09 990 .10 212 .10 433 .10 652 .10 871

012 234 455 674 892

035 256 477 696 914

057 278 499 718 936

079 300 521 740 958

101 322 542 761 979

123 344 564 783 001

145 366 586 805 023

168 389 608 827 045

190 411 630 849 066

2.15 2.16 2.17 2.18 2.19

.24 188 .24 351 .24 514 .24 676 .24 838

204 368 530 692 854

221 384 546 708 870

237 400 563 724 886

253 416 579 741 902

270 433 595 757 918

286 449 611 773 934

302 465 627 789 950

319 481 644 805 966

335 498 660 821 983

1.45 1.46 1.47 1.48 1.49

.11 088 .11 304 .11 520 .11 734 .11 947

110 326 541 755 968

131 347 562 776 989

153 369 584 798 010

175 390 605 819 032

196 412 627 840 053

218 433 648 862 074

239 455 669 883 095

261 476 691 904 116

283 498 712 925 138

2.20 2.21 2.22 2.23 2.24

.24 999 .25 159 .25 319 .25 479 .25 638

015 175 335 495 654

031 191 351 511 669

047 207 367 526 685

063 223 383 542 701

079 239 399 558 717

095 255 415 574 733

111 271 431 590 749

127 287 447 606 765

143 303 463 622 780

1.50 1.51 1.52 1.53 1.54

.12 159 .12 370 .12 580 .12 789 .12 997

180 391 601 810 018

201 412 622 831 039

222 433 643 852 060

243 454 664 872 080

264 475 685 893 101

286 496 706 914 122

307 517 726 935 142

328 538 747 956 163

349 559 768 977 184

2.25 2.26 2.27 2.28 2.29

.25 796 .25 954 .26 112 .26 269 .26 425

812 970 127 284 441

828 986 143 300 456

844 001 159 316 472

859 017 175 331 488

875 033 190 347 503

891 049 206 363 519

907 064 222 378 534

923 080 237 394 550

938 096 253 409 566

1.55 1.56 1.57 1.58 1.59

.13 205 .13 411 .13 616 .13 820 .14 024

225 431 636 841 044

246 452 657 861 064

266 472 677 881 085

287 493 698 902 105

308 513 718 922 125

328 534 739 942 145

349 554 759 963 165

370 575 780 983 186

390 595 800 003 206

2.30 2.31 2.32 2.33 2.34

.26 581 .26 737 .26 892 .27 046 .27 200

597 752 907 062 216

612 768 923 077 231

628 783 938 092 246

643 799 954 108 262

659 814 969 123 277

675 830 984 139 292

690 845 000 154 308

706 861 015 169 323

721 876 031 185 338

1.60 1.61 1.62 1.63 1.64

.14 226 .14 428 .14 628 .14 828 .15 027

246 448 648 848 047

267 468 668 868 067

287 488 688 888 087

307 508 708 908 107

327 528 728 928 126

347 548 748 948 146

367 568 768 968 166

387 588 788 988 186

408 608 808 007 206

2.35 2.36 2.37 2.38 2.39

.27 354 .27 507 .27 660 .27 812 .27 964

369 522 675 827 979

384 538 690 842 994

400 553 705 857 009

415 568 721 873 024

430 583 736 888 039

446 599 751 903 054

461 614 766 918 070

476 629 781 933 085

492 644 797 948 100

1.65 1.66 1.67 1.68 1.69

.15 225 .15 423 .15 619 .15 814 .16 009

245 442 638 834 028

265 462 658 853 048

284 481 678 873 067

304 501 697 892 087

324 521 717 912 106

344 540 736 931 125

363 560 756 951 145

383 580 775 970 164

403 599 795 990 184

2.40 2.41 2.42 2.43 2.44

.28 115 .28 266 .28 416 .28 566 .28 716

130 281 431 581 730

145 296 446 596 745

160 311 461 611 760

175 326 476 626 775

190 341 491 641 790

205 356 506 656 805

220 371 521 671 820

236 386 536 686 835

251 401 551 701 850

1.70 1.71 1.72 1.73 1.74

.16 203 .16 396 .16 588 .16 780 .16 970

222 415 607 799 989

242 434 626 818 008

261 454 646 837 027

280 473 665 856 046

299 492 684 875 065

319 511 703 894 084

338 531 722 913 103

357 550 741 932 122

377 569 760 951 141

2.45 2.46 2.47 2.48 2.49

.28 865 .29 013 .29 162 .29 309 .29 457

879 028 176 324 471

894 043 191 339 486

909 058 206 353 501

924 073 221 368 515

939 087 235 383 530

954 102 250 398 545

969 117 265 412 559

984 132 280 427 574

998 147 295 442 589

PUMPING OF LIQUIDS AND GASES

10-43

Values of X for Normal Air and Perfect Diatomic Gases (Continued )

TABLE 10-13 r

0

1

2

3

4

5

6

7

8

9

r

0

1

2

3

4

5

6

7

8

9

2.50 2.51 2.52 2.53 2.54

0.29 604 .29 750 .29 896 .30 042 .30 187

618 765 911 056 202

633 779 925 071 216

647 794 940 085 231

662 808 954 100 245

677 823 969 114 260

691 838 984 129 274

706 852 998 144 289

721 867 013 158 303

735 881 027 173 318

2.75 2.76 2.77 2.78 2.79

0.33 147 .33 284 .33 420 .33 556 .33 692

161 297 434 570 706

174 311 448 584 719

188 325 461 597 733

202 338 475 611 746

215 352 488 624 760

229 366 502 638 773

243 379 516 651 787

256 393 529 665 801

270 407 543 679 814

2.55 2.56 2.57 2.58 2.59

.30 332 .30 476 .30 620 .30 764 .30 907

346 491 635 778 921

361 505 649 793 936

375 520 663 807 950

390 534 678 821 964

404 548 692 836 979

419 563 707 850 993

433 577 721 864 007

448 592 735 879 021

462 606 750 893 036

2.80 2.81 2.82 2.83 2.84

.33 828 .33 963 .34 097 .34 232 .34 366

841 976 111 245 379

855 990 124 259 393

868 003 138 272 406

882 017 151 285 419

895 030 165 299 433

909 044 178 312 446

922 057 191 326 459

936 070 205 339 473

949 084 218 352 486

2.60 2.61 2.62 2.63 2.64

.31 050 .31 193 .31 335 .31 476 .31 618

064 207 349 490 632

079 221 363 505 646

093 235 377 519 660

107 249 391 533 674

121 264 405 547 688

136 278 420 561 702

150 292 434 575 716

164 306 448 589 730

178 320 462 603 744

2.85 2.86 2.87 2.88 2.89

.34 500 .34 633 .34 766 .34 899 .35 031

513 646 779 912 044

526 660 793 925 058

540 673 806 939 071

553 686 819 952 084

566 700 832 965 097

580 713 846 978 110

593 726 859 991 124

606 739 872 005 137

620 753 886 018 150

2.65 2.66 2.67 2.68 2.69

.31 759 .31 899 .32 039 .32 179 .32 318

773 913 053 193 332

787 927 067 207 346

801 941 081 221 360

815 955 095 235 374

829 969 109 249 388

843 983 123 262 402

857 997 137 276 416

871 011 151 290 429

885 025 165 304 443

2.90 2.91 2.92 2.93 2.94

.35 163 .35 295 .35 426 .35 557 .35 688

176 308 439 570 701

190 321 452 584 714

203 334 466 597 727

216 347 479 610 740

229 361 492 623 753

242 374 505 636 767

255 387 518 649 780

269 400 531 662 793

282 413 544 675 806

2.70 2.71 2.72 2.73 2.74

.32 457 .32 596 .32 734 .32 872 .33 010

471 610 748 886 023

485 624 762 900 037

499 637 776 913 051

513 651 789 927 065

527 665 803 941 078

540 679 817 955 092

554 693 831 968 106

568 707 845 982 119

582 720 858 996 133

2.95 2.96 2.97 2.98 2.99

.35 819 .35 949 .36 079 .36 208 .36 337

832 962 092 221 350

845 975 105 234 363

858 988 118 247 376

871 001 131 260 389

884 014 144 273 402

897 027 157 286 415

910 040 169 299 428

923 053 182 312 440

936 066 195 324 453

r

0

1

2

3

4

5

6

7

8

9

3.0 3.1 3.2 3.3 3.4

0.3647 .3774 .3898 .4020 .4139

0.3659 .3786 .3911 .4032 .4150

0.3672 .3799 .3923 .4044 .4162

0.3685 .3811 .3935 .4056 .4174

0.3698 .3824 .3947 .4068 .4186

0.3711 .3836 .3959 .4080 .4197

0.3723 .3849 .3971 .4091 .4209

0.3736 .3861 .3984 .4103 .4220

0.3749 .3874 .3996 .4115 .4232

0.3761 .3886 .4008 .4127 .4244

3.5 3.6 3.7 3.8 3.9

.4255 .4369 .4481 .4591 .4698

.4267 .4380 .4492 .4602 .4709

.4278 .4392 .4503 .4612 .4720

.4290 .4403 .4514 .4623 .4730

.4301 .4414 .4525 .4634 .4741

.4313 .4425 .4536 .4645 .4752

.4324 .4437 .4547 .4656 .4762

.4335 .4448 .4558 .4666 .4773

.4347 .4459 .4569 .4677 .4783

.4358 .4470 .4580 .4688 .4794

4.0 4.1 4.2 4.3 4.4

.4804 .4908 .5010 .5110 .5209

.4815 .4918 .5020 .5120 .5219

.4825 .4928 .5030 .5130 .5228

.4835 .4939 .5040 .5140 .5238

.4846 .4949 .5050 .5150 .5248

.4856 .4959 .5060 .5160 .5258

.4867 .4970 .5070 .5170 .5267

.4877 .4980 .5080 .5179 .5277

.4887 .4990 .5090 .5189 .5287

.4898 .5000 .5100 .5199 .5296

4.5 4.6 4.7 4.8 4.9

.5306 .5401 .5495 .5588 .5679

.5316 .5411 .5505 .5597 .5688

.5325 .5420 .5514 .5606 .5697

.5335 .5430 .5523 .5616 .5706

.5344 .5439 .5533 .5625 .5715

.5354 .5449 .5542 .5634 .5724

.5363 .5458 .5551 .5643 .5733

.5373 .5467 .5560 .5652 .5742

.5382 .5477 .5570 .5661 .5751

.5392 .5486 .5579 .5670 .5760

5.0 5.1 5.2 5.3 5.4

.5769 .5858 .5945 .6031 .6116

.5778 .5867 .5954 .6040 .6125

.5787 .5875 .5962 .6048 .6133

.5796 .5884 .5971 .6057 .6142

.5805 .5893 .5980 .6065 .6150

.5814 .5902 .5988 .6074 .6159

.5822 .5910 .5997 .6082 .6167

.5831 .5919 .6006 .6091 .6175

.5840 .5928 .6014 .6099 .6184

.5849 .5936 .6023 .6108 .6192

5.5 5.6 5.7 5.8 5.9

.6200 .6283 .6365 .6446 .6525

.6209 .6291 .6373 .6454 .6533

.6217 .6300 .6381 .6462 .6541

.6225 .6308 .6389 .6470 .6549

.6234 .6316 .6397 .6478 .6557

.6242 .6324 .6405 .6486 .6565

.6250 .6332 .6413 .6494 .6573

.6258 .6340 .6421 .6502 .6581

.6267 .6349 .6430 .6509 .6588

.6275 .6357 .6438 .6517 .6596

6.0 6.1 6.2 6.3 6.4

.6604 .6682 .6759 .6835 .6910

.6612 .6690 .6767 .6843 .6918

.6620 .6697 .6774 .6850 .6925

.6628 .6705 .6782 .6858 .6933

.6635 .6713 .6789 .6865 .6940

.6643 .6721 .6797 .6873 .6948

.6651 .6729 .6805 .6880 .6955

.6659 .6736 .6812 .6888 .6963

.6666 .6744 .6820 .6895 .6970

.6674 .6752 .6827 .6903 .6978

6.5 6.6 6.7 6.8 6.9

.6985 .7058 .7131 .7203 .7274

.6992 .7065 .7138 .7210 .7281

.7000 .7073 .7145 .7217 .7288

.7007 .7080 .7153 .7224 .7295

.7014 .7087 .7160 .7232 .7302

.7021 .7095 .7167 .7239 .7309

.7028 .7102 .7174 .7246 .7316

.7036 .7110 .7181 .7253 .7323

.7043 .7117 .7189 .7260 .7330

.7050 .7124 .7196 .7267 .7338

7.0 7.1 7.2 7.3 7.4

.7345 .7414 .7483 .7552 .7620

.7352 .7421 .7490 .7559 .7626

.7359 .7428 .7497 .7565 .7633

.7366 .7435 .7504 .7572 .7640

.7373 .7442 .7511 .7579 .7646

.7380 .7449 .7518 .7586 .7653

.7386 .7456 .7524 .7592 .7660

.7393 .7463 .7531 .7599 .7666

.7400 .7470 .7538 .7606 .7673

.7407 .7477 .7545 .7613 .7680

10-44

TRANSPORT AND STORAGE OF FLUIDS Values of X for Normal Air and Perfect Diatomic Gases (Concluded )

TABLE 10-13 r

0

1

2

3

4

5

6

7

8

9

7.5 7.6 7.7 7.8 7.9

0.7687 .7753 .7819 .7884 .7949

0.7693 .7760 .7825 .7890 .7955

0.7700 .7766 .7832 .7897 .7961

0.7706 .7773 .7838 .7903 .7968

0.7713 .7779 .7845 .7910 .7974

0.7720 .7786 .7851 .7916 .7981

0.7726 .7792 .7858 .7923 .7987

0.7733 .7799 .7864 .7929 .7993

0.7740 .7806 .7871 .7936 .8000

0.7746 .7813 .7877 .7942 .8006

8.0 8.1 8.2 8.3 8.4

.8013 .8076 .8139 .8201 .8263

.8019 .8082 .8145 .8207 .8269

.8025 .8089 .8151 .8214 .8275

.8032 .8095 .8158 .8220 .8281

.8038 .8101 .8164 .8226 .8288

.8044 .8108 .8170 .8232 .8294

.8051 .8114 .8176 .8238 .8300

.8057 .8120 .8183 .8245 .8306

.8063 .8126 .8189 .8251 .8312

.8070 .8133 .8195 .8257 .8318

8.5 8.6 8.7 8.8 8.9

.8324 .8385 .8445 .8505 .8564

.8330 .8391 .8451 .8511 .8570

.8336 .8397 .8457 .8517 .8576

.8343 .8403 .8463 .8523 .8582

.8349 .8409 .8469 .8529 .8588

.8355 .8415 .8475 .8535 .8594

.8361 .8421 .8481 .8541 .8600

.8367 .8427 .8487 .8547 .8605

.8373 .8433 .8493 .8552 .8611

.8379 .8439 .8499 .8558 .8617

9.0 9.1 9.2 9.3 9.4

.8623 .8681 .8739 .8797 .8854

.8629 .8687 .8745 .8802 .8859

.8635 .8693 .8751 .8808 .8865

.8641 .8699 .8757 .8814 .8871

.8646 .8705 .8762 .8819 .8876

.8652 .8710 .8768 .8825 .8882

.8658 .8716 .8774 .8831 .8888

.8664 .8722 .8779 .8837 .8893

.8670 .8728 .8785 .8842 .8899

.8676 .8734 .8791 .8848 .8905

9.5 9.6 9.7 9.8 9.9

.8910 .8966 .9022 .9077 .9132

.8916 .8972 .9028 .9083 .9138

.8921 .8977 .9033 .9088 .9143

.8927 .8983 .9039 .9094 .9149

.8933 .8989 .9044 .9099 .9154

.8938 .8994 .9050 .9105 .9159

.8944 .9000 .9055 .9110 .9165

.8949 .9005 .9061 .9116 .9170

.8955 .9011 .9066 .9121 .9176

.8961 .9016 .9072 .9127 .9181

10.0 10.1 10.2 10.3 10.4

.9187 .9241 .9295 .9348 .9401

.9192 .9246 .9300 .9353 .9406

.9198 .9252 .9305 .9358 .9411

.9203 .9257 .9311 .9364 .9417

.9208 .9262 .9316 .9369 .9422

.9214 .9268 .9321 .9374 .9427

.9219 .9273 .9327 .9380 .9432

.9225 .9278 .9332 .9385 .9438

.9230 .9284 .9337 .9390 .9443

.9235 .9289 .9343 .9396 .9448

10.5 10.6 10.7 10.8 10.9

.9453 .9506 .9558 .9609 .9660

.9459 .9511 .9563 .9614 .9665

.9464 .9516 .9568 .9619 .9671

.9469 .9521 .9573 .9625 .9676

.9474 .9526 .9578 .9630 .9681

.9480 .9532 .9583 .9635 .9686

.9485 .9537 .9589 .9640 .9691

.9490 .9542 .9594 .9645 .9696

.9495 .9547 .9599 .9650 .9701

.9500 .9552 .9604 .9655 .9706

11.0 11.1 11.2 11.3 11.4

.9711 .9762 .9812 .9862 .9912

.9716 .9767 .9817 .9867 .9916

.9721 .9772 .9822 .9872 .9921

.9726 .9777 .9827 .9877 .9926

.9732 .9782 .9832 .9882 .9931

.9737 .9787 .9837 .9887 .9936

.9742 .9792 .9842 .9892 .9941

.9747 .9797 .9847 .9897 .9946

.9752 .9802 .9852 .9902 .9951

.9757 .9807 .9857 .9907 .9956

11.5 11.6 11.7 11.8 11.9 12.0

.9961 1.0010 1.0058 1.0107 1.0155 1.0203

.9966 1.0015 1.0063 1.0112 1.0160 1.0207

.9971 1.0019 1.0068 1.0116 1.0164 1.0212

.9975 1.0024 1.0073 1.0121 1.0169 1.0217

.9980 1.0029 1.0078 1.0126 1.0174 1.0222

.9985 1.0034 1.0083 1.0131 1.0179 1.0226

.9990 1.0039 1.0087 1.0136 1.0184 1.0231

.9995 1.0044 1.0092 1.0140 1.0188 1.0236

1.0000 1.0049 1.0097 1.0145 1.0193 1.0241

1.0005 1.0054 1.0102 1.0150 1.0198 1.0245

Printed by permission of Compressed Air Data. Taken from Moss and Smith, Engineering Computations for Air and Gases, Trans. Am. Soc. Mech. Engrs., vol. 52, 1930, paper APM-52-8. For nozzles r = p1/p2. For compressors and exhausters r = p2 /p1. r

X

r

X

r

X

r

X

r

X

r

X

r

X

r

X

r

X

12.5 13.0 13.5 14.0 14.5

1.0428 1.0666 1.0887 1.1103 1.1314

15.0 15.5 16.0 16.5 17.0

1.1520 1.1720 1.1916 1.2108 1.2295

17.5 18.0 18.5 19.0 19.5

1.2479 1.2659 1.2835 1.3008 1.3189

20.0 20.5 21.0 21.5 22.0

1.3345 1.3509 1.3669 1.3828 1.3983

22.5 23.0 23.5 24.0 24.5

1.4136 1.4287 1.4435 1.4581 1.4725

25.0 25.5 26.0 26.5 27.0

1.4867 1.5006 1.5144 1.5280 1.5414

27.5 28.0 28.5 29.0 29.5

1.5546 1.5678 1.5794 1.5933 1.6059

30.0 30.5 31.0 31.5 32.0

1.6183 1.6306 1.6434 1.6547 1.6666

32.5 33.0 33.5 34.0 34.5

1.6783 1.6899 1.7014 1.7127 1.7240

Values of X from 12.5 to 34.5 calculated by Ingersoll-Rand Co.

PUMPING OF LIQUIDS AND GASES In U.S. customary units, 1 × 10−2NsQ1p1 Ns 苶G苶+ 苶苶1 − 1) hpad = ᎏᎏ (兹X d Ns

苶G苶+ 苶苶1 T2 = T1 兹X

(10-80b) (10-81)

To be able to decide which type of compressor would best fit the job, we should first divide the compressors into three main categories: positive displacement, centrifugal, and axial flow. In general terms, positive displacement compressors are used for high pressure and low flow characteristics; centrifugal compressors are used for medium to high pressure delivery and medium flow; and axial flow compressors are low pressure and high flow. Compressor Selection To select the most satisfactory compression equipment, engineers must consider a wide variety of types, each of which offers peculiar advantages for particular applications. Among the major factors to be considered are flow rate, head or pressure, temperature limitations, method of sealing, method of lubrication, power consumption, serviceability, and cost. To be able to decide which compressor best fits the job, the engineer must analyze the flow characteristics of the units. The following dimensionless numbers describe the flow characteristics. Reynolds number is the ratio of the inertia forces to the viscous forces ρVD NRe = ᎏ (10-82) µ where ρ is the density of the gas, V is the velocity of the gas, D is the diameter of the impeller, and µ is the viscosity of the gas. The specific speed compares the adiabatic head and flow rate in geometrically similar machines at various speeds. 苶 N兹Q Ns = ᎏ (10-83) 314 H ad where N is the speed of rotation of the compressor, Q is the volume flow rate, and H is the adiabatic head. The specific diameter compares head and flow rates in geometrically similar machines at various diameters DH1/4 DS = ᎏ (10-84) 苶 兹Q The flow coefficient is the capacity of the flow rate of the machine Q1 φ = ᎏ3 (10-85) ND

The pressure coefficient is the pressure or the pressure rise of the machine H Ψ=ᎏ (10-86) N 2D2 In selecting the machines of choice, the use of specific speed and diameter best describe the flow. Figure 10-67 shows the characteristics of the three types of compressors. Other considerations in chemical plant service such as problems with gases which may be corrosive or have abrasive solids in suspension must be dealt with. Gases at elevated temperatures may create a potential explosion hazard, while air at the same temperatures may be handled quite normally; minute amounts of lubricating oil or water may contaminate the process gas and so may not be permissible, and for continuous-process use, a high degree of equipment reliability is required, since frequent shutdowns for inspection or maintenance cannot be tolerated. FANS AND BLOWERS Fans are used for low pressures where generally the delivery pressure is less than 3.447 kPa (0.5 lb/in2), and blowers are used for higher pressures. However, they are usually below delivery pressures of 10.32 kPa (1.5 lbf/in2). These units can either be centrifugal or the axial-flow type. Fans and blowers are used for many types of ventilating work such as air-conditioning systems. In large buildings, blowers are often used due to the high delivery pressures needed to overcome the pressure drop in the ventilation system. Most of these blowers are of the centrifugal type. Blowers are also used to supply draft air to boilers and furnaces. Fans are used to move large volumes of air or gas through ducts, supplying air for drying, conveying material suspended in the gas stream, removing fumes, condensing towers and other highflow, low-pressure applications. Axial-flow fans are designed to handle very high flow rates and low pressure. The disc-type fans are similar to those of a household fan. They are usually for general circulation or exhaust work without ducts. The so-called propeller-type fans with blades that are aerodynamically designed (as seen in Fig. 10-68) can consist of two or more stages. The air in these fans enters in an axial direction and leaves in an axial direction. The fans usually have inlet guide vanes followed by a rotating blade, followed by a stationary (stator) blade. Centrifugal Blowers These blowers have air or gases entering in the axial direction and being discharged in the radial direction. These blowers have 3 types of blades, radial or straight blades, forward curved blades, and backward curved blades (Figs. 10-69–10-71).

Compressor coverage chart based on the normal range of operation of commercially available types shown. Solid lines: use left ordinate, head. Broken lines: use right ordinate, pressure. To convert cubic feet per minute to cubic meters per hour, multiply by 1.699; to convert feet to meters, multiply by 0.3048; and to convert pounds-force per square inch to kilopascals, multiply by 6.895; (°F − 32)% = °C.

FIG. 10-67

10-45

10-46

FIG. 10-68

TRANSPORT AND STORAGE OF FLUIDS

Straight-blade, or steel-plate, fan.

Forward-Curved Blade Blowers These blowers discharge the gas at a very high velocity. The pressure supplied by this blower is lower than that produced in the other two blade characteristics. The number of blades in such a rotor can be large—up to 50 blades—and the speed is high—usually 3600–1800 rpm in 60-cycle countries and 3000–1500 rpm in 50-cycle countries. Backward-Curved Blade Blowers These blowers are used when a higher discharge pressure is needed. It is used over a wide range of applications. Both the forward and backward curved blades do have much higher stresses than the radial bladed blower. The centrifugal blower produces energy in the air stream by the centrifugal force and imparts a velocity to the gas by the blades. Forward curved blades impart the most velocity to the gas. The scrollshaped volute diffuses the air and creates an increase in the static pressure by reducing the gas velocity. The change in total pressure occurs in the impeller—this is usually a small change. The static pressure is increased both in the impeller and the diffuses section. Operating efficiencies of the fan range from 40–80 percent. The discharge total pressure is the summation of the static pressure and the velocity head. The power needed to drive the fan can be computed as follows. Power (kw) = 2.72 × 10−5QP

(10-87)

where Q is the fan volume (m3/hr) and P is the total discharge pressure in cm of water column. In U.S. customary units, FIG. 10-69

hp = 1.57 × 10−4Qp

Forward-curved blade, or “scirocco”-type, fan.

FIG. 10-70

Backward-curved-blade fan.

FIG. 10-71

Two-stage axial-flow fan.

Radial blade blowers as seen in Fig. 10-68 are usually used in largediameter or high-temperature applications. The blades being radial in direction have very low stresses as compared to the backward or forward curve blades. The rotors have anywhere between 4 to 12 blades and usually operate at low speeds. These fans are used in exhaust work especially for gases at high temperature and with suspensions in the flow stream.

(10-88)

where hp is the fan power output, hp; Q is the fan volume, ft3/min; and p is the fan-operating pressure, inches water column. air power output Efficiency = ᎏᎏ (10-89) shaft power input Fan Performance The performance of a centrifugal fan varies with changes in conditions such as temperature, speed, and density of the gas being handled. It is important to keep this in mind in using the catalog data of various fan manufacturers, since such data are usually based on stated standard conditions. Corrections must be made for variations from these standards. The usual variations are as follows: When speed varies, (1) capacity varies directly as the speed ratio, (2) pressure varies as the square of the speed ratio, and (3) horsepower varies as the cube of the speed ratio. When the temperature of air or gas varies, horsepower and pressure vary inversely as the absolute temperature, speed and capacity being constant. See Fig. 10-72. When the density of air or gas varies, horsepower and pressure vary directly as the density, speed and capacity being constant.

FIG. 10-72

Approximate characteristic curves of various types of fans.

PUMPING OF LIQUIDS AND GASES

10-47

COMPRESSORS Compressors are used to handle large volumes of gas at pressure increases from 10.32 kPa (1.5lbg/in2) to several hundred kPa (lbg/in2). We can divide compressors into two major categories: 1. Continuous-flow compressors. a. Centrifugal compressors b. Axial flow compressors 2. Positive displacement compressors a. Rotary compressors b. Reciprocating compressors Continuous-Flow Compressors Continuous-flow compressors are machines where the flow is continuous, unlike positive displacement machines where the flow is fluctuating. Continuous-flow compressors are also classified as turbomachines. These types of machines are widely used in the chemical and petroleum industry for many services. They are also used extensively in many other industries such as the iron and steel industry, pipeline boosters, and on offshore platforms for reinjection compressors. Continuous-flow machines are usually much smaller in size and produce much less vibration than their counterpart, positive displacement units. Centrifugal Compressors The flow in a centrifugal compressor enters the impeller in an axial direction and exits in a radial direction. In a typical centrifugal compressor, the fluid is forced through the impeller by rapidly rotating impeller blades. The velocity of the fluid is converted to pressure, partially in the impeller and partially in the stationary diffusers. Most of the velocity leaving the impeller is converted into pressure energy in the diffuser as shown in Fig. 10-73. It is normal practice to design the compressor so that half the pressure rise takes place in the impeller and the other half in the diffuser. The diffuser consists of a vaneless space, a vane that is tangential to the impeller, or a combination of both. These vane passages diverge to convert the velocity head into pressure energy. Centrifugal compressors in general are used for higher pressure ratios and lower flow rates compared to lower-stage pressure ratios and higher flow rates in axial compressors. The pressure ratio in a single-stage centrifugal compressor varies depending on the industry and application. In the petrochemical industry the single stage pressure ratio is about 1.2:1. Centrifugal compressors used in the aerospace

FIG. 10-73

Pressure and velocity through a centrifugal compressor.

industry, usually as a compressor of a gas turbine, have pressure ratios between 3:1 to as high as 9:1 per stage. In the petrochemical industry, the centrifugal compressors consist mainly of casings with multiple stages. In many instances, multiple casings are also used, and to reduce the power required to drive these multiple casings, there are intercoolers between them. Each casing can have up to 9 stages. In some cases, intercoolers are also used between single stages of compressor to reduce the power required for compression. These compressors are usually driven by gas turbines, steam turbines, and electric motors. Speed-increasing gears may be used in conjunction with these drivers to obtain the high speeds at which many of these units operate. Rotative speeds of as high as 50,000 rpm are not uncommon. Most of the petrochemical units run between 9,000–15,000 rpm. The compressor’s operating range is between two major regions as seen in Fig. 10-74, which is a performance map of a centrifugal compressor. These two regions are surge, which is the lower flow limit of

10

ηc denotes efficiency related to total exhaust pressure and total inlet pressure

Re10 = 106 6

3

Ns =

N Q1 3/4 Had

Ds =

1/4 DHad Q1

ηc

.30 .6.60 ηc .700 .80 .85

Ds 1

N = rpm Q = ft3/sec Had = ft • lbf/lbm D = ft

φ ad

.6

= 02 .0 04 .0

.3

1 .0 2 .0 4 .0

.1

.2

.4

1

φad =

V2abs u2

.1 10

30

60

100

300 Ns

600

1000

3000

6000 10000

Centrifugal compressor map. (Balje, O. E., “A Study of Reynolds Number Effects in Turbomachinery,” Journal of Engineering for Power, ASME Trans., vol. 86, series A, p. 227).

FIG. 10-74

10-48

TRANSPORT AND STORAGE OF FLUIDS

stable operation, and choke or stonewall, which is the maximum flow through the compressor at a given operating speed. The centrifugal compressor’s operating range between surge and choke is reduced as the pressure ratio per stage is increased or the number of stages are added. A compressor is said to be in surge when the main flow through the compressor reverses its direction. Surge is often symptomized by excessive vibration and a large audible sound. This flow reversal is accompanied with a very violent change in energy, which causes a reversal of the thrust force. The surge process is cyclic in nature and if allowed to cycle for some time, irreparable damage can occur to the compressor. In a centrifugal compressor, surge is usually initiated at the exit of the impeller or at the diffuser entrance for impellers producing a pressure ratio of less than 3:1. For higher pressure ratios, the initiation of surge can occur in the inducer. A centrifugal compressor impeller can have three types of blades at the exit of the impeller. These are forward-curved, backward-curved, and radial blades. Forward-curved blades are not often used in a centrifugal compressor’s impeller because of the very high-velocity discharge at the compressor that would require conversion of the high velocity to a pressure head in the diffuser, which is accompanied by high losses. Radial blades are used in impellers of high pressure ratio since the stress levels are minimal. Backward-curved blades give the highest efficiency and the largest operating margin of any of the various types of blades in an impeller. Most centrifugal compressors in the petrochemical industry use backward-curved impellers because of the higher efficiency and larger operating range. Process compressors have impellers with very low pressure ratio impellers and thus large surge-to-choke margins. The common method of classifying process-type centrifugal compressors is based on the number of impellers and the casing design. Sectionalized casing types have impellers that are usually mounted on the extended motor shaft, and similar sections are bolted together to obtain the desired number of stages. Casing material is either steel or cast iron. These machines require minimum supervision and maintenance and are quite economic in their operating range. The sectionalized casing design is used extensively in supplying air for combustion in ovens and furnaces. The horizontally split type have casings split horizontally at the midsection and the top. The bottom halves are bolted and doweled together. This design type is preferred for large multistage units. The internal parts such as shaft, impellers, bearings, and seals are readily accessible for inspection and repairs by removing the top half. The casing material is cast iron or cast steel. Barrel casings are used for high pressures in which the horizontally split joint is inadequate. This type of compressor consists of a barrel into which a compressor bundle of multiple stages is inserted. The bundle is itself a horizontally split casing compressor. Compressor Configuration To properly design a centrifugal compressor, one must know the operating conditions—the type of gas, its pressure, temperature, and molecular weight. One must also know the corrosive properties of the gas so that proper metallurgical selection can be made. Gas fluctuations due to process instabilities must be pinpointed so that the compressor can operate without surging. Centrifugal compressors for industrial applications have relatively low pressure ratios per stage. This condition is necessary so that the compressors can have a wide operating range while stress levels are kept at a minimum. Because of the low pressure ratios for each stage, a single machine may have a number of stages in one “barrel” to achieve the desired overall pressure ratio. Figure 10-75 shows some of the many configurations. Some factors to be considered when selecting a configuration to meet plant needs are: 1. Intercooling between stages can considerably reduce the power consumed. 2. Back-to-back impellers allow for a balanced rotor thrust and minimize overloading the thrust bearings. 3. Cold inlet or hot discharge at the middle of the case reduces oil-seal and lubrication problems. 4. Single inlet or single discharge reduces external piping problems. 5. Balance planes that are easily accessible in the field can appreciably reduce field-balancing times.

Four post, three cooling points Series flow, cooling point

Parallel flow, suction in ends

Series flow (basic compressor)

Parallel flow, suction in center

Series flow, one cooling point

Series flow, two cooling points

Series flow, one cooling point suction on ends, cool ends

Series flow, one cooling point, suction in center, warm ends

Series flow, with double flow inlet and side stream

FIG. 10-75

Various configurations of centrifugal compressors.

6. Balance piston with no external leakage will greatly reduce wear on the thrust bearings. 7. Hot and cold sections of the case that are adjacent to each other will reduce thermal gradients and thus reduce case distortion. 8. Horizontally split casings are easier to open for inspection than vertically split ones, reducing maintenance time. 9. Overhung rotors present an easier alignment problem because

PUMPING OF LIQUIDS AND GASES shaft-end alignment is necessary only at the coupling between the compressor and driver. 10. Smaller, high-pressure compressors that do the same job will reduce foundation problems but will have greatly reduced operational range. Impeller Fabrication Centrifugal-compressor impellers are either shrouded or unshrouded. Open, shrouded impellers that are mainly used in single-stage applications are made by investmentcasting techniques or by three-dimensional milling. Such impellers are used, in most cases, for the high-pressure-ratio stages. The shrouded impeller is commonly used in the process compressor because of its low pressure ratio stages. The low tip stresses in this application make it a feasible design. Figure 10-76 shows several fabrication techniques. The most common type of construction is seen in A and B where the blades are fillet-welded to the hub and shroud. In B, the welds are full penetration. The disadvantage in this type of construction is the obstruction of the aerodynamic passage. In C, the blades are partially machined with the covers and then butt-welded down the middle. For backward lean-angled blades, this technique has not been very successful, and there has been difficulty in achieving a smooth contour around the leading edge. D illustrates a slot-welding technique and is used where bladepassage height is too small (or the backward lean-angle too high) to permit conventional fillet welding. In E, an electron-beam technique is shown. Its major disadvantage is that electron-beam welds should preferably be stressed in tension but, for the configuration of E, they are in shear. The configurations of G through J use rivets. Where the rivet heads protrude into the passage aerodynamic performance is reduced. Riveted impellers were used in the 1960s—they are very rarely used now. Elongation of these rivets occurs at certain critical surge conditions and can lead to major failures. Materials for fabricating these impellers are usually low-alloy steels, such as AISI 4140 or AISI 4340, AISI 4140 is satisfactory for most applications; AISI 4340 is used for large impellers requiring higher strengths. For corrosive gases, AISI 410 stainless steel (about 12 percent chromium) is used. Monel K-500 is employed in halogen gas atmospheres and oxygen compressors because of its resistance to sparking. Titanium impellers have been applied to chlorine service. Aluminum-alloy impellers have been used in great numbers, especially at lower temperatures (below 300°F). With new developments in aluminum alloys, this range is increasing. Aluminum and titanium are sometimes selected because of their low density. This low density can cause a shift in the critical speed of the rotor, which may be advantageous. Axial Flow Compressors Axial flow compressors are used mainly as compressors for gas turbines. They are also used in the steel industry as blast furnace blowers and in the chemical industry for large nitric acid plants. They are mainly used for applications where the head required is low and the flow large. Figure 10-77 shows a typical axial-flow compressor. The rotating element consists of a single drum to which are attached several rows of decreasing-height blades having airfoil cross sections. Between each rotating blade row is a stationary blade row. All blade angles and areas are designed precisely for a given performance and high efficiency. The use of multiple stages permits overall pressure incrases up to 30:1. The efficiency in an axial flow compressor is higher than the centrifugal compressor. Pressure ratio per casing can be comparable with those of centrifugal equipment, although flow rates are considerably higher for a given casing diameter because of the greater area of the flow path. The pressure ratio per stage is less than in a centrifugal compressor. The pressure ratio per stage in industrial compressors is between 1:05–1:15 per stage and for aeroturbines 1.1–1.2 per stage. The axial flow compressors used in gas turbines vary depending on the type of turbines. The industrial-type gas turbine has an axial flow compressor of a rugged construction. These units have blades that have low aspect ratio (R = blade height/blade chord) with minimum streamline curvation, and the shafts are support on sleeve-type bearings. The industrial gas turbine compressor has also a lower pressure ratio per stage (stage = rotor + stationary blade), giving a low blade loading. This also gives a larger operating range than its counterpart

(a) Fillet weld

(b) Full penetration fillet weld

(c) Butt weld

(d) Slot weld

(e) Electron beam weld

(f) Machined

(g) Riveted FIG. 10-76

(h) Riveted

(i) Riveted

10-49

(j) Riveted

Several fabrication techniques for centrifugal impellers.

the aero axial gas turbine compressor but considerably less than the centrifugal compressor. The axial flow compressors in aero gas turbines are heavily loaded. The aspect ratio of the blades, especially the first few stages, can be as high as 4.0, and the effect of streamline curvature is substantial. The streamline configuration is a function of the annular passage area, the camber and thickness distribution of the blade, and the flow angles at the inlet and outlet of the blades. The shafts on these units are supported on antifriction bearings (roller or ball bearings). The operation of the axial flow compressor is a function of the rotational speed of the blades and the turning of the flow in the rotor. The stationary blades (stator) are used to diffuse the flow and convert the velocity increased in the rotor to a pressure increase. One rotor and one stator make up a stage in a compressor. One additional row of fixed blades (inlet guide vanes) is frequently used at the compressor inlet to ensure that air enters the first stage rotors at the desired angle. In addition to the stators, another diffuser at the exit of the compressor further diffuses the gas and, in the case of gas turbines, controls its velocity entering the combustor. The axial flow compressor has a much smaller operating range “Surge to Choke” than its counterpart in the centrifugal compressor. Because of the steep characteristics of the head/flow capacity curve, the surge point is usually within 10 percent of the design point. The axial flow compressor has three distinct stall phenomena. Rotating stall and individual blade stall are aerodynamic phenomena. Stall flutter is an aeroelastic phenomenon. Rotating stall (propagating stall) consists of large stall zones covering several blade passages and propagates in the direction of the rotor and at some fraction of rotor speed. The number of stall zones and the propagating rates vary considerably. Rotating stall is the most prevalent type of stall phenomenon. Individual blade stall occurs when all the blades around the compressor annulus stall simultaneously without the occurrence of the stall propagation mechanism. The phenomena of stall flutter is caused by self-excitation of the blade and is aeroelastic. It must be distinguished from classic flutter, since classic flutter is a coupled torsional-flexural vibration that occurs when the freestream velocity over an airfoil section reaches a certain critical velocity. Stall flutter, on the other hand, is a phenomenon that occurs due to the stalling of the flow around a blade. Blade stall causes Karman vortices in the airfoil wake. Whenever the frequency of the vortices coincides with the natural frequency of airfoil, flutter will occur. Stall flutter is a major cause of compressor-blade failure.

10-50

TRANSPORT AND STORAGE OF FLUIDS

FIG. 10-77

Axial-flow compressor. (Courtesy of Allis-Chalmers Corporation.)

Positive Displacement Compressors Positive displacement compressors are machines that are essentially constant volume machines with variable discharge pressures. These machines can be divided into two types: 1. Rotary compressors 2. Reciprocating compressors Many users consider rotary compressors, such as the “Rootes”-type blower, as turbomachines because their behavior in terms of the rotor dynamics is very close to centrifugal and axial flow machinery. Unlike the reciprocating machines, the rotary machines do not have a very high vibration problem but, like the reciprocating machines, they are positive displacement machines. Rotary Compressors Rotary compressors are machines of the positive-displacement type. Such units are essentially constantvolume machines with variable discharge pressure. The volume can be varied only by changing the speed or by bypassing or wasting some of the capacity of the machine. The discharge pressure will vary with the resistance on the discharge side of the system. A characteristic curve typical of the form produced by these rotary units is shown in Fig. 10-78. Rotary compressors are generally classified as of the straight-lobe type, screw type, sliding-vane type, and liquid-piston type. Straight-Lobe Type This type is illustrated in Fig. 10-79. Such units are available for pressure differentials up to about 83 kPa (12 lbf/in2) and capacities up to 2.549 × 104 m3/h (15,000 ft3/min). Sometimes multiple units are operated in series to produce higher pressures; individual-stage pressure differentials are limited by the shaft deflection, which must necessarily be kept small to maintain rotor and casing clearance.

Screw-Type This type of rotary compressor, as shown in Fig. 10-80, is capable of handling capacities up to about 4.248 × 104 m3/h (25,000 ft3/min) at pressure ratios of 4:1 and higher. Relatively smalldiameter rotors allow rotative speeds of several thousand rev/min. Unlike the straight-lobe rotary machine, it has male and female rotors whose rotation causes the axial progression of successive sealed cavities. These machines are staged with intercoolers when such an

10-78 Approximate performance curves for a rotary positivedisplacement compressor. The safety valve in discharge line or bypass must be set to operate at a safe value determined by construction. FIG.

PUMPING OF LIQUIDS AND GASES

FIG. 10-79

Two-impeller type of rotary positive-displacement blower.

arrangement is advisable. Their high-speed operation usually necessitates the use of suction- and discharge-noise suppressors. The bearings used are sleeve-type bearings. Due to the side pressures experienced, tilting pad bearings are highly recommended. Sliding-Vane Type This type is illustrated in Fig. 10-81. These units are offered for operating pressures up to 0.86 MPa (125 lbf/in2) and in capacities up to 3.4 × 103 m3/h (2000 ft3/min). Generally, pressure ratios per stage are limited to 4:1. Lubrication of the vanes is required, and the air or gas stream therefore contains lubricating oil.

FIG. 10-80

Screw-type rotary compressor.

FIG. 10-81

Sliding-vane type of rotary compressor.

10-51

Liquid-Piston Type This type is illustrated in Fig. 10-82. These compressors are offered as single-stage units for pressure differentials up to about 0.52 MPa (75 lbf/in2) in the smaller sizes and capacities up to 6.8 × 103 m3/h (4000 ft3/min) when used with a lower pressure differential. Staging is employed for higher pressure differentials. These units have found wide application as vacuum pumps on wet-vacuum service. Inlet and discharge ports are located in the impeller hub. As the vaned impeller rotates, centrifugal force drives the sealing liquid against the walls of the elliptical housing, causing the air to be successively drawn into the vane cavities and expelled against discharge pressure. The sealing liquid must be externally cooled unless it is used in a once-through system. A separator is usually employed in the discharge line to minimize carryover of entrained liquid. Compressor capacity can be considerably reduced if the gas is highly soluble in the sealing liquid. The liquid-piston type of compressor has been of particular advantage when hazardous gases are being handled. Because of the gasliquid contact and because of the much greater liquid specific heat, the gas-temperature rise is very small. Reciprocating Compressors Reciprocating compressors are used mainly when high-pressure head is required at a low flow. Reciprocating compressors are furnished in either single-stage or multistage types. The number of stages is determined by the required compressor ratio p2 /p1. The compression ratio per stage is generally limited to 4, although low-capacity units are furnished with compression ratios of 8 and even higher. Generally, the maximum compression ratio is determined by the maximum allowable discharge-gas temperature. Single-acting air-cooled and water-cooled air compressors are available in sizes up to about 75 kW (100 hp). Such units are available in one, two, three, or four stages for pressure as high as 24 MPa (3500 lbf/in2). These machines are seldom used for gas compression because of the difficulty of preventing gas leakage and contamination of the lubricating oil. The compressors most commonly used for compressing gases have a crosshead to which the connecting rod and piston rod are connected. This provides a straight-line motion for the piston rod and permits simple packing to be used. Figure 10-83 illustrates a simple single-stage machine of this type having a double-acting piston. Either single-acting (Fig. 10-84) or double-acting pistons (Fig. 10-85) may be used, depending on the size of the machine and the number of stages. In some machines double-acting pistons are used in the first stages and single-acting in the later stages. On multistage machines, intercoolers are provided between stages. These heat exchangers remove the heat of compression from the gas and reduce its temperature to approximately the temperature existing at the compressor intake. Such cooling reduces the volume of gas going to the high-pressure cylinders, reduces the power required for compression, and keeps the temperature within safe operating limits. Figure 10-86 illustrates a two-stage compressor end such as might be used on the compressor illustrated in Fig. 10-83.

FIG. 10-82

Liquid-piston type of rotary compressor.

10-52

TRANSPORT AND STORAGE OF FLUIDS

FIG. 10-83

Typical single-stage, double-acting water-cooled compressor.

FIG. 10-84 Two-stage single-acting opposed piston in a single step-type cylinder.

Compressors with horizontal cylinders such as illustrated in Figs. 10-83 to 10-86 are most commonly used because of their accessibility. However, machines are also built with vertical cylinders and other arrangements such as right-angle (one horizontal and one vertical cylinder) and V-angle. Compressors up to around 75 kW (100 hp) usually have a single center-throw crank, as illustrated in Fig. 10-83. In larger sizes compressors are commonly of duplex construction with cranks on each end of the shaft (see Fig. 10-87). Some large synchronous motordriven units are of four-corner construction; i.e., they are of doubleduplex construction with two connecting rods from each of the two crank throws (see Fig. 10-88). Steam-driven compressors have one or more steam cylinders connected directly by piston rod or tie rods to the gas-cylinder piston or crosshead. Valve Losses Above piston speeds of 2.5 m/s (500 ft/min), suction and discharge valve losses begin to exert significant effects on the actual internal compression ratio of most compressors, depending on the valve port area available. The obvious results are high temperature rise and higher power requirements than might be expected. These effects become more pronounced with higher-molecular-weight gases. Valve problems can be a very major contributor to down time experienced by these machines. Control Devices In many installations the use of gas is intermit-

FIG. 10-85

Typical double-acting compressor piston and cylinder.

tent, and some means of controlling the output of the compressor is therefore necessary. In other cases constant output is required despite variations in discharge pressure, and the control device must operate to maintain a constant compressor speed. Compressor capacity, speed, or pressure may be varied in accordance with requirements. The nature of the control device will depend on the function to be regulated. Regulation of pressure, volume, temperature, or some other factor determines the type of regulation required and the type of the compressor driver. The most common control requirement is regulation of capacity. Many capacity controls, or unloading devices, as they are usually termed, are actuated by the pressure on the discharge side of the compressor. A falling pressure indicates that gas is being used faster than it is being compressed and that more gas is required. A rising pressure indicates that more gas is being compressed than is being used and that less gas is required. An obvious method of controlling the capacity of a compressor is to

PUMPING OF LIQUIDS AND GASES

FIG. 10-86

10-53

Two-stage double-acting compressor cylinders with intercooler. FIG. 10-87

vary the speed. This method is applicable to units driven by variablespeed drivers such as steam pistons, steam turbines, gas engines, diesel engines, etc. In these cases the regulator actuates the steamadmission or fuel-admission valve on the compressor driver and thus controls the speed. Motor-driven compressors usually operate at constant speed, and other methods of controlling the capacity are necessary. On reciprocating compressors discharging into receivers, up to about 75 kW (100 hp), two types of control are usually available. These are automatic-start-and-stop control and constant-speed control. Automatic-start-and-stop control, as its name implies, stops or starts the compressor by means of a pressure-actuated switch as the gas demand varies. It should be used only when the demand for gas will be intermittent. Constant-speed control should be used when gas demand is fairly constant. With this type of control, the compressor runs continuously but compresses only when gas is needed. Three methods of unloading the compressor with this type of control are in common use: (1) closed suction unloaders, (2) open inlet-valve unloaders, and (3) clearance unloaders. The closed suction unloader consists of a pressure-actuated valve which shuts off the compressor intake. Open inlet-valve unloaders (see Fig. 10-89) operate to hold the compressor inlet valves open and thereby prevent compression. Clearance unloaders (see Fig. 10-90) consist of pockets or small reservoirs which are opened when unloading is desired. The gas is compressed into

FIG. 10-88

Four-corner four-stage compressor (plan view).

Duplex two-stage compressor (plan view).

them on the compression stroke and reexpands into the cylinder on the return stroke, thus preventing the compression of additional gas. It is sometimes desirable to have a compressor equipped with both constant-speed and automatic-start-and-stop control. When this is done, a switch allows immediate selection of either type. Motor-driven reciprocating compressors above about 75 kW (100 hp) in size are usually equipped with a step control. This is in reality a variation of constant-speed control in which unloading is accomplished in a series of steps, varying from full load down to no load. Three-step control (full load, one-half load, and no load) is usually accomplished with inlet-valve unloaders. Five-step control (full load, three-fourths load, one-half load, one-fourth load, and no load) is accomplished by means of clearance pockets (see Fig. 10-91). On some machines, inlet-valve and clearance-control unloading are used in combination. Although such control devices are usually automatically operated, manual operation is satisfactory for some services. When manual operation is provided, it often consists of a valve or valves to open and close clearance pockets. In some cases, a movable cylinder head is provided for variable clearance in the cylinder (see Fig. 10-92). When no capacity control or unloading device is provided, it is necessary to provide bypasses between the inlet and discharge in order that the compressor can be started against no load (see Fig. 10-93). Nonlubricated Cylinders Most compressors use oil to lubricate

10-54

TRANSPORT AND STORAGE OF FLUIDS

FIG. 10-89

Inlet-valve unloader.

Actual indicator diagram of a two-stage compressor showing the operation of clearance control at five load points.

FIG. 10-91

FIG. 10-90

Clearance-control cylinder. (Courtesy of Ingersoll-Rand.)

the cylinder. In some processes, however, the slightest oil contamination is objectionable. For such cases a number of manufacturers furnish a “nonlubricated” cylinder (see Fig. 10-94). The piston on these cylinders is equipped with piston rings of graphitic carbon or Teflon* as well as pads or rings of the same material to maintain proper clearance between the piston and the cylinder. Plastic packing of a type that requires no lubricant is used on the stuffing box. Although oilwiper rings are used on the piston rod where it leaves the compressor frame, minute quantities of oil might conceivably enter the cylinder on the rod. If even such small amounts of oil are objectionable, an extended cylinder connecting piece can be furnished. This simply * ®Du Pont tetrafluoroethylene fluorocarbon resin.

lengthens the piston rod enough so that no portion of the rod can alternately enter the frame and the cylinder. In many cases, a small amount of gas leaking through the packing is objectionable. Special connecting pieces are furnished between the cylinder and the frame, which may be either single-compartment or double-compartment. These may be furnished gastight and vented back to the suction or filled with a sealing gas or fluid and held under a slight pressure. High-Pressure Compressors There is a definite trend in the chemical industry toward the use of high-pressure compressors with discharge pressures of from 34.5 to 172 MPa (5000 to 25,000 lbf/in2) and with capacities from 8.5 × 103 to 42.5 × 103 m3/h (5000 to 25,000 ft3/min). These require special design, and a complete knowledge of the characteristics of the gas is necessary. In most cases, these types of applications use the barrel-type centrifugal compressor. The gas usually deviates considerably from the perfect-gas laws, and in many cases temperature or other limitations necessitate a thor-

FIG. 10-92 Sectional view of a cylinder equipped with a hand-operated valve lifter on one end and a variable-volume clearance pocket at other end.

PUMPING OF LIQUIDS AND GASES

FIG. 10-95

FIG. 10-93 Bypass arrangement for a single-stage compressor. On multistage machines, each stage is bypassed in a similar manner. Such an arrangement is necessary for no-load starting.

ough engineering study of the problem. These compressors usually have five, six, seven, or eight stages, and the cylinders must be properly proportioned to meet the various limitations involved and also to balance the load among the various stages. In many cases, scrubbing or other processing is carried on between stages. High-pressure cylinders are steel forgings with single-acting plungers (see Fig. 10-95). The compressors are usually designed so that the pressure load against the plunger is opposed by one or more single-acting pistons of the lower pressure stages. Piston-rod packing is usually of the segmental-ring metallic type. Accurate fitting and correct lubrication are very important. High-pressure compressor valves are designed for the

Piston equipped with carbon piston and wearing rings for a nonlubricated cylinder.

Forged-steel single-acting high-pressure cylinder.

conditions involved. Extremely high-grade engineering and skill are necessary. Piston-Rod Packing Proper piston-rod packing is important. Many types are available, and the most suitable is determined by the gas handled and the operating conditions for a particular unit. There are many types and compositions of soft packing, semimetallic packing, and metallic packing. In many cases, metallic packing is to be recommended. A typical low-pressure packing arrangement is shown in Fig. 10-96. A high-pressure packing arrangement is shown in Fig. 10-97. When wet, volatile, or hazardous gases are handled or when the service is intermittent, an auxiliary packing gland and soft packing are usually employed (see Fig. 10-98). Metallic Diaphragm Compressors (Fig. 10-99) These are available for small quantities [up to about 17 m3/h (10 ft3/min)] for compression ratios as high as 10:1 per stage. Temperature rise is not a

FIG. 10-96

FIG. 10-94

10-55

Typical packing arrangements for low-pressure cylinders.

FIG. 10-97 Typical packing arrangement, using metallic packing, for highpressure cylinders.

10-56

TRANSPORT AND STORAGE OF FLUIDS

FIG. 10-98

Soft packing in an auxiliary stuffing box for handling gases.

serious problem, as the large wall area relative to the gas volume permits sufficient heat transfer to approach isothermal compression. These compressors possess the advantage of having no seals for the process gas. The diaphragm is actuated hydraulically by a plunger pump. EJECTORS An ejector is a simplified type of vacuum pump or compressor which has no pistons, valves, rotors, or other moving parts. Figure 10-100 illustrates a steam-jet ejector. It consists essentially of a nozzle which discharges a high-velocity jet across a suction chamber that is connected to the equipment to be evacuated. The gas is entrained by the steam and carried into a venturi-shaped diffuser which converts the velocity energy into pressure energy. Figure 10-101 shows a largesized ejector, sometimes called a booster ejector, with multiple nozzles. Nozzles are devices in subsonic flow that have a decreasing area and accelerate the flow. They convert pressure energy to velocity energy. A minimum area is reached when velocity reaches sonic flow. In supersonic flow, the nozzle is an increasing area device. A diffuser in subsonic flow has an increasing area and converts velocity energy into pressure energy. A diffuser in supersonic flow has a decreasing area. Two or more ejectors may be connected in series or stages. Also, a

FIG. 10-100

Typical steam-jet ejector.

FIG. 10-101

Booster ejector with multiple steam nozzles.

number of ejectors may be connected in parallel to handle larger quantities of gas or vapor. Liquid- or air-cooled condensers are usually used between stages. Liquid-cooled condensers may be of either the direct-contact (barometric) or the surface type. By condensing vapor the load on the following stage is reduced, thus minimizing its size and reducing consumption of motive gas. Likewise, a precondenser installed ahead of an ejector reduces its size and consumption if the suction gas contains vapors that are condensable at the temperature condition available. An aftercondenser is frequently used to condense vapors from the final stage, although this does not affect ejector performance. Ejector Performance The performance of any ejector is a function of the area of the motive-gas nozzle and venturi throat, pressure of the motive gas, suction and discharge pressures, and ratios of specific heats, molecular weights, and temperatures. Figure 10-102, based on the assumption of constant-area mixing, is useful in evaluating single-stage-ejector performance for compression ratios up to 10 and area ratios up to 100 (see Fig. 10-103 for notation). For example,* assume that it is desired to evacuate air at 2.94 lbf/in2 with a steam ejector discharging to 14.7 lbf/in2 with available steam FIG. 10-99 High-pressure, low-capacity compressor having a hydraulically actuated diaphragm. (Pressure Products Industries.)

* All data are given in U.S. customary units since the charts are in these units. Conversion factors to SI units are given on the charts.

PUMPING OF LIQUIDS AND GASES

FIG. 10-102

10-57

Design curves for optimum single-stage ejectors. [DeFrate and Hoerl, Chem. Eng. Prog., 55, Symp. Ser. 21, 46 (1959).]

pressure of 100 lbf/in2. Entering the chart at p03 /p0b = 5.0, at p0b /p0a = 2.94/100 = 0.0294 the optimum area ratio is 12. Proceeding horizontally to the left, wb /wa is approximately 0.15 lb of air per 1 lb of steam. This value must be corrected for the temperature and molecularweight differences of the two fluids by Eq. (10-90). 苶苶M w/wa = wb/wa兹T 0a苶苶T b /苶苶M 0b苶苶a

(10-90)

In addition, there are empirical correction factors which should be applied. Laboratory tests show that for ejectors with constant-area mixing the actual entrainment and compression ratios will be approximately 90 percent of the calculated values and even less at very small values of p0b /p0a. This compensates for ignoring wall friction in the mixing section and irreversibilities in the nozzle and diffuser. In theory, each point on a given design curve of Fig. 10-102 is associated with an optimum ejector for prevailing operating conditions. Adjacent points on the same curve represent theoretically different ejectors for the new conditions, the difference being that for each ratio of p0b /p0a there is an optimum area for the exit of the motive-gas nozzle. In practice, however, a segment of a given curve for constant A2 /At represents the performance of a single ejector satisfactorily for estimating purposes, provided that the suction pressure lies within 20 to 130 percent of the design suction pressure and the motive pressure within 80 to 120 percent of design motive pressure. Thus the curves can be used to

select an optimum ejector for the design point and to estimate its performance at off-design conditions within the limits noted. Final ejector selection should, of course, be made with the assistance of a manufacturer of such equipment. Uses of Ejectors For the operating range of steam-jet ejectors in vacuum applications, see the subsection “Vacuum Systems.” The choice of the most suitable type of ejector for a given application depends upon the following factors: 1. Steam pressure. Ejector selection should be based upon the minimum pressure in the supply line selected to serve the unit. 2. Water temperature. Selection is based on the maximum water temperature. 3. Suction pressure and temperature. Overall process requirements should be considered. Selection is usually governed by the minimum suction pressure required (the highest vacuum). 4. Capacity required. Again overall process requirements should be considered, but selection is usually governed by the capacity required at the minimum process pressure. Ejectors are easy to operate and require little maintenance. Installation costs are low. Since they have no moving parts, they have long life, sustained efficiency, and low maintenance cost. Ejectors are suitable for handling practically any type of gas or vapor. They are also suitable for handling wet or dry mixtures or gases containing sticky or solid matter such as chaff or dust. Ejectors are available in many materials of construction to suit process requirements. If the gases or vapors are not corrosive, the diffuser is usually constructed of cast iron and the steam nozzle of stainless steel. For more corrosive gases and vapors, many combinations of materials such as bronze, various stainless-steel alloys, and other corrosion-resistant metals, carbon, and glass can be used. VACUUM SYSTEMS

FIG. 10-103

Notation for Fig. 10-102.

Figure 10-104 illustrates the level of vacuum normally required to perform many of the common manufacturing processes. The attainment of various levels is related to available equipment in Fig. 10-105. Vacuum Equipment The equipment shown in Fig. 10-105 has been discussed elsewhere in this section with the exception of the dif-

10-58

TRANSPORT AND STORAGE OF FLUIDS

FIG. 10-104

Vacuum levels normally required to perform common manufacturing processes. (Courtesy of Compressed Air

magazine.)

fusion pump. Figure 10-106 depicts a typical design. A liquid of low absolute vapor pressure is boiled in the reservoir. The vapor is ejected at high velocity in a downward direction through multiple jets and is condensed on the walls, which are cooled by the surrounding coils. Molecules of the gas being pumped enter the vapor stream and are driven downward by collisions with the vapor molecules. The gas molecules are removed through the discharge line by a backing pump such as a rotary oil-sealed unit. Diffusion pumps operate at very low pressures. The ultimate vacuum attainable depends somewhat upon the vapor pressure of the pump liquid at the temperature of the condensing surfaces. By providing a cold trap between the diffusion pump and the region being evacuated, pressures as low as 10−7 mmHg absolute are achieved in

this manner. Liquids used for diffusion pumps are mercury and oils of low vapor pressure. Silicone oils have excellent characteristics for this service. SEALING OF ROTATING SHAFTS Seals are very important and often critical components in large rotating machinery especially on high-pressure and high-speed equipment. The principal sealing systems used between the rotor and stationary elements fall into two main categories: (1) noncontacting seals and (2) face seals. These seals are an integral part of the rotating system, they affect the dynamic operating characteristics of the machine. The stiffness and damping factors will be changed by the seal geometry and

PUMPING OF LIQUIDS AND GASES

FIG. 10-105

10-59

Vacuum levels attainable with various types of equipment. (Courtesy of Compressed Air magazine.)

pressures. In operation the rotating shafts have both radial and axial movement. Therefore any seal must be flexible and compact to ensure maximum sealing minimum effect on rotor dynamics. Noncontact Seals Noncontact seals are used extensively in gas

FIG. 10-106 Typical diffusion pump. (Courtesy of Compressed Air magazine.)

service in high speed rotating equipment. These seals have good mechanical reliability and minimum impact on the rotor dynamics of the system. They are not positive sealing. There are two types of noncontact seals: (1) labyrinth seals and (2) ring seals. Labyrinth Seals The labyrinth is one of the simplest of the many sealing devices. It consists of a series of circumferential strips of metal extending from the shaft or from the bore of the shaft housing to form a cascade of annular orifices. Labyrinth seal leakage is greater than that of clearance bushings, contact seals, or filmriding seals. The major advantages of labyrinth seals are their simplicity, reliability, tolerance to dirt, system adaptability, very low shaft power consumption, material selection flexibility, minimal effect on rotor dynamics, back diffusion reduction, integration of pressure, lack of pressure limitations, and tolerance to gross thermal variations. The major disadvantages are the high leakage, loss of machine efficiency, increased buffering costs, tolerance to ingestion of particulates with resulting damage to other critical items such as bearings, the possibility of the cavity clogging due to low gas velocities or back diffusion, and the inability to provide a simple seal system that meets OSHA or EPA standards. Because of some of the foregoing disadvantages, many machines are being converted to other types of seals. Labyrinth seals are simple to manufacture and can be made from conventional materials. Early designs of labyrinth seals used knifeedge seals and relatively large chambers or pockets between the knives. These relatively long knives are easily subject to damage. The modern, more functional, and more reliable labyrinth seals consist of sturdy, closely spaced lands. Some labyrinth seals are shown in Fig. 10-107. Figure 10-107a is the simplest form of the seal. Figure

10-60

TRANSPORT AND STORAGE OF FLUIDS

10-107b shows a grooved seal; it is more difficult to manufacture but produces a tighter seal. Figures 10-107c and 10-107d are rotating labyrinth-type seals. Figure 10-107e shows a simple labyrinth seal with a buffered gas for which pressure must be maintained above the process gas pressure and the outlet pressure (which can be greater than or less than the atmospheric pressure). The buffered gas produces a fluid barrier to the process gas. The eductor sucks gas from the vent near the atmospheric end. Figure 10-107f shows a buffered, stepped labyrinth. The step labyrinth gives a tighter seal. The match-

Low pressure

High pressure Leakage path

Stationary

Rotating surface (steel)

(a) Simplest design. (Labyrinth materials: aluminum, bronze, babbitt or steel)

Low pressure

Stationary sleeve

High pressure

ing stationary seal is usually manufactured from soft materials such as babbitt or bronze, while the stationary or rotating labyrinth lands are made from steel. This composition enables the seal to be assembled with minimal clearance. The lands can therefore cut into the softer materials to provide the necessary running clearances for adjusting to the dynamic excursions of the rotor. To maintain maximum sealing efficiency, it is essential that the labyrinth lands maintain sharp edges in the direction of the flow. Leakage past these labyrinths is approximately inversely propor-

Low pressure

Stationary

Rotating surface (steel)

High pressure Leakage path

(b) More difficult to manufacture but produces a tighter seal. (Same material as in a.)

Low pressure

High pressure Stationary sleeve

Leakage path

Rotating surface

Rotating surface (steel) (c) Rotating labyrinth type, before operation. (Sleeve material: babbitt, aluminum, nonmetallic or other soft materials)

(d) Rotating labyrinth, after operation. Radial and axial movement of rotor cuts grooves in sleeve material to simulate staggered type shown in b.

, , , , , ,, ,,, ,, ,,,,, Buffered (or barrier) inlet Pb

Process Pp

Inlet Pi

Atmosphere Pa

Process Pp

Outlet Po

Atmosphere Pa

Straight Stepped labyrinth labyrinth Shaft

(e) Buffered combination labyrinth FIG. 10-107

Various configurations of labyrinth seals.

(f) Buffered-vented straight labyrinth

PUMPING OF LIQUIDS AND GASES tional to the square root of the number of labyrinth lands. This translates into the following relationship if leakage is to be cut in half in a four labyrinth seal: The number of labyrinths would have to be increased to 16. The Elgi leakage formula can be modified and written as: (g/ Vo)(Po − Pn) mᐉ = AK ᎏᎏ n + ln (Pn /Po)

冤

冥

1/2

(10-91)

where mᐉ = leakage A = leakage Area of single throttling K = labyrinth constant (K = .9 for straight labyrinths, K = .75 for staggered labyrinths) Po = absolute pressure before the labyrinth Pn = absolute pressure after the last labyrinth Vo = specific volume before the labyrinth n = number of lands The leakage of a labyrinth seal can be kept to a minimum by providing: (1) minimum clearance between the seal lands and the seal sleeve, (2) sharp edges on the lands to reduce the flow discharge coefficient, and (3) grooves or steps in the flow path for reducing dynamic head carryover from stage to stage. The labyrinth sleeve can be flexibly mounted to permit radial motion for self-aligning effects. In practice, a radial clearance of under 0.008 is difficult to achieve. Ring Seals The restrictive ring seal is essentially a series of sleeves in which the bores form a small clearance around the shaft. Thus, the leakage is limited by the flow resistance in the restricted area and controlled by the laminar or turbulent friction. There are two types of ring seals: (1) fixed seal rings and (2) floating seal rings. The floating rings permit a much smaller leakage; they can be either the segmented type as shown in Fig. 10-108a or the rigid type as shown in Fig. 10-108b. Fixed Seal Rings The fixed-seal ring consists of a long sleeve affixed to a housing in which the shaft rotates with small clearances. Long assemblies must be used to keep leakage within a reasonable limit. Long seal assemblies aggravate alignment and rubbing problems, thus requiring shafts to operate below their capacity. The fixed bushing seal operates with appreciable eccentricity and, combined with large clearances, produces large leakages, thus making this kind of seal impractical where leakage is undesirable. Floating Seal Rings Clearance seals that are free to move in a radial direction are known as floating seals. The floating characteris-

Metal chamber

Process gas (inlet) PL

,,, ,

Circumferential groove

Moderate clearance

Radial vent

°C

°F

38 93 149 177 218 260 371 552 829

100 200 300 350 425 500 700 1025 1525

Packing may not provide a completely leak-free seal. With shaft surface speeds less than approximately 2.5 m/s (500 ft/min), the packing may be adjusted to seal completely. However, for higher speeds some leakage is required for lubrication, friction reduction, and cooling. Application of Packing Coils and spirals are cut to form closed or nearly closed rings in the stuffing box. Clearance between ends should be sufficient to allow for fitting and possible expansion due to increased temperature or liquid absorption of the packing while in operation.

Segmented carbon seal

Floating-type restrictive ring seal.

a)

Extension spring

Segmented metal retainer Key

Small radial clearance

(a)

FIG. 10-108

Flax Cotton Duck and rubber Rubber Metallic (lead-based) Asbestos 1003 Asbestos 204 Metallic (aluminum-based) Metallic (copper-based)

Flange gasket

Atmosphere (outlet) Po End plate

tics permit them to move freely, thus avoiding severe rubs. Due to differential thermal expansion between the shaft and bushing, the bushings should be made of material with a higher coefficient of thermal expansion. This is achieved by shrinking the carbon into a metallic retaining ring with a coefficient of expansion that equals or exceeds that of the shaft material. It is advisable in high shearing applications to lock the bushings against rotation. Buildup of dirt and other foreign material lodged between the seal ring and seat will create an excessive spin and damage on the floating seal ring unit. It is therefore improper to use soft material such as babbitt and silver as seal rings. Packing Seal A common type of rotating shaft seal consists of packing composed of fibers which are first woven, twisted, or braided into strands and then formed into coils, spirals, or rings. To ensure initial lubrication and to facilitate installation, the basic materials are often impregnated. Common materials are asbestos fabric, braided and twisted asbestos, rubber and duck, flax, jute, and metallic braids. The so-called plastic packings can be made up with varying amounts of fiber combined with a binder and lubricant for high-speed applications. Maximum temperatures that base materials of packings withstand and still give good service are as follows:


, ,

,,

,  Seal housing

10-61

Outlet

Bearing

, , , ,,, ,, ,, , , ,, ,, ,,,, ,, ,,, ,, ,, ,, ,,,,, ,,,,,,, ,, ,, Buffer inlet Pa

Sensing pressure Ps

Process Pp

Po

(b)

10-62

TRANSPORT AND STORAGE OF FLUIDS

The correct form of the ring joint depends on materials and service requirements. Braided and flexible metallic packings usually have butt or square joints (Fig. 10-109a). With other packing material, service experience indicates that rings cut with bevel or skive joints (Fig. 10-109b) are more satisfactory. A slight advantage of the bevel joint over the butt joint is that the bevel permits a certain amount of sliding action, thus absorbing a portion of ring expansion. In the manufacture of packings, the proper grade and type of lubricant is usually impregnated for each service for which the packing is recommended. However, it may be desirable to replenish the lubricant during the normal life of the packing. Lack of lubrication causes packing to become hard and lose its resiliency, thus increasing friction, shortening packing life, and increasing operating costs. An effective auxiliary device frequently used with packing and rotary shafts is the seal cage (or lantern ring), shown in Fig. 10-110. The seal cage provides an annulus around the shaft for the introduction of a lubricant, oil, grease, etc. The seal cage is also used to introduce liquid for cooling, to prevent the entrance of atmospheric air, or to prevent the infiltration of abrasives from the process liquid. The chief advantage of packing over other types of seals is the ease with which it can be adjusted or replaced. Most equipment is designed so that disassembly of major components is not required to

(a)

(b) FIG. 10-109

Butt (a) and skive (b) joints for compression packing rings.

FIG. 10-110

Seal cage or lantern ring. (Courtesy of Crane Packing Co.)

remove or add packing rings. The major disadvantages of a packingtype seal are (1) short life, (2) requirement for frequent adjustment, and (3) need for some leakage to provide lubrication and cooling. Mechanical Face Seals This type of seal forms a running seal between flat precision-finished surfaces. It is an excellent seal against leakages. The sealing surfaces are planes perpendicular to the rotating shaft, and the forces that hold the contact faces are parallel to the shaft axis. For a seal to function properly, there are four sealing points: 1. Stuffing box face 2. Leakage down the shaft 3. Mating ring in the gland plate 4. Dynamic faces Mechanical Seal Selection There are many factors that govern the selection of seals. These factors apply to any type of seal: 1. Product 2. Seal environment 3. Seal arrangement 4. Equipment 5. Secondary packing 6. Seal face combinations 7. Seal gland plate 8. Main seal body Product Physical and chemical properties of the liquid or gas being sealed places constraints on the type of material, design, and arrangement of the seal. Pressure. Pressure affects the choice of material and whether balanced or unbalanced seal design can be used. Most unbalanced seals are good up to 100 psig stuffing box pressure. Over 100 psig, balanced seals should be used. Temperature. The temperature of the liquid being pumped is important because it affects the seal face material selection as well as the wear life of the seal face. Lubricity. In any mechanical seal design, there is rubbing motion between the dynamic seal faces. This rubbing motion is often lubricated by the fluid being pumped. Most seal manufacturers limit the speed of their seals to 90 ft/sec (30 m/sec). This is primarily due to centrifugal forces acting on the seal, which tends to restrict the seal’s axial flexibility. Abrasion. If there are entrained solids in the liquid, it is desirable to have a flushed single inside type with a face combination of very hard material. Corrosion. This affects the type of seal body: what spring material, what face material, and what type of elastomer or gasket material. The corrosion rate will affect the decision of whether to use a single or multiple spring design because the spring can usually tolerate a greater amount of corrosion without weakening it appreciably. Seal Environment The design of the seal environment is based on the product and the four general parameters that regulate it: 1. Pressure control 2. Temperature control 3. Fluid replacement 4. Atmospheric air elimination Seal Arrangement There are four types of seal arrangements: 1. Double seals are standard with toxic and lethal products, but maintenance problems and seal design contribute to poor reliability. The double face-to-face seal may be a better solution. 2. Do not use a double seal in dirty service—the inside seal will hang up. 3. API standards for balanced and unbalanced seals are good guidelines; too low a pressure for a balanced seal may encourage face lift-off. 4. Arrangement of the seal will determine its success more than the vendor. Over 100 arrangements are available. Equipment The geometry of the pump or compressor is very important in seal effectiveness. Different pumps with the same shaft diameter and the total differential head can present different sealing problems. Secondary Packing Much more emphasis should be placed on secondary packing especially if Teflon is used. A wide variation in performance is seen between various seal vendors, depending on seal arrangement there can be difference in mating ring packing.

PUMPING OF LIQUIDS AND GASES Seal Face Combinations The dynamic of seal faces is better understood today. Seal-face combinations have come a long way in the past 8–10 years. Stellite is being phased out of the petroleum and petrochemical applications. Better grades of ceramic are available, cost of tungsten has come down, and relapping of tungsten are available near most industrial areas. Silicon carbide is being used in abrasive service. Seal Gland Plate The seal gland plate is caught in between the pump vendor and the seal vendor. Special glands should be furnished by seal vendors, especially if they require heating, quenching, and drain with a floating-throat bushing. Gland designs are complex and may have to be revisited, especially if seals are changed. Main Seal Body The term seal body makes reference to all rotating parts on a pusher seal, excluding shaft packing and seal ring. In many cases it is the chief reason to avoid a particular design for a particular service. Basically, most mechanical seals have the following components as seen in Fig. 10-111. 1. Rotating seal ring 2. Stationary seal ring 3. Spring devices to provide pressure 4. Static seals A loading device such as a spring is needed to ensure that in the event of loss or hydraulic pressure the sealing surfaces are kept closed. The amount of the load on the sealing area is determined by the degree of “seal balance.” Figure 10-113 shows what seal balance means. A completely balanced seal is when the only force exerted on the sealing surfaces is the spring force; i.e., hydraulic pressure does not act on the sealing surface. The type of spring depends on the space available, loading characteristics, and the seal environment. Based on these considerations, either a single or multiple spring can be used. In small axial space, belleville springs, finger washers, or curved washers can be used. Shaft-sealing elements can be split up into two groups. The first type may be called pusher-type seals and includes the O-ring, V-ring, U-cup, and wedge configurations. Figure 10-116 shows some typical pusher-type seals. The second type is the bellow-type seals, which differ from the pusher-type seals in that they form a static seal between themselves and the shaft. Internal and External Seals Mechanical seals are classified broadly as internal or external. Internal seals (Fig. 10-112) are installed with all seal components exposed to the fluid sealed. The advantages of this arrangement are (1) the ability to seal against high pressure, since the hydrostatic force is normally in the same direction as the spring force; (2) protection of seal parts from external mechanical damage; and (3) reduction in the shaft length required. For high-pressure installations, it is possible to balance partially or fully the hydrostatic force on the rotating member of an internal seal by using a stepped shaft or shaft sleeve (Fig. 10-113). This method of

FIG. 10-111

Mechanical-seal components.

FIG. 10-112

10-63

Internal mechanical seal.

relieving face pressure is an effective way of decreasing power consumption and extending seal life. When abrasive solids are present and it is not permissible to introduce appreciable quantities of a secondary flushing fluid into the process, double internal seals are sometimes used (Fig. 10-114). Both sealing faces are protected by the flushing fluid injected between them even though the inward flow is negligible. External seals (Fig. 10-115) are installed with all seal components protected from the process fluid. The advantages of this arrangement are that (1) fewer critical materials of construction are required, (2) installation and setting are somewhat simpler because of the exposed position of the parts, and (3) stuffing-box size is not a limiting factor. Hydraulic balancing is accomplished by proper proportioning of the seal face and secondary seal diameters.

FIG. 10-113

Balanced internal mechanical seal.

FIG. 10-114

Internal bellows-type double mechanical seal.

10-64

TRANSPORT AND STORAGE OF FLUIDS

,,
,,, , , , , , ,, , ,, , ,, ,,, ,, ,, ,,, ,, , , ,, ,, ,, , , , , ,,, , ,
,,, ,,,, ,,,
 Seal oil

11

Suction gas chamber

4

Process gas

5

8

FIG. 10-115

External mechanical seal.

7

2

1

9

Throttle bushings (Fig. 10-117) are commonly used with single internal or external seals when solids are present in the fluid and the inflow of a flushing fluid is not objectionable. These close-clearance bushings are intended to serve as flow restrictions through which the maintenance of a small inward flow of flushing fluid prevents the entrance of a process fluid into the stuffing box. A typical complex seal utilizes both the noncontact and mechanical aspects of sealing. Figure 10-118 shows such a seal with its two major elements. This type of seal will normally have buffering via a labyrinth seal and a positive shutdown device. For shutdown, the carbon ring is tightly sandwiched between the rotating seal ring and the stationary sleeve with gas pressure to prevent gas from leaking out when no oil pressure is available. In operation seal oil pressure is about 30–50 psi over the process gas pressure. The high-pressure oil enters the top and completely fills the seal cavity. A small percentage is forced across the carbon ring seal faces. The rotative speed of the carbon ring can be anywhere between zero and full rotational speed. Oil crossing the seal faces contacts the process gas and therefore “contaminated oil.” The contaminated oil

FIG. 10-116

Various types of shaft-sealing elements.

1. Rotating carbon ring 2. Rotating seal ring 3. Stationary sleeve 4. Spring retainer 5. Spring 6. Gas and contaminated oil drain

FIG. 10-118

3

6

10

7. Floating babbitt-faced steel ring 8. Seal wiper ring 9. Seal oil drain line 10. Buffer gas injection port 11. Bypass orifice

Mechanical contact shaft seal.

leaves through the contaminated oil drain to a degassifier for purification. The majority of the oil flows through the uncontaminated seal oil drain. Materials Springs and other metallic components are available in a wide variety of alloys and are usually selected on the basis of temperature and corrosion conditions. The use of a particular mechanical seal is frequently restricted by the temperature limitations of the organic materials used in the static seals. Most elastomers are limited to about 121°C (250°F). Teflon will withstand temperatures of 260°C (500°F) but softens appreciably above 204°C (400°F). Glass-filled Teflon is dimensionally stable up to 232 to 260°C (450 to 500°F). One of the most common elements used for seal faces is carbon. Although compatible with most process media, carbon is affected by strong oxidizing agents, including fuming nitric acid, hydrogen chloride, and high-temperature air [above 316°C (600°F)]. Normal mating-face materials for carbon are tungsten or chromium carbide, hard steel, stainless steel, or one of the cast irons. Other sealing-face combinations that have been satisfactory in corrosive service are carbide against carbide, ceramic against ceramic, ceramic against carbon, and carbon against glass. The ceramics have also been mated with the various hard-facing alloys. When selecting seal materials the possibility of galvanic corrosion must also be considered. BEARINGS

FIG. 10-117

External mechanical seal and throttle bushing.

Many factors enter into the selection of the proper design for bearings. Some of these factors are: 1. Shaft speed range. 2. Maximum shaft misalignment that can be tolerated. 3. Critical speed analysis and the influence of bearing stiffness on this analysis 4. Loading of the compressor impellers 5. Oil temperatures and viscosity 6. Foundation stiffness 7. Axial movement that can be tolerated 8. Type of lubrication system and its contamination 9. Maximum vibration levels that can be tolerated

PUMPING OF LIQUIDS AND GASES Types of Bearings Figure 10-119 shows a number of different types of journal bearings. A description of a few of the pertinent types of journal bearings is given here: 1. Plain journal. Bearing is bored with equal amounts of clearance (on the order of one and one-half to two thousands of an inch per inch of journal diameter) between the journal and bearing. 2. Circumferential grooved bearing. Normally the oil groove is half the bearing length. This configuration provides better coolings but reduces load capacity by dividing the bearing into two parts. 3. Cylindrical bore bearings. Another common bearing type used in turbines. It has a split construction with two axial oil-feed grooves at the split. 4. Pressure or pressure dam. Used in many places where bearing stability is required, this bearing is a plain journal bearing with a pressure pocket cut in the unloaded half. This pocket is approximately h of an inch deep with a width 50 percent of the bearing length. This groove or channel covers an arc of 135° and terminates abruptly in a sharp-edge-edge dam. The direction or rotation is such that the oil is pumped down the channel toward the sharp edge. Pressure dam bearings are for one direction of rotation. They can be used in conjunction with cylindrical bore bearings as shown in Fig. 10-119. 5. Lemon bore or elliptical. This bearing is bored with shims split line, which are removed before installation. The resulting shape approximates an ellipse with the major axis clearance approximately

Bearing type

Load capacity

twice the minor axis clearance. Elliptical bearings are for both directions of rotation. 6. Three-lobe bearing. The three-lobe bearing is not commonly used in turbomachines. It has a moderate load-carrying capacity and can be operated in both directions. 7. Offset halves. In principle, this bearing acts very similar to a pressure dam bearing. Its load-carrying capacity is good. It is restricted to one direction of rotation. 8. Tilt-pad bearings. This bearing is the most common bearing type in today’s machines. It consists of several bearing pads posed around the circumference of the shaft. Each pad is able to tilt to assume the most effective working position. This bearing also offers the greatest increase in fatigue life because of the following advantages: • Thermal conductive backing material to dissipate heat developed in oil film. • A thin babbitt layer can be centrifugally cast with a uniform thickness of about 0.005 inch. Thick babbitts greatly reduce bearing life. Babbitt thickness in the neighborhood of .01 reduce the bearing life by more than half. • Oil film thickness is critical in bearing stiffness calculations. In a tilting-pad bearing, one can change this thickness in a number of ways: (a) change the number of pads; (b) direct the load on or in between the pads; (c) change the axial length of pad. Suitable direction of rotation

Resistance to half-speed whirl

Stiffness and damping

Worst

Moderate

Cylindrical bore Good

Cylindrical bore with dammed groove Good

Moderate

Good

Moderate

Lemon bore

Increasing Three lobe Moderate

Good

Good

Excellent

Offset halves

Tilting pad Moderate

FIG. 10-119

10-65

Comparison of general bearing types.

Best

Good

10-66

TRANSPORT AND STORAGE OF FLUIDS

Bearing type

Load capacity

Suitable Tolerance Tolerance direction of changing of Space of rotation load/speed misalignment requirement

Plain washer

Taper land Bidirectional

Unidirectional Tilting pad Bidirectional Unidirectional

FIG. 10-120

Poor

Good

Moderate

Compact

Moderate

Poor

Poor

Compact

Good

Poor

Poor

Compact

Good

Good

Good

Greater

Good

Good

Good

Greater

Comparison of thrust-bearing types.

The previous list contains some of the most common types of journal bearings. They are listed in the order of growing stability. All of the bearings designed for increased stability are obtained at higher manufacturing costs and reduced efficiency. The antiwhirl bearings all impose a parasitic load on the journal, which causes higher-power losses to the bearings and in turn requires higher oil flow to cool the bearing. Thrust Bearings The most important function of a thrust bearing is to resist the unbalanced force in a machine’s working fluid and to maintain the rotor in its position (within prescribed limits). A complete analysis of the thrust load must be conducted. As mentioned earlier, compressors with back-to-back rotors reduce this load greatly on thrust bearings. Figure 10-120 shows a number of thrust-bearing types. Plain, grooved thrust washers are rarely used with any continuous load, and their use tends to be confined to cases where the thrust load is very short duration or possibly occurs at standstill or low speed only. Occasionally, this type of bearing is used for light loads (less than 50 lb/in2), and in these circumstances the operation is probably hydrodynamic due to small distortions present in the nominally flat bearing surface. When significant continuous loads have to be taken on a thrust washer, it is necessary to machine into the bearing surface a profile to generate a fluid film. This profile can be either a tapered wedge or occasionally a small step. The tapered-land thrust bearing, when properly designed, can take and support a load equal to a tilting-pad thrust bearing. With perfect alignment, it can match the load of even a self-equalizing tilting-pad thrust bearing that pivots on the back of the pad along a radial line. For variable-speed operation, tilting-pad thrust bearings as shown in Fig. 10-121 are advantageous when compared to conventional taperland bearings. The pads are free to pivot to form a proper angle for lubrication over a wide speed range. The self-leveling feature equalizes individual pad loadings and reduces the sensitivity to shaft misalignments that may occur during service. The major drawback of this bearing type is that standard designs require more axial space than a nonequalizing thrust bearing. The thrust-carrying capacity can be greatly improved by maintaining pad flatness and removing heat from the loaded zone. By the use of high thermal conductivity backing materials with proper thickness

,, ,, ,, , , ,, ,, ,, , , ,, ,,

Nonequalizing tilting pad thrust bearing with radial pivot (a)

(a)

FIG. 10-121

,,, ,, , ,,, ,, , ,,,

Nonequalizing tilting pad thrust bearing with ball pivot

(b)

(b)

,, ,, ,, , , ,, ,, ,, , , ,, ,, Self-equalizing thrust bearing ( c)

(c)

Various types of thrust bearings.

and proper support, the maximum continuous thrust limit can be increased to 1000 psi or more. This new limit can be used to increase either the factor of safety and improve the surge capacity of a given size bearing or reduce the thrust bearing size and consequently the losses generated for a given load. Since the higher thermal conductivity material (copper or bronze) is a much better bearing material than the conventional steel backing, it is possible to reduce the babbitt thickness to .010–.030 inch. Embedded thermocouples and RTDs will signal distress in the bearing if properly positioned. Temperature-monitoring systems have been found to be more accurate than axial-position indicators, which tend to have linearity problems at high temperatures.

PROCESS-PLANT PIPING

10-67

400 Steel back surface

Steel back 5 ′′ 16 below

250

Cu alloy back surface

Cu alloy back 5 ′′ below 16

200 Oil outlet 150 Oil inlet 100

0

FIG. 10-122

200

400

600 800 1000 Unit thrust load, PSI

1200

1400

17-inch bearing ∆ power loss, horsepower

Temperature, °F

300

300

15-inch bearing

200

100 12-inch bearing

Thrust-bearing temperature characteristics.

In a change from steel-backing to copper-backing, a different set of temperature limiting criteria should be used. Figure 10-122 shows a typical set of curves for the two backing materials. This chart also shows that drain oil temperature is a poor indicator of bearing operating conditions because there is very little change in drain oil temperature from low load to failure load. Thrust-Bearing Power Loss The power consumed by various thrust bearing types is an important consideration in any system. Power losses must be accurately predicted so that turbine efficiency can be computed and the oil supply system properly designed. Figure 10-123 shows a typical power consumption in thrust bearings as a function of unit speed. The total power loss is usually about 0.8–10 percent of the total rate power of the unit. New vector lube bearings reduce the horsepower loss by as much as 30 percent. In large vertical pumps, thrust bearings take not only the load caused by

0

0

2

4 10 6 8 Shaft speed, RPM × 10–3

12

FIG. 10-123 Difference in total-power-loss data test minus catalog frictional losses versus shaft speed for 6 × 6 pad double-element thrust bearings.

the fluid but also the load caused by the weight of the entire assembly (shaft and impellers). In some large pumps these could be about 60 ft (20 m) high and weigh 16 tons. The thrust bearing for such a pump is over 5 ft (1.7 meters) in diameter with each thrust pad weighing over 110 lb or (50 kg). In such cases, the entire pump assembly is first floated before the unit is started.

PROCESS-PLANT PIPING CODES AND STANDARDS Units: Pipe and Tubing Sizes and Ratings In this subsection pipe and tubing sizes are generally quoted in units of inches. To convert inches to millimeters, multiply by 25.4. Ratings are given in pounds. To convert pounds to kilograms, multiply by 0.454. Pressure-Piping Code The code for pressure piping (ANSI B31) consists of a number of sections which collectively constitute the code. Table 10-14 shows the status of the B31 code as of December 1980. The sections are published as separate documents for simplicity and convenience. The sections differ extensively. The Chemical Plant and Petroleum Refinery Piping Code (ANSI B31.3) is a section of ANSI B31. It was derived from a merging of the code groups for chemical-plant (B31.6) and petroleum-refinery (B31.3) piping into a single committee. Some of the significant requirements of ANSI B31.3, Petroleum Refinery Piping (1980 edition), are summarized in the following presentation, which is aimed primarily at welded and seamless construction. Where the word “code” is used in this subsection of the Handbook without other identification, it refers to the B31.3 section of ANSI B31. The code has been extensively quoted in this subsection of the Handbook with the permission of the publisher. The code is published by and copies are available from the American Society of Mechanical Engineers (ASME), 345 East 47th Street, New York, New York 10017. References to the ASME code are to the ASME Boiler and Pressure Vessel Code, also published by the American Society of Mechanical Engineers.

National Standards The American National Standards Institute (ANSI) and the American Petroleum Institute (API) have established dimensional standards for the most widely used piping components. Lists of these standards as well as specifications for pipe and fitting materials and testing methods of the American Society for Testing and Materials (ASTM), American Welding Society (AWS) specifications, and standards of the Manufacturers Standardization Society of the Valve and Fittings Industry (MSS) can be found in the ANSI B31 code sections. Many of these standards contain pressure-temperature ratings which will be of assistance to engineers in their design function. The use of published standards does not eliminate the need for engineering judgment. For example, although the code calculation formulas recognize the need to provide an allowance for corrosion, the standard rating tables for valves, flanges, fittings, etc., do not incorporate a corresponding allowance. The introduction to the code sets forth engineering requirements deemed necessary for the safe design and construction of piping systems. While safety is the basic consideration of the code, this factor alone will not necessarily govern final specifications for any pressure piping system. Designers are cautioned that the code is not a design handbook and does not do away with the need for competent engineering judgment. Governmental Regulations: OSHA Sections of the ANSI B31 code have been adopted with certain reservations or revisions by some state and local authorities as local codes. The specific requirements for piping systems in certain services have been promulgated as Occupational Safety and Health Act

10-68

TRANSPORT AND STORAGE OF FLUIDS

TABLE 10-14

Status of ANSI B31 Code for Pressure Piping

Standard number and designation B31.1.0

Scope and application

Remarks*

Power Piping

For all piping in steamgenerating stations

Latest issue: 1980

Fuel Gas Piping

For fuel gas for steamgenerating stations and industrial buildings

Latest issue: 1968

B31.3 Chemical Plant and Petroleum Refinery Piping

For all piping within the property limits of facilities engaged in the processing or handling of chemical, petroleum, or related products unless specifically excluded by the code

Latest issue: 1980

B31.4 Liquid Petroleum Transportation Piping Systems

For liquid crude or refined products in cross-country pipe lines

Latest issue: 1979

B31.5 Refrigeration Piping

For refrigeration piping in packaged units and commercial or public buildings

Latest issue: 1974

B31.7 Nuclear Power Piping

For fluids whose loss from the system could cause radiation hazard to plant personnel or the general public

Withdrawn; see ASME Boiler and Pressure Vessel Code, Sec. 3

B31.8 Gas Transmission and Distribution Systems

For gases in crosscountry pipe lines as well as for city distribution lines

Latest issue: 1975

B31.2

*Addenda are issued at intervals between publication of complete editions. Information on the latest issues can be obtained from the American Society of Mechanical Engineers, 345 East 47th Street, New York, N.Y. 10017.

(OSHA) regulations. These rules and regulations will presumably be revised and supplemented from time to time and may include specific requirements not contemplated in Sec. B31.3. CODE CONTENTS AND SCOPE The code prescribes minimum requirements for the materials, design, fabrication, assembly, support, erection, examination, inspection, and testing of piping systems subject to pressure or vacuum. The scope of the piping covered by B31.3 is illustrated in Fig. 10-124. It applies to all fluids including fluidized solids and to all services except as noted in the figure. Some of the more significant requirements of ANSI B31.3 (1980 edition) have been summarized and incorporated in this section of the Handbook. For a more comprehensive treatment of code requirements engineers are referred to the B31.3 code and the standards referenced therein. PIPE-SYSTEM MATERIALS The selection of material to resist deterioration in service is outside the scope of the B31.3 code (see Sec. 23). Experience has, however, resulted in the following material considerations extracted from the code with the permission of the publisher, the American Society of Mechanical Engineers, New York. General Considerations Considerations to be evaluated when selecting piping materials are (1) possible exposure to fire with respect to the loss of strength, degradation temperature, melting point, or combustibility of the pipe or support material; (2) ability of thermal insulation to protect the pipe from fire; (3) susceptibility of the pipe to brittle failure, possibly resulting in fragmentation hazards, or failure from thermal shock when exposed to fire or fire-fighting measures; (4) susceptibility of the piping material to crevice corrosion in stag-

nant confined areas (screwed joints) or adverse electrolytic effects if the metal is subject to contact with a dissimilar metal; (5) the suitability of packing, seals, gaskets, and lubricants or sealants used on threads as well as compatibility with the fluid handled; and (6) the refrigerating effect of a sudden loss of pressure on volatile fluids in determining the lowest expected service temperature. Specific Material Precautions Metals The following characteristics are to be evaluated when applying certain metals in piping: 1. Irons: cast, malleable, and high silicon (14.5 percent). Their lack of ductility and their sensitivity to thermal and mechanical shock. 2. Carbon steel and low- and intermediate-alloy steels a. The possibility of embrittlement when handling alkaline or strong caustic fluids. b. The possible conversion of carbides to graphite during longtime exposure to temperature above 427°C (800°F) of carbon steels, plain nickel steel, carbon-manganese steel, manganese-vanadium steel, and carbon-silicon steel. c. The possible conversion of carbides to graphite during longtime exposure to temperatures above 468°C (875°F) of carbonmolybdenum steel, manganese-molybdenum-vanadium steel, and chromium-vanadium steel. d. The advantages of silicon-killed carbon steel (0.1 percent silicon minimum) for temperatures above 480°C (900°F). e. The possibility of hydrogen damage when piping material is exposed to hydrogen or to aqueous acid solutions under certain temperature-pressure conditions. f. The possibility of deterioration when piping material is exposed to hydrogen sulfide. 3. High-alloy (stainless) steels a. The possibility of stress-corrosion cracking of austenitic stainless steels exposed to media such aa chlorides and other halides either internally or externally. The latter can result from improper selection or application of thermal insulation. b. The susceptibility to intergranular corrosion of austenitic stainless steels after sufficient exposure to temperatures between 427 and 871°C (800 and 1600°F) unless stabilized or low-carbon grades are used. c. The susceptibility to intercrystalline attack of austenitic stainless steels on contact with zinc or lead above their melting points or with many lead and zinc compounds at similarly elevated temperatures. d. The brittleness of ferritic stainless steels at room temperature after service at temperatures above 370°C (700°F). 4. Nickel and nickel-base alloys a. The susceptibility to grain boundary attack of nickel and nickelbase alloys not containing chromium when exposed to small quantities of sulfur at temperatures above 315°C (600°F). b. The susceptibility to grain boundary attack of nickel-base alloys containing chromium at temperatures above 595°C (1100°F) under reducing conditions and above 760°C (1400°F) under oxidizing conditions. c. The possibility of stress-corrosion cracking of nickel-copper alloy (70 Ni-30 Cu) in hydrofluoric acid vapor if the alloy is highly stressed or contains residual stresses from forming or welding. 5. Aluminum and aluminum alloys a. The compatibility with aluminum of thread compounds used in aluminum threaded joints to prevent seizing and galling. b. The possibility of corrosion from concrete, mortar, lime, plaster, or other alkaline materials used in buildings or other structures. c. The susceptibility of alloys 5154, 5087, 5083, and 5456 to exfoliation or intergranular attack; and the upper temperature limit of 65°C (150°F) to avoid such deterioration. 6. Copper and copper alloys a. The possibility of dezincification of brass alloys. b. The susceptibility to stress-corrosion cracking of copper-based alloys. c. The possibility of unstable acetylide formation when exposed to acetylene.

PROCESS-PLANT PIPING

10-69

FIG. 10-124 Scope of piping covered by the Chemical Plant and Petroleum Refinery Piping Code, ANSI B31.3 (From ASME Chemical Plant and Petroleum Refinery Piping Code, ANSI B31.3—1980; reproduced with permission of the publisher, the American Society of Mechanical Engineers, New York.)

10-70

TRANSPORT AND STORAGE OF FLUIDS

7. Titanium and titanium alloys. The possibility of deterioration of titanium and its alloys above 315°C (600°F). 8. Zirconium and zirconium alloys. The possibility of deterioration of zirconium and zirconium alloys above 315°C (600°F). 9. Tantalum. Above 300°C (570°F), the possibility of reactivity of tantalum with all gases except the inert gases. Below 300°C (570°F), the possibility of embrittlement of tantalum by nascent (monatomic) hydrogen (but not molecular hydrogen). Nascent hydrogen is produced by galvanic action or as a product of corrosion by certain chemicals. Nonmetals The following are specific considerations to be evaluated when applying certain nonmetals in piping: 1. Thermoplastics a. If thermoplastic piping is used above ground for compressed air or other compressed gases, special precautions should be observed. In determining the needed safeguarding for such services, the energetics and the specific failure mechanism need to be evaluated. Encasement of the plastic piping in shatter-resistant material may be considered. b. Table 10-15 lists recommended minimum and maximum temperature limits for thermoplastic pipe materials. c. Table 10-16 lists minimum and maximum temperature limits for thermoplastic materials used as nonpressure retaining linings. 2. Reinforced thermosetting resins. Table 10-17 lists the normally accepted maximum temperature limits for reinforced-thermosettingTABLE 10-15

Temperature Limits for Thermoplastic Pipe* Recommended temperature limits Minimum

Material (generic type) Acrylonitrile-butadiene-styrene (ABS) Cellulose acetate butyrate (CAB) Chlorinated polyether Polyacetal Polyethylene PE 1404 PE 2305 PE 2306 PE 3306 PE 3406 Polypropylene Poly (vinyl chloride) PVC 1120 PVC 1220 PVC 2110 PVC 2112 PVC 2116 PVC 2120 Chlorinated poly (vinyl chloride) (CPVC, 4120) Poly (vinylidene chloride) Poly (vinylidene fluoride) Nylon Polybutylene Poly (phenylene oxide) (POP 2125)

Maximum

°F

°C

°F

°C

−30 0 0 0

−34 −18 −18 −18

180 140 210 170

82 60 99 77

−30 −30 −30 −30 −30 30

−34 −34 −34 −34 −34 −01

100 120 140 160 180 210

38 49 60 71 82 99

0 0 0 0 0 0

−18 −18 −18 −18 −18 −18

150 150 130 130 150 150

66 66 54 54 66 66

0 40 0 −30 0

−18 4 −18 −34 −18

210 160 275 180 210

99 71 135 82 99

30

−01

210

99

*Extracted from the Chemical Plant and Petroleum Refinery Piping Code, ANSI B31.3–1980, with permission of the publisher, the American Society of Mechanical Engineers, New York. These recommendations are for low-pressure applications with water and other fluids that do not significantly affect the properties of the particular thermoplastic. The upper temperature limits are reduced at higher pressures, depending on the combination of fluid and expected service life. Lower temperature limits are affected more by installation, environment, and safeguarding than by strength. Because of low thermal conductivity, temperature gradients through the pipe wall may be substantial. Tabulated limits apply where more than half the wall thickness is at or above the stated temperature. These recommendations apply only to products covered by ASTM standards listed in Appendix A, Table 3, of the code. Manufacturers should be consulted for temperature limits on the specific types and kinds of plastic not covered by those ASTM standards.

TABLE 10-16 as Linings*

Temperature Limits for Thermoplastics Used

Material (generic type) Poly (tetrafluorethylene) Poly (fluorinated ethylene propylene) Poly (vinylidene chloride) Poly (vinylidene fluoride) Polypropylene Poly (perfluoroalkoxy)

Minimum temperature

Maximum temperature

°F

°C

°F

°C

−325

−198

500

260

−325

−198

400

204

0

−18

175

79

0 0 −325

−18 −18 −198

275 225 500

135 107 260

*Extracted from the Chemical Plant and Petroleum Refinery Piping Code, ANSI B31.3–1980, with permission of the publisher, the American Society of Mechanical Engineers. Listed temperature limits apply to lining material only. Rules for establishing temperature limits for components being listed are covered elsewhere in this code. These temperature limits are based on material tests and do not necessarily reflect evidence of successful use as piping-component linings at these temperatures. The designer should contact the manufacturer for specific applications, particularly as temperature limits are approached.

resin materials. The minimum recommended temperature is −29°C (−20°F) in all cases. 3. Asbestos cement. The normally accepted temperature limits for asbestos cement piping are −18°C (0°F) minimum and 93°C (200°F) maximum. 4. Borosilicate glass and impregnated graphite. Their lack of ductility and sensitivity to thermal and mechanical shock should be taken into account. METALLIC PIPE SYSTEMS: CARBON STEEL AND STAINLESS STEEL The ferrous-metal piping systems comprising wrought carbon and alloy steels including stainless steels are the most widely used and the most completely covered by national standards. Pipe and Tubing Pipe and tubing are divided into two main classes, seamless and welded. Seamless pipe, as a trade designation, refers to pipe made by forging a solid round, piercing it by simultaneously rotating and forcing it over a piercer point and further reducing it by rolling and drawing. However, seamless pipe and tubing are also produced by extrusion, casting into static or centrifugal molds, and by forging and boring. Seamless pipe has the same kilopascal (poundsforce per square inch) strength throughout the wall. Pierced seamless pipe frequently has the inside surface eccentric to the outside surface, resulting in nonuniform wall thickness. Welded pipe is made from rolled strips formed into cylinders and seam-welded by various methods. The welds are credited with 60 to 100 percent of the strength of the pipe wall depending on welding and inspection procedures. Larger diameters and lower ratios of wall thickness to diameter can be obtained in welded pipe than can be TABLE 10-17 Temperature Limits for Reinforced Thermosetting Resins*† Maximum temperature Material (generic type)

°C

°F

Epoxy, glass-fiber-reinforced Polyester, glass-fiber-reinforced Furan, glass-fiber-reinforced Furan, carbon-reinforced

149 93 93 93

300 200 200 200

*Extracted from the Chemical Plant and Petroleum Refinery Piping Code, ANSI B31.3–1980, with permission of the publisher, the American Society of Mechanical Engineers, New York. †Minimum recommended temperature of all materials is −29°C (−20°F).

PROCESS-PLANT PIPING obtained in seamless pipe (other than cast pipe). Uniform wall thickness is obtained. Hydrostatic testing does not reveal very short lengths of partially completed weld. This presents a possibility that small leaks may develop prematurely when corrosive fluids are being handled or the pipe is exposed to external corrosion. The weld must be taken into account in developing procedures for bending, flaring, and expanding the welded pipe. Additional thickness and additional size and wall-thickness combinations are available as tubing. Two common classifications of tubing are “pressure” and “mechanical.” Wall thickness (gauge) is specified as either “average wall” or “minimum wall.” Minimum wall is more costly than average wall, and because of closer wall-thickness and diametral tolerance, both gauge systems make pressure tubing more costly than pipe. However, average-wall carbon steel electricresistance-welded tubing, sizes 2r, 27⁄8, 3a, and 4a in outside diameter, produced from coiled strip on progressive forming rolls and electromagnetically rather than pressure-tested, competes vigorously with pipe. Table 10-18 gives standard size and wall-thickness combinations together with capacity and weight. Joints Pipe must be joined to pipe and to other components. Optimum design requires a minimum of assembly labor and provides the same resistance possessed by the pipe to (1) internal pressure as regards both rupture and leakage, (2) bending moments arising from spanning long distances between supports or from thermal expansion in piping containing offsets, (3) axial strain arising from internal pressure acting on changes in direction, blanks or closed valves, or thermal contraction in straight runs, and (4) rupture or leakage in event of fire. However, joints in pipe buried in the soil, where the position of each length and component is fixed, need provide the same resistance as the pipe to internal pressure only; in event of earth settlement, the joints may be required to yield to resulting bending moments without leakage. Also, in piping subject to thermal expansion and contraction, some joints may be required to yield to resulting bending moments and axial strains without leakage. The ideal pipe joint is free from changes in any dimension of the flow passage or direction of flow which would increase pressure drop or prevent complete drainage. It is free from crevices in which corrosion might be accelerated. It would require a minimum of labor to disassemble. Required frequency for disassembling the joint must be considered in making the selection. Generally speaking, joints which are easy to disassemble are deficient in one or more of the other requirements of the ideal joint. Most joints involve modifications of the components being joined; those with the desired modifications can usually be purchased. Welded Joints The most widely used joint in piping systems is the butt-weld joint (Fig. 10-125). In all ductile pipe metals which can be welded, pipe, elbows, tees, laterals, reducers, caps, valves, flanges, and V-clamp joints are available in all sizes and wall thicknesses with ends prepared for butt welding. Joint strength equal to the original pipe (except for work-hardened pipes which are annealed by the welding), unimpaired flow pattern, and generally unimpaired corrosion resistance more than compensate for the necessary careful alignment, skilled labor, and equipment required. Plain-end pipe used for socket-weld joints (Fig. 10-126) is available in all sizes, but fittings and valves with socket-weld ends are limited to sizes 3 in and smaller, for which the extra cost of the socket is outweighed by much easier alignment and less skill needed in welding. The joint is not so resistant to bending stress as the butt-welded joint but is otherwise equal, except that for some fluids the crevice between the pipe and the socket may promote corrosion. ANSI B16.11—1973,

10-71

Forged Steel Fittings, Socket-Welding and Threaded, requires that the wall thickness of the socket must be equal to or greater than 1.25 times the minimum pipe wall. Branch Welds These welds eliminate the purchase of tees and require no more weld metal than tees (Fig. 10-127). If the branch approaches the size of the run, careful end preparation of the branch pipe is required and the run pipe is weakened by the branch weld. See subsection “Pressure Design of Metallic Components: Wall Thickness” for rules for reinforcement. Reinforcing pads and fittings are commercially available. Use of the fittings facilitates visual inspection of the branch weld. See subsection “Welding, Brazing, or Soldering” for rules for welded joints. Threaded Joints Pipe with taper-pipe-thread ends (Fig. 10-128), per ANSI B2.1, is available 12 in and smaller, subject to minimumwall limitations. Fittings and valves with taper-pipe-thread ends are available in most pipe metals. Principal use of threaded joints is in sizes 2 in and smaller, in metals for which the most economically produced walls are thick enough to withstand considerable pressure and corrosion after reduction in thickness due to threading. For threaded joints over 2 in, assembly labor size and cost of tools increase rapidly. Careful alignment, required at the start of assembly and during rotation of the components, as well as variation in length produced by diametral tolerances in the threads, severely limits preassembly of the components. Threading is not a precise machining operation, and filler materials known as “pipe dope” are necessary to block the spiral leakage path.

(a)

(b)

(c) Branch welds. (a) Without added reinforcement. (b) With added reinforcement. (c) Angular branch.

FIG. 10-127 FIG. 10-125

Butt weld.

FIG. 10-126

Socket weld.

10-72

TRANSPORT AND STORAGE OF FLUIDS

TABLE 10-18

Properties of Steel Pipe

Nominal pipe size, in

Outside diameter, in

f

Cross-sectional area

Circumference, ft, or surface, ft2/ft of length

Schedule no.

Wall thickness, in

Inside diameter, in

Metal, in2

Flow, ft

Outside

0.405

10S 40ST, 40S 80XS, 80S

0.049 .068 .095

0.307 .269 .215

0.055 .072 .093

0.00051 .00040 .00025

0.106 .106 .106

d

0.540

10S 40ST, 40S 80XS, 80S

.065 .088 .119

.410 .364 .302

.097 .125 .157

.00092 .00072 .00050

r

0.675

10S 40ST, 40S 80XS, 80S

.065 .091 .126

.545 .493 .423

.125 .167 .217

a

0.840

5S 10S 40ST, 40S 80XS, 80S 160 XX

.065 .083 .109 .147 .188 .294

.710 .674 .622 .546 .464 .252

e

1.050

5S 10S 40ST, 40S 80XS, 80S 160 XX

.065 .083 .113 .154 .219 .308

1

1.315

5S 10S 40ST, 40S 80XS, 80S 160 XX

1d

1.660

1a

Capacity at 1-ft/s velocity

Inside

U.S. gal/ min

lb/h water

Weight of plain-end pipe, lb/ft

0.0804 .0705 .0563

0.231 .179 .113

115.5 89.5 56.5

0.19 .24 .31

.141 .141 .141

.107 .095 .079

.412 .323 .224

206.5 161.5 112.0

.33 .42 .54

.00162 .00133 .00098

.177 .177 .177

.143 .129 .111

.727 .596 .440

363.5 298.0 220.0

.42 .57 .74

.158 .197 .250 .320 .385 .504

.00275 .00248 .00211 .00163 .00117 .00035

.220 .220 .220 .220 .220 .220

.186 .176 .163 .143 .122 .066

1.234 1.112 0.945 0.730 0.527 0.155

617.0 556.0 472.0 365.0 263.5 77.5

.54 .67 .85 1.09 1.31 1.71

.920 .884 .824 .742 .612 .434

.201 .252 .333 .433 .572 .718

.00461 .00426 .00371 .00300 .00204 .00103

.275 .275 .275 .275 .275 .275

.241 .231 .216 .194 .160 .114

2.072 1.903 1.665 1.345 0.917 0.461

1036.0 951.5 832.5 672.5 458.5 230.5

0.69 0.86 1.13 1.47 1.94 2.44

.065 .109 .133 .179 .250 .358

1.185 1.097 1.049 0.957 0.815 0.599

.255 .413 .494 .639 .836 1.076

.00768 .00656 .00600 .00499 .00362 .00196

.344 .344 .344 .344 .344 .344

.310 .287 .275 .250 .213 .157

3.449 2.946 2.690 2.240 1.625 0.878

1725 1473 1345 1120 812.5 439.0

0.87 1.40 1.68 2.17 2.84 3.66

5S 10S 40ST, 40S 80XS, 80S 160 XX

.065 .109 .140 .191 .250 .382

1.530 1.442 1.380 1.278 1.160 0.896

0.326 0.531 0.668 0.881 1.107 1.534

.01277 .01134 .01040 .00891 .00734 .00438

.435 .435 .435 .435 .435 .435

.401 .378 .361 .335 .304 .235

5.73 5.09 4.57 3.99 3.29 1.97

2865 2545 2285 1995 1645 985

1.11 1.81 2.27 3.00 3.76 5.21

1.900

5S 10S 40ST, 40S 80XS, 80S 160 XX

.065 .109 .145 .200 .281 .400

1.770 1.682 1.610 1.500 1.338 1.100

0.375 0.614 0.800 1.069 1.429 1.885

.01709 .01543 .01414 .01225 .00976 .00660

.497 .497 .497 .497 .497 .497

.463 .440 .421 .393 .350 .288

7.67 6.94 6.34 5.49 4.38 2.96

3835 3465 3170 2745 2190 1480

1.28 2.09 2.72 3.63 4.86 6.41

2

2.375

5S 10S 40ST, 40S 80ST, 80S 160 XX

.065 .109 .154 .218 .344 .436

2.245 2.157 2.067 1.939 1.687 1.503

0.472 0.776 1.075 1.477 2.195 2.656

.02749 .02538 .02330 .02050 .01552 .01232

.622 .622 .622 .622 .622 .622

.588 .565 .541 .508 .436 .393

12.34 11.39 10.45 9.20 6.97 5.53

6170 5695 5225 4600 3485 2765

1.61 2.64 3.65 5.02 7.46 9.03

2a

2.875

5S 10S 40ST, 40S 80XS, 80S 160 XX

.083 .120 .203 .276 .375 .552

2.709 2.635 2.469 2.323 2.125 1.771

0.728 1.039 1.704 2.254 2.945 4.028

.04003 .03787 .03322 .02942 .02463 .01711

.753 .753 .753 .753 .753 .753

.709 .690 .647 .608 .556 .464

17.97 17.00 14.92 13.20 11.07 7.68

8985 8500 7460 6600 5535 3840

2.48 3.53 5.79 7.66 10.01 13.69

3

3.500

5S 10S 40ST, 40S 80XS, 80S 160 XX

.083 .120 .216 .300 .438 .600

3.334 3.260 3.068 2.900 2.624 2.300

0.891 1.274 2.228 3.016 4.213 5.466

.06063 .05796 .05130 .04587 .03755 .02885

.916 .916 .916 .916 .916 .916

.873 .853 .803 .759 .687 .602

27.21 26.02 23.00 20.55 16.86 12.95

13,605 13,010 11,500 10,275 8430 6475

3.03 4.33 7.58 10.25 14.32 18.58

3s

4.0

5S 10S 40ST, 40S 80XS, 80S

.083 .120 .226 .318

3.834 3.760 3.548 3.364

1.021 1.463 2.680 3.678

.08017 .07711 .06870 .06170

1.047 1.047 1.047 1.047

1.004 0.984 0.929 0.881

35.98 34.61 30.80 27.70

17,990 17,305 15,400 13,850

3.48 4.97 9.11 12.50

4

4.5

5S 10S 40ST, 40S 80XS, 80S

.083 .120 .237 .337

4.334 4.260 4.026 3.826

1.152 1.651 3.17 4.41

.10245 .09898 .08840 .07986

1.178 1.178 1.178 1.178

1.135 1.115 1.054 1.002

46.0 44.4 39.6 35.8

23,000 22,200 19,800 17,900

3.92 5.61 10.79 14.98

2

PROCESS-PLANT PIPING Properties of Steel Pipe (Continued )

TABLE 10-18

Nominal pipe size, in

10-73

Outside diameter, in

Schedule no. 120 160 XX

Cross-sectional area

Circumference, ft, or surface, ft2/ft of length

Wall thickness, in

Inside diameter, in

Metal, in2

Flow, ft

Outside

0.438 .531 .674

3.624 3.438 3.152

5.58 6.62 8.10

0.07170 .06647 .05419

1.178 1.178 1.178

2

Capacity at 1-ft/s velocity

Inside

U.S. gal/ min

lb/h water

Weight of plain-end pipe, lb/ft

0.949 0.900 0.825

32.2 28.9 24.3

16,100 14,450 12,150

19.00 22.51 27.54

5

5.563

5S 10S 40ST, 40S 80XS, 80S 120 160 XX

.109 .134 .258 .375 .500 .625 .750

5.345 5.295 5.047 4.813 4.563 4.313 4.063

1.87 2.29 4.30 6.11 7.95 9.70 11.34

.1558 .1529 .1390 .1263 .1136 .1015 .0900

1.456 1.456 1.456 1.456 1.456 1.456 1.456

1.399 1.386 1.321 1.260 1.195 1.129 1.064

69.9 68.6 62.3 57.7 51.0 45.5 40.4

34,950 34,300 31,150 28,850 25,500 22,750 20,200

6.36 7.77 14.62 20.78 27.04 32.96 38.55

6

6.625

5S 10S 40ST, 40S 80XS, 80S 120 160 XX

.109 .134 .280 .432 .562 .719 .864

6.407 6.357 6.065 5.761 5.501 5.187 4.897

2.23 2.73 5.58 8.40 10.70 13.34 15.64

.2239 .2204 .2006 .1810 .1650 .1467 .1308

1.734 1.734 1.734 1.734 1.734 1.734 1.734

1.677 1.664 1.588 1.508 1.440 1.358 1.282

100.5 98.9 90.0 81.1 73.9 65.9 58.7

50,250 49,450 45,000 40,550 36,950 32,950 29,350

7.60 9.29 18.97 28.57 36.39 45.34 53.16

8

8.625

5S 10S 20 30 40ST, 40S 60 80XS, 80S 100 120 140 XX 160

.109 .148 .250 .277 .322 .406 .500 .594 .719 .812 .875 .906

8.407 8.329 8.125 8.071 7.981 7.813 7.625 7.437 7.187 7.001 6.875 6.813

2.915 3.941 6.578 7.265 8.399 10.48 12.76 14.99 17.86 19.93 21.30 21.97

.3855 .3784 .3601 .3553 .3474 .3329 .3171 .3017 .2817 .2673 .2578 .2532

2.258 2.258 2.258 2.258 2.258 2.258 2.258 2.258 2.258 2.258 2.258 2.258

2.201 2.180 2.127 2.113 2.089 2.045 1.996 1.947 1.882 1.833 1.800 1.784

173.0 169.8 161.5 159.4 155.7 149.4 142.3 135.4 126.4 120.0 115.7 113.5

86,500 84,900 80,750 79,700 77,850 74,700 71,150 67,700 63,200 60,000 57,850 56,750

9.93 13.40 22.36 24.70 28.55 35.64 43.39 50.95 60.71 67.76 72.42 74.69

10

10.75

5S 10S 20 30 40ST, 40S 80S, 60XS 80 100 120 140, XX 160

.134 .165 .250 .307 .365 .500 .594 .719 .844 1.000 1.125

10.482 10.420 10.250 10.136 10.020 9.750 9.562 9.312 9.062 8.750 8.500

4.47 5.49 8.25 10.07 11.91 16.10 18.95 22.66 26.27 30.63 34.02

.5993 .5922 .5731 .5603 .5475 .5185 .4987 .4729 .4479 .4176 .3941

2.814 2.814 2.814 2.814 2.814 2.814 2.814 2.814 2.814 2.814 2.814

2.744 2.728 2.685 2.655 2.620 2.550 2.503 2.438 2.372 2.291 2.225

269.0 265.8 257.0 252.0 246.0 233.0 223.4 212.3 201.0 188.0 177.0

134,500 132,900 128,500 126,000 123,000 116,500 111,700 106,150 100,500 94,000 88,500

15.19 18.65 28.04 34.24 40.48 54.74 64.43 77.03 89.29 104.13 115.64

12

12.75

5S 10S 20 30 ST, 40S 40 XS, 80S 60 80 100 120, XX 140 160

0.156 0.180 0.250 0.330 0.375 0.406 0.500 0.562 0.688 0.844 1.000 1.125 1.312

12.438 12.390 12.250 12.090 12.000 11.938 11.750 11.626 11.374 11.062 10.750 10.500 10.126

6.17 7.11 9.82 12.88 14.58 15.74 19.24 21.52 26.07 31.57 36.91 41.09 47.14

.8438 .8373 .8185 .7972 .7854 .7773 .7530 .7372 .7056 .6674 .6303 .6013 .5592

3.338 3.338 3.338 3.338 3.338 3.338 3.338 3.338 3.338 3.338 3.338 3.338 3.338

3.26 3.24 3.21 3.17 3.14 3.13 3.08 3.04 2.98 2.90 2.81 2.75 2.65

378.7 375.8 367.0 358.0 352.5 349.0 338.0 331.0 316.7 299.6 283.0 270.0 251.0

189,350 187,900 183,500 179,000 176,250 174,500 169,000 165,500 158,350 149,800 141,500 135,000 125,500

20.98 24.17 33.38 43.77 49.56 53.52 65.42 73.15 88.63 107.32 125.49 139.67 160.27

14

14

5S 10S 10 20 30, ST 40 XS 60 80 100 120 140 160

0.156 0.188 0.250 0.312 0.375 0.438 0.500 0.594 0.750 0.938 1.094 1.250 1.406

13.688 13.624 13.500 13.376 13.250 13.124 13.000 12.812 12.500 12.124 11.812 11.500 11.188

6.78 8.16 10.80 13.42 16.05 18.66 21.21 25.02 31.22 38.49 44.36 50.07 55.63

1.0219 1.0125 0.9940 0.9750 0.9575 0.9397 0.9218 0.8957 0.8522 0.8017 0.7610 0.7213 0.6827

3.665 3.665 3.665 3.665 3.665 3.665 3.665 3.665 3.665 3.665 3.665 3.665 3.665

3.58 3.57 3.53 3.50 3.47 3.44 3.40 3.35 3.27 3.17 3.09 3.01 2.93

459 454 446 438 430 422 414 402 382 360 342 324 306

229,500 227,000 223,000 219,000 215,000 211,000 207,000 201,000 191,000 180,000 171,000 162,000 153,000

23.07 27.73 36.71 45.61 54.57 63.44 72.09 85.05 106.13 130.85 150.79 170.21 189.11

16

16

5S 10S 10

0.165 0.188 0.250

15.670 15.624 15.500

8.21 9.34 12.37

1.3393 1.3314 1.3104

4.189 4.189 4.189

4.10 4.09 4.06

601 598 587

300,500 299,000 293,500

27.90 31.75 42.05

10-74

TRANSPORT AND STORAGE OF FLUIDS

TABLE 10-18

Nominal pipe size, in

Properties of Steel Pipe (Concluded )

Outside diameter, in

Cross-sectional area

Circumference, ft, or surface, ft2/ft of length

Wall thickness, in

Inside diameter, in

Metal, in2

Flow, ft

Outside

20 30, ST 40, XS 60 80 100 120 140 160

0.312 0.375 0.500 0.656 0.844 1.031 1.219 1.438 1.594

15.376 15.250 15.000 14.688 14.312 13.938 13.562 13.124 12.812

15.38 18.41 24.35 31.62 40.19 48.48 56.61 65.79 72.14

1.2985 1.2680 1.2272 1.1766 1.1171 1.0596 1.0032 0.9394 0.8953

4.189 4.189 4.189 4.189 4.189 4.189 4.189 4.189 4.189

Schedule no.

2

Capacity at 1-ft/s velocity

Inside

U.S. gal/ min

lb/h water

Weight of plain-end pipe, lb/ft

4.03 3.99 3.93 3.85 3.75 3.65 3.55 3.44 3.35

578 568 550 528 501 474 450 422 402

289,000 284,000 275,000 264,000 250,500 237,000 225,000 211,000 201,000

52.27 62.58 82.77 107.50 136.61 164.82 192.43 223.64 245.25

18

18

5S 10S 10 20 ST 30 XS 40 60 80 100 120 140 160

0.165 0.188 0.250 0.312 0.375 0.438 0.500 0.562 0.750 0.938 1.156 1.375 1.562 1.781

17.670 17.624 17.500 17.376 17.250 17.124 17.000 16.876 16.500 16.124 15.688 15.250 14.876 14.438

9.25 10.52 13.94 17.34 20.76 24.16 27.49 30.79 40.64 50.28 61.17 71.82 80.66 90.75

1.7029 1.6941 1.6703 1.6468 1.6230 1.5993 1.5763 1.5533 1.4849 1.4180 1.3423 1.2684 1.2070 1.1370

4.712 4.712 4.712 4.712 4.712 4.712 4.712 4.712 4.712 4.712 4.712 4.712 4.712 4.712

4.63 4.61 4.58 4.55 4.52 4.48 4.45 4.42 4.32 4.22 4.11 3.99 3.89 3.78

764 760 750 739 728 718 707 697 666 636 602 569 540 510

382,000 379,400 375,000 369,500 364,000 359,000 353,500 348,500 333,000 318,000 301,000 284,500 270,000 255,000

31.43 35.76 47.39 58.94 70.59 82.15 93.45 104.67 138.17 170.92 207.96 244.14 274.22 308.50

20

20

5S 10S 10 20, ST 30, XS 40 60 80 100 120 140 160

0.188 0.218 0.250 0.375 0.500 0.594 0.812 1.031 1.281 1.500 1.750 1.969

19.624 19.564 19.500 19.250 19.000 18.812 18.376 17.938 17.438 17.000 16.500 16.062

11.70 13.55 15.51 23.12 30.63 36.21 48.95 61.44 75.33 87.18 100.3 111.5

2.1004 2.0878 2.0740 2.0211 1.9689 1.9302 1.8417 1.7550 1.6585 1.5763 1.4849 1.4071

5.236 5.236 5.236 5.236 5.236 5.236 5.236 5.236 5.236 5.236 5.236 5.236

5.14 5.12 5.11 5.04 4.97 4.92 4.81 4.70 4.57 4.45 4.32 4.21

943 937 930 902 883 866 826 787 744 707 665 632

471,500 467,500 465,000 451,000 441,500 433,000 413,000 393,500 372,000 353,500 332,500 316,000

39.78 46.06 52.73 78.60 104.13 123.11 166.40 208.87 256.10 296.37 341.09 397.17

24

24

5S 10, 10S 20, ST XS 30 40 60 80 100 120 140 160

0.218 0.250 0.375 0.500 0.562 0.688 0.969 1.219 1.531 1.812 2.062 2.344

23.564 23.500 23.250 23.000 22.876 22.624 22.062 21.562 20.938 20.376 19.876 19.312

16.29 18.65 27.83 36.90 41.39 50.39 70.11 87.24 108.1 126.3 142.1 159.5

3.0285 3.012 2.948 2.885 2.854 2.792 2.655 2.536 2.391 2.264 2.155 2.034

6.283 6.283 6.283 6.283 6.283 6.283 6.283 6.283 6.283 6.283 6.283 6.283

6.17 6.15 6.09 6.02 5.99 5.92 5.78 5.64 5.48 5.33 5.20 5.06

1359 1350 1325 1295 1281 1253 1192 1138 1073 1016 965 913

679,500 675,000 662,500 642,500 640,500 626,500 596,000 569,000 536,500 508,000 482,500 456,500

55.37 63.41 94.62 125.49 140.68 171.29 238.35 296.58 367.39 429.39 483.12 542.13

30

30

5S 10, 10S ST 20, XS 30

0.250 0.312 0.375 0.500 0.625

29.500 29.376 29.250 29.000 28.750

23.37 29.10 34.90 46.34 57.68

4.746 4.707 4.666 4.587 4.508

7.854 7.854 7.854 7.854 7.854

7.72 7.69 7.66 7.59 7.53

2130 2110 2094 2055 2020

1,065,000 1,055,000 1,048,000 1,027,500 1,010,000

79.43 98.93 118.65 157.53 196.08

5S, 10S, and 40S are extracted from Stainless Steel Pipe, ANSI B36.19—1976, with permission of the publisher, the American Society of Mechanical Engineers, New York. ST = standard wall, XS = extra strong wall, XX = double extra strong wall, and Schedules 10 through 160 are extracted from Wrought-Steel and WroughtIron Pipe, ANSI B36.10—1975, with permission of the same publisher. Decimal thicknesses for respective pipe sizes represent their nominal or average wall dimensions. Mill tolerances as high as ⫾12a percent are permitted. Plain-end pipe is produced by a square cut. Pipe is also shipped from the mills threaded, with a threaded coupling on one end, or with the ends beveled for welding, or grooved or sized for patented couplings. Weights per foot for threaded and coupled pipe are slightly greater because of the weight of the coupling, but it is not available larger than 12 in or lighter than Schedule 30 sizes 8 through 12 in, or Schedule 40 6 in and smaller. To convert inches to millimeters, multiply by 25.4; to convert square inches to square millimeters, multiply by 645; to convert feet to meters, multiply by 0.3048; to convert square feet to square meters, multiply by 0.0929; to convert pounds per foot to kilograms per meter, multiply by 1.49; to convert gallons to cubic meters, multiply by 3.7854 × 10−3; and to convert pounds to kilograms, multiply by 0.4536.

PROCESS-PLANT PIPING

FIG. 10-128

10-75

Taper pipe thread.

Threads notch the pipe and cause loss of strength and fatigue resistance. Enlargement and contraction of the flow passage at threaded joints creates turbulence; sometimes corrosion and erosion are aggravated at the point where the pipe has already been thinned by threading. The tendency of pipe wrenches to crush pipe and fittings limits the torque available for tightening threaded joints. For low-pressure systems, a slight rotation in the joint may be used to impart flexibility to the system, but this same rotation, unwanted, may cause leaks to develop in higher-pressure systems. In some metals, galling occurs when threaded joints are disassembled. Straight Pipe Threads These are confined to light-weight couplings in sizes 2 in and smaller (Fig. 10-129). Manufacturers of threaded pipe ship it with such couplings installed on one end of each pipe. The joint obtained is inferior to that obtained with taper threads. The code limits the joint shown in Fig. 10-129 to 1.0 MPa (150 lbf/in2) gauge maximum, 182°C (360°F) maximum, and to nonflammable, nontoxic fluids. When both components of a threaded joint are of weldable metal, the joint may be seal-welded as shown in Fig. 10-130. Seal welds may be used only to prevent leakage of threaded joints. They are not considered as contributing any strength to the joint. This type of joint is limited to new construction and is not suitable as a repair procedure, since pipe dope in the threads would interfere with welding. This method provides tight joints with a minimum of welding labor. When threaded joints used to join materials with widely different coefficients of thermal expansion are subject to temperature cycling, seal welding may be needed to prevent leakage. To assist in assembly and disassembly of both threaded and welded systems, union joints (Fig. 10-131) are used. They comprise metalto-metal seats drawn together by a shouldered straight thread nut and are available both in couplings for joining two lengths of pipe and on the ends of some fittings. On threaded piping systems in which disassembly is not contemplated, union joints installed at intervals permit future further tightening of threaded joints. Tightening of heavy unions yields tight joints even if the pipe is slightly misaligned at the start of tightening. Flanged Joints For sizes larger than 2 in when disassembly is contemplated, the flanged joint (Fig. 10-132) is the most widely used.

FIG. 10-129

Taper pipe to straight coupling thread.

FIG. 10-130

Taper pipe thread seal-welded.

FIG. 10-131

Union.

Figures 10-133 and 10-134 illustrate the wide variety of types and facings available. Though flanged joints consume a large volume of metal, precise machining is required only on the facing. Flanged joints do not impose severe diametral tolerances on the pipe. Careful alignment prior to assembly of flat-face and raised-face flanges is not required, and the necessary wrenches are far smaller than those for screwed assembly for the same size of pipe. Manufacturers offer flanged-end pipe in only a few metals. Otherwise, flanges are attached to pipe by various types of joints (Fig. 10-133). The lap joint involves a modification of the pipe which may be formed from the pipe itself or by welding a ring or a lap-joint stub end to it. Flanged-end fittings and valves are available in all sizes of most pipe metals. Welding-neck flanges provide joints as strong as the pipe under all types of static and cycling loading. Slip-on, socket-weld, and lap-joint flanges provide joints as strong as the pipe under static loading but have lower resistance to cyclic stresses (see Table 10-54). Lap-joint flanges avoid the necessity of orienting flanges so that vertical and horizontal centerlines are halfway between bolt holes and permit orientation of the stems of flanged valves at any angle needed to provide clearance. The tolerance is f in in the bolt holes; the necessity of making sure that the gasket does not protrude into the flow channel results in some disturbance of the flow pattern when flat-face and raised-face flanges are used. This can be eliminated by using weldingneck or socket-weld flanges with male-and-female or tongue-andgroove facings. Dimensions of alloy and carbon steel and cast-iron pipe flanges with flat and raised faces are given in Tables 10-19 to 10-25 (see Fig. 10-134). The dimensions were extracted from Cast-Iron Pipe Flanges and Flanged Fittings, ANSI B16.1—1975, and Steel Pipe Flanges and Flanged Fittings, ANSI 16.5—1977, with the permission of the publisher, the American Society of Mechanical Engineers, New York. Against cast-iron flanged fittings or valves, steel pipe flanges are often preferred to cast-iron flanges because they permit welded rather than

FIG. 10-132

Flanged joint.

10-76

TRANSPORT AND STORAGE OF FLUIDS

FIG. 10-133

screwed assembly to the pipe and because cast-iron pipe flanges, not being reinforced by the pipe, are not so resistant to abuse as flanges cast integrally on cast-iron fittings. Facing of flanges for alloy and carbon steel pipe and fittings is shown in Fig. 10-134; 125-lb cast-iron pipe and fitting flanges have flat faces, which with full-face gaskets minimize bending stresses; 250-lb cast-iron pipe and fitting flanges have 1.5-mm ( 1⁄18-in) raised faces (wider than on steel flanges) for the same purpose. Carbon steel and ductile- (nodular-) iron lap-joint flanges are widely used as backup flanges with stub ends in piping systems of austenitic stainless steel and other expensive materials to reduce costs (see Fig. 10-133). The code prohibits the use of ductile-iron flanges at temperatures above 343°C (650°F). When the type of facing affects the length through the hub dimension of flanges, correct dimensions for commonly used facings can be determined from the dimensional data in Fig. 10-134. Gaskets Gaskets must resist corrosion by the fluids handled. The more expensive male-and-female or tongue-and-groove facings may be required to seat hard gaskets adequately. With these facings the gasket generally cannot blow out. Flanged joints, by placing the gasket material under heavy compression and permitting only edge attack by the fluid handled, can use gasket materials which in other joints might not satisfactorily resist the fluid handled. The finish of flange facings varies with the manufacturer. For raised-face or male mating surfaces the finish usually consists of a continuous spiral groove formed by a round-nosed tool or a V tool (serrated finish). Female surfaces are smooth-finished (i.e., without definite tool markings). Other finishes are concentric-grooved, lapped, or mirror (cold-water). The latter two are usually for application without gaskets. In general, for 300-lb ANSI and lower-rated flanges compressed asbestos-sheet gaskets are used [400°C (750°F) maximum]. The metalasbestos spiral-wound type is used for higher pressure and temperature services [593°C (1100°F) maximum], including services involving cyclic or difficultly contained fluids. The development of substitutes for asbestos in gaskets is being actively pursued because of the health hazard associated with asbestos. Metal-TFE and metal-graphite spiral-wound gaskets are available and may seal better than metalasbestos gaskets. Spiral-wound gaskets are also used widely in highpressure steam services. Spiral-wound gaskets should preferably be used with a smooth-flange finish.

Types of carbon and alloy steel flanges.

TABLE 10-19

Dimensions of Class 150-lb Flanges for Use with Steel Pipe* All dimensions in inches Length through hub Outside diameter of flange

Thickness of flange, minimum

Diameter of bolt circle

a e

1 1d 1a

3.50 3.88 4.25 4.62 5.00

0.44 0.50 0.56 0.62 0.69

2.38 2.75 3.12 3.50 3.88

2 2a 3 3a 4

6.00 7.00 7.50 8.50 9.00

0.75 0.88 0.94 0.94 0.94

5 6 8 10 12

10.00 11.00 13.50 16.00 19.00

14 16 18 20 24

21.00 23.50 25.00 27.50 32.00

Nominal pipe size

Diameter of bolts

Lap joint

Welding neck

ANSI B16.1, screwed (125-lb)

0.62 0.62 0.69 0.81 0.88

1.88 2.06 2.19 2.25 2.44

0.69 0.81 0.88

1.00 1.12 1.19 1.25 1.31

1.00 1.12 1.19 1.25 1.31

2.50 2.75 2.75 2.81 3.00

1.00 1.12 1.19 1.25 1.31

8 8 8 12 12

1.44 1.56 1.75 1.94 2.19

1.44 1.56 1.75 1.94 2.19

3.50 3.50 4.00 4.00 4.50

1.44 1.56 1.75 1.94 2.19

12 16 16 20 20

2.25 2.50 2.69 2.88 3.25

3.12 3.44 3.81 4.06 4.38

5.00 5.00 5.50 5.69 6.00

2.25 2.50 2.69 2.88 3.25

No. of bolts

Threaded slip-on socket welding

a a a a a

4 4 4 4 4

0.62 0.62 0.69 0.81 0.88

4.75 5.50 6.00 7.00 7.50

v v v v v

4 4 4 8 8

0.94 1.00 1.12 1.19 1.25

8.50 9.50 11.75 14.25 17.00

e e e 7⁄ 8 7⁄ 8

1.38 1.44 1.56 1.69 1.88

18.75 21.25 22.75 25.00 29.50

1 1 1f 1f 1d

*Dimensions from ANSI B16.5—1977, unless otherwise noted. To convert inches to millimeters, multiply by 25.4.

PROCESS-PLANT PIPING

10-77

Flange facings, illustrated on welding-neck flanges. (On small male-and-female facings the outside diameter of the male face is less than the outside diameter of the pipe, so this facing does not apply to screwed or slip-on flanges. A similar joint can be made with screwed flanges and threaded pipe by projecting the pipe through one flange and recessing it in the other. However, pipe thicker than Schedule 40 is required to avoid crushing gaskets.) To convert inches to millimeters, multiply by 25.4.

FIG. 10-134

TABLE 10-20

Dimensions of Class 300 Flanges for Use with Steel Pipe* All dimensions in inches Length through hub Outside diameter of flange

Thickness of flange, minimum

Diameter of bolt circle

Diameter of bolts

No. of bolts

Threaded slip-on socket welding

a e 1 1d 1a

3.75 4.62 4.88 5.25 6.12

0.56 0.62 0.69 0.75 0.81

2.62 3.25 3.50 3.88 4.50

a v v v e

4 4 4 4 4

0.88 1.00 1.06 1.06 1.19

2 2a 3 3a 4

6.50 7.50 8.25 9.00 10.00

0.88 1.00 1.12 1.19 1.25

5.00 5.88 6.62 7.25 7.88

v e e e e

8 8 8 8 8

5 6 8 10 12

11.00 12.50 15.00 17.50 20.50

1.38 1.44 1.62 1.88 2.00

9.25 10.62 13.00 15.25 17.75

e e 7⁄ 8

14 16 18 20 24

23.00 25.50 28.00 30.50 36.00

2.12 2.25 2.38 2.50 2.75

20.25 22.50 24.75 27.00 32.00

Nominal pipe size

Lap joint

Welding neck

ANSI B16.1, screwed (Class 250)

0.88 1.00 1.06 1.06 1.19

2.06 2.25 2.44 2.56 2.69

0.88 1.00 1.12

1.31 1.50 1.69 1.75 1.88

1.31 1.50 1.69 1.75 1.88

2.75 3.00 3.12 3.19 3.38

1.25 1.43 1.56 1.62 1.75

1 1f

8 12 12 16 16

2.00 2.06 2.44 2.62 2.88

2.00 2.06 2.44 3.75 4.00

3.88 3.88 4.38 4.62 5.12

1.88 1.94 2.19 2.38 2.56

1f 1d 1d 1d 1a

20 20 24 24 24

3.00 3.25 3.50 3.75 4.19

4.38 4.75 5.12 5.50 6.00

5.62 5.75 6.25 6.38 6.62

2.69 2.88

*Dimensions from ANSI B16.5—1977, unless otherwise noted. To convert inches to millimeters, multiply by 25.4.

10-78

TRANSPORT AND STORAGE OF FLUIDS

TABLE 10-21

Dimensions of Class 400 Steel Flanges* All dimensions in inches

Nominal pipe size

Outside diameter of flange

Thickness of flange, minimum

Diameter of bolt circle

a e 1 1d 1a

Length through hub Diameter of bolts

No. of bolts

Threaded slip-on socket welding

Lap joint

Welding neck

2.00 2.12 2.25 2.69 2.88 3.12

2.00 2.12 2.25 2.69 4.00 4.25

3.50 4.00 4.06 4.62 4.88 5.38

3.31 3.69 3.88 4.00 4.50

4.62 5.00 5.38 5.75 6.25

5.88 6.00 6.50 6.62 6.88

Use Class 600 dimensions in these sizes.

2 2a 3 3a 4 5 6 8 10 12

10.00 11.00 12.50 15.00 17.50 20.50

1.38 1.50 1.62 1.88 2.12 2.25

7.88 9.25 10.62 13.00 15.25 17.75

⁄ ⁄ ⁄

1 1d 1d

8 8 12 12 16 16

14 16 18 20 24

23.00 25.50 28.00 30.50 36.00

2.38 2.50 2.62 2.75 3.00

20.25 22.50 24.75 27.00 32.00

1d 1r 1r 1a 1e

20 20 24 24 24

78 78 78

*Dimensions from ANSI B16.5—1977. To convert inches to millimeters, multiply by 25.4.

The spiral-wound type furnished with a solid metallic ring on the outside to limit gasket compression provides protection against blowout when used with raised facing. Metal-Ring-Joint Facing This is the most costly facing. The ring must be softer than the flange and is usually a softer grade of the same metal as the flange. It is used where other gasket materials are destroyed by the fluid being handled. In event of fire, it does not leak. Because the surfaces that the gasket contacts are below the flange face, it is the least likely facing to be damaged in handling. Compared with raised or smooth faces, it is more difficult to disassemble because the flanges can be separated only in the axial direction. Bolting Bolting requirements for ANSI flanged joints are preTABLE 10-22

sented in the code. For joining two steel flanges, by reference to ANSI B16.5, Steel Pipe Flanges and Flanged Fittings, the code requires alloy steel bolting, except that bolting for 150- and 300-lb flanges at 204°C (400°F) and lower may be made of ASTM A307 Grade B lowcarbon externally threaded fasteners. The code limits this exception to −29°C (−20°F) minimum. Steel 150-lb flanges may be bolted to cast-iron valves, fittings, or other cast-iron piping components having either Class 125 cast integral or screwed flanges. If such construction is used, it is preferred that the 1.5-mm (g-in) raised face on steel flanges be removed. If the raised face is removed and a flat-ring gasket extending to the inner edge of the bolt holes is used, the bolting shall not be stronger than

Dimensions of Class 600 Steel Flanges* All dimensions in inches Length through hub

Outside diameter of flange

Thickness of flange, minimum

Diameter of bolt circle

Diameter of bolts

No. of bolts

Threaded slip-on socket welding

Lap joint

Welding neck

a e 1 1d 1a

3.75 4.62 4.88 5.25 6.12

0.56 0.62 0.69 0.81 0.88

2.62 3.25 3.50 3.88 4.50

a v v v e

4 4 4 4 4

0.88 1.00 1.06 1.12 1.25

0.88 1.00 1.06 1.12 1.25

2.06 2.25 2.44 2.62 2.75

2 2a 3 3a

6.50 7.50 8.25 9.00

1.00 1.12 1.25 1.38

5.00 5.88 6.62 7.25

v e e 7⁄ 8

8 8 8 8

1.44 1.62 1.81 1.94

1.44 1.62 1.81 1.94

2.88 3.12 3.25 3.38

4 5 6 8 10 12

10.75 13.00 14.00 16.50 20.00 22.00

1.50 1.75 1.88 2.19 2.50 2.62

8.50 10.50 11.50 13.75 17.00 19.25

7⁄ 8 1 1 1f 1d 1d

8 8 12 12 16 20

2.12 2.38 2.62 3.00 3.38 3.62

2.12 2.38 2.62 3.00 4.38 4.62

4.00 4.50 4.62 5.25 6.00 6.12

14 16 18 20 24

23.75 27.00 29.25 32.00 37.00

2.75 3.00 3.25 3.50 4.00

20.75 23.75 25.75 28.50 33.00

1e 1a 1v 1v 17⁄ 8

20 20 20 24 24

3.69 4.19 4.62 5.00 5.50

5.00 5.50 6.00 6.50 7.25

6.50 7.00 7.25 7.50 8.00

Nominal pipe size

*Dimensions from ANSI B16.5—1977. To convert inches to millimeters, multiply by 25.4.

PROCESS-PLANT PIPING TABLE 10-23

10-79

Dimensions of Class 900 Steel Flanges* All dimensions in inches

Nominal pipe size

Outside diameter of flange

Thickness of flange, minimum

Diameter of bolt circle

a e 1 1d

Length through hub Diameter of bolts

No. of bolts

Threaded slip-on socket welding

Lap joint

Welding neck

Use Class 1500 dimensions in these sizes.

1a 2 2a 3 4

9.50 11.50

1.50 1.75

7.50 9.25

7⁄ 8 1f

8 8

2.12 2.75

2.12 2.75

4.00 4.50

5 6 8 10 12

13.75 15.00 18.50 21.50 24.00

2.00 2.19 2.50 2.75 3.12

11.00 12.50 15.50 18.50 21.00

1d 1f 1r 1r 1r

8 12 12 16 20

3.12 3.38 4.00 4.25 4.62

3.12 3.38 4.50 5.00 5.62

5.00 5.50 6.38 7.25 7.88

14 16 18 20 24

25.25 27.75 31.00 33.75 41.00

3.38 3.50 4.00 4.25 5.50

22.00 24.25 27.00 29.50 35.50

1a 1v 17⁄ 8 2 2a

20 20 20 20 20

5.12 5.25 6.00 6.25 8.00

6.12 6.50 7.50 8.25 10.50

8.38 8.50 9.00 9.75 11.50

*Dimensions from ANSI B16.5—1977. To convert inches to millimeters, multiply by 25.4.

carbon steel per ASTM A307 Grade B; if a full-face gasket is used, the bolting may be heat-treated carbon steel or alloy steel (ASTM A193). If the raised face of the steel flange is not removed, the bolting shall not be stronger than carbon steel ASTM A307 Grade B. Steel 300-lb flanges may be bolted to cast-iron valves, fittings, or other cast-iron piping components having either Class 250 cast-iron integral or screwed flanges, without any change in the raised face on either flange. If such construction is used, the bolting shall not be stronger than carbon steel, ASTM A307 Grade B. Cast-iron 25-lb and Class 125 integral or screwed companion flanges may be used with a full-face gasket or with a flat-ring gasket extending to the inner edge of the bolts. When a full-face gasket is

TABLE 10-24

used, the bolting may be of heat-treated carbon steel or alloy steel (ASTM A193). When a flat-ring gasket is used, the bolting shall not be stronger than carbon steel, per ASTM A307 Grade B. When two Class 250 cast-iron integral or screwed companion flanges having 1.5-mm (g-in) raised faces are bolted together, the bolting shall not be stronger than carbon steel, per ASTM A307 Grade B. Other Types of Piping Joints Packed-gland joints (Fig. 10-135) require no special end preparation of pipe but do require careful control of the diameter of the pipe. Thus the supplier of the pipe should be notified when packed-gland joints are to be used. Cast- and ductileiron pipe, fittings, and valves are available with the bell cast on one or

Dimensions of Class 1500 Steel Flanges* All dimensions in inches Length through hub

Outside diameter of flange

Thickness of flange, minimum

Diameter of bolt circle

Diameter of bolts

No. of bolts

Threaded slip-on socket welding

Lap joint

Welding neck

a e

1 1d

4.75 5.12 5.88 6.25

0.88 1.00 1.12 1.12

3.25 3.50 4.00 4.38

e e 7⁄ 8 7⁄ 8

4 4 4 4

1.25 1.38 1.62 1.62

1.25 1.38 1.62 1.62

2.38 2.75 2.88 2.88

1a 2 2a 3 4

7.00 8.50 9.62 10.50 12.25

1.25 1.50 1.62 1.88 2.12

4.88 6.50 7.50 8.00 9.50

1 1f 1d

4 8 8 8 8

1.75 2.25 2.50 2.88 3.56

1.75 2.25 2.50 2.88 3.56

3.25 4.00 4.12 4.62 4.88

5 6 8 10 12

14.75 15.50 19.00 23.00 26.50

2.88 3.25 3.62 4.25 4.88

11.50 12.50 15.50 19.00 22.50

1a 1r 1v 17⁄ 8 2

8 12 12 12 16

4.12 4.69 5.62 6.25 7.12

4.12 4.69 5.62 7.00 8.62

6.12 6.75 8.38 10.00 11.12

14 16 18 20 24

29.50 32.50 36.00 38.75 46.00

5.25 5.75 6.38 7.00 8.00

25.00 27.75 30.50 32.75 39.00

2d 2a 2e 3 3a

16 16 16 16 16

9.50 10.25 10.88 11.50 13.00

11.75 12.25 12.88 14.00 16.00

Nominal pipe size

1 ⁄

78

*Dimensions from ANSI B15.5—1977. To convert inches to millimeters, multiply by 25.4.

10-80

TRANSPORT AND STORAGE OF FLUIDS

TABLE 10-25

Dimensions of Class 2500 Steel Flanges* All dimensions in inches Length through hub

Outside diameter of flange

Thickness of flange, minimum

Diameter of bolt circle

Diameter of bolts

No. of bolts

Threaded

Lap joint

Welding neck

a e

1 1d 1a

5.25 5.50 6.25 7.25 8.00

1.19 1.25 1.38 1.50 1.75

3.50 3.75 4.25 5.12 5.75

e e 7⁄ 8 1 1f

4 4 4 4 4

1.56 1.69 1.88 2.06 2.38

1.56 1.69 1.88 2.06 2.38

2.88 3.12 3.50 3.75 4.38

2 2a 3 4

9.25 10.50 12.00 14.00

2.00 2.25 2.62 3.00

6.75 7.75 9.00 10.75

1 1f 1d 1a

8 8 8 8

2.75 3.12 3.62 4.25

2.75 3.12 3.62 4.25

5.00 5.62 6.62 7.50

16.50 19.00 21.75 26.50 30.00

3.62 4.25 5.00 6.50 7.25

12.75 14.50 17.25 21.25 24.38

1e 2 2 2a 2e

8 8 12 12 12

5.12 6.00 7.00 9.00 10.00

5.12 6.00 7.00 9.00 10.00

9.00 10.75 12.50 16.50 18.25

Nominal pipe size

5 6 8 10 12

*Dimensions from ANSI B16.5—1977. To convert inches to millimeters, multiply by 25.4.

more ends. Glands, bolts, and gaskets are shipped with the pipe. Couplings equipped with packed glands at each end, known as Dresser couplings, are available in several metals. The joints can be assembled with small wrenches and unskilled labor, in limited space, and if necessary, under water. Packed-gland joints are designed to take the same hoop stress as the pipe. They do not resist bending moments or axial forces tending to separate the joints but yield to them to an extent indicated by the vendor’s allowable-angular-deflection and end movement specifications. Further angular or end movement produces leakage, but end movement can be limited by harnessing or bridling with a combination of rods and welded clips or clamps, or by anchoring to existing or new structures. The crevice between the bell and the spigot may promote corrosion. The joints are widely used in underground lines. They are not affected by limited earth settlement, and friction of the earth prevents end separation. When disassembly by moving pipe axially is not practical, packed-joint couplings which can be slid entirely onto one of the two lengths joined are available. However, the tendency of the packing to adhere to the pipe makes this difficult. Poured joints (Fig. 10-136) require no special end preparation of the pipe or diametral control. They are used for brittle materials. Pipe, fittings, and valves are furnished with the bells cast on one or more ends. The pouring compound may be molten, or chemical-setting, or merely compacted; these choices are listed in descending order of ability to hold pressure. These joints cannot absorb angular or axial movement without leaking. Disassembly for maintenance is accomplished by cutting the pipe and reassembly by the use of a coupling with a bell at each end. Push-on joints (Fig. 10-137) require diametral control of the end of the pipe. They are used for brittle materials. Pipe, fittings, and valves are furnished with the bells cast on one or more ends. Considerable force is required to push the spigot through the O ring; this is reduced by the extension on the O ring, which causes the friction of the pipe to elongate the cross section of the main portion of the O ring. Push-on joints do not resist bending moments or axial forces tending to separate the joints but yield to them to an extent limited by the

FIG. 10-135

Poured joint.

vendor’s allowable-angular-deflection and end-movement specifications. End movement can be limited by harnessing or bridling with a combination of rods and clamps, or by anchoring to existing or new structures. The joints are widely used on underground lines. They are not affected by limited earth settlement, and friction of the earth prevents end separation. A lubricant is used on the O ring during assembly. After this disappears, the O ring bonds somewhat to the spigot and disassembly is very difficult. Disassembly for maintenance is accomplished by cutting the pipe and reassembly by use of a coupling with a packed-gland joint on each end. Expanded joints (Fig. 10-138) are confined to the smaller pipe sizes of ductile metals. A smooth finish is required on the outside of the pipe and on the faces of the ridges inside the bore. Pipe and bore must have the same coefficient of thermal expansion. Furthermore, it is essential that the pipe metal have a lower yield point than the metal

FIG. 10-136

Packed-gland joint.

FIG. 10-137

Push-on joint.

PROCESS-PLANT PIPING

FIG. 10-138

Expanded joint.

into which it is expanded, except in cases in which the metal into which it is expanded is a thin cylinder temporarily backed by clamped heavy semicylindrical metal shells with a high yield point. An expanding tool is required, one for each size of pipe. After completion of the joint, it is difficult to determine whether the increase in the inside diameter of the pipe represents permanent stretch of the bore of the mating part or flow of metal into the grooves of the bore. An excess of the latter results in excessive thinning of the tube, while an insufficiency of the latter may cause the pipe to pull out of the bore under axial loading. In a variation, the expanded joint is combined with a flared joint to increase resistance to axial load. These joints are used to attach unions and Lovekin flanges to pipe. For alloy piping, composite Lovekin flanges in which the bore and raised face portion are made of the alloy, retained in the steel balance of the flange by an offset, are available. Grooved joints (Fig. 10-139) are divided into two classes, cut grooves and rolled grooves. Rolled grooves are preferred because,

compared with cut grooves, they are easier to form and reduce the metal wall less. However, they slightly reduce the flow area. They are limited to thin walls of ductile material, while cut grooves, because of their reduction of the pipe wall, are limited to thick walls. In the larger pipe sizes, some commonly used wall thicknesses are too thick for rolled grooves but too thin for cut grooves. The thinning of the walls impairs resistance to corrosion and erosion but not to internal pressure, because the thinned area is reinforced by the coupling. Control of outside diameter is important. Permissible minus tolerance is limited, since it impairs the grip of the couplings. Plus tolerance makes it necessary to cut the cut grooves more deeply, increasing the thinning of the wall. Plus tolerance is not a problem with rolled grooves, since they are confined to walls thin enough so that the couplings can compress the pipe. Pipe is available from vendors already grooved and also with heavier-wall grooved ends welded on. Grooved joints resist axial forces tending to separate the joints. Angular deflection, up to the limit specified by the vendor, may be used to absorb thermal expansion and to permit the piping to be laid on uneven ground. Compared with flanged joints, grooved joints will not pull misaligned pipe into alignment, and thus they require more support, but otherwise they require less labor for handling, assembly, and disassembly. Gaskets are self-sealing against both internal and external pressure and are available in a wide variety of elastomers. However, successful performance of an elastomer as a flange gasket does not necessarily mean equally satisfactory performance in a grooved joint, since exposure to the fluid in the latter is much greater and hardening has a greater unfavorable effect. It is customary to use couplings which are resistant to corrosion by the fluid in the pipe, but couplings which would contaminate the fluid may be used. V-clamp joints (Fig. 10-140) are attached to the pipe by butt-weld or expanded joints. Theoretically, there is only one relative position of the parts in which the conical surfaces of the clamp are completely in contact with the conical surfaces of the stub ends. In actual practice, there is considerable flexing of the stub ends and the clamp; also complete contact is not required. This permits use of elastomeric gaskets

(a) FIG. 10-139

(b)

Grooved joint. (a) Section. (b) End view.

(a) FIG. 10-140

10-81

V-clamp joint. (a) Section. (b) End view.

(b)

10-82

TRANSPORT AND STORAGE OF FLUIDS

(a) FIG. 10-141

Seal-ring joint. (Courtesy of Gray Tool Co.)

as well as metal gaskets. Fittings are also available with integral conical shouldered ends. Conical ends vary from machined forgings to roll-formed tubing, and clamps vary from machined forgings to bands to which several roll-formed channels are attached at their centers by spot welding. A hinge may be inserted in the band as a substitute for one of the draw bolts. Latches may also be substituted for draw bolts. Compared with flanges, V-clamp joints use less metal, require less labor for assembly, and are less likely to leak under wide-range rapid temperature cycling. However, they are more susceptible to failure or damage from overtightening. They are widely used for high-alloy piping subject to periodic cleaning or relocation. Manufactured as forgings, they are used in carbon steel with metal gaskets for very high pressures. They resist both axial strain and bending moments. Each size of each type of joint is customarily rated by the vendor for both internal pressure and bending moment. Seal-ring joints (Fig. 10-141) consist of hubs attached to the pipe, generally by welding. The joint is proprietary and sold under the registered trade name of Grayloc. The metal seal ring is in effect a selfenergizing gasket. This joint is widely used in petrochemical plants for service at the higher pressures. Valves and other accessories are manufactured with Grayloc hub ends. Pressure-seal joints (Fig. 10-142) are used for pressures of 4.4 MPa (600 lbf/in2) and higher. They use less metal than flanged joints but require much more machining of surfaces. There are several designs, in all of which increasing fluid pressure increases the

FIG. 10-142

(b)

force holding the sealing surfaces against each other. These joints are widely used as bonnet joints in carbon and alloy steel valves. Tubing Joints Flared-fitting joints (see Fig. 10-143) are used for ductile tubing when the ratio of wall thickness to the diameter is small enough to permit flaring without cracking the inside surface. The tubing must have a smooth interior surface. The three-piece type avoids torsional strain on the tubing and minimizes vibration fatigue on the flared portion of the tubing. More labor is required for assembly, but the fitting is more resistant to temperature cycling than other tubing fittings and is unlikely to be damaged by overtightening, and its efficiency is not impaired by repeated assembly and disassembly. Size is limited because of the large number of machined surfaces. The nut and, in the three-piece type, the sleeve need not be of the same material as the tubing. For these fittings, less control of tubing diameter is required. Compression-fitting joints (Fig. 10-144) are used for ductile tubing with thin walls. The outside of the tubing must be clean and smooth. Assembly consists only of inserting the tubing and tightening the nut. These are the least costly tubing fittings but are not resistant to vibration or temperature cycling. Bite-type-fitting joints (Fig. 10-145) are used when the tubing has too high a ratio of wall thickness to diameter for flaring, when the tubing lacks sufficient ductility for flaring, and for low assembly-labor cost. The outside of the tubing must be clean and smooth. Assembly consists in merely inserting the tubing and tightening the nut. The sleeve must be considerably harder than the tubing yet still ductile

Pressure-seal joint.

(a) FIG. 10-143

Flared-fitting joint. (a) Three-piece. (b) Two-piece.

(b)

PROCESS-PLANT PIPING enough to be diametrally compressed and must be as resistant to corrosion by the fluid handled as the tubing. The fittings are resistant to vibration but not to wide-range rapid temperature cycling. Compared with flared fittings, they are less suited for repeated assembly and disassembly, require closer diametral control of the tubing, and are more susceptible to damage from overtightening. They are widely used for oil-filled hydraulic systems at all pressures. O-ring seal joints (Fig. 10-146) are also used for applications requiring heavy-wall tubing. The outside of the tubing must be clean and smooth. The joint may be assembled repeatedly, and as long as the tubing is not damaged, leaks can usually be corrected by replacing the O ring and the antiextrusion washer. This joint is used extensively in oil-filled hydraulic systems. Soldered Joints (Fig. 10-147) These joints require precise control of the diameter of the pipe or tubing and of the cup in the fitting in order to cause the solder to draw into the clearance between the cup and the tubing by capillary action (Fig. 10-147). Extrusion provides this diametral control, and the joints are most widely used in copper. A 50 percent lead, 50 percent tin solder is used for tempera-

FIG. 10-144

Compression-fitting joint.

FIG. 10-145

Bite-type-fitting joint.

FIG. 10-146

O-ring seal joint. (Courtesy of the Lenz Co.)

FIG. 10-147

Soldered, brazed, or cemented joint.

10-83

tures up to 93°C (200°F). Careful cleaning of the outside of the tubing and inside of the cup is required. Heat for soldering is usually obtained from torches. The high conductivity of copper makes it necessary to use large flames for the larger sizes, and for this reason the location in which the joint will be made must be carefully considered. Soldered joints are most widely used in sizes 2 in and smaller for which heat requirements are less burdensome. Soldered joints should not be used in areas where plant fires are likely because exposure to fires results in rapid and complete failure of the joints. Properly made, the joints are completely impervious. The code permits the use of soldered joints only for Category D fluid service and then only if the system is not subject to severe cyclic condions. Silver Brazed Joints These are similar to soldered joints except that a temperature of about 600°C (1100°F) is required. A 15 percent silver, 80 percent copper, 5 percent phosphorus solder is used for copper and copper alloys, while 45 percent silver, 15 percent copper, 16 percent zinc, 24 percent cadmium solders are used for copper, copper alloys, carbon steel, and alloy steel. Silver-brazed joints are used for temperatures up to 200°C (400°F). Cast-bronze fittings and valves with preinserted rings of 15 percent silver, 80 percent copper, 5 percent phosphorus brazing alloy are available. Silver-brazed joints are used when temperature or the combination of temperature and pressure is beyond the range of soldered joints. They are also more reliable in the event of plant fires and are more resistant to vibration. If they are used for fluids that are flammable, toxic, or damaging to human tissue, appropriate safeguarding is required by the code. There are OSHA regulations governing the use of silver brazing alloys containing cadmium and other toxic materials. Bends and Fittings Directional changes in piping systems require bends and elbow fittings. Bends may be made cold or hot. The outside wall is thinned by an amount that varies with the procedure used. Subsequent annealing is required for some materials. To prevent wrinkling and excessive flattening, sand packing is required for hot bending, and sand packing or flexible mandrels may be necessary for cold bending, depending on the ratios of the outside diameter of the pipe to the centerline radius of the bend and to the wall thickness of the pipe. For bends with a centerline radius of five nominal pipe diameters, internal support is not required when the wall thickness is at least 6 percent of the outside diameter of the pipe. Wrinkled bends are made by progressively heating the pipe only on the side which will be the inside of the bend. Elbow fittings may be cast, forged, or hot- or cold-formed from short pieces of pipe or made by welding together pieces of miter-cut pipe. The thinning of pipe during the forming of elbows is compensated for by starting with heavier walls. Flow in bends and elbow fittings is more turbulent than in straight pipe, thus increasing corrosion and erosion. This can be countered by selecting a component with greater radius of curvature, thicker wall, or smoother interior contour, but this is seldom economical in miter elbows. Compared with elbow fittings, bends with a centerline radius of three or five nominal pipe diameters save the cost of joints and reduce pressure drop. Such bends are not suited for installation in a bank of pipes of unequal size when the bends are in the same plane as the bank. Flanged fittings are used when pipe is likely to be dismantled for frequent cleaning or extensive revision, for lined piping systems, or for seasonal insertion of blanks as a substitute for valves. They are also used in areas where welding is not permitted. Cast fittings are usually flanged. Table 10-26 gives dimensions for flanged fittings. Dimensions of carbon and alloy steel butt-welding fittings are shown in Table 10-27. Butt-welding fittings are available in the wall thicknesses shown in Table 10-18. Butt-welding elbows with short, straight pipe extensions at the ends are also available for insertion in slip-on flanges. Schedule 5 and Schedule 10 stainless-steel buttwelding fittings are also available with such extensions for expanding into stainless-steel hubs mechanically locked in carbon steel ANSI B16.5 dimension flanges. The use of expanded joints (Fig. 10-138) is restricted by the code. Forged fittings made by boring out solid forgings are available with socket-weld (Fig. 10-126) or with screwed ends in sizes through 4 in, but 2 in is the usual upper size limit for use. ANSI B16.11—1973 gives minimum dimensions for socket-weld 3000- and 6000-lb classes and

TABLE 10-26

Dimensions of Flanged Fittings* All dimensions in inches

Nominal pipe size

ANSI B16.5, Class 150 ANSI B16.1, Class 125 AA

BB

CC

EE

ANSI B16.5, Class 300 ANSI B16.1, Class 250 FF

GG

AA

BB

CC

EE

ANSI B16.5, Class 400 FF

GG

a e

AA

CC

EE

FF

ANSI B16.5, Class 600 GG

AA

CC

EE

FF

GG

3.25 3.75

2.00 2.50

5.75 6.75

1.75 2.00

5.00 5.00

Use Class 600 dimensions in these sizes

1 1d 1a

3.50 3.75 4.00

5.00 5.50 6.00

1.75 2.00 2.25

5.75 6.25 7.00

1.75 1.75 2.00

4.50 4.50 4.50

4.00 4.25 4.50

5.00 5.50 6.00

2.25 2.50 2.75

6.50 7.25 8.50

2.00 2.25 2.50

4.50 4.50 4.50

4.25 4.50 4.75

2.50 2.75 3.00

7.25 8.00 9.00

2.25 2.50 2.75

5.00 5.00 5.00

2 2a 3 3a 4

4.50 5.00 5.50 6.00 6.50

6.50 7.00 7.75 8.50 9.00

2.50 3.00 3.00 3.50 4.00

8.00 9.50 10.00 11.50 12.00

2.50 2.50 3.00 3.00 3.00

5.00 5.50 6.00 6.50 7.00

5.00 5.50 6.00 6.50 7.00

6.50 7.00 7.75 8.50 9.00

3.00 3.50 3.50 4.00 4.50

9.00 10.50 11.00 12.50 13.50

2.50 2.50 3.00 3.00 3.00

5.00 5.50 6.00 6.50 7.00

8.00

5.50

16.00

4.50

8.25

5.75 6.50 7.00 7.50 8.50

4.25 4.50 5.00 5.50 6.00

10.25 11.50 12.75 14.00 16.50

3.50 3.50 4.00 4.50 4.50

6.00 6.75 7.25 7.75 8.75

5 6 8 10 12

7.50 8.00 9.00 11.00 12.00

10.25 11.50 14.00 16.50 19.00

4.50 5.00 5.50 6.50 7.50

13.50 14.50 17.50 20.50 24.50

3.50 3.50 4.50 5.00 5.50

8.00 9.00 11.00 12.00 14.00

8.00 8.50 10.00 11.50 13.00

10.25 11.50 14.00 16.50 19.00

5.00 5.50 6.00 7.00 8.00

15.00 17.50 20.50 24.00 27.50

3.50 4.00 5.00 5.50 6.00

8.00 9.00 11.00 12.00 14.00

9.00 9.75 11.75 13.25 15.00

6.00 6.25 6.75 7.75 8.75

16.75 18.75 22.25 25.75 29.75

5.00 5.25 5.75 6.25 6.50

9.25 10.00 12.00 13.50 15.25

10.00 11.00 13.00 15.50 16.50

7.00 7.50 8.50 9.50 10.00

19.50 21.00 24.50 29.50 31.50

6.00 6.50 7.00 8.00 8.50

10.25 11.25 13.25 15.75 16.75

14 16 18 20 24

14.00 15.00 16.50 18.00 22.00

21.50 24.00 26.50 29.00 34.00

7.50 8.00 8.50 9.50 11.00

27.00 30.00 32.00 35.00 40.50

6.00 6.50 7.00 8.00 9.00

16.00 18.00 19.00 20.00 24.00

15.00 16.50 18.00 19.50 22.50

21.50 24.00 26.50 29.00 34.00

8.50 9.50 10.00 10.50 12.00

31.00 34.50 37.50 40.50 47.50

6.50 7.50 8.00 8.50 10.00

16.00 18.00 19.00 20.00 24.00

16.25 17.75 19.25 20.75 24.25

9.25 10.25 10.75 11.25 12.75

32.75 36.25 39.25 42.75 50.25

7.00 8.00 8.50 9.00 10.50

16.50 18.50 19.50 21.00 24.50

17.50 19.50 21.50 23.50 27.50

10.75 11.75 12.25 13.00 14.75

34.25 38.50 42.00 45.50 53.00

9.00 10.00 10.50 11.00 13.00

17.75 19.75 21.75 23.75 27.75

10-84

Nominal pipe size

ANSI B16.5, Class 900 AA

CC

EE

ANSI B16.5, Class 1500 FF

GG

AA

CC

EE

a e

1 1d 1a

4.25 4.50 5.00 5.50 6.00

3.00 3.25 3.50 4.00 4.25

9.00 10.00 11.00

2 2a 3 4 5

4.75 5.25 5.75 7.25 8.75

FF

ANSI B16.5, Class 2500 GG

AA

CC

EE

FF

GG

2.50 3.00 3.50

5.00 5.75 6.25

5.19 5.37 6.06 6.87 7.56

4.00 4.25 4.75

13.25 15.25 17.25 19.25 23.25

4.00 4.50 5.00 6.00 7.50

7.25 8.25 9.25 10.75 13.75

8.87 10.00 11.37 13.25 15.62

5.75 6.25 7.25 8.50 10.00

15.25 17.25 19.75 23.00 27.25

5.25 5.75 6.75 7.75 9.25

9.50 10.50 11.75 13.50 15.75

18.00 20.12 25.00 28.00

11.50 12.75 16.00 17.75

31.25 35.25 43.25 49.25

10.50 11.75 14.75 16.25

18.00 20.50 25.50 29.00

Use Class 1500 dimensions in these sizes

7.50 9.00 11.00

5.50 6.50 7.50

14.50 17.50 21.00

4.50 5.50 6.50

7.75 9.25 11.25

7.25 8.25 9.25 10.75 13.25

6 8 10 12 14

12.00 14.50 16.50 19.00 20.25

8.00 9.00 10.00 11.00 11.50

22.50 27.50 31.50 34.50 36.50

6.50 7.50 8.50 9.00 9.50

12.25 14.75 16.75 17.75 19.00

13.88 16.38 19.50 22.25 24.75

9.38 10.88 12.00 13.25 14.25

24.88 29.88 36.00 40.75 44.00

8.12 9.12 10.25 12.00 12.50

14.50 17.00 20.25 23.00 25.75

16 18 20 24

22.25 24.00 26.00 30.50

12.50 13.25 14.50 18.00

40.75 45.50 50.25 60.00

10.50 12.00 13.00 15.50

21.00 24.50 26.50 30.50

27.25 30.25 32.75 38.25

16.25 17.75 18.75 20.75

48.25 53.25 57.75 67.25

14.75 16.50 17.75 20.50

28.25 31.50 34.00 39.75

*Outline drawings show d-in (6.5-mm) raised face machined onto flange, as for ANSI B16.5 400-lb and higher. ANSI B16.1 250-lb and ANSI B16.5 150- and 300-lb have g-in (1.5-mm) raised face; ANSI B16.1 125-lb has no raised face. See Tables 10-19 through 10-25 for flange drillings. Dimensions for 400- and 600-lb fittings are identical for sizes a to 3a in inclusive. Dimensions for 900- and 1500-lb fittings are identical for sizes a to 2a in inclusive. To convert inches to millimeters, multiply by 25.4. The dimensions were extracted from Cast-Iron Pipe Flanges and Flanged Fittings, ANSI B16.1—1975, and Steel Pipe Flanges and Flanged Fittings, ANSI B16.5—1977, with permission of the publisher, the American Society of Mechanical Engineers, New York.

10-85

10-86

TRANSPORT AND STORAGE OF FLUIDS

TABLE 10-27

Butt-Welding Fittings* All dimensions in inches

Pipe size

A

K

a e (1) 1 1d 1a

1.50 1.12 1.50 1.88 2.25

1.88 1.69 2.19 2.75 3.25

2 2a 3 3a 4

3.00 3.75 4.50 5.25 6.00

5 6 8 10 12 14 16 18 20 24

A1

K1

B

O

O1

M, C

H

E†

G

F

F1

R‡

1.00 1.25 1.50

1.62 2.06 2.44

0.62 0.44 0.88 1.00 1.12

3.00 2.25 3.00 3.75 4.50

2.00 2.50 3.00

1.00 1.12 1.50 1.88 2.25

1.50 2.00 2.00 2.50

1.00 1.00 1.50 1.50 1.50

1.38 1.69 2.00 2.50 2.88

3.00 3.00 4.00 4.00 4.00

2.00 2.00 2.00 2.00 2.00

0.12 0.12 0.12 0.19 0.25

4.19 5.19 6.25 7.25 8.25

2.00 2.50 3.00 3.50 4.00

3.19 3.94 4.75 5.50 6.25

1.38 1.75 2.00 2.25 2.50

6.00 7.50 9.00 10.50 12.00

4.00 5.00 6.00 7.00 8.00

2.50 3.00 3.38 3.75 4.12

3.00 3.50 3.50 4.00 4.00

1.50 1.50 2.00 2.50 2.50

3.62 4.12 5.00 5.50 6.19

6.00 6.00 6.00 6.00 6.00

2.50 2.50 2.50 3.00 3.00

0.31 0.31 0.38 0.38 0.44

7.50 9.00 12.00 15.00 18.00

10.31 12.31 16.31 20.38 24.38

5.00 6.00 8.00 10.00 12.00

7.75 9.31 12.31 15.38 18.38

3.12 3.75 5.00 6.25 7.50

15.00 18.00 24.00 30.00 36.00

10.00 12.00 16.00 20.00 24.00

4.88 5.62 7.00 8.50 10.00

5.00 5.50 6.00 7.00 8.00

3.00 3.50 4.00 5.00 6.00

7.31 8.50 10.62 12.75 15.00

8.00 8.00 8.00 10.00 10.00

3.00 3.50 4.00 5.00 6.00

0.44 0.50 0.50 0.50 0.50

21.00 24.00 27.00 30.00 36.00

28.00 32.00 36.00 40.00 48.00

14.00 16.00 18.00 20.00 24.00

21.00 24.00 27.00 30.00 36.00

8.75 10.00 11.25 12.50 15.00

42.00 48.00 54.00 60.00 72.00

28.00 32.00 36.00 40.00 48.00

11.00 12.00 13.50 15.00 17.00

13.00 14.00 15.00 20.00 20.00

6.50 7.00 8.00 9.00 10.50

16.25 18.50 21.00 23.00 27.25

12.00 12.00 12.00 12.00 12.00

6.00 6.00 6.00 6.00 6.00

0.50 0.50 0.50 0.50 0.50

*Extracted from Wrought-Steel Butt-Welding Fittings, ANSI B16.9—1978, and from Wrought-Steel Butt-Welding Short-Radius Elbows and Returns, ANSI B16.28—1978, with permission of the publisher, the American Society of Mechanical Engineers, New York. A and B dimensions of 1.50 and 0.75 in respectively may be furnished for NPS e at the manufacturer’s option. O and K dimensions may likewise be furnished in 2.00 in and 3.00 in respectively. †For wall thicknesses greater than extra heavy, E is greater than shown here for sizes 2 in and larger. ‡For MSS SP-43 type B stub ends, which are designed to be backed up by slip-on flanges, R = h in for 4 in and smaller and g in for 6 through 12 in. To convert inches to millimeters multiply by 25.4.

for screwed 2000-, 3000-, and 6000-lb classes. It also contains pressure-temperature ratings for the classes in various ferrous alloys. The use of socket-weld and screwed fittings is restricted by the code. Steel forged fittings with screwed ends may be installed without pipe dope in the threads and seal-welded (Fig. 10-130) to secure bubble-tight joints with a minimum of welders’ labor. They are not subject to deformation by pipe wrenches, and such couplings, bushings, and plugs are often used with the screwed fittings below. ANSI B16.3—1977 gives dimensions of 150-lb malleable-iron screwed fittings through the 6-in size for 1.0 MPa (150 lbf/in2) saturated steam and 2.1 MPa (300 lbf/in2) at room temperature and for 300-lb malleable-iron screwed fittings through the 3-in size for 2.1 MPa (300 lbf/in2) steam at 290°C (550°F) or 7.0 MPa (1000 lbf/in2) at room temperature. These fittings are available with male threads or unions on one end for installation in confined spaces. Major use is in 150-lb elbows, tees, and reducers in sizes 2 in and smaller. They are less costly than forged fittings but cannot be seal-welded. The code does not permit the use of malleable iron in toxic service or in flammable service above either 150°C (300°F) or 2.76 MPa (400 lbf/in2) gauge. ANSI B16.4—1977 gives dimensions of 125-lb cast-iron screwed fittings through the 12-in size for 0.86 MPa (125 lb/in2) saturated steam and 1.2 MPa (175 lbf/in2) at 66°C (150°F) and of 250-lb cast-iron screwed fittings through the 12-in size for 1.72 MPa (250 lbf/in2) saturated steam and for 2.76 MPa (400 lbf/in2) at 66°C (150°F). The 125-lb fittings are made in regular 90° and 45° elbows, reducing elbows, regular and reducing tees, and crosses. The 250-lb fittings are made only

in straight sizes. Major use is in 125-lb elbows, tees, and reducers in low-pressure noncritical service. The code does not permit the use of cast iron in toxic service or aboveground within process unit limits for flammable-fluid service above 150°C (300°F) or 1.0 MPa (150 lbf/in2). Tees Tees may be cast, forged, or hot- or cold-formed from short pieces of pipe. Though it is impossible to have the same flow simultaneously through all three end connections, it is not economical to produce or stock the great variety of tees which accurate sizing of end connections requires. It is customary to stock only tees with the two end (run) connections of the same size and the branch connection either of the same size as the run connections or one, two, or three sizes smaller. Adjacent reducers or reducing elbow fittings are used for other size reductions. Branch connections (see subsection “Joints”) are often more economical than tees, particularly when the ratio of branch to run is small. Reducers Reducers may be cast, forged, or hot- or cold-formed from short pieces of pipe. End connections may be concentric or eccentric, that is, tangent to the same plane at one point on their circumference. For pipe supported by hangers, concentric reducers permit maintenance of the same hanger length; for pipe laid on structural steel, eccentric reducers permit maintaining the same elevation of top of steel. Eccentric reducers with the common tangent plane below permit complete drainage of branched horizontal piping systems through branches smaller than the main. With the common tangent plane above, they permit liquid flow in horizontal lines to sweep the line free of gas or vapor.

PROCESS-PLANT PIPING Reducing elbow fittings permit change of direction and concentric size reduction in the same fitting. Valves Valve bodies may be cast, forged, machined from bar stock, or fabricated from welded plate. Steel valves are available with screwed or socket-weld ends in the smaller sizes. Bronze and brass screwed-end valves are widely used for low-pressure service in steel systems. Table 10-28 gives contact-surface-of-face to contact-surfaceof-face dimensions for flanged ferrous valves and end-to-end dimensions for butt-welding ferrous valves. Drilling of end flanges is shown in Tables 10-19 to 10-25. Bolt holes are located so that the stem is equidistant from the centerline of two bolt holes. Even if removal for maintenance is not anticipated, flanged valves are frequently used instead of butt-welding-end valves because they permit insertion of blanks for isolating sections of a loop piping system. Ferrous valves are also available in nodular (ductile) iron, which has tensile strength and yield point approximately equal to cast carbon steel at temperatures of 343°C (650°F) and below and only slightly less elongation. Valves serve not only to regulate the flow of fluids but also to isolate piping or equipment for maintenance without interrupting other connected units. Valve design should keep pressure, temperature changes, and strain from connected piping from distorting or misaligning the sealing surfaces. The sealing surfaces should be of such material and design that the valve will remain tight over a reasonable service period. The principal types are named, described, compared, and illustrated with line diagrams in subsequent subsections. In the line diagrams, the operating stem is shown in solid black, direction of flow by arrows on a thin solid line, and motion of valve parts by arrows on a dotted line. Moving parts are drawn with solid lines in the nearly closed position and with dotted lines in the fully open position. Packing is represented by an X in a square. Gate Valves These valves are designed in two types (Fig. 10-148). The wedge-shaped-gate, inclined-seat type is most commonly used. The wedge gate is usually solid but may be flexible (partly cut into halves by a plane at right angles to the pipe) or split (completely cleft by such a plane). Flexible and split wedges minimize galling of the sealing surfaces by distorting more easily to match angularly misaligned seats. In the double-disk parallel-seat type, an inclined-plane device mounted between the disks converts stem force to axial force, pressing the disks against the seats after the disks have been positioned for closing. This gate assembly distorts automatically to match both angular misalignment of the seats and longitudinal shrinkage of the valve body on cooling. When shearing high-velocity flow of dense fluids, the gate assemblies shake violently, and for this service solid-wedge or flexible-

FIG. 10-148

Gate valve.

10-87

wedge valves are preferred. When valve operation is manual, small bypass valves installed in parallel with the main valve may be used to eliminate the shake problem and to minimize manual effort in opening and closing the valves. Double-disk parallel-seat valves should be installed with the stem essentially vertical. All wedge gate valves are equipped with tongue-and-groove guides to keep the gate sealing surfaces from clattering on the seats and marring them during opening and closing. Depending on the velocity and density of the fluid stream being sheared, these guiding surfaces may be as cast, machined, or hard-surfaced and ground. Gate valves may have nonrising stems, inside-screw rising stems, or outside-screw rising stems, listed in order of decreasing exposure of the stem threads to the fluid handled. Rising-stem valves require more space, but the position of the stem visually indicates the position of the gate. Indication is clearest on the outside-screw rising-stem valves, and on these the stem threads and thrust collars may be lubricated, reducing operating effort. The stem connection to the gate assembly prevents the stem from rotating. Gate valves are used to minimize pressure drop in the open position and to stop the flow of fluid rather than to regulate it. The problem, when the valve is closed, of pressure buildup in the bonnet from cold liquids expanding or chemical action between fluid and bonnet should be solved by a relief valve or by notching the upstream seat ring. Globe Valves (Fig. 10-149) These are designed as either insidescrew rising-stem or outside-screw rising-stem. Small valves generally are of the inside-screw type, while in larger sizes the outside-screw type is preferred. In most designs the disks are free to rotate on the stems; this prevents galling between the disk and the seat. In the larger sizes, with conical seats, this swivel may permit enough misalignment to prevent proper sealing between the disk and the seat. When the valve is close to an elbow on the upstream side, the swivel also permits uneven distribution of the fluid to spin the disk on the stem. Guides above the disk, below the disk, or both are used to prevent misalignment and spinning. Misalignment can also be prevented by the use of spherical seats and designing the disk so that the pressure point of the stem on the disk is at the center of the sphere. In some designs, spinning and misalignment are prevented by rigidly attaching the disk to the stem, preventing rotation of the stem by lugs which ride along the yoke, and using a yoke bushing as in outsidescrew-and-yoke gate valves. Large globe valves should be installed with stems vertical. Globe valves are preferably installed with the higher-pressure side connected to the top of the disk. Exceptions occur (1) when blocked flow caused by separation of the disk from the stem would damage equipment or (2) when the valve is installed in seldom-used vertical drain lines in which accumulation of rust, scale, or sludge might prevent opening the valve. Pressure drop through globe valves is much greater than that for gate valves. In Y-type globe valves, the stem and seat are at about 45° to the pipe instead of 90°. This reduces pressure drop but impairs alignment of seat and disk.

FIG. 10-149

Globe valve.

10-88

TRANSPORT AND STORAGE OF FLUIDS

TABLE 10-28

Dimensions of Valves* All dimensions in inches

Class 125 cast iron

Class 150 steel, MSS-SP-42 through 12-in size

Class 250 cast iron

Flanged end

Welding end

Gate

Gate

Gate

Solid wedge and double disk A

Solid wedge and double disk B

Globe and lift check A and B

Angle and lift check D and E

Swing check A and B

Solid wedge and double disk A

Globe, lift check, and swing check A

Angle and lift check D

d r a e 1

4 4 4d 4v 5

4 4 4d 4v 5

4 4 4d 4v 5

2 2 2d 2a 2e

4 4 4d 4v 5

1d 1a 2 2a 3

7 7a 8

7 7a 8

8 8a 9a

4 4d 4e

8 8a 9a

5a 6a 7 7a 8

5a 6a 8a 9a 11f

5a 6a 8 8a 9a

3 3d 4 4d 4e

5a 6a 8 8a 9a

8a 9a 11f

10a 11a 12a

5d 5e 6d

3a 4 5 6 8

8a 9 10 10a 11a

8a 9 10 10a 11a

11a 13 14 19a

5e 6a 7 9e

11a 13 14 19a

† 9 10 10a 11a

12 15 157⁄ 8 16a

11a 14 16 19a

5e 7 8 9e

11a 13 14 19a

12 15 157⁄ 8 16a 18

14 15e 17a 21 24a

7 77⁄ 8 8e 10a 12d

10 12 14 16 18

13 14 15 16 17

13 14 † † †

24a 27a 31 36

12d 13e 15a 18

24a 27a 31 † †

13 14 15 16 17

18 19e 22a 24 26

24a 27a 31 36

12d 13e 15a 18

24a 27a 31 † †

19e 22a 24 26 28

28 † †

14

20 24

18 20

† †

† †

18 20

28 32

Flanged end Gate

Nominal valve size

Solid wedge A

Globe and lift check A

Double disk A

Angle and lift check D

Swing check A

Flanged end and welding end

Flanged end and welding end

Flanged end and welding end

Gate

Angle and lift check D and E

6 7 8 8a 9

8a 9a 11f 12 15

6 8 10 12 14

157⁄ 8 16a 18 19e 30

16 18 20 22 24

33 36 39 43 45

1 1d 1a 2 2a 3 4 5

31

Class 600 steel

Solid wedge and double disk A and B 5a 6 6a§ 7§ 7a

† †

Class 400 steel

Globe and lift check A and B

a e

Flanged end

Class 300 steel

Gate

Nominal valve size

Flanged end and welding end

Gate

Double disk A and B

Globe, lift check, and swing check A and B

Angle and lift check D and E

Solid wedge A and B

3d 3e 4d 4a 4e

6a 7a 8a 9 9a

Swing check A and B

Solid wedge A and B

3 3a 4 4d 4a

8a 9 9a

6a 7a 8a 9 9a

8a 9 9a

6a 7a 8a 9 9a

10a 11a 12a 14 15e

5d 5e 6d 7 77⁄ 8

10a 11a 12a 14 15e

11a 13 14 16 18

11a 13 14 16 18

11a 13 14 16 18

5e 6a 7 8 9

17a 22 24a 28

8e 11 12d 14

17a 21 24a 28 †

19a 23a 26a 30 32a

19a 23a 26a 30 30a

19a 23a 26a 30 †

9e 11e 13d 15

† † † † †

35a 38a 41a 45 48a

35a 38a 41a 45 48a

† † † † †

Angle and lift check

Short pattern‡ B

Regular globe, regular lift check, swing check A and B

8a 9 9a

5d 5e 6

6a 7a 8a 9 9a

5d 5e 6

3d 3e 4d 4a 4e

11a 13 14 17 20

11a 13 14 17 20

7 8a 10 12 15

11a 13 14 17 20

7 8a 10 12 15

5e 6a 7 8a 10

22 26 31 33 35

22 26 31 33 35

18 23 28 32 35

22 26 31 33 †

18 23 28 32

11 13 15a 16a

39 43 47 51 55

39 43 47 51 55

39 43 47

† † † † †

Double disk A and B

55

Short pattern‡ globe, short pattern lift check B

Regular D and E

Short pattern E

4d 5 6 7 8a 10

PROCESS-PLANT PIPING TABLE 10-28

Dimensions of Valves (Concluded ) Class 900 steel

Class 1500 steel

Flanged end and welding end

Flanged end and welding end

Double disk A and B

Short pattern‡ B

Regular globe regular lift check, swing check A and B 9

4a

5a 6a 7 8a

10 11 12 14a

5 5a 6 7d

10 11 12 14a

14a

6 7 8a 10

16a 18a 21a 26a 27e

16a 18a 21a 26a 27e

10 12 16 19 22

16a 18a 21a 26a 27e

8d 9d 10e 13d 137⁄8

13 15a 18 19a

32e 39 44a 49a 54a

32e 39 44a 49a 54a

28 34 39 42 47

32e 39 44a 49a

16r 19a 22d 24e

60a 65a 76a

60a 65a 76a

53 58

Gate

Nominal valve size

10-89

Solid wedge A and B

e

Short pattern‡ globe, short pattern lift check B

Angle and lift check

Regular D and E

1 1d 1a 2

10 11 12 14a

14a

2a 3 4 5 6

16a 15 18 22 24

16a 15 18 22 24

10 12 14 17 20

16a 15 18 22 24

12 14 17 20

8d 7a 9 11 12

8 10 12 14 16

29 33 38 40a 44a

29 33 38 40a 44a

26 31 36 39 43

29 33 38 40a

26 31 36 39

14a 16a 19 20d

18 20 24

48 52 61

48 52 61

† † †

Short pattern E

Gate

Solid wedge A and B

Double disk A and B

Short pattern‡ B

Globe, lift check, swing check A and B 9

4a

5a 6a 7 8a

10 11 12 14a

5 5a 6 7d

Angle and lift check D and E

Class 2500 steel Flanged end and welding end Gate

Nominal valve size

Solid wedge A and B

Double disk A and B

Short pattern‡ B

Globe, lift check, swing check A and B

Angle and lift check B

a

10r

10r

5t

e 1 1d 1a

10e 12f 13e 15f

10e 12f 13e 15f

5r 6g 67⁄8 7z

2 2a 3 4 5

17e 20 22e 26a 31d

17e 20 22e 26a 31d

11 13 14a 18 21

17e 20 22e 26a 31d

87⁄8 10 11r 13d 15v

36 40d 50 56

36 40d 50 56

24 30 36 41 44

36 40d 50 56

18 20f 25 28

6 8 10 12 14 16 18

7b 9f 9f

49 55

NOTE: Outline drawings for flanged valves shown d-in raised face machined onto flange, as for 400-lb cast-steel valves; 150- and 300-lb cast-steel valves and 250-lb cast-iron valves have g-in raised faces; 125-lb cast-iron and 150-lb corrosion-resistant valves covered by MSS-SP-42 have no raised faces. *Extracted from Face-to-Face and End-to-End Dimensions of Ferrous Valves, ANSI B16.10—1973, with permission of the publisher, the American Society of Mechanical Engineers, New York. To convert inches to millimeters, multiply by 25.4. †Not shown in ANSI B16.10 but commercially available. ‡These dimensions apply to pressure-seal or flangeless bonnet valves only. §Solid wedge only.

10-90

FIG. 10-150

TRANSPORT AND STORAGE OF FLUIDS

FIG. 10-151

Diaphragm valve.

FIG. 10-152

Plug cock.

Angle valve.

Globe valves in horizontal lines prevent complete drainage. Seatwiper valves in which the disk may be rotated by a separate stem inside and concentric with the main stem are used to clear the seats of solid deposits. Angle Valves These valves are similar to globe valves; the same bonnet, stem, and disk are used for both (Fig. 10-150). They combine an elbow fitting and a globe valve into one component with a substantial saving in pressure drop. Flanged angle valves are easier to remove and replace than flanged globe valves. Diaphragm Valves These valves are limited to pressures of approximately 50 lbf/in2 (Fig. 10-151). The fabric-reinforced diaphragms may be made from natural rubber, from a synthetic rubber, or from natural or synthetic rubbers faced with Teflon* fluorocarbon resin. The simple shape of the body makes lining it economical. Elastomers have shorter lives as diaphragms than as linings because of flexing but still provide satisfactory service. Plastic bodies, which have low moduli of elasticity compared with metals, are practical in diaphragm valves since alignment and distortion are minor problems. These valves are excellent for fluids containing suspended solids and can be installed in any position. Models in which the dam is very low, reducing pressure drop to a negligible quantity and permitting complete drainage in horizontal lines, are available. However, drainage can be obtained with any model simply by installing it with the stem horizontal. The only maintenance required is replacement of the diaphragm, which can be done very quickly without removing the valve from the line. Plug Cocks These valves (Fig. 10-152) are limited to temperatures below 260°C (500°F) since differential expansion between the plug and the body results in seizure. The size and shape of the port divide these valves into different types. In order of increasing cost they are short venturi, reduced rectangular port; long venturi, reduced rectangular port; full rectangular port; and full round port. In lever-sealed plug cocks, tapered plugs are used. The plugs are raised by turning one lever, rotated by another lever, and reseated by the first lever. Lubricated plug cocks may use straight or tapered plugs. The tapered plugs may be raised slightly, to reduce turning effort, by injection of the lubricant, which also acts as a seal. Plastic is used in nonlubricated plug cocks as a body liner, a plug coating, or port seals in the body or on the plug. In plug cocks other than lever-sealed plug cocks, the contact area between plug and body is large, and gearing is usually used in sizes 6 in and larger to minimize operating effort. There are several leversealed plug cocks incorporating mechanisms which convert the rotary motion of a handwheel into sequenced motion of the two levers. For lubricated plug cocks, the lubricant must have limited viscosity change over the range of operating temperature, must have low solubility in the fluid handled, and must be applied regularly. There must be no chemical reaction between the lubricant and the fluid which * Du Pont TFE fluorocarbon resin.

would harden or soften the lubricant or contaminate the fluid. For these reasons, lubricated plug cocks are most often used when there are a large number handling the same or closely related fluids at approximately the same temperature. Lever-sealed plug cocks are used for throttling service. Because of the large contact area between plug and body, if a plug cock is operable, there is little likelihood of leakage when closed, and the handle position is a clearly visible indication of the valve position. Ball Valves (Figs. 10-153 and 10-154) These valves are limited to temperatures that have little effect on their plastic seats. Since the sealing element is a ball, its alignment with the axis of the stem is not essential to tight shutoff. In free-ball valves the ball is free to move axially. Pressure differential across the valve forces the ball in the closed position against the downstream seat and the latter against the body. In fixed-ball valves, the ball rotates on stem extensions, with the bearings sealed with O rings. Plastic seats may be compressed or springloaded against the ball and the body by the assembly of the valves, or they may be forced against the ball by pressure across the valve acting against O rings which seal between the seat and the body. Ball valves in which the ball and seats are inserted from above are known as top-entry ball valves. Replacement of seats is easiest in this

FIG. 10-153

Ball valve; free ball.

PROCESS-PLANT PIPING

FIG. 10-156

FIG. 10-154

Ball valve; fixed ball.

type. The others are known as split-body valves. Some of these incorporate bolted assembly which permits their use as joints for assembly of the piping. Replacement of seats in this type is easiest when the body consists of three pieces with the ball and the seats contained in the middle piece. For the larger sizes in high-pressure service, the fixed-ball type with O-ring seat seals requires less operating effort. However, these require two different plastic materials with resistance to the fluid and its temperature. Like plug cocks, ball valves may be either restrictedport or full-port, but the ports are always round and pressure drop is low. Butterfly Valves These valves (Fig. 10-155) occupy less space in the line than any other valves. Relatively tight sealing without excessive operating torque and seat wear is accomplished by a variety of methods, such as resilient seats, piston rings on the disk, and inclining the stem to limit contact between the portions of disk closest to the stem and the body seat to a few degrees of curvature. Fluid-pressure distribution tends to close the valve. For this reason, the smaller manually operated valves have a latching device on the handle, and the larger manually operated valves use worm gearing on the stem. This hydraulic unbalance is proportional to the pressure drop and, with line velocities exceeding 7.6 m/s (25 ft/s), is the principal component in the torque required to operate the valves. Compared with other valves for low-pressure drops, these valves can be operated by smaller hydraulic cylinders. In this service butterfly valves with insert bodies for bolting between existing flanges with bolts that pass by the body are the lowest-first-cost valve in pipe sizes 10 in and larger. Pressure drop is quite high compared with that of gate valves. Swing Check Valves These valves (Fig. 10-156) are used to prevent reversal of flow. Normal design is for use only in horizontal lines, where the force of gravity on the disk is at a maximum at the start of closing and at a minimum at the end of closing. Unlike most other valves, check valves are more likely to leak at low pressure than at high pressure, since fluid pressure alone forces the disk to conform to the seat. For this reason elastomers are often mounted on the disk. Swing check valves are available with low-cost insert bodies.

FIG. 10-155

Butterfly valve.

10-91

Swing check valve.

Lift Check Valves These valves (Figs. 10-157 to 10-159) are made in three styles. Vertical lift check valves are for installation in vertical lines, where the flow is normally upward; globe check valves are for use in horizontal lines; angle check valves are for installation where a vertical line with upward flow turns horizontal. Globe and angle check valves normally incorporate an integral dashpot above the disk to slow the motion of the disk and reduce wear. In vertical lift check valves, this feature is found only in the larger sizes. Springs may be incorporated in the dashpots to speed closing, but this increases the pressure drop. Lift checks should not be used when the fluid contains suspended solids. Tilting-Disk Check Valves These valves (Fig. 10-160) may be installed in a horizontal line or in lines in which the flow is vertically upward. The pivot point is located so that the distribution of pressure in the fluid handled speeds the closing but arrests slamming. Compared with swing check valves of the same size, pressure drop is less at low velocities but greater at high velocities.

FIG. 10-157

Lift check valve, vertical.

FIG. 10-158

Lift check valve, globe.

FIG. 10-159

Lift check valve, angle.

10-92

FIG. 10-160

TRANSPORT AND STORAGE OF FLUIDS

Tilting-disk check valve.

Closure at the instant of reversal of flow is most nearly attained in these valves. This timing of closure is not the whole solution to noise and shock at check valves. For example, if cessation of pressure at the inlet of a valve produces flashing of the decelerating stream downstream from the valve or if stoppage of flow is caused by a sudden closure of a valve some distance downstream from the check valve and the stoppage is followed by returning water hammer, slower closure may be necessary. For these applications, tilting-disk check valves are equipped with external dashpots. They are also available with low-cost insert bodies. Valve Trim Various alloys are available for valve parts such as seats, disks, and stems which must retain smooth finish for successful operation. The problem in seat materials is fivefold: (1) resistance to corrosion by the fluid handled and to oxidation at high temperatures, (2) resistance to erosion by suspended solids in the fluid, (3) prevention of galling (seizure at point of contact) by differences in material or hardness or both, (4) maintenance of high strength at high temperature, and (5) avoidance of distortion. All valve trim materials have coefficients of thermal expansion which exceed those of cast or forged carbon steel by 24 to 45 percent and tend to cause distortion of seats and disks. To some extent leakage from this cause is prevented by closing the valve more tightly. Inserting a ring of high-temperature elastomer or plastic, either in or alongside the trim metal in the seat or disk, prevents leakage from this cause. CAST IRON, DUCTILE IRON, AND HIGH-SILICON IRON Cast Iron and Ductile Iron Cast iron and ductile iron provide more metal for less cost than steel in piping systems and are widely TABLE 10-29

used in low-pressure services in which internal and external corrosion may cause a considerable loss of metal. They are widely used for underground water distribution. Cement lining is available at a nominal cost for handling water causing tuberculation. Ductile iron has an elongation of 10 percent or more compared with essentially nil elongation for cast iron and has for all practical purposes supplanted cast iron as a cast piping material. It is usually centrifugally cast in rapidly revolving molds. This manufacturing method improves tensile strength and reduces porosity. Ductile-iron pipe is manufactured to ANSI A21.51—1976 and is available in nominal sizes from 3 through 54 in. Wall thicknesses are specified by seven standard thickness classes. Table 10-29 gives the outside diameter and standard thickness for various rated water working pressures for centrifugally cast ductile-iron pipe. The required wall thickness for underground installations increases with internal pressure, depth of laying, and weight of vehicles operating over the pipe. It is reduced by the degree to which the soil surrounding the pipe provides uniform support along the pipe and around the lower 180°. Tables are provided in ANSI A21.51 for determining wall-thickness-class recommendations for various installation conditions. The poured joint (Fig. 10-135) has been almost entirely superseded by the mechanical joint (Fig. 10-136) and the push-on joint (Fig. 10-137), which are better suited to wet trenches, bad weather, and unskilled labor and minimize strain on the pipe from ground settlement. Lengths vary between 5 and 6 m (between 18 and 20 ft), depending on the supplier. Stock fittings are designed for 1.72-MPa (250-lbf/in2) cast iron or 2.41-MPa (350-lbf/in2) ductile iron in sizes through 12 in and for 1.0- and 1.72-MPa (150- and 250-lbf/in2) cast iron or 2.41-MPa (350-lbf/in2) ductile iron in sizes 14 in and larger. Stock fittings include 22a° and 11d° bends. Ductile-iron pipe is also supplied with flanges that match the dimensions of Class 125 flanges shown in ANSI B16.1 (see Table 10-19). These flanges are assembled to the pipe barrel by threaded joints. High-Silicon Iron Duriron is a high-silicon iron containing approximately 14.5 percent silicon and 0.85 percent carbon. Durichlor is a special high-silicon iron containing appreciable amounts of molybdenum. These alloys are available in the cast form only. Pipe and fittings are cast with the upset ends being joined by split flanges. Integrally cast flanged pipe is also available. Allowable working pressures cannot be

Dimensions of Ductile-Iron Pipe* Standard thickness for internal pressure† Rated water working pressure, lbf/in2‡ 150

200

250

300

350

Pipe size, in.

Outside diameter, in

Thickness, in

Thickness class

Thickness, in

Thickness class

Thickness, in

Thickness class

Thickness, in

Thickness class

Thickness, in

Thickness class

3 4 6 8 10

3.96 4.80 6.90 9.05 11.10

0.25 0.26 0.25 0.27 0.29

51 51 50 50 50

0.25 0.26 0.25 0.27 0.29

51 51 50 50 50

0.25 0.26 0.25 0.27 0.29

51 51 50 50 50

0.25 0.26 0.25 0.27 0.29

51 51 50 50 50

0.25 0.26 0.25 0.27 0.29

51 51 50 50 50

12 14 16 18 20

13.20 15.30 17.40 19.50 21.60

0.31 0.33 0.34 0.35 0.36

50 50 50 50 50

0.31 0.33 0.34 0.35 0.36

50 50 50 50 50

0.31 0.33 0.34 0.35 0.36

50 50 50 50 50

0.31 0.33 0.34 0.35 0.36

50 50 50 50 50

0.31 0.33 0.34 0.35 0.39

50 50 50 50 51

24 30 36 42 48

25.80 32.00 38.30 44.50 50.80

0.38 0.39 0.43 0.47 0.51

50 50 50 50 50

0.38 0.39 0.43 0.47 0.51

50 50 50 50 50

0.38 0.43 0.48 0.53 0.58

50 51 51 51 51

0.41 0.47 0.53 0.59 0.65

51 52 52 52 52

0.44 0.51 0.58 0.65 0.72

52 53 53 53 53

54

57.10

0.57

50

0.57

50

0.65

51

0.73

52

0.81

53

*Extracted from the American National Standard for Ductile-Iron Pipe, Centrifugally Cast in Metal Molds or Sand-Lined Molds, for Water or Other Liquids, ANSI A21.51—1976, with permission of the publisher, the American Society of Mechanical Engineers, New York. †To convert from inches to millimeters, multiply by 25.4; to convert pounds-force per square inch to megapascals, multiply by 0.006895. ‡These pipe walls are adequate for the rated working pressure plus a surge allowance of 100 lbf/in2. For the effect of laying conditions and depth of bury, see ANSI A21.51.

PROCESS-PLANT PIPING TABLE 10-30

High-Silicon Iron Pipe* Split flanged ends

Size, inside diam., in 1 1a 2 2a 3 4 6 8 10 12 15

Bell-and-spigot ends

Outside diam., in

Wall thickness, in

Standard† length, ft

Weight per piece, lb

1e 2d 2e 3d 37⁄ 8

r r r r q

3 3 4 5 5

15 18 32 45 62

47 ⁄ 8 7 9d 11a 14

q a v e 1

5 5 6 6 6

100 180 265 433 694

Outside diam., in

Wall thickness, in

Standard† length, ft

Weight per piece, lb

2f 2v

b b

3 4

20 30

3x

11 32

⁄

5

68

4v 6x 9 11d 13d

b 11 ⁄ 32 a v v

5 5 5 5 5

89 133 232 341 463

⁄

5

680

16e

7 8

*The Duriron Co. †Laying lengths; lengths less than standard are available. NOTE: To convert inches to millimeters, multiply by 25.4; to convert feet to meters, multiply by 0.3048; and to convert pounds to kilograms, multiply by 0.4536.

stated in the manner customary for other types of pipe because of such variables as thermal shock, pulsating pressures, and the corrosive fluids being handled. Although rupture does not occur below 2.76MPa (400-lbf/in2) pressure in sizes up to and including 6 in, 0.3 MPa (50 lbf/in2) is a normal recommendation, even though the pipe has been used for pressure considerably in excess of that figure. Table 10-30 lists sizes 1 to 12 in, and larger sizes can be obtained. Bell-and-spigot pipe is produced in the weights and dimensions shown in Table 10-30; fittings are available. New hubless pipe utilizing TFE gaskets and stainless steel clamps to make a mechanical joint are now available (the trade name is Duriron MJ). The coefficient of linear expansion of these alloys in the temperature range of 21 to 100°C (70 to 212°F) is 12.2 × 10−6/°C (6.8 × 10−6/°F), which is slightly above that of cast iron (National Bureau of Standards). Since these alloys have practically no elasticity, it is necessary to use expansion joints in relatively short pipe lines. Connections for flanged pipe, fittings, valves, and pumps are made to 125-lb American Standard drilling. The use of high-silicon iron in flammable-fluid service or in Category M fluid service is prohibited by the code. NONFERROUS-METAL PIPING SYSTEMS Aluminum Seamless aluminum pipe and tube are produced by extrusion in essentially pure aluminum and in several alloys; 6-, 9-, and 12-m (20-, 30-, and 40-ft) lengths are available. Alloying and mill treatment improve physical properties, but welding reduces them. Essentially pure aluminum has an ultimate tensile strength of 65.5 MPa (9500 lbf/in2) subject to a slight increase by mill treatment which is lost during welding. Alloy 6061, which contains 0.25 percent copper, 0.6 percent silicon, 1 percent magnesium, and 0.25 percent chromium, has an ultimate tensile strength of 124 MPa (18,000 lbf/in2) in the annealed condition, 262 MPa (38,000 lbf/in2), milltreated as 6061-T6, and 165 MPa (24,000 lbf/in2) at welded joints. Extensive use is made of alloy 1060, which is 99.6 percent pure aluminum, for hydrogen peroxide; of alloy 3003, which contains 1.2 percent manganese, for high-purity chemicals; and of alloys 6063 and 6061 for many other services. Alloy 6063 is the same as 6061 minus the chromium and has slightly lower mechanical properties. Aluminum is not embrittled by low temperatures and is not subject to external corrosion when exposed to normal atmospheres. At 200°C (400°F) its strength is less than half that at room temperature. It is attacked by alkalies, by traces of copper, nickel, mercury, and other heavy-metal ions, and by prolonged contact with wet insulation. It suffers from galvanic corrosion when coupled to copper, nickel, or lead-

10-93

base alloys but not when coupled to galvanized iron or austenitic stainless steel. Aluminum pipe is stocked in 3003, 6061, and 6063 Schedule 40 through 10 in, Schedule 30, 8 through 10 in, and standard-weight 12-in size. It is also stocked in 6063 as Schedule 5 through 6 in and Schedule 10 through 8 in (see Table 10-18). Threaded aluminum fittings are seldom recommended for process piping. Wrought fittings with welding ends (see Table 10-27 for dimensions) and with grooved joint ends are available. Wrought 6061-T6 flanges with dimensions per Table 10-19 are also available. Cast flanges and flanged fittings, sand-cast as alloy B214, 3.8 percent magnesium alloy with 90-MPA (13,000-lbf/in2) yield strength, or permanent mold cast as alloy 356-T6, 7 percent silicon, 0.3 percent magnesium alloy with 185-MPa (27,000-lbf/in2) yield strength are available, but consideration must be given to the fact that the modulus of elasticity of aluminum is only slightly more than one-third that of ferrous alloys. See Table 10-26 for dimensions. Aluminum-body diaphragm and ball valves are used extensively. Copper and Copper Alloys Seamless copper, bronze, brass, copper-nickel-alloy, and copper-silicon-alloy pipe and tubing are produced by extrusion. Tubing is available in outside-diameter sizes from g to 16 in and in a range of wall thicknesses varying from 0.005 in for the smallest tubing to 0.75 in for the 16-in size. Tubing is usually specified by outside diameter and wall thickness. Seamless copper tubing is sold in water-tubing sizes (ASTM B88 and B306). These sizes are identified by a “standard” size designation dimensionally f in less than the nominal outside diameter. The tubing is also sold as outside-diameter copper tubing (ASTM B280). Copper tubing is widely used in offices and laboratories for water, steam tracing, pneumatic control systems, compressed air, refrigeration, and inert-gas piping. Connections are made with flared-fitting joints (Fig. 10-143), compression-fitting joints (Fig. 10-144), bitetype-fitting joints (Fig. 10-145), and soldered or brazed joints (Fig. 10147). Figure 10-147 is most economical for e-in size and larger. Ease of handling and bending favors the use of copper; it will usually survive a freeze-up without failure. Copper water tubing ASTM B88 with dimensions and tolerances as given in Table 10-31 is available drawn or annealed in straight lengths of 6.1 m (20 ft) in types K, L, and M through 8-in size. Type K is available in 5.5-m (18-ft) lengths in 10-in size and 3.6-m (12-ft) lengths in 12-in size. Type L is available in 6.1-m (20-ft) lengths in 10-in size and 5.5-m (18-ft) lengths in 12-in size. Type M is available in 6.1-m (20-ft) lengths through 12-in size. All three types are available in 18.3-m (60-ft) or 30-m (100-ft) coils in sizes up to 1 in, in 18.3-m (60-ft) coils in 1d- and 1a-in sizes, and in 12.2- or 13.7-m (40- or 45-ft) coils in 2-in size. DWV tubing, ASTM B280, is available in 6-m (20-ft) straight lengths in the following size-wall combinations: 1d in, 0.040-in wall; 1a in, 0.042-in wall; 2 in, 0.042-in wall; 3 in, 0.045-in wall; 4 in, 0.058-in wall; 5 in, 0.072-in wall; and 6 in, 0.083-in wall. DWV is available only in drawn temper. Outside-diameter copper tubing B280 is available in annealed or drawn temper, depending on size; it is used for refrigeration field service, automotive applications, and general service. Dimensions and tolerances are shown in Table 10-32. Drawn temper is available in 6.1-m (20-ft) straight lengths; annealed temper, in 15.2-m (50-ft) coils. Too high a temperature or too long a heating period when silverbrazing ruins red-brass solder-joint fittings more quickly than wroughtcopper fittings. The former are available in larger sizes. Yellow brass fails from dezincification in some waters. Red-brass and bronze valves are available with female solder-joint ends for soldered copper-tubing piping systems. Copper pipe is available per ASTM B42 with dimensions as in Table 10-33. Butt-welding fittings (Table 10-27) are available to fit copper pipe, as are screwed fittings per ANSI B16.15, but solderend fittings of approximately the same dimensions as the screwed fittings and silver-brazing alloy comprise the usual method of assembly. Red-brass or bronze valves with ends identical to the fittings are available. Flanges and flanged fittings are seldom used, since soldered or silver-brazed joints can be melted apart and reassembled.

TABLE 10-31

Copper Water Tubing—Types K, L, M (ASTM B88)* Average outside diameter tolerance, in†

Nominal wall thickness and tolerances, in

Theoretical weight, lb/ft

Nominal outside diameter, in

Annealed

Drawn

Wall thickness

Tolerance‡

Wall thickness

Tolerance‡

Wall thickness

Tolerance‡

Type K

Type L

Type M

d r a v e

0.375 0.500 0.625 0.750 0.875

0.002 0.0025 0.0025 0.0025 0.003

0.001 0.001 0.001 0.001 0.001

0.035 0.049 0.049 0.049 0.065

0.004 0.004 0.004 0.004 0.0045

0.030 0.035 0.040 0.042 0.045

0.0035 0.0035 0.0035 0.0035 0.004

§ 0.025 0.028 § 0.032

§ 0.0025 0.0025 § 0.003

0.145 0.269 0.344 0.418 0.641

0.126 0.198 0.285 0.362 0.455

§ 0.145 0.204 § 0.328

1 1d 1a 2 2a

1.125 1.375 1.625 2.125 2.625

0.0035 0.004 0.0045 0.005 0.005

0.0015 0.0015 0.002 0.002 0.002

0.065 0.065 0.072 0.083 0.095

0.0045 0.0045 0.005 0.007 0.007

0.050 0.055 0.060 0.070 0.080

0.004 0.0045 0.0045 0.006 0.006

0.035 0.042 0.049 0.058 0.065

0.0035 0.0035 0.004 0.006 0.006

0.839 1.04 1.36 2.06 2.93

0.655 0.884 1.14 1.75 2.48

0.465 0.682 0.940 1.46 2.03

3 3a 4 5 6 8

3.125 3.625 4.125 5.125 6.125 8.125

0.005 0.005 0.005 0.005 0.005 0.006

0.002 0.002 0.002 0.002 0.002 +0.002 −0.004

0.109 0.120 0.134 0.160 0.192 0.271

0.007 0.008 0.010 0.010 0.012 0.016

0.090 0.100 0.110 0.125 0.140 0.200

0.007 0.007 0.009 0.010 0.011 0.014

0.072 0.083 0.095 0.109 0.122 0.170

0.006 0.007 0.009 0.009 0.010 0.014

4.00 5.12 6.51 9.67 13.9 25.9

3.33 4.29 5.38 7.61 10.2 19.3

2.68 3.58 4.66 6.66 8.92 16.5

10

10.125

0.008

+0.002 −0.006

0.338

0.018

0.250

0.016

0.212

0.015

40.3

30.1

25.6

12

12.125

0.008

+0.002 −0.006

0.405

0.020

0.280

0.018

0.254

0.016

57.8

40.4

36.7

Standard size, in

Type K

Type L

Type M

*Copyright American Society for Testing and Materials, 1916 Race Street, Philadelphia, Pa. 19103; reprinted/adapted with permission. To convert inches to millimeters, multiply by 25.4; to convert pounds per foot to kilograms per meter, multiply by 1.49. †The average outside diameter of a tube is the average of the maximum and minimum outside diameter, as determined at any one cross section of the tube. ‡Maximum deviation at any one point. §Indicates that the material is not generally available or that no tolerance has been established. TABLE 10-32

Copper Outside-Diameter Tubing for Refrigeration Field Service and Automotive and General Service (ASTM B280)* For mechanical or soldered fittings Tolerances†

Standard size, in

Outside diameter, in (mm)

Wall thickness, in (mm)

f t d b r

0.125 (3.18) 0.187 (4.75) 0.250 (6.35) 0.312 (7.92) 0.375 (9.52)

0.030 (0.762) 0.030 (0.762) 0.030 (0.762) 0.032 (0.813) 0.032 (0.813)

0.0347 (0.0516) 0.0575 (0.0856) 0.0804 (0.120) 0.109 (0.162) 0.134 (0.199)

0.002 0.002 0.002 0.002 0.002

(0.051) (0.051) (0.051) (0.051) (0.051)

0.003 (0.076) 0.0025 (0.064) 0.0025 (0.064) 0.0025 (0.064) 0.0025 (0.064)

a v e e 7⁄ 8

0.500 (12.7) 0.625 (15.9) 0.750 (19.1) 0.750 (19.1) 0.875 (22.3)

0.032 (0.813) 0.035 (0.889) 0.035 (0.889) 0.042 (1.07) 0.045 (1.14)

0.182 0.251 0.305 0.362 0.455

(0.271) (0.373) (0.454) (0.539) (0.677)

0.002 (0.051) 0.002 (0.051) 0.0025 (0.064) 0.0025 (0.064) 0.003 (0.076)

0.0025 (0.064) 0.0030 (0.076) 0.0035 (0.089) 0.0035 (0.089) 0.004 (0.10)

1f 1r 1v

1.125 (28.6) 1.375 (34.9) 1.625 (41.3)

0.050 (1.27) 0.055 (1.40) 0.060 (1.52)

0.665 (0.975) 0.884 (1.32) 1.14 (1.70)

0.0035 (0.089) 0.004 (0.10) 0.0045 (0.11)

0.004 (0.10) 0.0045 (0.11) 0.0045 (0.11)

r a v e 7⁄ 8

0.375 (9.52) 0.500 (12.7) 0.625 (15.9) 0.750 (19.1) 0.875 (22.3)

0.030 (0.762) 0.035 (0.889) 0.040 (1.02) 0.042 (1.07) 0.045 (1.14)

0.126 0.198 0.285 0.362 0.455

(0.187) (0.146) (0.424) (0.539) (0.677)

0.001 0.001 0.001 0.001 0.001

0.0035 (0.089) 0.0035 (0.089) 0.0035 (0.089) 0.0035 (0.089) 0.004 (0.10)

1f 1r 1v 2f 2v

1.125 (28.6) 1.375 (34.9) 1.625 (41.3) 2.125 (54.0) 2.625 (66.7)

0.050 (1.27) 0.055 (1.40) 0.060 (1.52) 0.070 (1.78) 0.080 (2.03)

0.655 0.884 1.14 1.75 2.48

(0.975) (1.32) (1.70) (2.60) (3.69)

0.0015 (0.038) 0.0015 (0.038) 0.002 (0.051) 0.002 (0.051) 0.002 (0.051)

0.004 (0.10) 0.0045 (0.11) 0.0045 (0.11) 0.006 (0.15) 0.006 (0.15)

3f 3v 4f

3.125 (79.4) 3.625 (92.1) 4.125 (105)

0.090 (2.29) 0.100 (2.54) 0.110 (2.79)

3.33 4.29 5.38

(4.96) (6.38) (8.01)

0.002 (0.051) 0.002 (0.051) 0.002 (0.051)

0.007 (0.18) 0.007 (0.18) 0.009 (0.23)

Weight, lb/ft (kg/m)

Average outside diameter, plus and minus, in (mm)‡

Wall thickness, plus and minus, in (mm)

For coil

For straight lengths (applicable to drawn-temper tube only) (0.025) (0.025) (0.025) (0.025) (0.025)

*Copyright American Society for Testing and Materials, 1916 Race Street, Philadelphia, Pa. 19103; reprinted/adapted with permission. †The tolerances listed represent the maximum deviation at any point. ‡The average outside diameter of a tube is the average of the maximum and minimum outside diameters as determined at any one cross section of the tube.

PROCESS-PLANT PIPING TABLE 10-33 Standard pipe size, in

10-95

Copper and Red-Brass Pipe (ASTM B42 and B43)*: Standard Dimensions, Weights, and Tolerances Nominal outside diameter, in (mm)

Average outside diameter tolerances, in (mm), all minus†

Nominal wall thickness, in (mm)

Tolerance, in (mm)‡

Regular pipe

Theoretical weight, lb/ft (kg/m) Red brass

Theoretical weight, lb/ft (kg/m) Copper

f d r a e

0.405 (10.3) 0.540 (13.7) 0.675 (17.1) 0.840 (21.3) 1.050 (26.7)

0.004 (0.10) 0.004 (0.10) 0.005 (0.13) 0.005 (0.13) 0.006 (0.15)

0.062 (1.57) 0.082 (2.08) 0.090 (2.29) 0.107 (2.72) 0.114 (2.90)

0.004 (0.10) 0.005 (0.13) 0.005 (0.13) 0.006 (0.15) 0.006 (0.15)

0.253 (0.376) 0.447 (0.665) 0.627 (0.933) 0.934 (1.39) 1.27 (1.89)

0.259 (0.385) 0.457 (0.680) 0.641 (0.954) 0.955 (1.42) 1.30 (1.93)

1 1d 1a 2 2a

1.315 (33.4) 1.660 (42.2) 1.900 (48.3) 2.375 (60.3) 2.875 (73.0)

0.006 (0.15) 0.006 (0.15) 0.006 (0.15) 0.008 (0.20) 0.008 (0.20)

0.126 (3.20) 0.146 (3.71) 0.150 (3.81) 0.156 (3.96) 0.187 (4.75)

0.007 (0.18) 0.008 (0.20) 0.008 (0.20) 0.009 (0.23) 0.010 (0.25)

1.78 (2.65) 2.63 (3.91) 3.13 (4.66) 4.12 (6.13) 5.99 (8.91)

1.82 (2.71) 2.69 (4.00) 3.20 (4.76) 4.22 (6.28) 6.12 (9.11)

3 3a 4 5 6

3.500 (88.9) 4.000 (102) 4.500 (114) 5.562 (141) 6.625 (168)

0.010 (0.25) 0.010 (0.25) 0.012 (0.30) 0.014 (0.36) 0.016 (0.41)

0.219 (5.56) 0.250 (6.35) 0.250 (6.35) 0.250 (6.35) 0.250 (6.35)

0.012 (0.30) 0.013 (0.33) 0.014 (0.36) 0.014 (0.36) 0.014 (0.36)

8.56 (12.7) 11.2 (16.7) 12.7 (18.9) 15.8 (23.5) 19.0 (28.3)

8.76 (13.0) 11.4 (17.0) 12.9 (19.2) 16.2 (24.1) 19.4 (28.9)

8.625 (219) 10.750 (273) 12.750 (324)

0.020 (0.51) 0.022 (0.56) 0.024 (0.61)

0.312 (7.92) 0.365 (9.27) 0.375 (9.52)

0.022 (0.56) 0.030 (0.76) 0.030 (0.76)

30.9 (46.0) 45.2 (67.3) 55.3 (82.3)

31.6 (47.0) 46.2 (68.7) 56.5 (84.1)

f d r a e

0.405 (10.3) 0.540 (13.7) 0.675 (17.1) 0.840 (21.3) 1.050 (26.7)

0.004 (0.10) 0.004 (0.10) 0.005 (0.13) 0.005 (0.13) 0.006 (0.15)

0.100 (2.54) 0.123 (3.12) 0.127 (3.23) 0.149 (3.78) 0.157 (3.99)

0.006 (0.15) 0.007 (0.18) 0.007 (0.18) 0.008 (0.20) 0.009 (0.23)

0.363 (0.540) 0.611 (0.909) 0.829 (1.23) 1.23 (1.83) 1.67 (2.48)

0.371 (0.552) 0.625 (0.930) 0.847 (1.26) 1.25 (1.86) 1.71 (2.54)

1 1d 1a 2 2a

1.315 (33.4) 1.660 (42.2) 1.900 (48.3) 2.375 (60.3) 2.875 (73.0)

0.006 (0.15) 0.006 (0.15) 0.006 (0.15) 0.008 (0.20) 0.008 (0.20)

0.182 (4.62) 0.194 (4.93) 0.203 (5.16) 0.221 (5.61) 0.280 (7.11)

0.010 (0.25) 0.010 (0.25) 0.011 (0.28) 0.012 (0.30) 0.015 (0.38)

2.46 (3.66) 3.39 (5.04) 4.10 (6.10) 5.67 (8.44) 8.66 (12.9)

2.51 (3.73) 3.46 (5.15) 4.19 (6.23) 5.80 (8.63) 8.85 (13.2)

3 3a 4 5 6

3.500 (88.9) 4.000 (102) 4.500 (114) 5.562 (141) 6.625 (168)

0.010 (0.25) 0.010 (0.25) 0.012 (0.30) 0.014 (0.36) 0.016 (0.41)

0.304 (7.72) 0.321 (8.15) 0.341 (8.66) 0.375 (9.52) 0.437 (11.1)

0.016 (0.41) 0.017 (0.43) 0.018 (0.46) 0.019 (0.48) 0.027 (0.69)

11.6 (17.3) 14.1 (21.0) 16.9 (25.1) 23.2 (34.5) 32.2 (47.9)

11.8 (17.6) 14.4 (21.4) 17.3 (25.7) 23.7 (35.3) 32.9 (49.0)

8.625 (219) 10.750 (273)

0.020 (0.51) 0.022 (0.56)

0.500 (12.7) 0.500 (12.7)

0.035 (0.89) 0.040 (1.0)

48.4 (72.0) 61.1 (90.9)

49.5 (73.7) 62.4 (92.9)

8 10 12

Extra strong pipe

8 10

*Copyright American Society for Testing and Materials, 1916 Race Street, Philadelphia, Pa. 19103; reprinted/adapted with permission. All tolerances are plus and minus except as otherwise indicated. †The average outside diameter of a tube is the average of the maximum and minimum outside diameters as determined at any one cross section of the tube. ‡Maximum deviation at any one point.

Threadless copper pipe, thinner than ASTM B42, is available with dimensions as in Table 10-34. Solder-end fittings similar to ANSI B16.15 screwed fittings and solder-end valves are used with this pipe. Copper pipe is attacked by water originating in granite substrata, and for this reason red-brass pipe per ASTM B43 with red-brass screwed or solder-end fittings is sometimes used in its place. 70 percent copper, 30 percent nickel and 90 percent copper, 10 percent nickel ASTM B466 are available as seamless pipe and welding fittings for handling brackish water in Schedule 10 and regular copper pipe thicknesses. Copper-silicon alloy (96 percent cooper, 3 percent silicon, 1 percent manganese), per ASTM B315, is furnished as seamless pipe and welding fittings in Schedule 10 and regular and extra-strong copper pipe thicknesses. It is easier to weld than copper. Lead and Lead-Lined Steel Pipe Lead and lead-lined steel pipe have been essentially eliminated as piping materials owing to health hazards in fabrication and installation and to environmental objections. Lead has been replaced by suitable plastic, reinforced plastic, plastic-lined steel, or high-alloy materials. Magnesium Extruded magnesium tubing is available per ASTM B217–58 alloyed with aluminum, manganese, or zinc. Ultimate and

yield strengths at 204°C (400°F) are about one-half those at room temperature. Outside-diameter range is d through 8 in. Wall thickness ranges from a minimum of 0.028 in to a maximum of 0.031 in for the d-in diameter and from a minimum of 0.250 in to a maximum of 1.0 in for the 8-in diameter. Nickel and Nickel Alloys A wide range of ferrous and nonferrous nickel and nickel-bearing alloys are available. They are usually selected because of their improved resistance to chemical attack or their superior resistance to the effects of high temperature. In general terms their cost and corrosion resistance are somewhat a function of their nickel content. The 300 Series stainless steels are the most generally used. Some other frequently used alloys are listed in Table 10-35 together with their nominal compositions. For metallurgical and corrosion resistance data, see Sec. 28. Titanium Pipe per ASTM B337 is available welded or seamless via one of the following processes: extrusion, centrifugal casting, machining of bar stock, or powder compaction; Schedule 5S, 10S, 40S, and 80S, f- through 24-in size. Extruded and drawn tubing per ASTM B338 is available from d-in outside diameter, 0.020- through 0.083-in wall, up through 3-in outside diameter. Cast welding fittings, flanges, and valves are also available. Titanium is used at temperatures

10-96

TRANSPORT AND STORAGE OF FLUIDS

TABLE 10-34

Hard-Drawn Copper Threadless Pipe (ASTM B302)* Nominal dimensions, in (mm)

Standard pipe size, in

Tolerances, in (mm) Cross-sectional area of bore, in2 (cm2)

Nominal weight, lb/ft (kg/m)

Wall thickness, plus and minus

Outside diameter

Inside diameter

d r a e

0.540 (13.7) 0.675 (17.1) 0.840 (21.3) 1.050 (26.7)

0.410 (10.4) 0.545 (13.8) 0.710 (18.0) 0.920 (23.4)

0.065 (1.65) 0.065 (1.65) 0.065 (1.65) 0.065 (1.65)

0.132 (0.852) 0.233 (1.50) 0.396 (2.55) 0.665 (4.29)

0.376 (0.559) 0.483 (0.719) 0.613 (0.912) 0.780 (1.16)

0.004 (0.10) 0.004 (0.10) 0.005 (0.13) 0.005 (0.13)

0.0035 (0.089) 0.004 (0.10) 0.004 (0.10) 0.004 (0.10)

1 1d 1a 2 2a 3

1.315 (33.4) 1.660 (42.2) 1.900 (48.3) 2.375 (60.3) 2.875 (73.0) 3.500 (88.9)

1.185 (30.1) 1.530 (38.9) 1.770 (45.0) 2.245 (57.0) 2.745 (69.7) 3.334 (84.7)

0.065 (1.65) 0.065 (1.65) 0.065 (1.65) 0.065 (1.65) 0.065 (1.65) 0.083 (2.11)

1.10 1.84 2.46 3.96 5.92 8.73

0.989 (1.47) 1.26 (1.87) 1.45 (2.16) 1.83 (272) 2.22 (3.30) 3.45 (5.13)

0.005 (0.13) 0.006 (0.15) 0.006 (0.15) 0.007 (0.18) 0.007 (0.18) 0.008 (0.20)

0.004 0.004 0.004 0.006 0.006 0.007

(0.10) (0.10) (0.10) (0.15) (0.15) (0.18)

3a 4 5 6 8

4.000 (102) 4.500 (114) 5.562 (141) 6.625 (168) 8.625 (219)

3.810 (96.8) 4.286 (109) 5.298 (135) 6.309 (160) 8.215 (209)

0.095 (2.41) 0.107 (2.72) 0.132 (3.40) 0.158 (4.01) 0.205 (5.21)

11.4 14.4 22.0 31.3 53.0

(73.5) (92.9) (142) (202) (342)

4.52 5.72 8.73 12.4 21.0

(6.73) (8.51) (13.0) (18.5) (31.2)

0.008 (0.20) 0.010 (0.25) 0.012 (0.30) 0.014 (0.36) 0.018 (0.46)

0.007 0.009 0.010 0.010 0.014

(0.18) (0.23) (0.25) (0.25) (0.36)

10.750 (273) 12.750 (324)

10.238 (260) 12.124 (308)

0.256 (6.50) 0.313 (7.95)

82.3 115

(531) (742)

32.7 47.4

(48.7) (70.5)

0.018 (0.46) 0.018 (0.46)

0.016 (0.41) 0.020 (0.51)

10 12

Wall thickness

Average outside diameter, all minus†

(7.10) (11.9) (15.9) (25.5) (38.2) (56.3)

*Copyright American Society for Testing and Materials, 1916 Race Street, Philadelphia, Pa. 19103; reprinted/adapted with permission. †The average outside diameter of a tube is the average of the maximum and minimum outside diameters, as determined at any one cross section of the tube.

TABLE 10-35

Common Nickel and Nickel-Bearing Alloys

Common trade name or registered trademark

Code designation

Type 304 stainless steel Type 316 stainless steel b

Carpenter 20cb Incoloy 800c Incoloy 825c

d

Ni-Cr-Fe-Mo-Cu-Cb stabilized Ni-Fe-Cr Ni-Fe-Cr-Mo-Cu

Alloy no.

ASTM specification (pipe)

Nominal composition, %

S30400

A312

9

19

70

0.08

2.0

S31600

A312

11

18

2.5

66.5

0.08

2.0

N08020 N08800 N08825

B464 B407 B423

33 32.5 42

20 21 21.5

2.5 3

38.5 46 30

0.06 0.05 0.03

0.5 0.2

2.0 0.8 0.5

e

16 28

5 2

0.02 0.02

0.08 0.1

1 1

2.5 8 1.2

0.05 0.08 0.20

0.2 0.2 0.2

0.2 0.5 1

0.2 31.5

0.2 19.5

0.08 0.05

1

1.5

2

Ni

Cr

Mo

Hastelloy C-276 Hastelloy B-2d

Ni-Mo-Cr low carbon Ni-Mo

N10276 N10001

B575 B333e

54 64

15 1

Inconel 625c Inconel 600c Monel 400c

Ni-Cr-Mo-Cb Ni-Cr-Fe Ni-Cu

N06625 N06600 N04400

B444 B167 B165

61 76 66

21.5 15.5

9

Nickel 200c Hastelloy Gd

Ni Ni-Cr-Fe-Mo-Cu

N02200 N06007

B161 B622

99+ 42

22.2

6.5

Fe

Ca

Sia

Mn

Cu

Cb

3 0.4 2.2

1

Co

W

2.5

4

2.5a

1a

4

2.2 f

a

Maximum. Registered trademark, Carpenter Technology Corp. Registered trademark, Huntington Alloys, Inc. d Registered trademark, Cabot Corp. e Plate. f Cb + Ta. b c

up to 315°C (600°F). It is extremely notch-sensitive. Titanium alloys such as 6 Al-4V, with higher tensile strengths than straight titanium, are available. Unfortunately, they lack the corrosion resistance and weldability of the unalloyed material. Zirconium (Tin 1.2 to 1.7 Percent) Tubing is available seamless ranging from a- outside diameter by 0.030-in wall to 8-in outside diameter by 0.4-in wall, and welded up through 30-in outside diameter by f-in wall. Cast valves and fittings are also available. Flexible Metal Hose Deeply corrugated thin brass, bronze, Monel, aluminum, and steel tubes are covered with flexible braidedwire jackets to form flexible metal hose. Both tube and braid are brazed or welded to pipe-thread, union, or flanged ends. Failures are often the result of corrosion of the braided-wire jacket or of a poor

jacket-to-fitting weld. Inside diameters range from f to 12 in. Maximum recommended temperature for bronze hose is approximately 230°C (450°F). Metal thickness is much less than for straight tube for the same pressure-temperature conditions; so accurate data on corrosion and erosion are required to make proper selection. NONMETALLIC PIPE AND LINED PIPE SYSTEMS Asbestos Cement Asbestos-cement pipe is seamless pipe made of silica and portland cement, compacted under heavy pressure, uniformly reinforced with asbestos fiber, and thoroughly cured. The interior surface is smooth, does not corrode, and does not tuberculate. Under normal conditions of operation, asbestos cement will handle

PROCESS-PLANT PIPING solutions within a pH range of 4.5 to 14. It is a brittle material and undergoes expansion on wetting. There are stringent OSHA regulations pertaining to the fabrication and use of asbestos-containing materials. The most widely used joints are push-on joints. This pipe is used extensively for underground water systems, for paper-mill slurries and wastes, and for mine water. The push-on joints limit the temperature to 65°C (150°F). The light weight of the pipe minimizes handling labor, but careful handling is required to avoid damage. This pipe is available with an epoxy lining which increases its corrosion resistance. Asbestos-cement fittings and valves are not available, but flanged fabricated-steel fittings lined with segments of asbestos-cement pipe and cement-lined cast-iron fittings with end bells for push-on joint to asbestos-cement pipe can be obtained. Adapters to regular cast-iron fittings are also available. When the pipe is installed aboveground, two guided supports per length of pipe are recommended, and when push-on joints are used, internal pressure thrusts at changes in direction, at reducers, at dead ends, and at valves must be resisted by braces. When poured flanges are used, expansion joints must be used also with braces to resist corresponding pressure thrust. Pressure Pipe This pipe is made in three classes corresponding to working pressures of 0.7, 1.0, and 1.4 MPa (100, 150, and 200 lbf/ 2 in ) (Table 10-36). Gravity Sewer Pipe This pipe is made in five classes for varying depths of bury, trench dimension, soil, and vehicular loading (Table 10-37).

TABLE 10-36

10-97

Impervious Graphite Impervious-graphite pipe, fittings, and valve bodies are made of electric-furnace graphite which, after extruding or molding, is rendered impervious by impregnation with synthetic resins. When impregnated with phenolic resin, it is resistant to most acids (including hydrofluoric), salts, and organic compounds. When impregnated with modified phenolic resin, it is resistant to strong alkalies and highly oxidizing materials. Ultimate tensile strength is low, 17.2 MPa (2500 lbf/in2), and the material is brittle, but the modulus of elasticity is only 15,168 MPa (2.2 × 106 lbf/in2). The material is highly resistant to thermal shock and is available with glass-cloth and resin armor for protection against physical abuse. Maximum continuous operating temperature is 170°C (340°F). Components are designed for operating pressure which increases from 0.3 MPa (50 lbf/in2) at 170°C (340°F) to 0.5 MPa (75 lbf/in2) at 21°C (70°F). Table 10-38 lists standard sizes of pipe; a-, e-, and 7⁄8-in sizes are heat-exchanger tubing, and standard fittings are not available for these sizes. Pipe is shipped threaded on request. National Form straight threads are used. Fittings made from the same material with the same thread form are available and include laps which can be screwed on the ends of pipe and stub ends which can be screwed into the fittings, both for the purpose of making flanged lap joints. All threaded joints are permanently bonded by special cements. Flanged joints use split cast-iron backup flanges which have 150-lb ANSI B16.5 bolting in sizes 6 in and smaller and 300-lb ANSI B16.5 bolting in sizes 8 in and larger. Asbestos sheet packing is used between the flange and the back of the lap to equalize bearing. Pipe can be sawed

Asbestos-Cement Pressure Pipe* Class 100†

Nominal size

Length, ft

Inside diam., in

Wall, in‡

4 6 8 10 12 14 16

13 13 13 13 13 13 13

3.95 5.85 7.85 9.85 11.70 13.59 15.50

0.35 .42 .47 .52 .64 .74 .83

Class 150† Wt., lb/ft§

Inside diam., in

Wall, in‡

6.3 10.6 15.8 21.8 29.7 38.9 48.8

3.95 5.85 7.85 10.00 12.00 14.00 16.00

0.43 .53 .63 .83 .96 1.11 1.23

Class 200† Wt., lb/ft§

Inside diam., in

Wall, in‡

Wt., lb/ft§

7.6 13.0 19.9 32.0 43.8 58.5 73.0

3.95 5.70 7.60 9.63 11.56 13.59 15.50

0.43 .60 .75 1.01 1.18 1.31 1.48

9.3 15.4 23.9 37.2 51.7 69.0 89.2

*Johns-Manville Co. †Equivalent to working pressure, lb/sq in. ‡Minimum thickness of machined end; balance of pipe is thicker. §Pipe plus push-on joint coupling. NOTE: To convert inches to millimeters, multiply by 25.4; to convert pounds per foot to kilograms per meter, multiply by 1.49; to convert pounds-force per square inch to megapascals, multiply by 0.00689; and to convert feet to meters, multiply by 0.3048.

TABLE 10-37

Asbestos-Cement Gravity Sewer Pipe* Class 1500†

Class 2400†

Class 3300†

Class 4000†

Class 5000†

Nominal size

Inside diam., in

Wall, in‡

Wt., lb/ft

Wall, in‡

Wt., lb/ft

Wall, in†

Wt., lb/ft

Wall, in‡

Wt., lb/ft

Wall, in‡

Wt., lb/ft

6 8 10 12 14

6.00 8.00 10.00 12.05 14.05

0.46 .51 .56 .61

8.5 12.6 17.6 22.8

0.49 .52 .58 .63 .68

9.5 13.3 18.9 24.3 30.3

0.57 .61 .68 .75 .81

11.1 15.6 22.0 28.8 35.8

0.75 .82 .89

24.3 31.5 39.3

0.85 0.93 1.00

27.6 35.8 44.3

16 18 20 24 30

16.05 18.05 20.05 24.05 30.05

.73 .77 .81 .89

37.0 43.6 50.7 66.4

.86 .91 .96 1.05 1.17

43.1 50.9 59.2 77.2 106.8

.95 1.01 1.06 1.16 1.30

47.6 56.5 65.4 85.3 118.8

1.07 1.13 1.19 1.30 1.45

53.7 63.3 73.6 95.8 132.7

36

36.05

1.42

155.0

1.59

173.8

Standard pipe length is 13 ft except 6 in Class 1500 is 10 ft and 8 in Class 1500 may also be 10 ft. *Johns-Manville Co. †Crushing strength per A.S.T.M. three-edge bearing method. ‡Thickness of wall of pipe excluding machined ends. Same coupling is used for all classes; it protects the machined ends from crushing loads. NOTE: To convert inches to millimeters, multiply by 25.4; to convert pounds per foot to kilograms per meter, multiply by 1.49; and to convert feet to meters, multiply by 0.3048.

10-98

TRANSPORT AND STORAGE OF FLUIDS

TABLE 10-38

Standard Sizes of Impervious Graphite Pipe*

Nominal pipe size, in

Inside diameter, in

Outside diameter, in

1 1a 2 2a 3

1 1a 2 2r 3

1a 2 2e 3 4

4 6 8f 10f

5d 7a 9x 1221⁄ 32

4 6 8 10

Wall thickness, in

Average weight, lb/ft

Maximum length, ft

d d r b a

9 9 9 9 9

v e ⁄ 1 ⁄

9 9 6 6

25 32 17 64

0.74 1.1 1.7 2.0 5.4 8.1 15.6 23.2 44.2

Circumference, ft, or surface, ft2/ft of length

Inside crosssectional area, ft2

Inside

Outside

.00545 .01227 .0218 .0308 .0491

0.262 .393 .524 .622 .785

0.393 .524 .687 .785 1.047

.0873 .1965 .360 .559

1.047 1.571 2.127 2.650

1.374 1.964 2.536 3.313

*Courtesy Union Carbide Corporation, Carbon Products Division. NOTE: To convert inches to millimeters, multiply by 25.4; to convert feet to meters, multiply by 0.3048; to convert pounds per foot to kilograms per meter, multiply by 1.49; and to convert square feet to square meters, multiply by 0.0929.

to length in the field and threaded with special tools. Synthetic elastomeric and Teflon gaskets are available. Diaphragm valves with impervious graphite bodies are available in sizes from 1 through 6 in. Maximum recommended support spacing is 2.7 m (9 ft), and valves should be supported independently. Cement-Lined Steel Cement-lined steel pipe is made by lining steel pipe with special cement. Its use prevents pickup of iron by the fluid handled, corrosion of the metal by brackish water, and growth of tuberculation. Threaded pipe in sizes from e to 4 in is stocked; however, cement-lined pipe in sizes smaller than 1a in is not considered practical for common use. The coefficients of expansion of iron and cement are nearly alike. Table 10-39 gives dimensions of cement-lined pipe. Cement-lined carbon steel pipe larger than 4 in is shipped with flanged or welding ends. Welding does not damage the lining, which forms a slag protecting the weld. Shop cement lining of carbon steel pipe is covered by AWWA C205. Cement-lined carbon steel buttwelding fittings and flanged cast-iron fittings are available. AWWA C602 includes cement lining of both cast-iron and carbon steel water lines in place. Chemical Ware Acidproof chemical-stoneware pipe and fittings withstand most acid, alkali, or other corrosives, the main exception being hydrofluoric acid. The range of sizes made with the bell-andspigot joint and with plain butt ends is shown in Table 10-40. Plain butt-end pipe is furnished with cemented-on flanges with ANSI B16.1 drilling or (for use in ventilating work in which the space is too limited for bell-and-spigot pipe) with a ring for joining with a steel band. Medium-pressure chemical-stoneware pipe armored with glass fiber reinforced with furan resin can be obtained. Flanges with ANSI B16.1 drilling bear against hubs formed from the armor. Fittings and plug valves with ends to match the various types of pipe are available. Vitrified-Clay Sewer Pipe This pipe is resistant to very dilute chemicals except hydrofluoric acid and is produced as standardstrength and extra-strength (ASTM C700). It is used for sewage, industrial waste, and storm water at atmospheric pressure. Elbows, Y branches, tees, reducers, and increasers are available. Assembly is by poured joints which allow for ample angular deflection. Joint comTABLE 10-39

Cement-Lined Carbon-steel Pipe*

Inside ThickInside ThickStanddiam. ness of Weight, Standdiam. ness of Weight ard pipe after lining, per ft, ard pipe after lining, per ft, size, in lining, in in lb size, in lining, in in lb e 1 1d 1a 2 2a

0.70 .90 1.20 1.40 1.80 2.20

0.06 .07 .08 .09 .10 .10

1.3 1.9 2.5 3.0 4.1 6.6

3 4 6 8 10 12

2.70 3.60 5.40 7.40 9.40 11.40

0.13 .16 .25 .25 .30 .30

8.3 12.0 24.0 32.0 43.0 55.0

*To convert inches to millimeters, multiply by 25.4; to convert pounds per foot to kilograms per meter, multiply by 1.49.

pounds are of the hot-pour type or the cold mastic type; both adhere tightly to the scored clay surfaces but remain flexible enough to prevent leakage in the event of earth settlement. Pipe is also available with bituminous or plastic material die-cast on the outside of the spigot and the inside of the bell. The interfaces are a snug fit cemented by applying a solvent to them at the time of assembly. Dimensions of pipe are given in Table 10-41. Choice between standard and extra strength is based on earth and vehicular loading. Concrete Unreinforced-concrete sewer pipe is made with poured joint ends in sizes from 4 to 24 in conforming to ASTM C14. Reinforced-concrete culvert, storm-drain, and sewer pipe is made with poured joint or push-on joint ends conforming to ASTM C76 in five classes of reinforcement area and wall thickness in sizes from 12 through 108 in. Essentially the same pipe, except that it has push-on joint ends only, is available for water pressures up to 0.31 MPa (45 lbf/in2) in sizes 12 through 96 in and lengths up through 5.6 m (16 ft) conforming to AWWA C302. For higher water pressures, a steel cylinder approximately 1.6 mm (g in) thick is embedded in the wall of the pipe, which prevents leakage through cracks, and to this there may be added prestressed circumferential reinforcing wire applied after the cylinder has been stiffened by cement lining. Such pipe is available in accordance with AWWA C300, sizes 20 through 96 in, for pressures 0.27 through 1.8 MPa (40 through 260 lbf/in2), and in accordance with AWWA C301, sizes 16 through 96 in. Push-on joints are used. Pipe is also available with steel lugs welded to the reinforcing cages and projecting through the outside surface of the pipe for “bridling.” This is known as “subaqueous pipe.” Concrete fittings are also available. ConTABLE 10-40 Chemical Stoneware: Bell-and-Spigot and Plain Butt-End Pipe* Inside diam., in

Outside diam., in

Wall thickness, in

2d 2e 4 5 6

r r a a a

6 8 10 12 14

7d 9a 11e 13e 15e

v e 7⁄ 8 7⁄ 8 7⁄ 8

15 16 18 20

17 18 20 22

1a 2 3 4 5

1 1 1 1

Standard lengths up to 5 ft. *Maurice A. Knight Co. To convert inches to millimeters, multiply by 25.4; to convert feet to meters, multiply by 0.3048. NOTE:

PROCESS-PLANT PIPING TABLE 10-41

10-99

Vitrified-Clay Sewer Pipe* Min. wall thickness

Nominal size

Min. laying length, ft

Min. outside diam. of barrel, in

4 6 8 10 12

2 2 2 2 2

47⁄ 8 7g 9d 11a 13e

q a z x c

z e 7⁄ 8 1g

16 18 21 24 27

3 3 3 3 3

17t 20v 24f 27a 31

y 1f 1b 1a 1x

1r 1e 2 2d 2a

30 33 36

3 3 3

34r 37v 40e

17⁄ 8 2 2g

2e 3 3d

Standard strength, in

Extra strength, in

*To convert inches to millimeters, multiply by 25.4.

crete piping systems can be lined with special salt-glazed vitrified-clay liner plates, joined with a die-cast asphalt joint. Concrete pressure pipe is competitive with cement-lined ductile iron for underground plant water systems. Glass Pipe and Tubing These are made from heat- and chemical-resistant borosilicate glass (e.g., Corning Glass Works No. 7740) ASTM C599. This glass is highly stable in acids and resists attack by alkalies in solutions in which pH is 8 or less. It is attacked by hydrofluoric acid and glacial phosphoric acid. Some important physical properties are: Modulus of elasticity Specific gravity Specific heat Thermal conductivity at 75°F

9,750,000 lb/in2 (67,224 MPa) 2.23 0.20 8.1 Btu/(h⋅ft2)(°F/in)[1.168 W/(m⋅K)]

Conical flanged glass pipe (Fig. 10-161) is made in the sizes shown in Table 10-42 and in lengths from 0.15 to 3 m (6 in to 10 ft). Maximum recommended working pressure is 0.3 MPa (50 lbf/in2) through 3-in size, 0.24 MPa (35 lbf/in2) for 4-in size, and 0.14 MPa (20 lbf/in2) for 6-in size. Maximum sudden temperature differential is 93°C (200°F) through 3-in size, 80°C (175°F) for 4-in size, and 71°C (160°F) for 6-in size. Maximum operating temperature is 232°C (450°F). A complete line of fittings is available, and special parts are made to order. Thermal-expansion stresses should be completely relieved by tied Teflon corrugated expansion joints and offsets. Temperature rating may be limited by joint design and materials. Hangers should be padded to avoid scratching pipe, should fit loosely, and should be located 0.3 m (1 ft) from each end of each 3-m (10-ft) length. TABLE 10-42 Pipe size, in (mm) 1 (25) 1a (38) 2 (51) 3 (76) 4 (102) 6 (152)

Conical flanged joint.

Glass pipe can be furnished with an epoxy-resin coating reinforced with woven glass fiber to protect it from abuse. Equipped with special ball couplings, this may be used for 1-MPa (150-lbf/in2) pressure. For very low pressures, beaded-end pipe equipped with single-bolt band-type couplings is available. Glass-Lined Steel Pipe This pipe is fully resistant to all acids except hydrofluoric and concentrated phosphoric acids at temperatures up to 121°C (250°F). It is also resistant to alkaline solutions at moderate temperatures. Glass-lined steel pipe can be used at temperatures up to 232°C (450°F) under some exposure conditions provided there are no excessive sudden temperature changes. The operating pressure rating of commonly available systems is 1 MPa (150 lbf/in2). The glass lining is approximately 1.6 mm (g in) thick. It is made by lining Schedule 40 steel pipe. Fittings are available in glass-lined cast iron, ductile iron, and steel. The fitting rating and recommended applications for fittings depend on the substrate material. Standard pipe sizes available are 1a through 8 in. Larger-diameter pipe up to 48 in is available on a custom-order basis. A range of standard lengths is generally available from stock in 1a- through 4-in sizes. See Table 10-43 for dimensional data. Special Pfaudler-design steel split flanges drilled to ANSI Class 150 dimensions are used for assembly of the system. Chemical-Porcelain Pipe Made of dense, nonporous material and fired at 1230°C (2250°F), chemical-porcelain pipe, fittings, and valves are inert to all acids except hydrofluoric but are not usually recommended for alkalies. Surfaces, except when ground for gasketing, are usually glazed for easy cleaning. Working pressures of 0.3 to 0.7 MPa (50 to 100 lbf/in2) are recommended for valves and piping. Temperatures of 200°C (400°F) or more can be used, but sudden thermal shocks must be avoided. Cast-iron flanges (ANSI B16.1, 125-lb bolt spacing) are permanently attached to the porcelain with high strength acid-resistant cement. Flanged chemical-porcelain 90° and 45° elbows, tees, crosses, reducers, caps, and globe valves of the Y pattern are available. Armored chemical porcelain is furnished with 1.5- to 2.4-mm- (g- to

Glass Pipe and Tubing: Conical Flanged Joint* Pipe outside diameter, in (mm)

Cone outside diameter, in (mm)

Wall thickness, in (mm)

1b ⫾ 0.016 (33 ⫾ 0.4) 127⁄ 32 ⫾ 0.020 (47 ⫾ 0.5) 211⁄ 32 ⫾ 0.040 (60 ⫾ 1.0) 313⁄ 32 ⫾ 0.56) (87 ⫾ 1.4) 417⁄ 32 ⫾ 0.068 (115 ⫾ 1.7) 621⁄ 32 ⫾ 0.075 (169 ⫾ 1.9)

1z ⫾ 0.016 (40 ⫾ 0.4) 2f ⫾ 0.016 (54 ⫾ 0.4) 2v ⫾ 0.020 (67 ⫾ 0.5) 325⁄ 32 ⫾ 0.031 (96 ⫾ 0.8) 523⁄ 64 ⫾ 0.016 (136 ⫾ 0.4) 7.553 ⫾ 0.016 (192 ⫾ 0.4)

5⁄ 32 ⫾ 0.016 (4.0 ⫾ 0.4) 11⁄ 64 ⫾ 0.016 (4.4 ⫾ 0.4) 11⁄ 64 ⫾ 0.020 (4.4 ⫾ 0.5) 13⁄ 64 ⫾ 0.021 (5.2 ⫾ 0.5) 17⁄ 64 ⫾ 0.025 (6.7 ⫾ 0.6) b ⫾ 0.040 (7.9 ⫾ 1.0)

*From Corning Glass Works. See Fig. 10-161. To convert feet to meters, multiply by 0.3048.

NOTE:

FIG. 10-161

Cone angle, °

Approximate weight per foot, lb (kg)

12

0.6 (0.27)

12

1.0 (0.45)

12

1.13 (0.51)

12

2.0 (0.91)

21

3.4 (1.5)

21

6.3 (2.9)

10-100

TRANSPORT AND STORAGE OF FLUIDS

TABLE 10-43

Glass-Lined Steel Pipe*

Size, in

Outside diameter, in

Approximate inside diameter, in

1a 2 3 4 6‡ 8‡

1.875 2.375 3.500 4.500 6.625 8.625

1.50 1.95 2.95 3.90 5.95 7.85

Range of standard lengths, in Minimum†

Maximum

3a 4 4a 4a 5 5a

120 120 120 120 120 120

*From Pfaudler Company, division of Sybron Corp. To convert inches to millimeters, multiply by 25.4. Standard-length pipe spools are available in the following increments of length: For lengths, in 3a–6 6–8 8–10 10–12 12–120

Standard lengths available in length increments, in a 2 1 2 6

†Spacers are available in a-in increments for making up lengths of less than the minimum spool length shown. ‡Spool lengths less than 120 in are available but are not standard.

3 ⁄32-in-) thick woven glass cloth impregnated with and bonded to the porcelain by plastic cement. The armor is continuous end to end and runs under the flanges. It prevents abuse from cracking the porcelain and, if the porcelain is cracked, prevents rupture. Fused Silica or Fused Quartz Containing 99.8 percent silicon dioxide, fused silica or fused quartz can be obtained as opaque or transparent pipe and tubing. The melting point is 1710°C (3100°F). Tensile strength is approximately 48 MPa (7000 lbf/in2); specific gravity is about 2.2. The pipe and tubing can be used continuously at temperatures up to 1000°C (1830°F) and intermittently up to 1500°C (2730°F). The material’s chief assets are noncontamination of most chemicals in high-temperature service, thermal-shock resistance, and high-temperature electrical insulating characteristics. Transparent tubing is available in inside diameters from 1 to 125 mm in a range of wall thicknesses. Satin-surface tubing is available in inside diameters from g to 2 in, and sand-surface pipe and tubing are available in a- to 24-in inside diameters and lengths up to 6 m (20 ft). Sand-surface pipe and tubing are obtainable in wall thicknesses varying from f to 1 in. Pipe and tubing sections in both opaque and transparent fused silica or fused quartz can be readily machineground to special tolerances for pressure joints or other purposes. Also, fused-silica piping and tubing can be reprocessed to meet special-design requirements. Manufacturers should be consulted for specific details. Wood and Wood-Lined Steel Pipe Douglas fir, white pine, redwood, and cypress are the most common woods used for wood pipe. Wood-lined steel pipe is suitable for temperatures up to 82°C (180°F) and for pressures from 1.4 MPa (200 lbf/in2) for the 4-in size, through 0.86 MPa (125 lbf/in2) for the 10-in size, to 0.7 MPA (100 lbf/ in2) for sizes larger than 10 in. For fume stacks and similar uses, woodstave pipe with rods on 0.3-m (1-ft) centers is most satisfactory because it permits periodic tightening. In recent years reinforced plastics have supplanted wood pipe in most applications. Plastic-Lined and Rubber-Lined Steel Pipe Use of a variety of polymeric materials as liners for steel pipe rather than as piping systems solves problems which the relatively low tensile strength of the polymer at elevated temperature and high thermal expansion, compared with steel, would produce. The steel outer shell permits much wider spacing of supports, reliable flanged joints, and higher pressure and temperature in the piping. The size range is 1 through 12 in. The systems are flanged with 125-lb cast-iron, 150-lb ductile-iron, and 150- and 300-lb steel flanges. The linings are factory-installed in both pipe and fittings. Lengths are available up to 6 m (20 ft). Lined ball, diaphragm, and check valves and plug cocks are available. One method of manufacture consists of inserting the liner into an

oversize, approximately Schedule 40 steel tube and swaging the assembly to produce iron-pipe-size outside diameter, firmly engaging the liner which projects from both ends of the pipe. Flanges are then screwed onto the pipe, and the projecting liner is hot-flared over the flange faces nearly to the bolt holes. In another method, the liner is pushed into steel pipe having cold-flared laps backed up by flanges at the ends and then hot-flared over the faces of the laps. Pipe lengths made by either method may be shortened in the field and reflared with special procedures and tools. Square and tapered spacers are furnished to adjust for small discrepancies in assembly. Saran Liners Saran (Dow Chemical Co.) polyvinylidene chloride liners have excellent resistance to hydrochloric acid. Maximum temperature is 80°C (175°F). Polypropylene Liners Polypropylene liners (Hercules Incorporated) are used in sulfuric acid service. At 10 to 30 percent concentration the upper temperature limit is 93°C (200°F). In the range of 50 to 93 percent concentration, this drops from 66 to 24°C (from 150 to 75°F). Kynar Liners Kynar (Pennwalt Chemicals Corp.) vinylidene fluoride liners are used for many chemicals, including bromine and 50 percent hydrochloric acid. TFE-, PFA-, and FEP-Lined Steel Pipe These are available in sizes from 1 through 12 in and in lengths through 6 m (20 ft). The liners are not affected by any concentration of acids, alkalies, or solvents, but vent holes or internal grooving is required in the steel pipe to release gases which permeate through the liners. Manufacturers should be consulted before use in vacuum service. Experience has determined that practical upper temperature limits are 204°C (400°F) for TFE (polytetrafluoroethylene) and PFA (perfluoroalkoxy) and 149°C (300°F) for FEP (fluoroethylene polymer); 150-lb and 300-lb ductile-iron or steel flanged lined fittings and valves are used. The nonadhesive properties of the liner make it ideal for handling sticky or viscous substances. Thickness of the lining varies from 1.5 to 3.8 mm (60 to 150 mil), depending on pipe size. Only flanged joints are used. Rubber-Lined Pipe This pipe is made in lengths up to 6 m (20 ft) with seamless, straight seam-welded and some types of spiralwelded pipe using various types of natural and synthetic adhering rubber. The type of rubber is selected to provide the most suitable lining for the specific service. In general, soft rubber is used for abrasion resistance, semihard for general service, and hard for the more severe service conditions. Multiple-ply lining and combinations of hard and soft rubber are available. The thickness of lining ranges from 3.2 to 6.4 mm (f to d in) depending on the service, the type of rubber, and the method of lining. Cast-steel, ductile-iron, and cast-iron flanged fittings are available rubber-lined. The fittings are usually purchased by the vendor since absence of porosity on the inner surface is essential. Pipe is flanged before rubber lining, and welding elbows and tees may be incorporated at one end of the length of pipe, subject to the conditions that the size of the pipe and the location of the fittings are such that the operator doing the lining can place a hand on any point on the interior surface of the fitting. Welds must be ground smooth on the inside, and a radius is required at the inner edge of the flange face. The rubber lining is extended out over the face of flanges. With hard-rubber lining, a gasket is required. With soft-rubber lining, coating or a polyethylene sheet is required in place of a gasket to avoid bonding of the lining of one flange to the lining on the other and to permit disassembly of the flanged joint. Also, for pressures over 0.86 MPa (125 lbf/in2), the tendency of soft-rubber linings to extrude out between the flanges may be prevented by terminating the lining inside the bolt holes and filling the balance of the space between the flange faces with a Masonite spacer of the proper thickness. Hardrubber-lined gate, diaphragm, and swing check valves are available. In the gate valves, stem, wedge assembly, and seat rings, and in the check valves, hinge pin, flapper arm, disk, and seat ring must be made of metal resistant to the solution handled. Plastic Pipe In contrast to other piping materials, plastic pipe is free from internal and external corrosion, is easily cut and joined, and does not cause galvanic corrosion when coupled to other materials. Allowable stresses and upper temperature limits are low. Normal operation is in the creep range. Fluids for which a plastic is not suited penetrate and soften it rather than dissolve surface layers. Coefficients

PROCESS-PLANT PIPING of thermal expansion are high. The use of thermoplastic pipe in flammable service aboveground is prohibited by the code. Support spacing must be much closer than for carbon steel. As temperature increases, the allowable stress for many plastic pipes decreases very rapidly, and heat from sunlight or adjacent hot uninsulated equipment has a marked effect. Successful economical underground use of plastic pipe does not necessarily indicate similar economies outdoors aboveground. Plastic tubing is widely used for instrument air-signal connections. Methods of joining include threaded joints with IPS dimensions, solvent-welded joints, heat-fused joints, and insert fittings. Schedules 40 and 80 (see Table 10-18) have been used as a source for standardized dimensions at joints. Some plastics are available in several grades with allowable stresses varying by a factor of 2 to 1. For the same plastic, a-in Schedule 40 pipe of the strongest grade may have 4 times the allowable internal pressure of the weakest grade of a 2-in Schedule 40 pipe. For this reason, the plastic-pipe industry is shifting to standard dimension ratios (approximately the same ratio of diameter to wall thickness over a wide range of pipe sizes). ASTM and the Plastics Pipe Institute, a division of the Society of the Plastics Industry, have established identifications for plastic pipe in which the first group of letters identifies the plastic, the two following numbers identify the grade of that plastic, and the last two numbers represent the design stress in the nearest lower (0.7-MPa (100-lbf/in2) unit at 23°C (73.4°F). Polyethylene Polyethylene (PE) pipe and tubing are available in sizes 42 in and smaller. They have excellent resistance at room temperature to salts, sodium and ammonium hydroxides, and sulfuric, nitric, and hydrochloric acids. Pipe and tubing are produced by extrusion from resins whose density varies with the manufacturing process. Physical properties and therefore wall thickness depend on the particular resin used. About 3 percent carbon black is added to provide resistance to ultraviolet light. Use of higher-density resin reduces splitting and pinholing in service and increases the strength of the material and the maximum service temperature. ASTM D2104 covers PE pipe in sizes a through 6 in, with IPS Schedule 40 outside and inside diameters for insert-fitting joints. ASTM D2239 covers five standard dimension ratios of pipe diameter to wall thickness in sizes a through 6 in, with IPS Schedule 40 inside diameter for insert-fitting joints. ASTM D2447 covers sizes a through 12 in, with IPS Schedule 40 and 80 outside and inside diameters for use with heat-fusion socket-type and butt-type fittings. ASTM D3035 covers standard dimension ratios of pipe sizes from a through 6 in with IPS outside diameters. All these specifications cover five PE materials (see Table 10-15). Hydrostatic design stresses within the recommended temperature limits are given in Appendix A, Table 3, of the code. The hydrostatic design stress is the maximum tensile hoop stress due to internal hydrostatic water pressure that can be applied continuously with a high degree of certainty that failure of the pipe will not occur. Biaxially oriented polyethylene (PEO) pipe (ASTM D3287) has a higher hydrostatic design stress than PE pipe. Polyethylene water piping is not damaged by freezing. Pipe and tubing 2 in and smaller are shipped in coils several hundred feet in length. Clamped-insert joints (Fig. 10-162) are used for flexible plastic pipe up through the 2-in size. Friction between the pipe and the spud is developed both by forcing the spud into the pipe and by tightening the clamp. For the larger sizes, which have thicker walls, these meth-

FIG. 10-162

Clamped-insert joint.

10-101

ods cannot develop adequate friction. The joints also have high pressure drop. Stainless-steel bands are available. Inserts are available in nylon, polypropylene, and a variety of metals. A significant use for PE and PP pipe is the technique of rehabilitating deteriorated pipe lines by lining them with plastic pipe. Lining an existing pipe with plastic pipe has a large cost advantage over replacing the line, particularly if replacement of the old line would require excavation. Polyvinyl chloride Polyvinyl chloride (PVC) and chlorinated polyvinyl chloride (CPVC) pipe and tubing are available in sizes 12 in and smaller for PVC and 4 in and smaller for CPVC. They have excellent resistance at room temperature to salts, ammonium hydroxide, and sulfuric, nitric, acetic, and hydrochloric acid but may be damaged by ketones, aromatics, and some chlorinated hydrocarbons. Five PVC pipe materials having characteristic chemical resistance, impact strength, and hydrostatic design stresses are included in the group of ASTM pipe specifications pertaining to PVC. While all these materials have a −18°C (0°F) minimum-recommended-temperature limit (see Table 10-15), Code PVC-1120 and Code PVC-1220 materials become brittle at and below 4°C (40°F). On the other hand, Code PVC-2110, Code PVC-2112, and Code PVC-2216 materials have higher impact resistance but a lower hydrostatic design stress at elevated temperatures. Code PVC-2120 has the best combination of both properties. Allowable hydrostatic design stresses are given in Appendix A, Table 3, of the code, although no stresses are provided for temperatures above 38°C (100°F). The hydrostatic design stresses at 23°C (73.4°F) are 13.8 MPA (2000 lbf/in2) for PVC-1120, PVC1220, and PVC-2120, 11.0 MPa (1600 lbf/in2) for PVC-2116 and CPVC-4116, 8.6 MPa (1250 lbf/in2) for PVC-2116, and 6.9 MPa (1000 lbf/in2) for PVC-2110. ASTM D1785 covers sizes from f through 12 in of PVC pipe in IPS Schedules 40, 80, and 120, except that Schedule 120 starts at a in and is not IPS for sizes from a through 3 in. ASTM D2241 covers the same size range but with IPS outside diameter and seven standard dimension ratios: 13.5, 17, 21, 26, 32.5, 41, and 64. ASTM D2513 covers pipe in sizes from a through 12 in in both IPS outside diameter and plastic-tubing diameters from d through 1e in with standard-dimension-ratio wall thicknesses. This product is intended for gas service. ASTM D2672 covers bell-end pipe in sizes from f through 8 in in IPS Schedule 40 and in IPS outside diameter and the same standard dimension ratios for wall thicknesses as in D2241. The pipe is intended to be joined by cementing. ASTM D2740 covers PVC-tubing diameters from a through 1d in with standard-dimension-ratio wall thicknesses. Solvent-cemented joints (Fig. 10-147) are standard, but screwed joints are sometimes used with Schedule 80 pipe. Cemented joints must not be disturbed for 5 min and achieve full strength in 1 day. Because of the difference in thermal expansion, joints between PVC pipe and metal pipe should be flanged, using a PVC flange on the PVC pipe and a full-face gasket. Flanges are available with ANSI B16.5 150-lb drilling. Ball valves, Y-type globe valves, and diaphragm valves are available in PVC. Polypropylene Polypropylene (PP) pipe and fittings have excellent resistance to most common organic and mineral acids and their salts, strong and weak alkalies, and many organic chemicals. They are available in sizes a through 6 in, in Schedules 40 and 80, but are not covered as such by ASTM specifications. Reinforced-Thermosetting-Resin (RTR) Pipe Glass-reinforced epoxy resin has good resistance to nonoxidizing acids, alkalies, salt water, and corrosive gases. The glass reinforcement is many times stronger at room temperature than plastics, does not lose strength with increasing temperature, and reinforces the resin effectively up to 149°C (300°F). (See Table 10-17 for temperature limits.) The glass reinforcement is located near the outside wall, protected from the contents by a thick wall of resin and protected from the atmosphere by a thin wall of resin. Stock sizes are 2 through 12 in. Pipe is supplied in 6- and 12-m (20- and 40-ft) lengths. It is more economical for long, straight runs than for systems containing numerous fittings. When the pipe is sawed to nonfactory lengths, it must be sawed very carefully to avoid cracking the interior plastic zone. A twocomponent cement may be used to bond lengths into socket couplings or flanges or cemented-joint fittings. Curing of the cement is temperature-sensitive; it sets to full strength in 45 min at 93°C (200°F), in

10-102

TRANSPORT AND STORAGE OF FLUIDS

TABLE 10-44 Typical Hanger-Spacing Ranges Recommended for Reinforced-Thermosetting-Resin Pipe Nominal pipe size, in

2

3

4

6

8

10

12

Hanger-spacing range, ft

5–8

6–9

6–10

8–11

9–13

10–14

11–15

NOTE:

Consult pipe manufacturer for recommended hanger spacing for the specific RTR pipe being used. Tabulated values are based on a specific gravity of 1.25 for the contents of the pipe. To convert feet to meters, multiply by 0.3048.

12 h at 38°C (100°F), and in 24 h at 10°C (50°F). Extensive use is made of shop-fabricated flanged preassemblies. Only flanged joints are used to bond to metallic piping systems. Compared with that of other plastics, the ratio of fitting cost to pipe cost is high. Cementedjoint fittings and flanged fittings are available. Flanged lined metallic valves are used. RTR is more flexible than metallic pipe and consequently requires closer support spacing. While the recommended spacing varies among manufacturers and with the type of product, Table 10-44 gives typical hanger-spacing ranges. The pipe fabricator should be consulted for recommended hanger spacing on the specific pipe-wall construction being used. Epoxy resin has a higher strength at elevated temperatures than polyester resins but is not as resistant to attack by some fluids. Some glass-reinforced epoxy-resin pipe is made with a polyester-resin liner. The coefficient of thermal expansion of glass-reinforced resin pipe is higher than that for carbon steel but much less than that for plastics. Glass-reinforced polyester is the most widely used reinforced-resin system. A wide choice of polyester resins is available. The bisphenol resins resist strong acids as well as alkaline solutions. The size range is 2 through 12 in; the temperature range is shown in Table 10-17. Diameters are not standardized. Adhesive-cemented socket joints and hand-lay-up reinforced butt joints are used. For the latter, reinforcement consists of layers of glass cloth saturated with adhesive cement. Haveg 41NA This is a proprietary thermoset plastic consisting of a phenol-formaldehyde resin and nonasbestos silicate fillers. It is furnished as pipe and fittings with several types of joints and is resistant to most acidic chemicals, especially hydrochloric acid. The standard joint uses split cast-iron flanges set in tapered grooves machined in the outside of the pipe. A facing and grooving tool is available. Standard lengths are 1.2 m (4 ft) in the a- and e-in sizes and 3 m (10 ft) in all other sizes. Flanges are drilled per ANSI B16.5, except that the bolt holes are smaller. Figure 10-163 shows pressure-temperature ratings for standard-wall pipe with standard joints. Pipe and fittings with cemented sleeve joints are also available for use when external corrosion might destroy cast-iron flanges. Y-type globe valves, diaphragm valves, and foot and check valves are available.

FIG. 10-163 Operating pressure-temperature ratings for Haveg 41NA and 61NA pipe and fittings. (°F − 32)% = °C; to convert pounds-force per square inch to kilopascals, multiply by 6.895; to convert inches to millimeters, multiply by 25.4.

Haveg 61NA A proprietary nonasbestos silicate-filled furfuryl alcohol-formaldehyde resin pipe, Haveg 61NA is highly resistant to most acids and, with some reservations, to sodium hydroxide. It is also resistant to many hydrocarbons, halogenated organic compounds, and organic acids. Its pressure-temperature ratings are shown in Figure 10-163. PIPING-SYSTEM DESIGN Safeguarding Safeguarding may be defined as the provision of protective measures as required to ensure the safe operation of a proposed piping system. General considerations to be evaluated should include (1) the hazardous properties of the fluid, (2) the quantity of fluid which could be released by a piping failure, (3) the effect of a failure (such as possible loss of cooling water) on overall plant safety, (4) evaluation of effects on a reaction with the environment (i.e., possibility of a nearby source of ignition), (5) the probable extent of exposure of operating or maintenance personnel, and (6) the relative inherent safety of the piping by virtue of materials of construction, methods of joining, and history of service reliability. Evaluation of safeguarding requirements might include engineered protection against possible failures such as thermal insulation, armor, guards, barricades, and damping for protection against severe vibration, water hammer, or cyclic operating conditions. Simple means to protect people and property such as shields for valve bonnets, flanged joints, and sight glasses should not be overlooked. The necessity for means to shut off or control flow in the event of a piping failure such as block valves or excess-flow valves should be examined. Classification of Fluid Services The code applies to piping systems as illustrated in Fig. 10-124, but two categories of fluid services are segregated for special consideration as follows: Category D fluid service is defined as “a fluid service to which all the following apply: (1) the fluid handled is nonflammable and nontoxic; (2) the design gage pressure does not exceed 150 psi (1.0 MPa); and (3) the design temperature is between −20°F. (−29°C.) and 360°F. (182°C.).” Category M fluid service is defined as “a fluid service in which a single exposure to a very small quantity of a toxic fluid, caused by leakage, can produce serious irreversible harm to persons on breathing or bodily contact, even when prompt restorative measures are taken.” The code assigns to the owner the responsibility for identifying those fluid services which are in Categories D and M. The design and fabrication requirements for Class M toxic-service piping are beyond the scope of this Handbook. See ANSI B31.3—1976, chap. VIII. Design Conditions Definitions of the temperatures, pressures, and various forces applicable to the design of piping systems are as follows: Design Pressure The design pressure of a piping system shall not be less than the pressure at the most severe condition of coincident pressure and temperature resulting in the greatest required component thickness or rating. Design Temperature The design temperature is the material temperature representing the most severe condition of coincident pressure and temperature. For uninsulated metallic pipe with fluid below 38°C (100°F), the metal temperature is taken as the fluid temperature. With fluid at or above 38°C (100°F) and without external insulation, the metal temperature is taken as a percentage of the fluid temperature unless a lower temperature is determined by test or calculation. For pipe, threaded and welding-end valves, fittings, and other components with a wall thickness comparable with that of the pipe, the percentage is 95 percent; for flanges and flanged valves and fittings, 90 percent; for lap-joint flanges, 85 percent; and for bolting, 80 percent. With external insulation, the metal temperature is taken as the fluid temperature unless service data, tests, or calculations justify lower values. For internally insulated pipe, the design metal temperature shall be calculated or obtained from tests. Ambient Influences If cooling results in a vacuum, the design must provide for external pressure or a vacuum breaker installed; also provision must be made for thermal expansion of contents trapped

PROCESS-PLANT PIPING between or in closed valves. Nonmetallic or nonmetallic-lined pipe may require protection when ambient temperature exceeds design temperature. Occasional variations of pressure or temperature, or both, above operating levels are characteristic of certain services. If the following criteria are met, such variations need not be considered in determining pressure-temperature design conditions. Otherwise, the most severe conditions of coincident pressure and temperature during the variation shall be used to determine design conditions. (Application of pressures exceeding pressure-temperature ratings of valves may under certain conditions cause loss of seat tightness or difficulty of operation. Such an application is the owner’s responsibility.) All the following criteria must be met: 1. The piping system shall have no pressure-containing components of cast iron or other nonductile metal. 2. Nominal pressure stresses shall not exceed the yield strength at temperature (see Table 10-49 and Sy data in ASME Code, Sec. VIII, Division 2). 3. Combined longitudinal stresses SL shall not exceed the limits established in the code (see pressure design of piping components for SL limitations). 4. The number of cycles (or variations) shall not exceed 7000 during the life of the piping system. 5. Occasional variations above design conditions shall remain within one of the following limits for pressure design: • When the variation lasts no more than 10 h at any one time and no more than 100 h per year, it is permissible to exceed the pressure rating or the allowable stress for pressure design at the temperature of the increased condition by not more than 33 percent. • When the variation lasts no more than 50 h at any one time and not more than 500 h per year, it is permissible to exceed the pressure rating or the allowable stress for pressure design at the temperature of the increased condition by not more than 20 percent. Dynamic Effects Design must provide for impact (hydraulic shock, etc.), wind (exposed piping), earthquake (see ANSI A58.1), discharge reactions, and vibrations (of piping arrangement and support). Weight considerations include (1) live loads (contents, ice, and snow), (2) dead loads (pipe, valves, insulation, etc.), and (3) test loads (test fluid). Thermal-expansion and -contraction loads occur when a piping system is prevented from free thermal expansion or contraction as a result of anchors and restraints or undergoes large, rapid temperature changes or unequal temperature distribution because of an injection of cold liquid striking the wall of a pipe carrying hot gas. Design Criteria: Metallic Pipe The code uses three different approaches to design, as follows: 1. It provides for the use of dimensionally standardized components at their published pressure-temperature ratings. 2. It provides design formulas and maximum stresses. 3. It prohibits the use of materials, components, or assembly methods in certain conditions. Components Having Specific Ratings These are listed in ANSI, API, and industry standards. These ratings are acceptable for design pressures and temperatures unless limited in the code. A list of component standards is given in Appendix E of the code. The following rating tables covering commonly used components have been extracted from the original document with permission of the publisher, the American Society of Mechanical Engineers, New York: Table 10-45 lists pressure-temperature ratings for flanges, flanged fittings, and flanged valves; and Table 10-46 lists hydrostatic-shell test pressures for flanges, flanged fittings, and flanged valves. Flanged joints, flanged valves in the open position, and flanged fittings may be subjected to system hydrostatic tests at a pressure not to exceed the hydrostatic-shell test pressure. Flanged valves in the closed position may be subjected to a system hydrostatic test at a pressure not to exceed 110 percent of the 100°F rating of the valve unless otherwise limited by the manufacturer. Pressure-temperature ratings for soldered and brazed coppertubing joints are given in Tables 10-47 and 10-48 respectively. Components without Specific Ratings Components such as pipe and butt-welding fittings are generally furnished in nominal

10-103

thicknesses. Fittings are rated for the same allowable pressures as pipe of the same nominal thickness and, along with pipe, are rated by the rules for pressure design and other provisions of the code. Pressure Design of Metallic Components: Wall Thickness External-pressure stress evaluation of piping is the same as for pressure vessels. But an important difference exists when one is establishing design pressure and wall thickness for internal pressure as a result of the ASME Boiler and Pressure Vessel Code’s requirement that the relief-valve setting be not higher than the design pressure. For vessels this means that the design is for a pressure 10 percent more or less above the intended maximum operating pressure to avoid popping or leakage from the valve during normal operation. However, on piping the design pressure and temperature are taken as the maximum intended operating pressure and coincident temperature combination which results in the maximum thickness. The temporary increased operating conditions listed under “Design Criteria” cover temporary operation at pressures that cause relief valves to leak or open fully. Allowable stresses for nearly 1000 materials are contained in the code. For convenience, the allowable stresses for commonly used materials have been extracted from the code and listed in Table 10-49. For straight metal pipe under internal pressure the formula for minimum required wall thickness tm is applicable for Do / t ratios greater than 6. The more conservative Barlow and Lamé equations may also be used. Equation (10-92) includes a factor Y varying with material and temperature to account for the redistribution of circumferential stress which occurs under steady-state creep at high temperature and permits slightly lesser thickness at this range. PDo tm = ᎏᎏ +C (10-92) 2(SE + PY) where (in consistent units) P = design pressure Do = outside diameter of pipe C = sum of allowances for corrosion, erosion, and any thread or groove depth. For threaded components the depth is h of ANSI B2.1, and for grooved components the depth is the depth removed (plus 1⁄64 in when no tolerance is specified). SE = allowable stress (see Table 10-49) S = basic allowable stress for materials, excluding casting, joint, or structural-grade quality factors E = quality factor. The quality factor E is one or the product of more than one of the following quality factors: casting quality factor Ec, joint quality factor Ej (see Fig. 10-164), and structural-grade quality factor Es of 0.92. Y = coefficient having value in Table 10-50 for ductile ferrous materials, 0.4 for ductile nonferrous materials, and zero for brittle materials such as cast iron tm = minimum required thickness, in, to which manufacturing tolerance must be added when specifying pipe thickness on purchase orders. [Most ASTM specifications to which mill pipe is normally obtained permit minimum wall to be 12a percent less than nominal. ASTM A155 for fusion-welded pipe permits minimum wall 0.25 mm (0.01 in) less than nominal plate thickness.] Pipe with t equal to or greater than D/6 or P/SE greater than 0.385 requires special consideration. In addition to establishing the wall thickness for internal pressure, the stress values in Table 10-49 control other portions of the design. The sum of the longitudinal stresses SL (in the corroded condition) due to internal pressure, weight of pipe and contents between supports, and other sustained loadings such as friction between a laid (not hung) long length of straight cold pipe and its supports when it is placed in service, shall not exceed the value of Sh. In this determination, for pipe with welded longitudinal seams, the longitudinal weld joint factor is disregarded. Also, when thermal-expansion or contraction strains are taken up primarily by bending or torsion, the local stresses so produced are limited to the following range designated as SA: SA = f(1.25Sc + 0.25Sh)

(10-93)

10-104

TRANSPORT AND STORAGE OF FLUIDS

TABLE 10-45 Material group Materials temperature, °F

Pressure-Temperature Ratings for Flanges, Flanged Fittings, and Flanged Valves of Typical Materials,4 lbf/in2 1.1

1.5 Carbon

Normal

C, aMo

285 260 230

265

1.9

1.10

1, 1dCr, aMo

2dCr, 1Mo

1.13 5Cr, aMo

2.1

2.2

2.3

2.6

2.7

Type 304

Type 316

Type 304L Type 316L

Type 309

Type 310

275 235 205 180

275 240 215 195

230 195 175 160 145 140 125 110 95 80

260 230 220 200 170 140 125 110 95 80

65

65 50 35 20 670 605 570 535 505 480 465 455 445 435 425 415 385

150-lb class −20 to 100 200 300 400 500 600 650 700 750 800 850 900 950 1000

290 260 230 200 170 140 125 110 95 80

170 140 125 110 95 80

65 50 35 20

65 50 35 20 300-lb class

−20 to 100 200 300 400 500 600 650 700 750 800 850 900 950 1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500

740 675 655 635 600 550 535 535 505 410 270 170 105 50

695 680 655 640 620

−20 to 100 200 300 400 500 600 650 700 750 800 850 900 950 1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500

990 900 875 845 800 730 715 710 670 550 355 230 140 70

925 905 870 855 830

280 165

750 710 675 660

750 715 675 650 640 605 590 570 530 510 485 450 380

225 140 95 50 35

270 200 115 105 55

1000 950 895 880

1000 955 905 865

750 750 730 705 665

500 440 355 260 190 140 105 70 45

720 600 530 470 435 415 410 405 400 395 390 385 375 325 310 260 195 155 110 85 60 50 35 25

720 620 560 515 480 450 445 430 425 415 405 395 385 365 360 325 275 205 180 140 105 75 60 40

600 505 455 415 380 360 350 345 335 330 320

960 800 705 630 585 555 545 540 530 525 520 510 500 430 410 345 260 205 145 110 85 65 45 30

960 825 745 685 635 600 590 575 565 555 540 525 515 485 480 430 365 275 245 185 140 100 80 55

800 675 605 550 510 480 470 460 450 440 430

335 290 225 170 130 100 80 60 45 30 25

350 335 290 245 205 160 120 80 55 40 25

400-lb class*

375 220

855 805 785 755 710 675 650 600 505 300 185 130 70 45

355 265 150 140 75

1000 1000 970 940 885

665 585 470 350 255 190 140 90 60

895 805 760 710 670 635 620 610 595 580 565 555 515 450 390 300 230 175 135 105 80 60 40 30

465 445 390 330 275 215 160 105 75 50 30

PROCESS-PLANT PIPING

10-105

TABLE 10-45 Pressure-Temperature Ratings for Flanges, Flanged Fittings, and Flanged Valves of Typical Materials,4 lbf/in2 (Continued ) Material group Materials temperature, °F

1.1

1.5

Carbon steel

1.9

1.10

1.13

Normal

C, aMo

1, 1dCr, aMo

2dCr, 1Mo

−20 to 100 200 300 400 500 600 650 700 750 800 850 900 950 1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500

1480 1350 1315 1270 1200 1095 1075 1065 1010 825 535 345 205 105

1390 1360 1305 1280 1245

1500 1425 1345 1315 1285

1500 1430 1355 1295 1280

1500 1500 1455 1410 1330

445 275 190 105 70

535 400 225 205 110

995 880 705 520 385 280 205 140 90

−20 to 100 200 300 400 500 600 650 700 750 800 850 900 950 1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500

2220 2025 1970 1900 1795 1640 1610 1600 1510 1235 805 515 310 155

2085 2035 1955 1920 1865

2250 2135 2020 1975 1925

2250 2150 2030 1945 1920

2250 2250 2185 2115 1995

670 410 290 155 105

805 595 340 310 165

1490 1315 1060 780 575 420 310 205 135

−20 to 100 200 300 400 500 600 650 700 750 800 850 900

3705 3375 3280 3170 2995 2735 2685 2665 2520 2060 1340 860

3470 3395 3260 3200 3105

3750 3560 3365 3290 3210

3750 3580 3385 3240 3200

5Cr, aMo

2.1

2.2

2.3

2.6

2.7

Type 304

Type 316

Type 304L Type 316L

Type 309

Type 310

1440 1200 1055 940 875 830 815 805 795 790 780 770 750 645 620 515 390 310 220 165 125 90 70 50

1440 1240 1120 1030 955 905 890 865 845 830 810 790 775 725 720 645 550 410 365 275 205 150 115 85

1200 1015 910 825 765 720 700 685 670 660 645

2160 1800 1585 1410 1310 1245 1225 1210 1195 1180 1165 1150 1125 965 925 770 585 465 330 245 185 145 105 70

2160 1860 1680 1540 1435 1355 1330 1295 1270 1245 1215 1180 1160 1090 1080 965 825 620 545 410 310 225 175 125

1800 1520 1360 1240 1145 1080 1050 1030 1010 985 965

3600 3000 2640 2350 2185 2075 2040 2015 1990 1970 1945 1920

3600 3095 2795 2570 2390 2255 2220 2160 2110 2075 2030 1970

3600 2530 2270 2065 1910 1800 1750 1715 1680 1645 1610

600-lb class*

560 330

1210 1175 1135 1065 1015 975 900 755

1345 1210 1140 1065 1010 955 930 910 895 870 850 830 775 670 585 445 345 260 200 160 115 90 60 50

700 665 585 495 410 325 240 160 110 75 50

900-lb class*

845 495

1815 1765 1705 1595 1525 1460 1350 1130

2015 1815 1705 1600 1510 1435 1395 1370 1340 1305 1275 1245 1160 1010 875 670 515 390 300 235 175 135 95 70

1050 1000 875 740 620 485 360 235 165 115 70

1500-lb class

3025 2940 2840 2660 2540 2435 2245

3750 3750 3640 3530 3325

2485 2195 1765

3360 3025 2845 2665 2520 2390 2330 2280 2230 2170 2125 2075

10-106

TRANSPORT AND STORAGE OF FLUIDS

TABLE 10-45 Pressure-Temperature Ratings for Flanges, Flanged Fittings, and Flanged Valves of Typical Materials,4 lbf/in2 (Continued ) Material group Materials temperature, °F

1.1

1.5

Carbon steel Normal

C, aMo

950 1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500

515 260

1405 825

−20 to 100 200 300 400 500 600 650 700 750 800 850 900 950 1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500

6170 5625 5470 5280 4990 4560 4475 4440 4200 3430 2230 1430 860 430

5785 5660 5435 5330 5180

1.9

1.10

1, 1dCr, aMo

1.13

2dCr, 1Mo

2.2

2.3

2.6

2.7

Type 304

Type 316

Type 304L Type 316L

Type 309

Type 310

1305 960 705 515 345 225

1870 1610 1545 1285 980 770 550 410 310 240 170 120

1930 1820 1800 1610 1370 1030 910 685 515 380 290 205

6000 5000 4400 3920 3640 3460 3400 3360 3320 3280 3240 3200 3120 2685 2570 2145 1630 1285 915 685 515 400 285 200

6000 5160 4660 4280 3980 3760 3700 3600 3520 3460 3380 3280 3220 3030 3000 2685 2285 1715 1515 1145 860 630 485 345

5Cr, aMo

2.1

1500-lb class (Cont.) 1885 1115 685 480 260 170

1340 995 565 515 275

6250 5930 5605 5485 5350

6250 5965 5640 5400 5330

6250 6250 6070 5880 5540

2230 1660 945 860 460

4145 3660 2945 2170 1600 1170 860 570 370

1930 1680 1460 1115 860 650 495 395 290 225 155 120

1750 1665 1460 1235 1030 805 600 395 275 190 120

2500-lb class

2345 1370

5040 4905 4730 4430 4230 4060 3745 3145 1860 1145 800 430 285

*For Group 1.1, do not use ASTM A181 Grade I or II materials. NOTES:

1. Ratings shown apply to other material groups when column-dividing lines have been omitted. 2. Temperature notes for all material groups in Table 10-45.

Material group 1.1 1.5 1.9 1.10 1.13 2.1 2.2 2.3 2.6 2.7

Materials3 (specification—grade)

See Notes

A105, A181-II, A216-WCB, A515-70 A516-70 A350-LF2, A537-C1.1 A182-F1, A204-A, A204-B, A217-WC1 A352-LC1 A182-F11, A182-F12, A387-11, C1.2 A217-WC6 A182-F22, A387-22, C1.2 A217-WC9 A182-F5a, A217-C5 A182-F304, A182-F304H A240-304, A351-CF8 A351-CF3 A182-F316, A182-F316H, A240-316 A240-317, A351-CF8M A351-CF3M A182-F304L, A240-304L A182-F316L, A240-316L A240-309S, A351-CH8, A351-CH20 A182-F310, A240-310S A351-CK20

a, h a, g d b, h d, m m, c j, m c m, j m n n, o f n, o n, o g f g n, o k, n n

5000 4220 3780 3440 3180 3000 2920 2860 2800 2740 2680

5600 5040 4740 4440 4200 3980 3880 3800 3720 3620 3540 3460 3220 2800 2430 1860 1430 1085 830 660 485 370 260 200

2915 2770 2430 2060 1715 1345 1000 660 460 315 200

PROCESS-PLANT PIPING

10-107

TABLE 10-45 Pressure-Temperature Ratings for Flanges, Flanged Fittings, and Flanged Valves of Typical Materials,4 lbf/in2 (Concluded ) a. Permissible but not recommended for prolonged use above about 800°F. b. Permissible but not recommended for prolonged use above about 850°F. c. Permissible but not recommended for prolonged use above about 1100°F. d. Not to be used over 650°F. f. Not to be used over 800°F. g. Not to be used over 850°F. h. Not to be used over 1000°F. i. Not to be used over 1050°F. j. Not to be used over 1100°F. k. For service temperatures 1050°F and above, assurance must be provided that grain size is not finer than ASTM No. 6. l. Only killed steel shall be used above 850°F. m. Use normalized and tempered material only. n. At temperatures over 1000°F, use only when the carbon content is 0.04 percent or higher. o. See ANSI B16.5 for heat treatment for service temperatures over 1000°F. p. The ratings at −20 to 100°F given for the materials covered shall also apply at lower temperatures. The ratings for low-temperature service of the cast and forged materials listed in ASTM A352 and A350 shall be taken the same as the −20 to 100°F ratings for carbon steel. q. Some of the materials listed in the rating tables undergo a decrease in impact resistance at temperatures lower than −20°F to such an extent as to be unable to resist safely shock loadings, sudden changes of stress, or high stress concentration. 3. See ANSI B16.5, Table 1A, for additional information and notes relating to specific materials. 4. Extracted from Steel Pipe Flanges and Flanged Fittings, ANSI B16.5—1977 and B16.34—1977, with permission of the publisher, the American Society of Mechanical Engineers, New York. 5. A product used under the jurisdiction of the ASME Boiler and Pressure Vessel Code and the ANSI Code for Pressure Piping B31.1 is subject to any limitation of those codes. This includes any maximum-temperature limitation for a material or a code rule governing the use of a material at a low temperature. 6. (°F − 32)5⁄ 9 = °C; to convert pounds-force per square inch to megapascals, multiply by 0.006895.

where Sc = S from Table 10-49 at a minimum (cold) metal temperature normally expected during operation or shutdown (See Note 13, Table 10-49) Sh = S from Table 10-49 at maximum (hot) metal temperature normally expected during operation or shutdown (See Note 13, Table 10-49) f = stress-range reduction factor for total number of full temperature cycles over expected life (See Table 10-51) When the anticipated number of cycles is substantially less than 7000, useful information can be obtained from ASME Boiler and Pressure Vessel Code, Sec. III, “Nuclear Vessels.” However, if the sum of longitudinal stresses SL enumerated is less than their stated limit Sh, the difference may be added to the term 0.25Sh in the equation limiting the stress range: SA = f [1.25(Sc + Sh) − SL]

(10-94)

For flanges of nonstandard dimensions or for sizes beyond the scope of the approved standards, design shall be in accordance with the requirements of the ASME Boiler and Pressure Vessel Code, Sec. VIII, except that requirements for fabrication, assembly, inspection testing, and the pressure and temperature limits for materials of the Piping Code are to prevail. Countermoment flanges of flat face or otherwise providing a reaction outside the bolt circle are permitted if

TABLE 10-46 Hydrostatic-Shell Test Pressures for Flanges, Flanged Fittings, and Flanged Valves of Typical Materials* Shell test pressures by class, lbf/in2 gauge

Material group no.

150

300

400

600

900

1500

2500

1.1 1.5 1.9 1.10 1.13

450 400 450 450 450

1125 1050 1125 1125 1125

1500 1400 1500 1500 1500

2225 2100 2250 2250 2250

3350 3150 3375 3375 3375

5575 5225 5625 5625 5625

9275 8700 9375 9375 9375

2.1 2.2 2.3 2.6 2.7

425 425 350 400 400

1100 1100 900 1025 1025

1450 1450 1200 1350 1350

2175 2175 1800 2025 2025

3250 3250 2700 3025 3025

5400 5400 4500 5050 5050

9000 9000 7500 8400 8400

*Extracted from Steel Pipe Flanges and Flanged Fittings, ANSI B16.5— 1977, with permission of the publisher, the American Society of Mechanical Engineers, New York. Test temperature not to exceed 125°F. (°F − 32) 5⁄ 9 = °C; to convert pounds-force per square inch to megapascals, multiply by 0.006895.

designed or tested in accordance with code requirements under pressure-containing components “not covered by standards and for which design formulas or procedures are not given.” In accordance with listed standards, blind flanges may be used at their pressure-temperature ratings. The minimum thickness of nonstandard blind flanges shall be the same as for a bolted flat cover, in accordance with the rules of the ASME Boiler and Pressure Vessel Code, Sec. VIII. Operational blanks shall be of the same thickness as blind flanges or may be calculated by the following formula (use consistent units): t = d 兹3苶P 苶/1 苶6 苶S 苶 where

(10-95)

d = inside diameter of gasket for raised- or flat (plain)-face flanges, or the gasket pitch diameter for retained gasketed flanges P = internal design pressure or external design pressure S = applicable allowable stress

Valves must comply with the applicable standards listed in Appendix E of the code and with the allowable pressure-temperature limits established thereby but not beyond the code-established service or materials limitations. Special valves must meet the same requirements as for countermoment flanges. The code contains no specific rules for the design of fittings other than as branch openings. Ratings established by recognized standards are acceptable, however. ANSI Standard B16.5 for steel-flanged fittings incorporates a 1.5 shape factor and thus requires the entire fitting to be 50 percent heavier than a simple cylinder in order to provide reinforcement for openings and/or general shape. ANSI B16.9 for butt-welding fittings, on the other hand, requires only that the fittings be able to withstand the calculated bursting strength of the straight pipe with which they are to be used. The thickness of pipe bends shall be determined as for straight pipe, provided the bending operation does not result in a difference between maximum and minimum diameters greater than 8 and 3 percent of the nominal outside diameter of the pipe for internal and external pressure respectively. The maximum allowable internal pressure for multiple miter bends shall be the lesser value calculated from Eqs. (10-96) and (10-97). These equations are not applicable when θ exceeds 22.5°. SEt t P = ᎏ ᎏᎏᎏ (10-96) r2 t + 0.643 tan θ 兹苶 r2苶t

冢

SEt R1 − r2 P = ᎏ ᎏᎏ r2 R1 − 0.5r2

冢

冣

冣

(10-97)

10-108

TRANSPORT AND STORAGE OF FLUIDS TABLE 10-47

Strength of Solder Joints*

Maximum recommended pressure-temperature ratings for solder joints made with copper tubing and wrought-copper and -bronze or cast-bronze solder-joint pressure fittings and using representative commercial solders Maximum working pressure, lbf/in2 Working temperatures, °F

f to 1 in, inclusive†

1d to 2 in, inclusive†

2a to 4 in, inclusive†

5 to 8 in, inclusive†

50-50 tin-lead solder‡

100 150 200 250

200 150 100 85

175 125 90 75

150 100 75 50

135 90 70 45

95-5 tin-antimony solder

100 150 200 250

500 400 300 200

400 350 250 175

300 275 200 150

270 250 180 135

Joining material used in joints

NOTE: For extremely low working temperatures (in the 0 to −200°F range) it is recommended that a joining material melting at or above 1100°F be used. (Joining materials with melting points in excess of 800°F are defined as “brazing” alloys by the American Welding Society.) See Table 10-48. *Extracted from ANSI B16.22—1973 with permission of the publisher, the American Society of Mechanical Engineers, New York. (°F − 32)5⁄ 9 = °C; to convert inches to millimeters, multiply by 25.4; to convert pounds-force per square inch to megapascals, multiply by 0.006895. †Standard water-tubing sizes. ‡ASTM B32.66T Alloy Grade 50A.

where nomenclature is the same as for straight pipe except as follows (see Fig. 10-165): t = pressure design thickness r2 = mean radius of pipe R1 = effective radius of miter bend, defined as the shortest distance from the pipe centerline to the intersection of the planes of adjacent miter joints θ = angle of miter cut, ° α = angle of change in direction at miter joint = 2θ, ° TABLE 10-48

Strength of Silver-Brazed Joints*

Maximum recommended pressure-temperature ratings for brazed joints made with copper tubing and copper or copper-alloy fittings and using representative commercial brazing alloys Outsidediameter size, in

lbf/in2 150°F (S = 5100 lbf/in2)

250°F (S = 4700 lbf/in2)

350°F (S = 4000 lbf/in2)

400°F (S = 3000 lbf/in2)

1790 1190 890 840 780

1650 1100 825 780 720

1400 940 700 660 615

1050 700 525 500 460

a v e 7⁄ 8 1f

680 615 535 490 420

625 565 495 450 390

530 480 420 385 330

400 360 315 290 250

1r 1v 2f 2v 3f

380 350 310 286 270

350 320 285 265 250

295 275 245 225 190

220 205 180 170 140

3v 4f 5f 6f

260 250 225 215

240 230 210 195

200 195 180 165

150 145 135 125

f t d b r

*Extracted from ANSI B16.41—January 1977 draft, with permission of the publisher, the American Society of Mechanical Engineers, New York. (°F − 32)5⁄ 9 = °C; to convert inches to millimeters, multiply by 25.4; to convert poundsforce per square inch to megapascals, multiply by 0.006895.

For compliance with the code, the value of R1 shall not be less than that given by Eq. (10-98): R1 = A/tan θ + D/2 (10-98) where A has the following empirical values (not valid in SI units): t, in

A

≤0.5 0.5 < t < 0.88 ≥0.88

1.0 2t (2t/3) + 1.17

Piping branch connections involve the same considerations as pressure-vessel nozzles. However, outlet size in proportion to piping header size is unavoidably much greater for piping. The current Piping Code rules for calculation of branch-connection reinforcement are similar to those of the ASME Boiler and Pressure Vessel Code, Sec. VIII, Division I (1980 edition) for a branch with axis at right angles to the header axis. If the branch connection makes an angle β with the header axis from 45 to 90°, the Piping Code requires that the area to be replaced be increased by dividing it by sin β. In such cases the half width of the reinforcing zone measured along the header axis is similarly increased, except that it may not exceed the outside diameter of the header. Some details of commonly used reinforced branch connections are given in Fig. 10-166. The rules provide that a branch connection has adequate strength for pressure if a fitting (tee, lateral, or cross) is in accordance with an approved standard and is used within the pressure-temperature limitations or if the connection is made by welding a coupling or half coupling (wall thickness not less than the branch anywhere in reinforcement zone or less than extra heavy or 3000 lb) to the run and provided the ratio of branch to run diameters is not greater than onefourth and that the branch is not greater than 2 in nominal diameter. Dimensions of extra-heavy couplings are given in the Steel Products Manual published by the American Iron and Steel Institute. In ANSI B16.11—1966, 2000-lb couplings were superseded by 3000-lb couplings. ANSI B31.3 states that the reinforcement area for resistance to external pressure is to be at least one-half of that required to resist internal pressure. The code provides no guidance for analysis but requires that external and internal attachments be designed to avoid flattening of the pipe, excessive localized bending stresses, or harmful thermal gradients, with further emphasis on minimizing stress concentrations in cyclic service.

PROCESS-PLANT PIPING

10-109

Longitudinal and spiral-weld joint factor Ej. NOTE: It is not permitted to increase the joint quality factor by additional examination for joints 1, 2, and 4a. (Extracted from ANSI B31.3—1980, with permission of the publisher, the American Society of Mechanical Engineers, New York.)

FIG. 10-164

The code provides design requirements for closures which are flat, ellipsoidal, spherically dished, hemispherical, conical (without transition knuckles), conical convex to pressure, toriconical concave to pressure, and toriconical convex to pressure. Openings in closures over 50 percent in diameter are designed as flanges in flat closures and as reducers in other closures. Openings of not over one-half of the diameter are to be reinforced as branch connections. Thermal Expansion and Flexibility: Metallic Piping ANSI B31.3 requires that piping systems have sufficient flexibility to prevent thermal expansion or contraction or the movement of piping supports or terminals from causing (1) failure of piping supports from overstress or fatigue; (2) leakage at joints; or (3) detrimental stresses or distortions in piping or in connected equipment (pumps, turbines, or valves, for example), resulting from excessive thrusts or movements in the piping. To assure that a system meets these requirements, the computed displacement–stress range SE shall not exceed the allowable stress range SA [Eqs. (10-93) and (10-94)], the reaction forces Rm [Eq. (10-105)] shall not be detrimental to supports or connected equipment, and movement of the piping shall be within any prescribed limits. Displacement Strains Strains result from piping being displaced from its unrestrained position: 1. Thermal displacements. A piping system will undergo dimensional changes with any change in temperature. If it is constrained from free movement by terminals, guides, and anchors, it will be displaced from its unrestrained position. 2. Reaction displacements. If the restraints are not considered rigid and there is a predictable movement of the restraint under load, this may be treated as a compensating displacement.

3. Externally imposed displacements. Externally caused movement of restraints will impose displacements on the piping in addition to those related to thermal effects. Such movements may result from causes such as wind sway or temperature changes in connected equipment. Total Displacement Strains Thermal displacements, reaction displacements, and externally imposed displacements all have equivalent effects on the piping system and must be considered together in determining total displacement strains in a piping system. Expansion strains may be taken up in three ways: by bending, by torsion, or by axial compression. In the first two cases maximum stress occurs at the extreme fibers of the cross section at the critical location. In the third case the entire cross-sectional area over the entire length is for practical purposes equally stressed. Bending or torsional flexibility may be provided by bends, loops, or offsets; by corrugated pipe or expansion joints of the bellows type; or by other devices permitting rotational movement. These devices must be anchored or otherwise suitably connected to resist end forces from fluid pressure, frictional resistance to pipe movement, and other causes. Axial flexibility may be provided by expansion joints of the slipjoint or bellows types, suitably anchored and guided to resist end forces from fluid pressure, frictional resistance to movement, and other causes. Displacement Stresses Stresses may be considered proportional to the total displacement strain only if the strains are well distributed and not excessive at any point. The methods outlined here and in the code are applicable only to such a system. Poor distribution of strains (unbalanced systems) may result from: 1. Highly stressed small-size pipe runs in series with large and relatively stiff pipe runs

TABLE 10-49

Allowable Stresses in Tension for Materials (4, 13, 28)* Specifications are ASTM unless otherwise indicated. Numbers in parentheses refer to notes at end of table.

Material

P no. (23)

Specification

Grade

Class

Factor, E

Minimum tensile strength, kip/in2

Minimum yield strength, kip/in2

Iron Centrifugally cast pipe FS-WW-P421c AWWA C106 AWWA C108 Carbon steel Seamless pipe and tubing A53 A53 A106 A106 A106 A120 A333 A333 API 5L API 5L API 5LX API 5LX API 5LX API 5LX

1 1 1 1 1 1 1 1 1 1 SP2 SP3 SP3 SP3

A B A B C

Electric-resistance-welded pipe A53 A53 A120 A135 A135 A333 A333 A587 API 5L API 5L API 5L API 5L API 5LX API 5LX API 5LX

1 1 1 1 1 1 1 1 1 1 1 SP2 SP3 SP3 SP3

A B

1 6 A B X42 X46 X52 X52 Type E Type E

A B 1 6 A25 A B X42 X46 X52 X52

Electric-fusion-welded pipe (straight seam) A570 GR A A134 1 A570 GR B A134 1 A570 GR C A134 1 A570 GR D A134 1 A570 GR E A134 1 Low- and intermediate-alloy steel Seamless pipe 3a Ni A333 e Cr, e Ni, Cu, Al A333 2d Ni A333 9 Ni A333 C, a Mo A335 5 Cr, a Mo A335 1d Cr, a Mo A335 2d Cr, 1 Mo A335

Type S Type S

I and II

48.0 60.0 48.0 60.0 70.0

30.0 35.0 30.0 35.0 40.0

55.0 60.0 48.0 60.0 60.0 63.0 66.0 72.0

30.0 35.0 30.0 35.0 42.0 46.0 52.0 52.0

Minimum temperature (18)

Minimum temperature to 100

200

300

400

8, 10, 17 8, 10, 17 8, 10, 17

−20 −20 −20

6.0 6.0 6.0

6.0 6.0 6.0

6.0 6.0 6.0

6.0 6.0 6.0

1, 2 1, 2 2 2 2 21 1, 2 2 1, 2 1, 2 37, 38 37, 38 37, 38 37, 38

−20 −20 −20 −20 −20 −20 −50 −50 −20 −20 −20 −20 −20 −20

16.0 20.0 16.0 20.0 23.3 12.0 18.3 20.0 16.0 20.0 20.0 21.0 22.0 24.0

16.0 20.0 16.0 20.0 23.3 11.4 18.3 20.0 16.0 20.0 20.0 21.0 22.0 24.0

16.0 20.0 16.0 20.0 23.3

−20 −20 −20 −20 −20 −50 −50 −20 −20 −20 −20 −20 −20 −20 −20

13.6 17.0 10.2 13.6 17.0 15.6 17.0 13.6 12.8 13.6 17.0 17.0 17.9 18.7 20.4

Notes

500

600

16.0 20.0 16.0 20.0 22.9

16.0 18.9 16.0 18.9 21.6

14.8 17.3 14.8 17.3 19.7

17.7 20.0 16.0 20.0 20.0 21.0 22.0 24.0

17.2 20.0 16.0 20.0 20.0 21.0 22.0 24.0

16.2 18.9 16.0 18.9

14.8 17.3 14.8 17.3

13.6 17.0 9.7 13.6 17.0 15.6 17.0 13.6 12.8 13.6 17.0 17.0 17.9 18.7 20.4

13.6 17.0

13.6 17.0

13.6 16.1

12.6 14.7

13.6 17.0 15.0 17.0 13.6 12.3 13.6 17.0 17.0 17.9 18.7 20.4

13.6 17.0 14.6 17.0 13.6 11.8 13.6 17.0 17.0 17.9 18.7 20.4

13.6 16.1 13.8 16.1 13.6

12.6 14.7 12.6 14.7 12.6

13.6 16.1

12.6 14.7

0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85

48.0 60.0

30.0 35.0

48.0 60.0 55.0 60.0 48.0 45.0 48.0 60.0 60.0 63.0 66.0 72.0

30.0 35.0 30.0 35.0 30.0 25.0 30.0 35.0 42.0 46.0 52.0 52.0

1, 2 1, 2 21 1, 2 1, 2 1, 2 2 1, 2 1, 2 1, 2 1, 2 37, 38 37, 38 37, 38 37, 38

0.74 0.74 0.74 0.74 0.74

45.0 49.0 52.0 55.0 58.0

25.0 30.0 33.0 40.0 42.0

5, 21 5, 21 5, 21 5, 21 5, 21

−20 −20 −20 −20 −20

11.1 12.1 12.8 13.6 14.3

10.5 11.4 12.1 12.8 13.5

10.0 10.9 11.6 12.2 12.9

65.0 60.0 65.0 100.0 55.0 60.0 60.0 60.0

35.0 35.0 35.0 75.0 30.0 30.0 30.0 30.0

21.7 20.0 21.7 31.7 18.3 20.0 20.0 20.0

19.6 19.1 19.6 31.7 18.3 18.1 18.7 18.5

19.6 18.2 19.6

18.7 17.3 18.7

17.8 16.4 17.6

16.8 15.5 16.8

40 3

−150 −150 −100 −320 −20 −20 −20 −20

17.5 17.4 18.0 18.0

16.9 17.2 17.5 17.9

16.3 17.1 17.2 17.9

15.7 16.8 16.7 17.9

20.0 20.0 16.7 20.0 20.0 20.0

20.0 20.0 16.7 20.0 20.0 20.0

20.0 20.0 16.7 20.0 20.0 20.0

18.7 18.7 15.8 20.0 20.0 19.3

17.5 17.5 14.8 20.0 20.0 17.9

16.4 16.4 14.0 19.2 19.2 17.0

20.0

20.0

20.0

19.3

17.9

17.0

9B 4 9A 11A-SG1 3 5 4 5

3 4 7 8 P1 P5 P11 P22

8 8 8 8 8 8

TP304 TP304H TP304L TP310 TP310 TP316

75.0 75.0 70.0 75.0 75.0 75.0

30.0 30.0 25.0 30.0 30.0 30.0

19, 24, 32 6, 19, 24, 32 14, 16

−425 −325 −425 −325 −325 −325

8

TP316H

75.0

30.0

16

−325

8

TP316L

70.0

25.0

−325

16.7

16.7

16.7

15.5

14.4

13.5

8 8

TP347 TP347H

75.0 75.0

30.0 30.0

7, 14

−425 −325

20.0 20.0

20.0 20.0

20.0 20.0

20.0 20.0

19.9 19.9

19.3 19.3

Centrifugally cast pipe 18Cr, 8Ni A451 18Cr, 10Ni, 2Mo A451 18Cr, 10Ni, Cb A451 15Cr, 13Ni, 2Mo, Cb A451 23Cr, 13Ni A451 23Cr, 13Ni A451

8 8 8 8 8 8

0.90 0.90 0.90 0.90 0.90 0.90

70.0 70.0 70.0 70.0 65.0 70.0

30.0 30.0 30.0 30.0 28.0 30.0

18.0 18.0 18.0 18.0 16.8 18.0

18.0 18.0 18.0

17.8 18.0 18.0

15.8 17.5 18.0

14.8 16.3 17.4

14.1 15.4 16.5

16.8 18.0

16.8 18.0

16.8 18.0

16.8 18.0

16.2 17.3

8 8 8 8

0.90 0.85 0.85 0.85

65.0 75.0 75.0 75.0

28.0 30.0 30.0 30.0

14, 15, 16 14, 15, 16 7, 14, 15 7, 11, 14, 15 11, 14, 15, 19 9, 11, 14, 15, 19, 24 14, 15, 19, 24 15, 16 15, 16 15

−425 −425 −325 −325 −325 −325

25Cr, 20Ni 18Cr, 8Ni 16Cr, 12Ni, 2Mo 18Cr, 10Ni, Cb

CPF8 CPF8M CPF8C CPF10MC CPH8 CPH10 or CPH 20 CPK20 TP304H TP316H TP347H

−325 −325 −325 −325

16.8 17.0 17.0 17.0

16.8 17.0 17.0 17.0

16.8 17.0 17.0 17.0

16.8 15.9 16.4 17.0

16.8 14.8 15.2 16.9

16.2 14.0 14.4 16.4

Stainless steel Seamless pipe and tubing 18Cr, 8Ni pipe A312 18Cr, 8Ni pipe A312 18Cr, 8Ni pipe A312 25Cr, 20Ni pipe A312 25Cr, 20Ni pipe A312 16Cr, 12Ni, 2Mo A312 pipe 16Cr, 12Ni, 2Mo A312 pipe 16Cr, 12Ni, 2Mo A312 pipe 18Cr, 10Ni, Cb pipe A312 18Cr, 10Ni, Cb pipe A312

10-110

A451 A452 A452 A452

7, 14, 16, 20 16

Metal temperature, °F (22) 650

700

750

800

850

900

950

1000

1050

1100

1150

1200

14.5 17.0 14.5 17.0 19.4

14.4 16.8 14.4 16.8 19.2

10.7 13.0 10.7 13.0 14.8

9.3 | 10.8 | 9.3 | 10.8 | 12.0 |

7.9 8.7 7.9 8.7

6.5 6.5 6.5 6.5

4.5 4.5 4.5 4.5

2.5 2.5 2.5 2.5

1.6 1.6 1.6 1.6

1.0 1.0 1.0 1.0

14.5 17.0 14.5 17.0

14.4 16.8 14.4 16.8

12.0 13.0 10.7 13.0

10.2 | 10.8 | 9.3 | 10.8 |

8.3 8.7 7.9 8.7

6.5 6.5 6.5 6.5

4.5 4.5 4.5 4.5

2.5 2.5 2.5 2.5

1.6 1.6 1.6 1.6

1.0 1.0 1.0 1.0

12.3 14.5

12.2 14.0

9.1 11.0

7.9 9.2

6.7 7.4

5.5 5.5

3.8 3.8

2.1 2.1

1.4 1.4

0.9 0.9

12.3 14.5 12.3 14.5 12.3

12.2 14.0 12.2 14.0

9.1 11.0 10.2 11.0

7.9 9.2 8.7 9.2

6.7 7.4 7.1 7.4

5.5 5.5 5.5 5.5

3.8 3.8 3.8 3.8

2.1 2.1 2.1 2.1

1.4

0.9

1.4 1.4

0.9 0.9

12.3 14.5

12.2 14.0

9.1 11.0

7.9 9.2

6.7 7.4

5.5 5.5

3.8 3.8

2.1 2.1

1.4 1.4

0.9 0.9

16.3 15.0 16.3

15.5

13.9

11.4

9.0

6.5

4.5

2.5

1.6

1.0

15.5

13.9

11.4

9.0

6.5

4.5

2.5

1.6

1.0

15.4 16.6 16.2 17.9

15.1 16.3 15.6 17.9

13.8 13.2 15.0 17.9

13.5 12.8 15.0 15.2

13.1 12.1 14.4 14.5

12.7 10.9 13.1 12.8

8.2 8.0 11.0 11.0

4.8 5.8 7.8 7.8

4.2 5.5 5.8

2.9 4.0 4.2

2.0 2.5 3.0

1.3 1.2 2.0

16.2 16.2 13.7 18.8 18.8 16.7

16.0 16.0 13.5 18.3 18.3 16.3

15.6 15.6 13.3 18.0 18.0 16.1

15.2 15.2 13.0 17.5 17.5 15.9

14.9 14.9 12.8 14.6 14.6 15.7

14.6 14.6 11.9 13.9 13.9 15.5

14.4 14.4 9.9 12.5 12.5 15.4

13.8 13.8 7.8 11.0 | 11.0 | 15.3

12.2 12.2 6.3 7.1 9.8 14.5

9.7 9.7 5.1 5.0 8.5 12.4

7.7 7.7 4.0 3.6 7.3 9.8

16.7

16.3

16.1

15.9

15.7

15.5

15.4

15.3

14.5

12.4

13.2

12.9

12.6

12.4

12.1

11.8

19.0 19.0

18.6 18.6

18.5 18.5

18.3 18.3

15.4 18.2

14.9 18.1

11.5

11.2

10.8

14.8 18.1

14.0 18.0

12.1 17.1

13.8 15.0 16.2

13.6 14.6 15.8

13.4 14.2 15.5

13.3 14.0 15.4

11.6 13.2 12.6

11.4 13.0 12.5

11.1 12.6 12.3

9.7 11.8 12.1

15.7 16.9

15.4 16.5

15.1 16.2

14.7 15.7

11.5 12.2

11.2 12.0

10.6 11.2

15.7 13.8 14.2 16.1

15.4 13.5 13.8 15.8

15.1 13.2 13.6 15.7

14.7 12.8 13.5 15.5

11.5 12.7 13.3 15.4

11.2 12.4 13.2 15.4

10.7 12.2 13.1 15.4

9.4 | 9.5 | 9.9 11.7 13.0 15.3

1250

1300

1350

1400

1450

1500

6.0 6.0 3.2 2.5 6.0 7.4

4.7 4.7 2.6 1.5 4.8 5.5

3.7 3.7 2.1 0.8 3.5 4.1

2.9 2.9 1.7 0.5 2.3 3.1

2.3 2.3 1.1 0.4 1.6 2.3

1.8 1.8 1.0 0.3 1.1 1.7

1.4 1.4 0.9 0.2 0.8 1.3

9.8

7.4

5.5

4.1

3.1

2.3

1.7

1.3

10.2

8.8

6.4

4.7

3.5

2.5

1.8

1.3

1.0

9.1 14.2

6.1 10.5

4.4 7.9

3.3 5.9

2.2 4.4

1.5 3.2

1.2 2.5

0.9 1.8

0.8 1.3

8.6 10.3 11.7

6.7 8.4 9.7

5.2 7.2 7.2

4.0 6.1 4.5

2.9 4.8 3.2

2.2 3.6 2.4

1.6 2.7 1.8

1.2 2.1 1.4

0.9 1.7 1.0

0.7 1.3 0.9

7.6 7.6

5.8 5.8

4.5 4.5

3.3 3.3

2.6 2.6

2.1 2.1

1.5 1.5

1.2 1.2

0.8 0.8

0.7 0.7

8.8 10.3 12.3 14.5

7.6 8.3 10.5 12.1

6.5 6.5 8.3 8.9

5.4 5.1 6.3 6.7

4.3 4.0 4.6 5.0

3.1 3.1 3.5 3.7

2.1 2.4 2.6 2.7

1.4 1.9 1.9 2.1

1.0 1.5 1.4 1.5

0.7 1.2 1.0 1.1

10-111

TABLE 10-49

Allowable Stresses in Tension for Materials (4, 13, 28) (Continued ) Specifications are ASTM unless otherwise indicated. Numbers in parentheses refer to notes at end of table.

Material

Specification

Specification

Material

Material

0.85 0.85 0.85 0.85 0.85 0.85 0.85

75.0 75.0 70.0 75.0 75.0 75.0 75.0

30.0 30.0 25.0 30.0 30.0 30.0 30.0

TP316H

0.85

75.0

30.0

TP316L

0.85

70.0

25.0

TP317

0.85

75.0

30.0

TP321 TP321H TP347 TP347H

0.85 0.85 0.85 0.85

75.0 75.0 75.0 75.0

30.0 30.0 30.0 30.0

31

Drawn

34 34

Annealed Annealed

Grade

43

600 (N06600)

Class

Size range, in

5 OD and under Over 5 OD 5 OD and under Over 5 OD 5 OD and under Over 5 OD

Ni, Fe, Cr

B407

45

800 H (N08800)

Ni, Cr, Mo, Cb Welded pipe Ni, Mo Ni, Mo Ni, Mo, Cr Ni, Mo, Cr Ni, Cr, Fe, Mo, Cu Ni, Cr, Mo, Fe Ni, Fe, Cr, Mo

B444

43

625 (N06625) B (N10001) B-2 (N10665) C-4 (N06455) C276 (N10276) G1 (N06007) X (N06002) 20-MOD (N08320)

Solution-annealed Solution-annealed Solution-annealed Solution-annealed Solution-annealed Solution-annealed Solution annealed

P no.

Aluminum alloy Seamless pipe and tubing B210 21 B210 21 B210 23 B210 23

10-112

600

19, 24, 32 19, 24, 32 6, 19, 24, 32 14, 16

−425 −325 −425 −325 −325 −325 −325

17.0 17.0 14.2 17.0 17.0 17.0 17.0

17.0 17.0 14.2 17.0 17.0 17.0 17.0

17.0 17.0 14.2 17.0 17.0 17.0 17.0

15.9 15.9 13.4 17.0 17.0 17.0 16.4

14.8 14.8 12.5 17.0 17.0 17.0 15.2

14.0 14.0 11.9 16.3 16.3 16.3 14.4

16

−325

17.0

17.0

17.0

16.4

15.2

14.4

−325

14.2

14.2

14.2

13.2

12.2

11.5

14, 16

−325

17.0

17.0

17.0

16.4

15.2

14.4

7, 14

−325 −325 −425 −325

17.0 17.0 17.0 17.0

17.0 17.0 17.0 17.0

17.0 17.0 17.0 17.0

15.8 15.8 17.0 17.0

14.7 14.7 16.9 16.9

13.9 13.9 16.4 16.4

14, 16 16

7, 14

f–2, inclusive

Ni, Cr, Fe

Specification

500

102, 120, 122 C10200, C12000, C12200 C10200, C12000, C12200 C70600 C71500

Annealed Annealed Annealed Annealed Annealed Annealed Hot-finished or hotfinished annealed Hot-finished or hotfinished annealed Cold-drawn solution annealed or hot-finished Annealed

45

400

Drawn Annealed

200 (N02200) 200 (N02200) 201 (N02201) 201 (N02201) 400 (N04400) 400 (N04400) 600 (N06600)

44 44 44 45

300

31 31

41 41 41 41 42 42 43

B619 B619 B619 B619 B619 B619 B619

200

Notes

Size range, in

Nickel and nickel alloy Seamless pipe and tubing Nickel B161 Nickel B161 Low-C Ni B161 Low-C Ni B161 Ni, Cu B165 Ni, Cu B165 Ni, Cr, Fe B167 B167

Minimum temperature to 100

Class

B466 B466

P Specifi- no. cation (23)

TP304 TP304H TP304L TP309 TP310 TP310 TP316

Class

Minimum temperature (18)

Temper

B88

Cu, Ni 90/10 Cu, Ni 70/30

Minimum yield strength, kip/in2

P no. (23), (30)

B42 B88

Copper tubing

Minimum tensile strength, kip/in2

Grade

Electric-fusion-welded pipe and tubing 18Cr, 8Ni pipe A312 8 18Cr, 8Ni pipe A312 8 18Cr, 8Ni pipe A312 8 23Cr, 12Ni pipe A312 8 25Cr, 20Ni pipe A312 8 25Cr, 20Ni pipe A312 8 16Cr, 12Ni, 2Mo A312 8 pipe 16Cr, 12Ni, 2Mo A312 8 pipe 16Cr, 12Ni, 2Mo A312 8 pipe 18Cr, 13Ni, 3Mo A312 8 pipe 18Cr, 10Ni, Ti pipe A312 8 18Cr, 10Ni, Ti pipe A312 8 18Cr, 10Ni, Cb pipe A312 8 18Cr, 10Ni, Cb pipe A312 8

Copper and copper alloy Seamless pipe and tubing Copper pipe Copper tubing

Factor, E

P no. (23)

Grade

Temper

1060 3003 6061 6061

0 0 T4 T4, T6 welded

Size range, in

0.018–0.500 0.010–0.500 0.025–0.500

Factor, E

Minimum tensile strength, kip/in2

Minimum yield strength, kip/in2

Notes

45.0

40.0

9, 27

−325

30.0

9.0

9, 29

−325

36.0 38.0 50.0

30.0 13.0 18.0

9, 27, 29 9 9

−325 −325 −325

Minimum Minimum Minimum tensile yield Minimum temperaFactor, strength, strength, temperature to E kip/in2 kip/in2 Notes ture 100

55.0 55.0 50.0 50.0 70.0 70.0

Minimum temperature (18)

200

300

400

500

600

15.0 12.0 12.0 10.0 28.0 25.0

−325 −325 −325 −325 −325 −325

10.0 8.0 8.0 6.7 18.7 16.7

10.0 10.0 10.0 10.0 10.0 8.0 8.0 8.0 8.0 8.0 7.7 7.5 7.5 7.5 7.5 6.4 6.3 6.2 6.2 6.2 16.4 15.4 14.8 14.8 14.8 14.7 13.7 13.2 13.2 13.2

5 OD and under

80.0

30.0

−325

20.0

20.0 20.0 20.0 20.0 20.0

Over 5 OD

75.0

25.0

−325

16.7

16.7 16.7 16.7 16.7 16.7

65.0 120.0

25.0 60.0

−325 −325

16.7 30.0

16.7 16.7 16.7 16.7 16.5 30.0 30.0 28.2 27.0 26.4

100.0 110.0 100.0 100.0 90.0 100.0 75.0

45.0 51.0 40.0 41.0 35.0 40.0 28.0

−325 −325 −325 −325 −325 −325 −325

25.5 23.4 21.2 23.2 19.1 22.6 15.9

25.5 23.4 21.2 23.2 19.1 20.5 15.9

0.85 0.85 0.85 0.85 0.85 0.85 0.85

Minimum tensile strength, kip/in2

Minimum yield strength, kip/in2

8.5 14.0 30.0 24.0

2.5 5.0 16.0

39 42

25.5 23.4 21.2 23.2 19.1 19.8 15.8

25.5 23.4 21.2 23.2 18.6 19.5 15.2

25.5 23.4 21.0 22.9 18.3 18.9 15.0

25.5 23.1 20.7 21.6 17.9 17.9 14.9

Metal temperature, °F (22)

Notes

Minimum temperature, (18)

Minimum temperature, to 100

150

200

250

300

350

400

26 26 12, 26 35

−452 −452 −452 −452

1.7 3.3 10.0 8.0

1.7 3.3 10.0 8.0

1.6 3.3 10.0 8.0

1.5 3.1 9.8 7.9

1.3 2.4 9.2 7.4

1.1 1.8 7.9 6.1

0.8 1.4 5.6 4.3

Metal temperature, °F (22) 650

700

750

800

850

900

950

1000

1050

1100

1150

1200

1250

1300

1350

1400

1450

1500

13.7 13.7 11.6 16.0 16.0 16.0 14.2

13.6 13.6 11.4 15.6 15.6 15.6 13.8

13.2 13.2 11.3 15.3 15.3 15.3 13.6

12.9 12.9 11.0 14.9 14.9 14.9 13.5

12.7 12.7 10.9 12.4 12.4 12.4 13.3

12.5 12.5 10.1 11.8 11.8 11.8 13.2

12.2 12.2 8.4 10.6 10.6 10.6 13.1

11.7 11.7 6.6 8.9 9.3 9.3 13.0

10.3 10.3 5.4 7.2 6.0 8.3 12.3

8.3 8.3 4.3 5.5 4.2 7.2 10.5

6.5 6.5 3.4 4.2 3.1 6.2 8.3

5.1 5.1 2.8 3.2 2.1 5.1 6.3

4.0 4.0 2.2 2.5 1.2 4.0 4.6

3.1 3.1 1.8 2.0 0.6 3.0 3.5

2.5 2.5 1.4 1.5 0.4 2.0 2.6

2.0 2.0 0.9 1.1 0.3 1.4 1.9

1.5 1.5 0.8 0.8 0.2 0.9 1.4

1.2 1.2 0.7 0.6 0.2 0.6 1.1

14.2

13.8

13.6

13.5

13.3

13.2

13.1

13.0

12.3

10.5

8.3

6.3

4.6

3.5

2.6

1.9

1.4

1.1

11.2

10.9

10.7

10.5

10.3

10.0

9.8

9.5

9.2

8.7

7.4

5.4

4.0

3.0

2.1

1.6

1.1

0.9

14.2

13.9

13.6

13.5

13.3

13.2

13.1

13.0

12.3

10.5

8.3

6.3

4.6

3.5

2.6

1.9

1.4

1.1

13.6 13.6 16.1 16.1

13.4 13.4 15.8 15.8

13.3 13.3 15.7 15.7

13.1 13.1 15.5 15.6

13.0 13.0 13.1 15.5

13.0 13.0 12.7 15.4

12.9 12.9 12.3 15.4

11.7 11.9 11.9 15.3

8.2 9.9 10.3 14.5

5.8 7.7 7.7 12.1

4.2 5.9 5.2 8.9

3.1 4.5 3.7 6.7

2.2 3.5 2.8 5.0

1.4 2.7 1.9 3.7

0.9 2.1 1.3 2.7

0.6 1.6 1.0 2.1

0.4 1.2 0.8 1.5

0.3 0.9 0.6 1.1

450

500

550

600

650

700

9.4

Metal temperature, °F (22) Minimum temperature to 100

150

200

250

300

350

400

15.0

11.2

11.2

11.2

11.0

10.3

4.2

6.0

6.0

5.9

5.8

5.0

3.8

2.5

1.5

0.8

12.0 8.7 12.0

9.0 8.3 11.6

8.7 8.1 11.3

8.3 8.0 11.0

8.0 7.8 10.8

5.0 7.7 10.6

2.5 7.5 10.3

1.5 7.3 10.1

0.8 7.2 9.9

7.0 9.8

6.0 9.6

9.5

1250

1300

1350

1400

1450

1500

4.5

3.6

3.0

2.5

Metal temperature, °F (22) 650

700

750

800

850

900

950

1000

1050

1100

1150

1200

7.5 6.2 14.8 13.2

7.4 6.2 14.8 13.2

7.3 6.1 14.6 13.0

7.2 5.9 14.2 12.7

5.8 5.8

4.5 4.5

3.7 3.7

3.0 3.0

2.4 2.4

2.0 2.0

1.5 1.5

1.2 1.2

20.0

20.0

20.0

20.0

19.6

16.0

10.6

7.0

4.5

3.0

2.2

2.0

16.7

16.7

16.7

16.7

16.5

15.9

10.6

7.0

4.5

3.0

2.2

2.0

16.0 26.0

15.7 26.0

15.4 26.0

15.3 26.0

15.1 26.0

14.8 26.0

14.6 26.0

14.4 26.0

13.7 26.0

13.5 26.0

11.2 21.0

8.4 13.2

6.9

5.4

25.0 23.1 20.5 21.0 17.8 17.6 14.9

25.5 23.0 20.4 20.4 17.8 17.3 14.9

24.5 22.9 20.0 20.0 17.6 17.0 14.8

23.5 22.8 19.5 19.5 17.4 16.8 14.6

19.2 17.2 16.7

18.9 17.0 16.7

18.7 16.6 16.3

18.5 16.1 15.8

15.3

14.9

12.3

9.6

8.0

6.5

10-113

TABLE 10-49

Allowable Stresses in Tension for Materials (4, 13, 28) (Continued ) Design stresses for bolting materials

Material

Specification

Grade

Size range, in

Minimum tensile strength, kip/in2

Minimum yield strength, kip/in2

Minimum temperature (18)

Notes

Carbon steel A307 A325 A194 A194

B

60.0 105.0

25 25

Alloy steel Cr, Mo

A193

B7

2a and under

125.0

105.0

Cr, 0.2Mo Cr, Mo, V C, Mo Cr, Mo

A193 A193 A194 A320

B7M B16 4 L7, L7A, L7B, L7C

2a and under 2a and under

100.0 125.0

80.0 105.0

2a and under

125.0

105.0

Stainless steel 12 Cr 304 solution-treated 316 solution treated 304 strain-hardened

A193 A193 A193 A193

B6 B8, Cl. 1 B8M, Cl. 1 B8, Cl. 2

4 and under

110.0 75.0 75.0 125.0 115.0 105.0

316 strain-hardened

A193

B8M, Cl. 2

14 Cr, 24 Ni

A453

660A/B

Material

Up to e e to 1 Over 1 to 1d Over 1d to 1a Up to e e to 1 Over 1 to 1d Over 1d to 1a

Specification

200

300

400

500

600

650

13.7 19.3

13.7 19.3

13.7 19.3

13.7 | 19.3 |

13.7 || 19.3 ||

19.3

19.3

−20 −20 −20 −50

22

1, 2 2H

Minimum temperature, to 100

−20

25.0

25.0

25.0

25.0

25.0

20.0 25.0

20.0 25.0

25.0 — 20.0 25.0

25.0

−50 −20

20.0 25.0

20.0 25.0

20.0 25.0

20.0 25.0

25 31

−150

25.0

25.0

25.0

25.0

20.0

20.0

20.0

85.0 30.0 30.0 100.0 80.0 65.0

19, 31 31, 32, 41 31, 32, 41 31, 32, 41 31, 32, 41 31, 32, 41

−20 −325 −325 −325 −325 −325

21.2 18.8 18.8 25.0 20.0 16.2

21.2 15.6 16.1

21.2 14.0 14.6

21.2 12.9 13.3

21.2 12.1 12.5

21.2 11.4 11.8

21.2 11.2 11.5

100.0 110.0 100.0 95.0

50.0 95.0 80.0 65.0

31, 32, 41 31, 32, 41 31, 32, 41 31, 32, 41

−325 −325 −325 −325

12.5 22.0 20.0 16.2

22.0 20.0 16.2

22.0 20.0 16.2

22.0 20.0 16.2

22.0 20.0 16.2

22.0 20.0 16.2

22.0 20.0 16.2

90.0 130.0

50.0 85.0

31, 32, 41 19, 31

−325 −20

12.5 21.3

12.5 20.7

12.5 20.5

12.5 20.4

12.5 20.3

12.5 20.2

12.5 20.2

Grade

Temper

33

Minimum strength, kip/in2

Minimum yield strength, kip/in2

Notes

Minimum temperature (18)

0.500–4.500 0.125–8.000

62.0 42.0

42.0 35.0

34, 35 34, 35

−325 −325

52.0 75.0 85.0 100.0 75.0

15.0 45.0 45.0 50.0 35.0

43

Over a to 1 Over a to 1 a to 1 Over a to 1

−325 −325 −325 −325 −325

65.0 50.0 80.0 84.0 80.0

40.0 10.0 40.0 50.0 35.0

Size range, in

Aluminum and aluminum-base alloy B211 B211

2024 6061

T4 T6, T651

Copper and copper-base alloy Cu, Si Cu, Si Al, Bronze Al, Bronze Al, Bronze

B98 B98 B150 B150 B150

C65500, C66100 C65100 C64200 C63000 C61400

Soft Bolt

Nickel and nickel-base alloy Nickel Low C, Ni Ni, Cu Ni, Cu Ni, Cr, Fe

B160 B160 B164 B164 B166

200 (N02200) 201 (N02100) 400 (N04400) 400 (N04400) 600 (N06600)

Cold-drawn Annealed hot-finished Hot-finished Cold-drawn stress-relieved Annealed

All except hexagonal over 2a

36

−325 −325 −325 −325 −325

NOTES:

Special note for the sixth edition: At this time, metric equivalents have not been provided for the allowable-stress tables of the piping code B31.3. They may be computed by the following relationships; (°F − 32) × 5⁄ 9 = °C; lbf/in2 (stress) × 6.895 × 10−3 = MPa. 1. For temperatures above 480°C (900°F) consider the advantages of killed steel. 2. Conversion of carbides to graphite may occur after prolonged exposure to temperatures over 425°C (800°F). 3. Conversion of carbides to graphite may occur after prolonged exposure to temperatures over 468°C (875°F). 4. In shaded areas, allowable-stress values which are printed in italics exceed two-thirds of the expected yield strength at temperature. All other allowable-stress values in shaded areas are equal to 90 percent of expected yield strength at temperature. See ANSI B31.3. 5. A quality factor of 92 percent is included for structural grade. 6. The higher stress values at 566°C (1050°F) and above for this material shall be used only when the steel has an austenitic micrograin size No. 6 or less (coarser grain) as defined in ASTM E112. Otherwise the lower stress values shall be used. 7. For temperatures above 538°C (1000°F), these stress values may be used only if the material has been heat-treated at a temperature of 1090°C (2000°F) minimum. 8. There are restrictions in the code on the use of this material. 9. For use in code piping at the stated allowable stresses, the tensile and yield strengths listed in these tables must be verified by tensile tests at the mill; such tests shall be specified in the purchase order. 10. Pressure-temperature ratings of cast and forged parts as published in standards referenced in this code section may be used for parts meeting requirements of these standards. Allowable stresses for castings and forgings, where listed, are for use in the design of special components not furnished in accordance with such standards. 11. Certain forms of this material, as stated in Table 10-57, must be impact-tested to qualify for service below −29°C (−20°F). Alternatively, if provisions for impact testing are included in the material specification as supplementary requirements and are invoked, the material may be used down to the temperature at which the test was conducted in accordance with the specification. 12. For welded construction with work-hardened grades, use the stresses for annealed material; for welded construction with precipitation-hardened grades, use the special allowable stresses for welded construction given in the tables. 13. SE values shown in this table for welded pipe include the joint quality factor Ei for the longitudinal weld as required by Fig. 10-164 and, when applicable, the structural-grade quality factor ES of 0.92. For some code computations, particularly with regard to expansion, flexibility, structural attachments, supports, and restraints, the longitudinal-joint quality factor Ej need not be considered. To determine the allowable stress S for use in code computations not utilizing the joint quality factor Ej divide the value SE shown in this table by the longitudinal-joint quality factor Ej tabulated in Fig. 10-164. 14. For temperatures above 38°C (100°F) these stress values apply only when the carbon content is 0.04 percent or higher. 15. Stress values shown include the casting quality factor shown in this table. Higher stress values can be used if special inspection is accomplished. 16. These unstabilized grades of stainless steel have an increasing tendency to intergranular carbide precipitation as the carbon content increases above 0.03 percent. 17. The allowable stress to be used for this gray-cast-iron material at its upper temperature limit of 232°C (450°F) is the same as that shown in the 204°C (400°F) column.

10-114

Metal temperature, °F (22) 700

750

800

850

900

950

1000

25.0

23.6

21.0

17.0

12.5

8.5

4.5

20.0 25.0

20.0 25.0

18.5 25.0

16.2 23.5

12.5 20.5

8.5 16.0

4.5 11.0

20.0

20.0

20.0

16.2

12.5

8.5

4.5

21.2 11.0 11.3

21.2 | 10.8 11.0

19.6 10.5 10.9

15.6 10.3 10.8

12.0 10.1 10.7

9.9 10.7

9.7 10.6

22.0 20.0 16.2

22.0 20.0 16.2

22.0 20.0 16.2

12.5 20.1

12.5 20.0

12.5 19.9

19.9

19.9

19.8

19.8

1050

1100

6.3

2.8

9.5 10.5

8.8 10.3

1150

1200

1250

1300

1350

1400

1450

1500

7.7 9.3

6.0 7.4

4.7 5.4

3.7 4.1

2.9 3.0

2.3 2.2

1.8 1.7

1.4 1.2

Metal temperature, °F (22) Minimum temperature, to 100

200

300

400

10.5 8.4

10.5 8.4

10.4 8.4

4.5 4.4

10.0 11.3 21.3 25.0 18.8

10.0 11.3 21.3 25.0 18.8

10.0 11.3 21.3 25.0 18.8

10.0 6.7 20.0 12.5 20.0

10.0 6.4 20.0 12.5 20.0

10.0 6.3 20.0 12.5 20.0

500

600

650

700

750

800

850

900

950

1000

1050

1100

1150

1200

20.8 25.0 18.4

12.6 20.7 16.1

9.9 12.0

8.5

6.0

10.0 6.2 20.0 12.5 20.0

10.0 6.2 20.0 12.5 20.0

10.0 6.2 20.0

6.2 20.0

6.2 19.2

6.0 18.5

5.9 14.5

5.8 8.5

4.8 4.0

3.7

3.0

2.4

2.0

1.5

1.2

20.0

19.8

19.6

19.4

19.1

18.7

16.0

10.6

7.0

4.5

3.0

2.2

2.0

18. The minimum temperature shown is that design minimum temperature for which the material is normally suitable without impact testing other than that required by the material specification. However, the use of a material at a design minimum temperature below −29°C (−20°F) is established by rules elsewhere in the code, including any necessary impact-test requirements. 19. These steels are intended for use at high temperatures; however, they may have low ductility and/or low impact properties at room temperature after being used above the temperature indicated by the single bar (|). 20. For pipe sizes NPS 8 and larger and for wall thicknesses of Schedule 140 or heavier, the minimum specification tensile strength is 483 MPa (70.0 kip/in2). 21. There are restrictions on the use of this material in the text of the code. 22. A single bar (|) in these stress tables indicates that there are conditions other than stress which affect usage above or below the temperature as described in other referenced notes. A double bar (||) after a tabled stress indicates that use of the material is prohibited above that temperature. 23. See ANSI B31.3 for a description of P-number groupings. 24. This material when used below −29°C (−20°F) requires impact testing if the carbon content is above 0.10 percent. 25. This is a product specification. No design stresses are necessary. Limitations on metal temperature for materials covered by this specification are:

Grades 1 and 2 Grade 2H Grade 3 Grade 4 Grade 6 Grade 8FA (see Note 24) Grades 8MA and 8TA Grades 8A and 8CA

°C

°F

−29 to 480 −45 to 595 −29 to 595 −100 to 595 −29 to 425 −29 to 425 −198 to 815 −254 to 815

−20 to 900 −50 to 1100 −20 to 1100 −150 to 1100 −20 to 800 −20 to 800 −325 to 1500 −425 to 1500

26. For use in code piping at the stated allowable stresses, the required minimum tensile and yield properties must be verified by tensile test at the mill. If such tests are not mandatory in the ASTM specification, they shall be specified in the purchase order. 27. After use above the temperature indicated by a single bar (|), use at a lower temperature shall be based on the stress values allowed for the annealed condition of the material.

10-115

10-116

TRANSPORT AND STORAGE OF FLUIDS

TABLE 10-49

Allowable Stresses in Tension for Materials (4, 13, 28) (Concluded )

28. The SE values in Table 10-49 are equal to the basic allowable stresses in tension S multiplied by a quality factor E (see subsection “Pressure Design of Metallic Components: Wall Thickness”). The design stress values for bolting materials are equal to the basic allowable stresses S. The stress values in shear shall be 0.80 times the allowable stresses in tension derived from tabulated values in Table 10-49 adjusted when applicable in accordance with Note 13. Stress values in bearing shall be twice those in shear. 29. Yield strengths listed are not included in ASTM specifications. The value shown is based on yield strengths of materials with similar characteristics. 30. The letter a indicates alloys which are not recommended for welding and which, if welded, must be individually qualified. The letter b indicates copper-base alloys which must be individually qualified. 31. These stress values are established from a consideration of strength only and will be satisfactory for average service. For bolted joints when freedom from leakage over a long period of time without retightening is required, lower stress values may be necessary as determined from the flexibility of the flange and bolts and corresponding relaxation properties. 32. For temperatures above 538°C (1000°F), these stress values apply only when the carbon content is 0.04 percent or higher. 33. For use at temperatures below −29 through −45°C (−20 through −50°F) this material must be quenched and tempered. 34. The stress values given for this material are not applicable when either welding or thermal cutting is employed. 35. For stress-relieved tempers (T351, T3510, T3511, T451, T4510, T4511, T651, T6510, T6511) stress values for material in the listed temper shall be used. 36. The maximum operating temperature is arbitrarily set at 260°C (500°F) because harder temper adversely affects design stress in the creep-rupture-temperature ranges. 37. Pipe produced to this specification is not intended for high-temperature service. The stress values apply to either nonexpanded or cold-expanded material in the as-rolled, normalized, or normalized and tempered condition. 38. Special P numbers SP-1, SP-2, and SP-3 of carbon steels are not included in P No. 1 because of a possible high-carbon-high-manganese combination which would require special consideration in qualification. Qualification of any high-carbon-high-manganese grade may be extended to other grades in its group. 39. Annealed at approximately 1150°C (2100°F). 40. If no welding is employed in the fabrication of piping from these materials, the allowable stress values may be increased to 230 MPa (33.3 kip/in2). 41. For all design temperatures, the maximum hardness shall be Rockwell C35 immediately under the thread roots. The hardness shall be taken on a flat area at least 3 mm (f in) across, prepared by removing threads. No more material than necessary shall be removed to prepare the area. Hardness determination shall be made at the same frequency as tensile tests. 42. The minimum tensile strength of the reduced section tensile specimen in accordance with QW-462.1 of ASME Code Sec. IX shall not be less than 758 MPa (110.0 kip/in2). 43. Copper-silicon alloys are not always suitable when exposed to certain media and high temperature, particularly above 100°C (212°F). Users should satisfy themselves that the alloy selected is satisfactory for the service for which it is to be used. *Table 10-49 and notes have been extracted from the Chemical Plant and Petroleum Refinery Piping Code, ANSI B31.3–1980, with permission of the publisher, the American Society of Mechanical Engineers, New York.

TABLE 10-50 Values of Coefficient Y When t Is Less Than D/6* Temperature, °C (°F)

Materials Ferritic steels Austenitic steels Other ductile metals Cast iron

485 (900) and lower

510 (950)

540 (1000)

560 (1050)

595 (1100)

620 (1150) and higher

0.4

0.5

0.7

0.7

0.7

0.7

0.4

0.4

0.4

0.4

0.5

0.7

0.4 0.0

0.4

0.4

0.4

0.4

0.4

*Extracted from ANSI B31.3—1980, with permission of the publisher, the American Society of Mechanical Engineers, New York. TABLE 10-51

Stress-Range Reduction Factors f *

Cycles, number

Factor, f

7000 and less 7000–14,000 14,000–22,000 22,000–45,000 45,000–100,000 Over 100,000

1.0 0.9 0.8 0.7 0.6 0.5

*Extracted from ANSI B31.3—1980, with permission of the publisher, the American Society of Mechanical Engineers, New York.

Types of reinforcement for branch connections. (From Kellogg, Design of Piping Systems, Wiley, New York, 1965.) FIG. 10-166

Nomenclature for miter bends. (Extracted from the Chemical Plant and Petroleum Refinery Code, ANSI B31.3—1976, with permission of the publisher, the American Society of Mechanical Engineers, New York.)

FIG. 10-165

2. Local reduction in size or wall thickness or local use of a material having reduced yield strength (for example, girth welds of substantially lower strength than the base metal) 3. A line configuration in a system of uniform size in which expansion or contraction must be absorbed largely in a short offset from the major portion of the run If unbalanced layouts cannot be avoided, appropriate analytical

PROCESS-PLANT PIPING methods must be applied to assure adequate flexibility. If the designer determines that a piping system does not have adequate inherent flexibility, additional flexibility may be provided by adding bends, loops, offsets, swivel joints, corrugated pipe, expansion joints of the bellows or slip-joint type, or other devices. Suitable anchoring must be provided. As contrasted with stress from sustained loads such as internal pressure or weight, displacement stresses may be permitted to cause limited overstrain in various portions of a piping system. When the system is operated initially at its greatest displacement condition, any yielding reduces stress. When the system is returned to its original condition, there occurs a redistribution of stresses which is referred to as self-springing. It is similar to cold springing in its effects. While stresses resulting from thermal strain tend to diminish with time, the algebraic difference in displacement condition and in either the original (as-installed) condition or any anticipated condition with a greater opposite effect than the extreme displacement condition remains substantially constant during any one cycle of operation. This difference is defined as the displacement-stress range, and it is a determining factor in the design of piping for flexibility. See Eqs. (10-93) and (10-94) for the allowable stress range SA and Eq. (10-100) for the computed stress range SE. Cold Spring Cold spring is the intentional deformation of piping during assembly to produce a desired initial displacement and stress. For pipe operating at a temperature higher than that at which it was installed, cold spring is accomplished by fabricating it slightly shorter than design length. Cold spring is beneficial in that it serves to balance the magnitude of stress under initial and extreme displacement conditions. When cold spring is properly applied, there is less likelihood of overstrain during initial operation; hence, it is recommended especially for piping materials of limited ductility. There is also less deviation from as-installed dimensions during initial operation, so that hangers will not be displaced as far from their original settings. Inasmuch as the service life of a system is affected more by the range of stress variation than by the magnitude of stress at a given time, no credit for cold spring is permitted in stress-range calculations. However, in calculating the thrusts and moments when actual reactions as well as their range of variations are significant, credit is given for cold spring. Values of thermal-expansion coefficients to be used in determining total displacement strains for computing the stress range are determined from Table 10-52 as the algebraic difference between the value at design maximum temperature and that at the design minimum temperature for the thermal cycle under analysis. Values for Reactions Values of thermal displacements to be used in determining total displacement strains for the computation of reactions on supports and connected equipment shall be determined as the algebraic difference between the value at design maximum (or minimum) temperature for the thermal cycle under analysis and the value at the temperature expected during installation. The as-installed and maximum or minimum moduli of elasticity, Ea and Em respectively, shall be taken as the values shown in Table 10-53. Poisson’s ratio may be taken as 0.3 at all temperatures for all metals. The allowable stress range for displacement stresses SA and permissible additive stresses shall be as specified in Eqs. (10-93) and (10-94) for systems primarily stressed in bending and/or torsion. For pipe or piping components containing longitudinal welds the basic allowable stress S may be used to determine SA. (See Table 10-49, Note 13.) Nominal thicknesses and outside diameters of pipe and fittings shall be used in flexibility calculations. In the absence of more directly applicable data, the flexibility factor k and stress-intensification factor i shown in Table 10-54 may be used in flexibility calculations in Eq. (10-101). For piping components or attachments (such as valves, strainers, anchor rings, and bands) not covered in the table, suitable stress-intensification factors may be assumed by comparison of their significant geometry with that of the components shown. Requirements for Analysis No formal analysis of adequate flexibility is required in systems which (1) are duplicates of successfully operating installations or replacements without significant change of systems with a satisfactory service record; (2) can readily be judged

10-117

adequate by comparison with previously analyzed systems; or (3) are of uniform size, have no more than two points of fixation, have no intermediate restraints, and fall within the limitations of empirical Eq. (10-99):* Dy (10-99) ᎏ2 ≤ K1 (L − U) where D = outside diameter of pipe, in (mm) y = resultant of total displacement strains, in (mm), to be absorbed by the piping system L = developed length of piping between anchors, ft (m) U = anchor distance, straight line between anchors, ft (m) K1 = 0.03 for U.S. customary units listed = 208.3 for SI units listed in parentheses 1. All systems not meeting these criteria shall be analyzed by simplified, approximate, or comprehensive methods of analysis appropriate for the specific case. 2. Approximate or simplified methods may be applied only if they are used in the range of configurations for which their adequacy has been demonstrated. 3. Acceptable comprehensive methods of analysis include analytical and chart methods which provide an evaluation of the forces, moments, and stresses caused by displacement strains. 4. Comprehensive analysis shall take into account stressintensification factors for any component other than straight pipe. Credit may be taken for the extra flexibility of such a component. In calculating the flexibility of a piping system between anchor points, the system shall be treated as a whole. The significance of all parts of the line and of all restraints introduced for the purpose of reducing moments and forces on equipment or small branch lines and also the restraint introduced by support friction shall be recognized. Consider all displacements over the temperature range defined by operating and shutdown conditions. Flexibility Stresses Bending and torsional stresses shall be computed using the as-installed modulus of elasticity Ea and then combined in accordance with Eq. (10-100) to determine the computed displacement stress range SE, which shall not exceed the allowable stress range SA [Eqs. (10-93) and (10-94).] 2 苶b苶 苶 SE = 兹S +苶S 4苶苶t2

(10-100)

where Sb = resultant bending stress, lbf/in2 (MPa) St = Mt /2Z = torsional stress, lbf/in2 (MPa) Mt = torsional moment, in⋅lbf (N⋅mm) Z = section modulus of pipe, in3 (mm3) The resultant bending stresses Sb to be used in Eq. (10-100) for elbows and miter bends shall be calculated in accordance with Eq. (10-101), with moments as shown in Fig. (10-167): 2 兹苶 (i苶 +苶(苶 io苶 Mo苶)2苶 iM苶 i) 苶 Sb = ᎏᎏ (10-101) Z 2 where Sb = resultant bending stress, lbf/in (MPa) ii = in-plane stress-intensification factor from Table 10-54 io = out-plane stress-intensification factor from Table 10-54 Mi = in-plane bending moment, in⋅lbf (N⋅mm) Mo = out-plane bending moment, in⋅lbf (N⋅mm) Z = section modulus of pipe, in3 (mm3)

The resultant bending stresses Sb to be used in Eq. (10-100) for branch connections shall be calculated in accordance with Eqs. (10-102) and (10-103), with moments as shown in Fig. 10-168. * WARNING: No general proof can be offered that this equation will yield accurate or consistently conservative results. It is not applicable to systems used under severe cyclic conditions. It should be used with caution in configurations such as unequal leg U bends (L/U > 2.5) or near-straight sawtooth runs, or for large thin-wall pipe (i ≥ 5), or when extraneous displacements (not in the direction connecting anchor points) constitute a large part of the total displacement. There is no assurance that terminal reactions will be acceptably low even if a piping system falls within the limitations of Eq. (10-99).

TABLE 10-52

Thermal-Expansion Coefficients, U.S. Customary Units, for Metals* Mean coefficient of linear thermal expansion between 70°F and indicated temperature, µin/(in⋅°F)

Temperature, °F

Carbon steel, carbonmolybdenum low-chromium (through 3 Cr Mo)

5 Cr Mo through 9 Cr Mo

Austenitic stainless steels, 18 Cr, 8 Ni

12 Cr 17 Cr 27 Cr

−325 −300 −275 −250 −225

5.00 5.07 5.14 5.21 5.28

4.70 4.77 4.84 4.91 4.98

8.15 8.21 8.28 8.34 8.41

−200 −175 −150 −125 −100

5.35 5.42 5.50 5.57 5.65

5.05 5.12 5.20 5.26 5.32

−75 −50 −25 0 25

5.72 5.80 5.85 5.90 5.96

50 70 100 125 150

Monel 67 Ni, 30 Cu

3a Nickel

Aluminum

4.30 4.36 4.41 4.47 4.53

5.55 5.72 5.89 6.06 6.23

4.76 4.90 5.01 5.15 5.30

8.47 8.54 8.60 8.66 8.75

4.59 4.64 4.70 4.78 4.85

6.40 6.57 6.75 6.85 6.95

5.38 5.45 5.51 5.56 5.62

8.83 8.90 8.94 8.98 9.03

4.93 5.00 5.05 5.10 5.14

6.01 6.07 6.13 6.19 6.25

5.67 5.73 5.79 5.85 5.92

9.07 9.11 9.16 9.20 9.25

5.19 5.24 5.29 5.34 5.40

175 200 225 250 275

6.31 6.38 6.43 6.49 6.54

5.98 6.04 6.08 6.12 6.15

9.29 9.34 9.37 9.41 9.44

5.45 5.50 5.54 5.58 5.62

300 325 350 375 400

6.60 6.65 6.71 6.76 6.82

6.19 6.23 6.27 6.30 6.34

9.47 9.50 9.53 9.56 9.59

425 450 475 500 525

6.87 6.92 6.97 7.02 7.07

6.38 6.42 6.46 6.50 6.54

550 575 600 625 650

7.12 7.17 7.23 7.28 7.33

6.58 6.62 6.66 6.70 6.73

10-118

25 Cr, 20 Ni

Gray cast iron

Bronze

Brass

70 Cu, 30 Ni

9.90 10.04 10.18 10.33 10.47

8.40 8.45 8.50 8.55 8.60

8.20 8.24 8.29 8.33 8.37

6.65 6.76 6.86 6.97 7.08

5.45 5.52 5.59 5.67 5.78

10.61 10.76 10.90 11.08 11.25

8.65 8.70 8.75 8.85 8.95

8.41 8.46 8.50 8.61 8.73

7.19 7.29 7.40 7.50 7.60

4.65 4.76 4.87 4.98 5.10

7.05 7.15 7.22 7.28 7.35

5.83 5.88 5.94 6.00 6.08

11.43 11.60 11.73 11.86 11.99

9.05 9.15 9.23 9.32 9.40

8.84 8.95 9.03 9.11 9.18

7.70 7.80 7.87 7.94 8.02

5.20 5.30 5.40 5.50 5.58

7.41 7.48 7.55 7.62 7.70

6.16 6.25 6.33 6.36 6.39

12.12 12.25 12.39 12.53 12.67

9.49 9.57 9.66 9.75 9.85

9.26 9.34 9.42 9.51 9.59

8.09 8.16 8.24 8.31 8.39

8.79 8.81 8.83 8.85

7.77 7.84 7.89 7.93 7.98

6.42 6.45 6.50 6.55 6.60

12.81 12.95 13.03 13.12 13.20

5.75 5.80 5.84 5.89

9.93 10.03 10.05 10.08 10.10

9.68 9.76 9.82 9.88 9.94

8.46 8.54 8.58 8.63 8.67

5.66 5.70 5.74 5.77 5.81

8.87 8.89 8.90 8.91 8.92

8.02 8.07 8.11 8.16 8.20

6.65 6.69 6.73 6.77 6.80

13.28 13.36 13.44 13.52 13.60

5.93 5.97 6.02 6.06 6.10

10.12 10.15 10.18 10.20 10.23

10.00 10.06 10.11 10.17 10.23

8.71 8.76 8.81 8.85 8.90

9.62 9.65 9.67 9.70 9.73

5.85 5.89 5.92 5.96 6.00

8.92 8.92 8.92 8.93 8.93

8.25 8.30 8.35 8.40 8.45

6.83 6.86 6.89 6.93 6.97

13.68 13.75 13.83 13.90 13.98

6.15 6.19 6.24 6.28 6.33

10.25 10.28 10.30 10.32 10.35

9.76 9.79 9.82 9.85 9.87

6.05 6.09 6.13 6.17 6.20

8.93 8.93 8.94 8.94 8.95

8.49 8.54 8.58 8.63 8.68

7.01 7.04 7.08 7.12 7.16

14.05 14.13 14.20

6.38 6.42 6.47 6.52 6.56

10.38 10.41 10.44 10.46 10.48

Ni-Fe-Cr

Ni-Cr-Fe

Ductile iron

7.13 7.20 7.25 7.30

5.66 5.74 5.82 5.87 5.92

7.90 8.01 8.12 8.24

7.35 7.40 7.44 7.48 7.52

5.97 6.02 6.08 6.14 6.20

8.35 8.46 8.57 8.69 8.80

7.56 7.60 7.63 7.67 7.70

6.25 6.31 6.37 6.43 6.48

10.29 10.35 10.41 10.47 10.53

8.82 8.85 8.87 8.90 8.92

7.72 7.75 7.77 7.80 7.82

6.57 6.66 6.75 6.85 6.88

10.58 10.64 10.69 10.75 10.81

8.95 8.97 9.00 9.02 9.05

7.85 7.88 7.90 7.92 7.95

6.92 6.95 6.98 7.02 7.04

TABLE 10-52

Thermal-Expansion Coefficients, U.S. Customary Units, for Metals* Concluded Mean coefficient of linear thermal expansion between 70°F and indicated temperature, µin/(in⋅°F) Carbon steel, carbonmolybdenum low-chromium (through 3 Cr Mo)

5 Cr Mo through 9 Cr Mo

Austenitic stainless steels, 18 Cr, 8 Ni

12 Cr 17 Cr 27 Cr

25 Cr, 20 Ni

Monel 67 Ni, 30 Cu

3a Nickel

675 700 725 750 775

7.38 7.44 7.49 7.54 7.59

6.77 6.80 6.84 6.88 6.92

9.90 9.92 9.95 9.99 10.02

6.23 6.26 6.29 6.33 6.36

8.95 8.96 8.96 8.96 8.96

8.73 8.78 8.83 8.87 8.92

800 825 850 875 900

7.65 7.70 7.75 7.79 7.84

6.96 7.00 7.03 7.07 7.10

10.05 10.08 10.11 10.13 10.16

6.39 6.42 6.46 6.49 6.52

8.97 8.97 8.98 8.99 9.00

925 950 975 1000 1025

7.87 7.91 7.94 7.97 8.01

7.13 7.16 7.19 7.22 7.25

10.19 10.23 10.26 10.29 10.32

6.55 6.58 6.60 6.63 6.65

1050 1075 1100 1125 1150

8.05 8.08 8.12 8.14 8.16

7.27 7.30 7.32 7.34 7.37

10.34 10.37 10.39 10.41 10.44

1175 1200 1225 1250 1275

8.17 8.19 8.21 8.24 8.26

7.39 7.41 7.43 7.45 7.47

1300 1325 1350 1375 1400

8.28 8.30 8.32 8.34 8.36

7.49 7.51 7.52 7.54 7.55

Temperature, °F

1425 1450 1475 1500

Gray cast iron

Bronze

Brass

7.19 7.22 7.25 7.29 7.31

6.61 6.65 6.70 6.74 6.79

10.50 10.52 10.55 10.57 10.60

8.96 9.01 9.06 9.11 9.16

7.34 7.37 7.40 7.43 7.45

6.83 6.87 6.92 6.96 7.00

9.05 9.10 9.15 9.18 9.20

9.21 9.25 9.30 9.34 9.39

7.47 7.49 7.52 7.55

7.05 7.10 7.14 7.19

6.68 6.70 6.72 6.74 6.75

9.22 9.24 9.25 9.29 9.33

10.46 10.48 10.50 10.51 10.53

6.77 6.78 6.80 6.82 6.83

10.54 10.56 10.57 10.59 10.60

6.85 6.86 6.88 6.89 6.90

10.64 10.68 10.72 10.77

Ni-Fe-Cr

Ni-Cr-Fe

Ductile iron

10.86 10.92 10.98 11.04 11.10

9.07 9.10 9.12 9.15 9.17

7.98 8.00 8.02 8.05 8.08

7.08 7.11 7.14 7.18 7.22

10.62 10.65 10.67 10.70 10.72

11.16 11.22 11.28 11.34 11.40

9.20 9.22 9.25 9.27 9.30

8.10

7.25 7.27 7.31 7.34 7.37

10.74 10.76 10.78 10.80 10.83

11.46 11.52 11.57 11.63 11.69

9.32 9.35 9.37 9.40 9.42

9.43 9.48 9.52 9.57 9.61

10.85 10.88 10.90 10.93 10.95

11.74 11.80 11.85 11.91 11.97

9.45 9.47 9.50 9.52 9.55

9.36 9.39 9.43 9.47 9.50

9.66 9.70 9.75 9.79 9.84

10.98 11.00

12.03 12.09

9.57 9.60 9.64 9.68 9.71

9.53 9.53 9.54 9.55 9.56

9.88 9.92 9.96 10.00 10.04

Aluminum

70 Cu, 30 Ni

7.41 7.44 7.47 7.50

9.75 9.79 9.83 9.86 9.90 9.94 9.98 10.01 10.05

*Extracted from the Chemical Plant and Petroleum Refinery Piping Code, ANSI B31.3—1980, with permission of the publisher, the American Society of Mechanical Engineers, New York. These data are for information, and it is not implied that materials are suitable for all the temperatures shown. (°F − 32) 5⁄ 9 = °C; to convert microinches per inch-degree Fahrenheit to meters per meter-degree Kelvin, multiply by 1.8.

10-119

10-120

TRANSPORT AND STORAGE OF FLUIDS

TABLE 10-53

Modulus of Elasticity, U.S. Customary Units, for Metals* E = Modulus of elasticity, lbf/in2 (multiply tabulated values by 106) Temperature, °F

Material

−325

−200

−100

70

200

300

400

500

600

700

800

900

1000

1100

1200

1300

1400

1500

Modulus of elasticity: ferrous materials Carbon steels with carbon content 0.30 percent or less, 3a Ni Carbon steels with carbon content above 0.30 percent Carbon-molydenum steels, low-chromium steels through 3 Cr Mo Intermediate-chromium steels (5 Cr Mo through 9 Cr Mo) Austenitic steels (TP304, 310, 316, 321, 347) Straight chromium steels (12 Cr, 17 Cr, 27 Cr) Gray cast iron

30.0

29.5

29.0

27.9

27.7

27.4

27.0

26.4

25.7

24.8

23.4

18.5

15.4

13.0

31.0

30.6

30.4

29.9

29.5

29.0

28.3

27.4

26.7

25.4

23.8

21.5

18.8

15.0

11.2

31.0

30.6

30.4

29.9

29.5

29.0

28.6

28.0

27.4

26.6

25.7

24.5

23.0

20.4

15.6

29.4

28.5

28.1

27.4

27.1

26.8

26.4

26.0

25.4

24.9

24.2

23.5

22.8

21.9

20.8

19.5

18.1

30.4

29.9

29.4

28.3

27.7

27.1

26.6

26.1

25.4

24.8

24.1

23.4

22.7

22.0

21.3

20.7

19.3

30.8

30.3

29.8

29.2

28.7

28.3

27.7

27.0

26.0

24.8

23.1

21.1

18.6

15.6

12.2

13.4

13.2

12.9

12.6

12.2

11.7

11.0

10.2

17.9

Modulus of elasticity: nonferrous materials E = Modulus of elasticity, lbf/in2 (multiply tabulated values by 106) Temperature, °F Material

−325

−200

−100

70

100

200

300

400

500

600

700

800

900

1000

1100

1200

Monel (67 Ni, 30 Cu) and (66 Ni, 29 Cu—Al) Copper-nickel (70 Cu, 30 Ni) Aluminum alloys Copper (99.98 percent Cu) Commercial brass (66 Cu, 34 Zn) Leaded tin bronze (88 Cu, 6 Sn, 1.5 Pb, 4.5 Zn)

26.8

26.6

26.4

26.0

26.0

26.0

25.8

25.6

25.4

24.7

23.1

21.0

18.6

16.0

14.3

13.0

21.6

21.5

21.2

20.9

20.6

20.3

20.0

19.7

19.4

11.3 17.0 15.0

10.9 16.7 14.7

10.6 16.5 14.5

10.1 16.0 14.0

10.0 15.8 13.9

9.8 15.6 13.7

9.5 15.4 13.5

8.7 15.1 13.0

7.7 14.7 12.7

14.2 12.2

13.7 11.8

14.2

13.8

13.5

13.0

12.9

12.7

12.4

12.0

11.7

11.3

10.9

*Extracted from the Chemical Plant and Petroleum Refinery Piping Code, ANSI B31.3—1980, with permission of the publisher, the American Society of Mechanical Engineers, New York. These data are for information, and it is not implied that materials are suitable for all the temperatures shown. (°F − 32) 5⁄ 9 = °C; to convert pounds-force per square inch to megapascals, multiply by 0.006895.

Moments in bends. (Extracted from the Chemical Plant and Petroleum Refinery Piping Code, ANSI B31.3—1976, with permission of the publisher, the American Society of Mechanical Engineers, New York.)

FIG. 10-167

Moments in branch connections. (Extracted from the Chemical Plant and Petroleum Refiner Piping Code, ANSI B31.3—1976, with permission of the publisher, the American Society of Mechanical Engineers, New York.)

FIG. 10-168

PROCESS-PLANT PIPING TABLE 10-54

Flexibility Factor k and Stress-Intensification Factor i * Stress intensification factor a,h In-plane, ii

Flexibility characteristic h

0.75 ᎏ h2/3

0.9 ᎏ h2/3

T 苶R1 ᎏ2 (r2)

1.52 ᎏ h5/6

0.9 ᎏ h2/3

0.9 ᎏ h2/3

Ts cot θ 苶 ᎏ ᎏ2 2 (r2)

Single miter benda,b or widely spaced miter bend s ≥ r2 (1 + tan θ)

1.52 ᎏ h5/6

0.9 ᎏ h2/3

0.9 ᎏ h2/3

1 + cot θ T 苶 ᎏᎏ 2 r2

Welding teea,b,f per ANSI B16.9 with rx > f Db Tc ≥ 1.5 苶 T

1

0.9 ᎏ h2/3

e io + d

T 苶 4.4 ᎏ r2

Reinforced fabricateda,b,e tee with pad or saddle

1

0.9 ᎏ h2.3

e io + d

(T 苶 + 1/2 tr)5/2 ᎏᎏ T 苶 3/2 r2

Unreinforceda,b fabricated tee

1

0.9 ᎏ h2/3

e io + d

T 苶 ᎏ r2

Extrudeda,b welding tee T Tc < 1.5 苶

1

0.9 ᎏ h2/3

e io + d

冢1 + ᎏr 冣 ᎏr苶

Welded-ina,b contour insert rx ≥ f Db Tc ≥ 1.5 苶 T

1

0.9 ᎏ h2/3

e io + d

T 苶 4.4 ᎏ r2

Brancha,b,g welded-on fitting (integrally reinforced)

1

0.9 ᎏ h2/3

0.9 ᎏ h2/3

T 苶 3.3 ᎏ r2

Flexibility factor k

Out-plane, io

Welding elbow a,b,c,f,i or pipe bend

1.65 ᎏ h

Closely spaced miter benda,b,c s < r2 (1 + tan θ)

Description

rx

T

2

2

Sketch

10-121

10-122

TRANSPORT AND STORAGE OF FLUIDS

TABLE 10-54

Flexibility Factor k and Stress-Intensification Factor i (Concluded )

Description Butt-welded joint, reducer, or weld-neck flange Double-welded slip-on flange Fillet welded joint or pocketweld flange Lap-joint flange (with ANSI B16.9 lap-joint stub) Screwed pipe joint or screwed flange Corrugated straight pipe or corrugated or creased bendd

Flexibility factor k

Stress-intensification factor i

1

1.0

1 1

1.2 1.3

1

1.6

1

2.3

5

2.5

a The flexibility factor k applies to bending in any plane. The flexibility factors k and stress intensification factors i shall not be less than unity; factors for torsion equal unity. Both factors apply over the effective arc length (shown by heavy centerlines in the sketches) for curved and miter bends and to the intersection point for tees. b The values of k and i can be read directly from Chart A by entering with the characteristic h computed from the formulas given above. Nomenclature is as follows: T 苶 = for elbows and miter bends, the nominal wall thickness of the fitting, in (mm) = for tees, the nominal wall thickness of the matching pipe, in Tc = the crotch thickness of tees, in (mm) 苶 tr = pad or saddle thickness, in (mm) θ = one-half angle between adjacent miter axes, ° r2 = mean radius of matching pipe, in (mm) R1 = bend radius of welding elbow or pipe bend, in (mm) rx = radius of curvature of external contoured portion of outlet, in, measured in the plane containing the axes of the run and branch. s = miter spacing at centerline, in (mm) Db = outside diameter of branch, in (mm) c When flanges are attached to one or both ends, the values of k and i shall be corrected by the factors C1, which can be read directly from Chart B, entering with the computed h. d Factors shown apply to bending. Flexibility factor for torsion equals 0.9. e When tr is > 1a T 苶, use h = 4(T 苶/r2). f Designers are cautioned that cast butt-welded fittings may have considerably heavier walls than that of the pipe with which they are used. Large errors may be introduced unless the effect of these greater thicknesses is considered. g Designers must assure themselves that this fabrication has a pressure rating equivalent to that of straight pipe. h A single intensification factor equal to 0.9/h2/3 may be used for both ii and io if desired. i In large-diameter thin-wall elbows and bends, pressure can significantly affect the magnitudes of k and i. To correct values from the table, P r2 7/3 R1 1/3 P r2 5/2 R1 2/3 Divide i by 1 + 3.25 ᎏ ᎏ (10-107) Divide k by 1 + 6 ᎏ ᎏ ᎏ ᎏ Ec t r2 Ec t r2 *Extracted from the Chemical Plant and Petroleum Refinery Piping Code, ANSI B31.3—1980, with permission of the publisher, the American Society of Mechanical Engineers, New York.

冤 冢 冣冢 冣 冢 冣 冥

For header (legs 1 and 2): 2 2 兹(i 苶苶 +苶(苶 ioM 苶苶 im苶 i) 苶 o)苶

Sb = ᎏᎏ Z

(10-102)

For branch (leg 3): 2 2 兹苶 (i苶 +苶(苶 io苶 M苶 im苶 i) 苶 o)苶 (10-103) Sb = ᎏᎏ Ze where Sb = resultant bending stress, lbf/in2 (MPa) Ze = effective section modulus for branch, in3 (mm3)

Ze = π r 22 Ts

(10-104)

r2 = mean branch cross-sectional radius, in (mm) Ts = effective branch wall thickness, in (mm) [lesser of 苶 Th and (io)(T 苶b)] Th = thickness of pipe matching run of tee or header exclu苶 sive of reinforcing elements, in (mm) Tb = thickness of pipe matching branch, in (mm) 苶 io = out-plane stress-intensification factor (Table 10-54) ii = in-plane stress-intensification factor (Table 10-54) Allowable stress range SA and permissible additive stresses shall be computed in accordance with Eqs. (10-93) and (10-94).

冤

冢 冣冢 冣 冢 冣 冥

Required Weld Quality Assurance Any weld at which SE exceeds 0.8 SA for any portion of a piping system and the equivalent number of cycles N exceeds 7000 shall be fully examined in accordance with the requirements for severe cyclic service (presented later in this section). Reactions: Metallic Piping Reaction forces and moments to be used in the design of restraints and supports and in evaluating the effects of piping displacements on connected equipment shall be based on the reaction range R for the extreme displacement conditions, considering the range previously defined for reactions and using Ea. The designer shall consider instantaneous maximum values of forces and moments in the original and extreme displacement conditions as well as the reaction range in making these evaluations. Maximum Reactions for Simple Systems For two-anchor systems without intermediate restraints, the maximum instantaneous values of reaction forces and moments may be estimated from Eqs. (10-105) and (10-106). 1. For extreme displacement conditions, Rm. The temperature for this computation is the design maximum or design minimum temperature as previously defined for reactions, whichever produces the larger reaction: 2C Em Rm = R 1 − ᎏ ᎏ (10-105) 3 Ea

冢

冣

PROCESS-PLANT PIPING where C = cold-spring factor varying from zero for no cold spring to 1.0 for 100 percent cold spring. (The factor w is based on experience, which shows that specified cold spring cannot be fully assured even with elaborate precautions.) Ea = modulus of elasticity at installation temperature, lbf/in2 (MPa) Em = modulus of elasticity at design maximum or design minimum temperature, lbf/in2 (MPa) R = range of reaction forces or moments (derived from flexibility analysis) corresponding to the full displacementstress range and based on Ea, lbf or in⋅lbf (N or N⋅mm) Rm = estimated instantaneous maximum reaction force or moment at design maximum or design minimum temperature, lbf or in⋅lbf (N or N⋅mm) 2. For original condition, Ra. The temperature for this computation is the expected temperature at which the piping is to be assembled. Ra = CR or

C1R, whichever is greater

(10-106)

where nomenclature is as for Eq. (10-105) and C1 = 1 − (Sh Ea /SE Em) = estimated self-spring or relaxation factor (use zero if value of C1 is negative) Ra = estimated instantaneous reaction force or moment at installation temperature, lbf or in⋅lbf (N or N⋅mm) SE = computed displacement-stress range, lbf/in2 (MPa). See Eq. (10-100). Sh = See Eq. (10-93). Maximum Reactions for Complex Systems For multianchor systems and for two-anchor systems with intermediate restraints, Eqs. (10-105) and (10-106) are not applicable. Each case must be studied to estimate the location, nature, and extent of local overstrain and its effect on stress distribution and reactions. Acceptable comprehensive methods of analysis are analytical, model-test, and chart methods, which evaluate for the entire piping system under consideration the forces, moments, and stresses caused by bending and torsion from a simultaneous consideration of terminal and intermediate restraints to thermal expansion and include all external movements transmitted under thermal change to the piping by its terminal and intermediate attachments. Correction factors, as provided by the details of these rules, must be applied for the stress intensification of curved pipe and branch connections and may be applied for the increased flexibility of such component parts. Brock [in Crocker (ed.), Piping Handbook, 5th ed., McGraw-Hill, New York, 1967, sec. 4] provides further data on methods of analysis. Expansion Joints All the foregoing applies to “stiff piping systems,” i.e., systems without expansion joints (see detail 1 of Fig. 10-169). When space limitations, process requirements, or other considerations result in configurations of insufficient flexibility, capacity

Flexibility classification for piping systems. (From Kellogg, Design of Piping Systems, Wiley, New York, 1965.)

FIG. 10-169

10-123

for deflection within allowable stress range limits may be increased successively by the use of one or more hinged bellows expansion joints, viz., semirigid (detail 2) and nonrigid (detail 3) systems, and expansion effects essentially eliminated by a free-movement joint (detail 4) system. Expansion joints for semirigid and nonrigid systems are restrained against longitudinal and lateral movement by the hinges with the expansion element under bending movement only and are known as “rotation” or “hinged” joints (see Fig. 10-170). Semirigid systems are limited to one plane; nonrigid systems require a minimum of three joints for two-dimensional and five joints for three-dimensional expansion movement. Joints similar to that shown in Fig. 10-170, except with two pairs of hinge pins equally spaced around a gimbal ring, achieve similar results with a lesser number of joints. Expansion joints for free-movement systems can be designed for axial or offset movement alone, or for combined axial and offset movements (see Fig. 10-171). For offset movement alone, the end load due to pressure and weight can be transferred across the joint by tie rods or structural members (see Fig. 10-172). For axial or combined movements, anchors must be provided to absorb the unbalanced pressure load and force bellows to deflect. Commercial bellows elements are usually light-gauge [of the order of (0.05 to 0.10 in) thick] and are available in stainless and other alloy steels, copper, and other nonferrous materials. Multi-ply bellows, bellows with external reinforcing rings, and toroidal contour bellows are available for higher pressures. Since bellows elements are ordinarily rated for strain ranges which involve repetitive yielding, predictable

Hinged expansion joint. (From Kellogg, Design of Piping Systems, Wiley, New York, 1965.)

FIG. 10-170

Action of expansion bellows under various movements. (From Kellogg, Design of Piping Systems, Wiley, New York, 1965.)

FIG. 10-171

10-124

TRANSPORT AND STORAGE OF FLUIDS

Constrained-bellows expansion joints. (From Kellogg, Design of Piping Systems, Wiley, New York, 1965.)

FIG. 10-172

performance is assured only by adequate fabrication controls and knowledge of the potential fatigue performance of each design. The attendant cold work can affect corrosion resistance and promote susceptibility to corrosion fatigue or stress corrosion; joints in a horizontal position cannot be drained and have frequently undergone pitting or cracking due to the presence of condensate during operation or offstream. For low-pressure essentially nonhazardous service, nonmetallic bellows of fabric-reinforced rubber or special materials are sometimes used. For corrosive service Teflon bellows may be used. Because of the inherently greater susceptibility of expansion bellows to failure from unexpected corrosion, failure of guides to control joint movements, etc., it is advisable to examine critically their design choice in comparison with a stiff system. Slip-type expansion joints (Fig. 10-173) substitute packing (ring or plastic) for bellows. Their performance is sensitive to adequate design with respect to guiding to prevent binding and the adequacy of stuffing boxes and attendant packing, sealant, and lubrication. Anchors must be provided for the unbalanced pressure force and for the friction forces to move the joint. The latter can be much higher than the elastic force required to deflect a bellows joint. Rotary packed joints, ball joints, and other special joints can absorb end load. Corrugated pipe and corrugated and creased bends are also used to decrease stiffness. Pipe Supports Loads transmitted by piping to attached equipment and supporting elements include weight, temperature- and pressureinduced effects, vibration, wind, earthquake, shock, and thermal expansion and contraction. The design of supports and restraints is based on concurrently acting loads (if it is assumed that wind and earthquake do not act simultaneously). Resilient and constant-effort-type supports shall be designed for maximum loading conditions including test unless temporary supports are provided.

FIG. 10-173 Slip-type expansion joint. (From Kellogg, Design of Piping Systems, Wiley, New York, 1965.)

Though not specified in the code, supports for discharge piping from relief valves must be adequate to withstand the jet reaction produced by their discharge. The code states further that pipe-supporting elements shall (1) avoid excessive interference with thermal expansion and contraction of pipe which is otherwise adequately flexible; (2) be such that they do not contribute to leakage at joints or excessive sag in piping requiring drainage; (3) be designed to prevent overstress, resonance, or disengagement due to variation of load with temperature; also, so that combined longitudinal stresses in the piping shall not exceed the code allowable limits; (4) be such that a complete release of the piping load will be prevented in the event of spring failure or misalignment, weight transfer, or added load due to test during erection; (5) be of steel or wrought iron; (6) be of alloy steel or protected from temperature when the temperature limit for carbon steel may be exceeded; (7) not be cast iron except for roller bases, rollers, anchor bases, etc., under mainly compression loading; (8) not be malleable or nodular iron except for pipe clamps, beam clamps, hanger flanges, clips, bases, and swivel rings; (9) not be wood except for supports mainly in compression when the pipe temperature is at or below ambient; and (10) have threads for screw adjustment which shall conform to ANSI B1.1. A supporting element used as an anchor shall be designed to maintain an essentially fixed position. To protect terminal equipment or other (weaker) portions of the system, restraints (such as anchors and guides) shall be provided where necessary to control movement or to direct expansion into those portions of the system that are adequate to absorb them. The design, arrangement, and location of restraints shall ensure that expansion-joint movements occur in the directions for which the joint is designed. In addition to the other thermal forces and moments, the effects of friction in other supports of the system shall be considered in the design of such anchors and guides. Anchors for Expansion Joints Anchors (such as those of the corrugated, omega, disk, or slip type) shall be designed to withstand the algebraic sum of the forces at the maximum pressure and temperature at which the joint is to be used. These forces are: 1. Pressure thrust, which is the product of the effective thrust area times the maximum pressure to which the joint will be subjected during normal operation. (For slip joints the effective thrust area shall be computed by using the outside diameter of the pipe. For corrugated, omega, or disk-type joints, the effective thrust area shall be that area recommended by the joint manufacturer. If this information is unobtainable, the effective area shall be computed by using the maximum inside diameter of the expansion-joint bellows.) 2. The force required to compress or extend the joint in an amount equal to the calculated expansion movement. 3. The force required to overcome the static friction of the pipe in expanding or contracting on its supports, from installed to operating position. The length of pipe considered should be that located between the anchor and the expansion joint. Support Fixtures Hanger rods may be pipe straps, chains, bars, or threaded rods which permit free movement for thermal expansion or contraction. Sliding supports shall be designed for friction and bearing loads. Brackets shall be designed to withstand movements due to friction in addition to other loads. Spring-type supports shall be designed for weight load at the point of attachment and to prevent misalignment, buckling, or eccentric loading of springs, and provided with stops to prevent spring overtravel. Compensating-type spring hangers are recommended for high-temperature and critical-service piping to make the supporting force uniform with appreciable movement. Counterweight supports shall have stops to limit travel. Hydraulic supports shall be provided with safety devices and stops to support load in the event of loss of pressure. Vibration dampers or sway braces may be used to limit vibration amplitude. The code requires that the safe load for threaded hanger rods be based on the root area of the threads. This, however, assumes concentric loading. When hanger rods move to a nonvertical position so that the load is transferred from the rod to the supporting structure via the edge of one flat of the nut on the rod, it is necessary to consider the root area to be reduced by one-third. If a clamp is connected to a vertical line to support its weight, it is recommended that shear lugs be

PROCESS-PLANT PIPING welded to the pipe, or that the clamp be located below a fitting or flange, to prevent slippage. Consideration shall be given to the localized stresses induced in the piping by the integral attachment. Typical pipe supports are shown in Fig. 10-174. Much piping is supported from structures installed for other pur-

FIG. 10-174

10-125

poses. It is common practice to use beam formulas for tubular sections to determine stress, maximum deflection, and maximum slope of piping in spans between supports. When piping is supported from structures installed for that sole purpose and those structures rest on driven piles, detailed calculations are usually made to determine max-

Typical pipe supports and attachments. (From Kellogg, Design of Piping Systems, Wiley, New York, 1965.)

10-126

TRANSPORT AND STORAGE OF FLUIDS

imum permissible spans. Limits imposed on maximum slope to make the contents of the line drain to the lower end require calculations made on the weight per foot of the empty line. To avoid interference with other components, maximum deflection should be limited to 25.4 mm (1 in). Pipe hangers are essentially frictionless but require taller pipesupport structures which cost more than structures on which pipe is laid. Devices that reduce friction between laid pipe subject to thermal movement and its supports are used to accomplish the following: 1. Reduce loads on anchors or on equipment acting as anchors. 2. Reduce the tendency of pipe acting as a column loaded by friction at supports to buckle sideways off supports. 3. Reduce nonvertical loads imposed by piping on its supports so as to minimize cost of support foundations. 4. Reduce longitudinal stress in pipe. Linear bearing surfaces made of fluorinated hydrocarbons or of graphite and also rollers are used for this purpose. Design Criteria: Nonmetallic Pipe In using a nonmetallic material, designers must satisfy themselves as to the adequacy of the material and its manufacture, considering such factors as strength at design temperature, impact- and thermal-shock properties, toxicity, methods of making connections, and possible deterioration in service. Rating information, based usually on ASTM standards or specifications, is generally available from the manufacturers of these materials. Particular attention should be given to provisions for the thermal expansion of nonmetallic piping materials, which may be as much as 5 to 10 times that of steel (Table 10-55). Special consideration should be given to the strength of small pipe connections to piping and equipment and to the need for extra flexibility at the junction of metallic and nonmetallic systems. Table 10-56 gives values for the modulus of elasticity for nonmetals; however, no specific stress-limiting criteria or methods of stress analysis are presented. Stress-strain behavior of most nonmetals differs considerably from that of metals and is less well-defined for mathematic analysis. The piping system should be designed and laid out so that flexural stresses resulting from displacement due to expansion, contraction, and other movement are minimized. This concept requires special attention to supports, terminals, and other restraints. Displacement Strains The concepts of strain imposed by restraint of thermal expansion or contraction and by external movement described for metallic piping apply in principle to nonmetals. Nevertheless, the assumption that stresses throughout the piping system can be predicted from these strains because of fully elastic behavior of the piping materials is not generally valid for nonmetals. In thermoplastics and some thermosetting resins, displacement strains are not likely to produce immediate failure of the piping but may result in detrimental distortion. Especially in thermoplastics, progressive deformation may occur upon repeated thermal cycling or on prolonged exposure to elevated temperature. In brittle nonmetallics (such as porcelain, glass, impregnated graphite, etc.) and some thermosetting resins, the materials show rigid behavior and develop high displacement stresses up to the point of sudden breakage due to overstrain. Elastic Behavior The assumption that displacement strains will produce proportional stress over a sufficiently wide range to justify an elastic-stress analysis often is not valid for nonmetals. In brittle nonmetallic piping, strains initially will produce relatively large elastic stresses. The total displacement strain must be kept small, however, since overstrain results in failure rather than plastic deformation. In plastic and resin nonmetallic piping strains generally will produce stresses of the overstrained (plastic) type even at relatively low values of total displacement strain. Further information on the design of thermoplastic piping can be found in the Plastics Pipe Institute’s Technical Report TR-21.

TABLE 10-55

FABRICATION, ASSEMBLY, AND ERECTION

elbows and lap-joint stub ends. The code requirements for welding processes and operators are essentially the same as covered in the subsection on pressure vessels (i.e., qualification to Sec. IX of the ASME Boiler and Pressure Vessel Code) except that welding processes are not restricted, the material grouping (P number) must be in accordance with Appendix A, and welding positions are related to pipe posi-

Welding, Brazing, or Soldering Code requirements dealing with fabrication are more detailed for welding than for other methods of joining, since welding is used not only to join two pipes end to end but also to fabricate fittings which replace seamless fittings such as

Thermal Expansion Coefficients: Nonmetals* Mean coefficients (divide table values by 106)

Material description

in/(in⋅°F)

Range, °F

mm/mm, °C

Range, °C

Thermoplastics Acetal AP2012 Acrylonitrile-butadienestyrene ABS 1208 ABS 1210 ABS 1316 ABS 2112 Cellulose acetate butyrate CAB MH08 CAB 5004 Chlorinated poly (vinyl chloride) CPVC 4120 Polybutylene PB 2110 Polyether, chlorinated Polyethylene PE 1404 PE 2305 PE 2306 PE 3306 PE 3406 Polyphenylene POP 2125 Polypropylene PP1110 PP1208 PP2105 Poly(vinyl chloride) PVC 1120 PVC 1220 PVC 2110 PVC 2112 PVC 2116 PVC 2120 Vinylidine fluoride Vinylidine/vinyl chloride

2 60 55 40 40

4

45–55

108 99 72 72

80 95

144 171

35 72 45

63 130 81

100 90 80 70 60

46–100 46–100 46–100 46–100 46–100

30

180 162 144 126 108

8–12

8–38 8–38 8–38 8–38 8–38

54

48 43 40

33–67

86 77 72

0–20

30 35 50 45 40 30 85

23–37 34–40

54 63 90 81 72 54 153

−5–+3 1–4

37–45

100

3–8

180

Reinforced thermosetting resins Asbestos-phenolic Asbestos-epoxy Asbestos-polyester Glass-epoxy, centrifugal-cast Glass-polyester, centrifugal-cast Glass-polyester, filament-wound Glass-polyester, hand lay-up Glass-epoxy, filamentwound

11–30 11–30 11–30

20–54 20–54 20–54

9–13

16–23

9–15

16–27

9–11

16–20

12–15

22–27

9–13

16–23

Other nonmetallic materials Borosilicate glass Impregnated graphite Hard rubber (Buna N)

1.8 2.4 40

3 4 72

*Extracted from the Chemical Plant and Petroleum Refinery Piping Code, ANSI B31.3—1980, with permission of the publisher, the American Society of Mechanical Engineers, New York. Individual compounds may vary from the values in the table by as much as 10 percent. Consult manufacturers for specific values for their products.

PROCESS-PLANT PIPING TABLE 10-56

Modulus of Elasticity: Nonmetals*

Material description Thermoplastics Acetal ABS, type 1210 ABS, type 1316 CAB PVC, type 1120 PVC, type 1220 PVC, type 2110 PVC, type 2116 Chlorinated PVC Chlorinated polyether PE, type 2306 PE, type 3306 PE, type 3406 Polypropylene Vinylidene/vinyl chloride Vinylidene fluoride Thermosetting resins, axially reinforced Epoxy-asbestos Phenolic-asbestos Epoxy-glass, centrifugally cast Epoxy-glass, filament-wound Polyester-glass, centrifugally cast Polyester-glass, hand lay-up Other Borosilicate glass Impregnated graphite Hard rubber (Buna N)

E, kip/in2 (73.4°F)

E, MPa (23°C)

410 250 340 120 420 410 340 380 420 160 90 130 150 120 100 120

2830 1725 2345 830 2895 2830 2345 2620 2895 1105 620 895 1035 825 690 825

1200 1200 1200–1900 1100–2000 1200–1900 800–1000

8280 8280 8280–13100 7580–13800 8280–13100 5510–6900

9800 2300 300

67,600 15,900 2070

*Extracted from the Chemical Plant and Petroleum Refinery Piping Code, ANSI B31.3—1980, with permission of the publisher, the American Society of Mechanical Engineers, New York.

tion. The code also permits one fabricator to accept welders or welding operators qualified by another employer without requalification when welding pipe by the same or equivalent procedure. Procedure qualification may include a requirement for low-temperature toughness tests. See Table 10-57. Filler metal is required to conform with the requirements of Sec. IX. Backing rings (of ferrous material), when used, shall be of weldable quality with sulfur limited to 0.05 percent. Backing rings of nonferrous and nonmetallic materials may be used provided they are proved satisfactory by procedure-qualification tests and provided their use has been approved by the designer. The code requires internal alignment within the dimensional limits specified in the welding procedure and the engineering design without specific dimensional limitations. Internal trimming is permitted for correcting internal misalignment provided such trimming does not result in a finished wall thickness before welding of less than required minimum wall thickness tm. When necessary, weld metal may be deposited on the inside or outside of the component to provide alignment or sufficient material for trimming. Table 10-58 is a digest of code requirements for the quality of welds. The defects referred to are illustrated in Fig. 10-175. The qualification of brazing procedures, brazers, or brazing operations is required in accordance with the requirements of Part QB, Sec. IX, ASME Code, except that for Category D fluid service at design temperatures not over 93°C (200°F). Such qualification is at the owner’s option. The clearance between surfaces to be joined by brazing or soldering shall be no larger than is necessary to allow complete capillary distribution of the filler metal. The only requirement for solderers is that they follow the procedure in the Copper Tube Handbook of the Copper Development Association. Bending and Forming Pipe may be bent to any radius for which the bend-arc surface will be free of cracks and substantially free of buckles. The use of bends which are creased or corrugated is permitted. Bending may be by any hot or cold method permissible within the radii and material characteristics of the pipe being bent. Postbend heat treatment may be required for bends in some mate-

10-127

rials; its necessity depends on the severity of the bend. The details of these requirements are spelled out in the code. Piping components may be formed by any suitable hot or cold pressing, rolling, forging, hammering, spinning, drawing, or other method. Thickness after forming shall not be less than required by design. Special rules cover the forming and pressure design verification of flared laps. Hot bending and hot forming shall be done within a temperature range consistent with material characteristics, end use, or postoperation heat treatment. The development of fabrication facilities for bending pipe to the radius of commercial butt-welding long-radius elbows and forming flared metallic (Van Stone) laps on pipe are important techniques in reducing welded-piping costs. These techniques save both the cost of the ell or stub end and the welding operation required to attach the fitting to the pipe. Preheating and Heat Treatment Preheating and postoperation heat treatment are used to avert or relieve the detrimental effects of the high temperature and severe thermal gradients inherent in the welding of metals. In addition, heat treatment may be needed to relieve residual stresses created during the bending or forming of metals. The code provisions shown in Tables 10-59 and 10-60 represent basic practices which are acceptable for most applications of welding, bending, and forming, but they are not necessarily suitable for all service conditions. The specification of more or less stringent preheating and heat-treating requirements is a function of those responsible for the engineering design. Joining Nonmetallic Pipe Thermoplastic piping may be joined by a qualified hot-gas welding procedure, a qualified solvent-cement procedure, or by a qualified heat-fusion procedure. The general welding and heat-fusion procedures are described in ASTM D-2657 and solvent-cement procedures in ASTM D-2855. Two other techniques, for flared joints and elastomeric-sealed joints, are described in ASTM D-3140 and D-3139, respectively. In joining reinforced thermosetting pipe it is particularly important that the pipe be cut without chipping or cracking it. It is also important to sand, file, or grind any mold-release agent from the surfaces to be cemented. Joints are built up layer by layer of adhesive-saturated reinforcement by following the manufacturer’s recommended procedure. Application of adhesive to the surfaces to be joined and assembly of these surfaces shall produce a continuous bond and provide an adhesive seal to protect the reinforcement from attack by the contents of the pipe. Unfilled or unbonded areas of the joint are considered defects and must be repaired. Assembly and Erection Flanged-joint faces shall be aligned to the design plane to within g in/ft (0.5 percent) maximum measured across any diameter, and flange bolt holes shall be aligned to within 3.2-mm (f-in) maximum offset. Flanged joints involving flanges with widely differing mechanical properties shall be assembled with extra care, and tightening to a predetermined torque is recommended. The use of flat washers under bolt heads and nuts is a code requirement when assembling nonmetallic flanges. It is preferred that the bolts extend completely through their nuts; however, a lack of complete thread engagement not exceeding one thread is permitted by the code. In assembling nonmetallic lined joints consideration must be given to the need and means for maintaining electrical continuity when static sparking could occur. The assembly of cast-iron bell-andspigot piping is covered in AWWA Standard C600. Screwed joints which are intended to be seal-welded shall be made up without any thread compound. EXAMINATION, INSPECTION, AND TESTING Examination and Inspection The code differentiates between examination and inspection. “Examination” applies to quality-control functions performed by personnel of the piping manufacturer, fabricator, or erector. “Inspection” applies to functions performed for the owner by the authorized inspector. The authorized inspector shall be designated by the owner and shall be the owner, an employee of the owner, an employee of an engineering or scientific organization, or an employee of a recognized insurance or inspection company acting as the owner’s agent. The inspector

10-128

TRANSPORT AND STORAGE OF FLUIDS

TABLE 10-57

Requirements for Low-Temperature Toughness Tests*

Type of Material Listed metallic materials

Ductile iron, malleable iron, Carbon steel, ASTM A36, ASTM A283

Column A

Column B

At or above minimum temperature listed in Table 10-49 or Table 10-15

Below minimum temperatures listed in Table 10-49 or Table 10-15

1. No additional requirements.

1. Shall not be used.

Base metal

Deposited weld metal and heat-affected zone (See Note 1)

2a. No additional requirements.

2b. When materials are fabricated or assembled by welding, the deposited weld metal and heataffected zone shall be impact-tested if the design temperature is below −29°C (−20°F) unless conditions conform to Note 2.

Austenitic stainless steel

3a. If (1) the carbon content by analysis is greater than 0.10 percent or (2) the material is not in the solution-heat-treated condition, then impact testing is required for design temperatures below −29°C (−20°F). See Note 2.

3b. When materials are fabricated or assembled by welding, the deposited weld metal shall be impact-tested for design temperature below −29°C (−20°F) unless conditions conform to Note 2.

3. The material shall be impacttested. See Note 2.

Austenitic ductile iron, ASTM A571

4a. No additional requirements.

4b. Welding not permitted.

4. The material shall be impacttested. This material shall not be used at design minimum temperatures lower than −196°C (−320°F). Welding is not permitted.

Aluminum alloy, copper, copper alloy, nickel, nickel alloy, unalloyed titanium

5a. No additional requirements.

5b. No additional requirements except that when the composition of the filler metal is outside the range of composition for the base metal, testing shall be in accordance with column B, item 5.

5. Low-temperature tests such as tensile elongation and sharp-notch tensile strength (compared with unnotched tensile strength) shall have been conducted to provide assurance to the designer that the material and the deposited weld metal are suitable at the design minimum temperatures.

All other carbon steel, lowintermediate, and high-alloy steels, ferritic steels

2. Except when conditions conform to Note 2, the material shall be heattreated to control its microstructure by a method appropriate to the material as outlined in the specification applicable to the product form and then impacttested. (See Note 1.) Deposited weld metal and heataffected zone shall be impacttested.

Listed nonmetallic materials

6. No additional requirements.

6. Below the recommended minimum temperatures, the designer shall have test results at or below the lowest expected service temperature which assure that the materials will have adequate toughness and are suitable at the design minimum temperatures.

Unlisted materials

Unlisted materials which conform to a published specification and are of composition, heat treatment, and product form comparable with those of listed materials shall be subject to the same requirements as the listed materials. All other unlisted materials conforming to a published specification shall be qualified as required by the applicable item in col. B.

*Extracted from the Chemical Plant and Petroleum Refinery Piping Code, ANSI B31.3—1980, with permission of the publisher, the American Society of Mechanical Engineers, New York. NOTE: These toughness-test requirements are in addition to tests required by the material specification. 1. Any tests and associated acceptance criteria which are part of the welding-procedure qualification for filler materials and heat-affected zone need not be repeated. 2. Impact testing is not required if the design temperature is below −29°C (−20°F) but at or above −46°C (−50°F) and the maximum operating pressure of the fabricated or assembled components will not exceed 25 percent of the maximum allowable design pressure at ambient temperature and the combined longitudinal stress due to pressure, dead weight, and displacement strain (see Par. 319.2.1) does not exceed 41 MPa (6000 lbf/in2).

shall not represent or be an employee of the piping erector, the manufacturer, or the fabricator unless the owner is also the erector, the manufacturer, or the fabricator. The authorized inspector shall have a minimum of 10 years’ experience in the design, fabrication, or inspection of industrial pressure piping. Each 20 percent of satisfactory work toward an engineering

degree accredited by the Engineers’ Council for Professional Development shall be considered equivalent to 1 year’s experience, up to 5 years total. It is the owner’s responsibility, exercised through the authorized inspector, to verify that all required examinations and testing have been completed and to inspect the piping to the extent necessary to be

PROCESS-PLANT PIPING TABLE 10-58

10-129

Limitations on Imperfections in Welds*

Imperfection† Cracks or lack of fusion Incomplete penetration Internal porosity Slag inclusions or elongated defects Undercutting Surface porosity and exposed slag inclusion (t-in nominal wall thickness and less) Concave root surface (suck-up) Weld reinforcement

When required examination is

Girth and miter-joint butt welds

Any 100% radiography Visual or random or spot radiography 100% radiography Random or spot radiography 100% radiography Random or spot radiography Any

Fillet, socket, seal, and reinforcement attachment welds

Longitudinal butt welds‡

Welded branch connections and fabricated laps

None permitted None permitted A

None permitted None permitted None permitted

None permitted NA NA

None permitted None permitted A and H

B C D

B C D

NA NA NA

B and H C and H D and H

E Lesser of h in or 苶 Tw/4 None permitted

E None permitted None permitted

NA Lesser of h in or 苶 Tw/4 None permitted

B and H Lesser of h in or 苶 Tw/4 None permitted

F

F

NA

F and H

G

G

G

G and H

NOTES:

NA: Not applicable. A: The lesser of h in or 0.2 苶 Tw deep. The total length of such imperfections shall not exceed 1.5 in (38 mm) in any 6 in (150 mm) of weld length. (See Fig. 10-175). B: An individual pocket of porosity shall not exceed the lesser of 苶 Tw/3 or f in in its greatest dimension. The total area of porosity projected radially through the weld shall not exceed an area equivalent to 3 times the area of a single maximum pocket allowable in any square inch (645 mm2) of projected weld area. C: An individual pocket of porosity shall not exceed the lesser of 苶 Tw/2 or f in in its greatest dimension. The total area of porosity projected radially through the weld shall not exceed an area equivalent to 3 times the area of a single maximum pocket allowable to any square inch (645 mm2) of projected weld area. D: The developed length of any single slag inclusion or elongated defect shall not exceed 苶 Tw/3. The total cumulative developed length of slag inclusions and/or elongated defects shall not exceed 苶 Tw in any 12 苶 Tw length of weld. The width of a slag inclusion shall not exceed the lesser of g in or 苶 Tw/3. E: The developed length of any single slag inclusion or elongated defect shall not exceed 2 苶 Tw. The total cumulative developed length of slag inclusions and/or elongated defects shall not exceed 4 苶 Tw in any 6-in length of weld. The width of a slag inclusion shall not exceed the lesser of f in or 苶 Tw/2. F: For single-sided welded joints, concavity of the root surface shall not reduce the total thickness of the joint, including reinforcement, to less than the thickness of the thinner of the components being jointed. G: External weld reinforcement and internal weld protrusion shall be fused with and shall merge smoothly into the component surface. The thickness of external weld reinforcement and internal weld protrusion (when no backing ring is used) shall not exceed the following: Wall thickness Tw, in 苶

Weld reinforcement or protrusion, in, maximum

d and under over d through a over a through 1 over 1 (25.4 mm)

g f 5⁄ 32 t

H: These requirements apply only to butt welds. *Extracted from the Chemical Plant and Petroleum Refinery Piping Code, ANSI B31.3—1980, with permission of the publisher, the American Society of Mechanical Engineers, New York. To convert inches to millimeters, multiply by 25.4. †See Fig. 10-175 for illustration of the defects. ‡This column applies to welds not made in accordance with a standard listed in Appendix A or Appendix E of the code.

satisfied that it conforms to all applicable requirements of the code and the engineering design. This verification may include certifications and records pertaining to materials, components, heat treatment, examination and testing, and qualifications of operators and procedures. The authorized inspector may delegate the performance of inspection to a qualified person. Inspection does not relieve the manufacturer, the fabricator, or the erector of responsibility for providing materials, components, and skill in accordance with requirements of the code and the engineering design, performing all required examinations, and preparing records of examinations and tests for the inspector’s use. Examination Methods The code establishes the types of examinations for evaluating various types of imperfections (see Table 10-61). Personnel performing examinations other than visual shall be qualified in accordance with applicable portions of SNT TC-1A, Recommended Practice for Nondestructive Testing Personnel Qualification and Certification. Procedures shall be qualified as required in Par. T-150, Art. 1, Sec. V of the ASME Code. Limitations on imperfections shall be in accordance with the engineering design but shall at least meet the requirements of the code (see Tables 10-58 and 10-59) for the specific type of examination. Repairs shall be made as applicable. Visual Examination This consists of observation of the portion of components, joints, and other piping elements that are or can be

exposed to view before, during, or after manufacture, fabrication, assembly, erection, inspection, or testing. The examination includes verification of code and engineering design requirements for materials and components, dimensions, joint preparation, alignment, welding or joining, supports, assembly, and erection. Visual examination shall be performed in accordance with Art. 9, Sec. V of the ASME Code. Magnetic-Particle Examination This examination shall be performed in accordance with Art. 7, Sec. V of the ASME Code. Liquid-Penetrant Examination This examination shall be performed in accordance with Art. 6, Sec. V of the ASME Code. Radiographic Examination The following definitions apply to radiography required by the code or by the engineering design: 1. “Random radiography” applies only to girth butt welds. It is radiographic examination of the complete circumference of a specified percentage of the girth butt welds in a designated lot of piping. 2. “100 percent radiography” applies only to girth butt welds unless otherwise specified in the engineering design. It is defined as radiographic examination of the complete circumference of all the girth butt welds in a designated lot of piping. If the engineering design specifies that 100 percent radiography shall include welds other than girth butt welds, the examination shall include the full length of all such welds.

10-130

TRANSPORT AND STORAGE OF FLUIDS

FIG. 10-175 Typical weld imperfections. (Extracted from the Chemical Plant and Petroleum Refinery Piping Code, ANSI B31.3—1976, with permission of the publisher, the American Society of Mechanical Engineers, New York.)

3. “Spot radiography” is the practice of making a single-exposure radiograph at a point within a specified extent of welding. Required coverage for a single spot radiograph is as follows: • For longitudinal welds, at least 150 mm (6 in) of weld length. • For girth, miter, and branch welds in piping 2a in NPS and smaller, a single elliptical exposure which encompasses the entire weld circumference, and in piping larger than 2a in NPS, at least 25 percent of the inside circumference or 150 mm (6 in), whichever is less. Radiography of components other than castings and of welds shall be in accordance with Art. 2, Sec. V of the ASME Code. Limitations on imperfections in components other than castings and welds shall be as stated in Table 10-58 for the degree of radiography involved. Ultrasonic Examination Ultrasonic examination of welds shall be in accordance with Art. 5, Sec. V of the ASME Code, except that the modifications stated in Par. 336.4.6 of the code shall be substituted for T-535.1(d)(2). Type and Extent of Required Examination The intent of examinations is to provide the examiner and the inspector with reasonable assurance that the requirements of the code and the engineering design have been met. For P-number 3, 4, and 5 materials,

examination shall be performed after any heat treatment has been completed. Examination Normally Required Piping not covered by Category D fluid service or severe cyclic conditions shall be examined as follows or to any greater extent specified in the engineering design. 1. Visual examination a. Sufficient materials and components, selected at random, to satisfy the examiner that they conform to specifications and are free from damage. b. At least 5 percent of fabrication. For welds, each welder’s or welding operator’s work shall be represented, though not necessarily each type of weld for each welder or welding operator. Limitations on imperfections shall be as stated in Table 10-58. c. 100 percent of fabrication for longitudinal welds other than those in components made to material specifications recognized in the code. Limitations on imperfections are those of Table 10-58. d. Random examination of the assembly of threaded, bolted, and other joints to satisfy the examiner that they conform to requirements. e. Random examination during erection of piping, including checking of alignments, supports, and cold spring.

TABLE 10-59

Preheat Temperatures*

Base-metal P number†

Weld-metal analysis A number‡

1

1

3

2, 11

4

3

5

4, 5

6

6

7

7

8 9A, 9B, 9C

8, 9 10

Base-material group Carbon steel Alloy steels Cr a% maximum

Minimum temperature Required °C

Recommended

°F

°C

°F

10 80 80 10 80 80

50 175 175 50 175 175

All

150§

300§

All

All

10

50

All

All

All

10

50

All All All

All All All

All All All

All All All

95 80 150

200 175 300

All All

All All

All All

All All

10 10

50 50

mm

in

MPa

kip/in2

<25.4 ≥25.4 All <12.7 ≥12.7 All

<1 ≥1 All
≤490 All >490 ≤490 All ≥490

≤71 All >71 ≤71 All >71

All

All

All

All

150

300

All

All

All

All

175

350

All

All

All

All

All

All

Alloy steels Cr > a% to 2% Alloy steels Cr 2d% to 10% High-alloy steels: martensitic High-alloy steels: ferritic High-alloy steels: austenitic Nickel alloy steels Mn-V steel Cr-V steel

10A 10B 11A Group 1 P21-P52

Minimum specified tensile strength, base metal

Nominal wall thickness

9% Ni steel

*Extracted from the Chemical Plant and Petroleum Refinery Piping Code, ANSI B31.3—1980, with permission of the publisher, the American Society Mechanical Engineers, New York. †P number from ASME Code, Sec. IX, Table QW-422. ‡A number from ASME Code, Sec. IX, Table QW-442. §Maximum interpass temperature 315°C (600°F). TABLE 10-60

Base-metal P number†

Requirements for Heat Treatment*

Weld-metal analysis A number‡

Nominal wall thickness Material group

1

1

Carbon steel

3

2, 11

Alloy steels Cr a% max

4

3

Alloy steels Cr > a% to 2%

5

4, 5

6

6

7

7

8 9A 9B 10A 10B 11A, Group 1

8, 9 10

Alloy steels Cr 2d% to 10% High-alloy steels: martensitic A240, Gr 429 High-alloy steels: ferritic High-alloy steels: austenitic Nickel alloy steels Mn-V steel Cr-V steel 9% Ni steel

mm

in

≤19 ≤e >19 >e ≤19 ≤e >19 >e All All ≤12.7 ≤a >12.7 >a All All ≤a and ≤3% Cr and ≤0.15% C >a or >3% Cr or >0.15% C

Minimum specified tensile strength, base metal

Metal temperature range

h/in, nominal wall§

Minimum time, h

1

1

Brinell hardness, maximum

MPa

kip/in

°C

°F

All

All

≤490 All >490 ≤490 All >490

≤71 All >71 ≤71 All >71

None 595–650 None 595–720 595–720 None 705–745 705–745

None 1100–1200 None 1100–1325 1100–1325 None 1300–1375 1300–1375

1 1

1 1

225 225

1 1

2 2

225 225

All All

All All

None 705–760

None 1300–1400

1

2

241

1 1

2 2

241 241

2

All All

All All

All All

All All

730–790 620–660

1350–1450 1150–1225

All

All

All

All

None

None

All ≤19 >19 ≤19 >19 All ≤12.7 >12.7 All ≤51 >51

All ≤e >e ≤e >e All ≤a >a All ≤2 >2

All All All All All >490 ≤490 All >490 All All

All All All All All >71 ≤71 All >71 All All

None None None None 595–635 1100–1175 None None 595–705 1100–1300 595–705 1100–1300 None None 595–730 1100–1350 595–730 1100–1350 None None 550–585 1025–1085 [Cooling rate > 150°C (300°F)/h to 315°C (600°F)]

a

1

1 1

1 1

225 225

1 1

1 1

225 225

1

1

*Extracted from the Chemical Plant and Petroleum Refinery Piping Code, ANSI B31.3—1980, with permission of the publisher, the American Society of Mechanical Engineers, New York. †P number from ASME Code, Sec. IX, Table QW-422, Special P numbers (SP-1, SP-2, SP-3) require special consideration in procedure qualification. The required thermal treatment shall be established by the engineering design and demonstrated by the procedure qualification. ‡A number from ASME Code, Sec. IX, Table QW-422. §For SI equivalent, h/mm, divide h/in by 25. 10-131

10-132

TRANSPORT AND STORAGE OF FLUIDS

TABLE 10-61 Types of Examination for Evaluating Imperfections* Type of examination

Type of imperfection

Visual

Crack Incomplete penetration Lack of fusion Weld undercutting Weld reinforcement Internal porosity External porosity Internal slag inclusions External slag inclusions Concave root surface

X X X X X

Liquid-penetrant or magneticparticle X

Ultrasonic or radiographic Random

100%

X X X

X X X

X

X

X X X

X

X

X

X

*Extracted from the Chemical Plant and Petroleum Refinery Piping Code, ANSI B31.3—1980, with permission of the publisher, the American Society of Mechanical Engineers, New York. For limitations on imperfections in welds see Table 10-58.

f. Examination of erected piping for evidence of damage that would require repair or replacement and for other evident deviations from the intent of the design. 2. Other examination. When piping is intended for service at temperatures above 186°C (366°F) or gauge pressures above 1.0 MPa (150 lbf/in2) as designated in the engineering design, a minimum of 5 percent of circumferential butt welds shall be examined fully by random radiography or ultrasonic examination. The welds to be examined shall be selected to ensure that the work product of each individual welder or welding operator doing the production welding is included. They shall also be selected to maximize coverage of intersections with longitudinal joints. A minimum of 38 mm (1a in) of the longitudinal welds shall be examined. In-process examination may be substituted for all or part of the radiographic or ultrasonic examination on a weldfor-weld basis if specified in the engineering design. 3. In-process examination. In-process examination comprises visual examination of the following as applicable: a. Joint preparation and cleanliness b. Preheating c. Fit-up and internal alignment prior to welding d. Weld position, electrode, and other variables specified by the welding procedure e. Condition of the root pass after cleaning (external and, where accessible, internal), aided by liquid-penetrant or magnetic-particle examination when specified in the engineering design f. Slag removal and weld condition between passes g. Appearance of the finished weld 4. Certification and records for components and materials. The examiner shall be assured, by examination of certification, records, or other evidence, that the materials and components are of the specified grades and that they have received required heat treatment, examination, and testing. The examiner shall provide the inspector with a certification that all quality-control requirements of the code and of the engineering design have been met. Category D Fluid-Service Piping This piping, as designated in the engineering design, shall be visually examined to the extent necessary to satisfy the examiner that components, materials, and work conform to the requirements of the code and the engineering design. Piping Subject to Severe Cyclic Conditions Piping for other than Category D fluids to be used under severe cyclic conditions shall be examined as follows or to any greater extent specified in the engineering design. 1. Visual examination a. All fabrication threaded, bolted, and other joints shall be examined. b. All piping erection shall be examined to verify dimensions and alignment. Supports, guides, and points of cold spring shall be checked to assure that movement of the piping under all conditions of

start-up, operation, and shutdown will be accommodated without binding or constraint. 2. Other examination. All circumferential butt welds and all fabricated branch connection welds comparable to Fig. 10-127 shall be examined by 100 percent radiography or (if specified in the engineering design) by ultrasonic examination. Limitations on imperfections are as specified in Table 10-58. The code also requires that a welding procedure which promotes a smooth, fully penetrated internal surface be employed and that the external surface of the completed weld be free of undercutting and finished to within 500 AARH. Socket welds and nonradiographed branch-connection welds shall be examined by magnetic-particle or liquid-penetrant methods. Impact Testing Materials conforming to ASTM specifications listed in the code may generally be used at temperatures down to the lowest temperature listed for that material in the stress table without additional testing. When welding or other operations are performed on these materials, additional low-temperature toughness tests may be required. The code requirements are listed in Table 10-57. Pressure Testing Prior to initial operation, installed piping shall be pressure-tested to assure tightness except as permitted for Category D fluid service described later. The pressure test shall be maintained for a sufficient time to determine the presence of any leaks but not less than 10 min. If repairs or additions are made following the pressure tests, the affected piping shall be retested except that, in the case of minor repairs or additions, the owner may waive retest requirements when precautionary measures are taken to assure sound construction. When pressure tests are conducted at metal temperatures near the ductile-to-brittle transition temperature of the material, the possibility of brittle fracture shall be considered. The test shall be hydrostatic, using water, with the following exceptions. If there is a possibility of damage due to freezing or if the operating fluid or piping material would be adversely affected by water, any other suitable liquid may be used. If a flammable liquid is used, its flash point shall not be less than 50°C (120°F), and consideration shall be given to the test environment. The hydrostatic-test pressure at any point in the system shall be as follows: 1. Not less than 1a times the design pressure. 2. For a design temperature above the test temperature, the minimum test pressure shall be as calculated by the following formula: PT = 1.5 PST /S

(10-108)

where PT = test hydrostatic gauge pressure, MPa (lbf/in2) P = internal design pressure, MPa (lbf/in2) ST = allowable stress at test temperature, MPa (lbf/in2) S = allowable stress at design temperature, MPa (lbf/in2) If the test pressure as so defined would produce a stress in excess of the yield strength at test temperature, the test pressure may be reduced to the maximum pressure that will not exceed the yield strength at test temperature. A preliminary air test at not more than 0.17-MPa (25-lbf/in2) gauge pressure may be made prior to hydrostatic test in order to locate major leaks. If hydrostatic testing is not considered practicable by the owner, a pneumatic test in accordance with the following procedure may be substituted, using air or another nonflammable gas. If the piping is tested pneumatically, the test pressure shall be 110 percent of the design pressure. Pneumatic testing involves a hazard owing to the possible release of energy stored in compressed gas. Therefore, particular care must be taken to minimize the chance of brittle failure of metals and thermoplastics. The test temperature is important in this regard and must be considered when material is chosen in the original design. Any pneumatic test shall include a preliminary check at not more than 0.17-MPa (25-lbf/in2) gauge pressure. The pressure shall be increased gradually in steps providing sufficient time to allow the piping to equalize strains during test and to check for leaks. If the test liquid in the system is subject to thermal expansion, precautions shall be taken to avoid excessive pressure. At the owner’s option, a piping system used only for Category D

PROCESS-PLANT PIPING fluid service as defined in the subsection “Classification of Fluid Service” may be tested at the normal operating conditions of the system during or prior to initial operation by examining for leaks at every joint not previously tested. A preliminary check shall be made at not more than 0.17-MPa (25-lbf/in2) gauge pressure when the contained fluid is a gas or a vapor. The pressure shall be increased gradually in steps providing sufficient time to allow the piping to equalize strains during testing and to check for leaks. Tests alternative to those required by these provisions may be applied under certain conditions described in the code. Piping required to have a sensitive leak test shall be tested by the gas- and bubble-formation testing method specified in Art. 10, Sec. V of the ASME Code or by another method demonstrated to have equal or greater sensitivity. The sensitivity of the test shall be at least (100 Pa⋅mL)/s [(103 atm⋅mL)/s] under test conditions. If a hydrostatic pressure test is used, it shall be carried out after the sensitive leak test. Records shall be kept of each piping installation during the testing. COMPARISON OF PIPING-SYSTEM COSTS Piping may represent as much as 25 percent of the cost of a chemicalprocess plant. The installed cost of piping systems varies widely with the materials of construction and the complexity of the system. A study of piping costs shows that the most economical choice of material for a simple straight piping run may not be the most economical for a complex installation made up of many short runs involving numerous fittings and valves. The economics also depends heavily on the pipe size and fabrication techniques employed. Fabrication methods such as bending to standard long-radius-elbow dimensions and machine-flaring lap joints have a large effect on the cost of fabricating pipe from ductile materials suited to these techniques. Cost reductions of as high as 35 percent are quoted by some custom fabricators utilizing advanced techniques. Figure 10-176 is based on data extracted from a comparison of the installed cost of piping systems of various materials published by the Dow Chemical Co. The chart shows the relative cost ratios for systems of various materials based on two installations, one consisting of 152 m (500 ft) of 2-in pipe in a complex piping arrangement and the other of 305 m (1000 ft) of 2-in pipe in a straight-run piping arrangement. Figure 10-176 is based on field-fabrication construction techniques using welding stubs, the method commonly used by contractors. A considerably different ranking would result from using other construction methods such as machine-formed lap joints and bends in place of welding elbows. Piping-cost experience shows that it is difficult to generalize and reflect accurate piping-cost comparisons. For an accurate comparison the cost for each type of material must be estimated individually on the basis of the actual fabrication and installation methods that will be used and the conditions anticipated for the proposed installation. FORCES OF PIPING ON PROCESS MACHINERY AND PIPING VIBRATION The reliability of process rotating machinery is affected by the quality of the process piping installation. Excessive external forces and moments upset casing alignment and can reduce clearance between motor and casing. Further, the bearings, seals, and coupling can be adversely affected, resulting in repeated failures that may be correctly diagnosed as misalignment, and may have excessive piping forces as the root causes. Most turbine and compressor manufacturers have prescribed specification or will follow NEMA standards for allowable nozzle loading. Prior to any machinery alignment procedure, it is imperative to check for machine pipe strain. This is accomplished by the placement of dial indicators on the shaft and then loosening the hold-down bolts. Movements of greater than 1 mil are considered indication of a pipe strain condition. This is an important practical problem area, as piping vibration can cause considerable downtime or even pipe failure. Pipe vibration is caused by: 1. Internal flow (pulsation)

10-133

2. Plant machinery (such as compressors, pumps) Pulsation can be problematic and difficult to predict. Pulsations are also dependent on acoustic resonance characteristics. When a pulsation frequency coincides with a mechanical or acoustic resonance, severe vibration can result. A common cause for pulsation is the presence of flow control valves or pressure regulators. These often operate with high pressure drops (i.e., high flow velocities), which can result in the generation of severe pulsation. Flashing and cavitation can also contribute. Modern-day piping design codes can model the vibration situation, and problems can thus be resolved in the design phases. HEAT TRACING OF PIPING SYSTEMS Heat tracing is used to maintain pipes and the material that pipes contain at temperatures above the ambient temperature. Two common uses of heat tracing are preventing water pipes from freezing and maintaining fuel oil pipes at high enough temperatures such that the viscosity of the fuel oil will allow easy pumping. Heat tracing is also used to prevent the condensation of a liquid from a gas and to prevent the solidification of a liquid metal. A heat-tracing system is often more expensive on an installed cost basis than the piping system it is protecting, and it will also have significant operating costs. A recent study on heat-tracing costs by a major chemical company showed installed costs of $31/ft to $142/ft and yearly operating costs of $1.40/ft to $16.66/ft. In addition to being a major cost, the heat-tracing system is an important component of the reliability of a piping system. A failure in the heat-tracing system will often render the piping system inoperable. For example, with a water freeze protection system, the piping system may be destroyed by the expansion of water as it freezes if the heat-tracing system fails. The vast majority of heat-traced pipes are insulated to minimize heat loss to the environment. A heat input of 2 to 10 watts per foot is generally required to prevent an insulated pipe from freezing. With high wind speeds, an uninsulated pipe could require well over 100 watts per foot to prevent freezing. Such a high heat input would be very expensive. Heat tracing for insulated pipes is generally only required for the period when the material in the pipe is not flowing. The heat loss of an insulated pipe is very small compared to the heat capacity of a flowing fluid. Unless the pipe is extremely long (several thousands of feet), the temperature drop of a flowing fluid will not be significant. The three major methods of avoiding heat tracing are: 1. Changing the ambient temperature around the pipe to a temperature that will avoid low-temperature problems. Burying water pipes below the frost line or running them through a heated building are the two most common examples of this method. 2. Emptying a pipe after it is used. Arranging the piping such that it drains itself when not in use, can be an effective method of avoiding the need for heat tracing. Some infrequently used lines can be pigged or blown out with compressed air. This technique is not recommended for commonly used lines due to the high labor requirement. 3. Arranging a process such that some lines have continuous flow can eliminate the need for tracing these lines. This technique is generally not recommended because a failure that causes a flow stoppage can lead to blocked or broken pipes. Some combination of these techniques may be used to minimize the quantity of traced pipes. However, the majority of pipes containing fluids that must be kept above the minimum ambient temperature are generally going to require heat tracing. Types of Heat-Tracing Systems Industrial heat-tracing systems are generally fluid systems or electrical systems. In fluid systems, a pipe or tube called the tracer is attached to the pipe being traced, and a warm fluid is put through it. The tracer is placed under the insulation. Steam is by far the most common fluid used in the tracer, although ethylene glycol and more exotic heat-transfer fluids are used. In electrical systems, an electrical heating cable is placed against the pipe under the insulation. Fluid Tracing Systems Steam tracing is the most common type of industrial pipe tracing. In 1960, over 95 percent of industrial tracing systems were steam traced. By 1995, improvements in electric

10-134

TRANSPORT AND STORAGE OF FLUIDS

Cost rankings and cost ratios for various piping materials. This figure is based on field-fabrication construction techniques using welding stubs, as this is the method most often employed by contractors. A considerably different ranking would result from using other construction methods, such as machinedformed lap joints, for the alloy pipe. °Cost ratio = (cost of listed item)/(cost of Schedule 40 carbon steel piping system, field-fabricated by using welding stubs). (Extracted with permission from Installed Cost of Corrosion Resistant Piping, copyright 1977, Dow Chemical Co.)

FIG. 10-176

PROCESS-PLANT PIPING heating technology increased the electric share to 30 to 40 percent, but steam tracing is still the most common system. Fluid systems other than steam are rather uncommon and account for less than 5 percent of tracing systems. Half-inch copper tubing is commonly used for steam tracing. Three-eighths-inch tubing is also used, but the effective circuit length is then decreased from 150 feet to about 60 feet. In some corrosive environments, stainless steel tubing is used, and occasionally standard carbon steel pipe (one half inch to one inch) is used as the tracer. In addition to the tracer, a steam tracing system (Fig. 10-177) consists of steam supply lines to transport steam from the existing steam lines to the traced pipe, a steam trap to remove the condensate and hold back the steam, and in most cases a condensate return system to return the condensate to the existing condensate return system. In the past, a significant percentage of condensate from steam tracing was simply dumped to drains, but increased energy costs and environmental rules have caused almost all condensate from new steam tracing systems to be returned. This has significantly increased the initial cost of steam tracing systems. Applications requiring accurate temperature control are generally limited to electric tracing. For example chocolate lines cannot be exposed to steam temperatures or the product will degrade and if caustic soda is heated above 150°F it becomes extremely corrosive to carbon steel pipes. For some applications, either steam or electricity is simply not available and this makes the decision. It is rarely economic to install a steam boiler just for tracing. Steam tracing is generally considered only when a boiler already exists or is going to be installed for some other primary purpose. Additional electric capacity can be provided in most situations for reasonable costs. It is considerably more expensive to supply steam from a long distance than it is to provide electricity. Unless steam is available close to the pipes being traced, the automatic choice is usually electric tracing. For most applications, particularly in processing plants, either steam tracing or electric tracing could be used, and the correct choice is dependent on the installed costs and the operating costs of the competing systems.

FIG. 10-177

Steam tracing system.

TABLE 10-62

10-135

Steam versus Electric Tracing* TIC

TOC

Temperature maintained

Steam

Electric

Ratio S/E

Steam

Electric

Ratio S/E

50°F 150°F 250°F 400°F

22,265 22,265 22,807 26,924

7,733 13,113 17,624 14,056

2.88 1.70 1.29 1.92

1,671 4,356 5,348 6,724

334 1,892 2,114 3,942

5.00 2.30 2.53 1.71

*Specifications: 400 feet of four-inch pipe, $25/hr labor, $0.07/kWh, $4.00/1,000# steam, 100-foot supply lines. TIC = total installed cost; TOC = total operating costs.

Economics of Steam Tracing versus Electric Tracing The question of the economics of various tracing systems has been examined thoroughly. All of these papers have concluded that electric tracing is generally less expensive to install and significantly less expensive to operate. Electric tracing has significant cost advantages in terms of installation because less labor is required than steam tracing. However, it is clear that there are some special cases where steam tracing is more economical. The two key variables in the decision to use steam tracing or electric tracing are the temperature at which the pipe must be maintained and the distance to the supply of steam and a source of electric power. Table 10-62 shows the installed costs and operating costs for 400 feet of four-inch pipe, maintained at four different temperatures, with supply lengths of 100 ft. for both electricity and steam and $25/hr labor. The major advantages of a steam tracing system are: 1. High heat output. Due to its high temperature, a steam tracing system provides a large amount of heat to the pipe. There is a very high heat transfer rate between the metallic tracer and a metallic pipe. Even with damage to the insulation system, there is very little chance of a low temperature failure with a steam-tracing system. 2. High reliability. Many things can go wrong with a steam tracing system but, very few of the potential problems lead to a heat tracing failure. Steam traps fail, but they usually fail in the open position,

10-136

TRANSPORT AND STORAGE OF FLUIDS

allowing for a continuous flow of steam to the tracer. Other problems such as steam leaks that can cause wet insulation are generally prevented from becoming heat-tracing failures by the extremely high heat output of a steam tracer. Also, a tracing tube is capable of withstanding a large amount of mechanical abuse without failure. 3. Safety. While steam burns are fairly common, there are generally fewer safety concerns than with electric tracing. 4. Common usage. Steam tracing has been around for many years and many operators are familiar with the system. Because of this familiarity, failures due to operator error are not very common. The weaknesses of a steam-tracing system are: 1. High installed costs. The incremental piping required for the steam supply system and the condensate return system must be installed, insulated, and, in the case of the supply system, additional steam traps are often required. The tracer itself is not expensive, but the labor required for installation is relatively high. Studies have shown that steam tracing systems typically cost from 50 to 150 percent more than a comparable electric tracing system. 2. Energy inefficiency. A steam tracing system’s total energy use is often more than twenty times the actual energy requirement to keep the pipe at the desired temperature. The steam tracer itself puts out significantly more energy than required. The steam traps use energy even when they are properly operating and waste large amounts of energy when they fail in the open position, which is the common failure mode. Steam leaks waste large amounts of energy, and both the steam supply system and the condensate return system use significant amounts of energy. 3. Poor temperature control. A steam tracing system offers very little temperature control capability. The steam is at a constant temperature (50 psig steam is 300°F) usually well above that desired for the pipe. The pipe will reach an equilibrium temperature somewhere between the steam temperature and the ambient temperature. However, the section of pipe against the steam tracer will effectively be at the steam temperature. This is a serious problem for temperaturesensitive fluids such as food products. It also represents a problem with fluids such as bases and acids, which are not damaged by high temperatures but often become extremely corrosive to piping systems at higher temperatures. 4. High maintenance costs. Leaks must be repaired and steam traps must be checked and replaced if they have failed. Numerous studies have shown that, due to the energy lost through leaks and failed steam traps, an extensive maintenance program is an excellent investment. Steam maintenance costs are so high that for low-

FIG. 10-178

Electrical heat tracing system.

temperature maintenance applications, total steam operating costs are sometimes greater than electric operating costs, even if no value is placed on the steam. Electric Tracing An electric tracing system (see Fig. 10-178) consists of an electric heater placed against the pipe under the thermal insulation, the supply of electricity to the tracer, and any control or monitoring system that may be used (optional). The supply of electricity to the tracer usually consists of an electrical panel and electrical conduit or cable trays. Depending on the size of the tracing system and the capacity of the existing electrical system, an additional transformer may be required. Advantages of Electric Tracing 1. Lower installed and operating costs. Most studies have shown that electric tracing is less expensive to install and less expensive to operate. This is true for most applications. However, for some applications, the installed costs of steam tracing are equal to or less than electric tracing. 2. Reliability. In the past, electric heat tracing had a welldeserved reputation for poor reliability. However, since the introduction of self-regulating heaters in 1971, the reliability of electric heat tracing has improved dramatically. Self-regulating heaters cannot destroy themselves with their own heat output. This eliminates the most common failure mode of polymer-insulated constant wattage heaters. Also, the technology used to manufacture mineral-insulated cables, high-temperature electric heat tracing, has improved significantly, and this has improved their reliability. 3. Temperature control. Even without a thermostat or any control system, an electric tracing system usually provides better temperature control than a steam tracing system. With thermostatic or electronic control, very accurate temperature control can be achieved. 4. Safety. The use of self-regulating heaters and ground leakage circuit breakers has answered the safety concerns of most engineers considering electric tracing. Self-regulating heaters eliminate the problems from high-temperature failures, and ground leakage circuit breakers minimize the danger of an electrical fault to ground, causing injury or death. 5. Monitoring capability. One question often asked about any heat-tracing system is, “How do I know it is working?” Electric tracing now has available almost any level of monitoring desired. The temperature at any point can be monitored with both high and low alarm capability. This capability has allowed many users to switch to electric tracing with a high degree of confidence.

PROCESS-PLANT PIPING

FIG. 10-179

10-137

Self-regulating heating cable. FIG. 10-180

6. Energy efficiency. Electric heat tracing can accurately provide the energy required for each application without the large additional energy use of a steam system. Unlike steam tracing systems, other parts of the system do not use significant amounts of energy. Disadvantages of Electric Tracing 1. Poor reputation. In the past, electric tracing has been less than reliable. Due to past failures, some operating personnel are unwilling to take a chance on any electric tracing. 2. Design requirements. A slightly higher level of design expertise is required for electric tracing than for steam tracing. 3. Lower power output. Since electric tracing does not provide a large multiple of the required energy, it is less forgiving to problems such as damaged insulation or below design ambient temperatures. Most designers include a 10 to 20 percent safety factor in the heat loss calculation to cover these potential problems. Also, a somewhat higher than required design temperature is often specified to provide an additional safety margin. For example, many water systems are designed to maintain 50°F to prevent freezing. Types of Electric Tracing Self-regulating electric tracing (see Fig. 10-179) is by far the most popular type of electric tracing. The heating element in a self-regulating heater is a conductive polymer between the bus wires. This conductive polymer increases its resistance as its temperature increases. The increase in resistance with temperature causes the heater to lower its heat output at any point where its temperature increases (Fig. 10-180). This self-regulating effect eliminates the most common failure mode of constant wattage electric heaters, which is destruction of the heater by its own heat output. Because self-regulating heaters are parallel heaters, they may be cut to length at any point without changing their power output per unit of length. This makes them much easier to deal with in the field. They may be terminated, teed, or spliced in the field with hazardousarea-approved components. MI Cables (mineral insulated cables, Fig. 10-181) are the electric heat tracers of choice for high-temperature applications. Hightemperature applications are generally considered to maintain temperatures above 250°F or exposure temperatures above 420°F where self-regulating heaters cannot be used. MI cable consists of one or two heating wires, magnesium oxide insulation (from whence it gets its

FIG. 10-181

Mineral insulated cable (MI cable).

Self regulation.

name), and an outer metal sheath. Today the metal sheath is generally inconel. This eliminates both the corrosion problems with copper sheaths and the stress cracking problems with stainless steel. MI cables can maintain temperatures up to 1200°F and withstand exposure to up to 1500°F. The major disadvantage of MI cable is that it must be factory-fabricated to length. It is very difficult to terminate or splice the heater in the field. This means pipe measurements are necessary before the heaters are ordered. Also, any damage to an MI cable generally requires a complete new heater. It’s not as easy to splice in a good section as with self-regulating heaters. Polymer-insulated constant wattage electric heaters are slightly cheaper than self-regulating heaters, but they are generally being replaced with self-regulating heaters due to inferior reliability. These heaters tend to destroy themselves with their own heat output when they are overlapped at valves or flanges. Since overlapping selfregulating heaters is the standard installation technique, it is difficult to prevent this technique from being used on the similar-looking constant-wattage heaters. SECT (skin-effect current tracing) is a special type of electric tracing employing a tracing pipe, usually welded to the pipe being traced, that is used for extremely long lines. With SECT tracing circuits, up to 10 miles can be powered from one power point. All SECT systems are specially designed by heat-tracing vendors. Impedance tracing uses the pipe being traced to carry the current and generate the heat. Less than 1 percent of electric heat-tracing systems use this method. Low voltages and special electrical isolation techniques are used. Impedance heating is useful when extremely high heat densities are required, like when a pipe containing aluminum metal must be melted from room temperature on a regular basis. Most impedance systems are specially designed by heat tracing vendors. Choosing the Best Tracing System Some applications require either steam tracing or electric tracing regardless of the relative economics. For example, a large line that is regularly allowed to cool and needs to be quickly heated would require steam tracing because of its much higher heat output capability. In most heat-up applications, steam tracing is used with heat-transfer cement, and the heat output is increased by a factor of up to 10. This is much more heat than would

10-138

TRANSPORT AND STORAGE OF FLUIDS

be practical to provide with electric tracing. For example, a half-inch copper tube containing 50 psig steam with heat transfer cement would provide over 1100 BTU/hr/ft to a pipe at 50°F. This is over 300 watts per foot or more than 15 times the output of a high-powered electric tracer. Table 10-62 shows that electric tracing has a large advantage in terms of cost at low temperatures and smaller but still significant advantages at higher temperatures. Steam tracing does relatively better at higher temperatures because steam tracing supplies significantly more power than is necessary to maintain a pipe at low temperatures. Table 10-62 indicates that there is very little difference between the steam tracing system at 50°F and the system at 250°F. However, the electric system more than doubles in cost between these two temperatures because more heaters, higher powered heaters, and higher temperature heaters are required. The effect of supply lengths on a 150°F system can be seen from Table 10-63. Steam supply pipe is much more expensive to run than

TABLE 10-63

Effect of Supply Lengths

Ratio of Steam TIC to Electric TIC Maintained at 150°F Electric supply length Steam supply length

40 feet

100 feet

300 feet

40 feet 100 feet 300 feet

1.1 1.9 4.9

1.0 1.7 4.2

0.7 1.1 2.9

electrical conduit. With each system having relatively short supply lines (40 feet each) the electric system has only a small cost advantage (10 percent, or a ratio of 1.1). This ratio is 2.1 at 50°F and 0.8 at 250°F. However, as the supply lengths increase, electric tracing has a large cost advantage.

STORAGE AND PROCESS VESSELS STORAGE OF LIQUIDS Atmospheric Tanks The term atmospheric tank as used here applies to any tank that is designed to be used within plus or minus several hundred pascals (a few pounds per square foot) of atmospheric pressure. It may be either open to the atmosphere or enclosed. Minimum cost is usually obtained with a vertical cylindrical shape and a relatively flat bottom at ground level. American Petroleum Institute (API) The institute has developed a series of atmospheric tank standards and specifications. Some of these are: API Specification 12B, Bolted Production Tanks API Specification 12D, Large Welded Production Tanks API Specification 12F, Small Welded Production Tanks API Standard 650, Steel Tanks for Oil Storage American Water Works Association (AWWA) The association has many standards dealing with water handling and storage. A list of its publications is given in the AWWA Handbook (annually). AWWA D100, Standard for Steel Tanks—Standpipes, Reservoirs, and Elevated Tanks for Water Storage, contains rules for design and fabrication. Although AWWA tanks are intended for water, they could be used for the storage of other liquids. Underwriters Laboratories Inc. has published the following tank standards: UL 58, Steel Underground Tanks for Flammable and Combustible Liquids UL 142, Steel Aboveground Tanks for Flammable and Combustible Liquids UL 58 covers horizontal steel tanks up to 190 m3 (50,000 gal), with a maximum diameter of 3.66 m (12 ft), and a maximum length of six diameters. Thickness and a number of design and fabrication details are given. UL 142 covers horizontal steel tanks up to 190 m3 (50,000 gal) (like UL 58), and vertical tanks up to 10.7-m (35-ft) height. Thickness and other details are given. The maximum diameter for a vertical tank is not specified. The Underwriters Standards overlap API, but include tanks that are too small for API Standards. Underwriters Standards are, however, not as detailed as API and therefore put more responsibility on the designer. They do not specify grades of steel other than requiring weldability. Designers should also place their own limits on the diameter (or thickness) of vertical tanks. They can obtain guidance from API. Posttensioned Concrete This material is frequently used for tanks to about 57,000 m3 (15 × 106 gal), usually containing water. Their design is treated in detail by Creasy (Prestressed Concrete Cylindrical Tanks, Wiley, New York, 1961). For the most economical design of

large open tanks at ground levels, he recommends limiting vertical height to 6 m (20 ft). Seepage can be a problem if unlined concrete is used with some liquids (e.g., gasoline). Elevated Tanks These can supply a large flow when required, but pump capacities need be only for average flow. Thus, they may save on pump and piping investment. They also provide flow after pump failure, an important consideration for fire systems. Open Tanks These may be used to store materials that will not be harmed by water, weather, or atmospheric pollution. Otherwise, a roof, either fixed or floating, is required. Fixed roofs are usually either domed or coned. Large tanks have coned roofs with intermediate supports. Since negligible pressure is involved, snow and wind are the principal design loads. Local building codes often give required values. Fixed-roof atmospheric tanks require vents to prevent pressure changes which would otherwise result from temperature changes and withdrawal or addition of liquid. API Standard 2000, Venting Atmospheric and Low Pressure Storage Tanks, gives practical rules for vent design. The principles of this standard can be applied to fluids other than petroleum products. Excessive losses of volatile liquids, particularly those with flash points below 38°C (100°F), may result from the use of open vents on fixed-roof tanks. Sometimes vents are manifolded and led to a vent tank, or the vapor may be extracted by a recovery system. An effective way of preventing vent loss is to use one of the many types of variable-volume tanks. These are built under API Standard 650. They may have floating roofs of the double-deck or the singledeck type. There are lifter-roof types in which the roof either has a skirt moving up and down in an annular liquid seal or is connected to the tank shell by a flexible membrane. A fabric expansion chamber housed in a compartment on top of the tank roof also permits variation in volume. Floating Roofs These must have a seal between the roof and the tank shell. If not protected by a fixed roof, they must have drains for the removal of water, and the tank shell must have a “wind girder” to avoid distortion. An industry has developed to retrofit existing tanks with floating roofs. Much detail on the various types of tank roofs is given in manufacturers’ literature. Figure 10-182 shows types. These roofs cause less condensation buildup and are highly recommended. Pressure Tanks Vertical cylindrical tanks constructed with domed or coned roofs, which operate at pressures above several hundred pascals (a few pounds per square foot) but which are still relatively close to atmospheric pressure, can be built according to API Standard 650. The pressure force acting against the roof is transmitted to the shell, which may have sufficient weight to resist it. If not, the uplift will act on the tank bottom. The strength of the bottom, however, is limited, and if it is not sufficient, an anchor ring or a heavy

STORAGE AND PROCESS VESSELS

FIG. 10-182

10-139

Some types of atmospheric storage tanks.

foundation must be used. In the larger sizes uplift forces limit this style of tank to very low pressures. As the size or the pressure goes up, curvature on all surfaces becomes necessary. Tanks in this category, up to and including a pressure of 103.4 kPa (15 lbf/in2), can be built according to API Standard 620. Shapes used are spheres, ellipsoids, toroidal structures, and circular cylinders with torispherical, ellipsoidal, or hemispherical heads. The ASME Pressure Vessel Code (Sec. VIII of the ASME Boiler and Pressure Vessel Code), although not required below 103.4 kPa (15 lbf/in2), is also useful for designing such tanks. Tanks that could be subjected to vacuum should be provided with vacuum-breaking valves or be designed for vacuum (external pressure). The ASME Pressure Vessel Code contains design procedures. Calculation of Tank Volume A tank may be a single geometrical element, such as a cylinder, a sphere, or an ellipsoid. It may also have a compound form, such as a cylinder with hemispherical ends or a combination of a toroid and a sphere. To determine the volume, each geometrical element usually must be calculated separately. Calculations for a full tank are usually simple, but calculations for partially filled tanks may be complicated. To calculate the volume of a partially filled horizontal cylinder refer to Fig. 10-183. Calculate the angle α in degrees. Any units of length can be used, but they must be the same for H, R, and L. The liquid volume α V = LR2 ᎏ − sin α cos α (10-109) 57.30

冢

冣

This formula may be used for any depth of liquid between zero and the full tank, provided the algebraic signs are observed. If H is greater than R, sin α cos α will be negative and thus will add numerically to α/57.30. Table 10-64 gives liquid volume, for a partially filled horizontal cylinder, as a fraction of the total volume, for the dimensionless ratio H/D or H/2R. The volumes of heads must be calculated separately and added to the volume of the cylindrical portion of the tank. The four types of heads most frequently used are the standard dished head,* torispherical or ASME head, ellipsoidal head, and hemispherical head. Dimensions and volumes for all four of these types are given in Lukens Spun Heads, Lukens Inc., Coatesville, Pennsylvania. Approximate volumes can also be calculated by the formulas in Table 10-65. Consistent units must be used in these formulas. A partially filled horizontal tank requires the determination of the partial volume of the heads. The Lukens catalog gives approximate volumes for partially filled (axis horizontal) standard ASME and ellipsoidal heads. A formula for partially filled heads, by Doolittle [Ind. Eng. Chem., 21, 322–323 (1928)], is V = 0.215 H (3R − H) 2

(10-110)

where in consistent units V = volume, R = radius, and H = depth of liquid. Doolittle made some simplifying assumptions which affect the volume given by the equation, but the equation is satisfactory for * The standard dished head does not comply with the ASME Pressure Vessel Code.

FIG. 10-183 Calculation of partially filled horizontal tanks. H = depth of liquid; R = radius; D = diameter; L = length; α = half of the included angle; and cos α = 1 − H/R = 1 − 2H/D.

determining the volume as a fraction of the entire head. This fraction, calculated by Doolittle’s formula, is given in Table 10-66 as a function of H/Di (H is the depth of liquid, and Di is the inside diameter). Table 10-66 can be used for standard dished, torispherical, ellipsoidal, and hemispherical heads with an error of less than 2 percent of the volume of the entire head. The error is zero when H/Di = 0, 0.5, and 1.0. Table 10-66 cannot be used for conical heads. When a tank volume cannot be calculated or when greater precision is required, calibration may be necessary. This is done by draining (or filling) the tank and measuring the volume of liquid. The TABLE 10-64

Volume of Partially Filled Horizontal Cylinders

H/D

Fraction of volume

H/D

Fraction of volume

H/D

Fraction of volume

H/D

Fraction of volume

0.01 .02 .03 .04 .05

0.00169 .00477 .00874 .01342 .01869

0.26 .27 .28 .29 .30

0.20660 .21784 .22921 .24070 .25231

0.51 .52 .53 .54 .55

0.51273 .52546 .53818 .55088 .56356

0.76 .77 .78 .79 .80

0.81545 .82625 .83688 .84734 .85762

.06 .07 .08 .09 .10

.02450 .03077 .03748 .04458 .05204

.31 .32 .33 .34 .35

.26348 .27587 .28779 .29981 .31192

.56 .57 .58 .59 .60

.57621 .58884 .60142 .61397 .62647

.81 .82 .83 .84 .85

.86771 .87760 .88727 .89673 .90594

.11 .12 .13 .14 .15

.05985 .06797 .07639 .08509 .09406

.36 .37 .38 .39 .40

.32410 .33636 .34869 .36108 .37353

.61 .62 .63 .64 .65

.63892 .65131 .66364 .67590 .68808

.86 .87 .88 .89 .90

.91491 .92361 .93203 .94015 .94796

.16 .17 .18 .19 .20

.10327 .11273 .12240 .13229 .14238

.41 .42 .43 .44 .45

.38603 .39858 .41116 .42379 .43644

.66 .67 .68 .69 .70

.70019 .71221 .72413 .73652 .74769

.91 .92 .93 .94 .95

.95542 .96252 .96923 .97550 .98131

.21 .22 .23 .24 .25

.15266 .16312 .17375 .18455 .19550

.46 .47 .48 .49 .50

.44912 .46182 .47454 .48727 .50000

.71 .72 .73 .74 .75

.75930 .77079 .78216 .79340 .80450

.96 .97 .98 .99 1.00

.98658 .99126 .99523 .99831 1.00000

10-140

TRANSPORT AND STORAGE OF FLUIDS TABLE 10-65

Volumes of Heads*

Type of head

Knuckle radius, rk

Standard dished

Approx. 3t

Torispherical or A.S.M.E. Torispherical or A.S.M.E. Ellipsoidal Ellipsoidal Hemispherical Conical

h

L

Volume

Approx. Di

0.06L

Di

Approx. 0.050D i3 + 1.65tD2i 0.0809D 3i

3t

Di

Approx. 0.513hD 2i

Di /4 Di /2

Di /2

πD 2i h/6 πD 3i /24 πD 3i /12 πh(D 2i + Did + d 2)/12

% Error

Remarks

⫾10 ⫾0.1 ⫾8 0 0 0 0

h varies with t

冧

rk must be the larger of 0.06L and 3t

Standard proportions Truncated cone h = height d = diameter at small end

*Use consistent units.

measurement may be made by weighing, by a calibrated fluid meter, or by repeatedly filling small measuring tanks which have been calibrated by weight. Container Materials and Safety Storage tanks are made of almost any structural material. Steel and reinforced concrete are most widely used. Plastics and glass-reinforced plastics are used for tanks up to about 230 m3 (60,000 gal). Resistance to corrosion, light weight, and lower cost are their advantages. Plastic and glass coatings are also applied to steel tanks. Aluminum and other nonferrous metals are used when their special properties are required. When expensive metals such as tantalum are required, they may be applied as tank linings or as clad metals. Some grades of steel listed by API and AWWA Standards are of lower quality than is customarily used for pressure vessels. The stresses allowed by these standards are also higher than those allowed by the ASME Pressure Vessel Code. Small tanks containing nontoxic substances are not particularly hazardous and can tolerate a reduced factor of safety. Tanks containing highly toxic substances and very large tanks containing any substance can be hazardous. The designer must consider the magnitude of the hazard. The possibility of brittle behavior of ferrous metal should be taken into account in specifying materials (see subsection “Safety in Design”). TABLE 10-66 Volume of Partially Filled Heads on Horizontal Tanks* H/Di

Fraction of volume

H/Di

Fraction of volume

H/Di

Fraction of volume

H/Di

Fraction of volume

0.02 .04 .06 .08 .10

0.0012 .0047 .0104 .0182 .0280

0.28 .30 .32 .34 .36

0.1913 .216 .242 .268 .295

0.52 .54 .56 .58 .60

0.530 .560 .590 .619 .648

0.78 .80 .82 .84 .86

0.8761 .8960 .9145 .9314 .9467

.12 .14 .16 .18 .20

.0397 .0533 .0686 .0855 .1040

.38 .40 .42 .44 .46

.323 .352 .381 .410 .440

.62 .64 .66 .68 .70

.677 .705 .732 .758 .784

.88 .90 .92 .94 .96

.9603 .9720 .9818 .9896 .9953

.22 .24 .26

.1239 .1451 .1676

.48 .50

.470 .500

.72 .74 .76

.8087 .8324 .8549

.98 1.00

.9988 1.0000

*Based on Eq. (10-110).

Volume 1 of National Fire Codes (National Fire Protection Association, Quincy, Massachusetts) contains recommendations (Code 30) for venting, drainage, and dike construction of tanks for flammable liquids. Container Insulation Tanks containing materials above atmospheric temperature may require insulation to reduce loss of heat. Almost any of the commonly used insulating materials can be employed. Calcium silicate, glass fiber, mineral wool, cellular glass, and plastic foams are among those used. Tanks exposed to weather must have jackets or protective coatings, usually asphalt, to keep water out of the insulation. Tanks with contents at lower than atmospheric temperature may require insulation to minimize heat absorption. The insulation must have a vapor barrier at the outside to prevent condensation of atmospheric moisture from reducing its effectiveness. An insulation not damaged by moisture is preferable. The insulation techniques presently used for refrigerated systems can be applied (see subsection “Low-Temperature and Cryogenic Storage”). Tank Supports Large vertical atmospheric steel tanks may be built on a base of about 150 cm (6 in) of sand, gravel, or crushed stone if the subsoil has adequate bearing strength. It can be level or slightly coned, depending on the shape of the tank bottom. The porous base provides drainage in case of leaks. A few feet beyond the tank perimeter the surface should drop about 1 m (3 ft) to assure proper drainage of the subsoil. API Standard 650, Appendix B, and API Standard 620, Appendix C, give recommendations for tank foundations. The bearing pressure of the tank and contents must not exceed the bearing strength of the soil. Local building codes usually specify allowable soil loading. Some approximate bearing values are:

Soft clay (can be crumbled between fingers) Dry fine sand Dry fine sand with clay Coarse sand Dry hard clay (requires a pick to dig it) Gravel Rock

kPa

Tons/ft2

100 200 300 300 350 400 1000–4000

1 2 3 3 3.5 4 10–40

For high, heavy tanks, a foundation ring may be needed. Prestressed concrete tanks are sufficiently heavy to require foundation rings. Foundations must extend below the frost line. Some tanks that are not flat-bottomed may also be supported by soil if it is suitably

STORAGE AND PROCESS VESSELS graded and drained. When soil does not have adequate bearing strength, it may be excavated and backfilled with a suitable soil, or piles capped with a concrete mat may be required. Spheres, spheroids, and toroids use steel or concrete saddles or are supported by columns. Some may rest directly on soil. Horizontal cylindrical tanks should have two rather than multiple saddles to avoid indeterminate load distribution. Small horizontal tanks are sometimes supported by legs. Most tanks must be designed to resist the reactions of the saddles or legs, and they may require reinforcing. Neglect of this can cause collapse. Tanks without stiffeners usually need to make contact with the saddles on at least 2.1 rad (120°) of their circumference. An elevated steel tank may have either a circle of steel columns or a large central steel standpipe. Concrete tanks usually have concrete columns. Tanks are often supported by buildings. Pond and Underground Storage Low-cost liquid materials, if they will not be damaged by rain or atmospheric pollution, may be stored in ponds. A pond may be excavated or formed by damming a ravine. To prevent loss by seepage, the soil which will be submerged may require treatment to make it sufficiently impervious. This can also be accomplished by lining the pond with concrete, plastic film, or some other barrier. Prevention of seepage is especially necessary if the pond contains material that could contaminate present or future water supplies. Underground Storage Investment in both storage facilities and land can often be reduced by underground storage. Porous media between impervious rocks are also used. Cavities can be formed in salt domes and beds by dissolving the salt and pumping it out. Geological formations suitable for some of these methods can be found in numerous locations. The most extensive application has been the storage of petroleum products, both liquid and gaseous, in the southwestern part of the United States. Chemicals have been handled in this way. Information on some installations is given in articles by Billue, Haight and Bernard, and Nixon [Pet. Refiner, 33, 108–116 (1954)]. Another useful reference is Relationships between Selected Physical Parameters and Cost Responses for the Deep-Well Disposal of Aqueous Industrial Wastes, Technical Report to the U.S. Public Health Service, EHE 07-6801, CRWR28, by the Center for Research in Water Resources, University of Texas, Austin, August 1968. It contains an extensive bibliography. Water is also stored underground when suitable formations are available. When an excess of surface water is available part of the time, the excess is treated, if required, and pumped into the ground to be retrieved when needed. Sometimes pumping is unnecessary, and it will seep into the ground. Underground chambers are also constructed in frozen earth (see subsection “Low-Temperature and Cryogenic Storage”). Underground tunnel or tank storage is often the most practical way of storing hazardous or radioactive materials. A cover of 30 m (100 ft) of rock or dense earth can exert a pressure of about 690 kPa (100 lbf/in2). STORAGE OF GASES Gas Holders Gas is sometimes stored in expandable gas holders of either the liquid-seal or dry-seal type. The liquid-seal holder is a familiar sight. It has a cylindrical container, closed at the top, and varies its volume by moving it up and down in an annular water-filled seal tank. The seal tank may be staged in several lifts (as many as five). Seal tanks have been built in sizes up to 280,000 m3 (10 × 106 ft3). The dry-seal holder has a rigid top attached to the sidewalls by a flexible fabric diaphragm which permits it to move up and down. It does not involve the weight and foundation costs of the liquid-seal holder. Additional information on gas holders can be found in Gas Engineers Handbook, Industrial Press, New York, 1966. Solution of Gases in Liquids Certain gases will dissolve readily in liquids. In some cases in which the quantities are not large, this may be a practical storage procedure. Examples of gases that can be handled in this way are ammonia in water, acetylene in acetone, and hydrogen chloride in water. Whether or not this method is used depends mainly on whether the end use requires the anhydrous or the liquid state. Pressure may be either atmospheric or elevated. The

10-141

solution of acetylene in acetone is also a safety feature because of the instability of acetylene. Storage in Pressure Vessels, Bottles, and Pipe Lines The distinction between pressure vessels, bottles, and pipes is arbitrary. They can all be used for storing gases under pressure. A storage pressure vessel is usually a permanent installation. Storing a gas under pressure not only reduces its volume but also in many cases liquefies it at ambient temperature. Some gases in this category are carbon dioxide, several petroleum gases, chlorine, ammonia, sulfur dioxide, and some types of Freon. Pressure tanks are frequently installed underground. Liquefied petroleum gas (LPG) is the subject of API Standard 2510, The Design and Construction of Liquefied Petroleum Gas Installations at Marine and Pipeline Terminals, Natural Gas Processing Plants, Refineries, and Tank Farms. This standard in turn refers to: 1. National Fire Protection Association (NFPA) Standard 58, Standard for the Storage and Handling of Liquefied Petroleum Gases 2. NFPA Standard 59, Standard for the Storage and Handling of Liquefied Petroleum Gases at Utility Gas Plants 3. NFPA Standard 59A, Standard for the Production, Storage, and Handling of Liquefied Natural Gas (LNG) The API Standard gives considerable information on the construction and safety features of such installations. It also recommends minimum distances from property lines. The user may wish to obtain added safety by increasing these distances. The term bottle is usually applied to a pressure vessel that is small enough to be conveniently portable. Bottles range from about 57 L (2 ft3) down to CO2 capsules of about 16.4 mL (1 in3). Bottles are convenient for small quantities of many gases, including air, hydrogen, nitrogen, oxygen, argon, acetylene, Freon, and petroleum gas. Some are one-time-use disposable containers. Pipe Lines A pipe line is not ordinarily a storage device. Pipes, however, have been buried in a series of connected parallel lines and used for storage. This avoids the necessity of providing foundations, and the earth protects the pipe from extremes of temperature. The economics of such an installation would be doubtful if it were designed to the same stresses as a pressure vessel. Storage is also obtained by increasing the pressure in operating pipe lines and thus using the pipe volume as a tank. Low-Temperature and Cryogenic Storage This type is used for gases that liquefy under pressure at atmospheric temperature. In cryogenic storage the gas is at, or near to, atmospheric pressure and remains liquid because of low temperature. A system may also operate with a combination of pressure and reduced temperature. The term “cryogenic” usually refers to temperatures below −101°C (−150°F). Some gases, however, liquefy between −101°C and ambient temperatures. The principle is the same, but cryogenic temperatures create different problems with insulation and construction materials. The liquefied gas must be maintained at or below its boiling point. Refrigeration can be used, but the usual practice is to cool by evaporation. The quantity of liquid evaporated is minimized by insulation. The vapor may be vented to the atmosphere (wasteful), it may be compressed and reliquefied, or it may be used. At very low temperatures with liquid air and similar substances, the tank may have double walls with the interspace evacuated. The wellknown Dewar flask is an example. Large tanks and even pipe lines are now built this way. An alternative is to use double walls without vacuum but with an insulating material in the interspace. Perlite and plastic foams are two insulating materials employed in this way. Sometimes both insulation and vacuum are used. Materials Materials for liquefied-gas containers must be suitable for the temperatures, and they must not be brittle. Some carbon steels can be used down to −59°C (−75°F), and low-alloy steels to −101°C (−150°F) and sometimes −129°C (−200°F). Below these temperatures austenitic stainless steel (AISI 300 series) and aluminum are the principal materials. Low temperatures involve problems of differential thermal expansion. With the outer wall at ambient temperature and the inner wall at the liquid boiling point, relative movement must be accommodated. Some systems for accomplishing this are patented. The Gaz

10-142

TRANSPORT AND STORAGE OF FLUIDS

Transport of France reduces dimensional change by using a thin inner liner of Invar. Another patented French system accommodates this change by means of the flexibility of thin metal which is creased. The creases run in two directions, and the form of the crossings of the creases is a feature of the system. Low-temperature tanks may be installed underground to take advantage of the insulating value of the earth. Frozen-earth storage is also used. The frozen earth forms the tank. Some installations using this technique have been unsuccessful because of excessive heat absorption. COST OF STORAGE FACILITIES Contractors’ bids offer the most reliable information on cost. Orderof-magnitude costs, however, may be required for preliminary studies. One way of estimating them is to obtain cost information from similar facilities and scale it to the proposed installation. Costs of steel storage tanks and vessels have been found to vary approximately as the 0.6 to 0.7 power of their weight [see Happel, Chemical Process Economics, Wiley, 1958, p. 267; also Williams, Chem. Eng., 54(12), 124 (1947)]. All estimates based on the costs of existing equipment must be corrected for changes in the price index from the date when the equipment was built. Considerable uncertainty is involved in adjusting data more than a few years old. Based on a survey in 1994 for storage tanks, the prices for fielderected tanks are for multiple-tank installations erected by the contractor on foundations provided by the owner. Some cost information on tanks is given in various references cited in Sec. 25. Cost data vary considerably from one reference to another. Prestressed (posttensioned) concrete tanks cost about 20 percent more than steel tanks of the same capacity. Once installed, however, concrete tanks require very little maintenance. A true comparison with steel would, therefore, require evaluating the maintenance cost of both types. BULK TRANSPORT OF FLUIDS Transportation is often an important part of product cost. Bulk transportation may provide significant savings. When there is a choice between two or more forms of transportation, the competition may result in rate reduction. Transportation is subject to considerable regulation, which will be discussed in some detail under specific headings. Pipe Lines For quantities of fluid which an economic investigation indicates are sufficiently large and continuous to justify the investment, pipe lines are one of the lowest-cost means of transportation. They have been built up to 1.22 m (48 in) or more in diameter and about 3200 km (2000 mi) in length for oil, gas, and other products. Water is usually not transported more than 160 to 320 km (100 to 200 miles), but the conduits may be much greater than 1.22 m (48 in) in diameter. Open canals are also used for water transportation. Petroleum pipe lines before 1969 were built to ASA (now ANSI) Standard B31.4 for liquids and Standard B31.8 for gas. These standards were seldom mandatory because few states adopted them. The U.S. Department of Transportation (DOT), which now has responsibility for pipe-line regulation, issued Title 49, Part 192—Transportation of Natural Gas and Other Gas by Pipeline: Minimum Safety Standards, and Part 195—Transportation of Liquids by Pipeline. These contain considerable material from B31.4 and B31.8. They allow generally higher stresses than the ASME Pressure Vessel Code would allow for steels of comparable strength. The enforcement of their regulations is presently left to the states and is therefore somewhat uncertain. Pipe-line pumping stations usually range from 16 to 160 km (10 to 100 miles) apart, with maximum pressures up to 6900 kPa (1000 lbf/ in2) and velocities up to 3 m/s (10 ft/s) for liquid. Gas pipe lines have higher velocities and may have greater spacing of stations. Tanks Tank cars (single and multiple tank), tank trucks, portable tanks, drums, barrels, carboys, and cans are used to transport fluids (see Figs. 10-184–10-186). Interstate transportation is regulated by the DOT. There are other regulating agencies—state, local, and pri-

vate. Railroads make rules determining what they will accept, some states require compliance with DOT specifications on intrastate movements, and tunnel authorities as well as fire chiefs apply restrictions. Water shipments involve regulations of the U.S. Coast Guard. The American Bureau of Shipping sets rules for design and construction which are recognized by insurance underwriters. The most pertinent DOT regulations (Code of Federal Regulations, Title 18, Parts 171–179 and 397) were published by R. M. Graziano (then agent and attorney for carriers and freight forwarders) in his tariff titled Hazardous Materials Regulations of the Department of Transportation (1978). New tariffs identified by number are issued at intervals, and interim revisions are sent out. Agents change at intervals. Graziano’s tariff lists many regulated (dangerous) commodities (Part 172, DOT regulations) for transportation. This includes those that are poisonous, flammable, oxidizing, corrosive, explosive, radioactive, and compressed gases. Part 178 covers specifications for all types of containers from carboys to large portable tanks and tank trucks. Part 179 deals with tank-car construction. An Association of American Railroads (AAR) publication, Specifications for Tank Cars, covers many requirements beyond the DOT regulations. Some additional details are given later. Because of frequent changes, it is always necessary to check the latest rules. The shipper, not the carrier, has the ultimate responsibility for shipping in the correct container. Tank Cars These range in size from about 7.6 to 182 m3 (2000 to 48,000 gal), and a car may be single or multiunit. The DOT now limits them to 130 m3 (34,500 gal) and 120,000 kg (263,000 lb) gross mass. Large cars usually result in lower investment per cubic meter and take lower shipping rates. Cars may be insulated to reduce heating or cooling of the contents. Certain liquefied gases may be carried in insulated cars; temperatures are maintained by evaporation (see subsection “Low-Temperature and Cryogenic Storage”). Cars may be heated by steam coils or by electricity. Some products are loaded hot, solidify in transport, and are melted for removal. Some lowtemperature cargoes must be unloaded within a given time (usually 30 days) to prevent pressure buildup. Tank cars are classified as pressure or general-purpose. Pressure cars have relief-valve settings of 517 kPa (75 lbf/in2) and above. Those designated as general-purpose cars are, nevertheless, pressure vessels and may have relief valves or rupture disks. The DOT specification code number indicates the type of car. For instance, 105A500W indicates a pressure car with a test pressure of 3447 kPa (500 lbf/in2) and a relief-valve setting of 2585 kPa (375 lbf/in2). In most cases, loading and unloading valves, safety valves, and vent valves must be in a dome or an enclosure. Companies shipping dangerous materials sometimes build tank cars with metal thicker than required by the specifications in order to reduce the possibility of leakage during a wreck or fire. The punching of couplers or rail ends into heads of tanks is a hazard. Older tank cars have a center sill or beam running the entire length of the car. Most modern cars have no continuous sill, only short stub sills at each end. Cars with full sills have tanks anchored longitudinally at the center of the sill. The anchor is designed to be weaker than either the tank shell or the doubler plate between anchor and shell. Cars with stub sills have similar safeguards. Anchors and other parts are designed to meet AAR requirements. The impact forces on car couplers put high stresses in sills, anchors, and doublers. This may start fatigue cracks in the shell, particularly at the corners of welded doubler plates. With brittle steel in cold weather, such cracks sometimes cause complete rupture of the tank. Large end radii on the doublers and tougher steels will reduce this hazard. Inspection of older cars can reveal cracks before failure. A difference between tank cars and most pressure vessels is that tank cars are designed in terms of the theoretical ultimate or bursting strength of the tank. The test pressure is usually 40 percent of the bursting pressure (sometimes less). The safety valves are set at 75 percent of the test pressure. Thus, the maximum operating pressure is usually 30 percent of the bursting pressure. This gives a nominal factor of safety of 3.3, compared with 4.0 for Division 1 of the ASME Pressure Vessel Code.

STORAGE AND PROCESS VESSELS

FIG. 10-184 Cost of shop-fabricated tanks in mid-1980 with d-in walls. Multiplying factors on carbon steel costs for other materials are: carbon steel, 1.0; rubber-lined carbon steel, 1.5; aluminum, 1.6; glass-lined carbon steel, 4.5; and fiber-reinforced plastic, 0.75 to 1.5. Multiplying factors on type 316 stainlesssteel costs for other materials are: 316 stainless steel, 1.0; Monel, 2.0; Inconel, 2.0; nickel, 2.0; titanium, 3.2; and Hastelloy C, 3.8. Multiplying factors for wall thicknesses different from d in are:

Thickness, in

Carbon steel

304 stainless steel

316 stainless steel

a e 1

1.4 2.1 2.7

1.8 2.5 3.3

1.8 2.6 3.5

To convert gallons to cubic meters, multiply by 3.785 × 10−3.

The DOT rules require that pressure cars have relief valves designed to limit pressure to 82.5 percent (with certain exceptions) of test pressure (110 percent of maximum operating pressure) when exposed to fire. Appendix A of AAR Specifications deals with the flow capacity of relief devices. The formulas apply to cars in the upright position with the device discharging vapor. They may not protect the car adequately when it is overturned and the device is discharging liquid. Appendix B of AAR Specifications deals with the certification of facilities. Fabrication, repairing, testing, and specialty work on tank cars must be done in certified facilities. The AAR certifies shops to build cars of certain materials, to do test work on cars, or to make certain repairs and alterations. Tank Trucks These trucks may have single, compartmented, or multiple tanks. Many of their requirements are similar to those for tank cars, except that thinner shells are permitted in most cases. Trucks for nonhazardous materials are subject to few regulations other than the normal highway laws governing all motor vehicles. But trucks carrying hazardous materials must comply with DOT regulations, Parts 173, 177, 178, and 397. Maximum weight, axle loading, and length are governed by state highway regulations. Many states have limits in the vicinity of 31,750 kg (70,000 lb) total mass, 14,500 kg (32,000 lb) for tandem axles, and 18.3 m (60 ft) or less overall length. Some allow tandem trailers. Truck cargo tanks (for dangerous materials) are built under Part 173 and Subpart J of Part 178, DOT regulations. This includes Specifications MC-306, MC-307, MC-312, and MC-331. MC-331 is required for compressed gas. Subpart J requires tanks for pressures above 345 kPa (50 lbf/in2) in one case and 103 kPa (15 lbf/in2) in another to be built according to the ASME Pressure Vessel Code. A particular issue of the code is specified.

10-143

FIG. 10-185 Cost of small field-erected tanks in mid-1980, including stairs, platforms, and a normal complement of nozzles. The carbon steel curve is for API Standard 650 tanks, and the others are for stainless-steel tanks for atmospheric pressure with flat bottoms. The curves are for tanks purchased in quantities of three or more at a Gulf Coast site. Multiplying factors for other materials are: 316 stainless steel, 1.5; Monel, 2.0; Inconel, 2.0; nickel, 2.0; titanium, 3.2; and Hastelloy C, 3.8. Allowances should be added to the factored costs as follows: 10 percent for stiffener rings, 20 percent for API Standard 620, 15 percent for quantity of one tank, 10 percent for quantity of two tanks, 15 percent of steel cost for congested working area, 50 percent of steel cost for an integral steel dike. To convert gallons to cubic meters, multiply by 3.785 × 10−3.

Because of the demands of highway service, the DOT specifications have a number of requirements in addition to the ASME Code. These include design for impact forces and rollover protection for fittings. Portable tanks, drums, or bottles are shipped by rail, ship, air, or truck. Portable tanks containing hazardous materials must conform to DOT regulations, Parts 173 and 178, Subpart H. Some tanks are designed to be shipped by trailer and transferred to railcars or ships (see following discussion). Marine Transportation Seagoing tankers are for high tonnage. The traditional tanker uses the ship structure as a tank. It is subdivided into a number of tanks by means of transverse bulkheads and a centerline bulkhead. More than one product can be carried. An elaborate piping system connects the tanks to a pumping plant which can discharge or transfer the cargo. Harbor and docking facilities appear to be the only limit to tanker size. The largest tanker size to date is about 500,000 deadweight tons. In the United States, tankers are built to specifications of the American Bureau of Shipping and the U.S. Coast Guard. Low-temperature liquefied gases are shipped in special ships with insulation between the hull and an inner tank. Poisonous materials are shipped in separate tanks built into the ship. This prevents tank leakage from contaminating harbors. Separate tanks are also used to transport pressurized gases. Barges are used on inland waterways. Popular sizes are up to 16 m (52a) wide by 76 m (250 ft) long, with 2.6 m (8a ft) to 4.3 m (14 ft) draft. Cargo requirements and waterway limitations determine design. Use of barges of uniform size facilitates rafting them together.

10-144

TRANSPORT AND STORAGE OF FLUIDS

FIG. 10-186 Cost of large field-erected tanks in mid-1990, including stairs, platforms, and a normal complement of nozzles; curve for carbon steel API Standard 650 tanks in quantities of three or more at a Gulf Coast site. For type 304 stainless steel, multiply cost by 2.5; and for type 316 stainless steel, multiply cost by 3.5. Allowances should be added to the factored costs as follows: 10 percent for stiffener rings, 20 percent for API Standard 620, 15 percent for quantity of one tank, 10 percent for quantity of two tanks, and 15 percent of carbon steel cost for a congested working area. To convert gallons to cubic meters, multiply by 3.785 × 10−3.

Portable tanks may be stowed in the holds of conventional cargo ships or special container ships, or they may be fastened on deck. Container ships have guides in the hold and on deck which hold boxlike containers or tanks. The tank is latched to a trailer chassis and hauled to shipside. A movable gantry, sometimes permanently installed on the ship, hoists the tank from the trailer and lowers it into the guides on the ship. This system achieves large savings in labor, but its application is sometimes limited by lack of agreement between ship operators and unions. Portable tanks for regulated commodities in marine transportation must be designed and built under Coast Guard regulations (see discussion under “Pressure Vessels”). Materials of Construction for Bulk Transport Because of the more severe service, construction materials for transportation usually are more restricted than for storage. Most large pipe lines are constructed of steel conforming to API Specification 5L or 5LX. Most tanks (cars, etc.) are built of pressure-vessel steels or AAR specification steels, with a few of aluminum or stainless steel. Carbon steel tanks may be lined with rubber, plastic, nickel, glass, or other materials. In many cases this is practical and cheaper than using a stainlesssteel tank. Other materials for tank construction may be proposed and used if approved by the appropriate authorities (AAR and DOT). PRESSURE VESSELS This discussion of pressure vessels is intended as an overview of the codes most frequently used for the design and construction of pressure vessels. Chemical engineers who design or specify pressure vessels should determine the federal and local laws relevant to the problem and then refer to the most recent issue of the pertinent code or standard before proceeding. Laws, codes, and standards are frequently changed. A pressure vessel is a closed container of limited length (in contrast to the indefinite length of piping). Its smallest dimension is considerably larger than the connecting piping, and it is subject to pressures above 7 or 14 kPa (1 or 2 lbf/in2). It is distinguished from a boiler, which in most cases is used to generate steam for use external to itself.

Code Administration The American Society of Mechanical Engineers has written the ASME Boiler and Pressure Vessel Code, which contains rules for the design, fabrication, and inspection of boilers and pressure vessels. The ASME Code is an American National Standard. Most states in the United States and all Canadian provinces have passed legislation which makes the ASME Code or certain parts of it their legal requirement. Only a few jurisdictions have adopted the code for all vessels. The others apply it to certain types of vessels or to boilers. States employ inspectors (usually under a chief boiler inspector) to enforce code provisions. The authorities also depend a great deal on insurance company inspectors to see that boilers and pressure vessels are maintained in a safe condition. The ASME Code is written by a large committee and many subcommittees, composed of engineers appointed by the ASME. The Code Committee meets regularly to review the code and consider requests for its revision, interpretation, or extension. Interpretation and extension are accomplished through “code cases.” The decisions are published in Mechanical Engineering. Code cases are also mailed to those who subscribe to the service. A typical code case might be the approval of the use of a metal which is not presently on the list of approved code materials. Inquiries relative to code cases should be addressed to the secretary of the ASME Boiler and Pressure Vessel Committee, American Society of Mechanical Engineers, New York. A new edition of the code is issued every 3 years. Between editions, alterations are handled by issuing semiannual addenda, which may be purchased by subscription. The ASME considers any issue of the code to be adequate and safe, but some government authorities specify certain issues of the code as their legal requirement. Inspection Authority The National Board of Boiler and Pressure Vessel Inspectors is composed of the chief inspectors of states and municipalities in the United States and Canadian provinces which have made any part of the Boiler and Pressure Vessel Code a legal requirement. This board promotes uniform enforcement of boiler and pressure-vessel rules. One of the board’s important activities is providing examinations for, and commissioning of, inspectors. Inspectors so qualified and employed by an insurance company, state, municipality, or Canadian province may inspect a pressure vessel and permit it to be stamped ASME—NB (National Board). An inspector employed by a vessel user may authorize the use of only the ASME stamp. The ASME Code Committee authorizes fabricators to use the various ASME stamps. The stamps, however, may be applied to a vessel only with the approval of the inspector. The ASME Boiler and Pressure Vessel Code consists of eleven sections as follows: I. Power Boilers II. Materials a. Ferrous b. Nonferrous c. Welding rods, electrodes, and filler metals d. Properties III. Rules for Construction of Nuclear Power Plant Components IV. Heating Boilers V. Nondestructive Examination VI. Rules for Care and Operation of Heating Boilers VII. Guidelines for the Care of Power Boilers VIII. Pressure Vessels IX. Welding and Brazing Qualifications X. Fiber-Reinforced Plastic Pressure Vessels XI. Rules for Inservice Inspection of Nuclear Power Plant Components Pressure vessels (as distinguished from boilers) are involved with Secs. II, III, V, VIII, IX, X, and XI. Section VIII, Division I, is the Pressure Vessel Code as it existed in the past (and will continue). Division 2 was brought out as a means of permitting higher design stresses while ensuring at least as great a degree of safety as in Division 1. These two divisions plus Secs. III and X will be discussed briefly here. They refer to Secs. II and IX. ASME Code Section VIII, Division 1 Most pressure vessels used in the process industry in the United States are designed and constructed in accordance with Sec. VIII, Division 1 (see Fig. 10-187). This division is divided into three subsections followed by appendixes.

STORAGE AND PROCESS VESSELS

FIG. 10-187

ASME Code Sec. VIII, Division 1: applicable paragraphs for design and construction details. (Courtesy of Missouri Boiler and Tank Co.).

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Introduction The Introduction contains the scope of the division and defines the responsibilities of the user, the manufacturer, and the inspector. The scope defines pressure vessels as containers for the containment of pressure. It specifically excludes vessels having an internal pressure not exceeding 103 kPa (15 lbf/in2) and further states that the rules are applicable for pressures not exceeding 20,670 kPa (3000 lbf/in2). For higher pressures it is usually necessary to deviate from the rules in this division. The scope covers many other less basic exclusions, and inasmuch as the scope is occasionally revised, except for the most obvious cases, it is prudent to review the current issue before specifying or designing pressure vessels to this division. Any vessel which meets all the requirements of this division may be stamped with the code U symbol even though exempted from such stamping. Subsection A This subsection contains the general requirements applicable to all materials and methods of construction. Design temperature and pressure are defined here, and the loadings to be considered in design are specified. For stress failure and yielding, this section of the code uses the maximum-stress theory of failure as its criterion. This subsection refers to the tables elsewhere in the division in which the maximum allowable tensile-stress values are tabulated. The basis for the establishment of these allowable stresses is defined in detail in Appendix P; however, as the safety factors used were very important in establishing the various rules of this division, it is noted that the safety factors for internal-pressure loads are 4 on ultimate strength and 1.6 or 1.5 on yield strength, depending on the material. For external-pressure loads on cylindrical shells, the safety factors are 3 for both elastic buckling and plastic collapse. For other shapes subject to external pressure and for longitudinal shell compression, the safety factors are 4 for both elastic buckling and plastic collapse. Longitudinal compressive stress in cylindrical elements is limited in this subsection by the lower of either stress failure or buckling failure. Internal-pressure design rules and formulas are given for cylindrical and spherical shells and for ellipsoidal, torispherical (often called ASME heads), hemispherical, and conical heads. The formulas given assume membrane-stress failure, although the rules for heads include consideration for buckling failure in the transition area from cylinder to head (knuckle area). Longitudinal joints in cylinders are more highly stressed than circumferential joints, and the code takes this fact into account. When forming heads, there is usually some thinning from the original plate thickness in the knuckle area, and it is prudent to specify the minimum allowable thickness at this point. Unstayed flat heads and covers can be designed by very specific rules and formulas given in this subsection. The stresses caused by pressure on these members are bending stresses, and the formulas include an allowance for additional edge moments induced when the head, cover, or blind flange is attached by bolts. Rules are provided for quick-opening closures because of the risk of incomplete attachment or opening while the vessel is pressurized. Rules for braced and stayed surfaces are also provided. External-pressure failure of shells can result from overstress at one extreme or from elastic instability at the other or at some intermediate loading. The code provides the solution for most shells by using a number of charts. One chart is used for cylinders where the shell diameter-to-thickness ratio and the length-to-diameter ratio are the variables. The rest of the charts depict curves relating the geometry of cylinders and spheres to allowable stress by curves which are determined from the modulus of elasticity, tangent modulus, and yield strength at temperatures for various materials or classes of materials. The text of this subsection explains how the allowable stress is determined from the charts for cylinders, spheres, and hemispherical, ellipsoidal, torispherical, and conical heads. Frequently cost savings for cylindrical shells can result from reducing the effective length-to-diameter ratio and thereby reducing shell thickness. This can be accomplished by adding circumferential stiffeners to the shell. Rules are included for designing and locating the stiffeners. Openings are always required in pressure-vessel shells and heads. Stress intensification is created by the existence of a hole in an other-

wise symmetrical section. The code compensates for this by an areareplacement method. It takes a cross section through the opening, and it measures the area of the metal of the required shell that is removed and replaces it in the cross section by additional material (shell wall, nozzle wall, reinforcing plate, or weld) within certain distances of the opening centerline. These rules and formulas for calculation are included in Subsec. A. When a cylindrical shell is drilled for the insertion of multiple tubes, the shell is significantly weakened and the code provides rules for tube-hole patterns and the reduction in strength that must be accommodated. Fabrication tolerances are covered in this subsection. The tolerances permitted for shells for external pressure are much closer than those for internal pressure because the stability of the structure is dependent on the symmetry. Other paragraphs cover repair of defects during fabrication, material identification, heat treatment, and impact testing. Inspection and testing requirements are covered in detail. Most vessels are required to be hydrostatic-tested (generally with water) at 1a times the maximum allowable working pressure. Some enameled (glass-lined) vessels are permitted to be hydrostatic-tested at lower pressures. Pneumatic tests are permitted and are carried to at least 1d times the maximum allowable working pressure, and there is provision for proof testing when the strength of the vessel or any of its parts cannot be computed with satisfactory assurance of accuracy. Pneumatic or proof tests are rarely conducted. Pressure-relief-device requirements are defined in Subsec. A. Set point and maximum pressure during relief are defined according to the service, the cause of overpressure, and the number of relief devices. Safety, safety relief, relief valves, rupture disk, breaking pin, and rules on tolerances for the relieving point are given. Testing, certification, and installation rules for relieving devices are extensive. Every chemical engineer responsible for the design or operation of process units should become very familiar with these rules. The pressure-relief-device paragraphs are the only parts of Sec. VIII, Division 1, that are concerned with the installation and ongoing operation of the facility; all other rules apply only to the design and manufacture of the vessel. Subsection B This subsection contains rules pertaining to the methods of fabrication of pressure vessels. Part UW is applicable to welded vessels. Service restrictions are defined. Lethal service is for “lethal substances,” which are defined as poisonous gases or liquids of such a nature that a very small amount of the gas or the vapor of the liquid mixed or unmixed with air is dangerous to life when inhaled. It is stated that it is the user’s responsibility to advise the designer or manufacturer if the service is lethal. All vessels in lethal service shall have all butt-welded joints fully radiographed, and when practical, joints shall be butt-welded. All vessels fabricated of carbon or lowalloy steel shall be postweld-heat-treated. Low-temperature service is defined as being below −29°C (−20°F), and impact testing of many materials is required. The code is restrictive in the type of welding permitted. Unfired steam boilers with design pressures exceeding 345 kPa (50 lbf/in2) have restrictive rules on welded-joint design, and all butt joints require full radiography. Pressure vessels subject to direct firing have special requirements relative to welded-joint design and postweld heat treatment. This subsection includes rules governing welded-joint designs and the degree of radiography, with efficiencies for welded joints specified as functions of the quality of joint. These efficiencies are used in the formulas in Subsec. A for determining vessel thicknesses. Details are provided for head-to-shell welds, tube sheet-to-shell welds, and nozzle-to-shell welds. Acceptable forms of welded staybolts and plug and slot welds for staying plates are given here. Rules for the welded fabrication of pressure vessels cover welding processes, manufacturer’s record keeping on welding procedures, welder qualification, cleaning, fit-up alignment tolerances, and repair of weld defects. Procedures for postweld heat treatment are detailed. Checking the procedures and welders and radiographic and ultrasonic examination of welded joints are covered. Requirements for vessels fabricated by forging in Part UF include

STORAGE AND PROCESS VESSELS unique design requirements with particular concern for stress risers, fabrication, heat treatment, repair of defects, and inspection. Vessels fabricated by brazing are covered in Part UB. Brazed vessels cannot be used in lethal service, for unfired steam boilers, or for direct firing. Permitted brazing processes as well as testing of brazed joints for strength are covered. Fabrication and inspection rules are also included. Subsection C This subsection contains requirements pertaining to classes of materials. Carbon and low-alloy steels are governed by Part UCS, nonferrous materials by Part UNF, high-alloy steels by Part UHA, and steels with tensile properties enhanced by heat treatment by Part UHT. Each of these parts includes tables of maximum allowable stress values for all code materials for a range of metal temperatures. These stress values include appropriate safety factors. Rules governing the application, fabrication, and heat treatment of the vessels are included in each part. Part UHT also contains more stringent details for nozzle welding that are required for some of these high-strength materials. Part UCI has rules for cast-iron construction, Part UCL has rules for welded vessels of clad plate as lined vessels, and Part UCD has rules for ductile-iron pressure vessels. A relatively recent addition to the code is Part ULW, which contains requirements for vessels fabricated by layered construction. This type of construction is most frequently used for high pressures, usually in excess of 13,800 kPa (2000 lbf/in2). There are several methods of layering in common use: (1) thick layers shrunk together; (2) thin layers, each wrapped over the other and the longitudinal seam welded by using the prior layer as backup; and (3) thin layers spirally wrapped. The code rules are written for either thick or thin layers. Rules and details are provided for all the usual welded joints and nozzle reinforcement. Supports for layered vessels require special consideration, in that only the outer layer could contribute to the support. For lethal service only the inner shell and inner heads need comply with the requirements in Subsec. B. Inasmuch as radiography would not be practical for inspection of many of the welds, extensive use is made of magnetic-particle and ultrasonic inspection. When radiography is required, the code warns the inspector that indications sufficient for rejection in single-wall vessels may be acceptable. Vent holes are specified through each layer down to the inner shell to prevent buildup of pressure between layers in the event of leakage at the inner shell. Mandatory Appendixes These include a section on supplementary design formulas for shells not covered in Subsec. A. Formulas are given for thick shells, heads, and dished covers. Another appendix gives very specific rules, formulas, and charts for the design of boltedflange connections. The nature of these rules is such that they are readily programmable for a digital computer, and most flanges now are designed by using computers. One appendix includes only the charts used for calculating shells for external pressure discussed previously. Jacketed vessels are covered in a separate appendix in which very specific rules are given, particularly for the attachment of the jacket to the inner shell. Other appendixes cover inspection and quality control. Nonmandatory Appendixes These cover a number of subjects, primarily suggested good practices and other aids in understanding the code and in designing with the code. Several current nonmandatory appendixes will probably become mandatory. Figure 10-188 illustrates a pressure vessel with the applicable code paragraphs noted for the various elements. Additional important paragraphs are referenced at the bottom of the figure. ASME Code Section VIII, Division 2 Paragraph A-100e of Division 2 states: “In relation to the rules of Division 1 of Section VIII, these rules of Division 2 are more restrictive in the choice of materials which may be used but permit higher design stress intensity values to be employed in the range of temperatures over which the design stress intensity value is controlled by the ultimate strength or the yield strength; more precise design procedures are required and some common design details are prohibited; permissible fabrication procedures are specifically delineated and more complete testing and inspection are required.” Most Division 2 vessels fabricated to date have been large or intended for high pressure and, therefore, expensive when

10-147

ASME Code Sec. VIII, Division 1: applicable paragraphs for design and construction details. (Courtesy of Missouri Boiler and Tank Co.)

FIG. 10-188

the material and labor savings resulting from smaller safety factors have been greater than the additional engineering, administrative, and inspection costs. The organization of Division 2 differs from that of Division 1. Part A This part gives the scope of the division, establishes its jurisdiction, and sets forth the responsibilities of the user and the manufacturer. Of particular importance is the fact that no upper limitation in pressure is specified and that a user’s design specification is required. The user or the user’s agent shall provide requirements for intended operating conditions in such detail as to constitute an adequate basis for selecting materials and designing, fabricating, and inspecting the vessel. The user’s design specification shall include the method of supporting the vessel and any requirement for a fatigue analysis. If a fatigue analysis is required, the user must provide information in sufficient detail so that an analysis for cyclic operation can be made. Part AM This part lists permitted individual construction materials, applicable specifications, special requirements, design stress-intensity values, and other property information. Of particular importance are the ultrasonic-test and toughness requirements. Among the properties for which data are included are thermal conductivity and diffusivity, coefficient of thermal expansion, modulus of elasticity, and yield strength. The design stress-intensity values include a safety factor of 3 on ultimate strength at temperature or 1.5 on yield strength at temperature. Part AD This part contains requirements for the design of vessels. The rules of Division 2 are based on the maximum-shear theory of failure for stress failure and yielding. Higher stresses are permitted when wind or earthquake loads are considered. Any rules for determining the need for fatigue analysis are given here. Rules for the design of shells of revolution under internal pressure differ from the Division 1 rules, particularly the rules for formed heads when plastic deformation in the knuckle area is the failure criterion. Shells of revolution for external pressure are determined on the same criterion, including safety factors, as in Division 1. Reinforcement for openings uses the same area-replacement method as Division 1; however, in many cases the reinforcement metal must be closer to the opening centerline. The rest of the rules in Part AD for flat heads, bolted and studded connections, quick-actuating closures, and layered vessels essentially duplicate Division 1. The rules for support skirts are more definitive in Division 2.

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Part AF This part contains requirements governing the fabrication of vessels and vessel parts. Part AR This part contains rules for pressure-relieving devices. Part AI This part contains requirements controlling inspection of vessel. Part AT This part contains testing requirements and procedures. Part AS This part contains requirements for stamping and certifying the vessel and vessel parts. Appendixes Appendix 1 defines the basis used for defining stress-intensity values. Appendix 2 contains external-pressure charts, and Appendix 3 has the rules for bolted-flange connections; these two are exact duplicates of the equivalent appendixes in Division 1. Appendix 4 gives definitions and rules for stress analysis for shells, flat and formed heads, and tube sheets, layered vessels, and nozzles including discontinuity stresses. Of particular importance are Table 4-120.1, “Classification of Stresses for Some Typical Cases,” and Fig. 4-130.1, “Stress Categories and Limits of Stress Intensity.” These are very useful in that they clarify a number of paragraphs and simplify stress analysis. Appendix 5 contains rules and data for stress analysis for cyclic operation. Except in short-cycle batch processes, pressure vessels are usually subject to few cycles in their projected lifetime, and the endurance-limit data used in the machinery industries are not applicable. Curves are given for a broad spectrum of materials, covering a range from 10 to 1 million cycles with allowable stress values as high as 650,000 lbf/in2. This low-cycle fatigue has been developed from strain-fatigue work in which stress values are obtained by multiplying the strains by the modulus of elasticity. Stresses of this magnitude cannot occur, but strains do. The curves given have a factor of safety of 2 on stress or 20 on cycles. Appendix 6 contains requirements of experimental stress analysis, Appendix 8 has acceptance standards for radiographic examination, Appendix 9 covers nondestructive examination, Appendix 10 gives rules for capacity conversions for safety valves, and Appendix 18 details quality-control-system requirements. The remaining appendixes are nonmandatory but useful to engineers working with the code. General Considerations Most pressure vessels for the chemical-process industry will continue to be designed and built to the rules of Sec. VIII, Division 1. While the rules of Sec. VIII, Division 2, will frequently provide thinner elements, the cost of the engineering analysis, stress analysis and higher-quality construction, material control, and inspection required by these rules frequently exceeds the savings from the use of thinner walls. Additional ASME Code Considerations ASME Code Sec. III: Nuclear Power Plant Components This section of the code includes vessels, storage tanks, and concrete containment vessels as well as other nonvessel items. ASME Code Sec. X: Fiberglass–Reinforced-Plastic Pressure Vessels This section is limited to four types of vessels: bag-molded and centrifugally cast, each limited to 1,000 kPa (150 lbf/in2); filamentwound with cut filaments limited to 10,000 kPa (1500 lbf/in2); and filament-wound with uncut filaments limited to 21,000 kPa (3000 lbf/ in2). Operating temperatures are limited to the range from +66°C (150°F) to −54°C (−65°F). Low modulus of elasticity and other property differences between metal and plastic required that many of the procedures in Sec. X be different from those in the sections governing metal vessels. The requirement that at least one vessel of a particular design and fabrication shall be tested to destruction has prevented this section from being widely used. The results from the combined fatigue and burst test must give the design pressure a safety factor of 6 to the burst pressure. Safety in Design Designing a pressure vessel in accordance with the code will, under most circumstances, provide adequate safety. In the code’s own words, however, the rules “cover minimum construction requirements for the design, fabrication, inspection, and certification of pressure vessels.” The significant word is “minimum.” The ultimate responsibility for safety rests with the user and the designer. They must decide whether anything beyond code require-

ments is necessary. The code cannot foresee and provide for all the unusual conditions to which a pressure vessel might be exposed. If it tried to do so, the majority of pressure vessels would be unnecessarily restricted. Some of the conditions that a vessel might encounter are unusually low temperatures, unusual thermal stresses, stress ratcheting caused by thermal cycling, vibration of tall vessels excited by von Karman vortices caused by wind, very high pressures, runaway chemical reactions, repeated local overheating, explosions, exposure to fire, exposure to materials that rapidly attack the metal, containment of extremely toxic materials, and very large sizes of vessels. Large vessels, although they may contain nonhazardous materials, could, by their very size, create a serious hazard if they burst. The failure of the Boston molasses tank in 1919 killed 12 people. For pressure vessels which are outside code jurisdiction, there are sometimes special hazards in very-high-strength materials and plastics. There may be many others which the designers should recognize if they encounter them. Metal fatigue, when it is present, is a serious hazard. Section VIII, Division 1, mentions rapidly fluctuating pressures. Division 2 and Sec. III do require a fatigue analysis. In extreme cases vessel contents may affect the fatigue strength (endurance limit) of the material. This is corrosion fatigue. Although most ASME Code materials are not particularly sensitive to corrosion fatigue, even they may suffer an endurance limit loss of 50 percent in some environments. Highstrength heat-treated steels, on the other hand, are very sensitive to corrosion fatigue. It is not unusual to find some of these which lose 75 percent of their endurance in corrosive environments. In fact, in corrosion fatigue many steels do not have an endurance limit. The curve of stress versus cycles to failure (S/N curve) continues to slope downward regardless of the number of cycles. Brittle fracture is probably the most insidious type of pressurevessel failure. Without brittle fracture, a pressure vessel could be pressurized approximately to its ultimate strength before failure. With brittle behavior some vessels have failed well below their design pressures (which are about 25 percent of the theoretical bursting pressures). In order to reduce the possibility of brittle behavior, Division 2 and Sec. III require impact tests. The subject of brittle fracture has been understood only since about 1950, and knowledge of some of its aspects is still inadequate. A notched or cracked plate of pressure-vessel steel, stressed at 66°C (150°F), would elongate and absorb considerable energy before breaking. It would have a ductile or plastic fracture. As the temperature is lowered, a point is reached at which the plate would fail in a brittle manner with a flat fracture surface and almost no elongation. The transition from ductile to brittle fracture actually takes place over a temperature range, but a point in this range is selected as the transition temperature. One of the ways of determining this temperature is the Charpy impact test (see ASTM Specification E-23). After the transition temperature has been determined by laboratory impact tests, it must be correlated with service experience on full-size plates. The literature on brittle fracture contains information on the relation of impact tests to service experience on some carbon steels. A more precise but more elaborate method of dealing with the ductile-brittle transition is the fracture-analysis diagram. This uses a transition known as the nil-ductility temperature (NDT), which is determined by the drop-weight test (ASTM Standard E208) or the drop-weight tear test (ASTM Standard E436). The application of this diagram is explained in two papers by Pellini and Puzak (Trans. Am. Soc. Mech. Eng., 429 (October 1964); Welding Res. Counc. Bull. 88, 1963. Section VIII, Division 1, is rather lax with respect to brittle fracture. It allows the use of many steels down to −29°C (−20°F) without a check on toughness. Occasional brittle failures show that some vessels are operating below the nil-ductility temperature, i.e., the lower limit of ductility. Division 2 has resolved this problem by requiring impact tests in certain cases. Tougher grades of steel, such as the SA516 steels (in preference to SA515 steel), are available for a small price premium. Stress relief, steel made to fine-grain practice, and normalizing all reduce the hazard of brittle fracture. Nondestructive testing of both the plate and the finished vessel is important to safety. In the analysis of fracture hazards, it is important to know the size of the flaws that may be present in the completed ves-

STORAGE AND PROCESS VESSELS sel. The four most widely used methods of examination are radiographic, magnetic-particle, liquid-penetrant, and ultrasonic. Radiographic examination is either by x-rays or by gamma radiation. The former has greater penetrating power, but the latter is more portable. Few x-ray machines can penetrate beyond 300-mm (12-in) thickness. Ultrasonic techniques use vibrations with a frequency between 0.5 and 20 MHz transmitted to the metal by a transducer. The instrument sends out a series of pulses. These show on a cathode-ray screen as they are sent out and again when they return after being reflected from the opposite side of the member. If there is a crack or an inclusion along the way, it will reflect part of the beam. The initial pulse and its reflection from the back of the member are separated on the screen by a distance which represents the thickness. The reflection from a flaw will fall between these signals and indicate its magnitude and position. Ultrasonic examination can be used for almost any thickness of material from a fraction of an inch to several feet. Its use is dependent upon the shape of the body because irregular surfaces may give confusing reflections. Ultrasonic transducers can transmit pulses normal to the surface or at an angle. Transducers transmitting pulses that are oblique to the surface can solve a number of special inspection problems. Magnetic-particle examination is used only on magnetic materials. Magnetic flux is passed through the part in a path parallel to the surface. Fine magnetic particles, when dusted over the surface, will concentrate near the edges of a crack. The sensitivity of magneticparticle examination is proportional to the sine of the angle between the direction of the magnetic flux and the direction of the crack. To be sure of picking up all cracks, it is necessary to probe the area in two directions. Liquid-penetrant examination involves wetting the surface with a fluid which penetrates open cracks. After the excess liquid has been wiped off, the surface is coated with a material which will reveal any liquid that has penetrated the cracks. In some systems a colored dye will seep out of cracks and stain whitewash. Another system uses a penetrant that becomes fluorescent under ultraviolet light. Each of these four popular methods has its advantages. Frequently, best results are obtained by using more than one method. Magnetic particles or liquid penetrants are effective on surface cracks. Radiography and ultrasonics are necessary for subsurface flaws. No known method of nondestructive testing can guarantee the absence of flaws. There are other less widely used methods of examination. Among these are eddy-current, electrical-resistance, acoustics, and thermal testing. Nondestructive Testing Handbook [Robert C. McMaster (ed.), Ronald, New York, 1959] gives information on many testing techniques. The eddy-current technique involves an alternating-current coil along and close to the surface being examined. The electrical impedance of the coil is affected by flaws in the structure or changes in composition. Commercially, the principal use of eddy-current testing is for the examination of tubing. It could, however, be used for testing other things. The electrical-resistance method involves passing an electric current through the structure and exploring the surface with voltage probes. Flaws, cracks, or inclusions will cause a disturbance in the voltage gradient on the surface. Railroads have used this method for many years to locate transverse cracks in rails. The hydrostatic test is, in one sense, a method of examination of a vessel. It can reveal gross flaws, inadequate design, and flange leaks. Many believe that a hydrostatic test guarantees the safety of a vessel. This is not necessarily so. A vessel that has passed a hydrostatic test is probably safer than one that has not been tested. It can, however, still fail in service, even on the next application of pressure. Care in material selection, examination, and fabrication do more to guarantee vessel integrity than the hydrostatic test. The ASME Codes recommend that hydrostatic tests be run at a temperature that is usually above the nil-ductility temperature of the material. This is, in effect, a pressure-temperature treatment of the vessel. When tested in the relatively ductile condition above the nilductility temperature, the material will yield at the tips of cracks and flaws and at points of high residual weld stress. This procedure will

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actually reduce the residual stresses and cause a redistribution at crack tips. The vessel will then be in a safer condition for subsequent operation. This procedure is sometimes referred to as notch nullification. It is possible to design a hydrostatic test in such a way that it probably will be a proof test of the vessel. This usually requires, among other things, that the test be run at a temperature as low as and preferably lower than the minimum operating temperature of the vessel. Proof tests of this type are run on vessels built of ultrahigh-strength steel to operate at cryogenic temperatures. Other Regulations and Standards Pressure vessels may come under many types of regulation, depending on where they are and what they contain. Although many states have adopted the ASME Boiler and Pressure Vessel Code, either in total or in part, any state or municipality may enact its own requirements. The federal government regulates some pressure vessels through the Department of Transportation, which includes the Coast Guard. If pressure vessels are shipped into foreign countries, they may face additional regulations. Pressure vessels carried aboard United States-registered ships must conform to rules of the U.S. Coast Guard. Subchapter F of Title 46, Code of Federal Regulations, covers marine engineering. Of this, Parts 50 through 61 and 98 include pressure vessels. Many of the rules are similar to those in the ASME Code, but there are differences. The American Bureau of Shipping (ABS) has rules that insurance underwriters require for the design and construction of pressure vessels which are a permanent part of a ship. Pressure cargo tanks may be permanently attached and come under these rules. Such tanks supported at several points are independent of the ship’s structure and are distinguished from “integral cargo tanks” such as those in a tanker. ABS has pressure vessel rules in two of its publications. Most of them are in Rules for Building and Classing Steel Vessels. Standards of Tubular Exchanger Manufacturers Association (TEMA) give recommendations for the construction of tubular heat exchangers. Although TEMA is not a regulatory body and there is no legal requirement for the use of its standards, they are widely accepted as a good basis for design. By specifying TEMA standards, one can obtain adequate equipment without having to write detailed specifications for each piece. TEMA gives formulas for the thickness of tube sheets. Such formulas are not in ASME Codes. (See further discussion of TEMA in Sec. 11.) Vessels with Unusual Construction High pressures create design problems. The ASME Code Sec. VIII, Division 1, applies to vessels rated for pressures up to 20,670 kPa (3000 lbf/in2). Division 2 is unlimited. At high pressures, special designs not necessarily in accordance with the code are sometimes used. At such pressures, a vessel designed for ordinary low-carbon-steel plate, particularly in large diameters, would become too thick for practical fabrication by ordinary methods. The alternatives are to make the vessel of highstrength plate, use a solid forging, or use multilayer construction. High-strength steels with tensile strengths over 1380 MPa (200,000 lbf/in2) are limited largely to applications for which weight is very important. Welding procedures are carefully controlled, and preheat is used. These materials are brittle at almost any temperature, and vessels must be designed to prevent brittle fracture. Flat spots and variations in curvature are avoided. Openings and changes in shape require appropriate design. The maximum permissible size of flaws is determined by fracture mechanics, and the method of examination must assure as much as possible that larger flaws are not present. All methods of nondestructive testing may be used. Such vessels require the most sophisticated techniques in design, fabrication, and operation. Solid forgings are frequently used in construction for pressure vessels above 20,670 kPa (3000 lbf/in2) and even lower. Almost any shell thickness can be obtained, but most of them range between 50 and 300 mm (2 and 12 in). The ASME Code lists forging materials with tensile strengths from 414 to 930 MPa (from 60,000 to 135,000 lbf/in2). Brittle fracture is a possibility, and the hazard increases with thickness. Furthermore, some forging alloys have nil-ductility temperatures as high as 121°C (250°F). A forged vessel should have an NDT at least 17°C (30°F) below the design temperature. In operation, it should be slowly and uniformly heated at least to NDT before

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TRANSPORT AND STORAGE OF FLUIDS

it is subjected to pressure. During construction, nondestructive testing should be used to detect dangerous cracks or flaws. Section VIII of the ASME Code, particularly Division 2, gives design and testing techniques. As the size of a forged vessel increases, the sizes of ingot and handling equipment become larger. The cost may increase faster than the weight. The problems of getting sound material and avoiding brittle fracture also become more difficult. Some of these problems are avoided by use of multilayer construction. In this type of vessel, the heads and flanges are made of forgings, and the cylindrical portion is built up by a series of layers of thin material. The thickness of these layers may be between 3 and 50 mm (f and 2 in), depending on the type of construction. There is an inner lining which may be different from the outer layers. Although there are multilayer vessels as small as 380-mm (15-in) inside diameter and 2400 mm (8 ft) long, their principal advantage applies to the larger sizes. When properly made, a multilayer vessel is probably safer than a vessel with a solid wall. The layers of thin material are tougher and less susceptible to brittle fracture, have less probability of defects, and have the statistical advantage of a number of small elements instead of a single large one. The heads, flanges, and welds, of course, have the same hazards as other thick members. Proper attention is necessary to avoid cracks in these members. There are several assembly techniques. One frequently used is to form successive layers in half cylinders and butt-weld them over the previous layers. In doing this, the welds are staggered so that they do not fall together. This type of construction usually uses plates from 6 to 12 mm (d to a in) thick. Another method is to weld each layer separately to form a cylinder and then shrink it over the previous layers. Layers up to about 50-mm (2-in) thickness are assembled in this way. A third method of fabrication is to wind the layers as a continuous sheet. This technique is used in Japan. The Wickel construction, fabricated in Germany, uses helical winding of interlocking metal strip. Each method has its advantages and disadvantages, and choice will depend upon circumstances. Because of the possibility of voids between layers, it is preferable not to use multilayer vessels in applications where they will be subjected to fatigue. Inward thermal gradients (inside temperature lower than outside temperature) are also undesirable. Articles on these vessels have been written by Fratcher [Pet. Refiner, 34(11), 137 (1954)] and by Strelzoff, Pan, and Miller [Chem. Eng., 75(21), 143–150 (1968)]. Vessels for high-temperature service may be beyond the temperature limits of the stress tables in the ASME Codes. Section VIII, Division 1, makes provision for construction of pressure vessels up to 650°C (1200°F) for carbon and low-alloy steel and up to 815°C (1500°F) for stainless steels (300 series). If a vessel is required for temperatures above these values and above 103 kPa (15 lbf/in2), it would be necessary, in a code state, to get permission from the state authorities to build it as a special project. Above 815°C (1500°F), even the 300 series stainless steels are weak, and creep rates increase rapidly. If the metal which resists the pressure operates at these temperatures, the vessel pressure and size will be limited. The vessel must also be expendable because its life will be short. Long exposure to high temperature may cause the metal to deteriorate and become brittle. Sometimes, however, economics favor this type of operation. One way to circumvent the problem of low metal strength is to use a metal inner liner surrounded by insulating material, which in turn is confined by a pressure vessel. The liner, in some cases, may have perforations which will allow pressure to pass through the insulation and act on the outer shell, which is kept cool to obtain normal strength. The liner has no pressure differential acting on it and, therefore, does not need much strength. Ceramic linings are also useful for hightemperature work. Lined vessels are used for many applications. Any type of lining can be used in an ASME Code vessel, provided it is compatible with the metal of the vessel and the contents. Glass, rubber, plastics, rare metals, and ceramics are a few types. The lining may be installed separately, or if a metal is used, it may be in the form of clad plate. The cladding on plate can sometimes be considered as a stress-carrying part of the vessel.

A ceramic lining when used with high temperature acts as an insulator so that the steel outer shell is at a moderate temperature while the temperature at the inside of the lining may be very high. Ceramic linings may be of unstressed brick, or prestressed brick, or cast in place. Cast ceramic linings or unstressed brick may develop cracks and are used when the contents of the vessel will not damage the outer shell. They are usually designed so that the high temperature at the inside will expand them sufficiently to make them tight in the outer (and cooler) shell. This, however, is not usually sufficient to prevent some penetration by the product. Prestressed-brick linings can be used to protect the outer shell. In this case, the bricks are installed with a special thermosetting-resin mortar. After lining, the vessel is subjected to internal pressure and heat. This expands the steel vessel shell, and the mortar expands to take up the space. The pressure and temperature must be at least as high as the maximum that will be encountered in service. After the mortar has set, reduction of pressure and temperature will allow the vessel to contract, putting the brick in compression. The upper temperature limit for this construction is about 190°C (375°F). The installation of such linings is highly specialized work done by a few companies. Great care is usually exercised in operation to protect the vessel from exposure to unsymmetrical temperature gradients. Side nozzles and other unsymmetrical designs are avoided insofar as possible. Concrete pressure vessels may be used in applications that require large sizes. Such vessels, if made of steel, would be too large and heavy to ship. Through the use of posttensioned (prestressed) concrete, the vessel is fabricated on the site. In this construction, the reinforcing steel is placed in tubes or plastic covers, which are cast into the concrete. Tension is applied to the steel after the concrete has acquired most of its strength. Concrete nuclear reactor vessels, of the order of magnitude of 15-m (50-ft) inside diameter and length, have inner linings of steel which confine the pressure. After fabrication of the liner, the tubes for the cables or wires are put in place and the concrete is poured. Highstrength reinforcing steel is used. Because there are thousands of reinforcing tendons in the concrete vessel, there is a statistical factor of safety. The failure of 1 or even 10 tendons would have little effect on the overall structure. Plastic pressure vessels have the advantages of chemical resistance and light weight. Above 103 kPa (15 lbf/in2), with certain exceptions, they must be designed according to the ASME Code section (see “Storage of Gases”) and are confined to the three types of approved code construction. Below 103 kPa (15 lbf/in2), any construction may be used. Even in this pressure range, however, the code should be used for guidance. Solid plastics, because of low strength and creep, can be used only for the lowest pressures and sizes. A stress of a few hundred pounds-force per square inch is the maximum for most plastics. To obtain higher strength, the filled plastics or filamentwound vessels, specified by the code, must be used. Solid-plastic parts, however, are often employed inside a steel shell, particularly for heat exchangers. Graphite and ceramic vessels are used fully armored; that is, they are enclosed within metal pressure vessels. These materials are also used for boxlike vessels with backing plates on the sides. The plates are drawn together by tie bolts, thus putting the material in compression so that it can withstand low pressure. Vessel Codes Other Than ASME Different design and construction rules are used in other countries. Chemical engineers concerned with pressure vessels outside the United States must become familiar with local pressure-vessel laws and regulations. Boilers and Pressure Vessels, an international survey of design and approval requirements published by the British Standards Institution, Maylands Avenue, Hemel Hempstead, Hertfordshire, England, in 1975, gives pertinent information for 76 political jurisdictions. The British Code (British Standards) and the West German Code (A. D. Merkblätter) in addition to the ASME Code are most commonly permitted, although Netherlands, Sweden, and France also have codes. The major difference between the codes lies in factors of safety and in whether or not ultimate strength is considered. ASME Code, Sec. VIII, Division 1, vessels are generally heavier than vessels

STORAGE AND PROCESS VESSELS built to the other codes; however, the differences in allowable stress for a given material are less in the higher temperature (creep) range. Engineers and metallurgists have developed alloys to comply economically with individual codes. In West Germany, where design stress is determined from yield strength and creep-rupture strength and no allowance is made for ultimate strength, steels which have a very high yield-strength-to-ultimate-strength ratio are used. Other differences between codes include different bases for the design of reinforcement for openings and the design of flanges and heads. Some codes include rules for the design of heat-exchanger tube sheets, while others (ASME Code) do not. The Dutch Code (Grondslagen) includes very specific rules for calculation of wind loads, while the ASME Code leaves this entirely to the designer. There are also significant differences in construction and inspection rules. Unless engineers make a detailed study of the individual codes and keep current, they will be well advised to make use of responsible experts for any of the codes. Vessel Design and Construction The ASME Code lists a number of loads that must be considered in designing a pressure vessel. Among them are impact, weight of the vessel under operating and test conditions, superimposed loads from other equipment and piping, wind and earthquake loads, temperature-gradient stresses, and localized loadings from internal and external supports. In general, the code gives no values for these loads or methods for determining them, and no formulas are given for determining the stresses from these loads. Engineers must be knowledgeable in mechanics and strength of materials to solve these problems. Some of the problems are treated by Brownell and Young, Process Equipment Design, Wiley, New York, 1959. ASME papers treat others, and a number of books published by the ASME are collections of papers on pressure-vessel design: Pressure Vessels and Piping Design: Collected Papers, 1927–1959; Pressure Vessels and Piping Design and Analysis, four volumes; and International Conference: Pressure Vessel Technology, published annually. Throughout the year the Welding Research Council publishes bulletins which are final reports from projects sponsored by the council, important papers presented before engineering societies, and other reports of current interest which are not published in Welding Research. A large number of the published bulletins are pertinent for vessel designers. Care of Pressure Vessels Protection against excessive pressure is largely taken care of by code requirements for relief devices. Exposure to fire is also covered by the code. The code, however, does not provide for the possibility of local overheating and weakening of a vessel in a fire. Insulation reduces the required relieving capacity and also reduces the possibility of local overheating. A pressure-reducing valve in a line leading to a pressure vessel is not adequate protection against overpressure. Its failure will subject the vessel to full line pressure. Vessels that have an operating cycle which involves the solidification and remelting of solids can develop excessive pressures. A solid plug of material may seal off one end of the vessel. If heat is applied at that end to cause melting, the expansion of the liquid can build up a high pressure and possibly result in yielding or rupture. Solidification in connecting piping can create similar problems. Some vessels may be exposed to a runaway chemical reaction or even an explosion. This requires relief valves, rupture disks, or, in extreme cases, a barricade (the vessel is expendable). A vessel with a large rupture disk needs anchors designed for the jet thrust when the disk blows. Vacuum must be considered. It is nearly always possible that the contents of a vessel might contract or condense sufficiently to subject it to an internal vacuum. If the vessel cannot withstand the vacuum, it must have vacuum-breaking valves. Improper operation of a process may result in the vessel’s exceeding design temperature. Proper control is the only solution to this problem. Maintenance procedures can also cause excessive temperatures. Sometimes the contents of a vessel may be burned out with torches. If the flame impinges on the vessel shell, overheating and damage may occur. Excessively low temperature may involve the hazard of brittle

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fracture. A vessel that is out of use in cold weather could be at a subzero temperature and well below its nil-ductility temperature. In startup, the vessel should be warmed slowly and uniformly until it is above the NDT. A safe value is 38°C (100°F) for plate if the NDT is unknown. The vessel should not be pressurized until this temperature is exceeded. Even after the NDT has been passed, excessively rapid heating or cooling can cause high thermal stresses. Corrosion is probably the greatest threat to vessel life. Partially filled vessels frequently have severe pitting at the liquid-vapor interface. Vessels usually do not have a corrosion allowance on the outside. Lack of protection against the weather or against the drip of corrosive chemicals can reduce vessel life. Insulation may contain damaging substances. Chlorides in insulating materials can cause cracking of stainless steels. There are many ways in which a pressure vessel can suffer mechanical damage. The shells can be dented or even punctured, they can be dropped or have hoisting cables improperly attached, bolts can be broken, flanges are bent by excessive bolt tightening, gasket contact faces can be scratched and dented, rotating paddles can drag against the shell and cause wear, and a flange can be bolted up with a gasket half in the groove and half out. Most of these forms of damage can be prevented by care and common sense. If damage is repaired by straightening, as with a dented shell, it may be necessary to stress-relieve the repaired area. Some steels are susceptible to embrittlement by aging after severe straining. A safer procedure is to cut out the damaged area and replace it. The National Board Inspection Code, published by the National Board of Boiler and Pressure Vessel Inspectors, Columbus, Ohio, is helpful. Any repair, however, is acceptable if it is made in accordance with the rules of the Pressure Vessel Code. Pressure vessels should be inspected periodically. No rule can be given for the frequency of these inspections. Frequency depends on operating conditions. If the early inspections of a vessel indicate a low corrosion rate, intervals between inspections may be lengthened. Some vessels are inspected at 5-year intervals; others, as frequently as once a year. Measurement of corrosion is an important inspection item. One of the most convenient ways of measuring thickness (and corrosion) is to use an ultrasonic gauge. The location of the corrosion and whether it is uniform or localized in deep pits should be observed and reported. Cracks, any type of distortion, and leaks should be observed. Cracks are particularly dangerous because they can lead to sudden failure. Insulation is usually left in place during inspection of insulated vessels. If, however, severe external corrosion is suspected, the insulation should be removed. All forms of nondestructive testing are useful for examinations. Care in reassembling the vessel is particularly important. Gaskets should be properly located, particularly if they are in grooves. Bolts should be tightened in proper sequence. In some critical cases and with large bolts, it is necessary to control bolt tightening by torque wrenches, micrometers, patented bolt-tightening devices, or heating bolts. After assembly, vessels are sometimes given a hydrostatic test. Pressure-Vessel Cost and Weight The curves of Fig. 10-188 can be used for estimating cost (freight allowed) when a weight estimate is not available. The cost is based on some 1990 pressure-vessel costs. The prices are plotted as a function of vessel volume for average vessels 6.35 mm (d in) thick which are not of unusual design. Correction factors for other thicknesses are given. Complicated vessels could cost considerably more. Guthrie [Chem. Eng., 76(6), 114–142 (1969)] also gives pressure-vessel cost data. When vessels have complicated construction (large, heavy bolted connections, support skirts, etc.), it is preferable to estimate their weight and apply a unit cost in dollars per pound. Some data for vessels purchased in 1968 are plotted in Fig. 10-189. There is a variation of about 2 to 1 between the lowest and the highest costs. The unit FOB cost of carbon steel and type 304 stainless steel was found to vary as the −0.34 power of the weight. Stainless-steel vessels frequently include considerable carbon steel in the form of support skirts, brackets, legs, lap-joint flanges, bolts, etc. In calculating the equivalent weight of a stainless-steel vessel, each pound of carbon should be considered equivalent to 0.4 lb of stainless. Pressure-vessel weights are obtained by calculating the cylindrical

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TRANSPORT AND STORAGE OF FLUIDS TABLE 10-67 Factors for Estimating Diameters of Blanks for Formed Heads A.S.M.E. head Ellipsoidal head Hemispherical head

Ratio d/t

Blank diameter factor

Over 50 30–50 20–30 Over 20 10–20 Over 30 18–30 10–18

1.09 1.11 1.15 1.24 1.30 1.60 1.65 1.70

d = head diameter t = nominal minimum head thickness

TABLE 10-68

Extra Thickness Allowances for Formed Heads* Extra thickness, in A.S.M.E. and Ellipsoidal

Minimum head thickness, in

Head o.d. up to 150 in incl.

Head o.d. over 150 in

Hemispherical

Up to 0.99 1 to 1.99 2 to 2.99

g f d

f f d

t r v

*Lukens, Inc.

FIG. 10-189 Cost per pound of pressure vessels (1968). For carbon steel, C = 9.05 W−0.34; for type 304 stainless steel, C = 25.6 W−0.34; and for type 316 stainless steel, C = 34.2 W−0.34; where C = FOB cost in dollars per pound and W = weight in pounds. To convert pounds to kilograms, multiply by 0.454.

shell and heads separately and then adding the weights of nozzles and attachments. Steel weighs 0.283 lb/in3 and 40.7 lb/ft3 for 1-in plate. Metal in heads can be approximated by calculating the area of the blank (disk) used for forming the head. The required diameter of blank can be calculated by multiplying the head outside diameter by

the approximate factors given in Table 10-67. These factors make no allowance for the straight flange which is a cylindrical extension that is formed on the head. The blank diameter obtained from these factors must be increased by twice the length of straight flange, which is usually 1a to 2 in but can be up to several inches in length. Manufacturers’ catalogs give weights of heads. Forming a head thins it in certain areas. To obtain the required minimum thickness of a head, it is necessary to use a plate that is initially thicker. Table 10-68 gives allowances for additional thickness. Nozzles and flanges may add considerably to the weight of a vessel. Their weights can be obtained from manufacturers’ catalogs (Taylor Forge Division of Gulf & Western Industries, Inc., Tube Turns Inc., Ladish Co., Lenape Forge, and others). Other parts such as skirts, legs, support brackets, and other details must be calculated.

Section 11

Heat-Transfer Equipment

Richard L. Shilling, P.E., B.S.M., B.E.M.E., Manager of Engineering Development, Brown Fintube Company—a Koch Engineering Company; Member, American Society of Mechanical Engineers. (Shell-and-Tube Heat Exchangers, Hairpin/Double Pipe Heat Exchangers, Air Cooled Heat Exchangers, Heating and Cooling of Tanks, Fouling and Scaling, Heat Exchangers for Solids—significant contribution by Arthur D. Holt, Thermal Insulation—significant contribution by Herbert A. Moak) (Section Editor) Kenneth J. Bell, Ph.D., P.E., Regents Professor Emeritus, School of Chemical Engineering, Oklahoma State University; Member, American Institute of Chemical Engineers. (Thermal Design of Heat Exchangers, Condenser, Reboilers) Patrick M. Bernhagen, P.E., B.S.M.E., Senior Mechanical Engineer, Foster Wheeler USA Corporation, American Society of Mechanical Engineers. (Compact and Non-Tubular Heat Exchangers) Thomas M. Flynn, Ph.D., P.E., Cryogenic Engineer, President CRYOCO, Louisville, Colorado; Member, American Institute of Chemical Engineers. (Cryogenic Processes) Victor M. Goldschmidt, Ph.D., P.E., Professor of Mechanical Engineering, Purdue University, West Lafayette, Indiana. (Air Conditioning) Predrag S. Hrnjak, Ph.D., V.Res., Assistant Professor, University of Illinois at Ubana Champaign and Principal Investigator—U of I Air Conditioning and Refrigeration Center, Assistant Professor, University of Belgrade; Member, International Institute of Refrigeration, American Society of Heating, Refrigeration and Air Conditioning. (Refrigeration) F. C. Standiford, M.S., P.E., Member, American Institute of Chemical Engineers, American Chemical Society. (Thermal Design of Evaporators, Evaporators) Klaus D. Timmerhaus, Ph.D., P.E., Professor and President’s Teaching Scholar, University of Colorado, Boulder, Colorado; Fellow, American Institute of Chemical Engineers, American Society for Engineering Education, American Association for the Advancement of Science, Member, American Astronautical Society, National Academy of Engineering, Austrian Academy of Science, International Institute of Refrigeration, American Society of Heating, Refrigerating and Air Conditioning Engineers, American Society of Environmental Engineers, Engineering Society for Advancing Mobility on Land, Sea, Air, and Space, Sigma Xi, The Research Society. (Cryogenic Processes)

11-1

11-2

HEAT-TRANSFER EQUIPMENT

THERMAL DESIGN OF HEAT-TRANSFER EQUIPMENT Introduction to Thermal Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Approach to Heat-Exchanger Design . . . . . . . . . . . . . . . . . . . . . . . . . Overall Heat-Transfer Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mean Temperature Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Countercurrent or Cocurrent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . Reversed, Mixed, or Cross-Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Design for Single-Phase Heat Transfer . . . . . . . . . . . . . . . . . . Double-Pipe Heat Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Baffled Shell-and-Tube Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Design of Condensers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single-Component Condenser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multicomponent Condensers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Design of Reboilers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kettle Reboilers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vertical Thermosiphon Reboilers . . . . . . . . . . . . . . . . . . . . . . . . . . . . Forced-Recirculation Reboilers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Design of Evaporators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Forced-Circulation Evaporators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Long-Tube Vertical Evaporators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Short-Tube Vertical Evaporators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miscellaneous Evaporator Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heat Transfer from Various Metal Surfaces . . . . . . . . . . . . . . . . . . . . Effect of Fluid Properties on Heat Transfer . . . . . . . . . . . . . . . . . . . . Effect of Noncondensables on Heat Transfer . . . . . . . . . . . . . . . . . . . Batch Operations: Heating and Cooling of Vessels . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of External Heat Loss or Gain . . . . . . . . . . . . . . . . . . . . . . . . . Internal Coil or Jacket Plus External Heat Exchange. . . . . . . . . . . . . Equivalent-Area Concept. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonagitated Batches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Storage Tanks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Design of Tank Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maintenance of Temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heating and Cooling of Tanks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tank Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Teflon Immersion Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bayonet Heaters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . External Coils and Tracers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jacketed Vessels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extended or Finned Surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Finned-Surface Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . High Fins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low Fins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fouling and Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Control of Fouling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fouling Transients and Operating Periods . . . . . . . . . . . . . . . . . . . . . Removal of Fouling Deposits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fouling Resistances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Typical Heat-Transfer Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Design for Solids Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . Conductive Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contactive (Direct) Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convective Heat Transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radiative Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scraped-Surface Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11-4 11-4 11-4 11-4 11-4 11-5 11-5 11-5 11-7 11-11 11-11 11-12 11-13 11-13 11-13 11-13 11-13 11-13 11-14 11-15 11-15 11-16 11-17 11-18 11-18 11-18 11-18 11-19 11-19 11-19 11-19 11-19 11-20 11-20 11-20 11-20 11-20 11-20 11-21 11-22 11-22 11-22 11-22 11-22 11-22 11-23 11-23 11-23 11-23 11-24 11-24 11-24 11-24 11-24 11-28 11-30 11-30 11-33

TEMA-STYLE SHELL-AND-TUBE HEAT EXCHANGERS Types and Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TEMA Numbering and Type Designations. . . . . . . . . . . . . . . . . . . . . Functional Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Design Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selection of Flow Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Construction Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tube Bundle Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Principal Types of Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fixed-Tube-Sheet Heat Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . U-Tube Heat Exchanger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Packed-Lantern-Ring Exchanger. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outside-Packed Floating-Head Exchanger . . . . . . . . . . . . . . . . . . . . . Internal Floating-Head Exchanger . . . . . . . . . . . . . . . . . . . . . . . . . . . Pull-Through Floating-Head Exchanger . . . . . . . . . . . . . . . . . . . . . . . Falling-Film Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tube-Side Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tube-Side Header. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Special High-Pressure Closures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11-33 11-33 11-33 11-35 11-35 11-35 11-35 11-35 11-36 11-36 11-39 11-39 11-39 11-40 11-40 11-40 11-40 11-40 11-40

Tube-Side Passes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rolled Tube Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Welded Tube Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Double-Tube-Sheet Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shell-Side Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shell Sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shell-Side Arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Baffles and Tube Bundles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Segmental Baffles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rod Baffles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tie Rods and Spacers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Impingement Baffle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vapor Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tube-Bundle Bypassing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Longitudinal Flow Baffles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Corrosion in Heat Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Materials of Construction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bimetallic Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Clad Tube Sheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonmetallic Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shell-and-Tube Exchanger Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11-40 11-40 11-41 11-41 11-41 11-41 11-41 11-41 11-42 11-42 11-42 11-43 11-43 11-43 11-43 11-43 11-43 11-43 11-43 11-44 11-44 11-44 11-44

HAIRPIN/DOUBLE-PIPE HEAT EXCHANGERS Principles of Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Finned Double Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multitube Hairpins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11-46 11-46 11-46 11-47

AIR-COOLED HEAT EXCHANGERS Air Cooled Heat Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Forced and Induced Draft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tube Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tubing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Finned-Tube Construction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fan Drivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fan Ring and Plenum Chambers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Air-Flow Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Air Recirculation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trim Coolers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Humidification Chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evaporative Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Steam Condensers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Air-Cooled Overhead Condensers . . . . . . . . . . . . . . . . . . . . . . . . . . . . Air-Cooled Heat-Exchanger Costs. . . . . . . . . . . . . . . . . . . . . . . . . . . . Design Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11-47 11-47 11-48 11-48 11-49 11-49 11-49 11-50 11-50 11-50 11-50 11-50 11-50 11-51 11-51 11-51 11-51

COMPACT AND NONTUBULAR HEAT EXCHANGERS Compact Heat Exchangers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plate-and-Frame Exchangers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gasketed-Plate Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Welded- and Brazed-Plate Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . Spiral-Plate Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brazed-Plate-Fin Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plate-Fin Tubular Exchangers (PFE) . . . . . . . . . . . . . . . . . . . . . . . . . . . Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spiral-Tube Exchangers (STE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphite Heat Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications and Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cascade Coolers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bayonet-Tube Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Atmospheric Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonmetallic Heat Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PVDF Heat Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ceramic Heat Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Teflon Heat Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11-52 11-52 11-52 11-52 11-52 11-53 11-55 11-55 11-55 11-55 11-55 11-56 11-56 11-56 11-56 11-56 11-56 11-56 11-56 11-56 11-57 11-57 11-57 11-57 11-57 11-57 11-57 11-57 11-58 11-58

HEAT-TRANSFER EQUIPMENT HEAT EXCHANGERS FOR SOLIDS Equipment for Solidification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table Type. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Agitated-Pan Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vibratory Type. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Belt Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotating-Drum Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotating-Shelf Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equipment for Fusion of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Horizontal-Tank Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vertical Agitated-Kettle Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mill Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heat-Transfer Equipment for Sheeted Solids. . . . . . . . . . . . . . . . . . . . . Cylinder Heat-Transfer Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heat-Transfer Equipment for Divided Solids . . . . . . . . . . . . . . . . . . . . . Fluidized-Bed Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moving-Bed Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Agitated-Pan Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kneading Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shelf Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotating-Shell Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conveyor-Belt Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spiral-Conveyor Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Double-Cone Blending Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vibratory-Conveyor Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elevator Devices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pneumatic-Conveying Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vacuum-Shelf Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11-58 11-58 11-58 11-59 11-59 11-60 11-61 11-61 11-61 11-61 11-61 11-62 11-62 11-62 11-62 11-63 11-63 11-63 11-63 11-63 11-64 11-64 11-65 11-65 11-67 11-67 11-67

THERMAL INSULATION Insulation Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Conductivity (K Factor). . . . . . . . . . . . . . . . . . . . . . . . . . . . . Finishes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . System Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cryogenic High Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moderate and High Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . Economic Thickness of Insulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Thickness of Insulation. . . . . . . . . . . . . . . . . . . . . . . . Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Installation Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Method of Securing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Double Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Finish. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tanks, Vessels, and Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Method of Securing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Finish. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11-68 11-68 11-68 11-68 11-69 11-69 11-69 11-69 11-70 11-70 11-73 11-73 11-73 11-73 11-73 11-73 11-73 11-73 11-73 11-73 11-74

AIR CONDITIONING Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comfort Air Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Industrial Air Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ventilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Air-Conditioning Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Central Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Unitary Refrigerant-Based Air-Conditioning Systems . . . . . . . . . . . . . . Load Calculation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11-74 11-74 11-74 11-74 11-74 11-75 11-75 11-76

REFRIGERATION Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic Refrigeration Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanical Refrigeration (Vapor-Compression Systems) . . . . . . . . . . . Vapor-Compression Cycles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multistage Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cascade System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compressors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Positive-Displacement Compressors . . . . . . . . . . . . . . . . . . . . . . . . . . Centrifugal Compressors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Condensers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evaporators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11-76 11-76 11-76 11-77 11-77 11-78 11-79 11-79 11-79 11-80 11-83 11-83 11-84

11-3

System Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . System, Equipment, and Refrigerant Selection . . . . . . . . . . . . . . . . . Other Refrigerant Systems Applied in the Industry . . . . . . . . . . . . . . . . Absorption Refrigeration Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . Steam-Jet (Ejector) Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multistage Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Capacity Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Refrigerants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Secondary Refrigerants (Antifreezes or Brines) . . . . . . . . . . . . . . . . . Organic Compounds (Inhibited Glycols). . . . . . . . . . . . . . . . . . . . . . . Safety in Refrigeration Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11-85 11-87 11-88 11-88 11-89 11-93 11-93 11-93 11-94 11-95 11-96

CRYOGENIC PROCESSES Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties of Cryogenic Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties of Solids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structural Properties at Low Temperatures . . . . . . . . . . . . . . . . . . . . Thermal Properties at Low Temperatures . . . . . . . . . . . . . . . . . . . . . Electrical Properties at Low Temperatures. . . . . . . . . . . . . . . . . . . . . Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Refrigeration and Liquifaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Principles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expansion Types of Refrigerators . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miniature Refrigerators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermodynamic Analyses of Cycles. . . . . . . . . . . . . . . . . . . . . . . . . . . Process Equipment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heat Exchangers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expanders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Separation and Purification Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . Air-Separation Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Helium and Natural-Gas Systems Separation . . . . . . . . . . . . . . . . . . . Gas Purification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Storage and Transfer Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Insulation Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Types of Insulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Storage and Transfer Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cryogenic Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physiological Hazards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Materials and Construction Hazards . . . . . . . . . . . . . . . . . . . . . . . . . . Flammability and Explosion Hazards . . . . . . . . . . . . . . . . . . . . . . . . . High-Pressure Gas Hazards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11-96 11-96 11-96 11-97 11-97 11-97 11-97 11-98 11-98 11-98 11-100 11-100 11-100 11-101 11-101 11-102 11-102 11-103 11-103 11-104 11-104 11-104 11-105 11-106 11-106 11-106 11-106 11-106 11-106 11-107 11-107 11-107 11-107 11-107

EVAPORATORS Primary Design Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vapor-Liquid Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selection Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Product Quality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evaporator Types and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Forced-Circulation Evaporators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Short-Tube Vertical Evaporators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Long-Tube Vertical Evaporators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Horizontal-Tube Evaporators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miscellaneous Forms of Heating Surface . . . . . . . . . . . . . . . . . . . . . . Evaporators without Heating Surfaces . . . . . . . . . . . . . . . . . . . . . . . . Utilization of Temperature Difference . . . . . . . . . . . . . . . . . . . . . . . . . . Vapor-Liquid Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evaporator Arrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single-Effect Evaporators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermocompression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple-Effect Evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Seawater Evaporators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evaporator Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single-Effect Evaporators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermocompression Evaporators . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flash Evaporators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple-Effect Evaporators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evaporator Accessories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Condensers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vent Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Salt Removal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evaporator Operation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11-107 11-107 11-107 11-107 11-108 11-108 11-108 11-109 11-109 11-110 11-111 11-111 11-111 11-111 11-113 11-113 11-113 11-113 11-114 11-115 11-115 11-115 11-115 11-117 11-117 11-117 11-117 11-117 11-117 11-118

THERMAL DESIGN OF HEAT-TRANSFER EQUIPMENT INTRODUCTION TO THERMAL DESIGN Design methods for several important classes of process heat-transfer equipment are presented in the following portions of Sec. 11. Mechanical descriptions and specifications of equipment are given in this section and should be read in conjunction with the use of this material. It is impossible to present here a comprehensive treatment of heat-exchanger selection, design, and application. The best general references in this field are Hewitt, Shires, and Bott, Process Heat Transfer, CRC Press, Boca Raton, FL, 1994; and Schlünder (ed.), Heat Exchanger Design Handbook, Begell House, New York, 1983. Approach to Heat-Exchanger Design The proper use of basic heat-transfer knowledge in the design of practical heat-transfer equipment is an art. Designers must be constantly aware of the differences between the idealized conditions for and under which the basic knowledge was obtained and the real conditions of the mechanical expression of their design and its environment. The result must satisfy process and operational requirements (such as availability, flexibility, and maintainability) and do so economically. An important part of any design process is to consider and offset the consequences of error in the basic knowledge, in its subsequent incorporation into a design method, in the translation of design into equipment, or in the operation of the equipment and the process. Heat-exchanger design is not a highly accurate art under the best of conditions. The design of a process heat exchanger usually proceeds through the following steps: 1. Process conditions (stream compositions, flow rates, temperatures, pressures) must be specified. 2. Required physical properties over the temperature and pressure ranges of interest must be obtained. 3. The type of heat exchanger to be employed is chosen. 4. A preliminary estimate of the size of the exchanger is made, using a heat-transfer coefficient appropriate to the fluids, the process, and the equipment. 5. A first design is chosen, complete in all details necessary to carry out the design calculations. 6. The design chosen in step 5 is evaluated, or rated, as to its ability to meet the process specifications with respect to both heat transfer and pressure drop. 7. On the basis of the result of step 6, a new configuration is chosen if necessary and step 6 is repeated. If the first design was inadequate to meet the required heat load, it is usually necessary to increase the size of the exchanger while still remaining within specified or feasible limits of pressure drop, tube length, shell diameter, etc. This will sometimes mean going to multiple-exchanger configurations. If the first design more than meets heat-load requirements or does not use all the allowable pressure drop, a less expensive exchanger can usually be designed to fulfill process requirements. 8. The final design should meet process requirements (within reasonable expectations of error) at lowest cost. The lowest cost should include operation and maintenance costs and credit for ability to meet long-term process changes, as well as installed (capital) cost. Exchangers should not be selected entirely on a lowest-first-cost basis, which frequently results in future penalties. Overall Heat-Transfer Coefficient The basic design equation for a heat exchanger is dA = dQ/U ∆T (11-1)

where dA is the element of surface area required to transfer an amount of heat dQ at a point in the exchanger where the overall heattransfer coefficient is U and where the overall bulk temperature difference between the two streams is ∆ T. The overall heat-transfer coefficient is related to the individual film heat-transfer coefficients and fouling and wall resistances by Eq. (11-2). Basing Uo on the outside surface area A o results in 1 (11-2) Uo =




1/ho + Rdo + xAo /kw Awm + (1/hi + Rdi)Ao /Ai Equation (11-1) can be formally integrated to give the outside area required to transfer the total heat load QT: QT dQ (11-3) Ao =
0 Uo ∆T To integrate Eq. (11-3), Uo and ∆T must be known as functions of Q. For some problems, Uo varies strongly and nonlinearly throughout the exchanger. In these cases, it is necessary to evaluate Uo and ∆ T at several intermediate values and numerically or graphically integrate. For many practical cases, it is possible to calculate a constant mean overall coefficient Uom from Eq. (11-2) and define a corresponding mean value of ∆ Tm, such that




Ao = QT /Uom ∆ Tm

(11-4)

Care must be taken that Uo does not vary too strongly, that the proper equations and conditions are chosen for calculating the individual coefficients, and that the mean temperature difference is the correct one for the specified exchanger configuration. Mean Temperature Difference The temperature difference between the two fluids in the heat exchanger will, in general, vary from point to point. The mean temperature difference (∆ Tm or MTD) can be calculated from the terminal temperatures of the two streams if the following assumptions are valid: 1. All elements of a given fluid stream have the same thermal history in passing through the exchanger.* 2. The exchanger operates at steady state. 3. The specific heat is constant for each stream (or if either stream undergoes an isothermal phase transition). 4. The overall heat-transfer coefficient is constant. 5. Heat losses are negligible. Countercurrent or Cocurrent Flow If the flow of the streams is either completely countercurrent or completely cocurrent or if one or both streams are isothermal (condensing or vaporizing a pure component with negligible pressure change), the correct MTD is the logarithmic-mean temperature difference (LMTD), defined as (t′1 − t″2 ) − (t′2 − t1″) LMTD = ∆Tlm =


(11-5a) t′1 − t″2 ln

t2′ − t″1 for countercurrent flow (Fig. 11-1a) and (t′1 − t″1 ) − (t′2 − t″2 ) LMTD = ∆Tlm =


(11-5b) t′1 − t″1 ln

t2′ − t2″ for cocurrent flow (Fig. 11-1b) If U is not constant but a linear function of ∆T, the correct value of









* This assumption is vital but is usually omitted or less satisfactorily stated as “each stream is well mixed at each point.” In a heat exchanger with substantial bypassing of the heat-transfer surface, e.g., a typical baffled shell-and-tube exchanger, this condition is not satisfied. However, the error is in some degree offset if the same MTD formulation used in reducing experimental heat-transfer data to obtain the basic correlation is used in applying the correlation to design a heat exchanger. The compensation is not in general exact, and insight and judgment are required in the use of the MTD formulations. Particularly, in the design of an exchanger with a very close temperature approach, bypassing may result in an exchanger that is inefficient and even thermodynamically incapable of meeting specified outlet temperatures. 11-4

THERMAL DESIGN OF HEAT-TRANSFER EQUIPMENT

11-5

(a) FIG. 11-2 Diagram of a 1-2 exchanger (one well-baffled shell pass and two tube passes with an equal number of tubes in each pass).

(b) FIG. 11-1

Temperature profiles in heat exchangers. (a) Countercurrent. (b)

Cocurrent.

Uom ∆Tm to use in Eq. (11-4) is [Colburn, Ind. Eng. Chem., 25, 873 (1933)] U ″(t′ o 1 − t″ 2 ) − U′ o(t′2 − t″ 1) Uom ∆Tm =


U o″(t ′1 − t 2″) ln

U o′ (t ′2 − t 1″)





(11-6a)

for countercurrent flow, where U″o is the overall coefficient evaluated when the stream temperatures are t′1 and t″2 and U′o is evaluated at t′2 and t″1. The corresponding equation for cocurrent flow is U ″(t′ o 1 − t″ 1) − U′ o(t′2 − t″ 2) Uom ∆Tm =


(11-6b) Uo″(t ′1 − t 1″) ln

U ′o(t ′2 − t ″2) where U′o is evaluated at t′2 and t″2 and U″o is evaluated at t′1 and t″1. To use these equations, it is necessary to calculate two values of Uo.* The use of Eq. (11-6) will frequently give satisfactory results even if Uo is not strictly linear with temperature difference. Reversed, Mixed, or Cross-Flow If the flow pattern in the exchanger is not completely countercurrent or cocurrent, it is necessary to apply a correction factor FT by which the LMTD is multiplied to obtain the appropriate MTD. These corrections have been mathematically derived for flow patterns of interest, still by making assumptions 1 to 5 [see Bowman, Mueller, and Nagle, Trans. Am. Soc. Mech. Eng., 62, 283 (1940) or Hewitt, et al. op. cit.]. For a common flow pattern, the 1-2 exchanger (Fig. 11-2), the correction factor FT is given in Fig. 11-4a, which is also valid for finding FT for a 1-2 exchanger in which the shell-side flow direction is reversed from that shown in Fig. 11-2. Figure 11-4a is also applicable with negligible error to exchangers with one shell pass and any number of tube passes. Values of FT less than 0.8 (0.75 at the very lowest) are generally unacceptable because the exchanger configuration chosen is inefficient; the chart is difficult to read accurately; and even a small violation of the first assumption underlying the MTD will invalidate the mathematical derivation and lead to a thermodynamically inoperable exchanger. Correction-factor charts are also available for exchangers with more





than one shell pass provided by a longitudinal shell-side baffle. However, these exchangers are seldom used in practice because of mechanical complications in their construction. Also thermal and physical leakages across the longitudinal baffle further reduce the mean temperature difference and are not properly incorporated into the correction-factor charts. Such charts are useful, however, when it is necessary to construct a multiple-shell exchanger train such as that shown in Fig. 11-3 and are included here for two, three, four, and six separate identical shells and two or more tube passes per shell in Fig. 11-4b, c, d, and e. If only one tube pass per shell is required, the piping can and should be arranged to provide pure countercurrent flow, in which case the LMTD is used with no correction. Cross-flow exchangers of various kinds are also important and require correction to be applied to the LMTD calculated by assuming countercurrent flow. Several cases are given in Fig. 11-4f, g, h, i, and j. Many other MTD correction-factor charts have been prepared for various configurations. The FT charts are often employed to make approximate corrections for configurations even in cases for which they are not completely valid. THERMAL DESIGN FOR SINGLE-PHASE HEAT TRANSFER Double-Pipe Heat Exchangers The design of double-pipe heat exchangers is straightforward. It is generally conservative to neglect natural-convection and entrance effects in turbulent flow. In laminar flow, natural convection effects can increase the theoretical Graetz prediction by a factor of 3 or 4 for fully developed flows. Pressure drop is calculated by using the correlations given in Sec. 6. If the inner tube is longitudinally finned on the outside surface, the equivalent diameter is used as the characteristic length in both the Reynolds-number and the heat-transfer correlations. The fin effi-

Diagram of a 2-4 exchanger (two separate identical well-baffled shells and four or more tube passes).

FIG. 11-3

* This task can be avoided if a hydrocarbon stream is the limiting resistance by the use of the caloric temperature charts developed by Colburn [Ind. Eng. Chem., 25, 873 (1933)].

11-6

HEAT-TRANSFER EQUIPMENT

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

LMTD correction factors for heat exchangers. In all charts, R = (T1 − T2)/(t2 − t1) and S = (t2 − t1)/(T1 − t1). (a) One shell pass, two or more tube passes. (b) Two shell passes, four or more tube passes. (c) Three shell passes, six or more tube passes. (d) Four shell passes, eight or more tube passes. (e) Six shell passes, twelve or more tube passes. ( f) Cross-flow, one shell pass, one or more parallel rows of tubes. (g) Cross-flow, two passes, two rows of tubes; for more than two passes, use FT = 1.0. (h) Cross-flow, one shell pass, one tube pass, both fluids unmixed FIG. 11-4

THERMAL DESIGN OF HEAT-TRANSFER EQUIPMENT

(i)

11-7

(j)

FIG. 11-4 (Continued) LMTD correction factors for heat exchangers. In all charts, R = (T1 − T2)/(t2 − t1) and S = (t2 − t1)/(T1 − t1). (i) Cross-flow (drip type), two horizontal passes with U-bend connections (trombone type). ( j) Cross-flow (drip type), helical coils with two turns.

ciency must also be known to calculate an effective outside area to use in Eq. (11-2). Fittings contribute strongly to the pressure drop on the annulus side. General methods for predicting this are not reliable, and manufacturer’s data should be used when available. Double-pipe exchangers are often piped in complex series-parallel arrangements on both sides. The MTD to be used has been derived for some of these arrangements and is reported in Kern (Process Heat Transfer, McGraw-Hill, New York, 1950). More complex cases may require trial-and-error balancing of the heat loads and rate equations for subsections or even for individual exchangers in the bank. Baffled Shell-and-Tube Exchangers The method given here is based on the research summarized in Final Report, Cooperative Research Program on Shell and Tube Heat Exchangers, Univ. Del. Eng. Exp. Sta. Bull. 5 (June 1963). The method assumes that the shell-side heat transfer and pressure-drop characteristics are equal to those of the ideal tube bank corresponding to the cross-flow sections of the exchanger, modified for the distortion of flow pattern introduced by the baffles and the presence of leakage and bypass flow through the various clearances required by mechanical construction. It is assumed that process conditions and physical properties are known and the following are known or specified: tube outside diameter Do, tube geometrical arrangement (unit cell), shell inside diameter Ds, shell outer tube limit Dotl, baffle cut lc, baffle spacing ls, and number of sealing strips Nss. The effective tube length between tube sheets L may be either specified or calculated after the heat-transfer coefficient has been determined. If additional specific information (e.g., tube-baffle clearance) is available, the exact values (instead of estimates) of certain parameters may be used in the calculation with some improvement in accuracy. To complete the rating, it is necessary to know also the tube material and wall thickness or inside diameter. This rating method, though apparently generally the best in the open literature, is not extremely accurate. An exhaustive study by Palen and Taborek [Chem. Eng. Prog. Symp. Ser. 92, 65, 53 (1969)] showed that this method predicted shell-side coefficients from about 50 percent low to 100 percent high, while the pressure-drop range was from about 50 percent low to 200 percent high. The mean error for heat transfer was about 15 percent low (safe) for all Reynolds numbers, while the mean error for pressure drop was from about 5 percent low (unsafe) at Reynolds numbers above 1000 to about 100 percent high at Reynolds numbers below 10. Calculation of Shell-Side Geometrical Parameters 1. Total number of tubes in exchanger Nt. If not known by direct count, estimate using Eq. (11-84) or (11-85). 2. Tube pitch parallel to flow pp and normal to flow pn. These quantities are needed only for estimating other parameters. If a detailed drawing of the exchanger is available, it is better to obtain these other parameters by direct count or calculation. The pitches are described by Fig. 11-5 and read therefrom for common tube layouts. 3. Number of tube rows crossed in one cross-flow section Nc. Count from exchanger drawing or estimate from

Ds[1 − 2(lc /Ds)] Nc =

(11-7) pp 4. Fraction of total tubes in cross-flow Fc 1 Ds − 2 lc Ds − 2 lc Ds − 2lc Fc =
π + 2
sin cos−1
− 2 cos−1
(11-8) π Dotl Dotl Dotl Fc is plotted in Fig. 11-6. This figure is strictly applicable only to splitring, floating-head construction but may be used for other situations with minor error. 5. Number of effective cross-flow rows in each window Ncw 0.8lc Ncw =
(11-9) pp









Values of tube pitch for common tube layouts. To convert inches to meters, multiply by 0.0254. Not that Do, p′, pp, and pn have units of inches.

FIG. 11-5

11-8

HEAT-TRANSFER EQUIPMENT

FIG. 11-6 Estimation of fraction of tubes in cross-flow Fc [Eq. (11-8)]. To convert inches to meters, multiply by 0.0254. Note that lc and Ds have units of inches.

6. Cross-flow area at or near centerline for one cross-flow section Sm a. For rotated and in-line square layouts: Dotl − Do (11-10a) Sm = ls Ds − Dotl +
(p′ − Do) m2 (ft2) pn b. For triangular layouts: Dotl − Do Sm = ls Ds − Dotl +
(p′ − Do) m2 (ft2) (11-10b) p′ 7. Fraction of cross-flow area available for bypass flow Fbp (Ds − Dotl ) ls Fbp =

(11-11) Sm 8. Tube-to-baffle leakage area for one baffle Stb. Estimate from









Stb = bDo NT(1 + Fc ) m2 (ft2)

(11-12)

where b = (6.223)(10−4) (SI) or (1.701)(10−4) (U.S. customary). These values are based on Tubular Exchanger Manufacturers Association (TEMA) Class R construction which specifies h-in diametral clearance between tube and baffle. Values should be modified if extra tight or loose construction is specified or if clogging by dirt is anticipated. 9. Shell-to-baffle leakage area for one baffle Ssb. If diametral shellbaffle clearance δsb is known, Ssb can be calculated from 2l Ds δsb Ssb =
π − cos−1 1 −
c m2 (ft2) (11-13) 2 Ds −1 where the value of the term cos (1 − 2 lc /Ds) is in radians and is between 0 and π/2. Ssb is plotted in Fig. 11-7, based on TEMA Class R standards. Since pipe shells are generally limited to diameters below 24 in, the larger sizes are shown by using the rolled-shell specification. Allowance should be made for especially tight or loose construction. 10. Area for flow through window Sw. This area is obtained as the difference between the gross window area Swg and the window area occupied by tubes Swt:







FIG. 11-7 Estimation of shell-to-baffle leakage area [Eq. (11-13)]. To convert inches to meters, multiply by 0.0254; to convert square inches to square meters, multiply by (6.45)(10−4). Note that lc and Ds have units of inches.

Sw = Swg − Swt D s2 lc Swg =
cos−1 1 − 2
4 Ds







lc − 1−2
Ds

(11-14)



l 2
  1−1− D 2

c

m2 (ft2) (11-15)

s

Swg is plotted in Fig. 11-8. Swt can be calculated from Swt = (NT/8)(1 − Fc)π D o2

m2 (ft2)

(11-16)

11. Equivalent diameter of window Dw [required only if laminar flow, defined as (NRe)s ≤ 100, exists] 4Sw m (ft) (11-17) Dw =


(π/2)NT(1 − Fc) Do + Dsθb where θb is the baffle-cut angle given by 2 lc rad (11-18) θb = 2 cos−1 1 −
Ds 12. Number of baffles Nb L − 2 le Nb =
+ 1 (11-19) ls where le is the entrance/exit baffle spacing, often different from the central baffle spacing. The effective tube length L must be known to calculate Nb, which is needed to calculate shell-side pressure drop. In designing an exchanger, the shell-side coefficient may be calculated and the required exchanger length for heat transfer obtained before Nb is calculated. Shell-Side Heat-Transfer Coefficient Calculation 1. Calculate the shell-side Reynolds number (NRe)s.





(NRe)s = DoW/µbSm

(11-20)

where W = mass flow rate and µb = viscosity at bulk temperature. The arithmetic mean bulk shell-side fluid temperature is usually adequate

THERMAL DESIGN OF HEAT-TRANSFER EQUIPMENT

FIG. 11-10

11-9

Correction factor for baffle-configuration effects.

W k 2/3 µb 0.14 hk = jkc

(11-21)
Sm cµ µw where c is the specific heat, k is the thermal conductivity, and µw is the viscosity evaluated at the mean surface temperature. 4. Find the correction factor for baffle-configuration effects Jc from Fig. 11-10. 5. Find the correction factor for baffle-leakage effects Jl from Fig. 11-11. 6. Find the correction factor for bundle-bypassing effects Jb from Fig. 11-12 7. Find the correction factor for adverse temperature-gradient buildup at low Reynolds number Jr: a. If (NRe)s < 100, find J*r from Fig. 11-13, knowing Nb and (Nc + Ncw). b. If (NRe)s ≤ 20, Jr = J*r . c. If 20 < (NRe)s < 100, find Jr from Fig. 11-14, knowing J*r and (NRe)s. 8. Calculate the shell-side heat-transfer coefficient for the exchanger hs from

   

FIG. 11-8 Estimation of window cross-flow area [Eq. (11-15)]. To convert inches to meters, multiply by 0.0254. Note that lc and Ds have units of inches.

hs = hk Jc Jl Jb Jr

to evaluate all bulk properties of the shell-side fluid. For large temperature ranges or for viscosity that is very sensitive to temperature change, special care must be taken, such as using Eq. (11-6). 2. Find jk from the ideal-tube bank curve for a given tube layout at the calculated value of (NRe)s, using Fig. 11-9, which is adapted from ideal-tube-bank data obtained at Delaware by Bergelin et al. [Trans. Am. Soc. Mech. Eng., 74, 953 (1952) and the Grimison correlation [Trans. Am. Soc. Mech. Eng., 59, 583 (1937)]. 3. Calculate the shell-side heat-transfer coefficient for an ideal tube bank hk.

Correlation of j factor for ideal tube bank. To convert inches to meters, multiply by 0.0254. Note that p′ and Do have units of inches.

FIG. 11-9

FIG. 11-11

Correction factor for baffle-leakage effects.

(11-22)

11-10

HEAT-TRANSFER EQUIPMENT

FIG. 11-12

Correction factor for bypass flow.

(a)

(b) Correction of friction factors for ideal tube banks. (a) Triangular and rotated square arrays. (b) In-line square arrays.

FIG. 11-15 FIG. 11-13 Basic correction factor for adverse temperature gradient at low Reynolds numbers.

2. Shell-Side Pressure-Drop Calculation 1. Find fk from the ideal-tube-bank friction-factor curve for the given tube layout at the calculated value of (NRe)s, using Fig. 11-15a for triangular and rotated square arrays and Fig. 11-15b for in-line square arrays. These curves are adapted from Bergelin et al. and Grimison (loc. cit.).

Calculate the pressure drop for an ideal cross-flow section. fkW 2Nc µw 0.14 (11-23) ∆Pbk = b

ρSm2 µb

 

where b = (2.0)(10 ) (SI) or (9.9)(10 ) (U.S. customary). 3. Calculate the pressure drop for an ideal window section. If (NRe)s ≥ 100, W 2(2 + 0.6Ncw) (11-24a) ∆Pwk = b

SmSwρ where b = (5)(10−4) (SI) or (2.49)(10−5) (U.S. customary). If (NRe)s < 100, ls µbW Ncw W2 + b2
(11-24b) ∆Pwk = b1

+
SmSwρ p′ − Do Dw2 SmSwρ −3

−5





where b1 = (1.681)(10−5) (SI) or (1.08)(10−4) (U.S. customary), and b2 = (9.99)(10−4) (SI) or (4.97)(10−5) (U.S. customary). 4. Find the correction factor for the effect of baffle leakage on pressure drop Rl from Fig. 11-16. Curves shown are not to be extrapolated beyond the points shown. 5. Find the correction factor for bundle bypass Rb from Fig. 11-17. 6. Calculate the pressure drop across the shell side (excluding nozzles). Units for pressure drop are lbf/ft2. Ncw ∆Ps = [(Nb − 1)(∆ Pbk)Rb + Nb ∆Pwk]Rl + 2 ∆Pbk Rb 1 +
(11-25) Nc The values of hs and ∆Ps calculated by this procedure are for clean exchangers and are intended to be as accurate as possible, not conservative. A fouled exchanger will generally give lower heat-transfer rates, as reflected by the dirt resistances incorporated into Eq. (11-2), and higher pressure drops. Some estimate of fouling effects on pres-



Correction factor for adverse temperature gradient at intermediate Reynolds numbers.

FIG. 11-14



THERMAL DESIGN OF HEAT-TRANSFER EQUIPMENT

11-11

sure drop may be made by using the methods just given by assuming that the fouling deposit blocks the leakage and possibly the bypass areas. The fouling may also decrease the clearance between tubes and significantly increase the pressure drop in cross-flow. THERMAL DESIGN OF CONDENSERS

FIG. 11-16

Correction factor for baffle-leakage effect on pressure drop.

FIG. 11-17

Correction factor on pressure drop for bypass flow.

Single-Component Condensers Mean Temperature Difference In condensing a single component at its saturation temperature, the entire resistance to heat transfer on the condensing side is generally assumed to be in the layer of condensate. A mean condensing coefficient is calculated from the appropriate correlation and combined with the other resistances in Eq. (11-2). The overall coefficient is then used with the LMTD (no FT correction is necessary for isothermal condensation) to give the required area, even though the condensing coefficient and hence U are not constant throughout the condenser. If the vapor is superheated at the inlet, the vapor may first be desuperheated by sensible heat transfer from the vapor. This occurs if the surface temperature is above the saturation temperature, and a single-phase heat-transfer correlation is used. If the surface is below the saturation temperature, condensation will occur directly from the superheated vapor, and the effective coefficient is determined from the appropriate condensation correlation, using the saturation temperature in the LMTD. To determine whether or not condensation will occur directly from the superheated vapor, calculate the surface temperature by assuming single-phase heat transfer. U Tsurface = Tvapor −
(Tvapor − Tcoolant) (11-26) h where h is the sensible heat-transfer coefficient for the vapor, U is calculated by using h, and both are on the same area basis. If Tsurface > Tsaturation, no condensation occurs at that point and the heat flux is actually higher than if Tsurface ≤ Tsaturation and condensation did occur. It is generally conservative to design a pure-component desuperheatercondenser as if the entire heat load were transferred by condensation, using the saturation temperature in the LMTD. The design of an integral condensate subcooling section is more difficult, especially if close temperature approach is required. The condensate layer on the surface is on the average subcooled by onethird to one-half of the temperature drop across the film, and this is often sufficient if the condensate is not reheated by raining through the vapor. If the condensing-subcooling process is carried out inside tubes or in the shell of a vertical condenser, the single-phase subcooling section can be treated separately, giving an area that is added onto that needed for condensation. If the subcooling is achieved on the shell side of a horizontal condenser by flooding some of the bottom tubes with a weir or level controller, the rate and heat-balance equations must be solved for each section to obtain the area required. Pressure drop on the condensing side reduces the final condensing temperature and the MTD and should always be checked. In designs requiring close approach between inlet coolant and exit condensate (subcooled or not), underestimation of pressure drop on the condensing side can lead to an exchanger that cannot meet specified terminal temperatures. Since pressure-drop calculations in two-phase flows such as condensation are relatively inaccurate, designers must consider carefully the consequences of a larger-than-calculated pressure drop. Horizontal In-Shell Condensers The mean condensing coefficient for the outside of a bank of horizontal tubes is calculated from Eq. (5-93) for a single tube, corrected for the number of tubes in a vertical row. For undisturbed laminar flow over all the tubes, Eq. (5-97) is, for realistic condenser sizes, overly conservative because of rippling, splashing, and turbulent flow (Process Heat Transfer, McGraw-Hill, New York, 1950). Kern proposed an exponent of −j on the basis of experience, while Freon-11 data of Short and Brown (General Discussion on Heat Transfer, Institute of Mechanical Engineers, London, 1951) indicate independence of the number of tube rows. It seems reasonable to use no correction for inviscid liquids and Kern’s correction for viscous condensates. For a cylindrical tube bundle, where N varies, it is customary to take N equal to two-thirds of the maximum or centerline value.

11-12

HEAT-TRANSFER EQUIPMENT

Baffles in a horizontal in-shell condenser are oriented with the cuts vertical to facilitate drainage and eliminate the possibility of flooding in the upward cross-flow sections. Pressure drop on the vapor side can be estimated by the data and method of Diehl and Unruh [Pet. Refiner, 36(10), 147 (1957); 37(10), 124 (1958)]. High vapor velocities across the tubes enhance the condensing coefficient. There is no correlation in the open literature to permit designers to take advantage of this. Since the vapor flow rate varies along the length, an incremental calculation procedure would be required in any case. In general, the pressure drops required to gain significant benefit are above those allowed in most process applications. Vertical In-Shell Condensers Condensers are often designed so that condensation occurs on the outside of vertical tubes. Equation (5-88) is valid as long as the condensate film is laminar. When it becomes turbulent, Fig. 5-10 or Colburn’s equation [Trans. Am. Inst. Chem. Eng., 30, 187 (1933–1934) may be used. Some judgment is required in the use of these correlations because of construction features of the condenser. The tubes must be supported by baffles, usually with maximum cut (45 percent of the shell diameter) and maximum spacing to minimize pressure drop. The flow of the condensate is interrupted by the baffles, which may draw off or redistribute the liquid and which will also cause some splashing of free-falling drops onto the tubes. For subcooling, a liquid inventory may be maintained in the bottom end of the shell by means of a weir or a liquid-level-controller. The subcooling heat-transfer coefficient is given by the correlations for natural convection on a vertical surface [Eqs. (5-33a), (5-33b)], with the pool assumed to be well mixed (isothermal) at the subcooled condensate exit temperature. Pressure drop may be estimated by the shell-side procedure. Horizontal In-Tube Condensers Condensation of a vapor inside horizontal tubes occurs in kettle and horizontal thermosiphon reboilers and in air-cooled condensers. In-tube condensation also offers certain advantages for condensation of multicomponent mixtures, discussed in the subsection “Multicomponent Condensers.” The various in-tube correlations are closely connected to the twophase flow pattern in the tube [Chem. Eng. Prog. Symp. Ser., 66(102), 150 (1970)]. At low flow rates, when gravity dominates the flow pattern, Eq. (5-101) may be used. At high flow rates, the flow and heat transfer are governed by vapor shear on the condensate film, and Eq. (5-100a) is valid. A simple and generally conservative procedure is to calculate the coefficient for a given case by both correlations and use the larger one. Pressure drop during condensation inside horizontal tubes can be computed by using the correlations for two-phase flow given in Sec. 6 and neglecting the pressure recovery due to deceleration of the flow. Vertical In-Tube Condensation Vertical-tube condensers are generally designed so that vapor and liquid flow cocurrently downward; if pressure drop is not a limiting consideration, this configuration can result in higher heat-transfer coefficients than shell-side condensation and has particular advantages for multicomponent condensation. If gravity controls, the mean heat-transfer coefficient for condensation is given by Figs. 5-9 and 5-10. If vapor shear controls, Eq. (5-99a) is applicable. It is generally conservative to calculate the coefficients by both methods and choose the higher value. The pressure drop can be calculated by using the Lockhart-Martinelli method [Chem. Eng. Prog., 45, 39 (1945)] for friction loss, neglecting momentum and hydrostatic effects. Vertical in-tube condensers are often designed for reflux or knock-back application in reactors or distillation columns. In this case, vapor flow is upward, countercurrent to the liquid flow on the tube wall; the vapor shear acts to thicken and retard the drainage of the condensate film, reducing the coefficient. Neither the fluid dynamics nor the heat transfer is well understood in this case, but Soliman, Schuster, and Berenson [J. Heat Transfer, 90, 267–276 (1968)] discuss the problem and suggest a computational method. The Diehl-Koppany correlation [Chem. Eng. Prog. Symp. Ser. 92, 65 (1969)] may be used to estimate the maximum allowable vapor velocity at the tube inlet. If the vapor velocity is great enough, the liquid film will be carried upward; this design has been employed in a few cases in which only part of the stream is to be condensed. This veloc-

ity cannot be accurately computed, and a very conservative (high) outlet velocity must be used if unstable flow and flooding are to be avoided; 3 times the vapor velocity given by the Diehl-Koppany correlation for incipient flooding has been suggested as the design value for completely stable operation. Multicomponent Condensers Thermodynamic and Mass-Transfer Considerations Multicomponent vapor mixture includes several different cases: all the components may be liquids at the lowest temperature reached in the condensing side, or there may be components which dissolve substantially in the condensate even though their boiling points are below the exit temperature, or one or more components may be both noncondensable and nearly insoluble. Multicomponent condensation always involves sensible-heat changes in the vapor and liquid along with the latent-heat load. Compositions of both phases in general change through the condenser, and concentration gradients exist in both phases. Temperature and concentration profiles and transport rates at a point in the condenser usually cannot be calculated, but the binary cases have been treated: condensation of one component in the presence of a completely insoluble gas [Colburn and Hougen, Ind. Eng. Chem., 26, 1178–1182 (1934); and Colburn and Edison, Ind. Eng. Chem., 33, 457–458 (1941)] and condensation of a binary vapor [Colburn and Drew, Trans. Am. Inst. Chem. Eng., 33, 196–215 (1937)]. It is necessary to know or calculate diffusion coefficients for the system, and a reasonable approximate method to avoid this difficulty and the reiterative calculations is desirable. To integrate the point conditions over the total condensation requires the temperature, composition enthalpy, and flow-rate profiles as functions of the heat removed. These are calculated from component thermodynamic data if the vapor and liquid are assumed to be in equilibrium at the local vapor temperature. This assumption is not exactly true, since the condensate and the liquid-vapor interface (where equilibrium does exist) are intermediate in temperature between the coolant and the vapor. In calculating the condensing curve, it is generally assumed that the vapor and liquid flow collinearly and in intimate contact so that composition equilibrium is maintained between the total streams at all points. If, however, the condensate drops out of the vapor (as can happen in horizontal shell-side condensation) and flows to the exit without further interaction, the remaining vapor becomes excessively enriched in light components with a decrease in condensing temperature and in the temperature difference between vapor and coolant. The result may be not only a small reduction in the amount of heat transferred in the condenser but also an inability to condense totally the light ends even at reduced throughput or with the addition of more surface. To prevent the liquid from segregating, in-tube condensation is preferred in critical cases. Thermal Design If the controlling resistance for heat and mass transfer in the vapor is sensible-heat removal from the cooling vapor, the following design equation is obtained: A=




QT

0

1 + U′ZH /hsv

dQ U ′(Tv − Tc)

(11-27)

U′ is the overall heat-transfer coefficient between the vapor-liquid interface and the coolant, including condensate film, dirt and wall resistances, and coolant. The condensate film coefficient is calculated from the appropriate equation or correlation for pure vapor condensation for the geometry and flow regime involved, using mean liquid properties. ZH is the ratio of the sensible heat removed from the vapor-gas stream to the total heat transferred; this quantity is obtained from thermodynamic calculations and may vary substantially from one end of the condenser to the other, especially when removing vapor from a noncondensable gas. The sensible-heat-transfer coefficient for the vapor-gas stream hsv is calculated by using the appropriate correlation or design method for the geometry involved, neglecting the presence of the liquid. As the vapor condenses, this coefficient decreases and must be calculated at several points in the process. Tv and Tc are temperatures of the vapor and of the coolant respectively. This procedure is similar in principle to that of Ward [Petro/Chem. Eng., 32(11), 42–48 (1960)]. It may be nonconservative for condensing steam and

THERMAL DESIGN OF HEAT-TRANSFER EQUIPMENT other high-latent-heat substances, in which case it may be necessary to increase the calculated area by 25 to 50 percent. Pressure drop on the condensing side may be estimated by judicious application of the methods suggested for pure-component condensation, taking into account the generally nonlinear decrease of vapor-gas flow rate with heat removal. THERMAL DESIGN OF REBOILERS For a single-component reboiler design, attention is focused upon the mechanism of heat and momentum transfer at the hot surface. In multicomponent systems, the light components are preferentially vaporized at the surface, and the process becomes limited by their rate of diffusion. The net effect is to decrease the effective temperature difference between the hot surface and the bulk of the boiling liquid. If one attempts to vaporize too high a fraction of the feed liquid to the reboiler, the temperature difference between surface and liquid is reduced to the point that nucleation and vapor generation on the surface are suppressed and heat transfer to the liquid proceeds at the lower rate associated with single-phase natural convection. The only safe procedure in design for wide-boiling-range mixtures is to vaporize such a limited fraction of the feed that the boiling point of the remaining liquid mixture is still at least 5.5°C (10°F) below the surface temperature. Positive flow of the unvaporized liquid through and out of the reboiler should be provided. Kettle Reboilers It has been generally assumed that kettle reboilers operate in the pool boiling mode, but with a lower peak heat flux because of vapor binding and blanketing of the upper tubes in the bundle. There is some evidence that vapor generation in the bundle causes a high circulation rate through the bundle. The result is that, at the lower heat fluxes, the kettle reboiler actually gives higher heattransfer coefficients than a single tube. Present understanding of the recirculation phenomenon is insufficient to take advantage of this in design. Available nucleate pool boiling correlations are only very approximate, failing to account for differences in the nucleation characteristics of different surfaces. The Mostinski correlation [Eq. (5-102)] and the McNelly correlation [Eq. (5-103)] are generally the best for single components or narrow-boiling-range mixtures at low fluxes, though they may give errors of 40 to 50 percent. Experimental heat-transfer coefficients for pool boiling of a given liquid on a given surface should be used if available. The bundle peak heat flux is a function of tube-bundle geometry, especially of tube-packing density; in the absence of better information, the Palen-Small modification [Eq. (5-108)] of the Zuber maximum-heat-flux correlation is recommended. A general method for analyzing kettle reboiler performance is by Fair and Klip, Chem. Eng. Prog. 79(3), 86 (1983). It is effectively limited to computer application. Kettle reboilers are generally assumed to require negligible pressure drop. It is important to provide good longitudinal liquid flow paths within the shell so that the liquid is uniformly distributed along the entire length of the tubes and excessive local vaporization and vapor binding are avoided. This method may also be used for the thermal design of horizontal thermosiphon reboilers. The recirculation rate and pressure profile of the thermosiphon loop can be calculated by the methods of Fair [Pet. Refiner, 39(2), 105–123 (1960)]. Vertical Thermosiphon Reboilers Vertical thermosiphon reboilers operate by natural circulation of the liquid from the still through the downcomer to the reboiler and of the two-phase mixture from the reboiler through the return piping. The flow is induced by the hydrostatic pressure imbalance between the liquid in the downcomer and the two-phase mixture in the reboiler tubes. Thermosiphons do not require any pump for recirculation and are generally regarded as less likely to foul in service because of the relatively high two-phase velocities obtained in the tubes. Heavy components are not likely to accumulate in the thermosiphon, but they are more difficult to design satisfactorily than kettle reboilers, especially in vacuum operation. Several shortcut methods have been suggested for thermosiphon design, but they must generally be used with caution. The method due to Fair (loc. cit.), based upon two-phase flow correlations,

11-13

is the most complete in the open literature but requires a computer for practical use. Fair also suggests a shortcut method that is satisfactory for preliminary design and can be reasonably done by hand. Forced-Recirculation Reboilers In forced-recirculation reboilers, a pump is used to ensure circulation of the liquid past the heattransfer surface. Force-recirculation reboilers may be designed so that boiling occurs inside vertical tubes, inside horizontal tubes, or on the shell side. For forced boiling inside vertical tubes, Fair’s method (loc. cit.) may be employed, making only the minor modification that the recirculation rate is fixed and does not need to be balanced against the pressure available in the downcomer. Excess pressure required to circulate the two-phase fluid through the tubes and back into the column is supplied by the pump, which must develop a positive pressure increase in the liquid. Fair’s method may also be modified to design forced-recirculation reboilers with horizontal tubes. In this case the hydrostatic-headpressure effect through the tubes is zero but must be considered in the two-phase return lines to the column. The same procedure may be applied in principle to design of forced-recirculation reboilers with shell-side vapor generation. Little is known about two-phase flow on the shell side, but a reasonable estimate of the friction pressure drop can be made from the data of Diehl and Unruh [Pet. Refiner, 36(10), 147 (1957); 37(10), 124 (1958)]. No void-fraction data are available to permit accurate estimation of the hydrostatic or acceleration terms. These may be roughly estimated by assuming homogeneous flow. THERMAL DESIGN OF EVAPORATORS Heat duties of evaporator heating surfaces are usually determined by conventional heat and material balance calculations. Heating surface areas are normally, but not always taken as those in contact with the material being evaporated. It is the heat transfer ∆T that presents the most difficulty in deriving or applying heat-transfer coefficients. The total ∆T between heat source and heat sink is never all available for heat transfer. Since energy usually is carried to and from an evaporator body or effect by condensible vapors, loss in pressure represents a loss in ∆T. Such losses include pressure drop through entrainment separators, friction in vapor piping, and acceleration losses into and out of the piping. The latter loss has often been overlooked, even though it can be many times greater than the friction loss. Similarly, friction and acceleration losses past the heating surface, such as in a falling film evaporator, cause a loss of ∆T that may or may not have been included in the heat transfer ∆T when reporting experimental results. Boiling-point rise, the difference between the boiling point of the solution and the condensing point of the solvent at the same pressure, is another loss. Experimental data are almost always corrected for boiling-point rise, but plant data are suspect when based on temperature measurements because vapor at the point of measurement may still contain some superheat, which represents but a very small fraction of the heat given up when the vapor condenses but may represent a substantial fraction of the actual net ∆T available for heat transfer. A ∆T loss that must be considered in forced-circulation evaporators is that due to temperature rise through the heater, a consequence of the heat being absorbed there as sensible heat. A further loss may occur when the heater effluent flashes as it enters the vapor-liquid separator. Some of the liquid may not reach the surface and flash to equilibrium with the vapor pressure in the separator, instead of recirculating to the heater, raising the average temperature at which heat is absorbed and further reducing the net ∆T. Whether or not these ∆T losses are allowed for in the heattransfer coefficients reported depends on the method of measurement. Simply basing the liquid temperature on the measured vapor head pressure may ignore both—or only the latter if temperature rise through the heater is estimated separately from known heat input and circulation rate. In general, when calculating overall heat-transfer coefficients from individual-film coefficients, all of these losses must be allowed for, while when using reported overall coefficients care must be exercised to determine which losses may already have been included in the heat transfer ∆T. Forced-Circulation Evaporators In evaporators of this type in which hydrostatic head prevents boiling at the heating surface, heat-

11-14

HEAT-TRANSFER EQUIPMENT

transfer coefficients can be predicted from the usual correlations for condensing steam (Fig. 5-10) and forced-convection sensible heating [Eq. (5-50)]. The liquid film coefficient is improved if boiling is not completely suppressed. When only the film next to the wall is above the boiling point, Boarts, Badger, and Meisenberg [Ind. Eng. Chem., 29, 912 (1937)] found that results could be correlated by Eq. (5-50) by using a constant of 0.0278 instead of 0.023. In such cases, the course of the liquid temperature can still be calculated from known circulation rate and heat input. When the bulk of the liquid is boiling in part of the tube length, the film coefficient is even higher. However, the liquid temperature starts dropping as soon as full boiling develops, and it is difficult to estimate the course of the temperature curve. It is certainly safe to estimate heat transfer on the basis that no bulk boiling occurs. Fragen and Badger [Ind. Eng. Chem., 28, 534 (1936)] obtained an empirical correlation of overall heat-transfer coefficients in this type of evaporator, based on the ∆T at the heater inlet: In U.S. customary units U = 2020D0.57(Vs)3.6/L/µ0.25 ∆T 0.1

(11-28)

where D = mean tube diameter, Vs = inlet velocity, L = tube length, and µ = liquid viscosity. This equation is based primarily on experiments with copper tubes of 0.022 m (8/8 in) outside diameter, 0.00165 m (16 gauge), 2.44 m (8 ft) long, but it includes some work with 0.0127-m (a-in) tubes 2.44 m (8 ft) long and 0.0254-m (1-in) tubes 3.66 m (12 ft) long. Long-Tube Vertical Evaporators In the rising-film version of this type of evaporator, there is usually a nonboiling zone in the bottom section and a boiling zone in the top section. The length of the nonboiling zone depends on heat-transfer characteristics in the two zones and on pressure drop during two-phase flow in the boiling zone. The work of Martinelli and coworkers [Lockhart and Martinelli, Chem. Eng. Prog., 45, 39–48 (January 1949); and Martinelli and Nelson, Trans. Am. Soc. Mech. Eng., 70, 695–702 (August 1948)] permits a prediction of pressure drop, and a number of correlations are available for estimating film coefficients of heat transfer in the two zones. In estimating pressure drop, integrated curves similar to those presented by Martinelli and Nelson are the easiest to use. The curves for pure water are shown in Figs. 11-18 and 11-19, based on the assumption that the flow of both vapor and liquid would be turbulent if each were flowing alone in the tube. Similar curves can be prepared if one or both flows are laminar or if the properties of the liquid differ appreciably from the properties of pure water. The acceleration pressure drop ∆Pa is calculated from the equation

FIG. 11-19

Friction pressure drop in boiling flow. °C = (°F − 32)/1.8.

∆Pa = br2G2/32.2

(11-29)

where b = (2.6)(10 )(SI) and 1.0 (U.S. customary) and using r2 from Fig. 11-18. The frictional pressure drop is derived from Fig. 11-19, which shows the ratio of two-phase pressure drop to that of the entering liquid flowing alone. Pressure drop due to hydrostatic head can be calculated from liquid holdup R1. For nonfoaming dilute aqueous solutions, R1 can be estimated from R1 = 1/[1 + 2.5(V/L)(ρ1 /ρv)1/2]. Liquid holdup, which represents the ratio of liquid-only velocity to actual liquid velocity, also appears to be the principal determinant of the convective coefficient in the boiling zone (Dengler, Sc.D. thesis, MIT, 1952). In other words, the convective coefficient is that calculated from Eq. (5-50) by using the liquid-only velocity divided by R1 in the Reynolds number. Nucleate boiling augments convective heat transfer, primarily when ∆T’s are high and the convective coefficient is low [Chen, Ind. Eng. Chem. Process Des. Dev., 5, 322 (1966)]. Film coefficients for the boiling of liquids other than water have been investigated. Coulson and McNelly [Trans. Inst. Chem. Eng., 34, 247 (1956)] derived the following relation, which also correlated the data of Badger and coworkers [Chem. Metall. Eng., 46, 640 (1939); Chem. Eng., 61(2), 183 (1954); and Trans. Am. Inst. Chem. Eng., 33, 392 (1937); 35, 17 (1939); 36, 759 (1940)] on water: ρl 0.25 µg 0.23 0.34 NNu = (1.3 + b D)(NPr)0.9 (11-30)

l (NRe) l (NRe)g ρg µl where b = 128 (SI) or 39 (U.S. customary), NNu = Nusselt number based on liquid thermal conductivity, D = tube diameter, and the remaining terms are dimensionless groupings of liquid Prandtl number, liquid Reynolds number, vapor Reynolds number, and ratios of densities and viscosities. The Reynolds numbers are calculated on the basis of each fluid flowing by itself in the tube. Additional corrections must be applied when the fraction of vapor is so high that the remaining liquid does not wet the tube wall or when the velocity of the mixture at the tube exits approaches sonic velocity. McAdams, Woods, and Bryan (Trans. Am. Soc. Mech. Eng., 1940), Dengler and Addoms (loc. cit.), and Stroebe, Baker, and Badger [Ind. Eng. Chem., 31, 200 (1939)] encountered dry-wall conditions and reduced coefficients when the weight fraction of vapor exceeded about 80 percent. Schweppe and Foust [Chem. Eng. Prog., 49, Symp. Ser. 5, 77 (1953)] and Harvey and Foust (ibid., p. 91) found that “sonic choking” occurred at surprisingly low flow rates. The simplified method of calculation outlined includes no allowance for the effect of surface tension. Stroebe, Baker, and Badger (loc. cit.) found that by adding a small amount of surface7

   

FIG. 11-18

Acceleration losses in boiling flow. °C = (°F − 32)/1.8.

THERMAL DESIGN OF HEAT-TRANSFER EQUIPMENT

11-15

active agent the boiling-film coefficient varied inversely as the square of the surface tension. Coulson and Mehta [Trans. Inst. Chem. Eng., 31, 208 (1953)] found the exponent to be −1.4. The higher coefficients at low surface tension are offset to some extent by a higher pressure drop, probably because the more intimate mixture existing at low surface tension causes the liquid fraction to be accelerated to a velocity closer to that of the vapor. The pressure drop due to acceleration ∆Pa derived from Fig. 11-18 allows for some slippage. In the limiting case, such as might be approached at low surface tension, the acceleration pressure drop in which “fog” flow is assumed (no slippage) can be determined from the equation y(Vg − Vl )G2 ∆P′a =

gc where

(11-31)

y = fraction vapor by weight Vg, Vl = specific volume gas, liquid G = mass velocity

While the foregoing methods are valuable for detailed evaporator design or for evaluating the effect of changes in conditions on performance, they are cumbersome to use when making preliminary designs or cost estimates. Figure 11-20 gives the general range of overall long-tube vertical- (LTV) evaporator heat-transfer coefficients usually encountered in commercial practice. The higher coefficients are encountered when evaporating dilute solutions and the lower range when evaporating viscous liquids. The dashed curve represents the approximate lower limit, for liquids with viscosities of about 0.1 Pa⋅s (100 cP). The LTV evaporator does not work well at low temperature differences, as indicated by the results shown in Fig. 11-21 for seawater in 0.051-m (2-in), 0.0028-m (12-gauge) brass tubes 7.32 m (24 ft) long (W. L. Badger Associates, Inc., U.S. Department of the Interior, Office of Saline Water Rep. 26, December 1959, OTS Publ. PB 161290). The feed was at its boiling point at the vapor-head pressure, and feed rates varied from 0.025 to 0.050 kg/(s⋅tube) [200 to 400 lb/(h⋅tube)] at the higher temperature to 0.038 to 0.125 kg/ (s⋅tube) [300 to 1000 lb/(h⋅tube)] at the lowest temperature. Falling film evaporators find their widest use at low temperature differences—also at low temperatures. Under most operating conditions encountered, heat transfer is almost all by pure convection, with a negligible contribution from nucleate boiling. Film coefficients on the condensing side may be estimated from Dukler’s correlation, [Chem. Eng. Prog. 55, 62 1950]. The same Dukler correlation presents curves covering falling film heat transfer to nonboiling liquids that are equally applicable to the falling film evaporator [Sinek and Young, Chem. Eng. Prog. 58, No. 12, 74 (1962)]. Kunz and Yerazunis [ J. Heat Transfer 8, 413 (1969)] have

General range of long-tube vertical- (LTV) evaporator coefficients. °C = (°F − 32)/1.8; to convert British thermal units per hour-square foot-degrees Fahrenheit to joules per square meter-second-kelvins, multiply by 5.6783.

FIG. 11-20

FIG. 11-21 Heat-transfer coefficients in LTV seawater evaporators. °C = (°F − 32)/1.8; to convert British thermal units per hour-square foot-degrees Fahrenheit to joules per square meter-second-kelvins, multiply by 5.6783.

since extended the range of physical properties covered, as shown in Fig. 11-22. The boiling point in the tubes of such an evaporator is higher than in the vapor head because of both frictional-pressure drop and the head needed to accelerate the vapor to the tube-exit velocity. These factors, which can easily be predicted, make the overall apparent coefficients somewhat lower than those for nonboiling conditions. Figure 11-21 shows overall apparent heat-transfer coefficients determined in a falling-film seawater evaporator using the same tubes and flow rates as for the rising-film tests (W. L. Badger Associates, Inc., loc. cit.). Short-Tube Vertical Evaporators Coefficients can be estimated by the same detailed method described for recirculating LTV evaporators. Performance is primarily a function of temperature level, temperature difference, and viscosity. While liquid level can also have an important influence, this is usually encountered only at levels lower than considered safe in commercial operation. Overall heat-transfer coefficients are shown in Fig. 11-23 for a basket-type evaporator (one with an annular downtake) when boiling water with 0.051-m (2-in) outside-diameter 0.0028-m-wall (12-gauge), 1.22-m-(4-ft-) long steel tubes [Badger and Shepard, Chem. Metall. Eng., 23, 281 (1920)]. Liquid level was maintained at the top tube sheet. Foust, Baker, and Badger [Ind. Eng. Chem., 31, 206 (1939)] measured recirculating velocities and heat-transfer coefficients in the same evaporator except with 0.064-m (2.5-in) 0.0034-m-wall (10-gauge), 1.22-m- (4-ft-) long tubes and temperature differences from 7 to 26°C (12 to 46°F). In the normal range of liquid levels, their results can be expressed as 0.4 b(∆Tc)0.22N Pr Uc =

(11-32) 0.37 (Vg − Vl) where b = 153 (SI) or 375 (U.S. customary) and the subscript c refers to true liquid temperature, which under these conditions was about 0.56°C (1°F) above the vapor-head temperature. This work was done with water. No detailed tests have been reported for the performance of propeller calandrias. Not enough is known regarding the performance of the propellers themselves under the cavitating conditions usually encountered to permit predicting circulation rates. In many cases, it appears that the propeller does no good in accelerating heat transfer over the transfer for natural circulation (Fig. 11-23). Miscellaneous Evaporator Types Horizontal-tube evaporators operating with partially or fully submerged heating surfaces behave in much the same way as short-tube verticals, and heattransfer coefficients are of the same order of magnitude. Some test results for water were published by Badger [Trans. Am. Inst. Chem. Eng., 13, 139 (1921)]. When operating unsubmerged, their heat transfer performance is roughly comparable to the falling-film vertical tube evaporator. Condensing coefficients inside the tubes can be derived from Nusselt’s theory which, based on a constant-heat flux rather than a constant film ∆T, gives:

11-16

HEAT-TRANSFER EQUIPMENT

5 4

h/(K3ρ2g/µ2 ) 1/3

2

1000 400 200

NPr

6 4

1.0

NPr

100 60 40 20 10

3

2 0.6 1 0.4

Nus

selt

0.2 10

FIG. 11-22

20

50

100

200

500

1000 NRe = 4Γ/µ

2000

5000

10,000

20,000

50,000 100,000

Kunz and Yerazunis Correlation for falling-film heat transfer.

h (11-33a)

= 1.59(4Γ/µ)−1/3 (k3ρ2g/µ2)1/3 For the boiling side, a correlation based on seawater tests gave: h (11-33b)

= 0.0147(4Γ/µ)1/3(D)−1/3 (k3ρ2g/µ2)1/3 where Γ is based on feed-rate per unit length of the top tube in each vertical row of tubes and D is in meters. Heat-transfer coefficients in clean coiled-tube evaporators for seawater are shown in Fig. 11-24 [Hillier, Proc. Inst. Mech. Eng. (London), 1B(7), 295 (1953)]. The tubes were of copper. Heat-transfer coefficients in agitated-film evaporators depend primarily on liquid viscosity. This type is usually justifiable only for very viscous materials. Figure 11-25 shows general ranges of overall coefficients [Hauschild, Chem. Ing. Tech., 25, 573 (1953); Lindsey, Chem. Eng., 60(4), 227 (1953); and Leniger and Veldstra, Chem. Ing. Tech., 31, 493 (1959)]. When used with nonviscous fluids, a wipedfilm evaporator having fluted external surfaces can exhibit very high coefficients (Lustenader et al., Trans. Am. Soc. Mech. Eng., Paper 59SA-30, 1959), although at a probably unwarranted first cost. Heat Transfer from Various Metal Surfaces In an early work, Pridgeon and Badger [Ind. Eng. Chem., 16, 474 (1924)] published test results on copper and iron tubes in a horizontal-tube evaporator that indicated an extreme effect of surface cleanliness on heattransfer coefficients. However, the high degree of cleanliness needed for high coefficients was difficult to achieve, and the tube layout and

liquid level were changed during the course of the tests so as to make direct comparison of results difficult. Other workers have found little or no effect of conditions of surface or tube material on boiling-film coefficients in the range of commercial operating conditions [Averin, Izv. Akad. Nauk SSSR Otd. Tekh. Nauk, no. 3, p. 116, 1954; and Coulson and McNelly, Trans. Inst. Chem. Eng., 34, 247 (1956)]. Work in connection with desalination of seawater has shown that specially modified surfaces can have a profound effect on heattransfer coefficients in evaporators. Figure 11-26 (Alexander and Hoffman, Oak Ridge National Laboratory TM-2203) compares overall coefficients for some of these surfaces when boiling fresh water in 0.051-m (2-in) tubes 2.44-m (8-ft) long at atmospheric pressure in both upflow and downflow. The area basis used was the nominal outside area. Tube 20 was a smooth 0.0016-m- (0.062-in-) wall aluminum brass tube that had accumulated about 6 years of fouling in seawater service and exhibited a fouling resistance of about (2.6)(10−5) (m2 ⋅s⋅K)/ J [0.00015 (ft2 ⋅ h⋅°F)/Btu]. Tube 23 was a clean aluminum tube with 20 spiral corrugations of 0.0032-m (f-in) radius on a 0.254-m (10-in) pitch indented into the tube. Tube 48 was a clean copper tube that had 50 longitudinal flutes pressed into the wall (General Electric double-flute profile, Diedrich, U.S. Patent 3,244,601, Apr. 5, 1966). Tubes 47 and 39 had a specially patterned porous sintered-metal deposit on the boiling side to promote nucleate boiling (Minton, U.S.

Heat-transfer coefficients for water in short-tube evaporators. °C = (°F − 32)/1.8; to convert British thermal units per hour-square foot-degrees Fahrenheit to joules per square meter-second-kelvins, multiply by 5.6783.

FIG. 11-24

FIG. 11-23

Heat-transfer coefficients for seawater in coil-tube evaporators. °C = (°F − 32)/1.8; to convert British thermal units per hour-square foot-degrees Fahrenheit to joules per square meter-second-kelvins, multiply by 5.6783.

THERMAL DESIGN OF HEAT-TRANSFER EQUIPMENT

11-17

where b = 2100 (SI) or 1180 (U.S. customary). The boiling-side coefficient (based on actual inside area) for salt water in downflow may be approximated from the equation h = 0.035(k3ρ2g/µ2)1/3(4Γ/µ)1/3

Overall heat-transfer coefficients in agitated-film evaporators. °C = (°F − 32)/1.8; to convert British thermal units per hour-square foot-degrees Fahrenheit to joules per square meter-second-kelvins, multiply by 5.6783; to convert centipoises to pascal-seconds, multiply by 10−3.

FIG. 11-25

Patent 3,384,154, May 21, 1968). Both of these tubes also had steamside coatings to promote dropwise condensation—parylene for tube 47 and gold plating for tube 39. Of these special surfaces, only the double-fluted tube has seen extended services. Most of the gain in heat-transfer coefficient is due to the condensing side; the flutes tend to collect the condensate and leave the lands bare [Carnavos, Proc. First Int. Symp. Water Desalination, 2, 205 (1965)]. The condensing-film coefficient (based on the actual outside area, which is 28 percent greater than the nominal area) may be approximated from the equation k3ρ2g 1/3 µλ 1/3 q −0.833 h=b
(11-34a)

µ2 L A



    

FIG. 11-26 Heat-transfer coefficients for enhanced surfaces. °C = (°F − 32) /1.8; to convert British thermal units per hour-square foot-degrees Fahrenheit to joules per square meter-second-kelvins, multiply by 5.6783. (By permission from Oak Ridge National Laboratory TM-2203.)

(11-34b)

The boiling-film coefficient is about 30 percent lower for pure water than it is for salt water or seawater. There is as yet no accepted explanation for the superior performance in salt water. This phenomenon is also seen in evaporation from smooth tubes. Effect of Fluid Properties on Heat Transfer Most of the heattransfer data reported in the preceding paragraphs were obtained with water or with dilute solutions having properties close to those of water. Heat transfer with other materials will depend on the type of evaporator used. For forced-circulation evaporators, methods have been presented to calculate the effect of changes in fluid properties. For natural-circulation evaporators, viscosity is the most important variable as far as aqueous solutions are concerned. Badger (Heat Transfer and Evaporation, Chemical Catalog, New York, 1926, pp. 133–134) found that, as a rough rule, overall heat-transfer coefficients varied in inverse proportion to viscosity if the boiling film was the main resistance to heat transfer. When handling molasses solutions in a forcedcirculation evaporator in which boiling was allowed to occur in the tubes, Coates and Badger [Trans. Am. Inst. Chem. Eng., 32, 49 (1936)] found that from 0.005 to 0.03 Pa⋅s (5 to 30 cP) the overall heat-transfer coefficient could be represented by U = b/µ1.24 f , where b = 2.55 (SI) or 7043 (U.S. customary). Fragen and Badger [Ind. Eng. Chem., 28, 534 (1936)] correlated overall coefficients on sugar and sulfite liquor in the same evaporator for viscosities to 0.242 Pa⋅s (242 cP) and found a relationship that included the viscosity raised only to the 0.25 power. Little work has been published on the effect of viscosity on heat transfer in the long-tube vertical evaporator. Cessna, Leintz, and Badger [Trans. Am. Inst. Chem. Eng., 36, 759 (1940)] found that the overall coefficient in the nonboiling zone varied inversely as the 0.7 power of viscosity (with sugar solutions). Coulson and Mehta [Trans. Inst. Chem. Eng., 31, 208 (1953)] found the exponent to be −0.44, and Stroebe, Baker, and Badger (loc. cit.) arrived at an exponent of −0.3 for the effect of viscosity on the film coefficient in the boiling zone. Kerr (Louisiana Agr. Exp. Sta. Bull. 149) obtained plant data shown in Fig. 11-27 on various types of full-sized evaporators for cane sugar. These are invariably forward-feed evaporators concentrating to about 50° Brix, corresponding to a viscosity on the order of 0.005 Pa⋅s (5 cP) in the last effect. In Fig. 11-27 curve A is for short-tube verticals with central downtake, B is for standard horizontal tube evaporators, C is for Lillie evaporators (which were horizontal-tube machines with no liquor level but having recirculating liquor showered over the tubes), and D is for long-tube vertical evaporators. These curves show apparent coefficients, but sugar solutions have boiling-point rises low enough not to affect the results noticeably. Kerr also obtained the data shown in Fig.

FIG. 11-27 Kerr’s tests with full-sized sugar evaporators. °C = (°F − 32)/1.8; to convert British thermal units per hour-square foot-degrees Fahrenheit to joules per square meter-second-kelvins, multiply by 5.6783.

11-18

HEAT-TRANSFER EQUIPMENT and over the entire surface, (2) liquid flow rates are constant, (3) specific heats are constant for the process, (4) the heating or cooling medium has a constant inlet temperature, (5) agitation produces a uniform batch fluid temperature, (6) no partial phase changes occur, and (7) heat losses are negligible. The developed equations are as follows. If any of the assumptions do not apply to a system being designed, new equations should be developed or appropriate corrections made. Heat exchangers are counterflow except for the 1-2 exchangers, which are one-shell-pass, two-tube-pass, parallel-flow counterflow. Coil-in-Tank or Jacketed Vessel: Isothermal Heating Medium ln (T1 − t1)/(T1 − t2) = UAθ /Mc

(11-35)

Cooling-in-Tank or Jacketed Vessel: Isothermal Cooling Medium ln (T1 − t1)/(T2 − t1) = UAθ /MC

Effect of viscosity on heat transfer in short-tube vertical evaporator. To convert centipoises to pascal-seconds, multiply by 10−3; to convert British thermal units per hour-square foot-degrees Fahrenheit to joules per square meter-second-kelvins, multiply by 5.6783. FIG. 11-28

11-28 on a laboratory short-tube vertical evaporator with 0.44- by 0.61m (1e- by 24-in) tubes. This work was done with sugar juices boiling at 57°C (135°F) and an 11°C (20°F) temperature difference. Effect of Noncondensables on Heat Transfer Most of the heat transfer in evaporators does not occur from pure steam but from vapor evolved in a preceding effect. This vapor usually contains inert gases— from air leakage if the preceding effect was under vacuum, from air entrained or dissolved in the feed, or from gases liberated by decomposition reactions. To prevent these inerts from seriously impeding heat transfer, the gases must be channeled past the heating surface and vented from the system while the gas concentration is still quite low. The influence of inert gases on heat transfer is due partially to the effect on ∆T of lowering the partial pressure and hence condensing temperature of the steam. The primary effect, however, results from the formation at the heating surface of an insulating blanket of gas through which the steam must diffuse before it can condense. The latter effect can be treated as an added resistance or fouling factor equal to 6.5 × 10−5 times the local mole percent inert gas (in J−1⋅s⋅m2⋅K) [Standiford, Chem. Eng. Prog., 75, 59–62 ( July 1979)]. The effect on ∆T is readily calculated from Dalton’s law. Inert-gas concentrations may vary by a factor of 100 or more between vapor inlet and vent outlet, so these relationships should be integrated through the tube bundle. BATCH OPERATIONS: HEATING AND COOLING OF VESSELS Nomenclature (Use consistent units.) A = heat-transfer surface; C, c = specific heats of hot and cold fluids respectively; L0 = flow rate of liquid added to tank; M = mass of fluid in tank; T, t = temperature of hot and cold fluids respectively; T1, t1 = temperatures at beginning of heating or cooling period or at inlet; T2, t2 = temperature at end of period or at outlet; T0, t0 = temperature of liquid added to tank; U = coefficient of heat transfer; and W, w = flow rate through external exchanger of hot and cold fluids respectively. Applications One typical application in heat transfer with batch operations is the heating of a reactor mix, maintaining temperature during a reaction period, and then cooling the products after the reaction is complete. This subsection is concerned with the heating and cooling of such systems in either unknown or specified periods. The technique for deriving expressions relating time for heating or cooling agitated batches to coil or jacket area, heat-transfer coefficients, and the heat capacity of the vessel contents was developed by Bowman, Mueller, and Nagle [Trans. Am. Soc. Mech. Eng., 62, 283– 294 (1940)] and extended by Fisher [Ind. Eng. Chem., 36, 939–942 (1944)] and Chaddock and Sanders [Trans. Am. Inst. Chem. Eng., 40, 203–210 (1944)] to external heat exchangers. Kern (Process Heat Transfer, McGraw-Hill, New York, 1950, Chap. 18) collected and published the results of these investigators. The assumptions made were that (1) U is constant for the process

(11-35a)

Coil-in-Tank or Jacketed Vessel: Nonisothermal Heating Medium T1 − t1 WC K1 − 1 (11-35b) ln
=

θ T1 − t2 Mc K1 UA/WC where K1 = e Coil-in-Tank: Nonisothermal Cooling Medium T1 − t1 wc K2 − 1 (11-35c) ln
=

θ T2 − t1 MC K2 where K2 = eUA/wc External Heat Exchanger: Isothermal Heating Medium T1 − t1 wc K2 − 1 (11-35d) ln
=

θ T1 − t2 Mc K2 External Exchanger: Isothermal Cooling Medium T1 − t1 WC K1 − 1 (11-35e) ln
=

θ T2 − t1 MC K1 External Exchanger: Nonisothermal Heating Medium T1 − t1 K3 − 1 wWC (11-35f) ln
=


θ T1 − t2 M K3wc − WC where K3 = eUA(1/WC − 1/wc) External Exchanger: Nonisothermal Cooling Medium T1 − t1 K4 − 1 Wwc (11-35g) ln
=


θ T2 − t1 M K4wc − WC where K4 = eUA(1/WC − 1/wc External Exchanger with Liquid Continuously Added to Tank: Isothermal Heating Medium





























w K2 − 1 t1 − t0 −



(T1 − t1) L0 K2 ln


w K2 − 1 t2 − t0 −



(T1 − t2) L0 K2

 

 

M + L0θ w K2 − 1 =

+ 1 ln
M L0 K2

 

 

(11-35h)

If the addition of liquid to the tank causes an average endothermic or exothermic heat of solution, qsJ/kg (Btu/ lb) of makeup, it may be included by adding qs /c0 to both the numerator and the denominator of the left side. The subscript 0 refers to the makeup. External Exchanger with Liquid Continuously Added to Tank: Isothermal Cooling Medium W K1 − 1 T0 − T1 −



(T1 − t1) L0 K1 ln



W K1 − 1 T0 − T2 −



(T2 − t1) L0 K1

 

 

W K1 − 1 = 1−

L0 K1





M+Lθ  ln
M 0

(11-35i)

THERMAL DESIGN OF HEAT-TRANSFER EQUIPMENT The heat-of-solution effects can be included by adding qs /C0 to both the numerator and the denominator of the left side. External Exchanger with Liquid Continuously Added to Tank: Nonisothermal Heating Medium

ln

wWC(K 5 − 1)(T1 − t1) t0 − t1 +


L0(K 5WC − wc) wWC(K 5 − 1)(T1 − t2) t0 − t2 +


L0(K 5WC − wc ) M + L0θ wWC(K5 − 1) =

+ 1 ln
M L0 (K 5Wc − wc)





(11-35j)

where K5 = e(UA/wc)(1 − wc/WC) The heat-of-solution effects can be included by adding qs /c0 to both the numerator and the denominator of the left side. External Exchanger with Liquid Continuously Added to Tank: Nonisothermal Cooling Medium

ln

Wwc(K6 − 1)(T1 − t1) T0 − T1 −


L0(K6wc − WC) Wwc(K6 − 1)(T2 − t1) T0 − T2 −


L0(K6wc − WC) M + L0θ Wwc(K6 − 1) =

+ 1 ln
M L0(K6wc − WC)





(11-35k)

where K6 = e(UA/WC)(1 − WC/wc) The heat-of-solution effects can be included by adding qs /C0 to both the numerator and the denominator of the left side. Heating and Cooling Agitated Batches: 1-2 Parallel FlowCounterflow UA 1

2 +

1) 2 − S(R + 1 − R ln



=
2 wc R

+

1

2 +

1) 2 − S(R + 1 + R T1 − T2 wc R=
=
t′ − t WC

and

(11-35l)

t′ − t S=
T1 − t

2(K 7 − 1) S =



K 7 (R + 1 + R2 + 1) − (R + 1 − R2 + 1) External 1-2 Exchanger: Heating ln (T1 − t1)/(T1 − t2) = (Sw/M)θ

(11-35n)

External 1-2 Exchanger: Cooling ln (T1 − t1)/(T2 − t1) = S(wc/MC)θ

of the flow streams. Increasing flow rates to give values well above 0.8 is a matter of economic justification. If FT varies widely from one end of the range to the other, FT should be determined for one or more intermediate points. The average should then be determined for each step which has been established and the average of these taken for use in step 6. Effect of External Heat Loss or Gain If heat loss or gain through the vessel walls cannot be neglected, equations which include this heat transfer can be developed by using energy balances similar to those used for the derivations of equations given previously. Basically, these equations must be modified by adding a heat-loss or heat-gain term. A simpler procedure, which is probably acceptable for most practical cases, is to ratio UA or θ either up or down in accordance with the required modification in total heat load over time θ. Another procedure, which is more accurate for the external-heatexchanger cases, is to use an equivalent value for MC (for a vessel being heated) derived from the following energy balance: Q = (Mc)e(t2 − t1) = Mc(t2 − t1) + U′A′(MTD′ )θ

(11-35o)

The cases of multipass exchangers with liquid continuously added to the tank are covered by Kern, as cited earlier. An alternative method for all multipass-exchanger gases, including those presented as well as cases with two or more shells in series, is as follows: 1. Determine UA for using the applicable equations for counterflow heat exchangers. 2. Use the initial batch temperature T1 or t1. 3. Calculate the outlet temperature from the exchanger of each fluid. (This will require trial-and-error methods.) 4. Note the FT correction factor for the corrected mean temperature difference. (See Fig. 11-4.) 5. Repeat steps 2, 3, and 4 by using the final batch temperature T2 and t2. 6. Use the average of the two values for F, then increase the required multipass UA as follows: UA(multipass) = UA(counterflow)/FT In general, values of FT below 0.8 are uneconomical and should be avoided. FT can be raised by increasing the flow rate of either or both

(11-35p)

where Q is the total heat transferred over time θ, U′A′ is the heattransfer coefficient for heat loss times the area for heat loss, and MTD′ is the mean temperature difference for the heat loss. A similar energy balance would apply to a vessel being cooled. Internal Coil or Jacket Plus External Heat Exchanger This case can be most simply handled by treating it as two separate problems. M is divided into two separate masses M1 and (M − M1), and the appropriate equations given earlier are applied to each part of the system. Time θ, of course, must be the same for both parts. Equivalent-Area Concept The preceding equations for batch operations, particularly Eq. 11-35 can be applied for the calculation of heat loss from tanks which are allowed to cool over an extended period of time. However, different surfaces of a tank, such as the top (which would not be in contact with the tank contents) and the bottom, may have coefficients of heat transfer which are different from those of the vertical tank walls. The simplest way to resolve this difficulty is to use an equivalent area Ae in the appropriate equations where Ae = A bUb /Us + A tUt /Us + As

R2

+ 1) 2 − S(R + 1 −

2


= e(UA/wc) R

+ 1 = K7 (11-35m) 2 − S(R + 1 +

R2

+ 1)

11-19

(11-35q)

and the subscripts b, s, and t refer to the bottom, sides, and top respectively. U is usually taken as Us. Table 11-1 lists typical values for Us and expressions for Ae for various tank configurations. Nonagitated Batches Cases in which vessel contents are vertically stratified, rather than uniform in temperature, have been treated by Kern (op. cit.). These are of little practical importance except for tall, slender vessels heated or cooled with external exchangers. The result is that a smaller exchanger is required than for an equivalent agitated batch system that is uniform. Storage Tanks The equations for batch operations with agitation may be applied to storage tanks even though the tanks are not agitated. This approach gives conservative results. The important cases (nonsteady state) are: 1. Tanks cool; contents remain liquid. This case is relatively simple and can easily be handled by the equations given earlier. 2. Tanks cool, contents partially freeze, and solids drop to bottom or rise to top. This case requires a two-step calculation. The first step is handled as in case 1. The second step is calculated by assuming an isothermal system at the freezing point. It is possible, given time and a sufficiently low ambient temperature, for tank contents to freeze solid. 3. Tanks cool and partially freeze; solids form a layer of selfinsulation. This complex case, which has been known to occur with heavy hydrocarbons and mixtures of hydrocarbons, has been discussed by Stuhlbarg [Pet. Refiner, 38, 143 (Apr. 1, 1959)]. The contents in the center of such tanks have been known to remain warm and liquid even after several years of cooling. It is very important that a melt-out riser be installed whenever tank contents are expected to freeze on prolonged shutdown. The purpose is to provide a molten chimney through the crust for relief of thermal expansion or cavitation if fluids are to be pumped out or recirculated through an external exchanger. An external heat tracer, properly

11-20

HEAT-TRANSFER EQUIPMENT

TABLE 11-1

Typical Values for Use with Eqs. (11-36) to (11-44)* Application

Tanks on legs, outdoors, not insulated Tanks on legs, outdoors, insulated 1 in. Tanks on legs, indoors, not insulated Tanks on legs, indoors, insulated 1 in. Flat-bottom tanks,† outdoors, not insulated Flat-bottom tanks,† outdoors, insulated 1 in. Flat-bottom tanks, indoors, not insulated Flat-bottom tanks, indoors, insulated 1 in.

Fluid

Us

As

Oil Water at 150°F. Oil Water Oil Water Oil Water Oil Water Oil Water Oil Water Oil Water

3.7 5.1 0.45 0.43 1.5 1.8 0.36 0.37 3.7 5.1 0.36 0.37 1.5 1.8 0.36 0.37

0.22 At + Ab + As 0.16 At + Ab + As 0.7 At + Ab + As 0.67 At + Ab + As 0.53 At + Ab + As 0.35 At + Ab + As 0.8 At + Ab + As 0.73 At + Ab + As 0.22 At + As + 0.43 Dt 0.16 At + As + 0.31 Dt 0.7 At + As + 3.9 Dt 0.16 At + As + 3.7 Dt 0.53 At + As + 1.1 Dt 0.35 At + As + 0.9 Dt 0.8 At + As + 4.4 Dt 0.73 At + As + 4.5 Dt

*Based on typical coefficients. †The ratio (t − tg)(t − t′) assumed at 0.85 for outdoor tanks. °C = (°F − 32)/1.8; to convert British thermal units per hour-square foot-degrees Fahrenheit to joules per square meter-second-kelvins, multiply by 5.6783.

located, will serve the same purpose but may require more remelt time before pumping can be started.

qF Ac =

Uc(th − td)m

(11-39)

where F is a safety factor. THERMAL DESIGN OF TANK COILS The thermal design of tank coils involves the determination of the area of heat-transfer surface required to maintain the contents of the tank at a constant temperature or to raise or lower the temperature of the contents by a specified magnitude over a fixed time. Nomenclature A = area; A b = area of tank bottom; Ac = area of coil; Ae = equivalent area; As = area of sides; A t = area of top; A1 = equivalent area receiving heat from external coils; A2 = equivalent area not covered with external coils; Dt = diameter of tank; F = design (safety) factor; h = film coefficient; ha = coefficient of ambient air; hc = coefficient of coil; h h = coefficient of heating medium; hi = coefficient of liquid phase of tank contents or tube-side coefficient referred to outside of coil; hz = coefficient of insulation; k = thermal conductivity; kg = thermal conductivity of ground below tank; M = mass of tank contents when full; t = temperature; ta = temperature of ambient air; td = temperature of dead-air space; tf = temperature of contents at end of heating; tg = temperature of ground below tank; th = temperature of heating medium; t0 = temperature of contents at beginning of heating; U = overall coefficient; Ub = coefficient at tank bottom; Uc = coefficient of coil; Ud = coefficient of dead air to the tank contents; Ui = coefficient through insulation; Us = coefficient at sides; Ut = coefficient at top; and U2 = coefficient at area A 2. Typical coil coefficients are listed in Table 11-2. More exact values can be calculated by using the methods for natural convection or forced convection given elsewhere in this section. Maintenance of Temperature Tanks are often maintained at temperature with internal coils if the following equations are assumed to be applicable:

and

q = Us Ae(T − t′)

(11-36)

Ac = q/Uc(MTD)

(11-36a)

These make no allowance for unexpected shutdowns. One method of allowing for shutdown is to add a safety factor to Eq. 11-36a. In the case of a tank maintained at temperature with internal coils, the coils are usually designed to cover only a portion of the tank. The temperature td of the dead-air space between the coils and the tank is obtained from Ud A1(td − t) = U2 A2(t − t′)

(11-37)

q = Ud A1(td − t) + A1Ui (td − t′)

(11-38)

The heat load is The coil area is

Heating Heating with Internal Coil from Initial Temperature for Specified Time (11-40) Q = Wc(t f − to) Q tf + to 1 (11-41) Ac =
+ Us Ae
− t′

(F) θh 2 Uc [th − (tf + to)/2] where θ h is the length of heating period. This equation may also be used when the tank contents have cooled from tf to to and must be reheated to tf. If the contents cool during a time θc, the temperature at the end of this cooling period is obtained from Us Aeθc tf − t′ ln
=
(11-42) Wc to − t′ Heating with External Coil from Initial Temperature for Specified Time The temperature of the dead-air space is obtained from





 







Ud A1[td − 0.5(tf − to)] = U2 A2[0.5(tf − to) − t′] + Q/θ h

(11-43)

The heat load is q = Ui A1(td − t′ ) + U2 A2[0.5(tf − to) − t′] + Q/θ h

(11-44)

The coil area is obtained from Eq. 11-39. The safety factor used in the calculations is a matter of judgment based on confidence in the design. A value of 1.10 is normally not considered excessive. Typical design parameters are shown in Tables 11-1 and 11-2. HEATING AND COOLING OF TANKS Tank Coils Pipe tank coils are made in a wide variety of configurations, depending upon the application and shape of the vessel. Helical and spiral coils are most commonly shop-fabricated, while the hairpin pattern is generally field-fabricated. The helical coils are used principally in process tanks and pressure vessels when large areas for rapid heating or cooling are required. In general, heating coils are placed low in the tank, and cooling coils are placed high or distributed uniformly through the vertical height. Stocks which tend to solidify on cooling require uniform coverage of the bottom or agitation. A maximum spacing of 0.6 m (2 ft) between turns of 50.8-mm (2-in) and larger pipe and a close approach to the tank wall are recommended. For smaller pipe or for low-temperature heating media, closer spacing should be used. In the case of the com-

THERMAL DESIGN OF HEAT-TRANSFER EQUIPMENT TABLE 11-2

11-21

Overall Heat-Transfer Coefficients for Coils Immersed in Liquids U Expressed as Btu/(h ⋅ ft2 ⋅ °F)

Substance inside coil Steam Steam Steam Cold water Cold water Cold water Brine Cold water Water Water Steam Milk Cold water 60°F. water Steam and hydrogen at 1500 lb./sq. in. Steam 110–146 lb./ sq. in. gage Steam Cold water

Substance outside coil

Coil material

Water Sugar and molasses solutions Boiling aqueous solution Dilute organic dye intermediate Warm water Hot water Amino acids 25% oleum at 60°C. Aqueous solution 8% NaOH Fatty acid Water Hot water 50% aqueous sugar solution 60°F. water

Agitation

U

Lead Copper

Agitated None

Lead

Turboagitator at 95 r.p.m.

70 50–240 600 300

Wrought iron

Air bubbled into water surrounding coil 0.40 r.p.m. paddle stirrer 30 r.p.m. Agitated 500 r.p.m. sleeve propeller 22 r.p.m. None Agitation None Mild

Lead Wrought iron Lead Copper (pancake) Copper Lead Steel

150–300 90–360 100 20 250 155 96–100 300 105–180 50–60 100–165

Vegetable oil

Steel

None

23–29

Vegetable oil Vegetable oil

Steel Steel

Various Various

39–72 29–72

NOTES: Chilton, Drew, and Jebens [Ind. Eng. Chem., 36, 510 (1944)] give film coefficients for heating and cooling agitated fluids using a coil in a jacketed vessel. Because of the many factors affecting heat transfer, such as viscosity, temperature difference, and coil size, the values in this table should be used primarily for preliminary design estimates and checking calculated coefficients. °C = (°F − 32)/1.8; to convert British thermal units per hour-square foot-degrees Fahrenheit to joules per square meter-second-kelvins, multiply by 5.6783.

mon hairpin coils in vertical cylindrical tanks, this means adding an encircling ring within 152 mm (6 in) of the tank wall (see Fig. 11-29a for this and other typical coil layouts). The coils should be set directly on the bottom or raised not more than 50.8 to 152 mm (2 to 6 in), depending upon the difficulty of remelting the solids, in order to permit free movement of product within the vessel. The coil inlet should be above the liquid level (or an internal melt-out riser installed) to provide a molten path for liquid expansion or venting of vapors. Coils may be sloped to facilitate drainage. When it is impossible to do so and remain close enough to the bottom to get proper remelting, the coils should be blown out after usage in cold weather to avoid damage by freezing. Most coils are firmly clamped (but not welded) to supports. Supports should allow expansion but be rigid enough to prevent uncontrolled motion (see Fig. 11-29b). Nuts and bolts should be securely fastened. Reinforcement of the inlet and outlet connections through the tank wall is recommended, since bending stresses due to thermal expansion are usually high at such points. In general, 50.8- and 63.4-mm (2- and 2a-in) coils are the most economical for shop fabrication and 38.1- and 50.8-mm (1a- and 2-in) for field fabrication. The tube-side heat-transfer coefficient, high-pressure, or layout problems may lead to the use of smaller-size pipe. The wall thickness selected varies with the service and material. Carbon steel coils are often made from schedule 80 or heavier pipe to allow for corrosion. When stainless-steel or other high-alloy coils are not subject to corrosion or excessive pressure, they may be of schedule 5 or 10 pipe to keep costs at a minimum, although high-quality welding is required for these thin walls to assure trouble-free service.

(a)

(b)

Methods for calculating heat loss from tanks and the sizing of tank coils have been published by Stuhlbarg [Pet. Refiner, 38, 143 (April 1959)]. Fin-tube coils are used for fluids which have poor heat-transfer characteristics to provide more surface for the same configuration at reduced cost or when temperature-driven fouling is to be minimized. Fin tubing is not generally used when bottom coverage is important. Fin-tube tank heaters are compact prefabricated bundles which can be brought into tanks through manholes. These are normally installed vertically with longitudinal fins to produce good convection currents. To keep the heaters low in the tank, they can be installed horizontally with helical fins or with perforated longitudinal fins to prevent entrapment. Fin tubing is often used for heat-sensitive material because of the lower surface temperature for the same heating medium, resulting in a lesser tendency to foul. Plate or panel coils made from two metal sheets with one or both embossed to form passages for a heating or cooling medium can be used in lieu of pipe coils. Panel coils are relatively light in weight, easy to install, and easily removed for cleaning. They are available in a range of standard sizes and in both flat and curved patterns. Process tanks have been built by using panel coils for the sides or bottom. A serpentine construction is generally utilized when liquid flows through the unit. Header-type construction is used with steam or other condensing media. Standard glass coils with 0.18 to 11.1 m2 (2 to 120 ft2) of heattransfer surface are available. Also available are plate-type units made of impervious graphite. Teflon Immersion Coils Immersion coils made of Teflon fluorocarbon resin are available with 2.5-mm (0.10-in) ID tubes to

(c)

(d)

Typical coil designs for good bottom coverage. (a) Elevated inlet on spiral coil. (b) Spiral with recircling ring. (c) Hairpin with encircling ring. (d) Ring header type. FIG. 11-29a

11-22

HEAT-TRANSFER EQUIPMENT Spiral baffles, which are sometimes installed for liquid services to improve heat transfer and prevent channeling, can be designed to serve as reinforcements. A spiral-wound channel welded to the vessel wall is an alternative to the spiral baffle which is more predictable in performance, since cross-baffle leakage is eliminated, and is reportedly lower in cost [Feichtinger, Chem. Eng., 67, 197 (Sept. 5, 1960)]. The half-pipe jacket is used when high jacket pressures are required. The flow pattern of a liquid heat-transfer fluid can be controlled and designed for effective heat transfer. The dimple jacket offers structural advantages and is the most economical for high jacket pressures. The low volumetric capacity produces a fast response to temperature changes. EXTENDED OR FINNED SURFACES

FIG. 11-29b

Right and wrong ways to support coils. [Chem. Eng., 172 (May

16, 1960).]

increase overall heat-transfer efficiency. The flexible bundles are available with 100, 160, 280, 500, and 650 tubes with standard lengths varying in 0.6-m (2-ft) increments between 1.2 and 4.8 m (4 and 16 ft). These coils are most commonly used in metal-finishing baths and are adaptable to service in reaction vessels, crystallizers, and tanks where corrosive fluids are used. Bayonet Heaters A bayonet-tube element consists of an outer and an inner tube. These elements are inserted into tanks and process vessels for heating and cooling purposes. Often the outer tube is of expensive alloy or nonmetallic (e.g., glass, impervious graphite), while the inner tube is of carbon steel. In glass construction, elements with 50.8- or 76.2-mm (2- or 3-in) glass pipe [with lengths to 2.7 m (9 ft)] are in contact with the external fluid, with an inner tube of metal. External Coils and Tracers Tanks, vessels, and pipe lines can be equipped for heating or cooling purposes with external coils. These are generally 9.8 to 19 mm (r to e in) so as to provide good distribution over the surface and are often of soft copper or aluminum, which can be bent by hand to the contour of the tank or line. When necessary to avoid “hot spots,” the tracer is so mounted that it does not touch the tank. External coils spaced away from the tank wall exhibit a coefficient of around 5.7 W/(m2 ⋅°C) [1 Btu/(h⋅ft2 of coil surface⋅°F)]. Direct contact with the tank wall produces higher coefficients, but these are difficult to predict since they are strongly dependent upon the degree of contact. The use of heat-transfer cements does improve performance. These puttylike materials of high thermal conductivity are troweled or caulked into the space between the coil and the tank or pipe surface. Costs of the cements (in 1960) varied from 37 to 63 cents per pound, with requirements running from about 0.27 lb/ft of r-in outside-diameter tubing to 1.48 lb/ft of 1-in pipe. Panel coils require a to 1 lb/ft2. A rule of thumb for preliminary estimating is that the per-foot installed cost of tracer with cement is about double that of the tracer alone. Jacketed Vessels Jacketing is often used for vessels needing frequent cleaning and for glass-lined vessels which are difficult to equip with internal coils. The jacket eliminates the need for the coil yet gives a better overall coefficient than external coils. However, only a limited heat-transfer area is available. The conventional jacket is of simple construction and is frequently used. It is most effective with a condensing vapor. A liquid heat-transfer fluid does not maintain uniform flow characteristics in such a jacket. Nozzles, which set up a swirling motion in the jacket, are effective in improving heat transfer. Wall thicknesses are often high unless reinforcement rings are installed.

Finned-Surface Application Extended or finned surfaces are often used when one film coefficient is substantially lower than the other, the goal being to make ho Aoe ≈ hi Ai . A few typical fin configurations are shown in Fig. 11-30a. Longitudinal fins are used in double-pipe exchangers. Transverse fins are used in cross-flow and shell-and-tube configurations. High transverse fins are used mainly with low-pressure gases; low fins are used for boiling and condensation of nonaqueous streams as well as for sensible-heat transfer. Finned surfaces have been proven to be a successful means of controlling temperature driven fouling such as coking and scaling. Fin spacing should be great enough to avoid entrapment of particulate matter in the fluid stream (5mm Minimum Spacing). The area added by the fin is not as efficient for heat transfer as bare tube surface owing to resistance to conduction through the fin. The effective heat-transfer area is Aoe = A u f + Af Ω

(11-45)

The fin efficiency is found from mathematically derived relations, in which the film heat-transfer coefficient is assumed to be constant over the entire fin and temperature gradients across the thickness of the fin have been neglected (see Kraus, Extended Surfaces, Spartan Books, Baltimore, 1963). The efficiency curves for some common fin configurations are given in Figs. 11-30a and 11-30b. High Fins To calculate heat-transfer coefficients for cross-flow to a transversely finned surface, it is best to use a correlation based on experimental data for that surface. Such data are not often available, and a more general correlation must be used, making allowance for the possible error. Probably the best general correlation for bundles of finned tubes is given by Schmidt [Kaltetechnik, 15, 98–102, 370–378 (1963)]: 1/3 hDr /k = K(Dr ρV′max /µ)0.625Rf−0.375N Pr

FIG. 11-30a

Efficiencies for several longitudinal fin configurations.

(11-46)

THERMAL DESIGN OF HEAT-TRANSFER EQUIPMENT

FIG. 11-30b

Efficiencies for annular fins of constant thickness.

where K = 0.45 for staggered tube arrays and 0.30 for in-line tube arrays: Dr is the root or base diameter of the tube; V′max is the maximum velocity through the tube bank, i.e., the velocity through the minimum flow area between adjacent tubes; and Rf is the ratio of the total outside surface area of the tube (including fins) to the surface of a tube having the same root diameter but without fins. Pressure drop is particularly sensitive to geometrical parameters, and available correlations should be extrapolated to geometries different from those on which the correlation is based only with great caution and conservatism. The best correlation is that of Robinson and Briggs [Chem. Eng. Prog., 62, Symp. Ser. 64, 177–184 (1966)]. Low Fins Low-finned tubing is generally used in shell-and-tube configurations. For sensible-heat transfer, only minor modifications are needed to permit the shell-side method given earlier to be used for both heat transfer and pressure [see Briggs, Katz, and Young, Chem. Eng. Prog., 59(11), 49–59 (1963)]. For condensing on low-finned tubes in horizontal bundles, the Nusselt correlation is generally satisfactory for low-surface-tension [σ < (3)(10−6)N/m (30 dyn/cm)] condensates fins of finned surfaces should not be closely spaced for high-surfacetension condensates (notably water), which do not drain easily. The modified Palen-Small method can be employed for reboiler design using finned tubes, but the maximum flux is calculated from Ao, the total outside heat-transfer area including fins. The resulting value of qmax refers to Ao. FOULING AND SCALING Fouling refers to any change in the solid boundary separating two heat transfer fluids, whether by dirt accumulation or other means, which results in a decrease in the rate of heat transfer occurring across that boundary. Fouling may be classified by mechanism into six basic categories: 1. Corrosion fouling. The heat transfer surface reacts chemically with elements of the fluid stream producing a less conductive, corrosion layer on all or part of the surface. 2. Biofouling. Organisms present in the fluid stream are attracted to the warm heat-transfer surface where they attach, grow, and reproduce. The two subgroups are microbiofoulants such as slime and algae and macrobiofoulants such as snails and barnacles. 3. Particulate fouling. Particles held in suspension in the flow stream will deposit out on the heat-transfer surface in areas of sufficiently lower velocity. 4. Chemical reaction fouling (ex.—Coking). Chemical reaction of the fluid takes place on the heat-transfer surface producing an adhering solid product of reaction. 5. Precipitation fouling (ex.—Scaling). A fluid containing some dissolved material becomes supersaturated with respect to this mate-

11-23

rial at the temperatures seen at the heat-transfer surface. This results in a crystallization of the material which “plates out” on the warmer surface. 6. Freezing fouling. Overcooling of a fluid below the fluid’s freezing point at the heat-transfer surface causes solidification and coating of the heat-transfer surface. Control of Fouling Once the combination of mechanisms contributing to a particular fouling problem are recognized, methods to substantially reduce the fouling rate may be implemented. For the case of corrosion fouling, the common solution is to choose a less corrosive material of construction balancing material cost with equipment life. In cases of biofouling, the use of copper alloys and/or chemical treatment of the fluid stream to control organism growth and reproduction are the most common solutions. In the case of particulate fouling, one of the more common types, insuring a sufficient flow velocity and minimizing areas of lower velocities and stagnant flows to help keep particles in suspension is the most common means of dealing with the problem. For water, the recommended tubeside minimum velocity is about 0.9 to 1.0 m/s. This may not always be possible for moderate to high-viscosity fluids where the resulting pressure drop can be prohibitive. Special care should be taken in the application of any velocity requirement to the shellside of segmental-baffled bundles due to the many different flow streams and velocities present during operation, the unavoidable existence of high-fouling areas of flow stagnation, and the danger of flow-induced tube vibration. In general, shellsideparticulate fouling will be greatest for segmentally baffled bundles in the regions of low velocity and the TEMA-fouling factors (which are based upon the use of this bundle type) should be used. However, since the 1940’s, there have been a host of successful, low-fouling exchangers developed, some tubular and some not, which have in common the elimination of the cross-flow plate baffle and provide practically no regions of flow stagnation at the heat-transfer surface. Some examples are the plate and frame exchanger, the spiral plate exchanger, and the twisted tube exchanger, all of which have dispensed with baffles altogether and use the heat-transfer surface itself for bundle support. The general rule for these designs is to provide between 25 and 30 percent excess surface to compensate for potential fouling, although this can vary in special applications. For the remaining classifications—polymerization, precipitation, and freezing—fouling is the direct result of temperature extremes at the heat-transfer surface and is reduced by reducing the temperature difference between the heat-transfer surface and the bulk-fluid stream. Conventional wisdom says to increase velocity, thus increasing the local heat-transfer coefficient to bring the heat-transfer surface temperature closer to the bulk-fluid temperature. However, due to a practical limit on the amount of heat-transfer coefficient increase available by increasing velocity, this approach, although better than nothing, is often not satisfactory by itself. A more effective means of reducing the temperature difference is by using, in concert with adequate velocities, some form of extended surface. As discussed by Shilling (Proceedings of the 10th International Heat Transfer Conference, Brighton, U.K., 4, p. 423), this will tend to reduce the temperature extremes between fluid and heat transfer surface and not only reduce the rate of fouling but make the heat exchanger generally less sensitive to the effects of any fouling that does occur. In cases where unfinned tubing in a triangular tube layout would not be acceptable because fouling buildup and eventual mechanical cleaning are inevitable, extended surface should be used only when the exchanger construction allows access for cleaning. Fouling Transients and Operating Periods Three common behaviors are noted in the development of a fouling film over a period of time. One is the so-called asymptotic fouling in which the speed of fouling resistance increase decreases over time as it approaches some asymptotic value beyond which no further fouling can occur. This is commonly found in temperature-driven fouling. A second is linear fouling in which the increase in fouling resistance follows a straight line over the time of operation. This could be experienced in a case of severe particulate fouling where the accumulation of dirt during the time of operation did not appreciably increase velocities to mitigate the problem. The third, falling rate fouling, is neither linear nor asymptotic but instead lies somewhere between these two extremes.

11-24

HEAT-TRANSFER EQUIPMENT

The rate of fouling decreases with time but does not appear to approach an asymptotic maximum during the time of operation. This is the most common type of fouling in the process industry and is usually the result of a combination of different fouling mechanisms occurring together. The optimum operating period between cleanings depends upon the rate and type of fouling, the heat exchanger used (i.e. baffle type, use of extended surface, and velocity and pressure drop design constraints), and the ease with which the heat exchanger may be removed from service for cleaning. As noted above, care must be taken in the use of fouling factors for exchanger design, especially if the exchanger configuration has been selected specifically to minimize fouling accumulation. An oversurfaced heat exchanger which will not foul enough to operate properly can be almost as much a problem as an undersized exchanger. This is especially true in steam-heated exchangers where the ratio of design MTD to minimum achievable MTD is less than U_clean divided by U_fouled. Removal of Fouling Deposits Chemical removal of fouling can be achieved in some cases by weak acid, special solvents, and so on. Other deposits adhere weakly and can be washed off by periodic operation at very high velocities or by flushing with a high-velocity steam or water jet or using a sand-water slurry. These methods may be applied to both the shell side and tube side without pulling the bundle. Many fouling deposits, however, must be removed by positive mechanical action such as rodding, turbining, or scraping the surface. These techniques may be applied inside of tubes without pulling the bundle but can be applied on the shellside only after bundle removal. Even then there is limited access because of the tube pitch and rotated square or large triangular layouts are recommended. In many cases, it has been found that designs developed to minimize fouling often develop a fouling layer which is more easily removed. Fouling Resistances There are no published methods for predicting fouling resistances a priori. The accumulated experience of exchanger designers and users was assembled more than 40 years ago based primarily upon segmental-baffled exchanger bundles and may be found in the Standards of Tubular Exchanger Manufacturers Association (TEMA). In the absence of other information, the fouling resistances contained therein may be used. TYPICAL HEAT-TRANSFER COEFFICIENTS Typical overall heat-transfer coefficients are given in Tables 11-3 through 11-8. Values from these tables may be used for preliminary estimating purposes. They should not be used in place of the design methods described elsewhere in this section, although they may serve as a useful check on the results obtained by those design methods.



 

0.83 hDt /k = b1bD0.17 (cµ/k) t (DpG/µ)

Solids in divided form, such as powders, pellets, and lumps, are heated and/or cooled in chemical processing for a variety of objectives such as solidification or fusing (Sec. 11), drying and water removal (Sec. 20), solvent recovery (Secs. 13 and 20), sublimation (Sec. 17), chemical reactions (Sec. 20), and oxidation. For process and mechanical-design considerations, see the referenced sections. Thermal design concerns itself with sizing the equipment to effect the heat transfer necessary to carry on the process. The design equation is the familiar one basic to all modes of heat transfer, namely, (11-47)

where A = effective heat-transfer surface, Q = quantity of heat required to be transferred, ∆ t = temperature difference of the process, and U = overall heat-transfer coefficient. It is helpful to define the modes of heat transfer and the corresponding overall coefficient as Uco = overall heat-transfer coefficient for (indirect througha-wall) conduction, Uco = overall heat-transfer coefficient for the little-used convection mechanism, Uct = heat-transfer coefficient for the contactive mechanism in which the gaseous-phase heat carrier passes directly through the solids bed, and Ura = heat-transfer coefficient for radiation. There are two general methods for determining numerical values

(11-49a)

where b1 = 1.22 (SI) or 1.0 (U.S. customary), h = Uco = overall coefficient between the inner container surface and the fluid stream, D D 2 b = .2366 + .0092
p − 4.0672
p Dt Dt

 

THERMAL DESIGN FOR SOLIDS PROCESSING

A = Q/U ∆ t

for Uco, Ucv , Uct, and Ura. One is by analysis of actual operating data. Values so obtained are used on geometrically similar systems of a size not too different from the equipment from which the data were obtained. The second method is predictive and is based on the material properties and certain operating parameters. Relative values of the coefficients for the various modes of heat transfer at temperatures up to 980°C (1800°F) are as follows (Holt, Paper 11, Fourth National Heat Transfer Conference, Buffalo, 1960): Convective 1 Radiant 2 Conductive 20 Contactive 200 Because heat-transfer equipment for solids is generally an adaptation of a primarily material-handling device, the area of heat transfer is often small in relation to the overall size of the equipment. Also peculiar to solids heat transfer is that the ∆t varies for the different heat-transfer mechanisms. With a knowledge of these mechanisms, the ∆t term generally is readily estimated from temperature limitations imposed by the burden characteristics and/or the construction. Conductive Heat Transfer Heat-transfer equipment in which heat is transferred by conduction is so constructed that the solids load (burden) is separated from the heating medium by a wall. For a high proportion of applications, ∆t is the log-mean temperature difference. Values of Uco are reported in Secs. 11, 15, 17, and 19. A predictive equation for Uco is h 2ca (11-48) Uco =


h − 2ca/dm dm where h = wall film coefficient, c = volumetric heat capacity, dm = depth of the burden, and α = thermal diffusivity. Relevant thermal properties of various materials are given in Table 11-9. For details of terminology, equation development, numerical values of terms in typical equipment and use, see Holt [Chem. Eng., 69, 107 (Jan. 8, 1962)]. Equation (11-48) is applicable to burdens in the solid, liquid, or gaseous phase, either static or in laminar motion; it is applicable to solidification equipment and to divided-solids equipment such as metal belts, moving trays, stationary vertical tubes, and stationaryshell fluidizers. Fixed (or packed) bed operation occurs when the fluid velocity is low or the particle size is large so that fluidization does not occur. For such operation, Jakob (Heat Transfer, vol. 2, Wiley, New York, 1957) gives

 

 

 

D 3 D 4 + 18.229
p − 11.837
p (11.49b) Dt Dt Dp = particle diameter, Dt = vessel diameter, (note that Dp /Dt has units of foot per foot in the equation), G = superficial mass velocity, k = fluid thermal conductivity, µ = fluid viscosity, and c = fluid specific heat. Other correlations are those of Leva [Ind. Eng. Chem., 42, 2498 (1950)]: k DpG 0.90 D h = 0.813
e−6Dp /Dt
for
p < 0.35 (11-50a) Dt µ Dt









D k DpG 0.75 for 0.35 <
p < 0.60 (11-50b) h = 0.125

Dt µ Dt and Calderbank and Pogerski [Trans. Inst. Chem. Eng. (London), 35, 195 (1957)]: hDp /k = 3.6(DpG/µv )0.365

(11-51)

where v = fraction voids in the bed. A technique for calculating radial temperature gradients in a packed bed is given by Smith (Chemical Engineering Kinetics, McGraw-Hill, New York, 1956).

THERMAL DESIGN OF HEAT-TRANSFER EQUIPMENT TABLE 11-3

11-25

Typical Overall Heat-Transfer Coefficients in Tubular Heat Exchangers U = Btu/(°F ⋅ ft2 ⋅ h)

Shell side

Tube side

Design U

Includes total dirt

Liquid-liquid media Aroclor 1248 Cutback asphalt Demineralized water Ethanol amine (MEA or DEA) 10–25% solutions Fuel oil Fuel oil Gasoline Heavy oils Heavy oils Hydrogen-rich reformer stream Kerosene or gas oil Kerosene or gas oil Kerosene or jet fuels Jacket water Lube oil (low viscosity) Lube oil (high viscosity) Lube oil Naphtha Naphtha Organic solvents Organic solvents Organic solvents Tall oil derivatives, vegetable oil, etc. Water Water Wax distillate Wax distillate

Jet fuels Water Water Water or DEA, or MEA solutions Water Oil Water Heavy oils Water Hydrogen-rich reformer stream Water Oil Trichlorethylene Water Water Water Oil Water Oil Water Brine Organic solvents Water

100–150 10–20 300–500 140–200

0.0015 .01 .001 .003

15–25 10–15 60–100 10–40 15–50 90–120

.007 .008 .003 .004 .005 .002

25–50 20–35 40–50 230–300 25–50 40–80 11–20 50–70 25–35 50–150 35–90 20–60 20–50

.005 .005 .0015 .002 .002 .003 .006 .005 .005 .003 .003 .002 .004

Caustic soda solutions (10–30%) Water Water Oil

100–250

.003

200–250 15–25 13–23

.003 .005 .005

Water Dowtherm vapor Tall oil and derivatives

100–200 40–60 60–80

Tube side Dowtherm liquid Steam Water Water Oil Water Water or brine Water or brine Water Oil Water Oil Water Feed water No. 6 fuel oil No. 2 fuel oil Water Water Aromatic vapor-stream azeotrope

Includes total dirt

80–120 40–50 20–50 80–200 25–40

.0015 .0055 .003 .003 .004

100–200 20–60 50–120 30–65 20–30 50–75 20–30 80–120 400–1000 15–25 60–90 150–200 20–50

.003 .003 .003 .004 .005 .005 .005 .003 .0005 .0055 .0025 .003 .004

40–80

.005

40–80 10–50 20–40 5–20 80–125

.005 .005 .005 .005 .003

150–300 150–300 40–60

.0015 .0015 .0015

200–300 250–400

.0015 .0015

Gas-liquid media Air, N2, etc. (compressed) Air, N2, etc., A Water or brine Water or brine Water

Water or brine Water or brine Air, N2 (compressed) Air, N2, etc., A Hydrogen containing natural-gas mixtures

Anhydrous ammonia Chlorine Chlorine

Steam condensing Steam condensing Light heat-transfer oil Steam condensing Steam condensing

Vaporizers

Condensing vapor-liquid media Alcohol vapor Asphalt (450°F.) Dowtherm vapor

Shell side Dowtherm vapor Gas-plant tar High-boiling hydrocarbons V Low-boiling hydrocarbons A Hydrocarbon vapors (partial condenser) Organic solvents A Organic solvents high NC, A Organic solvents low NC, V Kerosene Kerosene Naphtha Naphtha Stabilizer reflux vapors Steam Steam Steam Sulfur dioxide Tall-oil derivatives, vegetable oils (vapor) Water

Design U

.002 .006 .004

Propane, butane, etc. Water

NC = noncondensable gas present. V = vacuum. A = atmospheric pressure. Dirt (or fouling factor) units are (h ⋅ ft2 ⋅ °F)/Btu. To convert British thermal units per hour-square foot-degrees Fahrenheit to joules per square meter-second-kelvins, multiply by 5.6783; to convert hours per square foot-degree Fahrenheit-British thermal units to square meters per second-kelvin-joules, multiply by 0.1761.

TABLE 11-4

Typical Overall Heat-Transfer Coefficients in Refinery Service Btu/(°F ⋅ ft2 ⋅ h)

Fluid A B C D E F G H J K

Propane Butane 400°F. end-point gasoline Virgin light naphtha Virgin heavy naphtha Kerosene Light gas oil Heavy gas oil Reduced crude Heavy fuel oil (tar)

API gravity

Fouling factor (one stream)

50 70 45 40 30 22 17 10

0.001 .001 .001 .001 .001 .001 .002 .003 .005 .005

Reboiler, steamheated

Condenser, watercooled*

160 155 120 140 95 85 70 60

95 90 80 85 75 60 50 45

Exchangers, liquid to liquid (tube-side fluid designation appears below)

Reboiler (heating liquid designated below)

C

G

H

C

G†

K

85 80 70 70 65 60 60 55 55 50

85 75 65 55 55 55 50 50 45 40

80 75 60 55 50 50 50 45 40 35

110 105 65 75 55

95 90 50 60 45 45 40 40

35 35 30 35 30 25 25 20

50

Condenser (cooling liquid designated below) D

F

G

J

80

55

40

30

50 50 45 40

35 35 30 30

30 30 30 20

75 70 70 70

Fouling factor, water side 0.0002; heating or cooling streams are shown at top of columns as C, D, F, G, etc.; to convert British thermal units per hour-square footdegrees Fahrenheit to joules per square meter-second-kelvins, multiply by 5.6783; to convert hours per square foot-degree Fahrenheit-British thermal units to square meters per second-kelvin-joules, multiply by 0.1761. *Cooler, water-cooled, rates are about 5 percent lower. †With heavy gas oil (H) as heating medium, rates are about 5 percent lower.

11-26

HEAT-TRANSFER EQUIPMENT

TABLE 11-5 Overall Coefficients for Air-Cooled Exchangers on Bare-Tube Basis Btu/(°F ⋅ ft2 ⋅ h)

where b = 0.29 (SI) or 11.6 (U.S. customary), cs = heat capacity of solid, ρs = particle density, η = fluidization efficiency (Fig. 11-31) and Nf = bed expansion ratio (Fig. 11-32). For internal walls, Wen and Fau give

Condensing

Coefficient

Liquid cooling

Coefficient

hi = bhG−0.37

Ammonia Freon-12 Gasoline Light hydrocarbons Light naphtha Heavy naphtha Reformer reactor effluent Low-pressure steam Overhead vapors

110 70 80 90 75 65

Engine-jacket water Fuel oil Light gas oil Light hydrocarbons Light naphtha Reformer liquid streams Residuum Tar

125 25 65 85 70

where b = 0.78 (SI) or 9 (U.S. customary), hi is the coefficient for internal walls, and h is calculated from Eq. (11-51a). Gmf, the minimum fluidizing velocity, is defined by 0.9 2 bρ1.1 g (ρs − ρg) Dp Gmf =

(11-51c) µ where b = (1.23)(10−2) (SI) or (5.23)(105) (U.S. customary). Wender and Cooper [Am. Inst. Chem. Eng. J., 4, 15 (1958)] developed an empirical correlation for internal walls: hDp /k k 0.43 DpG 0.23 cs 0.80 ρs 0.66
= bCR
(11-52a)


ρg 1 − v cρ µ cg where b = (3.51)(10−4) (SI) or 0.033 (U.S. customary) and CR = correction for displacement of the immersed tube from the axis of the vessel (see the reference). For external walls: hDp (11-52b)

= f(1 + 7.5e−x) kg(1 − v)(csρs /cgρg) where x = 0.44LH cs /Dtcg and f is given by Fig. 11-33. An important feature of this equation is inclusion of the ratio of bed depth to vessel diameter LH/Dt. For dilute fluidized beds on the shell side of an unbaffled tubular bundle Genetti and Knudsen [Inst. Chem. Eng. (London) Symp. Ser. 3,172 (1968)] obtained the relation:

Gas cooling

70 135 65 Operating pressure, lb./sq. in. gage

Pressure drop, lb./sq. in.

50 100 100 35 125 1000

0.1 to 0.5 2 5 1 3 5

Air or flue gas Hydrocarbon gas Ammonia reactor stream

70 15 7

Coefficient 10 20 30 35 55 80 85

Bare-tube external surface is 0.262 ft2/ft. Fin-tube surface/bare-tube surface ratio is 16.9. To convert British thermal units per hour-square foot-degrees Fahrenheit to joules per square meter-second-kelvins, multiply by 5.6783; to convert poundsforce per square inch to kilopascals, multiply by 6.895.

Fluidization occurs when the fluid flow rate is great enough so that the pressure drop across the bed equals the weight of the bed. As stated previously, the solids film thickness adjacent to the wall dm is difficult to measure and/or predict. Wen and Fau [Chem. Eng., 64(7), 254 (1957)] give for external walls: h = bk(cs ρs)0.4(Gη/µNf)0.36 TABLE 11-6

(11-51a)

 



(11-51b)

    

hDp 5φ(1 − v)
=




k ks s 1.1 Gmf 7/3 2 580 1 +







0.5 c  g  NRe D 1.5 G p s s g





    

where φ = particle surface area per area of sphere of same diameter. When particle transport occurred through the bundle, the heattransfer coefficients could be predicted by

Panel Coils Immersed in Liquid: Overall Average Heat-Transfer Coefficients* U expressed in Btu/(h ⋅ ft2 ⋅ °F) Design coefficients, considering usual fouling in this service

Clean-surface coefficients Hot side Heating applications: Steam Steam Steam Steam Steam Steam Steam Steam Steam High temperature hot water High temperature heat-transfer oil Dowtherm or Aroclor Cooling applications: Water Water Water Water Water Freon or ammonia Calcium or sodium brine

Cold side Watery solution Light oils Medium lube oil Bunker C or No. 6 fuel oil Tar or asphalt Molten sulfur Molten paraffin Air or gases Molasses or corn sirup Watery solutions Tar or asphalt Tar or asphalt Watery solution Quench oil Medium lube oil Molasses or corn sirup Air or gases Watery solution Watery solution

Natural convection

Forced convection

Natural convection

Forced convection

250–500 50–70 40–60 20–40

300–550 110–140 100–130 70–90

100–200 40–45 35–40 15–30

150–275 60–110 50–100 60–80

15–35 35–45 35–45 2–4 20–40 115–140

50–70 45–55 45–55 5–10 70–90 200–250

15–25 20–35 25–35 1–3 15–30 70–100

40–60 35–45 40–50 4–8 60–80 110–160

12–30

45–65

10–20

30–50

15–30

50–60

12–20

30–50

110–135 10–15 8–12 7–10

195–245 25–45 20–30 18–26

65–95 7–10 5–8 4–7

105–155 15–25 10–20 8–15

2–4 35–45 100–120

5–10 60–90 175–200

1–3 20–35 50–75

4–8 40–60 80–125

*Tranter Manufacturing, Inc. To convert British thermal units per hour-square foot-degrees Fahrenheit to joules per square meter-second-kelvins, multiply by 5.6783.

NOTE:

(11-53a)

THERMAL DESIGN OF HEAT-TRANSFER EQUIPMENT TABLE 11-7

11-27

Jacketed Vessels: Overall Coefficients Overall U* Fluid in vessel

Wall material

Btu/(h ⋅ ft2 ⋅ °F)

J/(m2 ⋅ s ⋅ K)

Steam Steam Steam Steam Steam

Water Aqueous solution Organics Light oil Heavy oil

Stainless steel Stainless steel Stainless steel Stainless steel Stainless steel

150–300 80–200 50–150 60–160 10–50

850–1700 450–1140 285–850 340–910 57–285

Brine Brine Brine Brine Brine

Water Aqueous solution Organics Light oil Heavy oil

Stainless steel Stainless steel Stainless steel Stainless steel Stainless steel

40–180 35–150 30–120 35–130 10–30

230–1625 200–850 170–680 200–740 57–170

Heat-transfer oil Heat-transfer oil Heat-transfer oil Heat-transfer oil Heat-transfer oil

Water Aqueous solution Organics Light oil Heavy oil

Stainless steel Stainless steel Stainless steel Stainless steel Stainless steel

50–200 40–170 30–120 35–130 10–40

285–1140 230–965 170–680 200–740 57–230

Steam Steam Steam Steam Steam

Water Aqueous solution Organics Light oil Heavy oil

Glass-lined CS Glass-lined CS Glass-lined CS Glass-lined CS Glass-lined CS

70–100 50–85 30–70 40–75 10–40

400–570 285–480 170–400 230–425 57–230

Brine Brine Brine Brine Brine

Water Aqueous solution Organics Light oil Heavy oil

Glass-lined CS Glass-lined CS Glass-lined CS Glass-lined CS Glass-lined CS

30–80 25–70 20–60 25–65 10–30

170–450 140–400 115–340 140–370 57–170

Heat-transfer oil Heat-transfer oil Heat-transfer oil Heat-transfer oil Heat-transfer oil

Water Aqueous solution Organics Light oil Heavy oil

Glass-lined CS Glass-lined CS Glass-lined CS Glass-lined CS Glass-lined CS

30–80 25–70 25–65 20–70 10–35

170–450 140–400 140–370 115–400 57–200

Jacket fluid

*Values listed are for moderate nonproximity agitation. CS = carbon steel.

jH = 0.14(NRe /φ)−0.68

(11-53b)

In Eqs. (11-53a) and (11-53b), NRe is based on particle diameter and superficial fluid velocity. Zenz and Othmer (see “Introduction: General References”) give an excellent summary of fluidized bed-to-wall heat-transfer investigations. Solidification involves heavy heat loads transferred essentially at a steady temperature difference. It also involves the varying values of liquid- and solid-phase thickness and thermal diffusivity. When these are substantial and/or in the case of a liquid flowing over a changing solid TABLE 11-8

External Coils; Typical Overall Coefficients* U expressed in Btu/(h ⋅ ft2 ⋅ °F)

Type of coil r in. o.d. copper tubing attached with bands at 24-in. spacing r in. o.d. copper tubing attached with bands at 24-in. spacing Panel coils

layer interface, Siegel and Savino (ASME Paper 67-WA/Ht-34, November 1967) offer equations and charts for prediction of the layer-growth time. For solidification (or melting) of a slab or a semi-infinite bar, initially at its transition temperature, the position of the interface is given by the one-dimensional Newmann’s solution given in Carslaw and Jaeger (Conduction of Heat in Solids, Clarendon Press, Oxford, 1959). Later work by Hashem and Sliepcevich [Chem. Eng. Prog., 63, Symp. Ser. 79, 35, 42 (1967)] offers more accurate second-order finite-difference equations. The heat-transfer rate is found to be substantially higher under conditions of agitation. The heat transfer is usually said to occur by com-

Coil spacing, in.† 2 3f 6d 12a or greater 2 3f 6d 12a or greater

Fluid in coil

Fluid in vessel

5 to 50 lb./sq. in. gage steam

Water under light agitation

50 lb./sq. in. gage steam

No. 6 fuel oil under light agitation

50 lb./sq. in. gage steam Water Water

Boiling water Water No. 6 fuel oil Water No. 6 fuel oil

Temp. range, °F.

U‡ without cement

U with heattransfer cement

158–210 158–210 158–210 158–210 158–258 158–258 158–240 158–238 212 158–212 228–278 130–150 130–150

1–5 1–5 1–5 1–5 1–5 1–5 1–5 1–5 29 8–30 6–15 7 4

42–46 50–53 60–64 69–72 20–30 25–38 30–40 35–46 48–54 19–48 24–56 15 9–19

*Data courtesy of Thermon Manufacturing Co. †External surface of tubing or side of panel coil facing tank. ‡For tubing, the coefficients are more dependent upon tightness of the coil against the tank than upon either fluid. The low end of the range is recommended. NOTE: To convert British thermal units per hour-square foot-degrees Fahrenheit to joules per square meter-second-kelvins, multiply by 5.6783; to convert inches to meters, multiply by 0.0254; and to convert pounds-force per square inch to kilopascals, multiply by 6.895.

11-28

HEAT-TRANSFER EQUIPMENT

TABLE 11-9 Thermal Properties of Various Materials as Affecting Conductive Heat Transfer

Material Air Water Double steel plate, sand divider Sand Powdered iron Magnetite iron ore Aerocat catalysts Table salt Bone char Pitch coke Phenolformaldehyde resin granules Phenolformaldehyde resin powder Powdered coal

Thermal conductivity, B.t.u./(hr.)(sq. ft.)(°F./ft.)

Volume specific heat, B.t.u./(cu. ft.)(°F.)

Thermal diffusivity, sq. ft./hr.

0.0183 0.3766

0.016 62.5

1.143 0.0755

0.207 0.207

19.1 19.1

0.0108 0.0108

0.0533

12.1

0.0044

0.212

63

0.0033

0.163 0.168 0.0877 0.333

20 12.6 16.9 16.2

0.0062 0.0133 0.0051 0.0198

0.0416

10.5

0.0042

0.070

10

0.0070

0.070

15

0.0047

To convert British thermal units per hour-square foot-degrees Fahrenheit to joules per meter-second-kelvins, multiply by 1.7307; to convert British thermal units per cubic foot-degrees Fahrenheit to joules per cubic meter-kelvins, multiply by (6.707)(104); and to convert square feet per hour to square meters per second, multiply by (2.581)(10−5).

bined conductive and convective modes. A discussion and explanation are given by Holt [Chem. Eng., 69(1), 110 (1962)]. Prediction of Uco by Eq. (11-48) can be accomplished by replacing α by αe, the effective thermal diffusivity of the bed. To date so little work has been performed in evaluating the effect of mixing parameters that few predictions can be made. However, for agitated liquid-phase devices Eq. (18-19) is applicable. Holt (loc. cit.) shows that this equation can be converted for solids heat transfer to yield Uco = a′cs Dt−0.3N0.7(cos ω)0.2

(11-54)

where Dt = agitator or vessel diameter; N = turning speed, r/min; ω = effective angle of repose of the burden; and a′ is a proportionality constant. This is applicable for such devices as agitated pans, agitated kettles, spiral conveyors, and rotating shells. The solids passage time through rotary devices is given by Saemann [Chem. Eng. Prog., 47, 508, (1951)]: θ = 0.318L sin ω/Sr NDt

(11-55a)

FIG. 11-32

Bed expansion ratio.

and by Marshall and Friedman [Chem. Eng. Prog., 45, 482–493, 573–588 (1949)]: θ = (0.23L/Sr N0.9Dt)  (0.6BLG/Fa)

Uco = a′cs αe

Fluidization efficiency.

(11-57)

with Uco having a nominal value of 114 J/(m2 ⋅s⋅K) [20 Btu/(h⋅ft2 ⋅°F)]. Contactive (Direct) Heat Transfer Contactive heat-transfer equipment is so constructed that the particulate burden in solid phase is directly exposed to and permeated by the heating or cooling medium (Sec. 20). The carrier may either heat or cool the solids. A large amount of the industrial heat processing of solids is effected by this mechanism. Physically, these can be classified into packed beds and various degrees of agitated beds from dilute to dense fluidized beds. The temperature difference for heat transfer is the log-mean temperature difference when the particles are large and/or the beds packed, or the difference between the inlet fluid temperature t3 and average exhausting fluid temperature t4, expressed ∆3t4, for small particles. The use of the log mean for packed beds has been confirmed by Thodos and Wilkins (Second American Institute of Chemical Engineers-IIQPR Meeting, Paper 30D, Tampa, May 1968). When fluid and solid flow directions are axially concurrent and particle size is small, as in a vertical-shell fluid bed, the temperature of the exiting solids t2 (which is also that of exiting gas t4) is used as ∆ 3 t2, as shown by Levenspiel, Olson, and Walton [Ind. Eng. Chem., 44, 1478 (1952)], Marshall [Chem. Eng. Prog., 50, Monogr. Ser. 2, 77 (1954)], Leva (Fluidization, McGraw-Hill, New York, 1959), and Holt (Fourth Int. Heat Transfer Conf. Paper 11, American Institute of Chemical Engineers-American Society of Mechanical Engineers, Buffalo, 1960). This temperature difference is also applicable for well-fluidized beds of small particles in cross-flow as in various vibratory carriers. The packed-bed-to-fluid heat-transfer coefficient has been investigated by Baumeister and Bennett [Am. Inst. Chem. Eng. J., 4, 69 (1958)], who proposed the equation m jH = (h/cG)(cµ/k)2/3 = aNRe

FIG. 11-31

(11-55b)

where the second term of Eq. (11-55b) is positive for counterflow of air, negative for concurrent flow, and zero for indirect rotary shells. From these equations a predictive equation is developed for rotaryshell devices, which is analogous to Eq. (11-54): b′csDt N0.9 Y Uco =

(11-56) (∆ t)L sin ω where θ = solids-bed passage time through the shell, min; Sr = shell slope; L = shell length; Y = percent fill; and b′ is a proportionality constant. Vibratory devices which constantly agitate the solids bed maintain a relatively constant value for Uco such that

(11-58)

where NRe is based on particle diameter and superficial fluid velocity. Values of a and m are as follows:

THERMAL DESIGN OF HEAT-TRANSFER EQUIPMENT

FIG. 11-33

11-29

f factor for Eq. (11-52b).

Dt/Dp (dimensionless)

a

m

10.7 16.0 25.7 >30

1.58 0.95 0.92 0.90

−0.40 −0.30 −0.28 −0.28

Glaser and Thodos [Am. Inst. Chem. Eng. J., 4, 63 (1958)] give a correlation involving individual particle shape and bed porosity. Kunii and Suzuki [Int. J. Heat Mass Transfer, 10, 845 (1967)] discuss heat and mass transfer in packed beds of fine particles. Particle-to-fluid heat-transfer coefficients in gas fluidized beds are predicted by the relation (Zenz and Othmer, op. cit.) hDp (11-59a)
= 0.017(DpGmf /µ)1.21 k where Gmf is the superficial mass velocity at incipient fluidization. A more general equation is given by Frantz [Chem. Eng., 69(20), 89 (1962)]: hDp /k = 0.015(DpG/µ)1.6(cµ/k)0.67 (11-59b) where h is based on true gas temperature. Bed-to-wall coefficients in dilute-phase transport generally can be predicted by an equation of the form of Eq. (5-50). For example, Bonilla et al. (American Institute of Chemical Engineers Heat Transfer Symp., Atlantic City, N.J., December 1951) found for 1- to 2-µm chalk particles in water up to 8 percent by volume that the coefficient on Eq. (5-50) is 0.029 where k, ρ, and c were arithmetic weighted averages and the viscosity was taken equal to the coefficient of rigidity. Farber and Morley [Ind. Eng. Chem., 49, 1143 (1957)] found the coefficient on Eq. (5-50) to be 0.025 for the upward flow of air transporting silica-alumina catalyst particles at rates less than 2 kg solids kg air (2 lb solids/lb air). Physical properties used were those of the transporting gas. See Zenz and Othmer (op. cit.) for additional details covering wider porosity ranges. The thermal performance of cylindrical rotating shell units is based upon a volumetric heat-transfer coefficient Q Uct =
(11-60a) Vr(∆ t) where Vr = volume. This term indirectly includes an area factor so that

thermal performance is governed by a cross-sectional area rather than by a heated area. Use of the heated area is possible, however: Q Q Uct =
or (11-60b)
(∆ 3 t2 )A (∆ 3 t4 )A For heat transfer directly to solids, predictive equations give directly the volume V or the heat-transfer area A, as determined by heat balance and airflow rate. For devices with gas flow normal to a fluidized-solids bed, Q A =

(11-61) ∆ tp(cρg)(Fg) where ∆tp = ∆3t4 as explained above, cρ = volumetric specific heat, and Fg = gas flow rate. For air, cρ at normal temperature and pressure is about 1100 J/(m3⋅K) [0.0167 Btu/(ft3⋅°F)]; so bQ A=
(11-62) (∆ 3 t4)Fg where b = 0.0009 (SI) or 60 (U.S. customary). Another such equation, for stationary vertical-shell and some horizontal rotary-shell and pneumatic-transport devices in which the gas flow is parallel with and directionally concurrent with the fluidized bed, is the same as Eq. (11-62) with ∆ 3 t4 replaced by ∆ 3 t2. If the operation involves drying or chemical reaction, the heat load Q is much greater than for sensibleheat transfer only. Also, the gas flow rate to provide moisture carry-off and stoichiometric requirements must be considered and simultaneously provided. A good treatise on the latter is given by Pinkey and Plint (Miner. Process., June 1968, p. 17). Evaporative cooling is a special patented technique that often can be advantageously employed in cooling solids by contactive heat transfer. The drying operation is terminated before the desired final moisture content is reached, and solids temperature is at a moderate value. The cooling operation involves contacting the burden (preferably fluidized) with air at normal temperature and pressure. The air adiabatically absorbs and carries off a large part of the moisture and, in doing so, picks up heat from the warm (or hot) solids particles to supply the latent heat demand of evaporation. For entering solids at temperatures of 180°C (350°F) and less with normal heat-capacity values of 0.85 to 1.0 kJ/(kg⋅K) [0.2 to 0.25 Btu/(lb⋅°F)], the effect can be calculated by: 1. Using 285 m3 (1000 ft3) of airflow at normal temperature and pressure at 40 percent relative humidity to carry off 0.45 kg (1 lb) of

11-30

HEAT-TRANSFER EQUIPMENT

water [latent heat 2326 kJ/kg (1000 Btu/lb)] and to lower temperature by 22 to 28°C (40 to 50°F). 2. Using the lowered solids temperature as t3 and calculating the remainder of the heat to be removed in the regular manner by Eq. (11-62). The required air quantity for (2) must be equal to or greater than that for (1). When the solids heat capacity is higher (as is the case for most organic materials), the temperature reduction is inversely proportional to the heat capacity. A nominal result of this technique is that the required airflow rate and equipment size is about two-thirds of that when evaporative cooling is not used. See Sec. 20 for equipment available. Convective Heat Transfer Equipment using the true convective mechanism when the heated particles are mixed with (and remain with) the cold particles is used so infrequently that performance and sizing equations are not available. Such a device is the pebble heater as described by Norton (Chem. Metall. Eng., July 1946). For operation data, see Sec. 9. Convective heat transfer is often used as an adjunct to other modes, particularly to the conductive mode. It is often more convenient to consider the agitative effect a performance-improvement influence on the thermal diffusivity factor α, modifying it to αe, the effective value. A pseudo-convective heat-transfer operation is one in which the heating gas (generally air) is passed over a bed of solids. Its use is almost exclusively limited to drying operations (see Sec. 12, tray and shelf dryers). The operation, sometimes termed direct, is more akin to the conductive mechanism. For this operation, Tsao and Wheelock [Chem. Eng., 74(13), 201 (1967)] predict the heat-transfer coefficient when radiative and conductive effects are absent by h = bG0.8

(11-63)

where b = 14.31 (SI) or 0.0128 (U.S. customary), h = convective heat transfer, and G = gas flow rate. The drying rate is given by h(Td − Tw) Kcv =

(11-64) λ 2 where Kcv = drying rate, for constant-rate period, kg/(m ⋅s) [lb/(h⋅ft2)]; Td and Tw = respective dry-bulb and wet-bulb temperatures of the air; and λ = latent heat of evaporation at temperature Tw. Note here that the temperature-difference determination of the operation is a simple linear one and of a steady-state nature. Also note that the operation is a function of the airflow rate. Further, the solids are granular with a fairly uniform size, have reasonable capillary voids, are of a firm texture, and have the particle surface wetted. The coefficient h is also used to predict (in the constant-rate period) the total overall air-to-solids heat-transfer coefficient Ucv by 1/Ucv = 1/h + x/k

(11-65)

where k = solids thermal conductivity and x is evaluated from z(Xc − Xo) x =

(11-65a) Xc − Xe where z = bed (or slab) thickness and is the total thickness when drying and/or heat transfer is from one side only but is one-half of the thickness when drying and/or heat transfer is simultaneously from both sides; Xo, Xc, and Xe are respectively the initial (or feed-stock), critical, and equilibrium (with the drying air) moisture contents of the solids, all in kg H2O/kg dry solids (lb H2O/lb dry solids). This coefficient is used to predict the instantaneous drying rate W dX Ucv(Td − Tw) −

=

(11-66) A dθ λ By rearrangement, this can be made into a design equation as follows: Wλ(dX/dθ) A = −

(11-67) Ucv(Td − Tw) where W = weight of dry solids in the equipment, λ = latent heat of evaporation, and θ = drying time. The reader should refer to the full reference article by Tsao and Wheelock (loc. cit.) for other solids conditions qualifying the use of these equations.

Radiative Heat Transfer Heat-transfer equipment using the radiative mechanism for divided solids is constructed as a “table” which is stationary, as with trays, or moving, as with a belt, and/or agitated, as with a vibrated pan, to distribute and expose the burden in a plane parallel to (but not in contact with) the plane of the radiant-heat sources. Presence of air is not necessary (see Sec. 12 for vacuum-shelf dryers and Sec. 22 for resublimation). In fact, if air in the intervening space has a high humidity or CO2 content, it acts as an energy absorber, thereby depressing the performance. For the radiative mechanism, the temperature difference is evaluated as ∆t = T e4 − T r4

(11-68)

where Te = absolute temperature of the radiant-heat source, K (°R); and Tr = absolute temperature of the bed of divided solids, K (°R). Numerical values for Ura for use in the general design equation may be calculated from experimental data by Q Ura =

(11-69) A(Te4 − Tr4) The literature to date offers practically no such values. However, enough proprietary work has been performed to present a reliable evaluation for the comparison of mechanisms (see “Introduction: Modes of Heat Transfer”). For the radiative mechanism of heat transfer to solids, the rate equation for parallel-surface operations is qra = b(Te4 − T r4)i f

(11-70)

where b = (5.67)(10 )(SI) or (0.172)(10 −8)(U.S. customary), qra = radiative heat flux, and if = an interchange factor which is evaluated from −8

1/if = 1/es + 1/er − 1

(11-70a)

where es = coefficient of emissivity of the source and er = “emissivity” (or “absorptivity”) of the receiver, which is the divided-solids bed. For the emissivity values, particularly of the heat source es, an important consideration is the wavelength at which the radiant source emits as well as the flux density of the emission. Data for these values are available from Polentz [Chem. Eng., 65(7), 137; (8), 151 (1958)] and Adlam (Radiant Heating, Industrial Press, New York, p. 40). Both give radiated flux density versus wavelength at varying temperatures. Often, the seemingly cooler but longer wavelength source is the better selection. Emitting sources are (1) pipes, tubes, and platters carrying steam, 2100 kPa (300 lbf/in2); (2) electrical-conducting glass plates, 150 to 315°C (300 to 600°F) range; (3) light-bulb type (tungsten-filament resistance heater); (4) modules of refractory brick for gas burning at high temperatures and high fluxes; and (5) modules of quartz tubes, also operable at high temperatures and fluxes. For some emissivity values see Table 11-10. For predictive work, where Ura is desired for sizing, this can be obtained by dividing the flux rate qra by ∆t: Ura = qra /(T e4 − T r4) = if b

(11-71)

where b = (5.67)(10−8) (SI) or (0.172)(10−8) (U.S. customary). Hence: Q A =

(11-72) Ura(T e4 − T r4) where A = bed area of solids in the equipment. Important considerations in the application of the foregoing equations are: 1. Since the temperature of the emitter is generally known (preselected or readily determined in an actual operation), the absorptivity value er is the unknown. This absorptivity is partly a measure of the ability of radiant heat to penetrate the body of a solid particle (or a moisture film) instantly, as compared with diffusional heat transfer by conduction. Such instant penetration greatly reduces processing time and case-hardening effects. Moisture release and other mass transfer, however, still progress by diffusional means. 2. In one of the major applications of radiative devices (drying),

THERMAL DESIGN OF HEAT-TRANSFER EQUIPMENT TABLE 11-10

11-31

Normal Total Emissivity of Various Surfaces A. Metals and Their Oxides Surface

Aluminum Highly polished plate, 98.3% pure Polished plate Rough plate Oxidized at 1110°F Aluminum-surfaced roofing Calorized surfaces, heated at 1110°F. Copper Steel Brass Highly polished: 73.2% Cu, 26.7% Zn 62.4% Cu, 36.8% Zn, 0.4% Pb, 0.3% Al 82.9% Cu, 17.0% Zn Hard rolled, polished: But direction of polishing visible But somewhat attacked But traces of stearin from polish left on Polished Rolled plate, natural surface Rubbed with coarse emery Dull plate Oxidized by heating at 1110°F Chromium; see Nickel Alloys for Ni-Cr steels Copper Carefully polished electrolytic copper Commercial, emeried, polished, but pits remaining Commercial, scraped shiny but not mirror-like Polished Plate, heated long time, covered with thick oxide layer Plate heated at 1110°F Cuprous oxide Molten copper Gold Pure, highly polished Iron and steel Metallic surfaces (or very thin oxide layer): Electrolytic iron, highly polished Polished iron Iron freshly emeried Cast iron, polished Wrought iron, highly polished Cast iron, newly turned Polished steel casting Ground sheet steel Smooth sheet iron Cast iron, turned on lathe Oxidized surfaces: Iron plate, pickled, then rusted red Completely rusted Rolled sheet steel Oxidized iron Cast iron, oxidized at 1100°F Steel, oxidized at 1100°F Smooth oxidized electrolytic iron Iron oxide Rough ingot iron

t, °F.*

Emissivity*

440–1070 73 78 390–1110 100

0.039–0.057 0.040 0.055 0.11–0.19 0.216

390–1110 390–1110

0.18–0.19 0.52–0.57

476–674 494–710 530

0.028–0.031 0.033–0.037 0.030

70 73 75 100–600 72 72 120–660 390–1110 100–1000

0.038 0.043 0.053 0.096 0.06 0.20 0.22 0.61–0.59 0.08–0.26

176

0.018

66

0.030

72 242

0.072 0.023

77 390–1110 1470–2010 1970–2330

0.78 0.57 0.66–0.54 0.16–0.13

440–1160

0.018–0.035

350–440 800–1880 68 392 100–480 72 1420–1900 1720–2010 1650–1900 1620–1810

0.052–0.064 0.144–0.377 0.242 0.21 0.28 0.435 0.52–0.56 0.55–0.61 0.55–0.60 0.60–0.70

68 67 70 212 390–1110 390–1110 260–980 930–2190 1700–2040

0.612 0.685 0.657 0.736 0.64–0.78 0.79 0.78–0.82 0.85–0.89 0.87–0.95

Surface Sheet steel, strong rough oxide layer Dense shiny oxide layer Cast plate: Smooth Rough Cast iron, rough, strongly oxidized Wrought iron, dull oxidized Steel plate, rough High temperature alloy steels (see Nickel Alloys) Molten metal Cast iron Mild steel Lead Pure (99.96%), unoxidized Gray oxidized Oxidized at 390°F Mercury Molybdenum filament Monel metal, oxidized at 1110°F Nickel Electroplated on polished iron, then polished Technically pure (98.9% Ni, + Mn), polished Electropolated on pickled iron, not polished Wire Plate, oxidized by heating at 1110°F Nickel oxide Nickel alloys Chromnickel Nickelin (18–32 Ni; 55–68 Cu; 20 Zn), gray oxidized KA-2S alloy steel (8% Ni; 18% Cr), light silvery, rough, brown, after heating After 42 hr. heating at 980°F NCT-3 alloy (20% Ni; 25% Cr.), brown, splotched, oxidized from service NCT-6 alloy (60% Ni; 12% Cr), smooth, black, firm adhesive oxide coat from service Platinum Pure, polished plate Strip Filament Wire Silver Polished, pure Polished Steel, see Iron Tantalum filament Tin—bright tinned iron sheet Tungsten Filament, aged Filament Zinc Commercial, 99.1% pure, polished Oxidized by heating at 750°F. Galvanized sheet iron, fairly bright Galvanized sheet iron, gray oxidized

t, °F.* 75 75

Emissivity* 0.80 0.82

73 73 100–480 70–680 100–700

0.80 0.82 0.95 0.94 0.94–0.97

2370–2550 2910–3270

0.29 0.28

260–440 75 390 32–212 1340–4700 390–1110

0.057–0.075 0.281 0.63 0.09–0.12 0.096–0.292 0.41–0.46

74

0.045

440–710

0.07–0.087

68 368–1844 390–1110 1200–2290

0.11 0.096–0.186 0.37–0.48 0.59–0.86

125–1894

0.64–0.76

70

0.262

420–914 420–980

0.44–0.36 0.62–0.73

420–980

0.90–0.97

520–1045

0.89–0.82

440–1160 1700–2960 80–2240 440–2510

0.054–0.104 0.12–0.17 0.036–0.192 0.073–0.182

440–1160 100–700

0.0198–0.0324 0.0221–0.0312

2420–5430 76

0.194–0.31 0.043 and 0.064

80–6000 6000

0.032–0.35 0.39

440–620 750 82 75

0.045–0.053 0.11 0.228 0.276

260–1160

0.81–0.79

1900–2560 206–520 209–362

0.526 0.952 0.959–0.947

B. Refractories, Building Materials, Paints, and Miscellaneous Asbestos Board Paper Brick Red, rough, but no gross irregularities Silica, unglazed, rough Silica, glazed, rough Grog brick, glazed See Refractory Materials below.

74 100–700 70 1832 2012 2012

0.96 0.93–0.945 0.93 0.80 0.85 0.75

Carbon T-carbon (Gebr. Siemens) 0.9% ash (this started with emissivity at 260°F. of 0.72, but on heating changed to values given) Carbon filament Candle soot Lampblack-waterglass coating

11-32

HEAT-TRANSFER EQUIPMENT

TABLE 11-10

Normal Total Emissivity of Various Surfaces (Concluded ) B. Refractories, Building Materials, Paints, and Miscellaneous Surface

Same Thin layer on iron plate Thick coat Lampblack, 0.003 in. or thicker Enamel, white fused, on iron Glass, smooth Gypsum, 0.02 in. thick on smooth or blackened plate Marble, light gray, polished Oak, planed Oil layers on polished nickel (lube oil) Polished surface, alone +0.001-in. oil +0.002-in. oil +0.005-in. oil Infinitely thick oil layer Oil layers on aluminum foil (linseed oil) Al foil +1 coat oil +2 coats oil Paints, lacquers, varnishes Snowhite enamel varnish or rough iron plate Black shiny lacquer, sprayed on iron Black shiny shellac on tinned iron sheet Black matte shellac Black lacquer Flat black lacquer White lacquer

t, °F.*

Emissivity*

260–440 69 68 100–700 66 72

0.957–0.952 0.927 0.967 0.945 0.897 0.937

70 72 70 68

0.903 0.931 0.895 0.045 0.27 0.46 0.72 0.82

212 212 212

0.087† 0.561 0.574

Surface Oil paints, sixteen different, all colors Aluminum paints and lacquers 10% Al, 22% lacquer body, on rough or smooth surface 26% Al, 27% lacquer body, on rough or smooth surface Other Al paints, varying age and Al content Al lacquer, varnish binder, on rough plate Al paint, after heating to 620°F Paper, thin Pasted on tinned iron plate On rough iron plate On black lacquered plate Plaster, rough lime Porcelain, glazed Quartz, rough, fused Refractory materials, 40 different poor radiators

t, °F.*

Emissivity*

212

0.92–0.96

212

0.52

212

0.3

212 70 300–600

0.27–0.67 0.39 0.35

66 66 66 50–190 72 70 1110–1830

0.924 0.929 0.944 0.91 0.924 0.932

good radiators 73 76 70 170–295 100–200 100–200 100–200

0.906 0.875 0.821 0.91 0.80–0.95 0.96–0.98 0.80–0.95

Roofing paper Rubber Hard, glossy plate Soft, gray, rough (reclaimed) Serpentine, polished Water

69 74 76 74 32–212



 



0.65 – 0.75 0.70 0.80 – 0.85 0.85 – 0.90 0.91 0.945 0.859 0.900 0.95–0.963

*When two temperatures and two emissivities are given, they correspond, first to first and second to second, and linear interpolation is permissible. °C = (°F − 32)/1.8. †Although this value is probably high, it is given for comparison with the data by the same investigator to show the effect of oil layers. See Aluminum, Part A of this table.

the surface-held moisture is a good heat absorber in the 2- to 7-µm wavelength range. Therefore, the absorptivity, color, and nature of the solids are of little importance. 3. For drying, it is important to provide a small amount of venting air to carry away the water vapor. This is needed for two reasons. First, water vapor is a good absorber of 2- to 7-µm energy. Second, watervapor accumulation depresses further vapor release by the solids. If the air over the solids is kept fairly dry by venting, very little heat is carried off, because dry air does not absorb radiant heat. 4. For some of the devices, when the overall conversion efficiency has been determined, the application is primarily a matter of computing the required heat load. It should be kept in mind, however, that there are two conversion efficiencies that must be differentiated. One measure of efficiency is that with which the source converts input energy to output radiated energy. The other is the overall efficiency that measures the proportion of input energy that is actually absorbed by the solids. This latter is, of course, the one that really matters. Other applications of radiant-heat processing of solids are the toasting, puffing, and baking of foods and the low-temperature roasting and preheating of plastic powder or pellets. Since the determination of heat loads for these operations is not well established, bench and pilot tests are generally necessary. Such processes require a fast input of heat and higher heat fluxes than can generally be provided by indirect equipment. Because of this, infrared-equipment size and space requirements are often much lower. Although direct contactive heat transfer can provide high temperatures and heat concentrations and at the same time be small in size, its use may not always be preferable because of undesired side effects such as drying, contamination, case hardening, shrinkage, off color, and dusting. When radiating and receiving surfaces are not in parallel, as in rotary-kiln devices, and the solids burden bed may be only intermittently exposed and/or agitated, the calculation and procedures become very complex, with photometric methods of optics requiring consideration. The following equation for heat transfer, which allows for convective effects, is commonly used by designers of hightemperature furnaces: qra = Q/A = bσ [(Tg /100)4 − (Ts /100)4]

(11-73)

where b = 5.67 (SI) or 0.172 (U.S. customary); Q = total furnace heat transfer; σ = an emissivity factor with recommended values of 0.74 for gas, 0.75 for oil, and 0.81 for coal; A = effective area for absorbing heat (here the solids burden exposed area); Tg = exiting-combustion-gas absolute temperature; and Ts = absorbing surface temperature. In rotary devices, reradiation from the exposed shell surface to the solids bed is a major design consideration. A treatise on furnaces, including radiative heat-transfer effects, is given by Ellwood and Danatos [Chem. Eng., 73(8), 174 (1966)]. For discussion of radiation heat-transfer computational methods, heat fluxes obtainable, and emissivity values, see Schornshort and Viskanta (ASME Paper 68-H 7-32), Sherman (ASME Paper 56-A-111), and the following subsection.

FIG. 11-34

Co., Inc.)

Scraper blade of scraped-surface exchanger. (Henry Vogt Machine

TEMA-STYLE SHELL-AND-TUBE HEAT EXCHANGERS SCRAPED-SURFACE EXCHANGERS Scraped-surface exchangers have a rotating element with springloaded scraper blades to scrape the inside surface (Fig. 11-34). Generally a double-pipe construction is used; the scraping mechanism is in the inner pipe, where the process fluid flows; and the cooling or heating medium is in the outer pipe. The most common size has 6-in inside and 8-in outside pipes. Also available are 3- by 4-in, 8- by 10-in, and 12- by 14-in sizes (in × 25.4 = mm). These double-pipe units are commonly connected in series and arranged in double stands. For chilling and crystallizing with an evaporating refrigerant, a 27-in shell with seven 6-in pipes is available (Henry Vogt Machine

11-33

Co.). In direct contact with the scraped surface is the process fluid which may deposit crystals upon chilling or be extremely fouling or of very high viscosity. Motors, chain drives, appropriate guards, and so on are required for the rotating element. For chilling service with a refrigerant in the outer shell, an accumulator drum is mounted on top of the unit. Scraped-surface exchangers are particularly suitable for heat transfer with crystallization, heat transfer with severe fouling of surfaces, heat transfer with solvent extraction, and heat transfer of highviscosity fluids. They are extensively used in paraffin-wax plants and in petrochemical plants for crystallization.

TEMA-STYLE SHELL-AND-TUBE HEAT EXCHANGERS TYPES AND DEFINITIONS TEMA-style shell-and-tube-type exchangers constitute the bulk of the unfired heat-transfer equipment in chemical-process plants, although increasing emphasis has been developing in other designs. These exchangers are illustrated in Fig. 11-35, and their features are summarized in Table 11-11. TEMA Numbering and Type Designation Recommended practice for the designation of TEMA-style shell-and-tube heat exchangers by numbers and letters has been established by the Tubular Exchanger Manufacturers Association (TEMA). This information from the sixth edition of the TEMA Standards is reproduced in the following paragraphs. It is recommended that heat-exchanger size and type be designated by numbers and letters. 1. Size. Sizes of shells (and tube bundles) shall be designated by numbers describing shell (and tube-bundle) diameters and tube lengths as follows: 2. Diameter. The nominal diameter shall be the inside diameter of the shell in inches, rounded off to the nearest integer. For kettle reboilers the nominal diameter shall be the port diameter followed by the shell diameter, each rounded off to the nearest integer. TABLE 11-11

3. Length. The nominal length shall be the tube length in inches. Tube length for straight tubes shall be taken as the actual overall length. For U tubes the length shall be taken as the straight length from end of tube to bend tangent. 4. Type. Type designation shall be by letters describing stationary head, shell (omitted for bundles only), and rear head, in that order, as indicated in Fig. 11-1. Typical Examples (A) Split-ring floating-heat exchanger with removable channel and cover, single-pass shell, 591-mm (23d-in) inside diameter with tubes 4.9 m (16 ft) long. SIZE 23–192 TYPE AES. (B) U-tube exchanger with bonnet-type stationary head, split-flow shell, 483mm (19-in) inside diameter with tubes 21-m (7-ft) straight length. SIZE 19–84 TYPE GBU. (C) Pull-through floating-heat-kettle-type reboiler having stationary head integral with tube sheet, 584-mm (23-in) port diameter and 940-mm (37-in) inside shell diameter with tubes 4.9-m (16-ft) long. SIZE 23/37–192 TYPE CKT. (D) Fixed-tube sheet exchanger with removable channel and cover, bonnettype rear head, two-pass shell, 841-mm (33s-in) diameter with tubes 2.4 m (8ft) long. SIZE 33–96 TYPE AFM. (E) Fixed-tube sheet exchanger having stationary and rear heads integral with tube sheets, single-pass shell, 432-mm (17-in) inside diameter with tubes 4.9-m (16-ft) long. SIZE 17–192 TYPE CEN.

Functional Definitions Heat-transfer equipment can be designated by type (e.g., fixed tube sheet, outside packed head, etc.) or by

Features of TEMA Shell-and-Tube-Type Exchangers*

Type of design T.E.M.A. rear-head type Relative cost increases from A (least expensive) through E (most expensive) Provision for differential expansion Removable bundle Replacement bundle possible Individual tubes replaceable Tube cleaning by chemicals inside and outside Interior tube cleaning mechanically Exterior tube cleaning mechanically: Triangular pitch Square pitch Hydraulic-jet cleaning: Tube interior Tube exterior Double tube sheet feasible Number of tube passes Internal gaskets eliminated

Internal floating head (split backing ring)

Fixed tube sheet

U-tube

Packed lantern-ring floating head

Outside-packed floating head

Pull-through floating head

L or M or N

U

W

S

P

T

B Expansion joint in shell No No Yes

A Individual tubes free to expand

C Floating head

E Floating head

D Floating head

E Floating head

Yes Yes Only those in outside row†

Yes Yes Yes

Yes Yes Yes

Yes Yes Yes

Yes Yes Yes

Yes Yes

Yes Special tools required

Yes Yes

Yes Yes

Yes Yes

Yes Yes

No No

No‡ Yes

No‡ Yes

No‡ Yes

No‡ Yes

No‡ Yes

Yes No Yes No practical limitations Yes

Special tools required Yes Yes Any even number possible Yes

Yes Yes No Limited to one or two passes Yes

Yes Yes No No practical limitations§ No

Yes Yes Yes No practical limitations Yes

Yes Yes No No practical limitations§ No

NOTE: Relative costs A and B are not significantly different and interchange for long lengths of tubing. *Modified from page a-8 of the Patterson-Kelley Co. Manual No. 700A, Heat Exchangers. †U-tube bundles have been built with tube supports which permit the U-bends to be spread apart and tubes inside of the bundle replaced. ‡Normal triangular pitch does not permit mechanical cleaning. With a wide triangular pitch, which is equal to 2 (tube diameter plus cleaning lane)/ 3 , mechanical cleaning is possible on removable bundles. This wide spacing is infrequently used. §For odd number of tube side passes, floating head requires packed joint or expansion joint.

11-34

HEAT-TRANSFER EQUIPMENT

FIG. 11-35

TEMA-type designations for shell-and-tube heat exchangers. (Standards of Tubular Exchanger Manufacturers Association, 6th ed., 1978.)

TEMA-STYLE SHELL-AND-TUBE HEAT EXCHANGERS function (chiller, condenser, cooler, etc.). Almost any type of unit can be used to perform any or all of the listed functions. Many of these terms have been defined by Donahue [Pet. Process., 103 (March 1956)]. Equipment Chiller Condenser Partial condenser

Final condenser

Cooler Exchanger Heater Reboiler

Thermosiphon reboiler Forced-circulation reboiler Steam generator Superheater Vaporizer Waste-heat boiler

Function Cools a fluid to a temperature below that obtainable if water only were used as a coolant. It uses a refrigerant such as ammonia or Freon. Condenses a vapor or mixture of vapors, either alone or in the presence of a noncondensable gas. Condenses vapors at a point high enough to provide a temperature difference sufficient to preheat a cold stream of process fluid. This saves heat and eliminates the need for providing a separate preheater (using flame or steam). Condenses the vapors to a final storage temperature of approximately 37.8°C (100°F). It uses water cooling, which means that the transferred heat is lost to the process. Cools liquids or gases by means of water. Performs a double function: (1) heats a cold fluid by (2) using a hot fluid which it cools. None of the transferred heat is lost. Imparts sensible heat to a liquid or a gas by means of condensing steam or Dowtherm. Connected to the bottom of a fractionating tower, it provides the reboil heat necessary for distillation. The heating medium may be either steam or a hot-process fluid. Natural circulation of the boiling medium is obtained by maintaining sufficient liquid head to provide for circulation. A pump is used to force liquid through the reboiler. Generates steam for use elsewhere in the plant by using the available high-level heat in tar or a heavy oil. Heats a vapor above the saturation temperature. A heater which vaporizes part of the liquid. Produces steam; similar to steam generator, except that the heating medium is a hot gas or liquid produced in a chemical reaction.

GENERAL DESIGN CONSIDERATIONS Selection of Flow Path In selecting the flow path for two fluids through an exchanger, several general approaches are used. The tubeside fluid is more corrosive or dirtier or at a higher pressure. The shell-side fluid is a liquid of high viscosity or a gas. When alloy construction for one of the two fluids is required, a carbon steel shell combined with alloy tube-side parts is less expensive than alloy in contact with the shell-side fluid combined with carbon steel headers. Cleaning of the inside of tubes is more readily done than cleaning of exterior surfaces. For gauge pressures in excess of 2068 kPa (300 lbf/in2) for one of the fluids, the less expensive construction has the high-pressure fluid in the tubes. For a given pressure drop, higher heat-transfer coefficients are obtained on the shell side than on the tube side. Heat-exchanger shutdowns are most often caused by fouling, corrosion, and erosion. Construction Codes “Rules for Construction of Pressure Vessels, Division 1,” which is part of Section VIII of the ASME Boiler and Pressure Vessel Code (American Society of Mechanical Engineers), serves as a construction code by providing minimum standards. New editions of the code are usually issued every 3 years. Interim revisions are made semiannually in the form of addenda. Compliance with ASME Code requirements is mandatory in much of the United States and Canada. Originally these rules were not prepared for heat exchangers. However, the welded joint between tube sheet and shell of the fixed-tube-sheet heat exchanger is now included. A nonmandatory

11-35

appendix on tube-to-tube-sheet joints is also included. Additional rules for heat exchangers are being developed. Standards of Tubular Exchanger Manufacturers Association, 6th ed., 1978 (commonly referred to as the TEMA Standards), serve to supplement and define the ASME Code for all shell-and-tube-type heat-exchanger applications (other than double-pipe construction). TEMA Class R design is “for the generally severe requirements of petroleum and related processing applications. Equipment fabricated in accordance with these standards is designed for safety and durability under the rigorous service and maintenance conditions in such applications.” TEMA Class C design is “for the generally moderate requirements of commercial and general process applications,” while TEMA Class B is “for chemical process service.” The mechanical-design requirements are identical for all three classes of construction. The differences between the TEMA classes are minor and were listed by Rubin [Hydrocarbon Process., 59, 92 (June 1980)]. Among the topics of the TEMA Standards are nomenclature, fabrication tolerances, inspection, guarantees, tubes, shells, baffles and support plates, floating heads, gaskets, tube sheets, channels, nozzles, end flanges and bolting, material specifications, and fouling resistances. Shell and Tube Heat Exchangers for General Refinery Services, API Standard 660, 4th ed., 1982, is published by the American Petroleum Institute to supplement both the TEMA Standards and the ASME Code. Many companies in the chemical and petroleum processing fields have their own standards to supplement these various requirements. The Interrelationships between Codes, Standards, and Customer Specifications for Process Heat Transfer Equipment is a symposium volume which was edited by F. L. Rubin and published by ASME in December 1979. (See discussion of pressure-vessel codes in Sec. 6.) Design pressures and temperatures for exchangers usually are specified with a margin of safety beyond the conditions expected in service. Design pressure is generally about 172 kPa (25 lbf/in2) greater than the maximum expected during operation or at pump shutoff. Design temperature is commonly 14°C (25°F) greater than the maximum temperature in service. Tube Bundle Vibration Damage from tube vibration has become an increasing problem as plate baffled heat exchangers are designed for higher flow rates and pressure drops. The most effective method of dealing with this problem is the avoidance of cross flow by use of tube support baffles which promote only longitudinal flow. However, even then, strict attention must be given the bundle area under the shell inlet nozzle where flow is introduced through the side of the shell. TEMA has devoted an entire section in its standards to this topic. In general, the mechanisms of tube vibration are as follows: Vortex Shedding The vortex-shedding frequency of the fluid in cross-flow over the tubes may coincide with a natural frequency of the tubes and excite large resonant vibration amplitudes. Fluid-Elastic Coupling Fluid flowing over tubes causes them to vibrate with a whirling motion. The mechanism of fluid-elastic coupling occurs when a “critical” velocity is exceeded and the vibration then becomes self-excited and grows in amplitude. This mechanism frequently occurs in process heat exchangers which suffer vibration damage. Pressure Fluctuation Turbulent pressure fluctuations which develop in the wake of a cylinder or are carried to the cylinder from upstream may provide a potential mechanism for tube vibration. The tubes respond to the portion of the energy spectrum that is close to their natural frequency. Acoustic Coupling When the shell-side fluid is a low-density gas, acoustic resonance or coupling develops when the standing waves in the shell are in phase with vortex shedding from the tubes. The standing waves are perpendicular to the axis of the tubes and to the direction of cross-flow. Damage to the tubes is rare. However, the noise can be extremely painful. Testing Upon completion of shop fabrication and also during maintenance operations it is desirable hydrostatically to test the shell side of tubular exchangers so that visual examination of tube ends can be made. Leaking tubes can be readily located and serviced. When leaks are determined without access to the tube ends, it is necessary to reroll or reweld all the tube-to-tube-sheet joints with possible damage to the satisfactory joints.

11-36

HEAT-TRANSFER EQUIPMENT

Testing for leaks in heat exchangers was discussed by Rubin [Chem. Eng., 68, 160–166 (July 24, 1961)]. Performance testing of heat exchangers is described in the American Institute of Chemical Engineers’ Standard Testing Procedure for Heat Exchangers, Sec. 1. “Sensible Heat Transfer in Shell-and-TubeType Equipment.”

tion is less costly than types B and M or A and L and still offers the advantage that tubes may be examined and replaced without disturbing the tube-side piping connections. There is no limitation on the number of tube-side passes. Shell-side passes can be one or more, although shells with more than two shellside passes are rarely used. Tubes can completely fill the heat-exchanger shell. Clearance between the outermost tubes and the shell is only the minimum necessary for fabrication. Between the inside of the shell and the baffles some clearance must be provided so that baffles can slide into the shell. Fabrication tolerances then require some additional clearance between the outside of the baffles and the outermost tubes. The edge distance between the outer tube limit (OTL) and the baffle diameter must be sufficient to prevent vibration of the tubes from breaking through the baffle holes. The outermost tube must be contained within the OTL. Clearances between the inside shell diameter and OTL are 13 mm (a in) for 635-mm-(25-in-) inside-diameter shells and up, 11 mm (q in) for 254- through 610-mm (10- through 24-in) pipe shells, and slightly less for smaller-diameter pipe shells. Tubes can be replaced. Tube-side headers, channel covers, gaskets, etc., are accessible for maintenance and replacement. Neither the shell-side baffle structure nor the blind gasket is accessible. During tube removal, a tube may break within the shell. When this occurs, it is most difficult to remove or to replace the tube. The usual procedure is to plug the appropriate holes in the tube sheets.

PRINCIPAL TYPES OF CONSTRUCTION Figure 11-36 shows details of the construction of the TEMA types of shell-and-tube heat exchangers. These and other types are discussed in the following paragraphs. Fixed-Tube-Sheet Heat Exchangers Fixed-tube-sheet exchangers (Fig. 11-36b) are used more often than any other type, and the frequency of use has been increasing in recent years. The tube sheets are welded to the shell. Usually these extend beyond the shell and serve as flanges to which the tube-side headers are bolted. This construction requires that the shell and tube-sheet materials be weldable to each other. When such welding is not possible, a “blind”-gasket type of construction is utilized. The blind gasket is not accessible for maintenance or replacement once the unit has been constructed. This construction is used for steam surface condensers, which operate under vacuum. The tube-side header (or channel) may be welded to the tube sheet, as shown in Fig. 11-35 for type C and N heads. This type of construc-

(a)

(b) Heat-exchanger-component nomenclature. (a) Internal-floating-head exchanger (with floating-head backing device). Type AES. (b) Fixed-tube-sheet exchanger. Type BEM. (Standard of Tubular Exchanger Manufacturers Association, 6th ed., 1978.)

FIG. 11-36

TEMA-STYLE SHELL-AND-TUBE HEAT EXCHANGERS

11-37

(c)

(d)

(e) (Continued) Heat-exchanger-component nomenclature. (c) Outside-packed floating-head exchanger. Type AEP. (d) U-tube heat exchanger. Type CFU. (e) Kettle-type floating-head reboiler. Type AKT. (Standard of Tubular Exchanger Manufacturers Association, 6th ed., 1978.)

FIG. 11-36

11-38

HEAT-TRANSFER EQUIPMENT

(f)

(Continued) Heat-exchanger-component nomenclature. ( f) Exchanger with packed floating tube sheet and lantern ring. Type AJW. (Standard of Tubular Exchanger Manufacturers Association, 6th ed., 1978.)

FIG. 11-36

Differential expansion between the shell and the tubes can develop because of differences in length caused by thermal expansion. Various types of expansion joints are used to eliminate excessive stresses caused by expansion. The need for an expansion joint is a function of both the amount of differential expansion and the cycling conditions to be expected during operation. A number of types of expansion joints are available (Fig. 11-37). a. Flat plates. Two concentric flat plates with a bar at the outer edges. The flat plates can flex to make some allowance for differential expansion. This design is generally used for vacuum service and gauge pressures below 103 kPa (15 lbf/in2). All welds are subject to severe stress during differential expansion. b. Flanged-only heads. The flat plates are flanged (or curved). The diameter of these heads is generally 203 mm (8 in) or more greater than the shell diameter. The welded joint at the shell is subject to the stress referred to before, but the joint connecting the heads is subjected to less stress during expansion because of the curved shape. c. Flared shell or pipe segments. The shell may be flared to connect with a pipe section, or a pipe may be halved and quartered to produce a ring. d. Formed heads. A pair of dished-only or elliptical or flanged and dished heads can be used. These are welded together or connected by a ring. This type of joint is similar to the flanged-only-head type but apparently is subject to less stress.

e. Flanged and flued heads. A pair of flanged-only heads is provided with concentric reverse flue holes. These heads are relatively expensive because of the cost of the fluing operation. The curved shape of the heads reduces the amount of stress at the welds to the shell and also connecting the heads. f. Toroidal. The toroidal joint has a mathematically predictable smooth stress pattern of low magnitude, with maximum stresses at sidewalls of the corrugation and minimum stresses at top and bottom. The foregoing designs were discussed as ring expansion joints by Kopp and Sayre, “Expansion Joints for Heat Exchangers” (ASME Misc. Pap., vol. 6, no. 211). All are statically indeterminate but are subjected to analysis by introducing various simplifying assumptions. Some joints in current industrial use are of lighter wall construction than is indicated by the method of this paper. g. Bellows. Thin-wall bellows joints are produced by various manufacturers. These are designed for differential expansion and are tested for axial and transverse movement as well as for cyclical life. Bellows may be of stainless steel, nickel alloys, or copper. (Aluminum, Monel, phosphor bronze, and titanium bellows have been manufactured.) Welding nipples of the same composition as the heat-exchanger shell are generally furnished. The bellows may be hydraulically formed from a single piece of metal or may consist of welded pieces. External insulation covers of carbon steel are often provided to protect the light-gauge bellows from damage. The cover also prevents insulation from interfering with movement of the bellows (see h). h. Toroidal bellows. For high-pressure service the bellows type of joint has been modified so that movement is taken up by thin-wall small-diameter bel-

TEMA-STYLE SHELL-AND-TUBE HEAT EXCHANGERS

FIG. 11-38

FIG. 11-37

Expansion joints.

lows of a toroidal shape. Thickness of parts under high pressure is reduced considerably (see f).

Improper handling during manufacture, transit, installation, or maintenance of the heat exchanger equipped with the thin-wallbellows type or toroidal type of expansion joint can damage the joint. In larger units these light-wall joints are particularly susceptible to damage, and some designers prefer the use of the heavier walls of formed heads. Chemical-plant exchangers requiring expansion joints most commonly have used the flanged-and-flued-head type. There is a trend toward more common use of the light-wall-bellows type. U-Tube Heat Exchanger (Fig. 11-36d) The tube bundle consists of a stationary tube sheet, U tubes (or hairpin tubes), baffles or support plates, and appropriate tie rods and spacers. The tube bundle can be removed from the heat-exchanger shell. A tube-side header (stationary head) and a shell with integral shell cover, which is welded to the shell, are provided. Each tube is free to expand or contract without any limitation being placed upon it by the other tubes. The U-tube bundle has the advantage of providing minimum clearance between the outer tube limit and the inside of the shell for any of the removable-tube-bundle constructions. Clearances are of the same magnitude as for fixed-tube-sheet heat exchangers. The number of tube holes in a given shell is less than that for a fixed-tube-sheet exchanger because of limitations on bending tubes of a very short radius. The U-tube design offers the advantage of reducing the number of joints. In high-pressure construction this feature becomes of considerable importance in reducing both initial and maintenance costs. The use of U-tube construction has increased significantly with the development of hydraulic tube cleaners, which can remove fouling residues from both the straight and the U-bend portions of the tubes. Mechanical cleaning of the inside of the tubes was described by John [Chem. Eng., 66, 187–192 (Dec. 14, 1959)]. Rods and conventional mechanical tube cleaners cannot pass from one end of the U tube to the other. Power-driven tube cleaners, which can clean both the straight legs of the tubes and the bends, are available. Hydraulic jetting with water forced through spray nozzles at high pressure for cleaning tube interiors and exteriors of removal bundles is reported by Canaday (“Hydraulic Jetting Tools for Cleaning Heat Exchangers,” ASME Pap. 58-A-217, unpublished).

11-39

Tank suction heater.

The tank suction heater, as illustrated in Fig. 11-38, contains a U-tube bundle. This design is often used with outdoor storage tanks for heavy fuel oils, tar, molasses, and similar fluids whose viscosity must be lowered to permit easy pumping. Uusally the tube-side heating medium is steam. One end of the heater shell is open, and the liquid being heated passes across the outside of the tubes. Pumping costs can be reduced without heating the entire contents of the tank. Bare tube and integral low-fin tubes are provided with baffles. Longitudinal fin-tube heaters are not baffled. Fins are most often used to minimize the fouling potential in these fluids. Kettle-type reboilers, evaporators, etc., are often U-tube exchangers with enlarged shell sections for vapor-liquid separation. The U-tube bundle replaces the floating-heat bundle of Fig. 11-36e. The U-tube exchanger with copper tubes, cast-iron header, and other parts of carbon steel is used for water and steam services in office buildings, schools, hospitals, hotels, etc. Nonferrous tube sheets and admiralty or 90-10 copper-nickel tubes are the most frequently used substitute materials. These standard exchangers are available from a number of manufacturers at costs far below those of custombuilt process-industry equipment. Packed-Lantern-Ring Exchanger (Fig. 11-36f) This construction is the least costly of the straight-tube removable bundle types. The shell- and tube-side fluids are each contained by separate rings of packing separated by a lantern ring and are installed at the floating tube sheet. The lantern ring is provided with weep holes. Any leakage passing the packing goes through the weep holes and then drops to the ground. Leakage at the packing will not result in mixing within the exchanger of the two fluids. The width of the floating tube sheet must be great enough to allow for the packings, the lantern ring, and differential expansion. Sometimes a small skirt is attached to a thin tube sheet to provide the required bearing surface for packings and lantern ring. The clearance between the outer tube limit and the inside of the shell is slightly larger than that for fixed-tube-sheet and U-tube exchangers. The use of a floating-tube-sheet skirt increases this clearance. Without the skirt the clearance must make allowance for tubehole distortion during tube rolling near the outside edge of the tube sheet or for tube-end welding at the floating tube sheet. The packed-lantern-ring construction is generally limited to design temperatures below 191°C (375°F) and to the mild services of water, steam, air, lubricating oil, etc. Design gauge pressure does not exceed 2068 kPa (300 lbf/in2) for pipe shell exchangers and is limited to 1034 kPa (150 lbf/in2) for 610- to 1067-mm- (24- to 42-in-) diameter shells. Outside-Packed Floating-Head Exchanger (Fig. 11-36c) The shell-side fluid is contained by rings of packing, which are compressed within a stuffing box by a packing follower ring. This construction was frequently used in the chemical industry, but in recent years usage has decreased. The removable-bundle construction accommodates differential expansion between shell and tubes and is used for shell-side service up to 4137 kPa gauge pressure (600 lbf/in2) at 316°C (600°F). There are no limitations upon the number of tube-side passes or upon the tube-side design pressure and temperature. The outside-packed floating-head exchanger was the most commonly used type of removable-bundle construction in chemical-plant service.

11-40

HEAT-TRANSFER EQUIPMENT

The floating-tube-sheet skirt, where in contact with the rings of packing, has fine machine finish. A split shear ring is inserted into a groove in the floating-tube-sheet skirt. A slip-on backing flange, which in service is held in place by the shear ring, bolts to the external floating-head cover. The floating-head cover is usually a circular disk. With an odd number of tube-side passes, an axial nozzle can be installed in such a floating-head cover. If a side nozzle is required, the circular disk is replaced by either a dished head or a channel barrel (similar to Fig. 11-36f ) bolted between floating-head cover and floating-tube-sheet skirt. The outer tube limit approaches the inside of the skirt but is farther removed from the inside of the shell than for any of the previously discussed constructions. Clearances between shell diameter and bundle OTL are 22 mm ( 7⁄8 in) for small-diameter pipe shells, 44 mm (1e in) for large-diameter pipe shells, and 58 mm (2g in) for moderatediameter plate shells. Internal Floating-Head Exchanger (Fig. 11-36a) The internal floating-head design is used extensively in petroleum-refinery service, but in recent years there has been a decline in usage. The tube bundle is removable, and the floating tube sheet moves (or floats) to accommodate differential expansion between shell and tubes. The outer tube limit approaches the inside diameter of the gasket at the floating tube sheet. Clearances (between shell and OTL) are 29 mm (1f in) for pipe shells and 37 mm (1q in) for moderatediameter plate shells. A split backing ring and bolting usually hold the floating-head cover at the floating tube sheet. These are located beyond the end of the shell and within the larger-diameter shell cover. Shell cover, split backing ring, and floating-head cover must be removed before the tube bundle can pass through the exchanger shell. With an even number of tube-side passes the floating-head cover serves as return cover for the tube-side fluid. With an odd number of passes a nozzle pipe must extend from the floating-head cover through the shell cover. Provision for both differential expansion and tube-bundle removal must be made. Pull-Through Floating-Head Exchanger (Fig. 11-36e) Construction is similar to that of the internal-floating-head split-backingring exchanger except that the floating-head cover bolts directly to the floating tube sheet. The tube bundle can be withdrawn from the shell without removing either shell cover or floating-head cover. This feature reduces maintenance time during inspection and repair. The large clearance between the tubes and the shell must provide for both the gasket and the bolting at the floating-head cover. This clearance is about 2 to 2a times that required by the split-ring design. Sealing strips or dummy tubes are often installed to reduce bypassing of the tube bundle. Falling-Film Exchangers Falling-film shell-and-tube heat exchangers have been developed for a wide variety of services and are described by Sack [Chem. Eng. Prog., 63, 55 (July 1967)]. The fluid enters at the top of the vertical tubes. Distributors or slotted tubes put the liquid in film flow in the inside surface of the tubes, and the film adheres to the tube surface while falling to the bottom of the tubes. The film can be cooled, heated, evaporated, or frozen by means of the proper heat-transfer medium outside the tubes. Tube distributors have been developed for a wide range of applications. Fixed tube sheets, with or without expansion joints, and outside-packed-head designs are used. Principal advantages are high rate of heat transfer, no internal pressure drop, short time of contact (very important for heat-sensitive materials), easy accessibility to tubes for cleaning, and, in some cases, prevention of leakage from one side to another. These falling-film exchangers are used in various services as described in the following paragraphs. Liquid Coolers and Condensers Dirty water can be used as the cooling medium. The top of the cooler is open to the atmosphere for access to tubes. These can be cleaned without shutting down the cooler by removing the distributors one at a time and scrubbing the tubes. Evaporators These are used extensively for the concentration of ammonium nitrate, urea, and other chemicals sensitive to heat when

minimum contact time is desirable. Air is sometimes introduced in the tubes to lower the partial pressure of liquids whose boiling points are high. These evaporators are built for pressure or vacuum and with top or bottom vapor removal. Absorbers These have a two-phase flow system. The absorbing medium is put in film flow during its fall downward on the tubes as it is cooled by a cooling medium outside the tubes. The film absorbs the gas which is introduced into the tubes. This operation can be cocurrent or countercurrent. Freezers By cooling the falling film to its freezing point, these exchangers convert a variety of chemicals to the solid phase. The most common application is the production of sized ice and paradichlorobenzene. Selective freezing is used for isolating isomers. By melting the solid material and refreezing in several stages, a higher degree of purity of product can be obtained. TUBE-SIDE CONSTRUCTION Tube-Side Header The tube-side header (or stationary head) contains one or more flow nozzles. The bonnet (Fig. 11-35B) bolts to the shell. It is necessary to remove the bonnet in order to examine the tube ends. The fixed-tubesheet exchanger of Fig. 11-36b has bonnets at both ends of the shell. The channel (Fig. 11-35A) has a removable channel cover. The tube ends can be examined by removing this cover without disturbing the piping connections to the channel nozzles. The channel can bolt to the shell as shown in Fig. 11-36a and c. The Type C and Type N channels of Fig. 11-35 are welded to the tube sheet. This design is comparable in cost with the bonnet but has the advantages of permitting access to the tubes without disturbing the piping connections and of eliminating a gasketed joint. Special High-Pressure Closures (Fig. 11-35D) The channel barrel and the tube sheet are generally forged. The removable channel cover is seated in place by hydrostatic pressure, while a shear ring subjected to shearing stress absorbs the end force. For pressures above 6205 kPa (900 lbf/in2) these designs are generally more economical than bolted constructions, which require larger flanges and bolting as pressure increases in order to contain the end force with bolts in tension. Relatively light-gauge internal pass partitions are provided to direct the flow of tube-side fluids but are designed only for the differential pressure across the tube bundle. Tube-Side Passes Most exchangers have an even number of tube-side passes. The fixed-tube-sheet exchanger (which has no shell cover) usually has a return cover without any flow nozzles as shown in Fig. 11-35M; Types L and N are also used. All removable-bundle designs (except for the U tube) have a floating-head cover directing the flow of tube-side fluid at the floating tube sheet. Tubes Standard heat-exchanger tubing is d, r, a, v, e, 1, 1d, and 1a in in outside diameter (in × 25.4 = mm). Wall thickness is measured in Birmingham wire gauge (BWG) units. (A comprehensive list of tubing characteristics and sizes is given in section 9, table D-7 of TEMA.) The most commonly used tubes in chemical plants and petroleum refineries are 19- and 25-mm (e- and 1-in) outside diameter. Standard tube lengths are 8, 10, 12, 16, and 20 ft, with 20 ft now the most common (ft × 0.3048 = m). Manufacturing tolerances for steel, stainless-steel, and nickelalloy tubes are such that the tubing is produced to either average or minimum wall thickness. Seamless carbon steel tube of minimum wall thickness may vary from 0 to 20 percent above the nominal wall thickness. Average-wall seamless tubing has an allowable variation of plus or minus 10 percent. Welded carbon steel tube is produced to closer tolerances (0 to plus 18 percent on minimum wall; plus or minus 9 percent on average wall). Tubing of aluminum, copper, and their alloys can be drawn easily and usually is made to minimum wall specifications. Common practice is to specify exchanger surface in terms of total external square feet of tubing. The effective outside heat-transfer surface is based on the length of tubes measured between the inner faces of tube sheets. In most heat exchangers there is little difference between the total and the effective surface. Significant differences are usually found in high-pressure and double-tube-sheet designs.

TEMA-STYLE SHELL-AND-TUBE HEAT EXCHANGERS Integrally finned tube, which is available in a variety of alloys and sizes, is being used in shell-and-tube heat exchangers. The fins are radially extruded from thick-walled tube to a height of 1.6 mm (g in) spaced at 1.33 mm (19 fins per inch) or to a height of 3.2 mm (f in) spaced at 2.3 mm (11 fins per inch). External surface is approximately 2a times the outside surface of a bare tube with the same outside diameter. Also available are 0.93-mm- (0.037-in-) high fins spaced 0.91 mm (28 fins per inch) with an external surface about 3.5 times the surface of the bare tube. Bare ends of nominal tube diameter are provided, while the fin height is slightly less than this diameter. The tube can be inserted into a conventional tube bundle and rolled or welded to the tube sheet by the same means, used for bare tubes. An integrally finned tube rolled into a tube sheet with double serrations and flared at the inlet is shown in Fig. 11-39. Internally finned tubes have been manufactured but have limited application.

11-41

The variables in tube-end welding were discussed in two unpublished papers (Emhardt, “Heat Exchanger Tube-to-Tubesheet Joints,” ASME Pap. 69-WA/HT-47; and Reynolds, “Tube Welding for Conventional and Nuclear Power Plant Heat Exchangers,” ASME Pap. 69-WA/HT-24), which were presented at the November 1969 meeting of the American Society of Mechanical Engineers. Tube-end rolling before welding may leave lubricant from the tube expander in the tube hole. Fouling during normal operation followed by maintenance operations will leave various impurities in and near the tube ends. Satisfactory welds are rarely possible under such conditions, since tube-end welding requires extreme cleanliness in the area to be welded. Tube expansion after welding has been found useful for low and moderate pressures. In high-pressure service tube rolling has not been able to prevent leakage after weld failure. Double-Tube-Sheet Joints This design prevents the passage of either fluid into the other because of leakage at the tube-to-tubesheet joints, which are generally the weakest points in heat exchangers. Any leakage at these joints admits the fluid to the gap between the tube sheets. Mechanical design, fabrication, and maintenance of double-tube-sheet designs require special consideration. SHELL-SIDE CONSTRUCTION

FIG. 11-39 Integrally finned tube rolled into tube sheet with double serrations and flared inlet. (Woverine Division, UOP, Inc.)

Longitudinal fins are commonly used in double-pipe exchangers upon the outside of the inner tube. U-tube and conventional removable tube bundles are also made from such tubing. The ratio of external to internal surface generally is about 10 or 15:1. Transverse fins upon tubes are used in low-pressure gas services. The primary application is in air-cooled heat exchangers (as discussed under that heading), but shell-and-tube exchangers with these tubes are in service. Rolled Tube Joints Expanded tube-to-tube-sheet joints are standard. Properly rolled joints have uniform tightness to minimize tube fractures, stress corrosion, tube-sheet ligament pushover and enlargement, and dishing of the tube sheet. Tubes are expanded into the tube sheet for a length of two tube diameters, or 50 mm (2 in), or tube-sheet thickness minus 3 mm (f in). Generally tubes are rolled for the last of these alternatives. The expanded portion should never extend beyond the shell-side face of the tube sheet, since removing such a tube is extremely difficult. Methods and tools for tube removal and tube rolling were discussed by John [Chem. Eng., 66, 77–80 (Dec. 28, 1959)], and rolling techniques by Bach [Pet. Refiner, 39, 8, 104 (1960)]. Tube ends may be projecting, flush, flared, or beaded (listed in order of usage). The flare or bell-mouth tube end is usually restricted to water service in condensers and serves to reduce erosion near the tube inlet. For moderate general process requirements at gauge pressures less than 2058 kPa (300 lbf/in2) and less than 177°C (350°F), tube-sheet holes without grooves are standard. For all other services with expanded tubes at least two grooves in each tube hole are common. The number of grooves is sometimes changed to one or three in proportion to tube-sheet thickness. Expanding the tube into the grooved tube holes provides a stronger joint but results in greater difficulties during tube removal. Welded Tube Joints When suitable materials of construction are used, the tube ends may be welded to the tube sheets. Welded joints may be seal-welded “for additional tightness beyond that of tube rolling” or may be strength-welded. Strength-welded joints have been found satisfactory in very severe services. Welded joints may or may not be rolled before or after welding.

Shell Sizes Heat-exchanger shells are generally made from standard-wall steel pipe in sizes up to 305-mm (12-in) diameter; from 9.5-mm (r-in) wall pipe in sizes from 356 to 610 mm (14 to 24 in); and from steel plate rolled at discrete intervals in larger sizes. Clearances between the outer tube limit and the shell are discussed elsewhere in connection with the different types of construction. The following formulae may be used to estimate tube counts for various bundle sizes and tube passes. The estimated values include the removal of tubes to provide an entrance area for shell nozzle sizes of one-fifth the shell diameter. Due to the large effect from other parameters such as design pressure/corrosion allowance, baffle cuts, seal strips, and so on, these are to be used as estimates only. Exact tube counts are part of the design package of most reputable exchanger design software and are normally used for the final design. Triangular tube layouts with pitch equal to 1.25 times the tube outside diameter: C = 0.75 (D/d) − 36; where D = Bundle O.D. d = Tube O.D. Range of accuracy: −24 ≤ C ≤ 24. 1 Tube Pass: Nt = 1298. + 74.86C + 1.283C2 − .0078C3 − .0006C4 2 Tube Pass: Nt = 1266. + 73.58C + 1.234C2 − .0071C3 − .0005C4 4 Tube Pass: Nt = 1196. + 70.79C + 1.180C2 − .0059C3 − .0004C4 6 Tube Pass: Nt = 1166. + 70.72C + 1.269C2 − .0074C3 − .0006C4

(11-74a) (11-74b) (11-74c) (11-74d)

Square tube layouts with pitch equal to 1.25 times the tube outside diameter: C = (D/d) − 36.; where D = Bundle O.D. d = Tube O.D. Range of accuracy: −24 ≤ C ≤ 24. 1 Tube Pass: Nt = 593.6 + 33.52C + .3782C2 − .0012C3 + .0001C4 2 Tube Pass: Nt = 578.8 + 33.36C + .3847C2 − .0013C3 + .0001C4 4 Tube Pass: Nt = 562.0 + 33.04C + .3661C2 − .0016C3 + .0002C4 6 Tube Pass: Nt = 550.4 + 32.49C + .3873C2 − .0013C3 + .0001C4

(11-75a) (11-75b) (11-75c) (11-75d)

Shell-Side Arrangements The one-pass shell (Fig. 11-35E) is the most commonly used arrangement. Condensers from single component vapors often have the nozzles moved to the center of the shell for vacuum and steam services. A solid longitudinal baffle is provided to form a two-pass shell (Fig. 11-35F). It may be insulated to improve thermal efficiency. (See further discussion on baffles). A two-pass shell can improve thermal effectiveness at a cost lower than for two shells in series. For split flow (Fig. 11-35G), the longitudinal baffle may be solid or perforated. The latter feature is used with condensing vapors. A double-split-flow design is shown in Fig. 11-35H. The longitudinal baffles may be solid or perforated. The divided flow design (Fig. 11-35J), mechanically is like the

11-42

HEAT-TRANSFER EQUIPMENT

one-pass shell except for the addition of a nozzle. Divided flow is used to meet low-pressure-drop requirements. The kettle reboiler is shown in Fig. 11-35K. When nucleate boiling is to be done on the shell-side, this common design provides adequate dome space for separation of vapor and liquid above the tube bundle and surge capacity beyond the weir near the shell cover.

fles, or for dirty fluids in which the dirt might settle out. Vertical-cut baffles are used for side-to-side flow in horizontal exchangers with condensing fluids or dirty fluids. Baffles are notched to assure complete drainage when the units are taken out of service. (These notches permit some bypassing of the tube bundle during normal operation.) Tubes are most commonly arranged on an equilateral triangular pitch. Tubes are arranged on a square pitch primarily for mechanical cleaning purposes in removable-bundle exchangers. Maximum baffle cut is limited to about 45 percent for single segmental baffles so that every pair of baffles will support each tube. Tube bundles are generally provided with baffles cut so that at least one row of tubes passes through all the baffles or support plates. These tubes hold the entire bundle together. In pipe-shell exchangers with a horizontal baffle cut and a horizontal pass rib for directing tubeside flow in the channel, the maximum baffle cut, which permits a minimum of one row of tubes to pass through all baffles, is approximately 33 percent in small shells and 40 percent in larger pipe shells. Maximum shell-side heat-transfer rates in forced convection are apparently obtained by cross-flow of the fluid at right angles to the tubes. In order to maximize this type of flow some heat exchangers are built with segmental-cut baffles and with “no tubes in the window” (or the baffle cutout). Maximum baffle spacing may thus equal maximum unsupported-tube span, while conventional baffle spacing is limited to one-half of this span. The maximum baffle spacing for no tubes in the window of single segmental baffles is unlimited when intermediate supports are provided. These are cut on both sides of the baffle and therefore do not affect the flow of the shell-side fluid. Each support engages all the tubes; the supports are spaced to provide adequate support for the tubes. Rod Baffles Rod or bar baffles have either rods or bars extending through the lanes between rows of tubes. A baffle set can consist of a baffle with rods in all the vertical lanes and another baffle with

BAFFLES AND TUBE BUNDLES The tube bundle is the most important part of a tubular heat exchanger. The tubes generally constitute the most expensive component of the exchanger and are the one most likely to corrode. Tube sheets, baffles, or support plates, tie rods, and usually spacers complete the bundle. Minimum baffle spacing is generally one-fifth of the shell diameter and not less than 50.8 mm (2 in). Maximum baffle spacing is limited by the requirement to provide adequate support for the tubes. The maximum unsupported tube span in inches equals 74 d0.75 (where d is the outside tube diameter in inches). The unsupported tube span is reduced by about 12 percent for aluminum, copper, and their alloys. Baffles are provided for heat-transfer purposes. When shell-side baffles are not required for heat-transfer purposes, as may be the case in condensers or reboilers, tube supports are installed. Segmental Baffles Segmental or cross-flow baffles are standard. Single, double, and triple segmental baffles are used. Baffle cuts are illustrated in Fig. 11-40. The double segmental baffle reduces crossflow velocity for a given baffle spacing. The triple segmental baffle reduces both cross-flow and long-flow velocities and has been identified as the “window-cut” baffle. Baffle cuts are expressed as the ratio of segment opening height to shell inside diameter. Cross-flow baffles with horizontal cut are shown in Fig. 11-36a, c, and f. This arrangement is not satisfactory for horizontal condensers, since the condensate can be trapped between baf-

(a)

(b)

(c) Baffle cuts. (a) Baffle cuts for single segmental baffles. (b) Baffle cuts for double segmental baffles. (c) Baffle cuts for triple segmental baffles. FIG. 11-40

TEMA-STYLE SHELL-AND-TUBE HEAT EXCHANGERS rods in all the horizontal lanes between the tubes. The shell-side flow is uniform and parallel to the tubes. Stagnant areas do not exist. One device uses four baffles in a baffle set. Only half of either the vertical or the horizontal tube lanes in a baffle have rods. The new design apparently provides a maximum shell-side heat-transfer coefficient for a given pressure drop. Tie Rods and Spacers Tie rods are used to hold the baffles in place with spacers, which are pieces of tubing or pipe placed on the rods to locate the baffles. Occasionally baffles are welded to the tie rods, and spacers are eliminated. Properly located tie rods and spacers serve both to hold the bundle together and to reduce bypassing of the tubes. In very large fixed-tube-sheet units, in which concentricity of shells decreases, baffles are occasionally welded to the shell to eliminate bypassing between the baffle and the shell. Metal baffles are standard. Occasionally plastic baffles are used either to reduce corrosion or in vibratory service, in which metal baffles may cut the tubes. Impingement Baffle The tube bundle is customarily protected against impingement by the incoming fluid at the shell inlet nozzle when the shell-side fluid is at a high velocity, is condensing, or is a twophase fluid. Minimum entrance area about the nozzle is generally equal to the inlet nozzle area. Exit nozzles also require adequate area between the tubes and the nozzles. A full bundle without any provision for shell inlet nozzle area can increase the velocity of the inlet fluid by as much as 300 percent with a consequent loss in pressure. Impingement baffles are generally made of rectangular plate, although circular plates are more desirable. Rods and other devices are sometimes used to protect the tubes from impingement. In order to maintain a maximum tube count the impingement plate is often placed in a conical nozzle opening or in a dome cap above the shell. Impingement baffles or flow-distribution devices are recommended for axial tube-side nozzles when entrance velocity is high. Vapor Distribution Relatively large shell inlet nozzles, which may be used in condensers under low pressure or vacuum, require provision for uniform vapor distribution. Tube-Bundle Bypassing Shell-side heat-transfer rates are maximized when bypassing of the tube bundle is at a minimum. The most significant bypass stream is generally between the outer tube limit and the inside of the shell. The clearance between tubes and shell is at a minimum for fixed-tube-sheet construction and is greatest for straight-tube removable bundles. Arrangements to reduce tube-bundle bypassing include: Dummy tubes. These tubes do not pass through the tube sheets and can be located close to the inside of the shell. Tie rods with spacers. These hold the baffles in place but can be located to prevent bypassing. Sealing strips. These longitudinal strips either extend from baffle to baffle or may be inserted in slots cut into the baffles. Dummy tubes or tie rods with spacers may be located within the pass partition lanes (and between the baffle cuts) in order to ensure maximum bundle penetration by the shell-side fluid. When tubes are omitted from the tube layout to provide entrance area about an impingement plate, the need for sealing strips or other devices to cause proper bundle penetration by the shell-side fluid is increased. Longitudinal Flow Baffles In fixed-tube-sheet construction with multipass shells, the baffle is usually welded to the shell and positive assurance against bypassing results. Removable tube bundles have a sealing device between the shell and the longitudinal baffle. Flexible light-gauge sealing strips and various packing devices have been used. Removable U-tube bundles with four tube-side passes and two shell-side passes can be installed in shells with the longitudinal baffle welded in place. In split-flow shells the longitudinal baffle may be installed without a positive seal at the edges if design conditions are not seriously affected by a limited amount of bypassing. Fouling in petroleum-refinery service has necessitated rough treatment of tube bundles during cleaning operations. Many refineries avoid the use of longitudinal baffles, since the sealing devices are subject to damage during cleaning and maintenance operations.

11-43

CORROSION IN HEAT EXCHANGERS Some of the special considerations in regard to heat-exchanger corrosion are discussed in this subsection. A more extended presentation in Sec. 23 covers corrosion and its various forms as well as materials of construction. Materials of Construction The most common material of construction for heat exchangers is carbon steel. Stainless-steel construction throughout is sometimes used in chemical-plant service and on rare occasions in petroleum refining. Many exchangers are constructed from dissimilar metals. Such combinations are functioning satisfactorily in certain services. Extreme care in their selection is required since electrolytic attack can develop. Carbon steel and alloy combinations appear in Table 11-12 “Alloys” in chemical- and petrochemical-plant service in approximate order of use are stainless-steel series 300, nickel, Monel, copper alloy, aluminum, Inconel, stainless-steel series 400, and other alloys. In petroleum-refinery service the frequency order shifts, with copper alloy (for water-cooled units) in first place and low-alloy steel in second place. In some segments of the petroleum industry copper alloy, stainless series 400, low-alloy steel, and aluminum are becoming the most commonly used alloys. Copper-alloy tubing, particularly inhibited admiralty, is generally used with cooling water. Copper-alloy tube sheets and baffles are generally of naval brass. Aluminum alloy (and in particular alclad aluminum) tubing is sometimes used in water service. The alclad alloy has a sacrificial aluminum-alloy layer metallurgically bonded to a core alloy. Tube-side headers for water service are made in a wide variety of materials: carbon steel, copper alloy, cast iron, and lead-lined or plastic-lined or specially painted carbon steel. Bimetallic Tubes When corrosive requirements or temperature conditions do not permit the use of a single alloy for the tubes, bimetallic (or duplex) tubes may be used. These can be made from almost any possible combination of metals. Tube sizes and gauges can be varied. For thin gauges the wall thickness is generally divided equally between the two components. In heavier gauges the more expensive component may comprise from a fifth to a third of the total thickness. The component materials comply with applicable ASTM specifications, but after manufacture the outer component may increase in hardness beyond specification limits, and special care is required during the tube-rolling operation. When the harder material is on the outside, precautions must be exercised to expand the tube properly. When the inner material is considerably softer, rolling may not be practical unless ferrules of the soft material are used. In order to eliminate galvanic action the outer tube material may be stripped from the tube ends and replaced with ferrules of the inner tube material. When the end of a tube with a ferrule is expanded or welded to a tube sheet, the tube-side fluid can contact only the inner tube material, while the outer material is exposed to the shell-side fluid. Bimetallic tubes are available from a small number of tube mills and are manufactured only on special order and in large quantities.

TABLE 11-12 Construction

Dissimilar Materials in Heat-Exchanger Relative use

Part Tubes Tube sheets Tube-side headers Baffles Shell

Relative cost

1

2

3

4

5

6

A

B

C

D

C

E

●

● ●

● ● ●

● ● ● ●

● ●

● ●

●

● ●

Carbon steel replaced by an alloy when ● appears. Relative use: from 1 (most popular) through 6 (least popular) combinations. Relative cost: from A (least expensive) to E (most expensive).

11-44

HEAT-TRANSFER EQUIPMENT

Clad Tube Sheets Usually tube sheets and other exchanger parts are of a solid metal. Clad or bimetallic tube sheets are used to reduce costs or because no single metal is satisfactory for the corrosive conditions. The alloy material (e.g., stainless steel, Monel) is generally bonded or clad to a carbon steel backing material. In fixed-tube-sheet construction a copper-alloy-clad tube sheet can be welded to a steel shell, while most copper-alloy tube sheets cannot be welded to steel in a manner acceptable to ASME Code authorities. Clad tube sheets in service with carbon steel backer material include stainless-steel types 304, 304L, 316, 316L, and 317, Monel, Inconel, nickel, naval rolled brass, copper, admiralty, silicon bronze, and titanium. Naval rolled brass and Monel clad on stainless steel are also in service. Ferrous-alloy-clad tube sheets are generally prepared by a weld overlay process in which the alloy material is deposited by welding upon the face of the tube sheet. Precautions are required to produce a weld deposit free of defects, since these may permit the process fluid to attack the base metal below the alloy. Copper-alloy-clad tube sheets are prepared by brazing the alloy to the carbon steel backing material. Clad materials can be prepared by bonding techniques, which involve rolling, heat treatment, explosive bonding, etc. When properly manufactured, the two metals do not separate because of thermalexpansion differences encountered in service. Applied tube-sheet facings prepared by tack welding at the outer edges of alloy and base metal or by bolting together the two metals are in limited use. Nonmetallic Construction Shell-and-tube exchangers with glass tubes 14 mm (0.551 in) in diameter and 1 mm (0.039 in) thick with tube lengths from 2.015 m (79.3 in) to 4.015 m (158 in) are available. Steel shell exchangers have a maximum design pressure of 517 kPa (75 lbf/in2). Glass shell exchangers have a maximum design gauge pressure of 103 kPa (15 lbf/in2). Shell diameters are 229 mm (9 in), 305 mm (12 in), and 457 mm (18 in). Heat-transfer surface ranges from 3.16 to 51 m2 (34 to 550 ft2). Each tube is free to expand, since a Teflon sealer sheet is used at the tube-to-tube-sheet joint. Impervious graphite heat-exchanger equipment is made in a variety of forms, including outside-packed-head shell-and-tube exchangers. They are fabricated with impervious graphite tubes and

tube-side headers and metallic shells. Single units containing up to 1300 m2 (14,000 ft2) of heat-transfer surface are available. Teflon heat exchangers of special construction are described later in this section. Fabrication Expanding the tube into the tube sheet reduces the tube wall thickness and work-hardens the metal. The induced stresses can lead to stress corrosion. Differential expansion between tubes and shell in fixed-tube-sheet exchangers can develop stresses, which lead to stress corrosion. When austenitic stainless-steel tubes are used for corrosion resistance, a close fit between the tube and the tube hole is recommended in order to minimize work hardening and the resulting loss of corrosion resistance. In order to facilitate removal and replacement of tubes it is customary to roller-expand the tubes to within 3 mm (f in) of the shellside face of the tube sheet. A 3-mm- (f-in-) long gap is thus created between the tube and the tube hole at this tube-sheet face. In some services this gap has been found to be a focal point for corrosion. It is standard practice to provide a chamfer at the inside edges of tube holes in tube sheets to prevent cutting of the tubes and to remove burrs produced by drilling or reaming the tube sheet. In the lower tube sheet of vertical units this chamfer serves as a pocket to collect material, dirt, etc., and to serve as a corrosion center. Adequate venting of exchangers is required both for proper operation and to reduce corrosion. Improper venting of the water side of exchangers can cause alternate wetting and drying and accompanying chloride concentration, which is particularly destructive to the series 300 stainless steels. Certain corrosive conditions require that special consideration be given to complete drainage when the unit is taken out of service. Particular consideration is required for the upper surfaces of tube sheets in vertical heat exchangers, for sagging tubes, and for shell-side baffles in horizontal units. SHELL-AND-TUBE EXCHANGER COSTS Basic costs of shell-and-tube heat exchangers made in the United States of carbon steel construction in 1958 are shown in Fig. 11-41.

FIG. 11-41 Costs of basic exchangers—all steel, TEMA Class R, 150 lbf/in2, 1958. To convert pounds-force per square inch to kilopascals, multiply by 6.895; to convert square feet to square meters, multiply by 0.0929; to convert inches to millimeters, multiply by 25.4; and to convert feet to meters, multiply by 0.3048.

TEMA-STYLE SHELL-AND-TUBE HEAT EXCHANGERS Cost data for shell-and-tube exchangers from 15 sources were correlated and found to be consistent when scaled by the Marshall and Swift index [Woods et al., Can. J. Chem. Eng., 54, 469–489 (December 1976)]. Costs of shell-and-tube heat exchangers can be estimated from Fig. 11-41 and Tables 11-13 and 11-14. These 1960 costs should be updated by use of the Marshall and Swift Index, which appears in each issue of Chemical Engineering. Note that during periods of high and low demand for heat exchangers the prices in the marketplace may vary significantly from those determined by this method. Small heat exchangers and exchangers bought in small quantities are likely to be more costly than indicated. Standard heat exchangers (which are in some instances off-theshelf items) are available in sizes ranging from 1.9 to 37 m2 (20 to 400 2 ft ) at costs lower than for custom-built units. Steel costs are approximately one-half, admiralty tube-side costs are two-thirds, and stainless costs are three-fourths of those for equivalent custom-built exchangers. Kettle-type-reboiler costs are 15 to 25 percent greater than for equivalent internal-floating-head or U-tube exchangers. The higher extra is applicable with relatively large kettle-to-port-diameter ratios

TABLE 11-13

11-45

and with increased internals (e.g., vapor-liquid separators, foam breakers, sight glasses). To estimate exchanger costs for varying construction details and alloys, first determine the base cost of a similar heat exchanger of basic construction (carbon steel, Class R, 150 lbf/in2) from Fig. 11-41. From Table 11-13, select appropriate extras for higher pressure rating and for alloy construction of tube sheets and baffles, shell and shell cover, and channel and floating-head cover. Compute these extras in accordance with the notes below the table. For tubes other than welded carbon steel, compute the extra by multiplying the exchanger surface by the appropriate cost per square foot from Table 11-14. When points for 20-ft-long tubes do not appear in Fig. 11-41, use 0.95 times the cost of the equivalent 16-ft-long exchanger. Length variation of steel heat exchangers affects costs by approximately $1 per square foot. Shell diameters for a given surface are approximately equal for U-tube and floating-head construction. Low-fin tubes (g-in-high fins) provide 2.5 times the surface per lineal foot. Surface required should be divided by 2.5; then use Fig. 11-41 to determine basic cost of the heat exchanger. Actual surface times extra costs (from Table 11-14) should then be added to determine cost of fin-tube exchanger.

Extras for Pressure and Alloy Construction and Surface and Weights* Percent of steel base price, 1500-lbf/in2 working pressure Shell diameters, in

Pressure† 300 lbf/in2 450 lbf/in2 600 lbf/in2 Alloy All-steel heat exchanger Tube sheets and baffles Naval rolled brass Monel 1d Cr, a Mo 4–6 Cr, a Mo 11–13 Cr (stainless 410) Stainless 304 Shell and shell cover Monel 1d Cr, a Mo 4–6 Cr, a Mo 11–13 Cr (stainless 410) Stainless 304 Channel and floating-head cover Monel 1d Cr, a Mo 4–6 Cr, a Mo 11–13 Cr (stainless 410) Stainless 304 Surface Surface, ft2, internal floating head, e-in OD by 1-in square pitch, 16 ft 0 in, tube‡ 1-in OD by 1d-in square pitch, 16-ft 0-in tube§ Weight, lb, internal floating head, 1-in OD, 14 BWG tube

12

14

16

18

20

22

24

27

30

33

36

39

42

7 18 28

7 19 29

8 20 31

8 21 33

9 22 35

9 23 37

10 24 39

11 27 40

11 29 41

12 31 32

13 32 44

14 33 45

15 35 50

100

100

100

100

100

100

100

100

100

100

100

100

100

14 24 6 19 21 22

17 31 7 22 24 27

19 35 7 24 26 29

21 37 7 25 27 30

22 39 8 26 27 31

22 39 8 26 27 31

22 40 8 26 27 31

22 40 8 25 27 31

22 41 9 25 27 30

23 41 10 25 27 30

24 41 10 26 27 30

24 41 10 26 27 31

25 42 11 26 28 31

45 20 28 29 32

48 22 31 33 34

51 24 33 35 36

52 25 35 36 37

53 25 35 36 38

52 25 35 36 37

52 24 34 35 37

51 22 32 34 35

49 20 30 32 33

47 19 28 30 31

45 18 27 29 30

44 17 26 27 29

44 17 26 27 28

40 23 36 37 37

42 24 37 38 39

42 24 38 39 39

43 25 38 39 39

42 24 37 38 38

41 24 36 37 37

40 23 34 35 36

37 22 31 32 33

34 21 29 30 31

32 21 27 28 29

31 21 26 27 28

40 20 25 26 26

30 20 24 25 26

251

302

438

565

726

890

1040

1470

1820

2270

2740

3220

3700

218

252

352

470

620

755

876

1260

1560

1860

2360

2770

3200

2750

3150

4200

5300

6600

7800

9400

11,500

14,300

17,600

20,500

24,000

29,000

*Modified from E. N. Sieder and G. H. Elliot, Pet. Refiner, 39(5), 223 (1960). †Total extra is 0.7 × pressure extra on shell side plus 0.3 × pressure extra on tube side. ‡Fixed-tube-sheet construction with e-in OD tube on y-in triangular pitch provides 36 percent more surface. §Fixed-tube-sheet construction with 1-in OD tube on 1d-in triangular pitch provides 18 percent more surface. For an all-steel heat exchanger with mixed design pressures the total extra for pressure is 0.7 × pressure extra on shell side plus 0.3 × pressure extra tube side. For an exchanger with alloy parts and a design pressure of 150 lbf/in2, the alloy extras are added. For shell and shell cover the combined alloy-pressure extra is the alloy extra times the shell-side pressure extra/100. For channel and floating-head cover the combined alloy-pressure extra is the alloy extra times the tube-side pressure extra/100. For tube sheets and baffles the combined alloy-pressure extra is the alloy extra times the higher-pressure extra times 0.9/100. (The 0.9 factor is included since baffle thickness does not increase because of pressure.) NOTE: To convert pounds-force per square inch to kilopascals, multiply by 6.895; to convert square feet to square meters, multiply by 0.0929; and to convert inches to millimeters, multiply by 25.4.

11-46

HEAT-TRANSFER EQUIPMENT TABLE 11-14

Base Quantity Extra Cost for Tube Gauge and Alloy Dollars per square foot e-in OD tubes

Carbon steel Admiralty (T-11) 1d Cr, a Mo (T-5) 4–6 Cr Stainless 410 welded Stainless 410 seamless Stainless 304 welded Stainless 304 seamless Stainless 316 welded Stainless 316 seamless 90-10 cupronickel Monel Low fin Carbon steel Admiralty 90-10 cupronickel NOTE:

1-in OD tubes

16 BWG

14 BWG

12 BWG

16 BWG

14 BWG

12 BWG

0 0.78 1.01 1.61 2.62 3.10 2.50 3.86 3.40 7.02 1.33 4.25

0.02 1.20 1.04 1.65 3.16 3.58 3.05 4.43 4.17 7.95 1.89 5.22

0.06 1.81 1.11 1.74 4.12 4.63 3.99 5.69 5.41 10.01 2.67 6.68

0 0.94 0.79 1.28 2.40 2.84 2.32 3.53 3.25 6.37 1.50 4.01

0.01 1.39 0.82 1.32 2.89 3.31 2.83 4.08 3.99 7.27 2.09 4.97

0.07 2.03 0.95 1.48 3.96 4.47 3.88 5.46 5.36 9.53 2.90 6.47

0.22 0.58 0.72

0.23 0.75 0.96

0.18 0.70 0.86

0.19 0.87 1.06

To convert inches to millimeters, multiply by 25.4.

HAIRPIN/DOUBLE-PIPE HEAT EXCHANGERS PRINCIPLES OF CONSTRUCTION Hairpin heat exchangers (often also referred to as “double pipes”) are characterized by a construction form which imparts a U-shaped appearance to the heat exchanger. In its classical sense, the term double pipe refers to a heat exchanger consisting of a pipe within a pipe, usually of a straight-leg construction with no bends. However, due to the need for removable bundle construction and the ability to handle differential thermal expansion while avoiding the use of expansion joints (often the weak point of the exchanger), the current U-shaped configuration has become the standard in the industry (Fig. 11-42). A further departure from the classical definition comes when more than one pipe or tube is used to make a tube bundle, complete with tubesheets and tube supports similar to the TEMA style exchanger. Hairpin heat exchangers consist of two shell assemblies housing a common set of tubes and interconnected by a return-bend cover referred to as the bonnet. The shell is supported by means of bracket assemblies designed to cradle both shells simultaneously. These brackets are configured to permit the modular assembly of many hairpin sections into an exchanger bank for inexpensive future-expansion capability and for providing the very long thermal lengths demanded by special process applications. The bracket construction permits support of the exchanger without fixing the supports to the shell. This provides for thermal movement of the shells within the brackets and prevents the transfer of thermal stresses into the process piping. In special cases the brackets may be welded to the shell. However, this is usually avoided due to the resulting loss of flexibility in field installation and equipment reuse at other sites and an increase in piping stresses. The hairpin heat exchanger, unlike the removable bundle TEMA styles, is designed for bundle insertion and removal from the return end rather than the tubesheet end. This is accomplished by means of removable split rings which slide into grooves machined around the outside of each tubesheet and lock the tubesheets to the external closure flanges. This provides a distinct advantage in maintenance since bundle removal

FIG. 11-42

Fin-tube Co.)

Double-pipe-exchanger section with longitudinal fins. (Brown

takes place at the exchanger end furthest from the plant process piping without disturbing any gasketed joints of this piping. FINNED DOUBLE PIPES The design of the classical single-tube double-pipe heat exchanger is an exercise in pure longitudinal flow with the shellside and tubeside coefficients differing primarily due to variations in flow areas. Adding longitudinal fins gives the more common double-pipe configuration (Table 11-15). Increasing the number of tubes yields the multitube hairpin. MULTITUBE HAIRPINS For years, the slightly higher mechanical-design complexity of the hairpin heat exchanger relegated it to only the smallest process requirements with shell sizes not exceeding 100 mm. In the early 1970s the maximum available sizes were increased to between 300 and 400 mm depending upon the manufacturer. At the present time, due to recent advances in design technology, hairpin exchangers are routinely produced in shell sizes between 50 (2 in) and 800 mm (30 in) for a wide range of pressures and temperatures and have been made in larger sizes as well. Table 11-16 gives common hairpin tube counts and areas for 19 mm (e in) O.D. tubes arranged on a 24 mm (y in) triangular tube layout. The hairpin width and the centerline distance of the two legs (shells) of the hairpin heat exchanger are limited by the outside diameter of the closure flanges at the tubesheets. This diameter, in turn, is a function of the design pressures. As a general rule, for low-tomoderate design pressures (less than 15 bar), the center-to-center distance is approximately 1.5 to 1.8 times the shell outside diameter, with this ratio decreasing slightly for the larger sizes. TABLE 11-15 Shell pipe O.D.

Double-Pipe Hairpin Section Data Inner pipe O.D.

Fin height

Fin count

Surface-areaper-unit length

mm

in

mm

in

mm

in

(max)

sq m/m

sq ft/ft

60.33 88.9 114.3 114.3 114.3 141.3 168.3

2.375 3.500 4.500 4.500 4.500 5.563 6.625

25.4 48.26 48.26 60.33 73.03 88.9 114.3

1.000 1.900 1.900 2.375 2.875 3.500 4.500

12.7 12.7 25.4 19.05 12.70 17.46 17.46

0.50 0.50 1.00 0.75 0.50 0.6875 0.6875

24 36 36 40 48 56 72

0.692 1.07 1.98 1.72 1.45 2.24 2.88

2.27 3.51 6.51 5.63 4.76 7.34 9.44

AIR-COOLED HEAT EXCHANGERS TABLE 11-16

Multitube Hairpin Section Data

Shell O.D.

Shell thickness

Tube count

Surface area for 6.1 m (20 ft.) nominal length

Size

mm

in

mm

in

19 mm

sq m

sq ft

03-MT 04-MT 05-MT 06-MT

88.9 114.3 141.3 168.3

3.500 4.500 5.563 6.625

5.49 6.02 6.55 7.11

0.216 0.237 0.258 0.280

5 9 14 22

3.75 6.73 10.5 16.7

40.4 72.4 113.2 179.6

08-MT 10-MT 12-MT 14-MT

219.1 273.1 323.9 355.6

8.625 10.75 12.75 14.00

8.18 9.27 9.53 9.53

0.322 0.365 0.375 0.375

42 68 109 136

32.0 52.5 84.7 107.

344.3 564.7 912.1 1159.

16-MT 18-MT 20-MT 22-MT

406.4 457.2 508.0 558.8

16.00 18.00 20.00 22.00

9.53 9.53 9.53 9.53

0.375 0.375 0.375 0.375

187 241 304 380

148. 191. 244. 307.

1594. 2054. 2622. 3307.

24-MT 26-MT 28-MT 30-MT

609.6 660.4 711.2 762.0

24.00 26.00 28.00 30.00

9.53 9.53 9.53 11.11

0.375 0.375 0.375 0.4375

463 559 649 752

378. 453. 529. 630.

4065. 4879. 5698. 6776.

One interesting consequence of this fact is the inability to construct a hairpin tube bundle having the smallest radius bends common to a conventional U-tube, TEMA shell, and tube bundle. In fact, in the larger hairpin sizes the tubes might be better described as curved rather than bent. The smallest U-bend diameters are greater than the outside diameter of shells less than 300 mm in size. The U-bend diameters are greater than 300 mm in larger shells. As a general rule, mechanical tube cleaning around the radius of a U-bend may be accomplished with a flexible shaft-cleaning tool for bend diameters greater than ten times the tube’s inside diameter. This permits the tool to pass around the curve of the tube bend without binding. In all of these configurations, maintaining longitudinal flow on both the shellside and tubeside allows the decision for placement of a fluid stream on either one side or the other to be based upon design efficiency (mass flow rates, fluid properties, pressure drops, and veloci-

11-47

ties) and not because there is any greater tendency to foul on one side than the other. Experience has shown that, in cases where fouling is influenced by flow velocity, overall fouling in tube bundles is less in properly designed longitudinal flow bundles where areas of low velocity can be avoided without flow-induced tube vibration. This same freedom of stream choice is not as readily applied when a segmental baffle is used. In those designs, the baffle’s creation of low velocities and stagnant flow areas on the outside of the bundle can result in increased shellside fouling at various locations of the bundle. The basis for choosing the stream side in those cases will be similar to the common shell and tube heat exchanger. At times a specific selection of stream side must be made regardless of tube-support mechanism in expectation of an unresolvable fouling problem. However, this is often the exception rather than the rule. DESIGN APPLICATIONS One benefit of the hairpin exchanger is its ability to handle high tubeside pressures at a lower cost than other removable-bundle exchangers. This is due in part to the lack of pass partitions at the tubesheets which complicate the gasketing design process. Present mechanical design technology has allowed the building of dependable, removable-bundle, hairpin multitubes at tubeside pressures of 825 bar (12,000 psi). The best known use of the hairpin is its operation in true countercurrent flow which yields the most efficient design for processes that have a close temperature approach or temperature cross. However, maintaining countercurrent flow in a tubular heat exchanger usually implies one tube pass for each shell pass. As recently as 30 years ago, the lack of inexpensive, multiple-tube pass capability often diluted the advantages gained from countercurrent flow. The early attempts to solve this problem led to investigations into the area of heat transfer augmentation. This familiarity with augmentation techniques inevitably led to improvements in the efficiency and capacity of the small heat exchangers. The result has been the application of the hairpin heat exchanger to the solution of unique process problems, such as dependable, once-through, convective boilers offering high-exit qualities, especially in cases of process-temperature crosses.

AIR-COOLED HEAT EXCHANGERS AIR-COOLED HEAT EXCHANGERS Atmospheric air has been used for many years to cool and condense fluids in areas of water scarcity. During the 1960s the use of air-cooled heat exchangers grew rapidly in the United States and elsewhere. In Europe, where seasonal variations in ambient temperatures are relatively small, air-cooled exchangers are used for the greater part of process cooling. In some new plants all cooling is done with air. Increased use of air-cooled heat exchangers has resulted from lack of available water, significant increases in water costs, and concern for water pollution. Air-cooled heat exchangers include a tube bundle, which generally has spiral-wound fins upon the tubes, and a fan, which moves air across the tubes and is provided with a driver. Electric motors are the most commonly used drivers; typical drive arrangements require a V belt or a direct right-angle gear. A plenum and structural supports are basic components. Louvers are often used: A bay generally has two tube bundles installed in parallel. These may be in the same or different services. Each bay is usually served by two (or more) fans and is furnished with a structure, a plenum, and other attendant equipment. The location of air-cooled heat exchangers must consider the large space requirements and the possible recirculation of heated air because of the effect of prevailing winds upon buildings, fired heaters, towers, various items of equipment, and other air-cooled exchangers. Inlet air temperature at the exchanger can be significantly higher than

the ambient air temperature at a nearby weather station. See AirCooled Heat Exchangers for General Refinery Services, API Standard 661, 2d ed., January 1978, for information on refinery-process aircooled heat exchangers. Forced and Induced Draft The forced-draft unit, which is illustrated in Fig. 11-43 pushes air across the finned tube surface. The fans are located below the tube bundles. The induced-draft design has the fan above the bundle, and the air is pulled across the finned tube surface. In theory, a primary advantage of the forced-draft unit is that less power is required. This is true when the air-temperature rise exceeds 30°C (54°F). Air-cooled heat exchangers are generally arranged in banks with several exchangers installed side by side. The height of the bundle aboveground must be one-half of the tube length to produce an inlet velocity equal to the face velocity. This requirement applies both to ground-mounted exchangers and to those pipe-rack-installed exchangers which have a fire deck above the pipe rack. The forced-draft design offers better accessibility to the fan for onstream maintenance and fan-blade adjustment. The design also provides a fan and V-belt assembly, which are not exposed to the hot-air stream that exits from the unit. Structural costs are less, and mechanical life is longer. Induced-draft design provides more even distribution of air across the bundle, since air velocity approaching the bundle is relatively low. This design is better suited for exchangers designed for a close approach of product outlet temperature to ambient-air temperature.

11-48

HEAT-TRANSFER EQUIPMENT

FIG. 11-43

Forced-draft air-cooled heat exchanger. [Chem. Eng., 114 (Mar. 27, 1978).]

Induced-draft units are less likely to recirculate the hot exhaust air, since the exit air velocity is several times that of the forced-draft unit. Induced-draft design more readily permits the installation of the aircooled equipment above other mechanical equipment such as pipe racks or shell-and-tube exchangers. In a service in which sudden temperature change would cause upset and loss of product, the induced-draft unit gives more protection in that only a fraction of the surface (as compared with the forceddraft unit) is exposed to rainfall, sleet, or snow. Tube Bundle The principal parts of the tube bundle are the finned tubes and the header. Most commonly used is the plug header, which is a welded box that is illustrated in Fig. 11-44. The finned tubes are described in a subsequent paragraph. The components of a tube bundle are identified in the figure. The second most commonly used header is a cover-plate header. The cover plate is bolted to the top, bottom, and end plates of the header. Removing the cover plate provides direct access to the tubes without the necessity of removing individual threaded plugs. Other types of headers include the bonnet-type header, which is constructed similarly to the bonnet construction of shell-and-tube

heat exchangers; manifold-type headers, which are made from pipe and have tubes welded into the manifold; and billet-type headers, made from a solid piece of material with machined channels for distributing the fluid. Serpentine-type tube bundles are sometimes used for very viscous fluids. A single continuous flow path through pipe is provided. Tube bundles are designed to be rigid and self-contained and are mounted so that they expand independently of the supporting structure. The face area of the tube bundle is its length times width. The net free area for air flow through the bundle is about 50 percent of the face area of the bundle. The standard air face velocity (FV) is the velocity of standard air passing through the tube bundle and generally ranges from 1.5 to 3.6 m/s (300 to 700 ft/min). Tubing The 25.4-mm (1-in) outside-diameter tube is most commonly used. Fin heights vary from 12.7 to 15.9 mm (0.5 to 0.625 in), fin spacing from 3.6 to 2.3 mm (7 to 11 per linear inch), and tube triangular pitch from 50.8 to 63.5 mm (2.0 to 2.5 in). Ratio of extended surface to bare-tube outside surface varies from about 7 to 20. The

Typical construction of a tube bundle with plug headers: (1) tube sheet; (2) plug sheet; (3) top and bottom plates; (4) end plate; (5) tube; (6) pass partition; (7) stiffener; (8) plug; (9) nozzle; (10) side frame; (11) tube spacer; (12) tube-support cross member; (13) tube keeper; (14) vent; (15) drain; (16) instrument connection. (API Standard 661.)

FIG. 11-44

AIR-COOLED HEAT EXCHANGERS 38-mm (1a-in) tube has been used for flue-gas and viscous-oil service. Tube size, fin heights, and fin spacing can be further varied. Tube lengths vary and may be as great as 18.3 m (60 ft). When tube length exceeds 12.2 m (40 ft), three fans are generally installed in each bay. Frequently used tube lengths vary from 6.1 to 12.2 m (20 to 40 ft). Finned-Tube Construction The following are descriptions of commonly used finned-tube constructions (Fig. 11-45). 1. Embedded. Rectangular-cross-section aluminum fin which is wrapped under tension and mechanically embedded in a groove 0.25  0.05 mm (0.010  0.002 in) deep, spirally cut into the outside surface of a tube. 2. Integral (or extruded). An aluminum outer tube from which fins have been formed by extrusion, mechanically bonded to an inner tube or liner. 3. Overlapped footed. L-shaped aluminum fin wrapped under tension over the outside surface of a tube, with the tube fully covered by the overlapped feet under and between the fins. 4. Footed. L-shaped aluminum fin wrapped under tension over the outside surface of a tube with the tube fully covered by the feet between the fins. 5. Bonded. Tubes on which fins are bonded to the outside surface by hot-dip galvanizing, brazing, or welding. Typical metal design temperatures for these finned-tube constructions are 399°C (750°F) embedded, 288°C (550°F) integral, 232°C (450°F) overlapped footed, and 177°C (350°F) footed. Tube ends are left bare to permit insertion of the tubes into appropriate holes in the headers or tube sheets. Tube ends are usually roller-expanded into these tube holes. Fans Axial-flow fans are large-volume, low-pressure devices. Fan diameters are selected to give velocity pressures of approximately 2.5 mm (0.1 in) of water. Total fan efficiency (fan, driver, and transmission device) is about 75 percent, and fan drives usually have a minimum of 95 percent mechanical efficiency. Usually fans are provided with four or six blades. Larger fans may have more blades. Fan diameter is generally slightly less than the width of the bay. At the fan-tip speeds required for economical performance, a large amount of noise is produced. The predominant source of noise is vortex shedding at the trailing edge of the fan blade. Noise control of aircooled exchangers is required by the Occupational Safety and Health Act (OSHA). API Standard 661 (Air-Cooled Heat Exchangers for Gen-

FIG. 11-45

Finned-tube construction.

11-49

eral Refinery Services, 2d ed., January 1978) has the purchaser specifying sound-pressure-level (SPL) values per fan at a location designated by the purchaser and also specifying sound-power-level (PWL) values per fan. These are designated at the following octave-band-center frequencies: 63, 125, 250, 1000, 2000, 4000, 8000, and also the dBa value (the dBa is a weighted single-value sound-pressure level). Reducing the fan-tip speed results in a straight-line reduction in air flow while the noise level decreases. The API Standard limits fan-tip speed to 61 m/s (12,000 ft/min) for typical constructions. Fan-design changes which reduce noise include increasing the number of fan blades, increasing the width of the fan blades, and reducing the clearance between fan tip and fan ring. Both the quantity of air and the developed static pressure of fans in air-cooled heat exchangers are lower than indicated by fan manufacturers’ test data, which are applicable to testing-facility tolerances and not to heat-exchanger constructions. The axial-flow fan is inherently a device for moving a consistent volume of air when blade setting and speed of rotation are constant. Variation in the amount of air flow can be obtained by adjusting the blade angle of the fan and the speed of rotation. The blade angle can be either (1) permanently fixed, (2) hand-adjustable, or (3) automatically adjusted. Air delivery and power are a direct function of blade pitch angle. Fan mounting should provide a minimum of one-half to threefourths diameter between fan and ground on a forced-draft heat exchanger and one-half diameter between tubes and fan on an induced-draft cooler. Fan blades can be made of aluminum, molded plastic, laminated plastic, carbon steel, stainless steel, and Monel. Fan Drivers Electric motors or steam turbines are most commonly used. These connect with gears or V belts. (Gas engines connected through gears and hydraulic motors either direct-connected or connected through gears are in use. Fans may be driven by a prime mover such as a compressor with a V-belt takeoff from the flywheel to a jack shaft and then through a gear or V belt to the fan. Direct motor drive is generally limited to small-diameter fans. V-belt drive assemblies are generally used with fans 3 m (10 ft) and less in diameter and motors of 22.4 kW (30 hp) and less. Right-angle gear drive is preferred for fans over 3 m (10 ft) in diameter, for electric motors over 22.4 kW (30 hp), and with steam-turbine drives.

11-50

HEAT-TRANSFER EQUIPMENT

Fan Ring and Plenum Chambers The air must be distributed from the circular fan to the rectangular face of the tube bundle. The air velocity at the fan is between 3.8 and 10.2 m/s (750 and 2000 ft/in). The plenum-chamber depth (from fan to tube bundle) is dependent upon the fan dispersion angle (Fig. 11-46), which should have a maximum value of 45°. The fan ring is made to commercial tolerances for the relatively large diameter fan. These tolerances are greater than those upon closely machined fan rings used for small-diameter laboratoryperformance testing. Fan performance is directly affected by this increased clearance between the blade tip and the ring, and adequate provision in design must be made for the reduction in air flow. API Standard 661 requires that fan-tip clearance be a maximum of 0.5 percent of the fan diameter for diameters between 1.9 and 3.8 m (6.25 and 12.5 ft). Maximum clearance is 9.5 mm (r in) for smaller fans and 19 mm (e in) for larger fans. The depth of the fan ring is critical. Worsham (ASME Pap. 59-PET27, Petroleum Mechanical Engineering Conference, Houston, 1959) reports an increase in flow varying from 5 to 15 percent with the same power consumption when the depth of a fan ring was doubled. The percentage increase was proportional to the volume of air and static pressure against which the fan was operating. When making a selection, the stall-out condition, which develops when the fan cannot produce any more air regardless of power input, should be considered. Air-Flow Control Process operating requirements and weather conditions are considered in determining the method of controlling air flow. The most common methods include simple on-off control, on-off step control (in the case of multiple-driver units), two-speedmotor control, variable-speed drivers, controllable fan pitch, manually or automatically adjustable louvers, and air recirculation. Winterization is the provision of design features, procedures, or systems for air-cooled heat exchangers to avoid process-fluid operating problems resulting from low-temperature inlet air. These include fluid freezing, pour point, wax formation, hydrate formation, laminar flow, and condensation at the dew point (which may initiate corrosion). Freezing points for some commonly encountered fluids in refinery service include: benzene, 5.6°C (42°F); p-xylene 15.5°C (55.9°F); cyclohexane, 6.6°C (43.8°F); phenol, 40.9°C (105.6°F); monoethanolamine, 10.3°C (50.5°F); and diethanolamine, 25.1°C (77.2°F). Water solutions of these organic compounds are likely to freeze in air-cooled exchangers during winter service. Paraffinic and olefinic gases (C1 through C4) saturated with water vapor form hydrates when cooled. These hydrates are solid crystals which can collect and plug exchanger tubes. Air-flow control in some services can prevent these problems. Cocurrent flow of air and process fluid during winter may be adequate

FIG. 11-46

Fan dispersion angle. (API Standard 661.)

to prevent problems. (Normal design has countercurrent flow of air and process fluid.) In some services when the hottest process fluid is in the bottom tubes, which are exposed to the lowest-temperature air, winterization problems may be eliminated. Following are references which deal with problems in lowtemperature environments: Brown and Benkley, “Heat Exchangers in Cold Service—A Contractor’s View,” Chem. Eng. Prog., 70, 59–62 (July 1974); Franklin and Munn, “Problems with Heat Exchangers in Low Temperature Environments,” Chem. Eng. Prog., 70, 63–67 (July 1974); Newell, “Air-Cooled Heat Exchangers in Low Temperature Environments: A Critique,” Chem. Eng. Prog., 70, 86–91 (October 1974); Rubin, “Winterizing Air Cooled Heat Exchangers,” Hydrocarbon Process., 59, 147–149 (October 1980); Shipes, “Air-Cooled Heat Exchangers in Cold Climates,” Chem. Eng. Prog., 70, 53–58 (July 1974). Air Recirculation Recirculation of air which has been heated as it crosses the tube bundle provides the best means of preventing operating problems due to low-temperature inlet air. Internal recirculation is the movement of air within a bay so that the heated air which has crossed the bundle is directed by a fan with reverse flow across another part of the bundle. Wind skirts and louvers are generally provided to minimize the entry of low-temperature air from the surroundings. Contained internal recirculation uses louvers within the bay to control the flow of warm air in the bay as illustrated in Fig. 11-47. Note that low-temperature inlet air has access to the tube bundle. External recirculation is the movement of the heated air within the bay to an external duct, where this air mixes with inlet air, and the mixture serves as the cooling fluid within the bay. Inlet air does not have direct access to the tube bundle; an adequate mixing chamber is essential. Recirculation over the end of the exchanger is illustrated in Fig. 11-48. Over-the-side recirculation also is used. External recirculation systems maintain the desired low temperature of the air crossing the tube bundle. Trim Coolers Conventional air-cooled heat exchangers can cool the process fluid to within 8.3°C (15°F) of the design dry-bulb temperature. When a lower process outlet temperature is required, a trim cooler is installed in series with the air-cooled heat exchanger. The water-cooled trim cooler can be designed for a 5.6 to 11.1°C (10 to 20°F) approach to the wet-bulb temperature (which in the United States is about 8.3°C (15°F) less than the dry-bulb temperature). In arid areas the difference between dry- and wet-bulb temperatures is much greater. Humidification Chambers The air-cooled heat exchanger is provided with humidification chambers in which the air is cooled to a close approach to the wet-bulb temperature before entering the finned-tube bundle of the heat exchanger. Evaporative Cooling The process fluid can be cooled by using evaporative cooling with the sink temperature approaching the wetbulb temperature.

FIG. 11-47 Contained internal recirculation (with internal louvers). [Hydrocarbon Process, 59, 148–149 (October 1980).]

AIR-COOLED HEAT EXCHANGERS TABLE 11-17

11-51

Air-Cooled Heat-Exchanger Costs (1970)

Surface (bare tube), sq. ft.

500

1000

2000

3000

5000

Cost for 12-row-deep bundle, dollars/square foot

9.0

7.6

6.8

5.7

5.3

Factor for bundle depth: 6 rows 4 rows 3 rows

1.07 1.2 1.25

1.07 1.2 1.25

1.07 1.2 1.25

1.12 1.3 1.5

1.12 1.3 1.5

Base: Bare-tube external surface 1 in. o.d. by 12 B.W.G. by 24 ft. 0 in. steel tube with 8 aluminum fins per inch v-in. high. Steel headers. 150 lb./sq. in. design pressure. V-belt drive and explosion-proof motor. Bare-tube surface 0.262 sq. ft./ft. Fin-tube surface/bare-tube surface ratio is 16.9. Factors: 20 ft. tube length 1.05 30 ft. tube length 0.95 18 B.W.G. admiralty tube 1.04 16 B.W.G. admiralty tube 1.12 NOTE: To convert feet to meters, multiply by 0.3048; to convert square feet to square meters, multiply by 0.0929; and to convert inches to millimeters, multiply by 25.4.

cooled heat exchangers have been discussed by J. E. Lerner [“Simplified Air Cooler Estimating,” Hydrocarbon Process., 52, 93–100 (February 1972)]. Design Considerations External recirculation with adequate mixing chamber. [Hydrocarbon Process, 59, 148–149 (October 1980).]

FIG. 11-48

Steam Condensers Air-cooled steam condensers have been fabricated with a single tube-side pass and several rows of tubes. The bottom row has a higher temperature difference than the top row, since the air has been heated as it crosses the rows of tubes. The bottom row condenses all the entering steam before the steam has traversed the length of the tube. The top row, with a lower temperature driving force, does not condense all the entering steam. At the exit header, uncondensed steam flows from the top row into the bottom row. Since noncondensable gases are always present in steam, these accumulate within the bottom row because steam is entering from both ends of the tube. Performance suffers. Various solutions have been used. These include orifices to regulate the flow into each tube, a “blow-through steam” technique with a vent condenser, complete separation of each row of tubes, and inclined tubes. Air-Cooled Overhead Condensers Air-cooled overhead condensers (AOC) have been designed and installed above distillation columns as integral parts of distillation systems. The condensers generally have inclined tubes, with air flow over the finned surfaces induced by a fan. Prevailing wind affects both structural design and performance. AOC provide the additional advantages of reducing ground-space requirements and piping and pumping requirements and of providing smoother column operation. The downflow condenser is used mainly for nonisothermal condensation. Vapors enter through a header at the top and flow downward. The reflux condenser is used for isothermal and small-temperaturechange conditions. Vapors enter at the bottom of the tubes. AOC usage first developed in Europe but became more prevalent in the United States during the 1960s. A state-of-the-art article was published by Dehne [Chem. Eng. Prog., 64, 51 (July 1969)]. Air-Cooled Heat-Exchanger Costs The cost data that appear in Table 11-17 are unchanged from those published in the 1963 edition of this Handbook. In 1969 Guthrie [Chem. Eng., 75, 114 (Mar. 24, 1969)] presented cost data for field-erected air-cooled exchangers. These costs are only 25 percent greater than those of Table 11-17 and include the costs of steel stairways, indirect subcontractor charges, and field-erection charges. Since minimal field costs would be this high (i.e., 25 percent of purchase price), the basic costs appear to be unchanged. (Guthrie indicated a cost band of plus or minus 25 percent.) Preliminary design and cost estimating of air-

1. Design dry-bulb temperature. The typically selected value is the temperature which is equaled or exceeded 2a percent of the time during the warmest consecutive 4 months. Since air temperatures at industrial sites are frequently higher than those used for these weather-data reports, it is good practice to add 1 to 3°C (2 to 6°F) to the tabulated value. 2. Air recirculation. Prevailing winds and the locations and elevations of buildings, equipment, fired heaters, etc., require consideration. All air-cooled heat exchangers in a bank are of one type, i.e., all forced-draft or all induced-draft. Banks of air-cooled exchangers must be placed far enough apart to minimize air recirculation. 3. Wintertime operations. In addition to the previously discussed problems of winterization, provision must be made for heavy rain, strong winds, freezing of moisture upon the fins, etc. 4. Noise. Two identical fans have a noise level 3 dBa higher than one fan, while eight identical fans have a noise level 9 dBa higher than a single fan. Noise level at the plant site is affected by the exchanger position, the reflective surfaces near the fan, the hardness of these surfaces, and noise from adjacent equipment. The extensive use of aircooled heat exchangers contributes significantly to plant noise level. 5. Ground area and space requirements. Comparisons of the overall space requirements for plants using air cooling versus water cooling are not consistent. Some air-cooled units are installed above other equipment—pipe racks, shell-and-tube exchangers, etc. Some plants avoid such installations because of safety considerations, as discussed later. 6. Safety. Leaks in air-cooled units are directly to the atmosphere and can cause fire hazards or toxic-fume hazards. However, the large air flow through an air-cooled exchanger greatly reduces any concentration of toxic fluids. Segal [Pet. Refiner, 38, 106 (April 1959)] reports that air-fin coolers “are not located over pumps, compressors, electrical switchgear, control houses and, in general, the amount of equipment such as drums and shell-and-tube exchangers located beneath them are minimized.” Pipe-rack-mounted air-cooled heat exchangers with flammable fluids generally have concrete fire decks which isolate the exchangers from the piping. 7. Atmospheric corrosion. Air-cooled heat exchangers should not be located where corrosive vapors and fumes from vent stacks will pass through them. 8. Air-side fouling. Air-side fouling is generally negligible. 9. Process-side cleaning. Either chemical or mechanical cleaning on the inside of the tubes can readily be accomplished. 10. Process-side design pressure. The high-pressure process fluid is always in the tubes. Tube-side headers are relatively small as

11-52

HEAT-TRANSFER EQUIPMENT

compared with water-cooled units when the high pressure is generally on the shell side. High-pressure design of rectangular headers is complicated. The plug-type header is normally used for design gauge pressures to 13,790 kPa (2000 lbf/in2) and has been used to 62,000 kPa (9000 lbf/in2). The use of threaded plugs at these pressures creates problems. Removable cover plate headers are generally limited to gauge pressures of 2068 kPa (300 lbf/in2). The expensive billet-type header is used for high-pressure service. 11. Bond resistance. Vibration and thermal cycling affect the bond resistance of the various types of tubes in different manners and thus affect the amount of heat transfer through the fin tube. 12. Approach temperature. The approach temperature, which is the difference between the process-fluid outlet temperature and the design dry-bulb air temperature, has a practical minimum of 8 to 14°C (15 to 25°F). When a lower process-fluid outlet temperature is required, an air-humidification chamber can be provided to reduce the inlet air temperature toward the wet-bulb temperature. A 5.6°C (10°F) approach is feasible. Since typical summer wet-bulb design temperatures in the United States are 8.3°C (15°F) lower than drybulb temperatures, the outlet process-fluid temperature can be 3°C (5°F) below the dry-bulb temperature. 13. Mean-temperature-difference (MTD) correction factor. When the outlet temperatures of both fluids are identical, the MTD correction factor for a 1:2 shell-and-tube exchanger (one pass shell

side, two or more passes tube side) is approximately 0.8. For a singlepass air-cooled heat exchanger the factor is 0.91. A two-pass exchanger has a factor of 0.96, while a three-pass exchanger has a factor of 0.99 when passes are arranged for counterflow. 14. Maintenance cost. Maintenance for air-cooled equipment as compared with shell-and-tube coolers (complete with cooling-tower costs) indicates that air-cooling maintenance costs are approximately 0.3 to 0.5 those for water-cooled equipment. 15. Operating costs. Power requirements for air-cooled heat exchangers can be lower than at the summer design condition provided that an adequate means of air-flow control is used. The annual power requirement for an exchanger is a function of the means of airflow control, the exchanger service, the air-temperature rise, and the approach temperature. When the mean annual temperature is 16.7°C (30°F) lower than the design dry-bulb temperature and when both fans in a bay have automatically controllable pitch of fan blades, annual power required has been found to be 22, 36, and 54 percent respectively of that needed at the design condition for three process services [Frank L. Rubin, “Power Requirements Are Lower for Air-Cooled Heat Exchangers with AV Fans,” Oil Gas J., 165–167 (Oct. 11, 1982)]. Alternatively, when fans have two-speed motors, these deliver one-half of the design flow of air at half speed and use only one-eighth of the power of the full-speed condition.

COMPACT AND NONTUBULAR HEAT EXCHANGERS COMPACT HEAT EXCHANGERS With equipment costs rising and limited available plot space, compact heat exchangers are gaining a larger portion of the heat exchange market. Numerous types use special enhancement techniques to achieve the required heat transfer in smaller plot areas and, in many cases, less initial investment. As with all items that afford a benefit there is a series of restrictions that limit the effectiveness or application of these special heat exchanger products. In most products discussed some of these considerations are presented, but a thorough review with reputable suppliers of these products is the only positive way to select a compact heat exchanger. The following guidelines will assist in prequalifying one of these. PLATE-AND-FRAME EXCHANGERS There are two major types gasketed and welded-plate heat exchangers. Each shall be discussed individually. GASKETED-PLATE EXCHANGERS (G. PHE) Description This type is the fastest growing of the compact exchangers and the most recognized (see Fig. 11-49). A series of corrugated alloy material channel plates, bounded by elastomeric gaskets are hung off and guided by longitudinal carrying bars, then compressed by large-diameter tightening bolts between two pressure retaining frame plates (cover plates). The frame and channel plates have portholes which allow the process fluids to enter alternating flow passages (the space between two adjacent-channel plates). Gaskets around the periphery of the channel plate prevent leakage to the atmosphere and also prevent process fluids from coming in contact with the frame plates. No interfluid leakage is possible in the port area due to a dual-gasket seal. The frame plates are typically epoxy-painted carbon-steel material and can be designed per most pressure vessel codes. Design limitations are in the Table 11-18. The channel plates are always an alloy material with 304SS as a minimum (see Table 11-18 for other materials). Channel plates are typically 0.4 to 0.8 mm thick and have corrugation depths of 2 to 10 mm. Special Wide Gap (WG PHE) plates are

available, in limited sizes, for slurry applications with depths of approximately 16 mm. The channel plates are compressed to achieve metal-to-metal contact for pressure-retaining integrity. These narrow gaps and high number of contact points which change fluid flow direction, combine to create a very high turbulence between the plates. This means high individual-heat-transfer coefficients (up to 14200 W/m2 °C), but also very high pressure drops per length as well. To compensate, the channel plate lengths are usually short, most under 2 and few over 3 meters in length. In general, the same pressure drops as conventional exchangers are used without loss of the enhanced heat transfer. Expansion of the initial unit is easily performed in the field without special considerations. The original frame length typically has an additional capacity of 15–20 percent more channel plates (i.e., surface area). In fact, if a known future capacity is available during fabrication stages, a longer carrying bar could be installed, and later, increasing the surface area would be easily handled. When the expansion is needed, simply untighten the carrying bolts, pull back the frame plate, add the additional channel plates, and tighten the frame plate. Applications Most PHE applications are liquid-liquid services but there are numerous steam heater and evaporator uses from their heritage in the food industry. Industrial users typically have chevron style channel plates while some food applications are washboard style. Fine particulate slurries in concentrations up to 70 percent by weight are possible with standard channel spacings. Wide-gap units are used with larger particle sizes. Typical particle size should not exceed 75 percent of the single plate (not total channel) gap. Close temperature approaches and tight temperature control possible with PHE’s and the ability to sanitize the entire heat transfer surface easily were a major benefit in the food industry. Multiple services in a single frame are possible. Gasket selection is one of the most critical and limiting factors in PHE usage. Table 11-19 gives some guidelines for fluid compatibility. Even trace fluid components need to be considered. The higher the operating temperature and pressure, the shorter the anticipated gasket life. Always consult the supplier on gasket selection and obtain an estimated or guaranteed lifetime. The major applications are, but not limited to, as follows: Temperature cross applications Close approaches

(lean/rich solvent) (fresh water/seawater)

COMPACT AND NONTUBULAR HEAT EXCHANGERS

11-53

Plate-and-frame heat exchanger. Hot fluid flows down between alternate plates, and cold fluid flows up between alternate plates. (Thermal Division, Alfa-Laval, Inc.)

FIG. 11-49

Viscous fluids Sterilized surface required Polished surface required Future expansion required Space restrictions Barrier coolant services Slurry applications

(emulsions) (food, pharmaceutical) (latex, pharmaceutical) (closed-loop coolers) (TiO2, Kaolin, precipitated calcium carbonate, and beet sugar raw juice)

Design Standard channel-plate designs, unique to each manufacturer, are developed with limited modifications of each plates’ corrugation depths and included angles. Manufacturers combine their different style plates to custom-fit each service. Due to the possible combinations, it is impossible to present a way to exactly size PHEs. However, it is possible to estimate areas for new units and to predict performance of existing units with different conditions (chevron-type channel plates are presented). The fixed length and limited corrugation included angles on channel plates makes the NTU method of sizing practical. (Waterlike fluids are assumed for the following examples). ∆ t of either side NTU =

(11-76) LMTD Most plates have NTU values of 0.5 to 4.0, with 2.0 to 3.0 as the most

common, (multipass shell and tube exchangers are typically less than 0.75). The more closely the fluid profile matches that of the channel plate, the smaller the required surface area. Attempting to increase the service NTU beyond the plate’s NTU capability causes oversurfacing (inefficiency). True sizing from scratch is impractical since a pressure balance on a channel-to-channel basis, from channel closest to inlet to furthest, must be achieved and when mixed plate angles are used; this is quite a challenge. Computer sizing is not just a benefit, it is a necessity for supplier’s selection. Averaging methods are recommended to perform any sizing calculations. From the APV heat-transfer handbook—Design & Application of Paraflow-Plate Heat Exchangers and J. Marriott’s article, “Where and How To Use Plate Heat Exchangers,” Chemical Engineering, April 5, 1971, there are the following equations for plate heat transfer. hDe Nu =
= 0.28 ∗ (Re)0.65 ∗ (Pr)0.4 k where De = 2 × depth of single-plate corrugation

(11-77)

W G =

(11-78) Np ∗ w ∗ De Width of the plate (w) is measured from inside to inside of the channel gasket. If not available, use the tear-sheet drawing width and sub-

11-54

HEAT-TRANSFER EQUIPMENT

TABLE 11-18

Compact Exchanger Applications Guide

Design conditions

G. PHE

W. PHE

WG. PHE

BHE

DBL

MLT

STE

CP

SHE

THE

Design temperature °C Minimum metal temp °C Design pressure MPa Inspect for leakage Mechanical cleaning Chemical cleaning Expansion capability Repair Temperature cross Surface area/unit m2 Holdup volume

165 −30 2.5 Yes Yes Yes Yes Yes Yes 1850 Low

150 −30 2.5 Partial Yes/no Yes Yes Yes/no Yes 900 Low

150 −30 0.7 Yes Yes Yes Yes Yes Yes 250 Low

185 −160 3.1 No No Yes No No Yes 50 Low

+500 −160 +20 Yes Yes Yes No Yes Yes 10 Med

+500 −160 +20 Yes Yes Yes No Yes Yes 150 Med

+500 −160 +20 Yes Yes/no Yes No Partial Yes 60 Low

450 −160 3.1 Partial Yes Yes No Partial Yes 275 Low

+400 −160 2.0 Yes Yes Yes No Partial Yes 450 Med

+500 −160 +20 Yes Yes Yes No Yes No* High High

Materials

G. PHE

W. PHE

WG. PHE

BHE

DBL

MLT

STE

CP

SHE

THE

No Yes Yes Yes Yes Yes Yes Yes Yes

No Yes Yes Yes Yes Yes Yes Yes No

No Yes Yes No No No No No No

No Yes No No No No No No No

Yes Yes Yes Yes Yes Yes Yes Yes No

Yes Yes Yes Yes Yes Yes Yes Yes No

Yes Yes Yes Yes Yes Yes Yes Yes No

Yes Yes Yes Yes Yes Yes Yes Yes No

Yes Yes Yes Yes Yes Yes Yes Yes No

Yes Yes Yes Yes Yes Yes Yes Yes Yes

G. PHE

W. PHE

WG. PHE

BHE

DBL

MLT

STE

CP

SHE

THE

A D A/B A/B B/D D D D D D D D A B

A A/D A/B A/B D D A A/B D C D C A B

A D A/B A/B A/B B D D D D B C A A

A A B D C D A A/B D D D A A D

A A A A A A A A A/D A/D A/D B A A

A A A A A/B B/C A A A A B B A A/B

A A A/B A/B C D B/C C A A A A A B/C

A A A/B A/B B B A A B/C B/C B/C B/C B B

A A A A A A A A A B B C A B

A A A A A/D A/D A A A A A/C A C A/D

Mild steel Stainless Titanium Hastalloy Nickel Alloy 20 Incoloy 825 Monel Impervious graphite Service Clean fluids Gasket incompatibility Medium viscosity High viscosity Slurries & pulp (fine) Slurries & pulp (coarse) Refrigerants Thermal fluids Vent condensers Process condenser Vacuum reboil/cond Evaporator Tight temp control High scaling

*Multipass Adapted from Alfa-Laval and Vicarb literature A—Very good C—Fair B—Good D—Poor

TABLE 11-19

Elastomer Selection Guide Uses

Avoid

Nitrile (NBR)

Oil resistant Fat resistant Food stuffs Mineral oil Water

Oxidants Acids Aromatics Alkalies Alcohols

Resin cured butyl (IIR)

Acids Lyes Strong alkalies Strong phosphoric acid Dilute mineral acids Ketones Amines Water

Fats and fatty acids Petroleum oils Chlorinated hydrocarbons Liquids with dissolved chlorine Mineral oil Oxygen rich demin. water Strong oxidants

Ethylene-propylene (EPDM)

Oxidizing agents Dilute acids Amines Water (Mostly any IIR fluid)

Oils Hot & conc. acids Very strong oxidants Fats & fatty acids Chlorinated hydrocarbons

Viton (FKM, FPM)

Water Petroleum oils Many inorganic acids (Most all NBR fluids)

Amines Ketones Esters Organic acids Liquid ammonia

tract two times the bolt diameter and subtract another 50 mm. For depth of corrugation ask supplier, or take the compressed plate pack dimension, divide by the number of plates and subtract the plate thickness from the result. The number of passages (Np) is the number of plates minus 1 then divided by 2. Typical overall coefficients to start a rough sizing are as below. Use these in conjunction with the NTU calculated for the process. The closer the NTU matches the plate (say between 2.0 and 3.0), the higher the range of listed coefficients can be used. The narrower (smaller) the depth of corrugation, the higher the coefficient (and pressure drop), but also the lower the ability to carry through any particulate. Water-water 5700–7400 W/(m2 °C) Steam-water 5700–7400 W/(m2 °C) Glycol/Glycol 2300–4000 W/(m2 °C) Amine/Amine 3400–5000 W/(m2 °C) Crude/Emulsion 400–1700 W/(m2 °C) Pressure drops typically can match conventional tubular exchangers. Again from the APV handbook an average correlation is as follows: 2fG2L ∆P =
(11-79) gρDe where f = 2.5 (GDe/µ)−0.3 g = gravitational constant Fouling factors are typically 1⁄10 of TEMA values or a percent oversurfacing of 10–20 percent is used. (“Sizing Plate Exchangers” Jeff Kerner, Chemical Engineering, November 1993).

COMPACT AND NONTUBULAR HEAT EXCHANGERSS LMTD is calculated like a 1 pass-1 pass shell and tube with no F correction factor required in most cases. Overall coefficients are determined like shell and tube exchangers; that is, sum all the resistances, then invert. The resistances include the hot-side coefficient, the cold-side coefficient, the fouling factor (usually only a total value not individual values per fluid side) and the wall resistance. WELDED- AND BRAZED-PLATE EXCHANGERS (W. PHE & BHE) The title of this group of plate exchangers has been used for a great variety of designs for various applications from normal gasketed-plate exchanger services to air-preheater services on fired heaters or boilers. The intent here is to discuss more traditional heat-exchanger designs, not the heat-recovery designs on fired equipment flue-gas streams. Many similarities exist between these products but the manufacturing techniques are quite different due to the normal operating conditions these units experience. To overcome the gasket limitations, PHE manufacturers have developed welded-plate exchangers. There are numerous approaches to this solution: weld plate pairs together with the other fluid-side conventionally gasketed, weld up both sides but use a horizonal stacking of plates method of assembly, entirely braze the plates together with copper or nickel brazing, diffusion bond then pressure form plates and bond etched, passage plates. Most methods of welded-plate manufacturing do not allow for inspection of the heat-transfer surface, mechanical cleaning of that surface, and have limited ability to repair or plug off damage channels. Consider these limitations when the fluid is heavily fouling, has solids, or in general the repair or plugging ability for severe services. One of the previous types has an additional consideration of the brazing material to consider for fluid compatibility. The brazing compound entirely coats both fluid’s heat-transfer surfaces. The second type, a Compabloc (CP) from Vicarb, has the advantage of removable cover plates, similar to air-cooled exchanger headers, to observe both fluids surface area. The fluids flow at 90° angles to each other on a horizonal plane. LMTD correction factors approach 1.0 for Compabloc just like the other welded and gasketed PHEs. Hydroblasting of Compabloc surfaces is also possible. The Compabloc has higher operating conditions than PHE’s or W-PHE. The performances and estimating methods of welded PHEs match those of gasketed PHEs in most cases, but normally the Compabloc, with larger depth of corrugations, can be lower in overall coefficient. Some extensions of the design operating conditions are possible with welded PHEs, most notably is that cryogenic applications are possible. Pressure vessel code acceptance is available on most units. SPIRAL-PLATE EXCHANGERS (SHE) Description The spiral-plate heat exchanger (SHE) may be one exchanger selected primarily on its virtues and not on its initial cost. SHEs offer high reliability and on-line performance in many severely fouling services such as slurries. The SHE is formed by rolling two strips of plate, with welded-on spacer studs, upon each other into clock-spring shape. This forms two passages. Passages are sealed off on one end of the SHE by welding a bar to the plates; hot and cold fluid passages are sealed off on opposite ends of the SHE. A single rectangular flow passage is now formed for each fluid, producing very high shear rates compared to tubular designs. Removable covers are provided on each end to access and clean the entire heat transfer surface. Pure countercurrent flow is achieved and LMTD correction factor is essentially = 1.0. Since there are no dead spaces in a SHE, the helical flow pattern combines to entrain any solids and create high turbulence creating a self-cleaning flow passage. There are no thermal-expansion problems in spirals. Since the center of the unit is not fixed, it can torque to relieve stress. The SHE can be expensive when only one fluid requires a highalloy material. Since the heat-transfer plate contacts both fluids, it is required to be fabricated out of the higher alloy. SHEs can be fabricated out of any material that can be cold-worked and welded.

11-55

The channel spacings can be different on each side to match the flow rates and pressure drops of the process design. The spacer studs are also adjusted in their pitch to match the fluid characteristics. As the coiled plate spirals outward, the plate thickness increases from a minimum of 2 mm to a maximum (as required by pressure) up to 10 mm. This means relatively thick material separates the two fluids compared to tubing of conventional exchangers. Pressure vessel code conformance is a common request. Applications The most common applications that fit SHE are slurries. The rectangular channel provides high shear and turbulence to sweep the surface clear of blockage and causes no distribution problems associated with other exchanger types. A localized restriction causes an increase in local velocity which aids in keeping the unit free flowing. Only fibers that are long and stringy cause SHE to have a blockage it cannot clear itself. As an additional antifoulant measure, SHEs have been coated with a phenolic lining. This provides some degree of corrosion protection as well, but this is not guaranteed due to pinholes in the lining process. There are three types of SHE to fit different applications: Type I is the spiral-spiral flow pattern. It is used for all heating and cooling services and can accommodate temperature crosses such as lean/rich services in one unit. The removable covers on each end allow access to one side at a time to perform maintenance on that fluid side. Never remove a cover with one side under pressure as the unit will telescope out like a collapsible cup. Type II units are the condenser and reboiler designs. One side is spiral flow and the other side is in cross flow. These SHEs provide very stable designs for vacuum condensing and reboiling services. A SHE can be fitted with special mounting connections for reflux-type ventcondenser applications. The vertically mounted SHE directly attaches on the column or tank. Type III units are a combination of the Type I and Type II where part is in spiral flow and part is in cross flow. This SHE can condense and subcool in a single unit. The unique channel arrangement has been used to provide on-line cleaning, by switching fluid sides to clean the fouling (caused by the fluid that previously flowed there) off the surface. Phosphoric acid coolers use pond water for cooling and both sides foul; water, as you expect, and phosphoric acid deposit crystals. By reversing the flow sides, the water dissolves the acid crystals and the acid clears up the organic fouling. SHEs are also used as oleum coolers, sludge coolers/heaters, slop oil heaters, and in other services where multipleflow-passage designs have not performed well. Design A thorough article by P.E. Minton of Union Carbide called “Designing Spiral-Plate Heat Exchangers,” appeared in Chemical Engineering, May 4, 1970. It covers the design in detail. Also an article in Chemical Engineering Progress titled “Applications of Spiral Plate Heat Exchangers” by A. Hargis, A. Beckman, and J. Loicano appeared in July 1967, provides formulae for heat-transfer and pressure-drop calculations. Spacings are from 6.35 to 31.75 mm (in 6.35 mm increments) with 9.5 mm the most common. Stud densities are 60 × 60 to 110 × 110 mm, the former the most common. The width (measured to the spiral flow passage), is from 150 to 2500 mm (in 150 mm increments). By varying the spacing and the width, separately for each fluid, velocities can be maintained at optimum rates to reduce fouling tendencies or utilize the allowable pressure drop most effectively. Diameters can reach 1500 mm. The total surface areas exceed 465 sqm. Materials that work harder are not suitable for spirals since hot-forming is not possible and heat treatment after forming is impractical. HDe Nu =
= 0.0315 (Re)0.8 (Pr)0.25 (µ/µw)0.17 k where

(11-80)

De = 2 × spacing Flow area = width × spacing

LV 2ρ ∆ P =

∗ 1.45 (1.45 for 60 × 60 mm studs) (11-81) 1.705E−03 LMTD and overall coefficient are calculated like in PHE section above.

11-56

HEAT-TRANSFER EQUIPMENT

BRAZED-PLATE-FIN HEAT EXCHANGER Brazed-aluminum-plate-fin heat exchangers (or core exchangers or cold boxes) as they are sometimes called, were first manufactured for the aircraft industry during World War II. In 1950, the first tonnage air-separation plant with these compact, lightweight, reversing heat exchangers began producing oxygen for a steel mill. Aluminum-platefin exchangers are used in the process and gas-separation industries, particularly for services below −45°C. Core exchangers are made up of a stack of rectangular sheets of aluminum separated by a wavy, usually perforated, aluminum fin. Two ends are sealed off to form a passage (see Fig. 11-50). The layers have the wavy fins and sealed ends alternating at 90° to each. Aluminum half-pipe-type headers are attached to the open ends to route the fluids into the alternating passages. Fluids usually flow at this same 90° angle to each other. Variations in the fin height, number of passages, and the length and width of the prime sheet allow for the core exchanger to match the needs of the intended service. Design conditions range in pressures from full vacuum to 96.5 bar g and in temperatures from −269°C to 200°C. This is accomplished meeting the quality standards of most pressure vessel codes. Applications are varied for this highly efficient, compact exchanger. Mainly it is seen in the cryogenic fluid services of air-separation plants, refrigeration trains like in ethylene plants, and in natural-gas processing plants. Fluids can be all vapor, liquid, condensing, or vaporizing. Multifluid exchangers and multiservice cores, that is one exchanger with up to 10 different fluids, are common for this type of product. Cold boxes are a group of cores assembled into a single structure or module, prepiped for minimum field connections. (Data obtained from ALTEC INTERNATIONAL. For detailed information refer to GPSA Engineering Handbook Section 9.) PLATE-FIN TUBULAR EXCHANGERS (PFE) Description These shell and tube exchangers are designed to use a group of tightly spaced plate fins to increase the shellside heat transfer performance as fins do on double-pipe exchangers. In this design, a series of very thin plates (fins), usually of copper or aluminum material, are punched to the same pattern as the tube layout, spaced very close together, and mechanically bonded to the tube. Fin spacing is 315–785 FPM (Fins Per Meter) with 550 FPM most common. The fin thicknesses are 0.24 mm for aluminum and 0.19 mm for copper. Surface-area ratios over bare prime-tube units can be 20:1 to 30:1. The cost of the additional plate-fin material, without a reduction in shell diameter in many cases, and increased fabrication has to be offset by the total reduction of plot space and prime tube-surface area. The more costly the prime tube or plot space cost, the better the payout for this design. A rectangular tube layout is normally used, no tubes in the window (NTIW). The window area (where no tubes are) of the plate-fins are cut out. This causes a larger shell diameter for a given tube count compared to conventional tubular units. A dome area on top and bottom of the inside of the shell has been created for the fluid to flow along the tube length. In order to exit the unit the fluid must flow across the plate-finned tube bundle with extremely low pressure loss. The units from the outside and from the tubeside appear like any conventional shell and tube exchanger. Applications Two principal applications are rotating equipment oil coolers and compressor inter- and after-coolers. Although seemingly different applications, both rely on the shellside finning to enhance the heat transfer of low heat-transfer characteristic fluids,

FIG. 11-50

Exploded view of a typical plate-fin arrangement. (Trane Co.)

viscous oils, and gases. By nature of the fluids and their applications, both are clean servicing. The tightly spaced fins would be a maintenance problem otherwise. Design The economics usually work out in the favor of gas coolers when the centrifugal machine’s flow rate reaches about 5000 scfm. The pressure loss can be kept to 7.0 kPa in most cases. When the ratio of Atht to Ashs is 20:1, is another point to consider these plate-fin designs. Vibration is practically impossible with this design, and uses in reciprocating compressors are possible due to this. Marine and hydraulic-oil coolers use these characteristics to enhance the coefficient of otherwise poorly performing fluids. The higher metallurgies in marine applications like 90/10 Cu-Ni afford the higher cost of plate-fin design to be offset by the less amount of alloy material being used. On small hydraulic coolers, these fins usually allow one to two size smaller coolers for the package and save skid space and initial cost. Always check on metallurgy compatibility and cleanliness of the shellside fluid! (Data provided by Bos-Hatten and ITT-Standard.) SPIRAL-TUBE EXCHANGERS (STE) Description These exchangers are typically a series of stacked helical-coiled tubes connected to manifolds, then inserted into a casing or shell. They have many advantages like spiral-plate designs, such as avoiding differential expansion problems, acceleration effects of the helical flow increasing the heat transfer coefficient, and compactness of plot area. They are typically selected because of their economical design. The most common form has both sides in helical flow patterns, pure countercurrent flow is followed and the LMTD correction factor approaches 1.0. Temperature crosses are possible in single units. Like the spiral-plate unit, different configurations are possible for special applications. Tube material includes any that can be formed into a coil, but usually copper, copper alloys, and stainless steel are most common. The casing or shell material can be cast iron, cast steel, cast bronze, fabricated steel, stainless, and other high-alloy materials. Units are available with pressure vessel code conformance. The data provided herein has been supplied by Graham Mfg. for their units called Heliflow. Applications The common Heliflow applications are tank-vent condensers, sample coolers, pump-seal coolers, and steam-jet vacuum condensers. Instant water heaters, glycol/water services, and cryogenic vaporizers use the spiral tube’s ability to reduce thermally induced stresses caused in these applications. Many other applications are well suited for spiral tube units but many believe only small surface areas are possible with these units. Graham Mfg. states units are available to 60 m2. Their ability to polish the surfaces, double-wall the coil, use finned coil, and insert static mixers, among others configurations in design, make them quite flexible. Tubeside design pressures can be up to 69000 kPa. A cross-flow design on the external surface of the coil is particularly useful in steam-jet ejector condensing service. These Heliflow units, can be made very cost-effective, especially in small units. The main differences, compared to spiral plate, is that the tubeside cannot be cleaned except chemically and that multiple flow passages make tubeside slurry applications (or fouling) impractical. Design The fluid flow is similar to the spiral-plate exchangers, but through parallel tube passages. Graham Mfg. has a liquid-liquid sizing pamphlet available from their local distributor. An article by M.A. Noble, J.S. Kamlani, and J.J. McKetta “Heat Transfer in Spiral Coils”, was published in Petroleum Engineer, April 1952 p. 723, discussing sizing techniques. The tubeside fluid must be clean or at least chemically cleanable. With a large number of tubes in the coil, cleaning of inside surfaces is not totally reliable. Fluids that attack stressed materials such as chlorides should be reviewed as to proper coil-material selection. Fluids that contain solids can be a problem due to erosion of relatively thin coil materials unlike the thick plates in spiral-plate units and multiple, parallel, fluid passages compared to a single passage in spiral-plate units.

COMPACT AND NONTUBULAR HEAT EXCHANGERS GRAPHITE HEAT EXCHANGERS Impervious graphite exchangers now come in a variety of geometries to suit the particular requirements of the service. They include cubic block form, drilled cylinder block, shell and tube, and plate and frame. Description Graphite is one of three crystalline forms of carbon. The other two are diamond and charcoal. Graphite has a hexagonal crystal structure, diamond is cubic, and charcoal is amorphous. Graphite is inert to most chemicals and resists corrosion attack. It is however porous and to be used, it must be impregnated with a resin sealer. Two main resins used are phenolic and PTFE with furan (one currently being phased out of production). Selection of resins include chemical compatibility, operating temperatures, and type of unit to be used. For proper selection, consult with a graphite supplier. Shell-and-tube units in graphite were started by Karbate in 1939. The European market started using block design in the 1940s. Both technologies utilize the high thermal conductivity of the graphite material to compensate for the poor mechanical strength. The thicker materials needed to sustain pressure do not adversely impede the heat transfer. Maximum design pressures range from 0.35 to 1.0 kPa depending on type and size of exchanger. Design temperature is dependent on the fluids and resin selection, the maximum is 230 °C. In all situations, the graphite heat transfer surface is contained within a metal structure or a shell (graphite lined on process side) to maintain the design pressure. For shell and tube units, the design is a packed floating tubesheet at both ends within a shell and channel. For stacked block design, the standardize blocks are glued together with special adhesives and compressed within a framework that includes manifold connections for each fluid. The cylindrical block unit is a combination of the above two with blocks glued together and surrounded by a pressure retaining shell. Pressure vessel code conformance of the units is possible due to the metallic components of these designs. Since welding of graphite is not possible, the selection and application of the adhesives used are critical to the proper operating of these units. Tube to tubesheet joints are glued since rolling of tubes into tubesheet is not possible. The packed channels and gasketed manifold connections are two areas of additional concern when selecting sealants for these units. Applications and Design The major applications for these units are in the acid-related industries. Sulfuric, phosphoric, and hydrochloric acids require either very costly metals or impervious graphite. Usually graphite is the more cost-effective material to be used. Applications are increasing in the herbicide and pharmaceutical industries as new products with chlorine and fluorine compounds expand. Services are coolers, condensers, and evaporators, basically all services requiring this material. Types of units are shell-and-tube, block-type (circular and rectangular), and plate-and-frame-type exchangers. The design of the shell-and-tube units are the same as any but the design characteristics of tubes, spacing, and thickness are unique to the graphite design. The block and plate and frame also can be evaluated using techniques previously addressed but again, the unique characteristics of the graphite materials require input from a reputable supplier. Most designs will need the supplier to provide the most cost-effective design for the immediate and future operation of the exchangers. Also, consider the entire system design as some condensers and/or evaporators can be integral with their associated column. CASCADE COOLERS Cascade coolers are a series of standard pipes, usually manifolded in parallel, and connected in series by vertically or horizontally oriented U-bends. Process fluid flows inside the pipe entering at the bottom and water trickles from the top downward over the external pipe surface. The water is collected from a trough under the pipe sections, cooled, and recirculated over the pipe sections. The pipe material can be any of the metallic and also glass, impervious graphite, and ceramics. The tubeside coefficient and pressure drop is as in any circular duct. The water coefficient (with Re number less than 2100) is calculated from the following equation by W.H. McAdams, T.B. Drew, and G.S. Bays Jr., from the ASME trans. 62, 627–631 (1940).

h= G′ = m= L= Do =

11-57

218 ∗ (G′/Do)1/3 (W/m2 °C) (11-82) m/(2L) water rate (kg/hr) length of each pipe section (meter) outside diameter of pipe (meter)

LMTD corrections are per Fig. 11-4 i or j depending on U-bend orientation. BAYONET-TUBE EXCHANGERS This type of exchanger gets its name from its design which is similar to a bayonet sword and its associated scabbard or sheath. The bayonet tube is a smaller-diameter tube inserted into a larger-diameter tube that has been capped at one end. The fluid flow is typically entering the inner tube, exiting, hitting the cap of the larger tube, and returning the opposite direction in the annular area. The design eliminates any thermal expansion problems. It also creates a unique nonfreezetype tube-side for steam heating of cryogenic fluids, the inner tube steam keeps the annulus condensate from freezing against the cold shellside fluid. This design can be expensive on a surface-area basis due to the need of a double channel design and only the outer tube surface is used to transfer heat. LMTD calculations for nonisothermal fluid are quite extensive and those applications are far too few to attempt to define it. The heat transfer is like the annular calculation of a double-pipe unit. The shellside is a conventional-baffled shell-andtube design. ATMOSPHERIC SECTIONS These consist of a rectangular bundle of tubes in similar fashion to air cooler bundles, placed just under the cooled water distribution section of a cooling tower. It, in essence, combines the exchanger and cooling tower into a single piece of equipment. This design is only practical for single-service cooler/condenser applications, and expansion capabilities are not provided. The process fluid flows inside the tubes and the cooling tower provides cool water that flows over the outside of the tube bundle. Water quality is critical for these applications to prevent fouling or corrosive attack on the outside of the tube surfaces and to prevent blockage of the spray nozzles. The initial and operating costs are lower than separate cooling tower and exchanger. Principal applications now are in the HVAC, Refrigeration and Industrial systems. Sometimes these are called “Wet Surface Air Coolers”. h = 1729 [(m2/hr)/face area m2]1/3

(11-83)

NONMETALLIC HEAT EXCHANGERS Another growing field is that of nonmetallic heat exchanger designs which typically are of the shell and tube or coiled-tubing type. The graphite units were previously discussed but numerous other materials are available. The materials include Teflon, PVDF, glass, ceramic, and others as the need arises. When using these types of products, consider the following topics and discuss the application openly with experienced suppliers. 1. The tube-to-tubesheet joint, how is it made? Many use “O” rings to add another material to the selection process. Preference should be given to a fusing technique of similar material. 2. What size tube or flow passage is available? Small tubes plug unless filtration is installed. Size of filtering is needed from the supplier. 3. These materials are very sensitive to temperature and pressure. Thermal or pressure shocks must be avoided. 4. Thermal conductivity of these materials is very low and affects the overall coefficient. When several materials are compatible, explore all of them, as final cost is not always the same as raw material costs. PVDF HEAT EXCHANGERS These shell-and-tube-type exchangers are similar to the Teflon designs but have some mechanical advantages over Teflon units. First

11-58

HEAT-TRANSFER EQUIPMENT

the tubes are available in 9.5 mm sizes which reduces the chances of plugging that are found in Teflon units with unfiltered fluids. Second, the material has higher strength even at lower temperatures almost double. Larger units are possible with PVDF materials. Tube to tubesheet joints, a weakness of most nonmetallic units, are fused by special techniques that do not severely affect the chemical suitability of the unit. Some nonmetallics use Teflon or “O” rings that add an extra consideration to material selection. The shell is usually a steel design and, like the graphite units before, can obtain pressure-vessel certification. CERAMIC HEAT EXCHANGERS These include glass, silicon carbide, and similar variations. Even larger tubes are available in these materials, up to 19-mm diameter. They have high thermal conductivities and are usually very smooth surfaces to resist fouling. Very high material/fluid compatibility is seen for these products, not many fluids are excluded. Brittleness is a consideration of these materials and a complete discussion of the service with an experienced supplier is warranted. The major selection criteria to explore is the use of “O” rings and other associated joints at tubesheet. The shell is steel in most cases.

TEFLON HEAT EXCHANGERS Teflon tube shell-and-tube heat exchangers (Ametek) made with tubes of chemically inert Teflon fluorocarbon resin are available. The tubes are 0.25-in OD by 0.20-in ID, 0.175-in OD by 0.160-in ID, or 0.125-in OD by 0.100-in ID (in × 25.4 equal mm). The larger tubes are primarily used when pressure-drop limitations or particles reduce the effectiveness of smaller tubes. These heat exchangers generally operate at higher pressure drops than conventional units and are best suited for relatively clean fluids. Being chemically inert, the tubing has many applications in which other materials corrode. Fouling is negligible because of the antistick properties of Teflon. The heat exchangers are of single-pass, countercurrent-flow design with removable tube bundles. Tube bundles are made of straight flexible tubes of Teflon joined together in integral honeycomb tube sheets. Baffles and O-ring gaskets are made of Teflon. Standard shell diameters are 102, 204, and 254 mm (4, 8, and 10 in). Tube counts range from 105 to 2000. Surface varies from 1.9 to 87 m2 (20 to 940 ft2). Tube lengths vary from 0.9 to 4.9 m (3 to 16 ft). At 37.8°C (100°F) maximum operating gauge pressures are 690 kPa (100 lbf/in2) internal and 379 kPa (55 lbf/in2) external. At 149°C (300°F) the maximum pressures are 207 kPa (30 lbf/in2) internal and 124 kPa (18 lbf/in2) external.

HEAT EXCHANGERS FOR SOLIDS This section describes equipment for heat transfer to or from solids by the indirect mode. Such equipment is so constructed that the solids load (burden) is separated from the heat-carrier medium by a wall; the two phases are never in direct contact. Heat transfer is by conduction based on diffusion laws. Equipment in which the phases are in direct contact is covered in other sections of this Handbook, principally in Sec. 20. Some of the devices covered here handle the solids burden in a static or laminar-flowing bed. Other devices can be considered as continuously agitated kettles in their heat-transfer aspect. For the latter, unit-area performance rates are higher. Computational and graphical methods for predicting performance are given for both major heat-transfer aspects in Sec. 10. In solids heat processing with indirect equipment, the engineer should remember that the heat-transfer capability of the wall is many times that of the solids burden. Hence the solids properties and bed geometry govern the rate of heat transfer. This is more fully explained earlier in this section. Only limited resultant (not predictive) and “experience” data are given here.

Agitated-Pan Type A natural evolution from the table type is a circular flat surface with jacketing on the underside for coolant flow and the added feature of a stirring means to sweep over the heattransfer surface. This device is the agitated-pan type (Fig. 11-51). It is a batch-operation device. Because of its age and versatility it still serves a variety of heat-transfer operations for the chemical-process industries. While the most prevalent designation is agitated-pan dryer (in this mode, the burden is heated rather than cooled), considerable use is made of it for solidification applications. In this field, it is particularly suitable for processing burdens that change phase (1) slowly, by “thickening,” (2) over a wide temperature range, (3) to an amorphous solid form, or (4) to a soft semigummy form (versus the usual hard crystalline structure). The stirring produces the end product in the desired divided-solids form. Hence, it is frequently termed a “granulator” or a “crystallizer.” A variety of factory-made sizes in various materials of construction are available. Initial cost is modest, while operating cost is rather high (as is true of all batch devices), but the ability to process “gummy” burdens and/or simultaneously effect two unit operations often yields an economical application.

EQUIPMENT FOR SOLIDIFICATION A frequent operation in the chemical field is the removal of heat from a material in a molten state to effect its conversion to the solid state. When the operation is carried on batchwise, it is termed casting, but when done continuously, it is termed flaking. Because of rapid heat transfer and temperature variations, jacketed types are limited to an initial melt temperature of 232°C (450°F). Higher temperatures [to 316°C (600°F)] require extreme care in jacket design and coolingliquid flow pattern. Best performance and greatest capacity are obtained by (1) holding precooling to the minimum and (2) optimizing the cake thickness. The latter cannot always be done from the heat-transfer standpoint, as size specifications for the end product may dictate thickness. Table Type This is a simple flat metal sheet with slightly upturned edges and jacketed on the underside for coolant flow. For many years this was the mainstay of food processors. Table types are still widely used when production is in small batches, when considerable batch-to-batch variation occurs, for pilot investigation, and when the cost of continuous devices is unjustifiable. Slab thicknesses are usually in the range of 13 to 25 mm (a to 1 in). These units are homemade, with no standards available. Initial cost is low, but operating labor is high.

FIG. 11-51 Heat-transfer equipment for solidification (with agitation); agitated-pan type.

HEAT EXCHANGERS FOR SOLIDS

FIG. 11-52

Heat-transfer equipment for batch solidification; vibrating-conveyor type. (Courtesy of Jeffrey Mfg. Co.)

Vibratory Type This construction (Fig. 11-52) takes advantage of the burden’s special needs and the characteristic of vibratory actuation. A flammable burden requires the use of an inert atmosphere over it and a suitable nonhazardous fluid in the jacket. The vibratory action permits construction of rigid self-cleaning chambers with simple flexible connections. When solidification has been completed and vibrators started, the intense vibratory motion of the whole deck structure (as a rigid unit) breaks free the friable cake [up to 76 mm (3 in) thick], shatters it into lumps, and conveys it up over the dam to discharge. Heat-transfer performance is good, with overall coefficient U of about 68 W/(m2 ⋅°C) [12 Btu/(h⋅ft2 ⋅°F)] and values of heat flux q in the order of 11,670 W/m2 [3700 Btu/(h⋅ft2)]. Application of timingcycle controls and a surge hopper for the discharge solids facilitates automatic operation of the caster and continuous operation of subsequent equipment. Belt Types The patented metal-belt type (Fig. 11-53a), termed the “water-bed” conveyor, features a thin wall, a well-agitated fluid side for a thin water film (there are no rigid welded jackets to fail), a stainless-steel or Swedish-iron conveyor belt “floated” on the water

with the aid of guides, no removal knife, and cleanability. It is mostly used for cake thicknesses of 3.2 to 15.9 mm (f to v in) at speeds up to 15 m/min (50 ft/min), with 45.7-m (150-ft) pulley centers common. For 25- to 32-mm (1- to 1d-in) cake, another belt on top to give twosided cooling is frequently used. Applications are in food operations for cooling to harden candies, cheeses, gelatins, margarines, gums, etc.; and in chemical operations for solidification of sulfur, greases, resins, soaps, waxes, chloride salts, and some insecticides. Heat transfer is good, with sulfur solidification showing values of q = 5800 W/m2 [1850 Btu/(h⋅ft2)] and U = 96 W/(m2 ⋅°C) [17 Btu/(h⋅ft2 ⋅°F)] for a 7.9-mm (b-in) cake. The submerged metal belt (Fig. 11-53b) is a special version of the metal belt to meet the peculiar handling properties of pitch in its solidification process. Although adhesive to a dry metal wall, pitch will not stick to the submerged wetted belt or rubber edge strips. Submergence helps to offset the very poor thermal conductivity through two-sided heat transfer. A fairly recent application of the water-cooled metal belt to solidification duty is shown in Fig. 11-54. The operation is termed pastilliz-

(a)

(b) Heat-transfer equipment for continuous solidification. (a) Cooled metal belt. (Courtesy of Sandvik, Inc.) (b) Submerged metal belt. (Courtesy of Sandvik, Inc.) FIG. 11-53

11-59

11-60

HEAT-TRANSFER EQUIPMENT

Heat-transfer equipment for solidification; belt type for the operation of pastillization. (Courtesy of Sandvik, Inc.)

FIG. 11-54

ing from the form of the solidified end product, termed “pastilles.” The novel feature is a one-step operation from the molten liquid to a fairly uniformly sized and shaped product without intermediate operations on the solid phase. Another development features a nonmetallic belt [Plast. Des. Process., 13 (July 1968)]. When rapid heat transfer is the objective, a glass-fiber, Teflon-coated construction in a thickness as little as 0.08 mm (0.003 in) is selected for use. No performance data are available, but presumably the thin belt permits rapid heat transfer while taking advantage of the nonsticking property of Teflon. Another development [Food Process. Mark., 69 (March 1969)] is extending the capability of belt solidification by providing use of subzero temperatures. Rotating-Drum Type This type (Fig. 11-55 a and b) is not an adaptation of a material-handling device (though volumetric material throughput is a first consideration) but is designed specifically for

heat-transfer service. It is well engineered, established, and widely used. The twin-drum type (Fig. 11-55b) is best suited to thin [0.4- to 6-mm (1⁄64 to d-in)] cake production. For temperatures to 149°C (300°F) the coolant water is piped in and siphoned out. Spray application of coolant water to the inside is employed for high-temperature work, permitting feed temperatures to at least 538°C (1000°F), or double those for jacketed equipment. Vaporizing refrigerants are readily applicable for very low temperature work. The burden must have a definite solidification temperature to assure proper pickup from the feed pan. This limitation can be overcome by side feeding through an auxiliary rotating spreader roll. Application limits are further extended by special feed devices for burdens having oxidation-sensitive and/or supercooling characteristics. The standard double-drum model turns downward, with adjustable roll spacing to control sheet thickness. The newer twin-drum model (Fig. 11-55b) turns upward and, though subject to variable cake thickness, handles viscous and indefinite solidification-temperature-point burden materials well. Drums have been successfully applied to a wide range of chemical products, both inorganic and organic, pharmaceutical compounds, waxes, soaps, insecticides, food products to a limited extent (including lard cooling), and even flake-ice production. A novel application is that of using a water-cooled roll to pick up from a molten-lead bath and turn out a 1.2-m- (4-ft-) wide continuous sheet, weighing 4.9 kg/m2 (1 lb/ft2), which is ideal for a sound barrier. This technique is more economical than other sheeting methods [Mech. Eng., 631 (March 1968)]. Heat-transfer performance of drums, in terms of reported heat flux is: for an 80°C (176°F) melting-point wax, 7880 W/m2 [2500 Btu/ (h⋅ft2)]; for a 130°C (266°F) melting-point organic chemical, 20,000 W/m2 [6500 Btu/(h⋅ft2)]; and for high- [318°C (604°F)] melting-point caustic soda (water-sprayed in drum), 95,000 to 125,000 W/m2 [30,000

(a)

(b)

(c)

Heat-transfer equipment for continuous solidification. (a) Single drum. (b) Twin drum. (c) Roto-shelf. (Courtesy of Buflovak Division, Blaw-Knox Food & Chemical Equipment, Inc.) FIG. 11-55

HEAT EXCHANGERS FOR SOLIDS to 40,000 Btu/(h ⋅ ft2)], with overall coefficients of 340 to 450 W/ (m2 ⋅°C) [60 to 80 Btu/(h⋅ft2 ⋅°F)]. An innovation that is claimed often to increase these performance values by as much as 300 percent is the addition of hoods to apply impinging streams of heated air to the solidifying and drying solids surface as the drums carry it upward [Chem. Eng., 74, 152 (June 19, 1967)]. Similar rotating-drum indirect heattransfer equipment is also extensively used for drying duty on liquids and thick slurries of solids (see Sec. 20). Rotating-Shelf Type The patented Roto-shelf type (Fig. 11-55c) features (1) a large heat-transfer surface provided over a small floor space and in a small building volume, (2) easy floor cleaning, (3) nonhazardous machinery, (4) stainless-steel surfaces, (5) good control range, and (6) substantial capacity by providing as needed 1 to 10 shelves operated in parallel. It is best suited for thick-cake production and burden materials having an indefinite solidification temperature. Solidification of liquid sulfur into 13- to 19-mm- (a- to e-in-) thick lumps is a successful application. Heat transfer, by liquid-coolant circulation through jackets, limits feed temperatures to 204°C (400°F). Heat-transfer rate, controlled by the thick cake rather than by equipment construction, should be equivalent to the belt type. Thermal performance is aided by applying water sprayed directly to the burden top to obtain two-sided cooling.

Only fair overall coefficient values may be expected, although heatflux values are good. Horizontal-Tank Type This type (Fig. 11-56a) is used to transfer heat for melting or cooking dry powdered solids, rendering lard from meat-scrap solids, and drying divided solids. Heat-transfer coefficients are 17 to 85 W/(m2 ⋅°C) [3 to 15 Btu/(h⋅ft2 ⋅°F)] for drying and 28 to 140 W/(m2 ⋅°C) [5 to 25 Btu/(h⋅ft2 ⋅°F)] for vacuum and/or solvent recovery. Vertical Agitated-Kettle Type Shown in Fig. 11-56b this type is used to cook, melt to the liquid state, and provide or remove reaction heat for solids that vary greatly in “body” during the process so that material handling is a real problem. The virtues are simplicity and 100 percent cleanability. These often outweigh the poor heat-transfer aspect. These devices are available from the small jacketed type illustrated to huge cast-iron direct-underfired bowls for calcining gypsum. Temperature limits vary with construction; the simpler jackets allow temperatures to 371°C (700°F) (as with Dowtherm), which is not true of all jacketed equipment. Mill Type Figure 11-56c shows one model of roll construction used. Note the ruggedness, as it is a power device as well as one for indirect heat transfer, employed to knead and heat a mixture of dry powdered-solid ingredients with the objective of reacting and reforming via fusion to a consolidated product. In this compounding operation, frictional heat generated by the kneading may require heat-flow reversal (by cooling). Heat-flow control and temperature-level considerations often predominate over heat-transfer performance. Power and mixing considerations, rather than heat transfer, govern. The tworoll mill shown is employed in compounding raw plastic, rubber, and rubberlike elastomer stocks. Multiple-roll mills less knives (termed calenders) are used for continuous sheet or film production in widths up to 2.3 m (7.7 ft). Similar equipment is employed in the chemical compounding of inks, dyes, paint pigments, and the like.

EQUIPMENT FOR FUSION OF SOLIDS The thermal duty here is the opposite of solidification operations. The indirect heat-transfer equipment suitable for one operation is not suitable for the other because of the material-handling rather than the thermal aspects. Whether the temperature of transformation is a definite or a ranging one is of little importance in the selection of equipment for fusion. The burden is much agitated, but the beds are deep.

(a)

(b)

(c)

Heat-transfer equipment for fusion of solids. (a) Horizontal-tank type. (Courtesy of Struthers Wells Corp.) (b) Agitated kettle. (Courtesy of Read-Standard Division, Capital Products Co.) (c) Double-drum mill. (Courtesy of Farrel-Birmingham Co.)

FIG. 11-56

11-61

11-62

HEAT-TRANSFER EQUIPMENT

HEAT-TRANSFER EQUIPMENT FOR SHEETED SOLIDS Cylinder Heat-Transfer Units Sometimes called “can” dryers or drying rolls, these devices are differentiated from drum dryers in that they are used for solids in flexible continuous-sheet form, whereas drum dryers are used for liquid or paste forms. The construction of the individual cylinders, or drums, is similar in most respects to that of drum dryers. Special designs are used to obtain uniform distribution of steam within large drums when uniform heating across the drum surface is critical. A cylinder dryer may consist of one large cylindrical drum, such as the so-called Yankee dryer, but more often it comprises a number of drums arranged so that a continuous sheet of material may pass over them in series. Typical of this arrangement are Fourdrinier-papermachine dryers, cellophane dryers, slashers for textile piece goods and fibers, etc. The multiple cylinders are arranged in various ways. Generally they are staggered in two horizontal rows. In any one row, the cylinders are placed close together. The sheet material contacts the undersurface of the lower rolls and passes over the upper rolls, contacting 60 to 70 percent of the cylinder surface. The cylinders may also be arranged in a single horizontal row, in more than two horizontal rows, or in one or more vertical rows. When it is desired to contact only one side of the sheet with the cylinder surface, unheated guide rolls are used to conduct the sheeting from one cylinder to the next. For sheet materials that shrink on processing, it is frequently necessary to drive the cylinders at progressively slower speeds through the dryer. This requires elaborate individual electric drives on each cylinder. Cylinder dryers usually operate at atmospheric pressure. However, the Minton paper dryer is designed for operation under vacuum. The drying cylinders are usually heated by steam, but occasionally single cylinders may be gas-heated, as in the case of the Pease blueprinting machine. Upon contacting the cylinder surface, wet sheet material is first heated to an equilibrium temperature somewhere between the wet-bulb temperature of the surrounding air and the boiling point of the liquid under the prevailing total pressure. The heat-transfer resistance of the vapor layer between the sheet and the cylinder surface may be significant. These cylinder units are applicable to almost any form of sheet material that is not injuriously affected by contact with steam-heated metal surfaces. They are used chiefly when the sheet possesses certain properties such as a tendency to shrink or lacks the mechanical strength necessary for most types of continuous-sheeting air dryers. Applications are to dry films of various sorts, paper pulp in sheet form, paper sheets, paperboard, textile piece goods and fibers, etc. In some cases, imparting a special finish to the surface of the sheet may be an objective. The heat-transfer performance capacity of cylinder dryers is not easy to estimate without a knowledge of the sheet temperature, which, in turn, is difficult to predict. According to published data, steam temperature is the largest single factor affecting capacity. Overall evaporation rates based on the total surface area of the dryers cover a range of 3.4 to 23 kg water/(h⋅m2) [0.7 to 4.8 lb water/(h⋅ft2)]. The value of the coefficient of heat transfer from steam to sheet is determined by the conditions prevailing on the inside and on the surface of the dryers. Low coefficients may be caused by (1) poor removal of air or other noncondensables from the steam in the cylinders, (2) poor removal of condensate, (3) accumulation of oil or rust on the interior of the drums, and (4) accumulation of a fiber lint on the outer surface of the drums. In a test reported by Lewis et al. [Pulp Pap. Mag. Can., 22 (February 1927)] on a sulfite-paper dryer, in which the actual sheet temperatures were measured, a value of 187 W/(m2 ⋅°C) [33 Btu/(h⋅ft2 ⋅°F)] was obtained for the coefficient of heat flow between the steam and the paper sheet. Operating-cost data for these units are meager. Power costs may be estimated by assuming 1 hp per cylinder for diameters of 1.2 to 1.8 m (4 to 6 ft). Data on labor and maintenance costs are also lacking. The size of commercial cylinder dryers covers a wide range. The individual rolls may vary in diameter from 0.6 to 1.8 m (2 to 6 ft) and up to 8.5 m (28 ft) in width. In some cases, the width of rolls decreases

throughout the dryer in order to conform to the shrinkage of the sheet. A single-cylinder dryer, such as the Yankee dryer, generally has a diameter between 2.7 and 4.6 m (9 and 15 ft). HEAT-TRANSFER EQUIPMENT FOR DIVIDED SOLIDS Most equipment for this service is some adaptation of a materialhandling device whether or not the transport ability is desired. The old vertical tube and the vertical shell (fluidizer) are exceptions. Material-handling problems, plant transport needs, power, and maintenance are prime considerations in equipment selection and frequently overshadow heat-transfer and capital-cost considerations. Material handling is generally the most important aspect. Material-handling characteristics of the divided solids may vary during heat processing. The body changes are usually important in drying, occasionally significant for heating, and only on occasion important for cooling. The ability to minimize effects of changes is a major consideration in equipment selection. Dehydration operations are better performed on contactive apparatus (see Sec. 12) that provides air to carry off released water vapor before a semiliquid form develops. Some types of equipment are convertible from heat removal to heat supply by simply changing the temperature level of the fluid or air. Other types require an auxiliary change. Others require constructional changes. Temperature limits for the equipment generally vary with the thermal operation. The kind of thermal operation has a major effect on heat-transfer values. For drying, overall coefficients are substantially higher in the presence of substantial moisture for the constant-rate period than in finishing. However, a stiff “body” occurrence due to moisture can prevent a normal “mixing” with an adverse effect on the coefficient. Fluidized-Bed Type Known as the cylindrical fluidizer, this operates with a bed of fluidized solids (Fig. 11-57). It is an indirect heat-transfer version of the contactive type in Sec. 17. An application disadvantage is the need for batch operation unless some short circuiting can be tolerated. Solids-cooling applications are few, as they can be more effectively accomplished by the fluidizing gas via the contactive mechanism that is referred to in Sec. 11. Heating applications are many and varied. These are subject to one shortcoming, which is the dissipation of the heat input by carry-off in the fluidizing gas. Heat-transfer performance for the indirect mode to solids has been outstanding, with overall coefficients in the range of 570 to 850 W/ (m2 ⋅°C) [100 to 150 Btu/(h⋅ft2 ⋅°F)]. This device with its thin film does for solids what the falling-film and other thin-film techniques do for fluids, as shown by Holt (Pap. 11, 4th National Heat-Transfer Conference, August 1960). In a design innovation with high heat-transfer capability, heat is supplied indirectly to the fluidized solids through

Heat-transfer equipment for divided solids; stationary verticalshell type. The indirect fluidizer.

FIG. 11-57

HEAT EXCHANGERS FOR SOLIDS the walls of in-bed, horizontally placed, finned tubes [Petrie, Freeby, and Buckham, Chem. Eng. Prog., 64(7), 45 (1968)]. Moving-Bed Type This concept uses a single-pass tube bundle in a vertical shell with the divided solids flowing by gravity in the tubes. It is little used for solids. A major difficulty in divided-solids applications is the problem of charging and discharging with uniformity. A second is poor heat-transfer rates. Because of these limitations, this tube-bundle type is not the workhorse for solids that it is for liquid and gas-phase heat exchange. However, there are applications in which the nature of a specific chemical reactor system requires indirect heating or cooling of a moving bed of divided solids. One of these is the segregation process which through a gaseous reaction frees chemically combined copper in an ore to a free copper form which permits easy, efficient subsequent recovery [Pinkey and Plint, Miner. Process., 17–30 (June 1968)]. The apparatus construction and principle of operation are shown in Fig. 11-58. The functioning is abetted by a novel heat-exchange provision of a fluidized sand bed in the jacket. This provides a much higher unit heat-input rate (coefficient value) than would the usual low-density hot-combustion-gas flow. Agitated-Pan Type This device (Fig. 11-52) is not an adaptation of a material-handling device but was developed many years ago primarily for heat-transfer purposes. As such, it has found wide application. In spite of its batch operation with high attendant labor costs, it is still used for processing divided solids when no phase change is occurring. Simplicity and easy cleanout make the unit a wise selection for handling small, experimental, and even some production runs when quite a variety of burden materials are heat-processed. Both heating and cooling are feasible with it, but greatest use has been for drying [see Sec. 12 and Uhl and Root, Chem. Eng. Prog., 63(7), 8 (1967)]. This device, because it can be readily covered (as shown in the illustration) and a vacuum drawn or special atmosphere provided, features versatility to widen its use. For drying granular solids, the heat-transfer rate ranges from 28 to 227 W/(m2 ⋅°C) [5 to 40 Btu/

Stationary vertical-tube type of indirect heat-transfer equipment with divided solids inside tubes, laminar solids flow and steady-state heat conditions.

FIG. 11-58

11-63

(h⋅ft2 ⋅°F)]. For atmospheric applications, thermal efficiency ranges from 65 to 75 percent. For vacuum applications, it is about 70 to 80 percent. These devices are available from several sources, fabricated of various metals used in chemical processes. Kneading Devices These are closely related to the agitated pan but differ as being primarily mixing devices with heat transfer a secondary consideration. Heat transfer is provided by jacketed construction of the main body and is effected by a coolant, hot water, or steam. These devices are applicable for the compounding of divided solids by mechanical rather than chemical action. Application is largely in the pharmaceutical and food-processing industries. For a more complete description, illustrations, performance, and power requirements, refer to Sec. 19. Shelf Devices Equipment having heated and/or cooled shelves is available but is little used for divided-solids heat processing. Most extensive use of stationary shelves is freezing of packaged solids for food industries and for freeze drying by sublimation (see Sec. 22). Rotating-Shell Devices These (see Fig. 11-59) are installed horizontally, whereas stationary-shell installations are vertical. Material-handling aspects are of greater importance than thermal performance. Thermal results are customarily given in terms of overall coefficient on the basis of the total area provided, which varies greatly with the design. The effective use, chiefly percent fill factor, varies widely, affecting the reliability of stated coefficient values. For performance calculations see Sec. 10 on heat-processing theory for solids. These devices are variously used for cooling, heating, and drying and are the workhorses for heat-processing divided solids in the largecapacity range. Different modifications are used for each of the three operations. The plain type (Fig. 11-59a) features simplicity and yet versatility through various end-construction modifications enabling wide and varied applications. Thermal performance is strongly affected by the “body” characteristics of the burden because of its dependency for material handling on frictional contact. Hence, performance ranges from well-agitated beds with good thin-film heat-transfer rates to poorly agitated beds with poor thick-film heat-transfer rates. Temperature limits in application are (1) low-range cooling with shell dipped in water, 400°C (750°F) and less; (2) intermediate cooling with forced circulation of tank water, to 760°C (1400°F); (3) primary cooling, above 760°C (1400°F), water copiously sprayed and loading kept light; (4) low-range heating, below steam temperature, hot-water dip; and (5) high-range heating by tempered combustion gases or ribbon radiant-gas burners. The flighted type (Fig. 11-59b) is a first-step modification of the plain type. The simple flight addition improves heat-transfer performance. This type is most effective on semifluid burdens which slide readily. Flighted models are restricted from applications in which softcake sticking occurs, breakage must be minimized, and abrasion is severe. A special flighting is one having the cross section compartmented into four lesser areas with ducts between. Hot gases are drawn through the ducts en route from the outer oven to the stack to provide about 75 percent more heating surface, improving efficiency and capacity with a modest cost increase. Another similar unit has the flights made in a triangular-duct cross section with hot gases drawn through. The tubed-shell type (Fig. 11-59c) is basically the same device more commonly known as a “steam-tube rotary dryer” (see Sec. 20). The rotation, combined with slight inclination from the horizontal, moves the shell-side solids through it continuously. This type features good mixing with the objective of increased heat-transfer performance. Tube-side fluid may be water, steam, or combustion gas. Bottom discharge slots in the shell are used so that heat-transfer-medium supply and removal can be made through the ends; these restrict wide-range loading and make the tubed type inapplicable for floody materials. These units are seldom applicable for sticky, soft-caking, scaling, or heat-sensitive burdens. They are not recommended for abrasive materials. This type has high thermal efficiency because heat loss is minimized. Heat-transfer coefficient values are: water, 34 W/ (m2 ⋅°C) [6 Btu/(h⋅ft2 ⋅°F)]; steam, same, with heat flux reliably constant at 3800 W/m2 [1200 Btu/(h⋅ft2)]; and gas, 17 W/(m2 ⋅°C) [3 Btu/ (h⋅ft2 ⋅°F)], with a high temperature difference. Although from the

11-64

HEAT-TRANSFER EQUIPMENT

(a)

(b)

(c)

(d) FIG. 11-59 Rotating shells as indirect heat-transfer equipment. (a) Plain. (Courtesy of BSP Corp.) (b) Flighted. (Courtesy of BSP Corp.) (c) Tubed. (d) Deep-finned type. (Courtesy of Link-Belt Co.)

preceding discussion the device may seem rather limited, it is nevertheless widely used for drying, with condensing steam predominating as the heat-carrying fluid. But with water or refrigerants flowing in the tubes, it is also effective for cooling operations. The units are custombuilt by several manufacturers in a wide range of sizes and materials. A few fabricators that specialize in this type of equipment have accumulated a vast store of data for determining application sizing. The patented deep-finned type in Fig. 11-59d is named the “Rotofin cooler.” It features loading with a small layer thickness, excellent mixing to give a good effective diffusivity value, and a thin fluidside film. Unlike other rotating-shell types, it is installed horizontally, and the burden is moved positively by the fins acting as an Archimedes spiral. Rotational speed and spiral pitch determine travel time. For cooling, this type is applicable to both secondary and intermediate cooling duties. Applications include solids in small lumps [9 mm (e in)] and granular size [6 mm and less (d to 0 in)] with no larger pieces to plug the fins, solids that have a free-flowing body characteristic with no sticking or caking tendencies, and drying of solids that have a low moisture and powder content unless special modifications are made for substantial vapor and dust handling. Thermal performance is very good, with overall coefficients to 110 W/(m2 ⋅°C) [20 Btu/(h⋅ft2 ⋅°F)], with one-half of these coefficients nominal for cooling based on the total area provided (nearly double those reported for other indirect rotaries). Conveyor-Belt Devices The metal-belt type (Fig. 11-55) is the only device in this classification of material-handling equipment that has had serious effort expended on it to adapt it to indirect heattransfer service with divided solids. It features a lightweight construction of a large area with a thin metal wall. Indirect-cooling applications have been made with poor thermal performance, as could be expected with a static layer. Auxiliary plowlike mixing devices, which are considered an absolute necessity to secure any worthwhile results for this service, restrict applications. Spiral-Conveyor Devices Figure 11-60 illustrates the major

adaptations of this widely used class of material-handling equipment to indirect heat-transfer purposes. These conveyors can be considered for heat-transfer purposes as continuously agitated kettles. The adaptation of Fig. 11-60d offers a batch-operated version for evaporation duty. For this service, all are package-priced and package-shipped items requiring few, if any, auxiliaries. The jacketed solid-flight type (Fig. 11-60a) is the standard lowcost (parts-basis-priced) material-handling device, with a simple jacket added and employed for secondary-range heat transfer of an incidental nature. Heat-transfer coefficients are as low as 11 to 34 W/ (m2 ⋅ °C) [2 to 6 Btu/(h⋅ft2 ⋅°F)] on sensible heat transfer and 11 to 68 W/(m2 ⋅°C) [2 to 12 Btu/(h⋅ft2 ⋅°F)] on drying because of substantial static solids-side film. The small-spiral–large-shaft type (Fig. 11-60b) is inserted in a solids-product line as pipe banks are in a fluid line, solely as a heattransfer device. It features a thin burden ring carried at a high rotative speed and subjected to two-sided conductance to yield an estimated heat-transfer coefficient of 285 W/(m2 ⋅°C) [50 Btu/(h⋅ft2 ⋅°F)], thereby ranking thermally next to the shell-fluidizer type. This device for powdered solids is comparable with the Votator of the fluid field. Figure 11-60c shows a fairly new spiral device with a medium-heavy annular solids bed and having the combination of a jacketed, stationary outer shell with moving paddles that carry the heat-transfer fluid. A unique feature of this device to increase volumetric throughput, by providing an overall greater temperature drop, is that the heat medium is supplied to and withdrawn from the rotor paddles by a parallel piping arrangement in the rotor shaft. This is a unique flow arrangement compared with the usual series flow. In addition, the rotor carries burden-agitating spikes which give it the trade name of Porcupine Heat-Processor (Chem. Equip. News, April 1966; and Uhl and Root, AIChE Prepr. 21, 11th National Heat-Transfer Conference, August 1967). The large-spiral hollow-flight type (Fig. 11-60d) is an adaptation, with external bearings, full fill, and salient construction points as

HEAT EXCHANGERS FOR SOLIDS

(a)

11-65

(c)

(b)

(d)

(e) Spiral-conveyor adaptations as heat-transfer equipment. (a) Standard jacketed solid flight. (Courtesy of Jeffrey Mfg. Co.) (b) Small spiral, large shaft. (Courtesy of Fuller Co.) (c) “Porcupine” medium shaft. (Courtesy of Bethlehem Corp.) (d) Large spiral, hollow flight. (Courtesy of Rietz Mfg. Co.) (e) Fluidized-bed large spiral, helical flight. (Courtesy of Western Precipitation Division, Joy Mfg. Co.) FIG. 11-60

shown, that is highly versatile in application. Heat-transfer coefficients are 34 to 57 W/(m2 ⋅°C) [6 to 10 Btu/(h⋅ft2 ⋅°F)] for poor, 45 to 85 W/(m2 ⋅°C) [8 to 15 Btu/(h⋅ft2 ⋅°F)] for fair, and 57 to 114 W/(m2 ⋅°C) [10 to 20 Btu/(h⋅ft2 ⋅°F)] for wet conductors. A popular version of this employs two such spirals in one material-handling chamber for a pugmill agitation of the deep solids bed. The spirals are seldom heated. The shaft and shell are heated. Another deep-bed spiral-activated solids-transport device is shown by Fig. 11-60e. The flights carry a heat-transfer medium as well as the jacket. A unique feature of this device which is purported to increase heat-transfer capability in a given equipment space and cost is the dense-phase fluidization of the deep bed that promotes agitation and moisture removal on drying operations. Double-Cone Blending Devices The original purpose of these devices was mixing (see Sec. 19). Adaptations have been made; so many models now are primarily for indirect heat-transfer processing. A jacket on the shell carries the heat-transfer medium. The mixing action, which breaks up agglomerates (but also causes some degradation), provides very effective burden exposure to the heat-transfer surface. On drying operations, the vapor release (which in a static bed is a slow diffusional process) takes place relatively quickly. To provide vapor removal from

the burden chamber, a hollow shaft is used. Many of these devices carry the hollow-shaft feature a step further by adding a rotating seal and drawing a vacuum. This increases thermal performance notably and makes the device a natural for solvent-recovery operations. These devices are replacing the older tank and spiral-conveyor devices. Better provisions for speed and ease of fill and discharge (without powered rotation) minimize downtime to make this batchoperated device attractive. Heat-transfer coefficients ranging from 28 to 200 W/(m2 ⋅°C) [5 to 35 Btu/(h⋅ft2 ⋅°F)] are obtained. However, if caking on the heat-transfer walls is serious, then values may drop to 5.5 or 11 W/(m2 ⋅°C) [1 or 2 Btu/(h⋅ft2 ⋅°F)], constituting a misapplication. The double cone is available in a fairly wide range of sizes and construction materials. The users are the fine-chemical, pharmaceutical, and biological-preparation industries. A novel variation is a cylindrical model equipped with a tube bundle to resemble a shell-and-tube heat exchanger with a bloated shell [Chem. Process., 20 (Nov. 15, 1968)]. Conical ends provide for redistribution of burden between passes. The improved heat-transfer performance is shown by Fig. 11-61. Vibratory-Conveyor Devices Figure 11-62 shows the various adaptations of vibratory material-handling equipment for indirect

11-66

FIG. 11-61

HEAT-TRANSFER EQUIPMENT

Performance of tubed, blender heat-transfer device.

heat-transfer service on divided solids. The basic vibratory-equipment data are given in Sec. 21. These indirect heat-transfer adaptations feature simplicity, nonhazardous construction, nondegradation, nondusting, no wear, ready conveying-rate variation [1.5 to 4.5 m/min (5 to 15 ft/min)], and good heat-transfer coefficient—115 W/(m2 ⋅°C) [20 Btu/(h⋅ft2 ⋅°F)] for sand. They usually require feed-rate and distribution auxiliaries. They are suited for heating and cooling of divided solids in powdered, granular, or moist forms but no sticky, liquefying, or floody ones. Terminal-temperature differences less than 11°C (20°F) on cooling and 17°C (30°F) on heating or drying operations are seldom practical. These devices are for medium and light capacities.

The heavy-duty jacketed type (Fig. 11-62a) is a special custombuilt adaptation of a heavy-duty vibratory conveyor shown in Fig. 11-60. Its application is continuously to cool the crushed material [from about 177°C (350°F)] produced by the vibratory-type “caster” of Fig. 11-53. It does not have the liquid dam and is made in longer lengths that employ L, switchback, and S arrangements on one floor. The capacity rate is 27,200 to 31,700 kg/h (30 to 35 tons/h) with heattransfer coefficients in the order of 142 to 170 W/(m2 ⋅°C) [25 to 30 Btu/(h⋅ft2 ⋅°F)]. For heating or drying applications, it employs steam to 414 kPa (60 lbf/in2). The jacketed or coolant-spraying type (Fig. 11-62b) is designed to assure a very thin, highly agitated liquid-side film and the same initial coolant temperature over the entire length. It is frequently employed for transporting substantial quantities of hot solids, with cooling as an incidental consideration. For heating or drying applications, hot water or steam at a gauge pressure of 7 kPa (1 lbf/in2) may be employed. This type is widely used because of its versatility, simplicity, cleanability, and good thermal performance. The light-duty jacketed type (Fig. 11-62c) is designed for use of air as a heat carrier. The flow through the jacket is highly turbulent and is usually counterflow. On long installations, the air flow is parallel to every two sections for more heat-carrying capacity and a fairly uniform surface temperature. The outstanding feature is that a wide range of temperature control is obtained by merely changing the heatcarrier temperature level from as low as atmospheric moisture condensation will allow to 204°C (400°F). On heating operations, a very good thermal efficiency can be obtained by insulating the machine and recycling the air. While heat-transfer rating is good, the heatremoval capacity is limited. Cooler units are often used in series with like units operated as dryers or when clean water is unavailable. Drying applications are for heat-sensitive [49 to 132°C (120° to 270°F)] products; when temperatures higher than steam at a gauge pressure of 7 kPa (1 lbf/in2) can provide are wanted but heavy-duty equipment is

(a)

(d) (b)

(c)

(e)

Vibratory-conveyor adaptations as indirect heat-transfer equipment. (a) Heavy-duty jacketed for liquid coolant or highpressure steam. (b) Jacketed for coolant spraying. (c) Light-duty jacketed construction. (d) Jacketed for air or steam in tiered arrangement. (e) Jacketed for air or steam with Mix-R-Step surface. (Courtesy of Jeffrey Mfg. Co.)

FIG. 11-62

HEAT EXCHANGERS FOR SOLIDS

11-67

too costly; when the jacket-corrosion hazard of steam is unwanted; when headroom space is at a premium; and for highly abrasive burden materials such as fritted or crushed glasses and porcelains. The tiered arrangement (Fig. 11-62d) employs the units of Fig. 11-62 with either air or steam at a gauge pressure of 7 kPa (1 lbf/in2) as a heat medium. These are custom-designed and built to provide a large amount of heat-transfer surface in a small space with the minimum of transport and to provide a complete processing system. These receive a damp material, resize while in process by granulators or rolls, finish dry, cool, and deliver to packaging or tableting. The applications are primarily in the fine-chemical, food, and pharmaceutical manufacturing fields. The Mix-R-Step type in Fig. 11-62e is an adaptation of a vibratory conveyor. It features better heat-transfer rates, practically doubling the coefficient values of the standard flat surface and trebling heatflux values, as the layer depth can be increased from the normal 13 to 25 and 32 mm (a to 1 and 1d in). It may be provided on decks jacketed for air, steam, or water spray. It is also often applicable when an infrared heat source is mounted overhead to supplement the indirect or as the sole heat source. Elevator Devices The vibratory elevating-spiral type (Fig. 11-63) adapts divided-solids-elevating material-handling equipment to heat-transfer service. It features a large heat-transfer area over a small floor space and employs a reciprocating shaker motion to effect transport. Applications, layer depth, and capacities are restricted, as burdens must be of such “body” character as to convey uphill by the microhopping-transport principle. The type lacks self-emptying ability. Complete washdown and cleaning is a feature not inherent in any other elevating device. A typical application is the cooling of a lowdensity plastic powder at the rate of 544 kg/h (1200 lb/h). Another elevator adaptation is that for a spiral-type elevating device developed for ground cement and thus limited to fine powdery burdens. The spiral operates inside a cylindrical shell, which is externally cooled by a falling film of water. The spiral not only elevates the material in a thin layer against the wall but keeps it agitated to achieve high heat-transfer rates. Specific operating data are not available [Chem. Eng. Prog., 68(7), 113 (1968)]. The falling-water film, besides being ideal thermally, by virtue of no jacket pressure very greatly reduces the hazard that the cooling water may contact the watersensitive burden in process. Surfaces wet by water are accessible for cleaning. A fair range of sizes is available, with material-handling capacities to 60 tons/h. Pneumatic-Conveying Devices See Sec. 21 for descriptions, ratings, and design factors on these devices. Use is primarily for transport purposes, and heat transfer is a very secondary consideration.

Applications have largely been for plastics in powder and pellet forms. By modifications, needed cooling operations have been simultaneously effected with transport to stock storage [Plast. Des. Process., 28 (December 1968)]. Heat-transfer aspects and performance were studied and reported on by Depew and Farbar (ASME Pap. 62-HT-14, September 1962). Heat-transfer coefficient characteristics are similar to those shown in Sec. 11 for the indirectly heated fluid bed. Another frequent application on plastics is a small, rather incidental but necessary amount of drying required for plastic pellets and powders on receipt when shipped in bulk to the users. Pneumatic conveyors modified for heat transfer can handle this readily. A pneumatic-transport device designed primarily for heat-sensitive products is shown in Fig. 11-64. This was introduced into the United States after 5 years’ use in Europe [Chem. Eng., 76, 54 (June 16, 1969)]. Both the shell and the rotor carry steam as a heating medium to effect indirect transfer as the burden briefly contacts those surfaces rather than from the transport air, as is normally the case. The rotor turns slowly (1 to 10 r/min) to control, by deflectors, product distribution and prevent caking on walls. The carrier gas can be inert, as nitrogen, and also recycled through appropriate auxiliaries for solvent recovery. Application is limited to burdens that (1) are fine and uniformly grained for the pneumatic transport, (2) dry very fast, and (3) have very little, if any, sticking or decomposition characteristics. Feeds can carry 5 to 100 percent moisture (dry basis) and discharge at 0.1 to 2 percent. Wall temperatures range from 100 to 170°C (212 to 340°F) for steam and lower for a hot-water-heat source. Pressure drops are in order of 500 to 1500 mmH2O (20 to 60 inH2O). Steam consumption approaches that of a contractive-mechanism dryer down to a low value of 2.9 kg steam/kg water (2.9 lb steam/lb water). Available burden capacities are 91 to 5900 kg/h (200 to 13,000 lb/h). Vacuum-Shelf Types These are very old devices, being a version of the table type. Early-day use was for drying (see Sec. 12). Heat transfer is slow even when supplemented by vacuum, which is 90 percent or more of present-day use. The newer vacuum blender and cone devices are taking over many applications. The slow heat-transfer rate is quite satisfactory in a major application, freeze drying, which is a sublimation operation (see Sec. 22 for description) in which the water must be retained in the solid state during its removal. Then slow diffusional processes govern. Another extensive application is freezing packaged foods for preservation purposes. Available sizes range from shelf areas of 0.4 to 67 m2 (4 to 726 ft2). These are available in several manufacturers’ standards, either as system components or with auxiliary gear as packaged systems.

Elevator type as heat-transfer equipment. (Courtesy of Carrier Conveyor Corp.)

FIG. 11-64 A pneumatic-transport adaptation for heat-transfer duty. (Courtesy of Werner & Pfleiderer Corp.)

FIG. 11-63

11-68

HEAT-TRANSFER EQUIPMENT

THERMAL INSULATION Materials or combinations of materials which have air- or gas-filled pockets or void spaces that retard the transfer of heat with reasonable effectiveness are thermal insulators. Such materials may be particulate and/or fibrous, with or without binders, or may be assembled, such as multiple heat-reflecting surfaces that incorporate air- or gasfilled void spaces. The ability of a material to retard the flow of heat is expressed by its thermal conductivity (for unit thickness) or conductance (for a specific thickness). Low values for thermal conductivity or conductance (or high thermal resistivity or resistance value) are characteristics of thermal insulation. Heat is transferred by radiation, conduction, and convection. Radiation is the primary mode and can occur even in a vacuum. The amount of heat transferred for a given area is relative to the temperature differential and emissivity from the radiating to the absorbing surface. Conduction is due to molecular motion and occurs within gases, liquids, and solids. The tighter the molecular structure, the higher the rate of transfer. As an example, steel conducts heat at a rate approximately 600 times that of typical thermal-insulation materials. Convection is due to mass motion and occurs only in fluids. The prime purpose of a thermal-insulation system is to minimize the amount of heat transferred. INSULATION MATERIALS Materials Thermal insulations are produced from many materials or combinations of materials in various forms, sizes, shapes, and thickness. The most commonly available materials fall within the following categories: Fibrous or cellular—mineral. Alumina, asbestos, glass, perlite, rock, silica, slag, or vermiculite. Fibrous or cellular—organic. Cane, cotton, wood, and wood bark (cork). Cellular organic plastics. Elastomer, polystyrene, polyisocyanate, polyisocyanurate, and polyvinyl acetate. TABLE 11-20

Cements. Insulating and/or finishing. Heat-reflecting metals (reflective). Aluminum, nickel, stainless steel. Available forms. Blanket (felt and batt), block, cements, loose fill, foil and sheet, formed or foamed in place, flexible, rigid, and semirigid. The actual thicknesses of piping insulation differ from the nominal values. Dimensional data of ASTM Standard C585 appear in Table 11-20. Thermal Conductivity (K Factor) Depending on the type of insulation, the thermal conductivity (K factor) can vary with age, manufacturer, moisture content, and temperature. Typical published values are shown in Fig. 11-65. Mean temperature is equal to the arithmetic average of the temperatures on both sides of the insulating material. Actual system heat loss (or gain) will normally exceed calculated values because of projections, axial and longitudinal seams, expansioncontraction openings, moisture, workers’ skill, and physical abuse. Finishes Thermal insulations require an external covering (finish) to provide protection against entry of water or process fluids, mechanical damage, and ultraviolet degradation of foamed materials. In some cases the finish can reduce the flame-spread rating and/or provide fire protection. The finish may be a coating (paint, asphaltic, resinous, or polymeric), a membrane (coated felt or paper, metal foil, or laminate of plastic, paper, foil or coatings), or sheet material (fabric, metal, or plastic). Finishes for systems operating below 2°C (35°F) must be sealed and retard vapor transmission. Those from 2°C (35°F) through 27°C (80°F) should retard vapor transmission (to prevent surface condensation), and those above 27°C (80°F) should prevent water entry and allow moisture to escape. Metal finishes are more durable, require less maintenance, reduce heat loss, and, if uncoated, increase the surface temperature on hot systems.

Thicknesses of Piping Insulation Insulation, nominal thickness

in mm Nominal ironpipe size, in

Outer diameter

1 25

1a 38

2 51

2a 64

3 76

3a 89

4 102

Approximate wall thickness in

mm

in

mm

in

mm

in

mm

in

mm

in

mm

in

mm

in

mm

a e 1 1d 1a

0.84 1.05 1.32 1.66 1.90

21 27 33 42 48

1.01 0.90 1.08 0.91 1.04

26 23 27 23 26

1.57 1.46 1.58 1.66 1.54

40 37 40 42 39

2.07 1.96 2.12 1.94 2.35

53 50 54 49 60

2.88 2.78 2.64 2.47 2.85

73 71 67 63 72

3.38 3.28 3.14 2.97 3.35

86 83 80 75 85

3.88 3.78 3.64 3.47 3.85

99 96 92 88 98

4.38 4.28 4.14 3.97 4.42

111 109 105 101 112

2 2a 3 3a 4

2.38 2.88 3.50 4.00 4.50

60 73 89 102 114

1.04 1.04 1.02 1.30 1.04

26 26 26 33 26

1.58 1.86 1.54 1.80 1.54

40 47 39 46 39

2.10 2.36 2.04 2.30 2.04

53 60 52 58 52

2.60 2.86 2.54 2.80 2.54

66 73 65 71 65

3.10 3.36 3.04 3.36 3.11

79 85 77 85 79

3.60 3.92 3.61 3.86 3.61

91 100 92 98 92

4.17 4.42 4.11 4.36 4.11

106 112 104 111 104

4a 5 6 7 8

5.00 5.56 6.62 7.62 8.62

127 141 168 194 219

1.30 0.99 0.96

33 25 24

1.80 1.49 1.46 1.52 1.52

46 38 37 39 39

2.30 1.99 2.02 2.02 2.02

58 51 51 51 51

2.86 2.56 2.52 2.52 2.65

73 65 64 64 67

3.36 3.06 3.02 3.15 3.15

85 78 77 80 80

3.86 3.56 3.65 3.65 3.65

98 90 93 93 93

4.48 4.18 4.15 4.15 4.15

114 106 105 105 105

9.62 10.75 11.75 12.75 14.00

244 273 298 324 356

1.52 1.58 1.58 1.58 1.46

39 40 40 40 37

2.15 2.08 2.08 2.08 1.96

55 53 53 53 50

2.65 2.58 2.58 2.58 2.46

67 66 66 66 62

3.15 3.08 3.08 3.08 2.96

80 78 78 78 75

3.65 3.58 3.58 3.58 3.46

93 91 91 91 88

4.15 4.08 4.08 4.08 3.96

105 104 104 104 101

1.46

37

1.96

50

2.46

62

2.96

75

3.46

88

3.96

101

9 10 11 12 14 Over 14, up to and including 36

THERMAL INSULATION

FIG. 11-66

11-69

Dewar flask.

on as the vapor retarder. Hairline cracks can develop, cells can break down, glass-fiber binders are absorbent, and moisture can enter at joints between all materials. Moderate and High Temperature [over 2∞C (36∞F)] Cellular or fibrous materials are normally used. See Fig. 11-68 for nominal

FIG. 11-65

Thermal conductivity of insulating materials.

SYSTEM SELECTION A combination of insulation and finish produces the thermalinsulation system. Selection of these components depends on the purpose for which the system is to be used. No single system performs satisfactorily from the cryogenic through the elevated-temperature range. Systems operating below freezing have a low vapor pressure, and atmospheric moisture is pushed into the insulation system, while the reverse is true for hot systems. Some general guidelines for system selection follow. Cryogenic [-273 to -101∞C (-459 to -150∞F)] High Vacuum This technique is based on the Dewar flask, which is a double-walled vessel with reflective surfaces on the evacuated side to reduce radiation losses. Figure 11-66 shows a typical laboratory-size Dewar. Figure 11-67 shows a semiportable type. Radiation losses can be further reduced by filling the cavity with powders such as perlite or silica prior to pulling the vacuum. Multilayer Multilayer systems consist of series of radiationreflective shields of low emittance separated by fillers or spacers of very low conductance and exposed to a high vacuum. Foamed or Cellular Cellular plastics such as polyurethane and polystyrene do not hold up or perform well in the cryogenic temperature range because of permeation of the cell structure by water vapor, which in turn increases the heat-transfer rate. Cellular glass holds up better and is less permeable. Low Temperature [-101 to -1∞C (-150 to +30∞F)] Cellular glass, glass fiber, polyurethane foam, and polystyrene foam are frequently used for this service range. A vapor-retarder finish with a perm rating less than 0.02 is required. In addition, it is good practice to coat all contact surfaces of the insulation with a vapor-retardant mastic to prevent moisture migration when the finish is damaged or is not properly maintained. Closed-cell insulation should not be relied

FIG. 11-67

Hydrogen bottle.

11-70

HEAT-TRANSFER EQUIPMENT low thermal conductivity. Underground systems are very difficult to keep dry permanently. Methods of insulation include factorypreinsulated pouring types and conventionally applied types. Corrosion can occur under wet insulation. A protective coating, applied directly to the metal surface, may be required. ECONOMIC THICKNESS OF INSULATION

FIG. 11-68

Optimal economic insulation thickness may be determined by various methods. Two of these are the minimum-total-cost method and the incremental-cost method (or marginal-cost method). The minimumtotal-cost method involves the actual calculations of lost energy and insulation costs for each insulation thickness. The thickness producing the lowest total cost is the optimal economic solution. The optimum thickness is determined to be the point where the last dollar invested in insulation results in exactly $1 in energy-cost savings (“ETI— Economic Thickness for Industrial Insulation,” Conservation Pap. 46, Federal Energy Administration, August 1976). The incremental-cost method provides a simplified and direct solution for the least-cost thickness. The total-cost method does not in general provide a satisfactory means for making most insulation investment decisions, since an economic return on investment is required by investors and the method does not properly consider this factor. Return on investment is considered by Rubin (“Piping Insulation—Economics and Profits,” in Practical Considerations in Piping Analysis, ASME Symposium, vol. 69, 1982, pp. 27–46). The incremental method used in this reference requires that each incremental a in of insulation provide the predetermined return on investment. The minimum thickness of installed insulation is used as a base for calculations. The incremental installed capital cost for each additional a in of insulation is determined. The energy saved for each increment is then determined. The value of this energy varies directly with the temperature level [e.g., steam at 538°C (1000°F) has a greater value than condensate at 100°C (212°F)]. The final increment selected for use is required either to provide a satisfactory return on investment or to have a suitable payback period. Recommended Thickness of Insulation Indoor insulation thickness appears in Table 11-21, and outdoor thickness appears in Table 11-22. These selections were based upon calcium silicate insulation with a suitable aluminum jacket. However, the variation in thickness for fiberglass, cellular glass, and rock wool is minimal. Fiberglass is available for maximum temperatures of 260, 343, and 454°C (500, 650, and 850°F). Rock wool, cellular glass, and calcium silicate are used up to 649°C (1200°F).

Insulating materials and applicable temperature ranges.

temperature range. Nonwicking insulation is desirable for systems operating below 100°C (212°F). Other Considerations Autoignition can occur if combustible fluids are absorbed by wicking-type insulations. Chloride stress corrosion of austenitic stainless steel can occur when chlorides are concentrated on metal surfaces at or above approximately 60°C (140°F). The chlorides can come from sources other than the insulation. Some calcium silicates are formulated to exceed the requirements of the MIL-I-24244A specification. Fire resistance of insulations varies widely. Calcium silicate, cellular glass, glass fiber, and mineral wool are fire-resistant but do not perform equally under actual fire conditions. A steel jacket provides protection, but aluminum does not. Traced pipe performs better with a nonwicking insulation which has

TABLE 11-21

Indoor Insulation Thickness, 80°F Still Ambient Air* Minimum pipe temperature, °F

Pipe size, in

Insulation thickness, in

Energy cost, $/million Btu 1

2

3

4

5

6

7

8

e

1a 2 2a 3

950

600

550

400 1100 1750

350 1000 1050

300 900 950

250 800 850

250 750 800 1200

1

1a 2 2a 3

1200

800

600 1200

500 1000

450 900 1200

400 800 1050 1100

350 700 1000 1150

300 700 900 950

1a

1a 2 2a 3

1100

750

550 1000

450 850 1050

400 700 900

400 650 800 1150

350 600 750 1100

300 500 650 1000

2

1a 2 2a 3

1050

700

500 1050 1100

450 850 950 1200

400 750 1000 1050

350 700 750 950

300 600 700 850

300 600 650 800

THERMAL INSULATION TABLE 11-21

Indoor Insulation Thickness, 80°F Still Ambient Air* (Concluded ) Minimum pipe temperature, °F

Pipe size, in

Insulation thickness, in

Energy cost, $/million Btu 1

2

3

4

5

6

7

8

3

1a 2 2a 3

950

650 1100

500 900 1050

400 700 850 1050

350 600 750 950

300 550 650 800

300 500 500 750

250 450 500 700

4

1a 2 2a 3 3a

950

600 1100

500 850 1200

400 700 1000 1050

350 600 850 900

300 550 750 800

300 500 700 750 1150

250 450 650 700 1050

6

1a 2 2a 3 3a 4

600

350 1100

300 850 900 1150

250 700 800 1000

250 600 650 850

200 550 600 750 1100

200 500 550 700 1000

200 500 550 600 900 1200

8

2 2a 3 3a 4

1000 1050

800 850

650 700 1050

550 600 900 1200

500 550 800 1100

450 500 750 1000 1150

400 450 700 900 1100

10

2 2a 3 3a 4

1100 1200

850 900 1050

700 750 900

650 700 750 1200

550 600 700 1050

500 550 600 950

450 500 550 900 1200

12

2 2a 3 3a 4 4a

1150

750 1000

600 800 1200

500 650 1000

400 550 900 1200

400 500 800 1100 1150 1200

350 450 700 1000 1050 1100

300 400 650 900 950 1000

14

2 2a 3 3a 4 4a

1050

650 1000

550 800 1100

450 650 950

400 550 800 1150 1200

350 500 700 1000 1050 1200

300 450 650 950 1000 1100

300 400 600 850 900 1000

16

2 2a 3 3a 4 4a

950

650 1000 1200

500 800 950

400 700 800

350 600 700 1150 1200

300 550 600 1050 1100 1150

300 500 550 950 1000 1050

300 450 500 850 900 950

18

2 2a 3 3a 4 4a

1000

650 950 1150

500 750 900

400 600 750

350 550 650 1200

350 500 550 1100 1150 1200

300 450 500 1000 1050 1100

300 400 500 900 950 1000

20

2 2a 3 3a 4 4a

1050

700 1000 1150

550 800 900

450 600 750

400 550 650

350 500 550 1100 1150

350 450 500 1000 1050 1200

300 400 500 950 1000 1100

24

2 2a 3 3a 4 4a

950

600 1150

500 900 1050

400 750 900

350 650 750 1100 1150

300 550 700 1000 1050 1150

300 500 600 900 950 1050

250 450 550 800 850 950

*Aluminum-jacketed calcium silicate insulation with an emissivity factor of 0.05. To convert inches to millimeters, multiply by 25.4, to convert dollars per 1 million British thermal units to dollars per 1 million kilojoules, multiply by 0.948, °C = 5/9 (°F − 32).

11-71

11-72

HEAT-TRANSFER EQUIPMENT TABLE 11-22

Outdoor Insulation Thickness, 7.5-mi/h Wind, 60°F Air* Minimum pipe temperature, °F Energy cost, $/million Btu

Pipe size, in

Thickness, in

1

2

3

4

5

6

7

8

e

1 1a 2 2a

450 800

300 500

250 400 1150 1100

250 300 950 1000

200 250 850 900

200 250 750 800

150 200 700 750

150 200 650 700

1

1 1a 2 2a 3

400 1000

300 650

250 500 1100

200 400 900 1200

200 350 800 1050 1100

150 300 700 950 1000

150 300 600 850 900

150 250 600 800 850

1a

1 1a 2 2a 3

350 900

250 600 100

200 450 850 1150

200 350 700 950

150 300 600 800 1200

150 300 550 750 1050

150 250 500 700 1000

150 250 450 600 900

2

1 1a 2 2a 3

350 900

250 550 1150

200 450 900 1000

150 400 750 850 1050

150 300 650 750 950

150 300 600 650 850

150 250 550 600 750

150 250 500 550 700

3

1 1a 2 2a 3 3a

300 750

200 500 950 1150

150 400 750 950 1150

150 300 600 750 1000

150 250 500 650 850

150 250 450 600 750

150 250 400 500 650

150 200 350 500 600 1150

4

1 1a 2 2a 3 3a

250 750

200 500 950

150 350 750 1050 1100

150 300 600 900 950

150 250 500 700 750

150 250 450 650 700 1200

150 200 400 600 650 1100

150 200 350 550 600 1000

6

1 1a 2 2a 3 3a 4 4a

250 450

150 300 900 1050

150 200 700 800 1050

150 200 600 650 900

150 150 500 600 750 1150

150 150 450 500 700 1050

150 150 400 450 600 950 1200

150 150 350 400 550 850 1150 1200

8

1 2 2a 3 3a 4

250

200 850 900

150 650 700 1100

150 550 600 950

150 450 500 800 1150

150 400 450 750 1000

150 350 400 700 950 1050

150 350 400 600 850 1000

10

2 2a 3 3a 4 4a

200

150 1000 1200

150 800 950

150 650 800

150 550 700 1100

150 500 600 1000

150 450 550 900 1150 1200

150 400 500 800 1050 1100

12

1a 2 2a 3 3a 4 4a

250 950

150 600 900

150 500 700 1100

150 400 550 900

150 350 500 800 1100 1150 1200

150 300 400 700 1000 1050 1100

150 250 400 650 900 950 1000

150 250 350 550 850 900 950

14

1a 2 2a 3 3a 4 4a

250 850

150 550 850

150 400 650 1000

150 350 550 850 1200

150 300 500 700 1000 1050

150 250 400 650 950 1000 1100

150 250 400 550 850 900 1000

150 250 400 500 800 850 950

THERMAL INSULATION TABLE 11-22

11-73

Outdoor Insulation Thickness, 7.5-mi/h Wind, 60°F Air* (Concluded) Minimum pipe temperature, °F Energy cost, $/million Btu

Pipe size, in 16

18

20

24

Thickness, in

1

2

3

4

5

6

7

8

1a 2 2a 3 3a 4 4a

250 800

150 500 900 1000

150 350 700 850

150 300 550 700 1200

150 300 500 600 1000 1100 1150

150 250 450 500 950 1000 1000

150 250 400 450 850 900 950

150 200 350 400 800 850 900

1a 2 2a 3 3a

250 850

150 550 800 1000

150 400 650 800

150 350 500 650

150 300 450 550 1100

150 250 400 500 1000

150 250 350 450 900

150 200 350 400 850

1a 2 2a 3 3a 4 4a

150 900

150 550 850 1000

150 450 650 800

150 350 550 650

150 300 450 550 1150 1200

150 300 400 500 1050 1100 1200

150 250 350 450 950 1000 1100

150 250 350 400 900 950 1050

1a 2 2a 3 3a 4 4a 5

150 800

150 500 950 1150

150 400 750 950

150 300 650 750 1150 1200

150 250 550 650 1000 1050

150 250 500 600 900 950 1050

150 200 450 550 800 850 950

150 200 400 500 750 800 850

*Aluminum-jacketed calcium silicate insulation with an emissivity factor of 0.05. To convert inches to millimeters, multiply by 25.4; to convert miles per hour to kilometers per hour, multiply by 1.609; and to convert dollars per 1 million British thermal units to dollars per 1 million kilojoules, multiply by 0.948; °C = 5/9 (°F − 32).

The tables were based upon the cost of energy at the end of the first year, a 10 percent inflation rate on energy costs, a 15 percent interest cost, and a present-worth pretax profit of 40 percent per annum on the last increment of insulation thickness. Dual-layer insulation was used for 3a-in and greater thicknesses. The tables and a full explanation of their derivation appear in a paper by F. L. Rubin (op. cit.). Alternatively, the selected thicknesses have a payback period on the last nominal a-in increment of 1.44 years as presented in a later paper by Rubin [“Can You Justify More Piping Insulation?” Hydrocarbon Process., 152–155 (July 1982)]. Example 1 For 24-in pipe at 371°C (700°F) with an energy cost of $4/million Btu, select 2-in thickness for indoor and 2a-in for outdoor locations. [A 2a-in thickness would be chosen at 399°C (750°F) indoors and 3a-in outdoors.] Example 2 For 16-in pipe at 343°C (650°F) with energy valued at $5/million Btu, select 2a-in insulation indoors [use 3-in thickness at 371°C (700°F)]. Outdoors choose 3-in insulation [use 3a-in dual-layer insulation at 538°C (1000°F)]. Example 3 For 12-in pipe at 593°C (1100°F) with an energy cost of $6/million Btu, select 3a-in thickness for an indoor installation and 4a-in thickness for an outdoor installation. INSTALLATION PRACTICE Pipe Depending on diameter, pipe is insulated with cylindrical half, third, or quarter sections or with flat segmental insulation. Fittings and valves are insulated with preformed insulation covers or with individual pieces cut from sectional straight pipe insulation. Method of Securing Insulation with factory-applied jacketing may be secured with adhesive on the overlap, staples, tape, or wire, depending on the type of jacket and the outside diameter. Insulation which has a separate jacket is wired or banded in place before the jacket (finish) is applied.

Double Layer Pipe expansion is a significant factor at temperatures above 600°F (316°C). Above this temperature, insulation should be applied in a double layer with all joints staggered to prevent excessive heat loss and high surface temperature at joints opened by pipe expansion. This procedure also minimizes thermal stresses in the insulation. Finish Covering for cylindrical surfaces ranges from asphaltsaturated or saturated and coated organic and asbestos paper, through laminates of such papers and plastic films or aluminum foil, to medium-gauge aluminum, galvanized steel, or stainless steel. Fittings and irregular surfaces may be covered with fabric-reinforced mastics or preformed metal or plastic covers. Finish selection depends on function and location. Vapor-barrier finishes may be in sheet form or a mastic, which may or may not require reinforcing, depending on the method of application, and additional protection may be required to prevent mechanical abuse and/or provide fire resistance. Criteria for selecting other finishes should include protection of insulation against water entry, mechanical abuse, or chemical attack. Appearance, lifecycle cost, and fire resistance may also be determining factors. Finish may be secured with tape, adhesive, bands, or screws. Fasteners which will penetrate vapor-retarder finishes should not be used. Tanks, Vessels, and Equipment Flat, curved, and irregular surfaces such as tanks, vessels, boilers, and breechings are normally insulated with flat blocks, beveled lags, curved segments, blankets, or spray-applied insulation. Since no general procedure can apply to all materials and conditions, it is important that manufacturers’ specifications and instructions be followed for specific insulation applications. Method of Securing On small-diameter cylindrical vessels, the insulation may be secured by banding around the circumference. On larger cylindrical vessels, banding may be supplemented with angleiron ledges to support the insulation and prevent slipping. On large flat and cylindrical surfaces, banding or wiring may be supplemented with various types of welded studs or pins. Breather springs may be required with bands to accommodate expansion and contraction.

11-74

HEAT-TRANSFER EQUIPMENT

Finish The materials are the same as for pipe and should satisfy the same criteria. Breather springs may be required with bands.

ADDITIONAL REFERENCES: ASHRAE Handbook and Product Directory: Fundamentals, American Society of Heating, Refrigerating and Air Condition-

ing Engineers, Atlanta, 1981. Turner and Malloy, Handbook of Thermal Insulation Design Economics for Pipes and Equipment, Krieger, New York, 1980. Turner and Malloy, Thermal Insulation Handbook, McGraw-Hill, New York, 1981.

AIR CONDITIONING INTRODUCTION Air Conditioning is the process of treating air so as to control simultaneously its temperature, humidity, cleanliness, and distribution to meet the requirements of the conditioned space. The portions relating only to temperature and humidity control will be discussed here. For detailed discussions of air cleanliness and distribution, refer, for example, to the current edition of the HVAC Applications volume of the A.S.H.R.A.E. Handbooks (ASHRAE, 1791 Tullie Circle, N.E., Atlanta, Ga.). Applications of air conditioning include the promotion of human comfort and the maintenance of proper conditions for the manufacture, processing, and preserving of material and equipment. Also, in industrial environments where, for economical or other reasons, conditions cannot be made entirely comfortable, air conditioning may be used for maintaining the efficiency and health of workers at safe tolerance limits. COMFORT AIR CONDITIONING Comfort is influenced by temperature, humidity, air velocity, radiant heat, clothing, and work intensity. Psychological factors may also influence comfort, but their discussion is beyond the scope of this handbook. The reader is referred to Chap. 42 of the HVAC Applications volume of the A.S.H.R.A.E. Handbooks for a full discussion of the control of noise, which must also be considered in air-conditioning design. Figure 5 in Chap. 8 of the HVAC Fundamentals volume of the A.S.H.R.A.E. Handbooks relates the variables of ambient temperature, dew point temperature (or humidity ratio) to comfort under clothing and activity conditions typical for office space occupancy. It also shows boundary values for ET*, the effective temperature index. This index combines temperature and moisture conditions into a single index representing the temperature of an environment at 50 percent relative humidity resulting in the same heat transfer from the skin as for the actual case. Hence, the ET* for 50 percent relative humidity is equal in value to the ambient dry-bulb temperature.

systems are used, it is necessary to introduce sufficient fresh air into the air-conditioned area to make up for that exhausted. Generally, low exhaust systems are used where the contaminant sources are concentrated and/or where the contaminant may be highly toxic. Where the contaminant comes from widely dispersed points, however, dilution ventilation is usually employed. Combinations of the two systems sometimes provide the least expensive installation. dilution ventilation is not appropriate for cases where large volumes of contaminant are released and cases where the employees must work near the contaminant source. The selection of dilution ventilation for cases with potential fire or smoke should be accompanied by careful study. Details for design of dilution ventilation systems are given in Chap. 25 of the HVAC Applications volume of the A.S.H.R.A.E. Handbooks. Chapter 27 of the same volume discusses industrial exhaust systems. Exhaust stacks should be high enough to adequately dispense the contaminated air and to prevent recirculation into fresh air intakes (Chap. 14 of the 1993 HVAC Fundamentals volume of the A.S.H.R.A.E. Handbooks). Depending on the contaminant and air pollution legislation, it may be necessary to reduce the contaminant emission rate by such methods as filtering, scrubbing, catalytic oxidation, or incineration. AIR-CONDITIONING EQUIPMENT Basically, an air-conditioning system consists of a fan unit which forces a mixture of fresh outdoor air and room air through a series of devices which act upon the air to clean it, to increase or decrease its temperature, and to increase or decrease its water-vapor content or humidity.

INDUSTRIAL AIR CONDITIONING Industrial buildings have to be designed according to their intended use. For instance, the manufacture of hygroscopic materials (paper, textiles, foods, etc.) will require relatively tight controls of relative humidity. On the other hand, the storage of furs will demand relatively low temperatures, while the ambient in a facility manufacturing refractories might be acceptable at notably higher temperatures. Chapter 12 of the HVAC Applications volume of the A.S.H.R.A.E. Handbooks provides extensive tables of suggested temperatures and humidities for industrial air conditioning. VENTILATION In the design of comfort air-conditioning systems, odors arising from occupants, cooking, or other sources must be controlled. this is accomplished by introducing fresh air or purified recirculated air in sufficient quantities to reduce odor concentrations to an acceptable level by dilution. Recommended fresh-air requirements for different types of buildings are called for in A.S.H.R.A.E. standard 62-1989 “Ventilation for Acceptable Indoor Air Quality.” These values range in the order of 15 to 30 cfm per person, according to application. In industrial air-conditioning systems, harmful environmental gases, vapors, dusts, and fumes are often encountered. These contaminants can be controlled by exhaust systems at the source, by dilution ventilation, or by a combination of the two methods. When exhaust

Typical central-station air-conditioning unit and control system. On a rising room wet-bulb temperature, the wet-bulb branch-line air pressure increases through the reverse-acting outdoor-air wet-bulb temperature-limit thermostat T1 to open gradually the maximum outdoor-air damper D1 and simultaneously closr return-air damper D2, then gradually open chilled-water valve V1. On a rising room dry-bulb temperature, the dry-bulb branch-line air pressure gradually increases to close reheat steam valve V2. When outdoor wetbulb temperature exceeds the set point of the outdoor-air wet-bulb-limit thermostat T1, which is set at the return-air wet-bulb temperature, this thermostat decreases branch-line pressure to close gradually maximum outdoor damper D1 and simultaneously open return-air damper D2. The reverse sequences are followed during the heating season. FIG. 11-69

AIR CONDITIONING In general, air conditioning equipment can be classified into two broad types: central (sometimes called field erected) and unitary (or packaged). CENTRAL SYSTEMS Figure 11-69 describes a typical central system. Either water or direct-expansion refrigerant coils or air washers may be used for cooling. Steam or hot-water coils are available for heating. Humidification may be provided by target-type water nozzles, pan humidifiers, air washers, or sprayed coils. Air cleaning is usually provided by cleanable or throwaway filters. Central-station air-conditioning units in capacities up to about 50,000 cu ft/min are available in prefabricated units. The principle types of refrigeration equipment used in large central systems are: Reciprocating (up to 300 hp); helical rotary (up to 750 tons); absorption (up to 2000 tons); and centrifugal (up to 10,000 tons). The drives for the reciprocating, rotary, and centrifugal compressors may be electric motors, gas or steam turbines, or gas or diesel engines. The heat rejected from the condensors usually calls for cooling towers or air-cooled condensors; in some cases evaporative cooling might be practical.

TABLE 11-23

11-75

UNITARY REFRIGERANT-BASED AIR-CONDITIONING SYSTEMS These systems include window-mounted air conditioners and heat pumps, outdoor unitary equipment, indoor unitary equipment, unitary self-contained systems, and commercial self-contained systems. These are described in detail in the HVAC Systems and Equipment volume of the A.S.H.R.A.E. Handbooks. A detailed analysis of the proposed installation is usually necessary to select the air conditioning equipment which is best in overall performance. Each type of air conditioner has its own particular advantages and disadvantages. Important factors to be considered in the selection of air conditioning equipment are degree of temperature and humidity control required, investment, owning, and operating costs, and space requirements. Another important factor is the building itself, that is, whether it is new or existing construction. For example, for existing buildings where it may be inadvisable to install air-supply ducts, the selfcontained or unit-type air conditioner may offer the greatest advantages in reduced installation costs. For large industrial processes where close temperature and humidity control are required, a central station system is usually employed.

Outdoor Design Temperatures* Winter

Summer

Winter

City-state

Dry bulb, °F.

Dry bulb, °F.

Wet bulb, °F.

Akron, Ohio Albany, N.Y. Albuquerque, N.M. Atlanta, Ga. Baltimore, Md. Billings, Mont. Birmingham, Ala. Bloomfield, N.J. Boise, Idaho Boston, Mass. Bridgeport, Conn. Buffalo, N.Y. Charleston, S.C. Chattanooga, Tenn. Chicago, Ill. Cincinnati, Ohio Cleveland, Ohio Columbus, Ohio Dallas, Tex. Dayton, Ohio Denver, Colo. Des Moines, Iowa Detroit, Mich. Duluth, Minn. East Orange, N.J. El Paso, Tex. Erie, Pa. Fitchburg, Mass. Flint, Mich. Fort Wayne, Ind. Fort Worth, Tex. Grand Rapids, Mich. Hartford, Conn. Houston, Tex. Indianapolis, Ind. Jacksonville, Fla. Jersey City, N.J. Kansas City, Mo. Lincoln, Nebr. Little Rock, Ark. Long Beach, Calif. Los Angeles, Calif. Louisville, Ky. Memphis, Tenn. Miami, Fla.

−5 −10 0 10 0 −25 10 0 −10 0 0 −5 15 10 −10 0 0 −10 0 0 −10 −15 −10 −25 0 10 −5 −10 −10 −10 10 −10 0 20 −10 25 0 −10 −10 5 35 35 0 0 35

95 93 95 95 95 90 95 95 95 92 95 93 95 95 95 95 95 95 100 95 95 95 95 93 95 100 93 93 95 95 100 95 93 95 95 95 95 100 95 95 90 90 95 95 91

75 75 70 76 78 66 78 75 65 75 75 73 78 76 75 78 75 76 78 78 64 78 75 73 75 69 75 75 75 75 78 75 75 80 76 78 75 76 78 78 70 70 78 78 79

Summer

City-state

Dry bulb, °F.

Dry bulb, °F.

Wet bulb, °F.

Milwaukee, Wis. Minneapolis, Minn. Nashville, Tenn. New Haven, Conn. New Orleans, La. New York, N.Y. Newark, N.J. Norfolk, Va. Oakland, Calif. Oklahoma City, Okla. Omaha, Nebr. Peoria, Ill. Philadelphia, Pa. Phoenix, Ariz. Pittsburgh, Pa. Portland, Me. Portland, Ore. Providence, R.I. Reno, Nev. Richmond, Va. Roanoke, Va. Rochester, N.Y. St. Louis, Mo. St. Paul, Minn. Salt Lake City, Utah San Antonio, Tex. San Francisco, Calif. Schenectady, N.Y. Scranton, Pa. Seattle, Wash. Shreveport, La. Sioux City, Iowa Spokane, Wash. Springfield, Mass. Syracuse, N.Y. Tampa, Fla. Toledo, Ohio Tucson, Ariz. Tulsa, Okla. Washington, D.C. Wichita, Kans. Wilmington, Del. Worcester, Mass. Youngstown, Ohio

−15 −20 0 0 20 0 0 15 30 0 −10 −10 0 0 −5 −5 10 0 −5 15 0 −5 0 −20 −10 20 35 −10 −5 15 20 −20 −15 −10 −10 30 −10 25 0 0 −10 0 0 −5

95 95 95 95 95 95 95 95 85 101 95 96 95 105 95 90 90 93 95 95 95 95 95 95 95 100 85 93 95 85 100 95 93 93 93 95 95 105 101 95 100 95 93 95

75 75 78 75 80 75 75 78 65 77 78 76 78 76 75 73 68 75 65 78 76 75 78 75 65 78 65 75 75 65 78 78 65 75 75 78 75 72 77 78 75 78 75 75

*Carrier, Cherne, Grant, and Roberts, Modern Air Conditioning, Heating, and Ventilating, 3d ed., p. 531, Pitman, New York, 1959.

11-76

HEAT-TRANSFER EQUIPMENT

LOAD CALCULATION First step in the solution of an air-conditioning problem is to determine the proper design temperature conditions. Since both outdoor and indoor temperatures greatly influence the size of the equipment, the designer must exercise good judgment in selecting the proper conditions for his/her particular case. Table 11-23 lists winter and summer outdoor temperature conditions in common use for comfort applications for various United States cities. For critical-process air conditioning, it may be desirable to use a different set of outdoor temperature conditions. However, it is seldom good practice to design for the extreme maximum or minimum outside conditions. (See the 1993 HVAC Fundamentals volume of the A.S.H.R.A.E. Handbooks). After the proper inside and summer outside temperature conditions for comfort and temperature conditions for process air conditioning have been selected, the next step is to calculate the space cooling load, which is made up of sensible heat and latent heat loads. The sensible heat load consists of (1) transmission through walls, roofs, floors, ceilings, and window glass, (2) solar and sky radiation, (3) heat gains from infiltration of outside air, (4) heat gains from people,

lights, appliances, and power equipment (including the supply-air fan motor), and (5) heat to be removed from materials or products brought in at higher than room temperature. The latent heat load includes loads due to moisture (1) given off from people, appliances, and products and (2) from infiltration of outside air. The space total heat load is the sum of the sensible heat load and latent heat load of the space. The total refrigeration load consists of the total space load plus the sensible and latent heat loads from the outside air introduced at the conditioning unit. The procedure for load calculation in nonresidential buildings should account for thermal mass (storage) effects as well as occupancy and other uses affecting the load. The load can in turn be strongly dependent on the nature of the building utilization; as an example, lightning might be a major component in the thermal load for a highrise office building causing a need for cooling even in winter days. There are various approaches to load calculation, some requiring elaborate computer models. Chapter 26 of the 1993 HVAC Fundamentals volume of the A.S.H.R.A.E. Handbooks presents a step-by-step outline of the current methods in practice for load calculation.

REFRIGERATION INTRODUCTION Refrigeration is a process where heat is transferred from a lower- to a higher-temperature level by doing work on a system. In some systems heat transfer is used to provide the energy to drive the refrigeration cycle. All refrigeration systems are heat pumps (“pumps energy from a lower to a higher potential”). The term heat pump is mostly used to describe refrigeration system applications where heat rejected to the condenser is of primary interest. There are many means to obtain refrigerating effect, but here three will be discussed: mechanical vapor refrigeration cycles, absorption and steam jet cycles due to their significance for industry. Basic Principles Since refrigeration is the practical application of the thermodynamics, comprehending the basic principles of thermodynamics is crucial for full understanding of refrigeration. Section 4 includes a through approach to the theory of thermodynamics. Since our goal is to understand refrigeration processes, cycles are of the crucial interest. The Carnot refrigeration cycle is reversible and consists of adiabatic (isentropic due to reversible character) compression (1-2), isothermal rejection of heat (2-3), adiabatic expansion (3-4) and isothermal addition of heat (4-1). The temperature-entropy diagram is shown in Fig. 11-70. The Carnot cycle is an unattainable ideal which serves as a standard of comparison and it provides a convenient guide to the temperatures that should be maintained to achieve maximum effectiveness. The measure of the system performance is coefficient of performance (COP). For refrigeration applications COP is the ratio of heat removed from the low-temperature level (Qlow) to the energy input (W):

Qlow COPR =
(11-84) W For the heat pump (HP) operation, heat rejected at the high temperature (Qhigh) is the objective, thus: Qhigh Q + W COPHP =
=
= COPR + 1 (11-85) W W For a Carnot cycle (where ∆Q = T∆s), the COP for the refrigeration application becomes (note than T is absolute temperature [K]): Tlow COPR =

(11-86) Thigh − Tlow and for heat pump application: Thigh COPHP =

(11-87) Thigh − Tlow The COP in real refrigeration cycles is always less than for the ideal (Carnot) cycle and there is constant effort to achieve this ideal value. Basic Refrigeration Methods Three basic methods of refrigeration (mentioned above) use similar processes for obtaining refrigeration effect: evaporation in the evaporator, condensation in the condenser where heat is rejected to the environment, and expansion in a flow restrictor. The main difference is in the way compression is being done (Fig. 11-71): using mechanical work (in compressor), thermal energy (for absorption and desorption), or pressure difference (in ejector).

Methods of transforming low-pressure vapor into high-pressure vapor in refrigeration systems (Stoecker, Refrigeration and Air Conditioning).

FIG. 11-71 FIG. 11-70

Temperature-entropy diagram of the Carnot cycle.

REFRIGERATION

FIG. 11-72

11-77

Basic refrigeration systems.

In the next figure (Fig. 11-72) basic refrigeration systems are displayed more detailed. More elaborated approach is presented in the text. MECHANICAL REFRIGERATION (VAPOR-COMPRESSION SYSTEMS) Vapor-Compression Cycles The most widely used refrigeration principle is vapor compression. Isothermal processes are realized through isobaric evaporation and condensation in the tubes. Standard vapor compression refrigeration cycle (counterclockwise Rankine cycle) is marked in Fig. 11-72a) by 1, 2, 3, 4. Work that could be obtained in turbine is small, and iturbine is substituted for an expansion valve. For the reasons of proper compressor function, wet compression is substituted for an compression of dry vapor. Although the T-s diagram is very useful for thermodynamic analysis, the pressure enthalpy diagram is used much more in refrigeration practice due to the fact that both evaporation and condensation are isobaric processes so that heat exchanged is equal to enthalpy difference ∆Q = ∆h. For the ideal, isentropic compression, the work could be also presented as enthalpy difference ∆W = ∆ h. The vapor compression cycle (Rankine) is presented in Fig. 11-73 in p-h coordinates. Figure 11-74 presents actual versus standard vapor-compression cycle. In reality, flow through the condenser and evaporator must be accompanied by pressure drop. There is always some subcooling in the condenser and superheating of the vapor entering the compressorsuction line, both due to continuing process in the heat exchangers and the influence of the environment. Subcooling and superheating are usually desirable to ensure only liquid enters the expansion device. Superheating is recommended as a precaution against droplets of liquid being carried over into the compressor.

There are many ways to increase cycle efficiency (COP). Some of them are better suited to one, but not for the other refrigerant. Sometimes, for the same refrigerant, the impact on COP could be different for various temperatures. One typical example is the use of a liquid-tosuction heat exchanger (Fig. 11-75). The suction vapor coming from the evaporator could be used to

FIG. 11-73

p-h diagram for vapor-compression cycle.

11-78

FIG. 11-74

HEAT-TRANSFER EQUIPMENT

Actual vapor-compression cycle compared with standard cycle.

FIG. 11-75 Refrigeration system with a heat exchanger to subcool the liquid from the condenser.

subcool the liquid from the condenser. Graphic interpretation in T-s diagram for such a process is shown in Fig. 11-76. The result of the use of suction line heat exchanger is to increase the refrigeration effect ∆Q and to increase the work by ∆W. The change in COP is then: Q + ∆Q ∆COP = COP′ − COP =

(11-88) (P + ∆P) − Q/P When dry, or superheated, vapor is used to subcool the liquid, the COP in R12 systems will increase, and decrease the COP in NH3 sys-

tems. For R22 systems it could have both effects, depending on the operating regime. Generally, this measure is advantageous (COP is improved) for fluids with high, specific heat of liquid (less-inclined saturated-liquid line on the p-h diagram), small heat of evaporation hfg, when vapor-specific heat is low (isobars in superheated regions are steep), and when the difference between evaporation and condensation temperature is high. Measures to increase COP should be studied for every refrigerant. Sometimes the purpose of the suction-line heat exchanger is not only to improve the COP, but to ensure that only the vapor reaches the compressor, particularly in the case of a malfunctioning expansion valve. The system shown in Fig. 11-75 is direct expansion where dry or slightly superheated vapor leaves the evaporator. Such systems are predominantly used in small applications because of their simplicity and light weight. For the systems where efficiency is crucial (large industrial systems), recirculating systems (Fig. 11-77) are more appropriate. Ammonia refrigeration plants are almost exclusively built as recirculating systems. The main advantage of recirculating versus direct expansion systems is better utilization of evaporator surface area. The diagram showing influence of quality on the local heat-transfer coefficients is shown in figure 11-90. It is clear that heat-transfer characteristics will be better if the outlet quality is lower than 1. Circulation could be achieved either by pumping (mechanical or gas) or using gravity (thermosyphon effect: density of pure liquid at the evaporator entrance is higher than density of the vapor-liquid mixture leaving the evaporator). The circulation ratio (ratio of actual mass flow rate to the evaporated mass flow rate) is higher than 1 and up to 5. Higher values are not recommended due to a small increase in heat-transfer rate for a significant increase in pumping costs. Multistage Systems When the evaporation and condensing pressure (or temperature) difference is large, it is prudent to separate compression in two stages. The use of multistage systems opens up the opportunity to use flash-gas removal and intercooling as measures to improve performance of the system. One typical two-stage system with two evaporating temperatures and both flash-gas removal and intercooling is shown in figure 11-78. The purpose of the flash-tank intercooler is to: (1) separate vapor created in the expansion process, (2) cool superheated vapor from compressor discharge, and (3) to eventually separate existing droplets at the exit of the mediumtemperature evaporator. The first measure will decrease the size of the low-stage compressor because it will not wastefully compress the portion of the flow which cannot perform the refrigeration and second will decrease the size of the high-stage compressor due to lowering the specific volume of the vapor from the low-stage compressor discharge, positively affecting operating temperatures of the high-stage compressor due to cooling effect. If the refrigerating requirement at a low-evaporating temperature is Ql and at the medium level is Qm, then mass flow rates (m1 and mm respectively) needed are: Ql Ql m1 =
=
(11-89) h1 − h8 h1 − h7 Qm mm =
h3 − h6

Refrigeration system with a heat exchanger to subcool the liquid from the condenser.

FIG. 11-76

FIG. 11-77

Recirculation system.

(11-90)

REFRIGERATION

11-79

FIG. 11-78 Typical two-stage system with two evaporating temperatures, flash-gas removal, and intercooling.

The mass flow rate at the flash-tank inlet mi consists of three components (mi = m1 + msup + mflash): m1 = liquid at pm feeding low temperature evaporator, msup = liquid at pm to evaporate in flash tank to cool superheated discharge, mflash = flashed refrigerant, used to cool remaining liquid. Vapor component is: mflash = xm ∗ mi

(11-91)

(1 − xm) ∗ mi = m1 + msup

(11-92)

and liquid component is: Liquid part of flow to cool superheated compressor discharge is determined by: Ql h2 − h3 h2 − h3 msup =
∗
= m1 ∗
(11-93) h1 − h8 h3 − h7 hfgm Since the quality xm is: h6 − h7 xm =
(11-94) h3 − h7 mass flow rate through condenser and high-stage compressor mh is finally: mh = mm + mi

Cascade System This is a reasonable choice in cases when the evaporating temperature is very low (below −60°C). When condensing pressures are to be in the rational limits, the same refrigerant has a high, specific volume at very low temperatures, requiring a large compressor. The evaporating pressure may be below atmospheric, which could cause moisture and air infiltration into the system if there is a leak. In other words, when the temperature difference between the medium that must be cooled and the environment is too high to be served with one refrigerant, it is wise to use different refrigerants in the high and low stages. Figure 11-80 shows a cascade system schematic diagram. There are basically two independent systems linked via a heat exchanger: the evaporator of the high-stage and the condenser of the low-stage system. EQUIPMENT Compressors These could be classified by one criteria (the way the increase in pressure is obtained) as positive-displacement and

(11-95)

The optimum intermediate pressure for the two-stage refrigeration cycles is determined as the geometric mean between evaporation pressure (pl) and condensing pressure (ph, Fig. 11-79): p pm = (sqrt)
h pl based on equal pressure ratios for low- and high-stage compressors. Optimum interstage pressure is slightly higher than the geometric mean of the suction and the discharge pressures, but, due to very flat optimum of power versus interstage pressure relation geometric mean, it is widely accepted for determining the intermediate pressure. Required pressure of intermediate-level evaporator may dictate interstage pressure other than determined as optimal. Two-stage systems should be seriously considered when the evaporating temperature is below −20°C. Such designs will save on power and reduce compressor discharge temperatures, but will increase initial cost.

 

FIG. 11-79 Pressure enthalpy diagram for typical two-stage system with two evaporating temperatures, flash-gas removal, and intercooling.

11-80

FIG. 11-80

HEAT-TRANSFER EQUIPMENT

Cascade system.

dynamic types as shown in Fig. 11-81 (see Sec. 10 for drawings and mechanical description of the various types of compressors). Positivedisplacement compressors (PDC) are the machines that increase the pressure of the vapor by reducing the volume of the chamber. Typical PDC are reciprocating (in a variety of types) or rotary as screw (with one and two rotors), vane, scroll, and so on. Centrifugal or turbocompressors are machines where the pressure is raised converting some of kinetic energy obtained by a rotating mechanical element which continuously adds angular momentum to a steadily flowing fluid, similar to a fan or pump. Generally, reciprocating compressors dominate in the range up to 300 kW refrigeration capacity. Centrifugal compressors are more accepted for the range over 500 kW, while screw compressors are in between with a tendency to go toward smaller capacities. The vane and the scroll compressors are finding their places primarily in very low capacity range (domestic refrigerators and the air conditioners), although vane compressors could be found in industrial compressors. Frequently, screw compressors operate as boosters, for the base load, while reciprocating compressors accommodate the variation of capacity, in the high stage. The major reason is for such design is the advantageous operation of screw compressors near full load and in design conditions, while reciprocating compressors seem to have better efficiencies at part-load operation than screw. Using other criteria, compressors are classified as open, semihermetic (accessible), or hermetic. Open type is characterized by shaft extension out of compressor where it is coupled to the driving motor. When the electric motor is in the same housing with the compressor mechanism, it could be either hermetic or accessible (semihermetic). Hermetic compressors have welded enclosures, not designed to be repaired, and are generally manufactured for smaller capacities (sel-

dom over 30 kW), while semihermetic or an accessible type is located in the housing which is tightened by screws. Semihermetic compressors have all the advantages of hermetic (no sealing of moving parts, e.g., no refrigerant leakage at the seal shaft, no external motor mounting, no coupling alignment) and could be serviced, but it is more expensive. Compared to other applications, refrigeration capacities in the chemical industry are usually high. That leads to wide usage of either centrifugal, screw, or high-capacity rotary compressors. Most centrifugal and screw compressors use economizers to minimize power and suction volume requirements. Generally, there is far greater use of open-drive type compressors in the chemical plants than in airconditioning, commercial, or food refrigeration. Very frequently, compressor lube oil systems are provided with auxiliary oil pumps, filters, coolers, and other equipment to permit maintenance and repair without shut down. Positive-Displacement Compressors Reciprocating compressors are built in different sizes (up to about one megawatt refrigeration capacity per unit). Modern compressors are high-speed, mostly direct-coupled, single-acting, from one to mostly eight, and occasionally up to sixteen cylinders. Two characteristics of compressors for refrigeration are the most important: refrigerating capacity and power. Typical characteristics are as presented in the Fig. 11-82. Refrigerating capacity Qe is the product of mass flow rate of refrigerant m and refrigerating effect R which is (for isobaric evaporation) R = hevaporator outlet − hevaporator inlet. Power P required for the compression, necessary for the motor selection, is the product of mass flow rate m and work of compression W. The latter is, for the isentropic compression, W = hdischarge − hsuction. Both of these characteristics could be calculated for the ideal (without losses) and for the actual compressor. Ideally, the mass flow rate is equal to the product of the compressor displacement Vi per unit time and the gas density ρ: m = Vi ∗ ρ. The compressor displacement rate is volume swept through by the pistons (product of the cylinder number n, and volume of cylinder V = stroke ∗ d 2 π/4) per second. In reality, the actual compressor delivers less refrigerant. Ratio of the actual flow rate (entering compressor) to the displacement rate is the volumetric efficiency ηva. The volumetric efficiency is

FIG. 11-82 FIG. 11-81

Types of refrigeration compressors.

pressor.

Typical capacity and power-input curves for reciprocating com-

REFRIGERATION less then unity due to: reexpansion of the compressed vapor in clearance volume, pressure drop (through suction and discharge valves, strainers, manifolds, etc.), internal gas leakage (through the clearance between piston rings and cylinder walls, etc.), valve inefficiencies, and due to expansion of the vapor in the suction cycle caused by the heat exchanged (hot cylinder walls, oil, motor, etc.). Similar to volumetric efficiency, isentropic (adiabatic) efficiency ηa is the ratio of the work required for isentropic compression of the gas to work input to the compressor shaft. The adiabatic efficiency is less than one mainly due to pressure drop through the valve ports and other restricted passages and the heating of the gas during compression. Figure 11-83 presents the compression on a pressure-volume diagram for an ideal compressor with clearance volume (thin lines) and actual (thick lines). Compression in an ideal compressor without clearance is extended using dashed lines to the points Id (end of discharge), line Id − Is (suction), and Is (beginning of suction). The area surrounded by the lines of compression, discharge, reexpansion and intake presents the work needed for compression. Actual compressor only appears to demand less work for compression due to smaller area in the p-V diagram. Mass flow rate for an ideal compressor is higher, which cannot be seen in the diagram. In reality, an actual compressor will have diabatic compression and reexpansion and higher-discharge and lower-suction pressures due to pressure drops in valves and lines. The slight increase in the pressure at the beginning of the discharge and suction is due to forces needed to initially open valves. When the suction pressure is lowered, the influence of the clearance will increase, causing in the extreme cases the entire volume to be used for reexpansion, which drives the volumetric efficiency to zero. There are various options for capacity control of reciprocating refrigeration compressors: 1. Opening the suction valves by some external force (oil from the lubricating system, discharge gas, electromagnets . . . ). 2. Gas bypassing—returning discharge gas to suction (within the compressor or outside the compressor).

11-81

3. Controlling suction pressure by throttling in the suction line. 4. Controlling discharge pressure. 5. Adding reexpansion volume. 6. Changing the stroke. 7. Changing the compressor speed. The first method is used most frequently. The next preference is for the last method, mostly used in small compressors due to problems with speed control of electrical motors. Other means of capacity control are very seldom utilized due to thermodynamic inefficiencies and design difficulties. Energy losses in a compressor, when capacity regulation is provided by lifting the suction valves, are due to friction of gas flowing in and out the unloaded cylinder. This is shown in Fig. 11-84 where the comparison is made for ideal partial load operation, reciprocating, and screw compressors. Rotary compressors are also PDC types, but where refrigerant flow rotates during compression. Unlike the reciprocating type, rotary compressors have a built-in volume ratio which is defined as volume in cavity when the suction port is closed (Vs = m ∗ vs) over the volume in the cavity when the discharge port is uncovered (Vd = m ∗ vd). Builtin volume ratio determines for a given refrigerant and conditions the pressure ratio which is: vs n pd (11-96)
=
ps vd where n represents the politropic exponent of compression. In other words, in a reciprocating compressor the discharge valve opens when the pressure in the cylinder is slightly higher than the pressure in the high-pressure side of the system, while in rotary compressors the discharge pressure will be established only by inlet conditions and built-in volume ratio regardless of the system discharge pressure. Very seldom are the discharge and system (condensing) pressure equal, causing the situation shown in Fig. 11-85. When condensing pressure (p) is lower than discharge (p2), shown as case (a), “over compression” will cause energy losses presented by the horn on the diagram. If the condensing pressure is higher, in the moment when the discharge port uncovers there will be flow of refrigerant

 

FIG. 11-83 Pressure-volume diagram of an ideal (thin line) and actual (thick line) reciprocating compressor.

11-82

HEAT-TRANSFER EQUIPMENT

Typical power-refrigeration capacity data for different types of compressors during partial, unloaded operation.

FIG. 11-84

backwards into the compressor, causing losses shown in Fig. 11-85b and the last stage will be only discharge without compression. The case when the compressor discharge pressure is equal to the condensing pressure is shown in the Fig. 11-85c. Double helical rotary (twin) screw compressors consist of two mating helically grooved rotors (male and female) with asymmetric profile, in a housing formed by two overlapped cylinders, with inlet and outlet ports. Developed relatively recently (in 1930s) the first twin screw compressors were used for air, and later (1950s) became popular for refrigeration. Screw compressors have some advantages over reciprocating compressors (fewer moving parts and more compact) but also some drawbacks (lower efficiency at off-design conditions, as discussed above, higher manufacturing cost due to complicated screw geometry, large separators and coolers for oil which is important as a sealant). Figure 11-86 shows the oil circuit of a screw compressor. Oil cooling could be provided by water, glycol, or refrigerant either in the heat-exchangerutilizing-thermosyphon effect or the using-direct-expansion concept. In order to overcome some inherent disadvantages, screw compressors have been initially used predominantly as booster (low-stage) compressors, and following development in capacity control and decreasing prices, they are widely used for high-stage applications. There are several methods for capacity regulation of screw compressors. One is variable speed drive, but a more economical first-cost concept is a slide valve that is used in some form by practically all

FIG. 11-85

ence.

screw compressors. The slide is located in the compressor casting below the rotors, allowing internal gas recirculation without compression. Slide valve is operated by a piston located in a hydraulic cylinder and actuated by high-pressure oil from both sides. When the compressor is started, the slide valve is fully open and the compressor is unloaded. To increase capacity, a solenoid valve on the hydraulic line opens, moving the piston in the direction of increasing capacity. In order to increase partload efficiency, the slide valve is designed to consist of two parts, one traditional slide valve for capacity regulation and other for built-in volume adjustment. Single screw compressors are a newer design (early 1960s) compared to twin screw compressors, and are manufactured in the range of capacity from 100 kW to 4 MW. The compressor screw is cylindrical with helical grooves mated with two star wheels (gaterotors) rotating in opposite direction from one another. Each tooth acts as the piston in the rotating “cylinder” formed by screw flute and cylindrical main-rotor casting. As compression occurs concurrently in both halves of the compressor, radial forces are oppositely directed, resulting in negligible netradial loads on the rotor bearings (unlike twin screw compressors), but there are some loads on the star wheel shafts. Scroll compressors are currently used in relatively small-sized installations, predominantly for residential air-conditioning (up to 50 kW). They are recognized for low-noise operation. Two scrolls (free-standing, involute spirals bounded on one side by a flat plate) facing each other form a closed volume while one moves in a controlled orbit around a fixed point on the other, fixed scroll. The suction gas which enters from the periphery is trapped by the scrolls. The closed volumes move radially inward until the discharge port is reached, when vapor is pressed out. The orbiting scroll is driven by a short-throw crank mechanism. Similar to screw compressors, internal leakage should be kept low, and is occurring in gaps between cylindrical surfaces and between the tips of the involute and the opposing scroll base plate. Similar to the screw compressor, the scroll compressor is a constantvolume-ratio machine. Losses occur when operating conditions of the compressor do not match the built-in volume ratio (see Fig. 11-85). Vane compressors are used in small, hermetic units, but sometimes as booster compressors in industrial applications. Two basic types are fixed (roller) or single-vane type and the rotating or multiple-vane type. In the single-vane type the rotor (called roller) is eccentrically placed in the cylinder so these two are always in contact. The contact line make the first separation between the suction and discharge chambers while the vane (spring-loaded divider) makes the second. In the multiple-vane compressors the rotor and the cylinder are not in the contact. The rotor has two or more sliding vanes which are held against the cylinder by centrifugal force. In the vane rotary compressors, no suction valves are needed. Since the gas enters the compressor continuously, gas pulsations are at minimum. Vane compressors have a high volumetric efficiency because of the small clearance volume and consequent low reexpansion losses. Rotary vane compressors have a low weight-to-displacement ratio, which makes them suitable for transport applications.

Matching compressor built-in pressure ratio with actual pressure differ-

REFRIGERATION

FIG. 11-87 FIG. 11-86

Oil cooling in a screw compressor.

Centrifugal Compressors These are sometimes called turbocompressors and mostly serve refrigeration systems in the capacity range 200 to 10,000 kW. The main component is a spinning impeller wheel, backwards curved, which imparts energy to the gas being compressed. Some of the kinetic energy converts into pressure in a volute. Refrigerating centrifugal compressors are predominantly multistage, compared to other turbocompressors, that produce high-pressure ratios. The torque T (Nm) the impeller ideally imparts to the gas is: T = m (utang.out rout − utang.in rin) where:

(11-97)

m (kg/s) = rout (m) = rin (m) = utang.out (m/s) =

mass flow rate radius of exit of impeller radius of exit of impeller tangential velocity of refrigerant leaving impeller utang.in (m/s) = tangential velocity of refrigerant entering impeller

When refrigerant enters essentially radially, u tang.in = 0 and torque becomes: T = m ∗ u tang.out ∗ rout

(11-98)

The power P (W), is the product of torque and rotative speed ω [l/s] so is P = T ∗ ω = m ∗ u tang.out ∗ rout ∗ ω

(11-99)

which for utang.out = rout ∗ ω becomes 2 P = m ∗ u tang.out

(11-100)

P = m ∗ ∆h

(11-101)

or for isentropic compression The performance of a centrifugal compressor (discharge to suctionpressure ratio vs. the flow rate) for different speeds is shown in Fig. 11-87. Lines of constant efficiencies show the maximum efficiency. Unstable operation sequence, called surging, occurs when compressors fails to operate in the range left of the surge envelope. It is characterized by noise and wide fluctuations of load on the compressor and the motor. The period of the cycle is usually 2 to 5 s, depending upon the size of the installation. The capacity could be controlled by: (1) adjusting the prerotation vanes at the impeller inlet, (2) varying the speed, (3) varying the condenser pressure, and (4) bypassing discharge gas. The first two methods are predominantly used. Condensers These are heat exchangers that convert refrigerant

11-83

Performance of the centrifugal compressor.

vapor to a liquid. Heat is transferred in three main phases: (1) desuperheating, (2) condensing, and (3) subcooling. In reality condensation occurs even in the superheated region and subcooling occurs in the condensation region. Three main types of refrigeration condensers are: air cooled, water cooled, and evaporative. Air-cooled condensers are used mostly in air-conditioning and for smaller-refrigeration capacities. The main advantage is availability of cooling medium (air) but heat-transfer rates for the air side are far below values when water is used as a cooling medium. Condensation always occurs inside tubes, while the air side uses extended surface (fins). The most common types of water-cooled refrigerant condensers are: (1) shell-and-tube, (2) shell-and-coil, (3) tube-in-tube, and (4) brazedplate. Shell-and-tube condensers are built up to 30 MW capacity. Cooling water flows through the tubes in a single or multipass circuit. Fixed-tube sheet and straight-tube construction are common. Horizontal layout is typical, but sometimes vertical is used. Heat-transfer coefficients for the vertical types are lower due to poor condensate drainage, but less water of lower purity can be utilized. Condensation always occurs on the tubes, and often the lower portion of the shell is used as a receiver. In shell-and-coil condensers water circulates through one or more continuous or assembled coils contained within the shell while refrigerant condenses outside. The tubes cannot be mechanically cleaned nor replaced. Tube-in-tube condensers could be found in versions where condensation occurs either in the inner tube or in the annulus. Condensing coefficients are more difficult to predict, especially in the cases where tubes are formed in spiral. Mechanical cleaning is more complicated, sometimes impossible, and tubes are not replaceable. Brazed-plate condensers are constructed of plates brazed together to make up an assembly of separate channels. The plates are typically stainless steel, wave-style corrugated, enabling high heat-transfer rates. Performance calculation is difficult, with very few correlations available. The main advantage is the highest performance/volume (mass) ratio and the lowest refrigerant charge. The last mentioned advantage seems to be the most important feature for many applications where minimization of charge inventory is crucial. Evaporative condensers (Fig. 11-88) are widely used due to lower condensing temperatures than in the air-cooled condensers and also lower than the water-cooled condenser combined with the cooling tower. Water demands are far lower than for water-cooled condensers. The chemical industry uses shell-and-tube condensers widely, although the use of air-cooled condensing equipment and evaporative condensers is on the increase. Generally, cooling water is of a lower quality then normal, having also higher mud and silt content. Sometimes even replaceable copper tubes in shell-and-tube heat exchangers are required. It is advisable to use cupronickel instead of copper tubes (when water is high in chlorides) and to use conservative water side velocities (less then 2 m/s for copper tubes).

11-84

HEAT-TRANSFER EQUIPMENT

FIG. 11-89 Typical values of the heat-rejection ratio of the heat rejected at the condenser to the refrigerating capacity. FIG. 11-88

Evaporative condenser with desuperheating coil.

Evaporative condensers are used quite extensively. In most cases commercial evaporative condensers are not totally suitable for chemical plants due to the hostile atmosphere which usually abounds in vapor and dusts which can cause either chemical (corrosion) or mechanical problems (plugging of spray nozzles). Air-cooled condensers are similar to evaporative in that the service dictates either the use of more expensive alloys in the tube construction or conventional materials of greater wall thickness. Heat rejected in the condenser QCd consists of heat absorbed in the evaporator QEevap and energy W supplied by the compressor: QCd = QEevap + W

(11-102)

For the actual systems, compressor work will be higher than for ideal for the isentropic efficiency and other losses. In the case of hermetic or accessible compressors where an electrical motor is cooled by the refrigerant, condenser capacity should be: QCd = QEevap + PEM

(11-103)

It is common that compressor manufacturers provide data for the ratio of the heat rejected at the condenser to the refrigeration capacity as shown in Fig. 11-89. The solid line represents data for the open compressors while the dotted line represents the hermetic and accessible compresors. The difference between solid and dotted line is due to all losses (mechanical and electrical in the electrical motor). Condenser design is based on the value: QCd = QEevap ∗ heat-rejection ratio

overfeed). In dry-expansion the evaporator’s outlet is dry or slightly superheated vapor. This limits the liquid feed to the amount that can be completely vaporized by the time it reaches the end of evaporator. In the liquid overfeed evaporator, the amount of liquid refrigerant circulating exceeds the amount evaporated by the circulation number. Decision on the type of the system to be used is one of the first in the design process. Direct-expansion evaporator is generally applied in smaller systems where compact design and the low first costs are crucial. Control of the refrigerant mass flow is then obtained by either a thermoexpansion valve or a capillary tube. Figure 11-90 suggests that the evaporator surface is the most effective in the regions with quality which is neither low nor high. In dry-expansion evaporators, inlet qualities are 10–20 percent, but when controlled by the thermoexpansion valve, vapor at the outlet is not only dry, but even superheated. In recirculating systems saturated liquid (x = 0) is entering the evaporator. Either the pump or gravity will deliver more refrigerant liquid then will evaporate, so outlet quality could be lower than one. The ratio of refrigerant flow rate supplied to the evaporator overflow rate of refrigerant vaporized is the circulation ratio, n. When n increases, the coefficient of heat transfer will increase due to the wetted outlet of the evaporator and the increased velocity at the inlet (Fig. 11-91). In the range of n = 2 to 4, the overall U value for air cooler increases roughly by 20 to 30 percent compared to the direct-expansion case. Circulation rates higher than four are not efficient.

(11-104)

Thermal and mechanical design of heat exchangers (condensers and evaporators) is presented earlier in this section. Evaporators These are heat exchangers where refrigerant is evaporated while cooling the product, fluid, or body. Refrigerant could be in direct contact with the body that is being cooled, or some other medium could be used as secondary fluid. Mostly air, water, or antifreeze are fluids that are cooled. Design is strongly influenced by the application. Evaporators for air cooling will have in-tube evaporation of the refrigerant, while liquid chillers could have refrigerant evaporation inside or outside the tube. The heat-transfer coefficient for evaporation inside the tube (vs. length or quality) is shown in the Fig. 11-90. Fundamentals of the heat transfer in evaporators, as well as design aspects, are presented in Sec. 11. We will point out only some specific aspects of refrigeration applications. Refrigeration evaporators could be classified according to the method of feed as either direct (dry) expansion or flooded (liquid

FIG. 11-90

Heat-transfer coefficient for boiling inside the tube.

REFRIGERATION

Effect of circulation ratio on the overall heat-transfer coefficient of an air-cooling coil.

FIG. 11-91

The price for an increase in heat-transfer characteristics is a more complex system with more auxiliary equipment: low-pressure receivers, refrigerant pumps, valves, and controls. Liquid refrigerant is predominantly pumped by mechanical pumps, however, sometimes gas at condensing pressure is used for pumping, in the variety of concepts. The important characteristics of the refrigeration evaporators is the presence of the oil. The system is contaminated with oil in the compressor, in spite of reasonably efficient oil separators. Some systems will recirculate oil, when miscible with refrigerant, returning it to the compressor crankcase. This is found mostly in the systems using halocarbon refrigerants. Although oils that are miscible with ammonia exist, immiscibles are predominantly used. This inhibits the ammonia systems from recirculating the oil. In systems with oil recirculation when halocarbons are used special consideration should be given to proper sizing and layout of the pipes. Proper pipeline configuration, slopes, and velocities (to ensure oil circulation under all operating loads) are essential for good system operation. When refrigerant is lighter than the oil in systems with no oil recirculation, oil will be at the bottom of every volume with a top outlet. Then oil must be drained periodically to avoid decreasing the performance of the equipment. It is essential for proper design to have the data for refrigerant-oil miscibility under all operating conditions. Some refrigerant-oil combinations will always be miscible, some always immiscible, but some will have both characteristics, depending on temperatures and pressures applied. Defrosting is the important issue for evaporators which are cooling air below freezing. Defrosting is done periodically, actuated predominantly by time relays, but other frost indicators are used (temperature, visual, or pressure-drop sensors). Defrost technique is determined mostly by fluids available and tolerable complexity of the system. Defrosting is done by the following mechanisms when the system is off: • Hot (or cool) refrigerant gas (the predominant method in industrial applications) • Water (defrosting from the outside, unlike hot gas defrost) • Air (only when room temperature is above freezing) • Electricity (for small systems where hot-gas defrost will be to complex and water is not available) • Combinations of above. System Analysis Design calculations are made on the basis of the close to the highest refrigeration load, however the system operates at the design conditions very seldom. The purpose of regulating devices is to adjust the system performance to cooling demands by decreasing the effect or performance of some component. Refrigeration systems have inherent self-regulating control which the engineer could rely on to a certain extent. When the refrigeration load starts to

11-85

decrease, less refrigerant will evaporate. This causes a drop in evaporation temperature (as long as compressor capacity is unchanged) due to the imbalance in vapor being taken by the compressor and produced by evaporation in evaporator. With a drop in evaporation pressure, the compressor capacity will decrease due to: (1) lower vapor density (lower mass flow for the same volumetric flow rate) and (2) decrease in volumetric efficiency. On the other hand, when the evaporation temperature drops, for the unchanged temperature of the medium being cooled, the evaporator capacity will increase due to increase in the mean-temperature difference between refrigerant and cooled medium, causing a positive effect (increase) on the cooling load. With a decrease in the evaporation temperature the heatrejection factor will increase causing an increase of heat rejected to the condenser, but refrigerant mass flow rate will decrease due to compressor characteristics. These will have an opposite effect on condenser load. Even a simplified analysis demonstrates the necessity for better understanding of system performance under different operating conditions. Two methods could be used for more accurate analysis. The traditional method of refrigeration-system analysis is through determination of balance points, while in recent years, system analysis is performed by system simulation or mathematical modeling, using mathematical (equation solving) rather than graphical (intersection of two curves) procedures. Systems with a small number of components such as the vapor-compression refrigeration system could be analyzed both ways. Graphical presentation, better suited for understanding trends is not appropriate for more complex systems, more detailed component description, and frequent change of parameters. There is a variety of different mathematical models tailored to fit specific systems, refrigerants, resources available, demands, and complexity. Although limited in its applications, graphical representation is valuable as the starting tool and for clear understanding of the system performance. Refrigeration capacity qe and power P curves for the reciprocating compressor are shown in Fig. 11-92. They are functions of temperatures of evaporation and condensation: and where qe (kW) = P (kW) = tevap (°C) = tcd (°C) =

qe = qe(tevap, tcd)

(11-105a)

P = P(tevap, tcd)

(11-105b)

refrigerating capacity power required by the compressor evaporating temperature condensing temperature.

Refrigerating capacity and power requirement for the reciprocating compressor.

FIG. 11-92

11-86

FIG. 11-93

HEAT-TRANSFER EQUIPMENT

FIG. 11-94

Heat-rejection ratio.

FIG. 11-95

Condenser evaporating effect.

Condenser performance.

A more detailed description of compressor performance is shown in the section on the refrigeration compressors. Condenser performance, shown in figure 11-93, could be simplified as: qcd = F(tcd − tamb) where

(11-105c)

qcd (kW) = capacity of condenser; F (kW/°C) = capacity of condenser per unit inlet temperature difference (F = U ∗ A); tamb (°C) = ambient temperature (or temperature of condenser cooling medium).

In this analysis F will be constant but it could be described more accurately as a function of parameters influencing heat transfer in the condenser (temperature, pressure, flow rate, fluid thermodynamical, and thermophysical characteristics . . . ). Condenser performance should be expressed as “evaporating effect” to enable matching with compressor and evaporator performance. Condenser “evaporating effect” is the refrigeration capacity of an evaporator served by a particular condenser. It is the function of the cycle, evaporating temperature, and the compressor. The “evaporating effect” could be calculated from the heat-rejection ratio qCd /qe: qCd qe =


(11-105d) heat-rejection ratio The heat-rejection rate is presented in Fig. 11-94 (or Fig. 11-89). Finally, the evaporating effect of the condenser is shown in Fig. 11-95. The performance of the condensing-unit (compressor and condenser) subsystem could be developed as shown in Fig. 11-96 by superimposing two graphs, one for compressor performance and the other for condenser evaporating effect. Evaporator performance could be simplified as: qe = Fevap(tamb − tevap) where

(11-106)

qe(kW) = evaporator capacity Fe (kW/°C) = evaporator capacity per unit inlet temperature difference tamb (°C) = ambient temperature (or temperature of cooled body or fluid).

The diagram of the evaporator performance is shown in the Fig. 11-97. The character of the curvature of the lines (variable heattransfer rate) indicates that the evaporator is cooling air. Influences of the flow rate of cooled fluid are also shown in this diagram; i.e., higher flow rate will increase heat transfer. The same effect could be shown

Balance points of compressor and condenser determines performance of condensing unit for fixed temperature of condenser cooling fluid (flow rate and heat-transfer coefficient are constant).

FIG. 11-96

REFRIGERATION

FIG. 11-97

Refrigerating capacity of evaporator.

in the condenser-performance curve. It is omitted only for the reasons of simplicity. The performance of the complete system could be predicted by superimposing the diagrams for the condensing unit and the evaporator, as shown in Fig. 11-98. Point 1 reveals a balance for constant flows and inlet temperatures of chilled fluid and fluid for condenser cooling. When this point is transferred in the diagram for the condensing unit in the Figs. 11-95 or 11-96, the condensing temperature could be determined. When the temperature of entering fluid in the evaporator tamb1 is lowered to tamb2 the new operating conditions will be determined by the state at point 2. Evaporation temperature drops from tevap1 to tevap2. If the evaporation temperature should be unchanged, the same reduction of inlet temperature could be achieved by reducing the capacity of the condensing unit from Cp to Cp*. The new operating point 3 shows reduction in capacity for ∆ due to the reduction in the compressor or the condenser capacity.

Performance of complete refrigeration system (1), when there is reduction in heat load (2), and when for the same ambient (or inlet in evaporator) evaporation temperature is maintained constant by reducing capacity of compressor/condenser part (3).

FIG. 11-98

11-87

Mathematical modeling is essentially the same process, but the description of the component performance is generally much more complex and detailed. This approach enables a user to vary more parameters easier, look into various possibilities for intervention, and predict the response of the system on different influences. Equationsolving does not necessarily have to be done by successive substitution or iteration as this procedure could suggest. System, Equipment, and Refrigerant Selection There is no universal rule which can be used to decide which system, equipment type, or refrigerant is the most appropriate for a given application. A number of variables influence the final-design decision: • Refrigeration load • Type of installation • Temperature level of medium to be cooled • Condensing media characteristics: type (water, air, . . . ), temperature level, available quantities • Energy source for driving the refrigeration unit (electricity, natural gas, steam, waste heat) • Location and space available (urban areas, sensitive equipment around, limited space . . . ) • Funds available (i.e. initial vs. run-cost ratio) • Safety requirements (explosive environment, aggressive fluids, . . . ) • Other demands (compatibility with existing systems, type of load, compactness, level of automatization, operating life, possibility to use process fluid as refrigerant) Generally, vapor compression systems are considered first. They can be used for almost every task. Whenever it is possible, prefabricated elements or complete units are recommended. Reciprocating compressors are widely used for lower rates, more uneven heat loads (when frequent and wider range of capacity reduction is required). They ask for more space and have higher maintenance costs then centrifugal compressors, but are often the most economical in first costs. Centrifugal compressors are considered for huge capacities, when the evaporating temperature is not too low. Screw compressors are considered first when space in the machine room is limited, when system is operating long hours, and when periods between service should be longer. Direct-expansions are more appropriate for smaller systems which should be compact, and where there are just one or few evaporators. Overfeed (recirculation) systems should be considered for all applications where first cost for additional equipment (surge drums, lowpressure receivers, refrigerant pumps, and accessories) is lower than the savings for the evaporator surface. Choice of refrigerant is complex and not straightforward. For industrial applications, advantages of ammonia (thermodynamical and economical) overcome drawbacks which are mostly related to possible low-toxic and panics created by accidental leaks when used in urban areas. Halocarbons have many advantages (not toxic, not explosive, odorless . . . ), but environmental issues and slightly inferior thermodynamical and thermophysical properties compared to ammonia or hydrocarbons as well as rising prices are giving the chance to other options. When this text was written the ozone-depletion issue was not resolved, R22 was still used but facing phaseout, and R134a was considered to be the best alternative for CFCs and HCFCs, having similar characteristics to the already banned R12. Very often, fluid to be cooled is used as a refrigerant in the chemical industry. Use of secondary refrigerants in combination with the ammonia centralrefrigeration unit is becoming a viable alternative in many applications. Absorption systems will be considered when there is low-cost lowpressure steam or waste heat available and evaporation temperature and refrigeration load are relatively high. Typical application range is for water chilling at 7–10°C, and capacities from 300 kW to 5 MW in a single unit. The main drawback is the difficulty in maintaining a tight system with the highly corrosive lithium bromide, and an operating pressure in the evaporator and the absorber below atmospheric. Ejector (steam-jet) refrigeration systems are used for similar applications, when chilled water-outlet temperature is relatively high, when relatively cool condensing water and cheap steam at 7 bar are available, and for similar high duties (0.3–5 MW). Even though these systems usually have low first and maintenance costs, there are not many steam-jet systems running.

11-88

HEAT-TRANSFER EQUIPMENT

OTHER REFRIGERATION SYSTEMS APPLIED IN THE INDUSTRY Absorption Refrigeration Systems Two main absorption systems are used in industrial application: lithium bromide-water and ammonia-water. Lithium bromide-water systems are limited to evaporation temperatures above freezing because water is used as the refrigerant, while the refrigerant in an ammonia-water system is ammonia and consequently it can be applied for the lower-temperature requirements. Single-effect indirect-fired lithium bromide cycle is shown in Fig. 11-99. The machine consists of five major components: Evaporator is the heat exchanger where refrigerant (water) evaporates (being sprayed over the tubes) due to low pressure in the vessel. Evaporation chills water flow inside the tubes that bring heat from the external system to be cooled. Absorber is a component where strong absorber solution is used to absorb the water vapor flashed in the evaporator. A solution pump sprays the lithium bromide over the absorber tube section. Cool water is passing through the tubes taking refrigeration load, heat of dilution, heat to cool condensed water, and sensible heat for solution cooling. Heat exchanger is used to improve efficiency of the cycle, reducing consumption of steam and condenser water. Generator is a component where heat brought to a system in a tube section is used to restore the solution concentration by boiling off the water vapor absorbed in the absorber. Condenser is an element where water vapor, boiled in the generator, is condensed, preparing pure water (refrigerant) for discharge to an evaporator. Heat supplied to the generator is boiling weak (dilute) absorbent solution on the outside of the tubes. Evaporated water is condensed on the outside of the condenser tubes. Water utilized to cool the condenser is usually cooled in the cooling tower. Both condenser and generator are located in the same vessel, being at the absolute pressure of about 6 kPa. The water condensate passes through a liquid trap and enters the evaporator. Refrigerant (water) boils on the evaporator tubes and cools the water flow that brings the refrigeration load.

FIG. 11-99

Refrigerant that is not evaporated flows to the recirculation pump to be sprayed over the evaporator tubes. Solution with high water concentration that enters the generator increases in concentration as water evaporates. The resulting strong, absorbent solution (solution with low water concentration) leaves the generator on its way to the heat exchanger. There the stream of high water concentration that flows to the generator cools the stream of solution with low water concentration that flows to the second vessel. The solution with low water concentration is distributed over the absorber tubes. Absorber and evaporator are located in the same vessel, so the refrigerant evaporated on the evaporator tubes is readily absorbed into the absorbent solution. The pressure in the second vessel during the operation is 7 kPa (absolute). Heat of absorption and dilution are removed by cooling water (usually from the cooling tower). The resulting solution with high water concentration is pumped through the heat exchanger to the generator, completing the cycle. Heat exchanger increases the efficiency of the system by preheating, that is, reducing the amount of heat that must be added to the high water solution before it begins to evaporate in the generator. The absorption machine operation is analyzed by the use of a lithium bromide-water equilibrium diagram, as shown in Fig. 11-100. Vapor pressure is plotted against the mass concentration of lithium bromide in the solution. The corresponding saturation temperature for a given vapor pressure is shown on the left-hand side of the diagram. The line in the lower right corner of the diagram is the crystallization line. It indicates the point at which the solution will begin to change from liquid to solid, and this is the limit of the cycle. If the solution becomes overconcentrated, the absorption cycle will be interrupted owing to solidification, and capacity will not be restored until the unit is desolidified. This normally requires the addition of heat to the outside of the solution heat exchanger and the solution pump. The diagram in Fig. 11-101 presents enthalpy data for LiBr-water solutions. It is needed for the thermal calculation of the cycle. Enthalpies for water and water vapor can be determined from the table of properties of water. The data in Fig. 11-101 are applicable to saturated or subcooled solutions and are based on a zero enthalpy of liquid water at 0°C and a zero enthalpy of solid LiBr at 25°C. Since

Two-shell lithium bromide-water cycle chiller.

REFRIGERATION

FIG. 11-100

11-89

Temperature-pressure-concentration diagram of saturated LiBr-water solutions (W. F. Stoecker and J. W. Jones: Refrigeration and Air-

Conditioning)

the zero enthalpy for the water in the solution is the same as that in conventional tables of properties of water, the water property tables can be used in conjunction with diagram in Fig. 11-100. Coefficient of performance of the absorption cycle is defined on the same principle as for the mechanical refrigeration: refrigeration rate useful effect COPabs =

=


heat input heat input at generator but it should be noted that here denominator for the COPabs is heat while for the mechanical refrigeration cycle it is work. Since these two forms of energy are not equal, COPabs is not as low (0.6–0.8) as it appears compared to COP for mechanical system (2.5–3.5). The double-effect absorption unit is shown in Fig. 11-102. All major components and operation of the double-effect absorption machine is similar to that for the single-effect machine. The primary generator, located in the vessel 1, is using an external heat source to evaporate water from dilute-absorbent (high water concentration) solution. Water vapor readily flows to the generator II where it is condensed on the tubes. The absorbent (LiBr) intermediate solution from generator I will pass through the heat exchanger on the way to the generator II where it is heated by the condensing water vapor. The throttling valve reduces pressure from vessel 1 (about 103 kPa absolute) to that of vessel 2. Following the reduction of pressure some water in the solution flashes to vapor, which is liquefied at the condenser. In the high temperature heat exchanger intermediate solution heats the weak (high water concentration) solution stream coming from the low temperature heat exchanger. In the low temperature heat exchanger strong solution is being cooled before entering the absorber. The absorber is on the same pressure as the evaporator. The double-effect absorption units achieve higher COPs than the single stage. The ammonia-water absorption system was extensively used until the fifties when the LiBr-water combination became popular. Figure 11-103 shows a simplified ammonia-water absorption cycle. The refrigerant is ammonia, and the absorbent is dilute aqueous solution of ammonia. Ammonia-water systems differ from water-lithium bromide equipment to accommodate major differences: Water (here absorbent) is also volatile, so the regeneration of weak water solution to strong water solution is a fractional distillation. Different refrigerant (ammonia) causes different, much higher pressures: about 1100– 2100 kPa absolute in condenser.

Ammonia vapor from the evaporator and the weak water solution from the generator are producing strong water solution in the absorber. Strong water solution is then separated in the rectifier producing (1) ammonia with some water vapor content and (2) very strong water solution at the bottom, in the generator. Heat in the generator vaporizes ammonia and the weak solution returns to absorber. On its way to the absorber the weak solution stream passes through the heat exchanger, heating strong solution from the absorber on the way to the rectifier. The other stream, mostly ammonia vapor but with some water-vapor content flows to the condenser. To remove water as much as possible, the vapor from the rectifier passes through the analyzer where it is additionally cooled. The remaining water escaped from the analyzer pass as liquid through the condenser and the evaporator to the absorber. Ammonia-water units can be arranged for single-stage or cascaded two-stage operation. The advantage of two-staging is it creates the possibility of utilizing only part of the heat on the higher and the rest on the lower temperature level but the price is increase in first cost and heat required. Ammonia-water and lithium bromide-water systems operate under comparable COP. The ammonia-water system is capable achieving evaporating temperatures below 0°C because the refrigerant is ammonia. Water as the refrigerant limits evaporating temperatures to no lower than freezing, better to 3°C. Advantage of the lithium bromide-water system is that it requires less equipment and operates at lower pressures. But this is also a drawback, because pressures are below atmospheric, causing air infiltration in the system which must be purged periodically. Due to corrosion problems, special inhibitors must be used in the lithium-bromide-water system. The infiltration of air in the ammonia-water system is also possible, but when evaporating temperature is below −33°C. This can result in formation of corrosive ammonium carbonate. Further readings: ASHRAE Handbook 1994 Refrigeration Systems and Applications; Bogart, M., 1981: Ammonia Absorption Refrigeration in Industrial Processes, Gulf Publishing Co. Houston; Stoecker W. F. and Jones J. W. 1982: Refrigeration and AirConditioning, McGraw-Hill Book Company, New York. Steam-Jet (Ejector) Systems These systems substitute an ejector for a mechanical compressor in a vapor compression system. Since refigerant is water, maintaining temperatures lower than the environment requires that the pressure of water in the evaporator must be

11-90

HEAT-TRANSFER EQUIPMENT

FIG. 11-101

Enthalpy of LiBr-water solutions (W. F. Stoecker and J. W. Jones: Refrigeration and Air-

Conditioning).

below atmospheric. A typical arrangement for the steam-jet refrigeration cycle is shown in Fig. 11-104. Main Components The main components of steam-jet refrigeration systems are: 1. Primary steam ejector. A kinetic device that utilizes the momentum of a high-velocity jet to entertain and accelerate a slowermoving medium into which it is directed. High-pressure steam is delivered to the nozzle of the ejector. The steam expands while flowing through the nozzle where the velocity increases rapidly. The velocity of steam leaving the nozzle is around 1200 m/s. Because of this high velocity, flash vapor from the tank is continually aspired into the moving steam. The mixture of steam and flash vapor then enters the diffuser section where the velocity is gradually reduced because of increasing cross-sectional area. The energy of the high-velocity steam compresses the vapor during its passage through the diffuser, raising it’s temperature above the temperature of the condenser cooling water. 2. Condenser. The component of the system where the vapor condenses and where the heat is rejected. The rate of heat rejected is: Qcond = (Ws + Ww) hfg

(11-107)

where: Qcond = Ws = Ww = hfg =

heat rejection (kW) primary booster steam rate (kg/s) flash vapor rate (kg/s) latent heat (kJ/kg)

The condenser design, surface area, and condenser cooling water quantity should be based on the highest cooling water temperature likely to be encountered. If the inlet cooling water temperature becomes hotter then the design, the primary booster (ejector) may cease functioning because of the increase in condenser pressure. Two types of condensers could be used: the surface condenser (shown in Fig. 11-104) and the barometric or jet condenser (Fig. 11-105). The surface condenser is of shell-and-tube design with water flowing through the tubes and steam condensed on the outside surface. In the jet condenser, condensing water and the steam being condensed are mixed directly, and no tubes are provided. The jet condenser can be barometric or a low-level type. The barometric condenser requires a height of ∼10 m above the level of the water in the hot well. A tailpipe of this length is needed so that condenser water and condensate can drain by gravity. In the low-level jet type, the tailpipe is eliminated, and it becomes necessary to remove the con-

REFRIGERATION

FIG. 11-102

Double-effect absorption unit.

FIG. 11-103

Simplified ammonia-water absorption cycle.

11-91

11-92

HEAT-TRANSFER EQUIPMENT

FIG. 11-104

Ejector (steam-jet) refrigeration cycle (with surface-type condenser).

denser water and condensate by pumping from the condenser to the hot well. The main advantages of the jet condenser are low maintenance with the absence of tubes and the fact that condenser water of varying degrees of cleanliness may be used. 3. Flash tank. This is the evaporator of the ejector system and is usually a large-volume vessel where large water-surface area is needed for efficient evaporative cooling action. Warm water returning from the process is sprayed into the flash chamber through nozzles (sometimes cascades are used for maximizing the contact surface, being less susceptible to carryover problems) and the cooled effluent is pumped from the bottom of the flash tank. When the steam supply to one ejector of a group is closed, some means must be provided to for preventing the pressure in the condenser and flash tank from equalizing through that ejector. A compartmental flash tank is frequently used for such purposes. With this

FIG. 11-105

Barometric condenser for steam-jet system.

arrangement, partitions are provided so that each booster is operating on its own flash tank. When the steam is shut off to any booster, the valve to the inlet spray water to that compartment also is closed. A float valve is provided to control the supply of makeup water to replace the water vapor that has flashed off. The flash tank should be insulated. Applications The steam-jet refrigeration is suited for: 1. Processes where direct vaporization is used for concentration or drying of heat-sensitive foods and chemicals, where, besides elimination of the heat exchanger, preservation of the product quality is an important advantage. 2. Enabling the use of hard or even sea water for heat rejection e.g. for absorption of gases (CO2, SO2, ClO2 . . . ) in chilled water (desorption is provided simultaniously with chilling) when a direct contact barometric condenser is used. Despite being simple, rugged, reliable, requiring low maintenance, low cost, and vibration free, steam-jet systems are not widely accepted in water chilling for air-conditioning due to characteristics of the cycle. Factors Affecting Capacity Ejector (steam-jet) units become attractive when cooling relatively high-temperature chilled water with a source of about 7 bar gauge waste steam and relatively cool condensing water. The factors involved with steam-jet capacity include the following: 1. Steam pressure. The main boosters can operate on steam pressures from as low as 0.15 bar up to 7 bar gauge. The quantity of steam required increases rapidly as the steam pressure drops (Fig. 11-106). The best steam rates are obtained with about 7 bar. Above this pressure the change in quantity of steam required is practically negligible. Ejectors must be designed for the highest available steam pressure, to take advantage of the lower steam consumption for various steam-inlet pressures. The secondary ejector systems used for removing air require steam pressures of 2.5 bar or greater. When the available steam pressure is lower than this, an electrically driven vacuum pump is used for either the final secondary ejector or for the entire secondary group. The secondary ejectors normally require 0.2–0.3 kg/h of steam per kW of refrigeration capacity. 2. Condenser water temperature. In comparison with other vapor-compression systems, steam-jet machines require relatively large water quantities for condensation. The higher the inlet-water temperature, the higher are the water requirements (Fig. 11-107).

REFRIGERATION

Effect of steam pressure on steam demand at 38°C condenser temperature (ASHRAE 1983 Equipment Handbook).

FIG. 11-106

FIG. 11-107

Steam demand versus condenser-water flow rate.

Steam demand versus chilled-water temperature for typical steam-jet system (ASHRAE 1983 Equipment Handbook).

FIG. 11-108

11-93

The condensing water temperature has an important effect on steam rate per refrigeration effect, rapidly decreasing with colder condenser cooling water. Figure 11-108 presents data on steam rate versus condenser water inlet for given chilled-water outlet temperatures and steam pressure. 3. Chilled-water temperature. As the chilled-water outlet temperature decreases, the ratio of steam/refrigeration effect decreases, thus increasing condensing temperatures and/or increasing the condensing-water requirements. Unlike other refrigeration systems, the chilled-water flow rate is of no particular importance in steam-jet system design, because there is, due to direct heat exchange, no influence of evaporator tube velocities and related temperature differences on heat-transfer rates. Widely varying return chilled-water temperatures have little effect on steamjet equipment. Multistage Systems The majority of steam-jet systems being currently installed are multistage. Up to five stage systems are in commercial operation. Capacity Control The simplest way to regulate the capacity of most steam vacuum refrigeration systems is to furnish several primary boosters in parallel and operate only those required to handle the heat load. It is not uncommon to have as many as four main boosters on larger units for capacity variation. A simple automatic on-off type of control may be used for this purpose. By sensing the chilled-water temperature leaving the flash tank, a controller can turn steam on and off to each ejector as required. Additionally, two other control systems which will regulate steam flow or condenser-water flow to the machine are available. As the condenser-water temperature decreases during various periods of the year, the absolute condenser pressure will decrease. This will permit the ejectors to operate on less steam because of the reduced discharge pressure. Either the steam flow or the condenser water quantities can be reduced in order to lower operating costs at other then design periods. The arrangement selected depends on cost considerations between the two flow quantities. Some systems have been arranged for a combination of the two, automatically reducing steam flow down to a point, followed by a reduction in condenser-water flow. For maximum operating efficiency, automatic control systems are usually justifiable in keeping operating cost to a minimum without excessive operator attention. In general, steam savings of about 10 percent of rated booster flow are realized for each 2.5°C reduction in condensing-water temperature below the design point. In some cases, with relatively cold inlet condenser water it has been possible to adjust automatically the steam inlet pressure in response to chilled-water outlet temperatures. In general, however, this type of control is not possible because of the differences in temperature between the flash tank and the condenser. Under usual conditions of warm condenser-water temperatures, the main ejectors must compress water vapor over a relatively high ratio, requiring an ejector with entirely different operating characteristics. In most cases, when the ejector steam pressure is throttled, the capacity of the jet remains almost constant until the steam pressure is reduced to a point at which there is a sharp capacity decrease. At this point, the ejectors are unstable, and the capacity is severely curtailed. With a sufficient increase in steam pressure, the ejectors will once again become stable and operate at their deign capacity. In effect, steam jets have a vapor-handling capacity fixed by the pressure at the suction inlet. In order for the ejector to operate along its characteristic pumping curve, it requires a certain minimum steam flow rate which is fixed for any particular pressure in the condenser. (For further information on the design of ejectors, see Sec. 6) Further reading and reference: ASHRAE 1983 Equipment Handbook; Spencer, E., 1961, New Development in Steam Vacuum Refrigeration, ASHRAE Transactions Vol. 67, p. 339 Refrigerants A refrigerant is any body or substance which acts as a cooling agent by absorbing heat from another body or substance which has to be cooled. Primary refrigerants are those that are used in the refrigeration systems, where they alternately vaporize and condense as they absorb or and give off heat respectively. Secondary refrigerants are heat transfer fluids or heat carriers. Refrigerant pairs in absorption systems are ammonia-water and lithium bromide-water,

11-94

HEAT-TRANSFER EQUIPMENT

while steam (water) is used as a refrigerant in ejector systems. Refrigerants used in the mechanical refrigeration systems are far more numerous. A list of the most significant refrigerants is presented in the ASHRAE Handbook Fundamentals. More data are shown in the Chap. 3 of this handbook—“Physical and Chemical Data.” Due to rapid changes in refrigerant, issue readers are advised to consult the most recent data and publications at the time of application. The first refrigerants were natural: air, ammonia, CO2, SO2, and so on. Fast expansion of refrigeration in the second and third quarters of the 20th century is marked by the new refrigerants, chlorofluorocarbons (CFC) and hydrochlorofluorocarbons (HCFC). They are halocarbons which contain one or more of three halogens chlorine, fluorine, and bromine (Fig. 11-109). These refrigerants introduced many advantageous qualities compared to most of the existing refrigerants: odorless, nonflamable, nonexplosive, compatible with the most engineering materials, reasonably high COP, and nontoxic. In the last decade, the refrigerant issue is extensively discussed due to the accepted hypothesis that the chlorine and bromine atoms from halocarbons released to the environment were using up ozone in the stratosphere, depleting it specially above the polar regions. Montreal Protocol and later agreements ban use of certain CFCs and halon compounds. It seems that all CFCs and most of the HCFCs will be out of production by the time this text will be published. Chemical companies are trying to develop safe and efficient refrigerant for the refrigeration industry and application, but uncertainty in CFC and HCFC substitutes is still high. When this text was written HFCs were a promising solution. That is true especially for the R134a which seems to be the best alternative for R12. Substitutes for R22 and R502 are still under debate. Numerous ecologist and chemists are for an extended ban on HFCs as well, mostly due to significant use of CFCs in production of HFCs. Extensive research is ongoing to find new refrigerants. Many projects are aimed to design and study refrigerant mixtures, both azeotropic (mixture which behaves physically as a single, pure compound) and zeotropic having desirable qualities for the processes with temperature glides in the evaporator and the condenser. Ammonia (R717) is the single natural refrigerant being used extensively (beside halocarbons). It is significant in industrial applications for its excellent thermodynamic and thermophysical characteristics. Many engineers are considering ammonia as a CFC substitute for various applications. Significant work is being done on reducing the refrigerant inventory and consequently problems related to leaks of this fluid with strong odor. There is growing interest in hydrocarbons in some contries, particularly in Europe. Indirect cooling (secondary refrigeration) is under reconsideration for many applications. Due to the vibrant refrigerant issue it will be a challenge for every engineer to find the best solution for the particular application, but basic principles are the same. Good refrigerant should be: • Safe: nontoxic, nonflamable, and nonexplosive • Environmentally friendly • Compatible with materials normally used in refrigeration: oils, metals, elastomers, etc. • Desirable thermodynamic and thermophysical characteristics: High latent heat Low specific volume of vapor Low compression ratio Low viscosity

FIG. 11-109

Halocarbon refrigerants.

Reasonably low pressures for operating temperatures Low specific heat of liquid High specific heat of vapor High conductivity and other heat transfer related characteristics Reasonably low compressor discharge temperatures Easily detected if leaking High dielectric constant Good stability Secondary Refrigerants (Antifreezes or Brines) These are mostly liquids used for transporting heat energy from the remote heat source (process heat exchanger) to the evaporator of the refrigeration system. Antifreezes or brines do not change state in the process, but there are examples where some secondary refrigerants are either changing state themselves, or just particles which are carried in them. Indirect refrigeration systems are more prevalent in the chemical industry than in the food industry, commercial refrigeration, or comfort air-conditioning. This is even more evident in the cases where a large amount of heat is to be removed or where a low temperature level is involved. Advantage of an indirect system is centralization of refrigeration equipment, which is specially important for relocation of refrigeration equipment in a nonhazardous area, both for people and equipment. Salt Brines The typical curve of freezing point is shown in Fig. 11-110. Brine of concentration x (water concentration is 1-x) will not solidify at 0°C (freezing temperature for water, point A). When the temperature drops to B, the first crystal of ice is formed. As the temperature decreases to C, ice crystals continue to form and their mixture with the brine solution forms the slush. At the point C there will be part ice in the mixture l2/(l1 + l2), and liquid (brine) l1/(l1 + l2). At point D there is mixture of m1 parts eutectic brine solution D1 [concentration m1/(m1 + m2)], and m2 parts of ice [concentration m2/(m1 + m2)]. Cooling the mixture below D solidifies the entire solution at the eutectic temperature. Eutectic temperature is the lowest temperature that can be reached with no solidification. It is obvious that further strengthening of brine has no effect, and can cause a different reaction—salt sometimes freezes out in the installations where concentration is too high. Sodium chloride, an ordinary salt (NaCl), is the least expensive per volume of any brine available. It can be used in contact with food and in open systems because of its low toxicity. Heat transfer coefficients are relatively high. However, its drawbacks are it has a relatively high freezing point and is highly corrosive (requires inhibitors thus must be checked on a regular schedule). Calcium chloride (CaCl2) is similar to NaCl. It is the second lowestcost brine, with a somewhat lower freezing point (used for temperatures as low as −37°C). Highly corrosive and not appropriate for direct contact with food. Heat transfer coefficients are rapidly reduced at temperatures below −20°C. The presence of magnesium salts in either sodium or calcium chloride is undesirable because they tend to form sludge. Air and carbon dioxide are contaminants and excessive aeration of the brine should be prevented by use of close systems. Oxygen, required for corrosion, normally comes from the atmosphere and dissolves in the brine solution. Dilute brines dissolve oxygen more readily and are generally more corrosive than concentrated brines. It is believed that even a closed brine system will not prevent the infiltration of oxygen. To adjust alkaline condition to pH 7.0–8.5 use caustic soda (to correct up to 7.0) or sodium dichromate (to reduce excessive alkalinity

REFRIGERATION

FIG. 11-110

11-95

Phase diagram of the brine.

below pH 8.5). Such slightly alkaline brines are generally less corrosive than neutral or acid ones, although with high alkalinity the activity may increase. If the untreated brine has the proper pH value, the acidifying effect of the dichromate may be neutralized by adding commercial flake caustic soda (76 percent pure) in quantity that corresponds to 27 percent of sodium dichromate used. Caustic soda must be thoroughly dissolved in warm water before it is added to the brine. Recommended inhibitor (sodium dichromate) concentrations are 2 kg/m3 of CaCl2 and 3.2 kg/m3 of NaCl brine. Sodium dichromate when dissolved in water or brine makes the solution acid. Steel, iron, copper, or red brass can be used with brine circulating systems. Calcium chloride systems are generally equipped with all-iron-and-steel pumps and valves to prevent electrolysis in event of acidity. Copper and red brass tubing are used for calcium chloride evaporators. Sodium chloride systems are using all-iron or all-bronze pumps. Organic Compounds (Inhibited Glycols) Ethylene glycol is colorless and practically odorless and completely miscible with water. Advantages are low volatility and relatively low corrosivity when properly inhibited. Main drawbacks are relatively low heat-transfer coefficients at lower temperatures due to high viscosities (even higher than for propylene glycol). It is somewhat toxic, but less harmful than methanol water solutions. It is not appropriate for food industry and should not stand in open containers. Preferably waters that are classified as soft and are low in chloride and sulfate ions should be used for preparation of ethylene glycol solution. Pure ethylene glycol freezes at −12.7°C. Exact composition and temperature for eutectic point are unknown, since solutions in this region turn to viscous, glassy mass that makes it difficult to determine the true freezing point. For the concentrations lower than eutectic, ice forms on freezing, while on the concentrated, solid glycol separates from the solution. Ethylene glycol normally has pH of 8.8 to 9.2 and should not be used below 7.5. Addition of more inhibitor can not restore the solution to original condition. Once inhibitor has been depleted, it is recommended that the old glycol be removed from the system and the new charge be installed. Propylene glycol is very similar to ethylene glycol, but it is not toxic

and is used in direct contact with food. It is more expensive and, having higher viscosity, shows lower heat transfer coefficients. Methanol water is an alcohol-base compound. It is less expensive than other organic compounds and, due to lower viscosity, has better heat transfer and pressure drop characteristics. It is used up to −35°C. Disadvantages are (1) considered more toxic than ethylene glycol and thus more suitable for outdoor applications (2) flammable and could be assumed to be a potential fire hazard. For ethylene glycol systems copper tubing is often used (up to 3 in), while pumps, cooler tubes, or coils are made of iron, steel, brass, copper, or aluminum. Galvanized tubes should not be used in ethylene glycol systems because of reaction of the inhibitor with the zinc. Methanol water solutions are compatible with most materials but in sufficient concentration will badly corrode aluminum. Ethanol water is a solution of denatured grain alcohol. Its main advantage is that it is nontoxic and thus is widely used in the food and chemical industry. By using corrosion inhibitors it could be made noncorrosive for brine service. It is more expensive than methanol water and has somewhat lower heat transfer coefficients. As an alcohol derivate it is flammable. Secondary refrigerants shown below, listed under their generic names, are sold under different trade names. Some other secondary refrigerants appropriate for various refrigeration application will be listed under their trade names. More data could be obtained from the manufacurer. Syltherm XLT (Dow Corning Corporation). A silicone polymer (Dimethyl Polysiloxane); recommended temperature range −70°C to 250°C; odorless; low in acute oral toxicity; noncorrosive toward metals and alloys commonly found in heat transfer systems. Syltherm 800 (Dow Corning Corporation). A silicone polymer (Dimethyl Polysiloxane); recommended temperature range −40°C to 400°C; similar to Syltherm XLT, more appropriate for somewhat higher temperatures; flash point is 160°C. D-limonene (Florida Chemicals). A compound of optically active terpene (C10H16) derived as an extract from orange and lemon oils; limited data shows very low viscosity at low temperatures—only one centipoise at −50°C; natural substance having questionable stability. Therminol D-12 (Monsanto). A synthetic hydrocarbon; clear liquid;

11-96

HEAT-TRANSFER EQUIPMENT

recommended range −40°C to 250°C; not appropriate for contact with food; precautions against ignitions and fires should be taken with this product; could be found under trade names Santotherm or Gilotherm. Therminol LT (Monsanto). Akylbenzene, synthetic aromatic (C10 H14); recommended range −70°C to −180°C; not appropriate for contact with food; precautions against ignitions and fire should be taken dealing with this product. Dowtherm J (Dow Corning Corporation). A mixture of isomers of an alkylated aromatic; recommended temperature range −70°C to 300°C; noncorrosive toward steel, common metals and alloys; combustible material; flash point 58°C; low toxic; prolonged and repeated exposure to vapors should be limited 10 ppm for daily exposures of eight hours. Dowtherm Q (Dow Corning Corporation). A mixture of dyphenylehane and alkylated aromatics; recommended temperature range −30°C to 330°C; combustible material; flash point 120°C; considered low toxic, similar to Dowtherm J. Safety in Refrigeration Systems This is of paramount importance and should be considered at every stage of installation. The design engineer should have safety as the primary concern by choosing suitable system and refrigerant: selecting components, choosing materials and thicknesses of vessels, pipes, and relief valves of pressure vessels, proper venting of machine rooms, and arranging the equipment for convenient access for service and maintenance (piping arrangements, valve location, machine room layout, etc.). He or she should conform to the stipulation of the safety codes, which is also important for the purpose of professional liability. During construction and installation, the installer’s good decisions

and judgments are crucial for safety, because design documentation never specifies all details. This is especially important when there is reconstruction or repair while the facility has been charged. During operation the plant is in the hands of the operating personnel. They should be properly trained and familiar with the installation. Very often, accidents are caused by an improper practice, such as making an attempt to repair when proper preparation is not made. Operators should be trained in first-aid procedures and how to respond to emergencies. Most frequently needed standards and codes are listed below, and the reader can find comments in: W. F. Stoecker: Industrial Refrigeration, Vol. 2. Ch. 12, Business News Publishing Co., Troy, MI. 1995; ASHRAE Handbook Refrigeration System and Applications, 1994, Ch. 51. Some important standards and codes on safety that a refrigeration engineer should consult are: ANSI/ASHRAE Standard 15-92—Safety Code for Mechanical Refrigeration, ASHRAE, Atlanta GA, 1992; ANSI/ASHRAE Standard 34-92—Number Designation of Refrigerants, ASHRAE, Atlanta GA, 1992; ANSI/ASME Boiler and Pressure Vessel Code, ASME, New York, 1989; ANSI/ASME Code for Pressure Piping, B31, B31.5–1987, ASME, New York, 1987; ANSI/IIAR 2— 1984, Equipment, Design and Installation of Ammonia Mechanical Refrigeration Systems, IIAR, Chicago, 1984; IIAR Minimum Safety Criteria for a Safe Ammonia Refrigeration Systems, Bulletin 109; IIAR, IIAR Start-up, Inspection, and Maintenance of Ammonia Mechanical Refrigeration Systems, Bulletin 110, Chicago, 1988; IIAR Recommended Procedures in Event of Ammonia Spills, Bulletin No. 106, IIAR, Chicago, 1977; A Guide to Good Practices for the Operation of an Ammonia Refrigeration System, IIAR Bulletin R1, 1983.

CRYOGENIC PROCESSES INTRODUCTION Cryogenics, the production of low temperatures, is a major business in the United States with an annual market in excess of 12 billion dollars. It is a very diverse supporting technology, a means to an end and not an end in itself. For example, the combined production of oxygen and nitrogen, obtained by the cryogenic separation of air, accounts for 15 percent of the total annual U.S. production of 2.77 × 1011 kg (611 billion pounds) of organics and inorganics (1993 C & EN Annual Report). Liquid hydrogen production, in the last four decades, has risen from laboratory quantities to a level of over 2.1 kg/s, first spurred by nuclear weapons development and later by the United States space program. Similarly, the space age increased the need for liquid helium by more than a factor of ten, requiring the construction of large plants to separate helium from natural gas by cryogenic means. The demands for energy have likewise accelerated the construction of large base-load liquefied natural gas (LNG) plants around the world and have been responsible for the associated domestic LNG industry of today with its use of peak shaving plants. Freezing as a means of preserving food dates back to 1840. However, today the food industry uses large quantities of liquid nitrogen for this purpose and as a refrigerant in frozen-food transport systems. In biological applications liquid-nitrogen cooled containers are routinely used to preserve whole blood, tissue, bone marrow, and animal semen for extended periods of time. Cryogenic surgery has become accepted in curing such involuntary disorders as Parkinson’s disease. Medical analysis of patients has increased in sophistication with the use of magnetic resonance imaging (MRI) which utilizes cryogenically cooled superconducting magnets. Finally, one must recognize the role that cryogenics plays in the chemical-processing industry with the recovery of valuable feedstocks from natural gas streams, upgrading the heat content of fuel gas, purification of various process and waste streams, the production of ethylene, and so on.

nitrogen, oxygen, argon, and methane. For example, the NIST Standard Reference Database 12, Version 3.0 includes a total of 34 thermophysical properties for seventeen fluids in the database. Cryodata Inc. provides a similar computer version for 28 pure fluids as well as for mixtures incorporating many of these fluids. However, a few peculiarities associated with the fluids of helium, hydrogen, oxygen, and air need to be noted below. Liquid helium-4 can exist in two different liquid phases: liquid helium I, the normal liquid, and liquid helium II, the superfluid, since under certain conditions the latter fluid acts as if it had no viscosity. The phase transition between the two liquid phases is identified as the lambda line and where this transition intersects the vapor-pressure curve is designated as the lambda point. Thus, there is no triple point for this fluid as for other fluids. In fact, solid helium can only exist under a pressure of 2.5 MPa or more. A unique property of hydrogen is that it can exist in two different molecular forms: orthohydrogen and parahydrogen. (This is also true for deuterium, an isotope of hydrogen with an atomic mass of 2.) The thermodynamic equilibrium composition of the ortho- and paravarieties is temperature dependent. The equilibrium mixture of 75 percent orthohydrogen and 25 percent parahydrogen at ambient temperatures is recognized as normal hydrogen. In contrast to other cryogenic fluids, liquid oxygen is slightly magnetic. It is also chemically reactive, particularly with hydrocarbon materials. Oxygen thus presents a safety problem and requires extra precautions in handling. Since air is a mixture of predominantly nitrogen, oxygen, and a host of lesser impurities, there has been less interest in developing precise thermodynamic properties. The only recent correlation of thermodynamic properties is that published by Vasserman, et al. (Barouch, Israel Program for Scientific Translations, Jerusalem, 1970), and is based on the principle of corresponding states because of the scarcity of experimental data.

PROPERTIES OF CRYOGENIC FLUIDS There are presently several database programs of thermodynamic properties data developed specifically for fluids commonly associated with low temperature processing including helium, hydrogen, neon,

PROPERTIES OF SOLIDS A knowledge of the properties and behavior of materials used in any cryogenic system is essential for proper design considerations. Often

CRYOGENIC PROCESSES the choice of materials for the construction of cryogenic equipment will be dictated by other considerations besides mechanical properties as, for example, thermal conductivity (heat transfer along a structural member), thermal expansivity (expansion and contraction during cycling between ambient and low temperatures), and density (mass of the system). Since properties at low temperatures are often significantly different from those at ambient temperature, there is no substitute for test data on a truly representative sample specimen when designing for the limit of effectiveness of a cryogenic material or structure. For example, some metals including elements, intermetallic compounds and alloys exhibit the phenomenon of superconductivity at very low temperatures. The properties that are affected when a material becomes superconducting include specific heat, thermal conductivity, electrical resistance, magnetic permeability, and thermoelectric effect. As a result, the use of superconducting metals in the construction of equipment for temperatures lower than 10 K needs to be evaluated carefully. (High temperature superconductors because of their brittle ceramic structure are presently not considered as construction materials.) Structural Properties at Low Temperatures It is most convenient to classify metals by their lattice symmetry for low temperature mechanical properties considerations. The face-centered-cubic (fcc) metals and their alloys are most often used in the construction of cryogenic equipment. Al, Cu Ni, their alloys, and the austenitic stainless steels of the 18-8 type are fcc and do not exhibit an impact ductileto-brittle transition at low temperatures. As a general rule, the mechanical properties of these metals with the exception of 2024-T4 aluminum, improve as the temperature is reduced. Since annealing of these metals and alloys can affect both the ultimate and yield strengths, care must be exercised under these conditions. The body-centered-cubic (bcc) metals and alloys are normally classified as undesirable for low temperature construction. This class includes Fe, the martensitic steels (low carbon and the 400-series stainless steels), Mo, and Nb. If not brittle at room temperature, these materials exhibit a ductile-to-brittle transition at low temperatures. Cold working of some steels, in particular, can induce the austeniteto-martensite transition. The hexagonal-close-packed (hcp) metals generally exhibit mechanical properties intermediate between those of the fcc and bcc metals. For example Zn encounters a ductile-to-brittle transition whereas Zr and pure Ti do not. The latter and their alloys with a hcp structure remain reasonably ductile at low temperatures and have been used for many applications where weight reduction and reduced heat leakage through the material have been important. However, small impurities of O, N, H, and C can have a detrimental effect on the low temperature ductility properties of Ti and its alloys. Plastics increase in strength as the temperature is decreased, but this is also accompanied by a rapid decrease in elongation in a tensile test and a decrease in impact resistance. Teflon and glass-reinforced plastics retain appreciable impact resistance as the temperature is lowered. The glass-reinforced plastics also have high strength-toweight and strength-to-thermal conductivity ratios. All elastomers, on the other hand, become brittle at low temperatures. Nevertheless, many of these materials including rubber, Mylar, and nylon can be used for static seal gaskets provided they are highly compressed at room temperature prior to cooling. The strength of glass under constant loading also increases with decrease in temperature. Since failure occurs at a lower stress when the glass surface contains surface defects, the strength can be improved by tempering the surface. Thermal Properties at Low Temperatures For solids, the Debye model developed with the aid of statistical mechanics and quantum theory gives a satisfactory representation of the specific heat with temperature. Procedures for calculating values of ΘD, the Debye characteristic temperature, using either elastic constants, the compressibility, the melting point, or the temperature dependence of the expansion coefficient are outlined by Barron (Cryogenic Systems, 2d ed., Oxford University Press, 1985, pp 24–29). Adequate prediction of the thermal conductivity for pure metals can be made by means of the Wiedeman-Franz law which states that the ratio of the thermal conductivity to the product of the electrical conductivity and the absolute temperature is a constant. This ratio for

11-97

high-conductivity metals extrapolates essentially to the Sommerfeld value of 2.449 × 10−8 W Ω/K2 at 0 K, but falls considerably below it at higher temperatures. High-purity aluminum and copper exhibit peaks in thermal conductivity between 20 to 50 K, but these peaks are rapidly suppressed with increased impurity levels and cold work of the metal. Some metals including Monel, Inconel, stainless steel, and structural and aluminum alloys show a steady decrease in thermal conductivity with a decrease in temperature. All cryogenic liquids except hydrogen and helium have thermal conductivities that increase as the temperature is decreased. For these two exceptions, the thermal conductivity decreases with a decrease in temperature. The kinetic theory of gases correctly predicts the decrease in thermal conductivity of all gases when the temperature is lowered. The expansion coefficient of a solid can be estimated with the aid of an approximate thermodynamic equation of state for solids which equates the thermal expansion coefficient β with the quantity γ Cv ρ/B where γ is the Grüneisen dimensionless ratio, Cv is the specific heat of the solid, ρ is the density of the material, and B is the bulk modulus. For fcc metals the average value of the Grüneisen constant is near 2.3. However, there is a tendency for this constant to increase with atomic number. Electrical Properties at Low Temperatures The electrical resistivity of most pure metallic elements at ambient and moderately low temperatures is approximately proportional to the absolute temperature. At very low temperatures, however, the resistivity (with the exception of superconductors) approaches a residual value almost independent of temperature. Alloys, on the other hand, have resistivities much higher than those of their constituent elements and resistance-temperature coefficients that are quite low. The electrical resistivity of alloys as a consequence is largely independent of temperature and may often be of the same magnitude as the room temperature value. Superconductivity The physical state in which all resistance to the flow of direct-current electricity disappears is defined as superconductivity. The Bardeen-Cooper-Schriefer (BCS) theory has been reasonably successful in accounting for most of the basic features observed of the superconducting state for low-temperature superconductors (LTS) operating below 23 K. The advent of the ceramic hightemperature superconductors (HTS) by Bednorz and Miller (Z. Phys. B64, 189, 1989) has called for modifications to existing theories which have not been finalized to date. The massive interest in the new superconductors that can be cooled with liquid nitrogen is just now beginning to make its way into new applications. Three important characteristics of the superconducting state are the critical temperature, the critical magnetic field, and the critical current. These parameters can be varied by using different materials or giving them special metallurgical treatments. The alloy niobium titanium (NbTi) and the intermetallic compound of niobium and tin (Nb3 Sn) are the most technologically advanced LTS materials presently available. Even though NbTi has a lower critical field and critical current density, it is often selected because its metallurgical properties favor convenient wire fabrication. In contrast, Nb3Sn is a very brittle material and requires wire fabrication under very well-defined temperature conditions. There are presently four families of high-temperature superconductors under investigation for practical magnet applications. Table 11-25 shows that all HTS are copper oxide ceramics even though the oxygen content may vary. However, this variation generally has little effect on the physical properties of importance to superconductivity. The most widely used development in HTS wire production is the powder-in-tube procedure with BSCCO ceramic materials. In this procedure very fine HTS powder, placed inside of a hollow silver tube, is fused as the tube length is mechanically increased to form a wire. Very high magnetic fields with this wire have been reported at 4 K; however, the performance degrades substantially above 20 to 30 K. HTS materials, because of their ceramic nature, are quite brittle. This has introduced problems relative to the winding of superconducting magnets. One solution is to first wind the magnet with the powder-in-tube wire before the ceramic powder has been bonded and then heat treat the desired configuration to form the final product. Another solution is to form the superconductor into such fine fila-

HEAT-TRANSFER EQUIPMENT

TABLE 11-25 Composition and Critical Temperature, Tc for HTS Materials Formula*

Accepted notation

Forms reported

Critical Tc, K

Y-Ba-Cu-O† Bi-Sr-Ca-Cu-O Tl-Ba-Ca-Cu-O Hg-Ba-Ca-Cu-O

YBCO BSCCO TBCCO HBCCO

123 124 2212 2223 Several 1201 1223

80–92 80–110 to 125 95–155‡

*Subscripts for compounds are not listed since there are generally several forms that can be produced (see column 3). †Other rare earths may be substituted for Y (yttrium) providing new compounds with somewhat different properties. ‡Highest Tc obtained while subjecting sample to external pressure.

Compressor Coolant

2

2 P2 Heat exchanger 3 Expansion valve 4

1 P1

Temperature

11-98

P1

P2

3

5 Liquid reservoir Q

ments that they remain sufficiently flexible even after the powder has been heat treated.

1

(a)

4

5 Entropy (b)

Refrigerator using simple J-T cycle.

REFRIGERATION AND LIQUEFACTION

FIG. 11-111

A process for producing refrigeration or liquefaction at cryogenic temperatures usually involves ambient compression of a process fluid with heat rejection to a coolant. During the compression process, the enthalpy and entropy of the fluid are decreased. At the cryogenic temperature where heat is absorbed, the enthalpy and entropy are increased. The reduction in temperature of the process fluid is usually accomplished by heat exchange with a colder fluid and then followed by an expansion. This expansion may take place using either a throttling device (isenthalpic expansion) with only a reduction in temperature or a work-producing device (isentropic expansion) in which both temperature and enthalpy are decreased. Because of liquid withdrawal, a liquefaction system experiences an unbalanced flow in the heat exchanger while a refrigeration system with no liquid withdrawal system usually operates with a balanced flow in the heat exchanger, except where a portion of the flow is diverted through the workproducing expander. Principles The performance of a real refrigerator is measured by the coefficient of performance, COP, defined as Q heat removed at low temperature COP =
=



(11-108) W net work input Another means of comparing the performance of a practical refrigerator is by the use of the figure of merit, FOM, defined as COP FOM =
(11-109) COPi where COP is the coefficient of performance of the actual refrigerator system and COPi is the coefficient of performance for the thermodynamically ideal system. For a liquefier, the FOM is generally specified as Wi /m ˙f FOM =
(11-110) W/m ˙f where Wi is the work of compression for the ideal cycle, W is the work of compression for the actual cycle, and m ˙ f is the mass rate liquefied in the ideal or actual cycle. The methods of refrigeration and/or liquefaction generally used include (1) vaporization of a liquid, (2) application of the JouleThomson effect in a gas, and (3) expansion of a gas in a workproducing engine. Normal commercial refrigeration generally is accomplished in a vapor-compression process. Temperatures to about 200 K can be obtained by cascading vapor-compression processes in which refrigeration is accomplished by liquid evaporation. Below this temperature, isenthalpic or isentropic expansions are generally used either singly or in combination. With few exceptions, refrigerators using these methods also absorb heat by vaporization of the liquid. If refrigeration is to be accomplished at a temperature range where no suitable liquid exists to absorb heat by evaporation, then a cold gas must be available to absorb the heat. This is generally accomplished by using a work-producing expansion engine. Expansion Types of Refrigerators A thermodynamic process

utilizing isenthalpic expansion to obtain cryogenic temperatures, and commonly referred to as the simple Linde or J-T cycle, is shown schematically with its corresponding temperature-entropy diagram in Fig. 11-111. The gaseous refrigerant is compressed at ambient temperature while essentially rejecting heat isothermally to a coolant. The compressed refrigerant is cooled countercurrently in a heat exchanger by the cold gas stream leaving the liquid reservoir before it enters the throttling valve. Upon expansion, Joule-Thomson cooling further reduces the temperature until, at steady state, a portion of the refrigerant is liquefied. For a refrigerator, the unliquefied fraction and the vapor formed by liquid evaporation from the absorbed heat Q are warmed in the heat exchanger before returning to the intake of the compressor. Assuming no heat inleaks, as well as negligible kinetic and potential energy changes in the fluid, the refrigeration duty Q is equivalent to m(h ˙ 1 − h2), where the subscripts refer to the locations shown on Fig. 11-111. Applying Eq. 11-108, the coefficient of performance for the ideal J-T refrigerator is given by h1 − h2 COP =


(11-111) T1[s1 − s2 − (h1 − h2)] For a simple J-T liquefier, the liquefied portion is continuously withdrawn from the reservoir and only the unliquefied portion of the fluid is warmed in the countercurrent heat exchanger and returned to the compressor. The fraction y that is liquefied is obtained by applying the first law to the heat exchanger, J-T valve, and liquid reservoir. This results in h1 − h2 y=
(11-112) h1 − hf where hf is the specific enthalpy of the liquid being withdrawn. Maximum liquefaction occurs when the difference between h1 and h2 is maximized. To account for heat inleak, qL, the relation needs to be modified to h1 − h2 − qL y =

(11-113) h1 − hf with a resultant decrease in the fraction liquefied. Refrigerants used in this process have a critical temperature well below ambient; consequently liquefaction by direct compression is not possible. In addition, the inversion temperature of the refrigerant must be above ambient temperature to provide initial cooling by the J-T process. Auxiliary refrigeration is required if the simple J-T cycle is to be used to liquefy neon, hydrogen, or helium whose inversion temperatures are below ambient. Liquid nitrogen is the optimum refrigerant for hydrogen and neon liquefaction systems, while liquid hydrogen is the normal refrigerant for helium liquefaction systems. To reduce the work of compression in this cycle a two-stage or dualpressure process may be used whereby the pressure is reduced by two successive isenthalpic expansions. Since the isothermal work of compression is approximately proportional to the logarithm of the pressure ratio, and the Joule-Thomson cooling is roughly proportional to

CRYOGENIC PROCESSES 300

Compressor

3 Expansion engine

P2 3 5

5

(a)

Natural gas

C3 200

C2

(b) FIG. 11-114

C2

200

C1

120

∆ Enthalpy (a)

Cold-gas refrigerator.

C3 250

150 C1

120

Entropy

Q

250

150

4 4

FIG. 11-112

Refrigerant

P1

Temperature, K

1 P1

300

1 Temperature, K

2 P2 Heat exchanger

2 Temperature

Coolant

11-99

∆ Enthalpy (b) b

Three- and nine-level cascade cycle cooling curve for natural

gas.

the pressure difference, there is a greater reduction in compressor work than in refrigerating performance for this dual-pressure process. In a work-producing expansion, the temperature of the process fluid is always reduced; hence, cooling does not depend on being below the inversion temperature prior to expansion. Additionally, the work-producing expansion results in a larger amount of cooling than in an isenthalpic expansion over the same pressure difference. In large systems utilizing expanders, the work produced during expansion is conserved. In small refrigerators, the energy from the expansion is usually expended in a gas or hydraulic pump, or other suitable device. A schematic of a simple cold-gas refrigerator using this expansion principle and the corresponding temperature-entropy diagram is shown in Fig. 11-112. Gas compressed isothermally at ambient temperature is cooled in a heat exchanger by gas being warmed on its return to the compressor intake. Further cooling takes place during the engine expansion. In practice this expansion is never truly isentropic, and is reflected by path 3-4 on the temperatureentropy diagram. This specific refrigerator produces a cold gas which absorbs heat from 4-5 and provides a method of refrigeration that can be used to obtain temperatures between those of the boiling points of the lower-boiling cryogens. It is not uncommon to utilize both the isentropic and isenthalpic expansions in a cycle. This is done to avoid the technical difficulties associated with the formation of liquid in the expander. The Claude or expansion engine cycle is an example of a combination of these meth-

Coolant

ods and is shown in Fig. 11-113 along with the corresponding temperature-entropy diagram. The mixed refrigerant cycle was developed to meet the need for liquefying large quantities of natural gas to minimize transportation costs of this fuel. This cycle resembles the classic cascade cycle in principle and may best be understood by referring to that cycle. In the latter, the natural gas stream after purification is cooled successively by vaporization of propane, ethylene, and methane. Each refrigerant may be vaporized at two or three pressure levels to increase the natural gas cooling efficiency, but at a cost of considerable increased process complexity. Cooling curves for natural gas liquefaction by the cascade process are shown in Fig. 11-114. It is evident that the cascade cycle efficiency can be improved by increasing the number of refrigerants employed. For the same refrigeration capacity, the actual work required for the nine-level cascade cycle depicted is approximately 80 percent of that required by the three-level cascade cycle. This increase in efficiency is achieved by minimizing the temperature difference between the refrigerant and the natural gas stream throughout each increment of the cooling curve. The mixed refrigerant cycle is a variation of the cascade cycle and involves the circulation of a single multicomponent refrigerant stream. The simplification of the compression and heat exchange services in such a cycle can offer potential for reduced capital expenditure over the conventional cascade cycle.

Compressor P2 2 1 P1 Heat exchanger

2

3

7

4

Temperature

Expansion engine

Expansion valve

1

P2

5

P1

3

7 5 6

6 Liquid reservoir

4

Entropy (b)

(b)

Q (a) FIG. 11-113

Claude cycle refrigerator utilizing both expansion processes.

11-100

HEAT-TRANSFER EQUIPMENT Cooling water

Natural gas feed

Separator-1

Heat exchanger-1

Compressor

Separator-2

Heat exchanger-2

Heat exchanger-3

To LKG storage FIG. 11-115

Mixed-refrigerant cycle.

Figure 11-115 shows the basic concepts for a mixed refrigerant cycle (Gaumer, Advances in Cryogenic Engineering Vol. 31, Plenum, New York, 1986, p. 1095). Variations of the cycle are proprietary with those cryogenic engineering firms that have developed the technology. However, all of the mixed refrigerant processes use a carefully prepared refrigerant mix which is repeatedly condensed, vaporized, separated, and expanded. Thus, these processes require more complete knowledge of the thermodynamic properties of gaseous mixtures than those required in the expander or classical cascade cycles. This is particularly evident when cooling curves similar to the one shown in Fig. 11-116 are desired. An inspection of the mixed refrigerant cycle also shows that these processes must routinely handle twophase flows in the heat exchangers. Miniature Refrigerators Expanded space and science projects have provided a need for miniature cryogenic coolers designated as cryocoolers. Such coolers provide useful cooling from a fraction of a watt to several watts at temperature levels from 1 to 90 K. These coolers are used to increase the sensitivity and signal-to-noise ratio of detectors by providing the required cryogenic operating temperatures as well as cooling the optical components to decrease the detector background radiation. The types of coolers developed to meet various specific requirements include solid cryogen coolers, radiative coolers, mechanical coolers, 3He adsorption coolers, adiabatic demagnetization refrigerators, and liquid helium storage systems. Mechanical 300

Temperature, K

C3 250 Natural gas 200 Mixed refrigerant (Stage 1) 150 120

Mixed refrigerant (Stage 2) ∆ Enthalpy

FIG. 11-116

natural gas.

Propane precooled mixed-refrigerant cycle cooling curve for

coolers are generally classified as regenerative or recuperative. Regenerative coolers use reciprocating components that move the working fluid back and forth through a regenerator. The recuperative coolers, on the other hand, use countercurrent heat exchangers to accomplish the heat transfer operation. The Stirling and GiffordMcMahon cycles are typically regenerative coolers while the JouleThomson and Brayton cycles are associated with recuperative coolers. The miniature split single-stage Stirling cooler developed by Philips Laboratories produces 5 W of cooling at 65 K with the aid of linear motors and magnetically moving parts. A smaller split Stirling cycle cooler that uses a stacked diaphragm spring rather than magnetic means to levitate the piston and displacer has been developed at Oxford University (Bradshaw, et al., Advances in Cryogenic Engineering, Vol. 31, Plenum, New York, 1986, p. 801). The promise of higher reliability has spurred interest in the pulse-tube refrigerator (PTR). In the latest version of this device, an orifice and reservoir have been added to the warm end of the pulse tube (OPTR) to permit control of the phase shift required for optimum resonance in the system. The Joule-Thomson cycle has also benefited from creative thinking. For example, Little (6th International Cryocooler Conference, Naval Postgraduate School, Monterey, CA, 1989, p. 3) has introduced a new method of fabricating J-T refrigerators using a photolithographic manufacturing process in which gas channels for the heat exchangers, expansion capillary, and liquid reservoir are etched on planar, glass substrates that are fused together to form the sealed refrigerator. These microminiature refrigerators have been made in a wide range of sizes and capacities. Mixtures of highly polar gases are receiving considerable attention for J-T cycles since the magnitude of the Joule-Thomson coefficient increases with nonideality of the gas. Little (op.cit) and Longsworth (8th International Cryocooler Conference, JPL, Vail, CO, 1994, in press) have shown considerable ingenuity with gas mixtures for J-T cycles, developing small, lightweight, low-cost, but reliable cryocoolers for a number of applications. Thermodynamic Analyses of Cycles The thermodynamic quality measure of either a piece of equipment or an entire process is its reversibility. The second law, or more precisely the entropy increase, is an effective guide to this degree of irreversibility. However, to obtain a clearer picture of what these entropy increases mean, it has become convenient to relate such an analysis to the additional work that is required to overcome these irreversibilities. The fundamental equation for such an analysis is W = Wrev + To

˙

m∆s

(11-114)

where the total work, W, is the sum of the reversible work, Wrev, plus a summation of the losses in availability for various unit operations in the analysis. Application of this method has been demonstrated numerically by Timmerhaus and Flynn (Cryogenic Process Engineering, Plenum Press, 1989, p. 175). Numerous analyses and comparisons of refrigeration and liquefaction cycles are available in the literature. Great care must be exercised in accepting these comparisons since it is quite difficult to put all processes on a strictly comparable basis. Many assumptions need to be made in the course of the calculations, and these can have considerable effect on the conclusions. Major factors upon which assumptions generally have to be made include heat leak, temperature differences in the exchangers, efficiencies of compressors and expanders, number of stages of compression, fraction of expander work recovered, state of expander exhaust, purity and condition of inlet gases, pressure drop due to fluid flow, and so on. In view of this fact, differences in power requirements of 10 to 20 percent can readily be due to differences in assumed variables and can negate the advantage of one cycle over another. Barron (Cryogenic Systems, 2d ed., Oxford University Press, New York, 1985, p. 94] has made an analysis of some of the more common liquefaction systems described earlier that emphasize this point rather well. PROCESS EQUIPMENT The equipment normally associated with cryogenic systems includes heat exchangers, compressors, expanders, throttling valves, and stor-

CRYOGENIC PROCESSES age containers. Since the reciprocal or centrifugal compressors used generally operate at ambient temperatures, their operating principles are not covered here but in Sec. 10. Storage containers, are discussed later in Sec. 11. Heat Exchangers Since most cryogens, with the exception of helium II behave as classical fluids, well-established principles of mechanics and thermodynamics at ambient temperature also apply for cryogens. Thus, similar conventional heat transfer correlations have been formulated for simple low-temperature heat exchangers. These correlations are described in terms of well-known dimensionless quantities such as the Nusselt, Reynolds, Prandtl, and Grashof numbers. Because of the need to operate more efficiently at low temperatures, the simple heat exchangers have generally been replaced with more sophisticated types. Guidance for the development of such units for low-temperature service include the following factors: 1. Small temperature differences between inlet and exit streams to enhance efficiency. 2. Large surface area-to-volume ratio to minimize heat leak. 3. High heat transfer to reduce surface area. 4. Low mass to minimize start-up time. 5. Multichannel capability to minimize the number of exchangers. 6. High-pressure capability to provide design flexibility. 7. Low or reasonable pressure drop to minimize compression requirements. 8. High reliability with minimal maintenance to reduce shutdowns. Problems sometimes occur in trying to minimize the temperature difference at the cold end of the heat exchanger, particularly if the specific heat of the warm fluid decreases with decreasing temperature as is the case with gaseous hydrogen. The selection of an exchanger for low-temperature operation is usually determined by the process-design requirements, mechanicaldesign limitations, and economic considerations. Laboratory needs are generally met by concentric tube and extended surface exchangers, while industrial needs are most often met by the coiled-tube, plate-fin, reversing, and regenerator types of exchangers. The coiled-tube heat exchanger offers unique advantages, especially when dealing with low-temperature design conditions where (1) simultaneous heat transfer between more than two streams is desired, (2) a large number of heat transfer units is required, and (3) high operating pressures are involved. Heat transfer for single-phase flow of either gas or liquid on the tubeside is generally well represented by either the Colburn correlation or modified forms of the Dittus-Boelter relationship. The shape of the cooling and warming curves in coiled-tube heat exchangers is affected by the pressure drop in both the tube and shellsides of the heat exchanger. This is particularly important for twophase flows of multicomponent systems. For example, an increase in pressure drop on the shellside causes boiling to occur at a higher temperature, while an increase in pressure drop on the tubeside will cause condensation to occur at a lower temperature. The net result is both a decrease in the effective temperature difference between the two streams and a requirement for additional heat transfer area to compensate for these losses. Plate-fin heat exchangers are about nine times as compact as conventional shell-and-tube heat exchangers with the same amount of surface area, weigh less than conventional heat exchangers, and withstand design pressures up to 6 MPa for temperatures between 4 and 340 K. Flow instability frequently becomes a limiting design parameter for plate-fin heat exchangers handling either boiling or condensing two-phase flows. This results in lower optimum economic mass flow velocities for plate-fin heat exchangers when compared with coiledtube heat exchangers. The use of fins or extended surfaces in plate-fin or similar exchangers greatly increases the heat transfer area. Calculations using finned surfaces are outlined earlier in Sec. 11. There are two basic approaches to heat-exchanger design for low temperatures: (1) the effectiveness-NTU approach and (2) the log-mean-temperature-difference (LMTD) approach. The LMTD approach is used most frequently when all the required mass flows are known and the area of the exchanger is to be determined. The effec-

11-101

tiveness-NTU approach is used more often when the inlet temperatures and the flow rates are specified for an exchanger with fixed area and the outlet temperatures are to be determined. Both methods are described earlier in Sec. 11. System performance in cryogenic liquefiers and refrigerators is directly related to the effectiveness of the heat exchangers used in the system. For example, the liquid yield for a simple J-T cycle as given by Eq. 11-112 needs to be modified to (h1 − h2) − (1 − ε)(h1 − hg) y =


(11-115) (h1 − hf) − (1 − ε)(h1 − hg) if the heat exchanger is less than 100 percent effective. Likewise, the heat exchanger ineffectiveness increases the work required for the system by an amount of ˙ = m(h ∆W ˙ 1 − hg)(1 − ε) (11-116) Uninterrupted operation of heat exchangers at low temperatures requires removal of essentially all impurities present in the streams that are to be cooled. Equipment is readily available for the satisfactory removal of these impurities by both chemical and physical methods, but at increased operating expense. Another effective method for also accomplishing this impurity removal utilizes reversing heat exchangers. Proper functioning of the reversing heat exchanger is dependent upon the relationship between the pressures and temperatures of the two streams. Since the pressures are generally fixed by other factors, the purification function of the heat exchanger is normally controlled by selecting the right temperature difference throughout the heat exchanger. To assure that reevaporation takes place, these differences must be such that the vapor pressure of the impurity is greater than the partial pressure of the impurity in the purging stream. Another type of reversing heat exchanger is the regenerator. As with all reversing heat exchangers, regenerators provide the simultaneous cooling and purification of gases in low-temperature processes. As noted earlier, reversing heat exchangers usually operate continuously. Regenerators do not operate continuously; instead, they operate by periodically storing heat in a packing during the first half of the cycle and then giving up this stored heat to the fluid during the second half of the cycle. Typically, a regenerator consists of two identical columns, which are packed with a porous solid material with a good heat capacity such as metal ribbon, through which the gases flow. The low cost of the packing material, its large surface-area-per-unit volume, and the low-pressure drops encountered provide compelling arguments for utilizing regenerators. However, the intercontamination of fluid streams, caused by the periodic flow reversals, and the problems associated with designing regenerators to handle three or more fluids, has restricted their use to simple fluids, and favored adoption of plate-fin reversing heat exchangers. Expanders The primary function of cryogenic expansion equipment is the reduction of the temperature of the gas being expanded to provide needed refrigeration. The expansion of a fluid to produce refrigeration may be carried out in two distinct ways: (1) in an expander where mechanical work is produced, and (2) in a JouleThomson valve where no work is produced. Mechanical Expanders Reciprocating expanders are very similar in concept and design to reciprocating compressors. Generally these units are used with inlet pressures of 4 to 20 MPa. These machines operate at speeds up to 500 rpm. The thermal efficiencies (actual enthalpy difference/maximum possible enthalpy difference) range from about 75 percent for small units to 85 percent for large machines. Turboexpanders have replaced reciprocating expanders in highpower installations as well as in small helium liquefiers. Sizes range from 0.75 to 7500 kW with flow rates up to 28 million m3/day. Today’s large-tonnage air-separation plants are a reality due to the development of highly reliable and efficient turboexpanders. These expanders are being selected over other cryogenic equipment because of their ability to condense ethane and heavier hydrocarbons. This type of expander usually weighs and costs less and requires less space and operating personnel. Turboexpanders can be classified as either axial or radial. Axial flow expanders have either impulse or reaction type blades and are suitable

11-102

HEAT-TRANSFER EQUIPMENT

for multistage expanders because they permit a much easier flow path from one stage to the next. However, radial turboexpanders have lower stresses at a given tip speed, which permits them to run at higher speeds. This results in higher efficiencies with correspondingly lower energy requirements. As a consequence, most turboexpanders built today are of the radial type. Joule-Thomson Valves The principal function of a J-T valve is to obtain isenthalpic cooling of the gas flowing through the valve. These valves generally are needle-type valves modified for cryogenic operation. They are an important component in most refrigeration systems, particularly in the last stage of the liquefaction process. JouleThomson valves also offer an attractive alternative to turboexpanders for small-scale gas-recovery applications.

Air Oxygen Nitrogen

V

SEPARATION AND PURIFICATION SYSTEMS The energy required to reversibly separate gas mixtures is the same as that necessary to isothermally compress each component in the mixture from the partial pressure of the gas in the mixture to the final pressure of the mixture. This reversible isothermal work is given by the familiar relation −Wi (11-117)
= T(s1 − s2) − (h1 − h2) m ˙ where s1 and h1 refer to conditions before the separation and s2 and h2 refer to conditions after the separation. For a binary system, and assuming a perfect gas for both components, Eq. (11-117) simplifies to PT −Wi PT (11-118)
= RT nA ln
+ nB ln
m ˙ pA pB where nA and nB are the moles of A and B in the mixture, pA and pB are the partial pressures of these two components in the mixture, and PT is the total pressure of the mixture. The figure of merit for a separation system is defined similar to that for a liquefaction system; see Eq. (11-110). If the mixture to be separated is essentially a binary, both the McCabe-Thiele and the Ponchon-Savarit methods outlined in Sec. 13, with the appropriate cryogenic properties, can be used to obtain the ideal number of stages required. It should be noted, however, that it is not satisfactory in the separation of air to treat it as a binary mixture of oxygen and nitrogen if high purity (99 percent or better) oxygen is desired. The separation of oxygen from argon is a more difficult separation than oxygen from nitrogen and would require correspondingly many more plates. In fact, if the argon is not extracted from air, only 95 percent oxygen would be produced. The other rare gas constituents of air-helium, neon, krypton, and xenon are present in such small quantities and have boiling points so far removed from those of oxygen and nitrogen that they introduce no important complications. Air-Separation Systems Of the various separation schemes available today, the simplest is known as the Linde single-column system shown in Fig. 11-117 and first introduced in 1902. In it, purified compressed air passes through a precooling heat exchanger (if oxygen gas is the desired product, a three-channel exchanger for air, waste nitrogen, and oxygen gas is used; if liquid oxygen is to be recovered from the bottom of the column, a two-channel exchanger for air and waste nitrogen is employed), then through a coil in the boiler of the rectifying column where it is further cooled (acting at the same time as the boiler heat source); following this, it expands essentially to atmospheric pressure through a J-T valve and reenters the top of the column with the liquid providing the required reflux. Rectification occurs as liquid and vapor on each plate establish equilibrium. If oxygen gas is to be the product, purified air need only be compressed to a pressure of 3 to 6 MPa; if the product is to be liquid oxygen, a compressor outlet pressure of 20 MPa is necessary. Note that the Linde single-column separation system is simply a J-T liquefaction system with a substitution of a rectification column for the liquid reservoir. However, any of the other liquefaction systems discussed earlier could have been used to furnish the liquid for the column. In a simple single-column process, although the oxygen purity is high, the nitrogen effluent stream is impure. The equilibrium vapor





FIG. 11-117

Linde single-column air separator.

concentration of the overhead nitrogen effluent for an initial liquid mixture of 21 percent oxygen/79 percent nitrogen at 100 kPa is about 6 to 7 percent oxygen. Thus, the nitrogen waste gas stream with such an impurity may only be usable as a purge gas for certain conditions. The impurity problem noted in the previous paragraph was solved by the introduction of the Linde double-column system shown in Fig. 11-118. Two rectification columns are placed one on top of the other (hence the name double-column system). In this system the liquid air is introduced at an intermediate point B into the lower column, and a condenser-evaporator at the top of the Air Oxygen Nitrogen C V3

A

E B

V2

V

D FIG. 11-118

Linde double-column air separator.

CRYOGENIC PROCESSES lower column makes the arrangement a complete reflux distillation column which delivers almost pure nitrogen at E. In order for the column simultaneously to deliver pure oxygen, the oxygen-rich liquid (about 45 percent O2) from the bottom boiler is introduced at an intermediate level C in the upper column. The reflux and rectification in the upper column produce pure oxygen at the bottom and pure nitrogen at the top provided all major impurities are first removed from the column. More than enough liquid nitrogen is produced in the lower column for the needed reflux in both columns. Since the condenser must condense nitrogen vapor by evaporating liquid oxygen, it is necessary to operate the lower column at a higher pressure, about 500 kPa, while the upper column is operated at approximately 110 kPa. This requires a reduction in the pressure of the fluids from the lower column as they are admitted to the upper column. In the cycle shown, gaseous oxygen and nitrogen are withdrawn at room temperature. Liquid oxygen could be withdrawn from point A and liquid nitrogen from point E, but in this case more refrigeration would be needed. Even the best modern low-temperature air separation plant has an efficiency only a small fraction of the theoretical optimum, that is, about 15 to 20 percent. The principal sources of inefficiency are threefold: (1) the nonideality of the refrigerating process, (2) the imperfection of the heat exchangers, and (3) losses of refrigeration through heat leak. Helium and Natural-Gas Systems Separation Helium is produced primarily by separation of helium-rich natural gas. The helium content of the natural gas from plants operated by the U.S. Bureau of Mines normally has varied from 1 to 2 percent while the nitrogen content of the natural gas has varied from 12 to 80 percent. The remainder of the natural gas is methane, ethane, and heavier hydrocarbons. A Bureau of Mines system for the separation of helium from natural gas is shown in Fig. 11-119. Since the major constituents of natural gas have boiling points very much different from that of helium, a distillation column is not necessary and the separation can be accomplished with condenser-evaporators. The need to obtain greater recoveries of the C2, C3, and C4’s in natural gas has resulted in the expanded use of low-temperature processing of these streams. The majority of the natural gas processing at low temperatures to recover light hydrocarbons is now accomplished using the turboexpander cycle. Feed gas is normally available from 1 to 10 MPa. The gas is first dehydrated to a dew point of 200 K and lower. After dehydration the feed is cooled with cold residue gas. Liquid produced at this point is separated before entering the expander and sent to the condensate stabilizer. The gas from the separator is

expanded in a turboexpander where the exit stream can contain as much as 20 wt % liquid. This two-phase mixture is sent to the top section of the stabilizer which separates the two phases. The liquid is used as reflux in this unit while the cold gas exchanges heat with the fresh feed and is recompressed by the expander-driven compressor. Many variations to this cycle are possible and have been used in actual plants. Gas Purification The nature and concentration of impurities to be removed depends on the type of process involved. For example, in the production of large quantities of oxygen, various impurities must be removed to avoid plugging of the cold process lines or to avoid buildup of hazardous contaminants. The impurities in air that would contribute most to plugging would be water and carbon dioxide. Helium, hydrogen, and neon, on the other hand, will accumulate on the condensing side of the condenser-reboiler located between the two separation columns and will reduce the rate of heat transfer unless removed by intermittent purging. The buildup of acetylene, however, can prove to be dangerous even though the feed concentration in the air is no greater than 0.04 ppm. Refrigeration purification is a relatively simple method for removing water, carbon dioxide, and certain other contaminants from a process stream by condensation or freezing. (Either regenerators or reversing heat exchangers may be used for this purpose since a flow reversal is periodically necessary to reevaporate and remove the solid deposits.) The effectiveness of this method depends upon the vapor pressure of the impurities relative to that of the major components of the process stream at the refrigeration temperature. Thus, assuming ideal gas behavior, the maximum impurity content in a gas stream after refrigeration would be inversely proportional to its vapor pressure. However, due to the departure from ideality at higher pressures, the impurity content will be considerably higher than predicted for the ideal situation. For example, the actual water vapor content in air will be over four times that predicted by ideal gas behavior at a temperature of 228 K and a pressure of 20 MPa. Purification by a solid adsorbent is one of the most common lowtemperature methods for removing impurities. Materials such as silica gel, carbon, and synthetic zeolites (molecular sieves) are widely used as adsorbents because of their extremely large effective surface areas. Most of the gels and carbon have pores of varying sizes in a given sample, but the synthetic zeolites are manufactured with closely controlled pore-size openings ranging from about 0.4 to 1.3 nm. This makes them even more selective than other adsorbents since it permits separation of gases on the basis of molecular size. Information needed in the design of low-temperature adsorbers N2 liquefier Crude He Crude He

A

B

Crude He D separator

F

N2 gas

H

G

Heat exchanger

Crude He Heat exchanger

He heat exchanger C Nat. gas

Compressed natural gas Natural Separator gas FIG. 11-119

11-103

E

We

95% N2

N2 He purifier

N2 + He in Pure He solution separator

Typical helium-separation plant as operated by the U.S. Bureau of Mines.

98.5% He

11-104

HEAT-TRANSFER EQUIPMENT

includes the equilibrium between the solid and gas and the rate of adsorption. Equilibrium data for the common systems generally are available from the suppliers of such material. The rate of adsorption is usually very rapid and the adsorption is essentially complete in a relatively narrow zone of the adsorber. If the concentration of the adsorbed gas is more than a trace, then heat of adsorption may also be a factor of importance in the design. (The heat of adsorption is usually of the same order or larger than the normal heat associated with the phase change.) Under such situations it is generally advisable to design the purification in two steps, that is, first removing a significant portion of the impurity either by condensation or chemical reaction and then completing the purification with a low-temperature adsorption system. A scheme combining the condensation and adsorption is shown in Fig. 11-120. In normal plant operation at least two adsorption purifiers are employed—one in service while the other being desorbed of its impurities. In some cases there is an advantage in using an additional purifier by placing this unit in series with the adsorption unit to provide a backup if impurities are not trapped by the first unit. The cooling of the purifier must be effected with some of the purified gas to avoid adsorption during this period. Experience in air separation plant operations and other cryogenic processing plants has shown that local freeze-out of impurities such as carbon dioxide can occur at concentrations well below the solubility limit. For this reason, the carbon dioxide content of the feed gas subject to the minimum operating temperature is usually kept below 50 ppm. The amine process and the molecular sieve adsorption process are the most widely used methods for carbon dioxide removal. The amine process involves adsorption of the impurity by a lean aqueous organic amine solution. With sufficient amine recirculation rate, the carbon dioxide in the treated gas can be reduced to less than 25 ppm. Oxygen is removed by a catalytic reaction with hydrogen to form water. STORAGE AND TRANSFER SYSTEMS Storage vessels range in type from low-performance containers, insulated by rigid foam or fibrous insulation where the liquid in the container boils away in a few hours, up to high-performance containers,

Liquid nitrogen

Dried gas

N2 vapor Refrigeration purifier Ice

,, ,, ,, ,, Adsorbent

Gas in N2 vapor Water, oil, etc.

LN2

Purified gas FIG. 11-120

Purifier using refrigeration and adsorption schemes in series.

insulated with multilayer insulations where less than 0.1 percent of the fluid contents is evaporated per day. In the more effective units, the storage container consists of an inner vessel which encloses the cryogenic fluid to be stored and an outer vessel or vacuum jacket. The latter maintains the vacuum necessary to make the insulation effective and at the same time serves as a vapor barrier to the migration of water and other condensibles to the cold surface of the inner vessel. Improvements have been made in the insulation used in these containers, but the vacuum-insulated double-walled Dewar is still the basic idea for high-performance cryogenic-fluid container designer. Insulation Principles The effectiveness of a liquefier or refrigerator is highly dependent upon the heat leak entering such a system. Since heat removal becomes more costly with a lowering in temperature as demonstrated by the Carnot limitation, most cryogenic systems employ some form of insulation to minimize the effect. The insulation strategy is to minimize radiative heat transfer, minimize convective heat transfer, and use only a minimum of solid conductance media. Factors considered in the selection of the most suitable insulation include its ruggedness, convenience, volume, weight, ease of fabrication and handling, and thermal effectiveness and cost. It is common practice to use an experimentally obtained apparent thermal conductivity to characterize the thermal effectiveness of various insulations. Typical ka values for insulations used in cryogenic service are listed in Table 11-26. TABLE 11-26 Representative Apparent Thermal Conductivity Values Type of insulation

ka, mW/m⋅K (77–300K)

Pure vacuum, 1.3 × 10−4 Pa Foam insulation Nonevacuated powders (perlite, silica aerogel) Evacuated powders and fibers (1.3 × 10−1 Pa) Opacified powdered insulations (1.3 × 10−1 Pa) Multilayer insulations (1.3 × 10−4 Pa)

5 26–35 19–44 1–2 3.5 × 10−1 1.7–4 × 10−2

Types of Insulation Cryogenic insulations have generally been divided into five general categories: high vacuum, multilayer insulation, powder, foam, and special insulations. Each is discussed in turn in the following sections. Vacuum Insulation Heat transport across an evacuated space (1.3 × 10−4 Pa or lower), is by radiation and by conduction through the residual gas. The heat transfer by radiation generally is predominant and can be approximated by −1 Q˙r 1 A1 1 (11-119)
= σ (T 42 − T 14)
+

− 1 A1 e1 A2 e2 where Q˙r /A1 is the heat transfer by radiation-per-unit area, σ is the Stefan-Boltzmann constant, and e is the emissivity of the surfaces. The subscript 2 refers to the hot surface and the subscript 1 refers to the cold surface. The bracketed term on the right-hand side of this relation is designated as the overall emissivity factor, Fe. The insertion of low-emissivity floating shields within the evacuated space can effectively reduce the heat transport by radiation. The effect of the shields is to greatly reduce the emissivity factor. For example, for N shields or (N + 2) surfaces, an emissivity of the outer and inner surface of eo, and an emissivity of the shields of es, the emissivity factor reduces to (N − 1)(2 − es) −1 1 1 2
+
− 1 +

(11-120) eo es es In essence, one properly located low-emissivity shield can reduce the radiant heat transfer to around one-half of the rate without the shield, two shields can reduce this to around one-fourth of the rate without the shield, and so on. Multilayer Insulation Multilayer insulation consists of alternating layers of highly reflecting material, such as aluminum foil or aluminized Mylar, and a low-conductivity spacer material or insulator, such as fiberglass mat or paper, glass fabric, or nylon net, all under high vacuum. When properly applied at the optimum density, this type of insulation can have an apparent thermal conductivity as low as 10 to 50 µW/m-K between 20 and 300 K.



 









CRYOGENIC PROCESSES For a highly evacuated (on the order of 1.3 × 10−4 Pa) multilayer insulation, heat is transferred primarily by radiation and solid conduction through the spacer material. The apparent thermal conductivity of the insulation material under these conditions may be determined from T 1 σeT23 T 2 ka =
hs +
1 +
1 1 +
1 (11-121) N/∆x 2−e T2 T2



  



where N/∆x is the number of complete layers (reflecting shield plus spacer) of the insulation-per-unit thickness, hs is the solid conductance of the spacer material, σ is the Stefan-Boltzmann constant, e is the effective emissivity of the reflecting shield, and T2 and T1 are the temperatures of the warm and cold sides of the insulation, respectively. It is evident that the apparent thermal conductivity can be reduced by increasing the layer density up to a certain point. It is not obvious from the above relation that a compressive load affects the apparent thermal conductivity and thus the performance of a multilayer insulation. However, under a compressive load the solid conductance increases much more rapidly than N/∆x resulting in an overall increase in ka. Plots of heat flux versus compressive load on a logarithmic scale result in straight lines with slopes between 0.5 and 0.67. The effective thermal conductivity values generally obtained in practice are at least a factor of two greater than the one-dimensional thermal conductivity values measured in the laboratory with carefully controlled techniques. This degradation in insulation thermal performance is caused by the combined presence of edge exposure to isothermal boundaries, gaps, joints, or penetrations in the insulation blanket required for structural supports, fill and vent lines, and high lateral thermal conductivity of these insulation systems. Powder Insulation A method of realizing some of the benefits of multiple floating shields without incurring the difficulties of awkward structural complexities is to use evacuated powder insulation. The penalty incurred in the use of this type of insulation, however, is a tenfold reduction in the overall thermal effectiveness of the insulation system over that obtained for multilayer insulation. In applications where this is not a serious factor, such as LNG storage facilities, and investment cost is of major concern, even unevacuated powderinsulation systems have found useful applications. The variation in apparent mean thermal conductivity of several powders as a function of interstitial gas pressure is shown in the familiar S-shaped curves of Fig. 11-121. 105

eres ph

olic

s

Note: Nitrogen and helium identify the interstitial gas in the powder.

L

ck bla

S ili c a

aer g

el

Di 103

m e a a ce rth ou s

kg, µW/mK

Ph

en

104

p am

at

o

Perlite Perlite

102 10–3

10–2

Silic

e aa

10–1

rog

el

T2 = 300K T1 = 76K Nitrogen

T2 = 76K T1 = 20K Helium 10 101 102 Pressure, pascals

103

104

Apparent mean thermal conductivities of several powder insulations as a function of interstitial gas pressure. FIG. 11-121

11-105

The apparent thermal conductivity of powder insulation at cryogenic temperatures is generally obtained from kg ka =

(11-122) 1 − Vr(1 − kg/ks) where kg is the thermal conductivity of the gas within the insulation, ks is the thermal conductivity of the powder, and Vr is the ratio of solid volume to the total volume. The amount of heat transport due to radiation through the powders can be reduced by the addition of metallic powders. A mixture containing approximately 40 to 50 wt % of a metallic powder gives the optimum performance. Foam Insulation Since foams are not homogeneous materials, their apparent thermal conductivity is dependent upon the bulk density of the insulation, the gas used to foam the insulation, and the mean temperature of the insulation. Heat conduction through a foam is determined by convection and radiation within the cells and by conduction in the solid structure. Evacuation of a foam is effective in reducing its thermal conductivity, indicating a partially open cellular structure, but the resulting values are still considerably higher than either multilayer or evacuated powder insulations. Data on the thermal conductivity for a variety of foams used at cryogenic temperatures have been presented by Kropschot (Cryogenic Technology, R. W. Vance, ed., Wiley, New York, 1963, p. 239). Of all the foams, polyurethane and polystryene have received the widest use at low temperatures. The major disadvantage of foams is that they tend to crack upon repeated thermal cycling and lose their insulation value. Storage and Transfer Systems In general, heat leak into a storage or transfer system for a cryogen is by (1) radiation and conduction through the insulation, and (2) conduction through any inner shell or transfer-line supports, piping leads, and access ports. Conduction losses are reduced by introducing long heat-leak paths, by making the cross sections for heat flow small, and by using materials with low thermal conductivity. Radiation losses, a major factor in the heat leak through insulations, are reduced with the use of radiation shields, such as multilayer insulation, boil-off vapor-cooled shields, and opacifiers in powder insulation. Several considerations must be met when designing the inner vessel. The material of construction selected must be compatible with the stored cryogen. Nine percent nickel steels are acceptable for the higher-boiling cryogens (T > 75 K) while many aluminum alloys and austenitic steels are usually structurally acceptable throughout the entire temperature range. Because of its high thermal conductivity, aluminum is not a recommended material for piping and supports that must cross the insulation space. A change to a material of lower thermal conductivity for this purpose introduces a transition joint of a dissimilar material. Since such transition joints are generally mechanical in nature, leaks into the vacuum space develop upon repeated temperature cycling. In addition, the larger thermal coefficient of expansion of aluminum can pose still further support and cooldown problems. Economic and cooldown considerations dictate that the shell of the storage container be as thin as possible. As a consequence, the inner container is designed to withstand only the internal pressure and bending forces while stiffening rings are used to support the weight of the fluid. The minimum thickness of the inner shell for a cylindrical vessel under such a design arrangement is given by Sec. VIII of the ASME Boiler and Pressure Vessel Code. Since the outer shell of the storage container is subjected to atmospheric pressure on one side and evacuated conditions going down to 1.3 × 10−4 Pa on the other, consideration must be given to provide ample thickness of the material to withstand collapsing or buckling. Failure by elastic instability is covered by the ASME Code, in which design charts are available for the design of cylinders and spheres subjected to external pressure. Stiffening rings are also used on the outer shell to support the weight of the inner container and its contents as well as maintaining the sphericity of the shell. The outer shell is normally constructed of carbon steel for economic reasons, unless aluminum is required to reduce the weight. Stainless-steel standoffs must be provided on the carbon steel outer shell for all piping penetrations to avoid direct contact with these penetrations when they are cold.

11-106

HEAT-TRANSFER EQUIPMENT

There are a variety of methods for supporting the inner shell within the outer shell and the cold transfer line within the outer line. Materials that have a high strength to thermal conductivity ratio are selected for these supports. Design of these supports for the inner shell must allow for shipping loads which may be several orders higher than inservice loads. Compression supports such as legs or pads may be used, but tension supports are more common. These may take the form of cables, welded straps, threaded bars, or a combination of these to provide restraint of the inner shell in several directions. Most storage containers for cryogens are designed for a 10 percent ullage volume. The latter permits reasonable vaporization of the contents due to heat leak without incurring too rapid a buildup of the pressure in the container. This, in turn, permits closure of the container for short periods of time to either avoid partial loss of the contents or to transport flammable or hazardous cryogens safely from one location to another. CRYOGENIC INSTRUMENTATION Even though the combined production of cryogenic nitrogen and oxygen exceeds the production of any other chemical in the United States, the cryogenic industry does not appear to warrant a separate product line of instruments for diagnostic and control purposes. Lowtemperature thermometry is the one exception. The general approach generally is that instruments developed for the usual CPI needs must be modified or accepted as is for cryogenic use. Quite often problems arise when instruments for normal service are subjected to low temperature use. Since some metals become brittle at low temperatures, the instrument literally falls apart. Elastomeric gaskets and seals contract faster with decreasing temperatures than the surrounding metal parts, and the seal often is lost. Even hermetically sealed instruments can develop pin holes or small cracks to permit cryogenic liquids to enter these cases with time. Warming the instrument causes the trapped liquid to vaporize, sometimes generating excessive gas pressure and failure of the case. Therefore, the first task in adapting normal instruments to cryogenic service is simply to give them a severe thermal shock by immersing them in liquid nitrogen repeatedly, and checking for mechanical integrity. This is the general issue; specific issues according to each type of measurement are discussed below. Pressure This parameter is usually measured by the flushmounted pressure transducer which consists of a force-summing device (bellow, diaphragm, bourdon tube, etc.) that translates the pressure into a displacement. The latter is then measured by an analog device (strain gage, piezeoelectric crystal, variable distance between capacitor plates, and the like). Since these elements are likely to be made of different materials (bronze diaphragm, stainless-steel case, semiconductor strain gage), each will react to the temperature change in a different way. This is especially serious during cooldown, when the transient nature of material and construction prohibits all of the pressure-gage elements from being at the same temperature at the same time. Under steady-state conditions it is often possible to provide some temperature compensation through the well-known instrument technique of common-mode-rejection. Such compensation is generally not successful during transient temperature fluctuations. Only two courses of action are open: (1) hand-check each type of pressure transducer for thermal noise by thermally shocking it with immersion in liquid nitrogen; and (2) simplify the pressure-transducer construction to eliminate differences between materials. Some success has been observed in the latter area by manufacturers who make very small pressure sensing elements from a single semiconductor chip. The miniature size of these devices helps to reduce or eliminate temperature gradients across the device. The single-element nature of the pressure-gage assembly reduces differences in materials of construction. Liquid Level The measurements for dense fluids such as liquid oxygen and liquid nitrogen are made in the conventional CPI approach using floats. Sight glasses cannot be used since radiation and thermal conduction would cause the cryogenic fluid within the sight glass to boil. The very light cryogens, liquid helium and liquid hydrogen, cannot sustain a float. Liquid hydrogen has the density of Styro-

foam,™ about 70 g/l, making floating devices impractical. Some electrical analog is used for hydrogen and helium, most frequently a linear concentric-tube electrical capacitor. The dielectric constant of cryogens is related to their density by the Clausius-Mosotti relation. As the liquid level rises, the greater dielectric constant of the liquid between the tubes causes the overall capacitance to vary in a linear fashion. For best accuracy, these capacitance liquid-level measuring devices should be calibrated in place. Flow The measurement of cryogenic fluids is most troublesome. Flow rate is not a natural physical parameter, like temperature, but is a derived quantity. A measurement of mass (or volume) must be made over a time interval to derive the flow rate. Because of this, any flow meter is only as good as its calibration. At this time, there is no national capability for calibrating cryogenic flowmeters. From data developed early in the nation’s space program, considerable confidence has been developed in turbine-type flowmeters and in pressure-drop-type flowmeters. If all the usual ASTM guidelines are followed for meter installation, and if adequate temperature corrections are applied to changes in dimensions, then such meters can have an accuracy of 1 percent of their water calibrations. For very small flow applications, the Coriolis meters are promising. Vortex shedding flow meters appear useful for very large flow rates. Nonetheless, an actual calibration on the cryogen of interest is the only proof of accuracy. Temperature The level of the temperature measurement (4 K, 20 K, 77 K, or higher) is the first issue to be considered. The second issue is the range needed (e.g., a few degrees around 90 K or 1 to 400 K). If the temperature level is that of air separation or liquefacting of natural gas (LNG), then the favorite choice is the platinum resistance thermometer (PRT). Platinum, as with all pure metals, has an electrical resistance that goes to zero as the absolute temperature decreases to zero. Accordingly, the lower useful limit of platinum is about 20 K, or liquid hydrogen temperatures. Below 20 K, semiconductor thermometers (germanium-, carbon-, or silicon-based) are preferred. Semiconductors have just the opposite resistancetemperature dependence of metals—their resistance increases as the temperature is lowered, as fewer valence electrons can be promoted into the conduction band at lower temperatures. Thus, semiconductors are usually chosen for temperatures from about 1 to 20 K. If the temperature range of interest is large, say 1 to 400 K, then diode thermometers are recommended. Diodes have other advantages compared to resistance thermometers. By contrast, diode thermometers are very much smaller and faster. By selection of diodes all from the same melt, they may be made interchangeable. That is, one diode has the same calibration curve as another, which is not always the case with either semiconductor or metallic-resistance thermometers. It is well known, however, that diode thermometers may rectify an ac field, and thus may impose a dc noise on the diode output. Adequate shielding is required. Special applications, such as in high-magnetic fields, require special thermometers. The carbon-glass and strontium-titinate resistance thermometers have the least magnetoresistance effects. Thermocouples are unsurpassed for making temperature-difference measurements. The thermoelectric power of thermocouple materials makes them adequate for use at liquid-air temperatures and above. At 20 K and below, the thermoelectric power drops to a few µV/K, and their use in this range is as much art as science. A descriptive flowchart has been prepared by Sparks (Materials at Low Temperatures, ASM, Metal Park, OH, 1983) to show the temperature range of cryogenic thermometers in general use today. Parese and Molinar (Modern Gas-Based Temperature and Pressure Measurements, Plenum, New York, 1992) provide details on gas- and vapor-pressure thermometry at these temperatures. SAFETY Past experience has shown that cryogenic fluids can be used safely in industrial environments as well as in typical laboratories provided all facilities are properly designed and maintained, and personnel handling these fluids are adequately trained and supervised. There are many hazards associated with cryogenic fluids. However, the principal

EVAPORATORS ones are those associated with the response of the human body and the surroundings to the fluids and their vapors, and those associated with reactions between the fluids and their surroundings. Physiological Hazards Severe cold “burns” may be inflicted if the human body comes in contact with cryogenic fluids or with surfaces cooled by cryogenic fluids. Damage to the skin or tissue is similar to an ordinary burn. Because the body is composed mainly of water, the low temperature effectively freezes the tissue—damaging or destroying it. The severity of the burn depends upon the contact area and the contact time with prolonged contact resulting in deeper burns. Cold burns are accompanied by stinging sensations and pain similar to those of ordinary burns. The ordinary reaction is to withdraw that portion of the body that is in contact with the cold surface. Severe burns are seldom sustained if withdrawal is possible. Cold gases may not be damaging if the turbulence in the gas is low, particularly since the body can normally adjust for a heat loss of 95 J/m2s for an area of limited exposure. If the heat loss becomes much greater than this, the skin temperature drops and freezing of the affected area may ensue. Freezing of facial tissue will occur in about 100 s if the heat loss is 2,300 J/m2s. Materials and Construction Hazards Construction materials for noncryogenic service usually are chosen on the basis of tensile strength, fatigue life, weight, cost, ease of fabrication, corrosion resistance, and so on. When working with low temperatures the designer must consider the ductility of the material since low temperatures, as noted earlier, have the effect of making some construction materials brittle or less ductile. Some materials become brittle at low temperatures but still can absorb considerable impact, while others become brittle and lose their impact strength. Flammability and Explosion Hazards In order to have a fire or an explosion requires the combination of an oxidant, a fuel, and an ignition source. Generally the oxidizer will be oxygen. The latter may be available from a variety of sources including leakage or spillage, condensation of air on cryogenically cooled surfaces below 90 K, and buildup, as a solid impurity in liquid hydrogen. The fuel may be almost any noncompatible material or flammable gas; compatible materials can also act as fuels in the presence of extreme heat (strong

11-107

ignition sources). The ignition source may be a mechanical or electrostatic spark, flame, impact, heat by kinetic effects, friction, chemical reaction, and so on. Certain combinations of oxygen, fuel, and ignition sources will always result in fire or explosion. The order of magnitude of flammability and detonability limits for fuel-oxidant gaseous mixtures of two widely used cryogens is shown in Table 11-27. TABLE 11-27 Flammability and Detonability Limits of Hydrogen and Methane Gas Mixture

Flammability Limits (mol %)

Detonability Limits (mol %)

H2-air H2-O2 CH4-air CH4-O2

4–75 4–95 5–15 5–61

20–65 15–90 6–14 10–50

High-Pressure Gas Hazards Potential hazards also exist in highly compressed gases because of the stored energy. In cryogenic systems such high pressures are obtained by gas compression during liquefaction or refrigeration, by pumping of liquids to high pressure followed by evaporation, and by confinement of cryogenic liquids with subsequent evaporation. If this confined gas is suddenly released through a rupture or break in a line, a significant thrust may be experienced. For example, the force generated by rupturing a 2.5-cm diameter valve located on a 13.9-MPa pressurized gas cylinder would be over 6670 N. SUMMARY It is obvious that the best designed facility is no better than the attention that is paid to safety. The latter is not considered once and forgotten. Rather, it is an ongoing activity that requires constant attention to every conceivable hazard that might be encountered. Because of its importance, safety, particularly at low temperatures, has received a large focus in the literature with its own safety manual prepared by NIST as well as by the British Cryogenics Council.

EVAPORATORS GENERAL REFERENCES: Badger and Banchero, Introduction to Chemical Engineering, McGraw-Hill, New York, 1955. Standiford, Chem. Eng., 70, 158–176 (Dec. 9, 1963). Testing Procedure for Evaporators, American Institute of Chemical Engineers, 1979. Upgrading Evaporators to Reduce Energy Consumption, ERDA Technical Information Center, Oak Ridge, Tenn., 1977.

PRIMARY DESIGN PROBLEMS Heat Transfer This is the most important single factor in evaporator design, since the heating surface represents the largest part of evaporator cost. Other things being equal, the type of evaporator selected is the one having the highest heat-transfer cost coefficient under desired operating conditions in terms of J/s⋅K (British thermal units per hour per degree Fahrenheit) per dollar of installed cost. When power is required to induce circulation past the heating surface, the coefficient must be even higher to offset the cost of power for circulation. Vapor-Liquid Separation This design problem may be important for a number of reasons. The most important is usually prevention of entrainment because of value of product lost, pollution, contamination of the condensed vapor, or fouling or corrosion of the surfaces on which the vapor is condensed. Vapor-liquid separation in the vapor head may also be important when spray forms deposits on the walls, when vortices increase head requirements of circulating pumps, and when short circuiting allows vapor or unflashed liquid to be carried back to the circulating pump and heating element. Evaporator performance is rated on the basis of steam economy— kilograms of solvent evaporated per kilogram of steam used. Heat is

required (1) to raise the feed from its initial temperature to the boiling temperature, (2) to provide the minimum thermodynamic energy to separate liquid solvent from the feed, and (3) to vaporize the solvent. The first of these can be changed appreciably by reducing the boiling temperature or by heat interchange between the feed and the residual product and/or condensate. The greatest increase in steam economy is achieved by reusing the vaporized solvent. This is done in a multiple-effect evaporator by using the vapor from one effect as the heating medium for another effect in which boiling takes place at a lower temperature and pressure. Another method of increasing the utilization of energy is to employ a thermocompression evaporator, in which the vapor is compressed so that it will condense at a temperature high enough to permit its use as the heating medium in the same evaporator. Selection Problems Aside from heat-transfer considerations, the selection of type of evaporator best suited for a particular service is governed by the characteristics of the feed and product. Points that must be considered are crystallization, salting and scaling, product quality, corrosion, and foaming. In the case of a crystallizing evaporator, the desirability of producing crystals of a definite uniform size usually limits the choice to evaporators having a positive means of circulation. Salting, which is the growth on body and heating-surface walls of a material having a solubility that increases with increase in temperature, is frequently encountered in crystallizing evaporators. It can be reduced or eliminated by keeping the evaporating liquid in close or frequent contact with a large surface area of crystallized solid. Scaling is the deposition and growth on body walls, and especially on heating surfaces, of a material undergoing an irreversible chemical

11-108

HEAT-TRANSFER EQUIPMENT

reaction in the evaporator or having a solubility that decreases with an increase in temperature. Scaling can be reduced or eliminated in the same general manner as salting. Both salting and scaling liquids are usually best handled in evaporators that do not depend on boiling to induce circulation. Fouling is the formation of deposits other than salt or scale and may be due to corrosion, solid matter entering with the feed, or deposits formed by the condensing vapor. Product Quality Considerations of product quality may require low holdup time and low-temperature operation to avoid thermal degradation. The low holdup time eliminates some types of evaporators, and some types are also eliminated because of poor heat-transfer characteristics at low temperature. Product quality may also dictate special materials of construction to avoid metallic contamination or a catalytic effect on decomposition of the product. Corrosion may also influence evaporator selection, since the advantages of evaporators having high heat-transfer coefficients are more apparent when expensive materials of construction are indicated. Corrosion and erosion are frequently more severe in evaporators than in other types of equipment because of the high liquid and vapor velocities used, the frequent presence of solids in suspension, and the necessary concentration differences.

(a)

(c)

(b)

EVAPORATOR TYPES AND APPLICATIONS Evaporators may be classified as follows: 1. Heating medium separated from evaporating liquid by tubular heating surfaces. 2. Heating medium confined by coils, jackets, double walls, flat plates, etc. 3. Heating medium brought into direct contact with evaporating liquid. 4. Heating by solar radiation. By far the largest number of industrial evaporators employ tubular heating surfaces. Circulation of liquid past the heating surface may be induced by boiling or by mechanical means. In the latter case, boiling may or may not occur at the heating surface. Forced-Circulation Evaporators (Fig. 11-122 a, b, c) Although it may not be the most economical for many uses, the forced-circulation (FC) evaporator is suitable for the widest variety of evaporator applications. The use of a pump to ensure circulation past the heating surface makes possible separating the functions of heat transfer, vapor-liquid separation, and crystallization. The pump withdraws liquor from the flash chamber and forces it through the heating element back to the flash chamber. Circulation is maintained regardless of the evaporation rate; so this type of evaporator is well suited to crystallizing operation, in which solids must be maintained in suspension at all times. The liquid velocity past the heating surface is limited only by the pumping power needed or available and by accelerated corrosion and erosion at the higher velocities. Tube velocities normally range from a minimum of about 1.2 m/s (4 ft/s) in salt evaporators with copper or brass tubes and liquid containing 5 percent or more solids up to about 3 m/s (10 ft/s) in caustic evaporators having nickel tubes and liquid containing only a small amount of solids. Even higher velocities can be used when corrosion is not accelerated by erosion. Highest heat-transfer coefficients are obtained in FC evaporators when the liquid is allowed to boil in the tubes, as in the type shown in Fig. 11-122a. The heating element projects into the vapor head, and the liquid level is maintained near and usually slightly below the top tube sheet. This type of FC evaporator is not well suited to salting solutions because boiling in the tubes increases the chances of salt deposit on the walls and the sudden flashing at the tube exits promotes excessive nucleation and production of fine crystals. Consequently, this type of evaporator is seldom used except when there are headroom limitations or when the liquid forms neither salt nor scale. By far the largest number of forced-circulation evaporators are of the submerged-tube type, as shown in Fig. 11-122b. The heating element is placed far enough below the liquid level or return line to the flash chamber to prevent boiling in the tubes. Preferably, the hydrostatic head should be sufficient to prevent boiling even in a tube that is plugged (and hence at steam temperature), since this prevents salting of the entire tube. Evaporators of this type sometimes have hori-

(d)

(e)

(f)

(j)

(h)

(i)

(j)

Evaporator types. (a) Forced circulation. (b) Submerged-tube forced circulation. (c) Oslo-type crystallizer. (d) Short-tube vertical. (e) Propeller calandria. ( f) Long-tube vertical. (g) Recirculating long-tube vertical. (h) Falling film. (i,j) Horizontal-tube evaporators. C = condensate; F = feed; G = vent; P = product; S = steam; V = vapor; ENT’T = separated entrainment outlet.

FIG. 11-122

zontal heating elements (usually two-pass), but the vertical single-pass heating element is used whenever sufficient headroom is available. The vertical element usually has a lower friction loss and is easier to clean or retube than a horizontal heater. The submerged-tube forcedcirculation evaporator is relatively immune to salting in the tubes, since no supersaturation is generated by evaporation in the tubes. The tendency toward scale formation is also reduced, since supersaturation in the heating element is generated only by a controlled amount of heating and not by both heating and evaporation. The type of vapor head used with the FC evaporator is chosen to suit the product characteristics and may range from a simple centrifugal separator to the crystallizing chambers shown in Fig. 11-122b and c. Figure 11-122b shows a type frequently used for common salt. It is designed to circulate a slurry of crystals throughout the system. Figure 11-122c shows a submerged-tube FC evaporator in which heating, flashing, and crystallization are completely separated. The crystallizing solids are maintained as a fluidized bed in the chamber below the vapor head and little or no solids circulate through the heater and

EVAPORATORS flash chamber. This type is well adapted to growing coarse crystals, but the crystals usually approach a spherical shape, and careful design is required to avoid production of tines in the flash chamber. In a submerged-tube FC evaporator, all heat is imparted as sensible heat, resulting in a temperature rise of the circulating liquor that reduces the overall temperature difference available for heat transfer. Temperature rise, tube proportions, tube velocity, and head requirements on the circulating pump all influence the selection of circulation rate. Head requirements are frequently difficult to estimate since they consist not only of the usual friction, entrance and contraction, and elevation losses when the return to the flash chamber is above the liquid level but also of increased friction losses due to flashing in the return line and vortex losses in the flash chamber. Circulation is sometimes limited by vapor in the pump suction line. This may be drawn in as a result of inadequate vapor-liquid separation or may come from vortices near the pump suction connection to the body or may be formed in the line itself by short circuiting from heater outlet to pump inlet of liquor that has not flashed completely to equilibrium at the pressure in the vapor head. Advantages of forced-circulation evaporators: 1. High heat-transfer coefficients 2. Positive circulation 3. Relative freedom from salting, scaling, and fouling Disadvantages of forced-circulation evaporators: 1. High cost 2. Power required for circulating pump 3. Relatively high holdup or residence time Best applications of forced-circulation evaporators: 1. Crystalline product 2. Corrosive solutions 3. Viscous solutions Frequent difficulties with forced-circulation evaporators: 1. Plugging of tube inlets by salt deposits detached from walls of equipment 2. Poor circulation due to higher than expected head losses 3. Salting due to boiling in tubes 4. Corrosion-erosion

Short-Tube Vertical Evaporators (Fig. 11-122d) This is one of the earliest types still in widespread commercial use. Its principal use at present is in the evaporation of cane-sugar juice. Circulation past the heating surface is induced by boiling in the tubes, which are usually 50.8 to 76.2 mm (2 to 3 in) in diameter by 1.2 to 1.8 m (4 to 6 ft) long. The body is a vertical cylinder, usually of cast iron, and the tubes are expanded into horizontal tube sheets that span the body diameter. The circulation rate through the tubes is many times the feed rate; so there must be a return passage from above the top tube sheet to below the bottom tube sheet. Most commonly used is a central well or downtake as shown in Fig. 11-122d. So that friction losses through the downtake do not appreciably impede circulation up through the tubes, the area of the downtake should be of the same order of magnitude as the combined cross-sectional area of the tubes. This results in a downtake almost half of the diameter of the tube sheet. Circulation and heat transfer in this type of evaporator are strongly affected by the liquid “level.” Highest heat-transfer coefficients are achieved when the level, as indicated by an external gauge glass, is only about halfway up the tubes. Slight reductions in level below the optimum result in incomplete wetting of the tube walls with a consequent increased tendency to foul and a rapid reduction in capacity. When this type of evaporator is used with a liquid that can deposit salt or scale, it is customary to operate with the liquid level appreciably higher than the optimum and usually appreciably above the top tube sheet. Circulation in the standard short-tube vertical evaporator is dependent entirely on boiling, and when boiling stops, any solids present settle out of suspension. Consequently, this type is seldom used as a crystallizing evaporator. By installing a propeller in the downtake, this objection can be overcome. Such an evaporator, usually called a pro-

11-109

peller calandria, is illustrated in Fig. 11-122e. The propeller is usually placed as low as possible to reduce cavitation and is shrouded by an extension of the downtake well. The use of the propeller can sometimes double the capacity of a short-tube vertical evaporator. The evaporator shown in Fig. 11-122e includes an elutriation leg for salt manufacture similar to that used on the FC evaporator of Fig. 11122b. The shape of the bottom will, of course, depend on the particular application and on whether the propeller is driven from above or below. To avoid salting when the evaporator is used for crystallizing solutions, the liquid level must be kept appreciably above the top tube sheet. Advantages of short-tube vertical evaporators: 1. High heat-transfer coefficients at high temperature differences 2. Low headroom 3. Easy mechanical descaling 4. Relatively inexpensive Disadvantages of short-tube vertical evaporators: 1. Poor heat transfer at low temperature differences and low temperature 2. High floor space and weight 3. Relatively high holdup 4. Poor heat transfer with viscous liquids Best applications of short-tube vertical evaporators: 1. Clear liquids 2. Crystalline product if propeller is used 3. Relatively noncorrosive liquids, since body is large and expensive if built of materials other than mild steel or cast iron 4. Mild scaling solutions requiring mechanical cleaning, since tubes are short and large in diameter

Long-Tube Vertical Evaporators (Fig. 11-122f, g, h) More total evaporation is accomplished in this type than in all others combined because it is normally the cheapest per unit of capacity. The long-tube vertical (LTV) evaporator consists of a simple one-pass vertical shell-and-tube heat exchanger discharging into a relatively small vapor head. Normally, no liquid level is maintained in the vapor head, and the residence time of liquor is only a few seconds. The tubes are usually about 50.8 mm (2 in) in diameter but may be smaller than 25.4 mm (1 in). Tube length may vary from less than 6 to 10.7 m (20 to 35 ft) in the rising film version and to as great as 20 m (65 ft) in the falling film version. The evaporator is usually operated single-pass, concentrating from the feed to discharge density in just the time that it takes the liquid and evolved vapor to pass through a tube. An extreme case is the caustic high concentrator, producing a substantially anhydrous product at 370°C (700°F) from an inlet feed of 50 percent NaOH at 149°C (300°F) in one pass up 22-mm- (8/8-in-) outside-diameter nickel tubes 6 m (20 ft) long. The largest use of LTV evaporators is for concentrating black liquor in the pulp and paper industry. Because of the long tubes and relatively high heat-transfer coefficients, it is possible to achieve higher single-unit capacities in this type of evaporator than in any other. The LTV evaporator shown in Fig. 11-122f is typical of those commonly used, especially for black liquor. Feed enters at the bottom of the tube and starts to boil partway up the tube, and the mixture of liquid and vapor leaving at the top at high velocity impinges against a deflector placed above the tube sheet. This deflector is effective both as a primary separator and as a foam breaker. In many cases, as when the ratio of feed to evaporation or the ratio of feed to heating surface is low, it is desirable to provide for recirculation of product through the evaporator. This can be done in the type shown in Fig. 11-122f by adding a pipe connection between the product line and the feed line. Higher recirculation rates can be achieved in the type shown in Fig. 11-122g, which is used widely for condensed milk. By extending the enlarged portion of the vapor head still lower to provide storage space for liquor, this type can be used as a batch evaporator. Liquid temperatures in the tubes of an LTV evaporator are far from uniform and are difficult to predict. At the lower end, the liquid is usually not boiling, and the liquor picks up heat as sensible heat. Since entering liquid velocities are usually very low, true heat-transfer coef-

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FIG. 11-123

HEAT-TRANSFER EQUIPMENT

Temperature variations in a long-tube vertical evaporator.

ficients are low in this nonboiling zone. At some point up the tube, the liquid starts to boil, and from that point on the liquid temperature decreases because of the reduction in static, friction, and acceleration heads until the vapor-liquid mixture reaches the top of the tubes at substantially vapor-head temperature. Thus the true temperature difference in the boiling zone is always less than the total temperature difference as measured from steam and vapor-head temperatures. Although the true heat-transfer coefficients in the boiling zone are quite high, they are partially offset by the reduced temperature difference. The point in the tubes at which boiling starts and at which the maximum temperature is reached is sensitive to operating conditions, such as feed properties, feed temperature, feed rate, and heat flux. Figure 11-123 shows typical variations in liquid temperature in tubes of an LTV evaporator operating at a constant terminal temperature difference. Curve 1 shows the normal case in which the feed is not boiling at the tube inlet. Curve 2 gives an indication of the temperature difference lost when the feed enters at the boiling point. Curve 3 is for exactly the same conditions as curve 2 except that the feed contained 0.01 percent Teepol to reduce surface tension [Coulson and Mehta, Trans. Inst. Chem. Eng., 31, 208 (1953)]. The surface-active agent yields a more intimate mixture of vapor and liquid, with the result that liquid is accelerated to a velocity more nearly approaching the vapor velocity, thereby increasing the pressure drop in the tube. Although the surface-active agent caused an increase of more than 100 percent in the true heat-transfer coefficient, this was more than offset by the reduced temperature difference so that the net result was a reduction in evaporator capacity. This sensitivity of the LTV evaporator to changes in operating conditions is less pronounced at high than at low temperature differences and temperature levels. The falling-film version of the LTV evaporator (Fig. 11-122h) eliminates these problems of hydrostatic head. Liquid is fed to the tops of the tubes and flows down the walls as a film. Vapor-liquid separation usually takes place at the bottom, although some evaporators of this type are arranged for vapor to rise through the tube countercurrently to the liquid. The pressure drop through the tubes is usually very small, and the boiling-liquid temperature is substantially the same as the vapor-head temperature. The falling-film evaporator is widely used for concentrating heat-sensitive materials, such as fruit juices, because the holdup time is very small, the liquid is not overheated during passage through the evaporator, and heat-transfer coefficients are high even at low boiling temperatures. The principal problem with the falling-film LTV evaporator is that of feed distribution to the tubes. It is essential that all tube surfaces be wetted continually. This usually requires recirculation of the liquid unless the ratio of feed to evaporation is quite high. An alternative to the simple recirculation system of Fig. 11-122h is sometimes used when the feed undergoes an appreciable concentration change and the product is viscous and/or has a high boiling point rise. The feed chamber and vapor head are divided into a number of liquor compartments, and separate pumps are used to pass the liquor through the various banks of tubes in series, all in parallel as to steam and vapor pressures. The actual distribution of feed to the individual tubes of a falling-film evaporator may be accomplished by orifices at the inlet to each tube, by a perforated plate above the tube sheet, or by one or more spray nozzles. Both rising- and falling-film LTV evaporators are generally unsuited to salting or severely scaling liquids. However, both are widely

used for black liquor, which presents a mild scaling problem, and also are used to carry solutions beyond saturation with respect to a crystallizing salt. In the latter case, deposits can usually be removed quickly by increasing the feed rate or reducing the steam rate in order to make the product unsaturated for a short time. The falling-film evaporator is not generally suited to liquids containing solids because of difficulty in plugging the feed distributors. However, it has been applied to the evaporation of saline waters saturated with CaSO4 and containing 5 to 10 percent CaSO4 seeds in suspension for scale prevention (Anderson, ASME Pap. 76-WA/Pwr-5, 1976). Because of their simplicity of construction, compactness, and generally high heat-transfer coefficients, LTV evaporators are well suited to service with corrosive liquids. An example is the reconcentration of rayon spin-bath liquor, which is highly acid. These evaporators employ impervious graphite tubes, lead, rubber-covered or impervious graphite tube sheets, and rubber-lined vapor heads. Polished stainless-steel LTV evaporators are widely used for food products. The latter evaporators are usually similar to that shown in Fig. 11-122g, in which the heating element is at one side of the vapor head to permit easy access to the tubes for cleaning. Advantages of long-tube vertical evaporators: 1. Low cost 2. Large heating surface in one body 3. Low holdup 4. Small floor space 5. Good heat-transfer coefficients at reasonable temperature differences (rising film) 6. Good heat-transfer coefficients at all temperature differences (falling film) Disadvantages of long-tube vertical evaporators: 1. High headroom 2. Generally unsuitable for salting and severely scaling liquids 3. Poor heat-transfer coefficients of rising-film version at low temperature differences 4. Recirculation usually required for falling-film version Best applications of long-tube vertical evaporators: 1. Clear liquids 2. Foaming liquids 3. Corrosive solutions 4. Large evaporation loads 5. High temperature differences—rising film, low temperature differences—falling film 6. Low-temperature operation—falling film 7. Vapor compression operation—falling film Frequent difficulties with long-tube vertical evaporators: 1. Sensitivity of rising-film units to changes in operating conditions 2. Poor feed distribution to falling-film units

Horizontal-Tube Evaporators (Fig. 11-122i) In these types the steam is inside and the liquor outside the tubes. The submergedtube version of Fig. 11-122i is seldom used except for the preparation of boiler feedwater. Low entrainment loss is the primary aim: the horizontal cylindrical shell yields a large disengagement area per unit of vessel volume. Special versions use deformed tubes between restrained tube sheets that crack off much of a scale deposit when sprayed with cold water. By showering liquor over the tubes in the version of Fig. 11-122f hydrostatic head losses are eliminated and heat-transfer performance is improved to that of the falling-film tubular type of Fig. 11-122h. Originally called the Lillie, this evaporator is now also called the spray-film or simply the horizontal-tube evaporator. Liquid distribution over the tubes is accomplished by sprays or perforated plates above the topmost tubes. Maintaining this distribution through the bundle to avoid overconcentrating the liquor is a problem unique to this type of evaporator. It is now used primarily for seawater evaporation. Advantages of horizontal-tube evaporators: 1. Very low headroom 2. Large vapor-liquid disengaging area—submerged-tube type

EVAPORATORS 3. Relatively low cost in small-capacity straight-tube type 4. Good heat-transfer coefficients 5. Easy semiautomatic descaling—bent-tube type Disadvantages of horizontal-tube evaporators: 1. Unsuitable for salting liquids 2. Unsuitable for scaling liquids—straight-tube type 3. High cost—bent-tube type 4. Maintaining liquid distribution—film type Best applications of horizontal-tube evaporators: 1. Limited headroom 2. Small capacity 3. Nonscaling nonsalting liquids—straight-tube type 4. Severely scaling liquids—bent-tube type

Miscellaneous Forms of Heating Surface Special evaporator designs are sometimes indicated when heat loads are small, special product characteristics are desired, or the product is especially difficult to handle. Jacketed kettles, frequently with agitators, are used when the product is very viscous, batches are small, intimate mixing is required, and/or ease of cleaning is an important factor. Evaporators with steam in coiled tubes may be used for small capacities with scaling liquids in designs that permit “cold shocking,” or complete withdrawal of the coil from the shell for manual scale removal. Other designs for scaling liquids employ flat-plate heat exchangers, since in general a scale deposit can be removed more easily from a flat plate than from a curved surface. One such design, the channel-switching evaporator, alternates the duty of either side of the heating surface periodically from boiling liquid to condensing vapor so that scale formed when the surface is in contact with boiling liquid is dissolved when the surface is next in contact with condensing vapor. Agitated thin-film evaporators employ a heating surface consisting of one large-diameter tube that may be either straight or tapered, horizontal or vertical. Liquid is spread on the tube wall by a rotating assembly of blades that either maintain a close clearance from the wall or actually ride on the film of liquid on the wall. The expensive construction limits application to the most difficult materials. High agitation [on the order of 12 m/s (40 ft/s) rotor-tip speed] and power intensities of 2 to 20 kW m2 (0.25 to 2.5 hp/ft2) permit handling extremely viscous materials. Residence times of only a few seconds permit concentration of heat-sensitive materials at temperatures and temperature differences higher than in other types [Mutzenberg, Parker, and Fischer. Chem. Eng., 72, 175–190 (Sept. 13, 1965)]. High feed-to-product ratios can be handled without recirculation. Economic and process considerations usually dictate that agitated thin-film evaporators be operated in single-effect mode. Very high temperature differences can then be used: many are heated with Dowtherm or other high-temperature media. This permits achieving reasonable capacities in spite of the relatively low heat-transfer coefficients and the small surface that can be provided in a single tube [to about 20 m2 (200 ft2)]. The structural need for wall thicknesses of 6 to 13 mm (d to a in) is a major reason for the relatively low heattransfer coefficients when evaporating water-like materials. Evaporators without Heating Surfaces The submergedcombustion evaporator makes use of combustion gases bubbling through the liquid as the means of heat transfer. It consists simply of a tank to hold the liquid, a burner and gas distributor that can be lowered into the liquid, and a combustion-control system. Since there are no heating surfaces on which scale can deposit, this evaporator is well suited to use with severely scaling liquids. The ease of constructing the tank and burner of special alloys or nonmetallic materials makes practical the handling of highly corrosive solutions. However, since the vapor is mixed with large quantities of noncondensable gases, it is impossible to reuse the heat in this vapor, and installations are usually limited to areas of low fuel cost. One difficulty frequently encountered in the use of submerged-combustion evaporators is a high entrainment loss. Also, these evaporators cannot be used when control of crystal size is important. Disk or cascade evaporators are used in the pulp and paper industry to recover heat and entrained chemicals from boiler stack gases and to effect a final concentration of the black liquor before it is

11-111

burned in the boiler. These evaporators consist of a horizontal shaft on which are mounted disks perpendicular to the shaft or bars parallel to the shaft. The assembly is partially immersed in the thick black liquor so that films of liquor are carried into the hot-gas stream as the shaft rotates. Some forms of flash evaporators require no heating surface. An example is a recrystallizing process for separating salts having normal solubility curves from salts having inverse solubility curves, as in separating sodium chloride from calcium sulfate [Richards, Chem. Eng., 59(3), 140 (1952)]. A suspension of raw solid feed in a recirculating brine stream is heated by direct steam injection. The increased temperature and dilution by the steam dissolve the salt having the normal solubility curve. The other salt remains undissolved and is separated from the hot solution before it is flashed to a lower temperature. The cooling and loss of water on flashing cause recrystallization of the salt having the normal solubility curve, which is separated from the brine before the brine is mixed with more solid feed for recycling to the heater. This system can be operated as a multiple effect by flashing down to the lower temperature in stages and using flash vapor from all but the last stage to heat the recycle brine by direct injection. In this process no net evaporation occurs from the total system, and the process cannot be used to concentrate solutions unless heating surfaces are added. UTILIZATION OF TEMPERATURE DIFFERENCE Temperature difference is the driving force for evaporator operation and usually is limited, as by compression ratio in vapor-compression evaporators and by available steam-pressure and heat-sink temperature in single- and multiple-effect evaporators. A fundamental objective of evaporator design is to make as much of this total temperature difference available for heat transfer as is economically justifiable. Some losses in temperature difference, such as those due to boiling point rise (BPR), are unavoidable. However, even these can be minimized, as by passing the liquor through effects or through different sections of a single effect in series so that only a portion of the heating surface is in contact with the strongest liquor. Figure 11-124 shows approximate BPR losses for a number of process liquids. A correlation for concentrated solutions of many inorganic salts at the atmospheric pressure boiling point [Meranda and Furter, J. Ch. and E. Data 22, 315-7 (1977)] is BPR = 104.9N 1.14 2

(11-123)

where N2 is the mole fraction of salts in solution. Correction to other pressures, when heats of solution are small, can be based on a constant ratio of vapor pressure of the solution to that of water at the same temperature. The principal reducible loss in ∆T is that due to friction and to entrance and exit losses in vapor piping and entrainment separators. Pressure-drop losses here correspond to a reduction in condensing temperature of the vapor and hence a loss in available ∆T. These losses become most critical at the low-temperature end of the evaporator, both because of the increasing specific volume of the vapor and because of the reduced slope of the vapor-pressure curve. Sizing of vapor lines is part of the economic optimization of the evaporator, extra costs of larger vapor lines being balanced against savings in ∆T, which correspond to savings in heating-surface requirements. It should be noted that entrance and exit losses in vapor lines usually exceed by severalfold the straight-pipe friction losses, so they cannot be ignored. VAPOR-LIQUID SEPARATION Product losses in evaporator vapor may result from foaming, splashing, or entrainment. Primary separation of liquid from vapor is accomplished in the vapor head by making the horizontal plan area large enough so that most of the entrained droplets can settle out against the rising flow of vapor. Allowable velocities are governed by the Souders-Brown equation: V = k

(ρ1

−

ρ

/ρv , in which k depends on v)

the size distribution of droplets and the decontamination factor F desired. For most evaporators and for F between 100 and 10,000, k 

11-112

HEAT-TRANSFER EQUIPMENT

FIG. 11-124

Boiling-point rise of aqueous solutions. °C = 5/9 (°F − 32).

0.245/(F − 50)0.4 (Standiford, Chemical Engineers’ Handbook, 4th ed., McGraw-Hill, New York, 1963, p. 11–35). Higher values of k (to about 0.15) can be tolerated in the falling-film evaporator, where most of the entrainment separation occurs in the tubes, the vapor is scrubbed by liquor leaving the tubes, and the vapor must reverse direction to reach the outlet. Foaming losses usually result from the presence in the evaporating liquid of colloids or of surface-tension depressants and finely divided solids. Antifoam agents are often effective. Other means of combating foam include the use of steam jets impinging on the foam surface, the removal of product at the surface layer, where the foaming agents seem to concentrate, and operation at a very low liquid level so that hot surfaces can break the foam. Impingement at high velocity against a baffle tends to break the foam mechanically, and this is the reason that the long-tube vertical, forced-circulation, and agitated-film evaporators are particularly effective with foaming liquids. Operating at lower temperatures and/or higher-dissolved solids concentrations may also reduce foaming tendencies. Splashing losses are usually insignificant if a reasonable height has been provided between the liquid level and the top of the vapor head. The height required depends on the violence of boiling. Heights of 2.4 to 3.6 m (8 to 12 ft) or more are provided in short-tube vertical evaporators, in which the liquid and vapor leaving the tubes are projected upward. Less height is required in forced-circulation evaporators, in which the liquid is given a centrifugal motion or is projected downward as by a baffle. The same is true of long-tube vertical evaporators, in which the rising vapor-liquid mixture is projected against a baffle. Entrainment losses by flashing are frequently encountered in an evaporator. If the feed is above the boiling point and is introduced above or only a short distance below the liquid level, entrainment losses may be excessive. This can occur in a short-tube-type evaporator if the feed is introduced at only one point below the lower tube

sheet (Kerr, Louisiana Agric. Expt. Stn. Bull. 149, 1915). The same difficulty may be encountered in forced-circulation evaporators having too high a temperature rise through the heating element and thus too wide a flashing range as the circulating liquid enters the body. Poor vacuum control, especially during startup, can cause the generation of far more vapor than the evaporator was designed to handle, with a consequent increase in entrainment. Entrainment separators are frequently used to reduce product losses. There are a number of specialized designs available, practically all of which rely on a change in direction of the vapor flow when the vapor is traveling at high velocity. Typical separators are shown in Fig. 11-122, although not necessarily with the type of evaporator with which they may be used. The most common separator is the cyclone, which may have either a top or a bottom outlet as shown in Fig. 11-122a and b or may even be wrapped around the heating element of the next effect as shown in Fig. 11-122f. The separation efficiency of a cyclone increases with an increase in inlet velocity, although at the cost of some pressure drop, which means a loss in available temperature difference. Pressure drop in a cyclone is from 10 to 16 velocity heads [Lawrence, Chem. Eng. Prog., 48, 241 (1952)], based on the velocity in the inlet pipe. Such cyclones can be sized in the same manner as a cyclone dust collector (using velocities of about 30 m/s (100 ft/s) at atmospheric pressure) although sizes may be increased somewhat in order to reduce losses in available temperature difference. Knitted wire mesh serves as an effective entrainment separator when it cannot easily be fouled by solids in the liquor. The mesh is available in woven metal wire of most alloys and is installed as a blanket across the top of the evaporator (Fig. 11-122d) or in a monitor of reduced diameter atop the vapor head. These separators have lowpressure drops, usually on the order of 13 mm (a in) of water, and collection efficiency is above 99.8 percent in the range of vapor velocities from 2.5 to 6 m/s (8 to 20 ft/s) [Carpenter and Othmer, Am. Inst. Chem.

EVAPORATORS Eng. J., 1, 549 (1955)]. Chevron (hook-and-vane) type separators are also used because of their higher-allowable velocities or because of their reduced tendency to foul with solids suspended in the entrained liquid. EVAPORATOR ARRANGEMENT Single-Effect Evaporators Single-effect evaporators are used when the required capacity is small, steam is cheap, the material is so corrosive that very expensive materials of construction are required, or the vapor is so contaminated that it cannot be reused. Single-effect evaporators may be operated in batch, semibatch, or continuousbatch modes or continuously. Strictly speaking, batch evaporators are ones in which filling, evaporating, and emptying are consecutive steps. This method of operation is rarely used since it requires that the body be large enough to hold the entire charge of feed and the heating element be placed low enough so as not to be uncovered when the volume is reduced to that of the product. The more usual method of operation is semibatch, in which feed is continually added to maintain a constant level until the entire charge reaches final density. Continuous-batch evaporators usually have a continuous feed and, over at least part of the cycle, a continuous discharge. One method of operation is to circulate from a storage tank to the evaporator and back until the entire tank is up to desired concentration and then finish in batches. Continuous evaporators have essentially continuous feed and discharge, and concentrations of both feed and product remain substantially constant. Thermocompression The simplest means of reducing the energy requirements of evaporation is to compress the vapor from a singleeffect evaporator so that the vapor can be used as the heating medium in the same evaporator. The compression may be accomplished by mechanical means or by a steam jet. In order to keep the compressor cost and power requirements within reason, the evaporator must work with a fairly narrow temperature difference, usually from about 5.5 to 11°C (10° to 20°F). This means that a large evaporator heating surface is needed, which usually makes the vapor-compression evaporator more expensive in first cost than a multiple-effect evaporator. However, total installation costs may be reduced when purchased power is the energy source, since the need for boiler and heat sink is eliminated. Substantial savings in operating cost are realized when electrical or mechanical power is available at a low cost relative to low-pressure steam, when only high-pressure steam is available to operate the evaporator, or when the cost of providing cooling water or other heat sink for a multipleeffect evaporator is high. Mechanical thermocompression may employ reciprocating, rotary positive-displacement, centrifugal, or axial-flow compressors. Positive-displacement compressors are impractical for all but the smallest capacities, such as portable seawater evaporators. Axial-flow compressors can be built for capacities of more than 472 m3/s (1 × 106 ft3/min). Centrifugal compressors are usually cheapest for the intermediate-capacity ranges that are normally encountered. In all cases, great care must be taken to keep entrainment at a minimum, since the vapor becomes superheated on compression and any liquid present will evaporate, leaving the dissolved solids behind. In some cases a vapor-scrubbing tower may be installed to protect the compressor. A mechanical recompression evaporator usually requires more heat than is available from the compressed vapor. Some of this extra heat can be obtained by preheating the feed with the condensate and, if possible, with the product. Rather extensive heat-exchange systems with close approach temperatures are usually justified, especially if the evaporator is operated at high temperature to reduce the volume of vapor to be compressed. When the product is a solid, an elutriation leg such as that shown in Fig. 11-122b is advantageous, since it cools the product almost to feed temperature. The remaining heat needed to maintain the evaporator in operation must be obtained from outside sources. While theoretical compressor power requirements are reduced slightly by going to lower evaporating temperatures, the volume of vapor to be compressed and hence compressor size and cost increase so rapidly that low-temperature operation is more expensive than high-temperature operation. The requirement of low temperature for fruit-juice concentration has led to the development of an evaporator

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employing a secondary fluid, usually Freon or ammonia. In this evaporator, the vapor is condensed in an exchanger cooled by boiling Freon. The Freon, at a much higher vapor density than the water vapor, is then compressed to serve as the heating medium for the evaporator. This system requires that the latent heat be transferred through two surfaces instead of one, but the savings in compressor size and cost are enough to justify the extra cost of heating surface or the cost of compressing through a wider temperature range. Steam-jet thermocompression is advantageous when steam is available at a pressure appreciably higher than can be used in the evaporator. The steam jet then serves as a reducing valve while doing some useful work. The efficiency of a steam jet is quite low and falls off rapidly when the jet is not used at the vapor-flow rate and terminal pressure conditions for which it was designed. Consequently multiple jets are used when wide variations in evaporation rate are expected. Because of the low first cost and the ability to handle large volumes of vapor, steam-jet thermocompressors are used to increase the economy of evaporators that must operate at low temperatures and hence cannot be operated in multiple effect. The steam-jet thermocompression evaporator has a heat input larger than that needed to balance the system, and some heat must be rejected. This is usually done by venting some of the vapor at the suction of the compressor. Multiple-Effect Evaporation Multiple-effect evaporation is the principal means in use for economizing on energy consumption. Most such evaporators operate on a continuous basis, although for a few difficult materials a continuous-batch cycle may be employed. In a multiple-effect evaporator, steam from an outside source is condensed in the heating element of the first effect. If the feed to the effect is at a temperature near the boiling point in the first effect, 1 kg of steam will evaporate almost 1 kg of water. The first effect operates at (but is not controlled at) a boiling temperature high enough so that the evaporated water can serve as the heating medium of the second effect. Here almost another kilogram of water is evaporated, and this may go to a condenser if the evaporator is a double-effect or may be used as the heating medium of the third effect. This method may be repeated for any number of effects. Large evaporators having six and seven effects are common in the pulp and paper industry, and evaporators having as many as 17 effects have been built. As a first approximation, the steam economy of a multiple-effect evaporator will increase in proportion to the number of effects and usually will be somewhat less numerically than the number of effects. The increased steam economy of a multiple-effect evaporator is gained at the expense of evaporator first cost. The total heat-transfer surface will increase substantially in proportion to the number of effects in the evaporator. This is only an approximation since going from one to two effects means that about half of the heat transfer is at a higher temperature level, where heat-transfer coefficients are generally higher. On the other hand, operating at lower temperature differences reduces the heat-transfer coefficient for many types of evaporator. If the material has an appreciable boiling-point elevation, this will also lower the available temperature difference. The only accurate means of predicting the changes in steam economy and surface requirements with changes in the number of effects is by detailed heat and material balances together with an analysis of the effect of changes in operating conditions on heat-transfer performance. The approximate temperature distribution in a multiple-effect evaporator is under the control of the designer, but once built, the evaporator establishes its own equilibrium. Basically, the effects are a number of series resistances to heat transfer, each resistance being approximately proportional to 1/UnAn. The total available temperature drop is divided between the effects in proportion to their resistances. If one effect starts to scale, its temperature drop will increase at the expense of the temperature drops across the other effects. This provides a convenient means of detecting a drop in heat-transfer coefficient in an effect of an operating evaporator. If the steam pressure and final vacuum do not change, the temperature in the effect that is scaling will decrease and the temperature in the preceding effect will increase. The feed to a multiple-effect evaporator is usually transferred from one effect to another in series so that the ultimate product concentration is reached only in one effect of the evaporator. In backward-feed

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HEAT-TRANSFER EQUIPMENT

operation, the raw feed enters the last (coldest) effect, the discharge from this effect becomes the feed to the next-to-the-last effect, and so on until product is discharged from the first effect. This method of operation is advantageous when the feed is cold, since much less liquid must be heated to the higher temperature existing in the early effects. It is also used when the product is so viscous that high temperatures are needed to keep the viscosity low enough to give reasonable heattransfer coefficients. When product viscosity is high but a hot product is not needed, the liquid from the first effect is sometimes flashed to a lower temperature in one or more stages and the flash vapor added to the vapor from one or more later effects of the evaporator. In forward-feed operation, raw feed is introduced in the first effect and passed from effect to effect parallel to the steam flow. Product is withdrawn from the last effect. This method of operation is advantageous when the feed is hot or when the concentrated product would be damaged or would deposit scale at high temperature. Forward feed simplifies operation when liquor can be transferred by pressure difference alone, thus eliminating all intermediate liquor pumps. When the feed is cold, forward feed gives a low steam economy since an appreciable part of the prime steam is needed to heat the feed to the boiling point and thus accomplishes no evaporation. If forward feed is necessary and feed is cold, steam economy can be improved markedly by preheating the feed in stages with vapor bled from intermediate effects of the evaporator. This usually represents little increase in total heating surface or cost since the feed must be heated in any event and shell-and-tube heat exchangers are generally less expensive per unit of surface area than evaporator heating surface. Mixed-feed operation is used only for special applications, as when liquor at an intermediate concentration and a certain temperature is desired for additional processing. Parallel feed involves the introduction of raw feed and the withdrawal of product at each effect of the evaporator. It is used primarily when the feed is substantially saturated and the product is a solid. An example is the evaporation of brine to make common salt. Evaporators of the types shown in Fig. 11-122b or e are used, and the product is withdrawn as a slurry. In this case, parallel feed is desirable because the feed washes impurities from the salt leaving the body. Heat-recovery systems are frequently incorporated in an evaporator to increase the steam economy. Ideally, product and evaporator condensate should leave the system at a temperature as low as possible. Also, heat should be recovered from these streams by exchange with feed or evaporating liquid at the highest possible temperature. This would normally require separate liquid-liquid heat exchangers, which add greatly to the complexity of the evaporator and are justifiable only in large plants. Normally, the loss in thermodynamic availability due to flashing is tolerated since the flash vapor can then be used directly in the evaporator effects. The most commonly used is a condensate flash system in which the condensate from each effect but the first (which normally must be returned to the boiler) is flashed in successive stages to the pressure in the heating element of each succeeding effect of the evaporator. Product flash tanks may also be used in a backward- or mixed-feed evaporator. In a forward-feed evaporator, the principal means of heat recovery may be by use of feed preheaters heated by vapor bled from each effect of the evaporator. In this case, condensate may be either flashed as before or used in a separate set of exchangers to accomplish some of the feed preheating. A feed preheated by last-effect vapor may also materially reduce condenser water requirements. Seawater Evaporators The production of potable water from saline waters represents a large and growing field of application for evaporators. Extensive work done in this field to 1972 was summarized in the annual Saline Water Conversion Reports of the Office of Saline Water, U.S. Department of the Interior. Steam economies on the order of 10 kg evaporation/kg steam are usually justified because (1) unit production capacities are high, (2) fixed charges are low on capital used for public works (i.e., they use long amortization periods and have low interest rates, with no other return on investment considered), (3) heat-transfer performance is comparable with that of pure water, and (4) properly treated seawater causes little deterioration due to scaling or fouling.

Figure 11-125a shows a multiple-effect (falling-film) flow sheet as used for seawater. Twelve effects are needed for a steam economy of 10. Seawater is used to condense last-effect vapor, and a portion is then treated to prevent scaling and corrosion. Treatment usually consists of acidification to break down bicarbonates, followed by deaeration, which also removes the carbon dioxide generated. The treated seawater is then heated to successively higher temperatures by a portion of the vapor from each effect and finally is fed to the evaporating surface of the first effect. The vapor generated therein and the partially concentrated liquid are passed to the second effect, and so on until the last effect. The feed rate is adjusted relative to the steam rate so that the residual liquid from the last effect can carry away all the salts in solution, in a volume about one-third of that of the feed. Condensate formed in each effect but the first is flashed down to the following effects in sequence and constitutes the product of the evaporator. As the feed-to-steam ratio is increased in the flow sheet of Fig. 11-125a, a point is reached where all the vapor is needed to preheat the feed and none is available for the evaporator tubes. This limiting case is the multistage flash evaporator, shown in its simplest form in Fig. 11-125b. Seawater is treated as before and then pumped through a number of feed heaters in series. It is given a final boost in temperature with prime steam in a brine heater before it is flashed down in series to provide the vapor needed by the feed heaters. The amount of steam required depends on the approach-temperature difference in the feed heaters and the flash range per stage. Condensate from the feed heaters is flashed down in the same manner as the brine. Since the flow being heated is identical to the total flow being flashed, the temperature rise in each heater is equal to the flash range in each flasher. This temperature difference represents a loss from the temperature difference available for heat transfer. There are thus two ways of increasing the steam economy of such plants: increasing the heating surface and increasing the number of stages. Whereas the number of effects in a multiple-effect plant will be about 20 percent greater than the steam economy, the number of stages in a flash plant will be 3 to 4 times the steam economy. However, a large number of stages can be provided in a single vessel by means of internal bulkheads. The heat-exchanger tubing is placed in the same vessel, and the tubes usually are continuous through a number of stages. This requires ferrules or special close tube-hole clearances where the tubes pass through the internal bulkheads. In a plant for a steam economy of 10, the ratio of flow rate to heating surface is usually such that the seawater must pass through about 152 m of 19-mm (500 ft of e-in) tubing before it reaches the brine heater. This places a limitation on the physical arrangement of the vessels. Inasmuch as it requires a flash range of about 61°C (110°F) to produce 1 kg of flash vapor for every 10 kg of seawater, the multistage flash evaporator requires handling a large volume of seawater relative to the product. In the flow sheet of Fig. 11-125b all this seawater must be deaerated and treated for scale prevention. In addition, the laststage vacuum varies with the ambient seawater temperature, and ejector equipment must be sized for the worst condition. These difficulties can be eliminated by using the recirculating multistage flash flow sheet of Fig. 11-125c. The last few stages, called the reject stages, are cooled by a flow of seawater that can be varied to maintain a reasonable last-stage vacuum. A small portion of the last-stage brine is blown down to carry away the dissolved salts, and the balance is recirculated to the heat-recovery stages. This arrangement requires a much smaller makeup of fresh seawater and hence a lower treatment cost. The multistage flash evaporator is similar to a multiple-effect forced-circulation evaporator, but with all the forced-circulation heaters in series. This has the advantage of requiring only one largevolume forced-circulation pump, but the sensible heating and shortcircuiting losses in available temperature differences remain. A disadvantage of the flash evaporator is that the liquid throughout the system is at almost the discharge concentration. This has limited its industrial use to solutions in which no great concentration differences are required between feed and product and to where the liquid can be heated through wide temperature ranges without scaling. A partial

EVAPORATORS

11-115

(a)

(b)

(c) Flow sheets for seawater evaporators. (a) Multiple effect (falling film). (b) Multistage flash (once-through). (c) Multistage flash (recirculating). FIG. 11-125

remedy is to arrange several multistage flash evaporators in series, the heat-rejection section of one being the brine heater of the next. This permits independent control of concentration but eliminates the principal advantage of the flash evaporator, which is the small number of pumps and vessels required. An unusual feature of the flash evaporator is that fouling of the heating surfaces reduces primarily the steam economy rather than the capacity of the evaporator. Capacity is not affected until the heat-rejection stages can no longer handle the increased flashing resulting from the increased heat input. EVAPORATOR CALCULATIONS Single-Effect Evaporators The heat requirements of a singleeffect continuous evaporator can be calculated by the usual methods of stoichiometry. If enthalpy data or specific heat and heat-of-solution data are not available, the heat requirement can be estimated as the sum of the heat needed to raise the feed from feed to product temperature and the heat required to evaporate the water. The latent heat of water is taken at the vapor-head pressure instead of at the product temperature in order to compensate partially for any heat of solution. If sufficient vapor-pressure data are available for the solution, methods are available to calculate the true latent heat from the slope of the Dühring line [Othmer, Ind. Eng. Chem., 32, 841 (1940)]. The heat requirements in batch evaporation are the same as those in continuous evaporation except that the temperature (and sometimes pressure) of the vapor changes during the course of the cycle. Since the enthalpy of water vapor changes but little relative to temperature, the difference between continuous and batch heat requirements is almost always negligible. More important usually is the effect of variation of fluid properties, such as viscosity and boiling-point rise, on heat transfer. These can only be estimated by a step-by-step calculation.

In selecting the boiling temperature, consideration must be given to the effect of temperature on heat-transfer characteristics of the type of evaporator to be used. Some evaporators show a marked drop in coefficient at low temperature—more than enough to offset any gain in available temperature difference. The condenser coolingwater temperature and cost must also be considered. Thermocompression Evaporators Thermocompression-evaporator calculations [Pridgeon, Chem. Metall. Eng., 28, 1109 (1923); Peter, Chimia (Switzerland), 3, 114 (1949); Petzold, Chem. Ing. Tech., 22, 147 (1950); and Weimer, Dolf, and Austin, Chem. Eng. Prog., 76(11), 78 (1980)] are much the same as single-effect calculations with the added complication that the heat supplied to the evaporator from compressed vapor and other sources must exactly balance the heat requirements. Some knowledge of compressor efficiency is also required. Large axial-flow machines on the order of 236-m3/s (500,000-ft3/min) capacity may have efficiencies of 80 to 85 percent. Efficiency drops to about 75 percent for a 14-m3/s (30,000-ft3/min) centrifugal compressor. Steam-jet compressors have thermodynamic efficiencies on the order of only 25 to 30 percent. Flash Evaporators The calculation of a heat and material balance on a flash evaporator is relatively easy once it is understood that the temperature rise in each heater and temperature drop in each flasher must all be substantially equal. The steam economy E, kg evaporation/kg of 1055-kJ steam (lb/lb of 1000-Btu steam) may be approximated from ∆T ∆T E = 1 −


(11-124) 1250 Y + R + ∆T/N where ∆T is the total temperature drop between feed to the first flasher and discharge from the last flasher. °C; N is the number of flash stages; Y is the approach between vapor temperature from the





11-116

HEAT-TRANSFER EQUIPMENT

first flasher and liquid leaving the heater in which this vapor is condensed. °C (the approach is usually substantially constant for all stages); and R. °C. is the sum of the boiling-point rise and the shortcircuiting loss in the first flash stage. The expression for the mean effective temperature difference ∆ t available for heat transfer then becomes ∆t =

∆T 1 − ∆T/1250 − RE/∆T N ln


1 − ∆ T 1250 − RE/∆T − E/N

(11-125)

Multiple-Effect Evaporators A number of approximate methods have been published for estimating performance and heatingsurface requirements of a multiple-effect evaporator [Coates and Pressburg, Chem. Eng., 67(6), 157 (1960); Coates, Chem. Eng. Prog., 45, 25 (1949); and Ray and Carnahan, Trans. Am. Inst. Chem. Eng., 41, 253 (1945)]. However, because of the wide variety of methods of feeding and the added complication of feed heaters and condensate flash systems, the only certain way of determining performance is by detailed heat and material balances. Algebraic solutions may be used, but if more than a few effects are involved, trial-and-error methods are usually quicker. These frequently involve trial-and-error within trial-and-error solutions. Usually, if condensate flash systems or feed heaters are involved, it is best to start at the first effect. The basic steps in the calculation are then as follows: 1. Estimate temperature distribution in the evaporator, taking into account boiling-point elevations. If all heating surfaces are to be equal, the temperature drop across each effect will be approximately inversely proportional to the heat-transfer coefficient in that effect. 2. Determine total evaporation required, and estimate steam consumption for the number of effects chosen. 3. From assumed feed temperature (forward feed) or feed flow (backward feed) to the first effect and assumed steam flow, calculate evaporation in the first effect. Repeat for each succeeding effect, checking intermediate assumptions as the calculation proceeds. Heat input from condensate flash can be incorporated easily since the condensate flow from the preceding effects will have already been determined. 4. The result of the calculation will be a feed to or a product discharge from the last effect that may not agree with actual requirements. The calculation must then be repeated with a new assumption of steam flow to the first effect. 5. These calculations should yield liquor concentrations in each effect that make possible a revised estimate of boiling-point rises. They also give the quantity of heat that must be transferred in each effect. From the heat loads, assumed temperature differences, and heat-transfer coefficients, heating-surface requirements can be determined. If the distribution of heating surface is not as desired, the entire calculation may need to be repeated with revised estimates of the temperature in each effect. 6. If sufficient data are available, heat-transfer coefficients under the proposed operating conditions can be calculated in greater detail and surface requirements readjusted. Such calculations require considerable judgment to avoid repetitive trials but are usually well worth the effort. Sample calculations are given in the American Institute of Chemical Engineers Testing Procedure for Evaporators and by Badger and Banchero, Introduction to Chemical Engineering, McGraw-Hill, New York, 1955. These balances may be done by computer but programming time frequently exceeds the time needed to do them manually, especially when variations in flow sheet are to be investigated. The MASSBAL program of SACDA, London, Ont., provides a considerable degree of flexibility in this regard. Another program, not specific to evaporators, is ASPEN PLUS by Aspen Tech., Cambridge, MA. Many such programs include simplifying assumptions and approximations that are not explicitly stated and can lead to erroneous results. Optimization The primary purpose of evaporator design is to enable production of the necessary amount of satisfactory product at the lowest total cost. This requires economic-balance calculations that may include a great number of variables. Among the possible variables are the following:

1. Initial steam pressure versus cost or availability. 2. Final vacuum versus water temperature, water cost, heattransfer performance, and product quality. 3. Number of effects versus steam, water, and pump power cost. 4. Distribution of heating surface between effects versus evaporator cost. 5. Type of evaporator versus cost and continuity of operation. 6. Materials of construction versus product quality, tube life, evaporator life, and evaporator cost. 7. Corrosion, erosion, and power consumption versus tube velocity. 8. Downtime for retubing and repairs. 9. Operating-labor and maintenance requirements. 10. Method of feeding and use of heat-recovery systems. 11. Size of recovery heat exchangers. 12. Possible withdrawal of steam from an intermediate effect for use elsewhere. 13. Entrainment separation requirements. The type of evaporator to be used and the materials of construction are generally selected on the basis of past experience with the material to be concentrated. The method of feeding can usually be decided on the basis of known feed temperature and the properties of feed and product. However, few of the listed variables are completely independent. For instance, if a large number of effects is to be used, with a consequent low temperature drop per effect, it is impractical to use a natural-circulation evaporator. If expensive materials of construction are desirable, it may be found that the forced-circulation evaporator is the cheapest and that only a few effects are justifiable. The variable having the greatest influence on total cost is the number of effects in the evaporator. An economic balance can establish the optimum number where the number is not limited by such factors as viscosity, corrosiveness, freezing point, boiling-point rise, or thermal sensitivity. Under present United States conditions, savings in steam and water costs justify the extra capital, maintenance, and power costs of about seven effects in large commercial installations when the properties of the fluid are favorable, as in black-liquor evaporation. Under governmental financing conditions, as for plants to supply fresh water from seawater, evaporators containing from 12 to 30 or more effects can be justified. As a general rule, the optimum number of effects increases with an increase in steam cost or plant size. Larger plants favor more effects, partly because they make it easier to install heat-recovery systems that increase the steam economy attainable with a given number of effects. Such recovery systems usually do not increase the total surface needed but do require that the heating surface be distributed between a greater number of pieces of equipment. The most common evaporator design is based on the use of the same heating surface in each effect. This is by no means essential since few evaporators are “standard” or involve the use of the same patterns. In fact, there is no reason why all effects in an evaporator must be of the same type. For instance, the cheapest salt evaporator might use propeller calandrias for the early effects and forcedcirculation effects at the low-temperature end, where their higher cost per unit area is more than offset by higher heat-transfer coefficients. Bonilla [Trans. Am. Inst. Chem. Eng., 41, 529 (1945)] developed a simplified method for distributing the heating surface in a multipleeffect evaporator to achieve minimum cost. If the cost of the evaporator per unit area of heating surface is constant throughout, then minimum cost and area will be achieved if the ratio of area to temperature difference A/∆T is the same for all effects. If the cost per unit area z varies, as when different tube materials or evaporator types are used, then zA/∆T should be the same for all effects. EVAPORATOR ACCESSORIES Condensers The vapor from the last effect of an evaporator is usually removed by a condenser. Surface condensers are employed when mixing of condensate with condenser cooling water is not desired. They are for the most part shell-and-tube condensers with vapor on the shell side and a multipass flow of cooling water on the

EVAPORATORS tube side. Heat loads, temperature differences, sizes, and costs are usually of the same order of magnitude as for another effect of the evaporator. Surface condensers use more cooling water and are so much more expensive that they are never used when a direct-contact condenser is suitable. The most common type of direct-contact condenser is the countercurrent barometric condenser, in which vapor is condensed by rising against a rain of cooling water. The condenser is set high enough so that water can discharge by gravity from the vacuum in the condenser. Such condensers are inexpensive and are economical on water consumption. They can usually be relied on to maintain a vacuum corresponding to a saturated-vapor temperature within 2.8°C (5°F) of the water temperature leaving the condenser [How, Chem. Eng., 63(2), 174 (1956)]. The ratio of water consumption to vapor condensed can be determined from the following equation: Water flow Hv − h2 (11-126)

=
Vapor flow h2 − h1 where Hv = vapor enthalpy and h1 and h2 = water enthalpies entering and leaving the condenser. Another type of direct-contact condenser is the jet or wet condenser, which makes use of high-velocity jets of water both to condense the vapor and to force noncondensable gases out the tailpipe. This type of condenser is frequently placed below barometric height and requires a pump to remove the mixture of water and gases. Jet condensers usually require more water than the more common barometric-type condensers and cannot be throttled easily to conserve water when operating at low evaporation rates. Vent Systems Noncondensable gases may be present in the evaporator vapor as a result of leakage, air dissolved in the feed, or decomposition reactions in the feed. When the vapor is condensed in the succeeding effect, the noncondensables increase in concentration and impede heat transfer. This occurs partially because of the reduced partial pressure of vapor in the mixture but mainly because the vapor flow toward the heating surface creates a film of poorly conducting gas at the interface. (See page 11-14 for means of estimating the effect of noncondensable gases on the steam-film coefficient.) The most important means of reducing the influence of noncondensables on heat transfer is by properly channeling them past the heating surface. A positive vapor-flow path from inlet to vent outlet should be provided, and the path should preferably be tapered to avoid pockets of low velocity where noncondensables can be trapped. Excessive clearances and low-resistance channels that could bypass vapor directly from the inlet to the vent should be avoided [Standiford, Chem. Eng. Prog., 75, 59–62 (July 1979)]. In any event, noncondensable gases should be vented well before their concentration reaches 10 percent. Since gas concentrations are difficult to measure, the usual practice is to overvent. This means that an appreciable amount of vapor can be lost. To help conserve steam economy, venting is usually done from the steam chest of one effect to the steam chest of the next. In this way, excess vapor in one vent does useful evaporation at a steam economy only about one less than the overall steam economy. Only when there are large amounts of noncondensable gases present, as in beet-sugar evaporation, is it desirable to pass the vents directly to the condenser to avoid serious losses in heat-transfer rates. In such cases, it can be worthwhile to recover heat from the vents in separate heat exchangers, which preheat the entering feed. The noncondensable gases eventually reach the condenser (unless vented from an effect above atmospheric pressure to the atmosphere or to auxiliary vent condensers). These gases will be supplemented by air dissolved in the condenser water and by carbon dioxide given off on decomposition of bicarbonates in the water if a barometric condenser is used. These gases may be removed by the use of a water-jettype condenser but are usually removed by a separate vacuum pump. The vacuum pump is usually of the steam-jet type if high-pressure steam is available. If high-pressure steam is not available, more expensive mechanical pumps may be used. These may be either a water-ring (Hytor) type or a reciprocating pump. The primary source of noncondensable gases usually is air dissolved in the condenser water. Figure 11-126 shows the dissolved-gas content of fresh water and seawater, calculated as equivalent air. The

FIG. 11-126

(°F − 32).

11-117

Gas content of water saturated at atmospheric pressure. °C = 5/9

lower curve for seawater includes only dissolved oxygen and nitrogen. The upper curve includes carbon dioxide that can be evolved by complete breakdown of bicarbonate in seawater. Breakdown of bicarbonates is usually not appreciable in a condenser but may go almost to completion in a seawater evaporator. The large increase in gas volume as a result of possible bicarbonate breakdown is illustrative of the uncertainties involved in sizing vacuum systems. By far the largest load on the vacuum pump is water vapor carried with the noncondensable gases. Standard power-plant practice assumes that the mixture leaving a surface condenser will have been cooled 4.2°C (7.5°F) below the saturation temperature of the vapor. This usually corresponds to about 2.5 kg of water vapor/kg of air. One advantage of the countercurrent barometric condenser is that it can cool the gases almost to the temperature of the incoming water and thus reduce the amount of water vapor carried with the air. In some cases, as with pulp-mill liquors, the evaporator vapors contain constituents more volatile than water, such as methanol and sulfur compounds. Special precautions may be necessary to minimize the effects of these compounds on heat transfer, corrosion, and condensate quality. They can include removing most of the condensate countercurrent to the vapor entering an evaporator-heating element, channeling vapor and condensate flow to concentrate most of the “foul” constituents into the last fraction of vapor condensed (and keeping this condensate separate from the rest of the condensate), and flashing the warm evaporator feed to a lower pressure to remove much of the foul constituents in only a small amount of flash vapor. In all such cases, special care is needed to properly channel vapor flow past the heating surfaces so there is a positive flow from steam inlet to vent outlet with no pockets, where foul constituents or noncondensibles can accumulate. Salt Removal When an evaporator is used to make a crystalline product, a number of means are available for concentrating and removing the salt from the system. The simplest is to provide settling space in the evaporator itself. This is done in the types shown in Fig. 11-122b, c, and e by providing a relatively quiescent zone in which the salt can settle. Sufficiently high slurry densities can usually be achieved in this manner to reach the limit of pumpability. The evaporators are usually placed above barometric height so that the slurry can be discharged intermittently on a short time cycle. This permits the use of high velocities in large lines that have little tendency to plug. If the amount of salts crystallized is on the order of a ton an hour or less, a salt trap may be used. This is simply a receiver that is connected to the bottom of the evaporator and is closed off from the evaporator periodically for emptying. Such traps are useful when insufficient

11-118

HEAT-TRANSFER EQUIPMENT

headroom is available for gravity removal of the solids. However, traps require a great deal of labor, give frequent trouble with the shutoff valves, and also can upset evaporator operation completely if a trap is reconnected to the evaporator without first displacing all air with feed liquor. EVAPORATOR OPERATION The two principal elements of evaporator control are evaporation rate and product concentration. Evaporation rate in single- and multipleeffect evaporators is usually achieved by steam-flow control. Conventional-control instrumentation is used (see Sec. 22), with the added precaution that pressure drop across meter and control valve, which reduces temperature difference available for heat transfer, not be excessive when maximum capacity is desired. Capacity control of thermocompression evaporators depends on the type of compressor; positive-displacement compressors can utilize speed control or variations in operating pressure level. Centrifugal machines normally utilize adjustable inlet-guide vanes. Steam jets may have an adjustable spindle in the high-pressure orifice or be arranged as multiple jets that can individually be cut out of the system. Product concentration can be controlled by any property of the solution that can be measured with the requisite accuracy and reliability. The preferred method is to impose control on rate of product

withdrawal. Feed rates to the evaporator effects are then controlled by their levels. When level control is impossible, as with the rising-film LTV, product concentration is used to control the feed rate—frequently by rationing of feed to steam with the ration reset by product concentration, sometimes also by feed concentration. Other controls that may be needed include vacuum control of the last effect (usually by air bleed to the condenser) and temperature-level control of thermocompression evaporators (usually by adding makeup heat or by venting excess vapor, or both as feed or weather conditions vary). For more control detail, see Measurement and Control in Water Desalination, N. Lior, ed., pp. 241–305, Elsevier Science Publ. Co., NY, 1986. Control of an evaporator requires more than proper instrumentation. Operator logs should reflect changes in basic characteristics, as by use of pseudo heat-transfer coefficients, which can detect obstructions to heat flow, hence to capacity. These are merely the ratio of any convenient measure of heat flow to the temperature drop across each effect. Dilution by wash and seal water should be monitored since it absorbs evaporative capacity. Detailed tests, routine measurements, and operating problems are covered more fully in Testing Procedure for Evaporators (loc. cit.) and by Standiford [Chem. Eng. Prog., 58(11), 80 (1962)]. By far the best application of computers to evaporators is for working up operators’ data into the basic performance parameters such as heat-transfer coefficients, steam economy, and dilution.

Section 12

Psychrometry, Evaporative Cooling, and Solids Drying

Charles G. Moyers, Ph D., P.E., Principal Engineer, Union Carbide Corporation, Fellow, American Institute of Chemical Engineers. (Section Editor, Evaporative Cooling and Solids Drying) Glenn W. Baldwin, M.S., P.E., Staff Engineer, Union Carbide Corporation, Member, American Institute of Chemical Engineers. (Psychrometry)

PSYCHROMETRY Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adiabatic-Saturation Temperature, or Constant-Enthalpy Lines. . . . Relation between Wet-Bulb and Adiabatic-Saturation Temperatures . . Example 1: Compare Wet-Bulb and Adiabatic-Saturation Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Use of Psychrometric Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples Illustrating Use of Psychrometric Charts . . . . . . . . . . . . . . Example 2: Determination of Moist Air Properties . . . . . . . . . . . . Example 3: Heating Air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 4: Evaporative Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 5: Cooling and Dehumidification . . . . . . . . . . . . . . . . . . . Example 6: Cooling Tower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 7: Recirculating Dryer. . . . . . . . . . . . . . . . . . . . . . . . . . . . Use of Psychrometric Charts at Pressures Other Than Atmospheric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 8: Determination of Air Properties. . . . . . . . . . . . . . . . . . Measurement of Humidity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dew-Point Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wet-Bulb Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanical Hygrometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EVAPORATIVE COOLING Principles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cooling-Tower Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 9: Calculation of Mass-Transfer Coefficient Group. . . . . Example 10: Application of Nomograph for Cooling-Tower Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanical-Draft Towers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 11: Application of Sizing and Horsepower Charts. . . . . . Example 12: Application of Sizing Chart. . . . . . . . . . . . . . . . . . . . . Cooling-Tower Operation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Water Makeup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 13: Calculation of Makeup Water. . . . . . . . . . . . . . . . . . . Fan Horsepower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pumping Horsepower. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12-3 12-3 12-4 12-4 12-4 12-4 12-4 12-4 12-5 12-5 12-5 12-12 12-13 12-13 12-13 12-13 12-13 12-13

12-14 12-14 12-15 12-15 12-15 12-16 12-17 12-17 12-17 12-17 12-17 12-18

Fogging and Plume Abatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Energy Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Natural-Draft Towers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 14: Duty Coefficient for a Hyperbolic Tower . . . . . . . . . Spray Ponds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 15: Cooling Capacity of a Spray Pond . . . . . . . . . . . . . . . Cooling Ponds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 16: Calculation of Cooling-Pond Size. . . . . . . . . . . . . . . .

12-18 12-19 12-20 12-21 12-21 12-21 12-23 12-23

SOLIDS-DRYING FUNDAMENTALS Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gas-Solids Contacting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gas-Solids Separations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Application of Psychrometry to Drying . . . . . . . . . . . . . . . . . . . . . . . . . . Humidity Charts for Solvent Vapors . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Conditions for Drying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Internal Mechanism of Liquid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . Capillary Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vapor Diffusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . External Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Periods of Drying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constant-Rate Period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Falling-Rate Period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Capillary Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Critical Moisture Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equilibrium Moisture Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimations for Total Drying Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analysis of Drying Data Tests on Plant Dryers . . . . . . . . . . . . . . . . . . . . Tests on Plant Dryers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12-25 12-25 12-26 12-26 12-27 12-29 12-29 12-31 12-31 12-31 12-31 12-31 12-31 12-31 12-32 12-33 12-33 12-33 12-34 12-34 12-34 12-36

SOLIDS-DRYING EQUIPMENT Classification of Dryers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Direct Dryers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12-36 12-36 12-1

12-2

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

Indirect Dryers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miscellaneous Dryers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selection of Drying Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Drying Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Batch Tray and Dryers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Field of Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Auxiliary Equipment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Direct-Heat Tray Dryers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Batch Through-Circulation Dryers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vacuum-Shelf Dryers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Batch Furnaces, Other Furnace Types, and Kilns . . . . . . . . . . . . . . . Continuous Tunnel Dryers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous Through-Circulation Dryers . . . . . . . . . . . . . . . . . . . . . . Design Methods for Continuous Tunnel Dryers. . . . . . . . . . . . . . . . . Performance and Cost Data for Continuous Tunnel Dryers . . . . . . . Continuous Furnaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotary Dryers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Field of Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Auxiliary Equipment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Direct-Heat Rotary Dryers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Direct-Heat Rotary Kilns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Indirect-Heat Rotary Steam-Tube Dryers. . . . . . . . . . . . . . . . . . . . . . Indirect-Heat Calciners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Direct-Heat Roto-Louvre Dryer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Agitated Dryers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Field of Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Batch Vacuum Rotary Dryers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbo-Tray Dryers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plate Dryers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison Data—Plate Dryers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conical Mixer Dryer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reactotherm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hearth Furnaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple-Hearth Furnaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gravity Dryers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fields of Application. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shaft Furnaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pellet Coolers and Dryers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spouted Beds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Direct-Heat Vibrating-Conveyor Dryers. . . . . . . . . . . . . . . . . . . . . . . . . Dispersion Dryers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pneumatic-Conveyor Dryers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ring Dryers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spray Dryers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Agitated Flash Dryers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12-38 12-38 12-38 12-38 12-42 12-42 12-42 12-42 12-42 12-44 12-44 12-45 12-47 12-47 12-48 12-48 12-49 12-52 12-52 12-52 12-52 12-53 12-56 12-60 12-62 12-64 12-64 12-64 12-64 12-65 12-66 12-68 12-69 12-69 12-71 12-71 12-71 12-71 12-71 12-72 12-73 12-75 12-75 12-76 12-77 12-77 12-80 12-81 12-90

Nomenclature and Units In this table, symbols used in this section are defined in a general way. SI and customary U.S. units are listed. Specialized symbols are either defined at the point of application or in a separate table of nomenclature. Symbol

Definition

SI units

U.S. customary units

A a C D D d E F G H h

m2 m2 J/kg m m2/s m V Dimensionless k/(m2⋅s) kg/kg J/(m2⋅s⋅K)

ft2 ft2 Btu/lbm ft ft2/s ft V Dimensionless lbm/(ft2⋅s) lbm/lbm Btu/(ft2⋅h⋅°F)

kg/(m2⋅s⋅atm) W/(m⋅K) m kg/mol l/s Pa Pa m3/s J/s m/m s °C K J/(m2⋅s⋅K)

lbm/(h⋅ft2⋅atm) Btu/(ft⋅h⋅°F) ft lbm/mol l/min lbf/in2 lbf/ft2 ft3/s Btu/h ft/ft s, h °F °R Btu/(ft2⋅h⋅°F)

u V V W

Area Area Heat capacity Diameter Diffusivity Thickness Potential difference Void fraction Mass velocity Humidity Heat-transfer coefficient Mass-transfer coefficient Thermal conductivity Length Molecular weight Rotational speed Pressure Pressure drop Flow rate Total heat flow Slope Time Temperature Absolute temperature Heat-transfer coefficient Velocity Velocity Volume Mass

m/s m/s m3 kg

ft/s ft/s ft3 lbm

⑀ θ λ µ ρ σ

Void fraction Time Latent heat Viscosity Density Surface tension

k K L M N P P Q Q S t t T U

Greek symbols Dimensionless s J/kg Pa⋅s kg/m3 N/m

Dimensionless s, h Btu/lbm lbm/(s⋅ft) lbm/ft3 dyn/cm

PSYCHROMETRY* GENERAL REFERENCES: Handbook of Fundamentals, American Society of Heating, Refrigerating and Air-Conditioning Engineers, New York, 1967. Quinn, “Humidity: The Neglected Parameter,” Test Eng. (July 1968). Treybal, Mass-Transfer Operations, 3d ed. McGraw-Hill, New York, 1980. Wexler, Humidity and Moisture, vol. I, Reinhold, New York, 1965. Zimmerman and Lavine, Psychrometric Charts and Tables, Industrial Research Service, 2d ed., Dover, N.H., 1964.

Psychrometry is concerned with determination of the properties of gas-vapor mixtures. The air-water vapor system is by far the system most commonly encountered. Principles involved in determining the properties of other systems are the same as with air-water vapor, with one major exception. Whereas the psychrometric ratio (ratio of heat-transfer coefficient to product of mass-transfer coefficient and humid heat, terms defined in the following subsection) for the air-water system can be taken as 1, the ratio for other systems in general does not equal 1. This has the effect of making the adiabatic-saturation temperature different from the wet-bulb temperature. Thus, for systems other than air-water vapor, calculation of psychrometric and drying problems is complicated by the necessity for point-to-point calculation of the temperature of the evaporating surface. For example, for the air-water system the temperature of the evaporating surface will be constant during the constant-rate drying period even though temperature and humidity of the gas stream change. For other systems, the temperature of the evaporating surface would change. TERMINOLOGY Terminology and relationships pertinent to psychrometry are: Absolute humidity H equals the pounds of water vapor carried by 1 lb of dry air. If ideal-gas behavior is assumed, H = M w p/[Ma(P − p)], where Mw = molecular weight of water; Ma = molecular weight of air; p = partial pressure of water vapor, atm; and P = total pressure, atm. When the partial pressure p of water vapor in the air at a given temperature equals the vapor pressure of water ps at the same temperature, the air is saturated and the absolute humidity is designated the saturation humidity Hs. Percentage absolute humidity (percentage saturation) is defined as the ratio of absolute humidity to saturation humidity and is given by 100 H/Hs = 100p(P − ps)/[ps(P − p)]. Percentage relative humidity is defined as the partial pressure of water vapor in air divided by the vapor pressure of water at the given temperature. Thus RH = 100p/ps. Dew point, or saturation temperature, is the temperature at which a given mixture of water vapor and air is saturated, for example, the temperature at which water exerts a vapor pressure equal to the partial pressure of water vapor in the given mixture. Humid heat cs is the heat capacity of 1 lb of dry air and the moisture it contains. For most engineering calculations, cs = 0.24 + 0.45H, where 0.24 and 0.45 are the heat capacities of dry air and water vapor, respectively, and both are assumed constant. Humid volume is the volume in cubic feet of 1 lb of dry air and the water vapor it contains. Saturated volume is the humid volume when the air is saturated. Wet-bulb temperature is the dynamic equilibrium temperature attained by a water surface when the rate of heat transfer to the surface by convection equals the rate of mass transfer away from the surface. At equilibrium, if negligible change in the dry-bulb temperature is assumed, a heat balance on the surface is

kg λ(ps − p) = hc(t − tw)

(12-1a)

where kg = mass-transfer coefficient, lb/(h⋅ft2⋅atm); λ = latent heat of vaporization, Btu/lb; ps = vapor pressure of water at wet-bulb temperature, atm; p = partial pressure of water vapor in the environment, atm; hc = heat-transfer coefficient, Btu/(h⋅ft2⋅°F); t = temperature of air-water vapor mixture (dry-bulb temperature), °F; and tw = wet-bulb temperature, °F. Under ordinary conditions the partial pressure and vapor pressure are small relative to the total pressure, and the wetbulb equation can be written in terms of humidity differences as Hs − H = (hc /λk′)(t − tw)

(12-1b)

where k′ = lb/(h⋅ft2) (unit humidity difference) = (Ma /Mw)kg = 1.6kg. Adiabatic-Saturation Temperature, or Constant-Enthalpy Lines If a stream of air is intimately mixed with a quantity of water at a temperature ts in an adiabatic system, the temperature of the air will drop and its humidity will increase. If ts is such that the air leaving the system is in equilibrium with the water, ts will be the adiabaticsaturation temperature, and the line relating the temperature and humidity of the air is the adiabatic-saturation line. The equation for the adiabatic-saturation line is Hs − H = (cs / λ)(t − ts)

(12-2)

RELATION BETWEEN WET-BULB AND ADIABATIC-SATURATION TEMPERATURES Experimentally it has been shown that for air-water systems the value of hc /k′cs, the psychrometric ratio, is approximately equal to 1. Under these conditions the wet-bulb temperatures and adiabaticsaturation temperatures are substantially equal and can be used interchangeably. The difference between adiabatic-saturation temperature and wet-bulb temperature increases with increasing humidity, but this effect is unimportant for most engineering calculations. An empirical formula for wet-bulb temperature determination of moist air at atmospheric pressure is presented by Liley [Int. J. of Mechanical Engineering Education, vol. 21, No. 2 (1993)]. For systems other than air-water vapor, the value of hc /k′cs may differ appreciably from unity, and the wet-bulb and adiabatic-saturation temperatures are no longer equal. For these systems the psychrometric ratio may be obtained by determining hc /k′ from heat- and masstransfer analogies such as the Chilton-Colburn analogy [Ind. Eng. Chem., 26, 1183 (1934)]. For low humidities this analogy gives µ/ρDv hc ᎏ = cs ᎏ k′ csµ/k

冢

冣

2/3

(12-3)

where cs = humid heat, Btu/(lb⋅°F); µ = viscosity, lb/(ft⋅h); ρ = density, lb/ft3; Dv = diffusivity, ft2/h; and k = thermal conductivity, Btu/(h⋅ft2) (°F/ft). All properties should be evaluated for the gas mixture. For the case of flow past cylinders, such as a wet-bulb thermometer, Bedingfield and Drew [Ind. Eng. Chem., 42, 1164 (1950)] obtained a correlation for their data on sublimation of cylinders into air and for the data of others on wet-bulb thermometers. For wet-bulb thermometers in air they give hc / k′ = 0.294(µ/ρDv)0.56

(12-4)

where the nomenclature is identical to that in Eq. (12-3). For evaporation into gases other than air, Eq. (12-3) with an exponent of 0.56 would apply. Application of these equations is illustrated in Example 1.

* The contribution of Eno Bagnoli, E. I. du Pont de Nemours & Co., to material that was used from the sixth edition is acknowledged. (Psychrometry) 12-3

12-4

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

Example 1: Compare Wet-Bulb and Adiabatic-Saturation Temperatures For the air-water system at atmospheric pressure, the measured values of dry-bulb and wet-bulb temperatures are 85 and 72°F respectively. Determine the absolute humidity and compare the wet-bulb temperature and adiabatic-saturation temperature. Assume that hc /k′ is given by Eq. (12-4). Solution. For relatively dry air the Schmidt number µ/ρDv is 0.60, and from Eq. (12-4) hc /k′ = 0.294(0.60)0.56 = 0.221. At 72°F the vapor pressure of water is 20.07 mmHg, and the latent heat of vaporization is 1051.6 Btu/lb. From Eq. (121b), [20.07/(760 − 20.07)](18/29) − H = (0.221/1051.6)(85 − 72), or H = 0.0140 lb water/lb dry air. The humid heat is calculated as cs = 0.24 + 0.45(0.0140) = 0.246. The adiabatic-saturation temperature is obtained from Eq. (12-2) as Hs − 0.0140 = (0.246/1051.6)(85 − ts) Values of Hs and ts are given by the saturation curve of the psychrometric chart, such as Fig. 12-2. By trial and error, ts = 72.1°F, or the adiabatic-saturation temperature is 0.1°F higher than the wet-bulb temperature.

USE OF PSYCHROMETRIC CHARTS Three charts for the air-water vapor system are given as Figs. 12-1 to 12-3 for low-, medium-, and high-temperature ranges. Figure 12-4 shows a modified Grosvenor chart, which is more familiar to the chemical engineer. These charts are for an absolute pressure of 1 atm. The corrections required at pressures different from atmospheric are given in Table 12-2. Figure 12-5 shows a psychrometric chart for combustion products in air. The thermodynamic properties of moist air are given in Table 12-1. Examples Illustrating Use of Psychrometric Charts In these examples the following nomenclature is used: t = dry-bulb temperatures, °F tw = wet-bulb temperature, °F td = dew-point temperature, °F H = moisture content, lb water/lb dry air

∆H = moisture added to or rejected from the air stream, lb water/lb dry air h′ = enthalpy at saturation, Btu/lb dry air D = enthalpy deviation, Btu/lb dry air h = h′ + D = true enthalpy, Btu/lb dry air hw = enthalpy of water added to or rejected from the system, Btu/lb dry air qa = heat added to the system, Btu/lb dry air qr = heat removed from system, Btu/lb dry air Subscripts 1, 2, 3, etc., indicate entering and subsequent states. Example 2: Determination of Moist Air Properties Find the properties of moist air when the dry-bulb temperature is 80°F and the wet-bulb temperature is 67°F. Solution. Read directly from Fig. 12-2 (Fig. 12-6 shows the solution diagrammatically). Moisture content H = 78 gr/lb dry air = 0.011 lb water/lb dry air Enthalpy at saturation h′ = 31.6 Btu/lb dry air Enthalpy deviation D = −0.1 Btu/lb dry air True enthalpy h = 31.5 Btu/lb dry air Specific volume v = 13.8 ft3/lb dry air Relative humidity = 51 percent Dew point td = 60.3°F

Example 3: Air Heating Air is heated by a steam coil from 30°F drybulb temperature and 80 percent relative humidity to 75°F dry-bulb temperature. Find the relative humidity, wet-bulb temperature, and dew point of the heated air. Determine the quantity of heat added per pound of dry air. Solution. Reading directly from the psychrometric chart (Fig. 12-2),

Psychrometric chart—low temperatures. Barometric pressure, 29.92 inHg. To convert British thermal units per pound to joules per kilogram, multiply by 2326; to convert British thermal units per pound dry air-degree Fahrenheit to joules per kilogram-kelvin, multiply by 4186.8; and to convert cubic feet per pound to cubic meters per kilogram, multiply by 0.0624.

FIG. 12-1

PSYCHROMETRY

12-5

Psychrometric chart—medium temperatures. Barometric pressure, 29.92 inHg. To convert British thermal units per pound dry air-degree Fahrenheit to joules per kilogram-kelvin, multiply by 4186.8; and to convert cubic feet per pound to cubic meters per kilogram, multiply by 0.0624.

FIG. 12-2

Relative humidity = 15 percent Wet-bulb temperature = 51.5°F Dew point = 25.2°F The enthalpy of the inlet air is obtained from Fig. 12-2 as h1 = h′1 + D1 = 10.1 + 0.06 = 10.16 Btu/lb dry air; at the exit, h2 = h′2 + D2 = 21.1 − 0.1 = 21 Btu/lb dry air. The heat added equals the enthalpy difference, or qa = ∆h = h2 − h1 = 21 − 10.16 = 10.84 Btu/lb dry air If the enthalpy deviation is ignored, the heat added qa is ∆h = 21.1 − 10.1 = 11 Btu/lb dry air, or the result is 1.5 percent high. Figure 12-7 shows the heating path on the psychrometric chart.

Example 4: Evaporative Cooling Air at 95°F dry-bulb temperature and 70°F wet-bulb temperature contacts a water spray, where its relative humidity is increased to 90 percent. The spray water is recirculated; makeup water enters at 70°F. Determine exit dry-bulb temperature, wet-bulb temperature, change in enthalpy of the air, and quantity of moisture added per pound of dry air. Solution. Figure 12-8 shows the path on a psychrometric chart. The leaving dry-bulb temperature is obtained directly from Fig. 12-2 as 72.2°F. Since the spray water enters at the wet-bulb temperature of 70°F and there is no heat added to or removed from it, this is by definition an adiabatic process and there will be no change in wet-bulb temperature. The only change in enthalpy is that from the heat content of the makeup water. This can be demonstrated as follows: Inlet moisture H1 = 70 gr/lb dry air Exit moisture H2 = 107 gr/lb dry air ∆H = 37 gr/lb dry air Inlet enthalpy h1 = h′1 + D1 = 34.1 − 0.22 = 33.88 Btu/lb dry air Exit enthalpy h2 = h′2 + D2 = 34.1 − 0.02 = 34.08 Btu/lb dry air

Then

Enthalpy of added water hw = 0.2 Btu/lb dry air (from small diagram, 37 gr at 70°F) qa = h2 − h1 + hw = 34.08 − 33.88 + 0.2 = 0

Example 5: Cooling and Dehumidification Find the cooling load per pound of dry air resulting from infiltration of room air at 80°F dry-bulb temperature and 67°F wet-bulb temperature into a cooler maintained at 30°F drybulb and 28°F wet-bulb temperature, where moisture freezes on the coil, which is maintained at 20°F. Solution. The path followed on a psychrometric chart is shown in Fig. 12-9. Inlet enthalpy h1 = h′1 + D1 = 31.62 − 0.1 = 31.52 Btu/lb dry air Exit enthalpy h2 = h′2 + D2 = 10.1 + 0.06 = 10.16 Btu/lb dry air Inlet moisture H1 = 78 gr/lb dry air Exit moisture H2 = 19 gr/lb dry air Moisture rejected ∆H = 59 gr/lb dry air Enthalpy of rejected moisture = −1.26 Btu/lb dry air (from small diagram of Fig. 12-2) Cooling load qr = 31.52 − 10.16 + 1.26 = 22.62 Btu/lb dry air Note that if the enthalpy deviations were ignored, the calculated cooling load would be about 5 percent low.

Example 6: Cooling Tower Determine water consumption and amount of heat dissipated per 1000 ft3/min of entering air at 90°F dry-bulb temperature and 70°F wet-bulb temperature when the air leaves saturated at 110°F and the makeup water is at 75°F. Solution. The path followed is shown in Fig. 12-10.

12-6

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

FIG. 12-3 Psychrometric chart—high temperatures. Barometric pressure, 29.92 inHg. To convert British thermal units per pound to joules per kilogram, multiply by 2326; to convert British thermal units per pound dry air-degree Fahrenheit to joules per kilogramkelvin, multiply by 4186.8; and to convert cubic feet per pound to cubic meters per kilogram, multiply by 0.0624.

PSYCHROMETRY

Humidity chart for air-water vapor mixtures. To convert British thermal units per pound to joules per kilogram, multiply by 2326; to convert British thermal units per pound dry air-degree Fahrenheit to joules per kilogramkelvin, multiply by 4186.8; and to convert cubic feet per pound to cubic meters per kilogram, multiply by 0.0624.

FIG. 12-4

FIG. 12-5 Revised form of high-temperature psychrometric chart for air and combustion products, based on poundmoles of water vapor and dry gases. [Hatta, Chem. Metall. Eng., 37, 64 (1930).]

12-7

12-8

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

TABLE 12-1

Thermodynamic Properties of Moist Air (Standard Atmospheric Pressure, 29.921 inHg) Condensed water Volume, cu. ft./lb. dry air

ss

Enthalpy, B.t.u./lb. hw

Entropy, B.t.u./ (lb.)(°F.) sw

Vapor press., in. Hg ps × 106

Temp. t, °F.

0.00000 .00000 .00000 .00000 .00000

−0.10300 −0.09901 −0.09508 −0.09121 −0.08740

−222.00 −220.40 −218.77 −217.12 −215.44

−0.4907 −0.4853 −0.4800 −0.4747 −0.4695

0.1009 .1842 .3301 .5807 1.004

−160 −155 −150 −145 −140

−0.08365 −0.07997

.00000 .00000

−0.08365 −0.07997

−213.75 −212.03

−0.4642 −0.4590

1.707 2.858

−135 −130

−30.057 −28.852 −27.648 −26.444 −25.239

−0.07634 −0.07277 −0.06924 −0.06577 −0.06234

.00000 .00000 .00000 .00000 .00000

−0.07634 −0.07277 −0.06924 −0.06577 −0.06234

−210.28 −208.52 −206.73 −204.92 −203.09

−0.4538 −0.4485 −0.4433 −0.4381 −0.4329

−24.036

−0.05897

.00000

−0.05897

−201.23

−0.4277

Entropy, B.t.u./(°F.)(lb. dry air)

Enthalpy, B.t.u./lb. dry air

Temp. t, °F.

Saturation humidity Hs × 108

va

vas

vs

ha

has

hs

sa

sas

−160 −155 −150 −145 −140

0.2120 .3869 .6932 1.219 2.109

7.520 7.647 7.775 7.902 8.029

0.000 .000 .000 .000 .000

7.520 7.647 7.775 7.902 8.029

−38.504 −37.296 −36.088 −34.881 −33.674

0.000 .000 .000 .000 .000

−38.504 −37.296 −36.088 −34.881 −33.674

−0.10300 −0.09901 −0.09508 −0.09121 −0.08740

−135 −130

3.586 6.000

8.156 8.283

.000 .000

8.156 8.283

−32.468 −31.262

.000 .000

−32.468 −31.262

0.9887 1.606 2.571 4.063 6.340

8.411 8.537 8.664 8.792 8.919

.000 .000 .000 .000 .000

8.411 8.537 8.664 8.792 8.919

−30.057 −28.852 −27.648 −26.444 −25.240

.000 .000 .000 .000 .001

9.772

9.046

.000

9.046

−24.037

.001

Hs × 107 −125 −120 −115 −110 −105 −100

ps × 105

Hs × 106 −95 −90 −85 −80 −75 −70 −65

1.489 2.242 3.342 4.930 7.196 10.40 14.91

−35

2.118 2.982 4.163 5.766 7.925 10.81

−125 −120 −115 −110 −105

4.666

−100

ps × 104 9.173 9.300 9.426 9.553 9.680

.000 .000 .000 .000 .000

9.173 9.300 9.426 9.553 9.680

−22.835 −21.631 −20.428 −19.225 −18.022

.002 .002 .003 .005 .007

−22.833 −21.629 −20.425 −19.220 −18.015

−0.05565 −0.05237 −0.04913 −0.04595 −0.04280

.00000 .00001 .00001 .00001 .00002

−0.05565 −0.05236 −0.04912 −0.04594 −0.04278

−199.35 −197.44 −195.51 −193.55 −191.57

−0.4225 −0.4173 −0.4121 −0.4069 −0.4017

0.7111 1.071 1.597 2.356 3.441

−95 −90 −85 −80 −75

9.806 9.932

.000 .000

9.806 9.932

−16.820 −15.617

.011 .015

−16.809 −15.602

−0.03969 −0.03663

.00003 .00005

−0.03966 −0.03658

−189.56 −187.53

−0.3965 −0.3913

4.976 7.130

−70 −65

Hs × 105 −60 −55 −50 −45 −40

0.4710 .7653 1.226 1.939 3.026

ps × 103 10.059 10.186 10.313 10.440 10.566

.000 .000 .001 .001 .001

10.059 10.186 10.314 10.441 10.567

−14.416 −13.214 −12.012 −10.811 −9.609

.022 .031 .043 .060 .083

−14.394 −13.183 −11.969 −10.751 −9.526

−0.03360 −0.03061 −0.02766 −0.02474 −0.02186

.00006 .00009 .00012 .00015 .00021

−0.03354 −0.03052 −0.02754 −0.02459 −0.02165

−185.47 −183.39 −181.29 −179.16 −177.01

−0.3861 −0.3810 −0.3758 −0.3707 −0.3655

10.693

.002

10.695

−8.408

.113

−8.295

−0.01902

.00028

−0.01874

−174.84

−0.3604

Hs × 104

1.0127 1.4258 1.9910 2.7578 3.7906

−60 −55 −50 −45 −40

5.1713

−35

ps × 102

−30 −25 −20 −15

1.464 1.969 2.630 3.491

10.820 10.946 11.073 11.200

.002 .004 .005 .006

10.822 10.950 11.078 11.206

−7.207 −6.005 −4.804 −3.603

.154 .207 .277 .368

−7.053 −5.798 −4.527 −3.235

−0.01621 −0.01342 −0.01067 −0.00796

.00038 .00051 .00068 .00089

−0.01583 −0.01291 −0.00999 −0.00707

−172.64 −170.42 −168.17 −165.90

−0.3552 −0.3500 −0.3449 −0.3398

0.70046 .94212 1.2587 1.6706

−30 −25 −20 −15

−10 −5

4.606 6.040

11.326 11.452

.008 .011

11.334 11.463

−2.402 −1.201

.487 .639

−1.915 −0.562

−0.00529 −0.00263

.00115 .00149

−0.00414 −0.00114

−163.60 −161.28

−0.3346 −0.3295

2.2035 2.8886

−10 −5

Hs × 103 0 5 10 15 20

0.7872 1.020 1.315 1.687 2.152

11.578 11.705 11.831 11.958 12.084

.015 .019 .025 .032 .042

11.593 11.724 11.856 11.990 12.126

0.000 1.201 2.402 3.603 4.804

.835 1.085 1.401 1.800 2.302

0.835 2.286 3.803 5.403 7.106

0.00000 .00260 .00518 .00772 .01023

.00192 .00246 .00314 .00399 .00504

0.00192 .00506 .00832 .01171 .01527

−158.93 −156.57 −154.17 −151.76 −149.31

−0.3244 −0.3193 −0.3141 −0.3090 −0.3039

3.7645 4.8779 6.2858 8.0565 10.272

0 5 10 15 20

25 30 32 32* 34

2.733 3.454 3.788 3.788 4.107

12.211 12.338 12.388 12.388 12.438

.054 .068 .075 .075 .082

12.265 12.406 12.463 12.463 12.520

6.005 7.206 7.686 7.686 8.167

2.929 3.709 4.072 4.072 4.418

8.934 10.915 11.758 11.758 12.585

.01273 .01519 .01617 .01617 .01715

.00635 .00796 .00870 .00870 .00940

.01908 .02315 .02487 .02487 .02655

−146.85 −144.36 −143.36 0.04 2.06

−0.2988 −0.2936 −0.2916 0.0000 .0041

13.032 16.452 18.035 18.037 19.546

25 30 32 32* 34

36 38 40 42 44

4.450 4.818 5.213 5.638 6.091

12.489 12.540 12.590 12.641 12.691

.089 .097 .105 .114 .124

12.578 12.637 12.695 12.755 12.815

8.647 9.128 9.608 10.088 10.569

4.791 5.191 5.622 6.084 6.580

13.438 14.319 15.230 16.172 17.149

.01812 .01909 .02005 .02101 .02197

.01016 .01097 .01183 .01275 .01373

.02828 .03006 .03188 .03376 .03570

4.07 6.08 8.09 10.09 12.10

.0081 .0122 .0162 .0202 .0242

ps 0.21166 .22904 .24767 .26763 .28899

36 38 40 42 44

NOTE: Compiled by John A. Goff and S. Gratch. See also Keenan and Kaye. Thermodynamic Properties of Air, Wiley, New York, 1945. Enthalpy of dry air taken as zero at 0°F. Enthalpy of liquid water taken as zero at 32°F. To convert British thermal units per pound to joules per kilogram, multiply by 2326; to convert British thermal units per pound dry air-degree Fahrenheit to joules per kilogram-kelvin, multiply by 4186.8; and to convert cubic feet per pound to cubic meters per kilogram, multiply by 0.0624. *Entrapolated to represent metastable equilibrium with undercooled liquid.

PSYCHROMETRY TABLE 12-1

12-9

Thermodynamic Properties of Moist Air (Standard Atmospheric Pressure, 29.921 inHg) (Continued ) Condensed water Volume, cu. ft./lb. dry air

Temp. t, °F.

Saturation humidity Hs × 108

va

46 48 50 52 54

6.578 7.100 7.658 8.256 8.894

12.742 12.792 12.843 12.894 12.944

56 58 60 62 64

9.575 10.30 11.08 11.91 12.80

66 68

13.74 14.75

vas

Entropy, B.t.u./(°F.)(lb. dry air)

Enthalpy, B.t.u./lb. dry air vs

ha

.134 .146 .158 .170 .185

12.876 12.938 13.001 13.064 13.129

11.049 11.530 12.010 12.491 12.971

12.995 13.045 13.096 13.147 13.197

.200 .216 .233 .251 .271

13.195 13.261 13.329 13.398 13.468

13.452 13.932 14.413 14.893 15.374

13.247 13.298

.292 .315

13.539 13.613

has 7.112 7.681 8.291 8.945 9.644

hs

sa

sas

ss

Enthalpy, B.t.u./lb. hw

Entropy, B.t.u./ (lb.)(°F.) sw

Vapor press., in. Hg ps × 106

Temp. t, °F.

18.161 19.211 20.301 21.436 22.615

.02293 .02387 .02481 .02575 .02669

.01478 .01591 .01711 .01839 .01976

.03771 .03978 .04192 .04414 .04645

14.10 16.11 18.11 20.11 22.12

.0282 .0321 .0361 .0400 .0439

.31185 .33629 .36240 .39028 .42004

46 48 50 52 54

10.39 11.19 12.05 12.96 13.94

23.84 25.12 26.46 27.85 29.31

.02762 .02855 .02948 .03040 .03132

.02121 .02276 .02441 .02616 .02803

.04883 .05131 .05389 .05656 .05935

24.12 26.12 28.12 30.12 32.12

.0478 .0517 .0555 .0594 .0632

.45176 .48558 .52159 .55994 .60073

56 58 60 62 64

15.855 16.335

14.98 16.09

30.83 32.42

.03223 .03314

.03002 .03213

.06225 .06527

34.11 36.11

.0670 .0708

.64411 .69019

66 68

.73915 .79112 .84624 .90470 .96665

70 72 74 76 78

Hs × 102 70 72 74 76 78

1.582 1.697 1.819 1.948 2.086

13.348 13.398 13.449 13.499 13.550

.339 .364 .392 .422 .453

13.687 13.762 13.841 13.921 14.003

16.816 17.297 17.778 18.259 18.740

17.27 18.53 19.88 21.31 22.84

34.09 35.83 37.66 39.57 41.58

.03405 .03495 .03585 .03675 .03765

.03437 .03675 .03928 .04197 .04482

.06842 .07170 .07513 .07872 .08247

38.11 40.11 42.10 44.10 46.10

.0746 .0784 .0821 .0859 .0896

80 82 84 86 88

2.233 2.389 2.555 2.731 2.919

13.601 13.651 13.702 13.752 13.803

0.486 .523 .560 .602 .645

14.087 14.174 14.262 14.354 14.448

19.221 19.702 20.183 20.663 21.144

24.47 26.20 28.04 30.00 32.09

43.69 45.90 48.22 50.66 53.23

0.03854 .03943 .04031 .04119 .04207

0.04784 .05105 .05446 .05807 .06189

0.08638 .09048 .09477 .09926 .10396

48.10 50.09 52.09 54.08 56.08

0.0933 .0970 .1007 .1043 .1080

1.0323 1.1017 1.1752 1.2529 1.3351

80 82 84 86 88

90 92 94 96 98

3.118 3.330 3.556 3.795 4.049

13.853 13.904 13.954 14.005 14.056

.692 .741 .795 .851 .911

14.545 14.645 14.749 14.856 14.967

21.625 22.106 22.587 23.068 23.548

34.31 36.67 39.18 41.85 44.68

55.93 58.78 61.77 64.92 68.23

.04295 .04382 .04469 .04556 .04643

.06596 .07025 .07480 .07963 .08474

.10890 .11407 .11949 .12519 .13117

58.08 60.07 62.07 64.06 66.06

.1116 .1153 .1188 .1224 .1260

1.4219 1.5135 1.6102 1.7123 1.8199

90 92 94 96 98

100 102 104

4.319 4.606 4.911

14.106 14.157 14.207

.975 1.043 1.117

15.081 15.200 15.324

24.029 24.510 24.991

47.70 50.91 54.32

71.73 75.42 79.31

.04729 .04815 .04900

.09016 .09591 .1020

.13745 .14406 .1510

68.06 70.05 72.05

.1296 .1332 .1367

1.9333 2.0528 2.1786

100 102 104

Hs × 10 106 108 110 112 114

0.5234 .5578 .5944 .6333 .6746

14.258 14.308 14.359 14.409 14.460

1.194 1.278 1.365 1.460 1.560

15.452 15.586 15.724 15.869 16.020

25.472 25.953 26.434 26.915 27.397

57.95 61.80 65.91 70.27 74.91

83.42 87.76 92.34 97.18 102.31

.04985 .05070 .05155 .05239 .05323

.1085 .1153 .1226 .1302 .1384

.1584 .1660 .1742 .1826 .1916

74.04 76.04 78.03 80.03 82.03

.1403 .1438 .1472 .1508 .1543

2.3109 2.4502 2.5966 2.7505 2.9123

106 108 110 112 114

116 118 120 122 124

.7185 .7652 .8149 .8678 .9242

14.510 14.561 14.611 14.662 14.712

1.668 1.782 1.905 2.034 2.174

16.178 16.343 16.516 16.696 16.886

27.878 28.359 28.841 29.322 29.804

79.85 85.10 90.70 96.66 103.0

107.73 113.46 119.54 125.98 132.8

.05407 .05490 .05573 .05656 .05739

.1470 .1562 .1659 .1763 .1872

.2011 .2111 .2216 .2329 .2446

84.02 86.02 88.01 90.01 92.01

.1577 .1612 .1646 .1681 .1715

3.0821 3.2603 3.4474 3.6436 3.8493

116 118 120 122 124

126 128 130 132 134

.9841 1.048 1.116 1.189 1.267

14.763 14.813 14.864 14.915 14.965

2.323 2.482 2.652 2.834 3.029

17.086 17.295 17.516 17.749 17.994

30.285 30.766 31.248 31.729 32.211

109.8 117.0 124.7 133.0 141.8

140.1 147.8 155.9 164.7 174.0

.05821 .05903 .05985 .06067 .06148

.1989 .2113 .2245 .2386 .2536

.2571 .2703 .2844 .2993 .3151

94.01 96.00 98.00 100.00 102.00

.1749 .1783 .1817 .1851 .1885

4.0649 4.2907 4.5272 4.7747 5.0337

126 128 130 132 134

136 138

1.350 1.439

15.016 15.066

3.237 3.462

18.253 18.528

32.692 33.174

151.2 161.2

183.9 194.4

.06229 .06310

.2695 .2865

.3318 .3496

104.00 106.00

.1918 .1952

5.3046 5.5878

136 138

140 142 144 146 148

0.1534 .1636 .1745 .1862 .1989

15.117 15.167 15.218 15.268 15.319

3.702 3.961 4.239 4.539 4.862

18.819 19.128 19.457 19.807 20.181

33.655 34.136 34.618 35.099 35.581

172.0 183.6 196.0 209.3 223.7

205.7 217.7 230.6 244.4 259.3

.06390 .06470 .06549 .06629 .06708

.3047 .3241 .3449 .3672 .3912

.3686 .3888 .4104 .4335 .4583

107.99 109.99 111.99 113.99 115.99

.1985 .2018 .2051 .2084 .2117

5.8838 6.1930 6.5160 6.8532 7.2051

140 142 144 146 148

150 152 154 156 158

.2125 .2271 .2430 .2602 .2788

15.369 15.420 15.470 15.521 15.571

5.211 5.587 5.996 6.439 6.922

20.580 21.007 21.466 21.960 22.493

36.063 36.545 37.026 37.508 37.990

239.2 255.9 273.9 293.5 314.7

275.3 292.4 310.9 331.0 352.7

.06787 .06866 .06945 .07023 .07101

.4169 .4445 .4743 .5066 .5415

.4848 .5132 .5438 .5768 .6125

117.99 119.99 121.99 123.99 125.99

.2150 .2183 .2216 .2248 .2281

7.5722 7.9550 8.3541 8.7701 9.2036

150 152 154 156 158

Hs

12-10

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

TABLE 12-1

Thermodynamic Properties of Moist Air (Standard Atmospheric Pressure, 29.921 inHg) (Concluded ) Condensed water

Temp. t, °F.

Saturation humidity Hs × 108

Volume, cu. ft./lb. dry air

Entropy, B.t.u./(°F.)(lb. dry air)

Enthalpy, B.t.u./lb. dry air

va

vas

vs

ha

has

hs

sa

sas

ss

Enthalpy, B.t.u./lb. hw

Entropy, B.t.u./ (lb.)(°F.) sw

Vapor press., in. Hg ps × 106

Temp. t, °F.

160 162 164 166 168

.2990 .3211 .3452 .3716 .4007

15.622 15.672 15.723 15.773 15.824

7.446 8.020 8.648 9.339 10.098

23.068 23.692 24.371 25.112 25.922

38.472 38.954 39.436 39.918 40.400

337.8 363.0 390.5 420.8 454.0

376.3 402.0 429.9 460.7 494.4

.07179 .07257 .07334 .07411 .07488

.5793 .6204 .6652 .7142 .7680

.6511 .6930 .7385 .7883 .8429

128.00 130.00 132.00 134.00 136.01

.2313 .2345 .2377 .2409 .2441

9.6556 10.125 10.614 11.123 11.652

160 162 164 166 168

170 172 174 176 178

.4327 .4682 .5078 .5519 .6016

15.874 15.925 15.975 16.026 16.076

10.938 11.870 12.911 14.074 15.386

26.812 27.795 28.886 30.100 31.462

40.882 41.364 41.846 42.328 42.810

490.6 531.3 576.5 627.1 684.1

531.5 572.7 618.3 669.4 726.9

.07565 .07641 .07718 .07794 .07870

.8273 .8927 .9654 1.047 1.137

.9030 .9691 1.0426 1.125 1.216

138.01 140.01 142.02 144.02 146.03

.2473 .2505 .2537 .2568 .2600

12.203 12.775 13.369 13.987 14.628

170 172 174 176 178

180 182 184 186 188

.6578 .7218 .7953 .8805 .9802

16.127 16.177 16.228 16.278 16.329

16.870 18.565 20.513 22.775 25.427

32.997 34.742 36.741 39.053 41.756

43.292 43.775 44.257 44.740 45.222

748.5 821.9 906.2 1004 1119

791.8 865.7 950.5 1049 1164

.07946 .08021 .08096 .08171 .08245

1.240 1.357 1.490 1.645 1.825

1.319 1.437 1.571 1.727 1.907

148.03 150.04 152.04 154.05 156.06

.2631 .2662 .2693 .2724 .2755

15.294 15.985 16.702 17.446 18.217

180 182 184 186 188

190 192 194 196 198

1.099 1.241 1.416 1.635 1.917

16.379 16.430 16.480 16.531 16.581

28.580 32.375 37.036 42.885 50.426

44.959 48.805 53.516 59.416 67.007

45.704 46.187 46.670 47.153 47.636

1255 1418 1619 1871 2195

1301 1464 1666 1918 2243

.08320 .08394 .08468 .08542 .08616

2.039 2.296 2.609 3.002 3.507

2.122 2.380 2.694 3.087 3.593

158.07 160.07 162.08 164.09 166.10

.2786 .2817 .2848 .2879 .2910

19.017 19.845 20.704 21.594 22.514

190 192 194 196 198

200

2.295

16.632

60.510

77.142

48.119

2629

2677

.08689

4.179

4.266

168.11

.2940

23.468

200

TABLE 12-2

Additive Corrections for H, h, and v When Barometric Pressure Differs from Standard Barometer Approximate altitude in feet −900

900

1800

2700

3700

4800

5900

∆p = +1

∆p = −1

∆p = −2

∆p = −3

∆p = −4

∆p = −5

∆p = −6

Wetbulb temp. tw

Sat. vapor press., in. Hg

∆Hs

∆h

∆Hs

∆h

∆Hs

∆h

−10 −8 −6 −4 −2

0.022 .025 .027 .030 .034

−0.10 −0.12 −0.13 −0.14 −0.16

−0.02 −0.02 −0.02 0.02 −0.02

0.11 .12 .14 .15 .17

0.02 .02 .02 .02 .02

0.23 .26 .29 .32 .35

0.03 .04 .04 .05 .05

0.36 .40 .44 .50 .55

0.05 .06 .07 .07 .08

0.50 .55 .62 .69 .76

0.07 .08 .09 .10 .11

0.64 .72 .80 .89 .99

0.10 .11 .12 .13 .15

0.81 .90 1.00 1.12 1.24

0.12 .13 .15 .17 .19

0 2 4 6 8

.038 .042 .046 .051 .057

−0.18 −0.20 −0.22 −0.24 −0.27

−0.03 −0.03 −0.03 −0.04 −0.04

.19 .21 .23 .26 .29

.03 .03 .03 .04 .04

.39 .44 .48 .54 .59

.06 .07 .07 .08 .09

.61 .68 .75 .83 .93

.09 .10 .11 .13 .14

.85 .94 1.05 1.16 1.28

.13 .14 .16 .18 .19

1.10 1.22 1.36 1.51 1.67

.17 .19 .21 .23 .25

1.38 1.53 1.70 1.89 2.09

.21 .23 .26 .29 .32

10 12 14 16 18

.063 .069 .077 .085 .093

−0.30 −0.33 −0.36 −0.40 −0.44

−0.04 −0.05 −0.05 −0.06 −0.07

.32 .35 .39 .43 .47

.05 .05 .06 .06 .07

.66 .73 .81 .89 .98

.10 .11 .12 .14 .15

1.03 1.13 1.25 1.38 1.53

.16 .17 .19 .21 .23

1.42 1.57 1.74 1.92 2.12

.22 .24 .26 .29 .32

1.85 2.04 2.26 2.49 2.75

.28 .31 .34 .38 .42

2.31 2.56 2.82 3.12 3.44

.35 .39 .43 .48 .53

20 22 24 26 28

.103 .113 .124 .137 .150

−0.49 −0.5 −0.6 −0.7 −0.7

−0.08 −0.08 −0.09 −0.10 −0.11

.52 .6 .6 .7 .8

.08 .09 .10 .11 .12

1.08 1.2 1.3 1.4 1.6

.17 .18 .20 .22 .24

1.68 1.9 2.1 2.3 2.5

.26 .29 .32 .35 .38

2.33 2.6 2.8 3.1 3.4

.36 .40 .43 .48 .52

3.03 3.4 3.7 4.1 4.5

.46 .52 .57 .63 .69

3.79 4.2 4.6 5.1 5.6

.58 .64 .71 .78 .86

30 32 34 36 38

.165 .180 .197 .212 .229

−0.8 −0.9 −0.9 −1.0 −1.1

−0.12 −0.13 −0.14 −0.15 −0.17

.8 .9 1.0 1.1 1.2

.13 .14 .15 .17 .18

1.7 1.9 2.1 2.2 2.4

.27 .29 .32 .35 .37

2.7 3.0 3.2 3.5 3.8

.42 .45 .49 .53 .58

3.8 4.1 4.4 4.8 5.2

.58 .63 .68 .74 .80

4.9 5.3 5.7 6.2 6.8

.75 .82 .88 .96 1.05

6.1 6.6 7.2 7.8 8.4

.92 1.01 1.11 1.20 1.30

40 42 44 46 48

.248 .268 .289 .312 .336

−1.2 −1.3 −1.4 −1.5 −1.6

−0.18 −0.20 −0.22 −0.23 −0.25

1.3 1.4 1.5 1.6 1.8

.20 .21 .23 .25 .27

2.6 2.8 3.1 3.3 3.6

.41 .44 .47 .51 .56

4.1 4.4 4.8 5.2 5.6

.63 .69 .74 .80 .87

5.7 6.1 6.7 7.2 7.8

.88 .94 1.04 1.11 1.21

7.4 8.0 8.7 9.4 10.2

1.14 1.23 1.34 1.45 1.58

9.2 10.0 10.8 11.7 12.6

1.42 1.54 1.67 1.81 1.95

50 52 54 56 58

.3624 .3903 .4200 .4518 .4856

−1.7 −1.9 −2.0 −2.2 −2.3

−0.27 −0.29 −0.31 −0.34 −0.37

1.9 2.0 2.2 2.4 2.5

.29 .32 .34 .37 .39

3.9 4.2 4.5 4.9 5.3

.60 .65 .70 .76 .82

6.1 6.5 7.0 7.6 8.2

.94 1.01 1.09 1.18 1.27

8.4 9.0 9.7 10.5 11.3

1.30 1.40 1.50 1.63 1.76

10.9 11.8 12.7 13.7 14.7

1.69 1.83 1.97 2.13 2.28

13.6 14.7 15.8 17.1 18.4

2.11 2.28 2.45 2.66 2.86

∆Hs

∆h

∆Hs

∆h

∆Hs

∆h

∆Hs

∆h

PSYCHROMETRY

12-11

Additive Corrections for H, h, and v When Barometric Pressure Differs from Standard Barometer (Concluded )

TABLE 12-2

Approximate altitude in feet −900

900

1800

2700

3700

4800

5900

∆p = +1

∆p = −1

∆p = −2

∆p = −3

∆p = −4

∆p = −5

∆p = −6

Wetbulb temp. tw

Sat. vapor press., in. Hg

∆Hs

∆h

∆Hs

∆h

∆Hs

∆h

∆Hs

∆h

∆Hs

∆h

∆Hs

∆h

∆Hs

∆h

60 62 64 66 68

.522 .560 .601 .644 .690

−2.5 −2.7 −2.9 −3.2 −3.4

−0.40 −0.43 −0.46 −0.50 −0.53

2.7 2.9 3.2 3.4 3.7

.42 .46 .49 .53 .57

5.7 6.1 6.5 7.1 7.6

.88 .95 1.02 1.10 1.18

8.8 9.5 10.2 11.0 11.8

1.37 1.48 1.59 1.72 1.84

12.2 13.2 14.2 15.3 16.4

1.90 2.05 2.21 2.38 2.56

15.9 17.1 18.4 19.8 21.3

2.47 2.66 2.87 3.09 3.32

19.9 21.4 23.1 24.8 26.7

3.09 3.33 3.60 3.87 4.16

70 72 74 76 78

.739 .791 .846 .905 .967

−3.7 −3.9 −4.2 −4.5 −4.9

−0.57 −0.61 −0.66 −0.71 −0.76

3.9 4.2 4.6 4.9 5.2

.61 .66 .71 .77 .82

8.1 8.7 9.4 10.0 10.8

1.27 1.36 1.46 1.57 1.69

12.7 13.6 14.6 15.7 16.9

1.98 2.13 2.28 2.46 2.65

17.6 18.8 20.2 21.7 23.3

2.75 2.94 3.16 3.39 3.65

22.9 24.6 26.4 28.3 30.5

3.58 3.84 4.14 4.42 4.77

28.7 30.9 33.1 35.5 38.2

4.48 4.82 5.18 5.56 5.98

80 82 84 86 88

1.032 1.102 1.175 1.253 1.335

−5.2 −5.6 −6.0 −6.4 −6.9

−0.82 −0.88 −0.94 −1.00 −1.08

5.6 6.0 6.4 6.9 7.4

.88 .94 1.00 1.08 1.16

11.6 12.5 13.3 14.3 15.3

1.82 1.96 2.10 2.24 2.40

18.1 19.5 20.9 22.3 23.9

2.84 3.06 3.28 3.50 3.75

25.1 27.0 28.9 30.9 33.1

3.93 4.24 4.54 4.85 5.20

32.7 35.1 37.7 40.4 43.2

5.13 5.51 5.92 6.34 6.79

41.0 44.0 47.2 50.6 54.2

6.43 6.90 7.41 7.94 8.51

90 92 94 96 98

1.422 1.514 1.610 1.712 1.820

−7.4 −7.9 −8.5 −9.1 −9.7

−1.16 −1.24 −1.34 −1.43 −1.53

7.9 8.5 9.1 9.8 10.4

1.24 1.34 1.43 1.54 1.64

16.5 17.6 18.9 20.2 21.7

2.59 2.77 2.98 3.18 3.42

25.7 27.5 29.5 31.5 33.8

4.04 4.33 4.64 4.96 5.33

35.6 38.2 41.0 43.8 47.0

5.60 6.01 6.46 6.90 7.41

46.4 49.8 53.4 57.2 61.3

7.29 7.83 8.41 9.01 9.67

58.2 62.5 67.0 71.7 76.8

9.15 9.83 10.55 11.30 12.11

100 102 104 106 108

1.933 2.053 2.179 2.311 2.450

−10.4 −11.1 −11.9 −12.8 −13.7

−1.64 −1.75 −1.88 −2.02 −2.17

11.2 12.0 12.8 13.7 14.7

1.77 1.90 2.02 2.17 2.33

23.2 24.8 26.6 28.6 30.6

3.66 3.92 4.20 4.52 4.84

36.3 38.9 41.6 44.6 47.7

5.73 6.14 6.58 7.06 7.55

50.4 54.1 57.9 62.1 66.5

7.95 8.54 9.15 9.82 10.53

65.7 70.5 75.5 81.1 87.0

10.37 11.13 11.93 12.83 13.77

82.5 88.5 94.8 101.7 109.1

13.02 13.98 14.98 16.09 17.27

110 112 114 116 118

2.597 2.751 2.913 3.082 3.260

−14.7 −15.7 −16.9 −18.0 −19.3

−2.33 −2.49 −2.68 −2.86 −3.07

15.8 16.9 18.1 19.4 20.8

2.50 2.68 2.87 3.08 3.31

32.8 35.2 37.7 40.4 43.3

5.20 5.58 5.98 6.42 6.88

51.3 55.0 58.9 63.2 67.8

8.13 8.72 9.50 10.03 10.77

71.3 76.4 82.0 88.0 94.4

11.30 12.11 13.01 13.97 15.00

93.1 99.9 107.3 115.1 123.5

14.75 15.84 17.03 18.28 19.63

117.0 125.9 135.0 144.7 155.4

18.54 19.96 21.42 22.98 24.73

120 122 124 126 128

3.448 3.644 3.850 4.065 4.291

−20.7 −22.2 −23.8 −25.6 −27.5

−3.29 −3.53 −3.79 −4.08 −4.39

22.4 24.0 25.8 27.6 29.7

3.56 3.82 4.11 4.40 4.74

46.6 50.0 53.7 57.7 62.0

7.41 7.96 8.55 9.20 9.89

72.8 78.2 84.0 90.3 97.1

11.58 12.45 13.38 14.39 15.49

101.4 109.0 117.1 125.9 135.5

16.13 17.35 18.65 20.07 21.61

132.7 142.6 153.3 165.0 177.6

21.10 22.70 24.42 26.30 28.33

167.1 179.6 193.2 208.0 224.0

26.58 28.58 30.77 33.15 35.73

130 132 134 136 138

4.527 4.775 5.034 5.305 5.588

−29.5 −31.8 −34.2 −36.8 −39.7

−4.71 −5.08 −5.47 −5.89 −6.36

32.0 34.4 37.1 40.0 43.2

5.11 5.50 5.93 6.40 6.92

66.7 71.8 77.4 83.4 90.0

10.64 11.47 12.37 13.34 14.41

104.5 112.6 121.4 130.9 141.4

16.68 17.99 19.41 20.94 22.64

145.9 157.2 169.6 183.1 197.8

23.29 25.11 27.12 29.30 31.67

191.4 206.3 222.7 240.5 260.1

30.55 32.96 35.60 38.48 41.65

241.5 260.6 281.4 304.2 329.3

38.55 41.63 44.99 48.67 52.73

140

5.884

−42.8

−6.86

46.5

8.45

97.3

15.59

152.8

24.48

214.0

34.29

281.6

45.12

356.8

57.17

t = dry-bulb temperature, °F tw = wet-bulb temperature, °F p = barometric pressure, inHg ∆p = pressure difference from standard barometer (inHg) H = moisture content of air, gr/lb dry air Hs = moisture content of air saturated at wet-bulb temperature (tw), gr/lb dry air ∆H = moisture-content correction of air when barometric pressure differs from standard barometer, gr/lb dry air ∆Hs = moisture-content correction of air saturated at wet-bulb temperature when barometric pressure differs from standard barometer, gr/lb dry air NOTE: To obtain ∆H reduce value of ∆Hs by 1 percent where t − tw = 24°F and correct proportionally when t − tw is not 24°F. h = enthalpy of moist air, Btu/lb dry air ∆h = enthalpy correction when barometric pressure differs from standard barometer, for saturated or unsaturated air, Btu/lb dry air v = volume of moist air, ft3/lb dry air H 0.754(t + 459.8) = ᎏᎏ 1 + ᎏ p 4360

冢

冣

Example At a barometric pressure of 25.92 with 220°F dry-bulb and 100°F wet-bulb temperatures, determine H, h, and v. ∆p = −4, and from table ∆Hs = 50.4. From note,

冢

冣

120 ∆H = ∆Hs − ᎏ × 0.01 × 50.4 = 50.4 − 2.5 = 47.9 24 Therefore H = 102 (from chart) + 47.9 = 149.9 gr/lb. dry air. From table ∆h = 7.95. Therefore, h = saturation enthalpy from chart + deviation + 7.95 = 71.7 − 2.0 + 7.95 = 77.65 Btu/lb dry air. From previous equation 0.754(220 + 459.7) 149.9 v = ᎏᎏ 1 + ᎏ = 20.43 ft3/lb dry air 25.92 4360

冢

冣

12-12

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

FIG. 12-10

Cooling tower.

Example 6: Cooling Tower (Continued)

FIG. 12-6

Diagram of psychrometric chart showing the properties of moist air.

Exit moisture H2 = 416 gr/lb dry air Inlet moisture H1 = 78 gr/lb dry air Moisture added ∆H = 338 gr/lb dry air Enthalpy of added moisture hw = 2.1 Btu/lb dry air (from small diagram of Fig. 12-3) If greater precision is desired, hw can be calculated as hw = (338/7000)(1)(75 − 32) = 2.08 Btu/lb dry air Enthalpy of inlet air h1 = h′1 + D1 = 34.1 − 0.18 = 33.92 Btu/lb dry air Enthalpy of exit air h2 = h′2 + D2 = 92.34 + 0 = 92.34 Btu/lb dry air Heat dissipated = h2 − h1 − hw = 92.34 − 33.92 − 2.08 = 56.34 Btu/lb dry air Specific volume of inlet air = 14.1 ft3/lb dry air (1000)(56.34) Total heat dissipated = ᎏᎏ = 3990 Btu/min 14.1

FIG. 12-7

Heating process.

Example 7: Recirculating Dryer A dryer is removing 100 lb water/h from the material being dried. The air entering the dryer has a dry-bulb temperature of 180°F and a wet-bulb temperature of 110°F. The air leaves the dryer at 140°F. A portion of the air is recirculated after mixing with room air having a dry-bulb temperature of 75°F and a relative humidity of 60 percent. Determine quantity of air required, recirculation rate, and load on the preheater if it is assumed that the system is adiabatic. Neglect heatup of the feed and of the conveying equipment. Solution. The path followed is shown in Fig. 12-11. Humidity of room air H1 = 0.0113 lb/lb dry air Humidity of air entering dryer H3 = 0.0418 lb/lb dry air Humidity of air leaving dryer H4 = 0.0518 lb/lb dry air Enthalpy of room air h1 = 30.2 − 0.3 = 29.9 Btu/lb dry air Enthalpy of entering air h3 = 92.5 − 1.3 = 91.2 Btu/lb dry air

FIG. 12-8

Spray or evaporative cooling.

FIG. 12-9

Cooling and dehumidifying process.

FIG. 12-11

Drying process with recirculation.

PSYCHROMETRY Enthalpy of leaving air h4 = 92.5 − 0.55 = 91.95 Btu/lb dry air Quantity of air required is 100/(0.0518 − 0.0418) = 10,000 lb dry air/h. At the dryer inlet the specific volume is 17.1 ft 3/lb dry air. Air volume is (10,000)(17.1)/ 60 = 2850 ft 3/min. Fraction exhausted is 0.0518 − 0.0418 X ᎏ = ᎏᎏ = 0.247 Wa 0.0518 − 0.0113 where X = quantity of fresh air and Wa = total air flow. Thus 75.3 percent of the air is recirculated. Load on the preheater is obtained from an enthalpy balance qa = 10,000(91.2) − 2470(29.9) − 7530(91.95) = 146,000 Btu/h

冢

Relative humidity = 100p/ps

To this must be added the correction of 2.75 read from Table 12-2 for ∆p = −4 and tw = 70°F, giving the true enthalpy as 33.92 + 2.75 = 36.67 Btu/lb dry air. The partial pressure of water vapor is calculated from Eq. (12-6) as HP 0.01362 × 25.92 p = ᎏᎏ = ᎏᎏ = 0.556 inHg (Mw /Ma) + H 0.622 + 0.01362 From a table of vapor pressure, this corresponds to a dew point of 61.8°F. Relative humidity is obtained from Eq. (12-7) as 100 p/ps = (100 × 0.556)/ 1.422 = 39.1 percent. The specific volume in cubic feet per pound of dry air is obtained from Eq. (12-5): 0.754(t + 460) HM v = ᎏᎏ 1 + ᎏa 25.92 Mw

冢

Use of Psychrometric Charts at Pressures Other Than Atmospheric The psychrometric charts shown as Figs. 12-1 through 12-4 and the data of Table 12-1 are based on a system pressure of 1 atm (29.92 inHg). For other system pressures, these data must be corrected for the effect of pressure. Additive corrections to be applied to the atmospheric values of absolute humidity and enthalpy are given in Table 12-2. The specific volume of moist air in cubic feet per pound of dry air can be determined for other pressures, if ideal-gas behavior is assumed, by the following equation: HM 0.754(t + 460) v = ᎏᎏ 1 + ᎏa (12-5) P Mw where v = specific volume, ft3/lb dry air; t = dry-bulb temperature, °F; P = pressure, inHg; H = absolute humidity, lb water/lb dry air; Ma = molecular weight of air, lb/(lb⋅mol); and Mw = molecular weight of water vapor, lb/(lb⋅mol). Relative humidity and dew point can be determined for other than atmospheric pressure from the partial pressure of water in the mixture and from the vapor pressure of water vapor. The partial pressure of water is calculated, if ideal-gas behavior is assumed, as HP p = ᎏᎏ (12-6) (Mw /Ma) + H where p = partial pressure of water vapor, inHg; P = total pressure, inHg; H = absolute humidity, lb water/lb dry air, corrected to the actual pressure; Ma = molecular weight of air, lb/(lb⋅mol); and Mw = molecular weight of water vapor, lb/(lb⋅mol). The dew point of the mixture is then read directly from a table of vapor pressures as the temperature corresponding to the calculated partial pressure. The relative humidity is obtained by dividing the calculated partial pressure by the vapor pressure of water at the dry-bulb temperature. Thus:

冣

(12-7)

where p = calculated partial pressure, inHg; and ps = vapor pressure at dry-bulb temperature, inHg. The preceding equations, which have assumed that both the air and the water vapor behave as ideal gases, are sufficiently accurate for most engineering calculations. If it is desired to remove the restriction that water vapor behave as an ideal gas, the actual density ratio should be used in place of the molecular-weight ratio in Eqs. (12-5) and (12-6). Since the Schmidt number, Prandtl number, latent heat of vaporization, and humid heat are all essentially independent of pressure, the adiabatic-saturation-temperature and wet-bulb-temperature lines will be substantially equal at pressures different from atmospheric. Example 8: Determination of Air Properties For a barometric pressure of 25.92 inHg (∆p = −4), a dry-bulb temperature of 90°F, and a wetbulb temperature of 70°F determine the following: absolute humidity, enthalpy, dew point, relative humidity, and specific volume. Solution. From Fig. 12-2, the moisture content is 78 gr/lb dry air = 0.0114 lb/lb dry air. From Table 12-2 at tw = 70°F and ∆p = −4 read ∆Hs = 17.6 gr/lb dry air (additive correction for air saturated at the wet-bulb temperature). ∆H = 17.6[1 − (20/24)(0.01)] = 17.4, or actual humidity is 78 + 17.4 = 95.4 gr/lb dry air, or 0.01362 lb/lb dry air. (See footnotes for Table 12-2.) The enthalpy is obtained from Fig. 12-2 as h = h′ + D = 34.1 − 0.18 = 33.92.

12-13

冣

0.754(90 + 460) 0.01362 = ᎏᎏ 1 + ᎏ 25.92 0.622

冢

冣

= 16.35 ft 3/lb dry air

MEASUREMENT OF HUMIDITY Dew-Point Method The dew point of wet air is measured directly by observing the temperature at which moisture begins to form on an artificially cooled polished surface. The polished surface is usually cooled by evaporation of a low-boiling solvent such as ether, by vaporization of a condensed permanent gas such as carbon dioxide or liquid air, or by a temperature-regulated stream of water. Although the dew-point method may be considered a fundamental technique for determining humidity, several uncertainties occur in its use. It is not always possible to measure precisely the temperature of the polished surface or to eliminate gradients across the surface. It is also difficult to detect the appearance or disappearance of fog; the usual practice is to take the dew point as the average of the temperatures when fog first appears on cooling and disappears on heating. Wet-Bulb Method Probably the most commonly used method for determining the humidity of a gas stream is the measurement of wet- and dry-bulb temperatures. The wet-bulb temperature is measured by contacting the air with a thermometer whose bulb is covered by a wick saturated with water. If the process is adiabatic, the thermometer bulb attains the wet-bulb temperature. When the wet- and dry-bulb temperatures are known, the humidity is readily obtained from charts such as Figs. 12-1 through 12-4. In order to obtain reliable information, care must be exercised to ensure that the wet-bulb thermometer remains wet and that radiation to the bulb is minimized. The latter is accomplished by making the relative velocity between wick and gas stream high [a velocity of 4.6 m/s (15 ft/s) is usually adequate for commonly used thermometers] or by the use of radiation shielding. Making sure that the wick remains wet is a mechanical problem, and the method used depends to a large extent on the particular arrangement. Again, as with the dew-point method, errors associated with the measurement of temperature can cause difficulty. For measurement of atmospheric humidities the sling psychrometer is widely used. This is composed of a wet- and dry-bulb thermometer mounted in a sling which is whirled manually to give the desired gas velocity across the bulb. In the Assmann psychrometer the air is drawn past the bulbs by a motor-driven fan. In addition to the mercury-in-glass thermometer, other temperature-sensing elements may be used for psychrometers. These include resistance thermometers, thermocouples, bimetal thermometers, and thermistors. Mechanical Hygrometers Materials such as human hair, wood fiber, and plastics have been used to measure humidity. These methods rely on a change in dimension with humidity. Electric hygrometers measure the electrical resistance of a film of moisture-absorbing materials exposed to the gas. A wide variety of sensing elements have been used. The gravimetric method is accepted as the most accurate humidity-measuring technique. In this method a known quantity of gas is passed over a moisture-absorbing chemical such as phosphorus pentoxide, and the increase in weight is determined.

12-14

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

EVAPORATIVE COOLING* GENERAL REFERENCES: ASHRAE Handbook and Product Directory: Equipment, American Society of Heating, Refrigerating and Air-Conditioning Engineers, Atlanta, 1983. Counterflow Cooling Tower Performance, Pritchard Corporation, Kansas City, Mo., 1957. Hensley, “Cooling Tower Energy,” Heat. Piping Air Cond. (October 1981). Kelley and Swenson, Chem. Eng. Prog., 52, 263 (1956). McAdams, Heat Transmission, 3d ed., McGraw-Hill, New York, 1954, pp. 356–365. Merkel, Z. Ver. Dtsch. Ing. Forsch., no. 275 (1925). The Parallel Path Wet-Dry Cooling Tower, Marley Co., Mission Woods, Kan., 1972. Performance Curves, Cooling Tower Institute, Houston, 1967. Plume Abatement and Water Conservation with Wet-Dry Cooling Tower, Marley Co., Mission Woods, Kan., 1973. Tech. Bull. R-54-P-5, R-58-P-5, Marley Co., Mission Woods, Kan., 1957. Wood and Betts, Engineer, 189 (4912), 377, (4913), 349 (1950). Zivi and Brand, Refrig. Eng., 64, 8, 31–34, 90 (1956).

PRINCIPLES The processes of cooling water are among the oldest known. Usually water is cooled by exposing its surface to air. Some of the processes are slow, such as the cooling of water on the surface of a pond; others are comparatively fast, such as the spraying of water into air. These processes all involve the exposure of water surface to air in varying degrees. The heat-transfer process involves (1) latent heat transfer owing to vaporization of a small portion of the water and (2) sensible heat transfer owing to the difference in temperature of water and air. Approximately 80 percent of this heat transfer is due to latent heat and 20 percent to sensible heat. Theoretical possible heat removal per pound of air circulated in a cooling tower depends on the temperature and moisture content of air. An indication of the moisture content of the air is its wet-bulb temperature. Ideally, then, the wet-bulb temperature is the lowest theoretical temperature to which the water can be cooled. Practically, the cold-water temperature approaches but does not equal the air wetbulb temperature in a cooling tower; this is so because it is impossible to contact all the water with fresh air as the water drops through the wetted fill surface to the basin. The magnitude of approach to the wetbulb temperature is dependent on tower design. Important factors are air-to-water contact time, amount of fill surface, and breakup of water into droplets. In actual practice, cooling towers are seldom designed for approaches closer than 2.8°C (5°F).

The water operating line is shown by line AB and is fixed by the inlet and outlet tower water temperatures. The air operating line begins at C, vertically below B and at a point having an enthalpy corresponding to that of the entering wet-bulb temperature. Line BC represents the initial driving force (h′ − h). In cooling water 1°F, the enthalpy per pound of air is increased 1 Btu multiplied by the ratio of pounds of water per pound of air. The liquid-gas ratio L/G is the slope of the operating line. The air leaving the tower is represented by point D. The cooling range is the projected length of line CD on the temperature scale. The cooling-tower approach is shown on the diagram as the difference between the cold-water temperature leaving the tower and the ambient wet-bulb temperature. The coordinates refer directly to the temperature and enthalpy of any point on the water operating line but refer directly only to the enthalpy of a point on the air operating line. The corresponding wetbulb temperature of any point on CD is found by projecting the point horizontally to the saturation curve, then vertically to the temperature coordinate. The integral [Eq. (12-8)] is represented by the area ABCD in the diagram. This value is known as the tower characteristic, varying with the L/G ratio. For example, an increase in entering wet-bulb temperature moves the origin C upward, and the line CD shifts to the right to maintain a constant KaV/L. If the cooling range increases, line CD lengthens. At a constant wet-bulb temperature, equilibrium is established by moving the line to the right to maintain a constant KaV/L. On the other hand, a change in L/G ratio changes the slope of CD, and the tower comes to equilibrium with a new KaV/L. In order to predict tower performance it is necessary to know the required tower characteristics for fixed ambient and water conditions. The tower characteristic KaV/L can be determined by integration. Normally used is the Chebyshev method for numerically evaluating the integral, whereby KaV ᎏ= L

冕

T1

T2

dT T1 − T2 1 1 1 1 ᎏ⬵ᎏ ᎏ+ᎏ+ᎏ+ᎏ hw − ha 4 ∆h1 ∆h2 ∆h3 ∆h4

冢

冣

COOLING-TOWER THEORY The most generally accepted theory of the cooling-tower heat-transfer process is that developed by Merkel (op. cit.). This analysis is based upon enthalpy potential difference as the driving force. Each particle of water is assumed to be surrounded by a film of air, and the enthalpy difference between the film and surrounding air provides the driving force for the cooling process. In the integrated form the Merkel equation is KaV ᎏ= L

冕

T1

T2

dT ᎏ h′ − h

(12-8)

where K = mass-transfer coefficient, lb water/(h⋅ft2); a = contact area, ft2/ft3 tower volume; V = active cooling volume, ft3/ft2 of plan area; L = water rate, lb/(h⋅ft2); h′ = enthalpy of saturated air at water temperature, Btu/lb; h = enthalpy of air stream, Btu/lb; and T1 and T2 = entering and leaving water temperatures, °F. The right-hand side of Eq. (12-8) is entirely in terms of air and water properties and is independent of tower dimensions. Figure 12-12 illustrates water and air relationships and the driving potential which exist in a counterflow tower, where air flows parallel but opposite in direction to water flow. An understanding of this diagram is important in visualizing the cooling-tower process.

FIG. 12-12

Cooling-tower process heat balance. (Marley Co.)

* The contribution of Robert W. Norris, Robert W. Norris and Associates, Inc., to material that was used from the sixth edition is acknowledged. (Evaporative Cooling)

EVAPORATIVE COOLING where hw = enthalpy of air-water vapor mixture at bulk water temperature, Btu/lb dry air ha = enthalpy of air-water vapor mixture at wet-bulb temperature, Btu/lb dry air ∆h1 = value of (hw − ha) at T2 + 0.1(T1 − T2) ∆h2 = value of (hw − ha) at T2 + 0.4(T1 − T2) ∆h3 = value of (hw − ha) at T1 − 0.4(T1 − T2) ∆h4 = value of (hw − ha) at T1 − 0.1(T1 − T2) Example 9: Calculation of Mass-Transfer Coefficient Group Determine the theoretically required KaV/L value for a cooling duty from 105°F inlet water, 85°F outlet water, 78°F ambient wet-bulb temperature, and an L/G ratio of 0.97. From air-water vapor-mixture tables, the enthalpy h1 of the ambient air at 78°F wet-bulb temperature is 41.58 Btu/lb. h2 (leaving air) = 41.58 + 0.97(105 − 85) = 60.98 Btu/lb T, °F.

hwater

hair

hw − ha

1/∆h

T2 = 85 T2 + 0.1(20) = 87 T2 + 0.4(20) = 93 T1 − 0.4(20) = 97 T1 − 0.1(20) = 103 T1 = 105

49.43 51.93 60.25 66.55 77.34 81.34

h1 = 41.58 h1 + 0.1L/G(20) = 43.52 h1 + 0.4L/G(20) = 49.34 h2 − 0.4L/G(20) = 53.22 h2 − 0.1L/G(20) = 59.04 h2 = 60.98

∆h1 = 8.41 ∆h2 = 10.91 ∆h3 = 13.33 ∆h4 = 18.30

0.119 0.092 0.075 0.055 0.341

KaV 105 − 85 ᎏ = ᎏ (0.341) = 1.71 L 4

A quicker but less accurate method is by the use of a nomograph (Fig. 12-13) prepared by Wood and Betts (op. cit.). Mechanical-draft cooling towers normally are designed for L/G ratios ranging from 0.75 to 1.50; accordingly, the values of KaV/L vary from 0.50 to 2.50. With these ranges in mind, an example of the use of the nomograph will readily explain the effect of changing variables.

12-15

Example 10: Application of Nomograph for Cooling-Tower Characteristics If a given tower is operating with 20°F range, a cold-water temperature of 80°F, and a wet-bulb temperature of 70°F, a straight line may be drawn on the nomograph. If the L/G ratio is calculated to be 1.0, then KaV/L may be established by a line drawn through L/G = 1.0 and parallel to the original line. The tower characteristic KaV/L is thus established at 1.42. If the wetbulb temperature were to drop to 50°F, KaV/K and L/G ratios may be assumed to remain constant. A new line parallel to the original will then show that for the same range the cold-water temperature will be 70°F.

The nomograph provides an approximate solution; degree of accuracy will vary with changes in cooling as well as from tower to tower. Once the theoretical cooling-tower characteristic has been determined by numerical integration or from the nomograph for a given cooling duty, it is necessary to design the cooling-tower fill and air distribution to meet the theoretical tower characteristic. The Pritchard Corporation (op. cit.) has developed performance data on various tower-fill designs. These data are too extensive to include here, and those interested should consult this reference. See also Baker and Mart (Marley Co., Tech. Bull. R-52-P-10, Mission Woods, Kan.) and Zivi and Brand (loc. cit.). MECHANICAL-DRAFT TOWERS Two types of mechanical-draft towers are in use today: the forceddraft and the induced-draft. In the forced-draft tower the fan is mounted at the base, and air is forced in at the bottom and discharged at low velocity through the top. This arrangement has the advantage of locating the fan and drive outside the tower, where it is convenient for inspection, maintenance, and repairs. Since the equipment is out of the hot, humid top area of the tower, the fan is not subjected to corrosive conditions. However, because of the low exit-air velocity, the forced-draft tower is subjected to excessive recirculation of the humid

FIG. 12-13 Nomograph of cooling-tower characteristics. [Wood and Betts, Engineer, 189(4912), 337 (1950).]

12-16

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

exhaust vapors back into the air intakes. Since the wet-bulb temperature of the exhaust air is considerably above the wet-bulb temperature of the ambient air, there is a decrease in performance evidenced by an increase in cold (leaving)-water temperature. The induced-draft tower is the most common type used in the United States. It is further classified into counterflow and cross-flow design, depending on the relative flow directions of water and air. Thermodynamically, the counterflow arrangement is more efficient, since the coldest water contacts the coldest air, thus obtaining maximum enthalpy potential. The greater the cooling ranges and the more difficult the approaches, the more distinct are the advantages of the counterflow type. For example, with an L/G ratio of 1, an ambient wet-bulb temperature of 25.5°C (78°F), and an inlet water temperature of 35°C (95°F), the counterflow tower requires a KaV/L characteristic of 1.75 for a 2.8°C (5°F) approach, while a cross-flow tower requires a characteristic of 2.25 for the same approach. However, if the approach is increased to 3.9°C (7°F), both types of tower have approximately the same required KaV/L (within 1 percent). The cross-flow-tower manufacturer may effectively reduce the tower characteristic at very low approaches by increasing the air quantity to give a lower L/G ratio. The increase in air flow is not necessarily achieved by increasing the air velocity but primarily by lengthening the tower to increase the air-flow cross-sectional area. It appears then that the cross-flow fill can be made progressively longer in the direction perpendicular to the air flow and shorter in the direction of the air flow until it almost loses its inherent potential-difference disadvantage. However, as this is done, fan power consumption increases. Ultimately, the economic choice between counterflow and crossflow is determined by the effectiveness of the fill, design conditions, and the costs of tower manufacture. Performance of a given type of cooling tower is governed by the ratio of the weights of air to water and the time of contact between water and air. In commercial practice, the variation in the ratio of air to water is first obtained by keeping the air velocity constant at about 350 ft/(min⋅ft2 of active tower area) and varying the water concentration, gal/(min⋅ft2 of tower area). As a secondary operation, air velocity is varied to make the tower accommodate the cooling requirement. Time of contact between water and air is governed largely by the time required for the water to discharge from the nozzles and fall through the tower to the basin. The time of contact is therefore obtained in a given type of unit by varying the height of the tower. Should the time of contact be insufficient, no amount of increase in the ratio of air to water will produce the desired cooling. It is therefore necessary to maintain a certain minimum height of cooling tower. When a wide approach of 8 to 11°C (15 to 20°F) to the wet-bulb temperature and a 13.9 to 19.4°C (25 to 35°F) cooling range are required, a relatively low cooling tower will suffice. A tower in which the water travels 4.6 to 6.1 m (15 to 20 ft) from the distributing system to the basin is sufficient. When a moderate approach and a cooling range of 13.9 to 19.4°C (25 to 35°F) are required, a tower in which the water travels 7.6 to 9.1 m (25 to 30 ft) is adequate. Where a close approach of 4.4°C (8°F) with a 13.9 to 19.4°C (25 to 35°F) cooling range is required, a tower in which the water travels from 10.7 to 12.2 m (35 to 40 ft) is required. It is usually not economical to design a cooling tower with an approach of less than 2.8°C (5°F), but it can be accomplished satisfactorily with a tower in which the water travels 10.7 to 12.2 m (35 to 40 ft). Figure 12-14 shows the relationship of the hot-water, cold-water, and wet-bulb temperatures to the water concentration.* From this, the minimum area required for a given performance of a welldesigned counterflow induced-draft cooling tower can be obtained. Figure 12-15 gives the horsepower per square foot of tower area required for a given performance. These curves do not apply to parallel or cross-flow cooling, since these processes are not so efficient as the counterflow process. Also, they do not apply when the approach to the cold-water temperature is less than 2.8°C (5°F). These charts should be considered approximate and for preliminary estimates only. Since many factors not shown in the graphs must be included in the

FIG. 12-14 Sizing chart for a counterflow induced-draft cooling tower, for induced-draft towers with (1) an upspray distributing system with 24 ft of fill or (2) a flume-type distributing system and 32 ft of fill. The chart will give approximations for towers of any height. (Ecodyne Corp.)

computation, the manufacturer should be consulted for final design recommendations. The cooling performance of any tower containing a given depth of filling varies with the water concentration. It has been found that maximum contact and performance are obtained with a tower having a water concentration of 2 to 5 gal/(min⋅ft2 of ground area). Thus the problem of calculating the size of a cooling tower becomes one of determining the proper concentration of water required to obtain the desired results. Once the necessary water concentration has been established, tower area can be calculated by dividing the gallons per minute circulated by the water concentration in gallons per minutesquare foot. The required tower size then is a function of the following: 1. Cooling range (hot-water temperature minus cold-water temperature) 2. Approach to wet-bulb temperature (cold-water temperature minus wet-bulb temperature) 3. Quantity of water to be cooled 4. Wet-bulb temperature 5. Air velocity through the cell 6. Tower height Example 11: Application of Sizing and Horsepower Charts To illustrate the use of the charts, assume the following conditions: Hot-water temperature T1, °F = 102 Cold-water temperature T2, °F = 78 Wet-bulb temperature tw, °F = 70 Water rate, gal/min = 2000 A straight line on Fig. 12-14, connecting the points representing the design water and wet-bulb temperatures, shows that a water concentration of 2 gal/ (min⋅ft2) is required. The area of the tower is calculated as 1000 ft2 (quantity of water circulated divided by water concentration). Fan horsepower is obtained from Fig. 12-15. Connecting the point representing 100 percent of standard tower performance with the turning point and extending this straight line to the horsepower scale show that it will require 0.041 hp/ft2 of actual effective tower area. For a tower area of 1000 ft2 41.0 fan hp is required to perform the necessary cooling. Suppose that the actual commercial tower size has an area of only 910 ft2. Within reasonable limits, the shortage of actual area can be compensated for by an increase in air velocity through the tower. However, this requires boosting fan horsepower to achieve 110 percent of standard tower performance. From Fig. 12-15, the fan horsepower is found to be 0.057 hp/ft2 of actual tower area, or 0.057 × 910 = 51.9 hp. On the other hand, if the actual commercial tower area is 1110 ft2, the cooling equivalent to 1000 ft2 of standard tower area can be accomplished with less air and less fan horsepower. From Fig. 12-15, the fan horsepower for a tower operating at 90 percent of standard performance is 0.031 hp/ft2 of actual tower area, or 34.5 hp.

This example illustrates the sensitivity of fan horsepower to small changes in tower area. The importance of designing a tower that is slightly oversize in ground area and of providing plenty of fan capacity becomes immediately apparent.

* See also London, Mason, and Boelter, loc. cit.; Lichtenstein, loc. cit.; Simpson and Sherwood, J. Am. Soc. Refrig. Eng., 52, 535, 574 (1946); Simons, Chem. Metall. Eng., 49(5), 138; (6), 83 (1942); 46, 208 (1939); and Hutchinson and Spivey, Trans. Inst. Chem. Eng., 20, 14 (1942).

EVAPORATIVE COOLING

12-17

Wm = We + Wd + Wb

(12-9)

where Wm = makeup water, Wd = drift loss, and Wb = blowdown [consistent units, m3/(h⋅gal⋅min)]. Evaporation loss can be estimated by the following equation: We = 0.00085 Wc(T1 − T2) where

FIG. 12-15 Horsepower chart for a counterflow induced-draft cooling tower. [Fluor Corp. (now Ecodyne Corp.)]

Example 12: Application of Sizing Chart Assume the same cooling range and approach as used in Example 11 except that the wet-bulb temperature is lower. Design conditions would then be Water rate, gal/min = 2000 Temperature range (T1 − T2), °F = 24 Temperature approach (T2 − tw),°F = 8 Hot-water temperature T1 , °F = 92 Cold-water temperature T2 , °F = 68 Wet-bulb temperature t w , °F = 60 From Fig. 12-14, the water concentration required to perform the cooling is 1.75 gal/(min·ft2), giving a tower area of 1145 ft2 versus 1000 ft2 for a 70°F wetbulb temperature. This shows that the lower the wet-bulb temperature for the same cooling range and approach, the larger is the area of the tower required and therefore the more difficult is the cooling job.

Figure 12-16 illustrates the type of performance curve furnished by the cooling-tower manufacturer. This shows the variation in performance with changes in wet-bulb and hot-water temperatures while the water quantity is maintained constant. COOLING-TOWER OPERATION Water Makeup Makeup requirements for a cooling tower consist of the summation of evaporation loss, drift loss, and blowdown. Therefore,

(12-10)

Wc = circulating-water flow, gal/min at tower inlet T1 − T2 = inlet-water temperature minus outlet-water temperature, °F

Drift is entrained water in the tower discharge vapors. Drift loss is a function of the drift-eliminator design, which typically varies between 0.1 and 0.2 percent of the water supplied to the tower. New developments in eliminator design make it possible to reduce drift loss well below 0.1 percent. Blowdown discards a portion of the concentrated circulating water due to the evaporation process in order to lower the system solids concentration. The amount of blowdown can be calculated according to the number of cycles of concentration required to limit scale formation. Cycles of concentration are the ratio of dissolved solids in the recirculating water to dissolved solids in the makeup water. Since chlorides remain soluble on concentration, cycles of concentration are best expressed as the ratio of the chloride content of the circulating and makeup waters. Thus, the blowdown quantities required are determined from Cycles of concentration = (We + Wb)/Wb

(12-11)

Wb = We /(cycles − 1)

(12-12)

or

Cycles of concentration involved with cooling-tower operation normally range from three to five cycles. Below three cycles of concentration, excessive blowdown quantities are required and the addition of acid to limit scale formation should be considered. Example 13: Calculation of Makeup Water Determine the amount of makeup required for a cooling tower with the following conditions: Inlet water flow, m3/h (gal/min) Inlet water temperature, °C (°F) Outlet water temperature, °C (°F) Drift loss, percent Concentration, cycles

2270 37.77 29.44 0.2 5

(10,000) (100) (85)

Evaporation loss We, m3/h = 0.00085 × 2270 × (37.77 − 29.44) × 1.8 = 28.9 We, gal/min = 127.5 Drift loss Wd, m3/h = 2270 × 0.002 = 4.54 Wd, gal/min = 20 Blowdown 28.9 We Wb, m3/h = ᎏ = ᎏ = 7.24 (S − 1) 4 Wb, gal/min = 31.9 Makeup Wm, m3/h = 28.9 + 4.54 + 7.24 = 40.7 Wm, gal/min = 179.4

FIG. 12-16

Typical cooling-tower performance curve.

Fan Horsepower In evaluating cooling-tower owning and operating costs, fan-horsepower requirements can be a significant factor. Large air quantities are circulated through cooling towers at exit velocities of about 10.2 m/s (2000 ft/min) maximum for induced-draft towers. Fan air-flow quantities depend upon tower-design factors, including such items as type of fill, tower configuration, and thermalperformance conditions. The effective output of the fan is static air horsepower (SAHP), which is obtained by the following equation:

12-18

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

Q(hs)(d) SAHP = ᎏᎏ (12-13) 33,000(12) 3 where Q = air volume, ft /min; hs = static head, in of water; and d = density of water at ambient temperature, lb/ft3. Cooling-tower fan horsepower can be reduced substantially as the ambient wet-bulb temperature decreases if two-speed fan motors are used. Theoretically, operating at half speed will reduce air flow by 50 percent while decreasing horsepower to one-eighth of full-speed operation. However, actual half-speed operation will require about 17 percent of the horsepower at full speed as a result of the inherent motor losses at lighter loads. Figure 12-17 shows a typical plot of outlet-water temperatures when a cooling tower is operated (1) in the fan-off position, (2) with the fan at half speed, and (3) with the fan at full speed. Note that at decreasing wet-bulb temperatures the water leaving the tower during half-speed operation could meet design water-temperature requirements of, say, 85°F. For example, for a 60°F wet-bulb, 20°F range, a leaving-water temperature slightly below 85°F is obtained with design water flow over the tower. If the fan had a 100-hp motor, 83 hp would be saved when operating it at half speed. In calculating savings, one should not overlook the advantage of having colder tower water available for the overall water-circulating system. Recent developments in cooling-tower fan energy management also include automatic variable-pitch propeller-type fans and invertertype devices to permit variable fan speeds. These schemes involve tracking the load at a constant leaving-water temperature. The variable-pitch arrangement at constant motor speed changes the pitch of the blades through a pneumatic signal from the leavingwater temperature. As the thermal load and/or the ambient wet-bulb temperature decreases, the blade pitch reduces air flow and less fan energy is required. Inverters make it possible to control a variable-speed fan by changing the frequency modulation. Standard alternating-current fan motors may be speed-regulated between 0 and 60 Hz. In using inverters for this application, it is important to avoid frequencies that would result in fan critical speeds. Even though tower-fan energy savings can result from these arrangements, they may not constitute the best system approach. Power-plant steam condensers and refrigeration units, for example, can take advantage of colder tower water to reduce power consumption. Invariably, these system savings are much larger than cooling-tower fan savings with constant leaving-water temperatures. A refrigeration-unit condenser can utilize inlet-water temperatures down to 12.8°C (55°F) to reduce compressor energy consumption by 25 to 30 percent.

Pumping Horsepower Another important factor in analyzing cooling-tower selections, especially in medium to large sizes, is the portion of pump horsepower directly attributed to the cooling tower. A counterflow type of tower with spray nozzles will have a pumping head equal to static lift plus nozzle pressure loss. A cross-flow type of tower with gravity flow enables a pumping head to equal static lift. A reduction in tower height therefore reduces static lift, thus reducing pump horsepower: gal/min(ht) Pump bhp = ᎏᎏᎏ (12-14) 3960 (pump efficiency) where ht = total head, ft. Fogging and Plume Abatement A phenomenon that occurs in cooling-tower operation is fogging, which produces a highly visible plume and possible icing hazards. Fogging results from mixing warm, highly saturated tower discharge air with cooler ambient air that lacks the capacity to absorb all the moisture as vapor. While in the past visible plumes have not been considered undesirable, properly locating towers to minimize possible sources of complaints has now received the necessary attention. In some instances, guyed high fan stacks have been used to reduce ground fog. Although tall stacks minimize the ground effects of plumes, they can do nothing about water-vapor saturation or visibility. The persistence of plumes is much greater in periods of low ambient temperatures. More recently, environmental aspects have caused public awareness and concern over any visible plume, although many lay persons misconstrue cooling-tower discharge as harmful. This has resulted in a new development for plume abatement known as a wet-dry coolingtower configuration. Reducing the relative humidity or moisture content of the tower discharge stream will reduce the frequency of plume formation. Figure 12-18 shows a “parallel path” arrangement that has been demonstrated to be technically sound but at substantially increased tower investment. Ambient air travels in parallel streams through the top dry-surface section and the evaporative section. Both sections benefit thermally by receiving cooler ambient air with the wet and dry air streams mixing after leaving their respective sections. Water flow is arranged in series, first flowing to the dry-coil section and then to the evaporation-fill section. A “series path” air-flow arrangement, in which dry coil sections can be located before or after the air traverses the evaporative section, also can be used. However, series-path air flow has the disadvantage of water impingement, which could result in coil scaling and restricted air flow. Wet-dry cooling towers incorporating these designs are being used for large-tower industrial applications. At present they are not available for commercial applications.

FIG. 12-18 FIG. 12-17

Typical plot of cooling-tower performance at varying fan speeds.

(Marley Co.)

Parallel-path cooling-tower arrangement for plume abatement.

EVAPORATIVE COOLING Energy Management With today’s emphasis on energy management, cooling towers have not been overlooked. During periods below 50°F ambient wet-bulb temperatures, cooling towers have the temperature capability to furnish chilled water directly to airconditioning systems. For existing refrigeration–cooling-tower systems, piping can be installed to bypass the chiller to allow tower effluent to flow directly to cooling coils. After heat has been removed from the air stream, water returns directly to the cooling tower. Water temperature leaving the cooling tower is controlled between 8.9 and 12.2°C (48 and 54°F), usually by cycling cooling-tower fans. Depending upon the cleanliness of the cooling-tower water, it may be necessary to install a side-stream or full-flow filter to minimize contamination of the normally closed chilled-water circuit. Figure 12-19 shows the general arrangement of this system. Substantial savings can be realized during colder months by eliminating refrigerationcompressor energy. Several other methods involving cooling towers have been used to reduce refrigeration energy consumption. These systems, as applied to centrifugal and absorption refrigeration machines, are known as thermocycle or free cooling systems. When water leaving the cooling tower is available below 10°C (50°F), the thermocycle system permits shutting down the compressor prime mover or reducing steam flow to an absorption unit. Figure 12-20 shows the arrangement for a centrifugal refrigeration unit.

FIG. 12-19

12-19

The thermocycle system can be operated only when condensing water is available at a temperature lower than the required chilledwater-supply temperature. Modifications for a centrifugal refrigeration unit include the installation of a small liquid-refrigerant pump, cooler spray header nozzles, and a vapor bypass line between the cooler and the condenser. Without the compressor operating, a thermocycle capacity up to 35 percent of the refrigeration-unit rating can be produced. The cooling-tower fan is operated at full speed to produce the coldest water temperature possible for a given ambient wet-bulb temperature. The large vapor bypass between the cooler and the condenser is opened along with the compressor suction damper or prerotational vanes. The heat removed from the chilled-water stream boils off refrigerant vapor from the cooler. This vapor flows mainly through the bypass line to the condenser, where it is condensed to a liquid. (Units having hot-gas bypass lines cannot be used for this purpose, as the pipe size is too small.) The liquid then returns to the cooler as in the normal refrigeration cycle. If the refrigeration unit contains internal float valves, they are bypassed manually or held open by an adjusting stem. Thermocycle capacity is a function of the temperature difference between the chilled-water outlet temperature leaving the cooler and the inlet condenser water. The cycle finally stops when these two temperatures approach each other and there is not sufficient vapor pressure difference to permit flow between the heat exchangers.

Cooling-tower water for direct cooling during winter months.

12-20

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

FIG. 12-20

Cooling-tower use on a thermocycle system during winter months.

Precise control of the outlet chilled-water temperature does not occur with thermocycle operation. This temperature is dependent on ambient wet-bulb-temperature conditions. Normally, during cold winter days little change occurs in wet-bulb temperatures, so that only slight water-temperature variations may occur. This would not be true of many spring and fall days, when relatively large climatic temperature swings can and do occur. Refrigeration units modified for free cooling do not include the liquid-refrigerant pump and cooler spray header nozzles. Without the cooler refrigerant agitation for improved heat transfer, this arrangement allows up to about 20 percent of rated capacity. Expected capacities for both thermocycle and free cooling are indicated in Fig. 12-21.

FIG. 12-21

In operating a cooling tower in the thermocycle or free-cooling mode, some precautions are necessary to minimize icing problems. These include fan reversals to circulate air down through the tower inlet louvers, proper water distribution, constant water flow over the tower, heat tracing of lines such as makeup lines as required, and maximum loading per tower cell. NATURAL-DRAFT TOWERS Natural-draft, or hyperbolic-type, towers have been in use since about 1916 in Europe and have become standard equipment for the watercooling requirements of British power stations. They are primarily

Thermocycle and free-cooling-system capacities.

EVAPORATIVE COOLING suited to very large cooling-water quantities, and the reinforcedconcrete structures used are as large as diameters of 80.7 m (265 ft) and heights of 103.6 m (340 ft). The design convenience obtained from the steady air flow of mechanical-draft towers is not realized in natural-draft-tower design. Air flow through a natural-draft tower is due largely to the difference in density between the cool inlet air and the warm exit air. The air leaving the stack is lighter than the ambient air, and a draft is created by chimney effect, thus eliminating the need for mechanical fans. McKelvey and Brooke (The Industrial Cooling Tower, Elsevier, New York, 1959, p. 108) note that natural-draft towers commonly operate at air-pressure differences in the region of 0.2-in water gauge when under full load. The mean velocity of the air above the tower packing is generally about 1.2 to 1.8 m/s (4 to 6 ft/s). The performance of the natural-draft tower differs from that of the mechanical-draft tower in that the cooling is dependent upon the relative humidity as well as on the wet-bulb temperature. The draft will increase through the tower at high-humidity conditions because of the increase in available static pressure difference to promote air flow against internal resistances. Thus the higher the humidity at a given wet bulb, the colder the outlet water will be for a given set of conditions. This fundamental relationship has been used to advantage in Great Britain, where relative humidities are commonly 75 to 80 percent. Therefore, it is important in the design stages to determine correctly and specify the density of the entering and effluent air in addition to the usual tower-design conditions of range, approach, and water quantity. The performance relationship to humidity conditions makes exact control of outlet-water temperature difficult to achieve with the natural-draft tower. Data for determining the size of natural-draft towers have been presented by Chilton [Proc. Inst. Elec. Eng., 99, 440 (1952)] and Rish and Steel (ASCE Symposium on Thermal Power Plants, October 1958). Chilton showed that the duty coefficient Dt of a tower is approximately constant over its normal range of operation and is related to tower size by an efficiency factor or performance coefficient Ct as follows: 苶t苶)/(Ct 兹C 苶t苶) Dt = (A 兹Z

(12-15)

where A = base area of tower, ft2, measured at pond sill level; and Zt = height of tower, ft, measured above sill level. The duty coefficient may be determined from the formula 苶t苶+ 苶苶.3 苶h 苶 (WL /Dt) = 90.59(∆h /∆T) 兹∆ 0苶1苶2苶4苶∆

(12-16)

where ∆h = change in total heat of the air passing through the tower, Btu/lb; ∆T = change of water temperature passing through tower, °F; ∆t = difference between air temperature leaving the packing and inlet dry-bulb temperature, °F; and WL = water load in the tower, lb/h. The air leaving the packing inside the tower is assumed to be saturated at a temperature halfway between the inlet- and outlet-water temperatures. A divergence between theory and practice of a few degrees in this latter assumption does not significantly affect the results, as the draft component depends on the ratio of the change of density to change of total heat and not on change of temperature alone. Example 14: Duty Coefficient for a Hyperbolic Tower Determine the duty coefficient for a hyperbolic tower operating with Temperature of water to tower, °F = 82 Leaving (recooled) water temperature, °F = 70 Temperature range ∆T, °F = 12 Dry-bulb air temperature t2, °F = 57 Aspirated (ambient) wet-bulb air temperature tw2, °F = 51.7 Water loading to tower WL, lb/h = 38,200,000 t2 =

57°

h2 = 21.3

19°

∆h

(lowering) it. By taking a Ct value of 5.0 and a tower height of 320 ft, the base 苶)/兹3 苶2苶0苶 = 34,600 ft2, or the internal base area of the tower will be (55,000)(5 兹5 diameter at sill level will be 210 ft. A ratio of height to base diameter of 3:2 is normally employed.

To determine how a natural-draft tower of any given duty coefficient will perform under varying conditions, Rish and Steel plotted the nomograph in Fig. 12-22. The straight line shown on the nomograph illustrates the conditions of Example 14. SPRAY PONDS Spray ponds provide an arrangement for lowering the temperature of water by evaporative cooling and, in so doing, greatly reduce the cooling area required in comparison with a cooling pond. A spray pond uses a number of nozzles which spray water into contact with the surrounding air. A well-designed spray nozzle should provide fine water drops but should not produce a mist which would be carried off as excessive drift loss. Table 12-3 provides design data which will assist in the layout of a spray pond. The pond should be placed with its long axis at right angles to the prevailing summer wind. A long, narrow pond is more effective than a square one, so that decreasing pond width and increasing pond length will improve performance. Performance can also be increased by decreasing the amount of water sprayed per unit of pond area, increasing the height and fineness of spray drops, and increasing nozzle height above the basin sides. Sufficient distance should be provided from the outer nozzles to keep spray from being carried over the sides of the basin. If it is not possible to provide 7.6 to 10.7 m (25 to 35 ft) of space, the pond should be enclosed with a louver fence, equal in height to the maximum height of the spray, to minimize drift loss. Also, during coldweather periods, fogging can occur from the spray pond, so that consideration should be given to possible hazards to roadways or buildings in the immediate vicinity. The physical designs and operating conditions of spray-pond installations vary greatly, and it is difficult to develop exact rating data that can be used for determining cooling performance in all cases. However, Fig. 12-23 shows the performance that can be obtained with a well-designed spray pond, based on a 21.1°C (70°F) wet-bulb temperature and a 2.2-m/s (5-mi/h) wind. This curve shows that a 3.3°C (6°F) approach to the wet bulb is possible at a 2.2°C (4°F) range, but at higher ranges the obtainable approach increases. If it is necessary to cool water through a large temperature range to a reasonably close approach, the spray pond could be staged. With this method, the water is initially sprayed, collected, and then resprayed in another part of a sectionalized pond basin. Figure 12-24 shows performance curves for a spray pond used in steam-condensing service at varying wet-bulb and range conditions. Spray-pond performance can be calculated within reasonable accuracy on the basis of the leaving wet-bulb temperature of the air passing through the spray-filled volume. The air temperature leaving cannot exceed the warm water to the pond, and the closeness of approach will depend on the pond layout. In calculating the cooling obtained, the spray-filled volume is figured from a height equal to the elevation of the nozzles above the pond surface plus 0.3 m (1 ft) for each 7-kPa (1bf/in2) nozzle pressure and a plan area extending 3 m (10 ft) beyond the outer nozzles. The air area involved is the projected area of a vertical plane through the filled volume and broadside to the direction of air movement. The horizontal distance that the air moves through the filled volume is considered the length of air travel. Example 15: Cooling Capacity of a Spray Pond Determine the cooling capacity of a spray pond operating at the following conditions:

t1 = (82° + 70°)/2 = 76° h1 = 39.8 (from Fig. 12-2) ∆t =

12-21

= 18.5

苶8苶.5 苶)苶 = 696 WL /Dt = 90.59(18.5/12) 兹1苶9苶苶 +苶.3 0苶1苶2苶4苶(1 Dt = 38,200,000/696 = 55,000 The performance coefficients usually attained have been about 5.2 for water loadings in excess of 750 lb/(h⋅ft2), though new types of packing are improving

Water flow, gal/min Spray-nozzle pressure, lb/in2 Water flow per nozzle, gal/min Effective area, length × width, ft2 Effective height, ft Wind velocity, ft/min

46,000 7 42.5 434 × 100 7 + 7 spray height 440

12-22

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

FIG. 12-22 Universal performance chart for natural-draft cooling towers. (Rish and Steel, ASCE Symposium on Thermal Power Plants, October 1958.)

Prevailing wind Ambient wet-bulb temperature, °F Water temperature in, °F

Broadside to pond 78 102

Effective air area = 434 × 14 = 6080 ft2 Air flow = 440 × 6080 = 2,680,000 ft3/min L = 46,000 × 8.33 = 384,000 lb water/min G = 2,680,000/14.3 = 187,500 lb air/min L/G = 384,000/187,500 = 2.05 h′ at 78°F wet-bulb temperature = 41.58 Btu/lb (from Table 12-1). Assume water temperature out = 92°F. TABLE 12-3

Spray-Pond Engineering Data and Design*

Recommendations

Usual

Minimum

Maximum

Nozzle capacity, gal./min. each Nozzles per 12-ft. length of pipe Height of nozzles above sides of basin, ft. Nozzle pressure, lb./sq. in. Size of nozzles and nozzle arms, in. Distance between spray lateral piping, ft. Distance of nozzles from side of pond, unfenced, ft. Distance of nozzles from side of pond, fenced, ft. Height of louver fence, ft. Depth of pond basin, ft. Friction loss per 100 ft. pipe, in. of water Design wind velocity, m.p.h.

35–50 5–6 7–8 5–7 2 25

10 4 2 4 1d 13

60 8 10 10 2a 38

25–35

20

50

12–18 12 4–5 1–3 5

10 6 2 — 3

25 18 7 6 10

*From Spray Pond Bull. SP-51, Marley Co., Mission Woods, Kan., p. 3.

FIG. 12-23

Spray-pond performance curve.

EVAPORATIVE COOLING

FIG. 12-24

Spray ponds: cooling curves for steam-condensing service.

L (h′2) air out − (h′1) air in h′2 − 41.58 ᎏ = ᎏᎏᎏᎏᎏ = 2.05 = ᎏᎏ G water temperature in − water temperature out 10 h′2 = 61.63 Btu/lb Corresponding wet-bulb temperature = 94°F air leaving pond. Approach possible to air leaving (from Table 12-4) = −2°F. Water temperature leaving spray pond = 94°F − 2°F = 92°F. Since leaving-water temperature checks assumption, spray pond is capable of cooling 46,000 gal/min from 102 to 92°F with 78°F wet-bulb temperature and 5-mi/h wind. Total of 1080 spray nozzles required at 42.5 gal/min each, nozzles at 7-lbf/in2 pressure.

12-23

at a relatively small investment by pushing up an earth dike 1.8 to 3.1 m (6 to 10 ft) high. For a successful pond installation, the soil must be reasonably impervious, and location in a flat area is desirable. Four principal heat-transfer processes are involved in obtaining cooling from an open pond. Heat is lost through evaporation, convection, and radiation and is gained through solar radiation. The required pond area depends on the number of degrees of cooling required and the net heat loss from each square foot of pond surface. Langhaar [Chem. Eng., 60(8), 194 (1953)] states that under given atmospheric conditions a body of water would eventually come to a temperature at which heat loss would equal heat gain. This temperature is referred to as the equilibrium temperature and is designated as E in Fig. 12-25, a nomograph of cooling-pond performance. The equilibrium temperature is greatly affected by the amount of solar radiation, which is usually not known very accurately and which varies throughout the day. If a pond has at least a 24-h holdup, then daily average weather conditions may be used. For practical purposes, it is recommended that the equilibrium temperature be taken as equal to normal river-water or lake temperature for the specified weather conditions. In order to cool to the equilibrium temperature, a pond of infinite size would be required for warm water. An approach of 1.7 to 2.2°C (3 to 4°F) is the lowest practicable in a pond of reasonable size. For a pond having more than a 24-h holdup, the leaving-water temperature will vary from the average by plus or minus 1.1°C (2°F) for a 0.9-m (5-ft) depth and 1.7°C (3°F) variation for a 0.9-m (3-ft) depth. The area of pond required for a given cooling load is almost independent of pond depth. A depth of at least 0.9 m (3 ft) appears advisable to prevent excessive channeling of flow with ponds having irregular bottoms and to avoid large day-to-night changes in outlet temperature. Factors considered to affect pond performance are air temperature, relative humidity, wind speed, and solar radiation. Items appearing to have only a minor effect include heat transfer between the earth and the pond, changing temperature and humidity of the air as it traverses the water, and rain. Figure 12-25 provides a rapid method of determining the pondarea requirements for a given cooling duty. D1 and D2 are the approaches to equilibrium for the entering and leaving water, °F; Vw is the wind velocity, mi/h; product PQ represents the area of the pond surface, ft2/(gal⋅min) of flow to the pond. The P factor assumes a pond with uniform flow, without turbulence, and with the water warmer than the air. Example 16: Calculation of Cooling-Pond Size Determine the required size of a cooling pond operating at the following conditions:

COOLING PONDS

Relative humidity, percent = 50 Wind velocity, mi/h = 5 Dry-bulb air temperature, °F = 68 Solar heat gain, Btu/(h⋅ft2) = 100 Water quantity, gal/min = 10,000

When large ground areas are available, cooling ponds offer a satisfactory method of removing heat from water. A pond may be constructed TABLE 12-4 Degree Adjustment to Be Applied to Leaving-Air Wet-Bulb Temperature to Find Cooled-Water Temperatures of Spray Ponds* Adjustment, °F. Entering Cooling wet-bulb Length of air travel, ft.‡ range, temp., † °F. °F. 100 50 25 10

80 70 60

−3 −2 −1.5

+2 +3 +3.5

+4 +5 +5.5

15

80 70 60

−5.0 −4.0 −3.5

+1 +2 +2.5

+5 +5.5 +6

20

80 70 60

−7 −6 −5.5

0 +1 +1.5

+6 +7 +7.5

Cooled-water temperature = wet-bulb temperature of leaving air plus values shown. *From “Heating, Ventilating, Air Conditioning Guide,” 38th ed., p. 598 American Society of Heating, Refrigerating and Air Conditioning Engineers 1960. †Wet-bulb temperature of air entering spray-filled volume. ‡Length of air travel through spray-filled volume.

TABLE 12-5 Maximum Expected Solar Radiation at Various North Latitudes* B.t.u./(hr.)(sq. ft.) 24-hr. avg. at north latitude Jan. 1 Feb. 1 Mar. 1 Apr. 1 May 1 June 1 July 1 Aug. 1 Sept. 1 Oct. 1 Nov. 1 Dec. 1

Noon value at north latitude

30°

35°

40°

45°

30°

35°

40°

45°

65 75 90 110 120 130 130 125 115 100 80 65

50 65 80 105 120 130 130 125 110 90 70 55

40 55 75 95 120 130 130 125 105 80 60 45

30 45 65 90 115 130 130 120 100 75 50 35

240 270 305 340 360 365 365 360 350 315 270 240

205 240 285 320 350 360 360 350 335 295 245 210

170 210 255 300 335 345 350 340 315 270 215 175

135 175 230 280 320 335 340 325 300 245 185 140

*Langhaar, Chem. Eng., 60(8), 194 (1953).

12-24

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

FIG. 12-25

Nomograph for determining cooling-pond performance and size. [Langhar, Chem. Eng., 60(8), 194 (1953).]

SOLIDS-DRYING FUNDAMENTALS Water inlet, °F = 110 Water outlet, °F = 90 From the nomograph, P = 68 and Q = 1.07, giving an area required of 73 ft2/ (gal⋅min). Area for 10,000 gal/min is thus 730,000 ft2, or 17 acres. With a depth of 5 ft, total volume of the pond would amount to a 45.5-h holdup, which is more than the 24-h holdup required to maintain a fairly constant discharge temperature throughout the day.

Table 12-5 presents typical values of solar radiation on a horizontal

12-25

surface, Btu/(h⋅ft2), based on analysis of Weather Bureau records for a number of stations throughout the United States. These are clear-day values, rarely exceeded even in the high arid regions. The normal or actual average monthly values are only 50 to 60 percent of the tabulated figures for most of the eastern United States and 80 to 90 percent in the arid southwest. Also, the solar radiation should be multiplied by the absorption coefficient for the pond, which appears to exceed 95 percent.

SOLIDS-DRYING FUNDAMENTALS* GENERAL REFERENCES: Cook and DuMont, Process Drying Practice, McGraw-Hill, 1991. Drying Technology—An International Journal, Marcel Dekker, N.Y., 1982. Hall, Dictionary of Drying, Marcel Dekker, N.Y., 1979. Keey, Introduction to Industrial Drying Operations, Pergamon, N.Y., 1978. Masters, Spray Drying Handbook, Wiley, N.Y., 1990. Mujumdar, Handbook of Industrial Drying, Marcel Dekker, N.Y., 1987. Nonhebel and Moss, Drying of Solids in the Chemical Industry, CRC Press, Ohio, 1971. van’t Land, Industrial Drying Equipment, Marcel Dekker, N.Y., 1991.

INTRODUCTION The material on solids drying is divided into two subsections, “SolidsDrying Fundamentals,” and “Solids-Drying Equipment.” In this introductory part some elementary definitions are given. In solids-gas contacting equipment, the solids bed can exist in any of the following four conditions. Static This is a dense bed of solids in which each particle rests upon another at essentially the settled bulk density of the solids phase. Specifically, there is no relative motion among solids particles (Fig. 12-26). Moving This is a slightly expanded bed of solids in which the particles are separated only enough to flow one over another. Usually the flow is downward under the force of gravity, but upward motion by mechanical lifting or agitation may also occur within the process vessel. In some cases, lifting of the solids is accomplished in separate equipment, and solids flow in the presence of the gas phase is downward only. The latter is a moving bed as usually defined in the petroleum industry. In this definition, solids motion is achieved by either mechanical agitation or gravity force (Fig. 12-27). Fluidized This is an expanded condition in which the solids particles are supported by drag forces caused by the gas phase passing through the interstices among the particles at some critical velocity. It is an unstable condition in that the superficial gas velocity upward is less than the terminal setting velocity of the solids particles; the gas

FIG. 12-26

FIG. 12-27

velocity is not sufficient to entrain and convey continuously all the solids. At the same time, there exist, within the stream of gas, eddies traveling at high enough velocities to lift the particles temporarily. Particle motion is continually upward and falling back. Specifically, the solids phase and the gas phase are intermixed and together behave like a boiling fluid (Fig. 12-28). Dilute This is a fully expanded condition in which the solids particles are so widely separated that they exert essentially no influence upon each other. Specifically, the solids phase is so fully dispersed in the gas that the density of the suspension is essentially that of the gas phase alone (Fig. 12-29). Commonly, this situation exists when the gas velocity at all points in the system exceeds the terminal settling velocity of the solids and the particles can be lifted and continuously conveyed by the gas; however, this is not always true. Gravity settling chambers such as prilling towers and countercurrent-flow spray dryers are two exceptions in which gas velocity is insufficient to entrain the solids completely. Gas-Solids Contacting Terms used in this section to describe the method by which gas may contact a bed of solids are the following: 1. Parallel flow. The direction of gas flow is parallel to the surface of the solids phase. Contacting is primarily at the interface between phases, with possibly some penetration of gas into the voids among the solids near the surface. The solids bed is usually in a static condition (Fig. 12-30).

Solids bed in static condition (tray dryer).

Moving solids bed in a rotary dryer with lifters.

FIG. 12-28

Fluidized solids bed.

FIG. 12-29

Solids in a dilute condition near the top of a spray dryer.

* The contribution of George A. Schurr, E. I. du Pont de Nemours & Co., to material that was used from the sixth edition is acknowledged. (Solids Drying)

12-26

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

FIG. 12-30

Parallel gas flow over a static bed of solids.

2. Perpendicular flow. The direction of gas flow is normal to the phase interface. The gas impinges on the solids bed. Again the solids bed is usually in a static condition (Fig. 12-31). 3. Through circulation. The gas penetrates and flows through interstices among the solids, circulating more or less freely around the individual particles (Fig. 12-32). This may occur when solids are in static, moving, fluidized, or dilute conditions. Three additional terms require definition. 1. Cocurrent gas flow. The gas phase and solids particles both flow in the same direction (Fig. 12-33). 2. Countercurrent gas flow. The direction of gas flow is exactly opposite to the direction of solids movement. 3. Cross-flow of gas. The direction of gas flow is at a right angle to that of solids movement, across the solids bed (Fig. 12-34). Because in a gas-solids-contacting operation heat transfer and mass transfer take place at the solids’ surfaces, maximum process efficiency can be expected with a maximum exposure of solids surface to the gas phase, together with thorough mixing of gas and solids. Both are important. Within any arrangement of particulate solids, gas is present in the voids among the particles and contacts all surfaces except at the points of particle contact. When the solids bed is in a static or slightly moving condition, however, gas within the voids is cut off from the main body of the gas phase. Some transfer of energy and mass may occur by diffusion, but it is usually insignificant.

Circulating gas impinging on a large solid object in perpendicular flow, in a roller-conveyor furnace.

FIG. 12-31

Gas passing through a bed of preformed solids, in through circulation on a perforated-apron conveyor.

FIG. 12-32

FIG. 12-33

conveyor.

Cocurrent gas-solids flow in a vertical-lift dilute-phase pneumatic

Cross-flow of gas and solids in a cascade-type gravity dryer. (LinkBelt Co., Multi-Louvre principle.)

FIG. 12-34

Equipment design and selection are governed by two factors: 1. Mechanical considerations 2. Solids flow and surface characteristics. The former usually involves process temperature or isolation. Solids surface characteristics are important in that they control the extent to which an operation is diffusion-limited, i.e., diffusion into and out of the pores of a given solids particle, not through the voids among separate particles. The size of the solids particles, the surface-to-mass ratio, is also important in the evaluation of surface characteristics and the diffusion problem. Gas-Solids Separations After the solids and gas have been brought together and mixed in a gas-solids contactor, it becomes necessary to separate the two phases. If the solids are sufficiently coarse and the gas velocity sufficiently low, it is possible to effect a complete gravitational separation in the primary contactor. Applications of this type are rare, however, and supplementary dust-collection equipment is commonly required. The recovery step may even dictate the type of primary contacting device selected. For example, when treating an extremely friable solid material, a deep fluidized-solids contactor might overload the collection system with fines, whereas the more gentle contacting of a traveling-screen contactor would be expected to produce a minimum of fines by attrition. Therefore, although gassolids separation is usually considered as separate and distinct from the primary contacting operation, it is usually desirable to evaluate the separation problem at the same time that contacting methods are evaluated. Definitions Drying generally refers to the removal of a liquid from a solid by evaporation. Mechanical methods for separating a liquid from a solid are not generally considered drying, although they often precede a drying operation, since it is less expensive and frequently easier to use mechanical methods than to use thermal methods. This subsection presents the theory and fundamental concepts of the drying of solids. Equipment commonly employed for the drying of solids is described both in this subsection in Sec. 12, where indirect heat transfer devices are discussed, and in Sec. 17 where fluidized beds are covered. Dryer control is discussed in Sec. 8. Excluding fluid beds this subsection contains mainly descriptions of direct-heat-transfer equipment. It also includes some indirect units; e.g., vacuum dryers, furnaces, steam-tube dryers, and rotary calciners. Generally accepted terminology and definitions are given alphabetically in the following paragraphs. Bound moisture in a solid is that liquid which exerts a vapor pressure less than that of the pure liquid at the given temperature. Liquid may become bound by retention in small capillaries, by solution in cell or fiber walls, by homogeneous solution throughout the solid, and by chemical or physical adsorption on solid surfaces. Capillary flow is the flow of liquid through the interstices and over the surface of a solid, caused by liquid-solid molecular attraction.

SOLIDS-DRYING FUNDAMENTALS Constant-rate period is that drying period during which the rate of water removal per unit of drying surface is constant. Critical moisture content is the average moisture content when the constant-rate period ends. Dry-weight basis expresses the moisture content of wet solid as kilograms of water per kilogram of bone-dry solid. Equilibrium moisture content is the limiting moisture to which a given material can be dried under specific conditions of air temperature and humidity. Falling-rate period is a drying period during which the instantaneous drying rate continually decreases. Fiber-saturation point is the moisture content of cellular materials (e.g., wood) at which the cell walls are completely saturated while the cavities are liquid-free. It may be defined as the equilibrium moisture content as the humidity of the surrounding atmosphere approaches saturation. Free-moisture content is that liquid which is removable at a given temperature and humidity. It may include bound and unbound moisture. Funicular state is that condition in drying a porous body when capillary suction results in air being sucked into the pores. Hygroscopic material is material that may contain bound moisture. Initial moisture distribution refers to the moisture distribution throughout a solid at the start of drying. Internal diffusion may be defined as the movement of liquid or vapor through a solid as the result of a concentration difference. Moisture content of a solid is usually expressed as moisture quantity per unit weight of the dry or wet solid. Moisture gradient refers to the distribution of water in a solid at a given moment in the drying process. Nonhygroscopic material is material that can contain no bound moisture. Pendular state is that state of a liquid in a porous solid when a continuous film of liquid no longer exists around and between discrete particles so that flow by capillary cannot occur. This state succeeds the funicular state. Unaccomplished moisture change is the ratio of the free moisture present at any time to that initially present. Unbound moisture in a hygroscopic material is that moisture in excess of the equilibrium moisture content corresponding to saturation humidity. All water in a nonhygroscopic material is unbound water. Wet-weight basis expresses the moisture in a material as a percentage of the weight of the wet solid. Use of a dry-weight basis is recommended since the percentage change of moisture is constant for all moisture levels. When the wet-weight basis is used to express moisture content, a 2 or 3 percent change at high moisture contents (above 70 percent) actually represents a 15 to 20 percent change in evaporative load. See Fig. 12-35 for the relationship between the dry- and wet-weight bases. APPLICATION OF PSYCHROMETRY TO DRYING In any drying process, if an adequate supply of heat is assumed, the temperature and rate at which liquid vaporization occurs will depend on the vapor concentration in the surrounding atmosphere.

12-27

In vacuum drying or other processes containing atmospheres of 100 percent vapor, the temperature of liquid vaporization will equal or exceed the saturation temperature of the liquid at the system pressure. (When a free liquid or wetted surface is present, drying will occur at the saturation temperature, just as free water at 101.325 kPa vaporizes in a 100 percent steam atmosphere at 100°C.) On the other hand, when evolved vapor is purged from the dryer environment by using a second (inert) gas, the temperature at which vaporization occurs will depend on the concentration of vapor in the surrounding gas. In effect, the liquid must be heated to a temperature at which its vapor pressure equals or exceeds the partial pressure of vapor in the purge gas. In the reverse situation, condensation will occur. In most drying operations, water is the liquid evaporated and air is the normally employed purge gas. For drying purposes, a psychrometric chart found very useful is that reproduced in Fig. 12-36. 1. The wet-bulb or saturation temperature line gives the maximum weight of water vapor that 1 kg of dry air can carry at the intersecting dry-bulb temperature shown on the abscissa at saturation humidity. The partial pressure of water in air equals the watervapor pressure at that temperature. The saturation humidity is defined by Hs = ps /(P − ps)18/28.9

(12-17)

where Hs = saturation humidity (kg /kg dry air), ps = vapor pressure of water at temperature ts, P = absolute pressure, and 18/28.9 = ratio of molecular weights of water (18) and air (28.9). Similarly, the humidity at any condition less than saturation is given by H = p/(P − p)18/28.9 2.

(12-18)

The percent relative humidity is defined by HR = 100(p/ps)

(12-19)

where p = partial pressure of water vapor in the air, ps = vapor pressure of water at the same temperature, and HR = percent relative humidity. 3. Humid volumes are given by the curves entitled “Volume m3/kg dry air.” The volumes are plotted as functions of absolute humidity and temperature. The difference between dry-air specific volume and humid-air volume at a given temperature is the volume of water vapor. 4. Enthalpy data are given on the basis of kilojoules per kilogram of dry air. Enthalpy-at-saturation data are accurate only at the saturation temperature and humidity. Enthalpy deviation curves permit enthalpy corrections for humidities less than saturation and show how the wet-bulb-temperature lines do not precisely coincide with constant-enthalpy, adiabatic cooling lines. 5. There are no lines for humid heats on Fig. 12-36. These may be calculated by Cs = 1.0 + 1.87H

(12-20)

where Cs = humid heat of moist air, kJ/(kg⋅K); 1.0 = specific heat of dry air, kJ/(kg⋅K), 1.87 = specific heat of water vapor, kJ/(kg⋅K); and H = absolute humidity, kg /kg dry air. 6. The wet-bulb-temperature lines represent also the adiabaticsaturation lines for air and water vapor only. These are based on the relationship Hs − H = (Cs / λ)(t − ts)

(12-21)

where Hs and ts = adiabatic saturation humidity and temperature respectively, corresponding to the air conditions represented by H and t, and Cs = humid heat for humidity H. The slope of the adiabaticsaturation curve is Cs /λ, where λ = latent heat of evaporation at ts. These lines show the relationship between the temperature and humidity of air passing through a continuous dryer operating adiabatically. The wet-bulb temperature is established by a dynamic equilibrium between heat and mass transfer when liquid evaporates from a small mass, such as the wet bulb of a thermometer, into a very large mass of gas such that the latter undergoes no temperature or humidity change. It is expressed by the relationship FIG. 12-35

Relationship between wet-weight and dry-weight bases.

hc(t − t w) = k′g λ(Hw − Ha)

(12-22)

12-28

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

FIG. 12-36

Psychrometric chart: properties of air and water-vapor mixtures from 20 to 120°C. (Carrier Corp.)

SOLIDS-DRYING FUNDAMENTALS where hc = heat-transfer coefficient by convection, J/(m2⋅s⋅K) [Btu/ (h⋅ft2⋅°F)]; t = air temperature, K; tw = wet-bulb temperature of air, K; k′g = mass-transfer coefficient, kg /(s⋅m2) (kg /kg) [lb/(h⋅ft2)(lb/lb)]; λ = latent heat of evaporation at tw, J/kg (Btu/lb); Hw = saturated humidity at t w = kg/kg of dry air; and Ha = humidity of the surrounding air, kg /kg of dry air. For air–water-vapor mixtures, it so happens that hc / k′g = Cs approximately, although there is no theoretical reason for this. Hence, since the ratio (Hw − Ha)/(tw − t) equals hc / k′g / λ, which represents the slope of the wet-bulb-temperature lines, it is also equal to Cs / λ, the slope of the adiabatic-saturation lines as shown previously. A given humidity chart is precise only at the pressure for which it is evaluated. Most air–water-vapor charts are based on a pressure of 1 atm. Humidities read from these charts for given values of wet- and dry-bulb temperature apply only at an atmospheric pressure of 760 mmHg. If the total pressure is different from 760 mmHg, the humidity at a given wet-bulb and dry-bulb temperature must be corrected according to the following relationship. 1 1 Ha = Ho + 0.622pw ᎏ − ᎏ (12-23) P − pw 760 − pw where Ha = humidity of air at pressure P, kg /kg of dry air; Ho = humidity of air as read from a humidity chart based on 760-mm pressure at the observed wet- and dry-bulb temperatures, kg /kg dry air, pw = vapor pressure of water at the observed wet-bulb temperature, mmHg; and P = the pressure at which the wet- and dry-bulb readings were taken. Similar corrections can be derived to correct specific volume, the saturation-humidity curve, and the relative-humidity curves.

冢

冣

HUMIDITY CHARTS FOR SOLVENT VAPORS Humidity charts for other solvent vapors may be prepared in an analogous manner. There is one important difference involved, however, in that the wet-bulb temperature differs considerably from the adiabatic-saturation temperatures for vapors other than water.

12-29

Figures 12-37 to 12-39 show humidity charts for carbon tetrachloride, benzene, and toluene. The lines on these charts have been calculated in the manner outlined for air–water vapor except for the wet-bulb-temperature lines. The determination of these lines depends on data for the psychrometric ratio hc / k′g, as indicated by Eq. (12-22). For the charts shown, the wet-bulb-temperature lines are based on the following equation: Hw − H = (αhc / λ w k′g)(t − t w)

(12-24)

where α = radiation correction factor, a value of 1.06 having been used for these charts. Values of hc / k′g, obtained from values of hc / k′gCs as presented by Walker, Lewis, McAdams, and Gilliland (Principles of Chemical Engineering, 3d ed., McGraw-Hill, New York, 1937), where Cs = humid heat of air with respect to the vapor involved, are as follows:

Material

Carbon tetrachloride

Benzene

Toluene

hc /k′gCs

0.51

0.54

0.47

A discussion of the theory of the relationship between hc and k′g may be found in the psychrometry part of this section. Because both theoretical and experimental values of hc / k′g apply only to dilute gas mixtures, the wet-bulb lines at high concentrations have been omitted. For a discussion of the precautions to be taken in making psychrometric determinations of solvent vapors at low solvent wet-bulb temperatures in the presence of water vapor, see the paper by Sherwood and Comings [Trans. Am. Inst. Chem. Eng., 28, 88 (1932)]. GENERAL CONDITIONS FOR DRYING Solids drying encompasses two fundamental and simultaneous processes: (1) heat is transferred to evaporate liquid, and (2) mass is

FIG. 12-37 Humidity chart for air-carbon tetrachloride vapor mixture. To convert British thermal units per pound to joules per kilogram, multiply by 2326; to convert British thermal units per pound dry air-degree Fahrenheit to joules per kilogram-kelvin, multiply by 4186.8; and to convert cubic feet per pound to cubic meters per kilogram, multiply by 0.0624.

Humidity chart for air-benzene-vapor mixture. To convert British thermal units per pound to joules per kilogram, multiply by 2326; to convert British thermal units per pound dry air-degree Fahrenheit to joules per kilogram-kelvin, multiply by 4186.8; and to convert cubic feet per pound to cubic meters per kilogram, multiply by 0.0624.

FIG. 12-38

Humidity chart for air-toluene-vapor mixture. To convert British thermal units per pound to joules per kilogram, multiply by 2326; to convert British thermal units per pound dry air-degree Fahrenheit to joules per kilogramkelvin, multiply by 4186.8; and to convert cubic feet per pound to cubic meters per kilogram, multiply by 0.0624.

FIG. 12-39

12-30

SOLIDS-DRYING FUNDAMENTALS transferred as a liquid or vapor within the solid and as a vapor from the surface. The factors governing the rates of these processes determine the drying rate. Commercial dryers differ fundamentally by the methods of heat transfer employed (see classification of dryers, Fig. 12-45). These industrial-dryer operations may utilize heat transfer by convection, conduction, radiation, or a combination of these. In each case, however, heat must flow to the outer surface and then into the interior of the solid. The single exception is dielectric and microwave drying, in which high-frequency electricity generates heat internally and produces a high temperature within the material and on its surface. Mass is transferred in drying as a liquid and vapor within the solid and as vapor from the exposed surfaces. Movement within the solid results from a concentration gradient which is dependent on the characteristics of the solid. A solid to be dried may be porous or nonporous. It can also be hygroscopic or nonhygroscopic. Many solids fall intermediately between these two extremes, but it is generally convenient to consider the solid to be one or the other. A study of how a solid dries may be based on the internal mechanism of liquid flow or on the effect of the external conditions of temperature, humidity, air flow, state of subdivision, etc., on the drying rate of the solids. The former procedure generally requires a fundamental study of the internal condition. The latter procedure, although less fundamental, is more generally used because the results have greater immediate application in equipment design and evaluation. INTERNAL MECHANISM OF LIQUID FLOW The structure of the solid determines the mechanism for which internal liquid flow may occur. These mechanisms can include (1) diffusion in continuous, homogeneous solids, (2) capillary flow in granular and porous solids, (3) flow caused by shrinkage and pressure gradients, (4) flow caused by gravity, and (5) flow caused by a vaporization-condensation sequence. In general, one mechanism predominates at any given time in a solid during drying, but it is not uncommon to find different mechanisms predominating at different times during the drying cycle. The study of internal moisture gradients establishes the particular mechanism which controls during the drying of a solid. The experimental determination of reliable moisture gradients is extremely difficult. Hougen, McCauley, and Marshall [Trans. Am. Inst. Chem. Eng., 36, 183 (1940)] discussed the conditions under which capillary and diffusional flow may be expected in a drying solid and analyzed the published experimental moisture-gradient data for the two cases. Their curves indicate that capillary flow is typified by a moisture gradient involving a double curvature and point of inflection (Fig. 12-40a) while diffusional flow is a smooth curve, concave downward (Fig. 12-40b), as would be predicted from the diffusion equations. They also showed that the liquid-diffusion coefficient is usually a function

(a) FIG. 12-40

(b)

Two types of internal moisture gradients obtained in drying solids.

12-31

of moisture content which decreases with decreasing moisture. The effect of variable diffusivity is illustrated in Fig. 12-40b, where the dashed line is calculated for constant diffusivity and the solid line is experimental for the case in which the diffusion coefficient is moisture-dependent. Thus, the integrated diffusion equations assuming constant diffusivity only approximate the actual behavior. These authors classified solids on the basis of capillary and diffusional flow: Capillary Flow Moisture which is held in the interstices of solids, as liquid on the surface, or as free moisture in cell cavities, moves by gravity and capillarity, provided that passageways for continuous flow are present. In drying, liquid flow resulting from capillarity applies to liquids not held in solution and to all moisture above the fiber-saturation point, as in textiles, paper, and leather, and to all moisture above the equilibrium moisture content at atmospheric saturations, as in fine powders and granular solids, such as paint pigments, minerals, clays, soil, and sand. Vapor Diffusion Moisture may move by vapor diffusion through the solid, provided that a temperature gradient is established by heating, thus creating a vapor-pressure gradient. Vaporization and vapor diffusion may occur in any solid in which heating takes place at one surface and drying from the other and in which liquid is isolated between granules of solid. Liquid Diffusion The movement of liquids by diffusion in solids is restricted to the equilibrium moisture content below the point of atmospheric saturation and to systems in which moisture and solid are mutually soluble. The first class applies to the last stages in the drying of clays, starches, flour, textiles, paper, and wood; the second class includes the drying of soaps, glues, gelatins, and pastes. External Conditions The principal external variables involved in any drying study are temperature, humidity, air flow, state of subdivision of the solid, agitation of the solid, method of supporting the solid, and contact between hot surfaces and wet solid. All these variables will not necessarily occur in one problem. PERIODS OF DRYING When a solid is dried experimentally, data relating moisture content to time are usually obtained. These data are then plotted as moisture content (dry basis) W versus time θ, as shown in Fig. 12-41a. This curve represents the general case when a wet solid loses moisture first by evaporation from a saturated surface on the solid, followed in turn by a period of evaporation from a saturated surface of gradually decreasing area, and, finally, when the latter evaporates in the interior of the solid. Figure 12-41a indicates that the drying rate is subject to variation with time or moisture content. This variation is better illustrated by graphically or numerically differentiating the curve and plotting dW/dθ versus W, as shown in Fig. 12-41b, or as dW/dθ versus θ, as shown in Fig. 12-41c. These rate curves illustrate that the drying process is not a smooth, continuous one in which a single mechanism controls throughout. Figure 12-41c has the advantage of showing how long each drying period lasts. The section AB on each curve represents a warming-up period of the solids. Section BC on each curve represents the constant-rate period. Point C, where the constant rate ends and the drying rate begins falling, is termed the critical-moisture content. The curved portion CD on Fig. 12-41a is termed the falling-rate period and, as shown in Fig. 12-41b and c, is typified by a continuously changing rate throughout the remainder of the drying cycle. Point E (Fig. 12-41b) represents the point at which all the exposed surface becomes completely unsaturated and marks the start of that portion of the drying cycle during which the rate of internal moisture movement controls the drying rate. Portion CE in Fig. 12-41b is usually defined as the first falling-rate drying period; portion DE, as the second falling-rate period. Constant-rate Period In the constant-rate period moisture movement within the solid is rapid enough to maintain a saturated condition at the surface, and the rate of drying is controlled by the rate of heat transferred to the evaporating surface. Drying proceeds by diffusion of vapor from the saturated surface of the material across a

12-32

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING of dryers in Fig. 12-45). Radiation is also effective in increasing the constant rate by augmenting the convection heat transfer and raising the surface temperature above the wet-bulb temperature. When the heat for evaporation in the constant-rate period is supplied by a hot gas, a dynamic equilibrium establishes the rate of heat transfer to the material and the rate of vapor removal from the surface: dw/dθ = h t A ∆t/λ = kg A ∆p

(12-25)

where dw/dθ = drying rate, kg water/s; ht = total heat-transfer coefficient, J/(m2⋅s⋅K) [Btu/(h⋅ft2⋅°F)]; A = area for heat transfer and evaporation, m2; λ = latent heat of evaporation at t s′, J/kg (Btu/lb); kg = mass-transfer coefficient, kg /(s⋅m2⋅atm) [(lb/(h⋅ft2⋅atm)]; ∆t = t − t′s, where t = gas (dry-bulb) temperature, K; p = ps − p, where ps = vapor pressure of water at surface temperature t′s, atm; and p = partial pressure of water vapor in the gas, atm. The magnitude of the constant rate depends upon three factors: 1. The heat- or mass-transfer coefficient 2. The area exposed to the drying medium 3. The difference in temperature or humidity between the gas stream and the wet surface of the solid All these factors are the external variables. The internal mechanism of liquid flow does not affect the constant rate. For drying calculations, it is convenient to express Eq. (12-25) in terms of the decrease in moisture content rather than in the quantity of water evaporated. For evaporation from a tray of wet material, if no change in volume during drying is assumed, Eq. (12-25) becomes dw/dθ = (h/ρs d λ)(t − t′s)

(12-26)

where dw/dθ = drying rate, kg water/(s⋅kg dry solids); ht = total heattransfer coefficient, J/(m2⋅s⋅K) [Btu/(h⋅ft2⋅°F)]; ρs = bulk density dry material, kg /m3; d = thickness of bed, m; λ = latent heat of vaporization, J/kg (Btu/lb); t = air temperature, K; and t ′s = evaporating surface temperature, K. (Note that dw/dθ is inherently negative.) A similar equation can be written for the through-circulation case: dW/dθ = (ht a/ρsλ(t − t′s)

FIG. 12-41

The periods of drying.

stagnant air film into the environment. The rate of mass transfer balances the rate of heat transfer, and the temperature of the saturated surface remains constant. The mechanism of moisture removal is equivalent to evaporation from a body of water* and is essentially independent of the nature of the solids. If heat is transferred solely by convection and in the absence of other heat effects, the surface temperature approaches the wet-bulb temperature. However, when heat is transferred by radiation, convection, or a combination of these and convection, the temperature at the saturated surface is between the wet-bulb temperature and the boiling point of water. Under these conditions, the rate of heat transfer is increased and a higher drying rate results. When heat is transferred to a wet solid by convection to hot surfaces and heat transfer by convection is negligible, the solids approach the boiling-point temperature rather than the wet-bulb temperature. This method of heat transfer is utilized in indirect dryers (see classification

(12-27)

where a = m2 of heat-transfer area/m3 of bed, 1/m; and other symbols are as in Eq. (12-26). The values of ρs and/or a must be known in order to use Eqs. (12-26) and (12-27). The value of a is difficult to estimate without experimental data. When the void fraction is known, a can sometimes be estimated from the following relationships: For spherical particles, 6(1 − F) a=ᎏ (12-28) (Dp)m For uniform cylindrical particles, 4(0.5D0 + Z)(1 − F) a = ᎏᎏᎏ (12-29) D0 Z where F = void fraction; (Dp)m = harmonic mean diameter of spherical particles, m; D0 = diameter of cylinder, m; and Z = height of cylinder, m. For cylindrical particles that are long relative to their diameter, the term 0.5D0 in Eq. (12-29) can be neglected. Falling-rate Period The falling-rate period begins at the critical moisture content when the constant-rate period ends. When the falling moisture content is above the critical moisture content, the whole drying process will occur under constant-rate conditions. If, on the other hand, the initial moisture content is below the critical moisture content, the entire drying process will occur in the falling-rate period. This period is usually divided into two zones: (1) the zone of unsaturated surface drying and (2) the zone where internal moisture movement controls. In the first zone, the entire evaporating surface can no longer be maintained and saturated by moisture movement within the solid. The drying rate decreases from the unsaturated portion, and hence the rate for the total surface decreases. Generally, the drying rate depends on factors affecting the diffusion of moisture away from the evaporating surface and those affecting the rate of internal moisture movement.

* The term “water” is used for convenience; this discussion applies equally to other liquids.

SOLIDS-DRYING FUNDAMENTALS As drying proceeds, the point is reached where the evaporating surface is unsaturated. The point of evaporation moves into the solid, and the dry process enters the second falling-rate period. The drying rate is now governed by the rate of internal moisture movement; the influence of external variables diminishes. This period usually predominates in determining the overall drying time to lower moisture content. LIQUID DIFFUSION Diffusion-controlled mass transfer is assumed when the vapor or liquid flow conforms to Fick’s second law of diffusion. This is stated in the unsteady-state-diffusion equation using mass-transfer notation as δc/δθ = DAB(δ2C/dx)

(12-30)

where c = concentration of one component in a two-component phase of A and B, θ = diffusion time, x = distance in the direction of diffusion, and DAB = binary diffusivity of the phase AB. This equation applies to diffusion in solids, stationary liquids, and stagnant gases. The diffusion equation for the falling-rate drying period for a slab can be derived from the diffusion equation if one assumes that the surface is dry or at an equilibrium moisture content and that the initial moisture distribution is uniform. For these conditions, the following equation is obtained: 2 2 W − We 8 n=∞ 1 (12-31) ᎏ = ᎏ2 冱 ᎏ e−(2n + 1) Dᐉθ(π/2d) Wc − We π n = 0 (2n + 1) where W, We, and Wc = average moisture content (dry basis) at any time, θ, at the start of the falling-rate period and in equilibrium with the environment respectively, kg /kg; Dᐉ = liquid diffusivity, m2/s; θ = time from start of falling-rate period, s; and d = one-half of the thickness of the solid layer through which diffusion occurs, m. When evaporation occurs from only one face, d = total thickness, m. Equation (12-31) assumes that Dᐉ is constant; however, Dᐉ is rarely constant but varies with moisture content, temperature, and humidity. For long drying times, Eq. (12-31) simplifies to a limiting form of the diffusion equation as 2 8 W − We (12-32) ᎏ = ᎏ2 [e−Dᐉθ(π/2d) ] Wc − We π Equation (12-32) may be differentiated to give the drying rate as dW π2Dᐉ (12-33) −ᎏ=ᎏ (W − We) dθ 4d 2 where dW/dθ = drying rate, kg /s. When Eq. (12-31) is plotted on semilogarithmic graph paper, a straight line is obtained for values of (W − We)/(Wc − We) < 0.6. It is in the straight-line portion that the approximate form [Eq. (12-33)] applies. Equations (12-31), (12-32), and (12-33) hold only for a slab-sheet solid whose thickness is small relative to the other two dimensions. For other shapes, reference should be made to Crank (The Mathematics of Diffusion, Oxford, London, 1956). An approximate equation for the falling-rate period may be obtained by integration of Eq. (12-33). This gives an equation for materials in which moisture movement is controlled by diffusion: Wc − We 4d 2 θf = ᎏ2 ln ᎏ (12-34) Dᐉπ W − We where θf = drying time in the falling-rate period. Diffusion equations may also be used to study vapor diffusion in porous materials. It should be clear that all estimates based on relationships that assume constant diffusivity are approximations. Liquid diffusivity in solids usually decreases with moisture concentration. Liquid and vapor diffusivity also change, and material shrinks during drying.

冤

冥

CAPILLARY THEORY If the porous size of a granular material is suitable, moisture may move from a region of high to one of low concentration as the result of capil-

12-33

lary action rather than by diffusion. The capillary theory assumes that a bed of nonporous spheres is composed of particles surrounding a space called a pore. These pores are connected by passages of various sizes. As water is progressively removed from the bed, the curvature of the water surface in the interstices of the top layer of spheres increases and a suction pressure resulting from curvature is set up. As the removal of water continues, the suction pressure attains a value in which air is drawn into the pore spaces between successive layers of spheres. This entry suction or suction potential is a measure of the resultant forces tending to draw water from the interior of the bed to the surface. For a pore formed by regularly packed nonporous spheres, the suction potential is given by Ps = xσ/rρg

(12-35)

where Ps = suction potential, m of water; σ = surface tension; dyn/m; ρ = density of water, kg /m3; g = 9.8 m/s2; r = sphere radius, m; and x is a packing factor equal to 12.9 for rhombohedral and 4.8 for cubical packing. As drying proceeds, the surface moisture evaporates, causing retreat of the surface menisci until the suction potential reaches a value given by Eq. (12-35). At this point, the pores of the surface will open, air will enter, and the moisture will redistribute itself with a slight lowering of the suction potential. As evaporation proceeds, the suction potential again increases until a slightly higher entry value is reached, when a further redistribution occurs. The drying rate curve (Fig. 12-41) can be analyzed in terms of capillary theory. In region BC, there is a loss of moisture with a gradual increase in suction and emptying of the bulk of the larger pores in the solid. In region CE, there is an increase in suction as the moisture content decreases and finer pores are opened. Section ED represents a condition in which moisture is being removed by vapor diffusion from the interior of the body, although there is still sufficient water in the bed to give rise to capillary forces. An approximate equation for use for materials in which moisture movement is controlled by capillary flow is given as Wc − We ρs dλ(Wc − We) θf = ᎏᎏ ln ᎏ ht(t − t′s) W − We

(12-36)

where θf = drying time in the falling-rate period. Table 12-6 gives an approximate classification of materials that obey Eqs. (12-34) and (12-36). CRITICAL MOISTURE CONTENT To use the preceding equations for estimating drying times in the falling-rate period, it is necessary to know values of critical moisture content Wc. Such values are difficult to obtain without making actual drying tests, which in themselves would give the required drying time and thereby obviate solving the equations. However, when drying tests are not feasible, some estimate of critical moisture content must be made. Values of critical moisture contents for some representative materials are given in Table 12-7 for drying by cross circulation and in Table 12-15 for drying by through circulation. The tabulated values are only approximate, since critical moisture content depends on the drying history. It appears that the constant-rate period ends when the moisture content at the surface reaches a specific value. Since the critical TABLE 12-6

Materials Obeying Eqs. (12-34) and (12-38)

Materials obeying Eq. (12-34)

Materials obeying (Eq. 12-38)

1. Single-phase solid systems such as soap, gelatin, and glue 2. Wood and similar solids below the fiber-saturation point 3. Last stages of drying starches, textiles, paper, clay, hydrophilic solids, and other materials when bound water is being removed

1. Coarse granular solids such as sand, paint pigments, and minerals 2. Materials in which moisture flow occurs at concentrations above the equilibrium moisture content at atmospheric saturation or above the fibersaturation point

12-34

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

TABLE 12-7 Approximate Critical Moisture Contents Obtained on the Air Drying of Various Materials, Expressed as Percentage Water on the Dry Basis

Material Barium nitrate crystals, on trays Beaverboard Brick clay Carbon pigment Celotex Chrome leather Copper carbonate (on trays) English china clay Flint clay refractory brick mix Gelatin, initially 400 % water Iron blue pigment (on trays) Kaolin Lithol red Lithopone press cake (in trays)

Niter cake fines, on trays Paper, white eggshell Fine book Coated Newsprint Plastic clay brick mix Poplar wood Prussian blue Pulp lead, initially 140% water Rock salt (in trays) Sand, 50–150 mesh Sand, 200–325 mesh Sand, through 325 mesh Sea sand (on trays)

Silica brick mix Sole leather Stannic tetrachloride sludge Subsoil, clay fraction 55.4% Subsoil, much higher clay content Sulfite pulp Sulfite pulp (pulp lap) White lead Whiting Wool fabric, worsted Wool, undyed serge

Thickness, in. 1.0 0.17 .62 1 0.44 .04 1–1.5 1 2.0 0.1–0.2 (wet) 0.25–0.75 1 0.25 .50 .75 1.0 0.0075 .005 .004 2.0 0.165 1.0 2.0 2.0 2.0 0.25 .5 .75 1.0 2.0 2.0 0.25 1 0.25–0.75 0.039 0.25–1.5

Critical moisture, % water, dry basis 7 Above 120 14 40 160 125 60 16 13 300 110 14 50 6.4 8.0 12.0 16.0 Above 16 41 33 34 60–70 19 120 40 Below 15 7 5 10 21 3 4.7 5.5 5.9 6.0 8 Above 90 180 21 35 60–80 110 11 6.9 31 8

moisture content is the average moisture through the material, its value depends on the rate of drying, the thickness of the material, and the factors influencing moisture movement and resulting gradients within the solid. As a result, the critical moisture content increases with increased drying rate and with increased thickness of the mass of material being dried. EQUILIBRIUM MOISTURE CONTENT In drying solids it is important to distinguish between hygroscopic and nonhygroscopic materials. If a hygroscopic material is maintained in contact with air at constant temperature and humidity until equilibrium is reached, the material will attain a definite moisture content. This moisture is termed the equilibrium moisture content for the specified conditions. Equilibrium moisture may be adsorbed as a surface film or condensed in the fine capillaries of the solid at reduced pressure, and its concentration will vary with the temperature and humidity of the surrounding air. However, at low temperatures, e.g., 15 to 50°C, a plot of equilibrium moisture content versus percent relative humidity is essentially independent of temperature. At zero humidity the equilibrium moisture content of all materials is zero.

Equilibrium moisture content depends greatly on the nature of the solid. For nonporous, i.e., nonhygroscopic, materials, the equilibrium moisture content is essentially zero at all temperatures and humidities. For organic materials such as wood, paper, and soap, equilibrium moisture contents vary regularly over wide ranges as temperature and humidity change. In the special case of the dehydration of hydrated inorganic salts such as copper sulfate, sodium sulfate, or barium chloride, temperature and humidity control is very important in obtaining the desired degree of moisture removal, and the proper conditions must be determined from data on the water of hydration or crystallization as a function of air temperature and humidity. Equilibrium moisture content of a solid is particularly important in drying because it represents the limiting moisture content for given conditions of humidity and temperature. If the material is dried to a moisture content less than it normally possesses in equilibrium with atmospheric air, it will return to its equilibrium value on storage unless special precautions are taken. Equilibrium moisture content of a hygroscopic material may be determined in a number of ways, the only requirement being a source of constant-temperature and constant-humidity air. Determination may be made under static or dynamic conditions, although the latter case is preferred. A simple static procedure is to place a number of samples in ordinary laboratory desiccators containing sulfuric acid solutions of known concentrations which produce atmospheres of known relative humidity. The sample in each desiccator is weighed periodically until a constant weight is obtained. Moisture content at this final weight represents the equilibrium moisture content for the particular conditions. The value of equilibrium moisture content, for many materials, depends on the direction in which equilibrium is approached. A different value is reached when a wet material loses moisture by desorption, as in drying, from that obtained when a dry material gains it by adsorption. For drying calculations the desorption values are preferred. In the general case, the equilibrum moisture content reached by losing moisture is higher than that reached by adsorbing it. Equilibrium moisture content can be measured dynamically by placing the sample in a U tube through which is drawn a continuous flow of controlled-humidity air. Again the sample is weighed periodically until a constant weight is reached. Properly humidified air for such a procedure can be obtained by bubbling dry air through a large volume of a saturated salt solution which produces a definite degree of saturation of the air. Care must be taken to ensure that the air and salt solution reach equilibrium. Values of the humidity over various salt solutions may be found in Table 12-8. Several manufacturers supply chambers with dial-in temperature and humidity conditions for airwater systems. ESTIMATIONS FOR TOTAL DRYING TIME Estimates of both the constant-rate and the falling-rate periods are needed to estimate the total drying time for a given drying operation. If estimates for these periods are available, the total drying time is estimated by summing as θt = θc + θf

(12-37)

where θt = total drying time, h; θc = drying time for constant-rate period, h; and θf = drying time for falling-rate period, h. The difficulty in estimating critical moisture content greatly reduces the number of drying cases in which calculation of a good estimate is possible. ANALYSIS OF DRYING DATA TESTS ON PLANT DRYERS When experiments are carried out to select a suitable dryer and to obtain design data, the effect of changes in various external variables is studied. These experiments should be conducted in an experimental unit that simulates the large-scale dryer from both the thermal and the material-handling aspects, and only material which is truly representative of full-scale production should be used. Data expressing moisture content in terms of elapsed time should be obtained and the results plotted as shown in Fig. 12-42. For purposes of analysis, the moisture-time curve must be differentiated

SOLIDS-DRYING FUNDAMENTALS TABLE 12-8

12-35

Maintenance of Constant Humidity

Solid phase

Max. temp., °C.

% humidity

H3PO4·aH2O ZnCl2·aH2O KC2H3O2 LiCl·H2O KC2H3O2 KF NaBr CaCl2·6H2O CaCl2·6H2O CaCl2·6H2O CrO3 CaCl2·6H2O CaCl2·6H2O K2CO3·2H2O K2CO3·2H2O Ca(NO3)2·4H2O NaHSO4·H2O Mg(NO3)2·6H2O NaClO3 Ca(NO3)2·4H2O Mg(NO3)2·6H2O NaBr·2H2O Mg(C2H3O2)·4H2O NaNO2 (NH4)2SO4 (NH4)2SO4 NaC2H3O2·3H2O Na2S2O3·5H2O NH4Cl NH4Cl NH4Cl KBr Tl2SO4 KHSO4 Na2CO3·10H2O K2CrO4 NaBrO3 Na2CO3·10H2O Na2SO4·10H2O Na2HPO4·12H2O NaF Pb(NO3)2 TlNO3 TlCl

24.5 20 168 20 20 100 100 24.5 20 18.5 20 10 5 24.5 18.5 24.5 20 24.5 100 18.5 18.5 20 20 20 108.2 20 20 20 20 25 30 20 104.7 20 24.5 20 20 18.5 20 20 100 20 100.3 100.1

9 10 13 15 20 22.9 22.9 31 32.3 35 35 38 39.8 43 44 51 52 52 54 56 56 58 65 66 75 81 76 78 79.5 79.3 77.5 84 84.8 86 87 88 92 92 93 95 96.6 98 98.7 99.7

For a more complete list of salts, and for references to the literature see “International Critical Tables,” vol. 1, p. 68.

graphically or numerically and the drying rates so obtained plotted to determine the nature and extent of the drying periods in the cycle. It is customary to plot drying rate versus moisture content as in Fig. 12-41b. Although instructive, this type of plot gives no information on duration of the drying periods. These are better shown by plots similar to Fig. 12-41c, in which drying rate is plotted as a function of time on either arithmetic or logarithmic coordinates. Logarithmic plots permit easy reading at low moisture contents or long times. In order to determine whether a simple relationship exists in the falling-rate period, the unaccomplished moisture change, defined as ratio of free moisture in the solid at time θ to total free moisture present at start of the falling-rate period (W − We)/(Wc − We), is plotted as a function of time on semilogarithmic paper. If a straight line is obtained such as curve B of Fig. 12-43 by using the upper scale of abscissa, either Eq. (12-34), for materials in which the moisture moves by diffusion, or Eq. (12-37), for materials in which the moisture movement is by capillary flow, may be applicable. If Eq. (12-37) applies, K1, the slope of the falling-rate drying curve, is related to the constant drying rate. The latter is calculated from Eq. (12-38) and can be compared with the measured value. If the slopes agree, the moisture movement is by capillary flow. If the slopes do not agree, the moisture movement is by diffusion and the slope of the line should equal π2Dᐉ/4d 2. The dependency of drying rate on material thickness must be established experimentally. With the effect of material thickness estab-

FIG. 12-42

Drying-time curves.

lished, liquid diffusivity can be calculated as indicated here. For this calculation, the theoretical values for an infinite slab are required. These are: Dᐉθ ᎏ d2

0.02

0.05

0.10

0.15

0.20

0.30

0.50

1.0

W − We ᎏ Wc − We

0.84

0.75

0.642

0.563

0.496

0.387

0.238

0.069

A plot of these values is shown as curve A in Fig. 12-43. If a straight line such as curve B of Fig. 12-43 represents the experimental data and if it has been established that the drying time varies inversely as the square of the thickness, the average liquid diffusivity can be obtained as follows. At a given value of (W − We)/(Wc − We), read the corresponding value of Dᐉθ/d 2 from curve A, Eq. (12-32), Fig. 12-43. At the same value of (W − We)/(Wc − We), read the corresponding experimental value of θ from curve B (upper scale). Then

FIG. 12-43

Analysis of drying data.

12-36

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

(Dᐉ θ/d 2) theoretical Dᐉavg = ᎏᎏᎏ (12-38) (θ/d 2) experimental where Dᐉavg = average experimental value of liquid diffusivity, m2/h. The value of diffusivity calculated from Eq. (12-38) must be recognized as an average value over the entire range of moisture change from (W − We)/(Wc − We) = 1 to the value (W − Wc)/(Wc − We) at which θ/d2 was evaluated. Further, Eq. (12-38) assumes that the theoretical curve is a straight line for all values of time. This is not true for values (W − We)/(Wc − We) less than 0.6. A more accurate value of Dᐉavg can be obtained by taking a ratio of slopes of the curves of Fig. 12-43. Thus the ratio of the slope of the experimental curve of unaccomplished moisture change versus drying time on a semilogarithmic plot [Eq. (12-32)] to the slope of the theoretical curve at the same unaccomplished moisture change, again on a semilogaritmic plot, equals the quantity Dᐉ/d 2. If d is known, Dᐉ can be evaluated. TESTS ON PLANT DRYERS Tests on plant-scale dryers are usually carried out to obtain design data for a specific material, to select a suitable dryer type, or to check present performance of an existing dryer with the objective of determining its capacity potential. In these tests overall performance data are obtained and the results used to make heat and material balances and to estimate overall drying rates or heat-transfer coefficients. Generally, the minimum data to be taken in order to calculate the performance of a dryer are: 1. Inlet and outlet moisture contents 2. Inlet and outlet gas temperatures 3. Inlet and outlet material temperatures 4. Feed rate 5. Gas rate 6. Inlet and outlet humidities 7. Retention time or time of passage through the dryer 8. Fuel consumption Whenever possible, moisture contents and temperatures should be measured at various points within the dryer. Typical experimental and calculated results of a drying test for a continuous adiabatic convection dryer are shown in Fig. 12-44. Test data as elaborate as those shown are not usually justified economically except when basic studies, aimed at clarifying the effect of operating variables, are being carried out in order to arrive at a reliable design procedure. The completeness of the information which is sought in any given test depends on the ultimate use of the data. In any case data for at least two sets of operating conditions are needed if a good analysis of dryer performance is to be made. Results of drying tests can be correlated empirically in terms of overall heat-transfer coefficient or length of a transfer unit as a function of operating variables. The former is generally applicable to all types of dryers, while the latter applies only in the case of continuous dryers. The relationship between these quantities is as follows. The number of transfer units in any direct dryer is given by Nt = (t1 − t2)/∆tm

(12-39)

where Nt = number of transfer units; t1 = inlet gas temperature, K; t2 = exit gas temperature; and ∆ tm = mean temperature difference between gas and solids through the dryer, K.

Typical results of dryer-performance tests. To convert British thermal units per hour-cubic foot-degree Fahrenheit to joules per cubic metersecond-kelvin, multiply by 1.73.

FIG. 12-44

The volumetric heat-transfer coefficient is given by wcs(t2 − t1) qd (12-40) Uv = ᎏ = ᎏᎏ Vd ∆ tm Ad Ld ∆ tm where Uv = volumetric heat-transfer coefficient, J/(m3⋅s⋅K) [Btu/(h⋅ft3⋅ °F)]; q = cross-sectional area of dryer, m2; and Ld = dryer length, m. The volumetric heat-transfer coefficients along the dryer are lower at the discharge end (Fig. 12-44) because of the internal resistance to moisture movement in the later stages of drying. When drying data are expressed in terms of overall performance, care and judgment should be exercised in extrapolating the results to other conditions, particularly conditions of different feed and product moisture. If, for example, the overall heat-transfer coefficients, from the data of Fig. 12-44, were used to predict a dryer design for reducing the product moisture below 10 percent, the design would be in error. Obviously, this problem can be circumvented by making sure that the final moisture in the experiments is below that desired in the product. In any capacity test to determine the potential of a plant dryer, the effects of the following variables should be studied: 1. Effect of increased temperature. This is often the simplest way to achieve increased capacity. 2. Effect of increased final moisture. Because of the marked increase in drying time required to dry to low moisture contents, the permissible maximum final moisture should always be established. 3. Effect of increasing air velocity should be determined. Frequently, higher air rates are necessary to provide the required additional heat at higher capacities. 4. Uniformity of air flow should be established. Air-flow maldistribution can seriously reduce dryer capacity and efficiency. 5. Possible benefits from air recirculation should be considered.

SOLIDS-DRYING EQUIPMENT CLASSIFICATION OF DRYERS Drying equipment may be classified in several ways. The two most useful classifications are based on (1) the method of transferring heat to the wet solids or (2) the handling characteristics and physical properties of the wet material. The first method of classification reveals differences in dryer design and operation, while the second method is

most useful in the selection of a group of dryers for preliminary consideration in a given drying problem. A classification chart of drying equipment on the basis of heat transfer is shown in Fig. 12-45. This chart classifies dryers as direct or indirect, with subclasses of continuous or batchwise operation. Direct Dryers The general operating characteristics of direct dryers are these:

SOLIDS-DRYING EQUIPMENT

FIG. 12-45

Classification of dryers, based on method of heat transfer. [Revised from Marshall, Heat, Piping Air Cond., 18, 71 (1946).]

12-37

12-38

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

1. Direct contacting of hot gases with the solids is employed for solids heating and vapor removal. 2. Drying temperatures may range up to 1000 K, the limiting temperature for most common structural metals. At the higher temperatures, radiation becomes an important heat-transfer mechanism. 3. At gas temperatures below the boiling point, the vapor content of gas influences the rate of drying and the final moisture content of the solid. With gas temperatures above the boiling point throughout, the vapor content of the gas has only a slight retarding effect on the drying rate and final moisture content. Thus, superheated vapors of the liquid being removed can be used for drying. 4. For low-temperature drying, dehumidification of the drying air may be required when atmospheric humidities are excessively high. 5. A direct dryer consumes more fuel per pound of water evaporated, the lower the final moisture content. 6. Efficiency increases with an increase in the inlet-gas temperature for a constant exhaust temperature. 7. Because large amounts of gas are required to supply all the heat for drying, dust-recovery equipment may be very large and expensive when drying very small particles. Indirect Dryers These differ from direct dryers with respect to heat transfer and vapor removal: 1. Heat is transferred to the wet material by conduction through a solid retaining wall, usually metallic. 2. Surface temperatures may range from below freezing in the case of freeze dryers to above 800 K in the case of indirect dryers heated by combustion products. 3. Indirect dryers are suited to drying under reduced pressures and inert atmospheres to permit the recovery of solvents and to prevent the occurrence of explosive mixtures or the oxidation of easily decomposed materials. 4. Indirect dryers using condensing fluids as the heating medium are generally economical from the standpoint of heat consumption, since they furnish heat only in accordance with the demand made by the material being dried. 5. Dust recovery and dusty materials can be handled more satisfactorily in indirect dryers than in direct dryers. Miscellaneous Dryers Infrared dryers depend on the transfer of radiant energy to evaporate moisture. The radiant energy is supplied electrically by infrared lamps, by electric resistance elements, or by incandescent refractories heated by gas. The latter method has the added advantage of convection heating. Infrared heating is not widely used in the chemical industries for the removal of moisture. Its principal use is in baking or drying paint films and in heating thin layers of materials. Dielectric dryers have not as yet found a wide field of application. Their fundamental characteristic of generating heat within the solid indicates potentialities for drying massive geometrical objects such as wood, sponge-rubber shapes, and ceramics. Power costs may range to 10 times the fuel costs of conventional methods. SELECTION OF DRYING EQUIPMENT 1. Initial selection of dryers. Select those dryers which appear best suited to handling the wet material and the dry product, which fit into the continuity of the process as a whole, and which will produce a product of the desired physical properties. This preliminary selection can be made with the aid of Table 12-9, which classifies the various types of dryers on the basis of the materials handled. 2. Initial comparison of dryers. The dryers so selected should be evaluated approximately from available cost and performance data. From this evaluation, those dryers which appear to be uneconomical or unsuitable from the standpoint of performance should be eliminated from further consideration. 3. Drying tests. Drying tests should be conducted in those dryers still under consideration. These tests will determine the optimum operating conditions and the product characteristics and will form the basis for firm quotations from equipment vendors. 4. Final selection of dryer. From the results of the drying tests and quotations, the final selection of the most suitable dryer can be made. A recent article describing dryer scale-up methodology in the

process industries was recently published by L. R. Genskow [Drying Technology—An International Journal, 12(1&2), 47 (1994)]. The important factors to consider in the preliminary selection of a dryer are the following: 1. Properties of the material being handled a. Physical characteristics when wet b. Physical characteristics when dry c. Corrosiveness d. Toxicity e. Flammability f. Particle size g. Abrasiveness 2. Drying characteristics of the material a. Type of moisture (bound, unbound, or both) b. Initial moisture content c. Final moisture content (maximum) d. Permissible drying temperature e. Probable drying time for different dryers 3. Flow of material to and from the dryer a. Quantity to be handled per hour b. Continuous or batch operation c. Process prior to drying d. Process subsequent to drying 4. Product qualities a. Shrinkage b. Contamination c. Uniformity of final moisture content d. Decomposition of product e. Overdrying f. State of subdivision g. Product temperature h. Bulk density 5. Recovery problems a. Dust recovery b. Solvent recovery 6. Facilities available at site of proposed installation a. Space b. Temperature, humidity, and cleanliness of air c. Available fuels d. Available electric power e. Permissible noise, vibration, dust, or heat losses f. Source of wet feed g. Exhaust-gas outlets The physical nature of the material to be handled is the primary item for consideration. A slurry will demand a different type of dryer from that required by a coarse crystalline solid, which, in turn, will be different from that required by a sheet material (Table 12-9). Following preliminary selection of suitable types of dryers, a highspot evaluation of the size and cost should be made to eliminate those which are obviously uneconomical. Information for this evaluation can be obtained from material presented under discussion of the various dryer types. When data are inadequate, preliminary cost and performance data can usually be obtained from the equipment manufacturer. In comparing dryer performance, the factors in the preceding list which affect dryer performance should be properly weighed. The possibility of eliminating or simplifying processing steps which precede or follow drying, such as filtration, grinding, or conveying, should be carefully considered. DRYING TESTS These tests should establish the optimum operating conditions, the ability of the dryer to handle the material physically, product quality and characteristics, and dryer size. The principal manufacturers of drying equipment are usually prepared to perform the required tests on dryers simulating their equipment. Occasionally, simple laboratory experiments can serve to reduce further the number of dryers under consideration. Once a given type and size of dryer has been installed, the product characteristics and drying capacity can be changed only within relatively narrow limits. Thus it is more economical and far more satisfac-

TABLE 12-9

Classification of Commercial Dryers Based on Materials Handled Granular, crystalline, or fibrous solids

Large solids, special forms and shapes

Slurries

True and colloidal solutions; emulsions. Examples: inorganic salt solutions, extracts, milk, blood, waste liquors, rubber latex, etc.

Pumpable suspensions. Examples: pigment slurries, soap and detergents, calcium carbonate, bentonite, clay slip, lead concentrates, etc.

Examples: filterpress cakes, sedimentation sludges, centrifuged solids, starch, etc.

100 mesh or less. Relatively free flowing in wet state. Dusty when dry. Examples: centrifuged precipitates, pigments, clay, cement.

Larger than 100 mesh. Examples: rayon staple, salt crystals, sand, ores, potato strips, synthetic rubber.

Examples: pottery, brick, rayon cakes, shotgun shells, hats, painted objects, rayon skeins, lumber.

Examples: paper, impregnated fabrics, cloth, cellophane, plastic sheets.

Examples: veneer, wallboard, photograph prints, leather, foam rubber sheets.

Vacuum freeze. Indirect type, batch or continuous operation

Usually used only for pharmaceuticals such as penicillin and blood plasma. Expensive. Used on heat-sensitive and readily oxidized materials

See comments under Liquids

See comments under Liquids

See comments under Liquids

See comments under Granular solids

Applicable in special cases such as emulsion-coated films

See comments under Granular solids

Pan. Indirect type, batch operation

Atmospheric or vacuum. Suitable for small batches. Easily cleaned. Solvents can be recovered. Material agitated while dried Not applicable, except when pumping slowly on dry “heel”

See comments under Liquids

See comments under Liquids

See comments under Liquids

Expensive. Usually used on pharmaceuticals and related products which cannot be dried successfully by other means. Applicable to fine chemicals Suitable for small batches. Easily cleaned. Material is agitated during drying, causing some degradation

Not applicable

Not applicable

Not applicable

May have application in special cases when pumping onto dry “heel”

Use is questionable. Material usually cakes to dryer walls and agitator. Solvents can be recovered

Suitable for nonsticking materials. Useful for large batches of heatsensitive materials and for solvent recovery

Not applicable

Not applicable

Not applicable

Screw conveyor and indirect rotary. Indirect type, continuous operation

Applicable with dry-product recirculation

Applicable with dry-product recirculation

Generally requires recirculation of dry product. Little dusting occurs

Not applicable

Not applicable

Not applicable

Fluid beds. Batch, continuous, direct, and indirect

Applicable only with inert bed or dry-solids recirculator Not applicable

See comments under Liquids

See comments under Liquids

Chief advantage is low dust loss. Well suited to most materials and capacities, particularly those requiring drying at steam temperature Suitable, if not too dusty

Useful for large batches of heatsensitive materials or where solvent is to be recovered. Product will suffer some grinding action. Dust collectors may be required. Low dust loss. Material must not stick or be temperature-sensitive

Suitable for crystals, granules, and short fibers

Not applicable

Use hot inert particles for contacting

Use hot inert particles for contacting

Not applicable

Not applicable

Suitable for freeflowing materials that can be conveyed on a vibrating tray

Not applicable

Not applicable

Not applicable

Vacuum rotary. Indirect type, batch operation

Vibrating tray. Indirect type, continuous operation

Suitable for freeflowing materials

Continuous sheets

Discontinuous sheets

Liquids

Type of dryer

Pastes and sludges

Free-flowing powders

12-39

TABLE 12-9

Classification of Commercial Dryers Based on Materials Handled (Concluded ) Free-flowing powders Not applicable

Granular, crystalline, or fibrous solids Not applicable

Large solids, special forms and shapes Not applicable

Slurries See comments under Liquids. Twin-drum dryers are widely used

Pastes and sludges Can be used only when paste or sludge can be made to flow. See comments under Liquids

Not applicable

Not applicable

Not applicable

Not applicable

Not applicable

Infrared. Batch or continuous operation

Only for thin films

See comments under Liquids

See comments under Liquids (only for thin layers)

Only for thin layers

Primarily suited to drying surface moisture. Not suited for thick layers

Specially suited for drying and baking paint and enamels

Dielectric. Batch or continuous operation

Very expensive

See comments under Liquids

See comments under Liquids

Very expensive

Very expensive

Rapid drying of large objects suited to this method

Tray and compartment. Direct type, batch operation

Not applicable

For very small batch production. Laboratory drying

Dusting may be a problem. See comments under Pastes and Sludges

Not applicable

Not applicable

Not applicable

Primarily useful for small objects

Not applicable

Not applicable

Tunnel. Continuous Tray. Direct type, continuous operation

Not applicable

Not applicable

Suited to batch operation. At large capacities, investment and operating costs are high. Long drying times Usually not suited for materials smaller than 30 mesh. Suited to small capacities and batch operation Essentially largescale, semicontinuous tray drying.

See comments under Granular solids

Batch throughcirculation. Direct type, batch operation

Suited to batch operation. At large capacities, investment and operating costs are high. Long drying times Suitable only if material can be preformed. Suited to batch operation. Shorter drying time than tray dryers Suitable for small and large-scale production.

Not applicable

Suited for leather, wallboard, veneer.

Continuous throughcirculation. Direct type, continuous operation

Not applicable

Only crystal filter dryer may be suited

Suitable for materials that can be preformed. Will handle large capacities. Rotolouvre requires dry-product recirculation

Not generally applicable, except Roto-louvre in certain cases

Suited to a wide variety of shapes and forms. Operation can be made continuous. Widely used Suited to smaller objects that can be loaded on each other. Can be used to convey materials through heated zones. Roto-louvre not suited.

Not applicable

Special designs are required. Suited to veneers. Rotolouvre not applicable

Applicable with dry-product recirculation

Suitable only if product does not stick to walls and

Suitable for most materials and capacities, pro-

Not applicable

Not applicable

Not applicable

Drum. Indirect type, continuous operation

Cylinder. Indirect type, continuous operation

Direct rotary. Direct type, continuous operation 12-40

Not applicable

See comments under Pastes and Sludges. Verticalturbo applicable

Usually not suited for materials smaller than 30 mesh. Material does not tumble, except in Rotolouvre dryer. Latter operates at higher temperatures Suitable for most materials at most capacities. Dust-

Continuous sheets Not applicable

Discontinuous sheets Not applicable

Liquids Single, double or twin. Atm. or vacuum operation. Product flaky and usually dusty. Maintenance costs may be high Not applicable

Suitable for thin or mechanically weak sheets which can be dried in contact with a heated surface. Special surface effects obtainable Usually used in conjunction with other methods. Useful when there are space limitations Applications for final stages of paper dryers

Suitable for materials which need not be dried flat and which will not be injured by contact with hot drum

Useful for laboratory work or in conjunction with other methods

Successful on foam rubber. Not fully developed on other materials See comments under Granular solids

TABLE 12-9

Classification of Commercial Dryers Based on Materials Handled (Concluded )

Liquids

Type of dryer

Applicable with dry-product recirculation True and colloidal solutions; emulsions. Examples: inorganic salt solutions, extracts, milk, blood, waste liquors, rubber latex, etc.

Slurries

Pumpable suspensions. Examples: pigment slurries, soap and detergents, calcium carbonate, bentonite, clay slip, lead concentrates, etc.

Pastes and sludges does not dust. Recirculation of product may prevent sticking Examples: filterpress cakes, sedimentation sludges, centrifuged solids, starch, etc.

Free-flowing powders vided that dusting is not too severe

Granular, crystalline, or fibrous solids

Large solids, special forms and shapes

Continuous sheets

Discontinuous sheets

ing or crystal abrasion will limit its use

100 mesh or less. Relatively free flowing in wet state. Dusty when dry. Examples: centrifuged precipitates, pigments, clay, cement.

Larger than 100 mesh. Examples: rayon staple, salt crystals, sand, ores, potato strips, synthetic rubber.

Examples: pottery, brick, rayon cakes, shotgun shells, hats, painted objects, rayon skeins, lumber.

Examples: paper, impregnated fabrics, cloth, cellophane, plastic sheets.

Examples: veneer, wallboard, photograph prints, leather, foam rubber sheets.

Usually requires recirculation of dry product to make suitable feed. Well suited to high capacities. Disintegration usually required Requires special pumping equipment to feed the atomizer. See comments under Liquids

Suitable for materials that are easily suspended in a gas stream and lose moisture readily. Well suited to high capacities

Not applicable

Not applicable

Not applicable

Not applicable

Suitable for materials that are easily suspended in a gas streatm. Well suited to high capacities. Product may suffer physical degradation Not applicable

Not applicable

Not applicable

Not applicable

Different types are available for different requirements. Suitable for drying without contacting hot surfaces Not applicable

Not applicable

Pneumatic conveying. Direct type, continuous operation

See comments under Slurries

Can be used only if product is recirculated to make feed suitable for handling

Spray. Direct type, continuous operation

Suited for large capacities. Product is usually powdery, spherical, and free-flowing. High temperatures can be used with heatsensitive materials. Product may have low bulk density

See comments under Liquids. Pressure-nozzle atomizers subject to erosion

Continuous sheeting. Direct type, continuous operation

Not applicable

Not applicable

Not applicable

Not applicable

Not applicable

Not applicable

Vacuum shelf. Indirect type, batch operation

Not applicable

Applicable for small-batch production

Suitable for batch operation, small capacities. Useful for heat-sensitive or readily oxidizable materials. Solvents can be recovered

See comments under Pastes and Sludges

Suitable for batch operation, small capacities. Useful for heat-sensitive or readily oxidizable materials. Solvents can be recovered.

See comments under Granular solids

See comments under Granular solids

12-42

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

tory to experiment in small-scale units than on the dryer that is finally installed. On the basis of the results of the drying tests that establish size and operating characteristics, formal quotations and guarantees should be obtained from dryer manufacturers. Initial costs, installation costs, operating costs, product quality, dryer operability, and dryer flexibility can then be given proper weight in final evaluation and selection. BATCH TRAY AND DRYERS Description A tray or compartment dryer is an enclosed, insulated housing in which solids are placed upon tiers of trays in the case of particulate solids or stacked in piles or upon shelves in the case of large objects. Heat transfer may be direct from gas to solids by circulation of large volumes of hot gas or indirect by use of heated shelves, radiator coils, or refractory walls inside the housing. In indirect-heat units, excepting vacuum-shelf equipment, circulation of a small quantity of gas is usually necessary to sweep moisture vapor from the compartment and prevent gas saturation and condensation. Compartment units are employed for the heating and drying of lumber, ceramics, sheet materials (supported on poles), painted and metal objects, and all forms of particulate solids. Field of Application Because of the high labor requirements usually associated with loading or unloading the compartments, batch compartment equipment is rarely economical except in the following situations: 1. A long heating cycle is necessary because the size of the solid objects or permissible heating temperature requires a long holdup for internal diffusion of heat or moisture. This case may apply when the cycle will exceed 12 to 24 h. 2. The production of several different products requires strict batch identity and thorough cleaning of equipment between batches. This is a situation existing in many small color-pigment-drying plants. 3. The quantity of material to be processed does not justify investment in more expensive, continuous equipment. This case would apply in many pharmaceutical-drying operations. Further, because of the nature of solids-gas contacting, which is usually by parallel flow and rarely by through circulation, heat transfer and mass transfer are comparatively inefficient. For this reason, use of tray and compartment equipment is restricted primarily to ordinary drying and heat-treating operations. Despite these harsh limitations, when the listed situations do exist, economical alternatives are difficult to develop. Auxiliary Equipment If noxious gases, fumes, or dust are given off during the operation, dust- or fume-recovery equipment will be necessary in the exhaust-gas system. Wet scrubbers are employed for the recovery of valuable solvents from dryers. In order to minimize heat losses, thorough insulation of the compartment with brick, asbestos, or other insulating compounds is necessary. Modern fabricated dryercompartment panels usually have 7.5 to 15 cm of blanket insulation placed between the internal and external sheet-metal walls. Doors and other access openings should be gasketed and tight. In the case of tray and truck equipment, it is usually desirable to have available extra trays and trucks so that they can be preloaded for rapid emptying and loading of the compartment between cycles. Air filters and gas dryers are occasionally employed on the inlet-air system for direct-heat units. Vacuum-shelf dryers require auxiliary stream jets or other vacuumproducing devices, intercondensers for vapor removal, and occasionally wet scrubbers or (heated) bag-type dust collectors. Uniform depth of loading in dryers and furnaces handling particulate solids is essential to consistent operation, minimum heating cycles, or control of final moisture. After a tray has been loaded, the bed should be leveled to a uniform depth. Special preform devices, noodle extruders, pelletizers, etc., are employed occasionally for preparing pastes and filter cakes so that screen bottom trays can be used and the advantages of through circulation approached. Control of tray and compartment equipment is usually maintained by control of the circulating-air temperature (and humidity) and rarely by solids temperature. On vacuum units, control of the absolute pressure and heating-medium temperature is utilized. In direct dryers, cycle controllers are frequently employed to vary the air temper-

ature or velocity across the solids during the cycle, e.g., high air temperatures may be employed during a constant-rate drying period while the solids surface remains close to the air wet-bulb temperature. During the falling-rate periods, this temperature may be reduced to prevent casehardening or other degrading effects caused by overheating the solids surfaces. In addition, higher air velocities may be employed during early drying stages to improve heat transfer; however, after surface drying has been completed, this velocity may need to be reduced to prevent dusting. Two-speed circulating fans are employed commonly for this purpose. Direct-Heat Tray Dryers Satisfactory operation of tray-type dryers depends on maintaining a constant temperature and a uniform air velocity over all the material being dried. Circulation of air at velocities of 1 to 10 m/s is desirable to improve the surface heat-transfer coefficient and to eliminate stagnant air pockets. Proper air flow in tray dryers depends on sufficient fan capacity, on the design of ductwork to modify sudden changes in direction, and on properly placed baffles. Nonuniform air flow is one of the most serious problems in the operation of tray dryers. Tray dryers may be of the tray-truck or the stationary-tray type. In the former, the trays are loaded on trucks which are pushed into the dryer; in the latter, the trays are loaded directly into stationary racks within the dryer. Trucks may be fitted with flanged wheels to run on tracks or with flat swivel wheels. They may also be suspended from and moved on monorails. Trucks usually contain two tiers of trays, with 18 to 48 trays per tier, depending upon the tray dimensions. Trays may be square or rectangular, with 0.5 to 1 m2 per tray, and may be fabricated from any material compatible with corrosion and temperature conditions. When the trays are stacked in the truck, there should be a clearance of not less than 4 cm between the material in one tray and the bottom of the tray immediately above. When material characteristics and handling permit, the trays should have screen bottoms for additional drying area. Metal trays are preferable to nonmetallic trays, since they conduct heat more readily. Tray loadings range usually from 1 to 10 cm deep. Steam is the usual heating medium, and a standard heater arrangement consists of a main heater before the circulating fan. When steam is not available or the drying load is small, electrical heat can be used. For temperatures above 450 K, products of combustion can be used, or indirect-fired air heaters. Air is circulated by propeller or centrifugal fans; the fan is usually mounted within or directly above the dryer. Above 450 K, external or water-cooled bearings become necessary. Total pressure drop through the trays, heaters, and ductwork is usually in the range of 2.5 to 5 cm of water. Air recirculation is generally in the order of 80 to 95 percent except during the initial drying stage of rapid evaporation. Fresh air is drawn in by the circulating fan, frequently through dust filters. In most installations, air is exhausted by a separate small exhaust fan with a damper to control air-recirculation rates. Prediction of Heat- and Mass-Transfer Coefficients in DirectHeat Tray Dryers In convection phenomena, heat-transfer coefficients depend on the geometry of the system, the gas velocity past the evaporating surface, and the physical properties of the drying gas. In estimating drying rates, the use of heat-transfer coefficients is preferred because they are usually more reliable than mass-transfer coefficients. In calculating mass-transfer coefficients from drying experiments; the partial pressure at the surface is usually inferred from the measured or calculated temperature of the evaporating surface. Small errors in temperature have negligible effect on the heattransfer coefficient but introduce relatively large errors in the partial pressure and hence in the mass-transfer coefficient. For many cases in drying, the heat-transfer coefficient can be expressed as hc = αGn/Dc p

(12-41)

where hc = heat-transfer coefficient, J/(m ⋅s⋅K)[Btu/(h⋅ft ⋅°F)]; G = mass velocity of drying gas, kg/(s⋅m2)[lb/(h⋅ft2)]; Dc = characteristic dimension of the system, m; and α, n, and p are empirical constants. When radiation and conduction effects are negligible, the constant rate of drying from a surface is thus given by the following heattransfer expression derived from Eqs. (12-26) and (12-41): 2

2

SOLIDS-DRYING EQUIPMENT dw/dθ = (αGnA/ λDcm)(t − t′s)

(12-42)

When the liquid is water and the drying gas is air, t′s is the wet-bulb temperature. In order to estimate drying rates from Eq. (12-42) values of the empirical constants are required for the particular geometry under consideration. For flow parallel to plane plates, exponent n has been reported to range from 0.35 to 0.8 [Chu, Lane, and Conklin, Ind. Eng. Chem., 45, 1856 (1953); Wenzel and White, Ind. Eng. Chem., 51, 275 (1958)]. The differences in exponent have been attributed to differences in flow pattern in the space above the evaporating surface. In the absence of applicable specific data, the heat-transfer coefficient for the parallel-flow case can be taken, for estimating purposes, as h = 8.8G0.8/Dc0.2

(12-43)

where the experimental data have been weighted in favor of an exponent of 0.8 in conformity with the usual Colburn j factor and average values of the properties of air at 370 K have been incorporated. Experimental data for drying from flat surfaces have been correlated by using the equivalent diameter of the flow channel or the length of the evaporating surface as the characteristic length dimension in the Reynolds number. However, the validity of one versus the other has not been established. The proper equivalent diameter probably depends at least on the geometry of the system, the roughness of the surface, and the flow conditions upstream of the evaporating surface. For most tray-drying calculations, the equivalent diameter (4 times the cross-sectional area divided by the perimeter of the flow channel) should be used. For air flow impinging normally to the surface from slots, nozzles, or perforated plates, the heat-transfer coefficient can be obtained from the data of Friedman and Mueller (Proceedings of the General Discussion on Heat Transfer, Institution of Mechanical Engineers, London, and American Society of Mechanical Engineers, New York, 1951, pp. 138–142). These investigators give hc = αG0.78

(12-44)

where the gas mass velocity G is based on the total heat-transfer area and α is dependent on the plate open area, hole or slot size, and spacing between the plate, nozzle, or slot and the heat-transfer surface. Most efficient performance is obtained with plates having open areas equal to 2 to 3 percent of the total heat-transfer area. The plate should be located at a distance equal to four to six hole (or equivalent) diameters from the heat-transfer surface.

(a)

Data from tests employing multiple slots, with a correction calculated for slot width, were reported by Korger and Kizek [Int. J. Heat Mass Transfer, London, 9, 337 (1966)]. Air impingement is commonly employed for drying sheets, film, thin slabs, and coatings. Another application in which it is used is as a secondary heat source on drum and can dryers (see Fig. 12-46). Determination of the Temperature of the Evaporating Surface in Direct-Heat Tray Dryers When radiation and conduction are negligible, the temperature of the evaporating surface approaches the wet-bulb temperature and is readily obtained from the humidity and dry-bulb temperature. Frequently, however, radiation and conduction cause the temperature of the evaporating surface to exceed the wetbulb temperature. When this occurs, the true surface temperature must be estimated. Under steady-state conditions the temperature of the evaporating surface increases until the rate of sensible heat transfer to the surface equals the rate of heat removed by evaporation from the surface. To calculate this temperature, it is convenient to modify Eq. (12-26) in terms of humidity rather than partial-pressure difference, as follows: kg(ps − p) = k′(Hs − H)

Example of the use of air impingement in drying as a secondary heat source on a double-drum dryer. [Chem. Eng., 197, ( June 19, 1967).]

FIG. 12-46

(12-45)

where k′ = mass-transfer coefficient, kg/(s⋅m2)(unit humidity difference), and k′ = pkg(Ma/Mw) is a suitable approximation at low humidities; kg = mass-transfer coefficient, kg/(s⋅m2⋅atm); Ma = molecular weight of air; Mw = molecular weight of the diffusing vapor; ps = vapor pressure of the liquid at the temperature of the evaporating surface, atm; p = partial pressure of vapor in air, atm; Hs = saturation humidity of the air at the temperature of the drying surface, kg/kg dry air; H = humidity of the drying air, kg/kg dry air; and p = total pressure, atm. For air-water mixtures k′ is approximately 1.6 kg at atmospheric pressure. A rate balance between evaporation and heat transfer when radiation occurs may be modified by means of the psychrometric ratio for air-water vapor mixtures to give: λ ht ⑀ (12-46) ᎏ (Hs − H) = (t − t′s) + ᎏ (tr − t′s) Cs hc where λ = latent heat of evaporation, J/kg at t′s; Cs = hc /k′, where hc = convection heat-transfer coefficient, J/(s⋅m2⋅K), and Cs = heat capacity of humid air, J/(kg dry air⋅K), as defined in the subsection “Application of Psychrometry to Drying”; t = temperature of drying gases, K; t′s = temperature of the wet surface, K; hc = radiation heat-transfer coefficient, J/(s⋅m2⋅K); hc = convection heat-transfer coefficient, J/(s⋅m2⋅K); tr = temperature of source radiating heat to the wet surface, K; and ⑀ = emissivity of surface receiving radiation. Equation (12-46) may be solved by trial and error or graphically to estimate the true values of Hs and t′s and, hence, the actual drying rate. The values of λ and hr depend on the value of t′s but can generally be considered constant over the range of temperatures usually encountered in air drying. Frequently, particularly in tray drying, heat arrives at the evaporating surface from the tray walls by conduction through the wet material. For this case, in which both radiation and conduction are significant, the total heat-transfer coefficient is given by Shepherd, Brewer, and Hadlock [Ind. Eng. Chem., 30, 388 (1938)] as Au ht = (hc + hr) 1 + ᎏᎏ (12-47) 1 + d(hc + hr)/k where ht = total heat-transfer coefficient, J/(s⋅m2⋅K); Au = ratio of outside unwetted surface to evaporating surface area; d = depth of material in tray, m; and k = thermal conductivity of the wet material, J/[(s⋅m2)(K/m)]. Note that hc must be corrected for emissivity of the surface. For insulated trays, the arithmetic average of inside and outside unwetted area should be used. Equation (12-47) assumes that all heat sources are at the same temperature and that the convection coefficients to the evaporating surface and to the unwetted portions of the tray are equal. When radiation occurs from a source at a different temperature, the radiation coefficient can be corrected to the same basis by multiplying by the ratio (t − t′s)/(tr − t′s), where t, t′s, and tr are the drying-gas, evaporating-surface, and radiator temperatures respectively.

冤

(b)

12-43

冥

12-44

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

A relationship for estimating the surface temperature t′s, based on the use of Eq. (12-47) to determine ht, is as follows: (Hs − H) = (htCs /λhc)(t − t′s)

(12-48)

Equation (12-48) can be solved numerically or graphically. Figure 12-47 indicates how Hs and t′s may be determined graphically on a humidity chart by the point of intersection on the saturation-humidity curve of a straight line of slope htCs /λhc passing through point (Hs, ts). Performance Data for Direct-Heat Tray Dryers A standard two-truck dryer is illustrated in Fig. 12-48. Adjustable baffles or a perforated distribution plate is normally employed to develop 0.3 to 1.3 cm of water-pressure drop at the wall through which air enters the truck enclosure. This will enhance the uniformity of air distribution, from top to bottom, among the trays. In three (or more) truck ovens, air-reheat coils may be placed between trucks if the evaporative load is high. Means for reversing air-flow direction may also be provided in multiple-truck units. Performance data on some typical tray and compartment dryers are tabulated in Table 12-10. These indicate that an overall rate of evaporation of 0.0025 to 0.025 kg water/(s⋅m2) of tray area may be expected from tray and tray-truck dryers. The thermal efficiency of this type of dryer will vary from 20 to 50 percent, depending on the drying temperature used and the humidity of the exhaust air. In drying to very low moisture contents under temperature restrictions, the thermal efficiency may be in the order of 10 percent. The major operating cost for a tray dryer is the labor involved in loading and unloading the trays. About two labor-hours are required to load and unload a standard two-truck tray dryer. In addition, about one-third to one-fifth of a

FIG. 12-47

Graphical estimation of surface temperature during constant-rate

period.

FIG. 12-48 Double-truck dryer. (A) Air-inlet duct. (B) Air-exhaust duct with damper. (C) Adjustable-pitch fan 1 to 15 hp. (D) Fan motor. (E) Fin heaters. (F) Plenum chamber. (G) Adjustable air-blast nozzles. (H) Trucks and trays. ( J) Turning vanes.

worker’s time is required to supervise the dryer during the drying period. Power for tray and compartment dryers will be approximately 1.1 kW per truck in the dryer. Maintenance will run from 3 to 5 percent of the installed cost per year. BATCH THROUGH-CIRCULATION DRYERS In one type of batch through-circulation dryer, heated air passes through a stationary permeable bed of the wet material placed on removable screen-bottom trays suitably supported in the dryer. This type is similar to a standard tray dryer except that hot air passes through the wet solid instead of across it. The pressure drop through the bed of material does not usually exceed about 2 cm of water. In another type, deep perforated-bottom trays are placed on top of plenum chambers in a closed-circuit hot-air-circulating system. In some food-drying plants, the material is placed in finishing bins with perforated bottoms; heated air passes up through the material and is removed from the top of the bin, reheated, and recirculated. The latter types involve a pressure drop through the bed of material of 1 to 8 cm of water at relatively low air rates. Table 12-11 gives performance data on three applications of batch through-circulation dryers. Batch through-circulation dryers are restricted in application to granular materials that permit free flow-through circulation of air. Drying times are usually much shorter than in parallel-flow tray dryers. Design methods are included in the subsection “Continuous Through-Circulation Dryers.” Vacuum-Shelf Dryers Vacuum-shelf dryers are indirect-heated batch dryers consisting of a vacuumtight chamber usually constructed of cast iron or steel plate, heated, supporting shelves within the chamber, a vacuum source, and usually a condenser. One or two doors are provided, depending on the size of the chamber. The doors are sealed with resilient gaskets of rubber or similar material (Fig. 12-49). Hollow shelves of flat steel plate are fastened permanently inside the vacuum chamber and are connected in parallel to inlet and outlet headers. The heating medium, entering through one header and passing through the hollow shelves to the exit header, is generally steam, ranging in pressure from 700 kPa gauge to subatmospheric pressure for low-temperature operations. Low temperatures can be provided by circulating hot water, and high temperatures can be obtained by circulating hot oil or Dowtherm. Some small dryers employ electrically heated shelves. The material to be dried is placed in pans or trays on the heated shelves. The trays are generally of metal to ensure good heat transfer between the shelf and the tray. Vacuum-shelf dryers may vary in size from 1 to 24 shelves, the largest chambers having overall dimensions of 6 m wide, 3 m long, and 2.5 m high. Vacuum is applied to the chamber and vapor is removed through a large pipe which is connected to the chamber in a manner such that, if the vacuum is broken suddenly, the inrushing air will not greatly disturb the bed of material being dried. This line leads to a condenser where moisture or solvent that has been vaporized is condensed. The noncondensable exhaust gas goes to the vacuum source, which may be a wet or dry vacuum pump or a steam-jet ejector. Vacuum-shelf dryers are used extensively for drying pharmaceuticals, temperature-sensitive or easily oxidizable materials, and materials so valuable that labor cost is insignificant. They are particularly useful for handling small batches of materials wet with toxic or valuable solvents. Recovery of the solvent is easily accomplished without danger of passing through an explosive range. Dusty materials may be dried with negligible dust loss. Hygroscopic materials may be completely dried at temperatures below that required in atmospheric dryers. The equipment is employed also for freeze-drying processes, for metallizing-furnace operations, and for the manufacture of semiconductor parts in controlled atmospheres. All these latter processes demand much lower operating pressures than do ordinary drying operations. Design Methods for Vacuum-Shelf Dryers Heat is transferred to the wet material by conduction through the shelf and bottom of the tray and by radiation from the shelf above. The critical moisture content will not be necessarily the same as for atmospheric tray drying [Ernst, Ridgway, and Tiller, Ind. Eng. Chem., 30, 1122 (1938)].

SOLIDS-DRYING EQUIPMENT TABLE 12-10

12-45

Manufacturer’s Performance Data for Tray and Tray-Truck Dryers* Material

Color

Chrome yellow

Toluidine red

Half-finished Titone

Type of dryer Capacity, kg product/h Number of trays Tray spacing, cm Tray size, cm Depth of loading, cm Initial moisture, % bone-dry basis Final moisture, % bone-dry basis Air temperature, °C Loading, kg product/m2 Drying time, h Air velocity, m/s Drying, kg water evaporated/(h⋅m2) Steam consumption, kg/kg water evaporated Total installed power, kW

2-truck 11.2 80 10 60 × 75 × 4 2.5 to 5 207 4.5 85–74 10.0 33 1.0 0.59 2.5 1.5

16-tray dryer 16.1 16 10 65 × 100 × 2.2 3 46 0.25 100 33.7 21 2.3 65 3.0 0.75

16-tray 1.9 16 10 65 × 100 × 2 3.5 220 0.1 50 7.8 41 2.3 0.41 — 0.75

3-truck 56.7 180 7.5 60 × 70 × 3.8 3 223 25 95 14.9 20 3.0 1.17 2.75 2.25

Color 2-truck 4.8 120 9 60 × 70 × 2.5 116 0.5 99 9.28 96 2.5 0.11 1.5

*Courtesy of Proctor & Schwartz, Inc.

During the constant-rate period, moisture is rapidly removed. Often 50 percent of the moisture will evaporate in the first hour of a 6- to 8-h cycle. The drying time has been found to be proportional to between the first and second power of the depth of loading. Shelf vacuum dryers operate in the range of 1 to 25 mmHg pressure. For sizeestimating purposes, a heat-transfer coefficient of 20 J/(m2⋅s⋅K) may be used. The area employed in this case should be the shelf area in direct contact with the trays. Trays should be maintained as flatly as possible to obtain maximum area of contact with the heated shelves. For the same reason, the shelves should be kept free from scale and rust. Air vents should be installed on steam-heated shelves to vent noncondensable gases. The heating medium should not be applied to the shelves until after the air has been evacuated from the chamber in order to reduce the possibility of the material’s overheating or boiling at the start of drying. Casehardening can sometimes be avoided by retarding the rate of drying in the early part of the cycle. Performance Data for Vacuum-Shelf Dryers The purchase price of a vacuum-shelf dryer depends upon the cabinet size and number of shelves per cabinet. For estimating purposes, typical prices (1985) and auxiliary-equipment requirements are given in Table 12-12. Installed cost of the equipment will be roughly 100 percent of the carbon steel purchase cost. The thermal efficiency of a vacuum-shelf dryer is usually on the order of 60 to 60 percent. Table 12-13 gives operating data for one TABLE 12-11 Dryers*

Performance Data for Batch Through-Circulation

Kind of material Capacity, kg product/h Number of trays Tray spacing, cm Tray size, cm Depth of loading, cm Physical form of product Initial moisture content, % dry basis Final moisture content, % dry basis Air temperature, °C Air velocity, superficial, m/s Tray loading, kg product/m2 Drying time, h Overall drying rate, kg water evaporated/(h⋅m2) Steam consumption, kg/kg water evaporated Installed power, kW

Granular polymer 122 16 43 91.4 × 104 7.0 Crumbs 11.1

Vegetable

Vegetable seeds

42.5 24 43 91.4 × 104 6 0.6-cm diced cubes 669.0

27.7 24 43 85 × 98 4 Washed seeds 100.0

0.1

5.0

9.9

88 1.0

77 dry-bulb 0.6 to 1.0

36 1.0

16.1 2.0 0.89

5.2 8.5 11.86

6.7 5.5 1.14

4.0

2.42

6.8

7.5

*Courtesy of Proctor & Schwartz, Inc.

19

19

organic color and two inorganic compounds. Labor may constitute 50 percent of the operating cost; maintenance, 20 percent. Annual maintenance costs amount to 5 to 10 percent of the total installed cost. Actual labor costs will depend on drying time, facilities for loading and unloading trays, etc. The power required for these dryers is only that for the vacuum system; for vacuums of 680 to 735 mmHg the power requirements are in the order of 0.06 to 0.12 kW/m2 tray surface. Batch Furnaces, Other Furnace Types, and Kilns Batch furnaces are employed mainly for the heat treating of metals, such as annealing, normalizing, and “drawing” (tempering), and for the drying and calcination of ceramic articles. Many specialized furnaces have been designed for these purposes and may be either batch or continuous in operation. Batch furnaces are used in chemical processing for the same purposes as batch tray and truck dryers, when the drying or process temperature exceeds that which can be tolerated by unlined metal walls; ordinary tray and truck dryers are rarely employed when the circulating-gas temperature will exceed 600 to 700 K. They are employed for small-batch calcination, thermal decompositions, and other chemical reactions; these are the same as those reactions performed on a larger scale in rotary kilns, hearth furnaces, and shaft furnaces. Design procedures and information on heat release in furnaces are given in Sec. 11. Tables indicating normal operating temperatures in various heating furnaces and the more common process furnaces are also included. Specialized designs of batch furnaces are shown in Figs. 12-50 to 12-53 and are described briefly in the following paragraphs. All may be heated by gas, oil, or electricity. Standard oven furnaces are similar in design to the small muffle furnace depicted in Fig. 12-53, but with the muffle housing eliminated. Forced-convection pit furnaces are employed for heat-treating small metal parts in bulk. Small pieces are suspended in a meshbottom basket, while larger pieces are placed on racks. Air heating is by means of Nichrome electric coils set in refractory walls around the periphery of the pit. A high-velocity fan beneath the basket circulates heated air up past the coils and then down through the basket. Some heat is radiated to the outer basket shell, but most is transferred by direct convection from the circulating gas to the solids. Car-bottom furnaces differ from standard types in that the charge is placed upon movable cars for running into the furnace enclosure. The top of the car is refractory-lined and forms the furnace hearth. The top only is exposed to heat, the lower metal structure being protected by the hearth brick, sand, and water seals at the sides and ends and by the circulation of cooling air around the car structure below the hearth. For use where floor space is limited elevator furnaces serve similar purposes. The rotary-hearth furnace consists of a heating chamber lined with refractory brick within which is an annular-shaped refractorylined rotating hearth. Around the periphery of the rotating hearth, sand or circulating liquid seals are employed to prevent air infiltration. It can be made semicontinuous in operation. The hearth speed can be

12-46

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

Vacuum-shelf dryer. (Stokes Equipment Division, Pennwalt Corp.)

FIG. 12-49

TABLE 12-12

Standard Vacuum-Shelf Dryers* Price/m2 (1995)

Shelf area, m2

Floor space, m2

Weight average, kg

Pump capacity, m3/s

Pump motor, kW

Condenser area, m2

Carbon steel

304 stainless steel

0.4–1.1 1.1–2.2 2.2–5.0 5.0–6.7 6.7–14.9 16.7–21.1

4.5 4.5 4.6 5.0 6.4 6.9

540 680 1130 1630 3900 5220

0.024 0.024 0.038 0.038 0.071 0.071

1.12 1.12 1.49 1.49 2.24 2.24

1 1 4 4 9 9

$110 75 45 36 27 22

$170 110 65 65 45 36

*Stokes Vacuum, Inc.

varied to meet changing requirements in size, weight, and load of the charge. For gas and oil heating, the burners fire from the sides of the chamber, tangentially to the hearth. Standard furnaces are usually direct-heated, in that the burner combustion gases circulate directly over the charge; occasionally the flame may be permitted to impinge on the charge. For bright annealing, tool hardening, powdered-metal sintering, and other work requirTABLE 12-13

Performance Data of Vacuum-Shelf Dryers

Material

Sulfur black

Loading, kg dry material/m2 Steam pressure, kPa gauge Vacuum, mmHg Initial moisture content, % (wet basis) Final moisture content, % (wet basis) Drying time, h Evaporation rate, kg/ (s⋅m2)

25 410 685–710 50 1 8 8.9 × 10−4

Calcium carbonate 17 410 685–710 50.3 1.15 7 7.9 × 10−4

Calcium phosphate 33 205 685–710 30.6 4.3 6 6.6 × 10−4

ing protection of the charge by special atmospheres, muffle-type furnaces are frequently employed. In these, the charge is separated from the burners and combustion gases by a refractory arch. Heat is transferred by hot-gas radiation and convection to the arch and by radiation from the arch to the charge. When used for ceramic heating, furnaces are called kilns. Operations include drying, oxidation, calcination, and vitrification. These kilns employ horizontal space burners with gaseous, liquid, or solid fuels. If product quality is not injured, ceramic ware may be exposed to flame and combustion gases; otherwise, muffle kilns are employed. Dutch ovens are used frequently for heat generation. Downdraft kilns are the most common type, being used for brick, pipe, tile, and stoneware. The name is derived from the direction of combustion-gas flow when contacting the charge. The gases then flow up inside the walls to the top of the kiln and chimney. Updraft kilns are similar except in direction of gas flow, which is upward past the charge. They are employed commonly for pottery burning. Stove kilns are variations of updraft kilns used for burning common brick. The kiln is built of green brick and covered with a layer of burned brick. It is completely dismantled after each burning. Clamp kilns are another variation of updraft kilns used for common brick and temporary in nature. They have no tops or flue systems but consist only of sidewalls with arched spaces for combustion.

SOLIDS-DRYING EQUIPMENT

FIG. 12-50

Pit furnace. (W. S. Rockwell Co.)

FIG. 12-51

Car-bottom furnace. (W. S. Rockwell Co.)

CONTINUOUS TUNNEL DRYERS Continuous tunnels are in many cases batch truck or tray compartments, operated in series. The solids to be processed are placed in trays or on trucks which move progressively through the tunnel in contact with hot gases. Operation is semicontinuous; when the tunnel is filled, one truck is removed from the discharge end as each new truck is fed into the inlet end. In some cases, the trucks move on tracks or monorails, and they are usually conveyed mechanically, employing chain drives connecting to the bottom of each truck. Schematic diagrams of three typical tunnel arrangements are shown in Fig. 12-54. Belt-conveyor and screen-conveyor tunnels are truly continuous in operation, carrying a layer of solids on an endless conveyor. Air flow can be totally cocurrent, countercurrent, or a combination of both as shown in Fig. 12-54. In addition, cross-flow designs

12-47

are employed frequently, with the heating air flowing back and forth across the trucks in series. Reheat coils may be installed after each cross-flow pass to maintain constant-temperature operation; large propeller-type circulating fans are installed at each stage, and air may be introduced or exhausted at any desirable points. Tunnel equipment possesses maximum flexibility for any combination of air flow and temperature staging. When handling granular, particulate solids which do not offer high resistance to air flow, perforated or screentype belt conveyors are employed with through circulation of gas to improve heat- and mass-transfer rates. In tunnel equipment, the solids are usually heated by direct contact with hot gases. In high-temperature operations, radiation from walls and refractory lining may be significant also. The air in a direct-heat unit may be heated directly or indirectly by combustion or, at temperature below 475 K, by finned steam coils. Applications of tunnel equipment are essentially the same as for batch tray and compartment units previously described, namely, practically all forms of particulate solids and large solid objects. In operation, they are more suitable for large-quantity production, usually representing investment and installation savings over (multiple) batch compartments. In the case of truck and tray tunnels, labor savings for loading and unloading are not significant compared with batch equipment. Belt and screen conveyors which are truly continuous represent major labor savings over batch operations but require additional investment for automatic feeding and unloading devices. Auxiliary equipment and the special design considerations discussed for batch trays and compartments apply also to tunnel equipment. For size-estimating purposes, tray and truck tunnels and furnaces can be treated in the same manner as discussed for batch equipment. Continuous Through-Circulation Dryers Continuous throughcirculation dryers operate on the principle of blowing hot air through a permeable bed of wet material passing continuously through the dryer. Dryer rates are high because of the large area of contact and short distance of travel for the internal moisture. The most widely used type is the horizontal conveying-screen dryer in which wet material is conveyed as a layer, 2 to 15 cm deep, on a horizontal mesh screen or perforated apron, while heated air is blown either upward or downward through the bed of material. Its drying characteristics were studied by Marshall and Hougen [Trans. Am. Inst. Chem. Eng., 38, 91 (1942)]. This dryer consists usually of a number of individual sections, complete with fan and heating coils, arranged in series to form a housing or tunnel through which the conveying screen travels. As shown in the sectional view in Fig. 12-55, the air circulates through the wet material and is reheated before reentering the bed. It is not uncommon to circulate the hot gas upward in the wet end and downward in the dry end. A portion of the air is exhausted continuously by one or two exhaust fans, not shown in the sketch, which handle air from several sections. Since each section can be operated independently, extremely flexible operation is possible, with high temperatures usually at the wet end, followed by lower temperatures; in some cases a unit with cooled or specially humidified air is employed for final conditioning. The maximum pressure drop that can be taken through the bed of solids without developing leaks or air bypassing is roughly 50 mm of water. Through-circulation drying requires that the wet material be in a state of granular or pelleted subdivision so that hot air may be readily blown through it. Many materials meet this requirement without special preparation. Others require special and often elaborate pretreatment to render them suitable for through-circulation drying. The process of converting a wet solid into a form suitable for through circulation of air is called preforming, and often the success or failure of this contacting method depends on the preforming step. Fibrous, flaky, and coarse granular materials are usually amenable to drying without preforming. They can be loaded directly onto the conveying screen by suitable spreading feeders of the oscillating-belt or vibrating type or by spiked drums or belts feeding from bins. When materials must be preformed, several methods are available, depending on the physical state of the wet solid. 1. Relatively dry materials such as centrifuge cakes can sometimes be granulated to give a suitably porous bed on the conveying screen.

12-48

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

FIG. 12-52

Rotary-hearth furnace. (W. S. Rockwell Co.)

2. Pasty materials can often be preformed by extrusion to form sphaghetti-like pieces, about 6 mm in diameter and several centimeters long. 3. Wet pastes that cannot be granulated or extruded may be predried and preformed on a steam-heated finned drum. Preforming on a finned drum may be desirable also in that some predrying is accomplished. 4. Thixotropic filter cakes from rotary vacuum filters that cannot be preformed by any of the above methods can often be scored by knives on the filter, the scored cake discharging in pieces suitable for through-circulation drying. 5. Material that shrinks markedly during drying is often reloaded during the drying cycle to 2 to 6 times the original loading depth. This is usually done after a degree of shrinkage which, by opening the bed, has destroyed the effectiveness of contact between the air and solids. 6. In a few cases, powders have been pelleted or formed in briquettes to eliminate dustiness and permit drying by through circula-

tion. Table 12-14 gives a list of materials classified by preforming methods suitable for through-circulation drying. Steam-heated air is the usual heat-transfer medium employed in these dryers, although combustion gases may be used also. Temperatures above 600 K are not usually feasible because of the problems of lubricating the conveyor, chain, and roller drives. Recirculation of air is in the range of 60 to 90 percent. Conveyors may be made of wiremesh screen or perforated-steel plate. The minimum practical screen opening size is about 30 mesh. Design Methods for Continuous Tunnel Dryers In actual practice, design of a continuous through-circulation dryer is best based upon data taken in pilot-plant tests. Loading and distribution of solids on the screen are rarely as nearly uniform in commercial installations as in test dryers; 50 to 100 percent may be added to the test drying time for commercial design. A mathematical method of a through-circulation dryer has been developed by Thygeson [Am. Inst. Chem. Eng. J., 16(5), 749 (1970)]. Results obtained by Gamson, Thodos, and Hougen [Trans. Am. Inst. Chem. Eng., 39, I (1943)] and Wilke and Hougen [ibid., 41, 444 (1945)] for the rates of adiabatic evaporation of water from packed beds of porous solids are applicable when drying gases flow upward to downward. Use of average additive properties of the drying gas leads to G0.59 DpG hc = 0.11 ᎏ for (12-49) ᎏ > 350 Dp0.41 µ DpG G0.49 for (12-50) hc = 0.15 ᎏ ᎏ < 350 Dp0.51 µ where µ = gas viscosity, lb/(ft⋅h); Dp = diameter of sphere having the same surface area as particle, ft; and G = mass velocity of drying gas, lb/(h⋅ft2) [to convert from pounds per foot-hour to newtons per second-square meter, multiply by 4.133 × 10−4; to convert from feet to meters, multiply by 0.3048; and to convert from pounds per hour-square foot to kilograms per second-square meter, multiply by 1.3562 × 10−3]. Performance and Cost Data for Continuous Tunnel Dryers Experimental performance data are given in Table 12-15 for numerand

FIG. 12-53

Small muffle furnace. (W. S. Rockwell Co.)

SOLIDS-DRYING EQUIPMENT

12-49

(a) Countercurrent Tunnel Dryer

(b) Parallel Current Tunnel Dryer

(c) Center Exhaust Tunnel Dryer FIG. 12-54

Three types of tunnel dryers. [Van Arsdale, Food Ind. 14 (10), 43 (1942).]

ous common materials. Performance data from several commercial through-circulation conveyor dryers are given in Table 12-16. Labor requirements vary depending on the time required for feed adjustments, inspection, etc. These dryers will consume from 0.9 to 1.1 kg of steam/kg of water evaporated. Thermal efficiency is a function of final moisture required and percent air recirculation. Conveying-screen dryers are fabricated with conveyor widths from 0.3- to 4.4-m sections 1.6 to 2.5 m long. Each section consists of a sheet-metal enclosure, insulated sidewalls and roof, heating coils, a circulating fan, inlet-air distributor baffles, a fines catch pan under the

FIG. 12-55 Section view of a continuous through-circulation conveyor dryer. (Proctor & Schwartz, Inc.)

conveyor, and a conveyor screeen (Fig. 12-56). Table 12-17 gives approximate purchase costs for equipment with type 304 stainlesssteel hinged conveyor screens and includes steam-coil heaters, fans, motors, and a variable-speed conveyor drive. Cabinet- and auxiliaryequipment fabrication is of aluminized steel or stainless-steel materials. Prices do not include temperature controllers, motor starters, preform equipment, or auxiliary feed and discharge conveyors. These may add $55,000 to $135,000 to the dryer purchase cost (1995 costs). A continuous conveyor dryer employing a combination of air impingement and through circulation is shown in Fig. 12-57. Continuous Furnaces Continuous furnaces are employed for the same general duties cited for batch furnaces. Units are gas, oil, or electrically heated and utilize direct circulation of combustion gases or muffles for heat transfer. Continuous furnaces frequently have an extension added for cooling the charge before exposure to atmospheric air. Conveyors may be of parallel-chain, mat, slat, woven wire-mesh belt, or cast-alloy type. Automatic tensioning devices are used to maintain belt tension during heating and cooling. The product may rest directly on the conveyor or on special supports built into it. Roller-conveyors are used for large pieces. Flame curtains are provided for sealing the ends and for protection of special treating atmospheres. The pusher-type furnace is relatively free from mechanical problems because all mechanical parts are located outside the hot zone. It employs a roller-conveyor usually and will handle charges weighing considerably more per square meter than a belt-conveyor furnace. Pushers are driven by electric motors, compressed air, or hydraulic systems and can be automatically timed and synchronized with door-

12-50

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

TABLE 12-14

Methods of Preforming Some Materials for Through-Circulation Drying

No preforming required Cellulose acetate Silica gel Scoured wool Sawdust Rayon waste Fluorspar Tapioca Breakfast food Asbestos fiber Cotton linters Rayon staple

TABLE 12-15

Scored on filter

Granulation

Extrusion

Finned drum

Starch Aluminum hydrate

Kaolin Cryolite Lead arsenate Cornstarch Cellulose acetate Dye intermediates

Calcium carbonate White lead Lithopone Titanium dioxide Magnesium carbonate Aluminum stearate Zinc stearate

Lithopone Zinc yellow Calcium carbonate Magnesium carbonate

Flaking on chilled drum

Briquetting and squeezing

Soap flakes

Soda ash Cornstarch Synthetic rubber

Experimental Through-Circulation Drying Data for Miscellaneous Materials Moisture contents, kg/kg dry solid

Material

Physical form

Initial

Critical

Final

Inlet-air temperature, K

Alumina hydrate Alumina hydrate Alumina hydrate Aluminum stearate Asbestos fiber Asbestos fiber Asbestos fiber Calcium carbonate Calcium carbonate Calcium carbonate Calcium carbonate Calcium stearate Calcium stearate Calcium stearate Cellulose acetate Cellulose acetate Cellulose acetate Cellulose acetate Clay Clay Cryolite Fluorspar Lead arsenate Lead arsenate Lead arsenate Lead arsenate Kaolin Kaolin Kaolin Kaolin Kaolin Lithopone (finished) Lithopone (crude) Lithopone Magnesium carbonate Magnesium carbonate Mercuric oxide Silica gel Silica gel Silica gel Soda salt Starch (potato) Starch (potato) Starch (corn) Starch (corn) Starch (corn) Titanium dioxide Titanium dioxide White lead White lead Zinc stearate

Briquettes Scored filter cake Scored filter cake 0.7-cm extrusions Flakes from squeeze rolls Flakes from squeeze rolls Flakes from squeeze rolls Preformed on finned drum Preformed on finned drum Extruded Extruded Extruded Extruded Extruded Granulated Granulated Granulated Granulated Granulated 1.5-cm extrusions Granulated Pellets Granulated Granulated Extruded Extruded Formed on finned drum Formed on finned drum Extruded Extruded Extruded Extruded Extruded Extruded Extruded Formed on finned drum Extruded Granular Granular Granular Extruded Scored filter cake Scored filter cake Scored filter cake Scored filter cake Scored filter cake Extruded Extruded Formed on finned drum Extruded Extruded

0.105 9.60 5.56 4.20 0.47 0.46 0.46 0.85 0.84 1.69 1.41 2.74 2.76 2.52 1.14 1.09 1.09 1.10 0.277 0.28 0.456 0.13 1.23 1.25 1.34 1.31 0.28 0.297 0.443 0.36 0.36 0.35 0.67 0.72 2.57 2.23 0.163 4.51 4.49 4.50 0.36 0.866 0.857 0.776 0.78 0.76 1.2 1.07 0.238 0.49 4.63

0.06 4.50 2.25 2.60 0.11 0.10 0.075 0.30 0.35 0.98 0.45 0.90 0.90 1.00 0.40 0.35 0.30 0.45 0.175 0.18 0.25 0.066 0.45 0.55 0.64 0.60 0.17 0.20 0.20 0.14 0.21 0.065 0.26 0.28 0.87 1.44 0.07 1.85 1.50 1.60 0.24 0.55 0.42 0.48 0.56 0.30 0.60 0.65 0.07 0.17 1.50

0.00 1.15 0.42 0.003 0.008 0.0 0.0 0.003 0.0 0.255 0.05 0.0026 0.007 0.0 0.09 0.0027 0.0041 0.004 0.0 0.0 0.0026 0.0 0.043 0.054 0.024 0.0006 0.0009 0.005 0.008 0.0033 0.0037 0.0004 0.0007 0.0013 0.001 0.0019 0.004 0.15 0.215 0.218 0.008 0.069 0.082 0.084 0.098 0.10 0.10 0.29 0.001 0.0 0.005

453 333 333 350 410 410 410 410 410 410 410 350 350 350 400 400 400 400 375 375 380 425 405 405 405 405 375 375 375 400 400 408 400 400 415 418 365 400 340 325 410 400 400 345 380 345 425 425 355 365 360

Depth of bed, cm 6.4 3.8 7.0 7.6 7.6 5.1 3.8 3.8 8.9 1.3 1.9 7.6 5.1 3.8 1.3 1.9 2.5 3.8 7.0 12.7 5.1 5.1 5.1 6.4 5.1 8.4 7.6 11.4 7.0 9.6 19.0 8.2 7.6 5.7 7.6 7.6 3.8 3.8–0.6 3.8–0.6 3.8–0.6 3.8 7.0 5.1 7.0 7.0 1.9 3.0 8.2 6.4 3.8 4.4

Loading, kg product/m2

Air velocity, m/s × 101

Experimental drying time, s × 10−2

60.0 1.6 4.6 6.5 13.6 6.3 4.5 16.0 25.7 4.9 5.8 8.8 5.9 4.4 1.4 2.7 4.1 6.1 46.2 100.0 34.2 51.4 18.1 22.0 18.1 26.9 44.0 56.3 45.0 40.6 80.7 63.6 41.1 28.9 11.0 13.2 66.5 3.2 3.4 3.5 22.8 26.3 17.7 26.4 27.4 7.7 6.8 16.0 76.8 33.8 4.2

6.0 11.0 11.0 13.0 9.0 9.0 11.0 11.5 11.7 14.3 10.2 5.6 6.0 10.2 12.7 8.6 5.6 5.1 10.2 10.7 9.1 11.6 11.6 10.2 9.4 9.2 9.2 12.2 10.16 15.2 10.6 10.2 9.1 11.7 11.4 8.6 11.2 8.6 9.1 9.1 5.1 10.2 9.4 7.4 7.6 6.7 13.7 8.6 11.2 10.2 8.6

18.0 90.0 108.0 36.0 5.6 3.6 2.7 12.0 18.0 9.0 12.0 57.0 42.0 24.0 1.8 7.2 10.8 18.0 19.2 43.8 24.0 7.8 18.0 24.0 36.0 42.0 21.0 15.0 18.0 12.0 30.0 18.0 51.0 18.0 17.4 24.0 24.0 15.0 63.0 66.0 51.0 27.0 15.0 54.0 24.0 15.0 6.3 6.0 30.0 27.0 36.0

SOLIDS-DRYING EQUIPMENT TABLE 12-16

12-51

Performance Data for Continuous Through-Circulation Dryers* Kind of material Inorganic pigment

Capacity, kg dry product/h

712 2

Approximate dryer area, m Depth of loading, cm Air temperature, °C Loading, kg product/m2 Type of conveyor, mm

22.11 3 120 18.8 1.59 by 6.35 slots Rolling extruder

Preforming method or feed Type and size of preformed particle, mm Initial moisture content, % bone-dry basis Final moisture content, % bone-dry basis Drying time, min Drying rate, kg water evaporated/(h⋅m2) Air velocity (superficial), m/s Heat source per kg water evaporated, steam kg/kg gas (m3/kg) Installed power, kW

6.35-diameter extrusions 120

Cornstarch 4536 66.42 4 115 to 140 27.3 1.19 by 4.76 slots Filtered and scored Scored filter cake 85.2

Fiber staple 1724 Stage A, Stage B 57.04 35.12

Charcoal briquettes 5443

Gelatin 295

Inorganic chemical 862

130 to 100 100 3.5 3.3 2.57-diameter holes, perforated plate Fiber feed

52.02 16 135 to 120 182.0 8.5 × 8.5 mesh screen Pressed

104.05 5 32 to 52 9.1 4.23 × 4.23 mesh screen Extrusion

Rolling extruder

Cut fiber

64 × 51 × 25

110

37.3

2-diameter extrusions 300

6.35-diameter extrusions 111.2

0.5

13.6

35 38.39

24 42.97

11 17.09

9

1.27 Gas 0.11

1.12 Steam 2.0

0.66 Steam 1.73

29.8

119.3

194.0

30.19 4 121 to 82 33 1.59 × 6.35 slot

5.3

11.1

1.0

105 22.95

192 9.91

70 31.25

1.12 Waste heat

1.27 Steam 2.83

1.27 Gas 0.13

179.0

41.03

82.06

*Courtesy of Proctor & Schwartz, Inc.

Section assembly of a continuous through-circulation conveyor dryer. (Proctor & Schwartz, Inc.)

FIG. 12-56

TABLE 12-17

Conveyor-Screen-Dryer Costs*

Length

2.4-m-wide conveyor

3.0-m-wide conveyor

7.5 m 15 m 22.5 m 30 m

$8600/m2 $6700/m2 $6200/m2 $5900/m2

$7110/m2 $5600/m2 $5150/m2 $4950/m2

*National Drying Machinery Company, 1996.

opening timers. For small solids, trays of perforated metal alloys are used to carry the product. These carriers ride through the tunnel on rollers, skid rails, and occasionally refractory skids, one tray pushing the next ahead. The charge may travel in a straight line or in counterflow movement in single or multiple chambers. In counterflow movement, heat from the outgoing solids is transferred directly to cold incoming solids, reducing heat losses and fuel requirements. Continuous conveyor ovens are employed also for drying refractory shapes and for drying and baking enameled pieces. In many of these latter, the parts are suspended from overhead chain conveyors. Ceramic tunnel kilns handling large irregular-shaped objects must be equipped for precise control of temperature and humidity conditions to prevent cracking and condensation on the product. The internal mechanism causing cracking when drying clay and ceramics has been studied extensively. Information on ceramic tunnel-kiln operation and design is reported fully in publications such as The American Ceramic Society Bulletin, Ceramic Industry, and Transactions of the British Ceramic Society.

Special conveyor dryer with air jets impinging on surface of bed on first pass. Dried material is crushed and passed again through dryer, with air going through the now-permeable bed. [Chem. Eng., 192 (June 19, 1967).]

FIG. 12-57

12-52

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

ROTARY DRYERS Description A rotary dryer consists of a cylinder, rotated upon suitable bearings and usually slightly inclined to the horizontal. The length of the cylinder may range from 4 to more than 10 times its diameter, which may vary from less than 0.3 to more than 3 m. Feed solids fed into one end of the cylinder progress through it by virtue of rotation, head effect, and slope of the cylinder and discharge as finished product at the other end. Gases flowing through the cylinder may retard or increase the rate of solids flow, depending upon whether gas flow is countercurrent or cocurrent with solids flow. Rotary dryers have been classified as direct, indirect-direct, indirect, and special types. The terms refer to the method of heat transfer, being “direct” when heat is added to or removed from the solids by direct exchange between flowing gas and solids and being “indirect” when the heating medium is separated from physical contact with the solids by a metal wall or tube. Only totally direct and totally indirect types will be discussed extensively here, as it must be recognized that an infinite number of variations between the two are possible. Their operating characteristics when performing heat- and mass-transfer operations make them suitable for the accomplishment of drying, chemical reactions, solvent recovery, thermal decompositions, mixing, sintering, and agglomeration of solids. The specific types included are the following: Direct rotary dryer (cooler). This is usually a bare metal cylinder, with or without flights. It is suitable for low- and medium-temperature operations, the operating temperature being limited primarily by the strength characteristics of the metal employed in fabrication. Direct rotary kiln. This is a metal cylinder lined on the interior with insulating block and/or refractory brick. It is suitable for hightemperature operations. Indirect steam-tube dryer. This is a bare metal cylinder provided with one or more rows of metal tubes installed longitudinally in the shell. It is suitable for operation up to available steam temperatures or in processes requiring water cooling of the tubes. Indirect rotary calciner. This is a bare metal cylinder surrounded on the outside by a fired or electrally heated furnace. It is suitable for operation at medium temperatures up to the maximum which can be tolerated by the metal wall of the cylinder, usually 650 to 700 K for carbon steel and 800 to 1025 K for stainless steel. Direct Roto-Louvre dryer. This is one of the more important special types, differing from the direct rotary unit in that true through circulation of gas through the solids bed is provided. Like the direct rotary, it is suitable for low- and medium-temperature operation. Field of Application Rotating equipment is applicable to batch or continuous processing solids which are relatively free-flowing and granular when discharged as product. Materials which are not completely free-flowing in their feed condition are handled in a special manner, either by recycling a portion of final product and premixing with the feed in an external mixer to form a uniform granular feed to the process or by maintaining a bed of free-flowing product in the cylinder at the feed end and, in essence, performing a premixing operation in the cylinder itself. A properly designed recycle process will permit processing of many forms of slurry and solution feeds in rotating vessels. Direct rotary kilns and indirect calciners without internal flights or other obstructions are often provided with hanging link chains. These may serve as surfaces upon which material can accumulate until it is no longer sticky, at which time it will break off as a granular solid and continue its movement through the cylinder. Scraper chains may also be provided on indirect calciners to maintain clean internal walls. As a general rule, the direct-heat units are the simplest and most economical in construction and are employed when direct contact between the solids and flue gases or air can be tolerated. Because the total heat load must be introduced or removed in the gas stream, large gas volumes and high gas velocities are usually required. The latter will be rarely less than 0.5 m/s in an economical design. Therefore, employment of direct rotating equipment with solids containing extremely fine particles is likely to result in excessive entrainment losses in the exit-gas stream. The indirect forms require only sufficient gas flow through the cylinder to remove vapors or otherwise complete the internal process.

In addition, these can be sealed for processes requiring special gas atmospheres and exclusion of outside air. Auxiliary Equipment On direct-heat rotating equipment, a combustion chamber is required for high temperatures and finned steam coils are used for low temperatures. If contamination of the product with combustion gases is undesirable on direct-heat units, indirect gas- or oil-fired air heaters may be employed to achieve temperatures in excess of available steam. The method of feeding rotating equipment depends upon material characteristics and the location and type of upstream processing equipment. When the feed comes from above, a chute extending into the cylinder may be employed. For sealing purposes or if gravity feed is not convenient, a screw feeder is normally used. On cocurrent direct-heat units, cold-water jacketing of the feed chute or conveyor may be desirable if it is contacted by the inlet hot-gas stream. This will prevent overheating of the metal wall with resultant scaling or overheating of heat-sensitive feed materials. Any type of solids conveyor may be suitable for recycle mixing; however, the most universally applicable is the double-shaft pug-milltype paddle mixer. This conveyor or mixer should be insulated to prevent excessive heat losses from the hot, dry recycle product. To ensure uniformity in the recycle operation, a surge storage reserve of recycle solids should be installed for startup purposes and in the event of interruption of product discharge from the cylinder. In recycle operations, 50 to 60 percent product recirculation is found economical in many instances. One method of feeding direct cocurrent drying equipment utilizes dryer exhaust gases to convey, mix, and predry wet feed. The latter is added to the exhaust gases, at high velocity, from the dryer. The wet feed, mixed with dust entrained from the dryer, separates from the exhaust gases in a cyclone and drops into the feed end of the cylinder. The technique combines pneumatic and rotary drying. High thermal efficiency results from two cocurrent-flow stages operating countercurrently. Pneumatic conveyors are frequently employed as both dry-product conveyors and coolers. Other types of cooling equipment often used are screw conveyors, vibrating conveyors, and direct or indirect rotating coolers. Dust entrained in the exit-gas stream is customarily removed in cyclone collectors. This dust may be discharged back into the process or separately collected. For expensive materials or extremely fine particles, bag collectors may follow a cyclone collector, provided fabric temperature stability is not limiting. When toxic gases or solids are present, the exit gas is at a high temperature, the gas is close to saturation as from a steam-tube dryer, or gas recirculation in a sealed system is involved, wet scrubbers may be used independently or following a cyclone. Cyclones and bag collectors in drying applications frequently require insulation and steam tracing. The exhaust fan should be located downstream from the collection system. Rotating equipment, except brick-lined vessels, operated above ambient temperatures is usually insulated to reduce heat losses. Exceptions are direct-heat units of bare metal construction operating at high temperatures, on which heat losses from the shell are necessary to prevent overheating of the metal. Insulation is particularly necessary on cocurrent direct-heat units. It is not unusual for product cooling or condensation on the shell to occur in the last 10 to 50 percent of the cylinder length if it is not well insulated. For best operation, the feed rate to rotating equipment should be closely controlled and uniform in quantity and quality. Because solids temperatures are difficult to measure and changes slowly detected, most rotating-equipment operations are controlled by indirect means. Inlet and exit gas temperatures are measured and controlled on direct-heat units such as direct dryers and kilns, steam temperature and pressure and exit-gas temperature and humidity are controlled on steam-tube units, and direct shell temperature measurements are taken on indirect calciners. Product temperature measurements are taken for secondary control purposes only in most instances. Equipment which is electrically driven and operated with metal temperatures exceeding 425 K should be provided with auxiliary drives or power sources. Loss of rotation of a heated calciner or hightemperature dryer carrying a heavy bed of hot solids will quickly result in sagging of the cylinder due to nonuniform cooling.

SOLIDS-DRYING EQUIPMENT Direct-Heat Rotary Dryers The direct-heat rotary dryer is usually equipped with flights on the interior for lifting and showering the solids through the gas stream during passage through the cylinder. These flights are usually offset every 0.6 to 2 m to ensure more continuous and uniform curtains of solids in the gas. The shape of the flights depends upon the handling characteristics of the solids. For free-flowing materials, a radial flight with a 90° lip is employed. For sticky materials, a flat radial flight without any lip is used. When materials change characteristics during drying, the flight design is changed along the dryer length. Many standard dryer designs employ flat flights with no lips in the first one-third of the dryer measured from the feed end, flights with 45° lips in the middle one-third, and flights with 90° lips in the final one-third of the cylinder. Spiral flights are usually provided in the first meter or so at the feed end to accelerate forward flow from under the feed chute or conveyor and to prevent leakage over the feed-end retainer ring into the gas seals. When cocurrent gas-solids flow is used, flights may be left out to the final meter or so at the exit end to reduce entrainment of dry product in the exit gas. Showering of wet feed at the feed end of a countercurrent dryer will, on the other hand, frequently serve as an effective means for scrubbing dry entrained solids from the gas stream before it leaves the cylinder. Some dryers are provided with sawtooth flights to obtain uniform showering, while others use lengths of chain, attached to the underside of the flights, to scrape over and knock the walls of the cylinder, thereby removing sticky solids which might normally adhere to it. In kilns, the chains may contribute significantly to heat transfer; however, their use contributes to maintenance costs when flights are present in direct dryers. Solids sticking on flights and walls are usually removed more efficiently by external shell knockers. In dryers of large cross section, internal elements or partitions are sometimes used to increase the effectiveness of material distribution and reduce dusting and impact grinding. Use of internal members increases the difficulty of cleaning and maintenance unless sufficient free area is left between partitions for easy access of a person. Some examples of the more common flight arrangements are shown in Fig. 12-58. Component arrangements of countercurrent direct rotary dryers are shown in Fig. 12-59, and those of a cocurrent unit in Fig. 12-60. Countercurrent flow of gas and solids gives greater heat-transfer efficiency with a given inlet-gas temperature, but cocurrent flow can be used more frequently to dry heat-sensitive materials at higher inlet-gas temperatures because of the rapid cooling of the gas during initial evaporation of surface moisture. A number of different methods are employed to seal the rotating cylinder and prevent gas leakage through the annular opening between the rotating cylinder and the stationary throat pieces. None is an effective solids seal, nor will any function satisfactorily as a gas seal if solids leakage over the retaining ring on the cylinder is permitted.

Three examples of ordinary gas seals are shown in Fig. 12-61. On direct rotary dryers, few gas seals are intended to be completely gastight, but by careful control of the internal pressure, generally between 0.25 and 2.5 mm of water below atmosphere, dusting to the outside is prevented and in-leakage of outside air is minimized. Figure 12-61 also illustrates three basic types of trunnion rollbearing assemblies. Antifriction pillow blocks are the most common on modern dryers; however, when the dryer load requires larger than a 12.7- to 15.2-cm-diameter bearing on the trunnion shaft, the deadshaft antifriction bearing is substituted. This represents a considerable cost saving compared with the larger pillow blocks. They are completely sealed and continuously bathed in lubricant. Pillow-block bushings are less often used. The thrust washers are difficult to seal against dust, and they draw more power. Thrust roll mountings are depicted also in Fig. 12-61. These are usually dead-shaft. Gases are forced through the cylinder by either an exhauster or an exhauster-blower combination. With the latter arrangement it is possible to maintain very precise control of internal pressure even when the total system pressure drop is high. When a low-pressure-drop air heater is employed, however, the exhauster alone is usually sufficient, as the major gas pressure losses are found in the exit-air ductwork and dust collectors. Use of a blower by itself to force gas through the cylinder is an unusual practice, because the internal pressure is above atmospheric and hot air and dust may be blown into the gas seals or out into the surrounding working areas. Special designs of direct rotary dryers, such as the Renneburg DehydrO-Mat (Edward Renneburg & Sons Co.), are constructed especially to provide lower retention during the falling-rate drying period for the escape of internal moisture from the solids. The DehydrO-Mat is a cocurrent dryer employing a small-diameter shell at the feed end, where rapid evaporation of surface moisture in the stream of initially hot gas is accomplished with low holdup. At the solids- and gas-exit end, the shell diameter is increased to reduce gas velocities and provide increased holdup for the solids while they are exposed to the partially cooled gas stream. The Louisville type P dryer is a cocurrent dryer developed for the drying of heat-sensitive polymers. It is designed for use on rather finely divided and bulky materials which are easily airborne since its basic design utilizes a discharge cone permitting pneumatic conveying of dried solids from the dryer. Its internal design provides additional retention time by slowing the progress of the material through the dryer cylinder, permitting a comparatively high velocity of the drying medium without excessive blow-through. The Louisville type H dryer is a modified cocurrent dryer with “flash-drying” characteristics. Its internal arrangements consist of alternating disks and doughnuts which give high differential velocities between the drying medium and the solids being processed to increase the heat transfer and hence the rate of moisture removal. Design Methods for Direct-Heat Rotary Dryers Direct drying in a direct-heat rotary dryer is best expressed as a heat-transfer mechanism as follows: Qt = UaV(∆t)m

FIG. 12-58

Alternative direct-heat rotary-dryer flight arrangements.

12-53

(12-51)

where Qt = total heat transferred, J/s; Ua = volumetric heat-transfer coefficient, J/(s⋅m3⋅K); V = dryer volume, m3; and (∆t)m = true mean temperature difference between the hot gases and material, K. When a considerable quantity of surface moisture is removed from the solids and the solids temperatures are unknown, a good approximation of (∆t)m is the logarithmic mean between the wet-bulb depressions of the drying air at the inlet and exit of the dryer. Data for evaluating Ua were developed by Miller [Trans. Am. Inst. Chem. Eng., 38, 841 (1942)], Friedman and Marshall [Chem. Eng. Prog., 45, 482, 573 (1949)], and Seaman and Mitchell [Chem. Eng. Prog., 50, 467 (1954)]. These authors all employed relationships that could be reduced to the form Ua = KGn/D, where K = a proportionality constant, G = gas mass velocity, D = diameter, and n = a constant. P. Y. McCormick [Chem. Eng. Prog., 58(6), 57 (1962)] compared all available data. The comparisons showed that flight geometry and shell speed should be accounted for in the value of K. He suggested that shell rotational speed and flight number and shape must affect the overall balance; however, data for evaluating these variables separately are not available. Also, it is not believed that the effect of gas velocity

12-54

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

FIG. 12-59

Component arrangements of a countercurrent direct-heat rotary dryer. (ABB Raymond/Bartlett-Snow TM.)

FIG. 12-60 Elevation of a 60-in-diameter by 30-ft-long direct-heat cocurrent rotary dryer. To convert inches to centimeters, multiply by 2.54; to convert feet to meters, multiply by 0.3048. (ABB Raymond/Bartlett-Snow TM.)

is toward reducing the surface film thickness around each particle; rather, an increase in gas velocity more likely breaks up more effectively the showering curtains of solids and thereby exposes more solids surface; i.e., the effect is to increase the a in Ua rather than U. The following relationship is recommended for commercial dryers now manufactured in the United States, which usually have a flight count per circle of 2.4 to 3.0 D and operate at shell peripheral speeds of 60 to 75 ft/min: Qt = (0.5G0.67/D)V ∆ tm

(12-52)

= 0.4LDG0.67 ∆ tm

(12-53)

where Qt = total heat transferred, Btu/h; L = dryer length, ft; D = dryer diameter, ft; G = gas mass velocity, lb/(h⋅ft2 of cross section); and ∆tm = log mean of the drying-gas wet-bulb depression at the inlet end and exit end of the dryer shell. Typical operating data for cocurrent rotary dryers are given in Table 12-18. (Note that the driving force ∆Tm must be based on wet-bulb depression and not on material temperatures. Use of material temperatures, particularly when the dry solids are superheated after drying, will yield conservative results.) In most cases, direct-heat rotary dryers are still sized on the basis of pilot-plant tests, because rarely is all the moisture to be removed truly “free” moisture, and residence time for diffusion is frequently needed.

A theoretical model simulating the operation of both cocurrent and countercurrent rotary dryers which relies on pilot and bench scale tests to determine parameters which describe solids transport and drying rate is described by Papadakis, Langrish, Kemp, and Bahu [Drying Technology—An International Journal, 12(1&2), 259 (1994)]. For maximum dryer heat-transfer efficiency, dryer fillage must be sufficient to load the lifting flights fully, as discussed later. Unless material characteristics limit the gas temperature, the inlet temperature is usually fixed by the heating medium employed; i.e., 400 to 450 K for steam or 800 to 1100 K for gas- and oil-fired burners. The proper exit-gas temperature is largely an economic function. Its value may be determined as follows: Nt = (t1 − t2)/(∆t)m

(12-54)

where Nt = number of heat-transfer units based upon the gas; t1 = initial-gas temperature, K; t2 = exit-gas temperature allowing for heat losses, K; and (∆t)m is as defined for Eq. (12-51). Equation (12-54) can be used to select an exit-gas temperature since it has been found (empirically) that rotary dryers are most economically operated between Nt = 1.5 and Nt = 2.5. The L/D (length-diameter) ratio found most efficient in commercial practice lies between 4 and 10. If the length calculated previously does not fall within these limits, another value of Nt which will place L/D in the proper range may be computed.

SOLIDS-DRYING EQUIPMENT

12-55

(b)

(c)

(a) Rotary-dryer components. (a) Alternative rotary gas seals. (b) Alternative trunnion roll bearings. (c) Thrust roll assembly. (ABB Raymond/Bartlett-Snow TM.)

FIG. 12-61

Rotary dryers usually operate with 10 to 15 percent of their volume filled with material. Lower fillage will be insufficient to utilize the lifters fully, while greater fillage creates the possibility of shortcircuiting feed solids across the top of the bed. Under normal fillage conditions, the dryer usually can be made to hold solids long enough to complete the removal of internal moisture. If the holdup in the dryer is not great enough, the time of passage may be too short to remove all internal moisture, or because of incomplete flight fillage performance may be erratic. The effect of fillage on retention time and uniformity in rotary dryers has been studied by Miskell and Marshall [Chem. Eng. Prog., 52, 1 (1956)]. Time of passage is defined as holdup divided by feed rate. It can be measured directly in rotary dryers if holdup and feed rate can be measured directly. Holdup cannot always be measured conveniently on large plant dryers, however, unless a period of shutdown occurs when the dryer can be discharged and its contents weighed. Other methods have been resorted to, one of which consists of adding a pound or two of an inert detectable solid or a radioisotope of a feed constituent to the feed and analyzing for it in the discharged product. The time required for the maximum concentration to occur represents the average time of passage. The time of passage in rotary dryers can be estimated by the relationships developed by Friedman and Marshall (op. cit.), as given here:

BLG 0.23L θ=ᎏ ⫾ 0.6 ᎏ SN0.9D F

(12-55)

B = 5(Dp)−0.5

(12-56)

where B = a constant depending upon the material being handled and approximately defined by Eq. (12-56); Dp = weight average particle size of material being handled, µm; F = feed rate to dryer, lb dry material/(h⋅ft2 of dryer cross section); θ = time of passage, min; S = slope, ft/ft; N = speed, r/min; L = dryer length, ft; G = air-mass velocity, lb/(h⋅ft2); and D = dryer diameter, ft. The plus sign refers to countercurrent flow and the negative sign to cocurrent flow. To convert from British thermal units per hour to watts, multiply by 0.293; to convert from pounds per hour-square foot of cross section to kilograms per second-square meter, multiply by 0.00135. Air-mass velocities in rotary dryers usually range from 0.5 to 5.0 kg /(s⋅m2). It is customary to employ the highest air velocity possible without serious dusting. The amount of dusting occurring during operation is a complex function of the material being dried, its physical state, the air velocity employed, the holdup in the dryer, the number of flights, the rate of rotation, and the construction of the breeching at the end of the dryer. It can be predicted accurately only by experimental tests. An air rate of 1.4 kg /(s⋅m2) can usually be safely used with 35-mesh solids. Information on the dusting of a number of

12-56

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

TABLE 12-18

Warm-Air Direct-Heat Cocurrent Rotary Dryers: Typical Performance Data*

Dryer size, m × m Evaporation, kg/h Work, 108 J/h Steam, kg/h at kg/m2 gauge Discharge, kg/h Exhaust velocity, m/min Exhaust volume, m3/min Exhaust fan, kW Dryer drive, kW Shipping weight, kg Price, FOB Chicago

1.219 × 7.62 136.1 3.61 317.5 408 70 63.7 3.7 2.2 7700 $158,000

1.372 × 7.621 181.4 4.60 408.2 522 70 80.7 3.7 5.6 10,900 $168,466

1.524 × 9.144 226.8 5.70 521.6 685 70 100.5 5.6 5.6 14,500 $173,066

1.839 × 10.668 317.5 8.23 725.7 953 70 144.4 7.5 7.5 19,100 $204,400

2.134 × 12.192 408.2 1.12 997.9 1270 70 196.8 11.2 14.9 35,800 $241,066

2.438 × 13.716 544.3 1.46 131.5 1633 70 257.7 18.6 18.6 39,900 $298,933

3.048 × 16.767 861.8 2.28 2041 2586 70 399.3 22.4 37.3 59,900 $393,333

*Courtesy of Swenson Process Equipment Inc. NOTE: Material: heat-sensitive solid Maximum solids temperature: 65°C Feed conditions: 25 percent moisture, 27°C Product conditions: 0.5 percent moisture, 65°C Inlet-air temperature: 165°C Exit-air temperature: 71°C Assumed pressure drop in system: 200 mm. System includes finned air heaters, transition piece, dryer, drive, product collector, duct, and fan. Prices are for carbon steel construction and include entire dryer system (November, 1994). For 304 stainless-steel fabrication, multiply the prices given by 1.5.

materials in a 0.3- by 2-m rotary dryer has been presented by Friedman and Marshall (ibid.). Rotary dryers operate at peripheral speeds of 0.25 to 0.5 m/s. Slopes of rotary-dryer shells vary from 0 to 8 cm/m. In some cases of cocurrent-flow operations, negative slopes have been used. The radial flight heights in a direct dryer will range from onetwelfth to one-eighth of the dryer diameter. The number of flights will range from 0.6 D to D, where D = diameter, m, for dryers larger than 0.6 m in diameter, and should be designed to carry and shower all the holdup and minimize any kiln action. Performance and Cost Data for Direct-Heat Rotary Dryers Table 12-18 gives estimating-price data for direct rotary dryers employing steam-heated air. Higher-temperature operations requiring combustion chambers and fuel burners will cost more. The total installed cost of rotary dryers including instrumentation, auxiliaries, allocated building space, etc., will run from 150 to 300 percent of the purchase cost. Simple erection costs average 10 to 20 percent of the purchase cost. Operating costs will include 5 to 10 percent of one worker’s time, plus power and fuel required. Yearly maintenance costs will range from 50 to 10 percent of total installed costs. Total power for fans, dryer drive, and feed and product conveyors will be in the range of 0.5 D2 to 1.0 D2. Thermal efficiency of a high-temperature direct-heat rotary dryer will range from 55 to 75 percent and, with steam-heated air, from 30 to 55 percent. A representative list of materials dried in direct-heat rotary dryers is given in Table 12-19. Direct-Heat Rotary Kilns One of the most important of the high-temperature process furnaces is the direct-fired rotary kiln. It replaces the ordinary rotary dryer when the wall temperature exceeds that which can be tolerated by a bare metal shell (650 to 700 K for carbon steel). Rotary-kiln shells are lined in part or for their entire length with a refractory brick to prevent overheating of the steel with resulting weakening. Occasionally two linings are used, the one next to the shell being an insulating brick. Insulation is infrequently used on the outside of the shell, and caution must be observed not to overheat the shell metal by this confinement. When wet feeds are applied to a kiln lining at the cold end, there may be leakage of liquid through the lining to the shell, which will cause trouble if the liquid is corrosive. The feed is introduced into the upper end of the kiln by various methods, i.e., inclined chutes, overhung screw conveyors, slurry pipes, etc. Sometimes ring dams or chokes of a refractory material are installed within the kiln to build a deeper bed at one or more points, thus changing the flow pattern. The hot product is discharged from the lower end of the kiln into quench tanks, onto conveyors, or into cooling devices which may or may not recover its heat content. These cooling and heat-recovery devices include rotating inclined cylinders, inclined slow-moving grates, shaking grates, etc. Some kilns have two or three diameters, part of the length being

one diameter and the remainder being another diameter. It is claimed that this arrangement increases kiln capacity, decreases fuel consumption, and improves product quality. Two types of kilns are depicted in Fig. 12-62. An enlarged cross section near the discharge end (and hotgas inlet) reduces the gas velocity and provides increased holdup for a “soaking” period at high temperature. The first rotary kilns used in the United States were very small, 2 by 20 m. Sizes gradually increased and seemed to stop for a period at a maximum size of 4 by 150 m. A few much larger units have been installed for cement production. Modern rotary-kiln shells are of all-welded construction. Riding rings are forged or cast steel; support rollers are forged or cast steel and, on rare occasions, tool steel. Main bearings are sleeve-type, normally bronze. Antifriction bearings are frequently used on very small kilns but never on large units. However, bearings on the pinion shafts are normally of the antifriction type. Gearing is single helical or spur; gear lubrication usually is an automatic spray type. Single drives are used up to 150 kW. Kilns requiring more than 150 kW may be equipped with dual drives, i.e., two driving TABLE 12-19 Representative Materials Dried in Direct-Heat Rotary Dryers* Moisture content, % (wet basis) Material dried High-temperature: Sand Stone Fluorspar Sodium chloride (vacuum salt) Sodium sulfate Ilmenite ore Medium-temperature: Copperas Ammonium sulfate Cellulose acetate Sodium chloride (grainer salt) Cast-iron borings Styrene Low-temperature: Oxalic acid Vinyl resins Ammonium nitrate prills Urea prills Urea crystals

Initial

Final

Heat efficiency, %

10 6 6 3

0.5 0.5 0.5 0.04

61 65 59 70–80

6 6

0.1 0.2

60 60–65

1 (moles) 0.10 0.5 0.06

55 50–60 51 35

6 5

0.5 0.1

50–60 45

5 30 4 2 3

0.2 1 0.25 0.2 0.1

29 50–55 30–35 20–30 50–55

7 3 60 25

*Taken from Chem. Eng., June 19, 1967, p. 190, Table III.

SOLIDS-DRYING EQUIPMENT

(a)

(b) FIG. 12-62

Rotary kilns.

pinions and two motors, both driving one bull gear. In this manner, the power load is split through two separate driving mechanisms, meshing with one and the same gear. Kiln inclination varies wtih processes from 2 to 6 cm/m. Speed of rotation also varies from very slow, i.e., a peripheral speed of 0.15 m/s for a TiO2 pigment kiln, or 0.22 m/s for a cement kiln, to 0.64 m/s for a unit calcining phosphate materials. Special features include the discharge end designed for air cooling or kilns that operate at high temperatures, such as cement, deadburned dolomite, and magnesia. Firing hoods are designed with retractable fronts and large side doors and are mounted on wheels. Internal heat recuperators are of numerous designs and are becoming more popular as fuel prices increase. Thermocouple collector rings are placed at various points on the shell for indicating and recording internal temperatures. Scoop systems are provided for introducing collected dust or, in some cases, a feed component through the shell at some intermediate point or points. Ports are installed in the shell for admitting combustion air at points beyond the hot zone; these are used in reducing kilns for burning carbon monoxide and volatiles from materials being processed. Firing may be accomplished at either end, depending on whether cocurrent or countercurrent flow of the charge and gases is desired. Sometimes a solid fuel is mixed with the charge and burned as it moves down the kiln. Gaseous, liquid, or powdered fuels may be used. The burner may be installed directly at the end of the kiln with combustion occurring inside it. In this case, the discharge-end housing usually consists of a fixed or movable kiln hood through which the fuel pipe enters the kiln. A center position for the fuel pipe is used when the flame is wanted off the charge. Some users prefer an off-center

(a)

(b)

position toward the trough between the charge chord and the descending kiln lining. The kiln and the hood (combustion chamber) usually have open ends which coincide with each other with the gap being closed by a sliding seal (Fig. 12-63). Sometimes a special offset chamber for the introduction of secondary tempering air is provided on dryers and kilns (Fig. 12-64). The exhaust gases are generally discharged into dust and fume knockdown equipment to avoid contamination of the atmosphere. Gas-cleaning equipment includes cyclones, settling chambers, scrubbing towers, and electrical precipitators. Heat-recovery devices are utilized both within and outside the kiln. These result in an increase in kiln capacity or a decrease in fuel consumption. Waste-heat boilers, grates, coil systems, and chains are used for this purpose. The feed end of a rotary kiln is partially closed by a ring-shaped feed head which retains the end brick and dams back flow of solids. On the discharge end, a brick retaining-ring casting is made up to suit the application. For low temperatures, segmental alloy-iron rings may be employed. For high-temperature processes, either segmental alloy-steel rings or kiln ends of the air-cooled type are employed; the latter provides longer life for both kiln end and the brick ring. Efficient air seals are essential for the controlled and economical operation of kilns. They reduce outside air entrance; certain types effectively prevent entrance of all outside air. The simplest type of air seal is a floating T-section ring mounted on a wearing pad around the feed end of the kiln and free to slide with expansion of the kiln shell. The web of the T ring is confined within circular retainer plates. Figure 12-65 shows two arrangements. The floating-type discharge-end air seal consists of a circular bar which floats on a wearing pad and which can be moved to provide the desired operating clearance between air seal and firing hood. The floating ring and the fixed portion of these seals can be furnished with renewable wearing surfaces. Air infiltration through this type of seal is usually less than 10 percent (Allis-Chalmers Corporation). For further reduction of air infiltration, lantern-ring-type floating seals, pressurized with inert gas or stack gases, are employed. Accelerated drying of slurries in the feed end of rotary kilns in wet-process operations is achieved by installation of hanging chains. Conveying spirals support suspended lengths of chain which are arranged in such a way that they form an effective pattern for drying. With the chain system, slurry is heated in three ways: by direct transfer from chains after suspension in hot gases, by lifting material into the path of hot gases, and by directing the flow of hot gas over the slurry bed in the space formed under the suspended chains. Frequently, the product forms into uniformly sized pellets which progress through the rest of the kiln in that form, resulting in improved heat transfer and reduced dust losses (Fig. 12-66). Design Methods for Rotary Kilns In rotary kilns, the material is not showered through the air stream but is retained in the lower part of the cylinder. Gas-solids contacting is much less efficient than in flighted units. Heat transfer is by radiation and convection from the flowing gas to the kiln brick and exposed bed surface and by radiation from the brick to the bed. For units employing separate combustion chambers, it can be assumed that at high temperatures the wall-film

(c)

(d)

Alternative kiln firing hoods. (a) Plain firing hood. (b) Hood for high temperatures. (c) Hood with enlarged combustion-air passage. (d) Hood with cooler connection. (Allis-Chalmers Corporation.)

FIG. 12-63

12-57

12-58

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

FIG. 12-64

Dryer firing hood with air-tempering chamber. (ABB Raymond/Bartlett-

Snow TM.)

1. Drying zone at feed end, when moisture is removed 2. Heating zone, where the charge is heated to the reaction temperature, i.e., the decomposition temperature for limestone or “burning” temperature for cement 3. Reaction zone, in which the charge is burned, decomposed, reduced, oxidized, etc. 4. Soaking zone, where the reacted charge is superheated or “soaked” at temperature or, if desired, cooled before discharge

(a)

(b)

Kiln-seal arrangements. (a) Single-floating-type feed end air seal. (b) Single-floating-type air seal on an air-cooled tapered feed end. (AllisChalmers Corp.) FIG. 12-65

FIG. 12-66

Kiln chain installation (patented). (Allis-Chalmers Corp.)

The rates of heat transfer in each zone will be different. Rotary kilns operate at various temperatures throughout their length. A graph of approximate gas and charge temperatures for wetprocess cement is shown in Fig. 12-67. The maximum charge temperature is 1700 to 1800 K; for the gases, 1800 to 1925 K. Overall heat-transfer rates have been estimated to be in the range of 25 to 60 KJ/(s⋅m3) on the basis of total kiln volume. Some commercial performance data for cement and lime kilns are shown in Table 12-20. Some of the other major uses of direct rotary kilns are in the following processes: Roasting. Rotary kilns are used for oxidizing and driving off sulfur and arsenic from various ores, including gold, silver, iron, etc. Temperatures employed will vary from 800 to 1600 K. Chloridizing. Silver ores are chloridized successfully in rotary kilns. Temperatures must be closely controlled between 1030 and 1090 K. Black ash. Barium sulfide (BaS) is produced by calcining a mixture of barite (BaSO4) and carbon at a temperature of 1350 K in continuous rotary kilns. Spodumene. A mixture of quartz, feldspar, and spodumene is being calcined in rotary kilns at 1475 K to produce lithium aluminum silicate. Vermiculite. A micaceous mineral is roasted to cause exfoliation for use as an insulating material. Revivification. Temperatures of 800 to 1030 K are used to revivify fuller’s and diatomaceous earth, although for some earths lower temperatures are employed.

resistance to convection heat transfer from the gas to the brick is limiting and that at any point the bed temperature approaches the wall temperature. Hence, the effective heat-transfer area is the inner kiln surface. For kilns under these conditions, the following empirical relationship is recommended for the convection heat-transfer coefficient from gas to brick: Us = 23.7G0.67

(12-57)

where Us = heat-transfer coefficient, J/(m2⋅s⋅K) [(Btu/(h⋅ft2 kiln surface⋅°F)]; and G = gas mass flow rate, kg /(s⋅m2 kiln cross section) [lb/(h⋅ft2)]. Equation (12-57) does not account for gas radiation at high temperature when the kiln charge can “see” the burner flame; hence, the method will yield a conservative design. When a kiln is fired internally, the major source of heat transfer is radiation from the flame and hot gases. This occurs directly to both the solids surface and the wall, and from the latter to the product by reradiation (with some conduction). Generally, a dry-feed kiln will have three zones of heating, and a wet-feed kiln will have four:

FIG. 12-67

Temperatures in rotary kiln on wet-process cement.

SOLIDS-DRYING EQUIPMENT TABLE 12-20

12-59

Typical Rotary-Kiln Installations* Nominal 24-hr. capacities‡ Usual No. of supports

Range of motor hp. to operate†

Dry process

Wet process

Lime sludge

Limestone

5 × 80 ft. 6 × 70 ft. 7 × 70 ft. 5 ft. 6 in. × 180 ft. 7 × 120 ft.

2 2 2 4 2

5–7.5 7.5–15 15–20 15–20 15–25

140 190 275 285 475

100 135 200 250 340

10 15 20 30 35

16 24 35 45 55

7 ft. 6 in. × 125 ft. 6 × 220 ft. 8 × 140 ft. 9 × 160 ft. 8 ft. 6 in. × 185 ft.

2 4 2 2 4

20–30 20–30 25–30 30–50 30–50

575 420 750 1100 1125

415 375 540 800 810

40 45 55 80 80

70 65 90 130 135

10 × 150 ft. 10 × 175 ft. 8 × 300 ft. 7 ft. 6 in. × 8 ft. 6 in. × 320 ft. 7 ft. 6 in. × 10 ft. × 8 ft. 6 in. × 300 ft.

2 2 5 5 5

40–75 50–75 50–75 50–75 50–75

1300 1500 1150 1175 1175

950 1100 1000 1020 1020

— — 110 115

145 155 160 165

10 × 11 × 175 ft. 10 ft. 6 in. × 185 ft. 11 × 175 ft. 8 ft. 6 in. × 10 ft. × 8 ft. 6 in. × 300 ft. 8 × 10 × 300 ft.

2 2 2 5 5

50–75 50–75 60–100 50–75 50–75

1650 1800 1850 1400 1425

1200 1300 1375 1200 1225

120 130 —

180 190 205

140

200

9 ft. 6 in. × 265 ft. 9 × 10 ft. 6 in. × 9 ft. × 325 ft. 10 ft. 6 in. × 250 ft. 9 ft. 6 in. × 11 ft. × 9 ft. 6 in. × 300 ft. 10 × 300 ft.

4 5 4 5 5

60–100 60–100 60–100 60–100 75–125

1500 1700 1750 1800 1900

1300 1500 1525 1550 1650

150

215

175

240

190

250

9 ft. 6 in. × 11 ft. × 9 ft. 6 in. × 375 ft. 11 × 300 ft. 11 ft. 6 in. × 300 ft. 10 ft. 6 in. × 375 ft. 11 ft. 3 in. × 360 ft.

6 5 4 5 5

75–125 75–125 100–150 100–150 125–175

2025 2400 2600 2700 2900

1800 2100 2250 2400 2500

225 240 250 275

300 320 325 350

11 ft. 6 in. × 475 ft. 12 × 500 ft.

7 8

150–250 200–300

4000 4600

3500 4000

375 425

450 500

Size, diam. × length

Portland cement, 376-lb. bbl.

Lime, net tons

*Allis-Chalmers Manufacturing Co. †Power requirements vary according to size of kiln, character of material handled, and method of operation. ‡Capacities indicated are conservative, and apply to normal operation at sea level. Corrections would apply at increased altitudes, and for differing methods of operation.

Zinc. Oxidized ores are calcined to drive off water of hydration and carbon dioxide. The sulfide ore is always roasted before smelting. Titanium oxide (TiO2). This is produced from ilmenite ore by mixing ore with carbon and heating in a rotary kiln. Also, the rotary kiln is used in the process of recovery of titanium oxide from hydrated titanium precipitate at about 1250 K. Roofing granules. Crushed quartz or sand of definite size is treated with various minerals, borax, soda ash, etc., and calcined at temperatures ranging from 1250 to 1600 K. Glass of different colors forms on the surface of the granules at various temperatures. An oxidizing or reducing flame is used to influence the final coloring. Alumina (Al2O3). Alumina is produced by calcining either bauxite or aluminum hydroxide in rotary kilns at temperatures from 1250 to 1600 K. In obtaining the highest-purity alumina, the bauxite is digested with alkali to remove impurities; the resultant aluminum hydroxide [Al2(OH)3], of approximately 200-mesh size, is then calcined in rotary kilns at 1350 K. Potassium salts. In this operation, potassium chloride (KCl) is introduced to the rotary kiln at a fineness of minus 100 mesh and containing 9 percent water. The salt is brought to the fusion temperature of 1048 K. Magnesium oxide. The natural minerals, i.e., magnesite (MgCO3), brucite [Mg(OH)2], etc., after being crushed to predetermined size, are calcined at temperatures varying from 1055 to 2000 K, depending upon whether a caustic or a dead-burned product (periclase) is being

produced. Magnesium hydroxide, recovered from seawater or salt brine, is also being treated in a similar manner except that it is added in the form of a sludge. Sodium aluminum sulfate. This product is now being successfully calcined in rotary kilns. In this process, the salt cake is broken up just before it enters the kiln. Calcination is for the purpose of driving off the combined water (45 percent) and sulfuric acid (3 percent). Temperatures employed are approximately 800 K. Phosphate rock. In this application, the rotary kiln is used to nodulize the fines in the ore and prepare them for electric-furnace operation. Ore under 5 cm in size and containing 50 percent or more minus 100 mesh is calcined. Ore nodulizes at approximately 1475 to 1500 K. Mercury. In recovering mercury from cinnabar ores, the ore is crushed to minus 1.5 cm and fed to rotary kilns, where it is calcined to over 800 K. Since the mercury exists as mercuric sulfide (HgS), the sulfur is oxidized to SO2 and the mercury vaporized. The gases are passed through cooling chambers, where the mercury condenses and is collected. Mercury vaporizes at 625 K. Gypsum. The rotary kiln is rapidly replacing the kettle in producing plaster of paris. Great care is required, as the temperatures for reaction are low and within narrow limits, 382 to 403 K. Gypsum (CaSO4⋅2H2O) is heated to drive off three-fourths of the water of crystallization to produce plaster of paris [(CaSO2)⋅H2O]. Any overheating drives off all the water, producing gypsite (CaSO4), which is unsatisfactory.

12-60

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

Clay. To produce lightweight aggregate for concrete, clay is calcined in rotary kilns. Temperatures employed vary from 1350 to 1600 K. The apparent density of the clay is reduced by 50 to 75 percent. Iron ores. Crushed iron ores are partially reduced in rotary kilns to obtain nodules which are used in blast-furnace charges. Manganese. Manganese ore, rhodochrosite, or manganese carbonate (MnCO2) is calcined at about 1525 K to produce the oxide (Mn3O4). When the oxide ore is available but is in a finely divided state, the rotary kiln is used only for nodulizing. Petroleum coke. In order to eliminate excess volatile matter, petroleum coke is calcined at temperatures of 1475 to 1525 K. This is a sensitive material, and temperature control is difficult to maintain. When it is desired to increase the capacity of an existing kiln installation, consideration should be given to the following changes: 1. Increase charge volume held in kiln. 2. Increase temperature and quantity of combustion gases. 3. Decrease quantity of air in excess of combustion needs. 4. Increase speed of rotation of kiln. 5. Install ring dams at intermediate and discharge points. 6. Increase capacity of feeding and discharge mechanisms. 7. Decrease moisture content of feed material. 8. Increase temperature of feed material. 9. Install chains or flights, etc., in feed end. 10. Preheat all combustion air. 11. Reduce leakage of cold air into kiln at hot end. 12. Increase stack draft by increasing height or by use of jets. 13. Install instrumentation to control the kiln at maximumcapacity conditions. The time of passage in rotary kilns (from which holdup can be calculated) can be estimated by the following formula (U.S. Bur. Mines Tech. Pap. 384, 1927): θ = 0.19L/NDS

(12-58)

where θ = time of passage in the kiln, min; L = kiln length, ft; N = rotational speed, r/min; S = slope of kiln, ft/ft; and D = diameter inside brick, ft. Other equations for estimating the time of passage employing internal dams and a discharge dam are given by Bayard [Chem. Metall. Eng., 52(3), 100–102 (1945)]. The total power required to drive a rotary kiln or a dryer with lifters can be calculated by the following formulas (courtesy of ABB Raymond). For a rotary kiln or calciner without lifters, N [18.85y (sin B)w + 0.1925DW + 0.33W] bhp = ᎏᎏᎏᎏᎏ (12-59) 100,000 For a rotary dryer or section of a kiln with lifters, N (4.75dw + 0.1925DW + 0.33W) bhp = ᎏᎏᎏᎏ (12-60) 100,000 where bhp = brake horsepower required (1 bhp = 0.75 kW); N = rotational speed, r/min; y = distance between centerline of kiln and the center of gravity of material bed, ft; B = angle of repose of material; W = total rotating load (equipment plus material), lb; w = live load (material), lb; D = riding-ring diameter, ft; and d = shell diameter, ft. For estimating purposes, let D = (d + 2). Drive motors should be of the high-starting-torque type and selected for 1.33 times maximum rotational speed. For two- or threediameter kilns, the brake horsepower for the several diameters should be calculated separately and summed. Auxiliary drives should be provided to maintain shell rotation in the event of power failure. These are usually gasoline or diesel engines. Thermal Efficiency of Rotary Kilns Kiln length is a major factor in determining thermal efficiency, and kilns with a high ratio of length to diameter have a greater thermal efficiency than those with a low ratio. The use of chains inside the kiln and of heat-recovery equipment on the gases and product leaving the kiln can increase substantially the thermal efficiency of a kiln installation. Efficiencies ranging from 45 to more than 80 percent have been reported. A reasonably satisfactory range based on present fuel prices and construction costs would be 65 to 75 percent utilization and recovery of the heat content of the fuel plus any heat of reaction of the charge. No distinction is made from an efficiency-calculation standpoint between the heat uti-

lized in the kiln and that recovered (or utilized) outside the kiln. With countercurrent flow of the combustion gases and the charge material, an exceptionally long kiln will give high efficiencies within itself. However, good economics may dictate that a shorter kiln be installed with a waste-heat boiler on the hot gases to obtain an equivalent thermal efficiency at a lower investment. The heat in the hot product usually is recovered as preheat in the combustion air. Size Segregation in Kilns When an assemblage of solid particles, not very closely screened, is rotated within a cylinder, the solids assume a lunar shape, as shown in Fig. 12-68. This causes serious size segregation. The finest sizes remain at the bottom, in contact with the hot brick. The coarser particles form the upper layer of the agitated mass. As the kiln completes a revolution, the exposed brick, in an upper position, absorbs sensible heat from the gas mass. As the heated brick completes its circuit, it passes under and is in conductive contact with the fine particles. These fines are thus effectively heated by direct solid-to-solid transfer. The larger particles are heated by direct radiation from gas and brick, and become adequately calcined. The particles of size intermediate between the fine and coarse remain, throughout a complete revolution, “sandwiched” between the coarse and fine layers and are protected from heat by the excellent insulation properties of these layers, thus perhaps escaping complete calcination. This factor of segregation is offset by some kiln operators who classify or screen the kiln feed so that only a narrow range of particle size is fed at one time. Also, faster kiln speeds which give a better agitation of the charge are used. Rotary kilns are usually operated with between 3 and 12 percent of their volume filled with material; 7 percent is considered normal. Cost Data for Kilns Purchase prices, weights, and horsepower requirements of typical units are given in Table 12-21. Installed costs will run to from 300 to 500 percent of purchase cost. Maintenance will average 5 to 10 percent of the total installed cost per year but is dependent largely on the life of the refractory lining. For estimating purposes refractory lining for a 2.7- to 3.4-mdiameter kiln costs $6000 to $15,000 per meter of kiln length installed (50 percent material, 50 percent labor). A discussion of retention time in rotary kilns is given in Brit. Chem. Eng., 27–29 ( January 1966). Rotary-kiln heat control is discussed in detail by Bauer [Chem. Eng., 193–200 (May 1954)] and Zubrzycki [Chem. Can., 33–37 (February 1957)]. Reduction of iron ore in rotary kilns is described by Stewart [Min. Congr. J., 34–38 (December 1958)]. The use of balls to improve solids flow is discussed in [Chem. Eng., 120–222 (March 1956)]. Brisbane examined problems of shell deformation [(Min. Eng., 210–212 (February 1956)]. Instrumentation is discussed by Dixon [Ind. Eng. Chem. Process Des. Dev., 1436–1441 ( July 1954)], and a mathematical simulation of a rotary kiln was developed by Sass [Ind. Eng. Chem. Process Des. Dev., 532–535 (October 1967)]. This last paper employed the empirical convection heattransfer coefficient given previously, and its use is discussed in later correspondence [ibid., 318–319 (April 1968)]. Indirect-Heat Rotary Steam-Tube Dryers Probably the most common type of indirect-heat rotary dryer is the steam-tube dryer (Fig. 12-69). Steam-heated tubes running the full length of the cylinder are fastened symmetrically in one, two, or three concentric rows inside the cylinder and rotate with it. Tubes may be simple pipe with condensate draining by gravity into the discharge manifold or bayonet-type. Bayonet-type tubes are also employed when units are used

FIG. 12-68

Size segregation of solids in a rotary kiln.

SOLIDS-DRYING EQUIPMENT TABLE 12-21 Approximate Purchase Costs and Weight of Rotary Kilns*

Kiln size, diameter × length

Total purchase price includes drive, burner, and controls (not including brick)†

8′0″ × 80′0″ 8′0″ × 140′0″ 8′0″ × 200′0″ 8′0″ × 300′0″ 9′0″ × 250′0″ 9′0″ × 300′0″ 10′0″ × 100′0″ 10′0″ × 150′0″ 10′0″ × 250′0″ 10′0″ × 300′0″ 10′0″ × 350′0″ 10′6″ × 175′0″ 10′6″ × 250′0″ 10′6″ × 350′0″ 11′0″ × 160′0″ 11′0″ × 250′0″ 11′0″ × 300′0″ 11′0″ × 350′0″ 11′0″ × 400′0″ 11′6″ × 160′0″ 11′6″ × 250′0″ 11′6″ × 350′0″ 11′6″ × 425′0″ 12′0″ × 250′0″ 12′0″ × 325′0″ 12′0″ × 400′0″ 12′0″ × 450′0″ 13′0″ × 500′0″ 14′0″ × 400′0″ 16′6″ × 600′0″

$ 448,000 600,000 960,000 1,240,000 1,373,000 1,545,000 682,000 965,000 1,502,000 1,682,000 1,779,000 1,182,000 1,677,000 1,942,000 1,213,000 1,670,000 1,858,000 2,344,000 2,544,000 1,251,000 1,768,000 2,393,000 2,676,000 1,837,000 2,598,000 2,645,000 3,570,000 4,388,000 4,155,000 8,190,000

*Courtesy of Fuller Co. †Prices for January 1982

as water-tube coolers. When handling sticky materials, one row of tubes is preferred. These are occasionally shielded at the feed end of the dryer to prevent buildup of solids behind them. Lifting flights are usually inserted behind the tubes to promote solids agitation. Wet feed enters the dryer through a chute or screw feeder. The

FIG. 12-69

Steam-tube rotary dryer.

12-61

product discharges through peripheral openings in the shell in ordinary dryers. These openings also serve to admit purge air to sweep moisture or other evolved gases from the shell. In practically all cases, gas flow is countercurrent to solids flow. To retain a deep bed of material within the dryer, normally 10 to 20 percent fillage, the discharge openings are supplied with removable chutes extending radially into the dryer. These, on removal, permit complete emptying of the dryer. Steam is admitted to the tubes through a revolving steam joint into the steam side of the manifold (Fig. 12-70). Condensate is removed continuously, by gravity through the steam joint to a condensate receiver and my means of lifters in the condensate side of the manifold. By employing simple tubes, noncondensables are continuously vented at the other ends of the tubes through Sarco-type vent valves mounted on an auxiliary manifold ring, also revolving with the cylinder. Vapors (from drying) are removed at the feed end of the dryer to the atmosphere through a natural-draft stack and settling chamber or wet scrubber. When employed in simple drying operations with 3.5 × 105 to 10 × 105 Pa steam, draft is controlled by a damper to admit only sufficient outside air to sweep moisture from the cylinder, discharging the air at 340 to 365 K and 80 to 90 percent saturation. In this way, shell gas velocities and dusting are minimized. When used for solvent recovery or other processes requiring a sealed system, sweep gas is recirculated through a scrubber-gas cooler and blower. Steam manifolds for pressures up to 10 × 105 Pa are of cast iron. For higher pressures, the manifold is fabricated from plate steel, staybolted, and welded. The tubes are fastened rigidly to the manifold face plate and are supported in a close-fitting annular plate at the other end to permit expansion. Packing on the steam neck is normally graphite-asbestos. Ordinary rotating seals are similar in design to those depicted in Fig. 12-61, with allowance for the admission of small quantities of outside air when the dryer is operated under a slight negative internal pressure. Steam-tube dryers are used for the continuous drying, heating, or cooling of granular or powdery solids which cannot be exposed to ordinary atmospheric or combustion gases. They are especially suitable for fine dusty particles because of the low gas velocities required for purging of the cylinder. Tube sticking is avoided or reduced by employing recycle, shell knockers, etc., as previously described; tube scaling by sticky solids is one of the major hazards to efficient operation. The dryers are suitable for drying, solvent recovery, and chemical reactions. Steam-tube units have found effective employment in

12-62

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

(a)

FIG. 12-71

Effect of steam pressure on the heat-transfer rate in steam-tube

dryers.

(b) Rotary steam joint for a standard steam-tube dryer. (a) Section of cast-steam manifold. (b) Section of manifold and steam joint.

FIG. 12-70

soda-ash production, replacing more expensive indirect-heat rotary calciners. Special types of steam-tube dryers employ packed and purged seals on all rotating joints, with a central solids-discharge manifold through the steam neck to reduce the seal diameter. This manifold contains the product discharge conveyor and a passage for the admission of sweep gas. Solids are removed from the shell by special volute lifters and dropped into the discharge conveyor. Units have been fabricated for operation at 76 mm of water, internal shell pressure, with no detectable air leakage. Design Methods for Indirect-Heat Rotary Steam-Tube Dryers Heat-transfer coefficients in steam-tube dryers range from 30 to 85 J/(m2⋅s⋅K). Coefficients will increase with increasing steam temperature because of increased heat transfer by radiation. In units carrying saturated steam at 420 to 450 K, the heat flux U∆T will range from 6300 J/(m2⋅s) for difficult-to-dry and organic solids and to 1890 to 3790 J/(m2⋅s) for finely divided inorganic materials. The effect of steam pressure on heat-transfer rates up to 8.6 × 105 Pa is illustrated in Fig. 12-71. Performance and Cost Data for Indirect-Heat Rotary SteamTube Dryers Table 12-22 contains data for a number of standard sizes of steam-tube dryers. Prices tabulated are for ordinary carbon steel construction. Installed costs will run from 150 to 300 percent of purchase cost. The thermal efficiency of steam-tube units will range from 70 to 90 percent, if a well-insulated cylinder is assumed. This does not allow for boiler efficiency, however, and is therefore not directly comparable with direct-heat units such as the direct-heat rotary dryer or indirect-heat calciner. Operating costs for these dryers include 5 to 10 percent of one person’s time. Maintenance will average 5 to 10 percent of total installed cost per year. Table 12-23 outlines typical performance data from three drying applications in steam-tube dryers. Indirect-Heat Calciners Indirect-heat rotary calciners, either batch or continuous, are employed for heat treating and drying at higher

temperatures than can be obtained in steam-heated rotating equipment. They generally require a minimum flow of gas to purge the cylinder, to reduce dusting and are suitable for gas-sealed operation with oxidizing, inert, or reducing atmospheres. Indirect calciners are widely utilized and some examples of specific applications are as follows: 1. Activating charcoal 2. Reducing mineral high oxides to low oxides 3. Drying and devolatilizing contaminated soils and sludges 4. Calcination of alumina oxide-based catalysts 5. Drying and removal of sulfur from cobalt, copper, and nickel 6. Reduction of metal oxides in a hydrogen atmosphere 7. Oxidizing and “burning off ” of organic impurities 8. Calcination of ferrites This unit consists essentially of a cylindrical retort, rotating within a stationary insulation lined furnace. The latter is arranged so that fuel combustion occurs within the annular ring between the retort and the furnace. The retort cylinder extends beyond both ends of the furnace. These end extensions carry the riding rings and drive gear. Material may be fed continuously at one end and discharged continuously at the other. Feeding and solids discharging are usually accomplished with screw feeders or other positive feeders to prevent leakage of gases into or out of the calciner. In some cases in which it is desirable to cool the product before removal to the outside atmosphere, the discharge end of the cylinder is provided with an additional extension, the exterior of which is waterspray-cooled. In cocurrent-flow calciners, hot gases from the interior of the heated portion of the cylinder are withdrawn through a special extraction tube. This tube extends centrally through the cooled section to prevent flow of gas near the cooled-shell surfaces and possible condensation. Frequently a separate cooler is used, isolated from the calciner by an air lock. To prevent sliding of solids over the smooth interior of the shell, agitating flights running longitudinally along the inside wall are frequently provided. These normally do not shower the solids as in a direct-heat vessel but merely prevent sliding so that the bed will turn over and constantly expose new surface for heat and mass transfer. To prevent scaling of the shell interior by sticky solids, cylinder scraper and knocker arrangements are occasionally employed. For example, a scraper chain is fairly common practice in soda-ash calciners, while knockers are frequently utilized on metallic-oxide calciners. Because indirect-heat calciners frequently require close-fitting gas seals, it is customary to support all parts on a self-contained frame, for sizes up to approximately 2 m in diameter. The furnace can employ electric heating elements or oil and/or gas burners as the heat source for the process. The hardware would be zoned down the length of the furnace to match the heat requirements of the process. Process control is normally by shell temperature, measured by thermocouples or radiation pyrometers. When a special gas atmosphere must be maintained inside the cylinder, positive rotary gas seals, with one or more pressurized and purged annular chambers, are employed. The diaphragm-type seal ABB Raymond (Bartlett-Snow TM) is suitable for pressures up to 5 cm of water, with no detectable leakage.

SOLIDS-DRYING EQUIPMENT TABLE 12-22

12-63

Standard Steam-Tube Dryers* Tubes

Size, diameter × length, m

No. OD (mm)

0.965 × 4.572 0.965 × 6.096 0.965 × 7.620 0.965 × 9.144 0.965 × 10.668 1.372 × 6.096 1.372 × 7.620 1.372 × 9.144 1.372 × 10.668 1.372 × 12.192 1.372 × 13.716 1.829 × 7.62 1.829 × 9.144 1.829 × 10.668 1.829 × 12.192 1.829 × 13.716 1.829 × 15.240 1.829 × 16.764 1.829 × 18.288 2.438 × 12.192 2.438 × 15.240 2.438 × 18.288 2.438 × 21.336 2.438 × 24.387

14 (114) 14 (114) 14 (114) 14 (114) 14 (114) 18 (114) 18 (114) 18 (114) 18 (114) 18 (114) 18 (114) 27 (114) 27 (114) 27 (114) 27 (114) 27 (114) 27 (114) 27 (114) 27 (114) 90 (114) 90 (114) 90 (114) 90 (114) 90 (114)

No. OD (mm)

18 (63.5) 18 (63.5) 18 (63.5) 18 (63.5) 18 (63.5) 18 (63.5) 27 (76.2) 27 (76.2) 27 (76.2) 27 (76.2) 27 (76.2) 27 (76.2) 27 (76.2) 27 (76.2)

m2 of free area

Dryer speed, r/min

Motor size, hp

Shipping weight, kg

Estimated price

21.4 29.3 36.7 44.6 52.0 58.1 73.4 88.7 104 119 135 118 143 167 192 217 242 266 291 394 492 590 689 786

6 6 6 6 6 4.4 4.4 5 5 5 5.5 4 4 4 4 4 4 4 4 3 3 3 3 3

2.2 2.2 3.7 3.7 3.7 3.7 3.7 5.6 5.6 5.6 7.5 5.6 5.6 7.5 7.5 11.2 11.2 14.9 14.9 11.2 14.9 14.9 22.4 29.8

5,500 5,900 6,500 6,900 7,500 10,200 11,100 12,100 13,100 14,200 15,000 19,300 20,600 22,100 23,800 25,700 27,500 29,300 30,700 49,900 56,300 63,500 69,900 75,300

$152,400 165,100 175,260 184,150 196,850 203,200 215,900 228,600 243,840 260,350 273,050 241,300 254,000 266,700 278,400 292,100 304,800 317,500 330,200 546,100 647,700 736,600 838,200 927,100

*Courtesy of Swenson Process Equipment Inc. (prices from November, 1994). Carbon steel fabrication; multiply by 1.75 for 304 stainless steel. TABLE 12-23

Steam-Tube Dryer Performance Data Class 1

Class 2

Class 3

Class of materials handled

High-moisture organic, distillers’ grains, brewers’ grains, citrus pulp

Pigment filter cakes, blanc fixe, barium carbonate, precipitated chalk

Description of class

Wet feed is granular and damp but not sticky or muddy and dries to granular meal 233

Wet feed is pasty, muddy, or sloppy; product is mostly hard pellets 100

Finely divided inorganic solids, water-ground mica, waterground silica, flotation concentrates Wet feed is crumbly and friable; product is powder with very few lumps 54

11

0.15

0.5

310–320 350–355 2 2250 860

280–290 380–410 1 1190 860

280–290 365–375 0.53 625 860

0.34

0.4

3.33

1.72

Normal moisture content of wet feed, % dry basis Normal moisture content of product, % dry basis Normal temperature of wet feed, K Normal temperature of product, K Evaporation per product, kg Heat load per lb product, kJ Steam pressure normally used, kPa gauge Heating surface required per kg product, m2 Steam consumption per kg product, kg

In general, the temperature range of operation for indirect-heated calciners can vary over a wide range, from 475 K at the low end to approximately 1475 K at the high end. All types of carbon steel, stainless, and alloy construction are used, depending upon temperature, process, and corrosion requirements. Fabricated-alloy cylinders can be used over the greater part of the temperature range; however, the greater creep-stress abilities of cast alloys makes their use desirable for the highest calciner-cylinder temperature applications. Design Methods for Calciners In indirect-heated calciners, heat transfer is primarily by radiation from the cylinder wall to the solids bed. The thermal efficiency ranges from 30 to 65 percent. By utilization of the furnace exhaust gases for preheated combustion air, steam production, or heat for other process steps, the thermal efficiency can be increased considerably. The limiting factors in heat transmission lie in the conductivity and radiation constants of the shell metal and solids bed. If the characteristics of these are known, equipment may be accurately sized by employing the Stefan-Boltzmann radiation equation. Apparent heat-transfer coefficients will range from 17 J/(m2⋅s⋅K) in low-temperature operations to 85 J/(m2⋅s⋅K) in high-temperature processes.

0.072 0.85

Cost Data for Calciners Power, operating, and maintenance costs are similar to those previously outlined for direct- and indirectheat rotary dryers. Estimating purchase costs for preassembled and frame-mounted rotary calciners with carbon steel and type 316 stainless-steel cylinders are given in Table 12-24 together with size, weight, and motor requirements. Sale price includes the cylinder, ordinary angle seals, furnace, drive, feed conveyor, burners, and controls. Installed cost may be estimated, not including building or foundation costs at up to 50 percent of the purchase cost. A layout of a typical continuous calciner with an extended cooler section is illustrated in Fig. 12-72. Small batch retorts, heated electrically or by combustion, are widely used as carburizing furnaces and are applicable also to chemical processes involving the heat treating of particulate solids. These are mounted on a structural-steel base, complete with cylinder, furnace, drive motor, burner, etc. Units are commercially available in diameters from 0.24 to 1.25 m and lengths of 1 to 2 m. Continuous retorts with helical internal spirals are employed for metal-heat-treating purposes. Precise retention control is maintained in these operations. Standard diameters are 0.33, 0.5, and 0.67 m with effective lengths up

12-64

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

TABLE 12-24

Indirect-Heat Rotary Calciners: Sizes and Purchase Costs*

Diameter, ft

Overall cylinder length

Heated cylinder length

Cylinder drive motor hp

Approximate Shipping weight, lb

Approximate sale price in carbon steel construction†

Approximate sale price in No. 316 stainless construction

4 5 6 7

40 ft 45 ft 50 ft 60 ft

30 ft 35 ft 40 ft 50 ft

7.5 10 20 30

50,000 60,000 75,000 90,000

$275,000 375,000 475,000 550,000

$325,000 425,000 550,000 675,000

* ABB Raymond (Bartlett-Snow™). † Prices for November, 1994.

FIG. 12-72 Gas-fired indirect-heat rotary calciner with a water-spray extended cooler and feeder assembly. (ABB Raymond/ Bartlett-Snow™.)

to 3 m. These vessels are employed in many small-scale chemicalprocess operations which require accurate control of retention. Their operating characteristics and applications are identical to those of the larger indirect-heat calciners. Direct-Heat Roto-Louvre Dryer One of the more important special types of rotating equipment is the Roto-Louvre dryer. As illustrated in Fig. 12-73, hot air (or cooling air) is blown through louvers in a double-wall rotating cylinder and up through the bed of solids. The latter moves continuously through the cylinder as it rotates. Constant turnover of the bed ensures uniform gas contacting for heat and mass transfer. The annular gas passage behind the louvers is partitioned so that contacting air enters the cylinder only beneath the solids bed. The number of louvers covered at any one time is roughly 30 percent. Because air circulates through the bed, fillages of 13 to 15 percent or greater are employed. Roto-Louvre dryers range in size from 0.8 to 3.6 m in diameter and from 2.5 to 11 m long. The largest unit is reported capable of evaporating 5500 kg/h of water. Hot gases from 400 to 865 K may be

FIG. 12-73 Link-Belt Roto-Louvre dryer. (Material Handling Systems Division, FMC Corp.)

employed. Because gas flow is through the bed of solids, high pressure drop, from 7 to 50 cm of water, may be encountered within the shell. For this reason, both a pressure inlet fan and an exhaust fan are provided in most applications to maintain the static pressure within the equipment as closely as possible to atmospheric. This prevents excessive in-leakage or blowing of hot gas and dust to the outside. For pressure control, one fan is usually operated under fixed conditions, with an automatic damper control on the other, regulated by a pressure detector-controller. In heating or drying applications, when cooling of the product is desired before discharge to the atmosphere, cool air is blown through a second annular space, outside the inlet hot-air annulus, and released through the louvers at the solids-discharge end of the shell. Roto-Louvre dryers are suitable for processing coarse granular solids which do not offer high resistance to air flow, require intimate gas contacting, and do not contain significant quantities of dust. Heat and mass transfer from the gas to the surface of the solids is extremely efficient; hence the equipment size required for a given duty is frequently less than required when an ordinary direct-heat rotary vessel with lifting flights is used. Purchase-price savings are partially balanced, however, by the more complex construction of the Roto-Louvre unit. A Roto-Louvre dryer will have a capacity roughly 1.5 times that of a single-shell rotary dryer of the same size under equivalent operating conditions. Because of the cross-flow method of heat exchange, the average ∆t is not a simple function of inlet and outlet ∆t’s. There are currently no published data which permit the sizing of equipment without pilot tests as recommended by the manufacturer. Three applications of Roto-Louvre dryers are outlined in Table 12-25. Installation, operating, power, and maintenance costs will be similar to those experienced with ordinary direct-heat rotary dryers. Thermal efficiency will range from 30 to 70 percent. AGITATED DRYERS Description An agitated dryer is defined as one on which the housing enclosing the process is stationary while solids movement is accomplished by an internal mechanical agitator. Many forms are in use, including batch and continuous versions. Field of Application Agitated dryers are applicable to process-

SOLIDS-DRYING EQUIPMENT TABLE 12-25 Manufacturer’s Performance Data for FMC LinkBelt Roto-Louvre Dryers* Material dried Dryer diameter Dryer length Moisture in feed, % wet basis Moisture in product, % wet basis Production rate, lb/h Evaporation rate, lb/h Type of fuel Fuel consumption Calorific value of fuel Efficiency, Btu, supplied per lb evaporation Total power required, hp

Ammonium sulfate

Foundry sand

Metallurgical coke

2 ft 7 in 10 ft 2.0

6 ft 4 in 24 ft 6.0

10 ft 3 in 30 ft 18.0

0.1

0.5

0.5

2500 50 Steam 255 lb/h 837 Btu/lb 4370

32,000 2130 Gas 4630 ft3/h 1000 Btu/ft3 2170

38,000 8110 Oil 115 gal/h 150,000 Btu/gal 2135

4

41

78

*Material Handling Systems Division, FMC Corp. To convert British thermal units to kilojoules, multiply by 1.06; to convert horsepower to kilowatts, multiply by 0.746.

ing solids which are relatively free-flowing and granular when discharged as product. Materials which are not free-flowing in their feed condition can be treated by recycle methods as described in the subsection “Rotary Dryers.” In general, agitated dryers have applications similar to those of rotating vessels. Their chief advantages compared with the latter lie in the fact that (1) large-diameter rotary seals are not required at the solids and gas feed and exit points because the housing is stationary, and for this reason gas-leakage problems are minimized. Rotary seals are required only at the points of entrance of the mechanical agitator shaft. (2) Use of a mechanical agitator for solids mixing introduces shear forces which are helpful for breaking up lumps and agglomerates. Balling and pelleting of sticky solids, an occasional occurrence in rotating vessels, can be prevented by special agitator design. The problems concerning dusting of fine particles in directheat units are identical to those discussed under “Rotary Dryers.”

FIG. 12-74

12-65

Batch Vacuum Rotary Dryers The more common type of vacuum rotary dryer consists of a stationary cylindrical shell, mounted horizontally, in which a set of agitator blades mounted on a revolving central shaft stirs the solids being treated. Heat is supplied by circulation of hot water, steam, or Dowtherm through a jacket surrounding the shell and, in larger units, through the hollow central shaft. The agitator is either a single discontinuous spiral or a double continuous spiral. The outer blades are set as closely as possible to the wall without touching, usually leaving a gap of 0.3 to 0.6 cm. Modern units occasionally employ spring-loaded shell scrapers mounted on the blades. The dryer is charged through a port at the top and emptied through one or more discharge nozzles at the bottom. Vacuum is applied and maintained by any of the conventional methods, i.e., steam jets, vacuum pumps, etc. Another type of vacuum rotary dryer consists of a rotating horizontal cylindrical shell, suitably jacketed. Vacuum is applied to this unit through hollow trunnions with suitable packing glands. Rotary glands must be used also for admitting and removing the heating medium from the jacket. The inside of the shell may have lifting bars, welded longitudinally, to assist agitation of the solids. The double-cone rotating vacuum dryer is a more common design. Although it is identical in operating design, the sloping walls of the cones permit more rapid emptying of solids when the dryer is in a stationary position. The older cylinder shape required continuous rotation during emptying to convey product to the discharge nozzles. As a result, a circular dust hood was frequently necessary to enclose the discharge-nozzle turning circle and prevent serious dust losses to the atmosphere during unloading. Several new designs of the doublecone type employ internal tubes or plate coils to provide additional heating surface. On all rotating dryers, the vapor-outlet tube is stationary; it enters the shell through a rotating gland and is fitted with an elbow and an upward extension so that the vapor inlet, usually protected by a felt dust filter, will be at all times near the top of the shell. A typical vacuum rotary dryer is illustrated in Fig. 12-74 and a double-cone vacuum dryer in Fig. 12-75. Vacuum is used in conjunction with drying or other chemical oper-

A typical vacuum dryer. (Blaw-Knox Food & Chemical Equipment, Inc.)

12-66

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

FIG. 12-75

Rotating (double-cone) vacuum dryer. (Stokes Vacuum, Inc.)

ations when low solids temperatures must be maintained because heat will cause damage to the product or change its nature, when air combines with the product as it is heated, causing oxidation or an explosive condition, when solvent recovery is required, and when materials must be dried to extremely low moisture levels. In vacuum processing and drying the objective is to create a large temperature-driving force between the jacket and the product. To accomplish this purpose at fairly low jacket temperatures, it is necessary to reduce the internal process pressure so that the liquid being removed will boil at a lower vapor pressure. It is not always economical, however, to reduce the internal pressure to extremely low levels because of the large vapor volumes thereby created. It is necessary to compromise on operating pressure, considering leakage, condensation problems, and the size of the vapor lines and pumping system. Very few vacuum dryers operate below 5 mmHg pressure on a commercial scale. Air in-leakage through gasket surfaces will be in the range of 0.2 kg/(h⋅linear m of gasketed surface) under these conditions. Design Methods for Batch Vacuum Rotary Dryers The rate of heat transfer from the heating medium through the dryer wall to the solids can be expressed by Q = UA∆ tm

(12-61)

where Q = heat flux, J/s (Btu/h); U = overall heat-transfer coefficient, J/(m2⋅s⋅K) [Btu/(h⋅ft2 jacket area⋅°F)]; A = total jacket area, m2; and

TABLE 12-26

∆ tm = log-mean-temperature driving force from heating medium to the solids, K. The overall heat-transfer rate is almost entirely dependent upon the film coefficient between the inner jacket wall and the solids, which depends to a large extent on the solids characteristics. Overall coefficients may range from 30 to 200 J/(m2⋅s⋅K), based upon total area if the dryer walls are kept reasonably clean. Coefficients as low as 5 or 10 may be encountered if caking on the walls occurs. For estimating purposes without tests, a reasonable coefficient for ordinary drying, and without taking the product to absolute dryness, may be assumed at U = 50 J/(m2⋅s⋅K) for rotary agitator dryers and 35 J/(m2⋅s⋅K) for rotating units. Vacuum dryers are usually filled to 50 to 65 percent of their total shell volume. Agitator speeds range from 3 to 8 r/min. Faster speeds yield a slight improvement in heat transfer but consume more power. Performance and Cost Data for Batch Vacuum Rotary Dryers Typical performance data for vacuum rotary dryers are given in Table 12-26. Size and cost data for rotary agitator units are given in Table 12-27. Data for double-cone units are in Table 12-28. Turbo-Tray Dryers The turbo-tray dryer is a continuous dryer consisting of a stack of rotating annular shelves in the center of which turbo-type fans revolve to circulate the air over the shelves. Wet material enters through the roof, falling onto the top shelf as it rotates beneath the feed opening. After completing one revolution, the mate-

Performance Data of Vacuum Rotary Dryers*

Material

Diameter × length, m

Initial moisture, % dry basis

Cellulose acetate Starch Sulfur black Fuller’s earth/mineral spirit

1.5 × 9.1 1.5 × 9.1 1.5 × 9.1 0.9 × 3.0

87.5 45–48 50 50

Steam pressure, Pa × 103

Agitator speed, r/min

Batch dry weight, kg

Final moisture, % dry basis

Pa × 103

Time, h

Evaporation, kg/(h⋅m2)

97 103 207 345

5.25 4 4 6

610 3630 3180 450

6 12 1 2

90–91 88–91 91 95

7 4.75 6 8

1.5 7.3 4.4 5.4

*Stokes Vacuum, Inc. TABLE 12-27 Diameter, m 0.46 0.61 0.91 0.91 1.2 1.5 1.5

Standard Rotary Vacuum Dryers* Purchase price (1995)

Length, m

Heating surface, m2

Working capacity, m3†

Agitator speed, r/min

Drive, kW

Weight, kg

Carbon steel

Stainless steel (304)

0.49 1.8 3.0 4.6 6.1 7.6 9.1

0.836 3.72 10.2 15.3 29.2 48.1 57.7

0.028 0.283 0.991 1.42 3.57 6.94 8.33

7a 7a 6 6 6 6 6

1.12 1.12 3.73 3.73 7.46 18.7 22.4

540 1,680 3,860 5,530 11,340 15,880 19,050

$ 43,000 105,000 145,000 180,000 270,000 305,000 330,000

$ 53,000 130,000 180,000 205,000 380,000 440,000 465,000

*Stokes Vacuum, Inc. Prices include shell, 50-lb/in2-gauge jacket, agitator, drive, and motor; auxiliary dust collectors, condensers. †Loading with product level on or around the agitator shaft.

SOLIDS-DRYING EQUIPMENT TABLE 12-28

12-67

Standard (Double-Cone) Rotating Vacuum Dryers* Purchase cost (1995)

Working capacity, m3

Total volume, m3

Heating surface, m2

Drive, kW

Floor space, m2

Weight, kg

Carbon steel

Stainless steel

0.085 0.283 0.708 1.42 2.83 4.25 7.08 9.20 11.3

0.130 0.436 1.09 2.18 4.36 6.51 10.5 13.9 16.0

1.11 2.79 5.30 8.45 13.9 17.5 *38.7 *46.7 *56.0

.373 .560 1.49 3.73 7.46 11.2 11.2 11.2 11.2

2.60 2.97 5.57 7.15 13.9 14.9 15.8 20.4 26.0

730 910 1810 2040 3860 5440 9070 9980 10,890

$ 32,400 37,800 50,400 97,200 198,000 225,000 324,000 358,000 378,000

$ 38,000 43,000 57,000 106,000 216,000 243,000 351,000 387,000 441,000

*Stokes Vacuum, Inc. Price includes dryer, 15-lb/in2 jacket, drive with motor, internal filter, and trunnion supports for concrete or steel foundations. Horsepower is established on 65 percent volume loading of material with a bulk density of 50 lb/ft3. Models of 250 ft3, 325 ft3, and 400 ft3 have extended surface area.

rial is wiped by a stationary wiper through radial slots onto the shelf below, where it is spread into a uniform pile by a stationary leveler. The action is repeated on each shelf, with transfers occurring once in each revolution. From the last shelf, material is discharged through the bottom of the dryer (Fig. 12-76). The steel-frame housing consists of removable insulated panels for access to the interior. All bearings and lubricated parts are exterior to the unit with the drives located under the housing. Parts in contact with the product may be of steel or special alloy. The trays can be of any sheet material, such as enameled steel, asbestos-cement composition board, or plastic-glass laminates. The rate at which each fan circulates air can be varied by changing the pitch of the fan blades. In final drying stages, in which diffusion controls or the product is light and powdery, the circulation rate is considerably lower than in the initial stage, in which high evaporation rates prevail. In the majority of applications, air flows through the

FIG. 12-76

Turbo-tray dryer. (Wyssmont Company, Inc.)

dryer upward in counterflow to the material. In special cases, required drying conditions dictate that air flow be cocurrent or both countercurrent and cocurrent with the exhaust leaving at some level between solids inlet and discharge. A separate cold-air-supply fan is provided if the product is to be cooled before being discharged. By virtue of its vertical construction, the turbo-type tray dryer has a stack effect, the resulting draft being frequently sufficient to operate the dryer with natural draft. Pressure at all points within the dryer is maintained close to atmospheric, as low as 0.1, usually less than 0.5 mm of water. Most of the roof area is used as a breeching, lowering the exhaust velocity to settle dust back into the dryer. Heaters can be located in the space between the trays and the dryer housing, where they are not in direct contact with the product, and thermal efficiencies up to 3500 kJ/kg (1500 Btu/lb) of water evaporated can be obtained by reheating the air within the dryer. Steam is the usual heating medium. The high cost of heating electrically generally restricts its use to relatively small equipment. For materials which have a tendency to foul internal heating surfaces, an external heating system is employed. The turbo-tray dryer can handle materials from thick slurries [1 million (N⋅s)/m2 (100,000 cP) and over] to fine powders. It is not suitable for fibrous materials which mat or for doughy or tacky materials. Thin slurries can often be handled by recycle of dry product. Filterpress cakes are granulated before feeding. Thixotropic materials are fed directly from a rotary filter by scoring the cake as it leaves the drum. Pastes can be extruded onto the top shelf and subjected to a hot blast of air to make them firm and free-flowing after one revolution. The turbo-tray dryer is manufactured in sizes from package units 2 m in height and 1.5 m in diameter to large outdoor installations 20 m in height and 11 m in diameter. Tray areas range from 1 for laboratory dryers to 1675 m2 for large-scale production in a single unit. The number of shelves in a tray rotor varies according to space available and minimum rate of transfer required, from as few as 12 shelves to as many as 58 in the largest units. Standard construction permits operating temperatures up to 615 K, and high-temperature heaters permit operation at temperatures up to 925 K. A recent innovation has enabled TURBO-Dryers to operate with very low inert gas make-up. Wyssmont has designed a tank housing that is welded up around the internal structure rather than the column-and-gasket panel design that has been the Wyssmont standard for many years. In field-erected units, the customer does the welding in the field; in packaged units, the tank-type welding is done in the shop. The tank-type housing finds particular application for operation under positive pressure. On the standard design, doors with explosion latches and gang latch operators are used. In the tank-type design, tight-sealing manway-type openings permit access to the interior. Tank-type housing designs have been requested when drying solvent wet materials and for applications where the material being dried is highly toxic and certainty is required that no toxic dust get out. Design Methods for Turbo-Tray Dryers The heat- and masstransfer mechanisms are similar to those in batch tray dryers, except that constant turning over and mixing of the solids significantly improves drying rates. Design must usually be based on previous installations or pilot tests by the manufacturer; apparent heat-transfer

12-68

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

Turbo-tray dryer in closed circuit for continuous drying with solvent recovery. (Wyssmont Company, Inc.)

FIG. 12-77

coefficients will range from 28 to 55 J/(m2⋅s⋅K) for dry solids to 68 to 115 J/(m2⋅s⋅K) for wet solids. Turbo-tray dryers have been employed successfully for the drying and cooling of calcium hypochlorite, urea crystals, calcium chloride flakes, and sodium chloride crystals. The Wyssmont “closed-circuit” system, as shown in Fig. 12-77, consists of the turbo-tray dryer with or without internal heaters, recirculation fan, condenser with receiver and mist eliminators, and reheater. Feed and discharge are through a sealed wet feeder and lock respectively. This method is used for continuous drying without leakage of fumes, vapors, or dust to the atmosphere. A unified approach for scaling up dryers such as turbo-tray, plate, conveyor, or any other dryer type that forms a defined layer of solids next to a heating source is described by C. G. Moyers [Drying Technology—An International Journal, 12(1 and 2), 393 (1994)]. Performance and Cost Data for Turbo-Tray Dryers Performance data for three applications of closed-circuit drying are included in Table 12-29. Operating, labor, and maintenance costs compare favorably with those of direct-heat rotating equipment. Plate Dryers The plate dryer is an indirect-heated, fully continuous dryer available for three modes of operation: atmospheric, gastight, or full vacuum. The dryer is of vertical design, with horizontal, heated plates mounted inside the housing. The plates are heated by either hot water, steam, or thermal oil, with operating temperatures up to 320°C possible. The product enters at the top and is conveyed through the dryer by a product-transport system consisting of a central-rotating shaft with arms and plows. (See dryer schematic, Fig. 12-78.) The thin product layer (approx. a in depth) on the surface of the plates, coupled with frequent product turnover by the conveying system, results in TABLE 12-29

short-retention times (approx. 5–40 min), true plug flow of the material, and uniform drying. The vapors are removed from the dryer by a small amount of heated purge gas or by vacuum. The material of construction of the plates and housing is normally stainless steel, with special metallurgies also available. The drive unit is located at the bottom of the dryer and supports the central-rotating shaft. Typical speed of the dryer is 1–7 rpm. Full-opening doors are located on two adjacent sides of the dryer for easy access to dryer internals. The plate dryer may vary in size from 5–35 vertically stacked plates with a heat-exchange area between 3.8–175 m2. The largest unit available has overall dimensions of 3 m (w) by 4 m (1) by 10 m (h). Depending upon the loose-bulk density of the material and the overall retention time, the plate dryer can process up to 5,000 kg/hr of wet product. The plate dryer is limited in its scope of applications only in the consistency of the feed material (the products must be friable, free flowing, and not undergo phase changes) and drying temperatures up to 320°C. Applications include specialty chemicals, pharmaceuticals, foods, polymers, pigments, etc. Initial moisture or volatile level can be as high as 65 percent and the unit is often used as a final dryer to take materials to a bone-dry state, if necessary. The plate dryer can also be used for heat treatment, removal of waters of hydration (bound moisture), solvent removal, and as a product cooler. The atmospheric plate dryer is a dust-tight system. The dryer housing is an octagonal, panel construction, with operating pressure in the range of ⫾0.5 kPa gauge. An exhaust-air fan draws the purge air through the housing for removal of the vapors from the drying process. The purge-air velocity through the dryer is in the range of 0.1–0.15 m/sec, resulting in minimal dusting and small dust filters for the exhaust air. The air temperature is normally equal to the plate temperature. The vapor laden exhaust air is passed through a dust filter or a scrubber (if necessary) and is discharged to the atmosphere. Normally, water is the volatile to be removed in this type of system. The gastight plate dryer, together with the components of the gasrecirculation system, forms a closed system. The dryer housing is semicylindrical and is rated for a nominal pressure of +5 kPa gauge. The flow rate of the recirculating purge gas must be sufficient to absorb the vapors generated from the drying process. The gas temperature must be adjusted according to the specific product characteristics and the type of volatile. After condensation of the volatiles, the purge gas (typically nitrogen) is recirculated back to the dryer via a blower and heat exchanger. Solvents such as methanol, toluene, and acetone are normally evaporated and recovered in the gastight system. The vacuum plate dryer is provided as part of a closed system. The vacuum dryer has a cylindrical housing and is rated for full-vacuum operation (typical pressure range 3–27 kPa absolute). The exhaust vapor is evacuated by a vacuum pump and is passed through a condenser for solvent recovery. There is no purge-gas system required for operation under vacuum. Of special note in the vacuum-drying system

Turbo-Tray Dryer Performance Data in Wyssmont Closed-Circuit Operations*

Material dried Dried product, kg/h Volatiles composition Feed volatiles, % wet basis Product volatiles, % wet basis Evaporation rate, kg/h Type of heating system Heating medium Drying medium Heat consumption, J/kg Power, dryer, kW Power, recirculation fan, kW Materials of construction Dryer height, m Dryer diameter, m Recovery system Condenser cooling medium Location Approximate cost of dryer (1995) Dryer assembly

Antioxidant 500 Methanol and water 10 0.5 53 External Steam Inert gas 0.56 × 106 1.8 5.6 Stainless-steel interior 4.4 2.9 Shell-and-tube condenser Brine Outdoor $225,000 Packaged unit

*Courtesy of Wyssmont Company, Inc.

Water-soluble polymer 85 Xylene and water 20 4.8 16 External Steam Inert gas 2.2 × 106 0.75 5.6 Stainless-steel interior 3.2 1.8 Shell-and-tube condenser Chilled water Indoor $115,000 Packaged unit

Antibiotic filter cake 2400 Alcohol and water 30 3.5 910 External Steam Inert gas 1.42 × 106 12.4 37.5 Stainless-steel interior 7.6 6.0 Direct-contact condenser Tower water Indoor $425,000 Field-erected unit

Petroleum coke 227 Methanol 30 0.2 302 External Steam Inert gas 1.74 × 106 6.4 15 Carbon steel 6.5 4.5 Shell-and-tube condenser Chilled water Indoor $225,000 Field-erected unit

SOLIDS-DRYING EQUIPMENT

12-69

Indirect-heated continuous plate dryer for atmospheric, gastight, or full-vacuum operation. (Krauss Maffei.) FIG. 12-78

are the vacuum feed and discharge locks, which allow for continuous operation of the plate dryer under full vacuum. Comparison Data—Plate Dryers Comparative studies have been done on products under both atmospheric and vacuum drying conditions. See Fig. 12-79. These curves demonstrate (1) the improvement in drying achieved with elevated temperature and (2) the impact to the drying process obtained with vacuum operation. Note that curve 4 at 90°C, pressure at 6.7 kPa absolute, is comparable to the atmospheric curve at 150°C. Also, the comparative atmospheric curve at 90°C requires 90 percent more drying time than the vacuum condition. The dramatic improvement with the use of vacuum is important to note for heat-sensitive materials. Drying Curve Product “N”

35%

TTB. .atmospheric plate dryer, vented VTT. .vacuum plate dryer, P = 6.7 KPA Dryer type

Moisture content

1 2 3

Plate Drying temp. time

1…TTB 90°C 76 min 2…TTB 110°C 60 min 3…TTB 127°C 50 min 4…VTT 90°C 40 min 5…TTB 150°C 37 min

4 5 0

0 Time

90 Min

Plate dryer drying curves demonstrating impact of elevated temperature and/or operation under vacuum. (Krauss Maffei.)

FIG. 12-79

The above drying curves have been generated via testing on a platedryer simulator. The test unit duplicates the physical setup of the production dryer, therefore linear scale-up from the test data can be made to the full-scale dryer. Because of the thin product layer on each plate, drying in the unit closely follows the normal type of drying curve in which the constant-rate period (steady evolution of moisture or volatiles) is followed by the falling-rate period of the drying process. This results in higher heat-transfer coefficients and specific drying capacities on the upper plates of the dryer as compared to the lower plates. The average specific drying capacity for the plate dryer is in the range of 2–20 kg /m2⋅hr (based on final dry product). Performance data for typical applications are shown on Table 12-30. Conical Mixer Dryer The conical mixer dryer is a batch-wise operating unit commonly used in the pharmaceutical and specialty chemical industries for the drying of solvent or water wet, free-flowing powders. The process area is a vertically oriented conical vessel with an internally mounted screw. Figure 12-80 shows a schematic of the bottom drive conical mixer dryer. The dryer utilizes the heatable, internal rotating screw to provide agitation of the batch of material and thus improve the heat and mass transfer of the process. Because it rotates around the full circumference of the vessel, the screw provides a self-cleaning effect for the heated vessel walls. The distinguishing feature of this dryer is the bottom-screw drive, as opposed to a top-drive unit, thus eliminating any mechanical drive components inside the vessel. The bottom-driven screw rotates about its own axis (speeds up to 100 rpm) and around the interior of the vessel (speeds up to 0.4 rpm). The screw is cantilevered in the vessel and requires no additional support (even in vessel sizes up to 20 m3 operating volume). The dryer is available in a variety of materials of construction, including SS 304 and 316, as well as Hastelloy. The heatable areas of the dryer are the vessel wall and the screw. The dryer makes maximum use of the product-heated areas—the filling volume of the vessel (up to the knuckle of the dished head) is the usable product loading. The top cover of the vessel is easily heated by either a half-pipe coil or heat tracing, which ensures that no vapor condensation will occur in the process area. In addition to the conical vessel heated area, heating the screw effectively increases the heat exchange area by 15–30 percent. This is accomplished via rotary joints at the base of the screw. The screw can be heated with the same

12-70

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING TABLE 12-30

Plate Dryer Performance Data for Three Applications*

Product Volatiles Production rate, dry Inlet volatiles content Final volatiles content Evaporative rate Heating medium Drying temperature Dryer pressure Air velocity Drying time, min Heat consumption, kcal/kg dry product Power, dryer drive Material of construction Dryer height Dryer footprint Location Dryer assembly Power, exhaust fan Power, vacuum pump

Plastic additive Methanol 362 kg/hr 30% 0.1% 155 kg/hr Hot water 70°C 11 kPa abs NA 24 350

Pigment Water 133 kg/hr 25% 0.5% 44 kg/hr Steam 150°C atmospheric 0.1 m/sec 23 480

Foodstuff Water 2030 kg/hr 4% 0.7% 70 kg/hr Hot water 90°C atmospheric 0.2 m/sec 48 100

3 kW SS 316L/316Ti 5m 2.6 m diameter Outdoors Fully assembled NA 20 kW

1.5 kW SS 316L/316Ti 2.6 m 2.2 m by 3.0 m Indoors Fully assembled 2.5 kW NA

7.5 kW SS 316L/316Ti 8.2 m 3.5 m by 4.5 m Indoors Fully assembled 15 kW NA

*Krauss Maffei

medium as the conical jacket (either hot water, steam, or thermal oil). Obviously, the impact of the heated screw will mean shorter batchdrying times, which yields higher productivity and better product quality due to shorter exposure to the drying temperature. The vessel cover is free of drive components, allowing space for additional process nozzles, manholes, explosion venting, etc., as well as a temperature lance for direct, continuous product-temperature

FIG. 12-80

measurement in the vessel. A dust filter is normally mounted directly on the top head of the dryer, thus allowing any entrapped dust to be pulsed back into the process area. Standard cloth-type dust filters are available, along with sintered metal filters. Because there are no drive components in the process area, the risk of batch failures due to contamination from gear lubricants is eliminated. Cleaning of the conical mixer dryer is facilitated with CIP systems and/or the vessel can be

Bottom drive conical mixer dryer. (Krauss Maffei.)

SOLIDS-DRYING EQUIPMENT completely flooded with water or solvents. The disassembly of the unit is simplified, as all work on removing the screw can be done without vessel entry. For disassembly, the screw is simply secured from the top and the drive components removed from the bottom of the dryer. Since the dryer is a batch-operating unit, it is commonly used in the pharmaceutical industry to maintain batch identification. In addition to pharmaceutical materials, the conical mixer dryer is used to dry polymers, additives, inorganic salts, and many other specialty chemicals. The conical mixer dryer is an indirect (conduction) dryer designed to operate under full vacuum. The heating medium is either hot water, steam, or thermal oil, with most applications in the temperature range of 50–150°C and pressures in the range of 3–30 kPa absolute. The vapors generated during the drying process are evacuated by a vacuum pump and passed through a condenser for recovery of the solvent. Design Methods for Conical Mixer Dryers Drying trials are conducted in small pilot dryers (50–100 liter batch units) to determine material handling and drying retention times. Variables such as drying temperature, vacuum level, and screw speed are analyzed during the test trials. Scale-up to larger units is done based upon the area to volume ratio of the pilot unit versus the production dryer. In most applications, the overall drying time in the production models is in the range of 2–24 hours. For design purposes, the heat-transfer coefficient for the conical mixer dryer is 60 W/m2⋅K. Reactotherm The reactotherm is a special thermal processor which has been designed to overcome the problems encountered with standard horizontal rotary-type dryers, such as crust formation on the heat-exchange surfaces and stalling of the rotor during drying. Incorporated in the design of the reactotherm is a high torque drive combined with rugged shaft construction to prevent rotor stall during processing; in addition, stationary mixing elements are installed in the process housing which continually clean the heat-exchange surfaces of the rotor to minimize any crust buildup and ensures an optimum heattransfer coefficient at all times. Because the unit can handle a wide range of product consistencies (dilute slurries, pastes, friable powders) the reactotherm can be used for processes such as reactions, mixing, drying, cooling, melting, sublimation, distilling, and vaporizing. The reactotherm is an indirect-heated thermal processor available in both batch and continuous design. Figure 12-81 shows a sketch of the continuous model. The unit is of horizontal design, with both the shell and internal rotor heated to provide high heat-transfer area/ product volume ratio. The units are conduction dryers, with operation in atmospheric, pressure and vacuum mode possible. Sizes range from 170 up to 10,000 liter operating volume with heat-exchange area from 3.1 m2 up to 103.8 m2. The typical filling volume of each unit is 75 percent of the available operating volume. The material of construction is either carbon steel or 300 series stainless steel, with optional special metallurgy available. In addition to many varied applications in the chemical industry, the reactotherm is widely used to process waste sludges which must be dried in order to landfill the solid components. In many of these cases, the solvents are valuable components of the waste stream and can be recovered and recycled. The housing of the continuous model consists of two fully enclosed, cylindrical, flanged-vessel sections. The standard design of the housing is suitable for full vacuum (3–30 kPa abs) or pressures up to 7 bar abs (higher design pressures available). The housing sections, including both end covers, are jacketed for heating or cooling. Hot water, steam, or thermal oil may be used as the heating medium, with maximum temperature up to 350°C possible. Flanged ports and nozzles are provided for product feed and discharge, vapor removal, inspection, and for the stationary mixing elements. The design of the continuous models is such that the ratio of vessel length to housing diameter is approximately 6:1, resulting in a long unit with narrow diameter. Bearing supports are provided at both ends of the unit for shaft support. The mixing bars in the continuous unit also convey the product through the unit, whereas the product filling level is determined by a weir flanged between the housing sections. The residence time and speed of the mixing shaft are specially matched to the requirements of the process. The batch-operating units consist of only one jacketed, cylindrical vessel with the necessary nozzles and ports for product inlet and out-

12-71

let, vapor, inspection, and mixing elements. The standard design of the housing is suitable for full vacuum or pressures up to 7 bar abs (higher design pressures available), with temperatures up to 350°C possible. The mixing bars are arranged such that a countercurrent flow in a radial direction of the product is achieved in the process vessel. For discharge of the material, a central port is provided at the bottom of the unit. The batch models up to 1080 liters volume are provided with cantilever-mounted mixing shafts. Above 1080 liters, the shaft is supported at both ends with bearings. The design of the batch models is such that the ratio of vessel length to housing diameter is approximately 2.5:1, resulting in a short unit with large diameter. Design Methods for Reactotherm Product trials are conducted in small pilot dryers (8–60 liter batch or continuous units) to determine material handling and process-retention times. Variables such as drying temperature, pressure level, and shaft speed are analyzed during the test trials. For design purposes, the heat-transfer coefficient for the reactotherm is in the range of 10 W/m2⋅K (light, free-flowing powders) up to 150 W/m2⋅K (dilute slurries). Hearth Furnaces A special design of a circular hearth furnace is the Mannheim furnace, in which sulfuric acid is reacted with sodium chloride to produce salt cake and hydrochloric acid. It consists of a refractory hearth, up to 6 m in diameter, with a silicon carbide arch. Hot flue gases are circulated around the muffle. The major portion of heat is transmitted through the arch and radiated to the product on the hearth. Feed materials are mixed and charged continuously to the center of the hearth, where they are stirred by underdriven rabble arms. The charge is gradually worked toward the periphery as the reaction generates hydrogen chloride gas. The gas is withdrawn through a separate duct to an absorption system. The salt cake is discharged at the periphery. Figure 12-82 shows a diagrammatic cross section of a Mannheim furnace. Combustion-chamber temperatures of about 1475 K are used for heating. The salt cake is discharged from the hearth at about 800 K. Multiple-Hearth Furnaces Multiple-hearth furnaces are known under various names: the Herreshoff, McDougall, Wedge, Pacific, etc. Figure 12-83 shows a general design. It consists of a number of annular-shaped hearths mounted one above the other. There are rabble arms on each hearth driven from a common center shaft. The feed is charged at the center of the upper hearth. The arms move the charge outward to the periphery, where it falls to the next hearth. Here it is moved again to the center, from which it falls to the next hearth. This continues down the furnace. The hollow center shaft is cooled internally by forced-air circulation. Burners may be mounted at any of the hearths, and the circulated air is used for combustion. These furnaces handle granular materials and provide a long countercurrent path between the flue gases and the charge material. Industrial sizes are built from 2 to 7 m in diameter and include 4 to 16 hearths. Total hearth areas range from 6.5 to 335 m2. The furnaces are used for roasting ores, drying and calcining lime, magnesite, and carbonate sludges, reactivation of decolorizing earths, and burning of sulfides to produce sulfur dioxide. The following is a partial list of applications: 1. Lime (a) from crushed limestone, (b) from oyster or sea shell, and (c) from dolomitic limestone 2. Lead and zinc; roasting of sulfides 3. Mercury from cinnabar ores by volatilization 4. Gold and silver: (a) chloridizing roast of gold-silver ore, and (b) removal of arsenic 5. Sulfuric acid from iron pyrites 6. Paint pigments; roasting of metallic oxides 7. Refractory clays; calcination of refractory clay to reduce shrinkage 8. Foundry sand; removal of carbon from used foundry sand 9. Fuller’s earth; calcination of fuller’s-earth material 10. Sewage disposal; calcination of sewage slurry Table 12-31 lists three specific applications with a brief description of the furnaces as to design and operating conditions. GRAVITY DRYERS Description A body of solids in which the particles, consisting of granules, pellets, beads, or briquettes, flow downward by gravity at

12-72

FIG. 12-81

FIG. 12-82

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

Continuous indirect-heated reactotherm. (Krauss Maffei.)

Mannheim-type mechanical hydrochloric acid furnace.

substantially their normal settled bulk density through a vessel in contact with gases is defined frequently as a moving bed. Moving-bed equipment finds application in blast furnaces, shaft furnaces, and petroleum refining. A gravity dryer consists of a stationary vertical, usually cylindrical housing with openings for the introduction of solids (at the top) and removal of solids (at the bottom). Gas flow is through the solids bed and may be cocurrent or countercurrent and, in some instances, crossflow. By definition, the rate of gas flow upward must be less than that required for fluidization. Fields of Application One of the major advantages of the gravity-bed technique is that it lends itself well to true intimate countercurrent contacting of solids and gases. This provides for efficient heat transfer and mass transfer. Gravity-bed contacting also permits the use of the solid as a heat-transfer medium, as in pebble heaters.

Gravity vessels are applicable to coarse granular free-flowing solids which are comparatively dust-free. The solids must possess physical properties in size and surface characteristics so that they will not stick together, bridge, or segregate during passage through the vessel. The presence of significant quantities of fines or dust will close the passages among the larger particles through which the gas must penetrate, increasing pressure drop. Fines may also segregate near the sides of the bed or in other areas where gas velocities are low, ultimately completely sealing off these portions of the vessel. The high efficiency of gas-solids contacting in gravity beds is due to the uniform distribution of gas throughout the solids bed; hence choice of feed and its preparation are important factors to successful operation. Preforming techniques such as pelleting and briquetting are employed frequently for the preparation of suitable feed materials. Gravity vessels are suitable for low-, medium-, and hightemperature operation; in the last case, the housing will be lined completely with refractory brick. Dust-recovery equipment is minimized in this type of operation since the bed actually performs as a dust collector itself, and dust in the bed will not, in a successful application, exist in large quantities. Other advantages of gravity beds include flexibility in gas and solids flow rates and capacities, variable retention times from minutes to several hours, space economy, ease of startup and shutdown, the potentially large number of contacting stages, and ease of control by using the inlet- and exit-gas temperatures. Maintenance of a uniform rate of solids movement downward over the entire cross section of the bed is one of the most critical operating problems encountered. For this reason gravity beds are designed to be as high and narrow as practical. In a vessel of large cross section, discharge through a conical bottom and center outlet will usually result in some degree of “ratholing” through the center of the bed. Flow through the center will be rapid while essentially stagnant pock-

SOLIDS-DRYING EQUIPMENT

FIG. 12-83

Pacific multiple-hearth furnace.

ets are left around the sides. To overcome this problem, multiple outlets are provided in the center and around the periphery; table unloaders, rotating plows, wide moving grates, and multiple-screw unloaders are employed; insertion of inverted cone baffles in the lower section of the bed, spaced so that flushing at the center is retarded, is also a successful method for improving uniformity of solids movement. Figure 12-84 illustrates a moving tray with multiple downspouts used to remove a precise amount of solids from each increment of area across the base of a gravity-bed reactor. The various pockets are filled at one extremity of its motion and emptied at the other. It is suitable primarily for fine nonabrasive solids. Figure 12-85 depicts a perforated-plate design, taking advantage of the flow characteristics and angle of repose of the solids to control the unloading rate. Still another design of this general type involves the use of a nest of inclined pipes, discharging into a common header, and placed to draw solids at geometrically spaced points across the base of the reactor. TABLE 12-31

FIG. 12-84

Gravity-bed reactor; solids-discharge mechanism.

FIG. 12-85

Perforated-tray type of reactor-discharge control.

12-73

Gas disengaging from the solids may represent another serious operating problem in a gravity bed. One method employs downspouts at the top for solids feeding while leaving an open space in the vessel above the downspout outlet for gas disengaging, as illustrated in Fig. 12-86a. Another uses a series of inverted V-shaped channels inserted into the top of the solids bed. Gas and vapor are collected and removed from under the V’s, while the solids flow over the top and around the channels (Fig. 12-86b). These methods for both gas and solids removal were developed originally for petroleum-refining catalytic reactors. Shaft Furnaces The oldest and most important application of the shaft furnace is the blast furnace used for the production of pig iron. Another use is in the manufacture of phosphorus from phosphate rock. Formerly lime was calcined exclusively in this type of furnace. Shaft furnaces are widely used also as gas producers. Chemicals are manufactured in shaft furnaces from briquetted mixtures of the reacting components. A shaft furnace is a vertical refractory-lined cylinder in which a stationary or descending column of solids is maintained and through

Applications of Multiple-Hearth Furnaces*

Product

Production rate

Furnace size

Mercury from cinnabar ore

225 tons ore/day (95% recovery)

(2) 18.0 ft. diam., 8 hearth furnaces

Lime from oyster shell

240 tons/day, shell (120 tons/day, lime)

Magnesium oxide from magnesium hydroxide

100 tons/day, 50% magnesium hydroxide slurry; yields 50 tons/day magnesium oxide

(1) 22 ft., 3 in. diam., 12 hearth furnaces (2) 22 ft., 3 in., 10 hearth furnaces

Special features Furnaces fired on hearths 3 to 7, inclusive; retention time of 1.0 hr.; furnaces are oil-fired with low-pressure atomizing air burners; all air, both primary and secondary, introduced through the burners; draft control by Monel cold-gas fans downstream from mercury condensers. Furnace walls of 4.5-in. firebrick, 9-in. insulation for 1550°F. operating temp. Furnace fired on hearths 4 to 10, inclusive

12-74

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

Countercurrent gas-solids flow at the top disengaging section of a moving-bed catalytic reactor.

FIG. 12-86a

FIG. 12-87

FIG. 12-86b Vapor disengaging tray at the top of a gravity-bed catalytic reactor. This design may also be employed for the addition of gas to a bed of solids.

which an ascending stream of hot gas is forced. Three methods of fuel application may be employed: (1) one in which a solid fuel is added alone or mixed with the reacting solids, (2) one in which the fuel is burned in a separate combustion chamber with the hot gases being blown into the furnace at some level of the column, (3) one in which the fuel is introduced and burned in the bottom of the shaft. For maximum heat economy, recovered exhaust heat is employed for preheating of the incoming solids and combustion air. The fuels used may be gas, oil, or pulverized coal. Bucket elevators, skip hoists, and cranes are used for top feeding of the furnace. Retention and downward flow are controlled by timing of the bottom discharge. Gases are propelled by a blower or by induced draft from a stack or discharge fan. In normal operation, the downward flow of solids and upward flow of gas are constant with time, maintaining ideal steady-state conditions. Figure 12-87 illustrates a shaft lime kiln. Design Methods for Shaft Furnaces The size and shape of the charge particles control the amount of surface over which heat may be transmitted to the particle and also the depth of penetration through which the heat must pass to reach the center of each particle. Also, this size and shape control the nature of the random packing in the shaft and the extent of voids for gas passage. As particle size is decreased, the surface area of the particles increases. At the same time, the depth of heat penetration decreases. Both these factors tend to improve performance. With small particle size, however, the charge column presents high resistance to the passage of gas. With closely screened material, the percentage of voids (usually 37 percent) is independent of particle size. With unscreened particles showing a wide variation in size, the void volume is decreased; irregularity in gas flow results. There is a large difference between the total surface of the particles (as determined by their size and shape) and the “effective surface” actually exposed to the passing gas stream. In practice, it has been estimated that as little as 10 to 25 percent of the total surface is effective in heat transfer when unscreened particles are treated. Irregular-shaped particles exhibit greater surface area than regularshaped cubes and spheres, the amount of this increase being possibly 25 percent. The effect of particle size and size distribution on effective surface, in a shaft employed for calcination of limestone, is shown in

Shaft furnace for lime production.

Fig. 12-88. Curve A shows the calculated surface based on an assumed 50 percent void volume and cubical-shaped particles. The B set of curves applies to such unscreened irregularly shaped particles as are usually encountered in practice. The laws governing the flow of fluids through packed beds given in Sec. 6 are applicable to shaft furnaces. Since the pressure drop in a bed is affected by the size and shape of the interstitial voids, the horizontal and vertical nonuniformity of the bed, the changes in gas composition during passage, and other operating factors, test data for a given material are necessary for proper design. In the case of limestone, Fig. 12-89 shows the effect of particle size on the gas-flow friction through the bed, assuming that the friction varies as the square of the gas mass velocity and inversely with the particle size, and utilizing

Curve A shows surface variations with stone size, 100 percent active surfaces. Curves in group B show the effect of irregular stone size.

FIG. 12-88

SOLIDS-DRYING EQUIPMENT

FIG. 12-89

Variation in gas friction with size of stone.

base points established during actual kiln operations. Information on the mathematical treatment of heat transfer in packed beds is included in Sec. 12. Pellet Coolers and Dryers Gravity beds are employed for the cooling and drying of extruded pellets and briquettes from sizeenlargement processes. The rotary cooler illustrated in Fig. 12-90 consists of a stationary steel tank having a wear cylinder at the top for entry of gas and solids (usually from a pneumatic conveyor), with air holes staggered around the outer wall to admit additional air for optimum circulation. The tank encloses a rotating cage for retention of the solids bed. This cage consists of an inner cylinder of wire mesh and an outer perforated shell. Air entering the tank is circulated in cross-flow through the pellet bed and discharges through the center column. Usually a rotary unloading gate and air lock are located underneath the cage. Another cross-flow design employs a rectangular housing, partitioned into three vertical sections (Fig. 12-91). Solids move downward in the two outer sections, while cooling or drying air is drawn through the louvered outer walls and through the solids bed and is discharged through the center section. Solids are discharged over a baffled shaking shoe. Units of this general type are used for drying wheat and other grain products and numerous forms of pelleted feeds. Gravitybed dryers are most suitable for drying of granular heat-sensitive products employing moderate air temperatures. These require extended holdup during the falling-rate drying period. Spouted Beds The spouted-bed technique was developed primarily for solids which are too coarse to be handled in fluidized beds. Although their applications overlap, the methods of gas-solids mixing

FIG. 12-90

12-75

are completely different. A schematic view of a spouted bed is given in Fig. 12-92. Mixing and gas-solids contacting are achieved first in a fluid “spout,” flowing upward through the center of a loosely packed bed of solids. Particles are entrained by the fluid and conveyed to the top of the bed. They then flow downward in the surrounding annulus as in an ordinary gravity bed, countercurrently to gas flow. The mechanisms of gas flow and solids flow in spouted beds were first described by Mathur and Gishler [Am. Inst. Chem. Eng. J., 1(2), 157–164 (1955)]. Drying studies have been carried out by Cowan [Eng. J., 41, 5, 60–64 (1958)], and a theoretical equation for predicting the minimum fluid velocity necessary to initiate spouting was developed by Madonna and Lama [Am. Inst. Chem. Eng. J., 4(4), 497 (1958)]. Investigations to determine maximum spoutable depths and to develop theoretical relationships based on vessel geometry and operating variables have been carried out by Lefroy [Trans. Inst. Chem. Eng., 47(5), T120–128 (1969)] and Reddy [Can. J. Chem. Eng., 46(5), 329–334 (1968)]. Gas flow in a spouted bed is partially through the spout and partially through the annulus. About 30 percent of the gas entering the system immediately diffuses into the downward-flowing annulus. Near the top of the bed, the quantity in the annulus approaches 66 percent of the total gas flow; the gas flow through the annulus at any point in the bed equals that which would flow through a loosely packed solids bed under the same conditions of pressure drop. Solids flow in the annulus is both downward and slightly inward. As the fluid spout rises in the bed, it entrains more and more particles, losing velocity and gas into the annulus. The volume of solids displaced by the spout is roughly 6 percent of the total bed. On the basis of experimental studies, Mathur and Gishler derived an empirical correlation to describe the minimum fluid flow necessary for spouting, in 3- to 12-in-diameter columns: D D 0.33 2gL(ps − pf) 0.5 u = ᎏp ᎏo (12-62) ᎏᎏ Dc Dc pf where u = superficial fluid velocity through the bed, ft/s; Dp = particle diameter, ft; Dc = column (or bed) diameter, ft; Do = fluid-inlet orifice diameter, ft; L = bed height, ft; ps = absolute solids density, lb/ft3; pf = fluid density, lb/ft3; and g = 32.2 ft/s2, gravity acceleration. To convert feet per second to meters per second, multiply by 0.305; to convert pounds per cubic foot to kilograms per cubic meter, multiply by 16. g = 9.8 m/s2 in SI units. The inlet orifice diameter, air rate, bed diameter, and bed depth were all found to be critical and interdependent: 1. In a given-diameter bed, deeper beds can be spouted as the gas-inlet orifice size is decreased. Using air, a 12-in-diameter bed containing 0.125- by 0.250-in wheat can be spouted at a depth of over 100 in with a 0.8-in orifice, but at only 20 in with a 2.4-in orifice.

Sprout, Waldron rotary cooler. (Sprout, Waldron Cos.)

冢 冣 冤

冥

12-76

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

FIG. 12-91

Vertical gravity-bed cooler with louvres. (Sprout, Waldron Co.)

Schematic diagram of a spouted bed. [Mathur and Gishler, Am. Inst. Chem. Eng. J., 1, 2, 157 (1955).]

FIG. 12-92

2. Increasing bed diameter increases spoutable depth. By employing a bed-orifice diameter ratio of 12 for air spouting, a 9-indiameter bed was spouted at a depth of 65 in while a 12-in-diameter bed was spouted at 95 in. 3. As indicated by Eq. (12-62) the superficial fluid velocity required for spouting increases with bed depth and orifice diameter and decreases as the bed diameter is increased. Employing wood chips, Cowan’s drying studies indicated that the volumetric heat-transfer coefficient obtainable in a spouted bed is at least twice that in a direct-heat rotary dryer. By using 20- to 30-mesh Ottawa sand, fluidized and spouted beds were compared. The volumetric coefficients in the fluid bed were 4 times those obtained in a spouted bed. Mathur dried wheat continuously in a 12-in-diameter spouted bed, followed by a 9-in-diameter spouted-bed cooler. A drying rate of roughly 100 lb/h of water was obtained by using 450 K inlet air. Six hundred pounds per hour of wheat was reduced from 16 to 26 percent to 4 percent moisture. Evaporation occurred also in the cooler by using sensible heat present in the wheat. The maximum drying-bed temperature was 118°F, and the overall thermal efficiency of the system was roughly 65 percent. Some aspects of the spouted-bed technique are covered by patent (U.S. Patent 2,786,280). Cowan reported that significant size reduction of solids occurred when cellulose acetate was dried in a spouted bed, indicating its possible limitations for handling other friable particles. DIRECT-HEAT VIBRATING-CONVEYOR DRYERS Information on vibrating conveyors and their mechanical construction is given in Sec. 21. The vibrating-conveyor dryer is a modified form of fluidized-bed equipment, in which fluidization is maintained by a combination of pneumatic and mechanical forces. The heating gas is introduced into a plenum beneath the conveying deck through ducts and flexible hose connections, passes up through a screen, perforated, or slotted conveying deck, through the fluidized bed of solids, and into an exhaust hood (Fig. 12-93). If ambient air is employed for cooling, the sides of the plenum may be open and a simple exhaust system

used; however, because the gas-distribution plate may be designed for several inches of water-pressure drop to ensure a uniform velocity distribution through the bed of solids, a combination pressure-blowerexhaust-fan system is desirable to balance the pressure above the deck with the outside atmosphere and prevent gas in-leakage or blowing at the solids feed and exit points. Units are fabricated in widths from 0.3 to 1.5 m. Lengths are variable from 3 to 50 m; however, most commercial units will not exceed a length of 10 to 16 m per section. Power required for the vibrating drive will be approximately 0.4 kW/m2 of deck. In general, this equipment offers an economical heat-transfer area for first cost as well as operating cost. Capacity is limited primarily by the air velocity which can be used without excessive dust entrainment. Table 12-32 shows limiting air velocities suitable for various solids particles. Usually, the equipment is satisfactory for particles larger than 100 mesh in size. [The use of indirect-heated conveyors eliminates the problem of dust entrainment, but capacity is limited by the heattransfer coefficients obtainable on the deck (see Sec. 11)]. When a stationary vessel is employed for fluidization, all solids being treated must be fluidized; nonfluidizable fractions fall to the bottom of the bed and may eventually block the gas distributor. The addition of mechanical vibration to a fluidized system offers the following advantages: 1. Equipment can handle nonfluidizable solids fractions. Although these fractions may drop through the bed to the screen, directional-throw vibration will cause them to be conveyed to the discharge end of the conveyor. Prescreening or sizing of the feed is less critical than in a stationary fluidized bed. 2. Because of mechanical vibration, incipient channeling is reduced. 3. Fluidization may be accomplished with lower pressures and gas velocities. This has been evidenced on vibratory units by the fact that fluidization stops when the vibrating drive is stopped. Vibrating-conveyor dryers are suitable for free-flowing solids containing mainly surface moisture. Retention is limited by conveying speeds which range from 0.02 to 0.12 m/s. Bed depth rarely exceeds 7 cm, although units are fabricated to carry 30- to 46-cm-deep beds; these also employ plate and pipe coils suspended in the bed to provide additional heat-transfer area. Vibrating dryers are not suitable for fibrous materials which mat or for sticky solids which may ball or adhere to the deck. For estimating purposes for direct-heat drying applications, it can be assumed that the average exit-gas temperature leaving the solids bed will approach the final solids discharge temperature on an ordinary unit carrying a 5- to 15-cm-deep bed. Calculation of the heat load and selection of an inlet-air temperature and superficial velocity (Table 12-32) will then permit approximate sizing, provided an approximation of the minimum required retention time can be made. Vibrating conveyors employing direct contacting of solids with hot, humid air have also been employed for the agglomeration of fine powders, chiefly for the preparation of agglomerated water-dispersible food products. Control of inlet-air temperature and dew point permits the uniform addition of small quantities of liquids to solids by con-

SOLIDS-DRYING EQUIPMENT

FIG. 12-93

Vibrating conveyor dryer. (Carrier Division, Rexnord Inc.)

densation on the cool incoming-particle surfaces. The wetting section of the conveyor is followed immediately by a warm-air-drying section and particle screening. DISPERSION DRYERS A gas-solids contacting operation in which the solids phase exists in a dilute condition is termed a dispersion system. It is often called a pneumatic system because, in most cases, the quantity and velocity of the gas are sufficient to lift and convey the solids against the force of gravity. Pneumatic systems may be distinguished by two characteristics: 1. Retention of a given solids particle in the system is on the average very short, usually no more than a few seconds. This means that any process conducted in a pneumatic system cannot be diffusioncontrolled. The reaction must be mainly a surface phenomenon, or the solids particles must be very small so that heat transfer and mass transfer from the interiors are essentially instantaneous. 2. On an energy-content basis, the system is balanced at all times; i.e., there is sufficient energy in the gas (or solids) present in the system at any time to complete the work on all the solids (or gas) present at the same time. This is significant in that there is no lag in response to control changes or in starting up and shutting down the system; no partially processed residual solids or gas need be retained between runs. It is for these reasons that pneumatic equipment is especially suitable for processing heat-sensitive, easily oxidized, explosive, or flammable materials which cannot be exposed to process conditions for extended periods. Gas flow and solids flow are usually cocurrent, one exception being a countercurrent-flow spray dryer. The method of gas-solids contactTABLE 12-32 Table for Estimating Maximum Superficial Air Velocities through Vibrating-Conveyor Screens* Velocity, m/s Mesh size

2.0 specific gravity

1.0 specific gravity

200 100 50 30 20 10 5

0.22 0.69 1.4 2.6 3.2 6.9 11.4

0.13 0.38 0.89 1.8 2.5 4.6 7.9

*Carrier Division, Rexnord Inc.

12-77

ing is best described as through circulation; however, in the dilute condition solids particles are so widely dispersed in the gas that they exhibit apparently no effect upon one another, and they offer essentially no resistance to the passage of gas among them. Pneumatic-Conveyor Dryers A pneumatic-conveyor dryer consists of a long tube or duct carrying a gas at high velocity, a fan to propel the gas, a suitable feeder for addition and dispersion of particulate solids in the gas stream, and a cyclone collector or other separation equipment for final recovery of solids from the gas. The solids feeder may be of any type; screw feeders, venturi sections, high-speed grinders, and dispersion mills are employed. For pneumatic conveyors, selection of the correct feeder to obtain thorough initial dispersion of solids in the gas is of major importance. For example, by employing an air-swept hammer mill in a drying operation, 65 to 95 percent of the total heat may be transferred within the mill itself if all the drying gas is passed through it. Fans may be of the induced-draft or the forced-draft type. The former is usually preferred because the system can then be operated under a slight negative pressure. Dust and hot gas will not be blown out through leaks in the equipment. Cyclone separators are preferred for low investment. If maximum recovery of dust or noxious fumes is required, the cyclone may be followed by a wet scrubber or bag collector. In ordinary heating and cooling operations, during which there is no moisture pickup, continuous recirculation of the conveying gas is frequently employed. Also, solvent-recovery operations employing continuously recirculated inert gas with intercondensers and gas reheaters are carried out in pneumatic conveyors. Pneumatic conveyors are suitable for materials which are granular and free-flowing when dispersed in the gas stream, so they do not stick on the conveyor walls or agglomerate. Sticky materials such as filter cakes may be dispersed and partially dried by an air-swept disintegrator in many cases. Otherwise, dry product may be recycled and mixed with fresh feed, and then the two dispersed together in a disintegrator. Coarse material containing internal moisture may be subjected to fine grinding in a hammer mill. The main requirement in all applications is that the operation be instantaneously completed; internal diffusion of moisture must not be limiting in drying operations, and particle sizes must be small enough so that the thermal conductivity of the solids does not control during heating and cooling operations. Pneumatic conveyors are rarely suitable for abrasive solids. Pneumatic conveying can result in significant particle-size reduction, particularly when crystalline or other friable materials are being handled. This may or may not be desirable but must be recognized if the system is selected. The action is similar to that of a fluid-energy grinder.

12-78

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

Pneumatic conveyors may be single-stage or multistage. The former is employed for evaporation of small quantities of surface moisture. Multistage installations are used for difficult drying processes, e.g., drying heat-sensitive products containing large quantities of moisture and drying materials initially containing internal as well as surface moisture. Typical single- and two-stage drying systems are illustrated in Figs. 12-94, 12-95a and b, and 12-96. Figure 12-94 incorporates a single-stage dryer with a second stage containing a cage-mill disintegrator. The second stage ensures complete drying after thorough dispersion of lumps and agglomerates. If disintegration is required to disperse the wet feed, the stages can be reversed, or disintegration can be employed in both stages. Systems of the type illustrated are employed for drying synthetic resins, of which low-pressure polyethylene and polypropylene are examples. Figure 12-96 illustrates a single-stage dryer employing a paddle mixer, recycle, and an ABB Raymond cage mill for fine grinding and

FIG. 12-95b Strong-Scott flash dryer with integral coarse-fraction classifier to separate undried particles for recycle. (Bepex Corp.)

Two-stage air-stream and cage mill, pneumatic-conveyor dryer. (ABB Raymond.)

FIG. 12-94

FIG. 12-95a Air-lift pneumatic-conveyor dryer; includes partial recycle of dry product and expanding tube and cone sections to provide longer holdup for coarse particles. (Bepex Corp.)

dispersion of the mixed feed in the air stream. These units are designed to handle filter and centrifuge cakes and other sticky or pasty feeds. Capacities are given in Table 12-33. Several typical products dried in pneumatic conveyors are described in Table 12-34. The air-stream type referred to is an ordinary single-stage dryer like the first stage of Fig. 12-94. Design Methods for Pneumatic-Conveyor Dryers Depending upon the temperature sensitivity of the product, inlet-air temperatures between 400 and 1000 K are employed. With a heat-sensitive solid, a high initial moisture content should permit use of a high inletair temperature. Evaporation of surface moisture takes place at essentially the wet-bulb air temperature. Until this has been completed, by which time the air will have cooled significantly, the surface-moisture film prevents the solids temperature from exceeding the wet-bulb temperature of the air. Pneumatic conveyors are used for solids having initial moisture contents ranging from 3 to 90 percent, wet basis. The air quantity required and solids-to-gas loading are fixed by the moisture load, the inlet-air temperature, and, frequently, the exit-air humidity. If the last is too great to permit complete drying, i.e., if the exit-air humidity is above that in equilibrium with the product at required dryness, the solids-to-gas loading must be reduced together with the inlet-air temperature. The gas velocity in the conveying duct must be sufficient to convey the largest particle. This may be calculated accurately by methods given in Sec. 12. For estimating purposes, a velocity of 25 m/s, calculated at the exit-air temperature, is frequently employed. If mainly surface moisture is present, the temperature driving force for drying will approach the log mean of the inlet- and exit-gas wet-bulb depressions. (The exit solids temperature will approach the exit-gas dry-bulb temperature.)

SOLIDS-DRYING EQUIPMENT

FIG. 12-96

12-79

Single-stage pneumatic-conveyor dryer. (ABB Raymond.)

Observation of operating conveyors indicates that the solids are rarely uniformly dispersed in the gas phase. With infrequent exceptions, the particles move in a laminar pattern, following a streamline along the duct wall where the flow velocity is at a minimum. Complete or even partial diffusion in the gas phase is rarely experienced even with low-specific-gravity particles. Air velocities may approach 20 to 30 m/s. It is doubtful, however, that even finer and lighter materials reach more than 80 percent of this speed, while heavier and larger TABLE 12-33

fractions may travel at much slower rates [Fischer, Mech. Eng., 81(11), 67–69 (1959)]. Very little information and operating data on pneumatic-conveyor dryers which would permit a true theoretical basis for design have been published. Therefore, firm design always requires pilot tests. It is believed, however, that the significant velocity effect in a pneumatic conveyor is the difference in velocities between gas and solids, which is the reason why a major part of the total drying actually occurs in the feed section.

Sizes and Capacities of ABB-Raymond Cage-Mill Flash-Drying Systems* Based upon 1200°F inlet temperatures†

Cyclone

A

B

C

D

E

F

G

Approximate weight, lb H

18 ft 0 in 16 ft 0 in 14 ft 0 in 12 ft 0 in 10 ft 0 in 9 ft 0 in 8 ft 0 in 7 ft 0 in 6 ft 0 in 5 ft 0 in 4 ft 0 in

18 ft 0 in 15 ft 0 in 14 ft 0 in 13 ft 0 in 11 ft 6 in 11 ft 0 in 10 ft 6 in 10 ft 0 in 9 ft 6 in 9 ft 0 in 8 ft 6 in

38 ft 0 in 34 ft 0 in 31 ft 0 in 29 ft 0 in 26 ft 0 in 24 ft 0 in 24 ft 0 in 23 ft 0 in 22 ft 0 in 20 ft 0 in 19 ft 0 in

20 ft 0 in 17 ft 4 in 14 ft 4 in 12 ft 6 in 10 ft 9 in 10 ft 3 in 9 ft 6 in 8 ft 9 in 8 ft 0 in 7 ft 3 in 6 ft 3 in

76 ft 0 in 66 ft 4 in 59 ft 4 in 54 ft 6 in 48 ft 3 in 45 ft 3 in 44 ft 0 in 41 ft 9 in 39 ft 6 in 36 ft 3 in 33 ft 9 in

19 ft 6 in 17 ft 0 in 16 ft 6 in 15 ft 3 in 13 ft 3 in 13 ft 3 in 12 ft 6 in 12 ft 6 in 11 ft 0 in 10 ft 9 in 10 ft 9 in

49 ft 6 in 45 ft 6 in 43 ft 4 in 40 ft 4 in 37 ft 8 in 36 ft 8 in 33 ft 11 in 32 ft 3 in 30 ft 9 in 28 ft 4 in 27 ft 1 in

10 ft 0 in 9 ft 9 in 9 ft 6 in 8 ft 2 in 7 ft 1 in 6 ft 11 in 6 ft 6 in 6 ft 3 in 5 ft 7 in 5 ft 5 in 4 ft 0 in

115,000 85,000 75,000 65,000 50,000 45,000 37,000 32,000 28,000 21,000 17,000

Evaporation capacity, lb water/h†

Price,‡ FOB shops

20,000 15,700 12,000 8,850 6,150 5,000 3,940 3,000 2,220 1,540 985

$550,000 475,000 400,000 375,000 325,000 300,000 285,000 270,000 250,000 230,000 210,000

*ABB Raymond †With inlet temperature of 600 to 700°F, consider the water evaporation to be one-half of that listed in the table. Considerably lower inlet temperatures are also frequently used for many materials. ‡Price based upon carbon steel construction (motors and secondary dust collectors by others). (1994)

12-80

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING TABLE 12-34

Typical Products Dried in Pneumatic-Conveyor Dryers* Material

Initial moisture, wet basis, %

Final moisture, wet basis, %

Remarks

Clay, acid-treated Coal, 1 cm × 0 Corn-gluten feed Kaolin (H2O washed and partially dried) Gluten (vital wheat) Clay, ball Gypsum, raw Pharmaceuticals Silica-gel catalyst Synthetic resin Carboxymethyl cellulose Sewage sludge

60 11.5 65 10 70 25 25 total 15 53 50 40 82

18 1.5 20 0.5 10 0.5 5 4 10 0.5 3 0

Cage-mill type Air-stream type Cage-mill type Imp-mill type; grind to 99.9%—325 mesh Imp-mill type; grind to—80 mesh Imp-mill type; grind to 95%—100 mesh Imp-mill type; grind and calcine to stucco Air-stream type Cage-mill type Two-stage system Imp-mill type; grind to 80%—200 mesh Cage-mill-type

*ABB Raymond

One manner in which size may be computed, for estimating purposes, is by employing a volumetric heat-transfer concept as used for rotary dryers. If it is assumed that contacting efficiency is in the same order as that provided by efficient lifters in a rotary dryer and that the velocity difference between gas and solids controls, Eq. (12-52) may be employed to estimate a volumetric heat-transfer coefficient. By assuming a duct diameter of 0.3 m (D) and a gas velocity of 23 m/s, if the solids velocity is taken as 80 percent of this speed, the velocity difference between the two would be 4.6 m/s. If the exit gas has a density of 1 kg/m3, the relative mass flow rate of the gas G becomes 4.8 kg/(s⋅m2); the volumetric heat-transfer coefficient is 2235 J/(m3⋅s⋅K). This is not far different from many coefficients found in commercial installations; however, it is usually not possible to predict accurately the actual difference in velocity between gas and solids. Furthermore, the coefficient is influenced by the solids-to-gas loading and particle size, which control the total solids surface exposed to the gas. Therefore, the figure given is only an approximation. For estimating purposes, the conveyor cross section is fixed by the assumed air velocity and quantity. The volume, hence the length, can then be calculated by the method just presented, employing the logmean air wet-bulb depression for the temperature driving force. A conveyor length >50 diameters is rarely required. Scale-up of pneumatic conveying dryers is outlined in a theoretical model which predicts dryer performance but which depends on uncertain parameters such as wall friction and agglomeration. [I. C. Kemp [Drying Technology—An International Journal, 12(1&2), 279 (1994)]. Pressure drop in the system may be computed by methods described in Sec. 6. To prevent excessive leakage into or out of the system, which may have a total pressure drop of 20 to 38 cm of water, rotary air locks or screw feeders are employed at the solids inlet and discharge. The conveyor and collector parts are thoroughly insulated to reduce heat losses in drying and other heating operations. Operating control is maintained usually by control of the exit-gas temperature, with the inlet-gas temperature varied to compensate for changing feed conditions. A constant solids feed rate must be maintained. Cost Data for Pneumatic-Conveyor Dryers Purchase costs vary widely; many pneumatic-conveyor installations are assembled units, each component being purchased from a different supplier. Representative prices are given in Table 12-33 for conventional pneumatic-conveyor dryers. These include a cage mill for disintegration and a primary cyclone collector. In general, pneumatic conveyors for similar duties will compete in cost with cocurrent rotary dryers. Space economics may reduce the total installed investment slightly below that of the rotary unit. Operating costs, thermal efficiency, etc., are similar to those of cocurrent rotary units sized for the same duty. When other operations, such as conveying, grinding, or classifying, are simultaneously performed, operating and investment costs may be reduced for the pneumatic-drying process itself by being partially written off on the secondary function. In this situation, a pneumatic conveyor becomes particularly attractive. Ring Dryers The ring dryer is a development of flash, or pneumatic-conveyor, drying technology, designed to increase the versatility of application of this technology and overcome many of its limitations.

One of the great advantages of flash drying is the very short retention time, typically no more than a few seconds. However, in a conventional flash dryer, residence time is fixed and this limits its application to materials in which the drying mechanism is not diffusion controlled and where a range of moisture within the final product is acceptable. The ring dryer offers two advantages over the flash dryer. Principally there is control of residence time by the use of an adjustable internal classifier which allows fine particles, which dry quickly, to leave, while larger particles, which dry slowly, have an extended residence time within the system. Second, the combination of the classifier with an internal mill can allow simultaneous grinding and drying with control of product particle size and moisture. Available with a range of different feed systems to handle a variety of applications, the ring dryer provide wide versatility. The essential difference between a conventional flash dryer and the ring dryer is the “manifold centrifugal classifier.” The manifold pro-

FIG. 12-97

Full manifold classifier for ring dryer. (Barr—Rosin)

SOLIDS-DRYING EQUIPMENT vides classification of the product about to leave the dryer using differential centrifugal force. The manifold, as shown in Fig. 12-97, uses the centrifugal effect of an air stream passing around the curve to concentrate the product into a moving layer, with the dense material on the bottom and the light material on the top. This enables the adjustable splitter blades within the manifold classifier to segregate the heavier, wetter material and return it for a further circuit of drying. Fine, dried material is allowed to leave the dryer with the exhaust air and pass to the product collection system. This selective extension of residence time ensures a more evenly dried material than possible from a conventional flash dryer. Many materials which have traditionally been regarded as difficult to dry can be processed to the required moisture content in a ring dryer. The recycle requirements of products in different applications can vary substantially depending upon the scale of operation, the ease of drying, and the finished-product specification. The location of reintroduction of undried material back into the drying medium has a significant impact upon the dryer performance and final-product characteristics. Three configurations of the ring dryer have been developed to offer flexibility in design and optimal performance: 1. Single-stage manifold—vertical configuration. The feed ring dryer (see Fig. 12-98) is similar to a flash dryer but incorporates a single-stage classifier, which diverts 40–60 percent of the product back to the feed point. The feed ring dryer is ideally suited for materials which are neither heat sensitive nor require a high degree of classification. An advantage of this configuration is that it can be manufactured to very large sizes to achieve high-evaporative capacities. 2. Full manifold—horizontal configuration. The full ring dryer (see Fig. 12-99) incorporates a multistage classifier which allows much higher recycle rates than the single-stage manifold. This configuration usually incorporates a disintegrator which provides adjustable amounts of product grinding depending upon speed and manifold setting. For sensitive or fine materials, the disintegrator can be omitted. Alternative feed locations are available to suit the material sensitivity and the final-product requirements. The full ring configuration gives a very high degree of control of both residence time and particle size, and is used for a wide variety of applications from small production rates of pharmaceutical and fine chemicals to large production rates of food products, bulk chemicals, and minerals. This is the most versatile configuration of the ring dryer. 3. P-type manifold—vertical configuration. The P ring dryer (see Fig. 12-100) incorporates a single-stage classifier and was developed specifically for use with heat-sensitive materials. The undried material is reintroduced into a cool part of the dryer in which it recirculates until it is dry enough to leave the circuit. An important element in optimizing the performance of a flash or ring dryer is the degree of dispersion at the feed point. Maximizing the product surface area in this region of highest evaporative driving force is a key objective in the design of this type of dryer. Ring dryers are fed using similar equipment to conventional flash dryers. Ring dryers with vertical configuration are normally fed by a flooded screw and a disperser which propels the wet feed into a high-velocity venturi, in which the bulk of the evaporation takes place. The full ring dryer normally employs an air-swept disperser or mill within the drying circuit to provide screenless grinding when required. Together with the manifold classifier this ensures a product with a uniform particle size. For liquid, slurry, or pasty feed materials, backmixing of the feed with a portion of the dry product will be carried out in order to produce a conditioned friable material. This further increases the versatility of the ring dryer, allowing it to handle sludge and slurry feeds with ease. Dried product is collected in either cyclones or baghouses depending upon the product-particle size. When primary collection is carried out in cyclones, secondary collection in a baghouse or scrubber is usually necessary in order to comply with environmental regulations. A rotary valve is used to provide an airlock at the discharge point. Screws are utilized to combine product from multiple cyclones or large baghouses. If required, a portion of the dried product is separated from the main stream and returned to the feed system for use as backmix. Design Methods for Ring Dryers Depending on the tempera-

12-81

ture sensitivity of the material to be processed, air inlet temperatures as high as 1400°F can be utilized. Even with heat-sensitive solids a high feed-moisture content may permit the use of a high air-inlet temperature since evaporation of surface moisture takes place at the wetbulb air temperature. Until the surface moisture has been removed it will prevent the solids temperature from exceeding the air wet-bulb temperature, by which time the air will generally have cooled significantly. Ring dryers have been used to process materials with feedmoisture contents between 2–95 percent, weight fraction. The product-moisture content has been controlled to values from 20 percent down to less than 1 percent. The air velocity required and air-tosolids ratio are determined by the evaporative load, the air-inlet temperature, and the exhaust-air humidity. Too high an exhaust-air humidity would prevent complete drying so a lower air-inlet temperature and air-to-solids ratio would be required. The air velocity within the dryer must be sufficient to convey the largest particle, or agglomerate. The air-to-solids ratio must be high enough to convey both the product and backmix, together with internal recycle from the manifold. For estimating purposes a velocity of 5000 ft/min, calculated at dryer exhaust conditions, is appropriate both for pneumatic-conveyor and ring dryers. Sizes, capacity, and costs for full manifold ring dryers are described on Table 12-35. Spray Dryers A spray dryer consists of a large cylindrical and usually vertical chamber into which material to be dried is sprayed in the form of small droplets and into which is fed a large volume of hot gas sufficient to supply the heat necessary to complete evaporation of the liquid. Heat transfer and mass transfer are accomplished by direct contact of the hot gas with the dispersed droplets. After completion of drying, the cooled gas and solids are separated. This may be accomplished partially at the bottom of the drying chamber by classification and separation of the coarse dried particles. Fine particles are separated from the gas in external cyclones or bag collectors. When only the coarse-particle fraction is desired for finished product, fines may be recovered in wet scrubbers; the scrubber liquid is concentrated and returned as feed to the dryer. Horizontal spray chambers are manufactured with a longitudinal screw conveyor in the bottom of the drying chamber for continuous removal of settled coarse particles. The principal use of spray dryers is for ordinary drying of water solutions and slurries. They are used also in combined drying and heat-treating operations, and for melt fusion and cooling of molten materials, e.g., ammonium nitrate prilling. The latter may be considered a solids size-enlargement process. Spray dryers are employed for wet-agglomeration processes to produce rapidly dispersible forms of concentrated food products, another form of size enlargement. In contacting performance, the spray dryer is similar to a pneumatic conveyor. It differs in application in that the feed material is usually a liquid solution, slurry, or paste capable of being dispersed in a fluidlike spray (rather than being composed of free-flowing particulate solids). Spray drying involves three fundamental unit processes: (1) liquid atomization, (2) gas-droplet mixing, and (3) drying from liquid droplets. Atomization is accomplished usually by one of three atomizing devices: (1) high-pressure nozzles, (2) two-fluid nozzles, and (3) high-speed centrifugal disks. With these atomizers, thin solutions may be dispersed into droplets as small as 2 µm. The largest drop sizes rarely exceed 500 µm (35 mesh). Because of the large total drying surface and small droplet sizes created, the actual drying time in a spray dryer is measured in seconds. Total residence of a particle in the system is on the average not more than 30 s. A review by Marshall [“Atomization and Spray Drying,” Chem. Eng. Prog. Monogr. Ser., 50, 2 (1954)] considers spray-drying theory in detail as well as the design and operating characteristics of modern spray dryers. A later survey of spray drying, which constitutes a good supplement to Marshall, was published by Masters [Ind. Eng. Chem., 60(10), 53–63 (1968)]. A more recent article summarizing scale-up of spray dryers has also been published by K. Masters [Drying Technology—An International Journal, 12(1&2), 235 (1994)]. Recently there has been considerable interest in the scale-up and analysis of spray dryers using computational fluid dynamics. D. E. Oakley [Drying Technology—An International Journal, 12(1&2), 217 (1994)]. Liquid atomization and dispersion are discussed in detail in Sec. 14. Atomizers commonly employed on spray dryers are described briefly in the following paragraphs.

12-82

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

FIG. 12-98

Feed-type ring dryer. (Barr—Rosin)

FIG. 12-99

Full manifold-type ring dryer. (Barr—Rosin) 12-83

12-84

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

FIG. 12-100

P-type ring dryer. (Barr—Rosin)

Special designs of spray dryers may provide for cooling air to enter around the chamber, closed systems for the recovery of solvents, and air sweepers or mechanical rakes to remove dry product from the walls and bottom of the chamber. Some are followed by pneumatic conveyors as depicted in Fig. 12-101, in which drying air is diluted with cool air for product cooling before separation. Spray dryers may

operate with cocurrent, mixed, or countercurrent flow of gas and solids. Inlet-gas temperatures may range from 425 to 1100 K. Pressure nozzles effect atomization by forcing the liquid under high pressure and with a high degree of spin through a small orifice. Pressures may range from 2700 to 69,000 kPa/m2, depending on the degree of atomization, capacity, and physical properties. Nozzle ori-

SOLIDS-DRYING EQUIPMENT TABLE 12-35

Sizes and Capacities of Full Ring Dryers*

Evaporative capacity (lb/hr)

Nominal area in plan (ft × ft)

Nominal height (ft)

Price FOB shops (US$)

1000 2000 5000 10000 15000 20000 40000

7 × 20 10 × 30 12 × 35 18 × 45 20 × 50 22 × 55 25 × 60

20 24 28 32 35 38 40

125,000 200,000 300,000 500,000 600,000 700,000 1,000,000

*Courtesy of Barr—Rosin. Based upon 1200°F air-inlet temperature. With an inlet temperature of 700°F the evaporative capacity would be half that given in the table. Prices are based upon carbon steel construction with cyclone collector (January, 1995). Motors and secondary dust collectors excluded

fices may range in size from 0.25- to 0.4-mm diameter, depending on the pressure desired for a given capacity and the degree of atomization required. For high pressures and when solids are in suspension in the liquid, the nozzle orifice will be subject to wear by erosion, and the orifice should be made of a hard alloy such as tungsten carbide or stellite. Maintenance on pressure nozzles is always a problem since erosion occurs with even the hardest inserts, and once the orifice has become scratched and nonuniform, good atomization is no longer possible. Likewise, incrustation and plugging by particles of foreign matter cause trouble. Piston pumps furnish the liquids at high pressure; erosion of the valves in these pumps is another maintenance problem. Spray characteristics of pressure nozzles depend on the pressure and nozzle-orifice size. Pressure affects not only the spray characteristics but also the capacity. If it is desired to reduce the amount of liquid sprayed by lowering the pressure, then the spray may become coarser. To correct this, a smaller orifice would be inserted, which might then require a higher pressure to produce the desired capacity, and a spray that would be finer than desired might result. Multiple nozzles tend to overcome this inflexible characteristic of pressure atomization, although several nozzles on a dryer complicate the chamber design and air-flow pattern and risk collision of particles, resulting in nonuniformity of spray and particle size. Two-fluid nozzles do not operate efficiently at high capacities and consequently are not used widely on plant-size spray dryers. Their chief advantage is that they operate at relatively low pressure, the liquid being 0 to 400 kPa/m2 pressure, while the atomizing fluid is usually no more than 700 kPa/m2 pressure. The atomizing fluid may be steam or air. Two-fluid nozzles have been employed for the dispersion of thick pastes and filter cakes not previously capable of being handled in ordinary atomizers [Baran, Ind. Eng. Chem., 56(10), 34–36 (1964); and Turba, Brit. Chem. Eng., 9(7), 457–460 (1964)]. Centrifugal disks atomize liquids by extending them in thin sheets which are discharged at high speeds from the periphery of the rapidly rotating, specially designed disk. The principal objectives in disk design are to ensure bringing the liquid to disk speed and to obtain a uniform drop-size distribution in the atomized liquid. Disk diameters range from 5 cm in small laboratory models to 35 cm for plant-size dryers. Disk speeds range from 3000 to 50,000 r/min. The high speed is generally used in small-diameter dryers. Usual speeds on plant-size dryers range from 4000 to 20,000 r/min, depending on disk diameter and the degree of atomization desired. The degree of atomization as a function of disk speed is affected by the product of disk diameter and speed, i.e., by peripheral speed as opposed to angular speed. Thus, a 13-cm disk operating at 30,000 r/min would be expected to atomize more finely than a 5-cm disk of the same design running at 50,000 r/min. Centrifugal-disk atomization is particularly advantageous for atomizing suspensions and pastes that erode and plug nozzles. Thick pastes can be handled if positive-pressure pumps are used to feed them to the disk. Disks are capable of operating over a wide range of feed rates and disk speeds without producing too variable a product. Centrifugal disks may be belt-driven, direct-driven by a high-speed electric motor powered by a frequency changer, or driven by a steam turbine. Direct

12-85

drive by an electric motor has advantages when very high speeds are required and when closely controlled speed variations are necessary. The life of high-speed bearings in centrifugal-disk atomizers depends on the conditions of operation. Average life may be 2000 h. A spare spray machine should be standard equipment. The particle-size distribution obtained by any one of the three methods of atomization depends on a number of factors. In general, the size distribution will depend on atomizer design, liquid properties, and degree of atomization. If the finest atomization possible is attempted, a limiting condition is approached, and the particle-size range, regardless of the method of atomization, will be narrow. This is particularly true of pressure nozzles, in which uniformity of size increases with pressure. On the other hand, for the production of a coarse product with a high percentage of large particles, the method of atomization will have a large effect on the particle-size distribution. Production of uniform coarse particles from centrifugal disks frequently can be obtained by careful design. One of the principal advantages of spray drying is the production of a spherical particle, which is usually not obtainable by any other drying method. This spherical particle may be solid or hollow, depending on the material, the feed condition, and the drying conditions. In general, aqueous solutions of materials such as soap, gelatin, and water-soluble polymers which form tough tenuous outer skins on drying will form hollow spherical particles when spray-dried. This is attributed to the formation of a casehardened outer surface on the particle which prevents liquid from reaching the surface from the particle interior. Because of high heat-transfer rates to the drops, the liquid at the center of the particle vaporizes, causing the outer shell to expand and form a hollow sphere. Sometimes the rate of vapor generation within the particle is sufficient to blow a hole through the wall of the spherical shell. Spherical particles may be obtained from true solutions or from slurries and may be produced by any of the previously described atomizers. The physical properties of spray-dried materials are subject to considerable variation, depending on the direction of flow of the inlet gas and its temperature, the degree and uniformity of atomization, the solids content of the feed, the temperature of the feed, and the degree of aeration of the feed. The properties of the product usually of greatest interest are (1) particle size, (2) bulk density, and (3) dustiness. The particle size is a function of atomizer-operating conditions and also of the solids content, liquid viscosity, liquid density, and feed rate. In general, particle size increases with solids content, viscosity, density, and feed rate. The bulk density of spray-dried solids is frequently the critical property subject to close control. The bulk density of material from a spray dryer may usually be increased by the following operating changes: (1) reducing droplet size, (2) reducing inlet-air temperature, (3) increasing air throughput, (4) increasing air turbulence, (5) employing countercurrent rather than cocurrent gas flow, and (6) effecting a wide range of size distribution from the atomizer. Chaloud et al. evaluated qualitatively the effects of operating variables on the bulk density of particles from detergent spray dryers [Chem. Eng. Prog., 53, 12, 593–596 (1957)]. A dusty product is caused by fine atomization or particle degradation after drying. Thin-wall hollow particles are susceptible to breakage during collection. Fine atomization and a high gas temperature contribute to high production rates in small drying chambers; they also generate fine particles and thin-wall spheres. Spray-drying installations yielding exceedingly fine and dusty products are often the result of an honest effort to design equipment for maximum capacity at a minimum investment. Large solids particles or heavy-wall spheres require longer drying cycles, hence larger drying chambers. Careful study in the pilot plant is necessary. In commercial installations, classification of particles and separation of a fine fraction from coarse product may be accomplished by countercurrent flow of gas and solids. The majority of spray dryers in commercial use employ cocurrent flow of gas and solids. Countercurrent-flow dryers are used primarily for drying soaps and detergents. Their classifying ability is useful in these applications. Air flow is upward, carrying entrained fines from the top of the chamber. The coarse product settles and is removed

12-86

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

(a)

(b) FIG. 12-101

(c)

Alternative chambers and gas solids contacting methods in spray dryers.

separately from the bottom of the chamber. Horizontal spray dryers always employ cocurrent flow of gas and solids. A swirling motion is imparted to the air to improve mixing. Many variations of air-flow patterns inside the drying chamber are employed commercially; most are intended primarily to produce turbulence and thorough mixing of gas and droplets and to achieve the most effective use of the chamber volume. The treatment of drying gas external to the drying chamber may take a variety of forms depending on whether indirect heating (Figs. 12-102 and 103) or direct heating (Figs. 12-104 and 105) is selected; and whether the system is open (Fig. 12-104), semiclosed (Figs. 12-102

and 105), or totally closed (Fig. 12-103); and if solvent is flammable, toxic, or total solvent recovery is required. Applications of Spray Dryers The major and most successful drying applications of spray dryers are for solutions, slurries, and pastes which (1) cannot be dewatered mechanically, (2) are heatsensitive and cannot be exposed to high-temperature atmospheres for long periods, or (3) contain ultrafine particles which will agglomerate and fuse if dried in other than a dilute condition. In other applications, spray drying is rarely competitive on a cost basis with two-step dewatering and solids-drying processes. The cost of bag collectors for solids recovery from large volumes of exit gas may double the cost of a spray-

SOLIDS-DRYING EQUIPMENT

FIG. 12-102

12-87

Semiclosed spray-drying cycle with indirect heating. (NIRO, Inc.)

FIG. 12-103

Closed spray-drying cycle with indirect heating. (NIRO, Inc.)

dryer installation. Additional costs must usually be justified on the basis of some improvement in product quality, such as particle form, size, flavor, color, or heat stability. Spray drying is applicable to heatsensitive products such as milk powders and other foods and pharmaceuticals because of the short contact time in the dryer hot zone. Further, the water film on the liquid drop protects the solids from high gas temperatures. Drying is carried out at essentially the dryingair wet-bulb temperature. Color pigments are examples of the class of products for which it is desired to maintain as closely as possible the

original solids particle size. Table 12-36 lists typical materials which have been successfully spray-dried. One other class of products particularly applicable to spray dryers is solids slurries, containing extremely fine particles, which is nonnewtonian in flow characteristics and remains fluid at very low moisture content. Certain classes of clays are found in this category. Also, spray dryers have been developed for encapsulation processes to convert liquid volatile flavors and perfumes to particulate solids forms [Maleeny, Soap Chem. Spec., 34, 1, 135–141 (1958)].

12-88

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING

FIG. 12-104

FIG. 12-105

Open spray-drying system with direct-fired heater. (NIRO, Inc.)

Semiclosed spray-drying cycle with direct heating. (NIRO, Inc.)

If the product in no way adheres to the dryer parts and simple cyclone collectors are sufficient for gas-solids separation, batch operation of a spray dryer may be considered. Otherwise, the time and costs for cleaning the large equipment parts make them rarely economical for other than continuous processing of a single material. Design Methods for Spray Dryers Design variables must be

established by experimental tests before final design of a chamber can be carried out. In general, chamber size, atomizer selection, and separation auxiliaries will be determined by the desired physical characteristics of the product. Drying by itself is rarely a problem. An installed spray dryer is relatively inflexible in meeting changing operating requirements while maintaining a constant production rate.

SOLIDS-DRYING EQUIPMENT TABLE 12-36 Some Materials That Have Been Successfully Spray-Dried in a 6-m-Diameter by 6-m-High Chamber with a Centrifugal-Disk Atomizer* Air temperature, K

For single-fluid pressure nozzles, a rule of thumb is employed: 3

Material

In

Out

Blood, animal Yeast Zinc sulfate Lignin Aluminum hydroxide Silica gel Magnesium carbonate Tanning extract Coffee extract Detergent A Detergent B Detergent C Manganese sulfate Aluminum sulfate Urea resin A Urea resin B Sodium sulfide Pigment

440 500 600 475 590 590 590 440 420 505 510 505 590 415 535 505 500 515

345 335 380 365 325 350 320 340 355 395 390 395 415 350 355 360 340 335

65 86 55 63 93 95 92 46 70 50 63 40 50 70 60 70 50 73

Evaporation rate, kg/s 5.9 8.2 10.0 6.9 19.4 16.9 18.2 5.2 3.8 5.0 6.2 2.6 5.5 1.7 3.8 1.9 2.0 13.2

*Courtesy of NIRO, Inc. 3 NOTE: The fan on this dryer handles about 5.2 m /s at outlet conditions. The outlet-air temperature includes cold air in-leakage, and the true temperature drop caused by evaporation must therefore be estimated from a heat balance.

Important variables which must be fixed before design of a commercial dryer are the following: 1. The form and particle size of product required 2. The physical properties of the feed: moisture, viscosity, density, etc. 3. The maximum inlet-gas and product temperatures Theoretical correlations of spray-dryer performance published by Gluckert [Am. Inst. Chem. Eng. J., 8(4), 460–466 (1962)] may be employed for the scale-up of laboratory dryers and, in some instances, for estimating dryer requirements in the absence of any tests. Several assumptions are necessary. 1. The largest droplets, which dry most slowly, are the limiting portion of the spray. They determine ultimate chamber dimensions and are employed for the evaluation. 2. The largest droplet in a spray population is 3 times the diameter of the average drop size [see Eq. (12-66)]. 3. A droplet Nusselt number = 2, corresponding to pure conduction (Reynolds number = 0) to infinity, is employed for evaluating the coefficient of heat transfer. 4. Drying conditions, because of turbulence and gas mixing, are uniform throughout the chamber; i.e., the entire chamber is at the gas exit temperature—this fact has been well established in many chambers except in the immediate zone of gas inlet and spray atomization. 5. The temperature driving force for drying is the difference between the drying-gas outlet temperature and, in the case of pure water, the gas wet-bulb temperature. In the case of a solution, the adiabatic saturation temperature of the pure saturated solution is employed rather than the wet-bulb temperature. Methods for calculating average and maximum drop sizes from various atomizers are given by Marshall (op. cit.). For pneumatic nozzles, an expression developed by Nukiyama and Tanasawa is recommended: 0.45 µ 1000QL 1.5 苶 1920兹α Xvs = ᎏ + 597 ᎏ (12-63) 苶 ᎏ 苶ρ 苶l苶 Qa 兹α 苶l苶 Va兹ρ

冢

QL = liquid volumetric flow rate Qa = air volumetric flow rate 苶P 苶 X vs = 500/兹∆ 苶

% water in feed

冣 冢

冣

where 苶 Xvs = average drop diameter, µm (a drop with the same volume-surface ratio as the total sum of all drops formed) α = surface tension, dyn/cm µ = liquid viscosity, P Va = relative velocity between air and liquid, ft/s ρl = liquid density, g /cm3

12-89

(12-64)

where ∆P = pressure drop across nozzle, lb/in2. For centrifugal disks, the relation of Friedman, Gluckert, and Marshall is employed [Chem. Eng. Prog., 48, 181 (1952)]: Dvs Γ 0.6 µ 0.2 αρlLw 0.1 (12-65) ᎏ = 0.4 ᎏ2 ᎏ ᎏ r ρl Nr Γ Γ2 where Dvs = average drop diameter, ft r = disk radius, ft Γ = spray mass velocity, lb/(min⋅ft of wetted disk periphery) ρl = liquid density, lb/ft3 N = disk speed, r/min µ = liquid viscosity, lb/(ft⋅min) α = surface tension, lb/min2 Lw = wetted disk periphery, ft

冢

冣 冢 冣 冢

冣

NOTE: All groups are dimensionless. To convert dynes per square centimeter to joules per square meter, multiply by 10−3; to convert poises to newton-seconds per square meter, multiply by 10−1; to convert feet per second to meters per second, multiply by 0.3048; to convert feet to meters, multiply by 0.3048; to convert pounds per minute-foot to kilograms per second-meter, multiply by 0.025; to convert pounds per cubic foot to kilograms per cubic meter, multiply by 16.019; to convert pounds per minute squared to kilograms per second squared, multiply by 1.26 × 10−4; to convert British thermal units per hour to kilojoules per second, multiply by 2.63 × 10−4; and to convert British thermal units per hour-square foot-degree Fahrenheit per foot to joules per square meter-second-kelvin per meter, multiply by 1.7307. Inspection of these relationships will show that the variables are difficult to specify in the absence of tests except when handling pure liquids—which in spray drying is rare indeed. The most useful method for employing these equations is to conduct small-scale drying tests in a chamber under conditions in which wall impingement and sticking are incipient. The maximum particle size can then be back-calculated by using the relationships given in the following paragraphs, and the effects of changing atomizing variables evaluated by using the preceding equations:

X 苶m = 3X 苶vs

(12-66)

where 苶 Xm = maximum drop diameter, µm. Gluckert gives the following relationships for calculating heat transfer under various conditions of atomization: Two-fluid pneumatic nozzles: 6.38Kf v2/3 ∆t ws ρa wa + ws Q = ᎏᎏ (12-67) ᎏ ᎏᎏ Dm2 ρs waVa wa Single-fluid pressure nozzles: 10.98Kf v2/3 ∆t ρt (12-68) Q = ᎏᎏ Ds ᎏ Dm2 ρs Centrifugal-disk atomizers: 4.19Kf (Rc − r/2)2 ∆t ws ρt (12-69) Q = ᎏᎏᎏ ᎏ D m2 ρs rN where Q = rate of heat transfer to spray, Btu/h Kf = thermal conductivity of gas film surrounding the droplet, Btu/(h⋅ft2)(°F⋅ft), evaluated at the average between dryer gas and drop temperature v = volume of dryer chamber, ft3 ∆t = temperature driving force (under terminal conditions described above), °F Dm = maximum drop diameter, ft ws = weight rate of liquid flow, lb/h ρs = density of liquid, lb/ft3

冪莦莦 冪莦

冪莦

12-90

PSYCHROMETRY, EVAPORATIVE COOLING, AND SOLIDS DRYING wa = weight rate of atomizing air flow, lb/h ρa = density of atomizing air, lb/ft3 Va = velocity of atomizing air at atomizer, ft/h Ds = diameter of pressure-nozzle discharge orifice, ft ρt = density of dryer gas at exit conditions, lb/ft3 Rc = radius of drying chamber with centrifugal disk, ft r = radius of disk, ft N = rate of disk rotation, r/h

For proper use of the equations, the chamber shape must conform to the spray pattern. With cocurrent gas-spray flow, the angle of spread of single-fluid pressure nozzles and two-fluid pneumatic nozzles is such that wall impingement will occur at a distance approximately four chamber diameters below the nozzle; therefore, chambers employing these atomizers should have vertical height-todiameter ratios of at least 4 and, more usually, 5. The discharge cone below the vertical portion should have a slope of at least 60°, to minimize settling accumulations, and is used entirely to accelerate gas and solids for entry into the exit duct. The critical dimension of a centrifugal-disk chamber is the diameter. Vertical height is usually 0.5 to 1.0 times the diameter; the large cone is needed mainly to accelerate to the discharge duct and prevent settling; it contributes little to drying capacity. Cost Data for Spray Dryers Drying chambers, ductwork, and cyclone separators are usually constructed of stainless steel. Savings of roughly 20 percent may be achieved on the total purchase cost by using carbon steel; the increasing tendency toward the use of heatresistant and corrosion-resistant plastic coatings (epoxy resins) makes the future appear promising for greater use of carbon steel construction. Wide differences in cost may be experienced in the selection of basic equipment. Air heaters vary in price range according to the selection of steam, electricity, direct-fired, and indirect-fired oil or gas heaters. Dust-collection equipment may consist of cyclone collectors or bag-type filters and may include a wet scrubber. Costs of nozzle and centrifugal atomizers are usually comparable. While the centrifugal atomizer requires mechanical gearing and motor drive, a highpressure nozzle requires a high-pressure pump, which will usually more than offset the cost of gearing and motor for the centrifugal atomizer. Auxiliary equipment which may be included comprises air filters, drying-chamber insulation, and mechanical or pneumatic cooling conveyors. A minimum of instrumentation consists of indicating and recording thermometers for inlet-air and outlet temperatures, an ammeter for atomizer motor drive (or a pressure gauge for nozzle atomization), a flowmeter, manometers, a high-temperature alarm, and a panelboard with push-button stations for all equipment. The drying process may be completely controlled automatically with some additional instrumentation. Spray dryers may operate under positive, negative, or neutral pressures. In general, pressure drop in a complete system will range from 15 to 50 cm of water, depending on duct size and separation equipment employed. Agitated Flash Dryers Agitated flash dryers produce fine powders from feeds with high solids contents, in the form of filter cakes, pastes, or thick, viscous liquids. Many continuous dryers are unable to dry highly viscous feeds. Spray dryers require a pumpable feed. Conventional flash dryers often require backmixing dry product to the feed in order to fluidize. Other drying methods for viscous pastes and filter cakes are well known, such as contact, drum, band and tray dryers. They all require long processing time, large floor space, high maintenance, and after treatment such as milling. The agitated flash dryer offers a number of process advantages: such as ability to dry pastes, sludges, and filter cakes to a homogeneous, fine powder in a single-unit operation; continuous operation; compact layout; effective heat and mass transfer—short drying times; negligible heat loss—high thermal efficiency; and easy access and cleanability. The agitated flash dryer (Fig. 12-106) consists of four major components: feed system, drying chamber, heater, and exhaust air system. Wet feed enters the feed tank which has a slow-rotating impeller to break up large particles. The level in the feed tank is maintained by a

Bag filter

Feed inlet

Product outlet

Feed tank

Drying chamber Feed dosing

Heater

FIG. 12-106

Agitated flash dryer with open cycle. (NIRO, Inc.)

level controller. The feed is metered at a constant rate into the drying chamber via a screw conveyor mounted under the feed tank. If the feed is shear thinning and can be pumped, the screw feeder can be replaced by a positive-displacement pump. The drying chamber is the heart of the system consisting of three important components: air disperser, rotating disintegrator, and drying section. Hot, drying air enters the air disperser tangentially and is introduced into the drying chamber as a swirling airflow. The swirling airflow is established by a guide-vane arrangement. The rotating disintegrator is mounted at the base of the drying chamber. The feed, exposed to the hot, swirling airflow and the agitation of the rotating disintegrator, is broken up and dried. The fine dry particles exit with the exhaust air and are collected in the bag filter. The speed of the rotating disintegrator controls the particles’ size. The outlet air temperature controls the product-moisture content. The drying air is heated either directly or indirectly, depending upon the feed material, powder properties, and available fuel source. The heat sensitivity of the product determines the drying air temperature. The highest possible value is used to optimize thermal efficiency. A bag filter is usually recommended for collecting the fine particles produced. The exhaust fan maintains a slight vacuum in the dryer, to prevent powder leakage into the surroundings. The appropriate process system is selected according to the feed and powder characteristics, available heating source, energy utilization, and operational health and safety requirements. Open systems use atmospheric air for drying. In cases where products pose a potential for dust explosion, plants are provided with pressure relief or suppression systems. For recycle systems, the drying system medium is recycled and the evaporated solvent recovered as condensate. There are two alternative designs: Self-inertizing mode, where oxygen content is held below 5 percent by combustion control at the heater. This is recommended for products with serious dustexplosion hazards; Inert mode, where nitrogen is the drying gas. This is used when an organic solvent is evaporated or product oxidation during drying must be prevented. Design Methods The size of the agitated flash dryer is based on the evaporation rate required. The operating temperatures are product-specific. Once established, they determine the airflow requirements. The drying chamber is designed based on air velocity (approximately 3–4 m/s) and residence time (product-specific).

Section 13

Distillation

J. D. Seader, Ph.D., Professor of Chemical Engineering, University of Utah, Salt Lake City, Utah; Fellow, American Institute of Chemical Engineers; Member, American Chemical Society; Member, American Society for Engineering Education. (Section Editor*) Jeffrey J. Siirola, Ph.D., Research Fellow, Eastman Chemical Company; Member, National Academy of Engineering; Fellow, American Institute of Chemical Engineers, American Chemical Society, American Association for Artificial Intelligence, American Society for Engineering Education. (Enhanced Distillation) Scott D. Barnicki, Ph.D., Senior Research Chemical Engineer, Eastman Chemical Company. (Enhanced Distillation)

CONTINUOUS-DISTILLATION OPERATIONS General Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equilibrium-Stage Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex Distillation Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Related Separation Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Synthesis of Multicomponent Separation Systems . . . . . . . . . . . . . . . . .

13-4 13-4 13-4 13-5 13-9

THERMODYNAMIC DATA Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phase Equilibrium Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphical K-Value Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analytical K-Value Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13-10 13-10 13-10 13-16

DEGREES OF FREEDOM AND DESIGN VARIABLES Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analysis of Elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analysis of Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Units and Complex Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13-22 13-22 13-23 13-24

SINGLE-STAGE EQUILIBRIUM-FLASH CALCULATIONS Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bubble Point and Dew Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Isothermal Flash . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adiabatic Flash . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Flash Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Three-Phase Flash . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex Mixtures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13-25 13-25 13-25 13-26 13-26 13-26 13-26

GRAPHICAL METHODS FOR BINARY DISTILLATION Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phase Equilibrium Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13-26 13-27

McCabe-Thiele Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Operating Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Condition of the Feed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equilibrium-Stage Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total-Column Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Feed-Stage Location. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minimum Stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minimum Reflux. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intermediate Reboilers and Condensers . . . . . . . . . . . . . . . . . . . . . . . Optimum Reflux Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Difficult Separations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stage Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miscellaneous Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13-27 13-27 13-28 13-29 13-29 13-32 13-32 13-32 13-32 13-32 13-32 13-34 13-34

APPROXIMATE MULTICOMPONENT DISTILLATION METHODS Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fenske-Underwood-Gilliland (FUG) Shortcut Method. . . . . . . . . . . . . Example 1: Calculation of FUG Method. . . . . . . . . . . . . . . . . . . . . . . Kremser Group Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 2: Calculation of Kremser Method. . . . . . . . . . . . . . . . . . . .

13-35 13-35 13-36 13-37 13-39

RIGOROUS METHODS FOR MULTICOMPONENT DISTILLATION-TYPE SEPARATIONS Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thiele-Geddes Stage-by-Stage Method for Simple Distillation . . . . . . . Example 3: Calculation of TG Method . . . . . . . . . . . . . . . . . . . . . . . . Equation-Tearing Procedures Using the Tridiagonal-Matrix Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tridiagonal-Matrix Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BP Method for Distillation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 4: Calculation of the BP Method . . . . . . . . . . . . . . . . . . . . .

13-39 13-40 13-40 13-43 13-44 13-45 13-46

* Certain portions of this section draw heavily on the work of Buford D. Smith, editor of this section in the fifth edition. 13-1

13-2

DISTILLATION

SR Method for Absorption and Stripping . . . . . . . . . . . . . . . . . . . . . . Example 5: Calculation of the SR Method . . . . . . . . . . . . . . . . . . . . . Simultaneous-Correction Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . Naphtali-Sandholm SC Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 6: Calculation of Naphtali-Sandholm SC Method . . . . . . . . Inside-Out Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 7: Calculation of Inside-Out Method . . . . . . . . . . . . . . . . . . Homotopy-Continuation Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stage Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rate-Based Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Material Balances (2C + 2 Equations) . . . . . . . . . . . . . . . . . . . . . . . . . Energy Balances (3 Equations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass-Transfer Rates (2C − 2 Equations). . . . . . . . . . . . . . . . . . . . . . . Summation of Mole Fractions (2 Equations) . . . . . . . . . . . . . . . . . . . Hydraulic Equation for Stage Pressure Drop (1 Equation) . . . . . . . . Interface Equilibrium (C Equations) . . . . . . . . . . . . . . . . . . . . . . . . . Example 8: Calculation of Rate-Based Distillation . . . . . . . . . . . . . . .

13-47 13-47 13-48 13-48 13-49 13-49 13-51 13-51 13-52 13-52 13-53 13-53 13-53 13-53 13-53 13-53 13-54

ENHANCED DISTILLATION Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Azeotropism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Residue Curve Maps and Distillation Region Diagrams . . . . . . . . . . . . Applications of RCM and DRD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extension to Batch Distillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Azeotropic Distillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exploitation of Homogeneous Azeotropes . . . . . . . . . . . . . . . . . . . . . Exploitation of Pressure Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . Exploitation of Boundary Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . Exploitation of Azeotropy and Liquid-Phase Immiscibility . . . . . . . . Design and Operation of Azeotropic Distillation Columns . . . . . . . . Extractive Distillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solvent Effects in Extractive Distillation. . . . . . . . . . . . . . . . . . . . . . . Extractive Distillation Design and Optimization . . . . . . . . . . . . . . . . Solvent Screening and Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extractive Distillation by Salt Effects . . . . . . . . . . . . . . . . . . . . . . . . .

13-54 13-54 13-56 13-58 13-66 13-68 13-68 13-69 13-72 13-73 13-73 13-75 13-75 13-75 13-76 13-77 13-79 13-81

Reactive Distillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation, Modeling, and Design Feasibility . . . . . . . . . . . . . . . . . . Mechanical Design and Implementation Issues . . . . . . . . . . . . . . . . . Process Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13-81 13-81 13-81 13-83 13-83

PETROLEUM AND COMPLEX-MIXTURE DISTILLATION Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Characterization of Petroleum and Petroleum Fractions . . . . . . . . . . . . Applications of Petroleum Distillation. . . . . . . . . . . . . . . . . . . . . . . . . . . Design Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 9: Simulation Calculation of an Atmospheric Tower . . . . . .

13-85 13-86 13-89 13-89 13-93

BATCH DISTILLATION Simple Batch Distillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Batch Distillation with Rectification . . . . . . . . . . . . . . . . . . . . . . . . . . . . Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Approximate Calculation Procedures for Binary Mixtures . . . . . . . . . . . Operating Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Batch Rectification at Constant Reflux . . . . . . . . . . . . . . . . . . . . . . . . Batch Rectification at Constant Overhead Composition . . . . . . . . . . Other Operating Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of Column Holdup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shortcut Methods for Multicomponent Batch Rectification . . . . . . . . . Calculational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rigorous Computer-Based Calculation Procedures . . . . . . . . . . . . . . Example 10: Calculation of Multicomponent Batch Distillation . . . . Rapid Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13-96 13-96 13-96 13-97 13-98 13-98 13-98 13-99 13-99 13-100 13-100 13-100 13-102 13-103

DYNAMIC DISTILLATION Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-104 Ideal Binary Distillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-104 Multicomponent Distillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-105 PACKED COLUMNS

DISTILLATION

13-3

Nomenclature and Units

Symbol A A C D E E E F F G G H H H K KC KD Kd L M M N N Nc Ni Nm Np Nr No N P P P sat Q Qc Qr Q R Rm S S S S T U V W

Definition Absorption factor Area Number of chemical species Distillate flow rate Deviation from set point Residual of heat-transfer expression Residual of phase equilibrium expression Feed flow rate Vector of stage functions Interlink flow rate Volume holdup of liquid Residual of energy balance Height of a transfer unit Enthalpy Vapor-liquid equilibrium ratio (K value) Controller gain Chemical equilibrium constant for dimerization Liquid-liquid distribution ratio Liquid flow rate Residual of component material balance Liquid holdup Number of transfer units Number of equilibrium stages Number of relationships Number of design variables Minimum number of equilibrium stages Number of phases Number of repetition variables Number of variables Rate of mass transfer Pressure Residual of pressure-drop expression Vapor pressure Heat-transfer rate Condenser duty Reboiler duty Residual of phase-equilibrium expression External-reflux ratio Minimum-reflux ratio Residual of mole-fraction sum Sidestream flow rate Stripping factor Vapor-sidestream ratio Temperature Liquid-sidestream rate Vapor flow rate Vapor-sidestream rate

SI units

U.S. customary units

m2

ft2

(kg⋅mol)/s

(lb⋅mol)/h

kW

Btu/h

(kg⋅mol)/s

(lb⋅mol)/h

(kg⋅mol)/s

(lb⋅mol)/h

(kg⋅mol)/s m3 kW m J/(kg⋅mol)

(lb⋅mol)/h ft3 Btu/h ft Btu/(lb⋅mol)

Symbol X a b d e f f f g g h h ᐉ p q

(kg⋅mol)/s (kg⋅mol)/s

(lb⋅mol)/h (lb⋅mol)/h

kg⋅mol

lb⋅mol

qc qr r s t t v w x y z

Definition Vector of stage variables Activity Component flow rate in bottoms Component flow rate in distillate Rate of heat transfer Fraction of feed leaving in bottoms Fugacity Function in homotopy expression Function in homotopy expression Residual of energy balance Height Homotopy function Component flow rate in liquid Pressure Measure of thermal condition of feed Condenser duty Reboiler duty Sidestream ratio Liquid-sidestream ratio Homotopy parameter Time Component flow rate in vapor Weight fraction Mole fraction in liquid Mole fraction in vapor Mole fraction in feed

U.S. customary units

SI units

(kg⋅mol)/s

(lb⋅mol)/h

(kg⋅mol)/s

(lb⋅mol)/h

kW

Btu/h

Pa

psia

kW m

Btu/h ft

(kg⋅mol)/s

(lb⋅mol)/h

kPa

psia

kW kW

Btu/h Btu/h

s (kg⋅mol)/s

h (lb⋅mol)/h

s

h

s

h

Greek symbols

(kg⋅mol)/s Pa Pa

(lb⋅mol)/h psia psia

Pa kW kW kW

psia Btu/h Btu/h Btu/h

α γ ε ξ η θ Θ Θ λ τ τ Φ φ ˆ Φ

(kg⋅mol)/s

(lb⋅mol)/h

K (kg⋅mol)/s (kg⋅mol)/s (kg⋅mol)/s

°R (lb⋅mol)/h (lb⋅mol)/h (lb⋅mol)/h

ΦA ΦS Ψ

Relative volatility Activity coefficient Convergence criterion Scale factor Murphree-stage efficiency Time for distillation Parameter in Underwood equations Holland theta factor Eigenvalue Sum of squares of residuals Feedback-reset time Fugacity coefficient of pure component Entrainment or occlusion ratio Fugacity coefficient in mixture Fraction of a component in feed vapor that is not absorbed Fraction of a component in entering liquid that is not stripped Factor in Gilliland correlation

GENERAL REFERENCES: Billet, Distillation Engineering, Chemical Publishing, New York, 1979. Fair and Bolles, “Modern Design of Distillation Columns,” Chem. Eng., 75(9), 156 (Apr. 22, 1968). Fredenslund, Gmehling, and Rasmussen, Vapor-Liquid Equilibria Using UNIFAC, a Group Contribution Method, Elsevier, Amsterdam, 1977. Friday and Smith, “An Analysis of the Equilibrium Stage Separation Problem—Formulation and Convergence,” Am. Inst. Chem. Eng. J., 10, 698 (1964). Hengstebeck, Distillation—Principles and Design Procedures, Reinhold, New York, 1961. Henley and Seader, Equilibrium-Stage Separation Operations in Chemical Engineering, Wiley, New York, 1981. Hoffman, Azeotropic and Extractive Distillation, Wiley, New York, 1964. Holland, Fundamentals and Modeling of Separation Processes, Prentice-Hall, Englewood Cliffs, N.J., 1975. Holland, Fundamentals of Multicomponent Distil-

lation, McGraw-Hill, New York, 1981. King, Separation Processes, 2d ed., McGraw-Hill, New York, 1980. Kister, Distillation Design, McGraw-Hill, New York, 1992. Kister, Distillation Operation, McGraw-Hill, New York, 1990. Robinson and Gilliland, Elements of Fractional Distillation, 4th ed., McGrawHill, New York, 1950. Rousseau, ed., Handbook of Separation Process Technology, Wiley-Interscience, New York, 1987. Seader, The B.C. (Before Computers) and A.D. of Equilibrium-Stage Operations, Chem. Eng. Educ., Vol. 14(2), (Spring 1985). Seader, Chem. Eng. Progress, 85(10), 41 (1989). Smith, Design of Equilibrium Stage Processes, McGraw-Hill, New York, 1963. Treybal, Mass Transfer Operations, 3d ed., McGraw-Hill, New York, 1980. Ullmann’s Encyclopedia of Industrial Chemistry, Vol. B3, VCH, Weinheim, 1988. Van Winkle, Distillation, McGraw-Hill, New York, 1967.

CONTINUOUS-DISTILLATION OPERATIONS GENERAL PRINCIPLES Separation operations achieve their objective by the creation of two or more coexisting zones which differ in temperature, pressure, composition, and/or phase state. Each molecular species in the mixture to be separated reacts in a unique way to differing environments offered by these zones. Consequently, as the system moves toward equilibrium, each species establishes a different concentration in each zone, and this results in a separation between the species. The separation operation called distillation utilizes vapor and liquid phases at essentially the same temperature and pressure for the coexisting zones. Various kinds of devices such as random or structured packings and plates or trays are used to bring the two phases into intimate contact. Trays are stacked one above the other and enclosed in a cylindrical shell to form a column. Packings are also generally contained in a cylindrical shell between hold-down and support plates. A typical tray-type distillation column plus major external accessories is shown schematically in Fig. 13-1. The feed material, which is to be separated into fractions, is introduced at one or more points along the column shell. Because of the difference in gravity between vapor and liquid phases, liquid runs down the column, cascading from tray to tray, while vapor flows up the column, contacting liquid at each tray. Liquid reaching the bottom of the column is partially vaporized in a heated reboiler to provide boil-up, which is sent back up the column. The remainder of the bottom liquid is withdrawn as bottoms, or bottom product. Vapor reaching the top of the column is cooled and condensed to liquid in the overhead condenser. Part of this liquid is returned to the column as reflux to provide liquid overflow. The remainder of the overhead stream is withdrawn as distillate, or overhead product. In some cases only part of the vapor is condensed so that a vapor distillate can be withdrawn. This overall flow pattern in a distillation column provides countercurrent contacting of vapor and liquid streams on all the trays through the column. Vapor and liquid phases on a given tray approach thermal, pressure, and composition equilibriums to an extent dependent upon the efficiency of the contacting tray. The lighter (lower-boiling) components tend to concentrate in the vapor phase, while the heavier (higher-boiling) components tend toward the liquid phase. The result is a vapor phase that becomes richer in light components as it passes up the column and a liquid phase that becomes richer in heavy components as it cascades downward. The overall separation achieved between the distillate and the bottoms depends primarily on the relative volatilities of the components, the number of contacting trays, and the ratio of the liquidphase flow rate to the vapor-phase flow rate. If the feed is introduced at one point along the column shell, the 13-4

column is divided into an upper section, which is often called the rectifying section, and a lower section, which is often referred to as the stripping section. These terms become rather indefinite in multiplefeed columns and in columns from which a liquid or vapor sidestream is withdrawn somewhere along the column length in addition to the two end-product streams. EQUILIBRIUM-STAGE CONCEPT Until recently, energy and mass-transfer processes in an actual distillation column were considered too complicated to be readily modeled in any direct way. This difficulty was circumvented by the equilibriumstage model, developed by Sorel in 1893, in which vapor and liquid streams leaving an equilibrium stage are in complete equilibrium with each other and thermodynamic relations can be used to determine the temperature of and relate the concentrations in the equilibrium streams at a given pressure. A hypothetical column composed of equilibrium stages (instead of actual contact trays) is designed to accomplish the separation specified for the actual column. The number of hypothetical equilibrium stages required is then converted to a number of actual trays by means of tray efficiencies, which describe the extent to which the performance of an actual contact tray duplicates the performance of an equilibrium stage. Alternatively and preferably, tray inefficiencies can be accounted for by using rate-based models that are described below. Use of the equilibrium-stage concept separates the design of a distillation column into three major steps: (1) Thermodynamic data and methods needed to predict equilibrium-phase compositions are assembled. (2) The number of equilibrium stages required to accomplish a specified separation, or the separation that will be accomplished in a given number of equilibrium stages, is calculated. (3) The number of equilibrium stages is converted to an equivalent number of actual contact trays or height of packing, and the column diameter is determined. Much of the third step is eliminated if a rate-based model is used. This section deals primarily with the second step. Section 14 covers the third step. Sections 3 and 4 cover the first step, but a summary of methods and some useful data are included in this section. COMPLEX DISTILLATION OPERATIONS All separation operations require energy input in the form of heat or work. In the conventional distillation operation, as typified in Fig. 13-1, energy required to separate the species is added in the form of heat to the reboiler at the bottom of the column, where the temperature is highest. Also, heat is removed from a condenser at the top of the column, where the temperature is lowest. This frequently results

CONTINUOUS-DISTILLATION OPERATIONS

Schematic diagram and nomenclature for a simple distillation column with one feed, a total overhead condenser, and a partial reboiler.

FIG. 13-1

in a large energy-input requirement and low overall thermodynamic efficiency, when the heat removed in the condenser is wasted. Complex distillation operations that offer higher thermodynamic efficiency and lower energy-input requirements have been under intense investigation. In some cases, all or a portion of the energy input is as work. Complex distillation operations may utilize single columns, as shown in Fig. 13-2 and discussed by Petterson and Wells [Chem. Eng., 84(20), 78 (Sept. 26, 1977)], Null [Chem. Eng. Prog., 72(7), 58 (1976)], and Brannon and Marple [Am. Inst. Chem. Eng. Symp. Ser. 76, 192, 10 (1980)], or two or more columns that are thermally linked as shown in Figs. 13-3 and 13-6 and discussed by Petterson and Wells (op. cit.) and Mah, Nicholas, and Wodnik [Am. Inst. Chem. Eng. J., 23, 651 (1977)]. In Fig. 13-2a, which is particularly useful when a large temperature difference exists between the ends of the column, interreboilers add heat at lower temperatures and/or intercondensers remove heat at higher temperatures. As shown in Fig. 13-2b, these intermediate heat exchangers may be coupled with a heat pump that takes energy from the intercondenser and uses shaft work to elevate this energy to a temperature high enough to transfer it to the interreboiler. Particularly when the temperature difference between the ends of the column is not large, any of the three heat-pump systems in Fig. 13-2c, d, and e that involve thermal coupling of the overhead condenser and bottoms reboiler might be considered to eliminate external heat transfer almost entirely, substituting shaft work as the prime energy input for achieving the separation. More complex arrangements are considered by Björn, Grén, and Ström [Chem. Eng. Process., 29, 185 (1991)]. Alternatively, the well-known multiple-

13-5

column or split-tower arrangement of Fig. 13-3a, which corresponds somewhat to the energy-saving concept employed in multieffect evaporation, might be used. The feed is split more or less equally among columns that operate in parallel, but at different pressures, in a cascade that decreases from left to right. With proper selection of column-operating pressure, this permits the overhead vapor from the higher-pressure column to be condensed in the reboiler of the lowerpressure column. External heat-transfer media are needed only for the reboiler of the first effect and the condenser of the last effect. Thus, for N effects, utility requirements are of the order 1/N of those for a conventional single-effect column. Wankat [Ind. Eng. Chem. Res., 32, 894 (1993)] develops a large number of more complex multieffect schemes, some of which show significant reductions in energy requirements. In another alternative, shown in Fig. 13-3b, the rectifying section may be operated at a pressure sufficiently higher than that of the stripping section such that heat can be transferred between any desired pairs of stages of the two sections. This technique, described by Mah et al. (op. cit.) and referred to as SRV (secondary reflux and vaporization) distillation, can result in a significant reduction in utility requirements for the overhead condenser and bottoms reboiler. When multicomponent mixtures are to be separated into three or more products, sequences of simple distillation columns of the type shown in Fig. 13-1 are commonly used. For example, if a ternary mixture is to be separated into three relatively pure products, either of the two sequences in Fig. 13-4 can be used. In the direct sequence, shown in Fig. 13-4a, all products but the heaviest are removed one by one as distillates. The reverse is true for the indirect sequence, shown in Fig. 13-4b. The number of possible sequences of simple distillation columns increases rapidly with the number of products. Thus, although only the 2 sequences shown in Fig. 13-4 are possible for a mixture separated into 3 products, 14 different sequences, one of which is shown in Fig. 13-5, can be synthesized when 5 products are to be obtained. As shown in a study by Tedder and Rudd [Am. Inst. Chem. Eng. J., 24, 303 (1978)], conventional sequences like those of Fig. 13-4 may not always be the optimal choice, particularly when species of intermediate volatility are present in large amounts in the feed or need not be recovered at high purity. Of particular interest are thermally coupled systems. For example, in Fig. 13-6a, an impure-vapor sidestream is withdrawn from the first column and purified in a side-cut rectifier, the bottoms of which is returned to the first column. The thermally coupled system in Fig. 13-6b, discussed by Stupin and Lockhart [Chem. Eng. Prog., 68(10), 71 (1972)] and referred to as Petlyuk towers, is particularly useful for reducing energy requirements when the initial feed contains close-boiling species. Shown for a ternary feed, the first column in Fig. 13-6b is a prefractionator, which sends essentially all of the light component and heavy component to the distillate and bottoms respectively, but permits the component of intermediate volatility to be split between the distillate and bottoms. Products from the prefractionator are sent to appropriate feed trays in the second column, where all three products are produced, the middle product being taken off as a sidestream. Only the second column is provided with condenser and reboiler; reflux and boil-up for the prefractionator are obtained from the second column. This concept is readily extended to separations that produce more than three products. Procedures for the optimal design of thermally coupled systems are presented by Triantafyllou and Smith [Trans. I. Chem. E., 70, Part A, 118 (1992)]. A scheme for combining thermal coupling with heat pumps is developed by Agrawal and Yee [Ind. Eng. Chem. Res., 33, 2717 (1994)]. RELATED SEPARATION OPERATIONS The simple and complex distillation operations just described all have two things in common: (1) both rectifying and stripping sections are provided so that a separation can be achieved between two components that are adjacent in volatility; and (2) the separation is effected only by the addition and removal of energy and not by the addition of any mass separating agent (MSA) such as in liquid-liquid extraction.

13-6

DISTILLATION

(b)

(a)

(c)

(d)

(e)

Complex distillation operations with single columns. (a) Use of intermediate heat exchangers. (b) Coupling of intermediate heat exchangers with heat pump. (c) Heat pump with external refrigerant. (d) Heat pump with vapor compression. (e) Heat pump with bottoms flashing.

FIG. 13-2

Sometimes, alternative single- or multiple-stage vapor-liquid separation operations, of the types shown in Fig. 13-7, may be more suitable than distillation for the specified task. A single-stage flash, as shown in Fig. 13-7a, may be appropriate if (1) the relative volatility between the two components to be separated is very large; (2) the recovery of only one component, without regard to the separation of the other components, in one of the two product streams is to be achieved; or (3) only a partial separation is to be made. A common example is the separation of light gases such as hydrogen and methane from aromatics. The desired temperature and pressure of a flash may be established by the use of heat exchangers, a valve, a compressor, and/or a pump upstream of the vessel used to separate the product vapor and liquid phases. Depending on the original condition of the feed, it may be partially condensed or partially vaporized in a so-called flash operation. If the recovery of only one component is required rather than a sharp separation between two components of adjacent volatility,

absorption or stripping in a single section of stages may be sufficient. If the feed is vapor at separation conditions, absorption is used either with a liquid MSA absorbent of relatively low volatility as in Fig. 13-7b or with reflux produced by an overhead partial condenser as in Fig. 13-7c. The choice usually depends on the ease of partially condensing the overhead vapor or of recovering and recycling the absorbent. If the feed is liquid at separation conditions, stripping is used, either with an externally supplied vapor stripping agent of relatively high volatility as shown in Fig. 13-7d or with boil-up produced by a partial reboiler as in Fig. 13-7c. The choice depends on the ease of partially reboiling the bottoms or of recovering and recycling the stripping agent. If a relatively sharp separation is required between two components of adjacent volatility, but either an undesirably low temperature is required to produce reflux at the column-operating pressure or an undesirably high temperature is required to produce boil-up, then refluxed stripping as shown in Fig. 13-7g or reboiled absorption

CONTINUOUS-DISTILLATION OPERATIONS

(a)

(b) FIG. 13-3

Complex distillation operations with two or more columns. (a) Multieffect distillation. (b) SRV distillation.

13-7

13-8

DISTILLATION

(a)

(a)

(b) FIG. 13-4 Distillation sequences for the separation of three components. (a) Direct sequence. (b) Indirect sequence.

(b) FIG. 13-6 Thermally coupled systems for separation into three products. (a) Fractionator with vapor sidestream and side-cut rectifier. (b) Petlyuk towers.

One of 14 different sequences for the separation of a 5-component mixture by simple distillation.

FIG. 13-5

as shown in Fig. 13-7f may be used. In either case, the choice of MSA follows the same consideration given for simple absorption and stripping. When the volatility difference between the two components to be separated is so small that a very large number of stages would be required, then extractive distillation, as shown in Fig. 13-7h, should be considered. Here, an MSA is selected that increases the volatility difference sufficiently to reduce the stage requirement to a reasonable number. Usually, the MSA is a polar compound of low volatility that leaves in the bottoms, from which it is recovered and recycled. It is introduced in an appreciable amount near the top stage of the column so as to affect the volatility difference over most of the stages. Some reflux to the top stage is utilized to minimize the MSA content in the distillate. An alternative to extractive distillation is azeotropic distillation, which is shown in Fig. 13-7i in just one of its many modes. In a common mode, an MSA that forms a heterogeneous minimum-

CONTINUOUS-DISTILLATION OPERATIONS

13-9

(a)

(b)

(f)

(c)

(g)

(d)

(h)

(e)

(i)

Separation operations related to distillation. (a) Flush vaporization or partial condensation. (b) Absorption. (c) Rectifier. (d) Stripping. (e) Reboiled stripping. ( f ) Reboiled absorption. (g) Refluxed stripping. (h) Extractive distillation. (i) Azeotropic distillation.

FIG. 13-7

boiling azeotrope with one or more components of the feed is utilized. The azeotrope is taken overhead, and the MSA-rich phase is decanted and returned to the top of the column as reflux. Numerous other multistaged configurations are possible. One important variation of a stripper, shown in Fig. 13-7d, is a refluxed stripper, in which an overhead condenser is added. Such a configuration is sometimes used to steam-strip sour water containing NH3, H2O, phenol, and HCN. All the separation operations shown in Fig. 13-7, as well as the simple and complex distillation operations described earlier, are referred to here as distillation-type separations because they have much in common with respect to calculations of (1) thermodynamic properties, (2) vapor-liquid equilibrium stages, and (3) column sizing. In fact, as will be evident from the remaining treatment of this section, the trend is toward single generalized digital-computer-program packages that compute many or all distillation-type separation operations. This section also includes a treatment of distillation-type separations from a rate-based point of view that utilizes principles of mass transfer and heat transfer. Section 14 also presents details of that subject as applied to absorption and stripping. SYNTHESIS OF MULTICOMPONENT SEPARATION SYSTEMS The sequencing of distillation columns and other types of equipment for the separation of multicomponent mixtures has received much attention in recent years. Although one separator of complex design can sometimes be devised to produce more than two products, more

often a sequence of two-product separators is preferable. Often, the sequence includes simple distillation columns. A summary of sequencing methods, prior to 1977, that can lead to optimal or nearoptimal designs, is given by Henley and Seader [op. cit.]. More recent methods for distillation-column sequencing are reviewed by Modi and Westerberg [Ind. Eng. Chem. Res., 31, 839 (1992)], who also present a more generally applicable method based on a marginal price that is the change in price of a separation operation when the separation is carried out in the absence of nonkey components. The synthesis of sequences that consider a wide range of separation operations in a knowledge-based approach is given by Barnicki and Fair for liquid mixtures [Ind. Eng. Chem. Res., 29, 421 (1990)] and for gas/vapor mixtures [Ind. Eng. Chem. Res., 31, 1679 (1992)]. A knowledge-based method is also given by Sheppard, Beltramini, and Motard [Chem. Eng. Comm., 106 (1991)] for the synthesis of distillation sequences that involve nonsharp separations where nonkey components distribute. The problem-decomposition approach of Wahnschafft, Le Rudulier, and Westerberg [Ind. Eng. Chem. Res., 32, 1121 (1993)] is directed to the synthesis of complex separation sequences that involve nonsharp splits and recycle, including azeotropic distillation. The method is applied using a computer-aided separation process designer called SPLIT. An expert system, called EXSEP, for the synthesis of solvent-based separation trains is presented by Brunet and Liu [Ind. Eng. Chem. Res., 32, 315 (1993)]. The use of ternary-composition diagrams and residue-curve maps, of the type made popular by Doherty and coworkers, is reviewed and evaluated for application to the synthesis of complex separation sequences by Fien and Liu [Ind. Eng. Chem. Res., 33, 2506 (1994)].

13-10

DISTILLATION

THERMODYNAMIC DATA INTRODUCTION Reliable thermodynamic data are essential for the accurate design or analysis of distillation columns. Failure of equipment to perform at specified levels is often attributable, at least in part, to the lack of such data. This subsection summarizes and presents examples of phase equilibrium data currently available to the designer. The thermodynamic concepts utilized are presented in the subsection “Thermodynamics” of Sec. 4. PHASE EQUILIBRIUM DATA For a binary mixture, pressure and temperature fix the equilibrium vapor and liquid compositions. Thus, experimental data are frequently presented in the form of tables of vapor mole fraction y and liquid mole fraction x for one constituent over a range of temperature T for a fixed pressure P or over a range of pressure for a fixed temperature. A compilation of such data, mainly at a pressure of 101.3 kPa (1 atm, 1.013 bar), for binary systems (mainly nonideal) is given in Table 13-1. More extensive presentations and bibliographies of such data may be found in Hala, Wichterle, Polak, and Boublik [Vapour-Liquid Equilibrium Data at Normal Pressures, Pergamon, Oxford (1968)]; Hirata, Ohe, and Nagahama [Computer Aided Data Book of Vapor-Liquid Equilibria, Elsevier, Amsterdam (1975)]; Wichterle, Linek, and Hala [Vapor-Liquid Equilibrium Data Bibliography, Elsevier, Amsterdam, 1973, Supplement I, 1976, Supplement II, (1979)]; Oe [Vapor-Liquid Equilibrium Data, Elsevier, Amsterdam (1989)]; Oe [Vapor-Liquid Equilibrium Data at High Pressure, Elsevier, Amsterdam (1990)]; Walas [Phase Equilibria in Chemical Engineering, Butterworth, Boston (1985)]; and, particularly, Gmehling and Onken [VaporLiquid Equilibrium Data Collection, DECHEMA Chemistry Data ser., vol. 1 (parts 1–10), (Frankfort, 1977)]. For application to distillation (a nearly isobaric process), as shown in Figs. 13-8 to 13-13, binary-mixture data are frequently plotted, for a fixed pressure, as y versus x, with a line of 45° slope included for reference, and as T versus y and x. In most binary systems, one of the components is more volatile than the other over the entire composition range. This is the case in Figs. 13-8 and 13-9 for the benzenetoluene system at pressures of both 101.3 and 202.6 kPa (1 and 2 atm), where benzene is more volatile than toluene. For some binary systems, one of the components is more volatile over only a part of the composition range. Two systems of this type, ethyl acetate–ethanol and chloroform-acetone, are shown in Figs. 13-10 to 13-12. Figure 13-10 shows that for two binary systems chloroform is less volatile than acetone below a concentration of 66 mole percent chloroform and that ethyl acetate is less volatile than ethanol below a concentration of 53 mole percent ethyl acetate. Above these concentrations, volatility is reversed. Such mixtures are known as azeotropic mixtures, and the composition in which the reversal occurs, which is the composition in which vapor and liquid compositions are equal, is the azeotropic composition, or azeotrope. The azeotropic liquid may be homogeneous or heterogeneous (two immiscible liquid phases). Many of the binary mixtures of Table 13-1 form homogeneous azeotropes. Non-azeotrope-forming mixtures such as benzene and toluene in Figs. 13-8 and 13-9 can be separated by simple distillation into two essentially pure products. By contrast, simple distillation of azeotropic mixtures will at best yield the azeotrope and one essentially pure species. The distillate and bottoms products obtained depend upon the feed composition and whether a minimum-boiling azeotrope is formed as with the ethyl acetate–ethanol mixture in Fig. 13-11 or a maximum-boiling azeotrope is formed as with the chloroform-acetone mixture in Fig. 13-12. For example, if a mixture of 30 mole percent chloroform and 70 mole percent acetone is fed to a simple distillation column, such as that shown in Fig. 13-1, operating at 101.3 kPa (1 atm), the distillate could approach pure acetone and the bottoms could approach the azeotrope.

An example of heterogeneous-azeotrope formation is shown in Fig. 13-13 for the water–normal butanol system at 101.3 kPa. At liquid compositions between 0 and 3 mole percent butanol and between 40 and 100 mole percent butanol, the liquid phase is homogeneous. Phase splitting into two separate liquid phases (one with 3 mole percent butanol and the other with 40 mole percent butanol) occurs for any overall liquid composition between 3 and 40 mole percent butanol. A minimum-boiling heterogeneous azeotrope occurs at 92°C (198°F) when the vapor composition and the overall composition of the two liquid phases are 75 mole percent butanol. For mixtures containing more than two species, an additional degree of freedom is available for each additional component. Thus, for a four-component system, the equilibrium vapor and liquid compositions are only fixed if the pressure, temperature, and mole fractions of two components are set. Representation of multicomponent vapor-liquid equilibrium data in tabular or graphical form of the type shown earlier for binary systems is either difficult or impossible. Instead, such data, as well as binary-system data, are commonly represented in terms of K values (vapor-liquid equilibrium ratios), which are defined by Ki = yi /xi

(13-1)

and are correlated empirically or theoretically in terms of temperature, pressure, and phase compositions in the form of tables, graphs, and equations. K values are widely used in multicomponentdistillation calculations, and the ratio of the K values of two species, called the relative volatility, αij = Ki /Kj

(13-2)

is a convenient index of the relative ease or difficulty of separating components i and j by distillation. Rarely is distillation used on a large scale if the relative volatility is less than 1.05, with i more volatile than j. GRAPHICAL K-VALUE CORRELATIONS As discussed in Sec. 4, the K value of a species is a complex function of temperature, pressure, and equilibrium vapor- and liquid-phase compositions. However, for mixtures of compounds of similar molecular structure and size, the K value depends mainly on temperature and pressure. For example, several major graphical K-value correlations are available for light-hydrocarbon systems. The easiest to use are the DePriester charts [Chem. Eng. Prog. Symp. Ser. 7, 49, 1 (1953)], which cover 12 hydrocarbons (methane, ethylene, ethane, propylene, propane, isobutane, isobutylene, n-butane, isopentane, n-pentane, n-hexane, and n-heptane). These charts are a simplification of the Kellogg charts [Liquid-Vapor Equilibria in Mixtures of Light Hydrocarbons, MWK Equilibrium Constants, Polyco Data, (1950)] and include additional experimental data. The Kellogg charts, and hence the DePriester charts, are based primarily on the BenedictWebb-Rubin equation of state [Chem. Eng. Prog., 47, 419 (1951); 47, 449 (1951)], which can represent both the liquid and the vapor phases and can predict K values quite accurately when the equation constants are available for the components in question. A trial-and-error procedure is required with any K-value correlation that takes into account the effect of composition. One cannot calculate K values until phase compositions are known, and those cannot be known until the K values are available to calculate them. For K as a function of T and P only, the DePriester charts provide good starting values for the iteration. These nomographs are shown in Fig. 13-14a and b. SI versions of these charts have been developed by Dadyburjor [Chem. Eng. Prog., 74(4), 85 (1978)]. The Kellogg and DePriester charts and their subsequent extensions and generalizations use the molar average boiling points of the liquid and vapor phases to represent the composition effect. An alternative measure of composition is the convergence pressure of the system, which is defined as that pressure at which the K values for all the components in an isothermal mixture converge to unity. It is analogous to the critical point for a pure component in the sense that the two

THERMODYNAMIC DATA TABLE 13-1

13-11

Constant-Pressure Liquid-Vapor Equilibrium Data for Selected Binary Systems Component

A

B

Mole fraction A in

Temperature, °C

Liquid

Vapor

Total pressure, kPa

Reference

Acetone

Chloroform

62.50 62.82 63.83 64.30 64.37 64.35 64.02 63.33 62.23 60.72 58.71 57.48

0.0817 0.1390 0.2338 0.3162 0.3535 0.3888 0.4582 0.5299 0.6106 0.7078 0.8302 0.9075

0.0500 0.1000 0.2000 0.3000 0.3500 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 0.9500

101.3

1

Acetone

Methanol

64.65 61.78 59.60 58.14 56.96 56.22 55.78 55.41 55.29 55.37 55.54 55.92 56.21

0.0 0.091 0.190 0.288 0.401 0.501 0.579 0.687 0.756 0.840 0.895 0.954 1.000

0.0 0.177 0.312 0.412 0.505 0.578 0.631 0.707 0.760 0.829 0.880 0.946 1.000

101.3

2

Acetone

Water

74.80 68.53 65.26 63.59 61.87 60.75 59.95 59.12 58.29 57.49 56.68 56.30

0.0500 0.1000 0.1500 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 0.9500

0.6381 0.7301 0.7716 0.7916 0.8124 0.8269 0.8387 0.8532 0.8712 0.8950 0.9335 0.9627

101.3

3

Carbon tetrachloride

Benzene

80.0 79.3 78.8 78.6 78.5 78.2 78.0 77.6 77.4 77.1

0.0 0.1364 0.2157 0.2573 0.2944 0.3634 0.4057 0.5269 0.6202 0.7223

0.0 0.1582 0.2415 0.2880 0.3215 0.3915 0.4350 0.5480 0.6380 0.7330

101.3

4

Chloroform

Methanol

63.0 60.9 59.3 57.8 55.9 54.7 54.0 53.7 53.5 53.5 53.7 54.4 55.2 56.3 57.9

0.040 0.095 0.146 0.196 0.287 0.383 0.459 0.557 0.636 0.667 0.753 0.855 0.904 0.937 0.970

0.102 0.215 0.304 0.378 0.472 0.540 0.580 0.619 0.646 0.655 0.684 0.730 0.768 0.812 0.875

101.3

5

Ethanol

Benzene

76.1 72.7 70.8 69.2 68.4 68.0 67.9 68.0 68.7 69.5

0.027 0.063 0.100 0.167 0.245 0.341 0.450 0.578 0.680 0.766

0.137 0.248 0.307 0.360 0.390 0.422 0.447 0.478 0.528 0.566

101.3

6

13-12

DISTILLATION

TABLE 13-1

Constant-Pressure Liquid-Vapor Equilibrium Data for Selected Binary Systems (Continued ) Component

A

B

Mole fraction A in

Temperature, °C

Liquid

Vapor

70.4 72.7 76.9

0.820 0.905 0.984

0.615 0.725 0.937

Total pressure, kPa

Reference

Ethanol

Water

95.5 89.0 86.7 85.3 84.1 82.7 82.3 81.5 80.7 79.8 79.7 79.3 78.74 78.41 78.15

0.0190 0.0721 0.0966 0.1238 0.1661 0.2337 0.2608 0.3273 0.3965 0.5079 0.5198 0.5732 0.6763 0.7472 0.8943

0.1700 0.3891 0.4375 0.4704 0.5089 0.5445 0.5580 0.5826 0.6122 0.6564 0.6599 0.6841 0.7385 0.7815 0.8943

101.3

7

Ethyl acetate

Ethanol

78.3 76.6 75.5 73.9 72.8 72.1 71.8 71.8 71.9 72.2 73.0 74.7 76.0 77.1

0.0 0.050 0.100 0.200 0.300 0.400 0.500 0.540 0.600 0.700 0.800 0.900 0.950 1.000

0.0 0.102 0.187 0.305 0.389 0.457 0.516 0.540 0.576 0.644 0.726 0.837 0.914 1.000

101.3

8

Ethylene glycol

Water

69.5 76.1 78.9 83.1 89.6 103.1 118.4 128.0 134.7 145.0 160.7

0.0 0.23 0.31 0.40 0.54 0.73 0.85 0.90 0.93 0.97 1.00

0.0 0.002 0.003 0.010 0.020 0.06 0.13 0.22 0.30 0.47 1.00

30.4

9

n-Hexane

Ethanol

78.30 76.00 73.20 67.40 65.90 61.80 59.40 58.70 58.35 58.10 58.00 58.25 58.45 59.15 60.20 63.50 66.70 68.70

0.0 0.0100 0.0200 0.0600 0.0800 0.1520 0.2450 0.3330 0.4520 0.5880 0.6700 0.7250 0.7650 0.8980 0.9550 0.9900 0.9940 1.0000

0.0 0.0950 0.1930 0.3650 0.4200 0.5320 0.6050 0.6300 0.6400 0.6500 0.6600 0.6700 0.6750 0.7100 0.7450 0.8400 0.9350 1.0000

101.3

10

Methanol

Benzene

70.67 66.44 62.87 60.20 58.64 58.02 58.10 58.47 59.90 62.71

0.026 0.050 0.088 0.164 0.333 0.549 0.699 0.782 0.898 0.973

0.267 0.371 0.457 0.526 0.559 0.595 0.633 0.665 0.760 0.907

101.3

11

THERMODYNAMIC DATA TABLE 13-1

13-13

Constant-Pressure Liquid-Vapor Equilibrium Data for Selected Binary Systems (Continued ) Component

A

B

Mole fraction A in

Temperature, °C

Liquid

Vapor

Total pressure, kPa

76.10 74.15 71.24 67.75 65.60 64.10 64.00 63.25 62.97 62.50 62.65 62.50 62.35 62.60 62.80 63.21 63.90

0.0125 0.0320 0.0800 0.1550 0.2510 0.3465 0.4020 0.4975 0.5610 0.5890 0.6220 0.6960 0.7650 0.8250 0.8550 0.9160 0.9550

Reference

0.0475 0.1330 0.2475 0.3650 0.4550 0.5205 0.5560 0.5970 0.6380 0.6560 0.6670 0.7000 0.7420 0.7890 0.8070 0.8600 0.9290

101.3

12

0.0 0.020 0.040 0.060 0.080 0.100 0.150 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 0.950 1.000

0.0 0.134 0.230 0.304 0.365 0.418 0.517 0.579 0.665 0.729 0.779 0.825 0.870 0.915 0.958 0.979 1.000

101.3

13

57.80 55.50 55.04 53.88 53.82 53.90 54.50 56.86

0.173 0.321 0.380 0.595 0.643 0.710 0.849 1.000

0.342 0.477 0.516 0.629 0.657 0.691 0.783 1.000

101.3

14

Methanol

Ethyl acetate

Methanol

Water

Methyl acetate

Methanol

1-Propanol

Water

100.00 98.59 95.09 91.05 88.96 88.26 87.96 87.79 87.66 87.83 89.34 92.30 97.18

0.0 0.0030 0.0123 0.0322 0.0697 0.1390 0.2310 0.3110 0.4120 0.5450 0.7300 0.8780 1.0000

0.0 0.0544 0.1790 0.3040 0.3650 0.3840 0.3970 0.4060 0.4280 0.4650 0.5670 0.7210 1.0000

101.3

15

2-Propanol

Water

100.00 97.57 96.20 93.66 87.84 84.28 82.84 82.52 81.52 81.45 81.19 80.77 80.73 80.58 80.52 80.46 80.55

0.0 0.0045 0.0069 0.0127 0.0357 0.0678 0.1330 0.1651 0.3204 0.3336 0.3752 0.4720 0.4756 0.5197 0.5945 0.7880 0.8020

0.0 0.0815 0.1405 0.2185 0.3692 0.4647 0.5036 0.5153 0.5456 0.5489 0.5615 0.5860 0.5886 0.6033 0.6330 0.7546 0.7680

101.3

16

100.0 96.4 93.5 91.2 89.3 87.7 84.4 81.7 78.0 75.3 73.1 71.2 69.3 67.5 66.0 65.0 64.5

13-14

DISTILLATION

TABLE 13-1

Constant-Pressure Liquid-Vapor Equilibrium Data for Selected Binary Systems (Continued ) Component

A

Tetrahydrofuran

B

Water

Mole fraction A in

Temperature, °C

Liquid

Vapor

Total pressure, kPa

81.32 81.85 82.39

0.9303 0.9660 1.0000

0.9010 0.9525 1.0000

Reference

73.00 66.50 65.58 64.94 64.32 64.27 64.23 64.16 63.94 63.70 63.54 63.53 63.57 63.64 63.87 64.29 65.07 65.39

0.0200 0.0400 0.0600 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.8200 0.8400 0.8600 0.9000 0.9400 0.9800 0.9900

0.6523 0.7381 0.7516 0.7587 0.7625 0.7635 0.7643 0.7658 0.7720 0.7831 0.8085 0.8180 0.8260 0.8368 0.8660 0.9070 0.9625 0.9805

101.3

17

Water

Acetic acid

118.3 110.6 107.8 105.2 104.3 103.5 102.8 102.1 101.5 100.8 100.8 100.5 100.2 100.0

0.0 0.1881 0.3084 0.4498 0.5195 0.5824 0.6750 0.7261 0.7951 0.8556 0.8787 0.9134 0.9578 1.0000

0.0 0.3063 0.4467 0.5973 0.6580 0.7112 0.7797 0.8239 0.8671 0.9042 0.9186 0.9409 0.9708 1.0000

101.3

18

Water

1-Butanol

117.6 111.4 106.7 102.0 101.0 98.5 96.7 95.2 93.6 93.1 93.0 92.9 92.9 93.2 95.2 96.8 100.0

0.0 0.049 0.100 0.161 0.173 0.232 0.288 0.358 0.487 0.551 0.580 0.628 0.927 0.986 0.993 0.996 1.000

0.0 0.245 0.397 0.520 0.534 0.605 0.654 0.693 0.739 0.751 0.752 0.758 0.758 0.760 0.832 0.883 1.000

101.3

19

Water

Formic acid

102.30 104.60 105.90 107.10 107.60 107.60 107.60 107.10 106.00 104.20 102.90 101.80 100.00

0.0405 0.1550 0.2180 0.3210 0.4090 0.4110 0.4640 0.5220 0.6320 0.7400 0.8290 0.9000 1.0000

0.0245 0.1020 0.1620 0.2790 0.4020 0.4050 0.4820 0.5670 0.7180 0.8360 0.9070 0.9510 1.0000

101.3

20

Water

Glycerol

278.8 247.0 224.0 219.2 210.0 202.5 196.5

0.0275 0.0467 0.0690 0.0767 0.0901 0.1031 0.1159

0.9315 0.9473 0.9563 0.9743 0.9783 0.9724 0.9839

101.3

21

THERMODYNAMIC DATA TABLE 13-1

Constant-Pressure Liquid-Vapor Equilibrium Data for Selected Binary Systems (Concluded ) Component

A

13-15

B

Temperature, °C 175.2 149.3 137.2 136.8 131.8 121.5 112.8 111.3 106.3 100.0

Mole fraction A in Liquid

Vapor

0.1756 0.3004 0.3847 0.3895 0.4358 0.5633 0.7068 0.7386 0.8442 1.0000

0.9899 0.9964 0.9976 0.9878 0.9976 0.9984 0.9993 0.9994 0.9996 1.0000

Total pressure, kPa

Reference

To convert degrees Celsius to degrees Fahrenheit, °C = (°F − 32)/1.8. To convert kilopascals to pounds-force per square inch, multiply by 0.145. Kojima, Kato, Sunaga, and Hashimoto, Kagaku Kogaku, 32, 337 (1968). Marinichev and Susarev, Zh. Prtkl. Khtm., 38, 378 (1965). 3 Kojima, Tochigi, Seki, and Watase, Kagaku Kogaku, 32, 149 (1968). 4 International Critical Tables, McGraw-Hill, New York, 1928. 5 Nagata, J. Chem. Eng. Data, 7, 367 (1962). 6 Ellis and Clark, Chem. Age India, 12, 377 (1961). 7 Carey and Lewis, Ind. Eng. Chem., 24, 882 (1932). 8 Chu, Getty, Brennecke, and Paul, Distillation Equilibrium Data, New York, 1950. 9 Trimble and Potts, Ind. Eng. Chem., 27, 66 (1935). 10 Sinor and Weber, J. Chem. Eng. Data, 5, 243 (1960). 11 Hudson and Van Winkle, J. Chem. Eng. Data, 14, 310 (1969). 12 Murti and Van Winkle, Chem. Eng. Data Ser., 3, 72 (1958). 13 Dunlop, M.S. thesis, Brooklyn Polytechnic Institute, 1948. 14 Dobroserdov and Bagrov, Zh. Prtkl. Kthm. (Leningrad), 40, 875 (1967). 15 Smirnova, Vestn. Leningr. Univ. Fiz. Khim., 81 (1959). 16 Kojima, Ochi, and Nakazawa, Int. Chem. Eng., 9, 342 (1964). 17 Shnitko and Kogan, J. Appl. Chem., 41, 1236 (1968). 18 Brusset, Kaiser, and Hoequel, Chim. Ind., Gente Chim. 99, 207 (1968). 19 Boublik, Collect. Czech. Chem. Commun., 25, 285 (1960). 20 Ito and Yoshida, J. Chem. Eng. Data, 8, 315 (1963). 21 Chen and Thompson, J. Chem. Eng. Data, 15, 471 (1970). NOTE: 1 2

Isobaric y-x curves for benzene-toluene. (Brian, Staged Cascades in Chemical Processing, Prentice-Hall, Englewood Cliffs, NJ, 1972.) FIG. 13-8

FIG. 13-9 Isobaric vapor-liquid equilibrium data for benzene-toluence. (Brian, Staged Cascades in Chemical Processing, Prentice-Hall, Englewood Cliffs, NJ, 1972.)

13-16

DISTILLATION

Liquid boiling points and vapor condensation temperatures for maximum-boiling azeotrope mixtures of chloroform and acetone at 101.3 kPa (1 atm) total pressure.

FIG. 13-12

Vapor-liquid equilibriums for the ethyl acetate–ethanol and chloroform-acetone systems at 101.3 kPa (1 atm).

FIG. 13-10

phases become indistinguishable. The behavior of a complex mixture of hydrocarbons for a convergence pressure of 34.5 MPa (5000 psia) is illustrated in Fig. 13-15. Two major graphical correlations based on convergence pressure as the third parameter (besides temperature and pressure) are the charts published by the Gas Processors Association (GPA, Engineering Data Book, 9th ed., Tulsa, 1981) and the charts of the American Petroleum Institute (API, Technical Data Book—Petroleum Refining, New York, 1966) based on the procedures from Hadden and Grayson [Hydrocarbon Process., Pet. Refiner, 40(9), 207 (1961)]. The former uses the method proposed by Hadden [Chem. Eng. Prog. Symp. Ser. 7, 49, 53 (1953)] for the prediction of convergence pressure as a function of composition. The basis for Hadden’s method is illustrated in Fig. 13-16, where it is shown that the critical loci for various mixtures of methane-propane-pentane fall within the area circumscribed by the three binary loci. (This behavior is not always typical of more nonideal systems.) The critical loci for the ternary mixtures vary linearly, at constant temperature, with weight percent propane on a methane-free basis. The essential point is that critical loci for mixtures are independent of the concentration of the lightest component in a mixture. This permits representation of a multicomponent mixture as a pseudo binary. The light component in this pseudo binary is the lightest species present (to a reasonable extent) in the multicomponent mixture. The heavy component is a pseudo substance whose critical temperature is an average of all other components in the multicomponent

Liquid boiling points and vapor condensation temperatures for minimum-boiling azeotrope mixtures of ethyl acetate and ethanol at 101.3 kPa (1 atm) total pressure.

mixture. This pseudocritical point can then be located on a P-T diagram containing the critical points for all compounds covered by the charts, and a critical locus can be drawn for the pseudo binary by interpolation between various real binary curves. Convergence pressure for the mixture at the desired temperature is read from the assumed loci at the desired system temperature. This method is illustrated in the left half of Fig. 13-17 for the methane-propane-pentane ternary. Associated K values for pentane at 104°C (220°F) are shown to the right as a function of mixture composition (or convergence pressure). The GPA convergence-pressure charts are primarily for alkane and alkene systems but do include charts for nitrogen, carbon dioxide, and hydrogen sulfide. The charts may not be valid when appreciable amounts of naphthenes or aromatics are present; the API charts use special procedures for such cases. Useful extensions of the convergence-pressure concept to more varied mixtures include the nomographs of Winn [Chem. Eng. Prog. Symp. Ser. 2, 48, 121 (1952)], Hadden and Grayson (op. cit.), and Cajander, Hipkin, and Lenoir [ J. Chem. Eng. Data, 5, 251 (1960)]. ANALYTICAL K-VALUE CORRELATIONS The widespread availability and utilization of digital computers for distillation calculations have given impetus to the development of analytical expressions for K values. McWilliams [Chem. Eng., 80(25), 138 (1973)] presents a regression equation and accompanying regression coefficients that represent the DePriester charts of Fig. 13-14. Regression equations and coefficients for various versions of the GPA convergence-pressure charts are available from the GPA.

FIG. 13-11

Vapor-liquid equilibrium data for an n-butanol–water system at 101.3 kPa (1 atm); phase splitting and heterogeneous-azeotrope formation. FIG. 13-13

THERMODYNAMIC DATA

13-17

(a) FIG. 13-14 K values (K = y/x) in light-hydrocarbon systems. (a) Low-temperature range. [DePriester, Chem. Eng. Prog. Symp. Sec. 7, 49, 1 (1953).]

Preferred analytical correlations are less empirical in nature and most often are theoretically based on one of two exact thermodynamic formulations, as derived in Sec. 4. When a single pressure-volumetemperature (PVT) equation of state is applicable to both vapor and liquid phases, the formulation used is ˆ iL/ Φ ˆ iV Ki = Φ (13-3) ˆ iV for ˆ iL for the liquid and Φ where the mixture fugacity coefficients Φ the vapor are derived by classical thermodynamics from the PVT expression. Consistent equations for enthalpy can similarly be derived. Until recently, equations of state that have been successfully applied to Eq. (13-3) have been restricted to mixtures of nonpolar compounds, namely, hydrocarbons and light gases. These equations include those of Benedict-Webb-Rubin (BWR), Soave (SRK) [Chem. Eng. Sci., 27, 1197 (1972)], who extended the remarkable RedlichKwong equation, and Peng-Robinson (PR) [Ind. Eng. Chem. Fundam., 15, 59 (1976)]. The SRK and PR equations belong to a family of so-called cubic equations of state. The Starling extension of the BWR equation (Fluid Thermodynamic Properties for Light Petroleum Systems, Gulf, Houston, 1973) predicts K values and enthalpies of the normal paraffins up through n-octane, as well as isobutane, isopentane, ethylene, propylene, nitrogen, carbon dioxide, and hydrogen sul-

fide, including the cryogenic region. Computer programs for K values derived from the SRK, PR and other equations of state are widely available in all computer-aided process design and simulation programs. The ability of the SRK correlation to predict K values even when the pressure approaches the convergence pressure is shown for a multicomponent system in Fig. 13-18. Similar results are achieved with the PR correlation. The Wong-Sandler mixing rules for cubic equations of state now permit such equations to be extended to mixtures of organic chemicals, as shown in a reformulated version by Orbey and Sandler [AIChE J., 41, 683 (1995)]. An alternative K-value formulation that has received wide application to mixtures containing polar and/or nonpolar compounds is Ki = γ iL ΦiL/ Φˆ iV (13-4) where different equations of state may be used to predict the purecomponent liquid fugacity coefficient Φ Li and the vapor-mixture fugacity coefficient, and any one of a number of mixture free-energy models may be used to obtain the liquid activity coefficient γ iL. At low to moderate pressures, accurate prediction of the latter is crucial to the application of Eq. (13-4). When either Eq. (13-3) or Eq. (13-4) can be applied, the former is generally preferred because it involves only a single equation of state

13-18

DISTILLATION

(b) (Continued) K values (K = y/x) in light-hydrocarbon systems. (b) High-temperature range. [DePriester, Chem. Eng. Prog. Symp. Sec. 7, 49, 1 (1953).]

FIG. 13-14

applicable to both phases and thus would seem to offer greater consistency. In addition, the quantity Φ Li in Eq. (13-4) is hypothetical for any components that are supercritical. In that case, a modification of Eq. (13-4) that uses Henry’s law is sometimes applied. For mixtures of hydrocarbons and light gases, Chao and Seader (CS) [AIChE, 7, 598 (1961)] applied Eq. (13-4) by using an empirical expression for Φ Li based on the generalized corresponding-states PVT correlation of Pitzer et al., the Redlich-Kwong equation of state for ˆ iV, and the regular solution theory of Scatchard and Hildebrand for Φ γ iL. The predictive ability of the last-named theory is exhibited in Fig. 13-19 for the heptane-toluene system at 101.3 kPa (1 atm). Five pure-component constants for each species (Tc, Pc, ω, δ, and vL) are required to use the CS method, which when applied within the restrictions discussed by Lenoir and Koppany [Hydrocarbon Process., 46(11), 249 (1967)] gives good results. Revised coefficients of Grayson and Streed (GS) (Pap. 20-P07, Sixth World Pet. Conf., Frankfurt, June, 1963) for the Φ Li expression permit application of

the CS correlation to higher temperatures and pressures and give improved predictions for hydrogen. Jin, Greenkorn, and Chao [AIChE J, 41, 1602 (1995)] present a revised correlation for the standard-state liquid fugacity of hydrogen, applicable from 200 to 730 K. For mixtures containing polar substances, more complex predictive equations for γ iL that involve binary-interaction parameters for each pair of components in the mixture are required for use in Eq. (13-4), as discussed in Sec. 4. Six popular expressions are the Margules, van Laar, Wilson, NRTL, UNIFAC, and UNIQUAC equations. Extensive listings of binary-interaction parameters for use in all but the UNIFAC equation are given by Gmehling and Onken (op. cit.). They obtained the parameters for binary systems at 101.3 kPa (1 atm) from best fits of the experimental T-y-x equilibrium data by setting ΦiV and ΦiL to their ideal-gas, ideal-solution limits of 1.0 and P sat/P respectively, with the vapor pressure P sat given by a threeconstant Antoine equation, whose values they tabulate. Table 13-2 lists their parameters for some of the binary systems included in

THERMODYNAMIC DATA

13-19

Effect of mixture composition upon K value for n-pentane at 104°C (220°F). K values are shown for various values of W, weight fraction propane on a methane-free basis for the methane-propane-pentane system. [Hadden, Chem. Eng. Prog. Symp. Sec. 7, 49, 58 (1953).] FIG. 13-17

FIG. 13-15 Typical variation of K values with total pressure at constant temperature for a complex mixture. Light hydrocarbons in admixture with crude oil. [Katz and Hachmuth, Ind. Eng. Chem., 29, 1072 (1937).]

Table 13-1, based on the binary-system activity-coefficient-equation forms given in Table 13-3. Consistent Antoine vapor-pressure constants and liquid molar volumes are listed in Table 13-4. The Wilson equation is particularly useful for systems that are highly nonideal but do not undergo phase splitting, as exemplified by the ethanol-hexane system, whose activity coefficients are shown in Fig. 13-20. For systems such as this, in which activity coefficients in dilute regions may

Comparison of experimental K-value data and SRK correlation. [Henley and Seader, Equilibrium-Stage Separation Operations in Chemical Engineering, Wiley, New York, 1981; data of Yarborough, J. Chem. Eng. Data, 17, 129 (1972).]

FIG. 13-18

Critical loci for a methane-propane-pentane system according to Hadden. [Chem. Eng. Prog. Symp. Sec. 7, 49, 53 (1953).] Parameter W is weight fraction propane on a methane-free basis. FIG. 13-16

13-20

DISTILLATION exceed values of approximately 7.5, the van Laar equation erroneously predicts phase splitting. Tables 13-1, 13-2, and 13-4 include data on formic acid and acetic acid, two substances that tend to dimerize in the vapor phase according to the chemical-equilibrium expression KD = PD /PM2 = 10A + B/T

(13-5)

where KD is the chemical-equilibrium constant for dimerization, PD and PM are partial pressures of dimer and monomer respectively in torr, and T is in K. Values of A and B for the first four normal aliphatic acids are:

Formic acid Acetic acid n-Propionic acid n-Butyric acid

A

B

−10.743 −10.421 −10.843 −10.100

3083 3166 3316 3040

As shown by Marek and Standart [Collect. Czech. Chem. Commun., 19, 1074 (1954)], it is preferable to correlate and utilize liquid-phase activity coefficients for the dimerizing component by considering separately the partial pressures of the monomer and dimer. For example, for a binary system of components 1 and 2, when only compound 1 dimerizes in the vapor phase, the following equations apply if an ideal gas is assumed: P1 = PD + PM

(13-6)

y1 = (PM + 2PD)/P

(13-7)

These equations when combined with Eq. (13-5) lead to the following equations for liquid-phase activity coefficients in terms of measurable quantities: Py1 1 + (1 + 4KDP1sat)0.5 γ1 = ᎏ ᎏᎏᎏ sat P1 x1 1 + [1 + 4KDPy1(2 − y1)]0.5




Liquid-phase activity coefficients for an n-heptane-toluene system at 101.3 kPa (1 atm). [Henley and Seader, Equilibrium-Stage Separation Operations in Chemical Engineering, Wiley, New York, 1981; data of Yerazunis et al., Am. Inst. Chem. Eng. J., 10, 660 (1964).]

FIG. 13-19



2{1 − y1 + [1 + 4KDPy1(2 − y1)]0.5} Py1 γ2 = ᎏ ᎏᎏᎏᎏ sat P2 x2 (2 − y1){1 + [1 + 4KDPy1(2 − y1)]0.5}



(13-8)



(13-9)

Detailed procedures, including computer programs for evaluating binary-interaction parameters from experimental data and then utiliz-

TABLE 13-2

Binary-Interaction Parameters* Margules

van Laar

Wilson (cal/mol)

System

A12 

A 21

A12

A21

(λ12 − λ11)

(λ21 − λ22)

Acetone (1), chloroform (2) Acetone (1), methanol (2) Acetone (1), water (2) Carbon tetrachloride (1), benzene (2) Chloroform (1), methanol (2) Ethanol (1), benzene (2) Ethanol (1), water (2) Ethyl acetate (1), ethanol (2) n-Hexane (1), ethanol (2) Methanol (1), benzene (2) Methanol (1), ethyl acetate (2) Methanol (1), water (2) Methyl acetate (1), methanol (2) 1-Propanol (1), water (2) 2-Propanol (1), water (2) Tetrahydrofuran (1), water (2) Water (1), acetic acid (2) Water (1), 1-butanol (2) Water (1), formic acid (2)

−0.8404 0.6184 2.0400 0.0948 0.8320 1.8362 1.6022 0.8557 1.9398 2.1411 1.0016 0.7923 0.9605 2.7070 2.3319 2.8258 0.4178 0.8608 −0.2966

−0.5610 0.5788 1.5461 0.0922 1.7365 1.4717 0.7947 0.7476 2.7054 1.7905 1.0517 0.5434 1.0120 0.7172 0.8976 1.9450 0.9533 3.2051 −0.2715

−0.8643 0.6184 2.1041 0.0951 0.9356 1.8570 1.6798 0.8552 1.9195 2.1623 1.0017 0.8041 0.9614 2.9095 2.4702 3.0216 0.4973 1.0996 −0.2935

−0.5899 0.5797 1.5555 0.0911 1.8860 1.4785 0.9227 0.7526 2.8463 1.7925 1.0524 0.5619 1.0126 1.1572 1.0938 1.9436 1.0623 4.1760 −0.2757

116.1171 −114.4047 344.3346 7.0459 −361.7944 1264.4318 325.0757 58.8869 320.3611 1666.4410 982.2689 82.9876 −93.8900 906.5256 659.5473 1475.2583 705.5876 1549.6600 −310.1060

−506.8519 545.2942 1482.2133 59.6233 1694.0241 266.6118 953.2792 570.0439 2189.2896 227.2126 −172.9317 520.6458 847.4348 1396.6398 1230.2080 1844.7926 111.6579 2050.2569 1180.8040

*Abstracted from Gmehling and Onken, Vapor-Liquid Equilibrium Data Collection, DECHEMA Chemistry Data ser., vol. 1 (parts 1–10), Frankfurt, 1977.

THERMODYNAMIC DATA

13-21

TABLE 13-3 Activity-Coefficient Equations in Binary Form for Use with Parameters and Constants in Tables 13-2 and 13-4 Type of equation

Adjustable parameters

Equations in binary form

Margules

A 12 A 21

ln γ1 = [A A12)x1]x 22 12 + 2(A 21 −  ln γ2 = [A A21)x2]x 12 21 + 2(A 12 − 

van Laar

A12

A21x2 ln γ1 = A12 ᎏᎏ A12 x1 + A21 x2

A21

A12x1 ln γ2 = A21 ᎏᎏ A12 x1 + A21 x2

λ12 − λ11

Λ12 Λ21 ln γ1 = −ln (x1 + Λ12 x2) + x2 ᎏᎏ − ᎏᎏ x1 + Λ12 x2 Λ21 x1 + x2



λ 21 − λ22

Λ12 Λ21 ln γ2 = −ln (x2 + Λ21 x1) − x1 ᎏᎏ − ᎏᎏ x1 + Λ12 x2 Λ21 x1 + x2



Wilson

v λ12 − λ11 v λ 21 − λ 22 where Λ12 = ᎏ exp − ᎏ Λ21 = ᎏ exp − ᎏ v1L v L2 RT RT L 2





L 1











2

2

 



vLi = molar volume of pure-liquid component i λ ij = interaction energy between components i and j, λ ij = λ ji

ing these parameters to predict K values and phase equilibria, are given in terms of the UNIQUAC equation by Prausnitz et al. (Computer Calculations for Multicomponent Vapor-Liquid and LiquidLiquid Equilibria, Prentice-Hall, Englewood Cliffs, N.J., 1980) and in terms of the UNIFAC group contribution method by Fredenslund, Gmehling, and Rasmussen (Vapor-Liquid Equilibria Using UNIFAC, Elsevier, Amsterdam, 1980). Both use the method of Hayden and O’Connell [Ind. Eng. Chem. Process Des. Dev., 14, 209 (1975)] to ˆ iV in Eq. (13-4). When the system temperature is greater compute Φ than the critical temperature of one or more components in the mixture, Prausnitz et al. utilize a Henry’s-law constant Hi,M in place of the product γ iL ΦiL in Eq. (13-4). Otherwise ΦiL is evaluated from vaporpressure data with a Poynting saturated-vapor fugacity correction. When the total pressure is less than about 202.6 kPa (2 atm) and all components in the mixture have a critical temperature that is greater

TABLE 13-4

ˆ iV = 1.0. Equation than the system temperature, then ΦiL = Pisat/P and Φ (13-4) then reduces to Ki = γ iLPisat/P

(13-10)

which is referred to as a modified Raoult’s-law K value. If, furthermore, the liquid phase is ideal, then γ iL = 1.0 and K i = Pisat/P

(13-11)

which is referred to as a Raoult’s-law K value that is dependent solely on the vapor pressure Pisat of the component in the mixture. The UNIFAC method is being periodically updated with new group contributions, with a recent article being that of Hansen et al. [Ind. Eng. Chem. Res., 30, 2352 (1991)].

Antoine Vapor-Pressure Constants and Liquid Molar Volume* vL, liquid molar volume, cm3/ g⋅mol

Species

A

B

C

Applicable temperature region, °C

Acetic acid Acetone Benzene 1-Butanol Carbon tetrachloride

8.02100 7.11714 6.87987 7.36366 6.84083

1936.010 1210.595 1196.760 1305.198 1177.910

258.451 229.664 219.161 173.427 220.576

18–118 (−13)–55 8–80 89–126 (−20)–77

57.54 74.05 89.41 91.97 97.09

Chloroform Ethanol Ethanol Ethyl acetate Formic acid

6.95465 7.58670 8.11220 7.10179 6.94459

1170.966 1281.590 1592.864 1244.951 1295.260

226.232 193.768 226.184 217.881 218.000

(−10)–60 78–203 20–93 16–76 36–108

80.67 58.68 58.68 98.49 37.91

n-Hexane Methanol Methyl acetate 1-Propanol 2-Propanol

6.91058 8.08097 7.06524 8.37895 8.87829

1189.640 1582.271 1157.630 1788.020 2010.320

226.280 239.726 219.726 227.438 252.636

(−30)–170 15–84 2–56 (−15)–98 (−26)–83

131.61 40.73 79.84 75.14 76.92

Tetrahydrofuran Water

6.99515 8.07131

1202.290 1730.630

226.254 233.426

23–100 1–100

81.55 18.07

Antoine constants†

*Abstracted from Gmehling and Onken, Vapor-Liquid Equilibrium Data Collection, DECHEMA Chemistry Data ser., vol. 1 (parts 1–10), Frankfurt, 1977. †Antoine equation is log Psat = A − B/(T + C) with Psat in torr and T in °C. NOTE: To convert degrees Celsius to degrees Fahrenheit, °F = 1.8°C + 32. To convert cubic centimeters per gram-mole to cubic feet per pound-mole, multiply by 0.016.

13-22

DISTILLATION

DEGREES OF FREEDOM AND DESIGN VARIABLES DEFINITIONS For separation processes, a design solution is possible if the number of independent equations equals the number of unknowns. Ni = Nv − Nc where Nv is the total number of variables (unknowns) involved in the process under consideration, Nc is the number of restricting relationships among the unknowns (independent equations), and Ni is termed the number of design variables. In the analogous phase-rule analysis, Ni is usually referred to as the degrees of freedom or variance. It is the number of variables that the designer must specify to define one unique operation (solution) of the process. The variables Ni with which the designer of a separation process must be concerned are: 1. Stream concentrations (e.g., mole fractions) 2. Temperatures 3. Pressures 4. Stream flow rates 5. Repetition variables Nr The first three are intensive variables. The fourth is an extensive variable that is not considered in the usual phase-rule analysis. The fifth is neither an intensive nor an extensive variable but is a single degree of freedom that the designer utilizes in specifying how often a particular element is repeated in a unit. For example, a distillation-column section is composed of a series of equilibrium stages, and when the designer specifies the number of stages that the section contains, he

or she utilizes the single degree of freedom represented by the repetition variable (Nr = 1.0). If the distillation column contains more than one section (such as above and below a feed stage), the number of stages in each section must be specified and as many repetition variables exist as there are sections, that is, Nr = 2. The various restricting relationships Nc can be classified as: 1. Inherent 2. Mass-balance 3. Energy-balance 4. Phase-distribution 5. Chemical-equilibrium The inherent restrictions are usually the result of definitions and take the form of identities. For example, the concept of the equilibrium stage involves the inherent restrictions that TV = TL and PV = PL where the superscripts V and L refer to the equilibrium exit streams. The mass-balance restrictions are the C balances written for the C components present in the system. (Since we will only deal with nonreactive mixtures, each chemical compound present is a phase-rule component.) An alternative is to write (C − 1) component balances and one overall mass balance. The phase-distribution restrictions reflect the requirement that fiV = fiL at equilibrium where f is the fugacity. This may be expressed by Eq. (13-1). In vapor-liquid systems, it should always be recognized that all components appear in both phases to some extent and there will be such a restriction for each component in the system. In vaporliquid-liquid systems, each component will have three such restrictions, but only two are independent. In general, when all components exist in all phases, the number of restricting relationships due to the distribution phenomenon will be C(Np − 1), where Np is the number of phases present. For the analysis here, the forms in which the restricting relationships are expressed are unimportant. Only the number of such restrictions is important. ANALYSIS OF ELEMENTS An element is defined as part of a more complex unit. The unit may be all or only part of an operation or the entire process. Our strategy will be to analyze all elements that appear in a separation process and determine the number of design variables associated with each. The appropriate elements can then be quickly combined to form the desired units and the various units combined to form the entire process. Allowance must of course be made for the connecting streams (interstreams) whose variables are counted twice when elements or units are joined. The simplest element is a single homogeneous stream. The variables necessary to define it are: N ce C−1 1 1 1 C+2

Concentrations Temperature Pressure Flow rate

There are no restricting relationships when the stream is considered only at a point. Henley and Seader (Equilibrium-Stage Separation Operations in Chemical Engineering, Wiley, New York, 1981) count all C concentrations as variables, but then have to include

 i

Liquid-phase activity coefficients for an ethanol–n-hexane system. [Henley and Seader, Equilibrium-Stage Separation Operations in Chemical Engineering, Wiley, New York, 1931; data of Sinor and Weber, J. Chem. Eng. Data, 5, 243–247 (1960).]

FIG. 13-20

xi = 1.0

or

 y = 1.0 i

i

as a restriction. A stream divider simply splits a stream into two or more streams of the same composition. Consider Fig. 13-21, which pictures the division of the condensed overhead liquid Lc into distillate D and

DEGREES OF FREEDOM AND DESIGN VARIABLES

13-23

These design variables can be utilized as follows:

FIG. 13-21

Specifications

Nie

Specification of Ln + 1 stream Specification of Vn − 1 stream Pressure of either leaving stream Heat leak q

C+2 C+2 1 1 2C + 6

Stream divider.

reflux LN + 1. The divider is permitted to operate nonadiabatically if desired. Three mass streams and one possible “energy stream” are involved; so

The results of the analyses for all the various elements commonly encountered in distillation processes are summarized in Table 13-5. Details of the analyses are given by Smith (Design of Equilibrium Stage Processes, McGraw-Hill, New York, 1967) and in a somewhat different form by Henley and Seader (op. cit.). ANALYSIS OF UNITS

Nve = 3(C + 2) + 1 = 3C + 7 Each mass stream contributes C + 2 variables, but an energy stream has only its rate q as a variable. The independent restrictions are as follows:

A “unit” is defined as a combination of elements and may or may not constitute the entire process. By definition Nvu = Nr +  N ie i

N ce Inherent T and P identities between LN + 1 and D Concentration identities between LN + 1 and D Mass balances Energy balance

2 C−1 C 1 2C + 2

The number of design variables for the element is given by N ie = N ve − N ce = (3C + 7) − (2C + 2) = C + 5 Specification of the feed stream Lc(C + 2 variables), the ratio LN + 1/D, the “heat leak” q, and the pressure of either stream leaving the divider utilizes these design variables and defines one unique operation of the divider. A simple equilibrium stage (no feed or sidestreams) is depicted in Fig. 13-22. Four mass streams and a heat-leak (or heat-addition) stream provide the following number of variables: N ve = 4(C + 2) + 1 = 4C + 9

N iu = Nvu − N cu

and

where N cu refers to new restricting relationships (identities) that may arise when elements are combined. N cu does not include any of the restrictions considered in calculating the N ie ’s for the various elements. It includes only the stream identities that exist in each interstream between two elements. The interstream variables (C + 2) were counted in each of the two elements when their respective N ie ’s were calculated. Therefore, (C + 2) new restricting relationships must be counted for each interstream in the combination of elements to prevent redundancy. The simple absorber column shown in Fig. 13-23 will be analyzed here to illustrate the procedure. This unit consists of a series of simple equilibrium stages of the type in Fig. 13-22. Specification of the number of stages N utilizes the single repetition variable and N vu = Nr +  N ei = 1 + N(2C + 6) i

since in Table 13-5 Nie = 2C + 6 for a simple equilibrium stage. There are 2(N − 1) interstreams, and therefore 2(N − 1)(C + 2) new identities (not previously counted) come into existence when elements are combined. Subtraction of these restrictions from N vu gives N iu, the design variables that must be specified. N iu = N vu − N cu = Nr +  N ie − N cu

Vapor and liquid streams Vn and Ln respectively are in equilibrium with each other by definition and therefore are at the same T and P. These two inherent identities when added to C-component balances, one energy balance, and the C phase-distribution relationships give

i

= [1 + N(2C + 6)] − [2(N − 1)(C + 2)] = 2C + 2N + 5

Nce = 2C + 3 Then

Nie = Nve − Nce = (4C + 9) − (2C + 3) = 2C + 6

FIG. 13-22

Simple equilibrium stage.

TABLE 13-5

e

Design Variables N i for Various Elements

Element

N ev

N ec

N ei

Homogeneous stream Stream divider Stream mixer Pump Heater Cooler Total condenser Total reboiler Partial condenser Partial reboiler Simple equilibrium state Feed stage Sidestream stage Adiabatic equilibrium flash Nonadiabatic equilibrium flash

C+2 3C + 7 3C + 7 2C + 5 2C + 5 2C + 5 2C + 5 2C + 5 3C + 7 3C + 7 4C + 9 5C + 11 5C + 11 3C + 6 3C + 7

0 2C + 2 C+1 C+1 C+1 C+1 C+1 C+1 2C + 3 2C + 3 2C + 3 2C + 3 3C + 4 2C + 3 2C + 3

C+2 C+5 2C + 6 C+4 C+4 C+4 C+4 C+4 C+4 C+4 2C + 6 3C + 8 2C + 7 C+3 C+4

13-24

DISTILLATION Specifications Pressure of either stream leaving each stage (including reboiler) Pressure of stream leaving condenser Pressure of either stream leaving reflux divider Heat leak for each stage (excluding reboiler) Heat leak for reflux divider Feed stream Reflux temperature Total number of stages N Number of stages below feed M Distillate rate D/F Reflux rate (LN + 1/D)

Niu N 1 1 N−1 1 C +2 1 1 1 1 1 C + 2N + 9

Other specifications often used in place of one or more of the last four listed are the fractional recovery of one component in either D or B and/or the concentration of one component in either D or B. OTHER UNITS AND COMPLEX PROCESSES In Table 13-6, the number of design variables is summarized for several distillation-type separation operations, most of which are shown in Fig. 13-7. For columns not shown in Figs. 13-1 or 13-7 that

FIG. 13-23

Simple absorption.

These might be used as follows: Niu

Specifications Two feed streams Number of stages N Pressure of either stream leaving each stage Heat leak for each stage

2C

+4 1

N N 2C + 2N + 5

A more complex unit is shown in Fig. 13-24, which is a schematic diagram of a distillation column with one feed, a total condenser, and a partial reboiler. Dotted lines encircle the six connected elements (or units) that constitute the distillation operation. The variables Nvu that must be considered in the analysis of the entire process are just the sum of the Nie’s for these six elements since here Nr = 0. Using Table 13-5, Element (or unit)

N uv =  Nie

Total condenser Reflux divider N − (M + 1) equilibrium stages Feed stage (M − 1) equilibrium stages Partial reboiler

C +4 C +5 2C + 2(N − M − 1) + 5 3C +8 2C + 2(M − 1) +5 C +4 10C + 2N + 27

i

Here, the two units of N − (M + 1) and (M − 1) stages are treated just like elements. Nine interstreams are created by the combination of elements; so Ncu = 9(C + 2) = 9C + 18 The number of design variables is Niu = C + 2N + 9 Nvu − Ncu = (10C + 2N + 27) − (9C + 18) One set of specifications that is particularly convenient for computer solutions is:

FIG. 13-24

reboiler.

Distillation column with one feed, a total condenser, and a partial

SINGLE-STAGE EQUILIBRIUM-FLASH CALCULATIONS involve additional feeds and/or sidestreams, add (C + 3) degrees of freedom for each additional feed (C + 2 to define the feed and 1 to designate the feed stage) and 2 degrees of freedom for each sidestream (1 for the sidestream flow rate and 1 to designate the sidestream-stage location). Any number of elements or units can be combined to form complex processes. No new rules beyond those developed earlier are necessary for their analysis. When applied to the thermally coupled distillation process of Fig. 13-6b, the result is Niu = 2(N + M) + C + 18. Further examples are given in Henley and Seader (op. cit.). An alternative method for determining the degrees of freedom for equipment and processes is given by Pham [Chem. Eng. Sci., 49, 2507 (1994)].

TABLE 13-6

13-25

Design Variables N iu for Separation Units Unit

N iu ∗

Distillation (partial reboiler–total condenser) Distillation (partial reboiler–partial condenser) Absorption Rectification (partial condenser) Stripping Reboiled stripping (partial reboiler) Reboiled absorption (partial reboiler) Refluxed stripping (total condenser) Extractive distillation (partial reboiler–total condenser)

C + 2N + 9 C + 2N + 6 2C + 2N + 5 C + 2N + 3 2C + 2N + 5 C + 2N + 3 2C + 2N + 6 2C + 2N + 9 2C + 2N + 12

∗ N includes reboiler, but not condenser.

SINGLE-STAGE EQUILIBRIUM-FLASH CALCULATIONS INTRODUCTION The simplest continuous-distillation process is the adiabatic singlestage equilibrium-flash process pictured in Fig. 13-25. Feed temperature and the pressure drop across the valve are adjusted to vaporize the feed to the desired extent, while the drum provides disengaging space to allow the vapor to separate from the liquid. The expansion across the valve is at constant enthalpy, and this fact can be used to calculate T2 (or T1 to give a desired T2 ). From Table 13-5 it can be seen that the variables subject to the designer’s control are C + 3 in number. The most common way to utilize these is to specify the feed rate, composition, and pressure (C + 1 variables) plus the drum temperature T2 and pressure P2. This operation will give one point on the equilibrium-flash curve shown in Fig. 13-26. This curve shows the relation at constant pressure between the fraction V/F of the feed flashed and the drum temperature. The temperature at V/F = 0.0 when the first bubble of vapor is about to form (saturated liquid) is the bubble-point temperature of the feed mixture, and the value at V/F = 1.0 when the first droplet of liquid is about to form (saturated liquid) is the dew-point temperature.

At the dew-point temperature yi still equals zi, and the relationship  xi =  zi /K i = 1.0 must be satisfied. As in the case of the bubble point, a trial-and-error search for the dew-point temperature at a specified pressure is involved. Or, if the temperature is specified, the dew-point pressure can be calculated. ISOTHERMAL FLASH The calculation for a point on the flash curve that is intermediate between the bubble point and the dew point is referred to as an isothermal-flash calculation because T2 is specified. Except for an ideal binary mixture, procedures for calculating an isothermal flash are iterative. A popular method is the following due to Rachford and Rice [ J. Pet. Technol., 4(10), sec. 1, p. 19, and sec. 2, p. 3 (October 1952)]. The component mole balance (Fzi = Vyi + Lxi), phasedistribution relation (Ki = yi /xi), and total mole balance (F = V + L) can be combined to give zi xi = ᎏᎏ (13-12) V 1 + ᎏᎏ (Ki − 1) F

BUBBLE POINT AND DEW POINT For a given drum pressure and feed composition, the bubble- and dew-point temperatures bracket the temperature range of the equilibrium flash. At the bubble-point temperature, the total vapor pressure exerted by the mixture becomes equal to the confining drum pressure, and it follows that  yi = 1.0 in the bubble formed. Since yi = K i xi and since the xi’s still equal the feed concentrations (denoted by zi’s), calculation of the bubble-point temperature involves a trial-anderror search for the temperature which, at the specified pressure, makes  K i zi = 1.0. If instead the temperature is specified, one can find the bubble-point pressure that satisfies this relationship.

FIG. 13-25

Equilibrium-flash separator.

Since  xi −  yi = 0,

Kizi yi = ᎏᎏ V 1 + ᎏᎏ (Ki − 1) F

(13-13)

zi(1 − Ki) f {V} =  ᎏᎏ =0 (13-14) V i 1 + ᎏᎏ (Ki − 1) F Equation (13-14) is solved iteratively for V/F, followed by the calculation of values of xi and yi from Eqs. (13-12) and (13-13) and L from the total mole balance. Any one of a number of numerical root-finding

FIG. 13-26

Equilibrium-flash curve.

13-26

DISTILLATION

procedures such as the Newton-Raphson, secant, false-position, or bisection method can be used to solve Eq. (13-14). Values of Ki are constants if they are independent of liquid and vapor compositions. Then the resulting calculations are straightforward. Otherwise, the K i values must be periodically updated for composition effects, perhaps after each iteration, using prorated values of xi and yi from Eqs. (13-12) and (13-13). Generally, the iterations are continued until the calculated value of V/F equals to within ⫾0.0005 the value of V/F that was used to initiate that iteration. When converged,  xi and  yi will each be very close to a value of 1, and, if desired, T1 can be computed from an energy balance around the valve if no heat exchanger is used. Alternatively, if T1 is fixed as mentioned earlier, a heat exchanger must be added before, after, or in place of the valve with the required heat duty being calculated from an energy balance. The limits of applicability of Eqs. (13-12) to (13-14) are the bubble point, at which V = 0 and xi = zi, and the dew point, at which L = 0 and yi = zi, at which Eq. (13-2) reduces to the bubble-point equation

K x =1 i i

(13-15)

i

and the dew-point equation yi

=1 ᎏ K i

(13-16)

i

For a binary feed, specification of the flash-drum temperature and pressure fixes the equilibrium-phase concentrations, which are related to the K values by x1 = (1 − K2)/(K1 − K2)

and y1 = (K1K2 − K1)/(K2 − K1)

The mole balance can be rearranged to V z1(K1 − K2)/(1.0 − K2) − 1.0 ᎏᎏ = ᎏᎏᎏ K1 − 1.0 F If K1 and K2 are functions of temperature and pressure only (ideal solutions), the flash curve can be calculated directly without iteration. ADIABATIC FLASH In Fig. 13-25, if P2 and the feed-stream conditions (i.e., F, zi, T1, P1) are known, then the calculation of T2, V, L, yi, and xi is referred to as an adiabatic flash. In addition to Eqs. (13-12) to (13-14) and the total mole balance, the following energy balance around both the valve and the flash drum combined must be included: HFF = HVV + HLL

(13-17)

Taking a basis of F = 1.0 mol and eliminating L with the total mole balance, Eq. (13-17) becomes f2{V, T2} = HF − V(HV − HL) − HL = 0

(13-18)

With T2 now unknown, Eq. (13-17) becomes zi(1 − Ki) (13-19) f1{V, T2} =  ᎏᎏ = 0 1 + V(Ki − 1) i A number of iterative procedures have been developed for solving Eqs. (13-18) and (13-19) simultaneously for V and T2. Frequently, and especially if the feed contains components of a narrow range of volatility, convergence is rapid for a tearing method in which a value of T2 is

assumed, Eq. (13-19) is solved iteratively by the isothermal-flash procedure, and, using that value of V, Eq. (13-18) is solved iteratively for a new approximation of T2, which is then used to initiate the next cycle until T2 and V converge. However, if the feed contains components of a wide range of volatility, it may be best to invert the sequence and assume a value for V, solve Eq. (13-19) for T2, solve Eq. (13-18) for V, and then repeat the cycle. If K values and/or enthalpies are sensitive to the unknown phase compositions, it may be necessary simultaneously to solve Eqs. (13-18) and (13-19) by a Newton or other suitable iterative technique. Alternatively, the two-tier method of Boston and Britt [Comput. Chem. Eng., 2, 109 (1978)], which is also suitable for difficult isothermal-flash calculations, may be applied. OTHER FLASH SPECIFICATIONS Flash-drum specifications in addition to (P2, T2) and (P2, adiabatic) are also possible but must be applied with care, as discussed by Michelsen [Comp. Chem. Engng., 17, 431 (1993)]. Most computer-aided process design and simulation programs permit a wide variety of flash specifications. THREE-PHASE FLASH Single-stage equilibrium-flash calculations become considerably more complex when an additional liquid phase can form, as from mixtures of water with hydrocarbons. Procedures for computing such situations are referred to as three-phase flash methods, which are given for the general case by Henley and Rosen (Material and Energy Balance Computations, Wiley, New York, 1968, chap. 8). When the two liquid phases are almost mutually insoluble, they can be considered separately and relatively simple procedures apply as discussed by Smith (Design of Equilibrium Stage Processes, McGraw-Hill, New York, 1963). Condensation of such mixtures may result in one liquid phase being formed before the other. Computer-aided process design and simulation programs all contain a Gibbs free-energy routine that can compute a three-phase flash by minimization of Gibbs free energy. Many difficult aspects of flash calculations are discussed by Michelsen [Fluid Phase Equil., 9, 1, 21 (1982)]. COMPLEX MIXTURES Feed analyses in terms of component concentrations are usually not available for complex hydrocarbon mixtures with a final normal boiling point above about 38°C (100°F) (n-pentane). One method of handling such a feed is to break it down into pseudo components (narrow-boiling fractions) and then estimate the mole fraction and K value for each such component. Edmister [Ind. Eng. Chem., 47, 1685 (1955)] and Maxwell (Data Book on Hydrocarbons, Van Nostrand, Princeton, N.J., 1958) give charts that are useful for this estimation. Once K values are available, the calculation proceeds as described above for multicomponent mixtures. Another approach to complex mixtures is to obtain an American Society for Testing and Materials (ASTM) or true-boiling point (TBP) curve for the mixture and then use empirical correlations to construct the atmospheric-pressure equilibrium-flash curve (EFV), which can then be corrected to the desired operating pressure. A discussion of this method and the necessary charts are presented in a later subsection entitled “Petroleum and Complex-Mixture Distillation.”

GRAPHICAL METHODS FOR BINARY DISTILLATION INTRODUCTION Multistage distillation under continuous, steady-state operating conditions is widely used in practice to separate a variety of mixtures. Table 13-7, taken from the study of Mix, Dweck, Weinberg, and Armstrong [Am. Inst. Chem. Eng. J. Symp. Ser. 76, 192, 10 (1980)] lists key components for 27 industrial distillation processes. The design of multiequilibrium-stage columns can be accomplished by graphical techniques when the feed mixture contains only two components. The x-y diagram

[McCabe and Thiele, Ind. Eng. Chem., 17, 605 (1925)] utilizes only equilibrium and mole-balance relationships but approaches rigorousness only for those systems in which energy effects on vapor and liquid rates leaving the stages are negligible. The enthalpy-concentration diagram [Ponchon, Tech. Mod., 13, 20, 55 (1921); and Savarit, Arts Metiers, 65, 142, 178, 241, 266, 307 (1922)] utilizes the energy balance also and is rigorous when enough calorimetric data are available to construct the diagram without assumptions.

GRAPHICAL METHODS FOR BINARY DISTILLATION TABLE 13-7 Key Components for Distillation Processes of Industrial Importance Key components

Typical number of trays

It is sometimes permissible to assume constant relative volatility in order to approximate the equilibrium curve quickly. Then by applying Eq. (13-2) to components 1 and 2, α = K1/K2 = y1x2 / x1y2

Hydrocarbon systems Ethylene-ethane Propylene-propane Propyne–1–3-butadiene 1–3 Butadiene–vinyl acetylene Benzene-toluene Benzene–ethyl benzene Benzene–diethyl benzene Toluene–ethyl benzene Toluene-xylenes Ethyl benzene–styrene o-Xylene–m-xylene

73 138 40 130 34, 53 20 50 28 45 34 130

Organic systems Methanol-formaldehyde Dichloroethane-trichloroethane Acetic acid–acetic anhydride Acetic anhydride–ethylene diacetate Vinyl acetate–ethyl acetate Ethylene glycol–diethylene glycol Cumene-phenol Phenol-acetophenone

23 30 50 32 90 16 38 39, 54

Aqueous systems HCN-water Acetic acid–water Methanol-water Ethanol-water Isopropanol-water Vinyl acetate–water Ethylene oxide–water Ethylene glycol–water

13-27

15 40 60 60 12 35 50 16

The availability of computers has decreased our reliance on graphical methods. Nevertheless, diagrams are useful for quick approximations and for demonstrating the effect of various design variables. The x-y diagram is the most convenient for these purposes, and its use is developed in detail here. The use of the enthalpy-concentration diagram is given by Smith (Design of Equilibrium Stage Processes, McGraw-Hill, New York, 1963) and Henley and Seader (EquilibriumStage Separation Operations in Chemical Engineering, Wiley, New York, 1981).

which, since x2 = 1 − x1 and y2 = 1 − y1, can be rewritten as x1α y1 = ᎏᎏ 1 + (α − 1)x1

(13-20)

for use in calculating points for the equilibrium curve. McCABE-THIELE METHOD Operating Lines The McCabe-Thiele method is based upon representation of the material-balance equations as operating lines on the y-x diagram. The lines are made straight (and the need for the energy balance obviated) by the assumption of constant molar overflow. The liquid-phase flow rate is assumed to be constant from tray to tray in each section of the column between addition (feed) and withdrawal (product) points. If the liquid rate is constant, the vapor rate must also be constant. The constant-molar-overflow assumption represents several prior assumptions. The most important one is equal molar heats of vaporization for the two components. The other assumptions are adiabatic operation (no heat leaks) and no heat of mixing or sensible heat effects. These assumptions are most closely approximated for closeboiling isomers. The result of these assumptions on the calculation method can be illustrated with Fig. 13-28, which shows two materialbalance envelopes cutting through the top section (above the top feed stream or sidestream) of the column. If Ln + 1 is assumed to be identical to Ln − 1 in rate, then Vn = Vn − 2 and the component material balance for both envelopes 1 and 2 can be represented by yn = (L/V)xn + 1 + (DxD /V)

(13-21)

where y and x have a stage subscript n or n + 1, but L and V need be identified only with the section of the column to which they apply. Equation (13-21) has the analytical form of a straight line where L/V is the slope and DxD /V is the y intercept at x1 = 0.

PHASE EQUILIBRIUM DATA Three types of binary equilibrium curves are shown in Fig. 13-27. The y-x diagram is almost always plotted for the component that is the more volatile (denoted by the subscript 1) in the region where distillation is to take place. Curve A shows the most usual case, in which component 1 remains more volatile over the entire composition range. Curve B is typical of many systems (ethanol-water, for example) in which the component that is more volatile at low values of x1 becomes less volatile than the other component at high values of x1. The vapor and liquid compositions are identical for the homogeneous azeotrope where curve B crosses the 45° diagonal. A heterogeneous azeotrope is formed with two liquid phases by curve C. An azeotrope limits the separation that can be obtained between components by simple distillation. For the system described by curve B, the maximum overhead-product concentration that could be obtained from a feed with x1 = 0.25 is the azeotropic composition. Similarly, a feed with x1 = 0.9 could produce a bottom-product composition no lower than the azeotrope. The phase rule permits only two variables to be specified arbitrarily in a binary two-phase system at equilibrium. Consequently, the curves in Fig. 13-27 can be plotted at either constant temperature or constant pressure but not both. The latter is more common, and data in Table 13-1 are for that case. The y-x diagram can be plotted in either mole, weight, or volume fractions. The units used later for the phase flow rates must, of course, agree with those used for the equilibrium data. Mole fractions, which are almost always used, are applied here.

FIG. 13-27 Typical binary equilibrium curves. Curve A, system with normal volatility. Curve B, system with homogeneous azeotrope (one liquid phase). Curve C, system with heterogeneous azeotrope (two liquid phases in equilibrium with one vapor phase).

13-28

DISTILLATION If the sidestream in Fig. 13-29 had been a feed, the balance for the section below the feed would be L′ DxD − FxF yn = ᎏ xn + 1 + ᎏᎏ (13-23) V′ V′ Similar equations can be written for the bottom section of the column. For the envelope shown in Fig. 13-30, ym = (L″/V″) xm + 1 − (BxB /V)

FIG. 13-28

Two material-balance envelopes in the top section of a distillation

column.

The effect of a sidestream withdrawal point is illustrated by Fig. 13-29. The material-balance equation for the column section below the sidestream is DxD + SxS L′ yn = ᎏ xn + 1 + ᎏᎏ (13-22) V′ V′ where the primes designate the L and V below the sidestream. Since the sidestream must be a saturated phase, V = V′ if a liquid side stream is withdrawn and L = L′ if it is a vapor.

FIG. 13-29 Material-balance envelope which contains two external streams D and S, where S represents a sidestream product withdrawn above the feed plate.

(13-24)

where the subscript m is used to identify the stage number in the bottom section. Equations such as (13-21) through (13-24) when plotted on the y-x diagram furnish a set of operating lines. A point on an operating line represents two passing streams, and the operating line itself is the locus of all possible pairs of passing streams within the column section to which the line applies. An operating line can be located on the y-x diagram if (1) two points on the line are known or (2) one point and the slope are known. The known points on an operating line are usually its intersection with the y-x diagonal and/or its intersection with another operating line. The slope L/V of the operating line is termed the internal-reflux ratio. This ratio in the operating-line equation for the top section of the column [see Eq. (13-21)] is related to the external-reflux ratio R = LN + 1/D by R L LN + 1 RD (13-25) ᎏ=ᎏ=ᎏ=ᎏ V VN (1 + R)D 1 + R when the reflux stream LN + 1 is a saturated liquid. Thermal Condition of the Feed The slope of the operating line changes whenever a feed stream or a sidestream is passed. To calculate this change, it is convenient to introduce a quantity q which is defined by the following equations for a feed stream F: L′ = L + qF

(13-26)

V = V′ + (1 − q)F

(13-27)

The primes denote the streams below the stage to which the feed is introduced. The q is a measure of the thermal condition of the feed and represents the moles of saturated liquid formed in the feed stage

Material-balance envelope around the bottom end of the column. The partial reboiler is equilibrium stage 1.

FIG. 13-30

GRAPHICAL METHODS FOR BINARY DISTILLATION per mole of feed. It takes on the following values for various possible feed thermal conditions. Subcooled-liquid feed: q > 1 Saturated-liquid feed: q = 1 Partially flashed feed: 1 > q > 0 Saturated-vapor feed: q = 0 Superheated-vapor feed: q < 0 The q value for a particular feed can be estimated from energy to convert 1 mol of feed to saturated vapor q = ᎏᎏᎏᎏᎏᎏ molar heat of vaporization Equations analogous to (13-26) and (13-27) can be written for a sidestream, but the q will be either 1 or 0 depending upon whether the sidestream is taken from the liquid or the vapor stream. The q can be used to derive the “q-line equation” for a feed stream or a sidestream. The q line is the locus of all points of intersection of the two operating lines, which meet at the feed-stream or sidestream stage. This intersection must occur along that section of the q line between the equilibrium curve and the y = x diagonal. At the point of intersection, the same y, x point must satisfy both the operating-line equation above the feed-stream (or sidestream) stage and the one below the feed-stream (or sidestream) stage. Subtracting one equation from the other gives for a feed stage (V − V′)y = (L − L′)x + FxF which when combined with Eqs. (13-26) and (13-27) gives the q-line equation xF q y=ᎏx−ᎏ (13-28) q−1 q−1 A q-line construction for a partially flashed feed is given in Fig. 13-31. It is easily shown that the q line must intersect the diagonal at xF. The slope of the q line is q/(q − 1). All five q-line cases are shown in Fig. 13-32. The derivation of Eq. (13-28) assumes a single-feed column and no sidestream. However, the same result is obtained for other column configurations. Typical q-line constructions for sidestream stages are shown in Fig. 13-33. Note that the q line for a sidestream must always intersect the diagonal at the composition (y1 or x1) of the sidestream. Figure 13-33 also shows the intersections of the operating lines with the diagonal construction line. The top operating line must always intersect the diagonal at the overhead-product composition xD. This

FIG. 13-31 Typical intersection of the two operating lines at the q line for a feed stage. The q line shown is for a partially flashed feed.

13-29

All five cases of q lines; (1) superheated-vapor feed, (2) saturatedvapor feed, (3) partially vaporized feed, (4) saturated-liquid feed, and (5) subcooled-liquid feed. Slope of q line = q/(q − 1).

FIG. 13-32

can be shown by substituting y = x in Eq. (13-21) and using V − L = D to reduce the resulting equation to x = xD. Similarly (except for columns in which open steam is introduced at the bottom), the bottom operating line must always intersect the diagonal at the bottomproduct composition xB. Equilibrium-Stage Construction The alternate use of the equilibrium curve and the operating line to “step off” equilibrium stages is illustrated in Fig. 13-34. The plotted portions of the equilibrium curve (curved) and the operating line (straight) cover the composition range existing in the column section shown in the lower right-hand corner. If yn and xn represent the compositions (in terms of the more volatile component) of the equilibrium vapor and liquid leaving stage n, then point (yn, xn) on the equilibrium curve must represent the equilibrium stage n. The operating line is the locus for compositions of all possible pairs of passing streams within the section and therefore a horizontal line (dotted) at yn must pass through the point (yn, xn + 1) on the operating line since yn and xn + 1 represent passing streams. Likewise, a vertical line (dashed) at xn must intersect the operating line at point (yn − 1, xn). The equilibrium stages above and below stage n can be located by a vertical line through (yn, xn + 1) to find (yn + 1, xn + 1) and a horizontal line through (yn − 1, xn) to find (yn − 1, xn − 1). It can be seen that one can work upward or downward through the column by alternating the use of equilibrium and operating lines. Total-Column Construction The graphical construction for an entire column is shown in Fig. 13-35. The process, pictured in the lower right-hand corner of the diagram, is an existing column with a number of actual trays equivalent to eight equilibrium stages. A partial reboiler (equivalent to an equilibrium stage) and a total condenser are used. This column configuration was analyzed earlier (see Fig. 13-24) and shown to have C + 2N + 9 design variables (degrees of freedom) which must be specified to define one unique operation. These may be used as follows as the basis for a graphical solution: Specifications

N ui

Stage pressures (including reboiler) Condenser pressure Stage heat leaks (except reboiler) Pressure and heat leak in reflux divider Feed stream Feed-stage location Total number of stages N One overhead purity Reflux temperature External-reflux ratio

N 1 N−1 2 C +2 1 1 1 1 1 C + 2N + 9

13-30

DISTILLATION

Typical construction for a sidestream showing the intersection of the two operating lines with the q line and with the x − y diagonal. (a) Liquid sidestream near the top of the column. (b) Vapor sidestream near the bottom of the column. FIG. 13-33

Illustration of how equilibrium stages can be located on the x-y diagram through the alternating use of the equilibrium curve and the operating line.

FIG. 13-34

GRAPHICAL METHODS FOR BINARY DISTILLATION

13-31

FIG. 13-35 Construction for a column with a bubble-point feed, a total condenser, and a partial reboiler.

Pressures can be specified at any level below the safe working pressure of the column. The condenser pressure will be set at 275.8 kPa (40 psia), and all pressure drops within the column will be neglected. The equilibrium curve in Fig. 13-35 represents data at that pressure. All heat leaks will be assumed to be zero. The feed composition is 40 mole percent of the more volatile component 1, and the feed rate is 0.126 (kg⋅mol)/s [1000 (lb⋅mol)/h] of saturated liquid (q = 1). The feed-stage location is fixed at stage 4 and the total number of stages at eight. The overhead purity is specified as xD = 0.95. The reflux temperature is the bubble-point temperature (saturated reflux), and the externalreflux ratio is set at R = 4.5. Answers are desired to the following two questions. First, what bottom-product composition xB will the column produce under these specifications? Second, what will be the top vapor rate VN in this operation, and will it exceed the maximum vapor-rate capacity for this column, which is assumed to be 0.252 (kg⋅mol)/s [2000 (lb⋅mol)/h] at the top-tray conditions? The solution is started by using Eq. (13-25) to convert the externalreflux ratio of 4.5 to an internal-reflux ratio of L/V = 0.818. The xD = 0.95 value is then located on the diagonal, and the upper operating line is drawn as shown in Fig. 13-35. If the xB value were known, the bottom operating line could be immediately drawn from the xB value on the diagonal up to its required intersection point with the upper operating line on the feed q line. In this problem, since the number of stages is fixed, the xB which gives a lower operating line that will require exactly eight stages must be found by trial and error. An xB value is assumed, and the resulting lower operating line is drawn. The stages can be stepped off by starting from either xB or xD; xB was used in this case. Note that the lower operating line is used until the fourth stage is passed, at which time the construction switches to the upper operating line. This is necessary because the vapor and liquid streams passing each other between the fourth and fifth stages must fall on the upper line.

The xB that requires exactly eight equilibrium stages is x1 = 0.026. An overall component balance gives D = 0.051 (kg⋅mol)/s [405 (lb⋅mol)/h]. Then, VN = VB = LN + 1 + D = D(R + 1) = 0.051(4.5 + 1.0) = 0.280 (kg⋅mol)/s [2230 (lb⋅mol)/h] which exceeds the column capacity of 0.252 (kg⋅mol)/s [2007 (lb⋅mol)/h]. This means that the column cannot provide an overheadproduct yield of 40.5 percent at 95 percent purity. Either the purity specification must be reduced, or we must be satisfied with a lower yield. If the xD = 0.95 specification is retained, the reflux rate must be reduced. This will cause the upper operating line to pivot upward around its fixed point of x = 0.95 on the diagonal. The new intersection of the upper line with the q line will lie closer to the equilibrium curve. The xB value must then move upward along the diagonal because the eight stages will not “reach” as far as formerly. The higher xB concentration will reduce the recovery of component 1 in the 95 percent overhead product. Another entire column with a partially vaporized feed, a liquidsidestream rate equal to D and withdrawn from the second stage from the top, and a total condenser is shown in Fig. 13-36. The specified concentrations are xF = 0.40, xB = 0.05, and xD = 0.95. The specified L/V ratio in the top section is 0.818. These specifications permit the top operating line to be located and the two top stages stepped off to determine the liquid-sidestream composition xS = 0.746. The operating line below the sidestream must intersect the diagonal at the “blend” of the sidestream and the overhead stream. Since S was specified to be equal to D in rate, the intersection point is (1.0)(0.746) + (1.0)(0.95) x = ᎏᎏᎏ = 0.848 1.0 + 1.0 This point plus the point of intersection of the two operating lines on the sidestream q line (vertical at xS = 0.746) permits the location of the middle operating line. (The slope of the middle operating line could

13-32

DISTILLATION

FIG. 13-36 Graphical solution for a column with a partially flashed feed, a liquid sidestream and a total condenser.

also have been used.) The lower operating line must run from the specified xB value on the diagonal to the required point of intersection on the feed q line. The stages are stepped off from the top down in this case. The sixth stage from the top is the feed stage, and a total of about 11.4 stages is required to reach the specified xB = 0.05. Fractional equilibrium stages have meaning. The 11.4 will be divided by a tray efficiency, and the rounding to an integral number of actual trays should be done after that division. For example, if the average tray efficiency for the process being modeled in Fig. 13-36 were 80 percent, then the number of actual trays required would be 11.4/0.8 = 14.3, which would be rounded to 15. Feed-Stage Location The optimum feed-stage location is that location which, with a given set of other operating specifications, will result in the widest separation between xD and xB for a given number of stages. Or, if the number of stages is not specified, the optimum feed location is the one that requires the lowest number of stages to accomplish a specified separation between xD and xB. Either of these criteria will always be satisfied if the operating line farthest from the equilibrium curve is used in each step as in Fig. 13-35. It can be seen from Fig. 13-35 that the optimum feed location would have been the fifth tray for that operation. If a new column were being designed, that would have been the designer’s choice. However, when an existing column is being modeled, the feed stage on the diagram should correspond as closely as possible to the actual feed tray in the column. It can be seen that a badly mislocated feed (a feed that requires one to remain with an operating line until it closely approaches the equilibrium curve) can be very wasteful insofar as the effectiveness of the stages is concerned. Minimum Stages A column operating at total reflux is diagramed in Fig. 13-37a. Enough material has been charged to the column to fill the reboiler, the trays, and the overhead condensate drum to their working levels. The column is then operated with no feed and with all the condensed overhead stream returned as reflux (LN + 1 = VN and D = 0). Also all the liquid reaching the reboiler is vaporized and returned to the column as vapor. Since F, D, and B are all zero, Ln + 1 = Vn at all points in the column. With a slope of unity (L/V = 1.0), the operating line must coincide with the diagonal throughout the col-

umn. Total-reflux operation gives the minimum number of stages required to effect a specified separation between xB and xD. Minimum Reflux The minimum-reflux ratio is defined as that ratio which if decreased by an infinitesimal amount would require an infinite number of stages to accomplish a specified separation between two components. The concept has meaning only if a separation between two components is specified and the number of stages is not specified. Figure 13-37b illustrates the minimum-reflux condition. As the reflux ratio is reduced, the two operating lines swing upward, pivoting around the specified xB and xD values, until one or both touch the equilibrium curve. For equilibrium curves shaped like the one shown, the contact occurs at the feed q line. Often an equilibrium curve will dip down closer to the diagonal at higher concentrations. In such cases, the upper operating line may make contact before its intersection point on the q line reaches the equilibrium curve. Wherever the contact appears, the intersection of the operating line with the equilibrium curve produces a pinch point which contains a very large number of stages, and a zone of constant composition is formed. Intermediate Reboilers and Condensers A distillation column of the type shown in Fig. 13-2a, operating with an interreboiler and an intercondenser in addition to a reboiler and a condenser, is diagramed with the solid lines in Fig. 13-38. The dashed lines correspond to simple distillation with only a bottoms reboiler and an overhead condenser. Total boiling and condensing heat loads are the same for both columns. As shown by Kayihan [Am. Inst. Chem. Eng. J. Symp. Ser. 76, 192, 1 (1980)], the addition of interreboilers and intercondensers increases thermodynamic efficiency but requires additional stages, as is clear from the positions of the operating lines in Fig. 13-38. Optimum Reflux Ratio The general effect of the operating reflux ratio on fixed costs, operating costs, and the sum of these is shown in Fig. 13-39. In ordinary situations, the minimum on the totalcost curve will generally occur at an operating reflux ratio of from 1.1 to 1.5 times the minimum R = LN + 1/D value, with the lower value corresponding to a value of the relative volatility close to 1. Difficult Separations Some binary separations may pose special problems because of extreme purity requirements for one or both products or because of a relative volatility close to 1. The y-x diagram

GRAPHICAL METHODS FOR BINARY DISTILLATION

(a)

(b) McCabe-Thiele diagrams for limiting cases. (a) Minimum stages for a column operating at total reflux with no feeds or products. (b) Minimum reflux for a binary system of normal volatility. FIG. 13-37

13-33

13-34

DISTILLATION A system with constant relative volatility can be handled conveniently by the equation of Smoker [Trans. Am. Inst. Chem. Eng., 34, 165 (1938)]. The derivation of the equation is shown, and its use is illustrated by Smith (op. cit.). Stage Efficiency The use of the Murphree plate efficiency is particularly convenient on y-x diagrams. The Murphree efficiency is defined for the vapor phase as η = (yn − yn − 1)/(y*n − yn − 1)

McCabe-Thiele diagram for columns with and without an interreboiler and an intercondenser. FIG. 13-38

is convenient for stepping off stages at extreme purities if it is plotted on log-log paper. The equilibrium curve at very low x1 values on ordinary graph paper can usually be assumed to be a straight line with an intercept term of zero which can be expressed as y = (y/x)x + 0.0 where the slope y/x is a constant. The necessity for knowing the slope is eliminated by taking the logarithm of both sides log y = log x + log (y/x) and plotting y versus x on a log-log plot to give a straight line with a slope of unity. The slope y/x is now an intercept term which need not be known. One point from the equilibrium curve is sufficient, therefore, to locate the equilibrium curve on the log-log plot. The operating line will be curved on the log-log plot and is located by plotting the appropriate material-balance equation. Both the equilibrium and the operating lines can be extended to any purity desired.

where y*n is the composition of the vapor that would be in equilibrium with the liquid leaving stage n and is the value read from the equilibrium curve. The yn − 1 and yn are the actual (nonequilibrium) values for vapor streams leaving the n − 1 and n stages respectively. Note that the yn − 1 and yn values assume that vapor streams are completely mixed and uniform in composition. An analogous efficiency can be defined for the liquid phase. The application of a 50 percent Murphree vapor-phase efficiency on a y-x diagram is illustrated in Fig. 13-40. A “pseudo-equilibrium” curve is drawn halfway (on a vertical line) between the operating lines and the true-equilibrium curve. The true-equilibrium curve is used for the first stage (the partial reboiler is assumed to be an equilibrium stage), but for all other stages the vapor leaving each stage is assumed to approach the equilibrium value y*n only 50 percent of the way. Consequently, the steps in Fig. 13-40 represent actual trays. Application of a constant efficiency to each stage as in Fig. 13-40 will not give, in general, the same answer as obtained when the number of equilibrium stages (obtained by using the true-equilibrium curve) is divided by the same efficiency factor. The prediction and use of stage efficiencies are described in detail in Sec. 14. Miscellaneous Operations The y-x diagrams for several other column configurations have not been presented here. The omitted items are partial condensers, rectifying columns (feed introduced to the bottom stage), stripping columns (feed introduced to the top stage), total reflux in the top section but not in the bottom section, multiple feeds, and introduction of open steam to the bottom stage to eliminate the reboiler. These configurations are discussed in Smith (op. cit.) and Henley and Seader (op. cit.), who also describe the more rigorous Ponchon-Savarit method, which is not included here.

Application of a 50 percent Murphree vapor-phase efficiency to each stage (excluding the reboiler) in the column. Each step in the diagram corresponds to an actual stage.

FIG. 13-40 FIG. 13-39

aration.

Location of the optimum reflux for a given feed and specified sep-

(13-29)

APPROXIMATE MULTICOMPONENT DISTILLATION METHODS

13-35

APPROXIMATE MULTICOMPONENT DISTILLATION METHODS INTRODUCTION Some approximate calculation methods for the solution of multicomponent, multistage separation problems continue to serve useful purposes even though computers are available to provide more rigorous solutions. The available phase equilibrium and enthalpy data may not be accurate enough to justify the longer rigorous methods. Or in extensive design and optimization studies, a large number of cases can be worked quickly and cheaply by an approximate method to define roughly the optimum specifications, which can then be investigated more exactly with a rigorous method. Two approximate multicomponent shortcut methods for simple distillation are the Smith-Brinkley (SB) method, which is based on an analytical solution of the finite-difference equations that can be written for staged separation processes when stages and interstage flow rates are known or assumed and the Fenske-Underwood-Gilliland (FUG) method, which combines Fenske’s total-reflux equation and Underwood’s minimum-reflux equation with a graphical correlation by Gilliland that relates actual column performance to total- and minimum-reflux conditions for a specified separation between two key components. Thus, the SB and FUG methods are rating and design methods respectively. Both methods work best when mixtures are nearly ideal. The SB method is not presented here, but is presented in detail in the sixth edition of Perry’s Chemical Engineers’ Handbook. Extensions of the SB method to nonideal mixtures and complex configurations are developed by Eckert and Hlavacek [Chem. Eng. Sci., 33, 77 (1978)] and Eckert [Chem. Eng. Sci., 37, 425 (1982)] respectively but are not discussed here. However, the approximate and very useful method of Kremser [Nat. Pet. News, 22(21), 43 (May 21, 1930)] for application to absorbers and strippers is discussed at the end of this subsection.

The particular value of αi is the effective value used in Eqs. (13-36) and (13-34) defined in terms of values for each stage in the column by αN = αNαN − 1 ⋅ ⋅ ⋅ α2α1

(13-34)

Equations (13-31) and (13-32) are rigorous relationships between the splits obtained for components i and r in a column at total reflux. However, the correct value of αi must always be estimated, and this is where the approximation enters. It is usually estimated from or

α = (αtopαbottom)1/2

(13-35)

α = (αtopαmiddleαbottom)1/3

(13-36)

A reasonably good estimate of the separation that will be accomplished in a plant column often can be obtained by specifying the split of one component (designated as the reference component r), setting Nm equal to from 40 to 60 percent of the number of equilibrium stages (not actual trays), and then using Eq. (13-32) to estimate the splits of all the other components. This is an iterative calculation because the component splits must first be arbitrarily assumed to give end compositions that can be used to give initial end-temperature estimates. The αtop and αbottom values corresponding to these end temperatures are used in Eq. (13-35) to give αi values for each component. The iteration is continued until the αi values do not change from trial to trial. The Underwood minimum-reflux equations of main interest are those that apply when some of the components do not appear in either the distillate or the bottoms products at minimum reflux. These equations are αi(xi,D)m = Rm + 1 (13-37) i ᎏ αi − ⌰ and

αixi,F

=1−q ᎏ α −⌰ i

(13-38)

i

FENSKE-UNDERWOOD-GILLILAND (FUG) SHORTCUT METHOD In this approach, Fenske’s equation [Ind. Eng. Chem., 24, 482 (1932)] is used to calculate Nm, which is the number of plates required to make a specified separation at total reflux, i.e., the minimum value of N. Underwood’s equations [J. Inst. Pet., 31, 111 (1945); 32, 598 (1946); 32, 614 (1946); and Chem. Eng. Prog., 44, 603 (1948)] are used to estimate the minimum-reflux ratio Rm. The empirical correlation of Gilliland [Ind. Eng. Chem., 32, 1220 (1940)] shown in Fig. 13-41 then uses these values to give N for any specified R or R for any specified N. Limitations of the Gilliland correlation are discussed by Henley and Seader (Equilibrium-Stage Separation Operations in Chemical Engineering, Wiley, New York, 1981). The following equation, developed by Molokanov et al. [Int. Chem. Eng., 12(2), 209 (1972)] satisfies the end points and fits the Gilliland curve reasonably well: N − Nm ᎏ = 1 − exp N+1

1 + 54.4Ψ

Ψ−1

ᎏ ᎏᎏ 11 + 117.2Ψ   Ψ  0.5

(13-30)

where Ψ = (R − Rm)/(R + 1). The Fenske total-reflux equation can be written as

ᎏx  = (α ) ᎏx  xi

i

xi

Nm

r

log or as

r

(13-31)

B

ᎏDBxxᎏ ᎏDBxxᎏ D

B

i

D r

Nm = ᎏᎏᎏ log αi B

(13-32)

where i is any component and r is an arbitrarily selected reference component in the definition of relative volatilities. αi = K i /K r = yi xr /yr xi

(13-33)

FIG. 13-41 Comparison of rigorous calculations with Gilliland correlation. [Henley and Seader, Equilibrium-Stage Separation Operations in Chemical Engineering, Wiley, New York, 1981; data of Van Winkle and Todd, Chem. Eng., 78(21), 136 (Sept. 20, 1971); data of Gilliland, Elements of Fractional Distillation, 4th ed., McGraw-Hill, New York, 1950; data of Brown and Martin, Trans. Am. Inst. Chem. Eng., 35, 679 (1939).]

13-36

DISTILLATION

The relative volatilities αi are defined by Eq. (13-33), Rm is the minimum-reflux ratio (LN + 1 /D)min, and q describes the thermal condition of the feed (e.g., 1.0 for a bubble-point feed and 0.0 for a saturatedvapor feed). The xi,F values are available from the given feed composition. The ⌰ is the common root for the top-section equations and the bottom-section equations developed by Underwood for a column at minimum reflux with separate zones of constant composition in each section. The common root value must fall between αhk and αlk, where hk and lk stand for heavy key and light key respectively. The key components are the ones that the designer wants to separate. In the butane-pentane splitter problem used in Example 1, the light key is n-C4 and the heavy key is i-C5. The αi values in Eqs. (13-37) and (13-38) are effective values obtained from Eq. (13-35) or Eq. (13-36). Once these values are available, ⌰ can be calculated in a straightforward iteration from Eq. (13-38). Since the (α − ⌰) difference can be small, ⌰ should be determined to four decimal places to avoid numerical difficulties. The (xi,D)m values in Eq. (13-37) are minimum-reflux values, i.e., the overhead concentration that would be produced by the column operating at the minimum reflux with an infinite number of stages. When the light key and the heavy key are adjacent in relative volatility and the specified split between them is sharp or the relative volatilities of the other components are not close to those of the two keys, only the two keys will distribute at minimum reflux and the (xi, D)m values are easily determined. This is often the case and is the only one considered here. Other cases in which some or all of the nonkey components distribute between distillate and bottom products are discussed in detail by Henley and Seader (op. cit.). The FUG method is convenient for new-column design with the following specifications: 1. R/Rm, the ratio of reflux to minimum reflux 2. Split on the reference component (usually chosen as the heavy key) 3. Split on one other component (usually the light key) However, the total number of equilibrium stages N, N/Nm, or the external-reflux ratio can be substituted for one of these three specifications. It should be noted that the feed location is automatically specified as the optimum one; this is assumed in the Underwood equations. The assumption of saturated reflux is also inherent in the Fenske and Underwood equations. An important limitation on the Underwood equations is the assumption of constant molar overflow. As discussed by Henley and Seader (op. cit.), this assumption can lead to a prediction of the minimum reflux that is considerably lower than the actual value. No such assumption is inherent in the Fenske equation. An exact calculational technique for minimum reflux is given by Tavana and Hansen [Ind. Eng. Chem. Process Des. Dev., 18, 154 (1979)]. A computer program for the FUG method is given by Chang [Hydrocarbon Process., 60(8), 79 (1980)]. The method is best applied to mixtures that form ideal or nearly ideal solutions. Example 1: Calculation of FUG Method A large butane-pentane splitter is to be shut down for repairs. Some of its feed will be diverted temporarily to an available smaller column, which has only 11 trays plus a partial reboiler. The feed enters on the middle tray. Past experience on similar feeds indicates that the 11 trays plus the reboiler are roughly equivalent to 10 equilibrium stages and that the column has a maximum top-vapor capacity of 1.75 times the feed rate on a mole basis. The column will operate at a condenser pressure of 827.4 kPa (120 psia). The feed will be at its bubble point (q = 1.0) at the feed-tray conditions and has the following composition on the basis of 0.0126 (kg⋅mol)/s [100 (lb⋅mol)/h]:

Component

FxF

C3 i-C4 n-C4 i-C5 n-C5

5 15 25 20 35 100

The original column normally has less than 7 mol percent i-C5 in the overhead and less than 3 mole percent n-C4 in the bottoms product when operating at a distillate rate of D/F = 0.489. Can these product purities be produced on the smaller column at D/F = 0.489? Pressure drops in the column will be neglected, and the K values will be read at 827 kPa (120 psia) in both column sections from the DePriester nomograph in Fig. 13-14b. When constant molar overflow is assumed in each section, the rates in pound-moles per hour in the upper and lower sections are as follows: Top section

Bottom section

D = (0.489)(100) = 48.9 V = (1.75)(100) = 175 L = 175 − 48.9 = 126.1 V/L = 1.388

B = 100 − 48.9 = 51.1 V′ = V = 175 L′ = L + F = 226.1 V′/L′ = 0.7739

L/L′ = 126.1/226.1 = 0.5577 R = 126.1/48.9 = 2.579

NOTE: To convert pound-moles per hour to kilogram-moles per second, multiply by 1.26 × 10−4.

Since the feed enters at the middle of the column, M = 5 and M + 1 = 6. Application of the FUG method is demonstrated on the splitter. Specifications necessary to model the existing column are: 1. N = 10, total number of equilibrium stages. 2. Optimum feed location (which may or may not reflect the actual location). 3. Maximum V/F at the top tray of 1.75. 4. Split on one component given in the following paragraphs. The solution starts with an assumed arbitrary split of all the components to give estimates of top and bottom compositions that can be used to get initial end temperatures. The αi’s evaluated at these temperatures are averaged with an assumed feed-stage temperature (assumed to be the bubble point of the feed) by using Eq. (13-36). The initial assumption for the split on i-C5 will be DxD/BxB = 3.15/16.85. As mentioned earlier, Nm usually ranges from 0.4N to 0.6N, and the initial Nm value assumed here will be (0.6)(10) = 6.0. Equation (13-32) can be rewritten as = α ᎏ = α ᎏᎏ Fx − Dx  16.85 DxD

F

6.0 i

D

3.15

6.0 i

(0.1869)

i

0.1869α6.0 i Dxi,D = ᎏᎏ Fxi,F 1 + 0.1869α6.0 i The evaluation of this equation for each component is as follows: or

Component

αi

C3 i-C4 n-C4 i-C5 n-C5

5.00 2.63 2.01 1.00 0.843

αi6.0 330 66 1.00 0.36

0.1869α i6.0

FxF

DxD

BxB

61.7 12.3 0.187 0.0672

5 15 25 20 35 100

5.0 14.8 25.1 3.15 2.20 48.25

0.0 0.2 1.9 16.85 32.80 51.75

The end temperatures corresponding to these product compositions are 344 K (159°F) and 386 K (236°F). These temperatures plus the feed bubble-point temperature of 358 K (185°F) provide a new set of αi’s which vary only slightly from those used earlier. Consequently, the D = 48.25 value is not expected to vary greatly and will be used to estimate a new i-C5 split. The desired overhead concentration for i-C5 is 7 percent; so it will be assumed that DxD = (0.07)(48.25) = 3.4 for i-C5 and that the split on that component will be 3.4/16.6. The results obtained with the new αi’s and the new i-C5 split are as follows: Component C3 i-C4 n-C4 i-C5 n-C5

α i6.0 322 68 1.00 0.415

0.2048α i6.0

FxF

DxD

BxB

xD

xB

65.9 13.9 0.205 0.085

5 15 25 20 35 100

5.0 14.8 23.3 3.4 2.7 49.2

0.0 0.2 1.7 16.6 32.3 50.8

0.102 0.301 0.473 0.069 0.055 1.000

0.000 0.004 0.033 0.327 0.636 1.000

The calculated i-C5 concentration in the overhead stream is 6.9 percent, which is close enough to the 7.0 figure for now. Table 13-8 shows subsequent calculations using the Underwood minimumreflux equations. The α and xD values in Table 13-8 are those from the Fenske

APPROXIMATE MULTICOMPONENT DISTILLATION METHODS TABLE 13-8

13-37

Application of Underwood Equations θ = 1.36

Component

xF

α

αxF

α−θ

C3 i-C4 n-C4 i-C5 n-C5

0.05 0.15 0.25 0.20 0.35 1.00

4.99 2.62 2.02 1.00 0.864

0.2495 0.3930 0.5050 0.2000 0.3024

3.63 1.26 0.66 −0.36 −0.496

θ = 1.365 αxF α−θ ᎏ α−θ

αxF ᎏ α−θ

0.0687 0.3119 0.7651 −0.5556 −0.6097 −0.0196

3.625 1.255 0.655 −0.365 −0.501

0.0688 0.3131 0.7710 −0.5479 −0.6036 +0.0014

xD

αxD

α−θ

0.102 0.301 0.473 0.069 0.055 1.000

0.5090 0.7886 0.9555 0.0690 0.0475

3.6253 1.2553 0.6553 −0.3647 −0.5007

αxD ᎏ α−θ 0.1404 0.6282 1.4581 −0.1892 −0.0949 1.9426 = Rm + 1

Interpolation gives θ = 1.3647. total-reflux calculation. As noted earlier, the xD values should be those at minimum reflux. This inconsistency may reduce the accuracy of the Underwood method, but to be useful a shortcut method must be fast, and it has not been shown that a more rigorous estimation of xD values results in an overall improvement in accuracy. The calculated Rm is 0.9426. The actual reflux assumed is obtained from the specified maximum top vapor rate of 0.022 (kg⋅mol)/s [175 (lb⋅mol)/h] and the calculated D of 49.2 (from the Fenske equation). LN + 1 = VN − D R = VN /D − 1 = 175/49.2 − 1 = 2.557 The Rm = 0.9426, R = 2.557, and N = 10 values are now used with the Gilliland correlation in Fig. 13-41 or Eq. (13-30) to check the initially assumed value of 6.0 for Nm. Equation (13-30) gives Nm = 6.95, which differs considerably from the assumed value. Repetition of the calculations with Nm = 7.0 gives R = 2.519, Rm = 0.9782, and a calculated check value of Nm = 6.85, which is close enough. The final-product compositions and the α values used are as follows: Component

αi

DxD

BxB

xD

xB

C3 i-C4 n-C4 i-C5 n-C5

4.98 2.61 2.02 1.00 0.851

5.00 14.91 24.16 3.48 2.23 49.78

0 0.09 0.84 16.52 32.87 50.32

0.1004 0.2996 0.4852 0.0700 0.0448 1.0000

0.0 0.0017 0.0168 0.3283 0.6532 1.0000

These results indicate that the 7 percent i-C5 in D and the 3 percent n-C4 in B concentrations obtained in the original column can easily be obtained on the smaller column. Unfortunately, this disagrees somewhat with the answers obtained from a rigorous computer solution as shown in the following comparison: xD

Consider the general adiabatic countercurrent cascade of Fig. 13-42 where v and ᐉ are molar component flow rates. Regardless of whether the cascade is an absorber or a stripper, components in the entering vapor will tend to be absorbed and components in the entering liquid will tend to be stripped. If more moles are stripped than absorbed, the cascade is a stripper; otherwise, the cascade is an absorber. The Kremser method is general and applies to either case. Application of component material-balance and phase equilibrium equations successively to stages 1 through N − 1, 1 through N − 2, etc., as shown by Henley and Seader (op. cit.), leads to the following equations originally derived by Kremser. For each component i, (vi)N = (vi)0(Φi)A + (ᐉi)N + 1[1 − (Φi)S]

(13-39)

(Ai)e − 1 (Φi)A = ᎏᎏ (13-40) (Ai)eN + 1 − 1 is the fraction of component i in the entering vapor that is not absorbed. where

(Si)e − 1 (Φi)S = ᎏᎏ (13-41) (Si)eN + 1 − 1 is the fraction of component i in the entering liquid that is not stripped. (Ai)e = (L/KiV)e

(13-42)

is the effective or average absorption factor for component i, and (Si)e = 1/(Ai)e

(13-43)

xB

Component

Rigorous

FUG

Rigorous

FUG

C3 i-C4 n-C4 i-C5 n-C5

0.102 0.299 0.473 0.073 0.053 1.000

0.100 0.300 0.485 0.070 0.045 1.000

0.0 0.006 0.037 0.322 0.635 1.000

0.0 0.002 0.017 0.328 0.653 1.000

KREMSER GROUP METHOD Starting with the classical method of Kremser (op. cit.), approximate group methods of increasing complexity have been developed to calculate groups of equilibrium stages for a countercurrent cascade, such as is used in simple absorbers and strippers of the type depicted in Fig. 13-7b and d. However, none of these group methods can adequately account for stage temperatures that are considerably higher or lower than the two entering-stream temperatures for absorption and stripping respectively when appreciable composition changes occur. Therefore, only the simplest form of the Kremser method is presented here. Fortunately, rigorous computer methods described later can be applied when accurate results are required. The Kremser method is most useful for making preliminary estimates of absorbent and stripping-agent flow rates or equilibrium-stage requirements. The method can also be used to extrapolate quickly results of a rigorous solution to a different number of equilibrium stages.

FIG. 13-42

or stripping.

General adiabatic countercurrent cascade for simple absorption

13-38

DISTILLATION

FIG. 13-43

Absorption and stripping factors. [W. C. Edmister, Am. Inst. Chem. Eng. J., 3, 165–171

(1957).]

is the effective or average stripping factor for component i. When the entering streams are at the same temperature and pressure and negligible absorption and stripping occur, effective component absorption and stripping factors are determined simply by enteringstream conditions. Thus, if K values are composition-independent, then (Ai)e = 1/(Si)e = (LN + 1/Ki{TN + 1, PN + 1}V0)

(13-44)

When entering-stream temperatures differ and/or moderate to appreciable absorption and/or stripping occurs, values of Ai and Si should be based on effective average values of L, V, and Ki in the cascade. However, even then Eq. (13-44) with TN + 1 replaced by (TN + 1 + T0)/2 may be able to give a first-order approximation of (Ai)e. In the case of an absorber, LN + 1 < Le and V0 > Ve will be compensated to some extent by Ki{(TN + 1 + T0)/2, P)} < Ki{Te, P}. A similar compensation, but in opposite directions, will occur in the case of a stripper.

RIGOROUS METHODS FOR MULTICOMPONENT DISTILLATION-TYPE SEPARATIONS

13-39

Equations (13-40) and (13-41) are plotted in Fig. 13-43. Components having large values of Ae or Se absorb or strip respectively to a large extent. Cooresponding values of ΦA and ΦS approach a value of 1 and are almost independent of the number of equilibrium stages. An estimate of the minimum absorbent flow rate for a specified amount of absorption from the entering gas of some key component K for a cascade with an infinite number of equilibrium stages is obtained from Eq. (13-40) as (LN + 1)min = K KV0[1 − (ΦK)A]

(13-45)

The corresponding estimate of minimum stripping-agent flow rate for a stripper is obtained as (V0)min = LN + 1[1 − (ΦK)S]/K K

(13-46)

Example 2: Calculation of Kremser Method For the simple absorber specified in Fig. 13-44, a rigorous calculation procedure as described below gives results in Table 13-9. Values of Φ were computed from componentproduct flow rates, and corresponding effective absorption and stripping factors were obtained by iterative calculations in using Eqs. (13-40) and (13-41) with N = 6. Use the Kremser method to estimate component-product rates if N is doubled to a value of 12. Assume that values of Ae and Se will not change with a change in N. Application of Eqs. (13-40), (13-41), and (13-39) gives the results in the last four columns of Table 13-10. Because of its small value of Ae, the extent of absorption of C1 is unchanged. For the other components, somewhat increased amounts of absorption occur. The degree of stripping of the absorber oil is essentially unchanged. Overall, only an additional 0.5 percent of absorption occurs. The greatest increase in absorption occurs for n-C4, to the extent of about 4 percent. TABLE 13-9

FIG. 13-44

Specifications for the absorber example.

Results of Calculations for Simple Absorber of Fig. 13-44 N = 6 (rigorous method)

N = 12 (Kremser method)

(lb ⋅ mol)/h Component

(vi)6

(ᐉi)1

(Φi)A

C1 C2 C3 nC4 nC5 Absorber oil Totals

147.64 276.03 105.42 1.15 0.0015 0.05 530.29

12.36 94.97 134.58 23.85 4.9985 164.95 435.71

0.9228 0.7460 0.4393 0.0460 0.0003

NOTE:

(Φi)S

(Ai)e

(lb ⋅ mol)/h (vi)12 (ᐉi)1

(Si)e

0.0772 0.2541 0.5692 1.3693 3.6 0.9997

147.64 275.98 103.46 0.16 0 0.05 527.29

0.0003

12.36 94.02 136.54 24.84 5.0 164.95 437.71

(Φi)A

(Φi)S

0.9228 0.7459 0.4311 0.0063 0.0 0.9997

To convert pound-moles per hour to kilogram-moles per hour, multiply by 0.454.

TABLE 13-10 Component C3 i-C4 n-C4 i-C5 n-C5

Top-Down Calculations for Example 3 ᐉ10 ᐉ10 R+1 A10 ᎏ ᎏ+1 d d 3.58 0.260 0.931 1.931 3.58 0.522 1.87 2.87 3.58 0.693 2.48 3.48 3.58 1.44 5.16 6.16 3.58 1.72 6.16 7.16

A9 0.232 0.450 0.590 1.22 1.46

ᐉ9 ᎏ d 0.448 1.29 2.05 7.52 10.5

ᐉ9 ᎏ+1 d 1.448 2.29 3.05 8.52 11.5

A8 0.212 0.405 0.526 1.05 1.23

ᐉ8 ᎏ d 0.307 0.927 1.60 8.95 14.1

ᐉ8 ᎏ+1 d 1.307 1.927 2.60 9.95 15.1

A7 0.198 0.371 0.484 0.936 1.09

ᐉ7 ᎏ d 0.259 0.715 1.26 9.31 16.5

RIGOROUS METHODS FOR MULTICOMPONENT DISTILLATION-TYPE SEPARATIONS INTRODUCTION Availability of large digital computers has made possible rigorous solutions of equilibrium-stage models for multicomponent, multistage distillation-type columns to an exactness limited only by the accuracy of the phase equilibrium and enthalpy data utilized. Time and cost requirements for obtaining such solutions are very low compared with the cost of manual solutions. Methods are available that can accurately solve almost any type of distillation-type problem quickly and efficiently. The material presented here covers, in some

detail, some of the more widely used computer algorithms as well as the classical Thiele-Geddes manual method. All are rating methods, in that the number of equilibrium stages and feed and withdrawal stages are specified. However, a successive-approximation design method that utilizes a rating method is given by Ricker and Grens [Am. Inst. Chem. Eng. J., 20, 238 (1974).] Those desiring further details are referred to the textbooks by Henley and Seader, King, and Holland cited under “General References” at the beginning of this section. These books, in turn, cite a myriad of references in chemical engineering journals. The mathematics involved is that of dealing

13-40

DISTILLATION

with sets of nonlinear algebraic equations. The general nature of the main mathematical problems is presented lucidly by Friday and Smith [“An Analysis of the Equilibrium Stage Separation Problem— Formulation and Convergence,” Am. Inst. Chem. Eng. J., 10, 698 (1964)]. THIELE-GEDDES STAGE-BY-STAGE METHOD FOR SIMPLE DISTILLATION Prior to the availability of digital computers, the most widely used manual methods for rigorous calculations of simple distillation were those of Lewis and Matheson (LM) [Ind. Eng. Chem., 24, 496 (1932)] and Thiele and Geddes (TG) [Ind. Eng. Chem., 25, 290 (1933)], in which the equilibrium-stage equations are solved one by one by using tearing techniques. The former is a design method, in which the number of stages is determined for a specified split between two key components. Thus, it is a rigorous analog of the FUG shortcut method. The TG method is a rating method in which distribution of components between distillate and bottoms is predicted for a specified number of stages. Thus, the TG method is a rigorous analog of the SB method. Both the LM and the TG methods suffer from numerical difficulties that can prevent convergence in certain cases. The stage-to-stage calculation used in the LM method proceeds from the top down and from the bottom up and is subject to large truncation-error buildup if the components differ widely in volatility. The TG method avoids that difficulty, but numerical instabilities arise as soon as the stage-to-stage calculation crosses a feed stage. Then, a difference term appears in the equations, and sometimes this results in a serious loss of significant digits, making the TG method basically unsuited for multiplefeed columns. All stage-to-stage methods that work from both ends of the column toward the middle suffer from two other disadvantages. First, the top-down and the bottom-up calculations must “mesh” somewhere in the column. Usually the mesh is made at a feed stage, and if more than one feed stage exists, a choice of mesh point must be made for each component. When the components vary widely in volatility, the same mesh point cannot be used for all components if serious numerical difficulties are to be avoided. Second, arbitrary procedures must be set up to handle nondistributed components. (A nondistributed component is one whose concentration in one of the end-product streams is smaller than the smallest number carried by the computer.) In the LM and TG equations, the concentrations for these components do not naturally take on nonzero values at the proper point as the calculations proceed through the column. Because of all these numerical difficulties, neither the LM nor the TG stage-by-stage method is commonly implemented in modern computer algorithms. Nevertheless, the TG method is very instructive and is developed in the following example. For a single narrow-boiling feed, the TG manual method is quite efficient.

K Stage

V

L

T

C3

i-C4

n-C4

i-C5

n-C5

10 9 8 7 6 5 4 3 2 1

175

126.1

163.5 178.5 191.3 202.0 210.0 216.4 221.7 226.3 230.3 234.0

2.77 3.10 3.40 3.63 3.84 4.00 4.15 4.28 4.36 4.42

1.38 1.60 1.78 1.94 2.06 2.21 2.28 2.36 2.43 2.50

1.04 1.22 1.37 1.49 1.60 1.73 1.80 1.88 1.94 1.99

0.500 0.590 0.685 0.770 0.825 0.895 0.925 0.965 1.000 1.030

0.420 0.495 0.585 0.660 0.702 0.765 0.800 0.835 0.870 0.890

 ↓

 ↓

226.1



↓

51.1

Stage compositions in the TG method are obtained by stage-to-stage calculations from both ends toward the feed stage. With reference to Fig. 13-1, the calculations work with the ratios vn /d, ᐉn/d, vm /b, and ᐉm /b instead of v or ᐉ directly. The working equations are derived as follows: In the rectifying section, the equilibrium relationship for component i at any stage n can be expressed in terms of component flow rate in the distillate d = DxD and component absorption factor An = Ln /KnVn. xn = yn /Kn Lnxn = (Ln /KnVn)Vnyn ᐉn = Anvn ᐉn/d = (vn /d)An

(13-47)

The general component-i balance around a section of stages from stage n to the top of the column is vn = ᐉn + 1 + d or

vn /d = (ᐉn + 1 /d) + 1

(13-48)

Increasing the subscripts in Eq. (13-47) by 1 and substituting for ᐉn + 1/d in Eq. (13-48) gives the following combined equilibrium and material-balance relationship for component i: vn /d = (vn + 1 /d)An + 1 + 1

(13-49)

Or, if vn /d is eliminated in Eq. (13-48) ᐉn + 1 ᐉn ᎏ = An ᎏ + 1 d d





(13-50)

Equation (13-50) is used to calculate, from the previous stage, the (ᐉ/d) ratio on each stage in the rectifying section. The assumed temperature and phaserate-profile assumptions conveniently fix all the An values for ideal solutions. The calculations are started by writing the equation for stage N: ᐉN ᐉN + 1 ᎏ = AN ᎏ + 1 d d





(13-51)

ᐉN + 1 /d = LN + 1/D = R

(13-52)

For a total condenser, xD = x N + 1 and

Example 3: Calculation of TG Method The TG method will be demonstrated by using the same example problem that was used above for the approximate methods. The example column was analyzed previously and found to have C + 2N + 9 design variables. The specifications to be used in this example were also listed at that time and included the total number of stages (N = 10), the feed-plate location (M = 5), the reflux temperature (corresponding to saturated liquid), the distillate rate (D = 48.9), and the top vapor rate (V = 175). As before, the pressure is uniform at 827 kPa (120 psia), but a pressure gradient could be easily handled if desired. A temperature profile plus a vapor-rate profile through the column must be assumed to start the procedure. These variables are referred to as tear variables and must be iterated on until convergence is achieved in which their values no longer change from iteration to iteration and all equations are satisfied to an acceptable degree of tolerance. Each iteration down and then up through the column is referred to as a column iteration. A set of assumed values of the tear variables consistent with the specifications, plus the component K values at the assumed temperatures, is as follows, using assumed end and middle temperatures and K values from Fig. 13-14b:

A knowledge of the reflux ratio (obtained from the specified distillate and top vapor rates) permits the calculation of (ᐉN/d)i from which (ᐉm − 1/d)i is obtained, etc. Equation (13-50) is applied to each stage in succession until the ratio ᐉM + 2/d in the overflow from the stage above the feed stage is obtained. The calculations are then switched to the stripping section. The equilibrium relationship for component i in the stripping section can be expressed in terms of component flow rate in the bottoms, b = BxB, and Sm = KmVm /Lm as ym = Km xm Vmym = (KmVm /Lm)Lm xm vm = Smᐉm vm /b = (ᐉm /b)Sm

(13-53)

RIGOROUS METHODS FOR MULTICOMPONENT DISTILLATION-TYPE SEPARATIONS

13-41

Combination with the material balance

gives

(ᐉm + 1/b) = vm /b + 1

(13-54)

(ᐉm + 1/b) = (ᐉm /b)Sm + 1

(13-55)

The bottom-up calculations are started by writing Eq. (13-55) for stage 1 as ᐉ2 /b = V1K1 /B + 1 = S1 + 1

(13-56)

The Sm values all are fixed by assumed temperature and phase-rate profiles. Equation (13-55) is applied to each of the stripping stages in sequence until the ratio ᐉM + 2 /b in the liquid entering the feed stage is obtained. The manner in which rectifying and stripping-section calculations are meshed at the feed stage depends upon the thermal condition of the feed. Figure 13-45 shows three possible ways in which fresh feed can affect the L and V rates between the feed stage and stage M + 2. The superscript bar denotes the stream rate when the stream enters a stage, while the lack of a bar denotes the rate when the stream leaves a stage. Top-down calculations for the example problem are shown in Table 13-10 and bottom-up calculations in Table 13-11. Top-down and bottom-up calculations have provided values of ᐉM + 2 /d and ᐉM + 2 /b respectively. For a bubblepoint feed,

(a)

vM + 1 = vM + 1

(b)

and a combination of Eqs. (13-48) and (13-54) provides for each component i b vM + 1 /d ᐉM + 2 /d + 1 ᎏ = ᎏ = ᎏᎏ d vM + 1 /b ᐉM + 2 /b − 1

(13-57)

The b/d ratios obtained from this equation can then be used to calculate the individual b and d values as follows. Since d + b = FxF

(13-58)

FxF d=ᎏ 1 + (b/d) b = (b/d)d

and

(13-59)

Calculated values of d from the first column iteration in the example problem are as follows: Component

ᐉ7 ᎏ+1 d

ᐉ7 ᎏ−1 b

b ᎏ d

FxF

d

C3 i-C4 n-C4 i-C5 n-C5

1.26 1.71 2.26 10.3 17.5

5450 175 47.7 2.46 1.54

0.000231 0.00977 0.0474 0.19 11.4

5 15 25 20 35

5.00 14.85 23.83 3.85 2.82 50.4

(c) Effect of feed on stream rates just above feed stage M + 1. (a) Subcooled or bubble-point feed. (b) Superheated or dew-point feed. (c) Partially flashed feed.

FIG. 13-45

⌰ method developed by Holland (Fundamentals of Multicomponent Distillation, McGraw-Hill, New York, 1981) and coworkers. A quantity ␪ is defined by FxF d′ = ᎏᎏ 1 + (b/d)⌰ where values of d′ are the ones that satisfy

(13-60)

 d′ = D

specified

i

The calculated D is 50.4 instead of 48.9. Before these incorrect d (and b) values are used to calculate the stage concentrations, followed by a new set of values of T, V, and L, convergence of the iteration is aided as follows by using the

TABLE 13-11 Component C3 i-C4 n-C4 i-C5 n-C5

Bottom-Up Calculations for Example 3 ᐉ2 S2 S1 ᎏ b 15.1 8.56 6.81 3.53 3.05

16.1 9.56 7.81 4.53 4.05

3.37 1.88 1.50 0.774 0.673

Comparison of Eqs. (13-57) and (13-60) shows that b′ = ⌰ (b/d)d′

(13-61)

ᐉ2 ᎏ S2 b

ᐉ3 ᎏ b

S3

ᐉ3 ᎏ S3 b

ᐉ4 ᎏ b

S4

54.3 18.0 11.7 3.51 2.73

55.3 19.0 12.7 4.51 3.73

3.31 1.83 1.45 0.747 0.646

183.0 34.8 18.4 3.37 2.41

184.0 35.8 19.4 4.37 3.41

3.21 1.76 1.39 0.716 0.619

Component

ᐉ4 ᎏ S4 b

ᐉ5 ᎏ b

S5

ᐉ5 ᎏ S5 b

ᐉ6 ᎏ b

S6

ᐉ6 ᎏ S6 b

ᐉ7 ᎏ b

C3 i-C4 n-C4 i-C5 n-C5

590.6 63.0 27.0 3.13 2.11

591.6 64.0 28.0 4.13 3.11

3.10 1.71 1.34 0.693 0.592

1834 109.4 37.5 2.86 1.84

1835 110.4 38.5 3.86 2.84

2.97 1.59 1.24 0.638 0.543

5450 175 47.7 2.46 1.54

5451 176 48.7 3.46 2.54

13-42

DISTILLATION

The value of ␪ is found by solving the following nonlinear equation, where D is the specified distillate rate: FxF D −  ᎏᎏ =0 (13-62) 1 + (b/d)⌰ i For the first column iteration, ⌰ = 1.25 satisfies this equation. The b/d values and the Θ value are used in Eqs. (13-60) and (13-61) to give the following corrected end concentrations: Component

b ᎏ d

C3 i-C4 n-C4 i-C5 n-C5

0.000231 0.00977 0.0474 4.19 11.4

Qr = DH LN + 1 + BH1L + Qc − FHF = (48.9)(10,750) + (51.1)(17,080) + 1,426,000 − 100(13,540) = 1,465,000 Btu/h (429.3 kW)

x1

A new set of values of Vm is obtained from energy balances around the bottom section of the column,

5.00 14.82 23.60 3.21 2.30 48.9

0.00144 0.181 1.40 16.79 32.7 51.1

0.102 0.303 0.482 0.066 0.047 1.000

0 0.004 0.027 0.329 0.640 1.000

Qr + B(HLM + 1 − HBV) Vm = ᎏᎏᎏ (13-67) HmV − HLm + 1 Similar balances around the top section yield a new set of values of Vn. Corresponding values of Ln and Lm are obtained by material balances around the top and bottom sections respectively. The new V, L, and T profiles are listed in Table 13-13. In this example, they do not differ much from the initial guesses in Table 13-10. It should be noted in Table 13-13 that it is not necessary to list two values of V, L, and T for the feed stage (stage 6) because the TC procedure gives a perfect match at the feed stage in each trial. This completes the first column iteration. The new temperature and flow-rate profiles (which would be used as the assumptions to begin the second column iteration) are compared in Fig. 13-46 with the final solution. Both profiles are moving toward the final result. Figure 13-47 shows the concentration profiles from the final solution. Note the discontinuities at the feed stage and the fact that feed-stage composition differs considerably from feed-stream composition. It can be seen in Fig. 13-47 from the n-C4 and i-C5 profiles that the separation between the keys improves rapidly with stage number; additional stages would be worthwhile. Convergence to the final solution is rapid with the TG method for narrowboiling feeds but may be slow for wide-boiling feeds. Generally, at least five column iterations are required. Convergence is obtained when successive sets of tear variables are identical to approximately four significant digits. This is accompanied by ⌰ = 1.0, x = normalized x, and nearly identical successive values of Qc as well as Qr.

(13-64)

i

Application of these equations gives the results in Table 13-12. A set of Tn is calculated from the normalized xn by bubble-point calculations. Corresponding values of yn are obtained from yn = Knxn. Once new xn and Tn are available, new values of Vn are calculated from energy balances by using data from Maxwell (Data Book on Hydrocarbons, Van Nostrand, Princeton, N.J., 1950). First, an estimate of condenser duty is computed from an energy balance around the condenser,

Stage Compositions from the First Trial of Example 3 ᐉ2 ᐉ3 x2 x3 x1 ᎏ b′ ᎏ b′ b b 0.000 0.004 0.027 0.329 0.640 1.000

(13-66)

xD

(ᐉm /b)b′ xm = ᎏᎏ  (ᐉm /b)b′

C3 i-C4 n-C4 i-C5 n-C5

(13-65)

b′

i

Component

= 175(18,900 − 10,750) = 1,426,000 Btu/h (417.9 kW) The reboiler duty Qr is obtained from an overall energy balance,

d′

Stage-to-stage calculations shown in Tables 13-10 and 13-11 provide ᐉ/d and ᐉ/b values for each stage. These are used in the following equations to calculate normalized liquid concentrations for each component at each stage: (ᐉn /d)d′ xn = ᎏᎏ (13-63)  (ᐉn /d)d′

TABLE 13-12

Qc = VN(HNV − H LN + 1)

0.0232 1.73 10.9 76.1 132.4 221.1

0.000 0.008 0.049 0.344 0.599 1.000

0.0796 3.44 17.8 75.7 122.0 219.0

0.000 0.016 0.081 0.346 0.557 1.000

ᐉ4 ᎏ b′ b

x4

ᐉ5 ᎏ b′ b

x5

ᐉ6 ᎏ b′ b

0.279 6.48 27.2 73.4 111.5 218.9

0.001 0.030 0.124 0.335 0.510 1.000

0.852 11.6 39.2 69.3 101.7 226.6

0.004 0.052 0.176 0.311 0.447 1.000

2.65 20.0 53.9 64.8 92.9 234.2

Component

x6

ᐉ7 ᎏd ′ d

x7

ᐉ8 ᎏ d′ d

x8

ᐉ9 ᎏ d′ d

x9

ᐉ10 ᎏ d′ d

x10

C3 i-C4 n-C4 i-C5 n-C5

0.011 0.085 0.230 0.277 0.397 1.000

1.295 10.6 29.7 29.9 37.9 109.4

0.012 0.097 0.271 0.273 0.347 1.000

1.535 13.7 37.8 28.7 32.4 114.1

0.013 0.120 0.331 0.252 0.284 1.000

2.240 19.1 48.4 24.1 24.1 118.0

0.019 0.162 0.410 0.204 0.204 1.000

4.66 27.7 58.5 16.6 14.2 121.7

0.038 0.228 0.481 0.136 0.117 1.000

TABLE 13-13

New Temperature and Rate Profiles from the First Trial of Example 3 HDL − HnL+ 1

D(HDL − HnL+ 1) + Qc

n

New T

10 9 8 7 6

160.0 175.0 186.0 194.0 200.0

0 −1220 −2190 −3010 −3490

1,426,000 1,367,000 1,319,000 1,279,000 1,256,000

m

New T

HmL + 1 − HBL

B(HmL + 1 − HBL) + Qr

5 4 3 2 1

211.0 220.0 228.0 233.5 237.5

−2480 −1780 −1130 −560 −210

1,338,000 1,374,000 1,407,000 1,438,000 1,454,000

HNV − HnL + 1 8150 7900 7830 7680 7700 HmV − HmL + 1 8200 8400 8560 8460 8410

V

L

175.0 172.9 168.5 166.5 163.1

124.0 119.6 117.6 114.2 214.3

V

L

163.2 163.6 164.4 170.0 172.9

214.7 215.5 221.1 224.0 51.1

RIGOROUS METHODS FOR MULTICOMPONENT DISTILLATION-TYPE SEPARATIONS

13-43

EQUATION-TEARING PROCEDURES USING THE TRIDIAGONAL-MATRIX ALGORITHM

Comparison of the assumed and calculated profiles from the first column iteration in Example 4 with the final computer solution.

FIG. 13-46

As seen earlier, the manual Thiele-Geddes method involves solving the equilibrium-stage equations one at a time. More powerful, flexible, and reliable computer programs are based on the application of sparse matrix methods for solving simultaneously all or at least some of the equations. For cases in which combined column feeds represent mixtures that boil within either a narrow range (typical of many distillation operations) or a wide range (typical of absorbers and strippers) and in which great flexibility of problem specifications is not required, equation-tearing procedures that involve solving simultaneously certain subsets of the equations can be applied. Two such equation-tearing procedures are the bubble-point (BP) method for narrow-boiling mixtures suggested by Friday and Smith (op. cit.) and developed in detail by Wang and Henke [Hydrocarbon Process., 45 (8), 155 (1966)], and the sum-rates (SR) method for wide-boiling mixtures proposed by Sujuta [Hydrocarbon Process., 40(12), 137 (1961)] and further developed by Burningham and Otto [Hydrocarbon Process., 46(10), 163 (1967)]. Both methods are described here. However, the BP method has been largely superseded by the more reliable and efficient simultaneous-correction and inside-out methods, which do, however, incorporate certain features of the BP method. Both the BP and SR methods start with the same primitive equations for the theoretical model of an equilibrium stage as presented next. Consider a general, continuous-flow, steady-state, multicomponent, multistage separation operation. Assume that phase equilibrium between an exiting vapor phase and a single exiting liquid phase is achieved at each stage, that no chemical reactions occur, and that neither of the exiting phases entrains the other phase. A general schematic representation of such a stage j is shown in Fig. 13-48. Entering stage j is a single- or two-phase feed at molal flow rate Fj, temperature TFj, and pressure PFj and with overall composition in mole fractions zi, j. Also entering stage j is interstage liquid from adjacent stage j − 1 above at molal flow rate Lj − 1, temperature Tj − 1, pressure Pj − 1, and mole fractions xi, j − 1. Similarly, interstage vapor from adjacent stage j + 1 below enters at molal flow rate Vj + 1, Tj + 1, Pj + 1 and mole fractions yi, j + 1. Heat is transferred from (+) or to (−) stage j at rate Qj to simulate a condenser, reboiler, intercooler, interheater, etc. Equilibrium vapor and liquid phases leave stage j at Tj and Pj and with mole fractions yi, j and xi, j respectively. The vapor may be partially withdrawn from the column as a sidestream at a molal flow rate Wj, with the remainder Vj sent to adjacent stage j − 1 above. Similarly, exiting liquid may be split into a sidestream at a molal flow rate of Uj, with the remainder Lj sent to adjacent stage j below. For each stage j, the following 2C + 3 component material-balance (M), phase-equilibrium (E), mole-fraction-summation (S), and energybalance (H) equations apply, where C is the number of chemical species: Lj − 1xi, j − 1 + Vj + 1yi, j + 1 + Fj zi, j − (Lj + Uj)xi, j − (Vj + Wj)yi, j = 0 yi, j − Ki, j xi, j = 0

y

i, j

(13-68) (13-69)

− 1.0 = 0

(13-70)

− 1.0 = 0

(13-71)

i

x

i, j

i

Lj − 1HLj − 1 + Vj + 1HVj + 1 + FjHFj − (Lj + Uj)HLj − (Vj + Wj)HVj − Qj = 0

(13-72)

In general, K values and molal enthalpies in these MESH equations are complex implicit functions of stage temperature, stage pressure, and equilibrium mole fractions: Ki, j = Ki, j{Tj, Pj, xj, yj} HVj = HVj{Tj, Pj, yj} FIG. 13-47 Concentration profiles from the final solution of Example 4. The points at N + 1 refer to the reflux composition, which is the same as the overhead vapor.

HLj = HLj{Tj, Pj, xj} where vectors xj and yj refer to all i values of xi, j and yi, j for the particular stage j. As shown in Fig. 13-49, a general countercurrent-flow col-

13-44

DISTILLATION

FIG. 13-48

General equilibrium stage.

umn of N stages can be formed from a collection of equilibrium stages of the type in Fig. 13-48. Note that streams L0, VN + 1, W1 and UN are zero and do not appear in Fig. 13-49. Such a column is represented by N(2C + 3) MESH equations in [N(3C + 10) + 1] variables, and the difference or [N(C + 7) + 1] variables must be specified. If the specified variables are the value of N and all values of zi, j, Fj, TFj, PFj, Pj, Uj, Wj, and Qj, then the remaining N(2C + 3) unknowns are all values of yi, j, xi, j, Lj, Vj, and Tj. In this case, Eqs. (13-73), (13-74), and (13-77) are nonlinear in the unknowns and the MESH equations can not be solved directly. Even if a different set of variable specifications is made, the MESH equations still remain predominantly nonlinear in the unknowns. For the BP method as applied to distillation, specified variables are those listed except that bottoms rate LN is specified rather than partial reboiler duty QN. This is equivalent by overall material balance to specifying vapor-distillate rate V1 in the case of a partial condenser or liquid-distillate rate U1 in the case of a total condenser. Also, reflux rate L1 is specified rather than condenser duty Q1. For the SR method as applied to absorption and stripping, the specified variables are those listed without exception.

Tridiagonal-Matrix Algorithm Both the BP and the SR equation-tearing methods compute liquid-phase mole fractions in the same way by first developing linear matrix equations in a manner shown by Amundson and Pontinen [Ind. Eng. Chem., 50, 730 (1958)]. Equations (13-69) and (13-68) are combined to eliminate yi, j and yi, j + 1 (however, the vector yj still remains implicitly in Ki, j): Lj − 1 xi, j − 1 + Vj + 1Ki, j + 1 xi, j + 1 + Fj zi, j − (Lj + Uj)xi, j − (Vj + Wj)Ki, j xi, j = 0

(13-73)

Next, Eq. (13-68) is summed over the C components and over stages 1 through j and combined with Eqs. (13-70), (13-71), and i zi, j − 1.0 = 0 to give a total material balance over stages 1 through j: j

Lj = Vj + 1 +

 (F

m

− Um − Wm) − V1

(13-74)

m=1

By combining Eq. (13-73) with Eq. (13-74), Lj is eliminated to give the following working equations for component material balances:

RIGOROUS METHODS FOR MULTICOMPONENT DISTILLATION-TYPE SEPARATIONS

13-45

into C sets, one for each component, and solved separately for values of xi, which pertains to all j values of xi, j for the particular species i. Each set of N equations is a special type of sparse matrix equation called a tridiagonal-matrix equation, which has the form shown in Fig. 13-50a for a five-stage example in which, for convenience, the subscript i has been dropped from the coefficients B, C, and D. For this type of sparse matrix equation, we can apply a highly efficient version of the gaussian elimination procedure called the Thomas algorithm, which avoids matrix inversion, eliminates the need to store the zero coefficients in the matrix, almost always avoids buildup of truncation errors, and rarely produces negative values of xi, j. The Thomas algorithm begins by a forward elimination, row by row starting down from the top row ( j = 1, the condenser stage), to give the following replacements shown in Fig. 13-50b. For row 1: p1 = C1/B1, q1 = D1/B1, B1 → 1, C1 → p1, D1 → q1 where → means “is replaced by.” For all subsequent rows: pj = Cj /(Bj − Aj pj − 1), qj = (Dj − A j qj − 1)/(Bj − A j pj − 1), Aj → 0, Bj → 1, Cj → pj, Dj → qj At the bottom row for component i, xN = qN. The remaining values of xj for species i are computed recursively by backward substitution: xj − 1 = qj − 1 − pj − 1xj

FIG. 13-49

BP Method for Distillation The bubble-point method for distillation, particularly when the components involved cover a relatively narrow range of volatility, proceeds iteratively by the following steps, where k is the iteration index for the entire distillation column. 1. Specify N and all values of zi, j, Fj, TFj, PFj, Pj, Uj, Wj, and Qj, except Q1 and QN. 2. Specify type of condenser. If total (U1 ≠ 0), compute LN from overall material balance; if partial (U1 = 0), specify V1 and compute LN from overall material balance. 3. Specify reflux rate L1, assuming no subcooling. 4. Compute V2 = V1 + L1 + U1 − F1. 5. Provide initial guesses (k = 0) or values of all tear variables Tj and Vj ( j > 2). Temperature guesses are readily obtained by linear interpolation between estimates of top- and bottom-stage temperatures. The bottom-stage temperature is estimated by making a bubble-point-temperature calculation by using an estimate of bottoms composition at the specified bottom-stage pressure. A similar calculation is made at the top stage by using an estimate of distillate composition; otherwise, for a partial condenser, a dew-point temperature calculation is made. An estimate of the vapor-rate profile is readily obtained by assuming constant molal overflow down the column. 6. Set index k = 1 to initiate the first column iteration. 7. Using specified stage pressures, current estimates of stage temperatures, and current estimates of stage vapor- and liquid-phase

General countercurrent cascade of N stages.

A j xi, j − 1 + Bi, j x i, j + Ci, j xi, j + 1 = Di, j

(13-75)

j−1

Aj = Vj +

where

 (F

m

− Wm − Um) − Vi

2≤j≤N

(13-76)

m=1





j

Bi, j = − Vj + 1 +

 (F

m

− Wm − Um) − V1 + Uj + (Vj + Wj)Ki, j

m=1

1 << j << N

(a)

(13-77)

Ci, j = Vj + 1Ki, j + 1

1 << j << N − 1

(13-78)

Di, j = −Fj zi, j

1 << j << N

(13-79)

The NC equations (13-75) are linearized in terms of the NC unknowns xi, j by selecting unknowns Vj and Tj as tear variables and using values of vectors xj and yj from the previous iteration to compute values of Ki, j for the current iteration. In this manner all values of Aj, Bi, j, and Ci, j can be estimated. Values of Di, j are fixed by feed specifications. Furthermore, the NC equations (13-75) can be partitioned

(b) Tridiagonal-matrix equation for a column with five theoretical stages. (a) Original equation. (b) After forward elimination.

FIG. 13-50

13-46

DISTILLATION

compositions, estimate all Ki, j values (for k = 1, initial estimates of stage phase compositions may be necessary if Ki, j values are sensitive to phase compositions). 8. Compute values of xi, j by solving Eqs. (13-75) through (13-79) by the tridiagonal-matrix algorithm once for each component. Unless all mesh equations are converged, i xi, j ≠ 1 for each stage j. 9. To force i xi, j = 1 at each stage j, normalize values by the replacement xi, j = xi, j / i xi, j. 10. Compute a new set of values of Tj(k) tear variables by computing, one at a time, the bubble-point temperature at each stage based on the specified stage pressure and corresponding normalized xi, j values. The equation used is obtained by combining Eqs. (13-69) and (13-70) to eliminate yi, j to give

K

{Tj, Pj, xj, yj} = 1.0

(13-80)

i, j

i

which is a nonlinear equation in Tj(k) and must be solved iteratively by some appropriate root-finding method, such as the Newton-Raphson or the Muller method. 11. Compute values of yi, j one at a time from Eq. (13-69). 12. Compute a new set of values of the Vj tear variables one at a time, starting with V3, from an energy-balance equation that is obtained by combining Eqs. (13-72) and (13-74), eliminating Lj − 1 and Lj to give (13-81) Vj = (C˜j − 1 − Ãj − 1Vj − 1)/B˜ j − 1 where

Ãj − 1 = HLj − 2 − HVj − 1 B˜ j − 1 = HVj − HLj − 1 j−2

C˜j − 1 =

  (F

m



− Wm − Um) − V1 (HLj − 1 − HLj − 2)

m=1

+ Fj − 1(HLj − 1 − HFj − 1)+ Wj − 1(HVj − 1 − HLj − 1) + Qj − 1

FIG. 13-51

Specifications for the calculation of distillation by the BP method.

(k)

13. Check to determine if the new sets of tear variables Tj and Vj(k) are within some prescribed tolerance of sets Tj(k − 1) and Vj(k − 1) used to initiate the current column iteration. A possible convergence criterion is N Tj(k) − Tj(k − 1) 2 N Vj(k) − Vj(k − 1) 2 +  ᎏᎏ ≤ 10−7N (13-82) ᎏᎏ  Tj(k) Vj(k) j=1 j=3 but Wang and Henke (op. cit.) use





N

 [T

(k)

j



− Tj(k − 1)]2 ≤ 0.01N



(13-83)

j=1

14. If the convergence criterion is met, compute values of Lj from Eq. (13-74) and values of Q1 and QN from Eq. (13-72). Otherwise, set k = k + 1 and repeat steps 7 to 14. Step 14 implies that if the calculations are not converged, values of Tj(k) computed in step 10 and values of Vj(k) computed in step 12 are used as values of the tear variables to initiate iteration k + 1. This is the method of successive substitution, which may require a large number of iterations and/or may result in oscillation. Alternatively, values of Tj(k) and Vj(k) can be adjusted prior to initiating iteration k + 1. Experience indicates that values of Tj should be reset if they tend to move outside of specified upper and lower bounds and that negative Vj values be reset to small positive values. Also, damping can be employed to prevent values of absolute Tj and Vj from changing by more than, say, 10 percent on successive iterations. Orbach and Crowe [Can. J. Chem. Eng., 49, 509 (1971)] show that the dominant eigenvalue method of adjusting values of Tj and Vj can generally accelerate convergence and is a worthwhile improvement to the BP method. Example 4: Calculation of the BP Method Use the BP method with the SRK equation-of-state for K values and enthalpy departures to compute stage temperatures, interstage vapor and liquid flow rates and compositions, and reboiler and condenser duties for the light-hydrocarbon distillation-column specifications shown in Fig. 13-51 with feed at 260 psia. The specifications are selected to obtain three products: a vapor distillate rich in C2 and C3, a vapor sidestream rich in n-C4, and a bottoms rich in n-C5 and n-C6. Initial estimates provided for the tear variables were as follows compared with final converged values (after 23 iterations), where numbers in parentheses are consistent with specifications:

Stage

T(0), °F

T(23), °F

V(0), (lb⋅mol)/h

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

110.00 121.87 133.75 145.62 157.50 169.37 181.25 193.12 205.00 216.87 228.75 240.62 252.50 264.37 276.25 288.12 300.00

124.4 141.0 157.4 172.6 184.9 194.5 202.6 211.1 221.9 229.9 236.5 242.9 250.2 258.1 267.4 278.1 290.5

(23) (173) 173 173 173 173 173 173 173 173 173 173 173 210 210 210 210

V(23), (lb⋅mol)/h 23.0 173.0 167.3 162.8 160.4 159.5 158.4 155.8 151.4 155.8 160.2 162.9 164.0 200.8 199.4 197.9 196.4

NOTE: To convert degrees Fahrenheit to degrees Celsius, °C = (°F − 32)/1.8. To convert pound-moles per hour to kilogram-moles per second, multiply by 1.26 × 10−4.

By employing successive substitution of the tear variables and the criterion of Eq. (13-83), convergence was achieved slowly, but without oscillation, in 23 iterations. Computed products are: Flow rate, (lb⋅mol)/h Component

Distillate

Sidestream

Bottoms

C2 C3 nC4 nC5 nC6

3.0 18.4 1.6 0.0 0.0 23.0

0.0 1.6 25.7 9.2 0.5 37.0

0.0 0.0 9.7 25.8 4.5 40.0

NOTE: To convert pound-moles per hour to kilogram-moles per second, multiply by 1.26 × 10−4.

RIGOROUS METHODS FOR MULTICOMPONENT DISTILLATION-TYPE SEPARATIONS Examination of interstage-composition results showed that maximum nC4 composition was achieved in the vapor leaving stage 12 rather than stage 13. Therefore, if the sidestream location were moved up one stage, a somewhat higher purity of nC4 could probably be achieved in that stream. Further improvement in purity of the sidestream as well as the other two products could be achieved by increasing the reflux rate and/or number of stages. Computed condenser and reboiler duties were 268,000 and 418,000 W (914,000 and 1,425,000 Btu/h) respectively.

SR Method for Absorption and Stripping As shown by Friday and Smith (op. cit.), when an attempt is made to apply the BP method to absorption, stripping, or distillation, in which the volatility range of the chemical components in the column is very wide, calculations of stage temperatures from Eq. (13-80) become very sensitive to liquid compositions. This generally causes very oscillatory excursions in temperature from iteration to iteration, making it impossible to obtain convergence. A very successful modification of the BP method for such cases is the sum-rates method, in which new stage temperatures are computed instead from the energy-balance equation. Interstage vapor rates are computed by material balance from new interstage liquid rates that are obtained by multiplying the previous interstage liquid rates by corresponding unnormalized liquid mole-fraction summations computed from the tridiagonal-matrix algorithm. The SR method proceeds by the following steps: 1. Specify N and all values of zi, j, Fj, TFj, PFj, Pj, Uj, Wj, and Qj. For an adiabatic operation, all Qj are zero. 2. Provide initial guesses (k = 0) for values of all tear variables Tj and Vj. Temperature guesses are readily obtained by linear interpolation between estimates of the top- and bottom-stage temperatures, taking the top as that of the liquid feed to the top stage and the bottom as that of the vapor feed to the bottom stage. An estimate of the vaporrate profile is readily obtained by assuming constant molal overflow working up from the bottom in using the specified vapor feed or feeds. Compute corresponding initial values of Lj from Eq. (13-74). 3. Same as step 6 of the BP method. 4. Same as step 7 of the BP method. 5. Same as step 8 of the BP method. 6. Compute a new set of values of Lj from the sum-rates equation: Lj(k + 1) = Lj(k)  xi, j

(13-84)

12. Same as step 13 of the BP method. 13. If the convergence criterion is not met, set k = k + 1 and repeat steps 4 to 13. With the SR method, convergence is often rapid even when successive substitution of Tj and Vj is used from one iteration to the next. Example 5: Calculation of the SR Method Use the SR method with the PR equation of state for K values and enthalpy departures. The oil was taken as n-dodecane. To compute stage temperatures and interstage vapor and liquid flow rates and compositions for absorber-column specifications shown in Fig. 13-52. Note that a secondary absorber oil is used in addition to the main absorber oil and that heat is withdrawn from the seventh theoretical stage. Initial estimates provided for the tear variables were as follows compared with final converged values obtained after five iterations: Stage

T(0), °F

T(5), °F

V(0), (lb⋅mol)/h

1 2 3 4 5 6 7 8

80.00 81.43 82.86 84.29 85.71 87.14 88.57 90.00

84.6 85.5 86.3 85.3 86.0 86.6 85.2 92.7

450 450 450 450 450 450 450 450

Vj = Lj − 1 − LN +

N

 (F

m

− Wm − Um)

V(5), (lb⋅mol)/h 323.9 367.7 371.3 374.7 388.6 393.0 398.2 410.4

NOTE: To convert degrees Fahrenheit to degrees Celsius, °C = (°F − 32)/1.8. To convert pound-moles per hour to kilogram-moles per second, multiply by 1.26 × 10−4.

Convergence was achieved rapidly in five iterations by using Eq. (13-88) as the criterion. Computed compositions for lean gas and rich oil are: Flow rate, (lb⋅mol)/h Component

Lean gas

Rich oil

C1 C2 C3 nC4 nC5 Oil

312.7 11.0 0.2 0.0 0.0 0.0 323.9

60.3 32.0 28.8 19.0 15.0 385.0 540.1

i

7. Compute a corresponding new set of Vj tear variables from the following total material balance, which is obtained by combining Eq. (13-74) with an overall material balance around the column:

13-47

NOTE: To convert pound-moles per hour to kilogram-moles per second, multiply by 1.26 × 10−4.

(13-85)

m=j

8. Same as step 9 of the BP method. 9. Same as step 11 of the BP method. 10. Normalize values of yi, j. 11. Compute a new set of values of the Tj tear variables by solving simultaneously the set of N energy-balance equations (13-72), which are nonlinear in the temperatures that determine the enthalpy values. When linearized by a Newton iterative procedure, a tridiagonal-matrix equation that is solved by the Thomas algorithm is obtained. If we set g j equal to Eq. (13-72), i.e., its residual, the linearized equations to be solved simultaneously are ∂g1 (r) ∂g1 (r) (13-86) ᎏ ∆T1 + ᎏ ∆ T2 = −g1(r) ∂T1 ∂T2

 

∂gj

ᎏ ∂T  j−1

(r)

 

∂gj ∆Tj − 1 + ᎏ ∂Tj

 

(r)

∂gj ∆Tj + ᎏ ∂Tj + 1





(r)

∆Tj + 1 = −g j(r) 2≤j≤N−1



∂gN ᎏ ∂TN − 1



(r)

∂gj ∆ TN − 1 + ᎏ ∂TN

 

(r)

∆TN = −g

(r) N

(13-87) (13-88)

where ∆Tj = T j(r + 1) − Tj(r), and thus T j(r + 1) = Tj(r) + ∆ Tj, and r is the iteration index. The partial derivatives depend upon the enthalpy correlations utilized and may be obtained analytically or numerically. Simultaneous Eqs. (13-86) to (13-88) are solved iteratively until corrections ∆Tj and, therefore, residual values of g j approach zero.

FIG. 13-52

method.

Specifications for the calculation of an absorber by the SR

13-48

DISTILLATION

Approximately 0.016 (kg⋅mol)/s [126 (lb⋅mol)/h] of vapor is absorbed with an energy liberation of about 198,000 W (670,000 Btu/h), 20 percent of which is removed by the intercooler on stage 7. The temperature profile departs from a smooth curve at stages 4 and 7, where secondary oil enters and heat is removed respectively.

where physical properties are not simplified:

SIMULTANEOUS-CORRECTION PROCEDURES

Let the order of corrections to the unknowns be according to stage number, which in terms of the corresponding unknowns is

The BP and SR tearing methods are generally successful only when applied respectively to the distillation of mixtures having a narrow boiling range and to absorbers and strippers. Furthermore, as shown earlier, specifications for these two tearing methods are very restricted. If one wishes to treat distillation of wide-boiling mixtures and other operations shown in Fig. 13-7 such as rectification, reboiled stripping, reboiled absorption, and refluxed stripping, it is usually necessary to utilize other procedures. One class of such procedures involves the solution of most or all of the MESH equations or their equivalent simultaneously by some iterative technique such as a Newton or a quasi-Newton method. Such simultaneous-correction (SC) methods are also useful for separations involving very nonideal liquid mixtures including extractive and azeotropic distillation or for cases in which considerable flexibility in specifications is desired. The development of an SC procedure involves a number of important decisions: (1) What variables should be used? (2) What equations should be used? (3) How should variables be ordered? (4) How should equations be ordered? (5) How should flexibility in specifications be provided? (6) Which derivatives of physical properties should be retained? (7) How should equations be linearized? (8) If Newton or quasi-Newton linearization techniques are employed, how should the Jacobian be updated? (9) Should corrections to unknowns that are computed at each iteration be modified to dampen or accelerate the solution or be kept within certain bounds? (10) What convergence criterion should be applied? Perhaps because of these many decisions, a large number of SC procedures have been published. Two quite different procedures that have achieved a significant degree of utilization in solving practical problems include the methods of Naphtali and Sandholm [Am. Inst. Chem. Eng. J., 17, 148 (1971)] and Goldstein and Stanfield [Ind. Eng. Chem. Process Des. Dev., 9, 78 (1970)]. The former procedure is of particular interest because, in principle, it can be applied to all cases. However, for situations involving large numbers of components, relatively small numbers of stages, and liquid solutions that are not too highly nonideal, the latter procedure is more efficient computationally. Naphtali-Sandholm SC Method This method employs the equilibrium-stage model of Figs. 13-48 and 13-49 but reduces the number of variables by 2N so that only N(2C + 1) equations in a like number of unknowns must be solved. In place of Vj, Lj, xi, j, and yi, j, component flow rates are used according to their definitions: vi, j = yi, jVj

(13-89)

ᐉi, j = xi, j Lj

(13-90)

In addition, sidestream flow rates are replaced with sidestream flow ratios by sj = Uj /Lj

(13-91)

Sj = Wj /Vj

(13-92)

The MESH equations (13-68) to (13-72) then become the MEH functions: Mi, j = ᐉi, j(1 + sj) + vi, j (1 + Sj) − ᐉi, j − 1 − vi, j + 1 − fi, j = 0 where

fi, j = Fj zi, j Ei, j = Ki, jᐉi, j

 v

a, j

a

Ⲑ

ᐉ

−v

a, j

a

i, j

(13-93)

(13-94)

i

i

− HLj − 1  ᐉi,j − HVj + 1  vi, j + 1 i

− HFj  fi, j − Qj i

=0

HLj = HLj{Tj,Pj,ᐉj}

where

(13-95)

X  = [X 1, X 2, . . . X j, . . . X N]T

(13-96)

Xj = [v1,j, v2, j, . . . vC, j, Tj, ᐉ1, j, ᐉ2, j, . . . ᐉC, j]T 

(13-97)

Let the order of the linearized MEH functions also be according to stage number, which in terms of the corresponding nonlinear functions is where

F  = [F  1, F 2, . . . F j, . . . F N]T

(13-98)

Fj = [Hj, M1, j, M2, j, . . . MC, j, E1,j, . . . EC, j]T 

(13-99)

Corrections to unknowns for the kth iteration are obtained from ∂F  −1 (k) (k) ∆X F (13-100) (k) = − ᎏ  ∂X The next approximations to the unknowns are obtained from

 

X(k + 1) =  X(k) + t ∆X  (k)

(13-101)

where t is a damping (0 < t < 1) or acceleration (t > 1) factor. By ordering the corrections to the unknowns and the linearized functions in this manner, the resulting Jacobian of partial derivatives of all functions with respect to all unknowns is of a very convenient sparse matrix form of block tridiagonal structure.

⫽⫽

  dF dX

⫽⫽

=



⫽

⫽

B1 A2 0* .. .. 0 0 0

C1 ⫽ B2 ⫽ A3

⫽

0 0 ⫽ C2 0 ⫽ ⫽ B 3† C 3‡

.. .. ..

.. ..

0 0

⫽

⫽

A *N − 1 B N − 1† ⫽ 0 AN

⫽

0 0 0 .. .. 0

C N − 1‡ ⫽ BN



(13-102) ⫽ ⫽

⫽

Blocks A j, B j, and C j are (2C + 1) by (2C + 1) submatrices of partial derivatives of the functions on stage j with respect to unknowns on stage j − 1, j, and j + 1 respectively. The solution to Eq. (13-100) is readily obtained by a matrix-algebra equivalent of the Thomas algorithm for a tridiagonal-matrix equation. Computer storage requirements are minimized by making the following replacements. Starting at top stage 1, using forward-block elimination, ⫽

⫽

⫽

⫽

C 1 → (B B 1)−1C 1, F B 1)−1 F 1 → (B 1 and

⫽

B 1 → I (the identity submatrix)

For stages j from 2 to (N − 1), ⫽

⫽

⫽⫽

⫽

C j → (B B j − A jC j − 1)−1C j, ⫽

⫽⫽

⫽

⫽

F j → (B j − A jC j − 1)−1(F  − A jF j − 1),

⫽

A j → 0,

Bj → I

For final stage N, ⫽

=0

Hj = HLj(1 + sj)  ᐉi, j + HVj(1 + Sj)  vi, j i

Ki,j = Ki,j{Tj,Pj,ᐉj,vj} HVj = HVj{Tj,Pj,vj}

⫽ ⫽

⫽

FN → (B N − A NC N − 1)−1(F  N − A NF N − 1),

⫽

A N → 0,

⫽

BN → I

This completes the forward steps to give ∆X N = −F N. Remaining values of corrections ∆  Xj are obtained by successive backward substitution ⫽ from ∆X j = −F j → −(F j − C jF j + 1). Matrix inversions are best done by LU decomposition. Efficiency is best for a small number of components C. The Newton iteration is initiated by providing reasonable guesses for all unknowns. However, these can be generated from guesses of just T, TN, and one interstage value of Fj or Lj. Remaining values of Tj are obtained by linear interpolation. By assuming constant molal over-

RIGOROUS METHODS FOR MULTICOMPONENT DISTILLATION-TYPE SEPARATIONS

13-49

flow, calculations are readily made of remaining values of Vj and Lj, from which initial values of vi, j and ᐉi, j are obtained from Eqs. (13-89) and (13-90) after obtaining approximations of xi, j and yi, j from steps 4, 5, 8, 9, and 10 of the SR method. Alternatively, a much cruder but often sufficient estimate of xi, j and yi, j is obtained by flashing the combined column feeds at average column pressure and a vapor-toliquid ratio that approximates the ratio of overhead plus vaporsidestream flows to bottoms plus liquid-sidestream flows. Resulting compositions are used as the initial estimate for every stage. At the conclusion of each iteration, convergence is checked by employing an approximate criterion such as H 2 τ =  ᎏj +  [(Mi, j)2 + (Ei, j)2] ≤ ⑀ (13-103) ξ j i where ξ is a scale factor that is of the order of the average molal heat of vaporization. If we take


 

⑀ = N(2C + 1)



 F  10 2 j

−10

(13-104)

j

converged values of the unknowns will generally be accurate, on the average, to from four or more significant digits. During early iterations, particularly when initial estimates of the unknowns are poor, τ and corrections to the unknowns will be very large. It is then preferred to utilize a small value of t in Eq. (13-101) so as to dampen changes to unknowns and prevent wild oscillations. However, the use of values of t much less than 0.25 may slow or prevent convergence. It is also best to reset to zero or small values any negative values of component flow rates before initiating the next iteration. When the neighborhood of the solution is reached, τ will often decrease by one or more orders of magnitude at each iteration, and it is best to set t = 1. Because the Newton method is quadratically convergent in the neighborhood of the solution, usually only three or four additional iterations will be required to reach the convergence criterion. Prior to that, it is not uncommon for τ to increase somewhat from one iteration to the next. If the Jacobian tends toward a singular condition, it may be necessary to restart the procedure with different initial guesses or adjust the Jacobian in some manner. Standard specifications for the Naphtali-Sandholm method are Qj (including zero values) at each stage at which heat transfer occurs and sidestream flow ratio sj or Sj (including zero values) at each stage at which a sidestream is withdrawn. However, the desirable block tridiagonal structure of the jacobian matrix can still be preserved when substitute specifications are made if they are associated with the same stage or an adjacent stage. For example, suppose that for a reboiled absorber, as in Fig. 13-7f, it is desired to specify a boil-up ratio rather than reboiler duty. Equation (13-95) for function HN is removed from the N(2C + 1) set of equations and is replaced by the equation ˜ N =  vi,N − (VN/LN)  ᐉi,N = 0 H i

(13-105)

i

where the value of (VN/LN) is specified. Following convergence of the calculations, QN is computed from the removed equation. All of the major computer-aided design and simulation programs have a simultaneous-correction algorithm. A Naphtali-Sandholm type of program, particularly suited for applications to distillation, extractive distillation, and azeotropic distillation, has been published by Fredenslund, Gmehling, and Rasmussen (Vapor-Liquid Equilibria Using UNIFAC, a Group Contribution Method, Elsevier, Amsterdam, 1977). Christiansen, Michelsen, and Fredenslund [Comput. Chem. Eng., 3, 535 (1979)] apply a modified Naphtali-Sandholm type of method to the distillation of natural-gas liquids, even near the critical region, using thermodynamic properties computed from the Soave-RedlichKwong equation of state. Block and Hegner [Am. Inst. Chem. Eng. J., 22, 582 (1976)] extended the Naphtali-Sandholm method to staged separators involving two liquid phases (liquid-liquid extraction) and three coexisting phases (three-phase distillation). Example 6: Calculation of Naphtali-Sandholm SC Method Use the Naphtali-Sandholm SC method to compute stage temperatures and interstage vapor and liquid flow rates and compositions for the reboiled-stripper specifications shown in Fig. 13-53. The specified bottoms rate is equivalent to removing most of the nC5 and nC6 and some of the nC7 in the bottoms.

Specifications for the calculation of a reboiled stripper by the Naphtali-Sandholm method.

FIG. 13-53

Calculations were made with the Grayson-Streed modification of the ChaoSeader method for K values and the Lee-Kesler method for enthalpy departures. Initial estimates for stage temperatures and flow rates were as follows, where numbers in parentheses are consistent with specifications:

(lb⋅mol)/h Stage

T, °F

V

L

1 8

130 250

(452.26)

550

NOTE: To convert degrees Fahrenheit to degrees Celsius, °C = (°F − 32)/1.8. To convert pound-moles per hour to kilogram-moles per second, multiply by 1.26 × 10−4.

For specified feed temperature and pressure, an isothermal flash of the feed gave 13.35 percent vaporization. Convergence was achieved in 3 iterations. Converged values of temperatures, total flows, and component flow rates are tabulated in Table 13-14. Computed reboiler duty is 1,295,000 W (4,421,000 Btu/h). Computed temperature, total vapor flow, and component flow profiles, shown in Fig. 13-54, are not of the shapes that might be expected. Vapor and liquid flow rates for nC4 change dramatically from stage to stage.

INSIDE-OUT METHODS The BP, SR, and SC methods described above expend a large percentage of their computational effort during each iteration in the calculation of K values, enthalpies, and derivatives thereof. An algorithm designed to significantly reduce that effort was developed by Boston and Sullivan [Can. J. Chem. Engr., 52, 52 (1974)]. The MESH equations are solved in an inner loop using simple, approximate equations for K values and enthalpies. The empirical constants in these equations are determined and infrequently updated from the more rigorous, but complex, K value and enthalpy correlations in an outer loop, using calculated compositions and temperatures from the inner loop. Thus, the method is referred to as the inside-out method. The iteration variables for the outer loop are the constants in the approximate thermodynamic-property equations in the inner loop. The iteration variables for the inner loop are related to stage j

13-50

DISTILLATION

TABLE 13-14

Converged Results for Reboiled Stripper of Fig. 13-53 Net flows

Stg

Temp °F

Pres psia

Liquid lb⋅mol/h

1 2 3 4 5 6 7 8

130.5 170.0 184.4 192.7 200.6 211.8 228.5 251.3

150.00 150.00 150.00 150.00 150.00 150.00 150.00 150.00

566.13 622.63 638.20 637.73 626.86 608.02 586.11

Vapor lb⋅mol/h

Feeds lb⋅mol/h

Product lb⋅mol/h

551.59

452.26

466.80 523.30 538.87 538.40 527.53 508.69 486.78

Tray compositions

99.33

Duties MMBtu/h

4.421

Tray compositions

Stage #1

130.49°F Vap lb⋅mol/h

150.00 psia Liq lb⋅mol/h

Stage #5

200.64°F Vap lb⋅mol/h

150.00 psia Liq lb⋅mol/h

Nitrogen Methane Ethane Propane n-Butane n-Pentane n-Hexane Total lb⋅mol/h

0.22000 59.51000 73.56996 153.18130 150.43202 12.75881 2.58982 452.2619

0.03033 4.72506 20.85994 115.69593 318.78543 70.16827 35.86041 566.1254

Nitrogen Methane Ethane Propane n-Butane n-Pentane n-Hexane Total lb⋅mol/h

0.00014 0.00159 0.21490 13.96836 434.63895 78.97008 10.60824 538.4022

0.00002 0.00010 0.04043 5.73731 410.72699 162.79074 47.56607 626.8616

Stage #2

170.01°F Vap lb⋅mol/h

150.00 psia Liq lb⋅mol/h

Stage #6

211.75°F Vap lb⋅mol/h

150.00 psia Liq lb⋅mol/h

Nitrogen Methane Ethane Propane n-Butane n-Pentane n-Hexane Total lb⋅mol/h

0.03033 4.72506 20.85991 115.65723 295.99744 24.70708 4.82023 466.7973

0.00517 0.36256 5.14611 66.94922 428.55569 83.84168 37.77268 622.6332

Nitrogen Methane Ethane Propane n-Butane n-Pentane n-Hexane Total lb⋅mol/h

0.00002 0.00010 0.04039 5.69862 387.93896 117.32958 16.52589 527.5336

0.00000 0.00001 0.00719 2.17265 329.74094 212.37617 63.72279 608.0197

Stage #3

184.42°F Vap lb⋅mol/h

150.00 psia Liq lb⋅mol/h

Stage #7

228.46°F Vap lb⋅mol/h

150.00 psia Liq lb⋅mol/h

Nitrogen Methane Ethane Propane n-Butane n-Pentane n-Hexane Total lb⋅mol/h

0.00517 0.36256 5.14608 66.91052 405.76770 38.38049 6.73250 523.3050

0.00085 0.02460 1.08705 31.95980 466.08041 99.77217 39.27149 638.1964

Nitrogen Methane Ethane Propane n-Butane n-Pentane n-Hexane Total lb⋅mol/h

0.00000 0.00001 0.00716 2.13396 306.95294 166.91501 32.68262 508.6917

0.00000 0.00000 0.00119 0.74298 227.83710 254.54031 102.98384 586.1054

Stage #4

192.68°F Vap lb⋅mol/h

150.00 psia Liq lb⋅mol/h

Stage #8

251.31°F Vap lb⋅mol/h

150.00 psia Liq lb⋅mol/h

Nitrogen Methane Ethane Propane n-Butane n-Pentane n-Hexane Total lb⋅mol/h

0.00085 0.02460 1.08701 31.92111 443.29239 54.31099 8.23131 538.8683

0.00014 0.00159 0.21493 14.00705 457.42691 124.43126 41.64842 637.7303

Nitrogen Methane Ethane Propane n-Butane n-Pentane n-Hexane Total lb⋅mol/h

0.00000 0.00000 0.00116 0.70429 205.04907 209.07915 71.94367 486.7773

stripping factors, K ijVj /Lj , for components i, which make use of volatility and energy parameters. Otherwise, the inner-loop calculations utilize computational features of the BP, SR, and SC methods, to compute stage temperatures, compositions, and flow rates. The inside-out method takes advantage of the following observations: (1) relative volatilities vary from iteration to iteration much less than the K values, (2) enthalpy of vaporization varies from iteration to iteration much less than phase enthalpies, and (3) component stripping factors combine effects of temperature and liquid and vapor flows at each stage. As an example of how the approximate thermodynamic-property equations are handled in the inner loop, consider the calculation of K values. The approximate models for nearly ideal liquid solutions are the following empirical Clausius-Clapeyron form of the K value in terms of a base or reference component, b, and the definition of the relative volatility, α.

0.00000 0.00000 0.00003 0.03869 22.78802 45.46117 31.04017 99.3281

K b, j = exp(Aj − Bj /Tj)

(13-106)

K i, j = αi, j K b, j

(13-107)

Values of A and B for the base component are back-calculated for each stage in the outer loop from a suitable K-value correlation (e.g. the SRK equation, which is also used to compute the K values of the other components on each of the other stages so that values of αij can be computed). The values of A, B, and α are passed from the outer loop to the inner loop, where they are used to formulate the phase equilibria equation: where

vi, j = α i, jSb, j li, j

(13-108)

Sb, j = K ij Vj /L j

(13-109)

The initial version of the inside-out method was developed for rapid calculations of simple and complex distillation, absorption, and

RIGOROUS METHODS FOR MULTICOMPONENT DISTILLATION-TYPE SEPARATIONS

13-51

lations, with T in °R, are determined from the SRK equation, with the following results: Stage

T, °F

A

B

Kb

1 2 3 4 5

65 95 118 142 165

6.870 6.962 7.080 7.039 6.998

3708 4031 4356 4466 4576

0.8219 0.7374 0.6341 0.6785 0.7205

Values of enthalpy constants for approximate equations are not tabulated here but are also computed for each stage based on the initial temperature distribution. In the inner-loop calculation sequence, component flow rates are computed from the MESH equations by the tridiagonal matrix method. The resulting bottoms-product flow rate deviates somewhat from the specified value of 50 lb⋅mol/h. However, by modifying the component stripping factors with a base stripping factor, Sb, in (13-109) of 1.1863, the error in the bottoms flow rate is reduced to 0.73 percent. The initial inside-loop error from the solution of the normalized energybalance equations, is found to be only 0.04624. This is reduced to 0.000401 after two iterations through the inner loop. At this point in the inside-out method, the revised column profiles of temperature and phase compositions are used in the outer loop with the complex SRK thermodynamic models to compute updates of the approximate K and H constants. Then only one inner-loop iteration is required to obtain satisfactory convergence of the energy equations. The K and H constants are again updated in the outer loop. After one inner-loop iteration, the approximate K and H constants are found to be sufficiently close to the SRK values that overall convergence is achieved. Thus, a total of only 3 outer-loop iterations and 4 inner-loop iterations are required. To illustrate the efficiency of the inside-out method to converge this example, the results from each of the three outer-loop iterations are summarized in the following tables:

T1

T2

T3

T4

T5

Initial guess 1 2 3

65 82.36 83.58 83.67

— 118.14 119.50 119.54

— 146.79 147.98 147.95

— 172.66 172.57 172.43

165 193.20 192.53 192.43

Outer-loop iteration

FIG. 13-54

Converged profiles for the reboiled stripper of Fig. 13-53.

stripping operations for hydrocarbon mixtures. However extensions and improvements in the method by Boston and coworkers [Comput. Chem. Engng., 2, 109 (1978), ACS Symp. Ser. No. 124, 135 (1980), Comput. Chem. Engng., 8, 105 (1984), and Chem. Eng. Prog., 86 (8), 45–54 (1990)] and by Russell [Chem. Eng., 90, (20), 53 (1983)] and Jelinek [Comput. Chem. Engng., 12, 195 (1988)] now make it possible to apply the method to reboiled absorption, reboiled stripping, extractive and azeotropic distillation, three-phase systems, reactive distillation, highly nonideal systems, and interlinked distillation systems with pumparounds, bypasses, and external heat exchangers. Inside-out methods are incorporated into most of the computer-aided process design and simulation programs and are now the methods of choice for design and simulation, as stated by Haas, who in Chap. 4 of Kister (op. cit.) presents details of two of the several inside-out algorithms. Example 7: Calculation of Inside-Out Method For the conditions of the simple distillation column shown in Fig. 13-55, obtain a converged solution by the inside-out method, using the SRK equation-of-state for thermodynamic properties (in the outer loop). A computer solution was obtained as follows. The only initial assumptions are a condenser outlet temperature of 65°F and a bottoms-product temperature of 165°F. The bubble-point temperature of the feed is computed as 123.5°F. In the initialization procedure, the constants A and B in (13-106) for inner-loop calcu-

Stage temperatures, °F

Outer-loop iteration

Specification 1 2 3

Total liquid flows, lb⋅moles/hr L1

L2

L3

L4

L5

100 100.00 100.03 100.0

— 89.68 89.83 89.87

— 187.22 188.84 188.96

— 189.39 190.59 190.56

— 50.00 49.99 50.00

Component flows in bottoms product, lb⋅moles/hr

Outer-loop iteration

C3

nC4

nC5

L5

1 2 3

0.687 0.947 0.955

12.045 12.341 12.363

37.268 36.697 36.683

50.000 49.985 50.001

From these tables, it is seen that the stage temperatures and total liquid flows are already close to the converged solution after only one outer-loop iteration. However, the composition of the bottoms product, specifically with respect to the lightest component, C3, is not close to the converged solution until after two iterations. The inside-out method does not always converge so dramatically, but is usually quite efficient.

HOMOTOPY-CONTINUATION METHODS Although the SC and inside-out methods are reasonably robust, they are not guaranteed to converge and sometimes fail, particularly for very nonideal liquid solutions and when initial guesses are poor. A much more robust, but more time-consuming, method is differential arclength homotopy continuation, the basic principles and applications of which are discussed by Wayburn and Seader [Comp. Chem. Engng., 11, 7–25 (1987); Proceedings Second Intern. Conf. Foundations of Computer-Aided Process Design, CACHE, Austin, TX, 765–

13-52

DISTILLATION Continuation methods, also called imbedding and path-following methods, were first applied to the solution of separation models involving large numbers of nonlinear equations by Salgovic, Hlavacek, and Ilavsky [Chem. Eng. Sci., 36, 1599 (1981)] and by Byrne and Baird [Comp. Chem. Engng., 9, 593 (1985)]. Since then, they have been applied successfully to problems involving interlinked distillation (Wayburn and Seader, op. cit.), azeotropic and three-phase distillation [Kovach, III and Seider, Comp. Chem. Engng., 11, 593 (1987)], and reactive distillation [Chang and Seader, Comp. Chem. Engng., 12, 1243 (1988)], when SC and inside-out methods have failed. Today, many computer-aided distillation-design and simulation packages include continuation techniques to make the codes more robust. STAGE EFFICIENCY

FIG. 13-55

Specifications for distillation column of Example 7.

862 (1984); AIChE monograph Series, AIChE, New York, 81, No. 15 (1985)]. Homotopy methods begin from a known solution of a companion set of equations and follow a path to the desired solution of the set of equations to be solved. In most cases, the path exists and can be followed. In one implementation, the set of equations to be solved, call it f(x), and the companion set of equations, call it g(x), are connected together by a set of mathematical homotopy equations: h(x,t) = t f(x) + (1 − t)g(x) = 0

(13-110)

where t is a homotopy parameter. An appropriate function is selected for g(x) such as f(x) − f (x0), where x0 are the initial guesses, which can be selected arbitrarily. At the beginning of the path, t = 0 and Eq. (13-110) becomes h(x,t) = f(x) − f(x0) = 0 or f(x) = f(x0). The homotopy parameter is then gradually moved from 0 to 1. At a value of t = 1, Eq. (13-110) becomes h(x,t) = f(x) = 0, which corresponds to the desired solution. The movement along the path is accomplished by a predictorcorrector continuation procedure, where the corrector is often a numerical Euler integration step of the differential-arclength form of Eq. (13-110) along the arclength of the path (rather than a step in t), as proposed by Klopfenstein [J. Assoc. Comput. Mach., 8, 366 (1961)]. The arclength of the path is preferred, over the homotopy parameter, as a continuation parameter because the path may make one or more turns in the homotopy parameter, making it difficult to take an integration step. The predictor step is accompanied by a truncation error that is reduced in the corrector step, which employs Newton’s method with Eq. (13-110) to return to the path. If the predictor steps along the path are not too large, the corrector steps always converge. Another implementation of homotopy-continuation methods is the use of problem-dependent homotopies that exploit some physical aspect of the problem. Vickery and Taylor [AIChE J., 32, 547 (1986)] utilized thermodynamic homotopies for K values and enthalpies to gradually move these properties from ideal to actual values so as to solve the MESH equations when very nonideal liquid solutions were involved. Taylor, Wayburn, and Vickery [I. Chem. E. Symp. Ser. No. 104, B305 (1987)] used a pseudo-Murphree efficiency homotopy to move the solution of the MESH equations from a low efficiency, where little separation occurs, to a higher and more reasonable efficiency.

The mathematical models presented earlier for rigorous calculations of multistage, multicomponent distillation-type separations assume that equilibrium with respect to both heat and mass transfer is attained at each stage. Unless temperature changes significantly from stage to stage, the assumption that vapor and liquid phases exiting from a stage are at the same temperature is generally valid. However, in most cases, equilibrium with respect to mass transfer is not a valid assumption. If all feed components have the same mass-transfer efficiency, the number of actual stages or trays is simply related to the number of equilibrium stages used in the modeling calculations by an overall stage efficiency. For distillation, as discussed in Sec. 14, this efficiency for well-designed trays typically varies from 40 to 120 percent; the higher value being achieved in some large-diameter towers because of a cross-flow effect. Efficiencies for absorption and extractive distillation can be lower than 40 percent. When it is desired to compute, with rigorous methods, actual rather than equilibrium stages, Eqs. (13-69) and (13-94) can be modified to include the Murphree vapor-phase efficiency η i, j, defined by Eq. (13-29). This is particularly desirable for multistage operations involving feeds containing components of a wide range of volatility and/or concentration, in which only a rectification (absorption) or stripping action is provided and all components are not sharply separated. In those cases, the use of a different Murphree efficiency for each component and each tray may be necessary to compute recovery accurately. Departures from the equilibrium-stage model may also occur when entrainment of liquid droplets in the rising vapor or occlusion of vapor in the liquid flow in the downcomer is significant. The former condition may occur at high vapor loading when flooding is approached. The latter condition is possible at high operating pressures when vapor and liquid densities are not drastically different. Entrainment and occlusion effects are not strictly due to mass-transfer inefficiency and are best taken into account by including entrainment terms in the modeling equations, as shown by Loud and Waggoner [Ind. Eng. Chem. Process Des. Dev., 17, 149 (1978)]. RATE-BASED MODELS Although the widely used equilibrium-stage models for distillation, described above, have proved to be quite adequate for binary and close-boiling, ideal and near-ideal multicomponent vapor-liquid mixtures, their deficiencies for general multicomponent mixtures have long been recognized. Even Murphree [Ind. Eng. Chem., 17, 747–750 and 960–964 (1925)], who formulated the widely used plate efficiencies that carry his name, pointed out clearly their deficiencies for multicomponent mixtures and when efficiencies are small. Later, Walter and Sherwood [Ind. Eng. Chem., 33, 493 (1941)] showed that experimentally measured efficiencies could cover an enormous range, with some values less than 10 percent, and Krishna et al. [Trans. Inst. Chem. Engr., 55, 178 (1977)] showed theoretically that the component mass-transfer coupling effects discovered by Toor [AIChE J., 3, 198 (1957)] could cause the rate of mass transfer for components having small concentration driving forces to be controlled by the other species, with the result that Murphree vapor efficiencies could cover the entire range of values from minus infinity to plus infinity.

RIGOROUS METHODS FOR MULTICOMPONENT DISTILLATION-TYPE SEPARATIONS The first major step toward the development of a more realistic rate-based (nonequilibrium) model for distillation was taken by Krishnamurthy and Taylor [AIChE J., 31, 449–465 (1985)]. More recently, Taylor, Kooijman, and Hung [Comp. Chem. Engng., 18, 205–217 (1994)] extended the initial development so as to add the effects of tray-pressure drop, entrainment, occlusion, and interlinks with other columns. In the augmented MESH equations, which they refer to as the MERSHQ equations, they replace the conventional mass and energy balances around each stage by two balances each, one for the vapor phase and one for the liquid phase. Each of the componentmaterial balances contains a term for the rate of mass transfer between the two phases; the energy balances contain a term for the rate of heat transfer between phases. Thus, each pair of phase balances is coupled by mass or heat-transfer rates, which are estimated from constitutive equations that account, in as rigorous a manner as possible, for bulk transport, species interactions, and coupling effects. The heat and mass-transfer coefficients in these equations are obtained from empirical correlations of experimental data and the Chilton-Colburn analogy. Equilibrium between the two phases is assumed at the phase interface. Thus, the rate-based model deals with both transport and thermodynamics. Although tray efficiencies are not part of the modeling equations, efficiencies can be backcalculated from the results of the simulation. Various options for vapor and liquid flow configurations are employed in the model, including plug flow and perfectly mixed flow on each tray. A schematic diagram of the nonequilibrium stage for the Taylor et al. model is shown in Fig. 13-56. Entering the stage are the following material streams: F V = vapor feed; FL = liquid feed; Vj + 1 = vapor from stage below together with fractional-liquid entrainment, φLj+ 1; Lj − 1 = liquid from stage above with fractional-vapor occlusion, φVj− 1; GV = vapor interlink; and GL = liquid interlink. Leaving the stage are the following material streams: Vj = vapor with fractional withdrawal as sidestream, rjV, and fractional-liquid entrainment, φjL; and Lj = liquid with fractional withdrawal as sidestream, rjL, and fractional-vapor occlusion, φVj . Also leaving the stage are heat-transfer streams, QjV and QjL. The rate of heat transfer from the vapor phase to the liquid phase is Ej and the rate of component mass transfer from the vapor phase to the liquid phase is Nij. The nonequilibrium-model equations for the stage in Fig. 13-56 are as follows in residual form, where i = component (i = 1 to C), j = stage number ( j = 1 to N), and ν = a stage in another column that supplies an interlink. Material Balances (2C + 2 Equations) Component for the vapor phase: MijV (1 + rjV + φ Vj )Vj yij − Vj + 1 yi, j + 1 − φ Vj − 1Vj − 1 yi, j − 1 − f Vij −

n



GVijν + Nij

ν=1

=0

i = 1, 2, . . . , c

(13-111)

Component for the liquid phase: n

L MijL (1 + rjL + φjL)Lj xij − Lj − 1xi, j − 1 − φLj+ 1Lj + 1xi, j + 1 − fijL −  Gijν − Nij ν=1

=0

i = 1, 2, . . . , c

(13-112)

V j

For the liquid phase: EjL (1 + rjL + φjL)LjHjL − Lj − 1H Lj− 1 − φLj+ 1Lj + 1H Lj+ 1 − FjLHjLF n

−  GLjν HLjν + QjL − ejL = 0 (13-116) ν=1

Continuity across the phase interface: EjI ejV − ejL = 0

(13-117)

Mass-Transfer Rates (2C - 2 Equations) Component in the vapor phase: RVij Nij − N Vij = 0

i = 1, 2, . . . , c − 1

(13-118)

i = 1, 2, . . . , c − 1

(13-119)

Component in the liquid phase: RLij Nij − NLij = 0

Summation of Mole Fractions (2 Equations) Vapor-phase interface: c

I SVI j  yi j − 1 = 0

(13-120)

i=1

Liquid-phase interface: c

M (1 + r + φ )Vj − Vj + 1 − φ V j

Schematic diagram of a nonequilibrium stage.

I SLI j  x ij − 1 = 0

Total for the vapor phase: V tj

FIG. 13-56

13-53

c

Vj − 1 − F − 

L j−1

V j

n

G

V ijν

+ Ntj

i=1 ν=1

=0

(13-113)

Total for the liquid phase: c

Mtj (1 + r jL + φ Lj )Lj − Lj − 1 − φLj + 1Lj + 1 − FjL −  L

n

G

L ijν

− Ntj

(13-114)

Energy Balances (3 Equations) For the vapor phase: EjV (1 + rjV + φ Vj )VjHjV − Vj + 1H Vj+ 1 − φVj− 1Vj − 1HVj− 1 − FjVHjVF −

n

G

V jν

ν=1

Hydraulic Equation for Stage Pressure Drop (1 Equation) Vapor-phase pressure drop: Pj pj − pj − 1 − (∆ pj − 1) = 0

(13-122)

Interface Equilibrium (C Equations) Component vaporliquid equilibrium:

i=1 ν=1

=0

(13-121)

i=1

HVjν + Q Vj + ejV = 0 (13-115)

QIij Kij x Iij − yIij = 0

i = 1, 2, . . . , c

(13-123)

Equations (13-111) to (13-114), (13-118) and (13-119), contain terms, Nij, for rates of mass transfer of components from the vapor phase to the liquid phase (rates are negative if transfer is from the liquid phase to the vapor phase). These rates are estimated from diffusive and bulk-flow contributions, where the former are based on interfacial area, average mole-fraction driving forces, and mass-

13-54

DISTILLATION

transfer coefficients that account for coupling effects through binarypair coefficients. Although the stage shown in Fig. 13-56 appears to apply to a trayed column, the model also applies for a section of a packed column. Accordingly, empirical correlations for the interfacial area and binary-pair mass-transfer coefficients cover bubble-cap trays, sieve trays, valve trays, dumped packings, and structured packings. The average mole-fraction driving forces for diffusion depend upon the assumed vapor and liquid-flow patterns. In the mixed-flow model, both phases are completely mixed. This is the simplest model and is usually suitable for small-diameter trayed columns. In the plugflow model, both phases move in plug flow. This model is applicable to packed columns and certain trayed columns. Equations (13-115) to (13-117) contain terms, ej, for rates of heat transfer from the vapor phase to the liquid phase. These rates are estimated from convective and bulk-flow contributions, where the former are based on interfacial area, average-temperature driving forces, and convective heat-transfer coefficients, which are determined from the Chilton-Colburn analogy for the vapor phase and from the penetration theory for the liquid phase. The K values (vapor-liquid equilibrium ratios) in Equation (13-123) are estimated from the same equation-of-state or activity-coefficient models that are used with equilibrium-stage models. Tray or packedsection pressure drops are estimated from suitable correlations of the type discussed by Kister (op. cit.). From the above list of rate-based model equations, it is seen that they total 5C + 6 for each tray, compared to 2C + 1 or 2C + 3 (depending on whether mole fractions or component flow rates are used for composition variables) for each stage in the equilibrium-stage model. Therefore, more computer time is required to solve the rate-based model, which is generally converged by an SC approach of the Newton type. A potential limitation of the application to design of a rate-based model compared to the equilibrium-stage model is that the latter can be computed independently of the geometry of the column because no transport equations are included in the model. Thus, the sizing of the column is decoupled from the determination of column operating conditions. However, this limitation of the early rate-based models has now been eliminated by incorporating a design mode that simultaneously designs trays and packed sections. A study of industrial applications by Taylor, Kooijman, and Woodman [IChemE. Symp. Ser. Distillation and Absorption 1992, A415– A427 (1992)] concluded that rate-based models are particularly desirable when simulating or designing: (1) packed columns, (2) systems with strongly nonideal liquid solutions, (3) systems with trace compo-

nents that need to be tracked closely, (4) columns with rapidly changing profiles, (5) systems where tray-efficiency data are lacking. Besides the extended model just described, a number of other investigators, as summarized by Taylor, Kooijman, and Hung (op. cit.), have developed rate-based models for specific applications and other purposes, including cryogenic distillation, crude distillation, vacuum distillation, catalytic distillation, three-phase distillation, dynamic distillation, and liquid-liquid extraction. Commercial computerized ratebased models are available in two simulation programs: RATEFRAC in ASPEN PLUS from Aspen Technology, Inc., Cambridge, Massachusetts and NEQ2 in ChemSep from R. Taylor and H. A. Kooijman of Clarkson University. Rate-based models could usher in a new era in trayed and packed-column design and simulation. Example 8: Calculation of Rate-Based Distillation The separation of 655 lb⋅mol/h of a bubble-point mixture of 16 mol % toluene, 9.5 mol % methanol, 53.3 mol % styrene, and 21.2 mol % ethylbenzene is to be carried out in a 9.84-ft diameter sieve-tray column having 40 sieve trays with 2-inch high weirs and on 24-inch tray spacing. The column is equipped with a total condenser and a partial reboiler. The feed will enter the column on the 21st tray from the top, where the column pressure will be 93 kPa, The bottom-tray pressure is 101 kPa and the top-tray pressure is 86 kPa. The distillate rate will be set at 167 lb⋅mol/h in an attempt to obtain a sharp separation between toluenemethanol, which will tend to accumulate in the distillate, and styrene and ethylbenzene. A reflux ratio of 4.8 will be used. Plug flow of vapor and complete mixing of liquid will be assumed on each tray. K values will be computed from the UNIFAC activity-coefficient method and the Chan-Fair correlation will be used to estimate mass-transfer coefficients. Predict, with a rate-based model, the separation that will be achieved and back-calculate from the computed tray compositions, the component vapor-phase Murphree-tray efficiencies. The calculations were made with the RATEFRAC program and comparisons were made with the companion RADFRAC program, which utilizes the insideout method for an equilibrium-based model. The rate-based model gave a distillate with 0.023 mol % ethylbenzene and 0.0003 mol % styrene, and a bottoms product with essentially no methanol and 0.008 mol % toluene. Murphree tray efficiencies for toluene, styrene, and ethylbenzene varied somewhat from tray to tray, but were confined mainly between 86 and 93 percent. Methanol tray efficiencies varied widely, mainly from 19 to 105 percent, with high values in the rectifying section and low values in the stripping section. Temperature differences between vapor and liquid phases leaving a tray were not larger than 5°F. Based on an average tray efficiency of 90 percent for the hydrocarbons, the equilibrium-based model calculations were made with 36 equilibrium stages. The results for the distillate and bottoms compositions, which were very close to those computed by the rate-based method, were a distillate with 0.018 mol % ethylbenzene and less than 0.0006 mol % styrene, and a bottoms product with only a trace of methanol and 0.006 mol % toluene.

ENHANCED DISTILLATION INTRODUCTION In distillation operations, separation results from differences in vaporand liquid-phase compositions arising from the partial vaporization of a liquid mixture or the partial condensation of a vapor mixture. The vapor phase becomes enriched in the more volatile components while the liquid phase is depleted of those same components. In many situations, however, the change in composition between the vapor and liquid phases in equilibrium becomes small (so-called “pinched condition”), and a large number of successive partial vaporizations and partial condensations is required to achieve the desired separation. Alternatively, the vapor and liquid phases may have identical compositions, because of the formation of an azeotrope, and no separation by simple distillation is possible. Several enhanced distillation-based separation techniques have been developed for close-boiling or low-relative-volatility systems, and for systems exhibiting azeotropic behavior. All of these special techniques are ultimately based on the same differences in the vapor and liquid compositions as ordinary distillation, but, in addition, they rely on some additional mechanism to further modify the vapor-liquid

behavior of the key components. These enhanced techniques can be classified according to their effect on the relationship between the vapor and liquid compositions: 1. Azeotropic distillation and pressure-swing distillation. Methods that cause or exploit azeotrope formation or behavior to alter the boiling characteristics and separability of the mixture. 2. Extractive distillation and salt distillation. Methods that primarily modify liquid-phase behavior to alter the relative volatility of the components of the mixture. 3. Reactive distillation. Methods that use chemical reaction to modify the composition of the mixture or, alternatively, use existing vapor-liquid differences between reaction products and reactants to enhance the performance of a reaction. AZEOTROPISM At low-to-moderate pressure ranges typical of most industrial applications, the fundamental composition relationship between the vapor and liquid phases in equilibrium can be expressed as a function of the total

ENHANCED DISTILLATION

13-55

system pressure, the vapor pressure of each pure component, and the liquid-phase activity coefficient of each component i in the mixture: yiP = xiγi P isat

(13-124)

In systems that exhibit ideal liquid-phase behavior, the activity coefficients, γi, are equal to unity and Eq. (13-124) simplifies to Raoult’s law. For nonideal liquid-phase behavior, a system is said to show negative deviations from Raoult’s law if γi < 1, and conversely, positive deviations from Raoult’s law if γi > 1. In sufficiently nonideal systems, the deviations may be so large the temperature-composition phase diagrams exhibit extrema, as shown in each of the three parts of Fig. 13-57. At such maxima or minima, the equilibrium vapor and liquid compositions are identical. Thus, yi = xi

for all i = 1, . . . n

(13-125)

and the system is said to form an azeotrope (from the Greek, meaning to boil unchanged). Azeotropic systems show a minimum in the T-x,y diagram when the deviations from Raoult’s law are positive (Fig. 13-57a) and a maximum in the T-x,y diagram when the deviations from Raoult’s law are negative (Fig. 13-57b). If at these two conditions, a single liquid phase is in equilibrium with the vapor phase, the azeotrope is homogeneous. If multiple liquid-phase behavior is exhibited at the azeotropic condition, the azeotrope is heterogeneous. For heterogeneous azeotropes, the vapor-phase composition is equal to the overall composition of the two (or more) liquid phases (Fig. 13-57c). Mixtures with only small deviations from Raoult’s law may form an azeotrope only if the components are close-boiling. As the boilingpoint difference between the components increases, the composition of the azeotrope shifts closer to one of the pure components (toward the lower-boiling pure component for minimum-boiling azeotropes, and toward the higher-boiling pure component for maximum-boiling azeotropes). Mixtures of components whose boiling points differ by more than about 30°C generally do not exhibit azeotropes distinguishable from the pure components even if large deviations from Raoult’s law are present. As a qualitative guide to liquid-phase activitycoefficient behavior, Robbins [Chem. Eng. Prog., 76 (10) 58 (1980)] developed a matrix of chemical families, shown in Table 13-15, which indicates expected deviations from Raoult’s law. The formation of two liquid phases within some temperature range for close-boiling mixtures is generally an indication that the system

FIG. 13-57 Schematic isobaric-phase diagrams for binary azeotropic mixtures. (a) Homogeneous minimum-boiling azeotropes.

13-57 (Continued) Schematic isobaric-phase diagrams for binary azeotropic mixtures. (b) Homogeneous maximum-boiling azeotrope. FIG.

will also exhibit a minimum-boiling azeotrope, since two liquid phases may form when deviations from Raoult’s law are extremely positive. The fact that immiscibility does occur, however, does not guarantee that the azeotrope will be heterogeneous. The azeotropic temperature is sometimes outside the range of temperatures at which a system exhibits two liquid phases. Moreover, the azeotropic composition may not necessarily fall within the composition range of the two-liquidphase region even when within the appropriate temperature range

13-57 (Continued) Schematic isobaric-phase diagrams for binary azeotropic mixtures. (c) Heterogeneous azeotrope. FIG.

13-56

DISTILLATION

TABLE 13-15 Solute class

Solute-Solvent Group Interactions Solvent class Group

1

2

3

4

5

6

7

8

9

10

11

12

0 0 − 0

0 0 − 0

− − 0 +

0 0 + 0

− − + −

− − 0 −

− 0 − −

− 0 − −

− 0 − −

− 0 − −

− − − 0

− − − −

−

−

+

−

0

+

−

−

−

+

−

−

− − −

− 0 0

0 − −

− − −

+ + +

0 + +

− 0 0

− 0 0

0 0 −

+ 0 +

0 0 −

0 − −

− −

0 0

+ +

− −

+ +

0 +

0 0

− −

0 −

+ 0

0 −

− −

+

+

+

0

+

0

0

−

0

+

0

0

+

+

+

+

+

0

+

+

+

+

0

0

H-donor 1 2 3 4

Phenol Acid, thiol Alcohol, water Active-H on multihalo paraffin

5

Ketone, amide with no H on N, sulfone, phosphine oxide Tertamine Secamine Pri amine, ammonia, amide with 2H on N Ether, oxide, sulfoxide Ester, aldehyde, carbonate, phosphate, nitrate, nitrite, nitrile, intramolecular bonding, e.g., o-nitro phenol Aromatic, olefin, halogen aromatic, multihalo paraffin without active H, monohalo paraffin

H-acceptor

6 7 8 9 10

11

Non-H-bonding 12 SOURCE:

Paraffin, carbon disulfide

Robbins, L. A., Chem. Eng. Prog., 76(10), 58–61 (1980), by permission.

for liquid-liquid behavior, as is for example the case for the methyl acetate-water and tetrahydrofuran-water systems. Homogeneous azeotropes that are completely miscible at all temperatures usually occur between species with very close boiling points and rather small liquid-phase nonidealities. Moreover, since strong positive deviations from Raoult’s law are required for liquid-liquid phase splitting, maximum-boiling azeotropes (γi < 1) are never heterogeneous. Additional general information on the thermodynamics of phase equilibria and azeotropy is available in Swietoslawski (Azeotropy and Polyazeotropy, Pergamon, London, 1963), Van Winkle (Distillation, McGraw-Hill, New York, 1967), Smith and Van Ness (Introduction to Chemical Engineering Thermodynamics, McGraw-Hill, New York, 1975), Wizniak [Chem. Eng. Sci., 38, 969 (1983)], and Walas (Phase Equilibria in Chemical Engineering, Butterworths, Boston, 1985). Horsley (Azeotropic Data-III, American Chemical Society, Washington, 1983) compiled an extensive list of binary and some ternary and higher experimental azeotropic boiling-point and composition data. Another source for azeotrope data and activity coefficient model parameters is the multivolume Vapor-Liquid Equilibrium Data Collection (DECHEMA, Frankfort 1977), a compendium of published experimental VLE data. Most of the data have been tested for thermodynamic consistency and have been fit to the Wilson, UNIQUAC, Van Laar, Margules, and NRTL equations. An extensive two-volume compilation of data for 18,800 systems involving 1,700 compounds, entitled Azeotropic Data by Gmehling et al., was published in 1994 by VCH Publishers, Deerfield Beach, Florida. A computational method for determining the temperatures and compositions of all azeotropes of a multicomponent mixture, from liquid-phase activity-coefficient correlations, by a differential arclength homotopy continuation method is given by Fidkowski, Malone, and Doherty [Computers and Chem. Eng., 17, 1141 (1993)]. RESIDUE CURVE MAPS AND DISTILLATION REGION DIAGRAMS The simplest form of distillation involves boiling a multicomponent liquid mixture batchwise in a single-stage still pot. At any instant in

time the vapor being generated and removed from the pot is assumed to be in equilibrium with the remaining liquid (assumed to be perfectly mixed) in the still. Because the vapor is richer in the more volatile components than the liquid, the composition and temperature of the liquid remaining in the still changes continuously over time and moves progressively toward less volatile compositions and higher temperatures until the last drop is vaporized. For some mixtures, this last composition is the highest-boiling pure component in the system. For other mixtures, this final composition may be a maximum-boiling azeotrope. For yet other systems, the final composition varies depending on the initial composition of the mixture charged to the still. A residue curve is a tracing of this change in perfectly mixed liquid composition for simple single-stage batch distillation with respect to time. Arrows are sometimes added, pointing in the direction of increasing time, increasing temperature, and decreasing volatility. Because simple, batch distillation can be described mathematically by dxi /d ξ = xi − yi

for all i = 1, . . . n

(13-126)

where ξ is a nonlinear time scale, residue curves may also be extrapolated backward in time to give more volatile compositions which would produce a residue equal to the specified initial composition. A residue curve map (RCM) is generated by varying the initial composition and extrapolating Eq. (13-126) both forward and backward in time [Doherty and Perkins, Chem. Eng. Sci., 33, 281 (1978)]. Unlike a binary y-x plot, relative-volatility information is not presented. Therefore, it is difficult to determine the ease of separation from a residue curve map alone. Residue curve maps can be constructed for mixtures of any number of components, but can be pictured graphically only for up to four components. For a binary mixture, a T-x,y diagram suffices; the system is simple enough that vapor-phase information can be included without confusion. With a ternary mixture, liquid-phase compositions are plotted on a triangular diagram, similar to that used in liquidliquid extraction. Four-component systems can be plotted in a 3-dimensional tetrahedron. The vertices of the triangular diagram or tetrahedron represent the pure components. Any binary, ternary, and quaternary azeotropes are placed at the appropriate compositions on the edges and/or interior of the triangle and tetrahedron.

ENHANCED DISTILLATION The simplest form of ternary RCM, as exemplified for the ideal normal-paraffin system of pentane-hexane-heptane, is illustrated in Fig. 13-58a, using a right-triangle diagram. Maps for all other nonazeotropic ternary mixtures are qualitatively similar. Each of the infinite number of possible residue curves originates at the pentane vertex, travels toward and then away from the hexane vertex, and terminates at the heptane vertex. The family of all residue curves that originate at one composition and terminate at another composition defines a region. Systems that do not involve azeotropes have only one region—the entire composition space. However, for many systems, not all residue curves originate or terminate at the same two compositions. Such systems will have more that one region. The demarcation between regions in which adjacent residue curves originate from different compositions or terminate at different compositions is called a separatrix. Separatrices are related to the existence of azeotropes. In the composition space for a binary system, the separatrix is a point (the azeotropic composition). With three components, the separatrix becomes a (generally curved) line, with four components the separatrix becomes a surface, and so on. All pure components and azeotropes in a system lie on region boundaries. Within each region, the most volatile composition on the boundary (either a pure component or a minimum-boiling azeotrope and the origin of all residue curves) is called the low-boiling node. The least-volatile composition on the boundary (again either a pure component or a maximum-boiling azeotrope and the terminus of all residue curves) is called the high-boiling node. All other pure components and azeotropes are called intermediate-boiling saddles (because no residue curves originate or terminate at these compositions). Adjacent regions may share nodes and saddles. Pure components and azeotropes are labeled as nodes and saddles as a result of the boiling points of all of the components and azeotropes in a system. If one species is removed, the labeling of all remaining pure components and azeotropes, particularly those that were saddles, may change. Region-defining separatrices always originate or terminate at saddle azeotropes, but never at saddle-pure components. Saddle-ternary azeotropes are particularly interesting because they are less obvious to determine experimentally (being neither minimum-boiling nor maximum-boiling), and have only recently begun to be recorded in the literature. (Gmehling et al., Azeotropic Data, VCH Publishers, Deer-

(Continued) Residue curve maps. (b) MEK-MIPK-water system containing two minumum-boiling binary azeotropes.

FIG. 13-58

field Beach, Florida, 1994). However, their presence in a mixture implies separatrices, which may have an important impact on the design of a separation system. Both methylethylketone (MEK) and methylisopropylketone (MIPK) form minimum-boiling azeotropes with water (Fig. 13-58b). In this ternary system, a separatrix connects the binary azeotropes and divides the RCM into two regions. The high-boiling node of Region I is pure water, while the low-boiling node is the MEK-water azeotrope.

(Continued) Residue curve maps. (c) Ethanol-cyclohexane-water system containing four minimum-boiling azeotropes and three distillation regions.

FIG. 13-58 FIG. 13-58

system.

Residue curve maps. (a) Nonazeotropic pentane-hexane-heptane

13-57

13-58

DISTILLATION

In Region II, the high- and low-boiling nodes are MIPK and the MEK-water azeotrope, respectively. The more complicated cyclohexane-ethanol-water system (Fig. 13-58c) has three separatrices and three regions, all of which share the ternary azeotrope as the lowboiling node. The liquid-composition profiles in continuous staged or packed distillation columns operating at infinite reflux and boilup are closely approximated by simple distillation-residue curves [Van Dongen and Doherty, Ind. Eng. Chem. Fundam., 24, 454 (1985)]. Residue curves are also indicative of many aspects of the general behavior of continuous columns operating at more practical reflux ratios. For example, to a first approximation, the composition of the distillate and bottoms of a single-feed, continuous distillation column lie on the same residue curve. Therefore, for systems with separatrices and multiple regions, distillation-composition profiles are constrained to lie in specific regions. The precise boundaries of these distillation regions are a function of reflux ratio, but they are closely approximated by the RCM separatrices. If a RCM separatrix exists in a system, a corresponding distillation boundary will also exist. Separatrices and distillation boundaries correspond exactly at all pure components and azeotropes. Residue curves can be constructed from experimental data or can be calculated analytically if equation-of-state or activity-coefficient expressions are available (e.g., Wilson binary-interaction parameters, UNIFAC groups). However, considerable information on system behavior can still be deduced from a simple semi-qualitative sketch of the RCM separatrices or distillation boundaries based only on pure component and azeotrope boiling-point data and approximate azeotrope compositions. Rules for constructing such qualitative distillation region diagrams (DRD) are given by Foucher et al. [Ind. Eng. Chem. Res., 30, 760–772, 2364 (1991)]. For ternary systems containing no more than one ternary azeotrope, and no more than one binary azeotrope between each pair of components, 125 such DRD are mathematically possible, although only a dozen or so represent most systems commonly encountered in practice. Figure 13.59 illustrates all of the 125 possible DRD for ternary systems. Azeotropes are schematically depicted generally to have equimolar composition, distillation boundaries are shown as straight lines, and the arrows on the distillation boundaries indicate increasing temperature. These DRD are indexed in Table 13-16 according to a temperature-profile sequence of position numbers, defined in a keyed-triangular diagram at the bottom of the table, arranged by increasing the boiling point. Positions 1, 3, and 5 are the pure components in order of decreasing volatility. Positions 2, 4, and 6 are binary azeotropes at the positions shown in the keyed triangle, and position 7 is the ternary azeotrope. Azeotrope position numbers are deleted from the temperature profile if the corresponding azeotrope is known not to exist. It should be noted that not every conceivable temperature profile corresponds to a thermodynamically consistent system, and such combinations have been excluded from the index. As is evident from the index, some DRD are consistent with more than one temperature profile. Also, some temperature profiles are consistent with more than one DRD. In such cases, the correct diagram for a system must be determined from residue curves obtained from experimental or calculated data. Schematic DRD shown in Fig. 13-59 are particularly useful in determining the implications of possibly unknown ternary saddle azeotropes by postulating position 7 at interior positions in the temperature profile. It should also be noted that some combinations of binary azeotropes require the existence of a ternary saddle azeotrope. As an example, consider the system acetone (56.4°C), chloroform (61.2°C), and methanol (64.7°C). Methanol forms minimum-boiling azeotropes with both acetone (54.6°C) and chloroform (53.5°C), and acetone-chloroform forms a maximum-boiling azeotrope (64.5°C). Experimentally there are no data for maximum or minimum-boiling ternary azeotropes. The temperature profile for this system is 461325, which from Table 13-16 is consistent with DRD 040 and DRD 042. However, Table 13-16 also indicates that the pure component and binary azeotrope data are consistent with three temperature profiles involving a ternary saddle azeotrope, namely 4671325, 4617325, and 4613725. All three of these temperature profiles correspond to DRD 107. Experimental residue curve trajectories for the acetone-

chloroform-methanol system, as shown in Fig. 13-60, suggest the existence of a ternary saddle azeotrope and DRD 107 as the correct approximation of the distillation regions. Ewell and Welch [Ind. Eng. Chem., 37, 1224 (1945)] confirm such a ternary saddle at 57.5°C. APPLICATIONS OF RCM AND DRD Residue curve maps and distillation region diagrams are very powerful tools for understanding all types of batch and continuous distillation operations, particularly when combined with other information such as liquid-liquid binodal curves. Applications include: 1. System visualization. Location of distillation boundaries, azeotropes, distillation regions, feasible products, and liquid-liquid regions. 2. Evaluation of laboratory data. Location and confirmation of saddle ternary azeotropes and a check of thermodynamic consistency of data. 3. Process synthesis. Concept development, construction of flowsheets for new processes, and redesign or modification of existing process flowsheets. 4. Process modeling. Identification of infeasible or problematic column specifications that could cause simulation convergence difficulties or failure, and determination of initial estimates of column parameters including feed-stage location, number of stages in the stripping and enriching sections, reflux ratio, and product compositions. 5. Control analysis/design. Analysis of column balances and profiles to aid in control system design and operation. 6. Process trouble shooting. Analysis of separation system operation and malfunction, examination of composition profiles, and tracking of trace impurities with implications for corrosion and process specifications. Material balances for mixing or continuous separation operations are represented graphically on triangular composition diagrams such as residue curve maps or distillation region diagrams by straight lines connecting pertinent compositions. Overall flow rates are found by the inverse-lever-arm rule. Distillation material balance lines are governed by two constraints: 1. The bottoms, distillate, and overall feed compositions must lie on the same straight line. 2. The bottoms and distillate compositions must lie (to a very close approximation) on the same residue curve. Since residue curves do not by definition cross separatrices, the distillate and bottoms compositions must be in the same distillation region with the mass balance line intersecting a residue curve in two places. Mass balance lines for mixing and for other separations not involving vapor-liquid equilibria, such as extraction and decantation, are of course not limited by distillation boundaries. For a given multicomponent mixture, a single-feed distillation column can be designed with sufficient stages, reflux, and material balance control to produce separations ranging from the direct mode of operation (low-boiling node taken as distillate) to the indirect mode (high-boiling node taken as bottoms). The bow-tie shaped set of reachable compositions for single-feed distillation is roughly bounded by the material balance lines corresponding to the sharpest direct separation and the sharpest indirect separation possible. The exact shape of the reachable composition space is further limited by the requirement that the distillate and bottoms lie on the same residue curve [Wahnschafft, et al., Ind. Eng. Chem. Res., 31, 2345 (1992)]. Since residue curves are deflected by saddles, it is generally not possible to obtain a saddle product (pure component or azeotrope) from a simple single-feed column. Consider the recovery of MIPK from an MEK-MIPK-water mixture. The bow-tie approximation of reachable compositions for several feeds are shown in Fig 13-61a and the exact reachable compositions are shown in Fig. 13-61b. From Feed F3, which is situated in a different distillation region than the desired product, pure MIPK cannot be obtained at all. With the upper edge of the bow-tie region for Feed F1 along the MEK-MIPK (water-free) face of the composition triangle, and part of the lower edge along the MEK-water (MIPK-free) face, there are conditions under which both the water in the bottoms MIPK product can be driven to low levels (high-product purity) and MIPK in

ENHANCED DISTILLATION

(a) FIG. 13-59

Distillation region diagrams for ternary mixtures.

13-59

13-60

DISTILLATION

(b) FIG. 13-59

(Continued) Distillation region diagrams for ternary mixtures.

ENHANCED DISTILLATION

(c) FIG. 13-59

(Continued) Distillation region diagrams for ternary mixtures.

13-61

13-62

DISTILLATION

(d) FIG. 13-59

(Continued) Distillation region diagrams for ternary mixtures.

ENHANCED DISTILLATION

(e) FIG. 13-59

(Continued) Distillation region diagrams for ternary mixtures.

13-63

13-64

DISTILLATION

(f) FIG. 13-59

(Continued) Distillation region diagrams for ternary mixtures.

the distillate can also be driven to low levels (high-product recovery), although achieving such an operation depends on having an adequate number of stages and reflux ratio. The bow-tie region for Feed F2 is significantly different, with the upper edge along the water-MIPK (MEK-free) face of the triangle and the lower edge along the distillation boundary. From this feed it is not possible to achieve a high-purity MIPK specification while simultaneously obtaining high MIPK recovery. If the column is operated to get a high purity of MIPK, then the material balance line runs into the distillation boundary. Alternatively, if the column is operated to obtain a high recovery of MIPK (by removing the MEK-water azeotrope as distillate), the material balance requires the bottoms to lie on the water-MIPK face of the triangle. The number of saddles in a particular distillation region can have significant impact on column-profile behavior, process stability, and convergence behavior in process simulation of the system. Referring to the MIPK-MEK-water system in Fig. 13-58b, Region I contains one saddle (MIPK-water azeotrope), while Region II contains two saddles (pure MEK and the MIPK-water azeotrope). These are threeand “four-sided” regions respectively. In a three-sided region, all residue curves track toward the solitary saddle. However, in a four (or more) sided region with saddles on either side of a node, some residue curves will tend to track toward one saddle, while others track toward another opposite saddle. For example, residue curve 1 in Region I originates from the MEK-water azeotrope low-boiling node and trav-

els first toward the single saddle of the region (MIPK-water azeotrope) before ending at the water high-boiling node. Likewise, residue curve 2 and all other residue curves in Region I follow the same general path. In Region II, residue curve 3 originates from the MEK-water azeotrope, travels toward the MIPK-water saddle azeotrope, and ends at pure MIPK. However, residue curve 4 follows a completely different path, traveling toward the pure MEK saddle before ending at pure MIPK. Some multicomponent columns have been designed for operation in four-sided regions with the feed composition adjusted so that both the high-boiling and low-boiling nodes can be obtained simultaneously as products. However, small perturbations in feed composition or reflux can result in feasible operation on many different residue curves that originate and terminate at these compositions. Multiple steady states and composition profiles that shift dramatically from tracking toward one saddle to the other are possible [Kovach, and Seider, AIChE J., 33, 1300 (1987). Consider a column operating in Region II of the MIPK-MEK-water diagram. Fig. 13-62 shows the composition and temperature profiles for the column operating at three different sets of operating conditions and two feed locations as given in Table 13-17. The desired product specification is 97 mol % MIPK, no more than 3 mol % MEK, and less than 10 ppm residual water. For Case A (Fig. 13-62a), the column profile tracks up the water-free side of the diagram. A pinched zone (area of little change in tray temperature and composition) occurs between the feed tray (tray

ENHANCED DISTILLATION TABLE 13-16

13-65

Temperature Profile—DRD # Table*

Temp. Profile

DRD #

Temp. Profile

DRD #

Temp. Profile

DRD #

Temp. Profile

DRD #

Temp. Profile

DRD #

Temp. Profile

DRD #

135 1325 1354 1356 1435 2135 6135 13254 13256 13524 13526 13542 13546 13547 13562 13564 13567 13725 13752 14325 027 14352 14356 14735 21354 21356 21435 24135 26135 41325 41735 42135 46135 61325 61354 61435 62135 64135 72135 76135 132546 132547 132564 132567 135246 135247 135264 135267 135426 135427 135462 135467 135624 135627 135642 135647 135724 135726 135742 135746 135762 135764 137254 137256

001 014 015 016 004 003 002 017 019 017 019 018 021 069 020 022 070 092 092 028

137524 137526 137542 137562 143256

099 100 099 100 035 036 078 078 036 035 078 035 037 073 097 097 097 097 093 056 034 072 033 076 046 096 046 096 011 096 044 011 011 095 036 036 037 097 097 097 097 093 045 096 012 096 040 042 042 098 012 098 038 038 039 077 094 094 041 040 098 043 041 013

624135 627135 641325

013 095 041 040 041 098 013 098 056 055 050 053 053 049 056 053 052 051 054 052 049 052 080 080 111 124 125 111 124 125 080 080 125 111 124 124 111 125 080 080 120 121 112 121 120 112 080 080 113 117 116 116 117 113 121 125 124 117 125 124 121 120 124

1357462

120 121 116 117 116 125 120 117 116 121 117 121 117 121 121 117 117 089 088 081 088 088 081 089 088 088 089 081 088 089 109 109 109 109 109 109 109 109 066 066 065 079 079 108 108 090 105 090 105 114 114 115 118 105 115 114 102 083 114 114 118 119 114 106

2671435

103 119 118 118 119 103 081 081 081 109 109 109 109 109 109 109 109 082 105 114 122 115 114 105 114 102 115 086 086 086 107 107 107 107 123 123 115 122 107 107 104 122 123 060 084 084 084 110 110 110 110 110 085 085 085 107 107 107 107 061 087 123 123 123

6247135

118 119 123 106 118 103 119 103 118 119 085 085 085 107 107 107 107 123 122 123 119 107 107 104 122 123 061 061 066 066 065 064 063 062 062 058 057 058 058 066 066 065 062 058 060 059 061 059 058 068 068 067 061 060 059 059 057 058 058 061 060 059 059 058

028 031 091 026 029 009 009 007 027 091 010 005 030 032 006 008 006 047 048 024 075 025 071 024 075 025 071 024 075 023 074 025 071 023 074 099 100 099 101 100 101 099 100

143257 143275 143526 143527 143562 143567 143725 143752 147325 147352 147356 174325 213546 213547 213564 213567 214356 214735 241356 241735 246135 247135 261354 261435 264135 267135 413256 413526 413562 413725 413752 417325 417352 417356 421356 421735 426135 427135 461325 461352 461735 462135 467135 613254 613524 613542 613547 613725 613752 614325 614735 621354 614352 621435

641352 641735 642135 647135 714325 721354 721356 721435 724135 726135 741325 742135 746135 761325 761354 761435 762135 764135 1325467 1325647 1325746 1325764 1352467 1352647 1352746 1352764 1354267 1354627 1354726 1354762 1356247 1356427 1356724 1356742 1357246 1357264 1357426

1357624 1357642 1372546 1372564 1375246 1375264 1375426 1375462 1375624 1375642 1432567 1432576 1432756 1435267 1435276 1435627 1435672 1435726 1435762 1437256 1437526 1437562 1473256 1473526 1473562 1743256 1743526 1743562 2135467 2135647 2135746 2135764 2143567 2147356 2413567 2417356 2461735 2467135 2471356 2476135 2613547 2614735 2641735 2647135 2671354

Ternary DRD table lookup procedure: 1. Classify a system by writing down each position number in ascending order of boiling points. • A position number is not written down if there is no azeotrope at that position. • The resulting sequence of numbers is known as the temperature profile. • Each temperature profile will have a minimum of three numbers and a maximum of seven numbers. • List multiple temperature profiles when you have incomplete azeotropic data. • All seven position numbers are shown on the diagram. 2. Using the table, look up the temperature profile(s) to find the corresponding DRD #. * Table 13-16 and Fig. 13-59 developed by Eric J. Peterson, Eastman Chemical Co.

2674135 4132567 4135267 4135627 4135726 4135762 4137256 4137526 4137562 4173256 4173526 4173562 4213567 4217356 4261735 4267135 4271356 4276135 4613257 4613275 4613527 4613725 4613752 4617325 4617352 4621735 4627135 4671325 4671352 4672135 4761325 6132547 6135247 6135427 6135724 6135742 6137254 6137524 6137542 6143257 6143275 6143527 6143725 6143752 6147325 6147352 6174325 6213547 6214735 6214735 6241735

6271354 6271435 6274135 6413257 6413275 6413527 6413725 6413752 6417325 6417352 6421735 6427135 6471325 6471352 6472135 6714325 6741325 7143256 7143526 7143562 7213546 7213564 7214356 7241356 7246135 7261354 7261435 7264135 7413256 7413526 7413562 7421356 7426135 7461325 7461352 7462135 7613254 7613524 7613542 7614325 7614352 7621354 7621435 7624135 7641325 7641352 7642135

13-66

DISTILLATION

Residue curves for acetone-chloroform-methanol system suggesting a ternary saddle azeotrope.

FIG. 13-60

4) and tray 18. The temperature remains constant at about 93°C throughout the pinch. Product specifications are met. When the feed composition becomes enriched in water, as with Case B, the column profile changes drastically (Fig. 13-62b). At the same reflux and boil-up, the column no longer meets specifications. The MIPK product is lean in MIPK and too rich in water. The profile now tracks generally up the left side of Region II. Note also the dramatic change in the temperature profile. A pinched zone still exists

FIG. 13-61

(Continued) MEK-MIPK-water system. (b) Exact-reachable

compositions.

between trays 4 and 18, but the tray temperature in the zone has dropped to 80°C (from 93°C). Most of the trays are required to move through the vicinity of the saddle. Typically, pinches (if they exist) occur close to saddles and nodes. In Case C (Fig. 13-62c), increasing the boil-up ratio to 6 brings the MIPK product back within specifications, but the production rate and recovery have dropped off. In addition, the profile has switched back to the right side of the region; the temperatures on trays in the pinched zone (trays 4–18) are back to 93°C. Such a drastic fluctuation in tray temperature with a relatively minor adjustment of the manipulated variable (boil-up in this case), can make control difficult. This is especially true if the control strategy involves maintaining a constant temperature on one of the trays between tray 4 and 18. If a tray is selected that exhibits wide temperature swings, the control system may have a difficult time compensating for disturbances. Such columns are also often difficult to model with a process simulator. Design algorithms often rely on perturbation of a variable (such as reflux or reboil) while checking for convergence of column heat and material balances. In situations where the column profile is altered drastically by minor changes in the perturbed variable, the simulator may be close to a feasible solution, but successive iterations may appear to be very far apart. The convergence routine may continue to oscillate between column profiles and never reach a solution. Likewise, when an attempt is made to design a column to obtain product compositions in different distillation regions, the simulation will never converge. EXTENSION TO BATCH DISTILLATION

MEK-MIPK-water system. (a) Approximate bow-tie reachable compositions by simple distillation.

FIG. 13-61

Although batch distillation is covered in a subsequent separate section, it is appropriate to consider the application of RCM and DRD to batch distillation at this time. With a conventional batch-rectification column, a charge of starting material is heated and fractionated, with a vapor product removed continuously. The composition of the vapor product changes continuously and at times drastically as the lighter component(s) are exhausted from the still. Between points of drastic change in the vapor composition, a “cut” is often made. Successive cuts can be removed until the still is nearly dry. The sequence, number, and limiting composition of each cut is dependent on the form of

ENHANCED DISTILLATION Methyl isopropyl ketone

Methyl isopropyl ketone Comp. 1. 2. 3.

1.0

Feed 0.1500 0.8000 0.0500

Distillate 0.6024 0.1649 0.2328

Bottoms 0.0263 0.9737 0.7210E-05

Reflux = Reboil = Q= Pressure (ATM) = Stages: Reboiler: Feed: Condenser:

0.8

0.1300E+02 0.3830E+01 0.1000E+01 0.1000E+01

Feed 0.1400 0.7800 0.0800

1 4 30

Distillate 0.4795 0.1535 0.3670

Bottoms 0.0471 0.9514 0.0015

Reflux = Reboil = Q= Pressure (ATM) = Stages: Reboiler: Feed: Condenser:

0.8

0.6

X2 0.4

0.4

0.2

0.2

0.0 0.0

0.2

0.6

0.4

0.8

X1

Water

0.0 0.0

1.0

1.0

1.0

0.8

0.8

0.6 0.4 0.2 0.0 0

0.1300E+02 0.3830E+01 0.1000E+01 0.1000E+01 1 4 30

0.6

0.4

0.8

X1

1.0 Methyl ethyl ketone

0.6 0.4 0.2

5

10 MEK

15 Stage no.

20

MIPK

25

0.0 0

30

90

Temperature (C)

90

90

80

10

15 Stage no.

20

10 MEK

95

5

5

Water

95

75 0

0.2

Water

Methyl ethyl ketone

Mole fraction

Mole fraction

Comp. 1. 2. 3.

1.0

X2

0.6

Temperature (C)

13-67

25

30

15 Stage no.

20

MIPK

25

30

25

30

Water

85

80 75 0

(a)

5

10

15 Stage no.

20

(b)

FIG. 13-62

Sensitivity of composition and temperature profiles for MEKMIPK-water system.

FIG. 13-62

the residue curve map and the composition of the initial charge to the still. As with continuous distillation operation, the set of reachable products (cuts) for a given charge to a batch distillation is constrained by the residue-curve-map separatrices, which cannot normally be crossed. Given a sufficient number of stages and reflux, the vapor composition can be made to closely approach direct-mode, continuous operation in which the lowest-boiling species is taken overhead. As the

low-boiling component is removed, the still composition moves along a straight material-balance line through the initial feed composition and the low-boiling node away from the initial composition until it reaches the edge of the composition triangle or a separatrix. The path then follows the edge or separatrix to the high-boiling node of the region. At each turn a new cut is taken. Examples for the acetonechloroform-methanol and MEK-water-MIPK systems are given in Fig. 13-63 [Bernot et al., Chem. Eng. Sci., 45, 1207 (1990)].

(Continued) Sensitivity of composition and temperature profiles for MEK-MIPK-water system.

13-68

DISTILLATION TABLE 13-17

Methyl isopropyl ketone Comp. 1. 2. 3.

1.0

Feed 0.1400 0.7800 0.0800

Distillate 0.4295 0.3038 0.2667

Reflux = Reboil = Q= Pressure (ATM) = Stages: Reboiler: Feed: Condenser:

0.8

Case

Reboil ratio

A

13

3.8

0.1300E+02 0.6000E+01 0.1000E+01 0.1000E+01

B

13

1 4 30

C

13

0.4

0.2

0.0 0.0

0.2

0.6

0.4

0.8

X1

Water

1.0 Methyl ethyl ketone

1.0

Mole fraction

0.8 0.6 0.4 0.2 0.0 0

5

10 MEK

15 Stage no.

20

MIPK

Sets of Operating Conditions for Fig. 13-62

Reflux ratio

X2

0.6

Bottoms 0.0159 0.9841 0.6194E-05

25

30

25

30

Feed composition

Distillate composition

Bottoms composition

MEK MIPK water

0.15 0.80 0.05

0.60 0.16 0.24

0.03 0.97 7 ppm

3.8

MEK MIPK water

0.14 0.78 0.08

0.48 0.15 0.37

0.05 0.95 20,000 ppm

6

MEK MIPK water

0.14 0.78 0.08

0.43 0.30 0.27

0.02 0.98 6.5 ppm

chemical and chemical processing industries for the separation of close-boiling, pinched, or azeotropic systems for which simple distillation is either too expensive or impossible. With an azeotropic feed mixture, presence of the azeotroping agent results in the formation of a more favorable azeotropic pattern for the desired separation. For a close-boiling or pinched feed mixture, the azeotroping agent changes the dimensionality of the system and allows separation to occur along a less-pinched path. Within the general heading of azeotropic distillation techniques, several approaches have been followed in devising azeotropic distillation flowsheets including: 1. Choosing an entrainer to give a residue curve map with specific distillation regions and node temperatures. 2. Exploiting changes in azeotropic composition with total system pressure. 3. Exploiting curvature of distillation region boundaries. 4. Choosing an entrainer to cause azeotrope formation in combination with liquid-liquid immiscibility. The first three of these are solely VLE-based approaches, involving a series of simple distillation operations and recycles. The final approach also relies on distillation (VLE), but also exploits another physical phenomena, liquid-liquid phase formation (phase splitting), to assist in entrainer recovery. This approach is the most powerful and versatile. Examples of industrial uses of azeotropic distillation grouped by method are given in Table 13-18.

Water

Temperature (C)

95 90

85

80 75 0

5

10

15 Stage no.

20

(c) (Continued) Sensitivity of composition and temperature profiles for MEK-MIPK-water system.

FIG. 13-62

AZEOTROPIC DISTILLATION Introduction The term azeotropic distillation has been applied to a broad class of fractional distillation-based separation techniques in that specific azeotropic behavior is exploited to effect a separation. The agent that causes the specific azeotropic behavior, often called the entrainer, may already be present in the feed mixture (a selfentraining mixture) or may be an added mass-separation agent. Azeotropic distillation techniques are used throughout the petro-

(a) FIG. 13-63

system.

Batch distillation paths. (a) Methanol-methyl propionate-water

ENHANCED DISTILLATION

(b) (Continued) Batch distillation paths. (b) Methanol-acetonechloroform system. FIG. 13-63

TABLE 13-18

13-69

The choice of the appropriate azeotropic distillation method and the resulting flowsheet for the separation of a particular mixture are strong functions of the separation objective. For example, it may be desirable to recover all constituents of the original feed mixture as pure components, or only some as pure components and some as azeotropic mixtures suitable for recycle. Not every objective may be obtainable by azeotropic distillation for a given mixture and portfolio of candidate entrainers. Exploitation of Homogeneous Azeotropes Homogeneous azeotropic distillation refers to a flowsheet structure in which azeotrope formation is exploited or avoided in order to accomplish the desired separation in one or more distillation columns. The azeotropes in the system either do not exhibit two-liquid-phase behavior or the liquid-phase behavior is not or cannot be exploited in the separation sequence. The structure of a particular sequence will depend on the geometry of the residue curve map or distillation region diagram for the feed mixture-entrainer system. Two approaches are possible: 1. Selection of an entrainer such that the desired products all lie within a single distillation region (the products may be pure components or azeotropic mixtures). 2. Selection of an entrainer such that although the desired products lie in different regions, some type of boundary-crossing mechanism is employed. As mentioned previously, ternary mixtures can be represented by 125 different residue curve maps or distillation region diagrams. However, feasible distillation sequences using the first approach can be developed for breaking homogeneous binary azeotropes by the addition of a third component only for those more restricted systems that do not have a distillation boundary connected to the azeotrope and for which one of the original components is a node. For example, from

Examples of Azeotropic Distillation

System

Type

Entrainer(s)

Remark

Exploitation of homogeneous azeotropes No known industrial examples Exploitation of pressure sensitivity THF-water Methyl acetate-methanol

Minimum boiling azeotrope Minimum boiling azeotrope

None None

Alcohol-ketone systems Ethanol-water

Minimum boiling azeotropes Minimum boiling azeotrope

None None

Hydrochloric acid-water

Maximum boiling azeotrope

Sulfuric acid

Nitric acid-water

Maximum-boiling azeotrope

Sulfuric acid

Ethanol-water

Minimum boiling azeotrope

Acetic acid-water

Pinched system

Butanol-water Acetic acid-water-vinyl acetate Methyl acetate-methanol

Minimum boiling azeotrope Pinched, azeotropic system Minimum boiling azeotrope

Diethoxymethanol-water-ethanol Pyridine-water Hydrocarbon-water

Minimum-boiling azeotropes Minimum-boiling azeotrope Minimum-boiling azeotrope

Alternative to extractive distillation Element of recovery system for alternative to production of methyl acetate by reactive distillation; alternative to azeotropic, extractive distillation Alternative to extractive distillation, salt extractive distillation, heterogeneous azeotropic distillation; must reduce pressure to less than 11.5kPa for azeotrope to disappear

Exploitation of boundary curvature Alternative to salt extractive distillation Alternative to salt extractive distillation

Exploitation of azeotropy and liquid phase immiscibility Cyclohexane, benzene, heptane, hexane, toluene, gasolene, diethyl ether Ethyl acetate, propyl acetate, diethyl ether, dichloroethane, butyl acetate Self-entraining Self-entraining Toluene, methyl isobutyl ketone

Self-entraining Benzene Self-entraining

Alternative to extractive distillation, pressure-swing distillation

Element of recovery system for alternative to production of methyl acetate by reactive distillation; alternative to extractive pressureswing distillation Alternative to extractive distillation

13-70

DISTILLATION

Fig. 13-59, the following eight residue curve maps are suitable for breaking homogeneous minimum-boiling azeotropes: DRD 002, 027, 030, 040, 051, 056, 060, and 061 as collected in Fig. 13-64a. To produce the necessary distillation region diagrams, an entrainer must be found that is either: (1) an intermediate boiler that forms no azeotropes (DRD 002), or (2) lowest boiling or intermediate boiling and forms a maximum-boiling azeotrope with the lower-boiling original component. In these cases, the entrainer may also optionally form a minimum-boiling azeotrope with the higher boiling of the original components or a minimum-boiling ternary azeotrope. In all cases, after the addition of the entrainer, the higher-boiling original component is a node and is removed as bottoms product from a first column

operated in the indirect mode with the lower-boiling original component recovered as distillate in a second column. The seven residue curve maps suitable for breaking homogeneous maximum-boiling azeotropes (DRD 028, 031, 035, 073, 078, 088, 089) are shown in Fig. 13-64b. In this case, the entrainer must form a minimum-boiling azeotrope with the higher-boiling original component and either a maximum-boiling azeotrope or no azeotrope with the lower-boiling original component. In all cases, after the addition of the entrainer, the lower-boiling original component is a node and is removed as distillate from a first column operated in the direct mode with the higher-boiling original component recovered as bottoms product in a second column.

(a) FIG. 13-64

Feasible distillation region diagrams for breaking homogeneous binary azeotrope A-B. (a) Low-boiling entrances.

ENHANCED DISTILLATION

13-71

(b) FIG. 13-64

(Continued) Feasible distillation region diagrams for breaking homogeneous binary azeotrope A-B. (b) Intermediate-boiling entrances.

In one sense, the restrictions on the boiling point and azeotrope formation of the entrainer act as efficient screening criteria for entrainer selection. Entrainers that do not show appropriate boiling-point characteristics can be discarded without detailed analysis. However, in another sense, although theoretically feasible, the above sequences suffer from serious drawbacks that limit their practical application. DRD 002 requires that the entrainer be an intermediate-boiling component that forms no azeotropes. Unfortunately these are often difficult criteria to meet, as any intermediate boiler will be closer-boiling to both of the original components and, therefore, will be more likely to be at least pinched or even form azeotropes. The remaining feasi-

ble distillation region diagrams require that the entrainer form a maximum-boiling azeotrope with the lower-boiling original component. Because maximum-boiling azeotropes are relatively rare, finding a suitable entrainer may be difficult. For example, the dehydration of organics that form homogeneous azeotropes with water is a common industrial problem. It is extremely difficult to find an intermediate-boiling entrainer that also does not form an azeotrope with water. Furthermore, the resulting separation is likely to be close-boiling or pinched throughout most of the column, requiring a large number of stages. However, consider the separation of valeric acid (187.0°C) and water. This system exhibits an azeotrope

13-72

DISTILLATION

(99.8°C). Ignoring for the moment potentially severe corrosion problems, formic acid (100.7°C), which is an intermediate boiler and which forms a maximum-boiling azeotrope with water (107.1°C), is a candidate entrainer (DRD 030, Fig. 13-65a). In the conceptual sequence shown in Fig. 13-65b, a recycle of the formic acid-water maximum-boiling azeotrope is added to the original valeric acid-water feed, which may be of any composition. Using the indirect mode of operation, high-boiling node valeric acid is removed in high purity and high recovery as bottoms in a first column, which by mass balance produces a formic acid-water distillate. This binary mixture is fed to a second column that produces pure water as distillate and the formic acid-water azeotrope as bottoms for recycle to the first column. The inventory of formic acid is an important optimization variable in this theoretically feasible but difficult separation scheme. Exploitation of Pressure Sensitivity The breaking of homogeneous azeotropes that are part of a distillation boundary (that is, into products in different distillation regions) requires that the boundary

be “crossed.” This may be done by mixing some external stream with the original feed stream in one region such that the resulting composition is in another region for further processing. However, the external stream must be completely regenerated, and mass-balance observed. For example, it is not possible to break a homogeneous binary azeotrope simply by adding one of the products to cross the azeotropic composition. The composition of many azeotropes varies with the system pressure (Horsley, Azeotropic Data-III, American Chemical Society, Washington, 1983 and Gmehling et al., Azeotropic Data, VCH Publishers, Deerfield Beach, Florida, 1994). This effect can be exploited to separate azeotropic mixtures by so-called pressure-swing distillation if at some pressure the azeotrope simply disappears, as for example does the ethanol-water azeotrope at pressures below 11.5 kPa. However, pressure sensitivity can still be exploited if the azeotropic composition and related distillation boundary change sufficiently over a moderate change in total system pressure. A composition in one region under one set of conditions, could be in a different region under a different set of conditions. A two-column sequence for separating a binary maximum-boiling azeotrope is shown in Fig. 13-66 for a system in which the azeotropic composition at pressure P1 is richer in component B than the azeotropic composition at pressure P2. The first column, operating at pressure P1, is fed a mixture of fresh feed and recycle stream from the second column such that the overall composition lies on the A side of the azeotropic composition at P1. Pure component A is recovered as distillate and a mixture near the azeotropic composition is produced as bottoms. The pressure of this bottoms stream is changed to P2 and fed to the second column. This feed is on the B side of the azeotropic composition at P2. Pure component B is now recovered as the distillate and the azeotropic bottoms composition is recycled to the first column. An analogous flowsheet can be used for separating binary-homogeneous minimum-boiling azeotropes. In this case the pure components are recovered as bot-

(a)

(b) Valeric acid-water separation with formic acid. (a) Mass balances on distillation region diagram. (b) Conceptual sequence. FIG. 13-65

FIG. 13-66 Conceptual sequence for separating maximum-boiling binary azeotrope with pressure swing distillation.

ENHANCED DISTILLATION toms in both columns and the distillate from each column is recycled to the other column. For pressure-swing distillation to be practical, the azeotropic composition must vary at least 5 percent, (preferably 10 percent or more) over a moderate pressure range (not more than ten atmospheres between the two pressures). With a very large pressure range, refrigeration may be required for condensation of the low-pressure distillate or an impractically high reboiler temperature may result in the high-pressure column. The smaller the variation of azeotrope composition over the pressure range, the larger will be the recycle streams between the two columns. In particular, for minimum-boiling azeotropes, the pressure-swing distillation approach requires high energy usage and high capital costs (large-diameter columns) because both recycled azeotropic compositions must be taken overhead. Often one lobe of an azeotropic VLE diagram is pinched regardless of pressure; and, therefore, one of the columns will require a large number of stages to produce the corresponding pure-component product. General information on pressure-swing distillation can be found in Van Winkle (Distillation, McGraw-Hill, New York, 1967), Wankat (Equilibrium-Staged Separations, Elsevier, New York, 1988), and Knapp and Doherty [Ind. Eng. Chem. Res., 31, 346 (1992)]. Only a relatively small fraction of azeotropes are sufficiently pressure sensitive for a pressure-swing process to be economical. Some applications include the minimum-boiling azeotrope tetrahydrofuran-water [Tanabe et al., U.S. Patent 4,093,633 (1978)], and maximum-boiling azeotropes of hydrogen chloride-water and formic acid-water (Horsley, Azeotropic Data-III, American Chemical Society, Washington, 1983). Since separatrices also move with pressure-sensitive azeotropes, the pressure-swing principle can also be used for overcoming distillation boundaries in multicomponent azeotropic mixtures. Exploitation of Boundary Curvature A second approach to boundary crossing exploits boundary curvature in order to produce compositions in different distillation regions. When distillation boundaries exhibit extreme curvature, it may be possible to design a column such that the distillate and bottoms are on the same residue curve in one distillation region, while the feed (which is not required to lie on the column-composition profile) is in another distillation region. In order for such a column to meet material-balance constraints (i.e., bottom, distillate, feed on a straight line), the feed must be located in a region where the boundary is concave. As an example, Van Dongen [Ph.D. Thesis, University of Massachusetts, (1983)] considered the separation of a methanol-methyl acetate mixture, which forms a homogeneous azeotrope, using n-hexane as an entrainer. The separatrices for this system (Fig. 1367a) are somewhat curved. A separation sequence that exploits this boundary curvature is shown in Fig. 13-67b. Recycled methanolmethyl acetate binary azeotrope and methanol-methyl acetate-hexane ternary azeotrope are added to the original feed F0 to produce a net-feed composition F1 for column C1 designed to lie on a line between pure methanol and the curved part of the boundary between Regions I and II. C1 is operated in the indirect mode producing the high-boiling node methanol as a bottoms product, and by mass balance, a distillate near the curved boundary. The distillate, although in Region I, becomes feed F2 to column C2 which is operated in the direct mode entirely in Region II, producing the low-boiling node ternary azeotrope as distillate and by mass balance, a methanolmethyl acetate mixture as bottoms. This bottoms mixture is on the opposite side of the methanol-methyl acetate azeotrope at the original feed F0. The bottoms from C2 is finally fed to binary distillation column C3, which produces pure methyl acetate as bottoms product and the methanol-methyl acetate azeotrope as distillate. The distillates from C2 and C3 are recycled to C1. The distillate and bottoms compositions for C2 lie on the same residue curve, and the composition profile lies entirely within Region II, even though its feed composition is in Region I. Exploitation of boundary curvature for breaking azeotropes is very similar to exploiting pressure sensitivity from a mass-balance point of view, and suffers from the same disadvantages. Separation schemes have large recycle flows, and in the case of minimum-boiling azeotropes, the recycle streams are distillates. However, in the case of maximum-boiling azeotropes, these recycles are underflows and

13-73

FIG. 13-67 Separation of methanol-methyl acetate by exploitation of distillation boundary curvature.

the economics are improved. One such application, illustrated in Fig. 13-68, is the separation of the nitric acid-water azeotrope by adding sulfuric acid. Recycled sulfuric acid is added to a nitric acid-water mixture near the azeotropic composition to produce a net feed in Region I. The first column, operated in the direct mode, produces a nitric-acid distillate and a bottoms product, by mass balance, near the distillation boundary. In this case, sulfuric acid associates with water so strongly and the separatrix is so curved and nearly tangent to the water-sulfuric acid edge of the composition diagram that the second column operating in the indirect mode in Region II, producing sulfuric acid as bottoms product also produces a distillate close enough to the water specification that a third column is not required (Thiemann et al., in Ullmann’s Encyclopedia of Industrial Chemistry, Fifth Edition, Volume A17, VCH Verlagsgesellschaft mbH, Weinheim, 1991). Exploitation of Azeotropy and Liquid-Phase Immiscibility One powerful and versatile separation approach exploits several physical phenomena simultaneously including enhanced vapor-liquid behavior, where possible, and liquid-liquid behavior to bypass difficult distillation separations. For example, the overall separation of closeboiling mixtures can be made easier by the addition of an entrainer that forms a heterogeneous minimum-boiling azeotrope with one (generally the lower-boiling) of the key components. Two-liquidphase formation provides a means of breaking this azeotrope, thus simplifying the entrainer recovery and recycle process. Moreover,

13-74

DISTILLATION to either column C1 or C2, depending on whether the feed is on the organic-rich side or the water-rich side of the azeotrope. The feed may also be added into the decanter directly if it doesn’t move the overall composition of the decanter outside of the two-liquid-phase region. Column C1 produces anhydrous butanol as a bottoms product and a composition close to the butanol-water azeotrope as the distillate. After condensation, the azeotrope rapidly phase separates in the decanter. The upper layer, consisting of 78 wt % butanol, is refluxed totally to C1 for further butanol recovery. The water layer, consisting of 92 wt % water, is fed to C2. This column produces pure water as a bottoms product and, again, a composition close to the azeotrope as distillate for recycle to the decanter. Sparged steam may be used in C2, saving the cost of a reboiler. A similar flowsheet can be used for dehydration of hydrocarbons and other species that are largely immiscible with water. A second example of the use of liquid-liquid immiscibilities in an azeotropic-distillation sequence is the separation of the ethanol-water minimum-boiling azeotrope. For this separation, a number of entrainers have been proposed, which are usually chosen to be immiscible with water, form a ternary minimum-boiling (preferably heterogeneous) azeotrope with ethanol and water (and, therefore, usually also binary minimum-boiling azeotropes with both ethanol and water). All such systems correspond to DRD 058, although the labeling of the vertices depends on whether the entrainer is lower boiling than ethanol, intermediate boiling, or higher boiling than water. The residue curve map for the case for cyclohexane as entrainer was illustrated in Fig. 13-58c. One three-column distillation sequence is shown in Fig. 13-70. Other two-, three-, or four-column sequences have been described by Knapp and Doherty (Kirk-Othmer Encyclopedia of Chemical Technology, Fourth Edition, Vol. 8, Wiley, New York, 1993). Fresh aqueous ethanol feed is first preconcentrated to nearly the azeotropic composition in column C3, while producing a water bottoms product. The distillate from C3 is sent to column C1, which is refluxed with the entire organic (entrainer-rich) layer, recycled from a decanter. Mixing of these two streams is the key to this sequence as it allows the overall feed composition to cross the distillation boundary into Region II. column C1 is operated to recover pure high-boiling node ethanol as a bottoms product and to produce a distillate close to the ternary azeotrope. If the ternary azeotrope is heterogeneous (as it

Separation of nitric acid-water system with sulfuric acid in a twocolumn sequence exploiting extreme boundary curvature. FIG. 13-68

since liquid-liquid tie lines are unaffected by distillation boundaries (and the separate liquid phases are often located in different distillation regions), liquid-liquid phase splitting is a powerful mechanism for crossing distillation boundaries. The phase separator is usually a simple decanter, but sometimes a multistage extractor is substituted. The decanter or extractor can also be replaced by some other non-VLEbased separation technique such as membrane permeation, chromatography, adsorption, or crystallization. In addition, sequences may include additional separation operations (distillations or other methods) for preconcentration of the feed mixture, entrainer recovery, and final-product purification. The simplest case of combining VLE and LLE is the separation of a binary heterogeneous azeotropic mixture. One example is the dehydration of 1-butanol, a self-entraining system, in which butanol (117.7°C) and water form a minimum-boiling heterogeneous azeotrope (93.0°C). As shown in Fig. 13-69, the fresh feed may be added

FIG. 13-69

lation.

Separation of butanol-water with heterogeneous azeotropic distil-

ENHANCED DISTILLATION

13-75

tors are capable of modeling the steady state and dynamic behavior of both homogeneous azeotropic distillation systems and those systems involving two-liquid phase behavior within the column, if accurate thermodynamic data and activity-coefficient or equation-of-state models are available. However, VLE and VLLE estimated or extrapolated from binary data or predicted from such methods as UNIFAC may not be able to accurately locate boundaries and predict the extent of liquid immiscibilities. Moreover, different activity-coefficient models fit to the same experimental data often give very different results for the shape of boundaries and liquid-liquid regions. Therefore the design of separation schemes relying on boundary curvature should not be attempted unless accurate, reliable experimental equilibrium data are available. Two liquid phases can occur within a column in the distillation of heterogeneous systems. Older references, for example Robinson and Gilliland (Elements of Fractional Distillation, McGraw-Hill, New York, 1950) state that the presence of two liquid phases in a column should be avoided as much as possible because performance may be reduced. However, more recent studies indicate that problems with two-phase flow have been overstated [Herron et al., AIChE J. 34, 1267 (1988) and Harrison, Chem. Eng. Prog., 86(11), 80 (1990)]. Based on case-history data and experimental evidence, there is no reason to expect unusual capacity or pressure-drop limitations, and standard correlations for these parameters should give acceptable results. Because of the violent nature of the gas/liquid/liquid mixing on trays, trayed column efficiencies are relatively unaffected by liquid-liquid phase behavior. The falling-film nature of gas/liquid/liquid contact in packing, however, makes that situation more uncertain. Reduced efficiencies may be expected in systems where one of the keys distributes between the phases. EXTRACTIVE DISTILLATION

FIG. 13-70 Three-column sequence for ethanol dehydration with cyclohexane (operating column C2 in the direct mode).

is in this case), it is sent to the decanter for phase separation. If the ternary azeotrope is homogeneous (as it is in the alternative case of ethyl acetate as the entrainer) the distillate is first mixed with water before being sent to the decanter. The inventory of entrainer is adjusted to allow C1 to operate essentially between two nodes, although such practice, as discussed previously, is relatively susceptible to instabilities from minor feed or reflux perturbations. Refluxing a fraction of the water-rich decanter layer results in an additional degree of freedom to mitigate against variability in the feed composition. The remaining portion of the water layer from the decanter is stripped of residual cyclohexane in column C2, which may be operated either in the direct mode (producing low-boiling node ternary azeotrope as distillate and, by mass balance, an ethanol-water bottoms for recycle to C3 or, in the indirect mode (producing high-boiling node water as bottoms and, by mass balance, a ternary distillate near the distillation boundary. The distillate may be recycled to the decanter, the top of C2, or C2 feed. Design and Operation of Azeotropic Distillation Columns Simulation and design of azeotropic distillation columns is a difficult computational problem, but one that is readily handled, in most cases, by widely available commercial computer process simulation packages [Glasscock and Hale, Chem. Eng., 101(11), 82 (1994)]. Most simula-

Introduction Extractive distillation is a partial vaporization process in the presence of a miscible, high-boiling, non-volatile massseparation agent, normally called the solvent, which is added to an azeotropic or nonazeotropic feed mixture to alter the volatilities of the key components without the formation of any additional azeotropes. Extractive distillation is used throughout the petrochemical- and chemical-processing industries for the separation of close-boiling, pinched, or azeotropic systems for which simple single-feed distillation is either too expensive or impossible. It can also be used to obtain products which are residue-curve saddles, a task not generally possible with single-feed distillation. Fig. 13-71 illustrates the classical implementation of an extractive distillation process for the separation of a binary system. The configuration consists of a double-feed extractive column and a solventrecovery column. The components A and B may have a low relative volatility or form a minimum-boiling azeotrope. The solvent is introduced into the extractive column at a high concentration a few stages below the condenser, but above the primary-feed stage. Since the solvent is chosen to be nonvolatile it remains at a relatively high concentration in the liquid phase throughout the sections of the column below the solvent-feed stage. One of the components, A (not necessarily the most volatile species of the original mixture), is withdrawn as an essentially pure distillate stream. Because the solvent is nonvolatile, at most a few stages above the solvent-feed stage are sufficient to rectify the solvent from the distillate. The bottoms product, consisting of B and the solvent, is sent to the recovery column. The distillate from the recovery column is pure B, and the solvent-bottoms product is recycled back to the extractive column. Extractive distillation works by the exploitation of the selective solvent-induced enhancements or moderations of the liquid-phase nonidealities of the components to be separated. The solvent selectively alters the activity coefficients of the components being separated. To do this, a high concentration of solvent is necessary. Several features are essential: 1. The solvent must be chosen to affect the liquid-phase behavior of the key components differently, otherwise no enhancement in separability will occur.

13-76

DISTILLATION recovered in the distillate. The possibility that the expected relative volatility may be reversed by the addition of solvent is entirely a function of the way the solvent interacts with and modifies the activity coefficients and, thus, the volatility of the components in the mixture. In normal applications of extractive distillation (i.e., pinched, closeboiling, or azeotropic systems), the relative volatilities between the light and heavy key components will be unity or close to unity. Assuming an ideal vapor phase and subcritical components, the relative volatility between the light and heavy keys of the desired separation can be written as the product of the ratios of the pure-component vapor pressures and activity-coefficient ratios whether the solvent is present or not: γL P Lsat αL,H = ᎏ (13-127) ᎏ PHsat γH where L and H denote the lower-boiling and higher-boiling pure component, respectively. The addition of the solvent has an indirect effect on the vaporpressure ratio. Because the solvent is high boiling and is generally added at a relatively high mole ratio to the primary-feed mixture, the temperature of an extractive-distillation process tends to increase over that of a simple distillation of the original binary mixture (unless the system pressure is lowered). The result is a corresponding increase in the vapor pressure of both key components. However, the rise in operating temperature generally does not result in a significant modification of the relative volatility, because the ratio of vapor pressures often remains approximately constant, unless the slopes of the vaporpressure curves differ significantly. The ratio of the vapor pressures typically remains greater than unity, following the “natural” volatility of the system. Since activity coefficients have a strong dependence on composition, the effect of the solvent on the activity coefficients is generally more pronounced. However, the magnitude and direction of change is highly dependent on the solvent concentration, as well as the liquidphase interactions between the key components and the solvent. The solvent acts to lessen the nonidealities of the key component whose liquid-phase behavior is similar to the solvent, while enhancing the nonideal behavior of the dissimilar key. The solvent and the key component that show most similar liquidphase behavior tend to exhibit little molecular interactions. These components form an ideal or nearly ideal liquid solution. The activity coefficient of this key approaches unity, or may even show negative deviations from Raoult’s law if solvating or complexing interactions occur. On the other hand, the dissimilar key and the solvent demonstrate unfavorable molecular interactions, and the activity coefficient of this key increases. The positive deviations from Raoult’s law are further enhanced by the diluting effect of the high-solvent concentration, and the value of the activity coefficient of this key may approach the infinite dilution value, often a very large number. The natural relative volatility of the system is enhanced when the activity coefficient of the lower-boiling pure component is increased by the solvent addition (γL /γ H increases and PLsat/PHsat > 1). In this case, the lower-boiling pure component will be recovered in the distillate as expected. In order for the higher-boiling pure component to be recovered in the distillate, the addition of the solvent must decrease the ratio γL /γH such that the product of the γL /γ H and PLsat/PHsat (i.e., αL,H) in the presence of the solvent is less than unity. Generally, the latter is more difficult and requires higher solvent-to-feed ratios. It is normally better to select a solvent that forces the lower-boiling component overhead. The effect of solvent concentration on the activity coefficients of the key components is shown in Fig. 13-72 for the system methanolacetone with either water or methylisopropylketone (MIPK) as solvent. For an initial-feed mixture of 50 mol % methanol and 50 mol % acetone (no solvent present), the ratio of activity coefficients of methanol and acetone is close to unity. With water as the solvent, the activity coefficient of the similar key (methanol) rises slightly as the solvent concentration increases, while the coefficient of acetone approaches the relatively large infinite-dilution value. With methylisopropylketone as the solvent, acetone is the similar key and its activity coefficient drops toward unity as the solvent concentration increases, while the activity coefficient of the methanol increases.

  

Typical extracting distillation sequence. Component A is less associated with the solvent.

FIG. 13-71

2. The solvent must be higher boiling than the key components of the separation and must be relatively nonvolatile in the extractive column, in order to remain largely in the liquid phase. 3. The solvent should not form additional azeotropes with the components in the mixture to be separated. 4. The extractive column must be a double-feed column, with the solvent feed above the primary feed; the column must have an extractive section. As a consequence of these restrictions, separations of binary systems by extractive distillation correspond to only two possible threecomponent distillation region diagrams, depending on whether the binary system is pinched or close boiling (DRD 001), or forms a minimum-boiling azeotrope (DRD 003). The addition of high-boiling solvents can also facilitate the breaking of maximum-boiling azeotropes (DRD 014), for example splitting the nitric acid-water azeotrope with sulfuric acid. However, as explained in the section on azeotropic distillation, this type of separation might better be characterized as exploitation of extreme boundary curvature rather than extractive distillation, as the important liquid-phase activity-coefficient modification occurs in the bottom of the column. Although many references show sulfuric acid being introduced high in the column, two separate feeds are in fact not required. Examples of industrial uses of extractive distillation grouped by distillation region diagram type are given in Table 13-19. Achievable compositions in dual-feed extractive distillation columns are very different from the bow-tie regions for single-feed columns. Either pure component (the higher-boiling of which is a saddle) for close-boiling systems, and either pure component (both of which are saddles) for minimum-boiling azeotropic systems can be obtained as distillate. Extractive distillation is generally only applicable to systems in which the components to be separated contain one or more different functional groups. Extractive distillation is usually uneconomical for separating stereoisomers, homologs, or homology or structural isomers containing the same functional groups, unless the differences in structure also contribute to significantly different polarity, dipole moment, or hydrophobic character. One such counter-example is the separation of ethanol from isopropanol, where the addition of methyl benzoate raises the relative volatility from 1.09 to 1.27 [Berg et al., Chem. Eng. Comm., 66, 1 (1988)]. Solvent Effects in Extractive Distillation In the distillation of ideal or nonazeotropic mixtures, the component with the lowest purecomponent boiling point is always recovered primarily in the distillate, while the highest boiler is recovered primarily in the bottoms. The situation is not as straightforward for an extractive-distillation operation. With some solvents, the component with the lower pure-component boiling point will be recovered in the distillate as in ordinary distillation. For another solvent, the expected order is reversed, and the component with the higher pure-component boiling point will be

ENHANCED DISTILLATION TABLE 13-19

13-77

Examples of Extractive Distillation, Salt Extractive Distillation

System

Type

Solvent(s)

Ethanol-water

Minimum-boiling azeotrope

Ethylene glycol, acetate salts for salt process Aniline Higher esters or alcohols, aromatics Propylene glycol

Benzene-cyclohexane Ethyl acetate-ethanol

Minimum-boiling azeotrope Minimum-boiling azeotrope

THF-water

Minimum-boiling azeotrope

Acetone-methanol Isoprene-pentane Pyridine-water Methyl acetate-methanol

Minimum-boiling azeotrope Minimum-boiling azeotrope Minimum-boiling azeotrope Minimum-boiling azeotrope

Water, aniline, ethylene glycol Furfural, DMF, acetonitrile Bisphenol Ethylene glycol monomethyl ether

C4 alkenes/C4 alkanes/ C4 dienes C5 alkenes/C5 alkanes/ C5 dienes Heptane isomerscyclohexane Heptane isomers-toluene

Close-boiling and minimumboiling azeotropes Close-boiling and minimumboiling azeotropes Close-boiling

Furfural, DMF, acetonitrile, n-methylpyrolidone Furfural, DMF, acetonitrile, n-methylpyrolidone Aniline, phenol

Close-boiling and minimumboiling azeotropes

Aniline, phenol

Vinyl acetate-ethyl acetate Propane-propylene

Close-boiling Close-boiling

Phenol, aromatics Acrylonitrile

Ethanol-isopropanol Hydrochloric acid-water

Close-boiling Maximum-boiling azeotrope

Nitric acid-water

Maximum-boiling azeotrope

Methyl benzoate Sulfuric acid, calcium chloride for salt process Sulfuric acid, magnesium nitrate for salt process

Remark Alternative to azeotropic distillation, pressure swing distillation Process similar for other alcohol-ester systems Alternative to pressure swing distillation

Element of recovery system for alternative to production of methyl acetate by reactive distillation; alternative to azeotropic, pressure, swing distillation

Alternative to simple distillation Alternative to simple distillation, adsorption Alternative to simple distillation Sulfuric acid process relies heavily on boundary curvature Sulfuric acid process relies heavily on boundary curvature

Several methods are available for determining whether the loweror higher-boiling pure component will be recovered in the distillate. For a series of solvent concentrations, the y-x phase diagram of the low-boiling and high-boiling keys can be plotted on a solvent-free basis. At a particular solvent concentration (dependent on the selected solvent and keys), the azeotropic point in the pseudobinary plot disappears at one of the pure component corners. The component corresponding to the corner where the azeotrope disappears is recovered in the distillate [Knapp and Doherty, in Kirk-Othmer Encyclopedia of Chemical Technology, Fourth Edition, Vol. 8, Wiley, New York (1993)]. LaRoche et al. [Can. J. Chem. Eng., 69, 1302 (1991)] present a related method in which the αL,H = 1 line is plotted on the ternary composition diagram. If the αL,H = 1 line intersects the lowerboiling pure component-solvent face, then the lower-boiling component will be recovered in the distillate and vice versa if the αL,H = 1 line

intersects the higher-boiling pure component-solvent face. A very simple method, if a rigorous residue curve map is available, is to examine the shape and inflection of the residue curves as they approach the pure solvent vertex. Whichever solvent-key component face the residue curves predominantly tend toward as they approach the solvent vertex is the key component that will be recovered in the bottoms with the solvent. In Fig. 13-73a, all residue curves approaching the water (solvent) vertex are inflected toward the methanol-water face, with the result that methanol will be recovered in the bottoms and acetone in the distillate. Alternatively, with MIPK as the solvent the residue curves (Fig. 13-73b), all residue curves show inflection toward the acetone-MIPK face, indicating that acetone will be recovered in the bottoms and methanol in the distillate. Extractive Distillation Design and Optimization Extractive distillation column composition profiles have a very characteristic

(a)

(b)

FIG. 13-72

Effect of solvent concentration on activity coefficients for acetone-methanol system. (a) water solvent. (b) MIPK solvent.

13-78

DISTILLATION

(a)

The variable that has the most significant impact on the economics of an extractive distillation is the solvent-to-feed (S/F) ratio. For closeboiling or pinched nonazeotropic mixtures, no minimum-solvent flow rate is required to effect the separation, as the separation is always theoretically possible (if not economical) in the absence of the solvent. However, the extent of enhancement of the relative volatility is largely determined by the solvent concentration and hence the S/F ratio. The relative volatility tends to increase as the S/F ratio increases. Thus, a given separation can be accomplished in fewer equilibrium stages. As an illustration, the total number of theoretical stages required as a function of S/F ratio is plotted in Fig. 13-75a for the separation of the nonazeotropic mixture of vinyl acetate and ethyl acetate using phenol as the solvent. For the separation of a minimum-boiling binary azeotrope by extractive distillation, there is clearly a minimum-solvent flow rate below which the separation is impossible (due to the azeotrope). For azeotropic separations, the number of equilibrium stages is infinite at or below (S/F)min and decreases rapidly with increasing solvent, and then may asymptote, or rise slowly. The relationship between the total number of stages and the S/F ratio for a given purity and recovery for the azeotropic acetone-methanol system with water as solvent is shown in Fig 13-75b. A rough idea of (S/F)min can be determined from a pseudobinary diagram or by plotting the αL,H = 1 line on a ternary diagram. The solvent composition at which the azeotrope disappears in a corner of the pseudobinary diagram is an indication of (S/F)min [LaRoche et al., Can. J. Chem. Eng., 69, 1302 (1991)]. Typically, operating S/F ratios for economically acceptable solvents is between two and five. Higher S/F ratios tend to increase the diameter of both the extractive column and the solvent-recovery columns, but reduce the required number of equilibrium stages and minimum-reflux ratio. Moreover, higher S/F ratios lead to higher reboiler temperatures, resulting in the use of higher-cost utilities, higher utility usages, and greater risk of degradation. Knight and Doherty [Ind. Eng. Chem. Fundam., 28, 564 (1989)] have published rigorous methods for computing minimum reflux for extractive distillation, with an operating reflux of 1.2 to 1.5 times the minimum value usually acceptable. Interestingly, unlike other forms of distillation, in extractive distillation the distillate purity or recovery does not increase monotonically with increasing reflux ratio for a given number of stages. Above a maximum-reflux ratio the separation can

(b) Residue curve maps for acetone-methanol systems. (a) With water. (b) With MIPK.

FIG. 13-73

shape on a ternary diagram. The composition profile for the separation of methanol-acetone with water is given in Fig. 13-74. Stripping and rectifying profiles start at the bottoms and distillate compositions respectively, track generally along the faces of the composition triangle, and then turn toward the high-boiling (solvent) node and low-boiling node, respectively. For a feasible single-feed design these profiles must cross at some point. However, in an extractive distillation they cannot cross. The extractive-section profile acts at the bridge between these two sections. Most of the key-component separation occurs in this section in the presence of high-solvent concentration.

Extractive distillation column composition profile for the separation of acetone-methanol with water.

FIG. 13-74

ENHANCED DISTILLATION

(a)

(b) FIG. 13-75 Number of theoretical stages versus solvent-to-feed ratio for extractive distillation. (a) Close-boiling vinyl acetate-ethyl acetate system with phenol solvent. (b) Azeotropic acetone-methanol system with water solvent.

no longer be achieved and the distillate purity actually decreases for a given number of stages [LaRoche et al., AIChE J., 38, 1309 (1992)]. The difference between Rmin and Rmax increases as the S/F ratio increases. Large amounts of reflux lowers the solvent concentration in the upper section of the column, degrading rather than enhancing column performance. Because the reflux ratio goes through a maximum, the conventional-control strategy of increasing reflux to maintain purity can be detrimental rather than beneficial. However Rmax generally occurs at impractically high reflux ratios and is typically not of major concern. The thermal quality of the solvent feed has no effect on the value of (S/F)min, but does affect the minimum reflux to some extent, especially as the (S/F) ratio increases. Rmax occurs at higher values of the reflux ratio as the upper-feed quality decreases; a subcooled upper feed provides additional refluxing capacity and less external reflux is required for the same separation. It is also sometimes advantageous to introduce the primary feed to the extractive distillation column as a vapor to help maintain a higher solvent concentration on the feed tray and the trays immediately below.

13-79

Robinson and Gilliland (Elements of Fractional Distillation, McGraw-Hill, New York, 1950), Smith (Design of Equilibrium Stage Processes, McGraw-Hill, New York, 1963), Van Winkle (Distillation, McGraw-Hill, New York, 1967), and Walas (Chemical Process Equipment, Butterworths, Boston, 1988) discuss rigorous stage-by-stage design techniques as well as shortcut and graphical methods for determining minimum stages, (S/F)min, minimum reflux, and the optimum locations of the solvent and primary feed points. More recently Knapp and Doherty [AIChE J., 40, 243 (1994)] have published columndesign methods based on geometric arguments and fixed-point analysis. Most commercial simulators are capable of solving multiple-feed extractive distillation heat and material balances, but do not include straightforward techniques for calculating (S/F)min, minimum or maximum reflux. Solvent Screening and Selection Choosing an effective solvent can have the most profound effect on the economics of an extractive distillation process. The approach most often adopted is to first generate a short list of potential solvents using simple qualitative screening and selection methods. Experimental verification is best undertaken only after a list of promising candidate solvents has been generated and some chance at economic viability has been demonstrated via preliminary process modeling. Solvent selection and screening approaches can be divided into two levels of analysis. The first level focuses on identification of functional groups or chemical families that are likely to give favorable solventkey component molecular interactions. The second level of analysis identifies and compares individual-candidate solvents. The various methods of analysis are described briefly and illustrated with an example of choosing a solvent for the methanol-acetone separation. First Level: Broad Screening by Functional Group or Chemical Family Homologous series. Select candidate solvents from the highboiling homologous series of both light and heavy key components. Favor homologs of the heavy key, as this tends to enhance the natural relative volatility of the system. Homologous components tend to form ideal solutions and are unlikely to form azeotropes [Scheibel, Chem. Eng. Prog. 44(12), 927 (1948)]. Robbins chart. Select candidate solvents from groups in the Robbins Chart (Table 13-15) that tend to give positive (or no) deviations from Raoult’s law for the key component desired in the distillate and negative (or no) deviations for the other key. Hydrogen-bonding characteristics. Select candidate solvents from groups that are likely to cause the formation of hydrogen bonds with the key component to be removed in the bottoms, or disruption of hydrogen bonds with the key to be removed in the distillate. Formation and disruption of hydrogen bonds are often associated with strong negative and positive deviations, respectively from Raoult’s law. Several authors have developed charts indicating expected hydrogen bonding interactions between families of compounds [Ewell et al., Ind. Eng. Chem., 36, 871 (1944), Gilmont et al., Ind. Eng. Chem., 53, 223 (1961), and Berg, Chem. Eng. Prog., 65(9), 52 (1969)]. Table 13-20 presents a hydrogen-bonding classification of chemical families and a summary of deviations from Raoult’s law. Polarity characteristics. Select candidate solvents from chemical groups that tend to show higher polarity than one key component or lower polarity than the other key. Polarity effects are often cited as a factor in causing deviations from Raoult’s law [Hopkins and Fritsch, Chem. Eng. Prog., 51(8), (1954), Carlson et al., Ind. Eng. Chem., 46, 350 (1954), and Prausnitz and Anderson, AIChE J., 7, 96 (1961)]. The general trend in polarity based on the functional group of a molecule is given in Table 13-21. The chart is best for molecules of similar size. A more quantitative measure of the polarity of a molecule is the polarity contribution to the three-term Hansen solubility parameter. A tabulation of calculated three-term solubility parameters is provided by Barton (CRC Handbook of Solubility Parameters and other Cohesion Parameters, CRC Press, Boca Raton, 1991), along with a groupcontribution method for calculating the three-term solubility parameters of compounds not listed in the reference. Second Level: Identification of Individual Candidate Solvents Boiling point characteristics. Select only candidate solvents that boil at least 30–40°C above the key components to ensure that the sol-

13-80

DISTILLATION

TABLE 13-20

Hydrogen Bonding Classification of Chemical Families

Class

Chemical family

H-Bonding, Strongly Associative (HBSA)

Water Primary amides Secondary amides

Polyacids Dicarboxylic acids Monohydroxy acids

Polyphenols Oximes Hydroxylamines

Amino alcohols Polyols

H-Bond Acceptor-Donor (HBAD)

Phenols Aromatic acids Aromatic amines Alpha H nitriles

Imines Monocarboxylic acids Other monoacids Peracids

Alpha H nitros Azines Primary amines Secondary amines

n-alcohols Other alcohols Ether alcohols

H-Bond Acceptor (HBA)

Acyl chlorides Acyl fluorides Hetero nitrogen aromatics Hetero oxygen aromatics

Tertiary amides Tertiary amines Other nitriles Other nitros Isocyanates Peroxides

Aldehydes Anhydrides Cyclo ketones Aliphatic ketones Esters Ethers

Aromatic esters Aromatic nitriles Aromatic ethers Sulfones Sulfolanes

π-Bonding Acceptor (π-HBA)

Alkynes Alkenes

Aromatics Unsaturated esters

H-Bond Donor (HBD)

Inorganic acids Active H chlorides

Active H fluorides Active H iodides

Active H bromides

Non-Bonding (NB)

Paraffins Nonactive H chlorides

Nonactive H fluorides Sulfides

Nonactive H iodides Disulfides

Nonactive H bromides Thiols

Deviations from Raoult’s Law H-Bonding classes

Type of deviations

Comments

HBSA + NB HBAD + NB

Alway positive dev., HBSA + NB often limited miscibility

H-bonds broken by interactions

HBA + HBD

Always negative dev.

H-bonds formed by interactions

HBSA + HBD HBAD + HBD

Always positive deviations, HBSA + HBD often limited miscibility

H-bonds broken and formed; dissociation of HBSA or HBAD liquid most important effect

HBSA + HBSA HBSA + HBAD HBSA + HBA HBAD + HBAD HBAD + HBA

Usually positive deviations; some give maximum-boiling azeotropes

H-bonds broken and formed

HBA + HBA HBA + NB HBD + HBD HBD + NB NB + NB

Ideal, quasi-ideal systems; always positive or no deviations; azeotropes, if any, minimum-boiling

No H-bonding involved

NOTE:

π-HBA is enhanced version of HBA.

vent is relatively nonvolatile and remains largely in the liquid phase. With this boiling point difference, the solvent should also not form azeotropes with the other components. Selectivity at infinite dilution. Rank candidate solvents according to their selectivity at infinite dilution. The selectivity at infinite dilution is defined as the ratio of the activity coefficients at infinite dilution of the two key components in the solvent. Since solvent effects tend to increase as solvent concentration increases, the infinite-dilution selectivity gives an upper bound on the efficacy of a solvent. Infinite-dilution activity coefficients can be predicted using such methods as UNIFAC, ASOG, MOSCED (Reid et al., Properties of Gases and Liquids, Fourth Edition, McGraw-Hill, New York, 1987). They can be found experimentally using a rapid gas-liquid chromatography method based on relative retention times in candidate solvents (Tassios in Extractive and Azeotropic Distillation, Advances in Chemistry Series 115, American Chemical Society, Washington, 1972) and they can be correlated to bubble-point data [Kojima and Ochi, J. Chem. Eng. Japan, 7(2), 71 (1974)]. DECHEMA [Vapor-Liquid Equilibrium Data Collection, Frankfort (1977)], has also published a compilation of experimental infinitedilution activity coefficients. Experimental measurement of relative volatility. Rank candidate solvents by the increase in relative volatility caused by the addition of the solvent. One technique is to experimentally measure the relative volatility of a fixed-composition key component-solvent mixture (often a 1/1 ratio of each key, with a 1/1 to 3/1 solvent/key ratio) for various solvents. [Carlson et al., Ind. Eng. Chem., 46, 350 (1954)]. The Oth-

mer equilibrium still is the apparatus of choice for these measurements [Zudkevitch, Chem. Eng. Comm., 116, 41 (1992)]. Methanol and acetone boil at 64.5°C and 56.1°C, respectively and form a minimum-boiling azeotrope at 55.3°C. The natural volatility of the system is acetone > methanol, so the favored solvents most likely will be those that cause the acetone to be recovered in the distillate. However, for the purposes of the example, a solvent that reverses the natural volatility will also be identified. First, examining the polarity of TABLE 13-21

Relative Polarities of Functional Groups

MOST POLAR

Water Organic acids Amines Polyols Alcohols Esters Ketones Aldehydes Ethers Aromatics Olefins Paraffins



  ↓ LEAST POLAR Effect of branching MOST POLAR LEAST POLAR

Normal Secondary Tertiary

ENHANCED DISTILLATION ketones and alcohols (Table 13-21), solvents favored for the recovery of methanol in the bottoms would come from groups more polar than methanol, such as acids, water, and polyols. Turning to the Robbins Chart (Table 13-15), favorable groups are amines, alcohols, polyols, and water since these show expected positive deviations for acetone and zero or negative deviations for methanol. For reversing the natural volatility, solvents should be chosen that are less polar than acetone, such as ethers, hydrocarbons, and aromatics. Unfortunately, both ethers and hydrocarbons are expected to give positive deviations for both acetone and methanol, so should be discarded. Halohydrocarbons and ketones are expected to give positive deviations for methanol and either negative or no deviations for acetone. The other qualitative indicators show that both homologous series (ketones and alcohols) look promising. Thus, after discounting halohydrocarbons for environmental reasons, the best solvents will probably come from alcohols, polyols, and water for recovering methanol in the bottoms and ketones for recovering acetone in the bottoms. Table 13-22 shows the boiling points and experimental or estimated infinite-dilution activity coefficients for several candidate solvents from the aforementioned groups. Methylethylketone boils too low, as does ethanol, and also forms an azeotrope with methanol. These two candidates can be discarded. Other members of the homologous series, along with water and ethylene glycol, have acceptable boiling points (at least 30°C higher than keys). Of these, water (the solvent used industrially) clearly has the largest effect on the activity coefficients, followed by ethylene glycol. Although inferior to water or ethylene glycol, both MIPK and MIBK would probably be acceptable for reversing the natural volatility of the system. Extractive Distillation by Salt Effects A second method of modifying the liquid-phase behavior (and thus the relative volatility) of a mixture in order to effect a separation is by the addition of a nonvolatile, soluble, ionic salt. The process is analogous to extractive distillation with a high-boiling liquid. In the simplest case, for the separation of a binary mixture, the salt is fed at the top of the column by dissolving it in the hot reflux stream before introduction into the column. In order to function effectively the salt must be adequately soluble in both components throughout the range of compositions encountered in the column. Since the salt is essentially completely nonvolatile, it remains in the liquid phase on each tray and alters the relative volatility throughout the length of the column. No rectification section is needed above the salt feed. The bottoms product is recovered from the salt solution by evaporation or drying, and the salt is recycled. The ions of a salt are typically capable of causing much larger and more selective effects on liquid-phase behavior than the molecules of a liquid solvent. As a result, salt-to-feed ratios less than 0.1 are typical. The use of a salting agent presents a number of problems not associated with a liquid solvent, such as the difficulty of transporting and metering a solid or saturated salt solution, slow mixing or dissolution rate of the salt, limits to solubility in the feed components, and potential for corrosion. However, in the limited number of systems for which an effective salt can be found, the energy usage, equipment size, capital investment, and ultimate separation cost can be significantly reduced compared to extractive distillation using a liquid solTABLE 13-22 Comparison of Candidate Solvents for Methanol/Acetone Extractive Distillation Boiling pt. (°C)

Azeotrope formation

∞ γ MeOH

∞ ∞ γ Acetone / γ MeOH

MEK MIPK MIBK

79.6 102.0 115.9

With MeOH No No

1.01 1.01 1.06

1.88 1.89 2.05

0.537 0.534 0.517

Ethanol 1-Propanol 1-Butanol Water

78.3 97.2 117.8 100.0

No No No No

1.85 1.90 1.93 11.77

1.04 1.20 1.33 2.34

1.78 1.58 1.45 5.03

EG

197.2

No

3.71

1.25

2.97

Solvent

∞ γ Acetone = 1.79 (in MeOH) ∞ γ MeOH = 1.81 (in acetone)

∞ γ Acetone

13-81

vent [Furter, Chem. Eng. Commun., 116, 35 (1992)]. Applications of salt extractive distillation include acetate salts to produce absolute ethanol, magnesium nitrate for the production of concentrated nitric acid as an alternative to the sulfuric-acid solvent process, and calcium chloride to produce anhydrous hydrogen chloride. Other examples are noted by Furter [Can. J. Chem. Eng., 55, 229 (1977)]. One problem limiting the consideration of salt extractive distillation is the fact that the performance and solubility of a salt in a particular system is difficult to predict without experimental data. Some recent advances have been made in modeling the VLE behavior of organicaqueous-salt solutions using modified UNIFAC, NRTL, UNIQUAC, and other approaches [Kumar, Sep. Sci. Tech., 28(1), 799 (1993)]. REACTIVE DISTILLATION Introduction Reactive distillation is a unit operation in which chemical reaction and distillative separation are carried out simultaneously within a fractional distillation apparatus. Reactive distillation may be advantageous for liquid-phase reaction systems when the reaction must be carried out with a large excess of one or more of the reactants, when a reaction can be driven to completion by removal of one or more of the products as they are formed, or when the product recovery or by-product recycle scheme is complicated or made infeasible by azeotrope formation. For consecutive reactions in which the desired product is formed in an intermediate step, excess reactant can be used to suppress additional series reactions by keeping the intermediate-species concentration low. A reactive distillation can achieve the same end by removing the desired intermediate from the reaction zone as it is formed. Similarly, if the equilibrium constant of a reversible reaction is small, high conversions can be achieved by use of a large excess of reactant. Alternatively, by Le Chatelier’s principle, the reaction can be driven to completion by removal of one or more of the products as they are formed. Typically, reactants can be kept much closer to stoichiometric proportions in a reactive distillation. When a reaction mixture exhibits azeotropism, the recovery of products and recycle of excess reagents can be quite complicated and expensive. Reactive distillation can provide a means of breaking azeotropes by altering or eliminating the condition for azeotrope formation in the reaction zone through the combined effects of vaporization-condensation and consumptionproduction of the species in the mixture. Alternatively, a reaction may be used to convert the species into components that are more easily distilled. In each of these situations, the conversion and selectivity often can be improved markedly, with much lower-reactant inventories and recycle rates, and much simpler recovery schemes. The capital savings can be quite dramatic. A list of applications of reactive distillation appearing in the literature is given in Table 13-23. Although reactive distillation has many potential applications, it is not appropriate for all situations. Since it is in essence a distillation process, it has the same range of applicability as other distillation operations. Distillation-based equipment is not designed to effectively handle solids, supercritical components (where no separate vapor and liquid phases exist), gas-phase reactions, or high-temperature or highpressure reactions such as hydrogenation, steam reforming, gasification, and hydrodealkylation. Simulation, Modeling, and Design Feasibility Because reaction and separation phenomena are closely coupled in a reactive distillation process, simulation and design is significantly more complex than that of sequential reaction and separation processes. In spite of the complexity, however, most commercial computer process modeling packages offer reliable and flexible routines for simulating steadystate reactive distillation columns, with either equilibrium or kinetically controlled reaction models. [Venkataraman et al., Chem. Eng. Prog., 86(6), 45 (1990)]. As with other enhanced distillation processes, the results are very sensitive to the thermodynamics model chosen and the accuracy of the VLE data used to generate model parameters. Of equal, if not more significance is the accuracy of data on reaction rate as a function of temperature. Very different conclusions can be drawn about the feasibility of a reactive distillation if the reaction is assumed to reach chemical equilibrium on each stage of the column or if the reaction is assumed to be kinetically controlled [Barbosa and Doherty, Chem. Eng. Sci., 43, 541 (1988)]. Tray holdup

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DISTILLATION

TABLE 13-23

Applications of Reactive Distillation Process

Reaction type

Methyl acetate from methanol and acetic acid General process for ester formation

Esterification Esterification

Dibutyl phthalate from butanol and phthalic acid Ethyl acetate from ethanol and butyl acetate Recovery of acetic acid and methanol from methyl acetate by-product of vinyl acetate production Nylon 6,6 prepolymer from adipic acid and hexamethylenediamine MTBE from isobutene and methanol TAME frompentenes and methanol Separation of close boiling 3- and 4-picoline by complexation with organic acids Separation of close-boiling meta and para xylenes by formation of tert-butyl meta-xyxlene Cumene from propylene and benzene General process for the alkylation of aromatics with olefins Production of specific higher and lower alkenes from butenes 4-Nitrochlorobenzene from chlorobenzene and nitric acid Production of methylal and high purity formaldehyde

Esterification Transesterification Hydrolysis Amidation

Jaswal and Pugi, U.S. Patent 3,900,450 (1975)

Etherification Etherification Acid-base

DeGarmo et al., Chem. Eng. Prog., 88(3), 43 (1992) Brockwell et al., Hyd. Proc., 70(9), 133 (1991) Duprat and Gau, Can. J. Chem. Eng., 69, 1320 (1991)

Transalkylation

Saito et al., J. Chem. Eng. Jap., 4, 37 (1971)

Alkylation Alkylation Diproportionation

Shoemaker and Jones, Hyd. Proc., 67(6), 57 (1987) Crossland, U.S. Patent 5,043,506 (1991) Jung et al., U.S. Patent 4,709,115 (1987)

Nitration

Belson, Ind. Eng. Chem. Res., 29, 1562 (1990) Masamoto and Matsuzaki, J. Chem. Eng. Jap., 27, 1 (1994)

and stage requirements are two important variables directly affected by the form of the reaction model chosen. When an equilibrium reaction occurs in a vapor-liquid system, the phase compositions depend not only on the relative volatility of the components in the mixture, but also on the consumption (and production) of species. Thus, the condition for azeotropy in a nonreactive system (yi = xi, for all i) no longer holds true in a reactive system and must be modified to include reaction stoichiometry: y1 − x1 yi − xi ᎏ=ᎏ v1 − vT x1 vi − vT xi

for all i = 1, n

(13-128)

n

vT =  vi

where

Reference Agreda et al., Chem. Eng. Prog., 86(2), 40 (1990) Simons, “Esterification” in Encyclopedia of Chemical Processing and Design, Vol 19, Dekker, New York, 1983 Berman et al., Ind. Eng. Chem., 40, 2139 (1948) Davies and Jeffreys, Trans. Inst. Chem. Eng., 51, 275 (1973) Fuchigami, J. Chem. Eng. Jap., 23, 354 (1990)

i=1

xi = mole fraction of component i in the liquid phase yi = mole fraction of component i in the vapor phase vi = stoichiometric coefficient of component i (negative for reactants, positive for products)

An equation identical to (13-129) defines the transformed mole fraction of component i in the vapor phase, Yi, where the terms in x are replaced by terms in y. The transformed variables describe the system composition with or without reaction and sum to unity as do xi and yi. The condition for azeotropy becomes Xi = Yi. Barbosa and Doherty have shown that phase and distillation diagrams constructed using the transformed composition coordinates have the same properties as phase and distillation region diagrams for nonreactive systems and similarly can be used to assist in design feasibility and operability studies [Chem. Eng. Sci., 43, 529, 1523, and 2377 (1988a,b,c)]. A residue curve map in transformed coordinates for the reactive system methanol-acetic acidmethyl acetate-water is shown in Fig. 13-76. Note that the nonreactive azeotrope between water and methyl acetate has disappeared, while the methyl acetate-methanol azeotrope remains intact. Only

Phase compositions that satisfy Eq. (13-128) are stationary points on a phase diagram and have been labeled reactive azeotropes by Barbosa and Doherty [Chem. Eng. Sci., 43, 529 (1988)]. At a reactive azeotrope the mass exchange between the vapor and liquid phase and the generation (or consumption) of each species is balanced such that the composition of neither phase changes. Reactive azeotropes show the same distillation properties as ordinary azeotropes and therefore affect what products are achievable. Reactive azeotropes are not easily visualized in conventional y-x coordinates but become apparent upon a transformation of coordinates which depends on the number of reactions, the order of each reaction (e.g., A + B ↔ C or A + B ↔ C + D), presence of nonreacting components, and the extent of reaction. The general vector-matrix form of the transform for C reacting components, with R reactions, and I nonreacting components has been derived by Ung and Doherty [Chem. Eng. Sci., 50, 23 (1995)]. For the transformed mole fraction of component i in the liquid phase, Xi, they give xi − viT(vRe f)−1xRe f , i = 1, . . . . C-R (13-129) Xi = ᎏᎏ 1 − vTTOT(vRe f)−1xRe f where viT = row vector of stoichiometric coefficients of component i for each reaction vRe f = square matrix of stoichiometric coefficients for R reference components in R reactions xRe f = column vector of mole fractions for the R reference components in the liquid phase vTTOT = row vector composed of the sum of the stoichiometric coefficients for each reaction





Residue curve map for the reactive system methanol-acetic acidmethyl acetate-water in chemical equilibrium.

FIG. 13-76

ENHANCED DISTILLATION those azeotropes containing either all the required reactants or products will be altered by the reaction (water and methyl acetate can back-react to form acetic acid and methanol, whereas methanol and methyl acetate cannot further react in the absence of either water or acetic acid). This reactive system consists of only one distillation region in which the methanol-methyl acetate azeotrope is the lowboiling and acetic acid is the high-boiling node. The situation becomes more complicated when the reaction is kinetically controlled and does not come to complete-chemical equilibrium under the conditions of temperature, liquid holdup, and rate of vaporization in the column reactor. Venimadhavan et al. [AIChE J., 40, 1814 (1994)] and Rev [Ind. Eng. Chem. Res., 33, 2174 (1994)] show that the existence and location of reactive azeotropes is a function of approach to equilibrium as well as the evaporation rate. Mechanical Design and Implementation Issues The choice of catalyst has a significant impact on the mechanical design and operation of the reactive column. The catalyst must allow the reaction to occur at reasonable rates at the relatively low temperatures and pressures common in distillation operations (typically less than 10 atmospheres and between 50°C and 250°C). Selection of a homogeneous catalyst, such as a high-boiling mineral acid, allows the use of more traditional tray designs and internals (albeit designed with allowance for high-liquid holdups). With a homogeneous catalyst, lifetime is not a problem, as it is added (and withdrawn) continuously. Alternatively, heterogeneous solid catalysts require either complicated mechanical means for continuous replenishment or relatively long lifetimes in order to avoid constant maintenance. As with other multiphase reactors, use of a solid catalyst adds an additional resistance to mass transfer from the bulk liquid (or vapor) to the catalyst surface, which may be the limiting resistance. The catalyst containment system must be designed to ensure adequate liquid-solid contacting and minimize bypassing. A number of specialized column internal designs, catalyst containment methods, and catalyst replenishment systems have been proposed for both homogeneous and heterogeneous catalysts. A partial list of these methods is given in Table 13-24. Heat management is another important consideration in the implementation of a reactive distillation process. Conventional TABLE 13-24

13-83

reactors for highly exothermic or endothermic reactions are often designed as modified shell-and-tube heat exchangers for efficient heat transfer. However, a trayed or packed distillation column is a rather poor mechanical design for the management of the heat of reaction. Although heat can be removed or added in the condenser or reboiler easily, the only mechanism for heat transfer in the column proper is through vaporization (or condensation). For highly exothermic reactions, a large excess of reactants may be required as a heat sink, necessitating high-reflux rates and larger-diameter columns to return the vaporized reactants back to the reaction zone. Often a prereactor of conventional design is used to accomplish most of the reaction and heat removal before feeding to the reactive column for final conversion, as exemplified in most processes for the production of tertiary amyl methyl ether (TAME) [Brockwell et al., Hyd. Proc., 70(9), 133 (1991)]. Highly endothermic reactions may require intermediate reboilers. None of these heat-management issues preclude the use of reactive distillation, but must be taken into account during the design phase. Comparison of heat of reaction and average heat of vaporization data for a system, as in Fig. 13-77, gives some indication of potential heat imbalances [Sundmacher et al., Chem. Eng. Comm., 127, 151 (1994)]. The heat neutral systems (−∆ Hreact ≈ ∆ Hvap(avg)) such as methyl acetate and other esters can be accomplished in one reactive column, whereas the MTBE and TAME processes, with higher heats of reaction than vaporization, often include an additional prereactor. One exception is the catalytic distillation process for cumene production, which is accomplished without a prereactor. Three moles of benzene reactant are vaporized (and refluxed) for every mole of cumene produced. The relatively high heat of reaction is advantageous in this case as it reduces the overall heat duty of the process by about 30 percent [Shoemaker and Jones, Hyd. Proc., 57(6), 57 (1987)]. Process Applications The production of esters from alcohols and carboxylic acids illustrates many of the principles of reactive distillation as applied to equilibrium-limited systems. The equilibrium constants for esterification reactions are usually relatively close to unity. Large excesses of alcohols must be used to obtain acceptable yields with large recycles. In a reactive-distillation scheme, the reac-

Catalyst Systems for Reactive Distillation Description

Application

Reference

Homogeneous catalysis Liquid-phase mineral-acid catalyst added to column or reboiler

Esterifications Dibutyl phlalate Methyl acetate

Keyes, Ind. Eng. Chem., 24, 1096 (1932) Berman et al., Ind. Eng. Chem., 40, 2139 (1948) Agreda et al., U.S. Patent 4,435,595 (1984)

Heterogeneous catalysis Catalyst-resin beads placed in cloth bags attached to fiberglass strip. Strip wound around helical stainless steel mesh spacer Ion exchange resin beads used as column packing

Etherifications Cumene

Smith et al., U.S. Patent 4,443,559 (1981) Shoemaker and Jones, Hyd. 57(6), 57 (1987)

Hydrolysis of methyl acetate

Molecular sieves placed in bags or porous containers Ion exchange resins formed into Raschig rings

Alkylation of aromatics MTBE

Granular catalyst resin loaded in corrugated sheet casings Trays modified to hold catalyst bed

Dimethyl acetals of formaldehyde MTBE

Distillation trays consturcted of porous catalytically active material and reinforcing resins Method described for removing or replacing catalyst on trays as a liquid slurry Catalyst bed placed in downcomer, designed to prevent vapor flow through bed Slotted plate for catalyst support designed with openings for vapor flow Ion exchanger fibers (reinforced ion exchange polymer) used as solid-acid catalyst High-liquid holdup trays designed with catalyst bed extending below tray level, perforated for vapor-liquid contact Catalyst bed placed in downcomer, in-line withdrawal/addition system

None specified

Fuchigami, J. Chem. Eng. Jap., 23, 354 (1990) Crossland, U.S. Patent 5,043,506 (1991) Flato and Hoffman, Chem. Eng. Tech., 15, 193 (1992) Zhang et al., Chinese Patent 1,065,412 (1992) Sanfilippo et al., Eur. Pat. Appl. EP 470,625 (1992) Wang et al., Chinese Patent 1,060,228 (1992)

None specified

Jones, U.S. Patent, 5,133,942 (1992)

Etherifications, alkylations

Asselineau, Eur. Pat. Appl. EP 547,939 (1993)

None specified

Evans and Stark, Eur. Pat. Appl. EP 571,163 (1993) Hirata et al., Jap. Patent 05,212,290 (1993)

Hydrolysis of methyl acetate None specified None specified

Yeoman et al., Int. Pat. Appl., WO 9408679 (1994) Carland, U.S. Patent, 5,308,451 (1994)

13-84

DISTILLATION ester-(alcohol) azeotrope. The esterification column is operated such that the low-boiling, water-laden azeotrope is taken as the distillation product. Upon cooling, the distillate separates into two liquid phases. The aqueous layer is steam-stripped, with the organics recycled to the decanter or reactor. The ester layer from the decanter contains some water and possibly alcohol. Part of this layer may be refluxed to the esterification column. The remainder is fed to a low-boiler column where the water-ester and alcohol-ester azeotropes are removed overhead and recycled to the decanter or reactor. The dry, alcoholfree ester is then optionally taken overhead in a final refining column. Methyl acetate cannot be produced in high purity using the simple esterification scheme described above. The methyl acetate-methanolwater system does not exhibit a ternary minimum-boiling azeotrope, the methyl acetate-methanol azeotrope is lower boiling than the watermethyl acetate azeotrope, a distillation boundary extends between these two binary azeotropes, and the heterogeneous region does not include either azeotrope, nor does it cross the distillation boundary. Consequently, the water of esterification cannot be removed effectively and methyl acetate cannot be separated from the methanol and water azeotropes by a simple decantation in the same manner as outlined above. Conventional sequential reaction-separation processes rely on large excesses of acetic acid to drive the reaction to higher conversion to methyl acetate, necessitating a capital- and energy-intensive acetic acidwater separation and large recycle streams. The crude methyl acetate product, contaminated with water and methanol, can be purified by a

FIG. 13-77 Similarity of heats of reaction and vaporization for compounds made by reactive distillation.

tion is driven to completion by removal of the water of esterification. The method used for removal of the water depends on the boiling points, compositions, and liquid-phase behavior of any azeotropes formed between the products and reactants and largely dictates the structure of the reactive-distillation flowsheet. When the ester forms a binary low-boiling azeotrope with water or a ternary alcohol-ester-water azeotrope and that azeotrope is heterogeneous (or can be moved into the two-liquid-phase region), the continuous flowsheet illustrated in Fig. 13-78 can be used. Such a flowsheet works for the production of ethyl acetate and higher homologs. In this process scheme, acetic acid and the alcohol are continuously fed to the reboiler of the esterification column, along with a homogeneous strong-acid catalyst. Since the catalyst is largely nonvolatile, the reboiler acts as the primary reaction site. The alcohol is usually fed in slight excess to ensure complete reaction of the acid and to compensate for alcohol losses through distillation of the water-

Flowsheet for exters which form a heterogeneous minimumboiling azeotrope with water.

FIG. 13-78

FIG. 13-79 Integrated reactive-extractive distillation column for the production of methyl acetate.

PETROLEUM AND COMPLEX-MIXTURE DISTILLATION

13-85

number of the enhanced distillation techniques such as pressure-swing distillation [Harrison, US Patent 2,704,271 (1955)], extractive distillation with ethylene glycol monomethylether as the solvent [Kumerle, German Patent 1,070,165 (1959)], or azeotropic distillation with an aromatic or ketone entrainer [Yeomans, Eur. Patent Appl. 060717 and 060719 (1982)]. The end result is a capital- and energy-intensive process typically requiring multiple reactors and distillation columns. The reactive-distillation process (Fig. 13-79) provides a mechanism for overcoming both the limitations on conversion due to chemical equilibrium as well as the difficulties in purification imposed by the water-methyl acetate and methanol-methyl acetate azeotropes [Agreda et al., Chem. Eng. Prog., 86(2), 40 (1990)]. Conceptually, this flowsheet can be thought of as four heat-integrated distillation columns (one of which is also a reactor) stacked on top of each other. The primary reaction zone consists of a series of countercurrent flashing stages in the middle of the column. Adequate residence time for the reaction is provided by high-liquid-holdup bubble-cap trays with specially designed downcomer sumps to further increase tray holdup. A nonvolatile homogeneous catalyst is fed at the top of the reactive section and exits with the underflow water by-product. The extractive-

distillation section, immediately above the reactive section, is critical in achieving high-methyl-acetate purity. As shown in Fig. 13-76, simultaneous reaction and distillation eliminates the water-methyl acetate azeotrope (and the distillation boundary of the nonreactive system). However, pure methyl acetate remains a saddle in the reactive system, and cannot be obtained as a pure component by simple reactive distillation. The acetic acid feed acts as a solvent in an extractive-distillation section placed above the reaction section, breaking the methanolmethyl acetate azeotrope, and yielding a pure methyl acetate distillate product. The uppermost rectification stages serve to remove any acetic acid from the methyl acetate product and the bottommost stripping section removes any methanol and methyl acetate from the water byproduct. The countercurrent flow of the reactants results in high local excesses at each end of the reactive section, even though the overall feed to the reactive column is stoichiometric. Therefore, the large excess of acetic acid at the top of the reactive section prevents methanol from reaching the distillate, while, similarly, methanol at the bottom of the reactive section keeps acetic acid from the water bottoms. Temperature and composition profiles for this reactive-extractive-distillation column are shown in Fig. 13-80a and b, respectively.

(a)

(b)

FIG. 13-80

Reactive extracting distillation for methyl acetate production. (a) Composition profile. (b) Temperature profile.

PETROLEUM AND COMPLEX-MIXTURE DISTILLATION INTRODUCTION Although the principles of multicomponent distillation apply to petroleum, synthetic crude oil, and other complex mixtures, this subject warrants special consideration for the following reasons: 1. Such feedstocks are of exceedingly complex composition, consisting, in the case of petroleum, of many different types of hydrocarbons and perhaps of inorganic and other organic compounds. The number of carbon atoms in the components may range from 1 to more than 50, so that the compounds may exhibit atmospheric-pressure boiling points from −162°C (−259°F) to more than 538°C (1000°F). In a given boiling range, the number of different compounds that exhibit only small differences in volatility multiplies rapidly with

increasing boiling point. For example, 16 of the 18 octane isomers boil within a range of only 12°C (22°F). 2. Products from the distillation of complex mixtures are in themselves complex mixtures. The character and yields of these products vary widely, depending upon the source of the feedstock. Even crude oils from the same locality may exhibit marked variations. 3. The scale of petroleum-distillation operations is generally large, and, as discussed in detail by Nelson (Petroleum Refinery Engineering, 4th ed., McGraw-Hill, New York, 1958) and Watkins (Petroleum Refinery Distillation, 2d ed., Gulf, Houston, 1979), such operations are common in several petroleum-refinery processes including atmospheric distillation of crude oil, vacuum distillation of bottoms residuum obtained from atmospheric distillation, main fractionation of gaseous

13-86

DISTILLATION

effluent from catalytic cracking of various petroleum fractions, and main fractionation of effluent from thermal coking of various petroleum fractions. These distillation operations are conducted in large pieces of equipment that can consume large quantities of energy. Therefore, optimization of design and operation is very important and frequently leads to a relatively complex equipment configuration. CHARACTERIZATION OF PETROLEUM AND PETROLEUM FRACTIONS Although much progress has been made in identifying the chemical species present in petroleum, it is generally sufficient for purposes of design and analysis of plant operation of distillation to characterize petroleum and petroleum fractions by gravity, laboratory-distillation curves, component analysis of light ends, and hydrocarbon-type analysis of middle and heavy ends. From such data, as discussed in the Technical Data Book—Petroleum Refining [American Petroleum Institute (API), Washington], five different average boiling points and an index of paraffinicity can be determined; these are then used to predict the physical properties of complex mixtures by a number of well-accepted correlations, whose use will be explained in detail and illustrated with examples. Many other characterizing properties or attributes such as sulfur content, pour point, water and sediment content, salt content, metals content, Reid vapor pressure, Saybolt Universal viscosity, aniline point, octane number, freezing point, cloud point, smoke point, diesel index, refractive index, cetane index, neutralization number, wax content, carbon content, and penetration are generally measured for a crude oil or certain of its fractions according to well-specified ASTM tests. But these attributes are of much less interest here even though feedstocks and products may be required to meet certain specified values of the attributes. Gravity of a crude-oil or petroleum fraction is generally measured by the ASTM D 287 test or the equivalent ASTM D 1298 test and may be reported as specific gravity (SG) 60/60°F [measured at 60°F (15.6°C) and referred to water at 60°F (15.6°C)] or, more commonly, as API gravity, which is defined as API gravity = 141.5/(SG 60/60°F) − 131.5 (13-130) Water, thus, has an API gravity of 10.0, and most crude oils and petroleum fractions have values of API gravity in the range of 10 to 80. Light hydrocarbons (n-pentane and lighter) have values of API gravity ranging upward from 92.8. The volatility of crude-oil and petroleum fractions is characterized in terms of one or more laboratory distillation tests that are summarized in Table 13-25. The ASTM D 86 and D 1160 tests are reasonably rapid batch laboratory distillations involving the equivalent of approximately one equilibrium stage and no reflux except for that caused by heat losses. Apparatus typical of the D 86 test is shown in Fig. 13-81 and consists of a heated 100-mL or 125-mL Engler flask containing a calibrated thermometer of suitable range to measure the temperature of the vapor at the inlet to the condensing tube, an inclined brass condenser in a cooling bath using a suitable coolant, and a graduated cylinder for collecting the distillate. A stem correction is not applied to the temperature reading. Related tests using similar apparatuses are the D 216 test for natural gasoline and the Engler distillation. TABLE 13-25

In the widely used ASTM D 86 test, 100 mL of sample is charged to the flask and heated at a sufficient rate to produce the first drop of distillate from the lower end of the condenser tube in from 5 to 15 min, depending on the nature of the sample. The temperature of the vapor at that instant is recorded as the initial boiling point (IBP). Heating is continued at a rate such that the time from the IBP to 5 volume percent recovered of the sample in the cylinder is 60 to 75 s. Again, vapor temperature is recorded. Then, successive vapor temperatures are recorded for from 10 to 90 percent recovered in intervals of 10, and at 95 percent recovered, with the heating rate adjusted so that 4 to 5 mL are collected per minute. At 95 percent recovered, the burner flame is increased if necessary to achieve a maximum vapor temperature referred to as the end point (EP) in from 3 to 5 additional min. The percent recovery is reported as the maximum percent recovered in the cylinder. Any residue remaining in the flask is reported as percent residue, and percent loss is reported as the difference between 100 mL and the sum of the percent recovery and percent residue. If the atmosphere test pressure P is other than 101.3 kPa (760 torr), temperature readings may be adjusted to that pressure by the Sidney Young equation, which for degrees Fahrenheit is T760 = TP + 0.00012(760 − P)(460 + TP)

(13-131)

Another pressure correction for percent loss can also be applied, as described in the ASTM test method. Results of a typical ASTM distillation test for an automotive gasoline are given in Table 13-26, in which temperatures have already been corrected to a pressure of 101.3 kPa (760 torr). It is generally assumed that percent loss corresponds to volatile noncondensables that are distilled off at the beginning of the test. In that case, the percent recovered values in Table 13-26 do not correspond to percent evaporated values, which are of greater scientific value. Therefore, it is common to adjust the reported temperatures according to a linear interpolation procedure given in the ASTM test method to obtain corrected temperatures in terms of percent evaporated at the standard intervals as included in Table 13-26. In the example, the corrections are not large because the loss is only 1.5 volume percent. Although most crude petroleum can be heated to 600°F (316°C) without noticeable cracking, when ASTM temperatures exceed 475°F (246°C), fumes may be evolved, indicating decomposition, which may cause thermometer readings to be low. In that case, the following correction attributed to S. T. Hadden may be applied: ∆ Tcorr = 10−1.587 + 0.004735T where

T = measured temperature, °F ∆ Tcorr = correction to be added to T, °F

At 500 and 600°F (260 and 316°C), the corrections are 6 and 18°F (3.3 and 10°C) respectively. As discussed by Nelson (op. cit.), virtually no fractionation occurs in an ASTM distillation. Thus, components in the mixture do distill one by one in the order of their boiling points but as mixtures of successively higher boiling points. The IBP, EP, and intermediate points have little theoretical significance, and, in fact, components boiling below the IBP and above the EP are present in the sample. Never-

Laboratory Distillation Tests

Test name

Reference

ASTM (atmospheric)

ASTM D 86

ASTM [vacuum, often 10 torr (1.3 kPa)]

ASTM D 1160

TBP [atmospheric or 10 torr (1.3 kPa)] Simulated TBP (gas chromatography) EFV (atmospheric, superatmospheric, or subatmospheric)

Nelson,* ASTM D 2892 ASTM D 2887 Nelson†

Main applicability Petroleum fractions or products, including gasolines, turbine fuels, naphthas, kerosines, gas oils, distillate fuel oils, and solvents that do not tend to decompose when vaporized at 760 mmHg Heavy petroleum fractions or products that tend to decompose in the ASTM D 86 test but can be partially or completely vaporized at a maximum liquid temperature of 750°F (400°C) at pressures down to 1 torr (0.13 kPa) Crude oil and petroleum fractions Crude oil and petroleum fractions Crude oil and petroleum fractions

*Nelson, Petroleum Refinery Engineering, 4th ed., McGraw-Hill, New York, 1958, pp. 95–99. †Ibid., pp. 104–105.

PETROLEUM AND COMPLEX-MIXTURE DISTILLATION

FIG. 13-82 FIG. 13-81 ASTM distillation apparatus; detail of distilling flask is shown in the upper figure.

theless, because ASTM distillations are quickly conducted, have been successfully automated, require only a small sample, and are quite reproducible, they are widely used for comparison and as a basis for specifications on a large number of petroleum intermediates and products, including many solvents and fuels. Typical ASTM curves for several such products are shown in Fig. 13-82. TABLE 13-26 Typical ASTM D 86 Test Results for Automobile Gasoline Pressure, 760 torr (101.3 kPa) Percent recovered basis (as measured) Percent recovered

T, °F

0(IBP) 5 10 20 30 40 50 60 70 80 90 95 EP

98 114 120 150 171 193 215 243 268 300 340 368 408

Percent evaporated basis (as corrected)

Percent evaporated

Percent evaporated

1.5 6.5 11.5 21.5 31.5 41.5 51.5 61.5 71.5 81.5 91.5 96.5

1.5 5 10 20 30 40 50 60 70 80 90 95

T, °F

Percent recovered

98 109 118 146 168 190 212 239 264 295 334 360 408

(IBP) 3.5 8.5 18.5 28.5 38.5 48.5 58.5 68.5 78.5 88.5 93.5 (EP)

NOTE: Percent recovery = 97.5; percent residue = 1.0; percent loss = 1.5. To convert degrees Fahrenheit to degrees Celsius, °C = (°F − 32)/1.8.

13-87

Representative ASTM D 86 distillation curves.

Data from a true-boiling-point (TBP) distillation test provides a much better theoretical basis for characterization. If the sample contains compounds that have moderate differences in boiling points such as in a light gasoline containing light hydrocarbons (e.g., isobutane, n-butane, isopentane, etc.), a plot of overhead-vapor-distillate temperature versus percent distilled in a TBP test would appear in the form of steps as in Fig. 13-83. However, if the sample has a higher average boiling range when the number of close-boiling isomers increases, the steps become indistinct and a TBP curve such as that in Fig. 13-84 results. Because the degree of separation for a TBP distillation test is much higher than for an ASTM distillation test, the IBP is lower and the EP is higher for the TBP method as compared with the ASTM method, as shown in Fig. 13-84. A standard TBP laboratory-distillation-test method has not been well accepted. Instead, as discussed by Nelson (op. cit., pp. 95–99),

FIG. 13-83 Variation of boiling temperature with percent distilled in trueboiling-point distillation of light hydrocarbons.

13-88

FIG. 13-84

DISTILLATION

Comparison of ASTM, TBP, and EFV distillation curves for kero-

sine.

batch distillation equipment that can achieve a good degree of fractionation is usually considered suitable. In general, TBP distillations are conducted in columns with 15 to 100 theoretical stages at reflux ratios of 5 or greater. Thus, the new ASTM D 2892 test method, which involves a column with from 14 to 17 theoretical stages and a reflux ratio of 5, essentially meets the minimum requirements. Distillate may be collected at a constant or a variable rate. Operation may be at 101.3-kPa (760 torr) pressure or at a vacuum at the top of the column as low as 0.067 kPa (0.5 torr) for high-boiling fractions, with 1.3 kPa (10 torr) being common. Results from vacuum operation are extrapolated to 101.3 kPa (760 torr) by the vapor-pressure correlation of Maxwell and Bonner [Ind. Eng. Chem., 49, 1187 (1957)], which is given in great detail in the API Technical Data Book—Petroleum Refining (op. cit.) and in the ASTM D 2892 test method. It includes a correction for the nature of the sample (paraffin, olefin, napthene, and aromatic content) in terms of the UOP characterization factor, UOP-K, as given by UOP-K = (TB)1/3/SG

define initial and final boiling points. The ASTM D 2887 standard test method is based on such a simulated distillation and is applicable to samples having a boiling range greater than 55°C (100°F) for temperature determinations as high as 538°C (1000°F). Typically, the test is conducted with a gas chromatograph having a thermal-conductivity detector, a programmed temperature capability, helium or hydrogen carrier gas, and column packing of silicone gum rubber on a crushedfire-brick or diatomaceous-earth support. It is important to note that simulated distillation does not always separate hydrocarbons in the order of their boiling point. For example, high-boiling multiple-ring-type compounds may be eluted earlier than normal paraffins (used as the calibration standard) of the same boiling point. Gas chromatography is also used in the ASTM D 2427 test method to determine quantitatively ethane through pentane hydrocarbons. A third fundamental type of laboratory distillation, which is the most tedious to perform of the three types of laboratory distillations, is equilibrium-flash distillation (EFV), for which no standard test exists. The sample is heated in such a manner that the total vapor produced remains in contact with the total remaining liquid until the desired temperature is reached at a set pressure. The volume percent vaporized at these conditions is recorded. To determine the complete flash curve, a series of runs at a fixed pressure is conducted over a range of temperature sufficient to cover the range of vaporization from 0 to 100 percent. As seen in Fig. 13-84, the component separation achieved by an EFV distillation is much less than by the ASTM or TBP distillation tests. The initial and final EFV points are the bubble point and the dew point respectively of the sample. If desired, EFV curves can be established at a series of pressures. Because of the time and expense involved in conducting laboratory distillation tests of all three basic types, it has become increasingly common to use empirical correlations to estimate the other two distillation curves when either the ASTM, TBP, or EFV curve is available. Preferred correlations given in the API Technical Data Book—Petroleum Refining (op. cit.) are based on the work of Edmister and Pollock [Chem. Eng. Prog., 44, 905 (1948)], Edmister and Okamoto [Pet. Refiner, 38(8), 117 (1959); 38(9), 271 (1959)], Maxwell (Data Book on

(13-132)

where TB = mean average boiling point, °R, which is the arithmetic average of the molal average boiling point and the cubic volumetric average boiling point. Values of UOP-K for n-hexane, 1-hexene, cyclohexene, and benzene are 12.82, 12.49, 10.99, and 9.73 respectively. Thus, paraffins with their lower values of specific gravity tend to have high values, and aromatics tend to have low values of UOP-K. A movement toward an international TBP standard is discussed by Vercier and Mouton [Oil Gas J., 77(38), 121 (1979)]. A crude-oil assay always includes a whole crude API gravity and a TBP curve. As discussed by Nelson (op. cit., pp. 89–90) and as shown in Fig. 13-85, a reasonably consistent correlation (based on more than 350 distillation curves) exists between whole crude API gravity and the TBP distillation curve at 101.3 kPa (760 torr). Exceptions not correlated by Fig. 13-85 are highly paraffinic or naphthenic crude oils. An alternative to TBP distillation is simulated distillation by gas chromatography. As described by Green, Schmauch, and Worman [Anal. Chem., 36, 1512 (1965)] and Worman and Green [Anal. Chem., 37, 1620 (1965)], the method is equivalent to a 100theoretical-plate TBP distillation, is very rapid, reproducible, and easily automated, requires only a small microliter sample, and can better

FIG. 13-85 Average true-boiling-point distillation curves of crude oils. (From W. E. Edmister, Applied Hydrocarbon Thermodynamics, vol. 1, 1st ed., 1961 Gulf Publishing Company, Houston, Texas, Used with permission. All rights reserved.)

PETROLEUM AND COMPLEX-MIXTURE DISTILLATION Hydrocarbons, Van Nostrand, Princeton, N.J., 1950), and Chu and Staffel [ J. Inst. Pet., 41, 92 (1955)]. Because of the lack of sufficiently precise and consistent data on which to develop the correlations, they are, at best, first approximations and should be used with caution. Also, they do not apply to mixtures containing only a few components of widely different boiling points. Perhaps the most useful correlation of the group is Fig. 13-86 for converting between ASTM D 86 and TBP distillations of petroleum fractions at 101.3 kPa (760 torr). The ASTM D 2889 test method, which presents a standard method for calculating EFV curves from the results of an ASTM D 86 test for a petroleum fraction having a 10 to 90 volume percent boiling range of less than 55°C (100°F), is also quite useful. APPLICATIONS OF PETROLEUM DISTILLATION Typical equipment configurations for the distillation of crude oil and other complex hydrocarbon mixtures in a crude unit, a catalyticcracking unit, and a delayed-coking unit of a petroleum refinery are shown in Figs. 13-87, 13-88, and 13-89. The initial separation of crude oil into fractions is conducted in two main columns, shown in Fig. 13-87. In the first column, called the atmospheric tower or topping still, partially vaporized crude oil, from which water, sediment, and salt have been removed, is mainly rectified, at a feed-tray pressure of no more than about 276 kPa (40 psia), to yield a noncondensable lighthydrocarbon gas, a light naphtha, a heavy naphtha, a light distillate (kerosine), a heavy distillate (diesel oil), and a bottoms residual of components whose TBP exceeds approximately 427°C (800°F). Alternatively, other fractions, shown in Fig. 13-82, may be withdrawn. To control the IBP of the ASTM D 86 curves, each of the sidestreams of the atmospheric tower and the vacuum and main fractionators of Figs. 13-87, 13-88, and 13-89 may be sent to side-cut strippers, which use a partial reboiler or steam stripping. Additional stripping by steam is commonly used in the bottom of the atmospheric tower as well as in the vacuum tower and other main fractionators. Additional distillate in the TBP range of approximately 427 to 593°C (800 to 1100°F) is recovered from bottoms residuum of the

13-89

atmospheric tower by rectification in a vacuum tower, also shown in Fig. 13-87, at the minimum practical overhead condenser pressure, which is typically 1.3 kPa (10 torr). Use of special low-pressure-drop trays or column packing permits feed-tray pressure to be approximately 5.3 to 6.7 kPa (40 to 50 torr) to obtain the maximum degree of vaporization. Vacuum towers may be designed or operated to produce several different products including heavy distillates, gas-oil feedstocks for catalytic cracking, lubricating oils, bunker fuel, and bottoms residua of asphalt (5 to 8° API gravity) or pitch (0 to 5° API gravity). The catalytic-cracking process of Fig. 13-88 produces a superheated vapor at approximately 538°C (1000°F) and 172 to 207 kPa (25 to 30 psia) of a TBP range that covers hydrogen to compounds with normal boiling points above 482°C (900°F). This gas is sent directly to a main fractionator for rectification to obtain products that are typically gas and naphtha [204°C (400°F) ASTM EP approximately], which are often fractionated further to produce relatively pure light hydrocarbons and gasoline; a light cycle oil [typically 204 to 371°C (400 to 700°F) ASTM D 86 range], which may be used for heating oil, hydrocracked, or recycled to the catalytic cracker; an intermediate cycle oil [typically 371 to 482°C (700 to 900°F) ASTM D 86 range], which is generally recycled to the catalytic cracker to extinction; and a heavy gas oil or bottom slurry oil. Vacuum-column bottoms, bottoms residuum from the main fractionation of a catalytic cracker, and other residua can be further processed at approximately 510°C (950°F) and 448 kPa (65 psia) in a delayed coker unit, as shown in Fig. 13-89 to produce petroleum coke and gas of TBP range that covers methane (with perhaps a small amount of hydrogen) to compounds with normal boiling points that may exceed 649°C (1200°F). The gas is sent directly to a main fractionator that is similar to the type used in conjunction with a catalytic cracker, except that in the delayed-coking operation the liquid to be coked first enters into and passes down through the bottom trays of the main fractionator to be preheated by and to scrub coker vapor of entrained coke particles and condensables for recycling to the delayed coker. Products produced from the main fractionator are similar, except for more unsaturated cyclic compounds, to those produced in a catalytic-cracking unit and include gas and coker naphtha, which are further processed to separate out light hydrocarbons and a coker naphtha that generally needs hydrotreating; and light and heavy coker gas oils, both of which may require hydrocracking to become suitable blending stocks. DESIGN PROCEDURES

FIG. 13-86 Relationship between ASTM and TBP distillation curves. (From W. C. Edmister, Applied Hydrocarbon Thermodynamics, vol. 1, 1st ed., 1961 Gulf Publishing Company, Houston, Texas. Used with permission. All rights reserved.)

Two general procedures are available for designing fractionators that process petroleum, synthetic crude oils, and complex mixtures. The first, which was originally developed for crude units by Packie [Trans. Am. Inst. Chem. Eng. J., 37, 51 (1941)], extended to main fractionators by Houghland, Lemieux, and Schreiner [Proc. API, sec. III, Refining, 385 (1954)], and further elaborated and described in great detail by Watkins (op. cit.), utilizes material and energy balances, with empirical correlations to establish tray requirements, and is essentially a hand-calculation procedure that is a valuable learning experience and is suitable for preliminary designs. Also, when backed by sufficient experience from previous designs, this procedure is adequate for final design. In the second procedure, which is best applied with a digital computer, the complex mixture being distilled is represented by actual components at the light end and by perhaps 30 pseudo components (e.g., petroleum fractions) over the remaining portion of the TBP distillation curve for the column feed. Each of the pseudo components is characterized by a TBP range, an average normal boiling point, an average API gravity, and an average molecular weight. Rigorous material-balance, energy-balance, and phase equilibrium calculations are then made by an appropriate equationtearing method as shown by Cecchetti et al. [Hydrocarbon Process., 42(9), 159 (1963)] or a simultaneous-correction procedure as shown, e.g., by Goldstein and Stanfield [Ind. Eng. Chem. Process Des. Dev., 9, 78 (1970) and Hess et al. [Hydrocarbon Process., 56(5), 241 (1977)]. Highly developed procedures of the latter type, suitable for preliminary or final design are included in most com-

13-90

DISTILLATION

FIG. 13-87

Crude unit with atmospheric and vacuum towers. [Kleinschrodt and Hammer, “Exchange Networks for Crude Units”. Chem. Eng. Prog., 79(7),

33 (1983).]

puter-aided steady-state process design and simulation programs as a special case of interlinked distillation, wherein the crude tower or fractionator is converged simultaneously with the sidecut-stripper columns. Regardless of the procedure used, certain initial steps must be taken for the determination or specification of certain product properties and yields based on the TBP distillation curve of the column feed, method of providing column reflux, column-operating pressure, type of condenser, and type of side-cut strippers and stripping requirements. These steps are developed and illustrated with several detailed examples by Watkins (op. cit.). Only one example, modified from one given by Watkins, is considered briefly here to indicate the approach taken during the initial steps. For the atmospheric tower shown in Fig. 13-90, suppose distillation specifications are as follows: Feed: 50,000 bbl (at 42 U.S. gal each) per stream day (BPSD) of 31.6° API crude oil. Measured light-ends analysis of feed: Component

Volume percent of crude oil

Ethane Propane Isobutane n-Butane Isopentane n-Pentane

0.04 0.37 0.27 0.89 0.77 1.13 3.47

Measured TBP and API gravity of feed, computed atmospheric pressure EFV (from API Technical Data Book), and molecular weight of feed: Volume percent vaporized

TBP, °F

EFV, °F

°API

Molecular weight

0 5 10 20 30 40 50 60 70 80

−130 148 213 327 430 534 639 747 867 1013

179 275 317 394 468 544 619 696 777 866

75.0 61.3 50.0 41.8 36.9 30.7 26.3 22.7 19.1

91 106 137 177 223 273 327 392 480

Product specifications: ASTM D 86, °F Desired cut

5%

50%

Overhead (OV) Heavy naphtha (HN) Light distillate (LD) Heavy distillate (HD) Bottoms (B)

278 398 546

314 453 589

NOTE:

95% 253 363 536

To convert degrees Fahrenheit to degrees Celsius, °C = (°F − 32)/1.8.

PETROLEUM AND COMPLEX-MIXTURE DISTILLATION

FIG. 13-88

Catalytic cracking unit. (New Horizons, Lummus Co., New York, 1954.)

FIG. 13-89

Delayed-coking unit. (Watkins, Petroleum Refinery Distillation, 2nd ed., Gulf, Houston, 1979.)

13-91

13-92

DISTILLATION

FIG. 13-90

Crude atmospheric tower.

TBP cut point between the heavy distillate and the bottoms = 650°F. Percent overflash = 2 volume percent of feed. Furnace outlet temperature = 343°C (650°F) maximum. Overhead temperature in reflux drum = 49°C (120°F) minimum. From the product specifications, distillate yields are computed as follows: From Fig. 13-86 and the ASTM D 86 50 percent temperatures, TBP 50 percent temperatures of the three intermediate cuts are obtained as 155, 236, and 316°C (311, 456, and 600°F) for the HN, LD, and HD respectively. The TBP cut points, corresponding volume fractions of crude oil, and flow rates of the four distillates are readily obtained by starting from the specified 343°C (650°F) cut point as follows, where CP is the cut point and T is the TBP temperature (°F): CPHD,B = 650°F (CPHD,B − THD50) = 650 − 600 = 50°F CPLD,HD = THD50 − 50 = 600 − 50 = 550°F (CPLD,HD − TLD50) = 550 − 456 = 94°F CPHN,LD = TLD50 − 94 = 456 − 94 = 362°F (CPHN,LD − THN50) = 362 − 311 = 51°F CPOV,HN = THN50 − 51 = 311 − 51 = 260°F These cut points are shown as vertical lines on the crude-oil TBP plot of Fig. 13-91, from which the following volume fractions and flow rates of product cuts are readily obtained: Desired cut

Volume percent of crude oil

BPSD

Overhead (OV) Heavy naphtha (HN) Light distillate (LD) Heavy distillate (HD) Bottoms (B)

13.4 10.3 17.4 10.0 48.9 100.0

6,700 5,150 8,700 5,000 24,450 50,000

FIG. 13-91

Example of crude-oil TBP cut points.

As shown in Fig. 13-92, methods of providing column reflux include (a) conventional top-tray reflux, (b) pump-back reflux from side-cut strippers, and (c) pump-around reflux. The latter two methods essentially function as intercondenser schemes that reduce the top-trayreflux requirement. As shown in Fig. 13-93 for the example being considered, the internal-reflux flow rate decreases rapidly from the top tray to the feed-flash zone for case a. The other two cases, particularly case c, result in better balancing of the column-reflux traffic. Because of this and the opportunity provided to recover energy at a moderate- to high-temperature level, pump-around reflux is the most commonly used technique. However, not indicated in Fig. 13-93 is the fact that in cases b and c the smaller quantity of reflux present in the upper portion of the column increases the tray requirements. Furthermore, the pump-around circuits, which extend over three trays each, are believed to be equivalent for mass-transfer purposes to only one tray each. Representative tray requirements for the three cases are included in Fig. 13-92. In case c heat-transfer rates associated with the two pump-around circuits account for approximately 40 percent of the total heat removed in the overhead condenser and from the two pump-around circuits combined. Bottoms and three side-cut strippers remove light ends from products and may utilize steam or reboilers. In Fig. 13-92 a reboiled stripper is utilized on the light distillate, which is the largest side cut withdrawn. Steam-stripping rates in side-cut strippers and at the bottom of the atmospheric column may vary from 0.45 to 4.5 kg (1 to 10 lb) of steam per barrel of stripped liquid, depending on the fraction of stripper feed liquid that is vaporized. Column pressure at the reflux drum is established so as to condense totally the overhead vapor or some fraction thereof. Flash-zone pressure is approximately 69 kPa (10 psia) higher. Crude-oil feed temper-

PETROLEUM AND COMPLEX-MIXTURE DISTILLATION

(a) FIG. 13-92

(b)

13-93

(c)

Methods of providing reflux to crude units. (a) Top reflux. (b) Pump-back reflux. (c) Pump-around reflux.

ature at flash-zone pressure must be sufficient to vaporize the total distillates plus the overflash, which is necessary to provide reflux between the lowest sidestream-product draw-off tray and the flash zone. Calculations are made by using the crude-oil EFV curve corrected for pressure. For the example being considered, percent vaporized at the flash zone must be 53.1 percent of the feed. Tray requirements depend on internal-reflux ratios and ASTM 5-95 gaps or overlaps, and may be estimated by the correlation of Packie (op. cit.) for crude units and the correlation of Houghland, Lemieux, and Schreiner (op. cit.) for main fractionators. Example 9: Simulation Calculation of an Atmospheric Tower The ability of a rigorous calculation procedure to simulate operation of an

FIG. 13-93

atmospheric tower with its accompanying side-cut strippers may be illustrated by comparing commercial-test data from an actual operation with results computed with the REFINE program of ChemShare Corporation, Houston, Texas. The tower configuration and plant-operating conditions are shown in Fig. 13-94. Light-component analysis and the TBP and API gravity for the feed are given in Table 13-27. Representation of this feed by pseudocomponents is given in Table 13-28 based on 16.7°C (30°F) cuts from 82 to 366°C (180°F to 690°F), followed by 41.7°C (75°F) and then 55.6°C (100°F) cuts. Actual tray numbers are shown in Fig. 13-94. Corresponding theoretical-stage numbers, which were determined by trial and error to obtain a reasonable match of computed- and measured-product TBP distillation curves, are shown in parentheses. Overall tray efficiency appears to be approximately 70 percent for the tower and 25 to 50 percent for the side-cut strippers. Results of rigorous calculations and comparison to plant data, when possible, are shown in Figs. 13-95, 13-96, and 13-97. Plant temperatures are in good

Comparison of internal-reflux rates for three methods of providing reflux.

13-94

DISTILLATION

TABLE 13-27 Light-Component Analysis and TBP Distillation of Feed for the Atmospheric Crude Tower of Fig. 13-94 Light-component analysis Component

Volume percent

Methane Ethane Propane n-Butane n-Pentane

0.073 0.388 0.618 0.817 2.05 TBP distillation of feed

NOTE:

API gravity

TBP, °F

Volume percent

80 70 57.5 45. 36. 29. 26.5 23. 20.5 17. 10. −4. −22.

−160. 155. 242. 377. 499. 609. 707. 805. 907. 1054. 1210. 1303. 1467

0.1 5. 10. 20. 30. 40. 50. 60. 70. 80. 90. 95. 100.

To convert degrees Fahrenheit to degrees Celsius, °C = (°F − 32)/1.8.

TABLE 13-28 Pseudo-Component Representation of Feed for the Atmospheric Crude Tower of Fig. 13-94 No.

Component name

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

Water Methane Ethane Propane n-Butane n-Pentane 131 ABP 180 ABP 210 ABP 240 ABP 270 ABP 300 ABP 330 ABP 360 ABP 390 ABP 420 ABP 450 ABP 480 ABP 510 ABP 540 ABP 570 ABP 600 ABP 630 ABP 660 ABP 690 ABP 742 ABP 817 ABP 892 ABP 967 ABP 1055 ABP 1155 ABP 1255 ABP 1355 ABP 1436 ABP

NOTE:

Molecular weight

Specific gravity

API gravity

(lb⋅mol)/h

18.02 16.04 30.07 44.09 58.12 72.15 83.70 95.03 102.23 109.78 118.52 127.69 137.30 147.33 157.97 169.37 181.24 193.59 206.52 220.18 234.31 248.30 265.43 283.37 302.14 335.94 387.54 446.02 509.43 588.46 665.13 668.15 643.79 597.05 246.90

1.0000 .3005 .3561 .5072 .5840 .6308 .6906 .7152 .7309 .7479 .7591 .7706 .7824 .7946 .8061 .8164 .8269 .8378 .8483 .8581 .8682 .8804 .8846 .8888 .8931 .9028 .9177 .9288 .9398 .9531 .9829 1.0658 1.1618 1.2533 .8887

10.0 339.5 265.8 147.5 110.8 92.8 73.4 66.3 62.1 57.7 54.9 52.1 49.4 46.6 44.0 41.8 39.6 37.4 35.3 33.4 31.5 29.2 28.5 27.7 26.9 25.2 22.7 20.8 19.1 17.0 12.5 1.3 −9.7 −18.6 27.7

.00 7.30 24.54 37.97 43.84 95.72 74.31 66.99 65.83 70.59 76.02 71.62 67.63 64.01 66.58 63.30 59.92 56.84 59.05 56.77 53.97 52.91 54.49 51.28 48.33 109.84 94.26 74.10 50.27 57.12 50.59 45.85 29.39 21.19 1922.43

To convert (lb⋅mol)/h to (kg⋅mol)/h, multiply by 0.454.

FIG. 13-94 Configuration and conditions for the simulation of the atmospheric tower of crude unit.

Comparison of computed stage temperatures with plant data for the example of Fig. 13-94.

FIG. 13-95

PETROLEUM AND COMPLEX-MIXTURE DISTILLATION agreement with computed values in Fig. 13-95. Computed sidestream-product TBP distillation curves are in reasonably good agreement with values converted from plant ASTM distillations as shown in Fig. 13-96. Exceptions are the initial points of all four cuts and the higher-boiling end of the heavy-distillate curve. This would seem to indicate that more theoretical stripping stages should be

added and that either the percent vaporization of the tower feed in the simulation is too high or the internal-reflux rate at the lower draw-off tray is too low. The liquid-rate profile in the tower is shown in Fig. 13-97. The use of two or three pump-around circuits instead of one would result in a better traffic pattern than that shown.

FIG. 13-96 Comparison of computed TBP curves with plant data for the example of Fig. 13-94.

FIG. 13-97

13-95

Liquid-rate profile for the example of Fig. 13-94.

13-96

DISTILLATION

BATCH DISTILLATION Batch distillation, which is the process of separating a specific quantity (the charge) of a liquid mixture into products, is used extensively in the laboratory and in small production units that may have to serve for many mixtures. When there are N components in the feed, one batch column will suffice where N − 1 simple continuous-distillation columns would be required. Many larger installations also feature a batch still. Material to be separated may be high in solids content, or it might contain tars or resins that would plug or foul a continuous unit. Use of a batch unit can keep solids separated and permit convenient removal at the termination of the process. SIMPLE BATCH DISTILLATION The simplest form of batch still consists of a heated vessel (pot or boiler), a condenser, and one or more receiving tanks. No trays or packing are provided. Feed is charged into the vessel and brought to boiling. Vapors are condensed and collected in a receiver. No reflux is returned. The rate of vaporization is sometimes controlled to prevent “bumping” the charge and to avoid overloading the condenser, but other controls are minimal. This process is often referred to as Rayleigh distillation. If we represent the moles of vapor by V, moles of liquid in the pot by M, the mole fraction of the more volatile component in this liquid by x, and the mole fraction of the same component in the vapor by y, a material balance yields −ydV = d(Mx)

(13-133)

Since dV = −dM, substitution and expansion give ydM = M dx + xdM

(13-134)

Rearranging and integrating give



xi M dx ln ᎏi = (13-135) ᎏ x Mf f y − x where subscript i represents the initial condition and f the final condition of the liquid in the still pot. Integration limits have been reversed to obtain a positive integral. If equilibrium is assumed between liquid and vapor, the right-hand side of Eq. (13-135) may be evaluated by plotting 1/(y − x) versus x and measuring the area under the curve between limits xi and xf. If the mixture is a binary system for which relative volatility α is constant or if an average value that will serve for the range considered can be found, then the relationship that defines relative volatility,

y/x α = ᎏᎏ (1 − y)/(1 − x)

(13-136)

can be substituted into Eq. (13-135) and a direct integration can be made: M 1 x f (1 − xi) 1−x ln ᎏf = ᎏ ln ᎏ + ln ᎏi Mi α−1 xi(1 − xf) 1 − xf

 







(13-137)

For any two components A and B of a multicomponent mixture, if constant α values are assumed for all pairs of components, −dMA/ −dMB = yA/yB = αA,B(xA /xB). When this is integrated, we obtain







MB( f ) MA( f ) ln ᎏ = αA,B ln ᎏ MA(i) MB(i)



(13-138)

where MA(i) and MA( f ) are the moles of component A in the pot before and after distillation and MB(i) and MB( f ) are the corresponding moles of component B. A typical application of a simple batch still might be distillation of an ethanol-water mixture at 101.3 kPa (1 atm). The initial charge is 100 mol of ethanol at 18 mole percent, and the mixture must be reduced to a maximum ethanol concentration in the still of 6 mole percent. By using equilibrium data interpolated from Table 13-1,

x

y

y−x

1/(y − x)

0.18 .16 .14 .12 .10 .08 .06

0.517 .502 .485 .464 .438 .405 .353

0.337 .342 .345 .344 .338 .325 .293

2.97 2.91 2.90 2.90 2.97 3.08 3.41

Plotting 1/(y − x) versus x and integrating graphically between the limits of 0.06 and 0.18 for x, the area under the curve is found to be 0.358. Then, ln (Mi /Mj) = 0.358, from which Mj = 100/1.43 = 70.0 mol. The liquid remaining consists of (70.0)(0.06) = 4.2 mol of ethanol and 65.8 mol of water. By material balance, the total distillate must contain (18.0 − 4.2) = 13.8 mol of alcohol and (82 − 65.8) = 16.2 mol of water. Total distillate is 30 mol, and distillate composition is 13.8/30 = 0.46 mole fraction ethanol. The simple batch still provides only one theoretical plate of separation. Its use is usually restricted to preliminary work in which products will be held for additional separation at a later time, when most of the volatile component must be removed from the batch before it is processed further, or for similar noncritical separations. BATCH DISTILLATION WITH RECTIFICATION To obtain products with a narrow composition range, a rectifying batch still is used that consists of a pot (or reboiler), a rectifying column, a condenser, some means of splitting off a portion of the condensed vapor (distillate) as reflux, and one or more receivers. Temperature of the distillate is controlled in order to return the reflux at or near the column temperature to permit a true indication of reflux quantity and to improve column operation. A subcooling heat exchanger is then used for the remainder of the distillate, which is sent to an accumulator or receiver. The column may also operate at elevated pressure or vacuum, in which case appropriate devices must be included to obtain the desired pressure. Equipment-design methods for batch-still components, except for the pot, follow the same principles as those presented for continuous units, but the design should be checked for each mixture if several mixtures are to be processed. It should also be checked at more than one point of a mixture, since composition in the column changes as distillation proceeds. Pot design is based on batch size and required vaporization rate. In operation, a batch of liquid is charged to the pot and the system is first brought to steady state under total reflux. A portion of the overhead condensate is then continuously withdrawn in accordance with the established reflux policy. Cuts are made by switching to alternate receivers, at which time operating conditions may be altered. The entire column operates as an enriching section. As time proceeds, composition of the material being distilled becomes less rich in the more volatile components, and distillation of a cut is stopped when accumulated distillate attains the desired average composition. CONTROL The progress of batch distillation can be controlled in several ways: 1. Constant reflux, varying overhead composition. Reflux is set at a predetermined value at which it is maintained for the run. Since pot liquid composition is changing, instantaneous composition of the distillate also changes. The progress of a binary separation is illustrated in Fig. 13-98. Variation with time of instantaneous distillate composition for a typical multicomponent batch distillation is shown in Fig. 13-99. The shapes of the curves are functions of volatility, reflux ratio, and number of theoretical plates. Distillation is continued until the average distillate composition is at the desired value. In the case of a binary, the overhead is then diverted to another receiver, and an intermediate cut is withdrawn until the remaining pot liquor meets the required specification. The intermediate cut is usually added to

BATCH DISTILLATION

13-97

off for a short period of time, after which the column is again returned to total-reflux operation. This cycle is repeated through the course of distillation. Another possibility is to optimize the reflux ratio in order to achieve the desired separation in a minimum of time. Complex operations may involve withdrawal of sidestreams, provision for intercondensers, addition of feeds to trays, and periodic charge addition to the pot. APPROXIMATE CALCULATION PROCEDURES FOR BINARY MIXTURES A useful method for a binary mixture employs an analysis based on the McCabe-Thiele graphical method. In addition to the usual assumptions of adiabatic column and equimolal overflow on the trays, the following procedure assumes negligible holdup of liquid on the trays, in the column, and in the condenser. As a first step in the calculation, the minimum-reflux ratio should be determined. In Fig. 13-100, point D, representing the distillate, is on the diagonal since a total condenser is assumed and xD = yD. Point F represents the initial condition in the still pot with coordinates xpi, ypi. Minimum internal reflux is represented by the slope of the line DF, (L/V)min = (yD − ypi)/(xD − xpi) Typical variation in distillate and reboiler compositions with amount distilled in binary batch distillation at a constant-reflux ratio. FIG. 13-98

(13-139)

where L is the liquid flow rate and V is the vapor rate, both in moles per hour. Since V = L + D (where D is distillate rate) and the externalreflux rate R is defined as R = L/D, then L/V = R/(R + 1) (13-140) (L/V)min or Rmin = ᎏᎏ (13-141) 1 − (L/V)min The condition of minimum reflux for an equilibrium curve with an inflection point P is shown in Fig. 13-102. In this case the minimum internal reflux is (L/V)min = (yD − yP)/(xD − xP)

(13-142)

The operating reflux ratio is usually 1.5 to 10 times the minimum. By using the ethanol-water equilibrium curve for 101.3 kPa (1 atm) pressure shown in Fig. 13-101 but extending the line to a convenient point for readability, (L/V)min = (0.800 − 0.695)/(0.800 − 0.600) = 0.52 and Rmin = 1.083.

FIG. 13-99 Distillate composition profile for a batch distillation of a fourcomponent mixture.

the next batch. For a multicomponent mixture, two or more intermediate cuts may be taken between product cuts. 2. Constant overhead composition, varying reflux. If it is desired to maintain a constant overhead composition in the case of a binary, the amount of reflux returned to the column must be constantly increased throughout the run. As time proceeds, the pot is gradually depleted of the lighter component. Finally, a point is reached at which the reflux ratio has attained a very high value. The receivers are then changed, the reflux is reduced, and an intermediate cut is taken as before. This technique can also be extended to a multicomponent mixture. 3. Other control methods. A cycling procedure can be used to set the pattern for column operation. The unit operates at total reflux until equilibrium is established. Distillate is then taken as total draw-

FIG. 13-100

Determination of minimum reflux for normal equilibrium curve.

13-98

DISTILLATION

FIG. 13-102 FIG. 13-101

Graphical method for constant-reflux operation.

Determination of minimum reflux for equilibrium curve with

inflection.

OPERATING METHODS Batch Rectification at Constant Reflux Using an analysis similar to the simple batch still, Smoker and Rose [Trans. Am. Inst. Chem. Eng., 36, 285 (1940)] developed the following equation: xpi M dxp ln ᎏi = (13-143) ᎏ xpf xD − xP Mf An overall component balance gives the average or accumulated distillate composition xD,avg Mi xpi − Mf xpf xD,avg = ᎏᎏ (13-144) Mi − Mf If the integral on the right side of Eq. (13-143) is labeled Q, the time θ in hours for distillation can be found by Mi(eQ − 1) θ = (R + 1) ᎏᎏ (13-145) VeQ An alternative equation is R+1 θ = ᎏ (Mi − Mf) (13-146) V Development of these equations is given by Block [Chem. Eng., 68, 88 (Feb. 6, 1961)]. The calculation process is illustrated in Fig. 13-102. Operating lines are drawn with the same slope but intersecting the 45° line at different points. The number of theoretical plates under consideration is stepped off to find equilibrium bottoms composition. In the figure, operating line L − 1 with slope L/V drawn from point D1 where the distillate composition is xD1 has an equilibrium pot composition of xp1-3 for three theoretical plates, xD2 has an equilibrium pot composition of xp2-3, etc.; performing a graphical integration of the right side of Eq. (13-143) permits calculation of xD,avg from Eq. (13-144). An iterative calculation is required to find the Mf that corresponds to the specified xD,avg. To illustrate the use of these equations, consider a charge of 520 mol of an ethanol-water mixture containing 18 mole percent ethanol to be distilled at 101.3 kPa (1 atm). Vaporization rate is 75 mol/h, and the product specification is 80 mole percent ethanol. Let L/V = 0.75, corresponding to a reflux ratio R = 3.0. If the system has seven theo-



retical plates, with the pot considered as one of these plates, find how many moles of product will be obtained, what the composition of the residue will be, and the time that the distillation will take. Using the vapor-liquid equilibrium data, plot a y-x diagram. Draw a number of operating lines at a slope of 0.75. Note the composition at the 45° intersection, and step off seven plates on each to find the equilibrium value of the bottoms. Some of the results are tabulated in the following table: xD

xp

xD − xp

1/(xD − xp)

0.800 .795 .790 .785 .780 .775

0.323 .245 .210 .180 .107 .041

0.477 .550 .580 .605 .673 .734

2.097 1.820 1.725 1.654 1.487 1.362

Use a trial procedure by integrating between xpi of 0.18 and various lower limits, and converge the procedure by graphing the results. It is found that xD,avg = 0.80 when xpf = 0.04, at which time the value of the integral = 0.205 = ln (Mi /Mf), so that Mf = 424 mol. Product = Mi − Mf = 520 − 424 = 96 mol. From Eq. (13-145), (4)(520)(e0.205 − 1) θ = ᎏᎏ = 5.2 h (75)(e0.205) Batch Rectification at Constant Overhead Composition Bogart [Trans. Am. Inst. Chem. Eng., 33, 139 (1937)] developed the following equation for this situation with column holdup assumed to be negligible: Mi(xD − xpi) xpi dxp θ = ᎏᎏ (13-147) ᎏᎏ2 xpf (1 − L/V)(xD − xp) V where the terms are defined as before. The quantity distilled can then be found by Mi(xpi − xpf) Mi − Mf = ᎏᎏ (13-148) xD − xpf The progress of a varying-reflux distillation is shown in Fig. 13-103. Distillate composition is held constant by increasing the reflux as pot composition becomes more dilute. Operating lines with varying slopes



BATCH DISTILLATION

FIG. 13-103

Graphical method for constant-composition operation.

(= L/V) are drawn from the distillate composition, and the appropriate number of plates is stepped off to find the corresponding bottoms composition. As an example, consider distilling at constant composition the same mixture that was used to illustrate constant reflux. The following table is compiled: L/V

xp

xD − xp

1/(1 − L/V)(xD − xp)2

0.600 .700 .750 .800 .850 .900

0.654 .453 .318 .143 .054 .021

0.147 .348 .483 .658 .747 .780

115.7 27.5 17.2 11.5 11.9 16.4

If the right-hand side of Eq. (13-147) is integrated graphically by using a limit for xpf of 0.04, the value of the integral is 1.615 and the time is (520)(0.800 − 0.180)(1.615) θ = ᎏᎏᎏ = 7.0 h 75 The quantity distilled can be found by Eq. (13-148): (520)(0.180 − 0.040) Mi − Mf = ᎏᎏᎏ = 96 mol 0.800 − 0.040 Other Operating Methods A useful control method for difficult industrial or laboratory distillations is cycling operation. The most common form of cycling control is operating the column at total reflux until equilibrium is established, taking off the complete distillate for a short period of time, and then returning to total reflux. An alternative scheme is to interrupt vapor flow to the column periodically by the use of a solenoid-operated butterfly valve in the vapor line from the pot. In both cases, equations necessary to describe the system are very complex, as shown by Schrodt et al. [Chem. Eng. Sci., 22, 759 (1967)]. The most reliable method for establishing the cycle relationships is by experimental trial on an operating column. Several investigators have also proposed that batch distillation can be programmed to attain time optimization by proper variation of the reflux ratio. A comprehensive discussion is presented by Coward [Chem. Eng. Sci., 22, 503 (1967)]. The choice of operating mode depends upon characteristics of the

13-99

specific system, the product specifications, and the engineer’s preference in setting up a control sequence. Probably the most direct and most common method is constant reflux. Operation can be regulated by a timed reflux splitter, a ratio controller, or simply a pair of rotameters. Since composition is changing with time, some way must be found to estimate the average accumulated-distillate composition in order to define the end point. This is no problem when the specification is not critical or the change in distillate composition is sharply defined. When the composition of the distillate changes slowly with time, the cut point is more difficult to determine. Operating with constant composition (varying reflux), the specification is automatically achieved if control can be linked to concentration or some concentration-sensitive physical variable. The relative advantage, ratewise, of the two systems depends upon the materials being separated and upon the number of theoretical plates in the column. Results of a comparison of distillation rates by using the same initial and final pot composition for the system benzene-toluene are given in Fig. 13-104. Typical control instrumentation is presented in an article by Block [Chem. Eng., 74, 147 (Jan. 16, 1967)]. Control procedures for reflux and vapor-cycling operation and for the time-optimal process are largely a matter of empirical trial. Effect of Column Holdup When the holdup of liquid on the trays and in the condenser is not negligible compared with the holdup in the pot, the distillate composition at constant-reflux ratio changes with time at a different rate than when the column holdup is negligible because of two separate effects. First, with an appreciable column holdup, composition of the charge to the pot will be higher in the light component than the pot composition at the start of the distillation; the reason for this is that before product takeoff begins, column holdup must be supplied, and its average composition is higher than that of the charge liquid from which it is supplied. Thus, when overhead takeoff begins, the pot composition is lower than it would be if there were no column holdup and separation is more difficult. The second effect of column holdup is to slow the rate of exchange of the components; the holdup exerts an inertia effect, which prevents compositions from changing as rapidly as they would otherwise, and the degree of separation is usually improved. As both these effects occur at the same time and change in importance during the course of distillation, it is difficult, without rigorous calculations, to predict whether the overall effect of holdup will be favorable or detrimental; it is equally difficult to estimate the magnitude of the holdup effect.

13-100

DISTILLATION

Although a number of studies were made and approximate methods developed for predicting the effect of liquid holdup in the period of the 1950s and 1960s, as summarized in the 6th edition of Perry’s Chemical Engineers’ Handbook, the complexity of the effect of liquid holdup is such that it is now best to use computer-based batchdistillation algorithms to determine the effect of holdup on a case-bycase basis. SHORTCUT METHODS FOR MULTICOMPONENT BATCH RECTIFICATION For preliminary studies of batch rectification of multicomponent mixtures, shortcut methods that assume constant molal overflow and negligible vapor and liquid holdup are useful. The method of Diwekar and Madhaven [Ind. Eng. Chem. Res., 30, 713 (1991)] can be used for constant reflux or constant overhead rate. The method of Sundaram and Evans [Ind. Eng. Chem. Res., 32, 511 (1993)] applies only to the case of constant reflux, but is easy to apply. Both methods employ the Fenske-Underwood-Gilliland (FUG) shortcut procedure at successive time steps. Thus, batch rectification is treated as a sequence of continuous, steady-state rectifications. CALCULATIONAL METHODS Rigorous Computer-Based Calculation Procedures It is obvious that a set of curves such as shown in Fig. 13-104 for a binary mixture is quite tedious to obtain by hand methods. The curves shown in Fig. 13-99 for a multicomponent batch distillation are extremely difficult to develop by hand methods. Therefore, since the early 1960s, when large digital computers became available, interest has been generated in developing rigorous calculation procedures for binary and multicomponent batch distillation. For binary mixtures of constant relative volatility, Huckaba and Danly [Am. Inst. Chem. Eng. J., 6, 335 (1960)] developed a computer program that assumed constant-mass tray holdups, adiabatic tray operation, and linear enthalpy relationships but did include energy balances around each tray and permitted use of nonequilibrium trays by means of specified tray efficiencies. Experimental data were provided to validate the simulation. Meadows [Chem. Eng. Prog. Symp. Ser. 46, 59, 48 (1963)] presented a multicomponent-batch-distillation model that included equations for energy, material, and volume balances around theoretical trays. The only assumptions made were perfect mixing on each tray, negligible vapor holdup, adiabatic operation, and constant-volume tray holdup. Distefano [Am. Inst. Chem. Eng. J., 14, 190 (1968)] extended the model and developed a computer-based-solution procedure that was used to simulate successfully several commercial batch-distillation columns. Boston et al. [Foundations of Computer-Aided Chemical Process Design, vol. II, ed. by Mah and Seider, American Institute of Chemical Engineers, New York, 1981, p. 203) further extended the model, provided a variety of practical sets of specifications, and utilized modern numerical procedures and equation formulations to handle efficiently the nonlinear and often stiff nature of the multicomponentbatch-distillation problem. The simpler model of Distefano is used here to illustrate this nonlinear and stiff nature. Consider the simple batch- or multicomponent-distillation operation in Fig. 13-105. The still consists of a pot or reboiler, a column with N theoretical trays or equivalent packing, and a condenser with an accompanying reflux drum. The mixture to be distilled is charged to the reboiler, to which heat is then supplied. Vapor leaving the top tray is totally condensed and drained into the reflux drum. Initially, no distillate is withdrawn from the system, but instead a total-reflux condition is established at a fixed overhead vapor rate. Then, starting at time t = 0, distillate is removed at a constant molal rate and sent to a receiver that is not shown in Fig. 13-105. Simultaneously, a fixed reflux ratio is established such that the overhead vapor rate is not changed from that at total reflux. Alternatively, heat input to the reboiler can be maintained constant and distillate rate allowed to vary accordingly. The equations of Distefano for a batch distillation operated in this manner are as follows (after minor rearrangement), where i,j refers to the ith of C components in the mixture and the f th of N theoretical plates.

FIG. 13-105 Schematic of a batch-distillation column. [Distefano, Am. Inst. Chem. Eng. J., 14, 190 (1963).]

Component mole balances for total-condenser-reflux drum, trays, and reboiler, respectively: dM L0 + D + ᎏᎏ0 dt dxi,0 xi,0 ᎏ = − ᎏᎏ M0 dt









V1Ki,1 + ᎏ xi,1 M0

i = 1 to C

(13-149)

j = 1 to N

(13-150)



dM Lj + Ki, jVj + ᎏᎏj dt dxi, j Lj − 1 xi, j ᎏ = ᎏ xi, j − 1 − ᎏᎏ M j dt Mj



 





Ki, j + 1Vj + 1 + ᎏᎏ xi, j + 1 Mj

i = 1 to C



dMN + 1 VN + 1Ki,N + 1 + ᎏᎏ dt dxi,N + 1 LN xi,N + 1 ᎏ = ᎏ xi,N − ᎏᎏᎏ M N+1 dt MN + 1







i = 1 to C

(13-151)

where L0 = RD. Total mole balance for total-condenser-reflux drum and trays respectively: dM V1 = D(R + 1) + ᎏ0 (13-152) dt dMj Lj = Vj + 1 + Lj − 1 − Vj − ᎏ dt Energy balance around jth tray:

j = 1 to N

(13-153)

BATCH DISTILLATION



1 dHL Vj + 1 = ᎏᎏ Vj(HVj − HLj) − Lj − 1(HLj − 1 − HLj) + Mj ᎏj (HVj + 1 − HLj) dt j = 2 to N + 1



(13-154)

where HV and HL are molar vapor and liquid enthalpies respectively. Phase equilibriums: yi, j = K i, j xi, j

i = 1 to C

j = 1 to N + 1

(13-155)

Mole-fraction sum:

y

i, j

i

=  K i, j xi, j = 1 .0

j = 0 to N + 1

(13-156)

i

Molar holdups in condenser-reflux drum, on trays, and in reboiler: M0 = G0 ρ0 Mj = Gj ρj

(13-157) j = 1 to N N

MN + 1 = M N + 1 −  Mj − 0

j=0

 Ddt t

(13-158)

0

where G is the constant-volume holdup, M0N + 1 is the initial molar charge to reboiler, and ρ is the liquid molar density. Energy balances around condenser and reboiler respectively: dHL0 Q0 = V1(HV1 − HL0) − M0 ᎏ dt

(13-159)

QN + 1 = VN + 1(HVN + 1 − HL N + 1) − LN(HL N − HLN + 1) + MN + 1



dHLN + 1 ᎏ dt



(13-160)

Equation (13-160) is replaced by the following overall energy-balance equation if QN + 1 is to be specified rather than D: N+1 d(MjHLj) dM QN + 1 − HV1 ᎏᎏ0 −  ᎏᎏ dt d t j=1 D = ᎏᎏᎏᎏ (13-161) (R + 1)HV1 − RHL0

 





With D and R specified, Eqs. (13-149) to (13-161) represent a coupled set of (2CN + 3C + 4N + 7) equations constituting an initial-value problem in an equal number of time-dependent unknown variables, namely, (CN + 2C)xi, j, (CN + C)yi, j, (N)Lj, (N + 1)Vj, (N + 2)Tj, (N + 2)Mj, Q0, and QN + 1, where initial conditions at t = 0 for all unknown variables are obtained by determining the total-reflux steady-state condition for specifications on the number of theoretical stages, amount and composition of initial charge, volume holdups, and molar vapor rate leaving the top stage and entering the condenser. Various procedures for solving Eqs. (13-149) to (13-161), ranging from a complete tearing method to solve the equations one at a time, as shown by Distefano, to a complete simultaneous method, have been studied. Regardless of the method used, the following considerations generally apply: 1. Derivatives or rates of change of tray and condenser-reflux drum liquid holdup with respect to time are sufficiently small compared with total flow rates that these derivatives can be approximated by incremental changes over the previous time step. Derivatives of liquid enthalpy with respect to time everywhere can be approximated in the same way. The derivative of the liquid holdup in the reboiler can likewise be approximated in the same way except when reflux ratios are low. 2. Ordinary differential Eqs. (13-149) to (13-151) for rates of change of liquid-phase mole fractions are nonlinear because the coefficients of xi, j change with time. Therefore, numerical methods of integration with respect to time must be employed. Furthermore, the equations may be difficult to integrate rapidly and accurately because they may constitute a so-called stiff system as considered by Gear (Numerical Initial Value Problems in Ordinary Differential Equations, Prentice Hall, Englewood Cliffs, N.J., 1971). The choice of time

13-101

step for simple explicit numerical procedures (such as the Euler and Runge-Kutta methods) of integrating sets of ordinary differential equations in initial-value problems may be governed by either stability or truncation-error considerations. Truncation errors in the dependent variables may be scarcely noticeable and generally accumulate gradually with time. Instability generally causes sudden and severe errors that are very noticeable. When the equations are stiff, stability controls and extremely small time steps may be necessary to prevent instability. A common measure of the severity of stiffness is the stiffness ratio |λ|max / |λ|min, where λ is an eigenvalue for the jacobian matrix of the set of ordinary differential equations. For Eqs. (13-149) to (13151), the jacobian matrix is tridiagonal if the equations and variables are arranged by stage (top down) for each component in order. For a general jacobian matrix pertaining to C components and N theoretical trays, as shown by Distefano [Am. Inst. Chem. Eng. J., 14, 946 (1968)]. Gerschgorin’s circle theorem (Varga, Matrix Iterative Analysis, Prentice Hall, Englewood Cliffs, N.J., 1962) may be employed to obtain bounds on the maximum and minimum absolute eigenvalues. Accordingly, dM Lj + Ki, jVj + ᎏᎏj dt Lj − 1 Ki, j + 1Vj + 1 |λ|max ≤ max ᎏ + ᎏᎏ + ᎏᎏ M j = 1, N j M M

  



j

j





The maximum absolute eigenvalue corresponds to the component with the largest K value (KL, j) and the tray with the smallest holdup. Therefore, if the derivative term and any variation in Lj, Vj, and Ki, j are neglected, Lj + Ki, jVj |λ|max 2 ᎏᎏ Mj





(13-162)

In a similar development, the minimum upper limit on the eigenvalue corresponds to the component with the largest K value and to the largest holdup, which occurs in the reboiler. Thus LN + KL,NVN |λ|min = ᎏᎏ MN + 1





(13-163)

Therefore, the lower bound on the stiffness ratio at the beginning of batch distillation is given approximately by



0

MN + 1 |λ|max ᎏ=2 ᎏ |λ|min MN



where MN + 1 and MN are the molar holdups in the reboiler initially and on the bottom tray respectively. In the sample problem presented by Distefano (ibid.) for the smallest charge, the approximate initialstiffness ratio is of the order of 250, which is not considered to be a particularly large value. Using an explicit integration method, almost 600 time increments, which were controlled by stability criteria, were required to distill 98 percent of the charge. At the other extreme of Distefano’s sample problems, for the largest initial charge, the maximum-stiffness ratio is of the order of 1500, which is considered to be a relatively large value. In this case, more than 10,000 time steps are required to distill 90 percent of the initial change, and the problem is better handled by a stiff integrator. In Distefano’s method, Eqs. (13-149) to (13-161) are solved with an initial condition of total reflux at L0 equal to D(R + 1) from the specifications. At t = 0, L0 is reduced so as to begin distillate withdrawal. The computational procedure is then as follows: 1. Replace Lj0 by Lj0 − D, but retain Vj0 and all other initial values from the total-reflux calculation. 2. Replace the holdup derivatives in Eqs. (13-149) to (13-151) by total-stage material-balance equations (e.g., dMj /dt = Vj + 1 + Lj − 1 − Vj − Lj) and solve the resulting equations one at a time by the predictor step of an explicit integration method for a time increment that is determined by stability and truncation considerations. If the mole fractions for a particular stage do not sum to 1, normalize them. 3. Compute a new set of stage temperatures from Eq. (13-156). Calculate a corresponding set of vapor-phase mole fractions from Eq. (13-155).

13-102

DISTILLATION

4. Calculate liquid densities, molar tray and condenser-reflux drum holdups, and liquor and vapor enthalpies. Determine holdup and enthalpy derivatives with respect to time by forward difference approximations. 5. From Eqs. (13-152) to (13-154) compute a new set of values of liquid and vapor molar flow rates. 6. Compute the reboiler molar holdup from Eq. (13-158). 7. Repeat steps 2 through 6 with a corrector step for the same time increment. Repeat again for any further predictor and/or predictor-corrector steps that may be advisable. Distefano (ibid.) discusses and compares a number of suitable explicit methods. 8. Compute condenser and reboiler heat-transfer rates from Eqs. (13-159) and (13-160). 9. Repeat steps 2 through 8 for subsequent time increments until the desired amount of distillate has been withdrawn. More flexible and efficient methods that can cope with stiffness in batch-distillation calculations utilize stable implicit integration procedures such as the method of Gear (op. cit.). Boston et al. (op. cit.) discuss such a method that also utilizes a two-tier equation-solving technique, referred to as the “inside-out” algorithm, that can handle both wide-boiling and narrow-boiling charges even when very nonideal mixtures are formed. In addition to the features of the Distefano model, the Boston et al. model permits multiple feeds, sidestream withdrawals, tray heat transfer, and vapor distillate and divides the batch-distillation process into a sequence of operation steps. At the beginning of each step, the reboiler may receive an additional charge, distillate or sidestream receivers may be dumped, and a feed reservoir may be refilled. Specifications for an operation step include feed, sidestream withdrawal, and tray heat-transfer rates. In addition, any two of the following five variables must be specified: reflux ratio, distillate rate, boil-up rate, condenser duty, and reboiler duty. An operation step is terminated when a specified criterion, selected from the following list, is reached: a time duration; a component purity in the reboiler, distillate, or distillate accumulator; an amount of material in the reboiler or distillate accumulator; or a reboiler or condenser temperature. The purity is specified to be met as the purity is increasing or decreasing. Finally, column configuration and operating conditions (number of stages, holdups, tray pressures, and feed, sidestream, and tray heat-transfer rates) can be changed at the beginning of each operation step. In addition, physical properties may be computed from a wide variety of correlations, including equation-of-state and activity-coefficient models. Example 10: Calculation of Multicomponent Batch Distillation A charge of 45.4 kg ⋅ mol (100 lb⋅mol) of 25 mole percent benzene, 50 mole percent monochlorobenzene (MCB), and 25 mole percent orthodichlorobenzene (DCB) is to be distilled in a batch still consisting of a reboiler, a column containing 10 theoretical stages, a total condenser, a reflux drum, and a distillate accumulator. Condenser-reflux drum and tray holdups are 0.0056 and

TABLE 13-29

0.00056 m3 (0.2 and 0.02 ft3) respectively. Pressures are 101.3, 107.6, 117.2, and 120.7 kPa (14.696, 15.6, 17, and 17.5 psia) at the condenser outlet, top stage, bottom stage, and reboiler respectively. Initially, the still is to be brought to total-reflux conditions at a boil-up rate of 45.4 (kg⋅mol)/h [100 (lb⋅mol)/h] leaving the reboiler. Then, at t = 0, the boil-up rate is to be increased to 90.8 (kg⋅mol)/h [200 (lb⋅mol)/h], and the reflux ratio is to be set at 3. The batch is then distilled in three steps. The first step, which is designed to obtain a benzene-rich product, is terminated when the mole fraction of benzene in the distillate being sent to the accumulator has dropped to 0.100 or when 2 h have elapsed. The purpose of the second operation step is to recover an MCB-rich product until the mole fraction of MCB in the distillate drops to 0.400 or 2 h have elapsed since the start of this step. The third step is to be terminated when the mole fraction of DCB in the reboiler reaches 0.98 or 2 h have elapsed since the start of this step. Ideal solutions and an ideal gas are assumed such that Raoult’s law can be used to obtain K values. Calculations are made by the method of Boston et al. (op. cit.). First, the total-reflux condition is computed by making several sets of stageto-stage calculations from the reboiler to the condenser. For the first set, the reboiler composition is assumed to be that of the initial charge. This composition is adjusted by material balance to initiate each subsequent set of calculations until convergence is achieved. Results are shown in Table 13-29. From these data, the initial-stiffness ratio is approximately [2(99.74)/0.01218] = 16,400. Thus the equations are quite stiff, and an implicit integration method is preferred. Detailed calculated conditions at the end of the first operation step are given in Table 13-30. A short summary of computed conditions for each of the three operation steps is given in Table 13-31. From Table 13-31, a total of 394 time increments were necessary to distill all but 22.08 lb⋅mol of the initial charge of 99.74 lb⋅mol following the establishment of total-reflux conditions. If this problem had to be solved by an explicit integrator, approximately 25,000 time increments would have been necessary. Instantaneous distillate (or reflux) composition as a function of total accumulated distillate for all three operation steps is plotted in Fig. 13-106. From these results, an alternative schedule of operation steps can be derived to obtain three relatively rich cuts and two intermediate cuts for recycle to the next batch. One example is as follows:

Composition, mole fractions Cut

Amount, lb⋅mol

Benzene

MCB

DCB

Benzene-rich Recycle 1 MCB-rich Recycle 2 DCB-rich Residual holdup Total

17.750 15.630 37.195 7.155 22.078 0.192 100.00

0.99553 0.44120 0.01164 0.00000 0.00000 0.00000 0.25000

0.00447 0.55880 0.98821 0.55430 0.02000 0.11980 0.50000

0.00000 0.00000 0.00015 0.44570 0.98000 0.88020 0.25000

NOTE:

To convert pound⋅moles to kilogram⋅moles, multiply by 0.454.

From these results, 22.98 lb⋅mol, or almost 23 percent of the charge, would be recycled for redistillation. All three products are at least 98 mole percent pure.

Total-Reflux Conditions x

Stage

T, °F

L, (lb⋅mol)/h

M, lb⋅mol

Benzene

Condenser 1 2 3 4 5 6 7 8 9 10 Reboiler

175.94 179.46 180.05 180.64 181.22 181.80 182.41 183.14 184.54 189.08 205.91 250.96

116.4 117.4 117.5 117.5 117.6 117.7 117.7 117.5 116.4 111.8 100.0 0.0

0.1307 0.01304 0.01303 0.01303 0.01302 0.01302 0.01301 0.01300 0.01295 0.01278 0.01218 99.74

1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.996 0.979 0.901 0.642 0.248

MCB 0.525 × 10−7 0.266 × 10−6 0.135 × 10−5 0.683 × 10−5 0.344 × 10−4 0.173 × 10−3 0.871 × 10−3 0.435 × 10−2 0.209 × 10−1 0.965 × 10−1 0.319 0.501

DCB 0.635 × 10−15 0.164 × 10−13 0.424 × 10−12 0.109 × 10−10 0.279 × 10−9 0.711 × 10−8 0.180 × 10−6 0.454 × 10−5 0.112 × 10−3 0.250 × 10−2 0.389 × 10−1 0.251

NOTE: Reboiler duty = 1,549,000 Btu/h. To convert degrees Fahrenheit to degrees Celsius, °C = (°F − 32)/1.8; to convert pound-moles per hour to kilogram-moles per hour, multiply by 0.454; and to convert pound-moles to kilogram-moles, multiply by 0.454.

BATCH DISTILLATION TABLE 13-30

13-103

Conditions at the End of the First Operation Step Time = 0.5963 h (lb⋅mol)/h

y

Stage

T, °F

V

L

M, lb⋅mol

Condenser 1 2 3 4 5 6 7 8 9 10 Reboiler

251.58 267.69 271.21 272.48 273.28 273.99 274.75 275.72 277.29 280.50 287.47 301.67

0 206.1 209.0 209.5 209.6 209.7 209.7 209.6 209.2 208.1 205.4 200.0

154.6 157.5 158.0 158.1 158.2 158.2 158.1 157.7 156.6 154.0 148.5 0

0.1113 0.01992 0.01088 0.01087 0.01086 0.01085 0.01084 0.01083 0.01080 0.01073 0.01057 66.40

NOTE:

Benzene 0.0994 0.0449 0.0331 0.0306 0.0301 0.0300 0.0300 0.0300 0.0301 0.0302 0.0304

MCB 0.901 0.955 0.967 0.969 0.970 0.969 0.967 0.963 0.949 0.912 0.829

x DCB

Benzene

MCB

DCB

0.293 × 10−5 0.101 × 10−4 0.312 × 10−4 0.943 × 10−4 0.282 × 10−3 0.839 × 10−3 0.249 × 10−2 0.731 × 10−2 0.211 × 10−1 0.577 × 10−1 0.141

0.100 0.276 × 10−1 0.121 × 10−1 0.884 × 10−2 0.818 × 10−2 0.806 × 10−2 0.803 × 10−2 0.801 × 10−2 0.794 × 10−2 0.772 × 10−2 0.720 × 10−2 0.634 × 10−2

0.900 0.972 0.988 0.991 0.991 0.991 0.989 0.982 0.964 0.915 0.803 0.617

0.292 × 10−5 0.124 × 10−4 0.405 × 10−4 0.124 × 10−3 0.373 × 10−3 0.111 × 10−2 0.329 × 10−2 0.969 × 10−2 0.280 × 10−1 0.770 × 10−1 0.189 0.376

To convert pound-moles per hour to kilogram-moles per hour, multiply by 0.454; to convert pound-moles to kilogram-moles, multiply by 0.454.

TABLE 13-31

Calculated Conditions for each Operation Step Operation step 1

Time of operation step, h Number of time increments Accumulated distillate Total lb⋅mol Mole fractions Benzene MCB DCB Reboiler holdup Total lb⋅mol Mole fractions Benzene MCB DCB Temperatures, °F Condenser outlet Reboiler outlet Heat duties, million Btu/h Condenser Reboiler

2

3

0.5963 201

0.7944 154

0.04828 39

33.38

41.99

2.361

0.7360 0.2640 0.45 × 10−6 66.40 0.0063 0.6172 0.3765 251.58 301.67 3.313 3.295

0.74 × 10−10 0.2872 0.7128

0.0103 0.9537 0.036 24.43

22.08

0.63 × 10 0.0448 0.9552

−11

308.60 362.63 3.472 3.469

0.20 × 10−12 0.0200 0.9800 330.17 366.39 3.456 3.433

NOTE: To convert degrees Fahrenheit to degrees Celsius, °C = (°F − 32)/1.8; to convert pound-moles to kilogram-moles, multiply by 0.454; and to convert British thermal units per hour to kilo-joules per hour, multiply by 1.055.

RAPID SOLUTION METHOD The quasi-steady-state method of Galindez and Fredenslund [Comput. Chem. Engng., 12, 281 (1988)] is a rapid alternative to the rigorous integration of the stiff differential equations. In their method, which is implemented in the CHEMCAD III process simulator of Chemstations, Inc., Houston, Texas, the transient conditions are simulated, without having to treat stiffness, by a succession of a finite number of continuous steady states of generally a few minutes of batch time each in duration. The calculations are started from an initialization at total-reflux conditions, taking into account holdup. Generally, computed results compare well with rigorous integration methods.

Distillate-composition profile for the multicomponent-batchdistillation example.

FIG. 13-106

13-104

DISTILLATION

DYNAMIC DISTILLATION INTRODUCTION

IDEAL BINARY DISTILLATION

As discussed in detail by Archer and Rothfus [Chem. Eng. Prog. Symp. Series, No. 36, 57, 2 (1961)], dynamic or transient behavior of a continuous-distillation operation is important in determining (1) startup and shutdown procedures, (2) transition path between steady states, (3) effect of upsets and fluctuations on controllability, (4) residence times and mass-transfer rates, and (5) operating strategies that may involve deliberate imposition of controlled cyclic fluctuations or oscillations, as summarized by Schrodt [Ind. Eng. Chem., 59(6), 58 (1967)]. Dynamic behavior may be studied with no controllers in the system to obtain a so-called open-loop response. Alternatively, controllers may be added for certain variables that are to be controlled by manipulating other variables to obtain a so-called closed-loop response. For this latter case, controllers of various levels of complexity [e.g., on-off, proportional (P), proportional with integral action (PI), and proportional with integral and derivative action (PID)] can be considered for various values of tuning parameters, and specific valves of known characteristics may be incorporated if desired.

Consider the closed-loop response during the dynamic distillation of an ideal binary mixture in the column shown in Fig. 13-107, under two assumptions of constant relative volatility at a value of 2.0 and constant molar vapor flow for a saturated liquid feed to tray Ns. Following the development by Luyben (op. cit.), it is not necessary to include energy-balance equations for each tray or to treat temperature and pressure as variables. Overhead vapor leaving top tray NT is totally condensed for negligible liquid holdup with condensate flowing to a reflux drum having constant and perfectly mixed molar liquid holdup MD. The reflux rate LNT + 1 is varied by a proportional-integral (PI) feedback controller to control distillate composition for a set point of 0.98 for the mole fraction xD of the light component. Holdup of reflux in the line leading back to the top tray is neglected. Under dynamic conditions, yNT may not equal xD. At the bottom of the column, a liquid sump of constant and perfectly mixed molar liquid holdup MB is provided. A portion of the liquid flowing from this sump passes to a thermosiphon reboiler, with the

FIG. 13-107

Binary distillation column; dynamic distillation of ideal binary mixture.

DYNAMIC DISTILLATION remainder taken as bottoms product at a molar flow rate B. Vapor boil-up generated in the reboiler is varied by a PI feedback controller to control bottoms’ composition with a set point of 0.02 for the mole fraction xB of the light component. Liquid holdups in the reboiler and lines leading from the sump are assumed to be negligible. The composition of the boil-up yB is assumed to be in equilibrium with xB. The liquid holdup Mn on each of the NT equilibrium trays is assumed to be perfectly mixed but will vary as liquid rates leaving the trays vary. Vapor holdup is assumed to be negligible everywhere. Tray molar vapor rates V vary with time but at any instant in time are everywhere equal. The dynamic material-balance and phase equilibrium equations corresponding to this description are as follows: All trays, n: dMn /dt = Fn + Ln + 1 − Ln

(13-164)

d ᎏ (Mnxn) = FnxFn + Ln + 1xn + 1 + Vyn − 1 − Lnxn − Vyn dt Ln + (Mn −  Mn)/β Ln = 

(13-165) (13-166)

αxn (13-167) yn = ᎏᎏ 1 + (α − 1)xn Where Fn is nonzero only for tray Ns, y and x refer to the light component only such that the corresponding mole fractions for the heavy component are (1 − y) and (1 − x), L Mn are the initial steadyn and  state values, and β is a constant that depends on tray hydraulics. For the condenser-reflux-drum combination: D = V − LNT + 1

(13-168)

MD(dxD /dt) = VyNT − VxD B = L1 − V

(13-170)

MB(dxB/dt) = L1x1 − VyB − BxB

(13-171)

αxB yB = ᎏᎏ 1 + (α − 1)xB The two PI-controller equations are 1 V= V − KCB EB + ᎏ EB dt τB





(13-172)





(13-173)



1 L NT + 1 + K CD ED + ᎏ ED dt (13-174) L NT + 1 =  τD where  V and  LNT + 1 are initial values, K C and τ are respectively feedback-controller gain and feedback-reset time for integral action, and E is the error or deviation from the set point as given by EB = x Bset − xB

(13-175)

ED = x

(13-176)

set D

− xD

TABLE 13-32 Initial and Fixed Conditions, Controller and Hydraulic Parameters, and Disturbance for Ideal Binary Dynamic-Distillation Example Other initial conditions F  = 100 (lb⋅mol)/min xF = 0.50 L NT + 1 = 128.01 (lb⋅mol)/min  V  = 178.01 (lb⋅mol)/min D  = 50 (lb⋅mol)/min x Dset = 0.98 B  = 50 (lb⋅mol)/min x Bset = 0.02 Mn,n = 1 to N T = 10 lb⋅mol Fixed conditions NT = 20 NS = 10 MD = 100 lb⋅mol MB = 100 lb⋅mol Controller and hydraulic parameters K CB = KCD = 1000 (lb⋅mol)/min τB = 1.25 min τD = 5.0 min β = 0.1 min Disturbance at t = 0+ xF = 0.55

Initial liquid-phase compositions Tray

xn

Bottoms 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Distillate

0.02 0.035 0.05719 0.08885 0.1318 0.18622 0.24951 0.31618 0.37948 0.43391 0.47688 0.51526 0.56295 0.61896 0.68052 0.74345 0.80319 0.85603 0.89995 0.93458 0.96079 0.98

NOTE:

To convert pound-moles per minute to kilogram-moles per minute, multiply by 0.454; to convert pound-moles to kilogram-moles, multiply by 0.454.

(13-169)

For the reboiler:



13-105

These differential equations are readily solved, as shown by Luyben (op. cit.), by simple Euler numerical integration, starting from an initial steady state, as determined, e.g., by the McCabe-Thiele method, followed by some prescribed disturbance such as a step change in feed composition. Typical results for the initial steady-state conditions, fixed conditions, controller and hydraulic parameters, and disturbance given in Table 13-32 are listed in Table 13-33. MULTICOMPONENT DISTILLATION Open-loop behavior of multicomponent distillation may be studied by solving modifications of the multicomponent equations of Distefano [Am. Inst. Chem. Eng. J., 14, 190 (1968)] as presented in the subsection “Batch Distillation.” One frequent modification is to include an equation, such as the Francis weir formula, to relate liquid holdup on a tray to liquid flow rate leaving the tray. Applications to azeotropicdistillation towers are particularly interesting because, as discussed by and illustrated in the following example from Prokopakis and Seider

[Am. Inst. Chem. Eng. J., 29, 1017 (1983)], the steep concentration and temperature fronts can be extremely sensitive to small changes in reflux ratio, boil-up rate, product recovery and purity, and feed rate and composition. Consider azeotropic distillation to dehydrate ethanol with benzene. Initial steady-state conditions are as shown in Fig. 13-108. The overhead vapor is condensed and cooled to 298 K to form two liquid phases that are separated in the decanter. The organic-rich phase is returned to the top tray as reflux together with a portion of the waterrich phase and makeup benzene. The other portion of the water-rich phase is sent to a stripper to recover organic compounds. Ordinarily, vapor from that stripper is condensed and recycled to the decanter, but that coupling is ignored here. Equations for the decanter are as follows if it is assumed that (1) there are constant holdups in the decanter of both phases in the same ratio as the ratio of the flow rates leaving the decanter, (2) there is a constant decanter temperature, and (3) the two liquid phases in the decanter are in physical equilibrium and each is perfectly mixed. d (13-177) ᎏ [Md(xi)d] = VN yi,N − L0(xi)0 − Lw(xi)w dt dMd /dt = VN − L0 − Lw K di = (xi)0 /(xi)w = (γi)w /(γi)0

(13-178) (13-179)

where Md is the total molar holdup of both phases in the decanter and the total composition in the decanter is (xi)0 L0 + (xi)w Lw (xi)d = ᎏᎏ (13-180) L0 + Lw Combination of Eqs. (13-177), (13-178), and (13-180) gives d(xi)d VN (13-181) ᎏ = ᎏ [yi, N − (xi)d] dt Md These equations, together with those for the tower, constitute a socalled stiff system. They were solved by Prokopakis and Seider (op. cit.), following a prescribed disturbance, using an adaptive semiimplicit Runge-Kutta integration technique by which VN and the yi,N were obtained by integration of the equations for the tower. Then Eq. (13-181) was integrated to give (xi)d, which was used with Eqs.

13-106

DISTILLATION

TABLE 13-33

Results for Ideal Binary Dynamic-Distillation Example of Table 13-32* Flow rate, (lb⋅mol)/min

Mole fraction of light component in liquid Time, min

Bottoms

Stage 5

Stage 10

Stage 15

Distillate

Reflux

Boli-up

0.00 .50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00 6.50 7.00 7.50 8.00 8.50

0.02000 0.02014 0.02107 0.02217 0.02275 0.02268 0.02212 0.02132 0.02051 0.01987 0.01950 0.01939 0.01950 0.01972 0.01995 0.02012 0.02019 0.02016

0.18622 0.19670 0.21174 0.22038 0.22209 0.21881 0.21287 0.20639 0.20104 0.19777 0.19679 0.19766 0.19956 0.20162 0.20314 0.20380 0.20362 0.20289

0.47688 0.51310 0.52426 0.53026 0.53229 0.53141 0.52879 0.52560 0.52282 0.52109 0.52057 0.52106 0.52209 0.52320 0.52400 0.52434 0.52422 0.52381

0.74345 0.74940 0.76049 0.76847 0.77217 0.77222 0.76993 0.76672 0.76381 0.76196 0.76142 0.76198 0.76315 0.76438 0.76525 0.76557 0.76537 0.76484

0.98000 0.98000 0.98010 0.98034 0.98061 0.98076 0.98077 0.98065 0.98047 0.98030 0.98018 0.98014 0.98016 0.98022 0.98029 0.98034 0.98036 0.98035

128.01 128.01 127.91 127.64 127.33 127.11 127.02 127.07 127.19 127.32 127.42 127.45 127.41 127.33 127.24 127.15 127.09 127.07

178.01 178.16 179.31 181.06 182.65 183.69 184.10 183.99 183.55 182.98 182.47 182.14 182.02 182.08 182.25 182.43 182.56 182.61

*From Luyben, Process Modeling, Simulation, and Control for Chemical Engineers, McGraw-Hill, New York, 1973. NOTE: To convert pound-moles per minute to kilogram-moles per minute, multiply by 0.454.

(13-179) and (13-180) to obtain the equilibrium compositions (xi)0 and (xi)w leaving the decanter. The UNIQUAC equation was used with data from Gmehling and Onken (Vapor-Liquid Equilibrium Data Collection, DECHEMA Chemistry Data Ser., Vol. 1 (parts 1–10). Frankfurt, 1977) to obtain the activity coefficients needed in Eq. (13-179). Reboiler and decanter volumetric holdups were assumed constant at 1.0 m3 (35.3 ft3), and volumetric tray holdups were computed from Mn = (ρL)n An(hw j + hc j )

FIG. 13-108

(13-182)

where (ρL)n is the liquid density; An, the cross-sectional area of the active portion of the tray = 0.23 m2 (2.48 ft2); hw j , the weir height = 0.0254 m (0.0833 ft); and hc j , the weir crest, assumed to be constant at 0.00508 m (0.0167 ft). Accordingly, volumetric tray holdup was constant at 0.007 m3 (0.247 ft3). Assume that at t = 0+ the feed rate to tray 23 is disturbed by increasing it by 30 percent to 130 mol/min without a change in composition. The resulting ethanol liquid mole fraction on several trays is tracked in Fig. 13-109. Above tray 16, ethanol concentration remains very small. Below tray 9, ethanol concentration initially decreases fairly rapidly but

Initial steady state for dynamic azeotropic distillation of ethanol-water with benzene.

DYNAMIC DISTILLATION then increases slowly and steadies out at significantly higher values than at the initial steady state. Tray 10 is one of the last trays to reach the new steady-state condition, which takes somewhat more than 200 min. This may be compared with initial residence times in the decanter and reboiler of approximately 50 and 250 min respectively. The movement through the column of concentration fronts for all three components is shown in Fig. 13-109. For the first 5 to 10 min, below tray 16, benzene

and ethanol fronts shift downward. Then a reversal occurs, and the fronts shift upward until the new steady state is attained. The upward shift is expected because the increased feed rate increases the waterbenzene entrainer ratio. The duration of the initial, temporary downward shift is highly dependent on tray holdup and is due to “wash-out” with the feed liquid. This phenomenon is also observed in the dynamic studies of Peiser and Grover [Chem. Eng. Prog., 58(9), 65 (1962)].

(a) Responses after a 30 percent increase in the feed flow rate for the multicomponent-dynamic-distillation example of Fig. 13-100. Profiles of liquid mole fractions at several times.

FIG. 13-109a

(b) Responses after a 30 percent increase in the feed flow rate for the multicomponent-dynamic-distillation example of Fig. 13-100. Alcohol mole fractions on several trays. (Prokopakis and Seider, Am. Inst, Chem. Eng. J., 29, 1017 (1983).]

FIG. 13-109b

13-107

13-108

DISTILLATION

If the feed rate is decreased, the trends of curves in Fig. 13-109 are reversed. The disturbance of other variables such as feed composition, boil-up ratio, and recycle of water-rich effluent from the decanter produces similar shifts in the steep concentration fronts, indicating that azeotropic towers are among the most sensitive separation operations, for which dynamic studies are essential if reli-

able process control is to be developed. Such studies indicate the importance of adjusting aqueous-phase recycle and reboiler duty to diminish the movement of steep concentration fronts and the possibility of multiple regimes of operation including unstable regimes, as shown by Magnussen et al. [Inst. Chem. Eng. Symp. Ser. 56 (1979)].

PACKED COLUMNS Distillation-type separation operations may be conducted in packed rather than tray-type columns. In prior years, except for small columns, plate columns were heavily favored over packed columns. However, development of more efficient random or structured packing materials and the need to increase capacity, increase efficiency, or reduce pressure drop in many applications has resulted in more extensive use of packed columns in larger sizes in recent years. Both types of contacting devices are discussed extensively in Sec. 14 and by Billet (Distillation Engineering, Chemical Publishing, New York, 1979). Packed columns may employ dumped (random) packing, e.g., pall rings, or structured (ordered, arranged, or stacked) packing, e.g., knitted wire and corrugated and perforated sheets. Tray-type columns generally employ valve, sieve, or bubble-cap trays with downcomers. The choice between a packed column and a tray-type column is based mainly on economics when factors of contacting efficiency, loadability, and pressure drop must be considered. Packed columns must be provided with good initial distribution of liquid across the column cross section and redistribution of liquid at various height intervals that decrease with increasing column diameter. A wide variety of distributors and redistributors are available. Packed columns should be considered when: 1. Temperature-sensitive mixtures are to be separated. To avoid decomposition and/or polymerization, vacuum operation may then be necessary. The smaller liquid holdup and pressure drop theoretical stage of a packed column may be particularly desirable.

2. Ceramic or plastic (e.g., propylene) is a desirable material of construction from a noncorrosion and liquid-wettability standpoint. 3. Refitting of a tray-type column is desired to increase loading, increase efficiency, and/or decrease pressure drop. Structured packing is particularly applicable in this case. 4. Liquid rates are very low and/or vapor rates are high, in which case structured packing may be particularly desirable. 5. The mixture to be separated is clear, nonfouling, and free of solids, and cleaning of column internals will not be necessary. 6. The mixture to be separated tends to form foam, which collapses more readily in a packed column. 7. High recovery of a volatile component by a batch operation is required. Liquid holdup is much lower in a packed column. Packed columns are almost always used for column diameters less than 762 mm (30 in) but otherwise generally need not be considered when: 1. Multiple feeds, sidestreams, and/or intermediate condensers and/or intermediate reboilers are required or desirable. 2. A wide range of loadability (turndown ratio) is required. Valve trays are particularly desirable in this case. 3. Design data for separation of the particular or similar mixture in a packed column are not available. Design procedures are better established for tray-type columns than for packed columns. This is particularly so with respect to separation efficiency since tray efficiency can be estimated more accurately than packed height equivalent to a theoretical stage (HETP).

Section 14

Gas Absorption and Gas-Liquid System Design*

James R. Fair, Ph.D., P.E., Professor of Chemical Engineering, University of Texas; Fellow, American Institute of Chemical Engineers; Member, American Chemical Society, American Society for Engineering Education, National Society of Professional Engineers. (Section Editor, Absorption, Gas-Liquid Contacting) D. E. Steinmeyer, M.A., M.S., P.E., Distinguished Fellow, Monsanto Company; Fellow, American Institute of Chemical Engineers; Member, American Chemical Society. (Liquid-inGas Dispersions) W. R. Penney, Ph.D., P.E., Professor of Chemical Engineering, University of Arkansas; Member, American Institute of Chemical Engineers. (Gas-in-Liquid Dispersions) B. B. Crocker, S.M., P.E., Consulting Chemical Engineer; Fellow, American Institute of Chemical Engineers; Member, Air Pollution Control Association. (Phase Separation)

INTRODUCTION Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data Sources in the Handbook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equilibrium Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14-4 14-4 14-4 14-4 14-4

DESIGN OF GAS-ABSORPTION SYSTEMS General Design Procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selection of Solvent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selection of Solubility Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 1: Gas Solubility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculation of Liquid-to-Gas Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selection of Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Column Diameter and Pressure Drop. . . . . . . . . . . . . . . . . . . . . . . . . . . Computation of Tower Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selection of Stripper-Operating Conditions . . . . . . . . . . . . . . . . . . . . . . Design of Absorber-Stripper Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . Importance of Design Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14-5 14-5 14-5 14-5 14-5 14-6 14-6 14-6 14-6 14-6 14-6

PACKED-TOWER DESIGN Use of Mass-Transfer-Rate Expressions . . . . . . . . . . . . . . . . . . . . . . . . . Example 2: Packed Height Requirement . . . . . . . . . . . . . . . . . . . . . . . .

14-8 14-8

Use of Operating Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculation of Transfer Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stripping Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 3: Air Stripping of VOCs from Water . . . . . . . . . . . . . . . . . . . . Use of HTU and KGa Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Use of HETP Data for Absorber Design. . . . . . . . . . . . . . . . . . . . . . . . .

14-8 14-9 14-9 14-10 14-10 14-10

PLATE-TOWER DESIGN Graphical Design Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algebraic Method for Dilute Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algebraic Method for Concentrated Gases . . . . . . . . . . . . . . . . . . . . . . . Stripping Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tray Efficiencies in Plate Absorbers and Strippers . . . . . . . . . . . . . . . . . Example 4: Actual Plates for Steam Stripping. . . . . . . . . . . . . . . . . . . . .

14-11 14-11 14-11 14-11 14-12 14-12

HEAT EFFECTS IN GAS ABSORPTION Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effects of Operating Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equipment Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classical Isothermal Design Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classical Adiabatic Design Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rigorous Design Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14-12 14-13 14-13 14-14 14-14 14-14

* Much of the material on absorption is taken from Sec. 14 of the sixth edition, and credit is due to Dr. William M. Edwards, editor of that section. 14-1

14-2

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

Direct Comparison of Design Methods. . . . . . . . . . . . . . . . . . . . . . . . . . Example 5: Packed Absorber, Acetone into Water . . . . . . . . . . . . . . . . . Example 6: Solvent Rate for Absorption . . . . . . . . . . . . . . . . . . . . . . . . .

14-14 14-14 14-14

MULTICOMPONENT SYSTEMS Example 7: Multicomponent Absorption, Dilute Case. . . . . . . . . . . . . . Graphical Design Methods for Dilute Systems. . . . . . . . . . . . . . . . . . . . Algebraic Design Method for Dilute Systems. . . . . . . . . . . . . . . . . . . . . Example 8: Multicomponent Absorption, Concentrated Case. . . . . . . .

14-15 14-16 14-16 14-16

ABSORPTION WITH CHEMICAL REACTION Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Overall Design Strategy . . . . . . . . . . . . . . . . . . . . . . . . . Applicability of Physical Design Methods . . . . . . . . . . . . . . . . . . . . . . . . Traditional Design Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scaling Up from Laboratory or Pilot-Plant Data. . . . . . . . . . . . . . . . . . . Principles of Rigorous Absorber Design . . . . . . . . . . . . . . . . . . . . . . . . . Estimation of kL for Irreversible Reactions . . . . . . . . . . . . . . . . . . . . . . . Estimation of kL for Reversible Reactions . . . . . . . . . . . . . . . . . . . . . . . . Simultaneous Absorption of Two Reacting Gases . . . . . . . . . . . . . . . . . . Desorption with Chemical Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . Use of Literature for Specific Systems . . . . . . . . . . . . . . . . . . . . . . . . . .

14-17 14-17 14-17 14-18 14-19 14-20 14-21 14-22 14-22 14-23 14-23

GAS-LIQUID CONTACTING SYSTEMS Plate Columns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plate Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plate-Column Capacity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 9: Loading/Flooding of a Distillation Plate . . . . . . . . . . . . . Example 10: Entrainment Effect on Plate Efficiency. . . . . . . . . . . . . Plate Layouts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure Drop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 11: Pressure Drop, Sieve Plate . . . . . . . . . . . . . . . . . . . . . . . Phase Inversion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plate Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 12: Estimation of Plate Efficiency . . . . . . . . . . . . . . . . . . . . Packed Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Packed Columns versus Plate Columns. . . . . . . . . . . . . . . . . . . . . . . . Packed-Column Hydraulics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flooding and Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure Drop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 13: Packed Column Pressure Drop . . . . . . . . . . . . . . . . . . . Support Plates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid Holdup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maldistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . End Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interfacial Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Behavior of Various Systems and Packings . . . . . . . . . . . . . . . . . . . . . Distillation Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid-Dispersed Contactors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heat-Transfer Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical Transfer Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Countercurrent, Cocurrent, or Backmixed . . . . . . . . . . . . . . . . . . . . . Empirical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wetted-Wall Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gas-Liquid-Column Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cost of Internals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cost of Column. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14-24 14-24 14-25 14-28 14-29 14-29 14-30 14-33 14-34 14-34 14-36 14-38 14-38 14-39 14-40 14-41 14-42 14-44 14-46 14-46 14-48 14-50 14-50 14-51 14-52 14-52 14-52 14-54 14-54 14-55 14-55 14-56 14-56 14-56 14-58 14-58 14-58

Plates versus Packings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14-59 14-61

PHASE DISPERSION General References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid-in-Gas Dispersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid Breakup into Droplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Droplet Breakup—High Turbulence. . . . . . . . . . . . . . . . . . . . . . . . . . Liquid-Column Breakup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid-Sheet Breakup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Isolated Droplet Breakup—in a Velocity Field . . . . . . . . . . . . . . . . . . Droplet Size Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Atomizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hydraulic (Pressure) Nozzles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of Physical Properties on Drop Size . . . . . . . . . . . . . . . . . . . . . Effect of Pressure Drop and Nozzle Size . . . . . . . . . . . . . . . . . . . . . . Spray Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two-Fluid (Pneumatic) Atomizers. . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotary Atomizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pipeline Contactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Entrainment Due to Gas Bubbling/Jetting through a Liquid. . . . . . . Fog Condensation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spontaneous (Homogeneous) Nucleation . . . . . . . . . . . . . . . . . . . . . . Growth on Foreign Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dropwise Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gas-in-Liquid Dispersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Objectives of Gas Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theory of Bubble and Foam Formation . . . . . . . . . . . . . . . . . . . . . . . Characteristics of Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Methods of Gas Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equipment Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Axial Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14-62 14-62 14-62 14-62 14-62 14-62 14-62 14-63 14-63 14-63 14-63 14-64 14-66 14-66 14-66 14-66 14-66 14-67 14-68 14-68 14-68 14-69 14-69 14-69 14-70 14-72 14-73 14-76 14-79 14-80

PHASE SEPARATION Gas-Phase Continuous Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions: Mist and Spray . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gas Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Particle-Size Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Collection Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Procedures for Design and Selection of Collection Devices . . . . . . . Collection Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Energy Requirements for Inertial-Impaction Efficiency . . . . . . . . . . Collection of Fine Mists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fiber Mist Eliminators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electrostatic Precipitators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electrically Augmented Collectors . . . . . . . . . . . . . . . . . . . . . . . . . . . Particle Growth and Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Collectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous Phase Uncertain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid-Phase Continuous Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Types of Gas-in-Liquid Dispersions. . . . . . . . . . . . . . . . . . . . . . . . . . . Separation of Unstable Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Separation of Foam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physical Defoaming Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chemical Defoaming Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . Foam Prevention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Automatic Foam Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14-81 14-81 14-81 14-81 14-82 14-82 14-82 14-83 14-93 14-93 14-93 14-94 14-94 14-95 14-95 14-95 14-95 14-95 14-95 14-95 14-96 14-97 14-97 14-98 14-98

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

14-3

Nomenclature ae A Af A Ae, A′ c c′ Csb, Cs Cv d db dh do dpc dpsd dpa50 D D32 Dvm e E E Ea Eg Eoc Eog Emv f F FLG g gc G GM h hf hT H H′ H Hg Hog HOL HL H′ HETP k1 k2 k kG kL K KOG, KG KOL L LM

Effective interfacial area Cross sectional area Fractional open area Absorption factor Effective absorption factor (Edmister) Concentration Stokes-Cunningham correction factor for terminal settling velocity Flooding coefficient Discharge coefficient Diameter Bubble diameter Hole diameter Orifice diameter Cut size of a particle collected in a device, 50% mass efficiency Mass median size particle in the pollutant gas Aerodynamic diameter of a real median size particle Diffusion coefficient Sauter mean diameter Volume mean diameter Entrainment, mass liquid/mass gas Plate or stage efficiency, fractional Power dissipation per mass Murphree plate efficiency, with entrainment, gas concentrations, fractional Point efficiency, gas phase only, fractional Overall column efficiency, fractional Overall point efficiency, gas concentrations, fractional Murphree plate efficiency, gas concentrations, fractional Fractional approach to flood F-factor for gas loading Flow parameter Gravitational constant Conversion factor Gas phase mass velocity Gas phase molar velocity Pressure head Height of froth Height of contacting Henry’s law constant Henry’s law constant Height of a transfer unit Height of a gas phase transfer unit Height of an overall transfer unit, gas phase concentrations Height of an overall transfer unit, liquid phase concentrations Height of a liquid phase transgfer unit Henry’s law coefficient Height equivalent to a theoretical plate or stage First order reaction velocity constant Second order reaction velocity constant Individual phase mass transfer coefficient gas phase mass transfer coefficient liquid phase mass transfer coefficient Vapor-liquid equilibrium ratio Overall mass transfer coefficient, gas concentrations Overall mass transfer coefficient, liquid concentrations Liquid mass velocity Liquid molar mass velocity

m2/m3 m2 -/-/-/kg-moles/m3

ft2/ft3 ft2 -/-/-/lb-mol/ft3

-/m/s -/m m m m

-/ft/s -/ft ft ft ft

µm

ft

µm

ft

µm m2/s m m kg/kg -/W

ft ft2/s ft ft lb/lb -/btu/lb

Lw m M nA p P, pT q Q R Rh s S Se, S′ T TS U Ua Un Ut x x* y y* Z

-/-/-

-/-/-

-/-

-/-

-/-

-/-

-/-/m/s(kg/m3)0.5 -/m/s2 1.0 (kg⋅m/N⋅s2) kg/s-m2 kg-moles/s-m2 mm m m

-/-/ft/s(lb/ft3)0.5 -/ft/s2 32.2(lb⋅ft)/ (lbf ⋅s2) lb/hr-ft2 lb-mol/hr-ft2 ft ft ft

m m

ft ft

m

ft

m m kPa/mole fraction m

ft ft atm/mole fraction ft

1/s m3/(s⋅kmol)

1/sec ft3/(h⋅lb-mol)

m/s m/s m/s -/-

ft/sec ft/sec ft/sec -/-

m/s

ft/sec

m/s kg/m2-s kmoles/m2-s

ft/sec lb/ft2-s lb-mol/ft2-s

Weir length Slope of equilibrium curve = dy*/dx Molecular weight Rate of solute transfer Partial pressure Total pressure Volumetric flow rate of liquid Volumetric flow rate of gas or vapor Gas constant Hydraulic radius Length of corrugation side, structured packing Stripping factor Effective stripping factor (Edmister) Absolute temperature Tray or plate spacing Linear velocity of gas Velocity of gas through active area Velocity of gas through net area Superficial velocity of gas Mole fraction, liquid phase Liquid mole fraction, equilibrium condition Mole fraction, gas or vapaor phase Gas mole fraction, equilibrium condition Height, plate spacing

m -/kg/kmol kmol/s kPa kPa m3/s m3/s

ft -/lb/lb-mol lb-mol/s atm atm ft3/s ft3/s

m

ft

m -/-/K m m/s m/s m/s m/s -/-

ft -/-/°F ft ft/s ft/s ft/s ft/s -/-

-/-/-

-/-/-

m

ft

-/-/-/-/-/kg/(s⋅m) m -/-/Pa-s m m2/s -/s kg/m3 mN/m -/-

-/-/-/-/-/lb/(s-ft) ft -/-/lb/(ft-s) ft ft2/s -/s lb/ft3 dyn/cm -/-

Greek symbols α β ε φ γ Γ δ η λ µ µm ν π θ ρ σ ψ

Relative volatility Aeration factor Void fraction Relative froth density Activity coefficient Flow rate per length Effective film thickness Collection eficiency, fractional Stripping factor = m/(LM/GM) Absolute viscosity Microns Kinematic viscosity 3.1416. . . . Residence time Density Surface tension Fractional entrainment

Subscripts A AB B e i G L p w 1 2

Species A Species A diffusing through species B Species B Effective value Interface value Gas or vapor Liquid Particle water Tower bottom Tower top Dimensionless Groups

NFr NRe NSc NWe

Froude number = (UL2)/(Sg) Reynolds number = (SUge ρG)/(µG) Schmidt number = µ/(ρD) Weber number = (UL2 ρL S)/(σgc)

GENERAL REFERENCES: Astarita, G., Mass Transfer with Chemical Reaction, Elsevier, New York, 1967. Astarita, G., D. W. Savage and A. Bisio, Gas Treating with Chemical Solvents, Wiley, New York, 1983. Billet, R., Distillation Engineering, Chemical Publishing Co., New York, 1979. Danckwerts, P. V., GasLiquid Reactions, McGraw-Hill, New York, 1970. Distillation and Absorption 1987, Rugby, U.K., Institution of Chemical Engineers, 1988. Distillation and Absorption 1992, Rugby, U.K., Institution of Chemical Engineers, 1992. Hines,

A. L. and R. N. Maddox, Mass Transfer—Fundamentals and Applications, Prentice Hall, Englewood Cliffs, New Jersey, 1985. Kister, H. Z., Distillation Design, McGraw-Hill, New York, 1992. Lockett, M.J., Distillation Tray Fundamentals, Cambridge, U.K., Cambridge Unioversity Press, 1986. Kohl, A. L. and F. C. Riesenfeld, Gas Purification, 4th ed., Gulf, Houston, 1985. Sherwood, T. K., R. L. Pigford, C. R. Wilke, Mass Transfer, McGraw-Hill, New York, 1975. Treybal, R. E., Mass Transfer Operations, McGraw-Hill, New York, 1980.

INTRODUCTION Definitions Gas absorption is a unit operation in which soluble components of a gas mixture are dissolved in a liquid. The inverse operation, called stripping or desorption, is employed when it is desired to transfer volatile components from a liquid mixture into a gas. Both absorption and stripping, in common with distillation (Sec. 13), make use of special equipment for bringing gas and liquid phases into intimate contact. This section is concerned with the design of gasliquid contacting equipment, as well as with the design of absorption and stripping processes. Equipment Absorption, stripping, and distillation operations are usually carried out in vertical, cylindrical columns or towers in which devices such as plates or packing elements are placed. The gas and liquid normally flow countercurrently, and the devices serve to provide the contacting and development of interfacial surface through which mass transfer takes place. Background material on this mass transfer process is given in Sec. 5. Design Procedures The procedures to be followed in specifying the principal dimensions of gas absorption and distillation equipment are described in this section and are supported by several worked-out examples. The experimental data required for executing the designs are keyed to appropriate references or to other sections of the handbook. For absorption, stripping, and distillation, there are three main steps involved in design: 1. Data on the gas-liquid or vapor-liquid equilibrium for the system at hand. If absorption, stripping, and distillation operations are considered equilibrium-limited processes, which is the usual approach, these data are critical for determining the maximum possible separation. In some cases, the operations are are considered rate-based (see Sec. 13) but require knowledge of equilibrium at the phase interface. Other data required include physical properties such as viscosity and density and thermodynamic properties such as enthalpy. Section 2 deals with sources of such data. 2. Information on the liquid- and gas-handling capacity of the contacting device chosen for the particular separation problem. Such information includes pressure drop characteristics of the device, in order that an optimum balance between capaital cost (column cross section) and energy requirements might be achieved. Capacity and pressure drop characteristics of the available devices are covered later in this Sec. 14. 3. Determination of the required height of contacting zone for the separation to be made as a function of properties of the fluid mixtures and mass-transfer efficiency of the contacting device. This determination involves the calculation of mass-transfer parameters such as heights of transfer units and plate efficiencies as well as equilibrium or rate parameters such as theoretical stages or numbers of transfer units. An additional consideration for systems in which chemical reaction occurs is the provision of adequate residence time for desired reactions to occur, or minimal residence time to prevent undesired reactions from occurring. For equilibrium-based operations, the parameters for required height are covered in the present section. Data Sources in the Handbook Sources of data for the analysis or design of absorbers, strippers, and distillation columns are mani14-4

fold, and a detailed listing of them is outside the scope of the presentation in this section. Some key sources within the handbook are shown in Table 14-1. Equilibrium Data Finding reliable gas-liquid and vapor-liquid equilibrium data may be the most time-consuming task associated with the design of absorbers and other gas-liquid contactors, and yet it may be the most important task at hand. For gas solubility, an important data source is the set of volumes edited by Kertes et al., Solubility Data Series, published by Pergamon Press (1979 ff.). In the introduction to each volume, there is an excellent discussion and definition of the various methods by which gas solubility data have been reported, such as the Bunsen coefficient, the Kuenen coefficient, the Ostwalt coefficient, the absorption coefficient, and the Henry’s law coefficient. The fourth edition of The Properties of Gases and Liquids by Reid, Prausnitz and Poling (McGraw-Hill, New York, 1987) provides data and recommended estimation methods for gas solubility as well as the broader area of vapor-liquid equilibrium. Finally, the Chemistry Data Series by Gmehling et al., especially the title VaporLiquid Equilibrium Collection (DECHEMA, Frankfurt, Germany, 1979 ff.), is a rich source of data evaluated against the various models used for interpolation and extrapolation. Section 13 of this handbook presents a good discussion of equilibrium K values. TABLE 14-1 Directory to Key Data for Absorption and Gas-Liquid Contactor Design Type of data Phase equilibrium data Gas solubilities Pure component vapor pressures Equilibrium K values Thermal data Heats of solution Specific heats Latent heats of vaporization Transport property data Diffusion coefficients Liquids Gases Viscosities Liquids Gases Densities Liquids Gases Surface tensions Packed tower data Pressure drop and flooding Mass transfer coefficients HTU, physical absorption HTU with chemical reaction Height equivalent to a theoretical plate (HETP) Plate tower data Pressure drop and flooding Plate efficiencies Costs of gas-liquid contacting equipment

Section 2 2 13 2 2 2 2 2 2 2 2 2 2 14 5 5 14 14 14 14

DESIGN OF GAS-ABSORPTION SYSTEMS

14-5

DESIGN OF GAS-ABSORPTION SYSTEMS General Design Procedure The designer ordinarily is required to determine (1) the best solvent; (2) the best gas velocity through the absorber, namely the vessel diameter; (3) the height of the vessel and its internal members, which is the height and type of packing or the number of contacting trays; (4) the optimum solvent circulation through the absorber and stripper; (5) the temperatures of streams entering and leaving the absorber and the quantity of heat to be removed to account for heat of solution and other thermal effects; (6) the pressures at which the absorber and stripper will operate; and (7) the mechanical design of the absorption and stripping vessels (normally columns or towers), including flow distributors, packing supports, and so on. This section is concerned with all these choices. The problem presented to the designer of a gas-absorption unit usually specifies the following quantities: (1) gas flow rate; (2) gas composition, at least with respect to the component or components to be absorbed; (3) operating pressure and allowable pressure drop across the absorber; (4) minimum degree of recovery of one or more solutes; and, possibly, (5) the solvent to be employed. Items 3, 4, and 5 may be subject to economic considerations and therefore are sometimes left up to the designer. For determining the number of variables that must be specified in order to fix a unique solution for the design of an absorber one can use the same phase-rule approach described in Sec. 13 for distillation systems. Recovery of the solvent, sometimes by chemical means but more often by distillation, is almost always required, and the recovery system ordinarily is considered an integral part of the absorption-system process design. A more efficient solvent-stripping operation normally will result in a less costly absorber because of a smaller concentration of residual dissolved solute in the regenerated solvent; however, this may increase the overall cost of solvent recovery. A more detailed discussion of these and other economic considerations is presented later in this section. Selection of Solvent When choice is possible, preference is given to liquids with high solubilities for the solute; a high solubility reduces the amount of solvent to be circulated. The solvent should be relatively nonvolatile, inexpensive, noncorrosive, stable, nonviscous, nonfoaming, and preferably nonflammable. Since the exit gas normally leaves saturated with solvent, solvent loss can be costly and may present environmental contamination problems. Thus, low-cost solvents may be chosen over more expensive ones of higher solubility or lower volatility. Water generally is used for gases fairly soluble in water, oils for light hydrocarbons, and special chemical solvents for acid gases such as CO2, SO2, and H2S. Sometimes a reversible chemical reaction will result in a very high solubility and a minimum solvent rate. Data on actual systems are desirable when chemical reactions are involved, and those available are referenced later under “Absorption with Chemical Reaction.” Selection of Solubility Data Solubility values determine the liquid rate necessary for complete or economic solute recovery and so are essential to design. Equilibrium data generally will be found in one of three forms: (1) solubility data expressed either as solubility in weight or mole percent or as Henry’s-law coefficients, (2) purecomponent vapor pressures, or (3) equilibrium distribution coefficients (K values). Data for specific systems may be found in Sec. 2; additional references to sources of data are presented in this section. In order to define completely the solubility of a gas in a liquid, it generally is necessary to state the temperature, the equilibrium partial pressure of the solute gas in the gas phase, and the concentration of the solute gas in the liquid phase. Strictly speaking, the total pressure on the system also should be stated, but for low total pressures, less than about 507 kPa (5 atm), the solubility for a particular partial pressure of solute gas normally will be relatively independent of the total pressure of the system. For dilute concentrations of many gases and over a fairly wide range for some gases, the equilibrium relationship is given by Henry’s law, which relates the partial pressure developed by a dissolved solute A in a liquid solvent B by one of the following equations: pA = HxA (14-1)

PA = H′cA

or

(14-2)

where H is the Henry’s law coefficient expressed in kilopascals per mole-fraction solute in liquid and H′ is the Henry’s law coefficient expressed in kilopascals per kilomole per cubic meter. Although quite useful when it can be applied, this law should be checked experimentally to determine the accuracy with which it can be used. If Henry’s law holds, the solubility is defined by stating the value of the constant H (or H′) along with the temperature and the solute partial pressure for which it is to be employed. For quite a number of gases, Henry’s law holds very well when the partial pressure of the solute is less than about 100 kPa (1 atm). For partial pressures of the solute gas greater than 100 kPa, H seldom is independent of the partial pressure of the solute gas, and a given value of H can be used over only a narrow range of partial pressures. There is a strongly nonlinear variation of Henry’s-law constants with temperature as discussed by Schulze and Prausnitz [Ind. Eng. Chem. Fundam., 20, 175 (1981)]. Consultation of this reference is recommended before considering temperature extrapolations of Henry’s-law data. Additional data and information on the applicability of Henry’s-law constants can be found in the references cited earlier in the subsection “Directory to Key Gas-Absorption Data.” The use of Henry’s-law constants is illustrated by the following examples. Example 1: Gas Solubility It is desired to find out how much hydrogen can be dissolved in 100 weights of water from a gas mixture when the total pressure is 101.3 kPa (760 torr; 1 atm), the partial pressure of the H2 is 26.7 kPa (200 torr), and the temperature is 20°C. For partial pressures up to about 100 kPa the value of H is given in Sec. 3 as 6.92 × 106 kPa (6.83 × 104 atm) at 20°C. According to Henry’s law, xH2 = pH2 /HH2 = 26.7/6.92 × 106 = 3.86 × 10−6 The mole fraction x is the ratio of the number of moles of H2 in solution to the total moles of all constituents contained. To calculate the weights of H2 per 100 weights of H2O, one can use the following formula, where the subscripts A and w correspond to the solute (hydrogen) and solvent (water): x M 2.02 3.86 × 10 ᎏ 100 =
ᎏᎏ  ᎏ 100
ᎏ 1−x  M 1 − 3.86 × 10 18.02 A

A

A

w

−6

−6

= 4.33 × 10−5 weights H2 /100 weights H2O = 0.43 parts per million weight

Pure-component vapor pressures can be used for predicting solubilities for systems in which Raoult’s law is valid. For such systems pA = p°A xA, where p°A is the pure-component vapor pressure of the solute and pA is its partial pressure. Extreme care should be exercised when attempting to use pure-component vapor pressures to predict gas-absorption behavior. Both liquid-phase and vapor-phase nonidealities can cause significant deviations from the behavior predicted from pure-component vapor pressures in combination with Raoult’s law. Vapor-pressure data are available in Sec. 3 for a variety of materials. Whenever data are available for a given system under similar conditions of temperature, pressure, and composition, equilibrium distribution coefficients (K = y/x) provide a much more reliable tool for predicting vapor-liquid distributions. A detailed discussion of equilibrium K values is presented in Sec. 13. Calculation of Liquid-to-Gas Ratio The minimum possible liquid rate is readily calculated from the composition of the entering gas and the solubility of the solute in the exit liquor, saturation being assumed. It may be necessary to estimate the temperature of the exit liquid based on the heat of solution of the solute gas. Values of latent and specific heats and values of heats of solution (at infinite dilution) are given in Sec. 2. The actual liquid-to-gas ratio (solvent-circulation rate) normally will be greater than the minimum by as much as 25 to 100 percent and may be arrived at by economic considerations as well as by judgment and experience. For example, in some packed-tower applications involving very soluble gases or vacuum operation, the minimum quantity of solvent needed to dissolve the solute may be insufficient to keep the packing surface thoroughly wet, leading to poor distribution of the liquid stream.

14-6

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

When the solute concentration in the inlet gas is low and when nearly all the solute is being absorbed (this is the usual case), the approximation y1GM ⬟ x1LM ⬟ (y°1 /m)LM

(14-3)

leads to the conclusion that the ratio mGM/LM represents the fractional approach of the exit liquid to saturation with the inlet gas, i.e., mGM /LM ⬟ y°1 / y1

(14-4)

Optimization of the liquid-to-gas ratio in terms of total annual costs often suggests that the molar liquid-to-gas ratio LM /GM should be about 1.2 to 1.5 times the theoretical minimum corresponding to equilibrium at the rich end of the tower (infinite height), provided flooding is not a problem. This would be an alternative to assuming that LM /GM ⬟ m/0.7, for example. When the exit-liquor temperature rises owing to the heat of absorption of the solute, the value of m changes through the tower, and the liquid-to-gas ratio must be chosen to give reasonable values of m1GM /LM and m2GM /LM, where the subscripts 1 and 2 refer to the bottom and top of the absorption tower respectively. For this case the value of m2GM /LM will be taken to be somewhat less than 0.7, so that the value of m1GM /LM will not approach unity too closely. This rule-ofthumb approach is useful only when low solute concentrations and mild heat effects are involved. When the solute has a large heat of solution or when the feed gas contains high percentages of the solute, one should consider the use of internal cooling coils or intermediate external heat exchangers in a plate-type tower to remove the heat of absorption. In a packed tower, one could consider the use of multiple packed sections with intermediate liquid-withdrawal points so that the liquid could be cooled by external heat exchange. Selection of Equipment Packed columns usually are chosen for very corrosive materials, for liquids that foam badly, for either smallor large-diameter towers involving very low allowable pressure drops, and for small-scale operations requiring diameters of less than 0.6 m (2 ft). The type of packing is selected on the basis of resistance to corrosion, mechanical strength, capacity for handling the required flows, mass-transfer efficiency, and cost. Economic factors are discussed later in this section. Plate columns may be economically preferable for large-scale operations and are needed when liquid rates are so low that packing would be inadequately wetted, when the gas velocity is so low (owing to a very high L/G) that axial dispersion or “pumping” of the gas back down the (packed) column can occur, or when intermediate cooling is desired. Also, plate towers may have a better turndown ratio and are less subject to fouling by solids than are packed towers. Details on the operating characteristics of plate towers are given later in this section. Column Diameter and Pressure Drop Flooding determines the minimum possible diameter of the absorber column, and the usual design is for 60 to 80 percent of the flooding velocity. Maximum allowable pressure drop may be determined by the cost of energy for compression of the feed gas. For systems having a significant tendency to foam, the maximum allowable velocity will be lower than estimated flooding velocity, especially for plate towers. The safe range of operating velocities should include the velocity one would derive from economic considerations, as discussed later. Methods for predicting flooding velocities and pressure drops are given later in this section. Computation of Tower Height The required height of a gasabsorption or stripping tower depends on (1) the phase equilibria involved, (2) the specified degree of removal of the solute from the gas, and (3) the mass-transfer efficiency of the apparatus. These same considerations apply both to plate towers and to packed towers. Items 1 and 2 dictate the required number of theoretical stages (plate tower) or transfer units (packed tower). Item 3 is derived from the tray efficiency and spacing (plate tower) or from the height of one transfer unit (packed tower). Solute-removal specifications normally are derived from economic considerations. For plate towers, the approximate design methods described below may be used in estimating the number of theoretical stages, and the tray efficiencies and spacings for the tower can be specified on the basis of the information given later. Considerations involved in the rigorous design of theoretical stages for plate towers are treated in Sec. 13.

For packed towers, the continuous differential nature of the contact between gas and liquid leads to a design procedure involving the solution of differential equations, as described in the next subsection. It should be noted that the design procedures discussed in this section are not applicable to reboiled absorbers, which should be designed according to the methods described in Sec. 13. Caution is advised in distinguishing between systems involving pure physical absorption and those in which a chemical reaction can significantly affect design procedures. Selection of Stripper-Operating Conditions Stripping involves the removal of one or more volatile components from a liquid by contacting it with a gas such steam, nitrogen, or air. The operating conditions chosen for stripping normally result in a low solubility of the solute (i.e., a high value of m), so that the ratio mGM /LM will be larger than unity. A value of 1.4 may be used for rule-of-thumb calculations involving pure physical desorption. For plate-tower calculations the stripping factor S = KGM /LM, where K = y°/x, usually is specified for each tray. When the solvent from an absorption operation must be regenerated for recycling back to the absorber, one may employ a “pressureswing concept,” a “temperature-swing concept,” or a combination of both in specifying stripping conditions. In pressure-swing operation the temperature of the stripper is about the same as that of the absorber, but the stripping pressure is much lower. In temperatureswing operation the pressures are about equal, but the stripping temperature is much higher than the absorption temperature. In pressure-swing operation a portion of the dissolved gas may be “sprung” from the liquid by the use of a flash drum upstream of the stripping-tower feed point. This type of operation is discussed by Burrows and Preece [Trans. Inst. Chem. Eng., 32, 99 (1954)] and by Langley and Haselden [Inst. Chem. Eng. Symp. Ser. (London), no. 28 (1968)]. If the flashing of the feed liquid takes place inside the stripping tower, this effect must be accounted for in the design of the upper section in order to avoid overloading and flooding near the top of the tower. More often than not the rate at which residual absorbed gas can be driven from the liquid in a stripping tower is limited by the rate of a chemical reaction, in which case the liquid-phase residence time (and hence, the tower liquid holdup) becomes the most important design factor. Thus, many stripper-regenerators are designed on the basis of liquid holdup rather than on the basis of mass transfer rate. Approximate design equations applicable only to the case of pure physical desorption are developed later in this section for both packed and plate stripping towers. A more rigorous approach using distillation concepts may be found in Sec. 13. A brief discussion of desorption with chemical reaction is given in the subsection “Absorption with Chemical Reaction.” Design of Absorber-Stripper Systems The solute-rich liquor leaving a gas absorber normally is distilled or stripped to regenerate the solvent for recirculation back to the absorber, as depicted in Fig. 14-1. It is apparent that the conditions selected for the absorption step (e.g., temperature, pressure, LM /GM) will affect the design of the stripping tower, and, conversely, a selection of stripping conditions will affect the absorber design. The choice of optimum operating conditions for an absorber-stripper system therefore involves a combination of economic factors and practical judgments as to the operability of the system within the context of the overall process flow sheet. Note that in Fig. 14-1 the stripping vapor is provided by a reboiler; alternately, an extraneous stripping gas may be used. An appropriate procedure for executing the design of an absorberstripper system is to set up a carefully selected series of design cases and then evaluate the investment costs, the operating costs, and the operability of each case. Some of the economic factors that need to be considered in selecting the optimum absorber-stripper design are discussed later in the subsection “Economic Design of Absorption Systems.” Importance of Design Diagrams One of the first things a designer should try to do is lay out a carefully constructed equilibrium curve, y° = F(x), on an xy diagram, as shown in Fig. 14-2. A horizontal line corresponding to the inlet-gas composition y1 is then the locus of feasible outlet-liquor compositions, and a vertical line corresponding to the inlet-solvent-liquor composition x2 is the locus of feasible out-

DESIGN OF GAS-ABSORPTION SYSTEMS

(a)

14-7

(b)

FIG. 14-1 Gas absorber using a solvent regenerated by stripping. (a) Absorber. (b) Stripper.

let-gas compositions. These lines are indicated as y = y1 and x = x2 respectively on Fig. 14-2. For gas absorption, the region of feasible operating lines lies above the equilibrium curve; for stripping, the feasible region for operating lines lies below the equilibrium curve. These feasible regions are bounded by the equilibrium curve and by the lines x = x2 and y = y1. By inspection, one should be able to visualize those operating lines that are feasible and those that would lead to “pinch points” within the tower. Also, it is possible to determine if a particular proposed design for solute recovery falls within the feasible envelope. Once the design recovery for an absorber has been established, the operating curve can be constructed by first locating the point x2, y2 on the diagram. The intersection of the horizontal line corresponding to the inlet gas composition y1 with the equilibrium curve y° = F(x)

(a) FIG. 14-2

Design diagrams for (a) absorption and (b) stripping.

defines the theoretical minimum liquid-to-gas ratio for systems in which there are no intermediate pinch points. The operating line which connects this point with the point x2, y2 corresponds to the minimum value of LM /GM. The actual design value of LM /GM normally should be around 1.2 to 1.5 times this minimum. Thus, the actual design operating line for a gas absorber will pass through the point x2, y2 and will intersect the line y = y1 to the left of the equilibrium curve. For stripping one begins by using the design specification to locate the point x1, y1. Then the intersection of the vertical line x = x2 with the equilibrium curve y° = F(x) defines the theoretical minimum gas-toliquid ratio. The actual value of GM /LM is chosen to be about 20 to 50 percent higher than this minimum, so the actual design operating line will intersect the line x = x2 at a point somewhat below the equilibrium curve.

(b)

14-8

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

Design diagrams minimize the possibility of making careless mistakes and allow one to assess easily the effects of operating variable changes on the operability of the system relative to pinch points, etc. Whenever analytical calculations or computer programs are being used for the design of gas-absorption systems, the construction of

design diagrams based either on calculation results or on computer printouts may reveal problem areas or even errors in the design concept. It is strongly recommended that design diagrams be employed whenever possible.

PACKED-TOWER DESIGN Methods for estimating the height of the active section of counterflow differential contactors such as packed towers, spray towers, and falling-film absorbers are based on rate expressions representing mass transfer at a point on the gas-liquid interface and on material balances representing the changes in bulk composition in the two phases that flow past each other. The rate expressions are based on the interphase mass-transfer principles described in Sec. 5. Combination of such expressions leads to an integral expression for the number of transfer units or to equations related closely to the number of theoretical plates. The paragraphs which follow set forth convenient methods for using such equations, first in a general case and then for cases in which simplifying assumptions are valid. Use of Mass-Transfer-Rate Expression Figure 14-3 shows a section of a packed absorption tower together with the nomenclature that will be used in developing the equations which follow. In a differential section dh, we can equate the rate at which solute is lost from the gas phase to the rate at which it is transferred through the gas phase to the interface as follows: −d(GM y) = −GM dy − y dGM = NAa dh

(14-5)

When only one component is transferred, dGM = −NAa dh

(14-6)

Substitution of this relation into Eq. (14-5) and rearranging yields GM dy dh = − ᎏᎏ (14-7) NAa(1 − y) For this derivation we use the gas-phase rate expression NA = kG(y − yi) and integrate over the tower to obtain y1 GM dy hT = (14-8) ᎏᎏ y2 kG a(1 − y)(y − yi)



Multiplying and dividing by yBM place Eq. (14-8) into the HGNG format y1 GM yBM dy hT = ᎏ ᎏᎏ y2 kG ayBM (1 − y)(y − yi)

 





y1

yBM dy (14-9) ᎏᎏ = HG,av NG (1 − y)(y − yi) The general expression given by Eq. (14-8) is more complex than normally is required, but it must be used when the mass-transfer coefficient varies from point to point, as may be the case when the gas is not dilute or when the gas velocity varies as the gas dissolves. The values of yi to be used in Eq. (14-8) depend on the local liquid composition xi and on the temperature. This dependency is best represented by using the operating and equilibrium lines as discussed later. Example 3 illustrates the use of Eq. (14-8) for scrubbing chlorine from air with aqueous caustic solution. For this case one can make the simplifying assumption that yi, the interfacial partial pressure of chlorine over the aqueous caustic solution, is zero owing to the rapid and complete reaction of the chlorine after it dissolves. We note that the feed gas is not dilute. = HG,av

y2

Example 2: Packed Height Requirement Let us compute the height of packing needed to reduce the chlorine concentration of 0.537 kg/ (s⋅m2), or 396 lb/(h⋅ft2), of a chlorine-air mixture containing 0.503 mole-fraction chlorine to 0.0403 mole fraction. On the basis of test data described by Sherwood and Pigford (Absorption and Extraction, McGraw-Hill, 1952, p. 121) the value of kG ayBM at a gas velocity equal to that at the bottom of the packing is equal to 0.1175 kmol/(s⋅m3), or 26.4 lb⋅mol/(h⋅ft3). The equilibrium back pressure yi can be assumed to be negligible. Solution. By assuming that the mass-transfer coefficient varies as the 0.8 power of the local gas mass velocity, we can derive the following relation: 71y + 29(1 − y) 1 − y 0.8 Kˆ Ga = kG ayBM = 0.1175 ᎏᎏ ᎏ1 71y1 + 29(1 − y1) 1 − y where 71 and 29 are the molecular weights of chlorine and air respectively. Noting that the inert-gas (air) flow rate is given by G′M = GM(1 − y) = 5.34 × 10−3 kmol/(s⋅m2), or 3.94 lb⋅mol/(h⋅ft2), and introducing these expressions into the integral gives 0.503 dy 1 − y 0.8 hT = 1.82 ᎏ ᎏᎏᎏ 0.0403 29 + 42y (1 − y)2 ln [1/(1 − y)] This definite integral can be evaluated numerically by the use of Simpson’s rule to obtain hT = 0.305 m (1 ft).



 








Use of Operating Curve Frequently, it is not possible to assume that yi = 0 as in Example 2, owing to diffusional resistance in the liquid phase or to the accumulation of solute in the liquid stream. When the back pressure cannot be neglected, it is necessary to supplement the equations with a material balance representing the operating line or curve. In view of the countercurrent flows into and from the differential section of packing shown in Fig. 14-3, a steady-state material balance leads to the following equivalent relations:

FIG. 14-3

stripper.

Nomenclature for material balances in a packed-tower absorber or

d(GMy) = d(LMx) (14-10) dx dy G′M ᎏ2 = L′M ᎏ2 (14-11) (1 − y) (1 − x) where L′M = molar mass velocity of the inert-liquid component and G′M = molar mass velocity of the inert gas. LM, L′M, GM, and G′M are superficial velocities based on the total tower cross section. Equation (14-11) is the differential equation of the operating curve, and its integral around the upper portion of the packing is the equation for the operating curve

PACKED-TOWER DESIGN

14-9

x2 y y2 x G′M ᎏ − ᎏ = L′M ᎏ − ᎏ (14-12) 1 − y 1 − y2 1 − x 1 − x2 For dilute solutions in which the mole fractions of x and y are small, the total molar flows GM and LM will be very nearly constant, and the operating-curve equation is

individual measurements of HG and HL by Eq. (14-19). Correlations for predicting HG, HL, and HOG in nonreacting systems are presented in Sec. 5. yBM mGM xBM HOG = ᎏ HG + ᎏ ᎏ HL (14-19a) y°BM LM y°BM

GM(y − y2) = LM(x − x2)

xBM LM yBM (14-19b) HOL = ᎏ HL + ᎏ ᎏ HG x B°M mGM x B°M On occasion the changes in gas flow and in the mole fraction of inert gas are so small that the inclusion of terms such as (1 − y) and y°BM can be neglected or at least can be included in an approximate way. This leads to some of the simplified procedures described later. One such simplification was suggested by Wiegand [Trans. Am. Inst. Chem. Eng., 36, 679 (1940)], who pointed out that the logarithmic-mean mole fraction of inert gas y°BM (or yBM) is often very nearly equal to the arithmetic mean. Thus, substitution of the relation y°BM (1 − y°) + (1 − y) y − y° (14-20) ᎏ ⬟ ᎏᎏ = ᎏ + 1 (1 − y) 2(1 − y) 2(1 − y) into the equations presented earlier leads to the simplified forms y1 1 1−y dy NG = ᎏ ln ᎏ2 + (14-21) ᎏ y 2 1 − y1 2 y − yi









(14-13)

This equation gives the relation between the bulk compositions of the gas and liquid streams at each level in the tower for conditions in which the operating curve can be approximated by a straight line. Figure 14-4 shows the relationship between the operating curve and the equilibrium curve yi = F(xi) for a typical example involving solvent recovery, where yi and xi are the interfacial compositions (assumed to be in equilibrium). Once y is known as a function of x along the operating curve, yi can be found at corresponding points on the equilibrium curve by (y − yi)/(xi − x) = kL /kG = k′L ρ L /k′G pT = LM HG /GMHL

(14-14)

where LM = molar liquid mass velocity, GM = molar gas mass velocity, HL = height of one transfer unit based on liquid-phase resistance, and HG = height of one transfer unit based on gas-phase resistance. Thence, the integral in Eq. (14-8) can be evaluated. Calculation of Transfer Units In the general case the equations described above must be employed in calculating the height of packing required for a given separation. However, if the local mass-transfer coefficient kG ayBM is approximately proportional to the first power of the local gas velocity GM, then the height of one gas-phase transfer unit, defined as HG = GM /kGayBM, will be constant in Eq. (14-9). Similar considerations lead to an assumption that the height of one overall gas-phase transfer unit HOG may be taken as constant. The height of packing required is then calculated according to the relation hT = HG NG = HOG NOG

(14-15)

where NG = number of gas-phase transfer units and NOG = number of overall gas-phase transfer units. When HG and HOG are not constant, it may be valid to employ averaged values between the top and bottom of the tower and the relation hT = HG,av NG = HOG,av NOG In these equations, the terms NG and NOG are defined by y1 yBM dy NG = ᎏᎏ y2 (1 − y)(y − yi)



and by

NOG =



y1

y2

y°BM dy ᎏᎏ (1 − y)(y − y°)

(14-16)

(14-17) (14-18)

respectively. Equation (14-18) is the more useful one in practice: it requires either actual experimental HOG data or values estimated by combining

Relationship between equilibrium curve and operating curve in a packed absorber; computation of interfacial compositions.

FIG. 14-4





y1 1 1 − y2 dy (14-22) NOG = ᎏ ln ᎏ + ᎏ y2 y − y° 2 1 − y1 The second (integral) terms represent the numbers of transfer units for an infinitely dilute gas. The first terms, frequently amounting to only small corrections, give the effect of a finite level of gas concentration. The procedure for applying Eqs. (14-21) and (14-22) involves two steps: (1) evaluation of the integrals and (2) addition of the correction corresponding to the first (logarithmic) term. The discussion which follows deals only with the evaluation of the integral terms (first step). The simplest possible case occurs when (1) both the operating and the equilibrium lines are straight (i.e., there are dilute solutions), (2) Henry’s law is valid (y°/x = yi / xi = m), and (3) absorption heat effects are negligible. Under these conditions, the integral term in Eq. (14-20) may be computed by Colburn’s equation [Trans. Am. Inst. Chem. Eng., 35, 211 (1939)]: mGM 1 mGM y1 − mx2 NOG = ᎏᎏ ln 1 − ᎏ ᎏ + ᎏ (14-23) 1 − (mGM /LM) LM y2 − mx2 LM











Figure (14-5) is a plot of Eq. (14-23) from which the value of NOG can be read directly as a function of mGM /LM and the ratio of concentrations. This plot and Eq. (14-23) are equivalent to the use of a logarithmic mean of terminal driving forces, but they are more convenient because one does not need to compute the exit-liquor concentration x1. In many practical situations involving nearly complete cleanup of the gas, an approximate result can be obtained from the equations just presented even when solutions are concentrated or when absorption heat effects are present. In such cases the driving forces in the upper part of the tower are very much smaller than those at the bottom, and the value of mGM /LM used in the equations should be the ratio of the slopes of the equilibrium line m and the operating line LM /GM in the low-concentration range near the top of the tower. Another approach is to divide the tower arbitrarily into a lean section (near the top), where approximate methods are valid, and to deal with the rich section separately. If the heat effects in the rich section are appreciable, consideration could be given to installing cooling units near the bottom of the tower. In any event a design diagram showing the operating and equilibrium curves should be prepared to check on the applicability of any simplified procedure. Figure 14-8, presented in Example 6 is one such diagram for an adiabatic absorption tower. Stripping Equations Stripping, or desorption, involves the removal of a volatile component from the liquid stream by contact with an inert gas such as nitrogen or steam. In this case the change in concentration of the liquid stream is of prime importance, and it

14-10

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

is more convenient to formulate the rate equation analogous to Eq. (14-6) in terms of the liquid composition x. This leads to the following equations defining numbers of transfer units and heights of transfer units based on liquid-phase resistance: x1 xBM dx hT = HL (14-24) ᎏᎏ = HLNL x2 (1 − x)(xi − x)





x1

x°BM dx (14-25) ᎏᎏ = HOLNOL (1 − x)(x° − x) where, as before, subscripts 1 and 2 refer to the bottom and top of the tower respectively (see Fig. 14-3). In situations in which one cannot assume that HL and HOL are constant, these terms must be incorporated inside the integrals in Eqs. (14-24) and (14-25). and the integrals must be evaluated graphically or numerically (by using Simpson’s rule, for example). In the normal case involving stripping without chemical reactions, the liquid-phase resistance will dominate, making it preferable to use Eq. (14-25) in conjunction with the relation HL ⬟ HOL. The Wiegand approximations of the above integrals in which arithmetic means are substituted for the logarithmic means xBM and x°BM are x2 1 1−x dx NL = ᎏ ln ᎏ1 + (14-26) ᎏ x1 x − xi 2 1 − x2 hT = HOL

x2





x2 1 1 − x1 dx (14-27) NOL = ᎏ ln ᎏ + ᎏ x1 x − x° 2 1 − x2 In these equations, the first term is a correction for finite liquid-phase concentrations, and the integral term represents the numbers of transfer units required for dilute solutions. It would be very unusual in practice to find an example in which the first (logarithmic) term is of any significance in a stripper design. For dilute solutions in which both the operating and the equilibrium lines are straight and in which heat effects can be neglected, the integral term in Eq. (14-27) is

1 NOL = ᎏᎏ ln (1 − LM/mGM)

x − y /m L ᎏ +ᎏ 
1 − ᎏ mG 
x − y /m  mG  LM

M

2

1

1

1

M

(14-28)

M

This equation is identical in form to Eq. (14-23). Thus, Fig. 14-5 is applicable if the concentration ratio (x2 − y1/m)/(x1 − y1/m) is substituted for the abscissa and if the parameter on the curves is identified as LM /mGM. Example 3: Air Stripping of VOCs from Water A 0.45-m diameter packed column was used by Dvorack et al. [Environ. Sci. Tech. 20, 945 (1996)] for removing trichloroethylene (TCE) from wastewater by stripping with atmospheric air. The column was packed with 2.5-cm Pall rings, fabricated from polypropylene, to a height of 3.0 m. The TCE concentration in the entering water was 38 parts per million by weight (ppmw). A molar ratio of entering water to entering air was kept at 23.7. What degree of removal was to be expected? The temperatures of water and air were 20°C. Pressure was atmospheric. Solution. For TCE in water, the Henry’s law coefficient may be taken as 417 atm/mf at 20°C. In this low-concentration region, the coefficient is constant and equal to the slope of the equilibrium line m. The solubility of TCE in water, based on H = 417, is 2390 ppm. Because of this low solubility, the entire resistance to mass transfer resides in the liquid phase. Thus, Eq. (14-25) may be used to obtain NOL, the number of overall liquid phase transfer units. In the equation, the ratio xBM⋅/(1 − x) is unity because of the very dilute solution. It is necessary to have a value of HL for the packing used, at given flow rates of liquid and gas. Methods for estimating HL may be found in Sec. 5. Dvorack et al. found HOL = 0.8 m. Then, for hT = 3.0 m, NL = NOL = 3.0/0.8 = 3.75 transfer units. Transfer units may be calculated from Eq. 14-25, replacing mole fractions with ppm concentrations, and since the operating and equilibrium lines are straight, 38 − (ppm)exit NOL = ᎏᎏ = 3.75 ln 38/(ppm)exit Solving, (ppm)exit = 0.00151. Thus, the stripped water would contain 1.51 parts per billion of TCE.

Use of HTU and KGa Data In estimating the size of a commercial gas absorber or liquid stripper it is desirable to have data on the

Number of overall gas-phase mass-transfer units in a packed absorption tower for constant mGM /LM; solution of Eq. (14-23). (From Sherwood and Pigford, Absorption and Extraction, McGraw-Hill, New York, 1952.) FIG. 14-5

overall mass transfer coefficients (or heights of transfer units) for the system of interest, and at the desired conditions of temperature, pressure, solute concentration, and fluid velocities. Such data should be obtained in an apparatus of pilot-plant or semiworks size to avoid abnormalities of scaleup. It must be remembered that values of the mass-transfer parameters are dependent not only on the phase properties and mass throughput in the contactor but also on the type of device used. Within the packing category, there are both random and ordered (structured) type packing elements. Physical characteristics of these devices will be described later. When no KGa or HTU data are available, their values may be estimated by means of a generalized model. A summary of useful models is given in Section 5, Table 5-28. The values obtained may then be combined by the use of Eq. 14-19 to obtain values of HOG and HOL. This procedure is not valid, however, when the rate of absorption is limited by a chemical reaction. Use of HETP Data for Absorber Design Distillation design methods (see Sec. 13) normally involve determination of the number of theoretical equilibrium stages or plates N. Thus, when packed towers are employed in distillation applications, it is common practice to rate the efficiency of tower packings in terms of the height of packing equivalent to one theoretical plate (HETP). The HETP of a packed-tower section, valid for either distillation or dilute-gas absorption and stripping systems in which constant molal overflow can be assumed and in which no chemical reactions occur, is related to the height of one overall gas-phase mass-transfer unit HOG by the equation ln (mGM /LM) HETP = HOG ᎏᎏ (mGM /LM − 1)

(14-29)

For gas-absorption systems in which the inlet gas is concentrated, the correct equation is




y°BM HETP = ᎏ 1−y



av

ln (mGM /LM) HOG ᎏᎏ mGM /LM − 1

(14-30)

PLATE-TOWER DESIGN where the correction term y°BM /(1 − y) is averaged over each individual theoretical plate. The equilibrium compositions corresponding to each theoretical plate may be estimated by the methods described in the subsection “Plate-Tower Design.” These compositions are used in conjunction with the local values of the gas and liquid flow rates and

14-11

the equilibrium slope m to obtain values for HG, HL, and HOG corresponding to the conditions on each theoretical stage, and the local values of the HETP are then computed by Eq. (14-30). The total height of packing required for the separation is the summation of the individual HETPs computed for each theoretical stage.

PLATE-TOWER DESIGN The design of a plate tower for gas-absorption or gas-stripping operations involves many of the same principles employed in distillation calculations, such as the determination of the number of theoretical plates needed to achieve a specified composition change (see Sec. 13). Distillation differs from gas absorption in that it involves the separation of components based on the distribution of the various substances between a gas phase and a liquid phase when all the components are present in both phases. In distillation, the new phase is generated from the original feed mixture by vaporization or condensation of the volatile components, and the separation is achieved by introducing reflux to the top of the tower. In gas absorption, the new phase consists of an inert nonvolatile solvent (absorption) or an inert nonsoluble gas (stripping), and normally no reflux is involved. The following paragraphs discuss some of the considerations peculiar to gas-absorption calculations for plate towers and some of the approximate design methods that can be employed when simplifying assumptions are valid. Graphical Design Procedure Construction of design diagrams (xy diagrams showing the equilibrium and operating curves) should be an integral part of any design involving the distribution of a single solute between an inert solvent and an inert gas. The number of theoretical plates can be stepped off rigorously provided the curvatures of the operating and equilibrium lines are correctly accounted for in the diagram. This procedure is valid even though an insoluble inert gas is present in the gas phase and an inert nonvolatile solvent is present in the liquid phase. Figure 14-6 illustrates the graphical method for a three-theoreticalplate system. Note that in gas absorption the operating line is above the equilibrium curve, whereas in distillation this does not happen. In gas stripping, the operating line will be below the equilibrium curve. On Fig. 14-6, note that the stepping-off procedure begins on the operating line. The starting point xf ,y3 represents the compositions of the entering lean wash liquor and of the gas exiting from the top of the tower, as determined by the design specifications. After three steps one reaches the point x1,yf, representing the compositions of the solute-rich feed gas yf and of the solute-rich liquor leaving the bottom of the tower x1.

Algebraic Method for Dilute Gases By assuming that the operating and equilibrium curves are straight lines and that heat effects are negligible, Souders and Brown [Ind. Eng. Chem., 24, 519 (1932)] developed the following equation: (y1 − y2)/(y1 − y°2 ) = (AN + 1 − A)/(AN + 1 − 1)

(14-31)

where N = number of theoretical plates, y1 = mole-fraction solute in the entering gas, y2 = mole-fraction solute in the leaving gas, y°2 = mx2 = mole-fraction solute in equilibrium with the incoming solvent liquor (zero for a pure solvent), and A = absorption factor = LM /mGM. Note that the absorption factor is the reciprocal of the expression given in Eq. (14-4) for packed columns. When A = 1, Eq. (14-31) is indeterminate, and for this case the solution is given by (y1 − y2)/(y1 − y°2 ) = N/(N + 1)

(14-32)

Although Eq. (14-31) is convenient for computing the composition of the exit gas as a function of the number of theoretical stages, an alternative equation derived by Colburn [Trans. Am. Inst. Chem. Eng., 35, 211 (1939)] is more useful when the number of theoretical plates is the unknown: ln [(1 − A−1)(y1 − y°2 )/(y2 − y°2 ) + A−1] N = ᎏᎏᎏᎏ (14-33) ln (A) The numerical results obtained by using either Eq. (14-31) or Eq. (14-33) are identical. Thus, the two equations may be used interchangeably as the need arises. Comparison of Eqs. (14-33) and (14-23) shows that NOG /N = ln (A)/(1 − A−1)

(14-34)

thus revealing the close relationship between theoretical stages in a plate tower and mass-transfer units in a packed tower. Equations (14-23) and (14-33) are related to each other by virtue of the relation hT = HOG NOG = (HETP)N

(14-35)

Algebraic Method for Concentrated Gases When the feed gas is concentrated, the absorption factor, which is defined in general as A = LM /KGM where K = y°/x, can vary throughout the tower owing to changes in the equilibrium K values due to temperature increases. An approximate solution to this problem can be obtained by substitution of the “effective” absorption factors Ae and A′ derived by Edmister [Ind. Eng. Chem., 35, 837 (1943)] into the equation y1 − y2 1 (LMx)2 AeN + 1 − Ae ᎏ = 1 − ᎏ ᎏ ᎏᎏ y1 A′ (GMy)1 AeN + 1 − 1





(14-36)

where subscripts 1 and 2 refer to the bottom and top of the tower respectively and the absorption factors are defined by the equations A ) .2 Ae = A 1+ 05 − 0.5 1(2+

(14-37)

A′ = A1(A2 + 1)/(A1 + 1)

(14-38)

This procedure has been applied to the absorption of C5 and lighter hydrocarbon vapors into a lean oil, for example. Stripping Equations When the liquid feed is dilute and the operating and equilibrium curves are straight lines, the stripping equations analogous to Eqs. (14-31) and (14-33) are Graphical method for a three-theoretical-plate gas-absorption tower with inlet-liquor composition xj and inlet-gas composition yj. FIG. 14-6

(x2 − x1)/(x2 − x°1) = (SN + 1 − S)/(SN + 1 − 1) where x°1 = y1 /m; S = mGM /LM = A−1; and

(14-39)

14-12

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

ln [(1 − A)(x2 − x°1)/(x1 − x°1) + A] N = ᎏᎏᎏᎏ (14-40) ln (S) For systems in which the concentrations are large and the stripping factor S may vary along the tower, the following Edmister equations [Ind. Eng. Chem., 35, 837 (1943)] are applicable: 1 (GM y)1 SeN + 1 − Se x2 − x1 ᎏ = 1 − ᎏ ᎏ ᎏᎏ x2 S′ (LM x)2 SeN + 1 − 1

(14-41)

S + ) .2 1+ 05 − 0.5 Se = S 2(1

(14-42)

S′ = S2(S1 + 1)/(S2 + 1)

(14-43)



where



and the subscripts 1 and 2 refer to the bottom and top of the tower respectively. Equations (14-37) and (14-42) represent two different ways of obtaining an effective factor, and a value of Ae obtained by taking the reciprocal of Se from Eq. (14-42) will not check exactly with a value of Ae derived by substituting A1 = 1/S1 and A2 = 1/S2 into Eq. (14-37). Regardless of this fact, the equations generally give reasonable results for approximate design calculations. It should be noted that throughout this section the subscripts 1 and 2 refer to the bottom and to the top of the apparatus respectively regardless of whether it is an absorber or a stripper. This has been done to maintain internal consistency among all the equations and to prevent the confusion created in some derivations in which the numbering system for an absorber is different from the numbering system for a stripper. Tray Efficiencies in Plate Absorbers and Strippers Computations of the number of theoretical plates N assume that the liquid on each plate is completely mixed and that the vapor leaving the plate is in equilibrium with the liquid. In actual practice a condition of complete equilibrium cannot exist since interphase mass transfer requires a finite driving-force difference. This leads to the definition of an overall plate efficiency E = Ntheoretical / Nactual

(14-44)

which can be correlated to system design variables. Mass-transfer theory indicates that for trays of a given design the factors most likely to influence E in absorption and stripping towers are the physical properties of the fluids and the dimensionless ratio mGM/LM. Systems in which the mass transfer is gas-film-controlled may be expected to have plate efficiencies as high as 50 to 100 percent, whereas plate efficiencies as low as 1 percent have been reported for the absorption of gases of low solubility (large m) into solvents of relatively high viscosity. The fluid properties are represented by the Schmidt numbers of the gas and liquid phases. For gases, the Schmidt numbers normally are close to unity and are independent of temperature and pressure. Thus, the gas-phase mass-transfer coefficients are relatively independent of the system. By contrast, the liquid-phase Schmidt numbers range from about 102 to 104 and depend strongly on the temperature. The effect of temperature on the liquid-phase mass-transfer coefficient is related primarily to changes in the liquid viscosity with temperature, and this derives primarily from the strong dependency of the liquid-phase Schmidt number upon viscosity. Consideration of the preceding discussion in connection with the relationship between mass-transfer coefficients (see Sec. 5):

O’Connell correlation for overall column efficiency Eoc for absorption. To convert HP/µ in pound-moles per cubic foot-centipoise to kilogrammoles per cubic meter-pascal-second, multiply by 1.60 × 104. [O’Connell, Trans. Am. Inst. Chem. Eng., 42, 741 (1946).]

FIG. 14-7

1/KG = (1/kG + m/kL)

(14-45)

indicates that variations in the overall resistance to mass transfer in absorbers and strippers are related primarily to variations in the liquid-phase viscosity µ and to variations in the slope m. A correlation of the efficiency of plate absorbers in terms of the viscosity of the liquid solvent and the solubility of the solute gas was developed by O’Connell [Trans. Am. Inst. Chem. Eng., 42, 741 (1946)]. The O’Connell correlation for plate absorbers is presented here as Fig. 14-7. The best procedure for making plate-efficiency corrections (which obviously can be quite large) is to use experimental-test data from a prototype system that is large enough to be representative of an actual commercial tower. Example 4: Actual Plates for Steam Stripping The number of actual plates required for steam-stripping an acetone-rich liquor containing 0.573 mole percent acetone in water is to be estimated. The design overhead recovery of acetone is 99.9 percent, leaving 18.5 ppm weight of acetone in the stripper bottoms. The design operating temperature and pressure are 101.3 kPa and 94°C respectively, the average liquid-phase viscosity is 0.30 cP, and the average value of K = y°/x for these conditions is 33. By choosing a value of mGM /LM = S = A−1 = 1.4 and noting that the stripping medium is pure steam (i.e., x°1 = 0), the number of theoretical trays according to Eq. (14-40) is ln [(1 − 0.714)(1000) + 0.714] N = ᎏᎏᎏᎏ = 16.8 ln (1.4) The O’Connell parameter for gas absorbers is ρL/KMµL, where ρL is the liquid density, lb/ft3; µL is the liquid viscosity, cP; M is the molecular weight of the liquid; and K = y°/x. For the present design ρL /KMµL = 60.1/(33 × 18 × 0.30) = 0.337 and according to the O’Connell graph for absorbers (Fig. 14-7) the overall tray efficiency for this case is estimated to be 30 percent. Thus, the required number of actual trays is 16.8/0.3 = 56 trays.

HEAT EFFECTS IN GAS ABSORPTION Overview One of the most important considerations involved in designing gas-absorption towers is to determine whether or not temperatures will vary along the length of the tower because of heat effects, since the solubility of the solute gas normally depends strongly upon the temperature. When heat effects can be neglected, computation of the tower dimensions and required flows is relatively straight-

forward, as indicated by the simplified design methods discussed earlier for both packed and plate absorbers and strippers. When heat effects cannot be neglected, the computational problem becomes much more difficult. Heat effects that may cause temperatures to vary from point to point in a gas absorber are (1) the heat of solution of the solute

HEAT EFFECTS IN GAS ABSORPTION (including heat of condensation, heat of mixing, and heat of reaction), which can lead to a rise in the liquid temperature; (2) the heat of vaporization or condensation of the solvent; (3) the exchange of sensible heat between the gas and liquid phases; and (4) the loss of sensible heat from the fluids to internal or external cooling coils or to the atmosphere via the tower walls. Y. T. Shah (Gas-Liquid-Solid Reactor Design, McGraw-Hill, New York, 1979, p. 51) has reviewed the literature concerning heat effects in systems involving gas-liquid reactions and concludes that in the majority of the systems involving chemical reactions temperature effects are not very important. For some systems in which large amounts of heat may be liberated, there are compensating effects which decrease the effect on the rate of absorption. For example, increasing temperatures tend simultaneously to increase the rate of chemical reaction and to decrease the solubility of the reactant at the solvent interface. Systems in which compensating effects can occur include absorption of CO2 in amine solutions and absorption of CO2 in NaOH solutions. There are, however, a number of well-known systems in which heat effects definitely cannot be ignored. Examples include absorption of ammonia in water, dehumidification of air with concentrated H2SO4, absorption of HCl in water, and absorption of SO3 in H2SO4. Another interesting example is the absorption of acetone in water, in which the heat effects are mild but not negligible. Some very thorough and knowledgeable discussions of the problems involved in gas absorption with large heat effects have been presented by Coggan and Bourne [Trans. Inst. Chem. Eng., 47, T96, T160 (1969)], by Bourne, von Stockar, and Coggan [Ind. Eng. Chem. Process Des. Dev., 13, 115, 124 (1974)], and also by von Stockar and Wilke [Ind. Eng. Chem. Fundam., 16, 89 (1977)]. The first two of these references discuss plate-tower absorbers and include interesting experimental studies of the absorption of ammonia in water. The third reference discusses the design of packed-tower gas absorbers and includes a shortcut design method based on a semitheoretical correlation of rigorous calculation results. All these authors clearly demonstrate both theoretically and experimentally that when the solvent is volatile, the temperature inside an absorber can go through a maximum. They note that the least expensive of the solvents, water, is capable of exhibiting this unusual “hot-spot” behavior. From a designer’s point of view there are a number of different approaches to be considered in dealing with heat effects, depending on the requirements of the job at hand. For example, one can (1) add internal or external heat-transfer surface to remove heat from the absorber; (2) treat the process as if it were isothermal by arbitrarily assuming that the temperature of the liquid phase is everywhere the same and then add a design safety factor; (3) employ the classical adiabatic model, which assumes that the heat of solution manifests itself only as sensible heat in the liquid phase and that solvent vaporization is negligible; (4) use semitheoretical shortcut methods derived from rigorous calculations; and (5) employ rigorous design procedures requiring the use of a digital computer. For preliminary-screening work the simpler methods may be adequate, but for final designs one should seriously consider using a more rigorous approach. Effects of Operating Variables Conditions that can give rise to significant heat effects are (1) an appreciable heat of solution and (2) absorption of large amounts of solute in the liquid phase. The second of these conditions can arise when the solute concentration in the inlet gas is large, when the liquid flow rate is relatively low (small LM/GM), when the solubility of the solute in the liquid phase is high, and/or when the operating pressure is high. When the solute is absorbed very rapidly, the rate of heat liberation often is largest near the bottom of the tower. This has the effect of causing the equilibrium line to curve upward near the solute-rich end, although it may remain relatively straight near the lean end, corresponding to the temperature of the lean solvent. If the solute-rich gas entering the bottom of an absorption tower is cold, the liquid phase may be cooled somewhat by transfer of sensible heat to the gas. A much stronger cooling effect can occur when the solvent is volatile and the entering rich gas is not saturated with respect to the solvent. It is possible to experience a condition in which

14-13

solvent is being evaporated near the bottom of the tower and condensed near the top. Under these conditions there may develop a pinch point in which the operating and equilibrium curves approach each other at a point inside the tower. In the references cited previously, the authors discuss the influence of operating variables upon the performance of plate towers when large heat effects are involved. Some general observations are as follows: Operating Pressure Raising the pressure may increase the separation efficiency considerably. Calculations involving the absorption of methanol from water-saturated air showed that doubling the pressure doubled the concentration of methanol which could be tolerated in the feed gas while still achieving a preset concentration specification in the off gas. Temperature of Fresh Solvent The temperature of the entering solvent has surprisingly little influence upon the degree of absorption or upon the internal-temperature profiles in an absorber when the heat effects are due primarily to heat of solution or to solvent vaporization. In these cases the temperature profile in the liquid phase apparently is dictated solely by the internal-heat effects. Temperature and Humidity of Rich Gas Cooling and consequent dehumidification of the feed gas to an absorption tower can be very beneficial. A high humidity (or relative saturation with solvent) limits the capacity of the gas phase to take up latent heat and therefore is unfavorable to absorption. Thus, dehumidification of the inlet gas prior to introducing it into the tower is worth considering in the design of gas absorbers with large heat effects. Liquid-to-Gas Ratio The L/G ratio can have a significant influence on the development of temperature profiles in gas absorbers. High L/G ratios tend to result in less strongly developed temperature profiles owing to the high heat capacity of the liquid phase. As the L/G is increased, the operating line moves away from the equilibrium line and there is a tendency for more solute to be absorbed per stage. However, there is a compensating effect in that as more heat is liberated at each stage, the plate temperatures will tend to rise, causing an upward shifting of the equilibrium line. As the L/G is decreased, the concentration of solute tends to build up in the upper parts of the absorber, and the point of highest temperature tends to move upward in the tower until finally the maximum temperature develops only on the topmost plate. Of course, the capacity of the liquid phase to absorb solute falls progressively as the L/G is reduced. Number of Stages When the heat effects combine to produce an extended zone within the tower where little absorption is taking place (i.e., a pinch zone), the addition of plates to the tower will have no useful effect on separation efficiency. Solutions to these difficulties must be sought by increasing the solvent flow, introducing strategically placed coolers, cooling and dehumidifying the inlet gas, and/or raising the tower operating pressure. Equipment Considerations When the solute has a large heat of solution and the feed gas contains a large percentage of solute, as in the absorption of HCl in water, the effects of heat release during absorption may be so pronounced that the installation of heat-transfer surface to remove the heat of absorption may be as important as providing sufficient interfacial area for the mass-transfer process itself. The added heat-transfer surface may consist of internal cooling coils on the plates, or else the solvent may be withdrawn from a point intermediate in the tower and passed through an external heat exchanger (intercooler) for cooling. In many cases the rate of heat liberation is largest near the bottom of the tower, where solute absorption is more rapid, so that cooling surfaces or intercoolers are required only on the first few trays at the bottom of the column. Coggan and Bourne [Trans. Inst. Chem. Eng., 47, T96, T160 (1969)] found, however, that the optimal position for a single interstage cooler does not necessarily coincide with the position of the maximum temperature or with the center of a pinch. They found that in a 12-plate tower, two strategically placed interstage coolers tripled the allowable ammonia feed concentration for a given off-gas specification. For a case involving methanol absorption, it was found that more separation was possible in a 12-stage column with two intercoolers than in a simple column with 100 stages and no intercoolers.

14-14

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

In the case of HCl absorption, a shell-and-tube heat exchanger often is employed as a cooled wetted-wall vertical-column absorber so that the exothermic heat of reaction can be removed continuously as it is released into the liquid film. Installation of heat-exchange equipment to precool and dehumidify the feed gas to an absorber also deserves consideration in order to take advantage of the cooling effects created by vaporization of solvent in the lower sections of the tower. Classical Isothermal Design Method When the feed gas is sufficiently dilute, the exact design solution may be approximated by the isothermal one over a broad range of L/G ratios, since heat effects generally are less important when washing dilute-gas mixtures. The problem, however, is one of defining the term “sufficiently dilute” for each specific case. For a new absorption duty, the assumption of isothermal operation must be subjected to verification by the use of a rigorous design procedure. When heat-exchange surface is being provided in the design of an absorber, the isothermal design procedure can be rendered valid by virtue of the exchanger design specifications. With ample surface area and a close approach, isothermal operation can be guaranteed. For preliminary screening and feasibility studies or for rough cost estimates, one may wish to employ a version of the isothermal method which assumes that the liquid temperatures in the tower are everywhere equal to the inlet-liquid temperature. In their analysis of packed-tower designs, von Stockar and Wilke [Ind. Eng. Chem. Fundam. 16, 89 (1977)] showed that the isothermal method tended to underestimate the required depth of packing by a factor of as much as 1.5 to 2. Thus, for rough estimates one may wish to employ the assumption that the temperature is equal to the inlet-liquid temperature and then apply a design factor to the result. Another instance in which the constant-temperature method is used involves the direct application of experimental KGa values obtained at the desired conditions of inlet temperatures, operating pressure, flow rates, and feed-stream compositions. The assumption here is that, regardless of any temperature profiles that may exist within the actual tower, the procedure of “working the problem in reverse” will yield a correct result. One should be cautious about extrapolating such data very far from the original basis and be careful to use compatible equilibrium data. Classical Adiabatic Design Method The classical adiabatic method assumes that the heat of solution serves only to heat up the liquid stream and that there is no vaporization of solvent. This assumption makes it feasible to relate increases in the liquid-phase temperature to the solute concentration x by a simple enthalpy balance. The equilibrium curve can then be adjusted to account for the corresponding temperature rise on an xy diagram. The adjusted equilibrium curve will become more concave upward as the concentration increases, tending to decrease the driving forces near the bottom of the tower, as illustrated in Fig. 14-8 in Example 6. Colburn [Trans. Am. Inst. Chem. Eng., 35, 211 (1939)] has shown that when the equilibrium line is straight near the origin but curved slightly at its upper end, NOG can be computed approximately by assuming that the equilibrium curve is a parabolic arc of slope m2 near the origin and passing through the point x1, K1x1 at the upper end. The Colburn equation for this case is 1 NOG = ᎏᎏ 1 − m2GM /LM m2GM (1 − m2GM /LM)2 y1 − m2 x2 × ln ᎏᎏ ᎏ +ᎏ 1 − K1GM /LM y2 − m2 x2 LM










(14-46)

Comparisons by von Stockar and Wilke [Ind. Eng. Chem. Fundam., 16, 89 (1977)] between the rigorous and the classical adiabatic design methods for packed towers indicate that the simple adiabatic method underestimates the packing depths by as much as a factor of 1.25 to 1.5. Thus, when using the classical adiabatic method, one should consider the possible need to apply a design safety factor. A slight variation of the above method accounts for increases in the solvent content of the gas stream between the inlet and the outlet of the tower and assumes that the evaporation of solvent tends to cool the liquid. This procedure offsets a part of the temperature rise that

would have been predicted with no solvent evaporation and leads to the prediction of a shorter tower. Rigorous Design Methods A detailed discussion of rigorous methods for the design of packed and plate absorbers when large heat effects are involved is beyond the scope of this section. In principle, material and energy balances may be executed under the same constraints as for rigorous distillation calculations (see Sec. 13). The MESH equations are solved, but for absorption or stripping, convergence may be quite sensitive to the relatively large heat effects compared with distillation. Absorption-stripping programs are included in the software packages for process simulation. The paper of von Stockar and Wilke [Ind. Eng. Chem. Fundam. 16, 89 (1977)] presents an interesting shortcut method for the design of packed absorbers which closely approximates rigorous results. Direct Comparison of Design Methods The following problem, originally solved by Sherwood, Pigford, and Wilke (Mass Transfer, McGraw-Hill, New York, 1975, p. 616), was employed by von Stockar and Wilke as the basis for a direct comparison between the isothermal, adiabatic, semitheoretical shortcut, and rigorous design methods for estimating the height of packed towers. Example 5: Packed Absorber, Acetone into Water Inlet gas to an absorber consists of a mixture of 6 mole percent acetone in air saturated with water vapor at 15°C and 101.3 kPa (1 atm). The scrubbing liquor is pure water at 15°C, and the inlet gas and liquid rates are given as 0.080 and 0.190 kmol/s respectively. The liquid rate corresponds to 20 percent over the theoretical minimum as calculated by assuming a value of x1 corresponding to complete equilibrium between the exit liquor and the incoming gas. HG and HL are given as 0.42 and 0.30 m respectively, and the acetone equilibrium data at 15°C are pA0 = 19.7 kPa (147.4 torr), γA = 6.46, and mA = 6.46 × 19.7/101.3 = 1.26. The heat of solution of acetone is 7656 cal/gmol (32.05 kJ/gmol), and the heat of vaporization of solvent (water) is 10,755 cal/gmol (45.03 kJ/gmol). The problem calls for determining the height of packing required to achieve a 90 percent recovery of the acetone. The following table compares the results obtained by von Stockar and Wilke (op. cit.) for the various design methods:

Design method used

NOG

Packed height, m

Design safety factor

Rigorous Shortcut rigorous Classical adiabatic Classical isothermal

5.56 5.56 4.01 3.30

3.63 3.73 2.38 1.96

1.00 0.97 1.53 1.85

It should be clear from this example that there is considerable room for error when approximate design methods are employed in situations involving large heat effects, even for a case in which the solute concentration in the inlet gas was only 6 mole percent.

Example 6: Solvent Rate for Absorption Let us consider the absorption of acetone from air at atmospheric pressure into a stream of pure water fed to the top of a packed absorber at 25°C. The inlet gas at 35°C contains 2 percent by volume of acetone and is 70 percent saturated with water vapor (4 percent H2O by volume). The mole-fraction acetone in the exit gas is to be reduced to 1/400 of the inlet value, or 50 ppmv. For 100 kmol of feed-gas mixture, how many kilomoles of fresh water should be fed to provide a positivedriving force throughout the packing? How many transfer units will be needed according to the classical adiabatic method? What is the estimated height of packing required if HOG = 0.70 m? The latent heats at 25°C are 7656 kcal/kmol for acetone and 10,490 kcal/kmol for water, and the differential heat of solution of acetone vapor in pure water is given as 2500 kcal/kmol. The specific heat of air is 7.0 kcal/(kmol⋅K). Acetone solubilities are defined by the equation K = y°/x = γa pa /pT

(14-47)

where the vapor pressure of pure acetone in mmHg (torr) is given by (Sherwood et al., Mass Transfer, McGraw-Hill, New York, 1975, p. 537): pA0 = exp (18.1594 − 3794.06/T)

(14-48)

and the liquid-phase-activity coefficient may be approximated for low concentrations (x ≤ 0.01) by the equation γa = 6.5 exp (2.0803 − 601.2/T)

(14-49)

Typical values of acetone solubility as a function of temperature at a total pressure of 760 mmHg are shown in the following table:

MULTICOMPONENT SYSTEMS t, °C

25

30

35

40

γa pa, mmHg K = γa p0a /760

6.92 229 2.09

7.16 283 2.66

7.40 346 3.37

7.63 422 4.23

14-15

For dry gas and liquid water at 25°C, the following enthalpies are computed for the inlet- and exit-gas streams (basis, 100 kmol of gas entering): Entering gas: Acetone 2(2500 + 7656) = 20,312 kcal Water vapor 4(10,490) = 41,960 Sensible heat (100)(7.0)(35 − 25) = 7,000 69,272 kcal Exit gas (assumed saturated with water at 25°C): Acetone (2/400)(94/100)(2500) = 12 kcal 23.7 Water vapor 94 ᎏᎏ (10,490) = 31,600 760 − 23.7






31,612 kcal Enthalpy change of liquid = 69,272 − 31,612 = 37,660 kcal/100 kmol gas. Thus, ∆t = t1 − t2 = 37,660/18LM, and the relation between LM /GM and the liquid-phase temperature rise is LM /GM = (37,660)/(18)(100) ∆ t = 20.92/∆ t The following table summarizes the critical values for various assumed temperature rises: ∆ t, °C 0 2 3 4 5 6 7

LM/GM

K1

K1GM /LM

m2GM /LM

10.46 6.97 5.23 4.18 3.49 2.99

2.09 2.31 2.42 2.54 2.66 2.79 2.93

0. 0.221 0.347 0.486 0.636 0.799 0.980

0. 0.200 0.300 0.400 0.500 0.599 0.699

Evidently a temperature rise of 7°C would not be a safe design because the equilibrium line nearly touches the operating line near the bottom of the tower, creating a pinch. A temperature rise of 6°C appears to give an operable design, and for this case LM = 349 kmol per 100 kmol of feed gas.

FIG. 14-8

Design diagram for adiabatic absorption of acetone in water,

Example 6. The design diagram for this case is shown in Fig. 14-8, in which the equilibrium curve is drawn with a french curve so that the slope at the origin m2 is equal to 2.09 and passes through the point x1 = 0.02/3.49 = 0.00573 at y°1 = 0.00573 × 2.79 = 0.0160. The number of transfer units can be calculated from the adiabatic design equation, Eq. (14-46): (1 − 0.599)2 1 NOG = ᎏᎏ ln ᎏᎏ (400) + 0.599 = 14.4 (1 − 0.799) 1 − 0.599 The estimated height of tower packing by assuming HOG = 0.70 m and a design safety factor of 1.5 is





hT = (14.4)(0.7)(1.5) = 15.1 m (49.6 ft) For this tower, one should consider the use of two or more shorter packed sections instead of one long section.

MULTICOMPONENT SYSTEMS When no chemical reactions are involved in the absorption of more than one soluble component from an insoluble gas, the design conditions (pressure, temperature, and liquid-to-gas ratio) normally are determined by the volatility or the physical solubility of the least soluble component for which complete recovery is economical. Components of lower volatility (higher solubility) also will be recovered completely. The more volatile (i.e., less soluble) components will be only partially absorbed even though the effluent liquid becomes completely saturated with respect to these lighter substances. When a condition of saturation exists, the value of y1/y2 will remain finite even for an infinite number of plates or transfer units. This can be seen in Fig. 14-9, in which the asymptotes become vertical for values of mGM /LM greater than unity. If the amount of volatile component in the incoming fresh solvent is negligible, then the limiting value of y1/y2 for each of the highly volatile components is y1/y2 = S/(S − 1)

(14-50)

where S = mGM /LM and the subscripts 1 and 2 refer to the bottom and top of the tower respectively. When the gas stream is dilute, absorption of each constituent can be considered separately as if the other components were absent. The following example illustrates the use of this principle.

Example 7: Multicomponent Absorption, Dilute Case Air entering a tower contains 1 percent acetaldehyde and 2 percent acetone. The liquid-to-gas ratio for optimum acetone recovery is LM /GM = 3.1 mol/mol when the fresh-solvent temperature is 31.5°C. The value of y°/x for acetaldehyde has been measured as 50 at the boiling point of a dilute solution, 93.5°C. What will the percentage recovery of acetaldehyde be under conditions of optimal acetone recovery? Solution. If the heat of solution is neglected, y°/x at 31.5°C is equal to 50(1200/7300) = 8.2, where the factor in parentheses is the ratio of pureacetaldehyde vapor pressures at 31.5 and 93.5°C respectively. Since LM/GM is equal to 3.1, the value of S for the aldehyde is S = mGM/LM = 8.2/3.1 = 2.64, and y1 /y2 = S/(S − 1) = 2.64/1.64 = 1.61. The acetaldehyde recovery is therefore equal to 100 × 0.61/1.61 = 38 percent recovery. In concentrated systems the change in gas and liquid flow rates within the tower and the heat effects accompanying the absorption of all the components must be considered. A trial-and-error calculation from one theoretical stage to the next usually is required if accurate results are to be obtained, and in such cases calculation procedures similar to those described in Sec. 13 normally are employed. A computer procedure for multicomponent adiabatic absorber design has been described by Feintuch and Treybal [Ind. Eng. Chem. Process Des. Dev., 17, 505 (1978)]. Also see Holland, Fundamentals and Modeling of Separation Processes, Prentice Hall, Englewood Cliffs, N.J., 1975.

14-16

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

When two or more gases are absorbed in systems involving chemical reactions, the situation is much more complex. This topic is discussed later in the subsection “Absorption with Chemical Reaction.” Graphical Design Method for Dilute Systems The following notation for multicomponent absorption calculations has been adapted from Sherwood, Pigford, and Wilke (Mass Transfer, McGraw-Hill, New York 1975, p. 415): L Ms = moles of solvent per unit time G0M = moles of rich feed gas to be treated per unit time X = moles of one solute per mole of solute-free solvent fed to the top of the tower Y = moles of one solute in the gas phase per mole of rich feed gas to be treated Subscripts 1 and 2 refer to the bottom and top of the tower respectively, and the material balance for any one component may be written as L Ms (X − X2) = G0M (Y − Y2)

(14-51)

L Ms (X1 − X) = G0M (Y1 − Y)

(14-52)

or else as For the special case of absorption from lean gases with relatively large amounts of solvent, the equilibrium lines are defined for each component by the relation Y0 = K′X

(14-53)

Thus, the equilibrium line for each component passes through the origin with slope K′, where K′ = K(GM/G0M)/(LM/L Ms )

(14-54)

and K = y°/x. When the system is sufficiently dilute, K′ ⬟ K. The liquid-to-gas ratio L Ms /G0M is chosen on the basis of the solubility of the least soluble substance in the feed gas that must be absorbed completely. Each individual component will then have its own operating line with slope equal to L Ms /G0M (i.e., the operating lines for all the various components will be parallel to each other). A typical diagram for the complete absorption of pentane and heavier components from a lean gas mixture is shown in Fig. 14-9. The oil used as solvent for this case was assumed to be solute-free (i.e., X2 = 0), and the “key component,” butane, was identified as that component absorbed in appreciable amounts whose equilibrium line is most

nearly parallel to the operating lines (i.e., the K value for butane is approximately equal to L Ms /G0M). In Fig. 14-9, the composition of the gas with respect to components more volatile than butane will approach equilibrium with the liquid phase at the bottom of the tower. The gas composition with respect to components less volatile (heavier) than butane will approach equilibrium with the oil entering the tower, and since X2 = 0, the components heavier than butane will be completely absorbed. Four theoretical plates have been stepped off for the key component (butane) on Fig. 14-9 and are sufficient to give a 75 percent recovery of butane. The operating lines for the other components were drawn in with the same slope and were placed so as to give the same number of theoretical plates insofar as possible. The diagram of Fig. 14-9 shows that for the light components equilibrium is achieved easily in fewer than four theoretical plates and that for the heavier components nearly complete recovery is obtained in four theoretical plates. The diagram also shows that absorption of the light components takes place in the upper part of the tower and absorption of the heavier components takes place in the lower section of the tower. Algebraic Design Method for Dilute Systems The design method described above can be performed algebraically by employing the following modified version of the Kremser formula: Y1 − Y2 (A0)N + 1 − A0 (14-55) ᎏ = ᎏᎏ Y1 − mX2 (A0)N + 1 − 1 where for dilute gas absorption A0 = L Ms /mG0M and m ⬟ K = y0/x. The left-hand side of Eq. (14-55) represents the efficiency of absorption of any one component of the feed-gas mixture. If the solvent oil is denuded of solute so that X2 = 0, the left-hand side is equal to the fractional absorption of the component from the rich feed gas. When the number of theoretical plates N and the liquid and gas rates L Ms and GM0 have been fixed, the fractional absorption of each component may be computed directly and the operating lines need not be placed by trial and error as in the graphical approach described earlier. According to Eq. (14-55), when A0 is less than unity and N is large, (Y1 − Y2)/(Y1 − mX2) ⬟ A0

(14-56)

This equation can be employed for estimating the fractional absorption of the more volatile components whenever the value of A0 for the component is smaller than the value of A0 for the key component by a factor of 3 or more. When A0 is very much larger than unity and when N is large, the right-hand side of Eq. (14-55) becomes equal to unity. This signifies that the gas will leave the top of the tower in equilibrium with the incoming oil, and when X2 = 0, it corresponds to complete absorption of the component in question. Thus, the least volatile components may be assumed to be at equilibrium with the lean oil at the top of the tower. When A0 = 1, the right-hand side of Eq. (14-55) becomes indeterminate. The solution for this case is (Y1 − Y2)/(Y1 − mX2) = N/(N + 1)

(14-57)

For systems in which the absorption factor A0 for each component is not constant throughout the tower, an effective absorption factor for use in the equations just presented can be estimated by the Edmister formula 0 0(A .2 A0e = A 1+ 05 − 0.5 1 + 2 )

(14-58)

This procedure is a reasonable approximation only when no pinch points exist within the tower and when the absorption factors vary in a regular manner between the bottom and the top of the tower. Example 8: Multicomponent Absorption, Concentrated Case

Graphical design method for multicomponent systems; absorption of butane and heavier components in a solute-free lean oil.

FIG. 14-9

A hydrocarbon feed gas is to be treated in an existing four-theoretical-tray absorber to remove butane and heavier components. The recovery specification for the key component, butane, is 75 percent. The composition of the exit gas from the absorber and the required liquid-to-gas ratio are to be estimated. The feed-gas composition and the equilibrium K values for each component at the temperature of the (solute-free) lean oil are presented in the following table:

ABSORPTION WITH CHEMICAL REACTION Component

Mole %

K value

Methane Ethane Propane Butane Pentane C6 plus

68.0 10.0 8.0 8.0 4.0 2.0

74.137 12.000 3.429 0.833 0.233 0.065

For N = 4 and Y2 /Y1 = 0.25, the value of A0 for butane is found to be equal to 0.89 from Eq. (14-55) by using a trial-and-error method. The values of A0 for the other components are then proportional to the ratios of their K values to that of butane. For example, A0 = 0.89(0.833/12.0) = 0.062 for ethane. The values of A0 for each of the other components and the exit-gas composition as computed from Eq. (14-55) are shown in the following table:

14-17

Component

A0

Y2, mol/ mol feed

Exit gas, mole %

Methane Ethane Propane Butane Pentane C6 plus

0.010 0.062 0.216 0.890 3.182 11.406

67.3 9.4 6.3 2.0 0.027 0.0012

79.1 11.1 7.4 2.4 0.03 0.0014

The molar liquid-to-gas ratio required for this separation is computed as LsM /G0M = A0 × K = 0.89 × 0.833 = 0.74. We note that this example is the analytical solution to the graphical design problem shown in Fig. 14-9, which therefore is the design diagram for this system.

ABSORPTION WITH CHEMICAL REACTION Introduction Many present-day commercial gas absorption processes involve systems in which chemical reactions take place in the liquid phase. These reactions generally enhance the rate of absorption and increase the capacity of the liquid solution to dissolve the solute, when compared with physical absorption systems. A necessary prerequisite to understanding the subject of absorption with chemical reaction is the development of a thorough understanding of the principles involved in physical absorption, as discussed earlier in this section and in Section 5. There are a number of excellent references the subject, such as the book by Danckwerts (Gas-Liquid Reactions, McGraw-Hill, New York, 1970) and Astarita et al. (Gas Treating with Chemical Solvents, Wiley, New York, 1983). Recommended Overall Design Strategy When considering the design of a gas-absorption system involving chemical reactions, the following procedure is recommended: 1. Consider the possibility that the physical design methods described earlier in this section may be applicable. 2. Determine whether commercial design overall KˆG a values are available for use in conjunction with the traditional design method, being careful to note whether or not the conditions under which the KˆG a data were obtained are essentially the same as for the new design. Contact the various tower-packing vendors for information as to whether KˆG a data are available for your system and conditions. 3. Consider the possibility of scaling up the design of a new system from experimental data obtained in a laboratory-bench scale or a small pilot-plant unit. 4. Consider the possibility of developing for the new system a rigorous, theoretically based design procedure which will be valid over a wide range of design conditions. These topics are discussed in the subsections that follow. Applicability of Physical Design Methods Physical design methods such as the classical isothermal design method or the classical adiabatic design method may be applicable for systems in which chemical reactions are either extremely fast or extremely slow or when chemical equilibrium is achieved between the gas and liquid phases. If the liquid-phase reaction is extremely fast and irreversible, the rate of absorption may in some cases be completely governed by the gas-phase resistance. For practical design purposes one may assume (for example) that this gas-phase mass-transfer limited condition will exist when the ratio yi /y is less than 0.05 everywhere in the apparatus. From the basic mass-transfer flux relationship for species A (Sec. 5), NA = kG(y − yi) = kL(xi − x)

ubility), or the reaction-enhancement factor φ = kL /kL0 is very large, or both. As discussed later, the reaction-enhancement factor φ will be large for all extremely fast pseudo-first-order reactions and will be large for extremely fast second-order irreversible reaction systems in which there is a sufficiently large excess of liquid-phase reagent. When the rate of an extremely fast second-order irreversible reaction system A + νB → products is limited by the availability of the liquid-phase reagent B, then the reaction-enhancement factor may be estimated by the formula φ = 1 + B0/νci. In systems for which this formula is applicable, it can be shown that the interface concentration yi will be equal to zero whenever the ratio kg yν/kL0 B0 is less than or equal to unity. Figure 14-10 illustrates the gas-film and liquid-film concentration profiles one might find in an extremely fast (gas-phase mass-transfer limited) second-order irreversible reaction system. The solid curve for reagent B represents the case in which there is a large excess of bulkliquid reagent B0. The dashed curve in Fig. 14-10 represents the case in which the bulk concentration B0 is not sufficiently large to prevent the depletion of B near the liquid interface and for which the equation φ = 1 + B0/␯ci is applicable. Whenever these conditions on the ratio yi /y apply, the design can be based upon the physical rate coefficient kˆ G or upon the height of one gas-phase mass-transfer unit HG. The gas-phase mass-transfer limited condition is approximately valid, for instance, in the following systems: absorption of NH3 into water or acidic solutions, vaporization of water into air, absorption of H2O into concentrated sulfuric acid solutions, absorption of SO2 into alkali solutions, absorption of H2S from a dilute-

(14-59)

one can readily show that this condition on yi /y requires that the ratio x/xi be negligibly small (i.e., a fast reaction) and that the ratio mkG /kL = mkG /kL0 φ be less than 0.05 everywhere in the apparatus. The ratio mkG /kL0 φ will be small if the equilibrium back pressure of the solute over the liquid solution is small (i.e., small m; high reactant sol-

FIG. 14-10 Gas-phase and liquid-phase solute-concentration profiles for an extremely fast (gas-phase mass-transfer limited) irreversible reaction system A + ␯B → products.

14-18

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

gas stream into a strong alkali solution, absorption of HCl into water or alkaline solutions, or absorption of Cl2 into strong alkali. When liquid-phase chemical reactions are extremely slow, the gas-phase resistance can be neglected and one can assume that the rate of reaction has a predominant effect upon the rate of absorption. In this case the differential rate of transfer is given by the equation dnA = RA fHS dh = (k0La/ρL)(ci − c)S dh

(14-60)

where nA = rate of solute transfer, RA = volumetric reaction rate, a function of c and T, fH = fractional liquid volume holdup in tower or apparatus, S = tower cross-sectional area, h = vertical distance, k0L = liquid-phase mass-transfer coefficient for pure physical absorption, a = effective interfacial mass-transfer area per unit volume of tower or apparatus, ρL = average molar density of liquid phase, ci = solute concentration in liquid at gas-liquid interface, and c = solute concentration in bulk liquid. Although the right-hand side of Eq. (14-60) remains valid even when chemical reactions are extremely slow, the mass-transfer driving force may become increasingly small, until finally c ⬟ ci. For extremely slow first-order irreversible reactions, the following rate expression can be derived from Eq. (14-60): RA = k1c = k1ci /(1 + k1ρL fH/k0L a)

(14-61)

where k1 = first-order reaction rate coefficient. For dilute systems in countercurrent absorption towers in which the equilibrium curve is a straight line (i.e., yi = mxi) the differential relation of Eq. (14-60) is formulated as dnA = −GMS dy = k1cfHS dh

(14-62)

where GM = molar gas-phase mass velocity and y = gas-phase solute mole fraction. Substitution of Eq. (14-61) into Eq. (14-62) and integration lead to the following relation for an extremely slow first-order reaction in an absorption tower: y2 = y1 exp (−γ) where

k1ρL fHhT /mGM γ = ᎏᎏ (1 + k1ρL fH/kL0 a)

(14-63) (14-64)

In Eq. (14-63) the subscripts 1 and 2 refer to the bottom and the top of the tower respectively. The Hatta number NHa usually is employed as the criterion for determining whether or not a reaction can be considered extremely slow. For extremely slow reactions a reasonable criterion is 0 NHa = kD 1/k A L ≤ 0.3

(14-65)

where DA = liquid-phase diffusion coefficient of the solute in the sol-

vent. Figure 14-11 illustrates the concentration profiles in the gas and liquid films for the case of an extremely slow chemical reaction. We note that when the second term in the denominator of Eq. (14-64) is small, the liquid holdup in the tower can have a significant influence upon the rate of absorption if an extremely slow chemical reaction is involved. When chemical equilibrium is achieved quickly throughout the liquid phase (or can be assumed to exist), the problem becomes one of properly defining the physical and chemical equilibria for the system. It sometimes is possible to design a plate-type absorber by assuming chemical-equilibrium relationships in conjunction with a stage efficiency factor as is done in distillation calculations. Rivas and Prausnitz [Am. Inst. Chem. Eng. J., 25, 975 (1979)] have presented an excellent discussion and example of the correct procedures to be followed for systems involving chemical equilibria. Traditional Design Method The traditionally employed conventional procedure for designing packed-tower gas-absorption systems involving chemical reactions makes use of overall volumetric mass-transfer coefficients as defined by the equation K G′ a = nA /(h T SpT ∆ y°1 m )

K′Ga = KGa/pT = kLa/H Gas-phase and liquid-phase solute-concentration profiles for an extremely slow (kinetically limited) reaction system for which NHa is less than 0.3.

FIG. 14-11

(14-66)

where K′Ga = overall volumetric mass-transfer coefficient, nA = rate of solute transfer from the gas to the liquid phase, hT = total height of tower packing, S = tower cross-sectional area, pT = total system pressure, and ∆y°1 m is defined by the equation (y − y°)1 − (y − y°)2 (14-67) ∆y°1 m = ᎏᎏᎏ ln [(y − y°)1/(y − y°)2] in which subscripts 1 and 2 refer to the bottom and top of the absorption tower respectively, y = mole-fraction solute in the gas phase, and y° = gas-phase solute mole fraction in equilibrium with bulk-liquidphase solute concentration x. When the equilibrium line is straight, y° = mx. The traditional design method normally makes use of overall K′Ga values even when resistance to transfer lies predominantly in the liquid phase. For example, the CO2-NaOH system most commonly used for comparing the K′Ga values of various tower packings is a liquidphase-controlled system. When the liquid phase is controlling, extrapolation to different concentration ranges or operating conditions is not recommended since changes in the reaction mechanism can cause kL to vary unexpectedly and the overall KG′ a values do not explicitly show such effects. Overall K′Ga data may be obtained from tower-packing vendors for many of the established commercial gas-absorption processes. Such data often are based either upon tests in large-diameter test units or upon actual commercial operating data. Since extrapolation to untried operating conditions is not recommended, the preferred procedure for applying the traditional design method is equivalent to duplicating a previously successful commercial installation. When this is not possible, then a commercial demonstration at the new operating conditions may be required, or else one could consider using some of the more rigorous methods described later. Aside from the lack of an explicitly defined liquid-phase-resistance term, the limitations on the use of Eq. (14-66) are related to the fact that its derivation implicitly assumes that the system is dilute (yBM ⬟ 1) and that the operating and equilibrium curves are straight lines over the range of tower operation. Also, Eq. (14-66) is strictly valid only for the temperature and pressure at which the original test was run even though the total pressure pT appears in the denominator. The ambiguity of the total pressure effect can be seen by a comparison of the gas-phase- and liquid-phase-controlled cases: when the gas phase controls, the liquid-phase resistance is negligible and KGa = K′GapT is independent of the total pressure. For this case the coefficient K′Ga is inversely proportional to the total system pressure as shown in Eq. (14-66). On the other hand, when the liquid phase controls, the correct equation is (14-68)

where H is the Henry’s-law constant defined as H = pi /xi. This equation indicates that K′G a will be independent of the total system pressure as long as the Henry’s-law constant H does not depend on the

ABSORPTION WITH CHEMICAL REACTION

FIG. 14-12 Effects of reagent-concentration and reagent-conversion level upon the relative values of KGa in the CO2-NaOH-H2O system. [Adapted from Eckert et al., Ind. Eng. Chem., 59(2), 41 (1967).]

total pressure (this will be true only for relatively low pressures). On the basis of this comparison it should be clear that the effects of total system pressure upon K′Ga are not properly defined by Eq. (14-66), especially in cases in which the liquid-phase resistance cannot be neglected. In using Eq. (14-66), therefore, it should be understood that the numerical values of K′Ga will be a complex function of the pressure, the temperature, the type and size of tower packing employed, the liquid and gas mass flow rates, and the system composition (for example, the degree of conversion of the liquid-phase reactant). Figure 14-12 illustrates the influence of system composition and degree of reactant conversion upon the numerical values of K′Ga for the absorption of CO2 into sodium hydroxide solutions at constant conditions of temperature, pressure, and type of packing. An excellent experimental study of the influence of operating variables upon overall K′Ga values is that of Field et al. (Pilot-Plant Studies of the Hot Carbonate Process for Removing Carbon Dioxide and Hydrogen Sulfide, U.S. Bureau of Mines Bulletin 597, 1962).

Table 14-2 illustrates the observed variations in KˆGa values for different packing types and sizes for the CO2-NaOH system at a 25 percent reactant-conversion level for two different liquid flow rates. The lower rate of 2.7 kg/(s⋅m2) or 2000 lb/(h⋅ft2) is equivalent to 4 (U.S. gal/min)/ft2 and is typical of the liquid rates employed in fume scrubbers. The higher rate of 13.6 kg/(s⋅m2) or 10,000 lb/(h⋅ft2) is equivalent to 20 (U.S. gal/min)/ft2 and is more typical of absorption towers such as are used in CO2 removal systems, for example. We note also that two different gas velocities are represented in the table, corresponding to superficial velocities of 0.59 and 1.05 m/s (1.94 and 3.44 ft/s). Table 14-3 presents a typical range of KˆGa values for chemically reacting systems. The first two entries in the table represent systems that can be designed by the use of purely physical design methods, for they are completely gas-phase mass-transfer limited. To ensure a negligible liquid-phase resistance in these two tests, the HCl was absorbed into a solution maintained at less than 8 percent weight HCl and the NH3 was absorbed into a water solution maintained below pH 7 by the addition of acid. The last two entries in Table 14-3 represent liquid-phase mass-transfer limited systems. The effects of system pressure on these KˆGa values can be estimated as in Eq. (14-68) by noting that Kˆ G a = K G ay°BM = K′G ay°BM pT and recalling that (1) in gas-phase mass-transfer limited systems KˆG a = kˆG a and is independent of system pressure, and (2) for liquid-phase masstransfer limited systems in which H is constant the KˆG a values can be corrected to other pressures by the relation KˆG a at p2 = (KˆG a at p1) × p2 /p1. When both resistances are significant, it is advisable to employ experimentally derived corrections. In any case it is inadvisable to make large pressure corrections by these procedures without experimental verification. Scaling Up from Laboratory or Pilot-Plant Data For many years it has been thought by practitioners of the art of gas absorption that it would be impossible to carry out an absorption process in a laboratory apparatus or small-scale pilot plant in such a way that the data could be of use in the design of a commercial absorption unit. Indeed, even today most commercial gas-absorption units are designed primarily on the basis of prior commercial experience by using the traditional design methods described previously. Although duplication of a previous commercial design is by far the preferred method, this approach is of little value in developing a completely new process or in attempting to extrapolate an existing design to widely different operating conditions. Since the early 1960s there have been developed some excellent laboratory experimental techniques, which unfortunately have largely been ignored by the industry. A noteworthy exception was described by Ouwerkerk (Hydrocarbon Process., April 1978, pp. 89–94), in which it was revealed that both laboratory and small-scale pilot-plant data were employed as the basis for the design of an 8.5-m- (28-ft-) diameter commercial Shell Claus off-gas treating (SCOT) plate-type absorber. It is claimed that the cost of developing comprehensive design procedures can be kept to a minimum, especially in the development of a new process, by the use of these modern techniques.

ˆ Ga in a Chemically Reacting TABLE 14-2 Typical Effects of Packing Type, Size, and Liquid Rate on K ˆ Ga, kmol/(h◊m3) System, K L = 2.7 kg/(s⋅m2) Packing size, mm

25

38

50

Berl-saddle ceramic Raschig-ring ceramic Raschig-ring metal Pall-ring plastic Pall-ring metal Intalox-saddle ceramic Super-Intalox ceramic Intalox-saddle plastic Intalox-saddle metal Hy-Pak metal

30 27 29 29 37 34 37∗ 40∗ 43∗ 35

24 24 24 27 32 27

21 21 19 26∗ 27 22 26∗ 24∗ 30∗ 27∗

35∗ 32∗

14-19

L = 13.6 kg/(s⋅m2) 75–90

16 21∗ 16∗ 16∗ 24∗ 18∗

25

38

50

45 42 45 45 56 56 59∗ 56∗ 66∗ 54

38 34 35 42 51 43

32 30 27 38∗ 43 34 40∗ 37∗ 48∗ 42∗

58∗ 50∗

75–90

24 27∗ 26∗ 26∗ 37∗ 27∗

Data courtesy of the Norton Company. Operating conditions: CO2, 1 percent mole in air; NaOH, 4 percent weight (1 normal); 25 percent conversion to sodium carbonate; temperature, 24°C (75°F); pressure, 98.6 kPa (0.97 atm); gas rate = 0.68 kg/(s⋅m2) = 0.59 m/s = 500 lb/(h⋅ft2) = 1.92 ft/s except for values with asterisks, which were run at 1.22 kg/(s⋅m2) = 1.05 m/s = 900 lb/(h⋅ft2) = 3.46 ft/s superficial velocity; packed height, 3.05 m (10 ft); tower diameter, 0.76 m (2.5 ft). To convert table values to units of (lb⋅mol)/(h⋅ft3), multiply by 0.0624.

14-20

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

ˆ Ga Values for Various Chemically TABLE 14-3 Typical K Reacting Systems, kmol/(h◊m3) Gas-phase reactant

Liquid-phase reactant

ˆ Ga K

Special conditions

HCl NH3 Cl2 SO2 HF Br2 HCN HCHO HBr H2S SO2 CO2 Cl2

H2O H2O NaOH Na2CO3 H2O NaOH H2O H2O H2O NaOH H2O NaOH H2O

353 337 272 224 152 131 114 114 98 96 59 38 8

Gas-phase limited Gas-phase limited 8% weight solution 11% weight solution 5% weight solution Physical absorption 4% weight solution 4% weight solution Liquid-phase limited

Data courtesy of the Norton Company. Operating conditions (see text): 38-mm ceramic Intalox saddles; solute gases, 0.5–1.0 percent mole; reagent conversions = 33 percent; pressure, 101 kPa (1 atm); temperature, 16–24°C; gas rate = 1.3 kg/(s⋅m2) = 1.1 m/s; liquid rates = 3.4 to 6.8 kg/(s⋅m2); packed height, 3.05 m; tower diameter, 0.76 m. Multiply table values by 0.0624 to convert to (lb⋅mol)/(h⋅ft3).

In 1966, in a paper that now is considered a classic, Danckwerts and Gillham [Trans. Inst. Chem. Eng., 44, T42 (1966)] showed that data taken in a small stirred-cell laboratory apparatus could be used in the design of a packed-tower absorber when chemical reactions are involved. They showed that if the packed-tower mass-transfer coefficient in the absence of reaction (kL0) can be reproduced in the laboratory unit, then the rate of absorption in the laboratory apparatus will respond to chemical reactions in the same way as in the packed column even though the means of agitating the liquid in the two systems might be quite different. According to this method, it is not necessary to investigate the kinetics of the chemical reactions in detail, nor is it necessary to determine the solubilities or the diffusivities of the various reactants in their unreacted forms. To use the method for scaling up, it is necessary independently to obtain data on the values of the interfacial area per unit volume a and the physical mass-transfer coefficient kL0 for the commercial packed tower. Once these data have been measured and tabulated, they can be used directly for scaling up the experimental laboratory data for any new chemically reacting system. Danckwerts and Gillham did not investigate the influence of the gas-phase resistance in their study (for some processes gas-phase resistance may be neglected). However, in 1975 Danckwerts and Alper [Trans. Inst. Chem. Eng., 53, 34 (1975)] showed that by placing a stirrer in the gas space of the stirred-cell laboratory absorber, the gas-phase mass-transfer coefficient kˆ G in the laboratory unit could be made identical to that in a packed-tower absorber. When this was done, laboratory data obtained for chemically reacting systems having a significant gas-side resistance could successfully be scaled up to predict the performance of a commercial packed-tower absorber. If it is assumed that the values of kˆ G, kL0, and a have been measured for the commercial tower packing to be employed, the procedure for using the laboratory stirred-cell reactor is as follows: 1. The gas-phase and liquid-phase stirring rates are adjusted so as to produce the same values of kˆ G and kL0 as will exist in the commercial tower. 2. For the reaction system under consideration, experiments are made at a series of bulk-liquid and bulk-gas compositions representing the compositions to be expected at different levels in the commercial absorber (on the basis of a material balance). 3. The rates of absorption rA(ci,B0) are measured at each pair of gas and liquid compositions. For dilute-gas systems one form of the equation to be solved in conjunction with these experimental data is GM y1 dy hT = ᎏ (14-69) ᎏ a y2 rA where hT = height of commercial tower packing, GM = molar gas-phase mass velocity, a = effective interfacial area for mass transfer per unit



volume in the commercial tower, y = mole-fraction solute in the gas phase, and rA = experimentally determined rate of absorption per unit of exposed interfacial area. By using the series of experimentally measured rates of absorption, Eq. (14-69) can be integrated numerically to determine the height of packing required in the commercial tower. There are a number of different types of experimental laboratory units that could be used to develop design data for chemically reacting systems. Charpentier [ACS Symp. Ser., 72, 223–261 (1978)] has summarized the state of the art with respect to methods of scaling up laboratory data and tabulated typical values of the mass-transfer coefficients, interfacial areas, and contact times to be found in various commercial gas absorbers as well as in currently available laboratory units. The laboratory units that have been employed to date for these experiments were designed to operate at a total system pressure of about 100 kPa (1 atm) and at near-ambient temperatures. In practical situations, it may become necessary to design a laboratory absorption unit that can be operated either under vacuum or at elevated pressures and over a reasonable range of temperatures in order to apply the Danckwerts method. It would be desirable to reinterpret existing data for commercial tower packings to extract the individual values of the interfacial area a and the mass-transfer coefficients kˆ G and kL0 in order to facilitate a more general usage of methods for scaling up from laboratory experiments. Some progress in this direction has already been made, as discussed later in this section. In the absence of such data, it is necessary to operate a pilot plant or a commercial absorber to obtain kˆ G, kL0, and a as described by Ouwerkerk (op. cit.). Principles of Rigorous Absorber Design Danckwerts and Alper [Trans. Inst. Chem. Eng., 53, 34 (1975)] have shown that when adequate data are available for the kinetic-reaction-rate coefficients, the mass-transfer coefficients kˆ G and kL0, the effective interfacial area per unit volume a, the physical solubility or Henry’s-law constants, and the effective diffusivities of the various reactants, then the design of a packed tower can be calculated from first principles with considerable precision. For example, the packed-tower design equation for a dilute system in which gas-phase reactant A is being absorbed and reacted with liquid-phase reagent B is LM 0 rA a dh = ᎏ dBh = −GM dyh (14-70) νρL where rA = specific rate of absorption per unit interfacial area, a = interfacial area per unit volume of packing, h = height of packing, LM = molar liquid mass velocity, ν = number of moles of B reacting with 1 mol of A; ρL = average molar density of liquid phase, B0h = bulkliquid-phase reagent concentration (a function of h), GM = molar gasphase mass velocity, and yh = mole fraction A in gas phase (a function of h). For dilute systems it can be assumed that GM, LM, and ρL are constant, and it normally is assumed that the interfacial area a of the packing is constant and is equal to the value that would exist without reaction. This last assumption needs careful consideration, since different methods for measuring a may give different results. Sharma and Danckwerts [Br. Chem. Eng., 15(4), 522 (1970)] have reviewed various techniques for measuring interfacial areas. Under the above assumptions for dilute systems Eq. (14-70) can be integrated as follows: 0 LM B2 dB0h GM y1 dyh hT = ᎏ (14-71) ᎏ=ᎏ ᎏ 0 νρLa B1 rA a y2 rA





where hT = total height of packing and the subscripts 1 and 2 refer to the bottom and the top of the tower packing respectively. The specific absorption rate rA = rA(ci,B0) is a function of h and may be computed by combining the rate equation rA = kL(xi − x) = (kL/ρL)(ci − c)

(14-72)

with the material-balance, or operating-curve, equation GM(y − y2) = (LM /νρL)(B02 − B0)

(14-73)

ABSORPTION WITH CHEMICAL REACTION

14-21

and with the appropriate relation for computing the interfacial concentration xi of reactant A. In Eq. (14-72) the mass-transfer coefficient kL is the coefficient with chemical reaction; i.e., kL = φk0L. The interfacial concentration xi is computed by combining the equilibrium relation yi = mxi with the equation kG(y − yi) = kL(xi − x) to obtain y/m + (kL /mkG)x xi = ᎏᎏ (14-74) (1 + kL /mkG) According to Eq. (14-74), when kL is very large and the ratio kL /mkG is much larger than unity, xi − x = ykG /kL and the specific absorption rate is defined by the equation rA = kL(xi − x) = kGy

(14-75)

This is the gas-phase mass-transfer limited condition, which can be substituted into Eq. (14-71) to obtain the following equation for calculating the height of packing for a dilute system: hT = (GM / kGa) ln (y1 /y2) = HG ln (y1 /y2)

(14-76)

At the other extreme, when the ratio kL /mkG is much smaller than unity, the interfacial concentration of reactant A may be approximated by the equilibrium relation xi = y/m, and the specific absorption rate expression is rA = kL(xi − x) = kL(y/m − x)

(14-77)

For fast chemical reactions the reactant A is by definition completely consumed in the thin film near the liquid interface. Thus, x = 0, and rA = kLy/m = (kL /ρL)ci

(14-78)

This is known as the liquid-phase mass-transfer limited condition, as illustrated in Fig. 14-13. Inspection of Eqs. (14-71) and (14-78) reveals that for fast chemical reactions which are liquid-phase mass-transfer limited the only unknown quantity is the mass-transfer coefficient kL. The problem of rigorous absorber design therefore is reduced to one of defining the influence of chemical reactions upon kL. Since the physical masstransfer coefficient k0L is already known for many tower packings, it

Gas-phase and liquid-phase solute-concentration profiles for a liquid-phase mass-transfer limited reaction system in which NHa is larger than 3.

FIG. 14-13

often is convenient to work in terms of the ratio kL /kL0 as discussed in the following paragraphs. Estimation of kL for Irreversible Reactions Figure 14-14 illustrates the influence of either first- or second-order irreversible chemical reactions on the mass-transfer coefficient kL as developed by Van Krevelen and Hoftyzer [Rec. Trav. Chim., 67, 563 (1948)] and as later refined by Perry and Pigford and by Brian et al. [Am. Inst. Chem. Eng. J., 7, 226 (1961)]. First-order and pseudo-first-order reactions are represented by the upper curve in Fig. 14-14. We note that for first-order reactions when the Hatta number NHa is larger than about 3, the rate coefficient kL can be computed by the formula 0  = (k  )A kL = kD 1A 2BD

Influence of irreversible chemical reactions on the liquid-phase mass-transfer coefficient kL. [Adapted from Van Krevelen and Hoftyzer, Rec. Trav. Chim., 67, 563 (1948).]

FIG. 14-14

(14-79)

where kL = liquid-phase mass-transfer coefficient, k1 = first-orderreaction-rate coefficient, k2B0 = pseudo-first-order-reaction-rate coef-

14-22

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

ficient, and DA = diffusion coefficient of gaseous reactant A in the liquid phase. The parameter values for the curves of Fig. 14-14 originally were defined from film theory as (DB /DA)(B0/␯ci ) but later were refined by the results of penetration theory to the definition (φ∞ − 1), where D D (B0/␯ci) φ∞ = D A/ B + D B/A

(14-81)

For NHa £ 2: kL /kL0 = 1 + (φ∞ − 1){1 − exp [−(φ∞ − 1)−1]} exp [1 − 2/NHa]

(14-82)

where the Hatta number NHa is defined as 0 DA/kL0 NHa = kB 2

(14-83)

Equation (14-81) originally was reported by Porter [Trans. Inst. Chem. Eng., 44(1), T25 (1966)]. Equation (14-82) was derived by the author. The Van Krevelen-Hoftyzer relationship was tested experimentally for the second-order system in which CO2 reacts with either NaOH or KOH solutions by Nijsing et al. [Chem. Eng. Sci., 10, 88 (1959)]. Nijsing’s results for the NaOH system are shown in Fig. 14-15 and are in excellent agreement with the second-order-reaction theory. Indeed, these experimental results can be described very well by Eqs. (14-80) and (14-81) when values of ν = 2 and DA/DB = 0.64 are employed in the equations. For fast irreversible chemical reactions, therefore, the principles of rigorous absorber design can be applied by first establishing the effects of the chemical reaction on kL and then employing the appropriate material-balance and rate equations in Eq. (14-71) to perform the integration to compute the required height of packing. For an isothermal absorber involving a dilute system in which a liquid-phase mass-transfer limited first-order irreversible chemical reaction is occurring, the packed-tower design equation is derived as a) ln (y1/y2) hT = (mGM /kD 1A

(14-84)

For a dilute system in which the liquid-phase mass-transfer limited condition is valid, in which a very fast second-order reaction is involved, and for which NHa is very large, the equation D  + D D (B0/νci) kL/kL0 = φ∞ = D A/B B/A

mGM hT = ᎏ kL0 a

D ᎏ  D

y1

B A

y2

(14-80)

in which DB = diffusion coefficient of the liquid-phase reactant B and φ∞ = value of kL /kL0 for large values of NHa approaching infinity. For design purposes the entire graph of Fig. 14-14 can be represented by the following pair of equations: For NHa ≥ 2: kL /kL0 = 1 + (φ∞ − 1){1 − exp [−(NHa − 1)/(φ∞ − 1)]}

is valid and results in the following equation for computing the height of packing in a packed tower:

(14-85)

dy ᎏᎏ mB0 DB ᎏᎏ + y νρLDA

(14-86)

Evaluation of the integral in Eq. (14-86) requires a knowledge of the liquid-phase bulk concentration of B as a function of y. This relationship is obtained by means of a material balance around the tower, as shown in Eq. (14-73). Numerical integration by a quadrature method such as Simpson’s rule normally will be required for this calculation. Estimation of kL for Reversible Reactions When the reaction is of the form A A B, where B is a nonvolatile product and the equilibrium constant is defined by cB = KeqcA, the expressions for computing kL become extremely complex. A good discussion of this situation is given in Mass Transfer by Sherwood, Pigford, and Wilke (McGrawHill, New York, 1975, p. 317). Three limiting cases are listed below: 1. For very slow reactions, lim kL = kL0 k1 → 0 2.

(14-87)

For extremely fast reactions where Keq is very large,  lim kL = kD 1 A k1 → ∞ Keq = ∞

3.

(14-88)

For extremely fast reactions where Keq is finite, lim kL = (1 + Keq)kL0 k1 → ∞ Keq = finite

(14-89)

When one of these three conditions is applicable, the appropriate design equation can be obtained by substitution into Eq. (14-71), followed by integration of the resulting relationship. Some more complex situations involving reversible reactions are discussed in Mass Transfer (ibid., pp. 336–343). Simultaneous Absorption of Two Reacting Gases In multicomponent physical absorption the presence of one gas often does not affect the rates of absorption of the other gases. When chemical reactions in which two or more gases are competing for the same liquidphase reagent are involved, selectivity of absorption can be affected by

FIG. 14-15 Experimental values of kL /kL0 for absorption of CO2 into NaOH solutions at 20°C. [Data of Nijsing et al., Chem. Eng. Sci., 10, 88 (1959).]

GAS-LIQUID CONTACTING SYSTEMS the choice of design conditions, and the situation may become extremely complex from a designer’s point of view. The classic work on this subject is that of Ramachandran and Sharma [Trans. Inst. Chem. Eng., 49, 253 (1971)] and is recommended to those needing further details. The following references also are offered as a sampling of the literature on the subject: • CO2 and H2S. Danckwerts and Sharma, Chem. Eng. (London), CE244–280 (October 1966); Onda, et al., J. Chem. Eng. Japan, 5, 27 (1972); Rivas and Prausnitz, Am. Inst. Chem. Eng. J., 25, 975 (1979). • CO2 and SO2. Goettler and Pigford, Inst. Chem. Eng. Symp. Ser., 28, 1 (1968); Teramoto et al., Int. Chem. Eng., 18, 250 (1978). • SO2 and NO2. Takeuchi and Yamanaka, Ind. Eng. Chem. Process Des. Dev., 17, 389 (1978). Desorption with Chemical Reaction When chemical reactions are involved in a stripping operation, the design problem can become extremely complex. In fact, much less is known about this very important process than is known about absorption. A classic work on this subject is that of Shah and Sharma [Trans. Inst. Chem. Eng., 54, 1 (1976)], which is recommended to those in need of more details. In the subsection “Design of Gas-Absorption Systems” it was stated that more often than not the liquid-phase residence time and, hence, the liquid holdup are considered to be the most important design parameters for stripping towers. If Eq. (14-60) is redefined to represent the stripping process for an extremely slow liquid-phase reaction for which RA = k1c, then one finds that the liquid holdup will be a factor only when the ratio k1ρL fH/kL0 a is less than unity. Thus, one can ensure that the liquid-phase reaction rate is not limiting by increasing the temperature and the liquid holdup until this ratio is equal to or greater than unity. The preferred method at present is to base the design on prior commercial experience. Use of Literature for Specific Systems A large body of experimental data obtained in bench-scale laboratory units and in smalldiameter packed towers has been published since the early 1940s. One might wish to consider using such data for a particular chemically

14-23

reacting system as the basis for scaling up to a commercial design. Extreme caution is recommended in interpreting such data for the purpose of developing commercial designs, as extrapolations of this kind of information can lead to serious errors. Extrapolation to temperatures, pressures, or liquid-phase reagent conversions different from those that were employed by the original investigator definitely should be regarded with caution. Bibliographies presented in the general references listed at the beginning of this section are an excellent source of information on specific chemically reacting systems. Gas-Liquid Reactions by Danckwerts (McGraw-Hill, New York, 1970) contains a tabulation of references to specific chemically reacting systems. Gas Treating with Chemical Solvents by Astarita et al. (Wiley, New York, 1983) deals with the absorption of acid gases, and includes an extensive listing of patents. Gas Purification by Kohl and Riesenfeld (Gulf Publishing, Houston, 1985) presents data and references for many chemically reacting systems of current commercial interest. In searching for data on a particular system, a computerized search of Chemical Abstracts, Engineering Index, and National Technical Information Service (NTIS) data bases should seriously be considered. Although the NTIS computer contains only information published after 1970, one normally can assume that most pre-1970 publications of merit likely will be referenced in the bibliographies of current articles on the subject. The experimental data for the system CO2-NaOH-Na2CO3 are unusually well known as the result of the work of many experimenters. A serious study of the data and theory for this system therefore is recommended as the basis for developing a good understanding of the kind and quality of experimental information that is needed for design purposes. In addition to data on CO2, information can readily be found in the literature for the following systems: O2, Cl2, NH3, NO2, NO, SO2, SO3, H2S, COS, CS2, HCl, HBr, HCN, H2, COCl2, PCl3, olefins, dienes, and water vapor.

GAS-LIQUID CONTACTING SYSTEMS Gas-liquid contacting systems are utilized for transferring mass, heat, and momentum between the phases, subject to constraints of physical and chemical equilibrium. Process equipment for such systems is designed to achieve the appropriate transfer operations with a minimum expenditure of energy and capital investment. In this section emphasis is placed on the transfer of mass. Typical gas-liquid mass-transfer systems are: Distillation Flashing Rectification Absorption Stripping

Evaporation Humidification Dehumidification Dephlegmation Spray drying

Distillation is the separation of the constituents of a liquid mixture via partial vaporization of the mixture and separate recovery of vapor and residue. The process of vaporization is generally of a differential nature. Flashing is a distillation process in which the total vapor removed approaches phase equilibrium with the residue liquid. Rectification is the separation of the constituents of a liquid mixture by successive distillations (partial vaporizations and condensations) and is obtained via the use of an integral or differential process. Separations into effectively pure components may be obtained through this procedure. Stripping or desorption is the transfer of gas, dissolved in a liquid, into a gas stream. The term is also applied to that section of a fractionating column below the feed plate. Absorption is the transfer of a soluble component in a gas-phase mixture into a liquid absorbent whose volatility is low under process conditions.

Evaporation generally refers to the removal of water, by vaporization, from aqueous solutions of nonvolatile substances. Humidification and dehumidification refer to the transfer of water between a gas stream and a water stream. Dephlegmation, or partial condensation, refers to the process in which a vapor stream is cooled to a desired temperature such that a portion of the less volatile components of the stream is removed from the vapor by condensation. Spray drying is an extension of the evaporative process in which almost all the liquid is removed from a solution of a nonvolatile solid in the liquid. All these processes are, in common, liquid-gas mass-transfer operations and thus require similar treatment from the aspects of phase equilibrium and kinetics of mass transfer. The fluid-dynamic analysis of the equipment utilized for the transfer also is similar for many types of liquid-gas process systems. Process equipment utilized for liquid-gas contacting is based on a combination of operating principles of the three categories: Mode of flow of streams Countercurrent Cocurrent Cross-flow Gross mechanism of transfer Differential Integral Continuous phase Gas* Liquid

* In this section the terms “gas” and “vapor” are used interchangeably. The latter is often used in distillation, in which the gas phase is represented by an equilibrium vapor.

14-24

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

TABLE 14-4

Characteristics of Liquid-Gas Systems

Equipment designation

Mode of flow

Gross mechanism

Continuous phase

Primary process applications

Plate column Packed column

Cross-flow, countercurrent Countercurrent, cocurrent

Integral Differential

Liquid and/or gas Liquid and/or gas

Wetted-wall (falling-film) column Spray chamber

Countercurrent, cocurrent

Differential

Liquid and/or gas

Cocurrent, cross-flow, countercurrent

Differential

Gas

Cocurrent, countercurrent Complete mixing Cocurrent

Differential Integral Differential

Gas Liquid Liquid or gas

Absorption, rectification, stripping Absorption, rectification, stripping, humidification, dehumidification Absorption, rectification, stripping, evaporation Absorption, stripping, humidification, dehumidification Evaporation, dephlegmation Absorption Absorption, stripping

Heat exchanger Agitated vessel Line mixer

The combination of these characteristics utilized in the various types of process equipment is indicated in Table 14-4. PLATE COLUMNS Plate Types Plate columns utilized for liquid-gas contacting may be classified according to mode of flow in their internal contacting devices: 1. Cross-flow plates 2. Counterflow plates The cross-flow plate (Fig. 14-16a) utilizes a liquid downcomer and is more generally used than the counterflow plate (Fig. 14-16b) because of transfer-efficiency advantages and greater operating range. The liquid-flow pattern on a cross-flow plate can be controlled by placement

(b)

(a) FIG. 14-16

(a) Cross-flow plate (side view). (b) Countercurrent plate (side

view).

of downcomers in order to achieve desired stability and transfer efficiency. Commonly used flow arrangements are shown in Fig. 14-17. A guide for the tentative selection of flow pattern is given in Table 14-5. It should be noted that the fraction of column cross-sectional area available for gas dispersers (perforations, bubble caps) decreases when more than one downcomer is used. Thus, optimum design of the plate involves a balance between liquid-flow accommodation and effective use of cross section for gas flow. Most new designs of cross-flow plates employ perforations for dispersing gas into liquid on the plate. These perforations may be simple TABLE 14-5

Selection of Cross-Flow-Plate Flow Pattern* Range of liquid capacity, gal/min

Estimated tower diam., ft

Reverse flow

Crossflow

Double pass

Cascade double pass

3 4 6 8 10 12 15 20

0–30 0–40 0–50 0–50 0–50 0–50 0–50 0–50

30–200 40–300 50–400 50–500 50–500 50–500 50–500 50–500

400–700 500–800 500–900 500–1000 500–1100 500–1100

900–1400 1000–1600 1100–1800 1100–2000

*Bolles, chap. 14 in Smith, Design of Equilibrium Stage Processes, McGrawHill, New York, 1963. To convert feet to meters, multiply by 0.3048; to convert gallons per minute to decimeters per second (liters per second), multiply by 0.06309; and to convert gallons per minute to cubic meters per second, multiply by 6.309 × 10−5.

Common liquid-flow patterns, cross-flow plates. (Smith, Design of Equilibrium Stage Processes, McGraw-Hill, New York, 1963.)

FIG. 14-17

GAS-LIQUID CONTACTING SYSTEMS round orifices, or they may contain movable “valves” that provide variable orifices of noncircular shape. These perforated plates are called sieve plates (Fig. 14-18) or valve plates (Fig. 14-19). For sieve plates, liquid is prevented from flowing through the perforations by the flowing action of the gas; thus, when the gas flow is low, it is possible for some or all of the liquid to drain through the perforations and in effect bypass portions of the contacting zone. The valve plate is designed to minimize this drainage, or “weeping,” since the valve tends to close as the gas flow becomes lower, the total orifice area varying to maintain a dynamic-pressure balance across the plate. Historically the most common gas disperser for cross-flow plates has been the bubble cap. This device has a built-in seal which prevents liquid drainage at low gas-flow rates. Typical bubble caps are shown in Fig. 14-20. Gas flows up through a center riser, reverses flow under the cap, passes downward through the annulus between riser and cap, and finally passes into the liquid through a series of openings, or “slots,” in the lower side of the cap. Bubble caps were used almost exclusively as cross-flow-plate dispersers until about 1950, when they were largely displaced by simple or valve-type perforations. Many varieties of bubble-cap design were used (and therefore are extant in many operating columns), but in most cases bell caps of 75- to 150-mm (3- to 6-in) diameter were utilized. In counterflow plates, liquid and gas utilize the same openings for flow. Thus, there are no downcomers. Openings are usually simple round perforations in the 3- to 13-mm (d- to a-in) range (dual-flow plate) or long slots with widths of 6 to 13 mm (d to a in) (Turbogrid tray). The plate material can be corrugated (Ripple tray) to segregate partially gas and liquid flow. In general, gas and liquid flow in a pulsating fashion with a particular opening passing both gas and liquid in an intermittent fashion. A counterflow plate often used for contacting gases with liquids containing solids is the baffle plate, or “shower deck” (Fig. 14-21).

FIG. 14-21

14-25

Baffle plate (shower deck).

Typically the plate is half-moon in shape and is sloped slightly in the direction of liquid flow. Gas contacts the liquid as it showers from the plate, and a serrated lip or weir at the edge of the plate can be used to improve the distribution of liquid in the shower. The baffle plate operates with liquid dispersed and gas as the continuous phase and is used primarily in heat-transfer applications. In summary, the perforated plate with liquid cross-flow (the “sieve plate”) is the most common type specified for new designs. Schematic diagrams of such a plate are shown in Fig. 14-22. Nomenclature items are shown, with heights hli, hf, hlo, and hl referring to liquid entering, froth, liquid + froth leaving, and equivalent clear liquid averaged across the plate. For the plan view, area terms are as follows: At = tower total cross section; Aa = active area; Ad = area of one downcomer; An = net area for vapor flow (usually total cross section minus blocking downcomers); and Ah = area of holes or perforations. For the single cross-flow plate shown, At = Aa + 2Ad An = Aa + Ad = At − Ad

FIG. 14-18

Sieve-plate dispersers. To convert inches to millimeters, multiply

by 25.4.

FIG. 14-19

Valve-plate dispersers. To convert inches to millimeters, multiply

by 25.4.

(a) FIG. 14-20

(a) Circular or bell cap. (b) Tunnel cap.

(b)

When downcomers are sloped or when perforations do not occupy essentially all the area between the downcomers, these simple relations do not apply. However, their adaptation should be obvious from the geometry involved. The term “froth” in Fig. 14-22 suggests aeration in which the liquid phase is continuous. Under certain conditions there can be an inversion to a gas-continuous regime, or “spray.” The spray has its phase boundaries equivalent to the boundaries for froth shown in Fig. 14-22. Plate-Column Capacity The maximum allowable capacity of a plate for handling gas and liquid flow is of primary importance because it fixes the minimum possible diameter of the column. For a constant liquid rate, increasing the gas rate results eventually in excessive entrainment and flooding. At the flood point it is difficult to obtain net downward flow of liquid, and any liquid fed to the column is carried out with the overhead gas. Furthermore, the column inventory of liquid increases, pressure drop across the column becomes quite large, and control becomes difficult. Rational design calls for operation at a safe margin below this maximum allowable condition. Flooding may also be brought on by increasing the liquid rate while holding the gas rate constant. Excessive liquid flow can overtax the capacity of downcomers or other passages, with the ultimate result of increased liquid inventory, increased pressure drop, and the other characteristics of a flooded column. These two types of flooding are usually considered separately when a plate column is being rated for capacity. For identification purposes they are called entrainment flooding (or “priming”) and downflow flooding. When counterflow action is destroyed by either type, transfer efficiency is lost and reasonable design limits have been exceeded. Minimum allowable capacity of a column is determined by the need for effective dispersion and contacting of the phases. The types of plates differ in their ability to permit low flows of gas and liquid. A cross-flow sieve plate can operate at reduced gas flow down to a point where liquid drains through the perforations and gas dispersion is inadequate for good efficiency. Valve plates can be operated at very

14-26

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

Stable operating region, plates. (Smith, Design of Equilibrium Stage Processes, McGraw-Hill, New York, 1963.)

FIG. 14-23

Entrainment Flooding The early work of Souders and Brown [Ind. Eng. Chem., 26, 98 (1934)] based on a force balance on an average suspended droplet of liquid led to the definition of a capacity parameter Csb: L(ρ / ρ Csab = Unρ t − g)

(14-90)

where Un = linear gas velocity based on net area An, m/s ρG = gas density, kg/m3 ρL = liquid density, kg/m3 For cross-flow plates, net area is the column cross section less that area blocked by the downcomer or downcomers (Fig. 14-22). The vapor velocity in the net area represents an approach velocity and thus controls the level of liquid entrainment. For counterflow plates, net area is the same as the column cross section, since no downcomers are involved. Maximum allowable values of the capacity parameter are for a flooding condition and are designated Csbf . Experimental values have been correlated against a dimensionless flow parameter FLG as shown in Fig. 14-25. The flow parameter represents a ratio of liquid to vapor kinetic energies: L ρG 0.5 FLG = ᎏ ᎏ (14-91) G ρL Low values of FLG indicate vacuum operation, high values indicate operation at higher ressures or at high liquid/vapor loadings as in gas absorption. The liquid/gas ratio L/G is based on mass flow rates. The parameter serves as a criterion for two-phase flow characteristics on the plate, as discussed by Hofhuis and Zuiderweg [Inst. Chem. Engrs. Symp. Ser. No. 56, 2.2-1 (1979)]. Notations on Fig. 14-25 are from this source. The correlation in Fig. 14-25 is intended to cover the full range of flow parameters, with the low values of Csbf to the right likely to result from downcomer flow restrictions rather than excessive entrainment. The curves may be expressed in equation form as [Lygeros and Magoulas, Hydrocarbon Proc. 65(12), 43 (1986)]:




Sieve-plate diagram. (Smith, Design of Equilibrium Stage Processes. McGraw-Hill, New York, 1963.)

FIG. 14-22

low gas rates because of valve closing. Bubble-cap plates can be operated at very low gas rates because of their seal arrangement. All devices have a definite minimum gas rate below which there is inadequate dispersion for intimate contacting. Similarly, there are minimum liquid flows below which good distribution is not possible, although the reverse-flow plate (Fig. 14-17) can accommodate extremely low liquid flows. For all plate devices a qualitative capacity diagram is shown in Fig. 14-23. The shape and extent of the satisfactory operating zone in Fig. 14-23 vary according to type of plate device. As a specific example, Fig. 14-24 shows actual test data for two cross-flow plates in distillation service at total reflux. The abscissa parameter is called the F-factor and is a vapor kinetic energy term. The decline in efficiency of the sieve plate at low F-factors is evident and is the result of liquid falling through some of the holes (“weeping”). The decline in efficiency of both devices at high F-factors results from liquid entrainment. For new designs, it is the objective of the designer to predict the likely location of performance curves such as those shown.

0.842 Csbf = 0.0105 + 8.127 (10−4)(TS0.755)exp[−1.463 FLG ]

(14-92)

where TS = plate spacing, mm. Figure 14-25 or Eq. (14-92) may be used for sieve plates, valve plates, or bubble-cap plates. The value of the flooding vapor velocity must be considered as approximate, and prudent designs call for approaches to flooding of 75 to 85 percent. The value of the capacity parameter (ordinate term in Fig. 14-25) may be used to calculate the maximum allowable vapor velocity through the net area of the plate: σ 0.2 ρL − ρg 0.5 Unf = Csb,flood ᎏ (14-93) ᎏ 20 ρg where Unf = gas velocity through net area at flood, m/s Csbf = capacity parameter, m/s σ = liquid surface tension, mN/m (dyn/cm) ρL = liquid density, kg/m3 ρG = gas density, kg/m3







GAS-LIQUID CONTACTING SYSTEMS

14-27

1.0 Sieve-valve plate

Efficiency, E

0.8

Sieve plate

0.6 0.4 0.2 0

0.6

0.8

1.0

2.0 2.2 1.6 1.8 1.2 1.4 F-factor, Ut (ρG)0.5, m/s(kg/m3)0.5

2.4

2.6

2.8

FIG. 14-24 Performance of two crossflow plates operating at 0.13 bar pressure and total reflux. Test mixture: ethylbenzene/styrene. Spacing between plates is 0.50 m, and outlet weir height is 38 mm. Ut = superficial vapor velocity, ρG = vapor density. [Billet, Conrad, and Grubb, I. Chem. E. Symp. Ser. No. 32, 5, 111 (1969).]

Spray zone

Tray spacing, mm 900

0.10

600

0.08

450

0.06

300 225 150

CSB = UNF

L

V

V

0.5

( ρ ρ– ρ ) ( 20σ )

0.2

m/s


,


0.20

0.04

0.02 Mixed froth 0.01 0.01

.02

.05

Emulsion flow 0.1 .2 .5 Flow parameter = L/G (ρ /ρ )0.5

1.0

2.0

Flooding correlation for columns with crossflow plates (sieve, valve, bubble-cap). [Fair, Pet/Chem Eng 33(10), 45 (September 1961).]

FIG. 14-25

Figure 14-25 gives flooding-gas velocities to ⫾10 percent subject to the following restrictions: 1. System is low or non-foaming 2. Weir height is less than 15 percent of plate spacing 3. Sieve-plate perforations are 13 mm (a in) or less in diameter 4. Ratio of slot (bubble cap), perforation (sieve), or full valve opening (valve plate) area Ah to active area Aa is 0.1 or greater. Otherwise the value of Unf obtained from Fig. 14-25 should be corrected: Ah/Aa

Unf /Unf, Fig. 14-25

0.10 008 0 06

1.00 090 0.80

where Ah = total slot, perforated, or open-valve area on plate.

For counterflow plates, the curves of Fig. 14-25 may be used for open areas of 20 percent or greater. Plates with 15 percent open areas have about 85 percent of the curve values, and open areas of less than 15 percent are not recommended. For counterflow-plate columns of the segmental-baffle type, 50 percent cut, allowable Csbf values are about 15 percent greater than those shown in Fig. 14-25, when vertical spacings of the baffles are equal to the tray spacings shown. An alternate method for predicting the flood point of sieve and valve plates has been reported by Kister and Haas [Chem. Eng. Progr., 86(9), 63 (1990)] and is said to reproduce a large data base of measured flood points to within ⫾30 percent. It applies to entrainment flooding only (values of FLG less than about 0.5). The general predictive equation is Csbf = 0.0277(dh2 σ/ρL)0.125(ρG/ρL)0.1(TS/hcl)0.5 where

dh = hole diameter, mm σ = surface tension, mN/m (dyn/cm) ρG, ρL = vapor and liquid densities, kg/m3

(14-94)

14-28

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN TS = plate spacing, mm hcl = clear liquid height at the froth-to-spray transition, mm; obtained from: hc l = hcl,H2O (996/ρL)0.5(1 − n) dh0.833 0.497 A−0.791 f hc l,H2O = ᎏᎏᎏ 1 + 0.013 L−0.59 A−1.79 f n = 0.00091 dh /Af

(14-95) (14-96) (14-97)

In Eq. 14-96, L = m liquid downflow/(hr-m weir length) and Af = fractional hole area based on active (“bubbling”) area; for instance, Af = Ah /Aa. For valve trays, adaptations of Eqs. (14-94) to (14-97) are required: 3

4 × (area of opening of one fully open valve) dh = ᎏᎏᎏᎏᎏᎏ wetted perimeter of opening of one fully open valve

(14-98)

no. valves × (area of opening of one fully open valve) Af = ᎏᎏᎏᎏᎏᎏ active (bubbling) area

(14-99)

Example 9: Loading/Flooding of a Distillation Plate An available sieve plate column of 2.5-m diameter is being considered for an ethylbenzene/styrene separation. An evaluation of loading at the top plate will be made. Key dimensions of the single-crossflow plate are: Column cross section, m2 Downcomer area, m2 Net area, m2 Active area, m2 Hole area, m2 Hole diameter, mm Weir length, m Weir height, mm Plate spacing, mm

4.91 0.25 4.66 4.41 0.617 4.76 1.50 38 500

Entrainment Entrainment in a plate column is that liquid which is carried with the vapor from a plate to the plate above. It is detrimental in that the effective plate efficiency is lowered because liquid from a plate of lower volatility is carried to a plate of higher volatility, thereby diluting distillation or absorption effects. Entrainment is also detrimental when nonvolatile impurities are carried upward to contaminate the overhead product from the column. Many experimental studies of entrainment have been made, but few of them have been made under actual distillation conditions. The studies are often questionable because they are limited to the airwater system, and they do not use a realistic method for collecting and measuring the amount of entrainment. It is clear that the dominant variable affecting entrainment is gas velocity through the two-phase zone on the plate. Mechanisms of entrainment generation are discussed in the subsection “Liquid-in-Gas Dispersions.” For distillations, it is often of more interest to ascertain the effect of entrainment on efficiency than to predict the quantitative amount of liquid entrained. For this purpose, the correlation shown in Fig. 14-26 is useful. The parametric curves in the figure represent approach to the entrainment flood point as measured or as predicted by Fig. 14-25 or some other flood correlation. The abscissa values are those of the flow parameter discussed earlier. The ordinate values ψ are fractions of gross liquid downflow, defined as follows: e ψ=ᎏ (14-100) L+e where e = absolute entrainment of liquid, moles/time L = liquid downflow rate without entrainment, moles/time Figure 14-26 also accepts the validity of the Colburn equation [Ind. Eng. Chem., 28, 526 (1936)] for the effect of entrainment on efficiency:

Conditions and properties at the top plate are: Temperature, °C Pressure, torr Vapor flow, kg/h Vapor density, kg/m3 Liquid flow, kg/h Liquid density, kg/m3 Surface tension, mN/m

78 100 25,500 0.481 22,000 841 25

The dimensions and flow rates are scaled to represent the conditions shown in Fig. 14-24. Solution. The flow parameter FLG = 0.021 (Eq. 14-73). From Fig. 14-25, Csbf = 0.095 m/s. Then, 0.095 = 4.15 m/s Unf = ᎏᎏᎏ (0.481/841)0.5 (20/25)0.2 This gives a superficial F-factor at flood = Ut ρ0.5 G = 4.15 (4.66/4.91)(.481)0.5 = 2.73 m/s(kg/m3)0.5, or about 90 percent of a value extrapolated from Fig. 14-24. The alternate method of Kister and Haas may be applied to the same problem: 22,000 L = ᎏᎏ = 17.44 m3/h-m weir 841 × 1.50 0.617 Af = ᎏ = 0.14 4.41 By Eq. 14-96, hcl,H2O = 7.98 mm Eq. 14-97: n = 0.0309 Eq. 14-95: hcl = 8.66 mm Finally, by Eq. 14-94, Csbf = 0.0277[(4.762)(25/841)]0.125 × (0.481/841)0.1(500/8.66)0.5 = 0.0947 m/s about the same as the answer obtained from Fig. 14-25. For the design condition, F-factor is 2.08 m/s(kg/m3)0.5, or about 76 percent of flood. The proposed column is entirely adequate for the service required.

Entrainment correlation. L/G = liquid-gass mass ratio; and ρl and ρg = liquid and gas densities. [Fair, Pet./Chem. Eng., 33(10), 45 (September 1961).] FIG. 14-26

GAS-LIQUID CONTACTING SYSTEMS Ea 1 ᎏ = ᎏᎏ Emv 1 + Emv [ψ/(1 − ψ)]

(14-101)

where Emv = Murphree vapor efficiency [see Eq. (14-28)] Ea = Murphree vapor efficiency, corrected for recycle of liquid entrainment The Colburn equation is based on complete mixing on the plate. For incomplete mixing, e.g., liquid approaching plug flow on the plate, Rahman and Lockett [I. Chem. E. Symp. Ser. No. 61, 111 (1981)] and Lockett et al. [Chem. Eng. Sci., 38, 661 (1983)] have provided corrections for Eq. 14-44. Figure 14-26 and Eq. 14-94 may be used to evaluate the effects of entrainment on efficiency. Example 10: Entrainment Effect on Plate Efficiency For the data shown in Fig. 14-24, estimate the efficiency of the sieve plate at a superficial F-factor of 2.6 m/s(kg/m3). The data show a midrange dry efficiency of 0.7 (70 percent). They indicate a flood F-factor value of about 3.0. Thus, the approach to flood is 2.6/3.0 = 0.87 (87 percent). The data were taken at total reflux, and thus FLG = (0.481/841)0.5 = 0.024 (densities taken from Example 9). From Fig. 14-26, ψ = 0.19, and from Eq. (14-76): Ea 1 ᎏ = ᎏᎏᎏ = 0.87 Emv 1 + 0.70[0.19/(1 − 0.19)] Thus, the wet efficiency Ea = 0.87(0.70) = 0.61 (61 percent). Figure 14-24 shows that for F = 2.6, the measured efficiency is 0.55 (55 percent).

Weeping Liquid flow through sieve-plate perforations occurs when the gas pressure drop through the perforations is not sufficient to create bubble surface and support the static head of froth above the perforations. Weeping can be deleterious in that liquid tends to shortcircuit the primary contacting zones. On the other hand, some mass transfer to and from the weeping liquid occurs. Usual practice is to design so that deleterious weeping does not occur, based on a correlation such as that shown in Fig. 14-27. In Fig. 14-27, hd = head loss to gas flow through perforations, mm liquid [see Eq. (14-107)], and he = head loss due to bubble formation, mm liquid. The latter loss is based on the energy required for bubble formation: 4σ hb = ᎏ (14-102) dh with a convenient dimensional form for use in Fig. 14-27 being: σ (14-103) hσ = 409 ᎏ ρL dh where σ = surface tension, mN/m ρL = liquid density, kg/m3 dh = diameter of a perforation, mm he = head loss due to bubble formation, mm liquid






If design shows a condition above the appropriate curve of Fig. 14-27, weeping will not be deleterious to plate performance as measured by a drop in plate efficiency (as in Fig. 14-24 for the sieve plate).

FIG. 14-27 Weeping, sieve plates. To convert millimeters to inches, multiply by 0.0394. (Smith, Design of Equilibrium Stage Processes, McGraw-Hill, New York, 1963.)

14-29

Downflow Flooding Columns can flood because of their inability to handle large quantities of liquid. For crossbow plates this limit on liquid rate Is evidenced by downcomer backup to the plate above. To avoid downflow flooding one must size the column downcomers such that excessive backup does not occur. Downcomer backup is calculated from the pressure-balance equation hdc = ht + hw + how + hda + hhg

(14-104)

where hdc = height in downcomer, mm liquid ht = total pressure drop across the plate, mm liquid hw = height of weir at plate outlet, mm liquid how = height of crest over weir, mm liquid hda = head loss due to liquid flow under downcomer apron, mm liquid hhg = liquid gradient across plate, mm liquid The heights of head losses in Eq. (14-104) should be in consistent units, e.g., millimeters or inches of liquid under operating conditions on the plate. As noted, hdc is calculated in terms of equivalent clear liquid. Actually, the liquid in the downcomer may be aerated and actual backup is hdc h′dc = ᎏ (14-105) φdc where φdc is an average relative froth density (ratio of froth density to liquid density) in the downcomer. Design must not permit h′dc to exceed the value of plate spacing; otherwise, flooding can be precipitated. In fact, plate spacing may be determined by some safe approach to the calculated value of h′dc. The value of φdc depends upon the tendency for gas and liquid to disengage (froth to collapse) in the downcomer. For cases favoring rapid bubble rise (low gas density, low liquid viscosity, low system foamability) collapse is rapid, and clear liquid fills the bottom of the downcomer (Fig. 14-22). For such cases, it is usual practice to employ a value of φdc = 0.5. For cases favoring slow bubble rise (high gas density, high liquid viscosity, high system foamability), values of φdc = 0.2 to 0.3 should be used. As the critical point is approached in highpressure distillations and absorptions, special precautions with downcomer sizing are mandatory, and sloping of the downcomer apron may be used to provide additional disengaging surface (but at the expense of cross-sectional area for perforations). Even so, some gas can be expected to recycle under the downcomer apron. Plate Layouts Cross-flow plates, whether bubble-cap, sieve, or valve, are similar in layout (Fig. 14-28). Possible zones on each plate are: Active vapor-dispersion Peripheral stiffening and support Disengaging Distributing Downcomer The downcomer zones generally occupy 10 to 30 percent of the total cross section. For segmental downcomers, weir length ranges from 60 to 80 percent of the column diameter, so that the downcomer zone on each end of the plate occupies from 5 to 15 percent of the total cross section. The fraction of plate area occupied by disengaging and distributing zones ranges from 5 to 20 percent of the cross section. For most sieveplate designs, these zones are eliminated completely. The peripheral stiffening zone (tray ring) is generally 25 to 50 mm (1 to 2 in) wide and occupies 2 to 5 percent of the cross section, the fraction decreasing with increase in plate diameter. Periphery waste (Fig. 14-28) occurs primarily with bubble-cap trays and results from the inability to fit the cap layout to the circular form of the plate. Valves and perforations can be located close to the wall and little dead area results. Typical values of the fraction of the total cross-sectional area available for vapor dispersion and contact with the liquid for cross-flow plates with a chord weir equal to 75 percent of the column diameter are given in Table 14-6. The plate thickness of bubble-cap and sieve plates is generally established by mechanical design factors and has little effect on pressure drop. For a sieve plate, however, the plate is an integral component of the vapor-dispersion system, and its thickness is important.

14-30

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

(a) FIG. 14-28

(b)

Zone distribution. (a) Bubble-cap plate. (b) Sieve or valve plate.

For sieve plates, thickness is usually in the 10-to-14 U.S. standard gauge range of 3.58 to 1.98 mm, or 0.141 to 0.078 in. Hardness of metal, size of die, and limits on hole size (for process reasons) lead to the following thickness criterion: plate thickness 0.4 < ᎏᎏ < 0.7 hole diameter Bubble caps and valves are generally arranged on an equilateraltriangle layout. Center-to-center spacing should be great enough to minimize impact of vapor streams from adjacent units. Weights of valves on the same plate can be varied to control available flow area at different vapor loadings. Hole sizes for sieve plates range from 1 mm to 25 mm (0.04 to 1 in) diameter, with sizes in the 4- to 12-mm (0.16to 0.50-in) range being popular. Small holes have less entrainment but may present fouling problems. The smaller hole sizes may lead to punching problems, however, although 1–2 mm holes can be punched from aluminum plate materials. The spacing of the holes, usually on an equilateral-triangle basis, ranges from 2a to four diameters. Closer

spacings lead to excessive weeping, and greater spacings lead to excessive pressure drop and to entrainment because of high hole velocities. Countercurrent plates are of perforated or slotted construction and require no downcomers. The vapor and the liquid use the same openings, alternating on an intermittent basis. Layout of such plates is extremely simple. Types of such plates used commercially are Perforated (dual-flow) Slotted (Turbogrid) Perforated-corrugated (Ripple) The open area for these plates ranges from 15 to 30 percent of the total cross section compared with 5 to 15 percent for sieve plates and 8 to 15 percent for bubble-cap plates. Hole sizes range from 6 to 25 mm (d to 1 in), and slot widths from 6 to 12 mm (d to a in). The Turbogrid and Ripple plates are proprietary devices. Pressure Drop Methods for estimating fluid-dynamic behavior of crossflow plates are analogous, whether the plates be bubble-cap, sieve, or valve. The total pressure drop across a plate is defined by the general equation (see Fig. 14-29) ht = hd + h′L

(14-106)

where ht = total pressure drop, mm liquid hd = pressure drop across the dispersion unit (dry hole for sieve plates; dry valve for valve plates; dry cap, riser, and slot drop for bubble caps, mm liquid h′L = pressure drop through aerated mass over and around the disperser, mm liquid It is convenient and consistent to relate all of these pressure-drop terms to height of equivalent clear liquid (deaerated basis) on the plate, in either millimeters or inches of liquid. Pressure drop across the disperser is calculated by variations of the standard orifice equation: ρG hd = K1 + K2 ᎏ U h2 ρL




FIG. 14-29 Pressure-drop contributions for cross-flow plates. hd = pressure drop through cap or sieve, equivalent height of plate liquid; hw = height of weir; how = weir crest; hs = static liquid seal; hhg = hydraulic gradient; and hdc = loss under downcomer.

(14-107)

where Uh = linear gas velocity through risers (bubble caps) or perforations (sieve plate), m/s. For bubble caps, K1 is the drop through the slots and K2 is the drop through the riser, reversal, and annular areas. Equations for evaluating these terms for various bubble-cap designs are given by Bolles (in chap. 14 of Smith, Equilibrium Stage Processes, McGraw-Hill, New York, 1963), or may be found in previous editions of this handbook.

GAS-LIQUID CONTACTING SYSTEMS TABLE 14-6

Disperser Bubble-cap

Sieve

Representative Plate Efficiencies Column diameter, ft

Tray spacing, in

Methanol-water Ethyl benzene-styrene Cyclohexane-n-heptane

1.31 1.31 2.5 3.2 2.6 4.0

10.6 16.3 14 15.7 19.7 24

Cyclohexane-n-heptane

4.0

24

Benzene-toluene Toluene-isooctane

1.5 5.0

15.7 24

14.7 14.7 14.7 14.7 1.9 14.7 24 50 5 24 14.7 14.7

Methanol-water Ethanol-water Methanol-water Ethyl benzene-styrene Benzene-toluene Methyl alcohol-n-propyl alcoholsec-butyl alcohol Mixed xylenes + C8-C10 paraffins and naphthenes Cyclohexane-n-heptane

3.2 2.5 3.2 2.6 1.5

15.7 14 15.7 19.7 15.7

14.7 14.7 14.7 1.9 14.7

1.0 1.57 0.75 3.0

6.0

18

18

1.38

13.0 4.0

21 24

1.25 2.0 2.0 2.0 2.0 2.0 2.0 3.0 2.0 2.0 2.0 1.0 0.75 3.0 3.0

System Ethanol-water

Pressure, psia

n-heptane-toluene methanol-water isopropanol-water toluene-methylcyclohexane

4.0 4.0 4.0 4.0 4.0 1.5 2.0 2.0 2.0

24 24 24 24 24 15.7 13.6 13.6 13.6

25 5 24 5 165 165 300 400 14.7 14.7 14.7 14.7

Methanol-water Ethanol-water Ethyl benzene-styrene Cyclohexane-n-heptane n-Butane-isobutene

3.2 2.5 2.6 4.0 4.0

15.7 14 19.7 24 24

14.7 14.7 1.9 20 165

Isobutane-n-butane

Valve

14-31

Static submergence, in 1.18 1.18 1.2 1.0 0.2 0.25 4.25 0.6 1.5 0.4

Efficiency, % Emv*

Eoc†

83–87 84–97 80–85 90–95 55–68 65–90 65–90 65–90 65–85 75–100 70–80

Remarks

Ref. 1 2 3 4 5

Tunnel caps

7 8

60–80 70–90 75–85 90–100 70 60–75

10.8% open 10.4% 4.8% open 12.3% open 8% open 64

3 2 3 9 14 10

86 60–70 80 70–80 110 120 110 100 60–75 68–72% 59–63% 70–82%

14% open 14% open 8% open 14% open 8% open 8% open 8% open 8% open 10% open

70–80 75–85 75–85

14.7% open 50–96 104–121

6

Rect. valves Rect. valves

5 13 13 12 13 12 12 12 14 15 15 15 3 2 4 11 11

References 1. Kirschbaum, Z. Ver. Dtsch. Ing. Beih. Verfahrenstech., (5), 131 (1938); (3), 69 (1940). 2. Kirschbaum, Distillier-Rektifiziertechnik, 4th ed., Springer-Verlag, Berlin and Heidelberg, 1969. 3. Kastanek and Standart, Sep. Sci. 2, 439 (1967). 4. Billet and Raichle, Chem. Ing. Tech., 38, 825 (1966); 40, 377 (1968). 5. AIChE Research Committee, Tray Efficiency in Distillation Columns, final report, University of Delaware, Newark, 1958. 6. Raichle and Billet, Chem. Ing. Tech., 35, 831 (1963). 7. Zuiderweg, Verburg, and Gilissen, Proc. Intn. Symp., Brighton, England, 1960. 8. Manning, Marple, and Hinds, Ind. Eng. Chem., 49, 2051 (1957). 9. Billet, Proc. Intn. Symp., Brighton, England, 1970. 10. Mayfield, Church, Green, Lee, and Rasmussen, Ind. Eng. Chem., 44, 2238 (1952). 11. Fractionation Research, Inc. “Report of Tests of Nutter Type B Float Valve Tray,” July 2, 1964 from Nutter Engineering Co. 12. Sakata and Yanagi, Inst. Chem. Eng. Symp. Ser., no. 56, 3.2/21 (1979). 13. Yanagi and Sakata, Ind. Eng. Chem. Process Des. Dev., 21, 712 (1982). 14. Zuiderweg and Van der Meer, Chem. Tech. (Leipzig), 24, 10 (1972). 15. Korchinsky, Trans. I. Chem. E., 72, Part A, 472 (1994). *See Eq. (14-130). †See Eq. (14-127). NOTE: To convert feet to meters, multiply by 0.3048; to convert inches to centimeters, multiply by 2.54; and to convert psia to kilopascals, multiply by 6.895.

(14-108)

h′L = βhds (14-109) β = aeration factor, dimensionless hds = calculated height of clear liquid over the dispersers, mm (dynamic seal)

For valve plates, values of K1 and K2 depend on whether the valves are fully open. They also depend on the shape and weight of the valves. Vendors of valve plates make K1 and K2 data (or their equivalent) readily available. An analysis of valve plate pressure drop has been reported by Bolles [Chem. Eng. Progr. 72(9), 43 (1976)], and typical dry head loss data, shown in Fig. 14-31, are taken from that work. Pressure drop through the aerated liquid [h′L, in Eq. (14-106)] is calculated by

The aeration factor β has been determined for bubble-cap and sieve plates, and a representative correlation of its values is shown in Fig. 14-32. Values of β in the figure may be calculated from q β = 0.0825 ln ᎏ − 0.269 ln Fvh + 1.679 (14-110) Lw where Lw = weir length, m 3 0.5 Fvh = F-factor for flow through holes, Fvh = Uhρ0.5 G , m/s (kg/m )

For sieve plates, K1 = 0 and K2 = 50.8/Cv2. Values of Cv are taken from Fig. 14-30. Values from Fig. 14-30 may be calculated from Cv = 0.74 (Ah /Aa) + exp[0.29(tt /dh) − 0.56]

where




14-32

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

Aeration factor for pressure drop calculation, sieve plates. [Bolles and Fair, Encyclopedia of Chemical Processing and Design, vols. 16, 86. J. M. McKetta (ed.), Marcel Dekker, New York, 1982.]

FIG. 14-32

Discharge coefficients for gas flow, sieve plates. [Liebson, Kelley, and Bullington, Pet. Refiner, 36(3), 288 (1957).]

FIG. 14-30

For sieve and valve plates, hds = hw + how + 0.5hhg

(14-111)

where hw = weir height, mm how = height of crest over weir equivalent clear liquid, mm hhg = hydraulic gradient across plate height of equivalent clear liquid, mm The value of weir height crest how may be calculated from the Francis weir equation and its modifications for various weir types. For a segmental weir and for height in millimeters of clear liquid,




q how = 664 ᎏ Lw where q = liquid flow, m3/s Lw = weir length, m

2/3

(14-112)

For serrated weirs,




where

q′ how = 851 ᎏ tan θ/2 q′= liquid flow, m3/s per serration θ = angle of serration, degrees



0.4

(14-113)

For circular weirs,




q how = 44,300 ᎏ dw where q = liquid flow, m3/s dw = weir diameter, mm

0.704

(14-114)

As noted, the weir crest how is calculated on an equivalent clearliquid basis. A more realistic approach is to recognize that in general a froth or spray flows over the outlet weir (settling can occur upstream of the weir if a large “calming zone” with no dispersers is used). Bennett et al. [AIChE J., 29, 434 (1983)] allowed for froth overflow in a comprehensive study of pressure drop across sieve plates; their correlation for residual pressure drop h′L in Eq. (14-87) is represented by Eqs. (14-115) through (14-120): h′L = hL + h′σ

(14-115)

where h′L = pressure drop through the aerated liquid (= hL − h′σ), mm hL = effective clear-liquid height (liquid holdup), mm h′σ = pressure drop for surface generation, mm




472σ = ᎏ gρL

−ρ ) 
ᎏᎏ dσ  g(ρL

G

1/3

(14-116)

h

with σ = surface tension, mN/m. First, an effective froth density φe (dimensionless) is calculated: φe = exp(−12.55K 0.91 s ) Pressure drop for a valve plate, measured versus model prediction of Bolles [Chem. Eng. Progr. 72(9), 43 (1976)]. Reproduced with permission of the American Institute of Chemical Engineers. Copyright © 1976 AIChE. All rights reserved. FIG. 14-31

where φe = hL /hf hf = froth height, mm Ks = Ua.[ρg /(ρL − ρG)0.5 = Fva /(∆ρ)0.5

(14-117) (14-117a) (14-118)

GAS-LIQUID CONTACTING SYSTEMS Ua = vapor velocity through the active area, m/s Fva = F-factor based on active area, m/s(kg/m3)0.5 Then the liquid holdup is calculated: where

hL = φe [hw + 15,330C(q/φe)2/3]

(14-119)

C = 0.0327 + 0.0286 exp[−0.1378hw].

(14-120)

In these equations, the h terms and dh (perforation diameter) are in mm, densities are in kg/m3, surface tension is in mN/m, and flow rate q is in m3/s. The gravitational constant g is 9.81 m/s2. For total pressure drop across the plate, Eq. (14-96) is used in conjunction with Eq. (14-107) and Fig. 14-30. For a base of 302 data points covering a wide range of systems and conditions, Eqs. (14-115) through (14-120) gave an average error of ⫾0.35 percent. For a similar data base, Eqs. (14-106) and (14-109) together with Fig. 14-32 gave an average error of less than 5 percent. Although more difficult to use, the method of Bennett et al. is recommended when determination of pressure drop is of critical importance. Example 11: Pressure Drop, Sieve Plate For the conditions of Example 10, estimate the pressure drop for flow across one plate. The thickness of the plate metal is 2 mm and the hole diameter is 4.8 mm. The superficial F-factor is 2.08 m/s(kg/m3)1/2. Solution. Method A: Eqs. (14-106, 14-109, 14-110), where ht = hd + β(hw + how). For Fvs = 2.08, Fva = 2.32 and Fvh = 16.55. From Example 9, Lw = 1.50 m and hw = 38 mm. For a liquid rate of 22,000 kg/hr, q = 7.27(10−3) m3/s, and q/Lw = 4.8(10−3). By Eq. (14-110) or Fig. 14-32, β = 0.48. From Eq. (14-108) or Fig. 14-30, Cv = 0.78. Then, by Eq. (14-107), hd = 32.6 mm liquid. Using Eq. (14-112), how = 18.9 mm. Finally, ht = hd + β(hw + how) = 32.6 + 0.48(38 + 18.9) = 60.0 mm liquid. Method B: Bennett et al. ht = hd + hL + h′σ; hd = same as for Method A. Eq. (14-116): h′σ = [(472 × 25)/(9.81 × 841)] [(9.81 × 841)/(5 × 25)]1/3 = 5.47 mm; Eq. (14-118): Ks = Fva/∆ρ0.5 = 2.32/(841 − 0.481)0.5 = 2.32/(840.5)0.5 = 0.080 m/s; Eq. (14-117): φe = exp[−12.55(0.080]0.91 = 0.284 (effective froth density); Eq. (14-120): C = 0.0327 + 0.0286 exp[−0.1378(38)] = 0.0329; and Eq. (14-119): hL = 0.284[38 + 15,330(0.0329)(0.00727/0.284]2/3 = 23.23 mm. Finally, ht = hd + hL + h′σ = 32.6 + 23.23 + 5.47 = 61.3 mm liquid. When straight or serrated segmental weirs are used in a column of circular cross section, a correction may be needed for the distorted pattern of flow at the ends of the weirs, depending on liquid flow rate. The correction factor Fw from Fig. 14-33 is used directly in Eq. (14-112) or Eq. (14-119). Even when circular downcomers are utilized, they are often fed by the overflow from a segmental weir. When the weir crest over a straight segmental weir is less than 6 mm (d in), it is desirable to use a serrated (notched) weir to provide good liquid distribution. Inasmuch as fabrication standards permit the tray to be 3 mm (f in) out of level, weir crests less than 6 mm (d in) can result in maldistribution of liquid flow.

14-33

Loss under Downcomer The head loss under the downcomer apron, as millimeters of liquid, may be estimated from q 2 hda = 165.2 ᎏ (14-121) Ada 3 where q = volumetric flow of liquid, m /s and Ada = minimum area of flow under the downcomer apron, m2. Although the loss under the downcomer is small, the clearance is significant from the aspect of tray stability and liquid distribution. The seal between the top of the liquid on the plate and the bottom of the downcomer should range between 13 and 38 mm (a and 1a in). Hydraulic Gradient Hydraulic gradient, the head of liquid necessary to overcome the frictional resistance to liquid (froth) passage across the plate, is important for plate stability inasmuch as it is the only liquid head that varies across the length of passage. If the gradient is excessive, the upstream portion of the plate may be rendered inoperative because of increased resistance to gas flow caused by increased liquid head (Fig. 14-34). In general the empirical criterion for stable operation is hd > 2.5hhg. Sieve plates usually have negligible hydraulic gradient. Bubble-cap plates can have significant gradient because of the blockage by the caps. Valve plates presumably are intermediate, with hydraulicgradient characteristics approaching those of sieve plates. For bubble-cap plates, hydraulic-gradient must be given serious consideration. It is a function of cap size, shape, and density on the plate. Methods for analyzing bubble-cap gradient may be found in the chapter by Bolles (Smith, Design of Equilibrium Stage Processes, Chap. 14, McGraw-Hill, New York, 1963) or in previous edition of this handbook. The hydraulic gradient on sieve plates should be checked in cases of long flow path of liquid. Hughmark and O’Connell (Chem. Eng. Prog., 53(3), 127 (1957)] presented a correlation for determining sieve-plate hydraulic gradient. Although the correlation does not explicitly indicate an effect of gas velocity, the effect is implicit in the choice of friction factor. The gradient is predicted by the relationship 1000f Uf2 Lf hhg = ᎏᎏ (14-122) gR h where f = friction factor correlated against a Reynolds modulus as in pipe flow RhUf ρL NReh = ᎏ (14-123) µL as shown in Fig. 14-34. In Eqs. (14-122) and (14-123), Rh is the hydraulic radius of the aerated mass, defined as follows: hf Df cross section Rh = ᎏᎏ = ᎏᎏ (14-124) wetted perimeter 2hf + 1000Df




where Df is the arithmetic average between tower diameter and weir

FIG. 14-33 Correction for effective weir length. To convert gallons per minute to cubic meters per second, multiply by 6.309 × 10−5; to convert feet to meters, multiply by 0.3048. [Bolles, Pet. Refiner, 25, 613 (1946).]

14-34

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

FIG. 14-34 Friction factor for froth crossflow, sieve plates. To convert inches to millimeters, multiply by 25.4. (Smith, Design of Equilibrium Stage Processes, McGraw-Hill, New York, 1963.)

length (average width of flow path), and hf is froth height. The value of hf is estimated from Eq. (14-117a). Uf is the velocity of the aerated mass, m/s, and is the same as for the clear liquid: 1000q 1000q Uf = ᎏ = ᎏ hf φeDf hL Df

(14-125)

Other terms in Eqs. (14-122) through (14-125) are: g = acceleration of gravity, m/s2 Lf = length of flow path across plate, m q = liquid-flow rate, m3/s φe = froth density on plate, dimensionless µL = liquid viscosity, Pa⋅s or kg/(m⋅s) ρL = liquid density, kg/m3 Phase Inversion Normally the two-phase mixture on the plate is in the form of a bubbly, or aerated liquid. This liquid-continuous mixture is called a froth. Under high gas rates and low liquid rates, however, the regime can invert to a gas-continuous spray comprising a multitude of liquid droplets of varying diameter. Many studies of this froth-to-spray transition have been made, most of them with air and water. The results of one such study, useful for design purposes, are shown in Fig. 14-35. The spray is predicted to prevail above the appropriate curve. Reviews of phase inversion have been provided by Lockett [Distillation Tray Fundamentals, Cambridge Univ. Press, Cambridge, U.K., 1986] and Prado et al. [Chem. Eng. Progr., 83(3), 32 (1987)]. The latter combined experimental observations of inversion with evaluations of plate efficiency, and proposed the following relationship for determining the gas velocity through the active portion of the tray, at the inversion point:




q Ua* = C1 ρG−0.50 ρL0.692 σ0.06 A0.25 ᎏ f Lw

0.05

dh−0.1

(14-126)

where Ua* = gas velocity through active area at inversion, m/s ρG = gas density, kg/m3 ρL = liquid density, kg/m3 σ = surface tension, mN/m Af = hole/active area ratio q/Lw = liquid flow, m3/s⋅m weir dh = hole diameter, mm C1 = 0.0583 for 25.4-mm overflow weirs = 0.0568 for 50.4-mm overflow weirs = 0.0635 for 101.6-mm overflow weirs

FIG. 14-35 Transition from froth to spray regime for holes of various diameters. Values on curves are liquid loadings, m3/(h⋅m weir length). To convert cubic meters per hour-meter to cubic feet per hour-foot, multiply by 10.764; to convert (meters per second) (kilograms per cubic meter)1/2 to (feet per second) (pounds per cubic foot)1/2, multiply by 0.8197; and to convert millimeters to inches, multiply by 0.0394. [Loon, Pinczewski, and Fell, Trans. Inst. Chem. Eng., 51, 374 (1973).]

Figure 14-25 also provides a means for estimating whether spray or froth might prevail on the tray. As can be seen, low values of the flow parameter FLG, as for vacuum fractionators, can lead to the spray regime. Plate Efficiency The efficiency of a plate for mass transfer depends upon three sets of design parameters: 1. The system—composition and properties 2. Flow conditions—rates of throughput 3. Geometry—plate type and dimensions The designer has little control over the first set but can deal effectively with the other two. Ultimate concern is with overall column efficiency: Eoc = Nt /Na

(14-127)

or the ratio of theoretical plates to actual plates required to make the separation. In arriving at a value of Eoc for design, the designer may rely on plant test data or on judicious use of pilot-plant-efficiency measurements. If such direct information is not available, the designer must resort to predictive methods. Methods for predicting plate efficiency are of three general types: 1. Empirical methods 2. Direct scale-up from laboratory measurements 3. Theoretical or semitheoretical mass-transfer methods The first of these gives Eoc directly. The second gives a point efficiency [Eq. (18-30)]. The third involves the prediction of individual phase efficiencies. Empirical Predictive Methods Two empirical correlations which have found wide use are the one of Drickamer and Bradford [Trans. Am. Inst. Chem. Eng., 39, 319 (1943)] and a modification of it by O’Connell [Trans. Am. Inst. Chem. Eng., 42, 741 (1946)]. The latter is shown in Fig. 14-36, the Drickamer-Bradford data are included in the distillation plot. A semitheoretical method which gives overall efficiency is that of

GAS-LIQUID CONTACTING SYSTEMS

14-35

Overall column efficiency of 25-mm Oldershaw column compared with point efficiency of 1.22-m-diameter-sieve sieve-plate column of Fractionation Research, Inc. System = cyclohexane-n-heptane. [(Fair, Null, and Bolles, Ind. Eng. Chem. Process Des. Dev., 22, 53 (1982).]

FIG. 14-37

O’Connell correlation for overall column efficiency Eoc for distillation. To convert centipoises to pascal-seconds, multiply by 10−3. [O’Connell, Trans. Am. Inst. Chem. Eng., 42, 741 (1946).]

FIG. 14-36

Bakowski [Br. Chem. Eng., 8, 384, 472 (1963); 14, 945 (1969)]. It is based on the assumption that the mass-transfer rate for a component moving to the vapor phase is proportional to the concentration of the component in the liquid and to its vapor pressure. Also, the interfacial area is assumed proportional to liquid depth, and surface renewal rate is assumed proportional to gas velocity. The resulting general equation for binary distillation is 1 ᎏᎏ Eoc = (14-128) KM 1 + 3.7(104) ᎏᎏ h′ρlT where Eoc = overall column efficiency, fractional K = vapor-liquid equilibrium ratio, y°/x y° = gas-phase concentration at equilibrium, mole fraction x = liquid-phase concentration, mole fraction M = molecular weight h′ = effective liquid depth, mm ρl = liquid density, kg/m3 T = temperature, K For sieve or valve plates, h′ = hw, outlet weir height. For bubble-cap plates, h′ = height of static seal. The original references present validations against laboratory and small-commercial-column data. Modifications of the efficiency equation for absorption-stripping are also included. Direct Scale-Up of Laboratory Distillation Efficiency Measurements It has been found by Fair, Null, and Bolles [Ind. Eng. Chem. Process Des. Dev., 22, 53 (1983)] that efficiency measurements in 25- and 50-mm- (1- and 2-in-) diameter laboratory Oldershaw columns closely approach the point efficiencies [Eq. (14-129)] measured in large sieve-plate columns. A representative comparison of scales of operation is shown in Fig. 14-37. Note that in order to achieve agreement between efficiencies it is necessary to ensure that (1) the systems being distilled are the same, (2) comparison is made at the same relative approach to the flood point, (3) operation is at total reflux, and (4) a standard Oldershaw device (a small perforated-plate column with downcomers) is used in the laboratory experimentation. Fair et al. made careful comparisons for several systems, utilizing as large-scale information the published efficiency studies of Fractionation Research, Inc. Theoretical Predictive Methods The approach to equilibrium on a plate may be defined as the ratio of the actual change in gas composition as it passes through the plate to the change that would have occurred if the gas had reached a state of equilibrium with the liquid. If a point on plate n is considered, this definition leads to the point efficiency:

yn − yn − 1 Eog = ᎏᎏ (14-129) y°n − yn − 1 point where y°n is the gas concentration in equilibrium with liquid concentration at the point. This efficiency cannot exceed 1.0 (100 percent). If there are liquid-concentration gradients on the plate (i.e., plate liquid is not completely mixed), then y° will vary and Eog may vary from point to point on the plate. It should be noted that an analogous efficiency definition could be expressed on the basis of liquid concentrations. It should be noted also that vaporization efficiency (Holland, Fundamentals of Multicomponent Distillation, McGraw-Hill, New York, 1981) could be used:






Ev = yn /y°n

(14-130)

For the entire plate and for gas concentrations, the Murphree vapor efficiency is used: yn − yn − 1 Emv = ᎏᎏ (14-131) y°n − yn − 1 plate where y°n is gas concentration in equilibrium with the concentration of the liquid leaving the plate (flowing into the downcomer, for a crossflow plate). Because of concentration gradients in the liquid, Emv can exceed 100 percent. The best-established theoretical method for predicting Eoc is that of the AIChE (Bubble-Tray Design Manual, American Institute of Chemical Engineers, New York, 1958). It is based on the sequential prediction of point efficiency, Murphree efficiency, and overall column efficiency:






Eog → Emv → Eoc with suitable correction of Emv for entrainment. The AIChE model is the basis for the development which follows. On the basis of the two-film model for mass transfer, and relating all efficiencies to gas-phase concentrations (for convenience only; a similar development can be made on the basis of liquid concentrations), point efficiency can be expressed in terms of transfer units: Eog = 1 − e−Nog

(14-132)

where Nog = overall transfer units calculated from Eq. (14-133). 1 Nog = ᎏᎏ (14-133) 1/Ng + λ /Nᐉ where Ng = gas-phase transfer units Nᐉ = liquid-phase transfer units λ = mGm /Lm (stripping factor) m = slope of equilibrium curve Gm = gas rate, mol/s Lm = liquid rate, mol/s Transfer units are dimensionless and are defined further in Sec. 5. According to Eq. (14-133), the evaluation of point efficiencies reduces

14-36

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

to the prediction of point values of Ng and Nᐉ plus the evaluation of m, Gm, and Lm for the particular conditions under investigation. Gas-phase transfer units are obtained from Eq. (14-134) Ng = kgaθg

(14-134)

where kg = gas-phase mass-transfer coefficient, (kg⋅mol)/(s⋅m2) (kg⋅mol/m3) or m/s a = effective interfacial area for mass transfer, m2/m3 froth on plate θg = residence time of gas in froth zone, s The effect of increasing gas rate is to increase kg and decrease θg, with the result that Ng tends to be constant over a range of gas rates. For sieve trays, Chan and Fair [Ind. Eng. Chem. Proc. Des. Dev., 23, 814 (1983)] used a data bank of larger-scale distillation column efficiencies to deduce the following expression for the product kGa: 316DG1/2 (1030 f + 867f 2) kga = ᎏᎏᎏ h1/2 L where kg = gas-phase mass-transfer coefficient, m/s a = effective interfacial area, m2/m3 froth DG = gas-phase diffusion coefficient, m2/s f = approach to flood, fractional hL = liquid holdup on plate, mm

(14-135)

Note that the product of the mass-transfer coefficient and the interfacial area is a volumetric coefficient and obviates the need for a value of the interfacial area. While areas for mass transfer on plates have been measured, the experimental contacting equipment differed significantly from that used for commercial distillation or gas absorption, and the reported areas are considered unreliable for design purposes. For evaluating the residence time θG of gas in the froth, the volume of the froth is taken as Aa hf, where the height of the froth hf is obtained by first determining effective froth density [Eq. (14-117)]. The dimensionless froth density is defined by hL φ = ᎏᎏ (14-136) hf + ε(ρG/ρL) When

ρL >> ρG, φ ∼ hL/hf

and

ε∼1−φ

(14-137)

then, residence time(s) may be estimated as εhf Aa 1 − φhL Aa = ᎏᎏ (14-138) θG = ᎏ 103 Q 103 φQ where Q = volumetric flow of vapor through the plate, m3/s hf = froth height, mm = hL /φ Liquid phase transfer units are obtained from NL = kLaθL

(14-139)

where kL = liquid phase transfer coefficient, (kg⋅mol)/(s⋅m2) (kg⋅mol/m3) = m/s a = effective interfacial area for mass transfer, m2/m3 froth or spray on the plate θL = residence time of liquid in the froth or spray zone, s The mass-transfer coefficient of Eq. (14-139) is carried as a product with interfacial area (giving a volumetric mass transfer coefficient): Sieve plates: kLa = (3.875 × 108 DL)0.5 (0.40Ua ρ G0.5 + 0.17)

(14-140)

Bubble-cap plates: kLa = (4.127 × 108 DL)0.5 (0.21UaρG0.5 + 0.15)

(14-141)

where DL = liquid-phase diffusion coefficient, m2/s (see Sec. 3). The residence time of liquid in the froth is (1 − ε) hf Aa θL = ᎏᎏ 103 q where q = volumetric flow of liquid across the plate, m3/s.

(14-142)

In summary, the point efficiency Eog is computed from Eq. (14-132) using NoG from Eq. (14-113), NG from Eq. (14-134) and m based on the relative voatility of the system. For a binary mixture and a zone of constant relative volatility, αij m = ᎏᎏ (14-143) [1 + (αij − 1) xi]2 where αij = relative volatility, component i (more volatile material) relative to component j, and xi = mole fraction of i in the liquid. The method for estimating point efficiency, outlined here, is not the only approach available for sieve plates, and more mechanistic methods are under development. For example, Prado and Fair [Ind. Eng. Chem. Res., 29, 1031 (1990)] have proposed a method whereby bubbling and jetting are taken into account; however the method has not been validated for nonaqueous systems. Chen and Chuang [Ind. Eng. Chem. Res., 32, 701 (1993)] have proposed a more mechanistic model for predicting point efficiency, but it needs evaluation against a commercial scale distillation data bank. One can expect more development in this area of plate efficiency prediction. Example 12: Estimation of Plate Efficiency For the conditions of Examples 9 and 11, estimate the point efficiency of the tray. Additional property data: Relative volatility, ethylbenzene/styrene 1.40 Stripping factor, λ 1.17 Gas diffusion coefficient, m2/s 2.09 (10−5) 2 Liquid diffusion coefficient, m /s 3.74 (10−9) Solution Gas flow: 25,500/(3600 × 0.481) = 14.73 m3/s Liquid flow: 22,000/(3600 × 841) = 0.00727 m3/s Froth density (Example 11), φ = 0.284 Liquid holdup (Example 11), hL = 23.23 mm liquid Approach to flood (Example 9) = 0.74 = 74% Gas residence time in froth [Eq. (14-138)] = [(1 − 0.284)(23.23)(4.41)]/ [103 (0.284)(14.73)] = 0.0175 s Gas volumetric coefficient [Eq. (14-135)] = kGa = {316 (2.09 × 10−5)0.5 [1030(0.74) − 867(0.742)]}/(23.23)0.5 = 86.1 s−1 Number of gas-phase transfer units [Eq. (14-134)]: NG = kGaθG = 86.1(0.0175) = 1.51 Liquid residence time in froth [Eq. (14-142)] = [(0.284)(81.8)(4.41)]/103 (0.00727)] = 14.1 s Liquid volumetric coefficient [Eq. (14-140)] = [3.875(108)(3.74(10−9)]0.5 [0.40(3.34)(0.481)0.5 + 0.17] = 1.32 s−1 Number of liquid-phase transfer units [Eq. (14-139)]: NL = kLaθL = 1.32 (14.1) = 18.6 Number of overall transfer units [Eq. (14-133)]: NOG = 1/[1/1.51 + 1.17/18.6)] = 1.37 Point efficiency [Eq. (14-132)]: EOG = 1 − exp(−NOG) = 1 − exp(−1.37) = 0.75 (75%) This value of efficiency is slightly higher than the measured value (Fig. 14-24). Effects of Gas and Liquid Mixing As noted previously, it is necessary in most instances to convert point efficiency Eog to Murphree plate efficiency Emv. This is true because of incomplete mixing; only in small laboratory or pilot-plant columns, under special conditions, is the assumption Eog = Emv likely to be valid. For a crossflow plate with no liquid mixing there is plug flow of liquid. For this condition of liquid flow, Lewis [Ind. Eng. Chem., 28, 399 (1936)] analyzed effects of gas mixing on efficiency. He considered three cases: 1. Gas enters plate at uniform composition (gas completely mixed between plates). 2. Gas unmixed; liquid flows in the same direction on successive plates. 3. Gas unmixed; liquid flows in alternate direction on successive plates. Case 1 has found the widest application in practice and is represented by the relationship Emv =1/λ[exp (λEvg) − 1]

(14-144)

λ is defined as for Eq. (14-133). Equation (14-144) assumes the following in addition to the base conditions: 1. L/V is constant. 2. Slope of equilibrium curve m is constant. 3. Point efficiency is constant across the tray.

GAS-LIQUID CONTACTING SYSTEMS

14-37

2. Sieve plates have less back mixing than bubble-cap plates because of less obstruction to flow. 3. Increased gas rate increases turbulence and the degree of back mixing of liquid. An alternative approach is presented in the AIChE Bubble-Tray Design Manual and is based on an eddy-diffusion model. According to this model, Emv 1 − e−(n′ + N ) ᎏ = ᎏᎏᎏ Eog (n′ + NPe){1 + [(n′ + NPe)/n′]} Pe

en − 1 + ᎏᎏᎏ n′{1 + [n′/(n′ + NPe)]} NPe where n′ = ᎏ 2 Effect of length of liquid path on number of stages. To convert feet to meters, multiply by 0.3048. [O’Connell and Gautreaux, Chem. Eng. Prog., 51, 236 (1955).]

FIG. 14-38

Most plate columns operate under conditions such that gas is completely mixed as it flows between the plates, but few operate with pure plug flow of liquid. Departure from plug flow of liquid has been studied by Gautreaux and O’Connell [Chem. Eng. Prog., 51, 232 (1955)] by assuming that liquid mixing can be represented as occurring in a series of stages of completely mixed liquid. For this model, Eog n 1 Emv = ᎏ 1 + λ ᎏ − 1 (14-145) λ n




 

where n = number of stages occurring on the tray. An approximation of the number of stages can be obtained from Fig. 14-38 using the following criteria: 1. Increased liquid rate favors plug flow.

(14-146)

E − 1 ᎏ
 1 − 4λ N og

Pe

Zᐉ2 NPe = Péclet number (dimensionless) = ᎏ DEθl Zᐉ = length of liquid travel, m; and λ = stripping factor [see Eq. (14-133)]. The value of θl is calculated from Eq. (14-142). The term DE is an eddy-diffusion coefficient that is obtained from experimental measurements. For sieve plates, Barker and Self [Chem. Eng. Sci., 17, 541 (1962)] obtained the following correlation: DE = 6.675(10−3)Ua1.44 + 0.922(10−4)hᐉ − 0.00562

(14-147)

where DE = eddy-diffusion coefficient, m2/s Ua = gas velocity through active area of plate, m/s hᐉ = liquid holdup on plate (Eq. 18-16), mm For bubble-cap plates, the eddy-diffusion correlation in the AIChE Bubble-Tray Design Manual should be used. The graphical representation of Eq. (14-146) is indicated in Fig. 14-39, where as usual λ = mGm /Lm.

FIG. 14-39 Mixing curves. Pe = Péclet number (NPe). (Bubble-Tray Design Manual, American Institute of Chemical Engineers, New York, 1958.)

14-38

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

The dimensionless Péclet number in Eq. (14-146) is a key parameter in evaluating departure from plug flow of liquid across the plate. The higher its value, the greater the enhancement of point efficiency, as shown in Fig. 14-39. A long liquid flow path ZL, a high liquid flow rate (low qL, and a low amount of diffusive backmixing (low value of DE) contribute to the plug flow effect. For use in the Gautreaux/ O’Connell model [Eq. (14-145), Fig. 14-38)], it can be shown that the number of mixing stages n is approximately NPe + 2 n=ᎏ 2

(14-148)

Overall Column Efficiency Calculated values of Emv must be corrected for entrainment, if any, by the Colburn equation [Eq. (14-101)]. The resulting corrected efficiency Ea is then converted to column efficiency by the relationship of Lewis [Ind. Eng. Chem., 28, 399 (1936)]: log [1 + Ea(λ − 1)] N Eoc = ᎏ = ᎏᎏ Nt log λ

(14-149)

Comparison of Efficiency of Various Plates Several studies of various plates have been carried out under conditions such that direct and meaningful comparisons are possible. Required conditions include identical system, same pressure, same column diameter, and equivalent submergence. Standart and coworkers [Br. Chem. Eng., 11 (11), 1370 (1966); Sep. Sci., 2, 439 (1967)] used the methanol-water system at atmospheric pressure in a 1.0-m (3.3-ft) column. For a plate spacing of 0.4 m (15.7 in) they studied the following: 1. Bubble-cap plate—70-mm (2.75-in) round caps, 25.4-mm (1.0-in) submergence 2. Sieve plate—4.0-mm (5⁄32 in) holes, hole-active area = 0.048, 40mm (1.57-in) outlet weir 3. Turbogrid plate—4.6-mm (0.18-in) slot width, 14.7 percent open area 4. Ripple plate—2.85-mm (7⁄64 in) holes, 10.8 percent open area Efficiency data from the work are summarized in Fig. 14-40. Kirschbaum (Distillier-Rektifiziertechnik, 4th ed., Springer-Verlag, Berlin and Heidelberg, 1969) reported on studies of the ethanolwater system at atmospheric pressure, using several columns. For a 0.75-m (2.46-ft) column and 0.35-m (14-in) plate spacing, the following were covered: 1. Bubble-cap plate—90-mm (3.5-in) round caps, 30-mm (1.2-in) static submergence 2. Sieve plate—10-mm (25⁄64 -in) holes, hole/active area = 0.104, 25.4-mm (1.0-in) outlet weir 3. Valve plate—40-mm (1.57-in) holes, 45 valves per plate, 25.4mm (1.0-in) outlet weir. Efficiency data are given in Fig. 14-41. Billet and coworkers [Chem. Ing. Tech., 38, 825 (1966); Instn. Chem. Engrs. Symp. Ser. No. 32, 5, 111 (1969] used the ethylbenzene/ styrene system at 100 torr and a 0.8-m column with 500-mm plate spacing. Two weir heights were used, 19 and 38 mm. Operation was at

Plate efficiencies, ethanol-water. To convert (feet per second) (pounds per cubic foot)1/2 to (meters per second) (kilograms per cubic meter)1/2, multiply by 1.2199. (Kirschbaum, Destillier-Rektifiziertechnik, 4th ed., Springer-Verlag, Berlin and Heidelberg, 1969.)

FIG. 14-41

total reflux. A variety of plate devices were tested, and Fig. 14-42 shows typical results for the following: Curve 1 Curve 2 Curve 3 Curve 4

Bubble-cap plate, 35-mm weir Valve plate with 64 V-1 valves Sieve plate, 38-mm weir Sieve-valve plate, 38-mm weir, 49 valves, 140 holes

Testing of plates and other devices is carried out by Fractionation Research, Inc. for industrial sponsors. Some of the test data for sieve plates have been published for the cyclohexane/n-heptane and isobutane/n-butane systems. Representative data are shown in Fig. 14-43. These are taken from Sakata and Yanagi [Instn. Chem. Engrs. Symp. Ser. No. 56, 3.2/21 (1979)] and Yanagi and Sakata [Ind. Eng. Chem. Proc. Des. Devel., 21, 712 (1982)]. The column diameter was 1.2 m, tray spacing was 600 mm, and weir height was 50 mm. Work at the University of Manchester Institue of Science and Technology (UMIST) has resulted in several papers reporting efficiency data taken in a 0.6-m-diameter column. The systems methanol/water, isopropanoil/water, and toluene methylcyclohexane have been used. The results may be found in Lockett and Ahmed [Chem. Eng. Res. Des., 61, 110 (1983)], Korchinsky et al. [Trans. Chem. E., 72, 406 (1994)], and Korchinsky [ibid., 472] All the foregoing test programs involve distillation of well-defined mixtures under total reflux conditions. The primary value of the results is in the comparative data, but it should be emphasized that the design of each device was not necessarily optimized for the test conditions. Additional plate-efficiency data are listed in Table 14-6. PACKED COLUMNS

Plate efficiencies, methanol-water. To convert feet per second to meters per second, multiply by 0.3048. [Standart et al., Br. Chem. Eng., 11, 1370 (1966); Sep. Sci., 2, 439 (1967).] FIG. 14-40

Introduction Packed columns for gas-liquid contacting are used extensively for absorption, stripping, and distillation operations. Usually the columns are filled with a randomly oriented packing material, but for an increasing number of applications the packing is very care-

GAS-LIQUID CONTACTING SYSTEMS

14-39

∆ρ/nρ Pressure drop, mm w.g.

Efficiency, E

1.0 4 2

0.8

3

0.6

1

0.4 1

2 3

60 40 20 0 0.6

0.8

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 U v ρv, Vapor capacity factor, m–1/2 . s–1 . kg1/2

2.6

2.8

FIG. 14-42 Overall (Murphree) efficiency and pressure drop data for several devices using the same test mixture (ethylbenzene/styrene). See text for details. [Billet, Conrad, and Grubb, Instn. Chem. Engrs. Symp. Ser. No. 32, 5, 111 (1979).]

fully positioned in the column. The packed column is characteristically operated with counterflow of the phases. The packed column is a simple device compared with the plate column (Fig. 14-44). A typical column consists of a cylindrical shell containing a support plate for the packing material and a liquid distributing device designed to provide effective irrigation of the packing. Devices may be added to the packed bed to provide redistribution of liquid that might channel down the wall or otherwise become maldistributed. Several beds, each with liquid distributor and support device, may be used within the same column shell. For example, a distillation column with rectifying and stripping zones, requires a minimum of two beds. 180

Overall tray efficiency, %

14% 8% Isobutane-n-butane (1138 kPa) Cyclohexane-n-heptane (165 kPa) Cyclohexane-n-heptane (~30 kPa)

140

The key issue in packed column design is the selection of the packing material. Such material should provide effective contacting of the phases without excessive pressure drop. Many packings are commercially available, each possessing specific advantages with respect to cost, surface availability, interface regeneration, pressure drop, weight, and corrosion resistance. Packed beds may be divided into two categories: Those containing packing elements that are placed in the column in a random arrangement, usually by dumping; and those containing carefully installed elements designed specifically to fit the column dimensions. The former elements are called random, or dumped, packings. The latter are called ordered, or structured, packings. Typical random and structured packings are shown in Fig. 14-45. Packed Columns versus Plate Columns Packed columns are usually specified when plate devices would not be feasible because of undesirable fluid characteristics or some special design requirement. Conditions favoring packed columns are: 1. For columns less than 0.6-m (2.0-ft) diameter, packings are usually cheaper than plates unless alloy-metal packings are required. 2. Acids and many other corrosive materials can be handled in

100

60

20

0

1.0 2.0 Fs = Vsρv1/2, (m . s–1)(kg . m–3)1/2

3.0

FIG. 14-43 Overall (Murphree) efficiencies of sieve plates with hole/active area ratios of 0.08 and 0.14. Efficiency values greater than 1.0 (100%) result from crossflow effects (Figs. 14-38, 14-39). [Yanagi and Sakata, Ind. Eng. Chem., Proc. Des. Devel., 21, 712 (1982).] Reproduced with permission, copyright © 1982 American Chemical Society.

FIG. 14-44

Packed column (schematic).

14-40

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

(b)

(a)

(c)

(a) (d )

(e)

(f )

Representative random packings. Types (c), (e) and ( f) are the through-flow variety.

FIG. 14-45a

packed columns because construction can be of ceramic, carbon, or other resistant materials. 3. Packings often exhibit desirable efficiency-pressure-drop characteristics for critical vacuum distillations. 4. Liquids tending to foam may be handled more readily in packed columns because of the relatively low degree of liquid agitation by the gas. 5. Holdup of liquid can be quite low in packed columns, an advantage when the liquid is thermally sensitive. Conditions unfavorable to packed columns are: 1. If solids are present in the liquid or gas, plate columns can be designed to permit easier cleaning. 2. Some packing materials are subject to easy breakage during insertion into the column or resulting from thermal expansion and contraction. 3. High liquid rates can often be handled more economically in plate columns than in packed columns. 4. Cooling coils can be incorporated more readily into plate devices. 5. Low liquid rates lead to incomplete wetting of column packings, thus decreasing contacting efficiency. Packed-Column Hydraulics Pressure drop of a gas flowing upward through a packing countercurrently to liquid flow, is characterized graphically in Fig. 14-46. At very low liquid rates, the effective open cross section of the packing is not appreciably different from that of dry packing, and pressure drop is due to flow through a series of variable openings in the bed. Thus, pressure drop is proportional approximately to the square of the gas velocity, as indicated in the region AB. At higher liquid rates the effective open cross section is smaller because of the presence of liquid, and a portion of the energy of the gas stream is used to support an increasing quantity of liquid in the column (region A′B′). For all liquid rates, a zone is reached where pressure drop is proportional to a gas-flow-rate power distinctly higher than 2; this zone is called the loading zone, as indicated in Fig. 14-46. The increase in pressure drop is due to the rapid accumulation of liquid in the packing-void volume. As the liquid holdup increases, one of two changes may occur. If the packing is composed essentially of extended surfaces, the effective orifice diameter becomes so small that the liquid surface becomes continuous across the cross section of the column, generally at the top of the packing. Column instability occurs concomitantly with a rising continuous-phase liquid body in the column. The change in pressure drop is quite great with only a slight change in gas rate (condition C or C′). The phenomenon is called flooding and is analogous to entrainment flooding in a plate column.

(b) FIG. 14-45b Representative arranged-type packings: (a) Koch Sulzer, (b) Flexipac. (Courtesy Koch Engineering Co., Inc.)

FIG. 14-46

Pressure-drop characteristics of packed columns.

GAS-LIQUID CONTACTING SYSTEMS

FIG. 14-47

Efficiency characteristics of packed columns (total-reflux distil-

lation.)

If the packing surface is discontinuous in nature, a phase inversion occurs, and gas bubbles through the liquid. The column is not unstable and can be brought back to gas-phase continuous operation by merely reducing the gas rate. Analogously to the flooding condition, the pressure drop rises rapidly as phase inversion occurs. A stable operating condition beyond “flooding” (region CD or C′D′) for nonextended surface packing with the liquid as the continuous phase and the gas as the dispersed phase has been reported by Lerner and Grove [Ind. Eng. Chem., 43, 216 (1951)] and Teller [Chem. Eng., 61(9), 168 (1954)]. For total-reflux distillations carried out in packed columns, regions of loading and flooding are identified by their effects on mass-transfer efficiency, as shown in Fig. 14-47. Gas and liquid rate increase

together, and a point is reached at which liquid accumulates rapidly (point A) and effective surface for mass transfer decreases rapidly. Flooding and Loading Since flooding or phase inversion normally represents the maximum capacity condition for a packed column, it is desirable to predict its value for new designs. The first generalized correlation of packed-column flood points was developed by Sherwood, Shipley, and Holloway [Ind. Eng. Chem., 30, 768 (1938)] on the basis of laboratory measurements primarily on the airwater system. Later work with air and liquids other than water led to modifications of the Sherwood correlation, first by Leva [Chem. Eng. Prog. Symp. Ser., 50(10), 51 (1954)] and later in a series of papers by Eckert. The generalized flooding-pressure drop chart by Eckert [Chem. Eng. Progr., 66(3), 39 (1970)], included in the previous edition of this handbook, was modified and simplified by Strigle [Packed Tower Design and Applications, Gulf Publ. Co., Houston, 1994] as shown in Fig. 14-48. It includes pressure drop curves, as introduced by Leva [Chem. Eng. Progr. Symp. Ser. No. 10, 50, 51 (1954)], and is often called the generalized pressure drop correlation (GPDC). The ordinate scale term is a capacity parameter related to the Souders-Brown coefficient used for plate columns: 0.50 ρG Cs = Ut ᎏ Fp0.5 ν0.05 (14-150) (ρL − ρG)



where



Ut = superficial gas velocity, ft/s ρG, ρL = gas and liquid densities, lb/ft3 or kg/m3 Fp = packing factor, ft−1 ν = kinematic viscosity of liquid, cS

The abscissa scale term is the same flow parameter used for plates (dimensionless): L FLG = ᎏᎏ (14-151) G(ρG /ρL)0.5 There is not a specific flood curve; a pressure drop of 1.50 in H2O/ft is considered to represent an incipient flooding condition, although

Generalized pressure drop correlation of Eckert/Leva, as modified by Strigle. To convert inches H2O to mm H2O/m, multiply by 83.31. From Packed Tower Design and Applications by Ralph E. Strigle, Jr., copyright © 1994 by Gulf Publishing Co., Houston, Texas. Used with permission. All rights reserved.

FIG. 14-48

14-41

14-42

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

pressure drops at flooding have been measured in the range of 2.0–2.5 inches H2O/ft. The packing factor Fp is empirically determined for each packing type and size. Values of Fp, together with general dimensional data for individual packings, are given in Table 14-7. For pressure drop and flooding estimates, values of Fp should always be used with caution. A detailed analysis of the GPDC approach (Fig. 14-48) was made by Kister and Gill [Chem. Eng. Progr., 87(2), 32 (1991)] who amassed a large data bank, mostly for random packings. They found it necessary to use separate curves for each packing type and size and also to differentiate between aqueous and nonaqueous systems. An example of their work is shown in Fig. 14-49, for 2-inch (50 mm) metal Pall rings. Note that a packing factor of 27 ft−1 was used for this packing. A complete set of charts for both random and ordered packings is given in Chap. 10 of the book by Kister (Distillation Design, McGraw-Hill, New York, 1992). Pressure Drop Reference to pressure drop has already been made in connection with the GPDC (Fig. 14-48). For gas flow through dry packings, pressure drop may also be estimated by use of an orifice equation. For irrigated packings, pressure drop increases because of the presence of liquid, which effectively decreases the available cross section for gas flow (Fig. 14-46). In principle, there should be a method for correcting the dry pressure drop for the presence of water. This approach was used by Leva [Chem. Eng. Progr. Symp. Ser. No. 10, 50, 51 (1954)]. A more recent method by by Robbins [Chem. Eng. Progr., 87(1), 19 (1990)] utilizes the same approach and is described here. The total pressure drop is ∆Pt = ∆Pd + ∆PL

The term Fpd is a dimensionless dry packing factor, specific for a given packing type and size. Values of Fpd are given in Table 14-7. For operating pressures above atmospheric, and for certain packing sizes, Lf and Gf are calculated differently: Gf = 986Fs(Fpd /20)0.5 100.3ρG Lf = L(62.4/ρL)(Fpd /20) µ

Name

(14-157)

0.2 L

Fpd > 200

(14-158)

= L(62.4/ρL)(20/Fpd )0.5µL0.1

Fpd < 200

(14-159)

The Robbins equations require careful attention to dimensions. However, use of the equations has been simplified through the introduction of Fig. 14-50. The terms Lf and Gf are evaluated, and the ∆PL is obtained directly from the chart. Basic nomenclature for the Robbins method follows: C3 = 7.4 (10)−8 C4 = 2.7 (10)−5 Fpd = dry packing factor, dimensionless Fs = superficial F-factor for gas, Utρg0.5, ft/s(lb/ft3)0.5 G = gas mass velocity = lb/hr⋅ft2 Gf = gas loading factor, lb/hr⋅ft2 L = liquid mass velocity, lb/hr⋅ft2 Lf = liquid loading factor, lb/hr⋅ft2 ∆P = pressure drop, inches H2O/ft packing (× 83.3 = mm H2O/m packing) ρG = gas density, lb/ft3 ρL = liquid density, lb/ft3 µL = liquid viscosity, cP For ordered, or structured, packings, pressure-drop estimation methods have been reviewed by Fair and Bravo [Chem. Eng. Progr., 86(1), 19 (1990)]. It is not common practice to use the packing factor approach for predicting pressure drop or flooding. For operation below the loading point, the model of Bravo et al. [Hydrocarbon

Material

Nominal Surface % Packing size area, m2/m3 voids factor, 1/m

Vendor

Flexipac

S

1 2 3

558 223 134

91 93 96

108 72 52

Koch

Flexiramic

C

28 48 88

282 157 102

70 74 85

131 79 49

Koch

Gempak

S

4A 3A 2A

446 335 223

92 93 95

105 69 53

Glitsch

Intalox

S

1T 2T 3T

315 213 177

95 97 97

66 56 43

Norton

Max-Pak

S

—

229

95

39

Jaeger

Mellapak

S

125Y 125X* 250Y 250X* 350Y 500X*

125 125 250 250 350 500

97 97 95 95 93 91

33

Sulzer

B1-125 B1-250 BI-350 A3-500 BSH-250 BSH-500

125 250 350 500 250 500

97 95 93 91 95 91

Montz-Pak

S G E

(14-152)

where ∆Pt = total pressure drop, inches H2O per foot of packing ∆Pd = dry pressure drop = C3G 2f 10(C4 L f ) (14-153) ∆PL = pressure drop due to liquid presence = 0.4[Lf /20,000]0.1 [C3 G f2 10(C4 L f )]4 (14-154) Gf = gas loading factor = 986Fs(Fpd /20)0.5 (14-155) Lf = liquid loading factor = L(62.4/ρL)(Fpd /20)0.5µ0.1 (14-156) L

0.5

TABLE 14-7a Characteristics of Structured Packings

66 8 75 25 Julius Montz 72

Ralupak

S

250YC*

250

95

Raschig

Sulzer

G

AX* BX* CY

250 492 700

95 90 85

Sulzer 69

NOTES:

*60° corrugation angle (with the horizontal); all others 45°. Packing factors from Kister and Gill [Chem. Eng. Progr., 87(2), 32 (1991), and Houston AIChE meeting, March 19–23, 1995]. Materials of construction: C = ceramic; E = expanded metal; G = metal gauze; S = sheet metal. Vendors: Glitsch, Inc., Dallas, Texas; Koch Engineering Co., Wichita, Kansas; Jaeger Products, Inc., Houston, Texas; Julius Montz, Hilden, Germany; Raschig AG, Ludwigshafen, Germany; and Sulzer Bros., Winterthur, Switzerland.

Proc., 65(3), 45 (1986)] is preferred. To use this and alternate models, dimensional characteristics of structured packing must be defined. Figure 14-51 shows nomenclature and definitions of key dimensions. Not shown, but also important, is the angle the corrugations make with the horizontal (usually 45 or 60°). Then the Rocha et al. predictive equation is: 5 2 92.7 Uge 1 ∆Pt = 0.171 + ᎏ ρg ᎏ (14-160) ᎏᎏ Reg S 1 − Co FrL0.05



where









SUgeρg Reg = ᎏ µg Ug Uge = ᎏ (ε sin θ)

UL2 FrL = ᎏ (Sg) and where S = length of corrugation side Ug = superficial velocity of gas ε = void fraction of packing θ = corrugation angle (from horizontal), deg. UL = superficial liquid velocity g = gravitational constant Any consistent set of units may be used. This model applies in the region below the loading point, and it cannot predict the flood point because it does not include the effects of gas velocity on liquid holdup. The model of Stichlmair et al. [Gas

GAS-LIQUID CONTACTING SYSTEMS

14-43

TABLE 14-7b Characteristics of Random Packings Nominal size Material

mm

Number

Wall thickness, mm

Bed weight, kg/m3

Area, m2/m3

% voids

Packing factor Fp, m−1

Dry packing factor Fpd, m−1

Raschig rings

C

6 13 25 50 75

— — — — —

1.6 2.4 3.2 6.4 9.5

960 880 670 660 590

710 370 190 92 62

62 64 74 74 75

— 1900 587 213 121

5250 1705 492 230 —

Various

Raschig rings

M

19 25 50 75

— — — —

1.6 1.6 1.6 1.6

1500 1140 590 400

245 185 95 66

80 86 92 95

984 472 187 105

— 492 223 —

Various

Pall rings

M

16 25 38 50 90

— — — — —

0.40 0.51 0.64 0.81 —

480 415 385 270

205 130 115 92

92 94 95 96 97

256 183 131 89 59

262 174 91 79 46

Norton, Koch, Glitsch

— — — — — —

1 1.5 2.5 3 1A 3A

— — — — — —

389 234 195 58 71 40

250 144 123 103 185 74

96 97 98 98 94 96

131 95 72 46 98 39

102 — 79 43 92 33

Glitsch

6 13 25 38 50

— — — — —

— — — — —

900 865 720 640 625

900 465 250 150 105

60 62 68 71 72

2950 900 308 154 102

Koch

790 360 215 150

Norton

Name

Cascade mini rings (CMR)

M

P Berl saddles

C

Vendor

Intalox saddles

C

6 13 25 50 75

— — — — —

— — — — —

864 736 672 608 576

984 623 256 118 92

65 71 73 76 79

302 — 302 131 72

2720 613 208 121 66

Fleximax

M

— —

300 400

— —

— —

141 85

98 98

85 56

— —

Koch

Metal Intalox (IMTP)

M

25 40 50 70

— — — —

— — — —

352 237 150 130

230 154 98 56

97 97 98 98

134 79 59 39

141 85 56 —

Norton

Nutter rings

M

— — — —

0.30 0.45 0.45 0.50

178 173 145 133

168 96 83 66

98 98 66 98

98 59 52 43

89 56 49 36

Nutter

Pall rings

P

25 50 90 25 38 50

— — — — — —

— — — — — —

80 61 53

206 102 85

90 92 92

180 85 56 351 180 141

180 82 39 — — —

Norton

— —

1 2

— —

96 56

207 108

90 93

131 92

131 85

Norton

—

C

Intalox saddles

P

1 2 2.5 3.0

Snowflake

P

—

Nor-Pac

P

25 38 50

1 1.5 2.0

—

45

92

95

43

—

Norton

—

180 144 102

92 93 94

82 56 39

— — —

NSW

—

72 61 53

Tri-Pack

P

25 50

1 2

— —

72 53

180 102

92 94

82 39

— —

Jaeger

VSP

M

25 50

1 2

— —

352 296

206 112

98 96

105 69

— —

Jaeger

Tellerettes

P

25 50

1 2

— —

112 59

180 125

87 93

— —

— —

Ceilcote

M = metal, carbon steel. Other metals available. P = plastic, polypropylene. Other plastics available. Packing factor Fp from Kister and Gill [Chem. Eng. Progr., 87(2), 32 (1991) and Houston AIChE meeting, March 19–23, 1995]; Strigle, Packed Tower Design and Applications [Gulf Publishing Co., Houston, Texas, 1994]; dry packing factor Fpd from Robbins [Chem. Eng. Progr., 87(1), 19 (1990)]. Vendors: Ceilcote Co., Berea, Ohio; Glitsch, Inc., Dallas, Texas; Koch Engineering Co., Wichita, Kansas; Jaeger Products, Inc., Houston, Texas; NSW Corp., Roanoke, Virginia; Norton Co., Akron, Ohio; and Nutter Engineering Co., Tulsa, Oklahoma. NOTES:

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

Capacity parameter

14-44

3.0 2.8 2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0

Legend ∆P curves from top to bottom represent 1.5, 1.0, 0.50, 0.25, 0.10 inches H2O/ft Symbol ∆P, inches H2O/ft Symbol

Nonaqueous

1.5 1.0 0.5 0.25 0.10

0.01

0.05

0.10 Flow parameter

0.50

Flood MOC

1.00

Capacity parameter

Basic: Fp = 27 Pressure drop measured in inches H2O/ft 3.0 2.8 2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0

Legend ∆P curves from top to bottom represent 1.5, 1.0, 0.50, 0.25, 0.10 inches H2O/ft Symbol ∆P, inches H2O/ft Symbol 1.5 Flood 1.0 MOC 0.5 0.25 0.10

Aqueous

0.01

0.05

0.10 Flow parameter

0.50

1.00

Basic: Fp = 27 Pressure drop measured in inches H2O/ft FIG. 14-49 Pressure drop/flooding correlation of Kister and Gill for 2-inch metal Pall rings. The upper chart is for nonaqueous systems, the lower chart for aqueous systems. To convert inches H2O/ft to mm H2O/m, multiply by 83.31.

Sepn. Purif., 3, 19 (1989)] takes holdup into account and applies to random as well as structured packings. It is somewhat cumbersome to use and requires three constants for each packing type and size. Such constants have been evaluated, however, for a number of commonly used packings. A more recent pressure drop and holdup model, suitable for extension to the flood point, has been published by Rocha et al. [Ind. Eng. Chem. Research, 35, 1660 (1996)]. This model takes into account variations in surface texturing of the different brands of packing. Representative pressure drop data for random and structured packings are given in Figs. 14-52–14-54. Example 13: Packed Column Pressure Drop Air and water are flowing countercurrently through a bed of 2-inch metal Pall rings. The air mass velocity is 2.03 kg/s⋅m2 (1500 lbs/hr⋅ft2), and the liquid mass velocity is 12.20 kg/s⋅m2 (9000 lbs/hr⋅ft2). Calculate the pressure drop by the generalized pres-

sure drop (GPDC, Fig. 14-48) and the Robbins methods. Properties: ρG = 0.074 lbs/ft3; ρL = 62.4 lbs/ft3, µL = 1.0 cP, ν = 1.0 cS. The packing factor Fp = 27 ft−1. For Robbins, Fpd = 24 ft−1. The flow parameter FLG = L/G (ρG /ρL)0.5 = (9000/1500) (0.072/62.4)0.5 = 0.207. The F-factor = Fs = UtρG0.5 = G/(ρG0.5 3600) = 1500/[(0.074)0.5 (3600)] = 1.53 ft/s(lb/ft3)0.5 Using the GPDC method, the capacity parameter [by Eq. (14-150)] = Ut[ρG/(ρL − ρG)]0.5 Fp0.5 ν0.05, which is roughly equivalent to




Fs 0.5 0.05 1.53 Fp ν = ᎏ ᎏ ρL0.5 62.4

0.5

270.5(1.0)

= 1.01 Referring to Fig. 14-48, the intersection of the capacity parameter and the flow parameter lines gives a pressure drop of 0.38 inches H2O/ft packing. (The same result is obtained from the Kister/Gill chart, Fig. 14-49.) Using the Robbins method, Gf = 986Fs(Fpd /20)0.5 = 986(1.53)(24/20)0.5 = 1653. Lf = L (62.4/ρL)(Fpd /20)0.5 µ0.1 = 9000 (1.0)(1.095)(1.0) = 9859. Lf /Gf = 5.96. From Fig. 14-50, pressure drop = 0.35 in. H2O/ft packing.

2.0

Lf /Gf =

Generalized pressure drop correlation 100 60 40 20 10 5 2 1

0

S

GAS-LIQUID CONTACTING SYSTEMS

h

. DEQ = 4 RH = 4 S S 1 = S 2 2S

B

∆P inches water/ft packing

14-45

(

)

(for angle = 90°)

Triangular cross section

1.0 0.9 0.8 0.7 0.6

a. Flow channel cross section

0.5 0.4

Flow channel 0.3

0.2

0.1 400

2000 600 8001000 Gf, Gas loading factor

4000

6000

FIG. 14-50 Pressure drop correlation for random packings, as presented by Robbins. [Chem. Eng. Progr., 87(1), 19 (1990). Reproduced with permission of the American Institute of Chemical Engineers. Copyright © 1990 AIChE. All rights reserved.] To convert inches H2O/ft to mm H2O/m, multiply by 83.31.

b. Flow channel arrangement FIG. 14-51

Geometric properties of typical structured packings.

Pressure drop for metal pall rings, 0.024-in wall thickness (1-in size) and 0.036-in wall (2-in size). Metal Raschig rings have g-in wall. L = lb liquid/(h⋅ft2). To convert inches of water per foot to millimeters of water per meter, multiply by 83.31; to convert inches to millimeters, multiply by 25.4; and to convert pounds per hour-square foot to kilograms per second-square meter, multiply by 0.001356. [Eckert, Foote, and Huntington, Chem. Eng. Progr., 54(1), 70 (1958).]

FIG. 14-52

14-46

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

Pressure for metal Intalox saddles, sizes No. 25 (nominal 25 mm) and No. 50 (nominal 50 mm). Air-water system at atmospheric pressure, 760mm (30-in) column, bed height, 3.05 m (10 ft). L = liquid rate, kg/(s⋅m2). To convert kilograms per second-square meter to pounds per hour-square foot, multiply by 151.7; to convert pascals per meter to inches of water per foot, multiply by 0.1225. (Courtesy Norton Company, Akron, Ohio.)

FIG. 14-53

Support Plates While the primary purpose of a packing support is to retain a bed of packing without excessive restriction to gas and liquid flow, it also serves to distribute both streams. Unless carefully designed, the support plate can also cause premature column flooding. Thus, design of the support plate significantly affects column pressure drop and stable operating range. Two basic types of support plates may be utilized: 1. Countercurrent 2. Separate flow passages for liquid and gas The two types are indicated in Figs. 14-55, 14-56, and 14-57. The degree of open area on a support plate is the fraction of void inherent in the design of the plate minus that portion of the open area occluded by the packing. To avoid premature flooding, the net open area of the plate must be greater than that of the packing itself. With the countercurrent type of support plate the free area for gas flow can range up to 90 percent of the column cross-sectional area. However, such a plate is easily occluded by the packing pieces resting directly on it. The separate flow passage devices can be designed for free areas up

FIG. 14-54 Pressure drop for Flexipac packing, sizes No. 1 and No. 3. Airwater system at atmospheric pressure. Liquid rate in gallons per minute-square foot. To convert (feet per second) (pounds per cubic foot)1/2 to (meters per second) (kilograms per cubic meter)1/2, multiply by 1.2199; to convert gallons per minute-square foot to pounds per hour-square foot, multiply by 500; to convert inches of water per foot to millimeters of water per meter, multiply by 83.31; and to convert pounds per hour-square foot to kilograms per second-square meter, multiply by 0.001356. (Courtesy Koch Engineering Co., Wichita, Kansas.)

to 90 percent, and because of their geometry they will have very little occlusion by the packing. Liquid Holdup Three modes of liquid holdup in packed columns are recognized: 1. Static, hs 2. Total, ht 3. Operating, ho

FIG. 14-55

Packing supports (countercurrent).

GAS-LIQUID CONTACTING SYSTEMS

14-47

where h values are in volumes of liquid per total volume of bed. The effective void fraction under operating conditions is ε′ = ε − ht

(14-162)

Static holdup depends upon the balance between surface-tension forces tending to hold liquid in the bed and gravity or other forces that tend to displace the liquid out of the bed. Estimates of static holdup (for gravity drainage) may be made from the following relationship of Shulman et al. [Am. Inst. Chem. Eng. J., 1, 259 (1955)]:

FIG. 14-56

Support plate, cap type (separate flow or “gas-injection” type).

C1 µC2σC3 hs = 2.79 ᎏᎏ ρ0.37 ᐉ

(14-163)

where µᐉ = liquid viscosity, mPa⋅s σ = surface tension, mN/m ρᐉ = liquid density, kg/m3 and constants are

FIG. 14-57

Beam-type “gas-injection” support plate for large columns.

Static holdup is the amount of liquid remaining on packing that has been fully wetted and then drained. Total holdup is the amount of liquid on the packing under dynamic conditions. Operating holdup is the amount of liquid attributed to operation and is measured experimentally as the difference between total and static holdup. Thus, ht = ho + hs

(14-161)

Packing

C1

C2

C3

1.0-in carbon Raschig rings 1.0-in ceramic Raschig rings 1.0-in ceramic Berl saddles

0.086 0.00092 0.0055

0.02 0.02 0.04

0.23 0.99 0.55

For other packings and for the case in which static holdup is changed by gas flowing through the bed, the method of Dombrowski and Brownell [Ind. Eng. Chem., 46, 1207 (1954)], which correlates static holdup with a dimensionless capillary number, should be used. Typical total holdup data for packings are shown in Figs. 14-58 and 14-59. It should be noted that over much of the preloading range gas rate has little effect on holdup. Operating holdup may be estimated by the dimensionless equation of Buchanan [Ind. Eng. Chem. Fundam., 6, 400 (1967)]: µ′ᐉ uᐉ ho = 2.2 ᎏ gρᐉ dp2






1/3




uᐉ2 + 1.8 ᎏ gdp

where µ′ᐉ = liquid viscosity, Pa⋅s uᐉ = liquid superficial velocity, m/s g = gravitational constant, m/s2 ρᐉ = liquid density, kg/m3 dp = nominal packing size, m

Typical holdup data for random packings and the air-water system. The raschig rings are of ceramic material. To convert pounds per hour per ft2 to kilograms per second per m2, multiply by 0.001356; to convert inches to millimeters, muultiply by 25.4. [Shulman et al., AIChE J. 1, 247 (1955).] FIG. 14-58

1/2

(14-164)

14-48

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN structured packings. The holdup model developed by the latter investigators is shown in Fig. 14-60 in a representative case. It is clear that the modern, through-flow random packings and the structured packings have much less operating liquid holdup of liquid than do the older, traditional bluff-body packings. Operating holdup contributes effectively to mass-transfer rate, since it provides residence time for phase contact and surface regeneration via agglomeration and dispersion. Static holdup is limited in its contribution to mass-transfer rates, as indicated by Thoenes and Kramers [Chem. Eng. Sci., 8, 271 (1958)]. In laminar regions holdup in general has a negative effect on the efficiency of separation. Liquid Distribution Uniform initial distribution of liquid at the top of the packed bed is essential for efficient column operation. This is accomplished by a device that spreads the liquid uniformly across the top of the packing. Baker, Chilton, and Vernon [Trans. Am. Inst. Chem. Eng., 31, 296 (1935)] studied the influence of the bed itself as a distributor and found that a single-point distribution in a 305-mm (12-in) column with 19-mm (e-in) packing required 3.05 m (10 ft) of bed before achieving uniform distribution across the bed. They found also a tendency for liquid to migrate toward the column wall (Fig. 14-61), especially for ratios of column diameter to packing size less than 8. For a multipoint distribution, their recommendation was one liquid stream for each 194-cm2 (30-in2) column area. Eckert [Chem. Eng. Prog., 57, 54 (1961)] recommends the following:

Typical vendor data for liquid holdup of a structured packing, Gempak 2A. [Courtesy Glitsch, Inc., Dallas, Texas.]

Column diameter, m

Streams/m2

1.2 or greater 0.75 0.40

40 170 340

FIG. 14-59

The first term is a “film number”; the second is the Froude number. The equation applies to ring packings only operating below the load point and correlates all literature data to about ⫾20 percent. Liquid holdup has been studied by Stichlmair et al. [Gas Sepn. Purif., 3, 19 (1989)] for both random and structured packings, and by Rocha et al. [Ind. Eng. Chem. Res., 32, 641 (1993)] for a variety of

Silvey and Keller [I. Chem. E. Symp. Ser. No. 32, 4, 18 (1969)] found that a trough-type distributor with 34 streams/m2 gave good liquid distribution in a 1.2-m column, up to a bed height of 10.7 m. Ceramic raschig rings of 38 and 76 mm nominal size were used, and efficiency profiles were measured by means of intermediate bed samples. A plot of their data is shown in Fig. 14-62; for the type of plot, a straight line indicates uniform distribution. The plotting technique is based on the Fenske relationship for theoretical stages (Sec. 13). This

Comparison of measured and calculated values of liquid holdup for Gempak 2A structured packing, air-water system. [Rocha et al., Ind. Eng. Chem., 32, 641 (1993).] Reproduced with permission. Copyright © 1993 American Chemical Society. FIG. 14-60

GAS-LIQUID CONTACTING SYSTEMS

14-49

1.5

1.0

log

(1 x– x )

0.5

0

–0.5

–1.0 Packing height Liquid distribution in a 6-in column packed with a-in brokenstone packing. Increments of radius represent equal-annual-area segments of tower cross section. Central-point inlet. Water rate = 500 lb/(h⋅ft2). Air rate = 810 lb/(h⋅ft2). To convert pounds per hour-square foot to kilograms per secondsquare meter, multiply by 0.001356; to convert inches to centimeters, multiply by 2.54. (Data from Baker, Chilton, and Vernon, in Sherwood and Pigford, Absorption and Extraction, 2d ed., McGraw-Hill, New York, 1952.) FIG. 14-61

38 75

= 3.0; = 6.4;

= 5.5 = 10.7

FIG. 14-62 Comparison of composition profiles at different bed heights and two sizes of ceramic raschig rings. Column diameter = 1.2 m, cyclohexane/ n-heptane system at 1.65 bar and total reflux. [Silvey and Keller, I. Chem. E. Symp. Ser. No. 32, 1969.]

2.0

1.5 No. of stages/m

same distributor was found to be unsatisfactory by Shariat and Kunesh [Ind. Eng. Chem. Res., 34, 1273 (1995)] when through-flow pall rings were used. A comparison between the trough distributor and an orifice-riser distributor with 100 streams/m2 is shown in Fig. 14-63. In the normal range of operation, 30–70 percent more stages were obtained with the increased number of streams. The column, test mixture (cyclohexane/n-heptane) and operating pressure (1.65 bar) were the same as for the earlier Silvey-Keller tests. The through-flow random packings and the structured packings require careful attention to liquid distribution. Such packings cannot correct an initial poor distribution. Several of the packing vendors maintain elaborate test stands for conducting special distributor evaluations. In general, practical aspects of design limit the number of streams to 100–120 per m2, and these values have proven adequate for the newer packings. Kunesh et al. [Ind. Eng. Chem. Res., 26, 1845 (1987)] have an in-depth treatment of commercial-scale experiments that support designs of packed column liquid distributors. Several distributor types are available: trough, orifice-rise, and perforated pipe. Examples of these types are shown in Fig. 14-64. The trough distributor provides good distribution under widely varying flow rates of gas and liquid. The liquid may flow through simple V-notch weirs, or it may flow through tubes that extend from the troughs to near the upper level of the packing. Some deposition of solids can be accommodated. The orifice-riser distributor is designed to lay the liquid carefully onto the bed, with a minimum of contact with gas during the process. It can be designed to provide a large number of liquid streams, with the limit of sufficient liquid head to provide uniform liquid flow through the orifices. The gas risers must be designed to accommodate the expected variations in flow rate, often with a minimum of pressure drop. For very distribution-sensitive packings, it is necessary to include pour points in the vicinity of the column wall (to within 25 mm).

Bed height, m

Diameter of rings, mm

1.0

0.5 Tubed drip pan, 3.66-m bed, dry-packed Notched trough, 5.49-m bed, wet-packed 0.0 0.0

0.5

1.0

1.5

2.0

2.5

F , m/s(kg/m3)0.5 Efficiency of beds of 51 mm Pall rings with two different distributors. Column diameter = 1.2 m, cyclohexane/n-heptane system at 1.65 bar and total reflux. [Shariat and Kunesh, Ind. Eng. Chem. Res., 34, 1273 (1995).] Reproduced with permission. Copyright © 1995, American Chemical Society.

FIG. 14-63

14-50

FIG. 14-64

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

Typical liquid distributors. (Courtesy Norton Company, Akron,

2. The liquid moving more easily to the wall than vice versa. The resultant channeling along the wall may be accentuated by vapor condensing because of column heat losses. 3. The packing geometry inhibiting lateral distribution. 4. Void variations due to packing being improperly installed. 5. The column being out of vertical alignment. Cause 1 has been covered in the preceding subsection. Causes 4 and 5 can be handled through careful design and installation. Causes 2 and 3 bear additional discussion at this point. The effect of channeling on mass-transfer efficiency has been studied theoretically by Manning and Cannon [Ind. Eng. Chem., 49, 347 (1957)], Huber and Hiltbrunner [Chem. Eng. Sci., 21, 819 (1966)], and Meier and Huber (Proc. Intn. Symp. Distill., Brighton, England, 1970). Typical results are shown in Fig. 14-65. The effect of maldistribution can be severe, and efforts are being made to relate a maldistribution index to mass-transfer efficiency. Albright [Hydrocarbon Proc., 63, 173 (Sept. 1984)] developed a computer model to simulate effects of distributor design on bed distribution. Zuiderweg and coworkers [Hoek et al., Chem. Eng. Res. Des., 64, 431 (1986); Zuiderweg et al., I. Chem. E. Symp. Ser. No. 104, B247 (1987); Zuiderweg et al., Trans. I. Chem. E., 71, Part A, (1993)] have studied the effect of structured and random packed bed designs on maldistribution and, in turn, its effect of packed column efficiency. Liquid migration to the wall appears to be favored by small-column diameter-packing diameter ratios (for random packings) and can be corrected by the use of side wipers or redistributors. Inhibition of lateral dispersion can be caused by the geometry of certain types of structured packing, according to Huber and Hiltbrunner (op. cit.). With careful attention given to the five causes of maldistribution, it is possible to design commercial packed columns for heights of 8 to 9 m between redistributors. End Effects Analysis of the mass-transfer efficiency of a packed column should take into account that transfer which takes place outside the bed, i.e., at the ends of the packed sections. Inlet gas may very well contact exit liquid below the bottom support plate, and exit gas can contact liquid from some types of distributors (e.g., spray nozzles). The bottom of the column is the more likely place for transfer, and Silvey and Keller [Chem. Eng. Prog., 62(1), 68 (1966)] found that the

Ohio.)

For larger diameter columns, and for low liquid rates, the distributor must be almost exactly level (e.g., within 6 mm for a 3-m diameter) or all pour points will not function. On the other hand, the rises must be high enough to accommodate the backup caused by high liquid rates. The needed head can be estimated from the orifice equation, with a discharge coefficient of 0.5. In some cases the orifices discharge directly into tubes that extend to the packed bed (the “tubed drip-pan distributor”). The perforated pipe distributor comprises a central feed sump and pipes that branch out from the sump to provide the liquid discharge. The level in the sump varies with liquid total flow rate, and the size of the lateral pipes and their perforations must be determined carefully to ensure that the ends of the pipes are not starved for liquid. The orifices are typically 4 to 6 mm diameter, and can be subject to plugging if foreign matter is present. The pipes must be leveled carefully, especially for large diameter columns. Another type of distributor, not shown in Fig. 14-64, is the spray nozzle. It is usually not recommended for liquid distribution for two reasons. First, except for small columns, it is difficult to obtain a uniform spray pattern for the packing. The full-cone nozzle type is usually used, with the need for a bank of nozzles in larger columns. When there is more than one nozzle, the problem of overlap or underlap arises. A second reason for not using spray nozzles is their tendency toward entrainment by the gas, especially the smaller droplets in the spray size distribution. However, some mass transfer in the spray can be expected. Maldistribution Departure from uniform distribution of the phases in a packed column can be caused by: 1. The liquid distributor not dividing the liquid evenly over the column cross section.

Effect of liquid channeling on column efficiency for a system with a relative volatility of 1.07. Total number of theoretical plates N of 10, 20, 40, and 100 at top liquid composition X of 90 and 60 mole percent. [Manning and Cannon, Ind. Eng. Chem., 49, 347 (1957).]

FIG. 14-65

GAS-LIQUID CONTACTING SYSTEMS

(a)

14-51

(b)

Effective interfacial area based on data of Fellinger. (a) 1-in Raschig rings. (b) 1-in Berl saddles. To convert square feet per cubic foot to square meters per cubic meter, multiply by 3.28; to convert pounds per hoursquare foot to kilograms per second-square meter, multiply by 0.001356. [Shulman, Am. Inst. Chem. Eng. J., 1, 257 (1955).]

FIG. 14-67

Efficiency of FRI reboiler and space below bottom support plate. To convert pounds per square inch absolute to kilopascals, multiply by 6.8947; to convert (feet per second) (pounds per cubic foot)1/2, to (meters per second) (kilograms per cubic meter)1/2, multiply by 1.2199. [Silvey and Keller, Chem. Eng. Prog., 62(1), 68 (1966).] FIG. 14-66

reboiler plus the end effect could give up to two or more theoretical plates (Fig. 14-66). Interfacial Area The effective area of contact between gas and liquid is that area which participates in the gas-liquid mass-exchange process. This area may be less than the actual interfacial area because of stagnant pools where liquid reaches saturation and no longer participates in the transfer process. Effective area should not be confused with “wetted area.” While film flow of liquid across the packing surface is a contributor, effective area includes also contributions from rivulets, drippings, and gas bubbles. Because of this complex physical picture, effective interfacial area is difficult to measure directly. Weisman and Bonilla [Ind. Eng. Chem., 42, 1099 (1950)] determined effective area ai of 25-mm (1-in) Raschig rings indirectly through the relationship ai = (kgai)/kg. The kg data were obtained via evaporation of water from presaturated rings by Taecker and Hougen [Chem. Eng. Prog., 45, 188 (1949)] and the vaporization kgai data from McAdams et al. [Chem. Eng. Prog., 45, 241 (1949)] for the air-water irrigated system. The authors proposed that ai /at = 0.54 G0.31L0.07

area differs between vaporization and absorption, as shown in Fig. 14-68. For distillation, they found effective areas to be different for systems with different surface tensions (Fig. 14-69) but, upon making the surface-tension correction, concluded that areas for distillation are approximately the same as those for absorption. For structured packings, Rocha et al. [Ind. Eng. Chem. Res., 35, 1660 (1996)] adapted the model of Shi and Mersmann [Ger. Chem. Eng., 8, 87 (1985)] to predict effective wetted area: 29.12(WeLFrL)0.15S0.359 ai (14-166) ᎏ = FSE ᎏᎏᎏᎏ 0.2 0.6 at ReL ε (1 − 0.93 cos γ)(sin θ)0.3 where WeL = (UL2 ρLS)/σgc) = Weber number for liquid FrL = UL2 /Sg = Froude number for liquid S = side dimension of corrugation (Fig. 14-51), m ReL = (ULSρL)/µL ε = void fraction of packing γ = contact angle (degrees) θ = angle of corrugation channel with horizontal (degrees) FSE = factor for surface enhancement

(14-165)

for the range of liquid rates from 4 to 17 kg/(s⋅m2) [2950 to 12,537 lb/(h⋅ft2)], and where at = external surface area of the packing (Table 14-7). In Eq. (14-165), both gas and liquid rates, G and L, are in kg/ (s⋅m2). Areas are in consistent units. A greater dependency on liquid rate was reported by Shulman et al. [Am. Inst. Chem. Eng. J., 1, 253 (1955)], who obtained the effective area via vaporization of packing constructed of naphthalene and from calculated ammonia absorption data of Fellinger (Sc.D. thesis, Massachusetts Institute of Technology, 1941), taking account of liquid-phase resistance. On the basis of gross system conditions, the values obtained are indicated in Fig. 14-67 for 25-mm (1-in) Raschig rings and Berl saddles. These packing types in the 12-, 38-, and 50-mm (0.5-, 1.5-, and 2.0-in) sizes were also studied. Yoshida and Koyanagi [Am. Inst. Chem. Eng. J., 8, 309 (1962)] used the Weisman-Bonilla approach, accepting the Taecker-Hougen kg data and making their own kgai measurements under vaporization, absorption, and distillation conditions. They found that the effective

Effective areas for 25-mm Raschig rings. To convert pounds per hour-square foot to kilograms per second-square meter, multiply by 0.001356. [Yoshida and Koyanagi, Am. Inst. Chem. Eng. J., 8, 309 (1962).]

FIG. 14-68

14-52

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN 2. Convert theoretical stages to transfer units using Eq. (14-170). 3. Determine the height of an overall transfer unit HOG using methods given in Sec. 5, Table 5-28. For a multicomponent mixture, the key components are often used in determining HOG. 4. Calculate the required packed bed height by Eq. (14-169). Behavior of Various Systems and Packings For orientation purposes, it is helpful to have representative data available for packing performance under a variety of conditions. In the preceding edition of this handbook, extensive data on absorption/stripping systems were given. Emphasis was given to the following systems:

Effective or wetted area, ft2/ft3

100 80 60

Organic liquids Water

40 ted

Wet

area

ion

illat

20

ist or d

n atio

af

z are ori for tive ap c e f rea ea rv r a a o Ef f e ted 10 rea tiv Wet ea ec v f i f t 8 E ec Eff 6 1 2 4 6 8 10 20

tion orp s ab

Ammonia-air-water Air-water Sulfur dioxide-air-water Carbon dioxide-air-water

40

60

Liquid rate, ft3/ft2 hr Comparison of effective and wetted areas, 25-mm Raschig rings. To convert cubic feet per square foot-hour to cubic meters per square metersecond, multiply by 8.47 × 10−5; to convert square feet per cubic foot to square meters per cubic meter, multiply by 3.28. [Yoshida and Koyanagi, Am. Inst. Chem. Eng. J., 8, 309 (1962).] FIG. 14-69

Values of FSE range from 0.009 for a smooth metal surface with small perforations, to 0.029 for a pierced metal surface. The contact angle is a key parameter for wettability and was related to the surface tension by Shi and Mersmann. This accounts for the differences in wettability found by Yoshida and Koyanagi (Fig. 14-69). In general, aqueous systems will wet ceramic surfaces better than metal surfaces, unless the latter are oxidized. Mass Transfer Relationships for calculating rates of mass transfer between gas and liquid in packed absorbers, strippers, and distillation columns may be found in Sec. 5 and are summarized in Table 5-28. The two-resistance approach is used, with rates expressed as transfer units: HOG = HG + λHL

(14-167)

where HOG = height of an overall transfer unit, gas concentration basis, m HG = height of a gas-phase transfer unit, m HL = height of a liquid-phase transfer unit, m λ = m/(Lm/Gm) = slope of equilibrium line/slope of operating line The various models for predicting values of HG and HL are given in Sec. 5. The important parameters in the models include gas rate, liquid rate, gas and liquid properties (density, viscosity, surface tension, diffusivity), packing type and size, and overall bed dimensions. In design practice, a less rigorous parameter, HETP, is used as an index of packing efficiency. The HETP is the height of packed bed required to achieve a theoretical stage. The terms HOG and HETP may be related under certain conditions: ln λ HETP = HOG ᎏ (14-168) (λ − 1)



and since

The reader may refer to the data in the preceding edition. For the current work, emphasis will be given to one absorption system, carbon dioxide-air-caustic, and to several distillation systems. Carbon Dioxide-Air-Caustic System The vendors of packings have adopted this system as a “standard” for comparing the performance of different packing types and sizes. For tests, air containing 1.0 mol % CO2 is passed countercurrently to a circulating stream of sodium hydroxide solution. The initial concentration of NaOH in water is 1.0 N (4.0 wt %), and as the circulating NaOH is converted to sodium carbonate it is necessary to make a mass-transfer correction because of reduced mass-transfer rate in the liquid phase. The procedure has been described by Eckert et al. [Ind. Eng. Chem., 59(2), 41 (1967); Chem. Eng. Progr., 54(1), 790 (1958)]. An overall coefficient is measured using gas-phase (CO2) concentrations: moles CO2 absorbed KOGae = ᎏᎏᎏᎏᎏᎏ time-bed volume-partial pressure CO2 driving force (14-171) The coefficients are usually corrected to a hydroxide conversion of 25 percent at 24°C. For other conversions, Fig. 14-15 may be used. Reported values of KOGa for representative random packings are given in Table 14-8. The effect of liquid rate on the coefficient is shown in Figs. 14-70 and 14-71. While the carbon dioxide/caustic test method has become accepted, one should use the results with caution. The chemical reaction masks the effect of physical absorption, and the relative values in the table may not hold for other cases, especially distillation applications where much of the resistance to mass transfer is in the gas phase. Background on this combination of physical and chemical absorption may be found earlier in the present section, under “Absorption with Chemical Reaction.” Distillation Applications Packings are now routinely considered for distillation columns with diameters up to 10 m or more. The pressure drop advantages of the modern, through-flow random packTABLE 14-8

Overall Coefficients for Representative Packings CO2-air-caustic system



Zp = (HOG)(NOG) = (HETP)(Nt) NOG = Nt [ln λ /(λ − 1)]

Liquid and gas phases contributing; chemical reaction contributing Gas phase controlling Liquid and gas-phase controlling Liquid phase controlling

(14-169)

Ceramic raschig rings

(14-170)

Ceramic Intalox saddles

Equations (14-168) and (14-170) have been developed for binary mixture separations and hold for cases where the operating line and equilibrium line are straight. Thus, when there is curvature, the equations should be used for sections of the column where linearity can be assumed. When the equilibrium line and operating line have the same slope, HETP = HOG and NOG = Nt (theoretical stages). In practice, the following procedure is normally used: 1. Determine theoretical stages for the separation: binary or multicomponent.

Metal pall rings Metal Intalox saddles (IMTP®)

Nominal size, mm

Overall coefficient KOGa, kg⋅moles/(hr⋅m3⋅atm)

25 50 25 50 25 50 25 50

37.0 26.1 45.1 30.1 49.6 34.9 54.8 39.1

NOTE: Basis for reported values: CO2 concentration in inlet gas, 1.0 vol %; 1N NaOH solution in water, 25 percent NaOH conversion; temperature = 24°C; atmospheric pressure: gas rate = 1.264 kg/(s⋅m2); liquid rate = 6.78 kg/(s⋅m2). SOURCE: Strigle, R. L., Random Packings and Packed Towers, Gulf Publ. Co., Houston, 1987.

GAS-LIQUID CONTACTING SYSTEMS 10

10

5

5

KGO, lb-moles/hr, ft3, atm

2

3

KGO, lb-moles/hr, ft3, atm

gs rin all p 1′ gs g rin schi 1′ ra

3

1 .8 .6

14-53

s ll ring 2′ pa s g rin hig asc r ′ 2

2

1 .8 .6 .4

.4

.3

.3

.2

.2 0.1 100

0.1 100

500

1000

2000

500

1000

2000

5000

Liquid rate, lbm/(hr • ft 2 ) Overall mass transfer coefficients for carbon dioxide absorbed from air by 1N caustic solution. Packing = 1-inch metal raschig and pall rings. Air rate = 0.61 kg/s-m2 (450 lb/hr⋅ft2). To convert lb/hr⋅ft2 to kg/s⋅m2, multiply by 0.00136; to convert lb⋅moles/h⋅ft3 atm to kg⋅moles/s⋅m3 atm, multiply by 0.00445. [Eckert et al., Chem. Eng. Progr., 54(1), 70 (1958).] FIG. 14-70

ings and the ordered, structured packings have made vacuum column applications of particular interest. Test data for total reflux distillations have become available from several sources, such as Fractionation Research, Inc. and The University of Texas at Austin Separations Research Program. Examples of the data from these and other

FIG. 14-71 Overall mass transfer coefficients for carbon dioxide absorbed from air by 1N caustic solution. Packing = 2-inch metal raschig and pall rings. Air rate = 0.61 kg/s⋅m2 (450 lb/hr⋅ft2). To convert from lb/hr⋅ft2 to kg/s⋅m2, multiply by 0.00136. To convert from lb-moles/hr⋅ft3 atm to kg-moles/s⋅m3 atm, multiply by 0.0045. [Eckert et al., Chem. Eng. Progr., 54(1), 70 (1958).]

sources are shown in Figs. 14-72–14-75. The comparative efficiency of different sizes of random packing is shown in Fig. 14-72. As would be expected, the smaller packings have higher mass-transfer efficiency (lower values of HETP). However, their capacity is limited. The equivalent pressure drop data are shown in Fig. 14-73, where the smaller packings are shown to have higher pressure drop.

Fs, m/s (kg/m3)0.5 0.0 100

0.5

1.0

1.5

2.0

2.5

3.0 2.4

90 80

2.0

1.6

3.5 inch (88.9 mm)

50

1.2

40

2 inch (50.8 mm)

HETP, m

HETP, in

70 60

0.8

30 5/8 inch (15.9 mm) 20

0.4 10 0 0.0

1 inch (25.4 mm) 0.4

0.8

5000

Liquid rate, lbm/(hr • ft2 )

1.2

1.6

2.0

2.4

0.0 2.8

Fs, ft/s (lb/ft3)0.5 FIG. 14-72 HETP values for four sizes of metal pall rings, vacuum operation. Cyclohexane/n-heptane system, total reflux, 35 kPa (5.0 psia). Column diameter = 1.2 m (4.0 ft). Bed height = 3.7 m (12 ft). Distributor = tubed drip pan, 100 streams/m2. [Adapted from Shariat and Kunesh, Ind. Eng. Chem. Res., 34, 1273 (1995).] Reproduced with permission. Copyright © 1995 American Chemical Society.

14-54

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN Fs, m/s (kg/m3)0.5 0.5

1.0

1

m)

5/8

9m 15.

h(

inc

0.1

1.5

2.0

2.5

3.0

100

) ) mm mm 5.4 50.8 2 ( ( h c ch 2 in 1 in m) .9 m h (88 c in 3.5

10

1

0.01

0.001 0.0

0.4

0.8

1.2

1.6

2.0

Pressure drop, mm H2O/m

Pressure drop, inches H2O/ft

0.0 10

0.1 2.8

2.4

Fs, ft/s (lb/ft3)0.5 FIG. 14-73 Pressure-drop data for several sizes of pall rings; test conditions same as described for Fig. 14-72. [Shariat and Kunesh, Ind. Eng. Chem. Res., 34, 1273 (1995).] Reproduced with permission. Copyright © 1995 American Chemical Society.

Efficiency data for a representative structured packing at two column diameters are shown in Fig. 14-74. The Max-Pak® packing has a surface area of 246 m2/m3 (75 ft2/ft3). The same test mixture (cyclohexane/n-heptane) and operating pressure was used for both tests. It would appear that column diameter does not have an influence in this range of values (0.43 to 1.2 m). Efficiency and pressure drop data for Sulzer BX metal gauze structured packing and for three test mixtures are shown in Fig. 14-75. For the ethyl benzene/styrene test mixture, the effect of operating pressure is shown. The high viscosity mixture, propylene glycol/ethylene

glycol, has a significantly lower efficiency than the other mixtures but does not appear to have a lower capacity. LIQUID-DISPERSED CONTACTORS Introduction There are two types of gas-liquid contactors where the liquid is deliberately dispersed. In the most common, a spray nozzle is used to generate droplets. A second type is the pipeline contactor, where the entrainment generated by flowing gas generates the droplets.

40 1.2-m column 35

0.43-m column

HETP, in

30 25 20 15 10

0

0.5

1

1.5

2

2.5

3

Fs, ft/s (lb/ft3)0.5 FIG. 14-74 HETP values for Max-Pak structured packing, 35 kPa (5 psia), two column diameters. Cyclohexane/n-heptane system, total reflux. For 0.43 m (1.4 ft) column: perforated pipe distributor, 400 streams/m2, 3.05 m (10 ft) bed height. For 1.2 m (4.0 ft) column: tubed drip pan distributor, 100 streams/m2, 3.7 m (12 ft) bed height. Smaller column data, University of Texas/Austin; Larger column data, Fractionation Research, Inc. To convert (ft/s)(lb/ft3)0.5 to (m/s)(kg/m3)0.5, multiply by 1.2199. (Courtesy Jaeger Products, Inc., Houston, Texas.)

GAS-LIQUID CONTACTING SYSTEMS

14-55

Efficiency, stages/m = 1/HETP, m–1

12 10 8 6 4 2 0

Pressure drop, mm H2O/m

140 120 100 80

: 1,2-propylene glycol–ethylene glycol, 10 mm Hg : Ethyl benzene–styrene, 50 mm Hg : Ethyl benzene–styrene, 100 mm Hg : Methanol–ethanol, 760 mm Hg L/V = 1

60 40 20 0 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2

Uv ρv, Vapor capacity factor, m–1/2 . s–1 . kg1/2 FIG. 14-75 Efficiency and pressure drop data for Sulzer BX structured packing. Test mixtures and operating pressures shown on graph. Total reflux, 0.8 m (2.6 ft) column diameter, ca. 2 m (6.5 ft bed height, perforated pipe distributor, 400 streams/m2). To convert (m/s)(kg/m3)0.5 to (ft/s)(lb/ft3)0.5, multiply by 0.8197. [Billet, Conrad, and Grubb, I. Chem. E. Symp. Serv. 5(32), 111 (1969).]

Because of the minimal internals for solids to grow on, both types of contactors are common in fouling services. The spray nozzle based devices are also inherently low pressure drop. The disadvantage is that volumetric efficiency is usually much less than conventional trays or packed contactors. Applications are usually limited to cases when only a few transfer units or a single equilibrium stage is required. Since many of these applications tend to be in heattransfer service, the following discussion will be in terms of thermal properties and thermal measures of performance. Heat-Transfer Applications Heat-transfer analogs of common mass-transfer terms are: Tg,out − Tg,in No. of gas-phase transfer units = ᎏᎏ = Ng (14-172) (Tg − Ti)mean Tᐉ,out − Tᐉ,in No. of liquid-phase transfer units = ᎏᎏ = Nᐉ (Tᐉ − Ti)mean

(14-173)

Gcp Gas-phase volumetric transfer rate = ᎏ = hga Hg

(14-174)

k Diffusivity = ᎏ = αT ρcp where Tg = gas temperature Tᐉ = liquid temperature Ti = interface temperature Hg = height of a gas-phase transfer unit G = weight gas flow/area cp = specific heat ρ = density k = thermal conductivity αT = thermal diffusivity

(14-175)

Note that the relative performance of a device can be converted from a mass-transfer basis to a heat-transfer basis by introducing these analogies into the rate equations.

The conversion is simplest when Kga and Kᐉa are defined in terms of mole-fraction driving force: hga hᐉa cp αT 0.5 (14-176) ᎏ or ᎏ = ᎏ ᎏ Kga Kᐉa M D where hᐉ a, hga = volumetric heat-transfer rate Kᐉ a, Kga = volumetric mass-transfer rate cp = heat capacity M = molecular weight αT /D = ratio of thermal diffusivity to molecular diffusivity, dimensionless




For gases, αT /D is usually close to 1, since the same basic transfer mechanism exists. For liquids, αT /D is invariably much greater than 1. A simplified model yields the relation αT (14-177) ᎏ = 1.9 × 107(µ/T) D where µ is in Pa⋅s and T is in K. For a 1-cP (10−3⋅Pa⋅s) liquid at 310 K, this gives a ratio of 61. This means that a closer approach to equilibrium can be obtained for heat transfer applications that are liquidlimited (such as in condensation of a pure vapor), than can be obtained for mass transfer liquid-limited applications. Theoretical Transfer Model Transfer from single droplets is theoretically well defined for the gas side. For a droplet moving counter to a gas, interfacial area is (in consistent units) Gᐉ 6 a = ᎏᎏ ᎏ (14-178) ρᐉ(Uᐉ − Ug) Dd




where




a = interfacial area/volume Gᐉ = liquid-flow rate (weight/cross-sectional area) ρᐉ = liquid density Uᐉ = liquid velocity relative to the gas, often approximately the terminal velocity of droplets (see Sec. 6 for estimation) Ug = superficial gas velocity Dd = droplet diameter

14-56

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

The transfer coefficient is defined by (in consistent units) hg = (k/Dd) (2 + 0.6N N 0.5 Re

0.33 Pr

)

(14-179)

where hg = transfer coefficient k = thermal conductivity of gas NRe = Reynolds number = DdUt ρg /µg NPr = Prandtl number = (cpµ/k)g The volumetric coefficient hg a from the combination of Eqs. (14-178) and (14-179) is useful in defining the effect of variable changes but is limited in value because of its dependence on Dd. The product of area and coefficient obtained from a given mass of liquid is proportional to (1/Dd)2 for small diameters. The prime problem is that droplet-size estimating procedures are often no better than ⫾50 percent. A secondary problem is that there is no Dd that truly characterizes either the motion or transfer process for the whole spectrum of particle sizes present. See Eqs. (14-193) and (14-194). The corresponding theory for transfer in the liquid phase is even less certain. If one had static drops, the transfer would be 4(k/ρcp)π 2t Nᐉ = 0.5 + ᎏᎏ (dimensionless) (14-180) Dd2 where Nᐉ = liquid-phase transfer units [Eq. (14-173)] k = liquid thermal conductivity ρ = liquid density cp = liquid specific heat Dd = droplet diameter t = time of contact However, the static-drop assumption is usually extremely conservative. For example, the high interfacial velocity in the spray from nozzles yields a high degree of internal mixing and much higher transfer. Countercurrent, Cocurrent, or Backmixed Spray-chamber contactors rarely approach countercurrent performance. Backmixing is hard to prevent and often limiting. Backmixing is so severe that many designers simply limit spray-chamber performance to a single equilibrium stage regardless of height. For a direct-contact heattransfer device, this means that the temperatures of the exiting gas and liquid would be equal. The main cause of the high degree of backmixing is that there is no stabilizing pressure drop caused by packing of plates. Consequently the chief resistance to gas flow is the rain of drops. Anything less than perfect liquid distribution will induce a dodging action in the opposed vapor flow. The result is the development of large eddies and bypass streams. Other sources of deviation from countercurrent flow are large drops falling faster than small ones and liquid striking the walls About all that can be done is to take special care to obtain a uniform spray pattern with minimum collection at the walls. Empirical Approach Sprays Large units generally yield approximately one equilibrium stage even when in nominal counter current flow. See Masters, [Spray Drying Handbook, 5th ed., Wiley, New York, (1991)]. For smaller towers, less than one equilibrium stage is typical. Transfer units can be estimated from the data of Pigford and Pyle [Ind. Eng. Chem. 43, 1949 (1951)]. These data show the height of a gas-limited transfer unit to be 1 to 3 m (3 to 12 ft). Pigford and Pyle show much shorter heights for a liquid-limited transfer unit, in the range of 0.5 to 1 m (1.5 to 3 ft). The high liquid transfer rates result from the high energy dissipation in the liquid and the enhanced transfer at the time of droplet formation. The same behavior is indicated by Simpson and Lynn [Am. Inst. Chem. Eng. J., 23(5), 666 (1977)], who show a 75 to 95 percent approach to equilibrium stripping in a 1.4-m-tall spray contact in a liquid-limited system. Pipeline Flow (Quenching) For the case of pipeline quenching, the flows are cocurrent. How closely the gas temperature approaches the liquid depends on e−Ng, where Ng varies with 1/(droplet diameter)2. Since the predicted droplet diameter at high velocity pipeline flow varies with (1/velocity)1.2, as shown by Eq. (14-201), the volumetric performance is strongly dependent on velocity: Ng = constant⋅(velocity)2.4

(14-181)

An empirical point in support of the strong velocity dependence is the rule of thumb for quenching, requiring a high relative velocity [>60 m/s (200 ft/s)]. Venturi scrubbers are similar in that they need high velocity to achieve small droplets. They are primarily employed for mist and dust collection and are discussed further in the mist-collection portion of this section. WETTED-WALL COLUMNS Wetted-wall or falling-film columns have found application in masstransfer problems when high-heat-transfer-rate requirements are concomitant with the absorption process. Large areas of open surface are available for heat transfer for a given rate of mass transfer in this type of equipment because of the low mass-transfer rate inherent in wetted-wall equipment. In addition, this type of equipment lends itself to annular-type cooling devices. Gilliland and Sherwood [Ind. Eng. Chem., 26, 516 (1934)] found that, for vaporization of pure liquids in air streams for streamline flow, kgD PBM 0.44 (14-182) ᎏ ᎏ = 0.023N0.83 Re NSc Dg π where Dg = diffusion coefficient D = inside diameter of tube kg = mass-transfer coefficient, gas phase Note that the group on the left side of Eq. (14-182) is dimensionless. When turbulence promoters are used at the inlet-gas section, an improvement in gas mass-transfer coefficient for absorption of water vapor by sulfuric acid was observed by Greenewalt [Ind. Eng. Chem., 18, 1291 (1926)]. A falling off of the rate of mass transfer below that indicated in Eq. (14-182) was observed by Cogan and Cogan (thesis, Massachusetts Institute of Technology, 1932) when a calming zone preceded the gas inlet in ammonia absorption (Fig. 14-76). In work with the hydrogen chloride-air-water system, Dobratz, Moore, Barnard, and Meyer [Chem. Eng. Prog., 49, 611 (1953)] using a cocurrent-flow system found that kg ∝ G1.8 (Fig. 14-77) instead of the 0.8 power as indicated by the Gilliland equation. Heat-transfer coefficients were also determined in this study. The radical increase in heattransfer rate in the range of G = 30 kg/(s⋅m2) [20,000 lb/(h⋅ft2)] was similar to that observed by Tepe and Mueller [Chem. Eng. Prog., 43, 267 (1947)] in condensation inside tubes. Gaylord and Miranda [Chem. Eng. Prog., 53, 139M (1957)] using a multitube cocurrent-flow falling-film hydrochloric acid absorber for hydrogen chloride absorption found 1.66(10−5) DG Kg = ᎏᎏ (14-183) ᎏ Mm1.75 µ where Kg = overall mass-transfer coefficient, (kg⋅mol)/(s⋅m2⋅atm) Mm = mean molecular weight of gas stream at inlet to tube D = diameter of tube, m G = mass velocity of gas at inlet to tube, kg/(s⋅m2) µ = viscosity of gas, Pa⋅s




Note that the group DG/µ is dimensionless. This relationship also satisfied the data obtained for this system, with a single-tube falling-film unit, by Coull, Bishop, and Gaylor [Chem. Eng. Prog., 45, 506 (1949)]. The rate of mass transfer in the liquid phase in wetted-wall columns is highly dependent on surface conditions. When laminar-flow conditions prevail without the presence of wave formation, the laminarpenetration theory prevails. When, however, ripples form at the surface, and they may occur at a Reynolds number exceeding 4, a significant rate of surface regeneration develops, resulting in an increase in mass-transfer rate. If no wave formations are present, analysis of behavior of the liquidfilm mass transfer as developed by Hatta and Katori [ J. Soc. Chem. Ind., 37, 280B (1934)] indicates that DᐉΓ kᐉ = 0.422 ᎏ (14-184) ρBF2 2 1/3 where BF = (3uΓ/ρ g) Dᐉ = liquid-phase diffusion coefficient, m2/s



GAS-LIQUID CONTACTING SYSTEMS

14-57

Mass-transfer rates in wetted-wall columns having turbulence promoters. To convert pound-moles per hour-square foot-atmosphere to kilogram-moles per second-square meter-atmosphere, multiply by 0.00136; to convert pounds per hour-square foot to kilograms per second-square meter, multiply by 0.00136; and to convert inches to millimeters, multiply by 25.4. (Data of Greenewalt and Cogan and Cogan, Sherwood, and Pigford, Absorption and Extraction, 2d ed., McGraw-Hill, New York, 1952.)

FIG. 14-76

ρ = liquid density, kg/m3 Z = length of surface, m kᐉ = liquid-film-transfer coefficient, (kg⋅mol)/[(s⋅m2) (kg⋅mol)/m3] Γ = liquid-flow rate, kg/(s⋅m) based on wetted perimeter µ = viscosity of liquid, Pa⋅s g = gravity acceleration, 9.81 m/s2

When Z is large or Γ/ρBF is so small that liquid penetration is complete, kᐉ = 11.800 Dᐉ/BF (14-185) and Hᐉ = 0.95 ΓBF/Dᐉ (14-186) A comparison of experimental data for carbon dioxide absorption obtained by Hatta and Katori (op. cit.), Grimley [Trans. Inst. Chem. Eng., 23, 228 (1945)], and Vyazov [Zh. Tekh. Fiz. (U.S.S.R.), 10, 1519 (1940)] and for absorption of oxygen and hydrogen by Hodgson (S. M. thesis, Massachusetts Institute of Technology, 1949), Henley (B.S. thesis, University of Delaware, 1949), Miller (B.S. thesis, University of Delaware, 1949), and Richards (B.S. thesis, University of Delaware, 1950) was made by Sherwood and Pigford (Absorption and Extraction, McGraw-Hill, New York, 1952) and is indicated in Fig. 14-78. In general, the observed mass-transfer rates are greater than those predicted by theory and may be related to the development of surface rippling, a phenomenon which increases in intensity with increasing liquid path. Vivian and Peaceman [Am. Inst. Chem. Eng. J., 2, 437 (1956)] investigated the characteristics of the CO2-H2O and Cl2-HCl, H2O system in a wetted-wall column and found that gas rate had no effect on the liquid-phase coefficient at Reynolds numbers below 2200. Beyond this rate, the effect of the resulting rippling was to increase significantly the liquid-phase transfer rate. The authors proposed a behavior relationship based on a dimensional analysis but suggested caution in its application concomitant with the use of this type of relationship. Cognizance was taken by the authors of the effects of column length, one to induce rippling and increase of rate of transfer, one to increase time of exposure which via the penetration theory decreases the average rate of mass transfer in the liquid phase. The equation is kᐉh µᐉ 1/2 ρᐉ2 gh3 1/6 4Γ 0.4 ᎏ ᎏ = 0.433 ᎏ ᎏ (14-187) µ2ᐉ Dᐉ ρᐉDᐉ µᐉ




Mass-transfer coefficients versus average gas velocity—HCl absorption, wetted-wall column. To convert pound-moles per hour-square footatmosphere to kilogram-moles per second-square meter-atmosphere, multiply by 0.00136; to convert pounds per hour-square foot to kilograms per secondsquare meter, multiply by 0.00136; to convert feet to meters, multiply by 0.305; and to convert inches to millimeters, multiply by 25.4. [Dobratz et al., Chem. Eng. Prog., 49, 611 (1953).] FIG. 14-77







where Dᐉ = diffusion coefficient of solute in liquid g = gravity-acceleration constant h = length of wetted wall kᐉ = mass-transfer coefficient, liquid phase Γ = mass rate of flow of liquid µᐉ = viscosity of liquid ρᐉ = density of liquid The equation is dimensionless.

14-58

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

Liquid-film resistance in absorption of gases in wetted-wall columns. Theoretical lines are calculated for oxygen absorption in water at 55°F. To convert feet to meters, multiply by 0.3048; °C = 5⁄ 9 (°F − 32). (Sherwood and Pigford, Absorption and Extraction, 2d ed., McGraw-Hill, New York, 1952.) FIG. 14-78

The effect of chemical reaction in reducing the effect of variation of the liquid rate on the rate of absorption in the laminar-flow regime was illustrated by the evaluation of the rate of absorption of chlorine in ferrous chloride solutions in a wetted-wall column by Gilliland, Baddour, and White [Am. Inst. Chem. Eng. J., 4, 323 (1958)]. Flooding in Wetted-Wall Columns When gas and liquid are in counterflow in wetted-wall columns, flooding can occur at high gas rates. Diehl and Koppany [Chem. Eng. Prog., 65, Symp. Ser. 42, 77 (1969)] correlated flooding data from a number of sources, including their own work, and developed the following expression: σ 1/2 Uf = F1F2 ᎏ (14-188) ρg where Uf = flooding gas velocity, m/s F1 = 1.22 when 3.2 di /σ > 1.0 F1 = 1.22 (3.2 di/σ)0.4 when 3.2 di /σ < 1.0 F2 = (G/L)0.25 G/L = gas-liquid mass ratio di = inside diameter of column, mm σ = surface tension, mN/m (dyn/cm) ρg = gas density, kg/m3




The data covered column sizes up to 50-mm (2-in) diameter; the correlation should be used with caution for larger columns. GAS-LIQUID-COLUMN ECONOMICS Estimation of column costs for preliminary process evaluations requires consideration not only of the basic type of internals but also

of their effect on overall system cost. For a distillation system, for example, the overall system can include the vessel (column), attendant structures, supports, and foundations; auxiliaries such as reboiler, condenser, feed heater, and control instruments; and connecting piping. The choice of internals influences all these costs, but other factors influence them as well. A complete optimization of the system requires a full-process simulation model that can cover all pertinent variables influencing economics. Cost of Internals Installed costs of plates (trays) may be estimated from Fig. 14-79, with corrections for plate material taken from Table 14-9. For two-pass plates the cost is 15 to 20 percent higher. Approximate costs of dumped (random) packing materials may be obtrained from Table 14-10, but it should be recognized that, because of competition, there can be significant variations in these costs from vendor to vendor. Also, packings sold in very large quantities carry discounts. In 1995, costs of structured packings, made from sheet metal, averaged $90–$110 per cubic foot, but the need for special distributors and redistributors can double the cost of structured-packings on a volumetric basis. Note that for Fig. 14-79 and Table 14-9, the effective cost date is January 1990, with the Marshall and Swift cost index being taken as 904. As indicated above, packed column internals include liquid distributors, packing support plates, redistributors (as needed), and holddown plates (to prevent movement of packing under flow conditions). Costs of these internals for columns with random packing are given in Fig. 14-80, based on early 1976 prices, and a Marshall and Swift cost index of 460. Cost of Column The cost of the vessel, including heads, skirt, nozzles, and ladderways, is usually estimated on the basis of weight.

GAS-LIQUID CONTACTING SYSTEMS

14-59

Quantity factors apply to all types of trays

Installed cost, dollars/tray

1 2 3 4 5 6 7

3.00 2.80 2.65 2.50 2.30 2.15 2.00

8 9 10 11 12 13 14

1.80 1.65 1.50 1.45 1.40 1.35 1.30

15 16 17 18 19 20 29 30 39 40+

104

1.25 1.20 1.15 1.10 1.05 1.00 0.98 0.97

Bubble-cap trays (stainless steel)

Sieve trays (carbon steel)

103 Valve trays (carbon steel)

Valve trays (stainless steel)

Sieve trays or Bubble-cap trays (stainless steel) (carbon steel)

Stamped turbogrid trays (stainless steel) 102

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Tower diameter, ft Cost of trays in plate towers. Price includes tray deck, bubble caps, risers, downcomers, and structural-steel parts. The stainless steel designated is type 410 (Peters and Timmerhaus, Plant Design and Economics for Chemical Engineers, 4th ed., McGraw-Hill, New York, 1991). FIG. 14-79

Figure 14-81 provides early 1990 cost data for the shell and heads, and Fig. 14-82 provides 1990 cost data for connections. For very approximate estimates of complete columns, including internals, Fig. 14-83 may be used. As for Figs. 14-81 and 14-82, the cost index is 904. Plates versus Packings Bases for using packings instead of plates have been given earlier in this section. In many cases, either type of device may be used. For vacuum fractionations, the low pressure drop characteristics of throughflow random packings and structured packings tend to give them a distinct advantage over plates. On the other hand, structured packings and their many requirements, such as distributors, are more expensive than plates. For atmospheric

and pressure columns, the pressure-drop characteristics of packings are less important, and a decision may be made on installed cost and reliability of design methods. Kister et al. [Chem. Eng. Progr., 90(2), 23 (1994)] reported a study of the relative capacity and efficiency of plates, modern random packings, and structured packings. They found that, for each device optimally designed for the design requireTABLE 14-10 Costs of Tower Packings, Uninstalled, January, 1990 Prices in dollars per ft3, 100 ft3 orders, f.o.b. manufacturing plant Size, in, $/ft3

TABLE 14-9 Relative Fabricated Cost for Metals Used in Tray-tower Construction*

Materials of construction

Relative cost per ft2 of tray area (based on carbon steel = 1)

Sheet-metal trays Steel 4–6% chrome—a moly alloy steel 11–13% chrome type 410 alloy steel Red brass Stainless steel type 304 Stainless steel type 347 Monel Stainless steel type 316 Inconel Cast-iron trays

1 2.1 2.6 3 4.2 5.1 7.0 5.5 8.2 2.8

*Peters and Timmerhaus, Plant Design and Economics for Chemical Engineers, 4th ed., McGraw-Hill, New York, 1991. To convert cost per square foot to cost per square meter, multiply by 10.76.

Raschig rings Chemical porcelain Carbon steel Stainless steel Carbon Intalox saddles Chemical stoneware Chemical porcelain Polypropylene Berl saddles Chemical stoneware Chemical porcelain Pall rings Carbon steel Stainless steel Polypropylene

1

1a

2

3

12.8 36.5 155 52

10.3 23.9 117 46.2

9.4 20.5 87.8 33.9

7.8 16.8 — 31.0

17.6 18.8 21.2

13.0 14.1 —

11.8 12.9 13.1

10.7 11.8 7.0

27.0 33.5

21.0 21.5

— 15.6

— —

29.3 131 21.2

19.9 99.0 14.4

18.2 86.2 13.1

— —

Peters and Timmerhaus, Plant Design and Economics for Chemical Engineers, 4th ed., McGraw-Hill, New York, 1991. To convert cubic feet to cubic meters, multiply by 0.0283; to convert inches to millimeters, multiply by 25.4; and to convert dollars per cubic foot to dollars per cubic meter, multiply by 35.3.

14-60

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

(a)

(b)

Cost of internal devices for columns containing dumped packings. (a) Holddown plates and support plates. (b) Redistributors. (c) Liquid distributors. [Pikulik and Diaz, Chem. Eng., 84(21), 106 (Oct. 10, 1977).]

FIG. 14-80

FIG. 14-81 Fabricated costs and installation time of towers. Costs are for shell with two heads and skirt, but without trays, packing, or connections. (Peters and Timmerhaus, Plant Design and Economics for Chemical Engineers, 4th ed., McGraw-Hill, New York, 1991.)

FIG. 14-82 Approximate installed cost of steel-tower connections. Values apply to 2070-kPa connections. Multiply costs by 0.9 for 1035-kPa (150-lb) connections and by 1.2 for 4140-kPa (600-lb) connections. To convert inches to millimeters, multiply by 25.4; to convert dollars per inch to dollars per centimeter, multiply by 0.394. (Peters and Timmerhaus, Plant Design and Economics for Chemical Engineers, 4th ed., New York, McGraw-Hill, 1991.)

GAS-LIQUID CONTACTING SYSTEMS

FIG. 14-83 Cost of towers including installation and auxiliaries. To convert inches to millimeters, multiply by 25.4; to convert feet to meters, multiply by 0.305; and to convert dollars per foot to dollars per meter, multiply by 3.28. (Peters and Timmerhaus, Plant Design and Economics for Chemical Engineers, 4th ed., McGraw-Hill, New York, 1991.)

ments, a rough guide could be developed on the basis of flow parameter L/G (ρG /ρL)0.5 (ordinate scale in Figs. 14-25 and 14-26) and the following tentative conclusions could be drawn: Flow Parameter 0.02–0.1 1. Plates and random packings have much the same efficiency and capacity. 2. Structured packing efficiency is about 1.5 times that of plates or random packing. 3. At a parameter of 0.02, the structured packing has a 1.3–1.4 capacity advantage over random packing and plates. This advantage disappears as the parameter approaches 0.1. Flow Parameter 0.1–0.3 1. Plates and random packings have about the same efficiency and capacity. 2. Structured packing has about the same capacity as plates and random packings. 3. The efficiency advantage of structured packing over random packings and plates decreases from 1.5 to 1.2 as the parameter increases from 0.1 to 0.3 Flow Parameter 0.3–0.5 1. The loss of capacity of structured packing is greatest in this range. 2. The random packing appears to have the highest capacity and efficiency, and structured packing the least capacity and efficiency. Experience indicates that use of structured packings may not have advantages in the higher-pressure (higher-flow-parameter) region. Special considerations for the use of structured packing at higher pressures are detailed by Kurtz et al. [Chem. Eng. Progr., 87(2), 43 (1991)]. Optimization As stated previously, optimization studies should include the entire system. Such a study was made by Fair and Bolles [Chem. Eng., 75(9), 156 (1968)], using a light-hydrocarbon system and with the objective of defining optimum reflux ratio. Coolants used were at −87, −40, and +30°C (−125, −40, and +85°F), corresponding to different pressures of operation and associated different condens-

14-61

Optimum-reflux-ratio determination. °F = 9 /5 °C + 32. [Fair and Bolles, Chem. Eng., 75(4), 156 (1968).]

FIG. 14-84

ing temperatures. The results are shown in Fig. 14-84; the optimum reflux ratio is quite close to the calculated minimum reflux ratio. Colburn (chemical engineering lecture notes, University of Delaware, 1943) proposed that the optimum reflux ratio is N + [(C2/hGb + C3)/C1] + (dN/dR) Ropt = ᎏᎏᎏᎏ (14-189) dN/dR where R = external reflux ratio = L/D N = number of theoretical plates Gb = allowable vapor velocity in heat exchangers, (lb⋅mol)/(h⋅ft2) h = hours of operation C1 = amortization rate for tower, dollars/(ft2⋅plate⋅year) C2 = amortization rate for heat exchangers, dollars/(ft2⋅year) C3 = cost of utilities per mole of distillate, dollars/mol Happel [Chem. Eng., 65(14), 144 (1958)] using a modification of the Colburn relationship found that the optimum number of trays varies from 2 to 3 times the number at total reflux. Gilliland [Ind. Eng. Chem., 32, 1220 (1940)] from the establishment of an empirical relationship between reflux ratio and theoretical trays based on a study of existing columns indicated that Ropt − Rmin 0.1 < ᎏᎏ < 0.3 Rmin + 1 and correspondingly Nopt − Nmin 0.35 < ᎏᎏ < 0.52 Nmin + 1 The effect of utilities costs on optimum operation was noted by Kiguchi and Ridgway [Pet. Refiner, 35(12), 179 (1956)], who indicated that in petroleum-distillation columns the optimum reflux ratio varies between 1.1 and 1.5 times the minimum reflux ratio. When refrigeration is involved, 1.1Rmin < Ropt < 1.2Rmin, and when cooling-tower water is used in the condensers, 1.2Rmin < Ropt < 1.4Rmin.

14-62

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

PHASE DISPERSION GENERAL REFERENCES: For an overall discussion of gas-liquid breakup processes, see Brodkey, The Phenomena of Fluid Motions, Addison-Wesley, Reading, Massachusetts, 1967. For a discussion of atomization devices and how they work, see Masters, Spray Drying Handbook, 5th ed., Wiley, New York, 1991; and Lefebvre, Atomization and Sprays, Hemisphere, New York, 1989. A beautifully illustrated older source is Dombrowski and Munday, Biochemical and Biological Engineering Science, vol. 2, Academic Press, London, 1968, pp. 209–320. Bayvel and Orzechowski, Liquid Atomization, Taylor and Francis, Washington DC, 1993, provides additional background into atomizer design. For a survey on fog formation, see Amelin, Theory of Fog Formation, Israel Program for Scientific Translations, Jerusalem, 1967.

LIQUID-IN-GAS DISPERSIONS Liquid Breakup into Droplets There are four basic mechanisms for breakup of liquid into droplets: • Droplets in a field of high turbulence (i.e., high power dissipation per unit mass) • Simple jets at low velocity • Expanding sheets of liquid at relatively low velocity • Droplets in a steady field of high relative velocity These mechanisms coexist, and the one that gives the smallest drop size will control. The four mechanisms follow distinctly different velocity dependencies: 1. Breakup in a highly turbulent field (1/velocity)1.2. This appears to be the dominant breakup process in distillation trays in the spray regime, pneumatic atomizers, and high-velocity pipeline contactors. 2. Breakup of a low-velocity liquid jet (1/velocity)0. This governs in special applications like prilling towers and is often an intermediate step in liquid breakup processes. 3. Breakup of a sheet of liquid (1/velocity)0.67. This governs drop size in most hydraulic spray nozzles. 4. Single-droplet breakup at very high velocicty (1/velocity)2. This governs drop size in free fall as well as breakup when droplets impinge on solid surfaces. Droplet Breakup—High Turbulence This is the dominant breakup mechanism for many process applications. Breakup results from local variations in turbulent pressure that distort the droplet shape. Hinze [Am. Inst. Chem. Eng. J., 1, 289–295 (1953)] applied turbulence theory to obtain the form of Eq. (14-171) and took liquidliquid data to define the coefficient: Dmax = 0.725(σ/ρG)0.6/E0.4 where

E = (power dissipated)/mass σ = surface tension ρG = gas density

about 1.9 times the initial diameter of the jet as shown by Fig. 14-85. As shown, the actual breakup is quite close to prediction, although smaller satellite drops are also formed. The prime advantage of this type of breakup is the greater uniformity of drop size. For high-viscosity liquids, the drops are larger, as shown by Eq. (14-191): 3µᐉ D = 1.9Dj 1 + ᎏᎏ (14-191) (σρᐉDj)1/2 where D = diameter of droplet Dj = diameter of jet µᐉ = viscosity of liquid ρᐉ = density of liquid σ = surface tension of liquid





These units are dimensionally consistent; any set of consistent units can be used. As the velocity of the jet is increased, the breakup process changes and ultimately becomes a mix of various competing effects, such as the capture of small drops by bigger ones in the slowing jet and the “turbulent breakup” of the bigger drops. The high-velocity jet is occasionally used in process applications because of the very narrow spray angle (5–20°) and the high penetration into a gas it can give. The focused stream also aids erosion of a surface. Liquid-Sheet Breakup The basic principle of most hydraulic atomizers is to form a thin sheet that breaks via a variety of mechanisms to form ligaments of liquid which in turn yield chains of droplets. See Fig. 14-86. For a typical nozzle, the drop size varies with 1/(pressure drop)1/3. When (velocity)2 is substituted for pressure drop, droplet size is seen to vary with (velocity)−2/3. Isolated Droplet Breakup—in a Velocity Field Much effort has focused on defining the conditions under which an isolated drop will break in a velocity field. The criterion for the largest stable drop

(14-190)

length2/time3 mass/time2 mass/length3

Note that Dmax comes out with units of length. Since E typically varies with (gas velocity)3, this results in drop size dependence with (1/velocity)1.2. The theoretical requirement for use of Eq. (14-190) is that the microscale of turbulence << Dmax. This is satisfied in most gas systems. For example, in three cases, (microscale of turbulence)/Dmax distillation tray in spray regime pipeline @ 40 m/s and atmospheric pressure two-fluid atomizer using 100 m/s air

0.007 0.012 0.03

For these three applications, Eq. (14-190) gives good prediction of drop size when the design variables are used to calculate E, as illustrated by Eqs. (14-198) and (14-201). Liquid-Column Breakup Because of increased pressure at points of reduced diameter, the liquid column is inherently unstable. As a result, it breaks into small drops with no external energy input. Ideally, it forms a series of uniform drops with the size of the drops set by the fastest-growing wave. This yields a dominant droplet diameter

(a)

(b)

(c)

(a) Idealized jet breakup suggesting uniform drop diameter and no satellites. (b) and (c) Actual breakup of a water jet as shown by high-speed photographs. [From W. R. Marshall, “Atomization and Spray Drying,” Chem. Eng. Prog. Monogr. Ser., no. 2 (1954).]

FIG. 14-85

PHASE DISPERSION

(a)

(b) FIG. 14-86 Sheet breakup. (a) By perforation. [After Fraser et al., Am. Inst. Chem. Eng. J., 8(5), 672 (1962).] (b) By sinusoidal wave growth. [After Dombrowski and Johns, Chem. Eng. Sci., 18, 203 (1963).]

size is the ratio of aerodynamic forces to surface-tension forces defined by the Weber number, NWe (dimensionless): NWe crit = constant = [␳G (velocity)2(Dmax)/(σ)]

(14-192)

NWe crit for low-viscosity fluids commonly ranges from 10 to 20, with the larger value for a free-fall condition and the smaller for a sudden acceleration. High liquid viscosity also increases NWe crit. Droplet breakup via impingement appears to follow a similar relationship, but much less data is available. This type of breakup can result from impingement on equipment walls or compressor blades. In general, there is less tendency to shatter on wetted surfaces. Droplet Size Distribution Instead of the single droplet size implied by the discussion above, a spectrum of droplet sizes is produced. The most common ways to characterize this spectrum are: • Volume median (mass median) Dvm. This has no fundamental meaning but is easy to determine since it is at the midpoint of a cumulative-volume plot. • Sauter mean D32. This has the same ratio of surface to volume as the total drop population. It is typically 70 to 90 percent of Dvm. It is frequently used in transport processes and is used here to characterize drop size. • Maximum Dmax. This is the largest-sized particle in the population. It is typically 3 to 4 times D32 in turbulent breakup processes, per Walzel [International Chemical Engineering, 33, 46, (1993)]. It is the size directly calculated from the power/mass relationship. D32 is estimated from Dmax by D32 = 0.3⋅Dmax

(14-193)

and Dvm is estimated from it by Dvm = 0.4⋅Dmax

(14-194)

However, any average drop size is fictitious, and none is completely satisfactory. For example, there is no way in which the high surface and transfer coefficients in small drops can be made available to the larger drops. Hence, a process calculation based on a given droplet size describes only what happens to that size and gives at best an approximation to the total mass.

14-63

There are a variety of ways to describe the droplet population. Figures 14-88 and 14-90 illustrate one of the most common methods, the plot of cumulative volume against droplet size on log-normal graph paper. This satisfies the restraint of not extrapolating to a negative drop size. Its other advantages are that it is easy to plot, the results are easy to visualize, and it yields a nearly straight line at lower drop sizes. Cumulative volume over the range of 1 to 50 percent can also be shown to vary approximately as D2. This is equivalent to finding that the number of droplets of a given size is inversely proportional to the droplet area or the surface energy of the droplet. Atomizers The common need to disperse a liquid into a gas has spawned a large variety of mechanical devices. The different designs emphasize different advantages such as freedom from plugging, pattern of spray, small droplet size, uniformity of spray, high turndown ratio, and low power consumption. As shown in Table 14-11, most atomizers fall into three categories: 1. Pressure nozzles (hydraulic) 2. Two-fluid nozzles (pneumatic) 3. Rotary devices (spinning cups, disks, or vaned wheels) These share certain features such as relatively low efficiency and low cost relative to most process equipment. The energy required to produce the increase in area is typically less than 0.1 percent of the total energy consumption. This is because atomization is a secondary process resulting from high interfacial shear or turbulence. As droplet sizes decrease, this efficiency drops lower. Other types are available that use sonic energy (from gas streams), ultrasonic energy (electronic), and electrostatic energy, but they are less commonly used in process industries. See Table 14-11 for a summary of the advantages/disadvantages of the different type units. An expanded discussion is given by Masters [Spray Drying Handbook, Wiley, New York, (1991)]. Special requirements such as size uniformity in prilling towers can dictate still other approaches to dispersion. Here plates are drilled with many holes to develop nearly uniform columns. Commonly, the most important feature of a nozzle is the size of droplet it produces. Since the heat or mass transfer that a given dispersion can produce is often proportional to (1/Dd)2, fine drops are usually favored. On the other extreme, drops that are too fine will not settle, and a concern is the amount of liquid that will be entrained from a given spray operation. For example, if sprays are used to contact atmospheric air flowing at 1.5 m/s, drops smaller than 350 µm [terminal velocity = 1.5 m/s (4.92 ft/s)] will be entrained. Even for the relative coarse spray of the hollow-cone nozzle shown in Fig. 14-88, 7.5 percent of the total liquid mass will be entrained. Hydraulic (Pressure) Nozzles Manufacturers’ data such as shown by Fig. 14-88 are available for most nozzles for the air-water system. In Fig. 14-88, note the much coarser solid-cone spray. The coarseness results from the less uniform discharge. Effect of Physical Properties on Drop Size Because of the extreme variety of available geometries, no attempt to encompass this variable is made here. The suggested predictive route starts with airwater droplet size data from the manufacturer at the chosen flow rate. This drop size is then corrected by Eq. (14-195) for different viscosity and surface tension: Dvm, system σsystem ᎏ= ᎏ Dvm, water 73




µᐉ


ᎏ 1.0  0.25

0.25

(14-195)

where Dvm = volume median droplet diameter σ = surface tension, mN/m (dyn/cm) µᐉ = liquid viscosity, mPa⋅s (cP) The exponential dependencies in Eq. (14-195) represent averages of values reported by a number of studies with particular weight given to Lefebvre ([Atomization and Sprays, Hemisphere, New York, (1989)]. Since viscosity can vary over a much broader range than surface tension, it has much more leverage on drop size. For example, it is common to find an oil with 1000 times the viscosity of water, while most liquids fall within a factor of 3 of its surface tension. Liquid density is generally even closer to that of water, and since the data are not clear that a liquid density correction is needed, none is shown in Eq.

14-64

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

TABLE 14-11

Atomizer Summary

Types of atomizer

Design features

Pressure.

1. Hollow cone. a. Whirl chamber (see Fig. 1487a). b. Grooved core. 2. Solid cone (see Fig. 14-87b). 3. Fan (flat) spray.

Advantages

Disadvantages

Flow α(∆P/ρᐉ)1/2. Only source of energy is from fluid being atomized.

Simplicity and low cost.

Liquid leaves as conical sheet as a result of centrifugal motion of liquid. Air core extends into nozzle. Centrifugal motion developed by tangential inlet in chamber upstream of orifice. Centrifugal motion developed by inserts in chamber.

High atomization efficiency.

Similar to hollow cone but with insert to provide even distribution. Liquid leaves as a flat sheet or flattened ellipse.

a. Oval or rectangular orifice (see Fig. 14-87c). Numerous variants on cavity and groove exist. b. Deflector (see Fig. 14-87d).

Combination of cavity and orifice produces two streams that impinge within the nozzle. Liquid from plain circular orifice impinges on curved deflector.

c. Impinging jets (see Fig. 14-87e).

Limited tolerance for solids; uncertain spray with high-viscosity liquids; susceptible to erosion. Need for special designs (e.g., bypass) to achieve turndown. Concentrated spray pattern at cone boundaries.

Minimum opportunity for plugging. Smaller spray angle than 1a and ability to handle flows smaller than 1a. More uniform spatial pattern than hollow cone. Flat pattern is useful for coating surfaces and for injection into streams.

Coarser drops for comparable flows and pressure drops. Failure to yield same pattern with different fluids. Small clearances.

Minimal plugging.

Coarser drops.

Two jets collide outside nozzle and produce a sheet perpendicular to their plane.

Different liquids are isolated until they mix outside of orifice. Can produce a flat circular sheet when jets impinge at 180°.

Extreme care needed to align jets.

A portion of the liquid is recirculated after going through the swirl chamber. Conical sheet is developed by flow between orifice and poppet. Increased pressure causes poppet to move out and increase flow area.

Achieves uniform hollow cone atomization pattern with very high turndown (50:1). Simplest control over broad range.

Waste of energy in bypass stream. Added piping for spill flow.

Two-fluid (see Fig. 14-87h).

Gas impinges coaxially and supplies energy for breakup.

Because gas is also accelerated, efficiency is inherently lower than pressure nozzles.

Sonic.

Gas generates an intense sound field into which liquid is directed. Liquid is fed to a rotating surface and spreads in a uniform film. Flat disks, disks with vanes, and bowlshaped cups are used. Liquid is thrown out at 90° to the axis.

High velocities can be achieved at lower pressures because the gas is the high-velocity stream. Liquidflow passages can be large, and hence plugging can be minimized. Similar to two-fluid but with greater tolerance for solids. The velocity that determines drop size is independent of flow. Hence these can handle a wide range of rates. They can also tolerate very viscous materials as well as slurries. Can achieve very high capacity in a single unit; does not require a highpressure pump. Fine atomization, small size, and low injection velocity.

4. Nozzles with wider range of turndown. a. Spill (bypass) (see Fig. 14-87f). b. Poppet (see Fig. 14-87g).

Rotary wheels (see Fig. 14-87i) disks, and cups.

Ultrasound.

Liquid is fed over a surface vibrating at a frequency > 20 kHz.

(14-195). Vapor density also has an impact on dropsize but the impact is complex, involving conflicts of a number of effects, and vapor density is commonly omitted in atomizer dropsize correlations. Effect of Pressure Drop and Nozzle Size For a nozzle with a developed pattern, the average drop size can be estimated to fall with rising ∆P (pressure drop) by Eq. (14-196): ∆P2 D1 ᎏ= ᎏ D2 ∆P1




1/3

(14-196)

For similar nozzles and constant ∆P, the drop size will increase with nozzle size as indicated by Eq. (14-197):




Difficult to maintain proper clearances.

Similar to two-fluid. Mechanical complexity of rotating equipment. Radial discharge.

Low flow rate and need for ultrasound generator.

orifice diameter1 D1 ᎏ = ᎏᎏ D2 orifice diameter2



1/2

(14-197)

Once again, these relationships are averages of a number of reported values and are intended as rough guides. The normal operating regime is well below turbulent breakup velocity. However the data of Kennedy [ J. of Engineering for Gas Turbines and Power, 108, 191, (1986)] at very high pressure drop in large nozzles shows a shift to a higher dependence on pressure drop. This data suggests that turbulent droplet breakup can also be governing with hydraulic spray nozzles, although this is unusual.

PHASE DISPERSION

(a)

(c)

(b)

(d)

14-65

(e)

(h) (f)

(g) (i) FIG. 14-87 Charactersitic spray nozzles. (a) Whirl-chamber hollow cone. (b) Solid cone. (c) Oval-orifice fan. (d) Deflector jet. (e) Impinging jet. ( f ) Bypass. (g) Poppet. (h) Two-fluid. (i) Vaned rotating wheel.

Droplet-size distribution for three different types of nozzles. To convert pounds per square inch gauge to kilopascals, multiply by 6.89; to convert gallons per minute to cubic meters per hour, multiply by 0.227. (Spraying Systems Inc.)

FIG. 14-88

14-66

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

Spray Angle A shift to a smaller-angle nozzle gives slightly larger drops for a given type of nozzle because of the reduced tendency of the sheet to thin. Dietrich [Proc. 1st Int. Conf. Liq. Atomization Spray Systems, Tokyo, (1978)] shows the following: Angle Dvm, µm

25° 1459

50° 1226

65° 988

80° 808

95° 771

In calculating the impact point of spray, one should recognize that the spray angle closes in as the spray moves away from the nozzle. This is caused by loss of momentum of the spray to the gas. At some low flow, pressure nozzles do not develop their normal pattern but tend to approach solid streams. The required flow to achieve the normal pattern increases with viscosity. Two-Fluid (Pneumatic) Atomizers This general category includes such diverse applications as venturi atomizers and reactoreffluent quench systems in addition to two-fluid spray nozzles. Depending on the manner in which the two fluids meet, several of the breakup mechanisms may be applicable, but the final one is high-level turbulent rupture. As shown by Table 14-12, empirical correlations for two-fluid atomization show dependence on high gas velocity to supply atomizing energy, usually to a power dependence close to that for turbulent breakup. In addition, the correlations show a dependence on the ratio of gas to liquid and system dimension. TABLE 14-12 Exponential Dependence of Drop Size on Different Parameters in Two-Fluid Atomization Relative Surface Atomizer velocity tension Gas density 1 + L/G dimension Jasuja (empirical for small nozzle) El-Shanawany and Lefebvre (empirical for small nozzle) Tatterson, Dallman, and Hanratty (pipe flow) Hinze (turbulence theory) Steinmeyer (extension of turbulence theory)

−0.9

0.45

−0.45

0.5

0.55

−1.2

0.6

−0.7

1

0.40

−1

0.5

−0.5

−1.2 −1.2

0.6 0.6

−0.6 −0.6

0.5 0.4

0.4

Further differences from hydraulic nozzles (controlled by sheet and ligament breakup) are the stronger increase in drop size with increasing surface tension and decreasing gas density. The similarity of these correlations to the dependencies shown by Eq. (14-190) was noted by Steinmeyer [Chem. Eng. Progr., 91(7), 72 (1995)] who reformulated Hinze’s relationship, Eq. (14-190), into Eq. (14-198) by including atomizer variables. 0.4 1 + L 0.4 σ 0.6 D32 = 0.29 ᎏ (1/velocity)1.2 ᎏ Dnozzle (14-198) ρG G




where









σ = surface tension ρG = gas density L/G = mass ratio of liquid flow to gas flow Dnozzle = diameter of the air discharge

This is remarkably similar to the empirical two-fluid atomizer relationships of El-Shanawany and Lefebvre [ J. Energy, 4, 184 (1980)] and Jasuja [Trans. Am. Soc. Mech. Engr., 103, 514 (1981)]. For example, El-Shanawany and Lefebvre give a relationship for a prefilming atomizer: D32 = 0.0711(σ/ρG)0.6(1/velocity)1.2(1 + L/G)(Dnozzle)0.4(ρL/ρG)0.1 + 0.015[(µL)2/(σ × ρL)]0.5(Dnozzle)0.5(1 + L/G)

(14-199)

where µL is liquid viscosity. According to Jasuja, D32 = 0.17(σ/ρG)0.45(1/velocity)0.9(1 + L/G)0.5(Dnozzle)0.55 + viscosity term

(14-200)

(Eqs. 14-198, 14-199, and 14-200 are dimensionally consistent; any set of consistent units on the right-hand side yields the droplet size in units of length on the left-hand side.) The second, additive term carrying the viscosity impact in Eq. (14-199) is small at viscosities around 1 centipoise but can become controlling as viscosity increases. For example, for air at atmospheric pressure atomizing water, with nozzle conditions. Dnozzle = 0.076 m (3 inches) velocity = 100 m/s (328 ft/s) L/G = 1 For this case, Steinmeyer’s correlation becomes El-Shanawany, Eq. (14-199) predicts 76 microns with the viscosity term contributing less than 1 percent. With the same system and same L/G, but with an oil with 30 times water viscosity, Eq. (14-199) predicts 91 microns, with the viscosity term contributing 54 percent of the total. The measure values for water and oil cases were 70 and 95 microns, respectively. For comparison, Eq. (14-198) predicts 102 microns for the water case. Rotary Atomizers For rotating wheels, vaneless disks, and cups, there are three regimes of operation. At low rates, the liquid is shed directly as drops from the rim. At intermediate rates, the liquid leaves the rim as threads; and at the highest rate, the liquid extends from the edge as a thin sheet that breaks down similarly to a fan or hollow-cone spray nozzle. As noted in Table 14-12, rotary devices have many unique advantages such as the ability to handle high viscosity and slurries and produce small droplets without high pressures. The prime applications are in spray drying. See Masters [Spray Drying Handbook, Wiley, New York (1991)] for more details. Pipeline Contactors The power dissipation per unit mass for pipeline flow is similar to that for two-fluid nozzles. σ D32 = 0.79 ᎏ ρG

1

ᎏ velocity  0.6

1.2

(Dpipe)0.4

(14-201)

(The relation is dimensionally consistent; any set of consistent units on the right-hand side yields the droplet size in units of length on the lefthand side.) The relationship is similar to the empirical correlation of Tatterson, Dallman, and Hanratty [Am. Inst. Chem. Eng. J., 23(1), 68 (1977)] σ D32 ∼ ᎏ ρG




0.5

(1/velocity)1(Dpipe)0.5

Predictions from Eq. (14-201) align well with the Tatterson data. For example, for a velocity of 43 m/s (140 ft/s) in a 0.05-m (1.8-inch) equivalent diameter channel, Eq. (14-201) predicts D32 of 490 microns, compared to the measured 460 to 480 microns. Entrainment Due to Gas Bubbling/Jetting through a Liquid Entrainment generally limits the capacity of distillation trays and is commonly a concern in vaporizers and evaporators. Fortunately, it is readily controllable by simple inertial entrainment capture devices such as wire mesh pads in gravity separators. In distillation towers, entrainment lowers the tray efficiency, and 1 pound of entrainment per 10 pounds of liquid is sometimes taken as the limit for acceptable performance. However, the impact of entrainment on distillation efficiency depends on the relative volatility of the component being considered. Entrainment has a minor impact on close separations when the difference between vapor and liquid concentration is small, but this factor can be dominant for systems where the liquid concentration is much higher than the vapor in equilibrium with it (i.e., when a component of the liquid has a very low volatility, as in an absorber). As shown by Fig. 14-90, entrainment droplet sizes span a broad range. The reason for the much larger drop sizes of the upper curve is the short disengaging space. For this curve, over 99 percent of the entrainment has a terminal velocity greater than the vapor velocity. For contrast, in the lower curve the terminal velocity of the largest particle reported is the same as the vapor velocity. For the settling velocity to limit the maximum drop size entrained, at least 0.8 m (30 in) disengaging space is usually required. Note that even for the lower curve, less than 10 percent of the entrainment is in drops of less than

PHASE DISPERSION

(a) FIG. 14-89

(b)

(c)

(d)

14-67

(e)

Mechanism of the burst of an air bubble on the surface of water. [Newitt, Dombrowski, and Knellman, Trans. Inst. Chem. Eng., 32, 244 (1954).]

50 µm. The coarseness results from the relatively low power dissipation per mass on distillation trays. This means that it is relatively easy to remove by a device such as a wire mesh pad. Over 50 percent is typically captured by the underside of the next higher tray or by a turn in the piping leaving an evaporator. Conversely, though small on a mass basis, the smaller drops are extremely numerous. On a number basis, more than one-half of the drops in the lower curve are under 5 µm. These can serve as nuclei for fog condensation in downstream equipment. Entrainment can stem from a variety of sources. 1. Excessive foaming. This is a case of a gas-in-liquid dispersion (covered in the next subsection). 2. Droplets formed from the collapse of the bubble dome (see Fig. 14-89). These are virtually unavoidable. They are generally under 25 µm, which means that their terminal velocities are low and they are invariably entrained. Fortunately, because of their small size, they contribute little on a weight basis (<0.001 kg liquid/kg vapor), although they dominate on a number basis. 3. Droplets from the jet caused by liquid rushing to fill the cavity left by the bubble (see Fig. 14-89). These droplets range up to 1000 µm, their size depending on bubble size. This is important only at modest loadings. Once foam forms over the surface, drop ejection by this mode decreases sharply. 4. At higher vapor loads, the kinetic energy of the vapor rather than the bubble burst supplies the thrust for jets and sheets of liquid that are thrown up as well as the energy from breakup into “spray.” This yields much higher levels of entrainment. In distillation trays it is the most common limit to capacity. The major variable in setting entrainment (E, weight of liquid entrained per weight of vapor) is vapor velocity. As velocity is increased, the dependence of E on velocity steepens. In the lowest velocity regime, E is proportional to velocity. At values of E of about 0.001 (around 10 percent of flood), there is a shift to a region where the dependence is with (velocity)3–5. Near flood, the dependence rises to approximately (velocity)8. In this regime, the kinetic energy of the vapor dominates, and the bulk of the dispersion on the plate is often in the form of a coarse spray. Pinczewski and Fell [Trans. Inst. Chem Eng., 55, 46 (1977)] show that the velocity at which vapor jets onto the tray sets the droplet size, rather than the superficial tray velocity. A maximum superficial velocity formulation that incorporates φ, the fractional open area, is logical since the fractional open area sets the jet velocity. Stichlmair and Mersmann [Int. Chem. Eng., 18(2), 223 (1978)] give such a correlation: 2.5[φg(ρl − ρg)σ]1/4 Umax = ᎏᎏ (14-202) ρ1/2 g (The relation is dimensionally consistent; any set of consistent units on the right-hand side yields velocity units for the left-hand side.) Stichlmair uses the ratio of actual velocity to this maximum velocity together with a term that increases entrainment as the distance gets small between the liquid-vapor layer and the tray deck above. His correlation spans a 105 fold range in entrainment. He shows a sharp increase in entrainment at 60 percent of the maximum velocity and attributes the increase to a shift to the spray regime. Puppich and Goedecke [Chem. Eng. Tech., 10, 224 (1987)] test this correlation against a wide range of tray types and find generally good agreement. Bubble caps give about twice as much entrainment as the correlation predicts. Sieve trays give about half as much as the correlation predicts. Steinmeyer [Chem. Eng. Progr., 91(7), 72 (1995)] derived a corre-

lation for entrainment utilizing predicted drop size from a turbulent power dissipation per unit mass) breakup in the “spray” regime. His predicted drop size matched the data of Pinczewski and Fell. When this prediction is combined with the estimated fraction of the droplet population that is entrained, the entrainment prediction, Eq. (14203), results. Note that this matches the empirical experience in predicting entrainment varying with (velocity)8 ρG4 E = constant [V]8φ3 ᎏᎏ (14-203) [g(ρL − ρG)]2.5 σ1.5 If flood is defined as the velocity at which E equals 1, this yields (dimensionless): {φ0.375[g(ρL − ρG)]0.3125σ0.1875(Z/Zbase)0.125} Vflood = 1.25 × ᎏᎏᎏᎏᎏ ρG0.5

(14-204)

where Z is tray spacing in meters and Z/Zbase is the ratio of tray spacing to 0.3 m (1 ft). Fair’s empirical correlation for sieve and bubble-cap trays shown in Fig. 14-26 is similar. Note that Fig. 14-26 incorporates a velocity dependence (velocity)6–8 above 90 percent of flood for high-density systems. The correlation implicitly considers the tray design factors such as the open area, tray spacing, and hole diameter through the impact of these factors on percent of flood. The dependencies of all three correlations are remarkably close, as shown by Table 14-13 and the numeric prediction of flooding velocity is also close. Correlations can be extended to evaporators at lower velocities by assuming that E declines with (velocity)2 between 60 percent and 10 percent of the maximum velocity. At velocities below 10 percent of the maximum velocity, E can be assumed to change directly with velocity. Fog Condensation This is an entirely different way of forming dispersions. Here, the dispersion results from condensation of a vapor rather than mechanical breakup. The particle sizes are usually much finer (0.1 to 30 µm) and are designated as mist or fog. Fog particles grow because of excess saturation in the gas. Usually this means that the gas is supersaturated (i.e., it is below its dew point). Sometimes, fog can also grow on soluble foreign nuclei at partial pressures below saturation. Increased saturation can occur through a variety of routes: 1. Mixing of two saturated streams at different temperatures. This is commonly seen in the plume from a stack. Since vapor pressure is an exponential function of temperature, the resultant mixture of two saturated streams will be supersaturated at the mixed temperature. Uneven flow patterns and cooling in heat exchangers make this route to supersaturation difficult to prevent. 2. Increased partial pressure due to reaction. An example is the reaction of SO3 and H2O to yield H2SO4, which has much lower vapor pressure than its components. 3. Isoentropic expansion (cooling) of a gas, as in a steam nozzle.

TABLE 14-13 Dependency of Distillation Flood Velocity on Physical Properties and Tray Open Area φ ρL − ρG σ ρG

Power/mass

Fair

Stichlmair/Mersmann

0.375 0.3125 0.1875 0.5

0.44 0.5 0.2 0.5

0.5 0.25 0.25 0.5

14-68

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

4. Cooling of a gas containing a condensable vapor. Here the problem is that the gas cools faster than condensable vapor can be removed by mass transfer. These mechanisms can be observed in many common situations. For example, fog via mixing can be seen in the discharge of breath on a cold day. Fog via adiabatic expansion can be seen in the low-pressure area over the wing of an airplane landing on a humid summer day; and fog via condensation can be seen in the exhaust from an automobile air conditioner (if you follow closely enough behind another car to pick up the ions or NO molecules needed for nucleation). All of these occur at a very low supersaturation and appear to be keyed to an abundance of foreign nuclei. All of these fogs also quickly dissipate as heat or unsaturated gas is added. The supersaturation in condensers arises for two reasons. First, the condensable vapor is generally of higher molecular weight than the noncondensable gas. This means that the molecular diffusivity of the vapor will be much less than the thermal diffusivity of the gas. Restated, the ratio of NSc /NPr is greater than 1. The result is that a condenser yields more heat-transfer units dTg /(Tg − Ti) than mass-transfer units dYg /(Yg − Yi). Second, both transfer processes derive their driving force from the temperature difference between the gas Tg and the interface Ti. Each incremental decrease in interface temperature yields the same relative increase in temperature driving force. However, the interface vapor pressure can only approach the limit of zero. Because of this, for equal molecular and thermal diffusivities a saturated mixture will supersaturate when cooled. The tendency to supersaturate generally increases with increased molecular weight of the condensable, increased temperature differences, and reduced initial superheating. To evaluate whether a given condensing step yields fog requires rigorous treatment of the coupled heat-transfer and masstransfer processes through the entire condensation. Steinmeyer [Chem. Eng. Prog., 68(7), 64 (1972)] illustrates this, showing the impact of foreign-nuclei concentration on calculated fog formation. See Table 14-14. Note the relatively large particles generated for cases 1 and 2 for 10,000 foreign nuclei per cm3. These are large enough to be fairly easily collected. There have been very few documented problems with industrial condensers despite the fact that most calculate to generate supersaturation along the condensing path. The explanation appears to be a limited supply of foreign nuclei. Ryan et al. [Chem. Eng. Progr., 90(8), 83 (1994)] show that separate mass and heat transfer-rate modeling of an HCl absorber predicts 2 percent fog in the vapor. The impact is equivalent to lowering the stage efficiency to 20 percent. Spontaneous (Homogeneous) Nucleation This process is quite difficult because of the energy barrier associated with creation of the interfacial area. It can be treated as a kinetic process with the TABLE 14-14 Simulation of Three Heat Exchangers with Varying Foreign Nuclei Weight fraction, noncondensable Inlet Outlet Molecular weight Inert Condensable Temperature difference between gas and liquid interface, K Inlet Outlet Percent of liquid that leaves unit as fog Nuclei concentration in inlet particles/cm3 100 1,000 10,000 100,000 1,000,000 10,000,000 ∞ Fog particle size based on 10,000 nuclei/cm3 at inlet, µm

1

2

0.51 0.80

0.42 0.80

3 0.02 0.32

28 86

29 99

29 210

14 4

24 10

67 4

0.05 0.44 3.2 9.6 13.3 14.7 14.7 28

1.1 5.6 9.8 11.4 11.6

2.2 3.9 4.9 5.1

11.8 25

5.1 4

rate a very steep function of the supersaturation ratio (S = partial pressure of condensable per vapor pressure at gas temperature). For water, an increase in S from 3.4 to 3.9 causes a 10,000-fold increase in the nucleation rate. As a result, below a critical supersaturation (Scrit), homogeneous nucleation is slow enough to be ignored. Generally, Scrit is defined as that which limits nucleation to one particle produced per cubic centimeter per second. It can be estimated roughly by traditional theory (Theory of Fog Condensation, Israel Program for Scientific Translations, Jerusalem, 1967) using the following equation: M σ 3/2 Scrit = exp 0.56 ᎏ ᎏ (14-205) ρl T where σ = surface tension, mN/m (dyn/cm) ρl = liquid density, g/cm3 T = temperature, K M = molecular weight of condensable






Table 14-15 shows typical experimental values of Scrit taken from the work of Russel [ J. Chem. Phys., 50, 1809 (1969)]. Since the critical supersaturation ratio for homogeneous nucleation is typically greater than 3, it is not often reached in process equipment. Growth on Foreign Nuclei As noted above, foreign nuclei are often present in abundance and permit fog formation at much lower supersaturation. For example, 1. Solids. Surveys have shown that air contains thousands of particles per cubic centimeter in the 0.1-µm to 1-µm range suitable for nuclei. The sources range from ocean-generated salt spray to combustion processes. The concentration is highest in large cities and industrial regions. When the foreign nuclei are soluble in the fog, nucleation occurs at S values very close to 1.0. This is the mechanism controlling atmospheric water condensation. Even when not soluble, a foreign particle is an effective nucleus if wet by the liquid. Thus, a 1-µm insoluble particle with zero contact angle requires an S of only 1.001 in order to serve as a condensation site for water. 2. Ions. Amelin [Theory of Fog Condensation, Israel Program for Scientific Translations, Jerusalem, (1967)] reports that ordinary air contains even higher concentrations of ions. These ions also reduce the required critical supersaturation, but by only about 10 to 20 percent, unless multiple charges are present. 3. Entrained liquids. Production of small droplets is inherent in the bubbling process, as shown by Fig. 14-90. Values range from near zero to 10,000/cm3 of vapor, depending on how the vapor breaks through the liquid and on the opportunity for evaporation of the small drops after entrainment. As a result of these mechanisms, most process streams contain enough foreign nuclei to cause some fogging. While fogging has been reported in only a relatively low percent of process partial condensers, it is rarely looked for and volunteers its presence only when yield losses or pollution is intolerable. Dropsize Distribution Monodisperse (nearly uniform droplet size) fogs can be grown by providing a long retention time for growth. However, industrial fogs usually show a broad distribution, as in Fig. 14-91. Note also that for this set of data, the sizes are several orders of magnitude smaller than those shown earlier for entrainment and atomizers. The result, as discussed in a later subsection, is a demand for different removal devices for the small particles. While generally fog formation is a nuisance, it can occasionally be useful because of the high surface area generated by the fine drops. An example is insecticide application. TABLE 14-15 H2O C2H5OH CH4OH C6H6 CCl4 CHCl3 C6H5Cl

Experimental Critical Supersaturation Ratios Temperature, K°

Scrit

264 275 264 253 247 258 250

4.91 2.13 3.55 5.32 6.5 3.73 9.5

PHASE DISPERSION

14-69

Entrainment droplet-size distribution. To convert meters per second to feet per second, multiply by 3.28, to convert meters to feet multiply by 3.28.

FIG. 14-90

Particle-size distribution and mist loading from absorption tower in a contact H2SO4 plant. [Gillespie and Johnstone, Chem. Eng. Prog., 51(2), 74 (1955).]

FIG. 14-91

GAS-IN-LIQUID DISPERSIONS GENERAL REFERENCES: Comprehensive treatments of bubbles or foams are given by Akers, Foams: Symposium 1975, Academic Press, New York, 1973; Bendure, Tappi, 58, 83 (1975); Benfratello, Energ Elettr., 30, 80, 486 (1953); Berkman and Egloff, Emulsions and Foams, Reinhold, New York, 1941, pp. 1 12–152; Bikerman, Foams, Springer-Verlag, New York, 1975; Kirk-Othmer Encyclopedia of Chemical Technology, 4th ed., Wiley, New York, 1993, pp. 82–145; Haberman and Morton, Report 802, David W. Taylor Model Basin, Washington, 1953; Levich, Physicochemical Hydrodynamics, Prentice-Hall, Englewood Cliffs, NJ, 1962; and Soo, Fluid Dynamics of Multiphase Systems, Blaisdell, Waltham, Massachusetts, 1967. The formation of bubbles is comprehensively treated by Clift, Grace, and Weber, Bubbles, Drops and Particles, Academic, New York, 1978, Kumar and Kuloor, Adv. Chem. Eng, 8, 255–368 (1970) and Wilkinson and Van Dierendonck, Chem. Eng Sci., 49, 1429–1438 (1994). Design methods for units operation in bubble columns and stirred vessels are

covered by Atika and Yoshida, Ind. Eng Chem. Process Des. Dev., 13, 84 (1974); Calderbank, The Chem. Eng. (London), CE209 (October, 1967); and Mixing, vol. II, Academic, New York, 1967, pp. 1–111; Fair, Chem. Eng, 74, 67 (July 3, 1967); Jordan, Chemical Process Dev., Interscience, New York, 1968, part 1, pp. 111–175; Mersmann, Ger. Chem. Eng, 1, 1 (1978); Resnick and Gal-Or, Adv. Chem. Eng., 7, 295–395 (1968); Valentin, Absorption in Gas-Liquid Dispersions, E. & F. N. Spon, London, 1967; Tatterson, Fluid Mixing and Gas Dispersion in Agitated Tanks, McGraw-Hill, 1991; and Deckwer and Schumpe, Chem. Eng. Sci., 48, 889–991 (1993). The influence of surface-active agents on bubbles and foams is summarized in selected passages from Schwartz and Perry, Surface Active Agents, vol. 1, Interscience, New York, 1949; and from Schwartz, Perry, and Berch, Surface Active Agents and Detergents, vol. 2, Interscience, New York, 1958. See also Elenkov, Theor. Found Chem. Eng., 1, 1, 117 (1967); and Rubel, Antifoaming and Defoaming Agents, Noyes Data Corp., Park Ridge, NJ, 1972. A review of foam stability also is given by de Vries, Meded, Rubber Sticht. Delft. No. 328, 1957. Foam-separation methodology is discussed by Aguoyo and Lemlich, Ind. Eng. Chem. Process Des. Dev., 13, 153 (1974); and Lemlich, Ind. Eng Chem., 60, 16 (1968). The following reviews of specific applications of gas-to-liquid dispersions are recommended: Industrial fermentations Aiba, Humphrey, and Millis, Biochemical Engineering, Academic, New York, 1965. Finn, Bacteriol. Rev., 18, 254 (1954). Oldshue, “Fermentation Mixing Scale-Up Techniques,” in Biotechnology and Bioengineering, vol. 8, 1966, pp. 3–24. Aerobic oxidation of wastes: Eckenfelder and McCabe, Advances in Biological Waste Treatment, Macmillan, New York, 1963. Eckenfelder and O’Connor, Biological Waste Treatment, Pergamon, New York, 1961. McCabe and Eckenfelder, Biological Treatment of Sewage and Industrial Wastes, vol. 1, Reinhold, New York, 1955. Proceedings of Industrial Waste Treatment Conference, Purdue University, annually. Zlokarnik, Adv. Biochem. Eng., 11, 158–180 (1979). Cellular elastomers: Fling, Natural Rubber Latex and Its Applications: The Preparation of Latex Foam Products, British Rubber Development Board, London, 1954. Gould, in Symposium on Application of Synthetic Rubbers, American Society for Testing and Materials, Philadelphia, 1944, pp. 90–103. Firefighting foams: Perri, in Bikerman, op. cit., Chap. 12. Ratzer, Ind. Eng. Chem., 48, 2013 (1956). Froth-flotation methods and equipment: Booth, in Bikerman, op. cit., Chap. 13. Gaudin, Flotation, McGraw-Hill, New York, 1957. Taggart, Handbook of Mineral Dressing, Wiley, New York, 1945, Sec. 12, pp. 52–81. Tatterson, Fluid Mixing and Gas Dispersion in Agitated Tanks, McGraw-Hill, New Y ork, 1991.

Objectives of Gas Dispersion The dispersion of gas as bubbles in a liquid or in a plastic mass is effected for one of the following purposes: (1) gas-liquid contacting (to promote absorption or stripping,

14-70

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

with or without chemical reaction), (2) agitation of the liquid phase, or (3) foam or froth production. Gas-in-liquid dispersions also may be produced or encountered inadvertently, sometimes undesirably. Gas-Liquid Contacting Usually this is accomplished with conventional columns or with spray absorbers (see preceding subsection “Liquid-in-Gas Dispersions”). For systems containing solids or tar likely to plug columns, absorptions accomplished by strongly exothermic reactions, or treatments involving a readily soluble gas or a condensable vapor, however, bubble columns or agitated vessels may be used to your advantage. Agitation Agitation by a stream of gas bubbles (often air) rising through a liquid is often employed when the extra expense of mechanical agitation is not justified. Gas spargers may be used for simple blending operations involving a liquid of low volatility or for applications where agitator shaft sealing is difficult. Foam Production This is important in froth-flotation separations; in the manufacture of cellular elastomers, plastics, and glass; and in certain special applications (e.g., food products, fire extinguishers). Unwanted foam can occur in process columns, in agitated vessels, and in reactors in which a gaseous product is formed; it must be avoided, destroyed, or controlled. Berkman and Egloff (Emulsions and Foams, Reinhold, New York, 1941, pp. 112–152) have mentioned that foam is produced only in systems possessing the proper combination of interfacial tension, viscosity, volatility, and concentration of solute or suspended solids. From the standpoint of gas comminution, foam production requires the creation of small bubbles in a liquid capable of sustaining foam. Theory of Bubble and Foam Formation A bubble is a globule of gas or vapor surrounded by a mass or thin film of liquid. By extension, globular voids in a solid are sometimes called bubbles. Foam is a group of bubbles separated from one another by thin films, the aggregation having a finite static life. Although nontechnical dictionaries do not distinguish between foam and froth, a technical distinction is often made. A highly concentrated dispersion of bubbles in a liquid is considered a froth even if its static life is substantially nil (i.e., it must be dynamically maintained). Thus, all foams are also froths, whereas the reverse is not true. The term lather implies a froth that is worked up on a solid surface by mechanical agitation; it is seldom used in technical discussions. The thin walls of bubbles comprising a foam are called laminae or lamellae. Bubbles in a liquid originate from one of three general sources: (1) They may be formed by desupersaturation of a solution of the gas or by the decomposition of a component in the liquid; (2) They may be introduced directly into the liquid by a bubbler or sparger or by mechanical entrainment; and (3) They may result from the disintegration of larger bubbles already in the liquid. Generation Spontaneous generation of gas bubbles within a homogeneous liquid is theoretically impossible (Bikerman, Foams: Theory and Industrial Applications, Reinhold, New York, 1953, p. 10). The appearance of a bubble requires a gas nucleus as a void in the liquid. The nucleus may be in the form of a small bubble or of a solid carrying adsorbed gas, examples of the latter being dust particles, boiling chips, and a solid wall. A void can result from cavitation, mechanically or acoustically induced. Blander and Katz [AIChE J., 21, 833 (1975)] have thoroughly reviewed bubble nucleation in liquids. Theory permits the approximation of the maximum size of a bubble that can adhere to a submerged horizontal surface if the contact angle between bubble and solid (angle formed by solid-liquid and liquid-gas interfaces) is known [Wark, J. Phys. Chem., 37, 623 (1933); Jakob, Mech. Eng., 58, 643 (1936)]. Because the bubbles that actually rise from a surface are always considerably smaller than those so calculated and inasmuch as the contact angle is seldom known, the theory is not directly useful. Formation at a Single Orifice The formation of bubbles at an orifice or capillary immersed in a liquid has been the subject of much study, both experimental and theoretical. Bikerman (op. cit., Secs. 3 to 7), Valentin (op. cit., Chap. 2), Jackson (op. cit.), Soo (op. cit., Chap. 3), Fair (op. cit.), Kumer et al. (op. cit.), Clift et al. (op. cit.) and Wilkinson and Van Dierendonck [Chem. Eng. Sci., 49, 1429 (1994)] have presented reviews and analyses of this subject. There are three regimes of bubble production (Silberman in Pro-

ceedings of the Fifth Midwestern Conference on Fluid Mechanics, Univ. of Michigan Press, Ann Arbor, 1957, pp. 263–284): (1) singlebubble, (2) intermediate, and (3) jet. Single-Bubble Regime Bubbles are produced one at a time, their size being determined primarily by the orifice diameter do, the interfacial tension of the gas-liquid film σ, the densities of the liquid ρL and gas ρG, and the gravitational acceleration g according to the relation 1/3 db 6σ (14-206) ᎏ = ᎏᎏ do do2 (ρL − ρG) where Db is the bubble diameter. The bubble size is independent of gas flow rate; the frequency, therefore, is directly proportional to the gas flow rate. Equation (14-206) leads to ρL − ρG f = Qg × ᎏ (14-207) πdoσ where f is the frequency of bubble formation and Q is the volumetric rate of gas flow in consistent units. Equations (14-206) and (14-207) result from a balance of bubble buoyancy against interfacial tension. They include no inertia or viscosity effects. At low bubbling rates (<1/s), these equations are quite satisfactory. Van Krevelen and Hoftijzer [Chem. Eng. Prog., 46, 29 (1950)], Guyer and Peterhaus [Helv. Chim. Acta, 26, 1099 (1943)] and Wilkinson (op. cit.) report good agreement with Eq. (14-185) for water, transformer oil, ether, and carbon tetrachloride for vertically oriented orifices with 0.004 < D < 0.95 cm. If the orifice diameter becomes too large, the bubble diameter will be smaller than the orifice diameter, as predicted by Eq. (14-206), and instability results; consequently, stable, stationary bubbles cannot be produced. For bubbles formed in water, the orifice diameter that permits bubbles of about its own size is calculated as 0.66 cm. Davidson and Amick [AIChE J., 2, 337 (1956)] confirmed this estimate in their observation that stable bubbles in water were formed at a 0.64-cm orifice but could not be formed at a 0.79-cm orifice. For very thin liquids, Eqs. (14-206) and (14-207) are expected to be valid up to a gas-flow Reynolds number of 200 (Valentin, op. cit., p. 8). For liquid viscosities up to 100 cP, Datta, Napier, and Newitt [Trans. Inst. Chem. Eng., 28, 14 (1950)] and Siems and Kauffman [Chem. Eng. Sci., 5, 127 (1956)] have shown that liquid viscosity has very little effect on the bubble volume, but Davidson and Schuler [Trans. Instn. Chem. Eng., 38, 144 (1960)] and Krishnamurthi et al. [Ind. Eng. Chem. Fundam., 7, 549 (1968)] have shown that bubble size increases considerably over that predicted by Eq. (14-206) for liquid viscosities above 1000 cP. In fact, Davidson et al. (op. cit.) found that their data agreed very well with a theoretical equation obtained by equating the buoyant force to drag based on Stokes’ law and the velocity of the bubble equator at break-off: νQ 3/4 6 4π 1/4 db = ᎏ 15 × ᎏ (14-208) ᎏ π 3 2g where ν is the liquid kinematic viscosity and Q is the gas volumetric flow rate. This equation is dimensionally consistent. The relative effect of liquid viscosity can be obtained by comparing the bubble diameters calculated from Eqs. (14-206) and (14-208). If liquid viscosity appears significant, one might want to use the long and tedious method developed by Krishnamurthi et al. (op. cit.) that considers both surface-tension forces and viscous-drag forces. Intermediate Regime This regime extends approximately from a Reynolds number of 200 to one of 2100. As the gas flow through a submerged orifice increases beyond the limit of the single-bubble regime, the frequency of bubble formation increases more slowly, and the bubbles begin to grow in size. Between the two regimes there may indeed be a range of gas rates over which the bubble size decreases with increasing rate, owing to the establishment of liquid currents that nip the bubbles off prematurely. The net result can be the occurrence of a minimum bubble diameter at some particular gas rate [Mater, U.S. Bur. Mines Bull. 260 (1927) and Bikerman, op. cit., p. 4]. At the upper portion of this region, the frequency becomes very nearly constant with respect to gas rate, and the bubble size correspondingly increases with gas rate. The bubble size is affected primarily by (1) ori-













PHASE DISPERSION fice diameter, (2) liquid-inertia effects, (3) liquid viscosity, (4) liquid density, and (5) the relationship between the constancy of gas flow and the constancy of pressure at the orifice. Kumar et al. have done extensive experimental and theoretical work reported in Ind. Eng. Chem. Fundam., 7, 549 (1968); Chem. Eng. Sci, 24, part 1, 731; part 2, 749; part 3, 1711 (1969) and summarized in Adv. Chem. Eng., 8, 255 (1970). They, along with other investigators—Swope [Can. J Chem. Eng., 44, 169 (1972)], Tsuge and Hibino [J. Chem. Eng. Japan, 11, 307 (1972)], Pinczewski [Chem. Eng. Sci., 36, 405 (1981)], Tsuge and Hibino [Int. Chem. Eng., 21, 66 (1981)], and Takahashi and Miyahara [ibid., p. 224]—have solved the equations resulting from a force balance on the forming bubble, taking into account buoyancy, surface tension, inertia, and viscous-drag forces for both conditions of constant flow through the orifice and constant pressure in the gas chamber. The design method is complex and tedious and involves the solution of algebraic and differential equations. Although Mersmann [Ger. Chem. Eng., 1, 1 (1978)] claims that the results of Kumar et al. (loc. cit.) well fit experimental data, Lanauze and Harn [Chem. Eng. Sci., 29, 1663 (1974)] claim differently: Further, it has been shown that the mathematical formulation of Kumar’s model, including the condition of detachment, cord not adequately describe the experimental situation—Kumar’s model has several fundamental weaknesses, the computational simplicity being achieved at the expense of physical reality.

In lieu of careful independent checks of predictive accuracy, the results of the comprehensive theoretical work will not be presented here. Simpler, more easily understood predictive methods, for certain important limiting cases, will be presented. As a check on the accuracy of these simpler methods, it will perhaps be prudent to calculate the bubble diameter from the graphical representation by Mersmann (loc. cit.) of the results of Kumar et al. (loc. cit.). For conditions approaching constant flow through the orifice, a relationship derived by equating the buoyant force to the inertia force of the liquid [Davidson et al., Trans. Instn. Chem. Engrs., 38, 335 (1960)] (dimensionally consistent), 6Q6/5 db = 1.378 × ᎏ (14-209) πg3/5 fits experimental data reasonably well. Surface tension and liquid viscosity tend to increase the bubble size—at a low Reynolds number. The effect of surface tension is greater for large orifice diameters. The magnitude of the diameter increase due to high liquid viscosity can be obtained from Eq. (14-208). For conditions approaching constant pressure at the orifice entrance, which probably simulates most industrial applications, there is no independently verified predictive method. For air at near atmospheric pressure sparged into relatively inviscid liquids (11 ∼ 100 cP), the correlation of Kumar et al. [Can. J. Chem. Eng., 54, 503 (1976)] fits experimental data well. Their correlation is presented here as Fig. 14-92.

14-71

Wilkinson et al. (op. cit.) make the following observation about the effect of gas density on bubble size: “The fact that the bubble size decreases slightly for higher gas densities can be explained on the basis of a force balance.” Jet Regime With further rate increases, turbulence occurs at the orifice, and the gas stream approaches the appearance of a continuous jet that breaks up 7.6 to 10.2 cm above the orifice. Actually, the stream consists of large, closely spaced, irregular bubbles with a rapid swirling motion. These bubbles disintegrate into a cloud of smaller ones of random size distribution between 0.025 cm or smaller and about 1.25 cm, with a mean size for air and water of about 0.4 cm (Leibson et al., loc. cit.). According to Wilkinson et al. (op. cit.), jetting begins when ρg doU o2 NWe,g = ᎏ ≤2 (14-210) σ There are many contradictory reports about the jet regime, and theory, although helpful (see, for example, Siberman, loc. cit.), is as yet unable to describe the phenomena observed. The correlation of Kumar et al. (Fig. 14-92) is recommended for air-liquid systems. Formation at Multiple Orifices At high velocities, coalescence of bubbles formed at individual orifices occurs; Helsby and Tuson [Research (London), 8, 270 (1955)], for example, observed the frequent coalescence of bubbles formed in pairs or in quartets at an orifice. Multiple orifices spaced by the order of magnitude of the orifice diameter increase the probability of coalescence, and when the magnitude is small (as in a sintered plate), there is invariably some. The broken lines of Fig. 14-92 presumably represent zones of increased coalescence and relatively less effective dispersion as the gas rate through porous-carbon tubes is increased. Savitskaya [Kolloidn. Zh., 13, 309 (1951)] found that the average bubble size formed at the surface of a porous plate was such as to maintain constancy of the product of bubble specific surface and interfacial tension as the latter was varied by addition of a surfactant. Konig et al. [Ger. Chem. Eng., 1, 199 (1978)] produced bubble sizes varying from 0.5 to 4 mm by the use of two porous-plate spargers and one perforated-plate sparger with superficial gas velocities from 1 to 8 cm/s. The small bubble sizes were stabilized by adding up to 0.5 percent of various alcohols to water. At high-flow rates through perforated plates such as those that occur in distillation columns, Calderbank and Rennie [Trans. Instn. Chem. Engrs., 40, T3 (1962)]; Porter et al. [ibid., 45, T265 (1967)]; Rennie and Evans [Br. Chem. Eng, 7, 498 (1962)]; and Valentin (op. cit., Chap. 3) have investigated and discussed the effect of the flow conditions through the multiple orifices on the froths and foams that occur above perforated plates. Entrainment and Mechanical Disintegration Gas can be entrained into a liquid by a solid or a stream of liquid falling from the gas phase into the liquid, by surface ripples or waves, and by the vertical swirl of a mass of agitated liquid about the axis of a rotating agita-

FIG. 14-92 Bubble-diameter correlation for air sparged into relatively inviscid liquids. Db = bubble diameter, D = orifice diameter, Vo = gas velocity through sparging orifice, P = fluid density, and µ = fluid viscosity. [From Can. J. Chem. Eng., 54, 503 (1976).]

14-72

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

tor. Small bubbles probably form near the surface of the liquid and are caught into the path of turbulent eddies, whose velocity exceeds the terminal velocity of the bubbles. The disintegration of a submerged mass of gas takes place by the turbulent tearing of smaller bubbles away from the exterior of the larger mass or by the influence of surface tension on the mass when it is attenuated by inertial or shear forces into a cylindrical or disk form. A fluid cylinder that is greater in length than in circumference is unstable and tends to break spontaneously into two or more spheres. These effects account for the action of fluid attrition and of an agitator in the disintegration of suspended gas. Quantitative correlations for gas entrainment by liquid jets and in agitated vessels will be given later. Foams Two excellent reviews (Shedlovsky, op. cit.; Lemlich, op. cit.) covering the literature pertinent to foams have been published. A foam is formed when bubbles rise to the surface of a liquid and persist for a while without coalescence with one another or without rupture into the vapor space. The formation of foam, then, consists simply of the formation, rise, and aggregation of bubbles in a liquid in which foam can exist. The life of foams varies over many magnitudes—from seconds to years—but in general is finite. Maintenance of a foam, therefore, is a dynamic phenomenon. Gravitational force favors the separation of gas from liquid in a disperse system, causing the bubbles to rise to the liquid surface and the liquid contained in the bubble walls to drain downward to the main body of the liquid. Interfacial tension favors the coalescence and ultimate disappearance of bubbles; indeed, it is the cause of bubble destruction upon the rupture of the laminae. The viscosity of the liquid in a film opposes the drainage of the film and its displacement by the approach of coalescing bubbles. The higher the viscosity, the slower will be the film-thinning process; furthermore, if viscosity increases as the film grows thinner, the process becomes self-retarding. The viscosity of films appears to be greater than that of the main body of the parent liquid in many cases. Sometimes this is a simple temperature effect, the film being cooler because of evaporation; sometimes it is a concentration effect, with dissolved or fine suspended solids migrating to the interface and producing classical or anomalous increases in viscosity; at yet other times, the effect seems to occur without explanation. If the liquid laminae of a foam system can be converted to impermeable solid membranes, the film viscosity can be regarded as having become infinite, and the resulting solid foam will be permanent. Likewise, if the laminae are composed of a gingham plastic or a thixotrope, the foam will be permanently stable for bubbles whose buoyancy does not permit exceeding the yield stress. For other nonnewtonian fluids, however, and for all newtonian ones, no matter how viscous, the viscosity can only delay but never prevent foam disappearance. The popular theory, held since the days of Plateau, that foam life is proportional to surface viscosity and inversely proportional to interfacial tension, is not correct, according to Bikerman (op. cit., p. 161), who points out that it is contradicted by experiment. The idea that foam films drain to a critical thickness at which they spontaneously burst is also rejected by Bikerman. Foam stability, rather, is keyed to the existence of a surface skin of low interfacial tension immediately overlying a solution bulk of higher tension, latent until it is exposed by rupture of the superficial layer [Maragoni, Nuovo Cimento, 2 (5–6), 239 (1871)]. Such a phenomenon of surface elasticity, resulting from concentration differences between bulk and surface of the liquid, accounts for the ability of bubbles to be penetrated by missiles without damage. It is conceivable that films below a certain thickness no longer carry any bulk of solution and hence have no capacity to close surface ruptures, thus becoming vulnerable to mechanical damage that will destroy them. The Maragoni phenomenon is consistent also with the observation that neither pure liquids nor saturated solutions will sustain a foam, since neither extreme will allow the necessary differences in concentration between surface and bulk of solution. The specific ability of certain finely divided, insoluble solids to stabilize foam has long been known [Berkman and Egloff, op. cit., p. 133; and Bikerman, op. cit., Chap. 11]. Bartsch [Kolloidchem. Beih, 20, 1 (1925)] found that the presence of fine galena greatly extended the life of air foam in aqueous isoamyl alcohol, and the finer the solids, the

greater the stability. Particles on the order of 50 µm length extended the life from 17 seconds to several hours. This behavior is consistent with theory, which indicates that a solid particle of medium contact angle with the liquid will prevent the coalescence of two bubbles with which it is in simultaneous contact. Quantitative observations of this phenomenon are scanty. Berkman and Egloff explain that some additives increase the flexibility or toughness of bubble walls, rather than their viscosity, to render them more durable. They cite as illustrations the addition of small quantities of soap to saponin solutions or of glycerin to soap solution to yield much more stable foam. The increased stability with ionic additives is probably due to electrostatic repulsion between charged, nearly parallel surfaces of the liquid film, which acts to retard draining and hence rupture. Characteristics of Dispersion Properties of Component Phases As discussed in the preceding subsection, dispersions of gases in liquids are affected by the viscosity of the liquid, the density of the liquid and of the gas, and the interfacial tension between the two phases. They also may be affected directly by the composition of the liquid phase. Both the formation of bubbles and their behavior during their lifetime are influenced by these quantities as well as by the mechanical aspects of their environment. Viscosity and density of the component phases can be measured with confidence by conventional methods, as can the interfacial tension between a pure liquid and a gas. The interfacial tension of a system involving a solution or micellar dispersion becomes less satisfactory, because the interfacial free energy depends on the concentration of solute at the interface. Dynamic methods and even some of the so-called static methods involve the creation of new surfaces. Since the establishment of equilibrium between this surface and the solute in the body of the solution requires a finite amount of time, the value measured will be in error if the measurement is made more rapidly than the solute can diffuse to the fresh surface. Eckenfelder and Barnhart (Am. Inst. Chem. Engrs., 42d national meeting, Repr. 30, Atlanta, 1960) found that measurements of the surface tension of sodium lauryl sulfate solutions by maximum bubble pressure were higher than those by DuNuoy tensiometer by 40 to 90 percent, the larger factor corresponding to a concentration of about 100 ppm, and the smaller to a concentration of 2500 ppm of sulfate. Even if the interfacial tension is measured accurately, there may be doubt about its applicability to the surface of bubbles being rapidly formed in a solution of a surface-active agent, for the bubble surface may not have time to become equilibrated with the solution. Coppock and Meiklejohn [Trans. Instn. Chem. Engrs., 29, 75 (1951)] reported that bubbles formed in the single-bubble regime at an orifice in a solution of a commercial detergent had a diameter larger than that calculated in terms of the measured surface tension of the solution [Eq. (14-206)]. The disparity is probably a reflection of unequilibrated bubble laminae. One concerned with the measurement of gas-liquid interfacial tension should consult the useful reviews of methods prepared by Harkins [in Chap. 9 of Weissberger, Techniques of Organic Chemstry, 2d ed., vol. 1, part 2, Interscience, New York, 1949), Schwartz and coauthors [Surface Acttve Agents, vol. 1, Interscience, New York, 1949, pp. 263–271; Surface Active Agents and Detergents, vol. 2, Interscience, New York, 1958, pp. 389–391, 417–418], and by Adamson [Physical Chemistry of Surfaces, Interscience, New York, 1960]. Dispersion Characteristics The chief characteristics of gas-inliquid dispersions, like those of liquid-in-gas suspensions, are heterogeneity and instability. The composition and structure of an unstable dispersion must be observed in the dynamic situation by looking at the mixture, with or without the aid of optical devices, or by photographing it, preferably in nominal steady state; photographs usually are required for quantitative treatment. Stable foams may be examined after the fact of their creation if they are sufficiently robust or if an immobilizing technique such as freezing is employed [Chang et al., Ind. Eng Chem., 48, 2035 (1956)]. The rate of rise of bubbles has been discussed in many papers, including two that present good reviews of the subject [Benfratello, Energ Elettr., 30, 80 (1953); Haberman and Morton, Report 802:

PHASE DISPERSION David W. Taylor Model Basin, Washington, September 1953; Jackson, loc. cit.; Valentin, op. cit., Chap. 2; Soo, op. cit., Chap. 3; Calderbank, loc. cit., p. CE220; and Levich, op. cit., Chap. 8). A comprehensive and apparently accurate predictive method has been published [Jamialahamadi et al., Trans ICE, 72, part A, 119–122 (1994)]. Small bubbles (below 0.2 mm in diameter) are essentially rigid spheres and rise at terminal velocities that place them clearly in the laminar-flow region; hence their rising velocity may be calculated from Stokes’ law. As bubble size increases to about 2 mm, the spherical shape is retained, and the Reynolds number is still sufficiently small (<10) that Stokes’ law should be nearly obeyed. As bubble size increases, two effects set in, however, that alter the velocity. At about NRe = 100, a wobble begins that can develop into a helical path if the bubbles are not liberated too closely to one another [Houghton, McLean, and Ritchie, Chem. Eng. Sci., 7, 40 (1957); and Houghton et al., ibid., p. 111]. Furthermore, for bubbles in the range of 1 mm and larger (until distortion becomes serious) internal circulation can set in [Garner and Hammerton, Chem. Eng. Sci., 3, (1954); and Haberman and Morton, loc. cit.], and according to theoretical analyses by Hadamard and Rybczynski and given by Levich (op. cit.), the drag coefficient for a low-viscosity dispersed phase and a highviscosity continuous phase will approach two-thirds of the drag coefficient for rigid spheres, namely CD = 16/NRe. The rise velocity of a circulating bubble or drop will thus be 1.5 times that of a rigid sphere. Redfield and Houghton [Chem. Eng. Sci., 20, 131 (1965)] have found that CO2 bubbles rising in pure water agree with the theoretical solution for a circulating drop below NRe = 1. Many investigators (see Valentin, op. cit.) have found that extremely small quantities of impurities can retard or stop this internal circulation. In this behavior may lie the explanation of the fact that the addition of long-chain fatty acids to water to produce a concentration of 1.5 × 10−4 molar markedly reduces the rate of rise of bubbles [Stuke, Naturwissenschaften, 39, 325 (1952)]. Above diameters of about 2 mm, bubbles begin to change to ellipsoids, and above 1 cm they become lens-shaped, according to Davies and Taylor [Proc. Roy. Soc. (London), A200, 379 (1950)]. The rising velocity in thin liquids for the size range 1 mm < DB < 20 mm has been reported as 20 to 30 cm/s by Haberman and Morton (op. cit.) and Davenport, Richardson, and Bradshaw [Chem. Eng. Sci., 22, 1221 (1967)]. Schwerdtieger [ibid., 23, 937 (1968)] even found the same for argon bubbles rising in mercury. Surface-active agents have no effect on the rise velocity of bubbles larger than 4 mm in thin liquids (Davenport et al., loc. cit.). Above a Reynolds number of the order of magnitude of 1000, bubbles assume a helmet shape, with a flat bottom (Eckenfelder and Barnhart, loc. cit.; and Leibson et al., loc. cit.). After bubbles become large enough to depart from Stokes’ law at their terminal velocity, behavior is generally complicated and erratic, and the reported data scatter considerably. The rise can be slowed, furthermore, by a wall effect if the diameter of the container is not greater than 10 times the diameter of the bubbles, as shown by Uno and Kintner [AIChE J., 2, 420 (1956); and Collins, J. Fluid Mech., 28(1), 97 (1967)]. Work has been done to predict the rise velocity of large bubbles [Rippin and Davidson, Chem. Eng. Sci., 22, 217 (1967); Grace and Harrison, ibid., 1337; Mendelson, AIChE J., 13, 250 (1967); Cole, ibid., Lehrer, J. Chem. Eng. Japan, 9, 237; (1976) and Lehrer, AIChE J., 26, 170 (1980)]. The works of Lehrer present correlations that accurately predict rise velocities for a wide range of system properties. An excellent review of the technical literature concerning the rise of single bubbles and drops has been published by Clift, Grace, and Weber (Bubbles, Drops and Particles, Academic, New York, 1978). Mendelson has used a wave theory to predict the terminal velocity, and Cole has checked the theory with additional data. The other authors listed solved some simplified form of the Navier-Stokes equations. Jamialahmadi et al., loc. cit., have developed a single equation predictive method for bubble rise velocity, which covers the entire range of bubble diameters. When bubbles are produced in clouds, as by a porous disperser, their behavior during rising is further complicated by interaction among themselves. In addition to the tendency for small bubbles to coalesce and large ones to disintegrate, there are two additional

14-73

opposing influences on the rate of rise of bubbles of any particular size: (1) A “chimney effect” can develop in which a massive current upward appears at the axis of the bubble stream, leading to increased net bubble velocity; and (2) the proximity of the bubbles to one another can result in a hindered-settling condition, leading to reduced average bubble velocity. Figure 14-93 shows the data of Houghton et al. (op. cit.) for clouds of bubbles compared with their single-bubble data for pure water and seawater and of Peebles and Garber [Chem. Eng. Progr., 49, 88 (1953)] for acetic acid and ethyl acetate. The bubble clouds were produced with a sintered-glass plate of mean pore size (inferred from air wet-permeability data) of 81 µm. The difference between the curves for pure water and seawater again illustrates the significance of small concentrations of solute with respect to bubble behavior. In commercial bubble columns and agitated vessels coalescence and breakup are so rapid and violent that the rise velocity of a single bubble is meaningless. The average rise velocity can, however, be readily calculated from holdup correlations that will be given later. The quantitative examination of bubble systems is aided by the use of proper illumination and photography. The formation of bubbles at single sources often is sufficiently periodic to be stopped by stroboscopic light. Clouds of rising bubbles are more difficult to assess and require careful technique. Satisfactory photographic methods have been developed by Vermenlen, Williams, and Langlois [Chem. Eng. Progr., 51, 85 (1955)] and by Calderbank [Trans. Instn. Chem. Engrs., 36, 443 (1958)] and are described by these authors. Calderbank’s technique resulted in particularly precise measurements that permitted a good estimation of the surface area of the dispersed bubbles. Methods of Gas Dispersion The problem of dispersing a gas in a liquid may be attacked in several ways: (1) The gas bubbles of the desired size or which grow to the desired size may be introduced directly into the liquid; (2) a volatile liquid may be vaporized by either decreasing the system pressure or increasing its temperature; (3) a chemical reaction may produce a gas; or (4) a massive bubble or stream of gas is disintegrated by fluid shear and/or turbulence in the liquid. Spargers: Simple Bubblers The simplest method of dispersing gas in a liquid contained in a tank is to introduce the gas through an open-end standpipe, a horizontal perforated pipe, or a perforated plate at the bottom of the tank. At ordinary gassing rates (corresponding to the jet regime), relatively large bubbles will be produced regardless of the size of the orifices.

FIG. 14-93 Velocity of rising bubbles, singly and in clouds. To convert feet per second to meters per second, multiply by 0.305. [From Chem. Eng. Sci., 7, 48 (1957).]

14-74

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

Perforated-pipe or -plate spargers usually have orifices 3 to 12 mm in diameter. Effective design methods to minimize maldistribution are presented in the fifth edition of this handbook, p. 5–47, 1973, and by Knaebel [Chem. Eng., 116 (Mar. 9, 1981)]. For turbulent flow conditions into the sparger, the following relationship will allow design of a perforated-pipe sparger for a given degree of maldistribution provided Nh > 5 and length/diameter < 300. 1/4 dh (14-211) dp = 0.95(NhCv)1/2 × ᎏ ∆Uh /Uh where dp = pipe diameter, dh = sparging hole diameter, Nh = number of holes in sparger, Cv = orifice coefficient for sparger hole (see Chemical Engineers’ Handbook, 5th ed., pp. 5–13, 5–34), Uh = average velocity through sparger holes, ∆Uh = difference between maximum and minimum velocities through sparger holes, and ∆Uh /Uh = fractional maldistribution of flow through sparger holes. Simple spargers are used as agitators for large tanks, principally in the cement and oil industries. Kauffman [Chem. Metall. Eng., 37, 178–180 (1930)] reported the following air rates for various degrees of agitation in a tank containing 2.7 m (9 ft) of liquid:






Degree of agitation

Air rate, m3/(m2 tank cross section, min)

Moderate Complete Violent

0.0033 0.0066 0.016

For a liquid depth of 0.9 m (3 ft), Kauffman recommended that the listed rates be doubled. An air lift consisting of a sparger jetting into a draft tube with ports discharging at several heights has been recommended by Heiser [Chem. Eng., 55(1), 135 (1948)] for maintaining agitation in a heavy, coarse slurry, the level of which varies widely. The design is illustrated in Fig. 14-94. The ability of a sparger to blend miscible liquids might be described in terms of a fictitious diffusivity. Siemes did so, reporting that the agitation produced by a stream of bubbles rising in a tube with a superficial velocity of about 8.2 cm/s corresponded to an apparent diffusion coefficient as large as 75 cm2/s [Chem. Ing. Tech., 29, 727 (1957)]. The blending rate thus is several orders of magnitude higher than it would be by natural diffusive action. These results are

typical of subsequent investigations on back mixing, which will be discussed in more detail later. Lehrer [Ind. Eng. Chem. Process Des. Dev., 7, 226 (1968)] conducted liquid-blending tests with air sparging in a 0.61-m-diameter by 0.61-m-tall vessel and found that an air volume equal to about onehalf of the vessel volume gave thorough blending of inviscid liquids of equal viscosities. Using an analogy to mechanically agitated vessels in which equal tank turnovers give equal blend times, one would expect this criterion to be applicable to other vessel sizes. Liquids of unequal density would probably require somewhat more air. Open-end pipes, perforated plates, and ring- or cross-style perforated-pipe spargers are used without mechanical agitation to promote mass transfer, as in chlorinators and biological sewage treatment. In the “quiescent regime” (superficial gas velocity less than 4.57 to 6.1 cm/s [0.15 to 0.2 ft/s]) the previously mentioned spargers are usually operated at orifice Reynolds numbers in excess of 6000 in order to get small bubbles so as to increase the interfacial area and thus increase mass transfer. In the “turbulent regime” (superficial gas velocity greater than 4.57 to 6.1 cm/s), sparger design is not critical because a balance between coalescence and breakup is established very quickly according to Towell et al. [AIChE Symp. Ser. No. 10, 97 (1965)]. However, a reasonably uniform orifice distribution over the column cross section is desirable, and according to Fair [Chem. Eng., 74, 67 ( July 3, 1967); 207 ( July 17, 1967)] the orifice velocity should be less than 75 to 90 m/s. Porous Septa In the quiescent regime porous plates, tubes, disks, or other shapes that are made by bonding or sintering together carefully sized particles of carbon, ceramic, polymer, or metal are frequently used for gas dispersion, particularly in foam fractionators. The resulting septa may be used as spargers to produce much smaller bubbles than will result from a simple bubbler. Figure 14-95 shows a com-

Comparison of bubbles from a porous septum and from a perforated-pipe sparger. Air in water at 70°F. Grade 25 porous-carbon diffuser operating under a pressure differential of 13.7 in of water. FIG. 14-95a

FIG. 14-94 Slotted air lift for agitation of a variable-level charge. [From Chem. Eng., 55(1), 135 (1948).]

PHASE DISPERSION TABLE 14-16

14-75

Characteristics of Porous Septa Air-permeability data

Grade

Avg. % porosity

Avg. pore diam.

Diaphragm thickness, in

Pressure differential, in water

Air flow, cu ft/ (sq ft)(min)

Alundum porous alumina* P2220 P2120 P260 P236 P216

36 35 34

60 45 25

48 48 48

Extra fine Fine Medium fine Medium Medium coarse Coarse Extra coarse

26.0 28.8 31.1 33.7 33.8 34.5 36.5

25 60 164 240 720

1 1 1 1 1

2 2 2 2 2

0.35 2 15 40 110

National porous carbon† 33 58 120

1 1 1

2 2 2

2 13

Filtros porous silica‡ 55 110 130 150 200 250 300

1.5 1.5 1.5 1.5 1.5 1.5 1.5

2 2 2 2 2 2 2

1–3 4–8 9–12 13–20 21–30 31–59 60–100

Porous plastic§ Teflon Kel-F

9 15

0.125 0.125

1.38 1.38

5 13

Micro Metallic porous stainless steel§,¶ H G F E D C Comparison of bubbles from a porous septum and from a perforated-pipe sparger. Air in water at 70°F. Karbate pipe perforated with 1/16-in holes on 1-in centers. (National Carbon Co.) To convert inches to centimeters, multiply by 2.54; °C = 5⁄ 9 (°F − 32). FIG. 14-95b

parison of the bubbles emitted by a perforated-pipe sparger [0.16-cm orifices] and a porous carbon septum (120 µm pores). The gas flux through a porous septum is limited on the lower side by the requirement that, for good performance, the whole sparger surface should bubble more or less uniformly, and on the higher side by the onset of serious coalescence at the surface of the septum, resulting in poor dispersion. In the practical range of fluxes, the size of the bubbles produced depends on both the size of pores in the septum and the pressure drop imposed across it, being a direct function of both. Table 14-16 lists typical grades of porous carbon, silica, alumina, stainless steel (type 316), and polymers commercially available. Porous media are also manufactured from porcelain, glass, silicon carbide, and a number of metals: Monel, Inconel, nickel, bronze, Hastelloy C, Stellite L-605, gold, platinum, and many types of stainless steel. The air permeabilities of Table 14-16 indicate the relative flow resistances of the various grades to homogeneous fluid but may not be used in designing a disperser for submerged operation, for the resistance of a septum to the flow of gas increases when it is wet. The air permeabilities for water-submerged porous carbon of some of the grades listed in the table are shown in Fig. 14-96. The data were determined with septa 0.625 inches thick in water at 70°F. Comparable wet-permeability data for 1-in Alundum plates of two grades of fineness are given in Table 14-17. The gas rate at which coalescence begins to reduce the effectiveness of dispersion appears to depend not only on the pore size and pore structure of the dispersing medium but also on the liquid properties, liquid depth, agitation, and other features of the sparging environment; coalescence is strongly dependent on the concentration of

45 50 50 50 50 55

5 10 20 35 65 165

0.125 0.125 0.125 0.125 0.125 0.125

1.38 1.38 1.38 1.38 1.38 27.7

1.8 3 5 18 60 990

*Data by courtesy of Norton Co., Worcester, Mass. A number of other grades between the extremes listed are available. †Data by courtesy of National Carbon Co., Cleveland, Ohio. ‡Data by courtesy of Filtros Inc., East Rochester, N.Y. §Data by courtesy of Pall Corp., Glen Cove, N.Y. ¶Similar septa made from other metals are available.

surfactants capable of forming an electrical double layer and thus produce ionic bubbles, long-chain alcohols in water being excellent examples. For porous-carbon media, the manufacturer suggests that the best dispersion performance will result if the broken-line regions of Fig. 14-96 are avoided. For porous stainless-steel spargers, which

FIG. 14-96 Pressure drop across porous-carbon diffusers submerged in water at 70°F. To convert feet per minute to meters per second, multiply by 0.0051; to convert inches to millimeters, multiply by 25.4; °C = 5⁄ 9 (°F − 32). (National Carbon Co.)

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GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

TABLE 14-17 1 in Thick*

Wet Permeability of Alundum Porous Plates

Dry permeability at 2 in of water differential, cu ft/(min)(sq ft)

Pressure differential across wet plate, in of water

Air flow through wet plate, cu ft/(min)(sq ft)

20.67 21.77 22.86 23.90 4.02 4.14 4.22 4.27 4.30

2.0 3.0 4.0 5.0 1.0 2.0 3.0 4.0 5.0

4.3

55.0

*Data by courtesy of Norton Company, Worcester, Mass. To convert inches to centimeters, multiply by 2.54; to convert feet per minute to meters per second, multiply by 0.0051.

extend to a lower pore size than carbon, Micro Metallic Division, Pall Corp., recommends (Release 120A, 1959) a working limit of 8 ft/min (0.044 m/s) to avoid serious coalescence. This agrees with the data reported by Konig et al. (loc. cit.), in which 0.08 m/s was used and bubbles as small as 1 mm were produced from a 5-µm porous sparger. Slabs of porous material are installed by grouting or welding together to form a diaphragm, usually horizontal. Tubes are prone to produce coalesced gas at rates high enough to cause bubbling from their lower faces, but they have the advantage of being demountable for cleaning or replacement (U.S. Patent 2,328,655). Roe [Sewage Works J., 18, 878 (1945)] claimed that silicon carbide tubes are superior to horizontal plates, principally because of the wiping action of the liquid circulating past the tube. He reported respective maximum capacities of 2.5 and 1.5 cm2/s of gas/cm2 of sparger for a horizontal tube and a horizontal plate of the same material (unspecified grade). Mounting a flat-plate porous sparger vertically instead of horizontally seriously reduces the effectiveness of the sparger for three reasons: (1) The gas is distributed over a reduced cross section; (2) at normal rates, the lower portion of the sparger may not operate because of difference in hydrostatic head; and (3) there is a marked tendency for bubbles to coalesce along the sparger surface. Bone (M.S. thesis in chemical engineering, University of Kansas, 1948) found that the oxygen sulfite solution coefficient for a 3.2- by 10-cm rectangular porous carbon sparger was 26 to 41 percent lower for vertical than for horizontal operation of the sparger, the greatest reduction occurring when the long dimension was vertical. Precipitation and Generation Methods For a thorough understanding of the phenomena involved, bubble nucleation should be considered. A discussion of nucleation phenomena is beyond the scope of this Handbook; however, excellent coverages are presented by Blander and Katz. Boerma and Lankester [Chem. Eng. Sci., 23, 799 (1968)] have measured the surface aeration of a six-bladed disk-type turbine (NOTE: A well-designed pitched-blade turbine will give equal or better perfor-

FIG. 14-98

The Flotator dissolved-air flotation thickener. (Process Engineers, Inc., a division of Etmco Corp., now Envirotech Corporation.) FIG. 14-97

mance). In a fully baffled vessel, the optimum depth to obtain maximum gas dispersion was 15 percent of the liquid depth. In a vessel with baffles extending only halfway to the liquid surface the optimum impeller submergence increased with agitator speed because of the vortex formed. At optimum depth, the following correlation is recommended for larger vessels: impeller speed, rpm 2 5 impeller diameter, cm 4.5 Q (m3/s) = 0.3 ᎏᎏᎏ ᎏᎏᎏ 500 25.4









(14-214) Gas dispersion through the free surface by mechanical aerators is commonplace in aerobic waste-treatment lagoons. Surface aerators are generally of three types: (1) large-diameter flow-speed turbines operating just below the free surface of the liquid, often pontoon-mounted; (2) small-diameter high-speed (normally motor-speed) propellers operating in draft tubes, the units of which are always pontoonmounted; and (3) hollow-tube turbines (Fig. 14-101). An example of the turbine type is illustrated in Fig. 14-102 and the propeller type is illustrated in Fig. 14-103. There are several other styles of the turbine type; for instance, Mixing Equipment Co., Inc., uses an unshrouded 45° axial-flow turbine [see Dykman and Michel, Chem. Eng., 117 (Mar. 10, 1969)], and Infilco makes a unit that has a large-diameter vaned disk operating just below the free surface with a smallerdiameter submerged-disk turbine for additional solids suspension. Equipment Selection Ideally, selection of equipment to produce a gas-in-liquid dispersion should be made on the basis of a complete economic analysis. The design engineer and especially the pilot-plant engineer seldom have sufficient information or time to do

Aeration ejector. (Penberthy, a division of Houdaille Industries, Inc.)

PHASE DISPERSION

FIG. 14-99

Impingement aerator.

a complete economic analysis. In the following discussion, some guidelines are given as to what equipment might be feasible and what equipment might prove most economical. For producing foam for foam-separation processes, perforatedplate or porous-plate spargers are normally used. Mechanical agitators are often not effective in the light foams needed in foam fractionation. Dissolved-air flotation, based on the release of a pressurized flow in which oxygen was dissolved, has been shown to be effective some times for particulate removal when sparged air failed because the bubbles formed upon precipitation are smaller—down to 80 µm— than bubbles possible with sparging, typically 1000 µm [Grieves and Ettelt, AIChE J., 13, 1167 (1967)]. Mechanically agitated surface aerators such as the Wemco-Fagergren flotation unit are used extensively for ore flotation. To produce foam in batch processes, mechanical agitators are used almost exclusively. The gas can either be introduced through the free surface by the entraining action of the impeller or alternatively sparged beneath the impeller. In such batch operation, the liquid

FIG. 14-100

Wire whip.

Installation and dimensions of a tube stirrer: h/d = 1; H/D ≈ 1; D/δ = 10; A = 1.5 d w2 ; D/dN = 10; d/dN = 3; d/dri = 7.5; d/dra = 6. [Zlokarnik, Ullman’s Encyclopedia of Industrial Chemistry, Sec. 25, VCH, Weinheim, Germany, 1988.] FIG. 14-101

FIG. 14-102

The Cyclox surface aerator. (Cleveland Mixer Co.)

14-77

14-78

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

FIG. 14-103

Propeller-type surface aerator. (Ashbrook-Simon-Hartley Corp.)

level gradually rises as the foam is generated; thus, squatly impellers such as turbines are rapidly covered with foam and must almost always be sparged from below. Tall impellers such as the previously mentioned wire whips are especially well suited to entrain gas from the vapor space. For a new application, generally some experimentation with different impellers is necessary in order to get the desired fine final bubble size without getting frothing over initially. For producing foams continually, an aspirating venturi nozzle and restrictions in pipes such as baffles and metal gauzes are generally most economical. For gas absorption, the equipment possibilities are generally packed columns; plate distillation towers, possibly with mechanical agitation on every plate; deep-bed contactors (bubble columns or sparged lagoons); and mechanically agitated vessels or lagoons. Packed towers and plate distillation columns are discussed elsewhere. Generally these devices are used when a relatively large number of stages (more than two or three) is required to achieve the desired result practically. The volumetric mass-transfer coefficients and heights of transfer units for bubble columns and packed towers have been compared for absorption of CO2 into water by Houghton et al. [Chem. Eng. Sci., 7, 26 (1957)]. The bubble column will tolerate much higher vapor velocities, and in the overlapping region (superficial gas velocities of 0.9 to 1.8 cm/s), the bubble column has about three times higher masstransfer coefficient and about 3 times greater height of transfer unit. The liquid in a bubble column is, for practical purposes, quite well mixed; thus, chemical reactions and component separations requiring significant plug flow of the liquid cannot be carried out with bubble columns. Bubble columns and agitated vessels are the ideal equipment for processes in which the fraction of gas absorbed need not be great, possibly the gas can be recycled, and the liquid can or should be well mixed. The gas phase in bubble columns is not nearly so well back-mixed as the liquid, and often plug flow of the gas is a logical assumption, but in agitated vessels the gas phase is also well mixed. The choice of a bubble column or an agitated vessel depends primarily on the solubility of the gas in the liquid, the corrosiveness of the liquid (often a gas compressor can be made of inexpensive material, whereas a mechanical agitator may have to be made of exotic, expensive materials), and the rate of chemical reaction as compared with the mass-transfer rate. Bubble columns and agitated vessels are seldom used for gas absorption except in chemical reactors. As a general rule,

if the overall reaction rate is five times greater than the mass-transfer rate in a simple bubble column, a mechanical agitator will be most economical unless the mechanical agitator would have to be made from considerably more expensive material than the gas compressor. In bubble columns and simply sparged lagoons, selecting the sparger is a very important consideration. In the turbulent regime (superficial gas velocity greater than 4.6 to 6 cm/s), inexpensive perforated-pipe spargers should be used. Often the holes must be placed on the pipe bottom in order to make the sparger free-draining during operation. In the quiescent regime, porous septa will often give considerably higher overall mass-transfer coefficients than perforated plates or pipes because of the formation of tiny bubbles that do not coalesce. Chain and coworkers (First International Symposium on Chemical Microbiology, World Health Organization, Monograph Ser. 10, Geneva, 1952) claimed that porous disks are about twice as effective as open-pipe and ring spargers for the air oxidation of sodium sulfite. Eckenfelder [Chem. Eng. Progr., 52(7), 290 (1956)] has compared the oxygen-transfer capabilities of various devices on the basis of the operating power required to absorb a given quantity of O2. The installed cost of the various pieces of equipment probably would not vary sufficiently to warrant being including in an economic analysis. Surface mechanical aerators are not included in this comparison. Of the units compared, it appears that porous tubes give the most efficient power usage. Kalinske (Adv. Biol. Waste Treatment, 1963, p. 157) has compared submerged sparged aerators with mechanical surface aerators. He has summarized this comparison in Water Sewage Works, 33 (January 1968). He indicates that surface aerators are significantly more efficient than subsurface aeration, both for oxygen absorption and for gas-stripping operations. Zlokarnik and Mann (paper at Mixing Conf., Rindge, New Hampshire, August 1975) have found the opposite of Kalinske, i.e., subsurface diffusers, subsurface sparged turbines, and surface aerators compare approximately as 4:2:1 respectively in terms of O2 transfer efficiency; however, Zlokarnik [Adv. Biochem. Eng., 11, 157 (1979)] later indicates that the scale-up correlation used earlier might be somewhat inaccurate. When all available information is considered, it appears that with near-optimum design any of the aeration systems (diffusers, submerged turbines, or surface impellers) should give a transfer efficiency of at least 2.25 kg O2 /kWh. Thus, the final selection should probably be made primarily on the basis of operational reliability, maintenance, and capital costs.

PHASE DISPERSION Mass Transfer Mass transfer in plate and packed gas-liquid contactors has been covered earlier in this subsection. Attention here will be limited to deep-bed contactors (bubble columns and agitated vessels). Theory underlying mass transfer between phases is discussed in Sec. 5 of this handbook. To design deep-bed contactors for mass-transfer operations, one must have, in general, predictive methods for the following design parameters: • Flooding (for both columns and agitator impellers) • Holdup of gas phase • Agitator power requirements • Gas-phase and liquid-phase mass-transfer coefficients interfacial area • Interfacial resistance • Mean driving force for transfer In most cases, available methods are incomplete or unreliable, and some supporting experimental work is necessary. The methods given here should allow theoretical feasibility studies, help minimize experimentation, and permit a measure of optimization in final design. Flooding of Agitator Impellers A review of impeller flooding has been done by Sensel et al. [AIChE Symp. Series No. 283, 89 (1993)] and they have offered the following flooding correlation for a six-bladed disk-type turbine. Q for (ND)T −0.4 ≤ 1.6 (14-215a) ᎏ3 = 0.0675(ND)T −0.4 ND Q (14-215b) ᎏ3 = 0.0231[(ND)T −0.4]3.75 for (ND)T −0.4 ≥ 1.6 ND 3 Where Q = volumetric flow of gas, ft /s; N = impeller speed, rev/s; D = impeller diameter, ft; and T = tank diameter, ft. Gassed Impeller Power Sensel et al. (op. cit.) have developed the following correlation for six-bladed disk impellers.




24.54Q Pg ᎏ = 1 − (−0.000715 µL + 0.723)tanh ᎏ Po ND3

ND 
ᎏ g  2

0.25

(14-216)

where Pg = gassed power, Po = ungassed power, g = gravitational acceleration, and µL = liquid viscosity. Gas Holdup in Agitated Vessels Sensel et al. (op. cit.) have also developed the following correlation for six-bladed disk-type impellers: ND ND ρ ᎏ

ᎏ g 
µ 

Q ε = 0.105 ᎏ3 ND

2

0.5

2

L

0.1

(14-217)

L

Gas-Phase Mass-Transfer Coefficient This term is quite high in deep-bed contactors, normally leading to negligible gas-phase resistance. Interfacial Area This consideration in agitated vessels has been reviewed and summarized by Tatterson (op. cit.). Predictive methods for interfacial area are not presented here because correlations are given for the overall volumetric mass transfer coefficient liquid phase controlling mass transfer. Overall Mass-Transfer Coefficient Tatterson (op. cit.) and Zlokarnik (op. cit.) have summarized the literature covering overall mass-transfer coefficients. There is much scatter in the experimental data because the presence of surface-active agents and electrolytes have a significant effect on the mass transfer. The correlation of Van’t Riet [Ind. Eng Chem. Process Des. Dev., 18(3), 357 (1979)] is recommended: P 0.4 kLa = 0.026 ᎏ Us0.5 for water (14-218) V

14-79

mechanically aerated waste-treatment lagoons, overall oxygen transfer has been found to be from 0.5 to 3 times that for pure water from tests with typical sewage streams (Eckenfelder et al., op. cit., p. 105). One cannot quantitatively predict the effect of the various interfacial phenomena; thus, these phenomena will not be covered in detail here. The following literature gives a good general review of the effects of interfacial phenomena on mass transfer: Goodridge and Robb, Ind. Eng. Chem. Fund., 4, 49 (1965); Calderbank, Chem. Eng. (London), CE 205 (1967); Gal-Or et al., Ind. Eng. Chem., 61(2), 22 (1969); Kintner, Adv. Chem. Eng., 4 (1963); Resnick and Gal-Or, op. cit., p. 295; Valentin, loc. cit.; and Elenkov, loc. cit., and Ind. Eng. Chem. Ann. Rev. Mass Transfer, 60(1), 67 (1968); 60(12), 53 (1968); 62(2), 41 (1970). In the following outline, the effects of the various interfacial phenomena on the factors that influence overall mass transfer are given. Possible effects of interfacial phenomena are tabulated below: 1. Effect on continuous-phase mass-transfer coefficient a. Impurities concentrate at interface. Bubble motion produces circumferential surface-tension gradients that act to retard circulation and vibration, thereby decreasing the mass-transfer coefficient. b. Large concentration gradients and large heat effects (very soluble gases) can cause interfacial turbulence (the Marangoni effect), which increases the mass-transfer coefficient. 2. Effect on interfacial area a. Impurities will lower static surface tension and give smaller bubbles. b. Surfactants can electrically charge the bubble surface (produce ionic bubbles) and retard coalescence (soap stabilization of an oil-water emulsion is an excellent example of this phenomenon), thereby increasing the interfacial area. c. Large concentration gradients and large heat effects can cause bubble breakup. 3. Effect on mean mass-transfer driving force a. Relatively insoluble impurities concentrate at the interface, giving an interfacial resistance. This phenomenon has been used in retarding evaporation from water reservoirs. b. The axial concentration variation can be changed by changes in coalescence. The mean driving force for mass transfer is therefore changed. Gas Holdup (␧) in Bubble Columns With coalescing systems, holdup may be estimated from a correlation by Hughmark [Ind. Eng Chem. Process Des. Dev., 6, 218–220 (1967)] reproduced here as Fig. (14-104). For noncoalescing systems, with considerably smaller bub-




and




P kLa = 0.002 ᎏ V

0.7

Us0.2

for ionic mixtures

(14-219)

where P/V = power/volume, W/m3; Us = superficial gas velocity, m/s; kLa = volumetric mass-transfer coefficient, s−1 Interfacial Phenomena These can significantly affect overall mass transfer. In fermentation reactors, small quantities of surfaceactive agents (especially antifoaming agents) can drastically reduce overall oxygen transfer (Aiba et al., op. cit., pp. 153, 154), and in aerobic

FIG. 14-104

218 (1967).]

Gas holdup correlation. [Ind. Eng. Chem. Process Des. Dev., 6,

14-80

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

bles, ε can be as great as 0.6 at Usg = 0.05 m/s, according to Mersmann [Ger. Chem. Eng., 1, 1 (1978)]. It is often helpful to use the relationship between ε and superficial gas velocity (Usg) and the rise velocity of a gas bubble relative to the liquid velocity (Ur + UL, with UL defined as positive upward): Usg ε=ᎏ (14-220) Ur + UL Rise velocities of bubbles through liquids have been discussed previously. For a better understanding of the interactions between parameters, it is often helpful to calculate the effective bubble rise velocity Ur from measured valves of ε; for example, the data of Mersmann (loc. cit.) indicated ε = 0.6 for Usg = 0.05 m/s, giving Ur = 0.083 m/s, which agrees with the data reported in Fig. 14-43 for the rise velocity of bubble clouds. The rise velocity of single bubbles, for db ∼ 2 mm, is about 0.3 m/s, for liquids with viscosities not too different from water. Using this value in Eq. (14-220) and comparing with Fig. 14104, one finds that at low values of Usg, the rise velocity of the bubbles is less than the rise velocity of a single bubble, especially for smalldiameter tubes, but that the opposite occurs for large values of Usg. More recent literature regarding generalized correlational efforts for gas holdup is adequately reviewed by Tsuchiya and Nakanishi [Chem. Eng Sci., 47(13/14), 3347 (1992)] and Sotelo et al. [Int. Chem. Eng., 34(1), 82–90 (1994)]. Sotelo et al. (op. cit.) have developed a dimensionless correlation for gas holdup that includes the effect of gas and liquid viscosity and density, interfacial tension, and diffuser pore diameter. For systems that deviate significantly from the waterlike liquids for which Fig. 14-104 is applicable, their correlation (the fourth numbered equation in the paper) should be used to obtain a more accurate estimate of gas holdup. Mersmann (op. cit.) and Deckwer et al. (op. cit.) should also be consulted. Liquid-phase mass-transfer coefficients in bubble columns have been reviewed by Calderbank (“Mixing,” loc. cit.), Fair (Chem. Eng., loc. cit.), Mersmann [Ger. Chem. Eng. 1, 1 (1978), Int. Chem. Eng., 32(3) 397–405 (1991)], Deckwer et al. [Can. J. Chem. Eng, 58, 190 (1980)], Hikita et al. [Chem. Eng. J., 22, 61 (1981)] and Deckwer and Schumpe [Chem. Eng. Sci., 48(5), 889–911 (1993)]. The correlation of Ozturk, Schumpe, and Deckwer [AIChE J., 33, 1473–1480 (1987)] is recommended. Deckwer et al. (op. cit.) have documented the case for using the correlation: Ozturk et al. (1987) developed a new correlation on the basis of a modification of the Akita-Yoshida correlation suggested by Nakanoh and Yoshida (1980). In addition, the bubble diameter db rather than the column diameter was used as the characteristic length as the column diameter has little influence on kLa. The value of db was assumed to be approximately constant (db = 0.003 m). The correlation was obtained by nonlinear regression is as follows: kLa dB

µL

gρLd B2

gρL2 dB

= 0.62 ᎏ 
ᎏ 
ᎏ 
ᎏ µ σ D  (ρ D ) L

L

0.5

0.33

0.29

2 L

L

UG × ᎏ (gdB)0.5



ρG


ᎏ ρ  0.68

0.04

(14-221)

L

where kLa = overall mass-transfer coefficient, dB = bubble diameter = 0.003 m, DL = diffusivity of gas in liquid, ρ = density, µ = viscosity, σ = interfacial tension, g = gravitational acceleration.

As mentioned earlier, surfactants and ionic solutions significantly affect mass transfer. Normally, surface affects act to retard coalescence and thus increase the mass transfer. For example, Hikata et al. [Chem. Eng. J., 22, 61–69 (1981)] have studied the effect of KCl on mass transfer in water. As KCI concentration increased, the mass transfer increased up to about 35 percent at an ionic strength of 6 gm/l. Other investigators have found similar increases for liquid mixtures. Axial Dispersion Backmixing in bubble columns has been extensively studied. An excellent review article by Shah et al. [AIChE

J., 24, 369 (1978)] has summarized the literature prior to 1978. Works by Konig et al. [Ger. Chem. Eng., 1, 199 (1978)], Lucke et al. [Trans. Inst. Chem. Eng., 58, 228 (1980)], Riquarts [Ger. Chem. Eng., 4, 18 (1981)], Mersmann (op. cit.), Deckwer (op. cit.), Yang et al. [Chem. Eng. Sci., 47(9–11), 2859 (1992)], and Garcia-Calvo and Leton [Chem. Eng. Sci., 49(21), 3643 (1994)] are particularly useful references. Axial dispersion occurs in both the liquid and the gas phases. The degree of axial dispersion is affected by vessel diameter, vessel internals, gas superficial velocity, and surface-active agents that retard coalescence. For systems with coalescence-retarding surfactants the initial bubble size produced by the gas sparger is also significant. The gas and liquid physical properties have only a slight effect on the degree of axial dispersion, except that liquid viscosity becomes important as the flow regime becomes laminar. With pure liquids, in the absence of coalescence-inhibiting, surface-active agents, the nature of the sparger has little effect on the axial dispersion, and experimental results are reasonably well correlated by the dispersion model. For the liquid phase the correlation recommended by Deckwer et al. (op. cit.), after the original work by Baird and Rice [Chem. Eng. J., 9, 171(1975)] is as follows: EL gD 1/3 (14-222) ᎏ = 0.35 ᎏ (DUG) UG2 where EL = liquid-phase axial dispersion coefficient, UG = superficial velocity of the gas phase, D = vessel diameter, and g = gravitational acceleration. The recommended correlation for the gas-phase axial-dispersion coefficient is given by Field and Davidson (loc. cit.): UG 3.56 EG = 56.4 D1.33 ᎏ (14-223) ε where EG = gas-phase axial-dispersion coefficient, m2/s; D = vessel diameter, m; UG = superficial gas velocity, m/s; and ε = fractional gas holdup, volume fraction. The correlations given in the preceding paragraphs are applicable to vertical cylindrical vessels with pure liquids without coalescence inhibitors. For other vessel geometries such as columns of rectangular cross section, packed columns, and coiled tubes, the work of Shah et al. (loc. cit.) should be consulted. For systems containing coalescenceinhibiting surfactants, axial dispersion can be vastly different from that in systems in which coalescence is negligible. Konig et al. (loc. cit.) have well demonstrated the effects of surfactants and sparger type by conducting tests with weak alcohol solutions using three different porous spargers. With pure water, the sparger—and, consequently, initial bubble size—had little effect on back mixing because coalescence produced a dynamic-equilibrium bubble size not far above the sparger. With surfactants, the average bubble size was smaller than the dynamic-equilibrium bubble size. Small bubbles produced minimal back mixing up to ε ≈ 0.40; however, above ε ≈ 0.40 backmixing increased very rapidly as UG increased The rapid increase in back mixing as ε exceeds 0.40 was postulated to occur indirectly because a bubble carries upward with it a volume of liquid equal to about 70 percent of the bubble volume, and, for ε ≈ 0.40, the bubbles carry so much liquid upward that steady, uniform bubble rise can no longer be maintained and an oscillating, slugging flow develops, which produces fluctuating pressure at the gas distributor and the formation of large eddies. The large eddies greatly increase backmixing. For the air alcohol-water system, the minimum bubble size to prevent unsteady conditions was about 1, 1.5, and 2 mm for UG = 1, 3, and 5 cm/s, respectively. Any smaller bubble size produced increased backmixing. The results of Konig et al. (loc. cit.) clearly indicate that the interaction of surfactants and sparger can be very complex; thus, one should proceed very cautiously in designing systems for which surfactants significantly retard coalescence. Caution is particularly important because surfactants can produce either much more or much less backmixing than surfactant-free systems, depending on the bubble size, which, in turn, depends on the sparger utilized.







PHASE SEPARATION

14-81

PHASE SEPARATION Gases and liquids may be intentionally contacted as in absorption and distillation, or a mixture of phases may occur unintentionally as in vapor condensation from inadvertent cooling or liquid entrainment from a film. Regardless of the origin, it is usually desirable or necessary ultimately to separate gas-liquid dispersions. While separation will usually occur naturally, the rate is often economically intolerable and separation processes are employed to accelerate the step. GAS-PHASE CONTINUOUS SYSTEMS Practical separation techniques for liquid particles in gases are discussed. Since gas-borne particulates include both liquid and solid particles, many devices used for dry-dust collection (discussed in Sec. 17 under “Gas-Solids Separation”) can be adapted to liquid-particle separation. Also, the basic subject of particle mechanics is covered in Sec. 6. Separation of liquid particulates is frequently desirable in chemical processes such as in countercurrent-stage contacting because liquid entrainment with the gas partially reduces true countercurrency. Separation before entering another process step may be needed to prevent corrosion, to prevent yield loss, or to prevent equipment damage or malfunction. Separation before the atmospheric release of gases may be necessary to prevent environmental problems and for regulatory compliance. GENERAL REFERENCES G-1. Buonicore and Davis, eds., Air Pollution Engineering Manual, Van Nostrand Reinhold, New York, 1992. G-2. Calvert and Englund, eds., Handbook of Air Pollution Technology, Wiley, New York, 1984. G-3. Cheremisinoff, ed., Encyclopedia of Environmental Control Technology, vol. 2, Gulf Pub., Houston, 1989. G-4. McKetta, Unit Operations Handbook, vol. 1–2, Dekker, New York, 1992. G-5. Wark and Warner, Air Pollution: Its Origin and Control, 2d ed., Harper & Row, New York, 1981. G-6. Hesketh, Air Pollution Control, 1979; Fine Particles in Gaseous Media, Ann Arbor Science Pubs., Ann Arbor, MI, 1977. G-7. Stern, Air Pollution, 3d ed., vols. 3–5, Academic, Orlando, FL, 1976–77. G-8. Strauss, Industrial Gas Cleaning, 2d ed., Pergamon, New York, 1975. G-9. Theodore and Buonicore, Air Pollution Control Equipment; Selection, Design, Operation and Maintenance, Prentice Hall, Englewood Cliffs, NJ, 1982. G-10. Wang and Pereira, eds., Handbook of Environmental Engineering, vol. 1, Humana, Clifton, NJ 1979. G-11. Cheremisinoff and Young, Air Pollution Control and Design Handbook, parts 1–2, Dekker, New York, 1977. G-12. Nonhebel, Gas Purification Processes for Air Pollution Control, NewnesButterworth, London, 1972. Sampling R-1. Code of Federal Regulations, 40 (CFR 40), subchapter C—Air Programs, parts 50–99, Office of the Federal Register, Washington. R-2. Ref. G-11, part 1, pp. 65–121. R-3. Cooper and Rossano, Source Testing for Air Pollution Control, Environmental Science Services, Wilton, Connecticut, 1970. R-4. Ref. G-7, vol. 3, pp. 525–587. R-5. Methods of Air Sampling and Analysis, 2d Ed., American Public Health Assoc., Washington, 1977. R-6. Stockham and Fochtman, Particle Size Analysis, Ann Arbor Science Pubs., Ann Arbor, Michigan, 1977. R-7. Ref. G-2, Ch. 31, pp. 785–832. R-8. Ref. G-8, Ch. 2, pp. 39–79. Specific R-9. Calvert, Goldchmid, Leith, and Mehta, NTIS Publ. PB-213016, 213017, 1972. R-10. Calvert, J. Air Pollut. Control Assoc. 24, 929 (1974). R-11. Calvert, Chem. Eng., 84(18), 54 (1977). R-12. Calvert, Yung, and Leung, NTIS Publ. PB-248050, 1975. R-13. Calvert and Lundgren, J. Air Pollut. Control Assoc., 18, 677 (1968). R-14. Calvert, Lundgren, and Mehta, J. Air Pollut. Control Assoc., 22, 529 (1972). R-15. Yung, Barbarika, and Calvert, J. Air Pollut. Control Assoc., 27, 348, (1977). R-16. Katz, M.S. thesis, Pennsylvania State University, 1958.

R-17. York and Poppele, Chem. Eng. Prog., 59(6), 45 (1963). R-18. York, Chem. Eng. Prog., 50, 421 (1954). R-19. Ref. G-2, Ch. 10, pp. 215–248. References with the notation (R- ) are cited in the text.

Definitions: Mist and Spray Little standardization has been adopted in defining gas-borne liquid particles, and this frequently leads to confusion in the selection, design, and operation of collection equipment. Aerosol applies to suspended particulate, either solid or liquid, which is slow to settle by gravity and to particles from the submicrometer range up to 10 to 20 µm. Mists are fine suspended liquid dispersions usually resulting from condensation and ranging upward in particle size from around 0.1 µm. Spray refers to entrained liquid droplets. The droplets may be entrained from atomizing processes previously discussed under “Liquid-in-Gas Dispersions” in this section. In such instances, size will range from the finest particles produced up to a particle whose terminal settling velocity is equal to the entraining gas velocity if some settling volume is provided. Process spray is often created unintentionally, such as by the condensation of vapors on cold duct walls and its subsequent reentrainment, or from two-phase flow in pipes, gas bubbling through liquids, and entrainment from boiling liquids. Entrainment size distribution from sieve trays has been given by Cheng and Teller [Am. Inst. Chem. Eng. J., 7(2), 282 (1961)] and evaporator spray by Garner et al. [Trans. Inst. Chem. Eng., 32, 222 (1954)]. In general, spray can range downward in particle size from 5000 µm. There can be overlapping in size between the coarsest mist particles and the finest spray particles, but some authorities have found it convenient arbitrarily to set a boundary of 10 µm between the two. Actually, considerable overlap exists in the region of 5 to 40 µm. Table 14-18 lists typical ranges of particle size created by different mechanisms. The sizes actually entrained can be influenced by the local gas velocity. Figure 14-105 compares the approximate size range of liquid particles with other particulate material and the approximate applicable size range of collection devices. Figure 17-34 gives an expanded chart by Lapple for solid particles. Mist and fog formation has been discussed previously. Gas Sampling The sampling of gases containing mists and sprays may be necessary to obtain data for collection-device design, in which case particle-size distribution, total mass loading, and gas volume, temperature, pressure, and composition may all be needed. Other reasons for sampling may be to determine equipment performance, measure yield loss, or determine compliance with regulations. Location of a sample probe in the process stream is critical especially when larger particles must be sampled. Mass loading in one portion of a duct may be severalfold greater than in another portion as affected by flow patterns. Therefore, the stream should be sampled at a number of points. The U.S. Environmental Protection Agency (R-1) has specified 8 points for ducts between 0.3 and 0.6 m (12 and 24 in) and 12 points for larger ducts, provided there are no flow disturbances for eight pipe diameters upstream and two downstream from the sampling point. When only particles smaller than 3 µm are to be sampled, location and number of sample points are less critical since such particles remain reasonably well dispersed by brownian motion. However, some gravity settling of such particles and even gases of high density have been observed in long horizontal breeching. Isokinetic sampling (velocity at the probe inlet is equal to local duct velocity) is required to get a representative sample of particles larger than 3 µm (error is small for 4- to 5-µm particles). Sampling methods and procedures for mass loading have been developed (R-1 through R-8). TABLE 14-18

Particle Sizes Produced by Various Mechanisms

Mechanism or process

Particle-size range, µm

Liquid pressure spray nozzle Gas-atomizing spray nozzle Gas bubbling through liquid or boiling liquid Condensation processes with fogging Annular two-phase flow in pipe or duct

100–5000 1–100 20–1000 0.1–30 10–2000

14-82

FIG. 14-105

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

Particle classification and useful collection equipment versus

particle size.

Particle-Size Analysis Many particle-size-analysis methods suitable for dry-dust measurement are unsuitable for liquids because of coalescence and drainage after collection. Measurement of particle sizes in the flowing aerosol stream by using a cascade impactor is one of the better means. The impacting principle has been described by Ranz and Wong [Ind. Eng. Chem., 44, 1371 (1952)] and Gillespie and Johnstone [Chem. Eng. Prog., 51, 75F (1955)]. The Andersen, Sierra, and University of Washington impactors may be used if the sampling period is kept short so as not to saturate the collection substrate. An impactor designed specifically for collecting liquids has been described by Brink, Kennedy, and Yu [Am. Inst. Chem. Eng. Symp. Ser., 70(137), 333 (1974)]. Collection Mechanisms Mechanisms which may be used for separating liquid particles from gases are (1) gravity settling, (2) inertial (including centrifugal) impaction, (3) flow-line interception, (4) diffusional (brownian) deposition, (5) electrostatic attraction, (6) thermal precipitation, (7) flux forces (thermophoresis, diffusiophoresis, Stefan flow), and (8) particle agglomeration (nucleation) techniques. Equations and parameters for these mechanisms are given in Table 17-2. Most collection devices rarely operate solely with a single mechanism, although one mechanism may so predominate that it may be referred to, for instance, as an inertial-impaction device. After collection, liquid particles coalesce and must be drained from the unit, preferably without reentrainment. Calvert (R-12) has studied the mechanism of reentrainment in a number of liquid-particle collectors. Four types of reentrainment were typically observed: (1) transition from separated flow of gas and liquid to a two-phase region of separated-entrained flow, (2) rupture of bubbles, (3) liquid creep on the separator surface, and (4) shattering of liquid droplets and splashing. Generally, reentrainment increased with increasing gas velocity. Unfortunately, in devices collecting primarily by centrifugal and inertial impaction, primary collection efficiency increases with gas velocity; thus overall efficiency may go through a maximum as reentrainment overtakes the incremental increase in efficiency. Prediction of collection efficiency must consider both primary collection and reentrainment.

Procedures for Design and Selection of Collection Devices Calvert and coworkers (R-9 to R-12 and R-19) have suggested useful design and selection procedures for particulate-collection devices in which direct impingement and inertial impaction are the most significant mechanisms. The concept is based on the premise that the mass median aerodynamic particle diameter dp50 is a significant measure of the difficulty of collection of the liquid particles and that the collection device cut size dpc (defined as the aerodynamic particle diameter collected with 50 percent efficiency) is a significant measure of the capability of the collection device. The aerodynamic diameter for a particle is the diameter of a spherical particle (with an arbitrarily assigned density of 1 g/cm3) which behaves in an air stream in the same fashion as the actual particle. For real spherical particles of diameter dp, the equivalent aerodynamic diameter dpa can be obtained from the equation dpa = dp(ρpC′)1/2, where ρp is the apparent particle density (mass/ volume) and C′ is the Stokes-Cunningham correction factor for the particle size, all in consistent units. If particle diameters are expressed in micrometers, ρp can be in grams per cubic centimeter and C′ can be approximated by C′ = 1 + Ac(2λ /Dp), where Ac is a constant dependent upon gas composition, temperature, and pressure (Ac = 0.88 for atmospheric air at 20°C) and λ is the mean free path of the gas molecules (λ = 0.10 µm for 20°C atmospheric air). For other temperatures or pressures, or gases other than air, calculations using these more precise equations may be made: Ac = 1.257 + 0.4 exp [−1.1 (dp /2λ)] and λ = µg /0.499ρg × µm (where µg is the gas viscosity, kg/m⋅h; pg is gas density, g/cm3; and µm is the mean molecular speed, m/s. um = [8RuT/ πM]0.5, where Ru is the universal gas constant, 8.315 kJ/kg⋅mol⋅K; T is the gas absolute temperature, K; and M is the molar mass or equivalent molecular weight of the gas. (π is the usual geometric constant.) For test purposes (air at 25°C and 1 atm), pg = 1.183 kg/m, µg = 0.0666 kg/m⋅h, λ = 0.067 µm, and um = 467 m/s. For airborne liquid particles, the assumption of spherical shape is reasonably accurate, and ρp is approximately unity for dilute aqueous particles at ambient temperatures. C′ is approximately unity at ambient conditions for such particles larger than 1 to 5 µm, so that often the actual liquid particle diameter and the equivalent aerodynamic diameter are identical. When a distribution of particle sizes which must be collected is present, the actual size distribution must be converted to a mass distribution by aerodynamic size. Frequently the distribution can be represented or approximated by a log-normal distribution (a straight line on a log-log plot of cumulative mass percent of particles versus diameter) which can be characterized by the mass median particle diameter dp50 and the standard statistical deviation of particles from the median σg. σg can be obtained from the log-log plot by σg = Dpa50 /Dpe at 15.87 percent = Dpe at 84.13 percent/Dpa50. The grade efficiency η of most collectors can be expressed as a function of the aerodynamic particle size in the form of an exponential equation. It is simpler to write the equation in terms of the particle penetration Pt (those particles not collected), where the fractional penetration Pt = 1 − η, when η is the fractional efficiency. The typical collection equation is Pt = e(−AaDpaB)

(14-224)

where Aa and B are functions of the collection device. Calvert (R-12) has determined that for many devices in which the primary collection mechanism is direct interception and inertial impaction, such as packed beds, knitted-mesh collectors, zigzag baffles, target collectors such as tube banks, sieve-plate columns, and venturi scrubbers, the value of B is approximately 2.0. For cyclonic collectors, the value of B is approximately 0.67. The overall integrated penetration  Pt for a device handling a distribution of particle sizes can be obtained by W dW Pt = (14-225) ᎏ Pt  0 W where (dW/W) is the mass of particles in a given narrow size distribution and Pt is the average penetration for that size range. When the particles to be collected are log-normally distributed and the collection device efficiency can be expressed by Eq. (14-224), the required overall integrated collection efficiency  Pt can be related to the ratio of the device aerodynamic cut size Dpc to the mass median aerodynamic particle size Dpa50. This required ratio for a given distribution and




PHASE SEPARATION collection is designated RrL and these relationships are illustrated graphically in Fig. 14-106. For the many devices for which B is approximately 2.0, a simplified plot (Fig. 14-107) is obtained. From these figures, by knowing the desired overall collection efficiency and particle distribution, the value of RrL can be read. Substituting the mass median particle diameter gives the aerodynamic cut size required from the collection device being considered. Therefore, an experimental plot of aerodynamic cut size for each collection device versus operating parameters can be used to determine the device suitability. Collection Equipment Gravity Settlers Gravity can act to remove larger droplets. Settling or disengaging space above aerated or boiling liquids in a tank or spray zone in a tower can be very useful. If gas velocity is kept low, all particles with terminal settling velocities (see Sec. 6) above the gas

FIG. 14-106 Overall integrated penetration as a function of particle-size distribution and collector parameters. (Calvert, Yung, and Leung, NTIS Publ. PB248050, 1975.)

14-83

velocity will eventually settle. Increasing vessel cross section in the settling zone is helpful. Terminal velocities for particles smaller than 50 µm are very low and generally not attractive for particle removal. Laminar flow of gas in long horizontal paths between trays or shelves on which the droplets settle is another effective means of employing gravity. Design equations are given in Sec. 17 under “Gas-Solids Separations.” Settler pressure drop is very low, usually being limited to entrance and exit losses. Centrifugal Separation Centrifugal force can be utilized to enhance particle collection to several hundredfold that of gravity. The design of cyclone separators for dust removal is treated in detail in Sec. 17 under “Gas-Solids Separations,” and typical cyclone designs are shown in Fig. 17-43. Dimension ratios for one family of cyclones are given in Fig. 17-36. Cyclones, if carefully designed, can be more efficient on liquids than on solids since liquids coalesce on capture and are easy to drain from the unit. However, some precautions not needed for solid cyclones are necessary to prevent reentrainment. Tests by Calvert (R-12) show high primary collection efficiency on droplets down to 10 µm and in accordance with the efficiency equations of Leith and Licht [Am. Inst. Chem. Eng. Symp. Ser., 68(126), 196–206 (1972)] for the specific cyclone geometry tested if entrainment is avoided. Typical entrainment points are (1) creep along the gas outlet pipe, (2) entrainment by shearing of the liquid film from the walls, and (3) vortex pickup from accumulated liquid in the bottom (Fig. 14-108a). Reentrainment from creep of liquid along the top of the cyclone and down the outlet pipe can be prevented by providing the outlet pipe with a flared conical skirt (Fig. 14-108b), which provides a point from which the liquid can drip without being caught in the outlet gas. The skirt should be slightly shorter than the gas outlet pipe but extend below the bottom of the gas inlet. The cyclone inlet gas should not impinge on this skirt. Often the bottom edge of the skirt is V-notched or serrated. Reentrainment is generally reduced by lower inlet gas velocities. Calvert (R-12) reviewed the literature on predicting the onset of entrainment and found that of Chien and Ibele (ASME Pap. 62WA170) to be the most reliable. Calvert applies their correlation to a liquid Reynolds number on the wall of the cyclone, NRe,L = 4QL /hivL, where QL is the volumetric liquid flow rate, cm3/s; hi is the cyclone inlet height, cm; and vL is the kinematic liquid viscosity, cm2/s. He finds that the onset of entrainment occurs at a cyclone inlet gas velocity Vci, m/s, in accordance with the relationship in Vci = 6.516 − 0.2865 ln NRe,L. Reentrainment from the bottom of the cyclone can be prevented in several ways. If a typical long-cone dry cyclone is used and liquid is kept continually drained, vortex entrainment is unlikely. However, a vortex breaker baffle in the outlet is desirable, and perhaps a flat disk on top extending to within 2 to 5 cm (0.8 to 2 in) of the walls may be

(a)

(b) FIG. 14-107 Overall integrated penetration as a function of particle-size distribution and collector cut diameter when B = 2 in Eq. (14-224). (Calvert, Goldshmid, Leith, and Mehta, NTIS Publ. PB-213016, 213017, 1972.)

FIG. 14-108 (a) Liquid entrainment from the bottom of a vessel by centrifugal flow. (Rietema and Verver, Cyclones in Industry, Elsevier, Amsterdam, 1961.) (b) Gas-outlet skirt for liquid cyclones. (Stern et al., Cyclone Dust Collectors, American Petroleum Institute, New York, 1955.)

14-84

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

(a)

(b)

(c)

Typical separators using impingement in addition to centrifugal force. (a) Hi-eF purifier. (V. D. Anderson Co.) (b) Flick separator. (Wurster & Sanger, Inc.) (c) Type RA line separator. (Centrifix Corp., Bull. 220.) FIG. 14-109

beneficial. Often liquid cyclones are built without cones and have dished bottoms. The modifications described earlier are definitely needed in such situations. Stern, Caplan, and Bush (Cyclone Dust Collectors, American Petroleum Institute, New York, 1955) and Rietema and Verver (in Tengbergen, Cyclones in Industry, Elsevier, Amsterdam, 1961, chap. 7) have discussed liquid-collecting cyclones. As with dust cyclones, no reliable pressure-drop equations exist (see Sec. 17), although many have been published. A part of the problem is that there is no standard cyclone geometry. Calvert (R-12) experimentally obtained ∆P = 0.000513 ρg(Qg /hiWi)2(2.8hiwi /d 2o), where ∆P is in cm of water; ρg is the gas density, g/cm3; Qg is the gas volumetric flow rate, cm3/s; hi and wi are cyclone inlet height and width respectively, cm; and do is the gas outlet diameter, cm. This equation is in the same form as that proposed by Shepherd and Lapple [Ind. Eng. Chem., 31, 1246 (1940)] but gives only 37 percent as much pressure drop. Liquid cyclone efficiency can be improved somewhat by introducing a coarse spray of liquid in the cyclone inlet. Large droplets which are easily collected collide with finer particles as they sweep the gas stream in their travel to the wall. (See subsection “Wet Scrubbers” regarding optimum spray size.) Cyclones may also be operated wet to improve their operation on dry dust. Efficiency can be improved through reduction in entrainment losses since the dust particles become trapped in the water film. Collision between droplets and dust particles aids collection, and adequate irrigation can eliminate problems of wall buildup and fouling. The most effective operation is obtained by spraying countercurrently to the gas flow in the cyclone inlet duct at liquid rates of 0.7 to 2.0 L /m3 of gas. There are also many proprietary designs of liquid separators using centrifugal force, some of which are illustrated in Fig. 14-109. Many of these were originally developed as steam separators to remove entrained condensate. In some designs, impingement on swirl baffles aids separation. Impingement Separation Impingement separation employs direct impact and inertial forces between particles, the gas streamlines, and target bodies to provide capture. The mechanism is discussed in Sec. 17 under “Gas-Solids Separations.” With liquids, droplet coalescence occurs on the target surface, and provision must be made for drainage without reentrainment. Calvert (R-12) has studied droplet collection by impingement on targets consisting of banks of tubes, zigzag baffles, and packed and mesh beds. Figure 14-110 illustrates some other types of impingement-separator designs. In its simplest form, an impingement separator may be nothing more than a target placed in front of a flow channel such as a disk at the end of a tube. To improve collection efficiency, the gas velocity may be increased by forming the end into a nozzle (Fig. 14-110a). Particle collection as a function of size may be estimated by using the target-efficiency correlation in Fig. 17-39. Since target efficiency will be low for systems with separation numbers below 5 to 10 (small particles, low gas velocities), the mist will frequently be subjected to a number of targets in series as in Fig. 14-110c, d, and g. The overall droplet penetration is the product of penetration for

each set of targets in series. Obviously, for a distribution of particle sizes, an integration procedure is required to give overall collection efficiency. This target-efficiency method is suitable for predicting efficiency when the design effectively prevents the bypassing or shortcircuiting of targets by the gas stream and provides adequate time to accelerate the liquid droplets to gas velocity. Katz (R-16) investigated a jet and target-plate entrainment separator design and found the pressure drop less than would be expected to supply the kinetic energy both for droplet acceleration and gas friction. An estimate based on his results indicates that the liquid particles on the average were being accelerated to only about 60 percent of the gas velocity. The largest droplets, which are the easiest to collect, will be accelerated less than the smaller particles. This factor has a leveling effect on collection efficiency as a function of particle size so that experimental results on such devices may not show as sharp a decrease in efficiency with particle size as predicted by calculation. Such results indicate that in many cases our lack of predicting ability results, not from imperfections in the theoretical treatment, but from our lack of knowledge of velocity distributions within the system. Katz (R-16) also studied wave-plate impingement separators (Fig. 14-110b) made up of 90° formed arcs with an 11.1-mm (0.44-in) radius and a 3.8-mm (0.15-in) clearance between sheets. The pressure drop is a function of system geometry. The pressure drop for Katz’s system and collection efficiency for seven waves are shown in Fig. 14-111. Katz used the Souders-Brown expression to define a design velocity for the gas between the waves:  U = K ρ ρ/ρ l − g)g

(14-226)

K is 0.12 to give U in ms−1 (0.4 for ft/s), and ρl and ρg are liquid and gas densities in any consistent set of units. Katz found no change in efficiency at gas velocities from one-half to 3 times that given by the equation. Calvert (R-12) investigated zigzag baffles of a design more like Fig. 14-110e. The baffles may have spaces between the changes in direction or be connected as shown. He found close to 100 per collection for water droplets of 10 µm and larger. Some designs had high efficiencies down to 5 or 8 µm. Desirable gas velocities were 2 to 3.5 m/s (6.6 to 11.5 ft/s), with a pressure drop for a six-pass baffle of 2 to 2.5 cm (0.8 to 1.0 in) of water. On the basis of turbulent mixing, an equation was developed for predicting primary collection efficiency as a function of particle size and collector geometry: utenWθ η = 1 − exp − ᎏᎏ (14-227) 57.3Ug b tan θ where η is the fractional primary collection efficiency; ute is the drop terminal centrifugal velocity in the normal direction, cm/s; Ug is the superficial gas velocity, cm/s; n is the number of rows of baffles or bends; θ is the angle of inclination of the baffle to the flow path, °; W is the width of the baffle, cm; and b is the spacing between baffles in the same row, cm. For conditions of low Reynolds number (NRe,D <





PHASE SEPARATION

14-85

(a)

(c)

(d)

(b)

(e) (f) (g)

(i)

(h)

(j)

Typical impingement separators. (a) Jet impactor. (b) Wave plate. (c) Staggered channels. (Blaw-Knox Food & Chemical Equipment, Inc.) (d) Vane-type mist extractor. (Maloney-Crawford Tank and Mfg. Co.) (e) Peerless line separator. (Peerless Mfg. Co.) ( f) Strong separator. (Strong Carlisle and Hammond.) (g) Karbate line separator. (Union Carbide Corporation) (h) Type E horizontal separator. (Wright-Austin Co.) (i) PL separator. (Ingersoll Rand.) ( j) Wire-mesh demister. (Otto H. York Co.)

FIG. 14-110

0.1) where Stokes’ law applies, Calvert obtains the value for drop terminal centrifugal velocity of ute = d p2 ρpa/18 µg, where dp and ρp are the drop particle diameter, cm, and particle density, g/cm3, respectively; µg is the gas viscosity, P; and a is the acceleration due to centrifugal force. It is defined by the equation a = 2U g2 sin θ/W cos3 θ. For situations in which Stokes’ law does not apply, Calvert recommends substitution in the derivation of Eq. (14-227) for u of drag coefficients from

(a)

drag-coefficient data of Foust et al. (Principles of Unit Operations, Toppan Co., Tokyo, 1959). Calvert found that reentrainment from the baffles was affected by the gas velocity, the liquid-to-gas ratio, and the orientation of the baffles. Horizontal gas flow past vertical baffles provided the best drainage and lowest reentrainment. Safe operating regions with vertical baffles are shown in Fig. 14-112. Horizontal baffles gave the poorest drainage

(b)

Pressure drop and collection efficiency of a wave-plate separator. (a) Pressure drop. (b) Efficiency DE = clearance between sheets. (Katz, M.S. thesis, Pennsylvania State University, 1958.) FIG. 14-111

14-86

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

and the highest reentrainment, with inclined baffles intermediate in performance. Equation (14-228), developed by Calvert, predicts pressure drop across zigzag baffles. The indicated summation must be made over the number of rows of baffles present. i=n U′g Ap ∆P =  1.02 × 10−3 fDρg ᎏ (14-228) 2 At i=1 ∆P is the pressure drop, cm of water; ρg is the gas density, g/cm3; A p is the total projected area of an entire row of baffles in the direction of inlet gas flow, cm2; and At is the duct cross-sectional area, cm2. The value fD is a drag coefficient for gas flow past inclined flat plates taken from Fig. 14-113, while U′g is the actual gas velocity, cm/s, which is related to the superficial gas velocity Ug by Ug′ = Ug /cos θ. It must be noted that the angle of incidence θ for the second and successive rows of baffles is twice the angle of incidence for the first row. Most of Calvert’s work was with 30° baffles, but the method correlates well with other data on 45° baffles. The Karbate line separator (Fig. 14-110g) is composed of several layers of teardrop-shaped target rods of Karbate. A design flow constant K in Eq. (14-226) of 0.035 m/s (1.0 ft/s) is recommended by the manufacturer. Pressure drop is said to be 5a velocity heads on the basis of the superficial gas velocity. This value would probably increase at high liquid loads. Figure 14-114 gives the manufacturer’s reported grade efficiency curve at the design air velocity. The use of multiple tube banks as a droplet collector has also been studied by Calvert (R-12). He reports that collection efficiency for

FIG. 14-114 Collection efficiency of Karbate line separator, based on particles with a specific gravity of 1.0 suspended in atmospheric air with a pressure drop of 2.5 cm water gauge. (Union Carbide Corporation Cat. Sec. S-6900, 1960.)

closely packed tubes follows equations for rectangular jet impaction which can be obtained graphically from Fig. 14-115 by using a dimensional parameter β which is based on the tube geometry; β = 2 li /b, where b is the open distance between adjacent tubes in the row (orifice width) and li is the impaction length (distance between orifice and impingement plane), or approximately the distance between centerlines of successive tube rows. Note that the impaction parameter Kp is plotted to the one-half power in Fig. 14-115 and that the radius of the droplet is used rather than the diameter. Collection efficiency overall for a given size of particle is predicted for the entire tube bank by η = 1 − (1 − ηb)N

(14-229)

where ηb is the collection efficiency for a given size of particle in one stage of a rectangular jet impactor (Fig. 14-115) and N is the number of stages in the tube bank (equal to one less than the number of rows). For widely spaced tubes, the target efficiency ηg can be calculated from Fig. 17-39 or from the impaction data of Golovin and Putnam [Ind. Eng. Chem. Fundam., 1, 264 (1962)]. The efficiency of the overall tube banks for a specific particle size can then be calculated from the equation η = 1 − (1 − ηt a′/A)n, where a′ is the cross-sectional area of all tubes in one row, A is the total flow area, and n is the number of rows of tubes.

FIG. 14-112 Safe operating region to prevent reentrainment from vertical zigzag baffles with horizontal gas flow. (Calvert, Yung, and Leung, NTIS Publ. PB-248050, 1975.)

Experimental collection efficiencies of rectangular impactors. C′ is the Stokes-Cunningham correction factor; ρp, particle density, g/cm3; Ug, superficial gas velocity, approaching the impactor openings, cm/s; and µg, gas viscosity, P. [Calvert, Yung, and Leung, NTIS Publ. PB-248050; based on Mercer and Chow, J. Coll. Interface Sci., 27, 75 (1968).]

FIG. 14-115

Drag coefficient for flow past inclined flat plates for use in Eq. (14-228). (Calvert, Yung, and Leung, NTIS Publ. PB-248050; based on Fage and Johansen, Proc. R. Soc. (London), 116A, 170 (1927).]

FIG. 14-113

PHASE SEPARATION Calvert reports pressure drop through tube banks to be largely unaffected by liquid loading and indicates that Grimison’s correlations in Sec. 6 (“Tube Banks”) for gas flow normal to tube banks or data for gas flow through heat-exchanger bundles can be used. However, the following equation is suggested: ∆P = 8.48 × 10−3 nρgU′g2

(14-230)

where ∆P is cm of water; n is the number of rows of tubes; ρg is the gas density, g/cm3; and U′g is the actual gas velocity between tubes in a row, cm/s. Calvert did find an increase in pressure drop of about 80 to 85 percent above that predicted by Eq. (14-230) in vertical upflow of gas through tube banks due to liquid holdup at gas velocities above 4 m/s. The onset of liquid reentrainment from tube banks can be predicted from Fig. 14-116. Reentrainment occurred at much lower velocities in vertical upflow than in horizontal gas flow through vertical tube banks. While the top of the cross-hatched line of Fig. 14-116a predicts reentrainment above gas velocities of 3 m/s (9.8 ft/s) at high liquid loading, most of the entrainment settled to the bottom of the duct in 1 to 2 m (3.3 to 6.6 ft), and entrainment did not carry significant distances until the gas velocity exceeded 7 m/s (23 ft/s). Packed-Bed Collectors Many different materials, including coal, coke, broken solids of various types such as brick, tile, rock, and stone, as well as normal types of tower-packing rings, saddles, and special plastic shapes, have been used over the years in packed beds to remove entrained liquids through impaction and filtration. Separators using natural materials are not available as standard commercial units but are designed for specific applications. Coke boxes were used extensively in the years 1920 to 1940 as sulfuric acid entrainment separators (see Chemical Engineers’ Handbook, 5th ed., p. 18–87) but have now been largely superseded by more sophisticated and efficient devices. Jackson and Calvert [Am. Inst. Chem. Eng. J., 12, 1075 (1966)] studied the collection of fine fuel-oil-mist particles in beds of a-in glass spheres, Raschig rings, and Berl and Intalox saddles. The mist had a mass median particle diameter of 6 µm and a standard deviation of 2.0. The collection efficiency as a function of particle size and gas

(a)

velocity in a 355-mm- (14-in-) diameter by 152-mm- (6-in-) thick bed of Intalox saddles is given in Fig. 14-117. This and additional work have been generalized by Calvert (R-12) to predict collection efficiencies of liquid particles in any packed bed. Assumptions in the theoretical development are that the drag force on the drop is given by Stokes’ law and that the number of semicircular bends to which the gas is subjected, η1, is related to the length of the bed, Z (cm), in the direction of gas flow, the packing diameter, dc (cm), and the gas-flow channel width, b (cm), such that η1 = Z/(dc + b). The gas velocity through the channels, Ugb (cm/s), is inversely proportional to the bed free volume for gas flow such that Ugb = Ug [1/(ε − hb)], where Ug is the gas superficial velocity, cm/s, approaching the bed, ε is the bed void fraction, and hb is the fraction of the total bed volume taken up with liquid which can be obtained from data on liquid holdup in packed beds. The width of the semicircular channels b can be expressed as a fraction j of the diameter of the packing elements, such that b = jdc. These assumptions (as modified by G. E. Goltz, personal communication) lead to an equation for predicting the penetration of a given size of liquid particle through a packed bed: −π Z Pt = exp ᎏᎏ (14-231) ᎏ Kp 2( j + j 2)(ε − hb) dc




 

ρp d Ug (14-232) Kp = ᎏ 9µg dc Values of ρp and dp are droplet density, g/cm3, and droplet diameter, cm; µg is the gas viscosity, P. All other terms were defined previously. Table 14-19 gives values of j calculated from experimental data of Jackson and Calvert. Values of j for most manufactured packing appear to fall in the range from 0.16 to 0.19. The low value of 0.03 for coke may be due to the porosity of the coke itself. Calvert (R-12) has tested the correlation in cross-flow packed beds, which tend to give better drainage than countercurrent beds, and has found the effect of gas-flow orientation insignificant. However, the onset of reentrainment was somewhat lower in a bed of 2.5-cm where

2 p

(b)

Experimental results showing effect of gas velocity and liquid load on entrainment from (a) vertical tube banks with horizontal gas flow and (b) horizontal tube banks with upflow. To convert meters per second to feet per second, multiply by 3.281. (Calvert, Yung, and Leung, NTIS Publ. PB-248050.)

FIG. 14-116

14-87

14-88

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

Experimental collection efficiency. a-in Intalox saddles. To convert feet per second to meters per second, multiply by 0.3048; to convert centimeters to inches, multiply by 0.394; and to convert grams per cubic centimeter to pounds per cubic foot, multiply by 62.43. [Jackson and Calvert, Am. Inst. Chem. Eng. J., 12, 1975 (1968).]

FIG. 14-117

(1.0-in) pall rings with gas upflow [6 m/s (20 ft/s)] than with horizontal cross-flow of gas. The onset of reentrainment was independent of liquid loading (all beds were nonirrigated), and entrainment occurred at values somewhat above the flood point for packed beds as predicted by conventional correlations. In beds with more than 3 cm (1.2 in) of water pressure drop, the experimental drop with both vertical and horizontal gas flow was somewhat less than predicted by generalized packed-bed pressure-drop correlations. However, Calvert recommends these correlations for design as conservative. Calvert’s data indicate that packed beds irrigated only with the collected liquid can have collection efficiencies of 80 to 90 percent on mist particles down to 3 µm but have low efficiency on finer mist particles. Frequently, irrigated packed towers and towers with internals will be used with liquid having a wetting capability for the fine mist which must be collected. Tennessee Valley Authority (TVA) experiments with the collection of 1.0-µm mass median phosphoric acid mist in packed towers have shown that the strength of the circulating phosphoric acid is highly important [see Baskerville, Am. Inst. Chem. Eng. J., 37, 79 (1941); and p. 18–87, 5th ed. of the Handbook]. Hesketh (J. Air Pollut. Control Assoc., 24, 942 (1974)] has reported up to 50 percent improvement in collection efficiency in venturi scrubbers on fine particles with the addition of only 0.10 percent of a lowfoaming nonionic surfactant to the scrubbing liquid, and others have experienced similar results in other gas-liquid-contacting devices. Calvert (R-9 and R-10) has reported on the efficiency of various gasliquid-contacting devices for fine particles. Figure 14-118 gives the particle aerodynamic cut size for a single-sieve-plate gas scrubber as a function of sieve hole size dh, cm; hole gas velocity uh, m/s; and froth or foam density on the plate F, g/cm3. This curve is based on standard air and water properties and wettable (hydrophilic) particles. The cut diameter decreases with an increase in froth density, which must be predicted from correlations for sieve-plate behavior (see Fig. 14-32). Equation (14-231) can be used to calculate generalized design curves for collection in packed columns in the same fashion by finding parameters of packing size, bed length, and gas velocity which give collection efficiencies of 50 percent for various size particles. Figure (14-119) illustrates such a plot for three gas velocities and two sizes of packing.

Aerodynamic cut diameter for a single-sieve-plate scrubber as a function of hole size, hole-gas velocity, and froth density, F, g/cm3. To convert meters per second to feet per second, multiply by 3.281; to convert grams per cubic centimeter to pounds per cubic foot, multiply by 62.43. [Calvert, J. Air Pollut. Control Assoc., 24, 929 (1974).] FIG. 14-118

Wire-Mesh Mist Collectors Knitted mesh of varying density and voidage is widely used for entrainment separators. Its advantage is close to 100 percent removal of drops larger than 5 µm at superficial gas velocities from about 0.2 ms/s (0.6 ft/s) to 5 m/s (16.4 ft/s), depending somewhat on the design of the mesh. Pressure drop is usually no more than 2.5 cm (1 in) of water. A major disadvantage is the ease with which tars and insoluble solids plug the mesh. The separator can be made to fit vessels of any shape and can be made of any material which can be drawn into a wire. Stainless-steel and plastic fibers are most common, but other metals are sometimes used. Generally three basic types of mesh are used: (1) layers with a crimp in the same direction (each layer is actually a nested double layer); (2) layers with a crimp in

TABLE 14-19 Experimental Values for j, Channel Width in Packing as a Fraction of Packing Diameter Packing size cm

in

Type of packing

j

1.27 2.54 3.8 7.6–12.7

0.5 1.0 1.5 3–5

Berl and Intalox saddles, marbles, Raschig rings Berl and Intalox saddles, pall rings Berl and Intalox saddles, pall rings Coke

0.192 0.190 0.165 0.03

Aerodynamic cut diameter for a typical packed-bed entrainment separator as a function of packing size, bed depth, and three gas velocities: curve 1–1.5 m/s, curve 2–3.0 m/s, and curve 3–4.5 m/s. To convert meters to feet, multiply by 3.281; to convert centimeters to inches, multiply by 0.394. [Calvert, J. Air Pollut. Control Assoc., 24, 929 (1974).]

FIG. 14-119

PHASE SEPARATION alternate directions, which increases voidage, reduces sheltering and increases target efficiency per layer, and gives a lower pressure drop per unit length; and (3) spiral-wound layers which reduce pressure drop by one-third, but fluid creep may lead to higher entrainment. Some small manufacturers of plastic meshes may offer other weaves claimed to be superior. The filament size can vary from about 0.15 mm (0.006 in) for fine-wire pads to 3.8 mm (0.15 in) for some plastic fibers. Typical pad thickness varies from 100 to 150 mm (4 to 6 in), but occasionally pads up to 300 mm (12 in) thick are used. A typical wire diameter for standard stainless mesh is 0.28 mm (0.011 in), with a finished mesh density of 0.15 g/cm3 (9.4 lb/ft3). A lower mesh density may be produced with standard wire to give 10 to 20 percent higher flow rates. Figure 14-120 presents an early calculated estimate of mesh efficiency as a fraction of mist-particle size. Experiments by Calvert (R-12) confirm the accuracy of the equation of Bradie and Dickson ( Joint Symp. Proc. Inst. Mech. Eng./Yorkshire Br. Inst. Chem. Eng., 1969, pp. 24–25) for primary efficiency in mesh separators: η = 1 − exp(−2/3)πalη i)

(14-232)

where η is the overall collection efficiency for a given-size particle; l is the thickness of the mesh, cm, in the direction of gas flow; a is the surface area of the wires per unit volume of mesh pad, cm2/cm3; and η i, the target collection efficiency for cylindrical wire, can be calculated from Fig. 17-39 or the impaction data of Golovin and Putnam [Ind. Eng. Chem., 1, 264 (1962)]. The factor 2/3, introduced by Carpenter and Othmer [Am. Inst. Chem. Eng. J., 1, 549 (1955)], corrects for the fact that not all the wires are perpendicular to the gas flow and gives the projected perpendicular area. If the specific mesh surface area a is not available, it can be calculated from the mesh void area ε and the mesh wire diameter dw in cm, a = 4(1 − ε)/dw. York and Poppele (R-17) have stated that factors governing maximum allowable gas velocity through the mesh are (1) gas and liquid density, (2) liquid surface tension, (3) liquid viscosity, (4) specific wire surface area, (5) entering-liquid loading, and (6) suspended-solids content. York (R-18) has proposed application of the Souders-Brown equation [Eq. (14-226)] for correlation of maximum allowable gas velocity with values of K for most cases of 0.1067 m/s to give U in m/s (0.35 for ft/s). When liquid viscosity or inlet loading is high or the liquid is dirty, the value of K must be reduced. Schroeder (M.S. thesis, Newark College of Engineering, 1962) found lower values for K necessary when liquid surface tension is reduced such as by the presence of surfactants in water. Ludwig (Applied Process Design for Chemical and Petrochemical Plants, 2d ed., vol. I, Gulf, Houston, 1977, p. 157) recommends reduced K values of (0.061 m/s) under vacuum at an absolute pressure of 6.77 kPa (0.98 lbf/in2) and K = 0.082 m/s at 54 kPa (7.83 lbf/in2) absolute. Most manufacturers suggest setting the design velocity at three-fourths of the maximum velocity to allow for surges in gas flow. York and Poppele (R-17) have suggested that total pressure drop through the mesh is equal to the sum of the mesh dry pressure drop

plus an increment due to the presence of liquid. They considered the mesh to be equivalent to numerous small circular channels and used the D’Arcy formula with a modified Reynolds number to correlate friction factor (see Fig. 14-121) for Eq. (14-233) giving dry pressure drop. ∆Pdry = flaρgUg2 /981 ε3

(14-233)

where ∆P is in cm of water; f is from Fig. (14-121); ρg is the gas density, g/cm3; Ug is the superficial gas velocity, cm/s; and ε is the mesh porosity or void fraction; l and a are as defined in Eq. (14-232). Figure 14-121 gives data of York and Poppele for mesh crimped in the same and alternating directions and also includes the data of Satsangee, of Schuring, and of Bradie and Dickson. The incremental pressure drop for wet mesh is not available for all operating conditions or for mesh of different styles. The data of York and Poppele for wet-mesh incremental pressure drop, ∆PL in cm of water, are shown in Fig. 14-122 or parameters of liquid velocity L/A, defined as liquid volumetric flow rate, cm3/min per unit of mesh crosssectional area in cm2; liquid density ρL is in g/cm3. York generally recommends the installation of the mesh horizontally with upflow of gas as in Fig. 14-110f; Calvert (R-12) tested the mesh horizontally with upflow and vertically with horizontal gas flow. He reports better drainage with the mesh vertical and somewhat higher permissible gas velocities without reentrainment, which is contrary to past practice. With horizontal flow through vertical mesh, he found collection efficiency to follow the predictions of Eq. 14-232 up to 4 m/s (13 ft/s) with air and water. Some reentrainment was encountered at higher velocities, but it did not appear serious until velocities exceeded 6.0 m/s (20 ft/s). With vertical upflow of gas, entrainment was encountered at velocities above and below 4.0 m/s (13 ft/s), depending on inlet liquid quantity (see Fig. 14-123). Figure 14-124 illustrates the onset of entrainment from mesh as a function of liquid loading and gas velocity and the safe operating area recommended by Calvert. Measurements of dry pressure drop by Calvert gave values only about one-third of those predicted from Eq. (14-233). He found the pressure drop to be highly affected by liquid load. The pressure drop of wet mesh could be correlated as a function of Ug1.65 and parameters of liquid loading L/A, as shown in Fig. 14-125. As indicated previously, mesh efficiency drops rapidly as particles decrease in size below 5 µm. An alternative is to use two mesh pads in series. The first mesh is made of fine wires and is operated beyond the

Value of friction factor f for dry knitted mesh for Eq. (14-233). Values of York and Poppele [Chem. Eng. Prog., 50, 421 (1954)] are given in curve 1 for mesh crimped in the alternating direction and curve 2 for mesh crimped in the same direction. Data of Bradie and Dickson ( Joint Symp. Proc. Inst. Mech. Eng./Yorkshire Br. Inst. Chem. Eng., 1969, pp. 24–25) are given in curve 3 for layered mesh and curve 4 for spiral-wound mesh. Curve 5 is data of Satsangee (M.S. thesis, Brooklyn Polytechnic Institute, 1948) and Schurig (D.Ch.E. dissertation, Brooklyn Polytechnic Institute, 1946). (From Calvert, Yung, and Leung, NTIS Publ. PB-248050, 1975.)

FIG. 14-121

FIG. 14-120 Collection efficiency of wire-mesh separator; 6-in thickness, 98.6 percent free space, 0.006-in-diameter wire used for experiment points. Curves calculated for target area equal to 2 and 3 times the solids volume of packing. To convert inches to millimeters, multiply by 25.4.

14-89

14-90

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

(a)

(b)

Incremental pressure drop in knitted mesh due to the presence of liquid (a) with the mesh crimps in the same direction and (b) with crimps in the alternating direction, based on the data of York and Poppele [Chem. Eng. Prog., 50, 421 (1954)]. To convert centimeters per minute to feet per minute, multiply by 0.0328; to convert centimeters per second to feet per second, multiply by 0.0328. (From Calvert, Yung, and Leung, NTIS Publ. PB-248050, 1975.) FIG. 14-122

flood point. It results in droplet coalescence, and the second mesh, using standard wire and operated below flooding, catches entrainment from the first mesh. Coalescence and flooding in the first mesh may be assisted with water sprays or irrigation. Massey [Chem. Eng. Prog., 53(5), 114 (1959)] and Coykendall et al. [ J. Air Pollut. Control Assoc., 18, 315 (1968)] have discussed such applications. Calvert (R-12) presents data on the particle size of entrained drops from mesh as a function of gas velocity which can be used for sizing the secondary collector. A major disadvantage of this approach is high pressure drop, which can be in the range from 25 cm (10 in) of water to as high as 85 cm (33 in) of water if the mist is mainly submicrometer. Wet Scrubbers Scrubbers have not been widely used for the collection of purely liquid particulate, probably because they are generally more complex and expensive than impaction devices of the types previously discussed. Further, scrubbers are no more efficient than the former devices for the same energy consumption. However,

Experimental data of Calvert with air and water in mesh with vertical upflow, showing the effect of liquid loading on efficiency and reentrainment. To convert meters per second to feet per second, multiply by 3.281; to convert cubic centimeters per square centimeter-minute to cubic feet per square foot-minute, multiply by 0.0328. (Calvert, Yung, and Leung, NTIS Publ. PB-248050, 1975.)

scrubbers of the types discussed in Sec. 17 and illustrated in Figs. 17-48 to 17-55 can be used to capture liquid particles efficiently. Their use is primarily indicated when it is desired to accomplish simultaneously another task such as gas absorption or the collection of solid and liquid particulate mixtures. Table 20-41 [Chemical Engineers’ Handbook, 5th ed.)], showing

FIG. 14-123

Effect of gas and liquid rates on onset of mesh reentrainment and safe operating regions. To convert meters per second to feet per second, multiply by 3.281. (Calvert, Yung, and Leung, NTIS Publ. PB-248050, 1975.)

FIG. 14-124

PHASE SEPARATION

(a)

(b)

Experimental pressure measured by Calvert as a function of gas velocity and liquid loading for (a) horizontal gas flow through vertical mesh and (b) gas upflow through horizontal mesh. Mesh thickness was 10 cm with 2.8-mm wire and void fraction of 98.2 percent, crimped in alternating directions. To convert meters per second to feet per second, multiply by 3.281; to convert centimeters to inches, multiply by 0.394. (Calvert, Yung, and Leung, NTIS Publ. PB-248050, 1975.)

FIG. 14-125

the minimum size of particles collectible in different types of scrubbers at reasonably high efficiencies, is a good selection guide. Cyclonic spray towers (Fig. 17-52) can effectively remove liquid particles down to around 2 to 3 µm. Figures 20-112 and 20-113 (Chemical Engineers’ Handbook, 5th ed.), giving target efficiency between spray drop size and particle size as calculated by Stairmand or Johnstone and Roberts, should be considered in selecting spray atomization for the most efficient tower operation. Figure 14-126 gives calculated particle cut size as a function of tower height (or length) for vertical countercurrent spray towers and for horizontal-gas-flow, vertical-liquid-flow crosscurrent spray towers with parameters for liquid drop size. These curves are based on physical properties of standard air and water and should be used under conditions in which these are reasonable

(a)

approximations. Lack of uniform liquid distribution or liquid flowing down the walls can affect the performance, requiring empirical correction factors. Calvert (R-10) suggests that a correction factor of 0.2 be used in small-diameter scrubbers to account for the liquid on the walls, i.e., let QL /Qg = 0.2 (QL /Qg)actual. Many more complicated wet scrubbers employ a combination of sprays or liquid atomization, cyclonic action, baffles, and targets. These combinations are not likely to be more efficient than similar devices previously discussed that operate at equivalent pressure drop. The vast majority of wet scrubbers operate at moderate pressure drop [8 to 15 cm (3 to 6 in) of water or 18 to 30 cm (7 to 12 in) of water] and cannot be expected to have high efficiency on particles smaller than 10 µm or 3 to 5 µm respectively. Fine and submicrometer particles can be captured efficiently only in wet scrubbers having high energy input such as venturi scrubbers, two-phase eductor scrubbers, and flux-force-condensation scrubbers. Venturi Scrubbers One type of venturi scrubber is illustrated in Fig. 17-48. Venturi scrubbers have been used extensively for collecting fine and submicrometer solid particulate, condensing tars and mists, and mixtures of liquids and solids. To a lesser extent, they have also been used for simultaneous gas absorption, although Lundy [Ind. Eng. Chem., 50, 293 (1958)] indicates that they are generally limited to three transfer units. They have been used to collect submicrometer chemical incinerator fume and mist as well as sulfuric and phosphoric acid mists. The collection efficiency of a venturi scrubber is highly dependent on the throat velocity or pressure drop, the liquid-to-gas ratio, and the chemical nature of wettability of the particulate. Throat velocities may range from 60 to 150 m/s (200 to 500 ft/s). Liquid injection rates are typically 0.67 to 1.4 m3/1000 m3 of gas. A liquid rate of 1.0 m3 per 1000 m3 of gas is usually close to optimum, but liquid rates as high as 2.7 m3 (95 ft3) have been used. Efficiency improves with increased liquid rate but only at the expense of higher pressure drop and energy consumption. Pressure-drop predictions for a given efficiency are hazardous without determining the nature of the particulate and the liquid-to-gas ratio. In general, particles coarser than 1 µm can be collected efficiently with pressure drops of 25 to 50 cm of water. For appreciable collection of submicrometer particles, pressure drops of 75 to 100 cm (30 to 40 in) of water are usually required. When particles are appreciably finer than 0.5 µm, pressure drops of 175 to 250 cm (70 to 100 in) of water have been used. One of the problems in predicting efficiency and required pressure drop of a venturi is the chemical nature or wettability of the particulate, which on 0.5-µm-size particles can make up to a threefold difference in required pressure drop for its efficient collection. Calvert

(b)

Predicted spray-tower cut diameter as a function of sprayed length and spray droplet size for (a) verticalcountercurrent towers and (b) horizontal-cross-flow towers per Calvert [ J. Air Pollut. Control Assoc., 24, 929 (1974)]. Curve 1 is for 200-µm spray droplets, curve 2 for 500-µm spray, and curve 3 for 1000-µm spray. QL /QC is the volumetric liquid-to-gas ratio, L liquid/m3 gas, and uG is the superficial gas velocity in the tower. To convert liters per cubic meter to cubic feet per cubic foot, multiply by 10−3. FIG. 14-126

14-91

14-92

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN ∆P is the pressure drop, cm of water; ρᐉ and ρg are the density of the scrubbing liquid and gas respectively, g/cm3; Ug is the velocity of the gas at the throat inlet, cm/s; Qt /Qg is the volumetric ratio of liquid to gas at the throat inlet, dimensionless; lt is the length of the throat, cm; CDi is the drag coefficient, dimensionless, for the mean liquid diameter, evaluated at the throat inlet; and dl is the Sauter mean diameter, cm, for the atomized liquid. The atomized-liquid mean diameter must be evaluated by the Nukiyama and Tanasawa [Trans. Soc. Mech Eng. ( Japan), 4, 5, 6 (1937–1940)] equation: 0.45 µᐉ Qᐉ 1.5 0.0585 σᐉ 0.5 dᐉ = ᎏ ᎏ + 0.0597 ᎏ (14-237) ᎏ ρᐉ (σᐉρᐉ)0.5 Ug Qg where σᐉ is the liquid surface tension, dyn/cm; and µᐉ is the liquid viscosity; P. The drag coefficient CDi should be evaluated by the Dickinson and Marshall [Am. Inst. Chem. Eng. J., 14, 541 (1968)] correlation CDi = 0.22 + (24/NRei)(1 + 0.15 N 0.6 Rei). The Reynolds number, NRei, is evaluated at the throat inlet considerations as dᐉGg /µg. All venturi scrubbers must be followed by an entrainment collector for the liquid spray. These collectors are usually centrifugal and will have an additional pressure drop of several centimeters of water, which must be added to that of the venturi itself. Other Scrubbers A liquid-ejector venturi (Fig. 17-53), in which high-pressure water from a jet induces the flow of gas, has been used to collect mist particles in the 1- to 2-µm range, but submicrometer particles will generally pass through an eductor. Power costs for liquid pumping are high if appreciable motive force must be imparted to the gas because jet-pump efficiency is usually less than 10 percent. Harris [Chem. Eng. Prog., 42(4), 55 (1966)] has described their application. Two-phase eductors have been considerably more successful on capture of submicrometer mist particles and could be attractive in situations in which large quantities of waste thermal energy are available. However, the equivalent energy consumption is equal to that required for high-energy venturi scrubbers, and such devices are likely to be no more attractive than venturi scrubbers when the thermal energy is priced at its proper value. Sparks [ J. Air Pollut. Control Assoc., 24, 958 (1974)] has discussed steam ejectors giving 99 percent collection of particles 0.3 to 10 µm. Energy requirements were 311,000 J/m3(8.25 Btu/scf). Gardenier [ J. Air Pollut. Control Assoc., 24, 954 (1974)] operated a liquid eductor with high-pressure (6900- to 27,600kPa) (1000- to 4000-lbf/in2) hot water heated to 200°C (392°F) which flashed into two phases as it issued from the jet. He obtained 95 to 99 percent collection of submicrometer particulate. Figure 14-128 shows the water-to-gas ratio required as a function of particle size to achieve 99 percent collection. Effect of Gas Saturation in Scrubbing If hot unsaturated gas is introduced into a wet scrubber, spray particles will evaporate to cool and saturate the gas. The evaporating liquid molecules moving away from the target droplets will repel particles which might collide with them. This results in the forces of diffusiophoresis opposing particle




FIG. 14-127 Prediction of venturi-scrubber cut diameter for hydrophobic particles as functions of operating parameters as measured by Calvert [Calvert, Goldshmid, Leith, and Mehta, NTIS Publ. PB-213016, 213017, 1972; and Calvert, J. Air Pollut. Control Assoc., 24, 929 (1974).] uG is the superficial throat velocity, and ∆P is the pressure drop from converging to diverging section. To convert meters per second to feet per second, multiply by 3.281; to convert liters per cubic meter to cubic feet per cubic foot, multiply by 10−3; and to convert centimeters to inches, multiply by 0.394.

(R-9, R-10) has represented this effect by an empirical factor f, which is based on the hydrophobic ( f = 0.25) or hydrophilic ( f = 0.50) nature of the particles. Figure 14-127 gives the cut diameter of a venturi scrubber as a function of its operating parameters (throat velocity, pressure drop, and liquid-to-gas ratio) for hydrophobic particles. Figure 14-129 compares cut diameter as a function of pressure drop for an otherwise identically operating venturi on hydrophobic and hydrophilic particles. Calvert (R-9) gives equations which can be used for constructing cut-size curves similar to those of Fig. 14-127 for other values of the empirical factor f. Most real particles are neither completely hydrophobic nor completely hydrophilic but have f values lying between the two extremes. Phosphoric acid mist, on the basis of data of Brink and Contant [Ind. Eng. Chem., 50, 1157 (1958)] appears to have a value of f = 0.46. Unfortunately, no chemical-test methods have yet been devised for determining appropriate f values for a particulate in the laboratory. Pressure drop in a venturi scrubber is controlled by throat velocity. While some venturis have fixed throats, many are designed with variable louvers to change throat dimensions and control performance for changes in gas flow. Pressure-drop equations have been developed by Calvert (R-13, R-14, R-15), Boll [Ind. Eng. Chem. Fundam., 12, 40 (1973)], and Hesketh [J. Air Pollut. Control Assoc., 24, 939 (1974)]. Hollands and Goel [Ind. Eng. Chem. Fundam., 14, 16 (1975)] have developed a generalized pressure-drop equation. The Hesketh equation is empirical and is based upon a regression analysis of data from a number of industrial venturi scrubbers: ∆P = Ugt2 ρg A0.155 L0.78/1270 t




x = (3lt CDiρg /16dl ρl) + 1




(14-234)

where ∆P is the pressure drop, in of water; Ugt is the gas velocity in the throat, ft/s; ρg is the gas density, lb/ft3; At is the throat area, ft2; and L is the liquid-to-gas ratio, gal/1000 acf. Calvert (R-15) critiqued the many pressure-drop equations and suggested the following simplified equation as accurate to ⫾10 percent: 2ρᐉUg2 Qt 4− x2)0.5 ] ∆P = ᎏ (14-235) ᎏ [1 − x2 + (x 981gc Qg where



(14-236)

FIG. 14-128 Superheated high-pressure hot-water requirements for 99 percent collection as a function of particle size in a two-phase eductor jet scrubber. To convert gallons per 1000 cubic feet to cubic meters per 1000 cubic meters, multiply by 0.134. [Gardenier, J. Air Pollut. Control Assoc., 24, 954 (1974).]

PHASE SEPARATION

FIG. 14-129 Typical cut diameter as a function of pressure drop for various liquid-particle collectors. Curves 1a and b are single-sieve plates with froth density of 0.4 g/cm3; 1a has sieve holes of 0.5 cm and 1b holes of 0.3 cm. Curves 2a and b are for a venturi scrubber with hydrophobic particles (2a) and hydrophilic particles (2b). Curve 3 is an impingement plate, and curve 4 is a packed column with 2.5-cm-diameter packing. Curve 5 is a zigzag baffle collector with six baffles at θ = 30°. Curve 7 is for six rows of staggered tubes with 1-cm spacing between adjacent tube walls in a row. Curve 8 is similar, except that tube-wall spacing in the row is 0.3 cm. Curve 9 is for wire-mesh pads. To convert grams per cubic centimeter to pounds per cubic foot, multiply by 62.43; to convert centimeters to inches, multiply by 0.394. [Calvert, J. Air Pollut. Control Assoc., 24, 929 (1974); and Calvert, Yung, and Leung, NTIS Publ. PB-248050, 1975.]

4.0 3.0

Aerodynamic cut-diameter, dpca, µm

collection. Semrau and Witham (Air Pollut. Control Assoc. Prepr. 7530.1) investigated temperature parameters in wet scrubbing and found a definite decrease in the efficiency of evaporative scrubbers and an enhancement of efficiency when a hot saturated gas is scrubbed with cold water rather than recirculated hot water. Little improvement was experienced in cooling a hot saturated gas below a 50°C dew point. Energy Requirements for Inertial-Impaction Efficiency Semrau [ J. Air Pollut. Control Assoc., 13, 587 (1963)] proposed a “contacting-power” principle which states that the collecting efficiency of a given size of particle is proportional to the power expended and that the smaller the particle, the greater the power required. Mathematically expressed, NT = ∝ PT␥, where NT is the number of particulate transfer units achieved and PT is the total energy expended within the collection device, including gas and liquid pressure drop and thermal and mechanical energy added in atomizers. NT is further defined as NT = ln [1/(1 − η)], where η is the overall fractional collection efficiency. This was intended as a universal principle, but the constants ∝ and γ have been found to be functions of the chemical nature of the system and the design of the control device. Others have pointed out that the principle is applicable only when the primary collection mechanism is impaction and direct interception. Calvert (R-10, R-12) has found that plotting particle cut size versus pressure drop (or power expended) as in Fig. 18-129 is a more suitable way to develop a generalized energy-requirement curve for impaction devices. The various curves fall close together and outline an imaginary curve that indicates the magnitude of pressure drop required as particle size decreases bound by the two limits of hydrophilic and hydrophobic particles. By calculating the required cut size for a given collection efficiency, Fig. 14-129. can also be used as a guide to deciding between different collection devices. Subsequently, Calvert (R-19, p. 228) has combined mathematical modeling with performance tests on a variety of industrial scrubbers and has obtained a refinement of the power-input/cut-size relationship as shown in Fig. 14-130. He considers these relationships sufficiently reliable to use this data as a tool for selection of scrubber type and performance prediction. The power input for this figure is based solely on gas pressure drop across the device.

14-93

2.0

Packed column

Sieve plate 1.0 Mobile bed

0.5 Venturi

0.4 0.3

0.2 10

20

30

40

50

100

Gas pressure drop, cm of water across wet scrubber collection device Calvert’s refined particle cut-size/power relationship for particle inertial impaction wet collectors. Ref. (R-19) by permission.

FIG. 14-130

Collection of Fine Mists Inertial-impaction devices previously discussed give high efficiency on particles above 5 µm in size and often reasonable efficiency on particles down to 3 µm in size at moderate pressure drops. However, this mechanism becomes ineffective for particles smaller than 3 µm because of the particle gaslike mobility. Only impaction devices having extremely high energy input such as venturi scrubbers and a flooded mesh pad (the pad interstices really become miniature venturi scrubbers in parallel and in series) can give high collection efficiency on fine particles, defined as 2.5 or 3 µm and smaller, including the submicrometer range. Fine particles are subjected to brownian motion in gases, and diffusional deposition can be employed for their collection. Diffusional deposition becomes highly efficient as particles become smaller, especially below 0.2 to 0.3 µm. Table 14-20 shows typical displacement velocity of particles. Randomly oriented fiber beds having tortuous and narrow gas passages are suitable devices for utilizing this collection mechanism. (The diffusional collection mechanism is discussed in Sec. 17 under “GasSolids Separations.”) Other collection mechanisms which are efficient for fine particles are electrostatic forces and flux forces such as thermophoresis and diffusiophoresis. Particle growth and nucleation methods are also applicable. Efficient collection of fine particles is important because particles in the range of 2.0 to around 0.2 µm are the ones which penetrate and are deposited in the lung most efficiently. Hence, particles in this range constitute the largest health hazard. Fiber Mist Eliminators These devices are produced in various configurations. Generally, randomly oriented glass or polypropylene fibers are densely packed between reinforcing screens, producing fiber beds varying in thickness usually from 25 to 75 mm (1 to 3 in), although thicker beds can be produced. Units with efficiencies as high as 99.9 percent on fine particles have been developed (see Chemical Engineers’ Handbook, 5th ed., p. 18–88). A combination of mechanisms interacts to provide high overall collection efficiency. Particles larger than 2 to 3 µm are collected on the fibers by inertial impaction

14-94

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

TABLE 14-20

Brownian Movement of Particles*

Particle diameter, µm

Brownian displacement of particle, µm/s

0.1 0.25 0.5 1.0 2.5 5.0 10.0

29.4 14.2 8.92 5.91 3.58 2.49 1.75

*Brink, Can. J. Chem. Eng., 41, 134 (1963). Based on spherical water particles in air at 21°C and 1 atm.

and direct interception, while small particles are collected by brownian diffusion. When the device is designed to use this latter mechanism as the primary means, efficiency turndown problems are eliminated as collection efficiency by diffusion increases with residence time. Pressure drop through the beds increases with velocity to the first power since the gas flow is laminar. This leads to design capability trade-offs. As pressure drop is reduced and energy is conserved, capital increases because more filtering area is required for the same efficiency. Three series of fiber mist eliminators are typically available. A spray-catcher series is designed primarily for essentially 100 percent capture of droplets larger than 3 µm. The high-velocity type is designed to give moderately high efficiency on particles down to 1.0 µm as well. Both of these types are usually produced in the form of flat panels of 25- to 50-mm (1- to 2-in) thickness. The highefficiency type is illustrated in Fig. 14-131. As mist particles are collected, they coalesce into a liquid film which wets the fibers. Liquid is moved horizontally through the bed by the gas drag force and downward by gravity. It drains down the downstream retaining screen to the bottom of the element and is returned to the process through a liquid seal. Table 14-21 gives typical operating characteristics of the three types of collectors. The application of these devices to sulfuric acid plants and other process gases has been discussed by Brink (see Chemical Engineers’ Handbook, 5th ed., pp. 18–89, 18–90). Solid particulates are captured as readily as liquids in fiber beds but can rapidly plug the bed if they are insoluble. Fiber beds have frequently been used for mixtures of liquids and soluble solids and with soluble solids in condensing situations. Sufficient solvent (usually water) is atomized into the gas stream entering the collector to irrigate the fiber elements and dissolve the collected particulate. Such fiber beds have been used to collect fine fumes such as ammonium nitrate and ammonium chloride smokes, and oil mists from compressed air. Electrostatic Precipitators The principles and operation of electrical precipitators are discussed in Sec. 17 under “Gas-Solids Separations.” Precipitators are admirably suited to the collection of fine mists and mixtures of mists and solid particulates. Tube-type precipitators have been used for many years for the collection of acid mists and the removal of tar from coke-oven gas. The first practical installation of a precipitator by Cottrell was made on sulfuric acid mist in 1907. Most older installations of precipitators were tube-type rather than plate-type. However, recently two plate-type wet precipitators employing water sprays or overflowing weirs have been introduced by Mikropul Corporation [Bakke, J. Air Pollut. Control Assoc., 25, 163 (1975)] and by Fluid Ionics. Such precipitators operate on the principle of making all particles conductive when possible, which increases the particle migration velocity and collection efficiency. Under these conditions, particle dielectric strength becomes a much more imporTABLE 14-21

FIG. 14-131 Monsanto high-efficiency fiber-mist-eliminator element. (Monsanto Company.)

tant variable, and particles with a low dielectric constant such as condensed hydrocarbon mists become much more difficult to collect than water-wettable particles. Bakke (U.S.–U.S.S.R. Joint Work. Group Symp.: Fine Particle Control, San Francisco, 1974) has developed equations for particle charge and relative collection efficiency in wet precipitators that show the effect of dielectric constant. Wet precipitators can also be used to absorb soluble gases simultaneously by adjusting the pH or the chemical composition of the liquid spray. The presence of the electric field appears to enhance absorption. Wet precipitators have found their greatest usefulness to date in handling mixtures of gaseous pollutants and submicrometer particulate (either liquid or solid, or both) such as fumes from aluminum-pot lines, carbon anode baking, fiberglass-fume control, coke-oven and metallurgical operations, chemical incineration, and phosphate-fertilizer operations. Two-stage precipitators are used increasingly for moderate-volume gas streams containing nonconductive liquid mists which will drain from the collecting plates. Their application on hydrocarbon mists has been quite successful, but careful attention must be given to fire and explosion hazards. Electrically Augmented Collectors A new area for enhancing collection efficiency and lowering cost is the combining of electrostatic forces with devices using other collecting mechanisms such as impaction and diffusion. Cooper (Air Pollut. Control Assoc. Prepr. 7502.1) evaluated the magnitude of forces operating between charged and uncharged particles and concluded that electrostatic attraction is the strongest collecting force operating on particles finer than 2 µm. Nielsen and Hill [Ind. Eng. Chem. Fundam., 15, 149 (1976)] have quantified these relationships, and a number of practical devices have been demonstrated. Pilat and Meyer (NTIS Publ. PB-252653, 1976) have demonstrated up to 99 percent collection of fine particles in a two-stage spray tower in which the inlet particles and water spray are charged with opposite polarity. The principle has been applied to retrofitting existing spray towers to enhance collection. Klugman and Sheppard (Air Pollut. Control Assoc. Prepr. 75-30.3) have developed an ionizing wet scrubber in which the charged mist particles are collected in a grounded, irrigated cross-flow bed of Tellerette packing. Particles smaller than 1 µm have been collected

Operating Characteristics of Various Types of Fiber Mist Eliminators as Used on Sulfuric Acid Plants*

Controlling mechanism for mist collection Superficial velocity, m/s Efficiency on particles greater than 3 µm, % Efficiency on particles 3 µm and smaller, % Pressure drop, cm H2O

High efficiency

High velocity

Spray catcher

Brownian movement 0.075–0.20 Essentially 100 95–99+ 12–38

Impaction 2.0–2.5 Essentially 100 90–98 15–20

Impaction 2.0–2.5 Essentially 100 15–30 1.0–2.5

*Brink, Burggrabe, and Greenwell, Chem. Eng. Prog., 64(11), 82 (1968). To convert centimeters to inches, multiply by 0.394.

PHASE SEPARATION with 98 percent efficiency by using two units in series. Dembinsky and Vicard (Air Pollut. Control Assoc. Prepr. 78-17.6) have used an electrically augmented low-pressure [5 to 10 cm (2 to 4 in) of water] venturi scrubber to give 95 to 98 percent collection efficiency on submicrometer particles. Particle Growth and Nucleation Fine particles may be subjected to conditions favoring the growth of particles either through condensation or through coalescence. Saturation of a hot gas stream with water, followed by condensation on the particles acting as nuclei when the gas is cooled, can increase particle size and ease of collection. Addition of steam can produce the same results. Scrubbing of the humid gas with a cold liquid can bring diffusiophoresis into play. The introduction of cold liquid drops causes a reduction in watervapor pressure at the surface of the cold drop. The resulting vaporpressure gradient causes a hydrodynamic flow toward the drop known as Stefan flow which enhances the movement of mist particles toward the spray drop. If the molecular mass of the diffusing vapor is different from the carrier gas, this density difference also produces a driving force, and the sum of these forces is known as diffusiophoresis. A mathematical description of these forces has been presented by Calvert (R-9) and by Sparks and Pilat [Atmos. Environ., 4, 651 (1970)]. Thermal differences between the carrier gas and the cold scrubbing droplets can further enhance collection through thermophoresis. Calvert and Jhaseri [ J. Air Pollut. Control Assoc., 24, 946 (1974)]; and NTIS Publ. PB-227307, 1973)] have investigated condensation scrubbing in multiple-sieve plate towers. Submicrometer droplets can be coagulated through brownian diffusion if given ample time. The introduction of particles 50 to 100 times larger in diameter can enhance coagulation, but the addition of a broad range of particle sizes is discouraged. Increasing turbulence will aid coagulation, so fans to stir the gas or narrow, tortuous passages such as those of a packed bed can be beneficial. Sonic energy can also produce coagulation, especially the production of standing waves in the confines of long, narrow tubes. Addition of water and oil mists can sometimes aid sonic coagulation. Sulfuric acid mist [Danser, Chem. Eng., 57(5), 158 (1950)] and carbon black [Stokes, Chem. Eng. Prog., 46, 423 (1950)] have been successfully agglomerated with sonic energy. Frequently sonic agglomeration has been unsuccessful because of the high energy requirement. Most sonic generators have very poor energy-transformation efficiency. Wegrzyn et al. (U.S. EPA Publ. EPA-600/7-79-004C, 1979, p. 233) have reviewed acoustic agglomerators. Mednikov (U.S.S.R. Akad. Soc. Moscow, 1963) suggested that the incorporation of sonic agglomeration with electrostatic precipitation could greatly reduce precipitator size. Other Collectors Tarry particulates and other difficult-tohandle liquids have been collected on a dry, expendable phenol formaldehyde-bonded glass-fiber mat (Goldfield, J. Air Pollut. Control Assoc., 20, 466 (1970)] in roll form which is advanced intermittently into a filter frame. Superficial gas velocities are 2.5 to 3.5 m/s (8.2 to 11.5 ft/s), and pressure drop is typically 41 to 46 cm (16 to 18 in) of water. Collection efficiencies of 99 percent have been obtained on submicrometer particles. Brady [Chem. Eng. Prog., 73(8), 45 (1977)] has discussed a cleanable modification of this approach in which the gas is passed through a reticulated foam filter that is slowly rotated and solvent-cleaned. In collecting very fine (mainly submicron) mists of a hazardous nature where one of the collectors previously discussed has been used as the primary one (fiber-mist eliminators of the Brownian diffusion type and electrically augmented collectors are primarily recommended), there is the chance that the effluent concentration may still be too high for atmospheric release when residual concentration must be in the range of 1–2 µm. In such situations, secondary treatment may be needed. Probably removal of the residual mist by adsorption will be in order. See “Adsorption,” Sec. 16. Another possibility might be treatment of the remaining gas by membrane separation. A separator having a gas-permeable membrane that is essentially nonliquidpermeable could be useful. However, if the gas-flow volumes are appreciable, the device could be expensive. Most membranes have low capacity (requiring high membrane surface area) to handle high gas-permeation capacity. See “Membrane Processes,” Sec. 22. Continuous Phase Uncertain Some situations exist such as in

14-95

two-phase gas-liquid flow where the volume of the liquid phase may approach being equal to the volume of the vapor phase, and where it may be difficult to be sure which phase is the continuous phase. Svrcek and Monnery [Chem. Eng. Prog., 89(10), 53–60 (Oct. 1993)] have discussed the design of two-phase separation in a tank with gasliquid separation in the middle, mist elimination in the top, and entrained gas-bubble removal from the liquid in the bottom. Monnery and Svrcek [Chem. Eng. Prog., 90(9), 29–40 (Sept. 1994)] have expanded the separation to include multiphase flow, where the components are a vapor and two immiscible liquids and these are also separated in a tank. A design approach for sizing the gas-liquid disengaging space in the vessel is given using a tangential tank inlet nozzle, followed by a wire mesh mist eliminator in the top of the vessel for final separation of entrained mist from the vapor. Design approaches and equations are also given for sizing the lower portion of the vessel for separation of the two immiscible liquid phases by settling and separation of discontinuous liquid droplets from the continuous liquid phase. LIQUID-PHASE CONTINUOUS SYSTEMS Practical separation techniques for gases dispersed in liquids are discussed. Processes and methods for dispersing gas in liquid have been discussed earlier in this section, together with information for predicting the bubble size produced. Gas-in-liquid dispersions are also produced in chemical reactions and electrochemical cells in which a gas is liberated. Such dispersions are likely to be much finer than those produced by the dispersion of a gas. Dispersions may also be unintentionally created in the vaporization of a liquid. GENERAL REFERENCES: Adamson, Physical Chemistry of Surfaces, 4th ed., Wiley, New York, 1982. Akers, Foams, Academic, New York, 1976. Bikerman, Foams, Springer-Verlag, New York, 1973. Bikerman, et al., Foams: Theory and Industrial Applications, Reinhold, New York, 1953. Cheremisinoff, ed., Encyclopedia of Fluid Mechanics, vol. 3, Gulf Publishing, Houston, 1986. Kerner, Foam Control Agents, Noyes Data Corp, Park Ridge, NJ, 1976. Rubel, Antifoaming and Defoaming Agents, Noyes Data Corp., Park Ridge, NJ, 1972. Rosen, Surfactants and Interfacial Phenomena, 2d ed., Wiley, New York, 1989. Sonntag and Strenge, Coagulation and Stability of Disperse Systems, HalstedWiley, New York, 1972. Wilson, ed., Foams: Physics, Chemistry and Structure, Springer-Verlag, London, 1989. “Defoamers” and “Foams”, Encyclopedia of Chemical Technology, 4th ed., vols. 7, 11, Wiley, New York, 1993–1994.

Types of Gas-in-Liquid Dispersions Two types of dispersions exist. In one, gas bubbles produce an unstable dispersion which separates readily under the influence of gravity once the mixture has been removed from the influence of the dispersing force. Gas-liquid contacting means such as bubble towers and gas-dispersing agitators are typical examples of equipment producing such dispersions. More difficulties may result in separation when the gas is dispersed in the form of bubbles only a few micrometers in size. An example is the evolution of gas from a liquid in which it has been dissolved or released through chemical reaction such as electrolysis. Coalescence of the dispersed phase can be helpful in such circumstances. The second type is a stable dispersion, or foam. Separation can be extremely difficult in some cases. A pure two-component system of gas and liquid cannot produce dispersions of the second type. Stable foams can be produced only when an additional substance is adsorbed at the liquid-surface interface. The substance adsorbed may be in true solution but with a chemical tendency to concentrate in the interface such as that of a surface-active agent, or it may be a finely divided solid which concentrates in the interface because it is only poorly wetted by the liquid. Surfactants and proteins are examples of soluble materials, while dust particles and extraneous dirt including traces of nonmiscible liquids can be examples of poorly wetted materials. Separation of gases and liquids always involves coalescence, but enhancement of the rate of coalescence may be required only in difficult separations. Separation of Unstable Systems The buoyancy of bubbles suspended in liquid can frequently be depended upon to cause the bubbles to rise to the surface and separate. This is a special case of gravity settling. The mixture is allowed to stand at rest or is moved along a

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GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

TABLE 14-22

Terminal Velocity of Standard Air Bubbles Rising in Water at 20∞C*

Bubble diameter, µm

10

30

50

100

200

300

Terminal velocity, mm/s

0.061

0.488

1.433

5.486

21.95

49.38

*Calculated from Stokes’ law. To convert millimeters per second to feet per second, multiply by 0.003281.

flow path in laminar flow until the bubbles have surfaced. Table 14-22 shows the calculated rate of rise of air bubbles at atmospheric pressure in water at 20°C (68°F) as a function of diameter. It will be observed that the velocity of rise for 10-µm bubbles is very low, so that long separating times would be required for gas which is more finely dispersed. For liquids other than water, the rise velocity can be approximated from Table 14-22 by multiplying by the liquid’s specific gravity and the reciprocal of its viscosity (in centipoises). For bubbles larger than 100 µm, this procedure is erroneous, but the error is less than 15 percent for bubbles up to 1000 µm. More serious is the underlying assumption of Table 14-22 that the bubbles are rigid spheres. Circulation within the bubble causes notable increases in velocity in the range of 100 µm to 1 mm, and the flattening of bubbles 1 cm and larger appreciably decreases their velocity. However, in this latter size range the velocity is so high as to make separation a trivial problem. In design of separating chambers, static vessels or continuous-flow tanks may be used. Care must be taken to protect the flow from turbulence, which could cause back mixing of partially separated fluids or which could carry unseparated liquids rapidly to the separated-liquid outlet. Vertical baffles to protect rising bubbles from flow currents are sometimes employed. Unseparated fluids should be distributed to the separating region as uniformly and with as little velocity as possible. When the bubble rise velocity is quite low, shallow tanks or flow channels should be used to minimize the residence time required. Quite low velocity rise of bubbles due either to small bubble size or to high liquid viscosity can cause difficult situations. With low-viscosity liquids, separation-enhancing possibilities in addition to those previously enumerated are to sparge the liquid with large-diameter gas bubbles or to atomize the mixture as a spray into a tower. Large gas bubbles rising rapidly through the liquid collide with small bubbles and aid their coalescence through capture. Atomizing of the continuous phase reduces the distance that small gas bubbles must travel to reach a gas interface. Evacuation of the spray space can also be beneficial in promoting small-bubble growth and especially in promoting gas evolution when the gas has appreciable liquid solubility. Liquid heating will also reduce solubility. Surfaces in the settling zone for bubble coalescence such as closely spaced vertical or inclined plates or tubes are beneficial. When clean low-viscosity fluids are involved, passage of the undegassed liquid through a tightly packed pad of mesh or fine fibers at low velocity will result in efficient bubble coalescence. Problems have been experienced in degassing a water-based organic solution that has been passed through an electrolytic cell for chemical reaction in which extremely fine bubbles of hydrogen gas are produced in the liquid within the cell. Near-total removal of hydrogen gas from the liquid is needed for process safety. This is extremely difficult to achieve by gravity settling alone because of the fine bubble size and the need for a coalescing surface. Utilization of a fine fiber media is strongly recommended in such situations. A low-forward liquid flow through the media is desireable to provide time for the bubbles to attach themselves to the fiber media through Brownian diffusion. Spielman and Goren [Ind. Eng. Chem., 62(10), (1970)] reviewed the literature on coalescence with porous media and reported their own experimental results [Ind. Eng. Chem. Fundam., 11(1), 73 (1972)] on the coalescence of oil-water liquid emulsions. The principles are applicable to a gas-in-liquid system. Glass-fiber mats composed of 3.5-, 6-, or 12-µm diameter fibers, varying in thickness from 1.3 to 3.3 mm, successfully coalesced and separated 1- to 7-µm oil droplets at superficial bed velocities of 0.02 to 1.5 cm/s (0.00067 to 0.049 ft/s). In the deaeration of high-viscosity fluids such as polymers, the material is flowed in thin sheets along solid surfaces. Vacuum is applied to increase bubble size and hasten separation. The Versator (Cornell Machine Co.) degasses viscous liquids by spreading them

into a thin film by centrifugal action as the liquids flow through an evacuated rotating bowl. Separation of Foam Foam is a colloidal system containing relatively large volumes of dispersed gas in a relatively small volume of liquid. Foams are thermodynamically unstable with respect to separation into their components of gas and vapor, and appreciable surface energy is released in the bursting of foam bubbles. Foams are dynamic systems in which a third component produces a surface layer that is different in composition from the bulk of the liquid phase. The stabilizing effect of such components (often present only in trace amounts) can produce foams of troubling persistence in many operations. (Foams which have lasted for years when left undisturbed have been produced.) Bendure [TAPPI, 58(2), 83 (1975)], Keszthelyi [ J. Paint Technol., 46(11), 31 (1974)], Ahmad [Sep. Sci. 10, 649 (1975)], and Shedlovsky (“Foams,” Encyclopedia of Chemical Technology, 2d ed., Wiley, New York, 1966) have presented concise articles on the characteristics and properties of foams in addition to the general references cited at the beginning of this subsection. Foams can be a severe problem in chemical-processing steps involving gas-liquid interaction such as distillation, absorption, evaporation, chemical reaction, and particle separation and settling. It can also be a major problem in pulp and paper manufacture, oil-well drilling fluids, production of water-based paints, utilization of lubricants and hydraulic fluids, dyeing and sizing of textiles, operation of steam boilers, fermentation operations, polymerization, wet-process phosphoric acid concentration, adhesive production, and foam control in products such as detergents, waxes, printing inks, instant coffee, and glycol antifreeze. Foams, as freshly generated, are gas emulsions with spherical bubbles separated by liquid films up to a few millimeters in thickness. They age rapidly by liquid drainage and form polyhedrals in which three bubbles intersect at corners with angles of approximately 120°. During drainage, the lamellae become increasingly thinner, especially in the center (only a few micrometers thickness), and more brittle. This feature indicates that with some foams if a foam layer can be tolerated, it may be self-limiting, as fresh foam is added to the bottom of the layer with drained foam collapsing on the top. (A quick-breaking foam may reach its maximum life cycle in 6 s. A moderately stable foam can persist for 140 s.) During drainage, gas from small foam bubbles, which is at a high pressure, will diffuse into large bubbles so that foam micelles increase with time. As drainage proceeds, weak areas in the lamella may develop. However, the presence of a higher concentration of surfactants in the surface produces a lower surface tension. As the lamella starts to fail, exposing bulk liquid with higher surface tension, the surface is renewed and healed. This is known as the Marangoni effect. If drainage can occur faster than Marangoni healing, a hole may develop in the lamella. The forces involved are such that collapse will occur in milliseconds without concern for rupture propagation. However, in very stable foams, electrostatic surface forces (zeta potential) prevent complete drainage and collapse. In some cases, stable lamella thicknesses of only two molecules have been measured. Drainage rate is influenced by surface viscosity, which is very temperature-sensitive. At a critical temperature, which is a function of the system, a temperature change of only a few degrees can change a slow-draining foam to a fast-draining foam. This change in drainage rate can be a factor of 100 or more; thus increasing the temperature of foam can cause its destruction. An increase in temperature may also cause liquid evaporation and lamella thinning. As the lamellae become thinner, they become more brittle and fragile. Thus, mechanical deformation or pressure changes, which cause a change in gasbubble volume, can also cause rupture. Bendure indicates 10 ways to increase foam stability: (1) increase bulk liquid viscosity, (2) increase surface viscosity, (3) maintain thick

PHASE SEPARATION walls (higher liquid-to-gas ratio), (4) reduce liquid surface tension, (5) increase surface elasticity, (6) increase surface concentration, (7) reduce surfactant-adsorption rate, (8) prevent liquid evaporation, (9) avoid mechanical stresses, and (10) eliminate foam inhibitors. Obviously, the reverse of each of these actions, when possible, is a way to control and break foam. Physical Defoaming Techniques Typical physical defoaming techniques include mechanical methods for producing foam stress, thermal methods involving heating or cooling, and electrical methods. Combinations of these methods may also be employed, or they may be used in conjunction with chemical defoamers. Some methods are only moderately successful when conditions are present to reform the foam such as breaking foam on the surface of boiling liquids. In some cases it may be desirable to draw the foam off and treat it separately. Foam can always be stopped by removing the energy source creating it, but this is often impractical. Thermal Methods Heating is often a suitable means of destroying foam. As indicated previously, raising the foam above a critical temperature (which must be determined experimentally) can greatly decrease the surface viscosity of the film and change the foam from a slow-draining to a fast-draining foam. Coupling such heating with a mechanical force such as a revolving paddle to cause foam deformation is frequently successful. Other effects of heating are expansion of the gas in the foam bubbles, which increases strain on the lamella walls as well as requiring their movement and flexing. Evaporation of solvent may occur causing thinning of the walls. At sufficiently high temperatures, desorption or decomposition of stabilizing substances may occur. Placing a high-temperature bank of steam coils at the maximum foam level is one control method. As the foam approaches or touches the coil, it collapses. The designer should consider the fact that the coil will frequently become coated with solute. Application of radiant heat to a foam surface is also practiced. Depending on the situation, the radiant source may be electric lamps, Glowbar units, or gas-fired radiant burners. Hot gases from burners will enhance film drying of the foam. Heat may also be applied by jetting or spraying hot water on the foam. This is a combination of methods since the jetting produces mechanical shear, and the water itself provides dilution and change in foam-film composition. Newer approaches might include foam heating with the application of focused microwaves. This could be coupled with continuous or intermittent pressure fluctuations to stress lamella walls as the foam ages. Cooling can also destroy foam if it is carried to the point of freezing since the formation of solvent crystals destroys the foam structure. Less drastic cooling such as spraying a hot foam with cold water may be effective. Cooling will reduce the gas pressure in the foam bubbles and may cause them to shrink. This is coupled with the effects of shear and dilution mentioned earlier. In general, moderate cooling will be less effective than heating since the surface viscosity is being modified in the direction of a more stable foam. Mechanical Methods Static or rotating breaker bars or slowly revolving paddles are sometimes successful. Their application in conjunction with other methods is frequently better. As indicated in the theory of foams, they will work better if installed at a level at which the foam has had some time to age and drain. A rotating breaker works by deforming the foam, which causes rupture of the lamella walls. Rapidly moving slingers will throw the foam against the vessel wall and may cause impact on other foam outside the envelope of the slinger. In some instances, stationary bars or closely spaced plates will limit the rise of foam. The action here is primarily one of providing surface for coalescence of the foam. Wettability of the surface, whether moving or stationary, is frequently important. Usually a surface not wetted by the liquid is superior, just as is frequently the case of porous media for foam coalescence. However, in both cases there are exceptions for which wettable surfaces are preferred. Shkodin [Kolloidn. Zh., 14, 213 (1952)] found molasses foam to be destroyed by contact with a wax-coated rod and unaffected by a clean glass rod. Goldberg and Rubin [Ind. Eng. Chem. Process Des. Dev., 6 195 (1967)] showed in tests with a disk spinning vertically to the foam layer that most mechanical procedures, whether centrifugation, mixing, or blowing through nozzles, consist basically of the application of shear stress. Subjecting foam to an air-jet impact can also provide a source

14-97

of drying and evaporation from the film, especially if the air is heated. Other effective means of destroying bubbles are to lower a frame of metal points periodically into the foam or to shower the foam with falling solid particles. Pressure and Acoustic Vibrations These methods for rupturing foam are really special forms of mechanical treatment. Change in pressure in the vessel containing the foam stresses the lamella walls by expanding or contracting the gas inside the foam bubbles. Oscillation of the vessel pressure subjects the foam to repeated film flexing. Parlow [Zucker, 3, 468 (1950)] controlled foam in sugar-sirup evaporators with high-frequency air pulses. It is by no means certain that highfrequency pulsing is necessary in all cases. Lower frequency and higher amplitude could be equally beneficial. Acoustic vibration is a similar phenomenon causing localized pressure oscillation by using sound waves. Impulses at 6 kHz have been found to break froth from coal flotation [Sun, Min. Eng., 3, 865 (1958)]. Sonntag and Strenge (Coagulation and Stability of Disperse Systems, Halsted-Wiley, New York, 1972, p. 121) report foam suppression with high-intensity sound waves (11 kHz, 150 dB) but indicate that the procedure is too expensive for large-scale application. The Sontrifuge (Teknika Inc., a subsidiary of Chemineer, Inc.) is a commercially available low-speed centrifuge employing sonic energy to break the foam. Walsh [Chem. Process., 29, 91 (1966)], Carlson [Pap. Trade J., 151, 38 (1967)], and Thorhildsen and Rich [TAPPI, 49, 95A (1966)] have described the unit. Electrical Methods As colloids, most foams typically have electrical double layers of charged ions which contribute to foam stability. Accordingly, foams can be broken by the influence of an external electric field. While few commercial applications have been developed, Sonntag and Strenge (op. cit., p. 114) indicate that foams can be broken by passage through devices much like electrostatic precipitators for dusts. Devices similar to two-stage precipitators having closely spaced plates of opposite polarity should be especially useful. Sonntag and Strenge, in experiments with liquid-liquid emulsions, indicate that the colloid structure can be broken at a field strength of the order of 8 to 9 × 105 V/cm. Chemical Defoaming Techniques Sonntag and Strenge (op. cit., p. 111) indicate two chemical methods for foam breaking. One method is causing the stabilizing substances to be desorbed from the interface, such as by displacement with other more surface-active but nonstabilizing compounds. Heat may also cause desorption. The second method is to carry on chemical changes in the adsorption layer, leading to a new structure. Some defoamers may act purely by mechanical means but will be discussed in this subsection since their action is generally considered to be chemical in nature. Often chemical defoamers act in more than one way. Chemical Defoamers The addition of chemical foam breakers is the most elegant way to break a foam. Effective defoamers cause very rapid disintegration of the foam and frequently need be present only in parts per million. The great diversity of compounds used for defoamers and the many different systems in which they are applied make a brief and orderly discussion of their selection difficult. Compounds needed to break aqueous foams may be different from those needed for aqueous-free systems. The majority of defoamers are insoluble or nonmiscible in the foam continuous phase, but some work best because of their ready solubility. Lichtman (Defoamers, 3d ed., Wiley, New York, 1979) has presented a concise summary of the application and use of defoamers. Rubel (Antifoaming and Defoaming Agents, Noyes Data Corp., Park Ridge, N.J., 1972) has reviewed the extensive patent literature on defoamers. Defoamers are also discussed extensively in the general references at the beginning of this subsection. One useful method of aqueous defoaming is to add a nonfoam stabilizing surfactant which is more surface-active than the stabilizing substance in the foam. Thus a foam stabilized with an ionic surfactant can be broken by the addition of a very surface-active but nonstabilizing silicone oil. The silicone displaces the foam stabilizer from the interface by virtue of its insolubility. However, it does not stabilize the foam because its foam films have poor elasticity and rupture easily. A major requirement for a defoamer is cost-effectiveness. Accordingly, some useful characteristics are low volatility (to prevent strip-

14-98

GAS ABSORPTION AND GAS-LIQUID SYSTEM DESIGN

ping from the system before it is dispersed and does its work), ease of dispersion and strong spreading power, and surface attractionorientation. Chemical defoamers must also be selected in regard to their possible effect on product quality and their environmental and health suitability. For instance, silicone antifoam agents are effective in textile jet dyeing but reduce the fire retardancy of the fabric. Mineral-oil defoamers in sugar evaporation have been replaced by specifically approved materials. The tendency is no longer to use a single defoamer compound but to use a formulation specially tailored for the application comprising carriers, secondary antifoam agents, emulsifiers, and stabilizing agents in addition to the primary defoamer. Carriers, usually hydrocarbon oils or water, serve as the vehicle to support the release and spread of the primary defoamer. Secondary defoamers may provide a synergistic effect for the primary defoamer or modify its properties such as spreadability or solubility. Emulsifiers may enhance the speed of dispersion, while stabilizing agents may enhance defoamer stability or shelf life. Hydrophobic silica defoamers work on a basis which may not be chemical at all. They are basically finely divided solid silica particles dispersed in a hydrocarbon or silicone oil which serves as a spreading vehicle. Kulkarni [Ind. Eng. Chem. Fundam., 16, 472 (1977)] theorizes that this mixture defoams by the penetration of the silica particle into the bubble and the rupture of the wall. Table 14-23 lists major types of defoamers and typical applications. Other Chemical Methods These methods rely chiefly on destroying the foam stabilizer or neutralizing its effect through methods other than displacement and are applicable when the process will permit changing the chemical environment. Forms stabilized with alkali esters can be broken by acidification since the equivalent free acids do not stabilize foam. Foams containing sulfated and sulfonated ionic detergents can be broken with the addition of fatty-acid soaps and calcium salts. Several theories have been proposed. One suggests that the surfactant is tied up in the foam as double calcium salts of both the sulfonate and the soap. Another suggests that calcium soaps oriented in the film render it inelastic. Ionic surfactants adsorb at the foam interface and orient with the

TABLE 14-23

charged group immersed in the lamellae and their uncharged tails pointed into the gas stream. As the film drains, the charged groups, which repel each other, tend to be moved more closely together. The repulsive force between like charges hinders drainage and stabilizes the film. Addition of a salt or an electrolyte to the foam screens the repulsive effect, permits additional drainage, and can reduce foam stability. Foam Prevention Chemical prevention of foam differs from defoaming only in that compounds or mixtures are added to a stream prior to processing to prevent the formation of foam either during processing or during customer use. Such additives, sometimes distinguished as antifoam agents, are usually in the same chemical class of materials as defoamers. However, they are usually specifically formulated for the application. Typical examples of products formulated with antifoam agents are laundry detergents (to control excess foaming), automotive antifreeze, instant coffee, and jet-aircraft fuel. Foaming in some chemical processes such as distillation or evaporation may be due to trace impurities such as surface-active agents. An alternative to antifoam agents is their removal before processing such as by treatment with activated carbon [Pool, Chem. Process., 21(9), 56 (1958)]. Automatic Foam Control In processing materials when foam can accumulate, it is often desirable to measure the height of the foam layer continuously and to dispense defoamer automatically as required to control the foam. Other corrective action can also be taken automatically. Methods of sensing the foam level have included electrodes in which the electrical circuit is completed when the foam touches the electrode [Nelson, Ind. Eng. Chem., 48, 2183 (1956); and Browne, U.S. Patent 2,981,693, 1961], floats designed to rise in a foam layer (Carter, U.S. Patent 3,154,577, 1964), and change in power input required to turn a foam-breaking impeller as the foam level rises (Yamashita, U.S. Patent 3,317,435, 1967). Timers to control the duration of defoamer addition have also been used. Browne has suggested automatic addition of defoamer through a porous wick when the foam level reaches the level of the wick. Foam control has also been discussed by Kroll [Ind. Eng. Chem., 48, 2190 (1956)].

Major Types and Applications of Defoamers

Classification

Examples

Silicones

Dimethyl silicone, trialkyl and tetraalkyl silanes

Aliphatic acids or esters

Mostly high-molecular-weight compounds; diethyl phthalate; lauric acid Moderate- to high-molecular-weight monohydric and polyhydric alcohols; octyl alcohol; C-12 to C-20 alcohols; lauryl alcohol Alkali metal salts of sulfated alcohols, sulfonic acid salts; alkyl-aryl sulfonates; sodium lauryl sulfate

Alcohols Sulfates or sulfonates Amines or amides Halogenated compounds Natural products Fatty-acid soaps Inorganic compounds Phosphates Hydrophobic silica Sulfides or thio derivatives

Alkyl amines (undecyloctyl and diamyl methyl amine); polyamides (acyl derivatives of piperazine) Fluochloro hydrocarbons with 5 to 50 C atoms; chlorinated hydrocarbons Vegetable oils; waxes, mineral oils plus their sulfated derivatives (including those of animal oils and fats) Alkali, alkaline earth, and other metal soaps; sodium stearate; aluminum stearate Monosodium phosphate mixed with boric acid and ethyl carbonate, disodium phosphate; sodium aluminate, bentonite and other solids Alkyl-alkalene diphosphates; tributyl phosphate in isopropanol Finely divided silica in polydimethyl siloxane Metallic derivatives of thio ethers and disulfides, usually mixed with organic phosphite esters; long-chain alkyl thienyl ketones

Applications Lubricating oils; distillation; fermentation; jam and wine making; food processing Papermaking; wood-pulp suspensions; water-based paints; food processing Distillation; fermentation; papermaking; glues and adhesives Nonaqueous systems; mixed aqueous and nonaqueous systems; oil-well drilling muds; spent H3SO4 recovery; deep-fat frying Boiler foam; sewage foam; fermentation; dye baths Lubrication-oil and grease distillation; vegetable-protein glues Sugar extraction; glue manufacture; cutting oils Gear oils; paper stock; paper sizing; glue solutions Distillation; instant coffee; boiler feedwater; sugar extraction Petroleum-oil systems; foam control in soap solutions Aqueous foaming systems Lubricating oils; boiler water

Section 15

Liquid-Liquid Extraction Operations and Equipment

Lanny A. Robbins, Ph.D., Research Fellow, Dow Chemical Company; Member, American Institute of Chemical Engineers Roger W. Cusack, Vice President, Glitsch Process Systems, Inc.; Member, American Institute of Chemical Engineers

LIQUID-LIQUID EXTRACTION OPERATIONS Uses for Liquid-Liquid Extraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15-4 15-5

PHASE EQUILIBRIA Equilibrium Partition Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phase Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermodynamic Basis of Liquid-Liquid Equilibria . . . . . . . . . . . . . . . . Example 1: Partition Ratios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hydrogen-Bonding Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Equilibrium Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Desirable Solvent Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15-6 15-6 15-8 15-8 15-8 15-9 15-9

CALCULATION METHODS Single Stage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Crosscurrent Theoretical Stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Countercurrent Theoretical Stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Right-Triangular Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 2: Stage and Composition Calculation . . . . . . . . . . . . . . Shortcut Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 3: Shortcut Calculation, Case A. . . . . . . . . . . . . . . . . . . . Example 4: Shortcut Calculation, Case B. . . . . . . . . . . . . . . . . . . . Countercurrent Mass-Transfer-Unit Calculations. . . . . . . . . . . . . . . . . . Example 5: Number of Transfer Units . . . . . . . . . . . . . . . . . . . . . . Stage Efficiency and Height of a Theoretical Stage or Transfer Unit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fractionation Stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Countercurrent Liquid-Liquid Heat Transfer. . . . . . . . . . . . . . . . . . . . . LIQUID-LIQUID EXTRACTION EQUIPMENT Stagewise Equipment (Mixer-Settlers) . . . . . . . . . . . . . . . . . . . . . . . . . .

15-16 15-16 15-16 15-16 15-17 15-17 15-18 15-19 15-19 15-20 15-20 15-20 15-22

15-22

Rates of Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous-Phase Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dispersed-Phase Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overall Coefficients and Stage Efficiency . . . . . . . . . . . . . . . . . . . . . . Scale-Up of Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Settlers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Emulsions and Dispersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sedimentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coalescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gravity Settlers; Decanters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cyclones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Centrifuges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Settler Auxiliaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coalescers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Separating Membranes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electrical Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mixer-Settler Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overall Stage Efficiencies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous (Differential) Contact Equipment . . . . . . . . . . . . . . . . . . . . General Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Axial Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equipment Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spray Towers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Packed Towers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Perforated-Plate (Sieve-Plate) Towers. . . . . . . . . . . . . . . . . . . . . . . . . Mechanically Agitated Gravity Devices . . . . . . . . . . . . . . . . . . . . . . . . Rotary-Disk Contactors (RDC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lightnin Mixer (Oldshue-Rushton) Tower . . . . . . . . . . . . . . . . . . . . . Scheibel Extraction Towers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kühni Tower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Treybal Tower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Karr Reciprocating Plate Tower. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pulsed Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Centrifugal Extractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15-22 15-22 15-23 15-23 15-24 15-25 15-25 15-26 15-26 15-26 15-27 15-27 15-27 15-27 15-27 15-28 15-28 15-29 15-29 15-30 15-30 15-30 15-30 15-32 15-34 15-37 15-37 15-38 15-40 15-42 15-42 15-42 15-44 15-46

15-1

15-2

LIQUID-LIQUID EXTRACTION OPERATIONS AND EQUIPMENT

Nomenclature Symbol A° Ap At a ap B b C CO c cp D D′ d dF dFC di dO dp dpJ dr ds dt E E′ Ef Ed EMD EO e e F F′ fe fr g gc He Hor Hr

Definition Activity of solute Area of a drop Cross-sectional area of tower Specific interfacial surface between liquids Specific packing surface Ratio of total length to characteristic length Constant Constant Orifice coefficient Concentration Heat capacity Solute diffusivity Enhanced diffusivity Differential operator Packing size Critical packing size Impeller diameter Nozzle, perforation, orifice diameter Drop diameter, diameter of sphere of same volume per surface Drop diameter at jetting if no jet forms Diameter of rotor Diameter of stator-ring opening Tube or tank diameter Weight (or mass flow rate) of extract Weight (or mass flow rate) of extraction solvent alone in extract Fractional efficiency of a single stage (mixer-settlers) Longitudinal dispersion coefficient (differential extractors) Murphree dispersed-phase stage efficiency, fractional Overall stage efficiency of a cascade, fractional 2.7183 (napierian logarithm base) Extraction factor (slope of equilibrium line/slope of operating line) Weight (or mass flow rate) of feed Weight (or mass flow rate) of feed solvent alone in feed Weighting factor Weighting factor Local acceleration due to gravity Gravitational conversion factor Height of a transfer unit attributed to driving force in extract phase Height of a transfer unit based on overall driving force in raffinate concentrations Height of a transfer unit attributed to driving force in raffinate phase

SI units

U.S. customary units

— m2 m2 m2/m3

— ft2 ft2 ft2/ft3

m2/m3

ft2/ft3

Symbol Hto HETS h A° hC hD

Dimensionless kmol/m3

Dimensionless (lb⋅mol)/ft3

m2/s m2/s

ft2/h ft2/h

hσ

m m m m

ft ft ft ft

K

m

ft

m

ft

m m

ft ft

m kg (or kg/s)

ft lb (or lb/ h)

kg (or kg/s)

lb (or lb/ h)

ho

K

K* K′ KC

KD

k kt L L′

m2/s

ft2/h m m m* mCD m′

kg (or kg/s)

lb (or lb/ h)

kg (or kg/s)

lb (or lb/ h)

9.83 m/s2

4.18 E08 ft/h2

1(kg⋅m)/(N⋅s) m

4.18 E08 (lbm⋅ ft)/(lbf⋅h2) ft

m

ft

m

ft

m′ m′CD Ns Nf Noh Nor NPe NPo NRe

Definition Overall height of a transfer unit Height equivalent to a theoretical stage Head loss due to friction Activity of solute Contribution to h due to continuous phase Contribution to h due to dispersed phase Contribution to hD due to orifice Contribution to hD due to interfacial tension Overall mass-transfer coefficient Partition coefficient in weight fractions Partition coefficient in mole fractions Partition coefficient in Bancroft (weight-ratio) coordinates Mass transfer coefficient for overall driving force in continuous phase concentration units Mass transfer coefficient for overall driving force in dispersed phase concentration units Individual-phase masstransfer coefficient Thermal conductivity

SI units

U.S. customary units

m

ft

m

ft

m —

ft —

m

ft

m

ft

m

ft

m

ft 2

kmol/(s⋅m ) (kmol/m3) Dimensionless

(lb⋅mol)/(h⋅ft2) [(lb⋅mol)/ft3] Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

kmol/(s⋅m2) (kmol/m3)

(lb mol)/(h⋅ft2) [(lb mol)/ft3]

kmol/(s⋅m2) (kmol/m3)

(lb mol)/(h⋅ft2) [(lb mol)/ft3]

kmol/(s⋅m2) (kmol/m3) W/(m⋅K)

Superficial mass velocity Superficial molar mass velocity Slope of equilibrium distribution curve, dy/dx Slope of equilibrium line in Bancroft coordinates

kg/(s⋅m2) kmol/(s⋅m2)

(lb⋅mol)/(h⋅ft2) [(lb⋅mol)/ft3] Btu/[h⋅ft2⋅°F)/ ft] lbm/(h⋅ft2) (lb⋅mol)/(h⋅ft2)

Dimensionless

Dimensionless

Slope of equilibrium line in mole fractions Slope of equilibrium line continuous/dispersed phase Slope of equilibrium distribution curve, dcE /dcR Slope of equilibrium line in concentration, c, units Slope of equilibrium curve, dcC /dcD Impeller speed Flux of mass transfer Number of heat transfer units based on hot phase Number of mass transfer units based on overall driving force in raffinate concentration Péclet number for axial dispersion, VdF /Ed for packing Power number, Pgc /ρN 3d 5i Reynolds number; for pipe flow, dtVρav /uav; for an impeller, d 2i Nρav /uav; for drops, dpVt uc /ρc

Dimensionless

Dimensionless

Dimensionless

Dimensionless

(kmol/m3)/ (kmol/m3) Dimensionless

[(lb⋅mol)/ft3]/ [(lb⋅mol)/ft3] Dimensionless

(kmol/m3)/ (kmol/m3) r/s kmol/(s⋅m2) Dimensionless

[(lb⋅mol)/ft3]/ [(lb⋅mol)/ft3] r/h (lb⋅mol)/(h⋅ft2) Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless Dimensionless

Dimensionless Dimensionless

LIQUID-LIQUID EXTRACTION OPERATIONS AND EQUIPMENT

15-3

Nomenclature (Concluded ) Symbol NSc Nto NWe,i NWe,t NTS n nd P Q R R′ S S′ SPM T T t Uo V Vd VK VO V′O V′OJ VS Vt v vp W W′ X x x° Y y y° Z

Definition

SI units

U.S. customary units

Schmidt number, u/ρD Number of overall transfer units Impeller Weber number, ρd i3N2/σgc Pipe Weber number, ρdtV 2/ σgc Number of theoretical ¡(equilibrium) stages Number of orifices or perforations per plate Number of drops Power for one real stage Total flow rate Weight (or mass flow rate) of raffinate Weight (or mass flow rate) of feed solvent alone in raffinate Weight (or mass flow rate) of extraction-solvent stream Weight (or mass flow rate) of extraction-solvent alone Reciprocating speed, strokes/ minute Diameter of mixing vessel or extraction tower Temperature in hot (raffinate) phase Temperature in cold (extract) phase Overall heat-transfer coefficient Superficial velocity

Dimensionless Dimensionless

Dimensionless Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Zt Z′t z z

W m3/s kg (or kg/s)

(ft⋅lbf)/h ft3/h lb (or lb/h)

α

kg (or kg/s)

lb (or lb/h)

kg (or kg/s)

lb (or lb/h)

kg (or kg/s)

lb (or lb/h)

m

ft

Velocity in a down spout Characteristic velocity Velocity through an orifice or nozzle Velocity through an orifice or nozzle Jetting velocity Slip velocity Terminal settling velocity Liquid volume Drop volume Weight (or mass flow rate) of wash phase or stream Weight (or mass flow rate) of wash solvent alone Weight solute/weight feed solvent in feed (raffinate) phase Weight-fraction solute in feed (raffinate) phase Mole-fraction solute in feed (raffinate) phase Weight solute/weight extraction solvent in extract Weight-fraction solute in extract phase Mole-fraction solute in extract phase Height of liquid in vessel or mixer; for towers, height of packed section

W/(m2⋅K)

Btu/(h⋅ft2⋅°F)

m/s = m3/ (s⋅m2) m/s m/s m/s

ft/h = ft3/(h⋅ft2)

m/s

ft/h

m/s m/s m/s m3 m3 kg (or kg/s)

ft/h ft/h ft/h ft3 ft3 lb (or lb/h)

kg (or kg/s)

lb (or lb/h)

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

m

ft

ft/h ft/h ft/h

Symbol ZH Zi ZL

Definition

SI units

Height of the heavy phase in decanter Height of the interface in decanter Height of the light phase in decanter Distance between trays Distance between trays Distance Weight-fraction solute in mixture

U.S. customary units

m

ft

ft m m

ft

m m m Dimensionless

ft in ft Dimensionless

Greek symbols

γ ∆ ∆p ∆ρ δ
θ θC θF λ µ µ′ ν π ρ Σ σ σ′ φ ω

Relative separation factor (selectivity) Activity coefficient of solute Delta (or difference) mixture Pressure drop Difference in density Dimensionless amplitude for oscillating drops Fraction void volume in packed section Time of contact Time between coalescences Time of drop formation Eigenvalue Viscosity Viscosity Coalescence frequency, fraction of drops coalescing per time 3.1416 Density Summation Interfacial tension Interfacial tension Volume fraction of a liquid in a vessel or extractor’s void volume Vibration frequency for oscillating drops

Pa kg/m3

lbf/ft2 lbm/ft3

s s s

h h h

Pa⋅s Pa⋅s L/s

lbm/(ft⋅h) cP L/h

kg/m3

lbm/ft3

N/m N/m

lbf/ft dyn/cm

L/s

L/h

Additional subscripts av C D E e F f H L M max o plug R r s w 1,2,etc. 1 2

Average Continuous phase Dispersed phase Extract Extract phase or stream Flooding Feed phase or stream Heavy liquid Light liquid Mixture Maximum Organic Plug flow Raffinate Raffinate phase or stream Extraction solvent phase or stream Water or aqueous liquid Stream leaving stage 1,2,etc. Concentrated end Dilute end

GENERAL REFERENCES: Foust, Wenzel, Clump, Maus, and Anderson, Principles of Unit Operations, 2d ed., Wiley, New York, 1980. Schweitzer (ed.), Handbook of Separation Techniques for Chemical Engineers, 2d ed., McGraw-Hill, New York, 1988. Sorenson and Arlt, “Liquid-Liquid Equilibrium Data Collection,” DECHEMA, Frankfurt, Germany, Binary Systems vol V, part 1, 1979, Ternary Systems vol V, part 2, 1980, Ternary & Quaternary Systems, vol 5, part 3, 1980, Macedo and Rasmussen, Supplement 1, vol V, part 4, 1987. Treybal, Liquid Extraction, 2d ed., McGraw-Hill, New York, 1963. Treybal, MassTransfer Operations, 3d ed., McGraw-Hill, New York, 1980. Wisniak and Tamir, Liquid-Liquid Equilibrium and Extraction: A Literature Source Book, part A, Elsevier, Amsterdam, 1981. Lo, Baird, and Hanson, Handbook of Solvent Extraction, Wiley, New York, 1983. Astarita, Mass Transfer with Chemical Reactions, Elsevier, New York, 1967. Calderbank, “Mass Transfer,” in Uhl and Gray (eds.), Mixing, vol. 2, Academic, New York, 1967. Cremer and Davies, Chemical Engineering Practice, vol. 5, Academic, New York, 1958. Davies, “Mass Transfer and Interfacial Phenomena,” in Drew, Hoopes, and Vermeulen (eds.), Advances in Chemical Engineering, vol. 4, Academic, New York, 1963. Hanson (ed.), Recent Advances in Liquid-Liquid Extraction, Pergamon, New York, 1971. Hyman, Mixing and Agitation, ibid, vol. 3, 1962. Kalichevsky and Kobe, Petroleum Refining with Chemicals, Van Nostrand, Princeton, N.J., 1965. Kintner,

“Drop Phenomena Affecting Liquid Extraction,” in Drew et al. (eds.), Advances in Chemical Engineering, vol. 4, Academic, New York, 1963. Molyneux, Chemical Plant Design, vol. I, Butterworth, Washington, 1965. Olney and Miller, “Liquid Extraction,” in Acrivos (ed.) Modern Chemical Engineering, vol 1, Reinhold, New York, 1963. Olson and Stout, “Mixing and Chemical Reactions,” in Uhl and Gray (eds.), Mixing, vol. 2, Academic, New York, 1967. Reitma, “Segregation in Liquid-Liquid Dispersions,” in Drew et al. (eds.), Advances in Chemical Engineering, vol. 5, Academic, New York, 1964. Rod, Misek, and Sterbacek, Liquid Extraction, Statne Nakladatelstor Techniki Literatury, Prague, 1964. Sideman, “Direct-Contact Heat Transfer between Immiscible Liquids,” in Drew et al. (eds.), Advances in Chemical Engineering, vol. 6, Academic, New York, 1966. Treybal, Mechanically Aided Liquid Extraction, ibid., vol. 1, 1956. Ziolkowski, Liquid Extraction in the Chemical Industry, Gos. Nauchn. Tekln. Izd. Khim. Lit, Leningrad, 1963. Cusack, Fremeaux, and Glatz, “A Fresh Look at Liquid-Liquid Extraction,” part 1, Extraction Systems, Chemical Engineering, vol. 98, no. 2, p. 66–67, Feb. 1991. Cusack, and Fremeaux, part 2, “Inside the Extractor,” Chemical Engineering, vol. 98, no. 3, p. 132–138, Mar. 1991. Cusack and Karr, part 3, “Extractor Design and Specification,” Chemical Engineering, vol. 98, no. 4, p. 112–120, Apr. 1991.

LIQUID-LIQUID EXTRACTION OPERATIONS Liquid-liquid extraction is a process for separating components in solution by their distribution between two immiscible liquid phases. Such a process can also be simply referred to as liquid extraction or solvent extraction; however, the latter term may be confusing because it also applies to the leaching of a soluble substance from a solid. Since liquid-liquid extraction involves the transfer of mass from one liquid phase into a second immiscible liquid phase, the process can be carried out in many different ways. The simplest example involves the transfer of one component from a binary mixture into a second immiscible liquid phase. One example is liquid-liquid extraction of an impurity from wastewater into an organic solvent. This is analogous to stripping or absorption in which mass is transferred from one phase to another. Transfer of the dissolved component (solute) may be enhanced by the addition of “salting out” agents to the feed mixture or by adding “complexing” agents to the extraction solvent. Or in some cases a chemical reaction can be used to enhance the transfer, an example being the use of an aqueous caustic solution to remove phenolics from a hydrocarbon stream. A more sophisticated concept of liquid-liquid fractionation can be used in a process to separate two solutes completely. A primary extraction solvent is used to extract one of the solutes from a mixture (similar to stripping in distillation), and a wash solvent is used to scrub the extract free from the second solute (similar to rectification in distillation).

sensitive, like antibiotics, or relatively nonvolatile, like mineral salts, and liquid-liquid extraction may provide the most cost-effective separation process. However, the potential use of distillation should generally be evaluated carefully before considering liquid-liquid extraction. An extraction process usually requires (1) liquid-liquid extraction, (2) solvent recovery, and (3) raffinate desolventizing. Several examples of cost-effective liquid-liquid extraction processes include the recovery of acetic acid from water (Fig. 15-1), using ethyl ether or ethyl acetate as described by Brown [Chem. Eng. Prog., 59(10), 65 (1963)], or the recovery of phenolics from water as described by Lauer, Littlewood, and Butler [Iron Steel Eng., 46(5), 99 (1969)] with butyl acetate, or with isopropyl ether as described by Wurm [Glückauf, 12, 517 (1968)], or with methyl isobutyl ketone as described by Scheibel [“Liquid-Liquid Extraction,” in Perry & Weiss-

USES FOR LIQUID-LIQUID EXTRACTION Liquid-liquid extraction is used primarily when distillation is impractical or too costly to use. It may be more practical than distillation when the relative volatility for two components falls between 1.0 and 1.2. Likewise, liquid-liquid extraction may be more economical than distillation or steam-stripping a dissolved impurity from wastewater when the relative volatility of the solute to water is less than 4. In one case discussed by Robbins [Chem. Eng. Prog., 76 (10), 58 (1980)], liquid-liquid extraction was economically more attractive than carbonbed or resin-bed adsorption as a pretreatment process for wastewater detoxification before biotreatment. In other cases the components to be separated may be heat15-4

FIG. 15-1

Solvent extraction of acetic acid from water.

LIQUID-LIQUID EXTRACTION OPERATIONS burg (eds.), Separation and Purification, 3d ed., Wiley, New York, 1978, chap. 3]. The solvent is recovered by distillation, and the raffinate is desolventized by steam stripping. In some cases the extraction solvent may have a higher boiling point than the solute to achieve reduced energy consumption, but a buildup of heavies in the recycle solvent can create another problem. The Udex process (Fig. 15-2) is a cost-effective liquid-liquid fractionation process for the separation of aromatics from aliphatics as described by Grote [Chem. Eng. Prog., 54(8), 43 (1958)]. In this process the extraction solvent, diethylene or triethylene glycol, is recovered by steam distillation, and the raffinate and extract streams are desolventized by water extraction. Subsequent process modifications described by Symoniak, Ganju, and Vidueira [Hydrocarbon Process., 139 (September 1981)] use tetraethylene glycol as the extraction solvent and a mixture of light aliphatics and benzene as the wash solvent to the main extractor. Water condensate from the steam distillation is used to extract residual extraction solvent from the raffinate and extract streams, so distillation for drying the extraction solvent has been eliminated. Solids are removed from recycle extraction solvent by filtration, while acids and heavies are removed by a solid adsorbent bed. Other processes similar to this use sulfolane (tetrahydrothiophene-1,1-dioxide) or NMP (N-methyl-pyrrolidone) as the extraction solvent. Another example of a cost-effective liquid-liquid extraction process is the one used for recovery of uranium from ore leach liquors (Fig. 15-3). In this case the solvents, alkyl phosphates in kerosine, are recovered by liquid-liquid extraction using a strip solution, and the

FIG. 15-2

Udex process.

FIG. 15-3

Liquid-liquid extraction of uranium.

15-5

raffinate requires practically no desolventizing because the solubility of the solvents in water is extremely low. Most of the solvent loss occurs because of the entrainment of small droplets in the water. The economic utility of a liquid-liquid extraction process depends strongly on the solvent selected and on the procedures used for solvent recovery and raffinate desolventizing. After these matters have been considered, the selection and design of an extraction device or assembly can be considered in proper perspective. DEFINITIONS The feed to a liquid-liquid extraction process is the solution that contains the components to be separated. The major liquid component in the feed can be referred to as the feed solvent. Minor components in solution are often referred to as solutes. The extraction solvent, or just plain solvent, is the immiscible liquid added to a process for the purpose of extracting a solute or solutes from the feed. The extractionsolvent phase leaving a liquid-liquid contactor is called the extract. The raffinate is the liquid phase left from the feed after being contacted by the second phase. A wash solvent is a liquid added to a liquid-liquid fractionation process to wash or enrich the purity of a solute in the extract phase. A theoretical or equilibrium stage is a device or combination of devices that accomplishes the effect of intimately mixing two immiscible liquids until equilibrium concentrations are reached, then physically separating the two phases into clear layers. Crosscurrent extraction (Fig. 15-4) is a cascade, or series of stages, in which the raffinate R from one extraction stage is contacted with additional fresh solvent S in a subsequent stage. Countercurrent extraction (Fig. 15-5) is an extraction scheme in which the extraction solvent enters the stage or end of the extraction farthest from where the feed F enters and the two phases pass countercurrently to each other. The objective is to transfer one or more components from the feed solution F into the extract E. When a staged contactor is used, the two phases are mixed with droplets of one phase suspended in the other, but the phases are separated before leaving each stage. When a differential contactor is used, one of the phases can remain dispersed as droplets throughout the contactor as the phases pass countercurrently to each other. The dispersed phase is then allowed to coalesce at the end of the device before being discharged. Liquid-liquid fractionation, or fractional extraction (Fig. 15-6), is a sophisticated scheme for nearly complete separation of one solute from a second solute by liquid-liquid extraction. Two immiscible liquids travel countercurrently through a contactor, with the solutes being fed near the center of the contactor. The ratio of immiscibleliquid flow rates is operated so that one of the phases preferentially moves the first solute to one end of the contactor and the other phase moves the second solute to the opposite end of the contactor. Another way to describe the operation is that a primary solvent S preferentially extracts, or strips, the first solute from the feed F and a wash solvent

FIG. 15-4

Crosscurrent extraction.

FIG. 15-5

Countercurrent extraction.

15-6

FIG. 15-6

LIQUID-LIQUID EXTRACTION OPERATIONS AND EQUIPMENT W scrubs the extract free from the unwanted second solute. The second solute leaves the contactor in the raffinate stream. Dissociation extraction is the process of using chemical reaction to force a solute to transfer from one liquid phase to another. One example is the use of a sodium hydroxide solution to extract phenolics, acids, or mercaptans from a hydrocarbon stream. The opposite transfer can be forced by adding an acid to a sodium phenate stream to spring the phenolic back to a free phenol that can be extracted into an organic solvent. Similarly, primary, secondary, and tertiary amines can be protonated with a strong acid to transfer the amine into a water solution, for example, as an amine hydrochloride salt. Conversely, a strong base can be added to convert the amine salt back to free base, which can be extracted into a solvent. This procedure is quite common in pharmaceutical production. Fractionation dissociation extraction involves both the chemical reaction and the fractionation scheme for the separation of components by their difference in dissociation constants as described by Colby [in Hanson (ed.), Recent Advances in Liquid-Liquid Extraction, Pergamon, New York, 1971, chap. 4].

Liquid-liquid fractionation.

PHASE EQUILIBRIA The separation of components by liquid-liquid extraction depends primarily on the thermodynamic equilibrium partition of those components between the two liquid phases. Knowledge of these partition relationships is essential for selecting the ratio of extraction solvent to feed that enters an extraction process and for evaluating the masstransfer rates or theoretical stage efficiencies achieved in process equipment. Since two liquid phases that are immiscible are used, the thermodynamic equilibrium involves considerable evaluation of nonideal solutions. In the simplest case a feed solvent F contains a solute that is to be transferred into an extraction solvent S.

For low concentrations in which the equilibrium line is linear the value of K′ is equal to m. The value of K′ is one of the main parameters used to establish the minimum ratio of extraction solvent to feed solvent that can be employed in an extraction process. For example, if the partition ratio K′ is 4, then a countercurrent extractor would require 0.25 kg or more of extraction-solvent flow to remove all the solute from 1 kg of feedsolvent flow. The relative separation, or selectivity, α between two components, b and c, can be described by the ratio of the two partition ratios [Eq. (15-5)]. α(b/c) = K°b /K°c = K b /K c = K′b /K′c

EQUILIBRIUM PARTITION RATIOS The weight fraction of solute in the extract phase y divided by the weight fraction of solute in the raffinate phase x at equilibrium is called the partition ratio, K [Eq. (15-1)]. K = y/x

(15-1)

Thermodynamically the partition ratio K° is derived in mole fractions y° and x° [Eq. (15-2)]. K° = y°/x°

(15-2)

For shortcut calculations the partition ratio K′ in Bancroft [Phys. Rev., 3, 120 (1895)] coordinates using the weight ratio of solute to extraction solvent in the extract phase Y and the weight ratio of solute to feed solvent in the raffinate phase X is preferred [Eq. (15-3)]. K′ = Y/X

(15-3)

In shortcut calculations the slope of the equilibrium line in Bancroft (weight-ratio) coordinates m is also used [Eq. (15-4)]. m = dY/dX TABLE 15-1

(15-4)

(15-5)

This is analogous to relative volatility in distillation. PHASE DIAGRAMS Ternary-phase equilibrium data can be tabulated as in Table 15-1 and then worked into an electronic spreadsheet as in Table 15-2 to be presented as a right-triangular diagram as shown in Fig. 15-7. The weight-fraction solute is on the horizontal axis and the weight-fraction extraction-solvent is on the vertical axis. The tie-lines connect the points that are in equilibrium. For low-solute concentrations the horizontal scale can be expanded. The water-acetic acid-methylisobutylketone ternary is a Type I system where only one of the binary pairs, waterMIBK, is immiscible. In a Type II system two of the binary pairs are immiscible, i.e. the solute is not totally miscible in one of the liquids. Many immiscible-liquid systems exhibit a critical solution temperature beyond which the system no longer separates into two liquid phases. This is shown in Fig. 15-8, in which an increase in temperature can change a Type II system to a Type I system above the

Water–Acetic Acid–Methyl Isobutyl Ketone, 25°C*

Weight % in raffinate

Weight % in extract

Water

Acetic acid

MIBK

X

Water

Acetic acid

MIBK

Y

98.45 95.46 85.8 75.7 67.8 55.0 42.9

0 2.85 11.7 20.5 26.2 32.8 34.6

1.55 1.7 2.5 3.8 6.0 12.2 22.5

0 0.0299 0.1364 0.2708 0.3864 0.5964 0.8065

2.12 2.80 5.4 9.2 14.5 22.0 31.0

0 1.87 8.9 17.3 24.6 30.8 33.6

97.88 95.33 85.7 73.5 60.9 47.2 35.4

0 0.0196 0.1039 0.2354 0.4039 0.6525 0.9492

*From Sherwood, Evans, and Longcor [Ind. Eng. Chem., 31, 599 (1939)].

PHASE EQUILIBRIA

15-7

TABLE 15-2 Spreadsheet for Right Triangular Ternary Diagram of Water/Acetic Acid/MIBK Liquid-Liquid-Equilibrium Data at 25°C in Fig. 15-7 Wt. fraction Variable Water Phase Line and Top of Triangle MIBK Phase Line

Tie-line 1 Tie-line 2 Tie-line 3 Tie-line 4 Tie-line 5 Tie-line 6

Acetic acid X

MIBK Y1

0.0000 0.0285 0.1170 0.2050 0.2620 0.3280 0.3460 1.0000 0.0000 0.0187 0.0890 0.1730 0.2460 0.3080 0.3360 0.0285 0.0187 0.1170 0.0890 0.2050 0.1730 0.2620 0.2460 0.3280 0.3080 0.3460 0.3360

0.0155 0.0170 0.0250 0.0380 0.0600 0.1220 0.2250

MIBK Y2

1 − wf AA Y3 1.0000 0.9715 0.8830 0.7950 0.7380 0.6720 0.6540 0.0000

0.9788 0.9533 0.8570 0.7350 0.6090 0.4720 0.3540 0.0170 0.9533 0.0250 0.8570 0.0380 0.7350 0.0600 0.6090 0.1220 0.4720

FIG. 15-8 Effect of temperature on ternary liquid-liquid equilibrium. A = feed solvent, B = solute, and S = extraction solvent.

0.2250 0.3540

Data from Sherwood, Evans, and Longcor [Ind. Eng. Chem., 31, 599 (1939)].

ordinary rectangular-coordinate graph paper. The entire ternary system can be plotted in Bancroft (weight-ratio) concentrations on loglog graph paper as shown by Hand [ J. Phys. Chem., 34, 1961 (1930)], and the equilibrium line can often be correlated by three straight-line segments (Fig. 15-9 and Table 15-3). The plait-point composition for a Type I system can easily be found by using this Hand plot as shown by Treybal, Weber, and Daley [Ind. Eng. Chem., 38, 817 (1946)]. This type of plot is also helpful for extrapolation and interpolation when data are scarce. Multicomponent systems containing four or more components become difficult to display graphically. However, process-design calculations can often be made for the extraction of the component with the lowest partition ratio K′ and treated as a ternary system. The components with higher K′ values may be extracted more thoroughly from the raffinate than the solute chosen for design. Or computer calculations can be used to reduce the tedium of multicomponent, multistage calculations.

critical temperature of the solute and extraction-solvent binary system TBS. The system becomes totally miscible above the critical temperature of the feed solvent and extraction-solvent binary TAS. Occasionally a system can also have a lower critical solution temperature below which the system will be totally miscible. The methyl ethyl ketonewater binary system provides one example. Changes in pressure ordinarily have a negligible effect on liquid-liquid equilibrium. For graphical calculation of the number of theoretical stages in a ternary system the right-triangular diagram is more convenient to use than an equilateral triangle. The ternary equilibrium data are simply plotted on ordinary rectangular-coordinate graph paper with the weight fraction of the solute on the horizontal axis and the weight fraction of the extraction solvent on the vertical axis. For low-solute concentrations the horizontal scale can be expanded. For the McCabe-Thiele type of graphical calculations and shortcut methods, the Bancroft (weight-ratio) concentrations can be used on

1.0000

0.7000 0.6000 0.5000 0.4000

MIBK

layer

Tie lines

Wt fraction MIBK

0.9000 0.8000

0.3000 0.2000 0.1000 0.0000 0.0000

FIG. 15-7

Water layer

0.0500

0.1000

0.1500 0.2000 0.2500 Weight fraction acetic acid

Type I ternary diagram (water-acetic acid-MIBK).

0.3000

0.3500

15-8

LIQUID-LIQUID EXTRACTION OPERATIONS AND EQUIPMENT Example 1: Partition Ratios Let us estimate the partition ratio in weight fractions K for extracting low concentrations of acetone from water into chloroform. The solute is acetone, the feed solvent is water, and the extraction solvent is chloroform in this case. Gmehling and Onken (op. cit.) give the activity coefficient of acetone in water at infinite dilution γ ∞ as 6.74 at 25°C, depending on which set of vaporliquid equilibrium data is correlated. From Eqs. (15-1) and (15-7) the partition ratio at infinite dilution of solute can be calculated as follows: γ molecular weight of feed solvent 6.74 (18) K = r  =   = 3.4 γe molecular weight of extraction solvent 0.30 (119.4) Sorenson and Arlt (Liquid-Liquid Equilibrium Data Collection, DECHEMA, Frankfurt, Germany, 1979) report several sets of liquid-liquid equilibrium data for the system acetone-water-chloroform, but the lowest solute concentrations reported at 25°C were 3 weight percent acetone in the water layer in equilibrium with 9 weight percent acetone in the chloroform layer. This gives a partition ratio K of 3.0.

FIG. 15-9

Hand-type ternary diagram for water-acetic acid-methyl isobutyl

ketone.

TABLE 15-3 Correlation of Liquid-Liquid Equilibrium Data for Water-Acetic Acid-MIBK X

Y

0.93⋅X1.10

0.0299 0.1364 0.2708 0.3864 0.5964 0.8065

0.0196 0.1039 0.2354 0.4039 0.6525 0.9492

0.0195 0.1039 0.2210

1.27⋅X1.29

0.2355 0.3725 0.6519 0.9624

THERMODYNAMIC BASIS OF LIQUID-LIQUID EQUILIBRIA In a ternary liquid-liquid system, such as the acetic acid–water–MIBK system, all three components are present in both liquid phases. At equilibrium the activity A° of any component is the same in both phases by definition [Eq. (15-6)]. A°r = γr x° = A°e = γe y°

(15-6)

where A° = activity of solute γ = activity coefficient of solute r = raffinate phase e = extract phase Consequently, the partition ratio in mole-fraction units K° is a result of the ratio of activity coefficients in the two layers [Eq. (15-7)]. K° = y°/x° = γr /γe

(15-7)

The activity coefficient γ can be defined as the escaping tendency of a component relative to Raoult’s law in vapor-liquid equilibrium (see Sec. 4 in this handbook or Null, Phase Equilibrium in Process Design, Wiley-Interscience, 1970). Gmehling and Onken (Vapor-Liquid Equilibrium Data Collection, DECHEMA, Frankfurt, Germany, 1979) have reported a large collection of vapor-liquid equilibrium data along with correlations of the resulting activity coefficients. This can be used to predict liquid-liquid equilibrium partition ratios as shown in Example 1.

This example clearly shows good distribution because of a negative deviation from Raoult’s law in the extract layer. The activity coefficient of acetone is less than 1.0 in the chloroform layer. However, there is another problem because acetone and chloroform reach a maximumboiling-point azeotrope composition and cannot be separated completely by distillation at atmospheric pressure. A higher-boiling solvent, e.g., 1,1,2-trichloroethane, can be used which still gives acetone a negative deviation from Raoult’s law (γe = 0.732 at 2 mole percent acetone) but does not form a maximumboiling-point azeotrope according to Treybal, Weber, and Daley [Ind. Eng. Chem., 38, 817 (1946)]. An activity coefficient greater than 1.0 for a solute in solution is generally considered to be a positive deviation from Raoult’s law; i.e., the escaping tendency is higher than predicted by Raoult’s law. Likewise, an activity coefficient less than 1.0 is considered to be a negative deviation from Raoult’s law; i.e., the escaping tendency is lower than predicted by ideal-solution behavior. “Positive” and “negative” thus refer to the sign of the logarithm of the activity coefficient. HYDROGEN-BONDING INTERACTIONS Deviations from Raoult’s law in solution behavior have been attributed to many characteristics such as molecular size and shape, but the strongest deviations appear to be due to hydrogen bonding and electron donor-acceptor interactions. Robbins [Chem. Eng. Prog., 76 (10), 58 (1980)] presented a table of these interactions, Table 15-4, that provides a qualitative guide to solvent selection for liquid-liquid extraction, extractive distillation, azeotropic distillation, or even solvent crystallization. The activity coefficient in the liquid phase is common to all these separation processes. In Example 1 the solute, acetone, contains a ketone carbonyl group which is a hydrogen acceptor, i.e., solute class 5 according to Table 15-4. This solute is to be extracted from water with chloroform solvent which contains a hydrogen donor group, i.e., solvent class 4. The solute class 5 and solvent class 4 interaction in Table 15-4 is shown to give a negative deviation from Raoult’s law. A negative deviation reduces the activity of the solute in the solvent, which enhances the liquid-liquid partition ratio but also leads to maximum-boiling-point azeotropes. Among other classes of solvents shown in Table 15-4 that suppress the escaping tendency of a ketone are classes 1 and 2, i.e., phenolics and acids. Other ketones, i.e., solvent class 5, are shown to be compatible with acetone, i.e., solute class 5, and tend to give activity coefficients near 1.0, i.e., nearly zero deviation from Raoult’s law, and tend to be nonazeotropic. The solvent classes 6 through 12 tend to provide a hostile environment for acetone which increases the escaping tendency, i.e., give activity coefficients greater than 1.0 and tend to form minimumboiling-point azeotropes. Whenever positive deviations give activity coefficients greater than 7.4, then phase separation, i.e., two liquid phases, can result, as shown by Martin [Hydrocarbon Process., 241 (November 1975)]. Most of the classes in Table 15-4 are self-explanatory, but some can use additional definition. Class 4 includes halogenated solvents that have highly active hydrogens as described by Ewell, Harrison, and Berg [Ind. Eng. Chem., 36, 871 (1944)]. These are molecules that

PHASE EQUILIBRIA TABLE 15-4

15-9

Organic-Group Interactions Based on 900 Binary Systems* Solvent class

Solute class

1

2

3

4

5

6

7

8

9

10

11

12

0 0 + 0

− − + −

− − 0 −

− 0 − −

− 0 − −

− 0 + −

− 0 + −

+ + + 0

+ + + +

+

+

+

+

+

+

H donor groups 0 0 − 0

0 0 − 0

− − 0 +

1 2 3 4

Phenol Acid, thiol Alcohol, water Active H on multihalogen paraffin

5

Ketone, amide with no H on N, sulfone, phosphine oxide

−

−

+

−

0

+

6

Tertiary amine

−

−

0

−

+

0

+

+

0

+

0

0

7

Secondary amine

−

0

−

−

+

+

0

0

0

0

0

+

8

Primary amine, ammonia, amide with 2H on N

−

0

−

−

+

+

0

0

+

+

+

+

9

Ether, oxide, sulfoxide

−

0

+

−

+

0

0

+

0

+

0

+

10

Ester, aldehyde, carbonate, phosphate, nitrate, nitrite, nitrile, intramolecular bonding, e.g., o-nitrophenol

−

0

+

−

+

+

0

+

+

0

+

+

11

Aromatic, olefin, halogen aromatic, multihalogen paraffin without active H, monohalogen paraffin

+

+

+

0

+

0

0

+

0

+

0

0

12

Paraffin, carbon disulfide

+

+

0

+

+

+

+

0

0

H acceptor groups

Non-H-bonding groups +

+

+

*From Robbins, Chem. Eng. Prog., 76(10), 58–61 (1980), by permission.

have two or three halogen atoms on the same carbon as a hydrogen atom, such as methylene chloride, chloroform, 1,1-dichloroethane and 1,1,2,2-tetrachloroethane. Class 4 also includes molecules that have one halogen on the same carbon atom as a hydrogen atom and one or more halogen atoms on an adjacent carbon atom, such as 1,2-dichloroethane and 1,1,2-trichloroethane. Apparently the halogens interact intramolecularly to leave the hydrogen atom highly active. Monohalogen paraffins like methyl chloride and ethyl chloride are in class 11 along with multihalogen paraffins and olefins without active hydrogen such as carbon tetrachloride and perchloroethylene. Chlorinated benzenes are also in class 11 because they do not have halogens on the same carbon as a hydrogen atom. Intramolecular bonding on aromatics is another fascinating interaction which gives a net result that behaves much like an ester group, class 10. Examples of this include ortho-nitrophenol and orthohydroxybenzaldehyde (salicylaldehyde). The intramolecular hydrogen bonding is so strong between the hydrogen donor group (phenol) and the hydrogen acceptor group (nitrate or aldehyde) that the molecule ends up by acting as an ester. One result is its low solubility in hot water. By contrast, the para derivative is highly soluble in hot water. Table 15-4 gives a qualitative indication of interactions between classes of molecules but does not give quantitative differences within each class. Taft et al. [ J. Am. Chem. Soc., 91, 4801 (1969)] have quantified the strength of hydrogen acceptors. The quantitative prediction of activity coefficients for solutions is reviewed by Reid, Prausnitz, and Sherwood (The Properties of Gases and Liquids, 3d ed., McGrawHill, New York, 1977) for the UNIFAC method, the Perotti, Deal, and Derr [Ind. Eng. Chem., 51, 95 (1959)] method, and the analyticalsolution-of-groups (ASOG) method. Leo, Hansch, and Elkins [Chem. Rev., 71(6), 525 (1971)] also provide methods for predicting partition ratios for solutes between water and many solvents. Magnussen, Rasmussen, and Fredenslund [Ind. Eng. Chem. Process Des. Dev., 20(2), 331 (1981)] have presented a UNIFAC parameter table specifically for predicting liquid-liquid equilibrium. EXPERIMENTAL EQUILIBRIUM DATA Several large collections of experimental equilibrium data are now available for liquid-liquid systems. Sorenson and Arlt (Liquid-Liquid

Equilibrium Data Collection, DECHEMA, Frankfurt, Germany, 1979) have reported several volumes of data that have been correlated with activity-coefficient equations. Wisniak and Tamir (Liquid-Liquid Equilibrium and Extraction: A Literature Source Book, Elsevier, Amsterdam, 1980) have listed many references. Leo, Hansch, and Elkins [Chem. Rev., 71(6), 525 (1971)] have tabulated partition ratios for a large number of solutes between water and solvents. Table 15-5 gives a selected list of partition ratios. DESIRABLE SOLVENT PROPERTIES The following properties of a potential solvent should be considered before use in a liquid-liquid extraction process. 1. Selectivity. The relative separation, or selectivity, α of a solvent is the ratio of two components in the extraction-solvent phase divided by the ratio of the same components in the feed-solvent phase. The separation power of a liquid-liquid system is governed by the deviation of α from unity, analogous to relative volatility in distillation. A relative separation α of 1.0 gives no separation of the components between the two liquid phases. Dilute solute concentrations generally give the highest relative separation factors. 2. Recoverability. The extraction solvent must usually be recovered from the extract stream and also from the raffinate stream in an extraction process. Since distillation is often used, the relative volatility of the extraction-solvent to nonsolvent components should be significantly greater or less than unity. A low latent heat of vaporization is desirable for a volatile solvent. 3. Partition ratio. The partition ratio for a solute should preferably be large so that a low ratio of extraction solvent to feed can be used. 4. Capacity. This property refers to the loading of solute per weight of extraction solvent that can be achieved in an extract layer at the plait point in a Type I system or at the solubility limit in a Type II system. 5. Solvent solubility. A low solubility of extraction solvent in the raffinate generally leads to a high relative volatility in a raffinate stripper or a low solvent loss if the raffinate is not desolventized. A low solubility of feed solvent in the extract leads to a high relative separation and, generally, to low solute-recovery costs.

15-10

LIQUID-LIQUID EXTRACTION OPERATIONS AND EQUIPMENT TABLE 15-5

Selected List of Ternary Systems

Component A = feed solvent, component B = solute, and component S = extraction solvent. K1 is the partition ratio in weightfraction solute y/x for the tie line of lowest solute concentration reported. Ordinarily, K will approach unity as the solute concentration is increased. Temp., °C.

K1

Ref.

Aniline Aniline

25 25

1.290 0.0784

47 47

Propane

85 98.5

0.150 0.1272

46 46

Furfural Nitromethane

25 25

0.680 0.397

44 127

Furfural

45 80 115

0.980 1.100 1.062

11 11 11

β,β′-Iminodipropionitrile β,β′-Oxydipropionitrile

ca. 25 ca. 25

0.625 0.377

92 92

Ethylene glycol

25

0.190

10

Amyl acetate n-Butyl acetate Cyclohexane Ethyl acetate Ethyl butyrate Ethyl propionate

31 31 27 31 31 31

1.838 1.940 0.508 1.850 1.903 2.32

86 86 86 86 86 86

n-Heptane

30 50 70 30 50 70

Component B A = cetane Benzene n-Heptane A = cottonseed oil Oleic acid A = cyclohexane Benzene Benzene A = docosane 1,6-Diphenylhexane A = dodecane Methylnaphthalene Methylnaphthalene A = ethylbenzene Styrene A = ethylene glycol Acetone Acetone Acetone Acetone Acetone Acetone A = furfural Trilinolein Triolein A = glycerol Ethanol Ethanol A = n-heptane Benzene

Component S

n-Heptane

Benzene Carbon tetrachloride Ethylene glycol

Benzene Benzene Cyclohexane Cyclohexane

β,β′-thiodipropionitrile Triethylene glycol Aniline Benzyl alcohol

Cyclohexane Cyclohexane Ethylbenzene Ethylbenzene Ethylbenzene Ethylbenzene Methylcyclohexane Toluene

Dimethylformamide Furfural Dipropylene glycol β,β′-Oxydipropionitrile β,β′-Thiodipropionitrile Triethylene glycol Aniline Aniline

Toluene Toluene

Benzyl alcohol Dimethylformamide

Toluene Toluene Toluene Toluene Toluene m-Xylene o-Xylene p-Xylene A = n-hexane Benzene A = neo-hexane Cyclopentane A = methylcyclohexane Toluene

Dipropylene glycol Ethylene glycol Propylene carbonate β,β′-Thiodipropionitrile Triethylene glycol β,β′-Thiodipropionitrile β,β′-Thiodipropionitrile β,β′-Thiodipropionitrile

47.5 21.4 19.5 95 108 41.5

15 15 15 15 15 15

25 25

0.159 0.0667

62 63

25 125 25 25 25 0 15 20 30 25 25 25 25 25 0 13 20 40 0 0 20 25 25 20 25 25 25 25 25

0.300 0.316 0.350 0.351 0.0815 0.107 0.267 0.1320 0.0635 0.329 0.180 0.100 0.140 0.087 0.577 0.477 0.457 0.425 0.694 0.667 0.514 0.331 0.150 0.732 0.150 0.289 0.050 0.150 0.080

50 50 92 89 47 29 29 28 78 90 101 101 89 116 27 27 27 27 29 28 28 90 101 39 101 89 101 101 101

Ethylenediamine

20

4.14

23

Aniline

15 25

0.1259 0.311

96 96

Methylperfluorooctanoate

10 25

0.1297 0.200

58 58

PHASE EQUILIBRIA TABLE 15-5

Selected List of Ternary Systems (Continued )

Component B A = iso-octane Benzene Cyclohexane n-Hexane A = perfluoroheptane Perfluorocyclic oxide Perfluorocyclic oxide A = perfluoro-n-hexane n-Hexane n-Hexane A = perfluorotri-n-butylamine Iso-octane A = toluene Acetone A = triethylene glycol α-Picoline α-Picoline α-Picoline A = triolein Oleic acid A = water Acetaldehyde Acetaldehyde Acetaldehyde Acetaldehyde Acetaldehyde Acetic acid

Temp., °C.

K1

Ref.

Furfural Furfural Furfural

25 25 30

0.833 0.1076 0.083

44 44 78

Carbon tetrachloride n-Heptane

30 30

0.1370 0.329

58 58

Benzene Carbon disulfide

30 25

6.22 6.50

80 80

Nitroethane

25 31.5 33.7

3.59 2.36 4.56

119 119 119

Ethylene glycol

0 24

0.286 0.326

100 100

Methylcyclohexane Diisobutylene Mixed heptanes

20 20 20

3.87 0.445 0.317

14 14 14

Propane

85

0.138

46

n-Amyl alcohol Benzene Furfural Toluene Vinyl acetate Benzene

18 18 16 17 20 25 30 40 50 60 26.7 30

1.43 1.119 0.967 0.478 0.560 0.0328 0.0984 0.1022 0.0558 0.0637 1.613 0.705 0.391 0.349 0.1920 0.0549 0.178 0.0865 0.1573 0.706 1.325 0.284 0.379 0.800 0.907 0.323 0.286 0.85 0.510 0.310 0.787 0.312 0.1623 0.828 0.520 0.0167 0.343 0.858 0.248 0.429 1.273 0.690 0.930 1.058 0.657 0.755 0.0435 0.1805 0.638 0.0644 0.140 0.0549

74 74 74 74 81 43 38 38 38 38 102 45 67 73 91 54 70 72 17 91 102 75 75 75 30 73 73 104 87 75 102 114 114 76 75 85 75 75 31 75 67 66 75 83 97 75 77 75 67 131 91 54

Component S

Acetic acid Acetic acid

1-Butanol Butyl acetate

Acetic acid Acetic acid

Caproic acid Carbon tetrachloride

Acetic acid

Chloroform

Acetic acid Acetic acid Acetic acid Acetic acid Acetic acid Acetic acid Acetic acid Acetic acid Acetic acid Acetic acid Acetic acid Acetic acid Acetic acid

Creosote oil Cyclohexanol Diisobutyl ketone Di-n-butyl ketone Diisopropyl carbinol Ethyl acetate 2-Ethylbutyric acid 2-Ethylhexoic acid Ethylidene diacetate Ethyl propionate Fenchone Furfural Heptadecanol

Acetic acid Acetic acid Acetic acid Acetic acid Acetic acid Acetic acid

3-Heptanol Hexalin acetate Hexane Isoamyl acetate Isophorone Isopropyl ether

Acetic acid Acetic acid Acetic acid Acetic acid Acetic acid

Methyl acetate Methyl butyrate Methyl cyclohexanone Methylisobutyl carbinol Methylisobutyl ketone

Acetic acid Acetic acid Acetic acid Acetic acid Acetic acid

Monochlorobenzene Octyl acetate n-Propyl acetate Toluene Trichloroethylene

25 27 27.5 ca. 25 25 56.8 34 26.7 25–26 25–26 25–26 30 25 25 25 28 25–26 26.7 25 50 25 25–26 31 25–26 25–26 20 25–26 30 25–26 30 25 25–26 25 25–26 25 27 30

15-11

15-12

LIQUID-LIQUID EXTRACTION OPERATIONS AND EQUIPMENT TABLE 15-5

Selected List of Ternary Systems (Continued )

Component B

Component S

Acetic acid Acetone Acetone

Vinyl acetate Amyl acetate Benzene

Acetone Acetone Acetone

n-Butyl acetate Carbon tetrachloride Chloroform

Acetone Acetone Acetone Acetone Acetone Acetone Acetone Acetone Acetone Acetone Acetone Acetone Acetone Acetone Acetone Acetone

Dibutyl ether Diethyl ether Ethyl acetate Ethyl butyrate Ethyl propionate n-Heptane n-Hexane Methyl acetate Methylisobutyl ketone Monochlorobenzene Propyl acetate Tetrachloroethane Tetrachloroethylene 1,1,2-Trichloroethane Toluene Vinyl acetate

Acetone Allyl alcohol Aniline

Xylene Diallyl ether Benzene

Aniline

n-Heptane

Aniline

Methylcyclohexane

Aniline Aniline Aniline hydrochloride Benzoic acid iso-Butanol iso-Butanol iso-Butanol n-Butanol

Nitrobenzene Toluene Aniline Methylisobutyl ketone Benzene 1,1,2,2-Tetrachloroethane Tetrachloroethylene Benzene

n-Butanol tert-Butanol tert-Butanol

Toluene Benzene tert-Butyl hypochlorite

tert-Butanol 2-Butoxyethanol 2,3-Butylene glycol

Ethyl acetate Methylethyl ketone n-Butanol

2,3-Butylene glycol

Butyl acetate

2,3-Butylene glycol

Butylene glycol diacetate

2,3-Butylene glycol

Methylvinyl carbinol acetate

n-Butylamine 1-Butyraldehyde Butyric acid Butyric acid Cobaltous chloride Cupric sulfate Cupric sulfate Cupric sulfate p-Cresol Diacetone alcohol Diacetone alcohol Dichloroacetic acid 1,4-Dioxane Ethanol Ethanol

Monochlorobenzene Ethyl acetate Methyl butyrate Methylisobutyl carbinol Dioxane n-Butanol sec-Butanol Mixed pentanols Methylnaphthalene Ethylbenzene Styrene Monochlorobenzene Benzene n-Amyl alcohol Benzene

Ethanol

n-Butanol

Temp., °C. 28 30 15 30 45 30 25 25 25–26 30 30 30 30 25 25 30 25–26 25–26 30 25–26 30 25 25–26 20 25 25–26 22 25 50 25 50 25 50 25 25 25 26.7 25 25 25 25 35 30 25 0 20 40 60 20 25 26 50 26 50 26 75 26 50 75 25 37.8 30 30 25 30 30 30 35 25 25 25 25 25–26 25 25 20

K1 0.294 1.228 0.940 0.862 0.725 1.127 0.238 1.830 1.720 1.941 1.00 1.500 1.278 1.385 0.274 0.343 1.153 1.910 1.000 0.243 2.37 0.237 1.467 0.835 1.237 3.63 0.659 0.572 14.40 15.50 1.425 2.20 2.05 3.41 18.89 12.91 0.0540 76.9* 0.989 1.80 0.0460 1.263 2.12 1.176 0.401 0.1393 0.1487 0.200 0.539 1.74 3.05 0.597 0.893 0.0222 0.0326 0.1328 0.565 0.237 0.351 0.247 1.391 41.3 6.75 12.12 0.0052 0.000501 0.00702 0.000225 9.89 0.335 0.445 0.0690 1.020 0.598 0.1191 0.0536 3.00

Ref. 103 117 11 11 11 67 12 43 3 75 54 117 117 117 112 114 117 75 75 117 57 88 113 75 81 104 75 32 40 40 40 40 40 40 108 107 98 49 1 36 7 126 126 37 99 130 130 129 129 5 68 71 71 71 71 71 71 71 71 71 77 52 66 83 93 9 9 9 82 22 22 77 8 75 13 115 26

PHASE EQUILIBRIA TABLE 15-5

Selected List of Ternary Systems (Continued )

Component B

Component S

Ethanol Ethanol Ethanol Ethanol Ethanol

Cyclohexane Cyclohexene Dibutyl ether Di-n-propyl ketone Ethyl acetate

Ethanol Ethanol Ethanol Ethanol Ethanol Ethanol Ethanol Ethanol Ethanol Ethylene glycol Ethylene glycol Ethylene glycol Ethylene glycol Ethylene glycol Formic acid

Ethyl isovalerate Heptadecanol n-Heptane 3-Heptanol n-Hexane n-Hexanol sec-Octanol Toluene Trichloroethylene n-Amyl alcohol n-Butanol Furfural n-Hexanol Methylethyl ketone Chloroform

Formic acid Furfural

Methylisobutyl carbinol n-Butane

Furfural Furfural Hydrogen chloride Hydrogen chloride Hydrogen chloride Hydrogen chloride Hydrogen chloride Hydrogen chloride Hydrogen chloride Hydrogen chloride Hydrogen chloride Hydrogen chloride Hydrogen chloride Hydrogen chloride Hydrogen chloride Hydrogen chloride Hydrogen chloride Hydrogen fluoride Lactic acid Methanol Methanol

Methylisobutyl ketone Toluene iso-Amyl alcohol 2,6-Dimethyl-4-heptanol 2-Ethyl-1-butanol Ethylbutyl ketone 3-Heptanol 1-Hexanol 2-Methyl-1-butanol Methylisobutyl ketone 2-Methyl-1-pentanol 2-Methyl-2-pentanol Methylisopropyl ketone 1-Octanol 2-Octanol 1-Pentanol Pentanols (mixed) Methylisobutyl ketone iso-Amyl alcohol Benzene n-Butanol

Methanol Methanol Methanol Methanol

p-Cresol Cyclohexane Cyclohexene Ethyl acetate

Methanol Methanol

n-Hexanol Methylnaphthalene

Methanol Methanol Methanol Methanol Methyl-n-butyl ketone Methylethyl ketone

sec-Octanol Phenol Toluene Trichloroethylene n-Butanol Cyclohexane

Methylethyl ketone Methylethyl ketone Methylethyl ketone

Gasoline n-Heptane n-Hexane

Methylethyl ketone Methylethyl ketone Methylethyl ketone Methylethyl ketone Methylethyl ketone Methylethyl ketone Nickelous chloride

2-Methyl furan Monochlorobenzene Naphtha 1,1,2-Trichloroethane Trichloroethylene 2,2,4-Trimethylpentane Dioxane

Temp., °C.

K1

Ref.

25 25 25–26 25–26 0 20 70 25 25 30 25 25 28 28 25 25 20 27 25 20 30 25 56.9 30 51.5 79.5 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 0 15 30 45 60 35 25 25 0 20 28 25 35 28 25 25 27.5 37.8 25 30 25 25 25 37.8 25 25 26.7 25 25 25 25

0.0157 0.0244 0.1458 0.592 0.0263 0.500 0.455 0.392 0.270 0.274 0.783 0.00212 1.00 0.825 0.01816 0.0682 0.1159 0.412 0.315 0.275 0.0527 0.00445 0.0192 1.218 0.712 0.930 7.10 5.64 0.170 0.266 0.534 0.01515 0.0250 0.345 0.470 0.0273 0.502 0.411 0.0814 0.424 0.380 0.257 0.271 0.370 0.352 0.01022 0.600 0.479 0.510 1.260 0.682 0.313 0.0156 0.01043 0.0589 0.238 0.565 0.025 0.0223 0.584 1.333 0.0099 0.0167 53.4 1.775 3.60 1.686 1.548 1.775 2.22 84.0 2.36 0.885† 3.44 3.27 1.572 0.0017

118 124 75 75 5 5 41 13 114 94 76 111 56 56 122 16 59 85 18 59 85 72 17 83 42 42 19 53 21 21 21 79 21 21 21 79 21 21 79 21 21 21 21 79 128 4 65 65 65 65 65 82 125 124 5 5 55 82 82 55 82 60 54 52 48 85 64 112 112 52 109 68 6 68 68 64 93

15-13

15-14

LIQUID-LIQUID EXTRACTION OPERATIONS AND EQUIPMENT TABLE 15-5

Selected List of Ternary Systems (Continued )

Component B

Temp., °C.

Component S

Nicotine Phenol α-Picoline α-Picoline α-Picoline α-Picoline iso-Propanol iso-Propanol iso-Propanol iso-Propanol

Carbon tetrachloride Methylnaphthalene Benzene Diisobutylene Heptanes (mixed) Methylcyclohexane Benzene Carbon tetrachloride Cyclohexane Cyclohexene

iso-Propanol iso-Propanol

Diisopropyl ether Ethyl acetate

iso-Propanol iso-Propanol n-Propanol n-Propanol n-Propanol n-Propanol

Tetrachloroethylene Toluene iso-Amyl alcohol Benzene n-Butanol Cyclohexane

n-Propanol

Ethyl acetate

n-Propanol n-Propanol n-Propanol

n-Heptane n-Hexane n-Propyl acetate

n-Propanol Propionic acid Propionic acid Propionic acid Propionic acid Propionic acid Propionic acid Propionic acid Propionic acid Propionic acid Propionic acid Propionic acid Propionic acid Propionic acid Propionic acid Pyridine

Toluene Benzene Cyclohexane Cyclohexene Ethyl acetate Ethyl butyrate Ethyl propionate Hexanes (mixed) Methyl butyrate Methylisobutyl carbinol Methylisobutyl ketone Monochlorobenzene Tetrachloroethylene Toluene Trichloroethylene Benzene

Pyridine Pyridine Pyridine Sodium chloride Sodium chloride Sodium chloride Sodium chloride Sodium chloride Sodium chloride Sodium chloride Sodium chloride Sodium chloride Sodium chloride Sodium hydroxide Sodium nitrate Succinic acid

Monochlorobenzene Toluene Xylene iso-Butanol n-Ethyl-sec-butyl amine n-Ethyl-tert-butyl amine 2-Ethylhexyl amine 1-Methyldiethyl amine 1-Methyldodecyl amine n-Methyl-1,3-dimethylbutyl amine 1-Methyloctyl amine tert-Nonyl amine 1,1,3,3-Tetramethyl butyl amine iso-Butanol Dioxane Ethyl ether

Trimethyl amine

Benzene

*Concentrations in lb.-moles/cu. ft. †Concentrations in volume fraction. References: 1. Alberty and Washburn, J. Phys. Chem., 49, 4 (1945). 2. Baker, J. Phys. Chem., 59, 1182 (1955). 3. Bancroft and Hubard, J. Am. Chem. Soc., 64, 347 (1942). 4. Barbaudy, Compt. rend., 182, 1279 (1926). 5. Beech and Glasstone, J. Chem. Soc., 1938, 67.

25 25 20 20 20 20 25 20 25 15 25 35 25 0 20 25 25 25 37.8 37.8 25 35 0 20 37.8 37.8 20 35 25 30 31 31 30 26 28 31 30 30 26.7 30 31 31 30 15 25 25 45 60 25 25 25 25 32 40 30 39.1 30 30 30 30 30 25 25 15 20 25 25 70

K1 9.50 7.06 8.75 1.360 1.378 1.00 0.276 1.405 0.0282 0.0583 0.0682 0.1875 0.406 0.200 1.205 0.388 0.1296 3.34 0.650 3.61 0.1553 0.1775 1.419 1.542 0.540 0.326 1.55 2.14 0.299 0.598 0.1955 0.303 2.77 1.470 0.510 0.186 2.15 3.52 1.949* 0.513 0.167 0.515 0.496 2.19 3.00 2.73 2.49 2.10 2.10 1.900 1.260 0.0182 0.0563 0.1792 0.187 0.0597 0.693 0.0537 0.589 0.0318 0.072 0.00857 0.0246 0.220 0.198 0.1805 0.857 2.36

Ref. 34 82 14 14 14 14 69 25 123 124 124 124 35 5 5 7 121 20 61 61 123 123 5 5 61 61 106 106 2 57 84 84 87 87 87 84 66 83 49 57 84 84 57 110 105 120 110 110 77 120 120 36 24 24 24 24 24 24 24 24 24 36 95 33 33 33 51 51

6. Berg, Manders, and Switzer, Chem. Eng. Progr., 47, 11 (1951). 7. Bergelin, Lockhart, and Brown, Trans. Am. Inst. Chem. Engrs., 39, 173 (1943). 8. Berndt and Lynch, J. Am. Chem. Soc., 66, 282 (1944). 9. Blumberg, Cejtlin, and Fuchs, J. Appl. Chem., 10, 407 (1960).

PHASE EQUILIBRIA TABLE 15-5

15-15

Selected List of Ternary Systems (Concluded )

10. Boobar et al., Ind. Eng. Chem., 43, 2922 (1951). 11. Briggs and Comings, Ind. Eng. Chem., 35, 411 (1943). 12. Buchanan, Ind. Eng. Chem., 44, 2449 (1952). 13. Chang and Moulton, Ind. Eng. Chem., 45, 2350 (1953). 14. Charles and Morton, J. Appl. Chem., 7, 39 (1957). 15. Church and Briggs, J. Chem. Eng. Data, 9, 207 (1964). 16. Colburn and Phillips, Trans. Am. Inst. Chem. Engrs., 40, 333 (1944). 17. Conti, Othmer, and Gilmont, J. Chem. Eng. Data, 5, 301 (1960). 18. Conway and Norton, Ind. Eng. Chem., 43, 1433 (1951). 19. Conway and Phillips, Ind. Eng. Chem., 46, 1474 (1954). 20. Coull and Hope, J. Phys. Chem., 39, 967 (1935). 21. Crittenden and Hixson, Ind. Eng. Chem., 46, 265 (1954). 22. Crook and Van Winkle, Ind. Eng. Chem., 46, 1474 (1954). 23. Cumming and Morton, J. Appl. Chem., 3, 358 (1953). 24. Davison, Smith, and Hood, J. Chem. Eng. Data, 11, 304 (1966). 25. Denzler, J. Phys. Chem., 49, 358 (1945). 26. Drouillon, J. chim. phys., 22, 149 (1925). 27. Durandet and Gladel, Rev. Inst. Franc. Pétrole, 9, 296 (1954). 28. Durandet and Gladel, Rev. Inst. Franc. Pétrole, 11, 811 (1956). 29. Durandet, Gladel, and Graziani, Rev. Inst. Franc. Pétrole, 10, 585 (1955). 30. Eaglesfield, Kelly, and Short, Ind. Chemist, 29, 147, 243 (1953). 31. Elgin and Browning, Trans. Am. Inst. Chem. Engrs., 31, 639 (1935). 32. Fairburn, Cheney, and Chernovsky, Chem. Eng. Progr., 43, 280 (1947). 33. Forbes and Coolidge, J. Am. Chem. Soc., 41, 150 (1919). 34. Fowler and Noble, J. Appl. Chem., 4, 546 (1954). 35. Frere, Ind. Eng. Chem., 41, 2365 (1949). 36. Fritzsche and Stockton, Ind. Eng. Chem., 38, 737 (1946). 37. Fuoss, J. Am. Chem. Soc., 62, 3183 (1940). 38. Garner, Ellis, and Roy, Chem. Eng. Sci., 2, 14 (1953). 39. Gladel and Lablaude, Rev. Inst. Franc. Pétrole, 12, 1236 (1957). 40. Griswold, Chew, and Klecka, Ind. Eng. Chem., 42, 1246 (1950). 41. Griswold, Chu, and Winsauer, Ind. Eng. Chem., 41, 2352 (1949). 42. Griswold, Klecka, and West, Chem. Eng. Progr., 44, 839 (1948). 43. Hand, J. Phys. Chem., 34, 1961 (1930). 44. Henty, McManamey, and Price, J. Appl. Chem., 14, 148 (1964). 45. Hirata and Hirose, Kagaku Kogaku, 27, 407 (1963). 46. Hixon and Bockelmann, Trans. Am. Inst. Chem. Engrs., 38, 891 (1942). 47. Hunter and Brown, Ind. Eng. Chem., 39, 1343 (1947). 48. Jeffreys, J. Chem. Eng. Data, 8, 320 (1963). 49. Johnson and Bliss, Trans. Am. Inst. Chem. Engrs., 42, 331 (1946). 50. Johnson and Francis, Ind. Eng. Chem., 46, 1662 (1954). 51. Jones and Grigsby, Ind. Eng. Chem., 44, 378 (1952). 52. Jones and McCants, Ind. Eng. Chem., 46, 1956 (1954). 53. Knight, Trans. Am. Inst. Chem. Engrs., 39, 439 (1943). 54. Krishnamurty, Murti, and Rao, J. Sci. Ind. Res., 12B, 583 (1953). 55. Krishnamurty and Rao, J. Sci. Ind. Res., 14B, 614 (1955). 56. Krishnamurty and Rao, Trans. Indian Inst. Chem. Engrs., 6, 153 (1954). 57. Krishnamurty, Rao, and Rao, Trans. Indian Inst. Chem. Engrs., 6, 161 (1954). 58. Kyle and Reed, J. Chem. Eng. Data, 5, 266 (1960). 59. Laddha and Smith, Ind. Eng. Chem., 40, 494 (1948). 60. Mason and Washburn, J. Am. Chem. Soc., 59, 2076 (1937). 61. McCants, Jones, and Hopson, Ind. Eng. Chem., 45, 454 (1953). 62. McDonald, J. Am. Chem. Soc., 62, 3183 (1940). 63. McDonald, Kluender, and Lane, J. Phys. Chem., 46, 946 (1942). 64. Moulton and Walkey, Trans. Am. Inst. Chem. Engrs., 40, 695 (1944). 65. Mueller, Pugsley, and Ferguson, J. Phys. Chem., 35, 1314 (1931). 66. Murty, Murty, and Subrahmanyam, J. Chem. Eng. Data, 11, 335 (1966). 67. Murti, Venkataratnam, and Rao, J. Sci. Ind. Res., 13B, 392 (1954). 68. Newman, Hayworth, and Treybal, Ind. Eng. Chem., 41, 2039 (1949). 69. Olsen and Washburn, J. Am. Chem. Soc., 57, 303 (1935). 70. Othmer, Chem. Met. Eng., 43, 325 (1936). 71. Othmer, Bergen, Schlechter, and Bruins, Ind. Eng. Chem., 37, 890 (1945).

72. Othmer and Ku, J. Chem. Eng. Data, 4, 42 (1959). 73. Othmer and Serrano, Ind. Eng. Chem., 41, 1030 (1949). 74. Othmer and Tobias, Ind. Eng. Chem., 34, 690 (1942). 75. Othmer, White, and Treuger, Ind. Eng. Chem., 33, 1240 (1941). 76. Oualline and Van Winkle, Ind. Eng. Chem., 44, 1668 (1952). 77. Peake and Thompson, Ind. Eng. Chem., 44, 2439 (1952). 78. Pennington and Marwill, Ind. Eng. Chem., 45, 1371 (1953). 79. Pilloton, A.S.T.M. Spec. Tech. Publ., 238, 5 (1958). 80. Pliskin and Treybal, J. Chem. Eng. Data, 11, 49 (1966). 81. Pratt and Glover, Trans. Inst. Chem. Engrs. (London), 24, 52 (1946). 82. Prutton, Walsh, and Desar, Ind. Eng. Chem., 42, 1210 (1950). 83. Rao, Ramamurty, and Rao, Chem. Eng. Sci., 8, 265 (1958). 84. Rao and Rao, J. Appl. Chem., 6, 270 (1956). 85. Rao and Rao, J. Appl. Chem., 7, 659 (1957). 86. Rao and Rao, J. Sci. Ind. Res., 14B, 204 (1955). 87. Rao and Rao, J. Sci. Ind. Res., 14B, 444 (1955). 88. Rao and Rao, Trans. Indian Inst. Chem. Engrs., 7, 78 (1954–1955). 89. Rifai, Riv. Combust., 11, 811 (1957). 90. Rifai, Riv. Combust., 11, 829 (1957). 91. Saletore, Mene, and Warhadpande, Trans. Indian. Inst. Chem. Engrs., 2, 16 (1950). 92. Saunders, Ind. Eng. Chem., 43, 121 (1951). 93. Schott and Lynch, J. Chem. Eng. Data, 11, 215 (1966). 94. Schweppe and Lorah, Ind. Eng. Chem., 46, 2391 (1954). 95. Selikson and Ricci, J. Am. Chem. Soc., 64, 2474 (1942). 96. Serjian, Spurr, and Gibbons, J. Am. Chem. Soc., 68, 1763 (1946). 97. Sherwood, Evans, and Longcor, Ind. Eng. Chem., 31, 1144 (1939). 98. Sidgwick, Pickford, and Wilsdon, J. Chem. Soc., 99, 1122 (1911). 99. Simonsen and Washburn, J. Am. Chem. Soc., 68, 235 (1946). 100. Sims and Bolme, J. Chem. Eng. Data, 10, 111 (1965). 101. Skinner, Ind. Eng. Chem., 47, 222 (1955). 102. Skrzec and Murphy, Ind. Eng. Chem., 46, 2245 (1954). 103. Smith, J. Phys. Chem., 45, 1301 (1941). 104. Smith, J. Phys. Chem., 46, 229 (1942). 105. Smith, J. Phys. Chem., 46, 376 (1942). 106. Smith and Bonner, Ind. Eng. Chem., 42, 896 (1950). 107. Smith and Drexel, Ind. Eng. Chem., 37, 601 (1945). 108. Smith, Foecking, and Barber, Ind. Eng. Chem., 41, 2289 (1949). 109. Smith and La Bonte, Ind. Eng. Chem., 44, 2740 (1952). 110. Smith, Stibolt, and Day, Ind. Eng. Chem., 43, 190 (1951). 111. Taresenkov and Paul’sen, J. Gen. Chem. (U.S.S.R.), 7, 2143 (1937). 112. Treybal and Vondrak, Ind. Eng. Chem., 41, 1761 (1949). 113. Treybal, Weber, and Daley, Ind. Eng. Chem., 38, 817 (1946). 114. Upchurch and Van Winkle, Ind. Eng. Chem., 44, 618 (1952). 115. Varteressian and Fenske, Ind. Eng. Chem., 28, 928 (1936). 116. Varteressian and Fenske, Ind. Eng. Chem., 29, 270 (1937). 117. Venkataratnam, Rao, and Rao, Chem. Eng. Sci., 7, 102 (1957). 118. Vold and Washburn, J. Am. Chem. Soc., 54, 4217 (1932). 119. Vreeland and Dunlap, J. Phys. Chem., 61, 329 (1957). 120. Vriens and Medcalf, Ind. Eng. Chem., 45, 1098 (1953). 121. Washburn and Beguin, J. Am. Chem. Soc., 62, 579 (1940). 122. Washburn, Beguin, and Beckord, J. Am. Chem. Soc., 61, 1694 (1939). 123. Washburn, Brockway, Graham, and Deming, J. Am. Chem. Soc., 64, 1886 (1942). 124. Washburn, Graham, Arnold, and Transue, J. Am. Chem. Soc., 62, 1454 (1940). 125. Washburn and Spencer, J. Am. Chem. Soc., 56, 361 (1934). 126. Washburn and Strandskov, J. Phys. Chem., 48, 241 (1944). 127. Weck and Hunt, Ind. Eng. Chem., 46, 2521 (1954). 128. Weiser and Geankoplis, Ind. Eng. Chem., 47, 858 (1955). 129. Westwater, Ind. Eng. Chem., 47, 451 (1955). 130. Westwater and Audrieth, Ind. Eng. Chem., 46, 1281 (1954). 131. Woodman, J. Phys. Chem., 30, 1283 (1926).

15-16

LIQUID-LIQUID EXTRACTION OPERATIONS AND EQUIPMENT

6. Density. The difference in density between the two liquid phases in equilibrium affects the countercurrent flow rates that can be achieved in extraction equipment as well as the coalescence rates. The density difference decreases to zero at a plait point, but in some systems it can become zero at an intermediate solute concentration (isopycnic, or twin-density, tie line) and can invert the phases at higher concentrations. Differential types of extractors cannot cross such a solute concentration, but mixer-settlers can. 7. Interfacial tension. A high interfacial tension promotes rapid coalescence and generally requires high mechanical agitation to produce small droplets. A low interfacial tension allows drop breakup with low agitation intensity but also leads to slow coalescence rates. Interfacial tension usually decreases as solubility and solute concentration increase and falls to zero at the plait point (Fig. 15-10). 8. Toxicity. Low toxicity from solvent-vapor inhalation or skin contact is preferred because of potential exposure during repair of equipment or while connections are being broken after a solvent transfer. Also, low toxicity to fish and bioorganisms is preferred when extraction is used as a pretreatment for wastewater before it enters a biotreatment plant and with final effluent discharge to a stream or lake. Often solvent toxicity is low if water solubility is high.

Correlation of interfacial tension with mutual solubility for binary and ternary liquid mixtures. (From Treybal, Liquid Extraction, 2d ed., McGrawHill, New York, 1963.)

FIG. 15-10

CALCULATION METHODS SINGLE STAGE

COUNTERCURRENT THEORETICAL STAGES

An equilibrium, or theoretical, stage in liquid-liquid extraction as defined earlier is routinely utilized in laboratory procedures. A feed solution is contacted with an immiscible solvent to remove one or more of the solutes from the feed. This can be carried out in a separating funnel, or, preferably, in an agitated vessel that can produce droplets of about 1 mm in diameter. After agitation has stopped and the phases separate, the two clear liquid layers are isolated by decantation. The equilibrium distribution coefficient can be calculated by material balance, using the weight of the feed F, raffinate R, and extract E, plus the weight-fraction solute in the feed xf and raffinate xr, when the weight-fraction solute in the extraction solvent ys is zero [Eq. (15-8)].

The main objective for calculating the number of theoretical stages (or mass-transfer units) in the design of a liquid-liquid extraction process is to evaluate the compromise between the size of the equipment, or number of contactors required, and the ratio of extraction solvent to feed flow rates required to achieve the desired transfer of mass from one phase to the other. In any mass-transfer process there can be an infinite number of combinations of flow rates, number of stages, and degrees of solute transfer. The optimum is governed by economic considerations. The number of stages that are required can be kept to a minimum by selecting a solvent with a high partition ratio or by operating with a high ratio of extraction solvent to feed. However, a high solvent flow rate usually requires a high operating cost because of the cost of recovering the solvent. A high solvent flow rate should be carefully compared with an increase in capital cost for taller or more equipment to achieve more theoretical stages (or mass-transfer units) and reduce the required flow of solvent. The operating cost of an extractor is generally quite low in comparison with the operating cost of the solventrecovery distillation column. The other common objective for calculating the number of countercurrent theoretical stages (or mass-transfer units) is to evaluate the performance of liquid-liquid extraction test equipment in a pilot plant or to evaluate production equipment in an industrial plant. Most liquid-liquid extraction equipment in common use can be designed to achieve the equivalent of 1 to 8 theoretical countercurrent stages, with some designed to achieve 10 to 12 stages. Right-Triangular Method This method is a rigorous PonchonSavarit type of graphical technique for determining the number of countercurrent theoretical stages of a ternary system (Fig. 15-11). The horizontal axis is the concentration of solute in weight fractions x or y. The vertical axis is the weight fraction of extraction solvent. The weight fraction of feed solvent is simply the amount remaining so that all three weight fractions add up to 1.0. For the system water–acetic acid–MIBK in Fig. 15-11 the raffinate (water) layer is the solubility curve with low concentrations of MIBK, and the extract (MIBK) layer is the solubility curve with high concentrations of MIBK. The dashed lines are tie lines which connect the two layers in equilibrium as given in Table 15-1. Example 2 describes the right-triangular method of calculating the number of theoretical stages required.

冤

冥

ye R F xf K== −1 xr E R xr

(15-8)

However, an actual analysis of the weight-fraction solute in the extract ye and raffinate xr is preferred. CROSSCURRENT THEORETICAL STAGES After a single-stage liquid-liquid contact the phase remaining from the feed solution (raffinate) can be contacted with another quantity of fresh extraction solvent. This crosscurrent extraction scheme (Fig. 15-4) is an excellent laboratory procedure because the extract and raffinate phases can be analyzed after each stage to generate equilibrium data. Also, the feasibility of solute removal to low levels can be demonstrated. The number of crosscurrent stages N that are required to reach a specified raffinate composition, in Bancroft coordinates Xn, can be calculated directly if K′ is constant, the ratio of extraction solvent to feed solvent S′/F′ is kept constant, and fresh extraction solvent Ys = 0 (presaturated with feed solvent) is used in each stage [Eq. (15-9)]. log (Xf /Xn) N =  log (K′S′/F′ + 1)

(15-9)

The crosscurrent scheme is not generally economically attractive for large commercial processes because solvent usage is high and solute concentration in the combined extract is low.

CALCULATION METHODS

FIG. 15-11

Right-triangular graphical stages.

Example 2: Stage and Composition Calculation A 100-kg/h feed stream containing 20 weight percent acetic acid in water is to be extracted with 200 kg/h of recycle MIBK that contains 0.1 percent acetic acid and 0.01 percent water. The aqueous raffinate is to be extracted down to 1 percent acetic acid. How many theoretical stages will be required and what will the extract composition be? The solute concentration in the feed, xf = 0.20, in the raffinate, xr = 0.01, and in the extraction solvent, ys = 0.001, can be located on the diagram. Then the mix point zm can be calculated from the feed, F = 100 kg/h, and the solvent, S = 200 kg/h, entering the extractor [Eq. (15-10)]. zm = (Fxf + Sys)/(F + S)

(15-10)

The mix point, zm = 0.0673, falls on a straight line connecting xf and ys. The extract composition is then determined by drawing a straight line from xr through zm until the line intersects the extract line at the final extract composition, ye = 0.084. The delta point z∆ is then found at the intersection of two lines. One line connects the feed and extract compositions xf and ye. The other line connects the raffinate and solvent compositions xr and ys. The graphical stepping off of theoretical stages starts at the extract composition ye, and a tie line is drawn (parallel to the nearest one) to the raffinate composition leaving stage 1, x1 = ye/K = 0.084 (0.117/0.089) = 0.1104. The size of the extract stream can be calculated by the material balance E = (F + S)(zm − xr)/ ye − xr). A straight line is drawn between x1 and z∆ to find the extract composition leaving stage 2, y2 = 0.0415. Another tie line is drawn to find the raffinate composition leaving stage 2, x2, and the stepwise procedure continues until the final raffinate composition, xr = 0.01, is achieved. This requires four theoretical stages plus a fraction. Additional details on the derivation of this procedure are provided by Foust, Wenzel, Clump, Maus, and Anderson (Principles of Unit Operations, 2d ed, Wiley, New York, 1980) and Treybal (Mass-Transfer Operations, 3d ed., McGraw-Hill, New York, 1980).

Shortcut Methods These methods are often preferred for repetitive calculations of pilot-plant data and numerous design conditions. In distillation calculations the assumption of constant molar vapor and liquid flow rates gave rise to the McCabe-Thiele stepwise calculation method with straight operating lines and a curved equilibrium line. A similar concept can be achieved in liquid-liquid extraction by assuming a constant flow rate of feed solvent F′ and a constant flow rate of extraction solvent S′ through the extractor. The solute concentrations are then given as the weight ratio of solute to feed solvent X and the weight ratio of solute to extraction solvent Y, i.e., Bancroft coordi-

15-17

nates. These concentrations and coordinates will essentially give a straight operating line on an XY diagram for stages 2 through r − 1 in Fig. 15-12. Equilibrium data using these weight ratios have already been shown to follow straight-line segments on a log-log plot (see Fig. 15-9). The main problem, then, is to evaluate the primary ratio of extraction solvent to feed solvent passing through the extractor in stages 2 through r − 1. Robbins (“Liquid-Liquid Extraction,” in Schweitzer, Handbook of Separation Techniques for Chemical Engineers, McGraw-Hill, New York, 1979, sec. 1.9) reported that most liquid-liquid extraction systems can be treated as having either (A) immiscible solvents, (B) partially miscible solvents with a low solute concentration in the extract, or (C) partially miscible solvents with a high solute concentration in the extract. In case A the solvents are immiscible, so the rate of feed solvent alone in the feed stream F′ is the same as the rate of feed solvent alone in the raffinate stream R′. In like manner, the rate of extraction solvent alone is the same in the stream entering S′ as in the extract stream leaving E′ (Fig. 15-12). The ratio of extraction-solvent to feed-solvent flow rates is therefore S′/F′ = E′/R′. A material balance can be written around the feed end of the extractor down to any stage n (see Fig. 15-12) and then rearranged to a McCabe-Thiele type of operating line with a slope of F′/S′ [Eq. (15-11)]. F′ E′Ye − F′Xf Yn + 1 =  Xn +  (15-11) S′ S′ Similarly, the same operating line can be derived from a material balance around the raffinate end of the extractor up to stage n [Eq. (15-12)]. F′ S′Ys − R′Xr Yn =  Xn − 1 +  (15-12) S′ S′ The overall extractor material balance is given by Eq. (15-13). F′Xf + S′Ys − R′Xr Ye =  (15-13) E′ The end points of the operating line on an XY plot (Fig. 15-13) are Xr, Ys and Xf, Ye, and the number of theoretical stages can be stepped off graphically. The equilibrium curve is taken from the Hand type of correlation shown earlier (Fig. 15-9). When the equilibrium line is straight, its intercept is zero, and the operating line is straight, the number of theoretical stages can be calculated with one of the Kremser equations [Eqs. (15-14) and (15-15)]. When the intercept of the equilibrium line is not zero, the value of Ys /K′s should be used

FIG. 15-12

Countercurrent extraction cascade.

15-18

FIG. 15-13

LIQUID-LIQUID EXTRACTION OPERATIONS AND EQUIPMENT

Graphical calculation of countercurrent stages (Bancroft coordi-

nates).

instead of Ys /m, where K′s is the partition ratio in Bancroft coordinates at Ys. When Ᏹ ≠ 1.0, Xf − Ys /m 1 1 ln   1−  +  Xr − Ys /m Ᏹ Ᏹ N =  (15-14) ln Ᏹ

冤冢

When Ᏹ = 1.0,

冣冢

冣 冥

FIG. 15-14

Graphical solution to the Kremser equation. Xr = 0.01/0.99 = 0.01 kg acetic acid/kg water S′ = 200(1 − 0.001) = 199.8 kg MIBK/h Ys = 0.2/199.8 = 0.001 kg acetic acid/kg MIBK

If we assume R′ = F′ and E′ = S′, calculate Ye from Eq. (15-13):

Xf − Ys /m N =  − 1 Xr − Ys /m

(15-15)

The value of m is the slope of the equilibrium line dY/dX [Eq. (15-4)]. This is equal to K′ [Eq. (15-3)] at low concentrations where the equilibrium line is straight. The value of Ᏹ, the extraction factor, is calculated by dividing the slope of the equilibrium line m by the slope of the operating line F′/S′ [Eq. (15-16)]. Ᏹ = mS′/F′

(15-16)

The solution to the Kremser equation is shown graphically in Fig. 15-14. When a system responds with a constant number of theoretical stages N, the solute concentration in the raffinate X r can readily be evaluated as the result of changing the ratio of solvent to feed [Eqs. (15-17) and (15-18)]. When Ᏹ ≠ 1.0, Ᏹ−1 Xr − Ys /m (15-17)  =  ᏱN + 1 − 1 Xf − Ys /m When Ᏹ = 1.0, Xr − Ys /m 1 (15-18)  =  Xf − Ys /m N + 1 When the equilibrium line is not straight, Treybal (Liquid Extraction, 2d ed., McGraw-Hill, New York, 1963) recommends that the geometric mean value of m be used. The geometric mean of the slope of the equilibrium line at the concentration leaving the feed state m1 and at the raffinate concentration leaving the raffinate stage mr is 兹苶 m苶 1mr苶. Example 3: Shortcut Calculation, Case A Let us solve the problem in Example 2 by using the shortcut calculation method assuming immiscible solvents, case A. From the problem, F′ = 100(1 − 0.2) = 80 kg water/h Xf = 0.2/0.8 = 0.25 kg acetic acid/kg water

kg acetic acid 80(0.25) + (199.8)(0.001) − 80(0.01) Ye =  = 0.097  199.8 kg MIBK From the correlation of equilibrium data (Table 15-3), Y = 0.930(X)1.10, for X between 0.03 and 0.25. Calculate X1 = (0.097/0.930)1/1.10 = 0.128: m = dY/dX = (0.930)(1.10)(X)0.1, for X between 0.03 and 0.25 m1 = 0.833 at X = 0.128 mr = dY/dX = K′ = 0.656, for X below 0.03 K′s = 0.656 at Ys = 0.001 苶苶 Ᏹ = 兹m 1mr苶 S′/F′ = (0.739)(199.8)/80 = 1.85 N is determined from Fig. 15-14, Eq. (15-14), or the McCabe-Thiele type of plot (Fig. 15-13): ln

0.25 − 0.001/0.656 1 1  1 − 冣 + 冥 冤冢 0.01 − 0.001/0.656 冣冢 1.85 1.85

N =  = 4.3 ln 1.85 From solubility data at Y = 0.1039 (Table 15-1) the extract layer contains 5.4/85.7 = 0.0630 kg water/kg MIBK and ye = (0.097)/(1 + 0.097 + 0.063) = 0.084 weight-fraction acetic acid in the extract.

For cases B and C, Robbins (“Liquid-Liquid Extraction,” in Schweitzer, Handbook of Separation Techniques for Chemical Engineers, McGraw-Hill, New York, 1979, sec. 1.9) developed the concept of pseudo solute concentrations for the feed and solvent streams entering the extractor that will allow the Kremser equations to be used. In case B the solvents are partially miscible, and the miscibility is nearly constant through the extractor. This frequently occurs when all solute concentrations are relatively low. The feed stream is assumed to dissolve extraction solvent only in the feed stage and to retain the same amount throughout the extractor. Likewise, the extraction solvent is assumed to dissolve feed solvent only in the raffinate stage. With these assumptions the primary extraction-solvent rate moving through the extractor is assumed to be S′, and the primary feed-

CALCULATION METHODS solvent rate is assumed to be F′. The extract rate E′ is less than S′, and the raffinate rate R′ is less than F′ because of solvent solubilities. The slope of the operating line is F′/S′, just as in Eqs. (15-11) and (15-12), but only stages 2 through r − 1 will fall directly on the operating line. And one knows that X1 will be on the equilibrium line in equilibrium with Ye by definition (see Fig. 15-12). One can calculate a pseudo feed concentration Xf that will fall on the operating line [Eq. (15-11)] at Yn + 1 = Ye [Eq. (15-19)]. S′ − E′ B (15-19) X f = Xf +  Ye F′ Likewise, one knows that Yr will be on the equilibrium line with Xr (see Fig. 15-12). One can therefore calculate a pseudo concentration of solute in the inlet extraction solvent YsB that will fall on the operating line [Eq. (15-12)] where Xn − 1 = Xr [Eq. (15-20)]. F′ − R′ (15-20) YsB = Ys +  Xr S′ B For case B, the two pseudo inlet concentrations X f and YsB can be used in the Kremser equation with the actual value of Xr and Ᏹ = mS′/F′ to calculate rapidly the number of theoretical stages required. The graphical stepwise solution shown in Fig. 15-13 can also be used. B The operating line will go through points Xr, YsB and X f , Ye with a slope of F′/S′. In one example studied by Robbins [Chem. Eng. Prog., 76(10), 58 (1980)], the actual feed and extract compositions gave a point to the left of the equilibrium line on an XY graph like Fig. 15-13 because the solubility of the solvent was so high. But the use of the pseudo feed composition still gave an accurate calculation of the number of theoretical stages as confirmed by a right-triangular graphical calculation. Example 4: Shortcut Calculation, Case B Let us solve the problem in Example 2 by assuming case B. The solute (acetic acid) concentration is low enough in the extract so that we may assume that the mutual solubilities of the solvents remain nearly constant. The material balance can be calculated by an iterative method. From equilibrium data (Table 15-1) the extraction-solvent (MIBK) loss in the raffinate will be about 0.016/0.984 = 0.0163 kg MIBK/kg water, and the feedsolvent (water) loss in the extract will be about 5.4/85.7 = 0.0630 kg water/kg MIBK. First iteration: assume R′ = F′ = 80 kg water/h. Then, extraction solvent in raffinate = (0.0163)(80) = 1.30 kg MIBK/h. Estimate E′ = 199.8 − 1.3 = 198.5 kg MIBK/h. Then feed solvent in extract = (0.063)(198.5) = 12.5 kg water/h. Second iteration: calculate R′ = 80 − 12.5 = 67.5 kg water/h. E′ = 199.8 − (0.0163)(67.5) = 198.7 kg MIBK/h. Third iteration: converge R′ = 80 − (0.063)(198.7) = 67.5 kg water/h. Ye is calculated from the overall extractor material balance [Eq. (15-13)]: (80)(0.25) + (199.8)(0.001) − (67.5)(0.01) kg acetic acid Ye =  = 0.0983  198.7 kg MIBK 0.0983 Ye =  = 0.0846 1 + 0.0983 + 0.0630 weight fraction acetic acid in extract. From the correlation of equilibrium data (Table 15-3), Ye = 0.930(X)1.10, for X between 0.03 and 0.25 The raffinate composition leaving the feed (first stage) is calculated: X1 = (0.0983/0.930)1/1.10 = 0.130 m1 = dY/dX = (0.930)(1.10)(X)0.1 mr = dY/dX = K′ = 0.656 m1 = 0.834 at X1 = 0.13 mr = 0.656 at Xr = 0.01 K′s = 0.656 at Ys = 0.001 苶苶 Ᏹ = 兹m 1mr苶 S′/F′ = (0.740)(199.8)/80 = 1.85 B f

X is calculated from Eq. (15-19): (199.8 − 198.7)(0.0983)(0.0983) B X f = 0.25 +  = 0.251 80 YsB is calculated from Eq. (15-20): (80 − 67.5)(0.01) YsB = 0.001 +  = 0.0016 199.8

15-19

N is determined from Fig. 15-13, Eq. (15-14), or a McCabe-Thiele type of plot (Fig. 15-13) for case B. ln N=

0.251 − 0.0016/0.656

1 1 1 − 冣 + 冥 冤冢 0.01 − 0.0016/0.656 冣冢 1.85 1.85

ln 1.85 = 4.5 theoretical stages

A less frequent situation, case C, can occur when the solute concentration in the extract is so high that a large amount of feed solvent is dissolved in the extract stream in the “feed stage” but a relatively small amount of feed solvent (say one-tenth as much) is dissolved by the extract stream in the “raffinate stage.” The feed stream is assumed to dissolve the extraction solvent only in the feed stage just as in case B. But the extract stream is assumed to dissolve a large amount of feed solvent leaving the feed stage and a negligible amount leaving the raffinate stage. With these assumptions the primary feed-solvent rate is assumed to be R′, so the slope of the operating line for case C is R′/S′. Again the extract rate E′ is less than S′, and the raffinate rate R′ is less than F′. The pseudo feed concentration for case C, X fC, can be calculated from Eq. (15-21). F′ S′ − E′ X fC =  Xf +  Ye (15-21) R′ R′ And the value of Ys will fall on the operating line for case C. The extraction factor for case C is calculated from Eq. (15-22). ᏱC = mS′/R′

(15-22)

On an XY diagram for case C the operating line will go through points Xr, Ys and X fC, Ye with a slope of R′/S′ similar to Fig. 15-13. When using the Kremser equation for case C, one uses the pseudo feed concentration X fC from Eq. (15-21) and the stripping factor ᏱC from Eq. (15-22). One uses the raffinate concentration Xr and inlet solvent concentration Ys without modification. For the first time through a liquid-liquid extraction problem, the right-triangular graphical method may be preferred because it is completely rigorous for a ternary system and reasonably easy to understand. However, the shortcut methods with the Bancroft coordinates and the Kremser equations become valuable time-savers for repetitive calculations and for data reduction from experimental runs. The calculation of pseudo inlet compositions and the use of the McCabeThiele type of stage calculations lend themselves readily to programmable calculator or computer routines with a simple correlation of equilibrium data. COUNTERCURRENT MASS-TRANSFER-UNIT CALCULATIONS The concept of a mass-transfer unit was developed many years ago to represent more rigorously what happens in a differential contactor rather than a stagewise contactor. For a straight operating line and a straight equilibrium line with an intercept of zero, the equation for calculating the number of mass-transfer units based on the overall raffinate phase Nor is identical to the Kremser equation except for the denominator when the extraction factor is not equal to 1.0 [Eq. (15-23)]. When Ᏹ ≠ 1.0, Xf − Ys /m 1 1 ln   1 −  +  Xr − Ys /m Ᏹ Ᏹ Nor =  (15-23) 1 − 1/ Ᏹ

冤冢

冣冢

冣 冥

The number of mass-transfer units Nor is identical to the number of theoretical stages when the extraction factor Ᏹ is 1.0 [Eq. (15-24)]. When Ᏹ = 1.0, Nor = [(Xf − Ys /m)/(Xr − Ys /m)] − 1

(15-24)

The differences become pronounced when values of the extraction factor are high [Eq. (15-25)]. Nor = N ln Ᏹ/(1 − 1/ Ᏹ)

(15-25)

15-20

LIQUID-LIQUID EXTRACTION OPERATIONS AND EQUIPMENT

Even staged equipment may be modeled best by the number of masstransfer units when the extraction factor is much higher than 1.5, especially if the stage efficiencies are low. The response of solute concentration in the raffinate Xr to the solvent-to-feed ratio S′/F′ can be calculated by Eqs. (15-26) and (15-27) for a constant number of transfer units based on the overall raffinate phase Nor. When Ᏹ ≠ 1.0, 1 − 1/ Ᏹ Xr − Ys /m (15-26)  =  N (1 − 1/Ᏹ) − 1/ Ᏹ Xf − Ys /m e or When Ᏹ = 1.0, Xr − Ys /m 1 (15-27)  =  Xf − Ys /m Nor + 1 The solution to these equations is shown graphically in Fig. 15-15. Note that the raffinate composition is not reduced appreciably when the extraction factor Ᏹ is increased from 5 to infinity. This is true because mass transfer from the raffinate phase limits the performance. This is typical of the performance of many devices including actual staged equipment. However, if there is sufficient residence time in each stage of a staged device so that high stage efficiencies can be achieved, then the raffinate can be reduced substantially by increasing the extraction factor above 5 (see Fig. 15-14). However, the solute concentration in the extract stream would be quite dilute. Example 5: Number of Transfer Units Let us calculate the number of transfer units required to achieve the separation in Example 3. The solution to the problem is the same as in Example 3 except that the denominator is changed in the final equation [Eq. (15-25)]: ln 1.85 Nor = 4.5  = 6.0 transfer units 1 − 1/1.85 STAGE EFFICIENCY AND HEIGHT OF A THEORETICAL STAGE OR TRANSFER UNIT The overall stage efficiency of a staged extraction system is simply the number of theoretical stages divided by the number of actual stages times 100 [Eq. (15-28)]. Percent stage efficiency = 100N/number of actual stages

(15-28)

A similar term of number of transfer units per actual stage could also be envisioned. The height equivalent to a theoretical stage (HETS) in an extraction tower is simply the height of the tower Zt divided by the number of theoretical stages achieved [Eq. (15-29)]. HETS = Zt /N

(15-29)

Likewise, the height of a transfer unit based on raffinate-phase compositions Hor is the height of tower divided by the number of transfer units [Eq. (15-30)]. Hor = Zt /Nor

(15-30)

The contribution to the height of a transfer unit overall based on the raffinate-phase compositions is the sum of the contribution from the resistance to mass transfer in the raffinate phase Hr plus the contribution from the resistance to mass transfer in the extract phase He, divided by the extraction factor Ᏹ [Eq. (15-31)]. Hor = Hr + He / Ᏹ

(15-31)

At high extraction factors the height of a transfer unit is mostly dependent on the resistance to the transfer of solute from the raffinate phase. Prediction methods attempt to quantify the resistances to mass transfer in terms of the raffinate rate R and the extract rate E, per tower cross-sectional area At, and the mass-transfer coefficient in the raffinate phase kr and the extract phase ke, times the interfacial (droplet) mass-transfer area per volume of tower a [Eqs. (15-32) and (15-33)]. Hr = R/At kr a

(15-32)

He = E/At ke a

(15-33)

The mass-transfer coefficients depend on complex functions of diffusivity, viscosity, density, interfacial tension, and turbulence. Similarly, the mass-transfer area of the droplets depends on complex functions of viscosity, interfacial tension, density difference, extractor geometry, agitation intensity, agitator design, flow rates, and interfacial rag deposits. Only limited success has been achieved in correlating extractor performance with these basic principles. The lumped parameter Hor deals directly with the ultimate design criterion, which is the height of an extraction tower. FRACTIONATION STAGES One of the most sophisticated separations achievable by liquid-liquid extraction is fractionation. Two solutes can be separated almost completely by isolating one solute b into the extraction solvent S′ and another solute c into a wash solvent W′ (Fig. 15-16). The bottom section of a fractionation extraction is about the same as the countercurrent extractions described earlier, with the extraction solvent S′ entering the bottom and extracting, i.e., stripping, one of the solutes b almost completely from the raffinate R′. As the extract stream moves above the feed stage, it is contacted countercurrently with a wash solvent W′ that scrubs the unwanted solute c out of the extract stream. This in effect purifies the solute b that is being extracted. The stripping section and the washing (enriching) section will each have its own operating line on a McCabe-Thiele type of XY diagram (Fig. 15-17). The overall material balance must be met at the feed stage. For the case in which the extraction solvent can be assumed to be totally immiscible with the wash solvent and there is no solvent in the feed, the extraction factor Ᏹ must be greater than 1.0 for component b and less than 1.0 for component c [Eq. (15-34)]. Ᏹ = mS′/W′

(15-34)

For a symmetrical separation of component b from c, Brian (Staged Cascades in Chemical Processing, Prentice-Hall, Englewood Cliffs, N.J., 1972) reported that the ratio of wash solvent to extraction solvent W′/S′ should be set equal to the geometric mean of the two slopes of the equilibrium lines [Eq. (15-35)]. FIG. 15-15

Graphical solution to the mass-transfer-unit equations.

苶苶 W′/S′ = 兹m bmc苶

(15-35)

CALCULATION METHODS

FIG. 15-16

Liquid-liquid fractionation cascade.

The ratio of wash solvent to extraction solvent is the same in the enriching section as in the stripping section if no solvent is added in the feed. The degree of separation to be achieved can be chosen for the process design, such as 99 percent of component b into the extract stream and 99 percent of component c into the raffinate stream. Then the feed rate can be chosen so that the solute loadings in the extract stream and

FIG. 15-17

Graphical calculation of fractionation stages for solute b.

15-21

the raffinate stream are reasonable. This becomes especially critical near the feed stage, where the solute loadings are highest. An overall material balance can be calculated around the extractor, and then an XY plot can be constructed for each solute (Figs. 15-17 and 15-18). The solute concentrations at the raffinate end of the extractor, Xbr and Ybs, can be plotted for component b, and the operating line can be drawn with a slope of W′/S′ with no solvent in the feed. The solute concentrations at the extract end of the extractor, Xbw and Ybe, can also be plotted for component b with the enriching-section operating line also having the slope W′/S′ if no solvents were added with the feed. The theoretical stages can be stepped off for each section of the extractor by starting at the extract end, stage 1, and stepping toward the feed stage f, then restarting at the raffinate end, stage r, and stepping toward the feed stage f (Fig. 15-17). A similar procedure is repeated for component c (Fig. 15-18). The feed-stage number is found by matching the concentrations and stage number. This occurs at the point where the feed should be introduced (see Treybal, Mass-Transfer Operations, 3d ed., McGrawHill, New York, 1980). The procedure for matching concentrations is carried out by plotting the stage number on the vertical axis and the raffinate concentration X for each component (Fig. 15-19). The concentrations are matched when the rectangle HJLK can be drawn as shown. The number of stages in the wash section including the feed stage is determined from the position of line HJ. The total number of stages r is determined from the position of LK, which is also at the feed stage. The solute concentrations can be seen to be highest at the feed stage (Figs. 15-17 and 15-18). Also the solute concentrations increase as the number of theoretical stages is increased. For a given flow rate of feed, the flow rates of the solvents entering the extraction must be sufficiently high so that neither solubility limits nor a plait point is exceeded, nor a pinch point is reached between the operating lines and the equilibrium lines. The presence of solvents in the feed stream will change the slope of one or both of the operating lines, and several ratios of extraction solvent to wash solvent may have to be evaluated to find the optimum. The final optimization is usually carried out in pilot-plant equipment. Theoretically the use of solute reflux to the ends of the extraction cascade can reduce the number of theoretical stages required by a factor of 2 according to Brian (Staged Cascades in Chemical Processing, Englewood Cliffs, N.J., 1972), but again the amount of solvent flow rates may have to be increased to avoid a pinch point or plait point near the feed stage.

FIG. 15-18

Graphical calculation of fractionation stages for solute c.

15-22

LIQUID-LIQUID EXTRACTION OPERATIONS AND EQUIPMENT

FIG. 15-19

Matching concentrations at the feed stage.

FIG. 15-20

COUNTERCURRENT LIQUID-LIQUID HEAT TRANSFER Heat may be transferred between two insoluble liquids in countercurrent flow through an extractor, and the performance can be evaluated in the same general manner as in mass transfer (Fig. 15-20). For a differential contactor the number of overall heat-transfer units based on the hot phase Noh can be derived from the same equations used for the number of mass-transfer units based on the feed (raffinate) phase [Eq. (15-36)]. Noh =

冕

Tf

Tr

dT Zt ZtUoar At == T − t Noh FCr

(15-36)

where T = temperature of the hot (raffinate) phase, t = temperature of

Countercurrent heat transfer.

the cold (extract) phase, Zt = height of tower, Hoh = height of an overall heat-transfer unit based on the hot (raffinate) phase, Uo = overall heat-transfer coefficient, ar = heat-transfer surface of the droplets per volume, At = cross-sectional area of tower, F = hot feed rate, and Cr = heat capacity of raffinate and feed. The solution to the integral in Eq. (15-36) is identical with Eqs. (15-23) and (15-24), where Xf = Tf, Xr = Tr, Ys = Ts, Ᏹ = SCe/FCr, S = cold solvent rate, and Ce = heat capacity of solvent and extract. The slope of the equilibrium line m = dT/dt = 1.0 since t = T at equilibrium. The height of a heat transfer unit Hoh is reported by Von Berg [in C. Hanson (ed.), Recent Advances in LiquidLiquid Extraction, Pergamon, New York, 1971, chap. 11] to be shorter than the height of a mass-transfer unit Hor by a factor of 3 to 20. As an alternative, the Kremser equations [Eqs. (15-14) and (15-15)] could be used to calculate the number of theoretical heat-transfer stages.

LIQUID-LIQUID EXTRACTION EQUIPMENT Liquid-liquid contacting equipment may be generally classified into two categories: stagewise and continuous (differential) contact.

RATES OF MASS TRANSFER

KD aav, or the equivalent stage efficiency. The coefficients KC and KD are made up of the coefficients for the individual liquids, kC and kD: 1 1 1 1 1 m′CD (15-37)  =  + ;  =  +  KD kD m′CD kC KC kC kD The evidence is that the coefficients kC and kD and the interfacial area aav depend differently upon operating variables. For purposes of design, therefore, it is ultimately necessary to have separate information on the quantities kC, kD, and aav. The role of an additional surface resistance is emphasized by the studies of Kishinevski and Moehalova [Zh. Prikl. Khim., 33, 2049 (1960)]. Information on the coefficients is relatively undeveloped. They are evidently strongly influenced by rate of drop coalescence and breakup, presence of surface-active agents, “interfacial turbulence” (Marangoni effect), drop-size distribution, and the like, none of which can be effectively evaluated at this time. Continuous-Phase Coefficients There have been a large number of measurements of kC for solid particles and gas bubbles suspended in agitated liquids [for review, see Miller, Ind. Eng. Chem., 56(10), 18 (1964)]. A typical correlation of these data is that of Calderbank and Moo-Young [Chem. Eng. Sci., 16, 39 (1961)]:

Measurements simply of the extent of extraction in an agitated vessel lead to the overall “volumetric” mass-transfer coefficients, KC aav or

Schindler and Treybal [Am. Inst. Chem. Eng. J., 14, 790 (1968)], how-

STAGEWISE EQUIPMENT (MIXER-SETTLERS) The function of a stage is to contact the liquids, allow equilibrium to be approached, and to make a mechanical separation of the liquids. The contacting and separating correspond to mixing the liquids, and settling the resulting dispersion; so these devices are usually called mixer-settlers. The operation may be carried out in batch fashion or with continuous flow. If batch, it is likely that the same vessel will serve for both mixing and settling, whereas if continuous, separate vessels are usually but not always used. In principle, at least, any mixer may be coupled with any settler to provide the complete stage. There are several combinations which are especially popular. Continuously operated devices usually, but not always, place the mixing and settling functions in separate vessels. Batch-operated devices may use the same vessel alternately for the separate functions.

kCNSc2/3 = 0.13(PµCgc /vρ2)1/4

(15-38)

LIQUID-LIQUID EXTRACTION EQUIPMENT

15-23

ever, found that for liquid dispersions of ethyl acetate saturated with water, agitated in water by flat-blade turbine impellers, kC was appreciably larger than that given by Eq. (15-38) for baffled vessels and even higher for unbaffled vessels (no air-liquid interface). The increase was attributed to the rate of coalescence of the droplets as the dispersion emerged from the impeller and recirculated through the tank and to their redispersion at the impeller. It was described by an expression of the form kC = kS + C(DC /θC)0.5

(15-39)

where kS was calculated from Harriott’s data for small-diameter solids [Harriott, Am. Inst. Chem. Eng. J., 8, 93 (1962)]. The continuous phase was found to be completely back-mixed and of uniform composition throughout for both baffled and unbaffled vessels. Dispersed-Phase Coefficients There have been no direct measurements of kD for liquid dispersions in agitated vessels. If the drops are small (as they usually are), internal circulation causes them to behave like rigid spheres with an enhanced diffusivity D′D. In stirred vessels, the ratio D′D /DD has been estimated to lie in the range of about 1:2 [Olney, Am. Inst. Chem. Eng. J., 7, 348 (1961); Treybal Liquid Extraction, 2d ed., McGraw-Hill, New York, 1963]. For a pump-mix impeller (Fig. 15-28), Coughlin and von Berg [Chem. Eng. Sci., 21, 3 (1966)], on the other hand, estimate kD to be higher than that for circulating drops but not so large as that for oscillating drops (see below). These estimates do not take into account drop coalescence, interfacial turbulence, etc.; they are based on an assumed value for kC and measured overall coefficients. Overall Coefficients and Stage Efficiency If it is assumed that values of aav, kC, kD (and therefore KD) can somehow be estimated, the stage efficiency can be calculated through KD aavZ KD aavθ(VC + VD) EMD = 1 − exp −  = 1 − exp −  VD VD

冢

冣

冤

冥

(15-40)

See also Treybal [Am. Inst. Chem. Eng. J., 4, 202 (1958); 6, 5M (1960)] and Olander [Chem. Eng. Sci., 18, 47 (1963); 19, 275 (1964)]. The remaining discussion is confined to measured values of stage efficiency or volumetric overall coefficients. These are largely of value only for the particular systems studied. For this reason, one fairly complete study will be described, and the others will only be mentioned. Figure 15-21 summarizes the results for the extraction of nbutylamine from kerosine into water in a continuously operated mixer [T = 0.37 m (1.23 ft); Z = 0.48 m (1.562 ft)] fed cocurrently upward, with and without four wall baffles and with a variety of impellers [Overcashier, Kingsley, and Olney, Am. Inst. Chem. Eng. J., 2, 529 (1956)]. When unbaffled, the vessel was full and without an air-liquid interface. EO represents the overall countercurrent efficiency of a single stage. EO at zero agitator speed was 0.18 at a liquid residence time of 1.08 min. The improved performance in the absence of baffles may be attributed to the reduction in back mixing and to the reduced power requirement for a given impeller speed. In the absence of baffles, vertical location of the impeller is immaterial. With baffles, the best performance is given with the impeller at 0.667 Z from the bottom, the worst at 0.25 Z from the bottom. For the spiral turbine, wall baffles and stator-ring baffles produced the same power-efficiency relationship. Off-center unbaffled operation at a propeller was intermediate between centered baffled and centered unbaffled operation. The data for propellers, spiral turbines, and flat-blade turbines, di = 0.10 to 0.25 m (0.333 to 0.833 ft), in both unbaffled and baffled tanks, with a flow rate to produce a residence time θ = 0.18 h, kerosine-water ratio = 1.57 by volume, are empirically correlated by 0.318(1015)(d i /d t )b EO = 1 −  3.2 1.37 N Re N Po

(15-41)

where b = 0 for baffled operation and 1.6 for unbaffled operation. Other detailed studies are the following:

1. Hixson and Smith [Ind. Eng. Chem., 41, 973 (1949)]. Batch extraction of iodine from water into carbon tetrachloride; unbaffled vessels, propeller agitated. Log (1 − E) is linear with time. 2. Karr and Scheibel [Chem. Eng. Prog., 50, Symp. Ser. 10, 73 (1954)]. Continuous extraction of acetic acid between methyl isobutyl ketone and water, and

FIG. 15-21 Continuous extraction of n-butylamine from kerosine into water. T = 1.23 ft, Z = 1.56 ft, no air-liquid interface, impellers centered, VR /VE × 1.57, residence time × 1.08 min. To convert feet to meters, multiply by 0.3048; to convert inches to centimeters, multiply by 2.54; and to convert horsepower to kilowatts, multiply by 0.746. [Overcashier, Kingsley, and Olney, Am. Inst. Chem. Eng. J., 2, 529 (1956), with permission.]

xylene and water, and of acetone between xylene and water; unbaffled vessels consisting of the unpacked section of the extractor of Fig. 15-44. Rate of extraction is larger when organic liquid is dispersed in the extractant than with other arrangements. 3. Flynn and Treybal [Am. Inst. Chem. Eng. J., 1, 324 (1955)]. Continuous extraction of benzoic acid from toluene and kerosine into water; baffled vessels, turbine agitators. Stage efficiency is correlated with agitator energy per unit of liquid treated. 4. Mottel and Colvin (U.S. AEC DP-254, 1957). Continuous heat transfer between kerosine and water; vessel of Pump-Mix design (Fig. 15-28). 5. Ryon, Daley, and Lowrie [Chem. Eng. Prog., 55(10), 70, (1959), U.S. AFC ORNL-2951, 1960]. Continuous extraction of uranium from sulfate-ore-leach liquors and kerosine + tributyl phosphate and di(2-ethylhexyl)-phosphoric acid; baffled vessels, turbine agitated. There is strong evidence of the influence of a slow chemical reaction. 6. Ryon and Lowrie (U.S. AEC ORNL-3381, 1960). Batch and continuous extraction of uranium from aqueous sulfate solutions into kerosine + amines, stripping of extract with aqueous sodium carbonate; baffled vessels, turbine agitated. A detailed process study. 7. David and Colvin [Am. Inst. Chem. Eng. J., 7, 72 (1961)]. Continuous heat transfer between kerosine and water; unbaffled vessel. Open impellers (paddles and propellers) are better than closed (centrifugal and disk impellers) at the same tip speed. 8. Simard et al. [Can. J. Chem. Eng., 39, 229 (1961)]. Continuous extraction of uranium from aqueous nitrate solutions into kerosine + tributyl phosphate and from sulfate solutions containing tricaprylamine; unbaffled vessel, propeller agitated. Process details for high recovery and low reagent costs. 9. Rushton, Nagata, and Rooney [Am. Inst. Chem. Eng. J., 10, 298 (1964)].

15-24

LIQUID-LIQUID EXTRACTION OPERATIONS AND EQUIPMENT

Batch extraction of octanoic acid from water and corn syrup into xylene, paraffin oil, and their mixtures; baffled vessel, turbine impeller. KCaav proportional to N 2.1µC−0.6µD−0.55. 10. Coughlin and von Berg [Chem. Eng. Sci., 21, 3 (1966)]. Continuous heat transfer and extraction of ethylbutyric acid between kerosine and water; unbaffled vessel, Pump-Mix design (Fig. 15-28). Interfacial area measured.

Scale-Up of Mixers For the details associated with the design and scale-up of agitated vessels, the reader is referred to Section 18 which covers this topic in great detail. The intention here is to provide only some of the general principles involved which have particular application to liquid-liquid extraction. For extraction, the mixing usually takes place either in a vessel which also serves as the settler (these can be baffled or unbaffled), or a separate mixing compartment (usually baffled if there is a gas-liquid interface, and usually unbaffled if it is liquid filled). The most common impellers are the marine impeller or disc flatblade turbine; the flow patterns which typically result are illustrated in Fig. 15-22. The power for agitation of two-phase mixtures in vessels such as these is given by the curves in Fig. 15-23. At low levels of power input, the dispersed phase holdup in the vessel (φD) can be less than the value in the feed (φDF); it will approach the value in the feed as the agitation is increased. Treybal (Mass Transfer Operations, 3d ed., McGraw-Hill, New York, 1980) gives the following correlations for estimation of the dispersed phase holdup based on power and physical properties for disc flat-blade turbines: Baffled vessels, impeller power/vessel volume > 105 W/m3 = 2.2 ft lbf /ft3⋅s: 0.178 φD PqDµC2 0.300 µ3C ρC 0.0741  = 0.764    φDF vLσ3gc qDρC2 σgc ∆ρ

冢

冣 冢

冣 冢 冣

σ3ρCgc3   µC4 g

冢

(a)

µ 冣 冢 µ 冣 0.276

D

0.136

(15-42)

c

(b)

Unbaffled vessels, full, no gas-liquid surface, no vortex: 0.427 φD PqDµC2 0.247 µ3c ρC 0.430  = 3.39    φDF vLσ3gc qDρC2 σgc ∆ρ

冢

冣 冢

冣 冢 冣

σ3ρCgc3 0.401 µD 0.0987 (15-43)    µC4 g µC He then recommends that the following relationships be used to estimate the mixture average density and viscosity for the power calculations: Baffled vessels: ρM = ρCφC + ρDφD (15-44) µC 1.5µDφD µM =  1 +  (15-45) φC µC + µD Unbaffled vessels, no gas-liquid interface, no vortex:

冢

冣 冢 冣

冢

µM =

冦

冣

µw 6µoφo  1+ ; φw µo + µw

φw > 0.4

(15-46a)

µo 1.5µwφw  1− ; φo µo + µw

φw < 0.4

(15-46b)

冢

冢

冣

冣

In scale-up, there are three types of similarity to be considered: 1. Geometric similarity. Two vessels are geometrically similar if the ratio of all corresponding dimensions is the same 2. Kinematic similarity. Two vessels are kinematically similar if they are first geometrically similar and have the same ratio of velocities in corresponding positions of the vessel 3. Dynamic similarity. Two vessels are dynamically similar if they are first kinematically similar and all force ratios are equal in corresponding positions of the vessel For most liquid-liquid extraction applications, the mixing section is usually scaled up on the principle of geometric similarity, and the power is based on maintaining the same power per unit volume. Treybal [Chem. Eng. Prog., 62(9), 67 (1966)] demonstrates that, for geometrically similar vessels with equal holding time and power per unit volume, the stage efficiency for liquid extraction is likely to increase on scale-up, so this is generally a conservative approach. Because of the difficulty in obtaining good data on mass-transfer coefficients and interfacial area as outlined earlier, it is necessary that bench or pilot scale experiments be performed to obtain the data needed for scale-up. The usual procedure is to determine a suitable range of residence times at various power inputs for a given mixer geometry. Most extractions are mass-transfer limited, so relatively short residence times are adequate (in the range of 1–3 minutes). However in some cases (such as metal extractions), there is actually a reactive-extraction taking place, and residence time becomes more critical; times in the range of 10–15 minutes are not unusual. Besides looking at just the mixing, it is important at this time to also consider the settling time of the phases after mixing since this will impact on the settler design. Higher intensity of mixing may decrease the residence time for mass transfer, but at the same time create fine dispersions which are difficult to settle. With the batch data, Slater and Godfrey in Lo, Baird, and Hanson, Handbook of Solvent Extraction, Wiley, New York, 1983, recommend that an approach to equilibrium be used to provide the fundamental basis for scale-up; they define the approach to equilibrium (Ef) as: Ci − Ct E= (15-47) Ci − Ce It has been found that this data can be correlated for batch extraction using the following correlation:

(c)

1 − Eb = e(−kt b )

FIG. 15-22 Liquid agitation by a disc flat blade turbine in the presence of a gas-liquid interface (a) without wall baffles, (b) with wall baffles, and (c) in full vessels without a gas-liquid interface (continuous flow) and without baffles. [Courtesy Treybal, Mass Transfer Operations, 3rd ed., p. 148, McGraw-Hill, NY, (1980).]

Once the value of k is obtained from the batch data, it can be related to a continuous extraction via the correlation: ktc Ef =  (15-49) 1 + ktc

(15-48)

LIQUID-LIQUID EXTRACTION EQUIPMENT

15-25

Power for agitation impellers immersed in single-phase liquids, baffled vessels with a gas-liquid surface [except curves (c) and (g)]. Curves correspond to: (a) marine impellers, (b) flat-blade turbines, w = d1 /5, (c) disk flat-blade turbines with and without a gas-liquid surface, (d) curved-blade turbines, (e) pitched-blade turbines, (g) flat-blade turbines, no baffles, no gas-liquid interface, no vortex. FIG. 15-23

Notes on Fig. 15-23 1. The power P is only that imparted to the liquid by the impeller. It is not that delivered to the motor drive, which additionally includes losses in the motor and speed-reducing gear. These may total 30 to 40 percent of P. A stuffing box where the shaft enters a covered vessel causes additional losses. 2. All the curves are for axial impeller shafts, with liquid depth Z equal to the tank diameter dt. 3. Curves a to e are for open vessels, with a gas-liquid surface, fitted with four baffles, b = dt /10 to dt /12. 4. Curve a is for marine propellers, di /dt ≈ s, set a distance C = di or greater from the bottom of the vessel. The effect of changing di /dt is apparently felt only at very high Reynolds numbers and is not well established. 5. Curves b to e are for turbines located at a distance C = di or greater from the bottom of the vessel. For disk flat-blade turbines, curve c, there is essentially no effect of di /dt in the range 0.15 to 0.50. For open types, curve b, the effect of di /dt may be strong, depending upon the group nb/dt. 6. Curve g is for disk flat-blade turbines operated in unbaffled vessels filled with liquid, covered, so that no vortex forms. If baffles are present, the power characteristics at high Reynolds numbers are essentially the same as curve b for baffled open vessels, with only a slight increase in power. 7. For very deep tanks, two impellers are sometimes mounted on the same shaft, one above the other. For the flat-blade disk turbines, at a spacing equal to 1.5di or greater, the combined power for both will approximate twice that for a single turbine. 8. SOURCE: Treybal, Mass Transfer Operations, 3d ed., p. 152, McGraw-Hill, NY, 1963.

where

E = approach to equilibrium of a single-stage contact Ci = initial concentration Ct = concentration of time t Ce = concentration of equilibrium Eb = approach to equilibrium for a batch process Ef = approach to equilibrium for a continuous process tb = mixing time for a batch process tc = residence time for a continuous process

Having established the residence time and power input, the scaleup can be now done using the principle of geometric similarity together with equal power per unit volume discussed earlier. The above covers most conventional mixers; there is another class of mixers, called pump-mix impellers, where the impeller serves not only to mix the fluids, but also to move the fluids through the extraction stages. These are specialized designs, often used in the metals extraction industries. For these types of impellers, a knowledge of the power characteristics for pumping is required in addition to that for mixing. For a more detailed treatment of these special cases, the reader is referred to Lo et al. Recycling some of one of the settled liquids back to the agitated

vessel sometimes improves settling of the dispersion. In addition, the stage efficiency of a stirred vessel can be considerably enhanced by recycling the liquid favored by solute distribution, whereas recycling the other liquid reduces the stage efficiency. When solute distribution favors the dispersed phase and mass-transfer rates are poor, recycling the settled dispersed phase can result in minimizing the volume of a cascade of extraction vessels [Treybal, Ind. Eng. Chem. Fundam., 3, 185 (1964)]. See also Gel’perin et al. (Khim. Neft. Mashinostr., 1966, 23). Extractive reaction, in which a solvent extracts one of the products to enhance the yield, is considered by Piret, Trambouze, et al. [Am. Inst. Chem. Eng. J., 6, 394, 574 (1960); 7, 138 (1961)]. See also Schmitz and Amundsen [Chem. Eng. Sci., 18, 265, 415, 447 (1963)]. SETTLERS Emulsions and Dispersions The mixture of liquids leaving a mixer is a cloudy dispersion which must be settled, coalesced, and separated into its liquid phases in order to be withdrawn as separate liquids from a stage. For a dispersion to “break” into separate phases, both sedimentation and coalescence of the drops of the dispersed

15-26

LIQUID-LIQUID EXTRACTION OPERATIONS AND EQUIPMENT

phase must occur. Unstable dispersions usually have droplet diameters of about 1 mm or larger and settle rapidly. Stable dispersions, or emulsions, are generally characterized by droplet diameters of about 1 µm or less. The unstable dispersions are preferred in liquid-liquidextraction operations and chemical-reaction systems involving two liquid phases that ultimately need to be separated. Dispersions and emulsions are usually characterized by the terms water-in-oil (meaning aqueous liquid droplets dispersed in organic liquid continuous phase) and oil-in-water (organic droplets in aqueous liquid). Dual emulsions and liquid-membrane systems are those in which the continuous phase is also present as very small droplets within larger drops of the other liquid. See Becher, Emulsions: Theory and Practice, ACS Monogr. 175, Reinhold, New York, 1957; and Li and Shrier, in Li (ed.), Recent Developments in Separation Science, vol. I, CRC Press, Cleveland, 1972, p. 163. The “breaking” of a dispersion in a batch settler may be divided into two periods: (1) primary break, or rapid settling and coalescence of most of the dispersed phase, which often leaves a fog of very small droplets suspended as parts per million in the majority phase; and (2) secondary break, which represents the slow settling of the fog. Most industrial settlers are designed for the primary break since the slow secondary break would require much longer residence times. The small amount of entrainment to a subsequent stage seldom influences stage efficiency in a multistage cascade. However, for conserving solvent and desolventizing the effluent streams from the final stages of a cascade, it may be necessary to clarify as completely as possible, including the use of coalescers to eliminate secondary fog. Sedimentation Isolated droplets, settling or rising in a stagnant liquid under the force of gravity, generally move more rapidly than solid spheres. The rate of settling or rising is more rapid for large droplet size, large density difference between phases, and low viscosity of the continuous phase. Felix and Holder [Am. Inst. Chem. Eng. J., 1, 296 (1955)] show considerably shorter settling time of petroleum-oil dispersions in water and phenol by reducing the continuousphase viscosity simply by raising the temperature. Coalescence The coalescence of droplets can occur whenever two or more droplets collide and remain in contact long enough for the continuous-phase film to become so thin that a hole develops and allows the liquid to become one body. A clean system with a high interfacial tension will generally coalesce quite rapidly. Particulates and polymeric films tend to accumulate at droplet surfaces and reduce the rate of coalescence. This can lead to the buildup of a “rag” layer at the liquid-liquid interface in an extractor. Rapid drop breakup and rapid coalescence can significantly enhance the rate of mass transfer between phases.

(a)

(c)

Gravity Settlers; Decanters These are tanks in which a liquidliquid dispersion is continuously settled and coalesced and from which the settled liquids are continuously withdrawn. They can be either horizontal or vertical. Figure 15-24 shows some typical horizontal decanters. For an uninstrumented decanter the height of the heavyphase-liquid leg above the interface is balanced against the height of the light-liquid phase above the interface, Eq. 15-50. (Zh − Zi)ρh = (ZL − Zi)ρL

The velocity of the liquid entering the decanter should be kept low to minimize disturbance of the interface. Sometimes an impingement baffle, or “picket fence,” has been used. In other cases, opposing inlets as in Fig. 15-24c and d have been used. For an external jackleg shown in Fig. 15-24a the heavy-liquid takeoff requires a siphon break to prevent emptying the vessel by siphoning. Some problems can occur because of pressure drop through the outlet piping and variable levels under flow conditions. The horizontal weirs in Fig. 15-24b and the circular weirs in Fig. 15-24c can be designed for a very low crest height at maximum design flow rates. When rag builds up at the interface, sometimes it can be purged by withdrawing a small stream, filtering out the solids, and returning the liquids to the decanter. The decanter can also be instrumented with an interface detector and automatic control valve on the heavy-phase flow. The light phase can still overflow from the vessel. For general reviews, see Ingersoll [Pet. Refiner, 30(6), (1951)] and Hart [Pet. Process., 2, 282, 471, 513, 632 (1947)]. In the petroleum industry, settler volumes have frequently been sufficiently large so as to provide a holding time from 0.5 to 1.0 h, which in most cases is probably excessive and costly. For most thin liquids, in which unusual emulsification problems do not occur, 5 to 10 min is ample. The size of the settler seems to be set by the rate of flow per unit of horizontal cross-sectional area as well as holding time [Williams et al., Trans. Inst. Chem. Eng. (London), 36, 464 (1958)]; Ryon, Daley, and Lowrie [Chem. Eng. Prog., 55(10), 70 (1959)], for the settling of aqueous uranium solutions and kerosine-alkyl phosphate solvents, used decanters of the type shown in Fig. 15-24b. The depth of the decanter having been chosen, these authors recommend that the horizontal cross section for the prevailing flow rate be set at twice the value which would give a dispersion-band thickness equal to the depth of the tank. In this manner dispersions of 9.08 m3/h aqueous + 15.90 m3/h solvent (40 gal/min aqueous + 70 gal/min solvent) were successfully settled in a decanter of 1.4-min holding time. Gravity settlers, basically of the type shown in Fig. 15-24a, were used by Wilke et al. (UCRL-10625, 1963; UCRL-11182, 1964) to settle water and Aroclor (specific gravity, 1.36). The dispersion may

(b)

(d)

Gravity decanters. (a) External jackleg, side view. (b) Straight weirs, side view. (c) Circular weirs, side view. (d) Circular weirs, top view.

FIG. 15-24

(15-50)

LIQUID-LIQUID EXTRACTION EQUIPMENT occupy a wedge-shaped volume in the region of the interface at low flow rates instead of covering the entire interface as for higher flow rates. Important variables influencing the performance are (1) the value of ϕD in the entering liquid mixture, (2) whether the dispersion is introduced above or below the interface, and (3) the distance of an impact baffle from the inlet pipe. The length of the dispersion wedge for kerosine-water (dispersed) is proportional to VD /d p3 (Jeffreys and Pitt, paper at AIChE meeting, Salt Lake City, May 1967). Higher temperatures (to decrease viscosity) and longer residence times within each phase improve the settling of water (dispersed)-coconut fatty acids [Manchanda and Woods, Ind. Eng. Chem. Process Des. Dev., 7, 182 (1968)]. In the extraction of uranium from ore leach liquors with kerosinereagent solvents, there is a saving in the cost of thickeners and filters if the aqueous liquors are not clarified before extraction. If such slurries are extracted, however, it is necessary to increase the solventaqueous ratio in the extractor in order to make the organic phase continuous; otherwise, unsettleable emulsions are produced. Table 15-6 gives the data of Shaw and Long [Chem. Eng., 64(11), 251 (1957)] for settling areas required for such extractions. The high organic-aqueous ratios are obtained by recycling settled organic phase from the settler to the mixer. Entrainment of organic solvent with the settled solids represents a serious problem in such operations. TABLE 15-6 Settling of Aqueous Uranium Leach Liquors with Kerosine–Alkyl Phosphate Solvent*

Nature of aqueous feed

Organic-aqueous ratio required

Permissible settler flow rate, U.S. gal/(min⋅ft2) horizontal area

Clear liquor Slimes (5% solids) Dense pulps (50–60% solids)

4 8 10

1.4–1.6 0.6 0.3

*Shaw and Long, Chem. Eng., 64(11), 251 (1957). To convert gallons per minute–square foot to cubic meters per hour–square meter, multiply by 2.44.

CYCLONES

15-27

especially suited for the purpose at hand. In an extensive study, Sareen et al. [Am. Inst. Chem. Eng. J., 12, 1045 (1966)] found, in part, that (1) coalescence is promoted by decreased fiber diameter, (2) a minimum bed density is required to achieve complete coalescence, dependent upon the system characteristics, (3) wetting of the fibers by droplets of dispersed phase is not necessary for good coalescence, (4) a fibrous bed of medical cotton can be made to coalesce almost any kind of liquid dispersed in another except if σ′ < 3 mN/m (dyn/cm), (5) cotton fibers are best supported from collapse by mixing with fibers of glass or Dynel [see also Langdon et al., Petro/Chem Eng., 1963(11), 35], (6) the optimum bed thickness of a mixed bed depends on the ratio of cotton to support (0.75 in for 50 percent cotton), (7) the maximum velocity through the bed with effective coalescing increases with bed depth, but increased pressure drop causes redispersion, presumably at values depending upon the liquid system, and (8) some surfactants interfere with coalescence, but others do not. For tests on petroleumbrine emulsions and Fiberglas, see Burtis and Kirkbride [Trans. Am. Inst. Chem. Eng., 42, 413 (1946)] and Hayes et al. [Chem. Eng. Prog., 45, 235 (1949)]. Beds of granular solids such as sand, etc., and bats of excelsior, steel wool, and the like have also been used. Separating Membranes If the capillary size of a porous substance is very small, the liquid which preferentially wets the solid may flow through the capillaries readily but strong interfacial films block the capillaries for flow of nonwetting liquid. Sufficient pressure will cause disruption of the films and permit passage of the nonwetting liquid, but regulation of the pressure commensurate with the pore size permits perfect phase separation. Separating membranes of this type are made of a variety of materials such as porcelain, paper which has been coated with special resins, and the like and may be either hydrophilic or hydrophobic in character. They are made thin so as to permit maximum passage of the wetting liquid [see Jordan, Trans. Am. Soc. Mech. Eng., 77, 393 (1955); and Belk, Chem. Eng. Prog., 61(10), 72 (1965)]. In practice, the dispersion is usually first passed through a coalescer so as to permit settling of the bulk of the dispersed phase before the mixture is presented to the separating membrane, thus relieving the load on the membrane. Figure 15-25 shows a combination device containing coalescers and both hydrophobic and hydrophilic separating membranes. Coalescers

Cyclones have been suggested as simple means of enhancing by centrifugal force the rate of settling of liquid dispersions. Tepe and Woods (U.S. AEC AECD-2864, 1943) report a few data for the separation of isobutanol-water dispersions in such devices, but the results were poor. The most thorough studies are those of Simkin and Olney [Am. Inst. Chem. Eng. J., 2, 545 (1956)] and of Hitchon [At. Energy Res. Estab. (Gt. Brit.) CE/R-2777, 1959], who conclude that high extraction efficiencies (requiring high degrees of dispersion) and good clarification of both effluents cannot be obtained in one stage. Tepe and Woods (loc. cit.) also tried helical coils of pipe for separating isobutanol-water mixtures, with poor results. CENTRIFUGES Mechanical centrifuges, high-speed machines, have been used for many years for separating liquid-liquid dispersions, for example, in the separation of caustic solutions and oils in the soap-making process, more recently in uranium extractions, and in many others. By enhancing the settling rate (without, however, influencing coalescence), they reduce the settling time considerably. See, for example, Landis [Chem. Eng. Prog., 61(10), 58 (1965)]. For details see Section 19. SETTLER AUXILIARIES These include the use of coalescers, separating membranes, and electrical devices and the addition of emulsion-breaking reagents. These last are used for treating permanent emulsions and will not be discussed here. Coalescers The small drops of a fine dispersion may be caused to coalesce and thus become larger by passing the dispersion through a coalescer. The enlarged drops then settle more rapidly. Coalescers are mats, beds, or layers of porous or fibrous solids whose properties are

FIG. 15-25 Combination coalescer, settler, and membrane separator. (Courtesy of Selas Corporation of America.)

15-28

LIQUID-LIQUID EXTRACTION OPERATIONS AND EQUIPMENT

FIG. 15-26

Fuel-water separator. (Courtesy of Warner-Lewis Co. Division, Fram Corp.)

and separating membranes are fashioned in the form of hollow cylinders, and flow is radially through the wall. After passing through the coalescers, the bulk of the liquids settles in the vertical member of the device, and then the settled phases are passed through their respective separating membranes. Devices of this type are designed to handle 0.57 to 6.81 m3/h (150 to 1800 gal/h), delivering completely separated phases; and further settling is unnecessary. Figure 15-26 shows another design for removing dispersed water from jet fuel or gasoline, available in sizes to handle from 68 to 250 m3/h (300 to 1100 gal/min) and delivering clear effluents. In this case only a hydrophobic membrane is required [Redmon, Chem. Eng. Prog., 59(9), 87 (1963)]. Electrical Devices Subjecting electrically conducting emulsions or dispersions to high-voltage electric fields may cause rupture of the protective film about a droplet and thus induce coalescence. Dispersions of low conductivity are subject, in an electric field, to forces between particles resulting from acquired induced dipoles, which induce coalescence. These phenomena have been used particularly for the desalting of petroleum emulsified with brine, and for similar applications. Devices have been built to handle 828 m3/h (125,000 bbl/day) of crude oil, at costs of approximately 0.1 to 0.5 cent/bbl [Waterman, Chem. Eng. Prog., 61(10), 51 (1965)]. For a detailed study see Sjoblom and Goren [Ind. Eng. Chem. Fundam., 5, 519 (1966)] and Brown and Hanson [Trans. Faraday Soc., 61, 1754 (1965)]. Figure 15-27 shows schematically the flow through a typical device.

lifts. Provision is made for recycle of settled organic phase by overflow to the mixer, the amount of which can be adjusted by changing the height of the organic-overflow pipe. The vanadium extractor at c in the figure is a box type, built into a circular tank, dt = 9.8 m (32 ft), Z = 2.1 m (7 ft). The 0.46-m- (18-in-) diameter turbines draw 5.6 kW (7.5 hp). Other modifications of the box-type mixer-settler (Denver Equipment Company, Denver, Bull. A1-B6), with capacities of from 0.23- to 5700-m3/h (1- to 25,000-gal/min) liquid flow, have been extensively used in a great variety of metal separations in process metallurgy. These provide for intrastage recycle of liquids, particularly advantageous when very low solvent-feed ratios typical of good solvents must be used and when it is desired to make the minority liquid continuous in order to improve settling characteristics. See also Williams et al. [Trans. Inst. Chem. Eng. (London), 36, 464 (1958)] and Hanson and Kaye [Chem. Process Eng., 44, 27, 654 (1963); 45, 413 (1964)].

MIXER-SETTLER COMBINATIONS Any mixer and settler can be combined to produce a stage, and the stages in turn arranged in a multistage cascade. A great many commonly used arrangements have been developed in an effort to reduce or eliminate interstage pumping and to reduce costs generally. Only a few of the more commonly used types are mentioned here. A compact alternating arrangement of mixers and settlers has been adopted in many of the “box-type” extractors developed originally for processing radioactive solutions, but now used in principle for many processes, with literally dozens of modifications. An example is the Pump-Mix mixer-settler (Fig. 15-28), in which adjacent stages have common walls [Coplan, Davidson, and Zebroski, Chem. Eng. Prog., 50, 403 (1954)]. The impellers in this case pump as well as mix by drawing the heavy liquid upward through the hollow impeller shaft and discharging it at a higher level through the hollow impeller. These extractors or variants of them have been built not only in relatively large sizes but also in miniature for bench-scale work. Figure 15-29 represents still further modification for low cost [Hazen and Henrickson, Min. Eng., 994 (1957); Quinn, Trefoil (Denver Equipment Company) Bull. M4-B90, 1957]. At a and b in the figure, the settler is a circular tank dt = 4.9 m (16 ft), Z = 2.1 m (7 ft), with the mixing vessel, 1.2 by 1.2 m (4 by 4 ft), contained inside. Agitators are turbines, di = 0.46 m (1.5 ft), operated at 150 r/min (1.12 kW) and 200 r/min (2.02 kW). The aqueous feed is 22.7-m3/h (100-gal/min) uranium-bearing ore leach liquor, the organic solvent 4.5-m3/h (20-gal/min) alkyl phosphate solutions in kerosine. Adjacent stages are at 0.3-m (1-ft) elevation difference, allowing gravity flow of the aqueous liquor, while the organic phase is pumped in countercurrent by air

Internal circulation and electric field, Petreco Cylectric coalescer (schematic). [Waterman, Chem. Eng., 61(10), 51 (1965), with permission.] FIG. 15-27

LIQUID-LIQUID EXTRACTION EQUIPMENT

15-29

and centrifuge of each stage operate on a common shaft [Clark, U.S. AEC DP-752 (1962); Kisbaugh, ibid., DP-841 (1963)]. See also Goncharenko et al. (Tr. Vses. Khemosorbtsii Nauchn-Tekhn. Sovesch. Protessy Zhidkostnoi Ekstraktsii Khemosorbtsii, 2d, Leningrad, 1964, 75) and Berestovoi et al. (ibid., 171). Still another uses a cyclone (hydroclone) for separating [Whatley and Woods, U.S. AEC ORNL3533 (1964); Finsterwalder, ibid., ORNL-4088 (1967)]. The Graesser extractor (Coleby, U.S. Patent 3,017,253, 1962) is a horizontal shell filled with stratified settled liquids, with a series of buckets revolving around the inner periphery which rain droplets of one liquid through the other. It has been used primarily in Europe for easily emulsified liquids. Overall Stage Efficiencies The mixer-settler extractors described have generally produced overall stage efficiencies in excess of 80 percent, usually nearly 90 to 95 percent. CONTINUOUS (DIFFERENTIAL) CONTACT EQUIPMENT

Pump-Mix mixer-settler. [Coplan, Davidson, and Zebroski, Chem. Eng. Prog., 50, 403 (1954), with permission.]

FIG. 15-28

Equipment in this category is usually arranged for multistage countercurrent contact of the insoluble liquids, without repeated complete separation of the liquids from each other between stages or their equivalent. Instead, the liquids remain in continuous contact throughout their passage through the equipment. General Characteristics Countercurrent flow is maintained by virtue of the difference in densities of the liquids and either the force of gravity (vertical towers) or centrifugal force (centrifugal extractors). Only one of the liquids may be pumped through the equipment at any desired velocity. The maximum velocity for the second is then fixed; if it is attempted to exceed this limit, the second liquid will be rejected and the extractor will be flooded. It cannot be overemphasized that knowledge of the characteristics of such equipment is surprisingly underdeveloped. The number of quantities that influence the rate of extraction is very large, and many of them are not well understood. Most of the available data were taken

Vertical arrangement of the stages is desirable, since then a single drive may be used for agitators and the floor-space requirement of a cascade is reduced to that of a single stage. See, for example, Hanson and Kaye, loc. cit. In the Lurgi extractor, the mixer and settlers are in separate vertical shells interconnected with piping [Guccione, Chem. Eng., 78 (July 4, 1966)]. A great many other devices are known. The Fenske extractor [Fenske and Long, Chem. Eng. Prog., 51, 194 (1955); Ind. Eng. Chem., 53, 791 (1961); Ind. Eng. Chem. Fundam., 1, 152 (1962)] is a vertical stack of mixer-settler stages, with mixing done by a vertically moving reciprocating plate in each mixer. One very successful device, particularly in the extraction of radioactive solutions, uses a centrifuge instead of a settler to separate the mixed liquids, and the pump-mixer

(a)

(b) FIG. 15-29

(c)

Kerr-McGee multistage mixer-settler. (a) and (b) For uranium. (c) For vanadium extraction.

15-30

LIQUID-LIQUID EXTRACTION OPERATIONS AND EQUIPMENT

from small laboratory devices, frequently only a few inches in diameter and a few feet high. For these reasons the generalizations given here should be used only for very rough estimates, with allowance for generous factors of safety. Axial Dispersion The devices in this category are subject to axial (longitudinal) dispersion within both liquids or departure from strictly “plug,” countercurrent flow. The result of this axial mixing is to decrease the effective concentration driving force in the contactor as illustrated in Fig. 15-30. As a result, the towers must be taller than simple application of the plug-flow numbers of transfer units would indicate. The problem has been extensively studied by Sleicher [Am. Inst. Chem. Eng. J., 5, 145 (1959)] and Vermeulen et al. [U.S. AEC UCRL-3911, 1958; suppl., 1958; 10928, 1963; Ind. Eng. Chem. Fundam., 2, 113, 304 (1963); Chem. Eng. Prog., 62(9), 95 (1966)]. The two studies lead to essentially the same results although they are expressed somewhat differently. For a review, see Li and Zeigler [Ind. Eng. Chem., 59(3), 30 (1967)]. The subject of axial mixing has also been treated extensively by Pratt and Baird (Handbook of Solvent Extraction, Wiley, NY, 1983, pp. 197–247). It will not be possible to outline in detail here all the considerations taken into account; for these the original papers should be consulted. For present purposes the procedure to be used in design, as developed by Vermeulen et al., will be outlined. It is limited to cases in which flow rates, distribution coefficients, and mass-transfer coefficients are constant throughout the extractor. For the final design it is important to conduct pilot tests in equipment which closely resembles that being considered for the full-size plant, and at conditions which simulate the conditions expected in the full-size plant. 1. Obtain NtOR,plug from Colburn’s equation [Trans. Am. Inst. Chem. Eng., 35, 211 (1938)]: VR cR1 − cE2 /m′ VR 1 NtOR,plug =  ln  1− + (15-51) 1 − VR /m′VE cR2 − cE2 /m′ m′VE m′VE 2. Obtain HtOR (from data correlations, etc.) and Zplug:

冤冢

冣冢

冣

Zplug = NtOR,plug HtOR

冥

(15-52)

3. Solve Eqs. (15-53) to (15-55) together with Fig. 15-31 simultaneously by trial and error to obtain NtOR: 1/NtOR = 1/NtOR,plug − 1/N′tOR (15-53) ln (VR /m′VE) (15-54) N′tOR = (NPeB)E +  VR /m′VE − 1 Equation (15-54) is applicable only for cases in which NtORVR /m′VE and (NPeB)E ≥ 1.0. −1 1 VR /m′VE (15-55) (NPeB)E =  +  fRNPe,RB fENPe,EB 4. The final height of the effective portion of the tower, Z, is then

冢

Z = Zplug(NtOR /NtOR,plug)

冣

(15-56)

In these expressions, B = Z/d, NPe,E = dVE /EE, NPe,R = dVR /ER, where d = some characteristic length such as dF for packed towers or T for spray towers. EE and ER are the longitudinal dispersion coefficients, which must ultimately be deter-

Effect of axial mixing on concentration profiles in towers subject to axial mixing.

FIG. 15-30

Factors for Eq. 15-67. [Vermeulen et al., Chem. Eng. Prog., 62(9), 95 (1966), with permission.]

FIG. 15-31

mined experimentally. They are usually reported as EC, ED, NPe,C, and NPe,D, since they are more characteristic of the continuous or dispersed nature of the liquid than whether the liquid is extract or raffinate. For plug flow, E = 0: for complete mixing, E = ∞. In using these expressions, HtOR should represent data that have been corrected for axial dispersion; unfortunately, very few data have been so corrected. Rod [Br. Chem. Eng., 9, 300 (1964)] presents a graphical calculation suitable even for curvilinear equilibrium curves.

Devices that are stagelike in character (sieve trays, compartmented extractors, etc.) are perhaps better treated by a somewhat different procedure which space does not permit outlining here. See Sleicher [Am. Inst. Chem. Eng. J., 6, 529 (1960)], Miyauchi and Vermeulen [Ind. Eng. Chem. Fundam., 2, 304 (1963)], and Van der Laan [Chem. Eng. Sci., 7, 187 (1958)]. Equipment Classification Equipment can be broadly classified into the following categories, generally in order of increasing complexity of internal construction. Those most generally used are: I. Gravity-operated extractors A. No mechanical agitation 1. Spray towers 2. Packed towers 3. Perforated-plate (sieve-plate) towers B. Mechanically agitated extractors 1. Towers with rotating stirrers 2. Pulsed towers a. Liquid contents pulsed b. Reciprocating plates II. Centrifugal extractors Spray Towers These are simple gravity extractors, consisting of empty towers with provisions for introducing and removing liquids at the ends (see Fig. 15-32). The interface can be run above the top distributor, below the bottom distributor, or in the middle, depending on where the best performance is achieved. Because of severe axial back mixing, it is difficult to achieve the equivalent of more than one or two theoretical stages or transfer units on one side of the interface. For this reason they have only rarely been applied in extraction applications. Distributors Most spray columns operate with the drops being formed at the ends of jets from the dispersed phase inlet distributor. The orifices or nozzles for introducing the dispersed phase are usually not smaller than 0.13 cm (0.05 in) in diameter in order to avoid clogging, nor larger than 0.64 cm (0.25 in) in order to avoid formation of

LIQUID-LIQUID EXTRACTION EQUIPMENT

15-31

produced is also approximately at its maximum. Typical nozzle velocities are in the range of 0.1–0.25 m/s. Holdup and Flooding At this point it is useful to introduce the concepts of holdup and flooding in column contactors. It is normal practice to select the phase which preferentially wets the internals of the column as the continuous phase. This then allows the dispersed phase to exist as discrete droplets within the column. If the dispersed phase were to preferentially wet the internals, this could cause the dispersion to prematurely coalesce and pass through the column as rivulets or streams which would decrease interfacial area and therefore column efficiency. The volume of droplets within the contactor at any time is referred to as the operational holdup of the dispersed phase, generally expressed as a fraction of the contactor volume. In a countercurrent-type column contactor, stable operation is possible as long as the rate of arrival of droplets in any section does not exceed the coalescence rate at the main interface; once this value is exceeded, droplet backup will occur at the interface and slowly build back into the column active area, a condition known as flooding. This is an inoperable condition. Besides starting at the interface and building back into the column, flooding can also start in other sections of the column depending on local hydrodynamic conditions. If for any reason the velocity of the continuous phase is increased, this will increase the drag force on the droplets and cause the smaller droplets to rise (or fall) more slowly. As the continuous phase velocity is increased further, there is reached a point where a significant number of droplets stop rising (or falling), forming a dense region which eventually coalesces and forms a second interface in the column; this also is an inoperable condition. This same phenomenon can be caused at constant continuous phase velocity by inducing the formation of smaller droplets, such as by increased agitation. These smaller droplets can no longer overcome the continuous phase velocity drag force and holdup, thus inducing flooding. The concepts of slip velocity and characteristic velocity are useful in defining the flooding point and operational regions of different types of column contactors. The slip (or relative) velocity is given by the equation: Vd Vc Vs =  + (15-59) φd 1 − φd FIG. 15-32

Spray tower with both phases dispersed.

excessively large drops. They should be designed to eliminate wetting by the dispersed liquid. The following equation is recommended for calculating the velocity at which a jet appears: 3σ(1 − do /dvj) 0.3 V′oj =  (15-57) ρd do The value of dvj can be found from the relationship: do do dvj =  where φ =  (15-58) ρ + q(φ)m (σ/∆ρ g)0.5 where Voj = hole velocity where a jet appears σ = interfacial tension do = hole diameter dvj = drop diameter in jetting ρd = dispersed phase density ∆ρ = density difference g = gravitational constant φ, q, m = constants

冤

For

冥

φ < 0.785: ρ = 1.0, q = 0.485, m = 2 φ > 0.785: ρ = 0.12, q = 1.51, m = 1

The distributor should be sized so that the hole velocity is greater than the jet formation velocity. As the velocity is increased, the jet reaches a maximum length at which it breaks into drops of approximately uniform size. It has been found that at this velocity the drop surface area

From this has been derived the concept of characteristic velocity which is defined by the general equation: Vs Vk =  (15-60) (1 − φd) The value of Vk may be identified with the average terminal velocity of the droplets within the contactor. Each different type of contactor will have a different and unique characteristic velocity. As flooding is approached, the slip velocity continues to decrease until at the flood point is zero and the following relationship applies; δVc δVd (15-61)  =  =0 δφd f δφd f Applying the above to the relationship for slip velocity yields the following relationships at flooding:

冢 冣 冢 冣

Vdf = 2Vkφdf (1 − φdf )

(15-62)

Vcf = Vk(1 − φdf ) (1 − 2φdf )

(15-63)

2

[1 + 8(Vc /Vd)f]0.5 − 3 φdf =  4(Vc /Vd)f − 4 where Vs = slip velocity Vc = continuous phase velocity Vd = dispersed phase velocity φd = dispersed phase holdup Vk = characteristic velocity Vcf = continuous phase velocity of flooding Vdf = dispersed phase velocity of flooding

(15-64)

These relationships are not restricted to any type of contactor; they can be used to predict either the flooding velocity at a given holdup,

15-32

LIQUID-LIQUID EXTRACTION OPERATIONS AND EQUIPMENT

or the holdup at flooding. However, it requires the knowledge of the characteristic velocity under actual mass-transfer conditions, and this has not been easy to obtain. As a result, column contactors still require pilot tests for reliable scale-up to full size. There are also other factors to be aware of which can have a significant impact of the holdup and flooding characteristics. One of the most important of these is the direction of mass transfer. When transfer is from the dispersed to continuous phases, there occurs what is known as the Marangoni effect which causes rapid interdroplet coalescence with resulting decrease in holdup, sometimes by as much as 50 percent. Also, changes in phase densities, particularly in the continuous phase, can have significant impact on axial mixing with resulting effects on extraction efficiency. Finally, the presence of contaminents can affect interfacial properties (in particular interfacial tension) as well as inhibit mass transfer. Flooding can be estimated theoretically by setting ∂Vc /∂φd = ∂Vd /∂φd = 0 [Thornton, Chem. Eng. Sci., 5, 201 (1956)], using Eq. (15-61). On the basis of purely statistical comparison of observed and calculated data, the empirical correlation of Minard and Johnson [Chem. Eng. Prog., 48, 62 (1952)], slightly modified, is recommended: 10,000∆ρ0.28 Vcf =  (15-65) 0.5 2 0.5 0.56 0.5 [0.453µ0.075 ρ c c + d p ρ d (Qd /Qc ) ] Use U.S. customary units only in this equation. In sizing the column diameter, it is ususlly assumed that the continuous phase velocity will set at 40 percent of this value, and therefore the column diameter is calculated by: 4Qc dt =  (15-66) 0.4πVcf where dt = column diameter Qc = volumetric flow rate of the continuous phase Vcf = velocity of the continuous phase of flooding

冪莦

Mass Transfer As mentioned earlier, spray columns rarely develop more than 1 theoretical stage due to the axial mixing in the column. Nevertheless, it is necesary to determine what column height will give this theoretical stage. It is recommended by Cavers in Lo et al. Handbook of Solvent Extraction p. 323 and p. 327, John Wiley & Sons, New York, 1983 that the following equation be used to estimate the overall efficiency coefficient:

Kca =

g3 ∆ρ3 0.25 φd(1 − φd)  σρc2  1 (NSc,c)0.5 +  (NSc,d)0.5 m

冢

冣

(15-67)

where K c a = overall mass transfer capacity coefficient based on the continuous phase φd = dispersed phase holdup ∆ρ = density difference of phases σ = interfacial tension ρc = density of the continuous phase NSc,c = Schmidt number—continuous phase NSc,d = Schmidt number—dispersed phase m = distribution coefficient With this value, the height of a transfer unit, (HTU)oc can be estimated from: Vc (HTU)oc =  (15-68) Kca where (HTU)oc = overall height of a transfer unit based on the continuous phase Vc = continuous phase superficial velocity On top of this should be put a safety factor of 30 percent due to the unreliability of the correlations. There are not many data on the scale-up of spray columns from pilot to industrial scale, so these types of calculations must be used for

the column design. As mentioned earlier, because of its limitations, the spray column is only rarely used in industrial applications. Heat Transfer Heat-transfer rates are generally large despite severe axial dispersion, with Uaav frequently observed in the range 18.6 to 74.5 and even to 130 kW/(m3⋅K) [1000 to 4000 and even to 7000 Btu/(h⋅ft3⋅°F)][see Bauerle and Ahlert, Ind. Eng. Chem. Process Des. Dev., 4, 225 (1965); and Greskovich et al., Am. Inst. Chem. Eng. J., 13, 1160 (1967); Sideman, in Drew et al. (eds.), Advances in Chemical Engineering, vol. 6, Academic, New York, 1966, p. 207, reviewed earlier work]. In the absence of specific heat-transfer correlations, it is suggested that rates be estimated from mass-transfer correlations via the heat–mass-transfer analogy. Axial Dispersion For low values of φd and in the absence of interdrop coalescence, axial dispersion for the dispersed phase is evidently very small (Ed ∼ 0). For the continuous phase, low µc and φd, Vermeulen et al. [Chem. Eng. Prog., 62(9), 95 (1966)] reviewed the available data and concluded that, for dt = 3.6 to 15.2 cm (0.117 to 0.5 ft), Ec is given empirically by Ec = c(Vd dt )1/2

(15-69)

where c = 23.6 for U.S. customary units and 7.2 for SI units. For treatment of heat transfer, particularly for high values of φd when axial dispersion evidently is the controlling factor, see Letan and Kehat [Am. Inst. Chem. Eng. J., 11, 804 (1965); 14, 398 (1968)] and Mixon et al. [ibid., 13, 21 (1967)]. Packed Towers For a packed-tower liquid-liquid extractor the empty shell of a spray tower is filled with packing to reduce the vertical circulation of the continuous phase. The standard commercial packings used in vapor-liquid systems are also used in liquid-liquid systems. This includes Raschig and pall rings, Berl and Intalox saddles, and other random-dumped packings as well as the newer structured packings. The packing reduces the available free space for flow but also significantly reduces the height required for mass transfer. However, Nemunaitis, Eckert, Foote, and Rollinson [Chem. Eng. Prog., 67(11), 60 (1971)] reported little benefit from a packed height greater than 3.05 m (10 ft) and recommended redistributing the dispersed phase about every 1.52 to 3.05 m (5 to 10 ft) to generate new droplets and mass-transfer surfaces. From this perspective the packing allows a wider spacing between sieve plates than described for a conventional sieve-plate tower. The pieces of random-dumped packing should be no larger than one-eighth of the tower diameter to minimize the wall effect which gives larger voids at the wall. The packing support can be an open grid or multiarch support if the dispersed phase is distributed to the top of the bed. But the packing support may also be a sieve plate with multiple light-liquid risers if the heavy phase is to be redispersed onto a lower bed. Or the packing support may be a sieve plate with multiple heavy-phase downcomers if the light phase is to be dispersed up into the bed. The streams of dispersed phase should be far enough apart to avoid coalescence at the dispersion plate, and the dispersed phase should not preferentially wet the packing. If the droplets wet the packing, they will coalesce and stream along the packing as rivulets. Eckert [Hydrocarbon Process., 117 (March 1976)] recommends the use of packed towers when the interfacial tension is below 10 mN/m (dyn/cm). Holdup It is recognized that the dispersed-phase holdup may be placed in two categories: a smaller portion which is permanent and a larger portion, free, which moves through the packing and enters into mass-transfer operations when a solute is transferred between phases. Vignes [Chem. Ind. Genie Chim., 95, 307 (1966)] further classifies the moving holdup into “free” and “semifree.” The total is φd, which here refers to the volume of dispersed phase expressed as a fraction of the void space in the packed section. See Beckmann et al. [Am. Inst. Chem. Eng. J., 1, 426 (1955); 3, 223 (1957)]. What follows is a very brief summary of the extensive work of Pratt and his coworkers, Dell, Gayler, Lewis, Jones, Roberts, and White [Trans. Inst. Chem. Eng. (London), 29, 89, 110, 126 (1951); 31, 57, 69 (1953); Chem. Ind. (London), 1952, p. 358]. For the standard commercial packings of 1.27-cm (a-in) size and larger, at low values of Vd, φd varies linearly with Vd up to values of φd = 0.10. With further increase of Vd , φd increases sharply up to a “lower transition point,”

LIQUID-LIQUID EXTRACTION EQUIPMENT

15-33

resembling “loading” in gas-liquid contact. At still higher values of Vd an upper transition point occurs, the drops of dispersed phase tend to coalesce, and Vd can increase without a corresponding increase in φd. This regime ends in flooding. Drops of the dispersed phase reach a characteristic size after leaving the distributor nozzles regardless of their initial size. For each system there is a critical packing size above which the mean drop size is a minimum. For smaller packing, the drop size is larger (and the interfacial area smaller). The critical size of packing, usually 1.27 cm (a in) or more, is given by dFC = 2.42(σgc /∆ρg)0.5

(15-70)

For packing larger than dFC, the characteristic drop diameter, for liquids that are in concentration equilibrium, is given by dp = 0.92(σgc /∆ρg)0.5(Vkεφd /Vd )

(15-71)

For liquids that are not in concentration equilibrium and when an unequilibrated solute is present, the characteristic drop size will generally be larger. If the drops formed at the distributor nozzle are smaller than this, there may be a tendency to flood until they grow to size. Thornton [Ind. Chem., 39, 632 (1963)] finds that large drops decay in exponential fashion to their final size. It is therefore best to design the nozzles to give drop sizes which are larger than that given by Eq. (21-67). Vk is a characteristic drop velocity (at Vc = 0, Vd approaching 0), and is given by Fig. 15-33. Below the upper transition point, the holdup is given by Vd Vc (15-72)  +  = εVk(1 − φd) φd 1 − φd Additional holdup correlations are offered by Sitarmayya and Laddha [Chem. Eng. Sci., 13, 263 (1961)] and Ghosal et al. [Trans. Indian Inst. Chem. Eng., 11, 23 (1958–1959)]. The interfacial area is given by a = 6εφd /dp It is generally desirable to design for φd in the range 0.15 to 0.25 (the lower value for Vd /Vc < 0.5). Flooding Many correlations are available. By a comparison of the observed and calculated velocities at flooding for all available data, those of Crawford and Wilke [Chem. Eng. Prog., 47, 423 (1951)] and Hoffing and Lockhart [Chem. Eng. Prog., 50, 94 (1954)] are best and about equally effective. The Crawford-Wilke correlation is the simpler and is given in Fig. 15-34. Nemunaitis, Eckert, Foote, and Rollinson [Chem. Eng. Prog., 67(11), 60 (1971)] updated the correlation using packing factors. See also Dell and Pratt [Trans. Inst. Chem. Eng. (London), 29, 89, 270 (1951)], Fujita et al. [Chem. Eng. ( Japan), 17, 230 (1957)], Sakiadis and Johnson [Ind. Eng. Chem., 46, 1229 (1954)], and Kafarov and Dytnerskii [Zh. Prikl. Khim., 30, 1698 (1957)]. For very small packings, see Rao and Rao [Chem. Eng. Sci., 9, 170 (1958)] and Venkatoramen and Laddha [Am. Inst. Chem. Eng. J., 6, 355 (1960)]. It is recommended that flow rates be set at no more than 50 percent of the flooding values, less if the interfacial tension of the liquids is high. Mass Transfer Extraction rates for packed towers are usually excellently correlated for a given situation on the coordinate system of Fig. 15-35. Treybal [Chem. Eng. Prog., 62(9), 67 (1966)] has suggested means whereby overall HtO’s may be resolved into constituent

Characteristic drop velocity for packed towers, for equilibrium liquids dF > dFC and T > 0.25 ft. [Pratt, Ind. Chem., 31, 552 (1955), with permission.] FIG. 15-33

FIG. 15-34 Flooding in packed towers. Use only customary units in the variables. [Crawford and Wilke, Chem. Eng. Prog., 47, 423 (1951), with permission.]

Ht’s. In connection with the data on this figure, it should be noted that economical values of m′VE /VR will usually lie in the range between 1 and 2, so that overall heights of transfer units are not too unreasonable even for this high-interfacial-tension system. For lower interfacial tensions, HtOC will ordinarily be appreciably less. The number of variables that are known to influence the rate of extraction is exceedingly large, and includes at least the following: Size, shape, and material of packing Tower diameter Packing depth Dispersed-phase distributor design Which liquid is dispersed Direction of extraction, whether from dispersed to continuous, organic liquid to water, or the reverse Dispersed-phase holdup Flow rates and flow ratio of the liquids

FIG. 15-35 Extraction of diethylamine from water into toluene (dispersed) in towers packed with unglazed porcelain Raschig rings. To convert feet to meters, multiply by 0.3048; to convert inches to centimeters, multiply by 2.54. [Leibson and Beckman, Chem. Eng. Prog., 49, 405 (1953), with permission.]

15-34

LIQUID-LIQUID EXTRACTION OPERATIONS AND EQUIPMENT

Physical properties of the liquids Presence or absence of surface-active agents Although many attempts have been made to establish a method for estimating the extraction rates [see, for example, Ellis, Ind. Chem., 28, 483 (1952); Jeffreys and Ellis, Congr. Chem. Eng. Des., 1962, 65; and Treybal, Liquid Extraction, 2d ed., McGraw-Hill, New York, 1963], it is still most important to pilot-plant any new process. About the most that can be said is that, for a given system, packing, and method of operation, Htd should be practically constant for all flow rates up to transition and that Htc should vary roughly as C(Vc / Vd )n, where C and n are constants, and to both Ht’s correction must be applied on scale-up for axial dispersion [Treybal, Chem. Eng. Prog., 62(9), 67 (1966)]. Table 15-7 lists additional selected data sources. Axial Dispersion Vermeulen et al. [Chem. Eng. Prog., 62(9), 95 (1966)] summarized many of the data for packings. Their correlation for the continuous phase is shown in Fig. 15-36. For the dispersed phase, their correlation is given by TABLE 15-7 Selected Sources of Packed-Tower Mass-Transfer Data

System Water–acetic acid–ethyl acetate, cyclohexane, methylcyclohexane, ethyl acetate + benzene Water–acetic acid–methyl isobutyl ketone

Water-acetone-hydrocarbon

Tower diameter, in. 1 1.95 3

1.88 2–4

Water–adipic acid–ethyl ether Water–benzoic acid–carbon tetrachloride Water-diethylamine-toluene Water–ethyl acetate Water-methylisobutyl-carbinol Water–methyl ethyl ketone Water–propionic acid–methyl isobutyl ketone Acetone (aq.)–soybean oil, linseed oil Petroleum-furfural Toluene–heptane–diethylene glycol

6 1.95 3, 4, 6 3 4 4 4 1.88 2 2 1.2 1.4, 2.25

Packing

Ref.

0.25-in. saddles

b

0.23-in. rings 0.375-in. plastic spheres 0.375-in. plastic, ceramic rings 0.5-in. plastic, ceramic saddles 0.25-, 0.375-in. rings, 6-mm. beads 0.5-, 0.75-in. rings 0.5-, 0.75-in. rings, 0.375-in. spheres 0.25-in. rings

g j

0.25–1-in. rings 0.375-in. rings 0.5-in. rings 0.5-in. rings 0.5-in. rings 0.25-, 0.375-in. rings, 6-mm. beads 0.25-in. saddles, 0.5-in. rings 0.25-in. rings 0.16-in. rings Glass and brass rings

i m c n n o

k k o a e f

p

Axial dispersion for the continuous phase in packed towers. Spheres (0.75-in, ε = 0.32 to 0.41; 0.50-in, ε = 0.62), Raschig rings (0.50-in, ε = 0.62; 0.75-in, ε = 0.65), Berl saddles (1.0-in, ε = 0.67). Use customary units in the variables. [Vermeulen et al., Chem. Eng. Prog., 62(9), 95 (1966), with permission.]

FIG. 15-36

1.

Nonwetted carbon rings and wetted Berl saddles: Ed V log  = 0.046 c + 0.301 (15-73) Vd df Vd 2. Wetted ceramic rings: Vc Ed log  = 0.161  + 0.347 (15-74) Vd df Vd The measurements were made with kerosine or diisobutyl ketone dispersed in water. Additional work is reported by Komasawa et al. [Kagaku Kogaku, 30, 237, 450, 928, 1103 (1966); English version, 4, 288, 363 (1966); 5, 125, 182 (1967)], and Olbrich et al. [Trans. Inst. Chem. Eng. (London), 44, T207 (1966)]. GENERAL REFERENCES: Bussolari, Schiff, and Treybal, Ind. Eng. Chem., 45, 2413 (1953). Fujita and Tanizawa, Chem. Eng. (Japan), 17, 111 (1953). Garner, Ellis, and Hill, Am. Inst. Chem. Eng. J., 1, 185 (1955); Trans. Inst. Chem. Eng. (London), 34, 223 (1956). Major and Hertzog, Chem. Eng. Prog., 51, 17 (1955). Mayfield and Church, Ind. Eng. Chem., 44, 2253 (1952). Planovskii and Bulatov, Khim. Mashinostr., 1960(2), 10; (3), 9. Pyle, Duffey, and Colburn, Ind. Eng. Chem., 42, 1042 (1950).

Perforated-Plate (Sieve-Plate) Towers A schematic diagram for the most common design of perforated-plate, or sieve-plate, tower, arranged for light liquid dispersed, is shown in Fig. 15-37. The light liquid flows through the perforations of each plate and is thereby dispersed into drops which rise through the continuous phase. The continuous liquid flows horizontally across each plate and passes to the plate beneath through the down spout. For heavy liquid dispersed,

d l h

a Degaleesan and Laddha, Chem. Eng. Sci., 21, 199 (1966); Indian Chem. Eng., 8(1), 6 (1966). b Eaglesfield, Kelly, and Short, Ind. Chem., 29, 147, 243 (1953). c Gaylor and Pratt, Trans. Inst. Chem. Eng. (London), 31, 78 (1953). d Garwin and Barber, Pet. Refiner, 32(1), 144 (1953). e Gier and Hougen, Ind. Eng. Chem., 45, 1362 (1953). f Guyer, Guyer, and Mauli, Helv. Chim. Acta, 38, 790 (1955). g Guyer, Guyer, and Mauli, Helv. Chim. Acta, 38, 955 (1955). h Kishinevskii and Mochalova, Zh. Prikl. Khim., 33, 2344 (1960). i Liebson and Beckmann, Chem. Eng. Prog., 49, 405 (1953). j Moorhead and Himmelblau, Ind. Eng. Chem. Fundam., 1, 68 (1962). k Osmon and Himmelblau, J. Chem. Eng. Data, 6, 551 (1961). l Sef and Moretu, Nafta (Zagreb), 5, 125 (1954). m Shih and Kraybill, Ind. Eng. Chem. Process. Des. Dev., 5, 260 (1966). n Smith and Beckmann, Am. Inst. Chem. Eng. J., 4, 180 (1958). o Rao and Rao, J. Chem. Eng. Data, 6, 200 (1961). p Young and Sullans, J. Am. Oil Chem. Soc., 32, 397 (155). NOTE: To convert inches to centimeters, multiply by 2.54.

FIG. 15-37

persed.

Portion of a perforated-tray tower, arranged for light liquid dis-

LIQUID-LIQUID EXTRACTION EQUIPMENT the same design may be used, but turned upside down. The plates serve to eliminate essentially completely the vertical recirculation of continuous phase characteristic of the spray tower. Furthermore, extraction rates are enhanced by the repeated coalescence and redispersion into droplets of the dispersed phase. Towers of the simple design suggested by Fig. 15-38 have been used successfully in a great variety of services and for petroleum-refining processes have commonly been built to diameters of 3.66 m (12 ft). With careful design, these towers may have excellent flow capacities, and with systems of low interfacial tension equally excellent mass-transfer characteristics. Many variations in design have been suggested and tried, for example, the use of tower packing in the down spouts to prevent entrainment of dispersed phase, arrangements in which both liquids must pass through perforations at each plate, arrangements with vertical perforated plates, etc. As examples of these, see Bradley (U.S. Patent 2,642,341, 1953), Williams (U.S. Patent 2,652,316, 1953), Maycock and Hartwig (U.S. Patent 2,729,550, 1956), and Pohlenz (U.S. Patent

15-35

2,872,295, 1959). Data are available only for arrangements of the sort shown in Fig. 15-39. In general, caplike sieve plates, bubble caps, and vertical perforated plates have not been as satisfactory as horizontal plates. Sieve-Plate Design For best tray efficiency, it is well established that the dispersed phase must issue cleanly from the perforations. This requires that the material of the plates be preferentially wet by the continuous phase (requiring the use of plastics or plastic-coated plates in some instances) or that the dispersed phase issue from nozzles projecting beyond the plate surface. These may be formed by punching the holes and leaving the burr in place or otherwise forming the jets (see Mayfield and Church, loc. cit.). The liquid flowing at the larger volume rate should be dispersed. Perforations are usually 0.32 to 0.64 cm (f to d in) in diameter, set 1.27 to 1.81 cm (a to e in) apart, on square or triangular pitch. There appears to be relatively little effect of hole size on extraction rate, except that with systems of high interfacial tension smaller holes will

FIG. 15-38 Extraction rates for sieve-plate and modified bubble-plate columns. System: benzoic acidwater-toluene, except where noted. To convert feet to meters, multiply by 0.3048; to convert inches to centimeters, multiply by 2.54. [Allerton, Strom, and Treybal, Trans. Am. Inst. Chem. Eng., 39, 361 (1943); Row, Koffolt, and Withrow, ibid., 37, 559 (1941); Treybal and Dumoulin, Ind. Eng. Chem., 34, 709 (1942).]

15-36

LIQUID-LIQUID EXTRACTION OPERATIONS AND EQUIPMENT

produce somewhat better rates. The entire hole area is suitably set at 15 to 25 percent of the column cross section, subject, however, to check through calculations as outlined below. The velocity through the holes should be such that drops do not form slowly at the holes, but rather that the dispersed phase streams through the openings to be broken up into droplets at a slight distance from the plate. This generally requires average linear velocities through the holes of from 15.2 to 30.5 cm/s (0.5 to 1.0 ft/s). The plate area directly opposite down spouts is kept free of perforations. A scum or “interface-rag” bypass can be incorporated in the trays (see Mayfield and Church, op. cit.) at the expense of tray efficiency, or provision may be made for periodic withdrawal of accumulations through the side of the tower between plates. Down spouts (or up spouts) are best set flush with the plate from which they lead, with no weir as in gas-liquid contact. The velocity of the continuous phase in the down spout Vd, which sets the down-spout cross section, should be set at a value lower than the terminal velocity of some arbitrarily small droplet of dispersed phase, say, 0.08 or 0.16 cm (h or g in) in diameter; otherwise, recirculation of entrained dispersed phase around a plate will result in flooding. The down spouts should extend beyond the accumulated layer of dispersed phase on the plate. The depth of dispersed liquid h accumulating on each plate is determined by the pressure drop required for flow of the liquids, h = hC + hD

(15-75)

hD = hσ + hO

(15-76)

TABLE 15-8

Mass-Transfer Data for Perforated-Tray Towers System

Benzene–acetic acid–water

Benzene-acetone-water Benzene–benzoic acid–water Benzene–monochloroacetic acid–water Benzene–propionic acid–water Carbon tetrachloride–propionic acid– water Ethyl acetate–acetic acid–water Ethyl ether–acetic acid–water Gasoline–methyl ethyl ketone–water Kerosene-acetone-water Kerosene–benzoic acid–water Isopar-H–benzyl alcohol, methyl benzyl alcohol, acetophenone-water Methylisobutylcarbinol–acetic acid– water Methyl isobutyl ketone–adipic acid– water Methyl isobutyl ketone–butyric acid– water Pegasol–propionic acid–water Toluene–benzoic acid–water

For the dispersed phase, The available data indicate that, for the orifice effect, V′O2ρD (VO2 − VD2)ρD = (15-77) hO =  2 2g(0.67) ∆ρ 28.9 ∆ρ and that hσ to overcome interfacial-tension effects may be estimated for drop formation at a low velocity through the holes, hσ = 6σgc /dp0.1 ∆ρg

(15-78)

where dp0.1 = drop diameter produced by flow of dispersed phase at VO = 109 m/h (V′O = 0.03 m/s) [360 ft/h (VO′ = 0.1 ft/s)] through the perforations. At hole velocities of 0.3 m/s (1100 m/h) [1 ft/s (3600 ft/h)] or more, hσ should be omitted, and hD = hO. The head required for flow of continuous phase hC includes losses due to (1) friction in the down spout, which should be negligible, (2) contraction and expansion upon entering and leaving the down spout, and (3) two abrupt changes in direction. These total 4.5 velocity heads: hC = 4.5Vd2 ρC /2g ∆ρ

(15-79)

The distance between trays Zt should be larger than h, sufficient so that (1) the “streamers” of dispersed liquid from the holes break up into drops before coalescing into the layer of liquid on the next plate, (2) the linear velocity of continuous liquid is not greater than that in the down spout to avoid excessive entrainment, and (3) the tower may be entered through handholes or manholes in the sides for cleaning. Mass Transfer Mass-transfer rates may be expressed in terms of overall heights of transfer units and successfully correlated for any tower and system as in Fig. 15-38. No significance in terms of individual heights of transfer units for the separate phases should be given to the slope and intercept of such lines. The advantage gained by dispersing the liquid flowing at the larger rate, which results in low values for the abscissa of Fig. 15-38 and consequently low transfer-unit heights, is clear. Alternatively, since the plates resemble and basically behave in the manner of stages, the performance is frequently expressed in terms of stage efficiency, either overall EO for the entire tower or, more satisfactorily, as Murphree efficiencies for each tray. The system of Fig. 15-38 is one of high interfacial tension, so that the heights of transfer units are relatively high and stage efficiency low. For systems of low interfacial tension, on the other hand, stage efficiencies may be very much improved. Table 15-8 lists sources of mass-transfer data. Treybal (Liquid Extraction, 2d ed., McGraw-Hill, New York, 1963)

Toluene-diethylamine-water 2,2,4-Trimethylpentane–methyl ethyl ketone–water

Tower diameter, in.

Tray spacing, in.

Ref.

1.97 1.97 2.2 1.6 × 3.2 3 3 1.97 1.97 1.97

3.9–6.3 3.2–6.3 2.8–6.3 5.9 4, 8 4 3.9–6.3 3.2–6.3 3.9–6.3

t s r p m m t s t

2 8.63 3.75 3 3.63 2 × 12

8–24 4–7.2 4.5, 6 4, 8 4.75 24

j n k m a b

3

6

l

4.18

6

e

4.8

6–23

g

4.8 8.75 3.63 3.56 3 2.72 2 4.18 3.75

6–11 6 4.75 3–9 6 9 24 6 4.5, 6

g o a q l f j c, d k

a Allerton, Strom, and Treybal, Trans. Am. Inst. Chem. Eng., 39, 361 (1943). b Angelo and Lightfoot, Am. Inst. Chem. Eng. J., 14, 53 (1968). c Garner, Ellis, and Fosbury, Trans. Inst. Chem. Eng. (London), 31, 348 (1953). d Garner, Ellis, and Hill, Am. Inst. Chem. Eng. J., 1, 185 (1955). e Garner, Ellis, and Hill, Trans. Inst. Chem. Eng. (London), 34, 223 (1956). f Goldberger and Benenati, Ind. Eng. Chem., 51, 641 (1959). g Krishnamurty and Rao, Indian J. Technol., 5, 205 (1967). h Krishnamurty and Rao, Ind. Eng. Chem. Process Des. Dev., 7, 166 (1968). i Lodh and Rao, Indian J. Technol., 4, 163 (1966). j Mayfield and Church, Ind. Eng. Chem., 44, 2253 (1952). k Moulton and Walkey, Trans. Am. Inst. Chem. Eng., 40, 695 (1944). l Murali and Rao, J. Chem. Eng. Data, 7, 468 (1962). m Nandi and Ghosh, J. Indian Chem. Soc., Ind. News Ed., 13, 93, 103, 108 (1950). n Pyle, Duffey, and Colburn, Ind. Eng. Chem., 42, 1042 (1950). o Row, Koffolt, and Withrow, Trans. Am. Inst. Chem. Eng., 37, 559 (1941). p Shirotsuka and Murakami, Kagaku Kogaku, 30, 727 (1966). q Treybal and Dumoulin, Ind. Eng. Chem., 34, 709 (1942). r Ueyama and Koboyashi, Bull. Univ. Osaka Prefect., A7, 113 (1959). s Zheliznyak, Zh. Prikl. Khim., 40, 689 (1967). t Zheliznyak and Brounshtein, Zh. Prikl. Khim., 40, 584 (1967). NOTE: To convert inches to centimeters, multiply by 2.54.

has shown that good estimates of the rate of extraction, or stage efficiency, may be made by computing the rates of extraction for drop formation, drop rise (by computing dispersed-phase holdup and drop velocity and by considering the continuous phase to be of uniform concentration vertically), and drop coalescence (see the subsections “Single Drops Immersed in Immiscible Liquids” and “Spray Towers.” See also Skelland and Cornish [Can. J. Chem. Eng., 43, 302 (1965)]. Specifically, Angelo and Lightfoot [Am. Inst. Chem. Eng. J., 14, 531 (1968)] have had good success in applying the surface-stretch theory to drop formation and drop rise for oscillating drops on a perforatedtray extractor. Zheleznyak and Brounshtein [Zh. Prikl. Khim., 40, 584, 689 (1967)] have shown that if the mass-transfer resistance lies within the drop phase, the approach to equilibrium of that phase produced by an extractor is simply related to the approach reached in one section.

LIQUID-LIQUID EXTRACTION EQUIPMENT The following empirical expression (Treybal, Liquid Extraction, 2d ed., McGraw-Hill, New York, 1963) has been found to represent all the available data reasonably well, considering the great variety of circumstances and the considerable scatter in many of the original data: 89,500Z 0.5 VD 0.42 0.9Z′t0.5 VD 0.42 t EO =  =  (15-80)  σgc VC σ′ VC Use only U.S. customary units in this equation. Krishnamurty and Rao [Ind. Eng. Chem. Process Des. Dev., 7, 166 (1968)] suggest that Eq. (21–77) is improved if the right-hand side is multiplied by 0.1123/dO0.35. Mechanically Agitated Gravity Devices Owing to the usual small density differences between the contacted liquids, the energy available from simple counterflow under the force of gravity is insufficient to disperse one liquid in the other and to establish turbulence levels to the extent necessary for rapid mass transfer, particularly for systems of high interfacial tension. Application of energy, mechanically applied through stirring devices, pulsations, etc., assists. The devices of major importance are considered below in order of increasing complexity of design.

冢 冣

冢 冣

Rotary-Disk Contactors (RDC) GENERAL REFERENCES: Logsdail, Thornton, and Pratt, Trans. Inst. Chem. Eng. (London), 35, 301 (1957). Misek, Collect. Czech. Commun., 28, 426, 570, 1631 (1963); 32, 4018 (1967) (in English); Ratacni Diskove Extraktory a Jejich Vypocty, SNTL, Prague, 1964. Olney et al., Am. Inst. Chem. Eng. J., 8, 252 (1962); 10, 827 (1964). Reman et al., U.S. Patent 2,601,674 (1952); Chem. Eng. Prog., 51, 141 (1955); 62(9), 56 (1966); Joint Symposium: Scaling-Up Chemical Plant and Processes, London, 1957, p. 26.

Refer to Fig. 15-39. The tower is formed into compartments by horizontal doughnut-shaped or annular baffles, and within each compartment agitation is provided by a rotating, centrally located, horizontal disk. Somewhat similar devices have been known for some time. The features here are that the rotating disk is smooth and flat and of a diameter less than that of the opening in the stationary baffles, which facilitates fabrication and apparently improves extraction rates. The typical proportions of the internals of the RDC are as follows: ds /dt = 0.7 dr /dt = 0.6 Zt /dt—the following table applies For

0 < dt < 0.1 m 0.1 < dt < 1.0 m

FIG. 15-39

tems Inc.)

Zt = (dt)0.5 Zt /dt = 0.15

Rotating-disk (RDC) extractor. (Courtesy of Glitsch Process Sys-

1.0 < dt < 1.5 m

Zt /dt = 0.12

1.5 < dt < 2.5 m

Zt /dt = 0.10

2.5 < dt > 2.5 m

Zt /dt = 0.08

15-37

where ds = stator diameter dt = tower diameter dr = rotor diameter Zt = stage height The general proportions may be varied from one end of the tower to the other to accommodate changing liquid volumes and physical properties. These towers have been used in diameters ranging from a few inches for laboratory work up to 2.4 m (8 ft) in diameter by 12.2 m (40 ft) tall for purposes of deasphalting petroleum. Other commercial services include furfural extraction of lubricating oils, desulfurization of gasoline, phenol recovery from wastewaters, and many others. Columns up to 4.5 m in diameter and up to 50 m in height have been constructed. A reliable design procedure for new systems, without the necessity for laboratory work, is not yet established. The data available show that the flow capacity increases with (1) decreased rotor speed, (2) decreased diameter of rotating disks, (3) increased diameter of opening in the stationary baffles, and (4) increased compartment height. Logsdail et al. (loc. cit.) have proposed that the slip velocity of Eq. 15-83, in the absence of mass transfer, can be set equal to VK(1 − ϕD), where VK is a “characteristic” velocity which can be related to the liquid properties, speed of agitation, and tower geometry. Kung and Beckmann [Am. Inst. Chem. Eng. J., 7, 319 (1961)] and Olney et al. (loc. cit.) have also used this. Misek (loc. cit.), however, has had considerable success by setting the slip velocity equal to VK(1 − ϕD) exp [ϕD(z − 4.1)], where z is a “coalescence coefficient” which depends on the liquid properties. Evidently mass transfer has a profound effect, as a result of drop coalescence; variation in the flooding rate from −15 to +200 percent has been noted in the extraction of acetone to and from water, respectively, with organic solvents. See also Kagan et al., Izv. Vyssh. Uchebn. Zaved. Khim. Khim. Tekhnol., 9, 836 (1966). Dropsize distribution which has an important influence on axial dispersion in the dispersed phase has been studied extensively by Misek and Olney (loc. cit.). The value of HETS becomes smaller with (1) increased rotor speed but passes through a minimum, (2) increased diameter of rotating disks, (3) decreased diameter of stationary baffle opening, and (4) decreased compartment height. Reman and Olney [Chem. Eng. Prog., 51, 141 (1955)] show a correlation of stage height for two sizes of RDCs with the system water-kerosine-butylamine, as in Fig. 15-40. That such correlations cannot be general is indicated by these authors’ data on caustic extraction of gasoline, which show quite different

FIG. 15-40 Extraction in RDC columns, water-butylamine-kerosine (continuous). dt = 0.33 and 1.33 ft. Curve A: VD = 50.7, VC = 78.9 ft/h. Curve B: VD = 25.4, VC = 78.9 ft/h. Use customary units in the variables. [Data of Reman and Olney, Chem. Eng. Prog., 51, 141 (1955).]

15-38

LIQUID-LIQUID EXTRACTION OPERATIONS AND EQUIPMENT

curves. Logsdail, Thornton, and Pratt (loc. cit.) tentatively suggest that data can be correlated through 2β/3 HtOC g2ρC 1/3 µCg ∆ρ 2(β − 1)/3 (15-81)   φD = C   VC µC VK3 (1 − φD)3ρC ρC the constants C and β to be determined for each system. For toluenewater-acetone, β = 0.13; for butyl acetate-water-acetone, β = 0.4; in both cases, transfer was from water to organic solvent. For transfer in the reverse direction, VK could not be computed (see above). A large number of studies of axial mixing have been made [Gel’perin et al., Teor. Osn. Khim. Tekhnol., 1, 666 (1967); Kagan et al., Zh. Prikl. Khim., 39, 88 (1966); Miyauchi et al., Am. Inst. Chem. Eng. J., 12, 508 (1966); Stainthorp and Sudall, Trans. Inst. Chem. Eng. (London), 42, 198 (1964); Stemerding and Zuiderweg, Chem. Ing. Tech., 35, 844 (1963); and Strand et al., Am. Inst. Chem. Eng. J., 8, 252 (1962)]. Reman [Chem. Eng. Prog., 62(9), 56 (1966)] recommends, for the continuous phase in columns 0.08 to 2.13 m (3 in to 7 ft) in diameter,

冢 冣

冤

冥 冢 冣

EC = 0.5ZtVC + 0.012di NZt(ds /dt )2

(15-82)

For the dispersed phase firm relationships have not been established, but at high rotor speeds, ED may be 1 to 3 times EC. In any event, axial mixing for the liquid flowing at the lower rate becomes very severe for extreme flow ratios (>10). Costs are given by Clerk (Chem. Eng., 232 (Oct. 12, 1964). Several modifications of the design have appeared. Modifications of the rotors include perforation of the disk [Krishnara et al., Br. Chem. Eng., 12, 719 (1967)] and radially supported arc plates [Nakamura and Hiratsuka, Kagaku Kogaku, 30, 1003 (1966)]. An “asymmetric” modification, with off-center rotors and arrangement of settling spaces for the liquids between dispersions (Misek, loc. cit.) is available in Europe. As stated above, the design of an RDC contactor usually involves the performance of pilot tests due to the large number of factors which can influence performance. These pilot plant data must then be scaled-up to full commercial size. The following procedure is recommended. 1. Pilot plant tests are conducted using the actual plant materials since small amounts of contaminents can have significant effects on throughput and efficiency. These tests are usually conducted in columns ranging from 0.075–0.15 m diameter; the column height (and therefore number of compartments) should be sufficient to accomplish the separation desired; this may require several iterations on column height. 2. The column is run over a range of total throughputs (Vo + Vc) and agitation speeds; at each condition the concentrations of the streams are measured after equilibrium is reached; the holdup is also measured by stopping the agitation, isolating the column, and measuring the change in the interface level. The flooding point is determined at each specific throughput by increasing the agitation speed until the column floods. 3. From the above data, the combination of specific throughput and agitation speed which gives the optimum performance in terms of separation can be determined. At this condition the following relationships can be calculated: Vd Vc Slip Velocity: Vs =  + (15-83) φd 1 − φd (Ns)3(dr)5 Specific Power Input =  (Zt)(dt)2

(15-84)

where Ns = rotational speed dr = rotor diameter Zt = stage height dt = tower diameter Max Continuous Phase Velocity at Flooding Vse(−φdf ) Vcf =  [Vd /Vc]/φdf

(15-85)

where Vcf = velocity of the continuous phase at flooding φdf = holdup of the dispersed phase at flooding Vd = dispersed phase velocity Vs = slip velocity 4. For design, the slip velocity is derated to 70–80 percent of the calculated value to give some margin of safety; this sets the design value of the continuous phase velocity (Vc). The column cross sectional area (and therefore diameter) is set by Qc /Vc. With the diameter set, the other dimensions can be set using the ratios given above. 5. The rotor speed of the scaled up tower is based on maintaining the same specific power input number as used on the pilot column; it can be determined by substituting the specific values into the relationship: N s3R 5r  Z t d 2t 6. For the column height, the pilot plant data must be corrected for the effect of axial mixing. The height of a transfer unit (HTU) can be determined from the pilot plant data; to this must be added the height of a diffusion unit (HDU). This is done by determining the axial mixing coefficients of the continuous and dispersed phases according the the following relationships: Ec = 0.5VcZt + 0.012dr NsZt(ds /dt )2 (15-86) 4.2 × 105 Vd 3.3 Ed = Ec  (15-87)  d 2t φd where Ec = diffusion coefficient, continuous phase; Ed = diffusion coefficient, dispersed phase; Vc, Zt, dr, Ns, Zt, dt, φd = same as defined previously; see Nomenclature list. From these the continuous and dispersed phase Péclet numbers can be determined from the relationships: 1 Ec(1 − φd) (15-88)  =  (Pe)c ZtVc

冤

冢 冣冥

4.2 × 105 Vd 1 1 φdVc  =     d t2 (Pe)d (Pe)c (1 − φd)Vd φd

冤 冥冤

冥冤

冢 冣冥 3.3

(15-89)

where (Pe)c = Péclet number, continuous phase (Pe)d = Péclet number, dispersed phase The HDU is then calculated from the relationship: Zt Zt HDU =  + (15-90) (Pe)c (Pe)d And finally, the effective height of a transfer unit is calculated from: (HTU)eff = HTU + HDU where

(15-91)

HDU = height of a diffusion unit HTU = height of a transfer unit (HTU)eff = effective height of a transfer unit

Lightnin Mixer (Oldshue-Rushton) Tower GENERAL REFERENCES: Bibaud and Treybal, Am. Inst. Chem. Eng. J., 12, 472 (1966). Dykstra, Thompson, and Clouse, Ind. Eng. Chem., 50, 161 (1958). Gustison, Treybal, and Capps, Chem. Eng. Prog., 58, Symp. Ser. 39, 8 (1962). Gutoff, Am. Inst. Chem. Eng. J., 11, 712 (1965). Oldshue and Rushton, Chem. Eng. Prog., 48, 297 (1952). Miyauchi et al., Am. Inst. Chem. Eng. J., 12, 508 (1966).

The Oldshue-Rushton (Mixco) extractor is similar in construction to the RDC in the fact that it is a relatively open design, consisting of a series of compartments separated by horizontal stator baffles. The major difference from the RDC is that the height/diameter ratio of the compartments is greater, each compartment is fitted with vertical baffles, and the mixing is accomplished by means of a turbine impeller rather than a disc.

LIQUID-LIQUID EXTRACTION EQUIPMENT

15-39

FIG. 15-42 Extraction in Mixco columns, methyl isobutyl ketone-acetic acidwater (continuous). dt = 0.5 ft, Zt = 0.333 ft, X = flooded condition. To convert feet to meters, multiply by 0.348; to convert feet per hour to meters per hour, multiply by 0.3048.

Mixco (Oldshue-Rushton) extractor.

FIG. 15-41

Refer to Fig. 15-41. The extractor is an extension of the simple baffled mixing vessel into a multistage column. Although commercial application has been made, data are scarce and are limited to towers of small diameter. The preferred proportions are Zt = 0.5d t , ds > di. For water (continuous) and toluene or kerosine (dispersed), in a tower with dt = 0.152 m (0.5 ft), Zt = 0.082 m (0.27 ft), di = 0.051 m (0.1667 ft), dispersed-phase holdup is given by Eq. (15-70) with Vs = Vk(1 − φd ) and the following relationship by Wong (M.Ch.E. thesis, New York University, 1963): Vk µc /σgc = 1.77(10−4)(g/di N 2)(∆ρ/ρC)0.9

(15-92)

For the same liquids axial mixing is described by (Bibaud and Treybal, loc. cit.) Ecφc /Vc Zt = −0.1400 + 0.0268(di Nφc /Vc ) d N ρc dN  = 0.393(10−8)  Ed σgc 2 i

冢

3 i

2

(15-93)

ρ Nρ  冣 冢 ∆ρ 冣 冢 µ 冣 1.54

c

4.18

d

2 i

c

scale and using the same mass velocities of flow. Bibaud (loc. cit.) found that, for butylamine extracted from kerosine (dispersed) into water, the extraction rates corrected for axial mixing in either phase were described by assuming the drops to be rigid spheres, with Thornton’s correlation [Ind. Chem., 39, 298 (1963)] for drop size. A somewhat related design has been studied by Nagata, Eguchi, and coworkers [Chem. Eng. ( Japan), 17, 20 (1953); 20, 2 (1956); Mem. Fac. Eng., Kyoto Univ., 19, 102 (1957); Kagaku Kogaku, 22, 483 (1958)]. This column is characterized by the relatively small, separate openings between compartments for passage of liquids and the eccentric location of the impeller shaft. In a pilot-plant column, dt = 0.3 m (0.983 ft), phenol was extracted from water [Vc = 11.6 m/h (38.1 ft/h)] into benzene [Vd = 6.4 m/h (21 ft/h)] at a stage efficiency of 0.618. Because of the above limitations in prediction of column performance based on correlations, the design of an Oldshue-Rushton must also be based on pilot-plant tests. The minimum column diameter which can be used to give reliable scale-up data is 6 inches (150 mm); it is usuaslly fitted with stages 3 inches (75 mm) high, and the stage opening is 2.4 inches (60 mm). The column should be high enough to accomplish the complete extraction; if it is not it will be necessary to “rerun” the extract and raffinate phases to examine effects in the dilute regions of the column. The following procedure is followed: 1. The column is run over a range of total throughputs (Vo + Vc)

0.61

(15-94)

c

See also Miyauchi et al. (loc. cit.), who express the axial mixing in terms of interstage flow. For the continuous phase with no dispersedphase flow, see Bibaud and Treybal, and Gutoff (loc. cit.). Figure 15-42 presents some of the data of Oldshue and Rushton (loc. cit.) which show an optimum agitator speed for each configuration studied. The optimum would be expected to vary with physical properties of the liquids contacted. HETS is improved, although capacity is decreased, by smaller openings in the stationary baffles. The effect of stage openings of efficiency and throughput for the system MIBK-acetic acid-water in a 6-inch (150 mm) diameter column is shown in Table 15-9. In the more difficult (because of high interfacial tension) extraction of uranium between kerosine-diluted solvents and aqueous solutions, Dykstra et al. (loc. cit.) have also shown the development of an optimum impeller speed. Gustison et al. (loc. cit) have found it possible to correlate the stage efficiency with the ratio of flow rates (Vd /Vc) and the distribution coefficient, which varies considerably with concentration in the extraction of uranium. They also found it possible to scale up performance from 0.152- to 0.305-m (6- to 12-in) diameter geometrically, on the assumption that the continuous phase was thoroughly mixed in each compartment, by applying equal power per unit volume of liquids treated on the large and the small

TABLE 15-9 Effect of Size of Opening between Compartments* Compartment opening, mm

Maximum stage efficiency

0 54 82 152

100 83 52 38

Minimum HETS, mm

Flow rate, kg s−1 m−2

Constant flow rate 2560 3098 4953 6731

0† 2.9† 2.9 2.9

At maximum efficiency 0 54 82 152

100 83 67 38

2560 3098 3860 6731

0† 2.9† 5.4† 6.0b

*Typical data for operation with methyl isobutyl ketone, water, acetic acid; four stages; 101.6-mm stage height, 152-mm-diameter column; extraction, water → ketone. †Optimum flow rate. Oldshue in Lo, Baird, Hanson, Handbook of Solvent Extraction, p. 436, John Wiley & Sons, NY, 1983. Used with permission.

15-40

LIQUID-LIQUID EXTRACTION OPERATIONS AND EQUIPMENT

and agitation speeds; at each condition the concentrations of the streams are measured after equilibrium has been reached. The flooding point is determined at each throughput by increasing the agitation speed until the column floods. 2. From the above data, the condition of throughput and agitation speed which gives the optimum performance can be determined. 3. Based on this design-specific throughput and the required production column rates, the diameter of the commercial column can be calculated. The stage geometry is next set by maintaining geometric similarity to the pilot column. 4. Finally, the production column agitator speed is determined by maintaining the same power per unit volume as was used on the pilot column. The above approach will usually result in a conservative design, since the stage efficiency is usually much higher in the production column than in the pilot column. A comparison of the controlling parameters which exist in the pilot and production scales are depicted in Fig. 15-43. Scheibel Extraction Towers The original Scheibel tower design [Chem. Eng. Prog., 44, 681, 771 (1948); U.S. Patent 2,493,265, 1950] used knitted-mesh packed sections in a tower for coalescence with a centrally located impeller between the packed sections for drop breakup. Scheibel and Karr [Ind. Eng. Chem., 42, 1048 (1950)] presented data on a 0.305-m- (12-in-) diameter column of this design (Fig. 15-44) for systems which are difficult to extract because of high interfacial tension or easy because of low interfacial tension. Excellent values of HETS were obtained with a wide variety of conditions. Low throughput and ratios of flow rates greatly different from unity required high agitator speeds for best results. Both direction of extraction and which phase was dispersed influenced the rates. The liquids of Fig. 15-44 were also used in tests involving the mixing sections alone (see operating characteristics of mechanically agitated vessels). Honekamp and Burkhart [Ind. Eng. Chem. Process Des. Dev., 1, 176 (1962)] found very little change in drop size to occur within the knitted-wire mesh and measured extraction rates in the mesh zone for one system. A second Scheibel tower design [Am. Inst. Chem. Eng. J., 2, 74 (1956); U.S. Patent 2,850,362, 1958] reduced HETS and permitted more direct scale-up. The impellers are surrounded by stationary

Extraction in first Scheibel column. T = 0.94 ft, di = 0.333 ft, height of mixer section = 3 in, height of packed section = 9 in. To convert inches to centimeters, multiply by 2.54; to convert feet to meters, multiply by 0.3048; and to convert feet per hour to meters per hour, multiply by 0.3048. [Data of Scheibel and Karr, Ind. Eng. Chem., 42, 1048 (1950).]

FIG. 15-44

Curve

System

VD, ft/h

VC, ft/h

A

MIBK(C)–water(D,E)–acetic acid MIBK(D)–water(C,E)–acetic acid MIBK(C,E)–water(D)–acetic acid MIBK(C)–water(D,E)–acetic acid MIBK(C,E)–water(D)–acetic acid MIBK(D)–water(C,E)–acetic acid o–Xylene(D)–water(C,E)–acetone o–Xylene(D,E)–water(C)–acetone o–Xylene(C)–water(D,E)–acetone o–Xylene(C,E)–water(D)–acetone

41.7

41.7

41.7 23.2

41.7 23.2

25.9 22.1 25.9 21.1

17.3 21.2 17.3 22.1

B C D E F G

MIBK = methyl isobutyl ketone; C = continuous; D = dispersed; E = extractant.

Pilot scale

Full scale Full scale compared to pilot scale

Residence time Blend time, undispersed Interstage mixing, undisperesed Interstage mixing, disp. Concentration gradient, disp. Max. impeller zone shear rate Ave. impeller zone shear rate Ave. tank zone shear rate Turbulent shear rates

Higher Longer Different Different Higher Higher Lower Lower Different

Mixing factors compared for pilot and full scale. [Oldshue in Lo, Baird, and Hanson, Handbook of Solvent Extraction, John Wiley & Sons, NY, 1983. Used with permission.] FIG. 15-43

shroud baffles to direct the flow of droplets as they are discharged from the tips of the impellers. Data taken from a 0.305-m- (12-in-) diameter tower are shown in Fig. 15-46 and correlated in terms of the power applied per unit volume of liquids handled per compartment. For the impeller used, the power number at turbulent Reynolds numbers is NPo = 1.85. The data show that while packing in alternate sections may increase mass-transfer rates, it decreases flow capacity. For many industrial systems, the knitted mesh was not used because of fouling (Fig. 15-45). Towers up to 3 m (9.8 ft) in diameter are in service. A third design by Scheibel (U.S. Patent 3,389,970, 1968) uses closed impellers plus horizontal baffles in the tower. Scheibel (Ref. 2) has shown that the efficiency of a mixing stage can be correlated to the power per unit of throughput, and is related to the ratio of dispersed/continuous phase flow rates; this is shown in Fig. 15-47. This figure shows an optimum power input; below this value efficiency drops off due to reduced interfacial area; above this value efficiency decreases due to increased axial mixing of the continuous and dispersed phases. Scheibel has found that the power input can be correlated by the following equation: (di)5 ρ(Ns)3 P = 1.85  (15-95) gc where P = Power input per mixing stage

冤

冥

LIQUID-LIQUID EXTRACTION EQUIPMENT

15-41

FIG. 15-46 Extraction in second Scheibel column. T = 0.94 ft, di = 0.333 ft, height of packed section = height of mixer section = 2 in. Use customary units in the variables. [Data of Scheibel, Am. Inst. Chem. Eng. J., 2, 74 (1956).] FIG. 15-45 Second Scheibel extractor with horizontal baffles and no wiremesh packing between stages. [Reprinted with permission of Am. Inst. Chem. Eng. J., 2, 74 (1956)].

di = impeller diameter ρ = average stage density Ns = impeller rotational speed gc = gravitational constant As with the design of the other columns described above, the design of a Scheibel column must be based on pilot plant tests and scale-up. The following procedure is recommended: 1. Pilot tests are usually conducted in 0.075-m diameter columns; the column should contain a sufficient number of stages to complete the extraction; this may require several iterations on column height. 2. The column is run over a range of throughputs (Vd + Vc) and agitation speeds; at each condition the concentrations of the streams

Curve

System

A, B* C* D†

Methyl isobutyl ketone–water–acetic acid o–Xylene–water–acetic acid o–Xylene–water–phenol Methyl isobutyl ketone–water–acetic acid o–Xylene–water–acetic acid

*Alternate mixing and packed sections. †Packing omitted. Agitators in alternate and also every section.

are measured after equilibrium is reached (usually 3–5 turnovers of column volume). At each throughput the flood point is determined by increasing the agitation until flooding is induced. A minimum of three throughput ranges are examined in this manner. 3. From the above data, the combination of specific throughput and agitation speed which gives the optimum performance in terms of separation can be determined. This determines the design specific

Correlation of mixing-stage efficiency with power input and liquid flow rates. [Scheibel in Lo, Baird, Hanson, Handbook of Solvent Extraction, p. 428 John Wiley & Sons, NY, 1983. Used with permission.]

FIG. 15-47

15-42

LIQUID-LIQUID EXTRACTION OPERATIONS AND EQUIPMENT

throughput value (m3/m2-h) and agitation speed (RPM). 4. Unlike the RDC and Oldshue-Rushton columns where the specific throughput of the scaled-up version is the same as the pilot column, it is the characteristic of the Scheibel column that the throughput of the scaled-up column is on the order of 3–5 times greater than that acheived on the pilot column. The reason for this is that the restricted geometry of the 0.075 m diameter column limits throughput; these restrictions are removed in the scaled-up columns. 5. Once the column diameter is determined, the stage geometry can be fixed. The geometry of a stage is a complex function of the column diameter; in the pilot (0.075 m) column the stage height to diameter ratio is on the order of 1:3; on a 3-m diameter column it is on the order of 1:8. 6. The principle of the Scheibel Column scale-up is to maintain the efficiency of the stage. Therefore, the scaled-up column will have the same number of actual stages as the pilot column. The only difference is that the stages will be taller to take into account the effect of axial mixing. With the agitator dimensions determined, the speed is then calculated to give the same power input per unit of throughput. The scale-up of the Scheibel column is still considered proprietary, and therefore the vendor (Glitsch Process Systems Inc.) should be consulted for the final design. From pilot tests in 0.075-m diameter column, industrial columns up to 3 m in diameter and containing 90 actual stages have been provided. Because of its internal baffling which controls the mixing patterns on the stages, the Scheibel column has proven to be one of the more efficient extractors in terms of height of a theoretical stage; this makes it ideally suited for applications which require a large number of stages, or are located indoors with headroom restrictions. Holmes, Karr, and Cusack (Solvent Extraction and Ion Exchange, vol. 8, no. 3, pp. 515–528, 1990) have published results comparing the efficiency of the Scheibel column to other extractors on the system tolueneacetone-water. Kühni Tower The extraction towers designed at Kühni [see Mögli and Bühlmann, in Lo, Baird, and Hanson (eds.), Handbook of Solvent Extraction, Wiley-Interscience, New York, 1983, sec. 13.5]

use shrouded (closed) impellers on a central shaft in the tower (Fig. 15-48). The droplet size can be controlled by the speed and diameter of the impeller, while the circulation rate can be controlled by the design of the width of the impeller. A perforated plate between each stage can control the droplet holdup by the percentage of open area in the plate. Treybal Tower Treybal [U.S. Patent 3,325,255, 1967); Chem. Eng. Prog., 60(5), 77 (1964)] adapted a mixer-settler cascade in tower form in which the liquids are settled between stages. Karr Reciprocating Plate Tower Up to this point, the agitated columns presented have all imparted their energy to the fluids by means of rotating elements (discs or impellers). However, there is another class of agitated columns which impart their energy by means of reciprocating plates or pulsing of the liquids. This results in a more uniform drop-size distribution due to the fact that the shear forces are more uniform over the entire cross section of the column. The reciprocating plate extractor developed by Karr [Am. Inst. Chem. Eng. J., 5, 446 (1959)] is a mechanically agitated tower using dual-flow plates with 50 to 60 percent open area, mounted on a central shaft and reciprocated vertically (Fig. 15-49). Typical perforated plates and baffle plates for a 35 v-in (0.9-m) diameter column are shown in Fig. 15-50. A typical stroke length is 2.54 cm (1 in) with a speed of 10 to 400 strokes per minute and a plate spacing of 5 to 15 cm (2 to 6 in). Scale-up relationships by Karr (Sep. Sci. Technol., 15(4), 877 (1980)] show that HETS increases with tower diameter to the 0.38 power in the most difficult case. Laboratory columns of 2.54- and 5.08-cm- (1- and 2-in-) diameter are used to scale up to towers as large as 1.5 to 2.0-m (5 to 6.5-ft) in diameter. A high volumetric efficiency is

Schematic arrangement of the 900-mm (36-in) reciprocatingplate column. (Courtesy of Glitsch Process Systems Inc.)

FIG. 15-49 FIG. 15-48

Kühni tower.

LIQUID-LIQUID EXTRACTION EQUIPMENT Perforated plate for nominal 35 5/8′′ column

Baffle plate for nominal 35 1/8′′ column

Open area 58%

Tie rod holes

26 3/4′′ 35 1/4′′

35 1/4′′ Perforations, 9/16′′ dia. holes on 11/16′′ triangular pitch

Free area—58%

(a) FIG. 15-50

(b)

(a) Typical perforated plate. (b) Typical baffle plate.

achieved as measured by total volumetric throughput per crosssectional area divided by HETS. One of the chief characteristics of the Karr column is its highvolumetric efficiency when compared to other extractors. Volumetric efficiency is defined as: Vd + Vc Volumetric efficiency =  (15-96) HETS Karr, Holmes, and Cusack have given comparisons of the Karr column volumetric efficiency with other types of extractors. In Table 15-10 are data showing the values of HETS and volumetric efficiency over a range of column diameters from 1–36 in (0.025–0.9 m); Fig. 15-51

shows how the HETS varies with agitation, again over a range of diameters but at relatively constant total throughput. It was from these data that Karr and Lo developed the scale-up procedure for this type of column. As with the other extractors, the final design of a Karr column depends on the scale-up from a pilot test. The following procedure is recommended. 1. For scale-up up to 2 m in diameter, testing in a pilot column of 0.025 m is sufficient; if the anticipated scaled-up diameter is greater than 2 m, then the pilot tests should be conducted in a 0.050-m diameter. The column should be tall enough to accomplish the complete extraction; this may require several iterations on column height.

TABLE 15-10

Summary of Minimum HETS Values and Volumetric Efficiencies for a Reciprocating-Plate Column*

Column diameter in.

Amplitude, in.

Plate spacing, in.

15-43

Agitator speed, strokes/min

Extractant

Dispersed phase

Minimum HETS

Throughput, gal hr−1 ft−2

Volumetric efficiencies Vt/HETS, h−1

3.1 2.8 4.2 8.1 4.9 6.3 7.5 4.3 6.7 5.0 7.75 5.8 5.7 4.4 5.6 7.2 7.2 14.0 7.0 9.5 11.05 9.5 7.8 6.2

572 913 459 1030 600 1193 1837 548 1168 1172 1707 547 1167 599 1193 602 1200 1821 555 1170 1694 1179 595 1202

296 523 175 204 196 304 393 205 280 376 353 151 328 218 342 134 268 208 127 197 246 199 123 311

9.1 8.2 7.7 9.1 23.3 20.0

424 424 424 804 425 442

75 83 88 142 29† 36†

MIBK-acetic acid-water system 1

a

1

1

a

1

3

a a a a a a

1 1 2 1 1 2

a

1

a

1

a

1

a

1

a a

1 1

1 a d 1 1 1

1 1 1 2 1 1

12 (with baffle)

360 401 278 152 330 245 355 320 230 367 240 430 285 244 170 250 225 150 225 200 150 275 200 150

MIBK

Water

Water

MIBK

MIBK

Water

Water

Water

Water

Water

Water

MIBK

MIBK

MIBK

MIBK

Water

Water

Water

Water MIBK

MIBK MIBK

Xylene-acetic acid-water system 3 3 3 3 36 36

267 537 995 340 168 168

Water Water Water Water Water Xylene

Water Water Water Water Water Water

*Lo, Baird, Hanson, Handbook of Solvent Extraction, John Wiley & Sons, NY, p. 37, 1983. †Because of instrumentation limits, the maximum volumetric efficiencies have not been explored. Used with permission.

15-44

LIQUID-LIQUID EXTRACTION OPERATIONS AND EQUIPMENT height and agitation speed of the scaled-up column is then calculated from the following relationships:

500

where

HETS, in

100 60 40 30 20

1

4

2 10 8 6 5

3

0

100

200 400 500 600 300 Reciprocating speed, strokes/min

Effect of reciprocating speed on HETS, o-xylene-acetic acidwater system. (Lo and Prochazka in Lo et al., p. 377.)

FIG. 15-51

Curve No.

Column diam, in

Phase dispersed

Phase extractant

Double amplitude, in

Plate spacing, in

Total throughput gal/(h)(ft2)

1 2 3 4

36 36 3 3

Water Water Water Water

Water Xylene Water Water

1 1 1 a

1 1 1 1

425 442 424 424

Predicted minimum based on exponents of 0.36 in Eq. (15-98) and 0.14 in Eq. (15-99).

2. The column is first optimized with regard to plate spacing; what is desired is for the tendency to flood to be equal over the entire column length. If one particular section apopears to be limiting the throughput, then the plate spacing should be increased in this area; this will decrease the power input into that section. Likewise, in sections which appear to be undermixed, plate spacing should be decreased. It has been found that the following correlation can be used to estimate the relative plate spacing in the column: 1 l ∝  (15-97) (∆ρ)5/3(σ)3/2 where l = relative plate spacing ∆ρ = density difference of the two phases σ = interfacial tension 3. Once the plate spacing is optimized, the column is run over a range of total throughputs (Vd + Vc ) and agitation speeds. There should be a minimum of three throughput levels, and at each throughput three agitation speeds. After equilibrium is attained at each condition (usually 3–5 turnovers of column volume), samples are taken and separation measured. At each condition the flood point is also determined. It is a characteristic of the Karr column that on the small diameters, the optimum efficiency usually occurs just before the flood point. 4. From these data, plots are made of volumetric efficiency and agitation speed at each throughput level; from these plots the condition which gives the maximum volumetric efficiency is selected for scale-up. 5. The following parameters are kept constant on the scale-up: total throughput per unit area, plate spacing, and stroke length. The

Z2 / Z1 = (dt2 /dt1)0.38

(15-98)

(SPM2 /SPM1) = (dt1 /dt2 )0.14

(15-99)

Z1 = plate stack height in pilot column Z2 = plate stack height in scaled-up column dt1 = diameter of pilot column dt2 = diameter of scaled up column SPM1 = reciprocating speed of pilot column SPM2 = reciprocating speed of scaled-up column

6. For the scaled-up column, suitable baffle plates are required to control axial mixing. For the final column layout the equipment vendor (Glitsch Process Systems Inc.) should be consulted. The Karr column is particularly well suited for systems which tend to emulsify since its uniform shear characteristics tend to minimize emulsion formation. It is also particularly well suited for corrosive systems (since the plates can be constructed of non-metals) or for systems containing significant solids (due to its large open area and hole size on the plates). Slurries containing up to 30 percent solids have been successfully processed in Karr columns. Pulsed Columns These are extractors in which a rapid reciprocating motion of relatively short amplitude is applied to the liquid contents. The agitation so produced has been found to give improved rates of extraction. The principle originated with van Dijck (U.S. Patent 2,011,186, 1935). Because agitation was necessary to reduce tower heights and consequently the expense of massive shielding, and because pulsing provided a means of agitation not requiring moving parts, bearings, and the like in contact with highly corrosive, dangerously radioactive liquids, pulsed columns have been freely applied in the extraction and separation of metals from solutions of atomic energy operations. With very few exceptions, applications appear thus far to be limited to this area. There are two major types of columns: (1) ordinary (spray, packed, etc.) extractors on which pulsations are imposed and (2) a special sieve-plate design. Their characteristics are quite different. Pulsing Devices Refer to Fig. 15-52. At a, a reciprocating plunger or piston pump from which the check valves have been removed is connected to the space containing continuous phase, as shown. This arrangement suffers the disadvantages that (1) the corrosive liquid may be in direct contact with the piston and (2) too rapid pulsing, especially with volatile organic liquids, may cause cavitation. The pipe connecting column and pulser may be of any length to pass through shielding, barriers, and the like, but high pressure drop in the transfer pipe contributes to cavitation difficulties. An alternative arrangement using an air pulse is shown at b in the figure [Thornton, Chem. Eng. Prog., 50, Symp. Ser. 13, 39 (1954); U.S. Patent 2,818,324, 1957]. This keeps corrosive liquids out of contact with the pulsing device and obviates the cavitation problem but because of the compressibility of the gas requires greater application of pulsing power for the same results. For design, see Week and Knight [Ind. Eng. Chem. Process Des. Dev., 6, 480 (1967); 7, 156 (1968)]. For pulsing at the natural frequency of the column, Baird [Proc. Am. Inst. Chem. Eng.-Inst. Chem. Eng. Joint Meeting, London, 1956(6), 53] connected the liquid space to a volume of gas which acts as a spring. Flexible bellows or diaphragms of reinforced rubber, plastic, or metal in contact with the liquids may be flexed mechanically or by an electromagnetic transducer (Thornton, loc. cit.). If hydraulically activated, these may have a life of up to 30,000,000 cycles or more [Jealous and Johnson, Ind. Eng. Chem., 47, 1159 (1955)]. With suitable cam mechanisms, pulsations whose amplitude-time characteristics appear as sine, square, or sawtooth wave shapes are possible. Pressure at the pulsing device and the conditions for cavitation and “water hammer” may be estimated by the methods of Williams and Little [Trans. Inst. Chem. Eng. (London), 32, 174 (1954)] provided the pressure-drop characteristics of the tower internals are known. Jealous and Johnson (loc. cit) have had good success in computing the power required for pulsing. Since power requirement alternates, the use of a flywheel on the pulse mechanism to act as an energy reservoir is suggested as a means of reducing power requirements. Alterna-

LIQUID-LIQUID EXTRACTION EQUIPMENT

(a)

15-45

(b)

Effect of pulsing on extraction in a packed column: methyl isobutyl ketone-acetic acid-water (continuous). Tower diameter = 1.58 in, 27-in depth of d-in Raschig rings. VD = VC = 7.5 to 10. To convert inches to centimeters, multiply by 2.54. [Data of Chantry, von Berg, and Wiegandt, Ind. Eng. Chem., 47, 1153 (1955), with permission.] FIG. 15-53

FIG. 15-52 Pulsed columns: (a) Perforated-plate column with pump pulse generator. (b) Packed column with air pulser.

tively, two columns could be pulsed 180° out of phase with one pulse generator (Griffith, Jasney, and Tupper, U.S. AEC AECD-3440, 1952). Irvine (U.S. AEC ORNL-2377, 1957) devised a pulse pump to utilize part of the pulse energy. Concatenated columns (long extractors built as several short columns, with liquids led from one to the other in strictly countercurrent fashion) may be pulsed by a single pulse generator to advantage, since less power is required owing to reduced static head [Jealous and Lieberman, Chem. Eng. Prog., 52, 366 (1956)]. The following terms are generally used to describe the pulse action: Frequency is the rate of application of the pulse action, cycles/time. Amplitude is the linear distance between extreme positions of the liquid in the column (not of the pulser) produced by pulsing. Pulsed volume = amplitude × frequency × column crosssectional area = volumetric rate of movement of liquid, expressed as volume/time or volume/(time-area). Pulsed Spray Columns Billerbeck et al. [Ind. Eng. Chem., 48, 183 (1956)] applied pulsing to a laboratory [3.8-cm- (1.5-in-) diameter] column. At pulse amplitude 1.11 cm (q in), rates of mass transfer improved slightly with increased frequency up to 400 cycles/min, but the effect was relatively small. Shirotsuka [Kagaku Kogaku, 22, 687 (1958)] provides additional data. There is not believed to be commercial application. Pulsed Packed Columns Any of the ordinary packings may be used, although random packings tend to orient on pulsing, which may lead to channeling. For this reason, Thornton [Chem. Eng. Prog., 50, Symp. Ser. 13, 39 (1954); Br. Chem. Eng., 3, 247 (1958)] recommends fixed packing made from plates of corrugated expanded metal. Polyethylene packing, not wet by aqueous solutions, provides higher flow capacities and mass-transfer rates than ceramic (wetted) packing [Jackson, Holman, and Grove, Am. Inst. Chem. Eng. J., 8, 659 (1952)]. Pulsing reduces the size of dispersed-phase droplets, increases holdup, and increases interfacial area for mass transfer. There is a greater tendency toward emulsification, and maximum throughput is decreased, but HETS is reduced considerably, by the pulsing. Pulsing can be applied on existing nonpulsed packed towers to good masstransfer advantage, provided limiting flow rates are not exceeded. Figure 15-53 is perhaps typical of the results obtainable, although no generalizations have been devised for estimating the mass-transfer rates in the absence of experiment. For additional data, see Crico [Genie Chim., 73, 57 (1955)], Feich and Anderson [Ind. Eng. Chem., 44, 404 (1952)], Karpacheva et al. [Khim. Masinostr., 1959(3), 6; 1960(2), 13; Khim. Prom., 1960, 469], Honda et al. [Kagaku Kikai, 21, 645 (1957); Kagaku Kogaku, 22, 97 (1958)], Oyama and Yam-

aguchi [Kagaku Kogaku, 22, 668 (1958)], Potnis et al. [Ind. Eng. Chem., 51, 645 (1959)], Widmer [Chem. Ing. Tech., 39, 900 (1967)]; Worall and Thwaites [Br. Chem. Eng., 10, 158 (1965)], Ziolkowski and Naumowicz [Chem. Stosow., 2, 457 (1958); 3, 475 (1959); 5, 363 (1961)]. A small perforated-plate column of conventional design was pulsed by Goldberger and Benenati [Ind. Eng. Chem., 51, 641 (1959)] with marked improvement in mass-transfer rates. Pulsed Sieve-Plate Columns The standard arrangement (see Fig. 15-52a) consists of a tower fitted with horizontal sieve plates which occupy the entire cross section of the columns. There are no down spouts as in ordinary sieve-plate columns. Typical arrangements use 0.32-cm- (f-in-) diameter perforations sufficient to provide 20 to 25 percent free space, with 5.08-cm (2-in) plate spacing, pulse amplitudes in the range 0.64 to 2.5 cm (0.25 to 1 in), and frequencies of 100 to 250 cycles/min, although the pulse characteristics will depend upon the system and flow rates under consideration. Plates are usually of metal, but Sobotik and Himmelblau [Am. Inst. Chem. Eng. J., 6, 619 (1960)] indicate that for certain services plates which are not wet by water (polyethylene) may be advantageous. Sege and Woodfield [Chem. Eng. Prog., 50, Symp. Ser. 13, 179 (1954)] provide a good description of the operational characteristics. Refer to Fig. 15-54. Since in many cases the perforations are too small to permit flow owing to interfacial tension of the liquids, the total pulsed volume must ordinarily approximate the volumetric rate of flow of the liquids [Edwards and Beyer, Am. Inst. Chem. Eng. J., 2, 148 (1956), show that slightly higher rates than Vd + Vc = pulsed volume may be obtained]. In region 1 of the figure, the column is flooded because of insufficient pulsed volume. In region 2, discrete layers of liquid appear between plates during the quiet portion of the pulse cycle. During upward pulsing, the light liquid is forced through the

FIG. 15-54 Pulsed column characteristics. [Sege and Woodfield, Chem. Eng. Prog., 50, Symp. Ser. 13, 179 (1954).]

15-46

LIQUID-LIQUID EXTRACTION OPERATIONS AND EQUIPMENT

perforations and forms drops which rise to the plate above. During downward pulsing, the heavy liquid behaves similarly. Flow is stable, but mass-transfer rates are generally poor. In region 3 there is little change in phase dispersion throughout the pulse cycle, and a fairly uniform dispersion of small droplets persists throughout. This region provides the best mass-transfer rates. Region 4 is characterized by irregular coalescence into fairly large drops, and periodic reversal of the continuous phase (local flooding). Extraction rates are generally poor. Further increase in frequency results in flooding owing to emulsification, region 5. Transition between regions is gradual and continuous, not abrupt. Excellent photographs of these phenomena are provided by Defives, Durandet, and Gladel [Rev. Inst. Fr. Pet. Ann. Combust. Liq., 11, 231 (1956)]. The literature is unusually large. In view of the fact that application of these extractors is almost entirely confined to processes related to atomic energy, only a brief listing of sources of data is presented here. Dispersed-phase holdup and flooding. Groenier, McAllister, and Ryon [U.S. AEC ORNL-3890, 1966; Chem. Eng. Sci., 22, 931 (1967)]; Babb et al. [Ind. Eng. Chem., 51, 1005 (1959); Ind. Eng. Chem. Process Des. Dev., 2, 38 (1963)]; Gel’perin et al. [Khim. Prom., 42, 607 (1966)]; Thornton and Logsdail [Trans. Inst. Chem. Eng. (London), 35, 316, 331 (1957)]. Longitudinal mixing. Babb et al. [Ind. Eng. Chem., 51, 1011 (1959); Ind. Eng. Chem. Process Des. Dev., 3, 210 (1964)]; Burger and Swift (U.S. AEC HW29010, 1953); Miyauchi et al. [Am. Inst. Chem. Eng. J., 11, 395 (1965); Kagaku Kogaku, 30, 895 (1966)]; Otake and Komasawa [ibid., 32(6), 19 (1968)]. Mass-transfer rates. Correlations are offered by Smoot, Mar, and Babb [Ind. Eng. Chem., 51, 1005 (1959); Ind. Eng. Chem. Fundam., 1, 93 (1962)] and Zwolkowski and Kubica [Chem. Stosow., Ser. B2, 392 (1965)].

FIG. 15-55

depending on size) causes radial counterflow of the liquids, which are then led out through the shaft. Materials of construction include steel, stainless steel, Hastelloy, and other corrosion-resistant alloys. The machines are particularly characterized by extremely low holdup of liquid per stage, and this led to their extensive use in the extraction of antibiotics, such as penicillin and the like, for which multistage extraction and phase separation must be done rapidly to avoid chemical destruction of the product under conditions of extraction. They have been used extensively in all phases of pharmaceutical manufacture and are increasingly being used in other fields: petroleum processing, both solvent refining and acid treating, dephenolization of wastewaters, extraction of uranium from ore leach liquors, as well as for clarification and phase-separation work. See Kaiser, Sewage Ind. Wastes, 27, 311 (1955); Podbielniak, Gavin, and Kaiser, J. Am. Oil Chem. Soc., 36, 238 (1959); Doyle and Rauch, Pet. Eng., 27(5), C-49 (1955); Anderson and Lau, Chem. Eng. Prog., 51, 507 (1955); Todd and Podbielniak, ibid., 61(5), 69 (1965); and Todd, ibid., 62(8), 119 (1966). The last contains data on interstage back mixing. Table 15-11 lists some of the characteristics of the machines. With a laboratory model [0.55 m (18 in) in diameter, 5.08 cm (2 in) wide, 18 concentric cylinders slotted at 180° intervals], Barson and Beyer [Chem. Eng. Prog., 49, 243 (1953)] obtained from two to eight ideal stages with isoamyl alcohol–boric acid–water at 5000 r/min. The number of stages increased with ratio of light-to-heavy-liquid flow but with varying position of the interface and consequently varying fraction of the machine devoted to light-liquid-dispersed. At constant flow rate, the number of stages was essentially independent of rotational speed. Jacobson and Beyer [Am. Inst. Chem. Eng. J., 2, 283 (1956)] obtained about the same results. Alexandre and Gentilini [Rev. Inst. Fr. Pet. Ann. Combust. Liq., 11, 389 (1956)] similarly obtained five

Controlled Cycling The compartmented character of sieve-plate columns described above lends itself particularly well to this technique, which is, however, not confined to these devices [Cannon, Oil Gas J., 51, 268 (1952); 55, 68 (1956); Szabo et al., Chem. Eng. Prog., 60(1), 66 (1964); Belter and Speaker, Ind. Eng. Chem. Process Des. Dev., 6, 36 (1967); Horn, ibid., 6, 30 (1967); Robinson and Engel, Ind. Eng. Chem., 59(3), 22 (1967); and Lövland, Ind. Eng. Chem. Process Des. Dev., 7, 65 (1968)]. A cycle is completed by the following sequence of events: (1) a light-phase flow period, during which the heavy phase does not flow; (2) a coalescing period, during which neither phase flows; (3) a heavy-phase flow period, during which the light phase does not flow; and (4) a repeat of the coalescing period. The net result can be an increased flow capacity (in the case of sieve-plate pulsed columns) and stage efficiency, such that the effect of 2N stages may be obtained with a column of N stages, provided the total holdup of each phase is displaced during each cycle. Centrifugal Extractors The force of gravity for counterflow of liquids of different density may be replaced and in effect increased (many thousandfold if desired) by centrifugal machines. These then become especially useful for handling liquids of low density difference and those with tendencies to form emulsions. Podbielniak Extractor (Podbielniak, U.S. Patent 2,044,996, 1935, and other patents) This is the most important of the group. Refer to Fig. 15-55. Rotation is about a horizontal shaft. The body of the extractor is a cylindrical drum containing concentric perforate cylinders. The liquids are introduced through the rotating shaft with the help of special mechanical seals; the light liquid is led internally to the drum periphery and the heavy liquid to the axis of the drum. Rapid rotation (up to several thousand revolutions per minute, TABLE 15-11

Podbielniak centrifugal extractor. (Courtesy of Baker Perkins Inc.)

Podbielniak Centrifugal Extractors* Over-all dimensions, in.

Model number

Width

A-1 B-10 D-18 D-36 E-48

16 55.5 76 94 113

Horsepower

Height

Length (incl. drive)

Total wt., lb.

Connected

12 33 45 45 59

30 67.5 85 85 107

150 2,700 8,600 10,250 21,500

3.0 7.5 15 25 40

Flow capacity, gal./min.

Continuous

Multistage extraction

Neutralization, acid treating, extraction of fermentation broths

2.5 6.7 10 15 22

1.0 30 150 300 500

0.5 30 75 150 300

*Courtesy Baker Perkins Inc. To convert inches to centimeters, multiply by 2.54; to convert pounds to kilograms, multiply by 0.454; to convert horsepower to kilowatts, multiply by 0.746; and to convert gallons per minute to cubic meters per hour, multiply by 0.227.

LIQUID-LIQUID EXTRACTION EQUIPMENT ideal stages with benzene–acetic acid–water, and 3.4 to 12.5 ideal stages with methyl isobutyl ketone–acetic acid–water. Anderson and Lau [Chem. Eng. Prog., 51, 507 (1955) describe a model handling 10 to 15 percent suspended solids in the liquids, and report a fraction to two ideal stages when extracting penicillin and chloromycetin, 7.04 to 8.71 m3/h (1860 to 2300 gal/ h) total flow rate. Quadronics (Liquid Dynamics) Extractor (Doyle et al., U.S. Patent 3,114,707, 1963, and others; paper at AIChE meeting, St. Louis, February 1968) This is a horizontally rotated device, a variant of the Podbielniak extractor, in which either fixed or adjustable orifices may be inserted as a package radially. These permit control of the mixing intensity as the liquids pass radially through the extractor. Flow capacities, depending on machine size, range from 0.34 to 340 m3/ h (1.5 to 1500 gal/min). Luwesta (Centriwesta) Extractor This is a development from

15-47

Coutor (U.S. Patent 2,036,924, 1936). See also Eisenlohr [Ind. Chem., 27, 271 (1951); Chem. Ing. Tech., 23, 12 (1951); Pharm. Ind., 17, 207 (1955); Trans. Indian Inst. Chem. Eng., 3, 7 (1949–1950)] and Husain et al. [Chim. Ind. (Milan), 82, 435 (1959)]. This centrifuge revolves about a vertical axis and contains three actual stages. It operates at 3800 r/min and handles approximately 4.92 m3/ h (1300 gal/ h) total liquid flow at 12-kW power requirement. Provision is made in the machine for the accumulation of solids separated from the liquids, for periodic removal. It is used, more extensively in Europe than in the United States, for the extraction of acetic acid, pharmaceuticals, and similar products. De Laval Extractor (Palmqvist and Beskow, U.S. Patent 3,108,953, 1959) This machine contains a number of perforated cylinders revolving about a vertical shaft. The liquids follow a spiral path about 25 m (82 ft) long, in countercurrent fashion radially, and mix when passing through the perforations. There are no published performance data.

Section 16

Adsorption and Ion Exchange

M. Douglas LeVan, Ph.D., Centennial Professor, Department of Chemical Engineering, Vanderbilt University; Member, American Institute of Chemical Engineers, American Chemical Society, International Adsorption Society. (Section Editor) Giorgio Carta, Ph.D., Professor, Department of Chemical Engineering, University of Virginia; Member, American Institute of Chemical Engineers, American Chemical Society, International Adsorption Society. Carmen M. Yon, M.S., Development Associate, UOP, Des Plaines, IL; Member, American Institute of Chemical Engineers. (Process Cycles, Equipment)

DESIGN CONCEPTS Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design Strategy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Characterization of Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adsorbent/Ion Exchanger Selection . . . . . . . . . . . . . . . . . . . . . . . . . . Fixed-Bed Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Practical Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16-4 16-5 16-5 16-5 16-6 16-7 16-7

ADSORBENTS AND ION EXCHANGERS Classifications and Characterizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . Adsorbents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ion Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16-8 16-8 16-8 16-9

SORPTION EQUILIBRIUM General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Surface Excess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classification of Isotherms by Shape . . . . . . . . . . . . . . . . . . . . . . . . . . Categorization of Equilibrium Models . . . . . . . . . . . . . . . . . . . . . . . . Heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Isosteric Heat of Adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dimensionless Concentration Variables . . . . . . . . . . . . . . . . . . . . . . . Single Component or Exchange. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flat-Surface Isotherm Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pore-Filling Isotherm Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ion Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Donnan Uptake. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Separation Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16-11 16-11 16-11 16-11 16-12 16-12 16-12 16-12 16-12 16-12 16-13 16-13 16-13 16-14 16-14

Multiple Components or Exchanges . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adsorbed-Solution Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Langmuir-Type Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Freundlich-Type Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equations of State. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ion Exchange—Stoichiometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass Action. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constant Separation-Factor Treatment . . . . . . . . . . . . . . . . . . . . . . . .

16-15 16-15 16-16 16-16 16-16 16-16 16-16 16-16

CONSERVATION EQUATIONS Material Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Energy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16-17 16-17

RATE AND DISPERSION FACTORS Transport and Dispersion Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . Intraparticle Transport Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . Extraparticle Transport and Dispersion Mechanisms . . . . . . . . . . . . . Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intraparticle Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pore Diffusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solid Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combined Pore and Solid Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . External Mass Transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Axial Dispersion in Packed Beds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rate Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Component Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear Driving Force Approximation . . . . . . . . . . . . . . . . . . . . . . . . . Combined Intraparticle Resistances . . . . . . . . . . . . . . . . . . . . . . . . . . Overall Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Axial Dispersion Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rapid Adsorption-Desorption Cycles . . . . . . . . . . . . . . . . . . . . . . . . . Determination of Controlling Rate Factor . . . . . . . . . . . . . . . . . . . . .

16-18 16-18 16-18 16-18 16-18 16-19 16-19 16-20 16-20 16-20 16-21 16-21 16-22 16-22 16-23 16-24 16-24 16-24

16-1

16-2

ADSORPTION AND ION EXCHANGE

BATCH ADSORPTION External Mass-Transfer Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solid Diffusion Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pore Diffusion Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combined Resistances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parallel Pore and Solid Diffusion Control . . . . . . . . . . . . . . . . . . . . . . External Mass Transfer and Intraparticle Diffusion Control . . . . . . . Bidispersed Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16-26 16-26 16-28 16-28 16-29 16-29 16-29

FIXED-BED TRANSITIONS Dimensionless System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Local Equilibrium Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single Transition System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple Transition System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constant Pattern Behavior for Favorable Isotherms . . . . . . . . . . . . . . . Asymptotic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Breakthrough Behavior for Axial Dispersion. . . . . . . . . . . . . . . . . . . . Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Square Root Spreading for Linear Isotherms . . . . . . . . . . . . . . . . . . . . . Complete Solution for Reaction Kinetics . . . . . . . . . . . . . . . . . . . . . . . . Numerical Methods and Characterization of Wave Shape . . . . . . . . . . .

16-30 16-30 16-30 16-31 16-31 16-32 16-34 16-36 16-36 16-36 16-37 16-37

CHROMATOGRAPHY Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modes of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elution Chromatography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frontal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Displacement Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Characterization of Experimental Chromatograms . . . . . . . . . . . . . . . . Method of Moments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Approximate Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tailing Peaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Prediction of Chromatographic Behavior . . . . . . . . . . . . . . . . . . . . . . . . Isocratic Elution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concentration Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16-38 16-38 16-38 16-39 16-39 16-40 16-40 16-40 16-41 16-41 16-42 16-42 16-43

Linear Gradient Elution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Displacement Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design for Trace Solute Separations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16-44 16-44 16-47

PROCESS CYCLES General Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature Swing Adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Cycle Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure-Swing Adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Cycle Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Purge/Concentration Swing Adsorption . . . . . . . . . . . . . . . . . . . . . . . . . Inert Purge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Displacement Purge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chromatography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ion Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parametric Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Adsorption Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Steam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Energy Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Energy Conservation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . Process Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16-48 16-48 16-49 16-49 16-49 16-50 16-50 16-51 16-51 16-51 16-52 16-52 16-53 16-55 16-55 16-55 16-55 16-55 16-55 16-56

EQUIPMENT Adsorption. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adsorber Vessel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Regeneration Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cycle Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous Countercurrent Systems. . . . . . . . . . . . . . . . . . . . . . . . . . Continuous Cross-Flow Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulated Continuous Countercurrent Systems. . . . . . . . . . . . . . . . . Ion Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16-56 16-56 16-56 16-58 16-59 16-60 16-61 16-62 16-64

ADSORPTION AND ION EXCHANGE

16-3

Nomenclature and Units a av A As b c cp cs C°pf Cs dp D De DL Dp Ds D0 D
F Fv h htu HETP HTU J k ka kc kf kn k′ K Kc K′ L m Mr Ms n ns N

Np NPe p P Pe q st Qi r, R rc rm rp rpore rs ℜ Re s Sc Sh

specific external surface area per unit bed volume, m2/m3 surface area per unit particle volume, m2/m3 particle surface area of solid, m2/kg chromatography peak asymmetry factor correction factor for resistances in series fluid-phase concentration, mol/m3 fluid pore fluid-phase concentration, mol/m3 fluid-phase concentration at particle surface, mol/m3 ideal gas heat capacity, J/(mol K) heat capacity of sorbent solid, J/(kg K) particle diameter, m fluid-phase diffusion coefficient, m2/s equivalent diffusion coefficient, m2/s axial dispersion coefficient, m2/s pore diffusion coefficient, m2/s adsorbed-phase (solid, surface, particle, or micropore) diffusion coefficient, m2/s diffusion coefficient corrected for thermodynamic driving force, m2/s ionic self-diffusion coefficient, m2/s fractional approach to equilibrium volumetric flow rate, m3/s enthalpy, J/mol; reduced height equivalent to theoretical plate reduced height equivalent to a transfer unit height equivalent to theoretical plate, m height equivalent to a transfer unit, m mass-transfer flux relative to molar average velocity, mol/(m2·s); J function rate coefficient, s−1 forward rate constant for reaction kinetics, m3/(mol s) rate coefficient based on fluid-phase concentration driving force, m3/(kg·s) external mass-transfer coefficient, m/s rate coefficient based on adsorbed-phase concentration driving force, s−1 retention factor isotherm parameter molar selectivity coefficient rational selectivity coefficient bed length, m isotherm exponent molecular weight, kg/kmol mass of adsorbent, kg adsorbed-phase concentration, mol/kg adsorbent ion-exchange capacity, g-equiv/kg number of transfer or reaction units; kf aL /(εvref) for external mass transfer; 15(1 − ε)εpDpL/(εv refR2p) for pore diffusion; 15ΛDsL /(εv refR2p) for solid diffusion; knΛL /(εvref) for linear drivingforce approximation; kacrefΛL /[(1 − R)εvref] for reaction kinetics number of theoretical plates vrefL /DL , bed Peclet number (number of dispersion units) partial pressure, Pa pressure, Pa particle-based Peclet number, dpv/DL isosteric heat of adsorption, J/mol amount of component i injected with feed, mol separation factor; particle radial coordinate, m column internal radius, m hydrodynamic radius of molecule, m particle radius, m pore radius, m radius of subparticles, m gas constant, Pa⋅m3/(mol K) Reynolds number based on particle diameter, dpεv/ν UNILAN isotherm parameter Schmidt number, ν/D Sherwood number, kf dp /D

t tc tf tr T uf us, usol v Vf W x y z

time, s cycle time, s feed time, s chromatographic retention time, s absolute temperature, K fluid-phase internal energy, J/mol stationary-phase and sorbent solid internal energy, J/kg interstitial velocity, m/s extraparticle fluid volume, m3 volume adsorbed as liquid, m3; baseline width of chromatographic peak, s adsorbed-phase mole fraction; particle coordinate, m fluid-phase mole fraction bed axial coordinate, m; ionic valence Greek letters

α β

εp εb γ Γ κ Λ Λ∞ µ µ0 µ1 ν Ω ϕ π ψ Ψ ρ ρb ρp ρs σ2 τ τ1 τp ξ ζ

separation factor scaling factor in Polanyi-based models; slope in gradient elution chromatography peak width at half height, s void fraction of packing (extraparticle); adsorption potential in Polanyi model, J/mol particle porosity (intraparticle void fraction) total bed voidage (inside and outside particles) activity coefficient surface excess, mol/m2 Boltzmann constant partition ratio ultimate fraction of solute adsorbed in batch fluid viscosity, kg/(m s) zero moment, mol s/m3 first moment, s kinematic viscosity, m2/s cycle-time dependent LDF coefficient volume fraction or mobile-phase modulator concentration, mol/m3 spreading pressure, N/m LDF correction factor mechanism parameter for combined resistances subparticle radial coordinate, m bulk density of packing, kg/m3 particle density, kg/m3 skeletal particle density, kg/m3 second central moment, s2 dimensionless time throughput parameter tortuosity factor particle dimensionless radial coordinate dimensionless bed axial coordinate

a f i, j tot

adsorbed phase fluid phase component index total

− ^ * e ref s SM TM 0 0′ ∞

an averaged concentration a combination of averaged concentrations dimensionless concentration variable equilibrium reference saturation service mark trademark initial fluid concentration in batch initial adsorbed-phase concentration in batch final state approached in batch

∆ ε

Subscripts

Superscripts

GENERAL REFERENCES 1. Adamson, Physical Chemistry of Surfaces, Wiley, New York, 1990. 2. Barrer, Zeolites and Clay Minerals as Adsorbents and Molecular Sieves, Academic Press, New York, 1978. 3. Breck, D. W., Zeolite Molecular Sieves, Wiley, New York, 1974. 4. Cheremisinoff and Ellerbusch, Carbon Adsorption Handbook, Ann Arbor Science, Ann Arbor, 1978. 5. Dorfner (ed.), Ion Exchangers, W. deGruyter, Berlin, 1991. 6. Dyer, An Introduction to Zeolite Molecular Sieves, Wiley, New York, 1988. 7. EPA, Process Design Manual for Carbon Adsorption, U.S. Envir. Protect. Agency., Cincinnati, 1973. 8. Gembicki, Oroskar, and Johnson, “Adsorption, Liquid Separation” in KirkOthmer Encyclopedia of Chemical Technology, 4th ed., Wiley, 1991. 9. Guiochon, Golsham-Shirazi, and Katti, Fundamentals of Preparative and Nonlinear Chromatography, Academic Press, Boston, Massachusetts, 1994. 10. Gregg and Sing, Adsorption, Surface Area and Porosity, Academic Press, New York, 1982. 11. Helfferich, Ion Exchange, McGraw-Hill, New York, 1962; reprinted by University Microfilms International, Ann Arbor, Michigan. 12. Helfferich and Klein, Multicomponent Chromatography, Marcel Dekker, New York, 1970. 13. Jaroniec and Madey, Physical Adsorption on Heterogeneous Solids, Elsevier, New York, 1988. 14. Karge and Ruthven, Diffusion in Zeolites and Other Microporous Solids, Wiley, New York, 1992. 15. Keller, Anderson, and Yon, “Adsorption” in Rousseau (ed.), Handbook of Separation Process Technology, Wiley-Interscience, New York, 1987. 16. Rhee, Aris, and Amundson, First-Order Partial Differential Equations:

17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.

Volume 1. Theory and Application of Single Equations; Volume 2. Theory and Application of Hyperbolic Systems of Quasi-Linear Equations, Prentice Hall, Englewood Cliffs, New Jersey, 1986, 1989. Rodrigues, LeVan, and Tondeur (eds.), Adsorption: Science and Technology, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1989. Rudzinski and Everett, Adsorption of Gases on Heterogeneous Surfaces, Academic Press, San Diego, 1992. Ruthven, Principles of Adsorption and Adsorption Processes, Wiley, New York, 1984. Ruthven, Farooq, and Knaebel, Pressure Swing Adsorption, VCH Publishers, New York, 1994. Sherman and Yon, “Adsorption, Gas Separation” in Kirk-Othmer Encyclopedia of Chemical Technology, 4th ed., Wiley, 1991. Streat and Cloete, “Ion Exchange,” in Rousseau (ed.), Handbook of Separation Process Technology, Wiley, New York, 1987. Suzuki, Adsorption Engineering, Elsevier, Amsterdam, 1990. Tien, Adsorption Calculations and Modeling, Butterworth-Heinemann, Newton, Massachusetts, 1994. Valenzuela and Myers, Adsorption Equilibrium Data Handbook, Prentice Hall, Englewood Cliffs, New Jersey, 1989. Vermeulen, LeVan, Hiester, and Klein, “Adsorption and Ion Exchange” in Perry, R. H. and Green, D. W. (eds.), Perry’s Chemical Engineers’ Handbook (6th ed.), McGraw-Hill, New York, 1984. Wankat, Large-Scale Adsorption and Chromatography, CRC Press, Boca Raton, Florida, 1986. Yang, Gas Separation by Adsorption Processes, Butterworth, Stoneham, Massachusetts, 1987. Young and Crowell, Physical Adsorption of Gases, Butterworths, London, 1962.

DESIGN CONCEPTS INTRODUCTION Adsorption and ion exchange share so many common features in regard to application in batch and fixed-bed processes that they can be grouped together as sorption for a unified treatment. These processes involve the transfer and resulting equilibrium distribution of one or more solutes between a fluid phase and particles. The partitioning of a single solute between fluid and sorbed phases or the selectivity of a sorbent towards multiple solutes makes it possible to separate solutes from a bulk fluid phase or from one another. This section treats batch and fixed-bed operations and reviews process cycles and equipment. As the processes indicate, fixed-bed operation with the sorbent in granule, bead, or pellet form is the predominant way of conducting sorption separations and purifications. Although the fixed-bed mode is highly useful, its analysis is complex. Therefore, fixed beds including chromatographic separations are given primary attention here with respect to both interpretation and prediction. Adsorption involves, in general, the accumulation (or depletion) of solute molecules at an interface (including gas-liquid interfaces, as in foam fractionation, and liquid-liquid interfaces, as in detergency). Here we consider only gas-solid and liquid-solid interfaces, with solute distributed selectively between the fluid and solid phases. The accumulation per unit surface area is small; thus, highly porous solids with very large internal area per unit volume are preferred. Adsorbent surfaces are often physically and/or chemically heterogeneous, and bonding energies may vary widely from one site to another. We seek to promote physical adsorption or physisorption, which involves van der Waals forces (as in vapor condensation), and retard chemical adsorption or chemisorption, which involves chemical bonding (and often dissociation, as in catalysis). The former is well suited for a regenera16-4

ble process, while the latter generally destroys the capacity of the adsorbent. Adsorbents are natural or synthetic materials of amorphous or microcrystalline structure. Those used on a large scale, in order of sales volume, are activated carbon, molecular sieves, silica gel, and activated alumina [Keller et al., gen. refs.]. Ion exchange usually occurs throughout a polymeric solid, the solid being of gel-type, which dissolves some fluid-phase solvent, or truly porous. In ion exchange, ions of positive charge in some cases (cations) and negative charge in others (anions) from the fluid (usually an aqueous solution) replace dissimilar ions of the same charge initially in the solid. The ion exchanger contains permanently bound functional groups of opposite charge-type (or, in special cases, notably weak-base exchangers act as if they do). Cation-exchange resins generally contain bound sulfonic acid groups; less commonly, these groups are carboxylic, phosphonic, phosphinic, and so on. Anionic resins involve quaternary ammonium groups (strongly basic) or other amino groups (weakly basic). Most ion exchangers in large-scale use are based on synthetic resins—either preformed and then chemically reacted, as for polystyrene, or formed from active monomers (olefinic acids, amines, or phenols). Natural zeolites were the first ion exchangers, and both natural and synthetic zeolites are in use today. Ion exchange may be thought of as a reversible reaction involving chemically equivalent quantities. A common example for cation exchange is the familiar water-softening reaction Ca++ + 2NaR A CaR2 + 2Na+ where R represents a stationary univalent anionic site in the polyelectrolyte network of the exchanger phase.

DESIGN CONCEPTS TABLE 16-1

16-5

Classification of Sorptive Separations

Type of interaction Adsorption

Basis for separation Equilibrium

Rate Molecular sieving

Examples Numerous purification and recovery processes for gases and liquids Activated carbon-based applications Desiccation using silica gels, aluminas, and zeolites Oxygen from air by PSA using 5A zeolite Nitrogen from air by PSA using carbon molecular sieve Separation on n- and iso-parafins using 5A zeolite Separation of xylenes using zeolite

Ion exchange (electrostatic)

Equilibrium

Deionization Water softening Rare earth separations Recovery and separation of pharmaceuticals (e.g., amino acids, proteins)

Ligand exchange

Equilibrium

Chromatographic separation of glucose-fructose mixtures with Ca-form resins Removal of heavy metals with chelating resins Affinity chromatography

Solubility

Equilibrium

Partition chromatography

None (purely steric)

Equilibrium partitioning in pores

Size exclusion or gel permeation chromatography

Table 16-1 classifies sorption operations by the type of interaction and the basis for the separation. In addition to the normal sorption operations of adsorption and ion exchange, some other similar separations are included. Applications are discussed in this section in “Process Cycles.” Example 1: Surface Area and Pore Volume of Adsorbent A simple example will show the extent of internal area in a typical granular adsorbent. A fixed bed is packed with particles of a porous adsorbent material. The bulk density of the packing is 500 kg/m3, and the interparticle void fraction is 0.40. The intraparticle porosity is 0.50, with two-thirds of this in cylindrical pores of diameter 1.4 nm and the rest in much larger pores. Find the surface area of the adsorbent and, if solute has formed a complete monomolecular layer 0.3 nm thick inside the pores, determine the percent of the particle volume and the percent of the total bed volume filled with adsorbate. From surface area to volume ratio considerations, the internal area is practically all in the small pores. One gram of the adsorbent occupies 2 cm3 as packed and has 0.4 cm3 in small pores, which gives a surface area of 1150 m2/g (or about 1 mi2 per 5 lb or 6.3 mi2/ft3 of packing). Based on the area of the annular region filled with adsorbate, the solute occupies 22.5 percent of the internal pore volume and 13.5 percent of the total packed-bed volume. DESIGN STRATEGY The design of sorption systems is based on a few underlying principles. First, knowledge of sorption equilibrium is required. This equilibrium, between solutes in the fluid phase and the solute-enriched phase of the solid, supplants what in most chemical engineering separations is a fluid-fluid equilibrium. The selection of the sorbent material with an understanding of its equilibrium properties (i.e., capacity and selectivity as a function of temperature and component concentrations) is of primary importance. Second, because sorption operations take place in batch, in fixed beds, or in simulated moving beds, the processes have dynamical character. Such operations generally do not run at steady state, although such operation may be approached in a simulated moving bed. Fixed-bed processes often approach a periodic condition called a periodic state or cyclic steady state, with several different feed steps constituting a cycle. Thus, some knowledge of how transitions travel through a bed is required. This introduces both time and space into the analysis, in contrast to many chemical engineering operations that can be analyzed at steady state with only a spatial dependence. For good design, it is crucial to understand fixed-bed performance in relation to adsorption equilibrium and rate behavior. Finally, many practical aspects must be included in design so that a process starts up and continues to perform well, and that it is not so overdesigned that it is wasteful. While these aspects are processspecific, they include an understanding of dispersive phenomena at the bed scale and, for regenerative processes, knowledge of aging characteristics of the sorbent material, with consequent changes in sorption equilibrium.

Characterization of Equilibria Phase equilibrium between fluid and sorbed phases for one or many components in adsorption or two or more species in ion exchange is usually the single most important factor affecting process performance. In most processes, it is much more important than mass and heat transfer rates; a doubling of the stoichiometric capacity of a sorbent or a significant change in the shape of an isotherm would almost always have a greater impact on process performance than a doubling of transfer rates. A difference between adsorption and ion exchange with completely ionized resins is indicated in the variance of the systems. In adsorption, part of the solid surface or pore volume is vacant. This diminishes as the fluid-phase concentration of solute increases. In contrast, for ion exchange the sorbent has a fixed total capacity and merely exchanges solutes while conserving charge. Variance is defined as the number of independent concentration variables in a sorption system at equilibrium—that is, variables that one can change separately and thereby control the values of all others. Thus, it also equals the difference between the total number of concentration variables and the number of independent relations connecting them. Numerous cases arise in which ion exchange is accompanied by chemical reaction (neutralization or precipitation, in particular), or adsorption is accompanied by evolution of sensible heat. The concept of variance helps greatly to assure correct interpretations and predictions. The working capacity of a sorbent depends on fluid concentrations and temperatures. Graphical depiction of sorption equilibrium for single component adsorption or binary ion exchange (monovariance) is usually in the form of isotherms [ni = ni(ci) or ni (pi) at constant T] or isosteres [pi = pi (T ) at constant ni]. Representative forms are shown in Fig. 16-1. An important dimensionless group dependent on adsorption equilibrium is the partition ratio (see Eq. 16-125), which is a measure of the relative affinities of the sorbed and fluid phases for solute. Historically, isotherms have been classified as favorable (concave downward) or unfavorable (concave upward). These terms refer to the spreading tendencies of transitions in fixed beds. A favorable isotherm gives a compact transition, whereas an unfavorable isotherm leads to a broad one. Example 2: Calculation of Variance In mixed-bed deionization of a solution of a single salt, there are 8 concentration variables: 2 each for cation, anion, hydrogen, and hydroxide. There are 6 connecting relations: 2 for ion exchange and 1 for neutralization equilibrium, and 2 ion-exchanger and 1 solution electroneutrality relations. The variance is therefore 8 − 6 = 2. Adsorbent/Ion Exchanger Selection Guidelines for sorbent selection are different for regenerative and nonregenerative systems. For a nonregenerative system, one generally wants a high capacity and a strongly favorable isotherm for a purification and additionally high selectivity for a separation. For a regenerative system, high overall capacity and selectivity are again desired, but needs for cost-effective

16-6

ADSORPTION AND ION EXCHANGE

Cold La rg i

all

ln pi

ni

en

Sm

Hot

ni

0 0

ci or pi

I/T

Isotherms (left) and isosteres (right). Isosteres plotted using these coordinates are nearly straight parallel lines, with deviations caused by the dependence of the isosteric heat of adsorption on temperature and loading.

FIG. 16-1

regeneration leading to a reasonable working capacity influence what is sought after in terms of isotherm shape. For separations by pressure swing adsorption (or vacuum pressure swing adsorption), generally one wants a linear to slightly favorable isotherm (although purifications can operate economically with more strongly favorable isotherms). Temperature-swing adsorption usually operates with moderately to strongly favorable isotherms, in part because one is typically dealing with heavier solutes and these are adsorbed fairly strongly (e.g., organic solvents on activated carbon and water vapor on zeolites). Exceptions exist, however; for example, water is adsorbed on silica gel and activated alumina only moderately favorably, with some isotherms showing unfavorable sections. Equilibria for ion exchange separations generally vary from moderately favorable to moderately unfavorable; depending on feed concentrations, the alternates often exist for the different steps of a regenerative cycle. Other factors in sorbent selection are mechanical and chemical stability, mass transfer characteristics, and cost. Fixed-Bed Behavior The number of transitions occurring in a fixed bed of initially uniform composition before it becomes saturated by a constant composition feed stream is generally equal to the variance of the system. This introductory discussion will be limited to single transition systems. Methods for analysis of fixed-bed transitions are shown in Table 16-2. Local equilibrium theory is based solely of stoichiometric concerns and system nonlinearities. A transition becomes a “simple wave” (a gradual transition), a “shock” (an abrupt transition), or a combination of the two. In other methods, mass-transfer resistances are incorporated. The asymptotic behavior of transitions under the influence of massTABLE 16-2

transfer resistances in long, “deep” beds is important. The three basic asymptotic forms are shown in Fig. 16-2. With an unfavorable isotherm, the breadth of the transition becomes proportional to the depth of bed it has passed through. For the linear isotherm, the breadth becomes proportional to the square root of the depth. For the favorable isotherm, the transition approaches a constant breadth called a constant pattern. Design of nonregenerative sorption systems and many regenerative ones often relies on the concept of the mass-transfer zone or MTZ, which closely resembles the constant pattern [Collins, Chem. Eng. Prog. Symp. Ser. No. 74, 63, 31 (1974); Keller et al., gen. refs.]. The length of this zone (depicted in Fig. 16-3) together with stoichiometry can be used to predict accurately how long a bed can be utilized prior to breakthrough. Upstream of the mass-transfer zone, the adsorbent is in equilibrium with the feed. Downstream, the adsorbent is in its initial state. Within the mass-transfer zone, the fluid-phase concentration drops from the feed value to the initial, presaturation state. Equilibrium with the feed is not attained in this region. As a result, because an adsorption bed must typically be removed from service shortly after breakthrough begins, the full capacity of the bed is not utilized. Obviously, the broader that the mass-transfer zone is, the greater will be the extent of unused capacity. Also shown in the figure is the length of the equivalent equilibrium section (LES) and the length of equivalent unused bed (LUB). The length of the MTZ is divided between these two. Adsorption with strongly favorable isotherms and ion exchange between strong electrolytes can usually be carried out until most of the stoichiometric capacity of the sorbent has been utilized, corresponding to a thin MTZ. Consequently, the total capacity of the bed is

Methods of Analysis of Fixed-Bed Transitions

Method

Purpose

Approximations

Local equilibrium theory

Shows wave character—simple waves and shocks Usually indicates best possible performance Better understanding

Mass and heat transfer very rapid Dispersion usually neglected If nonisothermal, then adiabatic

Mass-transfer zone

Design based on stoichiometry and experience

Isothermal MTZ length largely empirical Regeneration often empirical

Constant pattern and related analyses

Gives asymptotic transition shapes and upper bound on MTZ

Deep bed with fully developed transition

Full rate modeling

Accurate description of transitions Appropriate for shallow beds, with incomplete wave development General numerical solutions by finite difference or collocation methods

Various to few

DESIGN CONCEPTS ciref

cifeed

Equilibrium section

16-7

MTZ

ci

ci

0 0 0

ciref

ci

cifeed

L

z LES

LUB

ci

0 ciref

Stoichiometric front 0 0

ci

0 0

z

L

cifeed z

Limiting fixed-bed behavior: simple wave for unfavorable isotherm (top), square-root spreading for linear isotherm (middle), and constant pattern for favorable isotherm (bottom). [From LeVan in Rodrigues et al. (eds.), Adsorption: Science and Technology, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1989; reprinted with permission.]

ci

FIG. 16-2

0 0 practically constant regardless of the composition of the solution being treated. The effluent concentration history is the breakthrough curve, also shown in Fig. 16-3. The effluent concentration stays at or near zero or a low residual concentration until the transition reaches the column outlet. The effluent concentration then rises until it becomes unacceptable, this time being called the breakthrough time. The feed step must stop and, for a regenerative system, the regeneration step begins. Two dimensionless variables play key roles in the analysis of single transition systems (and some multiple transition systems). These are the throughput parameter [see Eq. (16-129)] and the number of transfer units (see Table 16-13). The former is time made dimensionless so that it is equal to unity at the stoichiometric center of a breakthrough curve. The latter is, as in packed tower calculations, a measure of mass-transfer resistance. Cycles Design methods for cycles rely on mathematical modeling (or empiricism) and often extensive pilot plant experiments. Many cycles can be easily analyzed using the methods described above applied to the collection of steps. In some cycles, however, especially those operated with short cycle times or in shallow beds, transitions may not be very fully developed, even at a periodic state, and the complexity may be compounded by multiple sorbates. A wide variety of complex process cycles have been developed. Systems with many beds incorporating multiple sorbents, possibly in layered beds, are in use. Mathematical models constructed to analyze such cycles can be complex. With a large number of variables and nonlinear equilibria involved, it is usually not beneficial to make all

t

Bed profiles (top and middle) and breakthrough curve (bottom). The bed profiles show the mass-transfer zone (MTZ) and equilibrium section at breakthrough. The stoichiometric front divides the MTZ into two parts with contributions to the length of equivalent equilibrium section (LES) and the length of equivalent unused bed (LUB). FIG. 16-3

variables in such models dimensionless; doing so does not help appreciably in making comparisons with other largely dissimilar systems. If dimensionless variables are used, these usually begin with a dimensionless bed length and a dimensionless time, which is often different from the throughput parameter. Practical Aspects There are a number of process-specific concerns that are accounted for in good design. In regenerable systems, sorbents age, losing capacity because of fouling by heavy contaminants, loss of surface area or crystallinity, oxidation, and the like. Masstransfer resistances may increase over time. Because of particle shape, size distribution, or column packing method, dispersion may be more pronounced than would normally be expected. The humidity of an entering stream will usually impact a solvent recovery application. Safety, including the possibility of a fire, may be a concern. For gasphase adsorption, scale-up from an isothermal laboratory column to a nonisothermal pilot plant column to a largely adiabatic process column requires careful judgment. If the MTZ concept is utilized, the length of the MTZ cannot be reliably determined solely from knowledge on other systems. Experience plays the key role in accounting for these and other such factors.

16-8

ADSORPTION AND ION EXCHANGE

ADSORBENTS AND ION EXCHANGERS CLASSIFICATIONS AND CHARACTERIZATIONS Adsorbents Table 16-3 classifies common adsorbents by structure type and water adsorption characteristics. Structured adsorbents take advantage of their crystalline structure (zeolites and silicalite) and/or their molecular sieving properties. The hydrophobic (nonpolar surface) or hydrophilic (polar surface) character may vary depending on the competing adsorbate. A large number of zeolites have been identified, and these include both synthetic and naturally occurring (e.g., mordenite and chabazite) varieties. TABLE 16-3

Classification of Common Adsorbents Amorphous

Structured

Hydrophobic

Activated carbon Polymers

Carbon molecular sieves Silicalite

Hydrophilic

Silica gel Activated alumina

Common zeolites: 3A (KA), 4A (NaA), 5A (CaA), 13X (NaX), Mordenite, Chabazite, etc.

The classifications in Table 16-3 are intended only as a rough guide. For example, a carbon molecular sieve is truly amorphous but has been manufactured to have certain structural, rate-selective properties. Similarly, the extent of hydrophobicity of an activated carbon will depend on its ash content and its level of surface oxidation. Zeolites are crystalline aluminosilicates. Zeolitic adsorbents have had their water of hydration removed by calcination to create a structure with well-defined openings into crystalline cages. The molecular sieving properties of zeolites are based on the size of these openings. Two crystal types are common: type A (with openings formed by 4 sodalite cages) and type X or Y (with openings formed by 6 sodalite cages). Cations balancing charge and their locations determine the size of opening into a crystal unit cell. Nominal openings sizes for the most common synthetic zeolites are 0.3 nm for KA, 0.4 nm for NaA, 0.5 nm for CaA, and 1.0 nm for NaX. Further details, including effective molecular diameters, are widely available [Barrer; Breck; Ruthven; Yang, gen. refs.]. Many adsorbents, particularly the amorphous adsorbents, are characterized by their pore size distribution. The distribution of small pores is usually determined by analysis, using one of several available methods, of a cryogenic nitrogen adsorption isotherm, although other probe molecules are also used. Russell and LeVan [Carbon, 32, 845 (1994)] compare popular methods using a single nitrogen isotherm measured on activated carbon and provide numerous references. The distribution of large pores is usually determined by mercury porisimetry [Gregg and Sing, gen. refs.]. Table 16-4 shows the IUPAC classification of pores by size. Micropores are small enough that a molecule is attracted to both of the opposing walls forming the pore. The potential energy functions for these walls superimpose to create a deep well, and strong adsorption results. Hysteresis is generally not observed. (However, water vapor adsorbed in the micropores of activated carbon shows a large hysteresis loop, and the desorption branch is sometimes used with the Kelvin equation to determine the pore size distribution.) Capillary condensation occurs in mesopores and a hysteresis loop is typically found. Macropores form important paths for molecules to diffuse into a particle; for gas-phase adsorption, they do not fill with adsorbate until the gas phase becomes saturated. TABLE 16-4

Classification of Pore Sizes

Type

Slit Width* (w)

Characteristic

Micropore† Mesopore Macropore

w < 2 nm 2 nm < w < 50 nm w > 50 nm

Superimposed wall potentials Capillary condensation Effectively flat walled until p → Ps

*Or pore diameter. †Further subdivided into ultramicropores and supermicropores (Gregg and Sing, gen. refs.).

Ion Exchangers Ion exchangers are classified according to (1) their functionality and (2) the physical properties of the support matrix. Cation and anion exchangers are classified in terms of their ability to exchange positively or negatively charged species. Strongly acidic and strongly basic ion exchangers are ionized and thus are effective at nearly all pH values (pH 0–14). Weakly acidic exchangers are typically effective in the range of pH 5–14. Weakly basic resins are effective in the range of pH 0–9. Weakly acidic and weakly basic exchangers are often easier to regenerate, but leakage due to incomplete exchange may occur. Chelating resins containing iminodiacetic acid form specific metal complexes with metal ions with complex stability constants that follow the same order as those for EDTA. However, depending on pH, they also function as weak cation exchangers. The achievable ion-exchange capacity depends on the concentration of ionogenic groups and their availability as an exchange site. The latter is a function of the support matrix. Polymer-based, synthetic ion-exchangers known as resins are available commercially in gel type or truly porous forms. Gel-type resins are not porous in the usual sense of the word, since their structure depends upon swelling in the solvent in which they are immersed. Removal of the solvent usually results in a collapse of the threedimensional structure, and no significant surface area or pore diameter can be defined by the ordinary techniques available for truly porous materials. In their swollen state, gel-type resins approximate a true molecular-scale solution. Thus, we can identify an internal porosity εp only in terms of the equilibrium uptake of water or other liquid. When crosslinked polymers are used as the support matrix, the internal porosity so defined varies in inverse proportion to the degree of crosslinking, with swelling and therefore porosity typically being more pronounced in solvents with a high dielectric constant. The ion held by the exchanger also influences the resin swelling. Thus, the size of the resin particles changes during the ion-exchange process as the resin is changed from one form to another, and this effect is more dramatic for resins with a lower degree of crosslinking. The choice of degree of crosslinking is dependent on several factors including: the extent of swelling, the exchange capacity, the intraparticle diffusivity, the ease of regeneration, and the physical and chemical stability of the exchanger under chosen operating conditions. The concentration of ionogenic groups determines the capacity of the resin. Although the capacity per unit mass of dry resin is insensitive to the degree of crosslinking, except for very highly crosslinked resins, the exchange capacity per unit volume of swollen resin increases significantly with degree of crosslinking, so long as the mesh size of the polymer network allows the ions free access to functional groups within the interior of the resin. The degree of crosslinking also affects the rate of ion exchange. The intraparticle diffusivity decreases nearly exponentially with the mesh size of the matrix. As a result, resins with a lower degree of crosslinking are normally required for the exchange of bulky species, such as organic ions with molecular weight in excess of 100. The regeneration efficiency is typically greater for resins with a lower degree of crosslinking. Finally, the degree of crosslinking also affects the long-term stability of the resin. Strongly acidic and strongly basic resins are subject to irreversible oxidative degradation of the polymer and thermal and oxidative decomposition of functional groups. Generally, more highly crosslinked resins are less prone to irreversible chemical degradation but they may be subject to osmotic breakage caused by volume changes that occur during cyclic operations. In general, experience shows that an intermediate degree of crosslinking is often preferred. However, readers are referred to manufacturers’ specifications for resin stability data at different operating conditions. Truly porous, synthetic ion exchangers are also available. These materials retain their porosity even after removal of the solvent and have measurable surface areas and pore size. The term macroreticular is commonly used for resins prepared from a phase separation technique, where the polymer matrix is prepared with the addition of a liquid that is a good solvent for the monomers, but in which the polymer is insoluble. Matrices prepared in this way usually have the appearance of a conglomerate of gel-type microspheres held together to

ADSORBENTS AND ION EXCHANGERS form an interconnected porous network. Macroporous resins possessing a more continuous gellular structure interlaced with a pore network have also been obtained with different techniques and are commercially available. Since higher degrees of crosslinking are typically used to produce truly porous ion-exchange resins, these materials tend to be more stable under highly oxidative conditions, more attrition-resistant, and more resistant to breakage due to osmotic shock than their gel-type counterparts. Moreover, since their porosity does not depend entirely on swelling, they can be used in solvents with low dielectric constant where gel-type resins can be ineffective. Porous ion-exchange resins are also useful for the recovery and separation of high-molecular-weight substances such as proteins or colloidal particles. Specialty resins with very large pore sizes (from 30 nm to larger than 1000 nm) are available for these applications. In general, compared to gel-type resins, truly porous resins typically have somewhat lower capacities and can be more expensive. Thus, for ordinary ion-exchange applications involving small ions under nonharsh conditions, gel-type resins are usually preferred.

TABLE 16-5

16-9

PHYSICAL PROPERTIES Selected data on commercially available adsorbents and ion exchangers are given in Tables 16-5 and 16-6. The purpose of the tables is twofold: to assist the engineer or scientist in identifying materials suitable for a needed application, and to supply typical physical property values. Excellent sources of information on the characteristics of adsorbents or ion exchange products for specific applications are the manufacturers themselves. The names, addresses, and phone numbers of suppliers may be readily found in most libraries in sources such as the Thomas Register and Dun & Bradstreet. Additional information on adsorbents and ion exchangers is available in many of the general references and in several articles in Kirk-Othmer Encyclopedia of Chemical Technology. A recent comprehensive summary of commercial ion-exchangers, including manufacturing methods, properties, and applications, is given by Dorfner (Ion Exchangers, de Gruyter, New York, 1991).

Physical Properties of Adsorbents

Material and uses Aluminas Low-porosity (fluoride sorbent) High-porosity (drying, separations) Desiccant, CaCl2-coated Activated bauxite Chromatographic alumina Silicates and aluminosilicates Molecular sieves Type 3A (dehydration) Type 4A (dehydration) Type 5A (separations) Type 13X (purification) Silicalite (hydrocarbons) Dealumininated Y (hydrocarbons) Mordenite (acid drying) Chabazite (acid drying) Silica gel (drying, separations) Magnesium silicate (decolorizing) Calcium silicate (fatty-acid removal) Clay, acid-treated (refining of petroleum, food products) Fuller’s earth (same) Diatomaceous earth Carbons Shell-based Wood-based Petroleum-based Peat-based Lignite-based Bituminous-coal-based Synthetic polymer based (pyrolized) Carbon molecular sieve (air separation) Organic polymers Polystyrene (removal of organics, e.g., phenol; antibiotics recovery) Polyacrylic ester (purification of pulping wastewaters; antibiotics recovery) Phenolic (also phenolic amine) resin (decolorizing and deodorizing of solutions)

Size range, U.S. standard mesh†

G, S G

Internal porosity, %

Bulk dry density, kg/L

Average pore diameter, nm

Surface area, km2/kg

8–14, etc. Various

40 57

0.70 0.85

∼7 4–14

0.32 0.25–0.36

G G G, P, S

3–8, etc. 8–20, etc. 80–200, etc.

30 35 30

0.91 0.85 0.93

4.5 5

0.2

S, C, P

Various

∼30 ∼32 ∼34 ∼38

0.62–0.68 0.61–0.67 0.60–0.66 0.58–0.64 0.64–0.70 0.48–0.53 0.88 0.72 0.70–0.82 ∼0.50 ∼0.20

0.3 0.4 0.5 1.0 0.6 0.8 0.3–0.8 0.4–0.5 2–5

∼0.7 ∼0.7 ∼0.7 ∼0.6 ∼0.4 0.5–0.8

Shape* of particles

S, C, P S, C, P

Various Various

G, P G, P P

Various Various

G

38–48 ∼33 75–80

0.6–0.8 0.18–0.30 ∼0.1

4–8

0.85

G, P G

<200 Various

0.80 0.44–0.50

G G G, C G, C, P G, P G, P S

Various Various Various Various Various 8–30, 12–40 20–100 Various

60 ∼80 ∼80 ∼55 70–85 60–80 40–70 35–50

0.45–0.55 0.25–0.30 0.45–0.55 0.30–0.50 0.40–0.70 0.40–0.60 0.49–0.60 0.5–0.7

0.3–0.6

S

20–60

40–60

0.64

4–20

0.3–0.7

G, S

20–60

50–55

0.65–0.70

10–25

0.15–0.4

G

16–50

45

0.42

Sorptive capacity, kg/kg (dry) 0.20 0.25–0.33 0.22 0.1–0.2 ∼0.14

0.21–0.23 0.22–0.26 0.23–0.28 0.25–0.36 0.12–0.16 0.28–0.42 0.12 0.20 0.35–0.50

∼0.002 2 2 1–4 3 2–4

0.8–1.6 0.8–1.8 0.9–1.3 0.8–1.6 0.4–0.7 0.9–1.2 0.1–1.1

0.40 ∼0.70 0.3–0.4 0.5 0.3 0.4 0.5–0.20

0.08–0.12

0.45–0.55

*Shapes: C, cylindrical pellets; F, fibrous flakes; G, granules; P, powder; S, spheres. †U.S. Standard sieve sizes (given in parentheses) correspond to the following diameters in millimeters: (3) 6.73, (4) 4.76, (8) 2.98, (12) 1.68, (14) 1.41, (16) 1.19, (20) 0.841, (30) 0.595, (40) 0.420, (50) 0.297, (60) 0.250, (80) 0.177, (200) 0.074.

16-10

ADSORPTION AND ION EXCHANGE

TABLE 16-6

Physical Properties of Ion-Exchange Materials

Material Cation exchangers: strongly acidic Polystyrene sulfonate Homogeneous (gel) resin 4% cross-linked 6% cross-linked 8–10% cross-linked 12% cross-linked 16% cross-linked 20% cross-linked Porous structure 12–20% cross-linked Sulfonated phenolic resin Sulfonated coal Cation exchangers: weakly acidic Acrylic (pK 5) or methacrylic (pK 6) Homogeneous (gel) resin Macroporous Phenolic resin Polystyrene phosphonate Polystyrene iminodiacetate Polystyrene amidoxime Polystyrene thiol Cellulose Phosphonate Methylene carboxylate Greensand (Fe silicate) Zeolite (Al silicate) Zirconium tungstate Anion exchangers: strongly basic Polystyrene-based Trimethyl benzyl ammonium (type I) Homogeneous, 8% CL Porous, 11% CL Dimethyl hydroxyethyl ammonium (type II) Homogeneous, 8% CL Porous, 10% CL Acrylic-based Homogeneous (gel) Porous Cellulose-based Ethyl trimethyl ammonium Triethyl hydroxypropyl ammonium Anion exchangers: intermediately basic (pK 11) Polystyrene-based Epoxy-polyamine Anion exchangers: weakly basic (pK 9) Aminopolystyrene Homogeneous (gel) Porous Acrylic-based amine Homogeneous (gel) Porous Cellulose-based Aminoethyl Diethyl aminoethyl

Bulk wet density (drained), kg/L

Moisture content (drained), % by weight

Swelling due to exchange, %

0.75–0.85 0.76–0.86 0.77–0.87 0.78–0.88 0.79–0.89 0.80–0.90

64–70 58–65 48–60 44–48 42–46 40–45

10–12 8–10 6–8 5 4 3

S G G

0.81 0.74–0.85

50–55 50–60

4–6 7

120–150 50–90

S S G G, S S S S

0.70–0.75 0.67–0.74 0.70–0.80 0.74 0.75 ∼0.75 ∼0.75

45–50 50–55 ∼50 50–70 68–75 58 45–50

20–80 10–100 10–25 <40 <100 10

120 120 45–65 120 75 50 60

Shape* of particles

S

Maximum operating temperature,† °C

Operating pH range

120–150

0–14

Exchange capacity Dry, equivalent/kg

Wet, equivalent/L

5.0–5.5 4.8–5.4 4.6–5.2 4.4–4.9 4.2–4.6 3.9–4.2

1.2–1.6 1.3–1.8 1.4–1.9 1.5–2.0 1.7–2.1 1.8–2.0

0–14 0–14

4.5–5.0 2.0–2.5

1.5–1.9 0.7–0.9

4–14 0–14 3–14 3–14 1–11 1–13

8.3–10 ∼8.0 2.5 6.6 2.9 2.8 ∼5

3.3–4.0 2.5–3.5 1.0–1.4 3.0 0.7 0.8–0.9 2.0

0.18 0.75 1.0

1.3 0.85–0.95 1.15–1.25

1–5 40–45 ∼5

0 0 0

60 60 >150

6–8 6–8 2–10

∼7.0 ∼0.7 0.14 1.4 1.2

S S

0.70 0.67

46–50 57–60

∼20 15–20

60–80 60–80

0–14 0–14

3.4–3.8 3.4

1.3–1.5 1.0

S S

0.71 0.67

∼42 ∼55

15–20 12–15

40–80 40–80

0–14 0–14

3.8–4.0 3.8

1.2 1.1

S S

0.72 0.67

∼70 ∼60

∼15 ∼12

40–80 40–80

0–14 0–14

∼5.0 3.0–3.3

1.0–1.2 0.8–0.9

100 100

4–10 4–10

0.62 0.57

F F, P, G G G G

F

S S

0.75 0.72

∼50 ∼64

15–25 8–10

65 75

0–10 0–7

4.8 6.5

1.8 1.7

S S

0.67 0.61

∼45 55–60

8–12 ∼25

100 100

0–7 0–9

5.5 4.9

1.8 1.2

S S

0.72 0.72

∼63 ∼68

8–10 12–15

80 60

0–7 0–9

6.5 5.0

1.7 1.1

P P

*Shapes: C, cylindrical pellets; G, granules; P, powder; S, spheres. †When two temperatures are shown, the first applies to H form for cation, or OH form for anion, exchanger; the second, to salt ion.

1.0 ∼0.9

SORPTION EQUILIBRIUM Several densities and void fractions are commonly used. For adsorbents, usually the bulk density ρb, the weight of clean material per unit bulk volume as packed in a column, is reported. The dry particle density ρp is related to the (external) void fraction of packing ε by ρb ρp = ᎏ 1−ε

ρp ρs = ᎏ (16-2) 1 − εp For an adsorbent or ion exchanger, the wet density ρw of a particle is related to these factors and to the liquid density ρf by ρw = ρp + εpρf

(16-1)

The skeletal density ρs of a particle (or crystalline density for a pure chemical compound) is given in terms of internal porosity εp by

16-11

(16-3)

The total voidage εb in a packed bed (outside and inside particles) is εb = ε + (1 − ε)εp

,, ,, ,,

(16-4)

SORPTION EQUILIBRIUM

Forces Molecules are attracted to surfaces as the result of two types of forces: dispersion-repulsion forces (also called London or van der Waals forces) such as described by the Lennard-Jones potential for molecule-molecule interactions; and electrostatic forces, which exist as the result of a molecule or surface group having a permanent electric dipole or quadrupole moment or net electric charge. Dispersion forces are always present and in the absence of any stronger force will determine equilibrium behavior, as with adsorption of molecules with no dipole or quadrupole moment on nonoxidized carbons and silicalite. If a surface is polar, its resulting electric field will induce a dipole moment in a molecule with no permanent dipole and, through this polarization, increase the extent of adsorption. Similarly, a molecule with a permanent dipole moment will polarize an otherwise nonpolar surface, thereby increasing the attraction. For a polar surface and molecules with permanent dipole moments, attraction is strong, as for water adsorption on a hydrophilic adsorbent. Similarly, for a polar surface, a molecule with a permanent quadrupole moment will be attracted more strongly than a similar molecule with a weaker moment; for example, nitrogen is adsorbed more strongly than oxygen on zeolites (Sherman and Yon, gen. refs.). Surface Excess With a Gibbs dividing surface placed at the surface of the solid, the surface excess of component i, Γi (mol/m2), is the amount per unit area of solid contained in the region near the surface, above that contained at the fluid-phase concentration far from the surface. This is depicted in two ways in Fig. 16-4. The quantity adsorbed per unit mass of adsorbent is ni = Γi A

(16-6)

where A (m2/kg) is the surface area of the solid. For a porous adsorbent, the amount adsorbed in the pore structure per unit mass of adsorbent, based on surface excess, is obtained by the difference ∞ ni = ntot (16-6) i − Vp c i (mol/kg) is the total amount of component i contained where ntot i within the particle’s pore volume Vp (m3/kg), and ci∞ is the concentration outside of the particle. If thermodynamics is a concern and ni differs significantly from ntot i (as it will for weakly adsorbed species), then

∞

Γi = ∫ο (ci – ci∞) dx

ci (mol/m3)

GENERAL CONSIDERATIONS

it is important to consider adsorbed-phase quantities in terms of surface excesses. A second convention is the placement of an imaginary envelope around the outermost boundary of a porous particle, so that all solute and nonadsorbing fluid contained within the pores of the particle is considered adsorbed. Classification of Isotherms by Shape Representative isotherms are shown in Fig. 16-5, as classified by Brunauer and coworkers.

ci∞ , bulk concentration

0

cil Γi = (cil – ci∞) ∆xl

ci (mol/m3)

The quantity of a solute adsorbed can be given conveniently in terms of moles or volume (for adsorption) or ion-equivalents (for ion exchange) per unit mass or volume (dry or wet) of sorbent. Common units for adsorption are mol/(m3 of fluid) for the fluid-phase concentration ci and mol/(kg of clean adsorbent) for adsorbed-phase concentration ni. For gases, partial pressure may replace concentration. Many models have been proposed for adsorption and ion exchange equilibria. The most important factor in selecting a model from an engineering standpoint is to have an accurate mathematical description over the entire range of process conditions. It is usually fairly easy to obtain correct capacities at selected points, but isotherm shape over the entire range is often a critical concern for a regenerable process.

ci∞ ∆xl

0

0

x, distance from surface

Depictions of surface excess Γi . Top: The force field of the solid concentrates component i near the surface; the concentration ci is low at the surface because of short-range repulsive forces between adsorbate and surface. Bottom: Surface excess for an imagined homogeneous surface layer of thickness ∆x l. FIG. 16-4

16-12

ADSORPTION AND ION EXCHANGE

Type II

Type III

ni

Type I

pi

Psi

Psi

pi

Psi



Type V

ni

Type IV

pi

pi

Psi

pi

Psi

FIG. 16-5 Representative isotherm types. pi and P is are pressure and vapor pressure of the solute. [Brunauer, J. Am. Chem. Soc., 62, 1723 (1940); reprinted with permission.]

Curves that are concave downward throughout (type I) have historically been designated as “favorable,” while those that are concave upward throughout (type III) are “unfavorable.” Other isotherms (types II, IV, and V) have one or more inflection points. The designations “favorable” and “unfavorable” refer to fixed-bed behavior for the uptake step, with a favorable isotherm maintaining a compact wave shape. A favorable isotherm for uptake is unfavorable for the discharge step. This becomes particularly important for a regenerative process, in which a favorable isotherm may be too favorable for regeneration to occur effectively. Categorization of Equilibrium Models Historically, sorption equilibrium has been approached from different viewpoints. For adsorption, many models for flat surfaces have been used to develop explicit equations and equations of state for pure components and mixtures, and many of the resulting equations are routinely applied to porous materials. Explicit equations for pore filling have also been proposed, generally based on the Polanyi potential theory. Ion exchange adds to these approaches concepts of absorption or dissolution (absorption) and exchange reactions. Statistical mechanics and molecular dynamics contribute to our understanding of all of these approaches (Steele, The Interaction of Gases with Solid Surfaces, Pergamon, Oxford, 1974; Nicholson and Parsonage, Computer Simulation and the Statistical Mechanics of Adsorption, Academic Press, New York, 1982). Mixture models are often based on the adsorbed solution theory, which uses thermodynamic equations from vaporliquid equilibria with volume replaced by surface area and pressure replaced by a two-dimensional spreading pressure. Other approaches include lattice theories and mass-action exchange equilibrium. Heterogeneity Adsorbents and ion exchangers can be physically and chemically heterogeneous. Although exceptions exist, solutes generally compete for the same sites. Models for adsorbent heterogeneity have been developed for both discrete and continuous distributions of energies [Ross and Olivier, On Physical Adsorption, Interscience, New York, 1964; Jaroniec and Madey, Rudzinski and Everett, gen. refs.]. Isosteric Heat of Adsorption The most useful heat of adsorption for fixed-bed calculations is the isosteric heat of adsorption, which is given by the Clausius-Clapeyron type relation ∂ ln p q ist = ℜT 2 ᎏi ∂T



ni ,n j

where the nj can be dropped for single-component adsorption. q sti is positive by convention. If isosteres are straight lines when plotted as ln pi versus T −1 (see Fig. 16-1), then Eq. (16-7) can be integrated to give qsti ln pi = f(ni) − ᎏ (16-8) ℜT where f(ni) is an arbitrary function dependent only on ni. Many other heats of adsorption have been defined and their utility depends on the application (Ross and Olivier; Young and Crowell, gen. refs.). From Eq. (16-8), if a single isotherm is known in the pressure explicit form pi = pi(ni, T) and if q ist is known at least approximately, then equilibria can be estimated over a narrow temperature range using pi qist 1 1 ln ᎏ =ᎏ (const ni) (16-9) ᎏ−ᎏ ref pi ℜ T ref T Similarly, Eq. (16-9) is used to calculate the isosteric heat of adsorption from two isotherms. Experiments Sorption equilibria are measured using apparatuses and methods classified as volumetric, gravimetric, flow-through (frontal analysis), and chromatographic. Apparatuses are discussed by Yang (gen. refs.). Heats of adsorption can be determined from isotherms measured at different temperatures or measured independently by calorimetric methods. Dimensionless Concentration Variables Where appropriate, isotherms will be written here using the dimensionless system variables ci ni c*i = ᎏ n*i = ᎏ (16-10) ciref nref i where the best choice of reference values depends on the operation. In some cases, to allow for some preloading of the adsorbent, it will be more convenient to use the dimensionless transition variables (ci − c′i ) (ni − n′i ) c*i = ᎏ n*i = ᎏᎏ (16-11) (c″i − c′i ) (n″i − n′i ) where single and double primes indicate initial and final concentrations, respectively. Figure 16-6 shows n* plotted versus c* for a sample system. Superimposed are an upward transition (loading) and a downward transition (unloading), shown by the respective positions of (c′i, n′i ) and (c″i , n″i ).

(16-7)



SINGLE COMPONENT OR EXCHANGE The simplest relationship between solid-phase and fluid-phase concentrations is the linear isotherm ni = K i ci

ni = K′i pi

or

(16-12)

niref

ci′′ (or ci′) ni′′ (or ni′) ni ni*

ni′ (or ni′′) 0 0

c*i ci′ (or ci′′) ci

ciref

FIG. 16-6 Isotherm showing concentration variables for a transition from (c′i, n′i) to (c″i , n″i ).

SORPTION EQUILIBRIUM Thermodynamics requires that a linear limit be approached in the Henry’s law region for all isotherm equations. Flat Surface Isotherm Equations The classification of isotherm equations into two broad categories for flat surfaces and pore filling reflects their origin. It does not restrict equations developed for flat surfaces from being applied successfully to describe data for porous adsorbents. The classical isotherm for a homogeneous flat surface, and most popular of all nonlinear isotherms, is the Langmuir isotherm nisK i ci ni = ᎏ (16-13) 1 + K i ci s where n i is the monolayer capacity approached at large concentrations and K i is an equilibrium constant. These parameters are often determined by plotting 1/ni versus 1/ci. The derivation of the isotherm assumes negligible interaction between adsorbed molecules. The classical isotherm for multilayer adsorption on a homogeneous, flat surface is the BET isotherm [Brunauer, Emmett, and Teller, J. Am. Chem. Soc., 60, 309 (1938)] nisK i pi ni = ᎏᎏᎏᎏ (16-14) [1 + K i pi − (pi /Pis)][1 − (pi /P is)] where pi is the pressure of the adsorbable component and P is is its vapor pressure. It is useful for gas-solid systems in which condensation is approached, fitting type-II behavior. For a heterogeneous flat surface, a classical isotherm is the Freundlich isotherm ni = K i cimi

(16-15)

where mi is positive and generally not an integer. The isotherm corresponds approximately to an exponential distribution of heats of adsorption. Although it lacks the required linear behavior in the Henry’s law region, it can often be used to correlate data on heterogeneous adsorbents over wide ranges of concentration. Several isotherms combine aspects of both the Langmuir and Freudlich equations. One that has been shown to be effective in describing data mathematically for heterogeneous adsorbents is the Tóth isotherm [Acta Chim. Acad. Sci. Hung., 69, 311 (1971)] nispi ni = ᎏᎏ (16-16) [(1/K i) + pimi]1/mi This three-parameter equation behaves linearly in the Henry’s law region and reduces to the Langmuir isotherm for m = 1. Other wellknown isotherms include the Radke-Prausnitz isotherm [Radke and Prausnitz, Ind. Eng. Chem. Fundam., 11, 445 (1972); AIChE J., 18, 761 (1972)] nisK i pi ni = ᎏᎏ (16-17) (1 + K i pi)m and the Sips isotherm [Sips, J. Chem. Phys., 16, 490 (1948); Koble and Corrigan, Ind. Eng. Chem., 44, 383 (1952)] or loading ratio correlation with prescribed temperature dependence [Yon and Turnock, AIChE Symp. Ser., 67(117), 75 (1971)] nis(K i pi)m ni = ᎏᎏ (16-18) 1 + (K i pi)m Another three-parameter equation that often fits data well and is linear in the Henry’s law region is the UNILAN equation [Honig and Reyerson, J. Phys. Chem., 56, 140 (1952)] ns 1 + K i esipi ni = ᎏi ln ᎏᎏ (16-19) 2si 1 + K i e−sipi which reduces to the Langmuir equation as si → 0. Equations of state are also used for pure components. Given such an equation written in terms of the two-dimensional spreading pressure π, the corresponding isotherm is easily determined, as described later for mixtures [see Eq. (16-42)]. The two-dimensional equivalent of an ideal gas is an ideal surface gas, which is described by πA = niℜT

(16-20)

which readily gives the linear isotherm, Eq. (16-12). Many more com-

16-13

plicated equations of state are available, including two-dimensional analogs of the virial equation and equations of van der Waals, RedlichKwong, Peng-Robinson, and so forth [Adamson, gen. refs.; Patrykiejewet et al., Chem. Eng. J., 15, 147 (1978); Haydel and Kobayashi, Ind. Eng. Chem. Fundam., 6, 546 (1967)]. Pore-Filling Isotherm Equations Most pore-filling models are grounded in the Polanyi potential theory. In Polanyi’s model, an attracting potential energy field is assumed to exist adjacent to the surface of the adsorbent and concentrates vapors there. Adsorption takes place whenever the strength of the field, independent of temperature, is great enough to compress the solute to a partial pressure greater than its vapor pressure. The last molecules to adsorb form an equipotential surface containing the adsorbed volume. The strength of this field, called the adsorption potential ε (J/mol), was defined by Polanyi to be equal to the work required to compress the solute from its partial pressure to its vapor pressure Ps ε = ℜT ln ᎏi (16-21) pi The same result is obtained by considering the change in chemical potential. In the basic theory, W (m3/kg), the volume adsorbed as saturated liquid at the adsorption temperature, is plotted versus ε to give a characteristic curve. Data measured at different temperatures for the same solute-adsorbent pair should fall on this curve. Using the method, it is possible to use data measured at a single temperature to predict isotherms at other temperatures. Data for additional, homologous solutes can be collapsed into a single “correlation curve” by defining a scaling factor β, the most useful of which has been V/V ref, the adsorbate molar volume as saturated liquid at the adsorption temperature divided by that for a reference compound. Thus, by plotting W versus ε/β or ε/ V for data measured at various temperatures for various similar solutes and a single adsorbent, a single curve should be obtained. Variations of the theory are often used to evaluate properties for components near or above their critical points [Grant and Manes, Ind. Eng. Chem. Fund., 5, 490 (1966)]. The most popular equations used to describe the shape of a characteristic curve or a correlation curve are the two-parameter DubininRadushkevich (DR) equation ε 2 W (16-22) ᎏ = exp −k ᎏ β W0 and the three-parameter Dubinin-Astakhov (DA) equation ε m W (16-23) ᎏ = exp −k ᎏ β W0 where W0 is micropore volume and m is related to the pore size distribution [Gregg and Sing, gen. refs.]. Neither of these equations has correct limiting behavior in the Henry’s law regime. Ion Exchange A useful tool is provided by the mass action law for describing the general exchange equilibrium in fully ionized exchanger systems as

 

  

  

zBA + zAB
A zB A
+ zAB where overbars indicate the ionic species in the ion exchanger phase and zA and zB are the valences of ions A and B. The associated equilibrium relation is of the form γA z γ
B z nA z cB z c K A,B = KA,B ᎏ (16-24) ᎏ = ᎏ ᎏ γ
A γB cA nB c where K A,B is the apparent equilibrium constant or molar selectivity coefficient, KA,B is the thermodynamic equilibrium constant based on activities, and the γs are activity coefficients. Often it is desirable to represent concentrations in terms of equivalent ionic fractions based on solution normality ctot and fixed exchanger capacity ns as c*i = zi ci /ctot and n*i = zi ni /ntot, where ctot =  zjcj and ntot =  zj nj = ns with the summations extended to all counter-ion species. A rational selectivity coefficient is then defined as ntot z − z n*A z c*B z c K′A,B = K A,B ᎏ = ᎏ (16-25) ᎏ ctot c*A n*B c For the exchange of ions of equal valence (zA = zB), K A,B and K′A,B are

      B

 

A

A

B

B

   B

A

A

16-14

ADSORPTION AND ION EXCHANGE

coincident and, to a first approximation, independent of concentration. When zA > zB, K′A,B decreases with solution normality. This reflects the fact that ion exchangers exhibit an increasing affinity for ions of lower valence as the solution normality increases. An alternate form of Eq. (16-25) is ntot 1 − n*A c n*A = K A,B ᎏ ᎏ ctot 1 − c*A When B is in excess (c*A, n*A → 0), this reduces to (zA − zB )/zB

 





zA /zB

c*A

(16-26)

ntot (z − z )/z c c*A = K Ac*A (16-27) n*A = K A,B ᎏ ctot c where the linear equilibrium constant K A = K A,B(ntot /ctot )(zA/zB) − 1 decreases with solution normality ctot when zA > zB or increases when zA < zB. Table 16-7 gives equilibrium constants for crosslinked cation and anion exchangers for a variety of counterions. The values given for cation exchangers are based on ion A replacing Li+, and those given for anion exchangers are based on ion A replacing Cl−. The selectivity for a particular ion generally increases with decreasing hydrated ion size and increasing degree of crosslinking. The selectivity coefficient for any two ions A and D can be obtained from Table 16-7 from values of KA,B and KD,B as

 

A

B

B

KA,D = KA,B /KD,B

(16-28)

The values given in this table are only approximate, but they are adequate for process screening purposes with Eqs. (16-24) and (16-25). Rigorous calculations generally require that activity coefficients be accounted for. However, for the exchange between ions of the same valence at solution concentrations of 0.1 N or less, or between any ions at 0.01 N or less, the solution-phase activity coefficients prorated to unit valence will be similar enough that they can be omitted. Models for ion exchange equilibria based on the mass-action law taking into account solution and exchanger-phase nonidealities with equations similar to those for liquid mixtures have been developed by several authors [see Smith and Woodburn, AIChE J., 24, 577 (1978); Mehablia et al., Chem. Eng. Sci., 49, 2277 (1994)]. Thermodynamicsbased approaches are also available [Soldatov in Dorfner, gen. refs.; Novosad and Myers, Can J. Chem. Eng., 60, 500 (1982); Myers and Byington in Rodrigues, ed., Ion Exchange Science and Technology, NATO ASI Series, No. 107, Nijhoff, Dordrecht, 1986, pp. 119–145] as TABLE 16-7

well as approaches for the exchange of macromolecules, taking into account steric-hindrance effects [Brooks and Cramer, AIChE J., 38, 12 (1992)]. Example 3: Calculation of Useful Ion-Exchange Capacity An 8 percent crosslinked sulfonated resin is used to remove calcium from a solution containing 0.0007 mol/l Ca+2 and 0.01 mol/l Na+. The resin capacity is 2.0 equiv/l. Estimate the resin capacity for calcium removal. From Table 16-7, we obtain KCa,Na = 5.16/1.98 = 2.6. Since ntot = 2 equiv/l and ctot = 2 × 0.0007 + 0.01 = 0.024 equiv/l, K′Ca,Na = 2.6 × 2/0.011 = 470. Thus, with c*Ca = 2 × 0.0007/0.011 = 0.13, Eq. (16-25) gives 470 = (n*Ca /0.13)[(1 − 0.13)/ (1 − n*Na)]2 or n*Ca = 0.9. The available capacity for calcium is nCa = 0.9 × 2.0/2 = 0.9 mol/l. Donnan Uptake The uptake of an electrolyte as a neutral ion pair of a salt is called Donnan uptake. It is generally negligible at low ionic concentrations. Above 0.5 g⋅equiv/l with strongly ionized exchangers (or at lower concentrations with those more weakly ionized), the resin’s fixed ion-exchange capacity is measurably exceeded as a result of electrolyte invasion. With only one coion species Y (matching the charge sign of the fixed groups in the resin), its uptake nY equals the total excess uptake of the counterion. Equilibrium is described by the mass-action law. For the case of a resin in A-form in equilibrium with a salt AY, the excess counterion uptake is given by [Helfferich, gen. refs., pp. 133–147] ns2 γAY 2
aw ␯ˆ / ␯ˆ 1/2 ns zY nY = ᎏ + (zY cY)2 ᎏ −ᎏ (16-29) ᎏ 4 γ
AY aw 2 where γs are activity coefficients, as water activities, and νs partial molar volumes. For dilute conditions, Eq. (16-29) predicts a squared dependence of nY on cY. Thus, the electrolyte sorption isotherm has a strong positive curvature. Donnan uptake is more pronounced for resins of lower degree of crosslinking and for counterions of low valence. Separation Factor By analogy with the mass-action case and appropriate for both adsorption and ion exchange, a separation factor r can be defined based on dimensionless system variables [Eq. (16-10)] by n*B /c*B c*i (1 − n*i ) r = ᎏᎏ or r=ᎏ (16-30) n*i (1 − c*i ) n*A /c*A This term is analogous to relative volatility or its reciprocal (or to an equilibrium selectivity). Similarly, the assumption of a constant sepa-



    AY

w

Equilibrium Constants for Polystyrene DVB Cation and Anion Exchangers Strong acid sulfonated cation exchangers (Li+ reference ion) Degree of crosslinking

Degree of crosslinking

Counterion

4% DVB

8% DVB

16% DVB

Counterion

4% DVB

8% DVB

16% DVB

Li+ H+ Na+ NH +4 K+ Rb+ Cs+ Ag+ Tl+

1.00 1.32 1.58 1.90 2.27 2.46 2.67 4.73 6.71

1.00 1.27 1.98 2.55 2.90 3.16 3.25 8.51 12.4

1.00 1.47 2.37 3.34 4.50 4.62 4.66 22.9 28.5

Mg++ Zn++ Co++ Cu++ Cd++ Ni++ Ca++ Pb++ Ba++

2.95 3.13 3.23 3.29 3.37 3.45 4.15 6.56 7.47

3.29 3.47 3.74 3.85 3.88 3.93 5.16 9.91 11.5

3.51 3.78 3.81 4.46 4.95 4.06 7.27 18.0 20.8

Strong base anion exchangers, 8% DVB (Cl− reference ion) Counterion

Type I resin*

Type II resin†

Counterion

Type I resin(a)

Type II resin(b)

Salicylate Iodide Phenoxide Nitrate Bromide Nitrite Bisulfite Cyanide

32 8.7 5.2 3.8 2.8 1.2 1.3 1.6

28 7.3 8.7 3.3 2.3 1.3 1.3 1.3

Cyanide Chloride Hydroxide Bicarbonate Formate Acetate Fluoride Sulfate

1.6 1.0 0.05–0.07 0.3 0.2 0.2 0.09 0.15

1.3 1.0 0.65 0.5 0.2 0.2 0.1 0.15

*Trimethylamine †Dimethyl-hydroxyethylamine Data from Bonner and Smith, J. Phys. Chem., 61, 326 (1957) and Wheaton and Bauman, Ind. Eng. Chem., 45, 1088 (1951).

SORPTION EQUILIBRIUM ration factor is a useful assumption in many sorptive operations. [It is constant for the Langmuir isotherm, as described below, and for massaction equilibrium with m = 1 in Eq. (16-24).] This gives the constant separation factor isotherm c*i n*i = ᎏᎏ (16-31) r + (1 − r)c*i The separation factor r identifies the equilibrium increase in n*i from 0 to 1, which accompanies an increase in c*i from 0 to 1. For a concentration change over only part of the isotherm, a separation factor R can be defined for the dimensionless transition variables [Eq. (16-11)]. This separation factor is n″i /c″i c*i (1 − n*i ) R=ᎏ = ᎏᎏ (16-32) n′i /c′i n*i (1 − c*i ) and gives an equation identical to Eq. (16-31) with R replacing r. Figure 16-7 shows constant separation factor isotherms for a range of r (or R) values. The isotherm is linear for r = 1, favorable for r < 1, rectangular (or irreversible) for r = 0, and unfavorable for r > 1. As a result of symmetry properties, if r is defined for adsorption of component i or exchange of ion A for B, then the reverse process is described by 1/r. The Langmuir isotherm, Eq. (16-13), corresponds to the constant separation factor isotherm with r = 1/(1 + K i c ref i )

(16-33)

for system variables Eq. (16-10) or r + (1 − r)(c′i /ciref ) R = ᎏᎏ (16-34) r + (1 − r)(c″i /ciref ) for transition variables [Eq. (16-11)]. Vermeulen et al. [gen. refs.] give additional properties of constant separation factor isotherms. Example 4: Application of Isotherms Thomas [Ann. N.Y. Acad. Sci., 49, 161 (1948)] provides the following Langmuir isotherm for the adsorption of anthracene from cyclohexane onto alumina: 22ci ni = ᎏ 1 + 375ci with ni in mol anthracene/kg alumina and ci in mol anthracene/l liquid. a. What are the values of K i and n is according to Eq. (16-13)? K i = 375 l/mol 22 nsi = ᎏ = 0.0587 mol/kg Ki

16-15

For a feed concentration of 8.11 × 10−4 mol/l, what is the value of r ? 1 r = ᎏᎏ = 0.766 [from Eq. (16-33)] 1 + Ki(8.11 × 10−4) c. If the alumina is presaturated with liquid containing 2.35 × 10−4 mol/l and the feed concentration is 8.11 × 10−4 mol/l, what is the value of R? b.

R = 0.834

[from Eq. (16-32) or (16-34)]

MULTIPLE COMPONENTS OR EXCHANGES When more than one adsorbed species or more than two ionexchanged species interact in some manner, equilibrium becomes more complicated. Usually, thermodynamics provides a sound basis for prediction. Adsorbed-Solution Theory The common thermodynamic approach to multicomponent adsorption treats adsorption equilibrium in a way analogous to fluid-fluid equilibrium. The theory has as its basis the Gibbs adsorption isotherm [Young and Crowell, gen. refs.], which is Adπ =  ni dµi (const T )

(16-35)

i

where µ is chemical potential. For an ideal gas (dµ = ℜTd ln pi), if it is assumed that an adsorbed solution is defined with a pure-component standard state (as is common for a liquid solution), then Eq. (16-35) can be integrated to give [Rudisill and LeVan, Chem. Eng. Sci., 47, 1239 (1992)] pi = γi xiP ref i (T, π)

(16-36)

where γi and xi are the adsorbed-phase activity coefficient and mole fraction of component i and P iref is the standard state, specified to be at the temperature and spreading pressure of the mixture. Equation (16-36) with γi = 1 provides the basis for the ideal adsorbed-solution theory [Myers and Prausnitz, AIChE J., 11, 121 (1965)]. The spreading pressure for a pure component is determined by integrating Eq. (16-35) for a pure component to obtain



ref

Pi πA ni (16-37) ᎏ= ᎏ dpi 0 ℜT pi where ni is given by the pure-component isotherm. Also, since  xi = 1, Eq. (16-36) with γi = 1 gives  (pi /P iref ) = 1. With no area change on mixing for the ideal solution, the total number of moles adsorbed per unit weight of adsorbent is determined using a two-dimensional form of Amagat’s law: 1 xi (16-38) ᎏ= ᎏ ntot niref where ntot =  ni and niref is given by the pure-component isotherm at Piref. Adsorbed-phase concentrations are calculated using ni = xintot. Generally, different values of π [or πA/(ℜT)] must be tried until the one is found that satisfies Eqs. (16-37) and  xi = 1.



Example 5: Application of Ideal Adsorbed-Solution Theory Consider a binary adsorbed mixture for which each pure component obeys the Langmuir equation, Eq. (16-13). Let n1s = 4 mol/kg, n2s = 3 mol/kg, K1 p1 = K 2 p2 = 1. Use the ideal adsorbed-solution theory to determine n1 and n2. Substituting the pure component Langmuir isotherm nsiKipi ni = ᎏ 1 + Ki pi into Eq. (16-37) and integrating gives

Constant separation factor isotherm as a function of the separation factor r (or interchangeably R). Each isotherm is symmetric about the perpendicular line connecting (0,1) and (1,0). Isotherms for r and 1/r are symmetric about the 45° line. FIG. 16-7

πA ᎏ = nis ln(1 + K iP iref ) ℜT which can be solved explicitly for KiP iref. Values are guessed for πA/(ℜT), values of KiP iref are calculated from the equation above, and  xi =  Kipi /(KiP iref ) = 1 is checked to see if it is satisfied. Trial and error gives πA/(ℜT) = 3.8530 mol/kg, K1P1ref = 1.6202, K2P 2ref = 2.6123, and x1 = 0.61720. Evaluating the purecomponent isotherms at the reference pressures and using Eq. (16-38) gives ntot = 2.3475 mol/kg, and finally ni = xintot gives n1 = 1.4489 mol/kg and n2 = 0.8986 mol/kg.

Other approaches to account for various effects have been developed. Negative deviations from Raoult’s law (i.e., γi < 1) are frequently

16-16

ADSORPTION AND ION EXCHANGE

found due to adsorbent heterogeneity [e.g., Myers, AIChE J., 29, 691 (1983)]. Thus, contributions include accounting for adsorbent heterogeneity [Valenzuela et al., AIChE J., 34, 397 (1988)] and excluded pore-volume effects [Myers, in Rodrigues et al., gen. refs.]. Several activity coefficient models have been developed to account for nonideal adsorbate-adsorbate interactions including a spreading pressuredependent activity coefficient model [e.g., Talu and Zwiebel, AIChE J., 32, 1263 (1986)] and a vacancy solution theory [Suwanayuen and Danner, AIChE J., 26, 68, 76 (1980)]. Langmuir-Type Relations For systems composed of solutes that individually follow Langmuir isotherms, the traditional multicomponent Langmuir equation, obtained via a kinetic derivation, is nis K i pi ni = ᎏᎏ (16-39) 1 +  K i pi This equation has been criticized on thermodynamic grounds because it does not satisfy the Gibbs adsorption isotherm unless all monolayer capacities nsi are equal. To satisfy the Gibbs adsorption isotherm for unequal monolayer capacities, explicit isotherms can be obtained in the form of a series expansion [LeVan and Vermeulen, J. Phys. Chem., 85, 3247 (1981)]. A two-term form is (n1s + n2s)K1 p1 (n1s − n2s )K1p1K2 p2 n1 = ᎏᎏ + ᎏᎏ ln(1 + K1p1 + K2 p2) 2(1 + K1 p1 + K2 p2) (K1p1 + K2 p2)2 (16-40) where the subscripts may be interchanged. Multicomponent forms are also available [Frey and Rodrigues, AIChE J., 40, 182 (1994)]. Example 6: Comparison of Binary Langmuir Isotherms Use the numerical values in Example 5 to evaluate the binary Langmuir isotherms given by Eqs. (16-39) and (16-40) and compare results with the exact answers given in Example 5. Equation (16-39) gives n1 = 1.3333 mol/kg and n2 = 1.0000 mol/kg for an average deviation from the exact values of approximately 10 percent. Equation (16-40) gives n1 = 1.4413 mol/kg and n2 = 0.8990 mol/kg for an average deviation of about 0.6 percent. Freundlich-Type Relations A binary Freundlich isotherm, obtained from the ideal adsorbed solution theory in loading-explicit closed form [Crittenden et al., Environ. Sci. Technol., 19, 1037 (1985)], is Nj= 1 (nj /mj) 1/mi ni ᎏᎏ ci = ᎏ (16-41)  jN= 1 nj K i /mi





Equations of State If an equation of state is specified for a multicomponent adsorbed phase of the form πA/(ℜT) = f(n1, n2, . . . ), then the isotherms are determined using [Van Ness, Ind. Eng. Chem. Fundam., 8, 464–473 (1969)] ∞ ∂[πA/(ℜT)] K i pi dA ln ᎏ = −1 ᎏ (16-42) ᎏᎏ A T,A,n j ni /Ms ∂ni A where, because integration is over A, Ms is mass of adsorbent, units for n and A are mol and m2 (rather than mol/kg and m2/kg), and K i pi / (ni /Ms) = 1 is the linear lower limit approached for the ideal surface gas [see Eqs. (16-12) and (16-20)]. Ion Exchange—Stoichiometry In most applications, except for some weak-electrolyte and some concentrated-solution cases, the following summations apply:

 



 zc = c i i

tot



= const

i



 zn = n i i

tot

= const

(16-43)

i

FIG. 16-8 Ideal mass-action equilibrium for three-component ion exchange with unequal valences. K′A,C = 8.06; K′B,C = 3.87. Duolite C-20 polystyrenesulfonate resin, with Ca as A, Mg as B, and Na as C. [Klein et al., Ind. Eng. Chem. Fund., 6, 339 (1967); reprinted with permission.]

For an N-species system, with N c*s (or n*s) known, the N n*s (or c*s) can be found by simultaneous solution of the N − 1 independent i,j combinations for Eq. (16-45) using Eq. (16-43); one n*j /c*j is assumed, the other values can be calculated using Eq. (16-45), and the sum of the trial n*s (or c*s) is compared with Eq. (16-44). Because an N-component system has N − 1 independent concentrations, a three-component equilibrium can be plotted in a plane and a four-component equilibrium in a three-dimensional space. Figure 16-8 shows a triangular plot of c* contours in equilibrium with the corresponding n* coordinates. Improved models for ion-exchange equilibria based on the massaction law that take into account solution and exchanger-phase nonidealities with equations similar to those for liquid mixtures have been developed by several authors including Smith and Woodburn [AIChE J., 24, 577 (1978)] and Mehablia et al. [Chem. Eng. Sci., 49, 2277 (1994)]. Thermodynamics-based approaches are also available [Soldatov in Dorfner, gen. refs.; Novosad and Myers, Can J. Chem. Eng., 60, 500 (1982); Myers and Byington in Rodrigues, ed., Ion Exchange: Science and Technology, NATO ASI Series, No. 107, Nijhoff, Dordrecht, 1986, pp. 119–145] as well as approaches for the exchange of macromolecules taking into account steric-hindrance effects [Brooks and Cramer, AIChE J., 38, 12 (1992)]. Constant Separation-Factor Treatment If the valences of all species are equal, the separation factor αij applies, where n*i c*j α ij = K ij = ᎏ c*i n*j

For a binary system, r = αBA = 1/αAB. The symbol r applies primarily to the process, while α is oriented toward interactions between pairs of solute species. For each binary pair, rij = αji = 1/αij. Equilibrium then is given explicitly by

In equivalent-fraction terms, the sums become

 c* = 1 i

i

 n* = 1 i

(16-44)

i

Mass Action Here the equilibrium relations, consistent with Eq. (16-25), are n* z j c*j z i K′ij = ᎏi (16-45) ᎏ c*i n*j

  

(16-46)

and

c*i α iNc*i n*i = ᎏ = ᎏ j αjic*j j αjNc*j

(16-47)

αNin*i n*i c*i = ᎏ = ᎏ j αijn*j j αNjn*j

(16-48)

For the constant separation factor case, the c* contours in a plot like Fig. 16-8 are linear.

RATE AND DISPERSION FACTORS

16-17

CONSERVATION EQUATIONS Material balances, often an energy balance, and occasionally a momentum balance are needed to describe an adsorption process. These are written in various forms depending on the specific application and desire for simplicity or rigor. Reasonably general material balances for various processes are given below. An energy balance is developed for a fixed bed for gas-phase application and simplified for liquid-phase application. Momentum balances for pressure drop in packed beds are given in Sec. 6. MATERIAL BALANCES At a microscale, a sorbable component exists at three locations—in a sorbed phase, in pore fluid, and in fluid outside particles. As a consequence, in material balances time derivatives must be included of terms involving ni, cpi (the pore concentration), and ci (the extraparticle concentration). Let
ni represent ni averaged over particle volume, and let c
pi represent cpi averaged over pore fluid volume. For batch or stirred tank processes, in terms of the mass of adsorbent Ms (kg), extraparticle volume of fluid Vf (m3), and volumetric flow rates Fv (m3/s) in and out of a tank, the material balance on component i is d(Vf ci) dˆn Ms ᎏi + ᎏ = Fv,incin,i − Fv,out ci (16-49) dt dt ni + (εp /ρp)c
pi =
ni + [(1 − ε)εp /ρb]c
pi (16-50) with nˆ i =


where ρp and ρb are particle and bulk densities, and ε and εp are void fraction (extra particle volume fraction) and particle porosity, respectively. For a fixed-bed process, the material balance for component i is ∂ˆn ∂c ∂(vci) ∂ ∂y ρb ᎏi + ε ᎏi + ε ᎏ = εDL ᎏ c ᎏi (16-51) ∂t ∂t ∂z ∂z ∂z where v is interstitial fluid velocity, DL is a Fickian axial dispersion coefficient, and yi = ci /c is the fluid-phase mole fraction of component i. An alternative form, grouping together fluid-phase concentrations rather than intraparticle concentrations, is ∂n
i + ε ∂cˆi + ε ∂(vci) = εD ∂ c ∂yi ρb ᎏ (16-52) ᎏ ᎏ b ᎏ L ᎏ ∂t ∂t ∂z ∂z ∂z where, noting Eq. (16-4), cˆi is defined by εbcˆi = εci + (1 − ε)εpc
pi (16-53) For moving-bed processes, we add a term to Eq. (16-51) to obtain ∂yi ∂nˆ i ∂nˆ i ∂ci ∂(vci) ∂ ρb ᎏ + vs ᎏ + ε ᎏ + ᎏ − DL ᎏ c ᎏ = 0 (16-54) ∂t ∂z ∂t ∂z ∂z ∂z where vs is the solid-phase velocity (opposite in sign to v for a countercurrent process).



 













ENERGY BALANCE Many different forms of the energy balance have been used in fixedbed adsorption studies. The form chosen for a particular study depends on the process considered (e.g., temperature swing adsorption or pressure swing adsorption) and on the degree of approximation that is appropriate. The energy balance for a general fixed-bed process, ignoring dispersion, is

2hw(T − Tw) ∂us ∂(εbcuf) ∂(εvchf) ρb ᎏ + ᎏ + ᎏ = − ᎏᎏ (16-55) ∂t ∂z ∂t rc where hw is a heat transfer coefficient for energy transfer with the column wall and rc is the radius of the column. The second term of Eq. (16-55) combines contributions from both pore and extraparticle fluid. Thermodynamic paths are necessary to evaluate the enthalpy (or internal energy) of the fluid phase and the internal energy of the stationary phase. For gas-phase processes at low and modest pressures, the enthalpy departure function for pressure changes can be ignored and a reference state for each pure component chosen to be ideal gas at temperature T ref, and a reference state for the stationary phase (adsorbent plus adsorbate) chosen to be adsorbate-free solid at T ref. Thus, for the gas phase we have



hf =  yihfi =  yi href fi + i

i





T

T ref

C°pfi dT

uf = hf − P/c

(16-56) (16-57)

and for the stationary phase us = usol + nua ≈ usol + nha usol = u + ref sol



(16-58)

T

T ref

Cs dT

(16-59)

The enthalpy of the adsorbed phase ha is evaluated along a path for which the gas-phase components undergo temperature change from Tref to T and then are adsorbed isothermally, giving ni 1 ha =  xi hfi − ᎏ  qist(ni , nj, T) dni (16-60) n i 0 i st The isoteric heat of adsorption q i is composition-dependent, and the sum of integrals Eq. (16-60) is difficult to evaluate for multicomponent adsorption if the isosteric heats indeed depend on loading. Because each isosteric heat depends on the loadings of all components, the sum must be evaluated for a path beginning with clean adsorbent and ending with the proper loadings of all components. If the isosteric heat of adsorption is constant, as is commonly assumed, then the energy balance (Eq. 16-55) becomes

  

∂T

−ρ q ρ C +  n C°  + ε cC°  ᎏ ∂t b

s

i

pfi

i

b

pf

st i

b

i

∂ni ∂(εbP) ᎏ−ᎏ ∂t ∂t

∂T 2hw(T − Tw) + εvcC°pf ᎏ = − ᎏᎏ (16-61) ∂z rc where Eq. (16-51) with DL = 0 has been used. Equation (16-61) is a popular form of the energy balance for fixed-bed adsorption calculations. Often the first summation on the left-hand side, which involves gas-phase heat capacities, is neglected, or the gas-phase heat capacities are replaced by adsorbed-phase heat capacities. Nonisothermal liquid-phase processes may be driven by changes in feed temperature or heat addition or withdrawal through a column wall. For these, heats of adsorption and pressure effects are generally of less concern. For this case a suitable energy balance is ∂T ∂T 2hw(T − Tw) (ρbCs + εbcC°pf) ᎏ + εvcC°pf ᎏ = − ᎏᎏ ∂t ∂z rc

(16-62)

RATE AND DISPERSION FACTORS The performance of adsorption processes results in general from the combined effects of thermodynamic and rate factors. It is convenient to consider first thermodynamic factors. These determine the process performance in a limit where the system behaves ideally; i.e. without mass transfer and kinetic limitations and with the fluid phase in perfect

piston flow. Rate factors determine the efficiency of the real process in relation to the ideal process performance. Rate factors include heatand mass-transfer limitations, reaction kinetic limitations, and hydrodynamic dispersion resulting from the velocity distribution across the bed and from mixing and diffusion in the interparticle void space.

16-18

ADSORPTION AND ION EXCHANGE

TRANSPORT AND DISPERSION MECHANISMS Figure 16-9 depicts porous adsorbent particles in an adsorption bed with sufficient generality to illustrate the nature and location of individual transport and dispersion mechanisms. Each mechanism involves a different driving force and, in general, gives rise to a different form of mathematical result. Intraparticle Transport Mechanisms Intraparticle transport may be limited by pore diffusion, solid diffusion, reaction kinetics at phase boundaries, or two or more of these mechanisms together. 1. Pore diffusion in fluid-filled pores. These pores are sufficiently large that the adsorbing molecule escapes the force field of the adsorbent surface. Thus, this process is often referred to as macropore diffusion. The driving force for such a diffusion process can be approximated by the gradient in mole fraction or, if the molar concentration is constant, by the gradient in concentration of the diffusing species within the pores. 2. Solid diffusion in the adsorbed phase. Diffusion in pores sufficiently small that the diffusing molecule never escapes the force field of the adsorbent surface. In this case, transport may occur by an activated process involving jumps between adsorption sites. Such a process is often called surface diffusion or, in the case of zeolites, micropore or intracrystalline diffusion. The driving force for the process can thus be approximated by the gradient in concentration of the species in its adsorbed state. Phenomenologically, the process is not distinguishable from that of homogeneous diffusion that occurs inside a sorbent gel or in a pore-filling fluid that is immiscible with the external fluid. The generic term solid diffusion is used here to encompass the general traits of these physically different systems. 3. Reaction kinetics at phase boundaries. Rates of adsorption and desorption in porous adsorbents are generally controlled by mass transfer within the pore network rather than by the kinetics of sorption at the surface. Exceptions are the cases of chemisorption and affinity-adsorption systems used for biological separations, where the kinetics of bond formation can be exceedingly slow. Intraparticle convection can also occur in packed beds when the adsorbent particles have very large and well-connected pores. Although, in general, bulk flow through the pores of the adsorbent particles is only a small fraction of the total flow, intraparticle convection can affect the transport of very slowly diffusing species such as macromolecules. The driving force for convection, in this case, is the

,, ,, ,, ,, ,, ,, ,, 1

5

Flow

3

2 4

5

Wall

General scheme of adsorbent particles in a packed bed showing the locations of mass transfer and dispersive mechanisms. Numerals correspond to numbered paragraphs in the text: 1, pore diffusion; 2, solid diffusion; 3, reaction kinetics at phase boundary; 4, external mass transfer; 5, fluid mixing.

FIG. 16-9

pressure drop across each particle that is generated by the frictional resistance to flow experienced by the fluid as this flows through the packed bed [Rodrigues et al., Chem. Eng. Sci., 46, 2765 (1991); Carta et al., Sep. Technol., 2, 62 (1992); Frey et al., Biotechnol. Progr., 9, 273 (1993); Liapis and McCoy, J. Chromatogr., 599, 87 (1992)]. Intraparticle convection can also be significant when there is a total pressure difference between the center of the particle and the outside, such as is experienced in pressurization and depressurization steps of pressure swing adsorption or when more gas is drawn into an adsorbent to equalize pressure as adsorption occurs from the gas phase within a porous particle [Lu et al., AIChE J., 38, 857 (1992); Lu et al., Gas Sep. Purif., 6, 89 (1992)]. Extraparticle Transport and Dispersion Mechanisms Extraparticle mechanisms are affected by the design of the contacting device and depend on the hydrodynamic conditions outside the particles. 4. External mass transfer between the external surfaces of the adsorbent particles and the surrounding fluid phase. The driving force is the concentration difference across the boundary layer that surrounds each particle, and the latter is affected by the hydrodynamic conditions outside the particles. 5. Mixing, or lack of mixing, between different parts of the contacting equipment. This may occur through the existence of a velocity distribution or dead zones in a packed bed or through inefficient mixing in an agitated contactor. In packed-bed adsorbers, mixing is often described in terms of an axial dispersion coefficient whereby all mechanisms contributing to axial mixing are lumped together in a single effective coefficient. Heat Transfer Since adsorption is generally accompanied by the evolution of heat, the rate of heat transfer between the adsorbent particles and the fluid phase may be important. In addition, heat transfer can occur across the column wall in small diameter beds and is important in energy applications of adsorption. In gas adsorption systems, even with highly porous particles, the controlling heat transfer resistance is generally associated with extraparticle transport [Lee and Ruthven, Can. J. Chem. Eng., 57, 65 (1979)], so that the temperature within the particles is essentially uniform. In liquid-phase adsorption, intraparticle and extraparticle heat transfer resistances are generally comparable. However, in this case the heat capacity of the fluid phase is sufficiently high that temperature effects may be negligible except in extreme cases. General discussions of heat-transfer effects in adsorbents and adsorption beds are found in Suzuki (gen. refs., pp. 187–208 and pp. 275–290) and in Ruthven (gen. refs., pp. 189–198 and pp. 215–219). INTRAPARTICLE MASS TRANSFER The phenomenological aspects of diffusional mass transfer in adsorption systems can be described in terms of Fick’s law: ∂c Ji = −Di(ci) ᎏi (16-63) ∂x This expression can be used to describe both pore and solid diffusion so long as the driving force is expressed in terms of the appropriate concentrations. Although the driving force should be more correctly expressed in terms of chemical potentials, Eq. (16-63) provides a qualitatively and quantitatively correct representation of adsorption systems so long as the diffusivity is allowed to be a function of the adsorbate concentration. The diffusivity will be constant only for a thermodynamically ideal system, which is only an adequate approximation for a limited number of adsorption systems. Results for several individual mechanisms will now be considered. The equations that follow refer to local conditions within the contacting equipment that may apply to the average concentrations in the neighborhood of a single adsorbent particle. Generally, they apply to particles that can be approximated as spherical and of a uniform size and properties. An appropriately chosen mean particle size must be used in these equations when dealing with adsorbents having a broad particle size distribution. The appropriate average depends on the controlling mass-transfer mechanism. For intraparticle mass-transfer mechanisms, the volume or mass-average particle size usually provides the best prediction.

RATE AND DISPERSION FACTORS Pore Diffusion When fluid transport through a network of fluidfilled pores inside the particles provides access for solute adsorption sites, the diffusion flux can be expressed in terms of a pore diffusion coefficient Dpi as: ∂cpi (16-64) Ji = −εp Dpi ᎏ ∂r Dpi is smaller than the diffusivity in a straight cylindrical pore as a result of the random orientation of the pores, which gives a longer diffusion path, and the variation in the pore diameter. Both effects are commonly accounted for by a tortuosity factor τp such that Dpi = Di /τp. In principle, predictions of the tortuosity factor can be made if the pore structure, pore size, and shape distributions are known (see Dullien, Porous Media: Fluid Transport and Pore Structure, Academic Press, NY, 1979). In some cases, approximate prediction can be obtained from the following equations. Mackie and Meares, Proc. Roy. Soc., A232, 498 (1955): (2 − εp)2 τp = ᎏ εp Wakao and Smith, Chem. Eng. Sci., 17, 825 (1962): 1 τp = ᎏ εp Suzuki and Smith, Chem. Eng. J., 3, 256 (1972):

(16-65a)

(16-65b)

τp = εp + 1.5(1 − εp)

(16-65c)

In practice, however, the predictive value of these equations is rather uncertain, and vastly different results are obtained from each. All of them, on the other hand, predict that τp increases as the porosity decreases. For catalyst particles, Satterfield (Heterogeneous Catalysis in Practice, McGraw-Hill, 1980) recommends the use of a value of τp = 4 when no other information is available, and this can be used for many adsorbents. In general, however, it is more reliable to treat the tortuosity as an empirical constant that is determined experimentally for any particular adsorbent. For adsorbent materials, experimental tortuosity factors generally fall in the range 2–6 and generally decrease as the particle porosity is increased. Higher apparent values may be obtained when the experimental measurements are affected by other resistances, while values much lower than 2 generally indicate that surface or solid diffusion occurs in parallel to pore diffusion. Ruthven (gen. refs.) summarizes methods for the measurement of effective pore diffusivities that can be used to obtain tortuosity factors by comparison with the estimated pore diffusion coefficient of the adsorbate. Molecular diffusivities can be estimated with the methods in Sec. 6. For gas-phase diffusion in small pores at low pressure, the molecular mean free path may be larger than the pore diameter, giving rise to Knudsen diffusion. Satterfield (Mass Transfer in Heterogeneous Catalysis, MIT, Cambridge, MA, 1970, p. 43), gives the following expression for the pore diffusivity: 1 3 πMri 1/2 1 −1 Dpi = ᎏ ᎏ ᎏ +ᎏ (16-66) τp 4rpore 2RT Di where rpore is the average pore radius, T the absolute temperature, and Mri the molecular weight. For liquid-phase diffusion of large adsorbate molecules, when the ratio λm = rm /rpore of the molecule radius rm to the pore radius is significantly greater than zero, the pore diffusivity is reduced by steric interactions with the pore wall and hydrodynamic resistance. When λm < 0.2, the following expressions derived by Brenner and Gaydos [J. Coll. Int. Sci., 58, 312 (1977)] for a hard sphere molecule (a particle) diffusing in a long cylindrical pore, can be used













9 D Dpi = ᎏi (1 − λm)−2 1 + ᎏ λm ln λm − 1.539λm τp 8

(16-67)

rm is the Stokes-Einstein radius of the solute that can be determined from the free diffusivity as

16-19

κT rm = ᎏ (16-68) 6πµDi where κ is the Boltzmann constant. When λm > 0.2, the centerline approximation [Anderson and Quinn, Biophys. J., 14, 130, (1974)] can be used instead of Eq. (16-67) D Dpi = ᎏi (1 − 2.1044λm + 2.089λm3 − 0.984λm5 ) × 0.865 (16-69) τp The 0.865 factor is used to match this equation to the Brenner and Gaydos expression for λm = 0.2. In these cases, the pore concentration Cpi is related to the external concentration Ci by the partition ratio (1 − λm)2. Solid Diffusion In the case of pore diffusion discussed above, transport occurs within the fluid phase contained inside the particle; here the solute concentration is generally similar in magnitude to the external fluid concentration. A solute molecule transported by pore diffusion may attach to the sorbent and detach many times along its path. In other cases, attachment can be essentially permanent, but in both cases, only detached molecules undergo transport. In contrast, the following four instances illustrate cases where diffusion of adsorbate molecules occurs in their adsorbed state within phases that are distinct from the pore fluid: 1. Movement of mobile adsorbed solute molecules along pore surfaces, without detaching 2. Transport in a homogeneously dissolved state, as for a neutral molecule inside a sorbent gel or in a pore filled with a liquid which is immiscible with the external fluid 3. Ion transport in charged ion-exchange resins 4. Advance of an adsorbate molecule from one cage to another within a zeolite crystal In these cases, the diffusion flux may be written in terms of the adsorbed solute concentration as ∂n Ji = −ρp Dsi ᎏi (16-70) ∂r The diffusion coefficient in these phases Dsi is usually considerably smaller than that in fluid-filled pores; however, the adsorbate concentration is often much larger. Thus, the diffusion rate can be smaller or larger than can be expected for pore diffusion, depending on the magnitude of the fluid/solid partition coefficient. Numerical values for solid diffusivities Dsi in adsorbents are sparse and disperse. Moreover, they may be strongly dependent on the adsorbed phase concentration of solute. Hence, locally conducted experiments and interpretation must be used to a great extent. Summaries of available data for surface diffusivities in activated carbon and other adsorbent materials and for micropore diffusivities in zeolites are given in Ruthven, Yang, Suzuki, and Karger and Ruthven (gen. refs.). Surface diffusivities are generally strongly dependent on the fractional surface coverage and increase rapidly at surface coverage greater than 80 percent [see for example Yang et al., AIChE J., 19, 1052 (1973)]. For estimation purposes, the correlation of Sladek et al. [Ind. Eng. Chem. Fundam., 13, 100 (1974)] can be used to predict surface diffusivities for gas-phase adsorption on a variety of adsorbents. Zeolite crystallite diffusivities for sorbed gases range from 10−7 to 10−14 cm2/s. These diffusivities generally show a strong increase with the adsorbate concentration that is accounted for by the Darken thermodynamic correction factor d ln a Dsi = D0i ᎏi (16-71) d ln ni where D0i is the corrected diffusivity, ai the thermodynamic activity of the species in the adsorbed phase, and ni the adsorbed phase solute concentration. Corrected diffusivities D0i calculated according to this equation are often found to be essentially independent of concentration. If the adsorption equilibrium isotherm obeys the Langmuir equation [Eq. (16-13)], Eq. (16-71) yields: n −1 Dsi = D0i 1 − ᎏis (16-72) ni The effect of temperature on diffusivities in zeolite crystals can be expressed in terms of the Eyring equation (see Ruthven, gen. refs.).





16-20

ADSORPTION AND ION EXCHANGE

In ion-exchange resins, diffusion is further complicated by electrical coupling effects. In a system with M counterions, diffusion rates are described by the Nernst-Planck equations (Helfferich, gen. refs.). Assuming complete Donnan exclusion, these equations can be written as: M−1 ∂zjnj 1 Ji = −ρp ᎏᎏ  D (16-73)
i, j ᎏ ∂r zi j = 1 with

D DM)z 2i ni
i(D
j −
D
i, j = − ᎏᎏ M
k zk2 nk k=1 D

(16-74a)

DM)z 2i ni D
i(D
i −
Di,i =
Di − ᎏᎏᎏ
M
k z k2 nk k=1 D

(16-74b)

which are dependent on the ionic self diffusivities
Di of the individual species. As a qualitative rule, ionic diffusivities of inorganic species in crosslinked polystyrene-DVB ion-exchange resins compared with those in water are 1:10 for monovalent ions, 1:100 for divalent ions, and 1:1000 for trivalent ions. Table 16-8 shows typical ionic diffusivities of inorganic ions in cation and anion exchange resins; larger organic ions, however, can have ionic diffusivities much smaller than inorganic ions of the same valence [see, for example, Jones and Carta, Ind. Eng. Chem. Research, 32, 117 (1993)]. For mixtures of unlike ions (the usual case), the apparent diffusivity will be intermediate between these values because of the electrical coupling effect. For a system with two counterions A and B, with charge zA and zB, Eqs. (16-73) and (16-74) reduce to: D
AD
B[zA2 nA + zB2 nB] ∂nA ∂nA JA = −ρpD ᎏ (16-75)
A, B ᎏ = −ρp ᎏᎏᎏ 2 z
AnA + zB2 D
BnB ∂r ∂r AD which shows that the apparent diffusivity D
A,B varies between DA when the ionic fraction of species A in the resin is very small
and
DB when the ionic fraction of A in the resin approaches unity, indicating that the ion present in smaller concentration has the stronger effect on the local interdiffusion rate. Combined Pore and Solid Diffusion In porous adsorbents and ion-exchange resins, intraparticle transport can occur with pore and solid diffusion in parallel. The dominant transport process is the faster one, and this depends on the relative diffusivities and concentrations in the pore fluid and in the adsorbed phase. Often, equilibrium between the pore fluid and the solid phase can be assumed to exist locally at each point within a particle. In this case, the mass-transfer flux is expressed by: ∂cpi dne ∂cpi Ji = − εp Dpi + ρp Dsi ᎏi ᎏ = −Dei(cpi) ᎏ (16-76) dci ∂r ∂r e where dn i /dci is the derivative of the adsorption isotherm and it has been assumed that at equilibrium cpi = ci. This equation suggests that





in such an adsorbent, pore and solid diffusivities can be obtained by determining the apparent diffusivity Dei for conditions of no adsorption (dnei /dci = 0) and for conditions of strong adsorption, where dn ei /dci is large. If the adsorption isotherm is linear over the range of experimental measurement: Dei = εp Dpi + ρp Ki Dsi

EXTERNAL MASS TRANSFER Because of the complexities encountered with a rigorous treatment of the hydrodynamics around particles in industrial contactors, mass transfer to and from the adsorbent is described in terms of a masstransfer coefficient kf. The flux at the particle surface is: Ni = kf (ci − cis)

(16-78)

s i

where ci and c are the solute concentrations in the bulk fluid and at the particle surface, respectively. kf can be estimated from available correlations in terms of the Sherwood number Sh = kf dp /Di and the Schmidt number Sc = ν/Di. For packed-bed operations, the correlations in Table 16-9 are recommended. A plot of these equations is given in Fig. 16-10 for representative ranges of Re and Sc with ε = 0.4. External mass-transfer coefficients for particles suspended in agitated contactors can be estimated from equations in Levins and Glastonbury [Trans. Instn. Chem. Eng., 50, 132 (1972)] and Armenante and Kirwan [Chem. Eng. Sci., 44, 2871 (1989)]. AXIAL DISPERSION IN PACKED BEDS The axial dispersion coefficient [cf. Eq. (16-51)] lumps together all mechanisms leading to axial mixing in packed beds. Thus, the axial dispersion coefficient must account not only for molecular diffusion and convective mixing but also for nonuniformities in the fluid velocity across the packed bed. As such, the axial dispersion coefficient is best determined experimentally for each specific contactor. The effects of flow nonuniformities, in particular, can be severe in gas systems when the ratio of bed-to-particle diameters is small; in liquid systems when viscous fingering occurs as a result of large viscosity gradients in the adsorption bed; when very small particles (<50 µm) are used, such as in high performance liquid chromatography systems; and in large-diameter beds. A lower bound of the axial

TABLE 16-8 Self Diffusion Coefficients in Polystyrene-divinylbenzene Ion Exchangers (units of 10-7 cm2/s)* 0.3 °C

Temperature Crosslinking, %

4

6

8

25 °C 12

16

4

6

8

12

16

Cation exchangers (sulfonated): Dowex 50 Na+ Cs+ Ag+ Zn2+ La3+

6.7

3.4 6.6 2.62 0.21 0.03

0.30

1.15

0.66 1.11 1.00 0.03 0.002

14.1

9.44 13.7 6.42 0.63 0.092

2.40 3.10 2.75 0.14 0.005

2.04

0.26

Anion exchangers (dimethyl hydroxyethylamine): Dowex 2 Cl− Br− I− BrO −3 WO 2− 4 PO 3− 4

(1.8)

1.25 1.50 0.35 1.76 0.60 0.16

0.63

(16-77)

Thus, a plot of the apparent diffusivity versus the linear adsorption equilibrium constant should be linear so long as Dpi and Dsi remain constant. In a particle having a bidispersed pore structure comprising spherical adsorptive subparticles of radius rs forming a macroporous aggregate, separate flux equations can be written for the macroporous network in terms of Eq. (16-64) and for the subparticles themselves in terms of Eq. (16-70) if solid diffusion occurs.

0.06

*Data from Boyd and Soldano, J. Am. Chem. Soc., 75, 6091 (1953).

(4.3)

3.54 3.87 1.33 4.55 1.80 0.57

RATE AND DISPERSION FACTORS

16-21

Recommended Correlations for External Mass Transfer Coefficients in Adsorption Beds (Re = evdp /n, Sc = n/D)

TABLE 16-9

Equation

Re

 

0.5

Re Sh = 1.15 ᎏ Sc0.33 ε Sh = 2.0 + 1.1 Re0.6Sc0.33

Phase

Re > 1

Ref.

Gas/liquid

Carberry, AIChE J., 6, 460 (1960)

3 < Re < 10

Gas/liquid

Wakao and Funazkri, Chem. Eng. Sci., 33, 1375 (1978)

Re < 40

Liquid

Kataoka et al., J. Chem. Eng. Japan, 5, 132 (1972)

1.09 Sh = ᎏ Re0.33Sc0.33 ε

0.0015 < Re < 55

Liquid

Wilson and Geankoplis, Ind. Eng. Chem. Fundam., 5, 9 (1966)

0.25 Sh = ᎏ Re0.69Sc0.33 ε

55 < Re < 1050

Liquid

Wilson and Geankoplis, Ind. Eng. Chem. Fundam., 5, 9 (1966)

1−ε Sh = 1.85 ᎏ ε





4

0.33

Re0.33Sc0.33

dispersion coefficient can be estimated for well-packed beds from correlations that follow. Neglecting flow nonuniformities, the contributions of molecular diffusion and turbulent mixing arising from stream splitting and recombination around the sorbent particles can be considered additive [Langer et al., Int. J. Heat and Mass Transfer, 21, 751 (1978)]; thus, the axial dispersion coefficient DL is given by: (Re)(Sc) dpv DL (16-79) ᎏ = γ1 + γ2 ᎏ = γ1 + γ2 ᎏ ε Di Di or, in terms of a particle-based Peclet number (Pe = d pv/DL), by: γ1ε 1 (16-80) ᎏ = ᎏ + γ2 Pe (Re)(Sc) The first term in Eqs. (16-79) and (16-80) accounts for molecular diffusion, and the second term accounts for mixing. For the first term, Wicke [Ber. Bunsenges, 77, 160 (1973)] has suggested: γ1 = 0.45 + 0.55ε

(16-81)

which, for typical void fractions, ε = 0.35 − 0.5 gives γ1 = 0.64 − 0.73 (Ruthven, gen. refs.). Expressions for the axial mixing term, γ2 in Eq. (16-79) are given in Table 16-10. The expression of Wakao and Funazkri includes an axial diffusion term, γ1, that varies from 0.7 for nonporous particles to 20/ε, depending on the intraparticle masstransfer mechanism. For strongly adsorbed species, Wakao and Funazkri suggest that the effective axial dispersion coefficient is much

larger than that predicted on the basis of nonporous, nonadsorbing particles. The Gunn expression includes a term σv2 accounting for deviations from plug flow. σv2 is defined as the dimensionless variance of the distribution of the ratio of velocity to average velocity over the cross section of the bed. The parameter values included in this equation are valid for spherical particles. Values for nonspherical particles can be found in the original reference. Figure 16-11 compares predicted values of DL /Di for σv = 0 and ε = 0.4 with Sc = 1 (gases at low pressure), and Sc = 1000 (liquids), based on the equations in Table 16-10. Correlations for axial dispersion in beds packed with very small particles (<50 µm) that take into account the holdup of liquid in a bed are discussed by Horvath and Lin [J. Chromatogr., 126, 401 (1976)]. RATE EQUATIONS Rate equations are used to describe interphase mass transfer in batch systems, packed beds, and other contacting devices for sorptive processes and are formulated in terms of fundamental transport properties of adsorbent and adsorbate. General Component Balance For a spherical adsorbent particle: ∂cpi 1 ∂ ∂n εp ᎏ + ρp ᎏi = ᎏ2 ᎏ (−r 2 Ni) (16-82) ∂t ∂t r ∂r For particles that have no macropores, such as gel-type ion-exchange resins, or when the solute holdup in the pore fluid is small, εp may be

Sherwood number correlations for external mass-transfer coefficients in packed beds for ε = 0.4 (adapted from Suzuki, gen. refs.).

FIG. 16-10

16-22

ADSORPTION AND ION EXCHANGE

TABLE 16-10

Coefficients for Axial Dispersion Correlations in Packed Beds Based on Eq. (16-79)

γ1 0.73 Nonporous particles: 0.7 Porous particles: ≤20/ε 1

0.714

γ2 13γ1ε 0.5 1 + ᎏ ReSc



Ref.

−1



Edwards and Richardson, Chem. Eng. Sci., 23, 109 (1968)

0.5

Wakao and Funazkri, Chem. Eng. Sci., 33, 1375 (1978)

3 π2ε(1 − ε) ᎏ ε + ᎏᎏ ln (ReSc) 6 4

Koch and Brady, J. Fluid Mech., 154, 399 (1985)

1 σ v2 −ᎏ ᎏ + (1 + σv2) γ(1 − p)2 + γ 2 p(1 − p)3 e γ p(1 − p) − 1 2 with γ = 0.043 ReSc/(1 − ε) p = 0.33 exp (−24/Re) + 0.17





taken as zero. Ignoring bulk flow terms, the fluxes Ni and Ji are equal. In this case, coupling the component balance with the flux expressions previously introduced gives the rate equations in Table 16-11. Typical boundary conditions are also included in this table. Linear Driving Force Approximation Simplified expressions can also be used for an approximate description of adsorption in terms of rate coefficients for both extraparticle and intraparticle mass transfer controlling. As an approximation, the rate of adsorption on a particle can be written as: ∂nˆ i (16-83) ᎏ = kf(ni, ci) ∂t where k is a rate coefficient, and the function f(ni, ci) is a driving force relationship. The variables kc and kn are used to denote rate coefficients based on fluid-phase and adsorbed-phase concentration driving forces, respectively. Commonly used forms of this rate equation are given in Table 16-12. For adsorption bed calculations with constant separation factor systems, somewhat improved predictions are obtained using correction factors ψs and ψp defined in Table 16-12 is the partition ratio defined by Eq. (16-25). The linear driving force (LDF) approximation is obtained when the driving force is expressed as a concentration difference. It was originally developed to describe packed-bed dynamics under linear equilibrium conditions [Glueckauf, Trans. Far. Soc., 51, 1540 (1955)]. This form is exact for a nonlinear isotherm only when external mass transfer is controlling. However, it can also be used for nonlinear sys-

FIG. 16-11



Gunn, Chem. Eng. Sci., 2, 363 (1987)

tems with pore or solid diffusion mechanisms as an approximation, since it provides qualitatively correct results. Alternate driving force approximations, item 2B in Table 16-12, for solid diffusion, and item 3B in Table 16-12, for pore diffusion, provide somewhat more accurate results in constant pattern packed-bed calculations with pore or solid diffusion controlling for constant separation factor systems. The reaction kinetics approximation is mechanistically correct for systems where the reaction step at pore surfaces or other fluid-solid interfaces is controlling. This may occur in the case of chemisorption on porous catalysts and in affinity adsorbents that involve very slow binding steps. In these cases, the mass-transfer parameter k is replaced by a second-order reaction rate constant ka. The driving force is written for a constant separation factor isotherm (column 4 in Table 16-12). When diffusion steps control the process, it is still possible to describe the system by its apparent second-order kinetic behavior, since it usually provides a good approximation to a more complex exact form for single transition systems (see “Fixed Bed Transitions”). Combined Intraparticle Resistances When solid diffusion and pore diffusion operate in parallel, the effective rate is the sum of these two rates. When solid diffusion predominates, mass transfer can be represented approximately in terms of the LDF approximation, replacing kn in column 2 of Table 16-12 with 15ψsDsi 15(1 − ε)ψpεp Dpi knc = ᎏ + ᎏᎏ r p2 Λr p2

Axial dispersion coefficient correlations for well-packed beds for ε = 0.4.

(16-84)

RATE AND DISPERSION FACTORS TABLE 16-11

16-23

Rate Equations for Description of Mass Transfer in Spherical Adsorbent Particles Mechanism

Flux equation

A. Pore diffusion

Rate equation ∂cpi ∂cpi ∂ni 1 ∂ εp ᎏ + ρp ᎏ = ᎏ2 ᎏ εpDpir 2 ᎏ ∂t ∂t ∂r r ∂r



16-64



(∂cpi /∂r)r = 0 = 0, (εpDpi∂cpi /∂r)r = r p = kf (ci − cpi|r = r ) p

or (cpi)r = r = ci for no external resistance p

B. Solid diffusion

∂ni 1 ∂ ∂ni ᎏ = ᎏ2 ᎏ Dsir 2 ᎏ ∂t ∂r r ∂r



16-70



(∂ni /∂r)r = 0 = 0, (ρpDsi∂ni /∂r)r = rp = kf (ci − cis) or (ni)r = rp = nie (ci) for no external resistance C. Parallel pore and solid diffusion (local equilibrium between pore and adsorbed phase)

dnie ∂cpi

1 ∂

ᎏ = ᎏ ᎏ r ε D ε + ρ ᎏ dc  ∂t r ∂r

16-76

p

p

2

2

i

p

pi

dnie ∂cpi + ρpDsi ᎏ ᎏ dci ∂r

 

(∂cpi /∂r)r = 0 = 0, [(εpDpi + ρpDsi dn /dci)∂cpi /∂r]r = rp = kf (ci − cpi|r = r ) e i

p

or (cpi)r = r = ci for no external resistance p

D. Diffusion in bidispersed particles (no external resistance)

∂ni 1 ∂ ∂ni ᎏ = ᎏ2 ᎏ Dsiρ2 ᎏ , (∂ni /∂ρ)ρ = 0 = 0, (ni)ρ = rs = nie(cpi) ∂t ρ ∂ρ ∂ρ



16-64 and 16-70

3 ni(r,t) = ᎏ3
rs



rs



ρ2ni dρ

0

∂n ∂cpi
i = 1 ∂ ε D r2 ∂cpi , (∂c /∂r) = 0, (c ) εp ᎏ + ρp ᎏ ᎏ2 ᎏ p pi ᎏ pi r=0 pi r = rp = ci ∂t ∂t ∂r r ∂r



TABLE 16-12



Expressions for Rate Coefficient k and Driving Force Relationships for Eq. 16-83 Mechanism

Expression for rate coefficient, k

1. External film kf a 3(1 − ε)kf kc = ᎏ = ᎏᎏ ρb ρb rp c − cie

A. Linear driving force (LDF)

ref Rcref n
i /n ci − ᎏᎏ ref 1 − (R − 1)n
i /n

LDF for constant R

2. Solid diffusion

3. Pore diffusion

4. Reaction kinetics

15ψsDsi kn = ᎏ r 2p

15ψp(1 − ε)εpDpi kn = ᎏᎏ Λr 2p

ka

nie −
ni

nei −
ni

—

nrefci /cref −n ᎏᎏ
i R + (R − 1)ci /cref

nrefci /cref −
ni ᎏᎏ R + (R − 1)ci /cref

—

Correction factors ψ for constant R

—

0.894 ᎏᎏ 1 − 0.106R0.5

0.775 ᎏᎏ 1 − 0.225R0.5

—

B. Alternate driving force for constant R

—

ne2 n2i i −
ᎏ 2n i


nei −
ni ᎏᎏᎏ ref 0.5 [1 − (R − 1)n
i /n ]

—

—

0.590 ᎏᎏ 1 − 0.410R0.5

0.548 ᎏᎏ 1 − 0.452R0.5

—

Correction factors ψ for alternate driving force

C. Reaction kinetics for constant R

—

—

—

ref ci(nref −
ni) − Rn
i(c − ci) ᎏᎏᎏ 1−R

References: 1A. Beaton and Furnas, Ind. Eng. Chem., 33, 1500 (1941); Michaels, Ind. Eng. Chem., 44, 1922 (1952) 2A,3A. Glueckauf and Coates, J. Chem. Soc., 1315 (1947); Trans. Faraday Soc., 51, 1540 (1955); Hall et al., Ind. Eng. Chem. Fundam., 5, 212 (1966) 2B. Vermeulen, Ind. Eng. Chem., 45, 1664 (1953) 3B. Vermeulen and Quilici, Ind. Eng. Chem. Fundam., 9, 179 (1970) 4C. Hiester and Vermeulen, Chem. Eng. Progr., 48, 505 (1952)

When pore diffusion predominates, use of column 3 in Table 16-12 is preferable, with kcn replacing kn. For particles with a bidispersed pore structure, the masstransfer parameter kn in the LDF approximation (column 2 in Table 16-12) can be approximated by the series-combination of resistances as: r 2s 1 1 Λr p2 ᎏc = ᎏ ᎏᎏ + ᎏ kn bs 15(1 − ε)ψpεpDpi 15ψs Dsi





(16-85)

where bs is a correction to the driving force that is described below. In the limiting cases where the controlling resistance is diffusion through

the particle pores or diffusion within the subparticles, the rate coefficients kn = 15(1 − ε)ψpεp Dpi /Λr p2 and kn = 15ψsDsi /r s2 are obtained. Overall Resistance With a linear isotherm (R = 1), the overall mass transfer resistance is the sum of intraparticle and extraparticle resistances. Thus, the overall LDF coefficient for use with a particleside driving force (column 2 in Table 16-12) is: 1 Λrp 1 (16-86) ᎏo = ᎏᎏ + ᎏc kn 3(1 − ε)kf kn ρb 1 ρbrp or (16-87) ᎏo = ᎏᎏ + ᎏc kc 3(1 − ε)kf Λkn for use with a fluid-phase driving force (column 1 in Table 16-12).

16-24

ADSORPTION AND ION EXCHANGE

In either equation, knc is given by Eq. (16-84) for parallel pore and surface diffusion or by Eq. (16-85) for a bidispersed particle. For nearly linear isotherms (0.7 < R < 1.5), the same linear addition of resistance can be used as a good approximation to predict the adsorption behavior of packed beds, since solutions for all mechanisms are nearly identical. With a highly favorable isotherm (R → 0), however, the rate at each point is controlled by the resistance that is locally greater, and the principle of additivity of resistances breaks down. For approximate calculations with intermediate values of R, an overall transport parameter for use with the LDF approximation can be calculated from the following relationship for solid diffusion and film resistance in series Λrp r 2p bs ᎏᎏ + ᎏ = ᎏo 3(1 − ε)kf 15ψs Dsi kn bf Λ =ᎏ (16-88) ρb kco bs and bf are correction factors that are given by Fig. 16-12 as a function of the separation factor R and the mechanism parameter 10ψsDsi Λ 1 Ψ=ᎏ (16-89) ᎏ Di Sh Axial Dispersion Effects In adsorption bed calculations, axial dispersion effects are typically accounted for by the axial diffusionlike term in the bed conservation equations [Eqs. (16-51) and (16-52)]. For nearly linear isotherms (0.5 < R < 1.5), the combined effects of axial dispersion and mass-transfer resistances on the adsorption behavior of packed beds can be expressed approximately in terms of an apparent rate coefficient kc for use with a fluid-phase driving force (column 1, Table 16-12): 1 ρb DL 1 + ᎏo (16-90) ᎏ=ᎏᎏ ε v2 kc kc which extends the linear addition principle to combined axial dispersion and mass-transfer resistances. Even for a highly nonlinear isotherm (R = 0.33), the linear addition principle expressed by this equation provides a useful approximation except in the extreme case of low mass-transfer resistance and large axial dispersion, when DLρp kco /v2ε >> 5 [Garg and Ruthven, Chem. Eng. Sci., 30, 1192 (1975)]. However, when the isotherm is irreversible (R → 0), the linear addition principle breaks down and axial dispersion has to be taken into account by explicit models (see “Fixed Bed Transitions”). Rapid Adsorption-Desorption Cycles For rapid cycles with particle diffusion controlling, when the cycle time tc is much smaller than the time constant for intraparticle transport, the LDF approximation becomes inaccurate. The generalized expression ∂n ˆi ni) (16-91) ᎏ = Ωkn(nei −
∂t can be used for packed-bed calculations when the parameter Ω is defined to be a function of the cycle time such that the amount of solute adsorbed and desorbed during a cycle is equal to that obtained by solution of the complete particle diffusion equations. Graphical and analytical expressions for Ω in the case of a single particle, usable for very short beds, are given by Nakao and Suzuki [ J. Chem. Eng. Japan, 16, 114 (1983)] and Carta [Chem. Eng. Sci., 48, 622 (1993)]. With equal adsorption and desorption times, ta = td = tc /2, Ω approaches the value π2/15 for long cycle times and the asymptote Ω = 1.877/ t
k c
n for short cycle times [Alpay and Scott, Chem. Eng. Sci., 47, 499 (1992)]. However, other results by Raghavan et al. [Chem. Eng. Sci., 41, 2787 (1986)] indicate that a limiting constant value of Ω (larger than 1) is obtained for very short cycles, when calculations are carried out for beds of finite length. Determination of Controlling Rate Factor The most important physical variables determining the controlling dispersion factor are particle size and structure, flow rate, fluid- and solid-phase diffusivities, partition ratio, and fluid viscosity. When multiple resistances and axial dispersion can potentially affect the rate, the spreading of a concentration wave in a fixed bed can be represented approximately

in terms of the single rate parameter k. In customary separationprocess calculations, the height of an adsorption bed can be calculated approximately as the product of the number of transfer units times the height of one fluid-phase transfer unit (HTU). The HTU is related to the LDF rate parameters kc and kn by: εv εv HTU = ᎏ = ᎏ ρbkc Λkn

(16-92)

Figure 16-13 is a plot of the dimensionless HTU (htu = HTU/dp) multiplied times the correction factor bf (between 1 and 2) as a function of the dimensionless velocity (Re)(Sc) = εvdp /D and a ratio of the controlling diffusivity to the fluid-phase diffusivity, generated on the basis of results of Vermeulen et al. (gen. ref.) using typical values of the individual physical factors likely to be found in adsorption beds. This figure can be used to determine the controlling rate factor from a knowledge of individual physical parameters. If fluid-side effects control, the dimensionless HTU is given by the bottom curve (dotted for gas and solid for liquid-phase systems). If particle-side diffusivities control, the dimensionless HTU is given by a point above the lower envelope on the appropriate diffusional contour (through the ψs, the contour value depends slightly on the separation factor R). If pore and solid diffusion occur in parallel, the reciprocal of the HTU is the sum of the reciprocals of the HTU values for the two mechanisms. Near the intersections of the diffusional contours with the envelope, the dimensionless HTU is the sum of the HTU values for fluid-side and particle-side resistances. Example 7: Estimation of Rate Coefficient An adsorption bed is used to remove methane from a methane-hydrogen mixture at 10 atm (abs.) (10.1 bar) and 25°C (298 K), containing 10 mol % methane. Activated carbon particles having a mean diameter dp = 0.17 cm, a surface area A = 1.1 × 107 cm2/s, a bulk density ρb = 0.509 g/cm3, a particle density ρp = 0.777 g/cm3, and a skeletal density ρs = 2.178 g/cm3 is used as the adsorbent. Based on data of Grant et al. [AIChE J., 8, 403 (1962)], adsorption equilibrium is represented by n = 2.0 × 10−3KApA/(1 + KApA) mol/g adsorbent, with KA = 0.346 atm−1. Estimate the rate coefficient and determine the controlling rate factor for a superficial velocity of 30 cm/s. 1. The intraparticle void fraction is εp = (0.777−1 − 2.178−1)/(0.777−1) = 0.643 and the extraparticle void fraction is ε = (0.509−1 − 0.777−1)/(0.509−1) = 0.345. The pore radius is estimated from rp = 2εp /(Aρp) = 1.5 × 10−7 cm. 2. The fluid phase diffusivity is D = 0.0742 cm2/s. The pore diffusivity is estimated from Eq. (16-66) with a tortuosity factor τp = 4; Dp = 1.45 × 10−3 cm2/s. 3. The fluid-side mass transfer coefficient is estimated from Fig. 16-10. For these conditions, ν = 0.108 cm2/s, Re = 30 × 0.17/0.108 = 47, and Sc = 0.108/0.0742 = 1.5. From Fig. 16-10 or equations in Table 16-9, Sh ∼ 13. 4. The isotherm parameters based on the feed concentration are R = 1/ (1 + KApA) = 0.4 and Λ = 0.509 × 7.68 × 10−4/4.09 × 10−5 = 9.56. For pore diffusion, ψp = 0.961 from item 3A in Table 16-12. Thus, (1 − ε)ψpεpDp /D = (1 − 0.345) × 0.961 × 0.643 × 1.45 × 10−3/0.0742 = 7.9 × 10−3. From Fig. 16-13 at ReSc = 69, b htu ∼ 150. b is found from Fig. 16-12. However, since the mechanism parameter Ψ is very small, b ∼ 1. Thus, kn = εv/(htu dpΛ) = 0.12 s−1. This value applies to the driving force nei −
ni. Since pore diffusion is dominant, this value is very close to the value kn = 15(1 − ε)ψpεpDp /(rp2Λ) = 0.13 s−1 obtained directly from Table 16-12. It should be noted that surface diffusion is neglected in this estimation. Its occurrence could significantly affect the overall mass transfer rate (see Suzuki, gen. refs., pp. 70–85). Example 8: Estimation of Rate Coefficient Estimate the rate coefficient for flow of a 0.01-M water solution of NaCl through a bed of cation exchange particles in hydrogen form with ε = 0.4. The superficial velocity is 0.2 cm/s and the temperature is 25°C. The particles are 600 µm in diameter, and the diffusion coefficient of sodium ion is 1.2 × 10−5 cm2/s in solution and 9.4 × 10−7 cm2/s inside the particles (cf. Table 16-8). The bulk density is 0.7 g dry resin/cm3 of bed, and the capacity of the resin is 4.9 mequiv/g dry resin. The mass action equilibrium constant is 1.5. 1. Estimate the fluid-side mass transfer coefficient; Re = εvdp /ν = 0.2 × 0.06/0.00913 = 1.3, Sc = ν/D = 0.00913/1.2 × 10−5 = 761. From Fig. 16-10 or Table 16-9, Sh ∼ 23. Thus, kf = D Sh/dp = 4.5 × 10−3 cm/s. 2. From the equilibrium constant, R = 1/KNa, H = 0.67. Thus, from Table 16-12, item 2A, ψs = 0.979. Using nref = 4.9 mequiv/g and cref = 0.01 mmole/cm3, Λ = ρbnref/cref = 343. Thus, ψsDsΛ/D = 26 and the external mass-transfer resistance is controlling (cf. Fig. 16-13). 3. The rate coefficient for use with a fluid-phase driving force is kc = 3(1 − ε)kf /(ρbrp) = 0.39 cm3/(g⋅s).

BATCH ADSORPTION

16-25

2.5 bs: solid resistance predominates

bf: fluid resistance predominates

2.0

R=0

bs or bf

1.5

0.1 1.25 0.2 0.3 0.4 0.5

1.1

1.0 0.1

0.2

0.3 0.4 0.5

1

2

3

4

5

10

10(1 – ε)Ψp εp Dp 10ΛΨs Ds Ψ= or D • Sh D • Sh Correction factors for addition of mass-transfer resistances, relative to effective overall solid phase or fluid phase rates, as a function of the mechanism parameter. Each curve corresponds to both bs and bf over its entire range.

FIG. 16-12

Effect of ReSc group, distribution ratio, and diffusivity ratio on height of a transfer unit. Dotted lines for gas and solid lines for liquid-phase systems.

FIG. 16-13

BATCH ADSORPTION In this section, we consider the transient adsorption of a solute from a dilute solution in a constant-volume, well-mixed batch system or, equivalently, adsorption of a pure gas. The solutions provided can approximate the response of a stirred vessel containing suspended adsorbent particles, or that of a very short adsorption bed. Uniform, spherical particles of radius rp are assumed. These particles, initially of uniform adsorbate concentration, are assumed to be exposed to a step change in concentration of the external fluid.

In general, solutions are obtained by coupling the basic conservation equation for the batch system, Eq. (16-49) with the appropriate rate equation. Rate equations are summarized in Table 16-11 and 16-12 for different controlling mechanisms. Solutions are provided for external mass-transfer control, intraparticle diffusion control, and mixed resistances for the case of constant Vf and Fv, in = Fv, out = 0. The results are in terms of the fractional approach to equilibrium F = (nˆ i − nˆ 0′i )/(ˆn∞i − nˆ 0′i ), where nˆ 0′i and nˆ ∞i are

16-26

ADSORPTION AND ION EXCHANGE

the initial and ultimate solute concentrations in the adsorbent. The solution concentration is related to the amount adsorbed by the material balance ci = c0i − (nˆ i − nˆ 0′i )Ms /Vf . Two general cases are considered: (1) adsorption under conditions of constant or nearly constant external solution concentration (equivalent to infinite fluid volume); and (2) adsorption in a batch with finite volume. In the latter case, the fluid concentration varies from c0i to c∞i when equilibrium is eventually attained. Λ∞ = (c0i − c∞i )/c0i = Ms(ˆn ∞i − nˆ 0i )/(Vf c0i ) is a partition ratio that represents the fraction of adsorbate that is ultimately adsorbed. It determines which general case should be considered in the analysis of experimental systems. Generally, when Λ∞ ≥ 0.1, solutions for the second case are required. EXTERNAL MASS-TRANSFER CONTROL The intraparticle concentration is uniform, and the rate equation is given by column 1 in Table 16-12. For a Langmuir isotherm with negligible solute accumulation in the particle pores, the solution for an infinite fluid volume: (1 − R)(1 − n /n )F − R ln (1 − F ) = (3kf t/rp)(c /ρ n ) 0′ i

0 i

0 i

0 p i

(16-93)

where n = n = n K c /(1 + K c ) is the adsorbate concentration in the particle at equilibrium with the fluid concentration. The predicted behavior is shown in Fig. 16-14 for n0′i = 0. In the irreversible limit (R = 0), F increases linearly with time; and in the linear limit (R = 1), 1 − F decreases exponentially with time. For a finite fluid volume (Λ∞ > 0), the fractional approach to equilibrium is given by: b′(1 − R∞) 1 (2c′F − b′ − q′)(−b′ + q′) 1 − ᎏᎏ ᎏ ln ᎏᎏᎏ 2c′ q′ (2c′F − b′ + q′)(−b′ − q′) 0 i



∞ i

s i

0 i i

0 i i



1 − R∞ 3kf t c0i − ᎏ ln (1 − b′F + c′F 2) = ᎏ ᎏ 2c′ rp ρpn∞i where

(16-94)

1 − R∞ b′ = ᎏ0 + Λ∞ 1−R

(16-95a)

c′ = Λ∞(1 − R∞)

(16-95b)

q′ = (b′2 − 4c′ )0.5 1 R0 = ᎏ0 1 + Ki c i

(16-95c) (16-95d)

1 R∞ = ᎏ (16-95e) 1 + Kc∞i 0 The predicted behavior is shown in Fig. 16-15 for R = 0.5 with different values of Λ∞. SOLID DIFFUSION CONTROL For a constant diffusivity and an infinite fluid volume the solution is: 6 ∞ 1 n2 π2Dsi t F = 1 − ᎏ2  ᎏ2 exp − ᎏ (16-96) π n=1 n rp2 For short times, this equation does not converge rapidly. The following approximations can be used instead (Helfferich and Hwang, in Dorfner, gen. refs., pp. 1277–1309): 6 Dsit 0.5 F=ᎏ ᎏ , F < 0.2 (16-97) π rp



   

6 Dst 0.5 3Dsi t , F < 0.8 (16-98) F=ᎏ ᎏ −ᎏ π rp rp2 For values of F > 0.8, the first term (n = 1) in Eq. (16-96) is generally sufficient. If the controlling resistance is diffusion in the subparticles of a bidispersed adsorbent, Eq. (16-96) applies with rs, replacing rp. For a finite fluid volume the solution is: ∞ exp (−p2n Dsi t/rp2) F = 1 − 6  ᎏᎏᎏ (16-99) ∞ ∞ ∞ 2 n = 1 9Λ /(1 − Λ ) + (1 − Λ )pn where the pns are the positive roots of tan pn 3 (16-100) ᎏ = ᎏᎏ pn 3 + (1/Λ∞ − 1)pn2 The predicted behavior is shown in Fig. 16-16. F is calculated from Eq. (16-96) for Λ∞ = 0 and from Eq. (16-99) for Λ∞ > 0. Significant deviations from the Λ∞ = 0 curve exist for Λ∞ > 0.1. For nonconstant diffusivity, a numerical solution of the conservation equations is generally required. In molecular sieve zeolites, when equilibrium is described by the Langmuir isotherm, the concentration dependence of the intracrystalline diffusivity can often be approximated by Eq. (16-72). The relevant rate equation is: ∂ni D0i ∂ r2 ∂ni (16-101) ᎏ=ᎏ ᎏ ᎏs ᎏ 2 ∂t r ∂r 1 − ni /ni ∂r



Constant separation factor batch adsorption curves for external mass-transfer control with an infinite fluid volume and n0i = 0.

FIG. 16-14





BATCH ADSORPTION

16-27

FIG. 16-15 Constant separation factor batch adsorption curves for external mass-transfer control with a finite fluid volume, n 0′i = 0 and R0 = 0.5.

A numerical solution of this equation for a constant surface concentration (infinite fluid volume) is given by Garg and Ruthven [Chem. Eng. Sci., 27, 417 (1972)]. The solution depends on the value of λ = (n0i − n0′i )/(nsi − n0′i ). Because of the effect of adsorbate concentration on the effective diffusivity, for large concentration steps adsorption is faster than desorption, while for small concentration steps, when Ds can be taken to be essentially constant, adsorption and desorption curves are mirror images of each other as predicted by Eq. (16-96); see Ruthven, gen. refs., p. 175. In binary ion-exchange, intraparticle mass transfer is described by Eq. (16-75) and is dependent on the ionic self diffusivities of the exchanging counterions. A numerical solution of the corresponding conservation equation for spherical particles with an infinite fluid volume is given by Helfferich and Plesset [ J. Chem. Phys., 66, 28, 418

(1958)]. The numerical results for the case of two counterions of equal valence where a resin bead, initially partially saturated with A, is completely converted to the B form, is expressed by: F = {1 − exp [π2( f1(α′)τD + f2(α′)τD2 + f3(α′)τD3 )]}1/2 with f1(α′) = −(0.570 + 0.430α′

0.775 −1

)

(16-103a)

f2(α′) = (0.260 + 0.782α′)−1

(16-103b)

f3(α′) = −(0.165 + 0.177α′)−1

(16-103c)

where τD =
DAt/r and α′ = 1 + (D
A /D
B − 1)n /n for 0.1 ≤ α′ ≤ 10. The predicted behavior is shown in Fig. 16-17. When α′ = 1 (equal ion 0′ diffusivities or nA ∼ 0), the solution coincides with Eq. (16-96). For α′ ≠ 1, the exchange rate is faster or slower depending on which coun2 p

Batch adsorption curves for solid diffusion control. The curve for Λ∞ = 0 corresponds to an infinite fluid volume (adapted from Ruthven, gen. refs., with permission). FIG. 16-16

(16-102)

0′ A

s

16-28

ADSORPTION AND ION EXCHANGE

Batch ion exchange for two equal-valence counterions. The exchanger is initially uniformly loaded with ion A in concentration n0′A and is completely converted to the B form. α′ = 1 + (D
A /D
B − 1)nA0′/ns. FIG. 16-17

terion is initially present in the ion exchanger and on the initial level of saturation. For an initially fully saturated particle, the exchange rate is faster when the faster counterion is initially in the resin, with the difference in rate becoming more important as conversion from one form to the other progresses. Helfferich (gen. refs., pp. 270–271) gives explicit expressions for the exchange of ions of unequal valence. PORE DIFFUSION CONTROL The rate equation is given by item A in Table 16-11. With pore fluid and adsorbent at equilibrium at each point within the particle and for a constant diffusivity, the rate equation can be written as: εpDpi 1 ∂ ∂cpi ∂cpi (16-104) ᎏ = ᎏᎏ ᎏ ᎏ r2 ᎏ εp + ρp dnei /dci r 2 ∂r ∂t ∂r





For a linear isotherm (ni = K i ci), this equation is identical to the conservation equation for solid diffusion, except that the solid diffusivity Dsi is replaced by the equivalent diffusivity Dei = εpDpi /(εp + ρpKi). Thus, Eqs. (16-96) and (16-99) can be used for pore diffusion control with infinite and finite fluid volumes simply by replacing Dsi with Dei. When the adsorption isotherm is nonlinear, a numerical solution is generally required. For a Langmuir system with negligible solute holdup in the pore fluid, item A in Table 16-11 gives: εpDpi 1 ∂ ∂ni ∂ni r2 (16-105) ᎏ=ᎏ ᎏ2 ᎏ ᎏᎏ ᎏ s ρ n K ∂t ∂r (1 − ni /nsi)2 ∂r p i i r





This equation has the same form of that obtained for solid diffusion control with Dsi replaced by the equivalent concentration-dependent diffusivity Dei = εpDpi /[ρpn siKi(1 − ni /nsi)2]. Numerical results for the case of adsorption on an initially clean particle are given in Fig. 16-18 for different values of λ = n0i /nis = 1 − R. The uptake curves become increasingly steeper, as the nonlinearity of the isotherm, measured by the parameter λ, increases. The desorption curve shown for a particle with n0′i /nsi = 0.9 shows that for the same step in concentration, adsorption occurs much more quickly than desorption. This difference, however, becomes smaller as the value of λ is reduced and in the linear region of the adsorption isotherm (λ → 0), adsorption and desorption curves are mirror images. The solution in Fig. 16-18 is applicable to a nonzero initial adsorbent loading by redefining λ as (n0i − n0′i )/(nsi − n0′)

and the dimensionless time variable as [εp Dpit/ρp(1 − n0′i /nsi)2 nsiK ir 2p]1/2 (Ruthven, gen. refs.). In the irreversible limit (R < 0.1), the adsorption front within the particle approaches a shock transition separating an inner core into which the adsorbate has not yet penetrated from an outer layer in which the adsorbed phase concentration is uniform at the saturation value. The dynamics of this process is described approximately by the shrinking-core model [Yagi and Kunii, Chem. Eng. (Japan), 19, 500 (1955)]. For an infinite fluid volume, the solution is: εpDpit ci0 1 1 1 ᎏ (16-106) ᎏ = ᎏ − ᎏ F − ᎏ (1 − F)2/3 rp2 ρpnsi 2 3 2 or, in explicit form [Brauch and Schlunder, Chem. Eng. Sci., 30, 540 (1975)]: 3 12εpDpi t c0i 1 π 1 F = 1 − ᎏ + cos ᎏ + ᎏ cos−1 1 − ᎏ (16-107) ᎏs 2 rp 2 3 3 ρpni









For a finite fluid volume with 0 < Λ ≤ 1, the solution is [Teo and Ruthven, Ind. Eng. Chem. Process Des. Dev., 25, 17 (1986)]: εpDpit c0i ᎏ (16-108) ᎏ = I2 − I1 rp2 ρpnsi where ∞

2η − λ′ 2 − λ′ 1 λ′3 + η3 λ′ + 1 1 I1 = ᎏ tan−1 ᎏ − tan−1 ᎏ + ᎏ ln ᎏ ᎏ λ′Λ∞
3 λ′ 3
λ′ 3
6λ′Λ∞ λ′3 + 1 λ′ + η









 3

(16-109a) 1 λ′3 + η3 ln ᎏ I2 = ᎏ ∞ 3Λ λ′3 + 1

(16-109b)

η = (1 − F )1/3

(16-109c)





1 −1 λ′ = ᎏ Λ∞

1/3

(16-109d)

COMBINED RESISTANCES In general, exact analytic solutions are available only for the linear (R = 1) and irreversible limits (R → 0). Intermediate cases require

BATCH ADSORPTION

16-29

Constant separation factor batch adsorption curves for pore diffusion control with an infinite fluid volume. λ is defined in the text.

FIG. 16-18

numerical solution or use of approximate driving force expressions (see “Rate and Dispersion Factors”). Parallel Pore and Solid Diffusion Control With a linear isotherm, assuming equilibrium between the pore fluid and the solid adsorbent, batch adsorption can be represented in terms of an equivalent solid diffusivity Dei = (εpDpi + ρp Dsi)/(εp + ρp Ki). Thus, Eqs. (16-96) and (16-99) can be used for this case with Dsi replaced by Dei. External Mass Transfer and Intraparticle Diffusion Control With a linear isotherm, the solution for combined external mass transfer and pore diffusion control with an infinite fluid volume is (Crank, Mathematics of Diffusion, 2d ed., Clarendon Press, 1975): ∞ 6Bi2 exp [−(p2nεp Dpi t/r p2)/(εp + ρp Ki)] F = 1 −  ᎏᎏᎏᎏ pn2 [p2n + Bi(Bi − 1)] n=1

(16-110)

where Bi = kf rp /εp Dpi is the Biot number and the pns are the positive roots of pn cot pn = 1 − Bi

For a finite fluid volume the solution is (Teo and Ruthen): εpDpi t c0i 1 ᎏ ᎏ = 1 − ᎏ I2 − I1 rp2 ρpnsi Bi



18 F=1−ᎏ β + 3α

2 p

where the pns are the positive roots of 1 − Λ∞ pn2 3 − ᎏᎏ ᎏᎏ Λ∞ Bi tan pn ᎏᎏᎏ ᎏ= 1 − Λ∞ (Bi − 1)pn2 pn 3 + ᎏᎏ ᎏᎏ Λ∞ Bi

(16-113)

These expressions can also be used for the case of external mass transfer and solid diffusion control by substituting Dsi for εpDpi /(εp + ρpKi) and kf rp /(ρp KiDsi) for the Biot number. In the irreversible limit, the solution for combined external resistance and pore diffusion with infinite fluid volume is (Yagi and Kunii): εpDpi t c0i 1 1 1 1 ᎏ ᎏ = ᎏ − ᎏ 1 − ᎏ F − ᎏ (1 − F )2/3 rp2 ρpnsi 2 3 Bi 2





(16-114)

n2π2

  ᎏ p  ∞

∞

m=1 n=1

4 n, m

exp(−p2n, m Dsi t/r 2s ) ⫻ ᎏᎏᎏᎏᎏ β cot pn, m α + ᎏᎏ 1 + ᎏᎏ (pn, m cot pn, m − 1) 2 pn, m

(16-111)

exp [−(p ε Dpi t/r )/(εp + ρp Ki)] F = 1 − 6  ᎏᎏᎏᎏᎏᎏ 9Λ∞ p2 p4n n=1 + (1 − Λ∞)pn2 − (5Λ∞ + 1) ᎏᎏn + (1 − Λ∞) ᎏᎏ ᎏᎏ ∞ 1−Λ Bi Bi2 (16-112) 2 n p

(16-115)

where I1 and I2 are given be Eqs. (16-109a) and (16-109b). Bidispersed Particles For particles of radius rp comprising adsorptive subparticles of radius rs that define a macropore network, conservation equations are needed to describe transport both within the macropores and within the subparticles and are given in Table 16-11, item D. Detailed equations and solutions for a linear isotherm are given in Ruthven (gen. refs., p. 183) and Ruckenstein et al. [Chem. Eng. Sci., 26, 1306 (1971)]. The solution for a linear isotherm with no external resistance and an infinite fluid volume is:



For a finite fluid volume the solution is: ∞





(16-116)

where the pn, m values are the roots of the equation

and

αp2n, m − n2π2 = β(pn, m cot pn, m − 1)

(16-117)

Dsi /r 2s α = ᎏ2 Dpi /rp

(16-118a)

3αρp Ki β=ᎏ εp

(16-118b)

In these equations, Dsi is the diffusivity in the subparticles, and Dpi is the diffusivity in the pore network formed by the subparticles. For large Ki values, the uptake curve depends only upon the value of the parameter β representing the ratio of characteristic time constants for diffusion in the pores and in the subparticles. For small β values, diffusion in the subparticles is controlling and the solution coincides with Eq. (16-96) with rs replacing rp. For large β values, pore diffusion is controlling, and the solution coincides with Eq. (16-96) with εpDpi /(εp + ρp Ki) replacing Dsi. Lee [AIChE J., 24, 531 (1978)] gives the solution for batch adsorption with bidispersed particles for the case of a finite fluid volume.

16-30

ADSORPTION AND ION EXCHANGE

FIXED-BED TRANSITIONS As discussed in “Design Concepts,” a large fraction of adsorption and ion-exchange processes takes place in fixed beds. Two classical methods for analyzing fixed-bed transitions are described here. First, local equilibrium theory is presented. In this, all mass-transfer resistances are ignored to focus on the often dominating role of isotherm shape. Second, results of constant pattern analysis are presented. This gives the maximum breadth to which a mass-transfer zone will spread for various rate mechanisms. It is therefore conservative for design purposes. Both of these methods pertain to behavior in deep beds. For shallow beds, the equations given below must be solved for the particular case of interest.

If the system under consideration involves use of the sorbent for only a single feed step or reuse after uniform regeneration, as in many applications with activated carbons and ion exchangers, then one of two paths is often followed at this point to simplify Eq. (16-124) further. The second term on the left-hand side of the equation is often assumed to be negligibly small (usually a good assumption), and time is redefined as τ τ1 = ᎏ (16-127) Λ to give

DIMENSIONLESS SYSTEM For the methods, we consider Eq. (16-52), the material balance for a fixed bed, written in the form ∂n ∂ci ∂(vci) ∂ ∂y
i + ε ᎏ ρb ᎏ +εᎏ = εDL ᎏ c ᎏi (16-119) b ∂t ∂t ∂z ∂z ∂z where it has been assumed that DL is constant and that cˆi ≈ ci (or that the second term in the balance is small compared to the first—usually a good assumption). Dimensionless variables can be defined for time, the axial coordinate, and velocity: εvreft τ=ᎏ (16-120) L





z ζ=ᎏ (16-121) L v v* = ᎏ (16-122) vref ref where L is bed length, v is the interstitial velocity at the bed inlet, and τ is equal to the number of empty bed volumes of feed passed into the bed. The material balance becomes ∂n ∂ci ∂(v*ci) 1 ∂ ∂y
i + ε ᎏ ρb ᎏ + ᎏ = ᎏ ᎏ c ᎏi (16-123) b ∂τ ∂τ ∂ζ NPe ∂ζ ∂ζ ref where NPe = εv L/DL is a Peclet number for the bed or a number of dispersion units. Equation (16-123) or a similar equation is often the material balance used in nonisothermal problems, in problems involving adsorption of nontrace components, and in calculations of cycles. For a trace, isothermal system, we have v* = 1, and using the dimensionless system variables for concentrations [Eq. (16-10)], Eq. (16-123) becomes ∂n ∂c*i ∂c* 1 ∂2c*i
*i + ε ᎏ Λᎏ + ᎏi = ᎏ ᎏ (16-124) b ∂τ ∂τ ∂ζ NPe ∂ζ2 where Λ is the partition ratio defined by ρbn ref i Λ=ᎏ (16-125) c ref i This important dimensionless group is the volumetric capacity of the bed for the sorbable component divided by the concentration of the sorbable component in the feed. The stoichiometric capacity of the bed for solute is exactly equal to Λ empty bed volumes of feed (to saturate the sorbent at the feed concentration) plus a fraction of a bed volume of feed to fill the voids outside and inside the particles. Alternatively, we also obtain Eq. (16-124) using the dimensionless transition variables for concentrations [Eq. (16-11)], but now the partition ratio in the first term of Eq. (16-124) pertains to the transition and is given by n″i − n′i Λ = ρb ᎏ (16-126) c″i − c′i Equation (16-124) is a commonly used form of material balance for a fixed-bed adsorber.





∂n ∂c*i 1 ∂2c*i
*i + ᎏ =ᎏᎏ ᎏ ∂τ1 ∂ζ NPe ∂ζ2

(16-128)

Alternatively, in the absence of axial dispersion, a variable of the form τ − εbζ (16-129) τ1 = ᎏ Λ can be defined to reduce Eq. (16-124) directly to ∂n ∂c*i
*i + ᎏ =0 ᎏ ∂τ1 ∂ζ

(16-130)

The variable τ1 defined by Eq. (16-127) or (16-129) is a throughput parameter, equal to unity (hence, the “1” subscript) at the time when the stoichiometric center of the concentration wave leaves the bed. This important group, in essence a dimensionless time variable, essentially determines the location of the stoichiometric center of the transition in the bed at any time. LOCAL EQUILIBRIUM THEORY In local equilibrium theory, fluid and sorbed phases are assumed to be in local equilibrium with one another at every axial position in the bed. Thus, because of uniform concentrations, the overbar on n*i is not necessary and we have cˆi ≈ ci [note Eqs. (16-52) and (16-119)]. Single Transition System For a system described by a single material balance, Eq. (16-130) gives dτ1 ∂c*i /∂ζ dn*i ᎏ=−ᎏ=ᎏ dζ ∂c*i /∂τ1 dc*i

(16-131)

where dτ1/dζ is the reciprocal of a concentration velocity. Equation (16-131) is the equation for a simple wave (or gradual transition or proportionate pattern). If a bed is initially uniformly saturated, then dτ1 /dζ = τ1 /ζ. Thus, for the dimensionless system, the reciprocal of the velocity of a concentration is equal to the slope of the isotherm at that concentration. Furthermore, from Eq. (16-131), the depth of penetration of a given concentration into the bed is directly proportional to time, so the breadth of a simple wave increases in direct proportion to the depth of its penetration into the bed (or to time). Thus, for the simple wave, the length of the MTZ is proportional to the depth of the bed through which the wave has passed. Consideration of isotherm shape indicates that a simple wave occurs for an unfavorable dimensionless isotherm (d 2n*i /dc*i 2 > 0), for which low concentrations will go faster than high concentrations. Equation (16-131) also pertains to a linear isotherm, in which case the wave is called a contact discontinuity, because it has neither a tendency to spread nor sharpen. If mass-transfer resistance is added to the consideration of wave character for unfavorable isotherms, the wave will still asymptotically approach the simple wave result given by Eq. (16-131). For a favorable isotherm (d 2n*i /dc*i 2 < 0), Eq. (16-131) gives the impossible result that three concentrations can coexist at one point in the bed (see example below). The correct solution is a shock (or abrupt transition) and not a simple wave. Mathematical theory has been developed for this case to give “weak solutions” to conservation laws. The form of the solution is change in flux shock speed = ᎏᎏᎏᎏ change in accumulated quantity

FIXED-BED TRANSITIONS

Example 9: Transition Types For the constant separation-factor isotherm given by Eq. (16-31), determine breakthrough curves for r = 2 and r = 0.5 for transitions from c *i = 0 to c*i = 1. Using Eq. (16-131), we obtain τ1 r ᎏ = ᎏᎏ ζ [r + (1 − r)c*i ]2 This equation, evaluated at ζ = 1, is plotted for r = 2 and r = 0.5 in Fig. 16-20. Clearly, the solution for r = 0.5 is not physically correct. Equation (16-132), with dτ1 /dζ = τ1 /ζ, is applied to this case to give the shock indicated by the dashed line. Alternatively, we could have obtained bed profiles by evaluating equations at τ1 = const.

c*i

1

0 0

where the derivatives are taken along the path of a transition (i.e., directional derivatives).

FIG. 16-19

Path in isotherm plane for a combined wave (After Tudge).

τ1

2

1

Multiple Transition System Local equilibrium theory for multiple transitions begins with some combination of material and energy balances, written ∂n ∂c ∂(v*ci) ρb ᎏi + εb ᎏi + ᎏ =0 (i = 1, 2, . . . ) (16-133) ∂τ ∂τ ∂ζ ∂(cuf) ∂(v*chf) ∂us (16-134) ρb ᎏ + εb ᎏ + ᎏ = 0 ∂τ ∂τ ∂ζ which are Eq. (16-123) written with no dispersion and Eq. (16-55) written for an adiabatic bed. For a simple wave, application of the method of characteristics (hodograph transformation) gives d(ρbus + εbcuf) dτ d(ρbn1 + εbc1) d(ρbn2 + εbc2) ᎏ = ᎏᎏ = ᎏᎏ = . . . = ᎏᎏ (16-135) d(v*c1) d(v*c2) d(v*chf) dζ

1

c*i

where the changes are jump discontinuities across the shock. The reciprocal of this equation, using Eq. (16-130), is dτ1 ∆n*i (16-132) ᎏ=ᎏ dζ ∆ c*i where the differences are taken across the shock. It is also possible to have a combined wave, which has both gradual and abrupt parts. The general rule for an isothermal, trace system is that in passing from the initial condition to the feed point in the isotherm plane, the slope of the path must not decrease. If it does, then a shock chord is taken for that part of the path. Referring to Fig. 16-19, for a transition from (0,0) to (1,1), the dashes indicate shock parts, which are connected by a simple wave part between points P1 and P2.

16-31

0 0 FIG. 16-20

1

τ2

2

Breakthrough curves for r = 2 (top) and r = 0.5 (bottom) for Exam-

ple 9.

If a simple wave is not possible on physical grounds, then it (or part of it) is replaced by a shock, given by ∆(ρbus + εbcuf) dτ ∆(ρbn1 + εbc1) ∆(ρbn2 + εbc2) ᎏ = ᎏᎏ = ᎏᎏ = . . . = ᎏᎏ (16-136) ∆(v*c1) ∆(v*c2) ∆(v*chf) dζ Extensions When more than two conservation equations are to be solved simultaneously, matrix methods for eigenvalues and left eigenvectors are efficient [Jeffrey and Taniuti, Nonlinear Wave Propagation, Academic Press, New York, 1964; Jacob and Tondeur, Chem. Eng. J., 22, 187 (1981), 26, 41 (1983); Davis and LeVan, AIChE J., 33, 470 (1987); Rhee et al., gen. refs.]. Nontrace isothermal systems give the “adsorption effect” (i.e., significant change in fluid velocity because of loss or gain of solute). Criteria for the existence of simple waves, contact discontinuities, and shocks are changed somewhat [Peterson and Helfferich, J. Phys. Chem., 69, 1283 (1965); LeVan et al., AIChE J., 34, 996 (1988); Frey, AIChE J., 38, 1649 (1992)]. Local equilibrium theory also pertains to adsorption with axial dispersion, since this mechanism does not disallow existence of equilibrium between stationary and fluid phases across the cross section of the bed [Rhee et al., Chem. Eng. Sci., 26, 1571 (1971)]. It is discussed below in further detail from the standpoint of the constant pattern. Example 10: Two-Component Isothermal Adsorption Two components present at low mole fractions are adsorbed isothermally from an inert fluid in an initially clean bed. The system is described by ρb = 500 kg/m3, εb = 0.7, and the binary Langmuir isotherm n isKici (i = 1, 2) ni = ᎏᎏ 1 + K1c1 + K2c2 3 s s with n1 = n 2 = 6 mol/kg, K1 = 40 m /mol, and K2 = 20 m3/mol. The feed is c1 = c2 = 0.5 mol/m3. Find the bed profile. Using the isotherm to calculate loadings in equilibrium with the feed gives n1 = 3.87 mol/kg and n2 = 1.94 mol/kg. An attempt to find a simple wave solution for this problem fails because of the favorable isotherms (see the next example for the general solution method). To obtain the two shocks, Eq. (16-136) is written

16-32

ADSORPTION AND ION EXCHANGE

dτ ∆(ρbn1 + εbc1) ∆(ρbn2 + εbc2) ᎏ = ᎏᎏ = ᎏᎏ ∆c1 ∆c2 dζ The concentration of one of the components will drop to zero in the shock nearest the bed inlet. If it is component 1, then using feed values and the equation above, that shock would be at ρbn1 + εbc1 ρbn1 τ ᎏ = ᎏᎏ ≈ ᎏ = 3870 c1 ζ c1 Similarly, if the second component were to disappear in the first shock, we would have τ /ζ = 1940. Material balance considerations require that we accept the shorter distance, so component 1 disappears in the first shock. The concentrations of component 2 on the plateau downstream of the first shock are then calculated from τ ∆(ρbn2 + εbc2) ρb∆n2 ᎏ = ᎏᎏ ≈ ᎏ = 3870 ∆c2 ζ ∆c2 and its pure component isotherm, giving c2 = 0.987 mol/m3 and n2 = 5.71 mol/kg. The location of this shock is determined using these concentrations and τ ρbn2 + εbc2 ρbn2 ᎏ = ᎏᎏ ≈ ᎏ c2 ζ c2 which gives τ /ζ = 2890. The bed profile is plotted in Fig. 16-21 using ζ/τ as the absissa. (This example can also be worked with the h-transformation described in this section under chromatography.)

Example 11: Adiabatic Adsorption and Thermal Regeneration An initially clean activated carbon bed at 320 K is fed a vapor of benzene in nitrogen at a total pressure of 1 MPa. The concentration of benzene in the feed is 6 mol/m3. After the bed is uniformly saturated with feed, it is regenerated using benzene-free nitrogen at 400 K and 1 MPa. Solve for both steps. For simplicity, neglect fluid-phase accumulation terms and assume constant mean heat capacities for stationary and fluid phases and a constant velocity. The system is described by ρbn1sK1c1 ρbn1 = ᎏ 1 + K1c1 ρbn1s = 2750 mol/m3 K1 = 3.88 × 10−8 T
exp [q1st /(ℜT)] m3/(mol K1/2) ρbCsm = 850 kJ/(m3 K) cCpfm = 11.3 kJ/(m3 K) q1st = 43.5 kJ/mol T ref = 320 K Extensive analysis has been made of this system [Rhee et al., Chem. Eng. J., 1, 241 (1970) and gen. refs.; LeVan, in Rodrigues et al., gen. refs.]. To obtain the concentration and temperature profiles, the two transitions are first assumed to be gradual. Equation (16-135) is written in the form dus τ dn1 ᎏ = ρb ᎏ = ρb ᎏ ζ dc1 d(chf) where, from Eqs. (16-56) to (16-60), ρbus = ρbCsm(T − T ref ) − ρbn1q1st chf = cCpfm(T − T ref )

c2*

1.0

c*i 0.5

0 0

2

3

4

5

ζ/τ × 104 FIG. 16-21

ple 10.

τ ρbCsm ρbq1st ∂n1 ∂n1 dc1 ∂n1 ∂n1 dT ᎏ = ρb ᎏ + ᎏ ᎏ = ᎏ − ᎏ ᎏ + ᎏ ᎏ ζ ∂c1 ∂T dc1 cCpfm cCpfm ∂T ∂c1 dT









Solving the rightmost equality for the directional derivative using the quadratic formula gives dc1 −b ⫾
b2

−
4
ad ᎏ = ᎏᎏ dT 2a with

ρbq1st ∂n1 a= ᎏ ᎏ (cCpfm) ∂c1





ρq ∂n1 ρbCsm ρb∂n1 b= ᎏ ᎏ−ᎏ+ᎏ (cCpfm) ∂T (cCpfm) ∂c1



st b 1



ρb∂n1 d=ᎏ ∂T The plus sign corresponds to the downstream transition and the minus sign to the upstream one. This equation is solved along each path beginning at the respective end points—the initial condition of the bed for the downstream transition and the feed condition for the upstream transition. If either path fails to evolve continuously in the expected direction, the difference form, from Eq. (16-136), ∆n1 dτ ∆us ᎏ = ρb ᎏ = ρb ᎏ dζ ∆c1 ∆(chf ) is used for that path (or part thereof, if appropriate); two solutions of this equation pass through each composition point, and care must be taken to ensure that the correct path is taken. The two correct paths found intersect to give the composition and temperature of an intermediate plateau region. Letting c1ref = 6 mol/m3, the isotherm gives ρbn1ref = 2700 mol/m3, and the partition ratio is Λ = 450. In the figures, Γ(k) and Σ(k) symbolize simple waves and shocks, respectively, with k = 1 downstream and k = 2 upstream. Adiabatic Adsorption The construction is shown in Figs. 16-22 and 16-23. The first path begins at the initial condition, point A (T = 320 K, c1 = 0 mol/m3). Since ∂n1 /∂T = 0 there, we obtain dc1/dT = 0 and τ/ζ = ρbCsm /(cCpfm) = 75.2 (or τ/Λ = 0.167 at ζ = 1), corresponding to a pure thermal wave along the c1 = 0 axis of Fig. 16-22. The second path begins at the feed condition, point B (T = 320 K, c1 = 6 mol/m3). A Γ(2) fails, so a Σ(2) is calculated and plotted in the two figures. The Γ(1) and Σ(2) intersect at point C (T = 348 K, c1 = 0 mol/m3) Breakthrough curves of temperature and concentration are shown in Fig. 16-24. Thermal Regeneration The bed, initially at point B, is fed with pure nitrogen. The feed, at point D (T = 400 K, c1 = 0 mol/m3), provides a successful Γ(2). The Γ(1) from point B fails, so a Σ(1), which differs imperceptibly from Γ(1), is determined. The Σ(1) and Γ(2) meet at point E (T = 320.8 K, c1 = 8.93 mol/m3), where ρbn1 = 2716 mol/m3, which is greater than the initial loading, indicating “roll-up,” a term attributed to Basmadjian et al. [Ind. Eng. Chem. Process Des. Dev., 14, 328 (1975)], of the adsorbed-phase concentration. Breakthrough curves are shown in Fig. 16-25. The Σ(1) leaves the bed at τ/Λ = 0.011. The Γ(2) begins to emerge at τ/Λ = 0.169, but regeneration is not complete until τ/Λ = 2.38. It is possible (with lower initial bed temperature, higher initial loading, or higher regeneration temperature or pressure) for the transition paths to contact the saturated vapor curve in Fig. 16-22 rather than intersect beneath it. For this case, liquid benzene condenses in the bed, and the effluent vapor is saturated during part of regeneration [Friday and LeVan, AIChE J., 30, 679 (1984)].

CONSTANT PATTERN BEHAVIOR FOR FAVORABLE ISOTHERMS

c1* & c2*

1

This equation is expanded in terms of c1 and T to obtain

Bed profiles for two-component isothermal adsorption, Exam-

With a favorable isotherm and a mass-transfer resistance or axial dispersion, a transition approaches a constant pattern, which is an asymptotic shape beyond which the wave will not spread. The wave is said to be “self-sharpening.” (If a wave is initially broader than the constant pattern, it will sharpen to approach the constant pattern.) Thus, for an initially uniformly loaded bed, the constant pattern gives the maximum breadth of the MTZ. As bed length is increased, the constant pattern will occupy an increasingly smaller fraction of the bed. (Square-root spreading for a linear isotherm gives this same qualitative result.) The treatment here is restricted to the Langmuir or constant separation factor isotherm, single-component adsorption, dilute systems, isothermal behavior, and mass-transfer resistances acting alone. References to extensions are given below. Different isotherms have been considered, and the theory is well understood for general isotherms.

FIXED-BED TRANSITIONS

FIG. 16-22

FIG. 16-23

eration.

Transition paths in c1, T plane for adiabatic adsorption and thermal regeneration.

Transition paths in isotherm plane for adiabatic adsorption and thermal regen-

16-33

16-34

ADSORPTION AND ION EXCHANGE

FIG. 16-24

Breakthrough curves for adiabatic adsorption.

FIG. 16-25

Breakthrough curves for thermal regeneration.

Asymptotic Solution Rate equations for the various masstransfer mechanisms are written in dimensionless form in Table 16-13 in terms of a number of transfer units, N = L/HTU, for particle-scale mass-transfer resistances, a number of reaction units for the reaction kinetics mechanism, and a number of dispersion units, NPe, for axial dispersion. For pore and solid diffusion, ξ = r/rp is a dimensionless radial coordinate, where rp is the radius of the particle. If a particle is bidisperse, then rp can be replaced by rs, the radius of a subparticle. For preliminary calculations, Fig. 16-13 can be used to estimate N for use with the LDF approximation when more than one resistance is important. In constant pattern analysis, equations are transformed into a new coordinate system that moves with the wave. Variables are changed from (ζ, τ1) to (ζ − τ1, τ1). The new variable ζ − τ1 is equal to zero at the stoichiometric center of the wave. Equation (16-130) for a bed

with no axial dispersion, when transformed to the (ζ − τ1, τ1) coordinate system, becomes ∂c*i ∂n ∂n
*i + ᎏ
*i + ᎏ −ᎏ =0 ∂(ζ − τ1) ∂τ1 ∂(ζ − τ1)

(16-137)

The constant pattern is approached as the τ1 dependence in this equation disappears. Thus, discarding the derivative with respect to τ1 and integrating, using the condition that
n*i and c*i approach zero as N(ζ − τ1) → ∞ [or approach unity as N(ζ − τ1) → −∞], gives simply n
*i = c*i

(16-138)

For adsorption with axial dispersion, the material balance transforms to

FIXED-BED TRANSITIONS TABLE 16-13

16-35

Constant Pattern Solutions for Constant Separation Factor Isotherm (R < 1)

Mechanism

Dimensionless rate equation1

N

Constant pattern

Refs.

Numerical

A

Numerical

B

∂n
*i = N (c* − c e*) ᎏ i i ∂τ1

1 (1 − c*i )R ᎏ ln ᎏ − 1 = N (ζ − τ1) 1−R c*i

C

knΛL ᎏ εvref

∂n
*i ᎏ = N (nei * −
n*i ) ∂τ1

1 1 − c*i + 1 = N (ζ − τ1) ᎏ ln ᎏ 1−R c*i R

D

kacrefΛL ᎏᎏ (1 − R)εvref

∂n
*i = N [(1 − n*)c* − Rn*(1 − c*)] ᎏ i
i i
i ∂τ1

1 1 − c*i ᎏ ln ᎏ = N (ζ − τ1) 1−R c*i

E

Pore diffusion

15(1 − ε)εpDpL ᎏᎏ εvref R p2

∂c*i ∂n*i N 1 ∂ ᎏ = ᎏ ᎏ2 ᎏ ξ2 ᎏ ∂τ1 ∂ξ 15 ξ ∂ξ

Solid diffusion

15ΛDsL ᎏ εvrefR 2p

∂n*i ∂n*i N 1 ∂ ᎏ = ᎏ ᎏ2 ᎏ ξ2 ᎏ ∂τ1 ∂ξ 15 ξ ∂ξ

External mass transfer

kfaL ᎏ εvref

Linear driving force Reaction kinetics

 

    



 

1 − c*i vrefL 1 Eq. (16-128) = N (ζ − τ1) F ᎏ ᎏ ln ᎏ DL 1−R c*i R 1: Dimensional rate equations are given in Table 16-11 and 16-12. A: Hall et al., Ind. Eng. Chem. Fundam., 5, 212 (1966). B: Hall et al., Ind. Eng. Chem. Fundam., 5, 212 (1966); Garg and Ruthven, Chem. Eng. Sci., 28, 791, 799 (1973). C: Michaels, Ind. Eng. Chem., 44, 1922 (1952); Miura and Hashimoto, J. Chem. Eng. Japan, 10, 490 (1977). D: Glueckauf Coates, J. Chem. Soc., 1947, 1315 (1947); Vermeulen, Advances in Chemical Engineering, 2, 147 (1958); Hall et al., Ind. Eng. Chem. Fundam., 5, 212 (1966); Miura and Hashimoto, J. Chem. Eng. Japan, 10, 490 (1977). E: Walter, J. Chem. Phys., 13, 229 (1945); Hiester and Vermeulen, Chem. Eng. Progress, 48, 505 (1952). F: Acrivos, Chem. Eng. Sci., 13, 1 (1960); Coppola and LeVan, Chem. Eng. Sci., 36, 967 (1981).



Axial dispersion

∂n*i ∂ 2c*i ∂n*i ∂c*i 1 − ᎏ + ᎏ + ᎏ = ᎏ ᎏ2 ∂(ζ − τ1) ∂τ1 ∂(ζ − τ1) NPe ∂(ζ − τ1)

(16-139)

The partial derivative with respect to τ1 is discarded and the resulting equation integrated once to give 1 dc*i −n*i + c*i = ᎏ ᎏ NPe d(ζ − τ1)

(16-140)

After eliminating n*i or c*i using the adsorption isotherm, Eq. (16-140) can be integrated directly to obtain the constant pattern. For other mechanisms, the particle-scale equation must be integrated. Equation (16-140) is used to advantage. For example, for external mass transfer acting alone, the dimensionless rate equation in Table 16-13 would be transformed into the (ζ − τ1, τ1) coordinate system and derivatives with respect to τ1 discarded. Equation (16-138) is then used to replace c*i with n
*i in the transformed equation. Furthermore, for this case there are assumed to be no gradients within the particles, so we have n
*i = n*i . After making this substitution, the transformed equation can be rearranged to dn*i −ᎏ = d(ζ − τ1) (16-141) n*i − c*i Since n*i and c*i are related by the adsorption isotherm, Eq. (16-141) can be integrated. The integration of Eq. (16-140) or (16-141) as an indefinite integral will give an integration constant that must be evaluated to center the transition properly. The material balance depicted in Fig. 16-26 is used. The two shaded regions must be of equal area if the stoichiometric center of the transition is located where the throughput parameter is unity. Thus, we have



0

−∞

(1 − n*i ) d(ζ − τ1) =



∞

n*i d(ζ − τ1)



can be neglected. If fluid-phase accumulation is important, then the transition for axial dispersion can be centered by taking into account the relative quantities of solute held in the fluid and adsorbed phases. Constant pattern solutions for the individual mechanisms and constant separation factor isotherm are given in Table 16-13. The solutions all have the expected dependence on R—the more favorable the isotherm, the sharper the profile. Figure 16-27 compares the various constant pattern solutions for R = 0.5. The curves are of a similar shape. The solution for reaction kinetics is perfectly symmetrical. The curves for the axial dispersion fluid-phase concentration profile and the linear driving force approximation are identical except that the latter occurs one transfer unit further down the bed. The curve for external mass transfer is exactly that for the linear driving force approximation turned upside down [i.e., rotated 180° about c*i = n*i = 0.5, N(ζ − τ1) = 0]. The linear driving force approximation provides a good approximation for both pore diffusion and surface diffusion. Because of the close similarity in shape of the profiles shown in Fig. 16-27 (as well as likely variations in parameters; e.g., concentrationdependent surface diffusion coefficient), a controlling mechanism cannot be reliably determined from transition shape. If reliable correlations are not available and rate parameters cannot be measured in independent experiments, then particle diameters, velocities, and other factors should be varied and the observed impact considered in relation to the definitions of the numbers of transfer units.

(16-142)

0

Integrating Eq. (16-142) by parts gives

 (N − NT) dn* = 0 1

i

(16-143)

0

For all mechanisms except axial dispersion, the transition can be centered just as well using c*i, because of Eq. (16-138). For axial dispersion, the transition should be centered using n*i, provided the fluid-phase accumulation term in the material balance, Eq. (16-124),

FIG. 16-26 Material balance for centering profile [ from LeVan in Rodrigues et al. (eds), Adsorption: Science and Technology, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1989; reprinted with permission.]

16-36

ADSORPTION AND ION EXCHANGE

FIG. 16-27 Constant pattern solutions for R = 0.5. Ordinant is c*i or n*i except for axial dispersion for which individual curves are labeled: a, axial dispersion; b, external mass transfer; c, pore diffusion (spherical particles); d, surface diffusion (spherical particles); e, linear driving force approximation; f, reaction kinetics. [from LeVan in Rodrigues et al. (eds.), Adsorption: Science and Technology, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1989; reprinted with permission.]

Breakthrough Behavior for Axial Dispersion Breakthrough behavior for adsorption with axial dispersion in a deep bed is not adequately described by the constant pattern profile for this mechanism. Equation (16-128), the partial differential equation of the second order Fickian model, requires two boundary conditions for its solution. The constant pattern pertains to a bed of infinite depth—in obtaining the solution we apply the downstream boundary condition c*i → 0 as NPeζ → ∞. Breakthrough behavior presumes the existence of a bed outlet, and a boundary condition must be applied there. The full mathematical model for this problem is Eq. (16-128) with boundary conditions 1 ∂c* c*i − ᎏ ᎏi = 1 at ζ=0 (16-144) NPe ∂ζ ∂c*i at ζ=1 (16-145) ᎏ=0 ∂ζ and an initial condition. Equation (16-144) specifies a constant flux at the bed inlet, and Eq. (16-145), the Danckwerts-type boundary condition at the bed outlet, is appropriate for fixed-bed adsorption, provided that the partition ratio is large. The solution to this model for a deep bed indicates an increase in velocity of the fluid-phase concentration wave during breakthrough. This is most dramatic for the rectangular isotherm—the instant the bed becomes saturated, the fluid-phase profile jumps in velocity from that of the adsorption transition to that of the fluid, and a near shocklike breakthrough curve is observed [Coppola and LeVan, Chem. Eng. Sci., 36, 967 (1981)]. Extensions Existence, uniqueness, and stability criteria have been developed for the constant pattern [Cooney and Lightfoot, Ind. Eng. Chem. Fundam., 4, 233 (1965); Rhee et al., Chem. Eng. Sci., 26, 1571 (1971); Rhee and Amundson, Chem. Eng. Sci., 26, 1571 (1971), 27, 199 (1972), 29, 2049 (1974)].

The rectangular isotherm has received special attention. For this, many of the constant patterns are developed fully at the bed inlet, as shown for external mass transfer [Klotz, Chem. Revs., 39, 241 (1946)], pore diffusion [Vermeulen, Adv. Chem. Eng., 2, 147 (1958); Hall et al., Ind. Eng. Chem. Fundam., 5, 212 (1966)], the linear driving force approximation [Cooper, Ind. Eng. Chem. Fundam., 4, 308 (1965)], reaction kinetics [Hiester and Vermeulen, Chem. Eng. Progress, 48, 505 (1952); Bohart and Adams, J. Amer. Chem. Soc., 42, 523 (1920)], and axial dispersion [Coppola and LeVan, Chem. Eng. Sci., 38, 991 (1983)]. Multiple mass-transfer resistances have been considered in many studies [Vermeulen, Adv. in Chem. Eng., 2, 147 (1958); Vermeulen et al., Ruthven, gen. refs.; Fleck et al., Ind. Eng. Chem. Fundam., 12, 95 (1973); Yoshida et al., Chem. Eng. Sci., 39, 1489 (1984)]. Treatments of constant pattern behavior have been carried out for multicomponent adsorption [Vermeulen, Adv. in Chem. Eng., 2, 147 (1958); Vermeulen et al., Ruthven, gen. refs.; Rhee and Amundson, Chem. Eng. Sci., 29, 2049 (1974); Cooney and Lightfoot, Ind. Eng. Chem. Fundam., 5, 25 (1966); Cooney and Strusi, Ind. Eng. Chem. Fundam., 11, 123 (1972); Bradley and Sweed, AIChE Symp. Ser. No. 152, 71, 59 (1975)]. The behavior is such that coexisting compositions advance through the bed together at a uniform rate; this is the coherence concept of Helfferich and coworkers [gen. refs.]. Nontrace systems have been considered [Sircar and Kumar, Ind. Eng. Chem. Proc. Des. Dev., 22, 271 (1983)]. Constant patterns have been developed for adiabatic adsorption [Pan and Basmadjian, Chem. Eng. Sci., 22, 285 (1967); Ruthven et al., Chem. Eng. Sci., 30, 803 (1975); Kaguei et al., Chem. Eng. Sci., 42, 2964 (1987)]. The constant pattern concept has also been extended to circumstances with nonplug flows, with various degrees of rigor, including flow profiles in tubes [Sartory, Ind. Eng. Chem. Fundam., 17, 97 (1978); Tereck et al., Ind. Eng. Chem. Res., 26, 1222 (1987)], wall effects [Vortmeyer and Michael, Chem. Eng. Sci., 40, 2135 (1985)], channeling [LeVan and Vermeulen in Myers and Belfort (eds.), Fundamentals of Adsorption, Engineering Foundation, New York (1984), pp. 305–314, AIChE Symp. Ser. No. 233, 80, 34 (1984)], networks [Avilés and LeVan, Chem. Eng. Sci., 46, 1935 (1991)], and general structures of constant cross section [Rudisill and LeVan, Ind. Eng. Chem. Res., 29, 1054 (1991)]. SQUARE ROOT SPREADING FOR LINEAR ISOTHERMS The simplest isotherm is n*i = c*i, corresponding to R = 1. For this isotherm, the rate equation for external mass transfer, the linear driving force approximation, or reaction kinetics, can be combined with Eq. (16-130) to obtain ∂n ∂c*i
*i = − ᎏ = N(c*i −
n*i ) (16-146) ᎏ ∂τ1 ∂ζ The solution to this equation, with initial condition n*i = 0 at τ1 = 0 and boundary condition c*i = 1 at ζ = 0, originally obtained for an analogous heat transfer case [Anzelius, Z. Angew Math. Mech., 6, 291 (1926); Schumann, J. Franklin Inst., 208, 405 (1929)], is c*i = J(Nζ, Nτ1)

n*i = 1 − J(Nτ1, Nζ)

(16-147)

where the J function is [Hiester and Vermeulen, Chem. Eng. Prog., 48, 505 (1952)] J(s, t) = 1 −

e s

−t − ξ

I0(2 tξ
) dξ

(16-148)

0

where I0 is the modified Bessel function of the first kind of order zero. This linear isotherm result can be generalized to remove the assumption that cˆ *i ≈ c*I if the throughput parameter is redefined as τ − εζ τ1 = ᎏᎏ (1 − ε)(ρp Ki + εp) The J function is plotted in Fig. 16-28 and tables are available (e.g., Sherwood et al., Mass Transfer, McGraw-Hill, New York, 1975). Vermeulen et al. (gen. refs.) discuss several approximations of the J function. For large arguments it approaches

FIXED-BED TRANSITIONS

FIG. 16-28

16-37

Plot of the function J(s, t) defined by Eq. (16-148).

1 J(s, t) = ᎏ erfc [ s
− t
] (16-149) 2 A derivation for particle-phase diffusion accompanied by fluid-side mass transfer has been carried out by Rosen [J. Chem. Phys., 18, 1587 (1950); ibid., 20, 387 (1952); Ind. Eng. Chem., 46, 1590 (1954)] with a limiting form at N > 50 of 1
N c*i = ᎏ erfc ᎏ (ζ − τ1) (16-150) 2 2 For axial dispersion in a semi-infinite bed with a linear isotherm, the complete solution has been obtained for a constant flux inlet boundary condition [Lapidus and Amundson, J. Phys. Chem., 56, 984 (1952); Brenner, Chem. Eng. Sci., 17, 229 (1962); Coates and Smith, Soc. Petrol. Engrs. J., 4, 73 (1964)]. For large N, the leading term is 1
N c*i = ᎏ erfc ᎏ (ζ − τ1) (16-151) 2 2
τ1 All of these solutions are very similar and show, for large N, a wave with breadth proportional to the square root of the bed depth through which it has passed.









COMPLETE SOLUTION FOR REACTION KINETICS In general, full time-dependent analytical solutions to differential equation-based models of the above mechanisms have not been found for nonlinear isotherms. Only for reaction kinetics with the constant separation factor isotherm has a full solution been found [Thomas, J. Amer. Chem. Soc., 66, 1664 (1944)]. Referred to as the Thomas solution, it has been extensively studied [Amundson, J. Phys. Colloid Chem., 54, 812 (1950); Hiester and Vermeulen, Chem. Eng. Progress, 48, 505 (1952); Gilliland and Baddour, Ind. Eng. Chem., 45, 330 (1953); Vermeulen, Adv. in Chem. Eng., 2, 147 (1958)]. The solution to Eqs. (16-130) and (16-130) for the same boundary conditions as Eq. (16-146) is

J(RNζ, Nτ1) c*i = ᎏᎏᎏᎏᎏ J(RNζ, Nτ1) + e−(R − 1)N(ζ − τ1)[1 − J(Nζ, RNτ1)] 1 − J(Nτ1, RNζ) (16-152) n*i = ᎏᎏᎏᎏᎏ J(RNζ, Nτ1) + e−(R − 1)N(ζ − τ1)[1 − J(Nζ, RNτ1)] The solution gives all of the expected asymptotic behaviors for large N—the proportionate pattern spreading of the simple wave if R > 1, the constant pattern if R < 1, and square root spreading for R = 1. NUMERICAL METHODS AND CHARACTERIZATION OF WAVE SHAPE For the solution of sophisticated mathematical models of adsorption cycles including complex multicomponent equilibrium and rate expressions, two numerical methods are popular. These are finite difference methods and orthogonal collocation. The former vary in the manner in which distance variables are discretized, ranging from simple backward difference stage models (akin to the plate theory of chromatography) to more involved schemes exhibiting little numerical dispersion. Collocation methods are often thought to be faster computationally, but oscillations in the polynomial trial function can be a problem. The choice of best method is often the preference of the user. For both the finite difference and collocation methods a set of coupled ordinary differential equations results which are integrated forward in time using the method of lines. Various software packages implementing Gear’s method are popular. The development of mathematical models is described in several of the general references [Guiochon et al., Rhee et al., Ruthven, Ruthven et al., Suzuki, Tien, Wankat, and Yang]. See also Finlayson [Numerical Methods for Problems with Moving Fronts, Ravenna Park, Washington, 1992; Holland and Liapis, Computer Methods for Solving Dynamic Separation Problems, McGraw-Hill, New York, 1982; Villadsen and Michelsen, Solution of Differential Equation Models by

16-38

ADSORPTION AND ION EXCHANGE

Polynomial Approximation, Prentice Hall, Englewood Cliffs, New Jersey, 1978]. For the characterization of wave shape and breakthrough curves, three methods are popular. The MTZ method [Michaels, Ind. Eng. Chem., 44, 1922 (1952)] measures the breadth of a wave between two chosen concentrations (e.g., c*i = 0.05 and 0.95 or c*i = 0.01 and 0.99). Outside of a laboratory, the measurement of full breakthrough curves is rare, so the breadth of the MTZ is often estimated from an independently determined stoichiometric capacity and a measured small

concentration in the “toe” of the breakthrough curve. A second method for characterizing wave shape is by the slope of the breakthrough curve at its midheight (i.e., c*i = 0.5 [Vermeulen et al., gen. refs.]). The use of moments of the slope of breakthrough curves is a third means for characterization. They can often be used to extract numerical values for rate coefficients for linear systems [Ruthven, Suzuki, gen. refs.; Nauman and Buffham, Mixing in Continuous Flow Systems, Wiley-Interscience, New York, 1983]. The method of moments is discussed further in the following part of this section.

CHROMATOGRAPHY CLASSIFICATION Chromatography is a sorptive separation process where a portion of a solute mixture (feed) is introduced at the inlet of a column containing a selective adsorbent (stationary phase) and separated over the length of the column by the action of a carrier fluid (mobile phase) that is continually supplied to the column following introduction of the feed. The mobile phase is generally free of the feed components, but may contain various other species introduced to modulate the chromatographic separation. The separation occurs as a result of the different partitioning of the feed solutes between the stationary phase; the separated solutes are recovered at different times in the effluent from the column. Chromatography is used both in analysis of mixtures and in preparative and process-scale applications. It can be used for both trace-level and for bulk separations both in the gas and the liquid phase. Modes of Operation The classical modes of operation of chromatography as enunciated by Tiselius [Kolloid Z., 105, 101 (1943)] are: elution chromatography, frontal analysis, and displacement development. Basic features of these techniques are illustrated in Fig.

16-29. Often, each of the different modes can be implemented with the same equipment and stationary phase. The results are, however, quite different in the three cases. Elution Chromatography The components of the mobile phase supplied to the column after feed introduction have less affinity for the stationary phase than any of the feed solutes. Under trace conditions, the feed solutes travel through the column as bands or zones at different velocities that depend only on the composition of the mobile phase and the operating temperature and that exit from the column at different times. Two variations of the technique exists: isocratic elution, when the mobile phase composition is kept constant, and gradient elution, when the mobile phase composition is varied during the separation. Isocratic elution is often the method of choice for analysis and in process applications when the retention characteristics of the solutes to be separated are similar and not dramatically sensitive to very small changes in operating conditions. Isocratic elution is also generally practical for systems where the equilibrium isotherm is linear or nearly linear. In all cases, isocratic elution results in a dilution of the separated products.

Modes of operation of chromatography for the separation of a mixture of two components A and B. Figures on the left represent a schematic of the column with sample passing through it. Top diagrams show column at end of feed step. Figures on the right show the corresponding effluent concentrations. C is either the eluent or the displacer (adapted from Ettre, 1980).

FIG. 16-29

CHROMATOGRAPHY In gradient elution, the eluting strength of the mobile phase is gradually increased after supplying the feed to the column. In liquid chromatography, this is accomplished by changing the mobile phase composition. The gradient in eluting strength of the mobile phase that is generated in the column is used to modulate the separation allowing control of the retention time of weakly and strongly retained components. A similar effect can be obtained in gas chromatography by modulating the column temperature. In either case, the column has to be brought back to the initial conditions before the next cycle is commenced. Generally, gradient elution is best suited for the separation of complex mixtures that contain both species that interact weakly with the stationary phase and species that interact strongly. Since the eluting strength of the mobile phase is adjusted continuously, weakly retained components of a mixture are separated in the initial phase when the relative elution strength of the mobile phase is low, while strongly retained components are separated later in the gradient when the elution strength is high. In addition, the technique is used to obtain reproducible chromatographic separations when the solute retention characteristics are extremely sensitive to the operating conditions, as in the case of the chromatography of biopolymers, such as proteins. These molecules are often found to be very strongly retained in an extremely small range of mobile phase compositions and completely unretained elsewhere, making it practically impossible to obtain reproducible isocratic separations. Frontal Analysis The feed mixture to be separated is continuously supplied to the column where the mixture components are competitively retained by the stationary phase. These solutes are partially separated in a series of fronts, preceded downstream by the least strongly retained species forming a pure component band, and upstream by the feed mixture. The technique is best suited for the removal of strongly adsorbed impurities present in trace amounts from an unretained or weakly retained product of interest. In this case, a large amount of feed can be processed before the impurities begin to break through. When this point is reached, the bed is washed to remove any desired product from the interstitial voids, and the adsorbent is regenerated. The method can only provide a single component in pure form, but avoids product dilution completely. Multicomponent separations require a series of processing steps; either a

series of frontal analysis separations, or a combination of elution and displacement separations. Example 10 illustrates bed concentration profiles for the frontal analysis separation of two components. Displacement Development The column is partially loaded with the feed mixture as in frontal analysis, usually for conditions where all solutes of interest are strongly and competitively retained by the stationary phase. The feed supply is then stopped and a mobile phase containing the displacer, a component that has an affinity for the stationary phase stronger than any of the feed components, is fed to the column. The advancement of the displacer front through the column causes desorption of the feed components and their competitive readsorption downstream of the displacer front. As in frontal chromatography, the less strongly retained species tends to migrate faster down the column concentrating in a band farthest from the displacer front, while the most strongly adsorbed solute tends to move more slowly concentrating in a band adjacent to the displacer front. If the column is sufficiently long, all feed components eventually become distributed into a pattern of adjacent pure component bands where each upstream component acts as a displacer for each downstream species located in the band immediately downstream. When this occurs, all bands in the displacement train move at the same velocity which is equal to the velocity of the displacer front and the bed concentration profile is called an isotachic pattern. The various operational steps of a displacement development separation are shown in Fig. 16-30. Ideally, the separated species exit the column as adjacent rectangular bands in order of increasing affinity for the stationary phase as shown in this figure. In practice, dispersion effects result in a partial mixing of adjacent bands requiring recycling of portions of the effluent that do not meet purity requirements. Following separation, the displacer has to be removed from the column with a suitable regenerant and the initial conditions of the column restored before the next cycle. Column regeneration may consume a significant portion of the cycle, when removal of the displacer is difficult. Displacement chromatography is suitable for the separation of multicomponent bulk mixtures. For dilute multicomponent mixtures it allows a simultaneous separation and concentration. Thus, it permits the separation of compounds with extremely low separation factors without the excessive dilution that would be obtained in elution techniques.

Operational steps in displacement chromatography. The column, initially equilibrated with a carrier solvent at time 0, is loaded with feed until time tF and supplied with displacer for a time tD + tE. Development of the displacement train occurs during the time tD and elution of the separated products ends at time tE. tR is the time required to remove the displacer from the column and restore the initial conditions. Components are numbered in order of decreasing affinity for the stationary phase. [Reference: Horvath et al., J. Chromatogr., 218, 365 (1981). Reprinted with permission of J. Chromatogr.] FIG. 16-30

16-39

16-40

ADSORPTION AND ION EXCHANGE

Other modes of operation, including recycle and flow reversal schemes and continuous chromatography, are discussed in Ganetsos and Barker (Preparative and Production Scale Chromatography, Marcel Dekker, New York, 1993).

The number of plates, Np, and the height equivalent to a theoretical plate, HETP, are defined as measures of dispersion effects as: L µ21 Np = ᎏ = ᎏ (16-157) HETP σ2

CHARACTERIZATION OF EXPERIMENTAL CHROMATOGRAMS

σ2L HETP = ᎏ (16-158) µ21 A high number of plates and a low HETP indicate a high column efficiency. Higher moments can also be computed and used to define the skewness of the response peak. However, difficulties often arise in such computations as a result of drifting of the detection system. In practice, experimental peaks can be affected by extracolumn retention and dispersion factors associated with the injector, connections, and any detector. For linear chromatography conditions, the apparent response parameters are related to their corresponding true column value by

Method of Moments The first step in the analysis of chromatographic systems is often a characterization of the column response to small pulse injections of a solute under trace conditions in the Henry’s law limit. For such conditions, the statistical moments of the response peak are used to characterize the chromatographic behavior. Such an approach is generally preferable to other descriptions of peak properties which are specific to Gaussian behavior, since the statistical moments are directly correlated to equilibrium and dispersion parameters. Useful references are Schneider and Smith [AIChE J., 14, 762 (1968)], Suzuki and Smith [Chem. Eng. Sci., 26, 221 (1971)], and Carbonell et al. [Chem. Eng. Sci., 9, 115 (1975); 16, 221 (1978)]. The relevant moments are: µ0 =



∞

ci dt

(16-153)



(16-154)

0

1 µ1 = ᎏ µ0

∞

cit dt

0



∞

1 ci(t − µ1)2 dt (16-155) σ2 = ᎏ µ0 0 where ci is the peak profile. µ0 represents the area, µ1 the mean residence time, and σ2 the variance of the response peak. Moments can be calculated by numerical integration of experimental profiles. The retention factor is defined as a dimensionless peak locator as: µ1 − µ01 k′i = ᎏ (16-156) µ01 where µ10 is the first moment obtained for an inert tracer which is excluded from the stationary phase. k′i is a partition ratio representing the equilibrium ratio of the amount of solute in the stationary phase (including any pores) and the amount in the external mobile phase. Another commonly used definition of the retention factor uses as a reference the first moment of an inert that has access to all the pores.

inj conn µapp + µ1det 1 = µ1 + µ1 + µ1

σ

2app

=σ +σ 2

2inj

+σ

2conn

(16-159)

+σ

2det

(16-160)

Approximate Methods For certain conditions, symmetrical, Gaussian-like peaks are obtained experimentally. Such peaks may be empirically described by: Qi /Fv t − tRi ci = ᎏ exp − ᎏ σ
2
π σ 2




Q /F N N t ᎏ exp − ᎏ  ᎏ − 1 

 =ᎏ t 2π 2 t 2

i

v

p

Ri

2

p

Ri

(16-161) where Qi is the amount of solute injected, Fv is the volumetric flow rate, and tRi is the peak apex time. The relationships between the moments and other properties of such peaks are shown in Fig. 16-31. For such peaks, approximate calculations of the number of plates can be done with the following equations: tRi 2 Np = 5.54 ᎏ (16-162) ∆

  t N = 16 ᎏ  W Ri

2

(16-163)

p



cmax i tRi Np = 2π ᎏ µ0

FIG. 16-31 Properties of a Gaussian peak. cimax is the peak height; tRi, the peak apex time; σ, the standard deviation; ∆, the peak width at midheight; and W, the distance between the baseline intercepts of the tangents to the peak.



2

(16-164)

CHROMATOGRAPHY where ∆ is the peak width at half peak height and W is the distance between the baseline intercepts of the tangents to the inflection points of the peak. In general, Gaussian behavior can be tested by plotting the cumut lative fractional recovery 0 ci dt/µ0 versus time on probability-linear coordinates; if the plot is linear, Gaussian behavior is confirmed. For nearly Gaussian peaks, calculations of Np based on Eqs. (16-162) to (16-164) provide results close to those obtained with a rigorous calculation of moments. When deviations from Gaussian behavior are significant, however, large errors can be obtained using these expressions. Tailing Peaks Tailing peaks can be obtained experimentally when the column efficiency is very low, when there are large extracolumn dispersion effects, when the stationary phase is heterogeneous (in the sense that it contains different adsorption sites), or when the adsorption equilibrium deviates from the Henry’s law limit. Asymmetrical tailing peaks can some times be described empirically by an exponentially modified Gaussian (EMG) defined as the convolute integral of a Gaussian constituent with mean time tG and standard deviation σG and an exponential decay with time constant τG [Grushka, Anal. Chem., 44, 1733 (1972)]: Qi /Fv ci = ᎏ σGτG
2
π



∞

0

t − tG − t′ exp − ᎏ σG
2





2



t′ − ᎏ dt′ τG

(16-165)

Although a numerical integration is required to compute the peak profile, the moments are calculated directly as µ1 = tG + τG and σ2 = σG2 + τG2 and the peak skew as: 1 Peak skew = ᎏ3 µ0σ



∞

0

2(τG /σG)3 ci(t − µ1)3 dt ∼ ᎏᎏ [1 + (τG /σG)2]3/2

(16-166)

For EMG peaks, peak skew increases with the ratio τG /σG. Figure 16-32 illustrates the characteristics of such a peak calculated for τG /σG = 1.5. In general, with τG /σG > 1 (peak skew > 0.7), a direct calculation of the moments is required to obtained a good approximation of the true value of Np, since other methods give a large error (Yau et al., Moderns Size-Exclusion Liquid Chromatography, Wiley, New York, 1979). Alternatively, Eq. (16-165) can be fitted to experimental peaks to determine the optimum values of tG, σG, and τG. In practice, the calculation of peak skew for highly tailing peaks is rendered difficult by baseline errors in the calculation of third moments. The peak asymmetry factor, As = b/a, at 10 percent of peak height (see Fig. 16-32) is thus frequently used. An approximate relationship between peak skew and As for tailing peaks, based on data in Yau et al. is: Peak skew = [0.51 + 0.19/(As − 1)]−1. Values of As < 1.25

(corresponding to peak skew < 0.7) are generally desirable for an efficient chromatographic separation. Resolution The chromatographic separation of two components, A and B, under trace conditions with small feed injections can be characterized in terms of the resolution, Rs. For nearly Gaussian peaks: 2(tR, A − tR, B) ∆tR Rs = ᎏᎏ ∼ᎏ WA + WB 4σAB

(16-167)

where ∆tR is the difference in retention time of the two peaks and σAB = (σA + σB)/2 is the average of their standard deviations. When Eq. (16-161) is applicable, the resolution for two closely spaced peaks is approximated by: 1 α − 1
ki′ Rs = ᎏ ᎏ ᎏ N
p
2 α + 1 1 +
k′i α − 1 k′A ∼ᎏᎏ N

p, 4 1 + k′A

for α ∼ 1

(16-168)

where α = k′A /k′B and
k′ = (k′A + k′B)/2. Equation (16-168) shows that the resolution is the result of independent effects of the separation selectivity (α), column efficiency (Np), and capacity (k
′). Generally, peaks are essentially completely resolved when Rs = 1.5 (>99.5 percent separation). In practice, values of Rs ∼ 1, corresponding to 98 percent separation, are often considered adequate. The preceding equations are accurate to within about 10 percent for feed injections that do not exceed 40 percent of the final peak width. For large, rectangular feed injections, the baseline width of the response peak is approximated by: W ∼ 4σ + tF

(16-169)

where 4σ is the baseline width obtained with a small pulse injection and tF is the duration of feed injection. In this case, the resolution is defined as (see Ruthven, gen. refs., pp. 324–331): tR, A − tR, B − tF Rs = ᎏᎏ 4σAB

(16-170)

For strongly retained components (k
′ >> 1), the number of plates required to obtain a given resolution with a finite feed injection is approximated by: 2 α+1 2 tF Np = 4R2s ᎏ 1+ᎏ (16-171) α−1 4σABRs





Exponentially modified Gaussian peak with τG /σG = 1.5. The graph also shows the definition of the peak asymmetry factor As at 10 percent of peak height.

FIG. 16-32

16-41



16-42

ADSORPTION AND ION EXCHANGE

PREDICTION OF CHROMATOGRAPHIC BEHAVIOR The conservation equations and the rate models described in the “Rate and Dispersion Factors” can normally be used for a quantitative description of chromatographic separations. Alternatively, plate models can be used for an approximate prediction, lumping together all dispersion contributions into a single parameter, the HETP or the number of plates [Sherwood et al., Mass Transfer, McGraw-Hill, New York, 1975, p. 576; Dondi and Guiochon, Theoretical Advancements in Chromatography and Related Techniques, NATO-ASI, Series C: Mathematical and Physical Sciences, vol. 383, Kluwer, Dordrecht, 1992, pp. 1–61]. Exact analytic solutions are generally available for linear isocratic elution under trace conditions (see Dondi and Guiochon, ibid., Ruthven [gen. refs., pp. 324–335], Suzuki [gen. refs., pp. 224– 243]). Other cases generally require numerical solution (see Guiochon et al., gen. refs.) or approximate treatments with simplified rate models. Isocratic Elution In the simplest case, feed with concentration cFi is applied to the column for a time tF followed by the pure carrier fluid. Under trace conditions, for a linear isotherm with external masstransfer control, the linear driving force approximation or reaction kinetics (see Table 16-12), solution of Eq. (16-146) gives the following expression for the dimensionless solute concentration at the column outlet: c*i = J(N, Nτ1) − J(N, Nτ′1) (16-172) where N is the number of transfer units given in Table 16-13 and τ1 = (εvt/L − ε)/[(1 − ε)(ρp Ki + εp)] the throughput parameter (see “Square Root Spreading for Linear Isotherms” in “Fixed Bed Transitions”). τ′1 represents the value of τ1 with time measured from the end of the feed step. Thus, the column effluent profile is obtained as the difference between a breakthrough profile for a feed started at t = 0 and another for a feed started at t = tF. The behavior predicted by this equation is illustrated in Fig. 16-33 with N = 80. τF = (εvtF /L)/[(1 − ε)(ρp Ki + εp)] is the dimensionless duration of the feed step and is equal to the amount of solute fed to the column divided by the sorption capacity. Thus, at τF = 1, the column has been supplied with an amount of solute equal to the stationary phase capacity. The graph shows the transition from a case where complete saturation of the bed occurs before elution (τF = 1) to incomplete saturation as τF is progressively reduced. The lower curves with τF ≤ 0.4 are seen to be nearly Gaussian and centered at a dimensionless time τm ∼ (1 − τF /2). Thus, as τF → 0, the response curve approaches a Gaussian centered at τ1 = 1.

When τF is small (<<0.4), the solution for a feed pulse represented by a Dirac’s delta function at ζ = 0 can be used in lieu of Eq. (16-172). In terms of the dimensionless concentration c*i = ci /cFi (Sherwood et al., ibid., pp. 571–577): τF N −N −Nτ1 c*i = ᎏ e e I1(2N
τ1) (16-173)
τ1 where I1 is the Bessel function of imaginary argument. When N is larger than 5, this equation is approximated by: τF c*i = ᎏ 2
π

N exp [−N(
τ1 − 1)2]

ᎏ ᎏᎏᎏ

 τ (τ ) 1

(16-174)

The behavior predicted by Eq. (16-174) is shown in Fig. 16-34 as c*i /(τF N
) versus τ1 for different values of N. For N > 50, the response peak is symmetrical, the peak apex occurs at τ1 = 1, and the dimensionless peak height is c*i max = τF
N
/4
π. A comparison of this equation with Eq. (16-172) with N = 80 is shown in Fig. 16-33 for τF = 0.05. The moments of the response peak predicted by Eq. (16-173) are 1−ε L L µ1 = ᎏ 1 + ᎏ (εp + ρp Ki) = ᎏ (1 + k′i ) (16-175) ε v v



2 1







2

2µ k′i (16-176) σ2 = ᎏ ᎏ N 1 + k′i where k′i = (1 − ε)(εp + ρp Ki)/ε. Correspondingly, the number of plates and the HETP are: N 1 + k′ 2 Np = ᎏ ᎏi (16-177) 2 k′i









k′i 2 2L (16-178) HETP = ᎏ ᎏ N 1 + k′i Since the term (1 + k′i )/k′i approaches unity for large k′-value, the number of plates is equal to one half the number of transfer units for a strongly retained component. For these conditions, when Np = N/2, Eq. (16-174) and Eq. (16-161) produce the same peak retention time, peak spreading, and predict essentially the same peak profile. In the general case of axially dispersed plug flow with bidispersed particles, the first and second moment of the pulse response are [Haynes and Sarma, AIChE J., 19, 1043 (1973)]: 1−ε L L µ1 = ᎏ 1 + ᎏ (εp + ρp Ki) = ᎏ (1 + k′i ) ε v v



Elution curves under trace linear equilibrium conditions for different feed loading periods and N = 80. Solid lines, Eq. (16-172); dashed line, Eq. (16-174) for τF = 0.05.

FIG. 16-33

1/4

1



(16-179)

CHROMATOGRAPHY

FIG. 16-34

16-43

Elution curves under trace linear equilibrium conditions with a pulse feed from

Eq. (16-174).

2LDL σ2 = ᎏ (1 + k′i)2 v3 r p2 r 2s ρpKi L εk′i 2 rp +2ᎏᎏ ᎏ + ᎏ + ᎏᎏ ᎏ (16-180) 2 1 − ε v 3kf 15εp Dpi (εp + ρp Ki) 15Dsi Correspondingly, the number of plates and the plate height are: 2ε v 2DL k′i 2 Np = ᎏ + ᎏ ᎏ ᎏ 1 − ε L 1 + k′i vL











r p2 rp ρpKi r 2s ⫻ ᎏ + ᎏ + ᎏᎏ ᎏ 2 3kf 15εpDpi (εp + ρpKi) 15Dsi





2εv 2DL k′i HETP = ᎏ + ᎏ ᎏ 1 − ε 1 + k′i v

−1



(16-181)



2

r p2 r 2s rp ρpKi ⫻ ᎏ + ᎏ + ᎏᎏ ᎏ (16-182) 2 3kf 15εpDpi (εp + ρpKi) 15Dsi In dimensionless form, a reduced HETP, h = HETP/dp, analogous to the reduced HTU (cf. Fig. 16-13), is obtained as a function of the dimensionless velocity ReSc: b h = ᎏ + a + cReSc (16-183) ReSc





where b = 2εγ1

(16-184a)

a = 2γ2 1 1 k′i c=ᎏᎏ ᎏ 30 1 − ε 1 + k′i



 2

(16-184b) 10 τp r 2s Dpi ρp Ki ᎏ + ᎏ + ᎏᎏ2 ᎏ ᎏ Sh εp (εp + ρp Ki) Dsi r p2



(16-184c) Equation 16-183 is qualitatively the same as the van Deemter equation [van Deemter and Zuiderweg, Chem. Eng. Sci., 5, 271 (1956)] and is equivalent to other empirical reduced HETP expressions such as the Knox equation [Knox, J. Chromatogr. Sci., 15, 352 (1977)]. The Sherwood number, Sh, is estimated from Table 16-9, and the dispersion parameters γ1 and γ2 from Table 16-10 for well-packed columns. Typical values are a = 1–4 and b = 0.5–1. Since HETP ∼ 2HTU, Fig. 16-13 can also be used for approximate calculations. Concentration Profiles In the general case but with a linear isotherm, the concentration profile can be found by numerical inversion of the Laplace-domain solution of Haynes and Sarma [see

Lenhoff, J. Chromatogr., 384, 285 (1987)] or by direct numerical solution of the conservation and rate equations. For the special case of no-axial dispersion with external mass transfer and pore diffusion, an explicit time-domain solution, useful for the case of time-periodic injections, is also available [Carta, Chem. Eng. Sci., 43, 2877 (1988)]. In most cases, however, when N > 50, use of Eq. (16-161), or (16-172) and (16-174) with N = 2Np calculated from Eq. (16-181) provides an approximation sufficiently accurate for most practical purposes. When the adsorption equilibrium is nonlinear, skewed peaks are obtained, even when N is large. For a constant separation-factor isotherm with R < 1 (favorable), the leading edge of the chromatographic peak is steeper than the trailing edge. When R > 1 (unfavorable), the opposite is true. Figure 16-35 portrays numerically calculated chromatographic peaks for a constant separation factor system showing the effect of feed loading on the elution profile with R = 0.5 (τ1 − τF = [εv(t − tF)/ L − ε]/Λ). When the dimensionless feed time τF = 1, the elution curve comprises a sharp leading profile reaching the feed concentration followed by a gradual decline to zero. As τF is reduced, breakthrough of the leading edge occurs at later times, while the trailing edge, past the peak apex, continues to follow the same profile. As the amount of feed injected approaches zero, mass-transfer resistance reduces the solute concentration to values that fall in the Henry’s law limit of the isotherm, and the peak retention time gradually approaches the value predicted in the infinite dilution limit for a linear isotherm. For high feed loads, the shape of the diffuse trailing profile and the location of the leading front can be predicted from local equilibrium theory (see “Fixed Bed Transitions”). This is illustrated in Fig. 16-35 for τF = 0.4. For the diffuse profile (a “simple wave”), Eq. (16-131) gives: 1/ 2 R 1 c*i = ᎏ ᎏᎏ −1 (16-185) 1 − R R(τ1 − τF) Thus, the effluent concentration becomes zero at τ1 − τF = 1/R. The position of the leading edge (a “shock front”) is determined from Eq. (16-132): 1 τ1s = τF + ᎏ [1 −
(1
−
R
)τF
]2 (16-186) R and the peak highest concentration by [Golshan-Shirazi and Guiochon, Anal. Chem., 60, 2364 (1988)]:
(1
−
R
)τF
R c*i max = ᎏ ᎏᎏ (16-187) 1 − R 1 − (1

−
R)τ

F







16-44

ADSORPTION AND ION EXCHANGE

Elution curves under trace conditions with a constant separation factor isotherm for different feed loadings and N = 80. Solid lines, rate model; dashed line, local equilibrium theory for τF = 0.4.

FIG. 16-35

The local equilibrium curve is in approximate agreement with the numerically calculated profiles except at very low concentrations when the isotherm becomes linear and near the peak apex. This occurs because band-spreading, in this case, is dominated by adsorption equilibrium, even if the number of transfer units is not very high. A similar treatment based on local equilibrium for a two-component mixture is given by Golshan-Shirazi and Guiochon [ J. Phys. Chem., 93, 4143 (1989)]. Prediction of multicomponent nonlinear chromatography accounting for rate factors requires numerical solution (see Guiochon et al., gen. refs., and “Numerical Methods and Characterization of Wave Shape” in “Fixed Bed Transitions”). Linear Gradient Elution Analytical solutions are available for special cases under trace conditions with a linear isotherm. Other situations normally require a numerical solution. General references are Snyder [in Horvath (ed.), High Performance Liquid Chromatography: Advances and Perspectives, vol. 1, Academic Press, 1980, p. 208], Antia and Horvath [ J. Chromatogr., 484, 1 (1989)], Yamamoto et al. [Ion Exchange Chromatography of Proteins, Marcel Dekker, 1988], Guiochon et al. (gen. refs.). The most commonly used gradients are linear gradients where the starting solvent is gradually mixed with a second gradient-forming solvent at the column entrance to yield a volume fraction ϕ of the mobile phase modulator that increases linearly with time: ϕ = ϕ0 + βt

(16-188)

When the mobile phase modulator is a dilute solute in a solvent, as when the modulator is a salt, ϕ indicates molar concentration. Under trace conditions, the retention of the modulator in the column is independent of the presence of any solutes. The modulator concentration at the column exit is approximated by L ϕ = ϕ0 + β t − ᎏ (1 + k′M ) (16-189) v where k′M is the retention factor of the modulator. For a small feed injection, the modulator concentration ϕR at which a feed solute is eluted from the column is obtained from the following integral relationship ϕR βL dϕ G(ϕR) = ᎏ = (16-190) ᎏᎏ ϕ0 v k′i(ϕ) − k′M where k′i(ϕ) is the solute retention factor as a function of the mobile







phase modulator concentration. Note that the solute retention time in the column is affected by the steepness of the gradient at the column entrance. k′i(ϕ) can be obtained experimentally from isocratic elution experiments at different ϕ values, or from linear gradient elution experiments where the ratio G = βL/v is varied. In the latter case, the retention factor is obtained by differentiation of Eq. (16-190) from k′i (ϕR) = k′M + (dG/dϕR)−1. Table 16-14 gives explicit expressions for chromatographic peak properties in isocratic elution and linear gradient elution for two cases. In reversed-phase chromatography (RPC), the mobile phase modulator is typically a water-miscible organic solvent, and the stationary phase is a hydrophobic adsorbent. In this case, the logarithm of solute retention factor is commonly found to be linearly related to the volume fraction of the organic solvent. In ion-exchange chromatography (IEC), the mobile phase modulator is typically a salt in aqueous solution, and the stationary phase is an ion-exchanger. For dilute conditions, the solute retention factor is commonly found to be a power-law function of the salt normality [cf. Eq. (16-27) for ion-exchange equilibrium]. Band broadening is also affected by the gradient steepness. This effect is expressed in Table 16-14 by a band compression factor C, which is a function of the gradient steepness and of equilibrium parameters. Since C < 1, gradient elution yields peaks that are sharper than those that would be obtained in isocratic elution at ϕ = ϕR. Other cases, involving an arbitrary relationship between the solute retention factor and the modulator concentration can be handled analytically using the approaches of Frey [Biotechnol. Bioeng., 35, 1055 (1990)] and Carta and Stringfield [ J. Chromatogr., 605, 151 (1992)]. Displacement Development A complete prediction of displacement chromatography accounting for rate factors requires a numerical solution since the adsorption equilibrium is nonlinear and intrinsically competitive. When the column efficiency is high, however, useful predictions can be obtained with the local equilibrium theory (see “Fixed Bed Transitions”). For constant-separation factor systems, the h-transformation of Helfferich and Klein (gen. refs.) or the method of Rhee et al. [AIChE J., 28, 423 (1982)] can be used [see also Helfferich, Chem. Eng. Sci., 46, 3320 (1991)]. The equations that follow are adapted from Frenz and Horvath [AIChE J., 31, 400 (1985)] and are based on the h-transformation. They refer to the separation of a mixture of M − 1

CHROMATOGRAPHY

16-45

TABLE 16-14 Expressions for Predictions of Chromatographic Peak Properties in Linear Gradient Elution Chromatography under Trace Conditions with a Small Feed Injection and Inlet Gradient Described by j = jo + bt (Adapted from Refs. A and B). Parameter

Isocratic

Dependence of retention factor on modulator concentration

—

Gradient elution—RPC (Ref. A)

L tR = ᎏ (1 + k′o) v

Retention time

Gradient elution—IEC (Ref. B)

k′ − k′M = αse−Sϕ

k′ − k′M = αzϕ−z





L 1 SβL tR = ᎏ (1 + k′M) + ᎏ ln ᎏ (k′o − k′M) + 1 Sβ v v





L 1 αZβL tR = ᎏ (1 + k′M) + ᎏ ᎏ (Z + 1) + ϕ oZ + 1 β v v







ϕR = ϕo

L ϕR = ϕo + β tR − ᎏ (1 + k′M) v

L ϕR = ϕo + β tR − ᎏ (1 + k′M) v

Retention factor at peak elution

k′R = k′o

k′o + (k′o − k′M)SβL/v k′R = ᎏᎏᎏ 1 + (k′o − k′M)SβL/v

αZβL k′R = αZ ᎏ (Z + 1) + ϕoZ + 1 v L/v σ = C ᎏ (1 + k′R)
NpR


L/v σ = ᎏ (1 + k′o)
NP


L/v σ = C ᎏ (1 + k′R)
NpR


Band compression factor

—

(1 + p + p2/3)1/2 C = ᎏᎏ 1+p (k′o − k′M)(1 + k′M)SβL/v p = ᎏᎏᎏ 1 + k′o



C=



ϕo −ᎏ β



Mobile phase composition at column exit

Peak standard deviation

1/(Z + 1)



M
′


+ k′M

M′ < 0.25

for

3.22M′ ᎏᎏ 1 + 3.13M′ 1

−Z/(Z + 1)

for

0.25 < M′ < 0.25

for

M′ > 120.25

1 1 + k′R Z + 1 M′ = ᎏ ᎏ ᎏ 2 1 + k′M Z References: A. Snyder in Horvath (ed.), High Performance Liquid Chromatography: Advances and Perspectives, vol. 1, Academic Press, 1980, p. 208. B. Yamamoto, Biotechnol. Bioeng., 48, 444 (1995). Solute equilibrium parameters: αs,S for RPC and αZ,Z for IEC Solute retention factor for initial mobile phase: k′o Retention factor of mobile phase modulator: k′M Plate number obtained for k′ = k′R: NpR

components with a displacer (component 1) that is more strongly adsorbed than any of the feed solutes. The multicomponent Langmuir isotherm [Eq. (16-39)] is assumed valid with equal monolayer capacities, and components are ranked numerically in order of decreasing affinity for the stationary phase (i.e., K1 > K2 > ⋅ ⋅ ⋅ KM). The development of component bands is predicted by mapping the trajectories of the h-function roots hi, which are obtained from the solution of M K i ci (16-191) ᎏ=1  i = 1 hαi, 1 − 1 where α i, j = K i /K j. Trivial roots also exist for each component with zero concentration. In displacement chromatography, for each of the transitions at the column entrance, a new set of M roots is generated. These roots are given in Table 16-15. The change from the initial solvent (carrier) to feed changes M − 1 roots, generating M − 1 boundaries or transitions that move through the column. These boundaries are all selfsharpening, characterized by upstream h″i values larger than the corresponding downstream h′i values. The switch from feed to displacer changes M roots, generating M boundaries. All of these boundaries are self-sharpening, except the boundary associated with the transition from hMF to α1, M + 1 that can be self-sharpening or diffuse depending on the relative value of these

roots. Note that only one root changes value across a particular boundary. The adjusted propagation velocity of each self-sharpening boundary is: dζ usi = ᎏ = h′i h″i Pi (16-192) dτ1 i−1

Pi =  h′j j=1

M+1

M



h″j

j=i+1



α j, 1

(16-193)

j=1

and that of each diffuse boundary: uhi = h2iPi

(16-194)

Solution

i

hi

i

hi

where ζ = z/L and τ1 = (εvt/L − εbζ)/Λ with Λ = ρbnD1 /c1D equal to the partition ratio for the displacer. The root trajectories are mapped as follows on the ζ-τ1 plane: 1. The starting points are the beginning and end of the feed step. From these points trajectories are traced according to Eqs. (16-192) and (16-194). 2. At the points of intersection of trajectories, a change in boundary velocity occurs: When two boundaries associated with a change in the same root number intersect, they combine into a single boundary with velocity given by Eq. (16-192) or (16-194), intermediate in value between those of the intersecting trajectories. When two trajectories associated with different root numbers intersect, they both continue beyond the point of intersection with new velocities calculated from Eqs. (16-192) and (16-194), reflecting any change in root values. 3. After constructing the trajectories, the following equation is used to calculate the band profiles ∏ Mi= 1 (hiαj, 1 − 1) cj = ᎏᎏ (16-195) Kj ∏ Mi= 1 (αj,i − 1)

Carrier Feed Displacer

1, 2, ⋅⋅⋅⋅ M 1 2, 3, ⋅⋅⋅⋅ M

α1,1, ⋅⋅⋅⋅ α1,M α1,1 α1,2, ⋅⋅⋅⋅ α1,M

— 2, ⋅⋅⋅⋅ M 1

— h2F, ⋅⋅⋅⋅hMF* α1,M + 1 = 1 + K1c1D

Example 12: Calculation of Band Profiles in Displacement Chromatography An equimolar mixture of two components (concentra-

TABLE 16-15 Concentrations and h-Function Roots for Displacement Chromatography of a Mixture of M-1 Components Numbered in Order of Decreasing Affinity for the Stationary Phase (Adapted from Frenz and Horvath, 1985). ci = 0

ci > 0

*Roots calculated from the solution of Eq. (16-191).

tions c2F = c3F = 1 arbitrary unit) is separated with a displacer with concentration c1F = 2. The equilibrium isotherm is:

16-46

ADSORPTION AND ION EXCHANGE n sKici ni = ᎏᎏᎏ 1 + K1c1 + K2c2 + K3c3

(i = 1, 2, 3)

with K1 = 2, K2 = 1, and K3 = 0.667. The dimensionless feed time is τF = 0.2. The separation factors used for calculating the trivial roots are α1, 1 = 1, α1, 2 = 2, α1, 3 = 3, α1, 4 = 1 + K1c1D = 5. The remaining h-function roots are calculated by solving Eq. (16-191) for the carrier, feed, and displacer concentrations. The results are summarized in Table 16-16. TABLE 16-16

Values of h-Roots for Example 16-12

State

h1

h2

h3

Carrier Feed Displacer

1 1 2

2 2.44 3

3 6.56 5

The switch from carrier to feed generates self-sharpening boundaries, since the h″i s (upstream) are larger than the h′i s (downstream). The switch from feed to displacer generates two self-sharpening

boundaries (corresponding to h1 and h2) and a diffuse boundary (corresponding to h3) since h″3 < h′3 . The velocity of each boundary is calculated from Eq. (16-192) for self-sharpening boundaries and from Eq. (16-194) for diffuse boundaries. The root trajectories are shown in Fig. 16-36 (top). From one boundary to the next, the root values change one at a time from the values corresponding to the carrier to the values corresponding to the displacer. Concentration profiles calculated from Eq. (16-195) are shown in Fig. 16-36 (bottom) for two different values of ζ. The concentrations in the z and t coordinates are easily reconstructed from the definitions of τ1 and ζ. In general, for a constant separation factor system, complete separation is obtained when: 1 c1D > ᎏ (α1, M − 1) (16-196) K1 Then, band profiles eventually develop into the isotachic pattern of pure component bands moving at the velocity of the displacer front. In Example 12, this occurs at ζ = 0.765.

FIG. 16-36 Dimensionless time-distance plot for the displacement chromatography of a binary mixture. The darker lines indicate self-sharpening boundaries and the thinner lines diffuse boundaries. Circled numerals indicate the root number. Concentration profiles are shown at intermediate dimensionless column lengths ζ = 0.43 and ζ = 0.765. The profiles remain unchanged for longer column lengths.

CHROMATOGRAPHY The isotachic concentrations c Ii in the fully developed train are calculated directly from the single component isotherms using: n1e n2e nMe (16-197) ᎏI = ᎏI = ⋅ ⋅ ⋅ = ᎏ c1 c2 cMI 1 cjI = c1D − ᎏ (1 − αj, 1) (16-198) Kj The graphical interpretation of Eq. (16-197) is shown in Fig. 16-37 for the conditions of Example 12. An operating line is drawn from the origin to the point of the pure displacer isotherm at c1 = c1D. For displacement to occur, the operating line must cross the pure component isotherms of the feed solutes. The product concentrations in the isotachic train are found where the operating line crosses the isotherms. When this condition is met, the feed concentrations do not affect the final product concentrations. Analyses of displacement chromatography by the method of characteristics with non-Langmuirian systems is discussed by Antia and Horvath [ J. Chromatogr., 556, 199 (1991)] and Carta and Dinerman [AIChE J., 40, 1618 (1994)]. Optimization studies and analyses by computer simulations are discussed by Jen and Pinto [ J. Chromatogr., 590, 47 (1992)], Katti and Guiochon [ J. Chromatogr., 449, 24 (1988)], and Phillips et al. [ J. Chromatogr., 54, 1 (1988)]. or

DESIGN FOR TRACE SOLUTE SEPARATIONS The design objectives of the analyst and the production line engineer are generally quite different. For analysis the primary concern is typically resolution. Hence operating conditions near the minimum value of the HETP or the HTU are desirable (see Fig. 16-13). In preparative chromatography, however, it is generally desirable to reduce capital costs by maximizing the productivity, or the amount of feed processed per unit column volume, subject to specified purity requirements. This reduction, however, must be balanced against operating costs that are determined mainly by the mobile phase flow rate and the pressure drop. In practice, preparative chromatography is often carried out under overload conditions, i.e., in the nonlinear region of the adsorption isotherm. Optimization under these conditions is discussed in Guiochon et al. (gen. refs.). General guidelines for trace-level, isocratic binary separations, in the Henry’s law limit of the isotherm are: 1. The stationary phase is selected to provide the maximum selectivity. Where possible, the retention factor is adjusted (by varying the mobile phase composition, temperature, or pressure) to an optimum value that generally falls between 2 and 10. Resolution is adversely affected when k′ << 2, while product dilution and separation time

16-47

increase greatly when k′ >> 10. When this is not possible for all feed components and large differences exist among the k′-values of the different solutes, gradient elution should be considered. 2. The average feed mixture charging rate, molar or volumetric, is fixed by the raw material supply or the demand for finished product. 3. The value of Np required to achieve a desired resolution is determined by Eq. (16-168) or (16-171). Since N = L/HTU ∼ 2Np = 2L/HETP, Fig. 16-13 or Eq. (16-183) can be used to determine the range of the dimensionless velocity ReSc that maximizes Np for a given particle diameter and column length. 4. The allowable pressure drop influences the choice of the particle size and helps determine the column length. Equations for estimating the pressure drop in packed beds are given in Section 6. 5. When only a few solutes are separated, they may occupy only a small portion of the total column volume at any given instant. In such cases, the productivity is improved by cyclic feed injections, timed so that the most strongly retained component from an injection elutes just before the least strongly retained component from the following injection (see Fig. 16-57). For a mixture of two components with
k′ > 1, when the same resolution is maintained between bands of the same injections and bands of successive injections, the cycle time tC and the plate number requirement are: 2L tC = 2(tR,A − tR,B) = ᎏ (k′A − k′B) (16-199) v α+1 2 (16-200) Np = 4R2s ᎏ (1 − 2φ)−2 α−1 where φ = tF /tC is the fraction of the cycle time during which feed is supplied to the column. The productivity, P = volume of feed/(time × bed volume), is: 1 α − 1 2 εv P = ᎏ2 ᎏ ᎏ φ(1 − 2φ)2 4R s α + 1 HETP









ReSc 1 α−1 2 D = ᎏ2 ᎏ ᎏ2i ᎏᎏᎏ φ(1 − 2φ)2 (16-201) 4R s α + 1 dp b/ReSc + a + cReSc For a given resolution, P is maximized when φ = 1/6 (i.e., feed is supplied for one sixth of the cycle time), and by the use of small particle sizes. The function ReSc/(b/ReSc + a + cReSc) generally increases with ReSc, so that productivity generally increases with the mobile phase velocity. For typical columns, however, this function is within about 10 percent of its maximum value (∼ 1/c) when ReSc is in the range 30–100. Thus, increasing the velocity above this range must be balanced against the costs associated with the higher pressure drop.



 

FIG. 16-37 Schematic showing the intersection of the operating line with the pure-component isotherms in displacement chromatography. Conditions are the same as in Fig. 16-36.



16-48

ADSORPTION AND ION EXCHANGE

PROCESS CYCLES GENERAL CONCEPTS Some applications of adsorption and ion exchange can be achieved by sorbent-fluid contact in batch equipment. Batch methods are well adapted to laboratory use and have also been applied on a larger scale in several specific instances. In a batch run for either adsorption or ion exchange, a sorbent is added to a fluid, completely mixed, and subsequently separated. Batch treatment is adopted when the capacity and equilibrium of the sorbent are large enough to give nearly complete sorption in a single step, as in purifying and decoloration of laboratory preparations with carbons and clays. Batch runs are useful in the measurement of equilibrium isotherms and also of adsorptive diffusion rate. Batch tests or measurements are often conducted in portions of adsorbent or ion-exchange material intended for larger-scale use. For example, either the equilibrium sorption of a solid and/or its ultimate sorption capacity can be determined in this way. The solid is first equilibrated with a fluid at the concentration of interest; after separating the phases, gravimetric and chemical analyses can be used to determine the sorbed amount and composition. Like the laboratory batch use, some commercial applications use the adsorbent on a throwaway basis. Reasons for using sorption nonregeneratively are usually: (1) low cost of the sorbent, (2) high value of the product, (3) very low dosage (sorbent-to-fluid ratio), and (4) difficulty in desorbing the sorbates. Magnesium perchlorate and barium oxide are used for drying, iron sponge (hydrated iron oxide on wood chips) is used to remove hydrogen sulfide, and sodium or potassium hydroxide is applied to removal of sulfur compounds or carbon dioxide. In wastewater treatment, powdered activated carbon (PAC) is added to enhance biological treatment but is not regenerated; instead, it remains with the sludge. Silica gel is used as a desiccant in packaging. Activated carbon is used in packaging and storage to adsorb other chemicals for preventing the tarnishing of silver, retarding the ripening or spoiling of fruits, “gettering” (scavenging) out-gassed solvents from electronic components, and removing odors. Synthetic zeolites, or blends of zeolites with silica gel, are used in dual-pane windows to adsorb water during initial dry-down and any in-leakage and to adsorb organic solvents emitted from the sealants during their cure; this prevents fogging between the sealed panes that could result from the condensation of water or the solvents [Ausikaitis, Glass Digest, 61, 69–78 (1982)]. Activated carbon is used to treat recirculated air in office buildings, apartments, and manufacturing plants using thin filter-like frames to treat the large volumes of air with low pressure drop. On a smaller scale, activated carbon filters are in kitchen hoods, air conditioners, and electronic air purifiers. On the smallest scale, gas masks containing carbon or carbon impregnated with promoters are used to protect individual wearers from industrial odors, toxic chemicals, and gas-warfare chemicals. Activated carbon fibers have been formed into fabrics for clothing to protect against vesicant and percutaneous chemical vapors [Macnair and Arons in Cheremisinoff and Ellerbusch (eds.), gen. refs.]. Ion exchangers are sometimes used on a throwaway basis also. In the laboratory, ion exchangers are used to produce deionized water, purify reagents, and prepare inorganic sols. In medicine, they are used as antacid, for sodium reduction, for the sustained release of drugs, in skin-care preparations, and in toxin removal. Although there are many practical applications for which the sorbent is discarded after one use, most applications of interest to chemical engineers involve the removal of adsorbates from the sorbent (i.e., regeneration). This allows the adsorbent to be reused and the adsorbates to be recovered. The maximum efficiency that a cyclic adsorption process can approach for any given set of operating conditions is defined by the adsorptive loading in equilibrium with the feed fluid. There are several factors that reduce the practical (or “operating”) adsorption: mass-transfer resistance (see above), deactivation (see above), and incomplete regeneration (or desorption). The severity of regeneration influences how closely the dynamic capacity of an adsorbent resembles that of fresh, virgin material. Regeneration, or reversal of the

adsorption process, requires a reduction in the driving force for adsorption. This is accomplished by increasing the equilibrium driving force for the adsorbed species to desorb from the solid to the surrounding fluid. TEMPERATURE SWING ADSORPTION A temperature-swing, or thermal-swing, adsorption (TSA) process cycle is one in which desorption takes place at a temperature much higher than adsorption. The elevation of temperature is used to shift the adsorption equilibrium and affect regeneration of the adsorbent. Figure 16-38 depicts a simplified (and ideal) TSA cycle. The feed fluid containing an adsorbate at a partial pressure of p1 is passed through an adsorbent at temperature T1. This adsorption step continues until the equilibrium loading n1 is achieved with p1. Next the adsorbent temperature is raised to T2 (heating step) so that the partial pressure in equilibrium with n1 is increased to p2, creating a partial pressure driving force for desorption into any fluid containing less than p2 of the adsorbate. By means of passing a purge fluid across the adsorbent, adsorbate is swept away, and the equilibrium proceeds down the isotherm to some point such as p1, n2. (This point need not coincide with the feed partial pressure; it is selected for illustrative purposes. Also, in some applications, roll-up of the adsorbed-phase concentration can occur during heating such that in some regions of the bed p2 reaches the condensation pressure of the component, causing a condensed liquid phase to form temporarily in particles [Friday and LeVan, AIChE J., 31, 1322 (1985)].) During a cooling step, the adsorbent temperature is returned to T1. The new equilibrium p3, n2 represents the best-quality product that can be produced from the adsorbent at a regenerated loading of n2 in the simplest cycle. The adsorption step is now repeated. The differential loading, n1 − n2, is the maximum loading that can be achieved for a TSA cycle operating between a feed containing p1 at temperature T1, regeneration at T2, and a product containing a partial pressure p3 of the adsorbate. The regeneration fluid will contain an average partial pressure between p2 and p1 and will therefore have accomplished a concentration of the adsorbate in the regenerant fluid. For liquid-phase adsorption, the partial pressure can be replaced by the fugacity of the adsorbate. Then, the entire discussion above is applicable whether the regeneration is by a fluid in the gas or liquid phase. In a TSA cycle, the heating step must provide the thermal energy necessary to raise the adsorbate, adsorbent, and adsorber temperatures, to desorb the adsorbate, and to make up for heat losses. TSA regeneration is classified as (1) heating-limited (or stoichiometriclimited) when transfer of energy to the system is limiting, or (2) stripping-limited (or equilibrium-limited) when transferring adsorbate away is limiting. Heating is accomplished by either direct contact of the adsorbent by the heating medium (external heat exchange to a purge gas) or indirect means (heating elements, coils, or panels inside the adsorber). Direct heating is the most commonly used, especially for stripping-limited heating. Indirect heating can be considered for stripping-limited heating, but the complexity of indirect heating limits its practicality to heating-limited regeneration where purge gas is in short supply. Microwave fields [Benchanaa, Lallemant, Simonet-Grange, and Bertrand, Thermochim. Acta, 152, 43–51 (1989)] and dielectric fields [Burkholder, Fanslow, and Bluhm, Ind. Eng. Chem. Fundam., 25, 414–416 (1986)] are also used to supply indirect heating. Because high temperatures can be used, resulting in thorough desorption, TSA cycles are characterized by low residual loadings and thus high operating loadings. These high capacities at low concentrations allow long cycle times for reasonably sized adsorbers (hours to days). Long cycle time is needed because particles of adsorbent respond slowly to changes in gas temperature. Most applications of TSA are for systems in which adsorbates are present at low concentration (purification), such as drying, and in which species are strongly adsorbed, such as sweetening, CO2 removal, and pollution control.

PROCESS CYCLES

FIG. 16-38

16-49

Ideal temperature swing cycle. (Reprinted with permission of UOP.)

Other Cycle Steps Besides the necessary adsorption and heating steps, TSA cycles may employ additional steps. A purge or sweep gas removes the thermally desorbed components from the adsorbent, and cooling returns it to adsorption temperature. Although the cooling is normally accomplished as a separate step after the heating, sometimes adsorption is started on a hot bed. If certain criteria are met [Basmadjian, Can. J. Chem. Eng., 53, 234–238 (1975)], the dynamic adsorption efficiency is not significantly affected by the lack of a cooling step. For liquid-phase adsorption cycles when the unit is to treat a product of significant value, there must be a step to remove the liquid from the adsorbent and one to displace any liquid regenerant thoroughly before filling with the valuable fluid. Because adsorbents and ion exchangers are porous, some retention of the product is unavoidable but needs to be minimized to maximize recovery. When regeneration is by a gas, removal and recovery is accomplished by a drain (or pressure-assisted drain) step using the gas to help displace liquid from the sorbent before heating. When the regenerant is another liquid, the feed or product can be displaced out of the adsorbent. When heating and cooling are complete, liquid feed must be introduced again to the adsorbent with a corresponding displacement of the gas or liquid regenerant. In ion exchange, these steps for draining and filling are commonly referred to as “sweetening off” and “sweetening on,” respectively. Applications Drying is the most common gas-phase application for TSA. The natural gas, chemical, and cryogenics industries all use adsorbents to dry streams. Zeolites, activated alumina, and silica gel are used for drying pipeline natural gas. Alumina and silica gel are used because they have higher equilibrium capacity and are more easily regenerated with waste-level heat [Crittenden, Chem. Engr., 452, 21–24 (1988); Goodboy and Fleming, Chem. Eng. Progr., 80, 63–68 (1984); Ruthven, Chem. Eng. Progr., 84, 42–50 (1988)]. The low dewpoint that can be achieved with zeolites is especially important when drying cryogenic-process feed-streams to prevent freeze-up. Zeolites dry natural gas before liquefaction to liquefied natural gas (LNG) and before ethane recovery utilizing the cryogenic turboexpander process [Anderson in Katzer (ed.), Molecular Sieves—II, Am. Chem. Soc. Symp. Ser., 40, 637–649 (1977); Brooking and Walton, The Chem. Engr., 257, 13–17 (1972)]. The feed air to be cryogenically separated into N2, O2, and argon is purified of both water and CO2 with 13X zeolites using TSA cycles. Zeolites, silica gel, and activated alumina are used to dry synthesis gas, inert gas, hydrocracker gas, rare gases, and reformer recycle H2. Because 3A and pore-closed 4A zeolites sizeselectively adsorb water but exclude hydrocarbons, they are used extensively to dry reactive streams such as cracked gas in order to prevent coke formation on the adsorbent. This molecular sieving increases the recovery of hydrocarbons by reducing the coadsorption

that would otherwise cause them to be desorbed and lost with the water. Another area of application for TSA processes is in sweetening. H2S, mercaptans, organic sulfides and disulfides, and carbonyl sulfide all must be removed from natural gas, H2, biogas, and refinery streams in order to prevent corrosion and catalyst poisoning. Natural gas feed to steam methane reforming is sweetened in order to protect the sulfur-sensitive, low-temperature shift catalyst. Well-head natural gas is treated by TSA to prevent pipeline corrosion using 4A zeolites to remove sulfur compounds without the coadsorption of CO2 that would cause shrinkage. Sweetening and drying of refinery hydrogen streams are needed to prevent poisoning of reformer catalysts. Adsorption can be used to dry and sweeten these in the same unit operation. TSA processes are applied to the removal of many inorganic pollutants. CO2 is removed from base-load and peak-shaving natural-gas liquefaction facilities using 4A zeolite in a TSA cycle. The Sulfacid and Hitachi fixed-bed processes, the Sumitomo and BF moving-bed processes, and the Westvaco fluidized-bed process all use activated carbon adsorbents to remove SO2 from flue gases and sulfuric acid plant tail gases [Juentgen, Carbon, 15, 273–283 (1977)]. Activated carbon with a catalyst is used by the Unitaka process to remove NOx by reacting with ammonia, and activated carbon has been used to convert NO to NO2, which is removed by scrubbing. Mercury vapor from air and other gas streams is removed and recovered by activated carbon impregnated with elemental sulfur; the Hg is then recovered by thermal oxidation in a retort [Lovett and Cunniff, Chem. Eng. Progr., 70, 43–47 (1974)]. Applications for HCl removal from Cl2, chlorinated hydrocarbons, and reformer catalyst gas streams use TSA with mordenite and clinoptilolite zeolites [Dyer, gen. refs., pp. 102–105]. Activated aluminas are also used for HCl adsorption as well as fluorine and boron-fluorine compounds from alkylation processes [Crittenden, ibid.]. PRESSURE-SWING ADSORPTION A pressure-swing adsorption (PSA) process cycle is one in which desorption takes place at a pressure much lower than adsorption. Reduction of pressure is used to shift the adsorption equilibrium and affect regeneration of the adsorbent. Figure 16-39 depicts a simplified pressure-swing cycle. Feed fluid containing an adsorbate at a molar concentration of y1 = p1 /P1 is passed through an adsorbent at conditions T1, P1, and the adsorption step continues until the equilibrium loading n1 is achieved with y1. Next, the total pressure is reduced to P2 during the depressurization (or blowdown) step. Now, although the partial pressure in equilibrium with n1 is still p1, there is a concentration driv-

16-50

ADSORPTION AND ION EXCHANGE

ing force of y2 = p1 /P2 > y1 for desorption into any fluid containing less than y2. By passing a fluid across the adsorbent in a purge step, adsorbate is swept away, and the equilibrium proceeds down the isotherm to some point such as y1, n2. (The choice of y1 is arbitrary and need not coincide with feed composition.) At this time, the adsorbent is repressurized to P1. The new equilibrium y3, n2 represents the best quality product that can be produced from the adsorbent at a regenerated loading of n2. The adsorption step is now repeated. The differential loading, n1 − n2, is the maximum loading that can be achieved for a pressure-swing cycle operating between a feed containing y1 and a product containing a molar concentration y3 of the adsorbate. The regeneration fluid will contain an average concentration between y2 and y1 and will therefore have accomplished a concentration of the adsorbate in the regenerant gas. There is no analog for a liquid-phase PSA process cycle. Thus, in a PSA process cycle, regeneration is achieved by a depressurization that must reduce the partial pressure of the adsorbates to allow desorption. These cycles operate at constant temperature, requiring no heating or cooling steps. Rather, they use the exothermic heat of adsorption remaining in the adsorbent to supply the endothermic heat needed for desorption. Pressure-swing cycles are classified as: (1) PSA, which, although used broadly, usually swings between a high superatmospheric and a low superatmospheric pressure; (2) VSA (vacuum-swing adsorption), which swings from a superatmospheric pressure to a subatmospheric pressure; and (3) PSPP (pressure-swing parametric pumping) and RPSA (rapid pressure-swing adsorption), which operate at very fast cycle times such that significant pressure gradients develop in the adsorbent bed (see the subsection on parametric pumping). Otherwise, the broad principles remain the same. Low pressure is not as effective in totally reversing adsorption as is temperature elevation unless very high feed to purge pressure ratios are applied (e.g., deep vacuum). Therefore, most PSA cycles are characterized by high residual loadings and thus low operating loadings. These low capacities at high concentrations require that cycle times be short for reasonably sized beds (seconds to minutes). These short cycle times are attainable because particles of adsorbent respond quickly to changes in pressure. Major uses for PSA processes include purification as well as applications where contaminants are present at high concentration (bulk separations). Other Cycle Steps A PSA cycle may have several other steps in addition to the basic adsorption, depressurization, and repressurization. Cocurrent depressurization, purge, and pressure-equalization steps are normally added to increase efficiency of separation and recovery of product. At the end of the adsorption step, the more weakly adsorbed species have been recovered as product, but there is still a significant amount held up in the bed in the inter- and intra-

FIG. 16-39

particle void spaces. A cocurrent depressurization step can be added before the blowdown step, which is countercurrent to adsorption. This increases the amount of product produced each cycle. In some applications, the purity of the more strongly adsorbed components has also been shown to be heavily dependent on the cocurrent depressurization step [Cen and Yang, Ind. Eng. Chem. Fundam., 25, 758–767 (1986)]. This cocurrent blowdown is optional because there is always a countercurrent one. Skarstrom developed criteria to determine when the use of both is justified [Skarstrom in Li, Recent Developments in Separation Science, vol. II, CRC Press, Boca Raton, pp. 95– 106 (1975)]. Additional stripping of the adsorbates from the adsorbent and purging of them from the voids can be accomplished by the addition of a purge step. The purge can begin toward the end of the depressurization or immediately afterward. Purging is accomplished with a flow of product countercurrent to adsorption to provide a lower residual at the product effluent end of the bed. The repressurization step that returns the adsorber to feed pressure and completes the steps of a PSA cycle should be completed with pressure equalization steps to conserve gas and compression energy. Portions of the effluent gas during depressurization, blowdown, and enrichment purge can be used for repressurization to reduce the quantity of feed or product gas needed to pressurize the beds. The most efficient cycle is one that most closely matches available pressures and adsorbate concentration to the appropriate portion of the bed at the proper point in the cycle. Applications PSA cycles are used primarily for purification of wet gases and of hydrogen. One of the earliest applications was the original Skarstrom two-bed cycle (adsorption, countercurrent blowdown, countercurrent purge, and cocurrent repressurization) to dry air stream to less than 1 ppm H2O [Skarstrom, ibid.]. Instrument-air dryers still use a PSA cycle similar to Skarstrom’s with activated alumina or silica gel [Armond, in Townsend, The Properties and Applications of Zeolites, The Chemical Society, London, pp. 92–102 (1980)]. The hydrocarbon exclusion by small-pore zeolites allows PSA to achieve a 10 to 30 K dewpoint depression in air-brake compressors, even at high discharge air temperatures in the presence of compressor oil [Ausikaitis, in Katzer, Molecular Sieves—II, Am. Chem. Soc. Symp. Ser., 40, pp. 681–695 (1977)]. The high-purity hydrogen employed in processes such as hydrogenation, hydrocracking, and ammonia and methanol production is produced by PSA cycles with adsorbent beds compounded of activated carbon, zeolites, and carbon molecular sieves [Martin, Gotzmann, Notaro, and Stewart, Adv. Cryog. Eng., 31, 1071–1086 (1986)]. The impurities to be removed include ammonia, carbon oxides, nitrogen, oxygen, methane, and heavier hydrocarbons. In order to be able to produce purities as high as 99.9999

Ideal pressure swing cycle. (Reprinted with permission of UOP.)

PROCESS CYCLES percent, systems such as a UOPTM PolybedTM separation unit use six to ten adsorbers with complex cycles (see below) [Fuderer and Rudelstorfer, U.S. Patent 3,986,849 (1976)]. Air separation, methane enrichment, and iso/normal separations are the major bulk separations utilizing PSA; recovery of CO and CO2 are also growing uses. PSA process cycles are used to produce oxygen and/or nitrogen from air. Synthetic zeolites, clinoptilolite, mordenite, and carbon molecular sieves are all used in various PSA, VSA, and RPSA cycles. The 85 to 95 percent purity oxygen produced is employed for electric furnace steel, waste water treating, solid waste combustion, and kilns [Martin et al., ibid.]. Small PSA oxygen units are used for patients requiring inhalation therapy in the hospital and at home [Cassidy and Holmes, AIChE Symp. Ser., 80 (1984), pp. 68– 75] and for pilots on board aircraft [Tedor, Horch, and Dangieri, SAFE J., 12, 4–9 (1982)]. Lower purity oxygen (25 to 55 percent) can be produced to enhance combustion, chemical reactions, and ozone production [Sircar, in Rodrigues et al., gen. refs., pp. 285–321]. High purity nitrogen (up to 99.99 percent) for inert blanketing is produced in PSA and VSA processes using zeolites and carbon molecular sieves [Kawai and Kaneko, Gas Sep. & Purif., 3, 2–6 (1989); Ruthven, ibid.]. Methane is upgraded to natural gas pipeline quality by another PSA process. The methane is recovered from fermentation gases of landfills and wastewater purification plants and from poor-quality natural gas wells and tertiary oil recovery. Carbon dioxide is the major bulk contaminant but the gases contain water and other “garbage” components such as sulfur and halogen compounds, alkanes, and aromatics [Kumar and VanSloun, Chem. Eng. Progr., 85, 34–40 (1989)]. These impurities are removed by TSA using activated carbon or carbon molecular sieves and then the CO2 is adsorbed using a PSA cycle. The cycle can use zeolites or silica gel in an equilibrium-selective separation or carbon molecular sieve in a rate-selective separation [Kapoor and Yang, Chem. Eng. Sci., 44, 1723–1733 (1989); Richter, Erdoel Kohle, Erdgas, Petrochem., 40, 432–438 (1987)]. The pore-size selectivity of zeolite 5A is employed to adsorb straight-chain molecules while excluding branched and cyclic species in the UOP IsoSivSM process. This PSA process separates C5 to C9 range hydrocarbons into a normal-hydrocarbon fraction of better than 95 percent purity, and a higher-octane isomer fraction with less than 2 percent normals [Cassidy and Holmes, ibid.]. PURGE/CONCENTRATION SWING ADSORPTION A purge-swing adsorption cycle is usually considered to be one in which desorption takes place at the same temperature and total pressure as adsorption. Desorption is accomplished either by partialpressure reduction using an inert gas purge or by adsorbate displace-

FIG. 16-40

16-51

ment with another adsorbable component. Purge cycles operate adiabatically at nearly constant inlet temperature and require no heating or cooling steps. As with PSA, they utilize the heat of adsorption remaining in the adsorbent to supply the heat of desorption. Purge processes are classified as (1) inert or (2) displacement. Inert Purge In inert-purge desorption cycles, inert refers to the fact that the purge is not adsorbable at the cycle conditions. Inert purging desorbs the adsorbate solely by partial pressure reduction. Regeneration of the adsorbent can be achieved while maintaining the same temperature and pressure by the introduction of an inert purge fluid. Figure 16-40 depicts a simplified inert-purge swing cycle utilizing a nonadsorbing purge fluid. As before, the feed stream containing an adsorbate at a partial pressure of p1 is passed through an adsorbent at temperature T1, and the adsorption step continues until equilibrium n1 is achieved. Next the nonadsorbing fluid is introduced to reduce the partial pressure below p1 by dilution. Therefore, there is a partial pressure driving force for desorption into the purge fluid, and the equilibrium proceeds down the isotherm to the point p2, n2, where p2 represents the best quality product that can be produced from the adsorbent at a regenerated loading of n2. The adsorption step is now repeated, and the differential loading is n1 − n2. The regeneration fluid will contain an average partial pressure between p2 and p1, and the cycle will therefore not have accomplished a concentration of the adsorbate in the regenerant fluid. But it will have transferred the adsorbates to a fluid from which it may be more easily separated by means such as distillation. Like PSA cycles, inert-purge processes are characterized by high residual loadings, low operating loadings, and short cycle times (minutes). Bulk separations of contaminants not easily separable at high concentration and of weakly adsorbed components are especially suited to inert-purge-swing adsorption. Another version of UOP’s IsoSiv process employs H2 in an inert-purge cycle for separating C5 to C9 naphtha by adsorbing straight-chain molecules and excluding branched and cyclic species on size selective 5A zeolite [Cassidy and Holmes, ibid.]. Automobiles made in the United States have canisters of activated carbon to adsorb gasoline vapors lost from the carburetor or the gas tank during running, from the tank during diurnal cycling, and from carburetor hot-soak losses; the vapors are desorbed by an inert purge of air that is drawn into the carburetor as fuel when the engine is running [Clarke, Gerrard, Skarstrom, Vardi and Wade, S.A.E. Trans., 76, 824–842 (1968)]. UOP’s Adsorptive Heat Recovery drying system has been commercialized for drying azeotropic ethanol to be blended with gasoline into gasohol; the process uses a closed loop of N2 as the inert purge to desorb the water [Garg and Yon, Chem. Eng. Progr., 82, 54–60 (1986)]. Displacement Purge Isothermal, isobaric regeneration of the

Ideal inert-purge swing cycle. (Reprinted with permission of UOP.)

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ADSORPTION AND ION EXCHANGE

adsorbent can also be accomplished by using a purge fluid that can adsorb. In displacement-purge stripping, displacement refers to the displacing action of the purge fluid caused by its ability to adsorb at the cycle conditions. Figure 16-41 depicts a simplified displacementpurge swing cycle utilizing an adsorbable purge. Again, the feed stream containing an adsorbate at a partial pressure of p1 is passed through an adsorbent at temperature T1, and the adsorption step continues until equilibrium n1 is achieved. Next the displacement fluid, B, is introduced. The presence of another adsorbable species reduces the adsorptivity of the key adsorbate, A. Therefore, there exists a partial pressure driving force for desorption into the purge fluid, and the equilibrium proceeds down the isotherm to some point such as p1, n2 (again arbitrary.) Next the adsorbent is recharged with a fluid that contains no component B, shifting the effective isotherm to that where the equilibrium of component A is p3. The new equilibrium p3, n2 represents the best quality product that can be produced from the adsorbent at a regenerated loading of n2. The adsorption step is now repeated. The differential loading (n1 − n2) is the maximum loading that can be achieved for a pressure-swing cycle operating between a feed containing y1 and a product containing a partial pressure p3 of the adsorbate. The regeneration fluid will contain an average partial pressure between p2 and p1 and will therefore have accomplished a concentration of the adsorbate in the regenerant gas. Displacementpurge cycles are not as dependent on the heat of adsorption remaining on the adsorbent, because the adsorption of purge releases most or all of the energy needed to desorb the adsorbate. It is best if the adsorbate is more selectively adsorbed than the displacement purge, so that the adsorbates can easily desorb the purge fluid during adsorption. The displacement purge must be carefully selected, because it contaminates both the product stream and the recovered adsorbate, and requires separation for recovery (e.g., distillation). Displacement-purge processes are more efficient for less selective adsorbate/adsorbent systems, while systems with high equilibrium loading of adsorbate will require more purging [Sircar and Kumar, Ind. Eng. Chem. Proc. Des. Dev., 24, 358–364 (1985)]. Several displacement-purge-swing processes have been commercialized for the separation of branched and cyclic C10–C18 from normals using the molecular-size selectivity of 5A zeolite: Exxon’s Ensorb, UOP’s IsoSiv, Texaco Selective Finishing (TSF), Leuna Werke’s Parex, and the Shell Process [Ruthven, ibid.]. All use a purge of normal paraffin or light naphtha with a carbon number of two to four less than the feed stream except for Ensorb, which uses ammonia [Yang, gen. refs.]. UOP has also developed a similar process, OlefinSiv, which separates isobutylene from normal butenes with displacement purge and a sizeselective zeolite [Adler and Johnson, Chem. Eng. Progr., 75, 77–79

FIG. 16-41

(1979)]. Solvent extraction to regenerate activated carbon is another example of a displacement-purge cycle; the adsorbent is then usually steamed to remove the purge fluid [Martin and Ng, Water Res., 18, 59–73 (1984)]. The best use of solvent regeneration is for water phase adsorption where the separation of water from carbon would use too much steam and where purge and water are easily separated; and for vapor-phase where the adsorbate is highly nonvolatile but soluble. Air Products has developed a process for separating ethanol/water on activated carbon using acetone as a displacement agent and adding a water rinse to improve the recovery of two products [Sircar, U.S. Patent 5,026,482 (1991)]. Displacement-purge forms the basis for most simulated continuous countercurrent systems (see hereafter) such as the UOP SorbexSM processes. UOP has licensed close to one hundred Sorbex units for its family of processes: ParexSM to separate p-xylene from C8 aromatics, MolexSM for n-paraffin from branched and cyclic hydrocarbons, OlexSM for olefins from paraffin, SarexSM for fructose from dextrose plus polysaccharides, CymexSM for p- or m-cymene from cymene isomers, and CresexSM for p- or m-cresol from cresol isomers. Toray Industries’ AromaxSM process is another for the production of p-xylene [Otani, Chem. Eng., 80(9), 106–107, (1973)]. Illinois Water Treatment [Making Waves in Liquid Processing, Illinois Water Treatment Company, IWT Adsep System, Rockford, IL, 6(1), (1984)] and Mitsubishi [Ishikawa, Tanabe, and Usui, U.S. Patent 4,182,633 (1980)] have also commercialized displacement-purge processes for the separation of fructose from dextrose. Chromatography Chromatography is a sorptive separation technique that allows multicomponent separations in both gas and liquid phase. As a preparative tool, it is often used as a displacement-purge process, although many applications employ an inert-displacement mode, especially for use in analysis. General characteristics and operating modes are discussed in a separate part of this section. ION EXCHANGE Except in very small-scale applications, ion-exchangers are used in cyclic operations involving sorption and desorption steps. A typical ion-exchange cycle used in water-treatment applications involves (a) backwash—used to remove accumulated solids obtained by an upflow of water to expand (50–80 percent expansion is typical) and fluidize the exchanger bed; (b) regeneration—a regenerant is passed slowly through the used to restore the original ionic form of the exchanger; (c) rinse—water is passed through the bed to remove regenerant from the void volume and, in the case of porous exchangers, from the resin pores; (d) loading—the fresh solution to be treated is passed through

Ideal displacement-purge swing cycle. (Reprinted with permission of UOP.)

PROCESS CYCLES

washing separates the usually lighter anion exchanger from the usually denser cation exchanger. The column typically has a screened distributor at the interface between the two exchangers, so that they may be separately regenerated without removing them from the column. The most common cycle (Fig. 16-43) permits sequential regeneration of the two exchangers, first with alkali flowing downward through the anion exchanger to the interface distributor and then acid flowing downward from the interface distributor through the cation exchanger. After regeneration and rinsing, the exchangers are remixed by compressed air. To alleviate the problem of intermixing of the two different exchangers and chemical penetration through the wrong one, an inert material of intermediate density can be used to provide a buffer zone between layers of cation and anion exchangers. When recovery of the sorbed solute is of interest, the cycle is modified to include a displacement step. In the manufacture of pharmaceuticals, ion-exchangers are used extensively in recovery and separation. Many of these compounds are amphoteric and are positively or negatively charged depending on the solution pH. Thus, using for example a cation exchanger, loading can be carried out at a low pH and displacement at a high pH. Differences in selectivity for different species can be used to carry out separations during the displacement [Carta et al., AIChE Symp. Ser., 84, 54–61 (1988)]. Multibed cycles are also used to facilitate integration with other chemical process operations. Fig. 16-44 shows a two-bed ion exchange system using both cation and anion exchangers to treat and recover chromate from rinse water in plating operations. The cation exchanger removes trivalent chromium, while the anion exchanger removes hexavalent chromium as an anion. Regeneration of the cation exchanger with sulfuric acid produces a concentrated solution of trivalent chromium as the sulfate salt. The hexavalent chromium is eluted from the anion exchanger with sodium hydroxide in a concentrated solution. This solution is recycled to the plating tank by passing it through a second cation exchange column in hydrogen form to convert the sodium chromate to a dilute chromic acid solution that is concentrated by evaporation.

(b)

(a)

Ion-exchanger regeneration. (a) Conventional. Acid is passed downflow through the cation-exchange resin bed. (b) Counterflow. Regenerant solution is introduced upflow with the resin bed held in place by a dry layer of resin. FIG. 16-42

the bed until leakage begins to occur. Water softening is practiced in this way with a cation exchange column in sodium form. At the low ionic strength used in the loading step, calcium and magnesium are strongly preferred over sodium, allowing nearly complete removal. Since the selectivity for divalent cations decreases sharply with ionic concentration, regeneration is carried out effectively with a concentrated sodium chloride solution. Removal of sulfates from boiler feed water is done by similar means with anion exchangers in chloride form. Many ion-exchange columns operate downflow and are regenerated in the same direction (Fig. 16-42a). However, a better regeneration and lower leakage during loading can be achieved by passing the regenerant countercurrently to the loading flow. Specialized equipment is available to perform countercurrent regeneration (see “Equipment” in this section). One approach (Fig. 16-42b) is to apply a vacuum to remove the regenerant at the top of the bed. Complete deionization with ion-exchange columns is the classical method of producing ultrapure water for boiler feed, in electronics manufacture, and for other general uses in the chemical and allied industries. Deionization requires use of two exchangers with opposite functionality to remove both cations and anions. These can be in separate columns, packed in adjacent layers in the same column, or, more frequently, in a mixed bed. In the latter case, the two exchangers are intimately mixed during the loading step. For regeneration, back-

(a)

(b)

16-53

PARAMETRIC PUMPING The term parametric pumping was coined by Wilhelm et al. [Wilhelm, Rice, and Bendelius, Ind. Eng. Chem. Fundam., 5, 141–144 (1966)] to describe a liquid-phase adsorption process in which separation is achieved by periodically reversing not only flow but also an intensive thermodynamic property such as temperature, which influences adsorptivity. Moreover, they considered the concurrent cycling of pressure, pH, and electrical and magnetic fields. A lot of research and development has been conducted on thermal, pressure, and pH driven cycles, but to date only gas-phase pressure-swing parametric pumping has found much commercial acceptance.

(c)

(d)

(e)

FIG. 16-43 Principles of mixed-bed ion exchange. (a) Service period (loading). (b) Backwash period. (c) Caustic regeneration. (d) Acid regeneration. (e) Resin mixing.

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ADSORPTION AND ION EXCHANGE

FIG. 16-44 Multicomponent ion-exchange process for chromate recovery from plating rinse water. (Adapted from Rhom & Haas with permission.)

Flow diagram of desiccant cooling cycle. [Reprinted with permission of American Society of Heating, Refrigeration and Air Conditioning Engineers, Inc. (ASHRAE). Reference: Collier, Cohen, and Slosberg in Harrimam, Desiccant Cooling and Dehumidification, ASHRAE, Atlanta, 1992.]

FIG. 16-45

PROCESS CYCLES Temperature Two modes of temperature parametric-pumping cycles have been defined—direct and recuperative. In direct mode, an adsorbent column is heated and cooled while the fluid feed is pumped forward and backward through the bed from reservoirs at each end. When the feed is a binary fluid, one component will concentrate in one reservoir and one in the other. In recuperative mode, the heating and cooling takes place outside the adsorbent column. Parametric pumping, thermal and pH modes, have been widely studied for separation of liquid mixtures. However, the primary success for separating gas mixtures in thermal mode has been the separation of propane/ethane on activated carbon [Jencziewski and Myers, Ind. Eng. Chem. Fundam., 9, 216–221 (1970)] and of air/SO2 on silica gel [Patrick, Schrodt, and Kermode, Sep. Sci., 7, 331–343 (1972)]. The difficulty with applying the thermal mode to gas separation is that in a fixed-volume gas-pressure increases during the hot step, which defeats the desorption purpose of this step. No thermal parametricpumping cycle has yet been practiced commercially. Pressure Another approach to parametric pumping is accomplished by pressure cycling of an adsorbent. An adsorbent bed is alternately pressurized with forward flow and depressurized with backward flow through the column from reservoirs at each end. Like TSA parametric pumping, one component concentrates in one reservoir and one in the other. The pressure mode of parametric pumping has been called pressure-swing parametric pumping (PSPP) and rapid pressure swing adsorption (RPSA). It was developed to minimize process complexity and investment at the expense of product recovery. RPSA is practiced in single-bed [Keller and Jones in Flank, Adsorption and Ion Exchange with Synthetic Zeolites, 135 (1980), pp. 275–286] and multiple-bed [Earls and Long, U.S. Patent number 4,194,892, 1980] implementations. Adsorbers are short (about 0.3 to 1.3 m), and particle sizes are very small (about 150 to 400 mm). The total cycle time including adsorption, dead time, countercurrent purge, and sometimes a second dead time, ranges from a few to about 30 seconds. The feature of RPSA that differentiates it from traditional PSA is the existence of axial pressure profiles throughout the cycle much as temperature gradients are present in TSA parametricpumping. Whereas PSA processes have essentially constant pressure through the bed at any given time, the flow resistance of the very small adsorbent particles produce substantial pressure drops in the bed. These pressure dynamics are key to the attainment of separation performance. RPSA has been commercialized for the production of oxygen and for the recovery of ethylene and chlorocarbons (the selectively adsorbed species) in an ethylene-chlorination process while purging nitrogen (the less selectively adsorbed specie). OTHER ADSORPTION CYCLES Steam When steam is used for regeneration of activated carbon, it is desorbing by a combination thermal swing and displacement purge (described earlier in this section). The exothermic heat released when the steam is adsorbed supplies the thermal energy much more efficiently than is possible with heated gas purging. Slightly superheated steam at about 130°C is introduced into the bed countercurrent to adsorption; for adsorbates with high boiling points, the steam temperature must be higher. Adsorbates are desorbed and purged out of the bed with the steam. Steam and desorbates then go to a condenser for subsequent separation. The water phase can be further cleaned by air stripping, and the sorbate-laden air from that stripper can be recycled with the feed to the adsorption bed. Steam regeneration is most commonly applied to activated carbon that has been used in the removal and/or recovery of solvents from gases. At volatile organic compound (VOC) concentration levels from 500 to 15,000 ppm, recovery of the VOC from the stream used for regeneration is economically justified. Below about 500 ppm, recovery is not economically justifiable, but environmental concerns often dictate adsorption followed by destruction. While activated carbon is also used to remove similar chemicals from water and wastewater, regeneration by steam is not usual. The reason is that the watertreatment carbon contains 1 to 5 kg of water per kg of adsorbent that must be removed by drying before regeneration or an excessive amount of superheated steam will be needed. In water treatment,

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there can also be significant amounts of nonvolatile compounds that do not desorb during steam regeneration and that residual will reduce the adsorption working capacity. There is a growing use of reticulated styrene-type polymeric resins for VOC removal from air [Beckett, Wood, and Dixon, Environ. Technol., 13, 1129–1140 (1992); Heinegaard, Chem.-Ing.-Tech., 60, 907–908 (1988)]. LeVan and Schweiger [in Mersmann and Scholl (eds.), Fundamentals of Adsorption, United Engineering Trustees, New York (1991), pp. 487–496] tabulate reported steam utilizations (kg steam/kg adsorbate recovered) for a number of processes. Energy Applications Desiccant cooling is a means for more efficiently providing air conditioning for enclosures such as supermarkets, ice rinks, hotels, and hospitals. Adsorbers are integrated with evaporative and electric vapor compression cooling equipment into an overall air handling system. Air conditioning is comprised of two cooling loads, latent heat for water removal and sensible heat for temperature reduction. The energy savings derive from shifting the latent heat load from expensive compression cooling (chilling) to cooling tower load. Early desiccant cooling used adsorption wheels (see hereafter) impregnated with the hygroscopic salt, LiCl. More recently, these wheels are being fabricated with zeolite and/or silica gel. They are then incorporated into a system such as the example shown in Fig. 16-45 [Collier, Cohen, and Slosberg, in Harrimam, Desiccant Cooling and Dehumidification, ASHRAE, Atlanta (1992)]. Process air stream 6, to be conditioned, passes through the adsorbent wheel, where it is dried. This is a nonisothermal process due to the release of heat of adsorption and transfer of heat from a wheel that may be above ambient temperature. The dry but heated air (7) is cooled in a heat exchanger that can be a thermal wheel. This stream (8) is further cooled, and the humidity adjusted back up to a comfort range by direct contact evaporative cooling to provide supply air. Regeneration air stream 1, which can be ambient air or exhausted air, is evaporatively cooled to provide a heat sink for the hot, dry air. This warmed air (3) is heated to the desired temperature for regeneration of the adsorbent wheel and exhausted to the atmosphere. Many other combinations of drying and cooling are used to accomplish desiccant cooling [Belding, in Proceedings of AFEAS Refrigeration and Air Conditioning Workshop, Breckenridge, CO (June 23–25, 1993)]. Heat pumps are another developing application of adsorbents. Zeolite/water systems have been evaluated as a means for transferring heat from a low temperature to a higher, more valuable level. Both natural (chabazite and clinoptilolite) and synthetic (NaX and high silica NaY) zeolites have favorable properties for sorption heat pumps. Data have demonstrated that hydrothermally stable Na-mordenite and dealuminated NaY can be used with water in chemical heat pumps to upgrade 100°C heat sources by 50 to 80°C using a 20°C heat sink [Fujiwara, Suzuki, Shin, and Sato, J. Chem. Eng. Japan, 23, 738–743 (1990)]. Other work has shown that integration of two adsorber beds can achieve heating coefficients of performance of 1.56 for the system NaX/water, upgrading 150°C heat to 200°C with a 50°C sink [Douss, Meunier, and Sun, Ind. Eng. Chem. Res., 27, 310–316 (1988)]. Energy Conservation Techniques The major use of energy in an adsorption cycle is associated with the regeneration step, whether it is thermal energy for TSA or compression energy for PSA. Since the regeneration energy per pound of adsorbent tends to be constant, the first step in minimizing consumption is to maximize the operating loading. When the mass-transfer zone (MTZ) is a large portion of the adsorber relative to the equilibrium section, the fraction of the bed being fully utilized is small. Most fixed-bed adsorption systems have two adsorbers so that one is on stream while the other is being regenerated. One means of improving adsorbent utilization is to use a lead/trim (or cascade, or merry-go-round) cycle. Two (or more) adsorbent beds in series treat the feed. The feed enters the lead bed first and then the trim bed. The trim bed is the one that has most recently been regenerated. The MTZ is allowed to proceed through the lead beds but not to break through the trim bed. In this way the lead bed can be almost totally utilized before being regenerated. When a lead bed is taken out of service, the trim bed is placed in lead position, and a regenerated bed is placed in the trim position. A thermal pulse cycle is a means of conserving thermal energy in

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heating-limited desorption. A process cycle that is heat-limited needs only a very small time (dwell) at temperature to achieve satisfactory desorption. If the entire bed is heated before the cooling is begun, every part of the bed will dwell at temperature for the entire time it takes the cooling front to traverse the bed. Thus, much of the heat in the bed at the start of cooling would be swept from the bed. Instead, cooling is begun before any heat front has exited the bed, creating a thermal pulse that moves through the bed. The pulse expends its thermal energy for desorption so that only a small temperature peak remains at the end of the regeneration and no excess heat has been wasted. If the heating step is stripping-limited, a thermal pulse is not applicable. A series cool/heat cycle is another way in which the heat that is purged from the bed during cooling can be conserved. Sometimes the outlet fluid is passed to a heat sink where energy is stored to be reused to preheat heating fluid, or cross exchanged against the purge fluid to recover energy. However, there is also a process cycle that accomplishes the same effect. Three adsorbers are used, with one on adsorption, one on heating, and one on cooling. The regeneration fluid flows in series, first to cool the bed just heated and then to heat the bed to be desorbed. Thus all of the heat swept from the adsorber during heating can be reused to reduce the heating requirement. Unlike thermal pulse, this cycle is applicable to both heat- and strippinglimited heating. Process Selection The preceding sections present many process cycles and their variations. It is important to have some guidelines for design engineers to narrow their choice of cycles to the most economical for a particular separation. Keller and coworkers [Keller et al., gen. refs.] have presented a method for choosing appropriate adsorpTABLE 16-18

TABLE 16-17

Process Descriptors

Number

Statement

1 2

Feed is a vaporized liquid or a gas. Feed is a liquid that can be fully vaporized at less than about 200°C. Feed is a liquid that cannot be fully vaporized at 200°C. Adsorbate concentration in the feed is less than about 3 wt %. Adsorbate concentration in the feed is between about 3 and 10 wt %. Adsorbate concentration in the feed is greater than about 10 wt %. Adsorbate must be recovered in high purity (> than 90–99% rejection of nonadsorbed material). Adsorbate can be desorbed by thermal regeneration. Practical purge or displacement agents cannot be easily separated from the adsorbate.

3 4 5 6 7 8 9

Keller, Anderson, and Yon in Rousseau (ed.), Handbook of Separation Process Technology, John Wiley & Sons, Inc., New York, 1987; reprinted with permission.

tion processes. Their procedure considers the economics of capital, energy, labor, and other costs. Although these costs can vary from site to site, the procedure is robust enough to include most scenarios. In Table 16-17, nine statements are made about the character of the separation being considered. The numbers of the statements that are true (i.e., applicable) are used in the matrix in Table 16-18. A “no” for any true statement under a given process should remove that process from further consideration. Any process having all “yes” answers for true statements deserves strong consideration. Entries other than “yes” or “no” provide a means of prioritizing processes when more than one cycle is satisfactory.

Process Selection Matrix Gas- or vapor-phase processes

Liquid-phase processes

Statement number, Table 16-17

Temperature swing

Inert purge

Displacement purge

Pressure swing

Chromatography

Temperature swing*

Simulated moving bed

Chromatography

1 2 3 4 5 6 7 8 9

Yes Not likely No Yes Yes No Yes Yes Maybe§

Yes Yes No Yes Yes Yes Yes No Not likely

Yes Yes No Not likely Yes Yes Yes No Not likely

Yes Yes No Not likely Yes Yes Maybe† No N/A

Yes Yes No Not likely Yes Yes Yes No Not likely

No Yes Yes Yes No No Yes Yes‡ Maybe§

No Yes Yes Not likely Yes Yes Yes No Not likely

No Yes Yes Maybe Yes Yes Yes No Not likely

*Includes powdered, fixed-bed, and moving-bed processes. †Very high ratio of feed to desorption pressure (>10:1) will be required. Vacuum desorption will probably be necessary. ‡If adsorbate concentration in the feed is very low, it may be practical to discard the loaded adsorbent or reprocess off-site. §If it is not necessary to recover the adsorbate, these processes are satisfactory. Keller, Anderson, and Yon in Rousseau (ed.), Handbook of Separation Process Technology, John Wiley & Sons, Inc., New York, 1987; reprinted with permission.

EQUIPMENT ADSORPTION General Design Adsorbents are used in adsorbers with fixed inventory, with intermittent solids flow, or with continuous-moving solids flow. The most common are fixed beds operating as batch units or as beds of adsorbent through which the feed fluid passes, with periodic interruption for regeneration. Total systems consist of pressure vessels or open tanks along with the associated piping, valves, controls, and auxiliary equipment needed to accomplish regeneration of the adsorbent. Gas treating equipment includes blowers or compressors with a multiplicity of paths to prevent dead-heading. Liquid treating equipment includes pumps with surge vessels as needed to assure continuous flow. Adsorber Vessel The most frequently used method of fluid-solid contact for adsorption operations is in cylindrical, vertical vessels, with

the adsorbent particles in a fixed and closely but randomly packed arrangement. The adsorbers must be designed with consideration for pressure drop and must contain a means of supporting the adsorbent and a means of assuring that the incoming fluid is evenly distributed to the face of the bed. There are additional design considerations for adsorbers when the streams are liquid and for high-performance separation applications using very small particles (<0.05 mm) such as in HPLC. For most large-scale processes, adsorbent particle size varies from 0.06 to 6 mm (0.0025 to 0.25 in), but the adsorbent packed in a fixed bed will have a fairly narrow particle size range. Pressure drop in adsorbers can be changed by changing the diameter to bed depth ratio and by changing the particle size (see Sec. 5). Adsorbent size also determines separation performance of adsorbent columns—increasing efficiency with decreasing particle size. In liquid-phase process-

EQUIPMENT ing, total cost of the adsorption step can sometimes be reduced by designing for overall pressure drops as large as 300 to 600 kPa (45 to 90 psi) because pumping is not the major utility cost. In special, highresolution applications (HPLC), pressure drops as high as 5000– 25,000 kPa (800–4000 psi) are sometimes used requiring special pumping and column hardware [Colin in Ganetsos and Barker, Preparative and Production Scale Chromatography, Marcel Dekker, New York, 1993, pp. 11–45]. However, the cost of compressing gases is significant. Since blowers are limited to about 5 kPa (20 in wc) of lift, atmospheric gas applications are typically designed with adsorbent pressure drops of 1 to 4 kPa (4 to 16 in wc). To keep compression ratio low, compressed gas adsorption pressure drops are 5 to 100 kPa (0.7 to 15 psi) depending on the pressure level. Besides influencing how much pressure drop is allowable, the operating pressure determines other design features. When adsorption and/or regeneration is to be performed at pressures above atmospheric, the adsorber vessels are designed like process pressure vessels (Fig. 16-46 [EPA, gen. refs.]). Their flow distributors can consume more gas momentum at higher pressure. On the other hand, for applications near atmospheric pressure, any pressure drop can be costly, and most design choices are made in the direction of minimizing head loss. Beds have large face areas and shallow depth. Many times, the choice is to fabricate a horizontal (horizontal axis) vessel where flows are radial rather than axial as in conventional vertical beds. Figure 16-47 [Leatherdale in Cheremisinoff and Ellerbusch, gen. refs.] depicts how a rectangular, shallow adsorber bed is oriented in a horizontal vessel. Flow distributors, especially for large units, are often

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elaborate in order to evenly divide the flow rather than consume precious head. There are two types of support systems used for fixed beds of adsorbent. The first is a progressive series of grid- and screen-layers. In this system, each higher layer screen has successively smaller openings, with the last small enough to prevent particles from passing through. Each lower layer has greater strength. A series of I-beams can be used to support a layer of subway grating that, in turn, supports several layers of screening. In other cases, special support grills such as JohnsonTM screens may rest on the I-beams or on clips at the vessel wall and thus directly support the adsorbent. The topmost screen must retain the original size particles and some reasonable size of broken particles. The second type of support is a graded system of particles such as ceramic balls or gravel that rests directly on the bottom of the adsorber. A typical system might consist of 100 mm (4 in) of 50 mm (2 in) diameter material, covered by succeeding layers of 25, 12, and 6 mm (1, a, d in) of support material for a 3 mm (f in) adsorbent. In water treatment, the support may actually start with filter blocks and have an upper layer of sand (see Fig. 16-48 [EPA, gen. refs.]). If flow is not evenly distributed throughout the bed of adsorbent, there will be less than maximum utilization of the adsorbent during adsorption and of the desorption fluid during regeneration. Incoming fluids from the nozzles is at a much higher velocity than the average through the bed and may have asymmetric momentum components due to the piping manifold. The simplest means of allowing flow to redistribute across the face of the bed is to employ ample plenum space above and below the fixed bed. A much more cost-effective

FIG. 16-46 Pressurized adsorber vessel. (Reprinted with permission of EPA. Reference: EPA, Process Design Manual for Carbon Adsorption, U.S. Envir. Protect. Agency., Cincinnati, 1973.)

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ADSORPTION AND ION EXCHANGE

Ambient pressure adsorber vessel. (Reprinted with permission of Ann Arbor Science. Reference: Leatherdale in Cheremisinoff and Ellerbusch, Carbon Adsorption Handbook, Ann Arbor Science, Ann Arbor, 1978.)

FIG. 16-47

method is to install simple baffle plates with symmetrically placed inlet and outlet nozzles. The solid, or perforated, baffles are designed to break the momentum of the incoming fluid and redistribute it to prevent direct impingement on the adsorbent. When graded bed support is installed at the bottom, the baffles should be covered by screening to restrain the particles. An alternative to screened baffles is slotted metal or JohnsonTM screen distributors. Shallow horizontal beds often have such a large flow area that multiple inlet and outlet nozzles are required. These nozzle headers must be carefully designed to assure balanced flow to each nozzle. In liquid systems, a single inlet may enter the vessel and branch into several pipes that are often perforated along their length (Fig. 16-46). Such “spiders” and “Christmas trees” often have holes that are not uniformly spaced and sized but are distributed to provide equal flow per bed area. Although allowable pressure loss with liquids is not a restricting factor, there are special considerations for liquid treating systems. Activated carbon adsorbers used in water and wastewater treatment are designed and constructed using the same considerations used for turbidity removal by sand or multilayer filters. A typical carbon bed is shown in Fig. 16-48. Such contactors for liquids at ambient pressure are often nothing more than open tanks or concrete basins with flow distribution simply an overflow weir. In liquid treating, the adsorbers must be designed with a means for liquid draining and filling occasionally or during every cycle when a gas is used for regeneration. Draining is by gravity, sometimes assisted by a 70–140 kPa (10–20 psig) pressure pad. Even with time to drain, there will be significant liquid holdup to recover. As much as 40 cc of liquid per 100 g of adsorbent is retained in the micro- and macropores and bridged between particles. When drain is cocurrent to adsorption, the drained liquid can be recovered as treated product. When drain is countercurrent, drained fluid must be returned to feed surge. Minimizing other holdup in dead volume is especially important for liquid separation processes such as chromatography, because it adversely affects product recovery and regeneration efficiency. In filling an adsorber, there must be sufficient time for any gas trapped in the pores to escape. The fill step is preferably upflow to sweep the vapor out and to prevent gas pockets that could cause product contamination, bed lifting, or flow maldistribution. In liquid upflow, the buoyancy force of the liquid plus the pressure drop must not exceed the gravitational forces if bed lifting is to be prevented. Because there is very little increase in pressure drop beyond the lifting (or fluidization) velocity, some liquid systems are designed with bed expansion to limit pressure drop. Upflowadsorption expanded-beds are also preferred when the liquid contains any suspended solids, so that the bed does not act as a filter and

become plugged. Since increased expansion causes the adsorbent to become increasingly well mixed, with accompanying drop in removal efficiency, expansion is usually limited to about 10 percent. Higher velocities also tend to cause too much particle turbulence, abrasion, attrition, and erosion. Regeneration Equipment Sometimes it is economically justified to remove the adsorbent from the adsorber when it is exhausted and have an outside contractor regenerate it rather than install on-site regeneration equipment. This is feasible only if the adsorbent can treat feed for weeks or months rather than only hours or days. In other cases, the process conditions during regeneration are so much more severe than those for adsorption that a single regenerator with materials of construction capable of handling the conditions is more costeffective than to make all adsorbers the expensive material. This is

FIG. 16-48 Adsorber vessel with graded support system. (Reprinted with permission of EPA. Reference: EPA, Process Design Manual for Carbon Adsorption, U.S. Envir. Protect. Agency., Cincinnati, 1973.)

EQUIPMENT

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FIG. 16-49 Two-bed TSA system with regeneration equipment. (Reprinted with permission of GPSA. Reference: Engineering Data Book, 10th ed., Gas Processors Suppliers Association, Tulsa, 1988, Sec. 20, p. 22.)

true for most water and wastewater treatment with thermally reactivated carbon. Otherwise, desorption is conducted in situ with any additional equipment connected to the adsorbers. Figure 16-49 [Engineering Data Book, 10th ed., Gas Processors Suppliers Association, Tulsa, 1988, Sec. 20, p. 22] depicts the flow scheme for a typical two-bed TSA dryer system showing the auxiliary equipment associated with regeneration. Some of the dry product gas is externally heated and used countercurrently to heat and desorb water from the adsorber not currently drying feed. The wet, spent regeneration gas is cooled; the water is condensed out; and the gas is recycled to feed for recovery. The thermal reactivation of spent activated carbon may require the same high temperatures and reaction conditions used for its manufacture. Although the exact conditions to be used for reactivation depend on the type of carbon and the adsorbates to be removed, the objective is to remove the adsorbed material without altering the carbon structure. This occurs in four stages: (a) drying, (b) desorption, (c) pyrolysis and carbonization, and (d) burnoff. Each of these steps is associated with a particular temperature range and is carried out in a multiple hearth furnace such as that depicted in Fig. 16-50. This sixhearth system is a typical configuration for water treatment carbons. The gas temperature ranges from 100°C on the top hearth to 950°C at the bottom. The rotating rabble arms rake the carbon from hearth to hearth. Wet, spent carbon is fed to the top, where it is dried by the top two hearths. Pyrolysis (desorption and decomposition) is accomplished on the next hearth. The bottom four hearths are for reactivation. Varying amounts of air, steam, and/or fuel are added to the different hearths to maintain the conditions established for the particular reactivation. A hearth area requirement of 0.1 m2/kg/hr (0.5 ft2/ lb/hr) is adequate for cost estimating, but there are also detailed design procedures available [von Dreusche in Cheremisinoff and Ellerbusch, gen. refs.]. In addition to the multiple-hearth furnace, the reactivation system is comprised of additional equipment to transport, store, dewater, and quench the carbon. A typical two-bed PSA dryer process flow scheme is very similar to Fig. 16-49 without the regeneration gas heater. A portion of the dry product gas is used countercurrently to purge the water from the adsorber not currently drying feed. Again, the wet, spent purge gas is cooled, and the water is condensed out and recycled to feed for recovery. If the cycle is VSA, then instead of the regeneration gas compressor, there is a vacuum compressor at the purge outlet upstream of the regeneration cooler. Some PSA cycles such as in air-brake dryers operate with one adsorber, a surge volume, and a three-way valve to

switch between pressurizing and countercurrent blowdown. The other extreme of complexity is demonstrated by the UOP nine-bed Polybed PSA H2 unit shown in Fig. 16-51 [Fuderer and Rudelstorfer, U.S. Patent number 3,986,849, 1976]. This cycle requires nine adsorbers sequenced by fifty-five valves to maximize purity and recovery. Cycle Control Valves are the heart of cycle control for cyclic adsorption systems. These on/off valves switch flows among beds so

Multiple hearth furnace for carbon reactivation. (Reprinted with permission of EPA. Reference: EPA, Process Design Manual for Carbon Adsorption, U.S. Envir. Protect. Agency., Cincinnati, 1973.)

FIG. 16-50

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ADSORPTION AND ION EXCHANGE

UOP nine-bed polybed PSA H2 unit: (a) flow scheme; (b) cycle diagram. (Reference: Fuderer and Rudelstorfer, U.S. Patent number 3,986,849, 1976.) FIG. 16-51

that external to the system it appears as if operation is continuous. In general, one valve is needed for each bed at each end for each step that is performed (e.g., for a two-bed system with an adsorption step plus heating and cooling step [carried out in the same direction and during the same step], only 2 × 2 × 2 = 8 valves would be needed [see Fig. 16-49]). In some cycles such as pressure-swing systems, it may be possible to use valves for more than one function (e.g., repressurization with feed gas using the same manifold as adsorption feed). Without multiple use, the cycle in Fig. 16-51 would need 9 × 2 × 5 = 90 valves instead of 55 to accommodate the five steps of adsorption, purge to product, blowdown, purge, and feed repressurization (even without the equalization). For some applications, 3- and 4-way valves can replace two and four valves, respectively. The ultimate integration of switching valves is the UOP rotary valve discussed below. For long step times (8 hours or more) it is possible for the valves to be manually

switched by operators. For most systems, it is advantageous for the opening and closing of the valves be controlled by automatic timers. The same controller can be responsible for maintaining flows and pressure, logic for proceeding only on completion of events, and safety bypass or shutdown. Automatic control can provide for a period of parallel flow paths to assure transitions. In some applications, process analyzers can interface with the controller to initiate bedswitching when adsorbate is detected breaking through into the effluent. Continuous Countercurrent Systems Most adsorption systems use fixed-bed adsorbers. However, if the fluid to be separated and that used for desorption can be countercurrently contacted by a moving bed of the adsorbent, there are significant efficiencies to be realized. Because the adsorbent leaves the adsorption section essentially in equilibrium with the feed composition, the inefficiency of the

EQUIPMENT

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mass-transfer zone is eliminated. The adsorption section only needs to contain a MTZ length of adsorbent compared to a MTZ plus an equilibrium section in a fixed bed. Likewise, countercurrent regeneration is more efficient. Since the adsorbent is moved from an adsorption chamber to another chamber for regeneration, only the regeneration section is designed for the often more severe conditions. Countercurrent adsorption can take advantage of an exceptionally favorable equilibrium in water softening; and the regeneration step can be made favorable by the use of relatively concentrated elutent. Continuous units generally require more headroom but much less footprint. The foremost problems to be overcome in the design and operation of continuous countercurrent sorption operations are the mechanical complexity of equipment and the attrition of the sorbent. An early example of a commercial countercurrent adsorption processes is the hypersorption process developed by Union Oil Company for the recovery of propane and heavier components from natural gas [Berg, Chem. Eng. Progr., 47, 585–590 (1951)]. Hypersorption used an activated carbon adsorbent moving as a dense bed continuously downward through a rising gas stream (Fig. 16-52 [Berg, ibid.]). However, this process proved to be less economical than cryogenic distillation and had excessive carbon losses resulting from attrition. The commercialization by Kureha Chemical Co. of Japan of a new, highly attrition-resistant, activated-carbon adsorbent, allowed development of a process employing fluidized-bed adsorption and moving bed desorption for removal of VOC compounds from air. The process has been marketed as GASTAKSM in Japan and as PuraSivSM HR (Fig. 16-53 [Anon., Chem. Eng., 84(18), 39–40 (1977)]) in the United States, and now as SOLDACS by Daikin Industries, Ltd. A

PuraSiv HR adsorber vessel. [Reprinted with permission of Chemical Engineering. Reference: Anon., Chem. Eng., 84(18), 39–40 (1977).]

FIG. 16-53

Hypersorption adsorber vessel [Reprinted with permission of AIChE. Reference: Berg, Chem. Eng. Progr., 47, 585–590 (1951).]

FIG. 16-52

similar process using beaded polymeric resin is offered by Chematur [Heinegaard, Chem.-Ing.-Tech., 60, 907–908 (1988)]. The recent discovery [Acharya and BeVier, U.S. Patent number 4,526,877, 1985] that the graphite coating of zeolites can dramatically improve attrition resistance without significantly impairing adsorption performance could allow the extension of moving-bed technology to other separations. A good review of continuous ion-exchange applications is presented by Wankat [gen. refs.]. Continuous Cross-Flow Systems There are at least three implementations of moving-bed adsorption that are cross flow rather than fixed beds or countercurrent flow: (1) panel beds, (2) adsorbent wheels, and (3) rotating annular beds. By cross flow is meant that the adsorbent is moving in a direction perpendicular to the fluid flow. All of these employ moving adsorbent—the first, a down-flowing solid; and the others, a constrained solid. Panel beds of activated carbon like those depicted in Fig. 16-54 [Lovett and Cunniff, Chem. Eng. Progr., 70(5), 43–47 (1974)] have been applied to odor control [Lovett and Cunniff, ibid.] and to the desulfurization of waste gas [Richter, Knoblauch, and Juntgen, Verfahrenstechnik (Mainz), 14, 338–342 (1980)]. The spent solid falls from the bottom panel into a load-out bin, and fresh regenerated carbon is added to the top; gas flows through as shown. The heart of an adsorbent wheel system is a rotating cylinder containing the adsorbent. Figure 16-55 illustrates two types: horizontal and vertical. In some adsorbent wheels, the adsorbent particles are placed in basket segments (a multitude of fixed beds) to form a hori-

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ADSORPTION AND ION EXCHANGE

FIG. 16-54 Cross-flow panel-bed adsorber. [Reprinted with permission of AIChE. Reference: Lovett and Cunniff, Chem. Eng. Progr., 70(5), 43–47 (1974).]

zontal wheel that rotates around a vertical axis. In other instances, the adsorbent is integral to the monolithic wheel or coated onto a metal, paper, or ceramic honeycomb substrate. These monolithic or honeycomb structures rotate around either a vertical or a horizontal axis. The gas to be treated usually flows through the wheel parallel to the axis of rotation, although some implementations use radial flow con-

(a) FIG. 16-55

of UOP.)

figurations. Most of the wheel is removing adsorbates. The remaining (smaller) portion of the wheel is undergoing thermal regeneration— usually countercurrently. The wheel constantly rotates to provide a continuous treated stream and a steady concentrated stream. Adsorbent wheels are most often used to treat ambient air because they have very low pressure drop. One application of wheels is the removal of VOC where the regeneration stream is usually sent to an incinerator for destruction of VOC. Another use is in desiccant cooling (see previously). They do suffer from low efficiency due to the short contact time, mechanical leakage at seals, and the tendency to allow the wheel to exceed breakthrough in order to get better adsorbent utilization. Some adsorbent wheels are operated in an intermittent manner such that the wheel periodically indexes to a new position; this is particularly true of radial flow wheels. The rotating annular bed system for liquid chromatographic separation (two-dimensional chromatography) is analogous to the horizontal adsorbent wheel for gases. Feed to be separated flows to a portion of the top face of an annular bed of sorbent. A displacement purge in the form of a solvent or carrier gas flows to the rest of the annulus. The less strongly adsorbed components travel downward through the sorbent at a higher rate. Thus, they will exit at the bottom of the annulus at a smaller angular distance from the feed sector. The more strongly adsorbed species will exit at a greater angle. The mechanical and packing complexities of such an apparatus have been overcome for a pressurized system by workers at the Oak Ridge National Laboratory shown in Fig. 16-56 [Canon, Begovich, and Sisson, Sep. Sci. Technol., 15, 655–678 (1980)]. Several potential applications are reviewed by Carta and Byers [Chromatographic and Membrane Processes in Biotechnology, NATO ASI Proceeding, Kluwer, 1991]. Simulated Continuous Countercurrent Systems Because of the problems associated with moving adsorbent in true continuous countercurrent mode, most commercial application of continuous countercurrent processes has been accomplished by using flow schemes that simulates the flow of adsorbent and process fluid. There are at least four types of implementation: (1) units designed to intermittently move the adsorbent rather than continuously, (2) continuously operating chromatographic columns to which the feed is periodically pulsed, (3) enhancement of the chromatographic separation by dividing the column and using recycles, and (4) moving the point of introduction of the feed and regeneration point along a series of beds. Pulsed beds of activated carbon are used in water and wastewater treatment systems. The adsorber tank is usually a vertical cylindrical pressure vessel, with fluid distributors at top and bottom, similar to the arrangement of an ion exchanger. The column is filled with granular carbon. Fluid flow is upward, and carbon is intermittently dis-

(b)

Adsorbent wheels for gas separation: (a) horizontal with fixed beds; (b) vertical monolith. (Reprinted with permission

EQUIPMENT

16-63

Annular bed for liquid separation. [Reprinted with permission of Marcel Dekker. Reference: Canon, Begovich, and Sisson, Sep. Sci. Technol., 15, 655–678 (1980).]

FIG. 16-56

placed downward by opening a valve at the bottom and injecting a measured slug of carbon into the top of the vessel. The exhausted slug (a small fraction of the total charge in order to approximate fully countercurrent operation very closely) is transferred to the sweeten-off tank, where residual product is displaced. It next is dewatered and fed to the regeneration furnace. The carbon is eventually returned to the adsorber where fluid flow is interrupted briefly to permit carbon transfer in, as in continuous ion exchanger systems. Liquid chromatography has been used commercially to separate glucose from fructose and other sugar isomers, for recovery of nucleic acids, and other uses. A patent to Sanmatsu Kogyo Co., Ltd [Yoritomi, Kezuka, and Moriya, U.S. Patent number 4,267,054, 1981] presents an improved chromatographic process that is simpler to build and operate than simulated moving-bed processes. Figure 16-57 [Keller et al., gen. refs.] diagrams its use for a binary separation. It is a displacement-purge cycle where pure component cuts are recovered, while cuts that contain both components are recycled to the feed end of the column. The UOP CyclesorbSM is another adsorptive separation process with semicontinuous recycle. It utilizes a series of chromatographic columns to separate fructose from glucose. A series of internal recycle streams of impure and dilute portions of the chromatograph are used to improve the efficiency [Gerhold, U.S. Patent numbers 4,402,832, 1983; and 4,478,721, 1984]. A schematic diagram of a six-vessel UOP Cyclesorb process is shown in Fig. 16-58 [Gembicki et al., gen. refs., p. 595]. The process has four external streams and four internal recycles: Dilute raffinate and impure extract are like displacement steps; and impure raffinate and dilute extract are recycled from the bottom of an adsorber to its top. Feed and desorbent are fed to the top of each column in a predetermined sequence. The switching of the feed and desorbent are accomplished by the same rotary valve used for Sorbex switching (see hereafter). A chromatographic profile is established in each column that is moving from top to bottom, and all portions of an adsorber are performing a useful function at any time.

The concept of a simulated moving bed (SMB) was originally used in a process developed and licensed under the name UOP Sorbex process [Broughton, Bieser, and Anderson, Pet. Int. (Milan), 23(3), p. 91 (1976); Broughton, Bieser, and Persak, Pet. Int. (Milan), 23(5), p. 36 (1976)]. The following discussion is based on that process, but the concepts can be generally applied. In a moving-bed system for continuous countercurrent effect, solids move continuously in a closed loop past fixed points of introduction and withdrawal of feed and regenerant. The same effect can be achieved by holding the bed stationary and periodically moving the points of introduction and withdrawal of the various streams. Shifting those points in the direction of fluid flow simulates the movement of solid in the opposite direction. Since moving the positions continuously is impractical, a finite number of access lines are provided to a limited series of adsorbent beds. The Sorbex commercial application of this concept is portrayed in Fig. 16-59, which shows the adsorbent as a stationary bed and the auxiliary distillation columns needed to separate the desorbent from raffinate and extract so that it can be recycled to the process. In this example, feed flows through the three beds of Zone 1, where the most strongly adsorbed component A is adsorbed and depleted from the raffinate. Desorbent flows through the three beds of Zone 3 to displace component A into the extract. The less strongly adsorbed component B is adsorbed slightly in the two beds of Zone 4 between raffinate outlet and desorbent, inlet to prevent it from contaminating extract in Zone 3. Component B is desorbed in the four beds of Zone 2 between the extract outlet and the feed inlet and most leaves with the raffinate. A pump draws liquid from the bottom outlet to the top inlet of the adsorbent chamber. All flows in the beds are downward, simulating an upward flow of solid. The four active port positions are likewise moved downward by the selection of the rotary valve. The rotary valve functions in the same manner as a multiport stopcock or chromatography valve, sequencing the four streams on the right to the lines connected to the inlet/outlet nozzles on the adsorber. The next position of the rotary valve will direct desorbent to

16-64

ADSORPTION AND ION EXCHANGE line 3, extract from 6, and feed to 10; and raffinate moves to the top of the column at line 1. The liquid flow rate in each of the four zones is different because of the addition or withdrawal of the various streams and is controlled by the circulating pump. The circulating pump must be programmed to pump at four different rates in order to keep the flow in each zone constant. The variables available for fine tuning performance are the cycle time, measured as the time required for one rotation of the rotary valve, and the liquid flow rate in Zones 2, 3, and 4. Chemical analyses at the liquid circulating pump can trace the performance of the entire bed and are used for changing operating conditions. Other versions of the SMB system have been used commercially by Toray Industries, Illinois Water Treatment and Mitsubishi (see “Displacement Purge”). ION EXCHANGE

Sanmatsu Kogyo chromatographic process. (Reprinted with permission of Wiley. Reference: Keller, Anderson, and Yon, Chap. 12 in Rousseau, Handbook of Separation Process Technology, John Wiley & Sons, Inc., New York, 1987.)

FIG. 16-57

A typical fixed-bed ion exchanger consists of a vertical cylindrical vessel of lined steel or stainless steel. Linings are usually of natural or synthetic rubber. Spargers are provided at the top and bottom, and frequently a separate distributor is used for the regenerant solution. The resin bed, consisting of a meter or more of ion-exchange particles, is supported by the screen of the bottom distributor. Externally, the unit is provided with a valve manifold to permit downflow loading, upflow backwash, regeneration, and rinsing. For deionization, a twoexchanger assembly comprising a cation and an anion exchanger, is a common configuration (Fig. 16-60). Column hardware designed to allow countercurrent, upflow regeneration of ion-exchange resins is available. An example is given in Fig. 16-61. During upflow of the regenerant, bed expansion is prevented by withdrawing the effluent through the application of vacuum. A layer of drained particles is formed at the top of the bed while the rest of the column functions in the usual way. Typical design data for fixed-bed ion-exchange columns are given in Table 16-19. These should be used for preliminary evaluation purposes only. Characteristic design calculations are presented and illustrated by Applebaum [Demineralization by Ion Exchange, Academic, New York, 1968]. Large amounts of data are available in published literature and in bulletins of manufacturers of ion exchangers. In general, however, laboratory testing and pilot-plant work is advisable to determine usable exchanger capacities, regenerant quantities,

UOP Cyclesorb process. (Reprinted with permission of John Wiley & Sons, Inc. Reference: Gembicki, Oroskar, and Johnson, “Adsorption, Liquid Separation,” in Kirk-Othmer Encyclopedia of Chemical Technology, 4th ed., Wiley, New York, 1991.)

FIG. 16-58

EQUIPMENT

16-65

UOP Sorbex process. (Reprinted with permission of John Wiley & Sons, Inc. Reference: Gembicki, Oroskar, and Johnson, “Adsorption, Liquid Separation,” in Kirk-Othmer Encyclopedia of Chemical Technology, 4th ed., John Wiley & Sons, Inc., New York, 1991.)

FIG. 16-59

exchanger life, and quality of product for any application not already thoroughly proven in other plants for similar conditions. Firms that manufacture ion exchangers and ion-exchange equipment will often cooperate in such tests. For larger-scale applications, a number of continuous or semicontinuous ion-exchange units are also available. The Higgins contactor (Fig. 16-62) was originally developed to recover uranium from leach slurries at the Oak Ridge National Laboratory. More recently, it has been adapted to a wide variety of applications, including large-volume water softening. The Asahi process (Fig. 16-63) is used principally for high-volume water treatment. The liquid to be treated is passed upward through a resin bed in the adsorption tank. The upward flow at 30–40 m/h [12–16 gal/(min ft2)] keeps the bed packed against the top. After a preset time, 10 to 60 min, the flow is interrupted for about 30 s, allowing the entire bed to drop. A small portion (10 percent or less) of the ionexchange resin is removed from the bottom of the adsorption tank and transferred hydraulically to the hopper feeding the regeneration tank.

FIG. 16-60

Typical two-bed deionizing system. (Infilco Degremont Inc.)

The process is then resumed. Meanwhile, regeneration is occurring by a similar flow system in the regeneration tank, from which the regenerated ion exchanger is transferred periodically to the hopper above the water-rinse tank. In the latter, the resin particles are fluidized to flush away fines and accumulated foreign matter before the resin is returned to the adsorption tank. Another continuous ion-exchange system is described by Himsley and Farkas (see Fig. 16-64). Such a system is used to treat 1590 m3/h (7000 gal/min) of uranium-bearing copper leach liquor using fiberglass-construction columns 3.7 m (12 ft) in diameter. The adsorption column is divided vertically into stages. Continuous transfer of batches of resin occurs from stage to stage without any interruption of flow. This is accomplished by pumping solution from one stage (A) to the stage (B) immediately above by means of external pumping in such a

FIG. 16-61

Internals of an upflow regenerated unit. (Infilco Degremont Inc.)

16-66

ADSORPTION AND ION EXCHANGE

TABLE 16-19

Design Data for Fixed-Bed Ion Exchanger*

Type of resin Weak acid cation Strong acid cation Weak and intermediate base anions Strong-base anions Mixed cation and strong-base anion (chemical-equivalent mixture)

Maximum and minimum flow, m/ h [gal/(min⋅ft2)]

Minimum bed depth, m (in)

Maximum operating temperatures, °C (°F)

Usable capacity, gequivalent/L†

20 max. (8) 3 min. (1) 30 max. (12) 3 min. (1)

0.6 (24)

120 (248)

0.5–2.0

0.6 (24)

120 (248)

17 max. (7)

0.75 (30)

40 (104)

0.8–1.5 0.5–1.0 0.7–1.4 0.8–1.4

110% theoretical (HCl or H2SO4) 80–250 NaCl 35–200 66° Bé. H2SO4 80–500 20° Bé. HCl 35–70 NaOH

0.75 (30)

50 (122)

0.35–0.7

70–140 NaOH

1.2 (47)

50 (122)

0.2–0.35 (based on mixture)

Same as cation and anion individually

3 min. (1) 17 max. (7) 3 min. (1) 40 max. (16)

Regenerant, g/L resin‡

*These figures represent the usual ranges of design for water-treatment applications. For chemical-process applications, allowable flow rates are generally somewhat lower than the maximums shown, and bed depths are usually somewhat greater. †To convert to capacity in terms of kilograins of CaCO3 per cubic foot of resin, multiply by 21.8. ‡To convert to pounds of regenerant per cubic foot of resin, multiply by 0.0625.

manner that the net flow through stage B is downward, carrying with it all of the resin in that stage B. When transfer of the ion exchanger is complete, the resin in stage C above is transferred downward in a similar manner. The process continues until the last stage (F) is empty. The regenerated resin is then transferred from the elution column to the empty stage (F). Elution of the sorbed product is carried out in the elution column in a moving packed bed mode. The countercurrent contact achieved in this column yields a concentrated eluate with a minimum consumption of regenerant. Disadvantages of these continuous countercurrent systems are associated primarily with the complexity of the equipment required and with the attrition resulting from the transport of the ion exchanger. An effective alternative for intermediate scale processes is the use of merry-go-round systems and SMB units employing only packed-beds with no movement of the ion-exchanger.

FIG. 16-62

Mode of operation of the Higgins contractor. (ORNL-LR-Dwg.

27857R.)

Himsley continuous ion-exchange system. (Himsley and Farkas, “Operating and Design Details of a Truly Continuous Ion Exchange System,” Soc. Chem. Ind. Conf., Cambridge, England, July 1976. Used by permission of the Society of Chemical Industry.)

FIG. 16-64

Asahi countercurrent ion-exchange process. [Gilwood, Chem. Eng., 74(26), 86 (1967); copyright 1967 by McGraw-Hill, Inc., New York. Excerpted with special permission of McGraw-Hill.]

FIG. 16-63

Section 17

Gas-Solid Operations and Equipment

Mel Pell, Ph.D., Senior Consultant, E. I. duPont de Nemours & Co.; Fellow, American Institute of Chemical Engineers. (Section Editor, Fluidized Bed Systems) James B. Dunson, M.S., Principal Consultant, E. I. duPont de Nemours & Co.; Member American Institute of Chemical Engineers; Registered Professional Engineer. (Delaware) (GasSolid Separations)

FLUIDIZED-BED SYSTEMS Gas-Solid Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Types of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phase Diagram (Zenz and Othmer). . . . . . . . . . . . . . . . . . . . . . . . . . . Phase Diagram (Grace) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Regime Diagram (Grace) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solids Concentration versus Height. . . . . . . . . . . . . . . . . . . . . . . . . . . Equipment Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minimum Fluidizing Velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Particulate Fluidization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vibrofluidization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design of Fluidized-Bed Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluidization Vessel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scale-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solids Mixing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gas Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Size Enlargement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Size Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Standpipes, Solids Feeders, and Solids Flow Control. . . . . . . . . . . . . Solids Discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dust Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 1: Length of Seal Leg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17-2 17-2 17-2 17-2 17-2 17-4 17-4 17-4 17-4 17-4 17-4 17-4 17-8 17-10 17-10 17-10 17-10 17-10 17-10 17-10 17-11 17-12 17-12 17-13

Uses of Fluidized Beds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chemical Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physical Contacting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17-14 17-14 17-17

GAS-SOLIDS SEPARATIONS Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Purpose of Dust Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties of Particle Dispersoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Particle Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Atmospheric-Pollution Measurements . . . . . . . . . . . . . . . . . . . . . . . . Process-Gas Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Particle-Size Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanisms of Dust Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Performance of Dust Collectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dust-Collector Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dust-Collection Equipment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gravity Settling Chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Impingement Separators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cyclone Separators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanical Centrifugal Separators . . . . . . . . . . . . . . . . . . . . . . . . . . . Particulate Scrubbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dry Scrubbing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fabric Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Granular-Bed Filters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Air Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electrical Precipitators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17-19 17-22 17-22 17-22 17-22 17-23 17-24 17-24 17-25 17-26 17-26 17-26 17-27 17-27 17-32 17-32 17-39 17-42 17-47 17-48 17-51

17-1

FLUIDIZED-BED SYSTEMS Fluidization, or fluidizing, converts a bed of solid particles into an expanded, suspended mass that has many properties of a liquid. This mass has zero angle of repose, seeks its own level, and assumes the shape of the containing vessel. Fluidized beds are used successfully in a multitude of processes both catalytic and noncatalytic. Among the catalytic uses are hydrocarbon cracking and re-forming, oxidation of naphthalene to phthalic anhydride, and ammoxidation of propylene to acrylonitrile. A few of the noncatalytic uses are roasting of sulfide ores, coking of petroleum residues, calcination of ores, incineration of sewage sludge, drying, and classification. Considerable effort and interest are now centered in the areas of coal and waste combustion to raise steam. The size of solid particles which can be fluidized varies greatly from less than 1 µm to 6 cm (2a in). It is generally concluded that particles distributed in sizes between 150 µm and 10 µm are the best for smooth fluidization (least formation of large bubbles). Large particles cause instability and result in slugging or massive surges. Small particles (less than 20 µm) frequently, even though dry, act as if damp, forming agglomerates or fissures in the bed, or spouting. Adding finersized particles to a coarse bed or coarser-sized particles to a bed of fines usually results in better fluidization. The upward velocity of the gas is usually between 0.15 m/s (0.5 ft/s) and 6 m/s (20 ft/s). This velocity is based upon the flow through the empty vessel and is referred to as the superficial velocity. For details beyond the scope of this subsection, reference should be made to Kunii and Levenspiel, Fluidization Engineering, 2d ed., Butterworth Heinemann, Boston, 1991; Pell, Gas Fluidization, Elsevier, New York, 1990; D. Geldart (ed.), Gas Fluidization Technology, Wiley, New York, 1986; and the vast number of papers published in periodicals, transcripts of symposia, and the American Institute of Chemical Engineers symposium series. GAS-SOLID SYSTEMS Several workers in the field have systematized the various types of fluidization. Several of these types are discussed in the following subsections because each adds another dimension to the understanding of the phenomena. Types of Solids Geldart [Powder Technol., 7, 285–292 (1973)] has characterized four groups of solids that exhibit different properties when fluidized with a gas. Figure 17-1 shows the division of the classes as a function of mean particle size, d
sv, µm, and density difference, (ρs − ρf), g/cm3, where ρs = particle density and ρf = fluid density 1 d
sv = ᎏᎏ  (xi/dpi) where
dsv = surface volume diameter of particle and xi = weight fraction of particles of size dpi.
dsv is the preferred particle averaging method for fluid bed applications. When gas is passed upward through a bed of particles of groups A, B, or D, friction causes a pressure drop expressed by the CarmanKozeny fixed-bed correlation. As the gas velocity is increased, the pressure drop increases until it equals the weight of the bed divided by the cross-sectional area. This velocity is called minimum fluidizing velocity, Umf. When this point is reached, the bed of group A particles will expand uniformly until at some higher velocity gas bubbles will form (minimum bubbling velocity, Umb). For group B and group D particles Umf and Umb are essentially equal. Group C particles exhibit cohesive tendencies, and as the gas flow is further increased, usually “rathole”; the gas opens channels that extend from the gas distributor to the surface. If channels are not formed, the whole bed will lift as a piston. At higher velocities or with mechanical agitation or vibration, 17-2

FIG. 17-1 Powder-classification diagram for fluidization by air (ambient conditions). [From Geldart, Powder Technol., 7, 285–292 (1973).]

this type of particle will fluidize but with the appearance of clumps or clusters of particles. For all groups of powder (A, B, C, and D) as the gas velocity is further increased, bed density is decreased and turbulence increased. In smaller-diameter beds, especially with group B and D powders, slugging will occur as the bubbles increase in size to greater than half of the bed diameter. Bubbles grow by vertical and lateral merging. Bubbles also increase in size as the gas velocity is increased [Whitehead, in Davidson and Harrison (eds.), Fluidization, Academic, London and New York, 1971]. As the gas velocity is increased further and bubbles tend to disappear and streamers of solids and gas prevail, pressure fluctuations in the bed are greatly reduced. Further increase in velocity results in dilute-phase pneumatic transport. Phase Diagram (Zenz and Othmer) Zenz and Othmer (op. cit.) have graphically represented (Fig. 17-2) all gas-solid systems in which the gas is flowing counter to gravity as a function of pressure drop per unit of height versus velocity. Note that line OAB in Fig. 17-2 is the pressure-drop versus gas-velocity curve for a packed bed and BD the curve for a fluid bed. Zenz indicates an instability between D and H because with no solids flow all the particles will be entrained from the bed; however, if solids are added to replace those entrained, system IJ prevails. The area DHIJ will be discussed further. Phase Diagram (Grace) Grace [Can. J. Chem. Eng., 64, 353– 363 (1986); Fig. 17-3] has correlated the various types of gas-solid systems in which the gas is flowing counter to gravity in a status graph using the parameters of the Archimedes number (Ar) for the particle size and a nondimensional velocity (U*) for the gas effects. By means of this plot, the regime of fluidization can be predicted. Regime Diagram (Grace) Grace [Can. J. Chem. Eng., 64, 353– 363 (1986); Fig. 17-4] has also sketched the way a fluid bed appears in the different regimes of fluidization resulting from increasing velocity from packed-bed operation to a transport reactor. The area that Zenz and Othmer consider to be discontinuous is called the fast fluid bed regime. Note that since beds in turbulent and faster regimes operate above the terminal velocity of some or all of the particles, solids return is necessary to maintain a bed.

FIG. 17-2 Schematic phase diagram in the region of upward gas flow. W = mass flow solids, lb/(h ⋅ ft2); ε = fraction voids; ρp = particle density, lb/ft3; ρf = fluid density, lb/ft3; Cd = drag coefficient; Re = modified Reynolds number. (Zenz and Othmer, Fluidization and Fluid Particle Systems, Reinhold, New York, 1960.)

Key: OAB = packed bed BD = fluidized bed DH = slugging bed

IJ = cocurrent flow (dilute phase) ST = countercurrent flow (dense phase)

AC = packed bed (restrained at top) OEG = fluid only (no solids)

FH = dilute phase MN = countercurrent flow (dilute phase) VW = cocurrent flow (dense phase)

FIG. 17-3 Simplified fluid-bed status graph [From Grace, Can. J. Chem. Eng., 64, 353–363 (1986); sketches from Reh, Ger. Chem. Eng., 1, 319–329 (1978).]

Solids return

Solids return

GAS-SOLID OPERATIONS AND EQUIPMENT

Solids return

17-4

Gas Fixed bed

Particulate regime

Bubbling regime

Slug flow regime

Turbulent regime

Fast fluidization

Pneumatic conveying

Aggregative fluidization Increasing gas velocity FIG. 17-4

Fluidization regimes. [Adapted from Grace, Can J. Chem. Eng., 64, 353–363 (1986).]

Solids Concentration versus Height From the foregoing it is apparent that there are several regimes of fluidization. These are, in order of increasing gas velocity, particulate fluidization (Geldart group A), bubbling (aggregative), turbulent, fast, and transport. Each of these regimes has characteristic solids concentration profiles as shown in Fig. 17-5. Equipment Types Fluidized-bed systems take many forms. Figure 17-6 shows some of the more prevalent concepts with approximate ranges of gas velocities. Minimum Fluidizing Velocity Umf, the minimum fluidizing velocity, is frequently used in fluid-bed calculations and in quantifying one of the particle properties. This parameter is best measured in small-scale equipment at ambient conditions. The correlation by Wen and Yu [A.I.Ch.E.J., 610–612 (1966)] given below can then be used to back calculate dp. This gives a particle size that takes into account effects of size distribution and sphericity. The correlation can then be used to estimate Umf at process conditions. If Umf cannot be determined experimentally, use the expression below directly. Remf = (1135.7 + 0.0408Ar)0.5 − 33.7 d pρfUmf /µ where Remf =
Ar =
d p3ρf(ρs − ρf)g/µ2 d
p = 1/ (xi /dpi) The flow required to maintain a complete homogeneous bed of solids in which coarse or heavy particles will not segregate from the fluidized

FIG. 17-5

fluidization.

Solids concentration versus height above distributor for regimes of

portion is very different from the minimum fluidizing velocity. See Nienow and Chiba, Fluidization, 2d ed., Wiley, 1985, pp. 357–382, for a discussion of segregation or mixing mechanism as well as the means of predicting this; also see Baeyens and Geldart, Gas Fluidization Technology, Wiley, 1986, 97–122. Particulate Fluidization Fluid beds of Geldart class A powders that are operated at gas velocities above the minimum fluidizing velocity (Umf) but below the minimum bubbling velocity (Umb) are said to be particulately fluidized. As the gas velocity is increased above Umf, the bed further expands. Decreasing (ρs − ρf), dp and/or increasing µf increases the spread between Umf and Umb until at some point, usually at high pressure, the bed is fully particulately fluidized. Richardson and Zaki [Trans. Inst. Chem. Eng., 32, 35 (1954)] showed that U/Ui = εn, where n is a function of system properties, ε = void fraction, U = superficial fluid velocity, and Ui = theoretical superficial velocity from the Richardson and Zaki plot when ε = 1. Vibrofluidization It is possible to fluidize a bed mechanically by imposing vibration to throw the particles upward cyclically. This enables the bed to operate with either no gas upward velocity or reduced gas flow. Entrainment can also be greatly reduced compared to unaided fluidization. The technique is used commercially in drying and other applications [Mujumdar and Erdesz, Drying Tech., 6, 255– 274 (1988)], and chemical reaction applications are possible. See Sec. 12 for more on drying applications of vibrofluidization. DESIGN OF FLUIDIZED-BED SYSTEMS The use of the fluidization technique requires in almost all cases the employment of a fluidized-bed system rather than an isolated piece of equipment. Figure 17-7 illustrates the arrangement of components of a system. The major parts of a fluidized-bed system can be listed as follows: 1. Fluidization vessel a. Fluidized-bed portion b. Disengaging space or freeboard c. Gas distributor 2. Solids feeder or flow control 3. Solids discharge 4. Dust separator for the exit gases 5. Instrumentation 6. Gas supply Fluidization Vessel The most common shape is a vertical cylinder. Just as for a vessel designed for boiling a liquid, space must be provided for vertical expansion of the solids and for disengaging

FLUIDIZED-BED SYSTEMS

(a)

(b)

(c)

(d)

FIG. 17-7

(e)

(h)

(f)

(i)

17-5

(g)

(j)

FIG. 17-6 Fluidized-bed systems. (a) Bubbling bed, external cyclone, U < 20 × Umf. (b) Turbulent bed, external cyclone, 20 × Umf < U < 200 × Umf. (c) Bubbling bed, internal cyclones, U < 20 × Umf. (d) Turbulent bed, internal cyclones, 20 × Umf < U < 200 × Umf. (e) Circulating (fast) bed, external cyclones, U > 200 × Umf. ( f) Circulating bed, U > 200 × Umf. (g) Transport, U > UT. (h) Bubbling or turbulent bed with internal heat transfer, 2 × Umf < U < 200 × Umf. (i) Bubbling or turbulent bed with internal heat transfer, 2 × Umf < U < 100 × Umf. ( j) Circulating bed with external heat transfer, U > 200 × Umf.

splashed and entrained material. The volume above the bed is called the disengaging space. The cross-sectional area is determined by the volumetric flow of gas and the allowable or required fluidizing velocity of the gas at operating conditions. In some cases the lowest permissible velocity of gas is used, and in others the greatest permissible velocity is used. The maximum flow is generally determined by the carry-over or entrainment of solids, and this is related to the dimensions of the disengaging space (cross-sectional area and height). Bed Bed height is determined by a number of factors, either individually or collectively, such as: 1. Gas-contact time 2. L/D ratio required to provide staging 3. Space required for internal heat exchangers 4. Solids-retention time Generally, bed heights are not less than 0.3 m (12 in) or more than 15 m (50 ft). Although the reactor is usually a vertical cylinder, there is no real limitation on shape. The specific design features vary with operating

Noncatalytic fluidized-bed system.

conditions, available space, and use. The lack of moving parts lends toward simple, clean design. Many fluidized-bed units operate at elevated temperatures. For this use, refractory-lined steel is the most economical design. The refractory serves two main purposes: (1) it insulates the metal shell from the elevated temperatures, and (2) it protects the metal shell from abrasion by the bed and particularly the splashing solids at the top of the bed resulting from bursting bubbles. Depending on specific conditions, several different refractory linings are used [Van Dyck, Chem. Eng. Prog., 46–51 (December 1979)]. Generally, for the moderate temperatures encountered in catalytic cracking of petroleum, a reinforced-gunnite lining has been found to be satisfactory. This also permits the construction of larger units than would be permissible if self-supporting ceramic domes were to be used for the roof of the reactor. When heavier refractories are required because of operating conditions, insulating brick is installed next to the shell and firebrick is installed to protect the insulating brick. Industrial experience in many fields of application has demonstrated that such a lining will successfully withstand the abrasive conditions for many years without replacement. Most serious refractory wear occurs with coarse particles at high gas velocities and is usually most pronounced near the operating level of the fluidized bed. Gas leakage behind the refractory has plagued a number of units. Care should be taken in the design and installation of the refractory to reduce the possibility of the formation of “chimneys” in the refractories. A small flow of solids and gas can quickly erode large passages in soft insulating brick or even in dense refractory. Gas stops are frequently attached to the shell and project into the refractory lining. Care in design and installation of openings in shell and lining is also required. In many cases, cold spots on the reactor shell will result in condensation and high corrosion rates. Sufficient insulation to maintain the shell and appurtenances above the dew point of the reaction gases is necessary. Hot spots can occur where refractory cracks allow heat to permeate to the shell. These can sometimes be repaired by pumping castable refractory into the hot area from the outside. The violent motion of a fluidized bed requires ample foundations and sturdy supporting structure for the reactor. Even a relatively small differential movement of the reactor shell with the lining will materially shorten refractory life. The lining and shell must be designed as a unit. Structural steel should not be supported from a vessel that is subject to severe vibration. Freeboard and Entrainment The freeboard or disengaging height is the distance between the top of the fluid bed and the gas-exit

17-6

GAS-SOLID OPERATIONS AND EQUIPMENT

5

7.5 3.0 1.5 0.6

2.5 1.5

0.3

TDH, m

1.0

.15 0.5

.08

0.25 0.15 Bed diameter, m

0.1

.025

0.05 0.02 0.03

0.06

0.12

0.3

0.6

1.2

1.8

Gas velocity, u – umb, m/s FIG. 17-8

Estimating transport disengaging height (TDH).

nozzle in bubbling- or turbulent-bed units. The distinction between bed and freeboard is difficult to determine in fast and transport units (see Fig. 17-5). At least two actions can take place in the freeboard: classification of solids and reaction of solids and gases. As a bubble reaches the upper surface of a fluidized bed, the gas breaks through the thin upper envelope composed of solid particles entraining some of these particles. The crater-shaped void formed is rapidly filled by flowing solids. When these solids meet at the center of the void, solids are geysered upward. The downward pull of gravity and the upward pull of the drag force of the upward-flowing gas act on the particles. The larger and denser particles return to the top of the bed, and the finer and lighter particles are carried upward. The distance above the bed at which the entrainment becomes constant is the transport disengaging height, TDH. Cyclones and vessel gas outlets are usually located above TDH. Figure 17-8 graphically estimates TDH as a function of velocity and bed size. The higher the concentration of an entrainable component in the bed, the greater its rate of entrainment. Finer particles have a greater rate of entrainment than coarse ones. These principles are embodied in the method of Geldart (Gas Fluidization Tech., Wiley, 1986, pp. 123–153) via the equation, E(i) = K*(i)x(i), where E(i) = entrainment rate for size i, kg/m2 s; K*(i) = entrainment rate constant for particle size i; and x(i) = weight fraction for particle size i. K* is a function of operating conditions given by K*(i)/(Pf u) = 23.7 exp [−5.4 Ut(i)/U]. The composition and the total entrainment are calculating by summing over the entrainable fractions. An alternative is to use the method of Zenz as reproduced by Pell (Gas Fluidization, Elsevier, 1990, pp. 69–72). In batch classification, the removal of fines (particles less than any arbitrary size) can be correlated by treating as a second-order reaction K = (F/θ)[1/x(x − F)], where K = rate constant, F = fines removed in time θ, and x = original concentration of fines. Gas Distributor The gas distributor has a considerable effect on proper operation of the fluidized bed. Basically there are two types: (1) for use when the inlet gas contains solids and (2) for use when the inlet gas is clean. In the latter case, the distributor is designed to prevent back flow of solids during normal operation, and in many cases it is designed to prevent back flow during shutdown. In order to provide distribution, it is necessary to restrict the gas or gas and solids flow so that pressure drops across the restriction amount to from 0.5 kPa (2 in of water) to 20 kPa (3 lbf/in2). The bubbling action of the bed produces local pressure fluctua-

tions. A temporary condition of low pressure in one section of the bed due to bubbles can combine with higher pressure in another section with fewer bubbles to produce permanent flow maldistribution. The grid should be designed for a pressure drop of at least s the bed weight for upflow distributors and about 1⁄10 the bed weight for downflow spargers. In units with shallow beds such as dryers or where distribution is less crucial, lower distributor pressure drops can be used. When both solids and gases pass through the distributor, such as in catalytic-cracking units, a number of variations are or have been used, such as concentric rings in the same plane, with the annuli open (Fig. 17-9a); concentric rings in the form of a cone (Fig. 17-9b); grids of T bars or other structural shapes (Fig. 17-9c); flat metal perforated plates supported or reinforced with structural members (Fig. 17-9d); dished and perforated plates concave both upward and downward (Fig. 17-9e and f). The last two forms are generally more economical. In order to generate the required pressure drop, a high velocity through the grid openings may be needed. It is best to limit this velocity to less than 70 m/s to minimize attrition of the bed material. It is common industrial practice to put a shroud of pipe over the opening so that the velocity can be kept high for pressure drop but is reduced for entry to the bed. The technique is applied to both plate and pipe spargers. Pressure drop through a pipe or a drilled plate is given by: u2ρ f ∆P = ᎏᎏ 0.64 × 2g where u = velocity in the hole at inlet conditions ρf = fluid density in the hole at inlet conditions ∆P = pressure drop in consistent units (may be lbs/ft2) Experience has shown that the concave-upward type is a better arrangement than the concave-downward type, as it tends to increase the flow of gases in the outer portion of the bed. This counteracts the normal tendency of higher gas flows in the center of the bed. Structurally, distributors must withstand the differential pressure across the restriction during normal and abnormal flow. In addition, during a shutdown all or a portion of the bed will be supported by the distributor until sufficient back flow of the solids has occurred both to reduce the weight of solids above the distributor and to support some of this remaining weight by transmitting the force to the walls and bottom of the reactor. During startup considerable upward thrust can be exerted against the distributor as the settled solids under the distributor are carried up into the normal reactor bed.

FLUIDIZED-BED SYSTEMS

(a)

17-7

(b)

(a)

(c)

(d)

(e)

(f)

(b) FIG. 17-9

Gas distribution for gases containing solids.

When the feed gas is devoid of or contains only small quantities of fine solids, more sophisticated designs of gas distributors can be used to effect economies in initial cost and maintenance. This is most pronounced when the inlet gas is cold and noncorrosive. When this is the case, the plenum chamber gas distributor, and distributor supports can be fabricated of mild steel by using normal temperature design factors. The first commercial fluidized-bed ore roaster [Mathews, Trans. Can. Inst. Min. Metall., L11, 97 (1949)], supplied by the Dorr Co. (now Dorr-Oliver Inc.) in 1947 to Cochenour-Willans, Red Lake, Ontario, was designed with a mild-steel constriction plate covered with castable refractory to insulate the plate from the calcine and also to provide cones in which refractory balls were placed to act as ball checks. The balls eroded unevenly, and the castable cracked. However, when the unit was shut down by closing the air-control valve, the runback of solids was negligible because of bridging. If, however, the unit was shut down by deenergizing the centrifugal blower motor, the higher pressure in the reactor would relieve through the blower and fluidizing gas plus solids would run back through the constriction plate. Figure 17-10 illustrates two designs of gas inlets which have been successfully used to prevent flowback of solids. For best results, irrespective of the design, the gas flow should be stopped and pressurerelieved from the bottom upward through the bed. Some units have been built and successfully operated with simple slot-type distributors made of heat-resistant steel. This requires a heat-resistant plenum chamber but eliminates the frequently encountered problem of corrosion caused by condensation of acids and water vapor on the cold metal of the distributor. When the inlet gas is hot, such as in dryers or in the upper distributors of multibed units, ceramic arches or heat-resistant metal grates are generally used. Self-supporting ceramic domes have been in successful use for

Gas inlets designed to prevent backflow of solids. (a) Insert tuyere; (b) clubhead tuyere. (Dorr-Oliver, Inc.) FIG. 17-10

many years as gas distributors when temperatures range up to 1100°C. Some of these domes are fitted with alloy-steel orifices to regulate air distribution. However, the ceramic arch presents the same problem as the dished head positioned concave downward. Either the holes in the center must be smaller so that the sum of the pressure drops through the distributor plus the bed is constant across the whole cross section, or the top of the arch must be flattened so that the bed depth in the center and outside is equal. This is especially important when shallow beds are used. It is important to consider thermal effects in design of the grid to

Multiple-pipe gas distributor. [From Stemerding, de Groot, and Kuypers, Soc. Chem. Ind. J. Symp. Fluidization Proc., 35–46, London (1963).] FIG. 17-11

17-8

GAS-SOLID OPERATIONS AND EQUIPMENT

shell seal. Bypassing of the grid at the seal point is a common problem caused by situations such as uneven expansion of metal and ceramic parts, a cold plenum and hot solids in contact with the grid plate at the same time, and startup and shutdown scenarios. When the atmosphere in the bed is sufficiently benign, a spargertype distributor may be used. See Fig. 17-11. In some cases, it is impractical to use a plenum chamber under the constriction plate. This condition arises when a flammable or explosive mixture of gases is being introduced to the reactor. One solution is to pipe the gases to a multitude of individual gas inlets in the floor of the reactor. In this way it may be possible to maintain the gas velocities in the pipes above the flame velocity or to reduce the volume of gas in each pipe to the point at which an explosion can be safely contained. Another solution is to provide separate inlets for the different gases and depend on mixing in the fluidized bed. The inlets should be fairly close to one another, as lateral gas mixing in fluidized beds is poor. Much attention has been given to the effect of gas distribution on bubble growth in the bed and the effect of this on catalyst utilization, space-time yield, etc. It would appear that the best gas distributor would be a porous membrane. This type of distributor is seldom practical for commercial units because of both structural limitations and the need for absolutely clean gas. Practically, the limitations on hole spacing are dependent on particle size of solids, materials of construction, and type of distributor. If easily worked metals are used, punching, drilling, and welding are not expensive operations and permit the use of large numbers of holes. The use of tuyeres or bubble caps permits horizontal distribution of the gas so that a smaller number of gasinlet ports can still achieve good gas distribution. If a ceramic arch is used, generally only one hole per brick is permissible and brick dimensions must be reasonable. Scale-Up Bubbling or Turbulent Beds Scale-up of noncatalytic fluidized beds when the reaction is fast, as in roasting or calcination, is straightforward and is usually carried out on an area basis. Small-scale tests are made to determine physical limitations such as sintering, agglomeration, solids-holdup time required, etc. Slower (k < 1/s) catalytic or more complex reactions in which several gas interchanges are required are usually scaled up in several steps, from laboratory to commercial size. The hydrodynamics of gas-solids flow and contacting is quite different in small-diameter high-L/D fluid beds as compared with large-diameter moderate-L/D beds. In small-diameter beds, bubbles tend to be small and cannot grow larger than the vessel diameter. In larger, deeper units, bubbles can grow very large. The large

(a)

FIG. 17-12

(b)

(c)

Methods of providing staging in fluidized beds.

bubbles have less surface for mass transfer to the solids than the same volume of small bubbles. The large bubbles also rise through the bed more quickly. The size of a bubble as a function of height was given by Darton et al. [Trans. Inst. Chem. Eng., 55, 274–280 (1977)] as 0.8

0.54(u − umb)0.4(h + 4A t /No
) db = ᎏᎏᎏᎏ 0.2 g

where

db = bubble diameter, m h = height above the grid, m At /No = grid area per hole

Bubble growth will be limited by the containing vessel and the bubble hydrodynamic stability. Bubbles in group-B systems can grow to several meters in diameter. Bubbles in group-A materials with high fines may reach a maximum stable bubble size of only several cm. Furthermore, the solids and gas back mixing is much less in highL/D beds, either slugging or bubbling, as compared with low-L/D beds. Thus, the conversion or yield in large unstaged reactors is sometimes considerably lower than in small high-L/D units. To overcome some of the problems of scale-up, staged units are used (see Fig. 17-12). It is generally concluded than an unstaged 1-m- (40-in-) diameter unit will achieve about the same conversion as a large industrial unit. The validity of this conclusion is dependent on many variables, including bed depth, particle size, and size distribution, temperature, and pressure. A brief history of fluidization, fluidized-bed scale-up, and modeling will illustrate the problems. Fluidized beds were used in Europe in the 1920s to gasify coal. Scale-up problems either were insignificant or were not publicized. During World War II, catalytic cracking of oil to produce gasoline was successfully commercialized by scaling up from pilot-plant size (a few centimeters in diameter) to commercial size (several meters in diameter). It is fortunate that the kinetics of the cracking reactions are fast, that the ratio of crude oil to catalyst is determined by thermal balance and the required catalyst circulation rates, and that the crude feed point was in the plug-flow riser. The first experience of problems with scale-up was associated with the production of gasoline from natural gas by using the Fischer-Tropsch process. Some 0.10-m- (4-in-), 0.20-m- (8-in-), and 0.30-m- (12-in-) diameter pilot-plant results were scaled to a 7-m-diameter commercial unit, where the yield was only about 50 percent of that achieved in the pilot units. The FischerTropsch synthesis is a relatively slow reaction; therefore, gas-solid contacting is very important. Since this unfortunate experience or perhaps because of it, much effort has been given to the scale-up of fluidized beds. Many models have been developed; these basically are

(d)

(e)

(f)

FLUIDIZED-BED SYSTEMS

FIG. 17-13 Two-phase model according to May [Chem. Eng. Prog., 55, 12, 5, 49–55 (1959)] and Van Deemter [Chem. Eng. Sci., 13, 143–154 (1961)]. U = superficial velocity, Umf = minimum fluidizing velocity, E = axial dispersion coefficient, and Kbe = mass-transfer coefficient.

of two types, the two-phase model [May, Chem. Eng. Prog., 55, 12, 5, 49–55 (1959); and Van Deemter, Chem. Eng. Sci., 13, 143–154 (1961)] and the bubble model (Kunii and Levenspiel, Fluidization Engineering, Wiley, New York, 1969). The two-phase model according to May and Van Deemter is shown in Fig. 17-13. In these models all or most of the gas passes through the bed in plug flow in the bubbles which do not contain solids (catalyst). The solids form a dense suspension-emulsion phase in which gas and solids mix according to an axial dispersion coefficient (E). Cross flow between the two phases is predicted by a mass-transfer coefficient. Conversion of a gaseous reactant can be given by C/C0 = exp [−Na × Nr/(Na + Nr)] where C = the exit concentration, C0 = the inlet concentration, Na = diffusional driving force and Nr = reaction driving force. Conversion is determined by both reaction and diffusional terms. It is possible for reaction to dominate in a lab unit with small bubbles and for diffusion to dominate in a plant size unit. It is this change of limiting regime that makes scaleup so difficult. Refinements of the basic model and predictions of mass-transfer and axialdispersion coefficients are the subject of many papers [Van Deemter, Proc. Symp. Fluidization, Eindhoven (1967); de Groot, ibid.; Van Swaaij; and Zuidiweg, Proc. 5th Eur. Symp. React. Eng., Amsterdam, B9–25 (1972); DeVries, Van Swaaij, Mantovani, and Heijkoop, ibid., B9–59 (1972); Werther, Ger. Chem. Eng., 1, 243–251 (1978); and Pell, Gas Fluidization, Elsevier, 75–81 (1990). The bubble model (Kunii and Levenspiel, Fluidization Engineering, Wiley, New York, 1969; Fig. 17-14) assumes constant-sized bubbles (effective bubble size db) rising through the suspension phase. Gas is transferred from the bubble void to the mantle and wake at

FIG. 17-14 Bubbling-bed model of Kunii and Levenspiel. db = effective bubble diameter, CAb = concentration of A in bubble, CAc = concentration of A in cloud, CAe = concentration of A in emulsion, q = volumetric gas flow into or out of bubble, kbc = mass-transfer coefficient between bubble and cloud, and kce = mass-transfer coefficient between cloud and emulsion. (From Kunii and Levenspiel, Fluidization Engineering, Wiley, New York, 1969, and Krieger, Malabar, Fla., 1977.)

17-9

mass-transfer coefficient Kbc and from the mantle and wake to the emulsion phase at mass-transfer coefficient Kce. Experimental results have been fitted to theory by means of adjusting the effective bubble size. As mentioned previously, bubble size changes from the bottom to the top of the bed, and thus this model is not realistic though of considerable use in evaluating reactor performance. Several bubble models using bubbles of increasing size from the distributor to the top of the bed and gas interchange between the bubbles and the emulsion phase according to Kunii and Levenspiel have been proposed [Kato and Wen, Chem. Eng. Sci., 24, 1351–1369 (1969); and Fryer and Potter, in Keairns (ed.), Fluidization Technology, vol. I, Hemisphere, Washington, 1975, pp. 171–178]. There are several methods available to reduce scaleup loss. These are summarized in Fig. 17-15. The efficiency of a fluid bed reactor usually decreases as the size of the reactor increases. This can be minimized by the use of high velocity, fine solids, staging methods, and a high L/D. High velocity maintains the reactor in the turbulent mode, where bubble breakup is frequent and backmixing is infrequent. A fine catalyst leads to smaller maximum bubble sizes by promoting instability of large bubbles. Maintaining high L/D minimizes backmixing, as does the use of baffles in the reactor. By these techniques, Mobil was able to scale up its methanol to gasoline technology with little difficulty. [Krambeck, Avidan, Lee and Lo, A.I.Ch.E.J., 1727–1734 (1987)]. Another way to examine scaleup of hydrodynamics is to build a cold or hot scale model of the commercial design. Validated scaling criteria have been developed and are particularly effective for group B and D materials [Glicksman, Hyre and Woloshun, Powder Tech., 177–199 (1993)]. Circulating or Fast Beds The circulating or fast fluidized bed carries the principles given above to the maximum extent. Velocity is increased until solids are entrained from the bed at a massive rate. Thus, solids extend to the exit and may constitute up to 10 percent of the freeboard volume. There are no bubbles, mass transfer rates are high, and there is little backmixing. The high velocity means high gas throughput, minimizing reactor cost. Scaleup is less of a problem than with bubbling beds. The system is characterized by an external cyclone return system that is usually as large as the reactor itself. The axial solids density profile is relatively flat, as indicated in Figs. 17-4 and 17-5. There is a radial solids density profile called core annular flow. In the center of the reactor, gas flow may be double the average, and the solids are in dilute flow, traveling at Ug − Ut, their expected slip velocity. Near the wall, the solids are close to their minimum bubbling density and are likely to be in downflow. Engineering methods for evaluating the hydrodynamics of the circulating bed are given by Kunii and Levenspiel (Fluidization Engineering, 2d ed., Butterworth, 1991, pp. 195–

Reducing scaleup loss (From Krambeck, Avidan, Lee and Lo, A.I.Ch.E.J., 1727–1734, 1987.) FIG. 17-15

17-10

GAS-SOLID OPERATIONS AND EQUIPMENT

209) and Werther (Circulating Fluid Bed Technology IV, in press, 1994). Transport Transport units can be scaled up on the principles of pneumatic conveying. Mass and heat transfer can be predicted on both the slip velocity during acceleration and the slip velocity at full acceleration. The slip velocity is increased as the solids concentration is increased. Heat Transfer Heat-exchange surfaces have been used to provide means of removing or adding heat to fluidized beds. Usually, these surfaces are provided in the form of vertical tubes manifolded at top and bottom or in trombone shape manifolded exterior to the vessel. Other shapes such as horizontal bayonets have been used. In any such installations adequate provision must be made for abrasion of the exchanger surface by the bed. The prediction of the heat-transfer coefficient is covered in Secs. 5 and 11. Normally, the transfer rate is between 5 and 25 times that for the gas alone. Heat transfer from solids to gas and gas to solids usually results in a coefficient of about 6 to 20 J/(m2⋅s⋅K) [3 to 10 Btu/(h⋅ft2⋅°F)]. However, the large area of the solids per cubic foot of bed, 5000 m2/m3 (15,000 ft2/ft3) for 60-µm particles of 600 kg/m3 (40 lb/ft3) bulk density, results in the rapid approach of gas and solids temperatures. With a fairly good distributor, essential equalization of temperatures occurs within 2 to 6 cm (1 to 3 in) of the top of the distributor. Bed thermal conductivities in the vertical direction have been measured in the laboratory in the range of 40 to 60 kJ/(m2⋅s⋅K) [20,000 to 30,000 Btu/(h⋅ft2⋅°F⋅ft)]. Horizontal conductivities for 3-mm (s-in) particles in the range of 2 kJ/(m2⋅s⋅K) [1000 Btu/(h⋅ft2⋅°F⋅ft)] have been measured in large-scale experiments. Except in extreme L/D ratios, the temperature in the fluidized bed is uniform—generally the temperature at any point being within 5 K (10°F) of any other point. Temperature Control Because of the rapid equalization of temperatures in fluidized beds, temperature control can be accomplished in a number of ways. 1. Adiabatic. Control gas flow and/or solids feed rate so that the heat of reaction is removed as sensible heat in off gases and solids or heat supplied by gases or solids. 2. Solids circulation. Remove or add heat by circulating solids. 3. Gas circulation. Recycle gas through heat exchangers to cool or heat. 4. Liquid injection. Add volatile liquid so that the latent heat of vaporization equals excess energy. 5. Cooling or heating surfaces in bed. Solids Mixing Solids are mixed in fluidized beds by means of solids entrained in the lower portion of bubbles, and the shedding of these solids from the wake of the bubble (Rowe and Patridge, “Particle Movement Caused by Bubbles in a Fluidized Bed,” Third Congress of European Federation of Chemical Engineering, London, 1962). Thus, no mixing will occur at incipient fluidization, and mixing increases as the gas rate is increased. Naturally, particles brought to the top of the bed must displace particles toward the bottom of the bed. Generally, solids upflow is greater in the center of the bed and downward at the wall. At high ratios of fluidizing velocity to minimum fluidizing velocity, tremendous solids circulation from top to bottom of the bed assures rapid mixing of the solids. For all practical purposes, beds with L/D ratios of from 4 to 0.1 can be considered to be completely mixed continuous-reaction vessels insofar as the solids are concerned. Batch mixing using fluidization has been successfully employed in many industries. In this case there is practically no limitation to vessel dimensions. All the foregoing pertains to solids of approximately the same physical characteristics. There is evidence that solids of widely different characteristics will classify one from the other at certain gas flow rates [Geldart, Baeyens, Pope, and van de Wijer, Powder Technol., 30(2), 195 (1981)]. Two fluidized beds, one on top of the other, may be formed, or a lower static bed with a fluidized bed above may result. The latter frequently occurs when agglomeration takes place because of either fusion in the bed or poor dispersion of sticky feed solids.

Increased gas flows sometimes overcome the problem; however, improved feeding techniques or a change in operating conditions may be required. Another solution is to remove agglomerates either continuously or periodically from the bottom of the bed. Gas Mixing The mixing of gases as they pass vertically up through the bed has never been considered a problem. However, horizontal mixing is very poor and requires effective distributors if two gases are to be mixed in the fluidized bed. In bubbling beds operated at velocities of less than about 5 to 11 times Umf the gases will flow upward in both the emulsion and the bubble phases. At velocities greater than about 5 to 11 times Umf the downward velocity of the emulsion phase is sufficient to carry the contained gas downward. The back mixing of gases increases as U/Umf is increased until the circulating or fast regime is reached where the back mixing decreases as the velocity is further increased. Size Enlargement Under proper conditions, particles of solids can be caused to grow. That is sometimes advantageous and at other times disadvantageous. Growth is associated with the liquefaction or softening of some portion of the bed material (i.e., addition of soda ash to calcium carbonate feed in lime reburning, tars in fluidized-bed coking, or lead or zinc roasting cause agglomeration of dry particles in much the same way as binders act in rotary pelletizers). The motion of the particles, one against the other, in the bed results in spherical pellets. If the size of these particles is not controlled, segregation of the large particles from the bed will occur. Control can be achieved by crushing a portion of the bed product and recycling it to form nuclei for new growth. In drying solutions or slurries of solutions, the location of the feedinjection nozzle (spray nozzle) has a great effect on the size of particle formed in the bed. Also of importance are the operating temperature, relative humidity of the off gas, and gas velocity. Particle growth can occur as agglomeration or as an “onion skinning.” Size Reduction Three major size-reduction mechanisms occur in the fluidized bed. These are attrition, impact, and thermal decrepitation. Because of the random motion of the solids, some abrasion of the surface occurs. This is generally quite small, usually amounting to about 0.25 to 1 percent of the solids per day. In areas of high gas velocities, greater rates of attrition will occur as well as fracture of particles by impact. This type of jet grinding is employed in some of the coking units to control particle size [Dunlop, Griffin and Moser, J. Chem. Eng. Prog., 54, 39–43, (1958)]. It also occurs to a lesser degree at the point of gas introduction when the pressure drop to assure gas distribution is taken across an orifice or pipe that discharges directly into the bed. Thermal decrepitation occurs frequently when crystals are rearranged because of transition from one form to another or when new compounds are formed (i.e., calcination of limestone). Sometimes the strains in cases such as this are sufficient to reduce the particle to the basic crystal size. All these mechanisms will cause completion of fractures that were started before the introduction of the solids into the fluidized bed. Standpipes, Solids Feeders, and Solids Flow Control In the case of catalytic-cracking units in which the addition of catalyst is small and need not be steady, the makeup catalyst may be fed from pressurized hoppers into one of the conveying lines. The main solidsflow-control problem is to maintain balanced inventories of catalyst in and controlled flow from and to the reactor and regenerator. This flow of solids from an oxidizing atmosphere to a reducing one, or vice versa, usually necessitates stripping gases from the interstices of the solids as well as gases adsorbed by the particles. Steam is usually used for this purpose. The point of removal of the solids from the fluidized bed is usually under a lower pressure than the point of feed introduction into the carrier gas. The pressure is higher at the bottom of the solids draw-off pipe due to the relative flow of gas counter to the solids flow. The gas may either be flowing downward more slowly than the solids or upward. The standpipe may be fluidized, or the solids may be in moving packed bed flow with no expansion. Gas is introduced at the bottom (best for group B) or at about 3-m intervals along the standpipe (best for group A). The increasing pressure causes gas inside and between

FLUIDIZED-BED SYSTEMS

(a) FIG. 17-16

(b)

(c)

(d)

(e)

17-11

(f)

Solids-flow-control devices. (a) Slide valve. (b) Rotary valve. (c) Table feeder. (d) Screw feeder. (e) Cone valve. ( f ) L Valve.

the particles to be compressed. Unless aeration gas is added, the solids could defluidize and become a moving fixed bed with a lowerpressure head than that of fluidized solids. In any event, the pressure drop across the solids control valve should be designed for 3 psi or more to safely prevent backflow. See Zenz, Powder Tech., 105–113 (1986). Several designs of valves for solids flow control are used. These should be chosen with care to suit the specific conditions. Usually, block valves are used in conjunction with the control valves. Figure 17-16 shows schematically some of the devices used for solids flow control. Not shown in Fig. 17-16 is the flow-control arrangement used in the Exxon Research & Engineering Co. model IV catalytic-cracking units. This device consists of a U bend. A variable portion of regenerating air is injected into the riser leg. Changes in air-injection rate change the fluid density in the riser and thereby achieve control of the solids flow rate. Catalyst circulation rates of 1200 kg/s (70 tons/min) have been reported. When the solid is one of the reactants, such as in ore roasting, the flow must be continuous and precise in order to maintain constant conditions in the reactor. Feeding of free-flowing granular solids into a fluidized bed is not difficult. Standard commercially available solidsweighing and -conveying equipment can be used to control the rate and deliver the solids to the feeder. Screw conveyors, dip pipes, seal legs, and injectors are used to introduce the solids into the reactor proper (Fig. 17-16). Difficulties arise and special techniques must be used when the solids are not free-flowing, such as is the case with most filter cakes. One solution to this problem was developed at Cochenour-Willans. After much difficulty in attempting to feed a wet and sometimes frozen filter cake into the reactor by means of a screw feeder, experimental feeding of a water slurry of flotation concentrates was attempted. This trial was successful, and this method has been used in almost all cases in which the heat balance, particle size of solids, and other considerations have permitted. Gilfillan et al. ( J. Chem. Metall. Min. Soc. S. Afr., May 1954) and Soloman and Beal (Uranium in South Africa, 1946–56) present complete details on the use of this system for feeding. When slurry feeding is impractical, recycling of solids product to mix with the feed, both to dry and to achieve a better-handling material, has been used successfully. Also, the use of a rotary table feeder mounted on top of the reactor, discharging through a mechanical disintegrator, has been successful. The wet solids generally must be broken up into discrete particles of very fine agglomerates either by mechanical action before entering the bed or by rapidly vaporizing water. If lumps of dry or semidry solids are fed, the agglomerates do not break up but tend to fuse together. As the size of the agglomerate is many times the size of the largest individual particle, these agglomerates will segregate out of the bed, and in time the whole of the fluidized bed may be replaced with a static bed of agglomerates. Solids Discharge The type of discharge mechanism utilized is dependent upon the necessity of sealing the atmosphere inside the fluidized-bed reactor and the subsequent treatment of the solids. The simplest solids discharge is an overflow weir. This can be used only when the escape of fluidizing gas does not present any hazards due to

nature or dust content or when the leakage of gas into the fluidizedbed chamber from the atmosphere into which the bed is discharged is permitted. Solids will overflow from a fluidized bed through a port even though the pressure above the bed is maintained at a slightly lower pressure than the exterior pressure. When it is necessary to restrict the flow of gas through the opening, a simple flapper valve is frequently used. Overflow to combination seal and quench tanks (Fig. 17-17) is used when it is permissible to wet the solids and when disposal or subsequent treatment of the solids in slurry form is desirable. The FluoSeal is a simple and effective way of sealing and purging gas from the solids when an overflow-type discharge is used (Fig. 17-18). Either trickle (flapper) or star (rotary) valves are effective sealing devices for solids discharge. Each functions with a head of solids above it. Bottom of the bed discharge is also acceptable via a slide valve with a head of solids. Seal legs are frequently used in conjunction with solids-flowcontrol valves to equalize pressures and to strip trapped or adsorbed gases from the solids. The operation of a seal leg is shown schematically in Fig. 17-19. The solids settle by gravity from the fluidized bed into the seal leg or standpipe. Seal and/or stripping gas is introduced near the bottom of the leg. This gas flows both upward and downward. Pressures indicated in the illustration have no absolute value but are only relative. The legs are designed for either fluidized or settled solids. The L valve is shown schematically in Fig. 17-20. It can act as a seal and as a solids-flow control valve. However, control of solids rate is only practical for solids that deaerate quickly (Geldart B and D). The

FIG. 17-17 Quench tank for overflow or cyclone solids discharge. [Gilfillan et al., “The FluoSolids Reactor as a Source of Sulphur Dioxide,” J. Chem. Metall. Min. Soc. S. Afr. (May 1954).]

17-12

GAS-SOLID OPERATIONS AND EQUIPMENT

FIG. 17-20

FIG. 17-18

FIG. 17-19

Dorrco FluoSeal, type UA. (Dorr-Oliver Inc.)

Fluidized-bed seal leg.

L valve.

height at which aeration is added in Fig. 17-20 is usually one exit pipe diameter above the centerline of the exit pipe. For L-valve design equations, see Yang and Knowlton [Powder Tech., 77, 49–54 (1993)]. In the sealing mode, the leg is usually fluidized. Gas introduced below the normal solids level and above the discharge port will flow upward and downward. The relative flow in each direction is selfadjusting, depending upon the differential pressure between the point of solids feed and discharge and the level of solids in the leg. The length and diameter of the discharge spout are selected so that the undisturbed angle of repose of the solids will prevent discharge of the solids. As solids are fed into the leg, height H of solids increases. This in turn reduces the flow of gas in an upward direction and increases the flow of gas in a downward direction. When the flow of gas downward and through the solids-discharge port reaches a given rate, the angle of repose of the solids is upset and solids discharge commences. Usually, the level of solids above the point of gas introduction will float. When used as a flow controller, the vertical leg is best run in the packed bed mode. The solids flow rate is controlled by varying the aeration gas flow. In most catalytic-reactor systems, no solids removal is necessary as the catalyst is retained in the system and solids loss is in the form of fines that are not collected by the dust-recovery system. Dust Separation It is usually necessary to recover the solids carried by the gas leaving the disengaging space or freeboard of the fluidized bed. Generally, cyclones are used to remove the major portion of these solids (see “Gas-Solids Separation”). However, in a few cases, usually on small-scale units, filters are employed without the use of cyclones to reduce the loading of solids in the gas. For hightemperature usage, either porous ceramic or sintered metal has been employed. Multiple units must be provided so that one unit can be blown back with clean gas while one or more are filtering. Cyclones are arranged generally in any one of the arrangements shown in Fig. 17-21. The effect of cyclone arrangement on the height of the vessel and the overall height of the system is apparent. Details regarding cyclone design and collection efficiencies are to be found in another portion of this section. Discharging of the cyclone into the fluidized bed requires some care. It is necessary to seal the bottom of the cyclone so that the collection efficiency of the cyclone will not be impaired by the passage of appreciable quantities of gas up through the solids-discharge port. This is usually done by sealing the dip leg in the fluid bed. Experience has shown, particularly in the case of deep beds, that the bottom of the dip pipe must be protected from the action of large gas bubbles which, if allowed to pass up the leg, would carry quantities of fine solids up into the cyclone and cause momentarily high losses. This can be done by attaching a plate larger in diameter than the pipe to the bottom (see Fig. 17-22e). Example 1: Length of Seal Leg The length of the seal leg can be estimated as shown in the following example. Given: Fluid density of bed at 0.3-m/s (1-ft/s) superficial gas velocity = 1100 kg/m3 (70 lb/ft3). Fluid density of cyclone product at 0.15 m/s (0.5 ft/s) = 650 kg/m3 (40 lb/ft3). Settled bed depth = 1.8 m (6 ft) Fluidized-bed depth = 2.4 m (8 ft) Pressure drop through cyclone = 1.4 kPa (0.2 lbf/in2) In order to assure seal at startup, the bottom of the seal leg is 1.5 m (5 ft) above the constriction plate or submerged 0.9 m (3 ft) in the fluidized bed.

FLUIDIZED-BED SYSTEMS

(a)

(b)

(c)

(d)

17-13

(e)

Fluidized-bed cyclone arrangements. (a) Single-stage internal cyclone. (b) Two-stage internal cyclone. (c) Single-stage external cyclone; dust returned to bed. (d) Two-stage external cyclone; dust returned to bed. (e) Two-stage external cyclone; dust collected externally.

FIG. 17-21

The pressure at the solids outlet of a gas cyclone is usually about 0.7 kPa (0.1 lbf/in2) lower than the pressure at the discharge of the leg. Total pressure to be balanced by the fluid leg in the cyclone dip leg is (0.9 × 1100 × 9.81)/1000 + 1.4 + 0.7 = 11.8 Kpa [(3 × 70)/144 + 0.2 + 0.1 = 1.7 lb/in2] Height of solids in dip leg = (11.8 × 1000)/(650 × 9.81) = 1.9 m [(1.7 × 144)/40 = 6.1 ft]; therefore, the bottom of the separator pot on the cyclone must be at least 1.9 + 1.5 or 3.4 m (6.1 + 5 or 11.1 ft) above the gas distributor. To allow for upsets, changes in size distribution, etc., use 4.6 m (15 ft).

In addition to the open dip leg, various other devices have been used to seal cyclone solids returns, especially for second-stage cyclones. A number of these are shown in Fig. 17-22. One of the most frequently used is the trickle valve (17-22a). There is no general agreement as to whether this valve should discharge below the bed level or in the freeboard. In any event, the legs must be large enough to carry momentarily high rates of solids and must provide seals to overcome cyclone pressure drops as well as to allow for differences in fluid density of bed and cyclone products. It has been reported that, in the case of catalytic-cracking catalysts, the fluid density of the solids collected by the primary cyclone is essentially the same as that in the fluidized bed because the particles in the bed are so small, nearly all are entrained. However, as a general rule the fluidized density of solids collected by the first cyclone is less than the fluidized density of the bed. Each succeeding cyclone collects finer and less dense solids. As cyclones are less effective as the particle size decreases, secondary collection units are frequently required, i.e., filters, electrostatic precipitators, and scrubbers. When dry collection is not required, elimination of cyclones is possible if allowance is made for heavy solids loads in the scrubber (see “Gas-Solids Separations”; see also Sec. 14).

(a)

(b)

(c)

Instrumentation Temperature Measurement This is usually simple, and standard temperature-sensing elements are adequate for continuous use. Because of the high abrasion wear on horizontal protection tubes, vertical installations are frequently used. In highly corrosive atmospheres in which metallic protection tubes cannot be used, short, heavy ceramic tubes have been used successfully. Pressure Measurement Although successful pressure-measurement probes or taps have been fabricated by using porous materials, the most universally accepted pressure tap consists of a purged tube projecting into the bed as nearly vertically as possible. Minimum internal diameters are 1 to 2 cm (a to 1 in). A purge rate of at least 0.9 m/s (3 ft/s) is usually required. Pressure measurements taken at various heights in the bed are used to determine bed level. Bed density is determined directly from ∆P/L, the pressure drop inside the bed itself (∆P/L in units of weight/area × L). The overall bed weight is obtained from ∆P taken between the grid and the freeboard. Nominal bed height is the length required to have the overall weight at the measured density. Of course, splashing and entrainment will place solids well above the nominal bed height in most cases. The pressure-drop signal is noisy due to bubble effects and the generally statistical nature of fluid bed flow parameters. A fast fourier transform of the pressure drop signal transforms the perturbations to a frequency-versus-amplitude plot with a maximum at about 5 Hz and frequencies generally tailing off above 20 Hz. Changes in frequency and amplitude are associated with changes in the quality of the fluidization. Experienced operators can frequently predict performance from changes in the ∆P signal. Flow Measurement Measurement of flow rates of clean gases presents no problem. Flow measurement of dirty gases is usually

(d)

(e)

Cyclone solids-return seals. (a) Trickle valve. (Ducon Co., Inc.) (b) J valve. (c) L valve. (d) Fluid-seal pot. (e) “Dollar” plate. a, b, c, and d may be used above the bed; a and e are used below the bed.

FIG. 17-22

17-14

GAS-SOLID OPERATIONS AND EQUIPMENT

avoided. The flow of solids is usually controlled but not measured except externally to the system. Solids flows in the system are usually adjusted on an inferential basis (temperature, pressure level, catalyst activity, gas analysis, etc.). In many roasting operations the color of the calcine indicates solids feed rate. USES OF FLUIDIZED BEDS There are many uses of fluidized beds. A number of applications have become commercial successes; others are in the pilot-plant stage, and others in bench-scale stage. Generally, the fluidized bed is used for gas-solids contacting; however, in some instances the presence of the gas or solid is used only to provide a fluidized bed to accomplish the end result. Uses or special characteristics follow: I. Chemical reactions A. Catalytic B. Noncatalytic 1. Homogeneous 2. Heterogeneous II. Physical contacting A. Heat transfer 1. To and from fluidized bed 2. Between gases and solids 3. Temperature control 4. Between points in bed B. Solids mixing C. Gas mixing D. Drying 1. Solids 2. Gases E. Size enlargement F. Size reduction G. Classification 1. Removal of fines from solids 2. Removal of fines from gas H. Adsorption-desorption I. Heat treatment J. Coating Chemical Reactions Catalytic Reactions This use has provided the greatest impetus for use, development, and research in the field of fluidized solids. Some of the details pertaining to this use are to be found in the preceding pages of this section. Reference should also be made to Sec. 23. Cracking. The evolution of fluidized catalytic cracking (FCC) since the early 1940s has resulted in several configurations depending upon the particular use and designer. The high rate of transfer of solids between the regenerator and the reactor permits a balancing of the exothermic burning of carbon and tars in the regenerator and the endothermic cracking of petroleum in the reactor, so that temperature in both units can usually be controlled without resorting to auxiliary heat-control mechanisms. The high rate of catalyst circulation also permits the maintenance of the catalyst at a constantly high activity. The original regenerators were considered to be backmixed units. Newer systems have staged regenerators to improve conversion (see Fig. 17-23). The use of the riser reactor (transport or fast fluid bed) results in much lower gas and solids back mixing and an approach to plug flow. The first fluid catalytic-cracking unit was placed in operation in Baytown, Texas, in 1942. This was a low-pressure, 115- to 120-kPa (2- to 3-psig) unit operating in what is now called the turbulent fluidization mode, 1.2 to 1.8 m/s (4 to 6 ft/s). Even before the startup of the first model I, it was realized that by lowering the velocity, a dense, aggregative fluidized bed, 300 to 400 kg/m3 (20 to 25 lb/ft3), would be formed, allowing completion of reaction and regeneration. Pressure was increased to 240 to 320 kPa (20 to 30 psig). In the 1960s more active catalysts resulted in the use of riser cracking. Recently, heavier crude feedstocks have produced higher coke yields. This has necessitated addition of catalyst cooling to the regeneration step as shown in Fig. 17-24. Many companies participated in the development of the

FIG. 17-23

UOP fluid cracking unit. (Reprinted with permission of UOP.)

Modern FCC unit configured for high-efficiency regeneration and extra catalyst cooling. (Reprinted with permission of UOP. RCC is a service mark of Ashland Oil Inc.)

FIG. 17-24

FLUIDIZED-BED SYSTEMS cat cracker, including Exxon Research & Engineering Co., Universal Oil Products Companies, Kellogg Co., Texaco Development Corp., Gulf Research Development Co., and Shell Oil Company. Many of the companies provide designs and/or licenses to operate to others. For further details, see Luckenbach, Reichle, Gladrow, and Worley, “Cracking, Catalytic,” in McKetta (ed.), Encyclopedia of Chemical Processing and Design, vol. 13, Marcel Dekker, New York, 1981, pp. 1–132. Alkyl chlorides. Olefins are chlorinated to alkyl chlorides in a single fluidized bed. HCl reacts with O2 over a copper chloride catalyst to form chlorine. The chlorine reacts with the olefin to form the alkyl chloride. The process developed by the Shell Development Co. uses a recycle of catalyst fines in aqueous HCl to control the temperature [Chem. Proc., 16, 42 (1953)]. Phthalic anhydride. Naphthalene is oxidized by air to phthalic anhydride in a bubbling fluidized reactor. Even though the naphthalene feed is in liquid form, the reaction is highly exothermic. Temperature control is achieved by removing heat through vertical tubes in the bed to raise steam [Graham and Way, Chem. Eng. Prog., 58, 96 (January 1962)]. Acrylonitrile. Acrylonitrile is produced by reacting propylene, ammonia, and oxygen (air) in a single fluidized bed of a complex catalyst. Known as the SOHIO process, this process was first operated commercially in 1960. In addition to acrylonitrile, significant quantities of HCN and acetonitrile are also produced. This process is also exothermic. Temperature control is achieved by raising steam inside vertical tubes immersed in the bed [Veatch, Hydrocarbon Process. Pet. Refiner, 41, 18 (November 1962)]. Fischer-Tropsch synthesis. The scale-up of a bubbling-bed reactor to produce gasoline from CO and H2 was unsuccessful (see “Design of Fluidized-Bed Systems: Scale-Up”). However, Kellogg Co. developed a successful Fischer-Tropsch synthesis reactor based on a dilute-phase or transport-reactor concept. Kellogg, in its design, prevented gas bypassing by using the transport reactor and maintained temperature control of the exothermic reaction by inserting heat exchangers in the transport line. This process has been very successful and repeatedly expanded at the South African Synthetic Oil Limited (SASOL) plant in the Republic of South Africa, where politics and economics favor the conversion of coal to gasoline and other hydrocarbons. Refer to Jewell and Johnson, U.S. Patent 2,543,974, Mar. 6, 1951. Recently, the process has been modified to a simpler, less expensive turbulent bed catalytic reactor system [Silverman et al., Fluidization V, Engineering Foundation 1986, pp. 441–448). The first commercial fluidized bed polyethylene plant was constructed by Union Carbide in 1968. Modern units operate at 100°C and 32 MPa (300 psig). The bed is fluidized with ethylene at about 0.5 m/s and probably operates near the turbulent fluidization regime. The excellent mixing provided by the fluidized bed is necessary to prevent hot spots, since the unit is operated near the melting point of the product. A model of the reactor (Fig. 17-25) that couples kinetics to the hydrodynamics was given by Choi and Ray, Chem. Eng. Sci., 40, 2261, 1985. Additional catalytic processes. Nitrobenzene is hydrogenated to aniline (U.S. Patent 2,891,094). Melamine and isophthalonitrile are produced in catalytic fluidized-bed reactors. Badger has announced a process to produce maleic anhydride by the partial oxidation of butane (Schaffel, Chen, and Graham, “Fluidized Bed Catalytic Oxidation of Butane to Maleic Anhydride,” presented at Chemical Engineering World Congress, Montreal, 1981). Dupont has announced a circulating bed process for production of maleic anhydride (Contractor, Circulating Fluidized Bed Tech. II, Pergamon, 1988, pp. 467– 474). Mobil Oil has developed a commercial process to convert methanol to gasoline. (Grimmer et al., Methane Conversion, Elsevier, 1988, pp. 273–291). Noncatalytic Reactions Homogeneous reactions. Homogeneous noncatalytic reactions are normally carried out in a fluidized bed to achieve mixing of the gases and temperature control. The solids of the bed act as a heat sink or source and facilitate heat transfer from or to the gas or from or to heat-exchange surfaces. Reactions of this type include chlorination of hydrocarbons or oxidation of gaseous fuels.

FIG. 17-25

17-15

High-pressure polyethylene reactor.

Heterogeneous reactions. This category covers the greatest commercial use of fluidized beds other than petroleum cracking. The roasting of sulfide, arsenical, and/or antimonial ores to facilitate the release of gold or silver values; the roasting of pyrite, pyrrhotite, or naturally occurring sulfur ores to provide SO2 for sulfuric acid manufacture; and the roasting of copper, cobalt, and zinc sulfide ores to solubilize the metal values are the major metallurgical uses. Figure 17-26 shows basic items in the system. Thermally efficient calcination of lime dolomite and clay can be carried out in a multicompartment fluidized bed (Fig. 17-27). Fuels are burned in a fluidized bed of the product to produce the required heat. Bunker C oil, natural gas, and coal are used in commercial units. Temperature control is accurate enough to permit production of lime of very high availability with close control of slaking characteristics. Also, half calcination of dolomite is an accepted practice. The requirement of large crystal size for the limestone limits application. Smallsized crystals in the limestone result in low yields due to high dust losses.

FIG. 17-26

Single-stage FluoSolids roaster or dryer. (Dorr-Oliver, Inc.)

17-16

FIG. 17-27

GAS-SOLID OPERATIONS AND EQUIPMENT

FluoSolids multicompartment fluidized bed. (Dorr-Oliver, Inc.)

Phosphate rock is calcined to remove carbonaceous material before being digested with sulfuric acid. Several different fluidization processes have been commercialized for the direct reduction of hematite to high-iron, low-oxide products. Foundry sand is also calcined to remove organic binders and release fines. The calcination of Al(OH)3 to Al2O3 in a circulating fluidized process produces a high-grade product. The process combines the use of circulating, bubbling, and transport beds to achieve high thermal efficiency. See Fig. 17-28. An interesting feature of these high-temperature-calcination applications is the direct injection of either heavy oil, natural gas, or fine coal into the fluidized bed. Combustion takes place at well below flame temperatures without atomization. Considerable care in the design of the fuel- and air-supply system is necessary to take full advantage of the fluidized bed, which serves to mix the air and fuel. Coal can be burned in fluidized beds in an environmentally acceptable manner by adding limestone or dolomite to the bed to

react with the SO2 to form CaSO4. Because of moderate combustion temperature, about 800 to 900°C, NOx, which results from the oxidation of nitrogen compounds contained in the coal, is kept at a low level. NOx is increased by higher temperatures and higher excess oxygen. Two-stage air addition reduces NOx. Several concepts of fluidized-bed combustion have been or are being developed. Atmospheric fluidized-bed combustion (AFBC), in which most of the heat-exchange tubes are located in the bed, is illustrated in Fig. 17-29. This type of unit is most commonly used for industrial applications up to about 50 metric tons/h of steam generation. Larger units are generally of the circulating bed type as shown in Fig. 17-30. Circulating fluidized bed combustors have many advantages. The velocity is significantly higher for greater throughput. Since all the solids are recycled, fine limestone and coal can be fed, which gives better limestone utilization and greater laitude in specifying coal sizing. Because of erosion due to high velocity coarse solids, heat-transfer surface is usually not designed into the bottom of the combustion zone. Figure 17-30 shows a commercial 110 MWe unit. Pressurized fluidized-bed combustion (PFBC) is, as the name implies, operated at above atmospheric pressures. The beds and heat-transfer surface are stacked to conserve space and to reduce the size of the pressure vessel. This type of unit is usually conceived as a cogeneration unit. Steam raised in the boilers would be employed to drive turbines or for other uses, and the hot pressurized gases after cleaning would be let down through an expander coupled to a compressor to supply the compressed combustion air and/or electric generator. A 71 MWe unit is shown in Fig. 17-31. Also see Sec. 27, “Energy Sources: Conversion and Utilization.” Incineration The majority of over 400 units in operation are used for the incineration of biological sludges. These units can be designed to operate autogenously with wet sludges containing as little as 6 MJ/kg (2600 Btu/lb) heating value (Fig. 17-32). Depending on the calorific value of the feed, heat can be recovered as steam either by means of waste-heat boilers or by a combination of waste-heat boilers and the heat-exchange surface in the fluid bed. Several units are used for sulfite-paper-mill waste-liquor disposal. At least six units are used for oil-refinery wastes, which sometimes include a mixture of liquid sludges, emulsions, and caustic waste

Fluidized-bed steam generator at Georgetown University; 12.6kg/s (100,000-lb/h) steam at 4.75-MPa (675-psig) pressure. (From Georgetown Univ. Q. Tech. Prog. Rep. METC/DOE/10381/135, July–September 1980.)

FIG. 17-29 FIG. 17-28

Circulating-fluid-bed calciner. (Lurgi Corp.)

FLUIDIZED-BED SYSTEMS

FIG. 17-30

17-17

Side view of 110 MWe Nucla CFB boiler. (Source: Pyropower Corporation.)

[Flood and Kernel, Chem. Proc. (Sept. 8, 1973)]. Miscellaneous uses include the incineration of sawdust, carbon-black waste, pharmaceutical waste, grease from domestic sewage, spent coffee grounds, and domestic garbage. Toxic or hazardous wastes can be disposed of in fluidized beds by either chemical capture or complete destruction. In the former case, bed material, such as limestone, will react with halides, sulfides, metals, etc., to form stable compounds which can be landfilled. Contact times of up to 5 or 10 s at 1200 K (900°C) to 1300 K (1000°C) assure complete destruction of most compounds.

Physical Contacting Drying Fluidized-bed units for drying solids, particularly coal, cement, rock, and limestone, are in general acceptance. Economic considerations make these units particularly attractive when large tonnages of solids are to be handled. Fuel requirements are 3.3 to 4.2 MJ/kg (1500 to 1900 Btu/lb of water removed), and total power for blowers, feeders, etc., is about 0.08 kWh/kg of water removed. The maximum-sized feed is 6 cm (1a in) × 0 coal. One of the major advantages of this type of dryer is the close control of conditions so that a predetermined amount of free moisture may be left with the solids to

17-18

GAS-SOLID OPERATIONS AND EQUIPMENT

FIG. 17-32

Hot windbox incinerator/reactor with air preheating. (Dorr-

Oliver, Inc.)

Classification The separation of fine particles from coarse can be effected by use of a fluidized bed (see “Drying”). However, for economic reasons (i.e., initial cost, power requirements for compression of fluidizing gas, etc.), it is doubtful except in special cases if a fluidized-bed classifier would be built for this purpose alone. It has been proposed that fluidized beds be used to remove fine solids from a gas stream. This is possible under special conditions. Adsorption-Desorption An arrangement for gas fractionation is shown in Fig. 17-33. The effects of adsorption and desorption on the performance of fluidized beds are discussed elsewhere. Adsorption of carbon disulfide

FIG. 17-31

71 MWe PFBC unit. (From Steam, 40th ed., 29-9, Babcock &

Wilcox, 1992).

prevent dusting of the product during subsequent material-handling operations. The fluidized-bed dryer is also used as a classifier so that both drying and classification operations are accomplished simultaneously. Wall and Ash [Ind. Eng. Chem., 41, 1247 (1949)] state that, in drying 4.8-mm (−4 mesh) dolomite with combustion gases at a superficial velocity of 1.2 m/s (4 ft/s), the following removals of fines were achieved: Particle size

% removed

−65 + 100 mesh −100 + 150 mesh −150 + 200 mesh −200 + 325 mesh −325 mesh

60 79 85 89 89

FIG. 17-33

1953).]

Fluidized bed for gas fractionation. [Sittig, Chem. Eng. (May

GAS-SOLIDS SEPARATIONS vapors from air streams as great as 300 m3/s (540,000 ft3/min) in a 17-m- (53-ft-) diameter unit has been reported by Avery and Tracey (“The Application of Fluidized Beds of Activated Carbon to Recover Solvent from Air or Gas Streams,” Tripartate Chemical Engineering Conference, Montreal, Sept. 24, 1968). Heat Treatment Heat treatment can be divided into two types, treatment of fluidizable solids and treatment of large, usually metallic objects in a fluid bed. The former is generally accomplished in multicompartment units to conserve heat (Fig. 17-27). The heat treatment of large metallic objects is accomplished in long, narrow heated beds.

17-19

The objects are conveyed through the beds by an overhead conveyor system. Fluid beds are used because of the high heat-transfer rate and uniform temperature. See Reindl, “Fluid Bed Technology,” American Society for Metals, Cincinnati, Sept. 23, 1981; Fennell, Ind. Heat., 48, 9, 36 (September 1981). Coating Fluidized beds of thermoplastic resins have been used to facilitate the coating of metallic parts. A properly prepared, heated metal part is dipped into the fluidized bed, which permits complete immersion in the dry solids. The heated metal fuses the thermoplastic, forming a continuous uniform coating.

GAS-SOLIDS SEPARATIONS This subsection is concerned with the application of particle mechanics (see Sec. 5, “Fluid and Particle Mechanics”) to the design and application of dust-collection systems. It includes wet collectors, or

scrubbers, for particle collection. Scrubbers designed for purposes of mass transfer are discussed in Secs. 14 and 18. Equipment for removing entrained liquid mist from gases is described in Sec. 18.

Nomenclature Except where otherwise noted here or in the text, either consistent system of units (SI or U.S. customary) may be used. Only SI units may be used for electrical quantities, since no comparable electrical units exist in the U.S. customary system. When special units are used, they are noted at the point of use. Symbols

Definition

SI units

U.S. customary units

Bc Be

Width of rectangular cyclone inlet duct Spacing between wire and plate, or between rod and curtain, or between parallel plates in electrical precipitators Width of gravity settling chamber Dry scrubber pollutant gas equilibrium concentration over sorbent Dry scrubber pollutant gasinlet concentration Dry scrubber pollutant gasoutlet concentration Dust concentration in gas stream Specific heat of gas Specific heat of collecting body Specific heat of particle Diameter or other representative dimension of collector body or device Other characteristic dimensions of collector body or device Cyclone diameter Outside diameter of wire or discharge electrode of concentriccylinder type of electrical precipitator Diameter of cyclone gas exit duct Volume/surface-mean-drop diameter Diameter of particle Cut diameter, diameter of particles of which 50% of those present are collected Particle diameter of fraction number c′ Inside diameter of collecting tube of concentriccylinder type of electrical precipitator Diffusion coefficient for particle Decontamination index = log10[1/(1 − η)] Natural (napierian) logarithmic base Potential difference Potential difference required for corona discharge to commence Voltage across dust layer Potential difference required for sparking to commence Cyclone collection efficiency at actual loading Cyclone collection efficiency at low loading Effective friction loss across wetted equipment in scrubber Packed bed friction loss Conversion factor Local acceleration due to gravity Height of rectangular cyclone inlet duct Height of gravity settling chamber Electrical current per unit of electrode length

m m

ft ft

m

ft

g/m3 J/(k⋅K) J/(kg⋅K) J/(kg⋅K) m

Btu/(lbm⋅°F) Btu/(lbm⋅°F) Btu/(lbm⋅°F) ft

m m m

ft ft ft

m

ft

m m

ft ft

m m

ft ft

m2/s Dimensionless 2.718 . . . V V

ft2/s Dimensionless 2.718 . . .

Bs C* C1 C2 cd ch chb chp Db Db1,Db2 Dc Dd De Do Dp Dpth dP Dt Dv DI e E Ec Ed Es EL EO FE Fk gc gL Hc Hs I

Special units

grains/ft3

µm µm µm

V V

kPa

m/s2 m m A/m

in water 32.17 (lbm/lbf)(ft/s2) ft/s2 ft ft

17-20

GAS-SOLID OPERATIONS AND EQUIPMENT

Nomenclature (Continued ) Symbols j kρ

Definition

K1

Corona current density at dust layer Density of gas relative to its density at 0°C, 1 atm Thermal conductivity of gas Thermal conductivity of collecting body Thermal conductivity of particle Empirical proportionality constant for cyclone pressure drop or friction loss Resistance coefficient of “conditioned” filter fabric

K2

Resistance coefficient of dust cake on filter fabric

Ka

Proportionality constant, for target efficiency of a single fiber in a bed of fibers

Kc

Resistance coefficient for “conditioned” filter fabric

Kd

Resistance coefficient for dust cake on filter fabric

Ke

Electrical-precipitator constant

KF Ko

Resistance coefficient for clean filter cloth “Energy-distance” constant for electrical discharge in gases Stokes-Cunningham correction factor Thickness of fibrous filter or of dust layer on surface filter Length of collecting electrode in direction of gas flow Length of gravity settling chamber in direction of gas flow Natural logarithm (logarithm to the base e) Molecular weight Exponent Knudsen number = λm/Db Mach number Number of elementary electrical charges acquired by a particle Reynolds number = (DpρVo /µ) or (Dpρut /µ) Interaction number = 18 µ/KmρpDv Diffusional separation number Electrostatic-attraction separation number Electrostatic-induction separation number Flow-line separation number Gravitational separation number Inertial separation number Thermal separation number Number of transfer units = ln [1/(1 − η)] Number of turns made by gas stream in a cyclone separator Gas pressure drop Gas pressure drop in cyclone or filter Gauge pressure of water fed to scrubber Gas-phase contacting power Liquid-phase contacting power Mechanical contacting power Total contacting power Gas flow rate Gas flow rate Liquid flow rate Electrical charge on particle Radius; distance from centerline of cyclone separator; distance from centerline of concentric-cylinder electrical precipitator Time Absolute gas temperature Absolute temperature of collecting body

kt ktb ktp K

Km L Le Ls ln M n NKn NMa No NRe Nsc Nsd Nsec Nsei Nsf Nsg Nsi Nst Nt Ns ∆p ∆pi pF PG PL PM PT q QG QL Qp r tm T Tb

SI units

U.S. customary units

Special units

A/m2 Dimensionless

Dimensionless

Dimensionless

W/(m⋅K) W/(m⋅K) W/(m⋅K) Dimensionless

Btu/(s⋅ft⋅°F) Btu/(s⋅ft⋅°F) Btu/(s⋅ft⋅°F) Dimensionless

kPa/(m/min) kPa ᎏᎏ (m/min)(g/m2)

in water/(ft/min) in water ᎏᎏ (ft/min)(lbm/ft2)

Dimensionless

Dimensionless in water ᎏᎏ (ft/min)(cP) in water ᎏᎏ (ft/min)(gr/ft2)(cP)

s/m

s/ft in water ᎏᎏ (ft/min)(cP)

m Dimensionless m

Dimensionless ft

m m Dimensionless kg/mol Dimensionless Dimensionless Dimensionless Dimensionless

ft ft Dimensionless lbm/mol Dimensionless Dimensionless Dimensionless Dimensionless

Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless kPa

Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless lbf/ft2

kPa MJ/1000 m3 MJ/1000 m3 MJ/1000 m3 MJ/1000 m3 m3/s

ft3/s ft3/s ft3/s

C m

ft

K K

°R °R

Dimensionless

Dimensionless Dimensionless

in water in water lbf/in2 hp/(1000 ft3/min) hp/(1000 ft3/min) hp/(1000 ft3/min) hp/(1000 ft3/min) ft3/min gal/min

min

GAS-SOLIDS SEPARATIONS

17-21

Nomenclature (Concluded ) Symbols us ut vm vp Vf Vo Vs Vct w

Definition Velocity of migration of particle toward collecting electrode Terminal settling velocity of particle under action of gravity Average cyclone inlet velocity, based on area Ac Actual particle velocity Filtration velocity (superficial gas velocity through filter) Gas velocity Average gas velocity in gravity settling Tangential component of gas velocity in cyclone Loading of collected dust on filter

SI units

U.S. customary units

m/s m/s m/s m/s m/min m/s m/s m/s g/m2

ft/s ft/s ft/s ft/s

Special units

ft/s ft/s ft/min

ft/s ft/s ft/s lbm/ft2

gr/ft2

Greek symbols α γ δ δg δo δb δp ∆ ε εb εv ζ η ηo ηt λi λp µ µL ρ ρd ρL ρs ρ′ σ σavg σL φs

Empirical constant in equation of scrubber performance curve Empirical constant in equation of scrubber performance curve Dielectric constant Dielectric constant at 0°C, 1 atm Permittivity of free space Dielectric constant of collecting body Dielectric constant of particle Fractional free area (for screens, perforated plates, grids) Elementary electrical charge Characteristics potential gradient at collecting surface Fraction voids in bed of solids (δ − 1) = 1 + 2 ᎏ ranges from a value of 1 for materials (δ + 2) with a dielectric constant of 1 to 3 for conductors Collection efficiency, weight fraction of entering dispersoid collected Target efficiency of an isolated collecting body, fraction of dispersoid in swept volume collected on body Target efficiency of a single collecting body in an array of collecting bodies, fraction of dispersoid in swept volume collected on body Ionic mobility of gas Particle mobility = ue /Ᏹ Gas viscosity Liquid viscosity Gas density Resistivity of dust layer Liquid density True (not bulk) density of solids or liquid drops Density of gas relative to its density at 25°C, 1 atm Ion density Average ion density Liquid surface tension Particle shape factor = (surface of sphere)/ (surface of particle of same volume)

MJ

Potential gradient Potential gradient required for corona discharge to commence Average potential gradient in ionization stage Electrical breakdown constant for gas Average potential gradient in collection stage Potential gradient required for sparking to commence

 ᎏᎏ 100 ft /min  hp

3

Dimensionless Dimensionless F/m Dimensionless Dimensionless Dimensionless 1.60210 × 10−19 C V/m Dimensionless

−γ

3

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

(m/s)/(V/m) (m/s)/(V/m) Pa⋅s

lbm/(s⋅ft)

cP cP

Dimensionless

g/m3 Ω⋅m kg/m3 Dimensionless Number/m3 Number/m3 Dimensionless

Script symbols Ᏹ Ᏹc Ᏹi Ᏹo Ᏹp Ᏹs

−γ

ᎏ 1000 m 

V/m V/m V/m V/m V/m V/m

lb/ft3 lbm/ft3 lbm/ft3 Dimensionless

lbm/ft3 lbm/ft3 Dimensionless

Dimensionless

dyn/cm Dimensionless

17-22

GAS-SOLID OPERATIONS AND EQUIPMENT

GENERAL REFERENCES: Burchsted, Kahn, and Fuller, Nuclear Air Cleaning Handbook, ERDA 76-21, Oak Ridge, Tenn., 1976. Cadle, The Measurement of Airborne Particles, Wiley, New York, 1975. Davies, Aerosol Science, Academic, New York, 1966. Davies, Air Filtration, Academic, New York, 1973. Dennis, Handbook on Aerosols, ERDA TID-26608, Oak Ridge, Tenn., 1976. Drinker and Hatch, Industrial Dust, 2d ed., McGraw-Hill, New York, 1954. Friedlander, Smoke, Dust, and Haze, Wiley, New York, 1977. Fuchs, The Mechanics of Aerosols, Pergamon, Oxford, 1964. Green and Lane, Particulate Clouds: Dusts, Smokes, and Mists, Van Nostrand, New York, 1964. Lapple, Fluid and Particle Mechanics, University of Delaware, Newark, 1951. Licht, Air Pollution Control Engineering—Basic Calculations for Particle Collection, Marcel Dekker, New York, 1980. Liu, Fine Particles—Aerosol Generation, Measurement, Sampling, and Analysis, Academic, New York, 1976. Lunde and Lapple, Chem. Eng. Prog., 53, 385 (1957). Lundgren et al., Aerosol Measurement, University of Florida, Gainesville, 1979. Mercer, Aerosol Technology in Hazard Evaluation, Academic, New York, 1973. Nonhebel, Processes for Air Pollution Control, CRC Press, Cleveland, 1972. Shaw, Fundamentals of Aerosol Science, Wiley, New York, 1978. Stern, Air Pollution: A Comprehensive Treatise, vols. 3 and 4, Academic, New York, 1977. Strauss, Industrial Gas Cleaning, 2d ed., Pergamon, New York, 1975. Theodore and Buonicore, Air Pollution Control Equipment: Selection, Design, Operation, and Maintenance, Prentice-Hall, Englewood Cliffs, N.J., 1982. White, Industrial Electrostatic Precipitation, Addison-Wesley, Reading, Mass., 1963. White and Smith, High-Efficiency Air Filtration, Butterworth, Washington, 1964. ASME Research Committee on Industrial and Municipal Wastes, Combustion Fundamentals for Waste Incineration, American Society of Mechanical Engineers, 1974. Buonicore and Davis (eds.), Air Pollution Engineering Manual, Air & Waste Management Association, Van Nostrand Reinhold, 1992. Burchsted, Fuller, and Kahn, Nuclear Air Cleaning Handbook, ORNL for the U.S. Energy Research and Development Administration, NTIS Report ERDA 76-21, 1976. Dennis (ed.), Handbook on Aerosols, GCA for the U.S. Energy Research and Development Administration, NTIS Report TID26608, 1976. Stern, Air Pollution, 3d ed., Academic Press, 1977 (supplement 1986).

PURPOSE OF DUST COLLECTION Dust collection is concerned with the removal or collection of solid dispersoids in gases for purposes of: 1. Air-pollution control, as in fly-ash removal from power-plant flue gases 2. Equipment-maintenance reduction, as in filtration of engineintake air or pyrites furnace-gas treatment prior to its entry to a contact sulfuric acid plant 3. Safety- or health-hazard elimination, as in collection of siliceous and metallic dusts around grinding and drilling equipment and in some metallurgical operations and flour dusts from milling or bagging operations 4. Product-quality improvement, as in air cleaning in the production of pharmaceutical products and photographic film 5. Recovery of a valuable product, as in collection of dusts from dryers and smelters 6. Powdered-product collection, as in pneumatic conveying; the spray drying of milk, eggs, and soap; and the manufacture of highpurity zinc oxide and carbon black PROPERTIES OF PARTICLE DISPERSOIDS An understanding of the fundamental properties and characteristics of gas dispersoids is essential to the design of industrial dust-control equipment. Figure 17-34 shows characteristics of dispersoids and other particles together with the types of gas-cleaning equipment that are applicable to their control. Two types of solid dispersoids are shown: (1) dust, which is composed of particles larger than 1 µm; and (2) fume, which consists of particles generally smaller than 1 µm. Dusts usually result from mechanical disintegration of matter. They may be redispersed from the settled, or bulk, condition by an air blast. Fumes are submicrometer dispersoids formed by processes such as combustion, sublimation, and condensation. Once collected, they cannot be redispersed from the settled condition to their original state of dispersion by air blasts or mechanical dispersion equipment. The primary distinguishing characteristic of gas dispersoids is particle size. The generally accepted unit of particle size is the micrometer, µm. (Prior to the adoption of the SI system, the same unit was known as the micron and was designated by µ.) The particle size of a gas dispersoid is usually taken as the diameter of a sphere having the same

mass and density as the particle in question. Another common method is to designate the screen mesh that has an aperture corresponding to the particle diameter; the screen scale used must also be specified to avoid confusion. From the standpoint of collector design and performance, the most important size-related property of a dust particle is its dynamic behavior. Particles larger than 100 µm are readily collectible by simple inertial or gravitational methods. For particles under 100 µm, the range of principal difficulty in dust collection, the resistance to motion in a gas is viscous (see Sec. 6, “Fluid and Particle Mechanics”), and for such particles, the most useful size specification is commonly the Stokes settling diameter, which is the diameter of the spherical particle of the same density that has the same terminal velocity in viscous flow as the particle in question. It is yet more convenient in many circumstances to use the “aerodynamic diameter,” which is the diameter of the particle of unit density (1 g/cm3) that has the same terminal settling velocity. Use of the aerodynamic diameter permits direct comparisons of the dynamic behavior of particles that are actually of different sizes, shapes, and densities [Raabe, J. Air Pollut. Control Assoc., 26, 856 (1976)]. When the size of a particle approaches the same order of magnitude as the mean free path of the gas molecules, the settling velocity is greater than predicted by Stokes’ law because of molecular slip. The slip-flow correction is appreciable for particles smaller than 1 µm and is allowed for by the Cunningham correction for Stokes’ law (Lapple, op. cit.; Licht, op. cit.). The Cunningham correction is applied in calculations of the aerodynamic diameters of particles that are in the appropriate size range. Although solid fume particles may range in size down to perhaps 0.001 µm, fine particles effectively smaller than about 0.1 µm are not of much significance in industrial dust and fume sources because their aggregate mass is only a very small fraction of the total mass emission. At the concentrations present in such sources (e.g., production of carbon black) the coagulation, or flocculation, rate of the ultrafine particles is extremely high, and the particles speedily grow to sizes of 0.1 µm or greater. The most difficult collection problems are thus concerned with particles in the range of about 0.1 to 2 µm, in which forces for deposition by inertia are small. For collection of particles under 0.1 µm, diffusional deposition becomes increasingly important as the particle size decreases. In a gas stream carrying dust or fume, some degree of particle flocculation will exist, so that both discrete particles and clusters of adhering particles will be present. The discrete particles composing the clusters may be only loosely attached to each other, as by van der Waals forces [Lapple, Chem. Eng., 75(11), 149 (1968)]. Flocculation tends to increase with increases in particle concentration and may strongly influence collector performance. PARTICLE MEASUREMENTS Measurements of the concentrations and characteristics of dust dispersed in air or other gases may be necessary (1) to determine the need for control measures, (2) to establish compliance with legal requirements, (3) to obtain information for collector design, and (4) to determine collector performance. Atmospheric-Pollution Measurements The dust-fall measurement is one of the common methods for obtaining a relative longperiod evaluation of particulate air pollution. Stack-smoke densities are often graded visually by means of the Ringelmann chart. Plume opacity may be continuously monitored and recorded by a photoelectric device which measures the amount of light transmitted through a stack plume. Equipment for local atmospheric-dust-concentration measurements fall into five general types: (1) the impinger, (2) the hot-wire or thermal precipitator, (3) the electrostatic precipitator, (4) the filter, and (5) impactors and cyclones. The filter is the most widely used, in the form of either a continuous tape, or a number of filter disks arranged in an automatic sequencing device, or a single, shortterm, high-volume sampler. Samplers such as these are commonly used to obtain mass emission and particle-size distribution. Impactors and small cyclones are commonly used as size-discriminating samplers and are usually followed by filters for the determination of the finest

GAS-SOLIDS SEPARATIONS

FIG. 17-34

17-23

Characteristics of particles and particle dispersoids. (Courtesy of the Stanford Research Institute; prepared by C. E. Lapple.)

fraction of the dust (Lundgren et al., Aerosol Measurement, University of Florida, Gainesville, 1979; and Dennis, Handbook on Aerosols, U.S. ERDA TID-26608, Oak Ridge, Tenn., 1976). Process-Gas Sampling In sampling process gases either to

determine dust concentration or to obtain a representative dust sample, it is necessary to take special precautions to avoid inertial segregation of the particles. To prevent such classification, a traverse of the duct may be required, and at each point the sampling nozzle must face

17-24

GAS-SOLID OPERATIONS AND EQUIPMENT

directly into the gas stream with the velocity in the mouth of the nozzle equal to the local gas velocity at that point. This is called “isokinetic sampling.” If the sampling velocity is too high, the dust sample will contain a lower concentration of dust than the mainstream, with a greater percentage of fine particles; if the sampling velocity is too low, the dust sample will contain a higher concentration of dust with a greater percentage of coarse particles [Lapple, Heat. Piping Air Cond., 16, 578 (1944); Manual of Disposal of Refinery Wastes, vol. V, American Petroleum Institute, New York, 1954; and Dennis, op. cit.]. Particle-Size Analysis Methods for particle-size analysis are shown in Fig. 17-34, and examples of size-analysis methods are given in Table 17-1. More detailed information may be found in Lapple, Chem. Eng., 75(11), 140 (1968); Lapple, “Particle-Size Analysis,” in Encyclopedia of Science and Technology, 5th ed., McGraw-Hill, New York, 1982; Cadle, The Measurement of Airborne Particles, Wiley, New York, 1975; Lowell, Introduction to Powder Surface Area, 2d ed., Wiley, New York, 1993; and Allen, Particle Size Measurement, 4th ed, Chapman and Hall, London, 1990. Particle-size distribution may be presented on either a frequency or a cumulative basis; the various methods are discussed in the references just cited. The most common method presents a plot of particle size versus the cumulative weight percent of material larger or smaller than the indicated size, on logarithmic-probability graph paper. For determination of the aerodynamic diameters of particles, the most commonly applicable methods for particle-size analysis are those based on inertia: aerosol centrifuges, cyclones, and inertial impactors (Lundgren et al., Aerosol Measurement, University of Florida, Gainesville, 1979; and Liu, Fine Particles—Aerosol Generation, Measurement, Sampling, and Analysis, Academic, New York, 1976). Impactors are the most commonly used. Nevertheless, impactor measurements are subject to numerous errors [Rao and Whitby, Am. Ind. Hyg. Assoc. J., 38, 174 (1977); Marple and Willeke, “Inertial Impactors,” in Lundgren et al., Aerosol Measurement; and Fuchs, “Aerosol Impactors,” in Shaw, Fundamentals of Aerosol SciTABLE 17-1

ence, Wiley, New York, 1978]. Reentrainment due to particle bouncing and blowoff of deposited particles makes a dust appear finer than it actually is, as does the breakup of flocculated particles. Processing cascade-impactor data also presents possibilities for substantial errors (Fuchs, The Mechanics of Aerosols, Pergamon, Oxford, 1964) and is laborious as well. Lawless (Rep. No. EPA-600/7-78-189, U.S. EPA, 1978) discusses problems in analyzing and fitting cascadeimpactor data to obtain dust-collector efficiencies for discrete particle sizes. The measured diameters of particles should as nearly as possible represent the effective particle size of a dust as it exists in the gas stream. When significant flocculation exists, it is sometimes possible to use measurement methods based on gravity settling. For dust-control work, it is recommended that a preliminary qualitative examination of the dust first be made without a detailed particle count. A visual estimate of particle-size distribution will often provide sufficient guidance for a preliminary assessment of requirements for collection equipment. MECHANISMS OF DUST COLLECTION The basic operations in dust collection by any device are (1) separation of the gas-borne particles from the gas stream by deposition on a collecting surface; (2) retention of the deposit on the surface; and (3) removal of the deposit from the surface for recovery or disposal. The separation step requires (1) application of a force that produces a differential motion of a particle relative to the gas and (2) a gas retention time sufficient for the particle to migrate to the collecting surface. The principal mechanisms of aerosol deposition that are applied in dust collectors are (1) gravitational deposition, (2) flow-line interception, (3) inertial deposition, (4) diffusional deposition, and (5) electrostatic deposition. Thermal deposition is only a minor factor in practical dustcollection equipment because the thermophoretic force is small. Table 17-2 lists these six mechanisms and presents the characteristic

Particle Size Analysis Methods and Equipment

Method Quantitative image analysis

Sieves: Punched Plate Woven wire Micromesh Sieving machines (Air jet, sonic wet and dry)

Brand names American Innovation Videometric, Analytical Measuring Systems Quickstep & Optomax, Artec Omnicon Automatix, Boeckeler, Buehler Omnimat, Compix Imaging Systems, Data Teanslation— Global Lab, Image, Hamamatsu C-1000, Hitech Olympus Cue-3, Joyce-Loebl Magiscan, Leco AMF System, Leico Quantimet, LeMont Oasys, Millipore_MC, Nachet 1500, Nicon Microphot, Oncor Instrument System, Optomax V, Outokumpu Imagist, Shapespeare Juliet, Tracor Northern, Carl Zeiss Videoplan Many Many Buckbee Mears, Veco, Endecottes Alpine, ATM, Gilson, Gradex, Hosokawa, Retsch, Seishin

Size range

Sample size, g

1–1000 µm

0.001

4–100 mm

500

20 µm–125 mm 5–500 + µm

25–200 1–5

Sedimentation

Pipette—Gilson, Photosedimentometer—gravitational, Paar; centrifugal, Joyce Loebl, Brookhaven, Horiba, Seishin, Shimadzu; x-ray absorption—gravitational, Quantachrome, Micromeretics; centrifugal, Brookhaven

Gravitational 1–100 µm Centrifugal 0.05–5 mm

0.1–5+ g

Classification Field scanning (light)

Air-Bahco, Water-Warmain Cyclosizer Cilas, Coulter, Insitec, Fritsch, Horiba, Leeds & Northrup (Microtrac), Malvern, Nitto, Seishin, Shimadzu, Sympatec

5–500 µm 0.04–3500 µm

Field scanning (ultrasonics)

Sympatec OPUS, Pen-Kem

5–100 g <1 g (wet) >20 g (on-line) On-line

Stream scanning

Brinkmann, Climet, Coulter, Dantec, Erdco, Faley, Flowvision, Hiac/Royco, Kowa, Lasentec, Malvern, Met One, Particle Measuring Systems, Polytec, Procedyne, Rion, Spectrex

0.2–10,000 µm

0.1–10 g (also on-line)

Zeta potential Photon correlation Spectroscopy

Zeta Plus, Micromeretics, Zeta sizer Malvern, Nicomp, Brookhaven, Coulter, Photol

0.001–30 µm

0.1–1 µm

NOTE: This table was compiled with the assistance of T. Allen, DuPont Particle Science and Technology, and is not intended to be comprehensive. Many other fine suppliers of particle analysis equipment are available.

GAS-SOLIDS SEPARATIONS TABLE 17-2

17-25

Summary of Mechanisms and Parameters in Aerosol Deposition Deposition mechanism measureable in terms of

Deposition

Origin of force field

 

Physical gradient*

Dp Nsf = ᎏ Db

Inertial deposition

Velocity gradient

KmρsDp2 Vo Nst = ᎏᎏ 18µDb

Diffusional deposition

Concentration gradient

Dv Nsd = ᎏ VoDb

Gravity settling

Elevation gradient

Electrostatic precipitation

Electric-field gradient§

Flow-line interception*

a. Attraction b. Induction

Thermal precipitation

Temperature gradient

Specific modifying parameters

Basic parameter







Nsf2 Nsc = ᎏ Nst Nsd





 

18µ † = ᎏ KmρpDv



KmQp⑀b Nsec = ᎏ µDpVo



δp − 1 Nsei = ᎏ δp + 2



δp, δb¶





Surface accommodation

Dp2 δo ⑀b 2

K  ᎏᎏ µD V  m

b

o

Nst = T − Tb ᎏ T

Geometry: (Db1 /Db), (Db2 /Db), etc. ⑀v ∆

Flow pattern: NRe‡ NMa NKn

 

ut Nsg = ᎏ Vo



System parameters

µ ᎏ KmρDbVo



kt ᎏ 2kt + ktp



(Tb /T), (Tp/T),† (NPr), (ktp /kt), (ktb /kt),¶ (chp /ch), (chb /ch)¶

SOURCE: Lunde and Lapple, Chem. Eng. Prog., 53, 385 (1957). *This has also commonly been termed “direct interception” and in conventional analysis would constitute a physical boundary condition imposed upon the particle path induced by action of other forces. By itself it reflects deposition that might result with a hypothetical particle having finite size but no mass or elasticity. †This parameter is an alternative to Nsf, Nsi, or Nad and is useful as a measure of the interactive effect of one of these on the other two. It is comparable with the Schmidt number. ‡When applied to the inertial deposition mechanism, a convenient alternative is (Kmρs /18ρ) = Nsi /(Nsf2 NRe). §In cases in which the body charge distribution is fixed and known, ⑀b may be replaced with Qbs /δo. ¶Not likely to be significant contributions.

parameters of their operation [Lunde and Lapple, Chem. Eng. Prog., 53, 385 (1957)]. The actions of the inertial-deposition, flow-lineinterception, and diffusional-deposition mechanisms are illustrated in Fig. 17-35 for the case of a collecting body immersed in a particleladen gas stream. Two other deposition mechanisms, in addition to the six listed, may be in operation under particular circumstances. Some dust particles may be collected on filters by sieving when the pore diameter is less than the particle diameter. Except in small membrane filters, the sieving mechanism is probably limited to surface-type filters, in which a layer of collected dust is itself the principal filter medium. The other mechanism appears in scrubbers. When water vapor diffuses from a gas stream to a cold surface and condenses, there is a net hydrodynamic flow of the noncondensable gas directed toward the surface. This flow, termed the Stefan flow, carries aerosol particles to the condensing surface (Goldsmith and May, in Davies, Aerosol Science, Academic, New York, 1966) and can substantially improve the performance of a scrubber. However, there is a corresponding Stefan flow directed away from a surface at which water is evaporating, and this will tend to repel aerosol particles from the surface. In addition to the deposition mechanisms themselves, methods for preliminary conditioning of aerosols may be used to increase the effectiveness of the deposition mechanisms subsequently applied. One such conditioning method consists of imposing on the gas highintensity acoustic vibrations to cause collisions and flocculation of the aerosol particles, producing large particles that can be separated by simple inertial devices such as cyclones. This process, termed “sonic (or acoustic) agglomeration,” has attained only limited commercial acceptance. Another conditioning method, adaptable to scrubber systems, consists of inducing condensation of water vapor on the aerosol particles as nuclei, increasing the size of the particles and making them more susceptible to collection by inertial deposition.

Most forms of dust-collection equipment use more than one of the collection mechanisms, and in some instances the controlling mechanism may change when the collector is operated over a wide range of conditions. Consequently, collectors are most conveniently classified by type rather than according to the underlying mechanisms that may be operating. PERFORMANCE OF DUST COLLECTORS The performance of a dust collector is most commonly expressed as the collection efficiency η, the weight ratio of the dust collected to the dust entering the apparatus. However, the collection efficiency is usually related exponentially to the properties of the dust and gas and the operating conditions of most types of collectors and hence is an insensitive function of the collector operating conditions as its value approaches 1.0. Performance in the high-efficiency range is better expressed by the penetration 1 − η, the weight ratio of the dust escaping to the dust entering. Particularly in reference to collection of radioactive aerosols, it is common to express performance in terms of the reciprocal of the penetration 1/(1 − η), which is termed the “decontamination factor.” The number of transfer units Nt, which is equal to ln [1/(1 − η)] in the case of dust collection, was first proposed for use by Lapple (Wright, Stasny, and Lapple, “High Velocity Air Filters,” WADC Tech. Rep. 55-457, ASTIA No. AD-142075, October 1957) and is more commonly used than the DI. Because of the exponential form of the relationship between efficiency and process variables for most dust collectors, the use of Nt (or DI) is particularly suitable for correlating collector performance data. In comparing alternative collectors for a given service, a figure of merit is desirable for ranking the different devices. Since power consumption is one of the most important characteristics of a collector, the ratio of Nt to power consumption is a useful criterion. Another is the ratio of Nt to capital investment.

17-26

GAS-SOLID OPERATIONS AND EQUIPMENT

FIG. 17-35

Particle deposition on collector bodies.

DUST-COLLECTOR DESIGN In dust-collection equipment, most or all of the collection mechanisms may be operating simultaneously, their relative importance being determined by the particle and gas characteristics, the geometry of the equipment, and the fluid-flow pattern. Although the general case is exceedingly complex, it is usually possible in specific instances to determine which mechanism or mechanisms may be controlling. Nevertheless, the difficulty of theoretical treatment of dust-collection phenomena has made necessary simplifying assumptions, with the introduction of corresponding uncertainties. Theoretical studies have been hampered by a lack of adequate experimental techniques for verification of predictions. Although theoretical treatment of collector performance has been greatly expanded in the period since 1960, few of the resulting performance models have received adequate experimental confirmation because of experimental limitations. The best-established models of collector performance are those for fibrous filters and fixed-bed granular filters, in which the structures and fluid-flow patterns are reasonably well defined. These devices are also adapted to small-scale testing under controlled laboratory conditions. Realistic modeling of full-scale electrostatic precipitators and scrubbers is incomparably more difficult. Confirmation of the models has been further limited by a lack of monodisperse aerosols that can be generated on a scale suitable for testing equipment of substantial sizes. When a polydisperse test dust is used, the particle-size distributions of the dust both entering and leaving a collector must be determined with extreme precision to avoid serious errors in the determination of the collection efficiency for a given particle size.

The design of industrial-scale collectors still rests essentially on empirical or semiempirical methods, although it is increasingly guided by concepts derived from theory. Existing theoretical models frequently embody constants that must be evaluated by experiment and that may actually compensate for deficiencies in the models. DUST-COLLECTION EQUIPMENT Gravity Settling Chambers The gravity settling chamber is probably the simplest and earliest type of dust-collection equipment, consisting of a chamber in which the gas velocity is reduced to enable dust to settle out by the action of gravity. Its simplicity lends it to almost any type of construction. Practically, however, its industrial utility is limited to removing particles larger than 325 mesh (43-µm diameter). For removing smaller particles, the required chamber size is generally excessive. Gravity collectors are generally built in the form of long, empty, horizontal, rectangular chambers with an inlet at one end and an outlet at the side or top of the other end. By assuming a low degree of turbulence relative to the settling velocity of the dust particle in question, the performance of a gravity settling chamber is given by ut Ls utBsLs η=ᎏ =ᎏ (for η ⬉ 1.0) (17-1) HsVs q where Vs = average gas velocity. Expressing ut in terms of particle size (equivalent spherical diameter), the smallest particle that can be completely separated out corresponds to η = 1.0 and, assuming Stokes’ law, is given by

GAS-SOLIDS SEPARATIONS Dp,min =

18µH V ᎏᎏ

g L (ρ − ρ) s

L

=

17-27

s

s

s

18µq ᎏᎏ

g B L (ρ − ρ) L

s

s

(17-2)

s

where ρ = gas density and ρs = particle density. For a given volumetric air-flow rate, collection efficiency depends on the total plan cross section of the chamber and is independent of the height. The height need be made only large enough so that the gas velocity Vs in the chamber is not so high as to cause reentrainment of separated dust. Generally Vs should not exceed about 3 m/s (10 ft/s). Horizontal plates arranged as shelves within the chamber will give a marked improvement in collection. This arrangement is known as the Howard dust chamber (Fume Arrester, U.S. Patent 896,111, 1908). The disadvantage of the unit is the difficulty of cleaning owing to the close shelf spacing and warpage at elevated temperatures. The pressure drop through a settling chamber is small, consisting primarily of entrance and exit losses. Because low gas velocities are used, the chamber is not subject to abrasion and may therefore be used as a precleaner to remove very coarse particles and thus minimize abrasion on subsequent equipment. Impingement Separators Impingement separators are a class of inertial separators in which particles are separated from the gas by inertial impingement on collecting bodies arrayed across the path of the gas stream, as shown on Fig. 17-35. Fibrous-pad inertial impingement separators for the collection of wet particles are the main application in current technology, as is described in Sec. 18 and Fig. 17-48. With the growing need for very high performance dust collectors, there is little application anymore for impingement collectors that catch large amounts of dry dust. Cyclone Separators The most widely used type of dustcollection equipment is the cyclone, in which dust-laden gas enters a cylindrical or conical chamber tangentially at one or more points and leaves through a central opening (Fig. 17-36). The dust particles, by virtue of their inertia, will tend to move toward the outside separator wall, from which they are led into a receiver. A cyclone is essentially a settling chamber in which gravitational acceleration is replaced by centrifugal acceleration. At operating conditions commonly employed, the centrifugal separating force or acceleration may range from 5 times gravity in very large diameter, low-resistance cyclones, to 2500 times gravity in very small, high-resistance units. The immediate entrance to a cyclone is usually rectangular. Fields of Application Within the range of their performance capabilities, cyclone collectors offer one of the least expensive means of dust collection from the standpoint of both investment and operation. Their major limitation is that unless very small units are used, their efficiency is low for collection of particles smaller than 5 µm. Although cyclones may be used to collect particles larger than 200 µm, gravity settling chambers or simple inertial separators (such as gas-reversal chambers) are usually satisfactory and less subject to abrasion. In special cases in which the dust is highly flocculated or high dust concentrations (over 230 g/m3, or 100 gr/ft3) are encountered, cyclones will remove dusts having small particle sizes. In certain instances efficiencies as high as 98 percent have been attained on dusts having ultimate particle sizes of 0.1 to 2.0 µm because of the predominant effect of flocculation. Cyclones are used to remove both solids and liquids from gases and have been operated at temperatures as high as 1000°C and pressures as high as 50,700 kPa (500 atm). Flow Pattern In a cyclone the gas path involves a double vortex with the gas spiraling downward at the outside and upward at the inside. When the gas enters the cyclone, its velocity undergoes a redistribution so that the tangential component of velocity increases with decreasing radius as expressed by Vct ∼ r−n. The spiral velocity in a cyclone may reach a value several times the average inlet-gas velocity. Theoretical considerations indicate that n should be equal to 1.0 in the absence of wall friction. Actual measurements [Shepherd and Lapple, Ind. Eng. Chem., 31, 972 (1939); 32, 1246 (1940)], however, indicate that n may range from 0.5 to 0.7 over a large portion of the cyclone

FIG. 17-36

Cyclone-separator proportions.

radius. Ter Linden [Inst. Mech. Eng. J., 160, 235 (1949)] found n to be 0.52 for tangential velocities measured in the cylindrical portion of the cyclone at positions ranging from the radius of the gas-outlet pipe to the radius of the collector. Although the velocity approaches zero at the wall, the boundary layer is sufficiently thin that pitot-tube measurements show relatively high tangential velocities there, as shown in Fig. 17-37. The radial velocity Vr is directed toward the center throughout most of the cyclone, except at the center, where it is directed outward. Superimposed on the “double spiral,” there may be a “double eddy” [Van Tongeran, Mech. Eng., 57, 753 (1935); and Wellmann, Feuerungstechnik, 26, 137 (1938)] similar to that encountered in pipe coils. Measurements on cyclones of the type shown in Fig. 17-36 indicate, however, that such double-eddy velocities are small compared with the spiral velocity (Shepherd and Lapple, op. cit.). Recent analyses of flow patterns can be found in Hoffman et al., Powder Tech., 70, 83 (1992); and Trefz and Muschelknautz, Chem. Eng. Tech., 16, 153 (1993). Cyclone Efficiency The methods described below for pressure drop and efficiency calculations were given by Zenz in Manual on Disposal of Refinery Wastes—Atmospheric Emissions, chap. 11 (1975), American Petroleum Institute Publ. 931 and improved by the Particulate Solid Research Inc. (PSRI), Chicago. Cyclones work by using centrifugal force to increase the gravity field experienced by the solids. They then settle to the wall under the influence of their increased weight. Settling is improved as the path the solids traverse under centrifugal flow is increased. This path is

17-28

GAS-SOLID OPERATIONS AND EQUIPMENT

FIG. 17-37 Variation of tangential velocity and radial velocity at different points in a cyclone. [Ter Linden, Inst. Mech. Eng. J., 160, 235 (1949).]

equated with the number of spirals the solids make in the cyclone barrel. Figure 17-38 gives the number of spirals Ns as a function of the maximum velocity in the cyclone. The maximum velocity may be seen either at the inlet or outlet depending on design. The equation for Dpth, the theoretical size particle removed by the cyclone, is 9µg Bc Dpth = ᎏᎏ πNsvin(ρp − ρg) When consistent units are used, the particle size will either be in meters or feet. The equation contains effects of cyclone size, velocity, viscosity, and density of solids. In practice, a design curve as given in Fig. 17-39 uses Dpth as the size at which 50 percent of solids of a given size are collected by the cyclone. The material entering the cyclone is divided into fractional sizes, and the collection efficiency for each size is determined. The total efficiency of collection is the sum of the collection efficiencies of the cuts. The above applies for very dilute systems, usually on the order of 1 grain/ft3, or 2.3 g/m3 where a grain equals 1/7000 of a pound. When an appreciable amount of solids are present, the efficiency increases



FIG. 17-39

Single particle collection efficiency curve. (Courtesy of PSRI,

Chicago.)

dramatically. This may be due to the coarse particles colliding with fines as they settle, which takes the fines to the wall more quickly. Other explanations are that the solids have a lower drag coefficient or tend to flocculate in multiparticle environments. At very high loadings, it is believed the gas simply cannot hold that much solid material in suspension at high gravities, and the bulk of the solids simply “condenses” out of the gas stream. The phenomenon is represented by Figs. 17-40 and 17-41 for Geldart-type A and B solids, respectively (see beginning of Sec. 17). The initial efficiency of a particle size cut is found on the chart, and the parametric line is followed to the proper overall solids loading. The efficiency for that cut size is then read from the graph. Pressure Drop Pressure drop is first determined by summing five flow losses associated with the cyclone. 1. Inlet contraction, 2 ∆P = 0.5ρg(vin2 − v vessel + Kvin2 )

where K is taken from Table 17-3. Using SI units gives the pressure drop in Pa. In English units, the factor of 32.2 for g must be included. This loss is primarily associated with cyclones inside a vessel. If the cyclone is connected outside a vessel, the dp tap may measure acceleration, and this term should not be used for total dp. TABLE 17-3 K versus Area Ratio

2.

Area ratio

K

0 0.1 0.2 0.3 0.4

.50 .47 .43 .395 .35

Particle acceleration, ∆P = Lvin(vpin − vpvessel)

FIG. 17-38 Ns versus velocity—where the larger of either the inlet or outlet velocity is used.

For small particles, the velocity is taken as equal to the gas velocity and L is the loading, kg/m3. 3. Barrel friction. The inlet diameter din is taken as 4 × (inlet area)/ inlet perimeter. Then, 2 f ρgvin2πDc Ns ∆P = ᎏᎏ din where the Reynolds number for determining the friction factor f is based on the inlet area.

GAS-SOLIDS SEPARATIONS

17-29

FIG. 17-40 Effect of inlet loading on collection efficiency for Geldart Group A and Group C particles. (Courtesy of PSRI, Chicago.)

4.

Gas flow reversal, ρg vin2 ∆P = ᎏ 2

5.

Exit contraction, ∆P = 0.5ρg(v2exit − vc2 + Kv2exit)

where K is determined from Table 17-3 based on the area ratio of barrel and exit tube of the cyclone. The total pressure drop is the sum of the 5 individual pressure drops. However, the actual pressure drop observed turns out to be a function of the solids loading. The pressure drop is high when the gas is free of solids and then decreases as the solids loading increases up to about 3 kg/m3 (0.2 lb/ft3). The cyclone dp then begins to increase with

loading. The cause of the initial decline is that the presence of solids decreases the tangential velocity of the gas [Yuu, Chem. Eng. Sci., 33, 1573 (1978)]. Figure 17-42 gives the actual pressure drop based on the loading. When solids are absent, the observed pressure drop can be 2.5 times the calculated pressure drop. Cyclone Design Factors Cyclones are generally designed to meet specified pressure-drop limitations. For ordinary installations, operating at approximately atmospheric pressure, fan limitations generally dictate a maximum allowable pressure drop corresponding to a cyclone inlet velocity in the range of 8 to 30 m/s (25 to 100 ft/s). Consequently, cyclones are usually designed for an inlet velocity of 15 m/s (50 ft/s), though this need not be strictly adhered to. In the removal of dusts, collection efficiency can be changed by only a relatively small amount by a variation in operating conditions.

FIG. 17-41 Effect of inlet loading on collection efficiency (Geldart Group B and Group D) particles. (Courtesy of PSRI, Chicago.)

17-30

GAS-SOLID OPERATIONS AND EQUIPMENT

FIG. 17-42

Effect of cyclone inlet loading on pressure drop. (Courtesy of PSRI, Chicago.)

The primary design factor that can be utilized to control collection efficiency is the cyclone diameter, a smaller-diameter unit operating at a fixed pressure drop having the higher efficiency [Anderson, Chem. Metall., 40, 525 (1933); Drijver, Wärme, 60, 333 (1937); and Whiton, Power, 75, 344 (1932); Chem. Metall., 39, 150 (1932)]. Smalldiameter cyclones, however, will require a multiple of units in parallel for a specified capacity. In such cases the individual cyclones can discharge the dust into a common receiving hopper [Whiton, Trans. Am. Soc. Mech. Eng., 63, 213 (1941)]. The final design involves a compromise between collection efficiency and complexity of equipment. It is customary to design a single cyclone for a given capacity, resorting to multiple parallel units only if the predicted collection efficiency is inadequate for a single unit. Reducing the gas outlet diameter will increase both collection efficiency and pressure drop. To exit the cyclone, gas must enter the cyclonic flow associated with the outlet tube. If the outlet diameter is reduced, the outlet vortex increases in length to compensate. Therefore, when the outlet area is less than the inlet area, the length of the cyclone must increase. Too short a cyclone is associated with erosion of the cone and reentrainment of solids into the exit flow. Table 17-4 below gives this increase as a function of outlet-to-inlet area. The length is measured centrally along a cylinder 10 cm larger than the inner diameter of the outlet tube to prevent interference with the cone. If the cone interferes, the barrel must be lengthened. The minimum cone angle should be 60° or greater with steeper angles appropriate to materials that are cohesive. The inlet is usually rectangular and sometimes circular. In either case, projection of the flow path should never interfere with the outlet tube. If a very heavy solids loading is anticipated, the barrel diameter should be increased slightly. Collection efficiency is normally increased by increasing the gas throughput (Drijver, op. cit.). However, if the entering dust is flocculated, increased gas velocities may cause deflocculation in the cyclone, so that efficiency remains the same or actually decreases. Also, variations in design proportions that result in increased collection effiTABLE 17-4 Required Cyclone Length as a Function of Area Ratio Aout /Ain

Length below outlet tube/Dc

>1.0 0.8 0.6 0.4

2.0 2.2 2.6 3.2

ciency with dispersed dusts may be detrimental with flocculated dusts. Kalen and Zenz [Am. Inst. Chem. Eng. Symp. Ser., 70 (137), 388 (1974)] report that collection efficiency increases with increasing gas inlet velocity up to a minimum tangential velocity at which dust is either reentrained or not deposited because of saltation. Koch and Licht [Chem. Eng., 84(24), 80 (1977)] estimate that for typical cyclones the saltation velocity is consistent with cyclone inlet velocities in the range of 15 to 27 m/s (50 to 90 ft/s). C. E. Lapple (private communication) reports that in cyclone tests with talc dust collection efficiency increased steadily as the inlet velocity was increased up to the maximum of 52 m/s (170 ft/s). With ilmenite dust, which was much more strongly flocculated, efficiency decreased over the same inlet-velocity range. In later experiments with well-dispersed talc dust, collection efficiency continued to increase at inlet velocities up to the maximum used, 82 m/s (270 ft/s). Another effect of increasing the cyclone inlet gas velocity is that friable materials may disintegrate as they hit the cyclone wall at high velocity. Thus, the increase in efficiency associated with increased velocity may be more than lost due to generation of fine attrited material that the cyclone cannot contain. Cyclones in series may be justified under some circumstances: 1. The dust has a broad size distribution, including particles under 10 to 15 µm as well as larger and possibly abrasive particles. A large low-velocity cyclone may be used to remove the coarse particles ahead of a unit with small-diameter multiple tubes. 2. The dust is composed of fine particles but is highly flocculated or tends to flocculate in preceding equipment and in the cyclones themselves. Efficiencies predicted on the basis of ultimate particle size will be highly conservative. 3. The dust is relatively uniform, and the efficiency of the secondstage cyclone is not greatly lower than that of the first stage. 4. Dependable operation is critical. Second-stage or even thirdstage cyclones may be used as backup. A cyclone will operate equally well on the suction or pressure side of a fan if the dust receiver is airtight. Probably the greatest single cause for poor cyclone performance, however, is the leakage of air into the dust outlet of the cyclone. A slight air leak at this point can result in a tremendous drop in collection efficiency, particularly with fine dusts. For a cyclone under pressure, air leakage at this point is objectionable primarily because of the local dust nuisance created. For batch operation, an airtight hopper or receiver may be used. For continuous withdrawal of collected dust, a rotary star valve, a double-lock valve, or a screw conveyor may be used, the latter only with fine dusts. A collapsible open-ended rubber tube can be used for cyclones operating under slight negative pressure; mechanical flapgate valves and

GAS-SOLIDS SEPARATIONS fluidized seal legs can be used (see “Fluidized-Bed Systems: Solids Discharge”). Special pneumatic unloading devices can also be used with dusts. In any case it is essential that sufficient unloading and receiving capacity be provided to prevent collected material from accumulating in the cyclone. Generally cone-and-disk baffles, helical guide vanes, etc., placed inside a cyclone, will have a detrimental effect on performance. A few of these devices do have some merit, however, under special circumstances. Although an inlet vane will reduce pressure drop, it causes a correspondingly greater reduction in collection efficiency. Its use is recommended only when collection efficiency is normally so high as to be a secondary consideration and when it is desired to decrease the resistance of an existing cyclone system for purposes of increased airhandling capacity or when floor-space or headroom requirements are controlling factors. If an inlet vane is used, it is advantageous to increase the gas-exit-duct length inside the cyclone chamber. A disk or cone baffle located beneath the gas-outlet duct may be beneficial if air in-leakage at the dust outlet cannot be avoided. A heavy chain suspended from the gas-outlet duct has been found beneficial to minimize dust buildup on the cyclone walls. Such a chain should be suspended from a swivel so that it is free to rotate without twisting. At present there are no known devices that will recover the gas spiralvelocity energy in the gas-outlet duct. Substantially all devices that

FIG. 17-43

17-31

have been reported to reduce pressure drop do so by reducing spiral velocities in the cyclone chamber and consequently result in reduced collection efficiency. At low dust loadings the pressure in the dust receiver of a single cyclone will generally be lower than in the gas-outlet duct. Increased dust loadings will increase the pressure in the dust receiver. Such devices as cones, disks, and inlet vanes will generally cause the pressure in the dust receiver to exceed that in the gas-outlet duct. A cyclone will operate as well in a horizontal position as in a vertical position. However, departure from the normal vertical position results in an increasing tendency to plug the dust outlet. If the dust outlet becomes plugged, collection efficiency will, of course, be low. If the cyclone exit duct must be reduced to tie in with proposed duct sizes, the transition should be made at least five diameters downstream from the cyclone and preferably after a bend. In the event that the transition must be made closer to the cyclone, a Greek cross should be installed in the transition piece in order to avoid excessive pressure drop. Commercial Equipment Simple cyclones are available in a wide variety of shapes ranging from long, slender units to short, largediameter units. The body may be conical or cylindrical, and entrances may be involute or tangential and round or rectangular. In Fig. 17-43 are shown some of the special types of commercial cyclones. In the Multiclone a spiral motion is imparted to the gas by

Some commercial cyclone design alternatives. (Courtesy of PSRI, Chicago.)

17-32

GAS-SOLID OPERATIONS AND EQUIPMENT

annular vanes, and it is furnished in multiple units of 15.2- and 22.9-cm (6- and 9-in) diameter. Its largest field of application has been in the collection of fly ash from steam boilers. The tubes are commonly constructed of cast iron and other abrasion-resistant alloys. In addition to the conventional reverse-flow cyclones, some use is made of uniflow, or straight-through, cyclones, in which the gas and solids discharge at the same end (Fig. 17-44). These devices act as concentrators; the concentrated dust, together with 5 to 20 percent of the inlet gas, is discharged at the periphery, while the clean gas passes out axially. The purge gas and concentrated dust enter a conventional cyclone for final separation. The straight-through cyclones are usually multiple-tube units. Mechanical Centrifugal Separators A number of collectors in which the centrifugal field is supplied by a rotating member are commercially available. In the typical unit shown in Fig. 17-45, the exhauster or fan and dust collector are combined as a single unit. The blades are especially shaped to direct the separated dust into an annular slot leading to the collection hopper while the cleaned gas continues to the scroll. Although no comparative data are available, the collection efficiency of units of this type is probably comparable with that of the single-unit high-pressure-drop-cyclone installation. The clearances are smaller and the centrifugal fields higher than in a cyclone, but these advantages are probably compensated for by the shorter gas path and the greater degree of turbulence with its inherent reentrainment tendency. The chief advantage of these units lies in their compactness, which may be a prime consideration for large installations or plants requiring a large number of individual collectors. Caution should be exercised when attempting to apply this type of unit to a dust that shows a marked tendency to build up on solid surfaces, because of the high maintenance costs that may be encountered from plugging and rotor unbalancing. Particulate Scrubbers Wet collectors, or scrubbers, form a class of devices in which a liquid (usually water) is used to assist or accomplish the collection of dusts or mists. Such devices have been in use for well over 100 years, and innumerable designs have been or are offered commercially or constructed by users. Wet-film collectors logically

FIG. 17-44

(1949).]

Uniflow cyclone. [Ter Linden, Inst. Mech. Eng. J., 160, 233

Typical mechanical centrifugal separator. Type D Rotoclone (cutaway view). (American Air Filter Co., Inc.)

FIG. 17-45

form a separate subcategory of devices. They comprise inertial collectors in which a film of liquid flows over the interior surfaces, preventing reentrainment of dust particles and flushing away the deposited dust. Wetted-wall cyclones are an example [Stairmand, Trans. Inst. Chem. Eng., 29, 356 (1951)]. Wet-film collectors have not been studied systematically but can probably be expected to perform much as do equivalent dry inertial collectors, except for the benefit of reduced reentrainment. In particulate scrubbers, the liquid is dispersed into the gas as a spray, and the liquid droplets are the principal collectors for the dust particles. Depending on their design and operating conditions, particulate scrubbers can be adapted to collecting fine as well as coarse particles. Collection of particles by the drops follows the same principles illustrated in Fig. 17-35. Various investigations of the relative contributions of the various mechanisms have led to the conclusion that the predominant mechanism is inertial deposition. Flow-line interception is only a minor mechanism in the collection of the finer dust particles by liquid droplets of the sizes encountered in scrubbers. Diffusion is indicated to be a relatively minor mechanism for the particles larger than 0.1 µm that are of principal concern. Thermal deposition is negligible. Gravitational settling is ineffective because of the high gas velocities and short residence times used in scrubbers. Electrostatic deposition is unlikely to be important except in cases in which the dust particles or the water, or both, are being deliberately charged from an external power source to enhance collection. Deposition produced by Stefan flow can be significant when water vapor is condensing in a scrubber. Despite numerous claims or speculations that wetting of dust particles by the scrubbing liquid plays a major role in the collection process, there is no unequivocal evidence that this is the case. The issue is whether wetting is an important factor in the adherence of a particle to a collecting droplet upon impact. From the body of general experience, it can be inferred that wettable particles probably are not collected much, if any, more readily than nonwettable particles of the same size. However, the available experimental techniques have not been adequate to permit any direct test to resolve the question. Changing from a wettable to a nonwettable test aerosol or from one scrubbing liquid to another is virtually certain to introduce other (and possibly unknown) factors into the scrubbing process. The most informative experimental studies appear to be some by Weber [Staub, English trans., 28, 37 (November 1968); 29, 12 (July 1969)], who bombarded single drops of various liquids with dust particles at different velocities and studied the behavior at impact by means of highspeed photography. Dust particles hitting the drops were invariably retained by the latter, regardless of their wettability by the liquid used. The use of wetting agents in scrubbing water is equally controversial, and there has been no clear demonstration that it is beneficial.

GAS-SOLIDS SEPARATIONS A particulate scrubber may be considered as consisting of two parts: (1) a contactor stage, in which a spray is generated and the dust-laden gas stream is brought into contact with it; and (2) an entrainment separation stage, in which the spray and deposited dust particles are separated from the cleaned gas. These two stages may be separate or physically combined. The contactor stage may be of any form intended to bring about effective contacting of the gas and spray. The spray may be generated by the flow of the gas itself in contact with the liquid, by spray nozzles (pressure-atomizing or pneumatic-atomizing), by a motor-driven mechanical spray generator, or by a motor-driven rotor through which both gas and liquid pass. Entrainment separation is accomplished with inertial separators, which are usually cyclones or impingement separators of various forms. If properly designed, these devices can remove virtually all droplets of the sizes produced in scrubbers. However, reentrainment of liquid can take place in poorly designed or overloaded separators. Scrubber Types and Performance The diversity of particulate scrubber designs is so great as to defy any detailed and self-consistent system of classification based on configuration or principle of operation. However, it is convenient to characterize scrubbers loosely according to prominent constructional features, even though the modes of operation of different devices in a group may vary widely. A relationship of power consumption to collection efficiency is characteristic of all particulate scrubbers. Attaining increased efficiency requires increased power consumption, and the power consumption required to attain a given efficiency increases as the particle size of the dust decreases. Experience generally indicates that the power consumption required to provide a specific efficiency on a given dust does not vary widely even with markedly different devices. The extent to which this generalization holds true has not been fully explored, but the known extent is sufficient to suggest that the underlying collection mechanism may be essentially the same in all types of particulate scrubbers. Since some relationship of power consumption to performance appears to be a universal characteristic of particulate scrubbers, it is useful to characterize such devices broadly according to the source from which the energy is supplied to the gas-liquid contacting process. The energy may be drawn from (1) the gas stream itself, (2) the liquid stream, or (3) a motor driving a rotor. For convenience, devices in these classes may be termed respectively (1) gas-atomized spray scrubbers, (2) spray, or preformed-spray, scrubbers, and (3) mechanical scrubbers. In the spray scrubbers, all the energy may be supplied from the liquid, using a pressure nozzle, but some or all may be provided by compressed air or steam in a two-fluid nozzle or by a motor driving a spray generator. Particulate scrubbers may also be classed broadly into low-energy and high-energy scrubbers. The distinction between the two classes is arbitrary, since the devices are not basically different and the same device may fall into either class depending on the amount of power it consumes. However, some differences in configuration are sometimes necessary to adapt a device for high-energy service. No specific level of power consumption is commonly agreed upon as the boundary between the two classes, but high-energy scrubbers may be regarded as those using sufficient power to give substantial efficiencies on submicrometer particles. Scrubber Performance Models A number of investigators have made theoretical studies of the performance of venturi scrubbers and have sought to produce performance models, based on first principles, that can be used to design a unit for a given duty without recourse to experimental data other than the particle size and size distribution of the dust. Among these workers are Calvert [Am. Inst. Chem. Eng. J., 16, 392 (1970); J. Air Pollut. Control Assoc., 24, 929 (1974)], Boll [Ind. Eng. Chem. Fundam., 12, 40 (1973)], Goel and Hollands [Atmos. Environ., 11, 837 (1977)], and Yung et al. [Environ. Sci. Technol., 12, 456 (1978)]. Comparatively few efforts have been made to model the performance of scrubbers of types other than the venturi, but a number of such models are summarized by Yung and Calvert (U.S. EPA-600/8-78-005b, 1978). The various venturi-scrubber models embody a variety of assumptions and approximations. The solutions of the equations for particulate collection must in general be determined numerically, although

17-33

Calvert et al. [J. Air Pollut. Control Assoc., 22, 529 (1972)] obtained an explicit equation by making some simplifying assumptions and incorporating an empirical constant that must be evaluated experimentally; the constant may absorb some of the deficiencies in the model. Although other models avoid direct incorporation of empirical constants, use of empirical relationships is necessary to obtain specific estimates of scrubber collection efficiency. One of the areas of greatest uncertainty is the estimation of droplet size. Most of the investigators have assumed the effective drop size of the spray to be the Sauter (surface-mean) diameter and have used the empirical equation of Nukiyama and Tanasawa [Trans. Soc. Mech. Eng., Japan, 5, 63 (1939)] to estimate the Sauter diameter:
L
µL 1920σ Do = ᎏᎏ + 75.4 ᎏᎏ




L
6
.3 Voρ /
2
.3 σ Lρ
6 L /
2



 ᎏ  Q 0.45

1000QL

1.5

(17-3)

G

where Do = drop diameter, µm; Vo = gas velocity, ft/s; σL = liquid surface tension, dyn/cm; ρL = liquid density, lb/ft3; µL = liquid viscosity, cP; QL = liquid flow rate, ft3/s; and QG = gas flow rate, ft3/s. The Nukiyama-Tanasawa equation, which is not dimensionally homogeneous, was derived from experiments with small, internal-mix pneumatic atomizing nozzles with concentric feed of air and liquid (Lapple et al., “Atomization: A Survey and Critique of the Literature,” Stanford Res. Inst. Tech. Rep. No. 6, AD 821-314, 1967; Lapple, “Atomization,” in McGraw-Hill Encyclopedia of Science and Technology, 5th ed., vol. 1, McGraw-Hill, New York, 1982, p. 858). The effect of nozzle size on the drop size is undefined. Even within the range of parameters for which the relationship was derived, the drop sizes reported by various investigators have varied by twofold to threefold from those predicted by the equation (Boll, op. cit.). The NukiyamaTanasawa equation has, nevertheless, been applied to large venturi and orifice scrubbers with configurations radically different from those of the atomizing nozzles for which the equation was originally developed. Primarily because of the lack of adequate experimental techniques (particularly, the production of appropriate monodisperse aerosols), there has been no comprehensive experimental test of any of the venturi-scrubber models over wide ranges of design and operating variables. The models for other types of scrubbers appear to be essentially untested. Contacting Power Correlation A scrubber design method that has achieved wide acceptance and use is based on correlation of the collection efficiency with the power dissipated in the gas-liquid contacting process, which is termed “contacting power.” The method originated from an investigation by Lapple and Kamack [Chem. Eng. Prog., 51, 110 (1955)] and has been extended and refined in a series of papers by Semrau and coworkers [Ind. Eng. Chem., 50, 1615 (1958); J. Air Pollut. Control Assoc., 10, 200 (1960); 13, 587 (1963); U.S. EPA-650/2-74-108, 1974; U.S. EPA-600/2-77-234, 1977; Chem. Eng., 84(20), 87 (1977); and “Performance of Particulate Scrubbers as Influenced by Gas-Liquid Contactor Design and by Dust Flocculation,” EPA-600/9-82-005c, 1982, p. 43]. Other workers have made extensive independent studies of the correlation method [Walker and Hall, J. Air Pollut. Control Assoc., 18, 319 (1968)], and numerous studies of narrower scope have been made. The major conclusion from these studies is that the collection efficiency of a scrubber on a given dust is essentially dependent only on the contacting power and is affected to only a minor degree by the size or geometry of the scrubber or by the way in which the contacting power is applied. This contacting-power rule is strictly empirical, and the full extent of its validity has still not been explored. It has been best verified for the class of gas-atomized spray scrubbers, in which the contacting power is derived from the gas stream and takes the form of gas pressure drop. Tests of the equivalence of contacting power supplied from the liquid stream in pressure spray nozzles have been far less extensive and are strongly indicative but not yet conclusive. Evidence for the equivalence of contacting power from mechanically driven devices is also indicative but extremely limited in quantity. Contacting power is defined as the power per unit of volumetric gas flow rate that is dissipated in gas-liquid contacting and is ultimately converted to heat. In the simplest case, in which all the energy is

17-34

GAS-SOLID OPERATIONS AND EQUIPMENT

obtained from the gas stream in the form of pressure drop, the contacting power is equivalent to the friction loss across the wetted equipment, which is termed “effective friction loss,” FE. The pressure drop may reflect kinetic-energy changes rather than energy dissipation, and pressure drops that result solely from kinetic-energy changes in the gas stream do not correlate with performance. Likewise, any friction losses taking place across equipment that is operating dry do not contribute to gas-liquid contacting and do not correlate with performance. The gross power input to a scrubber includes losses in motors, drive shafts, fans, and pumps that obviously should be unrelated to scrubber performance. The effective friction loss, or “gas-phase contacting power,” is easily determined by direct measurements. However, the “liquid-phase contacting power,” supplied from the stream of scrubbing liquid, and the “mechanical contacting power,” supplied by a mechanically driven rotor, are not directly measurable; the theoretical power inputs can be estimated, but the portions of these quantities effectively converted to contacting power can only be inferred from comparison with gasphase contacting power. Such data as are available indicate that the contributions of contacting power from different sources are directly additive in their relation to scrubber performance. Contacting power is variously expressed in units of MJ/1000 m3 (SI), kWh/1000 m3 (meter-kilogram-second system), and hp/(1000 ft3/ min) (U.S. customary). Relationships for conversion to SI units are 1.0 kWh/1000 m3 = 3.60 MJ/1000 m3 1.0 hp/(1000 ft3/min) = 1.58 MJ/1000 m3 The gas-phase contacting power PG may be calculated from the effective friction loss by the following relationships: SI units: PG = 1.0FE (17-4) where FE = kPa. U.S. customary units: PG = 0.1575FE (17-5) where FE = in of water. The power input from a liquid stream injected with a hydraulic spray nozzle may usually be taken as approximately equal to the product of the nozzle feed pressure pF and the volumetric liquid rate. The liquid-phase contacting power PL may then be calculated from the following formulas: SI units: PL = 1.0pF(QL /QG) (17-6) where pF = kPa gauge, and QL and QG = m3/s. U.S. customary units: PL = 0.583pF(QL /QG)

(17-7)

where pG = lbf/in2 gauge, QL = gal/min, and QG = ft3/min. The correlation of efficiency data is based on the total contacting power PT, which is the sum of PG, PL, and any power PM that may be supplied mechanically by a power-driven rotor. In general, the liquid-to-gas ratio does not have an influence independent of contacting power on the collection efficiency of scrubbers of the venturi type. This is true at least of operation with liquid-to-gas ratios above some critical lower value. However, several investigations [Semrau and Lunn, “Performance of Particulate Scrubbers as Influenced by Gas-Liquid Contactor Design and by Dust Flocculation,” EPA-600/9-82-005c, 1982, p. 43; and Muir et al., Filtr. Sep., 15, 332 (1978)] have shown that at low liquid-to-gas ratios relatively poor efficiencies may be obtained at a given contacting power. Such regions of operation are obviously to be avoided. It has sometimes been asserted that multiple gas-liquid contactors in series will give higher efficiencies at a given contacting power than will a single contacting stage. However, there is little experimental evidence to support this contention. Lapple and Kamack (op. cit.) obtained slightly higher efficiencies with a venturi and an orifice in series than they did with a venturi alone. Muir and Mihisei [Atmos. Environ., 13, 1187 (1979)] obtained somewhat higher efficiencies on two redispersed dusts when using two venturis in series rather than

one. The improvement obtained with two-stage scrubbing was greatest with the coarser of the two dusts and was relatively small with the finer dust. Flocculation or deflocculation of the dusts may have been responsible for some of the behavior encountered. Semrau et al. (EPA-600/2-77-234, 1977) compared the performance of a four-stage multiple-orifice contactor with that of a single-orifice contactor, using well-dispersed aerosols generated from ammonium fluorescein, The multiple-orifice contactor gave about the same efficiency as the single-orifice in the upper range of contacting power but lower efficiencies in the lower range. The deviations in performance in this case were probably characteristic of the particular multiple-orifice contactor rather than of multistage contacting as such. Most scrubbers actually incorporate more than one stage of gasliquid contacting even though these may not be identical (e.g., the contactor and the entrainment separator). The preponderance of evidence indicates that multiple-stage contacting is not inherently either more or less efficient than single-stage contacting. However, twostage contacting may have practical benefits in dealing with abrasive or flocculated dusts. Some investigators have proposed, mostly on the basis of mathematical modeling, to optimize the design of scrubbers to obtain a given efficiency with a minimum power consumption (e.g., Goel and Hollands, op. cit.). In fact, no optimum in performance appears to exist; apart from some avoidable regions of unfavorable operation, increased contacting power yields increased efficiency. Scrubber Performance Curves The scrubber performance curve, which shows the relationship of scrubber efficiency to the contacting power, has been found to take the form Nt = αPTγ

(17-8)

where α and γ are empirical constants that depend primarily on the aerosol (dust or mist) collected. In a log-log plot of Nt versus PT, γ is the slope of the performance curve and α is the intercept at PT = 1. Figure 17-46 shows such a performance curve for the collection of coal fly ash by a pilot-plant venturi scrubber (Raben “Use of Scrubbers for Control of Emissions from Power Boilers,” United States– U.S.S.R. Symposium on Control of Fine-Particulate Emissions from Industrial Sources, San Francisco, 1974). The scatter in the data reflects not merely experimental errors but actual variations in the particle-size characteristics of the dust. Because the characteristics of an industrial dust vary with time, the scrubber performance curve necessarily must represent an average material, and the scatter in the data is frequently greater than is shown in Fig. 17-46. For best definition, the curve should cover as wide a range of contacting power as possible. Obtaining the data thus requires pilot-plant equipment with the flexibility to operate over a wide range of conditions. Because scrubber performance is not greatly affected by the size of the unit, it is feasible to conduct the tests with a unit handling no more than 170 m3/h (100 ft3/min) of gas.

FIG. 17-46 Performance of pilot-plant venturi scrubber on fly ash. Liquid-togas ratio, gal/1000 ft3: 䊊, 10; 䉭, 15; 䊐, 20. (Raben, United States–U.S.S.R. Symposium on Fine-Particulate Emissions from Industrial Sources, San Francisco, 1974.)

GAS-SOLIDS SEPARATIONS A clear interpretation of γ, the slope of the curve, is still lacking. Presumably, it should be related to the particle-size distribution of the dust. Because scrubbing preferentially removes the coarser particles, the fraction of the dust removed (or the increment of Nt) per unit of contacting power should decrease as the contacting power and efficiency increase, so that the value of γ should be less than unity. In fact, the value of γ has been less than unity for most dusts. Nevertheless, some data in the literature have displayed values of γ greater than unity when plotted on the transfer-unit basis, indicating that the residual fraction of dust became more readily collectible as contacting power and efficiency increased. More recent studies by Semrau et al. (EPA-650/2-74-108 and EPA-600/2-77-237) have revealed performance curves having two branches (typified by Fig. 17-47), the lower having a slope greater than unity and the upper a slope less than unity. This suggests that had the earlier tests been extended into higher contacting-power ranges, performance curves with flatter slopes might have appeared in those ranges. Among the aerosols that gave performance curves with γ > 1, the only obvious common characteristic was that a large fraction of each was composed of submicrometer particles. Cut-Power Correlation Another design method, also based on scrubber power consumption, is the cut-power method of Calvert [ J. Air Pollut. Control Assoc., 24, 929 (1974); Chem. Eng., 84(18), 54 (1977)]. In this approach, the cut diameter (the particle diameter for which the collection efficiency is 50 percent) is given as a function of the gas pressure drop or of the power input per unit of volumetric gas flow rate. The functional relationship is presented as a log-log plot of the cut diameter versus the pressure drop (or power input). In principle, the function could be constructed by experimentally determining scrubber performance curves for discrete particle sizes and then plotting the particle sizes against the corresponding pressure drops necessary to give efficiencies of 50 percent. In practice, Calvert and coworkers evidently have in most cases constructed the cut-power functions for various scrubbers by modeling (Yung and Calvert, U.S. EPA-600/8-78-005b, 1978). They show a variety of curves, whereas empirical studies have indicated that different types of scrubbers generally have about the same performance at a given level of power consumption. Condensation Scrubbing The collection efficiency of scrubbing can be increased by the simultaneous condensation of water vapor from the gas stream. Water-vapor condensation assists in particle removal by two entirely different mechanisms. One is the deposition of particles on cold-water droplets or other surfaces as the result of

Performance curve for orifice scrubber collecting ammonium fluorescin aerosol. (Semrau et al., EPA 600/2-77-237, 1977.)

FIG. 17-47

17-35

Stefan flow. The other is the condensation of water vapor on particles as nuclei, which enlarges the particles and makes them more readily collected by inertial deposition on droplets. Both mechanisms can operate simultaneously. However, for the buildup of particles by condensation to be effective, there must be adequate time for the particles to grow substantially before the principal gas-liquid-contacting operation takes place. Hence, if particle buildup is to be sought, the scrubber should be preceded by an appropriate gas-conditioning section. On the other hand, particle collection by Stefan flow can be induced simply by scrubbing the hot, humid gas with sufficient cold water to bring the gas below its initial dew point. Any practical method of inducing condensation on the dust particles will incidentally afford opportunities for the operation of the Stefan-flow mechanism. The hot gas stream must, of course, have a high initial moisture content, since the magnitude of the effects obtained is related to the quantity of water vapor condensed. Although there is a considerable body of literature on particle collection by condensation mechanisms, most of it is either theoretical or, if experimental, treats basic phenomena in simplified cases. Few studies have been made to determine what performance may be expected from condensation scrubbing under practical conditions in industrial applications. In a series of studies, Calvert and coworkers investigated several types of equipment for condensation scrubbing, generally emphasizing the use of the condensation center effect to build up the particles for collection by inertial deposition (Calvert and Parker, EPA-600/8-78-005c, 1978). From early estimates, they predicted that a condensation scrubber would require only about one-third or less of the power required by a conventional high-energy scrubber. A subsequent demonstration-plant scrubber system consisted of a directcontact condensing tower fed with cold water followed by a venturi scrubber fed with recirculated water (Chmielewski and Calvert, EPA600/7-81-148, 1981). The condensation and particle buildup took place in the cooling tower. In operation on humidified iron-foundrycupola gas, this system still required about 65 percent as much power as for conventional high-energy scrubbing. Semrau and coworkers [Ind. Eng. Chem., 50, 1615 (1958); J. Air Pollut. Control Assoc., 13, 587 (1963); EPA-650/2-74-108, 1974] investigated condensation scrubbing in pilot-plant studies in the field and, later, under laboratory conditions. Hot, humid gases were scrubbed directly with cold water under conditions that were favorable for the Stefan-flow mechanism but offered little or no opportunity for particle buildup. Some of the field studies indicated a contacting-power saving of as much as 50 percent for condensation scrubbing of Kraft-recovery-furnace fume. Laboratory tests on a predominantly submicrometer synthetic aerosol showed contactingpower savings of up to 40 percent with condensation scrubbing. In the scrubbing of hot gases with high water content, condensation reduces contacting power and affords a direct power saving through the reduction of the gas volume by cooling and water-vapor condensation, but it incurs other costs for power and equipment for heat transfer and water cooling. However, condensation scrubbing may offer a net economic advantage if recovery of low-level heat is practical. It should also be advantageous when a hot gas must be not only cleaned but cooled and dehumidified as well; examples are the cleaning of blast-furnace gas for use as fuel and of SO2-bearing waste gases for feed to a sulfuric acid plant. Entrainment Separation The entrainment separator is a critical element of a scrubber, since the collection efficiency of the scrubber depends on essentially total removal of the spray from the gas stream. The sprays generated in scrubbers are generally large enough in droplet size that they can be readily removed by properly designed inertial separators. Primary collection of the spray is seldom the critical limitation on separator performance, but reentrainment is a common problem. In dust scrubbers it is essential that the entrainment separator not be of a form readily subject to blockage by solids deposits and that it be readily cleared of deposits if they should occur. Cyclone separators are advantageous in this respect and are widely used with venturi contactors. However, they cannot readily be made integral with scrubbers of some other configurations, which can be more conveniently fitted with various forms of impingement separators. Although separator design can be important, the most common

17-36

GAS-SOLID OPERATIONS AND EQUIPMENT

cause of reentrainment is simply the use of excessive gas velocities, and few data are available on the gas-handling capacities of separators. In the absence of good data, there is a frequent tendency to underdesign separators in an effort to reduce costs. Venturi Scrubbers The venturi scrubber is one of the most widely used types of particulate scrubbers. The designs have become generally standardized, and units are manufactured by a large number of companies. Venturi scrubbers may be used as either high- or lowenergy devices but are most commonly employed as high-energy units. The units originally studied and used were designed to the proportions of the classical venturis used for metering, but since it was discovered that these proportions have no special merits, simpler and more practical designs have been adopted. Most “venturi” contactors in current use are in fact not venturis but variable orifices of one form or another. Any of a wide range of devices can be used, including a simple pipe-line contactor. Although the venturi scrubber is not inherently more efficient at a given contacting power than other types of devices, its simplicity and flexibility favor its use. It is also useful as a gas absorber for relatively soluble gases, but because it is a cocurrent contactor, it is not well suited to absorption of gases having low solubilities. Current designs for venturi scrubbers generally use the vertical downflow of gas through the venturi contactor and incorporate three features: (1) a “wet-approach” or “flooded-wall” entry section, to avoid dust buildup at a wet-dry junction; (2) an adjustable throat for the venturi (or orifice), to provide for adjustment of the pressure drop; and (3) a “flooded elbow” located below the venturi and ahead of the entrainment separator, to reduce wear by abrasive particles. The venturi throat is sometimes fitted with a refractory lining to resist abrasion by dust particles. The entrainment separator is commonly, but not invariably, of the cyclone type. An example of the “standard form” of venturi scrubber is shown in Fig. 17-48. The wet-approach entry section has made practical the recirculation of slurries. Various forms of adjustable throats, which may be under manual or automatic control,

FIG. 17-48

Venturi scrubber. (Neptune AirPol.)

permit maintaining a constant pressure drop and constant efficiency under conditions of varying gas flow. Self-Induced Spray Scrubbers Self-induced spray scrubbers form a category of gas-atomized spray scrubbers in which a tube or a duct of some other shape forms the gas-liquid-contacting zone. The gas stream flowing at high velocity through the contactor atomizes the liquid in essentially the same manner as in a venturi scrubber. However, the liquid is fed into the contactor and later recirculated from the entrainment separator section by gravity instead of being circulated by a pump as in venturi scrubbers. The scheme is well illustrated in Fig. 17-49a. A great many such devices using contactor ducts of various shapes, as in Fig. 17-49b are offered commercially. Although self-induced spray scrubbers can be built as high-energy units and sometimes are, most such devices are designed for only low-energy service. The principal advantage of self-induced spray scrubbers is the elimination of a pump for recirculation of the scrubbing liquid. However, the designs for high-energy service are somewhat more complex and less flexible than those for venturi scrubbers. Plate Towers Plate (tray) towers are countercurrent gasatomized spray scrubbers using one or more plates for gas-liquid contacting. They are essentially the same as, if not identical to, the devices used for gas absorption and are frequently employed in applications in which gases are to be absorbed simultaneously with the removal of dust. Except possibly in cases in which condensation effects are involved, countercurrent operation is not significantly beneficial in dust collection. The plates may be any of several types, including sieve, bubble-cap, and valve trays. The impingement baffle plate (Fig. 17-50) is commonly used for dust collection applications. Impingement on the baffles is not the controlling mechanism of particle collection; the principal collecting bodies are the droplets produced from the liquid by the gas as it flows through the perforations and around the baffles. The slot stage (Fig. 17-50) is in effect a miniature venturi contactor. Valve trays constitute multiple self-adjusting orifices that provide nearly constant gas pressure drop over considerable ranges of variation in gas flow. The gas pressure drop that can be taken across a single plate is necessarily limited, so that units designed for high contacting power must use multiple plates. Plate towers are more subject to plugging and fouling than venturitype scrubbers that have large passages for gas and liquid. Packed-Bed Scrubbers Packed-bed scrubbers of the types used for gas absorption may also be used for dust collection but are subject to plugging by deposits of insoluble solids. Random packings, such as dumped Raschig rings and Berl saddles, are most seriously affected by plugging. Regular packings, such as stacked grids, are better in dustcollection service. When both a gas and particulate matter are to be collected, it is advisable to use a primary-stage scrubber of the venturi or similar type to collect the particulate matter ahead of a packed gas absorber. Packed-bed scrubbers may be constructed for either vertical or horizontal gas flow. Vertical-flow units (packed towers) commonly use countercurrent flow of gas and liquid, although cocurrent flow is sometimes used. Packed scrubbers using horizontal gas flow usually employ cross-flow of liquid. Scrubber packings are too large to serve as collecting bodies for any except very large dust particles. In the collection of fine particles, the packings serve primarily to promote fluid turbulence that aids the deposition of the dust particles on droplets. In a packed tower operating below the flooding point, with most of the liquid flowing in films and little spray formation, the relative efficiency in collection of particles may possibly be lower than that of a venturi-type scrubber operating at the same contacting power. However, no data are available to resolve the question. Mobile-Bed Scrubbers Mobile-bed scrubbers (Fig. 17-51) are constructed with one or more beds of low-density spheres that are free to move between upper and lower retaining grids. The spheres are commonly 1.0 in (2.5 cm) or more in diameter and made from rubber or a plastic such as polypropylene. The plastic spheres may be solid or hollow. Gas and liquid flows are countercurrent, and the spherical packings are fluidized by the upward-flowing gas. The movement of

GAS-SOLIDS SEPARATIONS

(a) FIG. 17-49

17-37

(b)

Self-induced spray scrubbers. (a) Blaw-Knox Food & Chemical Equipment, Inc. (b) American Air Filter Co.,

Inc.

the packings is intended to minimize fouling and plugging of the bed. Mobile-bed scrubbers were first developed for absorbing gases from gas streams that also carry solid or semisolid particles. The spherical packings are too large to serve as effective targets for the deposition of fine dust particles. In dust-collection service, the packings actually serve as turbulence promoters, while the dust particles are collected primarily by the liquid droplets. The gas pressure drop through the scrubber may be increased by increasing the gas velocity, the liquid-to-gas ratio, the depth of the bed, the density of the packings, and the number of beds in series. In an experimental study, Yung et al. (EPA-600/7-79-071, 1979) determined that the collection efficiency of a mobile-bed scrubber was dependent only on the gas pressure drop and was not influenced independently by the gas velocity, the liquid-to-gas ratio, or the number of beds except as these factors affected the pressure drop. Yung et al. also reported that the mobile-bed scrubber was less efficient at a given pressure drop than scrubbers of the venturi type, but without offering comparable experimental supporting evidence. Spray Scrubbers Spray scrubbers consist of empty chambers of some simple form in which the gas stream is contacted with liquid droplets generated by spray nozzles. A common form is a spray tower, in which the gas flows upward through a bank or successive banks of spray nozzles. Similar arrangements are sometimes used in spray chambers with horizontal gas flow. Such devices have very low gas pressure drops, and all but a small part of the contacting power is derived from the liquid stream. The required contacting power is obtained from an appropriate combination of liquid pressure and flow rate. Most spray scrubber are low-energy units. Collection of fine particles is possible but may require very high liquid-to-gas ratios, liquid feed pressures, or both. Plugging of the nozzles can be a persistent maintenance problem. Entrainment separators are necessary to prevent carry-over of spray into the exit gas.

Cyclone Scrubbers The vessels of cyclone scrubbers are all in the form of cyclones, which provide for entrainment separation. However, the gas-liquid-contacting devices may be of either the gasatomized-spray or the preformed-spray type. The cyclone-spray scrubber shown in Fig. 17-52a has an axial spray tree, or manifold, equipped with hydraulic spray nozzles. Similar units are available with the spray nozzles mounted in the wall of the cyclone, discharging inward. This latter arrangement makes the nozzles more accessible for maintenance. In the cyclone scrubber shown in Fig. 17-52b, most of the gas-liquid contacting is accomplished in the swirl vanes, with the energy being supplied from the gas stream in the form of pressure drop. The swirl vanes serve the same function as do the trays in a tray tower. Higher contacting power can be provided by using additional sets of swirl vanes in series. Ejector-Venturi Scrubbers In the ejector-venturi scrubber (Fig. 17-53) the cocurrent water jet from a spray nozzle serves both to scrub the gas and to provide the draft for moving the gas. No fan is required, but the equivalent power must be supplied to the pump that delivers water to the ejector nozzle. The water must be supplied in sufficient volume and at high enough pressure to provide both adequate draft and enough contacting power for the required scrubbing operation. Considered as a gas pump, the ejector is not a very efficient device, but the dissipated energy that is not effective in pumping does serve in gas-liquid contacting. The energy equivalent to any gas pressure rise across the scrubber is not part of the contacting power (Semrau et al., EPA-600/2-77-234, 1974). The ejector-venturi scrubber is widely used as a gas absorber, but the combinations of water pressure and flow rate that are sufficient to provide the required draft usually do not also yield enough contacting power to give high collection efficiency on submicrometer particles. Other types of ejectors have been employed to provide higher contacting-power levels. In one, superheated water is discharged through

17-38

GAS-SOLID OPERATIONS AND EQUIPMENT

FIG. 17-50

Impingement-plate scrubber. (Peabody Engineering Corp.)

the nozzle, and part flashes to steam, increasing the mechanical energy available for scrubbing [Gardenier, J. Air Pollut. Control Assoc., 24, 954 (1974)]. Some units use two-fluid nozzles, with either compressed air or steam as the compressible fluid [Sparks, J. Air Pollut. Control Assoc., 24, 958 (1974)]. Most of the energy for gas movement and for atomizing the liquid and scrubbing the gas is derived from the compressed air or steam. In some ejector-venturi-scrubbers installations, part of the draft is supplied by a fan [Williams and Fuller, TAPPI, 60(1), 108 (1977)]. Mechanical Scrubbers Mechanical scrubbers comprise those devices in which a power-driven rotor produces the fine spray and the contacting of gas and liquid. As in other types of scrubbers, it is the droplets that are the principal collecting bodies for the dust particles. The rotor acts as a turbulence producer. An entrainment separator must be used to prevent carry-over of spray. Among potential mainte-

nance problems are unbalancing of the rotor by buildup of dust deposits and abrasion by coarse particles. The simplest commercial devices of this type are essentially fans upon which water is sprayed. The unit shown in Fig. 17-54 is adapted to light duty, and heavy dust loads are avoided to minimize buildup on the rotor. Fiber-Bed Scrubbers Fibrous-bed structures are sometimes used as gas-liquid contactors, with cocurrent flow of the gas and liquid streams. In such contactors, both scrubbing (particle deposition on droplets) and filtration (particle deposition on fibers) may take place. If only mists are to be collected, small fibers may be used, but if solid particles are present, the use of fiber beds is limited by the tendency of the beds to plug. For dust-collection service, the fiber bed must be composed of coarse fibers and have a high void fraction, so as to minimize the tendency to plug. The fiber bed may be made from metal or

GAS-SOLIDS SEPARATIONS

FIG. 17-51

Mobile-bed scrubber. (Air Correction Division, UOP.)

plastic fibers in the form of knitted structures, multiple layers of screens, or random-packed fibers. However, the bed must have sufficient dimensional stability so that it will not be compacted during operation. Lucas and Porter (U.S. Patent 3,370,401, 1967) developed a fiberbed scrubber in which the gas and scrubbing liquid flow vertically upward through a fiber bed (Fig. 17-55). The beds tested were composed of knitted structures made from fibers with diameters ranging from 89 to 406 µm. Lucas and Porter reported that the fiber-bed scrubber gave substantially higher efficiencies than did venturi-type scrubbers tested with the same dust at the same gas pressure drop. In similar experiments, Semrau (Semrau and Lunn, op. cit.) also found that a fiber-bed contactor made with random-packed steel-wool fibers gave higher efficiencies than an orifice contactor. However, there

(a) FIG. 17-52

17-39

were indications that the fiber bed would have little advantage in the collection of submicrometer particles, presumably because of the large fiber size feasible for dust-collection service. Despite their potential for increased collection efficiency, fiber-bed scrubbers have had only limited commercial acceptance for dust collection because of their tendency to become plugged. Their principal use has been in small units such as engine-intake-air cleaners, for which it is feasible to remove the fiber bed for cleaning at frequent intervals. Electrically Augmented Scrubbers In some types of wet collectors, attempts are made to apply the electrostatic-deposition mechanism by charging the dust particles, the water droplets, or both. The objective is to combine in a scrubber high efficiency in collecting fine particles and the moderate power consumption characteristic of an electrical precipitator. Successful devices of this type have been essentially wet electrical precipitators and should properly be discussed in that category (see “Electrical Precipitators”). So far, there has been no clear demonstration of a device that combines the small size, compactness, and high efficiency of a high-energy scrubber with the relatively low power consumption of an electrical precipitator. Dry Scrubbing Dry scrubbing is an umbrella term used to associate several different unit operations and types of hardware that can be used in combinations to accomplish the unit process of dry scrubbing. They all utilize scrubbing, in which mass transfer takes place between the gas phase and an active liquidlike surface, and they all discharge the resulting products separately as a gas and a solidlike “dry” product for reuse or disposal. It is helpful to recognize three phases in the development of systems using dry scrubbing. The first phase used mainly good contactors and relatively expensive sorbents, which were tailored to catch targeted compounds with high efficiency (>99 percent). The second phase used mainly minimal contactors and cheap sorbents in order to get enough sulfur dioxide capture (>60 percent) to justify using cheaper higher sulfur coal. The third phase is ongoing; it uses mainly better contactors, sorbents, and fabric filters in order to get not only high SO2 capture (>90 percent), but also substantial capture of other regulated emissions such as metals and dioxins. The principle of dry scrubbing was first recognized by Lamb and Wilson [I&EC, 11(5), 420 (1919)] and Wilson [I&EC, 12(10), 1000 (1920)] at MIT during their World War I program to develop better

(b)

Cyclone scrubbers. (a) Chemico Air Pollution Control Corp. (b) Ducon Co., Inc.

17-40

GAS-SOLID OPERATIONS AND EQUIPMENT

FIG. 17-55

Fibrous-bed scrubber. (Lucas and Porter, U.S. Patent 3,370,401,

1967.)

FIG. 17-53

Ejector-venturi scrubber. (Schutte & Koerting Division, Amtek,

Inc.)

FIG. 17-54

Mechanical scrubber. (American Air Filter Co., Inc.)

media for military gas masks. The preferred medium turned out to be a mixture of granular activated carbon with a new kind of porous granular hydrated lime, which they dubbed activated soda-lime. What activated it was a small but critical amount of sodium hydroxide, which was enough to maintain in use on French battlefields an equilibrium moisture film a few molecules thick on the surface of pores within lime particles. That film enhanced surface hydrolysis, diffusion, and neutralization of acid gases such as Cl2 and SO2, to the point that their removal became controlled by fast external gas-film diffusion, up to substantial utilization of the potential stoichiometric capacity. The better varieties of army activated soda-lime used in gas masks captured more than 99 percent of Cl2 or SO2 even after using up more than 10 percent of alkalinity. Gas-mask-canister sorbent beds typically used 8 × 14 mesh granules 10 cm deep, and tests were run with 1 percent acid gas at space velocities of 3000 to 7400 l/hr. Carbon dioxide was tested to higher penetrations; its capture was more than 50 percent even after using up more than 50 percent of available alkalinity. With good dry scrubbing sorbents, the controlling resistance for gas cleaning is external turbulent diffusion, which also depends on energy dissipated by viscous and by inertial mechanisms. It turns out to be possible to correlate mass-transfer rate as a function of the friction factor. It turns out that packed beds much less than a hundred particles thick behave as if they were well-stirred due to the entrance effect. Although it may seem odd that a packed bed can behave as if wellstirred, it typically takes at least a 100-particle bed depth in order for a plug-flow concentration wave to develop. Integration of the Ergun equation for well stirred flow gives ρDv(1 − εv)L C2 − C1 (17-9) ᎏ = Fk ᎏᎏ C* − C2 µεvDp Recognition of dry scrubbing as a mechanism useful in treating large gas flows first came in the 1960s as regulations were tightening on fly ash emissions from utility boilers. Dry scrubbing turns out to be the appropriate way to describe the interactions of sulfuric acid vapor with fly ash from pulverized coal combustion. On one hand, powdered sodium bicabonate was added to some flue gases in order to protect fabric filter bags from attack by sulfuric acid films on ash from high sulfur coal combustion. On the other hand, sulfur trioxide was added to other flue gases in order to generate sulfuric acid films that would improve the performance of electrostatic precipitators collecting inherently high resistivity ash from low sulfur coal. From Eq. (17-9),

GAS-SOLIDS SEPARATIONS it is readily apparent that the way to get more gas cleaning per unit pressure drop is to optimize the particle size and void fraction, and use relatively fewer particles operating at higher velocity. Such an arrangement runs out of stoichiometric capacity quickly, which leads naturally to transported bed contactors fed with fresh sorbent. The actual name “dry scrubbing” was first publicized by Teller [U.S. Patent no. 3,721,066 (1973)]. He worked both with classical Armytype soda-lime and with his patented water-activated form of the alkaline feldspar nepheline syenite as a flow agent and feedstock sorbent for HF and SO2 in hot, sticky fumes from glass melting furnaces. He claimed capture of more than 99 percent of 180 ppm HF and SO2 for more than 20 hours in a packed bed of 200 × 325 mesh hydrated nepheline syenite at 42,000/hr. Teller [U.S. Patent no. 3,997,652 (1976)] also patented a number of “chromatographic sorbents” using semivolatile liquid coatings similar to mobile phases used in capillary columns for gas chromatographs. For simple acid gases, Teller generally used a flash-drying-type venturi injector and pneumatic transport system to disperse fresh sorbent into hot gas to be treated and to provide time for turbulent mixing. He then typically used a baghouse to collect the loaded sorbent for inclusion with other feeds into the glass melting system. Aluminum producers were also early users of another type of dry scrubbing for the difficult cleaning problem of acid gases in hot, sticky fumes from aluminum reduction potlines. Bazhenov et al. [Tsvetn. Met.—Russia, (6) 44 (1975)] described the use of a fluidized bed of feedstock aluminum oxide to clean potline offgas in an arrangement where there was an integral freeboard baghouse. Loaded sorbent was purged from the bed for feed into the aluminum reduction system. In that anhydrous system there was no water film as such inside the pores, but there was high specific surface, having liquidlike mobile active sites with a strong affinity for binding fluoride. The potential for achieving significant gas cleaning at very high space velocities led to dry scrubbing being applied both for incremental desulfurization of flue gases from coal-fired utility boilers and for removal of trace acid gases from solid waste incineration. A number of different hardware arrangements were tested all the way to full scale in order to determine accurately the various effects that must be optimized in order to achieve minimum net cost to be carried by the rate-paying public. It was apparent from the very earliest tests that control of thin moisture films on the surface of reactive particles was the key to success. The main three competing arrangements, as compared by Statnick et 500 400

100 90 80 70 60 50 40

°F

00

∆T m

30

=8

0°F

50

0°F

20

3

4 5 6 7 89

2

3

4 5 6 7 89

Time for complete evaporation, sec FIG. 17-56

Effect of drop diameter on time for complete evaporation of water drops.

2

3

4 5 6 7 89

100

2

10–2

10–3

20

10–1

Initial drop diameter, microns

Zero relative velocity Terminal velocity

200

10

al. [4th Annual Pitt. Coal Conf. (1987)] involved slurry spray dryers, where lime and water were injected together, versus systems where the gas was humidified by water injection before or after injection of limey dry powder reagents. It turns out that there are tradeoffs among the costs of hardware, reagent, and water dispersion and reagent purchase and disposal. Systems where water evaporates in the presence of active particles are usually less expensive overall. The evaporation lifetimes of pure water droplets are described by Marshall and Seltzer [CEP, 46(10), 501 and 46(11), 575 (1950)] and by Duffie and Marshall [CEP, 49(8), 417 (1953) and 49(9), 480 (1953)]. Figure 17-56 is easy to use to estimate the necessary spray dispersion. For typical humidifier conditions of 300°F gas inlet temperature and a 20°F approach to adiabatic saturation at the outlet, a 60-micrometer droplet will require about 0.9 seconds to evaporate, while a 110-micrometer droplet will require about 3 seconds. If the time available for drying is no more than 3 seconds, it therefore follows that the largest droplets in a humidifier spray should be no larger than 100 micrometers. If the material being dried contains solids, droplet top size will need to be smaller due to slower evaporation rates. Droplets this small not only require considerable expensive power to generate, but since they have inherent penetration distances of less than a meter, they require expensive dispersion arrangements to get good mixing into large gas flows without allowing damp, sticky particles to reach walls. The economics seem to be better for systems where dry powdered fresh lime plus ground recycled lime is injected along with a relatively coarse spray which impinges on and dries out from the reagent, as described by Stouffer et al. [I&EC Res., 28(1) 20 (1989)]. Withum et al. [9th Ann. Pitt. Coal Prep. Util. Env. Control Contractors Conf. (1993)] describes an advanced version of that system that has been further optimized to the point that it is competitive with wet limestone scrubbing for >90 percent flue gas desulfurization. The most popular dry scrubbing systems for incinerators have involved the spray drying of lime slurries, followed by dry collection in electrostatic precipitators or fabric filters. Moller and Christiansen [Air Poll. Cont. Assoc. 84-9.5 (1984)] published data on early European technology. Moller et al. [U.S. Patent no. 4,889,698 (1989)] describe the newer extension of that technology to include both spraydryer absorption and dry scrubbing with powdered, activated carbon injection. They claim greatly improved removal of mercury, dioxins, and NOx.

Evaporation lifetimes for pure water drops

300

17-41

17-42

GAS-SOLID OPERATIONS AND EQUIPMENT

Optimized modern dry scrubbing systems for incinerator gas cleaning are much more effective (and expensive) than their counterparts used so far for utility boiler flue gas cleaning. Brinckman and Maresca [ASME Med. Waste Symp. (1992)] describe the use of dry hydrated lime or sodium bicarbonate injection followed by membrane filtration as preferred treatment technology for control of acid gas and particulate matter emissions from modular medical waste incinerators, which have especially high dioxin emissions. Kreindl and Brinckman [Air Waste Man. Assoc., paper 93-RP154.04 (1993)] describe the three conventional flue-gas cleaningprocess concepts (standard dry scrubbing, quasi-dry scrubbing, and wet scrubbing) as being inherently inadequate to meet the emerging European incinerator metal and organic emissions limits. Standard dry scrubbing can be upgraded by using powdered carbon along with lime, and switching to membrane filter media. Quasi-dry scrubbing can be upgraded by using powdered carbon along with lime spray drying [as also described by Moller (1989)], and switching to membrane filter media. Wet scrubbing can be upgraded by post-treatment following the wet scrubber with a circulating fluidized bed of granular activated carbon, again followed by a membrane filter. Control of metal and organic emissions by dry scrubbing involves more than the simple acid gas neutralization discussed so far. Capture of metals other than mercury is mainly a matter of using a fabric filter having the capability of retaining unagglomerated submicron particles that are relatively enriched in toxic metals such as cadmium. Use of membrane-coated filter media is almost essential to do that well. Capture of mercury and dioxins is mainly achieved with activated carbon, which is more effective when used in combination with chlorine inhibitors. Karasek [U.S. Patent no. 4,793,270 (1988)] describes the inhibiting effects of sulfur compounds, which is why dioxins are rarely a problem in flue gases from coal-fired combustion. Naikwadi and Karasek [Chemosphere, 27(1–3), 335 (1993)] describe the potent inhibiting effect of amines. That is presumably why German experience is finding that dioxins are rarely a problem in flue gases that have undergone some form of ammonia deNOx treatment, as described by Hahn [Chemosphere, 25(1–2), 57 (1992)]. Hahn, like Kreindl and Brinckman, also argues that classical wet scrubbers will need to be upgraded with modern dry scrubbers and fabric filters. Fabric Filters Fabric filters, commonly termed “bag filters” or “baghouses,” are collectors in which dust is removed from the gas stream by passing the dust-laden gas through a fabric of some type (e.g., woven cloth, felt, or porous membrane). These devices are “surface” filters in that dust collects in a layer on the surface of the filter medium, and the dust layer itself becomes the effective filter medium. The pores in the medium (particularly in woven cloth) are usually many times the size of the dust particles, so that collection efficiency is low until sufficient particles have been collected to build up a “precoat” in the fabric pores (Billings and Wilder, Handbook of Fabric Filter Technology, vol. I, EPA No. APTD-0690, NTIS No. PB-200648, 1979). During this initial period, particle deposition takes place mainly by inertial and flow-line interception, diffusion, and gravity. Once the dust layer has been fully established, sieving is probably the dominant deposition mechanism, penetration is usually extremely low except during the fabric-cleaning cycle, and only limited additional means remain for influencing collection efficiency by filter design. Filter design is related mainly to choices of gas filtration velocities and pressure drops and of fabric-cleaning cycles. Because of their inherently high efficiency on dusts in all particlesize ranges, fabric filters have been used for collection of fine dusts and fumes for over 100 years. The greatest limitation on filter application has been imposed by the temperature limits of available fabric materials. The upper limit for natural fibers is about 90°C (200°F). The major new developments in filter technology that have been made since 1945 have followed the development of fabrics made from glass and synthetic fibers, which has extended the temperature limits to about 230 to 260°C (450 to 500°F). The capabilities of available fibers to resist high temperatures are still among the most severe limitations on the possible applications of fabric filters. Gas Pressure Drops The filtration, or superficial face, velocities used in fabric filters are generally in the range of 0.3 to 3 m/min (1 to 10 ft/min), depending on the types of fabric, fabric supports, and

cleaning methods used. In this range, gas pressure drops conform to Darcy’s law for streamline flow in porous media, in which the pressure drop is directly proportional to the flow rate. The pressure drop across the fabric and the collected dust layer may be expressed (Billings and Wilder, op. cit.) by ∆p = K1Vf + K2wVf

(17-10)

where ∆p = kPa, or in of water; Vf = superficial velocity through filter, m/min, or ft/min; w = dust loading on filter, g/m2, or lbm/ft2; and K1 and K2 are resistance coefficients for the “conditioned” fabric and the dust layer respectively. The conditioned fabric is that fabric in which a relatively consistent dust load remains deposited in depth following cycles of filtration and cleaning. K1, expressed in units of kPa/(m/min) or in water/(ft/min), may be more than 10 times the value of the resistance coefficient for the original clean fabric. If the depth of the dust layer on the fabric is greater than about 0.2 cm (g in), corresponding to a fabric dust loading on the order of 500 g/m2 (0.1 lbm/ft2), the pressure drop across the fabric (including the dust in the pores) is usually negligible relative to that across the dust layer. The specific resistance coefficient for the dust layer K2 was originally defined by Williams et al. [Heat. Piping Air Cond., 12, 259 (1940)], who proposed estimating values of the coefficient by use of the Kozeny-Carman equation [Carman, Trans. Inst. Chem. Eng. (London), 15, 150 (1937)]. In practice, K1 and K2 are measured directly in filtration experiments. The K1 and K2 values can be corrected for temperature by multiplying by the ratio of the gas viscosity at the desired condition to the gas viscosity at the original experimental conditions. Values of K2 determined for certain dusts by Williams et al. (op. cit.) are presented in Table 17-5. Lapple (in Perry, Chemical Engineers’ Handbook, 3d ed., McGrawHill, New York, 1950) presents an alternative form of Eq. (17-10) in which the gas-viscosity term is explicit instead of being incorporated into the resistance coefficients: ∆pi = Kc µVf + Kd µwVf

(17-11)

where ∆pi = in water, µ = cP, Vf = ft/min, w = gr/ft2, K c = cloth resistance coefficient = (in water)/(cP)(ft/min), and K d = dust-layer resistance coefficient = (in water)/(cP)(gr/ft2)(ft/min). K d may be expressed in the same units, using the Kozeny-Carman equation: 160.0(1 − εv) Kd = ᎏᎏ (17-12) φ2s Dp2 ρs ε3v where Dp = µm and ρs = lbm/ft3. Data sufficient to permit reasonable predictions of K d from Eq. (17-12) are seldom available. The range in the values of K d that may be encountered in practice is illustrated in Fig. 17-57 in which availTABLE 17-5

Specific Resistance Coefficients for Certain Dusts* K2† for particle size less than

Dust Granite Foundry Gypsum Feldspar Stone Lampblack Zinc oxide Wood Resin (cold) Oats Corn

20 mesh

140 mesh

1.58 0.62

2.20 1.58

375 mesh

90 µm

45 µm

20 µm

2 µm

19.8 3.78 6.30 6.30

0.96

18.9 27.3 6.30 47.2 15.7‡ 6.30

0.62 1.58 0.62

25.2 1.58

9.60 3.78

11.0 8.80

*Data from Williams et al., Heat. Piping Air Cond., 12, 259–263 (1940). in water ∆p †K2 = ᎏi , ᎏᎏ . Vf w (ft/min)(lbm/ft2) NOTE: These data were obtained when filtering air at ambient conditions. For gases other than atmospheric air, the ∆pi values predicted from Table 17-5 should be multiplied by the actual gas viscosity divided by the viscosity of atmospheric air. ‡Flocculated material not dispersed; size actually larger.

GAS-SOLIDS SEPARATIONS able experimental determinations of K d reported in the literature for a variety of dusts are plotted against particle size. In most cases no accurate particle-size data were reported, and the curves represent the estimated range of particle size involved. The data of Williams et al. (op. cit.) are related to a wide variety of dusts, and only the approximate limits enclosing these data are shown. Also included are curves predicted from Eq. (17-12) for specific values of φs, ρs, and εv. It is apparent from these curves that smaller particles tend toward higher values of εv, which is consistent with the observation that fine dusts (particularly those smaller than 10 µm) have lower bulk densities than coarser fractions, apparently because of the effects of surface forces. For coarse dusts, K d varies approximately inversely as the square of the particle diameter, which implies that the void fraction (or bulk density) does not change with particle size. However, for particles finer than 10 µm, the value of K d appears to become constant, increased voids compensating for the reduction in size. In addition to the increased voidages encountered with small particles, slip flow also contributes to the relative constancy of the experimental values of Kd for particle sizes under 5 µm. Because of the assumptions underlying its derivation, the KozenyCarman equation is not valid at void fractions greater than 0.7 to 0.8 (Billings and Wilder, op. cit.). In addition, in situ measurement of the void fraction of a dust layer on a filter fabric is extremely difficult and has seldom even been attempted. The structure of the layer is dependent on the character of the fabric surface as well as on the characteristics of the dust, whereas the application of Eq. (17-12) implicitly assumes that K 2 is dependent only on the properties of the dust. A smooth fabric surface permits the dust to become closely packed, leading to a relatively high value of K2. If the surface is napped or has numerous extended fibrils, the dust cake formed will be more porous and have a lower value of K 2 [Billings and Wilder, op. cit.; Snyder and Pring, Ind. Eng. Chem., 47, 960 (1955); and K. T. Semrau, unpublished data, SRI International, Menlo Park, Calif., 1952–1953]. Equation (17-10) indicates that for filtration at a given velocity the pressure drop is a linear function of the fabric dust loading w. In some cases, particularly with smooth-surfaced fabrics, this is at least approx-

FIG. 17-57 Resistance factors for dust layers. Theoretical curves given are based on Eq. (20-78) for a shape factor of 0.5 and a true particle specific gravity of 2.0. [Williams, Hatch, and Greenburg, Heat. Piping Air. Cond., 12, 259 (1940); Mumford, Markson, and Ravese, Trans. Am. Soc. Mech. Eng., 62, 271 (1940); Capwell, Gas, 15, 31 (August 1939)].

17-43

imately the case, but in other instances the function displays an upward curvature with increases in w, indicating compression of the dust layer, the fabric, or both, and a consequent increase in K 2 (Snyder and Pring, op. cit.; Semrau, op. cit.). Several investigations have shown K 2 to be increased by increases in the filtration velocity [Billings and Wilder, op. cit.; Spaite and Walsh, Am. Ind. Hyg. Assoc. J., 24, 357 (1968)]. However, the various investigators do not agree on the magnitude of the velocity effects. Billings and Wilder suggest assuming as an approximation that K2 is directly proportional to the filtration velocity, but the actual relationship is probably dependent on the nature of the fabric and fabric surface, the characteristics of the dust, the dust loading on the fabric, and the pressure drop. Clearly, the factors determining K2 are far more complex than is indicated by a simple application of the Kozeny-Carman equation, and when possible, filter design should be based on experimental determinations made under conditions approximating those expected in the planned installation. Types of Filters Current fabric-filter designs fall into three types, depending on the method of cleaning used: (1) shaker-cleaned, (2) reverse-flow-cleaned, and (3) reverse-pulse-cleaned. The shakercleaned filter is the earliest form of bag filter (Fig. 17-58). The open lower ends of the bags are fastened over openings in the tube sheet that separates the lower dirty-gas inlet chamber from the upper cleangas chamber. The bag supports from which the bags are suspended are connected to a shaking mechanism. The dirty gas flows upward into the filter bags, and the dust collects on the inside surfaces of the bags. When the gas pressure drop rises to a chosen upper limit as the result of dust accumulation, the gas flow is stopped and the shaker is operated, giving a whipping motion to the bags. The dislodged dust falls into the dust hopper located below the tube sheet. If the filter is to be operated continuously, it must be constructed with multiple compartments, so that the individual compartments can be sequentially taken off line for cleaning while the other compartments continue in operation (Fig. 17-59). Shaker-cleaned filters are available as standard commercial units, although large baghouses for heavy-duty service are commonly custom-designed and -fabricated. The oval or round bags used in the standard units are usually 12 to 20 cm (5 to 8 in) in diameter and 2.5 to 5 m (8 to 17 ft) long. The large, heavy-duty baghouses may use bags up to 30 cm (12 in) in diameter and 9 m (30 ft) long. The bags must be made of woven fabrics to withstand the flexing and stretching involved in shaking. The fabrics may be made from natural fibers (cotton or wool) or synthetic fibers. Fabrics of glass or mineral fibers are generally too fragile to be cleaned by shaking and are usually used in reverse-flow-cleaned filters. Large units (other than custom units) are usually built up of standardized rectangular sections in parallel. Each section contains on the order of 1000 to 2000 ft2 of cloth, and the sections are assembled in the field to form a single filter housing. In this manner, the filter can be partitioned so that one or more sections at a time can be cut out of service for shaking or general maintenance. Ordinary shaker-cleaned filters may be shaken every d to 8 h, depending on the service. A manometer connected across the filter is useful in determining when the filter should be shaken. Fully automatic filters may be shaken as frequently as every 2 min, but bag maintenance will be greatly reduced if the time between shakings can be increased to 15 or 20 min without developing excessive pressure drop. Cleaning may be actuated automatically by a differentialpressure switch. It is essential that the gas flow through the filter be stopped when shaking in order to permit the dust to fall off. With very fine dust, it may even be necessary to equalize the pressure across the cloth [Mumford, Markson, and Ravese, Trans. Am. Soc. Mech. Eng., 62, 271 (1940)]. In practice this can be accomplished without interrupting the operation by cutting one section out of service at a time, as shown in Fig. 17-59. In automatic filters this operation involves closing the dampers, shaking the filter units either pneumatically or mechanically, sometimes with the addition of a reverse flow of cleaned gas through the filter, and lastly reopening the dampers. For compressed-air-operated automatic filters, this entire operation may take only 2 to 10 s. For ordinary mechanical filters equipped for automatic control, the operation may take as long as 3 min.

17-44

GAS-SOLID OPERATIONS AND EQUIPMENT

(a) FIG. 17-58

(b)

Typical shaker-type fabric filters. (a) Buell Norblo (cutaway view). (b) Wheelabrator-Frye Inc. (sectional

view).

FIG. 17-59 Three-compartment bag filter at various stages in the cleaning cycle. (Wheelabrator-Frye Inc.)

GAS-SOLIDS SEPARATIONS Equation (17-11) may be rewritten as ∆pi = Kd µcdV t

2 f m

(17-13)

where cd = dust concentration in dirty gas, gr/ft3; and tm = filtration time, min. This shows that the pressure drop due to dust accumulation varies as the square of the gas velocity through the filter. (The actual effect of velocity on pressure drop may be even greater in some instances.) Greater cloth area and reduced filtration velocity therefore afford substantial reductions in shaking frequency and in bag wear. Consequently, it is generally economical to be conservative in specifying cloth area. Shaker-cleaned filters are generally operated at filtration velocities of 0.3 to 2.5 m/min (1 to 8 ft/min) and at pressure drops of 0.5 to 1.5 kPa (2 to 6 in water). For very fine dusts or high dust concentrations, filtration velocities should not exceed 1 m/min (3 ft/min). For fine fumes and dusts in heavy-duty installations, filtration velocities of 0.3 to 0.6 m/min (1 to 2 ft/min) have long been accepted on the basis of operating experience. Cyclone precleaners are sometimes used to reduce the dust load on the filter or to remove large hot cinders or other materials that might damage the bags. However, reducing the dust load on the filter by this means may not reduce the pressure drop, since the increase in K2 produced by the reduction in average particle size may compensate for the decrease in the fabric dust loading. In filter operation, it is essential that the gas be kept above its dew point to avoid water-vapor condensation on the bags and resulting plugging of the bag pores. However, fabric filters have been used successfully in steam atmospheres, such as those encountered in vacuum dryers. In such cases, the housing is generally steam-traced. Reverse-flow-cleaned filters are generally similar to the shakercleaned filters except for the elimination of the shaker. After the flow of dirty gas has stopped, a fan is used to force clean gas through the bags from the clean-gas side. This flow of gas partly collapses the bags and dislodges the collected dust, which falls to the dust hopper. Rings are usually sewn into the bags at intervals along the length to prevent complete collapse, which would obstruct the fall of the dislodged dust. The principal applications of reverse-flow cleaning are in units using fiberglass fabric bags for dust collection at temperatures above 150°C (300°F). Collapsing and reinflation of the bags can be made sufficiently gentle to avoid putting excessive stresses on the fiberglass fabrics [Perkins and Imbalzano, “Factors Affecting the Bag Life Performance in Coal-Fired Boilers,” 3d APCA Specialty Conference on the User and Fabric Filtration Equipment, Niagara Falls, N.Y., 1978; and Miller, Power, 125(8), 78 (1981)]. As with shaker-cleaned filters, compartments of the baghouse are taken off line sequentially for bag cleaning. The gas for reverse-flow cleaning is commonly supplied in

FIG. 17-60 Reverse-pulse fabric filter: (a) filter cylinders; (b) wire retainers; (c) collars; (d) tube sheet; (e) venturi nozzle; ( f) nozzle or orifice; (g) solenoid valve; (h) timer; ( j) air manifold; (k) collector housing; (l) inlet; (m) hopper; (n) air lock; (o) upper plenum. (Mikropul Division, U.S. Filter Corp.)

17-45

an amount necessary to give a superficial velocity through the bags of 0.5 to 0.6 m/min (1.5 to 2.0 ft/min), which is the same range as the filtration velocities frequently used. In the reverse-pulse filter (frequently termed a reverse-jet filter), the filter bag forms a sleeve that is drawn over a wire cage, which is usually cylindrical (Fig. 17-60). The cage supports the fabric on the clean-gas side, and the dust is collected on the outside of the bag. A venturi nozzle is located in the clean-gas outlet from the bag. For cleaning, a jet of high-velocity air is directed through the venturi nozzle and into the bag, inducing a flow of cleaned gas to enter the bag and flow through the fabric to the dirty-gas side. The high-velocity jet is released in a sudden, short pulse (typical duration 100 ms or less) from a compressed-air line by a solenoid valve. The pulse of air and clean gas expands the bag and dislodges the collected dust. Rows of bags are cleaned in a timed sequence by programmed operation of the solenoid valves. The pressure of the pulse is sufficient to dislodge the dust without cessation of the gas flow through the filter unit. It has been a common practice to clean the bags on line (i.e., without stopping the flow of dirty gas into the filter), and reverse-pulse bag filters have been built without division into multiple compartments. However, investigations [Leith et al., J. Air Pollut. Control Assoc., 27, 636 (1977)] and experience have shown that, with online cleaning of reverse-pulse filters, a large fraction of the dust dislodged from the bag being cleaned may redeposit on neighboring bags rather than fall to the dust hopper. As a result, there is a growing trend to off-line cleaning of reverse-pulse filters. The baghouse is sectionalized so that the outlet-gas plenum serving the bags in a section can be closed off from the clean-gas exhaust, thereby stopping the flow of inlet gas through the bags. On the dirty-gas side of the tube sheet, the bags of the section are separated by partitions from the neighboring sections, where filtration is continuing. Sections of the filter are cleaned in rotation, as in shaker and reverse-flow filters. Some manufacturers are using relatively low-pressure air (100 kPa, or 15 lbf/in2, instead of 690 kPa, or 100 lbf/in2) and are eliminating the venturi tubes for clean-gas induction. Others have eliminated the separate jet nozzles located at the individual bags and use a single jet to inject a pulse into the outlet-gas plenum. Reverse-pulse filters are typically operated at higher filtration velocities (air-to-cloth ratios) than shaker or reverse-flow filters designed for the same duty. Filtration velocities may range from 1 to 4.5 m/min (3 to 15 ft/min), depending on the dust being collected, but for most dusts the commonly used range is about 1.2 to 2.5 m/min (4 to 8 ft/min). The frequency of cleaning is also dependent on the nature and concentration of the dust, with the intervals between pulses varying from about 2 to 15 min. The cleaning action of the pulse is so effective that the dust layer may be completely removed from the surface of the fabric. Consequently, the fabric itself must serve as the principal filter medium for at least a substantial part of the filtration cycle. Woven fabrics are unsuitable for such service, and felts of various types must be used. The bulk of the dust is still removed in a surface layer, but the felt ensures that an adequate collection efficiency is maintained until the dust layer has formed. Filter Fabrics The cost of the filter bags represents a substantial part of the erected cost of a bag filter—typically 5 to 20 percent, depending on the bag material [Reigel and Bundy, Power, 121(1), 68 (1977)]. The cost of bag repair and replacement is the largest component of the cost of bag-filter maintenance. Consequently, the proper choice of filter fabric is critical to both the technical performance and the economics of operating a filter. With the advent of synthetic fibers, it has become possible to produce fabrics having a wide range of properties. However, demonstrating the acceptability of a fabric still depends on experience with prolonged operation under the actual or simulated conditions of the proposed application. The choice of a fabric material for a given service is necessarily a compromise, since no single material possesses all the properties that may be desired. Following the choice of material, the type of fabric construction is critical. Two principal types of fabric are adaptable to filter use: woven fabrics, which are used in shaker and reverse-flow filters; and felts, which are used in reverse-pulse filters. The felts made from synthetic fibers are needle felts (i.e., felted on a needle loom) and are normally rein-

17-46

GAS-SOLID OPERATIONS AND EQUIPMENT

forced with a woven insert. The physical properties and air permeabilities of some typical woven and felt filter fabrics are presented in Tables 17-6 and 17-7. The “air permeability” of a filter fabric is defined as the flow rate of air in cubic feet per minute (at 70°F, 1 atm) that will pass through 1 ft2 of clean fabric under an applied differential pressure of a in water. The resistance coefficient KF of the clean fabric is defined by the equation in Table 17-6, which may be used to calculate the value of KF from the air permeability. If ∆pi is taken as 0.5 in water, µ as 0.0181 cP (the viscosity of air at 70°F and 1 atm), and Vf as the air permeability, then KF = 27.8/air permeability. Collection Efficiency The inherent collection efficiency of fabric filters is usually so high that, for practical purposes, the precise TABLE 17-6

level has not commonly been the subject of much concern. Furthermore, for collection of a given dust, the efficiency is usually fixed by the choices of filter fabric, filtration velocity, method of cleaning, and cleaning cycle, leaving few if any controllable variables by which efficiency can be further influenced. Inefficiency usually results from bags that are poorly installed, torn, or stretched from excessive dust loading and pressure drop. Of course, certain types of fabrics may simply be unsuited for filtration of a particular dust, but usually this will soon become obvious. Few basic studies of the efficiency of bag filters have been made. Increased dust penetration immediately following cleaning has been readily observed while the dust layer is being reestablished. However,

Resistance Factors and Air Permeabilities for Typical Woven Fabrics Pore size,* in

Cloth Osnaburg cotton Osnaburg cotton (soiled)‡ Drill cotton Cotton§ Cotton§ Cotton sateen (unnapped) Cotton sateen (unnapped) Cotton sateen (unnapped) Cotton sateen (unnapped) Wool Wool Wool, white§ Wool, black§ Wool§ Vinyon§ Nylon tackle twill Nylon sailcloth Nylon§ Nylon§ Asbeston§ Orlon§ Orlon§ Orlon§ Orlon§ Smoothtex nickel screen Glass Dacron Dacron Teflon

Weight, oz/yd2

Threads/in 32 × 28 32 × 28 68 × 40 46 × 56 104 × 68 96 × 56 96 × 64 96 × 60 96 × 56

0.01 0.01 0.007 0.005 0.004

Thread* diameter, in 0.02

40 × 50 36 × 32 28 × 30 30 × 26 37 × 37 72 × 196 130 × 130 37 × 37

5.28

0.01

6.88 8.23 10.2

0.009 0.01 0.012 0.011

11.5

0.014

0.010 0.007

72 × 72 74 × 38 (300 mesh) 32 × 28 60 × 40 76 × 48 76 × 70

0.03 5.8 13.4 8.7

KF†

Air permeability, (ft3/min)/ft2 at ∆pi = a in H2O

0.51 4.80 0.093 1.39 1.54 0.27 0.88 1.63 1.12 0.25 0.33 0.15 0.25 0.51 0.12 0.66 1.66 1.74 3.71 0.56 0.66 0.75 1.16 1.98 0.16 1.60 0.84 0.29 1.39

55 5.8 300 20 18 103 32 17 25 111 84 185 110 55 23 42 17 16 7.5 50 42 37 24 14 174 17 33 9.5 20

*Estimates based on microscopic examination. †Measured with atmospheric air. This value will be constant only for streamline flow, which is the case for values of ρVf /µ of less than approximately 100. KF = ∆pi/µVf where ∆pi = pressure drop, in water; µ = gas viscosity, cP; Vf = superficial gas velocity through cloth, ft/min; and ρ = gas density, lb/ft3. ‡Cloth, similar to previous one, that had been in service and contained dust in pores although free of surface accumulation. §Data from Pring, Air Pollution, McGraw-Hill, New York, 1952, p. 280. TABLE 17-7

Physical Properties of Selected Felts for Reverse-Pulse Filters

2

Fiber

Weight, oz/yd

Thickness, in

Wool Wool Orlon* Orlon* Orlon* Acrilan* Dynel* Dacron* Dacron* Dacron* Nylon* Arnel* Teflon Teflon

23.1 21.2 10.9 17.9 24 17.9 24 17.9 9.9 24 24 24 15.6 43.5

0.135 0.129 0.045 0.088 0.125 0.075 0.125 0.080 0.250 0.125 0.125 0.125 0.053 0.119

*These data courtesy of American Felt Co.

Breaking strength, lbf/in width

65 85 110 100 60 125 20 175 100 60

Elongation, % to rupture

18 18 60 22 80 22 150 80 100 80

Air permeability, (ft3/min)/ft2 at ∆pi = a in water

KF

27.1 29.8 20–25 15–20 10–20 15–20 30–40 15–20 200–225 20–30 30–40 30–40 82.5 21.6

1.03 0.93 1.11–1.39 1.39–1.85 1.39–2.78 1.39–1.85 0.70–0.93 1.39–1.85 0.11–0.14 0.93–1.39 0.70–0.93 0.70–0.93 0.34 1.29

GAS-SOLIDS SEPARATIONS field and laboratory studies have indicated that during the rest of the filtration cycle the effluent-dust concentration tends to remain constant regardless of the inlet concentration [Dennis, J. Air Pollut. Control Assoc., 24, 1156 (1974)]. In addition, there has been little indication that the penetration is strongly related to dust-particle size, except possibly in the low-submicrometer range. These observations appear to be generally consistent with sieving being the principal collection mechanism. Leith and First [ J. Air Pollut. Control Assoc., 27, 534 (1977); 27, 754 (1977)] studied the collection efficiency of reverse-pulse filters and concluded that once the dust cake has been established, “straightthrough” penetration by dust particles that pass through the filter without being stopped is negligible by comparison with penetration by dust that actually deposits initially and then “seeps” through the fabric to be reentrained into the exit air stream. They also noted that “pinholes” may form in the dust cake, particularly over pores between yarns in a woven fabric, and that particles may subsequently penetrate straight through at the pinholes. The formation of pinholes, or “cake puncture,” had been observed earlier by Stephan et al. [Am. Ind. Hyg. Assoc. J., 21, 1 (1960)], but without measurement of the associated loss of collection efficiency. When a supported flat filter medium with extremely fine pores (e.g., glass-fiber paper, membrane filter) was used, no cake puncture took place even with very high pressure differentials across the cake. However, puncture did occur when a cotton-sateen filter fabric was used as the cake support. The formation of pinholes with certain combinations of dusts, fabrics, and filtration conditions was also observed by Koscianowski et al. (EPA-600/7-78056, 1978). Evidently puncture occurs when the local cake structure is not strong enough to maintain a bridge over the aperture represented by a large pore and the portion of the cake covering the pore is blown through the fabric. This suggests that formation of pinholes will be highly dependent on the strength of the surface forces between particles that produce flocculation of dusts. The seepage of a dust through a filter is probably also closely related to the strength of the surface forces. Surface pores can be greatly reduced in size by coating what will become the dusty side of the filter fabric with a thin microporous membrane that is supported by the underlying fabric. That has the effect of decreasing the effective penetration, both by eliminating cake pinholes, and by preventing the seepage of dust that is dragged through the fabric by successive cleanings. A variety of different membrane-forming polymers can be used in compatible service. The most versatile and effective surface filtration membranes are microfibrous Teflon as already described by Brinckman and Maresca [ASME Med. Waste Symp. (1992)] in the section on dry scrubbing. Granular-Bed Filters Granular-bed filters may be classified as “depth” filters, since dust particles deposit in depth within the bed of granules. The granules themselves present targets for the deposition of particles by inertia, diffusion, flow-line interception, gravity, and electrostatic attraction, depending on the dust and filter characteristics and the operating conditions. Other deposition mechanisms are minor at most. Although it is physically possible under some circumstances for a dust layer to form on the inlet face of the filter, the practical limits of gas pressure drop will normally have been reached long before a surface dust layer can be established. Granular-bed filters may be divided into three classes: 1. Fixed-bed, or packed-bed, filters. These units are not cleaned when they become plugged with deposited dust particles but are broken up for disposal or simply abandoned. If they are constructed from fine granules (e.g., sand particles), they may be designed to give high collection efficiencies on fine dust particles. However, if such a filter is to have a reasonable operating life, it can be used only on a gas containing a low concentration of dust particles. 2. Cleanable granular-bed filters. In these devices provisions are made to separate the collected dust from the granules either continuously or periodically, so that the units can operate continuously on gases containing moderate to high dust concentrations. The necessity for cleaning and recycling the granules generally restricts the practical lower granule size to about 3 to 10 mm. This in turn makes it difficult to attain high collection efficiencies on fine particles with granule beds of reasonable depth and gas pressure drop.

17-47

3. Fluidized-bed filters. Fluidized beds of granules have received considerable study on theoretical and experimental levels but have not been applied on a practical commercial scale. Fixed Granular-Bed Filters Fixed-bed filters composed of granules have received considerable theoretical and experimental study [Thomas and Yoder, AMA Arch. Ind. Health, 13, 545 (1956); 13, 550 (1956); Knettig and Beeckmans, Can. J. Chem. Eng., 52, 703 (1974); Schmidt et al., J. Air Pollut. Control Assoc., 28, 143 (1978); Tardos et al., J. Air Pollut. Control Assoc., 28, 354 (1978); and Gutfinger and Tardos, Atmos. Environ., 13, 853 (1979)]. The theoretical approach is the same as that used in the treatment of deep-bed fibrous filters. Fibers for filter applications can be produced with diameters smaller than it is practical to obtain with granules. Consequently, most concern with filtration of fine particles has been focused on fibrousbed rather than granular-bed filters. However, for certain specialized applications granular beds have shown some superior properties, such as greater dimensional stability. Granular-bed filters of special design (deep-bed sand filters) have been used since 1948 for removing radioactive particles from waste air and gas streams in atomic energy plants (Lapple, “Interim Report—200 Area Stack Contamination,” U.S. AEC Rep. HDC-743, Oct. 11, 1948; Juvinall et al., “Sand-Bed Filtration of Aerosols: A Review of Published Information,” U.S. AEC Rep. ANL-7683, 1970; and Burchsted et al., Nuclear Air Cleaning Handbook, U.S. ERDA 76-21, 1976). The filter characteristics needed included high collection efficiency on fine particles, large dust-holding capacity to give long operating life, and low maintenance requirements. The sand filters are as much as 2.7 m (9 ft) in depth and are constructed in graded layers with about a 2:1 variation in the granule size from one layer to the next. The air-flow direction is upward, and the granules decrease in size in the direction of the air flow. The bottom layer is composed of rocks about 5 to 7.5 cm (2 to 3 in) in diameter, and granule sizes in successive layers decrease to 0.3 to 0.6 mm (50 to 30 mesh) in the finest layer. With superficial face velocities of about 1.5 m/min (5 ft/min), gas pressure drops of clean filters have ranged from 1.7 to 2.8 kPa (7 to 11 in water). Collection efficiencies of up to 99.98 percent with a polydisperse dioctyl phthalate aerosol of 0.7-µm mean diameter have been reported (Juvinall et al., op. cit.). Operating lives of 5 years or more have been attained. Cleanable Granular-Bed Filters The principal objective in the development of cleanable granular-bed filters is to produce a device that can operate at temperatures above the range that can be tolerated with fabric filters. In some of the devices, the granules are circulated continuously through the unit, then are cleaned of the collected dust and returned to the filter bed. In others, the granular bed remains in place but is periodically taken out of service and cleaned by some means, such as backflushing with air. A number of moving-bed granular filters have used cross-flow designs. One form of cross-flow moving-granular-bed filter, produced by the Combustion Power Company (Fig. 17-61), is currently in commercial use in some applications. The granular filter medium consists of f- to d-in (3- to 6-mm) pea gravel. Gas face velocities range from 30 to 46 m/min (100 to 150 ft/min), and reported gas pressure drops are in the range of 0.5 to 3 kPa (2 to 12 in water). The original form of the device [Reese, TAPPI, 60(3), 109 (1977)] did not incorporate electrical augmentation. Collection efficiencies for submicrometer particles were low, and the electrical augmentation was added to correct the deficiency (Parquet, “The Electroscrubber Filter: Applications and Particulate Collection Performance,” EPA-600/9-82-005c, 1982, p. 363). The electrostatic grid immersed in the bed of granules is charged to a potential of 20,000 to 30,000 V, producing an electric field between the grid and the inlet and outlet louvers that enclose the bed. No ionizing electrode is used to charge particles in the incoming gas; reliance is placed on the existence of natural charges on the dust particles. Individual dust particles commonly carry positive or negative charges even though the net charge on the dust as a whole is normally neutral. Depending on their charges, dust particles are attracted or repelled by the electrical field and are therefore caused to deposit on the rocks in the bed. Self et al. (“Electrical Augmentation of Granular Bed Filters,” EPA600/9-80-039c, 1980, p. 309) demonstrated in theoretical studies and

17-48

GAS-SOLID OPERATIONS AND EQUIPMENT TABLE 17-8

Average Atmospheric-Dust Concentrations* 1 gr/1000 ft3 = 2.3 mg/m3 Location

Dust concentration, gr/1000 ft3

Rural and suburban districts Metropolitan districts Industrial districts Ordinary factories or workrooms Excessive dusty factories or mines

0.02–0.2 0.04–0.4 0.1–2.0 0.2–4.0 4.0–400

*Heating Ventilating Air Conditioning Guide, American Society of Heating, Refrigerating and Air-Conditioning Engineers, New York, 1960, p. 77.

FIG. 17-61

Electrically augmented granular-bed filter. (Combustion Power

Company.)

laboratory experiments that such an augmentation system should yield substantial increases in the collection efficiency for fine particles if the particles carry significant charges. Significant improvements in the performance of the Combustion Power units with electrical augmentation have been reported by the manufacturer (Parquet, op. cit.). Another type of gravel-bed filter, developed by GFE in Germany, has had limited commercial application in the United States [Schueler, Rock Prod., 76(7), 66 (1973); 77(11), 39 (1974)]. After precleaning in a cyclone, the gas flows downward through a stationary horizontal filter bed of gravel. When the bed becomes loaded with dust, the gas flow is cut off, and the bed is backflushed with air while being stirred with a double-armed rake that is rotated by a gear motor. The backflush air also flows backward through the cyclone, which then acts as a dropout chamber. Multiple filter units are constructed in parallel so that individual units can be taken off the line for cleaning. The dust dislodged from the bed and carried by the backflush air is flocculated, and part is collected in the cyclone. The backflush air with the remaining suspended dust is cleaned in the other gravel-bed filter units that are operating on line. Performance tests made on one installation for the U.S. Environmental Protection Agency (EPA600/7-78-093, 1978) did not give clear results but indicated that collection efficiencies were low on particles under 2 µm and that some of the dust in the backflush air was redispersed sufficiently to penetrate the operating filter units. Air Filters The types of equipment previously described are intended primarily for the collection of process dusts, whereas air filters comprise a variety of filtration devices designed for the collection of particulate matter at low concentrations, usually atmospheric dust. The difference in the two categories of equipment is not in the principles of operation but in the adaptations required to deal with the dif-

ferent quantities of dust. Process-dust concentrations may run as high as several hundred grams per cubic meter (or grains per cubic foot) but usually do not exceed 45 g/m3 (20 gr/ft3). Atmospheric-dust concentrations that may be expected in various types of locations are shown in Table 17-8 and are generally below 12 mg/m3 (5 gr/1000 ft3). The most frequent application of air filters is in cleaning atmospheric air for building ventilation, which usually requires only moderately high collection-efficiency levels. However, a variety of industrial operations developed mostly since the 1940s require air of extreme cleanliness, sometimes for pressurizing enclosures such as clean rooms and sometimes for use in a process itself. Examples of applications include the manufacture of antibiotics and other pharmaceuticals, the production of photographic film, and the manufacture and assembly of semiconductors and other electronic devices. Air cleaning at the necessary efficiency levels is accomplished by the use of highefficiency fibrous filters that have been developed since the 1940s. Air filters are also used to protect internal-combustion engines and gas turbines by cleaning the intake air. In some locations and applications, the atmospheric-dust concentrations encountered are much higher than those normally encountered in air-conditioning service. High-efficiency air filters are sometimes used for emission control when particulate contaminants are low in concentration but present special hazards; cleaning of ventilation air and other gas streams exhausted from nuclear plant operations is an example. Air-Filtration Theory Current high-efficiency air- and gasfiltration methods and equipment have resulted largely from the development of filtration theory since about 1930 and particularly since the 1940s. Much of the theoretical advance was originally encouraged by the requirements of the military and atomic energy programs. The fibrous filter has served both as a practical device and as a model for theoretical and experimental investigation. Extensive reviews and new treatments of air-filtration theory and experience have been presented by Chen [Chem. Rev., 55, 595 (1955)], Dorman (“Filtration,” in Davies, Aerosol Science, Academic, New York, 1966), Pich (Theory of Aerosol Filtration by Fibrous and Membrane Filters, in ibid.), Davies (Air Filtration, Academic, New York, 1973), and Kirsch and Stechkina (“The Theory of Aerosol Filtration with Fibrous Filters,” in Shaw, Fundamentals of Aerosol Science, Wiley, New York, 1978). The theoretical treatment of filtration starts with the processes of dust-particle deposition on collecting bodies, as outlined in Fig. 1735 and Table 17-2. All the mechanisms shown in Table 17-2 may come into play, but inertial deposition, flow-line interception, and diffusional deposition are usually dominant. Electrostatic precipitation may become a major mechanism if the collecting body, the dust particle, or both, are charged. Gravitational settling is a minor influence for particles in the size range of usual interest. Thermal precipitation is nil in the absence of significant temperature gradients. Sieving is a possible mechanism only when the pores in the filter medium are smaller than or approximately equal to the particle size and will not be encountered in fibrous filters unless they are loaded sufficiently for a surface dust layer to form. The theoretical prediction of the efficiency of collection of dust particles by a fibrous filter consists of three steps (Chen, op. cit.): 1. Calculation of the target efficiency ηo of an isolated fiber in an air stream having a superficial velocity the same as that in the filter 2. Determining the difference between the target efficiency of the isolated fiber and that of an individual fiber in the filter array η t

GAS-SOLIDS SEPARATIONS

FIG. 17-62

17-49

Isolated fiber efficiency for combined diffusion and interception mechanism at NRe = 10−2. [Chen, Chem. Rev., 55, 595

(1955).]

3. Determining the collection efficiency of the filter η from the target efficiency of the individual fibers The results of computations of ηo for an isolated fiber are illustrated in Figs. 17-62 and 17-63. The target efficiency ηt of an individual fiber in a filter differs from ηo for two main reasons (Pich, op. cit.): (1) the average gas velocity is higher in the filter, and (2) the velocity field around the individual fibers is influenced by the proximity of neighboring fibers. The interference effect is difficult to determine on a purely theoretical basis and is usually evaluated experimentally. Chen (op. cit.) expressed the effect with an empirical equation: ηt = ηo[1 + Kα(1 − εv)]

(17-14)

This indicates that the target efficiency of the fiber is increased by the proximity of other fibers. The value of Kα averaged 4.5 for values of the void fraction εv, ranging from 0.90 to 0.99. Extending use of the equation to values of εv lower than 0.90 may result in large errors. The collection efficiency of the filter may be calculated from the fiber target efficiency and other physical characteristics of the filter (Chen, op. cit.): 4η t L(1 − εv) Nt = ᎏᎏ (17-15) πDb εv where Db = fiber diameter and L = filter thickness. The derivation of Eq. (17-15) assumes that (1) ηt is the same throughout the filter, (2) all fibers are of the same diameter Db, are cylindrical and are normal to the direction of the gas flow, (3) the fraction of the particles deposited in any one layer of fiber is small, and (4) the gas passing through the filter is essentially completely remixed after it leaves one layer of the filter and before it enters the next. The first assumption requires that Eq. (17-15) apply only for particles of a single size for which there are corresponding values of ηt, η, and Nt. For filters of high porosity, εv approaches unity and Eq. (17-15) reduces to the expression used by Wong et al. [ J. Appl. Phys., 27, 161 (1956)] and Thomas and Lapple [Am. Inst. Chem. Eng. J., 7, 203 (1961)]: 4ηt L(1 − εv) Nt = ᎏᎏ (17-16) πDb

Isolated fiber efficiency for combined inertia and interception mechanisms at NRe = 0.2. [Chen, Chem. Rev., 55, 595 (1955).]

FIG. 17-63

17-50

GAS-SOLID OPERATIONS AND EQUIPMENT

The foregoing procedure is commonly employed in reverse to determine or confirm fiber target efficiencies from the experimentally determined efficiencies of fibrous filter pads. Filtration theory assumes that a dust particle that touches a collector body adheres to it. This assumption appears to be valid in most cases, but evidence of nonadherence, or particle bouncing, has appeared in some instances. Wright et al. (“High Velocity Air Filters,” WADC TR 55-457, ASTIA Doc. AD-142075, 1957) investigated the performance of fibrous filters at filtration velocities of 0.091 to 3.05 m/s (0.3 to 10 ft/s), using 0.3-µm and 1.4-µm supercooled liquid aerosols and a 1.2-µm solid aerosol. The collection efficiencies agreed well with theoretical predictions for the liquid aerosols and apparently also for the solid aerosol at filtration velocities under 0.3 m/s (1 ft/s). But at filtration velocities above 0.3 m/s some of the solid particles failed to adhere. With a filter composed of 30-µm glass fibers and a filtration velocity of 9.1 m/s (30 ft/s), there were indications that 90 percent of the solid aerosol particles striking a fiber bounced off. Bouncing may be regarded as a defect in the particle-deposition process. However, particles that have been deposited in filters may subsequently be blown off and reentrained into the air stream (Corn, “Adhesion of Particles,” in Davies, Aerosol Science, Academic, New York, 1966; and Davies, op. cit.). The theories of filtration by a fibrous filter relate only to the initial efficiency of the clean filter in the “static” period of filtration before the deposition of any appreciable quantity of dust particles. The deposition of particles in a filter increases the number of targets available to intercept particles, so that collection efficiency increases as the filter loads. At the same time, the filter undergoes clogging and the pressure drop increases. No theory is available for dealing with the “dynamic” period of filtration in which collection efficiency and pressure drop vary with the loading of collected dust. The theoretical treatment of this filtration period is incomparably more complex than that for the “static” period. Investigators have noted that both the increase in collection efficiency and the increase in pressure drop are exponential functions of the loading of collected dust or are at least roughly so (Davies, op. cit.). Some empirical relationships have been derived for correlating data in particular instances. The dust particles collected by a fibrous filter do not deposit in uniform layers on fibers but tend to deposit preferentially on previously deposited particles (Billings, “Effect of Particle Accumulation in Aerosol Filtration,” Ph.D. dissertation, California Institute of Technology, Pasadena, 1966), forming chainlike agglomerates termed “dendrites.” The growth of dendritic deposits on fibers has been studied experimentally [Billings, op. cit.; Bhutra and Payatakes, J. Aerosol Sci., 10, 445 (1979)], and Payatakes and coworkers [Payatakes and Tien, J. Aerosol Sci., 7, 85 (1976); Payatakes, Am. Inst. Chem. Eng. J., 23, 192 (1977); and Payatakes and Gradon, Chem. Eng. Sci., 35, 1083 (1980)] have attempted to model the growth of dendrites and its influence on filter efficiency and pressure drop. Air-Filter Types Air filters may be broadly divided into two classes: (1) panel, or unit, filters; and (2) automatic, or continuous, filters. Panel filters are constructed in units of convenient size (commonly 20- by 20-in or 24- by 24-in face area) to facilitate installation, maintenance, and cleaning. Each unit consists of a cleanable or replaceable cell or filter pad in a substantial frame that may be bolted to the frames of similar units to form an airtight partition between the source of the dusty air and the destination of the cleaned air. Panel filters may use either viscous or dry filter media. Viscous filters are so called because the filter medium is coated with a tacky liquid of high viscosity (e.g., mineral oil and adhesives) to retain the dust. The filter pad consists of an assembly of coarse fibers (now usually metal, glass, or plastic). Because the fibers are coarse and the media are highly porous, resistance to air flow is low and high filtration velocities can be used. Dry filters are usually deeper than viscous filters. The dry filter media use finer fibers and have much smaller pores than the viscous media and need not rely on an oil coating to retain collected dust. Because of their greater resistance to air flow, dry filters must use lower filtration velocities to avoid excessive pressure drops. Hence, dry media must have larger surface areas and are usually pleated or arranged in the form of pockets (Fig. 17-64), generally sheets of cellulose pulp, cotton, felt, or spun glass.

(a) Sectional View

(c) (b) Cutaway View Typical dry filters. (a) Throwaway type, Airplex. (Davies Air Filter Corporation.) (b) Replaceable medium type, Airmat PL-24, cutaway view. (American Air Filter Co., Inc.) (c) Cleanable type, Amirglass sawtooth. (Amirton Company.)

FIG. 17-64

Automatic filters are made with either viscous-coated or dry filter media. However, the cleaning or disposal of the loaded medium is essentially continuous and automatic. In most such devices the air passes horizontally through a movable filter curtain. As the filter loads with dust, the curtain is continuously or intermittently advanced to expose clean media to the air flow and to clean or dispose of the loaded medium. Movement of the curtain can be provided by a hand crank or a motor drive. Movement of a motor-driven curtain can be actuated automatically by a differential-pressure switch connected across the filter. High-Efficiency Air Cleaning Air-filter systems for nuclear facilities and for other applications demanding extremely high standards of air purity require filtration efficiencies well beyond those attainable with the equipment described above. The Nuclear Air Cleaning Handbook (Burchsted et al., op. cit.) presents an extensive treatment of the requirements for and the design of such air-cleaning facilities. Much of the material is pertinent to high-efficiency air-filter systems for applications to other than nuclear facilities. HEPA (high-efficiency particulate air) filters were originally developed for nuclear and military applications but are now widely used and are manufactured by numerous companies. By definition, an HEPA filter is a “throwaway, extended-medium dry-type” filter having (1) a minimum particle-removal efficiency of not less than 99.97 percent for 0.3-µm particles, (2) a maximum resistance, when clean, of 1.0 in water when operated at rated air-flow capacity, and (3) a rigid casing extending the full depth of the medium (Burchsted et al., op. cit.). The filter medium is a paper made of submicrometer glass fibers in a matrix of larger-diameter (1- to 4-µm) glass fibers. An organic binder is added during the papermaking process to hold the fibers and give the paper added tensile strength. Filter units are made in several standard sizes (Table 17-9). TABLE 17-9

Standard HEPA Filters*

Face dimensions, in

Depth, less gaskets, in

Design air-flow capacity at clean-filter resistance of 1.0 in water (standard ft3/min)

24 × 24 24 × 24 12 × 12 8×8 8×8

11a 57⁄ 8 57⁄ 8 57⁄ 8 3g

1000 500 125 20 25

*Burchsted et al., Nuclear Air Cleaning Handbook, ERDA 76-21, Oak Ridge, Tenn., 1976.

GAS-SOLIDS SEPARATIONS Because HEPA filters are designed primarily for high efficiency, their dust-loading capacities are limited, and it is common practice to use prefilters to extend their operating lives. In general, HEPA filters should be protected from (1) lint, (2) particles larger than 1 to 2 µm in diameter, and (3) dust concentrations greater than 23 mg/m3 (10 gr/ 1000 ft3). Air filters used in nuclear facilities as prefilters and buildingsupply air filters are classified as shown in Table 17-10. The standard of the American Society of Heating, Refrigerating and AirConditioning Engineers (Method of Testing Air Cleaning Devices Used in General Ventilation for Removing Particulate Matter, ASHRAE 52-68, 1968) requires both a dust-spot (dust-stain) efficiency test made with atmospheric dust and a weight-arrestance test made with a synthetic test dust. A more precise comparison of the different groups of filters, based on removal efficiencies for particles of specific sizes, is presented in Table 17-11. Table 17-12 presents the relative performance of Group I, II, and III filters with respect to air-flow capacity, resistance, and dustholding capacity. The dust-holding capacities correspond to the manufacturers’ recommended maximum allowable increases in air-flow resistance. The values for dust-holding capacity are based on tests with a synthetic dust and hence are relative. The actual dust-holding capacity in a specific application will depend on the characteristics of the dust encountered. In some instances it may be appropriate to use two or more stages of precleaning in air-filter systems to achieve a desired combination of operating life and efficiency. In very dusty locations, inertial devices such as multiple small cyclones may be used as first-stage separators. Electrical Precipitators When particles suspended in a gas are exposed to gas ions in an electrostatic field, they will become charged and migrate under the action of the field. The functional mechanisms of electrical precipitation may be listed as follows: 1. 2.

Gas ionization Particle collection a. Production of electrostatic field to cause charging and migration of dust particles b. Gas retention to permit particle migration to a collection surface c. Prevention of reentrainment of collected particles d. Removal of collected particles from the equipment

TABLE 17-10

Classification of Common Air Filters*

Group

Efficiency

I

Low

II

Moderate

III

High

HEPA

Extreme

Filter type Viscous impingement, panel type Extended medium, dry type Extended medium, dry type Extended medium, dry type

Stain test efficiency, %

Arrestance, %

<20†

40–80†

20–60†

80–96†

60–98‡

96–99†

100§

100†

*Burchsted et al., Nuclear Air Cleaning Handbook, ERDA 76-21, Oak Ridge, Tenn., 1976. †Test using synthetic dust. ‡Stain test using atmospheric dust. §ASHRAE/52-68, American Society of Heating, Refrigerating and AirConditioning Engineers. TABLE 17-11 Comparison of Air Filters by Percent Removal Efficiency for Various Particle Sizes* Removal efficiency, %, for particle size of Group

Efficiency

0.3 µm

1.0 µm

5.0 µm

10.0 µm

I II III HEPA

Low Moderate High Extreme

0–2 10–40 45–85 99.97 min

10–30 40–70 75–99 99.99

40–70 85–95 99–99.9 100

90–98 98–99 99.9 100

*Burchsted et al., Nuclear Air Cleaning Handbook, ERDA 76-21, Oak Ridge, Tenn., 1976.

17-51

TABLE 17-12 Air-Flow Capacity, Resistance, and Dust-Holding Capacity of Air Filters*

Group

Efficiency

Air-flow capacity, ft3/(min⋅ft2 of frontal area)

I II III

Low Moderate High

300–500 250–750 250–750

Clean filter

Used filter

Dust-holding capacity, g/(1000 ft3⋅min of air-flow capacity)

0.05–0.1 0.1–0.5 0.20–0.5

0.3–0.5 0.5–1.0 0.6–1.4

50–1000 100–500 50–200

Resistance, in water

*Burchsted et al., Nuclear Air Cleaning Handbook, ERDA 76-21, Oak Ridge, Tenn., 1976.

There are two general classes of electrical precipitators: (1) singlestage, in which ionization and collection are combined; (2) two-stage, in which ionization is achieved in one portion of the equipment, followed by collection in another. Various types in each class differ essentially in the details by which each function is accomplished. The underlying theory presented in the following paragraphs assumes that the dust concentration is small, since only very incomplete evaluations for conditions of high dust concentration have been made. Field Strength Whereas the applied potential or voltage is the quantity commonly known, it is the field strength that determines behavior in an electrostatic field. When the current flow is low (i.e., before the onset of spark or corona discharge), these are related by the following equations for two common forms of electrodes: Parallel plates: Ᏹ = E/Be

(17-17)

Concentric cylinders (wire-in-cylinder): E Ᏹ = ᎏᎏ r ln (Dt /Dd)

(17-18)

The field strength is uniform between parallel plates, whereas it varies in the space between concentric cylinders, being highest at the surface of the central cylinder. After corona sets in, the current flow will become appreciable. The field strength near the center electrode will be less than given by Eq. (17-18) and that in the major portion of the clearance space will be greater and more uniform [see Eqs. (17-23) and (17-24)]. Potential and Ionization In order to obtain gas ionization it is necessary to exceed, at least locally, the electrical breakdown strength of the gas. Corona is the name applied to such a local discharge that fails to propagate itself. Sparking is essentially an advanced stage of corona in which complete breakdown of the gas occurs along a given path. Since corona represents a local breakdown, it can occur only in a nonuniform electrical field (Whitehead, Dielectric Phenomena— Electrical Discharge in Gases, Van Nostrand, Princeton, N.J., 1927, p. 40). Consequently, for parallel plates, only sparking occurs at a field strength or potential difference given by the empirical expressions

  

Ko Ᏹs = Ᏹo kρ 1 + ᎏ kρBe

(17-19)

Es = ᏱokρBe + KoᏱo

(17-20)

For air in the range of kρ Be from 0.1 to 2, Ᏹo = 111.2 and Ko = 0.048. Thornton [Phil. Mag., 28(7), 666 (1939)] gives values for other gases. For concentric cylinders (Loeb, Fundamental Processes of Electrical Discharge in Gases, Wiley, New York, 1939; Peek, Dielectric Phenomena in High-Voltage Engineering, McGraw-Hill, New York, 1929; and Whitehead, op. cit.), corona sets in at the central wire when K ᎏ 

kD

Ᏹc = Ᏹokρ 1 +



Ᏹo kρDd Ec = ᎏ 2

o

ρ

(17-21)

d

K D ᎏ  ln  ᎏ   1 +

kD D o

ρ

t

d

d

(17-22)

17-52

GAS-SOLID OPERATIONS AND EQUIPMENT

TABLE 17-13 in Pipe)

Sparking Potentials* (Small Wire Concentric Sparking potential,† volts

Pipe diameter, in

Peak

Root mean square

4 6 9 12

59,000 76,000 90,000 100,000

45,000 58,000 69,000 77,000

*Data reported by Anderson in Perry, “Chemical Engineers’ Handbook,” 2d ed., p. 1873, McGraw-Hill, New York, 1941. †For gases at atmospheric pressure, 100°F, containing water vapor, air, CO2, and mist, and negative-discharge-electrode polarity.

For air approximate values are Ᏹo = 110, Ko = 0.18. Corona, however, will set in only if (Dt /Dd) > 2.718. If this ratio is less than 2.718, no corona occurs, and only sparking will result, following the laws given by Eqs. (17-21) and (17-22) (Peek, op. cit.). In practice, precipitators are usually operated at the highest voltage practicable without sparking, since this increases both the particle charge and the electrical precipitating field. The sparking potential is generally higher with a negative charge on the discharge electrode and is less erratic in behavior than a positive corona discharge. It is the consensus, however, that ozone formation with a positive discharge is considerably less than with a negative discharge. For these reasons negative discharge is generally used in industrial precipitators, and a positive discharge is utilized in air-conditioning applications. In Table 17-13 are given some typical values for the sparking potential for the case of small wires in pipes of various sizes. The sparking potential varies approximately directly as the density of the gas but is very sensitive to the character of any material collected on the electrodes. Even small amounts of poorly conducting material on the electrodes may markedly lower the sparking voltage. For positive polarity of the discharge electrode, the sparking voltage will be very much lower. The sparking voltage is greatly affected by the temperature and humidity of the gas, as shown in Fig. 17-65. Current Flow Corona discharge is accompanied by a relatively small flow of electric current, typically 0.1 to 0.5 mA/m2 of collectingelectrode area (projected, rather than actual area). Sparking usually involves a considerably larger flow of current which cannot be tolerated except for occasional periods of a fraction of a second duration, and then only when suitable electrical controls are provided to limit the current. However, when suitable controls are provided, precipitators have been operated continuously with a small amount of sparking

to ensure that the voltage is in the correct range to ensure corona. Besides disruptive effects on the electrical equipment and electrodes, sparking will result in low collection efficiency because of reduction in applied voltage, redispersion of collected dust, and current channeling. Although an exact calculation can be made for the current flow for a direct-current potential applied between concentric cylinders, the following simpler expression, based on the assumption of a constant space charge or ion density, gives a good approximation of corona current [Ladenburg, Ann. Phys., 4(5), 863 (1930)]: 8λiE(E − Ec) I = ᎏᎏ D2t ln (Dt /Dd)

(17-23)

and the average space charge is given by (Whitehead, op. cit.) 4(E − Ec) σavg = ᎏᎏ πD2t ε

(17-24)

In the space outside the immediate vicinity of corona discharge, the field strength is sensibly constant, and an average value is given by
I/
λ
i
Ᏹ = 2

(17-25)

which applies if the potential difference is above the critical potential required for corona discharge so that an appreciable current flows. Ionic mobilities are given by Loeb (International Critical Tables, vol. 6, McGraw-Hill, New York, 1929, p. 107). For air at 0°C, 760 mmHg, λi = 624 (cm/s)/(statV/cm) for negative ions. Positive ions usually have a slightly lower mobility. Loeb (Basic Processes of Gaseous Electronics, University of California Press, Berkeley and Los Angeles, 1955, p. 53) gives a theoretical expression for ionic mobility of gases which is probably good to within ⫾50 percent: 100.0 λi = ᎏᎏ
g
−

)M kρ(δ 1


(17-26)

In general, ionic mobilities are inversely proportional to gas density. Ionic velocities in the usual electrostatic precipitator are on the order of 30.5 m/s (100 ft/s). Electric Wind By virtue of the momentum transfer from gas ions moving in the electrical field to the surrounding gas molecules, a gas circulation, known as the “electric” or “ionic” wind, is set up between the electrodes. For conditions encountered in electrical precipitators, the velocity of this circulation is on the order of 0.6 m/s (2 ft/s). Also, as a result of this momentum transfer, the pressure at the collecting electrode is slightly higher than at the discharge electrode (Whitehead, op. cit., p. 167). Charging of Particles [Deutsch, Ann. Phys., 68(4), 335 (1922); 9(5), 249 (1931); 10(5), 847 (1931); Ladenburg, op. cit.; and Mierdel, Z. Tech. Phys., 13, 564 (1932).] Three forces act on a gas ion in the vicinity of a particle: attractive forces due to the field strength and the ionic image; and repulsive forces due to the Coulomb effect. For spherical particles larger than 1-µm diameter, the ionic image effect is negligible, and charging will continue until the other two forces balance according to the equation πσεtλ i ζ Ᏹ Dp2 No = ᎏ ᎏᎏ (17-27) 1 + πσελ i t 4ε The ultimate charge acquired by the particle is given by





No = ζᏱDp2 /4ε



(17-28)

and is very nearly attained in a fraction of a second. For particles smaller than 1-µm diameter, the initial charging will occur according to Eq. (17-27). However, owing to the ionic-image effect, the ultimate charge will be considerably greater because of penetration resulting from the kinetic energy of the gas ions. For charging times of the order encountered in electrical precipitation, the ultimate charge acquired by spherical particles smaller than about 1-µm diameter may be approximated (⫾30 percent) by the empirical expression No = 3.4 × 103DpT Sparking potential for negative point-to-plane a-in (1.3-cm) gap as a function of moisture content and temperature of air at 1-atm (101.3-kPa) pressure. [Sproull and Nakada, Ind. Eng. Chem., 43, 1356 (1951).]

FIG. 17-65

(17-29)

Values of No for various sized particles are listed in Table 17-14 for 70°F, ζ = 2, and Ᏹ = 10 statV/cm. Particle Mobility By equating the electrical force acting on a

GAS-SOLIDS SEPARATIONS TABLE 17-14 Charge and Motion of Spherical Particles in an Electric Field For ζ = 2, and ε = εi = εp = 10 statV/cm

TABLE 17-15 Performance Data on Typical Single-Stage Electrical Precipitator Installations*

Particle diam., µ

Number of elementary electrical charges, N0

Particle migration velocity,* ue, ft/sec

0.1 .25 .5 1.0 2.5 5.0 10.0 25.0

10 25 50 105 655 2,620 10,470 65,500

0.27 .15 .12 .11 .26 .50 .98 2.40

NOTE:

particle to the resistance due to air friction, as expressed by Stokes’ law, the particle velocity or mobility may be expressed by 1. For particles larger than 1-µm diameter: ζDpᏱiKm ue λp = ᎏ = ᎏᎏ (17-30) 12πµ Ᏹp 2. For particles smaller than 1-µm diameter: 360KmεT ue (17-31) λp = ᎏ = ᎏ µ Ᏹp

 

 

For single-stage precipitators, Ᏹi and Ᏹp may be considered as essentially equal. It is apparent from Eq. (17-31) that the mobility in an electric field will be almost the same for all particles smaller than about 1-µm diameter, and hence, in the absence of reentrainment, collection efficiency should be almost independent of particle size in this range. Very small particles will actually have a greater mobility because of the Stokes-Cunningham correction factor. Values of ue are listed in Table 17-14 for 70°F, ζ = 2, and Ᏹ = Ᏹi = Ᏹp = 10 statV/cm. Collection Efficiency Although actual particle mobilities may be considerably greater than would be calculated on the basis given in the preceding paragraph because of the action of the electric wind in single-stage precipitators, the latter acts in a compensating fashion, and the overall effect of the electric wind is probably to provide an equalization of particle concentration between the electrodes similar to the action of normal turbulence (Mierdel, op. cit.). On this basis Deutsch (op. cit.) has derived the following equations for collection efficiency, the form of which had previously been suggested by Anderson on the basis of experimental data: (17-32)

For the concentric-cylinder (or wire-in-cylinder) type of precipitator, K e = 4Le /DtVe; for rod-curtain or wire-plate types, K e = Le /BeVe. Strictly speaking, Eq. (17-32) applies only for a given particle size, and the overall efficiency must be obtained by an integration process for a specific dust distribution, as described in the subsection “Cyclone Separators.” However, over limited ranges of performance conditions, Eq. (17-32) has been found to give a good approximation of overall collection efficiency, with the term for particle migration velocity representing an empirical average value. Such values, calculated from overall collection-efficiency measurements, are given in Table 17-15 for specific installations. For two-stage precipitators with close collecting-plate spacings (Figs. 20-152, 20-153), the gas flow is substantially streamline, and no electric wind exists. Consequently, with reentrainment neglected, collection efficiency may be expressed as [Penny, Electr. Eng, 56, 159 (1937)] η = ue L e /Ve Be

(17-33)

which holds for values of η ⬉ 1.0. In practice, however, extraneous factors may cause the actual efficiency to approach a relationship of the type given by Eq. (17-32). Application The theoretical considerations that have been expounded should be used only for order-of-magnitude estimates, since a number of extraneous factors may enter into actual performance. In actual installations rectified alternating current is em-

Gas volume, cu ft/ min

Average gas velocity, ft/sec

Collecting electrode area, sq ft

Over-all collection efficiency, %

Average particle migration velocity, ft/sec

Type of precipitator

Type of dust

Rod curtain

Smelter fume Gypsum from kiln Fly ash

180,000

108,000

6

10,900

91

.40

Cement

204,000

9.5

26,000

91

.31

Tulip type

To convert feet per second to meters per second, multiply by 0.3048.

η = 1 − e−(ueAe/q) = 1 − e−Keue

17-53

Perforated plate Rod curtain

25,000

6 3.5

44,400 3,800

85 99.7

0.13 .64

*Research-Cottrell, Inc. To convert cubic feet per minute to cubic meters per second, multiply by 0.00047; to convert feet per second to meters per second, multiply by 0.3048; and to convert square feet to square meters, multiply by 0.0929.

ployed. Hence the electric field is not fixed but varies continuously, depending on the waveform of the rectifier, although Schmidt and Anderson [Electr. Eng., 57, 332 (1938)] report that the waveform is not a critical factor. Allowances for high dust concentrations have not been fully studied, although Deutsch (op. cit.) has presented a theoretical approach. In addition, irregularities on the discharge electrode will result in local discharges. Such irregularities can readily result from dust incrustation on the discharge electrodes due to charging of particles with opposite polarity within the thin but appreciable flow or ionization layer surrounding this electrode. Very high dust loadings increase the potential difference required for corona and reduce the current due to the space charge of the particles. This tends to reduce the average particle charge and reduces collection efficiency. This can be compensated for by increasing the potential difference when high dust loadings are involved. Several investigators have attempted to modify the basic Deutsch equation so that it would more nearly describe precipitator performance. Cooperman (“A New Theory of Precipitator Efficiency,” Pap. 69-4, APCA meeting, New York, 1969) introduced correction factors for diffusional forces arising from variations in particle concentration along the precipitator length and also perpendicular to the collecting surface. Robinson [Atmos. Environ. 1(3), 193 (1967)] derived an equation for collection efficiency in which two erosion or reentrainment terms are introduced. An analysis of precipitator performance based on theoretical considerations was undertaken by the Southern Research Institute for the National Air Pollution Control Administration (Nichols and Oglesby, “Electrostatic Precipitator Systems Analysis,” AIChE annual meeting, 1970). A mathematical model was developed for calculating the particle charge, electric field, and collection efficiency based on the Deutsch-Anderson equation. The system diagram is shown in Fig. 17-66. This system-analysis method, using high-speed computers, makes it possible to analyze what takes place in each increment of precipitator length. Collection efficiency versus particle size is computed for each 1 ft (0.3 m) of gas travel, and the inlet particle-size distribution is modified accordingly. Computed overall efficiencies compare well with measured values on three precipitators. The model assumes that field charging is the only charging mechanism. The authors considered the addition of several refinements to the program: the influence of diffusion charging; reentrainment effects due to rapping and erosion; and loss of efficiency due to maldistribution of gas, dust resistivity, and gas-property effects. The modeling technique appeared promising, but much more work was needed before it could be used for design. The same authors prepared a general treatise (Oglesby and Nichols, A Manual of Electrostatic Precipitator Technology, parts I and II, Southern Research Institute, Birmingham, Ala., U.S. Government Publications PB196360, 196381, 1970). High-Pressure-High-Temperature Electrostatic Precipitation In general, increased pressure increases precipitation efficiency, although a somewhat higher potential is required, because it reduces ion mobility and hence increases the potential required for corona

17-54

GAS-SOLID OPERATIONS AND EQUIPMENT

Electrostatic-precipitator-system model. (Nichols and Oglesby, “Electrostatic Precipitator Systems Analysis,” AIChE annual meeting, 1970.)

FIG. 17-66

B Ec ᎏ = A ᎏᎏ ρ′
d
ρ
′/
2
D

and sparking. Increased temperature reduces collection efficiency because ion mobility is increased, lowering critical potentials, and because gas viscosity is increased, reducing migration velocities. Precipitators have been operated at pressures up to 5.5 MPa (800 psig) and temperatures to 800°C. The effect of increasing gas density on sparkover voltage has been investigated by Robinson [J. Appl. Phys., 40, 5107 (1969); Air Pollution Control, part 1, Wiley-Interscience, New York, 1971, chap. 5]. Figure 17-67 shows the effect of gas density on corona-starting and sparkover voltages for positive and negative corona in a pipe precipitator. The sparkover voltages are experimental and are given by the solid points. Experimental corona-starting voltages are given by the hollow points. The solid lines are corona-starting voltage curves calculated from Eq. (17-33). This is an empirical relationship developed by Robinson.

Ec is the corona-starting field, kV/cm. ρ′ is the relative gas density, equal to the actual gas density divided by the density of air at 25°C, 1 atm. Dd is the diameter of the ionizing wire, cm. A and B are constants which are characteristics of the gas. In dry air, A = 32.2 kV/cm and B = 8.46 kV/cm1/2. Agreement between experimental and calculated starting voltages is good for the case of positive corona, but in the case of negative corona the calculated line serves as an upper limit for the data. This lower-than-expected starting-voltage characteristic of negative corona is confirmed by Hall et al. [Oil Gas J., 66, 109 (1968)] in a report of an electrostatic precipitator which removes lubricating-oil mist from natural gas at 5.5 MPa (800 psig) and 38°C (100°F). The use of electrostatic precipitators at elevated pressure is expected to in-

(a) Positive Corona

(b) Negative Corona

(17-34)

Corona-starting and sparkover voltages for coaxial wire-pipe electrodes in air (25°C). Dt and Dd are the respective pipe and wire diameters. The voltage is unvarying direct current. (Robinson, Air Pollution Control, part 1, Wiley-Interscience, New York, 1971, chap. 5.) FIG. 17-67

GAS-SOLIDS SEPARATIONS

17-55

crease, because the method requires very low pressure drop [approximately 69 Pa (0.1 lbf/in2)]. This results from the fact that the electric separation forces are applied directly to the particles themselves rather than to the entire mass of the gas, as in inertial separators. The use of electrostatic precipitators at temperatures up to 400°C is well developed for the powerhouse fly-ash application, but in the range of 600 to 800°C they are still in the experimental phase. The U.S. Bureau of Mines has tested a pilot-scale tubular precipitator for fly ash. See Shale [Air Pollut. Control Assoc. J., 17, 159 (1967)] and Shale and Fasching (Operating Characteristics of a High-Temperature Electrostatic Precipitator, U.S. Bur. Mines Rep. 7276, 1969). It operated over a temperature range of 27 to 816°C (80 to 1500°F) and a pressure range of 552 kPa (35 to 80 psig). Initial collection efficiencies ranged from 90 to 98 percent at 793°C (1460°F), 552 kPa (80 psig), but continuous operation was not achieved because of excessive thermal expansion of internal parts. Resistivity Problems Optimum performance of electrostatic precipitators is achieved when the electrical resistivity of the collected dust is sufficiently high to result in electrostatic pinning of the particles to the collecting surface, but not so high that dielectric breakdown of the dust layer occurs as the corona current passes through it. The optimum resistivity range is generally considered to be from 108 to 1010 Ω⋅cm, measured at operating conditions. As the dust builds up on the collecting electrode, it impedes the flow of current, so that a voltage drop is developed across the dust layer: Ed = jρdLd

(17-35)

If Ed /Ld exceeds the dielectric strength of the dust layer, sparks occur in the deposit and form back-corona craters. Ions of both polarities are formed. Positive ions formed in the craters are attracted to the negatively charged particles in the gas stream, whose charge level is reduced so that collection efficiency decreases. Some of the positive ions neutralize part of the negative-space-charge cloud normally present near the wire, thereby increasing total current. Collection efficiency under these conditions will not correlate with total power input (Owens, E. I. du Pont de Nemours & Co. internal communication, 1971). Under normal conditions, collection efficiency is an exponential function of corona power (White, Industrial Electrostatic Precipitation, Addison-Wesley, Reading, Mass., 1963). With typical ion density in the range of 109/cm3, overall voltage gradient would be about 4000 V/cm, and current about 1 µA/cm2. Dielectric breakdown of the dust layer (at about 10,000 V/cm) would therefore be expected for dusts with resistivities above 1010 Ω⋅cm. Problems due to high resistivity are of great concern in fly-ash precipitation because air-pollution regulations require that coals have low (<1 percent) sulfur content. Figure 17-68 shows that the resistivity of low-sulfur coal ash exceeds the threshold of 1010 Ω⋅cm at common operating temperatures. This has resulted in the installation of a number of precipitators which have failed to meet guaranteed performance. This has occurred to an alarming extent in the United States but has also been encountered in Australia, where the sulfur content is typically 0.3 to 0.6 percent. Maartmann (Pap. EN-34F, 2d International Clean Air Congress, Washington, 1970) reports the installation of a number of precipitators which performed below guarantees, so that the Electricity Commission of New South Wales decided that each manufacturer wishing to bid on a new station must first make pilot tests to prove performance on the actual coal to be burned in that station. Problems of back corona and excessive sparking with lowsulfur coal usually require that the operating voltage be reduced. This reduces the migration velocity and leads to larger precipitators. Ramsdell (Design Criteria for Precipitators for Modern Central Station Power Plants, American Power Conference, Chicago, 1968) developed the curves in Fig. 17-69. They show the results of extensive field tests by the Consolidated Edison Co. In another paper (“Antipollution Program of Consolidated Edison Co. of New York,” ASCE, May 13–17, 1968), Ramsdell traces the remarkable growth in the size of precipitators required for high efficiency on low-sulfur coals. The culmination of this work was the precipitator at boiler 30 at Ravenswood Station, New York. Resistivity problems were avoided by operating at high temperature [343°C (650°F)]. The mechanical

Trends in resistivity of fly ash with variations in flue-gas temperature and coal sulfur content. °C = (°F − 32) × 5⁄ 9. (Oglesby and Nichols, A Manual of Electrostatic Precipitator Technology, part II, Southern Research Institute, Birmingham, Ala., 1970.)

FIG. 17-68

(cyclone) collector was installed after the precipitator to clean up puffs due to rapping. Maartmann (op. cit.) agrees that sulfur content is important but feels that it should not be the sole criterion for the determination of collecting surface. He points to specific collecting-surface requirements as high as 500 ft2/(1000 ft3⋅min) for 95 percent collection efficiency with high-resistivity Australian ash. Schmidt and Anderson (op. cit.) and Anderson [Physics, 3, 23 (July 1932)] claim that resistivity of the collected dust may be a controlling factor which is very sensitive to moisture. They state that an increase in relative humidity of 5 percent may double the precipitation rate because of its effect on the conductivity of the collected dust layer. Conditioning agents have been added to the flue gas to alter dust resistivity. Steam, sodium chloride, sulfur trioxide, and ammonia have all been successfully used. Research by Chittum and others [Schmidt, Ind. Eng. Chem., 41, 2428 (1949)] led to a theory of conditioning by alteration of the moisture-adsorption properties of dust surfaces. Chittum proposed that an intermediate chemical-adsorption film, which was strongly bound to the particle and which in turn strongly adsorbed water, would be an effective conditioner. This explains how acid conditioners, such as SO3, help resistivity problems associated with basic dusts, such as many types of fly ash, whereas ammonia is a good additive for acidic dusts such as alumina. Moisture alone can be used as a conditioning agent. This is shown in Fig. 17-65. Moisture is beneficial in two ways: it reduces the electrical resistivity of most dusts (an exception is powdered sulfur, which apparently does not absorb water), and it increases the voltage which may safely be employed without sparking, as shown in Fig. 17-65. Low resistivity can sometimes be a problem. If the resistivity is below 104 Ω⋅cm, the collected particles are so conductive that their charges leak to ground faster than they are replenished by the corona. The particles are no longer electrostatically pinned to the plate, and

17-56

GAS-SOLID OPERATIONS AND EQUIPMENT

(a)

(b)

Design curves for electrostatic precipitators for fly ash. Collection efficiency for various levels of percent sulfur in coal versus (a) specific collecting surface, and (b) bus sections per 100,000 ft3/min (4.7 m3/s). °C = (°F − 32) × 5⁄ 9. (Ramsdell, Design Criteria for Precipitators for Modern Central Station Power Plants, American Power Conference, Chicago, Ill., 1968.) FIG. 17-69

they may then be swept away and reentrained in the exit gas. The particles may even pick up positive charges from the collecting plate and then be repelled. Low-resistivity problems are common with dusts of high carbon content and may also occur in fly-ash precipitators which handle the ash from high-sulfur coal and operate at low gas temperatures. Low resistivity in this case results from excessive condensation of electrically conductive sulfuric acid. Single-Stage Precipitators The single-stage type of unit, commonly known as a Cottrell precipitator, is most generally used for dust or mist collection from industrial-process gases. The corona discharge is maintained throughout the precipitator and, besides providing initial ionization, serves to prevent redispersion of precipitated dust and recharges neutralized or discharged particle ions. Cottrell precipitators may be divided into two main classes, the so-called plate type (Fig. 17-70), in which the collecting electrodes consist of parallel plates, screens, or rows of rods, chains, or wires; and the pipe type (Fig. 17-71), in which the collecting electrodes consist of a nest of parallel pipes which may be square, round, or of any other shape. The discharge or precipitating electrodes in each case are wires or rods, either round or edged, which are placed midway between the collecting electrodes or in the center of the pipes and may be either parallel

Horizontal-flow plate precipitator used in a cement plant. (Western Precipitation Division, Joy Manufacturing Company.)

or perpendicular to the gas flow in the case of plate precipitators. When the collecting electrodes are screens or rows of rods or wires, the gases are usually passed parallel to the plane of each but may also be passed through it. In pipe precipitators, the gas flow is generally vertical up through the pipe, although downflow is not unusual. The pipe-type precipitator is usually used for the removal of liquid particles and volatilized fumes [Beaver, op. cit.; and Cree, Am. Gas J., 162, 27 (March 1945)], and the plate type is used mainly on dusts. In the pipe type, the discharge electrodes are usually suspended from an insulated support and kept taut by a weight at the bottom. Cree (op.

FIG. 17-70

FIG. 17-71

Blast-furnace pipe precipitator. (Research-Cottrell, Inc.)

GAS-SOLIDS SEPARATIONS

FIG. 17-72 Two-stage water-film pipe precipitator. (Western Precipitation Division, Joy Manufacturing Company.)

cit.) discusses the application of electrical precipitators to tar removal in the gas industry. Rapping Except when liquid dispersoids are being collected or, in the case of film precipitators, when a liquid is circulated over the

17-57

collecting-electrode surface (Fig. 17-72), thus continuously removing the precipitated material, the collected dust is dislodged from the electrodes either periodically or continuously by mechanical rapping or scraping, which may be performed automatically or manually. Automatic rapping with either impact-type or vibrator-type rappers is common practice. White (op. cit.) recommends fairly continuous rapping with magnetic-impulse rappers. Rapping with excessive force leads to dust reentrainment and possible mechanical failure of the plates, while insufficient rapping leads to excessive dust buildup with poor electrical operation and reduced collection efficiency. Intermittent rapping at intervals of an hour or more causes heavy puffs of reentrained dust. Sproull [Air Pollut. Control Assoc. J., 15, 50 (1965)] reports the importance of electrode acceleration and shows that it varies with the type of dust, whether the electrode is rapped perpendicularly (normally) to the plate or parallel to it. Figure 17-73 shows the accelerations required for rapping normally to the plate. Difficult dusts may require as much as 100 G acceleration for 90 percent removal, and even higher accelerations are required when the vibrating force is applied in the plane of the plate. Perforated-plate or rod-curtain precipitators are frequently rapped without shutting off the gas flow and with the electrodes energized. This procedure, however, results in a tendency for reentrainment of collected dust. Sectional or composite-plate collecting electrodes (sometimes known as hollow, pocket, or tulip electrodes) are used to minimize this tendency in the continuous removal of the precipitated material, provided that it is free-flowing. These are generally designed for vertical gas flow and comprise a collecting electrode containing a dead air space and provided with horizontal protruding slots that guide the dust into this space (see Fig. 17-74), although some types use horizontal flow. The choice of size, shape, and type of electrode is based on economic considerations and is usually determined by the characteristics of the gas and suspended matter and by mechanical considerations such as flue arrangement, the available space, and previous experience with the electrodes on similar problems. The spacing between collecting electrodes in plate-type precipitators and the pipe diameter

FIG. 17-73 Normal (perpendicular) rapping efficiency for various precipitated dust layers having about 0.03 g dust/cm2 (0.2 g dust/in2) as a function of maximum acceleration in multiples of g. Curve 1, fly ash, 200 or 300°F, power off. Curve 2, fly ash, 70°F, power off; also 200 or 300°F, power on. Curve 3, fly ash, 70°F, power on. Curve 4, cement-kiln feed, 300°F, power off. Curve 5, cement dust, 300°F, power off. Curve 6, same as 5, except power on. Curve 7, cement-kiln feed, 300°F, power on. Curve 8, cement dust, 200°F, power off. Curve 9, same as 8, except power on. Curve 10, cement-kiln feed, 200°F, power off. Curve 11, same as 10, except at 70°F. Curve 12, cement-kiln feed, 200°F, power on. Curve 13, cement-kiln feed, 70°F, power on. °C = (°F − 32) × 5⁄ 9. [Sproull, Air Pollut. Control Assoc. J., 15, 50 (1965).]

17-58

GAS-SOLID OPERATIONS AND EQUIPMENT

Vertical-flow heavy-duty plate precipitator. (Western Precipitation Division, Joy Manufacturing Company.)

FIG. 17-74

in pipe-type precipitators usually ranges from 15 to 38 cm (6 to 15 in). The smaller the spacing, the lower the necessary voltage and overall equipment size, but the greater the difficulties involved in maintaining proper alignment and resulting from disturbances due to collected material. Large spacings are usually associated with high dust concentration in order to minimize sparkover due to dust buildup. For very high dust concentrations, such as those encountered in fluid-catalyst plants, it is advantageous to use greater spacings in the first half of the precipitator than in the second half. Precipitators, especially of the plate type, are frequently built with groups of collecting electrodes in series in a common housing. Collecting electrodes are generally on the order of 0.9 to 1.8 m (3 to 6 ft) wide and 3 to 5.5 m (10 to 18 ft) high in plate-type precipitators and 1.8 to 4.6 m (6 to 15 ft) high in pipe types. It is essential for good collection efficiency that the gas be evenly distributed across the various electrode elements. Although this can be achieved by proper gas-inlet transitions and guide vanes, perforated plates or screens located on the upstream side of the electrodes are generally used for distribution. Perforated plates or screens located on the downstream side may be used in special cases. Electrical precipitators are generally designed for collection efficiency in the range of 90 to 99.9 percent. It is essential, however, that the units be properly maintained in order to achieve the required collection efficiency. Electric power consumption is generally 0.2 to 0.6 kW/(1000 ft3⋅min) of gas handled, and the pressure drop across the precipitator unit is usually less than 124 Pa (0.5 in water), ranging from 62 to 248 Pa (d to 1 in) and representing primarily distributor and entrance-exit losses. Applied potentials range from 30,000 to 100,000 V. Gas velocities and retention times are generally in the range of 0.9 to 3 m/s (3 to 10 ft/s) and 1 to 15 s respectively. Velocities are kept low in conventional precipitators to avoid reentrainment of dust. There are, however, precipitator installations on carbon black in which the precipitator acts to flocculate the dust so that it may be subsequently collected in multiple small-diameter cyclone collectors. By not attempting to collect the particles in the precipitator, higher velocities may be used with a correspondingly lower investment cost. Power Supply Electrical precipitators are generally energized by rectified alternating current of commercial frequency. The voltage is stepped up to the required value by means of a transformer and then

rectified. The rectifying equipment has undergone an evolution which began with the synchronous mechanical rectifier in 1904 and was followed by mercury-vapor rectifiers in the 1920s; the first solid-state selenium rectifiers were introduced about 1939. Silicon rectifiers are the latest and most widely used type, since they provide high efficiency and reliability. Automatic controls commonly are tied to voltage, current, spark rate, or some combination of these parameters. Modern precipitators use control circuits similar to those on Fig. 17-75. A high-voltage silicon rectifier is used together with a saturable reactor and means for limiting current and controlling voltage and/or spark rate. One popular method adjusts the voltage to give a specified sparking frequency (typically 50 to 150 sparks per minute per bus section). Half-wave rectification is sometimes used because of its lower equipment requirements and power consumption. It also has the advantage of longer decay periods for sparks to extinguish between current pulses. Electrode insulators must also be designed for a particular service. The properties of the dust or mist and gas determine their design as well as the physical details of the installation. Conducting mists require special allowances such as oil seals, energized shielding cups, or air bleeds. With saturated gas, steam coils are frequently used to prevent condensation on the electrodes. Typical applications in the chemical field (Beaver, op. cit.) include detarring of manufactured gas, removal of acid mist and impurities in contact sulfuric acid plants, recovery of phosphoric acid mists, removal of dusts in gases from roasters, sintering machines, calciners, cement and lime kilns, blast furnaces, carbon-black furnaces, regenerators on fluid-catalyst units, chemical-recovery furnaces in soda and sulfate pulp mills, and gypsum kettles. Figure 17-74 shows a verticalflow steel-plate-type precipitator similar to a type used for catalystdust collection in certain fluid-catalyst plants. A development of interest to the chemical industry is the tubular precipitator of reinforced-plastic construction (Wanner, Gas Cleaning Plant after TiO2 Rotary Kilns, technical bulletin, Lurgi Corp., Frankfurt, Germany, 1971). Tubes made of polyvinyl chloride plastic are reinforced on the outside with polyester-fiber glass. The use of modern economical materials of construction to replace high-maintenance materials such as lead has been long awaited for corrosive applications. Electrical precipitators are probably the most versatile of all types of dust collectors. Very high collection efficiencies can be obtained regardless of the fineness of the dust, provided that the precipitators are given proper maintenance. The chief disadvantages are the high initial cost and, in some cases, high maintenance costs. Furthermore, caution must be exercised with dusts that are combustible in the carrier gas. Two-Stage Precipitators In two-stage precipitators, corona discharge takes place in the first stage between two electrodes having a nonuniform field (see Fig. 17-76). This is generally obtained by a finewire discharge electrode and a large-diameter receiving electrode. In this stage the potential difference must be above that required for corona discharge. The second stage involves a relatively uniform electrostatic field in which charged particles are caused to migrate to a collecting surface. This stage usually consists of either alternately charged parallel plates or concentric cylinders with relatively close clearances compared with their diameters. The only voltage requirement in this stage is that no sparking occur, though higher voltages will result in increased collection efficiency. Since collection occurs in the absence of corona discharge, there is no way of recharging reentrained and discharged particles. Consequently, some means must be provided for avoiding reentrainment of particles from the collecting surface. It is also essential that there be sufficient time and mixing between the first and second stages to secure distribution of gas ions across the gas stream and proper charging of the dust particles. A unit is available in which electrostatic precipitation is combined with a dry-air filter of the type shown in Fig. 17-64b. In another unit an electrostatic field is superimposed on an automatic filter. In this case the ionizer wires are located on the leading face of the unit, and the collecting electrodes consist of alternate stationary and rotating parallel plates. Cleaning in this case is automatic and continuous. Although intended primarily for air-conditioning applications, these units have been successfully applied to the collection of relatively nonconducting mists such as oil. However, other process applications

GAS-SOLIDS SEPARATIONS

(a)

(b) Schematic circuits for silicon rectifier sets with saturable reactor control. (a) Full-wave silicon rectifier. (b) Half-wave silicon rectifier. (White, Industrial Electrostatic Precipitation, Addison-Wesley, Reading, Mass., 1963.) FIG. 17-75

have been limited largely to experimental installations. The large cost advantage of these units over the Cottrell precipitator lies in the smaller equipment size made possible by the close plate spacing, in the lower power consumption due to the two-stage operation, and primarily in the mass production of standardized units. In process applications, the close plate spacing is objectionable because of the relatively high dust concentrations involved. Special material or weight requirements for the structural members may eliminate the mass-production advantage except for individual wide applications. Alternating-Current Precipitators High-voltage alternating current may be employed for electrical precipitation. Corona discharge will result in a net rectification, provided that no spark gaps are used in series with the precipitator. However, the equipment capacity for a given efficiency is considerably lower than for direct current. In addition, difficulties due to induced high-frequency currents may be encountered. The simplicity of an ac system, on the other hand, has permitted very satisfactory adaptation for laboratory and sampling purposes [Drinker, Thomson, and Fitchet, J. Ind. Hyg., 5, 162 (September 1923)]. Some promising work with alternating current has been undertaken at the University of Karlsruhe. Lau [Staub, English ed., 29, 10 (1969)] and coworkers found that ac precipitators operated at 50 Hz were more effective than dc precipitators for dusts with resistivities higher than 1011 Ω⋅cm. An insulating screen covering the collecting electrode permitted higher-voltage operation without sparkover.

FIG. 17-76

Two-stage electrical-precipitation principle.

17-59

Section 18

Liquid-Solid Operations and Equipment

Donald A. Dahlstrom, Ph.D., Research Professor, Chemical and Fuels Engineering Department and Metallurgical Engineering Department, University of Utah; Member, National Academy of Engineering, American Institute of Chemical Engineers (AIChE), American Chemical Society (ACS), Society of Mining, Metallurgical, and Exploration Engineers (SME) of the American Institute of Mining, Metallurgical, and Petroleum Engineers (AIME), American Society of Engineering Education (ASEE). (Section Editor) Richard C. Bennett, B.S. Ch.E., Registered Professional Engineer, Illinois; Member, American Institute of Chemical Engineers (AIChE), President of Crystallization Technology, Inc.; Former President of Swenson Process Equipment, Inc. (Crystallization) Robert C. Emmett, Jr., B.S., Ch.E., Senior Process Consultant, EIMCO Process Equipment Co.; Member, American Institute of Chemical Engineers (AIChE), American Institute of Mining, Metallurgical, and Petroleum Engineers (AIME), Society of Mining, Metallurgical, and Exploration Engineers (SME. Gravity Sedimentation Operations; Filtration) Peter Harriott, Ph.D., Professor, School of Chemical Engineering, Cornell University; Member, American Institute of Chemical Engineering (AIChE), American Chemical Society (ACS). (Selection of a Solids-Liquid Separator) Tim Laros, M.S., Mineral Processing, Senior Process Consultant, EIMCO Process Equipment Co.; Member, Society of Mining, Metallurgy, and Exploration (SME of AIME). (Gravity Sedimentation Operations; Filtration) Wallace Leung, Sc.D., Director, Process Technology, Bird Machine Company; Member, American Filtration and Separation Society (Director). (Centrifuges) Chad McCleary, EIMCO Process Equipment Company, Process Consultant. (Vacuum filters) Shelby A. Miller, Ph.D., Retired Sr. Eng., Argonne National Laboratories; Member, American Association for the Advancement of Science (AAAS), American Chemical Society (ACS), American Institute of Chemical Engineering (AIChE), American Institute of Chemists, Filtration Society, New York Academy of Sciences; Registered Professional Engineer, New York. (Leaching) Booker Morey, Ph.D., Senior Consultant, SRI International; Member, Society of Mining, Metallurgy, and Exploration (SME of AIME), Filtration Society, Air and Waste Management, Association, Registered Professional Engineer, California and Massachusetts. (Expression) James Y. Oldshue, Ph.D., President, Oldshue Technologies International, Inc.; Member, National Academy of Engineering; Adjunct Professor of Chemical Engineering at Beijing Institute of Chemical Technology, Beijing, China; Member, American Chemical Society (ACS), American Institute of Chemical Engineering (AIChE), Traveler Century Club; Member of Executive Committee on the Transfer of Appropriate Technology for the World Federation of Engineering Organizations. (Agitation of Low-Viscosity Particle Suspensions)

18-1

18-2

LIQUID-SOLID OPERATIONS AND EQUIPMENT

George Priday, B.S., Ch.E., EIMCO Process Equipment Co.; Member, American Institute of Chemical Engineering (AIChE), Instrument Society of America (ISA). (Gravity Sedimentation Operations) Charles E. Silverblatt, M.S., Ch.E., Peregrine International Associates, Inc.; Consultant to WesTech Engineering, Inc.; American Institute of Mining, Metallurgical, and Petroleum Engineers (AIME). (Gravity Sedimentation Operations; Filtration) J. Stephen Slottee, M.S., Ch.E., Manager, Technology and Development, EIMCO Process Equipment Co.; Member, American Institute of Chemical Engineers (AIChE). (Filtration) Julian C. Smith, B. Chem., Ch.E., Professor Emeritus, Chemical Engineering, Cornell University; Member, American Chemical Society, (ACS), American Institute of Chemical Engineers (AIChE). (Selection of Solids-Liquid Separator) David B. Todd, Ph.D., President, Todd Engineering; Member, American Association for the Advancement of Science (AAAS), American Chemical Society (ACS), American Institute of Chemical Engineering (AIChE), American Oil Chemists Society (ADCS), Society of Plastics Engineers (SPE), and Society of the Plastics Industry (SPI); Registered Professional Engineer, Michigan. (Paste and Viscous Material Mixing)

PHASE CONTACTING AND LIQUID-SOLID PROCESSING: AGITATION OF LOW-VISCOSITY PARTICLE SUSPENSIONS Fluid Mixing Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-5 Introductory Fluid Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-6 Scale-up/Scale-down. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-6 Mixing Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-8 Small Tanks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-8 Close-Clearance Impellers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-8 Axial-Flow Impellers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-8 Radial-Flow Impellers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-9 Close-Clearance Stirrers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-9 Unbaffled Tanks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-9 Baffled Tanks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-10 Fluid Behavior in Mixing Vessels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-10 Impeller Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-10 Relationship between Fluid Motion and Process Performance . . . . . 18-11 Turbulent Flow in Stirred Vessels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-11 Fluid Velocities in Mixing Equipment. . . . . . . . . . . . . . . . . . . . . . . . . 18-11 Impeller Discharge Rate and Fluid Head for Turbulent Flow . . . . . 18-11 Laminar Fluid Motion in Vessels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-12 Vortex Depth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-12 Power Consumption of Impellers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-12 Design of Agitation Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-12 Selection of Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-12 Blending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-13 High-Viscosity Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-14 Chemical Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-15 Solid-Liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-15 Solid Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-16 Solid-Liquid Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-16 Gas-Liquid Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-16 Gas-Liquid Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-16 Gas-Liquid Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-17 Liquid-Gas-Solid Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-18 Loop Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-18 Liquid-Liquid Contacting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-18 Emulsions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-18 Stagewise Equipment: Mixer-Settlers . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-18 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-18 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-18 Mixer-Settler Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-19 Flow or Line Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-19 Mixing in Agitated Vessels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-21 Liquid-Liquid Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-22 Liquid-Liquid-Solid Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-22

Fluid Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jackets and Coils of Agitated Vessels . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid-Liquid-Gas-Solid Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computational Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18-22 18-22 18-23 18-23 18-24 18-24

PASTE AND VISCOUS-MATERIAL MIXING Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Batch Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Change-Can Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Helical-Blade Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Double-Arm Kneading Mixers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intensive Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Roll Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miscellaneous Batch Mixers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single-Screw Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rietz Extruder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Twin-Screw Extruders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Farrel Continuous Mixer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miscellaneous Continuous Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . Process Design Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scaling Up Mixing Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heating and Cooling Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heating Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cooling Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18-25 18-25 18-25 18-26 18-26 18-27 18-28 18-29 18-29 18-29 18-30 18-30 18-30 18-30 18-32 18-32 18-34 18-34 18-34 18-34

CRYSTALLIZATION FROM SOLUTION Principles of Crystallization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solubility and Phase Diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heat Effects in a Crystallization Process . . . . . . . . . . . . . . . . . . . . . . . Yield of a Crystallization Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 1: Yield from a Crystallization Process . . . . . . . . . . . . . . . . . . . Fractional Crystallization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 2: Yield from Evaporative Cooling . . . . . . . . . . . . . . . . . . . . . . Crystal Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometry of Crystal Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Purity of the Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coefficient of Variation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Crystal Nucleation and Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 3: Population, Density, Growth and Nucleation Rate . . . . . . . Crystallization Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mixed-Suspension, Mixed-Product-Removal Crystallizers. . . . . . . . .

18-35 18-35 18-35 18-36 18-36 18-36 18-37 18-37 18-37 18-38 18-38 18-38 18-39 18-43 18-44 18-44

Reaction-Type Crystallizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mixed-Suspension, Classified-Product-Removal Crystallizers . . . . . . Classified-Suspension Crystallizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scraped-Surface Crystallizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Batch Crystallization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recompression Evaporation-Crystallization . . . . . . . . . . . . . . . . . . . . Information Required to Specify a Crystallizer. . . . . . . . . . . . . . . . . . . . Crystallizer Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Crystallizer Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18-47 18-48 18-48 18-49 18-49 18-50 18-50 18-51 18-54

LEACHING Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Methods of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Leaching Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dispersed-Solids Leaching. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Screw-Conveyor Extractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tray Classifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selection or Design of a Leaching Process . . . . . . . . . . . . . . . . . . . . . . . Process and Operating Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . Extractor-Sizing Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18-55 18-55 18-55 18-55 18-55 18-56 18-57 18-58 18-58 18-58 18-58

GRAVITY SEDIMENTATION OPERATIONS Classification of Settleable Solids; Sedimentation Tests . . . . . . . . . . . . . Sedimentation-Test Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thickeners. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . High-Rate Thickeners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . High-Density Thickeners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Clarifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rectangular Clarifiers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Circular Clarifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Clarifier-Thickener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Industrial Waste Secondary Clarifiers . . . . . . . . . . . . . . . . . . . . . . . . . Tilted-Plate Clarifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solids-Contact Clarifiers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Components and Accessories for Sedimentation Units . . . . . . . . . . . . . Tanks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Drive-Support Structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Drive Assemblies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Feedwell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overflow Arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Underflow Arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sludge Bed Level Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flowmeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Density Gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbidity Gauges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Streaming Current Detectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous Countercurrent Decantation . . . . . . . . . . . . . . . . . . . . . . . . Flow-Sheet Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Underflow Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overflow Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interstage Mixing Efficiencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design or Stipulation of a Sedimentation Unit . . . . . . . . . . . . . . . . . . . . Selection of Type of Thickener or Clarifier . . . . . . . . . . . . . . . . . . . . . Materials of Construction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design Sizing Criteria. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Torque Rating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thickener Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Operating Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18-60 18-60 18-64 18-64 18-64 18-65 18-65 18-65 18-65 18-65 18-66 18-66 18-66 18-66 18-67 18-67 18-67 18-68 18-69 18-69 18-69 18-70 18-71 18-71 18-71 18-71 18-71 18-71 18-71 18-71 18-72 18-72 18-72 18-73 18-73 18-73 18-73 18-73 18-73 18-73 18-73

FILTRATION Definitions and Classification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Filtration Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous Filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Factors Influencing Small-Scale Testing . . . . . . . . . . . . . . . . . . . . . . . . . Vacuum or Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cake Discharge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Feed Slurry Temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cake Thickness Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Filter Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Representative Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Feed Solids Concentration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pretreatment Chemicals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18-74 18-74 18-74 18-75 18-75 18-75 18-75 18-75 18-75 18-75 18-76 18-76

LIQUID-SOLID OPERATIONS AND EQUIPMENT

18-3

Cloth Blinding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Homogeneous Cake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Agitation of Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Use of Steam or Hot Air. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Small-Scale Test Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Test Program. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bottom-Feed Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Top-Feed Procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Precoat Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dry Cake Weight vs. Thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dry Solids or Filtrate Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of Time on Flocculated Slurries . . . . . . . . . . . . . . . . . . . . . . . . Cake Moisture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cake Washing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wash Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Air Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scale-Up Factors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scale-Up on Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scale-Up on Cake Discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scale-Up on Actual Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overall Scale-Up Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Full-Scale Filter Performance Evaluation. . . . . . . . . . . . . . . . . . . . . . . . Filter Sizing Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 1: Sizing a Disk Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 2: Sizing a Drum Belt Filter with Washing . . . . . . . . . . . . . . . Horizontal Belt Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Batch Filtration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constant-Pressure Filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constant-Rate Filtration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variable-Pressure, Variable-Rate Filtration. . . . . . . . . . . . . . . . . . . . . Pressure Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compression-Permeability Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scaling Up Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Filter Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fabrics of Woven Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Metal Fabrics or Screens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressed Felts and Cotton Batting . . . . . . . . . . . . . . . . . . . . . . . . . . . . Filter Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rigid Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polymer Membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Granular Beds of Particulate Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . Filter Aids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diatomaceous Silica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Perlite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Filtration Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cake Filters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Batch Cake Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous Cake Filters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotary Drum Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Disk Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Horizontal Vacuum Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Filter Thickeners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Clarifying Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selection of Filtration Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Filter Prices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18-76 18-76 18-77 18-77 18-77 18-77 18-78 18-78 18-80 18-80 18-81 18-81 18-81 18-81 18-81 18-82 18-83 18-83 18-84 18-85 18-85 18-85 18-85 18-85 18-85 18-85 18-86 18-86 18-86 18-86 18-86 18-87 18-87 18-88 18-88 18-88 18-88 18-88 18-89 18-89 18-89 18-89 18-90 18-90 18-90 18-90 18-90 18-90 18-90 18-96 18-96 18-99 18-99 18-100 18-100 18-104 18-105

CENTRIFUGES Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Centripetal and Centrifugal Acceleration . . . . . . . . . . . . . . . . . . . . . . Solid-Body Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G-Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coriolis Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of Fluid Viscosity and Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . Sedimenting and Filtering Centrifuges . . . . . . . . . . . . . . . . . . . . . . . . Performance Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stress in the Centrifuge Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G-Force vs. Throughput. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Critical Speeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sedimenting Centrifuges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laboratory Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transient Centrifugation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tubular-Bowl Centrifuges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multichamber Centrifuges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Knife-Discharge Centrifugal Clarifiers . . . . . . . . . . . . . . . . . . . . . . . . Disk Centrifuges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Decanter Centrifuges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Screenbowl Centrifuges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18-106 18-106 18-106 18-106 18-107 18-107 18-107 18-107 18-107 18-108 18-109 18-110 18-110 18-110 18-111 18-112 18-112 18-112 18-113 18-113 18-115

18-4

LIQUID-SOLID OPERATIONS AND EQUIPMENT

Continuous Centrifugal Sedimentation Theory . . . . . . . . . . . . . . . . . Filtering Centrifuges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Batch Centrifuges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solid-Bottom Basket Centrifuges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Open-Bottom Basket Centrifuges . . . . . . . . . . . . . . . . . . . . . . . . . . . . Batch-Automatic Basket Centrifuges. . . . . . . . . . . . . . . . . . . . . . . . . . Continuous-Filtering Centrifuges . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theory of Centrifugal Filtration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selection of Centrifuges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sedimenting Centrifuges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Filtering Centrifuges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Purchase Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Installation Costs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maintenance Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Operating Labor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18-115 18-117 18-117 18-117 18-117 18-118 18-119 18-121 18-123 18-123 18-123 18-124 18-124 18-125 18-125 18-125

EXPRESSION Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-126 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-126

Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Screw Presses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Disk Presses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Roll Presses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Belt Presses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variable-Volume Filter Presses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tube Presses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SELECTION OF A SOLIDS-LIQUID SEPARATOR Preliminary Definition and Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminary Selections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Samples and Tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Establishing Process Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Representative Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simple Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modification of Process Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . Consulting the Manufacturer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Nomenclature Symbol c C Co do dp, max dt dt D Da Dj DT g gc h

Definition

SI units

H k Lp N NJS NRe Np NQ Nr Nt P Q T v v′ V Z

Specific heat Constant Orifice coefficient Orifice diameter Drop diameter Pipe diameter Tube diameter Impeller diameter Impeller diameter Diameter of jacketed vessel Tank diameter Acceleration Dimensional constant Local individual coefficient of heat transfer, equals dq/(dA)(∆T) Velocity head Thermal conductivity Diameter of agitator blade Agitator rotational speed Agitator speed for just suspension Da2Nρ/µ impeller Reynolds number Power number = (qcP)/ρN 3Da5 Impeller pumping coefficient = Q/NDa3 Impeller speed Impeller speed Power Impeller flow rate Tank diameter Average fluid velocity Fluid velocity fluctuation Bulk average velocity Liquid level in tank

γ ∆p µ µ µb µc µD µf µwt ρ ρ ρav ρc σ ΦD

Rate of shear Pressure drop across orifice Viscosity of liquid at tank temperature Stirred liquid viscosity Viscosity of fluid at bulk temperature Viscosity, continuous phase Viscosity of dispersed phase Viscosity of liquid at mean film temperature Viscosity at wall temperature Stirred liquid density Density of fluid Density of dispersed phase Density Interfacial tension Average volume fraction of discontinuous phase

18-126 18-126 18-126 18-126 18-126 18-126 18-127 18-127 18-130

U.S. customary units

J/(kg⋅k)

Btu/(lb⋅°F)

Dimensionless m m m m m m m m m/s2 gc = 1 when using SI units J/(m2⋅s⋅K)

Dimensionless in ft in ft ft ft ft ft ft/s2 32.2 (ft⋅lb)/(lbf⋅s2) Btu/(h⋅ft2⋅°F)

m J/(m⋅s⋅K) m s−1, (r/s) s−1 Dimensionless Dimensionless Dimensionless s−1 s−1 (N⋅m/s) m3/s m m/s m/s m/s m

ft (Btu⋅ft)/(h⋅ft2⋅°F) ft s−1, (r/s) s−1 Dimensionless Dimensionless Dimensionless s−1 s−1 ft⋅lbf /s ft3/s ft ft/s ft/s ft/s ft

s−1

s−1 lbf/ft2 lb/(ft⋅s) lb/(ft⋅s) lb/(ft⋅s) lb/(ft⋅s) lb/(ft⋅s) lb/(ft⋅s) lb/(ft⋅s) lb/ft3 lb/ft3 lb/ft3 lb/ft3 lbf/ft Dimensionless

Greek symbols

Pa⋅s Pa⋅s Pa⋅s Pa⋅s Pa⋅s Pa⋅s Pa⋅s g/m3 kg/m3 kg/m3 kg/m3 N/m Dimensionless

18-131 18-131 18-131 18-131 18-132 18-132 18-132 18-133 18-133

PHASE CONTACTING AND LIQUID-SOLID PROCESSING: AGITATION OF LOW-VISCOSITY PARTICLE SUSPENSIONS GENERAL REFERENCES: Harnby, N., M. F. Edwards, and A. W. Neinow (eds.), Mixing in the Process Industries, Butterworth, Stoneham, Mass., 1986. Lo, T. C., M. H. I. Baird, and C. Hanson, Handbook of Solvent Extraction, Wiley, New York, 1983. Nagata, S., Mixing: Principles and Applications, Kodansha Ltd., Tokyo, Wiley, New York, 1975. Oldshue, J. Y., Fluid Mixing Technology, McGraw-Hill, New York, 1983. Tatterson, G. B., Fluid Mixing and Gas Dispersion in Agitated Tanks, McGraw-Hill, New York, 1991. Uhl, V. W., and J. B. Gray (eds.), Mixing, vols. I and II, Academic Press, New York, 1966; vol. III, Academic Press, Orlando, Fla., 1992. Ulbrecht, J. J., and G. K. Paterson (eds.), Mixing of Liquids by Mechanical Agitation, Godon & Breach Science Publishers, New York, 1985. PROCEEDINGS: Fluid Mixing, vol. I, Inst. Chem. Eng. Symp., Ser. No. 64 (Bradford, England), The Institute of Chemical Engineers, Rugby, England, 1984. Mixing—Theory Related to Practice, AIChE, Inst. Chem. Eng. Symp. Ser. No. 10 (London), AIChE and The Institute of Chemical Engineers, London, 1965. Proc. First (1974), Second (1977), Third (1979), Fourth (1982), Fifth (1985), and Sixth (1988) European Conf. on Mixing, N. G. Coles (ed.), (Cambridge, England) BHRA Fluid Eng., Cranfield, England. Process Mixing, Chemical and Biochemical Applications, G. B. Tatterson, and R. V. Calabrese (eds.), AIChE Symp. Ser. No. 286, 1992.

FLUID MIXING TECHNOLOGY Fluid mixers cut across almost every processing industry including the chemical process industry; minerals, pulp, and paper; waste and water treating and almost every individual process sector. The engineer working with the application and design of mixers for a given process has three basic sources for information. One is published literature, consisting of several thousand published articles and several currently available books, and brochures from equipment vendors. In addition, there may be a variety of in-house experience which may or may not be cataloged, categorized, or usefully available for the process application at hand. Also, short courses are currently available in selected locations and with various course objectives, and a large body of experience and information lies in the hands of equipment vendors. In the United States, it is customary to design and purchase a mixer from a mixing vendor and purchase the vessel from another supplier. In many other countries, it is more common to purchase the vessel and mixer as a package from one supplier. In any event, the users of the mixer can issue a mechanical specification and determine the speed, diameter of an impeller, and power with in-house expertise. Or they may issue a processes specification which describes the engineering purpose of the mixing operation and the vendor will supply a description of the mixer process performance as well as prepare a mechanical design. This section describes fluid mixing technology and is referred to in other sections in this handbook which discuss the use of fluid mixing equipment in their various operating disciplines. This section does not describe paste and dough mixing, which may require planetary and extruder-type mixers, nor the area of dry solid-solid mixing. It is convenient to divide mixing into five pairs (plus three triplets and one quadruplicate combination) of materials, as shown in Table 18-1. These five pairs are blending (miscible liquids), liquid-solid, liquid-gas, liquid-liquid (immiscible liquids), and fluid motion. There are also four other categories that occur, involving three or four phases. One concept that differentiates mixing requirements is the difference between physical criteria listed on the left side of Table 18-1, in which some degree of sampling can be used to determine the character of the mixture in various parts in the tank, and various definitions of mixing requirements can be based on these physical descriptions. The other category on the right side of Table 18-1 involves chemical and mass-transfer criteria in which rates of mass

transfer or chemical reaction are of interest and have many more complexities in expressing the mixing requirements. The first five classes have their own mixing technologies. Each of these 10 areas has its own mixing technology. There are relationships for the optimum geometry of impeller types, D/ T ratios, and tank geometry. They each often have general, overall mixing requirements and different scale-up relationships based on process definitions. In addition, there are many subclassifications, some of which are based on the viscosity of fluids. In the case of blending, we have blending in the viscous region, the transition region, and the turbulent region. Since any given mixer designed for a process may be required to do several different parts of these 10 categories, it must be a compromise of the geometry and other requirements for the total process result and may not optimize any one particular process component. If it turns out that one particular process requirement is so predominant that all the other requirements are satisfied as a consequence, then it is possible to optimize that particular process step. Often, the only process requirement is in one of these 10 areas, and the mixer can be designed and optimized for that one step only. As an example of the complexity of fluid mixing, many batch processes involve adding many different materials and varying the liquid level over wide ranges in the tank, have different temperatures and shear rate requirements, and obviously need experience and expert attention to all of the requirements. Superimpose the requirements for sound mechanical design, including drives, fluid seals, and rotating shafts, means that the concepts presented here are merely a beginning to the overall, final design. A few general principles are helpful at this point before proceeding to the examination of equipment and process details. For any given impeller geometry, speed, and diameter, the impeller draws a certain amount of power. This power is 100 percent converted to heat. In low-viscosity mixing (defined later), this power is used to generate a macro-scale regime in which one typically has the visual observation of flow pattern, swirls, and other surface phenomena. However, these flow patterns are primarily energy transfer agents that transfer the power down to the micro scale. The macro-scale regime involves the pumping capacity of the impeller as well as the total circulating capacity throughout the tank and it is an important part of the overall mixer design. The micro-scale area in which the power is dissipated does not care much which impeller is used to generate the energy dissipation. In contrast, in high-viscosity processes, there is a continual progress of energy dissipation from the macro scale down to the micro scale. There is a wide variety of impellers using fluidfoil principles, which are used when flow from the impeller is predominant in the process requirement and macro- or micro-scale shear rates are a subordinate issue. Scale-up involves selecting mixing variables to give the desired performance in both pilot and full scale. This is often difficult (sometimes TABLE 18-1

Classification System for Mixing Processes

Physical

Components

Chemical, mass transfer

Blending Suspension Dispersion

Blending Solid-liquid Gas-liquid Solid-liquid-gas Liquid-liquid Liquid-liquid-solid Gas-liquid-liquid Gas-liquid-liquid-solid Fluid motion

Chemical reactions Dissolving, precipitation Gas absorption

Emulsions

Pumping

Extraction

Heat transfer 18-5

18-6

LIQUID-SOLID OPERATIONS AND EQUIPMENT

impossible) using geometric similarity, so that the use of nongeometric impellers in the pilot plant compared to the impellers used in the plant often allows closer modeling of the mixing requirements to be achieved. Computational fluid mixing allows the modeling of flow patterns in mixing vessels and some of the principles on which this is based in current techniques are included. INTRODUCTORY FLUID MECHANICS The fluid mixing process involves three different areas of viscosity which affect flow patterns and scale-up, and two different scales within the fluid itself: macro scale and micro scale. Design questions come up when looking at the design and performance of mixing processes in a given volume. Considerations must be given to proper impeller and tank geometry as well as the proper speed and power for the impeller. Similar considerations come up when it is desired to scale up or scale down, and this involves another set of mixing considerations. If the fluid discharge from an impeller is measured with a device that has a high-frequency response, one can track the velocity of the fluid as a function of time. The velocity at a given point in time can then be expressed as an average velocity v plus fluctuating component v′. Average velocities can be integrated across the discharge of the impeller, and the pumping capacity normal to an arbitrary discharge plane can be calculated. This arbitrary discharge plane is often defined as the plane bounded by the boundaries of the impeller blade diameter and height. Because there is no casing, however, an additional 10 to 20 percent of flow typically can be considered as the primary flow from an impeller. The velocity gradients between the average velocities operate only on larger particles. Typically, these larger-size particles are greater than 1000 µm. This is not a proven definition, but it does give a feel for the magnitudes involved. This defines macro-scale mixing. In the turbulent region, these macro-scale fluctuations can also arise from the finite number of impeller blades. These set up velocity fluctuations that can also operate on the macro scale. Smaller particles see primarily only the fluctuating velocity component. When the particle size is much less than 100 µm, the turbulent properties of the fluid become important. This is the definition of the physical size for micro-scale mixing. All of the power applied by a mixer to a fluid through the impeller appears as heat. The conversion of power to heat is through viscous shear and is approximately 2542 Btu/h/hp. Viscous shear is present in turbulent flow only at the micro-scale level. As a result, the power per unit volume is a major component of the phenomena of micro-scale mixing. At a 1-µm level, in fact, it doesn’t matter what specific impeller design is used to supply the power. Numerous experiments show that power per unit volume in the zone of the impeller (which is about 5 percent of the total tank volume) is about 100 times higher than the power per unit volume in the rest of the vessel. Making some reasonable assumptions about the fluid mechanics parameters, the root-mean-square (rms) velocity fluctuation in the zone of the impeller appears to be approximately 5 to 10 times higher than in the rest of the vessel. This conclusion has been verified by experimental measurements. The ratio of the rms velocity fluctuation to the average velocity in the impeller zone is about 50 percent with many open impellers. If the rms velocity fluctuation is divided by the average velocity in the rest of the vessel, however, the ratio is on the order of 5 percent. This is also the level of rms velocity fluctuation to the mean velocity in pipeline flow. There are phenomena in micro-scale mixing that can occur in mixing tanks that do not occur in pipeline reactors. Whether this is good or bad depends upon the process requirements. Figure 18-1 shows velocity versus time for three different impellers. The differences between the impellers are quite significant and can be important for mixing processes. All three impellers are calculated for the same impeller flow Q and the same diameter. The A310 (Fig. 18-2) draws the least power and has the least velocity fluctuations. This gives the lowest micro-scale turbulence and shear rate. The A200 (Fig. 18-3) shows increased velocity

Velocity fluctuations versus time for equal total pumping capacity from three different impellers.

FIG. 18-1

fluctuations and draws more power. The R100 (Fig. 18-4) draws the most power and has the highest micro-scale shear rate. The proper impeller should be used for each individual process requirement. Scale-up/Scale-down Two aspects of scale-up frequently arise. One is building a model based on pilot-plant studies that develop an understanding of the process variables for an existing full-scale mixing installation. The other is taking a new process and studying it in the pilot plant in such a way that pertinent scale-up variables are worked out for a new mixing installation. There are a few principles of scale-up that can indicate which approach to take in either case. Using geometric similarity, the macroscale variables can be summarized as follows: • Blend and circulation times in the large tank will be much longer than in the small tank.

FIG. 18-2

An A310 impeller.

PHASE CONTACTING AND LIQUID-SOLID PROCESSING

FIG. 18-4

FIG. 18-3

Pitched-blade turbine.

• Maximum impeller zone shear rate will be higher in the larger tank, but the average impeller zone shear rate will be lower; therefore, there will be a much greater variation in shear rates in a full-scale tank than in a pilot unit. • Reynolds numbers in the large tank will be higher, typically on the order of 5 to 25 times higher than those in a small tank. • Large tanks tend to develop a recirculation pattern from the impeller through the tank back to the impeller. This results in a behavior similar to that for a number of tanks in a series. The net result is that the mean circulation time is increased over what would be predicted from the impeller pumping capacity. This also increases the standard deviation of the circulation times around the mean. • Heat transfer is normally much more demanding on a large scale. The introduction of helical coils, vertical tubes, or other heat-transfer devices causes an increased tendency for areas of low recirculation to exist. • In gas-liquid systems, the tendency for an increase in the gas superficial velocity upon scale-up can further increase the overall circulation time. What about the micro-scale phenomena? These are dependent primarily on the energy dissipation per unit volume, although one must also be concerned about the energy spectra. In general, the energy dissipation per unit volume around the impeller is approximately 100 times higher than in the rest of the tank. This results in an rms velocity fluctuation ratio to the average velocity on the order of 10:1 between the impeller zone and the rest of the tank. Because there are thousands of specific processes each year that involve mixing, there will be at least hundreds of different situations requiring a somewhat different pilot-plant approach. Unfortunately, no set of rules states how to carry out studies for any specific program, but here are a few guidelines that can help one carry out a pilot-plant program.

18-7

Flat-blade turbine.

• For any given process, one takes a qualitative look at the possible role of fluid shear stresses. Then one tries to consider pathways related to fluid shear stress that may affect the process. If there are none, then this extremely complex phenomenon can be dismissed and the process design can be based on such things as uniformity, circulation time, blend time, or velocity specifications. This is often the case in the blending of miscible fluids and the suspension of solids. • If fluid shear stresses are likely to be involved in obtaining a process result, then one must qualitatively look at the scale at which the shear stresses influence the result. If the particles, bubbles, droplets, or fluid clumps are on the order of 1000 µm or larger, the variables are macro scale and average velocities at a point are the predominant variable. When macro-scale variables are involved, every geometric design variable can affect the role of shear stresses. They can include such items as power, impeller speed, impeller diameter, impeller blade shape, impeller blade width or height, thickness of the material used to make the impeller, number of blades, impeller location, baffle location, and number of impellers. Micro-scale variables are involved when the particles, droplets, baffles, or fluid clumps are on the order of 100 µm or less. In this case, the critical parameters usually are power per unit volume, distribution of power per unit volume between the impeller and the rest of the tank, rms velocity fluctuation, energy spectra, dissipation length, the smallest micro-scale eddy size for the particular power level, and viscosity of the fluid. • The overall circulating pattern, including the circulation time and the deviation of the circulation times, can never be neglected. No matter what else a mixer does, it must be able to circulate fluid throughout an entire vessel appropriately. If it cannot, then that mixer is not suited for the task being considered. Qualitative and, hopefully, quantitative estimates of how the process result will be measured must be made in advance. The evaluations must allow one to establish the importance of the different steps in a process, such as gas-liquid mass transfer, chemical reaction rate, or heat transfer. • It is seldom possible, either economically or timewise, to study every potential mixing variable or to compare the performance of many impeller types. In many cases, a process needs a specific fluid regime that is relatively independent of the impeller type used to generate it. Because different impellers may require different geometries to achieve an optimum process combination, a random choice of only one diameter of each of two or more impeller types may not tell what is appropriate for the fluid regime ultimately required. • Often, a pilot plant will operate in the viscous region while the commercial unit will operate in the transition region, or alternatively, the pilot plant may be in the transition region and the commercial unit in the turbulent region. Some experience is required to estimate the difference in performance to be expected upon scale-up. • In general, it is not necessary to model Z/ T ratios between pilot and commercial units. • In order to make the pilot unit more like a commercial unit in macro-scale characteristics, the pilot unit impeller must be designed

18-8

LIQUID-SOLID OPERATIONS AND EQUIPMENT

to lengthen the blend time and to increase the maximum impeller zone shear rate. This will result in a greater range of shear rates than is normally found in a pilot unit. MIXING EQUIPMENT There are three types of mixing flow patterns that are markedly different. The so-called axial-flow turbines (Fig. 18-3) actually give a flow coming off the impeller of approximately 45°, and therefore have a recirculation pattern coming back into the impeller at the hub region of the blades. This flow pattern exists to an approximate Reynolds number of 200 to 600 and then becomes radial as the Reynolds number decreases. Both the R100 and A200 impellers normally require four baffles for an effective flow pattern. These baffles typically are 1⁄12 of the tank diameter and width. Radial-flow impellers include the flat-blade disc turbine, Fig. 18-4, which is labeled an R100. This generates a radial flow pattern at all Reynolds numbers. Figure 18-17 is the diagram of Reynolds number/power number curve, which allows one to calculate the power knowing the speed and diameter of the impeller. The impeller shown in Fig. 18-4 typically gives high shear rates and relatively low pumping capacity. The current design of fluidfoil impellers includes the A310 (Fig. 18-2), as well as several other impellers of that type commonly referred to as high-efficiency impellers, hydrofoil, and other descriptive names to illustrate that they are designed to maximize flow and minimize shear rate. These impellers typically require two baffles, but are normally used with three, since three gives a more stable flow pattern. Since most industrial mixing processes involve pumping capacity and, to a lesser degree, fluid shear rate, the fluidfoil impellers are now used on the majority of the mixer installations. There is now an additional family of these fluidfoil impellers, which depend upon different solidity ratios to operate in various kinds of fluid mixing systems. Figure 18-5 illustrates four of these impellers. The solidity ratio is the ratio of total blade area to a circle circumscribing the impeller and, as viscosity increases, higher values of the solidity ratios are more effective in providing an axial flow pattern rather than a radial flow pattern. Also the A315-type provides an effective area of preventing gas bypassing through the hub of the impeller by having exceptionally wide blades. Another impeller of that type is the Prochem Maxflo T. Small Tanks For tanks less than 1.8 m in diameter, the clamp or flanged mounted angular, off-center axial-flow impeller without baffles should be used for a wide range of process requirements (refer to Fig. 18-14). The impellers currently used are the fluidfoil type. Since small impellers typically operate at low Reynolds numbers, often in the transition region, the fluidfoil impeller should be designed to give good flow characteristics over a range of Reynolds numbers, probably on the order of 50 to 500. The Z/T ratio should be 0.75 to 1.5. The volume of liquid should not exceed 4 m3. Close-Clearance Impellers There are two close-clearance impellers. They are the anchor impeller (Fig. 18-6) and the helical

FIG. 18-5

foil type.

The solidity ratio for four different impellers of the axial-flow fluid-

FIG. 18-6

Anchor impeller.

impeller (Fig. 18-7), which operate near the tank wall and are particularly effective in pseudoplastic fluids in which it is desirable to have the mixing energy concentrated out near the tank wall where the flow pattern is more effective than with the open impellers that were covered earlier. Axial-Flow Impellers Axial-flow impellers include all impellers in which the blade makes an angle of less than 90° with the plane of rotation. Propellers and pitched-blade turbines, as illustrated in Figs. 18-8 and 18-3, are representative axial-flow impellers.

FIG. 18-7

Helical mixer for high-viscosity fluid.

PHASE CONTACTING AND LIQUID-SOLID PROCESSING

FIG. 18-8

Marine-type mixing impeller.

Portable mixers may be clamped on the side of an open vessel in the angular, off-center position shown in Fig. 18-14 or bolted to a flange or plate on the top of a closed vessel with the shaft in the same angular, off-center position. This mounting results in a strong top-tobottom circulation. Two basic speed ranges are available: 1150 or 1750 r/min with direct drive and 350 or 420 r/min with a gear drive. The high-speed units produce higher velocities and shear rates (Fig. 18-9) in the impeller discharge stream and a lower circulation rate throughout the vessel than the low-speed units. For suspension of solids, it is common to use the gear-driven units, while for rapid dispersion or fast reactions the high-speed units are more appropriate. Axial-flow impellers may also be mounted near the bottom of the cylindrical wall of a vessel as shown in Fig. 18-10. Such side-entering agitators are used to blend low-viscosity fluids [<0.1 Pa⋅s (100 cP)] or to keep slowly settling sediment suspended in tanks as large as some 4000 m3 (106 gal). Mixing of paper pulp is often carried out by sideentering propellers. Pitched-blade turbines (Fig. 18-3) are used on top-entering agitator shafts instead of propellers when a high axial circulation rate is desired and the power consumption is more than 2.2 kW (3 hp). A pitchedblade turbine near the upper surface of liquid in a vessel is effective for rapid submergence of floating particulate solids. Radial-Flow Impellers Radial-flow impellers have blades which are parallel to the axis of the drive shaft. The smaller multiblade ones are known as turbines; larger, slower-speed impellers, with two or four blades, are often called paddles. The diameter of a turbine is normally

FIG. 18-9

18-9

High shear rate impeller.

FIG. 18-10

Side-entering propeller mixer.

between 0.3 and 0.6 of the tank diameter. Turbine impellers come in a variety of types, such as curved-blade and flat-blade, as illustrated in Fig. 18-4. Curved blades aid in starting an impeller in settled solids. For processes in which corrosion of commonly used metals is a problem, glass-coated impellers may be economical. A typical modified curved-blade turbine of this type is shown in Fig. 18-11. Close-Clearance Stirrers For some pseudoplastic fluid systems stagnant fluid may be found next to the vessel walls in parts remote from propeller or turbine impellers. In such cases, an “anchor” impeller may be used (Fig. 18-6). The fluid flow is principally circular or helical (see Fig. 18-7) in the direction of rotation of the anchor. Whether substantial axial or radial fluid motion also occurs depends on the fluid viscosity and the design of the upper blade-supporting spokes. Anchor agitators are used particularly to obtain improved heat transfer in high-consistency fluids. Unbaffled Tanks If a low-viscosity liquid is stirred in an unbaffled tank by an axially mounted agitator, there is a tendency for a swirling flow pattern to develop regardless of the type of impeller. Figure 18-12 shows a typical flow pattern. A vortex is produced owing to centrifugal force acting on the rotating liquid. In spite of the presence of a vortex, satisfactory process results often can be obtained in an unbaffled vessel. However, there is a limit to the rotational speed that may be used, since once the vortex reaches the impeller, severe air entrainment may occur. In addition, the swirling mass of liquid often generates an oscillating surge in the tank, which coupled with the deep vortex may create a large fluctuating force acting on the mixer shaft.

FIG. 18-11

Glass-steel impeller. (The Pfaudler Company.)

18-10

LIQUID-SOLID OPERATIONS AND EQUIPMENT

FIG. 18-12 Typical flow pattern for either axial- or radial-flow impellers in an unbaffled tank.

Vertical velocities in a vortexing low-viscosity liquid are low relative to circumferential velocities in the vessel. Increased vertical circulation rates may be obtained by mounting the impeller off center, as illustrated in Fig. 18-13. This position may be used with either turbines or propellers. The position is critical, since too far or too little off center in one direction or the other will cause greater swirling, erratic vortexing, and dangerously high shaft stresses. Changes in viscosity and tank size also affect the flow pattern in such vessels. Off-center mountings have been particularly effective in the suspension of paper pulp. With axial-flow impellers, an angular off-center position may be used. The impeller is mounted approximately 15° from the vertical, as shown in Fig. 18-14. The angular off-center position used with fluidfoil units is usually limited to impellers delivering 2.2 kW (3 hp) or less. The unbalanced fluid forces generated by this mounting can become severe with higher power. Baffled Tanks For vigorous agitation of thin suspensions, the tank is provided with baffles which are flat vertical strips set radially along the tank wall, as illustrated in Figs. 18-15 and 18-16. Four baffles are almost always adequate. A common baffle width is one-tenth to one-twelfth of the tank diameter (radial dimension). For agitating slurries, the baffles often are located one-half of their width from the vessel wall to minimize accumulation of solids on or behind them. For Reynolds numbers greater than 2000 baffles are commonly used with turbine impellers and with on-centerline axial-flow impellers. The flow patterns illustrated in Figs. 18-15 and 18-16 are quite different, but in both cases the use of baffles results in a large top-to-bottom circulation without vortexing or severely unbalanced fluid forces on the impeller shaft.

Flow pattern with a paper-stock propeller, unbaffled; vertical offcenter position.

FIG. 18-13

FIG. 18-14 Typical flow pattern with a propeller in angular off-center position without baffles.

In the transition region [Reynolds numbers, Eq. (18-1), from 10 to 10,000], the width of the baffle may be reduced, often to one-half of standard width. If the circulation pattern is satisfactory when the tank is unbaffled but a vortex creates a problem, partial-length baffles may be used. These are standard-width and extend downward from the surface into about one-third of the liquid volume. In the region of laminar flow (NRe < 10), the same power is consumed by the impeller whether baffles are present or not, and they are seldom required. The flow pattern may be affected by the baffles, but not always advantageously. When they are needed, the baffles are usually placed one or two widths radially off the tank wall, to allow fluid to circulate behind them and at the same time produce some axial deflection of flow. FLUID BEHAVIOR IN MIXING VESSELS Impeller Reynolds Number The presence or absence of turbulence in an impeller-stirred vessel can be correlated with an impeller Reynolds number defined Da2 Nρ NRe = ᎏ (18-1) µ

Typical flow pattern in a baffled tank with a propeller or an axialflow turbine positioned on center.

FIG. 18-15

PHASE CONTACTING AND LIQUID-SOLID PROCESSING

FIG. 18-16

Typical flow pattern in a baffled tank with a turbine positioned on

center.

where N = rotational speed, r/s; Da = impeller diameter, m (ft); ρ = fluid density, kg/m3 (lb/ft3); and µ = viscosity, Pa⋅s [lb/(ft⋅s)]. Flow in the tank is turbulent when NRe > 10,000. Thus viscosity alone is not a valid indication of the type of flow to be expected. Between Reynolds numbers of 10,000 and approximately 10 is a transition range in which flow is turbulent at the impeller and laminar in remote parts of the vessel; when NRe < 10, flow is laminar only. Not only is the type of flow related to the impeller Reynolds number, but also such process performance characteristics as mixing time, impeller pumping rate, impeller power consumption, and heat- and mass-transfer coefficients can be correlated with this dimensionless group. Relationship between Fluid Motion and Process Performance Several phenomena which can be used to promote various processing objectives occur during fluid motion in a vessel. 1. Shear stresses are developed in a fluid when a layer of fluid moves faster or slower than a nearby layer of fluid or a solid surface. In laminar flow, the shear stress is equal to the product of fluid viscosity and velocity gradient or rate of shear. Under laminar-flow conditions, shear forces are larger than inertial forces in the fluid. With turbulent flow, shear stress also results from the behavior of transient random eddies, including large-scale eddies which decay to small eddies or fluctuations. The scale of the large eddies depends on equipment size. On the other hand, the scale of small eddies, which dissipate energy primarily through viscous shear, is almost independent of agitator and tank size. The shear stress in the fluid is much higher near the impeller than it is near the tank wall. The difference is greater in large tanks than in small ones. 2. Inertial forces are developed when the velocity of a fluid changes direction or magnitude. In turbulent flow, inertia forces are larger than viscous forces. Fluid in motion tends to continue in motion until it meets a solid surface or other fluid moving in a different direction. Forces are developed during the momentum transfer that takes place. The forces acting on the impeller blades fluctuate in a random manner related to the scale and intensity of turbulence at the impeller. 3. The interfacial area between gases and liquids, immiscible liquids, and solids and liquids may be enlarged or reduced by these viscous and inertia forces when interacting with interfacial forces such as surface tension. 4. Concentration and temperature differences are reduced by bulk flow or circulation in a vessel. Fluid regions of different composition or temperature are reduced in thickness by bulk motion in which velocity gradients exist. This process is called bulk diffusion or Taylor diffusion (Brodkey, in Uhl and Gray, op. cit., vol. 1, p. 48). The turbulent and molecular diffusion reduces the difference between these regions. In laminar flow, Taylor diffusion and molecular diffusion are the mechanisms of concentration- and temperature-difference reduction. 5. Equilibrium concentrations which tend to develop at solidliquid, gas-liquid, or liquid-liquid interfaces are displaced or changed by molecular and turbulent diffusion between bulk fluid and fluid adjacent to the interface. Bulk motion (Taylor diffusion) aids in this mass-transfer mechanism also. Turbulent Flow in Stirred Vessels Turbulence parameters such as intensity and scale of turbulence, correlation coefficients, and

18-11

energy spectra have been measured in stirred vessels. However, these characteristics are not used directly in the design of stirred vessels. Fluid Velocities in Mixing Equipment Fluid velocities have been measured for various turbines in baffled and unbaffled vessels. Typical data are summarized in Uhl and Gray, op. cit., vol. 1, chap. 4. Velocity data have been used for calculating impeller discharge and circulation rates but are not employed directly in the design of mixing equipment. Impeller Discharge Rate and Fluid Head for Turbulent Flow When fluid viscosity is low and flow is turbulent, an impeller moves fluids by an increase in momentum from the blades which exert a force on the fluid. The blades of rotating propellers and turbines change the direction and increase the velocity of the fluids. The pumping rate or discharge rate of an impeller is the flow rate perpendicular to the impeller discharge area. The fluid passing through this area has velocities proportional to the impeller peripheral velocity and velocity heads proportional to the square of these velocities at each point in the impeller discharge stream under turbulentflow conditions. The following equations relate velocity head, pumping rate, and power for geometrically similar impellers under turbulent-flow conditions: Q = NQ NDa3 2

(18-2) 2 a

Np N D H=ᎏ NQ g

冢 冣

Da5 P = NpρN 3 ᎏ gc

(18-3) (18-4)

ρHQg P=ᎏ (18-5) gc where Q = impeller discharge rate, m3/s (ft3/s); NQ = discharge coefficient, dimensionless; H = velocity head, m (ft); Np = power number, dimensionless; P = power, (N⋅m)/s [(ft⋅lbf)/s]; gc = dimensional constant, 32.2 (ft⋅lb)/(lbf⋅s2)(gc = 1 when using SI units); and g = gravitational acceleration, m/s2 (ft/s2). The discharge rate Q has been measured for several types of impellers, and discharge coefficients have been calculated. The data of a number of investigators are reviewed by Uhl and Gray (op. cit., vol. 1, chap. 4). NQ is 0.4 to 0.5 for a propeller with pitch equal to diameter at NRe = 105. For turbines, NQ ranges from 0.7 to 2.9, depending on the number of blades, blade-height-to-impellerdiameter ratio, and impeller-to-vessel-diameter ratio. The effects of these geometric variables are not well defined. Power consumption has also been measured and correlated with impeller Reynolds number. The velocity head for a mixing impeller can be calculated, then, from flow and power data, by Eq. (18-3) or Eq. (18-5). The velocity head of the impeller discharge stream is a measure of the maximum force that this fluid can exert when its velocity is changed. Such inertia forces are higher in streams with higher discharge velocities. Shear rates and shear stresses are also higher under these conditions in the smallest eddies. If a higher discharge velocity is desired at the same power consumption, a smaller-diameter impeller must be used at a higher rotational speed. According to Eq. (18-4), at a given power level N ∝ Da−5/3 and NDa ∝ Da−2/3. Then, H ∝ Da−4/3 and Q ∝ Da4/3. An impeller with a high fluid head is one with high peripheral velocity and discharge velocity. Such impellers are useful for (1) rapid reduction of concentration differences in the impeller discharge stream (rapid mixing), (2) production of large interfacial area and small droplets in gas-liquid and immiscible-liquid systems, (3) solids deagglomeration, and (4) promotion of mass transfer between phases. The impeller discharge rate can be increased at the same power consumption by increasing impeller diameter and decreasing rotational speed and peripheral velocity so that N 3Da5 is a constant (Eq. 18-4)]. Flow goes up, velocity head and peripheral velocity go down, but impeller torque TQ goes up. At the same torque, N 2Da5 is constant, P ∝ Da−5/2, and Q ∝ Da1/2. Therefore, increasing impeller diameter at constant torque increases discharge rate at lower power consumption. At the same discharge rate, NDa3 is constant, P ∝ Da−4, and TQ ∝ Da−1.

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LIQUID-SOLID OPERATIONS AND EQUIPMENT

Therefore, power and torque decrease as impeller diameter is increased at constant Q. A large-diameter impeller with a high discharge rate is used for (1) short times to complete mixing of miscible liquid throughout a vessel, (2) promotion of heat transfer, (3) reduction of concentration and temperature differences in all parts of vessels used for constantenvironment reactors and continuous averaging, and (4) suspension of particles of relatively low settling rate. Laminar Fluid Motion in Vessels When the impeller Reynolds number is less than 10, the flow induced by the impeller is laminar. Under these conditions, the impeller drags fluid with it in a predominantly circular pattern. If the impeller blades curve back, there is a viscous drag flow toward the tips of these blades. Under moderateviscosity conditions in laminar flow, centrifugal force acting on the fluid layer dragged in a circular path by the rotating impeller will move fluid in a radial direction. This centrifugal effect causes any gas accumulated behind a rotating blade to move to the axis of impeller rotation. Such radial-velocity components are small relative to tangential velocity. For turbines at Reynolds numbers less than 100, toroidal stagnant zones exist above and below the turbine periphery. Interchange of liquid between these regions and the rest of the vessel is principally by molecular diffusion. Suspensions of fine solids may have pseudoplastic or plastic-flow properties. When they are in laminar flow in a stirred vessel, motion in remote parts of the vessel where shear rates are low may become negligible or cease completely. To compensate for this behavior of slurries, large-diameter impellers or paddles are used, with (Da /DT) > 0.6, where DT is the tank diameter. In some cases, for example, with some anchors, Da > 0.95 DT. Two or more paddles may be used in deep tanks to avoid stagnant regions in slurries. In laminar flow (NRe < 10), Np ∝ 1/NRe and P ∝ µN 2Da3 . Since shear stress is proportional to rotational speed, shear stress can be increased at the same power consumption by increasing N proportionally to Da−3/2 as impeller diameter Da is decreased. Fluid circulation probably can be increased at the same power consumption and viscosity in laminar flow by increasing impeller diameter and decreasing rotational speed, but the relationship between Q, N, and Da for laminar flow from turbines has not been determined. As in the case of turbulent flow, then, small-diameter impellers (Da < DT /3) are useful for (1) rapid mixing of dry particles into liquids, (2) gas dispersion in slurries, (3) solid-particle deagglomeration, and (4) promoting mass transfer between solid and liquid phases. If stagnant regions are a problem, large impellers must be used and rotational speed and power increased to obtain the required results. Small continuous-processing equipment may be more economical than batch equipment in such cases. Likewise, large-diameter impellers (Da > DT / 2) are useful for (1) avoiding stagnant regions in slurries, (2) short mixing times to obtain uniformity throughout a vessel, (3) promotion of heat transfer, and (4) laminar continuous averaging of slurries. Vortex Depth In an unbaffled vessel with an impeller rotating in the center, centrifugal force acting on the fluid raises the fluid level at the wall and lowers the level at the shaft. The depth and shape of such a vortex (Rieger, Ditl, and Novak, Chem. Eng. Sci., 34, 397 (1978)] depend on impeller and vessel dimensions as well as rotational speed. Power Consumption of Impellers Power consumption is related to fluid density, fluid viscosity, rotational speed, and impeller diameter by plots of power number (gc P/ρN 3Da5) versus Reynolds number (Da2 Nρ /µ). Typical correlation lines for frequently used impellers operating in newtonian liquids contained in baffled cylindrical vessels are presented in Fig. 18-17. These curves may be used also for operation of the respective impellers in unbaffled tanks when the Reynolds number is 300 or less. When NRe is greater than 300, however, the power consumption is lower in an unbaffled vessel than indicated in Fig. 18-17. For example, for a six-blade disk turbine with DT /Da = 3 and Da /Wi = 5, Np = 1.2 when NRe = 104. This is only about one-fifth of the value of Np when baffles are present. Additional power data for other impeller types such as anchors, curved-blade turbines, and paddles in baffled and unbaffled vessels are available in the following references: Holland and Chapman, op.

FIG. 18-17 Impeller power correlations: curve 1, six-blade turbine, Da /Wi = 5, like Fig. 18-4 but with six blades, four baffles, each DT /12; curve 2, verticalblade, open turbine with six straight blades, Da /Wi = 8, four baffles each DT /12; curve 3, 45° pitched-blade turbine like Fig. 18-3 but with six blades, Da /Wi = 8, four baffles, each DT /12; curve 4, propeller, pitch equal to 2Da, four baffles, each 0.1DT, also same propeller in angular off-center position with no baffles; curve 5, propeller, pitch equal to Da, four baffles each 0.1DT, also same propeller in angular off-center position as in Fig. 18-14 with no baffles. Da = impeller diameter, DT = tank diameter, gc = gravitational conversion factor, N = impeller rotational speed, P = power transmitted by impeller shaft, Wi = impeller blade height, µ = viscosity of stirred liquid, and ρ = density of stirred mixture. Any set of consistent units may be used, but N must be rotations (rather than radians) per unit time. In the SI system, gc is dimensionless and unity. [Curves 4 and 5 from Rushton, Costich, and Everett, Chem. Eng. Prog., 46, 395, 467 (1950), by permission; curves 2 and 3 from Bates, Fondy, and Corpstein, Ind. Eng. Chem. Process Des. Dev., 2, 310 (1963), by permission of the copyright owner, the American Chemical Society.]

cit., chaps. 2, 4, Reinhold, New York, 1966; and Bates, Fondy, and Fenic, in Uhl and Gray, op. cit., vol. 1, chap. 3. Power consumption for impellers in pseudoplastic, Bingham plastic, and dilatant nonnewtonian fluids may be calculated by using the correlating lines of Fig. 18-17 if viscosity is obtained from viscosityshear rate curves as described here. For a pseudoplastic fluid, viscosity decreases as shear rate increases. A Bingham plastic is similar to a pseudoplastic fluid but requires that a minimum shear stress be exceeded for any flow to occur. For a dilatant fluid, viscosity increases as shear rate increases. The appropriate shear rate to use in calculating viscosity is given by one of the following equations when a propeller or a turbine is used (Bates et al., in Uhl and Gray, op. cit., vol. 1, p. 149): For dilatant liquids, D 0.5 γ˙ = 13N ᎏa (18-6) DT For pseudoplastic and Bingham plastic fluids, γ˙ = 10N (18-7)

冢 冣

where γ˙ = average shear rate, s−1. The shear rate calculated from impeller rotational speed is used to identify a viscosity from a plot of viscosity versus shear rate determined with a capillary or rotational viscometer. Next NRe is calculated, and Np is read from a plot like Fig. 18-17. DESIGN OF AGITATION EQUIPMENT Selection of Equipment The principal factors which influence mixing-equipment choice are (1) the process requirements, (2) the flow properties of the process fluids, (3) equipment costs, and (4) construction materials required. Ideally, the equipment chosen should be that of the lowest total cost which meets all process requirements. The total cost includes depreciation on investment, operating cost such as power, and maintenance costs. Rarely is any more than a superficial evaluation based on this

PHASE CONTACTING AND LIQUID-SOLID PROCESSING principle justified, however, because the cost of such an evaluation often exceeds the potential savings that can be realized. Usually optimization is based on experience with similar mixing operations. Often the process requirements can be matched with those of a similar operation, but sometimes tests are necessary to identify a satisfactory design and to find the minimum rotational speed and power. There are no satisfactory specific guides for selecting mixing equipment because the ranges of application of the various types of equipment overlap and the effects of flow properties on process performance have not been adequately defined. Nevertheless, what is frequently done in selecting equipment is described in the following paragraphs. Top-Entering Impellers For vessels less than 1.8 m (6 ft) in diameter, a clamp- or flange-mounted, angular, off-center fluidfoil impeller with no baffles should be the initial choice for meeting a wide range of process requirements (Fig. 18-14). The vessel straight-sideheight-to-diameter ratio should be 0.75 to 1.5, and the volume of stirred liquid should not exceed 4 m3 (about 1000 gal). For suspension of free-settling particles, circulation of pseudoplastic slurries, and heat transfer or mixing of miscible liquids to obtain uniformity, a speed of 350 or 420 r/min should be stipulated. For dispersion of dry particles in liquids or for rapid initial mixing of liquid reactants in a vessel, an 1150- or 1750- r/min propeller should be used at a distance DT /4 above the vessel bottom. A second propeller can be added to the shaft at a depth Da below the liquid surface if the submergence of floating liquids or particulate solids is otherwise inadequate. Such propeller mixers are readily available up to 2.2 kW (3 hp) for off-center sloped-shaft mounting. Propeller size, pitch, and rotational speed may be selected by model tests, by experience with similar operations, or, in a few cases, by published correlations of performance data such as mixing time or heat transfer. The propeller diameter and motor power should be the minimum which meet process requirements. If agitation is required for a vessel less than 1.8 m (6 ft) in diameter and the same operations will be scaled up to a larger vessel ultimately, the equipment type should be the same as that expected in the larger vessel. Axial-Flow Fluidfoil Impellers For vessel volumes of 4 to 200 m3 (1000 to 50,000 gal), a turbine mixer mounted coaxially within the vessel with four or more baffles should be the initial choice. Here also the vessel straight-side-height-to-diameter ratio should be 0.75 to 1.5. Four vertical baffles should be fastened perpendicularly to the vessel wall with a gap between baffle and wall equal to DT / 24 and a radial baffle width equal to DT /12. For suspension of rapidly settling particles, the impeller turbine diameter should be DT / 3 to DT / 2. A clearance of less than oneseventh of the fluid depth in the vessel should be used between the lower edge of the turbine blade tips and the vessel bottom. As the viscosity of a suspension increases, the impeller diameter should be increased. This diameter may be increased to 0.6 DT and a second impeller added to avoid stagnant regions in pseudoplastic slurries. Moving the baffles halfway between the impeller periphery and the vessel wall will also help avoid stagnant fluid near the baffles. As has been shown, power consumption is decreased and turbine discharge rate is increased as impeller diameter is increased at constant torque (in the completely turbulent regime). This means that for a stipulated discharge rate, more efficient operation is obtained (lower power and torque) with a relatively large impeller operating at a relatively low speed (N ∝ Da−3). Conversely, if power is held constant, decreasing impeller diameter results in increasing peripheral velocity and decreasing torque. Thus at a stipulated power level the rapid, efficient initial mixing of reactants identified with high peripheral velocity can be achieved by a relatively small impeller operating at a relatively high speed (N ∝ Da−5/3). For circulation and mixing to obtain uniformity, the impeller should be located at one-third of the liquid depth above the vessel bottom unless rapidly settling material or a need to stir a nearly empty vessel requires a lower impeller location. Side-Entering Impellers For vessels greater than 4 m3 (1000 gal), a side-entering propeller agitator (Fig. 18-9) may be more economical than a top-mounted impeller on a centered vertical shaft.

18-13

For vessels greater than 38 m3 (10,000 gal), the economic attractiveness of side-entering impellers increases. For vessels larger than 380 m3 (100,000 gal), units may be as large as 56 kW (75 hp), and two or even three may be installed in one tank. For the suspension of slow-settling particles or the maintenance of uniformity in a viscous slurry of small particles, the diameter and rotational speed of a sideentering agitator must be selected on the basis of model tests or experience with similar operations. When abrasive solid particles must be suspended, maintenance costs for the submerged shaft seal of a side-entering propeller may become high enough to make this type of mixer an uneconomical choice. Jet Mixers Continuous recycle of the contents of a tank through an external pump so arranged that the pump discharge stream appropriately reenters the vessel can result in a flow pattern in the tank which will produce a slow mixing action [Fossett, Trans. Inst. Chem. Eng., 29, 322 (1951)]. Large Tanks Most large vessels (over 4 m3) require a heavy-duty drive. About two-thirds of the mixing requirements industrially involve flow, circulation, and other types of pumping capacity requirements, including such applications as blending and solid suspension. There often is no requirement for any marked level of shear rate, so the use of the fluidfoil impellers is most common. If additional shear rate is required over what can be provided by the fluidfoil impeller, the axial-flow turbine (Fig. 18-3) is often used, and if extremely high shear rates are required, the flat-blade turbine (Rushton turbine) (Fig. 18-4) is required. For still higher shear rates, there is an entire variety of high-shear-rate impellers, typified by that shown in Fig. 18-10 that are used. The fluidfoil impellers in large tanks require only two baffles, but three are usually used to provide better flow pattern asymmetry. These fluidfoil impellers provide a true axial flow pattern, almost as though there was a draft tube around the impeller. Two or three or more impellers are used if tanks with high D/T ratios are involved. The fluidfoil impellers do not vortex vigorously even at relatively low coverage so that if gases or solids are to be incorporated at the surface, the axial-flow turbine is often required and can be used in combination with the fluidfoil impellers also on the same shaft. BLENDING If the blending process is between two or more fluids with relatively low viscosity such that the blending is not affected by fluid shear rates, then the difference in blend time and circulation between small and large tanks is the only factor involved. However, if the blending involves wide disparities in the density of viscosity and surface tension between the various phases, then a certain level of shear rate may be required before blending can proceed to the required degree of uniformity. The role of viscosity is a major factor in going from the turbulent regime, through the transition region, into the viscous regime and the change in the role of energy dissipation discussed previously. The role of non-newtonian viscosities comes into the picture very strongly since that tends to markedly change the type of influence of impellers and determines the appropriate geometry that is involved. There is the possibility of misinterpretation of the difference between circulation time and blend time. Circulation time is primarily a function of the pumping capacity of the impeller. For axial-flow impellers, a convenient parameter, but not particularly physically accurate, is to divide the pumping capacity of the impeller by the cross-sectional area of the tank to give a superficial liquid velocity. This is sometimes used by using the total volume of flow from the impeller including entrainment of the tank to obtain a superficial liquid velocity. As the flow from an impeller is increased from a given power level, there will be a higher fluid velocity and therefore a shorter circulation time. This holds true when dealing with any given impeller. This is shown in Fig. 18-18, which shows that circulation time versus D/T decreases. A major consideration is when increasing D/T becomes too large and actually causes the curve to reverse. This occurs somewhere around 0.45, ⫾0.05, so that using impellers of D/ T ratios of 0.6 to 0.8

18-14

LIQUID-SOLID OPERATIONS AND EQUIPMENT

FIG. 18-18

Effect of D/T ratio on two different impellers on the circulation time and the

blend time.

is often counterproductive for circulation time. They may be useful for the blending or motion of pseudoplastic fluids. When comparing different impeller types, an entirely different phenomenon is important. In terms of circulation time, the phenomena shown in Figs. 18-18 and 18-19 still apply with the different impellers shown in Fig. 18-5. When it comes to blending another factor enters the picture. When particles A and B meet each other as a result of shear rates, there has to be sufficient shear stress to cause A and B to blend, react, or otherwise participate in the process. It turns out that in low-viscosity blending the actual result does depend upon the measuring technique used to measure blend time. Two common techniques, which do not exhaust the possibilities in reported studies, are to use an acid-base indicator and inject an acid or base into the system that will result in a color change. One can also put a dye into the tank and measure the time for color to arrive at uniformity. Another system is to put in a conductivity probe and inject a salt or other electrolyte into the system. With any given impeller type at constant power, the circulation time will increase with the D/ T ratio of the impeller. Figure 18-18 shows that both circulation time and blend time decrease as D/T increases. The same is true for impeller speed. As impeller speed is increased with any impeller, blend time and circulation time are decreased (Fig. 18-19). However, when comparing different impeller types at the same power level, it turns out that impellers that have a higher pumping capacity will give decreased circulation time, but all the impellers, regardless of their pumping efficiency, give the same blend time at the

same power level and same diameter. This means that circulation time must be combined with shear rate to carry out a blending experiment which involves chemical reactions or interparticle mixing (Fig. 18-20). For other situations in low-viscosity blending, the fluid in tanks may become stratified. There are few studies on that situation, but Oldshue (op. cit.) indicates the relationship between some of the variables. The important difference is that blend time is inversely proportional to power, not impeller flow, so that the exponents are quite different for a stratified tank. This situation occurs more frequently in the petroleum industry, where large petroleum storage tanks become stratified either by filling techniques or by temperature fluctuations. There is a lot of common usage of the terms blend time, mixing time, and circulation time. There are differences in concept and interpretation of these different “times.” For any given experiment, one must pick a definition of blend time to be used. As an example, if one is measuring the fluctuation of concentration after an addition of material to the tank, then one can pick an arbitrary definition of blending such as reducing the fluctuations below a certain level. This often is chosen as a fluctuation equal to 5% of the original fluctuation when the feed material is added. This obviously is a function of the size of the probe used to measure these fluctuations, which often is on the order of 500 to 1000 µm. At the micro-scale level, there really is no way to measure concentration fluctuations. Resort must be made to other qualitative interpretation of results for either a process or a chemical reaction study. High-Viscosity Systems All axial-flow impellers become radial flow as Reynolds numbers approach the viscous region. Blending in

Effect of impeller speed and power for the same diameter on circulation time and blend time for a particular impeller.

FIG. 18-20 At constant power and constant impeller diameter, three different impellers give the same blend time but different circulation times.

FIG. 18-19

PHASE CONTACTING AND LIQUID-SOLID PROCESSING the transition and low-viscosity system is largely a measure of fluid motion throughout the tank. For close-clearance impellers, the anchor and helical impellers provide blending by having an effective action at the tank wall, which is particularly suitable for pseudoplastic fluids. Figure 18-21 gives some data on the circulation time of the helical impeller. It has been observed that it takes about three circulation times to get one blend time being the visual uniformity of a dye added to the material. This is a macro-scale blending definition. Axial-flow turbines are often used in blending pseudoplastic materials, and they are often used at relatively large D/T ratios, from 0.5 to 0.7, to adequately provide shear rate in the majority of the batch particularly in pseudoplastic material. These impellers develop a flow pattern which may or may not encompass an entire tank, and these areas of motion are sometimes referred to as caverns. Several papers describe the size of these caverns relative to various types of mixing phenomena. An effective procedure for the blending of pseudoplastic fluids is given in Oldshue (op. cit.). Chemical Reactions Chemical reactions are influenced by the uniformity of concentration both at the feed point and in the rest of the tank and can be markedly affected by the change in overall blend time and circulation time as well as the micro-scale environment. It is possible to keep the ratio between the power per unit volume at the impeller and in the rest of the tank relatively similar on scale-up, but many details need to be considered when talking about the reaction conditions, particularly where they involve selectivity. This means that reactions can take different paths depending upon chemistry and fluid mechanics, which is a major consideration in what should be examined. The method of introducing the reagent stream can be projected in several different ways depending upon the geometry of the impeller and feed system. Chemical reactions normally occur in the micro-scale range. In turbulent flow, almost all of the power dissipation occurs eventually in the micro-scale regime because that is the only place where the scale of the fluid fluctuations is small enough that viscous shear stress exists. At approximately 100 µm, the fluid does not know what type of impeller is used to generate the power; continuing down to 10 µm and, even further, to chemical reactions, the actual impeller type is not a major variable as long as the proper macro-scale regime has been provided throughout the entire tank. The intensity of the mixing environment in the micro-scale regime can be related to a series of variables in an increasing order of complexity. Since all of the power is ultimately dissipated in the micro-scale regime, the power per unit volume throughout the tank is one measure of the overall measure of micro-scale mixing and the power dissipation at individual volumes in the tank is another way of expressing the influence. In general, the power per unit

Effect of impeller speed on circulation time for a helical impeller in the Reynolds number arranged less than 10.

FIG. 18-21

18-15

volume dissipated around an impeller zone can be 100 times higher than the power dissipated throughout the remainder of the tank. The next level of complexity is to look at the rms velocity fluctuation, which is typically 50 percent of the mean velocity around the impeller zone and about 5 percent of the mean velocity in the rest of the vessel. This means that the feed introduction point for either a single reactant or several reactants can be of extreme importance. It seems that the selectivity of competing or consecutive chemical reactions can be a function of the rms velocity fluctuations in the feed point if the chemical reactants remain constant and involve an appropriate relationship to the time between the rms velocity fluctuations. There are three common ways of introducing reagents into a mixing vessel. One is to let them drip on the surface. The second is to use some type of introduction pipe to bring the material into various parts of the vessel. The third is to purposely bring them in and around the impeller zone. Generally, all three methods have to be tried before determining the effect of feed location. Since chemical reactions are on a scale much below 1 µm, and it appears that the Komolgoroff scale of isotropic turbulence turns out to be somewhere between 10 and 30 µm, other mechanisms must play a role in getting materials in and out of reaction zones and reactants in and out of those zones. One cannot really assign a shear rate magnitude to the area around a micro-scale zone, and it is primarily an environment that particles and reactants witness in this area. The next level of complexity looks at the kinetic energy of turbulence. There are several models that are used to study the fluid mechanics, such as the Kε model. One can also put the velocity measurements through a spectrum analyzer to look at the energy at various wave numbers. In the viscous regime, chemical reactants become associated with each other through viscous shear stresses. These shear stresses exist at all scales (macro to micro) and until the power is dissipated continuously through the entire spectrum. This gives a different relationship for power dissipation than in the case of turbulent flow. Solid-Liquid The most-used technique to study solid suspension, as documented in hundreds of papers in the literature, is called the speed for just suspension, NJS. The original work was done in 1958 by Zwietering and this is still the most extensive range of variables, although other investigators have added to it considerably. This particular technique is suitable only for laboratory investigation using tanks that are transparent and well illuminated. It does not lend itself to evaluation of the opaque tanks, nor is it used in any study of large-scale tanks in the field. It is a very minimal requirement for uniformity, and definitions suggested earlier are recommended for use in industrial design. Some Observations on the Use of NJS With D/T ratios of less than 0.4, uniformity throughout the rest of the tank is minimal. In D/T ratios greater than 0.4, the rest of the tank has a very vigorous fluid motion with marked approach to complete uniformity before NJS is reached. Much of the variation in NJS can be reduced by using PJS, which is the power in the just-suspended state. This also gives a better feel for the comparison of various impellers based on the energy requirement rather than speed, which has no economic relevance. The overall superficial fluid velocity, mentioned earlier, should be proportional to the settling velocity of the solids if that were the main mechanism for solid suspension. If this were the case, the requirement for power if the settling velocity were doubled should be eight times. Experimentally, it is found that the increase in power is more nearly four times, so that some effect of the shear rate in macro-scale turbulence is effective in providing uplift and motion in the system. Picking up the solids at the bottom of the tank depends upon the eddies and velocity fluctuations in the lower part of the tank and is a different criterion from the flow pattern required to keep particles suspended and moving in various velocity patterns throughout the remainder of the vessel. This leads to the variables in the design equation and a relationship that is quite different when these same variables are studied in relation to complete uniformity throughout the mixing vessel. Another concern is the effect of multiple particle sizes. In general, the presence of fine particles will affect the requirements of suspen-

18-16

LIQUID-SOLID OPERATIONS AND EQUIPMENT

sion of larger particles. The fine particles act largely as a potential viscosity-increasing agent and give a similar result to what would happen if the viscosity of the continuous phase were increased. Another phenomenon is the increase in power required with percent solids, which makes a dramatic change at approximately 40 percent by volume, and then dramatically changes again as we approach the ultimate weight percent of settled solids. This phenomenon is covered by Oldshue (op. cit.), who describes conditions required for mixing slurries in the 80 to 100 percent range of the ultimate weight percent of settled solids. Solids suspension in general is not usually affected by blend time or shear-rate changes in the relatively low to medium solids concentration in the range from 0 to 40 percent by weight. However, as solids become more concentrated, the effect of solids concentration on power required gives a change in criterion from the settling velocity of the individual particles in the mixture to the apparent viscosity of the more concentrated slurry. This means that we enter into an area where the blending of non-newtonian fluid regions affects the shear rates and plays a marked role. The suspension of a single solid particle should depend primarily on the upward velocity at a given point and also should be affected by the uniformity of this velocity profile across the entire tank cross section. There are upward velocities in the tank and there also must be corresponding downward velocities. In addition to the effect of the upward velocity on a settling particle, there is also the random motion of the micro-scale environment, which does not affect large particles very much but is a major factor in the concentration and uniformity of particles in the transition and micro-scale size range. Using a draft tube in the tank for solids suspension introduces another, different set of variables. There are other relationships that are very much affected by scale-up in this type of process, as shown in Fig. 18-22. Different scale-up problems exist whether the impeller is pumping up or down within the draft tube. Solid Dispersion If the process involves the dispersion of solids in a liquid, then we may either be involved with breaking up agglomerates or possibly physically breaking or shattering particles that have a low cohesive force between their components. Normally, we do not think of breaking up ionic bonds with the shear rates available in mixing machinery. If we know the shear stress required to break up a particle, we can then determine the shear rate required from the machinery by various viscosities with the equation: Shear stress = viscosity (shear rate)

The shear rate available from various types of mixing and dispersion devices is known approximately and also the range of viscosities in which they can operate. This makes the selection of the mixing equipment subject to calculation of the shear stress required for the viscosity to be used. In the equation referred to above, it is assumed that there is 100 percent transmission of the shear rate in the shear stress. However, with the slurry viscosity determined essentially by the properties of the slurry, at high concentrations of slurries there is a slippage factor. Internal motion of particles in the fluids over and around each other can reduce the effective transmission of viscosity efficiencies from 100 percent to as low as 30 percent. Animal cells in biotechnology do not normally have tough skins like those of fungal cells and they are very sensitive to mixing effects. Many approaches have been and are being tried to minimize the effect of increased shear rates on scale-up. These include encapsulating the organism in or on microparticles and/or conditioning cells selectively to shear rates. In addition, traditional fermentation processes have maximum shear-rate requirements in which cells become progressively more and more damaged until they become motile. Solid-Liquid Mass Transfer There is potentially a major effect of both shear rate and circulation time in these processes. The solids can either be fragile or rugged. We are looking at the slip velocity of the particle and also whether we can break up agglomerates of particles which may enhance the mass transfer. When the particles become small enough, they tend to follow the flow pattern, so the slip velocity necessary to affect the mass transfer becomes less and less available. What this shows is that, from the definition of off-bottom motion to complete uniformity, the effect of mixer power is much less than from going to on-bottom motion to off-bottom suspension. The initial increase in power causes more and more solids to be in active communication with the liquid and has a much greater mass-transfer rate than that occurring above the power level for off-bottom suspension, in which slip velocity between the particles of fluid is the major contributor (Fig. 18-23). Since there may well be chemical or biological reactions happening on or in the solid phase, depending upon the size of the process participants, macro- or micro-scale effects may or may not be appropriate to consider. In the case of living organisms, their access to dissolved oxygen throughout the tank is of great concern. Large tanks in the fermentation industry often have a Z/T ratio of 2:1 to 4:1; thus, top-to-bottom blending can be a major factor. Some biological particles are facultative and can adapt and reestablish their metabolisms at different dissolved-oxygen levels. Other organisms are irreversibly destroyed by sufficient exposure to low dissolved-oxygen levels. GAS-LIQUID SYSTEMS Gas-Liquid Dispersion This involves physical dispersion of gas bubbles by the impeller, and the effect of gas flow on the impeller.

Relative change in solid-liquid mass-transfer ratio with three different suspension levels, i.e., on-bottom motion, off-bottom motion, and complete uniformity. FIG. 18-23

Typical draft tube circulator, shown here for down-pumping mode for the impeller in the draft tube.

FIG. 18-22

PHASE CONTACTING AND LIQUID-SOLID PROCESSING The observation of the physical appearance of a tank undergoing gasliquid mass transfer can be helpful but is not a substitute for masstransfer data on the actual process. The mixing vessel can have four regimes of visual comparisons between gas bubbles and flow patterns. A helpful parameter is the ratio between the power given up by the gas phase and the power introduced by the mixing impeller. In general, if the power in the gas stream (calculated as the expansion energy from the gas expanding from the sparging area to the top of the tank, shown in Fig. 18-24) is greater, there will be considerable blurping and entrainment of liquid drops by a very violent explosion of gas bubbles at the surface. If the power level is more than the expanding gas energy, then the surface action will normally be very coalescent and uniform by comparison, and the gas will be reasonably well distributed throughout the remainder of the tank. With power levels up to 10 to 100 times the gas energy, the impeller will cause a more uniform and vigorous dispersion of the gas bubbles and smaller gas bubbles in the vessel. In the 1960s and before, most gas-liquid operations were conducted using flat-blade turbines as shown in Fig. 18-4. These impellers required input of approximately three times the energy in the gas stream before they completely control the flow pattern. This was usually the case, and the mass-transfer characteristics were comparable to what would be expected. One disadvantage of the radial-flow impeller is that it is a very poor blending device so blend time is very long compared to that in pilot-scale experiments and compared to the fluidfoil impeller types often used currently. Using curvature of the blades to modify the tendency of gas bubbles to streamline the back of the flatblade turbine gives a different characteristic to the power drawn by the impeller at a given gas rate compared to no gas rate, but it seems to give quite similar mass transfer at power levels similar to those of the flatblade design. In order to improve the blending and solid-suspension characteristics, fluidfoil impellers (typified by the A315, Fig. 18-25) have been introduced in recent years and they have many of the advantages and some of the disadvantages of the flat-blade turbine. These impellers typically have a very high solidity ratio, on the order of 0.85 or more, and produce a strong axial downflow at low gas rate. As the gas rate increases, the flow pattern becomes more radial due to the upflow of the gas counteracting the downward flow of the impeller. Mass-transfer characteristics on large-scale equipment seem to be quite similar, but the fluidfoil impellers tend to release a larger-

Typical arrangement of Rushton radial-flow R100 flat-blade turbine with typical sparge ring for gas-liquid mass transfer.

FIG. 18-24

18-17

An impeller designed for gas-liquid dispersion and mass transfer of the fluidfoil type, i.e., A315.

FIG. 18-25

diameter bubble than is common with the radial-flow turbines. The blend time is one-half or one-third as long, and solid-suspension characteristics are better so that there have been notable improved process results with these impellers. This is particularly true if the process requires better blending and there is solid suspension. If this is not the case, the results from these impellers can be negative compared to radial-flow turbines. It is very difficult to test these impellers on a small scale, since they provide better blending on a pilot scale where blending is already very effective compared to the large scale. Caution is recommended if it is desirable to study these impellers in pilot-scale equipment. Gas-Liquid Mass Transfer Gas-liquid mass transfer normally is correlated by means of the mass-transfer coefficient K g a versus power level at various superficial gas velocities. The superficial gas velocity is the volume of gas at the average temperature and pressure at the midpoint in the tank divided by the area of the vessel. In order to obtain the partial-pressure driving force, an assumption must be made of the partial pressure in equilibrium with the concentration of gas in the liquid. Many times this must be assumed, but if Fig. 18-26 is obtained in the pilot plant and the same assumption principle is used in evaluating the mixer in the full-scale tank, the error from the assumption is limited.

Typical curve for mass transfer coefficient Kga as a function of mixer power and superficial gas velocity. FIG. 18-26

18-18

LIQUID-SOLID OPERATIONS AND EQUIPMENT

In the plant-size unit, Fig. 18-26 must be translated into a masstransfer-rate curve for the particular tank volume and operating condition selected. Every time a new physical condition is selected, a different curve similar to that of Fig. 18-27 is obtained. Typical exponents on the effect of power and gas rate on K g a tend to be around 0.5 for each variable, ⫾0.1. Viscosity markedly changes the picture and, usually, increasing viscosity lowers the mass-transfer coefficient. For the common application of waste treating and for some of the published data on biological slurries, data for kLa (shown in Fig. 18-28) is obtained in the literature. For a completely new gas or liquid of a liquid slurry system, Fig. 18-26 must be obtained by an actual experiment. Liquid-Gas-Solid Systems Many gas-liquid systems contain solids that may be the ultimate recipient of the liquid-gas-solid mass transfer entering into the process result. Examples are biological processes in which the biological solids are the user of the mass transfer of the mixing-flow patterns, various types of slurries reactors in which the solids either are being reactive or there may be extraction or dissolving taking place, or there may be polymerization or precipitation of solids occurring. Normally there must be a way of determining whether the masstransfer rate with the solids is the key controlling parameter or the gas-liquid mass transfer rate. In general, introduction of a gas stream to a fluid will increase the blend time because the gas-flow patterns are counterproductive to the typical mixer-flow patterns. In a similar vein, the introduction of a gas stream to a liquid-solid suspension will decrease the suspension uniformity because the gas-flow pattern is normally counterproductive to the mixer-flow pattern. Many times the power needed for the gasliquid mass transfer is higher than the power needed for solid suspension, and the effect of the gas flow on the solid suspensions are of little concern. On the other hand, if power levels are relatively low and solid-suspension characteristics are critical—examples being the case of activated sludge reactors in the waste-treating field or biological solid reactors in the hydrometallurgical field—then the effect of the gas-flow pattern of the mixing system can be quite critical to the overall design. Another common situation is batch hydrogenation, in which pure hydrogen is introduced to a relatively high pressure reactor and a decision must be made to recycle the unabsorbed gas stream from the top of the reactor or use a vortexing mode for an upper impeller to incorporate the gas from the surface. Loop Reactors For some gas-liquid-solid processes, a recirculating loop can be an effective reactor. These involve a relatively high horsepower pumping system and various kinds of nozzles, baffles, and turbulence generators in the loop system. These have power levels

Usually, the gas-liquid mass-transfer coefficient, Kga, is reduced with increased viscosity. This shows the effect of increased concentration of microbial cells in a fermentation process. FIG. 18-28

anywhere from 1 to 10 times higher than the power level in a typical mixing reactor, and may allow the retention time to be less by a factor of 1 to 10. LIQUID-LIQUID CONTACTING Emulsions Almost every shear rate parameter affects liquidliquid emulsion formation. Some of the effects are dependent upon whether the emulsion is both dispersing and coalescing in the tank, or whether there are sufficient stabilizers present to maintain the smallest droplet size produced for long periods of time. Blend time and the standard deviation of circulation times affect the length of time it takes for a particle to be exposed to the various levels of shear work and thus the time it takes to achieve the ultimate small particle size desired. The prediction of drop sizes in liquid-liquid systems is difficult. Most of the studies have used very pure fluids as two of the immiscible liquids, and in industrial practice there almost always are other chemicals that are surface-active to some degree and make the prediction of absolute drop sizes very difficult. In addition, techniques to measure drop sizes in experimental studies have all types of experimental and interpretation variations and difficulties so that many of the equations and correlations in the literature give contradictory results under similar conditions. Experimental difficulties include dispersion and coalescence effects, difficulty of measuring actual drop size, the effect of visual or photographic studies on where in the tank you can make these observations, and the difficulty of using probes that measure bubble size or bubble area by light or other sample transmission techniques which are very sensitive to the concentration of the dispersed phase and often are used in very dilute solutions. It is seldom possible to specify an initial mixer design requirement for an absolute bubble size prediction, particularly if coalescence and dispersion are involved. However, if data are available on the actual system, then many of these correlations could be used to predict relative changes in drop size conditions with changes in fluid properties or impeller variables. STAGEWISE EQUIPMENT: MIXER-SETTLERS

Example of a specific chart to analyze the total mass-transfer rate in a particular tank under a process condition obtained from basic Kga data shown in Fig. 18-25. FIG. 18-27

Introduction Insoluble liquids may be brought into direct contact to cause transfer of dissolved substances, to allow transfer of heat, and to promote chemical reaction. This subsection concerns the design and selection of equipment used for conducting this type of liquid-liquid contact operation. Objectives There are four principal purposes of operations involving the direct contact of immiscible liquids. The purpose of a particular contact operation may involve any one or any combination of the following objectives: 1. Separation of components in solution. This includes the ordinary objectives of liquid extraction, in which the constituents of a solution are separated by causing their unequal distribution between two insoluble liquids, the washing of a liquid with another to remove small

PHASE CONTACTING AND LIQUID-SOLID PROCESSING amounts of a dissolved impurity, and the like. The theoretical principles governing the phase relationships, material balances, and number of ideal stages or transfer units required to bring about the desired changes are to be found in Sec. 15. Design of equipment is based on the quantities of liquids and the efficiency and operating characteristics of the type of equipment selected. 2. Chemical reaction. The reactants may be the liquids themselves, or they may be dissolved in the insoluble liquids. The kinetics of this type of reaction are treated in Sec. 4. 3. Cooling or heating a liquid by direct contact with another. Although liquid-liquid-contact operations have not been used widely for heat transfer alone, this technique is one of increasing interest. Applications also include cases in which chemical reaction or liquid extraction occurs simultaneously. 4. Creating permanent emulsions. The objective is to disperse one liquid within another in such finely divided form that separation by settling either does not occur or occurs extremely slowly. The purpose is to prepare the emulsion. Neither extraction nor chemical reaction between the liquids is ordinarily sought. Liquid-liquid contacting equipment may be generally classified into two categories: stagewise and continuous (differential) contact. The function of a stage is to contact the liquids, allow equilibrium to be approached, and to make a mechanical separation of the liquids. The contacting and separating correspond to mixing the liquids, and settling the resulting dispersion; so these devices are usually called mixer-settlers. The operation may be carried out in batch fashion or with continuous flow. If batch, it is likely that the same vessel will serve for both mixing and settling, whereas if continuous, separate vessels are usually but not always used. Mixer-Settler Equipment The equipment for extraction or chemical reaction may be classified as follows: I. Mixers A. Flow or line mixers 1. Mechanical agitation 2. No mechanical agitation B. Agitated vessels 1. Mechanical agitation 2. Gas agitation II. Settlers A. Nonmechanical 1. Gravity 2. Centrifugal (cyclones) B. Mechanical (centrifuges) C. Settler auxiliaries 1. Coalescers 2. Separator membranes 3. Electrostatic equipment In principle, at least, any mixer may be coupled with any settler to provide the complete stage. There are several combinations which are especially popular. Continuously operated devices usually, but not always, place the mixing and settling functions in separate vessels. Batch-operated devices may use the same vessel alternately for the separate functions. Flow or Line Mixers Definition Flow or line mixers are devices through which the liquids to be contacted are passed, characterized principally by the very small time of contact for the liquids. They are used only for continuous operations or semibatch (in which one liquid flows continuously and the other is continuously recycled). If holding time is required for extraction or reaction, it must be provided by passing the mixed liquids through a vessel of the necessary volume. This may be a long pipe of large diameter, sometimes fitted with segmental baffles, but frequently the settler which follows the mixer serves. The energy for mixing and dispersing usually comes from pressure drop resulting from flow. There are many types, and only the most important can be mentioned here. [See also Hunter, in Dunstan (ed.), Science of Petroleum, vol. 3, Oxford, New York, 1938, pp. 1779–1797.] They are used fairly extensively in treating petroleum distillates, in vegetable-oil, refining, in extraction of phenol-bearing coke-oven liquors, in some metal extractions, and the like. Kalichevsky and Kobe (Petroleum Refining

18-19

with Chemicals, Elsevier, New York, 1956) discuss detailed application in the refining of petroleum. Jet Mixers These depend upon impingement of one liquid on the other to obtain a dispersion, and one of the liquids is pumped through a small nozzle or orifice into a flowing stream of the other. Both liquids are pumped. They can be used successfully only for liquids of low interfacial tension. See Fig. 18-29 and also Hunter and Nash [Ind. Chem., 9, 245, 263, 317 (1933)]. Treybal (Liquid Extraction, 2d ed., McGraw-Hill, New York, 1963) describes a more elaborate device. For a study of the extraction of antibiotics with jet mixers, see Anneskova and Boiko, Med. Prom. SSSR, 13(5), 26 (1959). Insonation with ultrasound of a toluene-water mixture during methanol extraction with a simple jet mixer improves the rate of mass transfer, but the energy requirements for significant improvement are large [Woodle and Vilbrandt, Am. Inst. Chem. Eng. J., 6, 296 (1960)]. Injectors The flow of one liquid is induced by the flow of the other, with only the majority liquid being pumped at relatively high velocity. Figure 18-30 shows a typical device used in semibatch fashion for washing oil with a recirculated wash liquid. It is installed directly in the settling drum. See also Hampton (U.S. Patent 2,091,709, 1933), Sheldon (U.S. Patent 2,009,347, 1935), and Ng (U.S. Patent 2,665,975, 1954). Folsom [Chem. Eng. Prog., 44, 765 (1948)] gives a good review of basic principles. The most thorough study for extraction is provided by Kafarov and Zhukovskaya [Zh. Prikl. Khim., 31, 376 (1958)], who used very small injectors. With an injector measuring 73 mm from throat to exit, with 2.48-mm throat diameter, they extracted benzoic acid and acetic acid from water with carbon tetrachloride at the rate of 58 to 106 L/h, to obtain a stage efficiency E = 0.8 to 1.0. Data on flow characteristics are also given. Boyadzhiev and Elenkov [Collect. Czech. Chem. Commun., 31, 4072 (1966)] point out that the presence of surface-active agents exerts a profound influence on drop size in such devices. Orifices and Mixing Nozzles Both liquids are pumped through constrictions in a pipe, the pressure drop of which is partly utilized to create the dispersion (see Fig. 18-31). Single nozzles or several in series may be used. For the orifice mixers, as many as 20 orifice plates each with 13.8-kPa (2-lb/in2) pressure drop may be used in series [Morell and Bergman, Chem. Metall. Eng., 35, 211 (1928)]. In the Dualayer process for removal of mercaptans from gasoline, 258 m3/h (39,000 bbl/day) of oil and treating solution are contacted with 68.9kPa (10-lb/in2) pressure drop per stage [Greek et al., Ind. Eng. Chem., 49, 1938 (1957)]. Holland et al. [Am. Inst. Chem. Eng. J., 4, 346 (1958); 6, 615 (1960)] report on the interfacial area produced between two immiscible liquids entering a pipe (diameter 0.8 to 2.0 in) from an orifice, γ D = 0.02 to 0.20, at flow rates of 0.23 to 4.1 m3/h (1 to 18 gal/min). At a distance 17.8 cm (7 in) downstream from the orifice, 0.158 0.117 dt 4 0.179 σ兹g苶ρ c苶苶 av aav = ᎏ (CO2 ∆p)0.75 ᎏ γ D0.878 (18-8) ᎏ −1 σgc µD dO where aav = interfacial surface, cm2/cm3; CO = orifice coefficient, dimensionless; dt = pipe diameter, in; dO = orifice diameter, in; gc =

冢

FIG. 18-29

Elbow jet mixer.

冣 冤冢 冣 冥

18-20

LIQUID-SOLID OPERATIONS AND EQUIPMENT the extraction of slurries, as in the extraction of uranyl nitrate from acid-uranium-ore slurries [Chem. Eng., 66, 30 (Nov. 2, 1959)]. Shaw and Long [Chem. Eng., 64(11), 251 (1957)] obtain a stage efficiency of 100 percent (E = 1.0) in a uranium-ore-slurry extraction with an open impeller pump. In order to avoid emulsification difficulties in these extractions, it is necessary to maintain the organic phase continuous, if necessary by recycling a portion of the settled organic liquid to the mixer. Agitated Line Mixer See Fig. 18-32. This device, which combines the features of orifice mixers and agitators, is used extensively in treating petroleum and vegetable oils. It is available in sizes to fit a- to 10-in pipe. The device of Fig. 18-33, with two impellers in separate stages, is available in sizes to fit 4- to 20-in pipe. Packed Tubes Cocurrent flow of immiscible liquids through a packed tube produces a one-stage contact, characteristic of line mixers. For flow of isobutanol-water* through a 0.5-in diameter tube packed with 6 in of 3-mm glass beads, Leacock and Churchill [Am. Inst. Chem. Eng. J., 7, 196 (1961)] find kCaav = c1 LC0.5LD kD aav = c2 LC0.75LD0.75

FIG. 18-30

Injector mixer. (Ayres, U.S. Patent 2,531,547, 1950.)

gravitational conversion factor, (32.2 lbm⋅ft)/(lbf⋅s2); ∆p = pressure drop across orifice, lbf/ft2; µD = viscosity of dispersed phase, lbm/(ft⋅s); ρav = density of dispersed phase, lbm/ft; and σ = interfacial tension, lbf/ft. See also Shirotsuka et al. [Kagaku Kogaku, 25, 109 (1961)]. Valves Valves may be considered to be adjustable orifice mixers. In desalting crude petroleum by mixing with water, Hayes et al. [Chem. Eng. Prog., 45, 235 (1949)] used a globe-valve mixer operating at 110- to 221-kPa (16- to 32-lb/in2) pressure drop for mixing 66 m3/h (416 bbl/h) oil with 8 m3/h (50 bbl/h) water, with best results at the lowest value. Simkin and Olney [Am. Inst. Chem. Eng. J., 2, 545 (1956)] mixed kerosine and white oil with water, using 0.35- to 0.62-kPa (0.05- to 0.09-lb/in2) pressure drop across a 1-in gate valve, at 22-m3/h (10-gal/min) flow rate for optimum separating conditions in a cyclone, but higher pressure drops were required to give good extractor efficiencies. Pumps Centrifugal pumps, in which the two liquids are fed to the suction side of the pump, have been used fairly extensively, and they offer the advantage of providing interstage pumping at the same time. They have been commonly used in the extraction of phenols from coke-oven liquors with light oil [Gollmar, Ind. Eng. Chem., 39, 596, 1947); Carbone, Sewage Ind. Wastes, 22, 200 (1950)], but the intense shearing action causes emulsions with this low-interfacial-tension system. Modern plants use other types of extractors. Pumps are useful in

FIG. 18-31

Orifice mixer and nozzle mixer.

(18-9) (18-10)

where c1 = 0.00178 using SI units and 0.00032 using U.S. customary units; and c2 = 0.0037 using SI units and 0.00057 using U.S. customary units. These indicate a stage efficiency approaching 100 percent. Organic-phase holdup and pressure drop for larger pipes similarly packed are also available [Rigg and Churchill, ibid., 10, 810 (1964)]. Pipe Lines The principal interest here will be for flow in which one liquid is dispersed in another as they flow cocurrently through a pipe (stratified flow produces too little interfacial area for use in liquid extraction or chemical reaction between liquids). Drop size of dispersed phase, if initially very fine at high concentrations, increases as the distance downstream increases, owing to coalescence [see Holland, loc. cit.; Ward and Knudsen, Am. Inst. Chem. Eng. J., 13, 356 (1967)]; or if initially large, decreases by breakup in regions of high shear [Sleicher, ibid., 8, 471 (1962); Chem. Eng. Sci., 20, 57 (1965)]. The maximum drop size is given by (Sleicher, loc. cit.)

* Isobutanol dispersed: LD = 3500 to 27,000; water continuous; LC = 6000 to 32,000 in pounds-mass per hour-square foot (to convert to kilograms per second-square meter, multiply by 1.36 × 10−3).

FIG. 18-32

Nettco Corp. Flomix.

PHASE CONTACTING AND LIQUID-SOLID PROCESSING

18-21

essentially the same as those for dispersing finely divided solids in liquids, which are fully discussed in Sec. 18. The following summary of operating characteristics of mechanically agitated vessels is confined to the data available on liquid-liquid contacting. Phase Dispersed There is an ill-defined upper limit to the volume fraction of dispersed liquid which may be maintained in an agitated dispersion. For dispersions of organic liquids in water [Quinn and Sigloh, Can. J. Chem. Eng., 41, 15 (1963)], C γ Do,max = γ ′ + ᎏ3 (18-13) N where γ ′ is a constant, asymptotic value, and C is a constant, both depending in an unestablished manner upon the system physical properties and geometry. Thus, inversion of a dispersion may occur if the agitator speed is increased. With systems of low interfacial tension (σ′ = 2 to 3 mN/m or 2 to 3 dyn/cm), γ D as high as 0.8 can be maintained. Selker and Sleicher [Can. J. Chem. Eng., 43, 298 (1965)] and Yeh et al. [Am. Inst. Chem. Eng. J., 10, 260 (1964)] feel that the viscosity ratio of the liquids alone is important. Within the limits in which either phase can be dispersed, for batch operation of baffled vessels, that phase in which the impeller is immersed when at rest will normally be continuous [Rodger, Trice, and Rushton, Chem. Eng. Prog., 52, 515 (1956); Laity and Treybal, Am. Inst. Chem. Eng. J., 3, 176 (1957)]. With water dispersed, dual emulsions (continuous phase found as small droplets within larger drops of dispersed phase) are possible. In continuous operation, the vessel is first filled with the liquid to be continuous, and agitation is then begun, after which the liquid to be dispersed is introduced. Uniformity of Mixing This refers to the gross uniformity throughout the vessel and not to the size of the droplets produced. For unbaffled vessels, batch, with an air-liquid interface, Miller and Mann [Trans. Am. Inst. Chem. Eng., 40, 709 (1944)] mixed water with several organic liquids, measuring uniformity of mixing by sampling the tank at various places, comparing the percentage of dispersed phase found with that in the tank as a whole. A power application of 200 to 400 W/m3 [(250 to 500 ft⋅lb)/(min⋅ft3)] gave maximum and nearly uniform performance for all. See also Nagata et al. [Chem. Eng. ( Japan), 15, 59 (1951)]. For baffled vessels operated continuously, no air-liquid interface, flow upward, light liquid dispersed [Treybal, Am. Inst. Chem. Eng. J., 4, 202 (1958)], the average fraction of dispersed phase in the vessel γ D,av is less than the fraction of the dispersed liquid in the feed mixture, unless the impeller speed is above a certain critical value which depends upon vessel geometry and liquid properties. Thornton and Bouyatiotis [Ind. Chem., 39, 298 (1963); Inst. Chem. Eng. Symp. Liquid Extraction, Newcastle-upon-Tyne, April 1967] have presented correlations of data for a 17.8-cm (7-in) vessel, but these do not agree with observations on 15.2- and 30.5-cm (6- and 12-in) vessels in Treybal’s laboratory. See also Kovalev and Kagan [Zh. Prikl. Khim., 39, 1513 (1966)] and Trambouze [Chem. Eng. Sci., 14, 161 (1961)]. Stemerding et al. [Can. J. Chem. Eng., 43, 153 (1965)] present data on a large mixing tank [15 m3 (530 ft3)] fitted with a marine-type propeller and a draft tube. Drop Size and Interfacial Area The drops produced have a size range [Sullivan and Lindsey, Ind. Eng. Chem. Fundam., 1, 87 (1962); Sprow, Chem. Eng. Sci., 22, 435 (1967); and Chen and Middleman, Am. Inst. Chem. Eng. J., 13, 989 (1967)]. The average drop size may be expressed as

冢 冣

FIG. 18-33

Lightnin line blender. (Mixing Equipment Co., Inc., with per-

mission.)

dp,maxρCV 2 ᎏᎏ σgc

µ V µ V ᎏ = C 冤1 + 0.7 冢 ᎏ 冣 冥 冪莦 σg σg C

0.7

D

c

(18-11)

c

where C = 43 (dt = 0.013 m or 0.0417 ft) or 38 (dt = 0.038 m or 0.125 ft), with dp,av = dp,max /4 for high flow rates and dp,max /13 for low velocities. Extensive measurements of the rate of mass transfer between n-butanol and water flowing in a 0.008-m (0.314-in) ID horizontal pipe are reported by Watkinson and Cavers [Can. J. Chem. Eng., 45, 258 (1967)] in a series of graphs not readily reproduced here. Length of a transfer unit for either phase is strongly dependent upon flow rate and passes through a pronounced maximum at an organic-water phase ratio of 0.5. In energy (pressure-drop) requirements and volume, the pipe line compared favorably with other types of extractors. Boyadzhiev and Elenkov [Chem. Eng. Sci., 21, 955 (1966)] concluded that, for the extraction of iodine between carbon tetrachloride and water in turbulent flow, drop coalescence and breakup did not influence the extraction rate. Yoshida et al. [Coal Tar (Japan), 8, 107 (1956)] provide details of the treatment of crude benzene with sulfuric acid in a 1-in diameter pipe, NRe = 37,000 to 50,000. Fernandes and Sharma [Chem. Eng. Sci., 23, 9 (1968)] used cocurrent flow downward of two liquids in a pipe, agitated with an upward current of air. The pipe has also been used for the transfer of heat between two immiscible liquids in cocurrent flow. For hydrocarbon oil-water, the heat-transfer coefficient is given by γ DN 6/5 Uaav d 2t We,t ᎏ = ᎏᎏᎏ kto kto vkto ᎏᎏ + ᎏᎏ 0.415ktC 0.173ktD

(18-12)

for γ D = 0 to 0.2. Additional data for γ D = 0.4 to 0.8 are also given. Data for stratified flow are given by Wilke et al. [Chem. Eng. Prog., 59, 69 (1963)] and Grover and Knudsen [Chem. Eng. Prog., 51, Symp. Ser. 17, 71 (1955)]. Mixing in Agitated Vessels Agitated vessels may frequently be used for either batch or continuous service and for the latter may be sized to provide any holding time desired. They are useful for liquids of any viscosity up to 750 Pa⋅s (750,000 cP), although in contacting two liquids for reaction or extraction purposes viscosities in excess of 0.1 Pa⋅s (100 cP) are only rarely encountered. Mechanical Agitation This type of agitation utilizes a rotating impeller immersed in the liquid to accomplish the mixing and dispersion. There are literally hundreds of devices using this principle, the major variations being found when chemical reactions are being carried out. The basic requirements regarding shape and arrangement of the vessel, type and arrangement of the impeller, and the like are

冱 ni dpi3 dp,av = ᎏ2 冱 ni d pi

(18-14)

and if the drops are spherical, 6γ D,av (18-15) aav = ᎏ dp,av The drop size varies locally with location in the vessel, being smallest at the impeller and largest in regions farthest removed from the impeller owing to coalescence in regions of relatively low turbulence

18-22

LIQUID-SOLID OPERATIONS AND EQUIPMENT

intensity [Schindler and Treybal, Am. Inst. Chem. Eng. J., 14, 790 (1968); Vanderveen, U.S. AEC UCRL-8733, 1960]. Interfacial area and hence average drop size have been measured by light transmittance, light scattering, direct photography, and other means. Typical of the resulting correlations is that of Thornton and Bouyatiotis (Inst. Chem. Eng. Symp. Liquid Extraction, Newcastle-upon-Tyne, April 1967) for a 17.8-cm- (7-in-) diameter baffled vessel, six-bladed flatblade turbine, di = 6.85 cm (0.225 ft), operated full, for organic liquids (σ′ = 8.5 to 34, ρD = 43.1 to 56.4, µD = 1.18 to 1.81) dispersed in water, in the absence of mass transfer, and under conditions giving nearly the vessel-average dp,av: dp,av σ2g c2 µ 4c g 0.62 ∆ρ 0.05 = 1 + 1.18φD ᎏ (18-16) ᎏ ᎏ ᎏ 0 0 2 dp d p µc g ∆ρ σ3g c3 ρc 0 where d p is given by (d 0p)3ρ C2 g P 3gc3 −0.32 ρCσ3gc3 0.14 = 29.0 ᎏ (18-17) ᎏ ᎏ 3 2 2 µc v ρ c µc g4 µ 4c g Caution is needed in using such correlations, since those available do not generally agree with each other. For example, Eq. (21-28) gives dp,av = 4.78(10−4) ft for a liquid pair of properties a′ = 30, ρC = 62.0, ρD = 52.0, µC = 2.42, µD = 1.94, γ D,av = 0.20 in a vessel T = Z = 0.75, a turbine impeller di = 0.25 turning at 400 r/min. Other correlations provide 3.28(10−4) [Thornton and Bouyatiotis, Ind. Chem., 39, 298 (1963)], 8.58(10−4) [Calderbank, Trans. Inst. Chem. Eng. (London), 36, 443 (1958)], 6.1(10−4) [Kafarov and Babinov, Zh. Prikl. Khim, 32, 789 (1959)], and 2.68(10−3) (Rushton and Love, paper at AIChE, Mexico City, September 1967). See also Vermeulen et al. [Chem. Eng. Prog., 51, 85F (1955)], Rodgers et al. [ibid., 52, 515 (1956); U.S. AEC ANL-5575 (1956)], Rodrigues et al. [Am. Inst. Chem. Eng. J., 7, 663 (1961)], Sharma et al. [Chem. Eng. Sci., 21, 707 (1966); 22, 1267 (1967)], and Kagan and Kovalev [Khim. Prom., 42, 192 (1966)]. For the effect of absence of baffles, see Fick et al. (U.S. AEC UCRL-2545, 1954) and Schindler and Treybal [Am. Inst. Chem. Eng. J., 14, 790 (1968)]. The latter have observations during mass transfer. Coalescence Rates The droplets coalesce and redisperse at rates that depend upon the vessel geometry, N, γ D,av, and liquid properties. The few measurements available, made with a variety of techniques, do not as yet permit quantitative estimates of the coalescence frequency v. Madden and Damarell [Am. Inst. Chem. Eng. J., 8, 233 0.5 (1962)] found for baffled vessels that v varied as N 2.2γ D,av , and this has generally been confirmed by Groothius and Zuiderweg [Chem. Eng. Sci., 19, 63 (1964)], Miller et al. [Am. Inst. Chem. Eng. J., 9, 196 (1963)], and Howarth [ibid., 13, 1007 (1967)], although absolute values of v in the various studies are not well related. Hillestad and Rushton (paper at AIChE, Columbus, Ohio, May 1966), on the other hand, find v to vary as N 0.73γ D,av for impeller Weber numbers NWe,i below 1.58 a certain critical value and as N −3.5γ D,av for higher Weber numbers. The influence of liquid properties is strong. There is clear evidence [Groothius and Zuiderweg, loc. cit.; Chem. Eng. Sci., 12, 288 (1960)] that coalescence rates are enhanced by mass transfer from a drop to the surrounding continuum and retarded by transfer in the reverse direction. See also Howarth [Chem. Eng. Sci., 19, 33 (1964)]. For a theoretical treatment of drop breakage and coalescence and their effects, see Valentas and Amundsen [Ind. Eng. Chem. Fundam., 5, 271, 533 (1966); 7, 66 (1968)], Gal-Or and Walatka [Am. Inst. Chem. Eng. J., 13, 650 (1967)], and Curl [ibid., 9, 175 (1963)]. In calculating the power required for mixers, a reasonable estimate of the average density and viscosity for a two-phase system is satisfactory. Solids are often present in liquid streams either as a part of the processing system or as impurities that come along and have to be handled in the process. One advantage of mixers in differential contact equipment is the fact that they can handle slurries in one or both phases. In many industrial leaching systems, particularly in the minerals processing industry, coming out of the leach circuit is a slurry with a desired material involved in the liquid but a large amount of solids contained in the stream. Typically, the solids must be separated out by filtration or centrifugation, but there has always been a desire to try a direct liquid-liquid extraction with an immiscible liquid contact with this often highly concentrated slurry leach solution. The major prob-

冢

冣冢

冢

冣 冢 冣

冣 冢

冣

lem with this approach is loss of organic material going out with the highly concentrated liquid slurry. Data are not currently available on the dispersion with the newer fluidfoil impellers, but they are often used in industrial mixer-settler systems to maintain dispersion when additional resonance time holdup is required, after an initial dispersion is made by a radial- or axial-flow turbine. Recent data by Calabrese5 indicates that the sauter mean drop diameter can be correlated by equation and is useful to compare with other predictions indicated previously. As an aside, when a large liquid droplet is broken up by shear stress, it tends initially to elongate into a dumbbell shape, which determines the particle size of the two large droplets formed. Then, the neck in the center between the ends of the dumbbell may explode or shatter. This would give a debris of particle sizes which can be quite different than the two major particles produced. Liquid-Liquid Extraction The actual configuration of mixers in multistage mixer-settlers and/or multistage columns is summarized in Section 15. A general handbook on this subject is Handbook of Solvent Extraction by Lowe, Beard, and Hanson. This handbook gives a comprehensive review of this entire operation as well. In the liquid-liquid extraction area, in the mining industry, coming out of the leach tanks is normally a slurry, in which the desired mineral is dissolved in the liquid phase. To save the expense of separation, usually by filtration or centrifugation, attempts have been made to use a resident pump extraction system in which the organic material is contacted directly with the slurry. The main economic disadvantage to this proposed system is the fact that considerable amounts of organic liquid are entrained in the aqueous slurry system, which, after the extraction is complete, is discarded. In many systems this has caused an economic loss of solvent into this waste stream. LIQUID-LIQUID-SOLID SYSTEMS Many times solids are present in one or more phases of a solid-liquid system. They add a certain level of complexity in the process, especially if they tend to be a part of both phases, as they normally will do. Approximate methods need to be worked out to estimate the density of the emulsion and determine the overall velocity of the flow pattern so that proper evaluation of the suspension requirements can be made. In general, the solids will behave as though they were a fluid of a particular average density and viscosity and won’t care much that there is a two-phase dispersion going on in the system. However, if solids are being dissolved or precipitated by participating in one phase and not the other, then they will be affected by which phase is dispersed or continuous, and the process will behave somewhat differently than if the solids migrate independently between the two phases within the process. FLUID MOTION Pumping Some mixing applications can be specified by the pumping capacity desired from the impeller with a certain specified geometry in the vessel. As mentioned earlier, this sometimes is used to describe a blending requirement, but circulation and blending are two different things. The major area where this occurs is in draft tube circulators or pump-mix mixer settlers. In draft tube circulators (shown in Fig. 18-22), the circulation occurs through the draft tube and around the annulus and for a given geometry, the velocity head required can be calculated with reference to various formulas for geometric shapes. What is needed is a curve for head versus flow for the impeller, and then the system curve can be matched to the impeller curve. Adding to the complexity of this system is the fact that solids may settle out and change the character of the head curve so that the impeller can get involved in an unstable condition which has various degrees of erratic behavior depending upon the sophistication of the impeller and inlet and outlet vanes involved. These draft tube circulators often involve solids, and applications are often for precipitation or crystallization in these units. Draft tube circulators

PHASE CONTACTING AND LIQUID-SOLID PROCESSING can either have the impeller pump up in the draft tube and flow down the annulus or just the reverse. If the flow is down the annulus, then the flow has to make a 180° turn where it comes back at the bottom of the tank into the draft tube again. This is a very sensitive area, and special baffles must be used to carefully determine how the fluid will make this turn since many areas of constriction are involved in making this change in direction. When pumping down the draft tube, flow normally makes a more troublefree velocity change to a flow going up the annulus. Since the area of the draft tube is markedly less than the area of the annulus, pumping up the draft tube requires less flow to suspend solids of a given settling velocity than does pumping down the draft tube. Another example is to eliminate the interstage pump between mixing and settling stages in the countercurrent mixer-settler system. The radial-flow impeller typically used is placed very close to an orifice at the bottom of the mixing tank and can develop heads from 12 to 18 in. All the head-loss terms in the mixer and settler circuit have to be carefully calculated because they come very close to that 12- to 18-in range when the passages are very carefully designed and streamlined. If the mixing tank gets much above 10 ft in depth, then the heads have to be higher than the 12- to 18-in range and special designs have to be worked on which have the potential liability of increasing the shear rate acting on the dispersed phase to cause more entrainment and longer settling times. In these cases, it is sometimes desirable to put the mixer system outside the actual mixer tank and have it operate in a single phase or to use multiple impellers, each one of which can develop a portion of the total head required. Heat Transfer In general, the fluid mechanics of the film on the mixer side of the heat transfer surface is a function of what happens at that surface rather than the fluid mechanics going on around the impeller zone. The impeller largely provides flow across and adjacent to the heat-transfer surface and that is the major consideration of the heat-transfer result obtained. Many of the correlations are in terms of traditional dimensionless groups in heat transfer, while the impeller performance is often expressed as the impeller Reynolds number. The fluidfoil impellers (shown in Fig. 18-2) usually give more flow for a given power level than the traditional axial- or radial-flow turbines. This is also thought to be an advantage since the heat-transfer surface itself generates the turbulence to provide the film coefficient and more flow should be helpful. This is true to a limited degree in jacketed tanks (Fig. 18-34), but in helical coils (Fig. 18-35), the

FIG. 18-35

Typical arrangement of helical coil at mixing vessel for heat transfer.

extreme axial flow of these impellers tends to have the first or second turn in the coil at the bottom of the tank blank off the flow from the turns above it in a way that (at the same power level) the increased flow from the fluidfoil impeller is not helpful. It best gives the same coefficient as with the other impellers and on occasion can cause a 5 to 10 percent reduction in the heat-transfer coefficient over the entire coil. JACKETS AND COILS OF AGITATED VESSELS Most of the correlations for heat transfer from the agitated liquid contents of vessels to jacketed walls have been of the form: hDj Lp2 Nrρ ᎏ=a ᎏ k µ

冢

冢

TABLE 18-2

b

1/3

m

(18-18)

w

cµ µ ᎏ 冣 冢ᎏ k 冣 冢µ 冣 0.62

1/3

b

0.14

(18-19)

w

Values of Constants for Use in Eq. (18-18)

Agitator Paddlea Pitched-blade turbineb Disk, flat-blade turbinec Propeller d Anchor b Anchor b Helical ribbone

Typical jacket arrangement for heat transfer.

µb

cµ ᎏ 冣 冢ᎏ k 冣 冢µ 冣

The film coefficient h is for the inner wall; Dj is the inside diameter of the mixing vessel. The term L p2 Nr ρ /µ is the Reynolds number for mixing in which Lp is the diameter and Nr the speed of the agitator. Recommended values of the constants a, b, and m are given in Table 18-2. A wide variety of configurations exists for coils in agitated vessels. Correlations of data for heat transfer to helical coils have been of two forms, of which the following are representative: hDj Lp2 Nt ρ ᎏ = 0.87 ᎏ k µ

FIG. 18-34

18-23

a

b

m

0.36 0.53 0.54 0.54 1.0 0.36 0.633

w w w w a w a

0.21 0.24 0.14 0.14 0.18 0.18 0.18

Range of Reynolds number 300–3 × 10 5 80–200 40–3 × 10 5 2 × 103 (one point) 10–300 300–40,000 8–10 5

a Chilton, Drew, and Jebens, Ind. Eng. Chem., 36, 510 (1944), with constant m modified by Uhl. b Uhl, Chem. Eng. Progr., Symp. Ser. 17, 51, 93 (1955). c Brooks and Su, Chem. Eng. Progr., 55(10), 54 (1959). d Brown et al., Trans. Inst. Chem. Engrs. (London), 25, 181 (1947). e Gluz and Pavlushenko, J. Appl. Chem. U.S.S.R., 39, 2323 (1966).

18-24

LIQUID-SOLID OPERATIONS AND EQUIPMENT

where the agitator is a paddle, the Reynolds number range is 300 to 4 × 105 [Chilton, Drew, and Jebens, Ind. Eng. Chem., 36, 510 (1944)], and hDo Lp2 Nrρ 0.67 cµ 0.37 Lp 0.1 Do 0.5 (18-20) ᎏ = 0.17 ᎏ ᎏ ᎏ ᎏ k µ k Dj Dj

冢

冣 冢 冣 冢 冣 冢 冣

where the agitator is a disk flat-blade turbine, and the Reynolds number range is 400 to (2)(105) [Oldshue and Gretton, Chem. Eng. Prog., 50, 615 (1954)]. The term Do is the outside diameter of the coil tube. The most comprehensive correlation for heat transfer to vertical baffle-type coils is for a disk flat-blade turbine over the Reynolds number range 103 to (2)(106): hDo Lp2 Nrρ ᎏ = 0.09 ᎏ k µ

冢

µ

ᎏ ᎏ 冣 冢ᎏk 冣 冢ᎏ D 冣 冢n 冣 冢µ 冣 0.65

cµ

1/3

Lp

1/3

j

2

b

0.2

0.4

(18-21)

f

where nb is the number of baffle-type coils and µf is the fluid viscosity at the mean film temperature [Dunlop and Rushton, Chem. Eng. Prog. Symp. Ser. 5, 49, 137 (1953)]. Chapman and Holland (Liquid Mixing and Processing in Stirred Tanks, Reinhold, New York, 1966) review heat transfer to low-viscosity fluids in agitated vessels. Uhl [“Mechanically Aided Heat Transfer,” in Uhl and Gray (eds.), Mixing: Theory and Practice, vol. I, Academic, New York, 1966, chap. V] surveys heat transfer to low- and highviscosity agitated fluid systems. This review includes scraped-wall units and heat transfer on the jacket and coil side for agitated vessels. LIQUID-LIQUID-GAS-SOLID SYSTEMS This is a relatively unusual combination, and one of the more common times it exists is in the fermentation of hydrocarbons with aerobic microorganisms in an aqueous phase. The solid phase is a microorganism which is normally in the aqueous phase and is using the organic phase for food. Gas is supplied to the system to make the fermentation aerobic. Usually the viscosities are quite low, percent solids is also modest, and there are no special design conditions required when this particular gas-liquid-liquid-solid combination occurs. Normally, average properties for the density of viscosity of the liquid phase are used. In considering that the role the solids play in the system is adequate, there are cases of other processes which consist of four phases, each of which involves looking at the particular properties of the phases to see whether there are any problems of dispersion, suspension, or emulsification.

FIG. 18-36

Laser scan.

FIG. 18-37

Laser scan.

COMPUTATIONAL FLUID DYNAMICS There are several software programs that are available to model flow patterns of mixing tanks. They allow the prediction of flow patterns based on certain boundary conditions. The most reliable models use accurate fluid mechanics data generated for the impellers in question and a reasonable number of modeling cells to give the overall tank flow pattern. These flow patterns can give velocities, streamlines, and localized kinetic energy values for the systems. Their main use at the present time is to look at the effect of making changes in mixing variables based on doing certain things to the mixing process. These programs can model velocity, shear rates, and kinetic energy, but probably cannot adapt to the actual chemistry of diffusion or masstransfer kinetics of actual industrial process at the present time. Relatively uncomplicated transparent tank studies with tracer fluids or particles can give a similar feel for the overall flow pattern. It is important that a careful balance be made between the time and expense of calculating these flow patterns with computational fluid dynamics compared to their applicability to an actual industrial process. The future of computational fluid dynamics appears very encouraging and a reasonable amount of time and effort put forth in this regard can yield immediate results as well as potential for future process evaluation. Figures 18-36, 18-37, and 18-38 show some approaches. Figure 18-36 shows velocity vectors for an A310 impeller. Figure 18-37 shows contours of kinetic energy of turbulence. Figure 18-38 uses a particle trajectory approach with neutral buoyancy particles.

PASTE AND VISCOUS-MATERIAL MIXING

18-25

Numerical fluid mechanics can define many of the fluid mechanics parameters for an overall reactor system. Many of the models break up the mixing tank into small microcells. Suitable material and masstransfer balances between these cells throughout the reactor are then made. This can involve long and massive computational requirements. Programs are available that can give reasonably acceptable models of experimental data taken in mixing vessels. Modeling the threedimensional aspect of a flow pattern in a mixing tank can require a large amount of computing power. Most modeling codes are a time-averaging technique. Depending upon the process, a time-dependent technique may be more suitable. Time-dependent modeling requires much more computing power than does time averaging.

FIG. 18-38

GENERAL REFERENCES 1. J. Y. Oldshue, “Mixing ’89,” Chemical Engineering Progress, 85(5): 33–42 (1989). 2. J. C. Middleton, Proc. 3d European Conf. on Mixing, 4/89, BHRA, pp. 15–36. 3. J. Y. Oldshue, T. A. Post, R. J. Weetman, “Comparison of Mass Transfer Characteristics of Radial and Axial Flow Impellers,” BHRA Proc. 6th European Conf. on Mixing, 5/88. 4. A. W. Neinow, B. Buckland, R. J. Weetman, Mixing XII Research Conference, Potosi, Mo., 8/89. 5. R. Calabrese et al., AIChE J. 32: 657, 677 (1986). 6. T. N. Zwietering, Chemical Engineering Science, 8(3): 244–253 (1958). 7. J. Y. Oldshue, Chemical Engineering Progress, “Mixing of Slurries Near the Ultimate Settled Solids Concentration,” 77(5): 95–98 (1981).

Laser scan.

PASTE AND VISCOUS-MATERIAL MIXING GENERAL REFERENCES: Dry Solids, Paste and Dough Mixing Equipment Testing Procedure, American Institute of Chemical Engineers, New York, 1979. Fischer, Chem. Eng. 69(3): 52 (1962). Irving and Saxton, “Mixing of High Viscosity Materials,” in Uhl and Gray, Mixing Theory and Practice, vol. 2, Academic, New York, 1967, chap. 8. Mohr, in Bernhardt, Processing of Thermoplastic Materials, Reinhold, New York, 1967, chap. 3. Parker, Chem. Eng. 71(12): 166 (1964). Rauwendaal, Mixing in Polymer Processing, Marcel Dekker, 1991. Tadmor and Gogos, Principles of Polymer Processing, Wiley, New York, 1979.

INTRODUCTION Thick mixtures with viscosities greater than 10 Pa⋅s are not readily mixed in conventional stirred pots with either propeller or turbine agitators. The high viscosity may be due to that of the matrix fluid itself, to a high slurry concentration, or to interactions between components. Because of the high viscosity, the mixing Reynolds number (Re = D2ρN/µ) may be well below 100. Mixing occurs as a consequence of laminar shearing and stretching forces, and turbulence plays no part. Relative motion of an agitator stretches and deforms the material between itself and the vessel wall. As a layer of fluid gets stretched into thinner layers, the striations diminish and the shear forces tear solid agglomerates apart until apparent homogeneity is obtained. When solids are present, it may be necessary to reduce the particle size of the solids, even down to submicron size, such as in pigment dispersion. Mixers for high-viscosity materials generally have a small high shear zone (to minimize dissipative heat effects) and rely on the impeller to circulate all of the mixer contents past the high shear zone. It may be necessary to take special steps to eliminate any stagnant zones or perhaps to avoid material riding around on mixing blades without being reincorporated into the matrix. Most paste and high-viscosity mixtures are nonnewtonian, with viscosity dropping with shear rate. Consequently, increasing impeller speed may be counterproductive, as the transmitted shear drops rapidly and an isolated hole may be created in the central mass without circulation of the bulk material in the remainder of the vessel.

The relative viscosities of the materials being mixed can be very important. Karam and Bellinger [Ind. Eng. Chem. Fund. 7: 576 (1981)] and Grace [Engr. Fndn. Mixing Research Conf. Andover, N.H., 1973] have shown that there is a maximum viscosity ratio (µ dispersed/µ continuous equal to about 4) above which droplets can no longer be dispersed by shear forces. The only effective dispersing mechanism for high-viscosity-ratio mixtures is elongational flow. Equipment for viscous mixing usually has a small clearance between impeller and vessel walls, a relatively small volume, and a high power per unit volume. Intermeshing blades or stators may be present to prevent material from cylindering on the rotating impeller. Blade shape can have a significant impact on the mixing process. A scraping profile will be useful if heat transfer is important, whereas a smearing profile will be more effective for dispersion. Ease of cleaning and ease of discharge may also be important. BATCH MIXERS Change-Can Mixers Change-can mixers are vertical batch mixers in which the container is a separate unit easily placed in or removed from the frame of the machine. They are available in capacities of about 4 to 1500 L (1 to 400 gal). The commonest type is the pony mixer. Separate cans allow the batch to be carefully measured or weighed before being brought to the mixer itself. The mixer also may serve to transport the finished batch to the next operation or to storage. The identity of each batch is preserved, and weight checks are easily made. Change cans are relatively inexpensive. A good supply of cans allows cleaning to be done in a separate department, arranged for efficient cleaning. In paint and ink plants, where mixing precedes milling or grinding and where there may be a long run of the same formulation and color, the cans may be used for an extended period without cleaning, as long as no drying out or surface oxidation occurs. In most change-can mixers, the mixing elements are raised from the can by either a vertical lift or a tilting head; in others, the can is

18-26

LIQUID-SOLID OPERATIONS AND EQUIPMENT

dropped away from the mixing elements. After separation the mixing elements drain into the can, and the blades can be wiped down. With the can out of the way, complete cleaning in the blades and their supports is simple. If necessary, blades may be cleaned by rotating them in solvent. Intimate mixing is accomplished in change-can mixers in two ways. One method is to have the mixing-unit assembly revolve in a planetary motion so that the rotating blades sweep the entire circumference of the can (Fig. 18-39). The other is to mount the can on a rotating turntable so that all parts of the can wall pass fixed scraper blades or the agitator blades at a point of minimum clearance. A type of heavy-duty planetary change-can mixer has been used extensively for processing critical solid-propellant materials. In a design offered by APV Chemical Machinery Inc., the mixer blades pass through all portions of the can volume, and the two blades wipe each other, thus assuring no unmixed portions. A dead spot under the mixer blades is avoided by having both blades off the centerline of the can. The cans fit tightly enough so that mixing can be achieved under vacuum or pressure. There is no contact between the glands and the material being mixed. Charging ports located in the housing directly above the mixing can make it possible to charge materials to the mixer with the can in the operating position. This type of mixer is available in sizes of 0.5 L (about 1 pt) to 1.6 m3 (420 gal), involving power input of 0.2 to 75 kW (0.25 to 100 hp). Helical-Blade Mixers Helical mixers are now available in a variety of configurations. The mixing element may be in the form of a conical or a cylindrical helix. It may be a ribbon spaced radially from the shaft by spokes or a screw consisting of a helical surface that is continuous from the shaft to the periphery of the helix. A venerable example of the latter type is the soap crutcher, in which the screw is mounted in a draft tube. Close screw-tube clearance and a high rotational speed result in rapid motion of the material and high shear. The screw lifts the material through the tube, and gravity returns it to the bottom of the tank. If the tank has well-rounded corners, this kind of mixer may be used for fibrous materials. Heavy paper pulp containing 16 to 18 percent solids is uniformly bleached in large mixers of this type.

FIG. 18-39

Change-can mixer. (Charles Ross & Son Co.)

A double helix shortens mixing time but requires more power. The disadvantages of the higher torque requirement are frequently offset by the better mixing and heat transfer. A vertical helical ribbon blender can be combined with an axial screw of smaller diameter (Fig. 18-25). Such mixers are used in polymerization reactions in which uniform blending is required but in which high-shear dispersion is not a factor. Addition of the inner flight contributes little more turnover in mixing newtonian fluids but significantly shortens the mixing time in nonnewtonian systems and adds negligibly to the impeller power [Coyle et al., Am. Inst. Chem. Eng. J., 15, 903 (1970)]. Double-Arm Kneading Mixers The universal mixing and kneading machine consists of two counterrotating blades in a rectangular trough curved at the bottom to form two longitudinal half cylinders and a saddle section (Fig. 18-40). The blades are driven by gearing at either or both ends. The oldest style empties through a bottom door or valve and is still in use when 100 percent discharge or thorough cleaning between batches is not an essential requirement. More commonly, however, double-arm mixers are tilted for discharge. The tilting mechanism may be manual, mechanical, or hydraulic. A variety of blade shapes has evolved. The mixing action is a combination of bulk movement, smearing, stretching, folding, dividing, and recombining as the material is pulled and squeezed against blades, saddle, and sidewalls. The blades are pitched to achieve end-to-end circulation. Rotation is usually such that material is drawn down over the saddle. Clearances are as close as 1 mm (0.04 in). The blades may be tangential or overlapping. Tangential blades are run at different speeds, with the advantages of faster mixing from constant change of relative position, greater wiped heat-transfer area per unit volume, and less riding of material above the blades. Overlapping blades can be designed to avoid buildup of sticky material on the blades. The agitator design most widely used is the sigma blade (Fig. 18-41a). The sigma-blade mixer is capable of starting and operating with either liquids or solids or a combination of both. Modifications in blade-face design have been introduced to increase particular effects, such as shredding or wiping. The sigma blade has good mixing action, readily discharges materials which do not stick to the blades, and is relatively easy to clean when sticky materials are being processed. The dispersion blade (Fig. 18-41b) was developed particularly to provide compressive shear higher than that achieved with standard

FIG. 18-40

Double-arm kneader mixer. (APV Baker Perkins Inc.)

PASTE AND VISCOUS-MATERIAL MIXING TABLE 18-3

18-27

Characteristics of Double-Arm Kneading Mixers* Typical supplied horsepower

(a)

(b)

(c)

Size number 4 6 8 11 12

Capacity, U.S. gal Working 0.7 2.3 4.5 10 20

Maximum 1 3.5 7 15 30

Sigma blade, MWOL blade

Dispersion blade

Floor space, ft

1 2 5 15 25

2 5 7.5 20 40

1×3 2×3 3×4 5×6 6×6

14 15 16 17 18

50 100 150 200 300

75 150 225 300 450

30 50 60 75 100

60 100 150 200 —

6×8 8 × 10 9 × 11 9 × 13 10 × 14

20 21 22 23

500 600 750 1000

750 900 1125 1500

150 175 225 300

— — — —

11 × 16 12 × 16 12 × 17 14 × 18

*Data from APV Baker Perkins, Inc. To convert feet to meters, multiply by 0.3048; to convert gallons to cubic meters, multiply by 3.78 × 10−3; and to convert horsepower to kilowatts, multiply by 0.746.

(d)

(e)

FIG. 18-41 Agitator blades for double-arm kneaders: (a) sigma; (b) dispersion; (c) multiwiping overlap; (d) single-curve; (e) double-naben. (APV Baker Perkins, Inc.)

sigma blades. The blade face wedges material between itself and the trough, rather than scraping the trough, and is particularly suited for dispersing fine particles in a viscous mass. Rubbery materials have a tendency to ride the blades, and a ram is frequently used to keep the material in the mixing zone. Multiwiping overlapping (MWOL) blades (Fig. 18-41c) are commonly used for mixtures which start tough and rubberlike, inasmuch as the blade cuts the material into small pieces before plasticating it. The single-curve blade (Fig. 18-41d ) was developed for incorporating fiber reinforcement into plastics. In this application, the individual fibers (e.g., sisal or glass) must be wetted with polymer without incurring undue fiber breakage. Many other blade designs have been developed for specific applications. The double-naben blade (Fig. 18-41e) is a good blade for mixes which “ride,” that is, form a lump which bridges across the sigma blade. Double-arm mixers are available from several suppliers (e.g., Paul O. Abbe, Inc.; Littleford Day; Jaygo, Inc.; Charles Ross & Son Co.; Teledyne Readco). Options include vacuum design, cored blades, jacketed trough, choice of cover design, and a variety of seals and packing glands. Power requirements vary from j to 2 hp/gal of capacity. Table 18-3 lists specifications and space requirements for typical tilting-type, double-ended-drive, double-arm mixers. The working capacity is generally at or near the top of the blades, and the total capacity is the volume contained when the mixer is filled level with the top of the trough. Figure 18-42 provides a guide for typical applications. Individual formulation changes may require more power than indicated in the figure. Parker [Chem. Eng., 72(18), 121 (1965)] has described in greater detail how to select double-arm mixers. Screw-Discharge Batch Mixers A variant of the sigma-blade mixer is now available with an extrusion-discharge screw located in the saddle section. During the mixing cycle the screw moves the material within the reach of the mixing blades, thereby accelerating the

mixing process. At discharge time, the direction of rotation of the screw is reversed and the mixed material is extruded through suitable die openings in the side of the machine. The discharge screw is driven independently of the mixer blades by a separate drive. Working capacities range from 4 to 3800 L (1 to 1000 gal), with up to 300 kW (400 hp). This type of kneader is offered by most of the double-arm-kneader manufacturers. It is particularly suitable when a heel from the prior batch can be left without detriment to succeeding batches. Intensive Mixers Banbury Mixer Preeminent in the field of high-intensity mixers, with power input up to 6000 kW/m3 (30 hp/gal), is the Banbury mixer, made by the Farrel Co. (Fig. 18-43). It is used mainly in the plastics and rubber industries. The top of the charge is confined by an air-operated ram cover mounted so that it can be forced down on the charge. The clearance between the rotors and the walls is extremely small, and it is

Typical application and power for double-arm kneaders. To convert horsepower per gallon to kilowatts per cubic meter, multiply by 197.3. [Parker, Chem. Eng. 72(18): 125 (1965); excerpted by special permission of the copyright owner, McGraw-Hill, Inc.]

FIG. 18-42

18-28

FIG. 18-43

LIQUID-SOLID OPERATIONS AND EQUIPMENT

Banbury mixer. (Farrel Co.)

here that the mixing action takes place. The operation of the rotors of a Banbury at different speeds enables one rotor to drag the stock against the rear of the other and thus help clean ingredients from this area. The extremely high power consumption of the machines operating at speeds of 40 r/min or lower calls for rotor shafts of large diameter. The combination of heavy shafts, stubby blades, close clearances, and the confined charge limits the Banbury mixer to small batches. The production rate is increased as much as possible by using powerful drives and rotating the blades at the highest speed that the material will stand. The friction produced in the confined space is great, and with heat-sensitive materials cooling may be the limiting factor. Recent innovations include a drop door, four-wing rotors which can provide 30 percent greater power than the older two-wing rotors, and separable gear housings. Equipment is available from laboratory size to a mixer capable of handling a 450-kg (1000-lb) charge and applying 2240 kW (3000 hp). High-Intensity Mixer. Mixers such as that shown in Fig. 18-44 combine a high shear zone with a fluidized vortex mixing action. Blades at the bottom of the vessel scoop the batch upward at peripheral speeds of about 40 m/s (130 ft/s). The high shear stress (to 20,000 s−1) and blade impact easily reduce agglomerates and aid intimate dispersion. Since the energy input is high [200 kW/m3 (about 8 hp/ft3)], even powdery material is heated rapidly. Mixers of this type are available in sizes from 4 to 2000 L (1 to 500 gal), consuming from 1.5 to 375 kW (2 to 500 hp). These mixers are particularly suited for rapid mixing of powders and granules with liquids, for dissolving resins or solids in liquids, or for removal of volatiles from pastes under vacuum. Scale-up is usually on the basis of maintaining constant peripheral velocity of the impeller. Roll Mills Roll mills can provide exceedingly high localized shear while retaining extended surface for temperature control. Two-Roll Mills These mills contain two parallel rolls mounted in a heavy frame with provision for accurately regulating the pressure and distance between the rolls. As one pass between the rolls does little blending and only a small amount of work, the mills are practically always used as batch mixers. Only a small amount of material is in the high-shear zone at any one time. To increase the wiping action, the rolls are usually operated at different speeds. The material passing between the rolls is returned to the feed point by the rotation of the rolls. If the rolls are at different temperatures, the material usually will stick to the hotter roll and return to the feed point as a thick layer.

FIG. 18-44 High-intensity mixer: (a) bottom scraper; (b) fluidizing tool; (c) horn tool; (d) flushmounted discharge valve. (Henschel Mixers America, Inc.)

PASTE AND VISCOUS-MATERIAL MIXING

18-29

At the end of the period of batch mixing, heavy materials may be discharged by dropping between the rolls, while thin mixes may be removed by a scraper bar pressing against the descending surface of one of the rolls. Two-roll mills are used mainly for preparing color pastes for the ink, paint, and coating industries. There are a few applications in heavyduty blending of rubber stocks, for which corrugated and masticating rolls are often used. Miscellaneous Batch Mixers Bulk Blenders Many of the mixers used for solids blending (Sec. 19) are also suitable for some liquid-solids blending. Ribbon blenders can be used for such tasks as wetting out or coating a powder. When the final paste product is not too fluid, other solidshandling equipment finds frequent use. Plow Mixers Plow mixers such as the Littleford (Littleford Day) and the Marion (Rapids Machinery Co.) machines can be used for either batch or continuous mixing. Plow-shaped heads arranged on the horizontal shaft rotate at high speed, hurling the material throughout the free space of the vessel. Additional intermixing and blending occur as the impellers plow through the solids bed. Special high-speed choppers (3600 r/min) can be installed to break up lumps and aid liquid incorporation. The choppers also disperse fine particles throughout viscous materials to provide a uniform suspension. The mixer is available in sizes of 40 L to 40 m3 (10 to 10,000 gal) of working capacity. Cone and Screw Mixers The Nauta mixer of Fig. 18-45 (Day Mixing) utilizes an orbiting action of a helical screw rotating on its own axis to carry material upward, while revolving about the centerline of the cone-shaped shell near the wall for top-to-bottom circulation. Reversing the direction of screw rotation aids discharge of pasty materials. Partial batches are mixed in the Nauta type of mixer as efficiently as full loads. These mixers, available in sizes of 40 L to 40 m3 (10 to 10,000 gal), achieve excellent low-energy blending, with some hydraulic shear dispersion. At constant speed, both mixing time and power scale up with the square root of volume. Pan Muller Mixers These mixers can be used if the paste is not too fluid or too sticky. The main application of muller mixers is now in the foundry industry, in mixing small amounts of moisture and binder materials with sand particles for both core and molding sand. In paste processing, pan-and-plow mixers are used principally for mixing putty and clay pastes, while muller mixers handle such diversified materials as clay, storage-battery paste, welding-rod coatings, and chocolate coatings. In muller mixers the rotation of the circular pan or of the plows brings the material progressively into the path of the mullers, where the intensive action takes place. Figure 18-46 shows one type of mixer, in which the mullers and plows revolve around a stationary turret in a stationary pan. The outside plow moves material from the crib wall to the path of the following muller; the inside plow moves it from the central turret to the path of the other muller. The mullers crush the material, breaking down lumps and aggregates. Standard muller mixers range in capacity from a fraction of a cubic foot to more than 1.8 m3 (60 ft3), with power requirements ranging from 0.2 to 56 kW (s to 75 hp). A continuous muller design employs two intersecting and communicating cribs, each with its own mullers and plows. At the point of intersection of the two crib bodies, the outside plows give an approximately equal exchange of material from one crib to the other, but material builds up in the first crib until the feed rate and the discharge rate of material from the gate in the second crib are equal. The residence time is regulated by adjusting the outlet gate. CONTINUOUS MIXERS Some of the batch mixers previously described can be converted for continuous processing. Product uniformity may be poor because of the broad residence time distribution. If the different ingredients can be accurately metered, various continuous mixers are available. Continuous mixers generally consist of a closely fitted agitator element rotating within a stationary housing. Single-Screw Mixers Because of the growth of the plastics industry, use of extruders such as that depicted in Fig. 18-47 are now

FIG. 18-45

Nauta mixer. (Littleford Day.)

very widespread. The quality and usefulness of the product may well depend on how uniformly the various additives, stabilizers, fillers, etc., have been incorporated. The extruder combines the process functions of melting the base resin, mixing in the additives, and developing the pressure required for shaping the product into pellets, sheet, or profiles. Dry ingredients, sometimes premixed in a batch blender, are fed into the feed throat where the channel depth is deepest. As the root diameter of the screw is increased, the plastic is melted by a combination of friction and heat transfer from the barrel. Shear forces can be very high, especially in the melting zone, and the mixing is primarily a laminar shearing action. Single-screw extruders can be built with a long length-to-diameter ratio to permit a sequence of process operations, such as staged addition of various ingredients. Capacity is determined by diameter, length, and power. Although the majority of extruders are in the 25to 200-mm-diameter range, much larger units have been made for specific applications such as polyethylene homogenization. Mixing

18-30

LIQUID-SOLID OPERATIONS AND EQUIPMENT

(a)

(b) FIG. 18-46 Pan muller: (a) plan view; (b) sectional elevation. [Bullock, Chem. Eng. Prog. 51: 243 (1955), by permission.]

enhancers (Fig. 18-48) are utilized to provide both elongational reorienting and shearing action to provide for both dispersive and distributive mixing. The maximum power (P, kW) being supplied for single-screw extruders varies with screw diameter (D, mm) approximately as follows: P = 5.3 × 10−3 D2.25 The power required for most polymer mixing applications varies from 0.15 to 0.3 kWh/kg. Rietz Extruder This extruder, shown schematically in Fig. 18-49, has orifice plates and baffles along the vessel. The rotor carries multiple blades with a forward pitch, generating the head for extrusion through the orifice plates as well as battering the material to break up agglomerates between the baffles. Typical applications include wet

FIG. 18-47

Single-screw extruder. (Davis Standard.)

granulation of pharmaceuticals, blending color in bar soap, and mixing and extruding cellulose materials. The Extructor is available in rotor diameters up to 600 mm (24 in) and in a power range of 5 to 112 kW (7 to 150 hp). Twin-Screw Extruders With two screws in a figure-eight barrel, advantage can be taken of the interaction between the screws as well as between the screw and the barrel. Twin-screw machines are used in continuous melting, mixing, and homogenizing of different polymers with various additives, or to carry out the intimate mixing required for reactions in which at least one of the components is of high viscosity. The screws can be tangential or intermeshing, and the latter either coor counterrotating. Tangential designs allow variability in channel depth and permit longer lengths. Counterrotating intermeshing screws provide a dispersive milling action between the screws and behave like a positive displacement device with the ability to generate pressure more efficiently than any other extruder. The most common type of twin-screw mixing extruder is the corotating intermeshing variety (APV, Berstorff, Davis Standard, Leistritz, Werner & Pfleiderer). The two keyed or splined shafts are fitted with pairs of slip-on kneading or conveying elements, as shown in Fig. 18-50. The arrays of these elements can be varied to provide a wide range of mixing effects. The barrel sections are also segmented to allow for optimum positioning of feed ports, vents, barrel valves, etc. The barrels may be heated electrically or with oil or steam and cooled with air or water. Each pair of kneading paddles causes an alternating compression and expansion effect which massages the contents and provides a combination of shearing and elongational mixing actions. The corotating twin-screw mixers are available in sizes ranging from 15 to 300 mm, with length-to-diameter ratios up to 50 and throughput capacities up to 25,000 kg/h. Screw speeds can be high (to 500 r/min in the smaller-production-size extruders), and residence times for mixing are usually under 2 min. Farrel Continuous Mixer This mixer (Fig. 18-51) consists of rotors similar in cross section to the Banbury batch mixer. The first section of the rotor acts as a screw conveyor, propelling the feed ingredients to the mixing section. The mixing action is a combination of intensive shear, between rotor and chamber wall, kneading between the rotors, and a rolling action of the material itself. The amount and quality of mixing are controlled by adjustment of speed, feed rates, and discharge-orifice opening. Units are available in five sizes with mixing-chamber volumes ranging up to 0.12 m3 (4.2 ft3). At 200 r/min, the power range is 5 to 2200 kW (7 to 3000 hp). Miscellaneous Continuous Mixers Trough-and-Screw Mixers These mixers usually consist of single or twin rotors which continually turn the feed material over as it

PASTE AND VISCOUS-MATERIAL MIXING

18-31

(a)

(b)

FIG. 18-49

Rietz Extruder. (Bepex Corporation.)

(c)

(d)

(e) Mixing enhancers for single-screw extruders: (a) Maddock (straight); (b) Maddock (tapered); (c) pineapple; (d) gear; (e) pin.

FIG. 18-48

progresses toward the discharge end. Some have been designed with extensive heat-transfer area. The continuous-screw Holo-Flite Processor (Fig. 18-52; Denver Sala) is used primarily for heat transfer, since the hollow screws present extended surface without contributing much shear. Two or four screws may be used. Bethlehem Corp.’s Porcupine Processor also has heat-transfer media going through the flights of the rotor, but the agitator flights are cut to provide a folding action on the process mass. Breaker-bar assemblies, consisting of fingers extending toward the shaft, are frequently used to improve agitation. Another type of trough-and-screw mixer with a large volume available for mixing is the AP Conti (List AG) machine shown in Fig. 1853. These self-cleaning-type mixers are particularly appropriate when

Intermeshing corotating twin screw extruder: (a) drive motor; (b) gearbox; (c) feed port; (d) barrel; (e) assembled rotors; ( f ) vent; (g) barrel valve; (h) kneading paddles; (i) conveying screws; ( j) splined shafts; (k) blister rings. (APV Chemical Machinery, Inc.)

FIG. 18-50

18-32

LIQUID-SOLID OPERATIONS AND EQUIPMENT

FIG. 18-51

Farrel continuous mixer. (Farrel Co.)

the product being handled goes through a sticky stage, which could plug the mixer or foul the heat-transfer surfaces. Pug Mills A pug mill contains one or two shafts fitted with short, heavy paddles, mounted in a cylinder or trough which holds the material being processed. In two-shaft mills the shafts are parallel and may be horizontal or vertical. The paddles may or may not intermesh. Clearances are wide so that there is considerable mass mixing. Unmixed or partially mixed ingredients are fed at one end of the machine, which is usually totally enclosed. The paddles push the material forward as they cut through it, and carry the charge toward the discharge end as it is mixed. Product may discharge through one or two open ports or through one or more extrusion nozzles which give roughly shaped, continuous strips. Automatic cutters may be used to make blocks from the strips. Pug mills are most used for mixing mineral and clay products. Kneadermaster This mixer (Patterson Industries, Inc.) is an adaptation of a sigma-blade mixer for continuous operation. Each two

pairs of blades establish a mixing zone, the first pair pushing materials toward the discharge end of the trough and the second pair pushing them back. Forwarding to the next zone is by displacement with more feed material. Control of mixing intensity is by variation in rotor speed. Cored blades supplement the heat-transfer area of the jacketed trough. Motionless Mixers An alternative to agitated equipment for continuous viscous mixing involves the use of stationary shaped diverters inside conduits which force the fluid media to mix themselves through a progression of divisions and recombinations, forming striations of ever-decreasing thickness until uniformity is achieved. Simple diverters, such as the Kenics static mixer (Chemineer, Inc.; Fig. 18-54), provide 2n layerings per n diverters. The power consumed by a motionless mixer in producing the mixing action is simply that delivered by a pump to the fluid which it moves against the resistance of the diverter conduit. For a given rate of pumping, it is substantially proportional to that resistance. When the diverter consists of several passageways, as in the Sulzer static mixer (Koch Engineering Co., Inc.) shown in Fig. 18-55, the number of layerings (hence, the rate of mixing) per diverter is increased, but at the expense of a higher pressure drop. The pressure drop, usually expressed as a multiple K of that of the empty duct, is strongly dependent upon the hydraulic radius of the divided flow passageway. The value of K, obtainable from the mixer supplier, can range from 6 to several hundred, depending on the Reynolds number and the geometry of the mixer. Motionless mixers continuously interchange fluid elements between the walls and the center of the conduit, thereby providing enhanced heat transfer and relatively uniform residence times. Distributive mixing is usually excellent; however, dispersive mixing may be poor, especially when viscosity ratios are high. PROCESS DESIGN CONSIDERATIONS

FIG. 18-52

Holo-Flite Processor. (Denver Sala.)

Scaling Up Mixing Performance Scale-Up of Batch Mixers The prime basis of scale-up of batch mixers has been equal power per unit volume, although the most desirable practical criterion is equal blending per unit time. As size is increased, mechanical-design requirements may limit the larger mixer to lower agitator speeds; if so, blend times will be longer in the larger

PASTE AND VISCOUS-MATERIAL MIXING

FIG. 18-53

AP Conti paste mixer. (LIST.)

mixer than in the smaller prototype. If the power is high, the lower surface-to-volume ratio as size is increased may make temperature buildup a limiting factor. Since the impeller in a paste mixer generally comes close to the vessel wall, it is not possible to add cooling coils. In some instances, the impeller blades can be cored for additional heattransfer area. Experience with double-arm mixers indicates that power is proportional to the product of blade radius, blade-wing depth, trough length, and average of the speeds of the two blades (Irving and Saxton, loc. cit.). The mixing time scales up inversely with blade speed. Goodness of mixing is dependent primarily on the number of revolutions that the blades have made. As indicated previously, the minimum possible mixing time may become dependent on heat-transfer rate. Frequently, the physical properties of a paste vary considerably during the mixing cycle. Even if one knew exactly how power depended upon density and viscosity, it might be better to predict the requirements for a large paste mixer from the power-time curve observed in the prototype mixer rather than to try to calculate or measure all intermediate properties during the processing sequence (i.e., the prototype mixer may be the best instrument to use to measure the effective viscosity). Scale-Up of Continuous Mixers Although scaling up on the basis of constant power per unit feed rate [kWh/kg or (hp⋅h)/lb] is usually a good first estimate, several other factors may have to be considered. As the equipment scale is increased, geometric similarity being

FIG. 18-54

18-33

Kenics static mixer. (Chemineer, Inc.)

at least approximated, there is a loss in surface-to-volume ratio. As size is increased, changing shear rate or length-to-diameter ratio may be required because of equipment-fabrication limitations. Furthermore, even if a reliable method of scaling up power exists, the determination of net power in small-scale test equipment is frequently difficult and inaccurate because of fairly large no-load power. As a matter of fact, geometrical similarity usually cannot be maintained exactly as the size of the model is increased. In single-screw extruders, for example, channel depth in the flights cannot be increased in proportion to screw diameter because the distribution of heat generated by friction at the barrel wall requires more time as channel depth becomes greater. With constant retention time, therefore, nonhomogeneous product would be discharged from the scaledup model. As the result of the departure from geometrical similarity, the throughput rate of single-screw extruders scales up with diameter

FIG. 18-55

Sulzer static mixer. (Koch Engineering Co., Inc.)

18-34

LIQUID-SOLID OPERATIONS AND EQUIPMENT

to the power 2.0 to 2.5 (instead of diameter cubed) at constant lengthto-diameter ratio and screw speed. The throughput rate of intermeshing twin-screw extruders (Fig. 18-56) and the Farrel continuous mixer (Fig. 18-51) is scaled up with diameter to about the 2.6 power. If the process can be operated adiabatically, the production capacity is scaled up as the cube of diameter since geometry, shear rate, residence time, and power input per unit volume all can be held constant. Residence-time distributions. For flow through a conduit, the extent of axial dispersion can be characterized either by an axialdiffusion coefficient or by analogy to a number of well-mixed stages in series. Retention time can control the performance of a mixing system. As the number of apparent stages increases, there is greater assurance that all the material will have the required residence time. Under conditions requiring uniform retention time, it is imperative that the feed streams be fed in the correct ratio on a time scale much shorter than the average residence time of the mixer; otherwise, a perturbation in the feed will produce a comparable perturbation in the product. The mixing impellers in continuous mixers can be designed to cover the full range from minimum axial mixing (plug flow) to maximum (to damp out feed irregularities). Residence-time distributions and effective Peclet numbers have been determined for a wide variety of twin-screw configurations [Todd and Irving, Chem. Eng. Prog., 65(9), 84 (1969)]. Conventional single-screw extruder mixers have Peclet numbers about equal to the length-to-diameter ratio, or an equivalent number of stages equal to one-half of that. HEATING AND COOLING MIXERS Heat Transfer Pastes are often heated or cooled by heat transfer through the walls of the container or hollow mixing arms. Good agitation, a large ratio of transfer surface to mixer volume, and frequent removal of material from the surface are essential for high rates of heat transfer. Sometimes evaporation of part of the mix is used for cooling. In most mixers, the metal wall has a negligible thermal resistance. The paste film, however, usually has high resistance. It is important, therefore, while minimizing the resistance of the heating or cooling medium, to move the paste up to and away from the smooth wall surface as steadily and rapidly as possible. This is best achieved by having the paste flow so as to follow a close-fitting scraper which wipes the film from the wall with each rotation. Typical overall heat-transfer coefficients are between 25 and 200 J/(m2⋅s⋅K) [4 to 35 Btu/(h⋅ft2⋅°F)]. Heating Methods The most economical heating method varies with plant location and available facilities. Direct firing is rarely used, since it does not permit good surface-temperature control and may cause scorching of the material on the vessel walls. Steam heating is the most widely used method. It is economical, safe, and easily controlled. With thin-wall mixers there must be automatic release of the vacuum that results when the pressure is reduced and the steam in the jacket condenses; otherwise, weak sections will collapse. Transferliquid heating using water, oil, special organic liquids, or molten inorganic salts permits good temperature control and provides insurance against overheating the processed material. Jackets for transfer-liquid heating usually must be baffled to provide good circulation. Higher temperatures can be achieved without requiring the heavy vessel construction otherwise required by steam. Electrical heating is accomplished with resistance bands or ribbons which must be electrically insulated from the machine body but in good thermal contact with it. The heaters must be carefully spaced to avoid a succession of hot and cold areas. Sometimes they are mounted in aluminum blocks shaped to conform to the container walls. Their effective temperature range is 150 to 500°C (about 300 to 930°F). Temperature control is precise, maintenance and supervision costs are low, and conversion of electrical energy to useful heat is almost 100 percent. The cost of electrical energy is usually large, however, and may be prohibitive. Frictional heat develops rapidly in some units such as a Banbury mixer. The first temperature rise may be beneficial in softening the materials and accelerating chemical reactions. High temperatures detrimental to the product may easily be reached, however, and pro-

vision for cooling or frequent stopping of the machine must be made. Frictional heating may be lessened by reducing the number of working elements, their area, and their speed. Cooling thus is facilitated, but at the expense of increased mixing time. Cooling Methods In air cooling, air may be blown over the machine surfaces, the area of which is best extended with fins. Air or cooled inert gas may also be blown over the exposed surface of the mix, provided contamination or oxidation of the charge does not result. Evaporation of excess water or solvent under vacuum or at atmospheric pressure provides good cooling. A small amount of evaporation produces a large amount of cooling. Removing too much solvent, however, may damage the charge. Direct addition of ice to the mixer provides rapid, convenient cooling, but the resulting dilution of the mix must be permissible. Addition of dry ice is more expensive but results in lower temperatures, the mix is not diluted, and the CO2 gas evolved provides a good inert atmosphere. Many mixers are cooled by circulation of water or refrigerants through jackets or hollow agitators. In general, this is the least expensive method, but it is limited by the magnitude of heat-transfer coefficient obtainable. Selection of Equipment If a new product is being considered, the preliminary study must be highly detailed. Laboratory or pilotplant work must be done to establish the controlling factors. The problem is then to select and install equipment which will operate for quantity production at minimum overall cost. Most equipment vendors have pilot equipment available on a rental basis or can conduct test runs in their own customer-demonstration facilities. One approach to proper equipment specification is by analogy. What current product is most similar to the new one? How is this material produced? What difficulties are being experienced? In other situations the following procedure is recommended: 1. List carefully all materials to be handled at the processing point and describe their characteristics, such as: a. How received at the processing unit: in bags, barrels, or drums, in bulk, by pipe line, etc.? b. Must storage and/or weighing be done at the site? c. Physical form. d. Specific gravity and bulk characteristics. e. Particle size or size range. f. Viscosity. g. Melting or boiling point. h. Corrosive properties. i. Abrasive characteristics. j. Is material poisonous? k. Is material explosive? l. Is material an irritant to skin, eyes, or lungs? m. Is material sensitive to exposure of air, moisture, or heat? 2. List pertinent data covering production: a. Quantity to be produced per 8-h shift. b. Formulation of finished product. c. What accuracy of analysis is required? d. Will changes in color, flavor, odor, or grade require frequent cleaning of equipment? e. Is this operation independent, or does it serve other process stages with which it must be synchronized? f. Is there a change in physical state during processing? g. Is there a chemical reaction? Is it endothermic or exothermic? h. What are the temperature requirements? i. What is the form of the finished product? j. How must the material be removed from the apparatus (by pumping, free flow through pipe or chute, dumping, etc.)? 3. Describe in detail the controlling characteristics of the finished product: a. Permanence of the emulsion or dispersion. b. Degree of blending of aggregates or of ultimate particles. c. Ultimate color development required. d. Uniformity of the dispersion of active ingredients, as in a drug product. e. Degree of control of moisture content for pumping extrusion, and so on. Preparation and Addition of Materials To ensure maximum production of high-grade mixed material, the preliminary preparation

CRYSTALLIZATION FROM SOLUTION of the ingredients must be correct and they must be added in the proper order. There are equipment implications to these considerations. Some finely powdered materials, such as carbon black, contain much air. If possible they should be compacted or wet out before being added to the mix. If a sufficient quantity of light solvent is a part of the formula, it may be used to wet the powder and drive out the air. If the powder cannot be wet, it may be possible to densify it somewhat by mechanical means. Removal of adsorbed gas under vacuum is sometimes necessary. Not uncommonly, critical ingredients that are present in small proportion (e.g., vulcanizers, antioxidants, and antiacids) tend to form aggregates when dry. Before entering the mixer, they should be fluffed, either by screening if the aggregates are soft or by passage through a hammer mill, roll mill, or muller if they are hard. The mixing time is cut down and the product is more uniform if all ingredients are freed from aggregates before mixing.

18-35

If any solids present in small amounts are soluble in a liquid portion of the mix, it is well to add them as a solution, making provision to distribute the liquid uniformly throughout the mass. When a trace of solid material which is not soluble in any other ingredients is to be added, it may be expedient to add it as a solution in a neutral solvent, with provision to evaporate the solvent at the end of the mixing cycle. It may be advisable to consider master batching, in which a lowproportion ingredient is separately mixed with part of some other ingredient of the mix, this premix then being added to the rest of the mix for final dispersion. Master batching is especially valuable in adding tinting colors, antioxidants, and the like. The master batch may be made up with laboratory accuracy, while at the mixing station weighing errors are minimized by the dilution of the important ingredient. Considerations such as these may make it desirable to consider automatic weighing and batch accumulation, metering of liquid ingredients, and automatic control of various time cycles.

CRYSTALLIZATION FROM SOLUTION GENERAL REFERENCES: AIChE Testing Procedures: Crystallizers, American Institute of Chemical Engineers, New York, 1970; Evaporators, 1961. Bennett, Chem. Eng. Prog., 58(9), 76 (1962). Buckley, Crystal Growth, Wiley, New York, 1951. Campbell and Smith, Phase Rule, Dover, New York, 1951. De Jong and Jancic (eds.), Industrial Crystallization, North-Holland Publishing Company, Amsterdam, 1979. “Crystallization from Solution: Factors Influencing Size Distribution,” Chem. Eng. Prog. Symp. Ser., 67(110), (1971). Mullin (ed.), Industrial Crystallization, Plenum, New York, 1976. Newman and Bennett, Chem. Eng. Prog., 55(3), 65 (1959). Palermo and Larson (eds.), “Crystallization from Solutions and Melts,” Chem. Eng. Prog. Symp. Ser., 65(95), (1969). Randolph (ed.), “Design, Control and Analysis of Crystallization Processes,” Am. Inst. Chem. Eng. Symp. Ser., 76(193), (1980). Randolph and Larson, Theory of Particulate Processes, Academic, New York, 2d ed., 1988. Rousseau and Larson (eds.), “Analysis and Design of Crystallization Processes,” Am. Inst. Chem. Eng. Symp. Ser., 72(153), (1976). Seidell, Solubilities of Inorganic and Metal Organic Compounds, American Chemical Society, Washington, 1965. Myerson (ed.), Handbook of Industrial Crystallization, Butterworth, 1993.

Crystallization is important as an industrial process because of the number of materials that are and can be marketed in the form of crystals. Its wide use is probably due to the highly purified and attractive form of a chemical solid which can be obtained from relatively impure solutions in a single processing step. In terms of energy requirements, crystallization requires much less energy for separation than do distillation and other commonly used methods of purification. In addition, it can be performed at relatively low temperatures and on a scale which varies from a few grams up to thousands of tons per day. Crystallization may be carried out from a vapor, from a melt, or from a solution. Most of the industrial applications of the operation involve crystallization from solutions. Nevertheless, crystal solidification of metals is basically a crystallization process, and much theory has been developed in relation to metal crystallization. This topic is so specialized, however, that it is outside the scope of this subsection, which is limited to crystallization from solution.

Solubility and Phase Diagrams Equilibrium relations for crystallization systems are expressed in the form of solubility data which are plotted as phase diagrams or solubility curves. Solubility data are ordinarily given as parts by weight of anhydrous material per 100 parts by weight of total solvent. In some cases these data are reported as parts by weight of anhydrous material per 100 parts of solution. If water of crystallization is present in the crystals, this is indicated as a separate phase. The concentration is normally plotted as a function of temperature and has no general shape or slope. It can also be reported as a function of pressure, but for most materials the change in solubility with change in pressure is very small. If there are two components in solution, it is common to plot the concentration of these two components on the X and Y axes and represent the solubility by isotherms. When three or more components are present, there are various techniques for depicting the solubility and phase relations in both threedimension and two-dimension models. For a description of these techniques, refer to Campbell and Smith (loc. cit.). Shown in Fig. 18-56 is a phase diagram for magnesium sulfate in water. The line p–a represents the freezing points of ice (water) from solutions of magne-

PRINCIPLES OF CRYSTALLIZATION Crystals A crystal may be defined as a solid composed of atoms arranged in an orderly, repetitive array. The interatomic distances in a crystal of any definite material are constant and are characteristic of that material. Because the pattern or arrangement of the atoms is repeated in all directions, there are definite restrictions on the kinds of symmetry that crystals can possess. There are five main types of crystals, and these types have been arranged into seven crystallographic systems based on the crystal interfacial angles and the relative length of its axes. The treatment of the description and arrangement of the atomic structure of crystals is the science of crystallography. The material in this discussion will be limited to a treatment of the growth and production of crystals as a unit operation.

FIG. 18-56 Phase diagram. MgSO4⋅H2O. To convert pounds to kilograms, divide by 2.2; K = (°F + 459.7)/1.8.

18-36

LIQUID-SOLID OPERATIONS AND EQUIPMENT

sium sulfate. Point a is the eutectic, and the line a–b–c–d–q is the solubility curve of the various hydrates. Line a–b is the solubility curve for MgSO4⋅12H2O, b–c is the solubility curve for MgSO4⋅7H2O, c–d is the solubility curve for MgSO4⋅6H2O, and d–q is the portion of the solubility curve for MgSO4⋅H2O. As shown in Fig. 18-57, the mutual solubility of two salts can be plotted on the X and Y axes with temperatures as isotherm lines. In the example shown, all the solution compositions corresponding to 100°C with solid-phase sodium chloride present are shown on the line DE. All the solution compositions at equilibrium with solid-phase KCl at 100°C are shown by the line EF. If both solid-phase KCl and NaCl are present, the solution composition at equilibrium can only be represented by point E, which is the invariant point (at constant pressure). Connecting all the invariant points results in the mixed-salt line. The locus of this line is an important consideration in making phase separations. There are numerous solubility data in the literature; the standard reference is by Seidell (loc. cit.). Valuable as they are, they nevertheless must be used with caution because the solubility of compounds is often influenced by pH and/or the presence of other soluble impurities which usually tend to depress the solubility of the major constituents. While exact values for any system are frequently best determined by actual composition measurements, the difficulty of reproducing these solubility diagrams should not be underestimated. To obtain data which are readily reproducible, elaborate pains must be taken to be sure the system sampled is at equilibrium, and often this means holding a sample at constant temperature for a period of from 1 to 100 h. While the published curves may not be exact for actual solutions of interest, they generally will be indicative of the shape of the solubility curve and will show the presence of hydrates or double salts. Heat Effects in a Crystallization Process The heat effects in a crystallization process can be computed by two methods: (1) a heat balance can be made in which individual heat effects such as sensible heats, latent heats, and the heat of crystallization can be combined into an equation for total heat effects; or (2) an enthalpy balance can be made in which the total enthalpy of all leaving streams minus the total enthalpy of all entering streams is equal to the heat absorbed from external sources by the process. In using the heat-balance method, it is necessary to make a corresponding mass balance, since the heat effects are related to the quantities of solids produced through the heat of crystallization. The advantage of the enthalpy-concentration-diagram method is that both heat and mass effects are taken into account simultaneously. This method has limited use because of the difficulty in obtaining enthalpyconcentration data. This information has been published for only a few systems.

With compounds whose solubility increases with increasing temperature there is an absorption of heat when the compound dissolves. In compounds with decreasing solubility as the temperature increases, there is an evolution of heat when solution occurs. When there is no change in solubility with temperature, there is no heat effect. The solubility curve will be continuous as long as the solid substance of a given phase is in contact with the solution, and any sudden change in the slope of the curve will be accompanied by a change in the heat of solution and a change in the solid phase. Heats of solution are generally reported as the change in enthalpy associated with the dissolution of a large quantity of solute in an excess of pure solvent. Tables showing the heats of solution for various compounds are given in Sec. 2. At equilibrium the heat of crystallization is equal and opposite in sign to the heat of solution. Using the heat of solution at infinite dilution as equal but opposite in sign to the heat of crystallization is equivalent to neglecting the heat of dilution. With many materials the heat of dilution is small in comparison with the heat of solution and the approximation is justified; however, there are exceptions. Relatively large heat effects are usually found in the crystallization of hydrated salts. In such cases the total heat released by this effect may be a substantial portion of the total heat effects in a cooling-type crystallizer. In evaporative-type crystallizers the heat of crystallization is usually negligible when compared with the heat of vaporizing the solvent. Yield of a Crystallization Process In most cases the process of crystallization is slow, and the final mother liquor is in contact with a sufficiently large crystal surface so that the concentration of the mother liquor is substantially that of a saturated solution at the final temperature in the process. In such cases it is normal to calculate the yield from the initial solution composition and the solubility of the material at the final temperature. If evaporative crystallization is involved, the solvent removed must be taken into account in determining the final yield. If the crystals removed from solution are hydrated, account must be taken of the water of crystallization in the crystals, since this water is not available for retaining the solute in solution. The yield is also influenced in most plants by the removal of some mother liquor with the crystals being separated from the process. Typically, with a product separated on a centrifuge or filter, the adhering mother liquor would be in the range of 2 to 10 percent of the weight of the crystals. The actual yield may be obtained from algebraic calculations or trial-and-error calculations when the heat effects in the process and any resultant evaporation are used to correct the initial assumptions on calculated yield. When calculations are made by hand, it is generally preferable to use the trial-and-error system, since it permits easy adjustments for relatively small deviations found in practice, such as the addition of wash water, or instrument and purge water additions. The following calculations are typical of an evaporative crystallizer precipitating a hydrated salt. If SI units are desired, kilograms = pounds × 0.454; K = (°F + 459.7)/1.8. Example 1: Yield from a Crystallization Process A 10,000-lb batch of a 32.5 percent MgSO4 solution at 120°F is cooled without appreciable evaporation to 70°F. What weight of MgSO4⋅7H2O crystals will be formed (if it is assumed that the mother liquor leaving is saturated)? From the solubility diagram in Fig. 18-56 at 70°F the concentration of solids is 26.3 lb MgSO4 per 100-lb solution. The mole weight of MgSO4 is 120.38. The mole weight of MgSO4⋅7H2O is 246.49. For calculations involving hydrated salts, it is convenient to make the calculations based on the hydrated solute and the “free water.” 246.94 0.325 weight fraction × ᎏ = 0.6655 MgSO4⋅7H2O in the feed solution 120.38

FIG. 18-57

Phase diagram, KCl − NaCl − H2O. K = °C + 273.2.

246.94 0.263 × ᎏ = 0.5385 MgSO4⋅7H2O in the mother liquor 120.38 Since the free water remains constant (except when there is evaporation), the final amount of soluble MgSO4⋅7H2O is calculated by the ratio of 0.538 lb MgSO4⋅7H2O ᎏᎏᎏ (1 − 0.538) lb free water

CRYSTALLIZATION FROM SOLUTION

Feed Mother liquor Yield

Total

MgSO4⋅7H2O

Free water

MgSO4⋅7H2O ᎏᎏ Free water

10,000 7249 2751

6655 3904* 2751

3345 3345

1.989 1.167

*3345 × (0.538/0.462) = 3904 A formula method for calculation is sometimes used where 100W0 − S(H0 − E) P = R ᎏᎏᎏ 100 − S(R − 1) where P = weight of crystals in final magma, lb R = mole weight of hydrate/mole weight of anhydrous = 2.04759 S = solubility at mother-liquor temperature (anhydrous basis) in lb per 100 lb solvent. [0.263/(1 − 0.263)] × 100 = 35.68521 W0 = weight of anhydrous solute in the original batch. 10,000(0.325) = 3250 lb H0 = total weight of solvent at the beginning of the batch. 10,000 − 3250 = 6750 lb E = evaporation = 0 (100)(3250) − 35.7(6750) P = 2.04 ᎏᎏᎏ = 2751 lb 100 − 35.7(2.04 − 1) Note that taking the difference between large numbers in this method can increase the chance for error.

Fractional Crystallization When two or more solutes are dissolved in a solvent, it is often possible to (1) separate these into the pure components or (2) separate one and leave the other in the solution. Whether or not this can be done depends on the solubility and phase relations of the system under consideration. Normally alternative 2 is successful only when one of the components has a much more rapid change in solubility with temperature than does the other. A typical example which is practiced on a large scale is the separation of KCl and NaCl from water solution. A phase diagram for this system is shown in Fig. 18-57. In this case the solubility of NaCl is plotted on the Y axis in parts per 100 parts of water, and the solubility of KCl is plotted on the X axis. The isotherms show a marked decrease in solubility for each component as the amount of the other is increased. This is typical for most inorganic salts. As explained earlier, the mixed-salt line is CE, and to make a separation of the solutes into the pure components it is necessary to be on one side of this line or the other. Normally a 95 to 98 percent approach to this line is possible. When evaporation occurs during a cooling or concentration process, this can be represented by movement away from the origin on a straight line through the origin. Dilution by water is represented by movement in the opposite direction. A typical separation might be represented as follows: Starting at E with a saturated brine at 100°C a small amount of water is added to dissolve any traces of solid phase present and to make sure the solids precipitated initially are KCl. Evaporative cooling along line HG results in the precipitation of KCl. During this evaporative cooling, part of the water evaporated must be added back to the solution to prevent the coprecipitation of NaCl. The final composition at G can be calculated by the NaCl/KCl/H2O ratios and the known amount of NaCl in the incoming solution at E. The solution at point G may be concentrated by evaporation at 100°C. During this process the solution will increase in concentration with respect to both components until point I is reached. Then NaCl will precipitate, and the solution will become more concentrated in KCl, as indicated by the line IE, until the original point E is reached. If concentration is carried beyond point E, a mixture of KCl and NaCl will precipitate. Example 2: Yield from Evaporative Cooling Starting with 1000 lb of water in a solution at H on the solubility diagram in Fig. 18-57, calculate the yield on evaporative cooling and concentrate the solution back to point H so the cycle can be repeated, indicating the amount of NaCl precipitated and the evaporation and dilution required at the different steps in the process. In solving problems of this type, it is convenient to list the material balance and the solubility ratios. The various points on the material balance are calculated by multiplying the quantity of the component which does not precipitate from solution during the transition from one point to another (normally the

18-37

NaCl in cooling or the KCl in the evaporative step) by the solubility ratio at the next step, illustrated as follows: Basis. 1000 lb of water at the initial conditions. Solubility ratios Solution component

KCl

NaCl

Water

KCl

NaCl

Water

H G(a) KCl yield Net evaporation I(b) E(c) NaCl yield Evaporation Dilution H′

343 194 149

270 270

1000 950

34.3 20.4

27.0 28.4

100 100

194 194

270 153 117

50 860 554

22.6 35.0

31.4 27.5

100 100

306 11 565

34.3

27.0

−100

194

153

The calculations for these steps are: a. 270 lb NaCl (100 lb water/28.4 lb NaCl) = 950 lb water 950 lb water (20.4 lb KCl/100 lb water) = 194 lb KCl b. 270 lb NaCl (100 lb water/31.4 lb NaCl) = 860 lb water 860 lb water (22.6 lb KCl/100 lb water) = 194 lb KCl c. 194 lb KCl (100 lb water/35.0 lb KCl) = 554 lb water 554 lb water (27.5 lb NaCl/100 lb water) = 153 lb NaCl Note that during the cooling step the maximum amount of evaporation which is permitted by the material balance is 50 lb for the step shown. In an evaporative-cooling step, however, the actual evaporation which results from adiabatic cooling is more than this. Therefore, water must be added back to prevent the NaCl concentration from rising too high; otherwise, coprecipitation of NaCl will occur. Inasmuch as only mass ratios are involved in these calculations, kilograms or any other unit of mass may be substituted for pounds without affecting the validity of the example.

Although the figures given are for a step-by-step process, it is obvious that the same techniques will apply to a continuous system if the fresh feed containing KCl and NaCl is added at an appropriate part of the cycle, such as between steps G and I for the case of dilute feed solutions. Another method of fractional crystallization, in which advantage is taken of different crystallization rates, is sometimes used. Thus, a solution saturated with borax and potassium chloride will, in the absence of borax seed crystals, precipitate only potassium chloride on rapid cooling. The borax remains behind as a supersaturated solution, and the potassium chloride crystals can be removed before the slower borax crystallization starts. Crystal Formation There are obviously two steps involved in the preparation of crystal matter from a solution. The crystals must first form and then grow. The formation of a new solid phase either on an inert particle in the solution or in the solution itself is called nucleation. The increase in size of this nucleus with a layer-by-layer addition of solute is called growth. Both nucleation and crystal growth have supersaturation as a common driving force. Unless a solution is supersaturated, crystals can neither form nor grow. Supersaturation refers to the quantity of solute present in solution compared with the quantity which would be present if the solution were kept for a very long period of time with solid phase in contact with the solution. The latter value is the equilibrium solubility at the temperature and pressure under consideration. The supersaturation coefficient can be expressed parts solute/100 parts solvent S = ᎏᎏᎏᎏᎏ ⭌ 1.0 (18-22) parts solute at equilibrium/100 parts solvent Solutions vary greatly in their ability to sustain measurable amounts of supersaturation. With some materials, such as sucrose, it is possible to develop a supersaturation coefficient of 1.4 to 2.0 with little danger of nucleation. With some common inorganic solutions such as sodium chloride in water, the amount of supersaturation which can be generated stably is so small that it is difficult or impossible to measure. Certain qualitative facts in connection with supersaturation, growth, and the yield in a crystallization process are readily apparent.

18-38

LIQUID-SOLID OPERATIONS AND EQUIPMENT

If the concentration of the initial solution and the final mother liquor are fixed, the total weight of the crystalline crop is also fixed if equilibrium is obtained. The particle-size distribution of this weight, however, will depend on the relationship between the two processes of nucleation and growth. Considering a given quantity of solution cooled through a fixed range, if there is considerable nucleation initially during the cooling process, the yield will consist of many small crystals. If only a few nuclei form at the start of the precipitation and the resulting yield occurs uniformly on these nuclei without secondary nucleation, a crop of large uniform crystals will result. Obviously, many intermediate cases of varying nucleation rates and growth rates can also occur, depending on the nature of the materials being handled, the rate of cooling, agitation, and other factors. When a process is continuous, nucleation frequently occurs in the presence of a seeded solution by the combined effects of mechanical stimulus and nucleation caused by supersaturation (heterogeneous nucleation). If such a system is completely and uniformly mixed (i.e., the product stream represents the typical magma circulated within the system) and if the system is operating at steady state, the particlesize distribution has definite limits which can be predicted mathematically with a high degree of accuracy, as will be shown later in this section. Geometry of Crystal Growth Geometrically a crystal is a solid bounded by planes. The shape and size of such a solid are functions of the interfacial angles and of the linear dimension of the faces. As the result of the constancy of its interfacial angles, each face of a growing or dissolving crystal, as it moves away from or toward the center of the crystal, is always parallel to its original position. This concept is known as the “principle of the parallel displacement of faces.” The rate at which a face moves in a direction perpendicular to its original position is called the translation velocity of that face or the rate of growth of that face. From the industrial point of view, the term crystal habit or crystal morphology refers to the relative sizes of the faces of a crystal. The crystal habit is determined by the internal structure and external influences on the crystal such as the growth rate, solvent used, and impurities present during the crystallization growth period. The crystal habit of commercial products is of very great importance. Long, needlelike crystals tend to be easily broken during centrifugation and drying. Flat, platelike crystals are very difficult to wash during filtration or centrifugation and result in relatively low filtration rates. Complex or twinned crystals tend to be more easily broken in transport than chunky, compact crystal habits. Rounded or spherical crystals (caused generally by attrition during growth and handling) tend to give considerably less difficulty with caking than do cubical or other compact shapes. Internal structure can be different in crystals that are chemically identical, even though they may be formed at different temperatures and have a different appearance. This is called polymorphism and can be determined only by X-ray diffraction. For the same internal structure, very small amounts of foreign substances will often completely change the crystal habit. The selective adsorption of dyes by different faces of a crystal or the change from an alkaline to an acidic environment will often produce pronounced changes in the crystal habit. The presence of other soluble anions and cations often has a similar influence. In the crystallization of ammonium sulfate, the reduction in soluble iron to below 50 ppm of ferric ion is sufficient to cause significant change in the habit of an ammonium sulfate crystal from a long, narrow form to a relatively chunky and compact form. Additional information is available in the patent literature and Table 18-4 lists some of the better-known additives and their influences. Since the relative sizes of the individual faces of a crystal vary between wide limits, it follows that different faces must have different translational velocities. A geometric law of crystal growth known as the overlapping principle is based on those velocity differences: in growing a crystal, only those faces having the lowest translational velocities survive; and in dissolving a crystal, only those faces having the highest translational velocities survive. For example, consider the cross sections of a growing crystal as in Fig. 18-58. The polygons shown in the figure represent varying stages in the growth of the crystal. The faces marked A are slow-growing

faces (low translational velocities), and the faces marked B are fastgrowing (high translational velocities). It is apparent from Fig. 18-58 that the faster B faces tend to disappear as they are overlapped by the slower A faces. Hartman and Perdok (1955) predicted that crystal habit or crystal morphology was related to the internal structure based on energy considerations and speculated it should be possible to predict the growth shape of crystals from the slice energy of different flat faces. Later, Hartman was able to predict the calculated attachment energy for various crystal species. Recently computer programs have been developed that predict crystal morphology from attachment energies. These techniques are particularly useful in dealing with organic or molecular crystals and rapid progress in this area is being made by companies such as Molecular Simulations of Cambridge, England. Purity of the Product If a crystal is produced in a region of the phase diagram where a single-crystal composition precipitates, the crystal itself will normally be pure provided that it is grown at relatively low rates and constant conditions. With many products these purities approach a value of about 99.5 to 99.8 percent. The difference between this and a purity of 100 percent is generally the result of small pockets of mother liquor called occlusions trapped within the crystal. Although frequently large enough to be seen with an ordinary microscope, these occlusions can be submicroscopic and represent dislocations within the structure of the crystal. They can be caused by either attrition or breakage during the growth process or by slip planes within the crystal structure caused by interference between screw-type dislocations and the remainder of the crystal faces. To increase the purity of the crystal beyond the point where such occlusions are normally expected (about 0.1 to 0.5 percent by volume), it is generally necessary to reduce the impurities in the mother liquor itself to an acceptably low level so that the mother liquor contained within these occlusions will not contain sufficient impurities to cause an impure product to be formed. It is normally necessary to recrystallize material from a solution which is relatively pure to surmount this type of purity problem. In addition to the impurities within the crystal structure itself, there is normally an adhering mother-liquid film left on the surface of the crystal after separation in a centrifuge or on a filter. Typically a centrifuge may leave about 2 to 10 percent of the weight of the crystals as adhering mother liquor on the surface. This varies greatly with the size and shape or habit of the crystals. Large, uniform crystals precipitated from low-viscosity mother liquors will retain a minimum of mother liquor, while nonuniform or small crystals precipitated from viscous solutions will retain a considerably larger proportion. Comparable statements apply to the filtration of crystals, although normally the amounts of mother liquor adhering to the crystals are considerably larger. It is common practice when precipitating materials from solutions which contain appreciable quantities of impurities to wash the crystals on the centrifuge or filter with either fresh solvent or feed solution. In principle, such washing can reduce the impurities quite substantially. It is also possible in many cases to reslurry the crystals in fresh solvent and recentrifuge the product in an effort to obtain a longer residence time during the washing operation and better mixing of the wash liquors with the crystals. Coefficient of Variation One of the problems confronting any user or designer of crystallization equipment is the expected particle-size distribution of the solids leaving the system and how this distribution may be adequately described. Most crystalline-product distributions plotted on arithmetic-probability paper will exhibit a straight line for a considerable portion of the plotted distribution. In this type of plot the particle diameter should be plotted as the ordinate and the cumulative percent on the log-probability scale as the abscissa. It is common practice to use a parameter characterizing crystal-size distribution called the coefficient of variation. This is defined as follows: PD16% − PD84% CV = 100 ᎏᎏ (18-23) 2PD50% where CV = coefficient of variation, as a percentage PD = particle diameter from intercept on ordinate axis at percent indicated

CRYSTALLIZATION FROM SOLUTION TABLE 18-4

Some Impurities Known to Be Habit Modifiers

Material crystallized Ba(NO2)2 CaSO4⋅2H2O CuSO4⋅5H2O KCl KClO4 K2CrO4 KH2PO4 KNO2 KNO3 K2SO4

LiCl⋅H2O MgSO4⋅7H2O Na2B4O7⋅10H2O Na2CO3⋅H2O NaCO3⋅NaHCO3⋅2H2O NaCl

NaClO3 NaNO3 Na2SO4

NH4Cl NH4ClO4 NF4F (NH4)NO3 (NH4)2HPO4 NH4H2PO4 (NH4)2SO4

ZnSO4⋅7H2O Adipic acid Fructose L-asparagine Naphthalene Pentaerythritol Sodium glutamate Sucrose Urea 1. 2. 3. 4. 5. 6. 7. 8.

18-39

Additive(s)

Effect

Mg, Te+4 Citric, succinic, tartaric acids Sodium citrate H2SO4 K4Fe(CN)6 Pb, Bi, Sn+2, Ti, Zr, Th, Cd, Fe, Hg, Mg Congo red (dye) Acid magenta (dye) Na2B4O7 Fe Acid magenta (dye) Pb, Th, Bi Acid magenta (dye) Cl, Mn, Mg, Bi, Cu, Al, Fe Cl3 (NH4)3Ce(NO3)6 Cr·Mn+2, Sn+2, Co, Ni, Fe+3 Borax Sodium oleate Casein, gelatin NaOH, Na2CO3 SO4= Ca+2 and Mg+2 D-40 detergent Na4Fe(CN)6, CdBr Pb, Mn+2, Bi, Sn+2, Ti, Fe, Hg Urea, formamide Tetraalkyl ammon. salts Polyethylene-oxy compounds Na2SO4, NaClO4 Acid green (dye) NH4SO4 @ pH 6.5 CdCl2 Alkyl aryl sulfonates Calgon Mn, Fe, Cu, Co, Ni, Cr Urea Azurine (dye) Ca Acid magenta (dye) H2SO4 Fe+3, Cr, Al, Sn Cr+3, Fe+3, Al+3 H2SO4 Oxalic acid, citric acid H3PO4, SO2 Borax Surfactant-SDBS Glucose, difructose L-glutamic acid Cyclohexane (solvent) Methanol (solvent) Sucrose Acetone (solvent) Lysine, CaO Raffinose, KCl, NaBr Biuret NH4Cl

Helps growth Helps growth Forms prisms Chunky crystals Inhibits growth, dendrites Helps growth Modifies the 102 face Modifies the 010 face Aids growth Helps growth Tabular crystals Helps growth Forms plates Helps growth Reduces growth rate Reduces growth rate Helps growth Aids growth Reduces growth & nuc. Promotes flat crystals Promotes chunky crystals Reduces L/D ratio Increase bulk density Aids growth Forms dendrites Helps growth Forms octahedra Helps growth & hardness Helps growth & hardness Tetrahedrons Flattened rhombahedra Large single crystals Inhibits growth Aids growth Aids growth Aid growth Forms octahedra Modifies the 102 face Helps growth Forms 010 face plates Reduces L/D ratio Helps growth Promotes needles Promotes needles Promotes chunky crystals Promotes chunky crystals Aids growth Aids growth Affects growth Affects growth Forms needles Forms plates Aids growth Forms plates Affects growth Modify growth rate Reduces L/D & aids growth Reduces L/D & aids growth

Concentration — Low — 0.3% 1000 ppm Low 50 ppm 50 ppm — Low Low 2000 ppm Low 1000 ppm 1000 ppm Low 5% 5 ppm — — 0.1–1.0% 400 ppm 20 ppm 100 ppm Low Low 1–100 ppm — — Low 1000 ppm — 100 ppm Low 22 ppm Low 1% 7% Traces 50 ppm 2–6% 1000 ppm 1000 ppm — 50–100 ppm — — —

References 1 5 5 4 1 6 6 1 1 7 1 6 1 4 4 1 1 2, 5 Canadian Patent 812,685 U.S. Patent 3,459,497 U.S. Patent 3,233,983 4 1 2 U.S. Patent 3,095,281 U.S. Patent 3,000,708 3 7 4 2 1 5 6 1 6 1 U.S. Patent 2,092,073 U.S. Patent 2,228,742 1 2 8 8 2 1 2 8

2–7% 5–10%

Gillman, The Art and Science of Growing Crystals, Wiley, New York, 1963. Mullin, Crystallization, Butterworth, London, 1961. Buckley, Crystal Growth, Wiley, New York, 1961. Phoenix, L., British Chemical Engineering, vol. II, no. 1 (Jan. 1966), pp. 34–38. Garrett, D. E., British Chemical Engineering, vol. I, no. 12 (Dec. 1959), pp. 673–677. Buckley, Crystal Growth, (Faraday Soc.) Butterworths, 1949, p. 249. Butchart and Whetstone, Crystal Growth, (Faraday Soc.) Butterworths, 1949, p. 259. Nyvlt, J., Industrial Crystallization, Verlag Chemie Publishers, New York, 1978, pp. 26–31.

In order to be consistent with normal usage, the particle-size distribution when this parameter is used should be a straight line between approximately 10 percent cumulative weight and 90 percent cumulative weight. By giving the coefficient of variation and the mean particle diameter, a description of the particle-size distribution is obtained which is normally satisfactory for most industrial purposes. If the product is removed from a mixed-suspension crystallizer, this coeffi-

cient of variation should have a value of approximately 50 percent (Randolph and Larson, op. cit., chap. 2). Crystal Nucleation and Growth Rate of Growth Crystal growth is a layer-by-layer process, and since growth can occur only at the face of the crystal, material must be transported to that face from the bulk of the solution. Diffusional

18-40

LIQUID-SOLID OPERATIONS AND EQUIPMENT tion level is very high or the solution is supersaturated in the absence of agitation. In commercial crystallization equipment, in which supersaturation is low and agitation is employed to keep the growing crystals suspended, the predominant mechanism is contact nucleation or, in extreme cases, attrition. In order to treat crystallization systems both dynamically and continuously, a mathematical model has been developed which can correlate the nucleation rate to the level of supersaturation and/or the growth rate. Because the growth rate is more easily determined and because nucleation is sharply nonlinear in the regions normally encountered in industrial crystallization, it has been common to assume

FIG. 18-58

B0 = ks i

Overlapping principle.

resistance to the movement of molecules (or ions) to the growing crystal face, as well as the resistance to integration of those molecules into the face, must be considered. As discussed earlier, different faces can have different rates of growth, and these can be selectively altered by the addition or elimination of impurities. If L is a characteristic dimension of a crystal of selected material and shape, the rate of growth of a crystal face that is perpendicular to L is, by definition, ∆L dL G ⬅ lim ᎏ = ᎏ (18-24) ∆L→O ∆t dt where G is the growth rate over time internal t. It is customary to measure G in the practical units of millimeters per hour. It should be noted that growth rates so measured are actually twice the facial growth rate. The delta L law. It has been shown by McCabe [Ind. Eng. Chem., 21, 30, 112 (1929)] that all geometrically similar crystals of the same material suspended in the same solution grow at the same rate if growth rate is defined as in Eq. (18-24). The rate is independent of crystal size, provided that all crystals in the suspension are treated alike. This generalization is known as the delta L law. Although there are some well-known exceptions, they usually occur when the crystals are very large or when movement of the crystals in the solution is so rapid that substantial changes occur in diffusion-limited growth of the faces. It is emphasized that the delta L law does not apply when similar crystals are given preferential treatment based on size. It fails also when surface defects or dislocations significantly alter the growth rate of a crystal face. Nevertheless, it is a reasonably accurate generalization for a surprising number of industrial cases. When it is, it is important because it simplifies the mathematical treatment in modeling real crystallizers and is useful in predicting crystal-size distribution in many types of industrial crystallization equipment. Important exceptions to McCabe’s growth-rate model have been noted by Bramson, by Randolph, and by Abegg. These are discussed by Canning and Randolph, Am. Inst. Chem. Eng. J., 13, 5 (1967). Nucleation The mechanism of crystal nucleation from solution has been studied by many scientists, and recent work suggests that—in commercial crystallization equipment, at least—the nucleation rate is the sum of contributions by (1) homogeneous nucleation and (2) nucleation due to contact between crystals and (a) other crystals, (b) the walls of the container, and (c) the pump impeller. If B0 is the net number of new crystals formed in a unit volume of solution per unit of time, B0 = Bss + Be + Bc

(18-25)

where Be is the rate of nucleation due to crystal-impeller contacts, Bc is that due to crystal-crystal contacts, and Bss is the homogeneous nucleation rate due to the supersaturation driving force. The mechanism of the last-named is not precisely known, although it is obvious that molecules forming a nucleus not only have to coagulate, resisting the tendency to redissolve, but also must become oriented into a fixed lattice. The number of molecules required to form a stable crystal nucleus has been variously estimated at from 80 to 100 (with ice), and the probability that a stable nucleus will result from the simultaneous collision of that large number is extremely low unless the supersatura-

(18-26)

where s, the supersaturation, is defined as (C − Cs), C being the concentration of the solute and Cs its saturation concentration; and the exponent i and dimensional coefficient k are values characteristic of the material. While Eq. (18-26) has been popular among those attempting correlations between nucleation rate and supersaturation, recently it has become commoner to use a derived relationship between nucleation rate and growth rate by assuming that G = k′s

(18-27)

whence, in consideration of Eq. (18-26), B0 = k″G i

(18-28)

where the dimensional coefficient k′ is characteristic of the material and the conditions of crystallization and k″ = k/(k′)i. Feeling that a model in which nucleation depends only on supersaturation or growth rate is simplistically deficient, some have proposed that contact nucleation rate is also a power function of slurry density and that B0 = knG i MT j

(18-29)

where MT is the density of the crystal slurry, g/L. Although Eqs. (18-28) and (18-29) have been adopted by many as a matter of convenience, they are oversimplifications of the very complex relationship that is suggested by Eq. (18-25); Eq. (18-29) implicitly and quite arbitrarily combines the effects of homogeneous nucleation and those due to contact nucleation. They should be used only with caution. In work pioneered by Clontz and McCabe [Chem. Eng. Prog. Symp. Ser., 67(110), 6 (1971)] and subsequently extended by others, contact nucleation rate was found to be proportional to the input of energy of contact, as well as being a function of contact area and supersaturation. This observation is important to the scaling up of crystallizers: at laboratory or bench scale, contact energy level is relatively low and homogeneous nucleation can contribute significantly to the total rate of nucleation; in commercial equipment, on the other hand, contact energy input is intense and contact nucleation is the predominant mechanism. Scale-up modeling of a crystallizer, therefore, must include its mechanical characteristics as well as the physiochemical driving force. Nucleation and Growth From the preceding, it is clear that no analysis of a crystallizing system can be truly meaningful unless the simultaneous effects of nucleation rate, growth rate, heat balance, and material balance are considered. The most comprehensive treatment of this subject is by Randolph and Larson (op. cit), who developed a mathematical model for continuous crystallizers of the mixedsuspension or circulating-magma type [Am. Inst. Chem. Eng. J., 8, 639 (1962)] and subsequently examined variations of this model that include most of the aberrations found in commercial equipment. Randolph and Larson showed that when the total number of crystals in a given volume of suspension from a crystallizer is plotted as a function of the characteristic length as in Fig. 18-59, the slope of the line is usefully identified as the crystal population density, n: ∆N dN n = lim ᎏ = ᎏ (18-30) ∆L→O ∆L dL where N = total number of crystals up to size L per unit volume of magma. The population density thus defined is useful because it char-

CRYSTALLIZATION FROM SOLUTION

FIG. 18-59

Determination of the population density of crystals.

acterizes the nucleation-growth performance of a particular crystallization process or crystallizer. The data for a plot like Fig. 18-60 are easily obtained from a screen analysis of the total crystal content of a known volume (e.g., a liter) of magma. The analysis is made with a closely spaced set of testing sieves, as discussed in Sec. 19, Table 19-6, the cumulative number of particles smaller than each sieve in the nest being plotted against the aperture dimension of that sieve. The fraction retained on each sieve is weighed, and the mass is converted to the equivalent number of particles by dividing by the calculated mass of a particle whose dimension is the arithmetic mean of the mesh sizes of the sieve on which it is retained and the sieve immediately above it. In industrial practice, the size-distribution curve usually is not actually constructed. Instead, a mean value of the population density for any sieve fraction of interest (in essence, the population density of the particle of average dimension in that fraction) is determined directly as ∆N/∆L, ∆N being the number of particles retained on the sieve and ∆L being the difference between the mesh sizes of the retaining sieve and its immediate predecessor. It is common to employ the units of (mm⋅L)−1 for n. For a steady-state crystallizer receiving solids-free feed and containing a well-mixed suspension of crystals experiencing negligible breakage, a material-balance statement degenerates to a particle balance (the Randolph-Larson general-population balance); in turn, it simplifies to dn n (18-31) ᎏ+ᎏ=0 dL Gt if the delta L law applies (i.e., G is independent of L) and the drawdown (or retention) time is assumed to be invariant and calculated as t = V/Q. Integrated between the limits n0, the population density of nuclei (for which L is assumed to be zero), and n, that of any chosen crystal size L, Eq. (18-31) becomes n L dn dL (18-32) ᎏ=− ᎏ n0 n 0 Gt

冕

or

冕

−L ln n = ᎏ + ln n0 Gt

(18-33a)

n = n0e−L/Gt

(18-33b)

A plot of ln n versus L is a straight line whose intercept is ln n0 and whose slope is −1/Gt. (For plots on base-10 log paper, the appropriate slope correction must be made.) Thus, from a given product sample of known slurry density and retention time it is possible to obtain the nucleation rate and growth rate for the conditions tested if the sample satisfies the assumptions of the derivation and yields a straight line. A number of derived relations which describe the nucleation rate, size distribution, and average properties are summarized in Table 18-5. If a straight line does not result (Fig. 18-60), at least part of the explanation may be violation of the delta L law (Canning and Ran-

18-41

dolph, loc. cit.). The best current theory about what causes sizedependent growth suggests what has been called growth dispersion or “Bujacian behavior” [Mullen (ed.), op. cit., p. 23]. In the same environment different crystals of the same size can grow at different rates owing to differences in dislocations or other surface effects. The graphs of “slow” growers (Fig. 18-60, curve A) and “fast” growers (curve B) sum to a resultant line (curve C), concave upward, that is described by Eq. (18-34) (Randolph, in deJong and Jancic, op. cit., p. 295): B0i n = 冱 ᎏ e(−L/G i t) (18-34) Gi Equation (18-31) contains no information about the crystallizer’s influence on the nucleation rate. If the crystallizer is of a mixedsuspension, mixed-product-removal (MSMPR) type, satisfying the criteria for Eq. (18-31), and if the model of Clontz and McCabe is valid, the contribution to the nucleation rate by the circulating pump can be calculated [Bennett, Fiedelman, and Randolph, Chem. Eng. Prog., 69(7), 86 (1973)]: ∞ I2 Be = Ke ᎏ ρG nL4 dL (18-35) 0 P where I = tip speed of the propeller or impeller, m/s ρ = crystal density, g/cm3 P = volume of crystallizer/circulation rate (turnover), m3/(m3/s) = s

冢 冣 冕

Since the integral term is the fourth moment of the distribution (m4), Eq. (18-35) becomes I2 Be = KeρG ᎏ m4 (18-36) P Equation (18-36) is the general expression for impeller-induced nucleation. In a fixed-geometry system in which only the speed of the circulating pump is changed and in which the flow is roughly proportional to the pump speed, Eq. (18-36) may be satisfactorily replaced with

冢 冣

Be = K ″e ρG(SR)3m4

(18-37)

where SR = rotation rate of impeller, r/min. If the maximum crystalimpeller impact stress is a nonlinear function of the kinetic energy, shown to be the case in at least some systems, Eq. (18-37) no longer applies. In the specific case of an MSMPR exponential distribution, the fourth moment of the distribution may be calculated as m4 = 4!n0(Gt)5

(18-38)

Substitution of this expression into Eq. (18-36) gives Be = kn0 G(SR )3LD5

(18-39)

where LD = 3Gt, the dominant crystal (mode) size. Equation (18-39) displays the competing factors that stabilize secondary nucleation in an operating crystallizer when nucleation is due mostly to impeller-crystal contact. Any increase in particle size produces a fifth-power increase in nucleation rate, tending to counteract the direction of the change and thereby stabilizing the crystal-size distribution. From dimensional argument alone the size produced in a mixed crystallizer for a (fixed) nucleation rate varies as (B0)1/3. Thus, this fifth-order response of contact nucleation does not wildly upset the crystal size distribution but instead acts as a stabilizing feedback effect. Nucleation due to crystal-to-crystal contact is greater for equal striking energies than crystal-to-metal contact. However, the viscous drag of the liquid on particle sizes normally encountered limits the velocity of impact to extremely low values. The assumption that only the largest crystal sizes contribute significantly to the nucleation rate by crystal-to-crystal contact permits a simple computation of the rate: Bc = Kc ρGm2j

(18-40)

where mj = the fourth, fifth, sixth, or higher moments of the distribution. A number of different crystallizing systems have been investigated by using the Randolph-Larson technique, and some of the published

TABLE 18-5 Common Equations for Population-Balance Calculations Systems with fines removal Name

Symbol

Units

Systems without fines removal

Fines stream

Product stream

Drawdown time (retention time)

t

h

t = V/Q

tF = Vliquid /QF

t = V/Q

Growth rate

G

mm/h

G = dL/dt

G = dL/dt

G = dL/dt

Volume coefficient

Kv

1/no. (crystals)

volume of one crystal K v = ᎏᎏᎏ L3

volume of one crystal K v = ᎏᎏᎏ L3

volume of one crystal K v = ᎏᎏᎏ L3

Population density

n

No. (crystals)/mm

n = dN/dL

n = dN/dL

n = dN/dL

Nuclei population density

no

No. (crystals)/mm

no = K M M jG i − 1

Population density

n

No. (crystals)/mm

n = noe−L/Gt

No. (crystals)/h

B0 = Gn = K M M G

None

L x=ᎏ Gt

Nucleation rate Dimensionless length Mass/unit volume (slurry density)

B0 x MT

g/L

j

B0 = G

i

n = no − Le /GtFe−L/Gt

冕

∞

4

MTF = Kvρ

nL3 dL

0

1, 3

o n

L xF = ᎏ , L0 → Lf GtF

Mt = K v ρ

1 2

nF = noe−L/GtF

o

References

冕

Lf

noe−L/GtF L3 dL

L x = ᎏ , Lf → L Gt MT = K v ρ

0

冕

1

∞

noe−L/GtFe−L/Gt L3 dL

1

Lf

Mt = K v ρ6 n (Gt) o

Cumulative mass to x ᎏᎏᎏ Total mass

Wx

None

Wx = 1 − e−x

Dominant particle

Ld

mm

Id = 3Gt

Average particle, weight

La

mm

Ia = 3.67Gt

冢

4

冣

x3 x2 ᎏ+ᎏ+x+1 6 2

e−x (x 3 + 3x 2 + 6x + 6) − 6 WF = ᎏᎏᎏ e−x C(x c3 + 3x c2 + 6x c + 6) − 6

冤

冢

冣冥

x3 x2 6K v ρnoe−Le /GtFGt)4 1 − e−x ᎏᎏ + ᎏᎏ + x + 1 6 2 W = ᎏᎏᎏᎏᎏ Slurry density M, g/L

5

when Lc ≈ 0, compared with La

Total number of crystals

NT

No./L

NT =

冕

∞

0

1. 2. 3. 4. 5. 6.

Randolph and Larson, Am. Inst. Chem. Eng. J., 8, 639 (1962). Timm and Larson, Am. Inst. Chem. Eng. J., 14, 452 (1968). Larson, private communication. Larson, Timm, and Wolff, Am. Inst. Chem. Eng. J., 14, 448 (1968). Larson and Randolph, Chem. Eng. Prog. Symp. Ser., 65(95), 1 (1969). Schoen, Ind. Eng. Chem., 53, 607 (1961).

18-42

n dl

6 NF =

冕

Lf

0

nF dL

NT =

冕

∞

Lf

n dL

1, 3

CRYSTALLIZATION FROM SOLUTION

18-43

An additional check can be made of the accuracy of the data by the relation MT = 6k v ρn0(Gt)4 = 450 g/L 1.335 g/cm3 19.78 MT = (6)(1.0) ᎏᎏ e [(0.0324)(3.38)]4 1000 mm3/cm3 MT = 455 g/L ≈ 450 g/L Had only the growth rate been known, the size distribution of the solids could have been calculated from the equation

冢

冣

x3 x2 Wf = 1 − e−x ᎏ + ᎏ + x + 1 6 2 where Wf is the weight fraction up to size L and x = L/Gt. FIG. 18-60

Population density of crystals resulting from Bujacian behavior.

growth rates and nucleation rates are included in Table 18-6. Although the usefulness of these data is limited to the conditions tested, the table gives a range of values which may be expected, and it permits resolution of the information gained from a simple screen analysis into the fundamental factors of growth rate and nucleation rate. Experiments may then be conducted to determine the independent effects of operation and equipment design on these parameters. Although this procedure requires laborious calculations because of the number of samples normally needed, these computations and the determination of the best straight-line fit to the data are readily programmed for digital computers. Example 3: Population, Density, Growth and Nucleation Rate Calculate the population density, growth, and nucleation rates for a crystal sample of urea for which there is the following information. These data are from Bennett and Van Buren [Chem. Eng. Prog. Symp. Ser., 65(95), 44 (1969)]. Slurry density = 450 g/L Crystal density = 1.335 g/cm3 Drawdown time t = 3.38 h Shape factor kv = 1.00 Product size: −14 mesh, +20 mesh 4.4 percent −20 mesh, +28 mesh 14.4 percent −28 mesh, +35 mesh 24.2 percent −35 mesh, +48 mesh 31.6 percent −48 mesh, +65 mesh 15.5 percent −65 mesh, +100 mesh 7.4 percent −100 mesh 2.5 percent n = number of particles per liter of volume 14 mesh = 1.168 mm, 20 mesh = 0.833 mm, average opening 1.00 mm Size span = 0.335 mm = ∆L (450 g/L)(0.044) n20 = ᎏᎏᎏᎏᎏᎏ (1.335/1000) g/mm3(1.003 mm3/particle)(0.335 mm)(1.0) n20 = 44,270 ln n20 = 10,698 Repeating for each screen increment: Screen size

Weight, %

kv

ln n

L, average diameter, mm

100 65 48 35 28 20

7.4 15.5 31.6 24.2 14.4 4.4

1.0 1.0 1.0 1.0 1.0 1.0

18.099 17.452 16.778 15.131 13.224 10.698

0.178 0.251 0.356 0.503 0.711 1.000

Plotting ln n versus L as shown in Fig. 18-61, a straight line having an intercept at zero length of 19.781 and a slope of −9.127 results. As mentioned in discussing Eq. (18-24), the growth rate can then be found. Slope = −1/Gt or −9.127 = −1/[G(3.38)] or and and

G = 0.0324 mm/h n0 B0 = Gn0 = (0.0324)(e19.781) = 12.65 × 106 ᎏ L⋅h La = 3.67(0.0324)(3.38) = 0.40 mm

L L x = ᎏᎏ = ᎏ (0.0324)(3.38) 0.1095

Screen size

L, mm

x

20 28 35 48 65 100

0.833 0.589 0.417 0.295 0.208 0.147

7.70 5.38 3.80 2.70 1.90 1.34

Wf*

Cumulative % retained 100 (1 − Wf )

Measured cumulative % retained

0.944 0.784 0.526 0.286 0.125 0.048

5.6 21.6 47.4 71.4 87.5 95.2

4.4 18.8 43.0 74.6 90.1 97.5

*Values of Wf as a function of x may be obtained from a table of Wick’s functions. Note that the calculated distribution shows some deviation from the measured values because of the small departure of the actual sample from the theoretical coefficient of variation (i.e., 47.5 versus 52 percent). Here it can be seen that the nucleation rate is a decreasing function of growth rate (and supersaturation). The physical explanation is believed to be the mechanical influence of the crystallizer on the growing suspension and/or the effect of Bujacian behavior. Had sufficient data indicating a change in n0 for various values of M at constant G been available, a plot of ln n0 versus ln M at corresponding G’s would permit determination of the power j.

Crystallizers with Fines Removal In Example 3, the product was from a forced-circulation crystallizer of the MSMPR type. In many cases, the product produced by such machines is too small for commercial use; therefore, a separation baffle is added within the crystallizer to permit the removal of unwanted fine crystalline material from the magma, thereby controlling the population density in the machine so as to produce a coarser crystal product. When this is done, the product sample plots on a graph of ln n versus L as shown in line P, Fig. 18-62. The line of steepest slope, line F, represents the particlesize distribution of the fine material, and samples which show this distribution can be taken from the liquid leaving the fines-separation baffle. The product crystals have a slope of lower value, and typically there should be little or no material present smaller than Lf, the size which the baffle is designed to separate. The effective nucleation rate for the product material is the intersection of the extension of line P to zero size. As long as the largest particle separated by the fines-destruction baffle is small compared with the mean particle size of the product, the seed for the product may be thought of as the particle-size distribution corresponding to the fine material which ranges in length from zero to Lf, the largest size separated by the baffle. The product discharged from the crystallizer is characterized by the integral of the distribution from size Lf to infinity: MT = kvρ

冕

∞

n0 exp (−Lf /Gtf) exp (L/GT) L3 dL

(18-41)

Lf

The integrated form of this equation is shown in Table 18-5. For a given set of assumptions it is possible to calculate the characteristic curves for the product from the crystallizer when it is operated at various levels of fines removal as characterized by Lf. This has been done for an ammonium sulfate crystallizer in Fig. 18-63. Also shown in that figure is the actual size distribution obtained. In calculating theoretical size distributions in accordance with the Eq. (18-41), it is

18-44

LIQUID-SOLID OPERATIONS AND EQUIPMENT

TABLE 18-6

Growth Rates and Kinetic Equations for Some Industrial Crystallized Products

Material crystallized

G, m/s × 108

Range t, h

Range MT, g/L

Temp., °C

Scale*

Kinetic equation for B0 no./(L⋅s)

(NH4)2SO4

1.67

3.83

150

70

P

B0 = 6.62 × 10−25 G 0.82 p−0.92 m 2.05 2

(NH4)2SO4

0.20

0.25

38

18

B

B0 = 2.94(1010)G1.03

(NH4)2SO4

—

0.20

—

34

B

B0 = 6.14(10−11)S R7.84M T0.98 G1.22

MgSO4⋅7H2O

3.0–7.0

—

—

25

B

B0 = 9.65(1012)M T0.67 G1.24

MgSO4⋅7H2O

—

—

Low

29

B

B0 = f (N, L4, N 4.2, S 2.5)

KCl

2–12

—

200

32

P

B0 = 7.12(1039)M T0.14 G4.99

KCl

3.3

1–2

100

37

B

B0 = 5.16(1022)MT0.91 G2.77

KCl

0.3–0.45

—

50–147

25–68

B

B0 = 5 × 10−3 G 2.78(MT TIP 2)1.2

KCr2O7

1.2–9.1

0.25–1

14–42

—

B

B0 = 7.33(104)M T0.6G 0.5

KCr2O7

2.6–10

0.15–0.5

20–100

26–40

B

B0 = 1.59(10−3)S R3 MT G 0.48

KNO3

8.13

0.25–0.050

10–40

20

B

B0 = 3.85(1016)M T0.5 G 2.06

K2SO4

—

0.03–0.17

1–7

30

B

B0 = 2.62(103)S R2.5 M T0.5G 0.54

K2SO4

2–6

0.25–1

2–20

10–50

B

K2SO4

0.8–1.6

—

—

—

B

NaCl

4–13

0.2–1

25–200

50

B

B0 = 1.92(1010)S R2 MT G 2

NaCl

—

0.6

35–70

55

P

NaCl

0.5

1–2.5

70–190

72

P

Citric acid

1.1–3.7

—

—

16–24

B

B0 = 8 × 1010N 2G 2 MT I2 0.98 B0 = 1.47(102) ᎏ m0.84 4 G P B0 = 1.09(1010)m40.084G 0.84

Fructose

0.1–0.25

—

—

50

B

—

Sucrose

—

—

—

80

B

B0 = 5 × 106 N 0.7M T0.3G 0.4

冢

冣

10900 B0 = 4.09(106) exp ᎏ MT G 0.5 RT G ᎏ = 1 + 2 L2/3 (L in µm) G0

冢 冣

Sugar

2.5–5

0.375

50

45

B

B0 = 4.38(106)M T1.01(∆C − 0.5)1.42

Urea

0.4–4.2

2.5–6.8

350–510

55

P

B0 = 5.48(10−1)M T−3.87G 1.66

Urea

—

—

—

3–16

B

B0 = 1.49(10−31)S R2.3 M T1.07 G −3.54

References† Bennett and Wolf, AIChE, SFC, 1979. Larsen and Mullen, J. Crystal Growth 20: 183 (1973). Youngquist and Randolph, AIChE J. 18: 421 (1972). Sikdar and Randolph, AIChE J. 22: 110 (1976). Ness and White, AIChE Symposium Series 153, vol. 72, p. 64. Randolph et al., AIChE J. 23: 500 (1977). Randolph et al., Ind. Eng. Chem. Proc. Design Dev. 20: 496 (1981). Qian et al., AIChE J. 33(10): 1690 (1987). Desari et al., AIChE J. 20: 43 (1974). Janse, Ph.D. thesis, Delft Technical University, 1977. Juraszek and Larson, AIChE J. 23: 460 (1977). Randolph and Sikdar, Ind. Eng. Chem. Fund. 15: 64 (1976). Jones, Budz, and Mullin, AIChE J. 33: 12 (1986). White, Bendig, and Larson, AIChE Mtg., Washington, D.C., Dec. 1974. Asselbergs, Ph.D. thesis, Delft Technical University, 1978. Grootscholten et al., Chem. Eng. Design 62: 179 (1984). Bennett et al., Chem. Eng. Prog. 69(7): 86 (1973). Sikdar and Randolph, AIChE J. 22: 110 (1976). Shiau and Berglund, AIChE J. 33: 6 (1987). Berglund and deJong, Separations Technology 1: 38 (1990). Hart et al., AIChE Symposium Series 193, vol. 76, 1980. Bennett and Van Buren, Chem. Eng. Prog. Symposium Series 95(7): 65 (1973). Lodaya et al., Ind. Eng. Chem. Proc. Design Dev. 16: 294 (1977).

*B = bench scale; P = pilot plant. †Additional data on many components are in Garside and Shah, Ind. Eng. Chem. Proc. Design Dev., 19, 509 (1980).

assumed that the growth rate is a constant, whereas in fact larger values of Lf will interact with the system driving force to raise the growth rate and the nucleation rate. Nevertheless, Fig. 18-63 illustrates clearly the empirical result of the operation of such equipment, demonstrating that the most significant variable in changing the particle-size distribution of the product is the size removed by the baffle. Conversely, changes in retention time for a given particle-removal size Lf make a relatively small change in the product-size distribution. It is implicit that increasing the value of Lf will raise the supersaturation and growth rate to levels at which mass homogeneous nucleation can occur, thereby leading to periodic upsets of the system or cycling [Randolph, Beer, and Keener, Am. Inst. Chem. Eng. J., 19, 1140 (1973)]. That this could actually happen was demonstrated experimentally by Randolph, Beckman, and Kraljevich [Am. Inst. Chem. Eng. J., 23, 500 (1977)], and that it could be controlled dynamically by regulating the fines-destruction system was shown by Beckman and Randolph [ibid., (1977)]. Dynamic control of a crystallizer with a fines-destruction baffle and fine-particle-detection equipment

employing a light-scattering (laser) particle-size-measurement instrument is described in U.S. Patent, 4,263,010 and 5,124,265. CRYSTALLIZATION EQUIPMENT Whether a vessel is called an evaporator or a crystallizer depends primarily on the criteria used in arriving at its sizing. In an evaporator of the salting-out type, sizing is done on the basis of vapor release. In a crystallizer, sizing is normally done on the basis of the volume required for crystallization or for special features required to obtain the proper product size. In external appearance, the vessels could be identical. Evaporators are discussed in Sec. 11. In the discussion which follows, crystallization equipment has been classified according to the means of suspending the growing product. This technique reduces the number of major classifications and segregates those to which Eq. (18-31) applies. Mixed-Suspension, Mixed-Product-Removal Crystallizers This type of equipment, sometimes called the circulating-magma

CRYSTALLIZATION FROM SOLUTION

FIG. 18-62

FIG. 18-61

Population density plot for Example 3.

crystallizer, is by far the most important in use today. In most commercial equipment of this type, the uniformity of suspension of product solids within the crystallizer body is sufficient for the theory [Eqs. (18-31) to (18-33b)] to apply. Although a number of different varieties and features are included within this classification, the equipment operating with the highest capacity is the kind in which the vaporization of a solvent, usually water, occurs.

FIG. 18-63

18-45

Plot of Log N against L for a crystallizer with fines removal.

Although surface-cooled types of MSMPR crystallizers are available, most users prefer crystallizers employing vaporization of solvents or of refrigerants. The primary reason for this preference is that heat transferred through the critical supersaturating step is through a boiling-liquid-gas surface, avoiding the troublesome solid deposits that can form on a metal heat-transfer surface. Forced-Circulation Evaporator-Crystallizer This crystallizer is shown in Fig. 18-64. Slurry leaving the body is pumped through a circulating pipe and through a tube-and-shell heat exchanger, where its temperature increases by about 2 to 6°C (3 to 10°F). Since this heating is done without vaporization, materials of normal solubility should produce no deposition on the tubes. The heated slurry, returned to the body by a recirculation line, mixes with the body slurry and raises its temperature locally near the point of entry, which causes boiling at the liquid surface. During the consequent cooling and

Calculated product-size distribution for a crystallizer operation at different fine-crystal-separation sizes.

18-46

LIQUID-SOLID OPERATIONS AND EQUIPMENT

Forced-circulation (evaporative) crystallizer. (Swenson Process Equipment, Inc.)

FIG. 18-64

vaporization to achieve equilibrium between liquid and vapor, the supersaturation which is created causes deposits on the swirling body of suspended crystals until they again leave via the circulating pipe. The quantity and the velocity of the recirculation, the size of the body, and the type and speed of the circulating pump are critical design items if predictable results are to be achieved. A further discussion of the parameters affecting this type of equipment is given by Bennett, Newman, and Van Buren [Chem. Eng. Prog., 55(3), 65 (1959); Chem. Eng. Prog. Symp. Ser., 65(95), 34, 44 (1969)]. If the crystallizer is not of the evaporative type but relies only on adiabatic evaporative cooling to achieve the yield, the heating element is omitted. The feed is admitted into the circulating line after withdrawal of the slurry, at a point sufficiently below the free-liquid surface to prevent flashing during the mixing process. Draft-Tube-Baffle (DTB) Evaporator-Crystallizer Because mechanical circulation greatly influences the level of nucleation within the crystallizer, a number of designs have been developed that use circulators located within the body of the crystallizer, thereby reducing the head against which the circulator must pump. This technique reduces the power input and circulator tip speed and therefore the rate of nucleation. A typical example is the draft-tube-baffle (DTB) evaporator-crystallizer (Swenson Process Equipment, Inc.) shown in Fig. 18-65. The suspension of product crystals in maintained by a large, slow-moving propeller surrounded by a draft tube within the body. The propeller directs the slurry to the liquid surface so as to prevent solids from short-circuiting the zone of the most intense supersaturation. Slurry which has been cooled is returned to the bottom of the vessel and recirculated through the propeller. At the propeller, heated solution is mixed with the recirculating slurry. The design of Fig. 18-65 contains a fines-destruction feature comprising the settling zone surrounding the crystallizer body, the circulating pump, and the heating element. The heating element supplies sufficient heat to meet the evaporation requirements and to raise the temperature of the solution removed from the settler so as to destroy

FIG. 18-65

Draft-tube-baffle (DTB) crystallizer. (Swenson Process Equip-

ment, Inc.)

any small crystalline particles withdrawn. Coarse crystals are separated from the fines in the settling zone by gravitational sedimentation, and therefore this fines-destruction feature is applicable only to systems in which there is a substantial density difference between crystals and mother liquor. This type of equipment can also be used for applications in which the only heat removed is that required for adiabatic cooling of the incoming feed solution. When this is done and the fines-destruction feature is to be employed, a stream of liquid must be withdrawn from the settling zone of the crystallizer and the fine crystals must be separated or destroyed by some means other than heat addition—for example, either dilution or thickening and physical separation. In some crystallization applications it is desirable to increase the solids content of the slurry within the body above the natural consistency, which is that developed by equilibrium cooling of the incoming feed solution to the final temperature. This can be done by withdrawing a stream of mother liquor from the baffle zone, thereby thickening the slurry within the growing zone of the crystallizer. This mother liquor is also available for removal of fine crystals for size control of the product. Draft-Tube (DT) Crystallizer This crystallizer may be employed in systems in which fines destruction is not needed or wanted. In such cases the baffle is omitted, and the internal circulator is sized to have the minimum nucleating influence on the suspension. In DTB and DT crystallizers the circulation rate achieved is generally much greater than that available in a similar forced-circulation crystallizer. The equipment therefore finds application when it is necessary to circulate large quantities of slurry to minimize supersaturation levels within the equipment. In general, this approach is required to obtain long operating cycles with material capable of growing on

CRYSTALLIZATION FROM SOLUTION the walls of the crystallizer. The draft-tube and draft-tube-baffle designs are commonly used in the production of granular materials such as ammonium sulfate, potassium chloride, photographic hypo, and other inorganic and organic crystals for which product in the range 8 to 30 mesh is required. Surface-Cooled Crystallizer For some materials, such as sodium chlorate, it is possible to use a forced-circulation tube-and-shell exchanger in direct combination with a draft-tube-crystallizer body, as shown in Fig. 18-66. Careful attention must be paid to the temperature difference between the cooling medium and the slurry circulated through the exchanger tubes. In addition, the path and rate of slurry flow within the crystallizer body must be such that the volume contained in the body is “active.” That is to say, crystals must be so suspended within the body by the turbulence that they are effective in relieving supersaturation created by the reduction in temperature of the slurry as it passes through the exchanger. Obviously, the circulating pump is part of the crystallizing system, and careful attention must be paid to its type and its operating parameters to avoid undue nucleating influences. The use of the internal baffle permits operation of the crystallizer at a slurry consistency other than that naturally obtained by the cooling of the feed from the initial temperature to the final mother-liquor temperature. The baffle also permits fines removal and destruction. With most inorganic materials this type of equipment produces crystals in the range 30 to 100 mesh. The design is based on the allowable rates of heat exchange and the retention required to grow the product crystals. Direct-Contact-Refrigeration Crystallizer For some applications, such as the freezing of ice from seawater, it is necessary to go to such low temperatures that cooling by the use of refrigerants is the only economical solution. In such systems it is sometimes impractical to employ surface-cooled equipment because the allowable temperature difference is so small (under 3°C) that the heat-exchanger surface becomes excessive or because the viscosity is so high that the mechanical energy put in by the circulation system requires a heat-removal rate greater than can be obtained at reasonable temperature differences. In such systems, it is convenient to admix the refrigerant with the slurry being cooled in the crystallizer, as shown in Fig. 18-67, so that the heat of vaporization of the refrigerant cools the slurry by direct contact. The successful application of such systems requires that the refrigerant be relatively immiscible with the mother liquor and be capable of separation, compression, condensation, and subsequent recycle into the crystallizing system. The operating pressures and temperatures chosen have a large bearing on power consumption. This technique has been very successful in reducing the problems associated with buildup of solids on a cooling surface. The use of direct-

Forced-circulation baffle surface-cooled crystallizer. (Swenson Process Equipment, Inc.)

FIG. 18-66

18-47

FIG. 18-67 Direct-contact-refrigeration crystallizer (DTB type). (Swenson Process Equipment, Inc.)

contact refrigeration also reduces overall process-energy requirements, since in a refrigeration process involving two fluids a greater temperature difference is required on an overall basis when the refrigerant must first cool some intermediate solution, such as calcium chloride brine, and that solution in turn cools the mother liquor in the crystallizer. Equipment of this type has been successfully operated at temperatures as low as −59°C (−75°F). Reaction-Type Crystallizers In chemical reactions in which the end product is a solid-phase material such as a crystal or an amorphous solid the type of equipment described in the preceding subsections or shown in Fig. 18-68 may be used. By mixing the reactants in a large circulated stream of mother liquor containing suspended solids of the equilibrium phase, it is possible to minimize the driving force created during their reaction and remove the heat of reaction through the vaporization of a solvent, normally water. Depending on the final particle size required, it is possible to incorporate a fines-destruction baffle as shown in Fig. 18-68 and take advantage of the control over particle size afforded by this technique. In the case of ammonium sulfate crystallization from ammonia gas and concentrated sulfuric acid, it is necessary to vaporize water to remove the heat of reaction, and this water so removed can be reinjected after condensation into the fines-destruction stream to afford a very large amount of dissolving capability. Other examples of this technique are where a solid material is to be decomposed by mixing it with a mother liquor of a different composition, as shown in Fig. 18-69. Carnallite ore (KCl⋅MgCl2⋅4H2O) can be added to a mother liquor into which water is also added so that decomposition of the ore into potassium chloride (KCl) crystals and magnesium chloride–rich mother liquor takes place. Circulated slurry in the draft tube suspends the product crystals as well as the incoming ore particles until the ore can decompose into potassium chloride crystals and mother liquor. By taking advantage of the fact that water must be added to the process, the fines-bearing mother liquor can be removed behind the baffle and then water added so that the finest particles are dissolved before being returned to the crystallizer body.

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LIQUID-SOLID OPERATIONS AND EQUIPMENT

FIG. 18-68

Swenson reaction type DTB crystallizer. (Swenson Process Equipment, Inc.)

Other examples of this technique involve neutralization reactions such as the neutralization of sulfuric acid with calcium chloride to result in the precipitation of gypsum. Mixed-Suspension, Classified-Product-Removal Crystallizers Many of the crystallizers just described can be designed for classifiedproduct discharge. Classification of the product is normally done by means of an elutriation leg suspended beneath the crystallizing body as shown in Fig. 18-66. Introduction of clarified mother liquor to the lower portion of the leg fluidizes the particles prior to discharge and selectively returns the finest crystals to the body for further growth. A relatively wide distribution of material is usually produced unless the elutriation leg is extremely long. Inlet conditions at the leg are critical if good classifying action or washing action is to be achieved. If an elutriation leg or other product-classifying device is added to a crystallizer of the MSMPR type, the plot of the population density

versus L is distorted in the region of largest sizes. Also the incorporation of an elutriation leg destabilizes the crystal-size distribution and under some conditions can lead to cycling. The theoretical treatment of both the crystallizer model and the cycling relations is discussed by Randolph, Beer, and Keener (loc. cit.). Although such a feature can be included on many types of classified-suspension or mixed-suspension crystallizers, it is most common to use this feature with the forcedcirculation evaporative-crystallizer and the DTB crystallizer. Classified-Suspension Crystallizer This equipment is also known as the growth or Oslo crystallizer and is characterized by the production of supersaturation in a circulating stream of liquor. Supersaturation is developed in one part of the system by evaporative cooling or by cooling in a heat exchanger, and it is relieved by passing the liquor through a fluidized bed of crystals. The fluidized bed may be contained in a simple tank or in a more sophisticated vessel arranged

CRYSTALLIZATION FROM SOLUTION

FIG. 18-69

18-49

Swenson atmospheric reaction–type DTB crystallizer. (Swenson Process Equipment, Inc.)

for a pronounced classification of the crystal sizes. Ideally this equipment operates within the metastable supersaturation field described by Miers and Isaac, J. Chem. Soc., 1906, 413. In the evaporative crystallizer of Fig. 18-70, solution leaving the vaporization chamber at B is supersaturated slightly within the metastable zone so that new nuclei will not form. The liquor contacting the bed at E relieves its supersaturation on the growing crystals and leaves through the circulating pipe F. In a cooling-type crystallization hot feed is introduced at G, and the mixed liquor flashes when it reaches the vaporization chamber at A. If further evaporation is required to produce the driving force, a heat exchanger is installed between the circulating pump and the vaporization changer to supply the heat for the required rate of vaporization. The transfer of supersaturated liquor from the vaporizer (point B, Fig. 18-69) often causes salt buildup in the piping and reduction of the operating cycle in equipment of this type. The rate of buildup can be reduced by circulating a thin suspension of solids through the vaporizing chamber; however, the presence of such small seed crystals tends to rob the supersaturation developed in the vaporizer, thereby lowering the efficiency of the recirculation system. An Oslo surface-cooled crystallizer is illustrated in Fig. 18-71. Supersaturation is developed in the circulated liquor by chilling in the cooler H. This supersaturated liquor is contacted with the suspension of crystals in the suspension chamber at E. At the top of the suspension chamber a stream of mother liquor D can be removed to be used for fines removal and destruction. This feature can be added on either type of equipment. Fine crystals withdrawn from the top of the suspension are destroyed, thereby reducing the overall number of crystals in the system and increasing the particle size of the remaining product crystals. Scraped-Surface Crystallizer For relatively small-scale applications a number of crystallizer designs employing direct heat exchange between the slurry and a jacket or double wall containing a cooling medium have been developed. The heat-transfer surface is scraped or agitated in such a way that the deposits cannot build up.

The scraped-surface crystallizer provides an effective and inexpensive method of producing slurry in equipment which does not require expensive installation or supporting structures. Double-Pipe Scraped-Surface Crystallizer This type of equipment consists of a double-pipe heat exchanger with an internal agitator fitted with spring-loaded scrapers that wipe the wall of the inner pipe. The cooling liquid passes between the pipes, this annulus being dimensioned to permit reasonable shell-side velocities. The scrapers prevent the buildup of solids and maintain a good film coefficient of heat transfer. The equipment can be operated in a continuous or in a recirculating batch manner. Such units are generally built in lengths to above 12 m (40 ft). They can be arranged in parallel or in series to give the necessary liquid velocities for various capacities. Heat-transfer coefficients have been reported in the range of 170 to 850 W/(m2⋅K) [30 to 150 Btu/(h⋅ft2⋅°F)] at temperature differentials of 17°C (30°F) and higher [Garrett and Rosenbaum, Chem. Eng., 65(16), 127 (1958)]. Equipment of this type is marketed as the Votator and the Armstrong crystallizer. Batch Crystallization Batch crystallization has been practiced longer than any other form of crystallization in both atmospheric tanks, which are either static or agitated, as well as in vacuum or pressure vessels. It is still widely practiced in the pharmaceutical and fine chemical industry or in those applications where the capacity is very small. The integrity of the batch with respect to composition and history can be maintained easily and the inventory management is more precise than with continuous processes. Batch crystallizers can be left unattended (overnight) if necessary and this is an important advantage for many small producers. In any batch process the common mode of operation involves charging the crystallizer with concentrated or near-saturated solution, producing supersaturation by means of cooling the batch or evaporating solvent from the batch, seeding the batch by means of injecting seed crystals into the batch or by allowing homogeneous nucleation to occur, reaching the final mother-liquor temperature and concentration by some time-dependent means of control, and stopping the cycle

18-50

FIG. 18-70

LIQUID-SOLID OPERATIONS AND EQUIPMENT

OSLO evaporative crystallizer.

so that the batch may be dumped into a tank for processing by successive steps, which normally include centrifugation, filtration, and/or drying. In some cases, a small “heal” of slurry is left in the batch crystallizer to act as seed for the next batch. Control of a batch crystallizer is almost always the most difficult part and very often is not practiced except to permit homogeneous nucleation to take place when the system becomes supersaturated. If control is practiced, it is necessary to have some means for determining when the initial solution is supersaturated so that seed of the appropriate size, quantity, and habit may be introduced into the batch. Following seeding, it is necessary to limit the cooling or evaporation in

the batch to that which permits the generated supersaturation to be relieved on the seed crystals. This means that the first cooling or evaporation following seeding must be at a very slow rate, which is increased nonlinearly in order to achieve the optimum batch cycle. Frequently, such controls are operated by cycle timers or computers so as to achieve the required conditions. Sugar, many pharmaceutical products, and many fine chemicals are produced this way. Shown in Fig. 18-72 is a typical batch crystallizer comprising a jacketed closed tank with top-mounted agitator and feed connections. The tank is equipped with a short distillation column and surface condenser so that volatile materials may be retained in the tank and solvent recycled to maintain the batch integrity. Provisions are included so that the vessel may be heated with steam addition to the shell or cooling solution circulated through the jacket so as to control the temperature. Tanks of this type are intended to be operated with a wide variety of chemicals under both cooling and solvent evaporation conditions. Recompression Evaporation-Crystallization In all types of crystallization equipment wherein water or some other solvent is vaporized to produce supersaturation and/or cooling, attention should be given to the use of mechanical vapor recompression, which by its nature permits substitution of electrical energy for evaporation and solvent removal rather than requiring the direct utilization of heat energy in the form of steam or electricity. A typical recompression crystallizer flowsheet is shown in Fig. 18-73, which shows a singlestage evaporative crystallizer operating at approximately atmospheric pressure. The amount of heat energy necessary to remove 1 kg of water to produce the equivalent in crystal product is approximately 550 kilocalories. If the water evaporated is compressed by a mechanical compressor of high efficiency to a pressure where it can be condensed in the heat exchanger of the crystallizer, it can thereby supply the energy needed to sustain the process. Then the equivalent power for this compression is about 44 kilocalories (Bennett, Chem. Eng. Progress, 1978, pp. 67–70). Although this technique is limited economically to those large-scale cases where the materials handled have a relatively low boiling point elevation and in those cases where a significant amount of heat is required to produce the evaporation for the crystallization step, it nevertheless offers an attractive technique for reducing the use of heat energy and substituting mechanical energy or electrical energy in those cases where there is a cost advantage for doing so. This technique finds many applications in the crystallization of sodium sulfate, sodium carbonate monohydrate, and sodium chloride. Shown in Fig. 18-74 is the amount of vapor compressed per kilowatt-hour for water vapor at 100°C and various ∆Ts. The amount of water vapor compressed per horsepower decreases rapidly with increasing ∆T and, therefore, normal design considerations dictate that the recompression evaporators have a relatively large amount of heat-transfer surface so as to minimize the power cost. Often this technique is utilized only with the initial stages of evaporation where concentration of the solids is relatively low and, therefore, the boiling-point elevation is negligible. In order to maintain adequate tube velocity for heat transfer and suspension of crystals, the increased surface requires a large internal recirculation within the crystallizer body, which consequently lowers the supersaturation in the fluid pumped through the tubes. One benefit of this design is that with materials of flat or inverted solubility, the use of recompression complements the need to maintain low ∆Ts to prevent fouling of the heat-transfer surface. INFORMATION REQUIRED TO SPECIFY A CRYSTALLIZER

FIG. 18-71

OSLO surface-cooled crystallizer.

The following information regarding the product, properties of the feed solution, and required materials of construction must be available before a crystallizer application can be properly evaluated and the appropriate equipment options identified. Is the crystalline material being produced a hydrated or an anhydrous material? What is the solubility of the compound in water or in other solvents under consideration, and how does this change with temperature? Are other compounds in solution which coprecipitate with the product being crystallized, or do these remain in solution, increasing in concentration until some change in product phase occurs? What will be the influence of impurities in the solution on the crystal habit, growth, and

CRYSTALLIZATION FROM SOLUTION

FIG. 18-72

18-51

Typical agitated batch crystallizer. (Swenson Process Equipment, Inc.)

nucleation rates? What are the physical properties of the solution and its tendency to foam? What is the heat of crystallization of the product crystal? What is the production rate, and what is the basis on which this production rate is computed? What is the tendency of the material to grow on the walls of the crystallizer? What materials of construction can be used in contact with the solution at various temperatures? What utilities will be available at the crystallizer location, and what are the costs associated with the use of these utilities? Is the final product to be blended or mixed with other crystalline materials or solids? What size of product and what shape of product are required to meet these requirements? How can the crystalline material be separated from the mother liquor and dried? Are there temperature requirements or wash requirements which must be met? How can these solids or mixtures of solids be handled and stored without undue breakage and caking? Another basic consideration is whether crystallization is best carried out on a batch basis or on a continuous basis. The present tendency in most processing plants is to use continuous equipment whenever possible. Continuous equipment permits adjusting of the operating variables to a relatively fine degree in order to achieve the best results in terms of energy usage and product characteristics. It allows the use of a smaller labor force and results in a continuous util-

ity demand, which minimizes the size of boilers, cooling towers, and power-generation facilities. It also minimizes the capital investment required in the crystallizer and in the feed-storage and productliquor-storage facilities. Materials that have a tendency to grow readily on the walls of the crytallizer require periodic washout, and therefore an otherwise continuous operation would be interrupted once or even twice a week for the removal of these deposits. The impact that this contingency may have on the processing-equipment train ahead of the crystallizer must be considered. The batch handling of wet or semidry crystalline materials is substantially more difficult than the storing and handling of dry crystalline materials. A batch operation has economic application only on a relatively small scale or when temperature or product characteristics require unusual precautions. CRYSTALLIZER OPERATION Crystal growth is a layer-by-layer process, and the retention time required in most commercial equipment to produce crystals of the size normally desired is on the order of 2 to 6 h. On the other hand, nucleation in a supersaturated solution can be generated in a fraction

18-52

LIQUID-SOLID OPERATIONS AND EQUIPMENT

FIG. 18-73

Swenson single-stage recompression evaporator. (Swenson Process Equipment, Inc.)

of a second. The influence of any upsets in operating conditions, in terms of the excess nuclei produced, is very short-term in comparison with the total growth period of the product removed from the crystallizer. In a practical sense, this means that steadiness of operation is much more important in crystallization equipment than it is in many other types of process equipment. It is to be expected that four to six retention periods will pass before the effects of an upset will be damped out. Thus, the recovery period may last from 8 to 36 h. The rate of nuclei formation required to sustain a given product size decreases exponentially with increasing size of the product. Although when crystals in the range of 100 to 50 mesh are produced, the system may react quickly, the system response when generating large crystals in the 14-mesh size range is quite slow. This is because a single pound of 150-mesh seed crystals is sufficient to provide the

total number of particles in a ton of 14-mesh product crystals. In any system producing relatively large crystals, nucleation must be carefully controlled with respect to all internal and external sources. Particular attention must be paid to preventing seed crystals from entering with the incoming feed stream or being returned to the crystallizer with recycle streams of mother liquor coming back from the filter or centrifuge. Experience has shown that in any given body operating at a given production rate, control of the magma (slurry) density is important to the control of crystal size. Although in some systems a change in slurry density does not result in a change in the rate nucleation, the more general case is that an increase in the magma density increases the product size through reduction in nucleation and increased retention time of the crystals in the growing bed. The reduction in supersaturation at longer retention times together with the smaller distance

CRYSTALLIZATION FROM SOLUTION

FIG. 18-74

18-53

Recompression evaporator horsepower as a function of overall ∆T.

between growing crystals, which lowers the driving force that is required to transport material from the liquid phase to the growing solids (propinquity effect), appears to be responsible for the larger product. A reduction in the magma density will generally increase nucleation and decrease the particle size. This technique has the disadvantage that crystal formation on the equipment surfaces increases because lower slurry densities create higher levels of supersaturation within the equipment, particularly at the critical boiling surface in a vaporization-type crystallizer. High levels of supersaturation at the liquid surface or at the tube walls in a surface-cooled crystallizer are the dominant cause of wall salting. Although some types of crystallizers can operate for several months continuously when crystallizing KCl or (NH4)2SO4, most machines have much shorter operating cycles. Second only to control of particle size, the extension of operating cycles is the most difficult operating problem to be solved in most installations. In the forced-circulation-type crystallizer (Fig. 19-43) primary control over particle size is exercised by the designer in selecting the circulating system and volume of the body. From the operating standpoint there is little that can be done to an existing unit other than supply external seed, classify the discharge crystals, or control the slurry

density. Nevertheless, machines of this type are frequently carefully controlled by these techniques and produce a predictable and desirable product-size distribution. When crystals cannot be grown sufficiently large in forcedcirculation equipment to meet product-size requirements, it is common to employ one of the designs that allow some influence to be exercised over the population density of the finer crystals. In the DTB design (Fig. 18-69) this is done by regulating the flow in the circulating pipe so as to withdraw a portion of the fines in the body in the amount of about 0.05 to 0.5 percent by settled volume. The exact quantity of solids depends on the size of the product crystals and on the capacity of the fines-dissolving system. If the machine is not operating stably, this quantity of solids will appear and then disappear, indicating changes in the nucleation rate within the circuit. At steady-state operation, the quantity of solids overflowing will remain relatively constant, with some solids appearing at all times. Should the slurry density of product crystals circulated within the machine rise to a value higher than about 50 percent settled volume, large quantities of product crystals will appear in the overflow system, disabling the fines-destruction equipment. Too high a circulating rate through the fines trap will produce this same result. Too low a flow through the fines circuit will remove insufficient particles and result in a

18-54

LIQUID-SOLID OPERATIONS AND EQUIPMENT

smaller product-size crystal. To operate effectively, a crystallizer of the type employing fines-destruction techniques requires more sophisticated control than does operation of the simpler forced-circulation equipment. The classifying crystallizer (Fig. 18-70) requires approximately the same control of the fines-removal stream and, in addition, requires control of the fluidizing flow circulated by the main pump. This flow must be adjusted to achieve the proper degree of fluidization in the suspension chamber, and this quantity of flow varies as the crystal size varies between start-up operation and normal operation. As with the draft-tube-baffle machine, a considerably higher degree of skill is required for operation of this equipment than of the forcedcirculation type. While most of the industrial designs in use today are built to reduce the problems due to excess nucleation, it is true that in some crystallizing systems a deficiency of seed crystals is produced and the prod-

uct crystals are larger than are wanted or required. In such systems nucleation can be increased by increasing the mechanical stimulus created by the circulating devices or by seeding through the addition of fine crystals from some external source. CRYSTALLIZER COSTS Because crystallizers can come with such a wide variety of attachments, capacities, materials of construction, and designs, it is very difficult to present an accurate picture of the costs for any except certain specific types of equipment, crystallizing specific compounds. This is illustrated in Fig. 18-75, which shows the prices of equipment for crystallizing two different compounds at various production rates, one of the compounds being produced in two alternative crystallizer modes. Installed cost (including cost of equipment and accessories, foundations and supporting steel, utility piping,

Equipment prices, FOB point of fabrication, for typical crystallizer systems. Prices are for crystallizer plus accessories including vacuum equipment. (Data supplied by Swenson Process Equipment, Inc., effective January, 1995.) FIG. 18-75

LEACHING process piping and pumps, electrical switchgear, instrumentation, and labor, but excluding cost of a building) will be approximately twice these price figures. It should be ever present in the reader’s mind that for every particular case the appropriate crystallizer manufacturers should be con-

18-55

sulted for reliable price estimates. Most crystallization equipment is custom-designed, and costs for a particular application may vary greatly from those illustrated in Fig. 18-75. Realistic estimation of installation costs also requires reference to local labor rates, sitespecific factors, and other case specifics.

LEACHING GENERAL REFERENCES: Coulson and Richardson, Chemical Engineering, 4th ed., vol. 2, Pergamon Press, Oxford, 1991. Duby, “Hydrometallurgy,” in KirkOthmer Encyclopedia of Chemical Technology, 4th ed., vol. 16, Wiley, New York, 1995, p. 338. McCabe, Smith, and Harriott, Unit Operations of Chemical Engineering, 5th ed., McGraw-Hill, New York, 1993, chaps. 17 and 20. Prubhudesal, in Schweitzer, Handbook of Separation Techniques for Chemical Engineers, McGraw-Hill, New York, 1979, sec. 5.1. Rickles, Chem. Eng. 72(6): 157 (1965). Schwartzberg (chap. 10, “Leaching—Organic Materials”) and Wadsworth (chap. 9, “Leaching—Metals Applications”) in Rousseau, Handbook of Separation Process Technology, Wiley, New York, 1987. Wakeman, “Extraction (Liquid-Solid)” in Kirk-Othmer Encyclopedia of Chemical Technology, 4th ed., vol. 10, Wiley, New York, 1993, p. 186.

DEFINITION Leaching is the removal of a soluble fraction, in the form of a solution, from an insoluble, permeable solid phase with which it is associated. The separation usually involves selective dissolution, with or without diffusion, but in the extreme case of simple washing it consists merely of the displacement (with some mixing) of one interstitial liquid by another with which it is miscible. The soluble constituent may be solid or liquid; and it may be incorporated within, chemically combined with, adsorbed upon, or held mechanically in the pore structure of the insoluble material. The insoluble solid may be massive and porous; more often it is particulate, and the particles may be openly porous, cellular with selectively permeable cell walls, or surface-activated. It is common practice to exclude from consideration as leaching the elution of surface-adsorbed solute. This process is treated instead as a special case of the reverse operation, adsorption. Also usually excluded is the washing of filter cakes, whether in situ or by reslurrying and refiltration. Because of its variety of applications and its importance to several ancient industries, leaching is known by a number of other names. Among those encountered in chemical engineering practice are extraction, solid-liquid extraction, lixiviation, percolation, infusion, washing, and decantation-settling. The stream of solids being leached and the accompanying liquid is known as the underflow; in hydrometallurgy practice it is called pulp. The solid content of the stream is sometimes called marc (particularly by oil seed processors). The stream of liquid containing the leached solute is the overflow. As it leaves the leaching process it has several optional names: extract, solution, lixiviate, leachate, or miscella. Mechanism The mechanism of leaching may involve simple physical solution or dissolution made possible by chemical reaction. The rate of transport of solvent into the mass to be leached, or of soluble fraction into the solvent, or of extract solution out of the insoluble material, or some combination of these rates may be significant. A membranous resistance may be involved. A chemical-reaction rate may also affect the rate of leaching. Inasmuch as the overflow and underflow streams are not immiscible phases but streams based on the same solvent, the concept of equilibrium for leaching is not the one applied in other mass-transfer separations. If the solute is not adsorbed on the inert solid, true equilibrium is reached only when all the solute is dissolved and distributed uniformly throughout the solvent in both underflow and overflow (or when the solvent is uniformly saturated with the solute, a condition never encountered in a properly designed extractor). The practical interpretation of leaching equilibrium is the state in which the overflow and underflow liquids are of the same composition; on a y-x diagram, the equilibrium line will be a straight line through the origin with a slope of unity. It is customary to calculate the number of ideal

(equilibrium) stages required for a given leaching task and to adjust the number by applying a stage efficiency factor, although local efficiencies, if known, can be applied stage by stage. Usually, however, it is not feasible to establish a stage or overall efficiency or a leaching rate index (e.g., overall coefficient) without testing small-scale models of likely apparatus. In fact, the results of such tests may have to be scaled up empirically, without explicit evaluation of rate or quasi-equilibrium indices. Methods of Operation Leaching systems are distinguished by operating cycle (batch, continuous, or multibatch intermittent); by direction of streams (cocurrent, countercurrent, or hybrid flow); by staging (single-stage, multistage, or differential-stage); and by method of contacting (sprayed percolation, immersed percolation, or solids dispersion). In general, descriptors from all four categories must be assigned to stipulate a leaching system completely (e.g., the Bollman-type extractor is a continuous hybrid-flow multistage sprayed percolator). Whatever the mechanism and the method of operation, it is clear that the leaching process will be favored by increased surface per unit volume of solids to be leached and by decreased radial distances that must be traversed within the solids, both of which are favored by decreased particle size. Fine solids, on the other hand, cause slow percolation rate, difficult solids separation, and possible poor quality of solid product. The basis for an optimum particle size is established by these characteristics. LEACHING EQUIPMENT It is classification by contacting method that provides the two principal categories into which leaching equipment is divided: (1) that in which the leaching is accomplished by percolation and (2) that in which particulate solids are dispersed into a liquid and subsequently separated from it. Each includes batch and continuous units. Materials which disintegrate during leaching are treated in equipment of the second class. A few designs of continuous machines fall in neither of these major classes. Percolation In addition to being applied to ores and rock in place and by the simple technique of heap leaching (usually on very large scale; see Wadsworth, loc. cit.); percolation is carried out in batch tanks and in continuous or dump extractors (usually on smaller scale). Batch Percolators The batch tank is not unlike a big nutsche filter; it is a large circular or rectangular tank with a false bottom. The solids to be leached are dumped into the tank to a uniform depth. They are sprayed with solvent until their solute content is reduced to an economic minimum and are then excavated. Countercurrent flow of the solvent through a series of tanks is common, with fresh solvent entering the tank containing most nearly exhausted material. In a typical oredressing operation the tanks are 53 by 20 by 5.5 m (175 by 67 by 18 ft) and extract about 8200 Mg (9000 U.S. tons) of ore on a 13-day cycle. Some tanks operate under pressure, to contain volatile solvents or increase the percolation rate. A series of pressure tanks operating with countercurrent solvent flow is called a diffusion battery. Continuous Percolators Coarse solids are also leached by percolation in moving-bed equipment, including single-deck and multideck rake classifiers, bucket-elevator contactors, and horizontal-belt conveyors. The Bollman-type extractor shown in Fig. 18-76 is a bucketelevator unit designed to handle about 2000 to 20,000 kg/h (50 to 500 U.S. tons/day) of flaky solids (e.g., soybeans). Buckets with perforated

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LIQUID-SOLID OPERATIONS AND EQUIPMENT

Bollman-type extractor. (McCabe, Smith, and Harriott, Unit Operations of Chemical Engineering, 5th ed., p. 616. Copyright 1993 by McGraw-Hill, Inc., New York. Used with permission of McGraw-Hill Book Company.)

FIG. 18-76

bottoms are held on an endless moving belt. Dry flakes, fed into the descending buckets, are sprayed with partially enriched solvent (“half miscella”) pumped from the bottom of the column of ascending buckets. As the buckets rise on the other side of the unit, the solids are sprayed with a countercurrent stream of pure solvent. Exhausted flakes are dumped from the buckets at the top of the unit into a paddle conveyor; enriched solvent, the “full miscella,” is pumped from the bottom of the casing. Because the solids are unagitated and because the final miscella moves cocurrently, the Bollman extractor permits the use of thin flakes while producing extract of good clarity. It is only partially a countercurrent device, however, and it sometimes permits channeling and consequent low stage efficiency. Perhaps for

this reason, it is being displaced in the oil extraction industry by horizontal basket, pan, or belt percolators (Schwartzberg, loc. cit.). In the horizontal-basket design, illustrated by the Rotocel extractor (Fig. 18-77), walled compartments in the form of annular sectors with liquid-permeable floors revolve about a central axis. The compartments successively pass a feed point, a number of solvent sprays, a drainage section, and a discharge station (where the floor opens to discharge the extracted solids). The discharge station is circumferentially contiguous to the feed point. Countercurrent extraction is achieved by feeding fresh solvent only to the last compartment before dumping occurs and by washing the solids in each preceding compartment with the effluent from the succeeding one. The Rotocel is simple and inexpensive, and it requires little headroom. This type of equipment is made by a number of manufacturers. Horizontal table and tilting-pan vacuum filters, of which it is the gravity counterpart, are used as extractors for leaching processes involving difficult solution-residue separation. The endless-belt percolator (Wakeman, loc. cit.) is similar in principle, but the successive feed, solvent spray, drainage, and dumping stations are linearly rather than circularly disposed. Examples are the de Smet belt extractor (uncompartmented) and the Lurgi frame belt (compartmented), the latter being a kind of linear equivalent of the Rotocel. Horizontal-belt vacuum filters, which resemble endless-belt extractors, are sometimes used for leaching. The Kennedy extractor (Fig. 18-78), also requiring little headroom, operates substantially as a percolator that moves the bed of solids through the solvent rather than the conventional opposite. It comprises a nearly horizontal line of chambers through each of which in succession the solids being leached are moved by a slow impeller enclosed in that section. There is an opportunity for drainage between stages when the impeller lifts solids above the liquid level before dumping them into the next chamber. Solvent flows countercurrently from chamber to chamber. Because the solids are subjected to mechanical action somewhat more intense than in other types of continuous percolator, the Kennedy extractor is now little used for fragile materials such as flaked oil seeds. Dispersed-Solids Leaching Equipment for leaching fine solids by dispersion and separation includes batch tanks agitated by rotating impellers or by air and a variety of continuous devices. Batch Stirred Tanks Tanks agitated by coaxial impellers (turbines, paddles, or propellers) are commonly used for batch dissolution of solids in liquids and may be used for leaching fine solids. Insofar as the controlling rate in the mass transfer is the rate of transfer of mate-

FIG. 18-77 Rotocel extractor. [Rickles, Chem. Eng. 72(6): 164 (1965). Used with permission of McGraw-Hill, Inc.]

LEACHING

FIG. 18-78

18-57

Kennedy extractor. (Vulcan Cincinnati, Inc.)

rial into or from the interior of the solid particles rather than the rate of transfer to or from the surface of particles, the main function of the agitator is to supply unexhausted solvent to the particles while they reside in the tank long enough for the diffusive process to be completed. The agitator does this most efficiently if it just gently circulates the solids across the tank bottom or barely suspends them above the bottom. The leached solids must be separated from the extract by settling and decantation or by external filters, centrifuges, or thickeners, all of which are treated elsewhere in Sec. 18. The difficulty of solids-extract separation and the fact that a batch stirred tank provides only a single equilibrium stage are its major disadvantages. Pachuca Tanks Ores of gold, uranium, and other metals are commonly batch-leached in large air-agitated vessels known as Pachuca tanks. A typical tank is a vertical cylinder with a conical bottom section usually with a 60° included angle, 7 m (23 ft) in diameter and 14 m (46 ft) in overall height. In some designs air is admitted from an open pipe in the bottom of the cone and rises freely through the tank; more commonly, however, it enters through a central vertical tube, characteristically about 46 cm (18 in) in diameter, that extends from the bottom of the tank to a level above the conical section—in some cases, almost to the liquid surface. Before it disengages at the liquid surface, the air induces in and above the axial tube substantial flow of pulp, which then finds its way down the outer part of the tank, eventually reentering the riser. The circulation rate in Pachuca tanks is discussed by Lamont [Can. J. Chem. Eng., 36, 153 (1958)]. Continuous Dispersed-Solids Leaching Vertical-plate extractor. Exemplified by the Bonotto extractor (Fig. 18-79), this consists of a column divided into cylindrical compartments by equispaced horizontal plates. Each plate has a radial opening staggered 180° from the openings of the plates immediately above and below it, and each is wiped by a rotating radial blade. Alternatively, the plates may be mounted on a coaxial shaft and rotated past stationary blades. The solids, fed to the top plate, thus are caused to fall to each lower plate in succession. The solids fall as a curtain into solvent which flows upward through the tower. They are discharged by a screw conveyor and compactor. Like the Bollman extractor, the Bonotto has been virtually displaced by horizontal belt or tray percolators for the extraction of oil seeds. Gravity sedimentation tanks. Operated as thickeners, these tanks can serve as continuous contacting and separating devices in which fine solids may be leached continuously. A series of such units properly connected permit true continuous countercurrent washing of fine solids. If appropriate, a mixing tank may be associated with each thickener to improve the contact between the solids and liquid being fed to that stage. Gravity sedimentation thickeners are described under “Gravity Sedimentation Operations.” Of all continuous leaching equipment, gravity thickeners require the most area, and they are limited to relatively fine solids. The Dorr agitator (Coulson and Richardson, loc. cit.) consolidates in one unit the principles of the thickener and the Pachuca tank. Resembling a rake-equipped thickener, it differs in that the rake is driven by a hollow shaft through which the solids-liquid suspension is lifted and circulated by an air stream. The rake moves the pulp to the center, where it can be entrained by the air stream. The unit may be operated batchwise or continuously.

Impeller-agitated tanks. These can be operated as continuous leaching tanks, singly or in a series. If the solids feed is a mixture of particles of different settling velocities and if it is desirable that all particles reside in the leaching tank the same lengths of time, design of a continuous stirred leach tank is difficult and uncertain. Screw-Conveyor Extractors One type of continuous leaching equipment, employing the screw-conveyor principle, is strictly speaking neither a percolator nor a dispersed-solids extractor. Although it is often classed with percolators, there can be sufficient agitation of the solids during their conveyance by the screw that the action differs from an orthodox percolation. The Hildebrandt total-immersion extractor is shown schematically in Fig. 18-80. The helix surface is perforated so that solvent can pass through countercurrently. The screws are so designed to compact

Bonotto extractor. [Rickles, Chem. Eng. 72(6): 163 (1965); copyright 1965 by McGraw-Hill, Inc., New York. Excerpted with special permission of McGraw-Hill.]

FIG. 18-79

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LIQUID-SOLID OPERATIONS AND EQUIPMENT

the solids during their passage through the unit. The design offers the obvious advantages of countercurrent action and continuous solids compaction, but there are possibilities of some solvent loss and feed overflow, and successful operation is limited to light, permeable solids. A somewhat similar but simpler design uses a horizontal screw section for leaching and a second screw in an inclined section for washing, draining, and discharging the extracted solids. In the De Danske Sukkerfabriker, the axis of the extractor is tilted to about 10° from the horizontal, eliminating the necessity of two screws at different angles of inclination. Sugar-beet cossettes are successfully extracted while being transported upward in a vertical tower by an arrangement of inclined plates or wings attached to an axial shaft. The action is assisted by staggered guide plates on the tower wall. The shell is filled with water that passes downward as the beets travel upward. This configuration is employed in the BMA diffusion tower (Wakeman, loc. cit.). Schwartzberg (loc. cit.) reports that screw-conveyor extractors, once widely employed to extract flaked oil seeds, have fallen into disuse for this application because of their destructive action on the fragile seed flakes. Tray Classifier A hybrid like the screw-conveyor classifier, the tray classifier rakes pulp up the sloping bottom of a tank while solvent flows in the opposite direction. The solvent is forced by a baffle to the bottom of the tank at the lower end before it overflows. The solids must be rugged enough to stand the stress of raking. SELECTION OR DESIGN OF A LEACHING PROCESS* At the heart of a leaching plant design at any level—conceptual, preliminary, firm engineering, or whatever—is unit-operations and process design of the extraction unit or line. The major aspects that are particular for the leaching operation are the selection of process and operating conditions and the sizing of the extraction equipment. Process and Operating Conditions The major parameters that must be fixed or identified are the solvent to be used, the temperature, the terminal stream compositions and quantities, leaching cycle (batch or continuous), contact method, and specific extractor choice. Choice of Solvent The solvent selected will offer the best balance of a number of desirable characteristics: high saturation limit and selectivity for the solute to be extracted, capability to produce * Portions of this subsection are adaptations from the still-pertinent article by Rickles (loc. cit.).

Hildebrandt extractor. (McCabe, Smith, and Harriott, Unit Operations of Chemical Engineering, 5th ed., p. 616. Copyright 1993 by McGrawHill, Inc., New York. Used with permission by McGraw-Hill Book Company.)

FIG. 18-80

extracted material of quality unimpaired by the solvent, chemical stability under process conditions, low viscosity, low vapor pressure, low toxicity and flammability, low density, low surface tension, ease and economy of recovery from the extract stream, and price. These factors are listed in an approximate order of decreasing importance, but the specifics of each application determine their interaction and relative significance, and any one can control the decision under the right combination of process conditions. Temperature The temperature of the extraction should be chosen for the best balance of solubility, solvent-vapor pressure, solute diffusivity, solvent selectivity, and sensitivity of product. In some cases, temperature sensitivity of materials of construction to corrosion or erosion attack may be significant. Terminal Stream Compositions and Quantities These are basically linked to an arbitrary given: the production capacity of the leaching plant (rate of extract production or rate of raw-material purification by extraction). When options are permitted, the degree of solute removal and the concentration of the extract stream chosen are those that maximize process economy while sustaining conformance to regulatory standards. Leaching Cycle and Contact Method As is true generally, the choice between continuous and intermittent operation is largely a matter of the size and nature of the process of which the extraction is a part. The choice of a percolation or solids-dispersion technique depends principally on the amenability of the extraction to effective, sufficiently rapid percolation. Type of Reactor The specific type of reactor that is most compatible (or least incompatible) with the chosen combination of the preceding parameters seldom is clearly and unequivocally perceived without difficulty, if at all. In the end, however, that remains the objective. As is always true, the ultimate criteria are reliability and profitability. Extractor-Sizing Calculations For any given throughput rate (which fixes the cross-sectional area and/or the number of extractors), the size of the units boils down to the number of stages required, actual or equivalent. In calculation, this resolves into determination of the number of ideal stages required and application of appropriate stage efficiencies. The methods of calculation resemble those for other mass-transfer operations (see Secs. 13, 14, and 15), involving equilibrium data and contact conditions, and based on material balances. They are discussed briefly here with reference to countercurrent contacting. Composition Diagrams In its elemental form, a leaching system consists of three components: inert, insoluble solids; a single nonadsorbed solute, which may be liquid or solid; and a single solvent.* Thus, it is a ternary system, albeit an unusual one, as already mentioned, by virtue of the total mutual “insolubility” of two of the phases and the simple nature of equilibrium. The composition of a typical system is satisfactorily presented in the form of a diagram. Those diagrams most frequently employed are a right-triangular plot of mass fraction of solvent against mass fraction of solute (Fig. 18-81a) and a plot suggestive of a Ponchon-Savarit diagram, with inerts taking the place of enthalpy (Fig. 18-81b). A third diagram, less frequently used, is a modified McCabe-Thiele plot in which the overflow solution (inerts-free) and the underflow solution (traveling out of a stage with the inerts) are treated as pseudo phases, the mass fraction of solute in overflow, y, being plotted against the mass fraction of solute in underflow, x. (An additional representation, the equilateral-triangular diagram frequently employed for liquidliquid ternary systems, is seldom used because the field of leaching data is confined to a small portion of the triangle.) With reference to Fig. 18-81 (both graphs), EF represents the locus of overflow compositions for the case in which the overflow stream contains no inert solids. E′F′ represents the overflow streams containing some inert solids, either by entrainment or by partial solubility in the overflow solution. Lines GF, GL, and GM represent the loci of underflow compositions for the three different conditions indicated on the diagram. In Fig. 18-81a, the constant underflow line GM is parallel to EF, the hypotenuse of the triangle, whereas GF passes through * The solubility of the inert, adsorption of solute on the inert, and complexity of solvent and extracted material can be taken into account if necessary. Their consideration is beyond the scope of this treatment.

GRAVITY SEDIMENTATION OPERATIONS

(a)

(b) FIG. 18-81 Composition diagrams for leaching calculations: (a) righttriangular diagram; (b) modified Ponchon-Savarit diagram.

the right-hand vertex representing 100 percent solute. In Fig. 18-81b, underflow line GM is parallel to the abscissa, and GF passes through the point on the abscissa representing the composition of the clear solution adhering to the inert solids. Compositions of overflow and underflow streams leaving the same stage are represented by the intersection of the composition lines for those streams with a tie line (AC, AC′, BD, BD′). Equilibrium tie lines (AC, BD) pass through the origin (representing 100 percent inerts) in Fig. 18-81a, and are vertical (representing the same inert-free solution composition in both streams) in Fig. 18-81b. For nonequilibrium conditions with or without adsorption or for equilibrium conditions with selective adsorption, the tie lines are displaced, such as AC′ and BD′. Point C′ is to the right of C if the solute concentration in the overflow solution is less than that in the underflow solution adhering to the solids. Unequal concentrations in the two solutions indicate insufficient contact time and/or preferential adsorption of one of the components on the inert solids. Tie lines such as AC′ may be considered as “practical tie lines” (i.e., they represent actual rather than ideal stages) if data on underflow and overflow composition have been

18-59

obtained experimentally under conditions simulating actual operation, particularly with respect to contact time, agitation, and particle size of solids. The illustrative construction lines of Fig. 18-81 have been made with the assumption of constant underflow. In the more realistic case of variable underflow, the points C, C′, D, D′ would lie along line GL. Like the practical tie lines, GL is a representation of experimental data. Algebraic Computation This method starts with calculation of the quantities and compositions of all the terminal streams, using a convenient quantity of one of the streams as the basis of calculation. Material balance and stream compositions are then computed for a terminal ideal stage at either end of an extraction battery (i.e., at Point A or Point B in Fig. 18-81), using equilibrium and solution-retention data. Calculations are repeated for each successive ideal stage from one end of the system to the other until an ideal stage which corresponds to the desired conditions is obtained. Any solid-liquid extraction problem can be solved by this method. For certain simplified cases it is possible to calculate directly the number of stages required to attain a desired product composition for a given set of feed conditions. For example, if equilibrium is attained in all stages and if the underflow mass rate is constant, both the equilibrium and operating lines on a modified McCabe-Thiele diagram are straight, and it is possible to calculate directly the number of ideal stages required to accommodate any rational set of terminal flows and compositions (McCabe, Smith, and Harriott, op. cit.): log [(yb − xb)/(ya − xa)] N = ᎏᎏᎏ (18-42) log [(yb − ya)/(xb − xa)] Even when the conditions of equilibrium in each stage and constant underflow obtain, Eq. (18-42) normally is not valid for the first stage because the unextracted solids entering that stage usually are not premixed with solution to produce the underflow mass that will leave. This is easily rectified by calculating the exit streams for the first stage and using those values in Eq. (18-42) to calculate the number of stages required after stage 1. Graphical Method This method of calculation is simply a diagrammatic representation of all the possible compositions in a leaching system, including equilibrium values, on which material balances across ideal (or, in some cases, nonideal) stages can be evaluated in the graphical equivalent of the stage-by-stage algebraic computation. It normally is simpler than the hand calculation of the algebraic solution, and it is viewed by many as helpful because it permits visualization of the process variables and their effect on the operation. Any of the four types of composition diagrams described above can be used, but modified Ponchon-Savarit or right-triangular plots (Fig. 18-81) are most convenient for leaching calculations. The techniques of graphical solution, in fact, are not unlike those for distillation and absorption (binary) problems using McCabeThiele, Ponchon-Savarit, and right-triangular diagrams and are similar to those described in Sec. 15 for solvent-extraction (ternary) systems. More detailed explanations of the application of the several graphical conventions to leaching are presented by: Coulson and Richardson, right triangle; Rickles, modified Ponchon-Savarit; McCabe, Smith, and Harriott, modified McCabe-Thiele; and Schwartzberg, equilateral ternary diagram; all in the publications cited as general references. (See also Treybal, Mass Transfer Operations, 3d ed., McGraw-Hill, New York, 1980.)

GRAVITY SEDIMENTATION OPERATIONS GENERAL REFERENCES: Albertson, Fluid/Particle Separation Journal, 7(1), 18 (1994). Coe and Clevenger, Trans. Am. Inst. Min. Eng., 55, 356 (1916). Comings, Pruiss, and DeBord, Ind. Eng. Chem., 46, 1164 (1954). Counselmann, Trans. Am. Inst. Min. Eng., 187, 223 (1950). Fitch, Ind. Eng. Chem., 58(10), 18 (1966). Trans. Soc. Min. Eng. AIME, 223, 129 (1962). Kynch, Trans. Faraday Soc., 48, 166 (1952). Pearse, Gravity Thickening Theories: A Review, Warren Spring Laboratory, Hertfordshire, 1977. Purchas, Solid/Liquid Separation Equipment Scale-Up, Uplands Press, Croydon, England, 1977. Sankey and Payne, Chemical Reagents in the Mineral Processing Industry, 245, SME

(1985). Talmage and Fitch, Ind. Eng. Chem., 47, 38 (1955). Wilhelm and Naide, Min. Eng. (Littleton, Colo.), 33, 1710 (1981).

Sedimentation is the partial separation or concentration of suspended solid particles from a liquid by gravity settling. This field may be divided into the functional operations of thickening and clarification. The primary purpose of thickening is to increase the concentration of suspended solids in a feed stream, while that of clarification is to

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LIQUID-SOLID OPERATIONS AND EQUIPMENT

remove a relatively small quantity of suspended particles and produce a clear effluent. These two functions are similar and occur simultaneously, and the terminology merely makes a distinction between the primary process results desired. Generally, thickener mechanisms are designed for the heavier-duty requirements imposed by a large quantity of relatively concentrated pulp, while clarifiers usually will include features that ensure essentially complete suspended-solids removal, such as greater depth, special provision for coagulation or flocculation of the feed suspension, and greater overflow-weir length. CLASSIFICATION OF SETTLEABLE SOLIDS; SEDIMENTATION TESTS The types of sedimentation encountered in process technology will be greatly affected not only by the obvious factors—particle size, liquid viscosity, solid and solution densities—but also by the characteristics of the particles within the slurry. These properties, as well as the process requirements, will help determine both the type of equipment which will achieve the desired ends most effectively and the testing methods to be used to select the equipment. Figure 18-82 illustrates the relationship between solids concentration, interparticle cohesiveness, and the type of sedimentation that may exist. “Totally discrete” particles include many mineral particles (usually greater in diameter than 20 µm), salt crystals, and similar substances that have little tendency to cohere. “Flocculent” particles generally will include those smaller than 20 µm (unless present in a dispersed state owing to surface charges), metal hydroxides, many chemical precipitates, and most organic substances other than true colloids. At low concentrations, the type of sedimentation encountered is called particulate settling. Regardless of their nature, particles are sufficiently far apart to settle freely. Faster-settling particles may collide with slower-settling ones and, if they do not cohere, continue downward at their own specific rate. Those that do cohere will form floccules of a larger diameter that will settle at a rate greater than that of the individual particles. There is a gradual transition from particulate settling into the zonesettling regime, where the particles are constrained to settle as a mass. The principal characteristic of this zone is that the settling rate of the mass, as observed in batch tests, will be a function of its solids concentration (for any particular condition of flocculation, particle density, etc.). The solids concentration ultimately will reach a level at which particle descent is restrained not only by hydrodynamic forces but also partially by mechanical support from the particles below; therefore,

Combined effect of particle coherence and solids concentration on the settling characteristics of a suspension.

FIG. 18-82

the weight of particles in mutual contact can influence the rate of sedimentation of those at lower levels. This compression, as it is termed, will result in further solids concentration because of compaction of the individual floccules and partial filling of the interfloc voids by the deformed floccules. Accordingly, the rate of sedimentation in the compression regime is a function of both the solids concentration and the depth of pulp in this particular zone. As indicated in Fig. 18-82, granular, nonflocculent particles may reach their ultimate solids concentration without passing through this regime. As an illustration, coarse-size (45 µm) the aluminum oxide trihydrate particles produced in the Bayer process would be located near the extreme left of Fig. 18-82. These solids settle in a particulate manner, passing through a zone-settling regime only briefly, and reach a terminal density or ultimate solids concentration without any significant compressive effects. At this point, the solids concentration may be as much as 80 percent by weight. The same compound, but of the gelatinous nature it has when precipitated in water treatment as aluminum hydroxide, would be on the extreme right-hand side of the figure. This flocculent material enters into a zone-settling regime at a low concentration (relative to the ultimate concentration it can reach) and gradually thickens. With sufficient pulp depth present, preferably aided by gentle stirring or vibration, the compression-zone effect will occur; this is essential for the sludge to attain its maximum solids concentration, around 10 percent. Certain fine-size (1- to 2-µm) precipitates of this compound will possess characteristics intermediate between the two extremes. A feed stream to be clarified or thickened can exist at any state represented within this diagram. As it becomes concentrated owing to sedimentation, it may pass through all the regimes, and the settling rate in any one may be the size-determining factor for the required equipment. Sedimentation-Test Procedures Determination of Clarification-Zone Requirements In the treatment of solids suspensions which are in the particulate-settling regime, the usual objective will be the production of a clear effluent and test methods limited to this type of settling will be the normal sizing procedure, although the area demand for thickening should be verified. With particulate or slightly flocculent matter, any method that measures the rate of particle subsidence will be suitable, and either long-tube or short-tube procedures (described later) may be used. If the solids are strongly flocculent and particles cohere easily during sedimentation, the long tube test will yield erratic data, with better clarities being observed in samples taken from the lower taps (i.e., clarity appears to improve at higher settling rates). In these instances, time alone usually is the principal variable in clarification, and a simple detention test is recommended. Long-Tube Method A transparent tube 2 to 4 m long and at least 100 mm in diameter (preferably larger), fitted with sampling taps every 200 to 300 mm, is used in this test. The tube is mounted vertically and filled with a representative sample of feed suspension. At timed intervals approximately 100-mL samples are withdrawn from successive taps, beginning with the uppermost one. The time intervals will be determined largely by the settling rate of the particles and should be chosen so that a series of at least four time intervals will produce samples that bracket the desired solids-removal target. Also, this procedure will indicate whether or not detention time is a factor in the rate of clarification. Typically, intervals may be 30 min long, the last series of samples representing the results obtainable with 2-h detention. The samples are analyzed for suspended-solids concentration by any suitable means, such as filtration through membranes or centrifugation with calibrated tubes. A plot is made of the suspended-solids concentration in each of the samples as a function of the nominal settling velocity of the sample, which is determined from the corresponding sample-tap depth divided by the elapsed time between the start of the test and the time of sampling. For each sampling series after the first, the depth will have changed because of the removal of the preceding samples, and this must be taken into account. With particulate solids having little or no tendency to cohere, the data points generally will fall on one line irrespective of the detention time. This indicates that the settling area

GRAVITY SEDIMENTATION OPERATIONS available in the basin will determine the degree of solids removal, and the depth will have little bearing on the results, except as it may affect clarification efficiency. Since the solids concentration in the 100-mL samples withdrawn at different depths does not correspond to the average concentration in the fluid above the sample point (which would be equivalent to the overflow from the clarifier at that particular design rate), an adjustment must be made in calculating the required area. Accordingly, an approximate average of the solids concentration throughout the column of liquid above a given tap can be obtained by summing the values obtained from all the higher taps and the one which is being sampled and dividing by the total number of taps sampled. This value is then used in a plot of overflow suspended-solids concentration versus nominal settling velocity. Areal efficiencies for properly designed clarifiers in which detention time is not a significant factor range from 65 to 80 percent, and the surface area should be increased accordingly to reduce the overflow rate for scale-up. Should the particles have a tendency to cohere slightly during sedimentation, each sampling time, representing a different nominal detention time in the clarifier, will produce different suspended-solids concentrations at similar rates. These data can be plotted as sets of curves of concentration versus settling rate for each detention time by the means just described. Scale-up will be similar, except that detention time will be a factor, and both depth and area of the clarifier will influence the results. In most cases, more than one combination of diameter and depth will be capable of producing the same clarification result. These data may be evaluated by selecting different nominal overflow rates (equivalent to settling rates) for each of the detention-time values, and then plotting the suspended-solids concentrations for each nominal overflow rate (as a parameter) against the detention time. For a specified suspended-solids concentration in the effluent, a curve of overflow rate versus detention time can be prepared from this plot and used for optimizing the design of the equipment. Short-Tube Method This test is suitable in cases in which detention time does not change the degree of particle flocculation and hence has no significant influence on particle-settling rates. It is also useful for hydroseparator tests where the sedimentation device is to be used for classification (see Sec. 19). A tube 50 to 75 mm in diameter and 300 to 500 mm long is employed. A sample placed in the tube is mixed to ensure uniformity, and settling is allowed to occur for a measured interval. At the end of this time, the supernatant liquid is siphoned off quickly down to a chosen level, and the collected sample is analyzed for suspended-solids concentration. The level selected usually is based on the relative expected volumes of overflow and underflow. The suspended-solids concentration measured in the siphoned sample will be equivalent to the “averaged” values obtained in the long-tube test, at a corresponding settling rate. In hydroseparator tests, it is necessary to measure solids concentrations and size distributions of both the supernatant sample withdrawn and the fraction remaining in the cylinder. The volume of the latter sample should be such as to produce a solids concentration that would be typical of a readily pumped underflow slurry. Detention Test This test utilizes a 1- to 4-L beaker or similar vessel. The sample is placed in the container, flocculated by suitable means if required, and allowed to settle. Small samples for suspended-solids analysis are withdrawn from a point approximately midway between liquid surface and settled solids interface, taken with sufficient care that settled solids are not resuspended. Sampling times may be at consecutively longer intervals, such as 5, 10, 20, 40, and 80 min. The suspended-solids concentration can be plotted on log-log paper as a function of the sampling (detention) time. A straight line usually will result, and the required static detention time t to achieve a certain suspended-solids concentration C in the overflow of an ideal basin can be taken directly from the graph. If the plot is a straight line, the data are described by the equation C = Ktm

18-61

Should the suspension contain a fraction of solids which can be considered “unsettleable,” the data are more easily represented by using the so-called second-order procedure. This depends on the data being reasonably represented by the equation 1 1 Kt = ᎏ − ᎏ (18-44) C − C∞ C0 − C∞ where C∞ is the unsettleable-solids concentration and C0 is the concentration of suspended solids in the unsettled (feed) sample. The residual-solids concentration remaining in suspension after a sufficiently long detention time (C∞) must be determined first, and the data then plotted on linear paper as the reciprocal concentration function 1/(C − C∞) versus time. Bulk-settling test. In cases involving detention time only, the overflow rate must be considered by other means. This is done by carrying out a settling test in which the solids are first concentrated to a level at which zone settling just begins. This is usually marked by a very diffuse interface during initial settling. Its rate of descent is measured with a graduated cylinder of suitable size, preferably at least 1 L, and the initial straight-line portion of the settling curve is used for specifying a bulk-settling rate. The design overflow rate generally should not exceed half of the bulk-settling rate. Detention efficiency. Conversion from the ideal basin sized by detention-time procedures to an actual clarifier requires the inclusion of an efficiency factor to account for the effects of turbulence and nonuniform flow. Efficiencies vary greatly, being dependent not only on the relative dimensions of the clarifier and the means of feeding but also on the characteristics of the particles. The curve shown in Fig. 18-83 can be used to scale up laboratory data in sizing circular clarifiers. The static detention time determined from a test to produce a specific effluent solids concentration is divided by the efficiency (expressed as a fraction) to determine the nominal detention time, which represents the volume of the clarifier above the settled pulp interface divided by the overflow rate. Different diameter-depth combinations are considered by using the corresponding efficiency factor. In most cases, area may be determined by factors other than the bulksettling rate, such as practical tank-depth limitations. Thickener-Basin Area The area requirements for thickeners frequently are based on the solids flux rates measured in the zonesettling regime. Theory holds that, for any specific sedimentation condition, a critical concentration which will limit the solids throughput rate will exist in the thickener. This critical concentration will be evidenced as a pulp bed of variable depth in which the solids concentration is fairly uniform from top to bottom. Since the underflow concentration usually will be higher than this, gradually increasing concentrations will be found with progressing depth in the region beneath this constant-concentration zone. As the concentration within this critical zone represents a steady-state condition, its vertical extent may vary continually, responding to minor changes in the feed rate, underflow withdrawal rate, or flocculant dosage. In thickeners operating at relatively high underflow concentrations, with long

(18-43)

where the coefficient K and exponent m are characteristic of the particular suspension.

Efficiency curve for scale-up of barch clarification data to determine nominal detention time in a continuous clarifier. FIG. 18-83

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LIQUID-SOLID OPERATIONS AND EQUIPMENT

solids-detention times and lower throughput, this zone generally will not be present. Many batch-test methods which are based on determining the solids flux rate at this critical concentration have been developed. Most methods recognize that as the solids enter compression, thickening behavior is no longer a function only of solids concentration. Hence, these methods attempt to utilize the “critical” point dividing these two zones and size the area on the basis of the settling rate of a layer of pulp at this concentration. The difficulty lies in discerning where this point is located on the settling curve. Many procedures have been developed, but two in particular have been more widely used: the Coe and Clevenger approach and the Kynch method as defined by Talmage and Fitch (op. cit.). The former requires measurement of the initial settling rate of a pulp at different solids concentrations varying from feed to final underflow value. The area requirement for each solids concentration tested is calculated by equating the net overflow rate to the corresponding interfacial settling rate, as represented by the following equation for the unit area: 1/Ci − 1/Cu Unit area = ᎏᎏ (18-45) vi where Ci is the solids concentration at the interfacial settling velocity vi and Cu is the underflow concentration, both concentrations being expressed in terms of mass of solids per unit volume of slurry. Using kg/L for the concentrations and m/day for the settling velocity yields a unit area value in m2/(ton/day). These unit area values, plotted as a function of the feed concentration, will describe a maximum value that can be used to specify the thickener design unit area for the particular underflow concentration Cu employed in Eq. (18-45). The method is applicable for unflocculated pulps or those in which the ionic characteristics of the solution produce a flocculent structure. If polymeric flocculants are used, the floccule size will be highly dependent on the feed concentration, and an approach based on the Kynch theory is preferred. In this method, the test is carried out at the expected feed solids concentration and is continued until underflow concentration is achieved in the cylinder. To determine the unit area, Talmage and Fitch (op. cit.) proposed an equation derived from a relationship equivalent to that shown in Eq. (18-45): tu Unit area = ᎏ (18-46) C0H0 where tu is the time, days; C0 is the initial solids concentration in the feed, t/m3; and H0 is the initial height of the slurry in the test cylinder, m. The term tu is taken from the intersection of a tangent to the curve at the critical point and a horizontal line representing the depth of pulp at underflow concentration. There are various means for selecting this critical point, all of them empirical, and the unit area value determined cannot be considered precise. The review by Pearse (op. cit.) presents many of the different procedures used in applying this approach to laboratory settling test data. Because this method in itself does not adequately address the effect of different underflow concentrations on the unit area, a thickener sized using this approach could be too small in a case where a relatively concentrated underflow is desired. Two other approaches avoid using the critical point by computing the area requirements from the settling conditions existing at the underflow concentration. The Wilhelm and Naide procedure (op. cit.) applies zone-settling theory (Kynch) to the entire thickening regime. Tangents drawn to the settling curve are used to calculate the settling velocity at all concentrations obtained in the test. This permits construction of a plot (Figure 18-84) showing unit area as a function of underflow concentration. A second, “direct” approach which yields a similar result, since it also takes compression into account, utilizes the value of settling time tx taken from the settling curve at a particular underflow concentration. This value is used to solve the Talmage and Fitch equation (18-46) for unit area. Compression bed depth will have a significant effect on the overall settling rate (increasing compression zone depth reduces unit area).

FIG. 18-84 Characteristic relationship between thickener unit area and underflow solids concentration (fixed flocculant dosage and pulp depth).

Therefore, in applying either of these two procedures it is necessary to run the test in a vessel having an average bed depth close to that expected in a full-scale thickener. This requires a very large sample, and it is more convenient to carry out the test in a cylinder having a volume of 1 to 4 liters. The calculated unit area value from this test can be extrapolated to full-scale depth by carrying out similar tests at different depths to determine the effect on unit area. Alternatively, an empirical relationship can be used which is effective in applying a depth correction to laboratory cylinder data over normal operating ranges. The unit area calculated by either the Wilhelm and Naide approach or the direct method is multiplied by a factor equal to (h/H)n, where h is the average depth of the pulp in the cylinder, H is the expected full-scale compression zone depth, usually taken as 1 m, and n is the exponent calculated from Fig. 18-85. For conservative design purposes, the minimum value of this factor that should be used is 0.25. It is essential to use a slow-speed (approximately 0.1 r/min) picket rake in all cylinder tests to prevent particle bridging and allow the sample to attain the underflow density which is obtainable in a fullscale thickener. Continuously operated, small-scale or pilot-plant thickeners, ranging from 75 mm diameter by 400 mm depth to several meters in diameter, are also effectively used for sizing full-scale equipment. This approach requires a significantly greater volume of sample, such as

Depth correction factor to be applied to unit areas determined with Wilhelm-Naide and “direct” methods. Velocity ratio calculated using tangents to settling curve at a particular settled solids concentration and at start of test.

FIG. 18-85

GRAVITY SEDIMENTATION OPERATIONS would be available in an operating installation or a pilot plant. Continuous units and batch cylinders will produce equivalent results if proper procedures are followed with either system. Thickener-Basin Depth The pulp depth required in the thickener will be greatly affected by the role that compression plays in determining the rate of sedimentation. If the zone-settling conditions define the area needed, then depth of pulp will be unimportant and can be largely ignored, as the “normal” depth found in the thickener will be sufficient. On the other hand, with the compression zone controlling, depth of pulp will be significant, and it is essential to measure the sedimentation rate under these conditions. To determine the compression-zone requirement in a thickener, a test should be run in a deep cylinder in which the average settling pulp depth approximates the depth anticipated in the full-scale basin. The average density of the pulp in compression is calculated and used in Eq. (18-47) to determine the required compression-zone volume: θc(ρs − ρl) V = ᎏᎏ (18-47) ρs(ρsl − ρl) 3 where V is the volume, m , required per ton of solids per day; θc is the compression time, days, required in the test to reach underflow concentration; and ρs, ρl, ρsl, are the densities of the solids, liquid, and slurry (average), respectively, ton/m3. This value divided by the average depth of the pulp during the period represents the unit area defined by compression requirements. If it exceeds the value determined from the zone-settling tests, it is the quantity to be used. The side depth of the thickener is determined as the sum of the depths needed for the compression zone and for the clear zone. Normally, 1.5 to 2 m of clear liquid depth above the expected pulp level in a thickener will be sufficient for stable, effective operation. When the location of the pulp level cannot be predicted in advance or it is expected to be relatively low, a thickener sidewall depth of 2 to 3 m is usually safe. Greater depth may be used in order to provide better clarity, although in most thickener applications the improvement obtained by this means will be marginal. Scale-up Factors Factors used in thickening will vary, but, typically, a 1.2 to 1.3 multiplier applied to the unit area calculated from laboratory data is sufficient if proper testing procedures have been followed and the samples are representative. Flocculation Flocculants are commonly used because of the considerable reduction in equipment size and capital cost that can be effected with very nominal reagent dosages. Selection of the reagents usually involves simple bench-scale comparison tests on small samples of pulp for rough screening, followed by larger-scale tests in cylinders or in continuous pilot-plant thickeners. Determination of the optimum dosage is complicated; it involves a number of economic factors such as reagent and capital costs, cost of a shutdown or loss in availability due to failure in the flocculation system, and consideration of possible future increases in plant capacity. Polymeric flocculants are available in various chemical compositions and molecular weight ranges, and they may be nonionic in character or may have predominantly cationic or anionic charges. The range of application varies; but, in general, nonionics are well suited to acidic suspensions, anionic flocculants work well in neutral or alkaline environments, and cationics are most effective on organic material and colloidal matter. Colloidal solids (e.g., as encountered in waste treatment) may require initial treatment with a chemical having strong ionic properties, such as acid, lime, alum, or ferric sulfate. The latter two will precipitate at neutral pH and produce a gelatinous, flocculent structure which further helps collect extremely small particles. Some cationic polymers also may be effective in flocculation of particles of this type. (This action is commonly termed coagulation.) Prolonged, gentle agitation improves the degree and rate of flocculation under these conditions. If the solids concentration is relatively low, e.g., <500 mg/L, results can usually be improved by recirculation of settled material to the flocculation zone to produce an optimum concentration for floc growth. With polymer flocculation of slurries, however, extended agitation after the addition of the polymer may be detrimental. The reagent should be added to the slurry under conditions which promote rapid

18-63

dispersion and uniform, complete mixing with a minimum of shear. In cylinder tests, this can be accomplished by simultaneously injecting and mixing flocculant with the slurry, using an apparatus consisting of a syringe, a tube, and an inverted rubber stopper. The stopper, with a diameter approximately three-fourths of that of the cylinder, provides sufficient turbulence as it is moved gently up and down through the sample to cause good blending of reagent and pulp. Flocculant solution preparation and use should take into account specific properties of these reagents. During mixing and distribution, excessive shear should be avoided since it can reduce reagent effectiveness and result in higher consumption. Flocculant solution should be prepared at as high a concentration—generally, 0.25 to 1%—as can be handled by the metering pump and agitator, and added to the pulp at the maximum dilution possible. Dry polymers require sufficient aging time after initial mixing with water in order to develop their full effectiveness. Liquid polymers—emulsions and dispersions—contain additives such as carriers, activators, and other components which can have an effect on the process and other equipment, and this should be investigated before their use. Care should be taken that the water employed for dissolving as well as for dilution will not affect reagent activity, as this could increase consumption. Prior to design of the system, laboratory tests should be conducted with both plant and tap water to determine if there is a detrimental effect using process water. Torque Requirements Sufficient torque must be available in the raking mechanism of a full-scale thickener to allow it to move through the slurry and assist solids movement to the underflow outlet. Granular, particulate solids that settle rapidly and reach a terminal solids concentration without going through any apparent compression or zone-settling region require a maximum raking capability, as they must be moved to the outlet solely by the mechanism. At the other end of the spectrum, extremely fine materials, such as clays and precipitates, require a minimum of raking, for most of the solids may reach the underflow outlet hydrodynamically. The rakes prevent a gradual buildup of some solids on the bottom, however, and the gentle stirring action from the rake arm often aids the thickening process. As the underflow concentration approaches its ultimate limit, the consistency will increase greatly, resulting in a higher raking requirement and an increase in torque. For most materials, the particle size lies somewhere between these two extremes, and the torques required in two properly designed thickeners of the same size but in distinct applications can differ greatly. Unfortunately, test methods to specify torque from small-scale tests are of questionable value, since it is difficult to duplicate actual conditions. Manufacturers of sedimentation equipment select torque ratings from experience with similar substances and will recommend a torque capability on this basis. Definitions of operating torque vary with the manufacturer, and the user should ask the supplier to specify the B-10 life for bearings and to reference appropriate mechanical standards for continuous operation of the selected gear set at specific torque levels. This will provide guidelines for plant operators and help avoid premature failure of the mechanism. Abnormal conditions above the normal operating torque are inevitable, and a thickener should be provided with sufficient torque capability for short-term operation at higher levels in order to ensure continuous performance. Underflow Pump Requirements Many suspensions will thicken to a concentration higher than that which can be handled by conventional slurry pumps. Thickening tests should be performed with this in mind, for, in general, the unit area to produce the maximum concentration that can be pumped is the usual design basis. Determination of this ultimate pumpable concentration is largely a judgmental decision requiring some experience with slurry pumping; however, the behavior of the thickened suspension can be used as an approximate guide to pumpability. The supernatant should be decanted following a test and the settled solids repulped in the cylinder to a uniform consistency. Repulping is done easily with a rubber stopper fastened to the end of a rigid rod. If the bulk of the repulped slurry can be poured from the cylinder when it is tilted 10 to 30° above the horizontal, the corresponding thickener underflow can be handled by most types of slurry pumps. But if the slurry requires cylinder shaking or other mechanical means for its removal, it should be diluted to a more fluid condition, if conventional pump systems are to be employed.

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LIQUID-SOLID OPERATIONS AND EQUIPMENT

THICKENERS The primary function of a continuous thickener is to concentrate suspended solids by gravity settling so that a steady-state material balance is achieved, solids being withdrawn continuously in the underflow at the rate they are supplied in the feed. Normally, an inventory of pulp is maintained in order to achieve the desired concentration. This volume will vary somewhat as operating conditions change; on occasion, this inventory can be used for storage of solids when feed and underflow rates are reduced or temporarily suspended. A thickener has several basic components: a tank to contain the slurry, feed piping and a feedwell to allow the feed stream to enter the tank, a rake mechanism to assist in moving the concentrated solids to the withdrawal points, an underflow solids-withdrawal system, and an overflow launder. The basic design of a bridge-supported thickener mechanism is illustrated in Fig. 18-86. High-Rate Thickeners Flocculants are commonly used in thickeners, and this practice has resulted in thickener classification as either conventional or high-rate. These designations can be confusing in that they imply that there is a sharp distinction between the two,

FIG. 18-86

which is not the case. The greater capacity expected from a high-rate thickener is due solely to the effective use of flocculant to maximize throughput, as shown in Fig. 18-87. In most applications, there is a threshold dosage at which a noticeable increase in capacity begins to occur. This effect will continue up to a limit, at which point the capacity will be a maximum unless a lower underflow concentration is accepted, as illustrated in Fig. 18-84. Since flocculant is usually added to a thickener either in the feed line or the feedwell, there are a number of proprietary feedwell designs which are used in high-rate thickeners in order to help optimize flocculation. De-aeration systems may be included in some cases to avoid air entrainment in the flocculated slurry. The other components of these units are not materially different from those of a conventional thickener. High-Density Thickeners Thickeners can be designed to produce underflows having a very high apparent viscosity, permitting disposal of waste slurries at a concentration that avoids segregation of fines and coarse particles or formation of a free-liquid pond on the surface of the deposit. This practice is applied in dry-stacking systems and underground paste-fill operations for disposal of mine tailings and similar materials. The thickener mechanism generally will require a special

Unit thickener with bridge-supported mechanism. (EIMCO Process Equipment Co.)

GRAVITY SEDIMENTATION OPERATIONS

FIG. 18-87 Unit area vs. flocculant dose, illustrating the relationship between conventional and high-rate thickeners.

rake design and have a torque capability 3 to 4 times, or more, the normal for that particular diameter. Underflow slurries usually will be at a solids concentration 5 to 10% lower than that of a vacuum filter cake formed from the same material. Special pumping requirements are necessary if the slurry is to be transferred a significant distance, with line pressure drop typically in the range of 3 to 4 kPa/m of pipeline. Design Features There are three classes of thickeners, each differentiated by its drive mechanism: (1) bridge-supported, (2) centercolumn supported, and (3) traction drives. The diameter of the tank will range from 2 to 150 m (6.5 to 492 ft), and the support structure often is related to the size required. These classes are described in detail in the subsection “Components and Accessories for Sedimentation Units.” Operation When operated correctly, thickeners require a minimum of attention and, if the feed characteristics do not change radically, can be expected to maintain design performance consistently. In this regard, it is usually desirable to monitor feed and underflow rates and solids concentrations, flocculant dosage rate, and pulp interface level, preferably with dependable instrumentation systems. Process variations are then easily handled by changing the principal operating controls—underflow rate and flocculant dose—to maintain stability. Starting up a thickener is usually the most difficult part of the operation, and there is more potential for mechanical damage to the mechanism at this stage than at any other time. In general, two conditions require special attention at this point: underflow pumping and mechanism torque. If possible, the underflow pump should be in operation as soon as feed enters the system, recirculating underflow slurry at a reduced rate if the material is relatively fine or advancing it to the next process step (or disposal) if the feed contains a considerable quantity of coarse solids, e.g., more than 20 percent + 75 µm particles. At this stage of the operation, coarse solids separate from the pulp and produce a difficult raking and pumping situation. Torque can rise rapidly if this material accumulates faster than it is removed. If the torque reaches a point where the automatic control system raises the rakes, it is usually preferable to reduce or cut off the feed completely until the torque drops and the rakes are returned to the lowest position. As the fine fraction of the feed slurry begins to thicken and accumulate in the basin, providing both buoyancy and fluidity, torque will drop and normal feeding can be continued. This applies whether the thickener tank is empty at start-up or filled with liquid. The latter approach contributes to coarse-solids raking problems but at the same time provides conditions more suited to good flocculation, with the result that the thickener will reach stable operation much sooner. As the solids inventory in the thickener reaches a normal level— usually about 0.5 to 1.0 m below the feedwell outlet—with underflow

18-65

slurry at the desired concentration, the torque will reach a normal operating range. Special note should be made of the torque reading at this time. Subsequent higher torque levels while operating conditions remain unchanged can almost always be attributed to island formation, and corrective action can be taken early, before serious problems develop. Island is the name given to a mass of semisolidified solids that have accumulated on or in front of the rakes, often as a result of excessive flocculant use. This mass usually will continue to grow in size, eventually producing a torque spike that can shut down the thickener and often resulting in lower underflow densities than would otherwise be achievable. An island is easily detected, usually by the higher-than-normal, gradually increasing torque reading. Probing the rake arms near the thickener center with a rigid rod will confirm this condition—the mass is easily distinguished by its cohesive, claylike consistency. At an early stage, the island is readily removed by raising the rakes until the torque drops to a minimum value. The rakes are then lowered gradually, a few centimeters at a time, so as to shave off the mass of solids and discharge this gelled material through the underflow. This operation can take several hours, and if island formation is a frequent occurrence, the procedure should be carried out on a regular basis, typically once a day, preferably with an automatic system to control the entire operation. Stable thickener performance can be maintained by carefully monitoring operating conditions, particularly the pulp interface level and the underflow rate and concentration. As process changes occur, the pulp level can vary; regulation of the underflow pumping rate will keep the level within the desired range. If the underflow varies in concentration, this can be corrected by adjusting the flocculant dosage. Response will not be immediate, of course, and care should be taken to make only small step changes at any one time. Procedures for use of automatic control are described in the section on instrumentation. CLARIFIERS Continuous clarifiers generally are employed with dilute suspensions, principally industrial process streams and domestic municipal wastes, and their primary purpose is to produce a relatively clear overflow. They are basically identical to thickeners in design and layout except that they employ a mechanism of lighter construction and a drive head with a lower torque capability. These differences are permitted in clarification applications because the thickened pulp produced is smaller in volume and appreciably lower in suspended solids concentration, owing in part to the large percentage of relatively fine (smaller than 10 µm) solids. The installed cost of a clarifier, therefore, is approximately 5 to 10 percent less than that of a thickener of equal tank size, as given in Fig. 18-94. Rectangular Clarifiers Rectangular clarifiers are employed primarily in municipal water and waste treatment plants, as well as in certain industrial plants, also for waste streams. The raking mechanism employed in many designs consists of a chaintype drag, although suction systems are used for light-duty applications. The drag moves the deposited pulp to a sludge hopper located on one end by means of scrapers fixed to endless chains. During their return to the sludge raking position, the flights may travel near the water level and thus act as skimming devices for removal of surface scum. Rectangular clarifiers are available in widths of 2 to 10 m (6 to 33 ft). The length is generally 3 to 5 times the width. The larger widths have multiple raking mechanisms, each with a separate drive. This type of clarifier is used in applications such as preliminary oilwater separations in refineries and clarification of waste streams in steel mills. When multiple units are employed, common walls are possible, reducing construction costs and saving on floor space. Overflow clarities, however, generally are not as good as with circular clarifiers, due primarily to reduced overflow weir length for equivalent areas. Circular Clarifiers Circular units are available in the same three basic types as single-compartment thickeners: bridge, center-column, and peripheral-traction. Because of economic considerations, the bridge-supported type is limited generally to tanks less than 20 m in diameter.

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LIQUID-SOLID OPERATIONS AND EQUIPMENT

A circular clarifier often is equipped with a surface-skimming device, which includes a rotating skimmer, scum baffle, and scum-box assembly. In sewage and organic-waste applications, squeegees normally are provided for the rake-arm blades, as it is desirable that the bottom be scraped clean to preclude accumulation of organic solids, with resultant septicity and flotation of decomposing material. Center-drive mechanisms are also installed in square tanks. This mechanism differs from the standard circular mechanism in that a hinged corner blade is provided to sweep the corners which lie outside the path of the main mechanism. Clarifier-Thickener Clarifiers can serve as thickeners, achieving additional densification in a deep sludge sump adjacent to the center that extends a short distance radially and provides adequate retention time and pulp depth to compact the solids to a high density. Drive mechanisms on this type of clarifier usually must have higher torque capability than would be supplied on a standard clarifier. Industrial Waste Secondary Clarifiers Many plants which formerly discharged organic wastes to the sewer have turned to using their own treatment facilities in order to reduce municipal treatment plant charges. For organic wastes, the waste-activated sludge process is a preferred approach, using an aeration basin for the bio-oxidation step and a secondary clarifier to produce a clear effluent and to concentrate the biomass for recycling to the basin. To produce an acceptable effluent and achieve sufficient concentration of the low-density solids that make up the biomass, certain design criteria must be followed. If pilot-plant data are unavailable, the design procedures proposed by Albertson (op. cit.) can be used to specify tank diameter, depth, feedwell dimensions, feed inlet configuration, and rake-blade design for a unit which will meet the specifications for many wasteactivated sludge plants. Typical design parameters recommended by Albertson include the following: Feed pipe velocity: ≤1.2 m/s. Energy-dissipating feed entry velocity (tangential): ≤0.5 m/s. Downward velocity from feedwell: ≤0.5–0.75 (peak) m/min. Feedwell depth: Entry port depth +1 m. Radial velocity below feedwell: ≤90% of downward velocity. Tank depth: Clear water zone above dense sludge blanket is determined largely by clarification requirements; typically 3–5 m. Tank diameter: Largely a function of clear water depth; the maximum overflow rate in m/h = 0.278 × clear water depth. Resulting underflow design settling velocity should be less than 1.0 m/h for 1% solids and 120 mL/g SVI. Tilted-Plate Clarifiers Lamella or tilted-plate separators have achieved increased use for clarification. They contain a multiplicity of plates inclined at 45 to 60° from the horizontal. Various feed methods are employed so that the influent passes into each inclined channel at about one-third of the vertical height from the bottom. This results in the solids having to settle only a short distance in each channel before sliding down the base to the collection zone beneath the plates. The clarified liquid passes in the opposite direction beneath the ceiling of each channel to the overflow connection. The area that is theoretically available for separation is equal to the sum of the projected areas or all channels on the horizontal plane. Figure 18-88 shows the horizontally projected area A, of a single channel in a clarifier of unit width. If X is the uniform distance between plates (measured perpendicularly to the plate surface), the clarifier will contain sin α/X channels per unit length and an effective collection area per unit clarifier length of A, sin α/X, where α is the angle of inclination of the plates to the horizontal. It follows that the total horizontally projected plate area per unit volume of sludge in the clarifier As is As = cos α/X

(18-48)

As α and X are decreased, As is increased. However, α must be larger than the angle of repose of the sludge so that it will slide down the plate, and the most common range is 55 to 60°. Plate spacing must be large enough to accommodate the opposite flows of liquid and sludge while reducing interference and preventing plugging and to provide enough residence time for the solids to settle to the bottom plate. Usual X values are 50 to 75 mm (2 to 3 in).

FIG. 18-88

Basic concept of the Lamella-type clarifier.

Many different designs are available, the major difference among them being in feed-distribution methods and plate configurations. Operating capacities range from 1 to 3 m3 of feed/h/m2 of projected horizontal area [0.4 to 1.2 gal/(min⋅ft)]. The principal advantage of the tilted-plate clarifier is the increased capacity per unit of plane area. Major disadvantages are an underflow solids concentration that generally is lower than in other gravity clarifiers and difficulty of cleaning when scaling or deposition occurs. The lower underflow composition is due primarily to the reduced compression-zone volume relative to the large settling area. When flocculants are employed, flocculating equipment and tankage preceding the separator are required, as the design does not permit internal flocculation. Solids-Contact Clarifiers When desirable, mixing, flocculation, and sedimentation all may be accomplished in a single tank. Of the various designs available, those employing mechanically assisted mixing in the reaction zone are the most efficient. They generally permit the highest overflow rate at a minimum chemical dosage while producing the best effluent quality. The unit illustrated in Fig. 18-89 consists of a combination dual drive which moves the rake mechanism at a very slow speed as it rotates a high-pumping-rate, low-shear turbine located in the top portion of a center reaction well at a very much higher speed. The influent, dosed with chemicals as it enters, is contacted with previously settled solids in a recirculation draft tube within the reaction well by means of the pumping action of the turbine, resulting in a thorough mixing of these streams. Owing to the higher concentration of solids being recirculated, all chemical reactions are more rapid and more nearly complete, and flocculation is improved. The mixture passes out of the contacting and reaction well into the clarification area, where the flocculated particles settle out. They are raked to the center to be used again in the recirculation process, with a small amount being discharged through the sludge pump. When floccules are too heavy to be circulated up through the draft tube (as in the case of metallurgical pulps), a modified design using external recirculation of a portion of the thickened underflow is chosen. These units employ a special mixing impeller in a large feed well with a small-diameter central outlet. Solids-contact clarifiers are advantageous for clarifying turbid waters or slurries that require coagulation and flocculation for the removal of bacteria, suspended solids, or color. Applications include softening water by lime addition; clarifying industrial-process streams, sewage, and industrial wastewaters; tertiary treatment for removal of phosphates, BOD5, and turbidity; and silica removal from geothermal brines or from surface water for cooling-tower makeup.

GRAVITY SEDIMENTATION OPERATIONS

FIG. 18-89

18-67

Reactor-clarifier of the high-rate solids-contact type. (EIMCO Process Equipment Co.)

COMPONENTS AND ACCESSORIES FOR SEDIMENTATION UNITS Sedimentation systems consist of a collection of components, each of which can be supplied in a number of variations. The basic components are the same, whether the system is for thickening or clarifying: tank, drive-support structure, drive unit and lifting device, rake structure, feedwell, overflow arrangement, underflow arrangement, instrumentation, and flocculation facilities. Tanks Tanks or basins are constructed of such materials as steel, concrete, wood, compacted earth, plastic sheeting, and soil cement. The selection of the materials of construction is based on cost, availability, topography, water table, ground conditions, climate, operating temperature, and chemical-corrosion resistance. Typically, industrial tanks up to 30 m (100 ft) in diameter are made of steel. Concrete generally is used in municipal applications and in larger industrial applications. Extremely large units employing earthen basins with impermeable liners have proved to be economical. Drive-Support Structures There are three basic drive mechanisms. These are (1) the bridge-supported mechanism, (2) the center-

column-supported mechanism, and (3) the traction-drive thickener containing a center-column-supported mechanism with the driving arm attached to a motorized carriage at the tank periphery. Bridge-Supported Thickeners These thickeners (Fig. 18-86) are common in diameters up to 30 m, the maximum being about 45 m (150 ft). They offer the following advantages over a center-columnsupported design: (1) ability to transfer loads to the tank periphery; (2) ability to give a denser and more consistent underflow concentration with the single draw-off point; (3) a less complicated lifting device; (4) fewer structural members subject to mud accumulation; (5) access to the drive from both ends of the bridge; and (6) lower cost for units smaller than 30 m in diameter. Center-Column-Supported Thickeners These thickeners are usually 20 m (65 ft) or more in diameter. The mechanism is supported by a stationary steel or concrete center column, and the raking arms are attached to a driving cage which rotates around the center column. Traction Thickeners These thickeners are most adaptable to tanks larger than 60 m (200 ft) in diameter. Maintenance generally is less difficult than with other types of thickeners, which is an advantage

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LIQUID-SOLID OPERATIONS AND EQUIPMENT

in remote locations. The installed cost of the traction thickener may be more than that of a center-driven unit primarily because of the cost of constructing the heavy concrete wall required to support the drive carriage. Disadvantages of the traction thickener are that (1) no practical lifting device can be used, (2) operation may be difficult in climates where snow and ice are common, and (3) the driving-torque effort must be transmitted from the tank periphery to the center, where the heaviest raking conditions occur. Drive Assemblies The drive assembly is the key component of a sedimentation unit. The drive assembly provides (1) the force to move the rakes through the thickened pulp and to move settled solids to the point of discharge, (2) the support for the mechanism which permits it to rotate, (3) adequate reserve capacity to withstand upsets and temporary overloads, and (4) a reliable control which protects the mechanism from damage when a major overload occurs. Drives usually have steel or iron main spur gears mounted on bearings, alloy-steel pinions, and either a bronze or a malleable iron worm gear driven by a hardened-steel worm or a planetary gear system. Direct-drive hydraulic systems are also employed. The gearing components preferably are enclosed for maximum service life. The drive typically includes a torque-measuring system with torque indicated on the mechanism and often transmitted to a remote indicator. If the torque becomes excessive, it can automatically activate such safeguards against structural damage as sounding an alarm, raising the rakes, and stopping the drive. Rake-Lifting Mechanisms These should be provided when abnormal thickener operation is probable. Abnormal thickener operation or excessive torque may result from insufficient underflow pumping, surges in the solids feed rate, excessive amounts of large particles, sloughing of solids accumulated between the rakes and the bottom of

FIG. 18-90

Cable-arm design. (Dorr-Oliver, Inc.)

the tank or on structural members of the rake mechanism, or miscellaneous obstructions falling into the thickener. The lifting mechanism may be set to raise the rakes automatically when a specific torque level (e.g., 40 percent of design) is encountered, continuing to lift until the torque returns to normal or until the maximum lift height is reached. Generally, corrective action must be taken to eliminate the cause of the upset. Once the torque returns to normal, the rake mechanism is lowered slowly to “plow” gradually through the excess accumulated solids until these are removed from the tank. Rake-lifting devices can be manual for small-diameter thickeners or motorized for larger ones. Manual rake-lifting devices consist of a handwheel and a worm to raise or lower the rake mechanism by a distance usually ranging from 30 to 60 cm (1 to 2 ft). Motorized rakelifting devices typically are designed to allow for a vertical lift of the rake mechanism of up to 90 cm (3 ft). A platform-type lifting device lifts the entire drive and rake mechanism up to 2.5 m (8 ft) and is used for applications in which excessive torque is most probable or when storage of solids in the thickener is desired. Figure 18-90 illustrates the cable-arm design. This design uses cables attached to a truss above or near the liquid surface to move the rake arms, which are hinged to the drive structure, allowing the rakes to be raised when excessive torque is encountered. A major advantage of this design is the relatively small surface area of the raking mechanism, which reduces the solids accumulation and downtime in applications in which scaling or island formation can occur. One disadvantage of this or any hinged-arm or other self-lifting design is that there is very little lift at the center, where the overload usually occurs. A further disadvantage is the difficulty of returning the rakes to the lowered position in settlers containing solids that compact firmly.

GRAVITY SEDIMENTATION OPERATIONS Rake Mechanism The rake mechanism assists in moving the settled solids to the point of discharge. It also aids in thickening the pulp by disrupting bridged floccules, permitting trapped fluid to escape and allowing the floccules to become more consolidated. Rake mechanisms are designed for specific applications, usually having two long rake arms with an option for two short rake arms for bridge-supported and center-column-supported units. Traction units usually have one long arm and three short arms. Figure 18-91 illustrates three types of rake-arm designs. The conventional design typically is used in bridge-supported units, while the dual-slope design is used for units of larger diameter. The “thixo post” design employs rake blades on vertical posts extending below the truss to keep the truss out of the thickest pulp and thereby prevent the collection of solids that might cause an “island” to form. Rake blades can have attached spikes or serrated bottoms to cut into solids that have a tendency to compact. Lifting devices typically are used with these applications. Rake-speed requirements depend on the type of solids entering the thickener. Peripheral speed ranges used are, for slow-settling solids, 3 to 8 m/min (10 to 25 ft/min); for fast-settling solids, 8 to 12 m/min (25 to 40 ft/min); and for coarse solids or crystalline materials, 12 to 30 m/min (40 to 100 ft/min). Feedwell The feedwell is designed to allow the feed to enter the thickener with minimum turbulence and uniform distribution while dissipating most of its kinetic energy. Feed slurry enters the feedwell, which is usually located in the center of the thickener, through a pipe or launder suspended from the bridge. To avoid excess velocity, an open launder normally has a slope no greater than 1 to 2 percent. Pulp should enter the launder at a velocity that prevents sanding at the inlet. With nonsanding pulps, the feed may also enter upward through the center column from a pipeline installed beneath the tank. The standard feedwell for a thickener is designed for a maximum vertical outlet velocity of about 1.5 m/min (5 ft/min). High turbidity

Rake-mechanism designs employed for specific duties. (EIMCO Process Equipment Co.)

FIG. 18-91

18-69

caused by short-circuiting the feed to the overflow can be reduced by increasing the depth of the feedwell. When overflow clarity is important or the solids specific gravity is close to the liquid specific gravity, deep feedwells of large diameter are used, and measures are taken to reduce the velocity of the entering feed slurry. Shallow feedwells may be used when overflow clarity is not important, the overflow rate is low, and/or solids density is appreciably greater than that of water. Some special feedwell designs used to dissipate entrance velocity and create quiescent settling conditions split the feed stream and allow it to enter the feedwell tangentially on opposite sides. The two streams shear or collide with one another to dissipate kinetic energy. When flocculants are used, often it will be found that the optimum solids concentration for flocculation is considerably less than the normal concentration, and significant savings in reagent cost will be made possible by dilution of the feed prior to flocculation. This can be achieved by recycling overflow or more efficiently by feedwell modifications, including openings in the feedwell rim. These will allow supernatant to enter the feedwell, and flocculant can be added at this point or injected below the surface of the pulp in the feedwell. Another effective means of achieving this dilution prior to flocculant addition is illustrated in Fig. 18-92. This approach utilizes the energy available in the incoming feed stream to achieve the dilution by momentum transfer and requires no additional energy expenditure to dilute this slurry by as much as three to four times. Overflow Arrangements Clarified effluent typically is removed in a peripheral launder located inside or outside the tank. The effluent enters the launder by overflowing a V-notch or level flat weir, or through submerged orifices in the bottom of the launder. Uneven overflow rates caused by wind blowing across the liquid surface in large thickeners can be better controlled when submerged orifices or V-notch weirs are used. Radial launders are used when uniform upward liquid flow is desired in order to improve clarifier detention efficiency. This arrangement provides an additional benefit in reducing the effect of wind, which can seriously impair clarity in applications that employ basins of large diameter. The hydraulic capacity of a launder must be sufficient to prevent flooding, which can cause short-circuiting of the feed and deterioration of overflow clarity. Standards are occasionally imposed on weir overflow rates for clarifiers used in municipal applications; typical rates are 3.5 to 15 m3/(h⋅m) [7000 to 30,000 gal/(day⋅ft)], and they are highly dependent on clarifier side-water depth. Industrial clarifiers may have higher overflow rates, depending on the application and the desired overflow clarity. Launders can be arranged in a variety of configurations to achieve the desired overflow rate. Several alternatives to improve clarity include an annular launder inside the tank (the liquid overflows both sides), radial launders connected to the peripheral launder (providing the very long weir that may be needed when abnormally high overflow rates are encountered and overflow clarity is important), and Stamford baffles, which are located below the launder to direct flow currents back toward the center of the clarifier. In many thickener applications, on the other hand, complete peripheral launders are not required, and no difference in either overflow clarity or underflow concentration will result through the use of launders extending over only a fraction (e.g., one-fifth) of the perimeter. For design purposes, a weir-loading rate in the range of 7.5 to 30.0 m3/(h⋅m) [10 to 40 gpm/ft] can be used, the higher values being employed with well-flocculated, rapidly settling slurries. The overflow launder required may occupy only a single section of the perimeter rather than consisting of multiple, shorter segments spaced uniformly around the tank. Underflow Arrangements Concentrated solids are removed from the thickener by use of centrifugal or positive displacement pumps or, particularly with large-volume flows, by gravity discharge through a flow control valve or orifice suitable for slurry applications. Due to the risk to the thickener operation of a plugged underflow pipe, it is recommended that duplicate underflow pipes and pumps be installed in all thickening applications. Provision to recycle underflow slurry back to the feedwell is also useful, particularly if solids are to be stored in the thickener.

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LIQUID-SOLID OPERATIONS AND EQUIPMENT

FIG. 18-92

E-Duc® Feed dilution system installed on 122-m-diameter thickener. (EIMCO Process Equipment Co.)

There are several basic underflow arrangements: (1) the underflow pump adjacent to the thickener sidewall with buried piping from the discharge cone, (2) the underflow pump under the thickeners or adjacent to the sidewall with the piping from the discharge cone in a tunnel, (3) the underflow pump adjacent to the thickener sidewall with a peripheral discharge from the tank sidewall, and (4) the underflow pump located in the center of the thickener on the bridge close to the drive mechanism, or at the perimeter, using piping up through the center column. Pump Adjacent to Thickener with Buried Piping This arrangement of buried piping from the discharge cone is the least expensive system but the most susceptible to plugging. It is used only when the solids do not compact to an unpumpable slurry and can be easily backflushed if plugging occurs. Typically, two or more underflow pipes are installed from the discharge cone to the underflow pump so that solids removal can continue if one of the lines plugs. Valves should be installed to permit flushing with water and compressed air in both directions to remove blockages. Tunnel A tunnel may be constructed under the thickener to provide access to the discharge cone when underflow slurries are difficult to pump and have characteristics that cause plugging. The underflow pump may be installed underneath the thickener or at the perimeter. Occasionally thickeners are installed on legs or piers, making tunneling for access to the center unnecessary. A tunnel or an elevated thickener is more expensive than the other underflow arrangements, but there are certain operational and maintenance advantages. Of course, the hazards of working in a tunnel (flooding and interrupted ventilation, for example) and related safety regulations must be considered. Peripheral Discharge Peripheral discharge sometimes is used to permit the reduced installation cost of a flat-bottom tank on compacted soil. Because more torque is required to rake the solids to the perimeter of the tank, this arrangement is not suitable for service involving coarse solids or solids that become nonfluid at high concentrations. Center-Column Pumping This arrangement may be used instead of a tunnel. Several designs are available. The commonest is a bridge-mounted pump with a suction line through a wet or dry center column. The pump selection may be limiting, requiring special attention to priming, net positive suction head, and the maximum

density that the pump can handle. One design has the underflow pump located in a room under the thickener mechanism and connected to openings in the column. Access is through the drive gear at the top of the column. INSTRUMENTATION Thickener control philosophies are usually based on the idea that the underflow density obtained is the most important performance criterion. The overflow clarity is also a consideration, but this is generally not as critical. Additional factors which must be considered are optimization of flocculant usage and protection of the raking mechanism. Automated control schemes employ one or more sets of controls, which will fit into three categories: (1) control loops which are used to regulate the addition of flocculant, (2) control loops to regulate the withdrawal of underflow, and (3) rake drive controls. Usually, the feed to a thickener is not controlled and most control systems have been designed with some flexibility to deal with changes in feed characteristics, such as an increase in volume or alteration in the nature of the solids themselves. In severe cases, some equalization of the feed is required in order to allow the control system to perform effectively. Flocculant addition rate can be regulated in proportion to the thickener volumetric feed rate or solids mass flow in a feed-forward mode, or in a feed-back mode on either rake torque, underflow density, settling solids (sludge) bed level, or solids settling rate. Of these, feedforward on mass flow or feed-back on bed level are probably the most common. In some feed-forward schemes, the ratio multiplier is trimmed by one of the other parameters. Underflow is usually withdrawn continuously on the bases of bed level, rake torque, or underflow solids concentration in a feed-back mode. Most installations incorporate at least two of these parameters in their underflow withdrawal control philosophy. For example, the continuous withdrawal may be based on underflow solids density with an override to increase the withdrawal rate if either the rake torque or the bed level reaches a preset value. In some cases, underflow withdrawal has been regulated in a feed-forward mode on the basis of thickener feed solids mass flow rate. Any automated underflow pumping scheme should incorporate a lower limit on volumetric flow rate as a safeguard against line pluggage.

GRAVITY SEDIMENTATION OPERATIONS It is also important to consider the level of the sludge bed in the thickener. Although this can be allowed to increase or decrease within moderate limits, it must be controlled enough to prevent solids from overflowing the thickener or from falling so low that the underflow density becomes too dilute. The settling slurry within the sludge bed is normally free flowing and will disperse to a consistent level across the thickener diameter. Rake drive controls protect the drive mechanism from damage and usually incorporate an alarm to indicate high torque with an interlock to shut down the drive at a higher torque level. They can have an automated rake raising and lowering feature with a device to indicate the elevation of the rakes. A complete automated control scheme incorporates controls from each of the three categories. It is important to consider the interaction of the various controls, especially of the flocculant addition and underflow withdrawal control loops, when designing a system. The lag and dead times of any feedback loops as well as the actual response of the system to changes in manipulated variables must be considered. For example, in some applications it is possible that excessive flocculant addition may produce an increase in the rake torque (due to island formation or viscosity increase) without a corresponding increase in underflow density. Additionally, sludge bed level sensors generally require periodic cleaning to produce a reliable signal. In many cases, it has not been possible to effectively maintain the sludge bed level sensors, requiring a change in the thickener control logic after start-up. Some manufacturers offer complete thickener control packages. Control philosophies for clarifiers are based on the idea that the overflow is the most important performance criterion. Underflow density or suspended solids content is a consideration, as is optimal use of flocculation and pH control reagents. Automated controls are of three basic types: (1) control loops that optimize coagulant, flocculant, and pH control reagent additions; (2) those that regulate underflow removal; and (3) rake drive controls. Equalization of the feed is provided in some installations, but the clarifier feed is usually not a controlled variable with respect to the clarifier operation. Automated controls for flocculating reagents can use a feedforward mode based on feed turbidity and feed volumetric rate, or a feed-back mode incorporating a streaming current detector on the flocculated feed. Attempts to control coagulant addition on the basis of overflow turbidity generally have been less successful. Control for pH has been accomplished by feed-forward modes on the feed pH and by feed-back modes on the basis of clarifier feedwell or external reaction tank pH. Control loops based on measurement of feedwell pH are useful for control in applications in which flocculated solids are internally recirculated within the clarifier feedwell. Automated sludge withdrawal controls are usually based on the sludge bed level. These can operate in on-off or continuous modes and will use either single-point or continuous sludge level indication sensors. In many applications, automated control of underflow withdrawal does not provide an advantage, since so few settled solids are produced that it is only necessary to remove sludge for a short interval once a day, or even less frequently. In applications in which the underflow is recirculated internally within the feedwell, it is necessary to maintain sufficient sludge inventory for the recirculation turbine to pull from. This could be handled in an automated system with a single-point low sludge bed level sensor in conjunction with a low-level alarm or pump shutoff solenoid. Some applications require continuous external recirculation of the underflow direct to the feedwell or external reaction tanks, and an automated control loop could be used to maintain recirculation based on flow measurement, with a manually adjusted setpoint. Control philosophies applied to continuous countercurrent decantation (CCD) thickeners are similar to those used for thickeners in other applications, but have emphasis on maintaining the CCD circuit in balance. It is important to prevent any one of the thickeners from pumping out too fast, otherwise an upstream unit could be starved of wash liquor while at the same too much underflow could be placed in a downstream unit too quickly, disrupting the operation of both units as well as reducing the circuit washing efficiency. Several control configurations have been attempted, and the more successful schemes

18-71

have linked the solids mass flow rate of underflow pumping to that of the upstream unit or to the CCD circuit solids mass feed rate. Wide variations in the solids feed rate to a CCD circuit will require some means of dampening these fluctuations if design wash efficiency is to be maintained. The following types of devices are commonly applied to measure the various operational parameters of thickeners and clarifiers. They have been used in conjunction with automatic valves and variablespeed pumps to achieve automatic operation as well as to simply provide local or remote indications. Sludge Bed Level Sensors Most of the more commonly used devices for detection of the sludge bed level operate on principles of ultrasound, gamma radiation, infrared or visible light, or simply with a float that is carefully weighted to float on the sludge bed level interface. There are basically two types of sensors, those that provide a single-point indication at a fixed height within the thickener or clarifier, and those that follow the sludge bed depth over a distance and provide a continuous indication of the level. Some devices are available that can provide a profile of the slurry concentration within the sludge bed. Flowmeters These are used to measure flocculant addition, underflow, and feed flow rates. For automatic control, the more commonly used devices are magnetic flowmeters and Doppler effect flowmeters. Density Gauges These are used to measure the density or suspended solids content of the feed and underflow streams. Gamma radiation devices are the most commonly used for automatic control, but ultrasonic devices are effective in the lower range of slurry density. Marcy pulp density scales are an effective manually operated device. A solids “mass flow” indication is usually obtained by combining a density gauge output with the output from a flowmeter. Turbidity Gauges These operate with visible light beams and detectors. They are used to monitor feed and effluent turbidity. Streaming Current Detectors These units produce a measurement closely related to the zeta potential of a suspension and are used successfully in optimizing the coagulant dose in clarification applications. CONTINUOUS COUNTERCURRENT DECANTATION The system of separation of solid-phase material from an associated solution by repeated stages of dilution and gravity sedimentation is adapted for many industrial-processing applications through an operation known as continuous countercurrent decantation (CCD). The flow of solids proceeds in a direction countercurrent to the flow of solution diluent (water, usually), with each stage composed of a mixing step followed by settling of the solids from the suspension. The number of stages ranges from 2 to as many as 10, depending on the degree of separation required, the amount of wash fluid added (which influences the final solute concentration in the first-stage overflow), and the underflow solids concentration attainable. Applications include processes in which the solution is the valuable component (as in alumina extraction), or in which purified solids are sought (magnesium hydroxide from seawater), or both (as frequently encountered in the chemical-processing industry and in base-metal hydrometallurgy). The factors which may make CCD a preferred choice over other separation systems include the following: rapidly settling solids, assisted by flocculation: relatively high ratio of solids concentration between underflow and feed; moderately high wash ratios allowable (2 to 4 times the volume of liquor in the thickened underflows); large quantity of solids to be processed; and the presence of fine-size solids that are difficult to concentrate by other means. A technical feasibility and economic study is desirable in order to make the optimum choice. Flow-Sheet Design Thickener-sizing tests, as described earlier, will determine unit areas, flocculant dosages, and underflow densities for the various stages. For most cases, unit areas will not vary significantly throughout the circuit; similarly, underflow concentrations should be relatively constant. In practice, the same unit area is generally used for all thickeners in the circuit to simplify construction. Equipment The equipment selected for CCD circuits may consist of multiple-compartment washing-tray thickeners or a train of

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LIQUID-SOLID OPERATIONS AND EQUIPMENT

individual unit thickeners. The washing-tray thickener consists of a vertical array of coaxial trays connected in series, contained in a single tank. The advantages of this design are smaller floor-area requirements, less pumping equipment and piping, and reduced heat losses in circuits operating at elevated temperatures. However, operation is generally more difficult and user preference has shifted toward unit thickeners despite the larger floor-area requirement and greater initial cost. Underflow Pumping Diaphragm pumps with open discharge are employed in some cases, primarily because underflow densities are readily controlled with these units. Disadvantages include the generally higher maintenance and initial costs than for other types and their inability to transfer the slurry any great distance. Large flows often are TABLE 18-7

best handled with variable-speed, rubber-lined centrifugal pumps, utilizing automatic control to maintain the underflow rate and density. Overflow Pumps These can be omitted if the thickeners are located at increasing elevations from first to last so that overflows are transferred by gravity or if the mixture of underflow and overflow is to be pumped. Overflow pumps are necessary, however, when maximum flexibility and control are sought. Interstage Mixing Efficiencies Mixing or stage efficiencies rarely achieve the ideal 100 percent, in which solute concentrations in overflow and underflow liquor from each thickener are identical. Part of the deficiency is due to insufficient blending of the two streams, and attaining equilibrium will be hampered further by heavily flocculated solids. In systems in which flocculants are used, inter